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# WorkingWiki: a MediaWiki-based platform for collaborative research
Lee Worden
McMaster University
Hamilton Ontario Canada
University of California Berkeley
Berkeley California United States
[email protected]
###### Abstract
WorkingWiki is a software extension for the popular MediaWiki platform that
makes a wiki into a powerful environment for collaborating on publication-
quality manuscripts and software projects. Developed in Jonathan Dushoff’s
theoretical biology lab at McMaster University and available as free software,
it allows wiki users to work together on anything that can be done by using
UNIX commands to transform textual “source code” into output. Researchers can
use it to collaborate on programs written in R, python, C, or any other
language, and there are special features to support easy work on LaTeX
documents. It develops the potential of the wiki medium to serve as a
combination collaborative text editor, development environment, revision
control system, and publishing platform. Its potential uses are open-ended —
its processing is controlled by makefiles that are straightforward to
customize — and its modular design is intended to allow parts of it to be
adapted to other purposes.
Copyright ©2011 by Lee Worden ([email protected]). This work is licensed
under the Creative Commons Attribution 3.0 license. The human readable license
can be found here: http://creativecommons.org/licenses/by/3.0.
## 1 Introduction
The remarkable success of Wikipedia as a collaboratively constructed
repository of human knowledge is strong testimony to the power of the wiki as
a medium for online collaboration. Wikis — websites whose content can be
edited by readers — have been adopted by great numbers of diverse groups
around the world hoping to compile and present their shared knowledge.
While several somewhat high-profile academic wiki projects have been launched
and later abandoned — the quantum physics community’s Qwiki [14] for example —
there have also been successful academic wiki projects that strongly suggest
that wikis can be a transformative tool for accelerating and amplifying the
power of research collaborations. In particular, the OpenWetWare wiki [4] has
proven itself to have staying power as a home for an extended research
community’s projects and data, and the Polymath project [11] provides
especially powerful inspiration regarding the power of online collaboration to
accelerate the process of mathematical discovery.
MediaWiki, the software behind Wikipedia and its sister projects, available
openly as free software, is especially powerful, full-featured, and stable,
and is widely used in academic and popular sites alike. It excels at managing
information organized into pages of text, as it is on Wikipedia. As such, it
is very useful for collaborative and public documentation of a research team’s
techniques and results, but not applicable for collaboration on the daily
research itself, which tends to involve writing of software tools for data
analysis and simulation, and production of manuscripts for publication, with
figures, tables, formulae and citations.
This paper describes a software package that extends MediaWiki, creating a
hybrid environment which combines the desirable features of the wiki system —
easy collaborative editing, recording of history and authorship, and instant
publication on the internet — with support for complex formats including
programming languages and LaTeX document formatting, making it possible to
collaborate simply and flexibly on the actual daily work of the lab and making
it simple to store and publish, in an integrated form, the results, process
and presentation of the research.
## 2 WorkingWiki
WorkingWiki [27] is a software extension for the popular MediaWiki system that
makes a wiki into a powerful environment for collaborating on publication-
quality manuscripts and software projects. The WorkingWiki extension allows
you to store “source files” in your wiki and develop, test, run and publish
them easily, along with the products of computations using those source files.
Examples include a project of five LaTeX files and six EPS images that compile
together into a single PDF file, or an R script that includes two other R
source files and produces a CSV data file and several EPS figures. The
WorkingWiki extension keeps track of when the source files have changed and
when to redo the processing to update the output, and how to display the
various file formats involved. The output files and images can be displayed in
wiki pages along with the source code, and can be used as inputs to further
computations.
##### Example: collaborating on a LaTeX document.
It’s very common for a group of scientific authors to write a paper by
emailing each other copies of .tex files daily or hourly. This is inconvenient
— in order to look at the paper you have to save the file into a directory and
compile it — and unreliable — it’s easy to get mixed up and lose someone’s
edits, or overwrite them with someone else’s copy of the file. One solution is
to use a revision control system such as Subversion [8] or Git [2] to manage
the source code, but if any authors are unwilling to take on the work of
learning to use the tool, they’re likely to fall back on emailing the file or
just dictating changes to someone else. WorkingWiki addresses this problem by
providing basic revision control features together with easy editing. Once the
.tex files, .bib files, and images are in the wiki, it’s easy for everyone to
edit and see the updated results, and the wiki keeps track of all the changes
and their authors, and makes it easy to review or undo them. It also provides
a convenient place to discuss changes, without having to put comments into the
manuscript itself, and can be used as a website to present the research to the
public.
##### Example: collaborative, reproducible lab science.
A research team can use WorkingWiki to archive experimental data (using the
wiki’s history features to record who uploaded which data sets when); develop
their data-processing scripts collaboratively in the wiki; construct the
scripts that produce figures and tables in the wiki; create the manuscript
that presents the results in the wiki; and finally export the manuscript as a
.tar.gz file ready to submit to a scientific journal. The wiki can then be
used to publish the data, source code, and manuscript to the world as is. This
process captures all the files needed to understand and reproduce the research
project, with its revision history intact, and in a form that is easy to
annotate and publish online. A research team developing simulation programs
rather than using experimental data can use WorkingWiki in the same way.
WorkingWiki is developed principally for research groups, but is likely to
have a variety of other uses as well for mathematicians, scientists, and
software developers. WorkingWiki provides some features of an integrated
development environment: it coordinates compiling (if necessary) and running
the code when relevant source files have changed, and displaying the results.
It provides some features of a revision control system: it uses MediaWiki’s
history features to record author, date/time, and content of every change to
the files and the wiki pages they are connected to, and it allows viewers to
export the source code to their workstations and work on it offline.
It integrates editing and running code with wiki editing. Source and product
files can be mixed freely into wiki pages’ text. Editing is fully
collaborative. The effects of changes to source files can be fully previewed
before saving to the wiki. The WorkingWiki-extended wiki is a simple, elegant
way to present a research team’s work to the world.
WorkingWiki has special features for translating LaTeX documents to HTML, for
display directly in the wiki page. WorkingWiki allows collaborators to edit
complete LaTeX documents collaboratively on the wiki, view the compiled
document in the wiki page, and export the documents’ source files to your
workstation when ready to submit or circulate. Using Bruce Miller’s LaTeXML
software [20], the rendered contents of a LaTeX document are made visible in
the wiki page, including figures, citations and equations (optionally using
MathML), as well as in the standard PDF format. The editing history of all
files is maintained, including authorship of each change. WorkingWiki’s LaTeX
handling works with documents that involve multiple files, stored on multiple
wiki pages. LaTeX \include, \bibliography, \includegraphics, and like commands
are supported. Filenames do not need to match page names.
WorkingWiki is extensively customizable, supporting collaborative development
and use of computer programs in any language. Images and other files created
by computer programs can be included directly in LaTeX documents and read by
other programs, and are updated automatically when the programs or source data
files are changed. The development environment can be customized by adding
default make rules, and in many other ways.
WorkingWiki supports reproducible and open research by allowing researchers to
collect all the files involved in a research project — data files, source
code, documentation, publications — in an accessible place where collaborators
can develop them together, and the public can be allowed to download the
entire project, to verify results and try their own experiments.
## 3 WorkingWiki in use
WorkingWiki operates on _source files_ that are stored in standard wiki pages.
Source files are collected into _projects_. Behind the scenes, WorkingWiki
maintains a cached working directory where it stores and processes the
project’s files. When an output file is called for in a wiki page or by other
means, WorkingWiki does its work by invoking make [23] to create or update the
file from the source files in its project before displaying it. In this way,
users can edit their code (or their data files, or .tex documents) by editing
the wiki, and run the code and view the output (the typeset version of the
paper, the updated version of the figure, the textual output of the program)
just by previewing or saving the page.
Figure 1 provides a simple example of the source text of a WorkingWiki-enabled
wiki page. Most of the text of this page is standard MediaWiki markup using
constructs such as `==`…`==` for section headers and `[[`…`]]` for links. A
WorkingWiki source file is defined by including it in an XML-style source-file
element. The text between the opening and closing tags of that element defines
the content of that file, and it is written to the corresponding file in the
project’s working directory and used to update the project’s output files as
needed. In this example, project names are not explicitly given, signaling the
software to use by default the project whose name is the name of the wiki
page. The assignment of files to projects can be made explicit by supplying a
project attribute along with the filename attribute in the opening tag.
Below the source file is an output file, represented by a self-closing
project-file element. When the MediaWiki parser encounters this tag and passes
it to WorkingWiki’s code, WorkingWiki synchronizes all the project’s source
files with their copies in the working directory, creates a subprocess to run
the Unix command make figure.png, and (assuming the make command succeeds)
retrieves the file and inserts the file into the HTML page that is the output
of the wiki’s parser. Thus simply viewing the page causes the output file to
be updated and displayed. Figure 2 shows what this wiki page looks like in the
web browser.
This example is especially simple because the steps to make the output from
the source code are controlled by a system-wide makefile installed in a
central location and used in all projects. This is not necessary: makefiles
can also be added to individual projects, in the same way as any other source
file, and in this way users can control the processing of any kind of files
and specify their dependencies.
`==R graphics example==`
`Here is a simple example of how to do a figure with R, using `
`[[Recipe_Book#R | the lalashan site’s custom rules for using R]].`
`The custom rules make it simple: just define a .R file:`
`<``source-file filename=example.R>`
`plot(function(x){-x*cos(x-1)}, -pi, pi, col="blue");`
`</``source-file>`
`and request its output using just the right filename:`
`<``project-file filename=example.Rout.png/>`
Figure 1: Source text for example WorkingWiki-enabled wiki page, illustrating
the use of the source-file and project-file tags. Figure 2: How the markup of
figure 1 appears in a wiki page.
WorkingWiki completely supports MediaWiki’s previewing feature: changes to
pages, including source files, can be tested and revised extensively before
saving them to the wiki. When a user previews a page that includes project
files, WorkingWiki updates them from the modified source files, in a separate
preview copy of the project’s working directory. When the user saves the
changes to the wiki, WorkingWiki merges the temporary files into the permanent
copy, to avoid unnecessary repetition of processing steps. 111When a directory
becomes large, these copy operations can become quite expensive.
Unfortunately, it’s necessary, because if we processed unsaved code in the
primary cache directory it could modify files in ways that would affect the
outcome of future processing steps, even if the previewed changes were never
saved. We partially address this problem by making it possible to split out
large or numerous project files into separate project directories that are
left uncopied provided the project’s authors promise that they are protected
from problematic side effects, but a more flexible solution would be
desirable. We are considering using the Btrfs filesystem’s copy-on-write file
storage capabilities [1] to make these copy and merge operations fast and
cheap.
Editing source code in a form field in a web browser is much less convenient
than editing files in full-featured editors like vi and emacs, but browser
addons It’s All Text [15] for Firefox and TextAid [9] for Chrome make it much
easier by allowing a page’s contents to be opened in a text editor of the
user’s choice and kept open while repeatedly submitting and previewing the
page or a section of the page. This gives users access to all the editor
features they are used to, such as syntax highlighting and smart indenting.
### 3.1 LaTeX features
WorkingWiki allows a source file to be displayed in a transformed form. For
instance, if a page’s editor writes `<``source-file filename="example.R"
display="example.Rout.png">`, the project file example.Rout.png is updated and
displayed in the page in place of the source code. This feature has not proved
very popular — it seems to be preferable to make source code visible in most
cases — but the use of _default_ display attributes is very useful with LaTeX
and related formats. Default display transformations can be defined by the
wiki’s administrators, and WorkingWiki comes with a small number of them
predefined. In particular, .tex files are by default transformed to
.latexml.html files for display, and WorkingWiki’s system-wide makefile
provides rules that make that transformation by using LaTeXML to process the
document into HTML for display. Additionally, if a user has opted to enable
MathML output and is using a MathML-compatible browser, WorkingWiki instead
provides a .latexml.xhtml version of the document which uses MathML for all
mathematical content. (This automatic detection and output switching is also
available to wiki users and administrators for custom HTML-producing
processes.)
When displaying a .tex file, WorkingWiki also provides a link that makes and
provides a PDF version of the document using either LaTeX or PDFLaTeX (or
other programs, if customized). This link can be redefined or additional links
can be added to the default behaviour and they can be changed on a file-by-
file basis by adding attributes to the source-file element.
In figure 3 is a screenshot of a LaTeX document in a wiki page, illustrating
how the XHTML version of the manuscript is embedded in the page, and the PDF
link at the right margin. Embedding the rendered form of the manuscript
directly in the wiki page allows for a comfortable cycle of editing,
previewing, and editing some more, which is comparable to the ease of editing
a manuscript’s source text in a text editor, processing it into DVI or PDF,
viewing, and editing again, especially when using an external text editor with
the wiki as described above.
Figure 3: A LaTeX document in a wiki page.
These features allow users to edit LaTeX documents (and other source files) in
the same way that wiki pages are edited: open an edit form and change the
source code; press the preview button and see what it looks like when
processed; edit some more until it is right, and then save.
The make rules that are provided with WorkingWiki automatically keep track of
dependencies on BibTeX files, figures, locally provided style files and
included tex files, so that the displayed manuscript is kept up to date when
any part of it is updated.
It is simple to use LaTeX conditionals to insert comments and conversations in
the code of a manuscript that are visible in the HTML version of the
manuscript but invisible in the PDF, providing an easy way to coordinate while
keeping a clean manuscript for submission.
### 3.2 Advanced features
#### 3.2.1 Inter-project dependencies
For advanced users, WorkingWiki supports sharing of data among multiple
projects, and takes steps to ensure dependency relationships are respected and
data integrity is protected when previewing or running background jobs (see
below).
This feature allows a number of useful strategies. General-use code can be
shared among multiple projects, by placing it in a ”library” project. Complex
projects can be organized by grouping related things together into separate
WorkingWiki projects, while allowing interaction between the different
components. Independent parts of a project can be isolated from one another. A
particularly important case is that a journal article for publication can be
housed in a separate project from the data and programs that provide its
content. This allows, on the one hand, the authors to maintain the dependency
relationships within the wiki that allow the manuscript’s figures and tables
to be automatically kept up to date when the data and programs change, and on
the other hand makes it simple to export the article’s source files in a neat
.tar.gz package for submission to the journal and leave the programs and data
behind.
#### 3.2.2 Interaction with external data
WorkingWiki’s back end is capable of processing data from multiple sources,
and the front end allows those projects to be integrated with projects
originating on the wiki. For instance, this author has a research project in
progress in which a complex simulation program is stored in two GitHub
repositories, pulled into two project directories in the wiki’s file cache,
compiled and run by code in a third project whose source files are housed on
the wiki, and the output files are stored in a fourth project that is created
on the wiki but doesn’t have any source files.
This external-project feature also allows interaction between projects housed
on different wikis — this is useful on our site at McMaster because we operate
many interconnected wikis, and store some general-use code on a central wiki
for use on others.
Project data can be exported to a user’s local disk in a .tar.gz package,
which includes the wiki’s centralized makefile and other supplementary data,
to allow running and developing the code offline. It can then be re-imported
into the wiki. There is also a command-line tool to pull project files from
wikis to a local directory. In the future there may be an interface to git
[2], allowing one to pull and push source code from and to the wiki storage as
if it were a (somewhat simplified) git repository. Given the flexibility of
the git client, this would effectively make it possible to migrate projects
easily between wiki storage and many other repositories.
#### 3.2.3 Background jobs
When certain computational steps are too slow to run on the spot, WorkingWiki
allows them to be run as background jobs, which run outside of the wikitext-
parsing process. A background job is created simply by specifying a make
target and requesting it be made in the background. Any background jobs that
have been created are listed at the top of all pages that interact with the
projects they involve, in a listing that provides their basic information and
status. Whether a job has succeeded or failed, a user can browse its files,
destroy it, or merge its output into the project’s primary working directory.
Running jobs can be browsed and killed.
In a standard installation, background jobs are run as Unix subprocesses on
the same processors as the web server (using nice and ionice at the discretion
of the site administrators), but there is a prototype in development to run
background jobs on computing clusters using GridEngine [5].
## 4 Design of the software
WorkingWiki is implemented as a MediaWiki extension, written in PHP and
augmented by a few JavaScript and CSS resources, makefiles, and small helper
programs. It is freely available under the GNU General Public License [3], and
is compatible with all versions of MediaWiki from 1.13 on.
The source-file and project-file tags are implemented as tag hooks, a standard
means of extending MediaWiki’s parser, and retrieval of binary files and
project management are provided by two special pages, another standard form
for extensions. Like MediaWiki, WorkingWiki itself provides a number of hooks
that can be used by other extensions to provide additional features or modify
the ones that are provided. It has not been tested in combination with all
other MediaWiki extensions, but there are no known conflicts.
WorkingWiki’s behavior can be extensively customized as is. Administrators can
modify the rules controlling how different file types are displayed, and
provide default transformations like the one from LaTeX to HTML and links like
the one from LaTeX to PDF. Custom make rules can be added, to make it easy for
users to write source code and transform it in standard ways, and the existing
make rules can be partially or completely overridden.
### 4.1 Separation of wiki from project engine
A wiki is a powerful tool that combines a number of important functions. It is
effectively a combined revision control system, integrated development
environment, markup parser for website content, and publishing platform for
web pages written in its markup language. WorkingWiki extends all of these
functions to a wider range of source material, making the wiki into a
combination revision control system, development environment, execution
environment, and publishing platform for the general case of executable
program text. Each of these functions is provided in more powerful forms by
other tools, but the power of the wiki medium is in combining them together in
an elegant, easy-to-use form.
An ideal situation would be to make it easy for end users to separate all
these functions in a mix-and-match way, for instance providing a development,
execution and publishing platform for data stored in a revision control system
of the user’s choice, or providing revision control, execution and publishing
but using a third-party tool for editing and previewing. This is not entirely
possible at present, but WorkingWiki is written with these separations in
mind.
In particular, while the revision control, development (e.g. editing and
previewing), and publishing functions are essentially provided by MediaWiki
once the source files’ contents are inserted into the stored pages and output
files are inserted into the output HTML, WorkingWiki’s execution environment
is entirely separate from MediaWiki’s code, and is designed as a completely
independent component.
This component, called ProjectEngine, is a standalone tool that stores files,
performs make operations, and serves up-to-date file contents. Written in PHP,
as are MediaWiki and WorkingWiki, it can be used as a component of a larger
program — it is incorporated in WorkingWiki in this way by default — and can
also run as a self-contained HTTP service. It can be thought of as similar to
a simple web server — whose primary function is to retrieve the contents of
files for clients — but one that can create and update its files using make
rules before serving them.
ProjectEngine supports updating and removing files; creating, destroying and
merging preview sessions by making a copy of ”persistent” files; and creating,
destroying, merging and tracking background jobs.
The project engine seems to be a simple and powerful concept, and one that may
have uses beyond this single wiki system. If nothing else, it can be used as a
back end for similar extensions for other wiki engines, and the author has
discussed this possibility with the author of Projects Wiki [19], a
WorkingWiki-inspired plugin for Dokuwiki.
### 4.2 Security considerations
There are, of course, risks involved in running a web server that includes a
project engine, which executes programs supplied by users. To a first
approximation, the risks can be partitioned into just a few categories:
overuse or destruction of server resources, access to sensitive data, denial
of service, and harmful output. All of these risks can be managed.
The first category, overuse or destruction of system resources, is fairly
broad. It includes scenarios from user-supplied code altering files on the
server, to programs that send voluminous spam emails to innocent people, to
infinite loops that consume excessive CPU time or fill up a disk partition.
These risks can be managed by use of a mandatory access control system such as
TOMOYO Linux [7] to restrict access to all system resources, from sensitive
files to use of the server’s network interfaces. Additionally, ProjectEngine
uses nice and ionice to prevent its processes from monopolizing CPU time and
disk access, and uses setrlimit() to limit the number of subprocesses a make
process can create and kills make processes after a limited time period. A
quota system can be used to limit the amount of disk space ProjectEngine’s
files can consume.
Mandatory access control is also effective at keeping user-supplied code from
reading sensitive system files, and WorkingWiki’s inputs are carefully
validated to prevent backdoor access to SQL data. WorkingWiki works with
MediaWiki’s access control features to ensure that a password-protected wiki
doesn’t reveal data to unauthorized users.
Denial of service attacks can include inputs that cause crashes in the
software, as well as inputs that consume inordinate resources. The latter
category has been covered. The former case can probably never be ruled out
with complete confidence, but in any case, when a wiki is password-protected,
unauthorized users have no means to interact with WorkingWiki and thus any
attacks from outside the user community must be directed at other services.
The case of harmful output is the least well accounted for at present: in
order to provide an HTML rendering of LaTeX documents, it’s necessary to allow
ProjectEngine jobs to produce HTML output to be passed on to the client, and
in order to support programmers it’s necessary to allow them to write programs
and custom make rules; in combination this means that users’ projects can
produce HTML output that does unwelcome things on the client side, such as
making calls to third-party websites that reveal information about users
logged in to the wiki. It may be possible to filter HTML output in a way that
allows only safe output, but this is currently not implemented in WorkingWiki.
Another possibility is to provide as an option a restricted set of WorkingWiki
features, for instance allowing users to edit LaTeX documents but not to
create makefiles; this might suffice to provide a system that could be safely
opened up to anonymous editors.
The current recommendation is to use WorkingWiki only on password-protected
wikis, restricting editing access to trusted users. We believe it is safe for
publicly readable wikis as long as only trusted users can edit. WorkingWiki is
very useful and reliable for semi-closed wikis in this way, and use in public
wikis more like Wikipedia may be possible in the future.
## 5 Examples of WorkingWiki in use
WorkingWiki’s home site [27], which is itself a WorkingWiki-enabled wiki,
provides a handful of example WorkingWiki projects, illustrating how to create
projects for LaTeX and for programming (the nomogram example [10] is
especially engaging).
One active research team is using it to analyze and visualize African survey
data related to HIV and female genital cutting. For that research a utility
project has been created that automates the process of downloading the raw
survey data from the provider’s web site, merging separate data sets together,
and transforming them into .RData files ready for processing in R. Another
utility project provides custom R functions for plotting the data, allowing
users to create visualizations of particular variables, including geographical
plots, by inserting brief scripts of only a few lines into their wiki pages.
The Dushoff lab has created a suite of make rules to streamline R programming
within WorkingWiki, making it easy to process data in steps by creating a
series of small R modules that operate on the data produced by earlier
modules, and interleaving these brief program snippets with the plots and
textual output that they produce, in a wiki page that documents the steps of
the processing. This mode of working with processing steps and their output is
similar to the interactive notebooks provided by Mathematica [22] or Sage [6].
The Dushoff lab has also developed custom processes and make rules for
automatically generating BibTeX data and browser-friendly reference lists from
PubMed and similar identifiers, making it easy to maintain citation data
within the wikis.
This author is conducting an experiment in open research by maintaining a
project on a publicly readable wiki. This project, which is in the early
proof-of-concept phase, combines simulation and mathematical analysis in
modeling collective search for a solution to a complex problem [24].
Another team is using it to investigate the behavior of spatially extended
threshold models like those described by Schelling [21] and Granovetter [12],
using a combination of python simulations and collaborative mathematical
analysis (both in WorkingWiki). Other teams are using WorkingWiki to study the
use of non-negative matrix factorization for community detection in marine
ecometagenomics data, the effect of complex contact network structure on
infectious disease dynamics, and spread of coexisting favorable mutations in
spatially localized populations of plants and animals.
Papers completed in WorkingWiki have been published in _Ecological Economics_
[25], _Theoretical Ecology_ [13], and _Journal of Mathematical Biology_ [16]
(and now in this proceedings [26]).
## 6 WorkingWiki and math wikis
WorkingWiki’s makefile rules are straightforward to extend or replace. Its
LaTeX features can be extended to additional formats: this author once created
a structure for working on Sweave documents in WorkingWiki in an afternoon (it
took a few minutes to write the rule to create .tex files automatically from
Sweave files, and a few hours to create a LaTeXML style file to make the
Sweave output look good in the browser). It should be straightforward to
extend WorkingWiki to process any number of specialized document types
conveniently; for instance, allowing users to edit sTex [17] documents and
render them automatically to PDF, browser-ready XHTML, and OMDoc [18] formats.
It could also be used to develop, store, and process documents in computer-
aided theorem-proving systems, or using any other tool that can be invoked
from a UNIX command line to process files. Its uses are open-ended, and may
prove very fruitful to explore.
## References
* [1] Btrfs (b-tree file system). https://btrfs.wiki.kernel.org/index.php/Main_Page. Retrieved July 8, 2011.
* [2] Git: the fast version control system. http://git-scm.com/.
* [3] GNU general public license, version 2. http://www.gnu.org/copyleft/gpl.html.
* [4] OpenWetWare.org. http://openwetware.org.
* [5] Oracle grid engine. (Formerly known as Sun Grid Engine.) http://en.wikipedia.org/wiki/Oracle_Grid_Engine. Retrieved July 8, 2011\.
* [6] Sage: open source mathematics software. http://www.sagemath.org. Retrieved July 8, 2011.
* [7] Tomoyo linux. http://tomoyo.sourceforge.jp/. Retrieved July 8, 2011.
* [8] Ben Collins-Sussman, Brian W. Fitzpatrick, and C. Michael Pilato. Version Control with Subversion. O’Reilly, first edition, June 2004. http://svnbook.red-bean.com/.
* [9] Wayne Davison. Textaid. https://chrome.google.com/webstore/detail/ppoadiihggafnhokfkpphojggcdigllp. Retrieved July 8, 2011.
* [10] Jonathan Dushoff. Nomogram. http://lalashan.mcmaster.ca/theobio/math/index.php/Nomogram. Retrieved July 8, 2011.
* [11] Timothy Gowers and Michael Nielsen. Massively collaborative mathematics. Nature, 461:879–881, 15 October 2009.
* [12] Mark Granovetter. Threshold models of collective behavior. American Journal of Sociology, 1978.
* [13] Daihai He, Jonathan Dushoff, Troy Day, Junling Ma, and David J. D. Earn. Mechanistic modelling of the three waves of the 1918 influenza pandemic. Theoretical Ecology, 4(2):283–288, 2011.
* [14] Hideo Mabuchi lab, Stanford University. Qwiki (quantum wiki). http://qwiki.stanford.edu.
* [15] Christian Höltje. It’s all text! https://addons.mozilla.org/en-US/firefox/addon/its-all-text/. Retrieved July 8, 2011.
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|
arxiv-papers
| 2012-12-10T07:10:51 |
2024-09-04T02:49:39.083147
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lee Worden",
"submitter": "Lee Worden",
"url": "https://arxiv.org/abs/1212.1986"
}
|
1212.2103
|
††thanks: This work is supported by DITANET (novel DIagnostic Techniques for
future particle Accelerators: A Marie Curie Initial Training NETwork), Project
Number ITN-2008-215080
# Interpretation of transverse tune spectra in a heavy-ion synchrotron at high
intensities
R. Singh GSI Helmholzzentrum für Schwerionenforschung mbH, Planckstrasse 1,
64291 Darmstadt, Germany Technische Universität Darmstadt,
Schlossgartenstr.8, 64289 Darmstadt, Germany O. Boine-Frankenheim GSI
Helmholzzentrum für Schwerionenforschung mbH, Planckstrasse 1, 64291
Darmstadt, Germany Technische Universität Darmstadt, Schlossgartenstr.8,
64289 Darmstadt, Germany O. Chorniy P. Forck R. Haseitl W. Kaufmann P.
Kowina K. Lang GSI Helmholzzentrum für Schwerionenforschung mbH,
Planckstrasse 1, 64291 Darmstadt, Germany T. Weiland Technische Universität
Darmstadt, Schlossgartenstr.8, 64289 Darmstadt, Germany
###### Abstract
Two different tune measurement systems have been installed in the GSI heavy-
ion synchrotron SIS-18. Tune spectra are obtained with high accuracy using
these fast and sensitive systems . Besides the machine tune, the spectra
contain information about the intensity dependent coherent tune shift and the
incoherent space charge tune shift. The space charge tune shift is derived
from a fit of the observed shifted positions of the synchrotron satellites to
an analytic expression for the head-tail eigenmodes with space charge.
Furthermore, the chromaticity is extracted from the measured head-tail mode
structure. The results of the measurements provide experimental evidence of
the importance of space charge effects and head-tail modes for the
interpretation of transverse beam signals at high intensity.
###### pacs:
29.20.-c,29.20.D-,41.85.-p,29.27.-a,41.75.Ak,51.75.Cn
## I Introduction
Accurate measurements of the machine tune and of the chromaticity are of
importance for the operation of fast ramping, high intensity ion synchrotrons.
In such machines the tune spread $\delta Q_{x,y}$ at injection energy due to
space charge and chromaticity can reach values as large as 0.5. In order to
limit the incoherent particle tunes to the resonance free region the machine
tune has be controlled with a precision better than $\Delta Q\approx 10^{-3}$.
In the GSI heavy-ion synchrotron SIS-18 there are currently two betatron tune
measurement systems installed. The frequency resolution requirements of the
systems during acceleration are specified as $10^{-3}$, but they provide much
higher resolution ($10^{-4}$) on injection and extraction plateaus. The Tune,
Orbit and Position measurement system (TOPOS) is primarily a digital position
measurement system which calculates the tune from the measured position
Kowina2010 . The Baseband Q measurement system (BBQ) conceived at CERN
performs a tune measurement based on the concept of diode based bunch envelope
detection Gasior2012 . The BBQ system provides a higher measurement
sensitivity than the TOPOS system. Passive tune measurements require high
sensitivity (Schottky) pick-ups, low noise electronics and long averaging time
to achieve reasonable signal-to-noise ratio. For fast tune measurements with
the TOPOS and BBQ systems, which both use standard pick-ups, the beam has to
be excited externally in order to measure the transverse beam signals.
For low intensities the theory of transverse signals from bunched beams and
the tune measurement principles from Schottky or externally excited signals
are well known Chattopadhyay1984 ; Boussard1995 . In intense, low energy
bunches the transverse signals and the tune spectra can be modified
significantly by the transverse space charge force and by ring impedances.
Previously, the effect of space charge on head-tail modes had been the subject
of several analytical and simulation studies Blaskiewicz1998 ; Boine-
Frankenheim2009 ; Burov2009 ; Balbekov2009 ; Kornilov2010 .
Recently, the modification of transverse signals from high intensity bunches
was observed in the SIS-18 for periodically excited beams Singh2012 and for
initially kicked bunches Kornilov2012 , where the modified spectra was
explained in terms of the space charge induced head-tail mode shifts.
This contribution aims to complement the previous studies and extract the
relevant intensity parameters from tune spectra measurements using the TOPOS
and BBQ tune measurement systems. Section II presents the frequency content of
transverse bunched beam signals briefly. Sec. III presents the space charge
and image current effects on tune measurements and respective theoretical
models. Sec. IV report on the experimental conditions and compares the
characteristics of two installations as well as the various excitation
methods. Sec. V presents the experimental results in comparison with the
theoretical estimates of various high intensity effects.
## II Transverse bunch signals
Theoretical and experimental work related to transverse Schottky signals and
beam transfer functions (BTFs) for bunched beam at low intensities can be
found in the existing literature Chattopadhyay1984 ; Sacherer1968 ;
Linnecar1981 . The transverse signal of a beam is generated by the beam’s
dipole moment
$d(t)=\sum_{j=1}^{N}x_{j}(t)I_{j}(t)$ (1)
where $x_{j}(t)$ is the horizontal/vertical offset and $I_{j}(t)$ the current
of the $j^{th}$ particle at the position of a pick-up (PU) in the ring. The
sum extends over all $N$ particles in the detector. The Schottky noise power
spectrum as a function of the frequency $f$ is defined as $S(f)=|d(f)|^{2}$
where $d(f)$ is the Fourier transformed PU signal. For a beam excited by an
external amplitude spectrum $G(f)$, the transverse response function is
defined as $r(f)=d(f)/G(f)$ Borer1979 .
If the transverse signal from a low intensity bunch is sampled with the
revolution period $T_{s}$, then the positive frequency spectrum consists of
one set of equidistant lines
$Q_{k}=Q_{0}+\Delta Q_{k},$ (2)
usually defined as baseband tune spectrum, where $Q_{0}$ is the fractional
part of the machine tune, $\Delta Q_{k}=\pm kQ_{s}$ are the synchrotron
satellites and $Q_{s}$ is the synchrotron tune.
For a single particle performing betatron and synchrotron oscillations, the
relative amplitudes of the satellites are Chattopadhyay1984
$\mid d_{k}\mid\sim\mid J_{k}(\chi/2)\mid$ (3)
where $\chi=2\xi\phi_{m}/\eta_{0}$ is the chromatic phase, $\xi$ is the
chromaticity, $\phi_{m}$ is the longitudinal oscillation amplitude of the
particle and $\eta_{0}$ the frequency slip factor. $J_{k}$ are the Bessel
functions of order $k$. In bunched beams, the relative height and width of the
lines depends on the bunch distribution. The relative height is also affected
by the characteristics of the external noise excitation or by the initial
transverse perturbation applied to the bunch. In the absence of transverse
nonlinear field components the width of each satellite with $k\neq 0$ is
determined by the synchrotron tune spread $\delta
Q_{k}\approx|k|Q_{s}\phi^{2}_{m}/16$. An example tune spectrum obtained from a
simulation code for a Gaussian bunch distribution is shown in Fig. 1 .
Figure 1: Baseband tune spectrum from a simulation code for a low intensity
bunch.
## III Tune spectrum for high intensity
At high beam intensities, the transverse space charge force together with the
coherent force caused by the beam pipe impedance will affect the motion of the
beam particles and also the tune spectrum. The space charge force induces an
incoherent tune shift $Q_{0}-\Delta Q_{sc}$ for a symmetric beam profile of
homogeneous density where
$\Delta
Q_{sc}=\frac{qI_{p}R}{4\pi\epsilon_{0}cE_{0}{\gamma_{0}}^{2}\beta_{0}^{3}\varepsilon_{x}}$
(4)
is the tune shift, $I_{p}$ the bunch peak current, $q$ the particle charge and
$E_{0}=\gamma_{0}mc^{2}$ the total energy. The relativistic parameters are
$\gamma_{0}$ and $\beta_{0}$, the ring radius is $R$ and the emittance of the
rms equivalent K-V distribution is $\varepsilon_{x}$. In the case of an
elliptic transverse cross-section the emittance $\varepsilon_{x}$ in Eq. 4
should be replaced by
$\frac{1}{2}\left(\varepsilon_{x}+\sqrt{\varepsilon_{x}\varepsilon_{y}\frac{Q_{0x}}{Q_{0y}}}\right)$
(5)
For the vertical plane the procedure is the same, with $x$ replaced by $y$.
The image currents and image charges induced in the beam pipe, assumed here to
be perfectly conducting, cause a purely imaginary horizontal impedance
$Z_{x}=-i\frac{Z_{0}}{2\pi(\beta_{0}\gamma_{0}b_{x})^{2}}$ (6)
and real coherent tune shift
$\Delta Q_{c}=-i\frac{qIR^{2}Z_{x}}{2Q_{x0}\beta_{0}E_{0}}$ (7)
For a round beam profile with radius $a$ and pipe radius $b$ the coherent tune
shift is smaller by $\Delta Q_{c}=\frac{a^{2}}{b^{2}}\Delta Q_{sc}$ than the
space charge tune shift. Therefore the contribution of the pipe is especially
important for thick beams $(a\sim b)$ at low or medium energies.
In the presence of incoherent space charge, represented by the tune shift
$\Delta Q_{sc}$, or pipe effects, represented by the real coherent tune shift
$\Delta Q_{c}$, the shift of the synchrotron satellites in bunches can be
reproduced rather well by Boine-Frankenheim2009
$\Delta Q_{k}=-\frac{{\Delta Q_{sc}+\Delta Q_{c}}}{2}\pm\sqrt{(\Delta
Q_{sc}-\Delta Q_{c})^{2}/4+(kQ_{s})^{2}}$ (8)
where the sign $+$ is used for $k>0$. For k=0 one obtains $\Delta
Q_{k=0}=-\Delta Q_{c}$. The above expression represents the head-tail
eigenmodes for an airbag bunch distribution in a barrier potential
Blaskiewicz1998 with the eigenfunctions
$\bar{x}(\phi)=\cos(k\pi\phi/\phi_{b})\exp(-i\chi\phi/\phi_{b})$ (9)
where $\bar{x}$ is the local transverse bunch offset,
$\chi=\xi\phi_{b}/\eta_{0}$ is the chromatic phase, $\phi_{b}$ is the full
bunch length and $\eta_{0}$ is slip factor. The head-tail mode frequencies
obtained from Eq. 8 are shown in Fig. 2 . In Ref. Blaskiewicz1998 the
analytic solution for the eigenvalues Eq. 8 is obtained from a simplified
approach, where the transverse space charge force is assumed to be constant
for all particles. This assumption is correct if there are only dipolar
oscillations. In Ref. Boine-Frankenheim2009 it is has been pointed out, that
in the presence of space charge there is an additional envelope oscillation
amplitude. For the negative-$k$ eigenmodes the envelope contribution dominates
and therefore those modes disappear from the tune spectrum. In Ref. Boine-
Frankenheim2009 Eq. 8 has been successfully compared to Schottky spectra
obtained from 3D self-consistent simulations for realistic bunch distributions
in rf buckets. Analytic and numerical solutions for Gaussian and other bunch
distribution valid for $q_{sc}\gg 1$ were presented in Burov2009 ;
Balbekov2009 .
In an rf bucket the synchrotron tune $Q_{s}$ is a function of the synchrotron
oscillation amplitude $\hat{\phi}$. For short bunches $Q_{s}$ corresponds to
the small-amplitude synchrotron tune
$Q_{s0}^{2}=\frac{qV_{0}h|\eta_{0}|}{2\pi m\gamma\beta^{2}c^{2}}$ (10)
where $V_{0}$ is the rf voltage amplitude and $h$ is the rf harmonic number.
Figure 2: Head-tail mode frequencies as a function of the space charge
parameter using Eq. 8 . The red curves represent the result obtained for
$q_{c}=q_{sc}/10$.
For head-tail modes the space charge parameter is defined as a ratio of the
space-charge tune shift ( Eq. 4 ) to the small-amplitude synchrotron tune,
$q_{sc}=\frac{\Delta Q_{sc}}{Q_{s0}}$ (11)
and the coherent intensity parameter as,
$q_{c}=\frac{\Delta Q_{c}}{Q_{s0}}$ (12)
An important parameter for head-tail bunch oscillations in long bunches is the
effective synchrotron frequency which will be different from the small-
amplitude synchrotron frequency in short bunches. For an elliptic bunch
distribution (parabolic bunch profile) with the bunch half-length
$\phi_{m}=\sqrt{5}\sigma_{l}$ (rms bunch length $\sigma_{l}$), one obtains the
approximate analytic expression for the longitudinal dipole tune boine_rf2005
,
$\frac{Q_{s1}}{Q_{s0}}=\sqrt{1-\frac{\sigma_{l}^{2}}{2}}$ (13)
Using $Q_{s1}$ instead of $Q_{s0}$ in Eq. 8 shows a much better agreement with
the simulation spectra for long bunches in rf buckets (see Ref. Kornilov2012
).
Figure 3: Simulation result for $q_{sc}=2$. Figure 4: Simulation result for
$q_{sc}=10$.
For Gaussian bunches with a bunching factor $B_{f}=0.3$ ($B_{f}=I_{0}/I_{p}$,
$I_{0}$ is the dc current), the transverse tune spectra obtained from PATRIC
simulations Boine-Frankenheim2009 for different space charge factors and thin
beams ($q_{c}=0$) are shown in Figures 1, 3 and 4. The dotted vertical lines
indicate the positions of the head-tail tune shifts obtained from Eq. 8 with
$Q_{s}=Q_{s1}$. For the low-$k$ satellites there is a good agreement between
Eq. 8 and the simulation results. Lines with $k>2$ can only barely be
identified in the simulation spectra. The positions of the satellites for
$k=0,1,2$ together with the predicted head-tail tune shifts from Eq. 8 are
shown in Fig. 5 . The error bars indicate the obtained widths of the peaks in
the tune spectra.
Figure 5: Head-tail tune shifts and their width obtained from the simulations
for $q_{c}=0$. The error bars indicate the widths of the lines.
It is important to notice that the simulations for moderate space charge
parameters ($q_{sc}\lesssim 10$) require a 2.5D self-consistent space charge
solver. The theoretical studies rely on the solution of the Möhl-Sch nauer
equation Moehl1995 , which assumes a constant space charge tune shift for all
transverse particle amplitudes. In contrast to the self-consistent results,
PATRIC simulation studies using the Möhl-Sch nauer equation gave tune spectra
with pronounced, thin satellites also for large $k$. In order to account for
the intrinsic damping of head-tail modes Burov2009 ; Balbekov2009 ;
Kornilov2010 , which is the main cause of the peak widths obtained from the
simulations, a self-consistent treatment is required.
For thick beams (here $q_{c}=0.15q_{sc}$, which corresponds to the conditions
at injection in the SIS-18) the positions of the synchrotron satellites
obtained from the simulation are indicated in Fig. 6 . Again, the error bars
indicate the widths of the peaks. From the plot we notice an increase in the
spacing between the $k=0,1,2$ satellites, relative to the analytic expression.
Also the peak width for $k=1,2$ does not shrink with increasing $q_{sc}$.
Figure 6: Head-tail tune shifts and their width obtained from the simulations
for $q_{c}=0.15q_{sc}$. The error bars indicate the widths of the peaks.
The tune spectra obtained for $q_{sc}=3$, $q_{sc}=5$ and $q_{sc}=10$ are shown
in Fig. 7 , Fig. 8 and Fig. 9 . One can observe that for thick beams (here
$a\approx 0.4b$) the $k=1$ peak remains very broad up to $q_{sc}=10$.
Figure 7: Tune spectrum obtained from the simulation for $q_{sc}=3$ and
$q_{c}=0.15q_{sc}$. Figure 8: Tune spectrum obtained from the simulation for
$q_{sc}=5$ and $q_{c}=0.15q_{sc}$. Figure 9: Tune spectrum obtained from the
simulation for $q_{sc}=10$ and $q_{c}=0.15q_{sc}$.
This observation is consistent with a very simplified picture for the upper
$q_{sc}$ threshold for the intrinsic Landau damping of head-tail modes. For a
Gaussian bunch profile, the maximum incoherent tune shift, including the
modulation due to the synchrotron oscillation is
$\Delta Q_{\max}=-\Delta Q_{sc}+kQ_{s}$ (14)
where $\Delta Q_{sc}$ is determined by Eq. 4. The minimum space charge tune
shift is (see Refs. Balbekov2009 ; Kornilov2010 )
$\Delta Q_{\min}=-\alpha q_{sc}Q_{s}+kQ_{s}$ (15)
where $\alpha$ is determined from the average of the space charge tune shift
along a synchrotron oscillation with the amplitude $\hat{\phi}=\phi_{m}$. For
a parabolic bunch we obtain $\alpha=0.5$. For a Gaussian bunch and
$\hat{\phi}=3\sigma_{l}$, we obtain $\alpha=0.287$. Each band of the
incoherent transverse spectrum has a lower boundary determined by the maximum
tune shift $\Delta Q_{\max}$ and an upper boundary determined by $\Delta
Q_{\min}$. Landau damping, in its very approximate treatment, requires an
overlap of the coherent peak with the incoherent band. The head-tail tune for
low $q_{sc}$ can be approximated as
$\Delta Q_{k}=-\frac{1}{2}\left(\Delta Q_{sc}+\Delta Q_{c}\right)+kQ_{s}$ (16)
The distance between the coherent peak and the upper boundary of the
incoherent band for fixed $k$ is
$\delta Q_{k}=\left(\frac{1}{2}-\alpha\right)\Delta Q_{sc}+\frac{1}{2}\Delta
Q_{c}$ (17)
For large $q_{sc}$ the head-tail modes with positive $k$ converge towards
$Q_{k}=-\Delta Q_{c}/2$. For a given $k$ the mode is still inside the
incoherent band if
$k\gtrsim\alpha q_{sc}-\frac{1}{2}q_{c}$ (18)
holds. In order to illustrate the above analysis, the incoherent band for
$k=2$ is shown in Fig. 10 (shaded area). For $q_{c}=0$ the coherent head-tail
mode frequency crosses the upper boundary of the band at $q_{sc}\approx 4.5$.
For $q_{c}=0.15q_{sc}$ the $k=2$ head-tail mode remains inside the incoherent
band until $q_{sc}\approx 12$. For the $k=1$ modes the above analysis leads to
thresholds of $q_{sc}\approx 2$ (thin beams) and $q_{sc}\approx 6$ for
$q_{c}=0.15q_{sc}$.
Figure 10: Head-tail mode frequencies as a function of the space charge
parameter. The grey shaded area indicates the incoherent band for a Gaussian
bunch and $k=2$. The red curves represent the head-tail mode frequencies
obtained for $q_{c}=0.15q_{sc}$.
## IV Measurement setup for transverse bunch signals
In this section, a brief description of the transverse beam excitation
mechanisms as well as the two different tune measurement systems, TOPOS and
BBQ, in the SIS-18 is given. Further the experimental set-up, typical beam
parameters and uncertainty analysis of the measured beam parameters is
discussed.
### IV.1 Transverse Beam Excitation
The electronics used for beam excitation consist of a signal generator
connected to two $25$ W amplifiers which feed power to $50\Omega$ terminated
stripline exciters as shown in Fig. 11 . Excitation types such as band limited
noise and frequency sweep are utilized at various power levels to induce
coherent oscillations.
#### IV.1.1 Band limited noise:
Band limited noise is a traditionally used beam excitation system for slow
extraction in the SIS-18. The RF signal is mixed with Direct Digital Synthesis
(DDS) generated fractional tune frequency, resulting in RF harmonics and their
respective tune sidebands. This signal is further modulated by a pseudo-random
sequence resulting in a finite band around the tune frequency. The width of
this band is controlled by the frequency of the pseudo-random sequence.
Typical bandwidth of band limited exciter is $\approx 5\%$ of the tune
frequency. There are two main advantages of this system; first it is an easily
tunable excitation source available during the whole acceleration ramp and
second, the band limited nature of this noise results in an efficient
excitation of the beam in comparison to white noise excitation. The main
drawback is the difficulty in correlating the resultant tune spectrum with the
excitation signal.
#### IV.1.2 Frequency sweep:
Frequency sweep (chirp/harmonic excitation) using a network analyzer for BTF
measurements is an established method primarily for beam stability analysis
Boussard1995 . However, using this method for tune measurements during
acceleration is not trivial, and thus the method is not suitable for tune
measurements during the whole ramp cycle. Nevertheless, this method offers
advantages compared to the previous excitation method for careful
interpretation of tune spectrum in storage mode, e.g., injection plateau or
extraction flat top. Thus frequency sweep is used during measurements at
injection plateau to compare and understand the dependence of tune spectra on
the type of excitation.
### IV.2 TOPOS
Following the beam excitation, the signals from each of the 12 shoe-box type
BPMs Forck2004 at SIS-18 pass through a high dynamic range (90 dB) and
broadband (100 MHz) amplifier chain from the synchrotron tunnel to the
electronics room, where the signals are digitized using fast 14 bit ADCs at
125 MSa/s. Bunch-by-bunch position is calculated from these signals using
FPGAs in real time and displayed in the control room. The spatial resolution
is $\approx$0.5 mm in bunch-by-bunch mode. Further details can be found in
Kowina2010 . Hence, TOPOS is a versatile system which provides accurate bunch-
by-bunch position and longitudinal beam profile. This information is analyzed
to extract non trivial parameters like betatron tune, synchrotron tune, beam
intensity evolution etc.
Figure 11: Scheme of the excitation system and the Tune, Orbit and Position
(TOPOS) measurement system.
### IV.3 BBQ
The BBQ system is a fully analog system and its front end is divided into two
distinct parts; a diode based peak detector and an analog signal processing
chain consisting of input differential amplifier and a variable gain filter
chain of 1 MHz bandwidth. The simple schematic of BBQ system configuration at
SIS-18 is shown in Fig. 12 and the detailed principle of operation can be
found in Ref. Gasior2012 .
Figure 12: BBQ: Baseband Q measurement system. Diode detectors (a) and signal
chain (b).
### IV.4 Comparison of TOPOS and BBQ
The sensitivity of BBQ has been measured to be $\approx$10-15 dB higher than
that of TOPOS under the present configuration. The main reason for the
difference is the relative bandwidth of the two systems and their tune
detection principles. In BBQ, the tune signal is obtained using analog
electronics (diode based peak detectors and differential amplifier)
immediately after the BPM plates, while in TOPOS position calculation is done
after passing the whole bunch signal through a wide bandwidth amplifier chain.
Even though the bunches are integrated to calculate position in TOPOS which
serves as a low pass filter, the net signal-to-noise ratio is still below BBQ.
Operations similar to BBQ could also be performed digitally in TOPOS to obtain
higher sensitivity but would require higher computation and development costs.
TOPOS can provide individual tune spectra of any of the four bunches in the
machine, while BBQ system provides ”averaged” tune spectra of all the bunches.
Both the systems have been benchmarked against each other. Tune spectra shown
in Sec. V are mostly from the BBQ system while TOPOS is primarily used for
time domain analysis, nevertheless this will be pointed out when necessary.
### IV.5 Beam parameters during the measurements
Experiments were carried out using N7+ and U73+ ion beams at the SIS-18
injection energy of 11.4 MeV/u. The data were taken during 600 ms long
plateaus. At injection energy space charge effects are usually strongest. Four
bunches are formed from the initially coasting beam during adiabatic RF
capture. The experiment was repeated for different injection currents. At each
intensity level several measurements were performed with different types and
levels of beam excitation in both planes. Tune measurements were done
simultaneously using the TOPOS and BBQ systems. The beam current and the
transverse beam profile are measured using the beam current transformer
Reeg2001 and the ionization profile monitor (IPM) Giacomini2004
respectively. An examplary transverse beam profile is shown in Fig. 13 . The
dipole synchrotron tune $(Q_{s1})$ is deduced using the residual longitudinal
dipole fluctuations of the bunches. $Q_{s1}$ has been used as an effective
synchrotron tune for all experimental results and will be referred as $Q_{s}$
from hereon. The momentum spread is obtained from longitudinal Schottky
measurements Caspers2008 . Important beam parameters during the experiment are
given in Tab. 1 and Tab. 2 . It is important to note that all the parameters
required for analytical determination of $q_{sc},q_{c}$ are recorded during
the experiments.
Table 1: Beam parameters during the $N^{7+}$ experiment Beam/Machine parameters | Symbols | Values
---|---|---
Atomic mass | $A$ | 238
Charge state | $q$ | 73
Kinetic energy | $E_{kin}$ | 11.4 MeV/u (measured
Number of particles | $N_{p}$ | $1,5,12\cdot 10^{8}$ (measured)
Tune | $Q_{x},Q_{y}$ | 4.31, 3.27 (set value)
Chromaticity | $\xi_{x},\xi_{y}$ | -0.94, -1.85 (set value)
Transverse emittance | $\epsilon_{x},\epsilon_{y}(2\sigma)$ | $45,22$ mm-mrad (measured)
Slip factor | $\eta$ | 0.94
Bunching factor | $B_{f}$ | 0.4 (measured)
Synchrotron tune | $Q_{s0},Q_{s1}$ | 0.007,0.0065 (measured)
Momentum spread | $\frac{\Delta p}{p}$ | 0.001 (measured)
Lattice parameter | $\beta_{x,trip}$, $\beta_{y,trip}$ | 5.49, 7.76
Table 2: Beam parameters during the $N^{7+}$ experiment Beam/Machine Parameters | Symbols | Values
---|---|---
Atomic mass | $A$ | 14
Charge state | $q$ | 7
Kinetic energy | $E_{kin}$ | 11.56 MeV/u (measured)
Number of particles | $N_{p}$ | $3,6,11,15\cdot 10^{9}$ (measured)
Tune | $Q_{x},Q_{y}$ | 4.16, 3.27 (set value)
Chromaticity | $\xi_{x},\xi_{y}$ | -0.94, -1.85 (set value)
Transverse emittance | $\epsilon_{x},\epsilon_{y}(2\sigma)$ | $33,12$ mm-mrad (measured)
Slip factor | $\eta$ | 0.94
Bunching factor | $B_{f}$ | 0.37 (measured)
Synchrotron tune | $Q_{s0},Q_{s1}$ | 0.006,0.0057 (measured)
Momentum spread | $\frac{\Delta p}{p}$ | 0.0015 (measured)
Lattice parameter | $\beta_{x,trip}$, $\beta_{y,trip}$ | 5.49, 7.76
From Tab. 1 and Tab. 2 one can estimate that in the measurements the head-tail
space charge and image current parameters were in the range $q_{sc}\lesssim
10$ and $q_{c}\lesssim 0.2q_{sc}$ for the horizontal and vertical planes.
Figure 13: Normalized transverse beam profile in vertical plane at
$N^{7+}=15\cdot 10^{8}$ ions. The dotted lines shows the normal distribution
for the rms width obtained by evaluation of beam profile around its centre.
## V Experimental Observations
Tune spectra measurements at different currents are presented and interpreted
in comparison with the predictions of Sec. III . The effect of different
excitation types and power on the transverse tune spectra is studied.
Transverse impedances in both planes are obtained from the coherent tune
shifts. Incoherent tune shifts are obtained from the relative frequency shift
of head-tail modes in accordance with Eq. 8 . Chromaticity is measured using
head-tail eigenmodes and the obtained relative height of the observed peaks
for different chromaticities is analyzed.
### V.1 Modification of the tune spectrum with intensity
Figure 14: Horizontal tune spectra for $U^{73+}$ ions and beam parameters
given in Tab. 1 (see text). The dashed lines indicate the head-tail tune
shifts from Eq. 8 .
Fig. 14 shows the horizontal tune spectra obtained with the BBQ system using
band-width limited noise at different intensities. Fig. 14 (a) shows the
horizontal tune spectrum at low intensity. Here the $k=1,0,-1$ peaks are
almost equidistant, which is expected for low intensity bunches. The space
charge parameter obtained using the beam parameters and Eq. 4 is
$q_{sc}\approx 0.15$. The vertical lines indicate the positions of the
synchrotron satellites obtained from Eq. 8 (with $Q_{s}=Q_{s1}$). Fig. 14 (b)
shows the tune spectrum at moderate intensity ($q_{sc}\approx 0.7$). The
$k=2,-2$ peaks can both still be identified. Fig. 14 (c) shows the tune
spectra at larger intensity ($q_{sc}\approx 1.7$). An additional peak appears
between the $k=0$ and $k=-1$ peaks which can be attributed to the mixing
product of diode detectors (since at this intensity $30-40V$ acts across the
diodes pushing it into the non-linear regime). The $k=0,1,2$ peaks can be
identified very well, whereas the amplitudes of the lines for negative $k$
already start to decrease (see Sec. III ). In the horizontal plane the effect
of the pipe impedance and the corresponding coherent tune shift can usually be
neglected because of the larger pipe diameter.
Figure 15: Vertical tune spectra for $N^{7+}$ ions and beam parameters given
in Tab. 2 . The dashed lines indicate the head-tail tune shifts from Eq. 8 .
Fig. 15 shows the vertical tune spectrum obtained by the BBQ system with band
limited noise excitation for $N^{7+}$ beams with $q_{sc}$ values larger than
$2$. Here the negative modes $(k<0)$ could not be resolved anymore. In the
vertical plane the coherent tune shift is larger due to the smaller SIS-18
beam pipe diameter ($q_{c}\approx q_{sc}/10$). The shift of the $k=0$ peak due
to the effect of the pipe impedance is clearly visible in Fig. 15 .
In the measurements the width of the peaks is determined by the cumulative
effect of non-linear synchrotron motion, non-linearities of the optical
elements, closed orbit distortion, tune fluctuation during the measurement
interval as well as due to the intrinsic Landau damping ( Sec. III ). From the
comparison to the simulations we conclude that the intrinsic Landau damping is
an important contribution to the width of the $k=1,2$ peaks.
### V.2 Determination of coherent and incoherent tune shifts
The coherent tune shift $\Delta Q_{c}$ can be obtained by measuring shift of
the $k=0$ line as a function of the peak bunch current as shown in Fig. 16 .
The transverse impedance is obtained by a linear least square error fit of the
measured shifts in both planes to Eq. 6 and Eq. 7 . The impedance values are
obtained in the horizontal and vertical planes at injection energy are $0.23$
M$\Omega/m^{2}$ and $1.8$ M$\Omega/m^{2}$ respectively, which agrees very well
with the expected values for the average beam pipe radii of the SIS-18.
Figure 16: Coherent tune shift obtained from the measurement for the
horizontal and for the vertical planes as a function of the peak beam current.
The dotted lines correspond to a linear least square error fit. The error bars
in the horizontal plane are due to uncertainties in the current measurements.
In the vertical plane errors result from the width of $k=0$ mode. The FFT
resolution is $\approx 5\cdot 10^{-4}$ and is always kept higher than the mode
width. Figure 17: The measured positions of the peaks in the horizontal tune
spectra for different $U^{73+}$ beam intensities together with the analytical
curves from Eq. 8 using the space charge tune shift estimated from the beam
parameters in Tab. 1 . The dotted line corresponds to the incoherent tune
shift. The error bars for the vertical plane correspond to the width of
measured modes (see text). In the horizontal plane the error bars are
estimated by the propagation of uncertainties (see text). Figure 18: This plot
shows the predicted shifts from analytical Eq. 8 and measured head-tail mode
frequencies are overlaid in vertical plane for $U^{73+}$ ion beam at various
current levels ( Tab. 1 ). Figure 19: This plot shows the predicted shifts
from analytical Eq. 8 and measured head-tail mode frequencies are overlaid in
horizontal plane for $N^{7+}$ ion beam at various current levels ( Tab. 2 ).
Figure 20: This plot shows the predicted shifts from analytical Eq. 8 and
measured head-tail mode frequencies are overlaid in vertical plane for
$N^{7+}$ ion beam at various current levels ( Tab. 2 ).
Figures 17, 18, 19 and 20 show the measured positions of the peaks in the tune
spectra for different intensities. In comparison the analytical curves (solid
lines) obtained from Eq. 8 for the head-tail tune shifts are plotted using
$q_{sc}$ estimated from the beam parameters in Tab. 1 and Tab. 2 for each
intensity. The error bars in the vertical plane ($\delta q_{k}=\frac{\Delta
Q_{k}}{Q_{s}}$) correspond to the $3$ dB width of the measured peak due to
accumulation of various effects (see subsection V.1 ). In the horizontal
plane, error bars ($\delta q_{sc}$) are estimated by propagation of parameter
uncertainties mentioned in Measurement uncertainties .
In this subsection, we introduce another space charge parameter $q_{sc,m}$
which is the measured space charge parameter using the following method. It is
not to be confused with $q_{sc}$ which is predicted for a given set of beam
parameters by Eq. 4 . The incoherent space charge tune shift can be determined
directly from the tune spectra by measuring the separation between the $k=0$
and $k=1$ peaks, i.e. $(q_{k01}=\frac{\Delta Q_{k01}}{Q_{s}}$) and fitting it
with the parameter $q_{sc}$ in the predictions from Eq. 8 . The value of
$q_{sc}$ for the best fit is denoted as $q_{sc,m}$.
$\displaystyle q_{sc,m}=\frac{1-q_{k01}^{2}}{\gamma q_{k01}}\hskip
142.26378pt0\leq q_{k01}\leq 1$ (19)
Eq. 19 is obtained by rearranging Eq. 8 for $k=0,1$ while
$\gamma=\frac{q_{sc}-q_{c}}{q_{sc}}$. The linearized absolute error on
measured $q_{sc,m}$ ($\delta q_{sc,m}$) is given by
$\displaystyle\delta q_{sc,m}=\frac{-(1+q_{k01}^{2})}{\gamma
q_{k01}^{2}}\cdot\delta q_{k01}$ (20)
$\delta q_{k01}$ is given by either the width of the $k=0,1$ lines or by the
frequency resolution of the system. In a typical tune spectra measurement
using data from 4000 turns $\delta q_{k01}\approx 0.04$. The absolute error is
a non-linear function of $q_{sc}$ in accordance to the Eq. 19 . It is possible
to define the upper limit of $q_{sc}$ where this method is still adequate
based on the system resolution and Eq. 19 . If we define a criterion that,
$q_{k01}\gtrsim\delta q_{k01}$ to resolve the head-tail modes. This gives the
limit to be $q_{sc}\lesssim 8$ where the measurement error is still within the
defined criterion.
Fig. 21 shows a plot of the predicted space charge tune shifts ($q_{sc}$)
versus the ones measured from the tune spectra using the above procedure
($q_{sc,m}$). For $q_{sc}\lesssim 3.5$ the space charge tune shifts measured
from the tune spectra are systematically lower by a factor $=0.74$ than the
predicted shifts. It is shown by the dotted line in Fig. 21 which is obtained
by total least squares fit of the measured data points. For larger $q_{sc}$
the factor decreases to $\approx 0.4$. Thus the method for measuring the
incoherent tune shift based on head-tail tune shifts is found to be
satisfactory only in the range $q_{sc}\lesssim 3.5$. A possible explanation is
the effect of the pipe impedance. Similar observations are made by the results
of self-consistent simulations in Sec. III , where for $q_{sc}\gtrsim 2$ the
separation of the $k=0$ and $k=1$ peaks observed is underestimated by Eq. 8
(see Fig. 6 ).
Figure 21: Combining the results from the Figures 17, 18, 19 and 20, a plot of
predicted $q_{sc}$ using Eq. 4 against measured $q_{sc}$ using the distance
between modes $k=0$ and $k=1$ is obtained.
### V.3 Effect of excitation parameters on tune spectrum
Figure 22 presents the tune spectra obtained from BBQ system at various
excitation power levels of band limited noise. The beam is excited with
$0.25,1.0$ and $2.25$ mW/Hz power spectral density on a bandwidth of $10$ KHz.
Signal-to-noise ratio (SNR) increases with excitation power whereas the
spectral position of various modes is independent of excitation power.
Figure 22: Tune spectra at $5\cdot 10^{8}U^{73+}$ particles with beam excited
at three different excitation amplitudes. The beam is excited with $0.25,1.0$
and $2.25$ mW/Hz power (blue, red and black respectively). The dotted lines
mark the relative positions of the head-tail modes. The frequencies of various
head-tail modes are unaffected.
Beam excitation using two other excitation types i.e. frequency sweep and
white noise is also performed to study the effect of excitation type on the
tune spectra. Figure 23 shows the tune spectra under same beam conditions for
different types of beam excitation obtained from the BBQ system. The
frequencies of various modes in the tune spectra are independent of the type
of excitation. The signal-to-noise ratio (SNR) is optimum for band limited
noise due to long averaging time compared to ”one shot” spectra from sweep
excitation.
### V.4 Time domain identification of head tail modes
Figure 24 shows the 2-D contour plot for frequency sweep excitation in
vertical plane obtained from the TOPOS system, where various head-tail modes
are individually excited as the excitation frequency crosses them. Frequency
sweep excitation allows resolving the transverse center of mass along the
bunch for various modes which helps in identifying each head-tail mode in time
domain. This serves as a direct cross-check for the spectral information and
leaves no ambiguity in identification of the order (k) of the modes. Figure 25
shows the corresponding transverse center of mass along the bunch for $k=0,1$
and $2$ at the excited time instances. This method works only with sweep
excitation and requires high signal-to-noise ratio in the time domain, which
amounts to higher beam current or high excitation power.
Figure 23: Horizontal tune spectra over time with $U^{73+}$ at $5\cdot 10^{8}$
particles with band limited noise (RF Noise), frequency sweep (chirp noise)
and white noise (top to bottom). The dotted lines mark the relative positions
of the head-tail modes. Figure 24: Vertical tune spectra with frequency sweep
over time at $N^{7+}$ with $1.5\cdot 10^{9}$ particles. The various modes get
individually excited as the sweep frequency (marked by the arrow) coincides
with the mode frequency. The $k=0,1,2$ modes are marked. Symmetric sidebands
due to coherent synchrotron oscillations are visible around the sweep
frequency spaced with $Q_{s}$ and are clearly distinguished from the head-tail
modes. Figure 25: Transverse center of mass along the bunch for $k=0,1$ and
$2$ modes corresponding to Figure 24 . Frequency sweep is used to excite the
beam which enables to resolve the distinct head-tail modes temporally. The rms
bunch length ($2\sigma_{rms}$) is indicated.
### V.5 Measurement of the chromaticity
As highlighted in the previous section, the frequency sweep allows to resolve
the different head-tail modes both spectrally and temporally. This procedure
can be used for the precise determination of the chromaticity by fitting the
analytical expression for the head-tail eigenfunction Eq. 9 to the measured
bunch offset, with the chromaticity ($\xi$) as the fit parameter as shown in
Fig. 26 for $k=0,1$ and $2$. The measured chromaticity is independent of the
order of head-tail eigenfunction used to estimate it.
The fitting method is shown in Eq. 21 ; the head-tail eigenfunction from Eq. 9
is multiplied with the beam charge profile $\hat{S(t)}$ and corrected for the
beam offset $\Delta{x}$ at the BPM where the signal is measured.
$\displaystyle
F(\phi,\xi,A)=\Delta{x}\cdot\hat{S(t)}\cdot(1+A\cdot\bar{x}(\phi))$ (21)
$\displaystyle E(\xi,A)=(M(\phi)-F(\phi,\xi))^{2}$
The fit error $E(\xi,A)$ is reduced as a function of independent variables;
chromaticity $\xi$ and head-tail mode amplitude $A$. The fit error gives the
goodness of the fit. It is used to determine the error bars on measured
chromaticity. This method has been utilized for the determination of
chromaticity at SIS-18 as shown in Fig. 27 . The set and the measured
chromaticity can be fitted by linear least squares to obtain the form
$\xi_{s,y}=1.187\xi_{m,y}+0.804$ as shown by red dashed line in Fig. 27 . Fig.
27 also shows a coherent tune shift due to change in sextupole strength which
is used to adjust the chromaticity. This is due to uncorrected orbit
distortions during these measurements. These chromaticity measurements agree
with the previous chromaticity measurements using conventional methods
Paret2009 .
It is also possible to determine the relative response amplitude of each head-
tail mode to the beam excitation both in time and in frequency domain with
TOPOS. Fig. 28 shows the tune spectrum obtained with sweep excitation for
different chromaticity values. The beam parameters are kept the same
($N^{7+},14\cdot 10^{9},q_{sc}\approx 10$). The spectral position and and
relative amplitude of each head-tail mode peak are confirmed using the time
domain information (see Figure 25 ). In Fig. 29 the single particle response
amplitudes for different $k$ ( Equation 3 ) are plotted as a function of the
chromaticity. The measured relative amplitudes are indicated by the colored
symbols. The comparison indicates that the simple single particle result (
Equation 3 ) describes quite well the dependence of the relative height of the
peaks obtained from the TOPOS measurement.
Figure 26: Analytical curves from Eq. 9 (dotted) are fitted to the local
transverse offset for the $k=0$ and $k=1$ modes with the head-tail phase shift
as the fit parameter. This method has been used for the precise determination
of the chromaticity. Figure 27: Chromaticity measurement using the method
shown in Figure 25 over the possible range of operation at SIS-18. The black
dashed line marks the measured natural vertical chromaticity of SIS-18. The
error bars for the chromaticity measurement are determined mainly by the
electronic noise in the amplifier chain of the TOPOS system. Figure 28:
Relative signal amplitudes of various head-tail modes are plotted with respect
to the measured chromaticity. The centroid of the spectrum moves toward higher
order modes with increase in chromaticity. Figure 29: Relative signal
amplitudes of various head-tail modes are plotted with respect to the
coefficient of Bessel’s function in Equation 3 $(\chi_{f}=\chi/2)$. All the
modes are relative to k=1 mode, which has a significant amplitude during the
whole measurement range.
## VI Application to tune measurements in SIS-18
In this section we will discuss the application of our results to tune
measurements in the SIS-18 and in the projected SIS-100, as part of the FAIR
project at GSI FAIR2010 . As shown in the previous section the relative
amplitudes of the synchrotron satellites in the tune spectra are primarily a
function of chromaticity and possibly the excitation mechanism. In order to
determine the coherent tune with high precision the position of the $k=0$ mode
has to be measured. Depending on the machine settings, if the relative height
of the $k=0$ peak with respect to the other modes is small, then the $k=0$
mode may not be visible at all. To estimate the bare tune frequency in this
case, the information of space charge parameter, coherent tune shift and
chromaticity are all simultaneously required with good precision.
Another important point is the tune measurement during acceleration. The space
charge parameter for $1\cdot 10^{10}Ar^{18+}$ stored ions in the SIS-18 from
injection to extraction reduces only by $\approx 20\%$ as shown in Fig. 30 .
The dynamic shift of head-tail modes during acceleration is shown in Fig. 31
obtained from TOPOS system under same conditions. The asymmetry of $k=1,-1$
modes around $k=0$ mode can only be understood in view of the space charge
effects predicted by Eq. 8 . Thus, a correct estimate of this parameter plays
an important role in understanding the tune spectra not only during dedicated
experiments on injection plateau, but also during regular operations.
The measurement time required to resolve the various head-tail modes ($\Delta
Q_{k}$) is a complex function of $Q_{s}$, $q_{sc}$, beam intensity and
excitation power. To give some typical numbers for SIS-18; on a measurement
time of 600 ms on the injection plateau, if one spectrum is obtained in
$\approx 20$ ms ($\approx 4000$ turns), an improvement of factor $\approx 6$
in SNR by averaging 30 spectra. Following the calculations in subsection V.2 ,
$q_{sc}\lesssim 8$ can be resolved under typical injection operations.
However, the constraints on measurement time are much higher during
acceleration, where the tune/revolution frequency increases due to
acceleration. This allows the measurement of a single spectrum typically only
over $500-1000$ turns (depends on ramp rate as well). There are no averaging
possibilities since the tune is moving during acceleration due to dynamic
changes in machine settings as seen in Fig. 31 . In addition, the synchrotron
tune reduces with acceleration making it practically very difficult to resolve
the fine structure of the head-tail modes for $q_{sc}\gtrsim 2$.
Figure 30: Change in synchrotron tune and space charge tune shift on
acceleration from injection to extraction for $Ar^{18+}$ with $1\cdot 10^{10}$
particles. The estimated space charge parameter reduces by $\approx 20\%$ in
this typical case from injection at 11.4 MeV/u to 300 MeV/u at extraction.
Figure 31: The movement of head-tail modes during acceleration. The head-tail
modes are getting closer since the synchrotron tune reduces with acceleration,
but the asymmetry of the $k=1$ and $k=-1$ modes around $k=0$ is maintained
throughout the ramp which depends primarily on the space charge parameter
$(q_{sc})$.
## VII Conclusion
Two complimentary tune measurement systems, TOPOS and BBQ, installed in the
SIS-18 are presented. Analytical as well as simulation models predict a
characteristic modification of the tune spectra due to space charge and image
current effects in intense bunches. The position of the synchrotron satellites
corresponds to the head-tail tune shifts and depends on the incoherent and the
coherent tune shifts. The modification of the tune spectra for different bunch
intensities has been observed in the SIS-18 at injection energy, using the
TOPOS and BBQ systems. From the measured spectra the coherent and incoherent
tune shift for bunched beams in SIS-18 at injection energies were obtained
experimentally using the analytic expression for the head-tail tune shifts.
Head-tail modes were individually excited and identified in time domain and
correlated with the spectral information. A novel method for determination of
chromaticity based on gated excitation of individual head tail mode is shown.
The dependence of the relative amplitudes of various head-tail modes on
chromaticity is also studied. The systems were compared against each other as
well as with different kinds of excitation mechanisms and their respective
powers. These measurements give a clear interpretation of tune spectra at all
stages during acceleration under typical operating conditions. The
understanding to tune spectra provides an important input to new developments
related to planned transverse feedback systems for SIS-18 and SIS-100. The
measurement systems also open new possibilities for detailed beam
investigations as demonstrated in this contribution.
## Acknowledgement
We thank Marek Gasior of CERN BI Group for the help in installation of BBQ
system and many helpful discussions on the subject. GSI operations team is
also acknowledged for setting up the machine. We also thank Klaus-Peter Ningel
from the GSI-RF group who helped setting up the amplitude ramp for the
adiabatic bunching.
## Appendix
### Calculation of bunching factor
Figure 32: A typical normalized longitudinal beam profile from TOPOS system at
injection. The length of the RF period is $\approx 1.2$ $\mu s$.
Fig. 32 shows a typical longitudinal profile of the bunch. If $S_{j}$ is the
amplitude at each time instance $j=1,..,N$. The bunching factor is calculated
by the Eq. A1 .
$B_{f}=\frac{\sum_{j=1}^{N}S_{j}}{max(S_{j})\cdot N}$ (A1)
where N $\approx 147$ is the number of samples in one RF period (at
injection). The TOPOS system samples the bunch at 125 MSa/s, thus the
difference between adjacent samples is 8ns.
### Measurement uncertainties
The calculation of $\Delta Q_{sc}$ from Eq. 4 has a dependence on measured
current, transverse beam profiles, longitudinal beam profiles and the twiss
parameters. The measurement uncertainty on each of these measured parameters
at GSI SIS-18 were commented in the detailed analysis in Franchetti2010 . Even
though some parameters and the associated uncertainties are correlated, any
correlations are neglected in present analysis. Uncertainties in each measured
parameter are propagated to find the error bars on the calculated and measured
incoherent tune shifts.
Reproducing from Ref. Franchetti2010 , the relative random uncertainty (std.
deviation) in beam profile width($\sigma_{x}$) measurements is given by Eq. A2
.
$\frac{\delta\sigma_{x}}{\sigma_{x}}=\frac{0.043}{B}+0.33\delta N_{i}$ (A2)
where $\Delta x=2.1$mm is the wire spacing and $\delta N_{i}=\frac{1}{2^{8}}$
is the ADC resolution of the IPM and B is defined as $\sigma_{x}/\Delta x$. If
the error bars are derived from $j$ measurements, the measured profile is
given by
$\sigma_{av,x}=\langle\sigma_{x}\rangle_{j},\delta\sigma_{av,x}=\sqrt{{\langle\sigma_{x}^{2}\rangle}_{j}-\langle\sigma_{x}\rangle^{2}_{j}+\langle{\delta}{\sigma_{x}\rangle}^{2}}$
(A3)
For each tune measurement at the given intensity and excitation power, 5-8
transverse beam profiles were measured, and the relative error is obtained
$\approx 5\%$ using Eq. A2 and Eq. A3 . The relative systematic error (bias)
in transverse beam width measurements is $<1\%$ Franchetti2010 and ignored in
this analysis.
The uncertainty in the injected current is dominated by fluctuations in the
source and the relative uncertainty is estimated to be $\approx 5\%$ based on
5-8 measurements at the same intensity settings for each measurement point.
Bunch length and bunching factor vary by $\approx 2-3\%$ due to long term beam
losses only under high intensity beam conditions. The maximum relative bias in
the lattice parameter $\beta$ is assumed to be $\approx 5\%$ at the IPM
location. Taking all the relative errors, uncertainty propagation using
familiar Measurement uncertainties gives relative error for estimated
incoherent tune shifts $\approx 12\%$.
$\displaystyle\frac{\delta\varepsilon_{av,x}}{\varepsilon_{av,x}}=\sqrt{4{(\frac{\delta\sigma_{av,x}}{\sigma_{av,x}})}^{2}+{(\frac{\delta\beta_{x}}{\beta_{x}})}^{2}}$
$\displaystyle\frac{\delta\Delta Q_{sc}}{\Delta
Q_{sc}}=\sqrt{{(\frac{\delta\varepsilon_{av,x}}{\varepsilon_{av,x}})}^{2}+{(\frac{\delta
I_{p}}{I_{p}})}^{2}}$
Tune measurements done by averaging over long intervals contribute to the
width of modes due to long term beam losses. Beam losses lead to change in
coherent tune especially in the vertical plane where the image current effects
are larger. This has been highlighted at appropriate sections in the text.
## References
* (1) P. Kowina et al., “Digital baseband tune determination”, Proc. of BIW’10, Santa Fe, U.S.A (2010)
* (2) M. Gasior, “High sensitivity tune measurement using direct diode detection”, Proc. of BIW’12, Virginia, U.S.A (2012)
* (3) S. Chattopadhyay, “Some fundamental aspects of fluctuations and coherence in charged particle beams in storage rings”, Super Proton Synchrotron Division, CERN 84-11, (1984)
* (4) D. Boussard, “Schottky Noise and Beam Transfer Function Diagnostics”, CERN Accelerator School; Fifth Advanced Accelerator Physics Course”, (1995)
* (5) M. Blaskiewicz, Phys. Rev. ST Accel. Beams 1, 044201 (1998)
* (6) O. Boine-Frankenheim, V. Kornilov, Phys. Rev. ST Accel. Beams, 12, 114201 (2009)
* (7) A. Burov, Phys. Rev. ST Accel. Beams 12, 044202 (2009)
* (8) V. Balbekov, Phys. Rev. ST Accel. Beams 12, 124402 (2009)
* (9) V. Kornilov, O. Boine-Frankenheim, Phys. Rev. ST Accel. Beams,13, 114201 (2010)
* (10) R. Singh et al., Proc. of BIW’12, Newport News, U.S.A (2012)
* (11) V. Kornilov, O. Boine-Frankenheim, Phys. Rev. ST Accel. Beams,15, 114201 (2012)
* (12) D. Möhl, Part. Accel. 50, 177 (1995)
* (13) F. J. Sacherer, “Transverse space-charge effects in circular particle accelerators”, UCRL-18454 (1968)
* (14) T. Linnecar and W. Scandale, Proc. of PAC 1981, p. 2147 (1981)
* (15) J. Borer, Proc. of PAC 1979 (1979)
* (16) K. Schindl, “Space Charge”, “Joint US-CERN-Japan-Russia School on Particle Accelerators” (1998)
* (17) O. Boine-Frankenheim, T. Shukla, Phys. Rev. ST Accel. Beams 8, 034201 (2005)
* (18) P. Forck, “Lecture notes on beam intrumentation” (2004)
* (19) H. Reeg et al., Proc. of DIPAC, Grenoble, France (2001)
* (20) T. Giacomini, Proc. of BIW’04, Knoxville, U.S.A (2004)
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|
arxiv-papers
| 2012-12-10T15:42:20 |
2024-09-04T02:49:39.094447
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Singh, O. Boine-Frankenheim, O. Chorniy, P. Forck, R. Haseitl, W.\n Kaufmann, P. Kowina, K. Lang, T. Weiland",
"submitter": "Rahul Singh",
"url": "https://arxiv.org/abs/1212.2103"
}
|
1212.2137
|
# Thermodynamics of Fractal Universe
Ahmad Sheykhi1,[email protected], Zeinab Teimoori 3 and Bin Wang
[email protected] 1 Physics Department and Biruni Observatory, College of
Sciences, Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran
3 Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman,
Iran
4 INPAC and Department of Physics, Shanghai Jiao Tong University, Shanghai
200240, China
###### Abstract
We investigate the thermodynamical properties of the apparent horizon in a
fractal universe. We find that one can always rewrite the Friedmann equation
of the fractal universe in the form of the entropy balance relation $\delta
Q=T_{h}d{S_{h}}$, where $\delta Q$ and $T_{h}$ are the energy flux and Unruh
temperature seen by an accelerated observer just inside the apparent horizon.
We find that the entropy $S_{h}$ consists two terms, the first one which obeys
the usual area law and the second part which is the entropy production term
due to nonequilibrium thermodynamics of fractal universe. This shows that in a
fractal universe, a treatment with nonequilibrium thermodynamics of spacetime
may be needed. We also study the generalized second law of thermodynamics in
the framework of fractal universe. When the temperature of the apparent
horizon and the matter fields inside the horizon are equal, i.e. $T=T_{h}$,
the generalized second law of thermodynamics can be fulfilled provided the
deceleration and the equation of state parameters ranges either as $-1\leq
q<0$, $-1\leq w<-1/3$ or as $q<-1$, $w<-1$ which are consistent with recent
observations. We also find that for $T_{h}=bT$, with $b<1$, the GSL of
thermodynamics can be secured in a fractal universe by suitably choosing the
fractal parameter $\beta$.
## I Introduction
Nowadays, it is a general belief that there is a deep connection between
thermodynamics and gravity. The story started with the discovery of black
holes thermodynamics in $1970$’s by Hawking and Bekenstein Haw1 ; Bek ; Haw ;
Bek1 . According to their discovery, a black hole can be regarded as a
thermodynamic system, with temperature and entropy proportional to its surface
gravity and horizon area, respectively. After that, people were speculating
that maybe there is a direct connection between thermodynamics and Einstein
equation, a hyperbolic second order partial differential equation for the
spacetime metric. In $1995$, Jacobson Jac was indeed able to derive the
Einstein equation from the requirement that the Clausius relation $\delta
Q=T\delta S$ holds for all local acceleration horizons through each spacetime
point, where $\delta S$ is one-quarter the horizon area change in Planck units
and $\delta Q$ and $T$ are the energy flux across the horizon and the Unruh
temperature seen by an accelerating observer just inside the horizon.
Jacobson’s derivation of the Einstein field equation from thermodynamics
opened a new window for understanding the thermodynamic nature of gravity.
After Jacobson, a lot of works have been done to disclose the profound
connection between gravity and thermodynamics. It was shown that the
gravitational field equations in a wide range of theories, can be rewritten in
the form of the first law of thermodynamics and vice versa Elin ; Cai1 ; Akb ;
Pad1 ; Pad2 ; Pad3 ; Par ; Kot ; Pad4 . The studies were also generalized to
the cosmological setup, where it was shown that the differential form of the
Friedmann equation in the Friedmann-Robertson-Walker (FRW) universe can be
transformed to the first law of thermodynamics on the apparent horizon Cai2 ;
Cai3 ; CaiKim ; Fro ; Dan ; Abdalla ; WANG ; Myu ; Cai4 ; Shey1 ; Shey2 ;
Shey3 .
On the other side, the second law of black hole mechanics expresses that the
total area of the event horizon of any collection of classical black holes can
never decrease, even if they collide and swallow each other. This is
remarkably similar to the second law of thermodynamics where the area is
playing the role of entropy. Note that the second law of black hole
thermodynamics can be violated if one take into account the quantum effect,
such as the Hawking radiation. To overcome this difficulty, Bekenstein Bek ;
Bek1 introduced the so-called total entropy $S_{\rm tot}$ which is defined as
$S_{\rm tot}=S_{h}+S_{m},$ (1)
where $S_{h}$ and $S_{m}$ are, respectively, the black hole entropy and the
entropy of the surrounding matter. According to Bekenstein’s argument, in
general, the total entropy should be a non decreasing function. This statement
is known as the generalized second law (GSL) of thermodynamics,
$\triangle S_{\rm tot}\geq 0.$ (2)
Besides, if thermodynamical interpretation of gravity near the apparent
horizon is a generic feature, one needs to verify whether the results may hold
not only for more general spacetimes but also for the other principles of
thermodynamics, especially for the GSL of thermodynamics. The GSL of
thermodynamics is a universal principle governing the evolution of the
universe. It was argued that in the accelerating universe the GSL is valid
provided the boundary of the universe is chosen the apparent horizon wang1 ;
wang2 ; Shey4 ; Shey5 ; Shey6 .
In this paper, we would like to extend the study to the fractal universe.
Fractal cosmology was recently proposed by Calcagni Calc1 ; Calc2 for a
power-counting renormalizable field theory living in a fractal spacetime. It
is interesting to see whether the Friedmann equation of a fractal universe can
be written in the form of the first law of thermodynamics. As we will see, in
a fractal universe, the Friedmann equation can be transformed to Clausius
relation, but a treatment with nonequilibrium thermodynamics of spacetime is
needed.
In the next section we review the basic equations in the framework of fractal
cosmology. In section III, we show that the Friedmann equation of a fractal
universe can be written in the form of the fundamental relation $\delta
Q=T_{h}dS_{h}$, where $\delta Q$ and $T_{h}$ are, respectively, the energy
flux and Unruh temperature seen by an accelerated observer just inside the
apparent horizon. In section IV, we check the validity of the GSL of
thermodynamics for a fractal cosmology. The last section is devoted to some
concluding remarks.
## II Fractal universe
The total action of Einstein gravity in a fractal spacetime is given by Calc1
; Calc2
$S=S_{G}+S_{m},$ (3)
where the gravitational part of the action is given by
$S_{G}=\frac{1}{16\pi G}\int
d\varrho(x)\sqrt{-g}(R-2\Lambda-\omega\partial_{\mu}\upsilon\partial^{\mu}\upsilon),$
(4)
and the matter part of the action is
$S_{m}=\int d\varrho(x)\sqrt{-g}\mathcal{L}_{m}.$ (5)
Here $g$ is the determinant of the dimensionless metric $g_{\mu\nu}$,
$\Lambda$ and $R$ are, respectively, the cosmological constant and Ricci
scalar. $\upsilon$ is the fractional function and $\omega$ is the fractal
parameter. The standard measure $d^{4}x$ replaced with a Lebesgue-Stieltjes
measure $d\varrho(x)$. The derivation of the Einstein equations goes almost
like in scalar-tensor models. Taking the variation of the action (3) with
respect to the FRW metric $g_{\mu\nu}$, one can obtain the Friedmann equations
in a fractal universe as Calc2
$H^{2}+\frac{k}{a^{2}}+H\frac{\dot{\upsilon}}{\upsilon}-\frac{\omega}{6}\dot{\upsilon}^{2}=\frac{8\pi
G}{3}\rho+\frac{\Lambda}{3},$ (6)
$\dot{H}+H^{2}-H\frac{\dot{\upsilon}}{\upsilon}+\frac{\omega}{3}\dot{\upsilon}^{2}-\frac{1}{2}\frac{\square\upsilon}{\upsilon}=-\frac{8\pi
G}{6}(\rho+3p)+\frac{\Lambda}{3},$ (7)
where $H={\dot{a}}/{a}$ is the Hubble parameter, $\rho$ and $p$ are the total
energy density and pressure of the ideal fluid composing the universe,
respectively. The curvature constant $k=0,1,-1$ corresponding to a flat,
closed and open universe, respectively. The continuity equation in a fractal
universe takes the form Calc2
$\dot{\rho}+\left(3H+\frac{\dot{\upsilon}}{\upsilon}\right)(\rho+p)=0.$ (8)
It is clear that for $\upsilon=1$, the standard Friedmann equations are
recovered. We further assume that only the time direction is fractal, while
spatial slices have usual geometry. Indeed, in the framework of fractal
cosmology, classically fractals can be timelike $[\upsilon=\upsilon(t)]$ or
even spacelike $[\upsilon=\upsilon(x)]$ (see Ref. Calc2 for details). These
two cases lead to different classical physics, but at quantum level all
configurations should be taken into account, so there is no quantum analogue
of space or timelike fractals. In this paper we take a timelike fractal. Thus,
those parameters that depend on time change and those parts that related to
$x$ remain fixed.
Assuming a timelike fractal profile $\upsilon=t^{-\beta}$ Calc2 , where
$\beta=4(1-\alpha)$ is the fractal dimension, the Friedmann equations (6) and
(7) in the absence of the cosmological constant can be written as
$H^{2}+\frac{k}{a^{2}}-\frac{\beta}{t}H-\frac{\omega\beta^{2}}{6t^{2(\beta+1)}}=\frac{8\pi
G}{3}\rho,$ (9)
$\dot{H}+H^{2}-\frac{\beta}{2t}H+\frac{\beta(\beta+1)}{2t^{2}}+\frac{\omega\beta^{2}}{3t^{2(\beta+1)}}=-\frac{8\pi
G}{6}(\rho+3p),$ (10)
while, the continuity equation (8) takes the form
$\dot{\rho}+\left(3H-\frac{\beta}{t}\right)(\rho+p)=0.$ (11)
From the definition of the fractional integral Calc2 ; Hilfer , we know that
$\alpha$ ranges as $0<\alpha\leqslant 1$. Thus for $\alpha=1$, we obtain
$\beta=0$ which physically means that the universe does not have any fractal
structure and one can recovers the well-known Friedmann equations in standard
cosmology. As one can see from Friedmann equations (9) and (10), we have no
limit $t\rightarrow 0$ for a timelike fractal profile, since in this case the
Friedmann equations diverge unless $\beta=0$. This implies that at the early
stages of the universe, we could not have the timelike fractal structure.
In the remaining part of this paper we show that the differential form of the
Friedmann equation (9) can be written in the form of the fundamental relation
$\delta Q=T_{h}dS_{h}$, where $S_{h}$ is the entropy associated with the
apparent horizon. We also investigate the validity of the GSL of
thermodynamics for the fractal universe surrounded by the apparent horizon.
## III First law of thermodynamics in fractal cosmology
For a homogenous and isotropic FRW universe the line elements can be written
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (12)
where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, and $h_{\mu\nu}$=diag
$(-1,a^{2}/(1-kr^{2}))$ is the two dimensional metric. The dynamical apparent
horizon, a marginally trapped surface with vanishing expansion, is determined
by the relation $h^{\mu\nu}\partial_{\mu}\tilde{r}\partial_{\nu}\tilde{r}=0$.
Straightforward calculation gives the apparent horizon radius for the FRW
universe as Shey5
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (13)
The associated temperature $T$ with the apparent horizon is given by
$T_{h}=\frac{1}{2\pi\tilde{r}_{A}},$ (14)
where $A=4\pi\tilde{r}_{A}^{2}$ is the apparent horizon area and we have
assumed the apparent horizon radius is fixed. We shall assume the matter
source in the fractal universe has a perfect fluid form with stress-energy
tensor
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (15)
where $\rho$ and $p$ are the energy density and pressure, respectively.
Assuming the total energy inside the apparent horizon is given by $E=\rho V$,
where $V=\frac{4}{3}\pi\tilde{r}_{A}^{3}$ is the volume of the 3-sphere of
radius $\tilde{r}_{A}$. Taking differential form of energy, we can obtain the
energy flux $dE=Vd\rho$. Here we have assumed the volume enveloped by the
apparent horizon is fixed during the infinitesimal internal of time $dt$. Thus
$-dE$ is actually just the heat flux $\delta Q$ in Jac crossing the apparent
horizon within an infinitesimal internal of time $dt$, it is not the change in
the matter energy inside the apparent horizon due to the volume change, so
there is no term of volume change. Hence we can write
$dE=V\dot{\rho}dt.$ (16)
Note that by taking $\upsilon=t^{-\beta}$, from the continuity equation (11)
we have $\rho\sim(a^{3}\upsilon(t))^{-(1+w)}$, which shows that the measure
weight $\upsilon$ is hidden in $\rho$ and thus it is not necessary to consider
its variation in Eq. (16) separately.
Substituting $\dot{\rho}$ from the continuity equation (11), we get
$dE=-4\pi
H\tilde{r}_{A}^{3}(\rho+p)dt+\frac{4\pi}{3}\frac{\beta}{t}(\rho+p)\tilde{r}_{A}^{3}dt.$
(17)
Differentiating Friedmann equation (9) with respect to the cosmic time $t$ and
using the continuity equation(11), we find
$\displaystyle
H\left(\dot{H}-\frac{k}{a^{2}}\right)-\frac{\beta}{2t}\dot{H}+\frac{\beta}{2t^{2}}H+\frac{\omega\beta^{2}(\beta+1)}{6t^{2\beta+3}}=$
$\displaystyle-4\pi GH(\rho+p)+\frac{4\pi
G}{3}\left(\frac{\beta}{t}\right)(\rho+p).$ (18)
Multiplying the factor $(-\tilde{r}_{A}^{3})$ on both sides of Eq.(III), we
reach
$\displaystyle-H\left(\dot{H}-\frac{k}{a^{2}}\right)\tilde{r}_{A}^{3}+\frac{\beta}{2t}\dot{H}\tilde{r}_{A}^{3}-\frac{\beta}{2t^{2}}H\tilde{r}_{A}^{3}-\frac{\omega\beta^{2}(\beta+1)}{6t^{2\beta+3}}\tilde{r}_{A}^{3}$
$\displaystyle=4\pi GH(\rho+p)r_{A}^{3}-\frac{4\pi
G}{3}\frac{\beta}{t}(\rho+p)\tilde{r}_{A}^{3}.$
Differentiating Eq.(13) with respect to the cosmic time $t$, we obtain
$\dot{\tilde{r}}_{A}=-H\left(\dot{H}-\frac{k}{a^{2}}\right)\tilde{r}_{A}^{3}.$
(20)
Substituting Eq.(20) into (III) we can rewrite it as
$\displaystyle\frac{1}{G}\left[d\tilde{r}_{A}-\beta\left(\frac{H}{2t^{2}}-\frac{\dot{H}}{2t}\right)\tilde{r}_{A}^{3}dt-\frac{\omega}{6}\frac{\beta^{2}(\beta+1)}{t^{2\beta+3}}\tilde{r}_{A}^{3}dt\right]$
(21) $\displaystyle=4\pi
H(\rho+p)\tilde{r}_{A}^{3}dt-\frac{4\pi}{3}\frac{\beta}{t}(\rho+p)\tilde{r}_{A}^{3}dt.$
Combining Eq. (17) with (21) and using the fact that $\delta Q=-dE$ is just
the energy flux crossing through the apparent horizon, we have
$\delta
Q=\frac{1}{G}\left[d\tilde{r}_{A}-\beta\left(\frac{H}{2t^{2}}-\frac{\dot{H}}{2t}\right)\tilde{r}_{A}^{3}dt-\frac{\omega}{6}\frac{\beta^{2}(\beta+1)}{t^{2\beta+3}}\tilde{r}_{A}^{3}dt\right].$
(22)
Eq. (22) can be further rewritten as
$\displaystyle\delta
Q=\frac{1}{2\pi\tilde{r}_{A}}\frac{2\pi\tilde{r}_{A}}{G}\left[d\tilde{r}_{A}-\beta\left(\frac{H}{2t^{2}}-\frac{\dot{H}}{2t}\right)\tilde{r}_{A}^{3}dt\right.$
$\displaystyle\left.-\frac{\omega}{6}\frac{\beta^{2}(\beta+1)}{t^{2\beta+3}}\tilde{r}_{A}^{3}dt\right].$
(23)
Using definition (14) for the temperature, one can see that Eq. (III) is just
the entropy balance relation,
$\delta Q=T_{h}dS_{h},$ (24)
provided we define
$d{S_{h}}=\frac{2\pi\tilde{r}_{A}d\tilde{r}_{A}}{G}+\frac{2\pi\beta}{G}\left[\left(\frac{\dot{H}}{2t}-\frac{H}{2t^{2}}\right)-\frac{\omega}{6}\frac{\beta(\beta+1)}{t^{2\beta+3}}\right]\tilde{r}_{A}^{4}dt.$
(25)
Integrating, we find
$\displaystyle
S_{h}=\frac{A}{4G}+\frac{2\pi\beta}{G}\int\left[\left(\frac{\dot{H}}{2t}-\frac{H}{2t^{2}}\right)-\frac{\omega\beta(\beta+1)}{6t^{2\beta+3}}\right]\tilde{r}_{A}^{4}dt.$
As one can see, in a fractal universe, the entropy $S_{h}$ associated with the
apparent horizon consists two parts, the first one obeys the usual area law
and the second part is the entropy term developed internally in the system as
a result of being out of equilibrium Elin ; noneq . The entropy production
rate vanishes for standard cosmology where $\beta=0$. It is worth mentioning
that even in standard cosmology one can still have non-equilibrium
thermodynamics depending on the assumptions Elin .
## IV GSL of thermodynamics in fractal universe
In this section we investigate the validity of the GSL of thermodynamics in a
region enclosed by the apparent horizon in the framework of the fractal
universe. Let us put $k=0$ for simplicity, so we have
${\tilde{r}}_{A}={1}/{H}$,
$\dot{H}=-{\dot{\tilde{r}}_{A}}/{\tilde{r}_{A}^{2}}$. The total entropy
associated with the apparent horizon, $S_{h}$, can be written
$\displaystyle
dS_{h}=\frac{2\pi}{G}\left[\tilde{r}_{A}d\tilde{r}_{A}-\frac{\beta}{2t}\tilde{r}_{A}^{2}d{\tilde{r}_{A}}-\frac{\beta}{2t^{2}}\tilde{r}_{A}^{3}dt\right.$
$\displaystyle\left.-\frac{\omega}{6}\frac{\beta^{2}(\beta+1)}{t^{2\beta+3}}\tilde{r}_{A}^{4}dt\right].$
(27)
Dividing Eq.(IV) by $dt$, we arrive at
$\displaystyle\dot{S}_{h}=\frac{2\pi}{G}\left[\tilde{r}_{A}\dot{\tilde{r}}_{A}-\frac{\beta}{2t}\tilde{r}_{A}^{2}\dot{\tilde{r}}_{A}-\frac{\beta}{2t^{2}}\tilde{r}_{A}^{3}\right.$
$\displaystyle\left.-\frac{\omega}{6}\frac{\beta^{2}(\beta+1)}{t^{2\beta+3}}\tilde{r}_{A}^{4}\right].$
(28)
Eq. (IV), can be written as
$\displaystyle\dot{S}_{h}=\frac{2\pi}{G}\tilde{r}_{A}\dot{\tilde{r}}_{A}\left(1-\frac{\beta}{2t}\tilde{r}_{A}\right)-\frac{2\pi}{G}\frac{\beta}{2t^{2}}\tilde{r}_{A}^{3}$
$\displaystyle-\frac{2\pi}{G}\frac{\omega}{6}\frac{\beta^{2}(\beta+1)}{t^{2\beta+3}}\tilde{r}_{A}^{4}.$
(29)
The Friedmann equation (9), for a flat universe, becomes
$\frac{1}{\tilde{r}_{A}^{2}}-\frac{\beta}{t}\frac{1}{\tilde{r}_{A}}-\frac{\omega\beta^{2}}{6t^{2(\beta+1)}}=\frac{8\pi
G}{3}\rho.$ (30)
Differentiating the above equation with respect to the cosmic time and using
the continuity equation(11), after some simplification, we get
$\displaystyle\frac{\dot{\tilde{r}}_{A}}{\tilde{r}_{A}^{3}}\left(1-\frac{\beta}{2t}\tilde{r}_{A}\right)-\frac{\beta}{2t^{2}}\frac{1}{\tilde{r}_{A}}-\frac{\omega}{6}\frac{\beta^{2}(\beta+1)}{t^{2\beta+3}}$
$\displaystyle=4\pi GH(\rho+p)-\frac{4\pi G}{3}\frac{\beta}{t}(\rho+p).$ (31)
Solving this equation for $\dot{\tilde{r}}_{A}$, we find
$\displaystyle\dot{\tilde{r}}_{A}=\left(1-\frac{\beta}{2t}\tilde{r}_{A}\right)^{-1}\tilde{r}_{A}^{3}\left[4\pi
GH(\rho+p)\right.$ $\displaystyle\left.-\frac{4\pi
G}{3}\frac{\beta}{t}(\rho+p)+\frac{\beta}{2t^{2}}\frac{1}{\tilde{r}_{A}}+\frac{\omega}{6}\frac{\beta^{2}(\beta+1)}{t^{2\beta+3}}\right].$
(32)
Substituting $\dot{\tilde{r}}_{A}$ from Eq.(IV) into (IV), we obtain
$\dot{S}_{h}=\frac{2\pi}{G}H^{-4}\left[4\pi GH(\rho+p)-\frac{4\pi
G}{3}\frac{\beta}{t}(\rho+p)\right],$ (33)
which can also be rewritten in the following form
$\dot{S}_{h}=8\pi^{2}H^{-3}(\rho+p)\left(1-\frac{\beta}{3Ht}\right),$ (34)
where we have used ${\tilde{r}}_{A}={H}^{-1}$ for the flat universe. Let us
discuss the two cases in which $\dot{S_{h}}\geq 0$. In the first case we
assume the dominant energy condition valid, $\rho+p\geq 0$, therefore
$\dot{S_{h}}\geq 0$, provided $\beta\leq 3Ht$. However, in an accelerating
universe the dominant energy condition may violate, $\rho+p<0$. In this case
$\dot{S_{h}}\geq 0$ provided $\beta\geq 3Ht$. However, as we will see below,
the GSL can be still fulfilled in an accelerating fractal universe.
For latter convenience we also calculate $T_{h}\dot{S_{h}}$,
$T_{h}\dot{S_{h}}=4\pi H^{-2}(\rho+p)\left(1-\frac{\beta}{3Ht}\right).$ (35)
Next, we study the GSL of thermodynamics, namely the time evolution of the
total entropy including the entropy $S_{h}$ associated with the apparent
horizon together with the matter field entropy $S_{\rm in}$ inside the
apparent horizon. The entropy of the universe inside the horizon can be
related to its energy and pressure in the horizon by the Gibbs equation Pavon2
$TdS_{\rm in}=d(\rho V)+pdV=Vd\rho+(\rho+p)dV.$ (36)
We assume the temperature of the perfect fluid inside the apparent horizon
scales as the temperature of the horizon, which for flat universe is
$T_{h}=H/(2\pi)$. In general, if the temperature of the horizon differs much
from that the fluid, then the energy would spontaneously flow between the
horizon and the fluid, something at variance with FRW universe Pavon2 ; Muba .
Thus we suppose that the temperature $T_{h}$ associated with the apparent
horizon is $T_{h}=bT$ wang2 , where $b$ is a real proportional constant. Since
at the present time the horizon temperature is lower than that of the CMB by
many orders of magnitude, we will not consider the case $b>1$. We limit
ourselves to the assumption of the local equilibrium hypothesis, that the
energy would not spontaneously flow between the horizon and the fluid. Indeed,
this will certainly be the situation at late times, that is when the universe
fluids and the horizon will have interacted for a long time, it is ambiguous
if it will be the case at early or intermediate times jamil1 . However, in
order to avoid nonequilibrium thermodynamical calculations, which would lead
to lack of mathematical simplicity and generality, the assumption of
equilibrium, although restricting, has widely accepted for studying the GSL in
the literature jamil1 . Thus, we follow this assumption and we notice that our
results are valid only at the late stages of the universe evolution where the
universe fluids and the horizon will interact for a long time.
Let us first consider the case where $b=1$. Physically, this means that we
have assumed during the infinitesimal internal time $dt$, the temperature of
the perfect fluid is equal to the temperature $T_{h}$ associated with the
apparent horizon. Therefore, from Gibbs equation, after using the continuity
equation (11), we get
$\displaystyle T\dot{S}_{\rm in}$ $\displaystyle=$ $\displaystyle-4\pi
H^{-2}(\rho+p)\left(1-\frac{\beta}{3Ht}\right)$ (37) $\displaystyle+4\pi
H^{-2}(\rho+p)\dot{\tilde{r}}_{A}.$
To check the GSL of thermodynamics, we have to examine the evolution of the
total entropy $S_{h}+S_{\rm in}$. Adding equations (35) and (37), we get
$T_{h}(\dot{S}_{h}+\dot{S}_{\rm in})=4\pi
H^{-2}(\rho+p)\dot{\tilde{r}}_{A}=A(\rho+p)\dot{\tilde{r}}_{A},$ (38)
where $A=4\pi H^{-2}$ is the apparent horizon area. The deceleration parameter
$q$ can be written as
$q=-1-\frac{\dot{H}}{H^{2}}.$ (39)
Using the fact that in flat FRW universe we have
$\dot{\tilde{r}}_{A}=-\dot{H}/H^{2}$, Eq.(38) can be written as
$T_{h}(\dot{S}_{h}+\dot{S}_{\rm in})=A\rho(1+w)(1+q),$ (40)
where $w={p}/{\rho}$ is the equation of state parameter. From the above
equation we see that for $(1+q)(1+w)\geq 0$ the GSL of thermodynamics is
fulfilled in a region enclosed by the apparent horizon. Let us consider two
cases separately. In the first case where $q\geq-1$ and $w\geq-1$, the GSL
holds. Recent observations from type Ia supernova show that our universe is
currently undergoing a phase of accelerated expansion. For an accelerating
universe we have $q<0$ and $w<-1/3$. Therefore, in an accelerating universe
the GSL is fulfilled for $-1\leq q<0$ and $-1\leq w<-1/3$ which are consistent
with recent observations. In the second case where $q<-1$ and $w<-1$, the GSL
is again fulfilled, but in this case the equation of state parameter crosses
the phantom line, $w=-1$, which again some cosmological data confirm it. Now
we consider the general case where $b<1$. Since at the present time the
horizon temperature is lower than that of the CMB by many orders of magnitude,
we will not consider the case $b>1$. Adding equations (35) and (37) with
$T_{h}=bT$, we obtain
$\displaystyle T_{h}(\dot{S}_{h}+\dot{S}_{\rm in})=4\pi
H^{-2}(\rho+p)\left(1-\dfrac{\beta}{3Ht}\right)(1-b)$ (41) $\displaystyle+4\pi
bH^{-2}(\rho+p)(1+q)$ $\displaystyle=$ $\displaystyle 4\pi
H^{-2}(\rho+p)\left[\left(1-\dfrac{\beta}{3Ht}\right)\left(1-b\right)+b(1+q)\right]$
$\displaystyle=$ $\displaystyle 4\pi
H^{-2}\rho(1+w)\left[1+bq-\dfrac{\beta}{3Ht}\left(1-b\right)\right].$
Equation (41) shows that for $w\geq-1$, the GSL can be secured provided that
$\displaystyle\beta(1-b)\leq 3Ht(1+bq).$ (42)
Thus, for $b<1$, Eq. (42) can be translated to
$\displaystyle\beta\leq 3Ht\left(\frac{1+bq}{1-b}\right).$ (43)
Since $\beta$ is a positive definite, that is $\beta>0$, hence the above
condition can also be written
$\displaystyle 0<\beta\leq 3Ht\left(\frac{1+bq}{1-b}\right).$ (44)
The right hand side of inequality (44) should be positive, which leads to
$q>-b^{-1}$. In an accelerating universe $q<0$, and hence the condition for
keeping the GSL in a fractal universe reduced to $-b^{-1}<q<0$.
On the other hand, if equation of state parameter crosses the phantom line,
which some observational data support it, then we have $w<-1$. In this case,
the GSL holds provided
$\displaystyle\beta(1-b)>3Ht(1+bq).$ (45)
For $b<1$, the GSL can be fulfilled if
$\displaystyle\beta>3Ht\left(\frac{1+bq}{1-b}\right).$ (46)
We conclude that the requirement of the GSL of thermodynamics in a fractal
universe leads to constraint on the fractal dimension parameter $\beta$. It is
worth noting that our local equilibrium hypothesis, which we used for checking
the GSL, only valid at the late time. Thus, the result obtained in this secion
have no $t\rightarrow 0$ limit.
Finally, it is instructive to discuss the sign of the entropy and temperature,
when the universe lies in the phantom phase $(\omega<-1)$. A lot of works have
been done in the literature, showing that in the absence of chemical potential
in phantom regime, the temperature must be negative, while the energy density
and the entropy should be positive Myn ; Gon . In Lim ; Pere a negative
chemical potential was assumed for the phantom fluid, and showed that the
temperature, entropy and density must be positive. In Sar it was shown that
with an arbitrary chemical potential the density and entropy are always
positive, while the temperature of a phantom universe ($w<-1$) is negative,
and that of quintessence universe ($w>-1$) is positive. The temperature
negativity can only be interpreted in the quantum framework Sar . In Pavon2
it was found that the phantom temperature is positive and its entropy
negative. Finally, in Bre ; Noj it was argued that one can describe the
phantom universe either with negative temperature and positive entropy, or
with negative entropy and positive temperature. Since the horizon temperature
is always positive, it is deduced that the universe temperature will be
positive even if it lies in the phantom phase. Thus we should have a negative
universe entropy in this case. Although the total entropy is always positive.
Because negative entropy of the universe ingredients is overcome by the
positive horizon entropy.
## V concluding remarks
In summary, in this paper we studied thermodynamics of the apparent horizon in
the framework of fractal cosmology. We found that the Friedmann equation of a
fractal universe can be transformed to the fundamental relation $\delta
Q=T_{h}d{S_{h}}$ on the apparent horizon, where $\delta Q$ and $T_{h}$ are the
energy flux and Unruh temperature seen by an accelerated observer just inside
the apparent horizon. We showed that the entropy $S_{h}$ consists two terms,
the first one which obeys the usual area law and the second part which is the
entropy production term and appears due to the nonequilibrium thermodynamics
of the fractal universe. This indicates that in a fractal universe, a
treatment with nonequilibrium thermodynamics of spacetime maybe needed. We
also investigated the time evolution of the total entropy including the
entropy $S_{h}$ associated with the apparent horizon together with the matter
field entropy $S_{\rm in}$ inside the apparent horizon. We assumed the
temperature of the apparent horizon is proportional to the matter fields
temperature inside the horizon, i.e. $T_{h}=bT$. We studied several cases
including $b<1$ and $b=1$, in which the GSL of thermodynamics can be secured
in a fractal universe. Interestingly enough, we found that for an accelerating
fractal universe the GSL can be preserved, at least for the late times where
the local equilibrium hypothesis holds. The fulfillment of the GSL of
thermodynamics in an accelerating fractal universe leads also to constraints
on the fractal dimension parameter $\beta$.
###### Acknowledgements.
We thank the anonymous referee for constructive comments which helped us to
improve the paper significantly. This work has been supported financially by
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) and also
by Shiraz University Research Council.
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|
arxiv-papers
| 2012-10-29T13:11:15 |
2024-09-04T02:49:39.105151
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi, Zeinab Teimoori and Bin Wang",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1212.2137"
}
|
1212.2276
|
_Journal of Fractional Calculus and Applications_ ,
Vol. 3(S). July, 11, 2012 (Proc. of the 4th. Symb. of Fractional Calculus and
Applications)
No. 6, PP. 1 – 8
ISSN: 2090-5858.
http://www.fcaj.webs.com/
# A New Technique for the Calculation of Effective Mesonic Potential at
Finite Temperature in the Logarithmic Quark-Sigma Model
M. Abu-Shady M. Abu-shady
Faculty of Science, Menofia University, Menofia, Egypt
[email protected]
###### Abstract.
The logarithmic sigma model describes the interactions between quarks via
sigma and pion exchanges. The effective mesonic potential is extended to the
finite temperature and it is numerically calculated using n-midpoint rule.
Meson properties such as the phase transition, the sigma and pion masses, and
the critical point temperature are examined as functions of temperature. The
obtained results are compared with other approaches. We conclude that the
calculated effective potential is successfully to predict the meson
properties.
###### Key words and phrases:
Finite-temperature field theory, Chiral Lagrangians, Midpoint
###### 1991 Mathematics Subject Classification:
81V05, 81V25, 82B26, 82B80
Proc. of the 4th. Symb. of Frac. Calcu. Appl. Faculty of Science Alexandria
University, Alexandria, Egypt July, 11, 2012
## 1\. Introduction
The study of matter at very high temperature and densities is of interest
because of its relevance to particle physics and astrophysics. According to
the standard big bang model, it is believed that a series of phase transitions
happened at the early stages of the evolution of universe. The QCD phase
transition being one of them. The lattice QCD and effective field theories are
two main approaches to calculate the phase transition and meson properties at
finite temperature [1]. This subject has been under intense theoretical study
using various effective field theory models, such as the Namu-Jona-Lasinio
Model [2-4], a linear sigma model [5-8]. One of the effective models in
describing baryon properties is the linear sigma model, which was suggested
earlier by Gell-Mann and Levy [9] to describe the nucleons interacting via
sigma $(\sigma)$ and pion $(\mathbf{\pi})$ exchanges. At finite temperature,
the model gives a good description of the phase transition by using the
Hartree approximation [1, 10-12] within the Cornwall–Jackiw–Tomboulis (CJT)
formalism [13]. However, there exists serious difficulty concerning the
renormalization of the CJT effective action in the Hartree approximation.
Baacke and Michalski [14] indicated that the phase transition can be obtained
beyond the large $N$ and Hartree approximation through the systematic
expansions are bases on the resummation scheme by Cornwall-Jackiw-Tombonlis
and 2PI scheme.
The aim of this work is to calculate the effective logarithmic mesonic
potential, the phase transition, and meson masses at finite temperature. The
logarithmic mesonic potential at zero temperature was suggested in Ref. [15]
to provide a good description of hadron properties. The used method is
different as a new technique in comparison with other works as in Refs. [5-8].
In addition, the pressure is investigated as a function temperature.
This paper is organized as follows: In Sec. 2, the linear sigma model at zero
temperature and finite temperature are explained briefly. The numerical
calculations and discussion of the results are presented in Sec. 3. Summary
and conclusion is presented in Sec. 4.
## 2\. The Logarithmic Quark-Sigma Model
### 2.1. The Logarithmic Potential at zero Temperature
The Lagrangian density of quark sigma model that describes the interactions
between quarks via the $\sigma-$and $\mathbf{\pi}-$meson exchange [15]. The
Lagrangian density is,
$L\left(r\right)=i\overline{\Psi}\partial_{\mu}\gamma^{\mu}\Psi+\frac{1}{2}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma+\partial_{\mu}\mathbf{\pi}.\partial^{\mu}\mathbf{\pi}\right)+g\overline{\Psi}\left(\sigma+i\gamma_{5}\mathbf{\tau}.\mathbf{\pi}\right)\Psi-U^{T(0)}(\sigma,\mathbf{\pi)},$
(1)
with
$U^{T(0)}\left(\sigma,\mathbf{\pi}\right)=\lambda_{1}^{2}(\sigma^{2}+\mathbf{\pi}^{2})-\lambda_{2}^{2}\log(\sigma^{2}+\mathbf{\pi}^{2})+m_{\pi}^{2}f_{\pi}\sigma,$
(2)
$U^{T(0)}\left(\sigma,\mathbf{\pi}\right)$ is the meson-meson interaction
potential where $\Psi,\sigma$ and $\mathbf{\pi}$ are the quark, sigma, and
pion fields, respectively. In the mean-field approximation the meson fields
are treated as time-independent classical fields. This means that we replace
power and products of the meson fields by corresponding powers and products of
their expectation values. The meson-meson interactions in Eq. (2) lead to
hidden chiral $SU(2)\times SU(2)$ symmetry with $\sigma\left(r\right)$ taking
on a vacuum expectation value
$\ \ \ \ \ \ \left\langle\sigma\right\rangle=-f_{\pi},$ (3)
where $f_{\pi}=93$ MeV is the pion decay constant. The final term in Eq. (2)
is included to break the chiral symmetry. It leads to partial conservation of
axial-vector isospin current (PCAC). The parameters $\lambda^{2},\nu^{2}$ can
be expressed in terms of $\ f_{\pi}$, the masses of mesons as,
$\lambda_{1}^{2}=\frac{1}{4}(m_{\sigma}^{2}+m_{\pi}^{2}),$ (4)
$\lambda_{2}^{2}=\frac{f_{\pi}^{2}}{4}(m_{\sigma}^{2}-m_{\pi}^{2}).$ (5)
### 2.2. The Effective Mesonic Potential at Finite-Temperature
In Eq. (2), The effective potential is extended to calculate the chiral
interacting mesons with quarks at finite temperature. The quarks are
considered a heat bath in local thermal equilibrium
$U_{eff}(\sigma,\mathbf{\pi,}T\mathbf{)}=U^{T(0)}(\sigma,\mathbf{\pi)-}24T\mathop{\displaystyle\int}\frac{d^{3}p}{\left(2\pi\right)^{3}}\ln(1+e^{\frac{-\sqrt{p^{2}+g^{2}(\sigma^{2}+\mathbf{\pi}^{2})}}{T}}),$
(6)
the first term is the potential in the tree level defines in Eq. (2) and the
second term is for the chiral meson fields interacts with quarks at finite
temperature and zero-chemical potential. Non-zero values of the chiral fields
in the chiral broken phase dynamically generate a quark mass $m_{q}=gf_{\pi}$.
The integration is taken over momentum volume (for details, see Ref. [16]).
## 3\. Numerical Calculations and Discussion of Results
### 3.1. Numerical Calculations
The purpose of this section is to calculate the effective mesonic potential
$\sigma$ and $\mathbf{\pi}$\- masses, and the pressure. We rewrite Eq. (6) as
follows
$U_{eff}(\sigma,\mathbf{\pi,}T\mathbf{)}=U^{T(0)}(\sigma,\mathbf{\pi)-}\frac{12T}{\pi^{2}}\int_{0}^{\infty}p^{2}dp(\ln(1+e^{\frac{-\sqrt{p^{2}+g^{2}(\sigma^{2}+\mathbf{\pi}^{2})}}{T}}).$
(7)
Eq. (7) is written in the dimensionless form as follows
$U_{eff}(\sigma,\mathbf{\pi,}T\mathbf{)}=f_{\pi}^{4}[U^{T(0)}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{)-}\frac{12T^{\prime}}{\pi^{2}}\int_{0}^{\infty}p^{{}^{\prime}2}(\ln
e^{{}^{\left(-\frac{1}{T^{\prime}}\sqrt{P^{\prime 2}+g^{2}(\sigma^{\prime
2}+\chi^{\prime 2})}\right)}}+1)dp^{\prime}],$ (8)
where
$U^{T(0)}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{)=}\lambda_{1}^{{}^{\prime}2}(\sigma^{\prime
2}+\mathbf{\pi}^{{}^{\prime}2})-\lambda_{2}^{{}^{\prime}2}\log
f_{\pi}^{2}(\sigma^{{}^{\prime}2}+\mathbf{\pi}^{\prime
2})+m_{\pi}^{{}^{\prime}2}\sigma^{\prime},$ (9)
and
$\lambda_{1}^{{}^{\prime}2}=\frac{1}{4}(m_{\sigma}^{{}^{\prime}2}+m_{\pi}^{{}^{\prime}2}),\lambda_{2}^{\prime
2}=\frac{1}{4}(m_{\sigma}^{\prime 2}-m_{\pi}^{{}^{\prime}2}),$ (10)
where
$\sigma^{\prime},\mathbf{\pi}^{\prime},T^{\prime},m_{\pi}^{{}^{\prime}},$and
$m_{\sigma}^{{}^{\prime}}$ are in unit of $f_{\pi}.$ Therefore
$U^{T(0)}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{)}$ is the
dimensionless form of $U^{T(0)}(\sigma,\mathbf{\pi)}$. Substituting
$p^{\prime}=-\ln y$ into Eq. (8), we obtain:
$\displaystyle U_{eff}(\sigma,\mathbf{\pi,}T\mathbf{)}$ $\displaystyle=$
$\displaystyle
f_{\pi}^{4}[U^{T(0)}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{)-}$
$\displaystyle-\frac{12T^{\prime}}{\pi^{2}}\int_{0}^{1}\frac{(\ln
y)^{2}}{y}\ln\left(\exp\left(-\frac{1}{T^{\prime}}\sqrt{(\ln
y)^{2}+g^{2}(\sigma^{\prime 2}+\mathbf{\pi}^{\prime
2})}\right)+1\right)dy\allowbreak],\ \ \ \ \ $
hence we can write the dimensionless form of
$U_{eff}(\sigma,\mathbf{\pi,}T\mathbf{)}$ as follows:
By using midpoint rule, we obtain the approximate integral as follows:
$U_{eff}(\sigma,\mathbf{\pi,}T\mathbf{)}=f_{\pi}^{4}[U^{T(0)}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{)-}\frac{12T^{\prime}}{\pi^{2}}A\ln\left(\exp\left(-\frac{1}{T^{\prime}}\sqrt{g^{2}\left(\sigma^{\prime
2}+\mathbf{\pi}^{\prime 2}\right)+B}\right)+1\right)],$ (12)
where
$A=\frac{1}{n}\sum_{i=0}^{n}\frac{1}{\frac{1}{n}i+\frac{1}{2n}}\ln^{2}\left(\frac{1}{n}i+\frac{1}{2n}\right)\allowbreak,\
\ \ B=\sum_{i=0}^{n}\ln^{2}\left(\frac{1}{n}i+\frac{1}{2n}\right).$ (13)
(for details, see Refs. [17, 18]). In Ref. [19], the authors applied the
second derivation of the effective potential respect to $\sigma^{\prime}$ and
$\pi^{\prime}$ to obtain the effective meson masses. Then, the first
derivative of the effective potential
$U_{eff}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{,}T^{\prime}\mathbf{)}$
is given by
$\frac{\partial
U_{eff}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{,}T\mathbf{)}}{\partial\sigma}=\frac{1}{f_{\pi}}\frac{\partial
U_{eff}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{,}T\mathbf{)}}{\partial\sigma^{\prime}}=f_{\pi}^{3}[\frac{\partial
U_{0}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{,}T\mathbf{)}}{\partial\sigma^{\prime}}\mathbf{-}\frac{12T^{\prime}}{\pi^{2}}\frac{d_{1}d_{2}}{d_{3}+d_{3}d_{2}}],$
(14)
where
$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle-Ag^{2}\sigma^{\prime},\
\ d_{2}=\exp\left(-\frac{1}{T^{\prime}}\sqrt{B+g^{2}(\sigma^{\prime
2}+\mathbf{\pi}^{\prime 2})}\right),\ \ $ $\displaystyle d_{3}\ $
$\displaystyle=$ $\displaystyle T^{\prime}\sqrt{B+g^{2}(\sigma^{\prime
2}+\mathbf{\pi}^{\prime 2})}.\TCItag{15}$ (4)
Then, we obtain the effective sigma mass as follows
$m_{\sigma}^{2}(T)=\frac{\partial^{2}U_{eff}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{,}T\mathbf{)}}{f_{\pi}^{2}\partial\sigma^{\prime
2}}=f_{\pi}^{2}[\frac{\partial^{2}U_{0}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{,}T\mathbf{)}}{\partial\sigma^{\prime
2}}\mathbf{-}\frac{12T}{\pi^{2}}^{\prime}\frac{\partial}{\partial\sigma^{\prime}}(\frac{d_{1}d_{2}}{d_{3}+d_{3}d_{2}})],$
(16)
$m_{\sigma}(T)=f_{\pi}[\frac{\partial^{2}U_{0}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{,}T\mathbf{)}}{\partial\sigma^{\prime
2}}\mathbf{-}\frac{12T^{\prime}}{\pi^{2}}\frac{\partial}{\partial\sigma^{\prime}}(\frac{d_{1}d_{2}}{d_{3}+d_{3}d_{2}})]^{\frac{1}{2}},$
(17)
where
$\displaystyle\frac{\partial}{\partial\sigma^{\prime}}(\frac{d_{1}d_{2}}{d_{3}+d_{3}d_{2}})$
$\displaystyle=$ $\displaystyle
d_{1}\frac{d_{2}^{\prime}}{d_{3}+d_{2}d_{3}}+d_{2}\frac{d_{1}^{\prime}}{d_{3}+d_{2}d_{3}}-$
(5) $\displaystyle
d_{1}\frac{d_{2}}{\left(d_{3}+d_{2}d_{3}\right)^{2}}\allowbreak\left(d_{3}^{\prime}+d_{2}d_{3}^{\prime}+d_{3}d_{2}^{\prime}\right),\TCItag{18}$
with
$d_{1}^{\prime}=\frac{\partial d_{1}}{\partial\sigma^{\prime}}=\allowbreak-
Ag^{2},$ (19)
$d_{2}^{\prime}=\frac{\partial
d_{2}}{\partial\sigma^{\prime}}=\allowbreak-\frac{1}{T^{\prime}}g^{2}\frac{\sigma^{\prime}}{\sqrt{B+g^{2}(\sigma^{\prime
2}+\mathbf{\pi}^{\prime
2})}}\exp\left(-\frac{1}{T}\sqrt{B+g^{2}(\sigma^{\prime
2}+\mathbf{\pi}^{\prime 2})}\right)\allowbreak,$ (20)
$d_{3}^{\prime}=\frac{\partial d_{3}}{\partial\sigma^{\prime}}=\allowbreak
Tg^{2}\frac{\sigma^{\prime}}{\sqrt{B+g^{2}(\sigma^{\prime
2}+\mathbf{\pi}^{\prime 2})}}.$ (21)
Similarly, we obtain the effective pion mass as follows
$m_{\pi}(T)=f_{\pi}[\frac{\partial^{2}U(\sigma^{\prime},\mathbf{\pi}^{\prime})}{\partial\mathbf{\pi}^{\prime
2}}-.\frac{12}{\pi^{2}}T^{\prime}\frac{\partial}{\partial\mathbf{\pi}^{\prime}}(\frac{d_{4}d_{2}}{d_{3}+d_{3}d_{2}})]^{\frac{1}{2}},$
(22)
where
$d_{4}=-Ag^{2}\mathbf{\pi}^{\prime},$ (23)
$\displaystyle\frac{\partial}{\partial\mathbf{\pi}^{\prime}}(\frac{d_{4}d_{2}}{d_{3}+d_{3}d_{2}})$
$\displaystyle=$ $\displaystyle
d_{2}\frac{d_{4}^{\prime}}{d_{3}+d_{2}d_{3}}+d_{4}\frac{d_{2}^{\prime}}{d_{3}+d_{2}d_{3}}-$
(6) $\displaystyle
d_{2}\frac{d_{4}}{\left(d_{3}+d_{2}d_{3}\right)^{2}}\allowbreak\left(d_{3}^{\prime}+d_{2}d_{3}^{\prime}+d_{3}d_{2}^{\prime}\right),\TCItag{24}$
where
$d_{4}^{\prime}=\frac{\partial
d_{4}}{\partial\mathbf{\pi}^{\prime}}=\allowbreak-Ag^{2},$ (25)
$d_{2}^{\prime}=\frac{\partial
d_{2}}{\partial\mathbf{\pi}^{\prime}}=-\frac{1}{T}g^{2}\frac{\mathbf{\pi}^{\prime}}{\sqrt{B+g^{2}(\sigma^{\prime
2}+\mathbf{\pi}^{\prime
2})}}\exp\left(-\frac{1}{T}\sqrt{B+g^{2}(\sigma^{\prime
2}+\mathbf{\pi}^{\prime 2})}\right),$ (26)
$d_{3}^{\prime}=\frac{\partial
d_{3}}{\partial\mathbf{\pi}^{\prime}}=\allowbreak
Tg^{2}\frac{\mathbf{\pi}^{\prime}}{\sqrt{B+g^{2}(\sigma^{\prime
2}+\mathbf{\pi}^{\prime 2})}}.$ (27)
In Ref. [16], the pressure is given in the dimensionless form as follow:
$P^{\prime}(\sigma^{\prime},\mathbf{\pi}^{\prime},T)=U^{T(0)}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{)}-U_{eff}(\sigma^{\prime},\mathbf{\pi}^{\prime}\mathbf{,T}^{\prime}\mathbf{),}$
(28)
## 4\. Results and Discussion
In Eq. (7), The integration is solved by the n-midpoint algorithm and the
index n is taken $n=1000$ to get a good accuracy for a numerically
integration. In Ref. [20], the authors used a different method for calculating
the integration in the effective potential and they obtained it as a series of
$M^{2}=m^{2}+\frac{\lambda}{2}\phi^{2}$ , where the $m,$ $\lambda$ are
parameters of the model. The difficulty in this potential is to determine a
critical temperature $T_{c}.$ The two terms were taken only in the expression
of the potential since the increase of terms more than two terms will be the
critical point is a complex value, leading $T_{c}$ is not a physics quantity.
In Refs. [1-4], authors used double bubble graphs instead of summing infinite
set of daisy and superdaisy graphs using the tree level propagators.
Therefore, the Feynman diagrams are needed. In the present work, The
n-midpiont rule is used to avoid the difficulties in the above approaches.
Hence, we did not need to apply Feynman diagrams. The parameters of the model
such as $m_{\pi}=140$ MeV, $m_{\sigma}=600$ MeV$,~{}f_{\pi}=93$ MeV, and
coupling constant $g$ at zero-temperature are used as the initial parameters
at the finite temperature. In addition, the effective pion and sigma masses
are obtained as a second derivative respect to meson field. This method is
used extensively in other works such as in the Ref. [19]. In this section, we
examine the meson properties and the phase transition which depend on the
calculation of the effective mesonic potential at finite temperature. We
replaced the normal potential in the linear sigma model [5] at zero
temperature by logarithmic mesonic potential [15]. In our previous work [15].
The logarithmic potential was successfully to predict hadron properties at
zero temperature.
In Figs. (1, 2, 3), we investigate the behavior of the sigma and pion masses
as functions of temperature. Moreover, the effect of the sigma mass and the
coupling constant $g$ on a critical point temperature in the presence of the
explicit symmetry breaking ($m_{\pi}\neq 0)$ and the chiral limit(
$m_{\pi}=0$) is investigated. In Fig. 1, the sigma and pion masses are plotted
as functions of temperature at the presence of the explicit symmetry breaking
term. The sigma mass decreases with increasing temperature and the pion mass
increases with increasing temperature. The two curves crossed at a critical
point temperature $T_{c}$ where the sigma and pion masses have the same
massive value. At $m_{\sigma}=600$ MeV, we find the critical point temperature
$T_{c}$ =233 MeV. By increasing the sigma mass up to $m_{\sigma}=700$ MeV
leads to $T_{c}$ = 309 MeV. In comparison with lattice QCD results [19], the
critical point temperature is found in the range $T_{c}=100\ $to 300 MeV.
Therefore the present result is in agreement with lattice QCD results. Nemoto
et al. [19] calculated the critical point temperature equal 230 MeV using the
original sigma model. Abu-shady [5] calculated the critical point temperature
equal 226 MeV in the original sigma model. Therefore, the present result is in
good agreement with Refs. [5, 19]. In Fig. 2, the sigma and pion masses are
plotted as functions of temperature in the chiral limit ( $m_{\pi}=0$). A
similar behavior is obtained as in the case of the explicit symmetry term
expect the pion mass equal zero at zero temperature. In comparison with
original sigma model as in Refs. [1, 5, 19], we found that the behavior of the
sigma and pion masses are in good agreement with Refs. [1, 5, 19]. In Fig. 3,
we examine the effect of coupling constant $g$ on the critical point
temperature. We find that an increase on the coupling constant $g$ leads to
decrease in the critical point temperature g = 3.76 to g = 4.48 corresponding
to $T_{c}=$ 243 MeV to $T_{c}=191$ MeV, respectively. In Fig. (4), the
effective potential is plotted as a function of temperature. We note that the
potential decreases with increasing temperature. In order to get more insight
into the nature of the phase transition and verify that the order phase
transition is a second-order phase transition. We calculate the effective
potential $U_{eff}(\Phi\mathbf{,}T\mathbf{)}$ as a function of phase
transition ($\Phi=\sqrt{\sigma^{2}+\mathbf{\pi}^{2}})$ as seen in Fig. (4).
The shape of the potential confirms that the phase transition is the second-
order. Since it exhibits can degenerate one minima at $\Phi\neq 0$. The
indication of the second-order phase transition has been reported in many
works [14, 21- 23]. Also, We note the strong increase in the temperature T
which unchanged the shape of the potential
Next, we need to examine the effect of finite temperature on the behavior of
pressure where the quarks takes as a heat bath. In Fig. (5), the pressure is
plotted as a function of temperature. The pressure increases with increasing
temperature. Also, we note that the pressure value at lower-values of
temperature is not sensitive in comparison with the pressure value at the
higher-values of temperature. Hence the effect of largest values of
temperature is more affected on the value of pressure. Berger and Christov
[24] found that the pressure increases with increasing temperature in hot
medium using the NJL model in the mean field approximation. The present
behavior is in agreement with Ref. [24].
## 5\. Summary and Conclusion
In this work, the effective mesonic potential is calculated by using
n-midpoint algorithm in the logarithmic quark model. The meson properties, the
phase transition, and the pressure are calculated using the effective mesonic
potential. We summarized the following points:The behavior of sigma and pion
masses as functions of temperature are investigated. A comparison with
original sigma model is presented. The increase of sigma mass
($m_{\sigma})~{}$leads to increase the critical point temperature. The
increase of the coupling constant ($g)$ leads to decrease the critical point
temperature. The phase transition is predicted as a second-order phase
transition which agrees with other works. The critical point temperature and
the pressure are calculated and are in agreement with other works. Therefore,
the calculated effective logarithmic potential is successful to predict the
meson masses, the phase transition, and the pressure at finite temperature.
## References
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|
arxiv-papers
| 2012-12-10T13:33:28 |
2024-09-04T02:49:39.117763
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Abu-shady",
"submitter": "M. Abu-shady",
"url": "https://arxiv.org/abs/1212.2276"
}
|
1212.2286
|
# Raman scattering in a Heisenberg $S=1/2$ antiferromagnet on the anisotropic
triangular lattice
Natalia B. Perkins Department of Physics, University of Wisconsin-Madison,
Madison, WI 53706, USA Gia-Wei Chern Department of Physics, University of
Wisconsin-Madison, Madison, WI 53706, USA Theoretical Division, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA Wolfram Brenig
Institute for Theoretical Physics, Technical University Braunschweig,
Mendelssohnstr. 3, 38106 Braunschweig, Germany
###### Abstract
We investigate the two-magnon Raman scattering from an anisotropic $S=1/2$
triangular Heisenberg antiferromagnet Cs2CuCl4. We find that the Raman
response is very sensitive to magnon-magnon interactions and to scattering
geometries, a feature that is in remarkable contrast with the polarization-
independent Raman signal from the isotropic triangular Heisenberg
antiferromagnet. Since a spin-liquid ground state gives rise to a similar
rotationally invariant Raman response, our results on the polarization
dependence of the scattering spectrum suggest that Raman spectroscopy provides
a useful probe, complementary to neutron scattering, of the ground-state
properties of Cs2CuCl4, particularly whether the time-reversal symmetry is
broken in the ground state.
## I Introduction
Recently Heisenberg antiferromagnets on the triangular lattice have attracted
considerable experimental and theoretical interest. Among them, Cs2CuCl4 has
been under particular scrutiny as it provides an interesting example of a
spatially anisotropic spin-$1/2$ triangular antiferromagnet. coldea1 ; coldea2
; isakov05 ; alicea05 ; yunoki06 ; veillette05 ; dalidovich06 ; starykh07 ;
starykh10 ; kresel11 Much of the interest in this compound stems from its
unusual, non-classical magnetic properties, arising from the competition
between the spatial anisotropy, Dzyaloshinskii-Moriya (DM) interactions, and
quantum fluctuations.
Extensive neutron scattering studies coldea1 ; coldea2 on the magnetic
properties of Cs2CuCl4 revealed several interesting features. First, despite
frustration and low-dimensionality, a long-range magnetic order develops at
low temperatures: the observed spin order is incommensurate and sets in at
temperatures below $T_{N}=0.62$ K. Magnetic excitations above this ground
state are also quite unusual. While the low-energy excitation spectrum
contains well-defined sharp modes, as expected for an ordered state, a broad
continuum is formed at intermediate and high energies. A number of theoretical
proposals have been made to explain the origin of this continuum. It has been
suggested that the existence of a continuum is an indication that the system
is proximate to a spin liquid phase that determines the behavior of
excitations except for low energies.isakov05 ; alicea05 ; yunoki06 ; starykh07
An alternative suggestion is that the continuum might originate from magnon-
magnon scattering which is enhanced in non-colinear magnets.veillette05 ;
dalidovich06
In view of the ambiguity in the interpretation of the neutron scattering data,
a complementary experimental analysis of the magnetic properties of Cs2CuCl4
by a different technique is highly desirable. A very effective and frequently
used experimental tool to study low-temperature properties of low-dimensional
quantum magnets is the two-magnon Raman scattering.gozar05 ; gingras06 ;
choi09 ; iliev10 ; wang11 ; chen11 The two-magnon Raman intensity is directly
related to the spectrum of two interacting magnons in a total spin zero state
at vanishingly small momentum and weighted by a form factor that is dependent
on the polarization of the incident light. It contains detailed information on
the two-magnon density of states and the magnon-magnon interactions.
Therefore, direct comparison of experimental spectra with those obtained from
theoretical analysis can lead to rather accurate estimates on values of the
superexchange and DM interactions. In addition, the analysis of the
polarization dependence of the magnetic Raman scatteringperkins08 ; cepas08
might shed some light on whether the ground state is ordered, as neutron
scattering experiments have suggested, or is actually in a spin liquid state,
as some theories suggest. A pronounced polarization dependence would indicate
that magnetically ordered state is the most probable candidate for the ground
state, and that the observed continuum is due to relatively strong
interactions between magnons. On the other hand, if the Raman scattering
depends weakly on the scattering geometry, the continuum in neutron scattering
might be due to unconventional excitations above a spin liquid ground state.
Figure 1: Triangular lattice with anisotropic spin exchanges.
In this paper, we carry out a theoretical analysis of two-magnon Raman
scattering from an anisotropic $S=1/2$ Heisenberg antiferromagnet (HAF) on a
triangular lattice (see Fig.1). To evaluate the two-magnon Raman spectra, we
use the well-established, semi-phenomenological Loudon-Fleury (LF)
approach.fleury By chosing model parameters relevant for the compound
Cs2CuCl4, we find that the spectral shape as a function of frequency is
sensitive to $1/S$ corrections of the magnon spectrum and is strongly modified
by the magnon-magnon interactions in the final state. The intensity of the
two-magnon peak also significantly depends on the scattering geometry which is
in contrast with the polarization independence of the magnetic Raman response
in isotropic HAF on triangular lattice.perkins08
The paper is organized as follows. In Sec. II, we present a model of Cs2CuCl4
and discuss its classical ground state, which is an incommensurate spin spiral
with a pitch vector determined by the competition of anisotropic nearest-
neighbor interactions. In Sec. III, we first review results of the one magnon
excitations in the anisotropic $S=1/2$ HAF to first order in $1/S$. We show
that although $1/S$-corrections are present in the whole Brillouin zone (BZ),
they are less drastic than in the isotropic case of the triangular lattice.
This is due DM interactions, which suppress quantum fluctuations and open a
gap at the ordering vector. In Sec. IV, we first review the LF formalism and
then use it to calculate the Raman spectra at various levels of approximation,
i.e., using only the bare magnon dispersion, using a magnon dispersion
renormalized to order 1/S, and with final state interactions included. We show
that the Raman profile is very sensitive to the magnon-magnon interactions and
to the scattering geometry. Finally, Sec. V presents a summary of the work.
## II Model
We start with the following spin-1/2 model Hamiltonian for Cs2CuCl4:
$H=\sum_{\langle ij\rangle}\bigl{[}J_{ij}\,{\bf S}_{i}\cdot{\bf
S}_{j}+\mathbf{D}_{ij}\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})\bigr{]},$ (1)
where $\langle ij\rangle$ refers to nearest-neighbor (NN) bonds on the
triangular lattice, and $J_{ij}$ and $\mathbf{D}_{ij}$ are the symmetric and
antisymmetric exchange constants. The antisymmetric spin exchange originating
from the relativistic spin-orbit interaction is also known as Dzyaloshinskii-
Moriya (DM) interaction. For Cs2CuCl4, it is customary to denote the exchange
constants $J_{ij}$ along the horizontal bonds, which form quasi-one-
dimensional chains, as $J$, and $J_{ij}$ along the zigzag bonds as
$J^{\prime}$. coldea1 ; coldea2 In this paper we consider the DM vectors in
the geometry suggested by the neutron-scattering work by Coldea et al..coldea2
In this geometry the DM interaction vanishes along the quasi-1D chains,
whereas on the zigzag bonds, the DM vectors are perpendicular to the
triangular plane $\mathbf{D}_{ij}=\pm(0,0,D)$ (Fig. 1). Experimental
measurements in high magnetic field have resulted in $J\approx 0.374$ meV,
$J^{\prime}\approx 0.128$ meV, and $D=0.02$ meV.coldea2
The classical ground state of Hamiltonian (1) is given by a spin spiral
$\mathbf{S}_{i}/S=\cos\bigl{(}\mathbf{Q}\cdot\mathbf{r}_{i})\,\hat{\mathbf{b}}+\sin\bigl{(}\mathbf{Q}\cdot\mathbf{r}_{i}\bigr{)}\,\hat{\mathbf{c}}$,
where the pitch vector $\mathbf{Q}=Q\,\hat{\mathbf{b}}$ depends on the ratio
$J^{\prime}/J$. In the isotropic case $J^{\prime}=J$, the ground state is the
well known 120∘ non-collinear magnetic order with $Q=2\pi/3$.Jolicoeur1989 ;
chubukov To understand the classical ground state in detail, we consider the
energy of the magnetic spiral:
$\displaystyle E_{0}(\mathbf{Q})=3NS^{2}J^{T}_{\mathbf{Q}},\quad\quad
J^{T}_{\mathbf{Q}}=J_{\mathbf{Q}}-D_{\mathbf{Q}},$ (2)
where
$\displaystyle J_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\Bigl{(}J\cos
k_{y}+2J^{\prime}\cos\frac{k_{y}}{2}\cos\frac{\sqrt{3}k_{z}}{2}\Bigr{)},$ (3)
$\displaystyle D_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle\frac{2D}{3}\sin\frac{k_{y}}{2}\cos\frac{\sqrt{3}k_{z}}{2}.$ (4)
Minimization with respect to $Q$ leads to the following three types of long-
range magnetic order: (1) At $J^{\prime}>2J$, the magnetic spiral reduces to
collinear Néel order with ferromagnetic ordering of the spins along the
chains. Along the $c$ direction, spins on adjacent chains are antiparallel to
each other. (2) At $0<J^{\prime}\leq 2J$, the pitch of the spiral is given by
$Q=2\arccos(-\frac{J^{\prime}}{2J})$, which varies from $2\pi\rightarrow\pi$
as we vary $J^{\prime}$ from $2J$ to 0. (3) For $J^{\prime}=0$, the system
degenerates into decoupled antiferromagnetic chains.
## III Large-$S$ expansion
The large-$S$ expansion about the classical spiral order can be significantly
simplified with a locally rotated frame of reference. chubukov The spin
components $S_{i}$ in a laboratory frame are related to those in the rotated
local frame through
$\displaystyle S^{x}_{i}$ $\displaystyle=$ $\displaystyle\tilde{S}^{x}_{i},$
$\displaystyle S^{y}_{i}$ $\displaystyle=$ $\displaystyle\tilde{S}^{y}_{i}\cos
Q-\tilde{S}^{z}_{i}\sin Q,$ (5) $\displaystyle S^{z}_{i}$ $\displaystyle=$
$\displaystyle\tilde{S}^{y}_{i}\sin Q+\tilde{S}^{z}_{i}\cos Q$
The spiral viewed from the rotated local frame corresponds to a simple
ferromagnetic order $\tilde{\mathbf{S}}_{i}=S\hat{\mathbf{z}}$. We then employ
the Holstein-Primakoff transformation:
$\displaystyle\tilde{S}^{z}_{i}$ $\displaystyle=$ $\displaystyle
S-a^{+}_{i}a^{\phantom{+}}_{i}$ (6) $\displaystyle\tilde{S}^{+}_{i}$
$\displaystyle=$
$\displaystyle(2S-a^{+}_{i}a^{\phantom{+}}_{i})^{1/2}\,a^{\phantom{+}}_{i}$
$\displaystyle\tilde{S}^{-}_{i}$ $\displaystyle=$ $\displaystyle
a^{+}_{i}(2S-a^{+}_{i}a^{\phantom{+}}_{i})^{1/2}~{}.$
The magnon operators $a^{\dagger}_{i}$ and $a_{i}$ describe excitations around
the spiral ground state. As we intend to study magnon interactions to first
order in $1/S$, we need to expand the Hamiltonian in Eq. (1) up to quartic
order in the boson operators:
$\displaystyle H=E_{0}+3JS(H_{2}+H_{3}+H_{4}).$ (7)
Introducing Fourier transform $a_{i}=\sum_{\bf k}a_{{\bf k}}\,e^{i{\bf
k}\cdot{\bf r}_{i}}/\sqrt{N}$, the explicit expression of the various terms in
the Hamiltonian reads
$\displaystyle
H_{2}=\sum_{\mathbf{k}}\Bigl{[}A_{\mathbf{k}}\,a^{\dagger}_{\mathbf{k}}\,a_{\mathbf{k}}+\frac{B_{\mathbf{k}}}{2}\bigl{(}a_{\mathbf{k}}\,a_{-\mathbf{k}}+a^{\dagger}_{\mathbf{k}}\,a^{\dagger}_{-\mathbf{k}}\bigr{)}\Bigr{]},$
(8) $\displaystyle
H_{3}=\frac{i}{2}\sqrt{\frac{1}{2NS}}\sum_{\\{\mathbf{k}_{i}\\}}(C_{\mathbf{1}}+C_{\mathbf{2}})\,\bigl{(}a^{\dagger}_{\mathbf{3}}a_{\mathbf{1}}a_{\mathbf{2}}-a^{\dagger}_{\mathbf{1}}a^{\dagger}_{\mathbf{2}}a_{\mathbf{3}}\bigr{)},$
(9) $\displaystyle
H_{4}=\frac{1}{8NS}\sum_{\\{\mathbf{k}_{i}\\}}\biggl{\\{}\Bigl{[}(A_{\mathbf{1}-\mathbf{3}}+A_{\mathbf{1}-\mathbf{4}}+A_{\mathbf{2}-\mathbf{3}}+A_{\mathbf{2}-\mathbf{4}})$
$\displaystyle-(B_{\mathbf{1}-\mathbf{3}}+B_{\mathbf{1}-\mathbf{4}}+B_{\mathbf{2}-\mathbf{3}}+B_{\mathbf{2}-\mathbf{4}})$
$\displaystyle-(A_{\mathbf{1}}+A_{\mathbf{2}}+A_{\mathbf{3}}+A_{\mathbf{4}})\Bigr{]}a^{\dagger}_{\mathbf{1}}a^{\dagger}_{\mathbf{2}}a_{\mathbf{3}}a_{\mathbf{4}}$
$\displaystyle-\frac{2}{3}\Bigl{(}B_{\mathbf{1}}+B_{\mathbf{2}}+B_{\mathbf{3}}\Bigr{)}\,\Bigl{(}a^{\dagger}_{\mathbf{1}}a^{\dagger}_{\mathbf{2}}a^{\dagger}_{\mathbf{3}}a_{\mathbf{4}}+a^{\dagger}_{\mathbf{4}}a_{\mathbf{1}}a_{\mathbf{2}}a_{\mathbf{3}}\Bigr{)}\biggr{\\}}.$
(10)
Here $\mathbf{1}\cdots\mathbf{4}$ denote $\mathbf{k}_{1}\cdots\mathbf{k}_{4}$,
and the summation in $H_{3}$ and $H_{4}$ is subject to momentum conservation
module a reciprocal lattice vector: $\sum_{i}\mathbf{k}_{i}=0$ mod
$\mathbf{G}$. The following functions are introduced:
$\displaystyle\begin{array}[]{c}A_{\mathbf{k}}=J_{\mathbf{k}}+\frac{1}{2}\bigl{(}J^{T}_{\mathbf{Q}+\mathbf{k}}+J^{T}_{\mathbf{Q}-\mathbf{k}}\bigr{)}-2J^{T}_{\mathbf{Q}},\\\
\\\
B_{\mathbf{k}}=\frac{1}{2}\bigl{(}J^{T}_{\mathbf{Q}+\mathbf{k}}+J^{T}_{\mathbf{Q}-\mathbf{k}}\bigr{)}-J_{\mathbf{k}},\quad\quad
C_{\mathbf{k}}=J^{T}_{\mathbf{Q}+\mathbf{k}}-J^{T}_{\mathbf{Q}-\mathbf{k}}.\end{array}$
(14)
Here, for convenience, we rescale the interactions $J_{\mathbf{k}}$ and
$J^{T}_{\mathbf{k}}$ with respect to $J$, which is assumed to be $J=1$.
*
Figure 2: Top: Renormalized magnon dispersion. Dotted line corresponds to the
linear spin-wave dispersion $E_{\bf k}$, while red and blue solid lines
correspond to the real and imaginary part $Re(Im){\tilde{E}}_{\bf k}$,
respectively, computed on a lattice of $252\times 252$ ${\bf k}$-points with
artificial line broadening of $\eta=0.003$. Middle and Bottom: 3D-plot of the
$Re{\tilde{E}}_{\bf k}$ and of the $Im{\tilde{E}}_{\bf k}$, respectively. The
spectrum is computed for $J=1$, while other parameters describing interactions
is Cs2CuCl4, are correspondingly rescaled. All energies are measured in units
of $3JS$.
The quadratic Hamiltonian $H_{2}$ can then be diagonalized by a Bogoliubov
transformation:
$\displaystyle a^{\phantom{\dagger}}_{\bf k}$ $\displaystyle=$ $\displaystyle
u_{\bf k}c^{\phantom{\dagger}}_{\bf k}+v_{\bf k}c_{-{\bf k}}^{\dagger}$ (15)
$\displaystyle a_{\bf k}^{\dagger}$ $\displaystyle=$ $\displaystyle u_{\bf
k}c_{\bf k}^{\dagger}+v_{\bf k}c^{\phantom{\dagger}}_{-{\bf k}}~{},$
where $c^{(\dagger)}_{\bf k}$ are operators for Bogoliubov quasiparticles. The
coherence coefficients
$\displaystyle u_{\bf k}=\sqrt{\frac{A_{\bf k}+E_{\bf k}}{2E_{\bf
k}}},\quad\quad v_{\bf k}=-\frac{B_{\bf k}}{|B_{\bf k}|}\sqrt{\frac{A_{\bf
k}-E_{\bf k}}{2E_{\bf k}}}~{}$ (16)
satisfy $u_{\bf k}^{2}-v_{\bf k}^{2}=1$, and
$\displaystyle E_{\bf k}=\sqrt{A_{\bf k}^{2}-B_{\bf k}^{2}}$ (17)
describes the quasiparticle dispersion (the energy of the quasiparticles is
$3JSE_{\mathbf{k}}$). The diagonalized Hamiltonian $H_{2}$ is given by
$\displaystyle H_{2}=E_{2}({\bf Q})+\sum_{\bf k}E_{\bf k}c_{\bf
k}^{\dagger}c_{\bf k}~{},$ (18)
where
$\displaystyle E_{2}({\bf Q})$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}\bigl{(}A_{\mathbf{k}}v_{\bf
k}^{2}+B_{\mathbf{k}}u_{\bf k}v_{\bf k}\bigr{)}$ (19) $\displaystyle=$
$\displaystyle-
NJ^{T}_{\mathbf{Q}}+\frac{1}{2}\sum_{\mathbf{k}}E_{\mathbf{k}}.$
gives $1/S$ correction to the classical ground state energy $E_{0}({\bf Q})$.
The $1/S$ correction to the ordering wave vector $\mathbf{Q}$ is determined by
minimizing the sum
$\displaystyle
E_{0}+E_{2}=3JS(S-1)NJ^{T}_{\mathbf{Q}}+\frac{3JS}{2}\sum_{\mathbf{k}}E_{\mathbf{k}}$
(20)
with respect to $Q$. The quantum correction is given by
$\displaystyle\Delta Q=\frac{-1}{\partial^{2}J^{T}_{Q}/\partial
Q^{2}}\,\frac{1}{N}\sum_{\mathbf{k}}\frac{A_{\mathbf{k}}-B_{\mathbf{k}}}{2E_{\mathbf{k}}}\,\frac{\partial
J^{T}_{\mathbf{Q}+\mathbf{k}}}{\partial Q}\bigg{|}_{Q_{0}},$ (21)
where $Q_{0}=2\arccos(-\frac{J^{\prime}}{2J})$ is the pitch of the classical
ground state.
The $1/S$ contribution from the quartic Hamiltonian can be obtained through a
mean-field decoupling of $H_{4}$. We first define
$\displaystyle G(\mathbf{k})=\langle
a^{\dagger}_{\mathbf{k}}\,a_{\mathbf{k}}\rangle,\quad F(\mathbf{k})=\langle
a_{\mathbf{k}}\,a_{-\mathbf{k}}\rangle=\langle
a^{\dagger}_{\mathbf{k}}\,a^{\dagger}_{-\mathbf{k}}\rangle.$ (22)
The quadratic Hamiltonian plus the decoupled $H_{4}$ can be expressed as:
$H_{2}+{\bar{H}}_{4}=3JS\sum_{\mathbf{k}}\Bigl{[}\bar{A}_{\mathbf{k}}\,a^{\dagger}_{\mathbf{k}}\,a_{\mathbf{k}}+\frac{\bar{B}_{{\mathbf{k}}}}{2}\,\bigl{(}a_{\mathbf{k}}\,a_{-\mathbf{k}}+a^{\dagger}_{\mathbf{k}}\,a^{\dagger}_{-\mathbf{k}}\bigr{)}\Bigr{]},$
(23)
where
$\displaystyle\bar{A}_{\mathbf{k}}$ $\displaystyle=$ $\displaystyle
A_{\mathbf{k}}+\frac{1}{NS}\sum_{\mathbf{q}}\Bigl{[}\Bigl{(}A_{\mathbf{k}-\mathbf{q}}+B_{\mathbf{k}-\mathbf{q}}$
(24)
$\displaystyle\,\,-A_{\mathbf{k}}-A_{\mathbf{q}}\Bigr{)}\,G(\mathbf{q})-\Bigl{(}B_{\mathbf{q}}+\frac{B_{\mathbf{k}}}{2}\Bigr{)}\,F(\mathbf{q})\Bigr{]},$
$\displaystyle\bar{B}_{\mathbf{k}}$ $\displaystyle=$ $\displaystyle
B_{\mathbf{k}}+\frac{1}{NS}\sum_{\mathbf{q}}\Bigl{[}-\Bigl{(}B_{\mathbf{k}}+\frac{B_{\mathbf{q}}}{2}\Bigr{)}\,G(\mathbf{q})$
(25)
$\displaystyle+\Bigl{(}A_{\mathbf{k}-\mathbf{q}}+B_{\mathbf{k}-\mathbf{q}}-\frac{A_{\mathbf{q}}}{2}-\frac{A_{\mathbf{k}}}{2}\Bigr{)}\,F(\mathbf{q})\Bigr{]}.$
The magnon spectrum renormalized by the quartic Hamiltonian $H_{4}$ becomes
$\displaystyle{\bar{E}}_{\bf
k}=\sqrt{\bar{A}_{\mathbf{k}}^{2}-\bar{B}_{\mathbf{k}}^{2}}=E_{\bf
k}+\Sigma^{(4)}_{\mathbf{k}}+\mathcal{O}(1/S^{2})~{},$ (26)
where $\Sigma^{(4)}(\mathbf{k})$ is self-energy correction of order $1/S$:
$\displaystyle\Sigma^{(4)}_{\mathbf{k}}$ $\displaystyle=$ $\displaystyle
A^{(4)}_{\bf k}(u^{2}_{\mathbf{k}}+v^{2}_{\mathbf{k}})+2B^{(4)}_{\bf
k}u_{\mathbf{k}}v_{\mathbf{k}}$ (27) $\displaystyle=$
$\displaystyle\bigl{(}A_{\mathbf{k}}\,A^{(4)}_{\mathbf{k}}-B_{\mathbf{k}}\,B^{(4)}_{\mathbf{k}}\bigr{)}/E_{\mathbf{k}}.$
To obtain the $1/S$ correction from the cubic Hamiltonian $H_{3}$, we follow
Ref. chubukov, and consider interactions between quasiparticles $c$,
$c^{\dagger}$:
$\displaystyle H_{3}$ $\displaystyle=$
$\displaystyle\frac{i}{4}\sqrt{\frac{1}{2NS}}\sum_{\\{\mathbf{k}_{i}\\}}\Bigl{[}\Phi_{1}(\mathbf{k}_{1},\mathbf{k}_{2};\mathbf{k}_{3})\,c^{\dagger}_{\mathbf{k}_{1}}c^{\dagger}_{\mathbf{k}_{2}}c_{\mathbf{k}_{3}}$
(28)
$\displaystyle+\frac{1}{3}\Phi_{2}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})\,c^{\dagger}_{\mathbf{k}_{1}}c^{\dagger}_{\mathbf{k}_{2}}c^{\dagger}_{\mathbf{k}_{3}}\Bigr{]}+\mbox{h.c.}$
The vertex functions are given by (for simplicity, we denote
$1\equiv\mathbf{k}_{1}$, $2\equiv\mathbf{k}_{2}$, etc.)
$\displaystyle\Phi_{1}(1,2;3)=\frac{\tilde{\Phi}_{1}(1,2;3)}{\sqrt{E_{1}E_{2}E_{3}}},\quad\Phi_{2}(1,2,3)=\frac{\tilde{\Phi}_{2}(1,2,3)}{\sqrt{E_{1}E_{2}E_{3}}}.$
(29)
where
$\displaystyle\tilde{\Phi}_{1}(1,2;3)=C_{1}f^{(1)}_{-}(f^{(2)}_{+}f^{(3)}_{+}+f^{(2)}_{-}f^{(3)}_{-})$
$\displaystyle+C_{2}f^{(2)}_{-}(f^{(3)}_{+}f^{(1)}_{+}+f^{(3)}_{-}f^{(1)}_{-})$
$\displaystyle+C_{3}f^{(3)}_{-}(f^{(1)}_{+}f^{(2)}_{+}-f^{(1)}_{-}f^{(2)}_{-})$
(30)
$\displaystyle\tilde{\Phi}_{2}(1,2,3)=C_{1}f^{(1)}_{-}(f^{(2)}_{+}f^{(3)}_{+}-f^{(2)}_{-}f^{(3)}_{-})$
$\displaystyle+C_{2}f^{(2)}_{-}(f^{(3)}_{+}f^{(1)}_{+}-f^{(3)}_{-}f^{(1)}_{-})$
$\displaystyle+C_{3}f^{(3)}_{-}(f^{(1)}_{+}f^{(2)}_{+}-f^{(1)}_{-}f^{(2)}_{-})$
(31)
and $f^{(\alpha)}_{\pm}=\sqrt{A_{\alpha}\pm B_{\alpha}}$ for $\alpha=1,2,3$.
The triplic contribution to the self-energy is
$\displaystyle\Sigma^{(3)}_{\mathbf{k}}=-\frac{1}{16NS}\Biggl{(}\sum_{\mathbf{k}_{1}+\mathbf{k}_{2}=\mathbf{k}}\frac{|\Phi^{(1)}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k})|^{2}}{E_{\mathbf{k}_{1}}+E_{\mathbf{k}_{2}}-E_{\mathbf{k}}+i\,\eta}+\sum_{\mathbf{k}_{1}+\mathbf{k}_{2}=-\mathbf{k}}\frac{|\Phi^{(2)}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k})|^{2}}{E_{\mathbf{k}_{1}}+E_{\mathbf{k}_{2}}+E_{\mathbf{k}}+i\,\eta}\Biggr{)}.$
(32)
The first term in Eq.(32) describes a virtual decay of a magnon into two-
particle intermediate states. Terms with three creation(annihilation)
operators, as will become clear in the next section, play no role in
evaluating the magnon interactions within the Raman response. We also note
that $\Sigma^{(3)}_{\mathbf{k}}$ is computed within so-called on-shell
approximation. In this approximation the self-energy is evaluated at the bare
magnon energy $E_{\mathbf{k}}$.
Finally, the magnon energy renormalized by both the quartic and the triplic
terms is given by
$\displaystyle{\tilde{E}}_{\mathbf{k}}=E_{\mathbf{k}}+\Sigma^{(4)}_{\mathbf{k}}+\Sigma^{(3)}_{\mathbf{k}}+\mathcal{O}(1/S^{2})~{}.$
(33)
In Fig. 2 we plot the renormalized magnon spectrum ${\tilde{E}}_{\mathbf{k}}$
for parameters relevant to Cs2CuCl4. One can see that the renormalization of
the spectrum is much less pronounced than in the case of the isotropic
triangular lattice.chubukov06 ; mike Moreover, the imaginary part of the
magnon energy, $Im{\tilde{E}}_{\mathbf{k}}$ almost vanishes in the whole BZ.
Thus, the life time of the quasi-particles is very large at almost any
momentum. This happens because the DM interaction opens a gap at the ordering
$\mathbf{Q}$ vector, which significantly suppresses quantum
fluctuations.dalidovich06 We note that, as corrections to the ordering vector
$\mathbf{Q}$, determined by Eq.(21), are small at those values of $D$ relevant
to our calculation, we use the bare value of the ordering wave vector
${\mathbf{Q}}_{0}$ for the remainder of the paper.
## IV Raman intensity
### IV.1 Loudon-Fleury formalism
Here we present the analysis of the two-magnon Raman scattering from the
anisotropic triangular lattice. We employ the LF approach which models the
interaction of light with spin degrees of freedom. The LF scattering operator
is given by the photon-induced super-exchange operatorfleury ; comment
$\displaystyle R=\sum_{i,\pm\bm{\delta}_{\mu}}(\hat{\bm{\varepsilon}}_{\rm
in}\cdot{\bm{\delta}}_{\mu})(\hat{\bm{\varepsilon}}_{\rm
out}\cdot{\bm{\delta}}_{\mu})\,J_{\mu}\,{\bf S}_{i}\cdot{\bf
S}_{i\pm\bm{\delta}_{\mu}}~{},$ (34)
weere $\bm{\delta}_{\mu}$ denote basic vectors of triangular lattice:
$\bm{\delta}_{1}=(1,0)$, $\bm{\delta}_{2}=(\frac{1}{2},\frac{\sqrt{3}}{2})$
and $\bm{\delta}_{3}=(-\frac{1}{2},\frac{\sqrt{3}}{2})$. $J_{\mu}$ defines the
Heisenberg exchange on the bond ${\bm{\delta}}_{\mu}$. Since $J_{\mu}$ is
anisotropic, the $C_{3v}$ symmetry of the triangular lattice is broken. Thus,
instead of using $C_{3v}$-irreducible representations ($A_{1}$, $A_{2}$ and
$E$) for characterization of the polarizations, we determine polarizations of
incoming and outgoing light as $\hat{\bm{\varepsilon}}_{\rm
in}=\cos\theta\,\hat{\mathbf{x}}+\sin\theta\,\hat{\mathbf{y}}$ and
$\hat{\bm{\varepsilon}}_{\rm
out}=\cos\phi\,\hat{\mathbf{x}}+\sin\phi\,\hat{\mathbf{y}}$, where $\theta$
and $\phi$ are defined with respect to the $x$-axis.
In terms of Bogoliubov quasi-particle $c$-operators, the LF scattering
operator (34) takes the following form:
$\displaystyle
R=\sum_{\mathbf{k}}M_{\mathbf{k}}(c_{\mathbf{k}}c_{-{\mathbf{k}}}+c_{\mathbf{k}}^{\dagger}c_{-{\mathbf{k}}}^{\dagger})\equiv
r^{-}+r^{+}~{},$ (35)
where $M_{\mathbf{k}}$ is bare Raman vertex, which is determined by the the
magnon spectrum and by scattering geometry. The expression for
$M_{\mathbf{k}}$ is given by
$\displaystyle
M_{\mathbf{k}}=F_{1}({\mathbf{k}},\theta,\phi)u_{\mathbf{k}}v_{\mathbf{k}}+F_{2}({\mathbf{k}},\theta,\phi)(u_{\mathbf{k}}^{2}+v_{\mathbf{k}}^{2})~{},$
(36)
where we introduced the following notations:
$\displaystyle\begin{array}[]{ll}&F_{1}({\mathbf{k}},\theta,\phi)=2S\sum_{\mu=1}^{3}f_{\mu}(\theta,\phi)\xi_{{\mu}{\mathbf{k}}}~{},\\\\[5.69046pt]
&F_{2}({\mathbf{k}},\theta,\phi)=S\sum_{\mu=1}^{3}f_{\mu}(\theta,\phi)\nu_{{\mu}{\mathbf{k}}}~{},\end{array}$
(39)
and
$\displaystyle\xi_{1{\mathbf{k}}}$ $\displaystyle=$ $\displaystyle\cos
k_{x}\bigl{(}1+\cos Q_{0}\bigr{)}-2\cos Q_{0},$
$\displaystyle\xi_{2{\mathbf{k}}}$ $\displaystyle=$
$\displaystyle\cos\Bigl{(}\frac{k_{x}}{2}+\frac{\sqrt{3}k_{y}}{2}\Bigr{)}\Bigl{(}1+\cos\frac{Q_{0}}{2}\Bigr{)}-2\cos\frac{Q_{0}}{2},$
$\displaystyle\xi_{3{\mathbf{k}}}$ $\displaystyle=$
$\displaystyle\cos\Bigl{(}\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2}\Bigr{)}\Bigl{(}1+\cos\frac{Q_{0}}{2}\Bigr{)}-2\cos\frac{Q_{0}}{2},$
$\displaystyle\nu_{1{\mathbf{k}}}$ $\displaystyle=$ $\displaystyle\cos
k_{x}\bigl{(}1-\cos Q_{0}\bigr{)},$ (40) $\displaystyle\nu_{2{\mathbf{k}}}$
$\displaystyle=$
$\displaystyle\cos\Bigl{(}\frac{k_{x}}{2}+\frac{\sqrt{3}k_{y}}{2}\Bigr{)}\Bigl{(}1-\cos\frac{Q_{0}}{2}\Bigr{)},$
$\displaystyle\nu_{3{\mathbf{k}}}$ $\displaystyle=$
$\displaystyle\cos\Bigl{(}\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2}\Bigr{)}\Bigl{(}1-\cos\frac{Q_{0}}{2}\Bigr{)}.$
The functions $f_{\mu}(\theta,\phi)\equiv(\hat{\bm{\varepsilon}}_{\rm
in}\cdot{\bm{\delta}}_{\mu})(\hat{\bm{\varepsilon}}_{\rm
out}\cdot{\bm{\delta}}_{\mu})$ are symmetry weighting factors along the three
basic vectors $\bm{\delta}_{\mu}$ of the triangular lattice.
The Raman intensity is obtained from the Fermi’s golden rule, which on the
imaginary frequency axis reads
$I(\omega_{m})\simeq\Im{\rm
m}\bigl{[}\int_{0}^{\beta}d\tau\,e^{i\omega_{m}\tau}\left\langle
T_{\tau}(R(\tau)R)\right\rangle\bigr{]}~{}.$ (41)
By analytically continuing Matsubara frequencies $\omega_{m}=2\pi mT$ onto the
real axis as $i\omega_{m}\rightarrow\Omega+i\eta$, we obtain the Raman
intensity as a function of the inelastic energy transfer $\Omega=\omega_{\rm
in}-\omega_{\rm out}$ of the incident photons. For the rest of the paper we
assume the temperature $T=1/\beta$ to be zero.
To order $1/S$, the Raman polarization operator $\left\langle
T_{\tau}(R(\tau)R)\right\rangle$ contains only terms like $\left\langle
T_{\tau}(r^{+}(\tau)r^{-})\right\rangle+\left\langle
T_{\tau}(r^{-}(\tau)r^{+})\right\rangle$, where $r^{\pm}$ are specified in Eq.
(35) and Fig. 5a). By Hermitian conjugation, it is sufficient to calculate
$J(\tau)=\left\langle T_{\tau}(r^{-}(\tau)r^{+})\right\rangle$, which is
depicted in Fig. 5b).
### IV.2 Raman intensity without final state interactions
Figure 3: Bare Raman intensity (solid lines) and Raman intensity including the
one-magnon renormalization of the spectrum (dashed lines) at different
polarizations of light described by $\phi$, and $\theta$ scattering angles.
Number of ${\mathbf{k}}$-points: $252\times 252$. The broadening parameter is
$\eta=0.003$. $\Omega$ is measured in units of $3JS$.
We now focus on the contribution from two-magnon Raman scattering, which can
be computed at different levels of approximation. We begin by first using only
the bare spin-wave dispersion. Then we continue by including renormalizations
of the one-magnon spectrum to $1/S$ order. Calculating the Raman polarization
operator $\left\langle T_{\tau}(R(\tau)R)\right\rangle$ with the bare
propagators of the Bogoliubov quasiparticles,
$G({\mathbf{k}},i\omega_{n})=1/(i\omega_{n}-E_{\mathbf{k}})$, where
$E_{\mathbf{k}}$ is the bare magnon spectrum gives the following expression
for the Raman intensity:
$\displaystyle I(\Omega)$ $\displaystyle\simeq$ $\displaystyle\Im{\rm
m}\,\Bigl{[}\int d^{2}{\mathbf{k}}\int d\omega
M_{\mathbf{k}}^{2}\frac{1}{\imath\omega-
E_{\mathbf{k}}}\frac{1}{\imath(\Omega-\omega)-E_{\bf k}}\Bigr{]}$
$\displaystyle=$ $\displaystyle-2\pi\,\Im{\rm m}\,\Bigl{[}\,\int
d^{2}{\mathbf{k}}\,M_{\mathbf{k}}^{2}\,\frac{1}{\Omega-2E_{\mathbf{k}}+\imath\eta}\Bigr{]}$
$\displaystyle=$ $\displaystyle 2\pi\eta\int
d^{2}{\mathbf{k}}\,M_{\mathbf{k}}^{2}\,\frac{1}{(\Omega-2E_{\mathbf{k}})^{2}+\eta^{2}}$
Fig. 3 shows the bare Raman spectra (solid lines) as functions of the
transferred photon frequencies $\Omega$ for three scattering geometries: (i)
$\theta=\pi/2,~{}\phi=0$, (ii) $\theta=2\pi/3,~{}\phi=0$ and (iii)
$\theta=0,~{}\phi=0$. For two of the polarizations (i) and (ii), the Raman
spectra show a similar profile and intensity: the Raman response exhibits a
peak at $\Omega\simeq 0.8-0.9$, and the location of this peak corresponds to
the twice the energy of the dominant van-Hove singularity of the one-magnon
dispersion, cf. (Fig. 2).
Figure 4: Variation of the maximum of the bare and renormalized intensities as
a function of the scattering angle $\theta$ computed at $\Omega=0.8$ and
$\Omega=0.6$, respectively.
The small Raman intensity $I(\Omega)$ at low energies is due to two reasons.
First, the magnon energy is gapless only at the zone center and has a gap
caused by the DM interaction at the ordering wave vector ${\mathbf{Q}}_{0}$.
Thus, only magnons with momentum near ${\bf k}\simeq 0$ can be excited by
photons with frequency $\Omega<2\,E({\mathbf{Q}}_{0})$. As one can see from
Fig. 2, the rather steep magnon spectrum in the vicinity of zone center gives
rise to a small density of states. Second, the form of $M_{\mathbf{k}}^{2}$ is
such that it selects mostly wavevectors $k\sim\pi$ where the gap resides.
In the $\theta=\phi=0$ geometry, the Raman response is non-vanishing but very
small for the whole energy range compared with intensities observed in other
geometries. The relative smallness can be understood by comparing this result
with the well-studied case of the square lattice, for which the LF operator in
the $A_{1g}$ geometry commutes with the Heisenberg Hamiltonian, and, as a
result, the Raman response vanishes. In the case of the anisotropic triangular
lattice, a non-commuting part of the LF operator (34) remains non-zero even
for the A1g geometry. This part leads to small Raman intensities, scaling with
the ratio $(J^{\prime}/J)^{2}$. Indeed, as we have discussed earlier, the
anisotropic triangular lattice can be viewed as an interpolation between the
square and the isotropic triangular lattice by varying $J^{\prime}/J$.
In Fig. 4, the two-magnon peak is shown as a function of the scattering angle
$\theta$. A strong dependence on the scattering geometry can be seen: the
largest peak intensity is observed at the $\theta=\pi/2,\phi=0$, i.e. cross-
polarization, which gradually decreases and reaches its minimum at $\theta=0$
or $\pi$ (Though not exactly, the peak intensity is roughly proportional to
$\sin^{2}\theta$). This polarization dependence is consistent with the fact
that the anisotropy in the present case resembles more that of a rectangular
than of an isotropic triangular geometry.
An angular-dependence analysis of the Raman spectrum on Cs2CuCl4 is highly
desirable as it could provide a potential diagnosis of whether a long-range
spin order develops in the ground state. If an angular dependance similar to
the one described above is observed, then the ground state is likely to be an
incommensurate spiral. veillette05 ; dalidovich06 On the other hand, a Raman
response that is independent of the scattering geometry might indicate a spin-
liquid ground state, cepas08 as proposed in some recent theories. isakov05 ;
alicea05 ; yunoki06 ; starykh07 Here some precautions are necessary. Since
the underlying Hamiltonian is spatially anisotropic, the spinon excitations of
the proposed spin-liquid phase might inherit the crystal anisotropy to some
degree, hence also giving rise to a polarization-dependent Raman signal. This
question should be further studied but we believe that even if this is the
case, the polarization dependence of the Raman response will be much weaker
than in the case of spiral magnetic order.
Next we incorporate $1/S$ corrections to the Raman spectrum. This can be
easily done by replacing the bare energy with the renormalized magnon energy
$\tilde{E}_{\mathbf{k}}$ (30) in the propagator
$G({\mathbf{k}},i\omega_{n})=1/(i\omega_{n}-{\tilde{E}}_{\mathbf{k}})$. The
renormalized Raman spectra are shown by dashed lines in Fig. 3. We can see
that in the renormalized spectrum the two-magnon peak appears at the energy
$\Omega\simeq 0.6$, which is slightly lower than the peak in the bare
spectrum. This is in contrast with the case of isotropic triangular lattice,
where the peak is shifted to higher energies. Once again it shows that the
Raman spectra on the anisotropic triangular lattice has features of the both
triangular and square lattices.
### IV.3 Raman intensity with final state interactions
Next we consider the final-state magnon-magnon interactions. Usually these are
not small, particularly for $S=1/2$. The effect of the final-state magnon-
magnon interactions can be taken into account by computing vertex corrections
to the bare Raman vertex. Here we consider only the leading $1/S$ corrections.
In this approximation, the vertex corrections can be obtained from an infinite
summation of ladder diagrams. These ladder diagrams are shown in Fig. 5c) in
terms of the two-particle (ir)reducible Raman vertex
$(\gamma({\mathbf{k}},{\mathbf{p}},\omega_{n},\omega_{o}))~{}\Gamma({\mathbf{k}},\omega_{n},\omega_{m})$.
These are related by the Bethe-Salpeter equation:
$\displaystyle\Gamma({\mathbf{k}},\omega_{n},\omega_{m})=r^{-}({\mathbf{k}})+\sum_{{\mathbf{p}},\omega_{o}}\gamma({\mathbf{k}},{\mathbf{p}},\omega_{n},\omega_{o})$
$\displaystyle
G({\mathbf{p}},\omega_{o}+\omega_{m})G(-{\mathbf{p}},-\omega_{o})\Gamma({\mathbf{p}},\omega_{o},\omega_{m})~{}.$
(43)
The two-particle irreducible vertex can be decomposed as in Fig. 5d):
$\gamma({\mathbf{k}},{\mathbf{p}},\omega_{n},\omega_{o})=\gamma_{3}({\mathbf{k}},{\mathbf{p}},\omega_{n},\omega_{o})+\gamma_{4}({\mathbf{k}},{\mathbf{p}})$.
The quartic vertex $\gamma_{4}({\mathbf{k}},{\mathbf{p}})$ is identical to the
two-particle-two-hole contribution from the $H_{4}$ term; its explicit
expression is given by
$\displaystyle\gamma_{4}({\mathbf{k}},{\mathbf{p}})=\frac{1}{4S}\Bigl{[}4\bigl{(}A_{0}-B_{0}+A_{{\mathbf{p}}+{\mathbf{k}}}-B_{{\mathbf{p}}+{\mathbf{k}}}+A_{\mathbf{p}}-A_{\mathbf{k}}\bigr{)}u_{\mathbf{p}}u_{\mathbf{k}}v_{\mathbf{p}}v_{\mathbf{k}}+$
$\displaystyle\bigl{(}A_{{\mathbf{p}}-{\mathbf{k}}}-B_{{\mathbf{p}}-{\mathbf{k}}}+A_{{\mathbf{p}}+{\mathbf{k}}}-B_{{\mathbf{p}}+{\mathbf{k}}}-A_{\mathbf{p}}-A_{\mathbf{k}}\bigr{)}\bigl{(}u_{\mathbf{p}}^{2}u_{\mathbf{k}}^{2}+v_{\mathbf{p}}^{2}v_{\mathbf{k}}^{2}\bigr{)}-$
(44)
$\displaystyle\bigl{(}2B_{{\mathbf{p}}}+B_{{\mathbf{k}}}\bigr{)}\bigl{(}u_{{\mathbf{p}}}^{2}+v_{{\mathbf{p}}}^{2}\bigr{)}u_{{\mathbf{k}}}v_{{\mathbf{k}}}-\bigl{(}2B_{{\mathbf{k}}}+B_{{\mathbf{p}}}\bigr{)}\bigl{(}u_{{\mathbf{k}}}^{2}+v_{{\mathbf{k}}}^{2}\bigr{)}u_{{\mathbf{p}}}v_{{\mathbf{p}}}\Bigr{]}$
where the functions $A_{\mathbf{k}}$ and $B_{\mathbf{k}}$ are given by Eqs.
(14).
The triplic vertex
$\gamma_{3}({\mathbf{k}},{\mathbf{p}},\omega_{n},\omega_{o})$ is obtained from
the product of two vertices of the cubic term $H_{3}$ and one intermediate
propagator, and can be written as
$\displaystyle\gamma_{3}({\mathbf{k}},{\mathbf{p}},\omega_{n},\omega_{o})=\frac{1}{32S}\,[\Phi_{1}({\mathbf{p}},{\mathbf{k}}-{\mathbf{p}};\mathbf{k})\Phi^{*}_{1}({\mathbf{p}},{\mathbf{k}}-{\mathbf{p}};\mathbf{k})G^{0}({\mathbf{k}}-{\bf
p},i\omega_{o}-i\omega_{n})+$
$\displaystyle\Phi_{1}({-\mathbf{p}},{\mathbf{p}}-{\mathbf{k}};-\mathbf{k})\Phi^{*}_{1}(-{\mathbf{p}},{\mathbf{p}}-{\mathbf{k}};-{\mathbf{k}})G^{0}({\mathbf{p}}-{\mathbf{k}},i\omega_{n}-i\omega_{o})]~{},$
(45)
where the functions $\Phi_{1}(1,2;3)$ and their complex conjugates are given
by Eqs. (29)-(III). To keep
$\gamma_{3}({\mathbf{k}},{\mathbf{p}},\omega_{n},\omega_{o})$ to leading order
in $1/S$, we retained only the zeroth order propagators $G^{0}$ for each
intermediate line. We further simplify the expression (IV.3) by assuming that
the dominant contribution to the frequency summations in the Bethe-Salpeter
equation (IV.3) comes from the mass-shell of the intermediate particle-
particle propagators. This corresponds to the substitution of the intermediate
frequencies by $-i\omega_{n}\approx E_{\mathbf{k}},~{}-i\omega_{o}\approx
E_{\mathbf{p}}~{}$. The simplified expression for the triplic vertex then
reads as
$\displaystyle\gamma_{3}({\mathbf{k}},{\mathbf{p}})$ $\displaystyle\simeq$
$\displaystyle\frac{1}{32S}\,\Phi_{1}({\mathbf{p}},{\mathbf{k}}-{\mathbf{p}};\mathbf{k})\,\Phi^{*}_{1}(-{\mathbf{p}},{\mathbf{p}}-{\mathbf{k}};-\mathbf{k})\vspace{0.2cm}$
(46) $\displaystyle\times$
$\displaystyle\frac{2E_{{\mathbf{k}}-{\mathbf{p}}}}{(E_{\mathbf{k}}-E_{\mathbf{p}})^{2}-E^{2}_{{\mathbf{k}}-{\mathbf{p}}}}~{}.$
The triplic vertex then depends only on momenta.
Figure 5: a) Bare Raman vertex $R$ from Eqn. (35); b) Raman susceptibility
bubble; c) The integral equation for the dressed Raman vertex $\Gamma$ in
terms of the irreducible magnon particle-particle vertex $\gamma$; d) Leading
order $1/S$ contributions to $\gamma$.
Next we perform the frequency summation over $\omega_{o}$ on the right hand
side of Eq. (IV.3) as well as the analytic continuation
$i\omega_{m}\rightarrow\Omega+i\eta\equiv z$. With this, the reducible vertex
$\Gamma$ in the latter equation turns into a function of ${\mathbf{p}}$ and
$z$ only, leading to
$\displaystyle\sum_{\mathbf{p}}L_{{\mathbf{k}},{\mathbf{p}}}(z)\Gamma_{\mathbf{p}}(z)=r^{-}({\mathbf{k}})$
(47) $\displaystyle
L_{{\mathbf{k}},{\mathbf{p}}}(z)=\delta_{{\mathbf{k}},{\mathbf{p}}}-\frac{\gamma({\mathbf{k}},{\mathbf{p}})}{z-2{\tilde{E}}_{\mathbf{p}}}~{},$
(48)
which is an integral equation with respect to momentum only. Finally, the
expression for the Raman intensity can be written as
$\displaystyle I(\Omega)$ $\displaystyle\simeq$
$\displaystyle\bigl{[}J(\Omega)-J(-\Omega)\bigr{]}~{},$ (49)
where
$\displaystyle J(\Omega)$ $\displaystyle=$ $\displaystyle\Im{\rm
m}\,\left[\sum_{\mathbf{k}}\frac{M_{\mathbf{k}}\,\Gamma_{\mathbf{k}}(\Omega+i\eta)}{\Omega+i\eta-2{\tilde{E}}_{\mathbf{k}}}\right]~{}.$
(50)
The fully renormalized intensity for the polarization with
$\theta=2\pi/3,~{}\phi=0$ is shown in Fig. 6 a). Despite the damping by the
vertex corrections, the two-magnon peak survives and is further shifted
towards lower energies. Apparently, the damping of the peak is less pronounced
compared with the isotropic triangular lattice case.perkins08 At the higher
energies, we also see the appearance of a broad continuum. We would like to
point out that both the peak and the broad continuum are observed in
geometries with almost perpendicular $\hat{\bm{\varepsilon}}_{\rm in}$ and
$\hat{\bm{\varepsilon}}_{\rm out}$. Although the width and intensities of the
peak and the profile of the continuum vary with the angle $\theta$, the
position of the two-magnon peak and the center of the continuum do not change
significantly for $\theta\sim\pi/2$. On the other hand, the Raman signal is
extremely weak for nearly parallel geometries ($\theta$, $\phi\sim 0)$,
similar to the case without vertex corrections (see Fig. 3).
In order to disentangle the contributions coming from the triplic and quartic
terms, we also compute the Raman spectrum with an irreducible vertex which
includes only the quartic part. The comparison between the Raman spectrum
computed with the full vertex and with the one containing only
$\gamma_{4}(\mathbf{k},\mathbf{p})$ is presented in Fig. 6 b). One observes
that vertex corrections due to the quartic term split the sharp two-magnon
peak into two peaks of comparable intensities, which are, however, about the
half of the intensity of the two magnon peak with final-state interactions.
The triplic term modifies these two peaks quite differently. The lower energy
peak is only weakly renormalized by the triplic term, while the higher energy
peak is damped more strongly and is transformed into the broad continuum.
Figure 6: Effect of final state interactions on Raman intensity for
$\theta=2\pi/3,\phi=0$. a) Bare intensity, an intensity computed with
renormalized magnon energies and intensity computed with included final state
interactions are shown by blue, green and red lines, respectively. Number of
${\mathbf{k}}$-points: $69\times 69$. The imaginary broadening is $\eta=0.03$.
$\Omega$ is measured in units of $3JS$. b) Comparison of the intensities
computed with corrections only due to the quartic vertex,
$\gamma_{4}({\mathbf{k}})$ ( brown solid line with diamonds), and with
corrections due to full vertex,
$\gamma_{3}({\mathbf{k}})+\gamma_{4}({\mathbf{k}})$ ( red solid line with
circles).
Finally we note that the direct comparison of these results with Fig. 3 should
be taken with a certain precaution, since the artificial line broadening in
Fig. 6 is larger by one order of magnitude. This is a consequence of a factor
of 16 less $\mathbf{k}$-points used in the latter case. This is because the
kernel $L_{{\mathbf{k}},{\mathbf{p}}}(z)$ in the integral equation (47) and
(48) is not sparse and has rank $N^{2}\times N^{2}$. Consequently, a moderate
lattice size gives rise to a rather large dimension for the kernel. In the
above calculations, we have chosen $N=69$, leading to a 4761$\times$4761
system which we have solved 100 times to account for 100 frequencies in the
interval $\Omega\in[0,2]$. We also note that the kernel has points of singular
behavior in $({\mathbf{k}},{\mathbf{p}})$-space, which stem from the
singularities of the Bogoliubov factors and from the energy denominators in
vertex functions. Here, we have chosen to regularize these points by cutting
off eventual singularities in $L_{{\mathbf{k}},{\mathbf{p}}}$. This can be
justified because the weight of these points is negligibly small compared with
the total number of points in the BZ. We have checked that this regularization
does not significantly effect the obtained spectra.
## V Summary
In summary, we have studied the two-magnon Raman scattering in the anisotropic
triangular Heisenberg antiferromagnet considering various levels of
approximation within a controlled $1/S$-expansion. We have shown that the
Raman profile is sensitive to the magnon-magnon interactions and to the
scattering geometry. The calculations indicate that the main effect of the
magnon-magnon interactions is on the shifting of the two-magnon peak towards
lower energies and on the formation of the broad continuum at the higher
energies. We have also shown that through exchange and DM interactions, the
spatial anisotropy of the lattice is transferred to the magnon dispersion of
the spiral magnetic order. This makes the two-magnon Raman scattering
anisotropic and very sensitive to the scattering geometry when the spiral
magnetic order is the ground state. Our results on the polarization dependence
of the spectrum suggest that Raman spectroscopy might be very useful to
resolve the ambiguity in the interpretation of the neutron scattering
experiments and to gain insight into the magnetic structure of Cs2CuCl4.
A final note. In this paper we used as an input the orientation of the DM
vector extracted from neutron scattering in Ref. coldea2 . Recent electron
spin resonance (ESR) measurements by Povarov et al. povarov11 suggested an
alternative orientation of the DM vectors in Cs2CuCl4, in which the strongest
DM interaction is along the spin chains. The direction of the DM vector is
important for magnetic properties of Cs2CuCl4 in the presence of the magnetic
field because in this case both the magnetic ground state and the excitation
spectra depend on the relative orientation of the external field and the
direction of the DM vector.starykh10 ; povarov11 ; smirnov12 However, the
direction of the DM vector does not affect Raman intensity, at least at a
qualitative level. The reasoning is that the change of the orientation of the
DM vector does not change the classical ground state of the model (1) and does
not change qualitatively the spectrum of spin wave excitations, as in both
geometries it’s main role is to open the gap.
Acknowledgement. N.P. acknowledges the support from NSF grant DMR-1005932.
G.W.C. acknowledges the the support of ICAM and NSF grant DMR-0844115. N.P.
also thank the hospitality of the visitors program at MPIPKS, where part of
the work on this manuscript has been done.
## References
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* (3) S. V. Isakov, T. Senthil, and Yong Baek Kim, Phys. Rev. B 72, 174417 (2005).
* (4) Jason Alicea, Olexei I. Motrunich, and Matthew P. A. Fisher, Phys. Rev. Lett. 95, 247203 (2005); Phys. Rev. B 73, 174430 (2006).
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* (19) N. Perkins and W. Brenig, Phys. Rev. B 77, 174412 (2008).
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* (27) Here we note that in the presence of the spin-orbit coupling giving rise to the DM interaction, the effective Hamiltonian Eq.(34) describing the interaction of light with magnons shall also contain an antisymmetric part.fleury ; lara06 This term contributes only to the one-magnon Raman response which we do not describe here. This is the reason why we did not include the antisymmetric term in Eq.(34).
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|
arxiv-papers
| 2012-12-11T03:19:40 |
2024-09-04T02:49:39.124239
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Natalia B. Perkins, Gia-Wei Chern and Wolfram Brenig",
"submitter": "Natalia Perkins",
"url": "https://arxiv.org/abs/1212.2286"
}
|
1212.2360
|
# Generalized Semi-Holographic Universe
Hui Li 111Electronic address: [email protected] Department of Physics, Yantai
University, 30 Qingquan Road, Yantai 264005, Shandong Province, P.R.China
Hongsheng Zhang222Electronic address: [email protected] Shanghai United
Center for Astrophysics (SUCA), Shanghai Normal University, 100 Guilin Road,
Shanghai 200234, P.R.China Yi Zhang333Electronic address:
[email protected] College of Mathematics and Physics, Chongqing University
of Posts and Telecommunications, Chongqing 400065, P.R.China
###### Abstract
We study the semi-holographic idea in context of decaying dark components. The
energy flow between dark energy and the compensating dark matter is
thermodynamically generalized to involve a particle number variable dark
component with non-zero chemical potential. It’s found that, unlike the
original semi-holographic model, no cosmological constant is needed for a
dynamical evolution of the universe. A transient phantom phase appears while a
non-trivial dark energy-dark matter scaling solution keeps at late time, which
evades the big-rip and helps to resolve the coincidence problem. For
reasonable parameters, the deceleration parameter is well consistent with
current observations. The original semi-holographic model is extended and it
also suggests that the concordance model may be reconstructed from the semi-
holographic idea.
holographic principle, dark energy, grand canonical ensemble, chemical
potential
###### pacs:
98.80.-k 95.36.+x 11.10.Lm
## I Introduction
The existence of dark energy is one of the most significant cosmological
discoveries over the last century acce . Various models of dark energy have
been proposed synthesis , such as a small positive cosmological constant,
quintessence, k-essence, phantom, quintom, etc., see Ref.review for a recent
review. Stimulated by the holographic principle, it’s conjectured that dark
energy problem may be a problem of quantum gravity, although its
characteristic energy scale is very lowcohn . According to the holographic
principlethsu , the true laws of inside any surface are actually a description
of how its image evolves on that surface. In the case of black hole, the event
horizon is a proper holography. When applying the holographic principle to the
universe, the event horizon is probably not a good candidate of holography
screen at least for two reasons: the identification of the cosmological event
horizon is causally problematic in that it depends on the future of the cosmic
evolution; a decelerating universe would have no holographic description,
since it has no event horizon at all! With reference to the seminal work of
Jacobsonjaco , several arguments have been put forward that the apparent
horizon should be a causal horizon and is associated with the gravitational
entropy and Hawking temperaturecai bak , and hence the right holography of
the universe. Apart from this key observation, other thermodynamical
properties of the dark energy and dark matter have yet to be found in
detailsentropyu . Recently, the semi-holographic dark energy modelsemi-holo
has been proposed to examine how the universe could evolve if a dark component
of the universe strictly obeys the holographic principle. It was found that,
based on the first law of thermodynamics, the existence of the other dark
component (dark matter) is compulsory, as a compensation of dark energy. In
that model, with a non-zero cosmological constant the standard cosmological
kinematics is recovered and the cosmological coincidence problem is also
alleviated with the existence of a stable Einstein-de Sitter scaling solution.
For all that, the properties of the semi-holographic model should be clarified
for further estimations from three points of view. First, only massless
particles (photons) exchange are considered. Or we treat dark matter and dark
energy as two closed systems. Surely, by nature they can be open systems, and
thus the particle numbers of the two systems can be variables. Second, the EOS
of dark energy is constrained to be $\omega_{de}<-1$. It certainly is in
accord with the latest cosmological observations and phenomenologically
reasonable. However, as a thermodynamics motivated dark energy model, the
construction would be founded on a more solid basis of thermodynamics and
statistical physics, and exactly in this sense, the well-known theoretical
difficulties does matter. As a matter of fact, the case of phantom energy with
$\omega<-1$ is ruled out because the total entropy of the dark component is
negativeLima2 . The equation of state (hereafter EOS) was restricted to the
interval $-1\leq\omega<-1/2$ and a fermionic nature to the dark energy
particles is favored. Thermodynamics argumentsGonzalez in favor of the
phantom hypothesis had to resort to an unusual assumption that the temperature
of a phantomlike fluid is always negative in order to keep its entropy
positive definite (as statistically required) or by arguing that the scalar
field representations of a phantom field has a negative kinetic term which
quantifies the translational kinetic energy of the associated fluid system.
Third, as is known, the possibility of a coupling between dark matter and dark
energy is often considered with three types in the literature, and those
interaction forms are always introduced phenomenologically or through the low
energy effective actionchaplygin : dark matter decaying into dark energyDM2DE
, dark energy decaying into dark matterDE2DM and interaction in both
directionsDM-DE . It’s worth noting that, due to the non-adiabaticity of the
holographic evolution of dark energy, the semi-holographic model obviously
introduced a dark matter/dark energy non-gravitational interaction (A
hypothetical dark energy property sometimes induces a non-gravitational
coupling between dark matter and dark energy. See HuiLi for another
interacting model of this kind.), and the interaction belongs to the last
category: dark matter dominated in the past and transferred energy to dark
energy; the latter begins to dominate at present, and in the near future, the
transfer would be reversed and dark energy decays to dark matter with a stable
scaling solution reached. The theoretical obstacle is that, provided that the
chemical potential of both components vanish, it is the decay of dark energy
into dark matter that was favored by the thermodynamical point of view and the
second law.
Owing to a series of systematic workLima2 Lima Gonzalez negtivemu negtivemu2 ,
thermodynamics and statistical arguments on dark components and the cosmic
evolution have come to several important conclusions which may kick off those
dilemma. By considering the existence of a non-zero chemical potential,
reanalysis of the thermodynamics and statistical properties of the dark energy
scenario supports that the temperature of dark energy fluids must be always
positive definite and that a negative chemical potential will recover the
phantom scenario without the need to appeal to negative temperaturesnegtivemu
. In addition, a bosonic nature of the dark energy component becomes possible.
If the chemical potential of at least one of the cosmic fluids is not zero,
the decay can occur from the dark matter to dark energy, without violation of
the second lawnegtivemu2 . From this point of view, the decay process between
dark components with non-zero chemical potential is not merely a theoretical
tool, but should be seriously considered as a prerequisite to turn the phantom
dark energy and the decay in both directions to be physically real hypotheses.
More importantly, due to the insufficiency in discussing the particle number
variable transferring process between dark components, the original semi-
holographic model should not be regarded as a complete thermodynamical
examination of the semi-holographic idea in cosmology. A general framework
dealing with the decay of the dark component particles is still in lack and
therefore well motivated. Particularly speaking, the first law of
thermodynamics would be applied in the context of grand canonical ensemble
rather than canonical ensemble describing the original semi-holographic model.
This paper is organized as follows: In the next section we will study the
semi-holographic model in grand canonical ensemble with the dark energy
chemical potential explicitly given. The dynamical analysis is left in section
III. We find that there exists a stable dark matter-dark energy scaling
solution at late time, which is helpful to address the coincidence problem.
After a close analysis and comparison between model behavior and the
observational facts, we present our conclusion and some discussions in section
IV.
## II the model
There is decisive evidence that our observable universe evolves adiabatically
after inflation in a comoving volume, that is, there is no energy-momentum
flow between different patches of the observable universe so that the universe
keeps homogeneous and isotropic after inflation. That is the reason why we can
use an FRW geometry to describe the evolution of the universe. In an
adiabatically evolving universe, the first law of thermodynamics equals the
continuity equation. In a comoving volume the first law reads,
$dU=TdS-pdV+\tilde{\mu}dN,$ (1)
where $U=\Omega_{k}\rho a^{3}$ is the energy in this volume, $T$ denotes
temperature, $S$ represents the entropy of this volume, $V$ stands for the
physical volume $V=\Omega_{k}a^{3}$, $\tilde{\mu}$ is the chemical potential
of the energy component with particle number $N$. Here, $\Omega_{k}$ is a
factor related to the spatial curvature. For spatially flat case
$\Omega_{0}=\frac{4}{3}\pi$, in this paper we only consider the spatially flat
model, $\rho$ is the energy density and $a$ denotes the scale factor. The last
term in the above equation indicates that the grand canonical ensemble instead
of the canonical ensemble for the original semi-holographic model is
considered. As a consequence, the framework allows decaying of the particles
in consideration.
The semi-holographic model concerns the possibility of a non-adiabatical dark
energy, where the term $TdS$ does not equal zero. Based on the investigations
in bak ; cai , the entropy in a comoving volume the entropy becomes,
$S_{c}=\frac{8\pi^{2}\mu^{2}}{H^{2}}\frac{a^{3}}{H^{-3}}={8\pi^{2}\mu^{2}}Ha^{3}.$
(2)
where $H$ is the Hubble parameter, $\mu$ denotes the reduced Planck mass. The
entropy has been reasonably assumed to be homogeneous in the observable
universe. As our observable universe evolves adiabatically after reheating,
the varying entropy of dark energy in a comoving volume should be compensated
by an entropy change of the other component to keep the total entropy constant
in a comoving volume.
With the above supposition and conventions the entropy of the dark energy
satisfies (2),
$S_{de}={8\pi^{2}\mu^{2}}Ha^{3}.$ (3)
Correspondingly, the entropy of dark matter in this comoving volume should be
$S_{dm}=C-S_{de},$ (4)
where $C$ is a constant, representing the total entropy of the comoving
volume.
The Friedmann equation will determine the evolution of our universe
$H^{2}=\frac{1}{3\mu^{2}}(\rho_{dm}+\rho_{de}+\Lambda),$ (5)
where H is the Hubble parameter, $\rho_{dm}$ denotes the density of non-baryon
dark matter, $\rho_{de}$ denotes the density of dark energy, and $\Lambda$ is
the cosmological constant (or vacuum energy). Partly because the partition of
baryon matter is very small and does little work in the late time universe,
and partly because the non-gravitational coupling between baryonic matter and
dark energy is highly constrained, we just omit the baryon matter component in
the discussion. The appearance of the cosmological constant in the expression
is just for a general discussion, and as can be seen below, it is not a
necessary component in the generalized semi-holographic model. As a
comparison, the cosmological constant has to be present in the original model,
or else the ratio of holographic dark energy and dark matter density would
always remain constant which is obviously not realistic.
Below we will assume the chemical potential of dark energy to be of the form
in Ref.negtivemu2
$\tilde{\mu}=-\tilde{\alpha}T$ (6)
with the efficient $\tilde{\alpha}$ a positive constant. As was shown by
Pereira, if the coefficient $\tilde{\alpha}>0$, a negative chemical potential
will make the phantom fluid thermodynamically consistent and a decay of dark
matter into dark energy possible. This assumption may help us highly broaden
the exploration of the cosmological parameter region motivated by
thermodynamics.
## III dynamical analysis
To investigate the evolution in a more detailed way, we take a dynamical
analysis of the universe.
The holographic principle requires that the temperature cai
$T=\frac{H}{2\pi}.$ (7)
By using (7), (5), (3), and using the particle number $N$ to be proportional
to the energy density, the first law of thermodynamics (1) becomes the
evolution equation of dark energy,
$\frac{2}{3}\rho_{de}^{\prime}=\frac{\rho_{dm}(1-w_{dm})-\rho_{de}(1+2\tilde{\alpha}Y+3w_{de})+2\Lambda}{1+\tilde{\alpha}Y},$
(8)
where a prime denotes the derivative with respect to $\ln a$, $w_{dm}$
indicates the EOS of dark matter, $w_{de}$ represents the EOS of dark energy
and $Y=\sqrt{\rho_{de}+\rho_{dm}+\Lambda}$. Similarly, we derive the evolution
equation of dark matter,
$\frac{2}{3}\rho_{dm}^{\prime}=\frac{\rho_{de}[-1+(1-2\tilde{\alpha}Y)w_{de}]-\rho_{dm}[3+2\tilde{\alpha}Y+(1+2\tilde{\alpha}Y)w_{dm}]-2\Lambda}{1+\tilde{\alpha}Y}.$
(9)
Clearly, the above equations will degenerate to the equations (8) and (9) of
the paper semi-holo when $\tilde{\alpha}=0$. For convenience we introduce two
new dimensionless functions to represent the densities,
$\displaystyle u\triangleq\frac{\rho_{dm}}{3\mu^{2}H_{0}^{2}},$ (10)
$\displaystyle v\triangleq\frac{\rho_{de}}{3\mu^{2}H_{0}^{2}},$ (11)
and a dimensionless cosmological constant
$\lambda\triangleq\frac{\Lambda}{3\mu^{2}H_{0}^{2}},$ (12)
where $H_{0}$ denotes the present Hubble parameter. Then the equation set (9),
(8) becomes
$\displaystyle\frac{2}{3}u^{\prime}=\frac{-u[3+2\sqrt{3}\alpha
y+(1+2\sqrt{3}\alpha yw_{dm})]+v[-1+(1-2\sqrt{3}\alpha
y)w_{de}]-2\lambda}{1+\sqrt{3}\alpha y},$ (13)
$\displaystyle\frac{2}{3}v^{\prime}=\frac{u(1-w_{dm})-v(1+2\sqrt{3}\alpha
y+3w_{de})+2\lambda}{1+\sqrt{3}\alpha y}$ (14)
respectively, where $y=\sqrt{u+v+\lambda}$ and $\alpha=\tilde{\alpha}\mu
H_{0}$ are both dimensionless. We note that the time variable does not appear
in the dynamical system (13) and (14) because time has been completely
replaced by scale factor.
Before presenting the numerical examples for special parameters we study the
analytical property of this system. And we will below take $\Lambda=0$ for
simplicity. The critical points of the dynamical system (13) and (14) are
given by
$u_{c}^{\prime}=v_{c}^{\prime}=0,$ (15)
which yields,
$u_{c}=0,~{}~{}~{}~{}v_{c}=0$ (16)
and
$u_{c}=\frac{(1+w_{de})[1+w_{de}(2+w_{dm})]^{2}}{\tilde{\alpha}^{2}(w_{de}-w_{dm})(1+w_{dm})^{2}},~{}~{}~{}~{}v_{c}=-\frac{[1+w_{de}(2+w_{dm})]^{2}}{\tilde{\alpha}^{2}(w_{de}-w_{dm})(1+w_{dm})}.$
(17)
The trivial solution corresponds to a future infinitely diluted universe and
is less interesting for the coincidence problem. On the other hand, the non-
trivial fixed point satisfies the below equality:
$\frac{u_{c}}{v_{c}}=-\frac{1+w_{\rm de}}{1+w_{\rm dm}},$ (18)
which is shared by the seminal model without chemical potential involved. So,
finally the universe enters a de Sitter phase, and the ratio of dark matter
over dark energy is independent of the chemical potential. Furthermore, there
are two reasonable cases for the non-trivial scaling solution: case I, $w_{\rm
de}<-1$ and $w_{\rm dm}>-1$; case II, $w_{\rm de}>-1$ and $w_{\rm dm}<-1$,
since we should require that the final densities of dark matter and dark
energy are both positive; as a matter of fact, the negative energy density may
appear at most as a transient phenomenon, and it can not be a physically
stable and permanent state.
To study the stability property of the fixed pointsstability , the evolution
equations should be perturbed to the first order and we list them in the
Appendix.
Then, the perturbation equations to the trivial critical point of 16 are
simplified as
$\displaystyle\frac{2}{3}(\delta\rho_{dm})^{\prime}=-\delta\rho_{dm}(3+w_{\rm
dm})+\delta\rho_{de}(-1+w_{\rm de}),$ (19)
$\displaystyle\frac{2}{3}(\delta\rho_{de})^{\prime}=\delta\rho_{dm}(1-w_{\rm
dm})-\delta\rho_{de}(1+3w_{\rm de}).$ (20)
And near this fixed point,the eigenvalues of the linearized system read
$\displaystyle
l_{1}=\frac{1}{2}(-4-3w_{de}-w_{dm}-\sqrt{-8w_{de}+9w_{de}^{2}+8w_{dm}-10w_{de}w_{dm}+w_{dm}^{2}}),$
(21) $\displaystyle
l_{2}=\frac{1}{2}(-4-3w_{de}-w_{dm}+\sqrt{-8w_{de}+9w_{de}^{2}+8w_{dm}-10w_{de}w_{dm}+w_{dm}^{2}}).$
(22)
Stability requires that all of real parts of the eigenvalues are less than
zero. Combined this requirement with the observational fact of
$w_{de}<-\frac{1}{3}$, we find
$-1<w_{dm}<1,~{}~{}~{}~{}-\frac{1}{2+w_{dm}}<w_{de}<w_{dm}.$ (23)
Figure 1: The plane v versus u. The initial conditions are taken for different
orbits. There is a stationary node at the origin of coordinates, which
attracts orbits in the first quadrant; however the orbits have to get through
the v-axis before reaching the node, and therefore the case may be
unrealistic.
A typical picture of the evolution is displayed in Figure 1. For illustration
we set $w_{de}=-0.4$, $w_{dm}=-0.1$ and $\alpha=0.9$. As is seen to us, the
orbits with initial values of energy densities lying in the first quadrant
evolve to the second quadrant with the energy density of dark matter negative,
and they will keep negative before the tracks reach the trivial critical point
at the origin of coordinates. Evidently, this case is physically problematic
and might be irrelevant to our universe.
Now let’s turn to the non-trivial fixed point. Analytic eigenvalues of the
corresponding linearized system to this critical point have not been found,
and we will explore the stability property of the scaling solution via
numerical method. Here we plot Figure 2 to display the properties of evolution
of the universe controlled by the dynamical system. In the figure, we set
$\lambda=0$, $w_{de}=-1.2$, $w_{dm}=-0.2$ and the positive coefficient
$\alpha=0.6$ for illustration. The physically meaningful region is just the
first quadrant and different initial value tracks evolve to the common non-
trivial fixed point still in the first quadrant, which indicates the existence
of the physically stable attractor.
Figure 2: The plane v versus u. (a)We consider the evolution of the universe.
The initial conditions are taken for different orbits. It’s clear that there
is a stationary node, which attracts orbits in the first quadrant. (b)Orbits
distributions around the node.
The most significant parameter from the viewpoint of observations is the
deceleration parameter $q$, which carries the total effects of cosmic fluids.
Using (5), (8), and (9) we obtain the deceleration parameter in this model
$q=-\frac{\ddot{a}}{a}\frac{1}{H^{2}}=\frac{1}{2}\frac{\rho_{dm}(1+3w_{dm})+\rho_{de}(1+3w_{de})-2\lambda}{\rho_{dm}+\rho_{de}+\lambda}.$
(24)
For a numerical example, we take the present dark matter partition $u_{0}=0.3$
and the present dark energy partition $v_{0}=0.7$, which is favored by present
observationsacce .
Figure 3: The evolutions of $q$ in the model (solid curve) and in $\Lambda$CDM
(dashed curve), respectively.
Figure 3 illuminates the evolution of deceleration parameter. As a simple
example we just set $w_{dm}=-0.2$, $w_{de}=-1.2$ and $\alpha=1.0$. The dark
matter dominates in the past, and since its effective EOS in that period is
larger than the assumed EOS, dark matter decays into dark energy. Only some
time ago, this decay made the dark matter stiffer and with the accumulation of
the dark energy, the universe endures a deceleration-acceleration transition.
The acceleration goes through the present era and, if the dimensionless
coefficient $\alpha$ is about $1$ or less, it will march for a phantom phase
in the near future. That is to say, the deceleration parameter can be smaller
than $-1$. However, the super-acceleration will not last for ever. The decay
process of dark matter into dark energy then reverses and dark energy begins
to transfer its energy to dark matter component. This conversion impedes the
super-acceleration and eventually the evolution of the universe is pulled back
to the Einstein-de Sitter track; therefore, the troublesome cosmic big-rip is
evadedlittlerip . From the figure one sees that current $q\sim-1$ and at the
high redshift region it goes to $0.5$, which is consistent with current
observations review ; WMAP . As a comparison, the evolution of the
deceleration parameter in a spatially flat $\Lambda$CDM is also plotted, in
which the density parameter of dark matter $\Omega_{dm}=0.3$ too.
The density evolution of the cosmic fluid does not depend on the assumed EOS
above, but the effective EOS. We define the effective EOS as the following
procedure. Supposing the dark matter evolves adiabatically itself, we obtain
its evolution from (1),
${d\rho_{dm}}+3(\rho_{dm}+p_{eff})\frac{da}{a}=0,$ (25)
where $p_{eff}$ denotes the effective pressure of dark matter. Then we obtain
$\displaystyle
w_{dme}\triangleq\frac{p_{eff}}{\rho_{dm}}=\frac{\frac{v}{u}[1-(1-2\alpha
y)w_{de}]+2\frac{\lambda}{u}+3+2\alpha y+(1+2\alpha y)w_{dm}}{2+2\alpha y}-1,$
(26) $\displaystyle w_{dee}=-\frac{\frac{u}{v}(1-w_{dm})-(1+2\alpha
y+3w_{de})+2\frac{\lambda}{v}}{2+2\alpha y}-1,$ (27)
which is variable in the evolution history of the universe. Note that we have
assumed both $w_{dm}$ and $w_{de}$ are constant from the beginning. Figure 4
displays the effective EOS of dark matter in which the same parameters are set
as in Figure 3. It shows that, although its EOS is clearly negative, the dark
matter behaves like cold dark matter until now. Only recently the dark matter
has become stiff which relates to the current cosmic acceleration era. In
Figure 5, different evaluations of the parameter $\alpha$ are taken and the
corresponding deceleration evolution illustrated. A smaller $\alpha$ denotes a
later deceleration-acceleration transition and a greater degree of
acceleration in the near future. As the coefficient gets bigger, the cosmic
evolution becomes closer to the concordant $\Lambda$CDM model and the
transition happens more gently. For all that, the universe will go back to the
same track of an everlasting de-Sitter phase with each of the energy densities
unchanged thereafter.
Figure 4: The effective EOS of dark matter $w_{dme}$ as a function of $\ln a$.
.
Figure 5: The evolutions of $q$ in semi-holographic model with different
values of chemical potential coefficient $\alpha$ ($\alpha=0.1,0.6,1.2$ from
thin to thick curves)and in $\Lambda$CDM (dashed curve), respectively.
The deceleration-acceleration transition happens abruptly in the original
semi-holographic model, and the present model in grand canonical ensemble is
prone to moderate the process. As the chemical potential coefficient changes,
the deceleration parameter tracks fill in the intermediate region between the
original semi-holographic model and the $\Lambda$CDM model. Therefore, in this
sense we can say that the particle decay picture extends the original semi-
holographic model moderately and reconstructs the concordance model with the
semi-holographic idea. The tracks may have or may not have a super-
acceleration phase which depends on the different values of $\alpha$.
## IV Conclusion and discussion
Based on the semi-holographic model inspired by holographic principle,
especially the previous studies of the relation between thermal dynamics and
general relativity, we find that, the semi-holographic dark energy model in
the grand canonical ensemble identification recovers the standard expansion
history and evolves to the Einstein-de Sitter final state through a possible
transient super-acceleration.
A phantom phase may appear in the near future without a big-rip in the end.
Assuming the total matter in a comoving volume to be evolved adiabatically, a
compensating dark matter component should exist and the non-gravitational
coupling between dark energy and dark matter is also compulsory. As is shown
in the above, this interaction yields a future attractor solution, which is a
stable scaling solution for the dark matter-dark energy system in proper
region of the parameters ($w_{\rm dm},~{}w_{\rm de}$). The final ratio of dark
matter to dark energy only depends on $w_{\rm dm},~{}w_{\rm de}$, and is
independent of the initial values of the densities of dark matter and dark
energy and the coefficient $\alpha$ of chemical potential. This result is
helpful to address the coincidence problem. The conversion of canonical
ensemble context to grand canonical ensemble one seems to deform the original
model to the well established $\Lambda$CDM model in the sense that different
tracks roughly fill in the zone between the two models in the $q-\ln a$ plane.
Therefore, the particle decay picture extends the original semi-holographic
model moderately and reconstructs the concordance model with the semi-
holographic idea. This grand canonical ensemble viewpoint opens the
possibility to deepen our understandings of dark energy thermodynamics and
statistical properties, and is closely related to previous systematic
application of the cosmological thermodynamics.
With numerical examples, it is illustrated that the deceleration parameter is
well consistent with observations. As a phenomenological model, the parameters
including $\alpha$ may be further constrained by observational data and
detailed investigations on global fitting of cosmological parameters will also
be interesting.
Acknowledgments. H. Li is supported by National Natural Science Foundation of
China under grant No. 10747155, H. Zhang is supported the Program for
Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of
Higher Learning, Shanghai Municipal Pujiang grant No. 10PJ1408100, and
National Natural Science Foundation of China under grant No. 11075106, and Y.
Zhang is supported by the National Natural Science Foundation of China under
grant Nos. 11005164, 11175270 and 10935013, the Distinguished Young Scholar
Grant 10825313, CQ CSTC under grant No. 2010BB0408, and CQ MEC under grant No.
KJTD201016.
## V The Appendix: The perturbation equations
The perturbation equations for the combination $(\rho_{de},\rho_{dm})$ are:
$\displaystyle\delta\rho_{de}^{\prime}=$
$\displaystyle\frac{3}{2}\delta\rho_{dm}\left[\frac{1-w_{dm}-\frac{\tilde{\alpha}\rho_{de}}{Y}}{1+\tilde{\alpha}Y}+\frac{\rho_{de}(1+2\tilde{\alpha}Y+3w_{de})\frac{\tilde{\alpha}}{2Y}-\rho_{dm}(1-w_{dm})\frac{\tilde{\alpha}}{2Y}}{(1+\tilde{\alpha}Y)^{2}}\right]$
(28)
$\displaystyle-\frac{3}{2}\delta\rho_{de}\left[\frac{\rho_{de}\frac{\tilde{\alpha}}{Y}+1+2\tilde{\alpha}Y+3w_{de}}{1+\tilde{\alpha}Y}+\frac{\rho_{dm}(1-w_{dm})\tilde{\alpha}-\rho_{de}(1+2\tilde{\alpha}Y+3w_{de})\tilde{\alpha}}{2Y(1+\tilde{\alpha}Y)^{2}}\right],$
$\displaystyle(\delta\rho_{dm})^{\prime}=$
$\displaystyle-\frac{3}{2}\delta\rho_{dm}\left[\frac{3+2\tilde{\alpha}Y+(1+2\tilde{\alpha}Y)w_{dm}+\rho_{dm}(1+w_{dm})\frac{\tilde{\alpha}}{Y}+\rho_{de}w_{de}\frac{\tilde{\alpha}}{Y}}{1+\tilde{\alpha}Y}\right.$
(29)
$\displaystyle-\left.\frac{\rho_{de}[1-(1-2\tilde{\alpha}Y)w_{de}]\frac{\tilde{\alpha}}{2Y}+\rho_{dm}[3+2\tilde{\alpha}Y+(1+2\tilde{\alpha}Y)w_{dm}]\frac{\tilde{\alpha}}{2Y}}{(1+\tilde{\alpha}Y)^{2}}\right]$
$\displaystyle-\frac{3}{2}\delta\rho_{de}\left[\frac{\rho_{de}w_{de}\frac{\tilde{\alpha}}{Y}+\rho_{dm}(1+w_{dm})\frac{\tilde{\alpha}}{Y}+1-(1-2\tilde{\alpha}Y)w_{de}}{1+\tilde{\alpha}Y}\right.$
$\displaystyle-\left.\frac{\rho_{de}[1-(1-2\tilde{\alpha}Y)w_{de}]\frac{\tilde{\alpha}}{2Y}+\rho_{dm}[3+2\tilde{\alpha}Y+(1+2\tilde{\alpha}Y)w_{dm}]\frac{\tilde{\alpha}}{2Y}}{(1+\tilde{\alpha}Y)^{2}}\right]$
with all the notations declared in the main text.
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|
arxiv-papers
| 2012-12-11T10:00:10 |
2024-09-04T02:49:39.134894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hui Li, Hongsheng Zhang, Yi Zhang",
"submitter": "Hongsheng Zhang",
"url": "https://arxiv.org/abs/1212.2360"
}
|
1212.2581
|
# Graphene single electron transistor as a spin sensor for magnetic adsorbates
J. W. González(1),F. Delgado(1), and J. Fernández-Rossier(1,2) (1)
International Iberian Nanotechnology Laboratory (INL), Av. Mestre José Veiga,
4715-330 Braga, Portugal
(2) Departamento de Física Aplicada, Universidad de Alicante, 03690 San
Vicente del Raspeig, Spain
###### Abstract
We study single electron transport through a graphene quantum dot with
magnetic adsorbates. We focus on the relation between the spin order of the
adsorbates and the linear conductance of the device. The electronic structure
of the graphene dot with magnetic adsorbates is modeled through numerical
diagonalization of a tight-binding model with an exchange potential. We
consider several mechanisms by which the adsorbate magnetic state can
influence transport in a single electron transistor: by tuning the addition
energy, by changing the tunneling rate and, in the case of spin polarized
electrodes, through magnetoresistive effects. Whereas the first mechanism is
always present, the others require that the electrode has either an energy or
spin dependent density of states. We find that graphene dots are optimal
systems to detect the spin state of a few magnetic centers.
## I Introduction
Graphene is a very promising candidate for high precision molecular sensing,
due to its extremely large surface to volume ratio, and its electrically
tunable large conductivity.Schedin07 ; Wernsdorfer_NL ; Pisana On the other
hand, being a zero-gap semiconductor with small mass and small density of
spinfull nuclei, makes graphene a material with potentially large spin
lifetime both, for carriers and host magnetic dopants. Pesin2012 Taken
together, these two ideas naturally lead to the use of graphene as a detector
of the spin state of extrinsic magnetic centers, in the form of magnetic
adatoms, vacancies and spinfull molecules. This connects with recently
reported Wernsdorfer_NL ; Candini-PRB-2011 experiments in which gated
graphene nanoconstrictions, operating in the single electron transport (SET)
regime, showed hysteresis in the linear conductance when a magnetic field is
ramped. This behavior was observed both when the molecular magnets were
intentionally deposited on graphene, as well as in carbon
nanotubes,Wernsdorfer_NatMat but also in the case of bare graphene
nanojunctions,Candini-PRB-2011 where some type of graphene local moments
Yazyev07 ; Palacios2008 ; Soriano2010 is probably playing a role.
Figure 1: (Color online) (a) Scheme of a graphene constriction with randomly
distributed magnetic centers. (b) Diagram with the system energy levels and
graphene density of states.
The graphene spin sensor experiments of Ref. Wernsdorfer_NL, , are performed
in the Coulomb Blockade regime, showing a vanishing linear conductance except
in the neighborhood of specific values of the gate voltage $V_{g}$. This means
that the graphene nanoconstriction is weakly coupled to the electrodes and has
a charging energy larger than the thermal energy ($T\sim 100$ mK). The height
of the linear conductance peaks is significantly smaller than
$G_{0}=2e^{2}/h$, the quantum of conductance. These conditions imply that
transport takes place in the sequential regime.Beenakker Thus, current flow
takes place due to sequential tunneling of electrons through the graphene
constriction, which we refer to as the central region in the rest of the
paper, and the entire device behaves like a single electron
transistor.Geim2008 ; review-dot
The aim of this work is to provide a theoretical background to understand how
the magnetic state of localized magnetic moments affects transport through the
graphene nanoconstriction in the SET regime. This is different from previous
works where the influence of the magnetic state of magnetic edges Fede2009
and adsorbed hydrogens Soriano2010 on the conductivity was studied in the
ballistic regime, with a central island strongly coupled to the electrodes,
and also in the diffusive regime,Castro-Neto as well as SET through graphene
islands with magnetic zigzag edges.Ezawa
The paper is organized as follows. In Sec. II we discuss a tight-binding
Hamiltonian for the graphene island exchanged coupled to the spins of the
magnetic adsorbates. The results of this microscopic calculation justify the
use of a simple single-orbital spin-split model for the SET, discussed in
section III, together with the possible mechanisms that enable the magnetic
sensing. Finally, conclusions are presented in Sec. IV.
## II Model for graphene island with magnetic adsorbates
### II.1 Hamiltonian
Our starting point is a microscopic model for electrons confined in a graphene
nanoisland which are exchanged coupled to the magnetic centers. The graphene
central island is described with a tight-binding Hamiltonian for the honeycomb
lattice contained in a rectangular stripe of dimensions $L_{x}\times L_{y}$.
In order to avoid the spin-polarized states formed at the the zigzag
edges,Yazyev ; JP_JR we impose periodic boundary conditions in one direction
so that the structure only has open edges of armchair type. The coupling to
the magnetic moments of the adsorbate molecules $\vec{m}(i)$ is then assumed
to be a local exchange $J$ or spin-dependent potential, affecting $N$ sites
randomly selected in the graphene central island. For simplicity, we consider
that the magnetic moments of the molecules are all oriented along the same
axis, which we choose as the spin quantization axis. These assumptions are
good approximations in the case of strongly uniaxial TbPC2
molecules.Wernsdorfer_NL Hence, we can write the Hamiltonian of the graphene
and adsorbates as
$\mathcal{H}=\mathcal{H}_{0}+J\sum_{i}m_{z}(i)S_{z}(i)+eV_{g}\left(N_{TOT}-\hat{N}\right),$
(1)
where $\mathcal{H}_{0}$ is the tight-binding Hamiltonian for $\pi$-electrons
in graphene considering nearest neighbor interactions, $J$ is the strength of
the exchange coupling between the graphene electrons and the magnetic moment
of the molecules, which can take two values, $m_{z}(i)=\pm 1$. Finally, the
last term in the Hamiltonian describes the electrostatic coupling of the total
charge of the dot, which can be either $0$ or $e$, given by the difference in
the number of electrons $\hat{N}$ and the number of carbon atoms $N_{\rm TOT}$
in the central island. $S_{z}(i)$ is the local spin density of the $p_{z}$
electrons in graphene at site $i$,
$S_{z}(i)=\frac{1}{2}\left(c^{\dagger}_{i\uparrow}c_{i\uparrow}-c^{\dagger}_{i\downarrow}c_{i\downarrow}\right),$
(2)
where $c^{\dagger}_{i\uparrow}$ creates an $\uparrow$ electron at the $p_{z}$
orbital of site $i$ of graphene. In the following we assume that magnetic
fields controlling the spin orientation of the adsorbates, are applied along
the plane of graphene so that it is a good approximation to neglect the
diamagnetic coupling to the graphene electrons.
There are several independent microscopic mechanisms for spin dependent
interaction between magnetic adsorbates and the graphene $\pi$ electrons that
can be modeled with Eq. (1). In the case of magnetic molecule such as TbPC2,
used in in Ref. Wernsdorfer_NL, , the magnetic Tb atom is separated from the
graphene electrons by the non-magnetic atoms of the molecule, and the most
likely mechanism for spin coupling is kinetic exchange.Anderson61 This
coupling will generate a local Kondo-exchange between graphene electrons and
the molecules.Schrieffer-Wolff More complicated scenarios, like coupling of
graphene electrons to unpaired electrons in the organic rings of the
molecules, would imply that every molecule affects several sites in graphene.
Direct dipolar coupling would also affect several sites per molecule, but the
average magnetic field created by a magnetic moment of 5 $\mu_{B}$ at 0.5 nm
on a disk with an area around 400 nm2, the graphene constriction area in Ref.
Wernsdorfer_NL, , is smaller than 1 $\mu$T, which would yield a negligible
maximal Zeeman coupling of neV per molecule.
### II.2 Relevant energy scales
The reported dimensions of the central region, $L_{x}\simeq L_{y}\simeq 20$
nm, lead to an energy spacing $\Delta\epsilon$ of the single particle spectrum
much larger than the temperature and the charging energy.Geim2008 ; review-dot
We also assume that the exchange induced shifts are smaller than the single
particle splitting. As a result, the effect of exchange is to shift the bare
energy levels, without mixing them. Thereby, we can safely assume that
electrons tunnel through just one of the single particle levels, which might
be spin-split due to exchange with molecules. We assume that the charge of the
central island fluctuates between $q=0$ and $q=-|e|$, and that the transport
level is the lowest unoccupied level of the central island spectrum. The
energy of the transport level reads:
$\epsilon_{T}^{\sigma}=\epsilon_{0}+\sigma\frac{\delta}{2}-|e|V_{g},$ (3)
where $\epsilon_{0}$ is the single particle electron level, $\sigma=\pm 1$
denotes the spin direction and $\delta$ is the magnitude of the spin
splitting, which is a functional of the magnetic landscape $\\{m_{z}\\}$.
Within our model, a given magnetic landscape is defined by the location $i$
and the magnetic state $m_{z}(i)$ of the $N$ magnetic adsorbates. In Figs. 2
(a, b), we plot the value of $\delta$ for all the possible magnetic states of
a given arrangement of $N=10$ atoms, for two different values of $J$. This
choice corresponds to the estimated number of molecules in Ref.
Wernsdorfer_NL, . These figures show a correlation between the magnitude of
the splitting $\delta$ and the total magnetization $M_{T}$. The dispersion of
$\delta$ for a fixed total magnetization $M_{T}$ is the outcome of indirect
exchange couplingBrey-Das-Sarma
For comparison with the experiments, it is worth considering two extreme
magnetic landscapes. At large external field, all the magnetic moments are
aligned, i.e., $m_{z}(i)=+1$. We refer to this as the ferromagnetic (FM)
landscape. At magnetic fields smaller than the coercive field of the magnetic
molecules, their average magnetization should be zero and thus,
$\sum_{i}m_{z}(i)=0$. We refer to these cases as non-magnetic (NM). In order
to sample the positional disorder we perform an average over positional
configurations, both for NM and FM cases. For a fixed spin choice
$\\{m_{z}\\}$ with $M_{T}=0$, an average over positional configurations yields
$\langle\delta\rangle_{\rm pos}=0$. The reciprocal statement is also true: for
a fixed positional configuration, an average over all the magnetic landscapes
with $M_{T}=0$ also yields an average $\langle\delta\rangle_{\rm spins}=0$.
Figure 2: Spin splitting $\delta$ for a sample of dimensions $L_{y}=N_{y}a$
and $L_{x}=N_{x}\sqrt{3}a$, with $N_{x}=15$ and $N_{y}=17$, being $a=2.46\AA$
the lattice constant of graphene. $\delta$ versus total magnetization $M_{T}$
for a single spatial distribution of $N=10$ magnetic impurities with $J=5$ meV
(a) and $J=10$ (b). The red solid line indicates the average value
$\langle\delta\rangle_{\rm pos}$. Average $\langle\delta\rangle_{\rm pos}^{\rm
FM}$ versus number of magnetic molecules $N$ for $J=5$ meV (c) and exchange
energy $J$ for $N=4$ (d). The error bars correspond to the standard deviation.
In Fig. 2 we plot the average $\langle\delta\rangle_{\rm pos}^{\rm
FM}-\langle\delta\rangle_{\rm pos}^{\rm NM}=\langle\delta\rangle_{\rm
pos}^{\rm FM}$ over $500$ realizations as a function of the number of
molecules $N$ [Fig. 2(c)] and as a function of the molecule electron exchange
$J$ [Fig. 2(d)]. We have also calculated $\langle\delta\rangle_{\rm pos}^{\rm
FM}$ fixing the number of magnetic centers $N$, the strength of the coupling
$J$ and changing $N_{\rm TOT}=N_{x}N_{y}$, the total number of carbon atoms in
the island. We find that the results of all these simulations can be
summarized in the following equation:
$\left\langle\delta\right\rangle_{\rm pos}^{\rm FM}\approx\frac{N}{N_{\rm
TOT}}J.$ (4)
Whereas this result has been obtained from exact numerical diagonalization of
the Hamiltonian, this dependence can be rationalized using first order
perturbation theory, which yields the spin dependent shift of the transport
level:
$\Delta\epsilon_{T}^{\sigma}=\frac{\sigma}{2}J\sum_{i}^{N}|\phi_{T}(i)|^{2}m_{i},$
(5)
where $\phi_{T}(i)$ is the $J=0$ wave function of the transport orbital. We
now use $|\phi_{T}(i)|^{2}\simeq\frac{1}{N_{\rm TOT}}$ so that we can
approximate:
$\Delta\epsilon_{T}^{\sigma}\simeq\frac{\sigma}{2}\frac{J}{N_{\rm TOT}}M_{T}$
(6)
Using the fact that $M_{T}=N$ for the FM configurations and $0$ for the NM
ones, we arrive to Eq. (4).
## III Spin-Split Single Orbital Model for SET
In this section we discuss SET across a central island with a single spin
split particle level. This is justified by the results of the previous
section. We obtain expressions for the current of the system and we discuss
the conditions under which the conductance depends on the magnetic state of
the single electron transport.
### III.1 Single electron transistor with a spin-split single orbital model
We consider single electron transport though a spin split single transport
level,Recher2000 with energy $\epsilon_{T}^{\sigma}$. We assume that the
occupation of the transport level can be either 0 or 1, the doubly occupied
configuration being much higher in energy. Within these approximations, the
transport level has three relevant many body states: uncharged, and the two
charged with $\uparrow$ or $\downarrow$ spins. In the zero-applied bias limit,
each of these states will be occupied according to the thermal equilibrium
distribution, which we denoted as $P_{0}$, $P_{\uparrow}$ and $P_{\downarrow}$
respectively. In the SET regime we are interested and within the linear
response ($eV_{bias}\ll k_{B}T$), transport will be enabled only when the
addition energy lies within the thermally broadened transport window defined
by the applied bias.
Under these approximations, the current flowing from the left electrode to the
central island is given by:
$I=e\sum_{\sigma}\left\\{P_{0}W_{0\to\sigma}^{L}-P_{\sigma}W_{\sigma\to
0}^{L}\right\\},$ (7)
where $W_{0\to\sigma}^{L}$ and $W_{\sigma\to 0}^{L}$ are rates for electron
tunneling from the left electrode to the dot and vice-versa. Continuity
equation ensures that this current is identical to the current flowing towards
the right electrode and, thereby, equal to the net current flow. The tunneling
rates for electron tunneling out of and into the dot Haug_Jauho_book_1996 are
given respectively by
$W_{\sigma\to
0}^{L}=\frac{2\pi}{\hbar}|T_{L}|^{2}\rho_{L}(\epsilon_{T}^{\sigma})\left[1-f(\Delta_{\sigma}+V_{bias})\right],$
(8)
and
$W_{0\to\sigma}^{L}=\frac{2\pi}{\hbar}|T_{L}|^{2}\rho_{L}(\epsilon_{T}^{\sigma})f(\Delta_{\sigma}+V_{bias}),$
(9)
where $T_{L}$ is the strength of the dot - left electrode coupling, and
$\Delta_{\sigma}\equiv\epsilon_{T}^{\sigma}-E_{F},$ (10)
are the spin dependent addition energies. Importantly, both $\delta$ and
$V_{g}$ appear on equal footing, as additive quantities in this equation. The
density of states of the left electrode, evaluated at the spin-dependent
transport level energy, is denoted by $\rho_{L}(\epsilon_{T}^{\sigma})$, while
$f(\epsilon_{\sigma})=\left(e^{\beta\Delta_{\sigma}}+1\right)^{-1}$ denotes
the Fermi function. The electrode Fermi energy $E_{F}$ is taken to change
linearly with the bias voltage $V_{bias}$. In the zero bias limit, the linear
conductance reads:
$G=G_{0}\sum_{\sigma}\left(P_{0}+P_{\sigma}\right)\frac{\Gamma_{\sigma}}{k_{B}T}{\rm
Sech}^{2}\left(\frac{\beta\Delta_{\sigma}}{2}\right),$ (11)
where
$\Gamma_{\sigma}/\hbar=\frac{2\pi}{\hbar}|T_{L}|^{2}\rho_{L}(\epsilon_{T}^{\sigma}),$
(12)
is the single particle tunneling rate between the electrode and the transport
level.
### III.2 Influence of magnetic state on conductance
From the above discussion it is apparent that, for a given gate potential
$V_{g}$ and temperature $T$, the linear conductance depends on the magnetic
landscape affecting the central island through two classes of independent
mechanisms, illustrated in Fig. 3:
1. 1.
The change of the addition energies $\Delta_{\sigma}$ which, as we show below,
would result in a lateral shift of the $G(V_{g})$ resonance curve [Fig. 4(a)].
2. 2.
The change of the electron lifetime $\Gamma=\sum_{\sigma}\Gamma_{\sigma}$,
that would result in a vertical resizing of the $G(V_{g})$ resonance curve
[Fig. 5(a)].
Figure 3: (Color online) (a) Scheme showing the transport level energy
splitting. Scheme of spin-dependence of transport due to: (b) detuning of the
transport level with respect to the electrode Fermi energy, (c)
magnetoresistance associated to spin polarized electrode(s), and (d) energy
dependent tunneling rates.
In the first mechanism the change in the magnetic state modifies the value of
$\delta$, which must have a similar effect than changing the gate potential.
It resembles the magneto-Coulomb effect,Ono ; Vanwees by which the applied
magnetic field changes the Fermi energy of the electrode, shifting the
$G(V_{g})$ curves. However, this first mechanism necessarily implies a change
of sign of the variation of $G$ as the gate potential is scanned along the
resonance (see top panel of Fig. 4). Importantly, this is not observed in the
experiments with magnetic molecules,Wernsdorfer_NL but it is observed in the
case of graphene nanoconstrictions.Candini-PRB-2011
Figure 4: (Color online) Conductance in units of $g_{0}=G_{0}\beta\Gamma$ as a
function of the gate voltage $eV_{g}$ and the level splitting $\delta$. Panel
(a) shows the normalized conductance as a function of the gate potential
$e\beta V_{g}$, where the labels corresponds to the different $\beta\delta$
values. Panel (b) shows the conductance as a function of $\beta\delta$ for
several values of the gate $e\beta V_{g}$. For the sake of clarity all curves
have been displaced by $0.1$. In (c, d) we present a contour plot of the dot
charge (defined as $Q=P_{\uparrow}+P_{\downarrow}$) and the dot magnetization
($m=P_{\uparrow}-P_{\downarrow}$) as a function of the gate voltage $eV_{g}$
and level splitting $\delta$.
Motivated by the behavior reported in Ref. Wernsdorfer_NL, , we pay attention
also to the second mechanism. For spin un-polarized transport, the change in
the transport energy level results in a change on tunneling rate
$\Gamma/\hbar$ only if the electrode density of states depends on energy,
which is exactly the case of graphene. For spin-polarized transport, the
relative orientation of the electrode and island magnetic moment give rise to
magneto-resistive effects that are accounted for by the changes in
$\Gamma_{\sigma}$
### III.3 Transport for constant tunneling rates
We now discuss our transport simulations for the graphene single electron
transistor spin sensor. We focus on the first spin sensing mechanism in a
single electron transistor: changes in spin splitting of the transport level
produce changes in addition energies $\Delta_{\sigma}$ (figure (3)b). For that
matter, we neglect both the energy and spin dependence of the tunneling rates
$\Gamma_{\sigma}$. In Fig. 4(a) we show the linear conductance, in units of
$g_{0}=\beta\Gamma G_{0}$, as a function of gate voltage, for several values
of the transport level splitting $\delta$, in units of $k_{B}T$. It is
apparent that the Coulomb Blockade peaks undergoes a lateral shift, as
expected from the fact that $V_{g}$ and $\delta$ appear on equal footing on
the spin dependent addition energies. At $\Delta=0$ the two spin channels
contribute. Therefore, as we increase $\delta$, the height of the conductance
peaks decreases, because one of the two spin channels is removed from the
transport window of width $k_{B}T$
In figure (4)b we plot the variations in the linear conductances as a function
of the spin splitting $\delta$, for several values of $V_{g}$. We see two
types of curves. For values of $V_{g}$ such that the transport level is
occupied, as we increase $\delta$ the transport level is pushed downwards,
away from the elastic transport window, switching off the transistor
conductance. In contrast, for $V_{g}$ such that the transport level is empty
for $\delta=0$, lying above the elastic transport window, ramping $\delta$
makes one of the two spin states of the transport level enter the transport
window, giving rise to the double peak structure. The fact that $\delta$ and
$V_{g}$ play analogous roles is illustrated in Figs. 4(c, d), where we show
the average magnetization and occupation of the transport level in the phase
diagram defined by these two variables.
### III.4 Transport with energy dependent tunneling rates
We now consider the second mechanism for spin sensing in a single electron
transistor: changes in spin splitting of the transport level produce changes
in the tunneling rates $\Gamma_{\sigma}$. This can happen for two reasons:
1. 1.
One of the electrodes is spin polarized, so that
$\Gamma_{\downarrow}\neq\Gamma_{\uparrow}$. Spin polarized transport is
sensitive to the product of the magnetic moment of electrode and central
island. This type of effect has been thoroughly discussed in the case of SET
with ferromagnetic electrodes.Barnas98 ; Seneor2007
2. 2.
The density of states of the electrode depends on energy. Thus, changes in the
transport level change $\Gamma_{\sigma}$, for both spins. This is a natural
scenario for graphene electrodes. dots_Ensslin ; Sols-Guinea
Let us consider first the case of spin polarized electrodes. We do the
assumption that the density of states are spin dependent but energy
independent: $\rho_{\sigma}=\rho_{0}(1+\sigma{\cal P})$, with ${\cal P}$ the
electrode polarization. In Fig. 5(a) we plot the linear conductance vs $V_{g}$
curves for several values of the splitting $\delta$, assuming a large value of
the electrode spin polarization ${\cal P}=0.9$. It is apparent that, on top of
the shift of the resonance curve whose origin was discussed in the previous
subsection, there is a change in the amplitude of the curve. Notice that
$G(\delta=k_{B}T)$ and $G(\delta=2k_{B}T)$ are smaller than $G(0)$ for all
values of $V_{g}$. In this specific sense, the gate-independent spin contrast
is similar to the experimental report with magnetic molecules.Wernsdorfer_NL
In Fig. 5(a) we show the linear conductance as a function of $\delta$ for
different values of $V_{g}$. It is apparent that, as opposed to the case of
non-magnetic electrodes shown in Fig. 5(b), the function $G(\delta$) is no
longer an even functions, reflecting the magneto-resistive behaviour.
Basically, transport is favored when the spin polarization of the electrode
and the central island are parallel.
Figure 5: (Color online) Normalized Conductance for ferromagnetic electrodes
as a function of the level splitting $\delta$ for several gate values. In (a,
c) the density of states depends of the polarization
$\rho=\rho_{0}(1+\sigma{\cal P})$, in this particular case we take a
polarization of ${\cal P}=0.9$. In (b, d) the electrode density of states is
linear with the energy,
$\rho(\Delta_{\sigma})=\rho(\epsilon_{0}+\sigma\delta/2-E_{D})$. For the sake
of clarity the conductance curves in (c, d) have been displaced by $0.2$ and
$4$ units respectively.
We now consider a non-spin polarized electrode with an energy dependent
density of states. This scenario occurs naturally in graphene. If we consider
idealized graphene electrodes, neglecting effects of interactions, disorder
and confinement, we have $\rho(\epsilon)=\rho_{0}|\epsilon-E_{D}|$, where
$E_{D}$ is the Dirac point. The $G(V_{g})$ curves, shown in Fig. 5(b) for
different values of $\delta$, shift and change amplitude. The shift is related
to the change of the addition energies, discussed in the previous subsection,
and the change in amplitude comes from the variation of the tunneling rate as
the transport level scans the energy dependent density of states of the
electrode.
In Fig. 5(d) we plot $G(\delta)$ for several values of $V_{g}$. The curves are
similar to the case with energy independent tunneling rates, except for the
dip at zero $\delta$ which occurs because we chose the bare transport level
right at Dirac point. This is the most favorable choice to maximize the effect
of energy dependence of $\Gamma$. From our results, and given the fact that
experimentally is not possible to put the Fermi energy arbitrarily close to
the Diract point,Geim-closest we find it unlikely that this effect is playing
a role in the experiments.
### III.5 Sensitivity of the single electron transistor spin sensor
We now discuss the sensitivity of the spin sensor based on the graphene single
electron transistor, as described by our model, neglecting changes in
$\Gamma$. From Fig. 4(a) we propose, as rule of thumb, that variations of
$\delta$ similar or larger than $k_{B}T$ can be resolved. Estimating $\delta$
from the case of fully spin polarized magnetic adsorbates, given in Eq. (4),
we find a relation between the minimal number of magnetic centers $N$ that can
be detected, and the temperature and exchange constant:
$\frac{N}{N_{\rm TOT}}>\frac{k_{B}T}{J}.$ (13)
It is apparent that decreasing the temperature, or increasing the spin-
graphene exchange coupling, increases the sensitivity of the device (makes it
possible to detect a smaller concentration of molecules). For instance, at
$100$ mK, and taking $N_{\rm TOT}\sim 15000$, which corresponds to an
approximate area of $400$ nm2, one could detect $10$ molecules for an exchange
coupling $J\gtrsim 15$ meV.
Recent reports have shown that it is possible to fabricate graphene nano
islands with lateral dimensions of 1nm.Barreiro2012 These dots have $N_{\rm
TOT}<100$. Thus, they would permit the detection of the spin of a single
magnetic adsorbate provided that $k_{B}T$ is kept hundred times smaller than
$J$. For $T=100$ mK this implies $J>1$ meV. Interestingly, in such a small dot
Coulomb Blockade persists even at room temperature, but increasing $k_{B}T$
keeping the sensitivity would require also to increase $J$.
## IV Discussion and conclusions
Because of its structural and electronic properties, graphene is optimal for a
spin sensor device. Being all surface, the influence of adsorbates on
transport should be larger than any other bulk material. Because of the linear
relation momentum and large Fermi velocity, energy level spacing in graphene
nano structures can easily be larger than the temperature, the tunneling
induced broadening, and the perturbations created by the adsorbates. One of
the consequences is that single electron transport takes place through a
single orbital level.
Our simulations show how the spin splitting $\delta$ of the transport level is
sensitive to the average magnetization of the magnetic adsorbates, which is
controlled by application of a magnetic field along the plane of graphene, to
avoid diamagnetic shifts. On the other hand, the linear conductance $G$ of the
single electron transistor depends on $\delta$, which accounts for the sensing
mechanism. More specifically, $G$ depends on $\delta$ due to either changes in
the spin dependent addition energies $\Delta_{\sigma}$ or changes in the
electrons lifetime $\Gamma_{\sigma}$. The first is independent of the nature
of the electrodes, whereas the second only happens if they are magnetic or
have an energy dependent density of states.
We have shown how, within an independent particle model and in the single
electron transport regime, the energy dependence of the graphene electrode
density of states can only be relevant if the transport energy level is fine
tuned to the Dirac point. However, this fine tuning is quite unlike to happen
in experimental conditionsGeim-closest . Still, the combined action of
disorder and Coulomb interaction could give rise to a so called Coulomb Gap in
the density of states of graphene, that might make the tunneling rates depend
on the energy.Coulomb-gap-graphene ; Coulomb-gap-graphene1 ; Coulomb-gap-
graphene2
Finally, we have assumed that both the edges of the graphene island and
graphene electrodes are non-magnetic. Our discussion of the effect of spin-
polarized electrodes on the transport properties of the device would be valid
for electrodes with ferromagnetic zigzag edges. A second possibility, out of
the scope of this work, is to consider a graphene single electron transistor
whose central island has ferromagnetic edges. This case has been already
studied. Ezawa
In conclusion, we have studied the mechanisms by which a graphene single
electron transistor could work as a sensor of the magnetic order of magnetic
atoms or molecules adsorbed on the graphene central region. Our work has been
motivated in part by recent experimental works, Wernsdorfer_NL ; Candini-
PRB-2011 . Whereas further work is still necessary to nail down the physical
mechanisms for the spin sensing principles underlying the experimental work,
our study provides a conceptual framework for graphene single electron
transistor spin sensors.
This work has been financially supported by MEC-Spain (Grant Nos.
FIS2010-21883-C02-01 and CONSOLIDER CSD2007-0010) as well as Generalitat
Valenciana, grant Prometeo 2012-11. We thank A. Candini for useful comments on
the manuscript.
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|
arxiv-papers
| 2012-12-11T18:40:52 |
2024-09-04T02:49:39.146602
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. W. Gonz\\'alez, F. Delgado, J. Fern\\'andez-Rossier",
"submitter": "Jhon W. Gonz\\'alez",
"url": "https://arxiv.org/abs/1212.2581"
}
|
1212.2682
|
††thanks: Corresponding author. [email protected]
# Electron correlation and spin-orbit coupling effects in US3 and USe3
Yu Yang LCP, Institute of Applied Physics and Computational Mathematics, P.O.
Box 8009, Beijing 100088, People’s Republic of China Ping Zhang LCP,
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009,
Beijing 100088, People’s Republic of China Center for Applied Physics and
Technology, Peking University, Beijing 100871, People’s Republic of China
###### Abstract
A systematic density functional theory (DFT) +$U$ study is conducted to
investigate the electron correlation and spin-orbit coupling (SOC) effects in
US3 and USe3. Our calculations reveal that inclusion of the $U$ term is
essential to get energy band gaps for them, indicating the strong correlation
effects for uranium 5$f$ electrons. Taking consideration of the SOC effect
results in small reduction on the electronic band gaps of US3 and USe3, but
largely changes the energy band shapes around the Fermi energy. As a result,
US3 has a direct band gap while USe3 has an indirect one. Our calculations
predict that both US3 and USe3 are antiferromagnetic insulators, in agreement
with corresponding experimental results. Based on our DFT+$U$ calculations, we
systematically present the ground-state electronic, mechanical and Raman
properties for US3 and USe3.
## Introduction
Actinide based materials possess interesting physical behaviors due to the
existence of strongly-correlated 5$f$ electrons and have attracted extensive
attentions Savrasov01 ; Albers01 ; Hecker04 ; Moore06 ; Prodan06 ; Prodan07 .
Different from actinide oxides which have been widely studied to reveal the
detailed configurations and correlation effects of the 5$f$ electrons Prodan06
; Prodan07 ; Dudarev97 ; Dudarev98 ; Dudarev00 ; Sun08JCP ; Sun08CPB ; Shi10 ;
WangBT10 ; Zhang10 ; Kern99 ; Petit03 ; Petit10 ; Wilkins06 ; Yin08 ; Jomard08
; Andersson09 ; Santini09 ; Sanati11 ; Dorado09 ; Dorado10 ; Nakamura10 ;
Meredig10 ; Geng10 ; Yasuoka12 ; Manley12 , none of the actinide chalcogenides
has ever received comparable concerns. Here we take US3 and USe3 as two
representatives to study the electron correlation and spin-orbit coupling
(SOC) effects of 5$f$ electrons in actinide chalcogenides. The other reason
for us to investigate their electronic structures is that they employ the
layered MX3 structure (with M to be a metal element, and X to be S, Se, or
Te), which can be used as models for studying electronic behaviors in
1-dimensional (1D) systems. The MX3 structure belongs to the space group of
P21/$m$, with its atomic organizations depicted in Fig. 1(a). In the
monoclinic lattice, the top two chalcogen (S , Se or Te) atoms form a tightly
bound pair suggesting that the MX3 compounds may be regarded as MX(X2)
Nouvel87 . The top X-X pair together with an underneath M and a more
underneath X atom form a triangular unit, which repeats along the $\vec{b}$
lattice direction forming a 1D chain, and two such triangle units form a unit
cell of the MX3 compounds in the ($\vec{a}$,$\vec{c}$) plane. In this way, the
MX3 compounds are always considered as 1D materials. For example, the metallic
ZrTe3 and NbSe3 exhibiting charge-density-wave transitions have been
thoroughly investigated in angle-resolved photoelectron spectroscopy (ARPES)
and optical experiments Schafer03 ; Perucchi041 ; Perucchi042 ; Yokoya05 ;
Pacile07 . In the present paper, by contrast, we address the electronic
structures of US3 and USe3, two semiconducting members of the MX3 family.
In studies of actinide based materials, one has to be careful for the electron
correlation and SOC effects of the actinide 5$f$ electrons. As an example, for
actinide dioxides conventional density functional theory (DFT) schemes that
apply the local density approximation (LDA) or the generalized gradient
approximation (GGA) underestimate the strong on-site Coulomb repulsion of the
5$f$ electrons and consequently fail to capture the insulating properties
Dudarev97 ; Zhang10 . Several approaches, the LDA/GGA+$U$, the hybrid density
functional of (Heyd, Scuseria, and Enzerhof) HSE, the self-interaction
corrected local spin-density (SIC-LSD), and the Dynamical Mean-Field Theory
(DMFT), have been developed to correct the pure LDA/GGA failures in
calculations of actinide materials. Among them the effective modification of
pure DFT by LDA/GGA+$U$ formalisms has been widely used in theoretical studies
of UO2 Dudarev97 ; Dudarev00 ; Sanati11 and PuO2 Sun08JCP ; Sun08CPB ;
Jomard08 ; Shi10 . The obtained structural parameters as well as the
electronic structure and phonon dispersion curves Zhang10 ; Manley12 accord
well with experiments. In our present work, the GGA+$U$ schemes due to Dudarev
et al. Dudarev97 ; Dudarev98 ; Dudarev00 are employed to study the electron
correlation and spin-orbit coupling effects in US3 and USe3, as well as the
two materials’ electronic, mechanical, and Raman properties.
The rest of the paper is organized as follows. In Sec. II the computational
method is briefly described. In Sec. III we present the results of the
physical properties of US3 and USe3, and discuss the electron correlation and
SOC effects of the uranium 5$f$ electrons. Finally in Sec. IV, we close our
paper with a summary of our main results.
## Calculation method
Our total-energy calculations are carried out by employing the plane-wave
basis pseudopotential method as implemented in Vienna ab initio simulation
package (VASP) VASP . The exchange and correlation effects are described with
the GGA approximation in the Perdew-Burke-Ernzerhof (PBE) form PBE1 ; PBE2 .
The projected augmented wave (PAW) method of Blöchl PAW is employed with the
frozen-core approximation. Electron wave function is expanded in plane waves
up to a cutoff energy of 400 eV and all atoms are fully relaxed until the
Hellmann-Feynman forces on them are less than 0.01 eV/Å. A 7$\times$9$\times$1
Monkhorst-Pack MPkpoints k-point mesh is employed for integration over the
Brillouin zone of US3 and USe3. The uranium
6$s^{2}$6$p^{6}$5$f^{3}$6$d^{1}$7$s^{2}$, sulphur 3$s^{2}$3$p^{6}$, and
selenium 4$s^{2}$4$p^{6}$ electrons are treated as valence electrons.
Noncollinear calculations are used when considering the spin-orbital coupling
effects. The strong on-site Coulomb repulsions among the localized uranium
5$f$ electrons are described by using the formalism formulated by Dudarev et
al. Dudarev97 ; Dudarev98 ; Dudarev00 . In this scheme, the total GGA energy
functional is of the form
$E_{{\rm GGA}+U}=E_{\rm GGA}+\frac{U-J}{2}\sum_{\sigma}[{\rm
Tr}\rho^{\sigma}-{\rm Tr}(\rho^{\sigma}\rho^{\sigma})],$ (1)
where $\rho^{\sigma}$ is the density matrix of $f$ states with spin $\sigma$,
while $U$ and $J$ are the spherically averaged screened Coulomb energy and the
exchange energy, respectively.
In this paper, the Coulomb $U$ is treated as a variable, while the exchange
energy is set to be a constant $J$=0.51 eV. This value of $J$ is in the ball
park of the commonly accepted one for uranium compounds Sanati11 and close to
the theoretically predicted value of 0.54 eV in UO2 Kotani92 . Since only the
difference between $U$ and $J$ is significant, we will henceforth label them
as one single parameter $U_{\rm eff}$=$U$-$J$, while keeping in mind that the
nonzero $J$ has been used during calculations.
The phonon frequencies at the Gamma point for US3 and USe3 are calculated by
using the density functional perturbation theory (DFPT). And their Raman-
active frequencies are obtained through symmetry analysis on the corresponding
vibration modes. Before DFPT calculations, the lattice constants and atomic
positions of US3 and USe3 are further optimized using a denser $k$-point mesh
of 9$\times$13$\times$3, and a finer force convergence criteria of 0.001 eV/Å.
A 2$\times$2$\times$1 supercell is subsequently used for DFPT calculations.
Since both US3 and USe3 are stacked along the $\vec{c}$ direction through van
der Waals interactions, and with relatively large lattice constants, we do not
extend the supercell along the $\vec{c}$ direction. During DFPT calculations
on the 2$\times$2$\times$1 supercell, the $k$-point mesh is set to be
5$\times$7$\times$5, and the energy convergence criteria is set to be 1e-6 eV.
## Results and discussion
Experimentally, the magnetic orderings of uranium trichalcogenides were
studied ever since 1961 Nouvel87 ; Trzebiatowski61 ; Suski76 ; Baran83 ;
Noel86 . The relatively large atomic distances between neighboring U atoms in
US3 and USe3 suggest that magnetic orderings may occur in these compounds
through super-exchange interactions of uranium 5$f$ electrons via the S or Se
ions Nouvel87 . And different measurements indicate that both US3 and USe3
crystals undergo AFM transitions at very low temperatures, with the magnetic
moments of the two uranium atoms within each unit cell opposite to each other
Nouvel87 ; Baran83 ; Noel86 . In the present GGA+$U$ study, we consider the
nonmagnetic (NM), ferromagnetic (FM), and antiferromagnetic (AFM) phases for
each choice of the $U_{\rm eff}$ value and then determine the lowest-energy
state by a subsequent total-energy comparison of these three phases. Our
calculated electronic energies for different magnetic phases of US3 and USe3
are all listed in Table I. One can see that within GGA formalism or the
GGA+$U$ formalism with a too small $U_{\rm eff}$ value, the FM state is more
stable for both US3 and USe3, in contradiction with experimental results.
Therefore, the electron correlation effect has to be accounted for to
correctly describe ground-state US3 and USe3. In the discussions that follow,
we will confine our reports to the AFM solutions for US3 and USe3.
The experimentally determined lattice parameters of US3 and USe3 are ($a$=5.37
Å, $b$=3.96 Å, $c$=9.94 Å) and ($a$=5.65 Å, $b$=4.06 Å, $c$=10.47 Å)
respectively Nouvel87 ; Gronvold68 ; Salem84 . Our calculated lattice
constants with different $U_{\rm eff}$ values are summarized in Table II,
together with the experimental results. We can see that different from what we
found for actinide dioxides Zhang10 ; Wang10 , the lattice constants of
actinide trichalcogenides do not change monotonically with the value of the
$U_{\rm eff}$ parameter. Besides, the lattice constants obtained in GGA
calculations are obviously too small in comparison with corresponding
experimental results, for both US3 and USe3. For US3, the value of $U_{\rm
eff}$ has to be as large as 6 eV to get reasonable lattice constants compared
with experimental results. Differently, a $U_{\rm eff}$ value of 4 eV is
enough to get reasonable lattice constants for USe3, and changing $U_{\rm
eff}$ from 4 to 6 eV has negligible effects on the obtained lattice constants.
Besides of the prominent changes in the atomic structure parameters, the most
dramatic improvement brought by the GGA+$U$ formalism when compared to the GGA
results is in the description of electronic structure properties. For this, we
have investigated the band structures in AFM phases for US3 and USe3 aiming at
revealing the fundamental influences of considering the on-site Coulomb
interaction. Figures 2(a) and 2(b) show the obtained local density of states
(LDOS) for the S1, S3, and U atoms in US3, and Se1, Se3, and U atoms in USe3
respectively. Without accounting for the on-site Coulomb repulsion, one can
see that the GGA calculations predict an incorrect metallic ground state by
nonzero occupation of uranium electrons at the Fermi energy ($E_{\rm f}$).
When switching on the $U_{\rm eff}$ parameter, as shown both in Figs. 2(a) and
2(b), the uranium electronic states in US3 and USe3 begin to split at $E_{\rm
f}$ and tend to form two peaks with the gap of $\Delta E$. The amplitude of
this gap increases with increasing $U_{\rm eff}$. Previous electric
resistivity measurements pointed out that USe3 did not conduct at room
temperatures Shlyk95 , thus a band gap at $E_{\rm f}$ should be contained in
its electronic structure. We can see from Fig. 2(b) that an energy band gap
appears only when $U_{\rm eff}$ is larger than or equal to 4 eV for USe3. For
US3, the value of $U_{\rm eff}$ has to be enlarged to 6 eV to open the band
gap at the Fermi energy. These results also imply that the electron
correlation strength might be different in US3 and USe3. From the last figure
in Figs. 2(a) and 2(b), we can see that the sulphur (selenium) and uranium
electronic states overlap with each other and contribute equally to the
valence band maximum (VBM) of US3 (USe3), while the conduction band minimum
(CBM) is composed of uranium electronic states. The orbital mixing between
sulphur (selenium) and uranium electronic states below the Fermi energy
indicates that there are covalent interactions between the sulphur (selenium)
and uranium atoms in US3 (USe3).
To further analyze the orbital-resolved electronic structures, we calculate
the projected density of states (PDOS) for the X-$np$ ($n$=3 (X=S) or 4
(X=Se)), U-5$f$ and U-6$p$ electronic states in UX3. Figures 3(a)-3(d) show
the obtained PDOS for US3 in different calculations, while the corresponding
PDOS results for USe3 are shown in Figs. 3(e)-3(h). The value of $U_{\rm eff}$
is chosen to be 6 eV in all GGA+$U$ and GGA+$U$+SOC calculations. One can see
that without considering the electron correlation effect, US3 and USe3 are
both wrongly predicted as metallic materials, which contradicts with
experimental results. After adding the $U$ parameter to describe the strong
on-site energy of U-5$f$ electrons, the energy gaps are opened for US3 and
USe3. We can see from Figs. 3(d) and 3(h) that further considering SOC has
little influences on the electronic states around the Fermi energy. For both
US3 and USe3, the SOC effect lies in the deep energy level, causing an energy
splitting of the U-6$p$ states. From the PDOS results, we can also see clear
difference between the X1- and X3-$np$ (n=3 (X=S) or 4(X=Se)) electronic
states. The PDOS for the electronic states of the X2 atom is very similar to
that of the X1 atom and thus is not shown here. The different electronic state
distribution of the X1(X2) and X3 atoms proves the theory of considering UX3
as UX(X2), and indicates the strong covalent bondings between the X1 and X2
atoms.
To analyze more carefully the SOC effects in US3 and USe3, we recalculate the
electronic structures of them with higher resolutions, with the reduced
Gaussian smearing width of only 0.02 eV and a PDOS resolution of 0.01 eV.
Figures 4(a) and 4(b) show the obtained PDOS results for uranium 5$f$
electrons of US3 and USe3, by using the GGA method, while Figs. 5(a) and 5(b)
show the obtained energy band structures of US3 and USe3 by using the GGA+$U$
method, respectively. All the GGA+SOC and GGA+$U$+SOC calculations are
noncollinear, with both SOC and orbital polarizations considered. And the
results before and after considering the SOC effects are shown in solid and
dotted lines respectively. From the PDOS results obtained by using the GGA
method, we can see that pure inclusion of SOC to the GGA method does not lead
to energy band gaps for US3 and USe3. This discovery is different from the
situation of CoO, where SOC can solely open a band gap Norman90 .
From the band structures of US3 shown in Fig. 5(a), we can see that both the
lowest unoccupied conduction and highest occupied valence bands distribute
along the $\Gamma$-Z direction. For the lowest conduction band, the SOC effect
causes an energy downshift of 0.05 eV along the $\Gamma$-Z direction.
Contrarily for its highest valence band, the SOC effect causes an energy
upshift of 0.03 eV at the $\Gamma$ point and an upshift of 0.07 eV at the Z
point. As a result, the energy band gap of US3 changes from indirect
($\Gamma\rightarrow$Z) into direct type (Z$\rightarrow$Z) after considering
the SOC effect, and the gap value reduces from 1.59 to be 1.48 eV. Different
from US3, the lowest conduction and highest valence bands of USe3 distributes
along the K1-B and B-$\Gamma$-Y directions respectively. For the lowest
conduction band along the K1-B direction, the SOC effect causes an energy
downshift of 0.04 eV at the K1 point and a downshift of 0.02 eV at the B
point. Correspondingly the CBM of USe3 moves from B to the K1 point. However,
since the VBM of USe3 is at some point along the B-$\Gamma$-Y direction
instead of at any high-symmetry $k$ points, the SOC effect does not change the
indirect band gap type. Overall the band gap of USe3 changes from 1.30 to be
1.23 eV after considering the SOC effect. Our studies reveal that the energy
band gap of USe3 is smaller than that of US3.
After systematically presenting the electronic structure results for US3 and
USe3, we now turn to their mechanical properties. The elastic constants are
obtained by solving the eigenvalues of their Hessian matrix, which is
calculated based on the Hooke’s law and small position changes on independent
S, Se and U atoms. The obtained elastic constants of US3 and USe3 in different
calculations by using the GGA and GGA+$U$ methods are both listed in Table
III. We can see that the elastic constants calculated by using the GGA
formalism for US3 and USe3 do not satisfy the stability criteria Nye85 ; Wu07
for monoclinic structures that the $C_{\rm 11}$, $C_{\rm 22}$, $C_{\rm 33}$,
$C_{\rm 44}$, $C_{\rm 55}$, $C_{\rm 66}$ should be all positive. This result
further proves that conventional GGA formalism fails for describing the
uranium trichalcogenides. Contrarily, the calculated elastic constants by
using the GGA+$U$ method for both US3 and USe3 satisfy the above stability
criteria, as well as the other six criteria for monoclinic structures Nye85 ;
Wu07 , proving the stable existence of US3 and USe3, and their strong electron
correlation effects.
Based on the elastic constants of US3 and USe3, we further calculate the Voigt
and Reuss bounds on their bulk ($B_{\rm V}$, $B_{\rm R}$) and shear moduli
($G_{\rm V}$, $G_{\rm R}$) Supple . And the bulk and shear moduli of US3 and
USe3 can be estimated by $B$=($B_{\rm V}$+$B_{\rm R}$)/2, and $G$=($G_{\rm
V}$+$G_{\rm R}$)/2. And the Poisson’s ratios can be calculated by
$\nu$=(3$B$-2$G$)/(6$B$+2$G$). The obtained mechanical properties for US3 and
USe3 are all listed in Table IV. We can see that the Voigt and Beuss bounds
obviously differ from each other for bulk and shear moduli. This result
reflects the structural anisotropy of US3 and USe3. Besides, the bulk and
shear moduli are both found to be larger for US3 than for USe3. The Poisson’s
ratio is calculated to be 0.19 and 0.22 for US3 and USe3 respectively.
Until now, there are no experimental reports on the mechanical properties of
US3 or USe3. Hence our theoretical values can be used as references for
further investigations or industrial applications of them. At another side,
the Raman properties of the MX3 group materials have been systematically
studied for a long time, and some of the characteristic peaks of US3 and USe3
were successfully observed in experiments Nouvel87 . Therefore, here we
further calculate the Raman properties of US3 and USe3. According to group
theory and the symmetry of them, both US3 and USe3 are predicted to have 12
Raman-active vibrational modes Nouvel87 ; Zwick79 : 8 Ag and 4 Bg modes
respectively. The symbols Ag and Bg represent for two different vibration
symmetries respectively: vibrations inside and out of the mirror plane.
Specifically, Ag corresponds to the vibrations where $dy$ is always 0 while Bg
corresponds to the vibrations where $dy$ is not 0.
Our Raman results of US3 and USe3 are obtained by symmetry analysis based on
the vibrational modes calculated within the GGA+$U$ method, where SOC effects
are considered. Figures 6(a) and 6(b) show our determined Raman frequencies
for US3 and USe3 respectively, together with the experimental results by G.
Nouvel et al. Nouvel87 . One can see that by using the GGA formalism for both
US3 and USe3, the Raman frequencies are totally confused compared with the
experimental results. At another side, the GGA+$U$ results on the Raman
frequencies are clearly reasonable in two obvious aspects: i) there exist a
highest single Raman frequency for both US3 and USe3, corresponding to the
molecular-like X2 mode vibrations; ii) the highest Raman frequency of US3 is
much larger than that of USe3, indicating that the S-S interaction is stronger
than Se-Se. When comparing all the Raman frequencies, there seems to be a
frequency shift of about 25$\sim$35 cm-1 between our GGA+$U$ and the
experimental results.
## Conclusion
By using the GGA and GGA+$U$ methods, we have systematically studied the
electronic, mechanical, and Raman properties of US3 and USe3, aiming to reveal
the underlying electron correlation and SOC effects. By comparing the
calculated lattice constants of US3 and USe3 with corresponding experimental
results, and monitoring their electronic structures with different $U_{\rm
eff}$ parameters, we conclude that a $U_{\rm eff}$ value of 6 eV is needed to
get reasonable lattice constants, and the right insulating properties for both
US3 and USe3. After considering SOC effects, the deep U-6$p$ band will be
split into two separate energy bands in both US3 and USe3. The SOC effects
also changes the energy band shapes around the Fermi energies and lead to the
fact that US3 is a direct band gap while USe3 is an indirect band gap
insulator. Because of the SOC effects, the energy band gaps of US3 and USe3
reduce by 0.11 and 0.07 eV respectively. Based on the obtained
antiferromagnetic ground states of US3 and USe3, we then systematically give
out their electrical, mechanical, and Raman properties. The VBM and CBM of US3
(USe3) are found to be contributed by 3$p$-5$f$ (4$p$-5$f$) hybrid and 5$f$
(5$f$) electronic states respectively. The elastic constants of US3 and USe3
are found to satisfy the stability criteria, and their Poisson’s ratios are
calculated to be 0.19 and 0.22 respectively. For the Raman properties, we
theoretically repeat the two important experimental observations that the S-S
or Se-Se dimer vibrations have the largest frequencies in US3 and USe3, and
the frequency of the S-S dimer vibration is larger than that of the Se-Se
dimer vibration.
## Acknowledgement
This work was supported by the National Natural Science Foundation of China
under Grants No. 10904004 and No. 90921003 and Foundations for Development of
Science and Technology of China Academy of Engineering Physics under Grants
No. 2011B0301060, No. 2011A0301016, and No. 2008A0301013.
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Table 1: Electronic energies per unit cell of US3 and USe3 in the nonmagnetic (NM), ferromagnetic (FM), and antioferromagnetic (AFM) states by using different calculation methods. All values are in units of eV. Methods | US3 | USe3
---|---|---
NM | FM | AFM | NM | FM | AFM
GGA | -55.33 | -55.55 | -55.47 | -50.48 | -50.87 | -50.79
GGA+$U$ ($U_{\rm eff}$=1 eV) | -53.30 | -53.82 | -53.78 | -48.46 | -49.30 | -49.31
GGA+$U$ ($U_{\rm eff}$=2 eV) | -51.34 | -52.50 | -52.24 | -46.57 | -48.14 | -48.17
GGA+$U$ ($U_{\rm eff}$=3 eV) | -49.68 | -51.53 | -51.53 | -45.07 | -47.72 | -47.72
GGA+$U$ ($U_{\rm eff}$=4 eV) | -48.42 | -50.85 | -50.85 | -44.50 | -47.26 | -47.35
GGA+$U$ ($U_{\rm eff}$=5 eV) | -46.55 | -50.74 | -50.74 | -43.95 | -46.79 | -47.26
GGA+$U$ ($U_{\rm eff}$=6 eV) | -48.39 | -51.68 | -51.69 | -43.48 | -46.52 | -46.85
Table 2: Lattice constants of US3 and USe3 obtained by using different
methods. All values are in units of Å.
Methods | US3 | USe3
---|---|---
$a$ | $b$ | $c$ | $a$ | $b$ | $c$
GGA | 5.11 | 3.83 | 9.25 | 5.50 | 3.99 | 10.22
GGA+$U$ ($U_{\rm eff}$=1 eV) | 5.12 | 3.82 | 9.26 | 5.52 | 3.95 | 10.30
GGA+$U$ ($U_{\rm eff}$=2 eV) | 5.13 | 3.82 | 9.28 | 5.52 | 3.94 | 10.42
GGA+$U$ ($U_{\rm eff}$=3 eV) | 5.09 | 3.74 | 9.61 | 5.50 | 3.87 | 10.98
GGA+$U$ ($U_{\rm eff}$=4 eV) | 5.09 | 3.75 | 9.65 | 5.71 | 4.02 | 10.90
GGA+$U$ ($U_{\rm eff}$=5 eV) | 5.06 | 3.82 | 9.61 | 5.73 | 4.04 | 10.52
GGA+$U$ ($U_{\rm eff}$=6 eV) | 5.44 | 3.92 | 9.76 | 5.73 | 4.04 | 10.60
Exp.a,b | 5.37 | 3.96 | 9.94 | 5.65 | 4.06 | 10.47
aReference Gronvold68,
bReference Salem84,
Table 3: Elastic constants of US3 and USe3 obtained in GGA and GGA+$U$+SOC calculations. The $U_{\rm eff}$ value is chosen to be 6 eV in both GGA+$U$+SOC calculations. All data are in units of GPa. | $C_{11}$ | $C_{22}$ | $C_{33}$ | $C_{44}$ | $C_{55}$ | $C_{66}$ | $C_{12}$ | $C_{13}$ | $C_{23}$ | $C_{45}$ | $C_{16}$ | $C_{26}$ | $C_{36}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
US3 (GGA) | -654 | -80 | -4132 | 452 | -332 | -273 | -923 | -2953 | -1766 | 32 | -511 | -640 | -522
US3 (GGA+$U$+SOC) | 963 | 1055 | 437 | 313 | 299 | 169 | 196 | 132 | 181 | -6 | -102 | -55 | -74
USe3 (GGA) | 946 | 583 | -107 | 363 | 309 | 7 | -37 | -349 | -65 | 3 | 171 | 24 | 198
USe3 (GGA+$U$+SOC) | 951 | 938 | 496 | 257 | 244 | 267 | 195 | 194 | 216 | -13 | -19 | -20 | -33
Table 4: Mechanical properties of US3 and USe3 including the bulk and shear moduli, their Voigt and Reuss bounds, and the Poisson’s ratio calculated by using the GGA+$U$ method. The $U_{\rm eff}$ value is chosen to be 6 eV. All moduli data are in units of GPa. | $B_{\rm V}$ | $B_{\rm R}$ | $G_{\rm V}$ | $G_{\rm R}$ | $B$ | $G$ | $\nu$
---|---|---|---|---|---|---|---
US3 | 386 | 299 | 286 | 245 | 342 | 266 | 0.19
USe3 | 400 | 365 | 272 | 257 | 382 | 265 | 0.22
List of captions
Fig.1 (Color online). (a) The atomic structures of MX3, where blue and dark
green balls representing for metal (M) and chalcogen (X) atoms respectively.
The monoclinic lattices are depicted by the dashed lines. (b) Depiction of the
Brillouin Zone for the monoclinic MX3 lattice.
Fig.2 (Color online). The local density of states (LDOS) for the S1, S3, and U
atoms in US3 (a) and Se1, Se2, and U atoms in USe3 (b) by using the GGA and
GGA+$U$ methods with $U_{\rm eff}$ ranging from 1 to 6 eV. The S1 (Se1) and S3
(Se3) atoms correspond to X1 and X3 atoms depicted in Fig. 1(a). The Fermi
energies are denoted by dashed lines.
Fig.3 (Color online). (a)-(d) The projected density of states (PDOS) for the
3$p$ electronic states of S1 and S3 atoms, and 5$f$, 6$p$ electronic states of
the U atom in US3. (e)-(h) The PDOS for the 4$p$ electronic states of Se1 and
Se3 atoms, and 5$f$, 6$p$ electronic states of the U atom in USe3. The S1
(Se1) and S3 (Se3) atoms correspond to X1 and X3 atoms depicted in Fig. 1(a).
The Fermi energies are denoted by dashed lines. The values of $U_{\rm eff}$
are chosen to be 6 eV in all GGA+$U$ and GGA+$U$+SOC calculations.
Fig.4 (Color online). The projected density of states for the uranium 5$f$
electronic states in US3 (a) and USe3 (b), by using the GGA method. The Fermi
energies are set to be zero and denoted by dashed lines. The results before
and after considering the spin-orbit coupling effects are shown in red solid
and blue dotted lines respectively.
Fig.5 (Color online). The electronic energy band structures for US3 (a) and
USe3 (b) by using the GGA+$U$ method. The values of $U_{\rm eff}$ are chosen
to be 6 eV. The Fermi energy and different $k$-points are denoted by the
dashed lines. The results before and after considering the spin-orbit coupling
effects are shown in solid and dotted lines respectively.
Fig.6 (Color online). The calculated frequencies at the $\Gamma$ point for
Raman-active vibrational modes of US3 (a) and USe3 (b) by using the GGA and
GGA+$U$ methods, together with the experimental results listed in Ref.
Nouvel87, . The values of $U_{\rm eff}$ are chosen to be 6 eV. The symmetry
for each vibration mode are also presented.
Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
|
arxiv-papers
| 2012-12-12T00:46:44 |
2024-09-04T02:49:39.160107
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu Yang, Wei Yang, and Ping Zhang",
"submitter": "Yu Yang",
"url": "https://arxiv.org/abs/1212.2682"
}
|
1212.2689
|
# Spinors and the Weyl Tensor Classification in Six Dimensions
Carlos Batista [email protected] Bruno Carneiro da Cunha
[email protected] Departamento de Física, Universidade Federal de Pernambuco,
50670-901 Recife - PE, Brazil
###### Abstract
A spinorial approach to 6-dimensional differential geometry is constructed and
used to analyze tensor fields of low rank, with special attention to the Weyl
tensor. We perform a study similar to the 4-dimensional case, making full use
of the $SO(6)$ symmetry to uncover results not easily seen in the tensorial
approach. Using spinors, we propose a classification of the Weyl tensor by
reinterpreting it as a map from 3-vectors to 3-vectors. This classification is
shown to be intimately related to the integrability of maximally isotropic
subspaces, establishing a natural framework to generalize the Goldberg-Sachs
theorem. We work in complexified spaces, showing that the results for any
signature can be obtained by taking the desired real slice.
Spinors, Six dimensions, Weyl tensor, Isotropic structures, Integrability,
Pure spinors, Goldberg-Sachs theorem, Mariot-Robinson theorem.
## I Introduction
Despite its incontestable importance to Physics, spinors have had difficulties
in being accepted as useful geometric tools in Physical theories, being
replaced by the usual vector approach whenever possible. However there is a
growing consensus that spinorial methods are very useful in a variety of field
theory applications in the cases where there is a priori knowledge that there
is no mass gap developed WardWells . Due to recent developments it became
clear recently that spinor techniques can help fully use of the geometrical
constraints, such as local Poincaré invariance scattering4d-1 . Building from
work in generic solutions for the equations of motion for massless fields of
arbitrary-spin, the perturbation theory for gauge fields can be restructured
by enforcing Poincaré invariance rather than locality of interactions
scattering4d-1 . This helps to reveal the integrability aspects of, for
instance, maximally supersymmetric gauge theories, where one knows the
renormalization group flow to be trivial. In general relativity spinors were
particularly useful in four dimensions to construct explicit solutions of
Einstein’s equation, as well as clearly stating sufficient conditions for
integrability. Despite all the successes, geometric applications of spinors in
higher dimensions is still very much constrained to supersymmetric
applications. Partly this is due to the inherent difficulty in dealing with
generic dimensions consistently in spinorial language, even the basic
definitions can be heavily dependent on the details of the symmetry group. In
this paper we will focus on the six-dimensional case, covering the basic
definitions and expanding on the relationship between differential geometry
and algebraic structures. Nice reviews of spinorial applications in
4-dimensional differential geometry and field theory can be found in
HuggettTod ; WardWells .
In the beginning of sixties Roger Penrose has started to use the spinor
language in four-dimensional general relativity PenroseSpinor ; Penrose . This
approach not only helped the search of new achievements in the field, but also
it was of great importance to understand better previously known results.
Causal structures are at the core of general relativity so it is natural that
spinors prove to be very useful in this subject, specially in problems where
the null directions are relevant. Some examples in which spinorial techniques
are much more elegant and concise than the usual tensor approach are the
Petrov classification and the proof of the Goldberg-Sachs and the Mariot-
Robinson theorems AdvancedGR ; Penrose . The intent of the present article is
to follow some of these steps and introduce the spinor language in six-
dimensional spaces, with the hope that this approach may clarify known results
and pave way for further understanding. It will be shown that not only this
clarification is achieved but also some new results are established here. In
particular it will be shown that this language is very suitable when dealing
with totally null structures, just as happens in four dimensions. Some
previous material about spinors in six dimensions, with field theoretical
applications in mind, can be found in Spinors in 6D . General aspects of
spinors in even dimensions were also used in Kerr-Robinson , where the higher-
dimensional Kerr theorem was investigated.
The Petrov classification is a scheme to classify the Weyl tensor in four
dimensions that was of great importance to the geometrical theory of
gravitation during the second half of last century. It can be used to
judiciously find new solutions to the Einstein’s equation, the main examples
being the Kerr solution Kerr and afterwards all other type $D$ vacuum
solutions typeD - Kinnersley . Of fundamental importance in the applicability
of this classification was the Goldberg-Sachs (GS) theorem Goldberg-Sachs , a
theorem that relates the integrability of maximally isotropic submanifolds and
the associated tangent bundle distribution in 4-dimensional manifolds to the
algebraic form of the Weyl tensor Plebanski2 ; art2 . During the last decade
many efforts have been made towards the creation of a higher-dimensional
version of this classification as well as the GS theorem. In 5D classification
a classification for the Weyl tensor in five dimensions was developed using
spinor techniques and applications were made. A classification scheme valid in
Lorentzian spaces of all dimensions was developed in CMPP , dubbed the CMPP
classification. Posterior work tried, with partial success, to generalize the
GS theorem making connections between the optical matrix and the algebraic
type of the Weyl tensor on this classification Durkee Reall ; M. Ortaggio-GS
theorem ; M. Orataggio- GSII ; M. Ortaggio-Robinson-Trautman . Later a more
geometrical approach towards a higher-dimensional generalization of the GS
theorem was taken in HigherGSisotropic1 ; HigherGSisotropic2 , this path is
based on the maximally isotropic distributions and will be of central
importance here. Due to the relevance of these topics, the present work will
deserve special attention towards the development of an useful classification
for the Weyl tensor and the link between it and a generalized GS theorem. In
six dimensions this was not attempted before using spinorial techniques,
although this is probably the most suitable approach to deal with isotropic
structures, at least in low dimensions.
In section II it will be shown how the tensors of the vector space
$\mathbb{C}\otimes\mathbb{R}^{6}\simeq\mathbb{C}^{6}$ can be expressed in
terms of spinors. Particularly the totally null subspaces will be proven to
have really simple representations in terms of spinorial subjects. Also a
classification scheme for six-dimensional bivectors is developed. Section III
will show how the results obtained in the Euclidian space can be applied to
the other signatures and the way to impose reality conditions in the spinor
formalism will be displayed . After this, section IV deals with the issue of
Weyl tensor algebraic classification from three perspectives: (i) it is
studied the most simple forms that this tensor can take in the spinor
approach, it turns out that these forms are too restrictive an should be of
little use; (ii) the spinor representation is used to show that the Weyl
tensor provides a natural map from self-dual 3-vectors to self-dual 3-vectors.
Such map is used to define a classification scheme for the Weyl tensor and
specially in the Euclidian signature this classification turns out be really
simple, since in this case the map can be diagonalized; (iii) the well known
CMPP classification will be expressed in terms of spinors and used to classify
some previously defined types. Section V will express the integrability
condition for a maximally isotropic distribution, found in reference
HigherGSisotropic2 , in an elegant and geometrical way in terms of pure
spinors. It will also be shown that the integrability of such distributions is
intimately related to the classification of the Weyl tensor based on the map
in the 3-vector space. At the end of the section the generalized Mariot-
Robinson theorem is briefly treated. Finally, section VI apply the results of
this article to two important cases, the Schwarzschild and (a suitable
analogue of) $pp$-wave 6-dimensional space-times. Appendix A presents a
refinement of the Segre classification which will be suitable for our
purposes, appendix B will show how to express specific basis of vectors,
bivectors and 3-vectors in terms of spinors, while appendix C will treat some
details about the six-dimensional Clifford algebra using the index notation.
In this paper the spaces will be assumed to be complexified and it will be
shown how the results on real spaces of arbitrary signature can be easily
extracted from the complex case. Apart from the Lorentzian signature, the pure
Euclidian case is of particular interest to string theory compactifications.
The notion of algebraically special spaces provides a general geometric
setting to study the phenomenon of reduced holonomy, and spinorial tools allow
a direct correspondence to supersymmetry. Also, there is some attention drawn
recently to the case of signature (4,2), which embeds ${\rm AdS}_{5}$
isometrically in six-dimensional flat space. There is hope that the tools
developed here will be suited to deformations of the spaces mentioned above
which still display enough algebraic structure to allow for analytical
investigation.
## II From $SO(6)$ to $SU(4)$
In this section we will present some relevant facts about the spinorial
representation of the $SO(6;\mathbb{R})$ tensors. To this end we shall
remember from the study of Clifford algebras that the universal covering group
of $SO(6;\mathbb{R})$ is $SPin(\mathbb{R}^{6})=SU(4)$ Lounesto . Indeed, the
latter is a double covering of the former, just as $SU(2)$ is a double
covering for $SO(3;\mathbb{R})$. So every tensor of $SO(6;\mathbb{R})$ can be
represented in terms of $SU(4)$ tensors. In an abuse of language, we will call
the latter “spinorial representations” of the $SO(6;\mathbb{R})$ tensors.
The most basic representations of the group $SU(4)$ are the four-dimensional
representations 4 and $\overline{\textbf{4}}$ defined by111Throughout this
article the following indices conventions will be used: $A,B,C,\ldots$ are the
spinorial indices and pertain to $\\{1,2,3,4\\}$; $\mu,\nu,\rho,\ldots$ are
coordinate indices of $\mathbb{R}^{6}$, pertaining to $\\{1,2,\ldots,6\\}$;
$a,b,c,\ldots$ are labels for a null frame of
$\mathbb{C}\otimes\mathbb{R}^{6}$ and take the values $\\{1,2,\ldots,6\\}$;
$i,j,k$ pertain to $\\{1,2,3\\}$; $r,s,t$ label a basis of (anti-)self-dual
3-vectors and runs from 1 to 10; $\varsigma,\upsilon,\kappa,\epsilon$ are
four-dimensional spinor indices and can take the values 1 or 2; $p,q$ label a
basis of Weyl spinors and pertain to $\\{1,2,3,4\\}$.:
$\textbf{4}:\;\;\zeta^{A}\stackrel{{\scriptstyle
U}}{{\longrightarrow}}U^{A}_{\phantom{A}B}\,\zeta^{B}\;\;\;\;\;;\;\;\;\;\;\overline{\textbf{4}}:\;\;\gamma_{A}\stackrel{{\scriptstyle
U}}{{\longrightarrow}}\overline{U}_{A}^{\phantom{A}B}\,\gamma_{B}\;.$
Where $U^{A}_{\phantom{A}B}$ is an unitary $4\times 4$ matrix of unit
determinant whose complex conjugate is $\overline{U}_{A}^{\phantom{A}B}$. A
list of the low dimensional irreducible representations of $SU(4)$ can be
found in Slansky . From now on we shall call the 4-dimensional complex vectors
$\zeta^{A}$ and $\gamma_{A}$ spinors. Note that if $\zeta^{A}$ transforms
according to the representation 4, then its complex conjugate,
$\overline{\zeta^{A}}$, will transform according to the representation
$\overline{\textbf{4}}$. So it is natural to write
$\overline{\zeta^{A}}=\overline{\zeta}_{A}$, that is, complex conjugation
lowers the upper spinor indices and raises the lower spinor indices. It is
also of fundamental importance to note that since $U^{A}_{\phantom{A}B}$ is an
unitary matrix then the contraction of an upper index with a lower index is
invariant by the group $SU(4)$:
$\zeta^{A}\gamma_{A}\;\;\stackrel{{\scriptstyle
U}}{{\longrightarrow}}\;\;\zeta^{A}\gamma_{A}\;,\;\textrm{Invariant by
$SU(4)$}\,.$
Defining $\varepsilon_{ABCD}$ to be the unique completely antisymmetric symbol
such that $\varepsilon_{1234}=1$, it follows that the contraction of it with
four arbitrary spinors, $\zeta^{A},\eta^{A},\varphi^{A}$ and $\xi^{A}$ is also
invariant under $SU(4)$:
$\varepsilon_{ABCD}\zeta^{A}\eta^{B}\varphi^{C}\xi^{D}\;\stackrel{{\scriptstyle
U}}{{\longrightarrow}}\;\det(U)\,\varepsilon_{EFGH}\zeta^{E}\eta^{F}\varphi^{G}\xi^{H}=\varepsilon_{ABCD}\zeta^{A}\eta^{B}\varphi^{C}\xi^{D}.$
(1)
Since a vector of $SO(6;\mathbb{R})$, $V^{\mu}$, has 6 degrees of freedom it
follows that its spinorial equivalent must transform in a six-dimensional
representation of $SU(4)$. The group $SU(4)$ has two representations with this
dimension, the representation $V^{AB}=-V^{BA}$, denoted by 6, and
representation $V_{AB}=-V_{BA}$, denoted by $\overline{\textbf{6}}$. But both
representations are equivalent, since one can be transformed into the other by
means of the symbol $\varepsilon_{ABCD}$. On the same vein, the bivectors of
$SO(6;\mathbb{R})$, $B_{\mu\nu}=-B_{\nu\mu}$, have 15 degrees of freedom, so
they must be in a 15-dimensional representation of $SU(4)$. The representation
15 of $SU(4)$ is given by $B^{A}_{\phantom{A}B}$ with
$B^{A}_{\phantom{A}A}=0$.
This last result about the bivectors establishes that it is possible to
associate to every bivector an operator in the space of spinors with vanishing
trace and whose action is $\chi^{A}\mapsto\chi^{\prime
A}=B^{A}_{\phantom{A}B}\chi^{B}$. So the bivectors of $SO(6;\mathbb{R})$ can
be classified according to the refined Segre type of the $4\times 4$ matrix
representation of this operator (see appendix A). As an example note that if
$B^{A}_{\phantom{A}B}=\chi^{A}\gamma_{B}$ with $\chi^{A}\gamma_{A}=0$ then, in
the notation of appendix A, the type of this bivector is $[|2,1,1]$. It should
be observed that such classification would be quite hard and counterintuitive
without spinors.
To find the spinorial equivalent of the 3-vectors of $SO(6;\mathbb{R})$,
$T_{\mu\nu\rho}=T_{[\mu\nu\rho]}$, is a bit more involved, since 3-vectors
have 20 degrees of freedom and there are a few ways to form a 20-dimensional
representation of $SU(4)$. Once the 3-vectors are obtained from the linear
combination of antisymmetric products of bivectors and vectors it follows from
the above results that we should take a look at the product of the
representations 15 and 6 of $SU(4)$. From Slansky we see that
$\textbf{15}\otimes\textbf{6}=\textbf{6}+\textbf{10}+\overline{\textbf{10}}+\textbf{64}$.
So the 3-vectors must be in the representation
$\textbf{10}+\overline{\textbf{10}}$ of $SU(4)$, where the representation 10
is given by $T^{AB}=T^{BA}$.
It will be of fundamental importance in this article to find the spinor
equivalent of a tensor of $SO(6;\mathbb{R})$ with the symmetries of a Weyl
tensor, $C_{\mu\nu\rho\sigma}=C_{[\mu\nu][\rho\sigma]}=C_{\rho\sigma\mu\nu}$,
$C_{\mu[\nu\rho\sigma]}=0$ and $C^{\mu}_{\phantom{\mu}\nu\mu\sigma}=0$. A
tensor with these symmetries has 84 degrees of freedom, which is the dimension
of the representation
$\Psi^{AB}_{\phantom{AB}CD}=\Psi^{(AB)}_{\phantom{AB}(CD)}$ with
$\Psi^{AB}_{\phantom{AB}CB}=0$. There are other ways to find a 84-dimensional
representations of $SU(4)$ but this is the only one that provides a natural
map of bivectors into bivectors, just as the Weyl tensor does. More explicitly
the equivalent of $B_{\mu\nu}\mapsto
B^{\prime}_{\mu\nu}=C_{\mu\nu\rho\sigma}B^{\rho\sigma}$ is
$B^{A}_{\phantom{A}B}\mapsto B^{\prime
A}_{\phantom{A}B}=\Psi^{AC}_{\phantom{AC}BD}B^{D}_{\phantom{D}C}$, note that
$B^{\prime A}_{\phantom{A}A}=0$, as it should be for a bivector. So the
spinorial equivalent of $C_{\mu\nu\rho\sigma}$ must be
$\Psi^{AB}_{\phantom{AB}CD}$. Table 1 sums up the equivalence relations
between tensors and spinors.
$SO(6)$ Tensor | Spinorial Representation | Symmetries
---|---|---
$V^{\mu}$ | $V^{AB}$ | $V^{AB}=-V^{BA}$
$S_{\mu\nu}$ | $S^{AB}_{\phantom{AB}CD}$ | $S^{AB}_{\phantom{AB}CD}=S^{[AB]}_{\phantom{AB}[CD]},\,S^{AB}_{\phantom{AB}CB}=0$
$B_{\mu\nu}$ | $B^{A}_{\phantom{A}B}$ | $B^{A}_{\phantom{A}A}=0$
$T_{\mu\nu\rho}$ | $(T^{AB},\widetilde{T}_{AB})$ | $T^{AB}=T^{BA},\widetilde{T}_{AB}=\widetilde{T}_{BA}$
$C_{\mu\nu\rho\sigma}$ | $\Psi^{AB}_{\phantom{AB}CD}$ | $\Psi^{AB}_{\phantom{AB}CD}=\Psi^{(AB)}_{\phantom{AB}(CD)},\Psi^{AB}_{\phantom{AB}CB}=0$
Table 1: Spinorial equivalent of $SO(6;\mathbb{R})$ tensors. $V^{\mu}$ is a
vector, $S_{\mu\nu}$ is a trace-less symmetric tensor, $B_{\mu\nu}$ is a
2-vector, $T_{\mu\nu\rho}$ is a 3-vector and $C_{\mu\nu\rho\sigma}$ is a
tensor with the symmetries of a Weyl tensor.
Given two $SO(6)$ vectors, $V_{1}^{\mu}$ and $V_{2}^{\mu}$, it follows that
the scalar product of them, $V_{1}^{\,\mu}V_{2\,\mu}$, is the unique object,
up to a multiplicative factor, invariant under $SO(6)$ and linear in $V_{1}$
and $V_{2}$. Now by equation (1) it follows that
$\varepsilon_{ABCD}V_{1}^{\,AB}V_{2}^{\,CD}$ is invariant by $SU(4)$. Since
this scalar is also linear in $V_{1}$ and $V_{2}$ then it must be proportional
to the scalar product of these vectors. The proportionality constant can be
arbitrarily chosen, in this article the choice will be $1/2$:
$V_{1}^{\,\mu}V_{2\,\mu}=\frac{1}{2}\varepsilon_{ABCD}V_{1}^{\,AB}V_{2}^{\,CD}=V_{1}^{\,AB}V_{2\,CD}\,.$
(2)
Where in the last equality it was defined that the lowering of an
antisymmetric pair of spinorial indices is done by means of the contraction
with $\frac{1}{2}\varepsilon_{ABCD}$. As an example of the use of
$\varepsilon$ let us workout the spinorial representation of a trace-less
symmetric tensor, $S_{\mu\nu}=S_{(\mu\nu)}$ and
$S^{\mu}_{\phantom{\mu}\mu}=0$. This tensor has two indices, so it is obtained
by the tensor product of two vector representations,
$P^{AB\,CD}=P^{[AB]\,[CD]}$. A pair of antisymmetric indices can be lowered by
means of $\frac{1}{2}\varepsilon$, yielding
$P^{AB}_{\phantom{AB}CD}=P^{[AB]}_{\phantom{AB}[CD]}$. This object can now be
split into a scalar, $P^{AB}_{\phantom{AB}AB}$, a bivector,
$(P^{AB}_{\phantom{AB}CB}-\frac{1}{4}\delta^{A}_{C}P^{DB}_{\phantom{DB}DB})$,
and an object $S^{AB}_{\phantom{AB}CD}=S^{[AB]}_{\phantom{AB}[CD]}$ with zero
trace, $S^{AB}_{\phantom{AB}CB}=0$. Since this last object has $20$ degrees of
freedom it must be the spinor equivalent of tensors like $S_{\mu\nu}$. Now it
is time to be more explicit in the relations between the tensors of
$SO(6;\mathbb{R})$ and the tensors of $SU(4)$.
### II.1 The Precise Identifications
Just as in four dimensions the Infeld-van der Waerden symbols,
$\sigma^{a}_{\;\kappa\dot{\epsilon}}$, establish a link between a vector and
its spinorial representation,
$v_{\kappa\dot{\epsilon}}=v_{a}\sigma^{a}_{\;\kappa\dot{\epsilon}}$, here, in
six dimensions, there is an analogous symbol, denoted by
$\Sigma_{\mu}^{\;AB}$. Given a vector of $\mathbb{C}\otimes\mathbb{R}^{6}$,
$V^{\mu}$, its spinorial equivalent is defined by
$V^{AB}=V^{\mu}\Sigma_{\mu}^{\;AB}$. Because of equation (2) these symbols
must obey to the following relation.
$g_{\mu\nu}=\frac{1}{2}\Sigma_{\mu}^{\;AB}\varepsilon_{ABCD}\Sigma_{\nu}^{\;CD}$
Where $g_{\mu\nu}$ is the metric of the vector space. It is useful to define a
null frame for $\mathbb{C}\otimes\mathbb{R}^{6}$,
$\\{e_{1},e_{2},e_{3},e_{4}=\theta^{1},e_{5}=\theta^{2},e_{6}=\theta^{3}\\}$,
defined to be such that the only non-zero inner products are
$g(e_{i},\theta^{j})=\frac{1}{2}\delta_{i}^{\,j}$, in particular all base
vectors are null. In this basis the symbols $\Sigma_{a}^{\;AB}$ can be given
by
$\Sigma_{1}^{\;AB}=\delta_{1}^{[A}\delta_{2}^{B]}\;\;;\;\;\Sigma_{2}^{\;AB}=\delta_{1}^{[A}\delta_{3}^{B]}\;\;;\;\;\Sigma_{3}^{\;AB}=\delta_{1}^{[A}\delta_{4}^{B]}\;\;;\;\;\Sigma_{4}^{\;AB}=\delta_{3}^{[A}\delta_{4}^{B]}\;\;;\;\;\Sigma_{5}^{\;AB}=\delta_{4}^{[A}\delta_{2}^{B]}\;\;;\;\;\Sigma_{6}^{\;AB}=\delta_{2}^{[A}\delta_{3}^{B]}\,.$
(3)
So given a vector $V=V^{a}e_{a}$ then its spinorial equivalent is
$V^{AB}=V^{a}\Sigma_{a}^{\;AB}$. It is also immediate to obtain the spinorial
representation of a trace-less symmetric tensor,
$S^{AB}_{\phantom{AB}CD}=\Sigma_{a}^{\;AB}S^{ab}\Sigma_{b}^{\;EF}\frac{1}{2}\varepsilon_{EFCD}$.
Now one can pose the question of how to convert the other tensors previously
seen to the spinorial language. The answer to this must be worked case by
case.
A bivector, $B_{\mu\nu}$, has two antisymmetric vectorial indices so that it
must admit a spinorial representation $\mathfrak{B}^{AB\,CD}$ such that
$\mathfrak{B}^{AB\,CD}=\mathfrak{B}^{[AB]\,[CD]}=-\mathfrak{B}^{CD\,AB}$.
Thus, for example, if $B=(V_{1}\wedge V_{2})$ then
$\mathfrak{B}^{AB\,CD}=V_{1}^{\,AB}V_{2}^{\,CD}-V_{2}^{\,AB}V_{1}^{\,CD}$.
Table 1 imposes that $\mathfrak{B}^{AB\,CD}$ must be constructed in terms of
$B^{A}_{\phantom{A}B}$. Up to an arbitrary scale factor there is only one way
to do this:
$\mathfrak{B}^{AB\,CD}=B^{[A}_{\phantom{A}E}\varepsilon^{B]ECD}-B^{[C}_{\phantom{C}E}\varepsilon^{D]EAB}\,.$
(4)
Where $\varepsilon^{ABCD}$ is the completely antisymmetric symbol such that
$\varepsilon^{1234}=1$. It is possible to invert the above relation and obtain
$B^{A}_{\phantom{A}B}$ from $\mathfrak{B}^{AB\,CD}$ by a contraction with
$\varepsilon_{ABCD}$:
$B^{A}_{\phantom{A}B}=\frac{1}{4}\mathfrak{B}^{AC\,DE}\varepsilon_{CDEB}\,.$
(5)
Given a bivector $B_{\mu\nu}$ we can obtain its spinorial equivalent by means
of the formula
$\mathfrak{B}^{AB\,CD}=\Sigma_{a}^{\;AB}B^{ab}\Sigma_{b}^{\;CD}$. Subsequent
application of equation (5) then yields $B^{A}_{\phantom{A}B}$.
Now let us work out the 3-vectors. This kind of tensor may be used to map a
vector into a bivector, $V^{\mu}\mapsto T_{\mu\nu\rho}V^{\rho}$, as well to
map a bivector into a vector, $B_{\mu\nu}\mapsto T^{\mu\nu\rho}B_{\nu\rho}$.
These maps must also be realizable in the spinorial language. An object that
provides these maps in the spinorial language has the form
$\tau^{ABC}_{\phantom{ABC}D}=\tau^{A[BC]}_{\phantom{A[BC]}D}$ with
$\tau^{ABC}_{\phantom{ABC}A}=0$, since contracting it with the vector $V_{BC}$
produces a bivector, while contracting it with the bivector
$B^{D}_{\phantom{D}A}$ results in a vector. From table 1 it follows that a
3-vector, $T_{\mu\nu\rho}$, is represented by the pair
$(T^{AB},\widetilde{T}_{AB})$, so from this pair it must be possible to
construct $\tau^{ABC}_{\phantom{ABC}D}$. Up to two arbitrary scale factors
there is only one natural definition:
$\tau^{ABC}_{\phantom{ABC}D}=T^{A[B}\delta_{D}^{C]}+\frac{1}{2}\widetilde{T}_{DE}\varepsilon^{EABC}\,.$
(6)
To know $\tau$ is equivalent to know the pair $(T,\widetilde{T})$, since the
above relation can be easily inverted to give:
$T^{AB}=\frac{2}{3}\tau^{ABC}_{\phantom{ABC}C}\;\;\;;\;\;\;\widetilde{T}_{AB}=\frac{1}{3}\varepsilon_{ACDE}\,\tau^{CDE}_{\phantom{CDE}B}\,.$
(7)
Since $T_{\mu\nu\rho}$ has three completely antisymmetric vector indices it
must admit a spinorial representation of the form
$\mathcal{T}^{AB\,CD\,EF}=\mathcal{T}^{[AB]\,[CD]\,[EF]}$ that is completely
antisymmetric by the permutation between each pair of indices $AB$, $CD$ and
$EF$. The object $\mathcal{T}$ should be constructed from $\tau$. Up to an
arbitrary scale factor there is only one way to do this respecting the
symmetries:
$\mathcal{T}^{AB\,CD\,EF}\;=\;\varepsilon^{GAB[C}\tau^{D]EF}_{\phantom{D]EF}G}\;+\;\varepsilon^{GCD[E}\tau^{F]AB}_{\phantom{F]AB}G}\;+\;\varepsilon^{GEF[A}\tau^{B]CD}_{\phantom{D]CD}G}\;+\\\
\;-\;\varepsilon^{GCD[A}\tau^{B]EF}_{\phantom{B]EF}G}\;-\;\varepsilon^{GEF[C}\tau^{D]AB}_{\phantom{D]AB}G}\;-\;\varepsilon^{GAB[E}\tau^{F]CD}_{\phantom{F]CD}G}\,.$
(8)
The inverse of this relation is:
$\tau^{ABC}_{\phantom{ABC}D}=\frac{1}{12}\mathcal{T}^{CB\,AE\,FG}\,\varepsilon_{EFGD}\,.$
(9)
Once given a 3-vector $T_{abc}$, its spinor equivalent $\mathcal{T}$ is
obtained by contraction with the conversion symbols of equation (3),
$\mathcal{T}^{AB\,CD\,EF}=\Sigma_{a}^{\;AB}\Sigma_{b}^{\;CD}\Sigma_{c}^{\;EF}\,T^{abc}$.
Then equation (9) enables the calculation of $\tau$ and, eventually, equation
(7) yields the pair $(T,\widetilde{T})$.
Just as done above for bivectors and 3-vectors it is important to find the
spinorial representation of the Weyl tensor that is obtained by means of the
conversions symbols of equation (3). Given $C_{\mu\nu\rho\sigma}$ its spinor
equivalent is
$\mathcal{C}^{AB\,CD\,EF\,GH}=\Sigma_{a}^{\;AB}\Sigma_{b}^{\;CD}\Sigma_{c}^{\;EF}\Sigma_{d}^{\;GH}C^{abcd}$.
This spinor equivalent must have the known symmetries of the Weyl tensor and,
because of table 1, should be expressed in terms of
$\Psi^{AB}_{\phantom{AB}CD}$. Up to an arbitrary scale there is only one way
to do this respecting these symmetries:
$\mathcal{C}^{AB\,CD\,EF\,GH}=\Psi^{\mathring{A}\breve{E}}_{\phantom{AE}IJ}\varepsilon^{I\mathring{B}CD}\varepsilon^{J\breve{F}GH}+\Psi^{\mathring{C}\breve{G}}_{\phantom{AE}IJ}\varepsilon^{I\mathring{D}AB}\varepsilon^{J\breve{H}EF}-\Psi^{\mathring{A}\breve{G}}_{\phantom{AE}IJ}\varepsilon^{I\mathring{B}CD}\varepsilon^{J\breve{H}EF}-\Psi^{\mathring{C}\breve{E}}_{\phantom{AE}IJ}\varepsilon^{I\mathring{D}AB}\varepsilon^{J\breve{F}GH}.$
(10)
Where, to simplify the notation, it was introduced the convention that two
indices with $\mathring{}$ or with $\breve{}$ should be anti-symmetrized,
$F^{\mathring{A}\mathring{B}}=F^{AB}-F^{BA}=F^{\breve{A}\breve{B}}$. Laborious
calculations show that $\mathcal{C}^{AB\,CD\,EF\,GH}$ satisfy the expected
constraints, $\mathcal{C}^{AB\,CD\,EF}_{\phantom{AB\,CD\,EF}AB}=0$ and
$\mathcal{C}^{AB\,CD\,EF\,GH}+\mathcal{C}^{AB\,EF\,GH\,CD}+\mathcal{C}^{AB\,GH\,CD\,EF}=0$.
The inverse of equation (10) is given by:
$\Psi^{AB}_{\phantom{AB}CD}=2^{-6}\;\mathcal{C}^{AI\,EF\,GH\,JB}\varepsilon_{IEFC}\,\varepsilon_{GHJD}\,.$
It is also useful to know how the contraction of the Weyl tensor and a
bivector is expressed in terms of spinors. Let
$B^{\prime}_{\mu\nu}=C_{\mu\nu\rho\sigma}B^{\rho\sigma}$. The immediate
analogue of this equation in terms of spinors is $\mathfrak{B}^{\prime
AB\,CD}=\mathcal{C}^{AB\,CD\,EF\,GH}\mathfrak{B}_{EF\,GH}$. Now using
equations (4) and (10) in the right hand side of the last equation and after
this using relation (5) we arrive at the important result:
$B^{\prime}_{\mu\nu}=C_{\mu\nu\rho\sigma}B^{\rho\sigma}\;\;\;\Leftrightarrow\;\;\;B^{\prime
A}_{\phantom{A}C}=-32\Psi^{AB}_{\phantom{AB}CD}\,B^{D}_{\phantom{D}B}\,.$ (11)
With all these tolls at hand we can face the important task of finding how the
isotropic structures are described in terms of spinors. It will be seen that
the spinor language is the most suitable to deal with these mathematical
objects.
### II.2 Isotropic Structures
A subspace of $\mathbb{C}\otimes\mathbb{R}^{6}$ is called isotropic or totally
null when any vector belonging to it has zero norm. In this subsection it will
be shown how the isotropic subspaces of one, two and three dimensions are
expressed in terms of spinors. The final result will show that these subspaces
have very simple spinorial representations.
Let $V^{\mu}$ be a vector tangent to a null direction of
$\mathbb{C}\otimes\mathbb{R}^{6}$, $V^{\mu}V_{\mu}=0$. Now let us find the
spinorial equivalent of the null vector $V$. Note that if
$V^{AB}=\zeta^{[A}\eta^{B]}$ then because of (2) the norm of $V$ is zero.
Conversely if $V$ is a null vector then it is possible to find spinors $\zeta$
and $\eta$ such that $V^{AB}=\zeta^{[A}\eta^{B]}$. To see this inductively
suppose that $V^{AB}=\zeta^{[A}\eta^{B]}+\chi^{[A}\xi^{B]}$, then imposing
that the norm of $V$ is zero by means of (2) we get
$\varepsilon_{ABCD}\zeta^{A}\eta^{B}\chi^{C}\xi^{D}=0$, so that the spinors
$\zeta,\eta,\chi$ and $\xi$ are not all linearly independent. Then, for
example, $\xi=a\zeta+b\eta+c\chi$. Substituting this expansion in the
definition of $V^{AB}$ we find $V^{AB}=(\zeta+b\chi)^{[A}(\eta-a\chi)^{B]}$.
Thus can be concluded that:
$V^{\mu}V_{\mu}=0\;\;\;\Leftrightarrow\;\;\exists\;\;\zeta^{A},\eta^{B}\;\;|\;\;V^{AB}=\zeta^{[A}\eta^{B]}\,.$
(12)
The simple bivector, $B^{\mu\nu}=V_{1}^{\,[\mu}V_{2}^{\,\nu]}$, is said to
generate the 2-dimensional subspace spanned by the vectors $V_{1}$ and
$V_{2}$. This bivector is called null if such subspace is isotropic, which
means that
$V_{1}^{\,\mu}V_{1\,\mu}=V_{1}^{\,\mu}V_{2\,\mu}=V_{2}^{\,\mu}V_{2\,\mu}=0$.
Using equation (12) it is simple matter to see that if $V_{1}$ and $V_{2}$
generate a totally null subspace then it is possible to find spinors
$\zeta,\eta$ and $\chi$ such that $V_{1}^{\,AB}=\zeta^{[A}\eta^{B]}$ and
$V_{2}^{\,AB}=\zeta^{[A}\chi^{B]}$. The spinorial representation of this null
bivector is then
$2\mathfrak{B}^{AB\,CD}=\zeta^{[A}\eta^{B]}\zeta^{[C}\chi^{D]}-\zeta^{[A}\chi^{B]}\zeta^{[C}\eta^{D]}$.
Inserting this relation into equation (5) we find that
$B^{A}_{\phantom{A}B}=\frac{1}{8}\zeta^{A}(\varepsilon_{BCDE}\zeta^{C}\eta^{D}\chi^{E})$.
This result can be put in the following form:
$B_{\mu\nu}\;\;\textrm{is a null
bivector}\;\;\;\Leftrightarrow\;\;\;\exists\;\;\zeta^{A},\gamma_{B}\;\;\textrm{such
that}\;\;\zeta^{A}\gamma_{A}=0\;\;|\;\;B^{A}_{\phantom{A}B}=\zeta^{A}\gamma_{B}\,.$
(13)
The isotropic subspace generated by the null bivector
$B^{A}_{\phantom{A}B}=\zeta^{A}\gamma_{B}$ is the one spanned by the vectors
of the form $V^{AB}=\zeta^{[A}\xi^{B]}$ with $\xi^{A}$ such that
$\xi^{A}\gamma_{A}=0$.
The simple 3-vector
$T^{\mu\nu\rho}=V_{1}^{\,[\mu}V_{2}^{\,\nu}V_{2}^{\,\rho]}$ is said to
generate the 3-dimensional subspace spanned by the vectors $V_{1}$, $V_{2}$
and $V_{3}$. A 3-vector is called null if it is simple and the subspace
generated by it is isotropic. In this case, by what was seen above, one of the
options arise: (i) There exists spinors $\zeta^{A}$, $\eta^{A}$, $\chi^{A}$
and $\xi^{A}$ such that $V_{1}^{\,AB}=\zeta^{[A}\eta^{B]}$,
$V_{2}^{\,AB}=\zeta^{[A}\chi^{B]}$ and $V_{3}^{\,AB}=\zeta^{[A}\xi^{B]}$; (ii)
There exists spinors $\zeta^{A}$, $\eta^{A}$ and $\chi^{A}$ such that
$V_{1}^{\,AB}=\zeta^{[A}\eta^{B]}$, $V_{2}^{\,AB}=\zeta^{[A}\chi^{B]}$ and
$V_{3}^{\,AB}=\eta^{[A}\chi^{B]}$. In the case (i) it can be shown that the
spinorial representation of the 3-vector generated by these null vectors is
$(T^{AB},\widetilde{T}_{AB})\propto(\zeta^{A}\zeta^{B},0)$, while in the case
(ii) we have $(T^{AB},\widetilde{T}_{AB})\propto(0,\gamma_{A}\gamma_{B})$,
with $\gamma_{A}=\varepsilon_{ABCD}\zeta^{B}\eta^{C}\chi^{D}$. So that we can
conclude:
$T_{\mu\nu\rho}\;\;\textrm{is a null
3-vector}\;\;\;\Leftrightarrow\;\;\;(T^{AB},\widetilde{T}_{AB})=\begin{cases}(\zeta^{A}\zeta^{B}\;,\;0)\;\;\textrm{or}\\\
(0\;,\;\gamma_{A}\gamma_{B})\,.\end{cases}$ (14)
If $(T^{AB},\widetilde{T}_{AB})=(\zeta^{A}\zeta^{B},0)$ then the isotropic
subspace generated by this 3-vector is the one spanned by the vectors
$V^{AB}=\zeta^{[A}\xi^{B]}$ for all spinors $\xi^{B}$. While in the case
$(T^{AB},\widetilde{T}_{AB})=(0,\gamma_{A}\gamma_{B})$ the isotropic subspace
is the one spanned by vectors $V^{AB}=\varphi^{[A}\xi^{B]}$ for all
$\varphi^{A}$ and $\xi^{A}$ such that
$\varphi^{A}\gamma_{A}=0=\xi^{A}\gamma_{A}$. Note that in a complexified six-
dimensional space the maximal dimension of an isotropic subspace is three. So
3-vectors like $(e_{1}\wedge e_{2}\wedge e_{3})$ generate maximally isotropic
subspaces, these subspaces will be of fundamental importance throughout this
article.
In general, not only in the null case, if the spinorial equivalent of a
3-vector is such that $\widetilde{T}_{AB}=0$ then the 3-vector is said to be
self-dual, meaning that
$\frac{1}{3!}\epsilon_{\mu\nu\rho\sigma\alpha\beta}T^{\sigma\alpha\beta}=iT_{\mu\nu\rho}$,
where $\epsilon$ is the volume form of the vector space. Analogously, if
$T^{AB}=0$ then the 3-vector is called anti-self-dual and obey to the equation
$\frac{1}{3!}\epsilon_{\mu\nu\rho\sigma\alpha\beta}T^{\sigma\alpha\beta}=-iT_{\mu\nu\rho}$.
One important message of this subsection is that the isotropic structures take
really simple forms in the spinorial language. Note that the simplest form
that a vector can take in the spinorial formalism is
$V^{AB}=\chi^{[A}\eta^{B]}$ and this is exactly the form of a null vector. The
simplest form that a bivector can take in the spinorial language is
$B^{A}_{\phantom{A}B}=\chi^{A}\gamma_{B}$ with $\chi^{A}\gamma_{A}=0$, which
is the form of a null bivector, that is a bivector “tangent” to a totally null
plane. To finish, note that the simplest forms that a 3-vector can take in
spinor formalism are just the ones that represent null 3-vectors.
## III Reality Conditions and the Signatures
A general spinor $\xi^{A}$ has four complex degrees of freedom, which are
acted from the left by the group $SU(4)$. So a general a vector
$V^{AB}=V^{[AB]}$ constructed out of the spinor representation will have six
complex degrees of freedom. If we want to take only the real vectors of
$\mathbb{R}^{6}$ we will just have to impose the reality condition
$\overline{V^{AB}}=V_{AB}$. This condition makes sense from the group point of
view because, as seen before, when the complex conjugate of a spinor is taken
its upper indices must be lowered. Analogously a bivector is real if
$\overline{B^{A}_{\phantom{A}B}}\equiv\overline{B}_{A}^{\phantom{A}B}=B^{B}_{\phantom{B}A}$.
For 3-vectors the reality condition is
$\overline{T^{AB}}\equiv\overline{T}_{AB}=\widetilde{T}_{AB}$, while the Weyl
tensor is real if
$\overline{\Psi^{AB}_{\phantom{AB}CD}}\equiv\overline{\Psi}_{AB}^{\phantom{AB}CD}=\Psi^{CD}_{\phantom{CD}AB}$.
For example, by equation (3) we see that the null vector $e_{1}$ has the
spinorial representation $e_{1}^{\,AB}=\delta_{1}^{[A}\delta_{2}^{B]}$. Let us
see if it is real. In one hand
$\overline{e_{1}^{\,AB}}=\delta_{1}^{[A}\delta_{2}^{B]}=\delta^{1}_{[A}\delta^{2}_{B]}$,
on the other
$e_{1\,AB}\equiv\frac{1}{2}\varepsilon_{ABCD}e_{1}^{\,AB}=\delta^{3}_{[A}\delta^{4}_{B]}$.
So the vector $e_{1}$ is not real, as could be predicted from the fact that in
the Euclidian signature the only vector that is real and null is the zero
vector.
### III.1 Changing the Signature
Up to now only the Euclidian signature was considered, now we can go further
and try to describe the six-dimensional spaces with other signatures in the
spinor language. In this subsection it will be shown that most part of the
above calculations can be carried to this more general situation. The only
important difference is that while in the Euclidian signature it is easy to
impose the reality condition using spinorial objects, in the other signatures
this kind of operation is more laborious.
All the above results were extracted essentially from the fact that
$SPin(\mathbb{R}^{6})=SU(4)$. To analyze the six-dimensional spaces with other
signatures, $\mathbb{R}^{5,1}$, $\mathbb{R}^{4,2}$ and $\mathbb{R}^{3,3}$, it
is useful to first study the space $\mathbb{C}^{6}$ and then choose
conveniently the real slice in order to obtain the wanted signature Trautman ;
art1 . According to Trautman , in the different signatures of a six-
dimensional vector space we must have the following reality conditions on a
null frame $\\{e_{i},\,e_{j+3}=\theta^{j}\\}$ such that the only non-zero
inner products are $g(e_{i},\theta^{j})=\frac{1}{2}\delta_{i}^{j}$:
$\begin{cases}\mathbb{R}^{6}\;\textrm{(Euclidian)}\rightarrow\;\;\overline{e_{1}}=\theta^{1}\;,\;\overline{e_{2}}=\theta^{2}\;,\;\overline{e_{3}}=\theta^{3}\\\
\\\
\mathbb{R}^{5,1}\;\textrm{(Lorentzian)}\rightarrow\;\;\overline{e_{1}}=e_{1}\;,\;\overline{\theta^{1}}=\theta^{1}\;,\;\overline{e_{2}}=\theta^{2}\;,\;\overline{e_{3}}=\theta^{3}\\\
\\\
\mathbb{R}^{4,2}\rightarrow\;\;\begin{cases}\overline{e_{1}}=e_{1}\;,\;\overline{\theta^{1}}=\theta^{1}\;,\;\overline{e_{2}}=e_{2}\;,\;\overline{\theta^{2}}=\theta^{2}\;,\;\overline{e_{3}}=\theta^{3}\\\
\overline{e_{1}}=-\theta^{1}\;,\;\overline{e_{2}}=\theta^{2}\;,\;\overline{e_{3}}=\theta^{3}\end{cases}\\\
\\\ \mathbb{R}^{3,3}\;\textrm{(Split)}\rightarrow\;\;\begin{cases}\textrm{Real
Basis}\\\
\overline{e_{1}}=e_{1}\;,\;\overline{\theta^{1}}=\theta^{1}\;,\;\overline{e_{2}}=-\theta^{2}\;,\;\overline{e_{3}}=\theta^{3}\,.\end{cases}\end{cases}$
(15)
The isometry group of $\mathbb{C}^{6}$ is $SO(6;\mathbb{C})$, so that the
group $SPin(\mathbb{C}^{6})$ is the complexification of the group
$SPin(\mathbb{R}^{6})=SU(4)$, thus $SPin(\mathbb{C}^{6})=SL(4;\mathbb{C})$. So
the tensors of $SO(6;\mathbb{C})$ can be expressed in terms of the
representations of the group $SL(4;\mathbb{C})$. This last group has the
following basic representations:
$\textbf{4}:\;\;\zeta^{A}\stackrel{{\scriptstyle
S}}{{\longrightarrow}}S^{A}_{\phantom{A}B}\,\zeta^{B}\;\;\;\;;\;\;\;\;\widetilde{\textbf{4}}:\;\;\widetilde{\gamma}_{A}\stackrel{{\scriptstyle
S}}{{\longrightarrow}}S^{-1\,B}_{\phantom{-1\,B}A}\,\widetilde{\gamma}_{B}\;\;\;\;;\;\;\;\;\overline{\textbf{4}}:\;\;\gamma_{\dot{A}}\stackrel{{\scriptstyle
S}}{{\longrightarrow}}\overline{S}_{\dot{A}}^{\phantom{A}\dot{B}}\,\gamma_{\dot{B}}\;\;\;;\;\;\;\widetilde{\overline{\textbf{4}}}:\;\;\widetilde{\zeta}^{\dot{A}}\stackrel{{\scriptstyle
S}}{{\longrightarrow}}\overline{S}^{-1\phantom{B}\dot{A}}_{\phantom{-1}\dot{B}}\,\widetilde{\zeta}^{\dot{B}}$
(16)
Where $S^{A}_{\phantom{A}B}$ is a complex $4\times 4$ matrix with unit
determinant, $S^{-1}$ is its inverse, $\overline{S}$ its complex conjugate and
$\overline{S}^{-1}$ is the inverse of $\overline{S}$. The dot over some
indices means that these indices are related to the complex conjugate of the
fundamental representation. For the group $SU(4)$ the dot was not necessary
because there the inverse of a group element is the transpose of the complex
conjugate, so that the representations 4 and
$\widetilde{\overline{\textbf{4}}}$ are equivalent to each other, as well as
the representations $\widetilde{\textbf{4}}$ and $\overline{\textbf{4}}$. But
on the group $SL(4;\mathbb{C})$ such equivalences are not verified. Note also
that the contractions $\zeta^{A}\widetilde{\gamma}_{A}$ and
$\widetilde{\zeta}^{\dot{A}}\gamma_{\dot{A}}$ are invariant under the group
$SL(4;\mathbb{C})$, but the contraction of a dotted index with an index
without dot is not invariant.
Except for the differences concerning the complex conjugation of spinors,
almost everything seen before in the Euclidian signature carries to this more
general situation. For example, the analogue of equation (1) is still valid,
since $\det(S)=1$. Also the associations of table 1 remain the same in the
complex case, but now there is no natural way of imposing reality conditions
in the spinor language. For example, $V^{AB}$ and
$\overline{V^{AB}}=\overline{V}_{\dot{A}\dot{B}}$ can not be directly compared
in the complexified case, since the former is in the representation 6 of
$SL(4;\mathbb{C})$, while the later is in the representation
$\overline{\textbf{6}}$. If a charge conjugation operator is introduced then
it is possible to make such comparison, but this path will not be taken here
because it does not seem to play a role for the classification problem.
Instead the imposition of reality conditions when the signature is not
Euclidian will then be done in the tensor formalism, by means of equation
(15). It should be remembered that a similar phenomena happens in four
dimensions, in which case it is easy to impose reality conditions when the
signature is Lorentzian, but more difficult for the other signatures.
## IV Algebraic Classification of the Weyl Tensor
There are various ways of classifying a tensor field, depending on which use
for the classification one has in mind. In four dimensions it is well known
that the Petrov classification scheme is very useful and many solutions of
Einstein’s equation were found with the help of this classification, the Kerr
solution being the most important example Kerr . From the geometrical point of
view this usefulness stems from the fact that in vacuum when the Weyl tensor
is algebraically special according to the Petrov classification the manifold
admits an integrable distribution of isotropic planes, this is the content of
the Goldberg-Sachs theorem art2 ; Goldberg-Sachs ; Plebanski2 ; Robinson
Manifolds . The intent of this section is to define classifications for the
Weyl tensor in six dimensions that could be related to the integrability of
isotropic structures, i.e., classifications that could be used to generalize,
in some extent, to six dimensions the Goldberg-Sachs theorem.
This is an active area of research and there are two main paths being taken.
The most popular approach is related to the well known CMPP classification
CMPP ; Coley_Review , a scheme to classify tensors by means of the so called
boost weights. Probably one of the biggest advantages of this path is that
such classification is really simple and easy to understand. Some progress
toward the generalization of the Goldberg-Sachs theorem using such
classification was made in Durkee Reall ; Pravda type D ; M. Orataggio- GSII
and references therein, there the optical scalars are at the core of the
treatment and the results are valid only for Lorentzian spaces. The second
approach relates the algebraic form of the Weyl tensor to maximally isotropic
structures HigherGSisotropic1 ; HigherGSisotropic2 , this is the path that
will be taken here. Using this second classification Taghavi-Chabert was able
to develop one interesting and promising generalization of Goldberg-Sachs
theorem, valid for manifolds of arbitrary signature and dimension
HigherGSisotropic2 . A short and nice review of some methods to classify the
Weyl tensor in higher dimensions and its applications can be found in Reall -
Review , while a longer review focusing on the CMPP classification is
available in Coley_Review .
But none of the cited articles made use of spinors in six dimensions, so that
this is the main originality of the present work. As proved in the preceding
sections the spinorial language is the most suitable one to deal with
isotropic structures, so that this article will shed light on useful facts
that were not evident in the tensor formalism. In subsection IV.1 the
symmetries of $\Psi^{AB}_{\phantom{AB}CD}$ will be used to define some
algebraically special forms to the Weyl tensor. Subsection IV.2 will show that
the Weyl tensor can be seen as an operator on the space of the 3-vectors, and
this fact will be used to define a natural generalization of the Petrov
classification. Finally, in IV.3 the boost weight classification will be
described in terms of spinors.
From now on it will be assumed in this paper that the six-dimensional spaces
treated so far are the tangent spaces of a six-dimensional manifold endowed
with a non-degenerate metric $g$. Unless otherwise stated the tangent bundle
will always be assumed to be complexified. The Weyl tensor is now the one
associated to the Levi-Civita connection of the metric $g$ and all results are
local.
### IV.1 Some Algebraically Special Cases
In four dimensions the spinor equivalent of the Weyl tensor is the pair
$(\Psi_{\varsigma\upsilon\kappa\epsilon}\,,\,\widetilde{\Psi}_{\dot{\varsigma}\dot{\upsilon}\dot{\kappa}\dot{\epsilon}})$.
These objects are completely symmetric in its four indices, and since the
indices can only take values 1 and 2 it follows that they can decomposed in
terms of principal spinors. This is how the Petrov classification arises in
the spinor approach Plebanski 1 ; AdvancedGR ; Penrose .
In six dimensions we can follow a similar path. Here the spinorial
representation of the Weyl tensor is $\Psi^{AB}_{\phantom{AB}CD}$, whose
symmetries are $\Psi^{AB}_{\phantom{AB}CD}=\Psi^{(AB)}_{\phantom{AB}(CD)}$ and
$\Psi^{AB}_{\phantom{AB}CB}=0$. Then some simple forms that are certainly
algebraically special are defined in table 2.
$(R,\widetilde{R})\rightarrow\Psi^{AB}_{\phantom{AB}CD}=\chi^{A}\chi^{B}\widetilde{\gamma}_{C}\widetilde{\gamma}_{D}\,;$ | $(R,\widetilde{S})\rightarrow\Psi^{AB}_{\phantom{AB}CD}=\chi^{A}\chi^{B}\widetilde{\gamma}_{(C}\widetilde{\xi}_{D)}$ ; | $(S,\widetilde{S})\rightarrow\Psi^{AB}_{\phantom{AB}CD}=\chi^{(A}\varphi^{B)}\widetilde{\gamma}_{(C}\widetilde{\xi}_{D)}$ ;
---|---|---
$(R,\widetilde{NS})\rightarrow\Psi^{AB}_{\phantom{AB}CD}=\chi^{A}\chi^{B}\widetilde{f}_{CD}$ ; | $(S,\widetilde{NS})\rightarrow\Psi^{AB}_{\phantom{AB}CD}=\chi^{(A}\varphi^{B)}\widetilde{f}_{CD}$ ;
---|---
$(R+\widetilde{R})\rightarrow\Psi^{AB}_{\phantom{AB}CD}=\chi^{A}\chi^{B}\widetilde{\gamma}_{C}\widetilde{\gamma}_{D}+\lambda\gamma^{A}\gamma^{B}\widetilde{\chi}_{C}\widetilde{\chi}_{D}$ ; | $(R\times\widetilde{R})\rightarrow\Psi^{AB}_{\phantom{AB}CD}=(\chi^{A}\chi^{B}+\varphi^{A}\varphi^{B})(\widetilde{\gamma}_{C}\widetilde{\gamma}_{D}+\widetilde{\xi}_{C}\widetilde{\xi}_{D})$.
---|---
Table 2: Some algebraically special types for the Weyl tensor. The labels
comes from: Repeated(R), Simple(S) and Non-Simple(NS). Here
$\chi^{A}\widetilde{\gamma}_{A}=\gamma^{A}\widetilde{\chi}_{A}=\chi^{A}\widetilde{\xi}_{A}=\varphi^{A}\widetilde{\gamma}_{A}=\varphi^{A}\widetilde{\xi}_{A}=\chi^{A}\widetilde{f}_{AB}=\varphi^{A}\widetilde{f}_{AB}=0$,
$\widetilde{f}_{AB}=\widetilde{f}_{BA}$ and
$\chi^{A}\widetilde{\chi}_{A}=\gamma^{A}\widetilde{\gamma}_{A}=1$. The types
$(S,\widetilde{R})$, $(NS,\widetilde{R})$ and $(NS,\widetilde{S})$ can
obviously be defined, as well as many other special types.
Before proceeding it is important to note that if the Weyl tensor is real and
the space has Euclidian signature then most of the special types defined in
table 2 are prohibited. More precisely only type $(R+\widetilde{R})$ is
allowed in this case. For example, suppose that the type is
$(R,\widetilde{NS})$, then the reality condition
$\overline{\Psi}_{AB}^{\phantom{AB}CD}=\Psi^{CD}_{\phantom{CD}AB}$ imposes
that $\widetilde{f}_{AB}=\overline{\chi}_{A}\overline{\chi}_{B}$, so that
$\Psi^{AB}_{\phantom{AB}CD}=\chi^{A}\chi^{B}\overline{\chi}_{C}\overline{\chi}_{D}$,
but this is not compatible with the traceless condition
$\Psi^{AB}_{\phantom{AB}CB}=0$, unless $\chi^{A}=0$. A similar phenomenon
happens in four dimensions, where some algebraic types for the Weyl tensor are
prohibited or allowed depending on the space signature art1 .
Now let us sum up some results concerning the algebraically special types
defined on table 2. To this end the tools of appendix B are used and the
spinor basis is conveniently chosen. Below the non-zero components of the Weyl
tensor will be displayed up to the trivial symmetries of this tensor.
* •
Type $(R,\widetilde{R})$:
$\Psi^{AB}_{\phantom{AB}CD}\propto\chi_{1}^{\,A}\chi_{1}^{\,B}\widetilde{\gamma}^{2}_{\,C}\widetilde{\gamma}^{2}_{\,D}\rightarrow$
The only non-zero component of the Weyl tensor is $C_{5656}$. Note that
because of equation (15) this type is not allowed in Lorentzian signature if
the Weyl tensor is real, since in this case $\overline{C_{5656}}=C_{2323}=0$.
* •
Type $(R,\widetilde{S})$:
$\Psi^{AB}_{\phantom{AB}CD}\propto\chi_{1}^{\,A}\chi_{1}^{\,B}\widetilde{\gamma}^{2}_{\,(C}\widetilde{\gamma}^{3}_{\,D)}\rightarrow$
The only non-zero component of the Weyl tensor is $C_{4656}$. If the Weyl
tensor is real this type is not possible in the Lorentzian signature.
* •
Type $(S,\widetilde{S})$:
$\Psi^{AB}_{\phantom{AB}CD}\propto\chi_{1}^{\,(A}\chi_{2}^{\,B)}\widetilde{\gamma}^{3}_{\,(C}\widetilde{\gamma}^{4}_{\,D)}\rightarrow$
The non-zero components of the Weyl tensor are $C_{5424}=-C_{6434}$. This type
is realizable in the Lorentzian signature even if the Weyl tensor is real. In
this case, using (15) we have $\overline{C_{5424}}=C_{2454}=C_{5424}$ and
$\overline{C_{6434}}=C_{3464}=C_{6434}$.
* •
Type $(R,\widetilde{NS})$:
$\Psi^{AB}_{\phantom{AB}CD}=\chi_{1}^{\,A}\chi_{1}^{\,B}\widetilde{f}_{CD}\rightarrow$
The non-zero components of the Weyl tensor are $C_{4545}$, $C_{4546}$,
$C_{4556}$, $C_{4646}$, $C_{4656}$ and $C_{5656}$. Also using equation (15) it
is easy to see that this type is not allowed when the signature is Lorentzian
and the Weyl tensor is real.
* •
Type $(S,\widetilde{NS})$:
$\Psi^{AB}_{\phantom{AB}CD}=\chi_{1}^{\,(A}\chi_{2}^{\,B)}\widetilde{f}_{CD}\rightarrow$
The non-zero components of the Weyl tensor are $C_{4543}$, $C_{4246}$ and
$C_{4542}=-C_{4643}$. This type is permitted when the Weyl tensor is real and
the signature is Lorentzian, in which case $\overline{C_{4543}}=C_{4246}$,
$\overline{C_{4542}}=C_{4542}$ and $\overline{C_{4643}}=C_{4643}$.
* •
Type $(R+\widetilde{R})$:
$\Psi^{AB}_{\phantom{AB}CD}\propto(\chi_{1}^{\,A}\chi_{1}^{\,B}\widetilde{\gamma}^{2}_{\,C}\widetilde{\gamma}^{2}_{\,D}+\lambda\chi_{2}^{\,A}\chi_{2}^{\,B}\widetilde{\gamma}^{1}_{\,C}\widetilde{\gamma}^{1}_{\,D})\rightarrow$
The non-zero components of the Weyl tensor are $C_{5656}$ and
$C_{2323}=\lambda C_{5656}$. By equation (15) if the Weyl tensor is real and
the signature is Euclidian or Lorentzian then
$\overline{C_{5656}}=C_{2323}=\lambda C_{5656}$, this implies that
$|\lambda|=1$. So this type is allowed in the Euclidian and Lorentzian
signatures for a real Weyl tensor if, and only if, $|\lambda|=1$.
* •
Type $(R\times\widetilde{R})$:
$\Psi^{AB}_{\phantom{AB}CD}\propto(\chi_{1}^{\,A}\chi_{1}^{\,B}+\alpha\,\chi_{2}^{\,A}\chi_{2}^{\,B})(\widetilde{\gamma}^{3}_{\,C}\widetilde{\gamma}^{3}_{\,D}+\lambda\widetilde{\gamma}^{4}_{\,C}\widetilde{\gamma}^{4}_{\,D})\rightarrow$
The non-zero components of the Weyl tensor are $C_{6464}$, $C_{2424}=\lambda
C_{6464}$, $C_{5454}=\alpha C_{6464}$ and $C_{3434}=\alpha\lambda C_{6464}$.
If the Weyl tensor is real and the signature is Lorentzian then equation (15)
imposes that $|\lambda|=|\alpha|=1$. So this type is allowed in the Lorentzian
signature for a real Weyl tensor only if $|\lambda|=|\alpha|=1$.
The main conclusion that stems from the above results is that these
algebraically special types are too special to be useful in general, since
they impose that most of the Weyl tensor components vanish. Although very
restrictive, in some specific situations these types may still have relevance.
For example, these special types can be used to find new solutions to
Einstein’s equation.
### IV.2 A Map from 3-vectors to 3-vectors
In its original form, Petrov classification have come from the analysis of the
Weyl tensor as map from the space of bivectors to the space of bivectors
Petrov ; art1 . This map has the important property of sending (anti-)self-
dual bivectors into (anti-)self-dual bivectors. This splits the Weyl operator
that acts on the 6-dimensional space of bivectors into the direct sum of two
operators that act in 3-dimensional spaces, $C^{\prime}=C^{\prime+}\oplus
C^{\prime-}$, thus restricting enormously the possible algebraic types that
this operator can have. Each of the two operators $C^{\prime+}$ and
$C^{\prime-}$ will have only 6 possible types, the Petrov types. In
particular, the null eigen-bivectors of the Weyl operator generate integrable
isotropic planes when the Ricci tensor vanishes art2 , this is the geometrical
content of the celebrated Goldberg-Sachs theorem. The intent of this section
is to generalize to six dimensions this approach to the Petrov classification
in a natural and geometrical way. This is important because it can lead to a
geometric generalization of the Goldberg-Sachs theorem.
In six dimensions one cannot talk about a self-dual bivector, since the Hodge
dual of a bivector is a 4-vector. But the 3-vectors can be self-dual or anti-
self-dual, as already commented in subsection II.2. Note also that, as proved
in HigherGSisotropic2 , the maximally isotropic subspaces are at the core of
the higher-dimensional generalization of the Goldberg-Sachs theorem. While in
four dimensions these subspaces are generated by null bivectors, in six
dimensions the maximally isotropic subspaces are generated by null 3-vectors.
So it would be interesting if in six dimensions it was possible to see the
Weyl operator as an operator in the space of 3-vectors. It would be of
particular help if the spaces of self-dual and anti-self-dual 3-vectors were
invariant by this operator, since this is the natural analog of what happens
in four dimensions. Indeed we will see in this section that such a map exists
and has the desired property.
In six dimensions the spinor equivalent of a 3-vector is the pair
$(T^{AB},\widetilde{T}_{AB})$, while the spinor equivalent of the Weyl tensor
is $\Psi^{AB}_{\phantom{AB}CD}$. Now it is immediate to see that we can
associate to the Weyl tensor the following operator that sends 3-vectors into
3-vectors222In four dimensions the Weyl tensor is represented by
$(\Psi_{\varsigma\upsilon\kappa\epsilon}\,,\,\widetilde{\Psi}_{\dot{\upsilon}\dot{\upsilon}\dot{\kappa}\dot{\epsilon}})$
and a bivector by
$(\phi_{\alpha\beta},\widetilde{\phi}_{\dot{\alpha}\dot{\beta}})$. The map of
bivectors into bivectors in four dimensions is then
$(\phi_{\varsigma\upsilon},\widetilde{\phi}_{\dot{\varsigma}\dot{\upsilon}})\mapsto(\Psi_{\phantom{\kappa\epsilon}\varsigma\upsilon}^{\kappa\epsilon}\phi_{\kappa\epsilon},\widetilde{\Psi}_{\phantom{\kappa\epsilon}\dot{\varsigma}\dot{\upsilon}}^{\dot{\kappa}\dot{\epsilon}}\widetilde{\phi}_{\dot{\kappa}\dot{\epsilon}})$.
The resemblance with six dimensions is a boon.:
$C:\;\;(T^{AB},\widetilde{T}_{AB})\;\mapsto\;(T^{\prime
AB},\widetilde{T^{\prime}}_{AB})=(\Psi^{AB}_{\phantom{AB}CD}T^{CD}\,,\,\Psi^{CD}_{\phantom{CD}AB}\widetilde{T}_{CD})\,.$
(17)
This map has the important property of preserving the subspaces of self-dual
and anti-self-dual 3-vectors. To see this note that if $\widetilde{T}_{AB}=0$
then $\widetilde{T^{\prime}}_{AB}=0$ and, analogously, if $T^{AB}=0$ then
$T^{\prime AB}=0$. This implies that we can write $C=C^{+}\oplus C^{-}$, where
$C^{+}$ gives zero when acts on anti-self-dual 3-vectors, while $C^{-}$ is
trivial when acts on self-dual 3-vectors. So the $20\times 20$ matrix that
represents $C$ can be split into two blocks $10\times 10$. The map $C^{+}$ is
defined by $T^{AB}\mapsto T^{\prime AB}=\Psi^{AB}_{\phantom{AB}CD}T^{CD}$ and
the operator $C^{-}$ has the action
$\widetilde{T}_{AB}\mapsto\widetilde{T^{\prime}}_{AB}=\Psi^{CD}_{\phantom{CD}AB}\widetilde{T}_{CD}$.
Up to now the resemblance between the map $C^{\prime}$ in four dimensions and
the operator $C$ in six dimensions was striking, but there exists one
fundamental difference between the two cases. While in the former case the
operators $C^{\prime+}$ and $C^{\prime-}$ are independent of each other (in
the general complex case), in six dimensions the operators $C^{+}$ and $C^{-}$
have the same degrees of freedom. To see this explicitly let us make use of
the 3-vector basis introduced in appendix B. If $C^{\pm}_{\;rs}$ are the
matrices that represent the operators $C^{\pm}$ in the space of (anti-)self-
dual 3-vectors then $C^{+}(T_{s})=C^{+}_{\;rs}T_{r}$ and
$C^{-}(\widetilde{T}^{s})=C^{-}_{\;rs}\widetilde{T}^{r}$. Since
$T_{r}^{\;AB}\widetilde{T}^{s}_{\;AB}=\delta^{s}_{r}$ then it follows that
$C^{+}_{\;rs}=\widetilde{T}^{r}_{\;AB}\Psi^{AB}_{\phantom{AB}CD}T_{s}^{\;CD}=C^{-}_{\;sr}$.
Note also that $C^{+}_{\;rr}=\Psi^{AB}_{\phantom{AB}AB}=0$, so the trace of
$C^{\pm}$ is zero. Thus _in six dimensions, for any signature, the operator
$C^{+}$ is the transpose of $C^{-}$ and both operators have vanishing trace_.
So while in four dimensions it makes sense to look for self-dual spaces(the
ones with $C^{\prime-}=0$ and non-trivial $C^{\prime+}$) Plebanski3 , in six
dimensions such concept cannot be introduced, since if $C^{-}=0$ then $C^{+}$
must also vanish.
The here proposed algebraic classification for the Weyl tensor amounts to
compute of the refined Segre type of the operator $C^{+}$ (see appendix A).
This operator is not arbitrary, in particular, as already observed, it must
have zero trace. The traceless condition corresponds to only one of the 16
restrictions imposed by the identity $\Psi^{AB}_{\phantom{AB}CB}=0$. The
remaining 15 conditions cannot be expressed in a form independent of the basis
of 3-vectors, but certainly restrict the possible Segre types that this
operator can have. This is an important point that needs further
clarification. Note also that the algebraic type of $C^{-}$ is always the same
of $C^{+}$, since one operator is the transpose of the other.
In the particular case of the Euclidian signature if the Weyl tensor is real
then it is very simple to see what are the possible algebraic types for
$C^{+}$. If in appendix B we choose the spinor basis to be
$\chi_{p}^{\,A}=\delta_{p}^{A}$ then the complex conjugate, in this signature,
of $T_{r}^{\;AB}$ is $\widetilde{T}^{r}_{\;AB}$ so that
$\overline{C^{+}_{\;rs}}=\overline{\widetilde{T}^{r}_{\;AB}\Psi^{AB}_{\phantom{AB}CD}T_{s}^{\;CD}}=T_{r}^{\;AB}\Psi^{CD}_{\phantom{CD}AB}\widetilde{T}^{s}_{\;CD}=C^{+}_{\;sr}$.
_Thus if the Weyl tensor is real and the signature is Euclidian then the
operator $C^{+}$ is Hermitean and, consequently, can be diagonalized_. Then
the algebraic types of $C^{+}$ depends only on the 10 eigenvalues of this
operator. Using the refined Segre classification defined in appendix A all
numbers inside the square bracket will be one, remaining just the freedom of
where to put the round brackets and of how many eigenvalues are zero.
As an example of the just defined classification note that if the Weyl tensor
is type $(NS,NS)$ or more special,
$\Psi^{AB}_{\phantom{AB}CD}=f^{AB}\widetilde{g}_{CD}\neq 0$, then it is always
possible to define a basis for the space of self-dual 3-vectors,
$\\{\mathbb{T}_{r}^{\;AB}\\}$, such that
$\mathbb{T}_{r}^{\;AB}\widetilde{g}_{AB}=\delta^{2}_{r}$ and
$\mathbb{T}_{1}^{\;AB}=f^{AB}$. In this basis the operator $C^{+}$ has the
following matrix representation:
$[C^{+}]\,=\,\textrm{Blockdiag}(\left[\begin{array}[]{cc}0&1\\\ 0&0\\\
\end{array}\right],0,0,0,0,0,0,0,0)\,.$
The refined Segre classification of this matrix is $[|2,1,1,1,1,1,1,1,1]$. In
particular, all types defined in last subsection but $(R+\widetilde{R})$ are
of the form $\Psi^{AB}_{\phantom{AB}CD}=f^{AB}\widetilde{g}_{CD}$, thus have
this same algebraic classification. It is easy to see that the type
$(R+\widetilde{R})$ with $\lambda\neq 0$ admits a basis in which
$[C^{+}]=\operatorname{diag}(\sqrt{\lambda},-\sqrt{\lambda},0,\ldots,0)$, so
that the algebraic type is $[1,1|1,1,1,1,1,1,1,1]$.
Besides the approach used in the present section there are two other ways to
represent the operator $C$. As seen in subsection II.1, the pair
$(T^{AB},\widetilde{T}_{AB})$ can be equivalently represented by an unique
object, $\tau^{ABC}_{\phantom{ABC}D}$. The map $C$ is given in terms of this
object by $\tau^{ABC}_{\phantom{ABC}D}\mapsto\tau^{\prime
ABC}_{\phantom{ABC}D}=-3\Psi^{AE}_{\phantom{AE}DF}\tau^{FBC}_{\phantom{ABC}E}$.
Note that $\tau^{\prime ABC}_{\phantom{ABC}D}$ has the necessary symmetries.
Since this map deals only with objects that have tensorial equivalents, the
Weyl tensor and the 3-vectors, then it must admit a tensor version. Indeed, in
the tensorial formalism this map is proportional to the following map:
$T_{\mu\nu\alpha}\,\mapsto\,T^{\prime}_{\mu\nu\alpha}=C^{\rho\sigma}_{\phantom{\rho\sigma}[\mu\nu}\,T_{\alpha]\rho\sigma}\,.$
Before moving on it is worth remembering that in six dimensions the Weyl
tensor can also be seen as an operator on the space of bivectors whose action
is $B_{\mu\nu}\mapsto
B^{\prime}_{\mu\nu}=C_{\mu\nu\alpha\beta}B^{\alpha\beta}$. We can
algebraically classify this bivector operator using the Segre classification,
just as we did with the operator $C$, and this can be used to refine and
enhance other forms of classification. An extensive analysis of the bivector
operator in higher dimensions was done in Bivector_Coley . In spinorial
language such map was given on equation (11) and hopefully the spinorial
language could be used enlighten the study of the bivector operator in six
dimensions.
### IV.3 The Boost Weight Classification
In this subsection the well known CMPP classification CMPP ; Coley_Review
will be explained and expressed in terms of spinors. This is important because
this is the simplest and most developed way to classify the Weyl tensor. This
classification is most fruitful and plain in the Lorentzian case, therefore in
the present subsection this signature will be assumed, although at the end
some comments will be made about other signatures.
Once introduced a null frame $\\{e_{i},\theta^{j}\\}$, as defined before, let
us define the boost transformation by:
$e_{1}\mapsto\lambda\,e_{1}\;;\;\;\theta^{1}\mapsto\lambda^{-1}\,\theta^{1}\;;\;\;e_{2}\mapsto
e_{2}\;;\;\;\theta^{2}\mapsto\theta^{2}\;;\;\;e_{3}\mapsto
e_{3}\;;\;\;\theta^{3}\mapsto\theta^{3}\,,$ (18)
with $\lambda$ a real number. Note that equation (15) says that in Lorentzian
signature the vectors $e_{1}$ and $\theta^{1}$ are real, while the others are
complex. This is the way to know which vectors of the null frame should be
transformed by the boost.
A component of a tensor is said to have boost weight $b$ if under the above
transformation it get multiplied by a factor of $\lambda^{b}$. For example,
the component $T_{123}$ of the 3-vector $T_{\mu\nu\rho}$ has boost weight one,
since
$T_{123}=T_{\mu\nu\rho}e_{1}^{\;\mu}e_{2}^{\;\nu}e_{3}^{\;\rho}\stackrel{{\scriptstyle
boost}}{{\longrightarrow}}\lambda\,T_{\mu\nu\rho}e_{1}^{\;\mu}e_{2}^{\;\nu}e_{3}^{\;\rho}$.
It is easy to see that the boost weights of the Weyl tensor components vary
from 2 to $-2$. The CMPP classification then follows by the possibility of
finding a null frame where the components of the Weyl tensor with certain $b$
vanish. Table 3 defines the main CMPP types.
CMPP Type | $O$ | $N$ | $III$ | $D$ | $II$ | $I$
---|---|---|---|---|---|---
Vanishing Components | All | $b=2,1,0,-1$ | $b=2,1,0$ | $b=2,1,-1,-2$ | $b=2,1$ | $b=2$
Table 3: Definition of the CMPP types. The second row says which components of
the Weyl tensor should vanish according to the boost weight, $b$. For example,
when the type is $I$ all components of boost weight two must vanish in some
null frame.
If the null frame $\\{e_{i},\theta^{j}\\}$ is such that all components of the
Weyl tensor with boost weight $b=2$ vanish then the null vector $e_{1}$ is
called a Weyl aligned null direction (WAND). In all dimensions this is
equivalent to $e_{1}$ being a principal null direction according to the Bel-
Debever criteria,
$e_{1\,[\alpha}C_{\mu]\nu\rho[\sigma}e_{1\,\beta]}e_{1}^{\;\nu}e_{1}^{\;\rho}=0$
CMPP-Bel . While in four dimensions there always exists a WAND, in higher
dimensions this is not true. Moreover, in more than four dimensions it is
possible to have a continuum of WANDs Reall - continuous WAND , while in four
dimensions at most four of these directions can exist. If the frame is such
that the components of the Weyl tensor with $b=2$ and $b=1$ vanish, then the
null vector $e_{1}$ is called a multiple WAND. In Durkee Reall it was proved
that a vacuum solution admits a multiple WAND if, and only if, there exists a
multiple WAND that is geodesic. Another interesting property of this
classification is that all known black hole solutions are type $D$.
In terms of the spinor basis defined in appendix B, the boost transformation,
(18), is given by:
$\begin{cases}\chi_{1}\mapsto\sqrt{\lambda}\,\chi_{1}\;;\;\;\chi_{2}\mapsto\sqrt{\lambda}\,\chi_{2}\;;\;\;\chi_{3}\mapsto\frac{1}{\sqrt{\lambda}}\,\chi_{3}\;;\;\;\chi_{4}\mapsto\frac{1}{\sqrt{\lambda}}\,\chi_{4}\\\
\widetilde{\gamma}^{1}\mapsto\frac{1}{\sqrt{\lambda}}\,\widetilde{\gamma}^{1}\;;\;\;\widetilde{\gamma}^{2}\mapsto\frac{1}{\sqrt{\lambda}}\,\widetilde{\gamma}^{2}\;;\;\;\widetilde{\gamma}^{3}\mapsto\sqrt{\lambda}\,\widetilde{\gamma}^{3}\;;\;\;\widetilde{\gamma}^{4}\mapsto\sqrt{\lambda}\,\widetilde{\gamma}^{4}\,.\end{cases}$
(19)
With this at hand it is easy matter to know the boost weight of
$\Psi^{AB}_{\phantom{AB}CD}$ components, next table summarizes this analysis.
Components of $\Psi^{AB}_{\;\;\;\;\;CD}$ with boost weight $b$
---
| $b=2$ | $b=1$ | $b=1$ | $b=0$ | $b=0$ | $b=0$ | $b=-1$ | $b=-1$ | $b=-2$
$AB$ | $33,34,44$ | $13,14,23,24$ | $33,34,44$ | $13,14,23,24$ | $33,34,44$ | $11,12,22$ | $11,12,22$ | $13,14,23,24$ | $11,12,22$
$CD$ | $11,12,22$ | $11,12,22$ | $13,14,23,24$ | $13,14,23,24$ | $33,34,44$ | $11,12,22$ | $13,14,23,24$ | $33,34,44$ | $33,34,44$
Table 4: The boost weight of the various components of
$\Psi^{AB}_{\;\;\;\;\;CD}.$
As examples note that the special types defined in subsection IV.1 are such
that $(S,\widetilde{S})$, $(S,\widetilde{NS})$ and $(R\times\widetilde{R})$
are type $N$ in CMPP classification, $(R+\widetilde{R})$ and
$(R,\widetilde{R})$ are type $D$, $(R,\widetilde{S})$ is type $III$ and
$(R,\widetilde{NS})$ is type $II$. Remember that only types
$(S,\widetilde{S})$, $(S,\widetilde{NS})$, $(R\times\widetilde{R})$ and
$(R+\widetilde{R})$ are allowed when the Weyl tensor is real and the signature
is Lorentzian.
Commonly when working with the CMPP classification it is assumed that the
signature is Lorentzian, this happens because in this case two non-orthogonal
null directions, $e_{1}$ and $\theta^{1}$, can be distinguished in the null
frame by the property of being real. But this classification can also be used
in other non-Euclidian signatures ColeyPSEUD ; Hervik-VSI , moreover in art1
it was argued that it can be extended to Euclidian and complex spaces.
## V Integrability of Maximally Isotropic Subspaces, the Goldberg-Sachs
Theorem
The article HigherGSisotropic2 generalized in a geometrical way the important
Goldberg-Sachs theorem to all dimensions greater than four. This
generalization states that if the Weyl tensor satisfy certain algebraic
conditions then the manifold admits an integrable distribution of maximally
isotropic subspaces. The intent of the present section is to express these
algebraic conditions in terms of spinors, this will prove to be very elegant
and appropriate. It will also be shown that just as the four-dimensional
Goldberg-Sachs theorem is intimately related to the map of bivectors into
bivectors provided by the Weyl tensor art2 , the six-dimensional version of
such theorem is connected to the map of 3-vectors into 3-vectors provided by
the Weyl tensor. Before proceeding it is worth mentioning that very recently
it was investigated in M. Orataggio- GSII the existence of two-dimensional
integrable isotropic structures when the optical matrix is constrained and the
Weyl tensor is type II on the CMPP classification. Integrable null
distributions were also briefly investigated in Hervik-VSI .
Let $N$ be a distribution of vector fields, over the complexified tangent
bundle, that generates maximally isotropic subspaces at every point. The
higher-dimensional version of the Goldberg-Sachs theorem says that in a Ricci-
flat333Actually the theorem is more general and is also valid if certain
components of the Ricci tensor are non-zero. Its original version is expressed
in a conformally invariant way in terms of the Cotton-York tensor
HigherGSisotropic2 . Here the vacuum condition is taken just for simplicity.
complex Riemannian manifold if the Weyl tensor is such that
$C_{\mu\nu\rho\sigma}\,V_{1}^{\mu}V_{2}^{\nu}V_{3}^{\rho}=0$ for all vectors
fields $V_{1}$, $V_{2}$ and $V_{3}$ tangent to $N$ and generic otherwise444In
HigherGSisotropic2 the proof given for this theorem requires that some
generality conditions are satisfied by the Weyl tensor, so the imposition of
“generic otherwise” is certainly sufficient, but it is not clear at all what
is the necessary requirement. For example, in the section 3.4.2 of reference
HigherGSisotropic1 some cases are shown in which the generality assumption
can be relaxed. Also, at section 5.3 of M. Ortaggio-GS theorem it is said
that in five dimensions there exists many cases where such generality
conditions can be neglected if the Ricci identity is used. As such, we will
ignore this requirement in the present discussion. then the distribution $N$
is locally integrable, i.e, an involution under the Lie bracket. The converse
of this theorem is not true, as exemplified in HigherGSisotropic2 . In six
dimensions let us suppose that $N$ is generated by $\\{e_{1},e_{2},e_{3}\\}$
or, equivalently, the distribution is generated by the null 3-vector
$(e_{1}\wedge e_{2}\wedge e_{3})$, then we have:
$\textrm{Integrability Condition for
$N$:}\;\;\;\;C_{ijka}\,=\,0\;\;\;\;\forall\;\;i,j,k\;\in\;\\{1,2,3\\}\;;\;a\;\in\;\\{1,2,\ldots,6\\}\,.$
(20)
To express this condition in terms of spinors it is necessary to use table 6
of appendix B. Using this we get, for example, that $C_{12ja}=0$ is equivalent
to the annihilation of $\Psi^{4A}_{\phantom{4A}11}$ and
$\Psi^{4B}_{\phantom{4B}1C}$ for all $A$ and for all $B\neq 1\neq C$. In
general we have that:
$C_{ijka}\,=\,0\;\;\Leftrightarrow\;\;\begin{cases}\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}\chi_{1}^{\,D}\widetilde{\eta}_{B}=0\\\
\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}\widetilde{\xi}_{A}\widetilde{\eta}_{B}=0\end{cases}\;\;\forall\;\;\widetilde{\xi},\,\widetilde{\eta}\;|\;\chi_{1}^{\,A}\widetilde{\xi}_{A}=\chi_{1}^{\,A}\widetilde{\eta}_{A}=0\,.$
(21)
In particular note that if we make $\chi^{A}=\chi_{1}^{\,A}$ in table 2 then,
except for $(R+\widetilde{R})$, all types defined there obey to the
integrability condition $C_{ijka}=0$, since in these types we have
$\Psi^{AB}_{\phantom{AB}CD}\chi^{C}=0$. Because of the identity
$\Psi^{AB}_{\phantom{AB}CB}=0$ it follows that the first condition on the
right hand side of equation (21) is a consequence of the second one, i.e.,
$C_{ijka}=0\Leftrightarrow\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}=0\;\;\forall\;A\neq
1\neq B$. It is possible to workout a more elegant form to express this
integrability condition using equation (35) of appendix B to arise at the
following result:
$C_{ijka}\,=\,0\;\;\Leftrightarrow\;\;(\,\varepsilon_{AEFG}\,\varepsilon_{BHIJ}\,\Psi^{GJ}_{\phantom{GJ}CD}\,)\,\chi_{1}^{\,A}\,\chi_{1}^{\,B}\,\chi_{1}^{\,C}=0\,.$
(22)
Two other useful relations that can be proved are:
$C_{ijab}\,=\,0\;\;\Leftrightarrow\;\;(\,\varepsilon_{AEFG}\,\Psi^{GJ}_{\phantom{GJ}CD}\,)\chi_{1}^{\,A}\,\chi_{1}^{\,C}=0\;\;\;;\;\;\;C_{iabc}\,=\,0\;\;\Leftrightarrow\;\;(\,\varepsilon_{AEFG}\,\Psi^{GJ}_{\phantom{GJ}CD}\,)\chi_{1}^{\,A}\,=0\,.$
(23)
To understand in which way the above equations are elegant it is necessary to
remember some important concepts about Clifford algebra and spinors. Given a
spinor $\widehat{\psi}$, the null space associated to it,
$N_{\widehat{\psi}}$, is the vector space generated by the vectors that
annihilate this spinor under the clifford action,
$e\,\in\,N_{\widehat{\psi}}\,\Leftrightarrow\,\textbf{e}(\widehat{\psi})=0$
(see appendix C). Because of the Clifford algebra, the vector space
$N_{\widehat{\psi}}$ is always totally null. A spinor is called pure if its
associated null space has the maximal dimension, such correspondence between
maximally isotropic subspaces and pure spinors is one to one up to a scale. It
is well known that in even dimensions less or equal to six all Weyl spinors
are pure, but in higher dimensions this is not true. Now using the conventions
of appendices B and C it is easy to see that the Weyl spinor of positive
chirality $\chi_{1}^{\,A}$ is the pure spinor associated to the maximally
isotropic vector subspace generated by $\\{e_{1},e_{2},e_{3}\\}$. So from what
was said before it follows that _if the Ricci tensor vanishes and the Weyl
tensor satisfy the generality condition referred inHigherGSisotropic2 then_:
$(\,\varepsilon_{AEFG}\,\varepsilon_{BHIJ}\,\Psi^{GJ}_{\phantom{GJ}CD}\,)\,\chi^{\,A}\,\chi^{\,B}\,\chi^{\,C}=0\;\;\;\Rightarrow\;\;\;\textrm{The
subspaces $N_{\widehat{\chi}}$ are locally integrable.}$ (24)
Where $N_{\widehat{\chi}}$ is the distribution of maximally isotropic
subspaces associated to the pure spinor field $\chi^{\,A}$. So this equation
expresses the integrability condition of totally null subspaces of maximal
dimension as an algebraic condition involving the pure spinor field related to
these subspaces and the spinorial equivalent of the Weyl tenor. This is very
similar to what happens in four dimensions, where in vacuum the null planes
associated to a pure spinor $\iota^{\epsilon}$ are integrable if, and only if,
$\Psi_{\varsigma\upsilon\kappa\epsilon}\iota^{\upsilon}\iota^{\kappa}\iota^{\epsilon}=0$
HuggettTod . Equation (24) is valid only for the case when the pure spinor has
positive chirality. In the case of a negative chirality spinor,
$\widetilde{\chi}_{A}$, the algebraic condition on the left hand side of this
equation must be substituted by
$(\varepsilon^{AEFG}\,\varepsilon^{BHIJ}\,\Psi^{CD}_{\phantom{CD}GJ})\,\widetilde{\chi}_{\,A}\widetilde{\chi}_{\,B}\widetilde{\chi}_{\,C}=0\,.$
(25)
As an application of these integrability conditions note from table 4 that, in
a Lorentzian manifold, when the Weyl tensor is type $N$ on the CMPP
classification then
$\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}=\Psi^{AB}_{\phantom{AB}CD}\chi_{2}^{\,C}=0$
and
$\Psi^{AB}_{\phantom{AB}CD}\widetilde{\gamma}^{3}_{\,A}=\Psi^{AB}_{\phantom{AB}CD}\widetilde{\gamma}^{4}_{\,A}$.
In particular this implies, using equations (24) and (25), that the spinors
$\chi_{1}$, $\chi_{2}$, $\widetilde{\gamma}^{3}$ and $\widetilde{\gamma}^{4}$
all obey to the integrability condition. The maximally isotropic distributions
associated to these pure spinors are respectively $\\{e_{1},e_{2},e_{3}\\}$,
$\\{e_{1},e_{5},e_{6}\\}$, $\\{e_{1},e_{5},e_{3}\\}$ and
$\\{e_{1},e_{2},e_{6}\\}$. Although the integrability conditions are satisfied
these distributions are not necessarily integrable, since the theorem of
reference HigherGSisotropic2 assumes that the Ricci tensor obey to certain
restrictions (Ricci-flat, for example) and that the Weyl tensor is generic,
which are not true in general. It would be interesting to investigate what are
the sharp necessary conditions that the Weyl tensor must obey in order to
guarantee the integrability of these four distributions in vacuum type $N$
space-times.
Before moving on some comments are in order. The null subspaces associated to
pure spinors of positive chirality are generated by self-dual 3-vectors, while
the negative chirality spinors are associated to null subspaces generated by
anti-self-dual 3-vectors. The difference between self-dual and anti-self-dual
3-vectors has no intrinsic meaning, since one type of 3-vector can be
converted into the other by a simple change of sign in the volume form of the
manifold. So in general we can assume that a null 3-vector is self-dual. Note
also that although the calculations in this section are focused on the
“integrability of the 3-vector” $(e_{1}\wedge e_{2}\wedge e_{3})$ it must be
kept in mind that given a null 3-vector it is always possible to choose a
frame in which this 3-vector takes the form $(e_{1}\wedge e_{2}\wedge e_{3})$.
As an aside it is worth pointing out that the integrability condition for the
existence of a complex structure on an Euclidian manifold is the existence of
a pure spinor that obeys to the left-hand-side of equation (24) and whose
complex conjugate obeys to equation (25).
### V.1 Integrability Condition and the Map from 3-vectors to 3-vectors -
Complex Case
The integrability condition of equation (21) can be equivalently expressed in
terms of the properties of the operator $C^{+}$ that act on the space of self-
dual 3-vectors. The first thing to note is that if $C_{ijka}=0$ then
$T_{1}^{\;AB}=\chi_{1}^{\,A}\chi_{1}^{\,B}$ is an eigen-3-vector of the
operator $C^{+}$. Indeed, if this integrability condition is satisfied then
equation (21) implies
$\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}\chi_{1}^{\,D}\widetilde{\gamma}^{p}_{\,B}=0$
for $p\neq 1$. This last equation is equivalent to the relation
$\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}\chi_{1}^{\,D}\propto\chi_{1}^{\,A}\chi_{1}^{\,B}$.
Now by means of equations (6), (8) and (33) it is possible to see that the
self-dual 3-vector $T_{1}^{\;AB}=\chi_{1}^{\,A}\chi_{1}^{\,B}$ is proportional
to $(e_{1}\wedge e_{2}\wedge e_{3})$. So we arrived at the following result:
_The integrability condition of the maximally isotropic subspaces generated by
the 3-vector $(e_{1}\wedge e_{2}\wedge e_{3})$, $C_{ijka}=0$, implies that
this 3-vector is an eigen-3-vector of the Weyl operator._ This generalizes the
part of the Goldberg-Sachs theorem stating that in a four-dimensional vacuum
manifold if a maximally isotropic distribution is integrable then the bivector
that generates it is an eigen-bivector of the Weyl tensor art2 .
It is also easy to see that the second condition on the right hand side of
equation (21) is equivalent to say that the subspace generated by the
3-vectors $\\{T_{1},T_{2},T_{3},T_{4}\\}$ defined in appendix B is invariant
by the operator $C^{+}$. So at the end we can conclude that _the integrability
condition $C_{ijka}=0$ is equivalent to imposing that the matrix
representation of operator $C^{+}$ in the basis $\\{T_{r}\\}$ of equation (36)
is of the following form_:
$[C^{+}]=\left[\begin{array}[]{cccccc}c_{11}&c_{12}&c_{13}&c_{14}&\ulcorner\phantom{4\times
6}&\urcorner\\\ 0&\ulcorner&&\urcorner&\quad\quad 4\times 6&\\\ 0&&3\times
3&&\quad\quad\textrm{Block}&\\\
0&\llcorner&&\lrcorner&\llcorner\phantom{4\times 6}&\lrcorner\\\
0&0&0&0&\ulcorner\phantom{4\times 6}&\urcorner\\\
\vdots&\vdots&\vdots&\vdots&\quad\quad 6\times 6&\\\
0&0&0&0&\llcorner\phantom{4\times 6}&\lrcorner\\\ \end{array}\right]$ (26)
### V.2 Integrability Condition and the Map from 3-vectors to 3-vectors -
Euclidian Case
Now let us suppose that the metric is real and with Euclidian signature, in
particular the Weyl tensor is real. In this case it was proved in subsection
IV.2 that the matrix representation of operator $C^{+}$ in the basis
$\\{T_{r}\\}$ is Hermitean. So, by using this property it follows that in this
situation the integrability condition $C_{ijka}=0$ is equivalent to say that
the matrix representation of $C^{+}$ in the basis $\\{T_{r}\\}$ is the one of
equation (26) with $c_{12}=c_{13}=c_{14}=0$ and with the $4\times 6$ block
vanishing. Also the $3\times 3$ and $6\times 6$ blocks must be Hermitean. Note
that it is certainly much more difficult and less geometrical to derive this
kind of result using the tensorial formalism instead of the spinor language.
From the geometrical point of view the annihilation of more components of
$C^{+}$ when the metric is real and the signature is Euclidian happens
because, by equation (15), if the subspace generated by $(e_{1}\wedge
e_{2}\wedge e_{3})$ is integrable then the subspace generated by
$(\overline{e_{1}\wedge e_{2}\wedge
e_{3}})=(\theta^{1}\wedge\theta^{2}\wedge\theta^{3})$ must also be integrable.
### V.3 Integrability Condition and the Map from 3-vectors to 3-vectors -
Lorentzian Case
In this subsection the metric is assumed to be real and Lorentzian, in
particular the Weyl tensor is real. In this case the equation $C_{ijka}=0$
implies that $\overline{C_{ijka}}=C_{\overline{ijka}}=0$. So the integrability
condition for the subspaces generated by $(e_{1}\wedge e_{2}\wedge e_{3})$
imposes the integrability condition for the subspaces generated by
$(\overline{e_{1}\wedge e_{2}\wedge
e_{3}})=(e_{1}\wedge\theta^{2}\wedge\theta^{3})$, where equation (15) was
used. The pure spinor associated to this last family of subspaces is
$\chi_{2}^{\,A}$, as can be verified by means of equations (33) and (39). So
if $C_{ijka}=0$ then the right hand side of equation (21) is also valid if we
put $\chi_{2}$ instead of $\chi_{1}$, thus the 3-vectors
$T_{1}^{\;AB}=\chi_{1}^{\,A}\chi_{1}^{\,B}$ and
$T_{5}^{\;AB}=\chi_{2}^{\,A}\chi_{2}^{\,B}$ are eigen-3-vectors of the
operator $C^{+}$ and the following equations are valid:
$\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}=0\;\;\;,\quad\;\;\;\Psi^{EF}_{\phantom{EF}CD}\chi_{2}^{\,C}=0\quad\quad\forall\;\;A\neq
1\neq B\;,\;E\neq 2\neq F.$ (27)
In particular these equations implies that
$\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}\chi_{2}^{\,D}=0$ for all $AB\neq
1\,2$, that is, $T_{2}^{\;AB}=\sqrt{2}\,\chi_{1}^{\,(A}\chi_{2}^{\,B)}$ is an
eigen-3-vector of $C^{+}$. In short, using the results of subsection V.1, we
conclude that _in Lorentzian signature if $C_{ijka}=0$ then the 3-vectors
$T_{1},T_{2}$ and $T_{5}$ are eigen-3-vectors of $C^{+}$ and the subspaces_
$\textrm{Span}\\{T_{1},T_{2},T_{3},T_{4}\\}$ _and_
$\textrm{Span}\\{T_{2},T_{5},T_{6},T_{7}\\}$ _are invariant under $C^{+}$._
It is interesting to note that in Lorentzian signature the intersection of the
maximally isotropic subspaces generated by $(e_{1}\wedge e_{2}\wedge e_{3})$
and $(\overline{e_{1}\wedge e_{2}\wedge e_{3}})$ defines the real and null
vector field $e_{1}$. Such vector field has special properties, for example,
if $C_{ijka}=0$ it is easy to verify that $e_{1}$ is a Weyl Aligned Null
Direction. Also, it is possible to prove that $e_{1}$ must be geodesic if
these subspaces are integrable Robinson Manifolds , which comes from the fact
that the integral surfaces of an integrable maximally isotropic distribution
are totally geodesic Mason-Chabert-KillingYano . But, differently from what
happens in four dimensions, in general the null congruence generated by
$e_{1}$ is not shear-free555In higher dimensions the shear-free condition is
much more restrictive than in four dimensions. One of the pioneering works to
show this in an specific example was reference FrolovMyers5D , where the
5-dimensional Myers-Perry black hole was studied and it was noted that the
principal null directions (according to the Bel-Debever criteria) are not
shear-free. Thus proving that the Lorentzian Goldberg-Sachs theorem could not
be trivially generalized to higher dimensions..
In vacuum four-dimensional Lorentzian spaces the Petrov type $D$ is
characterized by the existence of four integrable totally null distributions
of dimension two art2 . In order to generalize this concept, reference
HigherGSisotropic1 suggests that a Lorentzian Ricci-flat manifold of
dimension $d=2n$ should be said to have Petrov type $D$ if it admits $2^{n}$
maximally isotropic integrable distributions. Such definition is of relevance
because it was established in Mason-Chabert-KillingYano a relation between
the existence of a conformal Killing-Yano tensor and $2^{n}$ maximally
isotropic integrable distributions. In six dimensions the Petrov type $D$
condition is equivalent to the existence of a null frame such that the
distributions generated by the 3-vectors $(e_{1}\wedge e_{2}\wedge e_{3})$,
$(e_{1}\wedge\theta^{2}\wedge\theta^{3})$, $(\theta^{1}\wedge
e_{2}\wedge\theta^{3})$, $(\theta^{1}\wedge\theta^{2}\wedge e_{3})$,
$(\theta^{1}\wedge\theta^{2}\wedge\theta^{3})$, $(\theta^{1}\wedge e_{2}\wedge
e_{3})$, $(e_{1}\wedge\theta^{2}\wedge e_{3})$ and $(e_{1}\wedge
e_{2}\wedge\theta^{3})$ are all integrable. Since the pure spinors associated
to these distributions are respectively $\chi_{1}^{\,A}$, $\chi_{2}^{\,A}$,
$\chi_{3}^{\,A}$, $\chi_{4}^{\,A}$, $\widetilde{\gamma}^{1}_{\,A}$,
$\widetilde{\gamma}^{2}_{\,A}$, $\widetilde{\gamma}^{3}_{\,A}$ and
$\widetilde{\gamma}^{4}_{\,A}$ it follows that the integrability condition for
these distributions are that spinors $\chi_{1}$, $\chi_{2}$, $\chi_{3}$ and
$\chi_{4}$ obey to the left hand side of equation (24) while spinors
$\widetilde{\gamma}^{1}$, $\widetilde{\gamma}^{2}$, $\widetilde{\gamma}^{3}$
and $\widetilde{\gamma}^{4}$ obey to equation (25). But it was proved in this
subsection that if $\chi_{1}$ and $\chi_{2}$ obey equation (24) then
$T_{1}^{\;AB}=\chi_{1}^{\,A}\chi_{1}^{\,B}$,
$T_{2}^{\;AB}=\sqrt{2}\,\chi_{1}^{\,(A}\chi_{2}^{\,B)}$ and
$T_{5}^{\;AB}=\chi_{2}^{\,A}\chi_{2}^{\,B}$ are eigen-3-vectors of $C^{+}$. So
if $\chi_{1}$, $\chi_{2}$, $\chi_{3}$ and $\chi_{4}$ obey to equation (24)
then all 3-vectors of the base defined in (36) are eigen-3-vectors of the Weyl
operator. Thus we arrived at the result: _In a Ricci-flat Lorentzian manifold
if $C^{+}$ admits a diagonal matrix representation in a basis of the form
defined in (36) then the Weyl tensor is of Petrov type $D$ 666Ignoring the
generality assumption of HigherGSisotropic2 .._ It is interesting to note that
it was not necessary to use the integrability conditions for the distributions
generated by the anti-self-dual 3-vectors
$(\theta^{1}\wedge\theta^{2}\wedge\theta^{3})$, $(\theta^{1}\wedge e_{2}\wedge
e_{3})$, $(e_{1}\wedge\theta^{2}\wedge e_{3})$ and $(e_{1}\wedge
e_{2}\wedge\theta^{3})$ to establish this result.
### V.4 Generalized Mariot-Robinson Theorem
In a space of dimension $d=2n$ a simple $n$-form
$\textbf{N}=V_{1\,\mu_{1}}\textbf{d}x^{\mu_{1}}\wedge\ldots\wedge
V_{n\,\mu_{n}}\textbf{d}x^{\mu_{n}}$ is called null when the distribution
generated by it, Span$\\{V_{1}^{\,\mu_{1}},\ldots,V_{n}^{\,\mu_{n}}\\}$, is
isotropic. The generalized Mariot-Robinson theorem states that in $d=2n$
dimensions the maximally isotropic distribution generated by the null
$n$-vector $\textbf{N}^{\prime}\neq 0$ is integrable if, and only if, there
exists some function $h\neq 0$ such that $\textbf{N}=h\textbf{N}^{\prime}$
satisfies the equations $\textbf{d}\textbf{N}=0$ and
$\textbf{d}(\star\textbf{N})=0$, where d is the exterior derivative and
$\star\textbf{N}$ is the Hodge dual of N. This theorem was first proved in
four-dimensional Lorentzian spaces in Robinson , later it was extended to all
signatures in McIntoshII and finally it was generalized to all signatures and
all even dimensions on reference Kerr-Robinson , using spinor and twistor
calculus.
By this theorem and by what was shown in preceding sections it follows that in
a Ricci-flat six-dimensional manifold we can generally associate the existence
of a null 3-form that is closed and co-closed to the existence of a positive
chirality spinor field that obeys to the left hand side of equation (24), or
to the existence of a negative chirality spinor field obeying to equation
(25). In the former case the null $3$-form is self-dual, while in the later
case the $3$-form is anti-self-dual. The generalized Mariot-Robinson theorem
gives one more hint that while in four dimensions the bivectors are the
relevant objects for the Weyl tensor classification and related matters, in
manifolds of dimension $d=2n$ the $n$-forms are the important mathematical
objects art4 .
## VI Examples
Now let us work out the topics treated in this article on two important
examples for the general relativity theory, the Schwarzschild and a six-
dimensional analogue of the $pp$-wave space-times.
### VI.1 6D Schwarzschild
The six-dimensional Schwarzschild space-time is the spherically symmetric
vacuum solution of Einstein’s equation whose metric is
$\textrm{ds}^{2}=-f^{2}\textrm{dt}^{2}+f^{-2}\textrm{dr}^{2}+r^{2}\\{\textrm{d}\phi_{1}^{2}+\sin^{2}\phi_{1}[\textrm{d}\phi_{2}^{2}+\sin^{2}\phi_{2}\,(\textrm{d}\phi_{3}^{2}+\sin^{2}\phi_{3}\,\textrm{d}\phi_{4}^{2})]\\}\,,$
with $f^{2}=(1-\alpha r^{-3})$. A suitable null frame is defined by:
$\displaystyle
e_{1}=\frac{1}{2}\left(f\partial_{r}+f^{-1}\partial_{t}\right)\;\;;\;\;e_{2}=\frac{1}{2}\left(\frac{1}{r}\partial_{\phi_{1}}+\frac{i}{r\sin\phi_{1}}\partial_{\phi_{2}}\right)\;\;;\;\;e_{3}=\frac{1}{2}\left(\frac{1}{r\sin\phi_{1}\sin\phi_{2}}\partial_{\phi_{3}}+\frac{i}{r\sin\phi_{1}\sin\phi_{2}\sin\phi_{3}}\partial_{\phi_{4}}\right)\;\;;$
$\displaystyle
e_{4}=\frac{1}{2}\left(f\partial_{r}-f^{-1}\partial_{t}\right)\;\;;\;\;e_{5}=\frac{1}{2}\left(\frac{1}{r}\partial_{\phi_{1}}-\frac{i}{r\sin\phi_{1}}\partial_{\phi_{2}}\right)\;\;;\;\;e_{6}=\frac{1}{2}\left(\frac{1}{r\sin\phi_{1}\sin\phi_{2}}\partial_{\phi_{3}}-\frac{i}{r\sin\phi_{1}\sin\phi_{2}\sin\phi_{3}}\partial_{\phi_{4}}\right)\,.$
With these definitions it is straightforward to prove the following
commutation relations:
$\displaystyle[e_{1},e_{2}]=-\frac{f}{2r}e_{2}\;\;;\;\;[e_{1},e_{3}]=-\frac{f}{2r}e_{3}\;\;;\;\;[e_{1},e_{4}]=\frac{3\alpha}{4r^{4}}f^{-1}(e_{1}-e_{4})\;\;;$
$\displaystyle[e_{2},e_{3}]=-\frac{1}{2r}(\cot\phi_{1}+i\frac{\cot\phi_{2}}{\sin\phi_{1}})e_{3}\;\;;\;\;[e_{2},e_{4}]=\frac{f}{2r}e_{2}\;\;;\;\;[e_{2},e_{5}]=\frac{\cot\phi_{1}}{2r}(e_{2}-e_{5})\;\;;$
$\displaystyle[e_{2},e_{6}]=-\frac{1}{2r}(\cot\phi_{1}+i\frac{\cot\phi_{2}}{\sin\phi_{1}})e_{6}\;\;;\;\;[e_{3},e_{4}]=\frac{f}{2r}e_{3}\;\;;\;\;[e_{3},e_{6}]=\frac{\cot\phi_{3}}{2r\sin\phi_{1}\sin\phi_{2}}(e_{3}-e_{6})\,.$
The missing commutators can be obtained from the above equations using the
following reality conditions: $\overline{e_{1}}=e_{1}$,
$\overline{e_{4}}=e_{4}$, $\overline{e_{2}}=e_{5}$ and
$\overline{e_{3}}=e_{6}$. From these commutation relations it is easily seen
that there exists, at least, eight maximally isotropic integrable
distributions in the six-dimensional Schwarzschild space-time, they are:
$\\{e_{1},e_{2},e_{3}\\}$, $\\{e_{1},e_{5},e_{6}\\}$,
$\\{e_{4},e_{2},e_{6}\\}$, $\\{e_{4},e_{5},e_{3}\\}$,
$\\{e_{4},e_{5},e_{6}\\}$, $\\{e_{4},e_{2},e_{3}\\}$,
$\\{e_{1},e_{5},e_{3}\\}$ and $\\{e_{1},e_{2},e_{6}\\}$. So this space-time is
of Petrov type D according to the definition of subsection V.3.
Up to the trivial symmetries the non-zero components of the Weyl tensor are:
$C_{1414}=-\frac{3\alpha}{2r^{5}}\;;\;\;C_{1245}=C_{1346}=C_{1542}=C_{1643}=-\frac{3\alpha}{8r^{5}}\;;\;\;C_{2356}=C_{2552}=C_{2653}=C_{3636}=\frac{\alpha}{4r^{5}}\,.$
From this it is easily concluded that the CMPP type of this space-time is $D$.
Using table 6 of appendix B it can be shown that the spinor equivalent of the
Weyl tensor in the six-dimensional Schwarzschild space-time is:
$\displaystyle\Psi^{AB}_{\phantom{AB}CD}\,=\,-\frac{\alpha}{8r^{5}}[\chi_{1}^{\,A}\chi_{1}^{\,B}\widetilde{\gamma}^{1}_{\,C}\widetilde{\gamma}^{1}_{\,D}+\chi_{2}^{\,A}\chi_{2}^{\,B}\widetilde{\gamma}^{2}_{\,C}\widetilde{\gamma}^{2}_{\,D}+\chi_{3}^{\,A}\chi_{3}^{\,B}\widetilde{\gamma}^{3}_{\,C}\widetilde{\gamma}^{3}_{\,D}+\chi_{4}^{\,A}\chi_{4}^{\,B}\widetilde{\gamma}^{4}_{\,C}\widetilde{\gamma}^{4}_{\,D}]\;+$
$\displaystyle-2\frac{\alpha}{8r^{5}}[\chi_{1}^{\,(A}\chi_{2}^{\,B)}\widetilde{\gamma}^{1}_{\,(C}\widetilde{\gamma}^{2}_{\,D)}+\chi_{3}^{\,(A}\chi_{4}^{\,B)}\widetilde{\gamma}^{3}_{\,(C}\widetilde{\gamma}^{4}_{\,D)}]\;+$
$\displaystyle+3\frac{\alpha}{8r^{5}}[\chi_{1}^{\,(A}\chi_{3}^{\,B)}\widetilde{\gamma}^{1}_{\,(C}\widetilde{\gamma}^{3}_{\,D)}+\chi_{1}^{\,(A}\chi_{4}^{\,B)}\widetilde{\gamma}^{1}_{\,(C}\widetilde{\gamma}^{4}_{\,D)}+\chi_{2}^{\,(A}\chi_{3}^{\,B)}\widetilde{\gamma}^{2}_{\,(C}\widetilde{\gamma}^{3}_{\,D)}+\chi_{2}^{\,(A}\chi_{4}^{\,B)}\widetilde{\gamma}^{2}_{\,(C}\widetilde{\gamma}^{4}_{\,D)}]\,.$
Using this equation it is simple matter to prove that the spinors $\chi_{1}$,
$\chi_{2}$, $\chi_{3}$ and $\chi_{4}$ obey to equation (24), while spinors
$\widetilde{\gamma}^{1}$, $\widetilde{\gamma}^{2}$, $\widetilde{\gamma}^{3}$
and $\widetilde{\gamma}^{4}$ obey to equation (25), so that the integrability
condition is satisfied by eight different pure spinor fields. These pure
spinors are respectively the ones associated to the eight integrable maximally
isotropic distributions listed above. Note also that any spinor of the form
$\kappa=\chi_{1}+h\chi_{2}$, for any function $h$, obeys to the integrability
condition of equation (24), but the maximally isotropic distributions
associated to these pure spinors, $\\{e_{1},(e_{2}+he_{6}),(e_{3}-he_{5})\\}$,
are not integrable for $h\neq 0$. This happens because the Weyl tensor of the
Schwarzschild manifold does not obeys to the generality condition of Taghavi-
Chabert HigherGSisotropic2 .
Actually these eight distributions are not the only integrable maximally
isotropic distributions of the Schwarzschild space-time777The authors thank to
Marcello Ortaggio for drawing our attention to this possibility. Comments on
the same lines can also be found in section 8.3 of M. Orataggio- GSII , where
it was argued that Robinson-Trautman space-times with transverse spaces of
constant curvature admit infinitely many isotropic structures. See also the
footnote in the section 5.2 of reference M. Ortaggio-GS theorem .. Since the
4-sphere has a vanishing Weyl tensor it follows that it is conformally flat.
Thus the Schwarzschild metric can be written as
$\textrm{ds}^{2}=-f^{2}\textrm{dt}^{2}+f^{-2}\textrm{dr}^{2}+r^{2}g(y_{j})[dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2}+dy_{4}^{2}]$.
Taking $n_{1}=a^{i}\partial_{y_{i}}$ and $n_{2}=b^{j}\partial_{y_{j}}$ with
$a^{i}$ and $b^{j}$ constants it is simple matter to note that there are
infinitely many choices of $a^{i}$ and $b^{i}$ that make $\\{n_{1},n_{2}\\}$
an isotropic distribution. It is also immediate to note that such
distributions are integrable, so that the distributions
$\\{e_{1},n_{1},n_{2}\\}$ and $\\{e_{4},n_{1},n_{2}\\}$ are also integrable.
Thus arriving at the result that there exist infinitely many integrable
maximally isotropic distributions on this space-time.
Using the above expression for $\Psi^{AB}_{\phantom{AB}CD}$ it is easy to show
that the matrix representation of the self-dual part of the Weyl operator in
the basis $\\{T_{r}\\}$ defined on equation (36) is:
$[C^{+}]\,=\,-\frac{\alpha}{16r^{5}}\,\textrm{diag}(2,2,-3,-3,2,-3,-3,2,2,2)\,.$
So that the refined Segre classification of this operator is
$[(1,1,1,1,1,1),(1,1,1,1)|]$.
### VI.2 $pp$-Wave
Let us consider now the six-dimensional Kerr-Schild metric
$g_{\mu\nu}=\eta_{\mu\nu}+2Hk_{\mu}k_{\nu}$ with, $\eta_{\mu\nu}$ the metric
of the six dimensional Minkowski space-time and $k$ a null vector field with
respect to $\eta_{\mu\nu}$ such that $\partial_{\mu}k_{\nu}=0$ (the possible
position dependence is chosen to be entirely in $H$). Then assuming that
$k(H)=k^{\mu}\partial_{\mu}H=0$ it follows that the Ricci and the Riemann
tensors are respectively given by:
$R_{\mu\nu}\,=\,-(\eta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}H)k_{\mu}k_{\nu}\quad\quad;\quad\quad
R_{\mu\nu\rho\sigma}=2k_{\sigma}k_{[\mu}\partial_{\nu]}\partial_{\rho}H\,-\,2k_{\rho}k_{[\mu}\partial_{\nu]}\partial_{\sigma}H\,.$
Where $\eta^{\mu\nu}$ is the inverse of $\eta_{\mu\nu}$. If
$\\{x^{0},x^{1},x^{2},x^{3},x^{4},x^{5}\\}$ are cartesian coordinates for the
Minkowski metric then defining the null vector field $k$ to be
$k=\frac{1}{\sqrt{2}}(\partial_{x^{0}}+\partial_{x^{1}})$ and defining the
null coordinates $u=x^{0}-x^{1}$, $v=x^{0}+x^{1}$, $z=x^{2}+ix^{3}$ and
$w=x^{4}+ix^{5}$ then it follows that the metric of this space is given by:
$\textrm{ds}^{2}\,=\,H\textrm{d}u^{2}-\textrm{d}u\textrm{d}v+\textrm{d}z\textrm{d}\overline{z}+\textrm{d}w\textrm{d}\overline{w}$
A natural null frame to define is:
$e_{1}=\partial_{v}\;;\;\;e_{2}=\partial_{z}\;;\;\;e_{3}=\partial_{w}\;;\;\;e_{4}=-(\partial_{u}+H\partial_{v})\;;\;\;e_{5}=\partial_{\overline{z}}\;;\;\;e_{6}=\partial_{\overline{w}}\,.$
(28)
It is immediate to verify that the maximally isotropic distributions
$\\{e_{1},e_{2},e_{3}\\}$, $\\{e_{1},e_{5},e_{6}\\}$,
$\\{e_{1},e_{5},e_{3}\\}$ and $\\{e_{1},e_{2},e_{6}\\}$ are all integrable,
since the relevant commutators vanish. Note also that the vector
$k=\sqrt{2}e_{1}$ is covariantly constant so that the space-time is $pp$-wave
and Kundt Kerr-Schild . Now, for reasons of simplification, let us assume that
$\partial_{u}H=\partial_{w}H=\partial_{\overline{w}}H=\partial_{z}\partial_{\overline{z}}H=0$,
in which case the Ricci tensor vanishes and the only non-zero components of
the Weyl tensor on the null frame of equation (28) are:
$C_{2424}=-\frac{1}{2}\partial_{z}\partial_{z}H\quad\quad;\quad\quad
C_{5454}=-\frac{1}{2}\partial_{\overline{z}}\partial_{\overline{z}}H\,.$
Then the CMPP type of this Weyl tensor is $N$. From table 6 we see that
$C_{2424}=4\Psi^{22}_{\phantom{22}33}$ and
$C_{5454}=4\Psi^{11}_{\phantom{11}44}$, therefore the spinorial equivalent of
the Weyl tensor is:
$\Psi^{AB}_{\phantom{AB}CD}\,=\,\Theta\,\chi_{1}^{\,A}\chi_{1}^{\,B}\widetilde{\gamma}^{4}_{\,C}\widetilde{\gamma}^{4}_{\,D}\,+\,\widetilde{\Theta}\,\chi_{2}^{\,A}\chi_{2}^{\,B}\widetilde{\gamma}^{3}_{\,C}\widetilde{\gamma}^{3}_{\,D}\,.$
Where $\Theta=-\frac{1}{8}\partial_{\overline{z}}\partial_{\overline{z}}H$ and
$\widetilde{\Theta}=-\frac{1}{8}\partial_{z}\partial_{z}H$. Since
$\Psi^{AB}_{\phantom{AB}CD}\chi_{1}^{\,C}=\Psi^{AB}_{\phantom{AB}CD}\chi_{2}^{\,C}=0$
and
$\Psi^{AB}_{\phantom{AB}CD}\widetilde{\gamma}^{3}_{\,A}=\Psi^{AB}_{\phantom{AB}CD}\widetilde{\gamma}^{4}_{\,A}=0$
then it follows that the pure spinors $\chi_{1}$, $\chi_{2}$,
$\widetilde{\gamma}^{3}$ and $\widetilde{\gamma}^{4}$ obey to the
integrability condition of equations (24) and (25)888Remember that the fact
that these four pure spinors obey to the integrability condition is a general
property of the type $N$ space-times, see the paragraph after equation (25)..
Indeed, these are respectively the pure spinors associated to the four
integrable maximally isotropic distributions described right after equation
(28). Now redefining the frame of self-dual 3-vectors of equation (36) by
making the changes $T_{2}\leftrightarrow T_{10}$, $T_{3}\leftrightarrow T_{5}$
and $T_{4}\leftrightarrow T_{8}$ we find the following matrix representation
for the operator $C^{+}$:
$[C^{+}]\;=\;\textrm{Blockdiag}(\left[\begin{array}[]{cc}0&\Theta\\\ 0&0\\\
\end{array}\right],\left[\begin{array}[]{cc}0&\widetilde{\Theta}\\\ 0&0\\\
\end{array}\right],0,0,0,0,0,0)$
From this matrix representation it is seen that all eigenvalues of $C^{+}$
vanish and that the algebraic type of this operator according to the refined
Segre classification is $[|2,2,1,1,1,1,1,1]$.
## VII Conclusions
The basic tools of spinorial formalism in six dimensions were introduced using
index notation and it was shown how to represent low-rank tensors of
$SO(6;\mathbb{C})$ in this language. In particular, the spinorial form of
bivectors opens the possibility to a new algebraic classification for the
bivectors. Also the spinor representation of the Weyl tensor makes clear that
this tensor can be interpreted as a map from self-dual 3-vectors to self-dual
3-vectors and this was used to define an algebraic classification for the Weyl
tensor. Such map takes particular advantage of the relationship between
spinors and isotropic structures. It would be interesting, and hopefully
valuable, to investigate how the different CMPP types constrain this 3-vector
map, just as was done in Hervik-Discri with the bivector map. The approach
throughout this article was to work with complex numbers and when necessary
choose a real slice according to the space signature. In the spinorial
language the imposition of reality conditions was seen to be very simple when
the signature is Euclidian, but it will depend on a non-trivial charge
conjugation operator in the other signatures. A similar thing happens in four
dimensions, in which case the reality conditions are trivial to deal using
spinor formalism only when the signature is Lorentzian.
An important result of the present work is the equation (24), expressing the
integrability condition for a maximally isotropic distribution in terms of an
algebraic condition involving the Weyl “spinor” and the pure spinor associated
to such distribution. This integrability condition was also proved to be
nicely expressed in terms of restrictions on the map of 3-vectors into
3-vectors provided by the Weyl tensor. Particularly, it was proved that in
general if a null 3-vector generates an integrable distribution in a Ricci-
flat manifold then it is an eigen-3-vector of the Weyl operator. The same
result is valid in four dimensions if we put null bivectors instead of null
3-vectors. Finally, the Mariot-Robinson theorem was used to make a link
between the existence of a null $3$-form that is harmonic (closed and co-
closed) and the existence of a pure spinor obeying the integrability
condition.
In four dimensions there is an intimate relationship between the existence of
integrable null distributions and the integrability of Einstein’s equation,
which is somewhat hidden in usual treatments of algebraically special spaces.
The 6-dimensional case helps us state that this seems to be the right way of
thinking about integrability in higher dimensions. Indeed, throughout this
article it was pointed out various similarities between the 4 and the
6-dimensional cases. Thus the present work should have implications in finding
exact solutions for 6-dimensional backgrounds, with relevance for string
theory compactifications. In most cases, the Ricci-flat condition can be
relaxed to “conformally Ricci-flat”, which includes Einstein spaces. Also, the
relationship between algebraically special manifolds and special holonomy is
known and has helped generating new supersymmetric backgrounds
Gauntlett:2004hs . There is however a whole class of solutions which, despite
being non-supersymmetric, do not generate a mass gap, and thus are of interest
to string compactifications. These are obtained as the Wick rotation of
extremal black holes Guica:2008mu . Recently a correspondence has been made
between the extremality and the algebraic character of the Killing horizon
BGCC:ARQ , and this better understanding will surely yield new tools to find
interesting solutions for compactification studies. Lastly, there is the
prospect that a spinorial approach can be helpful on the calculation of
scattering amplitudes, as it was in four dimensions scattering4d-1 ;
scattering4d-2 , by making full use of 6-dimensional Poincaré invariance.
More work in the direction of the present paper is in progress. The next
natural step is to introduce an extension of the Levi-Civita connection in the
spinor bundle. This can help to understand better the subtleties of the
generalized Goldberg-Sachs theorem introduced by Taghavi-Chabert
HigherGSisotropic2 , in particular this approach can elucidate when the
generality assumption on the Weyl tensor is necessary. Moreover the spinorial
techniques can help the obtention of a link between integrability of isotropic
structures, not necessarily maximal, and restrictions on the optical matrix
(shear, twist and expansion). It is also important to further investigate the
classification scheme for the Weyl tensor defined in subsection IV.2 as well
as its relation with the optical matrix.
## Acknowledgments
Carlos Batista thanks to CNPq (Conselho Nacional de Desenvolvimento Científico
e Tecnológico) for the financial support.
## Appendix A Refined Segre Classification
The Segre classification is a widely known form to classify square matrices
using the Jordan canonical form Segre Type . In this appendix such
classification will be explained and refined.
By means of a similarity transformation every square matrix over the complex
field can be put in the so called Jordan canonical form:
$[M]\rightarrow[M]_{J}\,=[BMB^{-1}]\,=\,\textrm{Blockdiag}(J_{1},J_{2},\ldots,J_{n})\;\;\;;\;\;J_{k}=\left[\begin{array}[]{cccc}\lambda_{k}&1&0&\ldots\\\
0&\lambda_{k}&1&\\\ \vdots&&\ddots&1\\\ 0&\ldots&0&\lambda_{k}\\\
\end{array}\right]\;\;\;\textrm{or}\;\;J_{k}=\lambda_{k}.$ (29)
Where $\lambda_{k}$ is a complex number. The matrices $J_{k}$ are called the
Jordan blocks of $M$. Each block $J_{k}$ admits just one eigenvector and its
eigenvalue is $\lambda_{k}$. The Jordan canonical form of a matrix is unique
up to the ordering of the Jordan blocks.
Now given a square matrix $M$, its Segre type is a sequence of numbers inside
a square bracket, each number denoting the dimension of a Jordan block of this
matrix. The numbers associated to Jordan blocks with the same eigenvalue are
put inside a round bracket. For example, the matrix
$[M]_{J}\,=\,\left[\begin{array}[]{ccccc}\lambda&1&0&0&0\\\ 0&\lambda&0&0&0\\\
0&0&\alpha&0&0\\\ 0&0&0&\beta&1\\\ 0&0&0&0&\beta\\\ \end{array}\right]$
can have the following Segre types: (i) If
$\lambda\neq\alpha\neq\beta\neq\lambda$ then the type is $[2,2,1]$; (ii) If
$\lambda=\alpha\neq\beta$ or $\lambda\neq\alpha=\beta$ the Segre type is
$[(2,1),2]$; (iii) If $\lambda=\beta\neq\alpha$ then the type is [(2,2),1];
(iv) If $\lambda=\alpha=\beta$ the type of this matrix is $[(2,2,1)]$.
It is useful to refine this classification by making explicit which Jordan
blocks have zero eigenvalue. This can be done by positioning, in the Segre
classification, the numbers related to the zero eigenvalues on the right of
the other numbers, separating by a vertical bar. As an illustration suppose
that in the last example we have $\lambda=0$, then the matrix can take the
following types in this refined classification: (i) If
$\lambda\neq\alpha\neq\beta\neq\lambda$ then the type is $[2,1|2]$; (ii) If
$\lambda=\alpha\neq\beta$ the types is $[2|(2,1)]\equiv[2|2,1]$; (ii’) If
$\lambda\neq\alpha=\beta$ then the refined Segre type is $[(2,1)|2]$; (iii) If
$\lambda=\beta\neq\alpha$ then the type is [1|2,2]; (iv) If
$\lambda=\alpha=\beta$, all eigenvalues vanish and the type of the above
matrix is $[|(2,2,1)]\equiv[|2,2,1]$.
## Appendix B A Basis for the Spinor Space
The space of (Weyl)spinors of $\mathbb{C}^{6}$ is the vector space where the
representation 4 of $SL(4;\mathbb{C})$ act, this is a four-dimensional complex
space. Let $\\{\chi_{1}^{\,A},\chi_{2}^{\,A},\chi_{3}^{\,A},\chi_{4}^{\,A}\\}$
be a basis for this space and let us choose the following normalization
condition:
$\varepsilon_{ABCD}\,\chi_{1}^{\,A}\chi_{2}^{\,B}\chi_{3}^{\,C}\chi_{4}^{\,D}\,=\,1$
(30)
The dual basis is then given by:
$\widetilde{\gamma}^{1}_{\,A}=\varepsilon_{ABCD}\chi_{2}^{\,B}\chi_{3}^{\,C}\chi_{4}^{\,D}\;\;;\;\;\widetilde{\gamma}^{2}_{\,A}=-\,\varepsilon_{ABCD}\chi_{1}^{\,B}\chi_{3}^{\,C}\chi_{4}^{\,D}\;\;;\;\;\widetilde{\gamma}^{3}_{\,A}=\varepsilon_{ABCD}\chi_{1}^{\,B}\chi_{2}^{\,C}\chi_{4}^{\,D}\;\;;\;\;\widetilde{\gamma}^{4}_{\,A}=-\,\varepsilon_{ABCD}\chi_{1}^{\,B}\chi_{2}^{\,C}\chi_{3}^{\,D}$
(31)
The elements of the dual basis transforms according to the representation
$\widetilde{\textbf{4}}$ of $SL(4;\mathbb{C})$ and obey to the important
relation $\chi_{p}^{\,A}\widetilde{\gamma}^{q}_{\,A}=\delta^{q}_{p}$. Now we
can express a null frame, $\\{e_{i},e_{j+3}=\theta^{j}\\}$, in terms of the
spinor basis:
$e_{1}^{\,AB}=\chi_{1}^{\,[A}\chi_{2}^{\,B]}\;;\;\;e_{2}^{\,AB}=\chi_{1}^{\,[A}\chi_{3}^{\,B]}\;;\;\;e_{3}^{\,AB}=\chi_{1}^{\,[A}\chi_{4}^{\,B]}\;;\;\;\theta^{1\,AB}=\chi_{3}^{\,[A}\chi_{4}^{\,B]}\;;\;\;\theta^{2\,AB}=\chi_{4}^{\,[A}\chi_{2}^{\,B]}\;;\;\;\theta^{3\,AB}=\chi_{2}^{\,[A}\chi_{3}^{\,B]}$
(32)
Lowering the pair of spinorial indices of this basis by means of the relation
$V_{AB}\equiv\frac{1}{2}\varepsilon_{ABCD}V^{CD}$ it is seen that:
$e_{1\,AB}=\widetilde{\gamma}^{3}_{\,[A}\widetilde{\gamma}^{4}_{\,B]}\;;\;\;e_{2\,AB}=\widetilde{\gamma}^{4}_{\,[A}\widetilde{\gamma}^{2}_{\,B]}\;;\;\;e_{3\,AB}=\widetilde{\gamma}^{2}_{\,[A}\widetilde{\gamma}^{3}_{\,B]}\;;\;\;\theta^{1}_{\,AB}=\widetilde{\gamma}^{1}_{\,[A}\widetilde{\gamma}^{2}_{\,B]}\;;\;\;\theta^{2}_{\,AB}=\widetilde{\gamma}^{1}_{\,[A}\widetilde{\gamma}^{3}_{\,B]}\;;\;\;\theta^{3}_{\,AB}=\widetilde{\gamma}^{1}_{\,[A}\widetilde{\gamma}^{4}_{\,B]}$
(33)
Using equations (32) and (33) it is immediate to verify that the inner
products are in accordance with the ones of a null basis,
$e_{i}^{\,AB}e_{j\,AB}=0=\theta^{i\,AB}\theta^{j}_{\,AB}$ and
$e_{i}^{\,AB}\theta^{j}_{\,AB}=\frac{1}{2}\delta^{j}_{i}$. Note that the
specific representation of equation (3) is obtained by taking
$\chi_{p}^{\,A}=\delta_{p}^{A}$.
As previously seen, in the spinorial formalism a bivector
$B_{\mu\nu}=B_{[\mu\nu]}$ is equivalent to $B^{A}_{\phantom{A}B}$ with
$B^{A}_{\phantom{A}A}=0$. Defining $(e_{a}\wedge e_{b})\equiv(e_{a}\otimes
e_{b}-e_{b}\otimes e_{a})$ and using equation (5) it is possible to find the
spinorial representation of a bivector basis, the final result is summarized
in table 5. Now given two bivectors, $B^{\prime}$ and $B$, it follows, by
definition, that
$B^{\prime}_{\mu\nu}B^{\mu\nu}=\mathfrak{B}^{\prime}_{AB\,CD}\mathfrak{B}^{AB\,CD}$,
where $\mathfrak{B}$ was defined on subsection II.1. By means of equation (4)
it is simple matter then to prove that
$B^{\prime}_{\mu\nu}B^{\mu\nu}=-8B^{\prime
A}_{\phantom{A}B}B^{B}_{\phantom{B}A}$. Then using this result and equation
(11) we arrive at the following important equation:
$C_{abcd}=2^{6}\,(e_{a}\wedge
e_{b})^{A}_{\phantom{A}B}\,\Psi^{BC}_{\phantom{BC}AD}\,(e_{c}\wedge
e_{d})^{D}_{\phantom{D}C}\,.$ (34)
Table 5 together with the above equation enables us to express the components
of the Weyl tensor, $C_{abcd}$, in terms of the components of
$\Psi^{AB}_{\phantom{AB}CD}$. For example, $C_{1234}=2^{6}(e_{1}\wedge
e_{2})^{A}_{\phantom{A}B}\,\Psi^{BC}_{\phantom{BC}AD}\,(e_{3}\wedge\theta^{1})^{D}_{\phantom{D}C}=4\chi_{1}^{\,A}\widetilde{\gamma}^{4}_{\,B}\Psi^{BC}_{\phantom{BC}AD}\chi_{4}^{\,D}\widetilde{\gamma}^{2}_{\,C}=4\Psi^{42}_{\phantom{42}14}$.
The explicit form of all Weyl tensor’s components in a null frame in terms of
spinors is given in table 6.
$(e_{1}\wedge e_{2})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{1}^{\,A}\widetilde{\gamma}^{4}_{\,B}$ ; | $(e_{1}\wedge e_{3})^{A}_{\phantom{A}B}=\frac{1}{4}\chi_{1}^{\,A}\widetilde{\gamma}^{3}_{\,B}$ ; | $(e_{1}\wedge\theta^{2})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{2}^{\,A}\widetilde{\gamma}^{3}_{\,B}$ ; | $(e_{1}\wedge\theta^{3})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{2}^{\,A}\widetilde{\gamma}^{4}_{\,B}$ ;
---|---|---|---
$(e_{2}\wedge e_{3})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{1}^{\,A}\widetilde{\gamma}^{2}_{\,B}$, | $(e_{2}\wedge\theta^{1})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{3}^{\,A}\widetilde{\gamma}^{2}_{\,B}$ ; | $(e_{2}\wedge\theta^{3})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{3}^{\,A}\widetilde{\gamma}^{4}_{\,B}$ ; | $(e_{3}\wedge\theta^{1})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{4}^{\,A}\widetilde{\gamma}^{2}_{\,B}$ ;
$(e_{3}\wedge\theta^{2})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{4}^{\,A}\widetilde{\gamma}^{3}_{\,B}$ ; | $(\theta^{1}\wedge\theta^{2})^{A}_{\phantom{A}B}=\frac{1}{4}\chi_{4}^{\,A}\widetilde{\gamma}^{1}_{\,B}$ ; | $(\theta^{1}\wedge\theta^{3})^{A}_{\phantom{A}B}=-\frac{1}{4}\chi_{3}^{\,A}\widetilde{\gamma}^{1}_{\,B}$ ; | $(\theta^{2}\wedge\theta^{3})^{A}_{\phantom{A}B}=\frac{1}{4}\chi_{2}^{\,A}\widetilde{\gamma}^{1}_{\,B}$ ;
$(e_{1}\wedge\theta^{1})^{A}_{\phantom{A}B}=\frac{1}{8}[-\chi_{1}^{\,A}\widetilde{\gamma}^{1}_{\,B}-\chi_{2}^{\,A}\widetilde{\gamma}^{2}_{\,B}+\chi_{3}^{\,A}\widetilde{\gamma}^{3}_{\,B}+\chi_{4}^{\,A}\widetilde{\gamma}^{4}_{\,B}]$
;
$(e_{2}\wedge\theta^{2})^{A}_{\phantom{A}B}=\frac{1}{8}[-\chi_{1}^{\,A}\widetilde{\gamma}^{1}_{\,B}+\chi_{2}^{\,A}\widetilde{\gamma}^{2}_{\,B}-\chi_{3}^{\,A}\widetilde{\gamma}^{3}_{\,B}+\chi_{4}^{\,A}\widetilde{\gamma}^{4}_{\,B}]$
;
$(e_{3}\wedge\theta^{3})^{A}_{\phantom{A}B}=\frac{1}{8}[-\chi_{1}^{\,A}\widetilde{\gamma}^{1}_{\,B}+\chi_{2}^{\,A}\widetilde{\gamma}^{2}_{\,B}+\chi_{3}^{\,A}\widetilde{\gamma}^{3}_{\,B}-\chi_{4}^{\,A}\widetilde{\gamma}^{4}_{\,B}]$
;
$\chi_{1}^{\,A}\widetilde{\gamma}^{1}_{\,B}=2[\frac{1}{8}\delta^{A}_{B}-(e_{1}\wedge\theta^{1})^{A}_{\phantom{A}B}-(e_{2}\wedge\theta^{2})^{A}_{\phantom{A}B}-(e_{3}\wedge\theta^{3})^{A}_{\phantom{A}B}]$ ; | $\chi_{2}^{\,A}\widetilde{\gamma}^{2}_{\,B}=2[\frac{1}{8}\delta^{A}_{B}-(e_{1}\wedge\theta^{1})^{A}_{\phantom{A}B}+(e_{2}\wedge\theta^{2})^{A}_{\phantom{A}B}+(e_{3}\wedge\theta^{3})^{A}_{\phantom{A}B}]$ ;
---|---
$\chi_{3}^{\,A}\widetilde{\gamma}^{3}_{\,B}=2[\frac{1}{8}\delta^{A}_{B}+(e_{1}\wedge\theta^{1})^{A}_{\phantom{A}B}-(e_{2}\wedge\theta^{2})^{A}_{\phantom{A}B}+(e_{3}\wedge\theta^{3})^{A}_{\phantom{A}B}]$ ; | $\chi_{4}^{\,A}\widetilde{\gamma}^{4}_{\,B}=2[\frac{1}{8}\delta^{A}_{B}+(e_{1}\wedge\theta^{1})^{A}_{\phantom{A}B}+(e_{2}\wedge\theta^{2})^{A}_{\phantom{A}B}-(e_{3}\wedge\theta^{3})^{A}_{\phantom{A}B}]$ .
Table 5: The spinorial representation of a bivector basis. The last two lines
display the inverse relation of the 3 short lines at the center.
$C_{1212}=4\Psi^{44}_{\phantom{44}11}$ ;
$C_{1213}=-4\Psi^{34}_{\phantom{34}11}$ ;
$C_{1215}=4\Psi^{34}_{\phantom{34}12}$ ;
$C_{1216}=4\Psi^{44}_{\phantom{44}12}$ ;
$C_{1313}=4\Psi^{33}_{\phantom{33}11}$
---
$C_{1315}=-4\Psi^{33}_{\phantom{33}12}$ ;
$C_{1316}=-4\Psi^{34}_{\phantom{34}12}$ ;
$C_{1515}=4\Psi^{33}_{\phantom{33}22}$ ;
$C_{1516}=4\Psi^{34}_{\phantom{34}32}$ ;
$C_{1616}=4\Psi^{44}_{\phantom{44}22}$
$C_{1223}=4\Psi^{24}_{\phantom{24}11}$ ;
$C_{1225}=4(\Psi^{14}_{\phantom{14}11}+\Psi^{34}_{\phantom{34}31})=-4(\Psi^{44}_{\phantom{44}41}+\Psi^{24}_{\phantom{24}21})$
; $C_{1226}=4\Psi^{44}_{\phantom{44}13}$
---
$C_{1235}=4\Psi^{34}_{\phantom{34}14}$ ;
$C_{1236}=4(\Psi^{14}_{\phantom{14}11}+\Psi^{44}_{\phantom{44}41})=-4(\Psi^{24}_{\phantom{24}21}+\Psi^{34}_{\phantom{34}31})$
; $C_{1256}=-4\Psi^{14}_{\phantom{14}12}$
$C_{1323}=-4\Psi^{23}_{\phantom{23}11}$ ;
$C_{1325}=4(\Psi^{23}_{\phantom{23}21}+\Psi^{43}_{\phantom{43}41})=-4(\Psi^{13}_{\phantom{13}11}+\Psi^{33}_{\phantom{33}31})$
; $C_{1326}=-4\Psi^{43}_{\phantom{43}13}$
$C_{1335}=-4\Psi^{33}_{\phantom{33}14}$ ;
$C_{1336}=4(\Psi^{23}_{\phantom{23}21}+\Psi^{33}_{\phantom{33}31})=-4(\Psi^{13}_{\phantom{13}11}+\Psi^{43}_{\phantom{43}41})$
; $C_{1356}=4\Psi^{13}_{\phantom{13}12}$
$C_{1523}=4\Psi^{32}_{\phantom{32}12}$ ;
$C_{1525}=4(\Psi^{13}_{\phantom{13}12}+\Psi^{33}_{\phantom{33}32})=-4(\Psi^{23}_{\phantom{23}22}+\Psi^{43}_{\phantom{43}42})$
; $C_{1526}=4\Psi^{34}_{\phantom{34}23}$
$C_{1535}=4\Psi^{33}_{\phantom{33}24}$ ;
$C_{1536}=4(\Psi^{13}_{\phantom{13}12}+\Psi^{43}_{\phantom{43}42})=-4(\Psi^{23}_{\phantom{23}22}+\Psi^{33}_{\phantom{33}32})$
; $C_{1556}=-4\Psi^{13}_{\phantom{13}22}$
$C_{1623}=4\Psi^{24}_{\phantom{24}21}$ ;
$C_{1625}=4(\Psi^{14}_{\phantom{14}12}+\Psi^{34}_{\phantom{34}32})=-4(\Psi^{24}_{\phantom{24}22}+\Psi^{44}_{\phantom{44}42})$
; $C_{1626}=4\Psi^{44}_{\phantom{44}23}$
$C_{1635}=4\Psi^{34}_{\phantom{34}24}$ ;
$C_{1636}=4(\Psi^{14}_{\phantom{14}12}+\Psi^{44}_{\phantom{44}42})=-4(\Psi^{24}_{\phantom{24}22}+\Psi^{34}_{\phantom{34}32})$
; $C_{1656}=-4\Psi^{14}_{\phantom{14}22}$
$C_{1412}=4(\Psi^{14}_{\phantom{14}11}+\Psi^{24}_{\phantom{24}21})=-4(\Psi^{34}_{\phantom{34}31}+\Psi^{44}_{\phantom{44}41})$
;
$C_{1413}=4(\Psi^{33}_{\phantom{33}31}+\Psi^{43}_{\phantom{43}41})=-4(\Psi^{13}_{\phantom{13}11}+\Psi^{23}_{\phantom{23}21})$
---
$C_{1415}=4(\Psi^{13}_{\phantom{13}12}+\Psi^{23}_{\phantom{23}22})=-4(\Psi^{33}_{\phantom{33}32}+\Psi^{43}_{\phantom{43}42})$
;
$C_{1416}=4(\Psi^{14}_{\phantom{14}12}+\Psi^{24}_{\phantom{24}22})=-4(\Psi^{34}_{\phantom{34}32}+\Psi^{44}_{\phantom{44}42})$
$C_{1414}=4(\Psi^{11}_{\phantom{11}11}+\Psi^{22}_{\phantom{22}22}+2\Psi^{12}_{\phantom{12}12})=4(\Psi^{33}_{\phantom{33}33}+\Psi^{44}_{\phantom{44}44}+2\Psi^{34}_{\phantom{34}34})$
; $C_{1425}=4(\Psi^{23}_{\phantom{23}23}-\Psi^{14}_{\phantom{14}14})$
---
$C_{2525}=4(\Psi^{11}_{\phantom{11}11}+\Psi^{33}_{\phantom{33}33}+2\Psi^{13}_{\phantom{13}13})=4(\Psi^{22}_{\phantom{22}22}+\Psi^{44}_{\phantom{44}44}+2\Psi^{24}_{\phantom{24}24})$
; $C_{1436}=4(\Psi^{24}_{\phantom{24}24}-\Psi^{13}_{\phantom{13}13})$
$C_{3636}=4(\Psi^{11}_{\phantom{11}11}+\Psi^{44}_{\phantom{44}44}+2\Psi^{14}_{\phantom{14}14})=4(\Psi^{22}_{\phantom{22}22}+\Psi^{33}_{\phantom{33}33}+2\Psi^{23}_{\phantom{23}23})$
; $C_{2536}=4(\Psi^{34}_{\phantom{34}34}-\Psi^{12}_{\phantom{12}12})$
$C_{1423}=4(\Psi^{12}_{\phantom{12}11}+\Psi^{22}_{\phantom{22}21})=-4(\Psi^{32}_{\phantom{32}31}+\Psi^{42}_{\phantom{42}41})$
;
$C_{1426}=4(\Psi^{14}_{\phantom{14}13}+\Psi^{24}_{\phantom{24}23})=-4(\Psi^{34}_{\phantom{34}33}+\Psi^{44}_{\phantom{44}43})$
---
$C_{1435}=4(\Psi^{13}_{\phantom{13}14}+\Psi^{23}_{\phantom{23}24})=-4(\Psi^{33}_{\phantom{33}34}+\Psi^{43}_{\phantom{43}44})$
;
$C_{1456}=4(\Psi^{31}_{\phantom{31}32}+\Psi^{41}_{\phantom{41}42})=-4(\Psi^{11}_{\phantom{11}12}+\Psi^{21}_{\phantom{21}22})$
$C_{2523}=4(\Psi^{12}_{\phantom{12}11}+\Psi^{23}_{\phantom{23}13})=-4(\Psi^{22}_{\phantom{22}12}+\Psi^{24}_{\phantom{24}14})$
;
$C_{3623}=4(\Psi^{12}_{\phantom{12}11}+\Psi^{24}_{\phantom{24}14})=-4(\Psi^{22}_{\phantom{22}12}+\Psi^{23}_{\phantom{23}13})$
$C_{2526}=4(\Psi^{14}_{\phantom{14}13}+\Psi^{34}_{\phantom{34}33})=-4(\Psi^{24}_{\phantom{24}23}+\Psi^{44}_{\phantom{44}34})$
;
$C_{3626}=4(\Psi^{14}_{\phantom{14}13}+\Psi^{44}_{\phantom{44}34})=-4(\Psi^{24}_{\phantom{24}23}+\Psi^{34}_{\phantom{34}33})$
$C_{2535}=4(\Psi^{13}_{\phantom{13}14}+\Psi^{33}_{\phantom{33}34})=-4(\Psi^{23}_{\phantom{23}24}+\Psi^{34}_{\phantom{34}44})$
;
$C_{3635}=4(\Psi^{13}_{\phantom{13}14}+\Psi^{34}_{\phantom{34}44})=-4(\Psi^{23}_{\phantom{23}24}+\Psi^{33}_{\phantom{33}34})$
$C_{2556}=4(\Psi^{12}_{\phantom{12}22}+\Psi^{14}_{\phantom{14}24})=-4(\Psi^{11}_{\phantom{11}12}+\Psi^{13}_{\phantom{13}23})$
;
$C_{3656}=4(\Psi^{12}_{\phantom{12}22}+\Psi^{13}_{\phantom{13}23})=-4(\Psi^{11}_{\phantom{11}12}+\Psi^{14}_{\phantom{14}24})$
$C_{2323}=4\Psi^{22}_{\phantom{22}11}$ ;
$C_{2326}=4\Psi^{24}_{\phantom{24}13}$ ;
$C_{2335}=4\Psi^{23}_{\phantom{23}14}$ ;
$C_{2356}=-4\Psi^{12}_{\phantom{12}12}$ ;
$C_{5656}=4\Psi^{11}_{\phantom{11}22}$
---
$C_{2626}=4\Psi^{44}_{\phantom{44}33}$ ;
$C_{2635}=4\Psi^{34}_{\phantom{34}34}$ ;
$C_{2656}=-4\Psi^{14}_{\phantom{14}23}$ ;
$C_{3535}=4\Psi^{33}_{\phantom{33}44}$ ;
$C_{3556}=-4\Psi^{13}_{\phantom{13}24}$
$C_{1242}=-4\Psi^{24}_{\phantom{24}13}$ ;
$C_{1243}=-4\Psi^{24}_{\phantom{24}14}$ ;
$C_{1245}=-4\Psi^{14}_{\phantom{14}14}$ ;
$C_{1246}=4\Psi^{14}_{\phantom{14}13}$ ;
$C_{1342}=4\Psi^{23}_{\phantom{23}13}$
$C_{1343}=4\Psi^{23}_{\phantom{23}14}$ ;
$C_{1345}=4\Psi^{13}_{\phantom{13}14}$ ;
$C_{1346}=-4\Psi^{13}_{\phantom{13}13}$ ;
$C_{1542}=-4\Psi^{23}_{\phantom{23}23}$ ;
$C_{1543}=-4\Psi^{23}_{\phantom{23}24}$
$C_{1545}=-4\Psi^{13}_{\phantom{13}24}$ ;
$C_{1546}=4\Psi^{13}_{\phantom{13}23}$ ;
$C_{1642}=-4\Psi^{24}_{\phantom{24}23}$ ;
$C_{1643}=-4\Psi^{24}_{\phantom{24}24}$ ;
$C_{1645}=-4\Psi^{14}_{\phantom{14}24}$ ;
$C_{1646}=4\Psi^{14}_{\phantom{14}23}$
---
Table 6: This table displays the relation between Weyl tensor’s components in
a null frame and its spinorial equivalent. The missing components of the Weyl
tensor can be obtained by making the changes $1\leftrightarrow 4$,
$2\leftrightarrow 5$ and $3\leftrightarrow 6$ on the vectorial indices while
swapping the upper and the lower indices of $\Psi$. The first two rows of the
above table contain the components of the Weyl tensor with boost weight $b=2$,
the next ten rows present the components with $b=1$, the other rows have the
components with zero boost weight.
With the tolls introduced in this appendix it is simple matter to show that
$\chi_{1}^{\,D}\varepsilon_{DABC}=6\widetilde{\gamma}^{2}_{\,[A}\widetilde{\gamma}^{3}_{\,B}\widetilde{\gamma}^{4}_{\,C]}$.
Using this relation it is then possible to prove, after some algebra, the
following equality:
$\frac{1}{4}(\,\varepsilon_{AEFG}\,\varepsilon_{BHIJ}\,\Psi^{GJ}_{\phantom{GJ}CD}\,)\,\chi_{1}^{\,A}\,\chi_{1}^{\,B}\,\chi_{1}^{\,C}\,=\,\Psi^{44}_{\phantom{44}1D}\widetilde{\gamma}^{2}_{\,[E}\widetilde{\gamma}^{3}_{\,F]}\widetilde{\gamma}^{2}_{\,[H}\widetilde{\gamma}^{3}_{\,I]}\,+\,\Psi^{43}_{\phantom{43}1D}(\widetilde{\gamma}^{2}_{\,[E}\widetilde{\gamma}^{3}_{\,F]}\widetilde{\gamma}^{4}_{\,[H}\widetilde{\gamma}^{2}_{\,I]}+\widetilde{\gamma}^{4}_{\,[E}\widetilde{\gamma}^{2}_{\,F]}\widetilde{\gamma}^{2}_{\,[H}\widetilde{\gamma}^{3}_{\,I]})\,+\,\\\
+\,\Psi^{42}_{\phantom{42}1D}(\widetilde{\gamma}^{2}_{\,[E}\widetilde{\gamma}^{3}_{\,F]}\widetilde{\gamma}^{3}_{\,[H}\widetilde{\gamma}^{4}_{\,I]}+\widetilde{\gamma}^{3}_{\,[E}\widetilde{\gamma}^{4}_{\,F]}\widetilde{\gamma}^{2}_{\,[H}\widetilde{\gamma}^{3}_{\,I]})\,+\,\Psi^{33}_{\phantom{33}1D}\widetilde{\gamma}^{2}_{\,[E}\widetilde{\gamma}^{4}_{\,F]}\widetilde{\gamma}^{2}_{\,[H}\widetilde{\gamma}^{4}_{\,I]}\,+\,\\\
+\,\Psi^{32}_{\phantom{42}1D}(\widetilde{\gamma}^{2}_{\,[E}\widetilde{\gamma}^{4}_{\,F]}\widetilde{\gamma}^{4}_{\,[H}\widetilde{\gamma}^{3}_{\,I]}+\widetilde{\gamma}^{4}_{\,[E}\widetilde{\gamma}^{3}_{\,F]}\widetilde{\gamma}^{2}_{\,[H}\widetilde{\gamma}^{4}_{\,I]})\,+\,\Psi^{22}_{\phantom{22}1D}\widetilde{\gamma}^{3}_{\,[E}\widetilde{\gamma}^{4}_{\,F]}\widetilde{\gamma}^{3}_{\,[H}\widetilde{\gamma}^{4}_{\,I]}.$
(35)
Now let us define a basis for the space of 3-vectors:
$T_{1}^{\;AB}=\chi_{1}^{\,A}\chi_{1}^{\,B}\;;\;T_{2}^{\;AB}=\sqrt{2}\,\chi_{1}^{\,(A}\chi_{2}^{\,B)}\;;\;T_{3}^{\;AB}=\sqrt{2}\,\chi_{1}^{\,(A}\chi_{3}^{\,B)}\;;\;T_{4}^{\;AB}=\sqrt{2}\,\chi_{1}^{\,(A}\chi_{4}^{\,B)}\;;\;T_{5}^{\;AB}=\chi_{2}^{\,A}\chi_{2}^{\,B}\;;\;T_{6}^{\;AB}=\sqrt{2}\,\chi_{2}^{\,(A}\chi_{3}^{\,B)}\\\
T_{7}^{\;AB}=\sqrt{2}\,\chi_{2}^{\,(A}\chi_{4}^{\,B)}\;;\;T_{8}^{\;AB}=\chi_{3}^{\,A}\chi_{3}^{\,B}\;;\;T_{9}^{\;AB}=\sqrt{2}\,\chi_{3}^{\,(A}\chi_{4}^{\,B)}\;;\;T_{10}^{\;AB}=\chi_{4}^{\,A}\chi_{4}^{\,B}\quad\quad\quad\quad\\\
\widetilde{T}^{1}_{\;AB}=\widetilde{\gamma}^{1}_{\,A}\widetilde{\gamma}^{1}_{\,B}\;;\;\widetilde{T}^{2}_{\;AB}=\sqrt{2}\widetilde{\gamma}^{1}_{\,(A}\widetilde{\gamma}^{2}_{\,B)}\;;\;\widetilde{T}^{3}_{\;AB}=\sqrt{2}\widetilde{\gamma}^{1}_{\,(A}\widetilde{\gamma}^{3}_{\,B)}\;;\;\widetilde{T}^{4}_{\;AB}=\sqrt{2}\widetilde{\gamma}^{1}_{\,(A}\widetilde{\gamma}^{4}_{\,B)}\;;\;\widetilde{T}^{5}_{\;AB}=\widetilde{\gamma}^{2}_{\,A}\widetilde{\gamma}^{2}_{\,B}\;;\;\widetilde{T}^{6}_{\;AB}=\sqrt{2}\widetilde{\gamma}^{2}_{\,(A}\widetilde{\gamma}^{3}_{\,B)}\;\\\
\widetilde{T}^{7}_{\;AB}=\sqrt{2}\widetilde{\gamma}^{2}_{\,(A}\widetilde{\gamma}^{4}_{\,B)}\;;\;\widetilde{T}^{8}_{\;AB}=\widetilde{\gamma}^{3}_{\,A}\widetilde{\gamma}^{3}_{\,B}\;;\;\widetilde{T}^{9}_{\;AB}=\sqrt{2}\widetilde{\gamma}^{3}_{\,(A}\widetilde{\gamma}^{4}_{\,B)}\;;\;\widetilde{T}^{10}_{\;AB}=\widetilde{\gamma}^{4}_{\,A}\widetilde{\gamma}^{4}_{\,B}\quad\quad\quad\quad\quad\quad$
(36)
Where by $T_{r}^{\;AB}$ it is meant the 3-vector $(T_{r}^{\;AB},0)$, while
$\widetilde{T}^{r}_{\;AB}$ means $(0,\widetilde{T}^{r}_{\;AB})$. The ten
3-vectors $\\{T_{r}\\}$ form a basis for the space of self-dual 3-vectors,
while $\\{\widetilde{T}^{r}\\}$ is a basis for the space of anti-self-dual
3-vectors. It is easily seen that
$T_{r}^{\;AB}\widetilde{T}^{s}_{\;AB}=\delta^{s}_{r}$.
## Appendix C Clifford Algebra
The space of the Dirac spinors, $S$, is the 8-dimensional space spanned by the
spinors $\\{\chi_{p}^{\,A},\widetilde{\gamma}^{q}_{\,B}\\}$. The 4-dimensional
subspace generated by $\\{\chi_{p}^{\,A}\\}$ is the space of Weyl spinors of
positive chirality, $S^{+}$, while $\\{\widetilde{\gamma}^{q}_{\,B}\\}$ spans
the subspace of negative chirality Weyl spinors, $S^{-}$. So a Dirac spinor
can be written as $\widehat{\psi}=\psi^{A}+\widetilde{\psi}_{A}$, where
$\psi^{A}$ pertains to $S^{+}$ and $\widetilde{\psi}_{A}$ belongs to $S^{-}$.
The inner product of two Dirac spinors is defined by:
$(\widehat{\psi}_{1},\widehat{\psi}_{2})=\psi_{1}^{\,A}\,\widetilde{\psi}_{2\,A}-\psi_{2}^{\,A}\,\widetilde{\psi}_{1\,A}.$
(37)
Note that this inner product is skew-symmetric.
In the Clifford algebra formalism the vectors are seen as linear operators
that act on the space $S$, $\textbf{e}_{a}:S\rightarrow S$. These maps must
obey to the following relation:
$\textbf{e}_{a}\textbf{e}_{b}+\textbf{e}_{b}\textbf{e}_{a}=2g(e_{a},e_{b})\,\textbf{1}=2g_{ab}\,\textbf{1}.$
(38)
Where 1 is the identity operator on $S$. An explicit expression for these
operators in terms of (32) and (33) is given by:
$\textbf{e}_{a}=2(e_{a}^{\,AB}-e_{a\,AB})\,:\;\;\;\;\;\textbf{e}_{a}(\widehat{\psi})=2e_{a}^{\,AB}\widetilde{\psi}_{B}-2e_{a\,AB}\psi^{B}.$
(39)
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|
arxiv-papers
| 2012-12-12T02:12:44 |
2024-09-04T02:49:39.172287
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Batista and Bruno Carneiro da Cunha",
"submitter": "Carlos A. Batista da S. Filho",
"url": "https://arxiv.org/abs/1212.2689"
}
|
1212.2721
|
Holographic insulator/superconductor phase transition in Born-Infeld
electrodynamics
Nan Baia, Yi-Hong Gaoa, Bu-Guan Qia and Xiao-Bao Xua
a State Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics,
Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
[email protected], [email protected], [email protected], and [email protected]
We studied holographic insulator/superconductor phase transition in the
framework of Born-Infeld electrodynamics both numerically and analytically.
First we numerically study the effects of the Born-Infeld electrodynamics on
the phase transition, find that the critical chemical potential is not changed
by the Born-Infeld parameter. Then we employ the variational method for the
Sturm-Liouville eigenvalue problem to analytically study the phase transition.
The analytical results obtained are found to be consistent with the numerical
results.
## 1 Introduction
The AdS/CFT correspondence [1, 2, 3] which relates string theory on
asymptotically AdS spacetime to a conformal field theory on the boundary, has
been extensively applied in nuclear physics and condensed matter physics. In
particular the related developments of dual gravitational models which
describe insulator/superconductor phase transitions in condensed matter
physics have been proposed in [4, 5].
The Einstein-Maxwell theory coupled to a charged scalar field would be the
simplest model for holographic superconductor. In this model, there exists a
critical temperature $T_{c}$ [4], below which charged AdS black holes may have
non-zero scalar hair. In the context of AdS/CFT connection, a charged scalar
field is dual to an operator which carries the global $U(1)$ charge. Having
non-zero scalar hair indicates that the expectation value of dual operator is
non-zero which leads to the break of global $U(1)$ symmetry, thus can be
understood as a second order conductor/superconductor phase transition. The
behavior of AC conductivity in these phases [5] make the interpretation above
reliable.
There have been many discussions concerning holographic superconductor models
among which general relativity coupled to Maxwell field plus a charged scalar
field attracts most attention [6]-[18] and the related Lagrangian is
$\mathcal{L}=-\frac{1}{4}F^{2}-|\nabla_{\mu}\psi-
iqA_{\mu}\psi|^{2}-m^{2}|\psi|^{2},\qquad F=F_{\mu\nu}F^{\mu\nu}$ (1)
However,the non-linear extension of the original Maxwell electrodynamics to
Born-Infeld (BI) model [28] also caused intensive investigations [19]-[27],
since BI electrodynamics possesses many interesting physical properties as
finite total energy and the invariance under electromagnetic duality. In
string theory, BI action can also describe gauge fields which arise from
D-brane attached with open strings [29, 30]. The effects of BI electrodynamics
on the holographic superconductors in the background of Ads-Schwarzschild
black hole spacetime has been analyzed numerically in [22] and analytically in
[25].The results shows it is much more difficult to have scalar condensation
in BI electrodynamics which is similar to the one of holographic
superconductors in Einstein-Gauss-Bonnet gravity [8, 15], where the higher
curvature corrections make condensation harder.
The authors of [10] have constructed a model describing an
insulator/superconductor phase transition at zero temperature, using a five
dimensional AdS soliton in an Einstein-Maxwell-charged scalar field. The
generalization of holographic insulator/superconductor has been carried out
[31, 32, 33].The aim of the present article is to extend holographic
insulator/superconductor in the framework of Born-Infeld electrodynamics. The
main purpose here is to see what effect of Born-Infeld scale parameter on the
holographic insulator/superconductor. With both numerical and analytic method,
we find a bit unexpected result that the effect of Born-Infeld electrodynamics
is too small to be seen in our study, but it is consistent with [27], which
state that “the critical chemical potential is not changed by the coupling
parameter $b$”.
The paper is organized as follows. In section 2 we study holographic
insulator/super-conductor in the framework of Born-Infeld electrodynamics in
the probe limit. We numerically calculate the critical chemical potential
$\mu_{c}$, the condensations of the scalar operators $\langle\cal{O}\rangle$
and charge density $\rho$. We also study the conductivity of the theory. In
section 3 we reproduce the same result as the numerical one for the critical
chemical potential $\mu_{c}$ by the Sturm-Liouville analytical method. The
last section is devoted to conclusions.
## 2 Numerical analysis of the critical chemical potential $\mu_{c}$
In order to construct the holographic insulator/superconductor phase
transition, we consider the fixed background of five dimensional AdS soliton,
e.g. in the probe limit. The Ads soliton metric reads
$\displaystyle ds^{2}$
$\displaystyle=L^{2}\frac{dr^{2}}{f(r)}+r^{2}(-dt^{2}+dx^{2}+dy^{2})+f(r)d\chi^{2}\
,$ (2) $\displaystyle f(r)$ $\displaystyle=r^{2}-\frac{r_{0}^{4}}{r^{2}}$ (3)
where $L$ is the AdS radius and $r_{0}$ the tip of the soliton which is a
conical singularity in this solution. Essentially this solution can be
obtained from a five dimensional AdS black hole solution by making use of
double Wick rotations [10]. We can also set $L=1$ and $r_{0}$ = 1 without the
loss of generality.
We now consider a charged complex scalar field and an electric field in the
above background, we investigate Born-Infeld electrodynamics instead of
Maxwell’s therefore the corresponding Lagrangian density can be
$\displaystyle\mathcal{L}=\mathcal{L}_{BI}-|\nabla_{\mu}\psi-
iqA_{\mu}\psi|^{2}-m^{2}|\psi|^{2}$ (4)
where $\psi$ is a charged complex scalar field, $\mathcal{L}_{BI}$ is the
Lagrangian density of the Born-Infeld electrodynamics
$\displaystyle\mathcal{L}_{BI}=\frac{1}{b}\bigg{(}1-\sqrt{1+\frac{bF}{2}}\bigg{)}.\quad
F\equiv F_{\mu\nu}F^{\mu\nu}$ (5)
The BI coupling parameter $b$ describes the nonlinearity of electrodynamics.
The above BI lagrangian could be reduced to Maxwell lagrangian
$-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ naturally in the limit $b\rightarrow 0$ .
We further assume that the BI coupling parameter $b$ is small,since the Ads
Soliton of background spacetimes is the solution of the Einstein equation in
Einstein-Maxwell-scalar field theory.
To study the insulator/superconductor phase transition we will consider the
following ansatz [10]
$\displaystyle A_{\mu}=(\phi(r),0,0,0),\;\;\;\;\psi=\psi(r)$ (6)
The equations of motion are
$\displaystyle\psi^{\prime\prime}+\left(\frac{f^{\prime}}{f}+\frac{3}{r}\right)\psi^{\prime}+\left(\frac{q^{2}\phi^{2}}{r^{2}f}-\frac{m^{2}}{f}\right)\psi=0$
(7)
and
$\displaystyle\phi^{\prime\prime}+\left(\frac{1}{r}+\frac{f^{\prime}}{f}\right)\phi^{\prime}-\left(\frac{bf^{\prime}}{2r^{2}}+\frac{2bf}{r^{3}}\right)\phi^{\prime
3}-\frac{2q^{2}\psi^{2}}{f}\bigg{(}1-\frac{bf}{r^{2}}\phi^{\prime
2}\bigg{)}^{3/2}\phi=0$ (8)
To seek the solutions of the equations (7) and (8), we impose the boundary
condition at the tip $r=r_{0}$ and the boundary $r=\infty$. Near the boundary,
the scalar field $\psi(r)$ and the scalar potential $\phi(r)$ behave as
$\displaystyle\psi$
$\displaystyle=\frac{\psi_{-}}{r^{\lambda_{-}}}+\frac{\psi_{+}}{r^{\lambda_{+}}},$
(9) $\displaystyle\phi$ $\displaystyle=\mu-\frac{\rho}{r^{2}}$ (10)
where
$\displaystyle\lambda_{\pm}=2\pm\sqrt{4+m^{2}}$ (11)
is the conformal dimension of the condensation operator
$\langle\cal{O}\rangle$ belongs to the boundary field theory,and $\psi_{-}$
and $\psi_{+}$ are the vacuum expectation values for the boundary operator
$\langle\cal{O}\rangle$. $\mu$ and $\rho$ are the chemical potential and the
charge density,respectively.
For simplicity we will take $m^{2}=-\frac{15}{4}$, which is above the
Breitenlohner-Freedman bound [34], hence
$\psi=\frac{\psi_{-}}{r^{\frac{3}{2}}}+\frac{\psi_{+}}{r^{\frac{5}{2}}}$. In
order to have a stable boundary theory, we impose the boundary condition as
$\psi_{-}=0$ or $\psi_{+}=0$ [5]. In this paper, we shall set $\psi_{-}=0$ and
$\langle\cal{O}\rangle=\psi_{+}$.
At the tip, we have
$\displaystyle\psi$
$\displaystyle=\alpha_{0}+\alpha_{1}\log(r-r_{0})+\alpha_{2}(r-r_{0})+\dots,$
(12) $\displaystyle\phi$
$\displaystyle=\beta_{0}+\beta_{1}\log(r-r_{0})+\beta_{2}(r-r_{0})+\dots$ (13)
We can impose the Neumann boundary condition $\alpha_{1}=\beta_{1}=0$ to make
the physical quantities finite. Note that the equations of motion (7) and (8)
have the scaling symmetry,
$(\psi,\phi,\mu,q,b)\to(\lambda\psi,\lambda\phi,\lambda\mu,q/\lambda,b\lambda^{2})$.
Thus we will set $q=1$ in the following discussions. Introducing a new
variable $z=1/r$, we rewrite the equation of motion (7) and (8) into
$\displaystyle\psi^{\prime\prime}+\left(\frac{f^{\prime}}{f}-\frac{1}{z}\right)\psi^{\prime}+\left(\frac{\phi^{2}}{z^{2}f}-\frac{m^{2}}{z^{4}f}\right)\psi=0$
(14)
and
$\displaystyle\phi^{\prime\prime}+\left(\frac{f^{\prime}}{f}+\frac{1}{z}+\left(2f-\frac{zf^{\prime}}{2}\right)bz^{5}\phi^{\prime
2}\right)\phi^{\prime}-\frac{2\psi^{2}}{z^{4}f}\left(1-bfz^{6}\phi^{\prime
2}\right)^{3/2}\phi=0$ (15)
We use shooting method numerically calculate the critical potential $\mu_{c}$,
the condensations of the scalar operators $\langle\cal{O}\rangle$ and charge
density $\rho$ for $b=0.01$. Then we change Born-Infeld parameter $b$ of the
theory, choosing $b=0,0.1,0.2,0.3$, find that the result varied so small that
we can neglect the effect of Born-Infeld parameter $b$. It is a somewhat
surprising result to us. However, our analytic calculation support this.
Figure 1: Here we plot the behaviors of $\langle\cal{O}\rangle$ and charge
density $\rho$ for various values of chemical potential $\mu$. Here
$\mu_{c}=1.888$.
Now we holographically calculate the conductivity $\sigma(\omega)$ by
perturbing the gauge field [5]. To do so, we consider the following ansatz for
the gauge field along $x$ direction
$A_{x}=A_{x}(r)e^{-i\omega t}$ (16)
Neglecting the backreaction of the perturbational field on background metric,
one get the equation of motion of $A_{x}$ :
$\displaystyle\begin{split}A_{x}^{\prime\prime}+\left(\frac{f^{\prime}}{f}+\frac{1}{r}\right)A_{x}^{\prime}+\left(\frac{\omega^{2}}{r^{2}f}-\frac{2\psi^{2}}{f}\big{[}1+b(-\frac{f\phi^{\prime
2}}{r^{2}}-\frac{\omega^{2}A_{x}^{2}}{r^{4}}+\frac{fA_{x}^{\prime
2}}{r^{2}})\big{]}^{1/2}\right)A_{x}\\\
-\frac{b}{2f}\frac{1}{1+b(-\frac{f\phi^{\prime
2}}{r^{2}}-\frac{\omega^{2}A_{x}^{2}}{r^{4}}+\frac{fA_{x}^{\prime
2}}{r^{2}})}\big{[}\frac{2f\phi^{\prime
2}}{r^{3}}-\frac{f^{\prime}\phi^{\prime
2}}{f}-\frac{2f\phi^{\prime}\phi^{\prime\prime}}{r^{2}}+\frac{4\omega^{2}A_{x}^{2}}{r^{5}}-\\\
\frac{2\omega^{2}A_{x}A_{x}^{\prime}}{r^{4}}-\frac{2fA_{x}^{\prime
2}}{r^{3}}+\frac{f^{\prime}A_{x}^{\prime
2}}{r^{2}}+\frac{2fA_{x}^{\prime}A_{x}^{\prime\prime}}{r^{2}}\big{]}=0\end{split}$
(17)
Near the boundary($r\to\infty$), the behavior of the gauge field is
$A_{x}=A_{x}^{(0)}+\frac{A_{x}^{(1)}}{r^{2}}+\frac{A_{x}^{(0)}\omega^{2}}{2}\frac{\log\Lambda
r}{r^{2}}+\cdots$ (18)
the holographic conductivity is given as follows [11]
$\sigma(\omega)=\frac{-2iA_{x}^{(1)}}{\omega A_{x}^{(0)}}+\frac{i\omega}{2}$
(19)
Figure 2: The conductivity for operator $\langle\mathcal{O}\rangle$ for
different values of $b$.
In the figure 2 we have plotted the behavior of the conductivity for different
Born-Infield parameter $b$ when the condensation is non-zero, we find a pole
at $\omega\to 0$ suggesting that we have an infinite conductivity according to
the Kramers-Kronig relation as expected for the superconductor phase. We
observed the effect of Born-Infield parameter $b$ on the conductivity, the gap
frequency becomes larger as Born-Infield parameter $b$ increases. This is
different from those of Born-Infeld electrodynamics in Schwarzschild AdS black
hole spacetime[22].
## 3 Analytical calculation of the critical chemical potential $\mu_{c}$
From the results of the numerical calculations above, we found there exists a
critical chemical potential $\mu_{c}$. When chemical potential $\mu$ exceeds
this critical value,the solution is unstable and a scalar hair will emerge
which means the condensations take place.This is the superconductor phase.For
$\mu<\mu_{c}$,the scalar field is zero,which can be viewed as an insulator
phase.The critical chemical potential $\mu_{c}$ can be obtained by using
Sturm-Liouville (SL) eigenvalue method.
When $\mu\leq\mu_{c}$,the scalar field $\psi=0$,Eq.(15) reduces to
$\displaystyle\phi^{\prime\prime}+\left(\frac{f^{\prime}}{f}+\frac{1}{z}+\left(2f-\frac{zf^{\prime}}{2}\right)bz^{5}\phi^{\prime
2}\right)\phi^{\prime}=0$ (20)
By substituting $\phi^{\prime}$ to a new variable ,the equation will be
transformed into a first order ODE of Bernoulli type, see appendix A , and the
general solution is
$\displaystyle\begin{split}\phi\left(z\right)=&\frac{b^{2}\left(117z^{8}-306z^{4}+221\right)z^{9}}{1989}-\frac{2bC_{1}\left(77z^{12}-315z^{8}+495z^{4}-385\right)z^{3}}{1155}+\\\
&C_{1}^{2}\left(\frac{z^{13}}{13}-\frac{4z^{9}}{9}+\frac{6z^{5}}{5}-\frac{1}{3z^{3}}-4z\right)+C_{2}\\\
\end{split}$ (21)
where $C_{1}$ and $C_{2}$ are the integration constants. The boundary
condition (10) near the infinity $z=0$ imposes $C_{1}=0$ which keeps $\phi$ a
finite value and $C_{2}=\mu$. Thus,
$\displaystyle\phi\left(z\right)=\frac{b^{2}\left(117z^{8}-306z^{4}+221\right)z^{9}}{1989}+\mu$
(22)
When $\mu\rightarrow\mu_{c}$ ,we take $\phi\left(z\right)$ above into the
equation (14),thus the equation of motion for $\psi$ becomes
$\displaystyle\psi(z)\left(\frac{\left(\frac{b^{2}\left(117z^{8}-306z^{4}+221\right)z^{9}}{1989}+\mu\right)^{2}}{z^{2}f(z)}-\frac{m^{2}}{z^{4}f(z)}\right)+\left(\frac{f^{\prime}(z)}{f(z)}-\frac{1}{z}\right)\psi^{\prime}(z)+\psi^{\prime\prime}(z)=0$
(23)
As in [9],we introduce a trial function $F\left(z\right)$ near the boundary
$z=0$ which satisfies
$\displaystyle\psi(z)\sim\langle{\cal O}_{i}\rangle z^{\lambda_{i}}F(z),$ (24)
We impose the boundary condition for $F(z)$ as $F(0)=1$ and
$F^{\prime}(0)=0$.Thus the equation of motion for $F(z)$ is
$\displaystyle\begin{split}&F^{\prime\prime}(z)+\left(\frac{f^{\prime}(z)}{f(z)}+\frac{2}{z}\right)F^{\prime}(z)+\\\
&\left(\left(B^{2}\left(\frac{z^{17}}{17}-\frac{2z^{13}}{13}+\frac{z^{9}}{9}\right)+\mu\right)^{2}+\frac{3}{2}\left(zf^{\prime}(z)-f(z)\right)+\frac{3f(z)}{4}-\frac{m^{2}}{z^{2}}\right)\frac{F(z)}{z^{2}f(z)}=0\\\
\end{split}$ (25)
Following the standard procedure of Sturm-Liouville eigenvalue problem [35],we
introduce a new function $T(z)$
$\displaystyle T(z)=\left(1-z^{4}\right)z^{2\lambda_{i}-3}$ (26)
Multiplying $T(z)$ to both sides of the equation of $F(z)$ ,we have
$\displaystyle\begin{split}&\left(T(z)F^{\prime}(z)\right)^{\prime}+\frac{F(z)T(z)}{1-z^{4}}\left(\left(b^{2}\left(\frac{z^{17}}{17}-\frac{2z^{13}}{13}+\frac{z^{9}}{9}\right)+\mu\right)^{2}-\frac{m^{2}}{z^{2}}+\right.\\\
&\left.(\lambda_{i}-1)\lambda_{i}\left(\frac{1}{z^{2}}-z^{2}\right)+\lambda_{i}\left(\left(-\frac{2}{z^{3}}-2z\right)z+z^{2}-\frac{1}{z^{2}}\right)\right)=0\end{split}$
(27)
Furthermore we define another two functions $U$ and $V$ as
$\displaystyle\begin{split}U=&\frac{1}{1-z^{4}}\left(b^{4}\left(\frac{z^{17}}{17}-\frac{2z^{13}}{13}+\frac{z^{9}}{9}\right)^{2}+2b^{2}\mu\left(\frac{z^{17}}{17}-\frac{2z^{13}}{13}+\frac{z^{9}}{9}\right)-\frac{m^{2}}{z^{2}}+\right.\\\
&\left.(\lambda_{i}-1)\lambda_{i}\left(\frac{1}{z^{2}}-z^{2}\right)+\lambda_{i}\left(\left(-\frac{2}{z^{3}}-2z\right)z+z^{2}-\frac{1}{z^{2}}\right)\right)\end{split}$
(28) $\displaystyle V=T(z)/(1-z^{4})$ (29)
Then we can obtain the eigenvalue of $\mu^{2}$ by substituting $U$ and $V$
into the expression
$\displaystyle\mu^{2}=\frac{\int^{1}_{0}T\left(F^{\prime
2}-UF^{2}\right)dz}{\int^{1}_{0}VF^{2}dz},$ (30)
In order to estimate the minimum value of $\mu^{2}$,we assume the trial
function to be $F(z)=1-az^{2}$,where $a$ is a constant.In the normal case ,the
r.h.s of $\mu^{2}$ is an expression of $a$ and other parameters already given
as constants, e.g Born-Infeld parameter b in our case. But the problem we are
dealing with is a bit different for the expression of $\mu^{2}$ contains $\mu$
as well,
$\displaystyle\mu^{2}=\mu^{2}\left(a,b,\mu\right)$ (31)
To avoid this difficulty ,we suppose $\mu$ on the r.h.s of the above
expression to be $\tilde{\mu}$ at first, and consider $\tilde{\mu}$ as a new
constant parameter. Then we can compute the extreme value of $\mu^{2}$ as we
have done in the normal situation. But at this time the values of $a$
corresponding to the extremal eigenvalues of $\mu^{2}$ are related to the
parameter $\tilde{\mu}$, therefore,
$\displaystyle\mu^{2}=\mu^{2}\left(a(\tilde{\mu}),b,\tilde{\mu}\right)=\mu^{2}(\tilde{\mu},b)$
(32)
At last ,for consistency, $\tilde{\mu}$ is required to be $\mu$,and we obtain
an equation for $\mu$
$\displaystyle\mu_{m}^{2}=\mu_{m}^{2}(\mu_{m},b)$ (33)
where a subscript “m” is added to indicate this is an equation of extreme
value $\mu$. For the calculation details, see appendix B. The analytical
calculation shows the minimum eigenvalues of $\mu^{2}$ and the corresponding
values of $a$ for different Born-Infeld parameter factor $b$ are almost the
same,which is $a=0.330$ and $\mu_{min}=1.890$. We see the analytical results
fit perfectly with the numerical ones.
## 4 Conclusion
In this paper ,we considered the Born-Infeld electrodynamics in the AdS
soliton background.By numerical calculations , we found there exists a
critical chemical potential $\mu_{c}$.When the chemical potential $\mu$ is
lower then the critical chemical potential $\mu_{c}$,the AdS soliton
background is stable and dual field theory can be viewed as an insulator.When
$\mu>\mu_{c}$ ,the background begins to be unstable and a scalar hair will be
developed as the condensation, thus can be interpreted as the superconductor
phase.So the insulator and superconductor phase transition appeared in the
Maxwell electrodynamics also exists in its non-linear extension.We compute the
critical chemical potential $\mu_{c}$ both analytically and numerically and
the results of these two methods fit perfectly good.We also found the effect
of different Born-Infeld parameters are very small that even can be neglected.
## Acknowledgments
We are very grateful of M. Alishahiha and L. Li for their useful helps.
## Appendix A Bernoulli type first order ODE
The standard Bernoulli type ordinary equation takes the form
$\displaystyle y^{\prime}=f(t,y(t))=g(t)y(t)+h(t){y(t)}^{n}$ $\displaystyle
n\neq 1$ (34)
We make the substitution $\xi=\phi^{\prime}$ in the equation (20) which
reduces to
$\displaystyle\frac{d\xi}{dz}=-\left(\frac{f^{\prime}}{f}+\frac{1}{z}\right)\xi-\left(2f-\frac{zf^{\prime}}{2}\right)bz^{5}\xi^{3}$
(35)
It is a Bernoulli type ordinary equation and the idea to solve the equation
(34) is to convert it into a linear ODE. Make the substitution
$\displaystyle v(t)={y(t)}^{1-n}$ (36)
thus
$\displaystyle v(t)^{\prime}=(1-n){y(t)}^{-n}y(t)^{\prime}$ (37)
$\displaystyle y(t)^{\prime}=\frac{y(t)^{n}}{1-n}v(t)^{\prime}$ (38)
so
$\displaystyle v(t)^{\prime}=(1-n)g(t)v(t)+(1-n)h(t)$ (39)
which is a standard linear ODE,and can be solved by the standard procedure.
## Appendix B Computation of $\mu_{min}$
Substituting $U$,$V$ into expression (30) ,we get
$\displaystyle\mu^{2}=\frac{A(a)\tilde{\mu}+B(a)}{C(a)}$ $\displaystyle A(a)$
$\displaystyle=$
$\displaystyle-254592b^{2}\lambda_{i}\left(\lambda_{i}^{14}+141\lambda_{i}^{13}+8953\lambda_{i}^{12}+337449\lambda_{i}^{11}+8370439\lambda_{i}^{10}+\right.$
$\displaystyle\left.142959663\lambda_{i}^{9}+1704340979\lambda_{i}^{8}+13997457507\lambda_{i}^{7}+75168485084\lambda_{i}^{6}+\right.$
$\displaystyle\left.226063117896\lambda_{i}^{5}+135629702768\lambda_{i}^{4}-1262172448656\lambda_{i}^{3}-\right.$
$\displaystyle\left.2753135394624\lambda_{i}^{2}+1021968576000\lambda_{i}+2540624486400\right)\left(a^{2}\left(64\lambda_{i}^{6}+\right.\right.$
$\displaystyle\left.\left.2848\lambda_{i}^{5}+53040\lambda_{i}^{4}+517088\lambda_{i}^{3}+2740052\lambda_{i}^{2}+7415574\lambda_{i}+7977879\right)-\right.$
$\displaystyle\left.2a\left(64\lambda_{i}^{6}+2912\lambda_{i}^{5}+55888\lambda_{i}^{4}+570128\lambda_{i}^{3}+3209404\lambda_{i}^{2}+9320246\lambda_{i}+\right.\right.$
$\displaystyle\left.\left.10774995\right)+64\lambda_{i}^{6}+2976\lambda_{i}^{5}+58864\lambda_{i}^{4}+628992\lambda_{i}^{3}+3790660\lambda_{i}^{2}+\right.$
$\displaystyle\left.12084582\lambda_{i}+15758847\right)$ $\displaystyle C(a)$
$\displaystyle=$ $\displaystyle
3956121(\lambda_{i}-2)(\lambda_{i}-1)\lambda_{i}(\lambda_{i}+1)(\lambda_{i}+8)(\lambda_{i}+9)(\lambda_{i}+10)(\lambda_{i}+11)(\lambda_{i}+12)$
$\displaystyle(\lambda_{i}+13)(\lambda_{i}+14)(\lambda_{i}+15)(\lambda_{i}+16)(\lambda_{i}+17)(\lambda_{i}+18)(2\lambda_{i}+7)(2\lambda_{i}+9)$
$\displaystyle(2\lambda_{i}+11)(2\lambda_{i}+13)(2\lambda_{i}+15)(2\lambda_{i}+17)(2\lambda_{i}+19)\left(\frac{a^{2}}{\lambda_{i}+1}-\frac{2a}{\lambda_{i}}+\frac{1}{\lambda_{i}-1}\right)$
$\displaystyle B(a)$ $\displaystyle=$ $\displaystyle
a^{2}\left(8\lambda_{i}^{3}+180\lambda_{i}^{2}+1318\lambda_{i}+3135\right)\left(16\lambda_{i}^{13}+1504\lambda_{i}^{12}+61872\lambda_{i}^{11}+\right.$
$\displaystyle\left.1461236\lambda_{i}^{10}+21801935\lambda_{i}^{9}+212273937\lambda_{i}^{8}+1333101101\lambda_{i}^{7}+\right.$
$\displaystyle\left.4996108013\lambda_{i}^{6}+8010950487\lambda_{i}^{5}-11997217421\lambda_{i}^{4}-68501188673\lambda_{i}^{3}-\right.$
$\displaystyle\left.66321743349\lambda_{i}^{2}+59135273262\lambda_{i}+73109116080\right)\left(2\left(-8\left(64b^{4}-\right.\right.\right.$
$\displaystyle\left.\left.\left.35605089\right)\lambda_{i}^{5}-96\left(368b^{4}-85715955\right)\lambda_{i}^{4}-16\left(59872b^{4}-\right.\right.\right.$
$\displaystyle\left.\left.\left.7536410505\right)\lambda_{i}^{3}-48\left(258472b^{4}-19251803493\right)\lambda_{i}^{2}-\right.\right.$
$\displaystyle\left.\left.1728\left(39269b^{4}-1933664031\right)\lambda_{i}+3956121\lambda_{i}^{6}+3828259169280\right)+\right.$
$\displaystyle\left.3956121\left(\lambda_{i}^{5}+70\lambda_{i}^{4}+1940\lambda_{i}^{3}+26600\lambda_{i}^{2}+180384\lambda_{i}+483840\right)m^{2}\right)-$
$\displaystyle
2a\lambda_{i}\left(16\lambda_{i}^{11}+1632\lambda_{i}^{10}+73016\lambda_{i}^{9}+1880616\lambda_{i}^{8}+30716889\lambda_{i}^{7}+\right.$
$\displaystyle\left.329379372\lambda_{i}^{6}+2305033064\lambda_{i}^{5}+9928719744\lambda_{i}^{4}+21299033680\lambda_{i}^{3}-\right.$
$\displaystyle\left.4869591744\lambda_{i}^{2}-119590571520\lambda_{i}-169885900800\right)\left(3956121\left(8\lambda_{i}^{8}+\right.\right.$
$\displaystyle\left.\left.676\lambda_{i}^{7}+24482\lambda_{i}^{6}+495859\lambda_{i}^{5}+6136697\lambda_{i}^{4}+47461414\lambda_{i}^{3}+\right.\right.$
$\displaystyle\left.\left.223683648\lambda_{i}^{2}+586275651\lambda_{i}+652759965\right)\left(2\lambda_{i}(\lambda_{i}+2)+(\lambda_{i}+1)m^{2}\right)-\right.$
$\displaystyle\left.128b^{4}\left(64\lambda_{i}^{9}+5408\lambda_{i}^{8}+195792\lambda_{i}^{7}+3929640\lambda_{i}^{6}+46829160\lambda_{i}^{5}+\right.\right.$
$\displaystyle\left.\left.322610184\lambda_{i}^{4}+1147216028\lambda_{i}^{3}+1425431605\lambda_{i}^{2}-1194241044\lambda_{i}-\right.\right.$
$\displaystyle\left.\left.1751976837\right)\right)+\lambda_{i}\left(16\lambda_{i}^{12}+1792\lambda_{i}^{11}+88864\lambda_{i}^{10}+2567628\lambda_{i}^{9}+\right.$
$\displaystyle\left.47829603\lambda_{i}^{8}+599455521\lambda_{i}^{7}+5108576957\lambda_{i}^{6}+29104930589\lambda_{i}^{5}+\right.$
$\displaystyle\left.104893992911\lambda_{i}^{4}+206701115187\lambda_{i}^{3}+113193419679\lambda_{i}^{2}-\right.$
$\displaystyle\left.236408070717\lambda_{i}-223243908030\right)\left(3956121\left(8\lambda_{i}^{8}+612\lambda_{i}^{7}+19974\lambda_{i}^{6}+\right.\right.$
$\displaystyle\left.\left.362715\lambda_{i}^{5}+4001532\lambda_{i}^{4}+27406788\lambda_{i}^{3}+113514256\lambda_{i}^{2}+\right.\right.$
$\displaystyle\left.\left.259082880\lambda_{i}+248371200\right)\left(2\lambda_{i}+m^{2}\right)-128b^{4}\left(64\lambda_{i}^{8}+4832\lambda_{i}^{7}+\right.\right.$
$\displaystyle\left.\left.154832\lambda_{i}^{6}+2705144\lambda_{i}^{5}+27068168\lambda_{i}^{4}+140926760\lambda_{i}^{3}+\right.\right.$
$\displaystyle\left.\left.253029340\lambda_{i}^{2}-493864887\lambda_{i}-1682001090\right)\right)$
Taking $\tilde{\mu}=\mu$, $\lambda_{i}=\frac{5}{2}$ , b=0.01 and solving the
above equation, we get
$\displaystyle\mu_{min}=1.89037$
where we have discarded the negative value of $\mu_{min}$.
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|
arxiv-papers
| 2012-12-12T07:49:01 |
2024-09-04T02:49:39.193425
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nan Bai, Yi-Hong Gao, Guan-Bu Qi, Xiao-Bao Xu",
"submitter": "Nan Bai",
"url": "https://arxiv.org/abs/1212.2721"
}
|
1212.2753
|
# Congeniality Bounds on Quark Masses from Nucleosynthesis
M. Hossain Ali [email protected] Department of Applied Mathematics,
Rajshahi University, Rajshahi 6205, Bangladesh M. Jakir Hossain
[email protected] Department of Mathematics, Rajshahi University, Rajshahi
6205, Bangladesh Abdullah Shams Bin Tariq [email protected] Department of
Physics, Rajshahi University, Rajshahi 6205, Bangladesh
###### Abstract
The work of Jaffe, Jenkins and Kimchi [Phys. Rev. D79, 065014 (2009)] is
revisited to see if indeed the region of congeniality found in their analysis
survives further restrictions from nucleosynthesis. It is observed that much
of their congenial region disappears when imposing conditions required to
produce the correct and required abundances of the primordial elements as well
as ensure that stars can continue to burn hydrogen nuclei to form helium as
the first step in forming heavier elements in stellar nucleosynthesis. The
remaining region is a very narrow slit reduced in width from around 29 MeV
found by Jaffe et al. to only about 2.2 MeV in the difference of the
nucleon/quark masses. Further bounds on $\delta m_{q}/m_{q}$ seem to reduce
even this narrow slit to the physical point itself.
###### pacs:
14.65.Bt, 26.20.Cd, 98.80.Ft
A reasonably contemporary approach is to study, even without going into
anthropic arguments, the nature of alternative universes as one changes the
values of physical parameters. In the parameter space, one then looks for
regions that could be similar to our universe and may possibly be congenial to
the creation and sustenance of intelligent life Weinberg (1989); Tegmark
(1997, 1998); Tegmark and Rees (1998); Hogan (2000); Tegmark _et al._ (2006);
Jaffe _et al._ (2009); Donoghue _et al._ (2010). Bounds thus obtained may be
referred to as congeniality bounds.
In a recent work, Jaffe, Jenkins and Kimchi Jaffe _et al._ (2009) studied how
sensitive our universe would be to variations of quark masses. For this they
chose to study the variations of masses of the three lightest quarks $u$, $d$
and $s$, under the constraint that the sum of these masses, $m_{T}$ remained
fixed. They also studied variations of $m_{T}$.
Their basic idea was to find the two lightest baryons for any quark mass
combination and consider them to play the roles of the proton and neutron in
forming nuclei. In this process they also considered $\Lambda_{\mathrm{QCD}}$
to be an adjusted free parameter that they tuned to keep the average nucleon
mass at 940 MeV. They then studied the variation of nuclear stability and, in
light of this, tried to obtain the regions of the parameter space where
nuclear chemistry in a somewhat familiar form could be sustained.
The starting point is that the three light quark masses would be changed
keeping their sum $m_{T}$ fixed. This parameter space can be neatly shown in
the form of an equilateral triangle [Fig. 1] where the distances of a point
from the base and right and left sides are, respectively, the masses of the
up, down and strange quarks.
Figure 1: The model space of light quark masses for a fixed $m_{T}$ shown in
the form of a triangle where the distance from the three sides give the three
masses. Figure reproduced from Jaffe _et al._ (2009).
In this manner they identified congenial regions in a triangle parametrized in
terms of $x_{3}$ and $x_{8}$ [Fig. 2] defined as
$\displaystyle x_{3}$ $\displaystyle=$
$\displaystyle\frac{2m_{3}}{\sqrt{3}}\frac{100}{m_{T}^{\oplus}}=\frac{100\left(m_{u}-m_{d}\right)}{\sqrt{3}m_{T}^{\oplus}}$
(1) $\displaystyle x_{8}$ $\displaystyle=$
$\displaystyle\frac{2m_{8}}{\sqrt{3}}\frac{100}{m_{T}^{\oplus}}=\frac{100\left(m_{u}+m_{d}-2m_{s}\right)}{\sqrt{3}m_{T}^{\oplus}}$
(2)
Figure 2: The model space of light quark masses parametrized in terms of
$x_{3}$ and $x_{8}$ reproduced from Jaffe _et al._ (2009). The point labeled
‘us’ points to the physical value in our present universe and therefore has
coordinates $(x_{3}^{\oplus},x_{8}^{\oplus})$.
Here $m_{T}^{\oplus}$ is the sum of the light ($u$, $d$ and $s$) quark masses
in our universe. It is obvious that $x_{3}$ basically gives the isospin
splitting while $x_{8}$ is related to the breaking of $SU(3)_{f}$ due to the
mass of the strange quark. Their results can be summarized in Fig. 3, where
the congenial regions are indicated in green 111If you are reading a black and
white print, then green appears as lightly shaded and red as deep shaded. This
triangle is for $m_{T}^{\oplus}$, i.e. with $m_{T}$ as it is in the present
universe. They have also studied variations in $m_{T}$, but our work is
limited to commenting on the case of $m_{T}^{\oplus}$, understanding that the
same arguments qualitatively extend to other values of $m_{T}$. This is
further justified in the discussions near the end of this report.
Figure 3: (color online) Figure reproduced from Jaffe _et al._ (2009)
identifying congenial regions in the quark mass triangle with green bands. The
red and white regions are uncongenial and uncertain, respectively.
It is increasingly being understood that if there is complexity, fine-tuning
is inevitable Bradford . Even if one is not happy with anthropic arguments, we
simply cannot get away from fine-tuning. With this in mind, the first
impression that one has from Fig. 3, is that the congenial region seems to be
surprisingly large - allowing, around one order of magnitude variations in the
quark masses. Though this already involves intricate compensating adjustments
in $\Lambda_{\mathrm{QCD}}$ to keep the average ‘nucleon’ mass fixed.
However, one should appreciate the difficulty in setting up a new framework in
which a problem can be studied. From this perspective the authors of Jaffe
_et al._ (2009) should be commended for presenting, literally from scratch, a
setup for studying the congeniality bounds on quark masses. This setup can be
extended removing some of the constraints used in any further work. Indeed,
considering the significance of the work it was chosen first for a Viewpoint
article in Physics Perez (2009) and then went into a cover story in Scientific
American Jenkins and Perez (2010).
In our work, we remain within the provided setup, but extend the analysis to
bounds provided by nucleosynthesis. It should be noted here that whereas, on
the one hand, nuclear masses and stability expectedly vary comparatively
slowly with quark masses; on the other hand, the observed abundances of the
lightest nuclei hydrogen and helium provide much more stringent bounds on the
variation of nucleon masses. We report below how the congeniality triangle of
Fig. 3 is modified by the application of these constraints.
At the outset, the variation of the octet baryon masses as one traverses along
the borders of the triangle (Figs. 10 and 11 of Jaffe _et al._ (2009)) were
reproduced to gain confidence in our code and our understanding of the
framework. The fitted parameter values of $c_{T}$, $c_{3}$ and $c_{8}$ from
Table III of Ref. Jaffe _et al._ (2009) were used in the equation
$M_{B}=C_{0}+c_{T}x_{T}+c_{8}x_{8}+c_{3}x_{3}+\left\langle
B\left|H_{\mathrm{EM}}\right|B\right\rangle.$ (3)
This leads to
$\displaystyle M_{p}=C_{0}+3.68\>x_{T}+3.53\>x_{8}+1.24\>x_{3}+0.63$ (4)
$\displaystyle M_{n}=C_{0}+3.68\>x_{T}+3.53\>x_{8}-1.24\>x_{3}-0.13.$ (5)
The quantity occurring most in the analysis below being
$M_{n}-M_{p}=-2.48\>x_{3}-0.76.$ (6)
It may be noted here that using updated values of baryon masses from the
Particle Data Book changes the parameters very slightly and this is neglected
considering the qualitative nature of this work. After a clarification on the
adjustment of $C_{0}$ (corresponding to an adjustment of
$\Lambda_{\mathrm{QCD}}$) from the authors Jaffe _et al._ it was possible to
reproduce the figures.
Then the issue of further bounds from nucleosynthesis were studied. It is well
known that the observed abundances of the primordial nuclei hydrogen
(protons), helium (alpha particles) etc. are sensitively tied to the masses of
the nucleons Barrow and Tipler (1986); Hogan (2000). The slight difference in
the masses of the proton and neutron are responsible for the survival of
protons with the observed abundance. The (un)congenial regions of the triangle
is explored further under these constraints.
There are three cases that arise here:
### Case I: $x_{3}>x_{3}^{\oplus}$
Let us concentrate on the region on the upper right of the triangle with
$x_{3}$ values greater than at the point labeled ‘us’ on the right hand side
of the triangle.
Of the two nucleons, the neutron is heavier by about 1.3 MeV. If it was just
0.8 MeV less, that would bring it below the electron capture threshold for
protons, i.e. it would become energetically favourable for protons to capture
electrons and become neutrons. All the protons would have been converted to
neutrons in the Big Bang. The Universe would be full of neutrons and nothing
else. We would not be here. In words of Barrow and Tipler Barrow and Tipler
(1986)
> Without electrostatic forces to support them, solid bodies would collapse
> rapidly into neutron stars or black holes. Thus, the coincidence that allows
> protons to partake in nuclear reactions in the early universe also prevents
> them decaying by weak interactions. It also, of course, prevents the 75% of
> the Universe which emerges from nucelosynthesis in the form of protons from
> simply decaying away into neutrons. If that were to happen no atoms would
> ever have formed and we would not be here to know it. [Ref. Barrow and
> Tipler (1986), p. 400]
The same issue is also discussed by Hogan Hogan (2000)
> The $u$-$d$ mass difference in particular attracts attention because the $d$
> is just heavier enough than $u$ to overcome the electromagnetic energy
> difference to make the proton ($uud$) lighter than the neutron ($udd$) and
> therefore stable. On the other hand, if it were a little heavier still, the
> deuteron would be unstable and it would be difficult to assemble any nuclei
> heavier than hydrogen.
Therefore, it is necessary to have
$M_{n}-M_{p}\geq 0.5\>\mathrm{MeV}.$ (7)
This reduces the congenial region on the upper right of the physical point to
$x_{3}\leq-0.51.$ (8)
### Case II: $x_{3}<x_{3}^{\oplus}$
Now let us move to the bottem-left side of the triangle (left of the $x_{8}$
axis) where $x_{3}$ values are smaller than that at the ‘us’-labeled point,
again concentrating on the right hand side of the triangle.
The key reaction by which hydrogen ‘burns’ in stars such as the sun involves
the reaction
$\displaystyle p+p$ $\displaystyle\rightarrow$ $\displaystyle
d+e^{+}+\nu+0.42\,\mathrm{MeV}$ (9) $\displaystyle e^{+}+e^{-}$
$\displaystyle\rightarrow$ $\displaystyle 1\>\mathrm{MeV}$ (10)
So the total amount of energy released in this reaction is 1.42 MeV.
If the neutron mass was 1.42 MeV (0.15%) more than it is, this reaction would
not happen at all. It would need energy to make it go, rather than producing
energy. Deuterons are a key step in burning hydrogen to helium. Without them,
hydrogen would not burn, and there would be no long-lived stars and no stellar
nucleosynthesis to produce the remaining elements.
Therefore, it is necessary that
$M_{n}-M_{p}\leq 2.72\>\mathrm{MeV}.$ (11)
This reduces the congenial region on the lower left of the physical point to
$x_{3}\geq-1.4.$ (12)
These two conditions, thus, significantly reduce the congenial corridor from
$-12.9\leq x_{3}\leq 4.1$ (13)
to
$-1.4\leq x_{3}\leq-0.5.$ (14)
It may be noted here that the width of this region is of the same order as the
uncertainty in $x_{3}^{\oplus}$ itself due to uncertainties in the light quark
masses given by $x_{3}^{\oplus}=-1.17\pm 0.43$.
In fact it was a pleasant surprise to realise, rather late into our work, that
Hogan Hogan (2000) reached essentially similar conclusions which were
expressed in terms of the up-down quark mass difference, $\delta m_{d-u}$ and
Section IV of his review Hogan (2000) is a recommended read for anybody
interested in this issue. The approximately 1.4 + 0.8 = 2.2 MeV window of
variation that we find is in agreement with the allowed region in Fig. 1 of
the same Hogan (2000).
### Case III: Left half of the triangle
If we move to the left half of the triangle, we essentially replace the down
quark with a strange quark. We know that the $s$-quark is, in some ways, like
a heavy $d$-quark. In the left half of the triangle the $s$-quark is light and
the $d$-quark is heavy. As if they simply interchange positions. That is why
Jaffe _e_ t al. Jaffe _et al._ (2009) seem to find a symmetric congenial
region in the left of the triangle. The discussions for Cases I and II narrow
it down, but do not remove it.
However, let us now turn towards the coupling between $u$-$d$ and $u$-$s$. The
$u$-$d$ coupling is much stronger, whereas the $u$-$s$ coupling is suppressed.
This is described by the well-known Cabibbo angle $\theta_{C}$. Where the
$u$-$d$ coupling carries a factor $cos\,\theta_{C}$ and the $u$-$s$ coupling
carries a factor of $sin\,\theta_{C}$, the Cabibbo angle being about 13
degrees.
This is like the present world with a much weaker weak interaction. This the
case where the weak decay rate of neutrons is not strong enough to produce the
primordial neutron-proton abundance ratio of 1:6. Without this we are left
without enough protons, i.e. without enough hydrogen, which is key to both
stellar burning and biological life itself. Therefore we are left with only a
narrow region on the right [Fig. 4].
Figure 4: (color online) Fig. 3 adapted by the further restrictions imposed
leaving only a very narrow congenial slit in the bottom-right region of the
triangle.
The only remaining question is probably regarding the length of this narrow
region extending nearly up to the centre of the triangle. As one moves up this
narrow slit towards the centre, away from the physical point, the up-down
quarks become heavier keeping the down quark slightly heavier than the up.
Meanwhile the strange quark becomes lighter to keep $m_{T}$ fixed. The physics
considered here is probably not very sensitive to the strange quark mass. The
increase in the up-down masses is offset by the compensating adjustment in
$\Lambda_{\mathrm{QCD}}$ to keep the nucleon masses fixed. Therefore, the
length of this region could probably be an artifact of the simultaneous and
compensating tuning of quark masses and $\Lambda_{\mathrm{QCD}}$.
This indeed has been one of the conclusions in Jaffe _et al._ (2009) as
summarized more elegantly in Jenkins and Perez (2010); as well as Bradford
where, reviewing the alternative universe landscapes studied by Aguirre
(2001); Harnik _et al._ (2006); Adams (2008); Jaffe _et al._ (2009); Jenkins
and Perez (2010) it has been observed that if one is prepared to adjust
another parameter in a compensating manner, it might be possible to find other
regions in the parameter space that are also congenial. However, that does not
remove the fine-tuning problem, as the alternative values are still finely
tuned and this is inevitable to produce complexity as observed in our present
universe. Here most of the alternatives are removed and the narrow region
remains as a result of the compensating adjustments of
$\Lambda_{\mathrm{QCD}}$.
Indeed along the narrow region the sum of the two lightest quarks vary with
the strange quark mass going in the opposite direction to keep $m_{T}$ fixed.
If the effect of this could be quantified, it would probably be possible to
restrict even the length of the narrow region [See the Addendum].
For example, as noted by Hogan Hogan (2000),
> … the sum of the (up and down) quark masses controls the pion mass, so
> changing them alters the range of the nuclear potential and significantly
> changes nuclear structure and energy levels. Even a small change radically
> alters the history of nuclear astrophysics, for example, by eliminating
> critical resonances of nucleosynthesis needed to produce abundant carbon.
> 222An interesting additional note is that, here Hogan cites Hoyle, F., D. N.
> F. Dunbar, W. A. Wenzel, and W. Whaling, 1953, Phys. Rev. 92, 1095. This has
> been cited several times in different papers, sometimes with Phys. Rev.
> Lett. as the source, and a few times with the title “A state in C12
> predicted from astrophysical evidence”. However, we failed to find any such
> article and would appreciate any information on this reference. The nearest
> match was D. N. F. Dunbar, R. E. Pixley, W. A. Wenzel, and W. Whaling, 1953,
> Phys. Rev. 92, 649, an article on the resonance in 12C often dubbed the
> ‘Hoyle resonance’.
Here it should be added that a more up-to-date view is that the strongest
effect on the scalar scattering lengths and deuteron binding energy seem to be
due to the “sigma-resonance” exchange (or correlated two-pion scalar-isoscalar
exchange) dependence on $m_{\pi}$ Hanhart _et al._ (2008); Pelaez and Rios
(2010).
As mentioned at the outset, the analysis here has been limited to the case of
$m_{T}=m_{T}^{\oplus}$. It has been noted by Jaffe et al. Jaffe _et al._
(2009) that the widths of the two major congenial bands on the bottom-left and
bottom-right of the triangle are independent of $m_{T}$. Therefore, naturally
the further exclusions for $x_{3}>x_{3}^{\oplus}$, $x_{3}<x_{3}^{\oplus}$
reducing the width of the band should also apply to other values of $m_{T}$.
The exclusion of the left half of the triangle should also extend to other
values of $m_{T}$. Therefore, in summary, it can be expected that for all
values of $m_{T}$, after applying constraints from nucleosynthesis, there will
only remain a similar very narrow congenial band at the bottom-right of the
triangle.
An additional comment is due here on the possibilities of universes with
deuterons, sigma-hydrogen, or delta-helium playing the roles of hydrogen as
listed in Jenkins and Perez (2010) as a summary of Jaffe _et al._ (2009). The
point made here does not contradict that these could be stable lightest
elements. It is only pointed out that stability alone is not enough to produce
and sustain nuclear chemistry in a manner familiar to us. Correct primordial
abundances and conditions for sustained stellar burning provide constraints
that are much more difficult to satisfy. This probably calls for a closer
analysis of the other half in Jenkins and Perez (2010) related to possible
universes without any weak interaction of Harnik _et al._ (2006) where there
indeed has been a detailed discussion of these issues pertaining to
nucleosynthesis. However, that would have to be another project; whereas, this
work is focused on Jaffe _et al._ (2009).
In summary, it can be observed that, primordial nuclear abundances and
processes of stellar nucleosynthesis provide much more stringent constraints
on quark masses than nuclear stability. Using these constraints it is possible
to significantly reduce the congenial region in the space of light quark
masses.
## Addendum
Our attention has been drawn through referee comments to studies of the bounds
from nucleosynthesis Bedaque _et al._ (2011); Berengut _et al._ (2013), the
latter appearing after the initial submission of this paper, on $\delta
m_{q}/m_{q}$, where $m_{q}$ is the average of the light (up and down) quark
mass and $\delta m_{q}$ is the change in $m_{q}$ keeping $m_{u}/m_{d}$ fixed.
Coincidentally, along the length of the remaining narrow congenial region
$m_{u}/m_{d}$ is approximately constant. The latest value is $\left|\delta
m_{q}/m_{q}\right|<0.009$ Berengut _et al._ (2013). There are other values in
literature, but they are generally of the same order. Let us try to do a crude
estimate of the effect of this constraint. For $m_{q}\approx 3.8$ MeV, $\delta
m_{q}\approx 0.035$ MeV.
From eqs. 8 and 9, we get
$M_{N}=\left(M_{n}+M_{p}\right)/2=C_{0}+3.68\>x_{T}+3.53\>x_{8}+2.5,$ (15)
which given that $x_{T}$ is kept fixed, leads to
$\delta M_{N}=\delta C_{0}+3.53\>\delta x_{8}.$ (16)
Now, $x_{8}$ as defined in eq. 2 can be re-expressed in terms of $m_{q}$ as
$x_{8}=\frac{200\left(m_{q}-m_{s}\right)}{\sqrt{3}m_{T}^{\oplus}}.$ (17)
If $m_{T}$ is kept fixed then $\delta m_{s}=-\delta m_{q}$, leading to
$\delta x_{8}=\frac{400\left(\delta
m_{q}\right)}{\sqrt{3}m_{T}^{\oplus}}=0.08,$ (18)
where we have used $\delta m_{q}\approx 0.035$ MeV and $m_{T}^{\oplus}\approx
100$ MeV. The remaining congenial region is too small to show on a figure of
this scale. In fact the region $x_{8}=x_{8}^{\oplus}\pm 0.08$ is too small to
show on a plot of this scale and is also very small compared to the
uncertainty in the value of $x_{8}^{\oplus}$ itself $x_{8}^{\oplus}=-59.5\pm
1.1$ due to the uncertainties in the determination of the light quark masses.
However, one should remember our estimate is rather crude without appropriate
consideration of the uncertainties in $m_{q}$. Taking these into account will
increase the region, but keep it within the same order as the uncertainty in
$x_{8}^{\oplus}$ itself. In short there is practically no congenial region
outside $\left(x_{3}^{\oplus},x_{8}^{\oplus}\right)$.
###### Acknowledgements.
The authors are grateful to Robert L. Jaffe, Alejandro Jenkins and Itamar
Kimchi for their kind replies to our queries and helpful suggestions. MHA and
ASBT would also like to acknowledge the support of the Abdus Salam
International Centre for Theoretical Physics (ICTP), Trieste, Italy through a
Regular Associateship and a Junior Associateship, respectively.
## References
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* Tegmark and Rees (1998) M. Tegmark and M. J. Rees, Astrophys.J. 499, 526 (1998), arXiv:astro-ph/9709058 [astro-ph] .
* Hogan (2000) C. J. Hogan, Rev. Mod. Phys. 72, 1149 (2000), arXiv:astro-ph/9909295 [astro-ph] .
* Tegmark _et al._ (2006) M. Tegmark, A. Aguirre, M. J. Rees, and F. Wilczek, Phys.Rev. D73, 023505 (2006), arXiv:astro-ph/0511774 [astro-ph] .
* Jaffe _et al._ (2009) R. L. Jaffe, A. Jenkins, and I. Kimchi, Phys. Rev. D79, 065014 (2009), arXiv:0809.1647 [hep-ph] .
* Donoghue _et al._ (2010) J. F. Donoghue, K. Dutta, A. Ross, and M. Tegmark, Phys.Rev. D81, 073003 (2010), arXiv:0903.1024 [hep-ph] .
* Note (1) If you are reading a black and white print, then green appears as lightly shaded and red as deep shaded.
* (10) R. Bradford, Int. J. Theor. Phys. 50, 1577.
* Perez (2009) G. Perez, Physics 2, 21 (2009).
* Jenkins and Perez (2010) A. Jenkins and G. Perez, Sci. Am. 2010, 302 (2010).
* (13) R. L. Jaffe, A. Jenkins, and I. Kimchi, Private Communication.
* Barrow and Tipler (1986) J. D. Barrow and F. J. Tipler, _The Anthropic Cosmological Principle_ (Oxford University Press, Oxford, New York, 1986).
* Aguirre (2001) A. Aguirre, Phys.Rev. D64, 083508 (2001), arXiv:astro-ph/0106143 [astro-ph] .
* Harnik _et al._ (2006) R. Harnik, G. D. Kribs, and G. Perez, Phys.Rev. D74, 035006 (2006), arXiv:hep-ph/0604027 [hep-ph] .
* Adams (2008) F. C. Adams, JCAP 0808, 010 (2008), arXiv:0807.3697 [astro-ph] .
* Note (2) An interesting additional note is that, here Hogan cites Hoyle, F., D. N. F. Dunbar, W. A. Wenzel, and W. Whaling, 1953, Phys. Rev. 92, 1095. This has been cited several times in different papers, sometimes with Phys. Rev. Lett. as the source, and a few times with the title “A state in C12 predicted from astrophysical evidence”. However, we failed to find any such article and would appreciate any information on this reference. The nearest match was D. N. F. Dunbar, R. E. Pixley, W. A. Wenzel, and W. Whaling, 1953, Phys. Rev. 92, 649, an article on the resonance in 12C often dubbed the ‘Hoyle resonance’.
* Hanhart _et al._ (2008) C. Hanhart, J. R. Pelaez, and G. Rios, Phys.Rev.Lett. 100, 152001 (2008), arXiv:0801.2871 [hep-ph] .
* Pelaez and Rios (2010) J. R. Pelaez and G. Rios, Phys.Rev. D82, 114002 (2010), arXiv:1010.6008 [hep-ph] .
* Bedaque _et al._ (2011) P. F. Bedaque, T. Luu, and L. Platter, Phys.Rev. C83, 045803 (2011), arXiv:1012.3840 [nucl-th] .
* Berengut _et al._ (2013) J. Berengut, E. Epelbaum, V. Flambaum, C. Hanhart, U.-G. Meissner, _et al._ , Phys.Rev. D87, 085018 (2013), arXiv:1301.1738 [nucl-th] .
|
arxiv-papers
| 2012-12-12T10:00:12 |
2024-09-04T02:49:39.203961
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Hossain Ali, M. Jakir Hossain and Abdullah Shams Bin Tariq",
"submitter": "Abdullah Shams Bin Tariq",
"url": "https://arxiv.org/abs/1212.2753"
}
|
1212.2811
|
# Room temperature Rydberg Single Photon Source
M. M. Müller Institut für Quanteninformationsverarbeitung, Universität Ulm,
89081 Ulm, Germany A. Kölle R. Löw T. Pfau 5\. Physikalisches Institut,
Universität Stuttgart, 70550 Stuttgart, Germany T. Calarco Institut für
Quanteninformationsverarbeitung, Universität Ulm, 89081 Ulm, Germany S.
Montangero Institut für Quanteninformationsverarbeitung, Universität Ulm,
89081 Ulm, Germany
###### Abstract
We present an optimal protocol to implement a room temperature Rydberg single
photon source within an experimental setup based on micro cells filled with
thermal vapor. The optimization of a pulsed four wave mixing scheme allows to
double the effective Rydberg blockade radius as compared to a simple Gaussian
pulse scheme, releasing some of the constrains on the geometry of the micro
cells. The performance of the optimized protocol is improved by about 70% with
respect to the standard protocol.
###### pacs:
xxx
The quest to find the perfect hardware to support quantum information
processing is one of the main open problems to be solved to bridge the
distance between quantum information theory and technological applications.
Indeed the conditions to be met are very demanding and can be summarized in
the need of having a (many-body) quantum system perfectly isolated from the
environment but under perfect control at will. One of the promising candidates
to fulfill such stringent conditions are Rydberg atoms as their interaction
strength can be tuned by twelve orders of magnitude by simply exciting them
SaffmanReview . This very peculiar property is very difficult to be found in
other architectures, as interaction is usually either very weak or always-on,
and allowed for a fast development of their application in quantum
information: phase gates GaetanNatPhys09 ; IsenhowerPRL10 and recently a
single photon source based on Rydberg excitation blockade Kuzmich have been
demonstrated using ultracold atoms. Despite these exciting successes, the need
for ultracold temperatures still limits their applicability and thus recent
efforts have been made towards replacing ultracold atoms with thermal vapor
Loew . Indeed there has been considerable progress toward similar results,
demonstrating appropriate coherence times Kuebler , four wave mixing Koelle
and van-der-Waals interaction inprep . Thermal vapor cells have been shown to
act as a long lived quantum memory Budker as well as a probabilistic single
photon source Kuzmich . Thermal vapor cells have even been shown to support
entanglement between quantum states of the spin variables in two distant cells
Polzik . A global approach that combines quantum memories, single photon
sources, quantum logic operations and efficient detectors on one platform is
highly desirable e.g. to implement quantum repeater protocols like the one
proposel by DLCZ DLCZ .
Here we propose a scheme to obtain a single photon source using a thermal
vapor of Rubidium 87 and we study theoretically its feasibility. The proposal
stems from previous works originally thought for ultracold atoms where the
Rydberg blockade has been exploited to prepare a collective single excitation
state (a W-state) as a resource to accomplish different tasks SaffmanWalker ;
lukin . When the W-state decays it will emit a photon with directionality
given by the imprinted laser phases. This decay process has been widely
studied by Scully ; Pedersen ; Manassah ; Dudin ; Bariani leading to the
experimental realization of an ultracold single photon source in Kuzmich . We
exploit optimal control theory to achieve the best possible preparation of the
W-state under the experimental constraints in the more challenging regime at
room temperature. Quantum optimal control theory has been applied successfully
to solve this class of problems in few-body quantum systems in many different
setups Brif and recently it has been proven to be successful also to
optimally drive many-body quantum systems dynamics Doria ; Caneva . We exploit
this recently introduced technique to optimize the state preparation,
simulating the unitary dynamics of the atomic ensemble driven by two lasers by
means of the time-dependent Density Matrix Renormalization Group in its Matrix
Product State (MPS) formulation that allows to include long-range interactions
schollwoeck ; dummies ; MPS . This is the first application of optimal control
theory to many-body quantum systems with long range interactions.
Figure 1: Level Scheme of 87Rb and sketched interaction. The used states are
$|g\rangle=5^{2}S_{1/2}$, $|i\rangle=5^{2}P_{3/2}$, $|r\rangle=44D$,
$|e\rangle=5^{2}P_{1/2}$. Interaction occurs between Rydberg ($|r\rangle$)
states with potential $C_{6}/r^{6}$ and $C_{6}=h\cdot 5880\,\mathrm{MHz\,\mu
m^{6}}$ Singer .
We start our analyses with an ideal scenario of a frozen ordered
unidimensional system: under such conditions, optimal control theory allows to
find the best pulse to reach the W-state with very high fidelity. We then
extract the general features of the found optimal pulses using them as initial
guesses to drive random instances of atoms distributed within a three-
dimensional sphere and optimize them. The resulting pulses provide a robust
strategy to prepare the desired state despite the new disordered geometry and
system dimensionality. Finally, we study in details the effects of finite
temperature (e.g. Doppler shifts) on the properties of the single-photon
source and we show that we obtain a room temperature single photon source with
performances similar to those obtained in the cold atoms case.
## System setup -
We consider a hot rubidium vapor in a wedge shaped micro cell as recently
introduced in Kuebler ; Koelle . At room temperature we can expect a linewidth
of about $2\,$MHz for the 44D-state Adams . Fig. 1 shows the relevant levels
of the 87Rb atoms, where $\Omega_{i}$ are the Rabi frequencies of the lasers
and $\delta_{gi}$, $\delta_{ir}$ and $\delta_{re}$ are the detunings of the
corresponding transitions. By detuning $\Omega_{1}$ from the
$|g\rangle\leftrightarrow|i\rangle$ transition we can adiabatically eliminate
the level $|i\rangle$ in our theoretical model assuming
$\delta_{ig}\gg\Omega_{1},\Omega_{2},\delta_{ir}$ Lambdasystem . From now on
we consider the three level system $|g\rangle$, $|r\rangle$ and $|e\rangle$
and two driving lasers with Rabi frequencies
$\Omega_{gr}=-\frac{\Omega_{1}\Omega_{2}}{2\delta_{gi}}$ and
$\Omega_{re}=\Omega_{3}$. The density of the vapor can be varied by changing
the gas temperature. At $220^{\circ}\,$C this gives a density of 87Rb
($27.83\,\%$ of all Rubidium atoms) of $543\,\mathrm{atoms/\mu m^{3}}$. The
velocity distribution is Gaussian, corresponding to a thermal distribution.
The decay rates are $\tau_{1}=26.2\,\mathrm{ns}$ for the
$|i\rangle\leftrightarrow|g\rangle$ transition, and
$\tau_{2}=27.7\,\mathrm{ns}$ for the $|e\rangle\leftrightarrow|g\rangle$
transition Steck . As the single photon shall be emitted by the latter
transition, this sets also the time scale for the decay process. This also
limits the duration of the exciting laser pulses, that we set to be
$T_{0}=2.5\,$ns. Given the separated timescales with respect to the observed
coherence times of about a hundred ns, we simulate a closed, coherent system
as far as the state preparation is concerned and analyze the non-coherent part
of the system evolution separately. Finally, the Hamiltonian of the system
$H=\sum_{i}H_{loc,i}+H_{int}$ after the adiabatic elimination and choosing
$\Omega_{1}=\Omega_{2}$ is given by
$\displaystyle H_{loc,i}$
$\displaystyle=\frac{\Omega_{gr}}{2}(|g\rangle\langle r|+|r\rangle\langle
g|)+\frac{\Omega_{3}}{2}(|r\rangle\langle e|+|e\rangle\langle r|)$
$\displaystyle+\delta_{ir}|r\rangle\langle r|+\delta_{re}|e\rangle\langle
e|\,$ (1) $\displaystyle H_{int}$ $\displaystyle=\sum_{i\neq
j}\frac{C_{6}}{r_{ij}^{6}}|r_{i}r_{j}\rangle\langle r_{i}r_{j}|.$ (2)
If we change the pulse duration from $T_{0}$ to $T$ and at the same time we
scale the Rabi frequencies and detunings by $T_{0}/T$, while typical length
scales like the diameter $L_{0}$ of the system scale by
$L_{T}=L_{0}(T/T_{0})^{1/6}$ the Hamilton practically remains the same.
## State preparation -
The goal of the manipulation protocol is to prepare a collective single
excitation state
$|W\rangle=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\mathrm{e}^{\imath\vec{k}_{0}\vec{r}_{i}}|e_{i}\rangle$
(where $|e_{i}\rangle=|g...geg...g\rangle$ means atom $i$ in state $e$ and all
other atoms in state $g$), starting from the system’s ground state
$|\psi(0)\rangle=|gg\dots gg\rangle$.
Figure 2: Final infidelity $\epsilon$ for a Gaussian $\pi$-pulses (circles)
and optimal pulses (squares) in a one-dimensional $N$ atoms chain with lattice
constant $a_{0}=0.35\,\mathrm{\mu m}$ at pulse duration
$T_{0}=2.5\,\mathrm{ns}$ and total extension $L_{0}=a_{0}(N-1)$.
In order to engineer the preparation of the W-state we simulate the time
evolution of the system and optimize the pulse sequence. The time evolution is
obtained by a tensor-network algorithm including two-body long-range
interactions schollwoeck ; MPS . The optimization is performed by means of the
Chopped RAndom Bases (CRAB) optimal control technique for many-body quantum
systems dynamics recently introduced in Doria ; Caneva . The method is based
on the a priori reduction of the problem complexity via a proper truncation of
the accessible space used to describe the system wave function and the control
field. In particular, in the system introduced above, the Rabi frequencies of
two driving lasers $\Omega_{gr}(t)$ and $\Omega_{er}(t)$ are subject to
optimization by the CRAB optimization method. The pulses are expanded in a
truncated basis $\Omega_{i}=\sum_{j=1}^{N_{i}}c_{ij}f_{ij}(t)$ where the
$c_{ij}$ are the coefficients of the expansion and the basis functions
$f_{ij}(t)$ are chosen according to physical properties of the system. In
particular here we choose the first principal harmonics of the Fourier
expansion. The figure of merit to be minimized is the final infidelity of the
system state evolved under the pulses $\Omega$ with respect to the target
state $|W\rangle$, namely
$\epsilon=1-|\langle W|\psi(T)\rangle|^{2}.$ (3)
The multi-variable function defined by (3), $\epsilon\equiv\epsilon(c_{ij})$
is then minimized by means of direct-search methods Doria . Typical parameters
are $N_{i}=14$, $10^{4}$ iterations with $10^{4}$ Trotter steps and bound
dimension $10:40$ for the MPS simulation.
## Results -
We first concentrate on a one-dimensional chain of $N$ Rydberg atoms with
fixed positions. Despite its simplicity – as we shall show later – this model
already captures the essential features of the system dynamics as the long
range interactions effectively map this system in a particular instance of the
full 3D scenario that we consider later. An experimental realization can be
with optical lattices like in arimondo . The interaction between two excited
Rydberg atoms can act as a blockade for double excitations and it enhances the
driving Rabi frequency for the $|g\rangle\leftrightarrow|r\rangle$ transition
by a factor of $\sqrt{N}$ in an ensemble of $N$ atoms, as one excitation
suppresses $N-1$ other possible excitations GaetanNatPhys09 ; BlockadeRadius ;
lukin .
Figure 3: Left: Gaussian $\pi$-pulses guess. Middle: Pulse from optimization
of 4 atoms chain. Right: Guess optimized for 3D cloud. Red: $\Omega_{gr}$,
blue: $\Omega_{re}$.
Fig. 2 shows the final infidelity $\epsilon$ with respect to the $|W\rangle$
state obtained via Gaussian $\pi$-pulses (Fig. 3, left) of atoms in a 1D chain
with lattice spacing $a_{0}=0.35\,\mathrm{\mu m}$ for $T_{0}=2.5\,\mathrm{ns}$
and total extension $L_{0}=a_{0}(N-1)$. Notice that for different numbers of
atoms $N$ the Rabi frequency $\Omega_{gr}$ has been rescaled by $1/\sqrt{N}$
to correct for the $\sqrt{N}$ enhancement due to the Rydberg blockade effect.
From the figure it is clear that the Gaussian $\pi$-pulses fail for
$L_{0}>0.7\,\mathrm{\mu m}$ (corresponding to more than three atoms) where
$\Omega_{gr}$ gets comparable to the interaction strength. On the contrary the
optimized pulses yield infidelity $\epsilon<10^{-2}$ up to $L_{0}\approx
1.4\,\mathrm{\mu m}$, thus allowing to almost double the system size. So by
optimization we can outperform the blockade radius associated with the guess
pulse. This effect exceeds what one can expect by just stretching
$\Omega_{gr}$ over the whole operation time and thus lowering its bandwidth.
The presented pulses have only real Rabi frequencies. Allowing also for
complex values (via phase modulation) roughly results in a 15% relative
improvement of the state preparation error.
As stated in the introduction, our final goal is to prepare an ensemble of
atoms with (uniformly distributed) random positions in 3D space moving
randomly with thermally distributed velocities in the W-state. We thus now
consider a 3D cloud of $N$ frozen atoms at random positions. Differently from
the previous case, the interactions between atoms are random due to the
distance-dependent interaction terms and this means that we have to produce
very robust pulses “on average” at the cost of some fidelity loss for each
given sample of atoms. To this aim, we extract the relevant features of the
optimized pulses e.g. Fig. 3 (middle) and optimize only few parameters like
the height and the width of the two pulses. The resulting optimal pulses are
reported in the right panel of Fig. 3; improvement here is mainly due to the
reduction of the bandwidth. The $|g\rangle\leftrightarrow|r\rangle$ transition
is performed via a long flat pulse followed by a fast kick in the
$|r\rangle\leftrightarrow|e\rangle$ transition. Finally, we compare the
performance of the two schemes: the infidelity $\epsilon$ obtained by W-state
preparation with the optimized pulse and with the guess Gaussian $\pi$-pulses
averaged over different system samples (i.e. different instances of the random
positions). Fig. 4 shows the infidelity $\epsilon$ for clouds of ten atoms as
a function of the maximum cloud extension $L_{0}$ while in the inset of Fig. 4
$\epsilon$ is shown as a function of the number of atoms $N=7,8,\dots 11$ in
the sample. In both cases the errorbars correspond to the statistical noise
from 8 realizations of random atoms positions within a cloud of fixed maximum
diameter $L_{0}$. The optimized pulses clearly result in improved infidelities
and most importantly with almost no dependency on the number of atoms. This
result, as we shall show below, is corroborated also by our analytical
estimate of accuracy of the optimal state preparation. An additional
complexity aspect for the experimental realization of the proposed protocol
arises from the fact that the number $N$ of atoms in the ensemble depends on
the density and the excitation volume and can only be determined with Poisson
precision ($\pm\sqrt{N}$) changing a $\pi$-pulse roughly as
$\cos(\frac{\pi}{2}\sqrt{1\pm N})\approx 1-\frac{\pi^{2}}{32N}$. This very
weak dependence on the atom number $N$ of the final fidelity paves the way to
a successful experimental realization of the W-state at room temperature.
Figure 4: 3D cloud of 10 randomly distributed atoms: infidelity $\epsilon$ as
a function of the cloud’s diameter $L_{0}$ plotted for Gaussian $\pi$-pulses
(circles) and an optimized sequence (squares). The inset shows that $N$ has
almost no influence on $\epsilon$. The curves in the inset correspond to
$L_{0}=0.83\dots 1.46\,\mathrm{\mu m}$ and the optimized pulse.
We can provide an analytical estimate of the state preparation error under the
hypothesis that: a) the major deviation of the prepared state from the desired
W-state is given by the population in the two-excitations sector, b) each
Rydberg atom is approximately performing an independent Rabi oscillation
between $|g\rangle$ and $|r\rangle$ and c) the influence on an atom by another
nearby excited one is to detune the single atom dynamics. The detuning will
just be the van-der-Waals interaction between two Rydberg excited atoms
$V=C_{6}d_{ij}^{-6}$, with $d_{ij}$ the distance between atom $i$ and atom $j$
and thus the resulting Rabi oscillation for this single atom is then only
going up to having a fraction of
$\frac{\Omega_{gr}^{2}}{\Omega_{gr}^{2}+V^{2}}=1-\frac{1}{1+c^{2}d_{ij}^{12}}$
in the excited (Rydberg) state ($c=\Omega_{gr}/C_{6}$). This double excitation
part is missing in the population $P_{i}$ representing the first excitation
sector. Summing this up over all neighbors gives
Figure 5: 3D cloud of $N=10$ randomly distributed atoms: Errors in the state
preparation. $P_{i}$ calculated by MPS simulation is compared to the theory
curve from the $P(r)$ model.
$P_{i}\approx\frac{1}{N(N-1)}\sum_{j=1,j\neq
i}^{N}\frac{1}{1+c^{2}d_{ij}^{12}}\,.$ (4)
In the thermodynamical limit an analogous continuous expression can be
obtained by considering a homogeneous spherical cloud of radius $R$,
introducing $P(r)$ as the single atom excitation. This yields
$P(r)\approx\frac{1}{NV}\int\frac{\rho^{2}\sin^{2}\theta\,d\rho\,d\theta\,d\phi\,}{1+c^{2}(r^{2}+\rho^{2}-2r\rho\sin\theta\cos\phi)^{6}}$
(5)
with $r$ the radial position (distance from the center of mass of the cloud)
of atom $i$ in the cloud and integrating over the sphere’s volume. Fig. 5
shows the comparison between Eq. (5) and the $P_{i}$ obtained from a numerical
simulation of the dynamics of ten atoms. The good correspondence of this high
density limit theory and numerical results of the few body simulation supports
the previous finding of the very weak dependence of the results on the number
of atoms $N$, that is, considering only ten atoms already is enough to get the
major features of the state preparation.
If we now include in our theoretical description the fact that the atoms are
moving with a thermal distribution, we shall consider the lasers’ Doppler
shifts depending on the velocity of the atoms. Furthermore we shall analyze
the effects of a time-dependent interaction $V(t)$ and the fact that the atoms
might move out of the laser beam during state preparation, as at room
temperature an average atom moves about $0.5\,\mathrm{\mu m}$ in $T_{0}$ time.
For this time scale we can achieve good blockade within a sphere of radius
$0.5-0.55\,\mathrm{\mu m}$ (corresponding to $L_{0}=1-1.1\,\mathrm{\mu m}$).
As the pulse is robust with respect to random positions we expect that the
thermal motion will not affect drastically the results.
Finally, the Doppler effect will cause a widening of the excitation
populations with a Lorentzian shape $P_{i}^{D}\propto
1/(1+v_{i,\parallel}^{2}k^{2}/\Omega^{2})$ depending on the velocity
$v_{i,\parallel}$ of atom $i$ in the direction of the incoming lasers. The
Doppler shift enters as an additional term $(k_{1}\pm
k_{2})v_{i,\parallel}|r\rangle\langle r|$ into $H_{loc,i}$ (we neglect the
Doppler shift of the third laser since $\Omega_{re}\gg k_{re}v_{i}$). The wave
numbers $k_{i}$ of lasers 1 ($|g\rangle\leftrightarrow|i\rangle$ transition)
and 2 ($|i\rangle\leftrightarrow|r\rangle$ transition) are summed up or
subtracted depending on whether the lasers are parallel or anti parallel. We
consider a sphere of radius $0.53\,\mathrm{\mu m}$ and an atomic density and
velocity distribution corresponding to temperatures of $200-260\,^{\circ}$C,
that is approximately $N=200-1200$ atoms and a velocity distribution
characterized by a Gaussian width of $213-226\,\mathrm{m/s}$. We calculate the
approximate W-state $|\tilde{W}\rangle=\sum_{i}\alpha_{i}|e_{i}\rangle$ (where
the $\alpha_{i}$ contain the Doppler widening depending on the directions of
the lasers) obtained by our state preparation pulses by simulating the
evolution by $\Omega_{gr}$ for $N$ atoms assuming perfect blockade, thus
keeping track only of the ground state and the $N$ singly excited states and
constraining the system evolution to the ground state and the first excitation
sector. To model the decay of $|\tilde{W}\rangle$ we follow the full
exponential kernel description Pedersen ; Eberly ; Manassah ; Scully ; Dudin ;
Bariani and in addition we include also the motion of the particles.
Figure 6: Directionality $p$ as a function of temperature $\vartheta$ for
parallel lasers (red) and anti parallel lasers (black). Squares are values
from state preparation with optimized pulses, circles from Gaussian
$\pi$-pulses. The inset shows the emission cone on resonance for
$\vartheta=220^{\circ}$C and anti parallel lasers.
Fig. 6 shows the resulting directionality $p$ (the probability of photon
emission in the forward direction) as a function of the temperature, each
value averaged over 240 random realizations of a uniform position distribution
and Gaussian velocity distribution after a decay time of $100\,\mathrm{ns}$.
The inset shows the emission cone on resonance for $\vartheta=220\,^{\circ}$C.
The forward cone is defined to cover about $3\,\%$ of the solid angle
corresponding to a maximum deviation from the forward direction of
$0.3\,\mathrm{rad}$ (dashed lines). Values for $p$ are black for parallel and
red for anti parallel lasers on the $|g\rangle\leftrightarrow|i\rangle$ and
$|i\rangle\leftrightarrow|r\rangle$ transitions. Clearly, the highest
directionality is obtained for anti parallel lasers. Squares correspond to
state preparation with the optimized pulse. Circles mean state preparation
with Gaussian $\pi$-pulses - in this case we have to decrease the cloud radius
and thus deal only with about half as many atoms participating in the W-state
leading to a reduction of the directionality.
At higher temperatures samples are denser and thus more atoms can be excited,
resulting in higher directionality. However, the collision rate is enhanced
thus coherence times are decreased. Therefore the temperature has to be chosen
properly. The best choice in the experimental settings is assumed to be in the
range of $220^{\circ}\,$C, corresponding to an overall $70\,\%$ increase in
directionality compared to the guess pulse.
## Conclusions -
We have analyzed the possibility of constructing a single photon source using
a thermal vapor of Rubidium 87. The results, however, are also qualitatively
valid for other atoms like Cesium. By CRAB assisted pulse shaping using a
Matrix Product State code we showed how to increase the number of atoms in a
W-state by a factor of two compared to the initial guess which corresponds
roughly to an increase of 70 % in directionality of the emitted photon.
Choosing the exciting lasers as anti parallel gives an additional crucial
increase. We plan to experimentally test these predictions also to quantify
the effect of the collisions and laser focus not included in the present
description and to see how they compare to experiments with ultracold atoms
Kuzmich .
###### Acknowledgements.
The authors acknowledge support from EU grants AQUTE and MALICIA; SFB/TRR21;
QuOReP and we thank the bwGRiD project 111bwGRiD (www.bw-grid.de), member of
the German D-Grid initiative, funded by BMBF and MWK Baden-Württemberg. for
the computational resources.
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|
arxiv-papers
| 2012-12-12T13:23:12 |
2024-09-04T02:49:39.211569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. M. M\\\"uller, A. K\\\"olle, R. L\\\"ow, T. Pfau, T. Calarco, and S.\n Montangero",
"submitter": "Matthias M\\\"uller",
"url": "https://arxiv.org/abs/1212.2811"
}
|
1212.2832
|
# Assembling large, complex environmental metagenomes
Adina Chuang Howe1,2 Janet Jansson3,4 Stephanie A. Malfatti3 Susannah G.
Tringe3 James M. Tiedje1,2 and C. Titus Brown1,5∗
###### Abstract
The large volumes of sequencing data required to sample complex environments
deeply pose new challenges to sequence analysis approaches. _De novo_
metagenomic assembly effectively reduces the total amount of data to be
analyzed but requires significant computational resources. We apply two pre-
assembly filtering approaches, digital normalization and partitioning, to make
large metagenome assemblies more computationaly tractable. Using a human gut
mock community dataset, we demonstrate that these methods result in assemblies
nearly identical to assemblies from unprocessed data. We then assemble two
large soil metagenomes from matched Iowa corn and native prairie soils. The
predicted functional content and phylogenetic origin of the assembled contigs
indicate significant taxonomic differences despite similar function. The
assembly strategies presented are generic and can be extended to any
metagenome; full source code is freely available under a BSD license.
Microbiology and Molecular Genetics, Michigan State University, East Lansing,
MI, USA
Plant, Soil, and Microbial Sciences, Michigan State University, East Lansing,
MI, USA
Department of Energy (DOE) Joint Genome Institute, Walnut Creek, CA, USA
Lawrence Berkeley National Laboratory, Earth Sciences Division, Berkeley, CA,
USA
Computer Science and Engineering, Michigan State University, East Lansing, MI,
USA
Complex microbial communities operate at the heart of many crucial
terrestrial, aquatic, and host-associated processes, providing critical
ecosystem functionality that underpins much of biology [1, 2, 3, 4, 5, 6, 7].
These systems are difficult to study in situ, and we are only beginning to
understand their diversity and functional potential. Advances in DNA
sequencing now provide unprecedented access to the genomic content of these
communities via shotgun sequencing, which produces millions to billions of
short-read sequences [2, 4, 5]. Because shotgun sequencing samples communities
randomly, ultradeep sequencing is needed to detect rare species in
environmental samples, with an estimated 50 Tbp needed for an individual gram
of soil [8]. Both short read lengths and the large volume of sequencing data
pose new challenges to sequence analysis approaches. A single metagenomic
project can readily generate as much or more data than is in global reference
databases; for example, a human-gut metagenome sample containing 578 Gbp [5]
produced more than double the data in NCBI RefSeq (Release 56). Moreover,
short reads contain only minimal signal for homology searches and are error-
prone, limiting direct annotation approaches against reference databases. And
finally, the majority of genes sequenced from complex metagenomes typically
contain little or no similarity to experimentally studied genes, further
complicating homology analysis [1, 5].
_De novo_ assembly of raw sequence data offers several advantages over
analyzing the sequences directly. Assembly removes most random sequencing
errors and decreases the total amount of data to be analyzed. These resulting
assembled contigs are longer than sequencing reads and provide gene order.
Importantly, _de novo_ assembly does not rely on the existence of reference
genomes, thus allowing for the discovery of novel genomic elements. The main
challenge for metagenomic applications of _de novo_ assembly is that current
assembly tools do not scale to the high diversity and large volume of
metagenomic data: metagenomes from rumen, human gut, and permafrost soil
sequencing could only be assembled by discarding low abundance sequences prior
to assembly [2, 4, 5]. Traditional assemblers are designed for single genomes
whose abundance distribution and diversity content are typically simpler than
community metagenomes. Although many metagenome-specific assemblers have
recently been developed for community assembly, most of these assemblers do
not scale to extremely large samples [9].
Here, we combine two approaches, digital normalization and partitioning, to
tackle the problem of de novo metagenome assembly. Digital normalization
normalizes sequence coverage and reduces the dataset size by discarding reads
from high-coverage regions [10]. Subsequently, partitioning separates reads
based on transitive connectivity, resulting in easily assembled subsets of
reads [11, 12]. We evaluate these approaches by applying them to a human gut
mock community dataset, and find that these filtering methods result in
assemblies nearly identical to assemblies from the unprocessed dataset. Next,
we apply these approaches to the assembly of metagenomes from two matched
soils, 100-year cultivated Iowa agricultural soil and native Iowa prairie. We
compare the predicted functional capacities and phylogenetic origins of the
assembled contigs and conclude that despite significant phylogenetic
differences, the functions encoded in both soil data sets are similar. We also
show that virtually no strain-level heterogeneity is detectable within the
assembled reads.
## Results
## Data reduction results in similar assemblies
We evaluated the recovery of reference genomes from de novo metagenomic
assembly by comparing unfiltered traditional assembly to the the described
filtered assembly (Fig. 6; see Methods and Supplementary Information).
Initially, the abundance of genomes within the mock dataset was estimated
based on the reference genome coverage of sequence reads in the unfiltered
dataset. Coverage (excluding genomes with less than 3-fold coverage) ranged
from 6-fold to 2,000-fold (Supplementary Table 1 and Supplementary Fig. 2 and
3). Overall, the unfiltered dataset reads covered a total of 93% of the
reference genomes. Filtering removed 5.9 million reads, 40% of the total
(Table 1); the remaining reads covered 91% of the reference genomes (Table 1
and Supplementary Fig. 2 and 3).
We next compared the recovery of reference genomes in contigs assembled from
the original and filtered datasets. Using the Velvet assembler, we recovered
43% and 44% of the reference genomes, respectively. The assembly of the
original dataset contained 29,063 contigs and 38 million bp, while the
filtered assembly contained 30,082 contigs and 35 million bp (Table 2).
Comparable recoveries of references between original and filtered datasets
were also obtained with other assemblers (SOAPdenovo and Meta-IDBA, Table 2).
Overall, the unfiltered and filtered assemblies were very similar, sharing 95%
of genomic content. For the highest abundance references (the plasmids
NC_005008.1, NC_005007.1, and NC_005003.1), the unfiltered assembly recovered
significantly more of the original sequence; however, for the large majority
of genomes, the filtered assembly recovered similar (and sometimes greater)
amounts of the reference genomes (Supplementary Fig. 2 and 3). The
distribution of contig lengths in unfiltered and filtered assemblies were also
comparable (Supplementary Fig. 4).
We estimated the abundance of assembled contigs and reference genomes using
the mapped sequencing reads (Supplementary Fig. 5). Above a sequencing
coverage of five, the majority of reads which could be mapped to reference
genomes were included in the assembled contigs (Supplementary Fig. 3 and 4).
Below this threshold, reads could be mapped to reference genomes but were less
likely to be associated with assembled contigs. We next compared the
abundances of the reference genomes to the abundances of the contigs in the
unfiltered and filtered assemblies. The abundance estimations from the
filtered assembly were significantly closer to predicted abundances from
reference genomes (_n_ = 28,652; p-value = 0.032, see Supplementary
Information).
## Partitioning separates most reads by species
We next partitioned the filtered data set based on de Bruijn graph
connectivity and assembled each partition independently [11, 12]. The
resulting assemblies of unpartitioned and partitioned were more than 99%
identical. In the mock dataset, we identified 9 million reads in 85,818
disconnected partitions (Supplementary Fig. 6). Among these, only 2,359 (2.7%)
of the partitions contained reads originating from more than one genome,
indicating that partitioning separated reads from distinct species.
In general, reference genomes with high sequencing coverage were associated
with fewer partitions (Supplementary Table 1): a total of 112 partitions
contained reads from high abundance reference genomes (coverage above 25)
compared to 2,771 partitions associated with lower abundance genomes. This is
consistent with the observation that the main effect of low coverage is to
“break” connectivity in the assembly graph [13, 14].
To further evaluate the effects of partitioning, we introduced spiked reads
from _E. coli_ genomes into the mock community dataset. First, simulated reads
from a single genome (_E. coli_ strain E24377A, NC_009801.1 with 2%
substitution error and 10x coverage) were added to the mock community dataset
and then processed in the same way as the unfiltered mock dataset. We observed
similar amounts of data reduction after digital normalization and partitioning
(Table 1). Among the 81,154 partitioned sets of reads, we identified only
2,580 (3.2%) partitions containing reads from multiple genomes. A total of 424
partitions contained reads from the spiked _E. coli_ genome (201 partitions
contained _only_ spiked reads) and when assembled aligned to 99.5% of _E.
coli_ strain E24377A genome (4,957,067 of 4,979,619 bp) (Supplementary Fig.
6).
Next, we introduced five closely-related _E. coli_ strains into the mock
community dataset and performed the same analysis. Partitioning this “mix-
spiked” mock community dataset resulted in 81,425 partitions, of which 1,154
(1.4%) partitions contained reads associated with multiple genomes. Among the
partitions which contained reads associated with a single genome, 658
partitions contained reads originating from one of the spiked _E. coli_
strains. In partitions containing reads from more than one genome, 224
partitions contained reads from a spiked _E. coli_ strain and one other
reference genome (either another spiked strain or from the mock community
dataset) (Supplementary Fig. 7). We independently assembled the partitions
containing reads originating from the spiked _E. coli_ strains. Among the
resulting 6,076 contigs, all but three contigs originated from a spiked _E.
coli_ genome. The remaining three contigs were more than 99% similar to HMP
mock reference genomes (NC_000915.1, NC_003112.2, and NC_009614.1). The
contigs associated with _E. coli_ were aligned against the spiked reference
genomes, recovering greater than 98% of each of the five genomes. Many of
these contigs contained similarities to reads originating from multiple
genomes (Supp Fig. 8), and 3,075 contigs (51%) could be aligned to reads which
originated from more than one spiked genome. This result is comparable to the
fraction of contigs which are associated with multiple genomes in the
unfiltered data set, where 66% of 4,702 contigs associated with spiked reads
contain reads that originate from more than one spiked genome.
## Data reduction and partitioning enable the assembly of two soil metagenomes
We next applied these approaches to the de novo assembly of two soil
metagenomes. Iowa corn and prairie datasets (containing 1.8 billion and 3.3
billion reads, respectively) could not be assembled by Velvet in 500 GB of
RAM. A 75 million reads subset of the Iowa corn dataset alone required 110 GB
of memory, suggesting that assembly of the 3.3 billion read data set might
need as much as 4 TB of RAM (Supplementary Fig. 9). Applying the same
filtering approaches as described above, the Iowa corn and prairie datasets
were reduced to 1.4 billion and 2.2 billion reads, respectively, and after
partitioning, a total of 1.0 billion and 1.7 billion reads remained,
respectively. Prefiltering used 300 GB of RAM or less. The large majority of
k-mers in the soil metagenomes are relatively low-abundance (Fig. 2), and
consequently digital normalization did not remove as many reads in the soil
metagenomes as in the mock data set.
Based on the mock community dataset, we estimated that above a sequencing
depth of five, the large majority of sequences could be assembled into contigs
larger than 300 bp (Supplementary Fig. 1). Given the greater diversity
expected in the soil metagenomes, we normalized these datasets to a sequencing
depth of 20 (i.e., discarding redundant reads within dataset above this
coverage). After partitioning the filtered datasets, we identified a total
31,537,798 and 55,993,006 partitions (containing more than five reads) in the
corn and prairie datasets, respectively. For assembly, we grouped partitions
together into files containing 10 million reads. Data reduction and
partitioning were completed in less than 300 GB of RAM; once partitioned, each
group of reads could be assembled in less than 14 GB and 4 hours. This readily
enabled the usage of multiple assemblers and assembly parameters.
The final assembly of the corn and prairie soil metagenomes resulted in a
total of 1.9 million and 3.1 million contigs greater than 300 bp,
respectively, and a total assembly length of 912 million bp and 1.5 billion
bp, respectively. To estimate abundance of assembled contigs and evaluate
incorporation of reads, all quality-trimmed reads were aligned to assembled
contigs. Overall, for the Iowa corn assembly, 8% of single reads and 10% of
paired end reads mapped to the assembly. Among the paired end reads, 95.5% of
the reads aligned concordantly. In the Iowa prairie assembly, 10% of the
single reads and 11% of the paired end reads aligned to the assembled contigs,
and 95.4% of the paired ends aligned concordantly (Table 4). Based on these
mappings, we calculated read recovery in assembled contigs within the soil
metagenomes (Fig. 3). Overall, there is a positively skewed distribution of
read overage of all contigs from both soil metagenomes, biased towards a
coverage of less than ten-fold, and 48% and 31% of total contigs in Iowa corn
and prairie assemblies respectively had a median basepair coverage less than
10.
Among contigs, the presence of polymorphisms was examined by identifying the
amount of consensus obtained by reads mapped (Supplementary Information
Methods). For both the Iowa corn and prairie metagenomes, more than 99.9% of
contigs contained base calls which were supported by a 95% consensus from
mapped reads over 90% of their lengths, demonstrating an unexpectedly low
polymorphism rate (Supplementary Fig. 10).
## Annotation of the soil assemblies revealed similar functional profiles but
different taxonomy
We annotated assembled contigs through the MG-RAST pipeline, which was
modified to account for per-contig abundance. This annotation resulted in
2,089,779 and 3,460,496 predicted protein coding regions in the corn and
prairie metagenomes, respectively. The large majority of these regions did not
share similarity with any gene in the M5NR database – 61.8% in corn and 70.0%
in prairie. In total, 613,213 (29.3%) and 777,454 (22.5%) protein coding
regions were assigned to functional categories. The functional profile of
these annotated features against SEED subsystems were compared (Fig. 5). For
both the corn and prairie metagenomes, the most abundant functions in the
assembly were associated with the carbohydrate (e.g., central carbohydrate
metabolism and sugar utilization), amino acid (e.g., biosynthesis and
degradation), and protein (e.g., biosynthesis, processing, and modification)
metabolism subsystems. The subsystem profile of both metagenomes were very
similar while the taxonomic profile of the metagenomes based on the
originating taxonomy (phyla) was different (Fig. 4, Supp Methods). Within both
metagenomes, Proteobacteria were most abundant. In Iowa corn, Actinobacteria,
Bacteroidetes, and Firmicutes were the next most abundant, while in the Iowa
prairie, Acidobacteria, Bacteroidetes, and Actinobacteria were the next most
abundant. The Iowa prairie also had nearly double the fraction of
Verrucomicrobia than did Iowa corn.
## Discussion
The diversity and sequencing depth represented by the mock community is
extremely low compared to that of most environmental metagenomes; however, it
represents a simplified, unevenly sampled model for a metagenomic dataset
which enables the evaluation of analyses through the availability of source
genomes. For this dataset, the filtering approaches described above were
effective at reducing the dataset size without significant loss of assembly.
This strategic filtering takes advantage of the observed coverage “sweet spot”
at which point sufficient sequences are present for robust assembly (Supp Fig.
1). The normalization of sequences also resulted in more even coverage (Fig.
2), minimizing assembly problems caused by variable coverage. Additional
reduction of the dataset was achieved by the removal of high abundance
sequences [11].
The specific effects of filtering varied depending on differences between
reference genomes. Variable abundance and conserved regions in references had
an impact on filtered assembly recovery. The filtered assemblies of the three
plasmids of the _Staphylococcus epidermidis_ genome (NC_005008.1, NC_005007.1,
and NC_005003.1) were highly abundant (Supplementary Table 1) and shared
several conserved regions (90% identity over more than 290 bp). During
normalization, repetitive elements in these genomes would appear as high
coverage elements and be removed, as evidenced by a large difference in the
number of reads associated with NC_005008.1 in the unfiltered and filtered
datasets (supplementary Fig. 2). Consequently, the unfiltered dataset
contained more reads spanning these repetitive regions. This most likely
enabled assemblers to extend the assembly of these sequences and resulted in
the observed increased recovery of these genomes in the unfiltered assemblies.
This result, though rare among the mock reference genomes, identifies a
shortcoming of our approach, and indeed for most short-read assembly
approaches, related to repetitive regions and/or polymorphisms. For the soil
metagenomes our data reduction may have caused some information loss in
exchange for the ability to assemble previously intractable data sets.
Evaluation of the mock community dataset suggests that this information loss
is minimal overall and that our approaches result in a comparable assembly
whose abundance estimations are slightly improved.
Metagenomes contain many distinct genomes, which are largely disconnected from
each other but which sometimes share sequences due to conservation or lateral
transfer. Our prefiltering approach removes both common multi-genome elements
as well as artificial connectivity stemming from the sequencing process. As
shown above on the mock data set, the removal of these sequences does not
significantly alter the recovery of reference genomes through de novo
assembly: the resulting assemblies of unpartitioned and partitioned datasets
were nearly identical for the mock data. The large majority of these
partitions contained reads from a single reference genome, supporting our
previous hypothesis that most connected subgraphs contain reads from distinct
genomes [12]. As expected, high abundance, well-sampled genomes were found to
contain fewer partitions and low abundance, under-sampled genomes contained
more partitions, due to fragmentation of the assembly graph.
We further examined the recovery of sequences through partitioning by
computationally spiking in one or more _E. coli_ strains before applying
filtering and partitioning. When we spiked in a single _E. coli_ strain, we
could reassemble 99% of the original genome (Supplementary Fig. 6). When we
spiked in five closely related strains, we could recover the large majority of
the genomic content of these strains, albeit largely in chimeric contigs
(Supplementary Fig. 8). This result is not unexpected, as assemblies of the
unfiltered dataset resulted in a slightly higher fraction of assembled contigs
associated with multiple references. Overall, closely related sequences which
result from either repetitive or inter-strain polymorphisms challenge
assemblers, and our approaches are not specifically designed to target such
regions. However, the partitions resulting from our approach could provide a
much-reduced subset of sequences to be targeted for more sensitive assembly
approaches for highly variable regions (i.e. overlap-layout-consensus
approaches or abundance binning approaches [15]).
One valuable result of partitioning is that it subdivides our datasets into
sets of reads which can be assembled with minimal computational resources. For
the mock community dataset, this gain was small, reducing unfiltered assembly
at 12 GB and 4 hours to less than 2 GB and 1 hour. However, for the soil
metagenomes, previously impossible assemblies could be completed in less than
a day and in under 14 GB of memory enabling the usage of multiple assembly
parameters (e.g., k-length) and multiple assemblers (Velvet, SOAPdenovo, and
MetaIDBA).
The final assemblies of the corn and prairie soil metagenomes resulted in a
total of 1.9 million and 3.1 million contigs, respectively, and a total
assembly length of 912 million bp and 1.5 billion bp, respectively –
equivalent to $\approx$ 500 _E. coli_ genomes worth of DNA. We evaluated these
assemblies based on paired-end concordance, which showed that the majority of
the assembled contigs agreed with the raw sequencing data. Overall, there is a
positively skewed distribution of abundance of all contigs from both soil
metagenomes, biased towards an abundance of less than ten, indicative of the
low sequencing coverage of these metagenomes.
This study represents the largest published soil metagenomic sequencing effort
to date, and these assembly results demonstrate the enormous amount of
diversity within the soil. Even with this level of sequencing, millions of
putative genes were defined for each metagenome, with hundreds of thousands of
functions. More than half of the assembled contigs are not similar to anything
in known databases, suggesting that soil holds considerable unexplored
taxonomic and functional novelty. Among the protein coding sequences which
were annotated, comparisons of the two soil datasets suggests that the
functional profiles are more similar to one another than the complementing
phylogenetic profiles. This result supports previous hypotheses that despite
large diversity with two different soil systems, the microbial community
provides similar overall function [16, 17, 18, 19].
We present two strategies that readily enable the assembly of very large
environmental metagenomes by discarding redundancy and subdividing the data
prior to assembly. These strategies are generic and should be applicable to
any metagenome. We demonstrate their effectiveness by first evaluating them on
the assembly of a mock community metagenome, and then applying them to two
previously intractable soil metagenomes.
Partitioning is an especially valuable approach because it enables the
extraction of read subsets that should assemble together. These read
partitions are small enough that a variety of assembly, abundance analysis,
and polymorphism analysis techniques can be easily applied to them
individually.
The two soil assemblies also provide a deeper glimpse of the opportunities and
challenges of large scale environmental metagenomics in high-diversity systems
such as soil: we identified millions of novel putative proteins, most of
unknown function.
This project was supported by Agriculture and Food Research Initiative
Competitive Grant no. 2010-65205-20361 from the United States Department of
Agriculture, National Institute of Food and Agriculture and National Science
Foundation IOS-0923812, both to C.T.B. A.H. was supported by NSF Postdoctoral
Fellowship Award #0905961 and the Great Lakes Bioenergy Research Center
(Department of Energy BER DE-FC02-07ER64494). The work conducted by the U.S.
Department of Energy Joint Genome Institute is supported by the Office of
Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
We acknowledge the support of Krystle Chavarria and Regina Lamendella for
extraction of DNA from Great Prairie soil samples and the technical support of
Eddy Rubin and Tijana Glavina del Rio at the DOE JGI and John Johnson and Eric
McDonald at MSU HPC.
A.H. and C.T.B. designed experiments and wrote paper. A.H. performed
experiments and analyzed the data. J.J., S.T., and J.T. discussed results and
commented on manuscript. S.M. managed raw sequencing datasets.
The authors declare that they have no competing financial interests.
Correspondence should be addressed to C. Titus Brown ([email protected]).
## Tables
Table 1: The total number of reads in unfiltered, filtered (normalized and
high abundance (HA) k-mer removal), and partitioned datasets and the
computational resources required (memory and time).
| Unfiltered | Filtered | Partitioned
---|---|---|---
| Reads (Mbp) | Reads (Mbp) | Reads (Mbp)
HMP Mock | 14,494,884 (1,136) | 8,656,520 (636) | 8,560,124 (631)
HMP Mock Spike | 14,992,845 (1,137) | 8,189,928 (612) | 8,094,475 (607)
HMP Mock Multispike | 17,010,607 (1,339) | 9,037,142 (702) | 8,930,840 (697)
Iowa Corn | 1,810,630,781 (140,750) | 1,406,361,241 (91,043) | 1,040,396,940 (77,603)
Iowa Prairie | 3,303,375,485 (256,610) | 2,241,951,533 (144,962) | 1,696,187,797 (125,105)
| Unfiltered (GB / h) | Filtered and Partitioned (GB / h) |
HMP Mock | 4 / $<$2 | 4 / $<$2 |
HMP Mock Spike | 4 / $<$2 | 4 / $<$2 |
HMP Mock Multispike | 4 / $<$2 | 4 / $<$2 |
Iowa Corn | 188 / 83 | 234 / 120 |
Iowa Prairie | 258 / 178 | 287 / 310 |
Table 2: Assembly summary statistics (total contigs, total million bp assembly
length, maximum contig size bp) of unfiltered (UF) and filtered (F) or
filtered/partitioned (FP) datasets with Velvet (V) assembler. Assembly for UF
and FP datasets also shown for MetaIDBA (M) and SOAPdenovo(S) assemblers. Iowa
corn and prairie metagenomes could not be completed on unfiltered datasets.
| UF | F | FP | Assembler
---|---|---|---|---
HMP Mock | 29,063 / 38 / 146,795 | 30,082 / 35 / 90,497 | 30,115 / 35 / 90,497 | V
HMP Mock | 24,300 / 36 / 86,445 | - | 27,475 / 36 / 96,041 | M
HMP Mock | 36,689 / 37 / 32,736 | - | 29,295 / 37 / 58,598 | S
Iowa corn | N/A | N/A | 1,862,962 / 912/ 20,234 | V
Iowa corn | N/A | N/A | 1,334,841 / 623 / 15,013 | M
Iowa corn | N/A | N/A | 1,542,436 / 675 / 15,075 | S
Iowa prairie | N/A | N/A | 3,120,263 / 1,510 / 9,397 | V
Iowa prairie | N/A | N/A | 2,102,163 / 998 / 7,206 | M
Iowa prairie | N/A | N/A | 2,599,767 / 1,145 / 5,423 | S
Table 3: Assembly comparisons of unfiltered (UF) and filtered (F) or filtered/partitioned (FP) HMP mock datasets using different assemblers (Velvet (V), MetaIDBA (M) and SOAPdenovo (S)). Assembly content similarity is based on the fraction of alignment of assemblies and similarly, the coverage of reference genomes is based on the alignment of assembled contigs to reference genomes (RG). Assembly Comparison | Percent Similarity | RG Coverage | Assembler
---|---|---|---
UF vs. F | 95% | 43.3% / 44.5% | V
UF vs. FP | 95% | 43.3% / 44.4% | V
UF vs. FP | 93% | 46.5% / 45.4% | M
UF vs. FP | 98% | 46.2% / 46.4% | S
Table 4: Fraction of single-end (SE) and paired-end (PE) reads mapped to Iowa corn and prairie Velvet assemblies. | Iowa Corn Assembly | Iowa Prairie Assemby
---|---|---
Total Unfiltered Reads | 1,810,630,781 | 3,303,375,485
Total Unfiltered SE Reads | 141,517,075 | 358,817,057
SE aligned 1 time | 11,368,837 | 32,539,726
SE aligned $>$ 1 time | 562,637 | 1,437,284
% SE Aligned | 8.43% | 9.47%
Total Unfiltered PE Reads | 834,556,853 | 1,472,279,214
PE aligned 1 time | 54,731,320 | 110,353,902
PE aligned $>$ 1 time | 1,993,902 | 3,133,710
% PE Aligned Disconcordantly | 0.47% | 0.63%
% PE Aligned | 9.68% | 11.20%
## Figures
Figure 1: K-mer coverage of HMP mock community dataset before and after
filtering approaches.
Figure 2: K-mer coverage of Iowa corn and prairie metagenomes before and after
filtering approaches.
Figure 3: Coverage (median basepair recovered) distribution of assembled
contigs from Iowa corn soil (top) and Iowa prairie soil (bottom) metagenomes.
Figure 4: Phylogenetic distribution from SEED subsystem annotations for Iowa
corn and prairie metagenomes.
Figure 5: Functional distribution from SEED subsystem annotations for Iowa
corn and prairie metagenomes.
Figure 6: Flowchart describing methods for _de novo_ metagenomic assembly.
Using the HMP mock community dataset, alternative approaches for data
reduction and assembly were compared. (a) Disconnected subgraphs of the
assembly graph were partitioned. Most connected subgraphs contained reads and
contigs aligning to distinct genomes (Supplementary Fig. 6). (b) The genomic
content of all assemblies were found to be comparable in genomic content. (c)
Reads and assembled contigs could be aligned to reference genomes to determine
effectiveness of recovery. (d) The abundance of contigs (based on read
mapping) could be compared to estimated abundances of corresponding reference
genomes.
## Online Methods
## Assembly Pipeline
The entire assembly pipeline for the mock community is described in detail in
an IPython notebook available for download at
_http://nbviewer.ipython.org/urls/raw.github.com/ngs-docs/ngs-notebooks/
master/ngs-70-hmp-diginorm.ipynb_ and
_http://nbviewer.ipython.org/urls/raw.github.com/ngs-docs/ngs-notebooks/
master/ngs-71-hmp-diginorm.ipynb_. Soil assembly was performed with the same
pipeline and parameter changes as described in Supplementary Information. The
annotated metagenome for Iowa corn can be found at
_http://metagenomics.anl.gov/linkin.cgi?metagenome=4504797.3_ and Iowa prairie
at _http://metagenomics.anl.gov/linkin.cgi?metagenome=4504798.3_.
## Statistical Methods
The reference-based abundance (from reads mapped to reference genomes) and
assembly-based abundance (from reads mapped to contigs) of genomes were
compared. Using a one-directional, paired t-test of squared deviations, the
abundance estimates of the unfiltered and filtered assemblies were compared.
The mean and standard deviation of the abundances of unfiltered contigs,
filtered contigs, and reference genes were 6.8 +/- 7.1, 8.1 +/- 7.7, and 7.8
+/- 5.2, respectively. We expected the filtered assembly to have increased
accuracy due to a reduction of errors (e.g. normalization and high abundance
filtering) and used a one-sided t-test which indicated that abundance
estimations from the filtered assembly were significantly closer to predicted
abundances from reference genomes (n=28,652, p-value of 0.032).
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* [19] Konstantinidis, K. T. & Tiedje, J. M. Trends between gene content and genome size in prokaryotic species with larger genomes. _Proceedings of the National Academy of Sciences of the United States of America_ 101, 3160–3165 (2004).
|
arxiv-papers
| 2012-12-12T14:56:18 |
2024-09-04T02:49:39.219178
|
{
"license": "Public Domain",
"authors": "Adina Chuang Howe, Janet Jansson, Stephanie A. Malfatti, Susannah G.\n Tringe, James M. Tiedje, C. Titus Brown",
"submitter": "C. Titus Brown",
"url": "https://arxiv.org/abs/1212.2832"
}
|
1212.2928
|
040005 2012 M. C. Barbosa 040005
With the aim of establishing a criterion for identifying when a lipid bilayer
has reached steady state using the molecular dynamics simulation technique,
lipid bilayers of different composition in their liquid crystalline phase were
simulated in aqueous solution in presence of CaCl2 as electrolyte, at
different concentration levels. In this regard, we used two different lipid
bilayer systems: one composed by 288 DPPC (DiPalmitoylPhosphatidylCholine) and
another constituted by 288 DPPS (DiPalmitoylPhosphatidylSerine). In this
sense, for both type of lipid bilayers, we have studied the temporal evolution
of some lipids properties, such as the surface area per lipid, the deuterium
order parameter, the lipid hydration and the lipid-calcium coordination. From
their analysis, it became evident how each property has a different time to
achieve equilibrium. The following order was found, from faster property to
slower property: coordination of ions $\approx$ deuterium order parameter $>$
area per lipid $\approx$ hydration. Consequently, when the hydration of lipids
or the mean area per lipid are stable, we can ensure that the lipid membrane
has reached the steady state.
# A criterion to identify the equilibration time in lipid bilayer simulations
Rodolfo D. Porasso [inst1] J.J. López Cascales E-mail: [email protected]
[inst2]
(3 September 2012; 24 October 2012)
††volume: 4
99 inst1 Instituto de Matemática Aplicada San Luis (IMASL) - Departamento de
Física, Universidad Nacional de San Luis/CONICET, D5700HHW, San Luis,
Argentina. inst2 Universidad Politécnica de Cartagena, Grupo de Bioinformática
y Macromoléculas (BioMac) Aulario II, Campus de Alfonso XIII, 30203 Cartagena,
Murcia, Spain.
## 1 Introduction
Over the last few decades, different computational techniques have emerged in
different fields of science, some of them being extensively implemented and
used by a great number of scientists around the globe. Among others, the
Molecular Dynamics (MD) simulation is a very popular computational technique,
which is widely used to obtain insight with atomic detail of steady and
dynamic properties in the fields of biology, physics and chemistry. In this
regard, a critical aspect that must be identified in all the MD simulations is
related to the required equilibration time to achieve a steady state. This
point is crucial in order to avoid simulation artifacts that could lead to
wrong conclusions. Currently, with the increment of the computing power
accessible to different investigation groups, much longer simulation
trajectories are being carried out to obtain reliable information about the
systems, with the purpose of approaching the time scale of the experimental
phenomena. However, even when this fact is objectively desirable without
further objections, nowadays, much longer equilibration times are arbitrarily
being required by certain reviewers during the revision process. From our
viewpoint, this should be thoroughly revised due to the following two main
reasons: first, because it results in a limiting factor in the use of this
technique by other research groups which cannot access to very expensive
computing centers (assuming that authors provide enough evidence of the
equilibration of the system). Second, to avoid wasting expensive computing
time in the study of certain properties which do not require such long
equilibration times, once the steady state of the system has been properly
identified.
Phospholipid bilayers are of a high biological relevance, due to the fact that
they play a crucial role in the control of the diffusion of small molecules,
cell recognition, and signal transduction, among others. In our case, we have
chosen the PhosphatidylCholine (PC) bilayer because it has been very well
studied by MD simulations [1, 2, 3, 4, 5, 6, 7] and experimentally as well [8,
9, 10, 11, 12, 13, 14]. Furthermore, studies of the effects of different types
of electrolytes on a PC bilayer have also been studied, experimentally [15,
16, 17, 18, 19, 20, 21, 22, 23, 24] and by simulation [25, 26, 27, 28, 29, 30,
31, 32, 33].
As mentioned above, the Molecular Dynamics (MD) simulations have emerged
during the last decades as a powerful tool to obtain insight with atomic
detail of the structure and dynamics of lipid bilayers [34, 35, 36]. Several
MD simulations of membranes under the influence of different salt
concentrations have been carried out. One of the main obstacles related to
these studies has been the time scale associated to the binding process of
ions to the lipid bilayer. Considering the literature, a vast dispersion of
equilibration times associated to the binding of ions to the membrane has been
reported, where values ranging from 5 to 100 ns have been suggested for
monovalent and divalent cations [25, 27, 28, 29, 32, 37, 38]. In this regard,
we carried out four independent simulations of a lipid bilayer formed by 288
DPPC in aqueous solutions, for different concentrations of CaCl2 to provide an
overview of their equilibration times. Among other properties, the surface
area per lipid, the deuterium order parameters, lipid hydration and lipid-
calcium coordination were studied.
Finally, in order to generalize our results, a bilayer formed by 288 DPPS in
its liquid crystalline phase, in presence of CaCl2 at 0.25N, was simulated as
well .
## 2 Methodology
Different molecular Dynamics (MD) simulations of lipid bilayer formed by 288
DPPC were carried out in aqueous solutions for different concentrations of
CaCl2, from 0, up to 0.50 N. Furthermore, with the aim of generalizing our
results, a bilayer of 288 DPPS in presence of CaCl2 at 0.25 N was simulated as
well. Note that the concentration of CaCl2 in terms of normality is defined
as:
$\textrm{normality}=\frac{n_{\textrm{equivalent
grams}}}{l_{\textrm{solution}}}$ (1)
where $n_{\textrm{equivalent
grams}}=\frac{\textrm{gr(solute)}}{\textrm{equivalent weight}}$ and
$\textrm{equivalent weight}=\frac{\textrm{Molecular weight}}{n}$, being $n$
the charge of the ions in the solution.
In Table 1, the number of molecules that constitute each system, applying Eq.
(1), is summarized.
Type of Lipid | [CaCl2] N | Ca2+ | Cl- | Water
---|---|---|---|---
DPPC | 0 | 0 | 0 | 10068
DPPC | 0.06 | 5 | 10 | 10053
DPPC | 0.13 | 12 | 24 | 10032
DPPC | 0.25 | 23 | 46 | 9999
DPPC | 0.50 | 46 | 92 | 9930
DPPS | 0.25 | 204 | 120 | 26932
Table 1: The simulated bilayer systems. Note that the salt concentration is
given in normal units. The numerals describe the number of molecules contained
in the simulation box.
To build up the original system, a single DPPC lipid molecule, or DPPS lipid
(Fig. 1), was placed with its molecular axis perpendicular to the membrane
surface ($xy$ plane). Next, each DPPC, or DPPS, was randomly rotated and
copied 144 times on each leaflet of the bilayer. Finally, the gaps existing in
the computational box (above and below the phospholipid bilayer) were filled
using an equilibrated box containing 216 water molecules of the extended
simple point charge (SPC/E) [39] water model.
Figure 1: Structure and atom numbers for DPPC and DPPS used in this work.
Thus, the starting point of the first system of Table 1 was formed by 288 DPPC
in absence of CaCl2. Once this first system was generated, the whole system
was subjected to the steepest descent minimization process to remove any
excess of strain associated with overlaps between neighboring atoms of the
system. Thereby, the following DPPC systems in presence of CaCl2 were
generated as follows: to obtain a [CaCl${}_{2}]$=0.06 N, 15 water molecules
were randomly substituted by 5 Ca2+ and 10 Cl-. An analogous procedure was
applied to the rest of the systems, where 36, 69 and 138 water molecules were
substituted by 12, 23 and 46 Ca2+ and 24, 46 and 92 Cl-, to obtain a
[CaCl${}_{2}]$ concentration of 0.13 N, 0.25 N and 0.50 N, respectively.
Finally, the DPPS bilayer was generated following the same procedure described
above for the DPPC, starting from a single DPPS molecule and once the lipid
bilayer in presence of water passed the minimization process, 324 water
molecules were substituted by 204 Ca2+ and 120 Cl- (note that 144 of the 204
calcium ions were added to balance the negative charge associated with the
DPPS).
GROMACS 3.3.3 package [40, 41] was used in the simulations, and the properties
showed in this work were obtained using our own code. The force field proposed
by Egberts et al. [2] was used for the lipids, and a time step of 2 fs was
used as integration time in all the simulations. A cut-off of 1.0 nm was used
for calculating the Lennard-Jones interactions. The electrostatic interaction
was evaluated using the particle mesh Ewald method [42, 43]. The real space
interaction was evaluated using a 0.9 nm cut-off, and the reciprocal space
interaction using a 0.12 nm grid with a fourth-order spline interpolation. A
semi-isotropic coupling pressure was used for the coupling pressure bath, with
a reference pressure of 1 atm which allowed the fluctuation of each axis of
the computer box independently. For the DPPC bilayer, each component of the
system (i.e., lipids, ions and water) was coupled to an external temperature
coupling bath at 330 K, which is well above the transition temperature of 314
K [44, 45]. For DPPS bilayer, each component of the system was coupled to an
external temperature coupling bath at 350 K, which is above the transition
temperature [46, 47]. All the MD simulations were carried out using periodic
boundary conditions. The total trajectory length of each simulated system was
of 80 ns of MD simulation, where the coordinates of the system were recorded
every 5 ps for their appropriate analysis.
Finally, in order to study the effect of the temperature, only the case
corresponding to 0.25 N CaCl2 was investigated at two additional temperatures,
340 K and 350 K.
## 3 Results and discussion
### 3.1 Effect of the CaCl2 concentration
#### 3.1.1 Surface area per lipid
Surface area per lipid $\langle A\rangle$ is a property of lipid bilayers
which has been accurately measured from experiments [48]. The calculation of
mean area per lipid can be determined from the MD simulation as:
$\langle A\rangle=\frac{x\cdot y}{N}$ (2)
where $x$ and $y$ represent the box sizes in the direction $x$ and $y$
(perpendicular to the membrane surface) over the simulation, and $N$ is the
number of lipids contained in one leaflet, in our case $N=144$.
Figure 2: Running area per lipid at T = 330 K in presence of [CaCl2] at (A)
0.06 N, (B) 0.13 N, (C) 0.25 N, (D) 0.50 N and (E) 0.25 N (in this case, T =
350 K). Solid lines represent the mean area obtained from the last 70 ns of
the simulated trajectories (see text for further explanation). The type of
lipid is indicated in the legends.
Focusing on the study of the time evolution of the area per lipid, Figure 2
depicts the running surface area per lipid for different concentrations of
CaCl2 and type of lipid. In general, for the 5 bilayers formed by DPPC or
DPPS, the area per lipid achieved a steady state after 10 ns of simulation,
being this equilibration time almost independent of the concentration of CaCl2
and type of lipid which composed the membrane.
In absence of salt, an average area per lipid of $\langle A\rangle=0.663\pm
0.008$ nm2 was calculated from the last 70 ns of the simulated trajectory,
discarding the first 10 ns corresponding to the equilibration time. This value
agrees with experimental data, where values in a range from 0.55 to 0.72 nm2
have been measured [10, 11, 48, 49, 50, 51]. Table 2 shows the mean surface
area per lipid (again, after discarding the equilibration time of 10 ns) with
their corresponding error bar.
From the simulation results, a shrinking in the surface area per lipid with
the increment of the ionic strength of the solution is observed. This
shrinking is expected and attributed to the complexation of lipid molecules by
calcium, such as it has been pointed out in previous studies [28, 29, 52].
#### 3.1.2 Deuterium order parameter
The deuterium order parameter, $S_{CD}$, is measured from 2H-NMR experiments.
This parameter provides relevant information related to the disorder of the
hydrocarbon region in the interior of the lipid bilayers by measuring the
orientation of the hydrogen dipole of the methylene groups with respect to the
perpendicular axis to the lipid bilayer. Due to the fact that hydrogens of the
lipid methylene groups (CH2) have not been taken into account (in an explicit
way) in our simulations, the order parameter $-S_{CD}$ on the $i+1$ methylene
group was defined as the normal unitary vector to the vector defined from the
$i$ to the $i+2$ CH2 group and contained in the plane formed by the methylene
groups $i$, $i+1$ and $i+2$. Thus, the deuterium order parameter $-S_{CD}$ on
the $i-th$ of the CH2 group can be estimated by Molecular Dynamics simulations
as follows:
$-S_{CD}=\frac{1}{2}\langle{3\cos^{2}(\theta)-1}\rangle$ (3)
where $\theta$ is the angle formed between the unitary vector defined above
and the $z$ axis. The expression in brackets $\langle\dots\rangle$ denotes an
average over all the lipids and time. Hence, note that the $-S_{CD}$ can adopt
any value between -0.5 (corresponding to a parallel orientation to the
lipid/water interface) and 1 (oriented along the normal axis to the lipid
bilayer).
Type of | [CaCl2] N | $\langle A\rangle$ (nm2) | Hydration
---|---|---|---
Lipid | | | Number
DPPC | 0 | 0.663 $\pm$0.008 | 1.758 $\pm$0.009
DPPC | 0.06 | 0.658 $\pm$0.008 | 1.740 $\pm$0.009
DPPC | 0.13 | 0.651 $\pm$0.007 | 1.719 $\pm$0.010
DPPC | 0.25 | 0.641 $\pm$0.009 | 1.680 $\pm$0.015
DPPC | 0.50 | 0.628 $\pm$0.010 | 1.610 $\pm$0.015
DPPS | 0.25 | 0.522 $\pm$0.007 | 2.552 $\pm$0.010
Table 2: Area per lipid and lipid hydration number as a function of salt
concentration (see text for further explanation). Note that the salt
concentration is given in normal units. Error bars were calculated for each
system separately from subtrajectories of 10 ns length. Simulation temperature
= 330 K.
Figure 3 shows the running $-S_{CD}$ for different carbons of the DPPC and
DPPS tails and salt concentrations. Only the carbons which correspond to the
initial (hydrocarbons 2 and 6), the middle (hydrocarbon 10) and final
(hydrocarbons 13 and 15) methylene groups of the lipid tails were depicted in
this figure. Each point of the figure represents the average values of
$-S_{CD}$ over 5 ns of subtrajectory length, and the lines represent the mean
values calculated from the last 70 ns of the trajectories simulated. From this
figure, it is observed how in all the cases, the required equilibration time
is less than 10 ns of simulation time, independently of the salt concentration
and the type of lipid. Finally, it is noted that Figure 3 exhibits an increase
in the deuterium order parameters with the salt concentration, consistent with
the shrinking of the area per lipid described above.
Figure 3: Running deuterium order parameter, $-S_{CD}$, in presence of [CaCl2]
at (A) 0.06 N, (B) 0.13 N, (C) 0.25 N, (D) 0.50 N and (E) 0.25 N. DPPC
simulations were performed at 330 K and DPPS simulation temperature was 350 K.
Solid lines represent mean values $-S_{CD}$ obtained from the last 70 ns of
the simulated trajectories. The type of lipid is indicated in the legends.
Symbols: $\circ$ hydrocarbon 2; $\diamond$ hydrocarbon 6; $\triangleleft$
hydrocarbon 10; $+$ hydrocarbon 13 and $\times$ hydrocarbon 15. Note that the
error bars have the same size as symbol.
#### 3.1.3 Lipid hydration
To analyze the lipid hydration, the radial distribution function $g(r)$ of
water around one of the oxygens of the phosphate group (atom number 10 in Fig.
1 for DPPC and DPPS) was calculated. The radial distribution function
$g\left(r\right)$ is defined as follows:
$g(r)=\frac{N(r)}{4{\pi}r^{2}{\rho}\delta r}$ (4)
where $N\left(r\right)$ is the number of atoms in a spherical shell at
distance $r$ and thickness $\delta r$ from a reference atom. $\rho$ is the
density number taken as the ratio of atoms to the volume of the total
computing box.
From numerical integration of the first peak of the radial distribution
function, the hydration numbers can be estimated for different atoms of the
DPPC or DPPS. Figure 4 depicts the hydration number of phosphate oxygen (atom
10 in Fig. 1 for DPPC and DPPS) in presence of CaCl2, where each point
represents the average of 5 ns subtrajectory length. These results show how
this property reached a steady state for the cases (A), (B) and (E), after 10
ns of simulation. However, for the cases (C) and (D), 5 ns of extra simulation
trajectory were required to reach a steady state. Table 2 shows the hydration
numbers for the last 70 ns of the trajectory length, corresponding to four
concentrations of CaCl2 and both types of lipids, DPPC and DPPS. In this
regard, from Fig. 4, the significant lipid dehydration with the increment of
the ionic strength of the solution is evident, in good accordance with
previous results [52].
Figure 4: Hydration number of the phosphate oxygen (atom 10 in Fig. 1) along
the simulated trajectories in presence of [CaCl2] at (A) 0.06 N, (B) 0.13 N,
(C) 0.25 N, (D) 0.50 N (for DPPC T = 330 K) and (E) 0.25 N. In this case, T =
350 K. Solid lines represent the mean value of the hydration number calculated
from the last 70 ns of the simulated trajectories. The type of lipid is
indicated in the legends. Note that the error bars have the same size as
symbol.
#### 3.1.4 Phospholipid-calcium coordination
Some authors have reported how the lipid coordination by divalent cations
widely varies . Thus, on the one hand, some authors [25] have reported that
this is a very slow process, which requires about 85 ns of simulation time,
but, on the other hand, other authors [26] have suggested that this process
results much more rapid, taking less than 1 ns. In this sense, the
coordination of DPPC-Ca2+ was studied by monitoring the oxygen-calcium
coordination of the carbonyl oxygens (atoms 16 and 35 in Fig. 1) and phosphate
oxygens (atoms 9 and 10 of DPPC in Fig. 1), as a function of time. The left
column in Fig. 5 represents the oxygen coordination number, while the right
one depicts the percentage of calcium ions involved in the coordination
process with respect to the total number of calcium ions present in the
aqueous solution. Figure 5 shows how the DPPC coordination by calcium is a
quick process, taking less than 5 ns of simulation time to achieve a steady
state. The kinetic of this process appears to be related to the ratio between
calcium/lipid. Thus, after the first 5 ns of simulation time, the Ca–lipid
coordination presents some fluctuation along the rest of the simulated
trajectory. However, in Fig. 5 (A) and (B) (for the cases of lower
concentration), it is observed how the percentage of coordination fluctuates
between a 60% and a 100%. We consider that this broad fluctuation is related
to the limited sample size of our simulations that introduces a certain noise
in our results.
Figure 5: Left column represents the number of Ca2+ coordinated to lipids in
presence of [CaCl2] at (A) 0.06 N, (B) 0.13 N, (C) 0.25 N and (D) 0.50 N, for
T = 330 K. Right column shows the quantity of calcium ions coordinated to
lipids expressed in percentage, along the simulated trajectory.
### 3.2 Effect of temperature
This section focuses on the study of the role played by temperature on the
equilibration process. In this regard, only the system corresponding to a
concentration of 0.25 N in CaCl2 was studied, for a range of temperatures from
330 K to 350 K (all of them above the transition temperature of 314 K [44, 45]
for the DPPC).
Figure 6 shows the running area along the trajectory. In this case, it was
noticed how the systems achieve a steady state after a trajectory length of
roughly 10 ns, where Table 3 shows the mean area per lipid calculated from the
last 70 ns of simulation time. Figure 7 shows the deuterium order parameter of
the methylene groups along the lipid tails, calculated from Eq. (3). Figure 7,
on the one hand, clearly shows that for the three temperatures the systems
have reached the steady state before the first 10 ns of simulation. On the
other hand, it shows an increase in the disorder of the lipid tails with
temperature, which is closely related with the increase of the area per lipid,
such as it was pointed out above. Figure 8 depicts the results of the
hydration numbers of DPPC for the three temperatures studied, where the
equilibrated state was achieved after 10 ns of simulation time. Table 3
provides the calculated hydration numbers in the equilibrium, showing how the
lipid hydration remained invariable with the rising of the temperature.
Concerning the lipid-calcium coordination, Fig. 9 represents the lipid-calcium
coordination number, and the right column represents the calcium that
participates in the coordination expressed in percentage respect the total of
calcium ions in solution. From simulation, it becomes evident how calcium ions
required less than 5 ns to achieve an equilibrated state for the three
temperatures studied. In summary, for all the properties studied in this
section, a slight decrease in the equilibration time with the increasing
temperature was appreciated.
T (K) | $\langle A\rangle$ (nm2) | Hydration Number
---|---|---
330 | 0.642 $\pm$0.009 | 1.680 $\pm$ 0.010
340 | 0.650 $\pm$0.007 | 1.683 $\pm$ 0.020
350 | 0.666 $\pm$0.008 | 1.689 $\pm$ 0.015
Table 3: Area per lipid and the lipid hydration number as a function of
temperature. Error bars were calculated from subtrajectories of 10 ns length.
DPPC bilayer in presence of [CaCl2] = 0.25 N. Figure 6: Running area per lipid
for [CaCl2] = 0.25 N at different temperatures, (A) T = 330 K, (B) T = 340 K
and (C) T = 350 K. Solid lines represent the mean values obtained from the
last 70 ns of simulation. Figure 7: Deuterium Order Parameter, $-S_{CD}$,
along the simulated trajectory for a concentration of [CaCl2] = 0.25 N, for
the following temperatures: (A) T = 330 K, (B) T = 340 K and (C) T = 350 K.
Solid lines represent the average order parameter for the last 70 ns of
simulation. Note that the error bars have the same size as symbol. Figure 8:
Hydration number of phosphate oxygen (atom 10 in Fig. 1) along the simulated
trajectories for a [CaCl2] = 0.25 N at different temperatures: (A) T = 330 K,
(B) T = 340 K and (C) T = 350 K. Solid lines represent the average hydration
number for the last 70 ns of simulation. Note that the error bars have the
same size as symbol. Figure 9: Left column represents the number of calcium
ions involved in the lipid coordination along time for a concentration of
[CaCl2]= 0.25 N at different temperatures: (A) T = 330 K, (B) T = 340 K and
(C) T = 350 K. The right column shows the same information expressed as a
percentage of the total number of calcium ions in solution.
## 4 Conclusions
The present work deals with the simulation time required to achieve the steady
state for a lipid bilayer system in presence of CaCl2. In this regard, we
studied two different systems: one with DPPC and another one with DPPS
bilayer; both systems in presence of CaCl2 (at different level concentration).
The salt free case was also studied, as control. The analysis of various lipid
properties studied here indicates that some properties reach the steady state
more quickly than others. In this sense, we found that the area per lipid and
the hydration number are slower than the deuterium order parameter and the
coordination of cations. Consequently, to ensure that a system composed by a
lipid bilayer has reached a steady state, the criterion that we propose is to
show that the area per lipid or the hydration number have reached the
equilibrium.
From our results, two important aspects should be remarked:
1. 1.
The equilibration time is strongly dependent on the starting conformation of
the system. Wrong starting conformations will require much longer
equilibration times, even of one order of magnitude higher than the requested
from a more refined starting conformation.
2. 2.
Temperature is a critical parameter for reducing the equilibration time in our
simulations, due to the fact that higher temperatures increase the kinetic
processes, i.e., the sampling of the configurational space of the system.
###### Acknowledgements.
Authors wish to thank the assistance of the Computing Center of the
Universidad Politécnica de Cartagena (SAIT), Spain. RDP is member of ‘Carrera
del Investigador’, CONICET, Argentine.
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|
arxiv-papers
| 2012-12-12T18:53:42 |
2024-09-04T02:49:39.228868
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rodolfo D. Porasso, Jos\\'e J. L\\'opez Cascales",
"submitter": "Rodolfo Porasso",
"url": "https://arxiv.org/abs/1212.2928"
}
|
1212.2991
|
# Accelerating Inference: towards a full Language, Compiler and Hardware stack
Shawn Hershey, Jeff Bernstein, Bill Bradley, Andrew Schweitzer, Noah Stein,
Theo Weber, Ben Vigoda Emails are [email protected]. All authors are at
Lyric Labs at Analog Devices.
###### Abstract
We introduce Dimple, a fully open-source API for probabilistic modeling.
Dimple allows the user to specify probabilistic models in the form of
graphical models, Bayesian networks, or factor graphs, and performs inference
(by automatically deriving an inference engine from a variety of algorithms)
on the model. Dimple also serves as a compiler for GP5, a hardware accelerator
for inference.
## 1 Introduction
To a large extent, probabilistic graphical models unify a great number of
models from machine learning, statistical text processing, vision,
bioinformatics, and many other fields concerned with the analysis and
understanding of noisy, incomplete, or inconsistent data.
Graphical models alleviate the complexity inherent to large dimensional
statistical models (the so-called curse of dimensionality) by dividing the
problem into a series of logically (and statistically) independent components.
By factoring the problem into subproblems with known and simple
interdependencies, and by adopting a common language to describe each
subproblem, one can considerably simplify the task of creating complex
Bayesian models. Modularity can be taken advantage of further by leveraging
this modeling hierarchy over several levels (e.g. a submodel can also be
decomposed into a family of sub-submodels). Finally, by providing a framework
which abstracts the key concepts underlying classes of models, graphical
models allow the design of general algorithms which can be efficiently applied
across completely different fields, and systematically derived from a model
description.
These observations have driven people towards developing computer languages
and APIs to naturally specify complex probabilistic models, and automatically
derive a corresponding inference engine (with limited input from the user),
drastically reducing the prototyping time for probabilistic modeling. Many
such tools have appeared over the last few years; examples include Infer.NET
[3], the Bayes Net Toolbox [4], BUGS, and Church[2] 111Stochastic Matlab and
Church belong in fact to probabilistic programming, an even richer class of
probabilistic models, where the model is described by an arbitrary computer
program with calls to elementary random procedures..
## 2 Dimple
Dimple is an open source API which allows the user to construct models that
can be written as factor graphs. The model specification is separated from the
inference engine. Models can then be defined without a priori consideration of
which algorithm to use (or even knowing how present algorithms work). The
modular architecture makes the development of new inference engines and
schedulers easy, and enables the developer of a new algorithm to immediately
run and test it on the relevant examples and target benchmarks (Dimple
provides a number of such benchmarks). Dimple fully supports undirected and
directed factors, discrete and continuous variables, and has growing support
for parameter estimation (including generalized Baum-Welch). It supports
infinite streaming chains but does not yet support the more general infinite
graphs needed to support nonparametric Bayesian statistics, although some
preliminary work has been done to integrate Stochastic MATLAB[7] (a variant of
Church) and Dimple.
A crucial feature of the Dimple architecture is that it is very easy for
anyone to add new inference engines and apply them to any model expressible by
Dimple. It is also very easy to write a new model and choose to use any of the
solvers (or increasingly, to mix and match solvers).
A second crucial feature of Dimple is that it serves as a front-end and
compiler for hardware-accelerated inference. In parallel to developing Dimple,
Analog Devices, Inc. is also a hardware company aiming to build general
purpose accelerators for machine learning and probabilistic programming in
particular. A result of those efforts is a special purpose chip called GP5 -
an accelerator for sum-product and min-sum belief propagation in discrete
factor graphs.
At the time of writing this document, Dimple and Infer.NET have similar
features: an ability to express finite graphical models, multiple inference
engines (including Gibbs and Sum Product), support for both discrete and
continuous variables, and support for both directed and undirected graphs.
Unlike Infer.NET, Dimple does not yet provide an Expectation Propagation
inference engine, but the architecture would support such an addition. On the
other hand, Dimple does provide advantages over Infer.NET, such as giving
users the option to create their own inference engines and custom schedules
(the order in which messages are passes or samples are generated). Dimple
supports other solvers not present in Infer.net : particle belief propagation,
sample based belief propagation, among others. Dimple and Church differ more
deeply. Church can be used to model a wider range of probabilistic models that
include infinite probabilistic graphical models, but also more complex models.
Dimple, on the other hand, can provide more types of inference engines because
of the very same restrictions on the class of models it can describe. In some
cases, Dimple handles undirected factors and hard constraints (such as parity
checks) more naturally than Church does.
### 2.1 Code example
If data is encoded with a Low Density Parity Check code and then corrupted,
one can try to recover it using belief propagation. We provide an example of
MATLAB Dimple code for decoding. A typical implementation from scratch might
take a thousand lines of code [5].
⬇
1% Parity Check Matrix:
M = [1 1 1 0 0 0; 1 0 1 0 1 0; 0 0 1 1 0 1];
3%Instantiate a Factor Graph
My_Factor_Graph=FactorGraph();
5My_Factor_Graph.Solver = SumProduct ;
%Create random variables with a domain of [0 1]
7My_Variables=Bit(size(M,2),1);
%The factor function
9xorDelta= @(bits) mod(sum(bits),2) == 0;
%Iterate over check equations and add constraints from the check matrix.
11for i=1:size(M,1)
variable_indices = find(M(i,:));
13 My_Factor_Graph.addFactor(xorDelta, My_Variables(variable_indices));
end
15%Set the priors
My_Variables = a_vector_representing_observed_data;
17%Run inference
My_Factor_Graph.solve()
19%Display the posterior distribution
disp(My_Variables.Belief)
### 2.2 Selected features
In this section we will highlight some of Dimple’s features. A more
comprehensive set of features can be found in the documentation found on
GitHub. [1]
#### 2.2.1 Modeler Features
* •
Easy specification of undirected and directed potentials. Options include:
* –
Dense factors: any arbitrary function of the variables connected to the
factor. The following code generates a simple graph with two variables, $a$
and $b$ and a factor function $f(a,b)=e^{(-|a-b])}$ where $a$ and
$b\in\\{1,2,...10\\}$ and the prior on $a$ defines $P(a=10)=1$.
⬇
1%Create Factor Graph
fg = FactorGraph();
3%Define function to penalize distance
f = @(x,y) exp(-abs(x-y));
5%create two variables
a = Discrete(1:10);
7b = Discrete(1:10);
%Add an undirected factor encouraging sameness between the
9%variables.
fg.addFactor(f,a,b);
11%Set a hard input for a (100% certainty of a 10)
a.Input = [zeros(1,9) 1];
13%Run inference
fg.solve();
15%Spit out the posterior belief of b
disp(b.Belief’);
Here we see the resulting posterior probability of $b$ after running sum
product.
⬇
1ans =
3 0.0001
0.0002
5 0.0006
0.0016
7 0.0043
0.0116
9 0.0315
0.0856
11 0.2326
0.6321
* –
Sparse factors: specify only those factor function values which are nonzero.
Improves performance by orders of magnitude when some variables are
deterministic functions of others, for example. The following code defines a
factor function $f(a,b)=(a=b)$, enforcing the constraint that $a=b$ with equal
weight for every value.
⬇
1%Create Factor Graph
fg = FactorGraph();
3%Create two variables
a = Discrete(1:4);
5b = Discrete(1:4);
%Specify the domain indices of non-zero entries in the factor table.
7indices = [0 0;
1 1;
9 2 2;
3 3];
11%specify the weights for each row in the index table
weights = [1 1 1 1];
13%Create the factor
fg.addFactor(indices,weights,a,b);
15%Set a hard input
a.Input = [zeros(1,3) 1];
17%Run inference
fg.solve();
19%Print out posterior
disp(b.Belief);
After running inference, the following result is produced.
⬇
1ans =
3 0.0000 0.0000 0.0000 1.0000
* –
Nested factor graphs: once created, a factor graph can be instantiated within
larger factor graphs, increasing code modularity, modifiability, reusability,
and readability, in a paradigm similar to object oriented programming. The
following code snippet creates a graph that defines a soft 4-bit xor. The
parent graph nests this graph twice to create a toy error correcting code with
two constraints over 4 variables each.
⬇
1%%%%%%% Create the nested graph
b = Bit(4,1);
3fourBitXor = FactorGraph(b);
c = Bit();
5fourBitXor.addFactor(@xorDelta,[b(1:2); c]);
fourBitXor.addFactor(@xorDelta,[b(3:4); c]);
7
%%%%%%% Create the parent graph
9fg = FactorGraph();
a = Bit(6,1);
11fg.addFactor(fourBitXor,a([1 2 4 5]));
fg.addFactor(fourBitXor,a([2 3 4 6]));
13
%Set inputs and run inference
15a.Input = ones(6,1)*.9;
fg.solve();
17disp(a.Belief’);
After running inference, we see that all bits are pulled more strongly towards
1. Graphs can be nested recursively, although not infinitely.
⬇
1ans =
3 0.9654 0.9886 0.9654 0.9886 0.9654 0.9654
* •
Vectorized Graph Creation: Achieving fast speeds with MATLAB requires avoiding
loops. Dimple provides the ability to create large n-dimensional collections
of variables as well as large numbers of factors without loops. This feature
is different than streaming line graphs described later in this paper.
The following examples show just a few ways to vectorize graph creation.
* –
Chains
⬇
1N = 100;
b = Bit(N,1);
3fg = FactorGraph();
ft = FactorTable(rand(2),b.Domain,b.Domain);
5ft.normalize(1);
f = fg.addFactorVectorized(ft,b(1:end-1),b(2:end));
7f.DirectedTo = b(2:end);
This code snippet creates a directed Markov Model. It will result in the
following graphical model. Circles represent variables and squares represent
factors.
* –
Grids
⬇
1N = 100;
D = 75;
3b = Discrete(1:D,N,N);
fg = FactorGraph();
5similarity = @(x,y) exp(-abs(x-y));
fg.addFactorVectorized(similarity,b(1:end-1,:),b(2:end,:));
7fg.addFactorVectorized(similarity,b(:,1:end-1),b(:,2:end));
This code snippet creates the following graph and enforces similarity between
adjacent variables.
* –
Fully Connected Layers: There are many ways to use the addFactorVectorized
feature. The Dimple documentation provides more detail. However, we’ll include
just one more example to illustrate the flexibility of this feature. The
following code generates a graph with two layers where each node in layer 1 is
connected to every node in layer 2.
⬇
1N = 100;
layers = Bit(N,2);
3fg = FactorGraph();
similarity = @(x,y) exp(-abs(x-y));
5fg.addFactorVectorized(similarity,repmat(layers(:,1),1,N),repmat(layers(:,2)’,N,1));
The following image represents the graph. Factors are omitted for readability.
* •
Streaming line graphs: many graphical models have a 1d structure (for
instance, dynamic bayesian networks, kth order Markov Chains, HMMs and all
varieties of Kalman filters). Dimple provides the ability to create a template
that is repeated over a specified window size. Data can then be streamed
through this window. For predictive inference, this allows effectively to use
very large graphs while only using the memory required to carry sufficient
statistics (i.e. the memory usage does not grow with the size of the line
graph).
The following code creates a Markov Chain. The $rf.BufferSize=2$ code snippet
specifies that the sliding window should hold two instances of the nested
graph in memory so that some smoothing is performed in addition to prediction.
⬇
1b = Bit(2,1);
ng = FactorGraph(b);
3similarity = @(x,y) exp(-abs(x-y));
ng.addFactor(similarity,b(1),b(2));
5
fg = FactorGraph();
7b = DiscreteStream(0:1);
rf = fg.addRepeatedFactor(ng,b.getSlice(1),b.getSlice(2));
9rf.BufferSize = 2;
N = 100;
11dds = DoubleArrayDataSource(repmat([1 0],N,1));
b.DataSource = dds;
13
fg.initialize();
15while true
17 %The ”false” argument tells Dimple not to initialize the
%messages so that as the rolled up graph progresses
19 %we don’t lose information from the past.
fg.solve(false);
21
%Display belief of current variable
23 b.FirstVar.Belief
25 if ~fg.hasNext()
break;
27 end
fg.advance();
29end
* •
Cluster belief propagation - loopy belief propagation can be improved by
considering joint distribution over groups of variables (instead of marginal
distributions); the graph can also be made less loopy by converting two or
more factors into a single, larger factor.
* •
Multivariate Gaussians - Dimple provides support for Belief Propagation with
Multivariate Gaussians with linear and nonlinear operator factors. The Dimple
documentation provides more detail. Dimple also contains a set of demos, which
include a Kalman filter.
#### 2.2.2 Architecture and Inference Features
* •
Speed - Dimple’s Belief Propagation engine is well optimized for many problems
and we intend to optimize it further for others.
* •
Open source - Lyric has made Dimple open source with the hope of contributing
to the probabilistic programming community; how the community uses Dimple will
also help guide the development of future features, improvements, and
hardware.
* •
MATLAB and Java API, extendable to any language on top of JVM - Dimple is
written mostly in Java. This was chosen as a trade-off in speed vs. ease of
development. A MATLAB front end is provided for convenience.
* •
Customizable - The modular architecture allows all inference behavior to be
customizable; one can even write entirely new inference engines. Existing
inference engines include Sum Product, Min Sum, a k-best approximation for Sum
Product and Min Sum, Gaussian BP, Particle BP, and Gibbs sampling. As an
example of customization, Dimple provides specialized factors for efficiently
performing inference over variables with a finite field domain.
## 3 Hardware acceleration
### 3.1 GP5
As of this writing, Lyric Labs’ first prototype of an inference accelerator,
GP5, will be available in the next few months. All acceleration estimates are
based on the GP5 simulator, which is complete but not yet released publicly.
GP5 accelerates discrete Sum Product, Min Sum, and more generally, performs
highly efficient weighted sparse tensor inner products.
The GP5 is optimized to accelerate inference on graphs with large factor
tables. A factor function is defined over the cartesian product of the domains
of the connected variables; as a result, complexity of the factor inference is
of order $O(nd^{n})$ where d is the domain size of the connected variables and
n is the degree of the factor. GP5 is aimed at optimizing inference in factor
graphs where that factor complexity is relatively high.
The actual execution time on the GP5 is determined by a collection of
complicated and interlocking factors: I/O between the host processor and the
GP5, various memory sizes on the GP5 relative to the message and factor table
sizes of the factor graph, the scheduling of messages on the factor graph, the
degree of each factor, and the various asynchronous sub-processors on the GP5.
Their interactions may produce computational bottlenecks that prevent the user
from achieving peak performance.
The GP5 supports factor nodes that can connect to up to 16 variables and
variables can connect to up to 256 factors. The factor table cache is 256KB
and the maximum domain size of variables is 4096. Data can be moved in and out
of the GP5 at 18Gb/s, the chip will be $36mm^{2}$, and draws between 1 and 2
watts of power.
In a binary image denoising benchmark[6] (variables with binary domains,
factors with degree 16 connecting 4 by 4 image patches), GP5 is estimated to
be 3200x faster than Dimple running on an intel i7, for a fraction of the
power. The speedup is achieved by a large number of architectural
optimizations, as well as the utilization of multiple small cores to
parallelize inference on a single factor. Future versions could tile those
cores to further enhance parallelism (including running several factors in
parallel). This is the best acceleration we have achieved in any of our
benchmarks.
In the case of a stereo vision benchmark[6], the GP5 is estimated to be 100x
faster than Dimple running on an intel i7. The sterevision acceleration is
worse than the denoising benchmark acceleration because of the small degree of
the factor nodes in the stereo vision demo.
It is challenging to describe an expected acceleration across the range of
models that can be used with the GP5 because of the complexity of the
interlocking factors. As a general guide, the larger the factor table and the
larger the degree, the more acceleration the GP5 can achieve. The GP5
simulator, however, can provide a fairly accurate estimation of the expected
acceleration.
The following image shows the output of the Instruction Level Profiling Tool
included with the GP5 simulator.
### 3.2 Dimple as a Compiler for GP5
The GP5 hardware requires a compiler to translate a Dimple model to a set of
GP5 hardware instructions. The GP5 Compiler fits nicely within the Dimple
architecture, acting as just another inference engine; as such, it is
optimized for the GP5 hardware. The design problems associated with compiling
down to the GP5 are unique and hard: for normal compiler design, optimal
allocation of variables to registers is an NP-complete map coloring problem;
GP5 faces an even harder version of this due to the requirement of allocating
larger blocks of memory. Additionally, CISC compilers provide challenges
because different instructions might take different numbers of cycles. In
order to get maximum efficiency wins, timing of GP5 instructions varies over
orders of magnitude.
## 4 Future Improvements
Lyric has a long list of plans for improving Dimple. Some of the immediate
plans include speed improvements, more real variable support, integrating with
languages like Stochastic MATLAB (an early prototype already exists),
providing modelers in different languages, and tools to make it easier to
provide inference engines in different languages.
#### References
[1] S. Hershey, B. Vigoda, J. Bernstein, B. Bradley, T. Weber, A. Schweitzer,
N. Stein Dimple, Lyric Labs Analog Devices, 2012.
http://github.com/AnalogDevicesLyricLabs/dimple
[2] Noah D. Goodman, Vikash K. Mansinghka, Daniel M. Roy, Keith Bonawitz, and
Joshua B. Tenenbaum. Church: a language for generative models. Proc.
Uncertainty in Artificial Intelligence (UAI), 2008.
[3] T. Minka, J. Winn, J. Guiver, and D. Knowles Infer.NET 2.4, Microsoft
Research Cambridge, 2010. http://research.microsoft.com/infernet
[4] Kevin Murphy, Bayes Net Toolbox, http://code.google.com/p/bnt
[5] Shaikh Faisal Zaheer, LDPC Code Simulation,
http://www.mathworks.com/matlabcentral/fileexchange/8977-ldpc-code-simulation
[6] S. Hershey, J. Bernstein, B. Bradley, T. Weber dimpleDemos, Lyric Labs
Analog Devices, 2012. http://github.com/AnalogDevicesLyricLabs/dimpleDemos
[7] David Wingate, Andreas Stuhlmueller and Noah D. Goodman, Lightweight
implementations of probabilistic programming languages via transformational
compilation, Proc. of AISTATS, 2011.
|
arxiv-papers
| 2012-12-12T21:40:23 |
2024-09-04T02:49:39.237255
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shawn Hershey, Jeff Bernstein, Bill Bradley, Andrew Schweitzer, Noah\n Stein, Theo Weber, Ben Vigoda",
"submitter": "Shawn Hershey",
"url": "https://arxiv.org/abs/1212.2991"
}
|
1212.3107
|
[labelstyle=]
# What is the real category of sets?
Samuele Maschio
Abstract
According to Kreisel, category theory provides a powerful tool to organize
mathematics. A sample of this descriptive power is given by the categorical
analysis of the practice of ”classes as shorthands” in ZF set theory. In this
case category theory provides a natural way to describe the relation between
mathematics and metamathematics: if metamathematics can be described by using
categories (in particular syntactic categories), then the mathematical level
is represented by internal categories. Through this two-level interpretation
we can clarify the relation between classes and sets in ZF and, in particular,
we can present two equivalent categorical notions of definable set. Some
common sayings about set theory will be interpreted in the light of this
representation, emphasizing the distinction between naive and rigorous
sentences about sets and classes.
## 1 The spiny relation between categories and sets
The relation between sets and categories is a spiny topic. The set theoretical
foundations of category theory were discussed by many mathematicians, e.g. by
Feferman [5], Engler [4] and [12]. Viceversa, the categorical foundation of
set theory is not a clear matter. Categorical logic allows us to speak about
categorical models of mathematical theories and in particular about
categorical models of set theories, as for example $IZF$, $CZF$, $ZF$. In 1995
Joyal and Moerdijk in their _Algebraic Set Theory_ proposed a new technique to
guarantee the existence of internal models of $ZF$ and $IZF$ in a category.
Before their work, all proposed categorical models of set theory were based on
categories built starting from some standard models of set theory. Joyal and
Moerdijk’s approach has two main advantages:
1. 1.
it allows to prove the existence of an internal model of $IZF$ by simply
checking some purely categorical properties of a class of arrows;
2. 2.
it works for (relatively) arbitrary categories.
Their book gave rise to a large spread of works. It looked as algebraic set
theory could give not only a useful tool to mathematics, but also something
that could be interesting for foundational studies.
There is maybe nothing more suggestive and more imprecise than the claim that
_Category theory provides a foundation for mathematics_. We don’t mean that
this is not true, but we must come to an agreement on what is the meaning of
the words _foundation for mathematics_. Kreisel observed that category theory
provides a powerful tool to organize mathematics, and this is the way in which
most mathematicians involved in category theory think about category theory as
a foundation. There are also some attempts to use category theory to build a
based-on-functions theory of sets on which it is possible to found the entire
mathematical building; these are, for example, the axiomatic systems proposed
by Lawvere [11] and recalled by [16]. In this paper we adopt the attitude to
go in accordance with the first vision of _category theory as foundation_. We
think that category theory _describes_ mathematics in a very effective way. In
the following pages this descriptive power will be exemplified by using
category theory to clarify the role of classes in the practice of $ZF$ set
theory. Category theory (in both its usual and internal version) will be used
to give an account of the relation between mathematics and metamathematics.
## 2 Classes and ZF
Set theory is mainly $ZF$ (or better most of the set theorists studies $ZF$),
that is the classical first-order theory with equality whose language has
(countably many) individual variables, no functional symbols and a binary
relational symbol $\in$, and whose specific axioms are the following
1. 1.
$\forall x\forall y(\forall t(t\in x\leftrightarrow t\in y)\rightarrow x=y)$;
2. 2.
for every formula $P$ with free variables $p_{1},...,p_{k},t$,
$\forall p_{1}...\forall p_{k}\forall x\exists y\forall t(t\in
y\leftrightarrow(t\in x\wedge P));$
3. 3.
$\forall x\forall y\exists z\forall t(t\in z\leftrightarrow(t=x\vee t=y))$;
4. 4.
$\forall x\exists z\forall t(t\in z\leftrightarrow\exists y(t\in y\wedge y\in
x))$;
5. 5.
$\forall y\exists z\forall x(x\in z\leftrightarrow x\subseteq y)$;
6. 6.
for every formula $F$ with free variables $p_{1},..,p_{k},x,y$,
$\forall p_{1}...\forall p_{k}\forall z(\forall x(x\in
z\rightarrow\exists!yF)\rightarrow\exists z^{\prime}\forall x(x\in
z\rightarrow\exists y(y\in z^{\prime}\wedge F)));$
7. 7.
$\exists x(0\in x\wedge\forall t(t\in x\rightarrow\sigma(t)\in x))$;
8. 8.
$\forall x(x\neq 0\rightarrow\exists z(z\in x\wedge\forall t(t\in x\rightarrow
t\notin z)))$;
where as usual $x\subseteq y$ is a shorthand for $\forall t(t\in x\rightarrow
t\in y)$ and $\sigma(t)$ stays for $t\cup\left\\{t\right\\}$.
Set theorists work with sets, but they speak also about classes. However
classes don’t exist in $ZF$. The practice of set theorists consist in
considering classes as shorthands or formal writings. If $P$ is a formula with
a distinguished variable $x$, then a class is a formal writing
$\left\\{x|P(x)\right\\}$ and the expression
$t\in\left\\{x|P(x)\right\\}$
is simply a shorthand for $P[t/x]$. For example set theorists write $t\in{\cal
V}$, as a shorthand for $t\in\left\\{x|x=x\right\\}$, that is a shorthand for
$t=t$. They also write $t\in{\cal ON}$ as a shorthand for
$t\in\left\\{x|ON(x)\right\\}$, that is a shorthand for $ON(t)$ where $ON(x)$
is the formula that expresses the fact that $x$ is an ordinal, that is
$\forall s\forall s^{\prime}\forall s^{\prime\prime}((s\in t\wedge
s^{\prime}\in t\wedge s^{\prime\prime}\in t\wedge s\in s^{\prime}\wedge
s^{\prime}\in s^{\prime\prime})\rightarrow s\in s^{\prime\prime})\wedge$
$\wedge\forall s\forall s^{\prime}((s\in t\wedge s^{\prime}\in t\wedge s\neq
s^{\prime})\rightarrow(s\in s^{\prime}\vee s^{\prime}\in s))\wedge\forall
s(s\in t\rightarrow s\subseteq t).$
What is clear, per se, is the fact that classes are not mathematical, but
metamathematical objects. However these shorthands enjoy some of the
properties that are enjoyed by sets. I can give a meaning to the notion of
intersection of classes or to that of subclass. This accidental fact makes
some confusion breaking into! The metamathematical level and the mathematical
one are confused giving rise to impressive, but heavily incorrect sentences
like
$\textrm{"Sets are exactly those classes for which comprehension axiom is
true".}(*)$
It doesn’t matter the fact that sentences like these can be used as some sort
of _convincing argument_ to make students having an intuition of the notion of
set, these sentences look more dangerous, than useful.
The ease with which set theorists use classes (that are formal objects) as
some quasi-sets is due to the dual fact that their syntactical structure (and
in particular the use of connectives and quantifiers), as we have already
seen, allows this for many operations and that they know that there exists a
theory of classes (NBG) that is a conservative extension of ZFC. However
classes are _nothing more than syntactical objects_.
In the next sections we will see how the use of category theory can help us to
describe the relation between metamathematics and mathematics, and to
distinguish between real sentences about sets and external (naive) sentences
about sets.
## 3 The syntactic category of ZF
We proceed now defining the syntactic category of $ZF$, that we will denote
with $\mathbb{ZF}$. Its objects are formulas in context, i.e. formal writing
as
$\left\\{x_{1},..,x_{n}|P\right\\},$
where $x_{1},...,x_{n}$ is a (possibly empty) list of distinct variables and
$P$ is a formula with all free variables among $x_{1},...,x_{n}$. An arrow
from a formula in context $\left\\{\textbf{x}|P\right\\}$ to another
$\left\\{\textbf{y}|Q\right\\}$ is an equivalence class of formulas in context
$[\left\\{\textbf{x'},\textbf{y'}|F\right\\}]_{\equiv},$
where x’ is a list of variables that have the same lenght as x, and y’ is a
list of variables that has the same lenght as y, for which
1. 1.
$F\vdash_{ZF}P[\textbf{x}^{\prime}/\textbf{x}]\wedge
Q[\textbf{y}^{\prime}/\textbf{y}]$;
2. 2.
$F\wedge F[\textbf{y''}/\textbf{y'}]\vdash_{ZF}\textbf{y'}=\textbf{y''}$;
3. 3.
$P[\textbf{x}^{\prime}/\textbf{x}]\vdash_{ZF}\exists\textbf{y'}F$.
The equivalence relation is given by
$\left\\{\textbf{x'},\textbf{y'}|F\right\\}\equiv\left\\{\textbf{x''},\textbf{y''}|F^{\prime}\right\\}\textrm{
iff }\vdash_{ZF}F\leftrightarrow
F^{\prime}[\textbf{x'}/\textbf{x''},\textbf{y'}/\textbf{y''}].$
For composition it is sufficient to consider arrows
$[\left\\{\textbf{x'},\textbf{y'}|F\right\\}]_{\equiv}$ and
$[\left\\{\textbf{y'},\textbf{z'}|F^{\prime}\right\\}]_{\equiv}$ with all
distinct variables (this doesn’t determine a loss of generality). In this case
the composition is given by
$[\left\\{\textbf{x'},\textbf{z'}|\exists\textbf{y'}(F\wedge
F^{\prime})\right\\}]_{\equiv}.$
The category we obtain is regular and Boolean (see Johnstone [7] for a proof).
In particular we recall the fact that
$\left\\{\textbf{x}|P\right\\}\times\left\\{\textbf{y}|Q\right\\}\cong\left\\{\textbf{x},\textbf{y}|P\wedge
Q\right\\},$
where we supposed (without loss of generality) that all involved variables are
distinct. Moreover terminal objects are given for example by
$\left\\{\;|\forall x(x=x)\right\\},\;\left\\{x|\forall t(t\notin
x)\right\\}.$
We can also easily prove that $\mathbb{ZF}$ is extensive, thanks to the fact
that
$\vdash_{ZF}0\neq 1.$
Initial objects are given for example by
$\left\\{\;|\exists x(x\neq x)\right\\},\;\left\\{x|x\neq x\right\\}.$
$\mathbb{ZF}$ is also an exact category; the proof is based on _Scott’s trick_
(see Rosolini [18], Maschio [15]). Finally $\mathbb{ZF}$ has also a subobject
classifier that is given by
$[\left\\{x,x^{\prime}|x=0\wedge
x^{\prime}=1\right\\}]:\left\\{x|x=0\right\\}\rightarrow\left\\{x|x=0\vee
x=1\right\\}.$
## 4 Definable classes
The category of definable classes of $ZF$, that we denote with
$\mathbb{DCL}[ZF]$, is the full subcategory of $\mathbb{ZF}$ determined by all
those objects that have list of variables of lenght $1$. It is clear that this
is the category of classes, as they are intended in the practice of $ZF$.
The meaningful fact about $\mathbb{DCL}[ZF]$ is the fact that, although it is
a subcategory of $\mathbb{ZF}$, it is provable to be equivalent to it. This
result comes from the fact that in $ZF$ there is an available representation
for ordered pairs:
$x=<x_{1},x_{2}>\equiv^{def}x=\left\\{\left\\{x_{1}\right\\},\left\\{x_{1},x_{2}\right\\}\right\\}.$
Every object of $\mathbb{ZF}$, $\left\\{x_{1},..,x_{n}|P\right\\}$, is
isomorphic to
$\left\\{x|\exists x_{1}...\exists
x_{n}(x=<<...<x_{1},x_{2}>,...>,x_{n}>\wedge P)\right\\},$
where $x$ is a variable distinct from $x_{1},...,x_{n}$.
## 5 The categorical side of class operations
Before going on, we want to spend some words giving an example of the
categorical interpretation of the practice of operations between classes. This
is done to justify the adequacy of $\mathbb{DCL}[ZF]$ for the aim of
describing the category of formal classes. For this purpose we focus on
intersection. We consider two definable classes $\left\\{x|P\right\\}$ and
$\left\\{y|Q\right\\}$ (we can suppose $x$ and $y$ to be distinct without loss
of generality); we usually take their intersection to be $\left\\{x,|P\wedge
Q[x/y]\right\\}$. However this operation can be expressed in purely
categorical terms. In fact $\left\\{x|P\wedge Q[x/y]\right\\}$ is isomorphic
in $\mathbb{DCL}[ZF]$ to the object we obtain by considering the following
pullback.
## 6 Definable sets
Let’s now consider the full subcategory $\mathbb{DST}[ZF]$ of
$\mathbb{DCL}[ZF]$ determined by the objects $\left\\{x|P\right\\}$ for which
$\vdash_{ZF}\exists z\forall x(x\in z\leftrightarrow P).$
This is the category of sets we have in mind, when we think sentence $(*)$ is
true.
We have the following result:
###### Theorem 6.1.
$\mathbb{DST}[ZF]$ is a topos.
###### Proof.
Finite limits are exactly finite limits in $\mathbb{ZF}$: a terminal object is
given by $\left\\{x|x=0\right\\}$, equalizers are exactly equalizers in
$\mathbb{ZF}$, while the product of two definible sets $\left\\{x|P\right\\}$,
$\left\\{y|Q\right\\}$ is given by
$\left\\{z|\exists x\exists y(z=<x,y>\wedge P\wedge Q)\right\\}$
with the obvious projections (we are assuming $x$ and $y$ to be distinct
without loss of generality). Subobject classifier is exactly the subobject
classifier of $\mathbb{ZF}$, while exponentials
$\left\\{y|Q\right\\}^{\left\\{x|P\right\\}}$ are given by
$\left\\{f|Fun(f)\wedge\forall s(s\in dom(f)\leftrightarrow P(s))\wedge\forall
s^{\prime}(s^{\prime}\in ran(f)\rightarrow Q(s^{\prime}))\right\\},$
with evaluation arrow given by
$[\\{F,y|\exists f\exists x(F=<f,x>\wedge Fun(f)\wedge\forall s(s\in
dom(f)\leftrightarrow P(s))\wedge$ $\wedge\forall s^{\prime}(s^{\prime}\in
ran(f)\rightarrow Q(s^{\prime}))\wedge<x,y>\in f)\\}]_{\equiv}.$
∎
## 7 A system of axioms for algebraic set theory
In this section we present a list of axioms for algebraic set theory proposed
by Simpson [19]. Every axiomatization of algebraic set theory is based on a
pair $(\mathbb{C},{\cal S})$ in which $\mathbb{C}$ is a category and ${\cal
S}$ is a family of arrows of $\mathbb{C}$ (called family of _small maps_). The
following are the axioms:
1. 1.
$\mathbb{C}$ is a regular category;
2. 2.
The composition of two arrows in ${\cal S}$ is in ${\cal S}$;
3. 3.
Every mono is in ${\cal S}$;
4. 4.
(STABILITY) If $f\in{\cal S}$ and $f^{\prime}$ is a pullback of $f$, then
$f^{\prime}\in{\cal S}$;
5. 5.
(REPRESENTABILITY) For every $X$, there exists an object ${\cal P}_{\cal
S}(X)$ and an arrow $e_{X}:\in_{X}\rightarrow X\times{\cal P}_{\cal S}(X)$
with $\pi_{2}\circ e_{X}\in{\cal S}$ so that, for every $\psi:R\rightarrow
X\times Z$ with $\pi_{2}\circ\psi\in{\cal S}$, there exists a unique arrow
$\rho:Z\rightarrow{\cal P}_{\cal S}(X)$ that fits in a pullback as follows;
6. 6.
(POWERSET) $\sqsubseteq_{X}:\subseteq_{X}\rightarrow{\cal P}_{\cal
S}(X)\times{\cal P}_{\cal S}(X)$ satisfies
$\pi_{2}\circ\sqsubseteq_{X}\in{\cal S}$,
where $\sqsubseteq_{X}$ is determined by the following property:
_An arrow $f=<f_{1},f_{2}>:Z\rightarrow{\cal P}_{\cal S}(X)\times{\cal
P}_{\cal S}(X)$ factorizes through $\sqsubseteq_{X}$ if and only if,
considering the following couple of pullbacks_, _the arrow $\pi$ factorizes
through_ $\pi^{\prime}$.
We want also to recall some definitions:
###### Definition 7.1.
An object $U$ in a category $\mathbb{C}$ is _universal_ , if for every object
$X$ in $\mathbb{C}$, there exists a mono $j:X\rightarrow U$.
###### Definition 7.2.
An object $U$ in a regular category $\mathbb{C}$ with a class of small maps
${\cal S}$ is a _universe_ if there exists a mono $j:{\cal P}_{\cal
S}(U)\rightarrow U$.
###### Definition 7.3.
A $ZF$-_algebra_ for a regular category $\mathbb{C}$ with a class of small
maps ${\cal S}$ is an internal sup-semilattice $(U,\subseteq)$ together with
an arrow $\sigma:U\rightarrow U$, so that for every $\lambda:B\rightarrow U$
and for every $j:B\rightarrow A\in{\cal S}$, there exists
$sup_{j}(\lambda):A\rightarrow U$ so that for any
$j^{\prime}:B^{\prime}\rightarrow A$ and
$\lambda^{\prime}:B^{\prime}\rightarrow U$, once we consider the following
pullback, we have that
$sup_{j}(\lambda)\circ j^{\prime}\subseteq\lambda^{\prime}\textrm{ if and only
if }\lambda\circ\pi_{2}\subseteq\lambda^{\prime}\circ\pi_{1}.$
_Morphisms_ of $ZF$-algebras are morphisms between internal sup-semilattices
that preserve $sup_{j}(\lambda)$ along $j\in{\cal S}$ and commute with the
arrows $\sigma$. An _initial_ $ZF$-algebra is an initial object in the
category of $ZF$-algebras and morphisms between them.
## 8 Small maps
We now want to define a class of small maps in $\mathbb{DCL}[ZF]$: the class
${\cal S}$. An arrow is in ${\cal S}$ if and only if
$\vdash_{ZF}\forall y\exists z\forall x(F(x,y)\leftrightarrow x\in z).$
The class ${\cal S}$ is a class of small maps in the sense of Simpson [19]:
###### Lemma 8.1.
Every mono is in ${\cal S}$.
###### Proof.
This is obtained using axiom 2. to obtain the existence of an empty set and
axiom 3. to prove the existence of singletons. ∎
###### Lemma 8.2.
Compositions of arrows in ${\cal S}$ are in ${\cal S}$.
###### Proof.
This is obtained using axioms 6. and 4. ∎
###### Lemma 8.3.
The stability axiom is true for ${\cal S}$.
###### Proof.
This is obtained using axiom 6. ∎
###### Lemma 8.4.
The representability axiom is satisfied by ${\cal S}$.
###### Proof.
Fix a definible class $X=\left\\{x|P\right\\}$. The definible class ${\cal
P}_{\cal S}(X)$ is given by
$\left\\{y|\forall x(x\in y\rightarrow P)\right\\},$
while the definible class $\in_{X}$ is given by
$\left\\{z|\exists x\exists y(z=<x,y>\wedge\forall t(t\in y\rightarrow
P(t))\wedge x\in y)\right\\}$
and the arrow $e_{X}:\in_{X}\rightarrow X\times{\cal P}_{\cal S}(X)$ is given
by
$[\left\\{z,z^{\prime}|\exists x\exists y(z=<x,y>\wedge\forall t(t\in
y\rightarrow P(t))\wedge x\in y)\wedge z=z^{\prime}\right\\}]_{\equiv}.$
Now if the following arrow is so that $R(x,y)\vdash_{ZF}P(x)\wedge Q(y)$ and
$\vdash_{ZF}\forall y\exists y^{\prime}\forall x(R(x,y)\leftrightarrow x\in
y^{\prime})$, that means that it represents (without loss of generality) a
relation which has second component in ${\cal S}$, then its representing arrow
from $\left\\{y|Q\right\\}$ to ${\cal P}_{\cal S}(X)$ is given by
$[\left\\{y,y^{\prime}|Q\wedge\forall x(R(x,y)\leftrightarrow x\in
y^{\prime})\right\\}]_{\equiv}.$
∎
###### Lemma 8.5.
The powerset axiom is verified by ${\cal S}$.
###### Proof.
The subset relation for $\left\\{x|P\right\\}$ is given by the following arrow
${[\left\\{z,z^{\prime}|\exists y\exists y^{\prime}(z=<y,y^{\prime}>\wedge
y\subseteq y^{\prime}\wedge\forall x(x\in y^{\prime}\rightarrow P(x)))\wedge
z=z^{\prime}\right\\}]_{\equiv}}$
from $\left\\{z|\exists y\exists y^{\prime}(z=<y,y^{\prime}>\wedge y\subseteq
y^{\prime}\wedge\forall x(x\in y^{\prime}\rightarrow P(x)))\right\\}$ to
${\cal P}_{\cal S}(X)\times{\cal P}_{\cal S}(X)$.
This relation is in ${\cal S}$ by virtue of axiom 5. ∎
We now have that small definible classes are exactly those classes
$\left\\{x|P\right\\}$ for which the unique arrow to $1$, that is
$[\left\\{x,y|P\wedge y=0\right\\}]$, is small. This means that
$\vdash_{ZF}\forall y\exists z\forall x(x\in z\leftrightarrow(P(x)\wedge
y=0)).$
Now we know that
$\vdash_{ZF}\exists y(y=0)$
and so the previous condition is equivalent to say that
$\vdash_{ZF}\exists z\forall x(x\in z\leftrightarrow P(x)).$
This means that small definible classes are exactly definible sets. We have
also that $\left\\{x|N(x)\right\\}$ is a small definible class, where $N(x)$
is the formula saying that $x$ is a finite ordinal: this follows from axioms
7. and 2.
We finally have that $\left\\{x|x=x\right\\}$ is a universal definible class
(and so also a universe), because, for every definible class
$\left\\{x|P\right\\}$, the arrow $[\left\\{x,x^{\prime}|P\wedge
x=x^{\prime}\right\\}]_{\equiv}$ is a mono from it to
$\left\\{x|x=x\right\\}$.
To conclude note that we have an explicit (and obvious) representation for an
initial $ZF$-algebra: this is given by
$(\left\\{x|x=x\right\\},\sqsubseteq_{\left\\{x|x=x\right\\}},[\left\\{x,z|z=\left\\{x\right\\}\right\\}]_{\equiv}).$
If
$[\left\\{z,z^{\prime}|F(z,z^{\prime})\right\\}]_{\equiv}:\left\\{z|P\right\\}\rightarrow\left\\{z^{\prime}|Q\right\\}$
is small in $\mathbb{DCL}[ZF]$, and
$[\left\\{z,x|\lambda(z,x)\right\\}]_{\equiv}$ is an arrow from
$\left\\{z|P\right\\}$ to $\left\\{x|x=x\right\\}$, then
$sup_{[\left\\{z,z^{\prime}|F(z,z^{\prime})\right\\}]}([\left\\{z,x|\lambda(z,x)\right\\}]_{\equiv})$
is given by the arrow
$[\left\\{z^{\prime},x|Q\wedge\forall t(t\in x\leftrightarrow\exists
z(F(z,z^{\prime})\wedge\lambda(z,t)))\right\\}]_{\equiv}.$
## 9 Internal category theory
The notion of internal category is the generalization of the notion of small
category. Although we can define what is an internal category in an arbitrary
category, we prefer to consider a category $\mathbb{C}$ with all finite
limits. We have the following
###### Definition 9.1.
An _internal category_ of $\mathbb{C}$ is a sestuple
$(C_{0},C_{1},\delta_{0},\delta_{1},ID,\Box)$
in which $C_{0},C_{1}$ are objects of $\mathbb{C}$ and
$\delta_{0},\delta_{1}:C_{1}\rightarrow C_{0}$, $ID:C_{0}\rightarrow C_{1}$,
$\Box:C_{1}\times_{\Box}C_{1}\rightarrow C_{1}$
are arrows of $\mathbb{C}$, where the following is a pullback, that satisfy
the following requests
1. 1.
$\delta_{1}\circ ID=\delta_{0}\circ ID=id_{C_{0}}$;
2. 2.
$\delta_{0}\circ\Box=\delta_{0}\circ p_{0}$ and
$\delta_{1}\circ\Box=\delta_{1}\circ p_{1}$;
3. 3.
$\Box\circ\lceil ID\circ\delta_{0},id_{C_{1}}\rceil=\Box\circ\lceil
id_{C_{1}},ID\circ\delta_{1}\rceil=id_{C_{1}}$;
4. 4.
$\Box\circ\lceil\Box\circ p_{0},p_{1}\rceil=\Box\circ\lceil p_{0},\Box\circ
p_{1}\rceil\circ\lceil p_{0}\circ p_{0},\lceil p_{1}\circ
p_{0},p_{1}\rceil\rceil:(C_{1}\times_{\Box}C_{1})\times_{\Box}C_{1}\rightarrow
C_{0}$,
where we denote with $\lceil f,f^{\prime}\rceil$ the unique arrows that exist
for the definitions of pullback and where
$(C_{1}\times_{\Box}C_{1})\times_{\Box}C_{1}$ is the pullback of
$\delta_{1}\circ\Box$ and $\delta_{0}$.
Before going to the next section we show a way to externalize internal
categories. This is done in a very natural way by means of global elements.
###### Proposition 9.2.
If ${\cal C}=(C_{0},C_{1},\delta_{0},\delta_{1},ID,\Box)$ is an internal
category of $\mathbb{C}$, then the following is a category:
$\Gamma({\cal
C}):=(Hom(1,C_{0}),Hom(1,C_{1}),\delta_{0}\circ(-),\delta_{1}\circ(-),ID\circ(-),\Box\circ\lceil(-)_{1},(-)_{2}\rceil)$
###### Proof.
Every point of the definition of category follows immediately because of the
relative point in the definition of internal category. The proof is
straightforward. ∎
## 10 The real category of sets: ${\cal SET}$
We will now define an internal category of $\mathbb{ZF}$ (or equivalently of
$\mathbb{DCL}(ZF)$), called ${\cal SET}$. This category is given by the
following assignments:
1. 1.
${\cal SET}_{0}:=\left\\{x|x=x\right\\}$;
2. 2.
${\cal SET}_{1}:=\left\\{F|\exists f\exists z(F=<f,z>\wedge Fun(f)\wedge
ran(f)\subseteq z)\right\\}$;
3. 3.
$\delta_{0}:=[\left\\{F,x|\exists f\exists z(F=<f,z>\wedge Fun(f)\wedge
ran(f)\subseteq z\wedge dom(f)=x)\right\\}]_{\equiv}$;
4. 4.
$\delta_{1}:=[\left\\{F,z|\exists f(F=<f,z>\wedge Fun(f)\wedge ran(f)\subseteq
z)\right\\}]_{\equiv}$;
5. 5.
$ID:=[\left\\{x,F|\exists f(F=<f,x>\wedge\forall t(t\in
f\leftrightarrow\exists s(s\in x\wedge t=<s,s>)))\right\\}]_{\equiv}$
6. 6.
$\Box:=[\\{J,G|\exists f\exists f^{\prime}\exists z\exists
f^{\prime\prime}(J=<f,<f^{\prime},z>>\wedge Fun(f)\wedge Fun(f^{\prime})$
$\wedge ran(f)\subseteq dom(f^{\prime})\wedge ran(f^{\prime})\subseteq z\wedge
G=<f^{\prime\prime},z>\wedge$ $\wedge\forall t(t\in
f^{\prime\prime}\leftrightarrow(\exists s\exists s^{\prime}\exists
s^{\prime\prime}(<s,s^{\prime}>\in f\wedge<s^{\prime},s^{\prime\prime}>\in
f^{\prime}\wedge t=<s,s^{\prime\prime}>))))\\}]_{\equiv},$
once we easily realized that the object of composable arrows is given by
$\\{J|\exists f\exists f^{\prime}\exists z(J=<f,<f^{\prime},z>>\wedge
Fun(f)\wedge$ $\wedge Fun(f^{\prime})\wedge ran(f)\subseteq
dom(f^{\prime})\wedge ran(f^{\prime})\subseteq z)\\}.$
We have that
###### Theorem 10.1.
${\cal SET}$ is an internal category.
###### Proof.
This is a straightforward proof. ∎
We have also that this is an internal topos, as every construction for a topos
can be done in $\mathbb{ZF}$, as one can see (this is given by a well written
formal proof of the fact that sets and functions form a topos).
## 11 Global elements: names for sets
Now we have an internal category ${\cal SET}$; we want to study the category
$\Gamma({\cal SET})$. As follows directly from the definition we have that the
objects of $\Gamma({\cal SET})$ are the equivalence classes
$[\left\\{x|P(x)\right\\}]_{\equiv}$ of definable classes so that
$\vdash_{ZF}\exists!xP(x)$. Arrows of $\Gamma({\cal SET})$ are classes of
equivalence $[\left\\{f|P(f)\right\\}]_{\equiv}$ so that
$\vdash_{ZF}\exists!fP(f)$, and
$P(f)\vdash_{ZF}\exists f^{\prime}\exists z(f=<f^{\prime},z>\wedge
Fun(f^{\prime})\wedge ran(f^{\prime})\subseteq z).$
We have the following
###### Theorem 11.1.
$\mathbb{DST}(ZF)$ and $\Gamma({\cal SET})$ are equivalent.
###### Proof.
Consider the following functors:
1. 1.
$\textbf{P}:\mathbb{DST}(ZF)\rightarrow\Gamma({\cal SET})$ is given by
$\textbf{P}(\left\\{x|P(x)\right\\}):=[\left\\{z|\forall x(x\in
z\leftrightarrow P(x))\right\\}]_{\equiv}$
$\textbf{P}([\left\\{x,y|F(x,y)\right\\}]_{\equiv}):=$
$:=[\\{f^{\prime}|\exists f\exists z(f^{\prime}=<f,z>\wedge\forall t(t\in
f\leftrightarrow\exists x\exists y(t=<x,y>\wedge F(x,y)))\wedge$
$\wedge\forall y(y\in z\leftrightarrow Q(y)))\\}]_{\equiv}$
$\textbf{P}(\left\\{y|Q(y)\right\\}):=[\left\\{z|\forall y(y\in
z\leftrightarrow Q(y))\right\\}]_{\equiv}$.
2. 2.
$\textbf{P'}:\Gamma({\cal SET})\rightarrow\mathbb{DST}[ZF]$ is given by
$\textbf{P'}([\left\\{z|P(z)\right\\}]_{\equiv}):=\left\\{x_{0}|\exists
z(P(z)\wedge x_{0}\in z)\right\\}$
$\textbf{P'}([\left\\{f^{\prime}|Q(f^{\prime})\right\\}]_{\equiv}):=[\left\\{x,x^{\prime}|\exists
f^{\prime}(Q(f^{\prime})\wedge\exists z\exists
f(f^{\prime}=<f,z>\wedge<x,x^{\prime}>\in f))\right\\}]_{\equiv}$
$\textbf{P'}([\left\\{z|P^{\prime}(z)\right\\}]_{\equiv}):=\left\\{x_{0}|\exists
z(P^{\prime}(z)\wedge x_{0}\in z)\right\\}$,
where $x_{0}$ is a fixed variable (we can think of it as the first variable if
we consider variables of $ZF$ to be presented in a countable list).
It is immediate to see that $\textbf{P}\circ\textbf{P'}$ is the identity
functor for $\Gamma({\cal SET})$, while there is an natural isomorphism from
the identity functor of $\mathbb{DCL}[ZF]$ to $\textbf{P'}\circ\textbf{P}$,
that is given by the arrows
$[\left\\{x,x^{\prime}|P(x)\wedge
x=x^{\prime}\right\\}]_{\equiv}:\left\\{x|P(x)\right\\}\rightarrow\left\\{x_{0}|\exists
z(x_{0}\in z\wedge\forall x(x\in z\leftrightarrow P(x)))\right\\}$
∎
## 12 Final remarks
In our attempt to clarify the relation between (formal) classes and sets,
between metamathematics and mathematics, by means of a unique mathematical
structure, we started by introducing the syntactical category $\mathbb{ZF}$
(that we proved to be equivalent to the category of definable classes of
$ZF$). This category corresponds to the metamathematical level: its objects
are classes as are usually introduced in the set theorists’ practice. Moreover
this category has a full subcategory of some importance: the category of
definable sets, that is the category whose objects are those definable classes
$\left\\{x|P\right\\}$ for which
$\vdash_{ZF}\exists z\forall x(x\in z\leftrightarrow P).$
This is the naive category of sets. Obviously this is _not_ the real category
of sets. The _real_ category of sets is ${\cal SET}$. But this is not a
category: it is an _internal category_ in $\mathbb{ZF}$. The relation between
metamathematics and mathematics is exactly the relation between categories and
internal categories. Mathematical concepts are represented through internal
categories, external (or metamathematical) concepts are expressed at the
categorical level. The most interesting result shown in the previous sections
is that showing the equivalence of the two more natural ways to give an
external account of the notion of set. We proved that the category of
definable sets is equivalent to the category obtained by global sections on
${\cal SET}$: _classes that satisfy comprehension axiom are exactly those to
which I can give a name_.
## References
* [1] S.Awodey, C.Butz, A.Simpson, T.Streicher. Relating topos theory and set theory via categories of classes. Bulletin of Symbolic Logic, 13(3): 340-358, 2007.
* [2] Y.Bar-Hillel, A.A.Fraenkel, A.Levy. Foundations of Set Theory. North Holland, 1973.
* [3] E. Casari. Questioni di filosofia della matematica. Feltrinelli, 1964.
* [4] H.R.E.Engler. On the Problem of Foundations of Category Theory. Dialectica, 3(1), 1969.
* [5] S.Feferman. Categorial foundations and foundations of category theory. Logic, foundations of mathematics and computability theory: 149-169, 1977.
* [6] T.Jech. Set theory. The third millennium edition. Springer-Verlag, 2003.
* [7] P.Johnstone. Sketches of an elephant: a topos theory compendium, vol.2. Oxford University Press, 2002.
* [8] A.Joyal, I.Moerdijk, Algebraic Set Theory. Cambridge Unviersity Press, 1995.
* [9] G.Kreisel. Observations of popular discussions on foundations. Axiomatic Set Theory, American Mathematical Society: 183-190, 1971
* [10] K.Kunen Set Theory: An introduction to Independence Proofs. North Holland, 1983.
* [11] F.W.Lawvere. An elementary theoy of the category of sets (long version) with commentary. Reprints in Theory and Applications of Categories, 11: 1-35, 2005
* [12] G.Lolli. Categorie, universi e principi di riflessione. Bollati Boringhieri,1977
* [13] S.MacLane. Category theory for the working mathematicians. Springer-Verlag,1972
* [14] M.Makkai, G.E.Reyes. First order categorical logic. Springer, 1977.
* [15] S.Maschio. Aspects of internal set theory. PhD thesis, 2012.
* [16] C.McLarty. Exploring categorical structuralism. Philosophia Mathematica (3) Vol.12: 37-53, 2004.
* [17] G.Rosolini Una teoria di classi e insiemi. Lecture notes, 2008
* [18] G.Rosolini. La categoria delle classi definibili in IZF è un modello di AST. XXIV incontro di logica AILA, 2011.
* [19] A.Simpson. Elementary axioms for category of classes. Logic in Computer Science, pp. 77-85, 1999.
|
arxiv-papers
| 2012-12-13T10:08:19 |
2024-09-04T02:49:39.245922
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Samuele Maschio",
"submitter": "Samuele Maschio",
"url": "https://arxiv.org/abs/1212.3107"
}
|
1212.3112
|
# Geometric interpretation of phyllotaxis transition
Takuya Okabe Faculty of Engineering, Shizuoka University, 3-5-1 Johoku,
Hamamatsu 432-8561, Japan [email protected]
###### Abstract
The original problem of phyllotaxis was focused on the regular arrangements of
leaves on mature stems represented by common fractions such as 1/2, 1/3, 2/5,
3/8, 5/13, etc. The phyllotaxis fraction is not fixed for each plant but it
may undergo stepwise transitions during ontogeny, despite contrasting
observation that the arrangement of leaf primordia at shoot apical meristems
changes continuously. No explanation has been given so far for the mechanism
of the phyllotaxis transition, excepting suggestion resorting to genetic
programs operating at some specific stages. Here it is pointed out that
varying length of the leaf trace acts as an important factor to control the
transition by analyzing Larson’s diagram of the procambial system of young
cottonwood plants. The transition is interpreted as a necessary consequence of
geometric constraints that the leaf traces cannot be fitted into a fractional
pattern unless their length is shorter than the denominator times the
internode.
###### keywords:
Schimper-Braun’s law, Populus deltoides, Fibonacci numbers, golden ratio,
phyllotaxy
## 1 INTRODUCTION
The spiral arrangement of leaves on a stem, phyllotaxis, is represented by the
fraction of the circumference of the stem traversed by the spiral in passing
from one leaf to the next. Braun and Schimper discovered that leaves are lined
up in ranks parallel to the stem so that the fraction is literally represented
by a common fraction (Braun (1835)). In a 3/8 phyllotaxis, for instance, every
eighth leaf comes over one below it after three turns of the spiral, so that
eight straight ranks are visible along the stem (Fig. 1). To this day, the
following list of the most common fractions has been circulated in books and
websites for non-specialists; 1/2 for elm, lime and linden, 1/3 for beech and
hazel, 2/5 for oak, cherry, apple, holly and plum, 3/8 for poplar, rose and
pear, 5/13 for almond, etc. Such correspondence tables seem to have existed
already in the middle of the nineteenth century (Henfrey (1870)). Sometimes
willow is listed in 5/13 (Coxeter (1961)) and sometimes in 3/8 (Adam (2006)).
As a matter of fact, it had been remarked since early times that even an
individual plant sometimes makes transitions between different fractions
(Braun (1835)). Therefore, the phyllotaxis fraction is not a determined
characteristic of each species. Most notably, Larson (1980) has revealed the
manner in which the vascular system is rearranged through the phyllotaxis
transition by mapping arrangement of the leaf traces, the portion of vascular
bundles of leaves that resides in the stem (Fig. 2). By contrast, it has been
commonly accepted that the arrangement of leaf primordia in the bud or at the
shoot apical meristem does not conform to any fractional number; primordia do
not appear as radial rows. The angular divergence between successive primordia
stays close to a unique “ideal” angle of 137.5∘, which is the golden mean
0.3820 of 360∘ (Church (1920); Richards (1951)). The golden mean is a
mathematical limit of the sequence of the above fractions. Much attention has
been paid to mathematics of these numbers and to mechanisms of leaf primordia
formation by which the ideal angle is achieved and regulated (Adler et al.
(1997)). In contrast, the phyllotaxis transition has been left unexplained
without attracting interest from researchers. The fractional phyllotaxis and
the phyllotaxis transition are two sides of the same problem. A key point
noted in the present study is that phyllotaxis in the bud changes continuously
whereas phyllotaxis on the mature stem changes stepwise. Evidently, the
arrangement of primordia at the apical growing point must be a determining
factor of leaf arrangement on the mature stem. But what causes the transition
from 2/5 to 3/8, for instance? There must be another factor. Larson (1980) has
suggested that the transition is programmed in the plant to occur at specific
stages of ontogeny. In another view, phyllotaxis of primordium formation and
vascularization are controlled by some higher-level system (Romberger et al.
(1993)). Kuhlemeier (2007) remarks that virtually nothing is known about the
molecular mechanisms that underlie the transitions between different spiral
systems (e.g. 3/8 to 5/13 patterns), except that larger meristems seem to have
higher Fibonacci numbers. The present paper aims at pointing out that this
phenomenon is consistently explained by considering geometry of growing leaf
traces. By means of a full quantitative analysis, which surely is not standard
in this field of research, it is shown (1) that the positions of the
phyllotaxis transition are located based on the angular positions at which
leaf traces exit from the vascular cylinder and (2) that the phyllotaxis
transition is caused as a necessary consequence of change in size of leaf
traces relative to internode length.
Figure 1: A young poplar in a 3/8 phyllotaxis with eight vertical ranks of
leaves.
A caveat: this paper deals with the mechanism of the phyllotaxis transition in
spiral systems. Although a common keyword of “phyllotaxis” may suggest, it
should not be confused with mechanisms of phyllotactic primordium formation,
for which considerable advances have been made over the last decade
(Kuhlemeier (2007)). Models of the latter category deal with continuous
changes of apical meristems, while they are not concerned with the fractional
expression of a phyllotactic pattern.
## 2 MATERIAL AND METHODS
Figure 2: Diagram of the procambial system of a typical cottonwood plant
compiled by Larson (1980). The vascular cylinder is displayed as if unrolled
and laid flat. The ordinate is the Leaf Plastochron Index (LPI) for leaves,
whereas the abscissa corresponds to the angular coordinate in a full turn
about the stem axis. Each leaf is entered by three traces; a central
($\times$), right ($\blacktriangle$), and left ($\triangle$) traces.
Phyllotaxis orders, 1/2, 1/3, 2/5, 3/8 and 5/13, are indicated by the right
vertical axis. Figure 3: Vertical length of leaf traces in Fig. 2 is plotted
against the Leaf Plastochron Index. Orders of phyllotaxis, 2/5, 3/8 and 5/13,
are denoted at the same position as indicated in Fig. 2. See Fig. 2 for trace
designations. The phyllotaxis transitions from 2/5 to 3/8 and from 3/8 to 5/13
are caused by the traces indicated by a solid arrow and a dashed arrow,
respectively. Horizontal dashed lines at 5 and 8, separating the different
phyllotaxis regimes, are drawn for reference sake (see Results and
Discussion).
The analysis is based on a diagram of the procambial system of a cottonwood
plant (Populus deltoides) reconstructed by Larson (1980), which is reproduced
in Fig. 2. The vascular cylinder is displayed as if unrolled and laid flat.
The ordinate is the leaf plastochron index (LPI) for numbering leaves. Each
leaf has three traces: central, right and left traces exit the vascular
cylinder at positions denoted by symbols $\times$, $\triangle$ and
$\blacktriangle$, respectively. For more details, see Larson (1980) and
references cited therein.
By virtue of the convention to take LPI as the ordinate, the vertical scale of
the diagram is normalized such that differences in height between two
successive leaves, or internodes, are a unit of length in the vertical
direction. Therefore, the vertical component of a line segment in Fig. 2
represents not its actual length but an effective length relative to the
internode length, namely the length measured in internode units. Accordingly,
vertical lengths of the leaf traces in internode units are directly evaluated
by applying a digitizing ruler to the diagram. The lower end points of right
and left traces are located without ambiguity, for they are connected to other
types of traces. Lengths of central traces are fixed by decomposing the whole
pattern into clusters consisting of three adjacent traces, the right trace of
leaf $n$, the central trace of leaf $n+2$ and the left trace of leaf $n-1$,
where $n$ for LPI is an integer. For illustration purposes, the clusters from
$n=0$ to 7 are shown in Fig. 5. The length of the leaf traces thus obtained is
plotted against LPI in Fig. 3, where reading errors are of the order of an
internode at most. On the other hand, divergence angles are evaluated from the
horizontal, angular coordinates of the exit points of the leaf traces denoted
by the symbols. Evaluated angles show rhythmic, systematic variations around
the “ideal” angle, which are typically observed in a quantitative analysis
(Okabe (2012a)). The systematic variations, which are correlated with the
angular positions, are suppressed apparently by deleting from the diagram a
blank rectangular strip of a narrow width along the left ordinate. The width
of the strip is determined so as to minimize the standard deviation of
divergence angles for ten youngest central traces (LPI less than $-6$).
Results for the divergence angle thus corrected are shown in Fig. 4. The width
ratio of the deleted strip is 0.023, whereby the standard deviation of the
divergence angle is suppressed from 4.2∘ to 0.89∘. The correction is made for
ease of understanding implications of Fig. 4. It does not affect the results
discussed below qualitatively.
## 3 RESULTS
Figure 4: Divergence angle between the traces at LPI $n$ and $n+1$ is plotted
against LPI $n$. See Fig. 2 for trace designations. The inset shows the angle
between the central traces at LPI $n$ and $n+5$: for ideal patterns of 2/5,
3/8 and 5/13 orders, it should be zero, 45 and 28 degrees, respectively. The
positions at which the phyllotaxis transitions from 2/5 to 3/8 and from 3/8 to
5/13 occur are indicated by solid and dashed arrows, respectively.
Results in Fig. 3 indicate that the three traces grow in length steadily all
alike. This behavior is consistent with other quantities reported by Larson
(1980). According to Fig. 3, three traces at LPI 2 have almost the same length
of about five internodes. This means that the traces at LPI 2 extend down to
the height of LPI 7 (see Fig. 2). As indicated by the right ordinate of Fig.
2, the phyllotaxis order fraction changes from 2/5 through 3/8 to 5/13 as we
climb up the stem, or as LPI decreases. The order fractions are indicated at
the top of Fig. 3 at the same LPI coordinates as in Fig. 2 by Larson (1980).
Horizontal dashed lines at five and eight internodes in Fig. 3 are drawn to
separate regimes of different phyllotaxis orders (see below).
Divergence angle between the leaf traces at LPI $n$ and $n+1$ is plotted
against LPI $n$ in Fig. 4. The inset shows the angle between the central
traces at LPI $n$ and $n+5$, that is, the net angle of inclination of
5-parastichies. The angle should become zero, 45 and 28 degrees for 2/5, 3/8
and 5/13 ideal patterns, respectively; for instance, the ideal angle for 3/8
is $360\times 3/8\times 5=360\times 2-45$, which is congruent to $-45$. The
inset of Fig. 4 clearly indicates stepwise transitions between the three
distinct fraction regimes. Thus, the positions of the phyllotaxis transition
are located based on the exit points, or the bases, of the leaf traces, i.e.,
without inspecting internal changes in the vascular structure. This crucial
property for us is brought to light probably because the subject plants are
grown under controlled uniform conditions (Larson (1980).
The cause of the transition is traced back by close inspection based on the
quantitative results. The transition from 3/8 to 5/13, indicated by a dashed
arrow in the inset of Fig. 4, is caused by an irregular decrease of divergence
angle at LPI 2, which therefore is also indicated by a dashed arrow in the
main figure. Similarly, a solid arrow in the inset indicates the transition
from 2/5 to 3/8, which is ascribed to an increase of divergence angle at LPI
7, a solid arrow in the main figure.
In connection with the latter transition, it is worth a remark that divergence
angle in the 2/5 phyllotaxis regime is not held at an ideal constant value of
144∘ $(360\times 2/5=144)$. This means that five ranks (orthostichies) of a
real 2/5 pattern is not equally spaced. According to Fig. 4, two full turns
($360^{\circ}\times 2$) of the 2/5 pattern is divided roughly into unequal
parts of $140^{\circ}\times 4+160^{\circ}\times 1$, instead of a regular
spacing with $144^{\circ}\times 5$. The figure shows that the irregular shift
at LPI 7 is shared with LPI 12. This is just as expected for the 2/5
arrangement $(7+5=12)$. Similarly, a cycle of $360^{\circ}\times 3$ of a 3/8
pattern is divided into $137^{\circ}\times 7+120^{\circ}\times 1$, as
indicated by the dashed arrow. Thus, the quantitative analysis reveals that
divergence angle on a stem is a secondary property, as it is very unlikely
that the exceptional angles of $160^{\circ}$ and $120^{\circ}$ are intrinsic
to the plant. Accordingly, the pattern of Fig. 2 is to be viewed as a result
of secondary processes.
The final step is to identify the cause of the irregular shift in divergence
angle. For brevity, let the central ($\times$), right ($\blacktriangle$) and
left ($\triangle$) trace of LPI $n$ be denoted as $n$C, $n$R and $n$L,
respectively. At the transition from 2/5 to 3/8, the irregular shift of 7C
(solid arrow) is accompanied by 5R on the right side (Fig. 4). Inspection of
Fig. 2 reveals that this collective shift is caused as a result of 0R
intervening between 3R and 5R. This is illustrated in Fig. 5, a schematic
excerpt from Fig. 2. For this reason, the trace 0R is indicated by a solid
arrow in Fig. 3 as the very cause of the phyllotaxis transition from 2/5 to
3/8. A horizontal line at five is drawn in Fig. 3 to indicate that 0R
interferes with 5R if only the former length exceeds $5-0=5$ internodes (cf.
Fig. 5). Indeed, the filled triangle at the solid arrow in Fig. 3 lies well
above the horizontal line at five internodes. Similarly, the trace causing the
transition from 3/8 to 5/13 is indicated by a dashed arrow in Fig. 3, where
the upper horizontal line at eight is drawn as a threshold length for the
transition.
Figure 5: A schematic excerpt from Fig. 2 near the transition from 2/5 to
3/8. The right trace at LPI 5 is denoted as 5R, and 7C signifies the central
trace at LPI 7. According to Fig. 4, the traces 5R and 7C are shifted slightly
to the right as compared to the preceding (lower) traces. This figure shows
that the shift (solid arrow) is caused by a longer trace 0R intervening
between 3R and 5R, thereby the transition is initiated. If the length of 0R
were shorter than five internodes, 0R should have been aligned with 5R to keep
the 2/5 order, as the preceding 1R is with 6R.
## 4 DISCUSSION
Figure 6: Distinct patterns of 2/5 and 3/8 orders on the mature stem
cylinders (bottom) result from similar arrangements of leaf primordia at the
shoot apical meristems (top). Each dot represents a leaf node, through which a
dotted line is drawn to demarcate internodes. Leaf traces (line segments) are
aligned in a 2/5 phyllotaxis pattern if their length is greater than three and
less than five internodes (left), whereas a 3/8 phyllotaxis results if it is
greater than five and less than eight internodes (right).
The trace length represented in internode units is an important geometric
factor as it imposes constraints on possible fractional patterns to be
realized (Okabe (2011, 2012b)). The geometric effect invoked in interpreting
the above results is schematically illustrated in Fig. 6. During growth of the
shoot apex, the precursor of the vascular system associated with a leaf
primordium develops to become the leaf trace (Romberger et al. (1993)). Both
lengths of the leaf trace and internode change continuously during growth. In
this view, the phyllotaxis order to be realized on the mature stem is
determined depending on the trace length represented in internode units. Top
two patterns in Fig. 6 represent the initial arrangements of the most common
case of 137.5∘ angular divergence, whereas their growth rates in the radial
direction are different, i.e., the two patterns are characterized with
different plastochron ratios (Richards (1951)). At this point, the difference
is not a qualitative but a quantitative one. Leaf traces (line segments) in
the left pattern traverse about four internodes, while those in the right
pattern traverse about six internodes. As the figure shows, a qualitative
difference is brought about through the quantitative difference in the trace
length: geometric constraints tend to achieve either a 2/5 or 3/8 phyllotaxis
on the mature stem depending on whether the trace length is shorter or longer
than five internodes. Experimentally, the difference would be judged on
whether leaf trace 0 is identified or not in the cross section at the level of
node 5. In a similar manner, the threshold value of the trace length for the
transition from 3/8 to 5/13 order is eight internodes, the denominator of the
lower order fraction. These are indicated by horizontal dashed lines in Fig.
3, as already remarked. In terms of the threshold length, the phyllotaxis
transition is interpreted consistently without resort to operations of
elaborate mechanisms like genetic programs; the phyllotaxis transitions are
caused because changing length of the leaf trace happens to cross the
threshold values of five and eight internodes. As illustrated in Fig. 6, this
view provides a consistent explanation of empirical observation that large
meristems result in arrangements of higher order fractions. Unequal
distribution of divergence angle noted in Fig. 4 is circumstantial evidence of
a unique intrinsic angle close to 137.5∘ and secondary distortions therefrom.
Fibonacci numbers 5 and 8 enter as the threshold values because leaf 0 appears
close to leaves 5 and 8 (Fig. 6). This in turn is a mathematical consequence
of the golden-mean angle at the apex. Mathematically, any number can be
approximated by a common fraction with any assigned degree of accuracy. The
greater the denominator, the better the approximation. A fast-converging
sequence of approximate fractions is uniquely determined for a given number
(Hardy and Wright (1979)). For the golden mean 0.3820, it is 1/2, 1/3, 2/5,
3/8, 5/13, etc., the main sequence of phyllotaxis. The denominators of these
approximate fractions comprise the index differences of nearby leaves, namely
the threshold values for the trace length. Thus, the geometric interpretation
predicts a correlation between the fraction index of the phyllotaxis order and
the trace length represented in internode units. Indeed, it has been remarked
as a general rule; the higher phyllotaxis order is associated with the longer
leaf trace (Girolami (1953); Esau (1965)). To conclude, the plant in its
maturity, as it were, achieves rational approximations to a divergence angle
at the apex in conformity with the leaf-trace length in internode units.
Mathematically, the golden mean is the worst “approximable” real number in the
sense that the sequence of the approximate fractions converges most badly. In
this connection, it has been commonly mentioned, despite objections, that the
golden-mean divergence is advantageous because leaves are distributed most
evenly to sunlight. When viewed in the context of this study, the golden-mean
divergence distributes the leaf traces most efficiently to coordinate the
vascular system. Although “phyllotaxis” is the arrangement of leaves on a stem
according to dictionaries, the geometric view on the leaf-trace organization,
the arrangement of leaves in a stem, may shed a new light on the long-standing
problem of phyllotaxis in vascular plants.
## References
* Adam (2006) Adam, J., 2006. Mathematics in Nature: Modeling Patterns in the Natural World. Princeton University Press.
* Adler et al. (1997) Adler, I., Barabé, D., Jean, R. V., 1997. A history of the study of phyllotaxis. Annals of Botany 80, 231–244.
* Braun (1835) Braun, A., 1835. Dr. Carl Schimper’s Vorträge über die Möglichkeit eines wissenschaftlichen Verständnisses der Blattstellung, nebst Andeutung der hauptsächlichen Blattstellungsgesetze und Insbesondere der Neuentdeckten Gesetze der Aneinanderreihung von Cyclen Verschiedene. Flora 18, 145–191.
* Church (1920) Church, A. H., 1920. On the interpretation of phenomena of phyllotaxis. Botanical memoirs. Hafner Pub. Co.
* Coxeter (1961) Coxeter, H. S. M., 1961. Introduction to Geometry. Wiley, New York and London.
* Esau (1965) Esau, K., 1965. Vascular differentiation in plants. New York: Holt, Rinehart and Winston.
* Girolami (1953) Girolami, G., 1953. Relation between phyllotaxis and primary vascular organization in linum. American Journal of Botany 40, 618–625.
* Hardy and Wright (1979) Hardy, G., Wright, E., 1979. An Introduction to the Theory of Numbers. Oxford Science Publications. Clarendon Press.
* Henfrey (1870) Henfrey, A., 1870. An elementary course of botany: structural, physiological and systematic. J. Van Voorst.
* Kuhlemeier (2007) Kuhlemeier, C., 2007. Phyllotaxis. TRENDS in Plant Science 12, 143–150.
* Larson (1980) Larson, P. R., 1980. Interrelations between phyllotaxis, leaf development and the primary-secondary vascular transition in Populus deltoides. Annals of Botany 46, 757–769.
* Okabe (2011) Okabe, T., 2011. Physical phenomenology of phyllotaxis. Journal of Theoretical Biology 280, 63–75.
* Okabe (2012a) Okabe, T., 2012a. Systematic variations in divergence angle. Journal of Theoretical Biology 313, 20–41.
* Okabe (2012b) Okabe, T., 2012b. Vascular phyllotaxis transition and an evolutionary mechanism of phyllotaxis. http://arxiv.org/abs/1207.2838.
* Richards (1951) Richards, F. J., 1951. Phyllotaxis: its quantitative expression and relation to growth in the apex. Philosophical Transactions of the Royal Society of London. Series B 225, 509–564.
* Romberger et al. (1993) Romberger, J., Hejnowicz, Z., Hill, J., 1993. Plant structure: function and development : a treatise on anatomy and vegetative development, with special reference to woody plants. Springer-Verlag.
|
arxiv-papers
| 2012-12-13T10:16:32 |
2024-09-04T02:49:39.253619
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Takuya Okabe",
"submitter": "Takuya Okabe",
"url": "https://arxiv.org/abs/1212.3112"
}
|
1212.3377
|
# Systematic variations in divergence angle
Takuya Okabe Faculty of Engineering, Shizuoka University, 3-5-1 Johoku,
Hamamatsu 432-8561,Japan
###### Abstract
Practical methods for quantitative analysis of radial and angular coordinates
of leafy organs of vascular plants are presented and applied to published
phyllotactic patterns of various real systems from young leaves on a shoot tip
to florets on a flower head. The constancy of divergence angle is borne out
with accuracy of less than a degree. It is shown that apparent fluctuations in
divergence angle are in large part systematic variations caused by the invalid
assumption of a fixed center and/or by secondary deformations, while random
fluctuations are of minor importance.
###### keywords:
phyllotaxis; logarithmic spiral; parastichy lattice; Helianthus; Asteraceae
## 1 Introduction
It has long since been recognized that divergence angles between successive
leafy organs of vascular plants are accurately regulated at one of special
constant values. Deviations from the the constant angle are normally so small
that this botanical phenomenon, phyllotaxis, should rather be regarded as a
genuine subject of exact science. Besides the angular regularity, radial
coordinates also exhibit mathematical regularity.
In a polar coordinate system, position of the $n$-th leaf is specified by the
radial and angular coordinates $(r_{n},\theta_{n})$. The angular regularity is
expressed by the equation
$\theta_{n}=nd,$ (1)
where $d$ is a divergence angle. A spiral pattern is made when the radial
component $r_{n}$ is a monotonic function of $n$, i.e., $r_{n}$ preserves the
order of the leaf index $n$. Particularly important is a logarithmic spiral
given by
$r_{n}=a^{n},$ (2)
where $a$ is a constant, called the plastochron ratio. Leaf primordia at a
shoot tip appear to be arranged on a logarithmic spiral. Eq. (2) is a solution
of the differential equation of exponential growth,
$\frac{{\rm d}}{{\rm d}n}\log r_{n}=\log a.$ (3)
Thus, the logarithmic spiral is characterized by the constant growth rate
$\log a$. On the other side, a power-law spiral given by
$r_{n}=\sqrt{n}$ (4)
has been often used as a mathematical model of packed seeds on a sunflower
head (Vogel (1979); Ridley (1982); Rivier et al. (1984)). The equation (4) is
expected when all seeds have equal areas, i.e., for the area per seed,
$\frac{{\rm d}}{{\rm d}n}(\pi r_{n}^{2})={\rm const}.$ (5)
In mathematical studies, the relations (1), (2) and (4) are often accepted
without inquiring their empirical basis. From a physical standpoint, the
validity of the mathematical relations has to be checked against experiments
at all events. While embarking on quantitative assessment, however, we are
confronted with a serious methodological problem. First of all, a center of
the pattern, the origin of the polar coordinate system, must be located.
Unfortunately, it is not always possible to locate a fixed center properly and
objectively. The uncertainty of the central position becomes a fundamental
obstacle to assessing the empirical relations in a quantitative manner.
Quantitative analysis of phyllotaxis has not been made until recently.
Maksymowych and Erickson (1977) used a trigonometric method for evaluating
divergence angle from positions of three successive leaves. Their three-point
method is based on the exponential growth (2), and it has been developed by
Meicenheimer (1986) and Hotton (2003). On the other side, Rutishauser (1998)
has estimated the plastochron ratio $a$ from the geometrical mean
$({r_{m}}/{r_{n}})^{1/(m-n)}$ evaluated for selected leaves. The exponential
growth is verified by the observation that the mean is nearly constant. Ryan
et al. (1991) has assessed the validity of (4) for real sunflowers by indirect
means of evaluating the Euclidean distance between successive seeds,
$D_{n}^{2}=r_{n}^{2}+r_{n+1}^{2}-2r_{n}r_{n+1}\cos(\theta_{n}-\theta_{n+1})$.
This method has a merit of being independent of the choice of the center
position. They have attempted parameter fittings with various functional forms
of $r_{n}$. Matkowski et al. (1998) has investigated computational methods by
regarding the center of gravity as the center of the pattern. Hotton (2003)
has proposed to use the minimal variation center that minimizes the standard
deviation of divergence angles.
The main aim of this paper is to present practical methods for evaluating
radial and angular coordinates of phyllotactic units without assuming any
specific functional form of $r_{n}$ nor the existence of a fixed center of the
pattern. In Sec. 2.1, a floating center of divergence is defined by positions
of four consecutive leaves. In Sec. 2.2, a floating center of parastichy is
defined by four nearby points forming a unit cell of a parastichy lattice. For
this purpose, a systematic index system for leaves of a multijugate pattern is
proposed in Sec. 2.2.2. In Sec. 3, effects of uniform deformation are
considered. In Sec. 4, presented methods are applied to real representative
systems, which are arbitrarily chosen from previous works (Maksymowych and
Erickson (1977); Ryan et al. (1991); Rutishauser (1998); Hotton et al.
(2006)). In A, mathematical characteristics of the logarithmic spiral pattern
are examined carefully. To assist in interpreting the results in Sec. 4,
general mathematical formulas for the plastochron ratio $a$ of systems with
orthogonal parastichies are derived. In B, a practical method for numbering
packed florets of a high phyllotaxis pattern is presented. The method is based
on a number theoretic algorithm. To analyze phyllotaxis of Asteraceae, Hotton
et al. (2006) have introduced a primordia front based on the assumption that
the pattern has a definite center and the radius $r_{n}$ from the center is a
monotonic function of $n$. In B, a similar concept, a front circle on a
parastichy lattice, is introduced without referring to the center.
## 2 Floating center
### 2.1 Floating center of divergence
For the purpose of this paper, it is suffice to investigate phyllotactic
patterns projected on a plane. For three-dimensional analysis, see Hotton et
al. (2006) and references therein. An effect due to a slight inclination of
the normal axis is discussed in Sec. 3.
Figure 1: Given four consecutive points, $P_{1}$, $P_{2}$, $P_{3}$ and
$P_{4}$, a center of divergence $O_{23}$ is defined such that $\angle
P_{1}O_{23}P_{2}=\angle P_{2}O_{23}P_{3}=\angle P_{3}O_{23}P_{4}$. A dashed
curve $C_{2}$ is a locus of the point $O_{2}$ satisfying $\angle
P_{1}O_{2}P_{2}=\angle P_{2}O_{2}P_{3}$. Similarly, a dashed curve $C_{3}$ is
drawn by the point $O_{3}$ satisfying $\angle P_{2}O_{3}P_{3}=\angle
P_{3}O_{3}P_{4}$. The point where $C_{2}$ and $C_{3}$ intersect is $O_{23}$,
the floating center of divergence.
Consider four consecutive points $P_{1}$, $P_{2}$, $P_{3}$ and $P_{4}$ in the
same geometric plane. Given the first three points $P_{1}$, $P_{2}$ and
$P_{3}$, the point $O_{2}$ satisfying
$\angle P_{1}O_{2}P_{2}=\angle P_{2}O_{2}P_{3}$ (6)
defines a curve $C_{2}$ (Fig. 1). When $P_{2}P_{1}=P_{2}P_{3}$, or the
triangle $P_{1}P_{2}P_{3}$ is isosceles, $C_{2}$ bisects the vertex angle
$\angle P_{1}P_{2}P_{3}$. Similarly, for the three points $P_{2}$, $P_{3}$ and
$P_{4}$, we obtain a curve $C_{3}$ traced by the point $O_{3}$ satisfying
$\angle P_{2}O_{3}P_{3}=\angle P_{3}O_{3}P_{4}.$ (7)
The floating center of divergence $O_{23}$ is defined as the crossing point of
the two curves $C_{2}$ and $C_{3}$. By definition,
$\angle P_{1}O_{23}P_{2}=\angle P_{2}O_{23}P_{3}=\angle P_{3}O_{23}P_{4}.$
Thus, the three angles spanned by the four consecutive points define a
divergence angle,
$d_{2}\equiv\angle P_{2}O_{23}P_{3}.$ (8)
Distance from $O_{23}$ is denoted as
$r_{2}\equiv|\overrightarrow{O_{23}P_{2}}|.$
In general, the divergence angle $d_{n}$ and the radius $r_{n}$ are evaluated
from four consecutive points $P_{n-1}$, $P_{n}$, $P_{n+1}$ and $P_{n+2}$;
$d_{n}\equiv\angle P_{n}O_{nn+1}P_{n+1},$ (9)
$r_{n}\equiv|\overrightarrow{O_{nn+1}P_{n}}|.$ (10)
These quantities are compared with
$\displaystyle d_{n}^{(0)}$ $\displaystyle=$ $\displaystyle\angle
P_{n}O_{0}P_{n+1},$ (11)
and
$r_{n}^{(0)}=|\overrightarrow{O_{0}P_{n}}|,$ (12)
which are defined in terms of an arbitrarily chosen center $O_{0}$.
Given coordinates of the four points $P_{n}$ $(n=1,2,3,4)$, the position of
the floating center $O_{23}$ is parametrically represented by means of two
parameters $X$ and $Y$;
$\overrightarrow{P_{2}O_{23}}=X\overrightarrow{P_{2}P_{3}}+Y\overrightarrow{P_{2}P_{1}}.$
(13)
The two parameters are determined by the two equations (6) and (7), namely
$\frac{|\overrightarrow{O_{23}P_{1}}|^{2}+|\overrightarrow{O_{23}P_{2}}|^{2}-|\overrightarrow{P_{1}P_{2}}|}{|\overrightarrow{O_{23}P_{1}}||\overrightarrow{O_{23}P_{2}}|}=\frac{|\overrightarrow{O_{23}P_{2}}|^{2}+|\overrightarrow{O_{23}P_{3}}|^{2}-|\overrightarrow{P_{2}P_{3}}|}{|\overrightarrow{O_{23}P_{2}}||\overrightarrow{O_{23}P_{3}}|},$
(14)
and
$\frac{|\overrightarrow{O_{23}P_{2}}|^{2}+|\overrightarrow{O_{23}P_{3}}|^{2}-|\overrightarrow{P_{2}P_{3}}|}{|\overrightarrow{O_{23}P_{2}}||\overrightarrow{O_{23}P_{3}}|}=\frac{|\overrightarrow{O_{23}P_{3}}|^{2}+|\overrightarrow{O_{23}P_{4}}|^{2}-|\overrightarrow{P_{3}P_{4}}|}{|\overrightarrow{O_{23}P_{3}}||\overrightarrow{O_{23}P_{4}}|}$
(15)
by the cosine formula in trigonometry.
For example, consider vertex points $P_{n}$ ($n=1,2,3,4,5$) of a regular
pentacle, whose Cartesian coordinates are given by $(x_{n},y_{n})=(\cos
nd,\sin nd),$ where $d=2\pi\alpha$ in radians and $\alpha=\frac{2}{5}$. Let
the center $O_{23}$ of the four points $P_{n}$ ($n=1,2,3,4$) be represented as
(13). For this special case, it is not difficult to find the solution
$X=Y=\frac{1}{2+\tau}\simeq 0.2764$ by geometrical considerations. The golden
ratio
$\tau=\frac{\sqrt{5}+1}{2}$ (16)
is the irrational number quintessential to the phenomenon of phyllotaxis. The
estimate of $X\simeq Y\sim 0.3$ can be used as an initial guess for the
numerical search of solutions in general cases.
The manner in which the floating center floats around may be illustrated by
means of a line segment connecting the floating center and the middle point of
the middle two leaves defining the center, that is, the line segment
$O_{23}M_{23}$, where $M_{23}$ is the middle point of $P_{2}$ and $P_{3}$. Let
us call a graph of the line segments a divergence diagram. If the center is
fixed in space, all the line segments radiate from the fixed center. If it is
not, the lines cross with each other. See Fig. 4, for instance.
When a pattern with small divergence angles is deformed significantly, it may
happen that no center is defined because the two curves $C_{2}$ and $C_{3}$ do
not cross (Fig. 2). Even in such a case, a center can be defined formally by
selecting four points properly. Here it is remarked only that a pattern
consisting of more than three points may not have a definite center. In other
words, it is very special for numerous leaves comprising a phyllotactic
pattern to have a unique, fixed center. Assuming the fixed center is not at
all a trivial matter.
Figure 2: Four points $P_{i}$ ($i=1,2,3,4$) from a deformed Lucas pattern for
which two curves $C_{2}$ and $C_{3}$ as defined in Fig. 1 fail to cross. Thus,
it may happen that the center of divergence is not defined when divergence
angle is small and variable.
### 2.2 Floating center of parastichy
#### 2.2.1 Spiral phyllotaxis
Figure 3: From four nearby points $P_{n}$, $P_{n+q}$, $P_{n+q^{\prime}}$ and
$P_{n+q+q^{\prime}}$ making a tetragonal cell of a lattice of $q$ and
$q^{\prime}$ parastichies, a center of parastichy $O_{n}^{(q,q^{\prime})}$ is
determined by (17) and (18). By definition, four angles subtended by four
sides of the tetragon give a unique divergence angle against
$O_{n}^{(q,q^{\prime})}$, which is called the floating center of parastichy.
For a spiral pattern with one leave per each node, leaves are orderly indexed
by an integer $n$, called the plastochron. Four points for a center need not
be consecutive in plastochron. A divergence angle is evaluated from four
vertex points $P_{n}$, $P_{n+q}$, $P_{n+q^{\prime}}$ and $P_{n+q+q^{\prime}}$
of a unit cell of a parastichy lattice, where $q$ and $q^{\prime}$ are
opposite parastichy numbers. Parastichies of logarithmic spirals are discussed
in A, where it is shown that divergence angle $d=2\pi\alpha_{0}$ of an ideal
pattern satisfies the inequalities (56) in terms of two auxiliary integers $p$
and $p^{\prime}$ satisfying $pq^{\prime}-p^{\prime}q=1$, (57). With this in
mind, a center of parastichy $O_{n}^{(q,q^{\prime})}$ is defined such that
$q^{\prime}\angle P_{n}O_{n}^{(q,q^{\prime})}P_{n+q}+q\angle
P_{n}O_{n}^{(q,q^{\prime})}P_{n+q^{\prime}}=2\pi,$ (17)
and
$\angle P_{n}O_{n}^{(q,q^{\prime})}P_{n+q}=\angle
P_{n+q^{\prime}}O_{n}^{(q,q^{\prime})}P_{n+q+q^{\prime}}.$ (18)
where $\angle P_{n}O_{n}^{(q,q^{\prime})}P_{n+q}$ signifies the angle
subtended by line segments $O_{n}^{(q,q^{\prime})}P_{n}$ and
$O_{n}^{(q,q^{\prime})}P_{n+q}$. See Fig. 3. Then, four angles made by the
four points give a divergence angle;
$\displaystyle d_{n}^{(q,q^{\prime})}$ $\displaystyle=$
$\displaystyle\left(2\pi p-\angle P_{n}O_{n}^{(q,q^{\prime})}P_{n+q}\right)/q$
(19) $\displaystyle=$ $\displaystyle\left(2\pi p-\angle
P_{n+q^{\prime}}O_{n}^{(q,q^{\prime})}P_{n+q+q^{\prime}}\right)/q$
$\displaystyle=$ $\displaystyle\left(\angle
P_{n}O_{n}^{(q,q^{\prime})}P_{n+q^{\prime}}+2\pi p^{\prime}\right)/q^{\prime}$
$\displaystyle=$ $\displaystyle\left(\angle
P_{n+q}O_{n}^{(q,q^{\prime})}P_{n+q+q^{\prime}}+2\pi
p^{\prime}\right)/q^{\prime}.$
The rationale behind this definition may be understood by expressing the first
equation as $\angle P_{n}O_{n}^{(q,q^{\prime})}P_{n+q}=2\pi
p-d_{n}^{(q,q^{\prime})}q$, which is to be compared with the denominator of
(50). This is the net angle between the two successive points along a $q$
parastichy $P_{n}$ and $P_{n+q}$ when divergence angles between the two points
are equal to $d_{n}^{(q,q^{\prime})}$. The difference in the signs in front of
geometrical angles like $\angle P_{n}O_{n}^{(q,q^{\prime})}P_{n+q}$ in (19) is
due to the convention that the angles including $d_{n}^{(q,q^{\prime})}$ are
regarded as positive quantities.
Coordinates of the center $O_{n}^{(q,q^{\prime})}$ are represented in terms of
two parameters $r$ and $\theta$ as
$\overrightarrow{P_{n}O_{n}^{(q,q^{\prime})}}=r\cos\theta\overrightarrow{P_{n+q}P_{n}}+r\sin\theta\overrightarrow{P_{n+q^{\prime}}P_{n}}.$
(20)
Accordingly, (17) and (18) are regarded as the defining equations for $r$ and
$\theta$. As is clear from Fig. 3, it is generally assumed that $r>0$ and
$0<\theta<\frac{\pi}{2}$. By regarding $\theta$ as a given constant, the first
equation (17) determines $r$ for the given value of $\theta$, or $r$ is
determined as a function of $\theta$. Then, the parameter $\theta$ is fixed by
the second equation (18). Thus, the center $O_{n}^{(q,q^{\prime})}$ for given
$P_{n}$, $P_{n+q}$, $P_{n+q^{\prime}}$ and $P_{n+q+q^{\prime}}$ is fixed
numerically. The equations depend not only on the coordinates of the four
points, but also on the parastichy numbers $q$ and $q^{\prime}$, while either
$p$ or $p^{\prime}$ is needed to evaluate the divergence angle in (19).
A caveat: In the above, it is assumed that $P_{n}$ with the smallest index $n$
is positioned nearest to the center (Fig. 3). In the opposite convention,
$P_{n}$ is farther from the center than $P_{n+q+q^{\prime}}$. To adapt the
above results to this case, the four points $P_{n}$, $P_{n+q}$,
$P_{n+q^{\prime}}$ and $P_{n+q+q^{\prime}}$ should be replaced by
$P_{n+q+q^{\prime}}$, $P_{n+q^{\prime}}$, $P_{n+q}$ and $P_{n}$, respectively.
A set of equations (17), (18), and (19) is invariant by this replacement,
whereas (20) should be read as
$\overrightarrow{P_{n+q+q^{\prime}}O_{n}^{(q,q^{\prime})}}=r\sin\theta\overrightarrow{P_{n+q}P_{n+q+q^{\prime}}}+r\cos\theta\overrightarrow{P_{n+q^{\prime}}P_{n+q+q^{\prime}}}.$
(21)
In the biological literature, leaves on a stem are numbered in the order of
appearance, while primordia on an apex are counted in the opposite order.
While $d_{n}$ in (9) based on four distant points is insensitive to individual
displacements of the points, $d_{n}^{(q,q^{\prime})}$ in (19) based on nearby
points is insensitive to a collective displacement of the points. The latter
has a merit of wider applicability in practice. It is stressed again that
parastichies generally do not have a definite center; the center
$O_{n}^{(q,q^{\prime})}$ thus determined depends on the plastochron $n$ and
the parastichy pair $(q,q^{\prime})$.
#### 2.2.2 Multijugate phyllotaxis
A multijugate pattern bears more than one leaves at each node. There has been
no established way of systematize multijugate leaves, particularly because a
special index to distinguish leaves at a node is lacking. Therefore, in the
first place, a systematic index system for a multijugate pattern is proposed
below. Then, the center of parastichy is defined similarly as in the last
subsection.
A $J$-jugate pattern has $J$ leaves at a node. There are $J$ fundamental
spirals correspondingly. Each of $J$ leaves at the $n$-th node is specified by
polar coordinates $(r_{n,j},\theta_{n,j})$, where the jugacy index
$j=0,1,2,\cdots,J-1$ is introduced. Any leaf may be chosen as the origin
$(n,j)=(0,0)$ of the index system. Along with the plastochron $n$, the jugacy
index $j$ is set in order in the direction of the fundamental spirals. The
leaf with the index $(n,j)$ is referred to by the symbol $P_{n}^{j}$, or
simply denoted as $n^{j}$. For instance, see Fig. 13 for a trijugate system
with $J=3$. For convenience’ sake, the value range of the jugacy index $j$ is
extended to all integers; the coordinates $(r_{n,j},\theta_{n,j})$ are
regarded as periodic in $j$ with period of $J$, namely
$(r_{nj+J},\theta_{nj+J})=(r_{n,j},\theta_{n,j})$ for all $n$ and $j$.
Accordingly, divergence angle
$d_{n}^{j}=\theta_{n+1,j}-\theta_{n,j}$ (22)
is periodic in $j$. As implied by the notation, $d_{n}^{j}$ depends on the
leaf index $(n,j)$, whereas it becomes constant, $d_{n}^{j}=d$, for an ideal
pattern.
To put it concretely, let us consider an ideal multijugate system spiraling in
the positive direction, i.e., $d>0$. A pattern with a negative angle $d<0$ is
the mirror image of the positive counterpart. In an ideal $J$-jugate pattern,
the radial coordinate $r_{n}^{j}$ is independent of the jugacy index $j$,
$r_{n,j}=r_{n},$
whereas the angular coordinate is given by
$\theta_{n,j}=2\pi\left(n\alpha_{0}+\frac{j}{J}\right),$ (23)
where the divergence angle $d=2\pi\alpha_{0}$ is represented with a
dimensionless parameter $\alpha_{0}$. Without loss of generality,
$0<J\alpha_{0}<\frac{1}{2}$. For multijugate patterns $J>1$, the other
quantity of interest is displacement angle between neighboring leaves at a
node,
$\Delta_{n}^{j}=\theta_{n,j+1}-\theta_{n,j}-\frac{2\pi}{J}.$ (24)
For the ideal pattern with (23), $\Delta_{n}^{j}=0$. In general,
$\Delta_{n}^{j}$ fluctuates evenly about zero.
In this index system, a parastichy is specified by a set of numbers $(q,j)$,
increments of $(n,j)$. Conspicuous parastichies following nearby leaves are
shown to have $j=-p$, where $p$ is an integer near $J\alpha_{0}q$ (cf. below
(50)). Thus, the $q$ parastichy running through a leaf $P_{n}^{i}$ connects
the points $P_{n}^{i}$, $P_{n+q}^{i-p}$, $P_{n+2q}^{i-2p}$,
$P_{n+3q}^{i-3p},\cdots$. For instance, the trijugate pattern in Fig. 13 has
$J=3$ and $(q,p)=(3,1)$, $(5,2)$, etc. Note that a 3-parastichy
$1^{0}-4^{-1}-7^{-2}-10^{-3}-\cdots$ is equivalently represented as
$1^{0}-4^{2}-7^{1}-10^{0}-\cdots$, as the jugacy superscript $j$ is understood
modulo $J=3$ by the periodic extension prescribed above.
Having thus prepared, the method of the parastichy center in the last
subsection is applied. The divergence angle $d_{n}^{j(q,q^{\prime})}$ of a
multijugate system is defined by the four points $P_{n}^{j}$, $P_{n+q}^{j-p}$,
$P_{n+q^{\prime}}^{j-p^{\prime}}$, $P_{n+q+q^{\prime}}^{j-p-p^{\prime}}$;
(17), (18) and (19) should read
$q^{\prime}\angle P_{n}^{j}O_{n}^{(q,q^{\prime})}P_{n+q}^{j-p}+q\angle
P_{n}^{j}O_{n}^{(q,q^{\prime})}P_{n+q^{\prime}}^{j-p^{\prime}}=2\pi/J,$ (25)
$\angle P_{n}^{j}O_{n}^{(q,q^{\prime})}P_{n+q}^{j-p}=\angle
P_{n+q^{\prime}}^{j-p^{\prime}}O_{n}^{(q,q^{\prime})}P_{n+q+q^{\prime}},$ (26)
and
$\displaystyle d_{n}^{j(q,q^{\prime})}$ $\displaystyle=$
$\displaystyle\left(\frac{2\pi p}{J}-\angle
P_{n}^{j}O_{n}^{(q,q^{\prime})}P_{n+q}^{j-p}\right)/q$ (27) $\displaystyle=$
$\displaystyle\left(\frac{2\pi p}{J}-\angle
P_{n+q^{\prime}}^{j-p^{\prime}}O_{n}^{(q,q^{\prime})}P_{n+q+q^{\prime}}^{j-p-p^{\prime}}\right)/q$
$\displaystyle=$ $\displaystyle\left(\angle
P_{n}^{j}O_{n}^{(q,q^{\prime})}P_{n+q^{\prime}}^{j-p^{\prime}}+\frac{2\pi
p^{\prime}}{J}\right)/q^{\prime}$ $\displaystyle=$ $\displaystyle\left(\angle
P_{n+q}^{j-p}O_{n}^{(q,q^{\prime})}P_{n+q+q^{\prime}}^{j-p-p^{\prime}}+\frac{2\pi
p^{\prime}}{J}\right)/q^{\prime}.$
## 3 Effect of uniform deformation
Consider a regular spiral pattern in which position of the $n$-th leaf is
given by Cartesian coordinates $(x_{n},y_{n})$. The center $O_{0}$ of the
polar coordinate system $(r_{n}^{(0)},\theta_{n}^{(0)})$ is set at the origin.
The azimuthal angle $\theta_{n}^{(0)}$ is measured from the $x$-axis. As
typical cases for a phyllotactic pattern to lose track of the center, two
types of global deformation are considered. The deformation is characterized
by a single parameter representing strain $C$.
Uniform deformation of the first order transforms $(x_{n},y_{n})$ as follows;
$\displaystyle x_{n}$ $\displaystyle\longrightarrow$
$\displaystyle\left\\{\begin{array}[]{ll}(1+C_{1})x_{n}.&x_{n}\geq 0\\\
(1-C_{1})x_{n}.&x_{n}<0\\\ \end{array}\right.$ (30) $\displaystyle y_{n}$
$\displaystyle\longrightarrow$ $\displaystyle y_{n}.$ (31)
Uniform deformation of the second order makes
$\displaystyle x_{n}$ $\displaystyle\longrightarrow$
$\displaystyle(1-C_{2})x_{n},$ $\displaystyle y_{n}$
$\displaystyle\longrightarrow$ $\displaystyle y_{n}.$ (32)
The magnitude of deformation is represented by constants $C_{1}$ and $C_{2}$,
respectively, which are assumed significantly smaller than 1. In both cases,
the radial component $r_{n}^{(0)}\equiv\sqrt{x_{n}^{2}+y_{n}^{2}}$ and
divergence angle $\theta_{n}^{(0)}-\theta_{n-1}^{(0)}$ suffer modulation
periodic in the azimuthal angle $\theta_{n}^{(0)}$. The periods of modulation
for the first and the second order deformation are 360 and 180 degrees,
respectively.
The amplitude of modulation in the radius $r_{n}^{(0)}\pm\Delta r_{n}^{(0)}$
is given by
$\frac{\Delta r_{n}^{(0)}}{{r_{n}^{(0)}}}=C_{1}$ (33)
for the first order deformation, and
$\frac{\Delta r_{n}^{(0)}}{{r_{n}^{(0)}}}=\frac{C_{2}}{2}$ (34)
for the second order deformation.
Amplitude of modulation in divergence angle, ${\Delta d}^{(0)}$, may be
estimated as follows. Consider an isosceles triangle with the vertex angle
$d^{(0)}$, the base of length $2L$, and the height $H$. Then,
$\tan\frac{d^{(0)}}{2}=\frac{L}{H}.$
By the first order deformation, $d^{(0)}$ becomes $d^{(0)}\pm\Delta d^{(0)}$
when the height $H$ parallel to the $x$ axis is modified to $(1\mp C_{1})H$.
By the second order deformation, $d$ is modified to $d^{(0)}\pm\Delta d^{(0)}$
as $(H,L)$ become $((1-C_{2})H,L)$ and $(H,(1-C_{2})L)$. In either case,
$\tan\frac{d^{(0)}\pm\Delta d^{(0)}}{2}\simeq\frac{L}{H}(1\mp C),$ (35)
where $C$ is $C_{1}$ or $C_{2}$. Accordingly,
${\Delta d^{(0)}}\simeq C{\sin d^{(0)}}.$ (36)
In particular,
${\Delta d^{(0)}}\simeq 38.7C\ {\rm degrees}$ (37)
for $d^{(0)}=137.5$ degrees, and
${\Delta d^{(0)}}\simeq 56.5C\ {\rm degrees}$ (38)
for $d^{(0)}=99.5$ degrees.
The first order deformation may be caused by a tendency toward or against the
direction of the sun (Kumazawa and Kumazawa (1971)). The second order
deformation may be caused by various reasons, real or apparent. The most
possible would be due to uniform compression. It applies also to the case
where a pattern is observed from a direction oblique to the perpendicular
direction. If the angle of inclination is denoted as $\varphi$ in radian
measure, then $1-C_{2}=\cos\varphi$, or $C_{2}\simeq{\varphi^{2}}/{2}$.
## 4 Results
### 4.1 Example
Figure 4: An ideal pattern of a logarithmic spiral with the plastochron ratio
$a=1.073$ is compressed (a) by (31) with $C_{1}=0.05$, and (b) by (32) with
$C_{2}=0.05$. Leaves are represented with points with integer index $n$.
Radiating lines represent $O_{nn+1}M_{nn+1}$, where $O_{nn+1}$ is the floating
center of divergence for the four consecutive leaves from the $(n-1)$-th to
$(n+2)$-th leaves and $M_{nn+1}$ is the middle point of the $n$-th and
$(n+1)$-th leaves. It is called a divergence diagram in Sec. 2.1.
(a) $d^{(0)}$ and $d$ against $n$ (b) $r^{(0)}$ and $r$ against $n$
(c) $d^{(0)}$ and $d$ against $\theta$
(d) $\Delta r^{(0)}/r^{(0)}$ and $\Delta r/r$ against $\theta$
Figure 5: Results for the deformed pattern in Fig. 4. $d^{(0)}$ and $r^{(0)}$
are divergence angle and radial coordinate with respect to the center of the
original (undeformed) spiral pattern, while $d$ and $r$ are divergence angle
and radial coordinate with respect to the floating center of divergence. For
the horizontal axis, $n$ is the leaf index, and $\theta$ is the polar angle,
or azimuth. $\Delta r^{(0)}$ and $\Delta r$ are changes in $r^{(0)}$ and $r$
by the uniform deformation for $C_{1}=0.05$.
(a) $d^{(0)}$ and $d$ against $n$ (b) $r^{(0)}$ and $r$ against $n$
(c) $d^{(0)}$ and $d$ against $\theta$
(d) $\Delta r^{(0)}/r^{(0)}$ and $\Delta r/r$ against $\theta$
Figure 6: Results for the deformed pattern in Fig. 4 ($C_{2}=0.05$); cf. Fig.
5.
Figure 7: Divergence angles of the original (not deformed) pattern of Fig. 4
are measured against a misplaced center $O_{0}$ set at the middle point of
leaves 1 and leaf 5. The misplacement brings about apparent wild variations in
$d_{n}^{(0)}$ (open circles, left), which are correlated with azimuth
$\theta_{n}$ (right). Divergence angle $d_{n}$ evaluated with the floating
center of divergence (filled circles) does not depend on the wrong center
$O_{0}$; the original divergence angle is correctly retrieved without knowing
the true center.
Consider a regular pattern with $d=2\pi/\tau^{2}$ (137.5 degrees) and
$a=1.073$ for (1) and (2). This is an orthogonal $(5,8)$ system corresponding
to $i=5$ in (69). By deforming the pattern according to (31) with $C_{1}=0.05$
and (32) with $C_{2}=0.05$, patterns shown in Figs. 4 and 4 are obtained,
respectively. In Fig. 4, line segments radiating from the center represent the
divergence diagram explained at the end of Sec. 2.1. The fixed center $O_{0}$
for divergence angle $d^{(0)}$ and radius $r^{(0)}$ is set at the center of
the original pattern. Divergence angles $d^{(0)}$, $d$ and radii $r^{(0)}$,
$r$ for the pattern of Figs. 4 and 4 are shown in Figs. 5 and 6, respectively.
All quantities show characteristic variations. A type of variations as
exhibited by $d^{(0)}$ in Fig. 5(a) is frequently met in real systems. Fig.
5(c) shows that the variations are correlated with azimuth $\theta_{n}$. In
general, deformation has a larger effect on the divergence angles than on the
radii. Variations in the divergence angle $d$ by means of the floating center
in Sec. 2.1 are considerably suppressed compared with those of $d^{(0)}$. A
period of 180 degrees in Fig. 6(c) is the characteristic of uniform
compression in one direction, namely deformation of the second order.
Divergence angles facing the compressed direction increase apparently. For
$C_{2}=0.05$, (37) gives $\Delta d\simeq 1.9$ degrees, which corresponds to
the amplitudes of $d^{(0)}$ in Fig. 6(c) and $\Delta r^{(0)}/r^{(0)}$ in Fig.
6(d). The amplitudes of $d$ and $\Delta r/r$ are suppressed and enhanced
compared with $d^{(0)}$ and $\Delta r^{(0)}/r^{(0)}$, respectively. The
factors of suppression and enhancement are about the same value $\simeq 1.8$.
Even if a pattern has a definite center, misidentification of the center
causes apparently systematic variations in $d^{(0)}$. Fig. 7 is a result for
the undistorted pattern $C_{1}=C_{2}=0$, whereas the middle point of 1 and 5
is chosen as the nominal center $O_{0}$. The divergence $d$ based on the
floating center does not depend on the choice of $O_{0}$, whereas the
variations in $d^{(0)}$ exhibit a period of 360 degrees when plotted against
the azimuth $\theta$. Thus, the misplacement of the center can be
misinterpreted as the first order deformation.
These results indicate that systematic features of variations may not be
revealed unless they are plotted against the azimuth.
### 4.2 Suaeda vera
(a) Position of leaves $(x_{n},y_{n})$
(b) Divergence angles $d_{n}$ and $d^{(0)}_{n}$ against the leaf index $n$
(c) $d_{n}$ and $d^{(0)}_{n}$ against $\theta_{n}$
(d) $\log r_{n}$ and $\log r^{(0)}_{n}$ against $n$
Figure 8: Results for a normal phyllotaxis of young leaves of Suaeda vera,
whose pattern shown in (a) is taken after Fig. 1a of Rutishauser (1998).
Radiating lines in (a) represent a divergence diagram (cf. the caption of Fig.
4 and Sec. 2.1).
As a typical example of real systems, phyllotaxis of young leaves of Suaeda
vera is shown in Fig. 8(a) after Fig. 1a of Rutishauser (1998). The actual
size of the pattern is about 1mm in diameter. Leaf position is optically read
as marked in the original figure, which is a drawn figure based on a microtome
section. A leaf is not a point. It has shape and size. In this case, and
presumably in most past cases reported so far, position of a leaf is
represented by the position of the main vascular bundle when it is visually
discernible. This paper does not take up cases in which position of leaves may
not be specified unambiguously. A fixed center $O_{0}$ for $d^{(0)}$ in Figs.
8(b) and 8(c) is set according to the original figure. Statistical results for
divergence angle are $d^{(0)}=136.8\pm 5.8^{\circ}$ by the fixed center
$O_{0}$, $d=136.9\pm 1.8^{\circ}$ by the floating center of divergence, and
$d^{(5,8)}=137.0\pm 0.9^{\circ}$ by the floating center of parastichy. It is
remarkable that no significant deviation from the ideal limit angle of
$137.5^{\circ}$ is found, although the divergence diagram in Fig. 8(a)
indicates that the pattern does not have a well-defined center. In the
figures, $d_{n}^{(5,8)}$ for $n=15$ is omitted because no solution was found.
Rutishauser (1998) has remarked angular width of leaf arc, $i$, as another
quantity of interest. From the original figure, it is estimated as $i=66\pm
13^{\circ}$. Oddly enough, the standard deviation of the angular width of leaf
arc is larger than that of divergence angle. If the right tip of leaf arc is
regarded as a representative point of the leaf, $d^{(0)}=137.4\pm
6.0^{\circ}$. If the left tip is used instead, $d^{(0)}=136.1\pm 5.8^{\circ}$.
Thus, divergence angle does not depend on which part of leaf is regarded as a
representative point. The results indicate that absolute positions of the
leaves are not affected even though the angular width of each leaf arc
fluctuates widely.
Apparent lack of stability in nominal divergence angle $d^{(0)}$ (open circles
in Fig. 8(b)) is due to systematic variations of the type expected for the
first order deformation (Fig. 8(c)). The azimuth plot of $d$ in Fig. 8(c)
(closed circles) indicates a slight indication of peaks at $\theta\simeq
80^{\circ}$ and $\theta\simeq 80+180^{\circ}$, signifying deformation of the
second order.
A semi-log plot of $r_{n}$ and $r_{n}^{(0)}$ in Fig. 8(d) confirms the
exponential rule (2). Straight lines labeled with $(3,5)$, $(5,8)$ and
$(8,13)$ represent the exponential growth according to (69) for $i=4,5$ and 6,
respectively. In the original figure, leaves are in contact with each other
along three contact parastichies 3, 5 and 8. The growth rate $\log a$ has to
be estimated in order to evaluate the phyllotaxis index (P.I.) in (74). This
may pose a methodological problem. The radial growth $r_{n}$ of a real system
is neither continuous nor monotonic in $n$. There are systematic variations in
$r_{n}^{(0)}$ and $r_{n}$, particularly because the pattern does not have a
definite center. Accordingly, a bad method or a bad choice of leaves may give
an unwanted result $\log a<0$ for some values of $n$. Rutishauser (1998) has
evaluated $a\simeq 1.06$ based on two ratios for three leaves, namely 2, 18
and 31 in Fig. 8(a) (30, 14 and 1 in the original figure). In the presence of
systematic variations, a curve fitting method objectively gives a more
accurate result. As shown in Fig. 8(d), the least-squares fitting method with
the function $r_{0}\exp(-n\log a)$ gives $\log a=0.0580\pm 0.0014$
($a=1.0597\pm 0.0015$) and $\log a=0.0593\pm 0.0023$ ($a=1.0611\pm 0.0024$)
for $r_{n}^{(0)}$ and $r_{n}$, respectively. Therefore, P.I.=4.18 by (74). The
pattern is close to an orthogonal $(5,8)$ system with P.I.=4.
### 4.3 Effect of gibberellic acid on Xanthium
(a) $(x_{n},y_{n})$
(b) $d_{n}$ and $d^{(0)}_{n}$ against $n$
(c) $d_{n}$ and $d^{(0)}_{n}$ against $\theta_{n}$
(d) $\log r_{n}$ and $\log r^{(0)}_{n}$ against $n$
Figure 9: Results for a control shoot of Xanthium, after the bottom photo-
micrograph in Fig. 6 of Maksymowych and Erickson (1977). Compare with Fig. 10.
(a) $(x_{n},y_{n})$
(b) $d_{n}$ and $d^{(0)}_{n}$ against $n$
(c) $d_{n}$ and $d^{(0)}_{n}$ against $\theta_{n}$
(d) $\log r_{n}$ and $\log r^{(0)}_{n}$ against $n$
Figure 10: Results for a gibberellic acid treated shoot of Xanthium, after
the top photo-micrograph in Fig. 6 of Maksymowych and Erickson (1977). Compare
with Fig. 9.
Figs. 9 and 10 are results obtained after the bottom and top photomicrographs
in Fig. 6 of Maksymowych and Erickson (1977), which are control and
gibberellic acid (GA) treated vegetative shoots of Xanthium, respectively. The
mean and standard deviation of divergence angle are $d^{(0)}=137.73\pm
8.03^{\circ}$ and $d=137.74\pm 1.46^{\circ}$ for the former, while
$d^{(0)}=138.71\pm 19.93^{\circ}$ and $d=138.55\pm 3.09^{\circ}$ for the
latter. The pattern of the control shoot in Fig. 9(a) indicates a well-defined
center, while the treated shoot in Fig. 10(a) does not have a center. As a
result, $d^{(0)}$ of the treated plant shows wild fluctuations. Thus, the GA
treatment not only decreases the growth rate $\log a$ (the slope of Figs. 9(d)
and 10(d)), but destabilize divergence angle appreciably (Fig. 10(b)).
Nonetheless, the treatment does not affect the mean divergence angle. These
results are consistent with analysis by Maksymowych and Erickson (1977) based
on the exponential growth (2). The exponential growth is corroborated
qualitatively, if not quantitatively, as shown in Figs. 9(d) and 10(d), where
reference lines labeled with parastichy pairs of Fibonacci numbers
$(F_{i},F_{i+1})$ represent the exponential growth with the constant rate
according to (69).
### 4.4 Helianthus tuberosus
#### 4.4.1 Lucas system: (11,18)
Figure 11: Lucas phyllotaxis on a capitulum of Helianthus tuberosus after
Fig. 2 of Ryan et al. (1991).
(a) $d_{n}$ and $d^{(0)}_{n}$ against $n$
(b) $d_{n}$ and $d^{(0)}_{n}$ against $\theta_{n}$
(c) $\log r_{n}$ and $\log r^{(0)}_{n}$ against $n$
(d) $r_{n}$ and $r^{(0)}_{n}$ against $n$
Figure 12: Results for a capitulum of Helianthus tuberosus plotted in Fig.
11. (After Fig. 2 of Ryan et al. (1991)).
In Figs. 11 and 12, results are shown for a capitulum of Helianthus tuberosus
after Fig. 2 of Ryan et al. (1991). The size of the pattern is about 13mm in
diameter. Here and below, digitized position of florets is taken after Ryan et
al. (1991), although the numberings are different from the original figures.
This is a Lucas system with the limit divergence angle of
$\alpha_{0}=\frac{1}{3+\tau^{-1}},$ (39)
or $d=2\pi\alpha_{0}=99.5^{\circ}$. A nominal center $O_{0}$ is set at the
center of gravity, or by
$\sum_{n}\overrightarrow{O_{0}P_{n}}=\vec{0}.$ (40)
The center of gravity is a good choice for the fixed center especially for a
system comprising a large number of pattern units. Results for divergence
angle are $d^{(0)}=99.72\pm 5.35^{\circ}$, $d=99.42\pm 1.28^{\circ}$ and
$d^{(18,11)}=99.47\pm 0.15^{\circ}$. Remark the accuracy attained in
$d^{(18,11)}$. Thus, the center of parastichy comes into its own in a high
phyllotaxis pattern. In a closely packed pattern, young seeds in a central
region are amenable to irregular displacements caused by inward pressure due
to old seeds in an outer region. This is observed as wide fluctuations of
$d^{(0)}$ for $n>70$ in Fig. 12(a). As shown below, the secondary displacement
appears to be a primary source of variations in divergence angle of a packed
pattern. The divergence angles in Fig. 12(b) indicate systematic variations of
the compression type. The amplitude $\Delta d^{(0)}\simeq 4^{\circ}$ implies
$C_{2}\simeq 0.07$ by (38), although the deformation is apparently
indiscernible from Fig. 11. In general, deviations of $r_{n}$ from an
exponential dependence give rise to shifts in parastichy numbers. In Figs.
12(c) and 12(d), curves labeled with $(11,18)$ and $(7,11)$ represent the
exponential dependence (2) with (55) for $\frac{p}{q}=\frac{5}{18}$,
$\frac{p^{\prime}}{q^{\prime}}=\frac{3}{11}$ and $\frac{p}{q}=\frac{2}{7}$,
$\frac{p^{\prime}}{q^{\prime}}=\frac{3}{11}$, respectively. The limit
divergence angle of (39) is used for $\alpha_{0}$. The $n$-dependence of
$r_{n}$ is neither exponential (2), nor square root (4). It is rather close to
linear (Fig. 12(d)). Accordingly, conspicuous parastichies change continuously
from $(11,18)$ near the rim to $(7,11)$ near the center. The apparent up shift
of the parastichy pair caused by gradual decrease of $\log a$ is called rising
phyllotaxis. The pattern in Fig. 11 shows falling phyllotaxis, the downshift
of the parastichy pair from $(11,18)$ to $(7,11)$, and even down to $(4,7)$.
It should be remarked that new cell primordia arise from the rim towards the
center, not vice versa as often assumed wrongly in theoretical models (cf.
(4)). Mathematically, the shift of parastichy corresponds to the fact that the
function $\log r_{n}$ is convex upward as shown in Fig. 12(c). Despite the
gradual shift in parastichy numbers, divergence angle is not affected (Fig.
12(a)). If phyllotaxis is to be judged by the divergence angle, there is no
sign of change in phyllotaxis.
#### 4.4.2 Trijugate system: 3 (3,5)
Figure 13: A rare trijugate $3(3,5)$ phyllotaxis on a sectioned capitulum of
Helianthus tuberosus. Adapted after Fig. 3 of Ryan et al. (1991). Seeds are
indexed as $n^{j}$ according to the numbering system $(n,j)$ explained in Sec.
2.2.2.
Figure 14: Results for a capitulum of Helianthus tuberosus plotted in Fig. 13
(after Fig. 3 of Ryan et al. (1991)). (a) $d_{n}^{1(5,3)}$, $d_{n}^{1(0)}$ and
$d_{n}^{(0)}=(d_{n}^{1(0)}+d_{n}^{2(0)}+d_{n}^{3(0)})/3$, and (b)
$r_{n}^{1(0)}$, $r_{n}^{2(0)}$ and $r_{n}^{3(0)}$ are plotted against the leaf
index $n$. They are evaluated in terms of the nominal center $O_{0}$ set by
(40).
Figs. 13 and 14 are results for a rare trijugate pattern $(J=3)$ of Helianthus
tuberosus based on Fig. 3 of Ryan et al. (1991). The size of the pattern is
about 8mm in diameter. In terms of the nominal center $O_{0}$ set by (40), one
obtains $3d^{j(0)}=137.4\pm 14.7^{\circ}$, $137.7\pm 12.9^{\circ}$ and
$135.3\pm 19.2^{\circ}$ for $j=1,2$ and 3, respectively, where
$d_{n}^{j(0)}=\angle P_{n}^{j}O_{0}P_{n+1}^{j}$
is divergence angle depending on the jugacy index $j$. Note that $j=3$ is
equivalent to $j=0$. In Fig. 13, the seed point $P_{n}^{j}$ is denoted as
$n^{j}$. For every integer $n$, the trijugate pattern has three organs of
approximately 120 degrees apart from each other, which are distinguished by
the jugacy index $j$. When averaged over $j$, nominal divergence angle is
$3d^{(0)}=136.8\pm 9.0^{\circ}$, showing a large standard deviation. As shown
by $d_{n}^{1(0)}$ in Fig. 14, variations in divergence angle of the
multijugate system are substantially larger than that of a simple spiral
system.
The radial coordinate
$r_{n}^{j(0)}=|\overrightarrow{O_{0}P_{n}^{j}}|,$
depends linearly on $n$ with less remarkable fluctuations. In Fig. 14, curves
labeled with $3(3,5)$ and $3(5,8)$ are drawn by (70) for $J=3,i=4$ and
$J=3,i=5$, respectively. The results suggest that fluctuations are due to
displacements in the angular direction. Indeed, large fluctuations
$\Delta^{(0)}=0\pm 6.85^{\circ}$ are observed for (24). Multijugate systems
are characterized by this large angular fluctuations. For this reason, the
method of the center of divergence is barely applicable. By means of the
parastichy center in Sec. 2.2.2, we get $3d^{j(3,5)}=136.53\pm 2.55^{\circ}$,
$137.85\pm 2.88^{\circ}$ and $137.43\pm 3.78^{\circ}$ for $j=1,2$ and 3,
respectively. In total, $3d^{(3,5)}=137.28\pm 3.09^{\circ}$. The standard
deviations are significantly reduced but still an order of magnitude larger
than the spiral system of the last subsection. The absence of a center of
rotation in a real $J$-jugate system is manifested by the very fact that
expected rotational symmetry of order $J$ is badly approximate. This may not
be apparent at a glance of Fig. 13, but it is immediately checked if Fig. 13
is overlapped with itself after being rotated by 120 degrees, or, e.g., by
comparing three pairs of $(2^{0},7^{1})$, $(2^{1},7^{2})$ and $(2^{2},7^{0})$.
The rotated pattern does not coincide with the original one.
### 4.5 Helianthus annuus
Figure 15: Developing florets on a capitulum of Helianthus annuus after Fig.
4 of Ryan et al. (1991).
(a) $d_{n}$ and $d^{(0)}_{n}$ against $n$
(b) $d_{n}$ and $d^{(0)}_{n}$ against $\theta_{n}$
(c) $\log r_{n}$ and $\log r^{(0)}_{n}$ against $n$
(d) $r_{n}$ and $r^{(0)}_{n}$ against $n$
(e) $r_{n}^{2}$ against $n$
(f) $d_{n}$ against the phyllotaxis index
(g) $d_{n}$ against $\theta_{n}$
(h) $\Delta r_{n}/r_{n}$ against $\theta_{n}$
Figure 16: Results for a capitulum of Helianthus annuus in Fig. 15 (after
Fig. 4 of Ryan et al. (1991)). Figure 17: Displacement from regular position
of florets on the capitulum of Helianthus annuus in Fig. 15 (cf. Fig. 16).
After Fig. 4 of Ryan et al. (1991).
Figs. 15 and 16 are for developing florets on a capitulum of Helianthus annuus
after Fig. 4 of Ryan et al. (1991). The actual size is about 9 cm in diameter.
Numerical results for divergence angle are $d^{(0)}=137.49\pm 1.64^{\circ}$,
$d=137.49\pm 0.56^{\circ}$ and $d^{(34,21)}=137.50\pm 0.07^{\circ}$. In
numerical calculation, some inner seeds fail to give the solution for the
center of parastichy, owing to significant displacement from regular position.
Ryan et al. (1991) have noted large fluctuations of $d^{(0)}$, open circles in
Fig. 16(a). The present study reveals that the fluctuations are not random but
mainly systematic variations due to local deformation of the pattern, as
indicated by the azimuth plots in Figs. 16(b) and 16(g).
The radius in Fig. 16(e) is fitted with
$r_{n}=r_{0}\sqrt{1-bn+cn^{2}}$ (41)
to give $r_{0}=0.528\pm 0.002$ (arbitrary), $b=(4.88\pm 0.07)\times 10^{-3}$,
and $c=(5.01\pm 0.27)\times 10^{-6}$. Thus, the area per seed,
$\frac{{\rm d}}{{\rm d}(-n)}(\pi r_{n}^{2})=\pi r_{0}^{2}(b+2c(-n)),$
depends on the seed index $n$. The positive value of the coefficient $c$
signifies that inner seeds are smaller than outer seeds, as expected
physically. The relative change in size is ${2c}/{b}=0.002$, namely 0.2
percent per seed. Accordingly, the innermost seed for $n=287$ is estimated to
be about half the size of the seeds around the rim. This is indeed seen from
the original figure. Scatterings of divergence angles of inner seeds (Fig.
16(a)) suggest that the physical effect to reduce the size of seeds affects
their position and divergence angles accordingly. In terms of the continuous
monotonic function (41), the $n$-dependent plastochron ratio is evaluated as
$\log a=-\frac{{\rm d}}{{\rm d}n}\log r_{n}=\frac{b-2cn}{2({1-bn+cn^{2}})},$
(42)
with which the phyllotaxis index is calculated by (74). In Fig. 16(f), the
divergence angle $d_{n}$ is shown against the phyllotaxis index. Note that
$\rm P.I.=5,6$ and 7 represent $(8,13)$, $(13,21)$ and $(21,34)$ orthogonal
systems, respectively. As the capitulum develops, the phyllotaxis index
decreases and variance in $d$ increases accordingly (falling phyllotaxis).
However, the mean value of $d$ is constantly held at the limit divergence
angle. The deviation $\Delta r_{n}\equiv r_{n}-r_{0}\sqrt{1-bn+cn^{2}}$
normalized by $r_{n}$ is plotted against the azimuthal angle $\theta$ in Fig.
16(h). Variations in $r_{n}$ show a similar dependence of $\theta$ as $d_{n}$
in Fig. 16(g), namely the dependence by deformation of the second order (Sec.
3). Amplitudes of variations in Figs. 16(g) and 16(h) are compared with Figs.
6(c) and 6(d) for $C_{2}=0.05$. Displacements of seeds are explicitly shown in
Fig. 17, where each vector represents shift in position of a seed from its
regular position determined by the mean divergence angle, virtually equal to
the limit divergence angle. The figure indicates that (i) the pattern is
compressed horizontally (stretched vertically) and (ii) inner seeds are
displaced appreciably by inward pressure. These are consistent with the above
observations, namely systematic modulation of divergence angle and decrease in
size of inner seeds. The latter is also consistent with the observation that
seeds near the center are even aborted (Ryan et al. (1991)).
(41) is transformed to
$r_{n}=r_{0}\sqrt{(n_{c}-n)\left(1+\frac{n_{c}-n}{n_{0}}\right)}.$ (43)
By definition, an integer $n_{c}$ is an index number of the innermost seed at
the center, $r_{n_{c}}=0$. In practice, however, the index $n_{c}$ of the last
seed may not be determined without ambiguity, for it depends on position of a
nominal center and whether or not to count aborted seeds. Be that as it may,
the numerical order of the index system can be reversed by formally replacing
$n_{c}-n$ with $n$. Consequently, (43) is equivalent to
$r_{n}=r_{0}\sqrt{n\left(1+\frac{n}{n_{0}}\right)}.$ (44)
in the reversed index system. Fig. 16(e) shows that the radial component
$r_{n}$ is better fitted with (44) than the square-root dependence of (4).
Ryan et al. (1991) have investigated products of algebraic, logarithmic, and
exponential dependences of $r_{n}$. The functional form of (44) has not been
considered previously, although it has wider practical applicability. A new
constant ${n_{0}}$ in (44) roughly signals the index $n$ of a seed for which
the square-root dependence (4) begins to fail; $r_{n}\propto\sqrt{n}$ for
$n\ll n_{0}$ whereas $r_{n}\propto n$ for $n\gg n_{0}$. As inner seeds $n\sim
0$ are likely to be displaced significantly, the limit exponent $\epsilon$
$(\simeq 0.5)$ of the power dependence near the center $r_{n}\propto
n^{\epsilon}$ ($n\rightarrow 0$) would be of little physical significance. As
a rough estimate, we get $n_{0}\sim 800$ for this sample, while $n_{0}\sim 20$
for Fig. 14.
### 4.6 Artichoke
Figure 18: An artichoke capitulum after Fig. 9A of Hotton et al. (2006).
(a) $d_{n}$ and $d^{(0)}_{n}$ against $n$
(b) $d_{n}$ and $d^{(0)}_{n}$ against $\theta_{n}$
(c) $\log r_{n}$ and $\log r^{(0)}_{n}$ against $n$
(d) $r_{n}$ and $r^{(0)}_{n}$ against $n$
(e) $d_{n}$ against $\theta$
(f) $r_{n}^{2}$ against $n$
(g) $\Delta r_{n}/r_{n}$ and $\Delta r_{n}^{(0)}/r_{n}^{(0)}$ against
$\theta_{n}$
(h) Displacement from regular position
Figure 19: Results for the artichoke capitulum in Fig. 18 (after Fig. 9A of
Hotton et al. (2006)).
Figs. 18 and 19 are results for an artichoke capitulum based on Fig. 9A of
Hotton et al. (2006). The size is about 4mm in diameter. Fig. 18 closely
resembles Fig. 15, although the sizes differ by more than an order of
magnitude. Position of primordia, which are round shaped and closely packed,
is read from the original figure with sufficient accuracy. Numerical results
obtained are $d^{(0)}=137.35\pm 5.03^{\circ}$, $d=137.53\pm 0.52^{\circ}$ and
$d^{(34,21)}=137.50\pm 0.08^{\circ}$, all falling onto the limit divergence
angle of (66). The variations in $d^{(0)}$ are substantially suppressed in $d$
and $d^{(34,21)}$ based on the floating centers. The standard deviation of
$d^{(34,21)}$ is particularly noteworthy. The nominal center $O_{0}$ for
$d^{(0)}$ is determined so as to minimize the standard deviation of $d^{(0)}$.
Still, $d^{(0)}$ shows relatively large variations. If the center of gravity
is chosen for $O_{0}$, one obtains $d^{(0)}=137.39\pm 5.03^{\circ}$ instead.
Therefore, large deviations in $d^{(0)}$ are not due to misplacement of a
fixed center. As a matter of fact, Fig. 19(b) clearly indicates that
deviations in divergence angle are due to systematic variations. As a rule,
apparently wild variations of divergence angle that cannot be suppressed by a
judicious choice of a center should be suspected as a sign of no definite
center by deformation. Fig. 19(e) reveals an unexpected result that remnant
variations in $d_{n}$ are primarily not of random origin but of the
compression type with modulation of $\Delta d\simeq 1^{\circ}$. As shown in
Fig. 19(f), the radius $r_{n}$ shows a square-root dependence of (4), namely
$r_{n}=r_{0}\sqrt{1-bn}$. The radius $r_{n}^{(0)}$, unlike the angle
$d^{(0)}$, is generally insensitive to the choice of a nominal center $O_{0}$,
so that $r_{n}^{(0)}$ is not affected severely even if the nominal center
$O_{0}$ is chosen inappropriately. In Fig. 19(g), the deviation from the fit
$\Delta r_{n}\equiv r_{n}-r_{0}\sqrt{1-bn}$ is plotted against the azimuthal
angle $\theta$. The systematic variations in $r_{n}$ have a different
$\theta$-dependence from that of $d_{n}$ in Fig. 19(e). This is inexplicable
by uniform deformation considered in Sec. 3. As shown in Fig. 19(h),
displacement of primordia from their regular position is not uniform;
displacement vectors display local structures resembling vortexes of
incompressible fluid. Here again, it is concluded that the local displacement
of closely packed organs gives rise to the systematic variations of $r_{n}$
and divergence angle $d_{n}$.
## 5 Discussion
Discontent not only with untested abstractions in the mathematical literature
but also with vague expressions like “the largest available space” and “about
137.5∘” in the botanical literature underlies this work. It is often assumed
implicitly that a spiral pattern has a definite center. This is not
necessarily true, as remarked in Sec. 2.1. As shown in Sec. 2.2, each cell of
a parastichy lattice may have its own center. The center of parastichy is not
fixed in space, but it may float around in the pattern as primordia arise one
after another. The floating center can be the main source of apparent
variations of divergence angle. In the literature, a logical leap is often
made from a measured value of about 137.5 to the irrational number
$360/\tau^{2}$ without subjecting data to statistical analysis. The lack of
quantification is partly because the center against which to measure the
angles is uncertain. Probably for this reason, little attention has been
directed to the numerical accuracy of divergence angle. The standard deviation
of divergence angle conveys no less important information than the mean value.
According to an observation, the standard deviation decreases systematically
as the plant grows day by day while the mean is kept constant (Williams
(1974)). This behavior is not confirmed by snapshot patterns investigated in
this paper. The present work finds no significant drift of the mean divergence
angle, which puts possible mechanisms of phyllotaxis under constraint.
There are two cases where it is invalid to assume for a phyllotactic pattern
to have a fixed center. In one case, the position of center is displaced as a
leaf primordium is initiated. In the other case, leafy organs are displaced by
secondary effects after they are initiated at their regular position. The
effects of the former and the latter are taken into account by means of the
floating centers in Sec. 2.1 and Sec. 2.2, respectively. It is noted that the
former concept conforms with observation by Green and Baxter (1987), who have
made a general remark that the center of apical area moves as a new leaf is
(or leaves are) initiated. Their observation is verified concretely by the
methods and results of this paper. Green and Baxter (1987) note that a
trajectory of the moving center may be used as a diagnostic for phyllotactic
patterns, which is to be contrasted with classification by means of
geometrical unit of ideal patterns (Meicenheimer and Zagórska-Marek (1989)).
At a glance of an impressive phyllotactic pattern, our attention is apt to be
directed to Fibonacci numbers, because our eyes tend to follow conspicuous
parastichies. However, parastichies do not serve much for quantitative
purposes. Polar coordinates $(r_{n},\theta_{n})$ of pattern units cannot be
deduced unequivocally from parastichy numbers alone, although parastichy
numbers are determined uniquely from coordinates $(r_{n},\theta_{n})$. The
polar coordinates are the most proper tool to specify a two-dimensional
pattern quantitatively, only with which it can be investigated whether and how
the divergence angle $\theta_{n}-\theta_{n-1}$ and the radial coordinate
$r_{n}$ depend on the leaf index $n$. If a pattern is deformed to lose a well-
defined center, the meaning of divergence angle blurs apparently. Still,
parastichies may remain little affected owing to their insensitivity to
quantitative details. Even an oval-shaped capitulum preserves parastichy
numbers (Szymanowska-Pulka (1994)). The method of B for indexing primordia
remains valid irrespective of whether the pattern has a center or not, because
it is based solely on parastichies. Although a real system generally do not
possess a well-defined center, the method of the floating centers enables us
to infer coordinates $(r_{n},\theta_{n})$ of the original pattern, as shown in
the last section.
Figure 20: Real and fake phyllotaxis to illustrate accurate control of
divergence angle $d$. Densely packed florets arise spirally in numerical order
from the rim toward the center. Inner florets may be omitted without affecting
arrangement near the rim, that is at issue. (a) Normal phyllotaxis of a
$(21,34)$ pattern for $d=137.5^{\circ}$, quite common in nature. (b) Seemingly
plausible but actually improbable phyllotaxis of a $(21,29)$ pattern for
$d=136.8^{\circ}$. Plants distinguish the difference of less than 1∘ without
fail, whereas the human eye would find it difficult to tell (a) from (b). The
difference comes to the fore when the 21st floret arises on either side of the
0th floret; plants know the correct position (angle) in advance. Central
portions after, say, the 120th floret are overlapped nearly perfectly by a
rotation of about 90∘, so that they are indistinguishable by means of lower
parastichy numbers there.
The results of this work underline the universal characteristic of
phyllotaxis: divergence angle during steady growth is stably held at
$137.5^{\circ}$, or rarely $99.5^{\circ}$ and other special values. This is
not just a general tendency. As a pivotal empirical rule deduced from
experimental facts that lays the foundation of mathematical analysis and thus
underlies this intriguing phenomenon of phyllotaxis, the author believes that
it is appropriate to attribute the status of a natural law to the universality
of the accurate control of divergence angle (cf. A). To convince that this is
not exaggeration, let us take two examples. The first is given by high
phyllotaxis of florets on a capitulum. According to (56), a $(21,34)$ pattern
is obtained if and only if divergence angle $d$ is constrained within a narrow
range of $\frac{8}{21}<\frac{d}{360^{\circ}}<\frac{13}{34}$, or
$137.14^{\circ}<d<137.65^{\circ}$. Imagine what happens if divergence angle
had been out of the range of width of 0.5∘. As a mathematical consequence, it
is shown that $(21,29)$ or $(13,34)$ should obtain instead of the normal
phyllotaxis $(21,34)$ (Fig. 20). Neither of these anomalous patterns is
observed actually. Thus, not only the constancy of divergence angle but also
the accurate control of it to the special value of $137.5^{\circ}$ should be
taken seriously as an empirical fact. The point is not just that a normal
$(21,34)$ phyllotaxis is observed, but rather that anomalous patterns of
$(21,29)$ and $(13,34)$ phyllotaxis do not coexist, or mixed, with the normal
pattern. Thus, plants must know precisely the correct value of divergence
angle beforehand. The second example is given by low phyllotaxis of young
leaves on the apex. As shown in Sec. 4.2, divergence angle is independent of
sizes of leaves. This is an important observation, because it apparently
conflicts with the assumption of dynamical models that divergence angle is
variable depending on the size of leaves (Mirabet et al. (2012)). So to speak,
the whole is more than the sum of its parts. It seems rather appropriate to
regard divergence angle as a given constant parameter (Sadoc et al. (2012)).
Along with a review on various approaches to phyllotaxis, a model conforming
to these observations has been given previously (Okabe (2012)).
A remark is in order. In the ontogeny, divergence angle generally begins with
180∘ or 90∘ of a distichous or decussate pattern, and thereafter shows some
transient fluctuations before attaining toward a universal constant value.
Variations in the transient regime are substantially larger (typically more
than about $10^{\circ}$) than the small variations in the steady-growth regime
investigated throughout this paper. Therefore, the two variations are
distinguished quantitatively. The former may be understood as a natural result
of the limited numbers of apical meristem cells pushing and shoving with each
other. The latter variations, however, are too small to be explained as such;
actually, as remarked above, it appears rather that divergence angle is
regulated steadily at a predetermined value, despite any adventitious factors.
Generally speaking, living systems are so complex that large variations are by
far easier to understand. For a quantitative understanding, the riddle of the
small variations cannot be overemphasized. It should be investigated for its
own sake, separately from the large transitory fluctuations during ontogeny.
It is also a finding of this paper that apparent deviations from the universal
mean are not due to random errors as commonly expected, but they are actually
systematic variations. On the one hand, they are likely to be due to
directional correlation with respect to the sun (Kumazawa and Kumazawa
(1971)). For lack of space, it is only noted here that the azimuthal
correlation, as plotted in Fig. 8(c), is a common phenomenon confirmed quite
generally. On the other hand, systematic variations are caused also by local
displacement when organs are closely packed. Concerning the latter, Ryan et
al. (1991) have attached importance to persistent fluctuations of individual
divergence angles (open circles in Fig. 16(a)) as a real phenomenon. As
indicated in Figs. 16(b) and 16(g), the fluctuations systematically correlated
with the azimuth, the absolute direction, should be regarded as a subsidiary
phenomenon. Thus, one should be cautious not to take apparent fluctuations
literally as evidence for or against a theoretical model of phyllotaxis. This
is an important point because almost all experimental and theoretical
divergence angles reported so far have not taken into account the possibility
that the center for the angles is not fixed in space. Therefore, virtually all
the reported results have to be reexamined carefully. If variations turn out
to be due to misidentification of the center, they have no real significance.
If they reflect real effects, they act to disturb the regularity. The
disturbance, if any, conveys important information, which may be analyzed with
a model from an appropriate theoretical perspective.
As illustrated by Figs. 17 and 19(h), variations of divergence angle are
interpreted naturally as necessary consequences of local distortion due to
contact pressure between closely packed organs. The contact pressure may
affect phyllotaxis not only quantitatively but even qualitatively. Primordia
under high pressure from the surrounding primordia may fail to grow normally,
as illustrated by innermost florets on a capitulum in Fig. 17. In general,
organs in a crowded portion are most likely to be aborted, or obliged not to
arise normally. Ill effects due to an aborted primordium may propagate along a
parastichy of vascular connection, along which nutrients are transported. The
other possibility is that new primordia may happen to arise to fill in gaps
opened between existing primordia. The author suspects that aborted or
inserted primordia might appear as a crystallographic defect in an otherwise
regular parastichy lattice (Zagórska-Marek (1994)). Szymanowska-Pulka (1994)
has made an interesting observation that the parastichy number decreases quite
more often than it increases.
Individual divergence angles measured against a nominal center may appear to
vary very wildly. Nonetheless, it has been empirically known that the limit
divergence angle of 137.5∘ is immediately obtained by evaluating the average
of several successive angles covering a few full turns. This empirical fact is
explained consistently by the observation of the present work that the
apparent variations are not random but systematically correlated with the
azimuth. If the variations are of random origin, the standard deviation should
depend on the number of sample leaves. The present study has found no such
dependence. In contrast, the systematic variations correlated with the azimuth
are averaged away within a few full turns of the azimuth. For the samples
analyzed in this paper, the systematic variations are of the order of 1
degree. If this secondary effects of the directional correlation are
compensated for, the accuracy and stability of angular control by nature
should stand out all the more strikingly; it should turn out to be of the
order of 0.1 degree as a general rule. This is an unexpected result. As
remarked above, the numerical accuracy of this quantitative phenomena of
living organs should arrest more attention than eye-catching Fibonacci
integers do.
As shown by the results, the angular equation (1) with constant divergence
angle $d$ remains valid quite generally. In contrast, the radial equation (4)
is not valid quantitatively, whereas (2) holds true at the shoot apex in a
good approximation. Unlike the divergence angle $d$, however, the plastochron
ratio $a$ may be changed artificially (Maksymowych and Erickson (1977)) (Sec.
4.3). As a matter of fact, it has been reported that the plastochron ratio $a$
changes naturally during shoot development. In particular, the ratio decreases
appreciably during transition to flowering without changing the phyllotactic
sequence determined by the divergence angle $d$. These observations suggest
that the growth in the radial direction $r_{n}$ is controlled independently of
the angle $\theta_{n}$ (cf. (3) and (5)). In other words, the regularity in
$r_{n}$ may not be dealt with on the same basis as the regularity in
$\theta_{n}$. Indeed, the present work finds no correlation between the angle
$\theta_{n}$ and the radius $r_{n}$. This point has bearing on how to
determine the numerical order of leaves, namely the leaf index system $n$. Two
index systems based on the numerical orders of $r_{n}$ and $\theta_{n}$ need
not be identical. Owing to the regularity in $\theta_{n}$, the index $n$ is
orderly set according to the angle $\theta_{n}$. In contrast, the radius
$r_{n}$ in practice is not a monotonic function of $n$. Fig. 8(d) indicates
that $r_{n}$ decreases non-monotonically, or the order in $n$ (horizontal
axis) is not preserved for $r_{n}$ (vertical axis). Actually, this is normally
the case when the plastochron ratio $a$ is very close to 1 (high phyllotaxis),
for irregular fluctuations in $r_{n}$ outweigh regular changes by the
plastochron ratio $a$. Even the height order of successive leaves on a stem
may be interchanged (Fujita (1942)).
Excepting such minor irregularities, which are more or less expected for
living organisms, observations generally support validity of mathematical
description in terms of the polar coordinate system. This is not at all
trivial. Just as an elliptic orbit of a planet pins down a special point in
space, the sun at the origin, the general regularities manifested plainly by
means of the polar coordinate system signify the existence of a special
singular point, the origin of the coordinate system. Unlike the solar system,
there is no obvious sign at the origin of a phyllotactic pattern. Indeed this
was a motivation of the present work; even without assuming the center, a
point of the sort has been indicated definitely (see Fig. 9(a)). The origin or
the center should be identified with the growing point biologically, though it
is meant here in a narrower sense than in general botanical use. The author
speculates that biochemical properties of the singular point, the growing
center of a spiral pattern, might hold a key to understanding not only
mechanisms of radial growth but also regulation mechanisms of divergence
angle.
## Appendix A Logarithmic spiral and parastichy concept revisited
Logarithmic spirals in phyllotaxis has been investigated intensively by Church
(1904), van Iterson (1907), Richards (1951), Erickson (1983) and Jean (1994),
among others. The main purpose of this section is to derive the formulas used
in the main text, (55), (58), (69) and (70) for the plastochron ratio $a$ when
two opposite parastichies are orthogonal, as they have not been presented
before in these forms. The following derivation has a merit of transparency by
which a physical assumption and a mathematical approximation are distinguished
by excluding unnecessary assumptions and complications. As remarked below,
this distinction is essential to a clear understanding of empirical rules of
phyllotaxis and Richards’ phyllotaxis index. Before that, the prevalent
concept of parastichy must be made clear and definite. It is also an important
aim of this section to point out that conventional expositions are incorrect
or insufficient.
In the polar coordinate system $(r,\theta)$, a logarithmic spiral through a
point $(r,\theta)=(1,0)$ is given by
$r=e^{b\theta},$ (45)
where $b$ is a constant. The logarithmic spiral is characterized by the
property that the angle $\varphi$ between the radial vector from the origin
and the tangent vector is held constant at every point $(r,\theta)$ on the
spiral.
$r\frac{d\theta}{dr}=\tan\varphi.$ (46)
Hence it is also called the equiangular spiral. The angle $\varphi$ is related
to the coefficient $b$ in (45) by
$b=\cot\varphi.$ (47)
The fundamental spiral of phyllotaxis given by (1) and (2) forms a logarithmic
spiral with
$b=\frac{\log a}{d}=\frac{\log a}{2\pi\alpha_{0}}.$ (48)
In the second equation, a reduced divergence angle $\alpha_{0}$ is introduced
by
$d=2\pi\alpha_{0}.$ (49)
Owing to the periodicity in angle, one may assume either $0\leq\alpha_{0}<1$
or $-\frac{1}{2}<\alpha_{0}\leq\frac{1}{2}$ without loss of generality. In the
latter convention, the direction of the fundamental spiral is determined by
the sign of divergence angle. In what follows, it is assumed
$0\leq\alpha_{0}\leq\frac{1}{2}$, i.e., the fundamental spiral runs outward
counterclockwise. It is straightforward to adapt the following results to
spirals for $-\frac{1}{2}<\alpha_{0}\leq 0$.
The concept of parastichy is widely used in the literature. As pointed out
explicitly below, prevalent expositions are not satisfactory. According to a
general consensus, a spiral connecting leaves with the indices differing by an
integer $q$ is referred to as a $q$-parastichy. However, there are infinitely
many such spirals. Therefore, the term $q$-parastichy is not appropriate in
the first place.
As a $q$-parastichy, consider a logarithmic spiral $r=e^{b\theta}$ running
through the 0-th and $q$-th leaves, namely $(r,\theta)=(1,0)$ and $(a^{q},2\pi
q\alpha_{0})$. The former is met from the outset, whereas the latter
determines the coefficient $b$. Owing to the periodicity in angle, the polar
coordinates of the $q$-th leaf $(r_{q},\theta_{q})=(a^{q},2\pi q\alpha_{0})$
is equivalently represented as $(a^{q},2\pi q\alpha_{0}-2\pi p)$, where $p$ is
an arbitrary integer. Substituting the latter into the spiral equation
$r=e^{b\theta}$,
$b=\frac{\log a^{q}}{2\pi\left(q\alpha_{0}-{p}\right)}=\frac{\log
a}{2\pi\left(\alpha_{0}-\frac{p}{q}\right)}.$ (50)
As the spiral thus determined runs through all leaves with the index $n$
divisible by the integer $q$, it is a $q$-parastichy. The $q$-parastichy
depends on the integer $p$. There are as many $q$-parastichies as the number
of integers. The fundamental spiral with (48) is a special case of (50) for
$(p,q)=(0,1)$. In the literature, the most conspicuous, shortest one is
usually regarded as the $q$-parastichy. Therefore, the integer $p$ is
identified with the integer nearest to $q\alpha_{0}$. As a matter of fact,
this need not be the case.
In practice, it is necessary and sufficient to let $p$ be an integer
approximating a number $q\alpha_{0}$. The denominator of the middle expression
of (50) represents the net angle between two successive leaves on the
$q$-parastichy. To keep the angle within a reduced range of width $2\pi$, a
multiple of a full turn, $2\pi{p}$, is subtracted from the nominal angle of
$2\pi q\alpha_{0}$. Hence the integer $p$ is the winding number of the
fundamental spiral executed between consecutive leaves on the $q$-parastichy.
If the number $q\alpha_{0}$ is not an integer, $p$ is most properly chosen to
be either the largest previous or the smallest following integer to
$q\alpha_{0}$. In the former case, $q\alpha_{0}-p>0$, the $q$ parastichy
spirals in the same direction as the fundamental spiral. In the latter case,
$q\alpha_{0}-p<0$, the direction of the $q$ parastichy is opposite to the
fundamental spiral. If $q\alpha_{0}$ is an integer, then $p=q\alpha$, and
$\varphi=0$ by (47). In this special case, the parastichy is not a spiral
curve but a straight half-line radiating from the center. Therefore it is
especially called the orthostichy. It goes without saying that orthostichy in
the strict sense does not exist in real life, as it does not make sense to ask
whether $q\alpha_{0}-p$ is zero or not; it is equivalent to asking whether
$\alpha_{0}$ is an irrational or a rational number. In any case, the
parastichy depends not only on the parastichy number $q$ but also on the
winding number $p$, or more precisely on $\alpha_{0}-\frac{p}{q}$ whose sign
and magnitude uniquely determine the sense and slope angle of the parastichy.
As the parastichy, or (50), depends only on the fraction $\frac{p}{q}$, it is
sufficient to consider the irreducible pair of $p$ and $q$. That is to say, if
$p=0$, then $q=1$; otherwise, $p$ and $q$ are restricted to coprime integers,
i.e., $p$ and $q$ are not evenly divided by any integer greater than 1. By the
assumption $0\leq\alpha_{0}\leq\frac{1}{2}$, it is sufficient to consider
irreducible fractions between 0 and $\frac{1}{2}$. For the denominator $q$ up
to 8, they are
$\frac{p}{q}=\frac{0}{1},\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{2}{5},\frac{1}{6},\frac{1}{7},\frac{2}{7},\frac{3}{7},\frac{1}{8},\frac{3}{8}.$
(51)
Hence, parastichies for $q=5,7,8$ are not unique. By way of illustration, five
parastichies for $\frac{p}{q}=\frac{1}{5}$ and $\frac{2}{5}$ are shown in Fig.
21 for $\alpha_{0}=\frac{3}{10}$. It is not common to notice 5-parastichies in
the pattern of Fig. 21. This is because the pattern has more conspicuous
parastichies. A numerical measure of conspicuousness is given by the absolute
value of $\alpha_{0}-\frac{p}{q}$, which is a winding angle per leaf of the
parastichy. In a word, the less winding the parastichy is, the more
conspicuous it appears to the eye. In other words, the better the fraction
$\frac{p}{q}$ approximates the divergence angle $\alpha_{0}$, the more
conspicuous the parastichy is. To find conspicuous parastichies for a given
value of $\alpha_{0}$, a sequence of irreducible fractions arranged in the
numerical order, called a Farey sequence, is of much help. By arranging (51),
we obtain the Farey sequence of order $q=8$;
$\frac{0}{1},\frac{1}{8},\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{1}{4},\frac{2}{7},\frac{1}{3},\frac{3}{8},\frac{2}{5},\frac{3}{7},\frac{1}{2}.$
(52)
As $\alpha_{0}=\frac{3}{10}$ lies between $\frac{2}{7}$ and $\frac{1}{3}$ in
this sequence, it is properly understood that 3 and 7 parastichies make a
conspicuous pair for the pattern of Fig. 21.
Figure 21: A phyllotactic pattern for $\alpha_{0}=\frac{3}{10}$ and $a=1.1$.
Two distinct 5-parastichies connect the same set of points, 1, 6, 11, 16, 21;
the solid curve is a 5-parastichy for $(p,q)=(1,5)$, and the dashed curve is
another 5-parastichy for $(p,q)=(2,5)$. The latter spiral winds in the
direction opposite to the fundamental spiral.
In spite of the above remark, the conventional term of the $q$-parastichy is
used throughout this paper in order not to complicate matters by introducing a
new term like a ${}^{p}q$-parastichy. In practice, we get along without
encountering the ambiguity because the number $q\alpha_{0}$ always happens to
be nearly an integer, namely a Fibonacci number, by the empirical fact that
divergence angle $\alpha_{0}$ is given by a special irrational number (cf.
(66)). Therefore, the integer is always identified with $p$ unwittingly.
Another reason is that different sets of parastichies with a common parastichy
number never become conspicuous at the same time, because adjacent terms in a
Farey sequence have different denominators (Hardy and Wright (1979)). This is
a consequence of (57) and concretely checked by (52).
The second shortest $q$-parastichy may become relevant. For instance, consider
a case for $\alpha_{0}=\frac{1}{12}$ and $q=3$ (Swinton (2012)). The
3-parastichy with $p=0$ is actually not a 3-parastichy because $(p,q)=(0,3)$
is reducible to $(0,1)$, the fundamental spiral. Therefore, a parastichy for
$(p,q)=(1,3)$ should rather be referred to as the 3-parastichy. This example
refutes the prevalent statement that $p$ is the integer nearest to
$q\alpha_{0}$ (Jean (1994); Swinton (2012)).
Along with the $q$ parastichy with $b$ in (50), let us consider a
$q^{\prime}$-parastichy for another arbitrary integer $q^{\prime}$. In a
similar manner as above, a $q^{\prime}$-parastichy is given by a logarithmic
spiral with
$b^{\prime}=\frac{\log
a}{2\pi\left(\alpha_{0}-\frac{p^{\prime}}{q^{\prime}}\right)}$ (53)
for $b$ in (45), where $p^{\prime}$ is an integer approximating
$q^{\prime}\alpha_{0}$. According to (47), the coefficient $b$ and
$b^{\prime}$ for the $q$ and $q^{\prime}$ parastichy are related with the
slope angle $\varphi$ and $\varphi^{\prime}$ by $b=\cot\varphi$ and
$b^{\prime}=\cot\varphi^{\prime}$, respectively. When the $q$-parastichy and
the $q^{\prime}$-parastichy are mutually orthogonal, the identity
$\varphi+\varphi^{\prime}=\frac{\pi}{2}$ holds true. Accordingly,
$bb^{\prime}=\cot\varphi\cot\varphi^{\prime}=-1$, i.e.,
$\frac{\log a}{2\pi\left(\alpha_{0}-\frac{p}{q}\right)}\frac{\log
a}{2\pi\left(\alpha_{0}-\frac{p^{\prime}}{q^{\prime}}\right)}=-1,$ (54)
or
$\log
a=2\pi\sqrt{\left(\frac{p}{q}-\alpha_{0}\right)\left(\alpha_{0}-\frac{p^{\prime}}{q^{\prime}}\right)},$
(55)
which is real and positive. The inequality $bb^{\prime}<0$, which holds when
the two parastichies are opposite, signifies that $\alpha_{0}$ should lie in
between the two fractions $\frac{p}{q}$ and $\frac{p^{\prime}}{q^{\prime}}$.
Let the former fraction be numerically larger than the latter, without loss of
generality. Then,
$\frac{p^{\prime}}{q^{\prime}}<\alpha_{0}<\frac{p}{q}.$ (56)
In the case of $\alpha_{0}<0$, both $p$ and $p^{\prime}$ become not positive,
so that the inequalities in (56) should be reversed, whereas (55) remains
valid formally as it is.
Figure 22: A phyllotactic pattern with divergence angle
$\alpha_{0}=\tau^{-2}$ and the plastochron ratio $a$ in (55) for
$\frac{p}{q}=\frac{5}{13}$ and $\frac{p^{\prime}}{q^{\prime}}=\frac{1}{3}$.
This pattern has an orthogonal parastichy pair of $(q,q^{\prime})=(3,13)$,
i.e., 3-parastichies (dotted) and 13-parastichies (dashed) cross orthogonally.
Nonetheless, it is usually referred to as a $(8,13)$ phyllotaxis, because
8-parastichies (solid) with
$\frac{p^{\prime\prime}}{q^{\prime\prime}}=\frac{3}{8}$ are as conspicuous as
13-parastichies. As $pq^{\prime\prime}-p^{\prime\prime}q=1$ for
$(q^{\prime\prime},q)=(8,13)$ whereas $pq^{\prime}-p^{\prime}q=2$ for
$(q^{\prime},q)=(3,13)$, the former has a reason to be regarded as the
fundamental pair. Parastichies are imaginary constructs, though some may have
real observable effects.
A pair of parastichies is commonly referred to by the denominator pair
$(q,q^{\prime})$. The parastichy pair $(q,q^{\prime})$ is usually not chosen
so that the parastichies are nearly orthogonal; it is chosen to be
irreducible, i.e., such that the two limiting fractions in (56) constitute
successive terms of a Farey sequence. See Fig. 22. By a theorem of number
theory (Hardy and Wright (1979)), it is equivalent to stating that integer
pairs for a pair of parastichies $(p,q)$ and $(p^{\prime},q^{\prime})$ are
chosen to satisfy
$pq^{\prime}-p^{\prime}q=1.$ (57)
The identity is confirmed for adjacent fractions in (52). The irreducible pair
of parastichies should not be confused with the irreducible numbers $(p,q)$
for a $q$-parastichy discussed above. The former may be rephrased as
conspicuous, although the term carrying ordinary connotations might better be
avoided. In Fig. 22, the parastichy pair $(3,13)$ is not conspicuous
(irreducible), whereas not only $(8,13)$ but $(3,5)$, $(5,8)$, $(13,21)$ and
so on are conspicuous (irreducible). Thus, (57) should be considered as a
convenient prescription for good choices of the parastichy pair. Physically,
the left-hand side of (57) represents the number of leaf points in a unit cell
of the lattice spanned by $q$ and $q^{\prime}$ parastichies (Fig. 22). (57)
means not only that $q$ and $q^{\prime}$ are coprime integers but that $p$ and
$p^{\prime}$ are determined independently of $\alpha_{0}$. As a result, the
parastichy numbers do not depend on the divergence angle, insofar as the
condition (56) is met. Note that the phyllotactic pattern is completely
characterized by the two parameters, the plastochron ratio $a$ and the
divergence angle $\alpha_{0}$, which are not determined uniquely by parastichy
numbers alone. The relation between the divergence angle $\alpha_{0}$ and the
limiting fractions $\frac{p}{q}$ and $\frac{p^{\prime}}{q^{\prime}}$ to be
used for (55) is immediately read from a diagram presented in Okabe (2011,
2012).
An opposite parastichy pair of a multijugate system has the greatest common
divisor $J>1$, so that the pair is written $J(q,q^{\prime})$, where $q$ and
$q^{\prime}$ are coprime integers. Instead of (55), one obtains
$\log
a=\frac{2\pi}{J}\sqrt{\left(\frac{p}{q}-J\alpha_{0}\right)\left(J\alpha_{0}-\frac{p^{\prime}}{q^{\prime}}\right)},$
(58)
where $p$ and $p^{\prime}$ are integers near $J\alpha_{0}q$ and
$J\alpha_{0}q^{\prime}$. The divergence angle $\alpha_{0}$ of the $J$-jugate
system satisfies
$\frac{p^{\prime}}{q^{\prime}}<J\alpha_{0}<\frac{p}{q}.$ (59)
If $J(q,q^{\prime})$ is a conspicuous pair,
$Jq^{\prime}p-Jqp^{\prime}=J,$ (60)
or (57) holds true. Thus, the divergence angle $\alpha_{0}$ and the
plastochron ratio $a$ of a multijugate system may be better expressed in the
forms of $J\alpha_{0}$ and $a^{J}$ to compare them with a single spiral system
(Sec. 4.4.2).
In the above, no approximation has been made besides the basic equations (1)
and (2). Above all, the derivation is not based on any ideal assumption on
leaf shape, e.g., contiguous circles abundant in the phyllotaxis literature.
This is practically of crucial importance because any argument based on
‘perfectly circular leaves’ must meet with severe criticism from
experimentalists. As a matter of fact, parastichy is a property of a point
lattice, so that it is independent of shape and size of pattern units. If (57)
is to be regarded as the condition for a visible parastichy pair, this is what
Jean (1994) calls the fundamental theorem of phyllotaxis. But then, one must
be careful about what is visible (Swinton (2012)). See a remark below (57) on
the term irreducible.
The above formulation is general. According to number theory, linear
Diophantine equations, (57) and (60), are solved for any parastichy pair of
integers. If the parastichies are orthogonal, the plastochron ratio $a$ is
given by (55) or (58). Nevertheless, nature adopts only selected numbers as
the parastichy pair. Parastichy numbers of a normal phyllotaxis comprise the
Fibonacci sequence,
$F_{i}=1,1,2,3,5,8,\cdots$ (61)
for
$i=1,2,3,4,5,6,\cdots,$
respectively. The Fibonacci numbers satisfy the mathematical identity
$F_{i-1}F_{i}-F_{i-2}F_{i+1}=(-1)^{i}.$ (62)
Therefore, a solution of (57) is given by
$(q,q^{\prime},p,p^{\prime})=(F_{i+1},F_{i},F_{i-1},F_{i-2})$ (63)
if $i$ is an even integer, or by
$(q,q^{\prime},p,p^{\prime})=(F_{i},F_{i+1},F_{i-2},F_{i-1})$ (64)
if $i$ is odd. In either case, (55) gives
$\log
a=2\pi\sqrt{\left(\frac{F_{i-2}}{F_{i}}-\alpha_{0}\right)\left(\alpha_{0}-\frac{F_{i-1}}{F_{i+1}}\right)}$
(65)
for the orthogonal Fibonacci parastichy system of $(F_{i},F_{i+1})$. (65) is
valid insofar as $\alpha_{0}$ lies between $\frac{F_{i-2}}{F_{i}}$ and
$\frac{F_{i-1}}{F_{i+1}}$. Nonetheless, it is usually taken for granted that
divergence angle $\alpha_{0}$ is identified with the mathematical limit of
$\frac{F_{i-2}}{F_{i}}$ and $\frac{F_{i-1}}{F_{i+1}}$ as $i\rightarrow\infty$,
that is,
$\alpha_{0}=\frac{1}{2+\tau^{-1}}=\tau^{-2}$ (66)
(Richards (1951); Jean (1983)), where $\tau$ is the golden ratio given in
(16). The divergence angle of $d\simeq 137.5^{\circ}$ for (66) is called the
limit divergence angle of the main (Fibonacci) sequence. To show that (66) is
the limit of $\frac{F_{i-2}}{F_{i}}$, the fraction is expanded with respect to
$\tau^{-1}$ after the exact formula
${F_{i}}=\frac{\tau^{i}-(-\tau)^{-i}}{\sqrt{5}}$ (67)
is substituted for the denominator and numerator;
$\frac{F_{i-2}}{F_{i}}=\frac{\tau^{i-2}-(-\tau)^{-i+2}}{\tau^{i}-(-\tau)^{-i}}\simeq\tau^{-2}-(-1)^{i}\sqrt{5}\tau^{-2i}.$
(68)
The second term vanishes in the limit $i\rightarrow\infty$. Substituting (66)
and (68) into (65),
$\log a=\frac{2\pi\sqrt{5}}{\tau^{2i+1}}$ (69)
for the orthogonal Fibonacci system $(F_{i},F_{i+1})$ with the limit
divergence angle (66). This result is generalized to an orthogonal $J$-jugate
system $J(F_{i},F_{i+1})$ with the limit divergence angle
$\alpha_{0}=\tau^{-2}/J$,
$\log a=\frac{2\pi\sqrt{5}}{J\tau^{2i+1}}.$ (70)
In terms of common logarithms, (69) is expressed as
$\log_{10}\left(\log_{10}a\right)=-i{\log_{10}(\tau^{2})}+\log_{10}\left(\frac{2\pi\sqrt{5}}{\tau\log(10)}\right),$
(71)
which is transformed into
$i-1=\frac{\log_{10}\left(\frac{2\pi\sqrt{5}}{\tau\log(10)}\right)}{{\log_{10}(\tau^{2})}}-1-\frac{\log_{10}\left(\log_{10}a\right)}{\log_{10}(\tau^{2})}.$
(72)
Expressed in numbers,
$i-1=0.37918-2.39249\log_{10}\left(\log_{10}a\right).$ (73)
The right-hand side is the phyllotaxis index (P.I.) of Richards (1951). Thus,
it is proved that the phyllotaxis index is nothing but the index $i$ of the
Fibonacci system $(F_{i},F_{i+1})$ minus one. This is an integer by
definition. In natural logarithms, the index is given by
${\rm P.I.}=\frac{\log({2\pi\sqrt{5}}/\tau)}{\log\tau^{2}}-\frac{\log(\log
a)}{\log\tau^{2}}-1.$ (74)
P.I. may take any value if it is regarded as a function of $\log a$. In
applying the formula (74) to real systems, it should be kept in mind that
$\log a$ in the logarithm must be a positive number, namely $a>1$ by
assumption.
The above derivation clearly indicates that Richards’ phyllotaxis index is
based on the physical assumption (66) and the mathematical approximation (68).
This is not obvious from the derivations by Richards (1951) and Jean (1983).
Interestingly, it is mostly the latter approximation that turns out to be less
reliable, i.e., the rational number $\frac{F_{i-2}}{F_{i}}$ for small $i$
cannot be replaced by the irrational number $\tau^{-2}$. The former assumption
(66), or $d\simeq 137.5^{\circ}$, is empirically supported quite accurately,
as shown in the main text. In the author’s view, the regularity expressed by
(66), or $\theta_{n}=nd$ with $d=\pm 2\pi/\tau^{2}$, should be called the
fundamental law of phyllotaxis and treated particularly as such. Let us
incidentally remark that logarithmic spirals of phyllotaxis are far more
miraculous than similar spirals appearing in certain growing forms like
nautilus shells by the very fact that the angle $d$ is not only held constant
but specifically fixed at $2\pi/\tau^{2}=137.5^{\circ}$.
For a non-logarithmic spiral pattern, parastichy numbers will be shifted in a
Fibonacci sequence depending on part of the pattern. For instance, the square-
root spiral by (4) is formally regarded as having the plastochron ratio $a$
depending on the leaf index $n$, namely $\log a=\frac{1}{2(n-1)}\log n$ by
$r_{n}=a^{n-1}=\sqrt{n}$, when a reference scale is set as $r_{1}=1$.
Leaves on a stem may be represented in terms of a cylinder coordinate system
as
$\displaystyle z_{n}$ $\displaystyle=$ $\displaystyle nh,$ (75)
$\displaystyle\theta_{n}$ $\displaystyle=$ $\displaystyle nd,$
where $h$ is the length of an internode relative to the girth and $d$ is a
divergence angle. Accordingly, the fundamental helix is parametrically given
by
$z=\frac{b}{2\pi}\theta,$ (76)
where
$b=\frac{2\pi h}{d}=\frac{h}{\alpha_{0}}.$ (77)
It is easily shown that the angle $\varphi$ that the helix (76) makes with the
$z$ axis is expressed in terms of $b$ in (77) by the same equation as (47),
namely $b=\cot\varphi$, Therefore, one may follow the same derivation as (55)
to obtain
$h=\sqrt{\left(\frac{p}{q}-\alpha_{0}\right)\left(\alpha_{0}-\frac{p^{\prime}}{q^{\prime}}\right)}$
(78)
for the orthogonal parastichy system $(q,q^{\prime})$. Note that the
plastochron ratio $a$ of a spiral pattern on the apex and the internode length
$h$ of a cylindrical pattern on the stem are formally related by
$h=\frac{1}{2\pi}\log a.$ (79)
Although the cylindrical representation is frequently used in mathematical
studies, real systems do not obey (75) because the internode $h$, unlike $d$
and $a$, is not constant even approximately. (78) should be understood as a
reference result of an abstract model.
## Appendix B How to number sunflower seeds
To measure divergence angle, the sequential order of organs has to be
identified according to their age or plastochron. Sometimes this may involve
laborious tasks, particularly for a high phyllotaxis pattern like packed seeds
on a sunflower head. In principle, all seeds reachable from a reference seed
along parastichies are numbered by successively adding or subtracting
parastichy numbers. A drawback of this simple method is that a single
miscalculation spoils all the following numbering. Therefore, it is
practically of much help to have a systematic device. The following method
starts from choosing a layer of seeds that lie near a rim circle, which are
regarded as a “siege” to start numbering whole seeds inward. In what follows,
seeds are counted from the outermost toward the center.
$I$ | 0 | | 1 | | 2 | | 3 | | 4 |
---|---|---|---|---|---|---|---|---|---|---
mod($IR,Q$) | 0 | $\nearrow$ | 2 | $\nearrow$ | 4 | $\searrow$ | 1 | $\nearrow$ | 3 | $\searrow$
$Q=5$ | | 2 | | 2 | | 1 | | 2 | | 1
Table 1: Numbers to make a primordia front circle (see Fig. 23). This table
is for a parastichy pair of $(Q,Q^{\prime})=(5,8)$. In the middle, mod($IR,Q$)
denotes the remainder of $IR$ divided by $Q$, where $R=2Q-Q^{\prime}=2$ and an
integer $I$ runs from 0 to $Q-1=4$. The last row is a 1-2 sequence for $Q=5$,
which is obtained by positing either 2 or 1 depending on whether the remainder
increases or decreases.
Let us consider a pattern with opposed parastichies $(Q,Q^{\prime})$, where
integers $Q$ and $Q^{\prime}$ have no common divisor. For definiteness, $Q$
and $Q^{\prime}$ are assumed to satisfy $1<Q^{\prime}/Q<2$, which practically
holds true in most cases. (By this assumption, capital letters are used for
the parastichy numbers.) The bounding integer 2 plays a key role below. In
addition, an auxiliary integer $R=2Q-Q^{\prime}$ is introduced. For every
integer $I=0,1,2,\cdots Q-1$, calculate the remainder of the division of $IR$
by $Q$, which is denoted as mod($IR,Q$). See Table 1 for
$(Q,Q^{\prime})=(5,8)$ and $R=2Q-Q^{\prime}=2$. In the last line, either 1 or
2 is inserted between columns depending on whether the remainder mod($IR,Q$)
increases or decreases as $I$ increases by 1. The last 1 at the right bottom
is set because mod($2I,5$)$=0$ for $I=0$ on the left end is smaller than 3 on
the right end for $I=4$. There are $R$ ones and $Q-R$ twos, so that the
numbers add up to $R+2(Q-R)=Q^{\prime}$. Thus, the 1-2 sequence in the last
line of Table 1 represents a partition of $Q^{\prime}(=8)$ into $Q(=5)$ parts
(2+2+1+2+1=8).
With this table at hand, Fig. 23 explains how to fix initial seeds on a front
circle from which to start numbering the inner seeds. The sequence of ones and
twos in the last line of Table 1 represents the number of steps that have to
be made in the direction of $Q$ parastichies before shifting back one step in
the minus direction of $Q^{\prime}$ parastichies. Starting from a reference
seed numbered 0, the seed 2 is reached after two steps of $Q$ and minus one
step of $Q^{\prime}$. Then it goes to the seed 4 after the same steps, because
$2Q-Q^{\prime}=+2$. The next is the seed 1 by one step along $Q$ and one step
back along $Q^{\prime}$ ($4+Q-Q^{\prime}=1$). In this way, the front circle is
closed as it returns to the seed 0 after five shifts of $-Q^{\prime}$. The
front circle thus defined without reference to a nominal center does not agree
with the primordia front of Hotton et al. (2006), which is defined such that a
new primordium added to this front lies closer to the center than all
primordia belonging to the front.
Cartesian coordinates of seeds may be harvested from a digital photo image by
means of a digitizing software (Mitchell (2009); Kiisk (2011)). In so doing,
useful formulas are derived if one decides to pick a constant number of seeds
from each of $Q$ parastichies. Let this number be denoted as $X_{\rm max}$.
While collecting data in a one-dimensional sequence, the starting position in
each $Q$ parastichy has to be successively shifted according to the 1-2
sequence, as described in Fig. 23. Every seed in the sequential data is
assigned two-dimensional coordinates $(X,Y)$ set along the $(Q,Q^{\prime})$
parastichies, where $X=0,1,2,\cdots,X_{\rm max}-1$ and $Y=0,1,2,\cdots,Q-1$.
In the one-dimensional sequence, the order of the seed at $(X,Y)$ is given by
$N=X+X_{\rm max}Y,$ (80)
while the plastochron of the seed is
${\rm Plastochron\ Index:}\quad n=XQ+{\rm mod}(RY,Q),$ (81)
which is not to be confused with the phyllotaxis index in (74). (80) is
inverted to give
$\displaystyle X$ $\displaystyle=$ $\displaystyle{\rm mod}(N,X_{\rm max}),$
$\displaystyle Y$ $\displaystyle=$ $\displaystyle{\rm int}(N/X_{\rm max}),$
(82)
where ${\rm int}(N/X_{\rm max})=(N-{\rm mod}(N,X_{\rm max}))/X_{\rm max}$
means the largest integer that does not exceed $N/X_{\rm max}$. By
substituting (82) into (81), the index number $n$ of the $N$-th seed is
obtained without manual calculation. In (81), the initial seed at
$(X,Y)=(0,0)$ is set to have $n=0$. The initial seed is arbitrary but it
should be chosen properly so as to make the front circle as large as possible
while keeping the circle within the capitulum. Seeds remaining outside the
front are indexed with negative numbers. Once completed, the index system may
be renumbered at one’s discretion.
Figure 23: How to make an encircling layer of seeds on a parastichy lattice
with a parastichy pair of $(Q,Q^{\prime})=(5,8)$. See Table 1 and the text.
The above method of data collection makes use of the 1-2 sequence. If one were
interested not in numbering, but only in picking seeds to make a front circle,
then it is sufficient to know a periodic 1-2 sequence. The periodic 1-2
sequence has period $Q$; all sequences 22121, 12122, 12212, 21221 and 21212
for $Q=5$ are regarded as equivalent. A periodic 1-2 sequence may be deduced
recursively by simple rules to replace 2 with 21 and 1 with 2;
$\displaystyle 1$ $\displaystyle\rightarrow$ $\displaystyle 2.$ $\displaystyle
2$ $\displaystyle\rightarrow$ $\displaystyle 21,$ (83)
For the main sequence of phyllotaxis, starting from
$Q=3:\quad 2\quad 2\quad 1,$ (84)
one gets
$5:\quad 21\quad 21\quad 2,$
then
$8:\quad 21\ 2\quad 21\ 2\quad 21,$
and so on. The rules in (83) may remind Fibonacci’s original problem, in which
1 and 2 correspond to new and mature pair of rabbits. The 1-2 sequence in the
correct order is obtained by reversing the sequence before applying (83). It
goes as follows:
$3:221$ $122$ $5:22121$ $12122$ $8:22122121$ $12122122$ $13:2212212122121$
$1212212122122$ $21:221221212212212122121$ $121221212212212122122$
$34:2212212122122121221212212212122121$
The 1-2 sequence depends on $Q^{\prime}$ too. For instance, it is 21211 for
$(Q,Q^{\prime})=(5,7)$, while 22121 for $(Q,Q^{\prime})=(5,8)$. Given a 1-2
sequence for a pair, 1-2 sequences for other pairs in the same Fibonacci
sequence are routinely derived, according to the rule of reversal and
replacement. From 21211 for $(Q,Q^{\prime})=(5,7)$, the rule gives 2221221 for
the next pair $(Q,Q^{\prime})=(7,12)$ of the $(5,7)$ sequence, 5, 7, 12, 19,
$\cdots$. The result is confirmed by the method of Table 1.
Figure 24: Digitizing an image from Jean (1994) by way of illustration. See
the text for a procedure of making the encircling front layer. The direction
of the fundamental spiral is opposite to that of Fig. 20.
Returning to the subject, let us take the cover image of Jean (1994) as an
example. The first thing to do is to detect a shell layer of a well-defined
encircling lattice spanned by opposed parastichies. In Fig. 24, there are
$Q=34$ main parastichies involuting clockwise and $Q^{\prime}=55$ steeper
opposite parastichies ($R=13$). The ordered 1-2 sequence for $Q=34$ is given
just above. For simplicity, let us choose $X_{\rm max}=3$ for the thickness of
the front layer. A procedure is: (i) import the image into the digitizer. (ii)
Set a Cartesian coordinate system with proper scales of $x$ and $y$ axes.
(iii) Fix the outermost layer of the primordia front according to the 1-2
sequence. Check if the front closes properly (the top and bottom of Fig. 23).
(iv) Start clicking the seeds in sequence (the middle of Fig. 23). (v) Index
the digitized Cartesian coordinates by (81) and (82).
The parastichy pair easiest to follow with the eye is to be used as
$(Q,Q^{\prime})$, but the choice is not unique. The same index system is
obtained if a lower parastichy pair is used, namely $(Q^{\prime}-Q,Q)$,
$(2Q-Q^{\prime},Q^{\prime}-Q)$, etc.
Parastichy numbers of a multijugate system have a common divisor $J$, so that
they are expressed as $J(Q,Q^{\prime})$ in terms of coprime integers $Q$ and
$Q^{\prime}$. To close the primordia front of the $J$-jugate system, $Q$ steps
of a 1-2 sequence are repeated $J$ times. Meanwhile, the $Y$ coordinate runs
through $JQ$ integers from 0 to $JQ-1$. The plastochron index is given by
(81). The jugacy index of the numbering system proposed in Sec. 2.2.2 shifts
orderly along parastichies. For the seed at $(X,Y)$, it is given by
${\rm Jugacy\ Index:}\quad j={\rm mod}(-PX+P^{\prime}Y,J)$ (85)
by means of integers $P$ and $P^{\prime}$ satisfying
$PQ^{\prime}-P^{\prime}Q=\pm 1$.
## References
* Church (1904) Church, A. H., 1904. On the Relation of Phyllotaxis to Mechanical Laws. On the Relation of Phyllotaxis to Mechanical Laws. Williams & Norgate, London.
* Erickson (1983) Erickson, R. O., 1983. The geometry of phyllotaxis. In: Dale, J., Milthorpe, F. (Eds.), The Growth and functioning of leaves: proceedings of a symposium held prior to the thirteenth International Botanical Congress at the University of Sydney, 18-20 August 1981. Cambridge University Press, pp. 53–88.
* Fujita (1942) Fujita, T., 1942. Zur Kenntnis der Organstellungen im Pflanzenreich. Jpn. J. Bot. 12, 1–55.
* Green and Baxter (1987) Green, P. B., Baxter, D. R., 1987. Phyllotacticpatterns: Characterization by geometrical activity at the formative region. Journal of Theoretical Biology 128, 387–395.
* Hardy and Wright (1979) Hardy, G., Wright, E., 1979. An Introduction to the Theory of Numbers. Oxford Science Publications. Clarendon Press.
* Hotton (2003) Hotton, S., 2003. Finding the center of a phyllotactic pattern. Journal of Theoretical Biology 225 (1), 15 – 32.
* Hotton et al. (2006) Hotton, S., Johnson, V., Wilbarger, J., Zwieniecki, K., Atela, P., Golé, C., Dumais, J., 2006. The possible and the actual in phyllotaxis: Bridging the gap between empirical observations and iterative models. Journal of Plant Growth Regulation 25, 313–323.
* Jean (1983) Jean, R. V., 1983. Allometric relations in plant growth. Journal of Mathematical Biology 18, 189–200.
* Jean (1994) Jean, R. V., 1994. Phyllotaxis: A Systemic Study in Plant Morphogenesis. Cambridge Univ. Press, Cambridge, New York.
* Kiisk (2011) Kiisk, V., 2011. Digitizer. http://www.physic.ut.ee/ kiisk/files-eng.htm.
* Kumazawa and Kumazawa (1971) Kumazawa, M., Kumazawa, M., 1971. Periodic variations of the divergence angle, internode length and leaf shape, revealed by correlogram analysis. Phytomorphology 21, 376–389.
* Maksymowych and Erickson (1977) Maksymowych, R., Erickson, R. O., 1977. Phyllotactic change induced by gibberellic acid in xanthium shoot apices. American Journal of Botany 64, 33–44.
* Matkowski et al. (1998) Matkowski, A., Karwowski, R., Zagórska-Marek, B., 1998. Two algorithms of determining the middle point of the shoot apex by surrounding organ primordia positions and their usage for computer measurements of divergence angles. Acta Soc. Bot. Pol. 67, 151–159.
* Meicenheimer (1986) Meicenheimer, R. D., 1986. Role of parenchyma in Linum usitatissimum leaf trace patterns. American Journal of Botany 73, 1649–1664.
* Meicenheimer and Zagórska-Marek (1989) Meicenheimer, R. D., Zagórska-Marek, B., 1989. Consideration of the geometry of the phyllotaxic triangular unit and discontinuous phyllotactic transitions. Journal of Theoretical Biology 139, 359–368.
* Mirabet et al. (2012) Mirabet, V., Besnard, F., Vernoux, T., Boudaoud, A., 2012. Noise and robustness in phyllotaxis. PLoS Comput Biol 8, e1002389.
* Mitchell (2009) Mitchell, M., 2009. Engauge Digitizer - Digitizing software. http://digitizer.sourceforge.net/.
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* Richards (1951) Richards, F. J., 1951. Phyllotaxis: Its quantitative expression and relation to growth in the apex. Philos. Trans. R. Soc. B 225, 509–564.
* Ridley (1982) Ridley, J. N., 1982. Packing efficiency in sunflower heads. Math. Biosci. 58, 129–139.
* Rivier et al. (1984) Rivier, N., Occelli, R., Pantaloni, J., Lissowski, A., 1984. Structure of Bénard convection cells, phyllotaxis and crystallography in cylindrical symmetry. J. Phys. (Paris) 45, 49–63.
* Rutishauser (1998) Rutishauser, R., 1998. Plastochrone ratio and leaf arc as parameters of a quantitative phyllotaxis analysis in vascular plants. In: Jean, R. V., Barabé, D. (Eds.), Symmetry in plants. World Scientific, pp. 171–212.
* Ryan et al. (1991) Ryan, G. W., Rouse, J., Bursill, L., 1991. Quantitative analysis of sunflower seed packing. Journal of Theoretical Biology 147 (3), 303–328.
* Sadoc et al. (2012) Sadoc, J.-F., Rivier, N., Charvolin, J., 2012. Phyllotaxis: a non-conventional crystalline solution to packing efficiency in situations with radial symmetry. Acta Crystallographica Section A 68 (4), 470–483.
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* Szymanowska-Pulka (1994) Szymanowska-Pulka, J., 1994. Phyllotactic patterns in capitula of Carlina acaulis L. Acta Soc. Bot. Pol., 229–245.
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|
arxiv-papers
| 2012-12-14T00:40:44 |
2024-09-04T02:49:39.271049
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Takuya Okabe",
"submitter": "Takuya Okabe",
"url": "https://arxiv.org/abs/1212.3377"
}
|
1212.3428
|
# Rearrangements of interacting Fermi liquids
Rong-Yao Yang and Wei-Zhou Jiang 111 Email: [email protected] Department of
Physics, Southeast University, Nanjing 210000, China
###### Abstract
The stability condition of Landau Fermi liquid theory may be broken when the
interaction between particles is strong enough. In this case, the ground state
is reconstructed to have a particle distribution different from the Fermi-step
function. For specific instances, one case with the vector boson exchange and
another with the relativistic heavy-ion collision are taken into
consideration. With the vector boson exchange, we find that the relative weak
interaction strength can lead to the ground-state rearrangement as long as the
fermion mass is large enough. It is found that the relativistic heavy-ion
collision may also cause the ground-state rearrangement, affecting the
statistics of the collision system.
Landau Fermi liquid theory, ground-state instability, relativistic heavy-ion
collisions, vector bosons
###### pacs:
71.10.Hf, 24.10.Jv, 26.60.+c, 25.75.-q
††preprint:
## I Introduction
Since the high $T_{c}$ superconductor and quantum Hall effect were discovered
in 1980s, more and more systems which disobey the Landau Fermi liquid theory
have been springing up. It is known that the Landau Fermi liquid theory is not
valid for one-dimensional (1D) systems and strongly correlated 2D liquids
LDG63 ; And90 . Khodel et al. KVA90 ; ZMV99 demonstrated that a state with
the fermion condensation or multi-connected momentum distribution arises as
the stability condition is broken and accordingly the ground state is
rearranged. When the ground-state rearrangement occurs, systems can possess
novel properties that are characteristic of a non-Fermi-liquid behavior. Such
a rearrangement can also occur in nuclear ground states Ll79 ; Ho81 , and can
have possible implications in astrophysical issues. For instance, the direct
Urca process may be remodelled when the rearrangement occurs, so that the fast
cooling of neutron stars is probable VDN00 .
In the present work, we consider two extreme examples that may undergo a
ground-state rearrangement due to repulsive interactions. One is that the
Fermi liquid system with the weak repulsion provided by the vector boson
exchanges. This may has a correspondence to the aggregation of weakly
interacting massive particles in compact stars. In addition, it could exist
the fifth force whose strength is certainly very weak fa89 . At least, this
possibility has not been excluded completely. Recently, considering this in
terms of the exchange of the U-boson, a few groups found that the U-boson that
couples with nucleons weakly can have a significant effect on properties of
neutron stars kr09 ; WDH09 ; Zh11 . It is interesting to examine whether the
weak coupling can also cause the ground-state rearrangement in specific
systems. One may expect that the rearrangement could have observable effects
that may possibly be used to constrain the new force.
Another example regards the furious deceleration in the relativistic heavy-ion
collisions in the time scale of the typical strong interaction. According to
the equivalence principle of the general relativity, the deceleration or
acceleration is identical to a local gravitational field. As the usual
particle statistics neglects the space-time character, its consideration is of
special interest and significance Eda11 . The present investigation serves the
purpose to mimic fermion rearrangements in strongly curved space that exist
during spectacular supernova explosions and in the horizon of a black hole. We
will focus on determining the conditions of the occurrence of the ground-state
rearrangement in these two scenarios.
The paper is arranged as follows. In Sec. II, the stability condition of
ground states together with the ground-state rearrangement is demonstrated for
Fermi liquids. The numerical results are presented in Sec. III. Finally, the
summary is given.
## II Stability condition of ground states
### II.1 Necessary stability condition for the Fermi liquid
The ground-state stability condition can be established on the fact that the
variation of the ground state energy is positive for any admissible variations
of distribution. Namely, the excitation states have higher energy than the
ground state, which means that
${\delta E_{0}}=\int[\epsilon(k)-\mu]\delta
n(\textbf{k})\frac{d^{3}\textbf{k}}{(2\pi)^{3}},$ (1)
is positive when the conservative condition for the particle number is
satisfied: $\int{\delta}n(\textbf{k})\frac{d^{3}\textbf{k}}{(2\pi)^{3}}=0$
LD80 , where $\epsilon$ is the single-particle energy and $\mu$ is the
chemical potential. The Landau Fermi liquid theory tells us that the ground-
state distribution of a homogeneous Fermi liquid is a Fermi-step function
$n_{F}(k)=\theta(k_{F}-k)$ with $k_{F}$ being the Fermi momentum. Namely, the
states are filled below the Fermi momentum and are empty above it. The
stability condition is satisfied if the single-particle energy above the Fermi
surface is larger than that below it. Otherwise, it is violated. That is to
say,
$s(k)=\frac{\epsilon(k)-\epsilon(k_{F})}{k-k_{F}},$
needs to be positive for the whole momentum range. In deed, $\epsilon(k_{F})$
is the chemical potential $\mu$ when the temperature $T\simeq 0$. As
$\lim_{k{\rightarrow}k_{F}}s(k)=\frac{d\epsilon(k)}{dk}\mid_{k=k_{F}}$, the
ground state of Landau Fermi liquid theory becomes unstable provided that
$d\epsilon/dk$ is negative or zero in the vicinity of the Fermi momentum. In
this case, the mass of quasi-particles $M_{Q}=k_{F}/v_{F}$ is consistently
negative or infinity once the value of the group velocity $v_{F}\leq
0(v_{F}=\frac{d\epsilon(k)}{dk}\mid_{k=k_{F}}$).
In a word, we obtain a necessary stability condition for the Landau Fermi
liquid theory: $d\epsilon/dk$ has to be positive near $k_{F}$ (In this paper
we define the interval $0.95k_{F}\sim 1.05k_{F}$ as the nearby $k_{F}$ to
examine the stability condition).
With the breaking of stability condition, the ground state can only be
reconstructed by means of the rearrangement. In Fig. 1, we show as an example
the reconstructed ground-state distribution after the rearrangement.
Figure 1: Reconstructed ground-state distribution after the rearrangement. In
this case, the ground state is multi-connected.
### II.2 Stability condition for systems with vector boson exchanges
The first example that concerns the ground-state rearrangement is realized by
the addition of repulsion with the exchange of the vector boson. The addition
of repulsion may be supposed in strongly interacting nuclear matter or in
weakly interacting dark matter. We can examine the system stability by
analyzing its energy spectrum as above. In the relativistic Hartree-Fock (RHF)
approximation, the energy momentum distribution is given as
$\displaystyle\epsilon(k)$ $\displaystyle=$ $\displaystyle\sqrt{k^{2}+M^{\ast
2}}+\left[\frac{{\gamma}g_{\omega}^{2}}{2\pi^{2}m_{\omega}^{2}}\int_{0}^{\infty}p^{2}n(p)dp+\sum_{i=\omega,\rho}\frac{g_{i}^{2}c_{i}}{8\pi^{2}k}\int_{0}^{\infty}pn(p)\ln\frac{(k+p)^{2}+m_{i}^{2}}{(k-p)^{2}+m_{i}^{2}}dp\right]$
(2)
$\displaystyle+\frac{{\gamma}g_{v}^{2}}{2\pi^{2}m_{v}^{2}}\int_{0}^{\infty}p^{2}n(p)dp+\frac{g_{v}^{2}}{8\pi^{2}k}\int_{0}^{\infty}pn(p)\ln\frac{(k+p)^{2}+m_{v}^{2}}{(k-p)^{2}+m_{v}^{2}}dp$
where $M^{\ast}$ is the fermion effective mass which is density dependent and
moderately model-dependent, $m_{v}$ is the mass of the vector boson of
interest (e.g., the U boson Zh11 ), $\gamma$ is the degree of degeneracy which
is 4 with the spin and isospin considered, $g_{v}$ is the boson-fermion
coupling constant, and $c_{i}=1,3$ for the $\omega$ and $\rho$ meson,
respectively. The terms in square brackets represent the contribution of the
vector mesons $\omega$ and $\rho$ in symmetric nuclear matter. In Eq.(2),
besides the usual vector mesons $\omega$ and $\rho$ in the relativistic
nuclear models, we have included the additional vector boson (denoted by the
subscript $v$) in the original model to provide the additional repulsion that
may result in a limited modification to the nucleon effective mass.
Substituting Fermi-step momentum distribution into Eq.(2), we can obtain the
explicit expression of the derivative
$\displaystyle\frac{d\epsilon(k)}{dk}$ $\displaystyle=$
$\displaystyle\frac{k}{\sqrt{k^{2}+M^{\ast
2}}}-\sum_{i=\omega,\rho}\left(\frac{g_{i}^{2}c_{i}}{8\pi^{2}k}\int_{0}^{k_{F}}p[\frac{1}{k}\ln\frac{(k+p)^{2}+m_{i}^{2}}{(k-p)^{2}+m_{i}^{2}}-\frac{2(k+p)}{(k+p)^{2}+m_{i}^{2}}+\frac{2(k-p)}{(k-p)^{2}+m_{i}^{2}}]dp\right)$
(3)
$\displaystyle-\frac{g_{v}^{2}}{8\pi^{2}k}\int_{0}^{k_{F}}p[\frac{1}{k}\ln\frac{(k+p)^{2}+m_{v}^{2}}{(k-p)^{2}+m_{v}^{2}}-\frac{2(k+p)}{(k+p)^{2}+m_{v}^{2}}+\frac{2(k-p)}{(k-p)^{2}+m_{v}^{2}}]dp$
It must be noted that the direct Hartree terms in Eq.(2) have no contribution
to the derivative as they are independent of momentum $k$. As discussed
before, the momentum distribution of the ground state deviates from the step
function once $d\epsilon/dk\leq 0$ in the vicinity of the Fermi surface. When
the coupling constant $g_{v}$ is larger than some critical value $g_{vc}$, the
zero point of $d\epsilon/dk$ will appear near $k_{F}$. By numerical
integration, we can search the zero point for $d\epsilon/dk$ in the interval
$0.95k_{F}{\leq}k{\leq}1.05k_{F}$, and the minimum $g_{v}$ for possessing at
least one zero point is the critical coupling constant $g_{vc}$.
### II.3 Necessary stability condition for collision systems
It is well-known that the local effect of gravity on a physical system is
identical with the effect of linear acceleration of the system according to
Einstein’s equivalence principle, and vice versa. In accelerated systems, the
Hamiltonian comprises a term provided by the non-inertial effect
$H_{nin}=\frac{A^{2}}{c^{2}}\sum_{i}M_{i}z_{i}^{2},$ (4)
where $A$ is the acceleration of the system, $c$ is the velocity of light
(equal to 1 in the natural unit), $M$ is the mass of particle (usually the
effective mass $M^{\ast}$ is used instead), z is the particle’s coordinate
along the direction of acceleration of the reference frame (in the case of
collision, z equals to half the distance between colliding particles, i.e.
$z=r/2$ in the center of mass reference frame) BCA07 .
The non-inertial contribution in Eq.(4) has a form of the harmonic oscillator.
In the Green function method or Feynman diagrammatic approach, an explicit
expression of the propagator of the harmonic oscillator is mathematically
complicated, and the numerics seems to be not straightforward Os98 ; Ma00 . In
order to simplify this tedious issue, we suppose that the repulsive force
decelerating the particles during collision is provided by a postulated vector
meson exchange through a well-known Yukawa coupling. At the same time, we
assume that the step function distributions are applied to systems prior to
the collision. The non-inertial effect can then be described by the vector
meson exchange between nucleons which is sophisticatedly applied in modern
physics. Instead of the given non-inertial effect term, we can use Yukawa
potential $g^{2}e^{-rm_{v}}/4{\pi}r$, i.e., we let
$MA^{2}z^{2}=g^{2}e^{-rm_{v}}/4{\pi}r$, where $g$ is a dimensionless coupling
constant,$m_{v}$ is the mass of the postulated meson (generally in the same
order of magnitude with nucleon mass), $r$ is the distance between nucleons.
The propagator of Yukawa potential is very simple:
$F(k)=g^{2}/(k^{2}+m^{2}_{v})$. The single-particle energy for colliding
systems can be expressed as
$\displaystyle\epsilon(\textbf{k})$ $\displaystyle=$
$\displaystyle\sqrt{\textbf{k}^{2}+M^{\ast
2}}+\sum_{i=\omega,\rho}\left[\frac{g_{i}^{2}c_{i}}{8\pi^{2}k}\int_{0}^{\infty}pn(p)\ln\frac{(k+p)^{2}+m_{i}^{2}}{(k-p)^{2}+m_{i}^{2}}dp\right]$
(5)
$\displaystyle+\int\frac{g^{2}}{(\textbf{k-p})^{2}+m^{2}_{v}}n(\textbf{p})\frac{d^{3}\textbf{p}}{(2\pi)^{3}},$
where the direct Hartree terms are neglected because they do not contribute to
the momentum derivative that is used to determine the critical coupling
constant. Here, we quote the same coefficient $c_{i}$ as in Eq.(2) by assuming
the symmetric matter after the collision. This is a reasonable assumption to
neglect the effect of isospin asymmetry because the coupling strength of the
$\rho$ meson is much less than that of the $\omega$ meson and the isospin
asymmetry is usually not large for the colliding system. After some
straightforward calculations, one can find that the derivative $d\epsilon/dk$
is exactly the same as Eq.(3), provided $g_{v}$ and $g_{vc}$ are replaced by
$g$ and $g_{c}$ , respectively. With the given quantities $M^{\ast}$ and
$m_{v}$ at the density $\rho$, we adjust the coupling constant $g$ to exceed
some critical value $g_{c}$ where $d\epsilon/dk$ has the zero point in the
vicinity of the Fermi surface. Then we can find out the critical coupling
constant for the ground-state rearrangement.
## III Numerical results and discussions
To obtain the fermion distribution function, one needs first to solve the
single-particle energy $\epsilon(k)$. Once the rearrangement occurs, the
fermion distribution function becomes multi-connected and the numerical
realization is quite tricky due to the existence of sharp edges. We do not
elaborate the numerical details, and readers can be referred to Ref. ZMV99
for details. In the following, we discuss in turn the occurrence of ground-
state rearrangement in cases of the vector boson exchanges and relativistic
heavy-ion collisions.
### III.1 Rearrangement with vector boson exchanges
Here we first work on the relativistic model SLC JWZ07 . The additional vector
boson is added to this model to provide additional repulsion, and the nucleon
effective mass is obtained in the RHF approximation NJW86 . With the density-
dependent nucleon effective mass, we are able to investigate the rearrangement
self-consistently at various densities. In what follows, we will discuss the
rearrangement in a simple system that consists of dark matter in which the
fermion mass is taken as a free parameter by assuming that the dark matter is
just weakly interacting. In this case, we intend to see whether the
rearrangement can occur with the weak repulsion.
Figure 2: The critical coupling constant as a function of density. It is
solved in the RHF approximation with the parametrization of the model SLC with
the addition of the additional vector boson exchange.
After obtaining the nucleon effective mass in the RHF approximation, we can
calculate the critical coupling constant for the ground-state rearrangement.
In Fig. 2, the critical coupling constant is plotted as a function of density
for various choices of the vector boson mass: $m_{v}=0.001,0.010$ and
$0.100GeV$. As one can see from Fig. 2, the critical coupling constant
$g_{vc}$ ascends monotonically with increasing the density when $m_{v}$ is
relative small ($m_{v}=0.001,0.010GeV$), while for the large mass
($m_{v}=0.100GeV$), $g_{vc}$ increases fast in the subsaturation density
region, then goes down slowly with the successive increase of the density. In
all cases, the critical coupling constant saturates at high densities.
Recently, the vector boson beyond the standard model, dubbed the U-boson that
may accounts for the modification to the inverse square law of the Newtonian
gravity, has been introduced to compensate the excessively small pressure
suggested by the super-soft symmetry energy WDH09 . The super-soft equation of
state (EOS) can be successfully remedied by the repulsion provided by the
U-boson with the ratio of the coupling constant to the mass of vector boson
ranging from 7.07 $GeV^{-1}$ to 12.25 $GeV^{-1}$ WDH09 . Later on, successive
work indicates that the U-boson with the ratio around 10 $GeV^{-1}$ can even
remedy largely the deviation between the soft and stiff EOS’s and the
difference in the mass-radius relation of neutron stars as well Zh11 . We are
intrigued to see whether the ground-state rearrangement can occur with these
ratio parameters of the U-boson. Unfortunately, as displayed in the Fig. 2,
various $g_{vc}/m_{v}$ are all larger than 40 $GeV^{-1}$. Thus, it is usually
impossible for nucleon systems to rearrange its ground state with the regular
ratio $g_{v}/m_{v}$ that is around 10 $GeV^{-1}$, and we are not able to
anticipate the constraint from whether or not the ground-state rearrangement
occurs on the U-boson parameters. On the other hand, this also reveals the
fact that using the Landau Fermi liquid theory is appropriate for the usual
nuclear Fermi liquid.
Now, we discuss dark matter that is invisible by the strong, electro-weak but
gravitational interaction. The properties of dark matter are totally unknown
to date Be05 . For instance, the mass of dark matter candidates can be assumed
to vary from the magnitude of eV to several hundred of GeV. Supposing that
dense dark matter can form, we are going to examine the ground-state
rearrangement of dark matter by taking the mass of fermionic dark matter
candidates as a free parameter. In this way, we still use Eq.(3) by simply
neglecting the terms concerning the meson exchanges that characterize the
strong interactions. Specifically, we calculate the $g_{vc}/m_{v}$ at
$m_{v}=0.010GeV$ for various $M$ at the number density that equals to nuclear
saturation density. The results are listed in table 1. It is seen that the
$g_{vc}/m_{v}$ drops clearly with increasing the mass of the dark matter
candidate. For $M=300.00GeV$, the critical coupling constant becomes as small
as 0.11 which gives an interaction strength comparable to the electromagnetic
interaction. This reveals a fact that the matter consisting of superheavy
fermions can not be regarded as a simple Fermi liquid as long as a weak
repulsion exists. Supposing that the heavy dark matter candidates stack up in
the core of dense stars, the rearrangement may easily occur therein provided
the repulsion comes up with the exchange of weakly interacting bosons.
Table 1: Ratios of the critical coupling constant to the vector boson mass at various $M$ for $m_{v}$=0.010 GeV and $\rho=0.16fm^{-3}$ $M(GeV)$ | 0.10 | 0.50 | 1.00 | 5.00 | 10.00 | 50.00 | 100.00 | 200.00 | 300.00
---|---|---|---|---|---|---|---|---|---
$g_{vc}/m_{v}(GeV^{-1})$ | 352.44 | 248.15 | 183.29 | 83.27 | 58.91 | 26.35 | 18.63 | 13.18 | 10.76
### III.2 The case of collision
For the heavy-ion collision, one needs in principle to determine the density
of crashed matter that depends on the collision energy. However, it is
complicated and beyond the scope of present work. Considering the fact that
the critical coupling constant for the ground-state rearrangement does not
change much as the density rises up around the saturation density, see Fig. 2,
we demonstrate the rearrangement at saturation density. For numerical trials
at other densities, the results are qualitatively similar, and we will not
specify the numerical details. Again, we adopt the basic SLC parametrization
to include the non-inertial effect and work out the nucleon effective mass
self-consistently in the RHF.
Though the non-inertial effect is hypothetically replaced by the exchange of
vector meson, we do not have the constraint on its mass. In practical
calculation, we thus change the postulated meson mass from the lowest pion
mass to the one close to 1 GeV. At the same density, we can calculate
different critical coupling constants for different postulated meson masses.
For instance of $m_{v}=0.5478GeV$, we obtain that at $k\simeq 0.95k_{F}$ the
derivative $d\epsilon/dk$ is negative for $g>g_{c}=31.91$. The results are
tabulated in Table 2. We see that the critical coupling constant of the
postulated meson increases clearly with the mass, while the ratio of the
coupling constant to the mass changes much slowly.
Table 2: Critical coupling constants and accelerations for various masses of a postulated vector meson. Here, we take $\rho=0.16fm^{-3}$ and $R_{0}=10fm$. $m_{v}$ (GeV) | 0.1350 | 0.2626 | 0.4976 | 0.5478 | 0.6424 | 0.7800 | 0.8900 | 0.9578
---|---|---|---|---|---|---|---|---
$g_{c}$ | 7.19 | 12.89 | 27.81 | 31.91 | 40.57 | 55.44 | 69.29 | 78.71
$g_{c}/m_{v}$ (GeV${}^{-1})$ | 53.27 | 49.09 | 55.89 | 58.25 | 63.16 | 71.08 | 77.86 | 82.17
$A_{c}$(fm-1) | 0.0441 | 0.0538 | 0.0855 | 0.0946 | 0.114 | 0.147 | 0.179 | 0.200
As one can anticipate, to make sure the occurrence of rearrangement, the
inequality ${\mid}A{\mid}{\geq}A_{c}$ should hold, while $A_{c}$ is the
minimum acceleration to rearrange the ground state when the distance between
the colliding particles is given. Supposing the full stopping of the colliding
system leads eventually to a fireball with the radius of a few femtometres, we
estimate the $A_{c}$ by averaging the Yukawa-type coupling and the coupling of
the harmonic oscillator within such a sphere, namely,
$A_{c}^{2}=\left(\int^{R_{0}}_{0}d^{3}rg_{c}^{2}\frac{e^{-rm}}{r}\right)/{\int^{R_{0}}_{0}d^{3}r\pi
M^{\ast}r^{2}}.$ (6)
In this way, different critical decelerations for different postulated meson
masses can be estimated. Here, we use $\rho=0.16fm^{-3}$ and $R_{0}=10fm$ as
an example, and results are tabulated in Table 2. As one can see, for
$m=0.5478GeV$, with $g_{c}=31.91$, we get ${\mid}A{\mid}\geq 0.0946fm^{-1}$
for the occurrence of rearrangement ( ${\mid}A_{c}\mid\simeq 8.50\times
10^{30}m/s^{2}$). This critical deceleration is really tremendous. However, it
is not too difficult to reach such an enormous deceleration in relativistic
heavy-ion collisions. For instance, two nuclei, being accelerated almost up to
the light velocity, crash against each other to be in full stopping in the
time scale of a few femtometres, e.g., $5-10fm$, which is comparable to the
size of stopped matter. Then, the deceleration is $0.1-0.2fm^{-1}$ which gives
a reasonable and consistent magnitude of deceleration with those in Table 2.
The present estimate indicates that the non-inertial effect because of the
huge deceleration can lead to the ground-state rearrangement. Note that we do
not consider the effect of thermalization at the late stage of the collision.
The estimate also implies that as long as the gravitational field is
sufficiently strong, the ground-state rearrangement may lead to deviation from
the quasi-particle Fermi liquid. For the case of larger density formed by the
collision, we can get moderately smaller critical decelerations. However, the
conclusion does not alter much for various densities of matter.
## IV Summary
We have investigated the applicability of the Landau Fermi liquid theory and
found that the necessary stability condition for this theory may be broken in
some circumstances. Once the necessary stability condition is broken, the
ground state is rearranged to be different from the step function. In our
relativistic model, the nuclear Fermi liquid is stable with the usual strong
interaction strength and no rearrangement occurs. This is nevertheless in
sharp contrast with the system consisting of very heavy fermions that may be
regarded as dark matter candidates. We have found that it is possible for
heavy-fermion matter to rearrange the ground state even by exchanging the
weakly coupling vector boson. In the case of the relativistic heavy-ion
collisions, the ground state of the colliding systems can be rearranged due to
the strong non-inertial effect which changes the ways of the quasi-particle
occupation. This finding is potentially instructive in investigating the
fermion system during the supernova explosions and in the horizon of black
holes.
## Acknowledgements
The work was supported in part by the National Natural Science Foundation of
China under Grant Nos. 10975033 and 11275048 and the China Jiangsu Provincial
Natural Science Foundation under Grant No.BK20131286.
## References
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* (8) P. Fayet, Phys. Lett. B $\bf 227$, 127 (1989).
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|
arxiv-papers
| 2012-12-14T10:14:01 |
2024-09-04T02:49:39.286624
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rong-Yao Yang and Wei-Zhou Jiang",
"submitter": "Rongyao Yang",
"url": "https://arxiv.org/abs/1212.3428"
}
|
1212.3628
|
# Non-Markovian reversible diffusion-influenced
reactions in two dimensions
Thorsten Prüstel Laboratory of Systems Biology
National Institute of Allergy and Infectious Diseases
National Institutes of Health Martin Meier-Schellersheim Laboratory of
Systems Biology
National Institute of Allergy and Infectious Diseases
National Institutes of Health
###### Abstract
We investigate the reversible diffusion-influenced reaction of an isolated
pair in the presence of a non-Markovian generalization of the backreaction
boundary condition in two space dimensions. Following earlier work by Agmon
and Weiss, we consider residence time probability densities that decay slower
than an exponential and that are characterized by a parameter $0<\sigma\leq
1$. We calculate an exact expression for the probability $S(t|\ast)$ that the
initially bound particle is unbound, which is valid for arbitrary $\sigma$ and
for all times. Furthermore, we derive an approximate solution for long times.
We show that the ultimate fate of the bound state is complete dissociation, as
in the 2D Markovian case. However, the limiting value is approached quite
differently: Instead of a $\sim t^{-1}$ decay, we obtain $1-S(t|\ast)\sim
t^{-\sigma}\ln t$.
11footnotetext: Email: [email protected], [email protected]
## 1 Introduction
It has been shown that the ultimate fate of an isolated pair for the
reversible diffusion-influenced reaction is independent from the
dimensionality of space. In fact, the ultimate escape probability is unity in
one, two and three space dimensions [2, 3, 9, 12]. However, the dimensionality
does have an influence on how the limiting value for $S(t|\ast)$, which
denotes the probability that the initially bound particle is unbound, is
approached: Denoting the number of dimensions by $d$, the decay goes as
$1-S(t|\ast)\sim t^{-d/2}$, which implies for instance considerable
ramifications for the off-rate. This dependence from the dimension can be
traced back to the return to the origin probability of the underlying random
walk. In one and two dimensions (2D), Brownian motion is recurrent, i.e. the
return to the origin happens with probability one, while in 3D it is
transient, meaning that the non-return probability does not vanish.
It is reasonable that, apart from the diffusive motion of the particles, the
type of their interaction potentially influences the time-dependence of the
probability $S(t|\ast)$ and, in particular, its long-time behavior. Indeed, in
most cases one studied the problem by tacitly assuming a Markovian residence
time probability density for the bound state. However, Agmon and Weiss [4]
demonstrated that abandoning the Markovian assumption changes the time-
dependence of $S(t|\ast)$ in a substantial way. They found a phase transition
as a function of the parameter $\sigma$, which is a measure for the deviation
from the Markovian density (where $\sigma=1$ corresponds to the Markovian
limit). In one dimension, they found a first-order phase transition in which
the long-time behavior of $S(t|\ast)$ undergoes a discontinuous change at
$\sigma=1/2$. In contrast, in 3D they observed a second-order phase transition
at $\sigma=1$ and the ultimate fate is again complete dissociation,
independent from $\sigma$.
In what follows, we will extend the analysis presented in Ref. [4] to the 2D
case. We will investigate non-Markovian dissociation for an isolated pair of
reversibly binding particles which move in an infinitely extended two
dimensional plane. The case of two dimensions is special, because it is the
critical dimension with regard to recurrence and transience [14]. We will show
whether even weak recurrence is sufficient to obtain a first-order phase
transition.
We would like to point out that the consideration of non-Markovian
dissociation is not only theoretically interesting. In particular in view of
the intricate and heterogeneous composition of biological cell membranes [6],
generalizations of the Markovian behavior could potentially be significant for
more realistic descriptions of the corresponding local biochemistries.
The organization of the paper is as follows. First, we will introduce the
general theoretical context, especially the Smoluchowski equation and
backreaction boundary condition (BC). Next, we will discuss memory effects and
a non-Markovian generalization of the backreaction BC. In theses parts of our
presentation, we shall closely follow Agmon and Weiss [4]. Then, we will apply
the general formalism to the 2D case. We will obtain an integral
representation for $S(t|\ast)$ in the time domain, which is valid for all
times. Furthermore, we will present an approximate solution for the long-time
behavior of the probability $S(t|\ast)$. Finally, we discuss similarities and
differences to the non-Markovian 1D and 3D cases.
## 2 Smoluchowski equation and backreaction BC
Solutions of the diffusion (Smoluchowski) equation [15, 7, 13] can be employed
to study diffusion-influenced reactions of an isolated pair. The different
types of chemical reactions are taken into account by imposing certain
boundary conditions at the encounter distance. Typically, one considers the
following scenario [2, 3, 9, 12]. Two spherical (or rather disklike in 2D)
particles $A$ and $B$ with diffusion constants $D_{A}$ and $D_{B}$,
respectively, may associate when their separation equals the encounter
distance $a$ to form a bound molecule $AB$. Such a system may equivalently be
described as the diffusion of a point-like particle with diffusion constant
$D=D_{A}+D_{B}$ around a static disk with radius $a$.
The irreversible association reaction is described by the radiation BC that is
characterized by an intrinsic association constant $\kappa_{a}$ [8]. In the
reversible case, the bound state may dissociate again to form an unbound pair
$A+B$ and the radiation boundary condition has to be generalized
correspondingly. The so-called backreaction BC [2, 10, 11, 5, 3, 9, 12] has
been used extensively to take into account reversible association-dissociation
reactions, but in its conventional form it assumes a Markovian probability
density for the residence time [4]
$\psi(t)=\kappa_{d}e^{-\kappa_{d}t},$ (2.1)
or, equivalently, in the Laplace domain
$\tilde{\psi}(s)=\frac{\kappa_{d}}{\kappa_{d}+s}.$ (2.2)
Eq. (2.1) corresponds to the situation where the bound molecule can be
sufficiently well described by a single, discrete state. However, the bound
state is typically comprised of a large number of different internal states.
It follows that the associated probability density $\psi(t)$ for residence
time in the set of all possible bound states is given by a superposition of
exponentials, which correspond to the individual states
$\psi(t)=\sum^{\infty}_{n=0}\varphi_{n}e^{-\lambda_{n}t}.$ (2.3)
Only if the time scales set by the $\lambda_{n}$ parameters are sufficiently
separated, i.e. the decay of the ground state (=dissociation) is much slower
than the decay of all other internal states, Eq. (2.3) can be replaced by Eq.
(2.1) and the Markovian description is justified. However, if there is no
sufficient separation of the involved time scales, the effective description
of all internal states as one single state cannot be Markovian any more.
Furthermore, as already mentioned in the introduction, a non-Markovian
deviation could be caused by a complex environment, for instance in biological
cellular systems.
To proceed we consider the probability density function (PDF) $p(r,t|\ast)$,
which yields the probability to find the particle at a distance equal to $r$,
given that it was initially in the bound state. The diffusion (or
Smoluchowski) equation in 2D [4, 3] governs the time evolution of
$p(r,t|\ast)$
$\frac{\partial}{\partial t}p(r,t|\ast)=D\bigg{(}\frac{\partial^{2}}{\partial
r^{2}}p(r,t|\ast)+\frac{1}{r}\frac{\partial}{\partial
r}p(r,t|\ast)\bigg{)},\quad r\geq a.$ (2.4)
The initial condition is
$p(r,t=0|\ast)=0,\quad r>a.$ (2.5)
The PDF we are interested in is only defined for $r\geq a>0$ and one has to
impose a BC for $r=a$ specifying the behavior at the encounter distance. The
(Markovian) backreaction BC incorporates association and dissociation by
relating the diffusional flux at the surface of the ”interaction disc” and the
probability $S(t|\ast)$ that the initially bound particle is unbound at time
$t>0$ as follows [2, 3, 9]
$\displaystyle 2\pi aD\frac{\partial}{\partial
r}p(r,t|\ast)|_{r=a}=\kappa_{a}p(a,t|\ast)-\kappa_{d}[1-S(t|\ast)].$ (2.6)
Note that in 2D the diffusional flux is given by
$\displaystyle-J(a|\ast)=2\pi aD\frac{\partial}{\partial
r}p(r,t|\ast)|_{r=a},$ (2.7)
and the probability $S(t|\ast)$ is defined by
$\displaystyle S(t|\ast)$ $\displaystyle=$ $\displaystyle
2\pi\int^{\infty}_{a}p(r,t|\ast)rdr.$ (2.8)
Integrating the diffusion equation over space yields
$J(a|\ast)=\frac{\partial}{\partial t}S(t|\ast),$ (2.9)
which translates in the Laplace domain to
$\tilde{J}(a,s|\ast)=s\tilde{S}(s|\ast),$ (2.10)
where we have used that $S(0|\ast)$ vanishes. Hence, the BC Eq. (2.6) becomes
in the Laplace domain [4]
$\tilde{J}(a,s|\ast)=\frac{\kappa_{d}-s\kappa_{a}\tilde{p}(a,s|\ast)}{s+\kappa_{d}}.$
(2.11)
For our analysis, we will make use of the following central identity [4] that
relates in the Laplace domain the probability that the particle is not bound
to the PDF $\tilde{p}_{\text{ref}}(a,s|a)$, which corresponds to nonreactive
diffusion and solves the diffusion equation subject to a reflecting BC,
$\tilde{J}(a,s|a)=0$,
$s\tilde{S}(s|\ast)=\frac{\kappa_{d}}{\kappa_{d}+s[1+\kappa_{a}\tilde{p}_{\text{ref}}(a,s|a)]}.$
(2.12)
A salient feature of Eq. (2.12) is that it is valid in any dimension, thereby
allowing to find an expression for $\tilde{S}$ once
$\tilde{p}_{\text{ref}}(a,s|a)$ is known. In 2D, one has [12]
$\tilde{p}_{\text{ref}}(a,s|a)=\frac{1}{2\pi
D}\bigg{[}I_{0}(qa)K_{0}(qa)+K_{0}(qa)K_{0}(qa)\frac{I_{1}(qa)}{K_{1}(qa)}\bigg{]},$
(2.13)
where $I_{0}(z),I_{1}(z),K_{z}(x),K_{1}(z)$ refer to the modified Bessel
function of first and second kind and zero and first order, respectively [1,
Sect. 9.6]. Using $I_{0}(z)K_{1}(z)+I_{1}(z)K_{0}(z)=z^{-1}$ and combining
Eqs. (2.12) and (2.13) we arrive at
$s\tilde{S}(s|\ast)=\frac{\kappa_{d}qK_{1}(qa)}{qK_{1}(qa)(s+\kappa_{d})+\kappa_{a}/(2\pi
aD)sK_{0}(qa)}.$ (2.14)
To find an expression for the probability that the particle is unbound in the
time domain, we could invert the Laplace transform explicitly by calculating a
Bromwich contour integral. However, in the Markovian backreaction BC case, an
expression has already been derived by using alternatively the Green’s
function $p(r,t|t_{0})$ of the diffusion equation subject to a backreaction
BC. The obtained result is [12]
$S(t|*)=1-2\frac{\kappa_{a}\kappa_{d}}{\pi^{3}a^{2}D^{2}}\int^{\infty}_{0}e^{-Dtx^{2}}\frac{1}{\alpha^{2}(x)+\beta^{2}(x)}\frac{1}{x}dx,$
(2.15)
where
$\displaystyle\alpha(x)$ $\displaystyle=$
$\displaystyle(x^{2}-\kappa_{D})J_{1}(xa)+hxJ_{0}(xa),$ (2.16)
$\displaystyle\beta(x)$ $\displaystyle=$
$\displaystyle(x^{2}-\kappa_{D})Y_{1}(xa)+hxY_{0}(xa),$ (2.17) $\displaystyle
h$ $\displaystyle=$ $\displaystyle\frac{\kappa_{a}}{2\pi aD},$ (2.18)
$\displaystyle\kappa_{D}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{d}}{D}.$ (2.19)
$J_{n}(z),Y_{n}(z)$ denote the Bessel functions of first and second kind,
respectively [1, Sect. 9.1].
We now seek the non-Markovian generalization of Eq. (2.15). To this end, we
have to find the non-Markovian analogue of the term $\kappa_{d}[1-S(t|\ast)]$
in the BC Eq. (2.6).
### 2.1 Non-Markovian backreaction BC
In Ref. [4] it was shown that the backreaction BC can also be written in terms
of the probability density of the residence time as follows
$J(a,t|\ast)=-\kappa_{a}p(a,t|\ast)+\psi(t)+\kappa_{a}\int^{t}_{0}p(a,\tau|\ast)\psi(t-\tau)d\tau.$
(2.20)
Application of the Laplace transform leads to
$\tilde{J}(a,s|\ast)=\tilde{\psi}(s)-\kappa_{a}[1-\tilde{\psi}(s)]p(a,s|\ast).$
(2.21)
Similarly, the Laplace transform of $S(t|\ast)$ may be written as [4]
$s\tilde{S}(s|\ast)=\frac{\tilde{\psi}(s)}{1+\kappa_{a}[1-\tilde{\psi}(s)]\tilde{p}_{\text{ref}}(a,s|a)}.$
(2.22)
Here, two remarks are in order. First, upon using the Markovian version of the
residence time probability density Eq. (2.2) in Eqs. (2.21), (2.22), one
recovers Eqs. (2.11) and (2.12). On the other hand, starting from Eqs. (2.11)
and (2.12), one can simply substitute $\kappa_{d}$ by
$s\tilde{\psi}(s)/[1-\tilde{\psi}(s)]$ [3] to arrive at Eqs. (2.21), (2.22).
Next, we turn our attention to the concrete form of the residence time density
$\psi(t)$. In Ref. [4], the following choice of densities was employed
$\tilde{\psi}(s)=\frac{1}{1+(s\kappa^{-1})^{\sigma}},\quad 0<\sigma\leq 1.$
(2.23)
For a discussion and motivation for this particular form, we refer to [4].
Here, we only note that for $\sigma=1$ and by identifying $\kappa$ with
$\kappa_{d}$, one recovers Eq. (2.2). However, for $\sigma<1$, $\psi(t)$
decays slower than an exponential.
## 3 Non-Markovian survival probability in 2D
In what follows, we will use Eqs. (2.22), (2.13) and (2.23) to derive an
expression for $S(t|\ast)$ for arbitrary $0<\sigma\leq 1$ and all times. We
obtain
$\tilde{S}(s|\ast)=\frac{1}{s}\frac{\kappa^{\sigma}qK_{1}(qa)}{qK_{1}(qa)(s^{\sigma}+\kappa^{\sigma})+hs^{\sigma}K_{0}(qa)},$
(3.1)
where $q=\sqrt{s/D}$. The inversion theorem for the Laplace transformation can
be applied to find the corresponding expression of $\tilde{S}(s|\ast)$ in the
time domain
$S(t|\ast)=\frac{1}{2\pi
i}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{st}\,\tilde{S}(s|\ast)ds.$ (3.2)
To calculate the Bromwich contour integral, we first note that
$\tilde{S}(s|\ast)$ has a branch point at $s=0$. Therefore, we use the contour
of FIG. 1 with a branch cut along the negative real axis. We note that the
contribution from the small circle around the origin does not vanish, but
yields 1, which can be seen by the limiting forms of the modified Bessel
functions for small arguments [1, Eqs. (9.6.7)-(9.6.9)]. Therefore, we obtain
$\displaystyle\int^{\gamma+i\infty}_{\gamma-i\infty}e^{st}\,\tilde{S}(s|\ast)ds$
$\displaystyle=$ $\displaystyle 2\pi
i-\int_{\mathcal{C}_{2}}e^{st}\,\tilde{S}(s|\ast)ds-\int_{\mathcal{C}_{4}}e^{st}\,\tilde{S}(s|\ast)ds.$
(3.3)
It remains to calculate the integrals along $\mathcal{C}_{2},\mathcal{C}_{4}$.
To this end, we choose $s=Dx^{2}e^{i\pi}$ and use [7, Append. 3, Eqs. (25),
(26))]
$\displaystyle I_{n}(xe^{\pm\pi i/2})$ $\displaystyle=$ $\displaystyle e^{\pm
n\pi i/2}J_{n}(x),$ (3.4) $\displaystyle K_{n}(xe^{\pm\pi i/2})$
$\displaystyle=$ $\displaystyle\pm\frac{1}{2}\pi ie^{\mp n\pi
i/2}[-J_{n}(x)\pm iY_{n}(x)].$ (3.5)
It follows that
$\displaystyle\int_{\mathcal{C}_{2}}e^{st}\,\tilde{S}(s|\ast)ds=2\kappa^{\sigma}_{D}\int^{\infty}_{0}e^{-Dx^{2}t}\frac{[Y_{1}(xa)+iJ_{1}(xa)][\beta_{\sigma}(x)-i\alpha_{\sigma}(x)]}{\alpha_{\sigma}(x)^{2}+\beta_{\sigma}(x)^{2}}\frac{dx}{x},$
(3.6)
where $\kappa^{\sigma}_{D}=\kappa^{\sigma}/D^{\sigma}$ and we have defined
$\displaystyle\alpha_{\sigma}$ $\displaystyle=$ $\displaystyle
x^{2\sigma-1}\bigg{[}-\big{(}xJ_{1}(xa)+hJ_{0}(xa)\big{)}\cos(\pi\sigma)-\big{(}xY_{1}(xa)+hY_{0}(xa)\big{)}\sin(\pi\sigma)\bigg{]}$
(3.7) $\displaystyle-\kappa^{\sigma}_{D}J_{1}(xa),$
$\displaystyle\beta_{\sigma}$ $\displaystyle=$ $\displaystyle
x^{2\sigma-1}\bigg{[}\big{(}xJ_{1}(xa)+hJ_{0}(xa)\big{)}\sin(\pi\sigma)-\big{(}xY_{1}(xa)+hY_{0}(xa)\big{)}\cos(\pi\sigma)\bigg{]}$
(3.8) $\displaystyle-\kappa^{\sigma}_{D}Y_{1}(xa).$
We observe that $\alpha_{\sigma=1}=\alpha,\beta_{\sigma=1}=\beta$, cp. Eqs.
(2.16), (2.17).
To evaluate the integral along the contour $\mathcal{C}_{4}$ we choose
$p=Dx^{2}e^{-i\pi}$ and after an analogous calculation one finds that
$\int_{\mathcal{C}_{2}}e^{st}\,\tilde{S}(s|\ast)ds=-\bigl{(}\int_{\mathcal{C}_{4}}e^{st}\,\tilde{S}(s|\ast)ds\bigr{)}^{\star},$
where $\star$ means complex conjugation. Thus, we have
$\displaystyle
S(t|\ast)-1=-\frac{1}{\pi}\Im\biggl{(}\int_{\mathcal{C}_{2}}e^{st}\,\tilde{S}(s|\ast)ds\biggr{)},$
(3.9)
and arrive finally at the following expression for the probability to find the
initially bound particle unbound in the time domain
$\displaystyle
S(t|\ast)=1+\frac{2}{\pi}\kappa^{\sigma}_{D}\bigg{[}\frac{2h}{a\pi}\cos(\pi\sigma)Q_{1}+\sin(\pi\sigma)Q_{2}\bigg{]},$
(3.10)
where
$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}e^{-Dtx^{2}}\frac{x^{2\sigma-3}}{\alpha^{2}_{\sigma}(x)+\beta^{2}_{\sigma}(x)}dx,$
(3.11) $\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle-\int^{\infty}_{0}e^{-Dtx^{2}}\frac{x^{2\sigma-2}\Omega(x)}{\alpha^{2}_{\sigma}(x)+\beta^{2}_{\sigma}(x)}dx,$
(3.12)
and we have introduced
$\displaystyle\Omega(x)=x\big{[}J_{1}^{2}(xa)+Y_{1}^{2}(xa)\big{]}+h\big{[}J_{0}(xa)J_{1}(xa)+Y_{0}(xa)Y_{1}(xa)\big{]}.$
(3.13)
Eq. (3.10) shows that the parameter $\sigma$ appears as a sort of mixing angle
for the two qualitatively different contributions $Q_{1},Q_{2}$. We note that
the term $Q_{2}$ is foreign to the Markovian expression Eq. (2.15).
Correspondingly, the second term vanishes for $\sigma=1$, while one recovers
the Markovian limiting case Eq. (2.15) due to the first term $Q_{1}$ (we set
$\kappa:=\kappa_{d})$.
Furthermore, we may already conclude from Eq. (3.10) that in 2D the ultimate
escape probability is unity for all $\sigma$, like in the Markovian and in the
3D non-Markovian case. This behavior, however, is quite different to the non-
Markovian 1D case, where the escape probability depends on the parameter
$\sigma$ [4].
Finally, we are interested, how the limiting value is approached. To this end,
we derive the asymptotic behavior for $t\rightarrow\infty$. Starting point is
the expression for $\tilde{S}(s|\ast)$ Eq. (3.1). Using the series expansions
of the modified Bessel functions for small arguments [1, Eqs.
(9.6.10)-(9.6.13)], we obtain
$\tilde{S}(s|\ast)=\frac{1}{s}+\frac{ha}{\kappa^{\sigma}}s^{\sigma-1}\ln\bigg{(}\frac{1}{2}e^{\gamma_{E}}qa\bigg{)}-\frac{s^{\sigma-1}}{\kappa^{\sigma}}+\ldots,$
(3.14)
where $\gamma_{E}$ denotes Euler’s number $\gamma_{E}=0.5772156649\ldots$ [1,
Eqs. (6.1.3)]. For $\sigma=1$ we obtain in the time domain
$S(t|\ast)=1-\frac{\kappa_{a}}{\kappa_{d}}\frac{1}{4\pi Dt}+\ldots,$ (3.15)
which is the known Markovian result. If $0<\sigma<1$, we use [1, Eqs.
(29.3.99), (29.3.7)]
$\displaystyle\mathcal{L}^{-1}\bigg{[}\frac{1}{s^{k}}\ln s\bigg{]}$
$\displaystyle=$ $\displaystyle\frac{t^{k-1}}{\Gamma(k)}\bigg{[}\Psi(k)-\ln
t\bigg{]},\quad k>0,$ (3.16)
$\displaystyle\mathcal{L}^{-1}\bigg{[}\frac{1}{s^{k}}\bigg{]}$
$\displaystyle=$ $\displaystyle\frac{t^{k-1}}{\Gamma(k)},\quad k>0,$ (3.17)
where $\mathcal{L}^{-1},\Gamma,\Psi$ denote the inverse Laplace transform, the
gamma function and the digamma function [1, Eqs. (6.3.1)], respectively.
Hence, we find
$S(t|\ast)=1-\frac{\kappa_{a}}{\kappa^{\sigma}}\frac{1}{4\pi
D}\frac{t^{-\sigma}}{\Gamma(1-\sigma)}\ln
t+C\frac{t^{-\sigma}}{\Gamma(1-\sigma)}+\ldots,$ (3.18)
where we introduced the constant
$C=\frac{\kappa_{a}}{\kappa^{\sigma}}\frac{1}{2\pi
D}\bigg{[}\frac{\Psi(1-\sigma)}{2}+\ln\bigg{(}\frac{1}{2}e^{\gamma_{E}}D^{-1/2}a\bigg{)}\bigg{]}-\frac{1}{\kappa^{\sigma}}.$
(3.19)
We find again that, independent from the parameter $\sigma$, the ultimate fate
is complete dissociation, although the underlying random walk is recurrent. In
contrast, in 1D the parameter $\sigma$ decides if particles escape or remain
bound at long times. In this regard, the 2D case is more similar to the (non-
Markovian) 3D case. However, it is very different to the 3D case with respect
to how the limiting value is approached. In 3D, it was shown that the decay
goes as $t^{-\sigma}$ at $\sigma<1$, but as $t^{-3/2}$ at $\sigma=1$, which
means there is a second-order phase transition. However, in 2D, even the
functional form is different: While we observe a $t^{-\sigma}\ln t$ decay at
$\sigma<1$, we find $t^{-1}$ at $\sigma=1$.
### Acknowledgments
This research was supported by the Intramural Research Program of the NIH,
National Institute of Allergy and Infectious Diseases.
We would like to thank Bastian R. Angermann and Frederick Klauschen for
stimulating discussions.
Figure 1: Integration contour used in Eq. (3.3).
## References
* [1] M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1965.
* [2] N. Agmon. J. Chem. Phys., 81:2811, 1984.
* [3] N. Agmon and A. Szabo. J. Chem. Phys., 92:5270, 1990.
* [4] N. Agmon and G.H. Weiss. J. Chem. Phys., 91:6937, 1989.
* [5] Pines E. Agmon, N. and D. Huppert. J. Chem. Phys., 88:5631, 1988.
* [6] I. Bethani, S.S. Skanland, I. Dikic, and A. Acker-Palmer. EMBO J., 29:2677, 2010.
* [7] H.S. Carslaw and J.C. Jaeger. Conduction of Heat in Solids. Clarendon Press, New York, 1986.
* [8] F.C. Collins and G.E. Kimball. J. Colloid Sci., 4:425, 1949.
* [9] H. Kim and K.J. Shin. Phys. Rev. Lett., 82:1578, 1999.
* [10] E. Pines and D. Huppert. Chem. Phys. Lett., 126:88, 1986.
* [11] E. Pines, D. Huppert, and N. Agmon. J. Chem. Phys., 88:5620, 1988.
* [12] T. Prüstel and M. Meier-Schellersheim. J. Chem. Phys., 137:054104, 2012.
* [13] S. A. Rice. Diffusion Limited Reactions. Elsevier, New York, 1985.
* [14] D. Toussaint and F. Wilczek. J. Chem. Phys., 78:2642, 1983.
* [15] M. von Smoluchowski. Z. Phys. Chem., 92:129, 1917.
|
arxiv-papers
| 2012-12-14T22:28:01 |
2024-09-04T02:49:39.296834
|
{
"license": "Public Domain",
"authors": "Thorsten Pr\\\"ustel and Martin Meier-Schellersheim",
"submitter": "Thorsten Pr\\\"ustel",
"url": "https://arxiv.org/abs/1212.3628"
}
|
1212.3648
|
# The Metadata Anonymization Toolkit
Julien Voisin Belfort-Montbéliard University Institute
of Technology (UTMB), France
[email protected]
Christophe Guyeux, Jacques M. Bahi Computer Science Laboratory DISC
FEMTO-ST Institute, UMR 6174 CNRS
University of Franche-Comté
Besançon, France
{jacques.bahi, christophe.guyeux}@femto-st.fr
###### Abstract
This document summarizes the experience of Julien Voisin during the 2011
edition of the well-known _Google Summer of Code_. This project is a first
step in the domain of metadata anonymization in Free Software. This article is
articulated in three parts. First, a state of the art and a categorization of
usual metadata, then the privacy policy is exposed/discussed in order to find
the right balance between information lost and privacy enhancement. Finally,
the specification of the Metadata Anonymization Toolkit (MAT) is presented,
and future possible works are sketched.
###### Index Terms:
Metadata; Anonymization; Digital Forensics; Anonymity; Privacy; Side Channel
Information.
## I Introduction
The _Google Summer of Code_ (GSoC [1]) is an annual program organized by
Google, to promote the development of free software: students are paid by
_Google_ to work on open-source projects during the summer break. Julien
Voisin, principal author of this paper, is an undergraduate student in
computer science at the University of Technology of Belfort-Montbéliard
(University of Franche-Comté) who was selected in the 2011 GSoC session. This
article, written under the supervision of the other authors, summarizes his
experiences.
The project was to depersonalize or remove any personal information embedded
in media sent through the _Tor_ network [2]. _Tor_ , which stands for _The
Onion Router_ , is a network of virtual tunnels that allows people and groups
to improve their privacy and security on the Internet. It also enables
software developers to create new communication tools with built-in privacy
features. _Tor_ provides the foundation for a range of applications that allow
organizations and individuals to share information over public networks
without compromising their privacy. However, the desired anonymity can be
broken by the data sent through it, if it contains information on the sender.
For instance, metadata is frequently added during several data acquisition
processes, possibly without the user’s knowledge or consent. Obviously, such
information can be problematic when desiring to reach anonymity on the
Internet.
Metadata consists of information that characterizes digital data like
Microsoft Word documents, pictures, music files, and so on. In essence,
metadata does provide an excellent source of information on every facet of the
data that can be characterized, like who produced the document, when and where
it was produced, for which reasons, what the content of the media is, and so
on. Indeed, metadata within a file can reveal a lot of things about the author
of the document. For instance, cameras record the date on which any picture
has been taken and which camera has been used. Office documents like PDF or
those produced by LibreOffice/Microsoft Office automatically add authors and
company information into documents and spreadsheets. Not everybody is willing
to disclose such information on the web.
This paper presents the main achievement of a project realized by Julien
Voisin during his 2011 session of the Google Summer of Code (GSoC). This
project aimed at removing any personal information embedded in any given
media, leading to the development of a Metadata Anonymization Toolkit (MAT)
library111The MAT is available at https://mat.boum.org and is distributed
under the GPLv2 license. Although it offers a command line interface, the most
common usage is to use it by the mean of its GUI.. In this document, the MAT
is presented, and the necessary choices that have been made during its
realization are documented. For instance, such a tool is supposes to be able
to choose which information to remove. However, to determine which metadata
raises problems for privacy is not a trivial question. Some metadata is
clearly problematic, such as GPS coordinates, but others are less easy to tell
apart (like the _gid_ (Group IDentifier) giving the group of the file under
consideration). To solve this problem, the following politic has been chosen:
most of the time, the MAT library tries to remove all metadata that is not
mandatory to the file integrity.
The Metadata Anonymization Toolkit is already embedded in the Tails GNU/Linux
distribution [3]. Tails is a live CD or live USB that aims at preserving the
users privacy and anonymity in a friendly way. More specifically, Tails
delivers the three following services. Firstly, it helps to use the Internet
anonymously, from any location and on any computer: all connections to the
Internet are forced to go through the Tor network. Secondly, it leaves no
trace on the computer unless the user asks it explicitly. Lastly, up-to-date
cryptographic tools can be used to encrypt files, emails, and instant
messaging. Tails is not the only LiveCD dedicated to privacy. Incognito [4]
was a similar project in all aspects, but based on Gentoo instead of Debian.
The main developer has abandoned the project to join forces with the Tails
team in 2010. JonDo LiveCD [5] also deserves a mention, although it is more
focused on JonDonym (their home-made anonymity network) rather than than on
Tor and Amnesic features.
FIELD | EXAMPLE
---|---
File Name | example.jpg
File Modification Date/Time | 2012:03:02 16:20:52+01:00
File Type | JPEG
MIME Type | image/jpeg
JFIF Version | 1.02
Orientation | Horizontal (normal)
X Resolution | 72
Y Resolution | 72
Resolution Unit | inches
Software | Adobe Photoshop CS3 Windows
Modify Date | 2009:06:14 21:09:57
Exif Image Width | 1984
Exif Image Height | 5300
Compression | JPEG (old-style)
Thumbnail Offset | 332
Thumbnail Length | 7172
Current IPTC Digest | e8f15cf32fc118a1a27b67adc564d5ba
IPTC Digest | e8f15cf32fc118a1a27b67adc564d5ba
Copyright Flag | False
Photoshop Thumbnail | (Binary data 7172 bytes, use -b option to extract)
Photoshop Quality | 5
XMP Toolkit | Adobe XMP Core 4.1-c036 46.276720, Mon Feb 19 2007 22:40:08
Creator Tool | Adobe Photoshop CS3 Windows
Create Date | 2009:06:14 21:09:57+02:00
Metadata Date | 2009:06:14 21:09:57+02:00
Document ID | uuid:97B34EC31559DE1181FD86CC9CA57AAA
History | (empty)
Primary Platform | Microsoft Corporation
Image Size | 1984x5300
Thumbnail Image | (Binary data 7172 bytes, use -b option to extract)
TABLE I: Example of metadata embedded in a picture (recovered with ExifTool
[6])
The remainder of this student research work is organized as follows. In
Section II, the metadata under concern is introduced. Section III presents a
short state-of-the-art in current metadata forensics investigation. Section IV
analyses the different kinds of metadata and proposes to categorize them.
Section VI presents the MAT tool, focusing on the chosen privacy policy and
implemented features, whereas in Section VII future potential improvements are
addressed. This research work ends by a conclusion section, where the
contribution in the field of anonymity is summarized and intended future work
is presented.
## II Metadata Overview
### II-A Presentation
Metadata, also known as _data about data_ , is information that characterizes
or gives details on digital media like music, images, movies, Office files,
etc. There are two kinds of metadata: structural and descriptive.
On the one hand, structural metadata provides information about the internal
structure of the file. Such information, which is required to extract the
content from the binary representation, does not change for a given type of
file. For instance, digital cameras insert into each picture they produce: the
date, the camera model, the post-processing software used, and even, for some
high-end models, the GPS coordinates of the place where the images have been
taken! Office documents like PDF or LibreOffice/Microsoft Office generally
contains authors, operating system, company information, and even the history
of revisions into each document.
On the other hand, descriptive metadata provides unnecessary information about
the file or the data content. Its reason to be is to enrich the media with
secondary additional information like comments, creation date, and so on.
Obviously, it can also compromise the anonymity and privacy of users in a
network context, and previous works in the literature show that such
descriptive metadata can allow the authors to be tracked back [7, 8]. For
example, two zip files of the same size will have the same _compressed size_
metadata, but not necessarily the same _last modification date_. More
disquieting, some metadata is _hidden_ [9, 10]: it is added during the data
acquisition stage of the file creation process, possibly without the user’s
knowledge nor agreement. Not everybody is willing to accept the automatic
presence of such information whose existence or content is not always known or
precisely documented. Examples of such metadata are presented in the next
subsection, whereas in Sect. II-C some well-known case studies where metadata
has helped to find the author of a document are shown.
### II-B Example of metadata contents
Table I shows an example of the kind of metadata that can be found in a given
picture. This metadata has been obtained using _Exiftool_ , a Perl library/CLI
application written by Phil Harvey for reading/writing metadata, that supports
many file types [6]. It is possible to deduce at least the following facts
from these data:
The last file modification of the picture is 2012:03:02, but the internal
metadata shows that the picture has been modified on 2009:06:14, so the
picture may not have been created as such. The software used is Adobe
Photoshop CS3 on Microsoft Windows, this is why there is a Photoshop thumbnail
embedded in the medium. The IPTC digest222IPTC is a filestructure/type of
metadata, like XMP is both embedded and re-calculated, allowing a simple
integrity test to be performed.
The absence of a copyright flag may indicate that the picture was not created
by a professional. The XMP toolkit is the metadata normally used by Adobe, and
the date corresponds to the release of Photoshop CS3 (April 2007). The file’s
creation date and that in the metadata are the same, which may indicate that
the picture was created and not modified subsequently. The document ID allows
the tracking of the document. The absence of history is a little bit unsusual,
it cannot help to establish a precise chronology of the image’s creation
timeline. The primary platform reinforces the hypothese of a Windows platform.
Since the thumbnail also contains its own metadata, it is an additional vector
that may compromises anonymity.
### II-C Metadata against anonymity: some case studies
Metadata is generally not purposefully inserted in order to reveal any user’s
information to unauthorized observers, it is embedded with the purpose of
enriching the media. Metadata provides additional features and services, both
to the user and the applications he or she uses. Thus, removing all the
metadata that may potentially lead to personal information leaks can appear as
excessive: to refuse concrete services that metadata offers for largely
hypothetical threats may look unreasonable. However, metadata has already been
successfully used to discover the hidden authors of unauthenticated digital
documents published on the Internet, as illustrated by the two following
examples.
In February 2012, a hacker named Higinio (w0rmer) O. Ochoa III posted a link,
on his Twitter account, to a website disclosing data taken from various law
enforcement websites. On the bottom of the website was a picture of a woman’s
breast with a message destined to mock the authorities (Fig. 1). Unfortunately
for him, the photo was taken from an iPhone, which embeds GPS coordinates
among other metadata. These coordinates led the police to identify the girl in
the picture, who was the girlfriend of the hacker. He was sentenced to 27
months of prison [11, 12].
Another well-known example of arrest using metadata is the case of the
infamous “BTK serial killer”, namely Dennis Rader [13]. Some weeks before his
arrest, he asked the police if he could communicate with them using a floppy
disk, without being traced back to a particular computer. The police answered
by posting an advertising in the local newspaper, as instructed by Rader,
saying “Rex, it will be OK”. Some weeks later, such a disk was received at a
local television station. A forensic analysis (with the EnCase forensic
software [14]) revealed an erased Word document that contained the terms
“Christ Lutheran Church”, and that was last modified by “Dennis”. Dennis Rader
was the president of the local church congregation’s council. The BTK had
taken efforts to delete identifying information from the disk, but had printed
the file in his church, because his printer was down. The police lieutenant
Ken Landwehr, head of the multiagency task force in charge of the case, said
that “this clue was a determinant factor for his arrest. If he had just quit
and kept his mouth shut, we might never have connected the dots.”
A more recent example is the capture of John McAfee, which was probably made
possible with the help of metadata forensics. Sought by the justice of Belize
for questioning about the murder of his neighbour, McAfee fled to Guatemala.
He was accompanied by two journalists from Vice magazine, who took a picture
of him during his escape. Unfortunately for him, they forgot to wipe the
metadata : GPS Altitude : 7.1 m Above Sea Level GPS Latitude : 15 deg 39’
29.40" N GPS Longitude : 88 deg 59’ 31.80" W GPS Position : 15 deg 39’ 29.40"
N, 88 deg 59’ 31.80" W.
Such information probably led to his arrest by the Guatemalian authorities for
entering the country illegally.
Even if these case studies have concerned anonymity disclosure of criminals
demanded by police officers, everybody should have in mind that such
investigations can be conducted illegally by dictatorial governments, or
illegally by non-authorized parties. The individual right to privacy is
inalienable as recalled by the UN Declaration of Human Rights, the
International Convenant on Civil and Political Rights, and many other
international and regional treaties.
## III Digital Forensics and Anonymity: a State-of-the-art
Items of digital media are more easily modifiable than traditional media. Due
to the revolutions of personal computers and the Internet, digital media is
now of widespread use and can be found everywhere. Various media manipulation
programs have been developed over the last decades, and they have been
progressively simplified and are now accessible to everyone. These programs
and afferent consequences have necessitated the emergence of digital
forensics, a new discipline that flows initially from the need to address the
challenges arising from digital media manipulation. An important goal of this
recent discipline is to (in)validate the authenticity or integrity of media,
and it usually tries to answer the two following questions [15]. Firstly, “Who
were the media producers?” Secondly, “Has the media been manipulated, is it
faked?”
Figure 1: Higinio’s girlfriend (with GPS coordinates)
Most of the multimedia forensics tools that try to addess these questions work
on the data themselves, as removing the bullet-scratch of manipulation or
creation is most of the time a difficult signal processing task only
accessible to experts [16]. In usual forensics frameworks, statistical signal
processing analysis is combined with information available through metadata,
to respond to these kind of questions [17]. The best known software in
forensic investigation, which is nowadays a common law enforcement area, is
maybe EnCase [14], already referred to in this document. Another well known
tool is called COFEE (Computer Online Forensic Evidence Extractor [18]), a
software program developed by Microsoft to conduct forensic investigations,
which became famous when it was leaked on the Internet in November 2009.
Microsoft provides devices and free technical support to law enforcement
agencies all around the world. Although the majority of forensic software is
closed-source and expensive, some others are free, like the
DEFT333http://deftlinux.net - DEFT is a GNU/Linux liveCD running Windows
forensic freeware using Wine or CAINE444http://caine-live.net - A GNU/Linux
forensic liveCD, SLEUTH555http://sleuthkit.org - A Free library and collection
of tools to investigate disk images., and so on.
Oddly, the converse problem, namely metadata anonymization, is not a well
studied area of research in digital forensics. The little that has been
investigated is about _volunteer_ anonymization, which happens at the
_attacker_ side. For instance, Symantec has explored metadata anonymization
for file system images [19], following an approach described below. Symantec
has its own file system called _VxFS_. An interesting feature of this file
system is its ability to take a snapshot of metadata (no user data is
retained), for forensic analysis purposes, debugging, or troubleshooting. If
privacy is an important concern of the client, they can demand metadata
anonymization: all folder and file names will be hashed with the SHA-1
function. Doing so is space and time efficient, and maintains the file system
integrity (the output of SHA-1 has a fixed length, so if the computed hash is
too lengthy, characters are removed, whereas it is simply processed by segment
if this latter is too short). Obviously, this rather simplistic approach can
easily be bypassed, but it prevents unintentional information leaks. For
Internet exchanges, the network utility _Wireshark_ allows an anonymization of
network packets, whose headers can contain information about the transmitter
and the receiver. For more information, the reader is referred to [20] and the
references therein.
One can regret the relative absence of tools and research in the specific
field of user’s metadata anonymization. This has been an important motivation
to develop the MAT toolkit. However, among the very few publications about
metadata anonymization, some significant research can be highlighted. One of
the most interesting papers concerns the risks and countermeasures for PDF
publication files [21]. Although the approach focuses on _Adobe Acrobat
Professional_ , this article explores a wide-range of potential attack vectors
related to metadata, and associated counter-measures. The national library of
New Zealand has also developed a tool called the “Metadata Extraction Tool
[22]” to obtain metadata from various document formats, in order to
automatically perform analysis or classification on them. It is entirely
written in Java, and released under the terms of the Apache license.
Anti-forensic measures have not yet been largely investigated. Almost all
public tools are too simplistic, when not completely broken. But some are
worth being mentioned like the MAFIA (Metasploit Anti-Forensic Investigation
Arsenal), which provides a suite of efficient tools, like Timestomp (NTFS
timestamps removal). Unfortunately, these tools are far too complex for the
non-technical user.
Type | Extension | Support | Metadata | Removal method | Remaining Metadata
---|---|---|---|---|---
Portable | | | | Hachoir |
Network | .png | Full | Textual metadata + date | or ExifTool when available | None
Graphics | | | | |
Jpeg666Joint Photographic Experts Group | Jpeg .jpeg, .jpg | Partial | Comment + exif/photoshop/adobe | Hachoir or ExifTool if available | Canon Raw tags
Open Document | .odt, .odx, .ods, … | Full | A meta.xml file contains all metadata | Removal of the meta.xml file | None
Office Openxml | .docx, .pptx, .xlsx, … | Full | A docProps folder containing | Removal of the docProps folder | None
xml metadata files |
| .pdf | Full | A lot | Rendering of the PDF file on a | None
Portable | cairo surface with the help of |
Document | poppler, exporting the surface as |
Fileformat | a PDF. Metadata added during |
| the process is removed with |
| python-pdfrw. |
Tape ARchive | .tar, .tar.bz2, .tar.gz | Full | Tar metadata, metadata from | Extracting and processing | None
compressed files themselves, | each file, recompression
metadata added by archive | in a new archive, processing
to compressed files. | of the new archive.
Zip | .zip | Partial | Zip metadata, metadata from | Extractiing and processing of | Metadata added
compressed files themselves, | each file, recompression in | by zip itself
metadata added by archive | a new archive, processing | into internal files
to compressed files. | of the new archive. |
MPEG Audio | .mp3, .mp2, | Full | id3 | Hachoir | None
.mp1, .mpa | or Mutagen when available
Ogg | .ogg | Full | Vorbis | Hachoir | None
Vorbis | | | | or Mutagen when available |
Free Lossless | .flac | Full | Flac | Mutagen | None
Audio Codec | | | Vorbis | |
TABLE II: Supported file formats
## IV Investigating Metadata
### IV-A Different types of metadata
Items of metadata are of different kinds, depending on the way they are
produced. To have a clear understanding of their composition, we propose the
following classifications.
#### IV-A1 The contextual type
Physical or logical devices that produce documents usually insert metadata
into these produced files, mostly in order to enrich them. For instance,
authors and software names are embedded in _Office_ documents, whereas author,
interpreter, composer, and track number are usually inserted into multimedia
data. They are not added to identify the user or producer, but still
compromise their privacy. These types of metadata are generally documented, or
can be easily obtained by reverse engineering. We will show that the MAT can
handle them in most cases.
#### IV-A2 The watermark type
Some multimedia files embed a watermark directly in their data, usually for
copyright reasons. If this data reveals no information about the user, it is
categorized into this “watermark type”. The authors are not aware of any case
when information susceptible to revealling the users identity was embedded as
a watermark
#### IV-A3 The fingerprinting type
For transmission of confidential documents, _fingerprinting_ metadata (also
known as a _tattoos_) is often embedded inside the document, to enable
traceing back an eventual leaker. Contrary to the watermark type, this
metadata potentially reveals some information about the file user. Due to its
final purpose, this type of metadata is really difficult to detect and even
more difficult to remove without breaking file integrity. There are two sorts
of fingerprinting metadata: robust, and fragile.
##### Robust fingerprinting
The goal of robust fingerprinting is to insert hidden information that resists
content modification such as format conversion or resizing. Robust
fingerprinting is usually designed to be only removable by modifying the
content enough that it becomes completely unusable.
##### Fragile fingerprinting
The fragile type of fingerprinting is usually used to guarantee the integrity
of a document. The slightest modification of the file will break the
watermark. The goal is to prove that the file has not been altered or
modified.
Most of the proposed fingerprint removal solutions require a large database of
watermarked/non-watermarked files. They imply complex statistical signal
processing methods coupled with artificial intelligence tools like massive
fine-tuned learning machines.
The removal of a fingerprint from a single document remains an open problem,
which is out of the scope of this paper.
## V Threat model
The Metadata Anonymisation Toolkit adversary has a number of goals,
capabilities, and counter-attack types that can be used to guide us towards a
set of requirements for the MAT.
### V-A Adversary
#### V-A1 Goals
* •
Identifying the source of the document, since a document always has one.
Who/where/when/how was a picture taken, where was the document leaked from and
by whom, …
* •
Identify the author; in some cases documents may be anonymously authored or
created. In these cases, identifying the author is the goal.
* •
Identify the equipment/software used. If the attacker fails to directly
identify the author and/or source, his next goal is to determine the source of
the equipment used to produce, copy, and transmit the document. This can
include the model of camera used to take a photo, or which software was used
to produce an office document.
#### V-A2 Positioning
* •
The adversary created the document specifically for this user. This is the
strongest position for the adversary to have. In this case, the adversary is
capable of inserting arbitrary, custom watermarks specifically for tracking
the user. In general, MAT cannot defend against this adversary, but we list it
for completeness.
* •
The adversary created the document for a group of users. In this case, the
adversary knows that they attempted to limit distribution to a specific group
of users. They may or may not have watermarked the document for these users,
but they certainly know the format used.
* •
The adversary did not create the document, the weakest position for the
adversary to have. The file format is (most of the time) standard, nothing
custom is added: MAT should be able to remove all meta-information from the
file.
### V-B Requirements
#### V-B1 Processing
* •
The MAT should avoid interactions with information. Its goal is to remove
metadata, and the user is solely responsible for the information of the file.
* •
The MAT must warn when encountering an unknown format. For example, in a
zipfile, if MAT encounters an unknown format, it should warn the user, and ask
if the file should be added to the anonymized archive that is produced.
* •
The MAT must not add metadata, since its purpose is to anonymize files: every
added items of metadata decreases anonymity.
* •
The MAT must handle unknown/hidden metadata fields, like proprietary
extensions of open formats.
## VI Privacy and Anonymity Policy Implemented in the MAT
### VI-A Technical aspects
MAT stands for Metadata Anonymization Toolkit. It is designed to improve
anonymity of files published online. It consists of an extensible library, a
Command Line Interface (CLI), and a Graphic User Interface (GUI). The MAT
suite aims at providing, within the reach of anyone, software dedicated to
listing and removing metadata; for portability purposes, it is entirely
written in _Python_ [23], and is based on the _Hachoir_ library [24].
### VI-B Security and anonymity policy
To offer a reliable tool for metadata removal, one must first determine
criteria to decide whether a given field must be considered harmful or not.
This raises the question of choosing a security and anonymity policy. The
strategy used by the _MAT_ is to process all the metadata that can be removed:
any piece of the file that (1) is not a data, and (2) can be removed, is
considered as a threat and so is deleted.
This may seems rough, but it appears to the author of MAT as the best
solution: categorizing any possible metadata of every handled format is, on
the one hand, very subjective (for instance, is the _gid_ field of a file a
compromising metadata?), and on the other hand it is an intractable task in
practice. Additionally, doing so leaves the least possible amount of metadata
to the attacker. Even if the absence of any metadata also provides
information, the quantity of information leaked by this absence is obviously
lower than the quantity provided by remaining metadata.
### VI-C White list approach
Since the _MAT_ handles “usual” file formats, they are often documented ones,
even in the case of closed formats. Thus, a white list approach is possible.
Because the format structure is known, each unknown field is a non-standard
one that can be safely removed without breaking the file integrity.
A counterpart of this approach is that some information loss may occur if the
file is not well-documented, or if it has been saved using a non standard
extension. For example, Adobe use their own extension for PDFs. So files
produced with Adobe products, processed by the MAT, and finally read with any
Adobe products, may lose information during this process.
Unfortunately, some closed formats are too complex to be completely understood
by a reverse engineering study, and so the MAT library cannot handle them.
This is the case for the Microsoft Office pre-2003 formats ($.doc$, $.ppt$,
and so on), which are known to be complex and whose design is often reported
as disputable.
### VI-D Field Anonymization
Metadata fields are suppressed whenever possible. Otherwise, numerical data is
set to 0, dates to Epoch, and strings to an empty string. Filling fields with
random values or real-looking ones may seem to make sense. But this is a poor
strategy, as producing false data is not harmless. It requires an ad-hoc
algorithm that could be traced back to the owner or designer of this tailored
anonymizing algorithm. Additionally, potential input data of this algorithm
(seeds, PRNG parameters, and so on) can accidentally leak information.
Indeed, except the cryptographically secure ones (like ISAAC or BBS, see [25,
26, 27, 28]), all the commonly used pseudorandom number generators (PRNGs) are
quite biased. Several attacks on common PRNGs have been reported [29], making
it possible to determine the algorithm used, and even in some cases the seed,
when considering a sufficiently large sample of pseudorandom numbers generated
by an algorithm. So, using pseudorandom values instead of empty strings may
lead to a successful forensics attack, and to the discovery of some tools used
for this anonymization, revealing information on the user. Furthermore, a
simple source code inspection could allow an attacker to deduce all possible
values that the _MAT_ can generate, allowing either brute force or
probabilistic attacks. Contrarily, removing everything that is possible to
remove is fairly safe.
### VI-E Surface Rendering for PDF
The PDF format is quite complex: its specification document is more than 1300
pages long. Furthermore major PDF producers have developed their own
extensions. This is why such a format should be carefully cleaned part by
part.
Of course, usual metadata is well documented. But this format is so rich that
it is possible to embed virtually anything into a PDF: text, pictures,
javascript, or even videos. This is why the _MAT_ uses a clever trick to
handle PDF: rendering on cairo’s PDF surface. The rendering process is
comparable to a print: hyperlinks are broken, videos too, invisible metadata
is removed, which drastically reduces the risk of leaking the author’s
information.
### VI-F Supported formats and implemented features
The _MAT_ supports most of the “usual” formats, from pictures to Office
documents (see Table II for an exhaustive list of supported formats).
Concerning archive formats (noting that most of the Office documents are
zipped XML files, and thus these formats belong into this category), metadata
mostly exists in a simple file folder, which is easy to remove. Audio format
processing relies on Mutagen when available, and images processing relies on
Exiftool, again when available.
### VI-G The MAT output
The MAT’s output is intentionally minimal, since it is intended for non-
technical people. The goal of the "metadata listing" functionality is to give
a global view of present compromising metadata. Since this is the same picture
as the one studied previously, we can recognize some patterns.
## VII Conclusion and Future Work
This paper is a first step in the direction towards document metadata
processing for anonymity and privacy. Based on a study of current trends in
digital media forensics, the problem of metadata removal has been pointed out
as a potential vulnerability. A thorough analysis of different metadata has
allowed to propose a categorization in two criteria: usefulness for media
integrity, and anonymity and privacy threats. Finally, technical choices and
their implementation have been presented.
Potential future improvements include dealing with fingerprints, removing
sensor “bullet scratch” from digital media, and processing more file formats.
The major short term improvement for the next version of the MAT is the
handling of metadata related to file-system-timestamps. New file formats
should be supported in the near future. Furthermore, until now, the _MAT_ is
neither able to detect nor to remove fingerprints embedded in digital media.
This might not seem useful, as the insertion of a watermark is usually a
choice of user. However, more and more camera models insert, in every captured
image, a fingerprint which identifies camera brand and model (see [9, 10] and
therein references).
## References
* [1] Google, “Gsoc, the google summer of code,” http://www.google-melange.com/gsoc/homepage/google/gsoc2011, Jan. 2012.
* [2] Tor, “Tor, the onion router,” https://www.torproject.org/, Jan. 2012.
* [3] Tails, “Tails,” https://tails.boum.org, September 2012.
* [4] [Online]. Available: \url{https://tails.boum.org/news/new_project_name/index.en.html}
* [5] [Online]. Available: \url{https://anonymous-proxy-servers.net/en/jondo-live-cd.html}
* [6] P. Harvey, “Exiftool,” http://nschloe.blogspot.fr/2009/06/bibtex-how-to-cite-website_21.html, September 2012.
* [7] F. Buchholz and E. Spafford, “On the role of file system metadata in digital forensics,” _Digital Investigation_ , vol. 1, no. 4, pp. 298–309, 2004.
* [8] A. Castiglione, A. Santis, and C.Soriente, “Taking advantages of a disadvantage: Digital forensics and steganography using document metadata,” _Journal of Systems and Software_ , vol. 80, no. 5, pp. 750–764, 2007.
* [9] J.Ohmura and A. Kawasaki, “Digital camera capable of embedding an electronic watermark into image data,” US Patent 6 963 363, 2005.
* [10] B. Davis, G. Rhoads, and W. Conwell, “Authenticating metadata and embedding metadata in watermarks of media signals,” US Patent 7 209 571, 2007.
* [11] Dailymail, “Busted! fbi led to anonymous hacker after he posts picture of girlfriend’s breasts online,” http://www.dailymail.co.uk/news/article-2129257/Higinio-O-Ochoa-III-FBI-led-Anonymous-hacker-girlfriend-postspicture-breasts-online.html, April 2012.
* [12] CSI, “Embedded data, not breasts, brought down hacker,” http://www.csoonline.com/article/705170/embedded-data-not-breasts-brought-down-hacker, April 2012.
* [13] M. Hansen, “How the cops caught btk,” Abajournal, April 2006. [Online]. Available: http://www.abajournal.com/magazine/article/how_the_cops_caught_btk/
* [14] G. Software, “Encase forensic.” [Online]. Available: http://www.guidancesoftware.com/encase-forensic.htm
* [15] J. Redi, W. Taktak, and J.-L. Dugelay, “Digital image forensics: a booklet for beginners,” _Multimedia Tools and Applications_ , vol. 51, pp. 133–162, 2010\.
* [16] T. Lanh, K.-S. Chong, S. Emmanuel, and M. Kankanhalli, “A survey on digital camera image forensic methods,” in _IEEE International Conference on Multimedia and Expo_ , July 2007, pp. 16–19.
* [17] H. Blitzer and J.Jacobia, “Forensic digital imaging and photography.”
* [18] Microsoft, “Cofee , the computer online forensic evidence extractor tool.” [Online]. Available: https://cofee.nw3c.org/
* [19] Symantech, “Anonymizing filesystem metadata for analysis,” 2006.
* [20] Sharkfest’11, “A-11: Trace file anonymisation,” 2011.
* [21] N. Agency, “Hidden data and metadata in adobe pdf files: Publication risks and countermeasures,” Symantech, Tech. Rep., July 2008.
* [22] “Metadata extraction tool,” The National Library of New Zealand. [Online]. Available: \url{http://meta-extractor.sourceforge.net/}
* [23] “Python.” [Online]. Available: \url{http://www.python.org/}
* [24] “Hachoir.” [Online]. Available: \url{https://bitbucket.org/haypo/hachoir/wiki/Home}
* [25] Jenkins, “ISAAC,” in _IWFSE: International Workshop on Fast Software Encryption, LNCS_ , 1996.
* [26] L. Blum, M. Blum, and M. Shub, “A simple unpredictable pseudo-random number generator,” _SIAM Journal on Computing_ , vol. 15, pp. 364–383, 1986.
* [27] J. Bahi, X. Fang, C. Guyeux, and Q. Wang, “Evaluating quality of chaotic pseudo-random generators. application to information hiding,” _IJAS, International Journal On Advances in Security_ , vol. 4, no. 1-2, pp. 118–130, 2011.
* [28] J. Bahi, J.-F. Couchot, C. Guyeux, and Q. Wang, “Class of trustworthy pseudo random number generators,” in _INTERNET 2011, the 3-rd Int. Conf. on Evolving Internet_ , Luxembourg, Luxembourg, Jun. 2011, pp. 72–77.
* [29] D. Arroyo, G. Alvarez, and V. Fernandez, “On the inadequacy of the logistic map for cryptographic applications,” _X Reunión Española sobre Criptología y Seguridad de la Información (X RECSI)_ , vol. 1, pp. 77–82, 2008\.
|
arxiv-papers
| 2012-12-15T01:37:10 |
2024-09-04T02:49:39.304146
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Julien Voisin, Christophe Guyeux, Jacques M. Bahi",
"submitter": "Julien Voisin Mr",
"url": "https://arxiv.org/abs/1212.3648"
}
|
1212.3707
|
# Minimum Energy-Surface Required by Quantum Memory Devices
Wim van Dam [email protected] Department of Computer Science, Department of
Physics, University of California, Santa Barbara, CA 93106-5110, USA Hieu D.
Nguyen [email protected] Department of Physics, University of California,
Santa Barbara, CA 93106-5110, USA
###### Abstract
We address the question what physical resources are required and sufficient to
store classical information. While there is no lower bound on the required
energy or space to store information, we find that there is a nonzero lower
bound for the product $P=\langle E\rangle\langle r^{2}\rangle$ of these two
resources. Specifically, we prove that any physical system of mass $m$ and $d$
degrees of freedom that stores $S$ bits of information will have lower bound
on the product $P$ that is proportional to $d^{2}/m(\exp(S/d)-1)^{2}$. This
result is obtained in a non-relativistic, quantum mechanical setting and it is
independent from earlier thermodynamical results such as the Bekenstein bound
on the entropy of black holes.
###### pacs:
pacs numbers
## I Introduction.
Although information may seem abstract and elusive, it has made a number of
surprisingly concrete connections to physics. Originally prompted by Maxwell’s
demon, the link of information to thermodynamics has been made lucid by the
works of Bennett and Landauer (see dem90 for complete coverage). Physicists
and computer scientists alike have come to recognize that information does not
exist as an independent entity but is encoded in physical devices, and,
therefore, “Information is physical” has become a new mantra lan96 . Moreover,
the technological trend from the last few decades continually has been to
compress more information in less space hdd . Due to Moore’s law we sense that
we are near the end of the classical regime and entering into the quantum
regime where exciting possibilities and challenges abound. Thus, to understand
the ultimate limits to information storage based on the quantum laws of
physics may be of intellectual as well as practical significance. In this
paper we seek a general and robust result concerning information storage using
non-relativistic quantum theory. Abstracting away the details of hardware, we
compute the absolute minimal amount of physical resources required to store
information, and the result further illustrates its physical nature.
Typically, information is stored by preparing a device in a number of
different stationary states corresponding to different messages. The device is
effectively described by a mixed state of a certain entropy, and looking at a
number of simple devices such as particle in a box, harmonic oscillator, and
the hydrogen atom, we notice an interesting interplay among energy, surface
area, mass, and entropy (see Table 1). Keeping the constituent particles’
masses the same, one could save on energy or size but never both. For
instance, the particle-in-a-box model requires the same amount of space
regardless of the number of states used in the encoding but the energy
increases with the number of states. The hydrogen atom is an example of the
opposite extreme. There is an infinite number of states in a finite energy
window, roughly equal to $13.6$ eV, but the spatial size of the orbitals
increases without bound. However, if we could change the mass, then both
energy and space can be reduced arbitrarily. Since a number of variables could
affect the cost, it is necessary to hone in on a precise notion of the cost of
information storage. We will do so in the next section and determine the
optimal device based on a number of clearly stated plausible hypotheses. Then,
we proceed to compute the minimum cost and prove a lemma which is needed in
the calculation. Our main result is a Heisenberg-like uncertainty principle
governing energy, space, and entropy. We conclude with a comparison to rough
estimates existing in the literature as well as some thoughts upon its
implications for future technological advances.
System | | Energy | Surface Area | Product
---|---|---|---|---
| | | |
Particle of mass $m$ in a 1d box of width $L$ | | $\frac{\hbar^{2}\pi^{2}}{mL^{2}}\langle n^{2}\rangle$ | $L^{2}$ | $\frac{\hbar^{2}}{m}\langle n^{2}\rangle$
| | | |
Harmonic oscillator of mass $m$ and natural frequency $\omega$ | | $\hbar\omega\langle n\rangle$ | $\frac{\hbar}{m\omega}\langle n\rangle$ | $\frac{\hbar^{2}}{m}\langle n\rangle^{2}$
| | | |
Electronic states of Hydrogen atom | | $\frac{e^{4}m}{\hbar^{2}}\langle\frac{1}{n^{2}}\rangle$ | $\frac{\hbar^{4}}{m^{2}e^{4}}\langle n^{4}\rangle$ | $\frac{\hbar^{2}}{m}\langle n^{4}\rangle\langle 1/n^{2}\rangle$
| | | |
Table 1: A system is prepared in different stationary states, labelled by $n$,
according to a probability distribution whose Shannon entropy is the desired
amount of encoded information. Quoted are order-of-magnitude estimates of the
average energy and surface area, denoted by $\langle\cdot\rangle$, for a
number of simple systems sakurai . Averages such as $\langle n\rangle$,
$\langle n^{2}\rangle$, and $\langle n^{4}\rangle$ grow with increasing
entropy. Therefore, although energy and surface area show different behaviours
for different systems, their products invariably grow with entropy.
## II Definition of cost
Consider storing information in a device described by the Hamiltonian
$H_{\text{dev}}=-\frac{\hbar^{2}}{2m}\sum_{i=1}^{d}\frac{d^{2}}{dx_{i}^{2}}+V(\vec{x}),$
(1)
by preparing it in a stationary (energy eigenstate) $\Psi_{\alpha}$ with
probability $p_{\alpha}$. The amount of information thus encoded is
$S=-\sum\limits_{\alpha}p_{\alpha}\log p_{\alpha}$ (2)
and the device will be in the state
$\rho_{\text{dev}}=\sum\limits_{\alpha}p_{\alpha}\mathinner{|\Psi_{\alpha}\rangle}\mathinner{\langle\Psi_{\alpha}|}.$
(3)
We consider the average energy and size associated with such a mixed state the
cost in physical resources required by the device. More specifically, let
$\operatorname{Tr}{\rho_{\text{dev}}r^{2}}$ be the measure of spatial cost
where
$r^{2}:=\sum\limits_{i=1}^{d}x_{i}^{2}.$ (4)
To prevent arbitrary shifts due to displacements of the coordinate origin, we
require for all $i\in\\{1,\dots,d\\}$ that
$\operatorname{Tr}{\rho_{\text{dev}}x_{i}}=0$. Moreover, let
$\operatorname{Tr}{\rho_{\text{dev}}H_{\text{dev}}}$ be a measure of the cost
in energy, and similarly, to prevent meaningless shifts in the Hamiltonian, we
require the ground state (g.s.) energy to be zero,
$\langle H_{\text{dev}}\rangle_{\text{g.s.}}=0$ (5)
The cost of information storage specific to using $H_{\text{dev}}$ is the
joint product of energy and space
$C_{\text{dev}}:=\min_{\rho_{\text{dev}}}\operatorname{Tr}{\rho_{\text{dev}}H_{\text{dev}}}\operatorname{Tr}{\rho_{\text{dev}}r^{2}},$
(6)
subject to the constraint of Equation (2). Since the cost in Equation (6) is
specific to a device, to find the cost of information storage we need to
minimize over all devices having the same number of degrees of freedom and
particles’ mass as follows:
$P:=\min_{V(x)}C_{\text{dev}}$ (7)
over potentials that satisfy (5).
We choose to express the intuitive notion of cost in energy and space by
considering linear power in energy and quadratic in length, (6). One could
potentially capture the same intuition through other powers of the average $H$
and $r$ such as $\langle H^{s_{1}}\rangle\langle r^{s_{2}}\rangle$. Putting
the exponents inside or outside of the average may change the end result
slightly but should convey the same physics. Since the only dimensionful
parameters in our problem are $\hbar$ and $m$, if the cost is measured in
units of $J^{s_{1}}l^{s_{2}}$, where $J$ and $l$ stand for Joules and meters,
then the answer must be expressed in powers $t_{1}$ of $\hbar$ and $t_{2}$ of
$m$ such that
$(J\cdot s)^{t_{1}}kg^{t_{2}}=J^{s_{1}}l^{s_{2}},$ (8)
where $s$ and $kg$ stand for seconds and kilograms. Solving this equation
leads to these constraints on the exponents: $t_{1}=2s_{1}$, $t_{2}=-s_{1}$,
and $s_{2}=2s_{1}$. The first two equations guarantee that other powers of
energy and space admit a solution and therefore are also valid, but, as
implied by the last equation, they all are powers of the basic combination of
linear in energy and quadratic in space.
## III Optimal potential
For the purpose of computing a lower bound of $P$, we can relax the condition
that the mixed state be diagonal in the eigenstates of $H_{\text{dev}}$ and
consider instead the following functional of $V(\vec{x})$
$C[V(\vec{x})]:=\min\limits_{\rho}\left\\{\operatorname{Tr}{\rho
H_{\text{dev}}}\operatorname{Tr}{\rho r^{2}}\right\\},$ (9)
minimized over all density matrices satisfying the entropy condition
$S=-\operatorname{Tr}{\rho\log\rho}.$ (10)
Obviously, for a given $V(x)$,
$C_{\text{dev}}\geq C[V(x)]$ (11)
Hence,
$P\geq\min\limits_{V(x)}C[V(x)]$ (12)
with $V(x)$ subject to the same constraint of Equation $\eqref{eq:pot-const}$.
Let us start by observing that $C[V(\vec{x})]$ is rotationally invariant.
Under a coordinate rotation $\vec{y}=R\vec{x}$, with $R\in SO(d)$, we have
$\displaystyle C[V(R\vec{x})]$ (13)
$\displaystyle=\min\limits_{\rho}\left\\{\operatorname{Tr}{\rho\left(-\frac{\hbar^{2}}{2m}\sum_{i=1}^{d}\frac{d^{2}}{dx_{i}^{2}}+V(R\vec{x})\right)}\operatorname{Tr}{\rho
r^{2}}\right\\}$ (14)
$\displaystyle=\min\limits_{\rho}\left\\{\operatorname{Tr}{\rho\left(-\frac{\hbar^{2}}{2m}\sum_{i=1}^{d}\frac{d^{2}}{dy_{i}^{2}}+V(\vec{y})\right)}\operatorname{Tr}{\rho
r^{2}}\right\\}$ (15) $\displaystyle=C[V(\vec{x})]$ (16)
The first equality comes from applying the definition in Equation
$~{}\eqref{eq:defn-c}$; the second comes from using the $y$-coordinates
instead of $x$ and rotational invariance of kinetic energy and $r^{2}$;
finally, the third comes from applying the definition again. Thus, if we
assume a minimizing potential $V(\vec{x})$ exists, then $C[V(R\vec{x})]$ must
also be minimal by rotational invariance. Moreover, if we assume the optimal
potential to be also unique, then $V(R\vec{x})=V(\vec{x})$. Hence, by assuming
existence and uniqueness, we conclude the optimal potential must be a function
of only the hyper-radius $r$. As we try to find the potential which minimizes
$C[V(\vec{x})]$, we have in the problem only the parameters $\hbar$, $m$, and
$r$ to construct our optimal potential. By dimensional analysis one is led to
conjecture the solution to be $V(r)=-W\frac{\hbar^{2}}{2mr^{2}}$, where $W$ is
a dimensionless number possibly depending on $S$. The larger $W$, that is the
more attractive the potential, the smaller the orbitals and total energy, thus
allowing us to save on the total cost without compromise. However, the
attractive inverse square potential for $W>(1-d/2)^{2}$ is unstable as it has
negative infinity ground state energy, .i.e. “fall to the center” kw74 ; gr93
; sh31 ; landau . Thus, the optimal potential occurs at the maximum allowed
value of $W$ before it collapses, namely
$H_{\text{opt}}=-\frac{\hbar^{2}}{2m}\sum_{i=1}^{d}\frac{d^{2}}{dx_{i}^{2}}-\Big{(}1-\frac{d}{2}\Big{)}^{2}\frac{\hbar^{2}}{2mr^{2}}$
(17)
It remains a conjecture for the rest of our paper that $H_{\text{opt}}$ yields
the smallest cost possible.
## IV Minimum cost
Since $H_{\text{opt}}$ minimizes $C[V(x)]$, by Equations (9) and (12), we have
$P\geq\min\limits_{\rho}\left\\{\operatorname{Tr}{\rho
H_{\text{opt}}}\operatorname{Tr}{\rho r^{2}}\right\\}.$ (18)
To evaluate the right hand side, let us introduce $\kappa>0$ and define
$\tilde{C}:=\min\limits_{\rho}\operatorname{Tr}{\rho\left(H_{\text{opt}}+\frac{\kappa}{2}r^{2}\right)}$
(19)
with $\rho$ subject to the entropy constraint Equation $~{}\eqref{eq:entr-
const}$. The auxiliary variable $\kappa$ allows us to convert the cost in
space into an energy quantity so that in a single expression, $\tilde{C}$ is
the joint costs of energy and space in the form of a sum. As to be shown in
the subsection, we have
$\tilde{C}\geq\hbar\sqrt{\frac{\kappa}{m}}d(\exp{(S/d)}-1)$ (20)
Hence, for the mixed state obtaining the minimum of the right hand side in
Equation (19), we have
$\operatorname{Tr}{\rho
H_{\text{opt}}}\geq\hbar\sqrt{\frac{\kappa}{m}}d(\exp{(S/d)}-1)-\frac{\kappa}{2}\operatorname{Tr}{\rho
r^{2}},$ (21)
which yields
$\displaystyle\operatorname{Tr}{\rho H_{\text{opt}}}\operatorname{Tr}{\rho
r^{2}}\geq$
$\displaystyle\sqrt{\kappa}\left(\hbar\frac{d}{\sqrt{m}}(\exp{(S/d)}-1)-\frac{\sqrt{\kappa}}{2}\operatorname{Tr}{\rho
r^{2}}\right)\operatorname{Tr}{\rho r^{2}}$
for all non-negative values of $\kappa$. Since the right hand side is
quadratic in $\sqrt{\kappa}$, at the critical $\kappa$ the inequality yields
$\operatorname{Tr}{\rho H_{\text{opt}}}\operatorname{Tr}{\rho
r^{2}}\geq\frac{\hbar^{2}}{2m}d^{2}(\exp{(S/d)}-1)^{2},$ (22)
which is now free of the introduced variable $\kappa$. Because of inequality
(18), the cost of storing information satisfies the bound
$P\geq\frac{\hbar^{2}}{2m}d^{2}(\exp{(S/d)}-1)^{2},$ (23)
which is our main result.
### IV.1 Central Lemma
We now proceed to prove lemma (20). Simplifying the dimensionful quantities in
$\tilde{C}$ yields
$\tilde{C}=\hbar\sqrt{\kappa/m}\min\limits_{\rho}\operatorname{Tr}{\rho H},$
(24)
where
$H:=\frac{1}{2}\sum_{i=1}^{d}-\frac{d^{2}}{dq_{i}^{2}}-\frac{W}{q^{2}}+q^{2}$
(25)
is dimensionless as is $\vec{q}:=\vec{x}(m\kappa)^{1/4}$. Let
$\rho=\sum\limits_{i}p_{i}\mathinner{|\Psi_{i}\rangle}\mathinner{\langle\Psi_{i}|}$
(26)
with eigenvalues $p_{i}$ and $\Psi_{i}$ orthogonal and not necessarily
eigenstates of $H$. Thus, we would like to minimize
$\sum\limits_{i}p_{i}\left|\mathinner{\langle\Psi_{i}|}H\mathinner{|\Psi_{i}\rangle}\right|^{2}$
by varying $\mathinner{|\Psi_{i}\rangle}$ and $p_{i}$ subject to the entropy
condition (10). Obviously, we would have the smallest sum possible if
corresponding to the highest $p_{i}$ we have the smallest
$\left|\mathinner{\langle\Psi_{i}|}H\mathinner{|\Psi_{i}\rangle}\right|^{2}$
possible. Namely, it should be the ground state of $H$. Similarly, the next
most frequently used $\Psi_{i}$ should be that which yields the next least
expectation value of $H$. That is, it should be the first excited state.
Continuing this argument, we see that $\Psi_{i}$ are eigenstates of $H$ with
some yet to be determined distribution $p_{i}$. Optimization over $p_{i}$ with
the constraints of entropy and normalization can be treated with Lagrange
multipliers in a manner identical to that typically found in textbooks, thus
resulting in the Boltzmann distribution. In other words, the minimizing
density matrix of Equation (24) is diagonal in the eigenstates of $H$ with
$p_{i}\propto\exp{(-\beta E_{i})}$ with $E_{i}$ being eigen-energies of $H$
and $\beta$ a constant depending on $S$. The spectrum of $H$ can be computed
by casting the Laplacian into hyper-spherical coordinates and solving the
ensuing hypergeometric differential equation as in kw74 . Eigenstates are
characterized by two quantum numbers $n,l=0,1,2,\ldots$ and have energy
$E(n,l)=2n+\sqrt{l(l+d-2)},$ (27)
with degeneracy
$g(l)=\frac{(d+2l-2)(d+l-3)!}{l!(d-2)!}$ (28)
Thus, Equation (24) becomes
$\tilde{C}=\hbar\sqrt{\kappa/m}\frac{1}{Z}\sum\limits_{n=0}\sum\limits_{l=0}\exp{(-\beta
E(n,l))}g(l)E(n,l)$ (29)
where
$Z:=\sum\limits_{n=0}\sum\limits_{l=0}g(l)\exp{(-\beta E(n,l))}.$ (30)
Moreover, the separation of $n$ and $l$ in the eigen-energy equation allows
the partition function to be factorized
$Z=\underbrace{\sum\limits_{n=0}\exp{(-2\beta
n)}}_{Z_{n}}\underbrace{\sum\limits_{l=0}\exp{(-\beta\sqrt{l(l+d-2)})}g(l)}_{Z_{l}}$
(31)
Thus, although in Equation (24) $\tilde{C}$ in our context means the cost of
energy and space, it can also be interpreted as the internal energy of a
thermodynamic system composed of two uncoupled sub-systems $n$ and $l$. Let
$U_{n}$, $U_{l}$, $S_{n}$, and $S_{l}$ be internal energies and entropies of
the $n$ and $l$ subsystems respectively, and we have
$\displaystyle\tilde{C}$
$\displaystyle=\hbar\sqrt{\frac{\kappa}{m}}\left(U_{n}+U_{l}\right)\quad\mbox{and}$
(32) $\displaystyle S$ $\displaystyle=S_{n}+S_{l},$ (33)
Because $Z_{n}$, being a geometric sum, can be computed in closed form, we get
the following exact results for the thermodynamic quantities
$\displaystyle\beta$
$\displaystyle=\frac{1}{2}\log\left(1+\frac{2}{U_{n}}\right)$ (34)
$\displaystyle S_{n}$
$\displaystyle=\log\left(1+\frac{U_{n}}{2}\right)+\frac{U_{n}}{2}\log\left(1+\frac{2}{U_{n}}\right).$
(35)
While the $n$ sum is trivial, we can make progress with the $l$-sum only when
making approximations in certain regimes. In the following we will work in the
limits $d\gg 1$ and $\beta\ll 1$, which corresponds to high entropy. Using
Stirling’s formula and the method of steepest descent to compute the integral
version of the discrete sum, we get
$Z_{l}\approx 2\beta^{-d+1}$ (36)
with the error governed by
$|\log Z_{l}-\log
2\beta^{-d+1}|=\mathcal{O}\left(\frac{\beta}{d}\right)\quad\text{(See figure
1)}$ (37)
By the canonical equations yielding internal energy and entropy from the
partition function, we then obtain
$\displaystyle U_{l}$
$\displaystyle=\frac{d-1}{\beta}+\mathcal{O}\left(\frac{1}{d}\right)$ (38)
$\displaystyle S_{l}$
$\displaystyle=(d-1)\left(\log\frac{U_{l}}{d-1}+1\right)+\mathcal{O}\left(\frac{\beta}{d}\right)$
(39)
Since we work in the regime $\beta\ll 1$, Equation $~{}\eqref{eq:betaUn}$
implies $U_{n}\gg 1$. Hence, keeping only the dominant terms in the
$n$-subsystem, we obtain
$\displaystyle U_{n}$
$\displaystyle=\frac{1}{\beta}-1+\mathcal{O}\left(\beta\right)$ (40)
$\displaystyle S_{n}$
$\displaystyle=\log\left(1+\frac{U_{n}}{2}\right)+\mathcal{O}\left(1\right)$
(41)
When we combine the above results for the separate sub-systems and keep only
the dominant terms, we get
$\displaystyle\tilde{C}$
$\displaystyle=\hbar\sqrt{\frac{\kappa}{m}}\left(dU_{n}-(d-1)+\mathcal{O}\left(1/d\right)\right)\quad\mbox{and}$
(42) $\displaystyle S$ $\displaystyle=d\log U_{n}+\mathcal{O}\left(1\right),$
(43)
We now obtain the equation relating the amount of encoded information to the
associated cost,
$\frac{\tilde{C}}{d}=\hbar\sqrt{\frac{\kappa}{m}}\left(\exp{(S/d)}-\frac{d-1}{d}+\mathcal{O}\left(1/d^{2}\right)\right),$
(44)
which yields lemma (20) in the limit of large $d$.
(a) Error Figure 1: We computed $\log Z$ in two different ways for $d=10$. The
$\log Z$-plot is the result of carrying out the sum in Equation
$~{}\eqref{eq:part_sep}$ to $1600$ terms at which it seems stable. The
approximation-plot is Equation $~{}\eqref{eq:part_approx}$ resulting from the
method of steepest descent. The error plot shows the difference $\log Z-\log
2\beta^{-d+1}$, which is of the order $\beta/d$.
## V Conclusion.
Using non-relativistic quantum theory, we find the minimal physical resources
needed to store information based on a number of plausible hypotheses.
Defining the cost in the form of a product of energy and space, we proceed to
find the interaction potential that minimizes the cost when storing a given
amount of information. We assume such a potential exists and is rotationally
invariant, which also follows if assuming uniqueness of the optimal potential.
As our main result we arrive at Equation $~{}\eqref{eq:mainresult}$, which
provides a lower bound to any device’s energy and surface area when it encodes
$S$ amount of information in $d$ degrees of freedom in the limits $d,S\gg 1$.
According to our intuition we expect information to require no energy and
space, and this only appears so because the factor $\hbar/m$ is extremely
small in the classical regime. For comparison with some existing rough
estimates in the literature, let us consider a spherical memory device made up
of ordinary matter of total mass $1$ kilogram and volume $1$ litter. Since the
energy in our result is not the rest-mass energy but the excitation energies
relative to the ground, let us take the energy to be the ionization energy of
typical atoms, which is $10$ eV. Thus, our memory looks like a soup of many
electrons at a macroscopic distance from a positively charged center. Our
inequality yields $S/d\approx 20$, about $20$ bits per atom, which agrees with
Bekenstein’s estimate for electronic levels but falls far short compared to
$10^{6}$ when energy is taken to be the rest-mass energy bek84 . Hence, the
whole device can hold about $10^{31}$ bits, which is the same as Lloyd’s
ultimate laptop, llo00 ; llo02 . Moreover, the cost is proportional to the
number of degrees of freedom but exponential in the information density,
$S/d$, thus showing analog storage to be much more expensive than digital.
Given a device storing one bit of information, analog storage would keep the
same device and use its remaining internal states to store additional bits,
and this is shown to be exponentially more expensive than bringing in more
copies of the same device, which is digital storage. Our current state-of-the-
art technology stores gigabytes of information in a macroscopic device
consisting of an Avogadro number of entities; that is, we are operating at the
low density regime, $S/d\approx 10^{-14}$. As we try to create smaller devices
with greater information capacity, we are pushing $S/d$ to greater values.
However, our result implies this endeavour will become infeasible due to the
exponential scaling.
#### Acknowledgments.
This material is based upon work supported by the National Science Foundation
under Grant No. 0917244.
## References
* (1) _Maxwell’s Demon: Entropy, Information, Computing_ , ed. H.S. Leff, et al (1990).
* (2) ewR. Landauer, Phys. Lett. A $\mathbf{217}$, 188-193, (1996).
* (3) R.F. Freitas, W.W. Wilcke, IBM J. Res. Dev. $\mathbf{52}$, 439-447 (2008).
* (4) J.J. Sakurai, _Modern Quantum Mechanics_ (Addison-Wesley Publishing Company, Inc., 1994.
* (5) K. B. Wolf, J. of Math. Phys. $\mathbf{15}$, 2102 (1974).
* (6) K.S. Gupta and S.G. Rajeev, Phys. Rev. D. $\mathbf{48}$, 5940 (1993).
* (7) G. H. Shortley, Physical Review, $\mathbf{38}$, 120 (1931).
* (8) L.D. Landau _et al._ , $\mathit{QuantumMechanics}$(Pergamon Press, Inc., New York, 1958), Sec. 35.
* (9) J. D. Bekenstein, Phys. Rev. D $\mathbf{30}$, 1669-1679 (1984).
* (10) S. Lloyd, Nature $\mathbf{406}$, 1047-1054 (2000).
* (11) S. Lloyd, Phys. Rev. Lett. $\mathbf{88}$, 237901 (2002).
|
arxiv-papers
| 2012-12-15T17:45:12 |
2024-09-04T02:49:39.313644
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wim van Dam, Hieu D. Nguyen",
"submitter": "Hieu D. Nguyen",
"url": "https://arxiv.org/abs/1212.3707"
}
|
1212.3762
|
# Excitation Spectrum and Momentum Distribution of Bose-Hubbard Model with On-
site Two- and Three-body Interaction
Beibing Huang
Department of Experiment Teaching, Yancheng Institute of Technology, Yancheng,
224051, China
and Shaolong Wan
Institute for Theoretical Physics and Department of Modern Physics, University
of Science and
Technology of China, Hefei, 230026, China
Abstract
An effective action for Bose-Hubbard model with two- and three-body on-site
interaction in a square optical lattice is derived in the frame of a strong-
coupling approach developed by Sengupta and Dupuis. From this effective
action, superfluid-Mott insulator (MI) phase transition, excitation spectrum
and momentum distribution for two phases are calculated by taking into account
Gaussian fluctuation about the saddle-point approximation. In addition the
effects of three-body interaction are also discussed.
PACS number(s): 03.75.Hh, 73.43.Nq, 05.30.Jp
## 1 Introduction
The quantum phase transition from superfluid to Mott insulator (MI) has been
realized for cold bosonic atoms in a cubic optical lattice [1] and well
explained in the frame of Bose-Hubbard model (BHM) only with on-site two-body
interaction [2]: when the lattice potential is very weak, the kinetic energy
dominates and the system is a delocalized superfluid. By contrast if the
lattice potential is very strong, the interaction will forbid bosons to
hopping from one site to another site and the system is in the MI phase. Hence
for intermediate lattice potential we naturally expect the phase transition
from superfluid to MI phase. In fact many other phenomena of bosons in optical
lattices can also be well described by BHM [3, 4].
However, the work of Köhler [5] and Johnson et al. [6] pointed out that an
effective three-body interaction should be included to explain the
experimental criterion for collapse and revival of coherent matter waves [7,
8, 9]. In terms of a pure bosonic system, the effective three-body interaction
$g_{3}$ is related to the two-body scattering length $a_{s}$ and the dilute
gas parameter $\eta=\sqrt{\rho a_{s}^{3}}$ by $g_{3}\propto
a_{s}^{4}\ln{[C\eta^{2}]}$ [5, 10, 11, 12], where $\rho$ is the boson density
and the constant $C$ has been determined by applying a microscopic description
[5]. Generally in the current experiments the strength of three-body
interaction is much weaker than its two-body counterpart. In order to enhance
the ratio of three-body to two-body interactions, a feasible route is by the
induction of fermions. Considering the mixture of a BEC and a single component
Fermi gas, the Fermi degrees of freedom can be completely integrated out and
the effective three-body interaction can be produced [13, 14]. In this method
the interspecies interaction $g_{BF}$ not only induces three-body interaction
$g_{3}\propto g_{BF}^{3}$, but also weakens two-body interaction to
$g_{2}=g_{B}-Pg_{BF}^{2}$, where $g_{B}$ and $g_{2}$ are bosonic two-body
interaction before and after the integration of Fermi degrees of freedom and
$P$ is a constant [13]. Very explicitly by Feshbach resonance to tune
interspecies interaction $g_{BF}$ [15, 16], one should have enough space to
adjust the $g_{3}/g_{2}$. From this viewpoint it is very significant to
include three-body interaction into BHM.
As a matter of fact, some references [17, 18, 19, 20] have studied superfluid-
MI phase transition in the presence of two- and three-body on-site
interactions at the mean field level and found some interesting phenomena such
as the rotation of phase boundary around a fixed point and extension of MI.
Especially the reference [19] also studied the excitation spectrum in MI by
functional-integral method. Different from these works, in this paper, we
discuss the effects of three-body interaction on excitation spectrum and
momentum distribution both in the superfluid and MI phases on the same
footing, following a strong-coupling approach [21, 22]. From the experimental
viewpoint, the momentum distribution can be measured by imaging atom gas after
turning off the trap potential, while the excitation spectrum resulting from a
particle-hole excitation can be attained by applying a potential gradient to
the system in the MI phase. It is by probing the momentum distribution and
excitation spectrum that superfluid-MI phase transition is observed [1]. In
section 2 we derive an effective action by performing two successive Hubbard-
Stratonovich transformations of the intersite hopping term. In section 3
starting from this effective action, the phase diagram, excitation spectrum
and momentum distribution are calculated by taking into account Gaussian
fluctuations about the mean-field approximation as in the Bogoliubov theories
of the weakly interacting Bose gas. A brief conclusion is given in section 4.
Throughout this paper, we set $\hbar=k_{B}=1$, and take the lattice constant
unity.
## 2 Hamiltonian and the Effective Action
The system of bosons in an optical lattice with on-site two- and three-body
interaction can be described by the modified Bose-Hubbard model
$\displaystyle
H=-\sum_{i,j}t_{ij}a_{i}^{{\dagger}}a_{j}+\frac{U}{2}\sum_{i}n_{i}(n_{i}-1)+\frac{W}{6}\sum_{i}n_{i}(n_{i}-1)(n_{i}-2)-\mu\sum_{i}a_{i}^{{\dagger}}a_{i},$
(1)
where $a_{i}$ is the bosonic field operator, $n_{i}=a_{i}^{{\dagger}}a_{i}$ is
the particle number operator for bosons at the lattice site $i$. The intersite
hopping matrix $t_{ij}$ is nonzero (labeled $t$) only when lattice sites $i$
and $j$ are the nearest neighbors, $U$, $W$ are two- and three-body repulsive
interaction strength among bosons. In the light of the statement in the
introduction about the strengths of two- and three-body interactions, one
consider that the ratio $W/U$ is adjustable in a large parameter space. The
last term involving the chemical potential $\mu$ is added because it is very
convenient in the grand canonical ensemble. Without loss of generality a
square optical lattice is assumed.
In the imaginary-time functional integral formalism, the partition function of
the system is written as [23]
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\int\mathscr{D}[a^{\ast},a]e^{-S_{0}-S^{\prime}},$
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\sum_{i}a_{i}^{\ast}(\partial_{\tau}-\mu)a_{i}+\frac{U}{2}\sum_{i}n_{i}(n_{i}-1)+\frac{W}{6}\sum_{i}n_{i}(n_{i}-1)(n_{i}-2),$
$\displaystyle S^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\left[-t\sum_{<i,j>}a_{i}^{{\dagger}}a_{j}\right],$
(2)
with $\beta=1/T$ and $\tau$ imaginary time. $\mathscr{D}[\cdots]$ represents
the functional integral for field operators.
Carrying out Hubbard-Stratonovich transformation for $S^{\prime}$ and
integrating out $a$, the partition function
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\int\mathscr{D}[a^{\ast},a,b^{\ast},b]e^{-S_{0}-b_{i}^{\ast}t^{-1}_{ij}b_{j}+a^{\ast}_{i}b_{i}+b^{\ast}_{i}a_{i}},$
(3) $\displaystyle=$ $\displaystyle
Z_{0}\int\mathscr{D}[b^{\ast},b]e^{-b_{i}^{\ast}t^{-1}_{ij}b_{j}}<\sum_{n=0}^{\infty}\frac{(a^{\ast}_{i}b_{i}+b^{\ast}_{i}a_{i})^{n}}{n!}>_{0}$
$\displaystyle=$ $\displaystyle
Z_{0}\int\mathscr{D}[b^{\ast},b]e^{-b_{i}^{\ast}t^{-1}_{ij}b_{j}+V[b^{\ast},b]}$
where $t^{-1}$ denotes the inverse matrix of the intersite hopping matrix
$t_{ij}$, $Z_{0}=\int\mathscr{D}[a^{\ast},a]e^{-S_{0}}$ is the partition
function in the local limit ($t=0$), $<\cdots>_{0}$ means that the average is
taken with $S_{0}$. According to the linked cluster expansion [23], the
functional $V[b^{\ast},b]$ only includes contributions from linked diagrams.
Until the fourth order of $b$ and after a careful calculation
$\displaystyle V[b^{\ast},b]$ $\displaystyle=$
$\displaystyle-\int_{0}^{\beta}d\tau_{1}d\tau_{2}\sum_{i}b_{i}^{\ast}(\tau_{1})G_{1c}(\tau_{1},\tau_{2})b_{i}(\tau_{2})+$
(4)
$\displaystyle\frac{1}{4}\int_{0}^{\beta}d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\sum_{i}G_{2c}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})b_{i}^{\ast}(\tau_{1})b_{i}^{\ast}(\tau_{2})b_{i}(\tau_{4})b_{i}(\tau_{3}),$
where $G_{1c}(\tau_{1},\tau_{2})$
($G_{2c}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$) denote the connected local
single (two) particle(s) Green function. In fact, $V[b^{\ast},b]$ is the
generating functional of connected Green functions in the local limit if $b$
field is regarded as external source.
It is easily found that the relation between the Green function of $a$ field
$G_{a}$ and $G_{b}$ of $b$ field is $G_{a}=t^{-1}-t^{-1}G_{b}t^{-1}$. Hence
once $G_{b}$ is known, we can get $G_{a}$. Unfortunately Green function
$G_{a}$ obtained in this way is not physical since it leads in the superfluid
phase to a spectral function which is not normalized to unity [21]. To
overcome this difficulty, a second Hubbard-Stratonovich transformation was
performed to decouple intersite hopping term and the partition function
becomes
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\int\mathscr{D}[c^{\ast},c,b^{\ast},b]e^{c_{i}^{\ast}t_{ij}c_{j}-c^{\ast}_{i}b_{i}-b^{\ast}_{i}c_{i}+V[b^{\ast},b]}.$
(5)
Reference [21] has shown that the auxiliary field $c$ has the same correlation
functions as the original boson field $a$, hence below we will substitute $a$
for $c$. In (5) considering the second order term in $V[b^{\ast},b]$ as weight
and integrating out $b$ field, till the fourth order of $a$, the effective
action $S[a^{\ast},a]$ is
$\displaystyle S[a^{\ast},a]$ $\displaystyle=$
$\displaystyle-\int_{0}^{\beta}d\tau_{1}d\tau_{2}\sum_{i,j}a_{i}^{\ast}(\tau_{1})\left[G_{1c}^{-1}(\tau_{1},\tau_{2})+t_{ij}\delta(\tau_{1}-\tau_{2})\right]a_{j}(\tau_{2})$
(6)
$\displaystyle+\frac{1}{4}\int_{0}^{\beta}d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\sum_{i}\Gamma(\tau_{1},\tau_{2},\tau_{3},\tau_{4})a_{i}^{\ast}(\tau_{1})a_{i}^{\ast}(\tau_{2})a_{i}(\tau_{4})a_{i}(\tau_{3}),$
where we have discarded the contributions from all anomalous terms in the same
spirit of Ref.[22]. $\Gamma(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$ is the exact
two-particle vertex in the local limit
$\displaystyle\Gamma(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$ $\displaystyle=$
$\displaystyle-\int_{0}^{\beta}d\tau_{1}^{\prime}d\tau_{2}^{\prime}d\tau_{3}^{\prime}d\tau_{4}^{\prime}G_{2c}(\tau_{1}^{\prime},\tau_{2}^{\prime},\tau_{3}^{\prime},\tau_{4}^{\prime})\times$
(7) $\displaystyle
G_{1c}^{-1}(\tau_{1},\tau_{1}^{\prime})G_{1c}^{-1}(\tau_{2},\tau_{2}^{\prime})G_{1c}^{-1}(\tau_{4},\tau_{4}^{\prime})G_{1c}^{-1}(\tau_{3},\tau_{3}^{\prime}),$
and has a complicated time-dependence. Taking the static limit of $\Gamma$
into consideration and introducing a parameter
$\displaystyle
g=\frac{1}{2}\Gamma_{static}=-\frac{1}{2}\frac{\overline{G}_{2c}}{\overline{G}_{1c}^{4}},$
(8)
where $\overline{G}_{1c}$, $\overline{G}_{2c}$ denote the zero-frequency
component of Fourier transformation of $G_{1c}$, $G_{2c}$ respectively, we
finally obtain
$\displaystyle S[a^{\ast},a]$ $\displaystyle=$
$\displaystyle-\int_{0}^{\beta}d\tau_{1}d\tau_{2}\sum_{i,j}a_{i}^{\ast}(\tau_{1})\left[G_{1c}^{-1}(\tau_{1},\tau_{2})+t_{ij}\delta(\tau_{1}-\tau_{2})\right]a_{j}(\tau_{2})$
(9)
$\displaystyle+\frac{g}{2}\int_{0}^{\beta}d\tau\sum_{i}a_{i}^{\ast}a_{i}^{\ast}a_{i}a_{i}.$
In contrast to the original action (2), on the one hand the intersite hopping
has been renormalized by the exact local single particle Green function
$G_{1c}$, on the other hand the interaction $U$ has been substituted by the
exact local two-particle vertex. This action (9) is the starting point of our
analysis.
## 3 Excitation Spectrum and Momentum Distribution
In this section we decide the boundary of superfluid-MI phase transition,
excitation spectrum and momentum distribution for both superfluid phase and MI
at zero temperature. Before doing these, we first calculate local Green
functions $G_{1c}$ and $G_{2c}$ at zero temperature to determine the parameter
$g$.
In the local limit, the Hamiltonian (1) is diagonal about lattice sites and
its eigenstates are Fock states $|n>=(n!)^{-1/2}(a^{{\dagger}})^{n}|0>$ with
eigenvalue $\epsilon_{n}=(U/2)n(n-1)+(W/6)n(n-1)(n-2)-\mu n$. In order to
attain $g$ we firstly calculate one-particle Green function
$G_{1}(\tau)=-<\mathcal{T}_{\tau}a(\tau)a^{{\dagger}}(0)>$ and two-particle
Green function
$G_{2}(\tau_{1},\tau_{2};\tau_{3},\tau_{4}=0)=<\mathcal{T}_{\tau}a(\tau_{1})a(\tau_{2})a^{{\dagger}}(0)a^{{\dagger}}(\tau_{3})>$,
which are easily calculated using the closure relation
$\sum_{n=0}^{\infty}|n><n|=1$. Then $G_{1c}(\tau)=G_{1}(\tau)$ and
$G_{2c}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})=G_{2}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})-G_{1}(\tau_{1},\tau_{3})G_{1}(\tau_{2},\tau_{4})-G_{1}(\tau_{1},\tau_{4})G_{1}(\tau_{2},\tau_{3})$.
Making Fourier transformation, we obtain
$\displaystyle\overline{G}_{1c}(i\omega_{n})$ $\displaystyle=$
$\displaystyle\frac{-n_{0}}{i\omega_{n}+\epsilon_{n_{0}-1}-\epsilon_{n_{0}}}+\frac{n_{0}+1}{i\omega_{n}+\epsilon_{n_{0}}-\epsilon_{n_{0}+1}},$
(10) $\displaystyle\overline{G}_{2c}$ $\displaystyle=$
$\displaystyle-\frac{4(n_{0}+1)(n_{0}+2)}{(\epsilon_{n_{0}}-\epsilon_{n_{0}+2})(\epsilon_{n_{0}}-\epsilon_{n_{0}+1})^{2}}-\frac{4n_{0}(n_{0}-1)}{(\epsilon_{n_{0}}-\epsilon_{n_{0}-2})(\epsilon_{n_{0}}-\epsilon_{n_{0}-1})^{2}}$
(11) $\displaystyle+$
$\displaystyle\frac{4n_{0}(n_{0}+1)}{(\epsilon_{n_{0}}-\epsilon_{n_{0}+1})(\epsilon_{n_{0}}-\epsilon_{n_{0}-1})^{2}}+\frac{4n_{0}(n_{0}+1)}{(\epsilon_{n_{0}}-\epsilon_{n_{0}+1})^{2}(\epsilon_{n_{0}}-\epsilon_{n_{0}-1})}$
$\displaystyle+$
$\displaystyle\frac{4n_{0}^{2}}{(\epsilon_{n_{0}}-\epsilon_{n_{0}-1})^{3}}+\frac{4(n_{0}+1)^{2}}{(\epsilon_{n_{0}}-\epsilon_{n_{0}+1})^{3}},$
where $n_{0}$ is the occupation number of ground state in the local limit
which minimizes the eigenvalue $\epsilon_{n}$.
### 3.1 Superfluid-MI Phase Transition
To proceed further, we consider a quadratic expansion of the action (9) in
terms of fluctuation near the saddle-point value of $a$ field. Choosing
saddle-point value $\psi_{0}$ for $a$, that is
$a_{i}(\tau)\rightarrow\psi_{0}+\widetilde{a}_{i}(\tau)$, (9) becomes
$S=S_{0}+S_{1}+S_{2}$ with
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle-(\overline{G}_{1c}^{-1}+4t)\psi_{0}^{2}+\frac{g}{2}\psi_{0}^{4},$
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle-\psi_{0}(\overline{G}_{1c}^{-1}+4t-g\psi_{0}^{2})\int_{0}^{\beta}d\tau\sum_{i}[\widetilde{a}_{i}(\tau)+\widetilde{a}_{i}^{\ast}(\tau)],$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle-\int_{0}^{\beta}d\tau_{1}d\tau_{2}\sum_{i,j}\widetilde{a}_{i}^{\ast}(\tau_{1})\left[G_{1c}^{-1}(\tau_{1},\tau_{2})+t_{ij}\delta(\tau_{1}-\tau_{2})\right]\widetilde{a}_{j}(\tau_{2})$
(12)
$\displaystyle+\frac{g}{2}\psi_{0}^{2}\int_{0}^{\beta}d\tau\sum_{i}\widetilde{a}_{i}^{\ast}\widetilde{a}_{i}^{\ast}+\widetilde{a}_{i}\widetilde{a}_{i}+4\widetilde{a}_{i}^{\ast}\widetilde{a}_{i}.$
The constant $\psi_{0}$ is determined by requiring the coefficients of the
linear terms in $\widetilde{a}_{i}$ and $\widetilde{a}_{i}^{\ast}$ to vanish,
leading to the equation
$\displaystyle\overline{G}_{1c}^{-1}+4t-g\psi_{0}^{2}=0.$ (13)
It is well-known that a nonzero $\psi_{0}$ signifies superfluid of system,
therefore the boundary of superfluid-MI phase transition is
$\displaystyle\overline{G}_{1c}^{-1}+4t=0,$ (14)
which is in agreement with the results in [17, 18, 19, 20]. Fig. 1, the phase
diagram in the $t/U-\mu/U$ plane for different $W/U$ from (14), shows that the
effects of three-body interaction on the phase boundary are very dramatic. On
the one hand Mott lobes with $n_{0}=1$ remains unaltered in the presence of
$W$. On the other hand for Mott lobes with higher densities, with the increase
of $W$, their widths and heights gradually increase. In addition for small $W$
phase boundary has the same characteristics as the conventional superfluid- MI
phase transition that Mott lobes with higher occupation have smaller area.
While $W$ gradually increases and dominates two-body interaction $U$, Mott
lobes with higher occupation have bigger area. These phenomena can be
illustrated from the pure two-body and three-body interaction [17].
Figure 1: The phase diagram of superfluid-MI phase transition with different
three-body interaction $W$.
The excitation spectrum and momentum distribution can be deduced from the
second order action $S_{2}$. Introducing the Fourier transformation
$\displaystyle\widetilde{a}(\vec{k},i\omega_{n})=\frac{1}{\sqrt{\beta
N}}\int_{0}^{\beta}d\tau\sum_{i}\widetilde{a}_{i}(\tau)e^{-i\vec{k}\cdot\vec{R}_{i}+i\omega_{n}\tau},$
(15)
where $N$ is the total number of lattice sites, $\vec{R}_{i}$ is a lattice
vector for site $i$, $\omega_{n}=2n\pi/\beta$ is bosonic Matsubara frequency,
$S_{2}$ becomes
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\vec{k},\omega_{n}}\left[\widetilde{a}^{\ast}(\vec{k},i\omega_{n}),\widetilde{a}(-\vec{k},-i\omega_{n})\right]M_{2\times
2}\left[\widetilde{a}(\vec{k},i\omega_{n}),\widetilde{a}^{\ast}(-\vec{k},-i\omega_{n})\right]$
(16)
with
$\displaystyle M_{2\times 2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-G_{1c}^{-1}(i\omega_{n})+\varepsilon_{k}+2g\psi_{0}^{2}&g\psi_{0}^{2}\\\
\\\
g\psi_{0}^{2}&-G_{1c}^{-1}(-i\omega_{n})+\varepsilon_{-k}+2g\psi_{0}^{2}\end{array}\right)$
(20)
and $\varepsilon_{k}=-2t(\cos{k_{x}}+\cos{k_{y}})$ denoting the boson
dispersion in the absence of on-site interaction.
### 3.2 MI Phase
Figure 2: The excitation spectrum for quasiholes (a) and quasiparticles (b).
We choose $t/U=0.01$, $\mu/U=1.5$ and $W/U=0,2.0,4.0$ to make the system into
MI with the filling factor $n_{0}=2$. In (a) the effects of different $W$ is
very small. In (b) from bottom to top $W$ is increasing.
In the MI phase $\psi_{0}=0$, the matrix $M_{2\times 2}$ becomes diagonal and
$S_{2}$ is reduced into
$\displaystyle
S_{2}=\sum_{\vec{k},\omega_{n}}\widetilde{a}^{\ast}(\vec{k},i\omega_{n})\left[-G_{1c}^{-1}(i\omega_{n})+\varepsilon_{k}\right]\widetilde{a}(\vec{k},i\omega_{n}).$
(21)
The Green function
$g_{MI}(\vec{k},i\omega_{n})=-<\widetilde{a}(\vec{k},i\omega_{n})\widetilde{a}^{\ast}(\vec{k},i\omega_{n})>$
can be directly obtained
$g_{MI}^{-1}(\vec{k},i\omega_{n})=G_{1c}^{-1}(i\omega_{n})-\varepsilon_{{k}}$.
Using (10), we obtain
$\displaystyle
g_{MI}(\vec{k},i\omega_{n})=\frac{z_{k}}{i\omega_{n}-E_{k}^{+}}+\frac{1-z_{k}}{i\omega_{n}-E_{k}^{-}}$
(22)
with
$\displaystyle E_{k}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{n_{0}+1}-\epsilon_{n_{0}-1}+\varepsilon_{k}\pm\sqrt{\Delta_{k}}}{2},$
(23) $\displaystyle\Delta_{k}$ $\displaystyle=$
$\displaystyle(\epsilon_{n_{0}+1}-\epsilon_{n_{0}-1}+\varepsilon_{k})^{2}-4(\epsilon_{n_{0}-1}-\epsilon_{n_{0}})(\epsilon_{n_{0}}-\epsilon_{n_{0}+1})$
(24) $\displaystyle-$ $\displaystyle
4\varepsilon_{k}[n_{0}(\epsilon_{n_{0}}-\epsilon_{n_{0}+1})+(n_{0}+1)(\epsilon_{n_{0}}-\epsilon_{n_{0}-1})],$
$\displaystyle z_{k}$ $\displaystyle=$
$\displaystyle\frac{E_{k}^{+}-n_{0}(\epsilon_{n_{0}}-\epsilon_{n_{0}+1})-(n_{0}+1)(\epsilon_{n_{0}}-\epsilon_{n_{0}-1})}{E_{k}^{+}-E_{k}^{-}}.$
(25)
In (23), $E_{k}^{\mp}$ stand for excitations of quasiholes (removing particles
from sites) or quasiparticles (adding particles to sites), which is consistent
with the results in [17] and is shown in Fig.2(a) (Fig.2(b)). Very explicitly,
three-body interaction $W$ has little impacts on quasihole excitations, but it
has the large effects on quasiparticle excitation: with the increase of $W$,
quasiparticle energy becomes higher, hence its excitation also becomes
difficult.
Figure 3: The excitation spectrum for density fluctuation $\Xi_{k}$. We choose
$t/U=0.06$, $W/U=2.0$ and $\mu/U=1.5,5.5,10.5$ to make the system into MI with
filling factor $n_{0}=2,3,4$ respectively. From bottom to top $\mu$ is
increasing.
In Greiner’s experiment [1], Superfluid-MI phase transition is also detected
by applying a potential gradient to the system in the MI phase and probing the
excitation spectrum resulting from a particle-hole excitation. According to
the above results, we can find a first approximation for the dispersion of
particle-hole excitation (density fluctuations) $\Xi_{k}$ by subtracting
$E_{k}^{-}$ from $E_{k}^{+}$, which yields
$\displaystyle\Xi_{k}=E_{k}^{+}-E_{k}^{-}=\sqrt{\Delta_{k}}.$ (26)
Due to little impacts of three-body interaction $W$ on $E_{k}^{-}$, the
dependence of $\Xi_{k}$ on $W$ is the same as $E_{k}^{+}$. In Fig.3 we show
$\Xi_{k}$ for a fixed $W$, but different filling factor $n_{0}$, i.e.
different chemical potential $\mu$. We can see that in MI phases there is
always a band gap increasing with the filling factor, which is different from
the situation without three-body interaction where the gap is approximately
independent of the filling factor. It is from this perspective that reference
[20] proposed to observe weak three-body interaction for high filling MI.
Figure 4: The momentum distribution in MI phases with filling factor
$n_{0}=2.0$. Parameters are chosen as follows: $t/U=0.01$, $\mu/U=1.5$ and
$W/U=0$ in (a) $W/U=2.0$ in (b).
From the theorem of fluctuation and dissipation, the momentum distribution at
zero temperature $n_{k}=<\widetilde{a}^{\ast}(\vec{k})\widetilde{a}(\vec{k})>$
is
$\displaystyle n_{k}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{1}{e^{\beta\omega}-1}(-2)Im[g_{MI}(\vec{k},\omega+i0^{+})]$
(27) $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{d\omega}{e^{\beta\omega}-1}[z_{k}\delta(\omega-
E_{k}^{+})+(1-z_{k})\delta(\omega-E_{k}^{-})]=z_{k}-1,$
where the label ”Im” denotes the imaginary part. In deriving (27) we have
noted that quasiparticle dispersion $E_{k}^{+}$ is always greater than or
equal to zero and $E_{k}^{-}$ is always smaller than or equal to zero. Because
of this only the quasiholes give a contribution to the total density at zero
temperature. Fig.4 shows momentum distributions in MI phases and we find that
three-body interaction $W$ leads to not only flattening but also broadening of
momentum distribution around $\vec{k}=0$.
### 3.3 Superfluid Phase
In terms of the superfluid phase,
$\psi_{0}^{2}=(\overline{G}_{1c}^{-1}+4t)/g$. From the condition
$det(M_{2\times 2})=0$, we obtain four excitation branches $\pm E_{k}^{\pm}$
with
$\displaystyle E_{k}^{\pm 2}$ $\displaystyle=$
$\displaystyle-\frac{B_{k}}{2}\pm\frac{1}{2}(B_{k}^{2}-4C_{k})^{1/2},$
$\displaystyle B_{k}$ $\displaystyle=$ $\displaystyle
2h_{2}-2(\varepsilon_{k}+2g\psi_{0}^{2})h_{1}-(h_{3}-\varepsilon_{k}-2g\psi_{0}^{2})^{2}+g^{2}\psi_{0}^{4},$
$\displaystyle C_{k}$ $\displaystyle=$
$\displaystyle[h_{2}-(\varepsilon_{k}+2g\psi_{0}^{2})h_{1}]^{2}-g^{2}\psi_{0}^{4}h_{1}^{2},$
$\displaystyle h_{1}$ $\displaystyle=$
$\displaystyle(n_{0}+1)(\epsilon_{n_{0}-1}-\epsilon_{n_{0}})-n_{0}(\epsilon_{n_{0}}-\epsilon_{n_{0}+1}),$
$\displaystyle h_{2}$ $\displaystyle=$
$\displaystyle(\epsilon_{n_{0}-1}-\epsilon_{n_{0}})(\epsilon_{n_{0}}-\epsilon_{n_{0}+1}),$
$\displaystyle h_{3}$ $\displaystyle=$
$\displaystyle(\epsilon_{n_{0}-1}-\epsilon_{n_{0}})+(\epsilon_{n_{0}}-\epsilon_{n_{0}+1}).$
(28)
Compared to the Bogoliubov’s theory for weakly interacting Bose gases, where
only two excitation branches exist, there are two more excitation branches
$\pm E_{k}^{+}$ due to renormalization of the tunneling term by locally exact
single-particle Green function $\overline{G}_{1c}(i\omega)$. Fig.5 show
typical characters for $E_{k}^{\pm}$: $E_{k}^{+}$ is gapped while $E_{k}^{-}$
is gapless. Moreover we also find that a gap opens between $E_{k}^{+}$ and
$E_{k}^{-}$ when three-body interaction $W$ becomes strong. Here, we are only
interested in gapless excitations described by $\pm E_{k}^{-}$, which
originate from spontaneously broken gauge symmetry and describe the low energy
behaviors of the system. By expanding $E_{k}^{-}$ in the vicinity of
$\vec{k}=0$, we find a linear spectrum $E_{k}^{-}=vk$ with
$\displaystyle
v=\left[\frac{2t(\overline{G}_{1c}^{-1}+4t)}{\alpha^{2}+2\gamma(\overline{G}_{1c}^{-1}+4t)}\right]^{1/2},\alpha=\frac{h_{2}-h_{1}h_{3}}{h_{1}^{2}},\gamma=\frac{h_{1}^{2}+h_{2}-h_{1}h_{3}}{h_{1}^{3}}.$
(29)
The existence of such linear spectrum is consistent with the Landau’s
criterion of superfluidity [24]. Fig.6 shows the dependence of sound velocity
$v$ on three-body interaction $W$. It is should be noted that the system is in
the superfluid phase near $n_{0}=2$ MI phase for parameters $\mu$ and $t$
chosen in Fig.6. For different parameters $\mu$ and $t$, on the one hand $v$
generally is not monotonic as the function of $W$, on the other hand the
critical values $W_{c}$, where $v=0$ or the system enters into the MI phase,
are very different. The second character can be illustrated from the phase
diagram Fig.1: with the increase of $W$, the left boundary of MI lobe with
$n_{0}=2$ changes very slowly, but the right boundary changes very quickly.
Figure 5: The excitation spectrum in superfluid phase with parameters
$t/U=0.06$, $\mu/U=1.5$, $W/U=0.0$ in (a) and $W/U=2.0$ in (b).
From the excitation spectrum $\pm E_{k}^{\pm}$, the Green function
$g_{SF}(\vec{k},i\omega_{n})$ can be easily obtained
$\displaystyle
g_{SF}(\vec{k},i\omega_{n})=\frac{P_{+}}{i\omega_{n}+E_{k}^{+}}+\frac{P_{-}}{i\omega_{n}-E_{k}^{+}}+\frac{Q_{+}}{i\omega_{n}+E_{k}^{-}}+\frac{Q_{-}}{i\omega_{n}-E_{k}^{-}},$
(30)
where
$\displaystyle P_{\pm}$ $\displaystyle=$ $\displaystyle\frac{E_{k}^{+3}\mp
a_{2}E_{k}^{+2}-a_{1}E_{k}^{+}\mp
a_{0}}{2E_{k}^{+}(E_{k}^{+2}-E_{k}^{-2})},Q_{\pm}=\frac{E_{k}^{-3}\mp
a_{2}E_{k}^{-2}-a_{1}E_{k}^{-}\mp a_{0}}{2E_{k}^{-}(E_{k}^{-2}-E_{k}^{+2})},$
$\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle
h_{1}-h_{3}+\varepsilon_{k}+2g\psi_{0}^{2},a_{1}=h_{3}h_{1}-h_{2},a_{0}=h_{1}[h_{2}-h_{1}(\varepsilon_{k}+2g\psi_{0}^{2})].$
(31)
Figure 6: The sound velocity $v$ as the function of three-body interaction $W$
for (a) $\mu/U=1.2$, (b) $\mu/U=1.8$, (c) $t/U=0.06$.
Following the same steps in (27), the momentum distribution in the superfluid
phase is
$\displaystyle n_{k}=N\psi_{0}^{2}\delta_{k,0}-P_{+}-Q_{+},$ (32)
where the first term $N\psi_{0}^{2}\delta_{k,0}$ comes from the condensate and
it is this term that supplies a coherent peak in absorptive images of atom
gases. Fig.7 show the momentum distribution from the second and third terms.
In contrast to the situation in MI phases, three-body interaction $W$ only
broadens the width of the peak near $\vec{k}=0$.
Figure 7: The momentum distribution in superfluid phase. Parameters are chosen
as follows: $t/U=0.06$, $\mu/U=1.5$ and $W/U=0$ in (a) $W/U=2.0$ in (b).
## 4 Conclusions
In conclusion, an effective action for Bose-Hubbard model with two- and three-
body on-site interaction in a square optical lattice has been derived by
performing two successive Hubbard-Stratonovich transformations of the
intersite hopping term. The main advantage of this method is that superfluid
and MI phases can be analyzed on the same footing. Starting from this
effective action, superfluid-MI phase transition, excitation spectrum and
momentum distribution for two phases are calculated by taking into account
Gaussian fluctuation about the saddle-point approximation. In addition the
effects of three-body interaction are also discussed. We find that the sound
velocity in superfluid phase generally is not monotonic as the function of
three-body interaction, depending on the chemical potential and hopping term.
## Acknowledgement
The work was supported by National Natural Science Foundation of China under
Grant No. 11275108. The author Huang also thanks Foundation of Yancheng
Institute of Technology under Grant No. XKR2010007.
## References
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* [2] M. P. A. Fisher, P. B. Weichmann, G. Grinstein and D. S. Fisher, Phys. Rev. B 40, 546 (1989).
* [3] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).
* [4] D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams and P. Zoller Phys. Rev. Lett. 89, 040402 (2002).
* [5] T. Köhler, Phys. Rev. Lett. 89, 210404 (2002).
* [6] P. R. Johnson, E. Tiesinga, J. V. Porto and C. J. Williams, New J. Phys. 11, 093022 (2009).
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* [8] M. Anderlini, J. S. Strabley, J. Kruse, J. V. Porto and W. D. Phillips, J. Phys. B 39 S199 (2006).
* [9] J. S. Strabley, B. L. Brown, M. Anderlini, P. J. Lee, P. R. Johnson, W. D. Phillips and J. V. Porto, Phys. Rev. Lett. 98, 200405 (2007).
* [10] T. T. Wu, Phys. Rev. 115, 1390 (1959).
* [11] E. Braaten and A. Nieto, Eur. Phys. J. B 11, 143 (1999).
* [12] F. K. Abdullaev, A. Gammal, L. Tomio and T. Frederico, Phys. Rev. A 63, 043604 (2001).
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* [16] C. A. Stan, M. W. Zwierlein, C. H. Schunck, S. M. F. Raupach and W. Ketterle, Phys. Rev. Lett. 93, 143001 (2004).
* [17] B. L. Chen, X. B. Huang, S. P. Kou and Y. Zhang, Phys. Rev. A 78, 043603 (2008).
* [18] B. Huang and S. Wan, Phys. Lett. A 374, 4364 (2010).
* [19] K. Zhou, Z. Liang and Z. Zhang, Phys. Rev. A 82, 013634 (2010).
* [20] M. Singh, A. Dhar, T. Mishra, R. V. Pai and B. P. Das, arXiv:1203.1412
* [21] K. Sengupta and N. Dupuis, Phys. Rev. A 71, 033629 (2005).
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* [24] L. D. Landau, J. Phys. (Moscow) 5, 71 (1941).
|
arxiv-papers
| 2012-12-16T08:32:36 |
2024-09-04T02:49:39.321390
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Beibing Huang and Shaolong Wan",
"submitter": "Beibing Huang",
"url": "https://arxiv.org/abs/1212.3762"
}
|
1212.3802
|
# Galerkin method for linear Integral-Algebraic Equations of index $1$
B. Shiri Faculty of mathematical science, University of Tabriz, Tabriz - Iran
Faculty of mathematical science, University of Tabriz, Tabriz - Iran
[email protected]
###### Abstract.
In this paper, we study direct and indirect Galerkin method for solving linear
Integral-Algebraic Equations of index $1.$ Convergence of indirect method is
also analyzed.
###### Key words and phrases:
Galerkin method, Differential Index, Integral-Algebraic Equations.
## 1\. Introduction
Recently numerical solution for Integral-Algebraic Equation (IAE) has been
studied by many researchers [1]. In physical models with constrained time or
space variables, Algebraic Equations including Integral or Differential
Equations arise. Therefore, the investigation of mixed types of equations is
important. This can be seen by observing the number of papers that are about
Differential-Algebraic Equations, Integro Differential Equations, and so on.
One of these systems which mixes Volterra Integral Equations of first and
second type is the system of Integral- Algebraic Equations. However, we don t
find any application for this systems except [6, 7, 8, 9], but it seems that
they be very important in the future. In this paper we use the standard
Galerkin method, for the system of Integral Algebraic Equations of index 1. We
implement the Galerkin method by considering the structure of IAE of index 1
in two ways, direct and indirect and we analyze indirect method here. We
Consider the system
(1.1) $A(t)y(t)+\int_{0}^{t}k(t,s)y(s)ds=f(t),\ \ \ \ t\in I:=[0,T],$
where $A\in\mathbf{C}(I,\mathbb{R}^{r\times r}),$
$f\in\mathbf{C}(I,\mathbb{R}^{r})$ and
$k\in\mathbf{C}(\mathbb{D},\mathbb{R}^{r\times r})$ with
$\mathbb{D}:=\\{(t,s):0\leq s\leq t\leq T\\}.$ If $A(t)$ is nonsingular for
all $t\in I$ then multiplying (1.1) by $A.^{-1}$ changes it to a system of the
second kind Volterra Integral Equations that it’s theoretical and numerical
analysis have been almost completely investigated [3, 4]. Otherwise, if $A(t)$
is a singular matrix with constant rank for all $t\in I$, then, the system
(1.1) is an IAE. The classification of this system done by introducing index
notation. The differential index of differential-algebraic equation can be
extended to this system. The following extension is done by Gear in [5].
###### Definition 1.1.
We say the system (1.1) has index m if, m is the minimum possible number of
differentiating (1.1) to obtain a system of second kind Volterra Integral
equation.
For example the following system
(1.2) $x(t)=f_{1}(t)+\int_{0}^{t}k^{11}(t,s)x(s)+k^{12}(t,s)y(s)ds,$ (1.3)
$0=f_{2}(t)+\int_{0}^{t}k^{21}(t,s)x(s)+k^{22}(t,s)y(s)ds.$
with the functions $f_{i}$, $k^{ij}$, $i,j=1,2$ are sufficiently smooth and
$f_{2}(0)=0$, $|k^{22}(t,t)|\geq k_{0}>0$ for all $t\in[0,T];$ (1.2) and (1.3)
is the system of Integral-Algebraic Equations of index 1. For this system the
Piecewise Polynomial Collocation method is investigated by Kauthen in [2]. In
this paper we investigate the Galerkin method for this system.
## 2\. preliminary
The Galerkin method is a projection type method [3, 4] that we apply to IAE of
index 1. Let $\mathcal{K}:\mathbf{X}\rightarrow\mathbf{X}$ be an operator,
where $\mathbf{X}$ is a Hilbert space with orthonormal base
$\\{\phi_{i}\\}_{i=1}^{\infty}$ and $\mathbf{X}_{n}$ be an $n$-dimensional
subspace of X generated by $\\{\phi_{i}\\}_{i=1}^{n}.$ Then the system (1.2)
and (1.3) changes to
(2.1) $\mathcal{K}x=f.$
For using the Galerkin method’s we define the projection
$P_{n}:\mathbf{X}\rightarrow\mathbf{X}_{n}$ by $P_{n}x=\sum_{i=1}^{n}\langle
x,\phi_{i}\rangle\phi_{i}.$ Then it is clear that $\|P_{n}x-x\|\rightarrow 0$
as $n\rightarrow\infty$. We search $\mathbf{x}_{n}\in\mathbf{X}_{n}$,
$\mathbf{x}_{n}=\overline{x}\overline{\phi}$ with
$\overline{x}=(x_{1},\ldots,x_{n})$ and
$\overline{\phi}=(\phi_{1},\ldots,\phi_{n})^{T},$ such that
(2.2) $P_{n}\mathcal{K}\mathbf{x}_{n}=P_{n}f.$
The corresponding operator of the system (1.1) is defined by
(2.3)
$\mathcal{K}([x,y])=\left(x-\int_{0}^{t}k^{11}(t,s)x(s)+k^{12}(t,s)y(s)ds,\
-\int_{0}^{t}k^{21}(t,s)x(s)+k^{22}(t,s)y(s)ds\right)^{T},$
and the assumption that $f=(f_{1},f_{2})^{T}$,
$\mathbf{X}\subseteq(\mathcal{L}^{2}[0,T])$ and the functions $f_{i}$,
$k^{ij}$, $i,j=1,2$ are sufficiently smooth and $f_{2}(0)=0$,
$|k^{22}(t,t)|\geq k_{0}>0$ for all $t\in[0,T],$ guarantee us that
$\mathcal{K}x=f$ has a continuous solution as it follows. We first
differentiate equation (1.3) w. r. to t and we have
$\begin{split}0=&f^{{}^{\prime}}_{2}(t)+k^{21}(t,t)x(t)+k^{22}(t,t)y(t)\\\
&+\int_{0}^{t}\frac{\partial k^{21}(t,s)}{\partial t}x(s)+\frac{\partial
k^{22}(t,s)}{\partial t}y(s)ds.\end{split}$
Because $|k^{22}(t,t)|\geq k_{0}>0$ we obtain
(2.4)
$\begin{split}y(t)=&-\frac{f^{{}^{\prime}}_{2}(t)}{k^{22}(t,t)}-\frac{k^{21}(t,t)}{k^{22}(t,t)}x(t)\\\
&-\int_{0}^{t}\frac{1}{k^{22}(t,t)}\frac{\partial k^{21}(t,s)}{\partial
t}x(s)-\frac{1}{k^{22}(t,t)}\frac{\partial k^{22}(t,s)}{\partial
t}y(s)ds,\end{split}$
as a Volterra integral equations of the second kind for $y(t)$. Now it is easy
to show that this system has a unique solution (see [4]).
## 3\. Implementation and Convergence
We can implement the Galerkin method for two systems (1.2),(1.3) and
(1.2),(2.4). For the former the Galerkin method is called direct and for the
further, is called indirect method. For the second system, the convergence and
order of convergence can be obtained by similar arguments as discussed [6] and
[3]. In [3] the order of convergence has been explained for Fredholm integral
equation. Since, by using suitable kernel, each Volterra integral equation can
be changed to a Fredholm integral equation, Galerkin method for both are the
same. Hence the convergence order of the approximate solution is equal to the
convergence order of the best approximation in space $X_{n}$. This is
confirmed by numerical examples. Let $\\{V_{i}\\}_{i=1}^{\infty}$ be Legendre
polynomials as orthonormal base for $\mathcal{L}^{2}[-1,1]$. For
$\mathcal{L}^{2}[0,T]$ we use shifted Legendre polynomials
$\\{V_{T_{i}}\\}_{i=1}^{\infty}$,
$V_{T_{i}}(s)=\sqrt{\frac{2}{T}}V_{i}(\frac{2*s-T}{T})$, $s\in[0,T]$ and for
$(\mathcal{L}^{2}[0,T])^{2},$ we use
$\\{(V_{T_{i}},0),(0,V_{T_{i}})\\}_{i=1}^{\infty}$ with
$\langle(x,y),(z,t)\rangle=\langle x,z\rangle_{2}+\langle y,t\rangle_{2}$ and
$\langle x,y\rangle_{2}=\int_{0}^{T}x\overline{y}dt$. Also we suppose
$P_{n}([x,y])=\left(\sum_{i=1}^{n}x_{i}V_{T_{i}},\sum_{i=1}^{n}y_{i}V_{T_{i}}\right).$
For equations (1.2), (1.3) and (2.4) we have.
(3.1)
$\begin{split}\sum_{i=1}^{n}x_{i}V_{T_{i}}(t)=&f_{1}(t)+\sum_{i=1}^{n}x_{i}\int_{0}^{t}k^{11}(t,s)V_{T_{i}}(s)ds\\\
&+\sum_{i=1}^{n}y_{i}\int_{0}^{t}k^{12}(t,s)V_{T_{i}}(s)ds,\end{split}$ (3.2)
$0=f_{2}(t)+\sum_{i=1}^{n}x_{i}\int_{0}^{t}k^{21}(t,s)V_{T_{i}}(s)ds+\sum_{i=1}^{n}y_{i}\int_{0}^{t}k^{22}(t,s)V_{T_{i}}(s)ds,$
(3.3)
$\begin{split}\sum_{i=1}^{n}y_{i}V_{T_{i}}(t)=&-\frac{f^{{}^{\prime}}_{2}(t)}{k^{22}(t,t)}-\frac{k^{21}(t,t)}{k^{22}(t,t)}\sum_{i=1}^{n}x_{i}V_{T_{i}}(t)\\\
&-\sum_{i=1}^{n}x_{i}\int_{0}^{t}\frac{1}{k^{22}(t,t)}\frac{\partial
k^{21}(t,s)}{\partial t}V_{T_{i}}(s)ds\\\
&-\sum_{i=1}^{n}y_{i}\int_{0}^{t}\frac{1}{k^{22}(t,t)}\frac{\partial
k^{22}(t,s)}{\partial t}V_{T_{i}}(s)ds.\end{split}$
Now we can use (3.1),(3.2) for the first system and (3.1),(3.3) for the second
system introduced above. It is important for analysis to note that formula
(3.3) is derivative of (3.2). Now if we multiply equations (3.1)-(3.3) to
$V_{T_{j}}(t)$ and integrate on the $[0,T]$, we have obtained
(3.4)
$\begin{split}x_{j}=&\int_{0}^{T}f_{1}(t)V_{T_{j}}(t)dt+\sum_{i=1}^{n}x_{i}\int_{0}^{T}\int_{0}^{t}k^{11}(t,s)V_{T_{i}}(s)V_{T_{j}}(t)dsdt\\\
&+\sum_{i=1}^{n}y_{i}\int_{0}^{T}\int_{0}^{t}k^{12}(t,s)V_{T_{i}}(s)V_{T_{j}}(t)dsdt,\end{split}$
(3.5)
$\begin{split}0=&\int_{0}^{T}f_{2}(t)V_{T_{j}}(t)dt+\sum_{i=1}^{n}x_{i}\int_{0}^{T}\int_{0}^{t}k^{21}(t,s)V_{T_{i}}(s)V_{T_{j}}(t)dsdt\\\
&+\sum_{i=1}^{n}y_{i}\int_{0}^{T}\int_{0}^{t}k^{22}(t,s)V_{T_{i}}(s)V_{T_{j}}(t)dsdt,\end{split}$
(3.6)
$\begin{split}y_{j}=&-\int_{0}^{T}\frac{f^{{}^{\prime}}_{2}(t)}{k^{22}(t,t)}dt-\sum_{i=1}^{n}x_{i}\int_{0}^{T}\int_{0}^{t}\frac{1}{k^{22}(t,t)}\frac{\partial
k^{21}(t,s)}{\partial t}V_{T_{i}}(s)V_{T_{j}}(t)dsdt\\\
&-\sum_{i=1}^{n}x_{i}\int_{0}^{T}\frac{k^{21}(t,t)}{k^{22}(t,t)}V_{T_{i}}(t)V_{T_{j}}(t)dt\\\
&-\sum_{i=1}^{n}y_{i}\int_{0}^{T}\int_{0}^{t}\frac{1}{k^{22}(t,t)}\frac{\partial
k^{22}(t,s)}{\partial t}V_{T_{i}}(s)V_{T_{j}}(t)dsdt.\end{split}$
The direct and indirect Galerkin method for system (1.2),(1.3) are solving the
linear systems (3.4),(3.5) and (3.4),(3.6) respectively.
###### Lemma 3.1.
[4] Let $V$ and $W$ be normed spaces, with $W$ complete. Let
$K\in\mathcal{L}(V,W)$, let $\\{K_{n}\\}$ be a sequence of compact operators
in $\mathcal{L}(V,W),$ and assume $K_{n}\rightarrow K$ in $\mathcal{L}(V,W).$
Then K is compact.
###### Lemma 3.2.
[4] Let $V$ be a Banach space, and let $\\{P_{n}\\}$ be a family of bounded
projections on $V$ with
$P_{n}u\rightarrow u\ \ \ \mbox{as}\ \ \ n\rightarrow\infty,u\in V$
If $K:V\rightarrow V$ is compact, then
$\|K-P_{n}K\|\rightarrow 0\ \ \ \mbox{as}\ \ \ \ n\rightarrow\infty.$
###### Theorem 3.3.
[4] Assume $K:V\rightarrow V$ is bounded, with V a Banach space; and assume
$\lambda-K:V\rightarrow V$ is one to one and surjective. Further assume
$\|K-P_{n}K\|\rightarrow 0\ \mbox{as}\ n\rightarrow\infty.$
Then for all sufficiently large $n$, say, $n\geq N,$ the operator
$(\lambda-K)^{-1}$ exists as a bounded operator from $V$ to $V.$ Moreover, it
is uniformly bounded:
$\sup_{n\geq N}\|(\lambda-P_{n}K)^{-1}\|\leq\infty.$
For the solutions $u_{n}$ (n sufficiently large) of
$P_{n}(\lambda-K)u_{n}=P_{n}F$ and the solutions $u$ of $(\lambda-K)u=F,$ we
have
$u-u_{n}=\lambda(\lambda-P_{n}K)^{-1}(u-P_{n}u)$
and the two-sided error estimate
$\frac{|\lambda|}{\|\lambda-
P_{n}K\|}\|u-P_{n}u\|\leq\|u-u_{n}\|\leq|\lambda|\left\|(\lambda-
P_{n}K)^{-1}\right\|\|u-P_{n}u\|.$
We have following Theorem for indirect method.
###### Theorem 3.4.
Let for the system (1.2) and (1.3) the following conditions are satisfied:
1. (1)
$f_{i}\in C^{1}(0,T),\ K^{i,j}[t,s]\in C^{1}(\mathbb{D}),$ for all
$i,j=\\{1,2\\},$
2. (2)
$k^{22}(t,t)\geq k_{0}>0,$
where $\mathbb{D}:=\\{(t,s):0\leq s\leq t\leq T\\}.$ Then the convergence
order of the approximate solution is equal to the convergence order of the
best approximation in the space $X_{n}=\\{V_{T_{i}}\\}_{i=1}^{\infty}.$
###### Proof.
The space $\mathcal{L}^{2}[0,T]$ with $\mathcal{L}^{2}$ Norm is Banach space,
hence the Cartesian product on this space, i.e.,
$V=(\mathcal{L}^{2}[0,T])^{2},$ with introduced norm generate also Banach
space. Now we change the system (1.2) and (2.4) to the following system
(3.7) $(I-K)x=F,\ \ F=[F_{1},F_{2}]^{T},\ \ x=[x_{1},x_{2}]^{T},$
with
$F_{1}(t)=f_{1}(t),\ \ \ \ \ \
F_{2}(t)=-\frac{f^{{}^{\prime}}_{2}(t)}{k^{22}(t,t)}-\frac{k^{21}(t,t)}{k^{22}(t,t)}f_{1}(t),$
$K^{11}(t,s)=\left\\{\begin{array}[]{lc}k^{11}(t,s),&s\leq t,\\\
0,&s>t,\end{array}\right.\ \
K^{12}(t,s)=\left\\{\begin{array}[]{lc}k^{12}(t,s),&s\leq t,\\\
0,&s>t,\end{array}\right.\ \ $
$K^{21}(t,s)=\left\\{\begin{array}[]{lc}-\frac{k^{21}(t,t)}{k^{22}(t,t)}k^{11}(t,s)-\frac{1}{k^{22}(t,t)}\frac{\partial
k^{21}(t,s)}{\partial t},&s\leq t,\\\ 0,&s>t,\end{array}\right.$
$K^{22}(t,s)=\left\\{\begin{array}[]{lc}-\frac{k^{21}(t,t)}{k^{22}(t,t)}k^{12}(t,s)-\frac{1}{k^{22}(t,t)}\frac{\partial
k^{22}(t,s)}{\partial t},&s\leq t,\\\ 0.&s>t,\end{array}\right.$
and with the integral operator
$Kx=\int_{0}^{T}\left[\begin{array}[]{cc}K^{11}(t,s)&K^{12}(t,s)\\\
K^{21}(t,s)&K^{22}(t,s)\\\
\end{array}\right]\left[\begin{array}[]{c}x_{1}(s)\\\ x_{2}(s)\\\
\end{array}\right]ds.$
By introducing
$K_{n}x:=\int_{0}^{T}\left[\begin{array}[]{cc}K^{11}_{n}(t,s)&K^{12}_{n}(t,s)\\\
K^{21}_{n}(t,s)&K^{22}_{n}(t,s)\\\
\end{array}\right]\left[\begin{array}[]{c}x_{1}(s)\\\ x_{2}(s)\\\
\end{array}\right]ds,$
with continuous kernels
$K^{11}_{n}(t,s)=\left\\{\begin{array}[]{lc}k^{11}(t,s),&s\leq t,\\\
-n(k^{11}(t,t))((t+\frac{1}{n})-s),&s\in[t,t+\frac{1}{n}],\\\
0,&s>t+\frac{1}{n},\end{array}\right.$
$K^{12}_{n}(t,s)=\left\\{\begin{array}[]{lc}k^{12}_{n}(t,s),&s\leq t\\\
-n(k^{12}(t,t))((t+\frac{1}{n})-s),&s\in[t,t+\frac{1}{n}],\\\
0,&s>t+\frac{1}{n},\end{array}\right.\ \ $
$K^{21}_{n}(t,s)=\left\\{\begin{array}[]{lc}-\frac{k^{21}(t,t)}{k^{22}(t,t)}k^{11}(t,s)-\frac{1}{k^{22}(t,t)}\frac{\partial
k^{21}(t,s)}{\partial t},&s\leq t,\\\
-n(\frac{k^{21}(t,t)}{k^{22}(t,t)}k^{11}(t,t)+\frac{1}{k^{22}(t,t)}\frac{\partial
k^{21}(t,t)}{\partial t})((t+\frac{1}{n})-s),&s\in[t,t+\frac{1}{n}],\\\
0,&s>t+\frac{1}{n},\end{array}\right.$
$K^{22}_{n}(t,s)=\left\\{\begin{array}[]{lc}-\frac{k^{21}(t,t)}{k^{22}(t,t)}k^{12}(t,s)-\frac{1}{k^{22}(t,t)}\frac{\partial
k^{22}(t,s)}{\partial t},&s\leq t,\\\
-n(\frac{k^{21}(t,t)}{k^{22}(t,t)}k^{12}(t,t)+\frac{1}{k^{22}(t,t)}\frac{\partial
k^{22}(t,t)}{\partial t})((t+\frac{1}{n})-s),&s\in[t,t+\frac{1}{n}],\\\
0,&s>t+\frac{1}{n}.\end{array}\right.$
which is compact and using lemma 3.1, imply K is compact.
Applying the Galerkin method to the Fredholm system (3.7) leads to the linear
system equivalent with the system (3.4),(3.6). So it is enough to show that
for this system the convergence order of the approximate solution by the
Galerkin method is equal to the convergence order of the best approximation.
Since, the system (3.7) also is a volterra system, using the same line of [4],
$(I-K)$ is invertible so is surjective. The Lemma 3.2 provide the remainder
condition. Finally all conditions of the theorem 3.3 are satisfied and this
completes the proof. ∎
## 4\. Numerical results
.
Table 1. Numerical result obtained from (3.4),(3.5)
$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr n&2&4&6&8&10\\\
\hline\cr\|x_{n}-x\|&4.0e-002&3.0e-004&7.4e-007&9.4e-010&7.6e-013\\\
\hline\cr\|y_{n}-y\|&1.6e-001&2.2e-003&8.1e-006&1.4e-008&1.4e-011\\\
\hline\cr\end{array}$
.
Table 2. Numerical result obtained from (3.4),(3.6)
$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr n&2&4&6&8&10\\\
\hline\cr\|x_{n}-x\|&3.8e-002&2.6e-004&6.7e-007&8.8e-010&6.8e-013\\\
\hline\cr\|y_{n}-y\|&7.3e-002&5.1e-004&1.3e-006&1.7e-009&1.3e-012\\\
\hline\cr\end{array}$
.
Table 3. Logarithm of the error by best approximation using Legendre
polynomial)
$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr n&2&4&6&8&10\\\
\hline\cr\|P_{n}\sin(t)-\sin(t)\|&5.1e-002&3.3e-004&8.1e-007&1.0e-009&8.0e-013\\\
\hline\cr\|P_{n}\cos(t)-\cos(t)\|&7.8e-002&5.4e-004&1.4e-006&1.7e-009&1.3e-012\\\
\hline\cr\end{array}$
Figure 1. plots of convergence order by direct Galerkin method
Figure 2. plots of convergence order by indirect Galerkin method. We note that
the slope of this figures is same as the slope of figure 3 as claimed
Figure 3. plots of convergence order of best the approximation
In this section we illustrate the methods by numerical examples.
Example 1.
We consider (2.3) with $k^{11}=s+t$, $k^{12}=s^{2}+t^{2}$, $k^{21}=s-t^{2}$,
$k^{31}=s+t+1$, $f_{1}=-t-2sin(t)t^{2}+2sin(t)$ and
$f_{2}=t^{2}-2sin(t)+cos(t)t-cos(t)t^{2}+1-cos(t)-2sin(t)t$ where the exact
solutions are $x=sin(t)$ and $y=cos(t)$. Numerical results obtained by two
methods are in table 1 and 2 which show rapid convergence. In these tables
$\|f\|=\max_{T\in[0,T]}|f(t)|$. In the following figures, we also illustrate
$\log_{10}(\|x_{n}-x\|)$ and $\log_{10}(\|y_{n}-y\|)$ . For the second system,
we expect that this figure to be similar to the figure of
$\log_{10}(\|P_{n}x-x\|)$ and $\log_{10}(\|P_{n}y-y\|)$. This numerical
experiment shows that the second system is more accurate than the first. But
we need more computations in comparison with the first.
###### Remark 4.1.
In equations (3.4),(3.5) and (3.6), we find two type of integrals that should
be computed, single and double integrals. The integral of type
$\int_{0}^{T}f(t)dt$ can be computed by Gaussian Integration Methods (see
[11]) of the form
$\int_{-1}^{1}f(t)dt\simeq\sum_{i=1}^{n}w_{i}f(x_{i})$
and by changing of variable we have
(4.1) $\int_{0}^{T}f(t)dt\simeq\sum_{i=1}^{n}w_{i}f(Tx_{i}/2+T/2)$
where $x_{1},...,x_{n}$ are the roots of $V_{T_{(}n+1)},$ and
$w_{1},...,w_{n}$ are computed by the system expressed in (3.6.13) of [11]. An
approximate values of are computed as follows
$\int_{0}^{T}\int_{0}^{t}f(t,s)dsdt$ in this calculation is as it follows
(4.2)
$\int_{0}^{T}\int_{0}^{t}f(t,s)dsdt=\sum_{i=1}^{n}\sum_{j=1}^{n}w_{i}w_{j}f(\frac{Tx_{i}}{2}+\frac{T}{2},\frac{(\frac{Tx_{i}}{2}+\frac{T}{2})x_{i}}{2}+\frac{\frac{Tx_{i}}{2}+\frac{T}{2}}{2})$
Also we notice that the term $n$ is not fixed for different $P_{m}$, indeed in
our calculation $n=m$ that means for improving accuracy of the solution we
improve the accurate approximate solutions of integrals.
## References
* [1] Brunner, Hermann, Collocation methods for Volterra Integral and Related Functional Differential Equation, Cambridge Univercity Press, 2004.
* [2] J.-P. Kauthen, the numerical solution of Integral-Algebraic Equations of index $1$ by polynomial spline collocation methods, math. comp, 70, 236, 1503-1514, 2000.
* [3] K. Atkinson, The Numerical Solution of Integral Eauations of the Second Kind, Combridge Univercity Press, 1997.
* [4] K. Atkinson, W. Han Theoritical Numerical Analysis, A Functional Analysis Framework, Second Edition, Springer, 2005.
* [5] C.W. Gear, Differential algebraic equations, indices and integral algebraic equations, SIAM J. Numer. Anal. 27 (1990), 1527–1534.
* [6] J. Janno, L. V. Wolfersdorf, Identification of weakly singular memory kernels in heat conduction, Z. Angew. Math. Mech. 77 (1997), 243–257.
* [7] J. Janno, and L. v. Wolfersdorf, Inverse problems for identification of memory kernels in viscoelasticity, Math. Methods Appl. Sci. 20 (1997), 291–314.
* [8] B. Jumarhon, W. Lamb, S. McKee and T. Tang, A Volterra integral type method for solving a class of nonlinear initial-boundary value problems, Numer. Methods Partial Differential Equations 12 (1996), 265–281.
* [9] E. M. Kiss, Ein Beitrag zur Regularisierung und Diskretisierung des inversen Problems der Identifikation des Memorykerns in der Viskoelastizitat, Dissertation, Fakultat Mathematik und Informatik, TU Bergakademie Freiberg, (1999).
* [10] Zhengsu WAN, Yanping CHEN, Yunqing HUANG, Legendre spectral Galerkin method for second-kind Volterra integral equations, Front. Math. China 4(1), 181–193, 2009.
* [11] J. Stoer, R.Bulirsch, Introduction to Numerical Analysis, 2nd ed, Springer, 1996.
|
arxiv-papers
| 2012-12-16T16:10:48 |
2024-09-04T02:49:39.329291
|
{
"license": "Public Domain",
"authors": "B. Shiri",
"submitter": "Babak Shiri",
"url": "https://arxiv.org/abs/1212.3802"
}
|
1212.3807
|
# Preventing the tragedy of the commons through punishment of over-consumers
and encouragement of under-consumers
Irina Karevaa,b,d, Benjamin Morina,b, Georgy Karevc
a Mathematical, Computational Modeling Sciences Center,
PO Box 871904, Arizona State University, Tempe, AZ 85287
b School of Human Evolution and Social Changes,
Arizona State University, Tempe, AZ 85287
c Center for Biotechnology Information (NCBI)
National Institutes of Health, Bethesda, 20892
d Center of Cancer Systems Biology, Steward St. Elizabeth’s Medical Center
Tufts University School of Medicine, Boston, MA, 02135
###### Abstract
The conditions that can lead to the exploitative depletion of a shared
resource, i.e, the tragedy of the commons, can be reformulated as a game of
prisoner’s dilemma: while preserving the common resource is in the best
interest of the group, over-consumption is in the interest of each particular
individual at any given point in time. One way to try and prevent the tragedy
of the commons is through infliction of punishment for over-consumption and/or
encouraging under-consumption, thus selecting against over-consumers. Here,
the effectiveness of various punishment functions in an evolving consumer-
resource system is evaluated within a framework of a parametrically
heterogeneous system of ordinary differential equations (ODEs). Conditions
leading to the possibility of sustainable coexistence with the common resource
for a subset of cases are identified analytically using adaptive dynamics; the
effects of punishment on heterogeneous populations with different initial
composition are evaluated using the Reduction theorem for replicator
equations. Obtained results suggest that one cannot prevent the tragedy of the
commons through rewarding of under-consumers alone \- there must also be an
implementation of some degree of punishment that increases in a non-linear
fashion with respect to over-consumption and which may vary depending on the
initial distribution of clones in the population.
Keywords: tragedy of the commons, mathematical model, Reduction theorem,
adaptive dynamics
## Introduction
Ecological and social systems are complex and adaptive, composed of diverse
agents that are interconnected and interdependent [12, 15, 16]. Heterogeneity
within these systems often drives the evolution and adaptability of the system
components enabling them to withstand and recover from environmental
perturbations. However, it is also heterogeneity that makes the appearance and
short-term prosperity of over-consumers possible, which in turn can eventually
lead to exhaustion of the common resources (tragedy of the commons [4]) and
consequent collapse of the entire population, also known as evolutionary
suicide [6]).
Elinor Ostrom has focused on the question of avoiding the tragedy of the
commons from the point of view of collective decision making in small
fisheries [14]. She observed that the mutually satisfactory and functioning
institution of collective action could be developed in small communities where
the possible resource over-consumption by each individual was immediately
noticeable and punished, such as in water-monitoring systems instituted in
Spanish huertas, where each user was able to closely watch the other, and
where punishment for stealing scarce water was instituted not only in monetary
fines but also in public humiliation. This illustrates a first path to prevent
the tragedy of the commons: infliction of punishment/tax for over-consumption,
effectively selecting against over-consumers. For instance, Ostrom cites as an
example of a successful adaptive governance system rural villages in Japan,
where about 3 million hectares of forests and mountain meadows were
successfully managed even until now in no small part because “accounts were
kept about who contributed to what to make sure no household evaded its
responsibilities unnoticed. Only illness, family tragedy, or the absence of
able-bodied adults … were recognized as excuses for getting out of collective
labor… But if there was no acceptable excuse, punishment was in order” .
Another situation, when the tragedy was successfully avoided, is when the
community introduces some kind of “social currency”, where an individual is
rewarded for cooperation with social status [11, 18]. This is an example of a
second approach to preventing the tragedy of the commons: bestowing
reward/subsidy to under-consumers.
The dynamics of trade-offs between personal and population good have been
studied through classical game theory [13, 17, 5]. However, many of the
potentially relevant results have been obtained for largely homogeneous
populations of players. In this paper we approach the question of preventing
the tragedy of the commons through instituting punishment/reward systems in
heterogeneous populations using two recently developed methods for studying
evolving heterogeneous populations in equation-based models, namely adaptive
dynamics [1] and the Reduction theorem for replicator equations [7, 8].
This paper is organized as follows: first we give a derivation of a previously
studied model, modified to incorporate the effects of a punishment/reward
system on overall population survival. Next, we describe different approaches
for studying the effects of punishment and reward on population composition,
namely, adaptive dynamics and the Reduction theorem for replicator equations.
Then we apply the two techniques to a variety of punishment/reward structures,
obtaining both analytical and numerical results for sample populations that
differ in their initial composition with respect to over- and under-
consumers. We end the paper with a reformulation of the obtained results in
the context of the prisoner’s dilemma, as well as with a comparison of the two
modeling methods.
## Model Description
In order to answer the questions posed in the introduction, we first turn to a
modified version of a mathematical model of ‘niche construction’ proposed by
Krakauer et al.[10], where a population of consumers $x(t)$ interact with a
collectively constructed ‘niche’, represented as a common dynamical carrying
capacity $z(t)$. Rather than restricting the model to a specific physical
resource, we take the ‘niche’ to be a generalized abstract ‘resource’, such as
available nutrients, energy and other characteristics of the environment that
may affect both the individuals’ survival and fecundity (in case of financial
systems, the ’resource’ could be an inter-convertible unit such as money) that
each individual can invest or subtract from the ‘common good’. Different
individuals can contribute to the common dynamical carrying capacity, or they
can subtract from it at different rates. Since sufficient reduction of the
common ‘carrying capacity’ can indeed lead to population collapse, this model
allows to model effectively in a conceptual framework the question of the
effects of over-consumption on the survival of the population,
Within the frameworks of the model, each individual is characterized by his or
her own intrinsic value of resource consumption, denoted by parameter $c$; a
set of consumers that are characterized by the same value of $c$ is referred
to as $c$-clone. The total population of all consumers is
$N(t)=\sum_{\mathbb{A}}x_{c}$ if $c$ takes on discrete values and
$N(t)=\int_{\mathbb{A}}x_{c}dc$ if $c$ is continuous, where $\mathbb{A}$
denotes the range of possible values of parameter $c$.
The individuals $x_{c}(t)$ grow according to a logistic growth function, where
the carrying capacity is not constant but is a dynamic variable. Each
individual invests in resource restoration at a rate proportional to $(1-c)$;
an individual is considered to be an over-consumer if $c>1$. Moreover, each
individual is punished or rewarded according to a general continuous function
$f(c)\in C^{1}(\mathbb{\mathbb{R^{\dotplus}}})$ that directly affects the
fitness of each consumer-producer such that each clone is immediately punished
for over-consumption if $c>1$ or rewarded for under-consumption when $c<1$.
The dynamics of the resource is determined by a natural restoration rate
$\gamma$ and decay/loss rate $\delta z(t)$. Restoration process is accounted
for by the term $e(1-c)\frac{N/z}{1+N/z}$, where $z(t)$ is restored both
proportionally to each individual’s investment, accounted for by parameter
$c$, and to the total amount of ’resource’ consumed by the entire population,
represented by $N(t)/z(t)$.
This yields the following system of equations:
$\displaystyle\underbrace{x^{\prime}_{c}}_{\text{clones}}(t)$ $\displaystyle=$
$\displaystyle\underbrace{rx_{c}(t)}_{\text{intrinsic
growth}}\left(\underbrace{c}_{\text{consumption}}-\underbrace{\frac{N(t)}{kz(t)}}_{\text{dynamical
carrying
capacity}}\right)+\underbrace{x_{c}(t)f(c)}_{\text{punishment/reward}},$ (1)
$\displaystyle\underbrace{z^{\prime}(t)}_{\text{resource}}$ $\displaystyle=$
$\displaystyle\underbrace{\gamma}_{\text{restoration}}+\underbrace{e\frac{\sum\nolimits_{\mathbb{A}}x_{c}(t)(1-c)}{z(t)+N(t)}}_{\text{change
in resource caused by consumers}}-\underbrace{\delta
z(t)}_{\text{decay/inflation}}.$ (2)
The meaning of all the variables and parameters summarized in Table 1.
| Meaning | Range
---|---|---
$z(t)$ | abstract generalized ‘resource’ | $z\geq 0$
$c$ | rate of resource consumption | $c\in[c_{0},c_{f}]$
$x_{c}(t)$ | population of individuals competing for the resource | $x_{c}\geq 0$
$N(t)$ | total population size $N(t)=\intop_{\mathbb{\mathbb{A}}}x_{c}(t)dc$ | $N(t)\geq 0$
$r$ | individual proliferation rate | $r\geq 0$
$k$ | ‘resource’ conversion factor | $k\geq 0$
$e$ | efficiency of construction of common ‘niche’ | $e\geq 0$
$\gamma$ | intrinsic rate of ‘resource’ restoration independent of $x_{c}$ | $\gamma\geq 0$
$\delta$ | per capita rate of natural ‘resource’ decay | $\delta\geq 0$
$c_{0}$ | lower boundary value of parameter $c$ | $c_{0}\geq 0$
$c_{f}$ | upper boundary value of parameter $c$ | $c_{f}\geq 0$
Table 1: Summary of variables and parameters used in System 1. | $\gamma$ | $d$ | $e$ | $r$ | $k$ | $N_{0}$ | $z_{0}$ | $\mu$ | $c_{0}$ | $c_{f}$
---|---|---|---|---|---|---|---|---|---|---
set 1 | 1 | 1 | 1 | 1 | 1 | 0.6 | 0.1 | 10 | 0 | 2.5
set 2 | 7.72 | 22 | 1 | 1 | 1 | 0.6 | 0.1 | 10 | 0 | 9.12
Table 2: Sample parameter values.
The case, where $f(c)=0$ was previously completely studied in [9], both for
the case of a parametrically homogeneous and parametrically heterogeneous
system with respect to parameter $c$. Several transitional regimes were
identified with increasing parameter $c$, ranging from sustainable coexistence
with the common resource with ever decreasing domain of attraction to
sustained oscillatory regimes to collapse due to exhaustion of the common
resource. The results are summarized in Figure 1. In Domain 1, when the
parameter of resource consumption is small, the common carrying capacity
remains large, successfully supporting the entire population, since no
individual is taking more resource than they restore. In Domain 2, a parabolic
sector appears near the origin, decreasing the domain of attraction of the
non-trivial equilibrium point $A$. The population can still sustainably
coexist with the resource even with moderate levels of over-consumption but
the range of initial conditions, where it is possible, decreases. As the value
of $c$ is further increased, the range of possible parameter values that allow
sustainable coexistence with the common resource decreases and is now bounded
by the unstable limit cycle, which appears around point $A$ through a
catastrophic Hopf bifurcation in Domain 3, and via generalized Hopf
bifurcation in Domain 6. Finally, in Domain 4 and 5, population extinction is
inevitable due to extremely high over-consumption rates unsupportable by the
resource.
Figure 1: Bifurcation diagram of System (1) when $f(c)=0$ in the $(\gamma,c)$
and _(N,z)_ phase spaces for fixed positive parameters $e$ and $\delta$. The
non-trivial equilibrium point $A$ is globally stable in Domain 1; it shares
basins of attraction with equilibrium $O$ in Domains 2 and 3. The separatrix
of $O$ and the unstable limit cycle that contains point $A$, serve,
correspondingly, as the boundaries of the basins of attraction. Only
equilibrium $O$ is globally stable in Domains 4, which also contains also
unstable non-trivial $A$, and 5, which contains the elliptic sector. Domain 6
exists only for $\delta>5+\sqrt{24}$, where the stable limit cycle that is in
turn contained inside an unstable limit cycle, shares basins of attraction
with equilibrium $O$. Boundaries between Domains $K,S,H,Nul,C$ correspond
respectively to appearance of an attractive sector in a neighborhood of $O$,
appearance of unstable limit cycle containing $A$, change of stability of
equilibrium $A$ via Hopf bifurcations, disappearance of positive $A$ and
saddle-node bifurcation of limit cycles.
In this paper we will investigate the question of whether timely infliction of
punishment for over-consumption or rewarding under-consumption can prevent
such a collapse and consequently prevent the tragedy of the commons. We will
also investigate the degree of effectiveness of punishment depending on
initial composition of the population with respect to parameter $c$. Finally,
we will try to investigate the question of whether punishing those who over-
consume or rewarding those who under-consume will yield better results. These
questions will be addressed using two recently developed methods for modeling
parametrically heterogeneous populations, namely, adaptive dynamics [1] and
Reduction theorem for replicator equations [7, 8].
### Adaptive dynamics
Adaptive dynamics is a series of techniques that have been recently developed
to address questions of system invasibility by rare ‘mutant’ clones. The main
focus of this method is in evaluating a “mutant’s” ability to proliferate in
an environment set by the ‘resident’ population [1]. We use this method to
evaluate different punishment/reward functions and their efficiency in
preventing the tragedy of the commons. It is worth noting that a more general
theory of systems with inheritance, which allows studying the problem of
‘external stability’ of a population of ‘residents’ to invasion by a ‘mutant’
population, has been developed in [2]; it was later published in English in
[3].
Consider the equation for the dynamics of a rare mutant $x_{m}$ in an
environment set by the resident $x_{res}$. The total population size is
$x_{m}+x_{res}\approx x_{res}$, since $x_{m}$ is assumed to be present at such
low frequency that its contribution to the size of the entire population is
negligible.
Let $x_{res}^{*}$ satisfy $\frac{dx_{res}}{dt}=0$ , which implies that
$x_{res}^{*}=\frac{kz}{r}(f(c_{res})+rc_{res})$. Now introduce a mutant, such
that
$\frac{dx_{m}}{dt}=rx_{m}(c_{m}-\frac{x_{m}+x_{res}}{kz(t)})+x_{m}f(c_{m}).$
(3)
When the two subpopulations interact, the outcome of their interaction is
determined by the sign of the dominant eigenvalue of System
$\begin{split}\frac{dx_{res}}{dt}&=rx_{res}(c_{res}-\frac{x_{m}+x_{res}}{kz(t)})+x_{res}f(c_{res}),\\\
\frac{dx_{m}}{dt}&=rx_{m}(c_{m}-\frac{x_{m}+x_{res}}{kz(t)})+x_{m}f(c_{m})\\\
\frac{dz}{dt}&=\gamma+e\frac{x_{res}(1-c_{res})+x_{m}(1-c_{m})}{z+x_{m}+x_{res}}-\delta
z\end{split}$ (4)
The stability of the single positive non-trivial equilibrium point
$(x_{r}^{*},x_{m}^{*},z^{*})$ under the assumption that the mutant is rare,
i.e., when $x_{m}^{*}\approx 0$, is determined by the dominant eigenvalue of
System (4), which is given by $\lambda=r(c_{m}-c_{res})-f(c_{res})+f(c_{m})$:
the mutant will be able to invade if $\lambda>0$ and it will not succeed if
$\lambda<0$. Invasion fitness of the mutant, i.e., the expected growth rate of
a mutant in an environment set by the resident, is given by
$\underset{T\rightarrow\infty}{lim}\frac{1}{T}\int_{0}^{T}\left[r(c_{m}-\frac{x_{res}(t)}{kz(t)})+f(c_{m})\right]dt=r(c_{m}-\frac{1}{k}\overline{(\frac{x_{res}}{z})})+f(c_{m})=r(c_{m},E_{res}),$
(5)
where
$\overline{(\frac{x_{res}}{z})}=\underset{T\rightarrow\infty}{lim}\frac{1}{T}\int_{0}^{T}\frac{x_{res}(t)}{kz(t)}dt$.
The selection gradient, which is defined as the slope of invasiogamen fitness,
determines if the invasion will be successful (positive sign of the selection
gradient predicts successful invasion) and is then given by
$D(c_{m})=\frac{\partial}{\partial
c_{m}}r(c_{m},E_{res})|_{[c_{m}=c_{res}]}=r+f^{\prime}(c_{m}).$ (6)
These conditions allow answering a question of “long-term” invasibility, i.e.,
whether a mutant with a slightly difference value of $c_{m}$ will be able to
permanently invade the population of individuals, characterized by parameter
$c_{res}$. The points where selection gradient is zero are known as
evolutionarily singular strategies (ESS) and are denoted here as
$c_{res}^{*}$. Stability of $c_{res}^{*}$ for different types of
punishment/reward functions is discussed below; summary of some of the
possible types of $c_{res}^{*}$ is given in Table 3.
| Adaptive dynamics | Reduction theorem
---|---|---
Goal | Model evolution of parametrically heterogeneous populations
Assumptions | Clonal reproduction
| Separation of evolutionary and ecological time scales
Environment | Can be variable and affected by changes in population composition (also known as seascape, or a dancing landscape)
Population | Two types: invader and resident | Any number of types
| Small initial frequency of the invader | Initial population composition can be arbitrary
| Can introduce new “mutants” | All types must initially be present in the population, even if at a near-zero frequency
| Requires “Lotka-Volterra” type growth rates ($x^{\prime}=xF(t)$)
Purpose/ question | Invasion: can a mutant invade the resident population? | Evolution of a parametrically heterogeneous system over time due to natural selection
| Uses both theoretical analysis (bifurcation theory) and numerical solutions
Visual representation | Pairwise invasibility plot (PIP) | Bifurcation diagram of the corresponding parametrically homogeneous system
Table 3: A comparison of adaptive dynamics and the Reduction theorem
### Different types of punishment/reward functions
#### Case 1. Moderate punishment
Consider the case, when the punishment function is of the form
$f(c)=a\frac{1-c}{1+c}$. This functional form allows to incorporate moderate
levels of punishment/reward, the severity of which is determined by the value
of the parameter $a$.
The pairwise invasibility plot (PIP), which allows to visualize under what
relative values of $c_{m}$ and $c_{res}$ the dominant eigenvalue $\lambda$
changes its sign, can be seen on Figure 2b,c, for $a=1$ and $a=4$. Blue
regions correspond to the case when $\lambda(c_{res},c_{m})<0$, and
consequently the mutant cannot invade; red regions correspond to the case when
$\lambda(c_{res},c_{m})>0$, and the mutant can invade. The point of
intersection of the two curves corresponds to a convergence stable (CSS) but
evolutionarily unstable strategy, which can be invaded by “mutants” with large
enough values of $c_{m}$.
The selection gradient for this punishment function,
$D(c_{m})=r-\frac{2a}{(1+c_{m})^{2}}$, so the mutant with $c_{m}\approx c_{r}$
can invade when $a<\frac{r}{2}(1+c_{m})^{2}$ (see Figure 2 a).
Figure 2: Selection gradient and pairwise invasibility plots (PIPs) for
function of the type $f(c)=a\frac{1-c}{1+c}$. Blue regions correspond to
parameter values where the mutant cannot invade; red regions correspond to
parameter values where it can invade. (a) selection gradient, defined in
Equation (6) (b) PIP for $a=1$; the singular strategy $c_{res}^{*}$ is
evolutionarily unstable and convergence stable (c) PIP for $a=4$; the singular
strategy $c_{res}^{*}$ is evolutionarily unstable and convergence stable. This
punishment/reward function is not effective against aggressive over-consumers.
These results suggest that modest punishment can therefore protect only from
modest over-consumers. However, more aggressive over-consumers cannot be kept
of out the population, as the punishment is not severe enough.
#### Case 2. Severe punishment/generous reward
Now consider a case, when the punishment function is of the form
$f(c)=a(1-c)^{3}$, which allows to impose much more severe punishment on over-
consumers and more generous reward on under-consumers compared to the previous
case.
The PIP for this functional form can be seen on Figure 3 for $a=0.1$ and
$a=0.6$. Once again, blue regions correspond to the case when
$\lambda(c_{res},c_{m})<0$, and consequently the mutant cannot invade; red
regions correspond to the case when $\lambda(c_{res},c_{m})>0$, and the mutant
can invade.
For this type of punishment/reward function, there is a region where invader
can invade but unlike the previous case, there exists an upper boundary for
the possible values of successful $c_{m}$. Unlike in the previous case,
singular strategy is stable, which predictably implies that punishment needs
to be severe enough in order to be able to prevent invasion by over consumers.
Moreover, for large enough values of $a$, there can exist two singular
strategies, one evolutionarily stable and one evolutionarily unstable. In this
case, the more “altruistic” ESS, which corresponds to smaller values of
$c_{res}^{*}$, is unstable, probably because in this case the reward for
under-consumption is insufficiently large, and the punishment is not
sufficiently severe. The second ESS, which corresponds to larger values of
$c_{res}^{*}$ is evolutionarily stable, and so the mutant over-consumer cannot
invade.
For this punishment function, the selection gradient is
$D(c_{m})=r-3a(1-c_{m})^{2}$, so $D(c_{m})>0$ if either
$c_{m}>1-\sqrt{\frac{r}{3a}}$ or $c_{m}<1+\sqrt{\frac{r}{3a}}$. One can see
that there exists a region, where the mutant cannot invade even when the
enforcement of punishment, accounted for with parameter $a$, is quite small.
This can be interpreted as the rewards of over-consumption being too small
below the red invasibility zone, and the costs of punishment being too great
above the red invasibility zone.
Figure 3: Selection gradient and pairwise invasibility plots for function of
the type $f(c)=a(1-c)^{3}$. Blue regions correspond to parameter values where
the mutant cannot invade; red regions correspond to parameter values where it
can invade. (a) selection gradient, defined in Equation (6) (b) PIP for
$a=0.1$; the singular strategy $c_{res}^{*}$ is evolutionarily and convergence
stable (c) PIP for $a=0.6$; there seem to appear two singular strategies for
large enough values of $a$; one of the strategies is evolutionarily unstable,
as neither the reward is large enough for the under-consumers, nor the
punishment severe enough for over-consumers; the second strategy is
$c_{res}^{*}$ is evolutionarily and convergence stable. This punishment/reward
function is effective against aggressive over-consumers.
#### Case 3. Separating punishment and reward
Finally, consider the following punishment/reward function:
$f(c)=\rho(1-c^{\eta})$. This functional form allows to separate the impact of
punishment for over-consumption, which is increased or decreased depending on
the value of parameter $\eta$, and reward for under-consumption, which is
influenced primarily by the value of parameter $\rho$. In this case, the
dominant eigenvalue is given by
$\lambda=r(c_{m}-c_{res})-\rho(c_{m}^{\eta}-c_{res}^{\eta})$. As one can
clearly see, if for instance $r=\rho$, then $\lambda>0$ when $\eta<1$, and so
the mutant should be able to invade, since the punishment for over-consumption
is always less than its reproductive benefits; if $\eta>1$, the inequality is
reversed, and punishment overwhelms reproductive benefits. If $\eta=1$, then
everything is determined solely by the relative values of growth rate $r$ and
the reward parameter $\rho$. If $c_{m}\approx c_{res}$, then the mutant can
(or cannot) invade if $D(c_{m})=r-\rho\eta c_{m}^{(\eta-1)}$ is positive
(negative).
As one can see on Figure 4 and 5, this type of function behaves like case 1
for $\eta<1$ and like case 2 for $\eta>1$. These results reiterate the claim
that was made in the previous two cases: punishment needs to be severe enough
in order to successfully prevent invasion by over-consumers.
Figure 4: Pairwise invasibility plots for function type
$f(c)=\rho(1-c^{\eta}),$ $\eta=0.9$. The effectiveness of this function is the
same as was for case 1: punishment is not severe enough to keep over-consumers
out of the population. Figure 5: Pairwise invasibility plots for function type
$f(c)=\rho(1-c^{\eta}),$ $\eta=1.2$. The effectiveness of this function is the
same as was for case 2: punishment is sufficiently severe to keep over-
consumers out of the population.
### Modeling parametrically heterogeneous populations using the Reduction
theorem
Adaptive dynamics allows to obtain analytical results about system
susceptibility to invasion in the case when the invading “mutant” (in this
case an over-consumer with a higher value of $c_{m}$) is rare. However, what
if the initial mutant is not rare (invasion by a group)? What if there is more
than one type of mutant (the system did not have time to recalibrate before
the new mutant came along)? We attempt to address these questions through
application of the Reduction theorem for replicator equations.
Without incorporating heterogeneity one cannot study the effects of natural
selection on the system, and until recently any attempts to write such models
resulted in systems of immense dimensionality. However, the proposed approach
allows us to overcome this problem.
Assume that each individual in the population studied is characterized by its
own value of some intrinsic parameter, such as birth, death, resource
consumption rates, etc. (a set of individuals characterized by a particular
value of the parameter studied is referred to here as a clone). The
distribution of clones will change over time due to system dynamics, since
different clones impose different selective pressures on each other.
Consequently, the mean of the parameter studied also changes over time. The
mean can be computed at each time point using the moment generating function
of the initial distribution of clones in the population. The changes in the
mean of the parameter allows easy tracking of changes in population
composition in response to changes in micro-environment (such as varying
nutrient availability) or with respect to different initial distributions of
clones.
Now let us introduce an auxiliary variable $q(t)$, which satisfies
$q(t)^{\prime}=\frac{N(t)}{kz(t)}$, so that one may rewrite the system in the
following form:
$\displaystyle\frac{x_{c}^{\prime}(t)}{x_{c}(t)}$
$\displaystyle=r(c-q(t)^{\prime})+f(c),$ $\displaystyle z^{\prime}(t)$
$\displaystyle=p-dz(t)+e\frac{N(t)(1-E^{t}[c])}{N(t)+z(t)}.$
Then
$x_{c}(t)=x_{c}(0)e^{-q(t)+t(rc+f(c))}.$ (7)
Total population size is then given by
$N(t)=\int_{c\in\mathbb{A}}x_{c}(t)dc=N(0)e^{-q(t)}\int_{c}e^{t(rc+f(c))}P_{c}(0)dc,$
(8)
where $P_{c}(0)=\frac{x_{c}(0)}{N(0)}$, and the current-time pdf is given by
$P_{c}(t)=\frac{x_{c}(t)}{N(t)}=\frac{e^{t(rc+f(c))}P_{c}(0)}{\int
e^{t(rc+f(c))}P_{c}(0)dc}.$ (9)
Now, the expected value of $c$ can be calculated from the definition:
$E^{t}[c]=\int_{c}cP_{c}(t)=\frac{\int_{c}ce^{t(rc+f(c))}P_{c}(0)dc}{\int_{c}e^{t(rc+f(c))}P_{c}(0)dc}.$
(10)
The final system of equations becomes :
$\displaystyle z^{\prime}(t)$ $\displaystyle=$ $\displaystyle
p-dz+e\frac{N(t)(1-E^{t}[c])}{N(t)+z(t)},$ (11) $\displaystyle q^{\prime}(t)$
$\displaystyle=$ $\displaystyle\frac{N(t)}{kz(t)},$ (12)
where $N(t)$ and $E^{t}[c]$ are defined above.
The case, when $f(c)=0$ was investigated in [9]. The authors observed that
although it takes longer for a heterogeneous population to go extinct, tragedy
of the commons eventually happens if the maximum value of $c$ is large enough.
Moreover, one could observe transitional regimes as the population was
evolving towards being composed of increasingly efficient over-consumers. This
situation can be used to forecast upcoming crisis and start implementing
punishment functions.
In the proposed form, if the system is parametrically homogeneous system, i.e.
when $E^{t}[c]$ is constant, $f(c)$ can be factored into equation
$\frac{x^{\prime}}{x}=r(c+f(c)-\frac{bx}{kz})$, and consequently, the spectrum
of possible dynamical behaviors for this modified system should be
qualitatively the same compared to the initial model, analyzed in [9] and
summarized in Section 2. However, while in [9] nothing prevented increase of
$E^{t}[c]$ up to the maximum possible value, here we want to investigate
punishment/reward functions that will prevent the increase of $E^{t}[c]$ that
would otherwise drive the dynamics outside of the regions of sustainable
coexistence with the common resource.
## Results
We evaluated the effectiveness of several punishment/reward functions on the
dynamics of growth of a heterogeneous population of consumers supported by a
common dynamical carrying capacity, which can be increased or decreased by the
individuals themselves. We evaluated the effects of the same type of
punishment on populations with two different types of initial distributions of
clones, namely truncated exponential with parameter $\mu=10$, and Beta
distribution with parameters $\alpha=2\>,\beta=2$ and $\alpha=2,\>\beta=5$
(see Figure 6). Parameter values were chosen in such a way as to give
different shapes of the initial distribution of clones within the population
We hypothesized that the effectiveness of punishment will depend not only
intrinsic parameter values of the system but also on the initial composition
of the population. Specifically, we hypothesized that higher levels of
punishment/reward will be necessary for initial distributions, where
population composition is spread out further away from small values of $c,$
such as Beta distributions with parameters $\alpha=2,\>\beta=5$ and even more
so Beta distribution with parameters $\alpha=2,\>\beta=2$ (see Figure 6b)
compared to truncated exponential distribution, where fewer over consumers are
initially present in the population (see Figure 6a).
Figure 6: Initial distributions (a) truncated exponential with parameter
$\mu=10$ (b) Beta distribution with parameters $\alpha=2,\beta=2$ and
$\alpha=2,\beta=5$ .
First, we evaluated the effectiveness of the moderate punishment function of
type $f(c)=a\frac{1-c}{1+c}$; the severity of punishment is captured through
varying parameter $a$. The initial distribution was taken to be truncated
exponential with parameter $\mu=10$. We took parameter $a=0;\>0.5;\>1;2$ and
plotted the changes in $x_{c}(t)$ for various $c$ over time (Figure 7), as
well as the changes in the total population size and the amount of resource
(Figure 8). We observed that when the punishment imposed is moderate, over-
consumption could be avoided only when the value of $a$ was very high, i.e.
when punishment is imposed very severely.
Figure 7: (a) Truncated exponential, moderate punishment (case 1), set 1,
$a=0$. (b) Truncated exponential, case 1, set 1, $a=0.5$. (c) Truncated
exponential, case 1, set 1, $a=1$. (d) Truncated exponential, case 1, set 1,
$a=2$. Figure 8: Truncated exponential distribution, set 1, case 1, dynamics
of the total population size and total resource with respect to different
values of $a$ (different levels of severity of imposed punishment). One can
see that successful management of over consumers was possible only when
punishment implementation was very high.
Similar results were observed for the cases of Beta distribution with
parameters $\alpha=2,\>\beta=2$ and $\alpha=2,\>\beta=5$ (additional figures
are provided in Appendix). The value of $a$ that was necessary for successful
management of over-consumers varied depending on different initial
distributions, indicating that in order to be able to prevent the tragedy of
the commons, one needs to evaluate not only the type of punishment and the
severity of its enforcement but also match it to the composition of the
population, since one level of punishment can be effective for one
distribution of clones within a population of consumers and not another.
Noticeably, this kind of insight would be impossible to obtain using the
analytical methods of adaptive dynamics.
Next, we conducted the same set of numerical experiments for the severe
punishment\generous reward function $f(c)=a(1-c)^{3}$. We observed that the
value of $a$ that would correspond to successful restraint of over-consumers
was much lower than in the previous case for all initial distributions
considered here (see Figures 10, 9; additional examples are provided in
Appendix). The system was able to support individuals with higher values of
parameter $c$ present in the initial population than in the previous case. In
some cases we were also able to observe brief periods of oscillatory
transitional dynamical behavior before the system collapsed (see Figures 9 and
10).
Figure 9: Severe punishment/generous reward (case 2). Beta distribution with
parameters $[\alpha,\beta]=[2,2]$. (a) $a=0$, (b) $a=0.1$, (c) $a=0.17$, (d)
$a=0.2$. One can see the population going through transitional regimes before
collapse, when the punishment is implemented severely but not quite severely
enough (when $a=0.7$). This most probably corresponds to the expected value of
parameter $c$ going through region 3 in the phase parameter portrait of the
non-distributed system. Figure 10: Case 2, set 1. Beta distribution with
parameters $[2,2]$. Changes in total population size and of the common
resource over time.
Finally, we evaluated punishment function of the type $f(c)=\rho(1-c^{\eta}),$
where the intensity of punishment and reward are accounted for by parameters
$\eta$ and $\rho$ respectively. We observed that in order to evaluate the
expected effectiveness of the punishment/reward system one needs to not only
adjust parameters $\rho$ and $\eta$ (see Figure 11) to each particular case
considered but also be able to evaluate the expected range of parameter $c$
(see Figure 12), since one level of punishment may be appropriate for one set
of initial conditions but not another. For instance, as one can see on Figure
12, the time to collapse under fixed values of parameter $\rho$ and $\eta$ is
different for different initial distributions depending on the maximum value
of $c$ present in the initial population. Moreover, in accordance with our
hypothesis, indeed the time to collapse varies depending on the initial
distribution of the clones within the population, and the higher the frequency
of over-consumers is in the initial population, the worse the prognosis. For
the examples considered, population in which the clones are distributed
according to truncated exponential distribution is less likely to collapse due
to over-consumption than Beta distribution with parameters
$\alpha=2,\>\beta=5$ which in turn is slightly less prone to collapse than
Beta distribution with parameters $\alpha=2,\>\beta=2$ (Figure 6).
Figure 11: $f(c)=\rho(1-c^{\eta}),$ initial Beta distribution with parameters
$\alpha=2,\beta=2$, $c\in[0,3]$, $\eta=1.2$. Figure 12: The importance of
evaluating the range of possible values of $c_{f}$, illustrated for different
initial distribution. Punishment function is of the type
$f(c)=\rho(1-c^{\eta})$, where $\rho=0.6$, $\eta=1.2$. Initial distributions
are taken to be truncated exponential with parameter $\mu=10$, and Beta with
parameters $\alpha=2,\beta=2$ and $\alpha=2,\beta=5$; $\rho=0.6$, $\eta=1.2$.
The top row corresponds to $c\in[0,3]$; the bottom row corresponds to
$c\in[0,4]$.
### On the existence and properties of stationary distributions
The adaptive dynamics approach assumes that there exists a stable state of a
population composed from a single clone $x_{res}$. In the absence of
punishment, any initial distribution evolves in such a way, that
asymptotically only the clone having maximal possible value of the parameter
$c$ will persist. In contrast, introducing a punishment function allows for
the possibility of a stationary distribution of clones that is concentrated in
more than one point. Specifically, we show below that a population in a
stationary state may consist of up to 2 clones with punishment function of the
1st and 3rd type and of up to 3 clones with punishment function of the 2nd
type.
Assume that there exists a stationary distribution
$P_{c}(t)=\frac{x_{c}(t)}{N(t)}$ of clones $x_{c}$ that stabilizes over time,
which would occur when
$\frac{dP_{c}(t)}{dt}=\frac{x_{c}^{\prime}}{N}-\frac{x_{c}}{N^{2}}N^{\prime}=P_{c}(t)[rc+f(c)-E^{t}[rc]-E^{t}[f(c)]]=0,$
(13)
which holds only when
$rc+f(c)=E^{t}[rc]+E^{t}[f(c)].$ (14)
The left hand side of Equation (14) does not depend on time, while the right
hand side is independent of $c$, which implies that Equation (14) holds only
when
$rc+f(c)=K,$ (15)
where $K$ is a constant.
Recall that $\\{c_{i}\\}$ is the support of a probability distribution $P$ if
$P(c)>0$ for all $c_{i}$ and $P(c)=0$ otherwise. In our case, if there exists
a constant $K$ such that
$f(c_{i})=K-rc_{i},$ (16)
then to each $K$ corresponds a stationary distribution $P(c_{i})$, whose
support $\\{c_{i}\\}$ coincides with the solution to this equation. Therefore,
Equation (16) can be used to identify the values of $c$ that correspond to
distributed evolutionary steady states. (Note that a) any distribution
concentrated in a single point $c$ is clearly stationary, and b) if
$f(c)\equiv ac$, where $a$ is a constant, then every stationary distribution
is concentrated in a single point.)
It follows from Equation (16) that for a given punishment function $f(c)$, the
set of solutions to Equation (16) at fixed $K$ can be represented
geometrically as a set $S(K)$ of abscissas of points of intersection of the
curve $y=f(c)$ and the line $y=K-rc$. The constant $K$ is free, so by changing
K in such a way that the set $S(K)$ is not empty, we obtain all possible
supports of the stationary distributions (see Figure 13).
Figure 13: Steady states are concentrated at the points of intersection of
punishment function $y=f(c)$ and a line $y=K-rc$. Depending on the type of
punishment function and on the angle of the line $y=K-rc$, one may have up to
three clone types at the stationary state.
An important conclusion can be made from these figures alone. As one can see
in Figure 13a, for instance, the two points of intersection of $f(c)$ and
$y=K-rc$ that correspond to points of concentration of clones at a stationary
state, can be less than or greater than 1, depending on the parameter $r$,
which is the slope of the line $y=K-rc$. The case when one of the points of
intersection is less than one (“altruistic” under-consuming clones) and one is
greater than one (over-consumers) can be interpreted as a population of under-
consumers “supporting” the population of over-consumers at a stationary state.
Asymptotic behavior of the population, i.e., population survival or
extinction, will be determined by the dynamics of System (17), which is
derived below, and corresponding parameter values.
Once again, let $\\{c_{i}\\}$ be a support of stationary probability
distribution. Then $\frac{dx_{c_{i}}}{dt}/x_{c_{i}}=r(K-\frac{N}{kz}).$ The
right hand side does not depend on $i$, which implies that
$\ln(x_{c_{i}}(t))-\ln(x_{c_{j}}(t))=const,\>\text{for all }i,j$
and $\frac{x_{c_{i}}(t)}{x_{c_{j}}(t)}=\frac{x_{c_{i}}(0)}{x_{c_{j}}(0)}$
$\text{for all}\>t$. (Note that $i,\>j$ notation is used in this section to
distinguish the fact that we are no longer dealing with rare clones in a
resident population, and hence previously used $m,\>res$ notation is no longer
appropriate.)
The dynamics of System (4) in this case can be described by
$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle
rN(K-\frac{N}{kz})$ (17) $\displaystyle\frac{dz}{dt}$ $\displaystyle=$
$\displaystyle\gamma-\delta z+\frac{eN}{z+N}(1-E),$
where $E=E^{t}[c]=const$ and consequently does not depend on $t$.
Now let $A_{i}=\frac{x_{c_{i}}(0)}{\sum_{j}x_{c_{j}}(0)}$ be the initial
frequency of clones $x_{c_{i}}$; then $E=\sum_{i}A_{i}c_{i}$. Therefore, by
changing the initial values of $x(c_{i})$ we can vary the value of $E$ from
$min(c_{i})$ up to $max(c_{i})$. Stable equilibria $(N^{*},\>z^{*})$ of System
(17) can now be found from $N^{*}=kKz^{*}$ and $\gamma-\delta
z^{*}+\frac{eN^{*}}{z^{*}+N^{*}}(1-E)=\gamma-\delta z^{*}+B=0$, where
$B=\frac{(1-E)ekK}{1+ekK}=const$. Therefore, $z^{*}=\frac{\gamma+B}{\delta}$,
given that $\gamma+B>0$.
System (17) is very similar to the initial parametrically homogeneous model
with no punishment; the difference is that now the values of $K$ and $E$ are
not identical: the value $K$ is in one-to-one correspondence with the support
$\\{c_{i}\\}$, and the value of $E$ is defined by the support $\\{c_{i}\\}$
and initial values $\\{x_{c_{i}}(0)\\}$. _These results suggest that if the
system has a stationary distribution, then the dynamics of the resource would
be determined not by absolute sizes of the populations of ‘invaders’ and
‘residents’, but by their ratio._
Remark. In this work we are considering punishment functions, where the
support of the stationary distribution can consist of up to three points,
i.e., the system can be composed of up to three clones (see Figure 13b).
However, one can conceive of a punishment function that can support an
arbitrary number of clones in the population at the stationary distribution,
which correspond to points of intersection of the curve $y=f(c)$ and the line
$y=K-rc$. Such system is of course unlikely to survive for any significant
period of time unless the number of individuals in the clones with $c^{*}\gg
1$ is small enough. Such a case can be interpreted as a large number of under-
consumers supporting a very small number of over-consumers.
#### Example 1.
Consider the punishment function of the form $f(c)=\frac{a(1-c)}{1+c}$; the
points of tangency of the curve $y=\frac{a(1-c)}{1+c}$ and the line $y=K-rc$
are $c_{1,2}^{*}=\pm\sqrt{2a/r}-1$. In order for $c^{*}=\sqrt{2a/r}-1>0$,
condition $2a>r$ must be satisfied. At $c^{*}$, $f(c^{*})=\sqrt{2ar}-a$, so
the line $y=K-rc$ is tangent with $y=f(c)=\frac{a(1-c)}{1+c}$ at
$K=K^{*}=\sqrt{2ar}-A+r(\sqrt{2a/r}-1)$ (see dotted line in Figure 13a).
Hence, the line $y=K-rc$ intersects the curve $y=f(c)$ in two points (see
dashed red line in Figure13a) for all $K^{*}<K<a$.
For the purposes of illustration, take $a=1,\>r=0.4,\>K=0.7$ (see Figure 13a).
The stationary distribution is concentrated in two points $c_{1}=0.25$ and
$c_{2}=3$, so in its steady state, the population is composed both of under-
consumers and of over-consumers. The population dynamics is then described by
system
$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle
0.4N(0.7-\frac{N}{kz}),$ $\displaystyle\frac{dz}{dt}$ $\displaystyle=$
$\displaystyle\text{$\gamma$}-\delta z+\frac{eN}{z+N}(1-E),$
where $E\in[c_{1}=0.25,\>c_{2}=3]$. For these particular values of parameters,
the system will remain in the stable state if $\delta
z^{*}=\text{\leavevmode{g}}+\frac{eN^{*}}{z^{*}+N^{*}}(1-E)=\gamma+0.412e(1-E)>0$.
Otherwise, it will go to extinction.
#### Example 2.
Now consider the punishment function of the form $f(c)=a(1-c)^{3}$. The line
$y=K-rc$ is tangent to the curve $y=f(c)$ at $c_{1,2}^{*}=1\pm\sqrt{r/(3a)}$.
Denote $\sqrt{r/(3a)}=\psi$. Then $c_{i,2}^{*}=1\pm\psi$; the condition
$\psi<1$ needs to be satisfied in order for both roots to be positive.
Consequently,
$f(c_{1,2}^{*})=\mp\psi,$
$K_{1,2}^{*}=f(c_{2,1}^{*})+rc_{1,2}^{*}=r(1\pm\psi)\mp\psi^{3};$
and therefore
$K_{min}=r(1+\psi)-\psi^{3}<K<r(1-\psi)+\psi^{3}=K_{max}.$
There can exist up to 3 points of intersection of the curve $f(c)=a(1-c)^{3}$
and the line $y=K-rc$ depending on the value of $K$.
As one can see in Figure 13b, where
$a=r=1,\>\psi=\sqrt{1/3},\>c_{1,2}^{*}=1\pm\sqrt{1/3}$, if $K>K_{max}=1.3849,$
then there are no solutions to Equation (16); if $1<K<K_{max}$, then there
exist 2 solutions to Equation (16); if $0.6151=K_{min}<K<1$, then there can
exist 3 solutions. For instance, taking $K=0.8$ and solving equation
$f(c)=K-rc,$ i.e., $(1-c)^{3}=0.8-c$, we obtain the following solutions:
$c_{1}=0.121,$ $c_{2}=0.791$, $c_{3}=2.08$. So, in this case, at the steady
state, two ‘altruistic’ clones with $c_{1}=0.121$ and $c_{2}=0.791$ are
supporting over-consuming clones with $c_{3}=2.088$.
The overall system dynamics for this example are given by
$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle
rN(0.8-\frac{N}{kz}),$ $\displaystyle\frac{dz}{dt}$ $\displaystyle=$
$\displaystyle\text{$\gamma$}-\delta z+\frac{eN}{z+N}(1-E),$
where $E\in[c_{1}=0.121,c_{3}=2.088].$ If $k=1$, then at the stationary state
$N^{*}=0.8z^{*}$ and $\frac{N^{*}}{z^{*}+N^{*}}=\frac{0.8}{1.8}=0.444$. The
system will remain at a steady state if
$\delta z^{*}=\gamma+\frac{eN^{*}}{z^{*}+N^{*}}(1-E)=\gamma+0.444e(1-E)>0;$
Otherwise, it will go to extinction.
## Discussion
In this paper we studied the dynamics of a consumer-resource type system in
order to answer the question of whether infliction of punishment for over-
consumption and reward for under-consumption can successfully prevent, or at
least delay, the onset of the tragedy of the commons. We evaluated the
effectiveness of three types of punishment functions, as well as the effects
of punishment on populations with different initial composition of individuals
with respect to the levels of resource (over)consumption.
The proposed model was studied analytically in [9, 10] without incorporating
punishment/reward for over-/under- consumption. It describes the interactions
of a population of consumers, characterized by the value of an intrinsic
parameter $c$, with a common renewable resource in such a way that each
individual can either contribute to increasing the common dynamical carrying
capacity ($c<1$) or take more than they restore ($c>1$). As the value of $c$
increases, the population goes through a series of transitional regimes from
sustainable coexistence with the resource to oscillatory regime to eventually
committing evolutionary suicide through decreasing the common carrying
capacity to a level that can no longer support the population.
We began by identifying analytical conditions leading to the possibility of
sustainable coexistence with the common resource for a subset of cases using
adaptive dynamics. This method allows to address the question of whether a
mutant (in our case, an individual with a higher value of $c$) can invade a
homogeneous resident population of resource consumers. We evaluated the
effectiveness of three types of punishment functions: moderate punishment,
$f(c)=a\frac{1-c}{1+c}$; severe punishment, $f(c)=a(1-c)^{3}$, where the
parameter $a$ denotes the severity of implementation of punishment on
individuals with the corresponding value of parameter $c$, and function of the
type $f(c)=\rho(1-c^{\eta})$, which allows to separate the influence of reward
for under-consumption, primarily accounted for with parameter $\rho$, and
punishment for over-consumption, accounted for with parameter $\eta$.
We demonstrated that while moderate punishment/reward function can be
effective in keeping off moderate over-consumers at sufficiently high values
of $a$ (see Figures 7 and 8), the evolutionarily singular strategy in this
case is unstable, and thus eventually the population will be invaded by over-
consumers with large enough value of $c$. The severe punishment/generous
reward approach was uniformly effective, almost irrespective of the value of
$a$, and allowed invasion of moderate over consumers only in the small region
of $c\approx 1$, when the punishment is not yet severe enough to outweigh the
benefits of moderate over-consumption (see Figures 9 and 10), and the reward
does not yet provide sufficient payoffs in terms of higher growth rates. The
evolutionarily singular strategy in this case is stable, suggesting that
punishment is severe enough to prevent the tragedy of the commons.
Finally, we investigated a punishment/reward function of the type
$f(c)=\rho(1-c^{\eta})$ that allowed separating the influence of punishment
for over-consumption (parameter $\eta$) from that of rewarding under-
consumption (parameter $\rho$). This functional type behaves as a moderate
punishment/reward function for $\eta<1$, rendering it effective only for
moderate over-consumers, and it behaves like the severe punishment/reward
function for $\eta>1$; in the critical case $\eta=1$ the outcome of the
interactions between resident and invader populations is determined solely by
relative values of growth rate parameter $r$ and parameter $\rho$. Our results
suggest that just rewarding under-consumers is not enough to prevent invasion
by over consumers and hence one should not expect to be able to prevent the
tragedy of the commons through reward alone.
Adaptive dynamics techniques do not yet allow answering the question of system
invasibility when the mutant is not rare, such as in cases of migration and
consequent invasion by a group, or when the resident population is
inhomogeneous. We address these questions using the Reduction theorem for
replicator equations.
Assume that each individual consumer is characterized by his or her own value
of the intrinsic parameter $c$; a group of consumers with this value of $c$ is
referred to here as $c$-clones. This trait directly affects fitness, and thus
the distribution of clones will change over time due to system dynamics. The
clones will experience selective pressures not only from the external
environment, competing for the limited resource, but also from each other.
Consequently, the mean of the parameter will also change over time, affecting
system dynamics. The mean of the parameter can be computed at each time point
from the moment generating function of the initial distribution of clones,
which allows to evaluate the effectiveness of different types of
punishment/reward functions on population composition by tracking how the
distribution of clones changed over time with respect to parameter $c$.
We evaluated the effectiveness of the same three types of punishment/reward
functions on system evolution and calculated predicted the outcomes for
different initial distributions of clones within the population, which were
taken to be truncated exponential with parameter $\mu=10$, and Beta
distribution with parameters $\alpha=2,\>\beta=2$ and $\alpha=2,\>\beta=5$.
The initial distributions were chosen in such a way as to give significantly
different shapes of the initial probability density function; in applications
they should be matched to real data, when it is available. We observed that
the intensity of implementation of punishment\reward has to differ for
different initial distributions if one is to successfully stop over-
consumption, and so in order to be able to make any reasonable predictions one
needs to understand what the initial composition of the affected population
is. We hypothesized that the higher the frequency of over consumers in the
initial population, the more severe the punishment for over-consumption would
have to be, and the more generous the reward; specifically, for our examples,
we anticipated the prognosis to be the most favorable for initial truncated
exponential distribution, followed by Beta distribution with parameters
$\alpha=2,\>\beta=5$ and then finally $\alpha=2,\>\beta=2$. This effect
additionally implies that the results obtained analytically from adaptive
dynamics can only be relevant for a subset of cases, i.e. when the invader is
rare.
As anticipated, we observed that severe punishment/generous reward approach
was much more effective in preventing the tragedy of the commons than the
moderate punishment/reward function, particularly for the cases, when over-
consumers were present at higher frequencies (such as both Beta initial
distributions). Specifically, we observed that the level of implementation,
$a$, could be nearly ten times lower for severe punishment/generous reward
system as compared to the moderate punishment function ($a\approx 0.2$ vs
$a\approx 2$) in order to obtain the same effect of selecting against the
over-consumers, which can be a very important factor in cases when there are
large costs associated with implementation of such intervention systems. This
comes not only from the severity of punishment but also from the fact that
moderate punishment allows more time for the over-consumers to replicate, and
thus by the time the punishment has an appreciable effect, the population
composition had changed, and the moderate punishment will no longer be
effective. So, in punishment implementation one needs to take into account not
only the severity of punishment but also the time window that moderate
punishment may provide, allowing over-consumers to proliferate. Within the
frameworks of the proposed model, moderate implementation of more severe
punishment\reward system is more effective than severe implementation of
moderate punishment\reward.
Proposed here is just one way to try and modify individuals’ payoffs in order
to prevent resource over-consumption - through inflicting punishment and
reward that affects the growth rates of clones directly. This approach can be
modified depending on different situations, inflicting punishment or reward
based not just on the intrinsic value of $c$ but on total resource currently
available.
##### Adaptive dynamics and the Reduction theorem.
In order to address our questions we have used two recently developed methods
for modeling parametrically heterogeneous populations: adaptive dynamics [1]
and the Reduction theorem for replicator equations [7, 8]. Adaptive dynamics
allows addressing the question of rare “mutant” clone invasion using standard
bifurcation theory. The method allows to formulate analytical conditions,
which can be conveniently visualized using pairwise invasibility plots (PIPs).
However, the method does not allow addressing questions of system invasibility
by clones that are not rare, such as in cases of invasion by a group, or when
the resident population is heterogeneous. These questions can be addressed
using the Reduction theorem for replicator equations.
We consider the moment of “invasion” as the initial time moment; more
formally, we assume that all “invaders” must be present initially in the
population, falling within some known distribution; one can then see which
clone type(s) will be favored over time due to natural selection and visualize
evolutionary trajectories, which cannot be achieved using adaptive dynamics.
The outcome depends both on the initial distribution of the clones within the
population and on the initial state of the system, i.e., other intrinsic
properties of both individuals within the population and the resources, which
in this case were assumed to be fixed. However, the system of ODEs that
results from the transformation done using the Reduction theorem is typically
non-autonomous; hence no analytical conditions can typically be obtained using
standard bifurcation theory. The two methods therefore can complement each
other. A more detailed comparison of the two methods is in Table 3.
#### Tragedy of the commons as prisoner’s dilemma
The conditions that can lead to the tragedy of the commons can be reformulated
as a game of prisoner’s dilemma: in such systems preserving the common
resource is in the best interest of the group as a whole but over-consumption
is in the interest of each particular individual. In the cases where decisions
about the resource are made by each individual independently, tragedy of the
commons seems to be inevitable. However, a number of cases have been observed,
when the tragedy could be avoided if the individuals within the population
were able to communicate with each other [14]. The common thread that runs
through many of the available examples is that 1) in small groups the effects
of over-consumption are immediately noticeable, since within a small
population each individual’s actions are more visible than in larger
populations, and 2) punishment is enforced without delay.
In this paper we are dealing with a particular case of a game of prisoner’s
dilemma, where punishment/reward functions $f(c)$ affect individual payoffs in
such a way as to give the possibility to outweigh both the benefits of over-
consumption and the deleterious effects of under-consumption (see Figure 14).
We were able to demonstrate that preventing the tragedy of the commons through
solely rewarding under-consumers is unlikely. However, the effectiveness of
“intervention” is increased when both reward and punishment systems are
enforced, although they do need to be modified depending on population
composition, since the choices of individuals may be affected not only by
their immediate ‘payoffs’ but also by the actions of those surrounding them.
Therefore, if one is to try and avert the tragedy of the commons through
punishing over-consumption and rewarding under-consumption, one needs to not
only find an appropriate punishment/reward function but also calibrate it to
the current composition of each particular population.
Figure 14: Tragedy of the commons as prisoner’s dilemma. The tragedy can be
avoided if the immediate payoffs of all the players are modified through
appropriate punishment/reward functions.
## Acknowledgments
The authors wougameld like to thank Kalle Parvinen and John Nagy for their
help with adaptive dynamics. This project has been supported by grants from
the National Science Foundation (NSF - Grant DMPS-0838705), the National
Security Agency (NSA - Grant H98230-09-1-0104), the Alfred P. Sloan
Foundation, the Office of the Provost of Arizona State University and
Intramural Research Program of the NIH, NCBI. This material is also based upon
work partially supported by the National Science Foundation under Grant No.
DMS-1135663.
## References
* [1] S.A.H. Geritz, E. Kisdi, G. Mesze, and J.A.J. Metz. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, _Evolutionary ecology_ , 12(1): 35-57, 1997.
* [2] AN Gorban, _Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis_ , Nauka, Novosibirsk, 1984.
* [3] AN Gorban, Selection theorem for systems with inheritance, _Math. Model. Nat. Phenom_ , 2(4):1-45, 2007.
* [4] G. Hardin. The tragedy of the commons. _Journal of Natural Resources Policy Research_ , 1(3):243-253, 2009.
* [5] J. Hofbauer and K. Sigmund. Evolutionary game dynamics. _Bulletin of the American Mathematical Society_ , 40(4):479, 2003.
* [6] D. J Rankin and A. Lopez-Sepulcre. Can adaptation lead to extinction? _Oikos_ , 111(3):616-619, 2005.
* [7] G.P. Karev. On mathematical theory of selection: continuous time population dynamics._Journal of mathematical biology_ , 60(1):107-129, 2010.
* [8] G.P. Karev. Principle of minimum discrimination information and replica dynamics._Entropy_ , 12(7):1673-1695, 2010.
* [9] I. Kareva, F. Berezovskaya, and C. Castillo-Chavez. Transitional regimes as early warning signals in resource dependent competition models. _Mathematical Biosciences_ , 240(2): 114-123, 2012.
* [10] D.C. Krakauer, K.M. Page, and D.H. Erwin. Diversity, dilemmas, and monopolies of niche construction. _The American Naturalist_ , 173(1):26-40, 2009.
* [11] M. Milinski, D. Semmann, and H.J. Krambeck. Reputation helps solve the tragedy of the commons._Nature_ , 415(6870):424-426, 2002.
* [12] J.H. Miller and S.E. Page. _Complex adaptive systems: An introduction to computational models of social life_. Princeton University Press, 2007.
* [13] M.A. Nowak._Evolutionary dynamics: exploring the equations of life_. Belknap Press, 2006.
* [14] E. Ostrom. _Governing the commons: The evolution of institutions for collective action_. Cambridge University Press, 1990.
* [15] S.E. Page. _The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies (New Edition)_. Princeton University Press, 2008.
* [16] S.E. Page. _Diversity and complexity._ Princeton University Press, 2011.
* [17] T.L. Vincent and J.S. Brown. _Evolutionary game theory, natural selection, and Darwinian dynamics._ Cambridge University Press, 2005.
* [18] B. Vollan and E. Ostrom. Cooperation and the commons. _Science_ , 330(6006):923-924, 2010.
|
arxiv-papers
| 2012-12-16T17:28:41 |
2024-09-04T02:49:39.336141
|
{
"license": "Public Domain",
"authors": "Irina Kareva, Benjamin Morin, Georgy Karev",
"submitter": "Irina Kareva",
"url": "https://arxiv.org/abs/1212.3807"
}
|
1212.3853
|
Introduction The past decade has seen the rapid increase of content
distribution using the Internet as the medium of delivery Lab09. Users and
applications expect a low cost for content, but at the same time require high
levels of quality of service. However, providing content distribution at a low
cost is challenging. The major costs associated with meeting demand at a good
quality of service are (i) the high cost of hosting services on the managed
infrastructure of CDNs such as Akamai nor07,akamai, and (ii) the lost revenue
associated with the fact that digital content is easily duplicable, and hence
can be shared in an illicit peer-to-peer (P2P) manner that generates no
revenue for the content provider. Together, these factors have led content
distributors to search for methods of defraying costs.
One technique that is often suggested for defraying distribution costs is to
use legal peer-to-peer (P2P) networks to supplement provider distribution
pando,rawflow,RodTan06. It is well documented that the efficient use of P2P
methods can result in significant cost reductions from the perspective of ISPs
KarRod05,nor07; however there are substantial drawbacks as well. Probably the
most troublesome is that providers fear losing control of content ownership,
in the sense that they are no longer in control of the distribution of the
content and worry about feeding illegal P2P activity.
Thus, a key question that must be answered before we can expect mainstream
utilization of P2P approaches is: How can users that have obtained content
legally be encouraged to reshare it legally? Said in a different way, can
mechanisms be designed that ensure legitimate P2P swarms will dominate the
illicit P2P swarms?
In this paper, we investigate a “revenue sharing” approach to this issue. We
suggest that users can be motivated to reshare the content legally by allowing
them to share the revenue associated with future sales. This can be
accomplished through either a lottery scheme or by simply sharing a fraction
of the sale price. Recent work on using lotteries to promote societally
beneficial conduct MerPra09 suggests that such schemes could potentially see
wide spread adoption.
Such an approach has two key benefits: First, obviously, this mechanism
ensures that users are incentivized to join the legitimate P2P network since
they can profit from joining. Second, less obviously, this approach actually
damages the illicit P2P network. Specifically, despite the fact that content
is free in the illicit P2P network, since most users expect a reasonable
quality of service, if the delay in the illegitimate swarm is large they may
be willing to use the legitimate P2P network instead. Thus, by encouraging
users to reshare legitimately, we are averting them from joining the illicit
P2P network, reducing its capacity and performance; thus making it less likely
for others to use it.
The natural concern about a revenue sharing approach is that by sharing
profits with users, the provider is losing revenue. However, the key insight
provided by the results in this paper is that by discouraging users from
joining illicit P2P network, the increased share (possibly exponentially more)
of legitimate copies makes up for the cost of sharing revenue with end-users.
More specifically, the contribution of this paper is to develop and analyze a
model to explore the revenue sharing approach described above. Our model (see
Section s.model) is a fluid model that builds on work studying the capacity of
P2P content distribution systems. The key novel component of the model is the
competition for users among an illicit P2P system and a legal content
distribution network (CDN), which may make use of a supplementary P2P network
with revenue sharing. The main results of the paper (see Section s.results)
are Theorems thm:inefficient-thm:efficient-bass, which highlight the order-of-
magnitude gains in revenue extracted by the provider as a result of
participating in revenue sharing. Further, In addition to the analytic
results, to validate the insights provided by our asymptotic analysis of the
fluid model we also perform numerical experiments of the underlying finite
stochastic model. Tables tbl:ineff and tbl:eff summarize these experiments,
which highlight both that the results obtained in the fluid model are quite
predictive for the finite setting and that there are significant beneficial
effects of revenue sharing.
There is a significant body of prior work modeling and analyzing P2P systems.
Perhaps the most related work from this literature is the work that focuses on
server-assisted P2P content distribution networks
WanYeo07,SetApo07,LiuZha08,ChePon08,ShaJoh10,ParSha11A in which a central
server is used to “boost” P2P systems. This boost is important since pure P2P
systems suffer poor performance during initial stages of content distribution.
In fact, it is this initially poor performance that our revenue sharing
mechanism exploits to ensure that the legitimate P2P network dominates.
Two key differentiating factors of the current work compared to this work are:
(i) We model the impact of competition between legal and illegal swarms on the
revenue extraction of a content provider. (ii) Unlike most previous works on
P2P systems, we consider a time varying viral demand model for the evolution
of demand in a piece of content based on the Bass diffusion model (see Section
s.model). Thus, we model the fact that interest in content grows as interested
users contact others and make them interested.
With respect to (i), there has been prior work that focuses on identifying the
relative value of content and resources for different users gamenets,MisIon10.
For instance, gamenets deals with creating a content exchange that goes beyond
traditional P2P barter schemes, while MisIon10 attempts to characterize the
relative value of peers in terms of their impact on system performance as a
function of time. However, to the best of our knowledge, ours is the first
work that considers the question of economics and incentives in hybrid P2P
content distribution networks.
With respect to (ii), there has been prior work that considers fluid models of
P2P systems such as QiuSri04,MasVoj05,YanVec06. However, these all focus on
the performance evaluation of a P2P system with constant demand rate. As
mentioned above, a unique facet of our approach is that we explicitly make use
the transient nature of demand in our modeling. In the sense of explicitly
accounting for transient demand, the closest work to ours is ShaJoh10.
However, ShaJoh10 focuses only on jointly optimizing server and P2P usage in
the case of transient demand in order to obtain a target delay guarantee at
the lowest possible server cost.
The remainder of the paper is organized as follows. We first introduce the
details of our model in Section s.model. Then, Section s.results summarizes
analytic and numeric results, the proofs of which are included in the
appendix. Finally, Section s.conclusion provides concluding remarks.
|
arxiv-papers
| 2012-12-17T00:17:36 |
2024-09-04T02:49:39.348530
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vinod Ramaswamy, Sachin Adlakha, Srinivas Shakkottai and Adam Wierman",
"submitter": "Vinod Ramaswamy Pillai",
"url": "https://arxiv.org/abs/1212.3853"
}
|
1212.3989
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IJMC: INTERNATIONAL JOURNAL OF MATHEMATICAL COMBINATORICS (2010), Vol 2
#A07-14
GENERALIZATIONS OF POLY-BERNOULLI NUMBERS AND POLYNOMIALS
Hassan Jolany and M.R.Darafsheh, R.Eizadi Alikelaye
School of Mathematics,college of science,university of Tehran, Tehran, Iran
[email protected]
Received April 12, 2010: , Accepted: May 28, 2010
Keywords:Poly-Bernoulli polynomials,Euler polynomials
,generalized Euler polynomials,generalized poly-Bernoulli polynomials
Abstract
The Concepts of poly-Bernoulli numbers $B_{n}^{(k)}$ , poly-Bernoulli
polynomials $B_{n}^{k}{(t)}$ and the generalized poly-bernoulli numbers
$B_{n}^{(k)}(a,b)$ are generalized to $B_{n}^{(k)}(t,a,b,c)$ which is called
the generalized poly-Bernoulli polynomials depending on real parameters a,b,c
.Some properties of these polynomials and some relationships between
$B_{n}^{k}$ , $B_{n}^{(k)}(t)$ , $B_{n}^{(k)}(a,b)$ and $B_{n}^{(k)}(t,a,b,c)$
are established.
## 1 Introduction
In this paper we shall develop a number of generalizations of the poly-
Bernoulli numbers and polynomials , and obtain some results about these
generalizations.They are fundamental objects in the theory of special
functions.
Euler numbers are denoted with $B_{k}$ and are the coefficients of Taylor
expansion of the function $\frac{t}{e^{t}-1}$ ,as following;
$\frac{t}{e^{t}-1}=\sum_{k=0}^{\infty}B_{k}\frac{t^{k}}{k!}.$
The Euler polynomials $E_{n}(x)$ are expressed in the following series
$\frac{2e^{xt}}{e^{t}+1}=\sum_{k=0}^{\infty}E_{k}(x)\frac{t^{k}}{k!}.$
for more details, see $[1]$-$[4].$
In $[10]$, Q.M.Luo, F.Oi and L.Debnath defined the generalization of Euler
polynomials
$E_{k}(x,a,b,c)$ which are expressed in the following series :
$\frac{2c^{xt}}{b^{t}+a^{t}}=\sum_{k=0}^{\infty}E_{k}(x,a,b,c)\frac{t^{k}}{k!}.$
where $a,b,c\in\mathcal{Z}^{+}$.They proved that
$I)$ for $a=1$ and $b=c=e$
$E_{k}(x+1)=\sum_{j=0}^{k}\left({\begin{array}[]{l}k\\\
j\end{array}}\right)E_{j}(x),$ (1)
and
$E_{k}(x+1)+E_{k}(x)=2x^{k},$ (2)
$II)$ for $a=1$ and $b=c$ ,
$E_{k}(x+1,1,b,b)+E_{k}(x,1,b,b)=2x^{k}(\ln b)^{k}.$ (3)
In$[5]$, Kaneko introduced and studied poly-Bernoulli numbers which generalize
the classical Bernoulli numbers. Poly-Bernoulli numbers $B_{n}^{(k)}$ with
$k\in\mathcal{Z}$ and $n\in\mathcal{N}$, appear in the following power series:
$\frac{Li_{k}(1-e^{-x})}{1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}\frac{t^{n}}{n!},\hskip
14.22636pt(*)$
where $k\in\mathcal{Z}$ and
$Li_{k}(z)=\sum_{m=1}^{\infty}\frac{z^{m}}{m^{k}}.\hskip 17.07164pt|z|<1$
so for $k\leq 1$,
$Li_{1}(z)=-\ln(1-z),Li_{0}(z)=\frac{z}{1-z},Li_{-1}=\frac{z}{(1-z)^{2}},...\hskip
5.69054pt.$
Moreover when $k\geq 1$, the left hand side of $(*)$ can be wrriten in the
form of ”interated integrals”
$e^{t}\frac{1}{e^{t}-1}\int_{0}^{t}\frac{1}{e^{t}-1}\int_{0}^{t}...\frac{1}{e^{t}-1}\int_{0}^{t}\frac{t}{e^{t}-1}dtdt...dt$
$=\sum_{n=0}^{\infty}B_{n}^{(k)}\frac{t^{n}}{n!}.$
In the special case, one can see $B^{(1)}_{n}=B_{n}$.
###### Definition 1
The poly-Bernoulli polynomials,$B_{n}^{(k)}(t)$, are appeared in the expansion
of $\frac{Li_{k}(1-e^{-x})}{1-e^{-x}}e^{xt}$ as following,
$\frac{Li_{k}(1-e^{-x})}{1-e^{-x}}e^{xt}=\sum_{n=0}^{\infty}\frac{B_{n}^{(k)}(t)}{n!}x^{n}$
(4)
for more details, see $[6]-[11]$.
###### Proposition 1
(Kaneko theorem$[6]$) The Poly-Bernoulli numbers of non-negative index $k$
,satisfy the following
$B_{n}^{(k)}=(-1)^{n}\sum_{m=1}^{n+1}\frac{(-1)^{m-1}(m-1)!\left\\{{\begin{array}[]{l}n\\\
m-1\end{array}}\right\\}}{m^{k}},$ (5)
and for negative index k, we have
$B_{n}^{(-k)}=\sum_{j=0}^{min(n,k)}(j!)^{2}\left\\{{\begin{array}[]{l}n+1\\\
j+1\end{array}}\right\\}\left\\{{\begin{array}[]{l}k+1\\\
j+1\end{array}}\right\\},$ (6)
where
$\left\\{{\begin{array}[]{l}n\\\
m\end{array}}\right\\}=\frac{(-1)^{m}}{m!}\sum_{l=0}^{m}(-1)^{l}\left(\begin{array}[]{l}m\\\
l\end{array}\right)l^{n}\hskip 8.53581ptm,n\geq 0$ (7)
###### Definition 2
Let $a,b>0$ and $a\neq b$. The generalized poly-Bernoulli numbers
$B_{n}^{(k)}(a,b)$, the generalized poly-Bernoulli polynomials
$B_{n}^{(k)}(t,a,b)$ and the polynomial $B_{n}^{(k)}(t,a,b,c)$ are appeared in
the following series respectively;
$\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}=\sum_{n=0}^{\infty}\frac{B_{n}^{(k)}(a,b)}{n!}t^{n}\hskip
11.38109pt|t|<\frac{2\pi}{|\ln a+\ln b|},$ (8)
$\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}e^{xt}=\sum_{n=0}^{\infty}\frac{B_{n}^{(k)}(x,a,b)}{n!}t^{n}\hskip
11.38109pt|t|<\frac{2\pi}{|\ln a+\ln b|},$ (9)
$\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}c^{xt}=\sum_{n=0}^{\infty}\frac{B_{n}^{(k)}(x,a,b,c)}{n!}t^{n}\hskip
11.38109pt|t|<\frac{2\pi}{|\ln a+\ln b|},$ (10)
## 2 The main theorems
We present some recurrence formulae for generalized poly-Bernoulli
polynomials.
###### Theorem 1
Let $x\in\mathbb{R}$ and $n\geq 0$. For every positive real numbers a,b and c
such that $a\neq b$ and $b>a$ , we have
$B_{n}^{(k)}(a,b)=B_{n}^{(k)}\left(\frac{-\ln b}{\ln a+\ln b}\right)(\ln a+\ln
b)^{n},$ (11) $B_{j}^{(k)}(a,b)=\sum_{i=1}^{j}(-1)^{j-i}(\ln a+\ln b)^{i}(\ln
b)^{j-i}\left(\begin{array}[]{l}j\\\ i\end{array}\right)B_{j}^{(k)},$ (12)
$B_{n}^{(k)}(x;a,b,c)=\sum_{l=0}^{n}\left(\begin{array}[]{l}n\\\
l\end{array}\right)(\ln c)^{n-l}B_{l}^{(k)}(a,b)x^{n-l},$ (13)
$B_{n}^{(k)}(x+1;a,b,c)=B_{n}^{(k)}(x;ac,\frac{b}{c},c),$ (14)
$B_{n}^{(k)}(t)=B_{n}^{(k)}(e^{t+1},e^{-t}),$ (15) $B_{n}^{(k)}(x,a,b,c)=(\ln
a+\ln b)^{n}B_{n}^{(k)}(\frac{-\ln b+x\ln c}{\ln a+\ln b}).$ (16)
Proof:By applying definition 2, we have
(11)
$\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}=\sum_{n=0}^{\infty}\frac{B_{n}^{(k)}(a,b)}{n!}t^{n}$
$=\frac{1}{b^{t}}\left(\frac{Li_{k}(1-e^{-t\ln ab})}{1-e^{-t\ln
ab}}\right)=e^{-t\ln b}\left(\frac{Li_{k}(1-e^{-t\ln ab})}{1-e^{-t(\ln
ab)}}\right)$
$=\sum_{n=0}^{\infty}B_{n}^{(k)}\left(\frac{-\ln b}{\ln a+\ln b}\right)(\ln
a+\ln b)^{n}\frac{t^{n}}{n!}$
Therefore
$B^{(k)}_{n}(a,b)=B_{n}^{(k)}\left(\frac{-\ln b}{\ln a+\ln b}\right)(\ln a+\ln
b)^{n}.$
(12)
$\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}=\frac{1}{b^{t}}\left(\frac{Li_{k}(1-(ab)^{-t\
})}{1-e^{-t\ln ab}}\right)=\left(\sum_{k=0}^{\infty}\frac{(\ln
b)^{k}}{k!}(-1)^{k}t^{k}\right)\left(\sum_{n=0}^{\infty}B_{n}^{(k)}\frac{(\ln
a+\ln b)^{n}}{n!}t^{n}\right)$
$=\sum_{j=0}^{\infty}\left(\sum_{i=0}^{j}(-1)^{j-i}B_{i}^{(k)}\frac{(\ln a+\ln
b)^{i}}{i!(j-i)!}(\ln b)^{j-i}\right)t^{j}$
So we have
$B_{j}^{(k)}(a,b)=\sum_{i=0}^{j}(-1)^{j-i}(\ln a+\ln b)^{i}(\ln
b)^{j-i}\left(\begin{array}[]{l}j\\\ i\end{array}\right)B_{i}^{(k)}.$
(13)
$\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}c^{xt}=\sum_{n=0}^{\infty}B_{n}^{(k)}(x,a,b,c)\frac{t^{n}}{n!}$
$=\left(\sum_{l=0}^{\infty}B_{l}^{(k)}(a,b)\frac{t^{l}}{l!}\right)\left(\sum_{i=0}^{\infty}\frac{(\ln
c)^{i}t^{i}}{i!}x^{i}\right)$
$=\sum_{l=0}^{\infty}\sum_{i=0}^{l}\frac{(\ln
c)^{l-i}}{i!(l-i)!}B_{i}^{(k)}(a,b)x^{l-i}t^{l}$
$=\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\left(\begin{array}[]{l}n\\\
l\end{array}\right)(\ln
c)^{n-l}B_{l}^{(k)}(a,b)x^{n-l}\right)\frac{t^{n}}{n!}.$
(14)
$\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}c^{(x+1)t}=\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}c^{xt}.c^{t}$
$=\frac{Li_{k}(1-(ab)^{-t})}{\left(\frac{b}{c}\right)^{t}-(ac)^{-t}}c^{xt}=\sum_{n=0}^{\infty}B_{n}^{(k)}(x;ac,\frac{b}{c},c)\frac{t^{n}}{n!}.$
(15)
$\frac{Li_{k}(1-e^{-x})}{1-e^{-x}}e^{xt}=\frac{Li_{k}(1-e^{-x})}{e^{-xt}-e^{-x-xt}}=\frac{Li_{k}(1-e^{-x})}{(e^{-t})^{x}-(e^{1+t})^{-x}}$
so,
$B_{n}^{(k)}(t)=B_{n}^{(k)}(e^{t+1},e^{-t}).$
(16)we can write
$\sum_{n=0}^{\infty}B_{n}^{(k)}(x,a,b,c)\frac{t^{n}}{n!}=\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}c^{xt}=\frac{1}{b^{t}}\frac{Li_{k}(1-(ab)^{-t})}{(1-(ab)^{-t})}c^{xt}$
$=e^{t({-\ln b+x\ln c})}\left(\frac{Li_{k}(1-e^{-t\ln ab})}{1-e^{-t(\ln
ab)}}\right)=\sum_{n=0}^{\infty}(\ln a+\ln b)^{n}B_{n}^{(k)}\left(\frac{-\ln
b+x\ln c}{\ln a+\ln b}\right)\frac{t^{n}}{n!}$
so
$B_{n}^{(k)}(x,a,b,c)=(\ln a+\ln b)^{n}B_{n}^{(k)}\left(\frac{-\ln b+x\ln
c}{\ln a+\ln b}\right).$
###### Theorem 2
Let $x\in\mathbb{R}$ , $n\geq 0$. For every positive real numbers a,b such
that $a\neq b$ and $b>a>0$, we have
$B_{n}^{(k)}(x+y,a,b,c)=\sum_{l=0}^{\infty}\left(\begin{array}[]{l}n\\\
l\end{array}\right)(\ln c)^{n-l}B_{l}^{(k)}(x;a,b,c)y^{n-l}\newline \\\
=\sum_{l=0}^{n}\left(\begin{array}[]{l}n\\\ l\end{array}\right)(\ln
c)^{n-l}B_{l}^{(k)}(y,a,b,c)x^{n-l}.$ (17)
Proof: We have
$\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}c^{(x+y)t}=\sum_{n=0}^{\infty}B_{n}^{(k)}(x+y;a,b,c)\frac{t^{n}}{n!}$
$=\frac{Li_{k}(1-(ab)^{-t})}{b^{t}-a^{-t}}c^{xt}.c^{yt}=\left(\sum_{n=0}^{\infty}B_{n}^{(k)}(x;a,b,c)\frac{t^{n}}{n!}\right)\left(\sum_{i=0}^{\infty}\frac{y^{i}(\ln
c)^{i}}{i!}t^{i}\right)$
$=\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\left(\begin{array}[]{l}n\\\
l\end{array}\right)y^{n-l}(\ln
c)^{n-l}B_{l}^{(k)}(x,a,b,c)\right)\frac{t^{n}}{n!}$
so we get
$\frac{Li_{k}(1-(ab)^{t})}{b^{t}-a^{-t}}c^{(x+y)t}=\frac{Li_{k}(1-(ab)^{-t}}{b^{t}-a^{-t}}c^{yt}c^{xt}$
$=\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\left(\begin{array}[]{l}n\\\
l\end{array}\right)x^{n-l}(\ln
c)^{n-l}B_{l}^{(k)}(y,a,b,c)\right)\frac{t^{n}}{n!}.$
###### Theorem 3
Let $x\in\mathbb{R}$ and $n\geq 0$.For every positive real numbers a,b and c
such that $a\neq b$ and $b>a>0$, we have
$B_{n}^{(k)}(x;a,b,c)=\sum_{l=0}^{n}\left(\begin{array}[]{l}n\\\
l\end{array}\right)(\ln c)^{n-l}B_{l}^{(k)}\left(\frac{-\ln b}{\ln a+\ln
b}\right)(\ln a+\ln b)^{l}x^{n-l},$ (18)
$B_{n}^{(k)}(x;a,b,c)=\sum_{l=0}^{n}\sum_{j=0}^{l}(-1)^{l-j}\left(\begin{array}[]{l}n\\\
l\end{array}\right)\left(\begin{array}[]{l}l\\\ j\end{array}\right)(\ln
c)^{n-l}(\ln b)^{l-j}(\ln a+\ln b)^{j}B_{j}^{(k)}x^{n-k}.$ (19)
Proof:
$\hskip 28.45274pt(18)$By applying Theorems 1 and 2 we know ,
$B_{n}^{(k)}(x;a,b,c)=\sum_{l=0}^{n}\left(\begin{array}[]{l}n\\\
l\end{array}\right)(\ln c)^{n-l}B_{l}^{(k)}(a,b)x^{n-l}$
and
$B_{n}^{(k)}(a,b)=B_{n}^{(k)}(\frac{-lnb}{\ln a+\ln b})(\ln a+\ln b)^{n}$
The relation (18)will follow if we combine these formulae.
(19) proof is similar to(18).
Now,we give some results about derivatives and integrals of the generalized
poly-Bernoulli polynomials in the following theorem.
###### Theorem 4
Let $x\in\mathbb{R}$.If $a,b$ and $c>0$ , $a\neq b$ and $b>a>0$ ,For any non-
negative integer l and real numbers $\alpha$ and $\beta$ we have
$\frac{\partial^{l}B_{n}^{(k)}(x,a,b,c)}{\partial x^{l}}=\frac{n!}{(n-l)!}(\ln
c)^{l}B_{n-l}^{(k)}(x,a,b,c)$ (20)
$\int_{\alpha}^{\beta}B_{n}^{(k)}(x,a,b,c)dx=\frac{1}{(n+1)\ln
c}[B_{n+1}^{(k)}(\beta,a,b,c)-B_{n+1}^{(k)}(\alpha,a,b,c)]$ (21)
Proof: By applying (20) and (21) can be proved by induction on n.
In [9], GI-Sang Cheon investigated the classical relationship between
Bernoulli and Euler polynomials, in this paper we study the relationship
between the generalized poly-Bernoulli and Euler polynomials.
###### Theorem 5
For $b>0$ we have
$B_{n}^{(k_{1})}(x+y,1,b,b)=\frac{1}{2}\sum_{k=0}^{n}\left(\begin{array}[]{l}n\\\
k\end{array}\right)[B_{n}^{(k_{1})}(y,1,b,b)+B_{n}^{(k_{1})}(y+1,1,b,b)]E_{n-k}(x,1,b,b).$
Proof:We know
$B_{n}^{(k_{1})}(x+y,1,b,b)=\sum_{k=0}^{\infty}\left(\begin{array}[]{l}n\\\
k\end{array}\right)(\ln b)^{n-k}B_{k}^{(k_{1})}(y;1,b,b)x^{n-k}$
and
$E_{k}(x+y,1,b,b)+E_{k}(x,1,b,b)=2x^{k}(\ln b)^{k}$
So, We obtain $B_{n}^{(k_{1})}(x+y,1,b,b)=$
$\frac{1}{2}\sum_{k=0}^{n}\left(\begin{array}[]{l}n\\\ k\end{array}\right)(\ln
b)^{n-k}B_{k}^{(k_{1})}(y;1,b,b)\left[\frac{1}{(\ln
b)^{n-k}}(E_{n-k}(x,1,b,b)+E_{n-k}(x+1,1,b,b))\right]\ $
$=\frac{1}{2}\sum_{k=0}^{n}\left(\begin{array}[]{l}n\\\
k\end{array}\right)B_{k}^{(k_{1})}(y;1,b,b)\left[\left(E_{n-k}(x,1,b,b)+\sum_{j=0}^{n-k}\left(\begin{array}[]{l}n-k\\\
j\end{array}\right)E_{j}(x,1,b,b)\right)\right]\ $
$=\frac{1}{2}\sum_{k=0}^{n}\left(\begin{array}[]{l}n\\\
k\end{array}\right)B_{k}^{(k_{1})}(y;1,b,b)E_{n-k}(x,1,b,b)$
$+\frac{1}{2}\sum_{j=0}^{n}\left(\begin{array}[]{l}n\\\
j\end{array}\right)E_{j}(x;1,b,b)\sum_{k=0}^{n-j}\left(\begin{array}[]{l}n-j\\\
k\end{array}\right)B_{k}^{(k_{1})}(y,1,b,b)$
$=\frac{1}{2}\sum_{k=0}^{n}\left(\begin{array}[]{l}n\\\
k\end{array}\right)B_{k}^{(k_{1})}(y;1,b,b)E_{n-k}(x,1,b,b)+$
$\frac{1}{2}\sum_{j=0}^{n}\left(\begin{array}[]{l}n\\\
j\end{array}\right)B_{n-j}^{(k_{1})}(y+1;1,b,b)E_{j}(x,1,b,b)$
So we have
$B_{n}^{(k_{1})}(x+y,1,b,b)=\frac{1}{2}\sum_{k=0}^{n}\left(\begin{array}[]{l}n\\\
k\end{array}\right)[B_{n}^{(k_{1})}(y,1,b,b)+B_{n}^{(k_{1})}(y+1,1,b,b)]E_{n-k}(x,1,b,b).$
###### Corollary 1
In theorem 5, if we set $k_{1}=1$ and $b=e$, we obtain
$B_{n}(x)=\sum_{(k=0),(k\neq 1)}^{n}\left(\begin{array}[]{l}n\\\
k\end{array}\right)B_{k}E_{n-k}(x).$
For more detail see [7].
Acknowledgement:The authors would like to thank the referee for has valuable
comments concerning theorem 5.
## References
* [1] T.Arakawa and M.Kaneko:_On poly-Bernoulli numbers_ ,Comment.Math.univ.st.Pauli48, (1999),159-167
* [2] B.N. Oue and F.Qi:_Generalization of Bernoulli polynomials_ ,internat.j.math.ed.sci.tech 33, (2002),No,3,428-431
* [3] M.-S.Kim and T.Kim:_An explicit formula on the generalized Bernoulli number with order n_ ,Indian.j.pure and applied math 31, (2000),1455-1466
* [4] Hassan Jolany and M.R.Darafsheh:_Some another remarks on the generalization of Bernoulli and Euler polynomials_ ,Scientia magna Vol5,NO 3
* [5] M.Kaneko:_poly-Bernoulli numbers_ ,journal de theorides numbers de bordeaux. 9, (1997),221-228
* [6] Y.Hamahata,H.Masubuch:_Special Multi-Poly- Bernoulli numbers_ ,Journal of Integer sequences vol 10, (2007)
* [7] H.M.Srivastava and A.Pinter:_Remarks on some relationships Between the Bernoulli and Euler polynomials_ ,Applied .Math.Letter 17, (2004)375-380
* [8] Chad Brewbaker :_A combinatorial Interpretion of the poly-Bernoulli numbers and two fermat analogues_ , Integers journal 8, (2008)
* [9] GI-Sang Cheon:_A note on the Bernoulli and Euler polynomials_ ,Applied.Math.Letter 16, (2003),365-368
* [10] Q.M.Luo,F.Oi and L.Debnath _Generalization of Euler numbers and polynomials_ ,Int.J.Math.Sci (2003)3893-3901
* [11] Y.Hamahata,H.Masubuchi _Recurrence formulae for Multi-poly-Bernoulli numbers_ ,Integers journal 7(2007)
* [12] Hassan Jolany, _Explicit formula for generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters._ http://arxiv.org/pdf/1109.1387v1.pdf
|
arxiv-papers
| 2012-12-17T13:46:38 |
2024-09-04T02:49:39.359133
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hassan Jolany and M. R. Darafsheh, R. Eizadi Alikelaye",
"submitter": "Hassan Jolany",
"url": "https://arxiv.org/abs/1212.3989"
}
|
1212.4163
|
# Hyers–Ulam stability of higher-order Cauchy-Euler dynamic equations on time
scales
Douglas R. Anderson Department of Mathematics, Concordia College, Moorhead,
MN 56562 USA [email protected], [email protected], [email protected]
http://www.cord.edu/faculty/andersod/
###### Abstract.
We establish the stability of higher-order linear nonhomogeneous Cauchy-Euler
dynamic equations on time scales in the sense of Hyers and Ulam. That is, if
an approximate solution of a higher-order Cauchy-Euler equation exists, then
there exists an exact solution to that dynamic equation that is close to the
approximate one. Some examples illustrate the applicability of the main
results.
###### Key words and phrases:
Difference equations; quantum equations; non-homogeneous equations.
###### 2010 Mathematics Subject Classification:
34N05, 26E70, 39A10
## 1\. introduction
In 1940, Ulam [26] posed the following problem concerning the stability of
functional equations: give conditions in order for a linear mapping near an
approximately linear mapping to exist. The problem for the case of
approximately additive mappings was solved by Hyers [11] who proved that the
Cauchy equation is stable in Banach spaces, and the result of Hyers was
generalized by Rassias [23]. Alsina and Ger [1] were the first authors who
investigated the Hyers-Ulam stability of a differential equation.
Since then there has been a significant amount of interest in Hyers-Ulam
stability, especially in relation to ordinary differential equations, for
example see [8, 9, 12, 13, 14, 15, 19, 20, 22, 25]. Also of interest are many
of the articles in a special issue guest edited by Rassias [24], dealing with
Ulam, Hyers-Ulam, and Hyers-Ulam-Rassias stability in various contexts. Also
see Li and Shen [17, 18], Wang, Zhou, and Sun [27], and Popa et al [21, 6].
András and Mészáros [3] recently used an operator approach to show the
stability of linear dynamic equations on time scales with constant
coefficients, as well as for certain integral equations. Anderson et al [2,
Corollary 2.6] proved the following concerning second-order non-homogeneous
Cauchy-Euler equations on time scales:
###### Theorem 1.1 (Cauchy-Euler Equation).
Let $\lambda_{1},\lambda_{2}\in\mathbb{R}$ (or
$\lambda_{2}=\overline{\lambda_{1}}$, the complex conjugate) be such that
$t+\lambda_{k}\mu(t)\neq 0,\quad k=1,2$
for all $t\in[a,\sigma(b)]_{\mathbb{T}}$, where $a\in\mathbb{T}$ satisfies
$a>0$. Then the Cauchy-Euler equation
$x^{\Delta\Delta}(t)+\frac{1-\lambda_{1}-\lambda_{2}}{\sigma(t)}\;x^{\Delta}(t)+\frac{\lambda_{1}\lambda_{2}}{t\sigma(t)}\;x(t)=f(t),\quad
t\in[a,b]_{\mathbb{T}}$ (1.1)
has Hyers-Ulam stability on $[a,b]_{\mathbb{T}}$. To wit, if there exists
$y\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]_{\mathbb{T}}$ that satisfies
$\left|y^{\Delta\Delta}(t)+\frac{1-\lambda_{1}-\lambda_{2}}{\sigma(t)}\;y^{\Delta}(t)+\frac{\lambda_{1}\lambda_{2}}{t\sigma(t)}\;y(t)-f(t)\right|\leq\varepsilon$
for $t\in[a,b]_{\mathbb{T}}$, then there exists a solution
$u\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]{{}_{\mathbb{T}}}$ of (1.1) given
by
$u(t)=e_{\frac{\lambda_{1}}{t}}\left(t,\tau_{2}\right)y\left(\tau_{2}\right)+\int_{\tau_{2}}^{t}e_{\frac{\lambda_{1}}{t}}(t,\sigma(s))w(s)\Delta
s,\quad\text{any}\quad\tau_{2}\in[a,\sigma^{2}(b)]_{\mathbb{T}},$
where for any $\tau_{1}\in[a,\sigma(b)]_{\mathbb{T}}$ the function $w$ is
given by
$w(s)=e_{\frac{\lambda_{2}-1}{\sigma(s)}}(s,\tau_{1})\left[y^{\Delta}(\tau_{1})-\frac{\lambda_{1}}{\tau_{1}}y(\tau_{1})\right]+\int_{\tau_{1}}^{s}e_{\frac{\lambda_{2}-1}{\sigma(s)}}(s,\sigma(\zeta))f(\zeta)\Delta\zeta,$
such that $|y-u|\leq K\varepsilon$ on $[a,\sigma^{2}(b)]_{\mathbb{T}}$ for
some constant $K>0$.
The motivation for this work is to extend Theorem 1.1 to the general $n$th-
order Cauchy-Euler dynamic equation; we will show the stability in the sense
of Hyers and Ulam of the equation
$\sum_{k=0}^{n}\alpha_{k}M_{k}y(t)=f(t),\quad M_{0}y(t):=y(t),\quad
M_{k+1}y(t):=\varphi(t)\left(M_{k}y\right)^{\Delta}(t),\;k=0,1,\cdots,n-1.$
This is essentially [5, (2.14)] if $\varphi(t)=t$ and $f(t)=0$. Throughout
this work we assume the reader has a working knowledge of time scales as can
be found in Bohner and Peterson [4, 5], originally introduced by Hilger [10].
## 2\. Hyers-Ulam stability for higher-order Cauchy-Euler dynamic equations
In this section we establish the Hyers-Ulam stability of the higher-order non-
homogeneous Cauchy-Euler dynamic equation on time scales of the form
$\sum_{k=0}^{n}\alpha_{k}M_{k}y(t)=f(t),\quad M_{0}y(t):=y(t),\quad
M_{k+1}y(t):=\varphi(t)\left(M_{k}y\right)^{\Delta}(t),\;k=0,1,\cdots,n-1$
(2.1)
for given constants $\alpha_{k}\in\mathbb{R}$ with $\alpha_{n}\equiv 1$, and
for functions $\varphi,f\in\operatorname{C_{rd}}[a,b]_{\mathbb{T}}$, using the
following definition.
###### Definition 2.1 (Hyers-Ulam stability).
Let $\varphi,f\in\operatorname{C_{rd}}[a,b]_{\mathbb{T}}$ and
$n\in\mathbb{N}$. If whenever
$M_{k}x\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ satisfies
$\left|\sum_{k=0}^{n}\alpha_{k}M_{k}x(t)-f(t)\right|\leq\varepsilon,\quad
t\in[a,b]_{\mathbb{T}}$
there exists a solution $u$ of (2.1) with
$M_{k}u\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ for
$k=0,1,\cdots,n-1$ such that $|x-u|\leq K\varepsilon$ on
$[a,\sigma^{n}(b)]_{\mathbb{T}}$ for some constant $K>0$, then (2.1) has
Hyers-Ulam stability $[a,b]_{\mathbb{T}}$.
###### Remark 2.2.
Before proving the Hyers-Ulam stability of (2.1) we will need the following
lemma, which allows us to factor (2.1) using the elementary symmetric
polynomials [7] in the $n$ symbols $\rho_{1},\cdots,\rho_{n}$ given by
$\displaystyle s_{1}^{n}$ $\displaystyle=$ $\displaystyle
s_{1}(\rho_{1},\cdots,\rho_{n})=\sum_{i}\rho_{i}$ $\displaystyle s_{2}^{n}$
$\displaystyle=$ $\displaystyle
s_{2}(\rho_{1},\cdots,\rho_{n})=\sum_{i<j}\rho_{i}\rho_{j}$ $\displaystyle
s_{3}^{n}$ $\displaystyle=$ $\displaystyle
s_{3}(\rho_{1},\cdots,\rho_{n})=\sum_{i<j<k}\rho_{i}\rho_{j}\rho_{k}$
$\displaystyle s_{4}^{n}$ $\displaystyle=$ $\displaystyle
s_{4}(\rho_{1},\cdots,\rho_{n})=\sum_{i<j<k<\ell}\rho_{i}\rho_{j}\rho_{k}\rho_{\ell}$
$\displaystyle\vdots$ $\displaystyle s_{t}^{n}$ $\displaystyle=$
$\displaystyle
s_{t}(\rho_{1},\cdots,\rho_{n})=\sum_{i_{1}<i_{2}<\cdots<i_{t}}\rho_{i_{1}}\rho_{i_{2}}\cdots\rho_{i_{t}}$
$\displaystyle\vdots$ $\displaystyle s_{n}^{n}$ $\displaystyle=$
$\displaystyle
s_{n}(\rho_{1},\cdots,\rho_{n})=\rho_{1}\rho_{2}\rho_{3}\cdots\rho_{n}.$
In general, we let $s_{i}^{j}$ represent the $i$th elementary symmetric
polynomial on $j$ symbols. Then, given the $\alpha_{k}$ in (2.1), introduce
the characteristic values $\lambda_{k}\in\mathbb{C}$ via the elementary
symmetric polynomial $s_{t}^{n}$ on the $n$ symbols
$-\lambda_{1},\cdots,-\lambda_{n}$, where
$\alpha_{k}=s_{n-k}^{n}=s_{n-k}(-\lambda_{1},\cdots,-\lambda_{n})=\sum_{i_{1}<i_{2}<\cdots<i_{n-k}}(-1)^{n-k}\lambda_{i_{1}}\lambda_{i_{2}}\cdots\lambda_{i_{n-k}},\quad\alpha_{n}=s_{0}\equiv
1.$ (2.2)
###### Lemma 2.3 (Factorization).
Given $y,\varphi\in\operatorname{C_{rd}}[a,b]_{\mathbb{T}}$ and
$\alpha_{k}\in\mathbb{R}$ with $\alpha_{n}\equiv 1$, let
$M_{k}y\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$, where
$M_{0}y(t):=y(t)$ and $M_{k+1}y(t):=\varphi(t)\left(M_{k}y\right)^{\Delta}(t)$
for $k=0,1,\cdots,n-1$. Then we have the factorization
$\sum_{k=0}^{n}\alpha_{k}M_{k}y(t)=\prod_{k=1}^{n}\left(\varphi
D-\lambda_{k}I\right)y(t),\qquad n\in\mathbb{N},$ (2.3)
where the differential operator $D$ is defined via $Dx=x^{\Delta}$ for
$x\in\operatorname{C^{\Delta}_{rd}}[a,b]_{\mathbb{T}}$, and $I$ is the
identity operator.
###### Proof.
We proceed by mathematical induction on $n\in\mathbb{N}$, utilizing the
substitution defined in (2.2). For $n=1$,
$\sum_{k=0}^{n}\alpha_{k}M_{k}y(t)=\alpha_{0}M_{0}y(t)+\alpha_{1}M_{1}y(t)=s_{1}(-\lambda_{1})y(t)+1\cdot\varphi(t)y^{\Delta}(t)=\left(\varphi
D-\lambda_{1}I\right)y(t)$
and the result holds. Assume (2.3) holds for $n\geq 1$. Then we have
$\alpha_{n+1}\equiv 1$ and
$\displaystyle\sum_{k=0}^{n+1}\alpha_{k}M_{k}y(t)$ $\displaystyle=$
$\displaystyle\alpha_{0}y(t)+\sum_{k=1}^{n}\alpha_{k}M_{k}y(t)+M_{n+1}y(t)$
$\displaystyle=$ $\displaystyle
s_{n+1}^{n+1}y(t)+\sum_{k=1}^{n}s_{n+1-k}^{n+1}M_{k}y(t)+\varphi(t)\left(M_{n}y\right)^{\Delta}(t)$
$\displaystyle=$
$\displaystyle-\lambda_{n+1}s_{n}^{n}y(t)+\sum_{k=1}^{n}\left(s_{n+1-k}^{n}-\lambda_{n+1}s_{n-k}^{n}\right)M_{k}y(t)+\varphi(t)D\left(M_{n}y\right)(t)$
$\displaystyle=$
$\displaystyle-\lambda_{n+1}\left[s_{n}^{n}y(t)+\sum_{k=1}^{n}s_{n-k}^{n}M_{k}y(t)\right]+\sum_{k=1}^{n}s_{n+1-k}^{n}M_{k}y(t)+\varphi(t)D\left(M_{n}y\right)(t)$
$\displaystyle=$
$\displaystyle-\lambda_{n+1}\sum_{k=0}^{n}s_{n-k}^{n}M_{k}y(t)+\varphi(t)D\left(\sum_{k=1}^{n}s_{n+1-k}^{n}M_{k-1}y(t)+M_{n}y\right)(t)$
$\displaystyle=$
$\displaystyle-\lambda_{n+1}\sum_{k=0}^{n}s_{n-k}^{n}M_{k}y(t)+\varphi(t)D\left(\sum_{k=0}^{n-1}s_{n-k}^{n}M_{k}y(t)+M_{n}y\right)(t)$
$\displaystyle=$
$\displaystyle-\lambda_{n+1}\sum_{k=0}^{n}s_{n-k}^{n}M_{k}y(t)+\varphi(t)D\sum_{k=0}^{n}s_{n-k}^{n}M_{k}y(t)$
$\displaystyle=$
$\displaystyle\left(\varphi(t)D-\lambda_{n+1}I\right)\sum_{k=0}^{n}s_{n-k}^{n}M_{k}y(t)$
$\displaystyle=$
$\displaystyle\left(\varphi(t)D-\lambda_{n+1}I\right)\sum_{k=0}^{n}\alpha_{k}M_{k}y(t)$
$\displaystyle=$
$\displaystyle\left(\varphi(t)D-\lambda_{n+1}I\right)\prod_{k=1}^{n}\left(\varphi
D-\lambda_{k}I\right)y(t)$
and the proof is complete. ∎
###### Theorem 2.4 (Hyers-Ulam Stability).
Given $y,\varphi,f\in\operatorname{C_{rd}}[a,b]_{\mathbb{T}}$ with
$|\varphi|\geq A>0$ for some constant $A$, and $\alpha_{k}\in\mathbb{R}$ with
$\alpha_{n}\equiv 1$, consider (2.1) with
$M_{k}y\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ for
$k=0,\cdots,n-1$. Using the $\lambda_{k}$ from the factorization in Lemma 2.3,
assume
$\varphi(t)+\lambda_{k}\mu(t)\neq 0,\quad k=1,2,\cdots,n$ (2.4)
for all $t\in[a,\sigma^{n-1}(b)]_{\mathbb{T}}$. Then (2.1) has Hyers-Ulam
stability on $[a,b]_{\mathbb{T}}$.
###### Proof.
Let $\varepsilon>0$ be given, and suppose there is a function $x$, with
$M_{k}x\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$, that
satisfies
$\left|\sum_{k=0}^{n}\alpha_{k}M_{k}x(t)-f(t)\right|\leq\varepsilon,\quad
t\in[a,b]_{\mathbb{T}}.$
We will show there exists a solution $u$ of (2.1) with
$M_{k}u\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ for
$k=0,1,\cdots,n-1$ such that $|x-u|\leq K\varepsilon$ on
$[a,\sigma^{n}(b)]_{\mathbb{T}}$ for some constant $K>0$.
To this end, set
$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle\varphi
x^{\Delta}-\lambda_{1}x=\left(\varphi D-\lambda_{1}I\right)x$ $\displaystyle
g_{2}$ $\displaystyle=$ $\displaystyle\varphi
g_{1}^{\Delta}-\lambda_{2}g_{1}=\left(\varphi D-\lambda_{2}I\right)g_{1}$
$\displaystyle\vdots$ $\displaystyle g_{k}$ $\displaystyle=$
$\displaystyle\varphi g_{k-1}^{\Delta}-\lambda_{k}g_{k-1}=\left(\varphi
D-\lambda_{k}I\right)g_{k-1}$ $\displaystyle\vdots$ $\displaystyle g_{n}$
$\displaystyle=$ $\displaystyle\varphi
g_{n-1}^{\Delta}-\lambda_{n}g_{n-1}=\left(\varphi
D-\lambda_{n}I\right)g_{n-1}.$
This implies by Lemma 2.3 that
$g_{n}(t)-f(t)=\sum_{k=0}^{n}\alpha_{k}M_{k}x(t)-f(t),$
so that
$\left|g_{n}(t)-f(t)\right|\leq\varepsilon,\quad t\in[a,b]_{\mathbb{T}}.$
By the construction of $g_{n}$ we have $\left|\varphi
g_{n-1}^{\Delta}-\lambda_{n}g_{n-1}-f\right|\leq\varepsilon$, that is
$\left|g_{n-1}^{\Delta}-\frac{\lambda_{n}}{\varphi}g_{n-1}-\frac{f}{\varphi}\right|\leq\frac{\varepsilon}{|\varphi|}\leq\frac{\varepsilon}{A}.$
By [2, Lemma 2.3] and (2.4) there exists a solution
$w_{1}\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ of
$w^{\Delta}(t)-\frac{\lambda_{n}}{\varphi(t)}w(t)-\frac{f(t)}{\varphi(t)}=0,\quad\text{or}\quad\varphi(t)w^{\Delta}(t)-\lambda_{n}w(t)-f(t)=0,$
(2.5)
$t\in[a,b]_{\mathbb{T}}$, where $w_{1}$ is given by
$w_{1}(t)=e_{\frac{\lambda_{n}}{\varphi}}(t,\tau_{1})g_{n-1}(\tau_{1})+\int_{\tau_{1}}^{t}e_{\frac{\lambda_{n}}{\varphi}}(t,\sigma(s))\frac{f(s)}{\varphi(s)}\Delta
s,\quad\text{any}\quad\tau_{1}\in[a,\sigma(b)]_{\mathbb{T}},$
and there exists an $L_{1}>0$ such that
$|g_{n-1}(t)-w_{1}(t)|\leq L_{1}\varepsilon/A,\quad
t\in[a,\sigma(b)]_{\mathbb{T}}.$
Since $g_{n-1}=\varphi g_{n-2}^{\Delta}-\lambda_{n-1}g_{n-2}$, we have that
$|\varphi g_{n-2}^{\Delta}-\lambda_{n-1}g_{n-2}(t)-w_{1}(t)|\leq
L_{1}\varepsilon/A,\quad t\in[a,\sigma(b)]_{\mathbb{T}}.$
Again we apply [2, Lemma 2.3] to see that there exists a solution
$w_{2}\in\operatorname{C^{\Delta}_{rd}}[a,\sigma(b)]{{}_{\mathbb{T}}}$ of
$w^{\Delta}(t)-\frac{\lambda_{n-1}}{\varphi(t)}w(t)-\frac{w_{1}(t)}{\varphi(t)}=0,\quad\text{or}\quad\varphi(t)w^{\Delta}(t)-\lambda_{n-1}w(t)-w_{1}(t)=0,$
$t\in[a,\sigma(b)]_{\mathbb{T}}$, where $w_{2}$ is given by
$w_{2}(t)=e_{\frac{\lambda_{n-1}}{\varphi}}(t,\tau_{2})g_{n-2}(\tau_{2})+\int_{\tau_{2}}^{t}e_{\frac{\lambda_{n-1}}{\varphi}}(t,\sigma(s))\frac{w_{1}(s)}{\varphi(s)}\Delta
s,\quad\text{any}\quad\tau_{2}\in[a,\sigma^{2}(b)]_{\mathbb{T}},$
and there exists an $L_{2}>0$ such that
$|g_{n-2}(t)-w_{2}(t)|\leq L_{2}L_{1}\varepsilon/A^{2},\quad
t\in[a,\sigma^{2}(b)]_{\mathbb{T}}.$
Continuing in this manner, we see that for $k=1,2,\cdots,n-1$ there exists a
solution
$w_{k}\in\operatorname{C^{\Delta}_{rd}}[a,\sigma^{k-1}(b)]{{}_{\mathbb{T}}}$
of
$w^{\Delta}(t)-\frac{\lambda_{n-k+1}}{\varphi(t)}w(t)-\frac{w_{k-1}(t)}{\varphi(t)}=0,\quad\text{or}\quad\varphi(t)w^{\Delta}(t)-\lambda_{n-k+1}w(t)-w_{k-1}(t)=0,$
$t\in[a,\sigma^{k-1}(b)]_{\mathbb{T}}$, where $w_{k}$ is given by
$w_{k}(t)=e_{\frac{\lambda_{n-k+1}}{\varphi}}(t,\tau_{k})g_{n-k}(\tau_{k})+\int_{\tau_{k}}^{t}e_{\frac{\lambda_{n-k+1}}{\varphi}}(t,\sigma(s))\frac{w_{k-1}(s)}{\varphi(s)}\Delta
s,\quad\text{any}\quad\tau_{k}\in[a,\sigma^{k}(b)]_{\mathbb{T}},$ (2.6)
and there exists an $L_{k}>0$ such that
$|g_{n-k}(t)-w_{k}(t)|\leq\prod_{j=1}^{k}L_{j}\varepsilon/A^{k},\quad
t\in[a,\sigma^{k}(b)]_{\mathbb{T}}.$
In particular, for $k=n-1$,
$|g_{1}(t)-w_{n-1}(t)|\leq\prod_{j=1}^{n-1}L_{j}\varepsilon/A^{n-1},\quad
t\in[a,\sigma^{n-1}(b)]_{\mathbb{T}}$
implies by the definition of $g_{1}$ that
$\left|x^{\Delta}(t)-\frac{\lambda_{1}}{\varphi(t)}x(t)-\frac{w_{n-1}(t)}{\varphi(t)}\right|\leq\prod_{j=1}^{n-1}L_{j}\varepsilon/A^{n},\quad
t\in[a,\sigma^{n-1}(b)]_{\mathbb{T}}.$
Thus there exists a solution
$w_{n}\in\operatorname{C^{\Delta}_{rd}}[a,\sigma^{n-1}(b)]{{}_{\mathbb{T}}}$
of
$w^{\Delta}(t)-\frac{\lambda_{1}}{\varphi(t)}w(t)-\frac{w_{n-1}(t)}{\varphi(t)}=0,\quad\text{or}\quad\varphi(t)w^{\Delta}(t)-\lambda_{1}w(t)-w_{n-1}(t)=0,$
$t\in[a,\sigma^{n-1}(b)]_{\mathbb{T}}$, where $w_{n}$ is given by
$w_{n}(t)=e_{\frac{\lambda_{1}}{\varphi}}(t,\tau_{n})x(\tau_{n})+\int_{\tau_{n}}^{t}e_{\frac{\lambda_{1}}{\varphi}}(t,\sigma(s))\frac{w_{n-1}(s)}{\varphi(s)}\Delta
s,\quad\text{any}\quad\tau_{n}\in[a,\sigma^{n}(b)]_{\mathbb{T}},$ (2.7)
and there exists an $L_{n}>0$ such that
$|x(t)-w_{n}(t)|\leq K\varepsilon:=\prod_{j=1}^{n}L_{j}\varepsilon/A^{n},\quad
t\in[a,\sigma^{n}(b)]_{\mathbb{T}}.$ (2.8)
By construction,
$\displaystyle\left(\varphi D-\lambda_{1}I\right)w_{n}(t)$ $\displaystyle=$
$\displaystyle w_{n-1}(t)$ $\displaystyle\prod_{k=1}^{2}\left(\varphi
D-\lambda_{k}I\right)w_{n}(t)$ $\displaystyle=$ $\displaystyle\left(\varphi
D-\lambda_{2}I\right)w_{n-1}(t)=w_{n-2}(t)$ $\displaystyle\vdots$
$\displaystyle\prod_{k=1}^{n}\left(\varphi D-\lambda_{k}I\right)w_{n}(t)$
$\displaystyle=$ $\displaystyle\left(\varphi
D-\lambda_{n}I\right)w_{1}(t)\stackrel{{\scriptstyle\eqref{w1sol}}}{{=}}f(t)$
on $[a,\sigma^{n-1}(b)]_{\mathbb{T}}$, so that $u=w_{n}$ is a solution of
(2.1), with
$u\in\operatorname{C^{\Delta}_{rd}}[a,\sigma^{n-1}(b)]{{}_{\mathbb{T}}}$ and
$|x(t)-w_{n}(t)|\leq K\varepsilon$ for $t\in[a,\sigma^{n}(b)]_{\mathbb{T}}$ by
(2.8). Moreover, using (2.7) and (2.6), we have an iterative formula for this
solution $u=w_{n}$ in terms of the function $x$ given at the beginning of the
proof. ∎
## References
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|
arxiv-papers
| 2012-12-17T21:05:21 |
2024-09-04T02:49:39.368085
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Douglas R. Anderson",
"submitter": "Douglas R. Anderson",
"url": "https://arxiv.org/abs/1212.4163"
}
|
1212.4180
|
Measurement of the CP observables in $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ and
first observation of $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-}$
Steven R. Blusk
Department of Physics
Syracuse University
Syracuse, NY 13244, USA
Proceedings of CKM 2012, the 7th International Workshop on the CKM Unitarity
Triangle, University of Cincinnati, USA, 28th September - 2 October 2012
## 1 Introduction
A central goal of flavor physics is to measure the angle ${\gamma\equiv{\rm
arg}\left(-{V_{\rm ub}^{*}V_{\rm ud}\over V_{\rm cb}^{*}V_{\rm cd}}\right)}$
in the Cabibbo-Kobayashi-Maskawa (CKM) [1, 2] mixing matrix, which is
currently known to a precision of about 10-12o [3]. The theoretically cleanest
methods employ $B\rightarrow DK$ decays, where the sensitivity to $\gamma$
results from the interference between $b\rightarrow c$ and $b\rightarrow u$
transitions. Since both transitions are $\mathcal{O}(\lambda^{3}$) in the
Wolfenstein parameter [4], large CP violating asymmetries are expected. One
powerful class of methods utilize $B^{-}\rightarrow DK^{-}$ where the $D$ is
detected in either a CP eigenstate [7], a flavor-specific mode [6], or a
multi-body decay [8]. An advantage of these decays is that they do not require
knowledge of the $b$-hadron flavor at production (flavor tagging), and only
rely on measuring the time integrated rates. Another powerful method to
extract $\gamma$ is to perform a time-dependent analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ [9,
10, 11] and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$. Time-dependent analyses of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$$(\pi^{+}\pi^{-})$ are only possible at hadron colliders, and
are a unique capability of LHCb.
The time-dependent decay rates of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to a flavor-specific final
state, $f=D^{+}_{s}K^{-}$, is given by:
$\displaystyle\frac{d\Gamma_{B^{0}_{s}\rightarrow f}(t)}{dt}$
$\displaystyle=\frac{1}{2}|A_{f}|^{2}(1+|\lambda_{f}|^{2})e^{-\Gamma_{s}t}$
$\displaystyle\left[\cosh\left(\frac{\Delta\Gamma_{s}t}{2}\right)+D_{f}\sinh\left(\frac{\Delta\Gamma_{s}t}{2}\right)\right.$
(1) $\displaystyle\left.\phantom{\Big{(}}+C_{f}\cos\left(\Delta
m_{s}t\right)-S_{f}\sin\left(\Delta m_{s}t\right)\right.\bigg{]},$
$\displaystyle\frac{d\Gamma_{\overline{B}^{0}_{s}\rightarrow f}(t)}{dt}$
$\displaystyle=\frac{1}{2}|A_{f}|^{2}\left|\frac{p}{q}\right|^{2}(1+|\lambda_{f}|^{2})e^{-\Gamma_{s}t}$
$\displaystyle\left[\cosh\left(\frac{\Delta\Gamma_{s}t}{2}\right)+D_{f}\sinh\left(\frac{\Delta\Gamma_{s}t}{2}\right)\right.$
(2) $\displaystyle\left.\phantom{\Big{)}}-C_{f}\cos\left(\Delta
m_{s}t\right)+S_{f}\sin\left(\Delta m_{s}t\right)\right.\bigg{]},$
where $A_{f}$ is the decay amplitude $A(B^{0}_{s}\rightarrow f)$ and
$\lambda_{f}=(q/p)(\overline{A}_{f}/A_{f})=|\lambda_{f}|e^{i(\Delta-(\gamma-2\beta_{s}))}$.
Here, $|\lambda_{f}|$ and $\Delta$ are the relative magnitude and strong phase
difference between the $b\rightarrow u$ and $b\rightarrow c$ transitions, and
$2\beta_{s}$ is the phase of $B^{0}_{s}$ mixing. The complex coefficients $p$
and $q$ relate the $B^{0}_{s}$ meson mass eigenstates, $B_{H,L}$, to the
flavor eigenstates, $B^{0}_{s}$ and $\overline{B}^{0}_{s}$ via:
$\begin{array}[]{c}B_{L}=pB^{0}_{s}+q\overline{B}^{0}_{s}\\\
B_{H}=pB^{0}_{s}-q\overline{B}^{0}_{s}\end{array},\qquad|p|^{2}+|q|^{2}=1\,.$
(3)
Similar equations can be written for the $C\\!P$-conjugate decays, replacing
$A_{f}$ by $\overline{A}_{\overline{f}}=A(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow\overline{f})$,
$\lambda_{f}$ by
$\overline{\lambda}_{\overline{f}}=(p/q)(A_{\overline{f}}/\overline{A}_{\overline{f}})$,
$|p/q|^{2}$ by $|q/p|^{2}$, $C_{f}$ by $C_{\overline{f}}$, $S_{f}$ by
$S_{\overline{f}}$, and $D_{f}$ by $D_{\overline{f}}$. The $C\\!P$ asymmetry
observables $C_{f}$, $S_{f}$, $D_{f}$, $C_{\overline{f}}$, $S_{\overline{f}}$
and $D_{\overline{f}}$ are then given by
$\displaystyle
C_{f}=C_{\overline{f}}=\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}}\,,\quad\quad
S_{f}=\frac{2\mathcal{I}m(\lambda_{f})}{1+|\lambda_{f}|^{2}}\,,\quad\quad
D_{f}=\frac{2\mathcal{R}e(\lambda_{f})}{1+|\lambda_{f}|^{2}},\quad$
$\displaystyle
S_{\overline{f}}=\frac{2\mathcal{I}m(\overline{\lambda}_{\overline{f}})}{1+|\overline{\lambda}_{\overline{f}}|^{2}}\,,\quad\quad
D_{\overline{f}}=\frac{2\mathcal{R}e(\overline{\lambda}_{\overline{f}})}{1+|\overline{\lambda}_{\overline{f}}|^{2}}\,.$
(4)
Since CP violation in mixing is expected to be below the percent level, it
follows that $|q/p|=1$, $|\lambda_{f}|=|\overline{\lambda}_{\overline{f}}|$,
and consequently $C_{f}=C_{\overline{f}}$. Thus there are five observables
that depend on the 3 physics parameters of interest: $|\lambda_{f}|$, $\Delta$
and $\gamma-2\beta_{s}$. Similar expressions are applicable to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, however, there is a potential dilution due to
the varying strong phase across the $D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ Dalitz
plane.
In this article, we present the first measurements of these five CP
observables. First observations of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}_{s}\rightarrow D_{s1}(2536)^{+}\pi^{-}$
decays are also presented, along with measurements of their relative branching
fractions. All results are based on 1.0 fb-1 of integrated luminosity recorded
in 2011 by the LHCb experiment. More detailed documentation of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ analyses can be found in Refs. [12] and [13],
respectively.
## 2 Event Selection
Signal $D^{+}_{s}$ candidates are formed by reconstructing
$D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$,
$D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ and $D^{+}_{s}\rightarrow
K^{+}\pi^{-}\pi^{+}$. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates, only the $D^{+}_{s}\rightarrow
K^{+}K^{-}\pi^{+}$ decay is considered. The $D^{+}_{s}$ candidates are
required to form a good quality vertex, be spacially well separated from any
primary vertex (PV), and have an invariant mass consistent with the known
$D^{+}_{s}$ mass (within about 3 times the mass resolution). Multivariate
selection algorithms are employed to suppress the combinatorial background,
and typically have a signal efficiency of 80-90% while rejecting about 85% of
the combinatorial background. Invariant mass distributions for $D^{+}_{s}$
candidates are shown in Fig. 1 for the higher signal yield $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
decay, showing that clean signals are achievable even in the suppressed
$D^{+}_{s}$ decay modes.
Figure 1: Invariant mass distributions for $D^{+}_{s}$ candidates in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}$ data sample, for (left) $K^{+}K^{-}\pi^{+}$, (middle)
$K^{+}\pi^{-}\pi^{+}$, and (right) $\pi^{+}\pi^{-}\pi^{+}$ final states.
Tighter particle identification requirements are applied to the $K^{-}$ or
$K^{-}\pi^{+}\pi^{-}$ recoiling from the $D^{+}_{s}$ to suppress cross-feed
from the favored $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decays. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decays, the invariant mass of the
$\pi^{-}\pi^{+}\pi^{-}$ and $K^{-}\pi^{+}\pi^{-}$ systems are restricted to be
below 3000 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
## 3 Analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$
The invariant mass distributions for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$ are shown in Figs. 2 and 3. All three $D^{+}_{s}$ decay modes
have approximately equal $B^{0}_{s}$ mass resolutions, and are summed together
in these distributions. The signal shape is modeled as the sum of two Crystal
Ball [14] functions, with one exponential tail on each side of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal peak. A number of
specific backgrounds, due to either a missed particle (e.g. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\rho^{-}$,
with the $\pi^{0}$ undetected), a misidentified particle (e.g. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
reconstructed as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$), or
both (e.g. $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\rho^{-}$ reconstructed as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$) are
accounted for using either data or simulation to model the shape of these
backgrounds. From an unbinned extended maximum likelihood fit, $27,965\pm 395$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}$ and $1390\pm 98$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
signal events are selected.
Figure 2: Invariant mass distributions $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
candidates. The signal component is indicated by the dashed curve, and the
backgrounds are indicated by the various color-filled (shaded, in B/W) curves.
Figure 3: Invariant mass distributions $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
candidates. The signal component is indicated by the dashed curve, and the
backgrounds are indicated by the various color-filled (shaded, in B/W) curves.
The CP parameters are obtained by a fit to the decay time distribution of the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$ signal candidates. Two methods have been developed. The first,
referred to as sFit, uses sWeights [15] obtained from the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ mass
fit to statistically subtract the background contribution. The second method,
referred to as cFit, is a conventional two-dimensional fit to the
reconstructed mass and decay time. The advantage of the first method is that
there is no need to model the time distribution of all the backgrounds, as
they are statistically removed via the sWeights. The statistical subtraction,
as presented here, uses events in the full mass fit region, and the
subtraction of this background leads to a larger statistical uncertainty than
if just a narrow signal region is used. For this reason, the second method is
expected to give a smaller statistical uncertainty; however it requires an
accurate model of the time distributions of the backgrounds that enter into
the signal region. For the analysis presented here, the sFit provides the
nominal result, and the cFit is used as a cross-check.
The measurement of the CP parameters in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
requires a fit to the time-dependent decay rates. The fit accounts for (i) the
acceptance versus reconstructed decay time, (ii) the decay time resolution,
and (iii) the effective tagging efficiency. The functional form of the
acceptance function is determined from simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$,
and its parameters are determined in a fit to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
data, where the $B^{0}_{s}$ lifetime and mixing frequency, $\Delta m_{s}$, are
fixed to 1.51 ps and 17.69 ps-1 [17], respectively. The average decay time
resolution is about 50 fs, and is modeled by the sum of three Gaussian
functions, whose parameters are determined from simulation. The Gaussian width
parameters obtained from simulation are scaled up by 1.15 to account for
better resolution in the simulation than in data; this factor is determined by
comparing the width of the zero decay time component of prompt $D^{+}_{s}$
plus one random track in data and simulation. For the flavor tagging, only
opposite side (OS) taggers are currently used. These algorithms exploit the
correlation in flavor between the signal $b$ hadron at production, and the
other $b$ hadron in the event (referred to as the tag-$b$). In particular, the
charge of either an electron, a muon, or a kaon that does not come from any
$pp$ interaction vertex (or the signal $b$), or the charge of another
secondary vertex in the event, provide information on the flavor of the
tag-$b$ hadron. Because $b\overline{b}$ are produced in pairs, this translates
into a flavor determination of the signal $B^{0}_{s}$. The OS flavor tagging
algorithm was initially tuned using simulated decays, and then re-optimized
and calibrated to obtain the largest effective tagging efficiency using the
self-tagging $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $B^{0}\rightarrow D^{*-}\mu^{+}\nu$ decays in data. In
general, the performance of the OS tagging algorithms are independent of the
signal $b$-hadron decay, and have a combined effective tagging efficiency of
$\epsilon D^{2}=1.90\%$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$.
Further details of the tagging algorithms can be found in Ref [16].
In the fit to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$, the following parameters are fixed: $\Delta m_{s}=17.69$
ps-1, $\tau_{B_{s}}=1.51$ ps and
$\Delta\Gamma_{s}\equiv\Gamma_{s,L}-\Gamma_{s,H}=0.105$ ps-1 [17]. About 60%
of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$ candidates have no flavor tag; the time-dependent decay rates
for these untagged decays is given by the sum of the two expressions in Eq.
LABEL:eq:decay_rates_2, and the sensitivity to $\gamma$ enters through the
hyperbolic sine term. The decay time distribution of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
signal decays and projections of the fitted are shown in Fig. 4. The
projections show the four possible tagged decays, $B^{0}_{s}\rightarrow
D_{s}^{\pm}K^{\mp}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{\pm}K^{\mp}$,
as well as the untagged time-dependent decay rates $(B^{0}_{s},\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})\rightarrow D^{-}_{s}K^{+}$ and
$(B^{0}_{s},\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})\rightarrow
D^{+}_{s}K^{-}$.
Figure 4: Distribution of reconstruct decay time for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
signal decays (points with error bars), along with the results of the fit.
Projections of the decay rates versus the decay time for the four possible
flavor tagged decays, and the two untagged decays.
The fitted values for the CP parameters are
$\displaystyle C$ $\displaystyle=\phantom{-}1.01\pm 0.50\pm 0.23\,,$
$\displaystyle S_{f}$ $\displaystyle=-1.25\pm 0.56\pm 0.24\,,$ $\displaystyle
S_{\overline{f}}$ $\displaystyle=\phantom{-}0.08\pm 0.68\pm 0.28\,,$
$\displaystyle D_{f}$ $\displaystyle=-1.33\pm 0.60\pm 0.26\,,$ $\displaystyle
D_{\overline{f}}$ $\displaystyle=-0.81\pm 0.56\pm 0.26\,,$
where the first uncertainties are statistical and the second are systematic.
Several sources of systematic uncertainty have been considered. The dominant
sources are due to the precision on the effective flavor tagging efficiency
(0.16$\sigma_{\rm stat}$-0.23$\sigma_{\rm stat}$), variations in the
parameters that are fixed in the default fits (0.15$\sigma_{\rm
stat}$-0.42$\sigma_{\rm stat}$), and the correlation between the mass of
specific backgrounds and their reconstructed decay time (0.08$\sigma_{\rm
stat}$-0.27$\sigma_{\rm stat}$), where these uncertainties are expressed as a
fraction of the statistical error. These are the first measurements of the CP
parameters in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$. With additional data and analysis refinements, reduction in
both the statistical and systematic uncertainties are expected.
## 4 First observation of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-}$
The decay $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ can be analyzed in a similar way to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ to
measure the weak phase $\gamma$. While this decay has not yet been observed,
if one uses $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$
decays as a guide, it would naively be expected that its branching fraction is
1.5-2.0 times larger than $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$,
making this a potentially attractive decay mode to explore. The first step in
such an analysis is to firmly establish an observation of this decay and
measure its branching fraction (here, relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$). While searching for this decay, the decay
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ is also observed and its branching fraction is
measured relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$.
With the previously defined selections, Fig. 5 shows the invariant mass
distributions for (left) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates and (right) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ candidates. Significant $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signals are seen in both
spectra, and a $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal is
seen in the $D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ mass distribution. The main sources
of background are $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ (to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$), and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$, and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{*+}_{s}K^{-}\pi^{+}\pi^{-}$ (to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$). Their shapes are taken from simulation, with
parameters that are allowed to vary within their uncertainties. Yields of
$5683\pm 83$ $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, $216\pm 21$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $402\pm 33$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ are observed. After correcting for the relative
efficiencies, the ratio of branching fractions are measured to be
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(5.2\pm 0.5\pm 0.3)\times
10^{-2}$ $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$ $\displaystyle=0.54\pm 0.07\pm 0.07,$
where the uncertainties are statistical and systematic, respectively.
Figure 5: Invariant mass distribution for (left) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates and (right) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ candidates. The fitted signal (dashed lines) and
background shapes (shaded/hatched regions) are shown, as described in the
text.
These are the first observations of these decays. Since $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ has a branching fraction that is about twice
as large as $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}$, and ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-})\sim
0.09\times{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-})$
[18], it follows that ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})$ is at least as large as ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-})$,
or as much as 50% larger. The ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})$ is also sizeable, and is likely dominated by
contributions where an extra $s\overline{s}$ pair is produced in addition to
the weak decay (see Ref. [13] for more details).
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decay has also been analyzed to search for
intermediate excited $D_{sj}$ states. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates within 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal peak, the mass
difference, $\Delta M\equiv M(D^{+}_{s}\pi^{-}\pi^{+})-M(D^{+}_{s})$ is
computed for both $\pi^{-}\pi^{+}$ mass combinations. The resulting mass
difference spectrum is shown in Fig. 6. The signal is fit with a Breit-Wigner
convolved with a Gaussian resolution function whose width is fixed to the
expected $\Delta M$ resolution. A signal of $20.0\pm 5.1$ events is observed
with a $\Delta M$ value and width consistent with the $D_{s1}(2536)^{+}$
state. Applying corrections for the relative efficiency, the ratio of
branching fractions is measured to be
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}=(4.0\pm 1.0\pm 0.4)\times 10^{-3}.$
The excess of events is 5.9 standard deviations over the expected background,
thus establishing the first observation of this decay.
Figure 6: Distribution of the difference in invariant mass,
$M(D^{+}_{s}\pi^{-}\pi^{+})-M(D^{+}_{s})$, using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates within 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $B^{0}_{s}$ mass
(points) and in the upper $B^{0}_{s}$ mass sidebands (filled histogram). The
fit to the distribution is shown, as described in the text.
## 5 Summary
First measurements of the CP observables in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
decay have been reported. With the larger data sample recorded in 2012, and
the larger data set anticipated in the future, this decay will contribute
significantly to the determination of the weak phase $\gamma$. First
observations of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ are also reported. The former can be used in a
similar way to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ to
extract $\gamma$. After including $D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\pi^{+}$
and $D^{+}_{s}\rightarrow K^{-}\pi^{+}\pi^{-}$ decays, and reoptimizing the
selection for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ only, the yield in this mode more than doubles
with a comparable signal-to-background. The yield in this mode is therefore
expected to have about 35-40% of that obtained in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$. The
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-}$ decay is also observed for the first time, and its
branching fraction relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ is presented.
## Acknowledgements
I gratefully acknowledge support from the National Science Foundation, which
makes this research possible.
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* [16] R. Aaij et. al. (LHCb Collaboration), Eur. Phys. J. C72, 2022 (2012); LHCb-CONF-2012-026; Also see contribution by J. Wishahi in these proceedings.
* [17] Heavy Flavor Averaging Group, D. Asner et al., Averages of b-hadron, chadron, and tau-lepton Properties, arXiv:1010.1589, Online updates available at http://www.slac.stanford.edu/xorg/hfag/.
* [18] Particle Data Group, J. Beringer et al., Phys. Rev. D86, 010001 (2012).
|
arxiv-papers
| 2012-12-17T22:17:49 |
2024-09-04T02:49:39.374565
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Steven Blusk (for the LHCb Collaboration)",
"submitter": "Steven R. Blusk",
"url": "https://arxiv.org/abs/1212.4180"
}
|
1212.4255
|
# Higher order wave-particle duality
Jie-Hui Huang1, Sabine Wölk2, Shi-Yao Zhu1, and M. Suhail Zubairy1,2 1Beijing
Computational Science Research Center, Beijing, 100084, People’s Republic of
China
2Institute of Quantum Science and Engineering (IQSE) and Department of Physics
and Astronomy, Texas A$\&$M University, College Station, TX 77843-4242, USA
###### Abstract
The complementarity of single-photon’s particle-like and wave-like behaviors
can be described by the inequality $D^{2}+V^{2}\leq 1$, with $D$ being the
path distinguishability and $V$ being the fringe visibility. In this paper, we
generalize this duality relation to multi-photon case, where two new concepts,
higher order distinguishability and higher order fringe visibility, are
introduced to quantify the higher order particle-like and wave-like behaviors
of multi-photons.
###### pacs:
03.65.Ta, 42.50.Ar, 42.50.Xa, 07.60.Ly
## I Introduction
The complementarity principle, developed and introduced by Bohr in 1927 bohr ,
is fundamentally important in the quantum theory, which predicts that a
quantum system may exhibit different properties based on different measurement
schemes. As the most typical example of complementarity, wave-particle duality
has attracted much attention since the early days of the quantum theory
complementarity . By defining the particle-like knowledge of an object
governed by quantum mechanics as the distinguishability ($D$) of its passage
in a two-path interferometer, and the wave-like knowledge as the visibility
($V$) of the interference pattern behind the interferometer, a tradeoff
relation between an object’s particle-like and wave-like behaviors can be
established Wootters ; Glauber ; Yasin ; Rempe ; Englert ,
$\displaystyle D^{2}+V^{2}\leq 1,$ (1)
where the equal sign holds for pure state of single-particles. This duality
relation, already confirmed in many experiments experiments , is valid even
when the choice of measuring apparatus is delayed after the single-particle’s
entrance into the interferometer wheeler . Such a delayed-choice gedanken
experiment has been realized in experiments at single-photon level by Roch
group Roch . The scheme of quantum eraser eraser , with the experimental
realization reported in Ref.shiyh , provides another clear way to demonstrate
the exclusive relation between an object’s particle-like and wave-like
behaviors. Recently, a new optical device named quantum beam splitter (QBS) is
theoretically proposed in Ref. qbs1 ; qbs2 , and the wave-particle morphing
behaviors of single-photons in this quantum device is becoming a hot topic
qbs3 .
In this paper, we generalize the discussion on the duality of single-photons
and investigate the higher order duality relations of multi-photons, no matter
what kind of state, pure or mixed, is prepared for the multi-photons.
The organization of the paper is as follows: In section II we introduce two
new concepts, higher order distinguishability and visibility. In section III,
we derive an inequality for higher order duality. In section IV, we present a
physical interpretation on the higher order distinguishability and visibility
and in section V we present a measurement scheme for the higher order
visibility.
## II Higher order distinguishability and visibility
Before the concept of higher order duality is introduced, we first recall the
definitions of the particle-like information and the wave-like information for
single-photons. By feeding the interferometer with single-photons, the
particle-like information is usually quantified as the distinguishability
($D$) of single-photons’ passage along the two paths inside the interferometer
(see Fig. 1). Thus we can use an operator shanxi ,
$\displaystyle\hat{D}\equiv\frac{a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}}{\langle
a_{1}^{\dagger}a_{1}\rangle+\langle a_{2}^{\dagger}a_{2}\rangle},$ (2)
to describe the measurement of the particle-like information mentioned above.
Here $a_{1}^{\dagger}~{}(a_{1})$ and $a_{2}^{\dagger}~{}(a_{2})$ denote the
creation (annihilation) operators of the modes in path $1$ and $2$, and the
denominator $\langle a_{1}^{\dagger}a_{1}\rangle+\langle
a_{2}^{\dagger}a_{2}\rangle$ is for normalization. The wave-like information
is defined as the visibility ($V$) of the interference pattern after the
single-photons pass through the A interferometer, whose measurement is in
accord with the following operator,
$\displaystyle\hat{V}\equiv\left.\frac{a_{1}^{\dagger}a_{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}+a_{2}^{\dagger}a_{1}{\mkern 1.0mu\mathrm{e}\mkern 1.0mu}^{-{\mkern
1.0mu\mathrm{i}\mkern 1.0mu}\phi}}{\langle a_{1}^{\dagger}a_{1}\rangle+\langle
a_{2}^{\dagger}a_{2}\rangle}\right.$ (3)
By describing the distinguishability and visibility in terms of operators, the
particle-like and wave-like information of single-photons are then the modules
of the two expectation values, i.e., $D=|\langle\hat{D}\rangle|$ and
$V=|\langle\hat{V^{\prime}}\rangle|_{\text{max by }\phi}$, where the phase
parameter $\phi$, controlled by the phase shifter in the interferometer,
should be appropriately chosen to maximize the expectation value of the
operator (3).
The terms $a_{1}^{\dagger}a_{1}$ ($a_{2}^{\dagger}a_{2}$) in Eq. (2) and
$a_{1}^{\dagger}a_{2}$ in Eq. (3) are just the first order auto-correlation of
the field in path $1$ ($2$) and the first order coherence between the fields
in the two paths correlation , respectively. Therefore the distinguishability
and visibility defined in Eqs. (2) and (3) can be regarded as the difference
and coherence between the first order correlation function of the two fields
in the two paths $1$ and $2$.
Based on this viewpoint, we now introduce the concepts of $k$th-order
distinguishability,
$\displaystyle\hat{D}_{k}\equiv\frac{(a_{1}^{\dagger})^{k}a_{1}^{k}-(a_{2}^{\dagger})^{k}a_{2}^{k}}{\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle+\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle},$
(4a) and $k$th-order visibility,
$\displaystyle\hat{V}_{k}\equiv\frac{(a_{1}^{\dagger})^{k}a_{2}^{k}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}}{\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle+\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle},$
(4b)
the denominator
$\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle+\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle$
in both equations is again for normalization. Here we have used $k$th-order
auto-correlation $(a_{1}^{\dagger})^{k}a_{1}^{k}$
($(a_{2}^{\dagger})^{k}a_{2}^{k}$) and $k$th-order coherence
$(a_{1}^{\dagger})^{k}a_{2}^{k}$ to replace the first order auto-correlation
and the first order coherence used in Eqs. (2) and (3), which can now be
regarded as the special case of the definitions in (4) by setting $k=1$.
Similar to the above treatment on the first order distinguishability and
visibility, the $k$th-order particle-like information and $k$th-order wave-
like information are just the modules of the corresponding expectation values,
i.e., $D_{k}=|\langle\hat{D}_{k}\rangle|$ and
$V_{k}=|\langle\hat{V}_{k}\rangle|_{\text{max by }\phi}$. Here the phase
parameter $\phi$ should be appropriately chosen to maximize the visibility
$V_{k}$, whose measurement will be introduced in section V in more details.
## III Inequality for higher order duality
Now we define the $k$th-order duality as the sum of the squared $k$th-order
particle-like information and $k$th-order wave-like information, which is,
$\displaystyle D_{k}^{2}+V_{k}^{2}=$
$\displaystyle\left(\frac{\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle+\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle}{\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle+\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle}\right)^{2}$
(5)
$\displaystyle+4\frac{\left|\langle(a_{1}^{\dagger})^{k}a_{2}^{k}\rangle\right|^{2}-\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle}{\left(\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle+\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle\right)^{2}}.$
The Cauchy-Schwarz inequality predicts
$\displaystyle\left|\langle(a_{1}^{\dagger})^{k}a_{2}^{k}\rangle\right|^{2}\leq\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle.$
(6)
Therefore, the second term in Eq. (5) is negative or zero. This result leads
to the inequality for higher order duality,
$\displaystyle D_{k}^{2}+V_{k}^{2}\leq 1,$ (7)
which is the main conclusion in this paper. Although this inequality has a
similar formula to Eq. (1), it undoubtedly carries more information about the
wave-particle duality and helps deepen our understanding. In fact, we have
generalized the duality relation from single-photon fields to multi-photon
fields. In a typical duality experiment, if the interferometer (see Fig. 1) is
fed with multi-photons, besides single-photons, according to our conclusion,
the fields in the two paths have to obey not only the first order duality
relation (1), but also the higher order duality relation (7). Since $n$-photon
component in a field only contributes the $k$th-order correlation function and
the $k$th-order coherence with $k\leq n$, and takes no effect for the
$k^{\prime}$th-order correlation function or coherence if $k^{\prime}>n$, all
higher than $n$th-order duality information does not exist (sums up to zero)
for the case that at most $n$-photon component is found in the field. As a
consequence, for a state with $n$-photons there exist $n$ inequalities,
$D_{k}^{2}+V_{k}^{2}\leq 1$, with $k=1,2,\dots,n$. That is why we only need to
consider the first order distinguishability (2) and the first order visibility
(3) in the duality experiments with single-photons.
Figure 1: The distinguishability and visibility are measured in the open
(removing the beam splitter BS) and closed (employing the beam splitter BS)
interferometer, respectively.
From Eq. (5), it is easy to find that the equality sign in the higher order
duality relation (7) will be satisfied under the condition
$|\langle(a_{1}^{\dagger})^{k}a_{2}^{k}\rangle|^{2}=\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle$.
For example, the $k$th-order duality, if it exists, is saturated for
$k$-photon pure states, which is a generalization of the first order duality
relation $D_{1}^{2}+V_{1}^{2}=1$ for the single-photons in a pure state. The
condition on which the $k$th-order duality (7) achieves its maximum value
unity is usually very complicated if $n$-photon component with $n>k$ is
involved in the field, and
$|\langle(a_{1}^{\dagger})^{k}a_{2}^{k}\rangle|^{2}=\langle(a_{1}^{\dagger})^{k}a_{1}^{k}\rangle\langle(a_{2}^{\dagger})^{k}a_{2}^{k}\rangle$
is at present the only test equation we can obtain.
## IV Physical interpretation
The $k$th-order auto-correlation $(a_{i}^{\dagger})^{k}a_{i}^{k}$ (i=1,2) used
in the definition (4a) is equal to,
$\displaystyle(a_{i}^{\dagger})^{k}a_{i}^{k}=\prod_{j=0}^{k-1}(\hat{n}_{i}^{\dagger}-j),$
(8)
with the number operator $\hat{n}_{i}=a_{i}^{\dagger}a_{i}$. Imposing this
operator onto a number state $|n_{i}>$, we have,
$(a_{i}^{\dagger})^{k}a_{i}^{k}|n_{i}>=(\begin{array}[]{c}n\\\
k\end{array})|n_{i}>$, with the binomial coefficient $(\begin{array}[]{c}n\\\
k\end{array})=\frac{n!}{k!(n-k)!}$. Thus
$\langle(a_{i}^{\dagger})^{k}a_{i}^{k}\rangle$ can be regarded as the
combination number of picking out $k$ photons, disregarding order, in path
$i$, no matter what state is prepared for the optical field in this path. The
$k$th order distinguishability $D_{k}$ then has a very clear physical meaning,
i.e., the normalized difference between the $k$-combinations of the photons in
paths $1$ and $2$.
In the basis $|0_{1}0_{2}>,|0_{1}1_{2}>,\cdots,|n_{1}n_{2}>$, a general
quantum state for the photons in the interferometer can be described by a
density matrix,
$\displaystyle\rho=\left(\begin{array}[]{ccc}\rho_{11}{}&\cdots&\rho_{1(n+1)^{2}}\\\
\vdots{}&\ddots&\vdots\\\
\rho_{(n+1)^{2}1}{}&\cdots&\rho_{(n+1)^{2}(n+1)^{2}}\end{array}\right)$ (12)
Under this quantum state, we can directly write down the expectation value of
the $k$th order distinguishability,
$\displaystyle\langle
D_{k}\rangle=\frac{\sum_{i=k}^{n}\sum_{j=0}^{n}\left(\rho_{p_{i,j}p_{i,j}}-\rho_{q_{i,j}q_{i,j}}\right)\left(\begin{array}[]{c}i\\\
k\end{array}\right)}{\sum_{i=k}^{n}\sum_{j=0}^{n}\left(\rho_{p_{i,j}p_{i,j}}+\rho_{q_{i,j}q_{i,j}}\right)\left(\begin{array}[]{c}i\\\
k\end{array}\right)},$ (17)
with $p_{i,j}=i*n+i+j+1$ and $q_{i,j}=j*n+i+j+1$. Here we see that only the
diagonal elements contribute to the distinguishability.
Just as we already mentioned, the visibility in a duality experiment is
actually related to the coherence between the two paths. Thus the visibility
is determined by the off diagonal elements of the density matrix for the
photons in an interferometer. For example, for the operator
$(a_{1}^{\dagger})^{k}a_{2}^{k}{\mkern 1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern
1.0mu\mathrm{i}\mkern 1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}k\phi}$
used in the definition of the $k$th order visibility (4b), the off diagonal
elements
$<m^{\prime}_{1}m^{\prime\prime}_{2}|\rho|(m^{\prime}+k)_{1}(m^{\prime\prime}-k)_{2}>$
and
$<m^{\prime}_{1}m^{\prime\prime}_{2}|\rho|(m^{\prime}-k)_{1}(m^{\prime\prime}+k)_{2}>$
with $m^{\prime},m^{\prime\prime}\in[0,n]$ have contributions. The diagonal
elements play no role in the evaluation of the visibility. However, for a
density matrix, what we can directly measure in experiments are just the
diagonal elements, usually represented as the photon counting or higher order
coincidence counting. So we have to turn the information carried by the off
diagonal elements to diagonal elements. That is why a $50:50$ beam splitter is
to be employed at the output of the interferometer for the measurement of the
visibility. For more details on the measurement of higher order visibility,
please see section V.
Now we can conclude that the distribution of the photons in an interferometer
projected on the Fock states, i.e., the diagonal elements of the density
matrix in the Fock state basis, determines the photons’ particle information,
and the wave information relies on the off diagonal elements, no matter what
order distinguishability and visibility is considered.
## V Measurement scheme for the visibility
Compared with the measurement of the distinguishability $D_{k}$, which is
directly defined by the auto-correlation, the measurement of the fringe
visibility $V_{k}$, which depends on higher order coherence, is more
complicated. For $k=1$, there exist a straight forward way to measure $V_{1}$.
Suppose the beam splitter in Fig. 1 is active and the modes after the beam
splitter are denoted with $C$ and $D$, then the annihilation operators $c$ and
$d$ of the modes in these two paths are connected to the annihilation
operators $a_{1}$ and $a_{2}$ of the modes in paths $1$ and $2$ through the
relations, $c=\frac{1}{\sqrt{2}}(a_{1}+a_{2}e^{i\phi})$ and
$d=\frac{1}{\sqrt{2}}(a_{1}-a_{2}e^{i\phi})$, respectively, where the phase
difference $\phi$ between the two paths can be controlled in experiments by a
phase shifter (see Fig. 1). The counting difference between the two detectors
$D_{1}$ and $D_{2}$ (see Fig. 1) is equal to,
$\displaystyle\langle c^{\dagger}c-d^{\dagger}d\rangle_{\phi}=\langle
a_{1}^{\dagger}a_{2}{\mkern 1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern
1.0mu\mathrm{i}\mkern 1.0mu}\phi}+a_{2}^{\dagger}a_{1}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}\rangle,$ (18)
which is just the first order visibility defined in Eq. (3).
The second order correlation function, needed for $V_{2}$, can also be
measured in experiments based on current technology hongoumandel . However,
the measurement of the higher order visibility $V_{k}$ is complicated.
For a general description, we now suppose the two detectors, $D_{1}$ and
$D_{2}$, in Fig. 1 are ideal ones, so that all Fock states with arbitrary
photon number can be directly detected and distinguished. For the case of
$k=2$, we take the sum of the expectation value of the both detectors and
obtain,
$\begin{split}\langle(c^{\dagger})^{2}c^{2}+(d^{\dagger})^{2}d^{2}\rangle_{\phi}&=\frac{1}{2}\langle(a_{1}^{\dagger})^{2}a_{1}^{2}+(a_{2}^{\dagger})^{2}a_{2}^{2}+4a_{1}^{\dagger}a_{2}^{\dagger}a_{1}a_{2}\\\
&+(a_{1}^{\dagger})^{2}a_{2}^{2}{\mkern 1.0mu\mathrm{e}\mkern 1.0mu}^{2{\mkern
1.0mu\mathrm{i}\mkern 1.0mu}\phi}+(a_{2}^{\dagger})^{2}a_{1}^{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{-2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}\rangle.\end{split}$ (19)
Further retarding the phase shift $\phi$ between the two path $1$ and $2$ by
the value $\pi/2$, we obtain another similar relation,
$\begin{split}\langle(c^{\dagger})^{2}c^{2}+(d^{\dagger})^{2}d^{2}\rangle_{\phi+\pi/2}&=\frac{1}{2}\langle(a_{1}^{\dagger})^{2}a_{1}^{2}+(a_{2}^{\dagger})^{2}a_{2}^{2}+4a_{1}^{\dagger}a_{2}^{\dagger}a_{1}a_{2}\\\
&-(a_{1}^{\dagger})^{2}a_{2}^{2}{\mkern 1.0mu\mathrm{e}\mkern 1.0mu}^{2{\mkern
1.0mu\mathrm{i}\mkern 1.0mu}\phi}-(a_{2}^{\dagger})^{2}a_{1}^{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{-2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}\rangle.\end{split}$ (20)
In the following, we use two symbols $R^{\pm}_{k,\phi}$, whose values are in
principle obtainable in experiments, to replace the expectation values
$\langle(c^{\dagger})^{k}c^{k}\pm(d^{\dagger})^{k}d^{k}\rangle_{\phi}$. Thus
the equality (18) can be rewritten as $\langle a_{1}^{\dagger}a_{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}+a_{2}^{\dagger}a_{1}{\mkern 1.0mu\mathrm{e}\mkern 1.0mu}^{-{\mkern
1.0mu\mathrm{i}\mkern 1.0mu}\phi}\rangle=R^{-}_{1,\phi}$, and the quantity
$\langle(a_{1}^{\dagger})^{2}a_{2}^{2}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}+(a_{2}^{\dagger})^{2}a_{1}^{2}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-2{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}\phi}\rangle$, involved in both
equalities (19) and (20), can be described by
$\begin{split}\langle(a_{1}^{\dagger})^{2}a_{2}^{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}+(a_{2}^{\dagger})^{2}a_{1}^{2}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}\rangle=R^{+}_{2,\phi}-R^{+}_{2,\phi+\pi/2}.\end{split}$ (21)
The maximum value of $\langle(a_{1}^{\dagger})^{2}a_{2}^{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}+(a_{2}^{\dagger})^{2}a_{1}^{2}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-2{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}\phi}\rangle$ over the phase
factor $\phi$, required in the quantification of the second order visibility
$V_{2}$, can then be evaluated by,
$\begin{split}|\langle(a_{1}^{\dagger})^{2}&a_{2}^{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi}+(a_{2}^{\dagger})^{2}a_{1}^{2}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-2{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}\phi}\rangle|_{\text{max by
}\phi}^{2}=\\\
&\left(R^{+}_{2,\phi^{\prime}}-R^{+}_{2,\phi^{\prime}+\pi/2}\right)^{2}+\left(R^{+}_{2,\phi^{\prime}-\pi/4}-R^{+}_{2,\phi^{\prime}+\pi/4}\right)^{2},\end{split}$
(22)
where $(R^{+}_{2,\phi^{\prime}}-R^{+}_{2,\phi^{\prime}+\pi/2})$ is the real
part of the vector $2\langle(a_{1}^{\dagger})^{2}a_{2}^{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi^{\prime}}\rangle$ (see Eq. (21)), and
$(R^{+}_{2,\phi^{\prime}-\pi/4}-R^{+}_{2,\phi^{\prime}+\pi/4})$ is the real
part of the vector $2\langle(a_{1}^{\dagger})^{2}a_{2}^{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}(\phi^{\prime}-\pi/4)}\rangle$, which is equal to the imaginary part of
the vector $2\langle(a_{1}^{\dagger})^{2}a_{2}^{2}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{2{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}\phi^{\prime}}\rangle$. The absolute value of this quantity is
mathematically equivalent to the unnormalized visibility $V_{2}$, due to the
relation $|\langle(a_{1}^{\dagger})^{k}a_{2}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}k\phi}\rangle|_{\text{max by
}\phi}=2|\langle(a_{1}^{\dagger})^{k}a_{2}^{k}\rangle|$. The phase
$\phi^{\prime}$ can be arbitrarily chosen, because the modulus of a vector
should remain invariant under the rotation of the coordinate system.
In general, the term $\langle(a_{1}^{\dagger})^{k}a_{2}^{k}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}k\phi}\rangle$ can be determined
by adding and subtracting
$\langle(c^{\dagger})^{k}c^{k}\pm(d^{\dagger})^{k}d^{k}\rangle_{\phi}$ for $k$
different values of $\phi$. For example, for odd number of $k$, we have
$\begin{split}\langle(a_{1}^{\dagger})^{k}&a_{2}^{k}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}\rangle=\frac{2^{k-1}}{k}\sum_{m=0}^{k-1}R^{-}_{k,\phi+2m\pi/k}.\end{split}$
(23)
Accordingly, the maximum value of
$\langle(a_{1}^{\dagger})^{k}a_{2}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}k\phi}\rangle$ over the phase
factor $\phi$, used for the $k$th order visibility $V_{k}$, can be evaluated
by,
$\begin{split}|\langle(&a_{1}^{\dagger})^{k}a_{2}^{k}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}k\phi}\rangle|_{\text{max by
}\phi}^{2}=\left(\frac{2^{k-1}}{k}\right)^{2}\times\\\
&\left[\left(\sum_{m=0}^{k-1}R^{-}_{k,\phi^{\prime}+2m\pi/k}\right)^{2}+\left(\sum_{m=0}^{k-1}R^{-}_{k,\phi^{\prime}-\pi/(2k)+2m\pi/k}\right)^{2}\right],\end{split}$
(24)
where the phase $\phi^{\prime}$ can be arbitrarily chosen.
For even number of $k$, we have
$\begin{split}\langle(a_{1}^{\dagger})^{k}&a_{2}^{k}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}\rangle=\frac{2^{k-1}}{k}\sum_{m=0}^{k-1}(-1)^{m}R^{+}_{k,\phi+m\pi/k}.\end{split}$
(25)
The maximum value of $\langle(a_{1}^{\dagger})^{k}a_{2}^{k}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}k\phi}\rangle$ over the phase
factor $\phi$, used for the $k$th order visibility $V_{k}$, can be evaluated
by,
$\begin{split}&|\langle(a_{1}^{\dagger})^{k}a_{2}^{k}{\mkern
1.0mu\mathrm{e}\mkern 1.0mu}^{{\mkern 1.0mu\mathrm{i}\mkern
1.0mu}k\phi}+(a_{2}^{\dagger})^{k}a_{1}^{k}{\mkern 1.0mu\mathrm{e}\mkern
1.0mu}^{-{\mkern 1.0mu\mathrm{i}\mkern 1.0mu}k\phi}\rangle|_{\text{max by
}\phi}^{2}=\left(\frac{2^{k-1}}{k}\right)^{2}\times\\\
&\left[\left(\sum_{m=0}^{k-1}(-1)^{m}R^{+}_{k,\phi^{\prime}+m\pi/k}\right)^{2}+\left(\sum_{m=0}^{k-1}(-1)^{m}R^{+}_{k,\phi^{\prime}-\pi/(2k)+m\pi/k}\right)^{2}\right],\end{split}$
(26)
with an arbitrary phase factor $\phi^{\prime}$.
## VI Conclusions
The distinguishability of photons’ passage in the interferometer and the
visibility of the interference pattern after the interferometer can be
regarded as the first order particle-like information and the first order
wave-like information. By introducing the concepts of higher order
distinguishability and visibility for multi-photons, which are related to
higher order auto-correlation and coherence between the fields in the two
paths of the interferometer, we generalize the wave-particle duality relation
from the first order case to higher order case. We believe it to be a useful
tool for analyzing the duality experiments with the input of multi-photons, or
even a classical light. If we do the duality experiment by using different
light sources, the same results may be obtained if only the first order
duality is considered. However, we believe it will exhibit different results
for higher order duality information. The concept of higher order duality may
provide us more information about the duality experiments, especially with the
input of multi-photons, and accordingly helps us deepen the understanding on
the wave-particle duality.
## Acknowledgements
This research was supported by the National Basic Research Program of China
(Grant No. 2011CB922203 and 2012CB921603), the national Natural Science
Foundation of China (Grant No. 11174118 and 11174026), and the Natural Science
Foundation of Jiangxi Province under Grant No. 20114BAB212003. The research of
MSZ is supported by NPRP grant (No. 4-346-1-061) from Qatar National Research
Fund.
## References
* (1) N. Bohr, Naturwissenschaften 16, 245 (1928).
* (2) M. O. Scully and M. S. Zubairy, in _Quantum optics_ , (Cambridge University Press, 1997)
* (3) W. K. Wootters and W. H. Zurek, Phys. Rev. D 19, 473 (1979).
* (4) R. J. Glauber, Annals of the New York Academy of Sciences 480 (1), 336 (1986).
* (5) D. M. Greenberger and A. Yasin, Phys. Lett. A 128, 391 (1988).
* (6) S. Dürr and G. Rempe, Am. J. Phys. 68, 1021 (2000).
* (7) B. G. Englert, Phys. Rev. Lett. 77, 2154 (1996).
* (8) T. Pfau, S. Spälter, Ch. Kurtsiefer, C. R. Ekstrom, and J. Mlynek, Phys. Rev. Lett. 73, 1223 (1994); M. S. Chapman, T. D. Hammond, A. Lenef, J. Schmiedmayer, R. A. Rubenstein, E. Smith, and D. E. Pritchard, Phys. Rev. Lett. 75, 3783 (1995); E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, Nature (London) 391, 871-874 (1998); S. Dürr, T. Nonn and G. Rempe , Nature (London) 395, 33-37 (1998); P. Bertet, S. Osnaghi, A. Rauschenbeutel, G. Nogues, A. Auffeves, M. Brune, J. M. Raimond, and S. Haroche, Nature (London) 411, 166-170 (2001).
* (9) J. A. Wheeler, in _Quantum Theory and Measurement_ , J. A. Wheeler, W. H. Zurek, Eds. (Princeton Univ. Press, Princeton, NJ, 1984), pp. 182-213.
* (10) V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Science 315, 966 (2007); Phys. Rev. Lett. 100, 220402 (2008).
* (11) M. O. Scully and K. Drühl, Phys. Rev. A 25, 2208 (1982); M. O. Scully and H. Walther, Phys. Rev. A 39, 5229 (1989); M. O. Scully, B. G. Englert, and J. Schwinger, Phys. Rev. A 40 1775 (1989); M. O. Scully, B. G. Englert, and H. Walther, Nature 351, 111 (1991); Y. Aharonov and M. S. Zubairy, Science 307 875 (2005).
* (12) Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
* (13) R. Ionicioiu, D. R. Terno, Phys. Rev. Lett. 107, 230406 (2011).
* (14) M. Schirber, Physics 4, 102 (2011).
* (15) J. S. Tang, Y. L. Li, X. Y. Xu, G. Y. Xiang, C. F. Li, and G. C. Guo, Nat. Photonics 6, 602 (2012); A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, Science 338, 634-637 (2012); F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637-640 (2012).
* (16) G. Björk, J. Söderholm, A. Trifonov, T. Tsegaye, and A. Karlsson, Phys. Rev. A 60, 1874 (1999); P. Busch and C. Shilladay, Phys. Rep. 435, 1-31 (2006); H. Y. Liu, J. H. Huang, J. R. Gao, M. S. Zubairy, and S. Y. Zhu, Phys. Rev A 85, 022106 (2012).
* (17) R. J. Glauber, in _Quantum Optics and Electronics_ , Les Houches, ed. C. DeWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York 1965).
* (18) C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
|
arxiv-papers
| 2012-12-18T07:48:45 |
2024-09-04T02:49:39.382384
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jie-Hui Huang, Sabine W\\\"olk, Shi-Yao Zhu, and M. Suhail Zubairy",
"submitter": "Jie-Hui Huang",
"url": "https://arxiv.org/abs/1212.4255"
}
|
1212.4397
|
Measurements of $\Delta m_{d}$, $\Delta m_{s}$, and $\sin 2\beta$
with LHCb
Julian Wishahi on behalf of LHCb collaboration
Technische Universität Dortmund
Experimentelle Physik V
D-44221 Dortmund, GERMANY
Proceedings of CKM 2012, the 7th International Workshop on the CKM Unitarity
Triangle, University of Cincinnati, USA, 28 September – 2 October 2012
## 1 Introduction
The LHCb experiment [1] at the Large Hadron Collider (LHC) at CERN is
dedicated to the study of $b$ and $c$ flavour physics. It exploits the large
production cross-section of $b$ and $c$ hadrons in LHC’s $pp$ collisions. The
measurement of time-dependent decay rates in decays of neutral $b$ mesons,
i.e. $B^{0}$ and the $B^{0}_{s}$ mesons, gives access to a variety of
observables that are linked to the Cabibbo-Kobayashi-Maskawa quark mixing
matrix [3, 2]. The LHCb detector provides good decay time and impact parameter
resolution, and an excellent particle identification system to efficiently
reconstruct exclusive $B$ decay final states.
Precision measurements of the oscillation frequencies $\Delta m_{d}$ of
$B^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptB}{}^{0}$ and $\Delta m_{s}$
of $B^{0}_{s}$–$\kern 1.99997pt\overline{\kern-1.99997ptB}{}^{0}_{s}$ meson
mixing allow to constrain the apex of the CKM triangle while a measurement of
the time-dependent $C\\!P$-asymmetry in decays of $B^{0}\to J\\!/\\!\psi
K_{\text{S}}^{0}$ gives access to the CKM angle $\beta$. Hence, these
measurements give valuable input to tests of the unitarity of the CKM matrix.
## 2 Flavour tagging in LHCb
Measurements of time-dependent asymmetries in decays of neutral $B$ mesons
require flavour tagging, i.e. the knowledge on the production flavour of the
reconstructed decay. As $b$ quarks are predominantly produced as
$b\overline{}b$ pairs at LHCb, two classes of flavour tagging algorithms are
utilised in LHCb [4, 5]. The same-side taggers reconstruct the charge of the
hadronisation remnant of the signal $B$ meson, i.e. a $K^{\pm}$ from the
$B^{0}_{s}$ hadronisation or a $\pi^{\pm}$ for the $B^{0}$/$B^{\pm}$. The
opposite-side taggers reconstruct the flavour of the non-signal $b$ hadron by
identifying the charge of its decay products, like leptons from semileptonic
decays or kaons from $b\to c\to s$ decays.
Inefficiencies in the tagging, e.g. the choice of wrong tagging particle
tracks, have to be measured in the context of $C\\!P$ measurements. The
tagging efficiency $\epsilon_{\text{tag}}$ is the fraction of events with a
tagging decision, the mistag fraction $\omega$ represents the probability for
an incorrectly assigned tag. The overall tagging power that reflects the
statistical precision is given by $\epsilon_{\text{tag}}(1-2\omega)^{2}$.
The mistag fraction $\omega$ is calibrated on the self-tagging channel
$B^{+}\to J\\!/\\!\psi K^{+}$ for the opposite-side taggers. The mixing
analyses described below are used to measure the mistag probability of the
same side taggers. The tagging power of the combined opposite side taggers is
found to be around $2.3\%$, while the same-side tagger for $B^{0}_{s}$ mesons
(same-side kaon tagger) adds about $1\%$ of tagging power and the same-side
tagger for $B^{0}/B^{+}$ mesons (same-side pion tagger) adds around $0.5\%$.
## 3 Neutral $B$ meson mixing
Measurements of neutral $B$ meson mixing require a decay time dependent
analysis of the mixing behaviour. Reconstructed $B$ candidates are categorised
by their mixing state: they are defined as unmixed (mixed) if their production
flavour matches (does not match) the flavour at decay. This knowledge can be
used to determine the oscillation frequency $\Delta m_{q}$, with $q=d$ ($q=s$)
in the $B^{0}$ ($B^{0}_{s}$) system, by using the time-dependent mixing
asymmetry
$\displaystyle\mathcal{A}_{\text{mix}}(t)=\frac{N_{\text{unmixed}}(t)-N_{\text{mixed}}(t)}{N_{\text{unmixed}}(t)+N_{\text{mixed}}(t)}=\cos\left(\Delta
m_{q}t\right)\ ,$ (1)
where $N_{\text{(un)mixed}}(t)$ is the number of (un)mixed candidates at $B$
decay time $t$. Due to the imperfections in tagging, the accessible amplitude
of the asymmetry is reduced by $(1-2\omega)$. The amplitude is additionally
damped due to the decay time resolution. At LHCb, this dampening has to be
considered for measurements involving $B^{0}_{s}$ mixing but is negligible in
$B^{0}$ mixing related analyses.
### 3.1 Measurement of $\Delta m_{s}$ in $B^{0}_{s}\to D_{s}^{-}\pi^{+}$
decays
The measurement of the $B^{0}_{s}$ oscillation frequency $\Delta m_{s}$ at
LHCb is performed using $340\,\mathrm{pb^{-1}}$ of data collected in
$\sqrt{s}=7\,\mathrm{TeV}$ $pp$ collisions [6, 7]. Around 9100 $B^{0}_{s}\to
D_{s}^{-}\pi^{+}$ decays with subsequent decays of $D_{s}^{-}\to\phi\pi^{-}$
($\phi\to K^{+}K^{-}$), $D_{s}^{-}\to K^{\ast 0}(892)K^{-}$ ($K^{\ast
0}(892)\to K^{+}\pi^{-}$), and the non-resonant $D_{s}^{-}\to
K^{+}K^{-}\pi^{-}$ are reconstructed. The analysis is performed using the
combination of opposite-side taggers and the same-side kaon tagger as well as
using only the same-side kaon tagger. The resulting projection of the time-
dependent mixing asymmetry is shown in Fig. 1. A decay time resolution of
$45\,\mathrm{fs}$ was measured. The analysis yields the most precise
measurement of $\Delta m_{s}$ with
$\displaystyle\Delta m_{s}=17.725\pm 0.041\,\text{(stat.)}\pm
0.026\,\text{(syst.)}\mathrm{ps^{-1}}\ ,$
where the largest systematic uncertainty is related to the uncertainty of the
length scale.
Figure 1: Mixing asymmetry for $B^{0}_{s}$ signal candidates as a function of
decay time, modulo $\left(\frac{2\pi}{\Delta m_{s}}\right)$, for the fit using
only the same-side tagger (left) and the combination of opposite- and same-
side taggers (right). The fitted signal asymmetries are superimposed.
### 3.2 Measurement of $\Delta m_{d}$ in $B^{0}\to J\\!/\\!\psi K^{\ast 0}$
and $B^{0}\to D^{-}\pi^{+}$ decays
The basic analysis strategy for measuring $\Delta m_{d}$ [8] is very similar
to the analysis strategy for the measurement of $\Delta m_{s}$, however, as
the $B^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptB}{}^{0}$ oscillations
are about $35$ times slower than $B^{0}_{s}$–$\kern
1.99997pt\overline{\kern-1.99997ptB}{}^{0}_{s}$ oscillations, the decay time
resolution has an inferior role in this analysis. From a dataset of
$1.0\,\mathrm{fb^{-1}}$ collected at $\sqrt{s}=7\,\mathrm{TeV}$ $pp$
collisions about $88{,}000$ decays of $B^{0}\to D^{-}\pi^{+}$ ($D^{-}\to
K^{+}\pi^{-}\pi^{-}$) and approximately $39{,}000$ decays of $B^{0}\to
J\\!/\\!\psi K^{\ast 0}(892)$ ($J\\!/\\!\psi\to\mu^{-}\mu^{+}$, $K^{\ast
0}(892)\to K^{+}\pi^{-}$) are reconstructed. A combination of the opposite-
side taggers with the same-side pion tagger is used. The multi-dimensional fit
to the distributions of the reconstructed mass and the decay time yields
$\displaystyle\Delta m_{d}(B^{0}\to D^{-}\pi^{+})\,$ $\displaystyle=0.5178\pm
0.0061\,\text{(stat.)}\pm 0.0037\,\text{(syst.)}\mathrm{ps^{-1}}\mbox{ and}$
$\displaystyle\Delta m_{d}(B^{0}\to J\\!/\\!\psi K^{\ast 0})\,$
$\displaystyle=0.5096\pm 0.0114\,\text{(stat.)}\pm
0.0022\,\text{(syst.)}\mathrm{ps^{-1}}.$
The mixing asymmetries in the two channels and the resulting fit projections
are shown in Fig. 2. The largest systematic uncertainties are related to the
limited knowledge of the decay time distributions for the background
components in the fit.
Combining both channels results in the currently most precise single
measurement of this parameter,
$\displaystyle\Delta m_{d}=0.5156\pm 0.0051\,\text{(stat.)}\pm
0.0033\,\text{(syst.)}\mathrm{ps^{-1}}\ ,$
in full agreement with previous measurements [9].
Figure 2: Raw mixing asymmetry (black points) for (left) $B^{0}\to
D^{-}\pi^{+}$ and (right) $B^{0}\to J\\!/\\!\psi K^{\ast 0}$ candidates. The
solid black line is the projection of the mixing asymmetry of the combined
PDF.
## 4 Measurement of $\sin 2\beta$ in $B^{0}\to J\\!/\\!\psi K_{\text{S}}^{0}$
decays
The CKM observable $\sin 2\beta$ is one of the most precisely measured $C\\!P$
parameters. Its current world average is $\sin 2\beta=0.67\pm 0.02\pm 0.01$
[9], where the most precise measurements were performed by the $B$ factories
Belle and BaBar.
For the LHCb measurement of $\sin 2\beta$ [10] around $8{,}000$ $B^{0}\to
J\\!/\\!\psi K_{\text{S}}^{0}$ ($J\\!/\\!\psi\to\mu^{-}\mu^{+}$,
$K_{\text{S}}^{0}\to\pi^{+}\pi^{-}$) decays with tagging information from the
opposite side taggers are reconstructed in a data sample of
$1\,\mathrm{fb^{-1}}$. The parameter $\sin 2\beta$ is determined by measuring
the time-dependent $C\\!P$ asymmetry
$\displaystyle\mathcal{A}_{J\\!/\\!\psi K_{\text{S}}^{0}}(t)$
$\displaystyle=\frac{\Gamma(\kern
1.99997pt\overline{\kern-1.99997ptB}{}^{0}(t)\to J\\!/\\!\psi
K_{\text{S}}^{0})-\Gamma(B^{0}(t)\to J\\!/\\!\psi
K_{\text{S}}^{0})}{\Gamma(\kern
1.99997pt\overline{\kern-1.99997ptB}{}^{0}(t)\to J\\!/\\!\psi
K_{\text{S}}^{0})+\Gamma(B^{0}(t)\to J\\!/\\!\psi K_{\text{S}}^{0})}$
$\displaystyle=S_{J\\!/\\!\psi K_{\text{S}}^{0}}\sin(\Delta
m_{d}t)-C_{J\\!/\\!\psi K_{\text{S}}^{0}}\cos(\Delta m_{d}t),$ (2)
where $S_{J\\!/\\!\psi K_{\text{S}}^{0}}=\sqrt{1-C_{J\\!/\\!\psi
K_{\text{S}}^{0}}^{2}}\sin 2\beta$. A multi-dimensional fit of the data is
performed. Fig. 3 shows the background corrected asymmetry and the fit
projection.
Figure 3: Time-dependent asymmetry $(N_{\kern
1.39998pt\overline{\kern-1.39998ptB}{}^{0}}-N_{B^{0}})/(N_{\kern
1.39998pt\overline{\kern-1.39998ptB}{}^{0}}+N_{B^{0}})$. Here, $N_{B^{0}}$
($N_{\kern 1.39998pt\overline{\kern-1.39998ptB}{}^{0}}$) is the number of
$B^{0}\to J\\!/\\!\psi K_{\text{S}}^{0}$ decays with a $B^{0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptB}{}^{0}$) flavour tag. The data points are
obtained with the ${}_{s}Plot$ [11] technique, assigning signal weights to the
events based on a fit to the reconstructed mass distributions. The solid curve
is the signal projection of the PDF. The green shaded band corresponds to the
one standard deviation statistical error.
The measurement yields
$\displaystyle S_{J\\!/\\!\psi K_{\text{S}}^{0}}$ $\displaystyle=0.73\pm
0.07\text{\,(stat)}\pm 0.04\text{\,(syst)},$ $\displaystyle C_{J\\!/\\!\psi
K_{\text{S}}^{0}}$ $\displaystyle=0.03\pm 0.09\text{\,(stat)}\pm
0.01\text{\,(syst)},$
where the largest systematic uncertainties are related to the uncertainty of
the flavour tagging calibration and the background parameterisation. This is
the first significant measurement of $C\\!P$ violation in $B^{0}\to
J\\!/\\!\psi K_{\text{S}}^{0}$ decays at a hadron collider. The result is in
agreement with the world averages [9].
## 5 Conclusion
LHCb has collected $1.0\,\mathrm{fb^{-1}}$ of data from $pp$ collisions at a
centre-of-mass energy of $\sqrt{s}=7\,\mathrm{TeV}$. The time-dependent
measurements of $\Delta m_{s}$, $\Delta m_{d}$, and $\sin 2\beta$ with this
dataset demonstrate the excellent performance of LHCb in terms of signal
selection, decay time resolution, and flavour tagging.
## References
* [1] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [2] M. Kobayashi and T. Maskawa, $\mathit{C\\!P}$ violation in the renormalizable theory of weak interaction, Prog. Theor. Phys. 49 (1973) 652
* [3] N. Cabibbo, Unitary symmetry and leptonic decays, Phys. Rev. Lett. 10 (1963) 531
* [4] LHCb collaboration, R. Aaij et al., Opposite-side flavour tagging of B mesons at the LHCb experiment, Eur. Phys. J. C72 (2012) 2022, arXiv:1202.4979
* [5] LHCb collaboration, R. Aaij et al., Performance of flavour tagging algorithms optimised for the analysis of $B^{0}_{s}\to J\\!/\\!\psi\phi$, LHCb-CONF-2012-026
* [6] LHCb collaboration, R. Aaij et al., Measurement of the $B^{0}_{s}-\kern 1.99997pt\overline{\kern-1.99997ptB}{}^{0}_{s}$ oscillation frequency $\Delta m_{s}$ in $B^{0}_{s}\to D_{s}^{-}(3)\pi^{+}$ decays, Phys. Lett. B709 (2012) 177, arXiv:1112.4311
* [7] LHCb collaboration, R. Aaij et al., Measurement of $\Delta m_{s}$ in the decay $B^{0}_{s}\rightarrow D^{-}_{s}(K^{+}K^{-}\pi^{-})\pi^{+}$ using opposite-side and same-side flavour tagging algorithms, LHCb-CONF-2011-050
* [8] LHCb collaboration, R. Aaij et al., Measurement of the $B^{0}$–$\overline{B}^{0}$ oscillation frequency $\Delta m_{d}$ with the decays $B^{0}\to D^{-}\pi^{+}$ and $B^{0}\to J\ \psi K^{*0}$, LHCB-PAPER-2012-032, CERN-PH-EP-2012-315, arXiv:1210.6750
* [9] Heavy Flavour Averaging Group, Y. Amhis et al., Averages of b-hadron, c-hadron, and tau-lepton properties as of early 2012, arXiv:1207.1158
* [10] LHCb collaboration, R. Aaij et al., Measurement of the time-dependent $C\\!P$ asymmetry in $B^{0}\to J\\!/\\!\psi K_{\text{S}}^{0}$ decays, LHCB-PAPER-2012-035, CERN-PH-EP-2012-331, arXiv:1211.6093
* [11] M. Pivk and F. R. Le Diberder, sPlot: A statistical tool to unfold data distributions, Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
|
arxiv-papers
| 2012-12-18T15:48:07 |
2024-09-04T02:49:39.393835
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Julian Wishahi",
"submitter": "Julian Wishahi",
"url": "https://arxiv.org/abs/1212.4397"
}
|
1212.4446
|
# The Grammar Hammer of 2012111The title relates both to the folklore story of
a steel driving man named John Henry dying with a hammer in his hand instead
of losing to a steam drill [Nel08] and to a psychologist Abraham Maslow
stating that if the only tool you have is a hammer, it is tempting to treat
everything as if it were a nail [Mas62].
Vadim Zaytsev, [email protected]
Software Analysis and Transformation (SWAT) Team
Centrum Wiskunde & Informatica (CWI)
Amsterdam, The Netherlands
###### Contents
1. 1 Introduction
2. 2 Preliminaries
1. 2.1 Background notions
2. 2.2 Major contributions in a nutshell
3. 2.3 Selected minor contributions
4. 2.4 Motivation for this report
3. 3 Topics overview
1. 3.1 Guided grammar convergence
1. 3.1.1 Generalisation of production signatures
2. 3.1.2 History of attempted publication
2. 3.2 Grammar transformation languages
1. 3.2.1 XBGF
2. 3.2.2 $\Upxi$BGF
3. 3.2.3 NXBGF?
4. 3.2.4 EXBGF
5. 3.2.5 $\Delta$BGF?
3. 3.3 Metasyntax
1. 3.3.1 Notation specification
2. 3.3.2 Transforming metasyntaxes
3. 3.3.3 Notation-parametric grammar recovery
4. 3.3.4 Notation-driven grammar convergence
4. 3.4 Tolerance in parsing
5. 3.5 Megamodelling
1. 3.5.1 MegaL dissection
2. 3.5.2 Renarrating megamodels
6. 3.6 Grammar repository
7. 3.7 (Open) Notebook Science
8. 3.8 Minor topics
1. 3.8.1 Grammar mutation
2. 3.8.2 Iterative parsing
3. 3.8.3 Unparsing techniques
4. 3.8.4 Migration to git
5. 3.8.5 Turing machine programming
6. 3.8.6 Grammarware visualisation
7. 3.8.7 Wiki activity
8. 3.8.8 Colloquium organisation
4. 4 Venues
1. 4.1 Exercised venues
2. 4.2 Inspiring venues
5. 5 Concluding remarks
1. 5.1 Immediate results
2. 5.2 Special features
3. 5.3 Acknowledgements
## 1 Introduction
The purpose of this report is documenting personal research results of the
year 2012 in a form primarily intended for assessment of their scientific
merit as a foundation for future work, not for quantitative assessment of the
resulting publication record. This can be considered as an aggressive form of
self-archiving initiative [Har01] where scientific and engineering
contributions are not only logged, but also put in perspective by a separate
first class atomic scientific knowledge object. This report is mostly meant
for my SWAT colleagues. However, it is open for broad audience and meant to be
readable by any researcher with reasonable degree of familiarity with computer
science. It can be consumed as a self-contained document, but many details are
not pulled in from available referenced sources.
We start right away with a the overview of the field (§2.1) followed by brief
descriptions of major (§2.2) and minor (§2.3) contributions, followed by a
more elaborate motivation for creation of this document (§2.4). Next, all
research topics are laid out in detail one by one (§3). For the sake of
complexity, a separate overview of all involved venues (§4) is included. §5
concludes the report.
## 2 Preliminaries
### 2.1 Background notions
_Software language_ is a concept that generalises over programming languages,
markup languages, database schemata, data structures, abstract data types,
data types, modelling languages, ontologies, etc. Whenever we observe some
degree of _commitment to structure_ , we can identify it with a language,
which elements (symbols) can be separately defined and the allowed
combinations of them can be somehow specified. Studying software language
engineering is important because of possibly gained insights into relations
between the way such languages are defined and used in different technological
spaces (e.g., we can study data binding as a way to map a relational database
to an object model, or language convergence as a way to compare an XML schema
with a syntax definition).
_Formal grammars_ is a long-existing approach of dealing with languages —
originally context free grammars [Cho56] were mainly aimed at textual
programming languages [ASU85], but later other variants of grammars were
proposed, including keyword grammars [GM77], indexed grammars [Aho68],
lexicalised grammars [SAJ88], object grammars [SCL12], pattern grammars
[Gre96], array grammars [SSK72], puzzle grammars [Niv+91], picture grammars
[MS67], picture processing grammars [Cha70], tile grammars [RP05], grid
grammars [Dre+01], motion picture grammars [BAF97], pair grammars [Pra71],
triple graph grammars [Sch95], deterministic graph grammars [Cau07], string
adjunct grammars [JKY69], head grammars [Pol84], tree adjunct grammars
[JLT75], tree description grammars [Kal01], description tree grammars
[RVSW95], description tree substitution grammars [RWVS01], functional grammars
[Łuk77], Łukaszewicz universal grammars [Łuk82], two level grammars [Wij74],
van Wijngaarden grammars [Wij65], metamorphosis grammars [Col78], affix
grammars [Kos91], extended affix grammars [Mei90], attribute grammars [Knu90],
extended attribute grammars [WM83], definite clause grammars [PW86],
minimalist grammars [LR01], categorial grammars [Ajd35], type grammars
[Lam58], pregroup grammars [Lam08], Montague universal grammars [Mon70], logic
grammars [AD89], assumption grammars [DTL97], constraint handling grammars
[Chr05], abductive logic grammars [CD09], simple transduction grammars [LS68],
inversion transduction grammars [Wu97], range concatenation grammars [Bou98],
island grammars [DK99], bridge grammars [NNEH09], skeleton grammars [KL03],
permissive grammars [Kat+09], conjunctive grammars [Okh01], Boolean grammars
[Okh04], Peirce grammars [Böt01], transformational grammars [DeR74],
probabilistic grammars [Kor11], notional grammars [And91], analytic grammars
[For04], parsing schemata [Sik97], cooperating string grammar systems [CV+95],
cooperating array grammar systems [DFP95], cooperating puzzle grammar systems
[SSC06], etc222The earliest possible reference is given for each variant,
preferably from the programming language research field.. A grammar of a
software language, which specifies commitment to grammatical structure, is
called a _grammar in a broad sense_ [KLV05], even if in practice it defines a
metamodel or an API, thus not officially being a grammar at all. The
grammarware technological space is commonly perceived as mature and drained of
any scientific challenge, but provides many unsolved problems for researchers
who are active in that field.
For the last years, and specifically in 2012, I have focused my efforts on
using grammar-based techniques in the broad field of software language
engineering.
### 2.2 Major contributions in a nutshell
This section contains brief descriptions of the contributions of 2012 and some
statements about their usability and/or importance. Sections that contain
extended descriptions of the contributions with some level of technical
detail, are referenced in parenthesis.
Guided grammar convergence
(§3.1).
Grammar convergence is a lightweight verification method for establishing and
maintaining the correspondence between grammar knowledge ingrained in various
software artifacts [LZ09]. The method entails programming grammar
transformation steps with a general purpose grammar transformation operator
suite. It was acknowledged in [Cam10, p.34] as “a product-line approach to
provide […] an organised software structure”. Yet, the method had some weak
sides that inspired further investigation.
One of the biggest issues is maintenance of the grammar relationships. Once
they have been established by programming grammar transformation steps, it
becomes very hard to coevolve these steps with eventual changes in the source
grammars. An ideal solution would be a way to automatically reestablish
grammar relationships based on declarative constraints. This way is _guided
grammar convergence_ : instead of programming the transformations, we
construct an idealised “master grammar” that shows the most essential
properties of all grammars that are to be converged, and the transformation
steps are then derived automatically, guided by the structure of the master
grammar.
The transformation inference algorithm relies on the source grammars and their
metasyntax. This method was prototyped twice: in Python and in Rascal, and
tested successfully on 12 grammars in a broad sense obtained from different
technological spaces. It has not been properly published after being rejected
three times [Zay12g, Zay12h, Zay12i], but received encouraging feedback from
some of those venues and from one presentation [Zay12f].
Grammar transformation
(§3.2)
Grammar convergence, evolution, maintenance and any other activity that deals
with changes, can profit from expressing such changes in the functional way:
every step is represented as a function application, where a function is a
transformation operator such as _rename_ or _add_. The latest of such operator
suites has been developed in 2010 [Zay+08, XBGF Manual] and shown to be
superior to its alternatives [LZ11, §4].
During 2012, XBGF has been: reimplemented in Rascal, which led to extensive
testing and more systematic specification of operator semantics (§3.2.1);
extended for bidirectionality by pairing operators, introducing lacking ones
and abandoning unsuitable ones (§3.2.2);
experimentally extended for adaptability (§3.2.3);
extended by mining patterns of it usage (§3.2.4);
investigated for migration from the functional paradigm to the declarative one
(§3.2.5). Each of these initiatives is a nontrivial project complete with
conceptual motivation, programmed prototypes and obtained results (positive
for the first three, controversial for the fourth and decisively negative for
the last one).
Metasyntactic experiments
(§3.3)
Metasyntax as a language in which grammars are specified, was a topic briefly
touched in my PhD thesis [Zay10], but never officially published. In 2012, I
finally dedicated enough time and attention to engineer a proper prototype for
metasyntax specifications (§3.3.1) and their transformations (§3.3.2), as well
as to perform a series of experiments on metasyntax-driven grammar recovery
(§3.3.3) and convergence (§3.3.4). This area has now been exhaustively
covered, and the only possible future extensions must rely on going way beyond
textually specified context-free grammars.
To be completely frank, it should be noted here that most of the experiments
with metasyntax were done in the course of 2011 and were only polished,
presented and published in 2012 (which still required considerable effort).
Tolerant parsing overview
(§3.4)
Just like the grammar recovery paper came with an extensive related work
section which listed all grammar recovery initiatives in the last decade or
two [Zay12w, §2], a new parsing algorithm that I tried to propose (§3.8.2)
came with an extensive overview of all methods of tolerant parsing known to
grammarware engineers up to date (§3.4). While the iterative parsing method
was novel but ultimately dull and uninteresting, the overview itself was
received very warmly during the presentation on it [Zay12af]. One of the
reviewers of [Zay12m] has also advised to throw away the thing I thought was
the main contribution of the paper, and extend the thing I thought of as a
byproduct, into a longer journal article. While surprising at first, this
seems indeed like a reasonable course of action.
### 2.3 Selected minor contributions
Figure 1: The results of 2012, according to DBLP.
In the following sections, I will present a detailed overview of major
(§§3.1–3.7) and minor (§3.8) contributions, but the border between them is
naturally flexible. Thus, in the previous section introduced only four of the
best major ones, and this section will introduce several middleweight
contributions (“less major” mixed with “not so minor” ones).
Grammar mutation
(§3.8.1)
It has been noted in [Zay12o, Zay12q] that there is a separate group of
grammar changes that reside between traditional grammar transformations
(“rename X to Y”) and the grammar transformation operators (“rename”), which
was labelled as a grammar _mutation_ and formalised differently from them.
While the only truly important property of grammar mutation in the context of
[Zay12o, Zay12q] was that they are considerably harder to bidirectionalise, a
lot of useful grammar manipulations like “rename all uppercase nonterminals to
lowercase” or “eliminate all nonterminals unreachable from the root” belong to
the class of mutations, so it deserves to be studied closer. In [Zay12ag], I
have composed a list of 16 mutations identified in already published academic
papers or in publicly available grammarware source code, but the paper was not
accepted, so the topic remains only marginally explored.
Iterative parsing
(§3.8.2)
Starting from a fresh yet weird topic of what “the cloud” can mean for
grammarware engineering, I ended up proposing an algorithm for _parsing in the
cloud_ , which was not based on parallel parsing [Alb+94], but rather on
island grammars [Moo01, KL03]. The whole topic is questionable and only
suitable for a “wild ideas workshop”, as was nicely put by one of the
reviewers, but is still potentially of some interest. The paper containing the
algorithm was rejected twice [Zay12m, Zay12n] so far, and requires investing
more time in empirical validation at least, in order to increase the chances
of acceptance.
Unparsing in a broad sense
(§3.8.3)
I could not help noticing that parsing (i.e., mapping strings to graphs)
receives much more research attention than the reverse process of unparsing
(i.e., mapping graphs to strings). However, the only thing I did accomplish
this year was to collect a couple of references on existing research and make
a “new ideas” extended abstract [Zay12ah], which was classified as a “request
for discussion” and rejected. I am already prepared to give a discussion-
provoking presentation on this topic, but it requires much more effort to be
invested until more tangible results are obtained.
Megamodelling
(§3.5)
Megamodelling is higher abstraction level form of modelling that is concerned
with software languages and technologies and relations between them. This year
I have published some papers with megamodels in them [Zay12t, Zay12u, Zay12aa,
Zay12ab, Zay12j] and touched upon the topic in a range of presentations
[Zay12r, Zay12ac, Zay12z, Zay12v]. Much more work on this topic is planned for
2013.
Open notebook computer science
(§3.7)
Open notebook science is an open science paradigm of doing research in a
transparent way. It is already a fairly widely accepted methodology in areas
like chemistry [San08] and drug discovery [Sin08] and is generally perceived
as the next big step after open access [Llo08]. However, in computer science
and software engineering it has never been a tradition to keep a lab notebook,
and it takes quite some time to maintain it, with few apparently visible
benefits. I have been experimenting quite a lot with this idea, but finally
decided to come out to a bigger public with two presentations in 2012 [Zay12y,
Zay12ae]. In general, I believe this is a reasonable idea, and I will keep
practicing open notebook science myself, but it will take quite some effort to
put it carefully into words in order to publish, so I am not even sure it is
feasible to expect a publication in 2013.
### 2.4 Motivation for this report
Figure 2: The results of 2012, according to the self-archiver.
The progress of a scientist is traditionally measured by an outsider by the
papers that the scientist produces. According to DBLP, the main supplier of
bibliography lists currently, the year 2012 for me yielded the following
results (see the screencapture on Figure 1): one journal paper [Zay12q], one
conference proceedings paper [Zay12c], one preprint [Zay12j]. However, the
first one is an only slightly extended version of a workshop paper [Zay12o]
written mostly in 2011; the second one was written and accepted in 2011; and
the third one was intended to be a supplementary material for another paper
that is not yet accepted anywhere. Additionally, there are three more post-
proceedings papers in print [Zay12w, Zay12ab, Zay12u], which are already
finished and submitted and will eventually appear in the ACM Digital Library —
when they do, they will also be listed at DBLP under 2012, but at that time it
will be too late to write a year report.
What about the self-archiving initiative [Har01]? Luckily, I disclose
relatively large amounts of dark data [Goe07] about my research activities,
having an extensively linked daily updated website with an open notebook (see
§3.7) and many generated lists, including the current publishing progress, as
seen on the the screencapture on Figure 2. Even judging by the bare numbers,
one can already tell that this list contains much more information than the
DBLP list. However, it also has its problems: the “published” column contains
the works of previous years that happened to be delayed enough for the post-
proceedings to appear in January 2012 [FLZ12]; as well as mentions of drafts
planned for future publication (easily localised in the last column). It also
contains editorial work for non-mainstream venues [JZ12a, JZ12] which is of
much lesser relevance because there is no scientific value to it. What it does
not contain, is relations between all these papers: obviously some papers are
enhanced version of previously rejected drafts, but in order to figure them
out, one needs to read the open notebook at http://grammarware.net/opens or
analyse it automatically (no readily available tools are provided).
Personally, I can state that guided grammar convergence (see §3.1) is my top
result of the year. However, it has not (yet) been properly published. After
being rejected at ECMFA [Zay12g] and ICSM [Zay12h], it received very positive
reactions from POPL [Zay12i], yet was also deemed not mature enough for
publication. Still, having to figure out what are the limits of the proposed
methodology and how to describe it well, does not change the fact that this is
my best contribution of the year 2012.
Grammar transformation operator suites like XBGF (see §3.2.1), $\Upxi$BGF
(§3.2.2), EXBGF (§3.2.4), $\Delta$BGF (§3.2.5) and NBGF (§3.2.3) represent
massive amounts of work, but they are not publishable by themselves, if at
all. Still, each of them represents a milestone enabling further advances.
Engineering work that supports scientific research, has rarely been explicitly
noted and appreciated.
Quoting [San08]: _“The notebook is about publishing data as quickly as
possible. The paper is about synthesizing knowledge from all those results.”_
Hence, this report is aimed at synthesizing knowledge about the experiments
and achievements undertaken during the course of 2012 by me (possibly in
collaboration with someone else) within the NWO project 612.001.007,
“Foundations for a Grammar Laboratory”. It holds the most value for myself and
my project colleagues, but is also available for anyone interested in the
topics discussed: unlike open notebook entries, this report is a proper atomic
scientific knowledge object [Giu+10, Sim+11]. Only two topics directly
relevant to the project, are not included: one must remain hidden according to
the rules of the target venue, and for the other one the context and
consequences are not yet understood enough even for such a lightweight
presentation.
## 3 Topics overview
### 3.1 Guided grammar convergence
Let us consider two grammars in a broad sense [KLV05]. We say that they
represent one _intended_ software language, if there exists a complete
bidirectional mapping between language instances that commits to grammatical
structure of different grammars. For example, if a parser produces parse trees
that can always be converted to abstract syntax trees expected by a static
analysis tool and back, it means that they represent the same intended
language. As another example, consider an object model used in a tool that
stores its objects in an external database (XML or relational): the existence
of a bidirectional mapping between entries (trees or tables) in the database
and the objects in memory, means that they represent the same intended
language, even though they use very different ways to describe it. An
equivalence class spawned by this definition (i.e., a set of different
grammars of the same intended language) effectively forms a _grammarware
product line_ of products that perform different tasks on instances of the
same intended language: in that sense, for example, all Java-based tools form
a product line, if they agree on a language version and do not employ any
highly permissive methods that would shift them into a broader class. For the
sake of simplicity, let us focus on _grammar product lines_ : collections of
grammars of the same intended language. The relation between a grammar product
line and a grammarware product line is justified by research on automated
derivation of grammar-based tools like parsers, environments, documentation,
formatters and renovators from grammars [Kli93, SV99, Jon02, KLV05, Cam+10,
ZL11].
Suppose that we have two grammars: one that we call a _master grammar_ (a
specially pre-constructed abstract grammar of the intended language) and one
that we call a _servant grammar_ (a grammar derived from a particular language
implementation). In general, there are four phases of guided grammar
convergence, and they are presented in this section in the _reverse_ order.
First, we consider the simplest scenario when all mismatches are of
_structural_ nature. Then, we move on to a more complicated situation when a
_nominal_ matching between sets of nonterminals is unknown. Since this is
rather uncommon (most methods used in practice for imploding parse trees to
abstract syntax trees, from Popart [Wil97] to Rascal [KSV11], heavily rely on
equality of names), a new method for matching nonterminals has been developed.
In short, it comprises construction of production signatures for each
production rule in both grammars, and a search for equivalent and weakly
equivalent production rules with respect to those signatures. Once a name
resolution relation has been successfully built, a previously discussed
structural matching can be applied. We will also discuss _normalisations_ that
can transform any arbitrary grammar to a form easily consumable by our nominal
and structural matching algorithms. Finally, I will list additional problems
that indicate _grammar design_ decisions and therefore not affected by
normalisations. However, I describe how to automatically detect such issues
and to address them with grammar mutations.
#### Structural matching
Let us assume the simplest scenarios: the two input grammars have the same set
of nonterminals; neither of them has terminals; the starting nonterminal is
the same and that the sets of production rules are different but have the same
cardinality. These would be typical circumstances if, for example, the
grammars define two alternative abstract syntaxes for the same intended
language.
We can start from the roots of both grammars and traverse them synchronously
top-down, encountering only the following four circumstances:
Perfect match.
Convergence is trivially achieved.
Nonterminal vs. value.
By “values” I mean nonterminals that are built-in in the underlying framework
(e.g., “string”).
Sequence element permutations
can be automatically detected and converged.
Lists of symbols.
Many frameworks that have components with grammatical knowledge, have a notion
of a list or a repetition of symbols in their metalanguage.
It can be shown that these four are the only possibilities, and that their
resolution can be resolved.
#### Nominal resolution
In a more complicated scenario, let us consider the case of different
nonterminal sets in two input grammars, and for simplicity we assume that all
production rules are vertical (non-flat) and chained (if there is more than
one production rule for the same nonterminal, all of them are chain
productions — i.e., have one nonterminal as their right hand side). Next, we
define a _footprint_ of a nonterminal in an expression as follows:
$\pi_{n}(x)=\begin{cases}\\{1\\}&\text{if }x=n\\\ \\{?\\}&\text{if }x=n?\\\
\\{+\\}&\text{if }x=n^{+}\\\ \\{*\\}&\text{if }x=n^{*}\\\ \bigcup\limits_{e\in
L}\pi_{n}(e)&\text{if }x\text{ is a sequence }L\\\
\varnothing&\text{otherwise}\end{cases}$
By extension, we define a footprint of a nonterminal in a production rule as a
footprint of it in its right hand side:
$\pi_{n}(m\to e)=\pi_{n}(e)$
Based on that, we define a _production signature_ , or a prodsig, of a
production rule, by collecting all footprints of all nonterminals encountered
in its right hand side:
$\sigma(p)=\\{\langle
n,\pi_{n}(e)\rangle\>|\>n\in\mathbb{N},\>\pi_{n}(e)\not=\varnothing\\}$
Production rule Production signature $p_{1}$=(_program_ $\to$ _function_ +)
$\\{\langle\textit{function},\raisebox{0.80005pt}{$\scriptscriptstyle\mathord{+}$}\rangle\\}$
$p_{2}$=(_function_ $\to$ $str$ $str^{+}$ _expr_)
$\\{\langle\textit{expr},{1}\rangle,\langle
str,{1}\raisebox{0.80005pt}{$\scriptscriptstyle\mathord{+}$}\rangle\\}$
$p_{3}$=(_expr_ $\to$ $str$) $\\{\langle str,{1}\rangle\\}$ $p_{4}$=(_expr_
$\to$ $int$) $\\{\langle int,{1}\rangle\\}$ $p_{5}$=(_expr_ $\to$ _apply_)
$\\{\langle\mathit{apply},{1}\rangle\\}$ $p_{6}$=(_expr_ $\to$ _binary_)
$\\{\langle\mathit{binary},{1}\rangle\\}$ $p_{7}$=(_expr_ $\to$ _cond_)
$\\{\langle\mathit{cond},{1}\rangle\\}$ $p_{8}$=(_apply_ $\to$ $str$ _expr_ +)
$\\{\langle\textit{expr},\raisebox{0.80005pt}{$\scriptscriptstyle\mathord{+}$}\rangle,\langle
str,{1}\rangle\\}$ $p_{9}$=(_binary_ $\to$ _expr_ _operator_ _expr_)
$\\{\langle\textit{expr},{1}{1}\rangle,\langle\textit{operator},{1}\rangle\\}$
$p_{10}$=(_cond_ $\to$ _expr_ _expr_ _expr_)
$\\{\langle\textit{expr},{1}{1}{1}\rangle\\}$
Table 1: Production rules of the master grammar for FL, with their production
signatures.
A good example of how production signatures look like, is to be found on Table
1.
We say that two production rules are _prodsig-equivalent_ , if and only if
there is a unique match between tuple ranges of their signatures:
$p\bumpeq q\>\Longleftrightarrow\>\forall\langle
n,\pi\rangle\in\sigma(p),\>\exists!\langle m,\xi\rangle\in\sigma(q),\>\pi=\xi$
Similarly, a weak prodsig-equivalence $p\Bumpeq q$ is defined by dropping the
uniqueness constraint and weakening the equality constraint in the last
definition to footprint equivalence which disregards repetition kinds
($\scriptscriptstyle\mathord{+}$ is equivalent to $\scriptstyle\mathord{*}$).
Then it can be proven that for any two strongly prodsig-equivalent production
rules $p$ and $q$, $p\bumpeq q$, a _nominal resolution_ relationship has the
form of:
$p\diamond q=\sigma(p)\circ\overline{\sigma(q)}$
where $\rho_{1}\circ\rho_{2}$ is a composition of two relations in the classic
sense and $\overline{\rho}$ is the classic inverse of a relation. Moreover,
for any two weakly prodsig-equivalent production rules $p$ and $q$, $p\Bumpeq
q$, there is (at least one) nominal resolution relationship $p\diamond q$ that
satisfies the following:
$\displaystyle\forall\langle a,b\rangle\in p\diamond q:a=\omega\vee
b=\omega\>\vee$ $\displaystyle\exists\pi,\exists\xi,\pi\approx\xi,\langle
a,\pi\rangle\in\sigma(p),\langle b,\xi\rangle\in\sigma(q)$
and
$\forall\langle a,b\rangle\in p\diamond q,\forall\langle c,d\rangle\in
p\diamond q:a=c\Rightarrow b=d$
Where $\omega$ is used to explicitly denote unmatched nonterminals.
#### Abstract Normal Form
In order to fit any grammar into the conditions required by the previously
described matching techniques, we demand the following normalisation:
1. 1.
lack of labels for production rules
2. 2.
lack of named subexpressions
3. 3.
lack of terminal symbols
4. 4.
maximal outward factoring of inner choices
5. 5.
lack of horizontal production rules
6. 6.
lack of separator lists
7. 7.
lack of trivially defined nonterminals (with $\alpha$, $\varepsilon$ or
$\varphi$)
8. 8.
no mixing of chain and non-chain production rules
9. 9.
the nonterminal call graph is connected, and its top nonterminals are the
starting symbols of the grammar
It can be shown that transforming any grammar into its Abstract Normal Form is
in fact a grammar mutation (see §3.8.1). In the prototype, I have implemented
it to effectively generate bidirectional grammar transformation steps, so the
normalisation preserves any information that it needs to abstract from.
#### Grammar design mutation
Some grammar design smells (terminology per [Sto12]) like yaccification (per
[SV99, BSV98]) or layered expressions (per [LZ09]) have shown to be persistent
enough to survive all normalisations and cause problems for establishing
nominal and structural mappings. They can be identified and dealt with by
automated analyses and mutations, but so far I have to proof that they are the
only possible obstacles, and no guarantees about any other smells problematic
for guided grammar convergence.
#### 3.1.1 Generalisation of production signatures
The method of establishing nonterminal mappings of different grammars of the
same intended language, can be generalised as follows. Suppose that we have a
metalanguage. Without loss of generality, let us assume that each grammar
definition construct that is present in it, can be referred to by a single
symbol: “,”, “?”, “*”, etc and uses prefix notation. This metasyntactic
alphabet $\Lambda$ will form the foundation of our footprints and signatures.
Let us also assume that all metasymbols are unary or are encoded as unary,
except for two composition constructs: a sequential “,” and an alternative
“$\mathbf{|}$”, which take a list of symbols.
Then, a footprint of any nonterminal $n$ in an expression $x$ is a multiset of
metasymbols that are used for occurrences of $n$ within $x$:
$\pi_{n}(x)=\begin{cases}\\{1\\}&\text{if }x=n\\\ \\{\mu\\}&\text{if
}x=\mu(n),\mu\in\Lambda\\\ \bigcup\limits_{e\in L}\pi_{n}(e)&\text{if
}x=\raisebox{2.0pt}{{,}}(L)\\\ \varnothing&\text{otherwise, also if
}x=\mathbf{|}(L)\\\ \end{cases}$
Our previously given definition of a production signature can still be used
with this generally redefined footprints.
It is well known that language equivalence is undecidable. Any formulation of
the grammar equivalence problem, that is based on language equivalence, is
thus also undecidable. Grammar convergence [LZ09, LZ11] is a practically
reformulated grammar equivalence problem that uses automated grammar
transformation steps programmed by a human expert. By using these generalised
metasyntactic signatures, we can _infer converging transformation steps_
automatically, thus eliminating the weakest link of the present methodology.
However, this is not the only application of the generalisation.
The most trivial use of metasyntactic footprints and signatures would lie in
_grammarware metrics_. Research on software metrics applied to context-free
grammars has never been an extremely popular topic, but it did receive some
attention in the 1970s [Gru71], 1980s [Kel81] and even recently [PM04,
Čre+10]. Using quantitative aspects of metasyntactic footprints and signatures
(numbers of different footprints within the grammar, statistics on them, etc)
is possible and conceptually akin to using micropatterns [GM05] and
nanopatterns [Bat10], but nothing of this kind has ever been done for grammars
(in a broad sense or otherwise).
A different more advanced application of metasyntactic footprints and
signatures is the analysis of their usage by mining existing grammar
repositories like Grammar Zoo [Zay+08]. This can lead to not only improving
the quality of the grammars by increasing their utilisation of the
metalanguage functionality, but also to _validation of metalanguage design_.
The whole programming language community uses dialects and variations of BNF
[Bac60] and EBNF [Wir77], but their design has never been formally verified.
However, one may expect that introducing EBNF elements like symbol repetition
to BNF can be justified by analysing plain BNF grammars and finding many
occurrences of encoding them (“yaccification”, etc). It will also be
interesting to see what new features the EBNF lacks practically — none of the
existing proposals so far (ABNF [Ove05], TBNF [Man06], etc) were ever formally
validated.
#### 3.1.2 History of attempted publication
Initially, the idea of guided grammar convergence has emerged as a
contribution for ECMFA [Zay12g]. The level of contribution was praised by the
reviewers, but the paper itself was deemed inappropriate for a heavily model-
related venue. A bit later it was resubmitted after minor revision to ICSM
[Zay12h], where it was received even colder, presumably because the reviewers
were seeking a more practical side which was not demonstrated well enough.
After much more effort put into experiments, prototypes, auxiliary material
[Zay12j] and a complete rewrite of the paper itself, the method was submitted
to POPL [Zay12i]. It was unanimously rejected, but with very constructive and
encouraging reviews. In 2013, they will be taken into account when the paper
will be submitted again (the last time as a conference paper — otherwise I
will admit it to be impossible for me to explain this method within the common
limitations and go for a much longer self-contained journal submission).
In [Zay12l], I have attempted to sell the very act of validating the new
method of guided grammar convergence by letting it cover the older case study
done with contemporary grammar convergence, as a some sort of experimental
replication in a broad sense. The reviewers praised the nonconformism and
originality of the approach, and rejected the paper.
The generalisation of the method was proposed as an extended abstract to NWPT
[Zay12s], where the reviewers did not see any merit in it (which I personally
found strange since both ICSM and POPL reviewers insisted that various
components of the method like ANF and prodsigs must be treasured as standalone
contributions which applicability is much wider than the automated convergence
of grammars). Either my way of explaining was bad enough to obfuscate this
point, or I have terribly misunderstood their call for papers.
### 3.2 Grammar transformation languages
#### 3.2.1 XBGF
XBGF, standing for Transformation of BNF-like Grammar Format, is a domain-
specific language for automated programmable operator-based transformations of
grammars in a broad sense. It has been previously implemented in Prolog (which
was mostly done by Ralf Lämmel) and published as a part of a journal article
[LZ11, §4], as well as a separate online manual [Zay+08, XBGF Manual] — in
fact, just a byproduct of the research on language documentation [ZL11].
XBGF is essentially finished work: it is working, it is useful for
experiments, it has documentation, it has a test suite, etc. The only thing
that was added in the course of 2012 is the reimplementation of XBGF in Rascal
[KSV11]. Beside some metaprogramming, this reimplementation led to
streamlining some of the applicability preconditions and postcondition, which
could be viewed as a very minor scientific contribution.
#### 3.2.2 $\Upxi$BGF
If XBGF was read as “iks bee gee eff”, then $\Upxi$BGF is “ksee bee gee eff”,
its bidirectional counterpart. Inspired by the call for papers of BX’12 (The
First Workshop on Bidirectional Transformations, see §4.1), I was
experimenting with bidirectionality in the grammarware technological space,
and this language is what came out of it. 80% of the work for creating it
involved trivial coupling of grammar transformation operators like _chain_ and
_unchain_ , but the remaining 20% have provided a lot of fuel for thinking
about what seemed to be a polished and finished product. $\Upxi$BGF was
published as a part of online pre-proceedings [Zay12o], and then, after the
second round of reviews, as a journal article [Zay12q]. The only problem was
that the BX paper took off on its own, so the bidirectional grammar
transformation operator suite seems like one of many byproducts there. There
was a failed attempt to craft a paper that would be more focused on $\Upxi$BGF
(and other aspects of grammar transformation not covered sufficiently by the
BX submission), but a wrong venue was targeted, which resulted in desk
rejection [Zay12ag].
#### 3.2.3 NXBGF?
Another property of programmable grammar transformations that always bothered
me, was their rigidity: once written, they are hard to maintain and adapt, and
one little change in the original grammar (for example, when the extractor is
changed) can unexpectedly and unpredictably break (make defunct) some of the
transformation steps much later in the chain, and there is no method available
to detect the change impact. Analysing this problem led to an idea that was
originally in preparation for the FM+AM workshop (see §4.2), but was not ready
before the deadline, so it went to the Extreme Modelling Workshop instead,
where it received surprisingly warm reaction.
The idea is: negotiations. Whenever an error arises (usually an applicability
condition is not met), instead of failing the whole chain, try to recover by
negotiating the outcome with the data about near-failure and some external
entity (usually an oracle or a human operator). For example, when we want to
rename a nonterminal that does not exist, the transformation engine may seek
nonterminals with names similar to the required one, and try renaming them.
The idea of negotiated grammar transformations was published in the online
proceedings [Zay12t] and then in the ACM Digital Library [Zay12u], after which
I was invited to submit an extended version to a journal. This will soon lead
to a prototype implementation of such a system and perhaps to some interesting
experiments with it. If this advancement yields a yet another grammar
transformation operator suite, it may or may not be named “NXBGF”.
#### 3.2.4 EXBGF
| jls1 | jls2 | jls3 | jls12 | jls123 | r12 | r123 | Total
---|---|---|---|---|---|---|---|---
XBGF, LOC | 682 | 6774 | 10721 | 5114 | 2847 | 1639 | 3082 | 30859
EXBGF, LOC | 399 | 5509 | 7524 | 3835 | 2532 | 1195 | 2750 | 23744
| $-$42% | $-$19% | $-$30% | $-$25% | $-$11% | $-$27% | $-$11% | $-$23%
genXBGF, LOC | 516 | 5851 | 9317 | 4548 | 2596 | 1331 | 2667 | 26826
| $-$24% | $-$14% | $-$13% | $-$11% | $-$9% | $-$19% | $-$13% | $-$13%
XBGF, nodes | 309 | 3,433 | 5,478 | 2,699 | 1,540 | 786 | 1,606 | 15851
EXBGF, nodes | 177 | 2,726 | 3,648 | 1,962 | 1,377 | 558 | 1,446 | 11894
| $-$43% | $-$21% | $-$33% | $-$27% | $-$11% | $-$29% | $-$10% | $-$25%
genXBGF, nodes | 326 | 3,502 | 5,576 | 2,726 | 1,542 | 798 | 1,610 | 16080
| +6% | +2% | +2% | +1% | +0.1% | +2% | +0.3% | +1%
XBGF, steps | 67 | 387 | 544 | 290 | 111 | 77 | 135 | 1611
EXBGF, steps | 42 | 275 | 398 | 214 | 98 | 50 | 120 | 1197
…pure EXBGF | 27 | 104 | 162 | 80 | 30 | 34 | 44 |
…just XBGF | 15 | 171 | 236 | 134 | 68 | 16 | 76 |
| $-$37% | $-$29% | $-$27% | $-$26% | $-$12% | $-$35% | $-$11% | $-$26%
genXBGF, steps | 73 | 390 | 555 | 296 | 112 | 83 | 139 | 1648
| +9% | +1% | +2% | +2% | +1% | +8% | +2% | +2%
Table 2: Size measurements of the Java grammar convergence case study, done in
XBGF and in EXBGF. In the table, XBGF refers to the original transformation
scripts, EXBGF to the transformations in Extended XBGF, genXBGF measures XBGF
scripts generated from EXBGF. LOC means lines of code, calculated with wc -l;
nodes represent the number of nodes in the XML tree, calculated by XPath;
steps are nodes that correspond to transformation operators and not to their
arguments. Percentages are calculated against the XBGF scripts of the original
study.
Considerations about the state of XBGF led me to start cursory reexamination
of the available transformation scripts. The Java case study undertaken in
2009–2010 and published as a conference paper [LZ09a], a journal paper [LZ11]
and open source repository [Zay+08], provided me with plenty of them. Manual
ad hoc pattern recognition has resulted in development of a new operator
suite, with higher order operators such as _exbgf:pull-out_ , which would be
equivalent to a superposition of _xbgf:horizontal_ , _xbgf:factor_ ,
_xbgf:extract_ and _xbgf:vertical_. As shown on Table 2, size metrics show a
drop of 23–26% in Extended XBGF with respect to XBGF, but also the complexity
was obviously decreased. However, the results were not extremely convincing
and lacked real strength since only a few uses per high level operator were
found, and the new EXBGF language was not designed systematically. Besides all
that, the case study I have done, is, strictly speaking, about _refactoring_
XBGF scripts to Extended XBGF, so claims about usefulness of EXBGF for
_creating_ new transformation scripts, should be stated with caution.
EXBGF was first described as an idea as a part of [Zay12ag]. After its
rejection, it was developed further and laid out in much more detail in a
journal submission, which was also eventually rejected [Zay12l]. The fact that
I presented Extended XBGF first as a “trend” and then as an “experiment”,
perfectly reflects my point of view that it is not a solid contribution on its
own.
#### 3.2.5 $\Delta$BGF?
If there was one good outcome of getting a grammar transformation paper
[Zay12ag] rejected at a functional programming conference, then this is it: I
started contemplating how to specify them in a non-so-functional way. Having
recently been to a bidirectional transformations workshop helped, and I
started researching _tridirectional_ transformations (in fact, they quickly
turned multidirectional). The idea was clean and simple: do not specify
grammar changes as functions; instead, specify them as predicates. Such a
predicate would, for example, introduce a nominal binding between nonterminals
in different grammars — after which, the actual renaming steps can be easily
inferred from such a binding predicate.
Unfortunately, this idea was so beautiful in theory, but proven nearly
impossible in practice (or in detailed theory, for that matter). The main
problem lies with the order of execution: a functional grammar transformation
script specifies that order naturally, while a list of predicates does not. As
I found out the hard way, my prototypes were still clean and beautiful when
they dealt with one transformation step; reasonable tricks and extensions
could let me go up to three steps; beyond that some serious redesign was
needed; and so far I have not figured out how to overcome this.
### 3.3 Metasyntax
Whenever we have a software language, we can speak of its _syntax_ as a way it
allows and disallows structural combinations of elements: programming
languages rely on keywords and possibly layout conventions; spreadsheets have
ways of distinguishing between cells and referring to one from another; markup
languages have symbol sequences of special meaning; musical notes are arranged
on a grid; graphs must have uniquely identifiable nodes and edges connecting
exactly two each; etc. Then, a _metasyntax_ is a way of specifying this
syntax. In the classic programming language theory, languages are textual and
can be processed as sequences of lexems, and the metasyntax is Backus Normal
Form [Bac60], also called Backus Naur Form [Knu64], or its enhanced variant
Extended Backus Naur Form [Wir77]. Despite the fact that EBNF has been
standardised by ISO [ISO96], there is no agreement in the software language
engineering community on the exact variant of EBNF: some people just prefer
using “$:\equiv$” or “$\triangleq$” instead of “=” for esthetic reasons or
prefer separating production rules with double newlines for readability
reasons and for the sake of easy processing.
The idea was hinted in my PhD thesis in 2010 [Zay10], completely worked out in
2011 and was put to several good uses in 2012. These are listed in the
following subsections.
#### 3.3.1 Notation specification
The first step in treating metalanguages as first class entities is, of
course, encapsulating a particular metalanguage with a specification that
defines it. By extending the list of possible metasymbols from the ISO EBNF
standard [ISO96] and by reusing the empirically constructed Table 6.1 from my
thesis [Zay10, p.135], I was able to construct such a specification, which was
subsequently named EDD, for EBNF Dialect Definition. It was then turned into a
small nicely packaged paper for the PL track of SAC [Zay12c] — the very fact
that it was published separately, gave me a lot of freedom later, when I did
not feel like I need to introduce all the metasymbols all over again in each
work that followed.
#### 3.3.2 Transforming metasyntaxes
Figure 3: Components of a notation evolution: $\sigma$, a bidirectional
_notation specification transformation_ that changes the notation itself;
$\delta$, a _convergence relationship_ that can transform the notation
grammars; $\gamma$, a bidirectional _grammar adaptation_ that prepares a
beautified readable version of $N^{\prime}$. $\mu$, an unidirectional _coupled
grammar mutation_ that migrates the grammarbase according to notation changes;
possibly $\mu^{\prime}$, an unidirectional _coupled grammar mutation_ that
migrates the grammarbase according to the inverse of the intended notation
changes.
Once you have a notation specification as a first class entity, you can define
transformations on them. This was probably the first transformation language
that I have designed, where the main complexity was not in defining the
transformation operators as such, but rather in coupling them with the grammar
transformation steps that they imply. The transformation suite consisted of
just three operators:
rename-metasymbol$(s,v_{1},v_{2})$
where $s$ is the metasymbol and values $v_{1}$ and $v_{2}$ are strings
For example, we can decide to update the notation specification from using “:”
as a defining metasymbol to using “::=”. This is the most trivial
transformation, but also bidirectional by nature.
introduce-metasymbol$(s,v)$
where $s$ is the metasymbol and $v$ is its desired string value
For example, a syntactic notation can exist without terminator metasymbol, and
we may want to introduce one.
eliminate-metasymbol$(s,v)$
where $s$ is the metasymbol and $v$ is its current string value
Naturally, eliminate and introduce together form a bidirectional pair.
Specifying the current value of a metasymbol is not necessary, but enables
extra validation, as well as trivial bidirectionalisation.
Yet, the final megamodel of the infrastructure that did not even consider
language instances (only grammars and metasyntaxes) looked as complex as
Figure 3. The paper about evolution of metalanguages had a bidirectionality
flavour and was conditionally accepted at the BX workshop [Zay12o], and then
also for the journal special issue [Zay12q].
#### 3.3.3 Notation-parametric grammar recovery
In all previously published grammar recovery initiatives [BSV97, LV99, SV00,
LV01, LV01a, Läm05, Zay05, LZ09, Zay10, LZ11, Zay11a, Zay12ad] the step of
transforming the raw grammar-containing text obtained from the language manual
was either not automated (the grammar was re-typed from scratch in the
notation required by the target grammarware framework), or semi-automated
(comprised many rounds of test-driven improvement), or automated with a
throwaway tool (one that can not be reused unless the replication deploys
exactly the same EBNF dialect). Having a notation specification as a first
class entity, we can step up from throwaway tools to throwaway notation
specifications: at least they take minutes to create, not days.
Notation-parametric grammar recovery [Zay12x, Zay12w] was my best result of
2011, and this year it was officially published and put to several good uses.
These uses are not exactly publishable simply because grammar recovery from
(nearly) well-formed has become a trivial process itself, but there was one
story that was enabled by this triviality. The grammar of MediaWiki syntax,
for recovery of which I have a previously exposed preprint [Zay11a], is a
unique case of using multiple notations within one community-created grammar.
With any other recovery method, it would have been easier to just retype the
grammar again in a uniform fashion, but notation-parametric grammar recovery
allowed to treat all six different incoherent metalanguages with relative ease
and derive the final grammar from the inconsistent input. A continuation of
this topic was intended to be a published closure on the case of MediaWiki
grammar recovery, but was unfortunately rejected in the end [Zay12ad].
#### 3.3.4 Notation-driven grammar convergence
Grammar convergence was originally a lightweight verification method not
intended for full automation [LZ09]. However, seeing how many transformations
that were in fact converging grammars, it was possible to infer automatically
for the metalanguage evolution case study [Zay12s] (see also §3.3.2), I could
not help starting to wonder whether and to what extent it was possible to
drive the automated convergence process by the notation properties. The result
of that was the methodology of guided grammar convergence, which was already
covered by §3.1.
### 3.4 Tolerance in parsing
Originally, the “parsing in the cloud” paper [Zay12m, Zay12n] was intended to
present a useful crossing of the in-the-cloud and as-a-service paradigm with
the engineering discipline for grammarware. However, the related work digging
quickly got out of hand and turned into a contribution of its own. The
overview of many grammar-based techniques with some level of tolerance towards
their input data and its weak commitment to grammatical structure, was
presented at the PEM Colloquium [Zay12af] (see also §3.8.8), where it was
received very warm acceptance and led to many useful insights. It has been
advised to me both by reviewers and colleagues to put more effort into
demonstrative prototype and publish the overview with them separately from the
parsing algorithm (see §3.8.2) itself. This is among one of the planned
activities for 2013.
So far, at least the following tolerant parsing methods have been identified:
ad hoc lexical analysis [BSV00, KLV05a], hierarchical lexical analysis [MN95],
iterative lexical analysis [Cox03], fuzzy parsing [Kop97], parsing incomplete
sentences [Lan88], island grammars [DK99], lake grammars [Moo01], robust
multilingual parsing [SCD03], gap parsing [BN05], bridge grammars [NNEH09],
skeleton grammars [KL03], breadth-first parsing [Lee67, Oph97], grammar
relaxation [ASU85], agile parsing [Dea+03], permissive grammars [Kat+09],
hierarchical error repair [BH82], panic mode [ASU85], noncorrecting error
recovery [Ric85], precise parsing [AU72]. It remains to be seen whether they
form a straight spectrum from lexical analysis to strict syntactic analysis.
### 3.5 Megamodelling
In computer science, _modelling_ happens when a real artefact is represented
by its abstraction, which is then called a model; _metamodelling_ happens when
the structure of such models is analysed and expressed as a model for models,
or a metamodel; and _megamodelling_ happens when the infrastructure itself,
involving multiple models and metamodels, is modelled. The need for megamodels
is being advocated at least since 2004 [BJV04, FN04].
The current state of the art is: in the simplest cases, people do not need a
special formalism to state that, for example, “models A and B conform to the
metamodel C”; in somewhat more complicated scenarios scientists and engineers
tend to develop their own domain-specific _ad hoc megamodelling_ methodologies
and employ them in narrow domains; and in truly complex situations, any
existing approach only adds to complexity, overwhelming stakeholders with a
yet another view on the system architecture. However, at least one solid
business case was found for megamodelling: the problem of comparing different
technological spaces [KBA02]: for example, comparing the relations between XML
documents, schemata, data models and validators, with relations between object
models, source code and compilers.
At the University of Koblenz-Landau, the Software Languages Team is dedicated
to develop a general purpose megamodelling language called MegaL [FLV12].
After attending presentations about MegaL on several occasions, I have paid a
working visit to them in July. The consequences of that visit: I tried to use
MegaL for my own megamodelling needs on several occasions [Zay12t, Zay12u,
Zay12j], I have presented an extensive overview of currently existing ad hoc
megamodelling techniques (see §3.5.1), and I have proposed my own method of
dealing with overly complex megamodels (see §3.5.2).
#### 3.5.1 MegaL dissection
So far at least these previously existing ad hoc megamodelling approaches have
been spotted: ATL [Jou+08], UNCOL [Bra61, Con58], tombstone [MHW70],
grammarware megamodelling [KLV05], software evolution megamodelling [FN04],
evolution of software architectures [Gra07, Gra07a], MEGAF [Hil+10], global
model management [Vig+11], grammar convergence [LZ09, Zay10, Zay11], software
language engineering [Zay10, Zay+08], modelling language evolution framework
[MV11], metasyntactic evolution [Zay12o, Zay12q].
My superficial overview of them, comparing them with MegaL, was presented to
the MegaL designers in July [Zay12r], and my current research activities
include active collaboration with them with a paper presenting a unified model
for megamodelling in mind.
#### 3.5.2 Renarrating megamodels
Having seen enough presentations on megamodelling made me realise that they
are very easy to follow even for untrained people, unlike the resulting
megamodels that contain far too much detail and are very intimidating. So, my
take on this problem was introducing two operations: slicing (to make
megamodels smaller) and narrating (to traverse the elements in the megamodel).
If we have them, we can take the baseline megamodel that only experts can try
to understand, and cut it to consumable chunks bundled with the story that
introduces the remaining elements one by one and explaining each step. The
resulting paper was sent to a workshop on Multiparadigm Modelling, where it
was presented as a poster [Zay12z], published in online pre-proceedings
[Zay12aa] and is currently on its way to the post-proceedings in the ACM DL
[Zay12ab].
### 3.6 Grammar repository
My first project proposal ever, titled “Automated Reuse-driven Grammar
Restructuring”, was sent to the NWO Veni program in January, passed a rebuttal
phase in May and was finally rejected in July after informing me that it ended
up in the category “very good” [Zay12]. The idea described there was small and
elegant: mining grammarware repositories. While repository mining techniques
receive quite some attention nowadays, very few people actually have entire
repositories filled with grammars: let’s face it, they are omnipresent yet at
the same time scarce. However, I already have this initiative called Grammar
Zoo [Zay+08], which contains many grammars of languages big and small, and
armed with the arsenal of extraction tools developed in my PhD time, it can
grow even more. The goal of such mining is, of course, to reverse engineer
reusable grammar fragments and forward engineer the discipline of their
composition.
A paper advocating the need and the usefulness of the repository itself, was
written and submitted to a journal in November [Zay13]. The outcome will only
become known in 2013.
### 3.7 (Open) Notebook Science
Open Notebook Science is an open science paradigm of doing research in a
transparent way [Llo08]. It involves keeping a lab notebook that collects all
data and metadata on experiments, hypotheses, results, details and other
observations that occur during the research phase, so that after the final
objective is reached (or deemed unreachable), the complete path towards it can
be exposed and made publicly available for inspection, replication and reuse.
The open notebook approach is fairly well-known and somewhat popular in fields
like biology and chemistry [San08, Sin08], that strive on experimental
frameworks and traditionally involve lab notebooks, so in practice exercising
this approach had the only consequence of sharing the already existing
notebook and systematically referring to it from the papers. In computer
science, however, there are none to few adopters of this approach, mainly due
to the seeming complexity of the method and the amount of extra effort that is
needed to set up and to maintain such a lab notebook and the lack of positive
feedback from it in the form of community encouragement and peer
acknowledgement.
During the Software Freedom Day, I have given a presentation, explaining one
possible feasible way to start practicing open notebook science for computer
science and software engineering researchers, with the case study of myself
[Zay12y]. A couple of days later SL(E)BOK organisers have heard about it and
asked me to record a keynote presentation [Zay12ae] about that, linking open
access ideas with the existing research on “scientific knowledge objects”
(SKO) and on a “body of knowledge” [Giu+10, Sim+11].
In short, open notebook science strives to enable open access to atomic SKOs;
to expose all the dark data [Goe07] from failed experiments and unpublished
results; to self-archive [Har01] subatomic SKOs, which are relevant for the
final result, but smaller than a “publon”. Examples of subatomic SKOs include:
Commits to an open source repository;
Tweets on work-related subjects;
Quora answers on work-related topics;
Papers: preprints, reports, drafts, etc;
Presentations: slides, screencasts, etc;
Blog posts;
Wiki edits;
Exposed tools;
Documentation;
Shared raw data;
Auxiliary material.
As it has been pointed out to me by some of the attendees of both talks, the
topic of subatomic SKOs is bigger than just open notebook computer science,
because if I can show the usefulness of keeping a notebook of actions for a
researcher, it does not necessary mean that the notebook must be public to
profit from its traceability. The first comprehensive paper on this topic is
still in the process of being designed, but hopefully will be submitted
somewhere during the next year or two.
### 3.8 Minor topics
Additionally to the topics and achievements I consider major for 2012, there
are several lesser contributions: their are either topics that did not receive
enough attention to yield a solid major contribution (yet not insignificant
enough to be omitted from the report completely); or just not traditionally
considered worthy of mentioning (programming, engineering, organising effort).
One topic is intentionally hidden from this section, in order to prevent
jeopardising an upcoming submission to a strictly double blind peer reviewed
venue.
#### 3.8.1 Grammar mutation
In the paradigm of programmable grammar transformations, the semantics of each
of the transformation operators is bound to the operator itself, and may
require arguments to be provided before the actual input grammar. Such
partially evaluated operators (with all arguments provided, but no input
grammar yet) are treated as transformation steps, and their applicability
constraints only depend on the grammar: if they hold, the change takes place;
if they do not, an error occurs instead. In other words, the exact consequence
of the transformation step depends on operands, not on the grammar. However,
those applicability constraints can also be processed as filters: whatever
part of the grammar satisfies them, will be transformed — that way, the exact
change in the grammar depends on the grammar, not on the operands.
As an example, consider renaming grammatical symbols: “rename nonterminal”
itself is an operator. Its semantics can be expressed easily on the classic
definition of a grammar. If the input grammar is
$G=\langle\mathbb{N},\mathbb{T},\mathbb{P},S\rangle$, then the output must be
$G^{\prime}=\langle\mathbb{N}\cap\\{x\\}\cup\\{y\\},\mathbb{T},\mathbb{P}|_{x\to
y},S^{\prime}\rangle$
where $x$ and $y$ are operands; $S^{\prime}$ is $S$ unless $S=x$ and $y$
otherwise; and $A|_{x\to y}$ means substitution (for example, by term
rewriting). When $x$ and $y$ are provided, then $G^{\prime}$ above becomes
fully defined and yields meaningful results when applicability conditions
(e.g., $x\in\mathbb{N}$ and $y\not\in\mathbb{N}$) are satisfied. Renaming a
terminal symbol is specified similarly.
However, “renaming all lowercase nonterminals to uppercase” is not an operator
(or at least even it is made one, it will be of much higher level than the
simple “rename”), and it is not an atomic transformation step either: in fact,
it can lead to any number of changes in the grammar from $0$ to
$|\mathbb{N}|$, depending on $G$. This number absolutely cannot be known
before $G$ is provided.
This kind of grammar manipulation was identified first as a part of research
on bidirectional transformations [Zay12o, Zay12q, Zay12p, Zay12b] (because
they are not bidirectionalisable), where it received the name of “grammar
mutation”. Later there was an endeavour to compose a comprehensive list of
useful grammar mutations as a part of [Zay12ag], but it was rejected.
For the sake of providing a better overview of the current state of research
on grammar mutations, I collect all of them in the exhaustive list below. Note
that conceptually the same mutations may have been appearing under different
names in various sources: for example, the first mutation in the list, “remove
all terminal symbols”, has previously been known as a transformation “
_stripTs_ ” [LZ09, §5.3] and as a generator “ _striptxbgf_ ” [Zay10,
§4.9,§4.10.6.1].
Remove all terminal symbols
[LZ09, Zay12g, Zay12h, Zay12i, Zay10, Zay11]
A simple grammar mutation that is helpful when converging a concrete syntax
and an abstract syntax of the same intended language. While the abstract
syntax definition may have differently ordered parameters of some of its
constructs, and full convergence will require dealing them them and
rearranging the structure with (algebraic) semantic-preserving
transformations, we will certainly not encounter any terminal symbols and can
safely employ this mutation.
Remove all expression selectors
[LZ09, Zay12g, Zay12h, Zay12i, Zay10]
Named (selectable) subexpressions are encountered in many contexts, but the
choice of names for them is usually even more subjective than the naming
convention for the nonterminal symbols.
Remove all production labels
[Zay12g, Zay12h, Zay12i]
Technically, having production label is the same as making a selectable
subexpression out of the right hand side of a nonterminal definition. Still,
in some frameworks the semantics and/or the intended use for labels and for
selectors differ.
Disciplined rename
[Zay12o, Zay12q, Zay10, Zay11]
There are several different well-defined naming conventions for nonterminal
symbols in current practice of grammarware engineering, in particular
concerning multiword names. Enforcing a particular naming convention such as
making all nonterminal names uppercase or turning camelcased names into dash-
separated lowercase names, can be specified as a unidirectional grammar
mutation (one for each convention).
Reroot to top
[Zay11, Zay12g, Zay12h, Zay12i]
A top nonterminal is a nonterminal that is defined in the grammar but never
used [LV01a]. In many cases it is realistic to assume that the top
nonterminals are intended starting symbols (roots) of the grammar. A variation
of this mutation was used in §3.1 with an additional requirement that a top
nonterminal must not be a leaf in the relation graph. This is a rational
constraint since a leaf top nonterminal defines a separated component.
Eliminate top
[Zay10, Zay11]
In the situations when the root is known with certainly, we can assume all
other top (unused) nonterminals to be useless, since they are unreachable from
the starting symbol and are therefore not a part of the grammar.
Extract subgrammar
[Zay12g, Zay12h, Zay12i]
Alternatively, we can generalise the last mutation to a parametrised one:
given a grammar and a nonterminal (or a list of nonterminals), we can always
automatically construct another grammar with the given nonterminal(s) as
root(s) and the contents formed by all production rules of all nonterminals
reachable from the assumed root nonterminal(s). Constructing a subgrammar
starting with the already known roots will eliminate top nonterminals.
Make all production rules vertical
[Zay10, Zay11, Zay12g, Zay12h, Zay12i]
Vertical definitions contain several alternative production rules, while
horizontal ones have one with a top level choice. There are different
approaches known to handle this distinction, including complete transparency
(one form being a syntactic sugar of the other). For normalisation purposes or
for quick convergence of a consistently vertical grammar and a consistently
horizontal one, we can use this automated mutation.
Make all production rules horizontal
[Zay10, Zay11]
A similar grammar mutation is possible, yet much less useful in practice.
Distribute all factored definitions
[Zay12g, Zay12h, Zay12i]
Aggressive factoring a-la xbgf:distribute can also be discussed.Surfacing all
inner choices in a given grammar is a powerful normalisation technique.
Make all potentially horizontal rules vertical
[Zay10, Zay11]
Technically, this mutation is a superposition of distribution of all factored
definition and converting all resulting horizontal production rules to an
equivalent vertical form.
Deyaccify all yaccified nonterminals
[Zay10, Zay11]
A “yaccified” definition [Läm01, JM01] is named after YACC [Joh75], a compiler
compiler, the old versions of which required explicitly defined recursive
nonterminals — i.e., one would write A : B and A : A B, because in LALR
parsers like YACC left recursion was preferred to right recursion (contrary to
recursive descent parsers, which are unable to process left recursion directly
at all). The common good practice is modern grammarware engineering is to use
iteration metalanguage constructs such as B* for zero or more repetitions and
B+ for one or more — this way, the compiler compiler can make its own
decisions about the particular way of implementation, and will neither crash
nor perform any transformations behind the scenes. However, many grammars
[Zay+08] contain yaccified definitions, and usually the first step in any
transformation that attempts to reuse such grammars for practical purposes,
start with deyaccification, which can be easily automated.
Remove lazy nonterminals
[Zay10, Zay11]
Many grammars, in particular those that strive for better readability or for
generality, contain excessive number of nonterminals that are used only once
or chain production rules that are unnecessary for parsing and for many other
activities one can engage in with grammars. We have used an optimising
mutation that removes such elements with _xbgf:inline_ and _xbgf:unchain_ on
several occasions, including improving readability of automatically generated
grammars.
Normalise to ANF
[Zay12g, Zay12h, Zay12i]
The Abstract Normal Form (ANF) was introduced in §3.1 as means of limiting the
search space for guided grammar convergence. Technically, such normalisation
is equivalent to a superposition of removing all labels, removing all
selectors, removing all terminals, surfacing all inner choices, converting all
horizontal production rules to a vertical form, rerooting to top non-leaf
nonterminals and eliminating others unreachable from them. For conceptual
foundations of ANF the reader is redirected to the article where it was
proposed.
Fold all grouped subexpressions
[Zay12o, Zay12q]
In the context of metalinguistic evolution, we need to construct a coupled
mutation for the grammarbase, if the notation change contains retiring of a
metasyntactic construct that is in use. One of such constructs is the
possibility to group symbols together in an atomic subsequence — a feature
that is often taken for granted and therefore misused, improperly documented
or implemented. Naturally, eliminating grouped subexpressions entails folding
them to newly introduced nonterminals by means of _xbgf:extract_.
Explicitly encode all separator lists
[Zay12o, Zay12q]
Our internal representation of grammars for software languages, following many
other syntactic notations, contains a construct for defining separator lists.
For example: {A ","}+ is a syntactic sugar for A ("," A)* or (A ",")* A — all
three variants specify a comma-separated list of one or more As. When such a
construct needs to be retired from the notation, the coupled grammar mutation
must refactor its occurrences to explicitly encode separator lists with one of
the equivalent alternatives.
A full fledged paper shining enough light on grammar mutations, is still being
written and will hit the submission desks in 2013.
#### 3.8.2 Iterative parsing
As the main (intended) contribution of [Zay12m, Zay12n], I have proposed the
algorithm for iterative parsing. The basic idea is very simple: we take the
baseline grammar and skeletonise it as far as it can be automated, in such a
way that the relation between the “lakes” and the nonterminals in the baseline
grammar are preserved. Then, our parse tree will give the basic structure and
a number of watery fragments parsed with useless lake grammars (usually in a
form of “anything but newline” or “something in balanced out curly brackets”).
If needed, any of those lakes can be parsed further with a subgrammar of the
baseline grammar, with the new root being the nonterminal that corresponds to
the lake.
This parsing approach was being sold as “parsing in the cloud” in [Zay12m,
Zay12n], which was certainly not the best (even though the coolest) way to
look at it. Other applications for this form of lazy parsing can be found in
debugging (disambiguation, fault localisation) and other areas that
traditionally profit from laziness. This remains future work.
#### 3.8.3 Unparsing techniques
One of the most confusing paper that I have submitted anywhere in 2012, was
the one about unparsing techniques [Zay12ah]. Only after finishing writing it,
I have realised how big and overwhelming this topic is. The paper was
rightfully rejected after being classified as a “request for discussion”: a
much deeper survey of (some of) the presented topics must be composed sooner
or later, but it requires much careful consideration. I have not done much in
this topic after that, but there was at least one paper published recently
that explicitly considered unparsing [SCL12].
The starting idea is simple as a sunrise: there was a lot of effort put in
researching parsing techniques, so why not the opposite? The unparsing
techniques can be understood in a very broad sense: pretty-printing, syntax
highlighting, structural import yielding an editable textual representation,
bidirectional construction of equivalent views, etc.
Some papers consider _conservative pretty-printing_ as a way to preserve
peculiar layout pieces (like multiple spaces) during unparsing [Jon02, Ruc96].
This is a narrow application of a general idea of propagating layout through
transformations, which is a long-standing and a well-researched problem.
However, even the most conservative unparsers have the risk of introducing an
inconsistently formatted code fragment, if that code was originally introduced
by a source code manipulation technique and not produced by a parser. In other
words, replacing a GO TO statement with a WHILE loop should look differently,
depending on how the code around the introduced fragment was formatted.
Possibly, results from the grammar inference research field [SC12] can be
reused for recovering formatting rules in some reasonable proximity of the
code fragment in order to unparse it correctly and avoid code alienation.
Suppose not just one desired textual formatted representation of the language
instance exists, but several of them, which form a family, or a product line,
like the line of metalanguages considered in §3.3 in the context of
metasyntactic evolution. Following that example, suppose we are given a
grammar in some internal representation and a syntactic notation specification
[Zay12c], then it is somewhat trivial to construct an unparser that would
produce the same grammar in a textual form. In other words, such an unparser
should generate a text that, given a notation specification, can yield the
same grammar after automated notation-parametric grammar recovery [Zay12w,
§3]. However, other questions remain. How to find a minimal notation needed to
unparse a given grammar? How in general to validate compatibility of a given
grammar and a given notation? How to produce grammar transformations (see
§3.2) to make the grammar fit the notation, how to produce notation
transformations (see §3.3.2) to make the notation fit the grammar, and how to
negotiate to find a properly balanced outcome? These questions are not trivial
and require investigation. Unparser-completeness has recently been studied in
the context of template engines [ABS11].
Unparsing can also be viewed as commitment to grammatical structure [KLV05].
Can we recover grammars from them, compare and converge them with other
grammars of the same language that we would like synchronised (e.g., concrete
syntax definition intended for parsing, multiple abstract syntaxes for
performing various grammar-based analysis tasks, data models for
serialisation)? Are there some specific properties that such grammars always
possess? What is the minimal upper formalism for the baseline grammar from
which grammars for parsing and unparsing can be derived automatically with a
language-independent or language-parametric technology? These questions are
not trivial and require investigation.
Connecting to the topic of robust/tolerant parsing (see §3.4), we can consider
at least two kinds of techniques that as the opposite: incremental unparsing
and unparsing incomplete trees. By _incremental unparsing_ I mean a modular
technique for unparsing modified code fragments and combining them with the
previously unparsed versions of the unmodified code fragments. This is usually
not considered for simple cases, but is possibly worth investigating for large
scale scenarios (consider architectural modifications to an IT portfolio with
hundreds of millions lines of code in dozens of languages). By _unparsing
incomplete trees_ we define the process of unparsing structured
representations of incomplete language instances. Besides scenarios when this
technique is used together with tolerant/robust parsing (and then the lacking
information may be somehow propagated to the unparser anyway), there are also
other scenarios when the gaps are deliberately left out to be filled by the
unparser. In documentation generation, this is the way code examples can be
treated — for a sample implementation we refer to Rascal Tutor [Kli+12].
For construction of compiler compilers and similar grammarware with unparsing
facilities, there is a commonly encountered problem of bracket minimality for
avoiding constructions ambiguous for parsing: since brackets are there in the
text only to guide the parsing process, they are removed from the AST, so how
to put back as few of them as possible during unparsing? This is a typical
research question for the unparsing techniques field. One could also
investigate various ways to infer grammar adaptation steps needed to unparse
the given grammatical structure in order to guarantee the lack of ambiguities
if it is to be parsed again.
#### 3.8.4 Migration to git
Following the current trend of leaving old-fashioned open source farms in
favour of more modern 2.0 social coding websites, I have migrated the Software
Language Processing Repository from SourceForge to GitHub [Zay+08]. The
project was started in 2008 by Ralf Lämmel [Läm08] and quickly after that
become the main target for my efforts and the main repository for my code. As
of now (December 2012), it contains 954 revisions committed by me, 314 by Ralf
Lämmel, 44 by Tijs van der Storm and 28 by all other contributors combined.
This would have not been worth mentioning, if I did not migrate all my other
repositories to git as well, which enabled efficient linking to all of them
from the open notebook (see §3.7). For closed source repositories (like ones
used for writing papers) we use Atlassian BitBucket instead of GitHub.
#### 3.8.5 Turing machine programming
Two of my colleagues from Centrum Wiskunde & Informatica (CWI), Davy Landman
and Jeroen van den Bos, have built a physical Turing machine with a finite
tape and separate program space, from LEGO blocks [Bos+12]. We were all
passively yet encouragingly watching them do that and then watching with
excitement how the resulting machine could sum two and two in less than half
an hour. From the software perspective, they have created a kind of “Turing
assembly” DSL that consisted of commands for accessing bits on the tape,
moving the head and making decisions on the next command, and was translatable
into some real code that could run on the LEGO chip brick. Then, there was a
slightly more advanced DSL called “Turing level 2” developed on top of it,
enhanced with label names and repetition loops, as well as IDE support
features like a visualiser/simulator.
My spontaneous contribution to the project involved writing several programs
for the machine in this “Turing language level 2”, including copying of unary
numbers, incrementing them, performing various forms of addition and finally
multiplying two unary numbers. All these programs are publicly accessible at
the official repository: http://github.com/cwi-
swat/TuringLEGO/tree/master/examples.
#### 3.8.6 Grammarware visualisation
Various controversial thoughts on grammar recovery visualisation, related to
the previous body of works on grammar recovery both (co)authored by me [Zay05,
LZ09, Zay10, Zay11a, LZ11, Zay12c, Zay12w, Zay12ad] and the giants on
shoulders of which I was standing [BSV97, LV99, SV00, LV01, LV01a, KLV05],
yielded some experimental code, but no valuable stable results.
In a draft sent to the “new ideas” track of FSE 2012 [Zay12ai] to be rejected
there, I have argued that introducing or improving visualisation of processes
in grammarware engineering has at least these benefits:
Process comprehension:
it becomes easier to understand the process and to see what exactly is
happening when it is applied to certain input.
Process verification:
while complete formal verification of a sophisticated process with many
branches and underlying algorithms, may be a challenging task, it is
relatively easy to pursue lightweight verification methods. One of them comes
more or less for free when an experienced observer can see what is happening
and detect peculiarities naturally.
Process improvement:
observing a process does not only let one find mistakes in it, but also to get
familiar with bottlenecks and other problematic issues, which in turn will
help to suggest refinements and improvements.
Interactiveness:
there are many examples of processes which are impossible or unfeasibly hard
to automate completely, but for which reasonable automation schemes exist that
exercise “semi-automation” and require occasional feedback from a system
operator. The request-response loop for such feedback can be drastically
shortened in the case of interactive visualisations.
The point of the paper was well-received by the FSE NIER reviewers: nobody
tried to argue that visualisation techniques would be useless. However, I
obviously overestimated a contribution that I could make with providing a
“mile wide, inch deep” (a quote from one of the reviews) overview, so perhaps
a much later overview with the list of solid achieved results, would be in
order. For the sake of completeness of this report, I list the nine showcases
that were briefly described in the NIER submission below. Each item of this
list is a relatively low hanging fruit for an article or a series thereof.
Grammar recovery:
the state of the art in automated grammar recovery (see also §3.3.3) is to
work based on a set of appropriate heuristics [LZ11, Zay12w]. Proper
visualisation of them would help: dealing with some particularly tricky
notations; verifying that the heuristics do what they are intended to do;
collecting evidence and statistics on the use of certain heuristics; proposing
additional heuristics and other process improvements.
API-fication
is a term used in [KLV05] to describe a process of replacing low level API
calls for manipulating a data structure with more expressive and more
maintainable high level API calls generated from a grammar [JO04]. Thus, API-
fication is a form of grammar-aware software renovation where surfacing
grammar knowledge is a crucial contribution of the process. Visualising both
the API calls themselves and the improvement steps on them, can serve as a
motivation and even as a lightweight verification of API-fication.
Grammar transformation & convergence.
There are at least two commonly used ways to visualise a grammar: in a textual
form as (E)BNF; or as a syntax diagram (“railroad track”). Neither of them has
a designated visualisation notation for transformations.
Mapping between grammar notations
is of the biggest challenges in research on grammars in a broad sense, since
grammarware strives to cover such a big range of various structural
definitions. Mapping between EBNF dialects [Zay12q], X/O mapping [Läm07], O/R
mapping [O’N08], R/X mapping [Fer+02] and many other internotational mappings
exist along with intranotational techniques for grammar diffing, graph
comparison, nonterminal matching, model weaving, etc. Displaying matching
artefacts in a traceable way by metagrammarware tools is usually rather
limited and either display local (mis)matches or global statistics.
Grammarware coevolution.
Concurrent and coupled evolution of grammars and language instances [Cic+08],
of coexisting related grammars [Läm01], of grammars and language
transformations [CH06], of language design and implementation [D’H+01] are
special mixed cases of mapping and transformations (see last two sections),
where we would like to visualise both what kind of matches are made and what
kind of actions are inferred from them.
Grammar-based analysis
comprises syntactic analysis (parsing), but also similarly geared techniques
that never received enough attention. As an example, it would be great to have
something to demonstrate hierarchical lexical analysis [MN95] to the same
degree as [AMUFVI09] demonstrated for LL and LALR parsing.
Disambiguation
is a process of filtering a parse forest or reasoning about the origins of it,
in modern generalised parsing algorithms like SGLR [Vis97] or GLL [SJ10].
Visualising SGLR disambiguation [Bra+02] was implemented in the ASF+SDF Meta-
Environment as a part of parse tree rendering, so in fact it visualised the
ambiguities themselves and not the process of removing them, which was still
of considerable help. More recent GLL disambiguation algorithms [Bas10] were
expressed mostly in a textual form even within a PhD project entirely
dedicated to ambiguity detection [Bas11] — primarily because there is no clear
understanding of how exactly they would be useful to visualise.
Grammar-based testing
methods based on combinatorial (non-probabilistic) exploration of the software
language under test, have emerged from recent research [LS06, FLZ12].
Visualising coverage achieved by them and adjusting the visualisation with
each new test case should help both to keep track of the process by expressing
its progress, and to localise grammar fragments responsible for the failing
test cases.
Grammar inference
is a family of methods of inferring the grammar, partially or completely, from
the available codebase and even from code indentation [Čre+05, Nie+07, SC12].
Such inference is a complicated process based on heuristics and sometimes even
on search-based methods. As a consequence, each attempt at grammar inference
remains somehow unconnected to the rest of the research field: adoption of
such methods by scientists and engineers outside the original working group
happens rarely, if ever. One can think that a proper visualisation of such
process would help new users to get acquainted with a grammar reconstruction
system and tweak it to their needs.
Production rule | Prod. signature
---|---
$\mathrm{p}\left(\text{`'},\mathit{Expr},\mathit{Expr_{1}}\right)$ | $\\{\langle\mathit{Expr_{1}},1\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Expr},str\right)$ | $\\{\langle str,1\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Expr},\mathit{Expr_{2}}\right)$ | $\\{\langle\mathit{Expr_{2}},1\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Expr},\mathit{Expr_{3}}\right)$ | $\\{\langle\mathit{Expr_{3}},1\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Expr},int\right)$ | $\\{\langle int,1\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Function},\mathrm{seq}\left(\left[str,\raisebox{0.80005pt}{$\scriptstyle\mathord{*}$}\left(str\right),\mathit{Expr}\right]\right)\right)$ | $\\{\langle\mathit{Expr},1\rangle,\langle str,1{*}\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Program},\raisebox{0.80005pt}{$\scriptstyle\mathord{*}$}\left(\mathit{Function}\right)\right)$ | $\\{\langle\mathit{Function},{*}\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Expr_{1}},\mathrm{seq}\left(\left[str,\raisebox{0.80005pt}{$\scriptstyle\mathord{*}$}\left(\mathit{Expr}\right)\right]\right)\right)$ | $\\{\langle str,1\rangle,\langle\mathit{Expr},{*}\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Expr_{2}},\mathrm{seq}\left(\left[\mathit{Ops},\mathit{Expr},\mathit{Expr}\right]\right)\right)$ | $\\{\langle\mathit{Ops},1\rangle,\langle\mathit{Expr},11\rangle\\}$
$\mathrm{p}\left(\text{`'},\mathit{Expr_{3}},\mathrm{seq}\left(\left[\mathit{Expr},\mathit{Expr},\mathit{Expr}\right]\right)\right)$ | $\\{\langle\mathit{Expr},111\rangle\\}$
Table 3: The JAXB grammar in a broad sense: in fact, an object model obtained
by a data binding framework. Generated automatically by JAXB [FV99] from the
XML schema for the Factorial Language [Zay12j].
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Expr},\mathit{Expr_{1}}\right)$
$\displaystyle\bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{expression},\mathit{apply}\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Expr},str\right)$
$\displaystyle\bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{expression},str\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Expr},\mathit{Expr_{2}}\right)$
$\displaystyle\bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{expression},\mathit{binary}\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Expr},\mathit{Expr_{3}}\right)$
$\displaystyle\bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{expression},\mathit{conditional}\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Expr},int\right)$
$\displaystyle\bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{expression},int\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Function},\mathrm{seq}\left(\left[str,\raisebox{1.00006pt}{$\scriptstyle\mathord{*}$}\left(str\right),\mathit{Expr}\right]\right)\right)$
$\displaystyle\Bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{function},\mathrm{seq}\left(\left[str,\raisebox{1.00006pt}{$\scriptscriptstyle\mathord{+}$}\left(str\right),\mathit{expression}\right]\right)\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Program},\raisebox{1.00006pt}{$\scriptstyle\mathord{*}$}\left(\mathit{Function}\right)\right)$
$\displaystyle\Bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{program},\raisebox{1.00006pt}{$\scriptscriptstyle\mathord{+}$}\left(\mathit{function}\right)\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Expr_{1}},\mathrm{seq}\left(\left[str,\raisebox{1.00006pt}{$\scriptstyle\mathord{*}$}\left(\mathit{Expr}\right)\right]\right)\right)$
$\displaystyle\Bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{apply},\mathrm{seq}\left(\left[str,\raisebox{1.00006pt}{$\scriptscriptstyle\mathord{+}$}\left(\mathit{expression}\right)\right]\right)\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Expr_{2}},\mathrm{seq}\left(\left[\mathit{Ops},\mathit{Expr},\mathit{Expr}\right]\right)\right)$
$\displaystyle\Bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{binary},\mathrm{seq}\left(\left[\mathit{expression},\mathit{operator},\mathit{expression}\right]\right)\right)$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{Expr_{3}},\mathrm{seq}\left(\left[\mathit{Expr},\mathit{Expr},\mathit{Expr}\right]\right)\right)$
$\displaystyle\bumpeq$
$\displaystyle\mathrm{p}\left(\text{`'},\mathit{conditional},\mathrm{seq}\left(\left[\mathit{expression},\mathit{expression},\mathit{expression}\right]\right)\right)$
Figure 4: Matching of production rules with the Abstract Normal Form of the
JAXB-produced grammar on the left and the master grammar on the right
[Zay12j].
NB: the last item was written before the publication of the excellent grammar
inference field overview [SC12], which can also be seen as considering
visualisation in a very broad sense.
Another newer initiative which can be seen as grammarware process
visualisation, concerns guided convergence (see also §3.1). We can recall that
the whole process of the guided grammar convergence is rather complicated and
involves normalising the input grammar and going through several phases of
unification to ensure the final nonterminal mapping that looks like this
[Zay12j]:
$\displaystyle\mathit{jaxb}\>\diamond\>\mathit{master}=\>$
$\displaystyle\\{\langle\mathit{Expr_{2}},\mathit{binary}\rangle,$
$\displaystyle\langle\mathit{Expr_{3}},\mathit{conditional}\rangle,$
$\displaystyle\langle int,int\rangle,$
$\displaystyle\langle\mathit{Function},\mathit{function}\rangle,$
$\displaystyle\langle str,str\rangle,$
$\displaystyle\langle\mathit{Program},\mathit{program}\rangle,$
$\displaystyle\langle\mathit{Expr},\mathit{expression}\rangle,$
$\displaystyle\langle\mathit{Expr_{1}},\mathit{apply}\rangle,$
$\displaystyle\langle\mathit{Ops},\mathit{operator}\rangle\\}$
While preparing the main guided grammar submission, I have noticed that this
particular mapping, as well as the normalised grammar (Table 3) and the list
of weakly and strongly prodsig-equivalent production rules (Figure 4) can be
automatically produced by the convergence tool virtually without any
additional effort in a completely transparent, traceable, reliable and
reproducible fashion. This led to open publication of [Zay12j], an extended
appendix for the main guided grammar convergence paper, which was, except for
the two-page introduction, generated automatically, but is still readable and
useful.
#### 3.8.7 Wiki activity
While contributing to wiki websites is not usually considered an activity
worthy of tracking or mentioning in the academic sense, of the 72 wiki-
articles I have written in 2012 I can identify at least six that can be viewed
as (popular) scientific writing:
Grammar in a broad sense (11 kB + 1 figure)
Technological space (16 kB)
Megamodelling (6 kB + 2 figures)
Island grammar (12 kB)
Adriaan van Wijgaarden (21 kB)
Ninomiya Sontoku (Kinjiro) (10 kB)
#### 3.8.8 Colloquium organisation
Again, participating in organisation of various events is commonly considered
normal for a practicing academic researcher, but is never counted as a
scientific contribution. Not arguing with that, I am still happy to be able to
maintain the existing seminar culture of CWI (Centrum Wiskunde & Informatica,
my current employer) as a colloquium organiser of a series of events that have
been taken place continuously at least since 1997333http://event.cwi.nl/pem.
Over the course of 2012, 56 presentations were given in total as a part of
Programming Environment Meeting (PEM, mostly an inter-institutional outlet),
Software Engineering Meeting (SEM, mostly an internal group seminar) and a
special one-day event Symposium on Language Composability and Modularity
(SLaC’M, most trouble of organising which was taken by Tijs van der Storm).
These speakers have appeared at PEM, SEM and SLaC’M in 2012 (in chronological
order of their first appearance):
Dr. Vadim Zaytsev [Zay12k, Zay12b, Zay12af, Zay12d, Zay12e]
Atze van der Ploeg [Plo12a, Plo12]
Prof. Dr. Serge Demeyer [Dem12]
Dr. Alexander Serebrenik [Ser12]
Stella Pachidi [Pac12]
Dr. Tijs van der Storm [Sto12, Sto12b, Sto12a]
Michael Steindorfer [Ste12, Ste12a, Ste12b]
Dr. Antony Sloane [Slo12]
Riemer van Rozen [Roz12, Roz12a]
Jeroen van den Bos [Bos12a, LB12, Bos12]
Alex Loh [Loh12a, Loh12b, Loh12]
Dr. Daniel M. German [Ger12]
Dr. Michael Godfrey [God12]
Dr. Mark Hills [Hil12b, Hil12a, Hil12]
Davy Landman [LB12, Lan12]
Luuk Stevens [Ste12c]
Dr. Krzysztof Czarnecki [Cza12]
Prof. Dr. Magne Haveraaen [HB12, Hav12]
Dr. Anya Helene Bagge [HB12, Bag12]
Dr. Sunil Simon [Sim12]
Dr. T. B. Dinesh [Din12]
Dr. Jurgen Vinju [Vin12, Vin12a]
Dr. William R. Cook [Coo12]
Anastasia Izmaylova [Izm12]
Dr. Lennart Kats [Kat12]
Carel Bast, Wim Bast, Tom Brus [BBB12]
Tesfahun Tesfay [Tes12]
Dr. William B. Langdon [Lan12a]
Andrei Varanovich [Var12]
Dr. Joris Dormans [Dor12]
Sebastiaan Joosten [Joo12]
Dr. Magiel Bruntink [Bru12]
Dr. Patricia Lago [Lag12]
Prof. Dr. Frank Tip [Tip12]
Dr. Raphael Poss [Pos12]
Arjan Scherpenisse [Sch12]
## 4 Venues
Academic venues (mostly conferences, workshops and journals) are essential
components of the research process: publishing there means community
recognition; submitting eventually leads to receiving peer reviews; and even
reading calls for papers can be very inspiring and eye-opening. Below I list
two kinds of venues that contributed to my research in 2012: one list is for
those where I have submitted, the other one for the rest — I am deeply
grateful to all the reviewers and organisers of both kinds. The lists are not
meant to cover all possible venues for my field, just those directly relevant
to my activities this year.
### 4.1 Exercised venues
BX 2012 (ETAPS workshop)
I have been a “ _bx-curious_ ” person for quite a while, but BX 2012 was my
first venue to come out. A very inspiring call for
papers444http://www.program-transformation.org/BX12, excellent atmosphere
during the workshop, friendly and productive reviewers. A typical example of
an event that appreciates you preparing a dedicated paper for which this
becomes the one and only target venue. I submitted against all the odds
(December deadlines are rather stressful), got there against all the odds (had
to fly from ETAPS to SAC and then back) and still regretted nothing. I will
not attend BX 2013 (my grandmother has her 80th birthday on the day of the
workshop, and one has to set priorities), but I would if I could. Definitely
recommended for people at least marginally interested in this field [Cza+09].
SAC 2012 (PL track)
A yet another experimental submission in the sense that I did not know almost
anyone from the programme committee at that moment. However, I know people
from my technological space who published there, and the call for
papers555http://www.cis.uab.edu/bryant/sac2012 was inspiring, so I gave it a
try, and did not regret it. The whole conference is huge, so I was afraid that
attending would be unproductive, but I was proven wrong: if you know at least
a couple of people with similar research interests and stick to them all the
time, you will find many other similar researchers to talk to. I did not
submit anything to SAC 2013 due to bad planning (holidays right before the
deadline are unproductive), but I definitely will consider it very seriously
every year from now on.
LDTA 2012 (ETAPS worshop)
Trying to be a good programme committee member, I knew I have to attend, so I
have submitted the best result of 2011 there: the Grammar Hunter. I was also
pleased to see how the current call for
papers666http://ldta.info/ldta_2012_cfp.pdf positioned LDTA as “SLE, but with
more grammarware”. The future of LDTA remains to be determined, but it has
departed from ETAPS and will most probably join forces with SLE.
ECMFA 2012
The call for
papers777http://www2.imm.dtu.dk/conferences/ECMFA-2012/contributions/?page=cfp
made it look like I have a chance, so I submitted something that I believed to
be of good quality and of possible interest to the modelware researchers. One
of the reviewers said that the paper “clearly makes the most contribution of
any paper I read”, which was rather encouraging, but ended up with rejection.
In the end, I must conclude that I should have devoted this time to writing
for ICPC or one of the journal special issues with deadlines around early
spring.
TFP 2012
The call for papers888http://www-fp.cs.st-
andrews.ac.uk/tifp/TFP2012/TFP_2012/CFP12.txt looked challenging, but I really
liked the “trends” aspect of it, since most traditional conferences dislike
overview papers unless they are extremely strong and retrospective: there is
simply no place for overviews of the current trends, unless you are already in
the field and you systematically explore the “future work” sections of all
papers you come by. In contrast to BX, this was an example of a venue that did
not appreciate preparing a paper specifically for them on a topic relevant to
me. In less than two weeks after submission I have received a short
notification that it was judged to be out of scope. This was obviously not the
only reason since other (stronger, less “trendy”) papers from my technological
space like [SS12] were accepted, so I can only conclude that I have failed to
explain the link between grammar transformation and the functional programming
paradigm properly. Given the fact that I am not qualified to report on
“trends” in any other field, I doubt that I will try sending anything to this
venue in the future, but I surely do not discourage others to do so.
Personally for me, it would have been more more productive to pursue MoDELS
which had a competing deadline this year.
JUCS (journal)
The call for
papers999http://www.jucs.org/ujs/jucs/info/special_issues/sbcars_cfp.pdf made
it clear that this special issue is linked to a workshop where I did not
participate, but the call was open, and I answered. I cannot say that that was
very appreciated: the reviews for [Zay12ad] came very late (several months
after the notification deadline), were extremely short and discouraging.
SCAM 2012
This is the third time I have served as a programme committee member for SCAM,
where I have been invited after our paper with Ralf Lämmel got a best paper
award in 2009 [LZ09a]. I have never attended since that time, and received a
warning that I will not be included next year if I miss the event again. So,
putting date-conflicting events like SLE and CSMR aside, I did my best, which
for me meant submitting one paper to SCAM and one to the colocated ICSM (see
below). The topic chosen for SCAM (island grammars) seemed to be in scope of
the call for papers101010http://scam2012.cs.usask.ca/CFP, but the paper was
seen as weird and immature, and was hopelessly rejected. The reviews it
received were pretty helpful, even though one of the reviewers really hated
the “in the cloud” aspect (and that is exactly how I tried to sell it).
Apparently, putting some effort into submitting something has already been
noticed, since I have been, against all the odds, invited to the programme
committee again for SCAM 2013.
ICSM 2012
The call for papers111111http://selab.fbk.eu/icsm2012/download/cfp-
icsm2012.pdf came to my attention right after the rejection letter from ECMFA,
and I decided that ICSM would be a good venue for the guided grammar
convergence methodology (§3.1). Getting a paper there would also increase my
chances at going to SCAM (see above for the reasons). Reviews were rather
cold, but some of them (except one) useful nonetheless.
NordiCloud (WICSA/ECSA workshop)
Not really being an architecture researcher, I would have never considered
going to WICSA/ECSA, but the call for
contributions121212http://46.22.129.68/NordiCloud/?page_id=39 was out
precisely a couple of weeks after my SCAM rejection, and I was not feeling
enough energy to rewrite the island parsing paper completely, so NordiCloud
was a relatively cheap way for me to resubmit the same material after a minor
revision. It did not pay off: most of the reviewers were scared off just by
seeing a grammar-related submission.
FSE 2012 (NIER track)
The call for papers131313http://www.sigsoft.org/fse20/cfpNewIdeas.html came
out at a very busy time, but four page limit was easily reachable, so I have
submitted two papers on different new ideas. Unfortunately, they were indeed
more of idealistic proposals for discussing and considering certain aspects,
than usual “short papers” that are just normal papers at the early stage. Both
were hopelessly rejected, and I still want to find some venue for the future
that would be good for sharing and discussing fresh ideas — perhaps OBT? I
have to try to find out.
SoTeSoLa 2012 (summer school)
An experiment in “Research 2.0” driven mostly by Jean-Marie Favre and Ralf
Lämmel, this summer school was by far not a typical one. There was a lot of
innovations: submitting a one-page profile of yourself, making a one-minute
video about yourself, listening to lots of remote lectures, having a hackathon
distributed in time and space, registering at a social networking website,
etc. Not all of them very entirely successful: partly due to being ahead of
its time, partly due to other reasons, which are being dissected, analysed and
researched now by Jean-Marie Favre. I was involved in all kinds of activities
from the relatively early stage, and in the end it was officially classified
as serving as a “Social Media Chair” and a “Hackathon Lead Coordinator”. This
was not a publishing venue, and I did not give any invited lecture, but it was
fun to be a part of it.
SATToSE 2012 (seminar)
A non-publishing seminar series where I have given a presentation on
bidirectional grammar transformation [Zay12a]. The material presented there
was in a state somewhere between [Zay12q] and the planned future paper on
bidirectionalisation.
POPL 2013
The call for papers141414http://popl.mpi-sws.org/2013/popl2013-cfp.pdf was
concise and crunchy, but POPL is one of the venues that does not require much
advertisement. I have poured a lot of effort into [Zay12i], completely
redesigned the convergence process (see §3.1), reimplemented the prototype and
rewritten the paper with respect to [Zay12g, Zay12h]. In a way, it did pay
off: the paper was rejected, but the reviews were among the most useful that I
have received this year.
NWPT 2012
The call for papers151515http://nwpt12.ii.uib.no/call-for-papers was brought
to my attention by Anya Helene Bagge, a co-organiser of this workshop. In an
extended abstract that was submitted there, I apparently went overboard with
the required abstraction level and assumed level of grammatical knowledge, and
recent POPL rejection has possibly jeopardised the outsourcing of the
usefulness statement of the method. Reviews were curt and bleak.
EMSE (journal)
The call for
papers161616http://sequoia.cs.byu.edu/lab/?page=reser2013§ion=emseSpecialIssue
called for “experimental replications” and went to great length explaining how
important it is to be able to publish not just the experiments themselves, but
also replications thereof. I was immediately convinced, but decided to
reinterpret the definition of a replication. Instead of doing classical
empirical studies, I presented research activities (and in particular
prototype engineering) as experiments. That way, the replications were also
“experiments” in that sense that were intended to cover an older experiment
and could therefore be measured and assessed based on grounds of that
coverage. I could even find some related work on the topic in form of papers
that described the prototype development process itself. My paper was intended
to contain three case studies: (1) replicating the grammar convergence case
study of the Factorial Language from [LZ09] with the guided grammar
convergence methodology (see §3.1); (2) replicating a bigger grammar
convergence case study of Java from [LZ11] with more abstract and concise
Extended XBGF (see §3.2.4); (3) replicating both of these case studies with a
bidirectional $\Upxi$BGF (see §3.2.2). Due to insane amounts of work that this
turned out to be, only the first two replications have made it into a 42-page
long paper [Zay12l]. Only one of three reviewers was excited by my approach,
and all three agreed that the empirical software engineering journal is not
the right venue for such a report.
MPM 2012 (MoDELS worshop)
Basically, this venue was chosen _after_ I have written a paper. The text
underwent some polishing after the choice was made, but the topic was not
adjusted. I have had a nice idea of transforming megamodels in order to make a
good story out of them (see §3.5): a substantial contribution was not yet
there (and such work is still ongoing), but I wanted to expose it to the
public and to discuss it first. The call for
papers171717http://avalon.aut.bme.hu/mpm12/MPM12-CFP.pdf for MPM looked the
most inviting for this kind of cross-paradigm approach among all MoDELS
workshops, and indeed the reviewers found the paper weird yet acceptable, so I
was able to give a short presentation and hang my poster there [Zay12z].
XM 2012 (MoDELS worshop)
The topics list181818http://www.di.univaq.it/XM2012/page.php?page_id=13
provided by the organisers of this workshop was fascinating, and I desperately
wanted to submit anything, but eventually gave up to find the time it
deserved. Soon after that, the deadline was extended, and I had no other
choice than to write down the idea that was floating around in my head for a
while (see §3.2.3).
SCP (journal)
The call for systems191919http://www.win.tue.nl/~mvdbrand/SCP-EST was very
much in sync with what its guest editors have tried to achieve in the last
years, and I support them wholeheartedly in that. The Grammar Zoo, one of
essential parts of the SLPS [Zay+08], that did not receive a lot of my
attention in 2012, but that was always on my mind, was packaged and submitted
there both as an available system and as an important repository of
experimental systems in grammarware. The outcome will become known in 2013.
### 4.2 Inspiring venues
There have been many venues that I did not submit anything to, but not because
I did not want to. Their calls for papers gave me inspiration to work on
something, even though I was not productive enough to be able to fit into
their deadlines or produce anything of value at the required level.
MSR 2012
The mining challenge202020http://2012.msrconf.org/challenge.php of MSR looked
very interesting, so I looked at it, but since I was looking specifically for
grammars, it did not work out at all: only two grammars were found, and there
was no sensible way to connect them to the rest of the system. If more of them
could have been obtained written in a variety of EBNF dialects, it could have
become an interesting case study similar to [Zay12ad].
Laws and Principles of Software Evolution
The call for papers212121http://listserv.acm.org/scripts/wa-
acmlpx.exe?A2=ind1111&L=seworld&F=&S=&P=25841 for this special issue of JSME
looked tempting, so I even emailed the editors, asking for some additional
information. Unfortunately, the collaboration that I hoped to achieve with
other people, did not work out, and nothing was produced in time.
Success Stories in Model Driven Engineering
The call for papers222222http://www.di.univaq.it/ssmde came out at the time
when I was busy with all kinds of other initiatives. Besides that, this
special issue of SCP was actually looking for extended reports on already
published projects, and I was busy with new experiments. Possibly, a strong
“lessons learnt” kind of paper on grammar hunting would make sense, but I was
too immersed in new stuff at the time to go back. However, I have to admit
that when/if I finally sit down to write a comprehensive grammar recovery
paper (i.e., connecting §§3.3.3, 3.8.1, 3.8.6 and 3.6), it must go to either
SCP or SP&E.
CloudMDE (ECMFA 2012 workshop)
This was _the_ venue that gave me the eerie thought of writing a “parsing in
the cloud” paper (see §3.8.2). However, I was disheartened by the rejection of
[Zay12g] at ECMFA and decided to not submit anything to ECMFA
workshops232323V. Zaytsev (grammarware). “Yet another bridging attempt failed:
my grammar paper got rejected at @ecmfa2012. Now I will also go submit the
#CloudMDE draft elsewhere.” Tweet.
https://twitter.com/grammarware/status/189976445995593728. 11 April 2012,
9:21..
ICPC 2012
The call for papers242424http://icpc12.sosy-lab.org/CfP.pdf competes date-wise
with many other good venues, so this year ICPC just happened to not be among
the ones I have chosen as my targets.
RC 2012
The call for papers252525http://www.reversible-
computation.org/2012/index91b1.html?call_for_papers for the fourth workshop on
reversible computation gave me a lot of ideas and keyword pointers for the
bidirectionality topic. However, I did not feel confident enough to submit
anything. Anyway, thanks a lot and congratulations on becoming a conference in
2013!
CSCW 2013
This is not a typical venue for me, but I have a dream of eventually
submitting something wiki-related there. The call for
participation262626http://cscw.acm.org/participation_paper.html was as good as
it always is, and even better this year because they have introduced a new
rule concerning the paper size: 10 pages is no longer the _limit_ , it is
rather a _standard_. If your idea fits on smaller number of pages, your
reviewers have the right to complain if you try to bloat your submission. On
the other hand, if that is not enough, you can always make your paper longer,
but the contribution then needs to grow accordingly. I believe that with small
incremental and non-disruptive ideas like these, we could achieve modern
comfortable publishing models easier than with endeavours to revolutionise the
field.
PPDP/LOPSTR 2012
The calls for papers272727http://dtai.cs.kuleuven.be/events/PPDP2012/ppdp-
cfp.txt,282828http://costa.ls.fi.upm.es/lopstr12/cfp.pdf were both
interesting, but at my level I could not actually decide between the two
venues. I was working honestly toward the seemingly achievable goal (see
§3.2.5), but it turned out to be unachievable. Being insecure about my ability
to write a strong paper about negative results, I gave up.
FM+AM 2012 (SEFM workshop)
This was _the_ workshop292929http://ssfm.cs.up.ac.za/workshop/FMAM12.htm that
set my thoughts in the agile/extreme mode, which ultimately led to the paper
at XM (see §3.2.3) simply because I did not manage to complete the work before
the FM+AM deadline. Imagine my surprise when I found out that FM+AM was
cancelled due to the lack of good submissions!
WoSQ 2012 (FSE workshop)
The call for papers303030http://sites.google.com/site/wosq2012/cfp has led me
to believe that this would be a good possible venue for the paper on grammar
mutations (see §3.8.1). However, the time was too tight, and both of my NIER
submissions have been rejected, so an FSE workshop stopped looking that
attractive after all.
SQM 2012 (CSMR workshop)
The workshop313131http://sqm2012.sig.eu happened at the same time as I was
attending both ETAPS and SAC, so I could not possible be at the third place at
the same time as well, but I just want to name it as a relatively small venue
where I have enjoyed reviewing a couple of papers as a PC member (will be on
PC next year as well).
WCN 2012
The website323232http://www.wikimediaconferentie.nl is in Dutch, as the
conference itself. This was my second experience being a Program Chair (the
first one was with WCN 2011), and this time I counted: 842 emails needed to be
sent or answered by me in order for this conference to happen. Luckily, CWI
(my employer) did not mind since they could proudly list “one of theirs” to be
the PC at a venue where one of the keynote speakers is Jimmy Wales [Wal12].
## 5 Concluding remarks
### 5.1 Immediate results
Writing this extended year report has achieved at least three goals:
Streamlining new ideas.
Reexplaning (renarrating?) research ideas and putting them in perspective has
helped to crystallise them into publishable achievable objectives.
Knowledge dissemination.
This document can serve both as a scientific report for my colleagues and
superiors, and as an entrance point for people who want to get acquainted with
my results for other reasons.
Case study in self-archiving.
As I have said in the introduction, this report can be seen as advanced form
of self-archiving. It was a relatively big effort, compared to the traditional
“just put the PDF online” thing. Together with the open notebook initiative,
it stressed the paradigm and raised some questions yet to be answered (e.g.:
How to properly break one atomic SKO — essentially, a publication — into
subatomic ones to distinguish “I’ve done for the tool that was later described
in this paper” from “I’ve done this for the particular version of this paper,
which was later rejected”? What are all possible stages in the SKO
lifecycle?).
### 5.2 Special features
The presence of an open notebook.
A lot of claims about dates, continuations and amounts of effort, made on the
pages of this report, can be reformulated into queries on the open notebook,
and formally validated as such. For now these claims were intentionally done
in plain text because no reliable or traditionally acceptable infrastructure
exists for them (yet).
Open access window.
All the papers mentioned here, were put online immediately after their
submission (unless prohibited explicitly by the submission rules), and taken
down immediately after their rejection (if any). At this stage, I do not know
any better way to expose your research results to the public: official
acceptance can take months and years, during which one could have profited
from sharing the contents around.
Rejected material.
Not all rejected papers are rejected because they are inherently, irreparably
bad: some turn out to be out of scope, lacking some essential results or
simply not mature enough to be published (yet). With this report, I have
exposed most of the dark data concerning my rejected material.
Unfruitful attempts.
Also classified as dark data by [Goe07], but of an entirely different nature:
these are failed experiments: prototypes that have never made it to the point
of being ready to be described in a paper. There can be traces of such
unfruitful attempts in presentations and other subatomic SKOs before their
futility becomes apparent.
Venues.
Knowledge about workshops, conferences and journals seems to float around in
the academic community and is usually distributed as folklore, if at all.
There are many reasons for doing so, ranging from the lack of incentive to the
fear of occasional offence.
### 5.3 Acknowledgements
I am most grateful to the following people, without whom 2012 would not have
been the same:
Ronald de Wolf, Bas de Lange, Tijs van der Storm, Jean-Marie Favre — for
inviting me to present at various events.
Ralf Lämmel — for inviting me for a working visit to Koblenz and hosting me
there.
Felienne Hermans — for referring me to the matrix grammars, picture calculus
and adjacent topics.
Paul Klint, Jurgen Vinju, Tijs van der Storm, Sander Bohte — for contributing
to organising the PEM Colloquium.
T. B. Dinesh — for introducing me to the notion of renarration.
Frans Grijzenhout and Sandra Rientjes — for coorganising a conference with me.
Lodewijk Gelauff — for introducing me to the topic of structured data
extraction.
Mark van den Brand — for appreciating the input in the discussion on the
future of LDTA, and for putting trust in me for chairing its programme next
year.
Jurgen Vinju, Tijs van der Storm — for collaborating on a grammar-related
topic.
Ralf Lämmel, Andrei Varanovich, Jean-Marie Favre — for collaborating on
megamodelling topics.
Zinovy Diskin — for insightful discussions during ETAPS’12 and MoDELS’12 on
category theory and grammars in a broad sense.
Anonymous reviewers of BX’12, SAC’12, LDTA’12, ECMFA’12, ICSM’12, SCAM’12,
NordiCloud’12, FSE NIER’12, XM’12, MPM’12, NWPT’12, POPL’13, JUCS, EMSE and
NWO Veni programme — for the effort that they have put into considering,
assessing and reviewing my work.
All presenters at PEM and WCN — for their contributions, the only essential
component of the final product.
All unmentioned colleagues and uncounted conference contacts — for fruitful
discussions.
All people who tried to contact me through email — for patience and
unreasonable waiting times.
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* [Zay05] V. Zaytsev “Correct C# Grammar too Sharp for ISO” Extended abstract In _Participants Workshop, Part II of the Pre-proceedings of the International Summer School on Generative and Transformational Techniques in Software Engineering (GTTSE 2005)_ Braga, Portugal: Technical Report, TR-CCTC/DI-36, Universidade do Minho, 2005, pp. 154–155 URL: http://grammarware.net/writes/#Too-Sharp2005
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* [Zay11a] V. Zaytsev “MediaWiki Grammar Recovery” In _Computing Research Repository (CoRR)_ abs/1107.4661 ACM, 2011, pp. 1–47 URL: http://arxiv.org/abs/1107.4661
* [Zay12] V. Zaytsev “Automated Reuse-driven Grammar Restructuring” Classified as “very good” and rejected, 2012, pp. 1–17
* [Zay12a] V. Zaytsev “Bidirectional Grammar Transformations” Seminar presentation at the _Fifth Seminar Series on Advanced Techniques and Tools for Software Evolution (SATToSE)_ , 20 August, 2012 URL: http://grammarware.net/talks/#BGX2012
* [Zay12b] V. Zaytsev “Bidirectional Transformations and Grammarware” Colloquium presentation at the _Software Engineering Meeting_ , 3 February, 2012 URL: http://grammarware.net/talks/#SEM2012BX
* [Zay12c] V. Zaytsev “BNF WAS HERE: What Have We Done About the Unnecessary Diversity of Notation for Syntactic Definitions” In _Programming Languages Track, Volume II of the Proceedings of the 27th ACM Symposium on Applied Computing (SAC 2012)_ Riva del Garda, Italy: ACM, 2012, pp. 1910–1915 DOI: 10.1145/2245276.2232090
* [Zay12d] V. Zaytsev “Experimental Replications” Colloquium discussion at the _Programming Environment Meeting_ , 25 May, 2012 URL: http://grammarware.net/talks/#Replications2012
* [Zay12e] V. Zaytsev “Grammar Composition and Extension” Symposium presentation at the _Symposium on Language Composability and Modularity_ , 10 August, 2012 URL: http://grammarware.net/talks/#Decomposition2012
* [Zay12f] V. Zaytsev “Grammar Convergence” Presentation of recent research results at the _Software Languages Team_ , 25 July, 2012 URL: http://grammarware.net/talks/#Guided-Convergence2012
* [Zay12g] V. Zaytsev “Guided Grammar Convergence” Submitted to the Eighth European Conference on Modelling Foundations and Applications (ECMFA 2012). Rejected, 2012
* [Zay12h] V. Zaytsev “Guided Grammar Convergence” Submitted to the 28th IEEE International Conference on Software Maintenance (ICSM 2012). Rejected, 2012
* [Zay12i] V. Zaytsev “Guided Grammar Convergence” Submitted to the 40th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL 2013). Rejected, 2012
* [Zay12j] V. Zaytsev “Guided Grammar Convergence. Full Case Study Report. Generated by converge::Guided” In _ACM Computing Research Repository (CoRR)_ abs/1207.6541, 2012, pp. 1–44 URL: http://arxiv.org/abs/1207.6541
* [Zay12k] V. Zaytsev “History and Future of the PEM Colloquium” Colloquium discussion at the _Programming Environment Meeting_ , 20 January, 2012 URL: http://grammarware.net/talks/#PEM2012
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* [Zay12m] V. Zaytsev “Islands in the Cloud” Submitted to the 12th IEEE International Working Conference on Source Code Analysis and Manipulation (SCAM 2012). Rejected, 2012
* [Zay12n] V. Zaytsev “Islands in the Cloud” Submitted to the Nordic Symposium on Cloud Computing and Internet Technologies (NordiCloud 2012). Rejected, 2012
* [Zay12o] V. Zaytsev “Language Evolution, Metasyntactically” In _Pre-proceedings of the First International Workshop on Bidirectional Transformation (BX 2012)_ Institute of Cybernetics at Tallinn University of Technology, 2012 DOI: 10.6084/m9.figshare.91427
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* [Zay12q] V. Zaytsev “Language Evolution, Metasyntactically” In _Electronic Communications of the European Association of Software Science and Technology (EC-EASST)_ 49 EASST, 2012, pp. 1–17 URL: http://journal.ub.tu-berlin.de/eceasst/article/view/708
* [Zay12r] V. Zaytsev “Megamodelling Language Design: User Experiences and Ad Hoc Megamodelling” Local talk about ongoing research results in joint work at the _Software Languages Team_ , 18 July, 2012 URL: http://grammarware.net/talks/#MegaL2012
* [Zay12s] V. Zaytsev “Metasyntactic Footprints and Signatures” Submitted to the 24th Nordic Workshop on Programming Theory (NWPT). Rejected. Extended abstract, 2012, pp. 1–3
* [Zay12t] V. Zaytsev “Negotiated Grammar Transformation” In _Extreme Modeling Workshop (XM 2012)_ Dipartimento di Informatica, Università degli Studi dell’Aquila, 2012 URL: http://www.di.univaq.it/diruscio/sites/XM2012/xm2012_submission_11.pdf
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* [Zay12v] V. Zaytsev “Negotiated Grammar Transformation” Paper presentation at the _Extreme Modeling Workshop_ , 1 October, 2012 URL: http://grammarware.net/talks/#Negotiated-XM2012
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* [Zay12ai] V. Zaytsev “Visualization of Grammarware Engineering Processes” Submitted to the New Ideas Track of the 20th ACM SIGSOFT International Symposium on the Foundations of Software Engineering (FSE 2012 NIER). Rejected, 2012
* [Zay13] V. Zaytsev “Grammar Zoo: A Repository of Experimental Grammarware” Submitted to the Fifth Special issue on Experimental Software and Toolkits of Science of Computer Programming (SCP EST5). Under review, 2013, pp. 1–25 URL: http://grammarware.net/writes/#Zoo2013
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* [Zay+08] V. Zaytsev, R. Lämmel, T. van der Storm, L. Renggli and G. Wachsmuth “Software Language Processing Suite333333The authors are given according to the list of contributors at http://github.com/grammarware/slps/graphs/contributors.” http://grammarware.github.com. Contains, among other works: _XBGF Manual: BGF Transformation Operator Suite v.1.0_ (Zaytsev, 2010), http://grammarware.github.com/xbgf; _Grammar Zoo_ (Zaytsev, 2009–2012), http://grammarware.github.com/zoo; _Grammar Tank_ (Zaytsev, 2011–2012), http://grammarware.github.com/tank., 2008–2012
|
arxiv-papers
| 2012-12-17T19:42:16 |
2024-09-04T02:49:39.406103
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vadim Zaytsev",
"submitter": "Vadim Zaytsev",
"url": "https://arxiv.org/abs/1212.4446"
}
|
1212.4620
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-363 LHCb-PAPER-2012-036 21 January 2013
Measurement of the cross-section for $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ production in pp collisions at
$\sqrt{s}=7$$\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration†††Authors are listed on the following pages.
A measurement of the cross-section for $\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ is presented using data at
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ corresponding to an integrated
luminosity of 0.94$\mbox{\,fb}^{-1}$. The process is measured within the
kinematic acceptance $\mbox{$p_{\rm
T}$}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2<\eta<4.5$ for the
daughter electrons and dielectron invariant mass in the range
60–120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The cross-section is
determined to be
$\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-})=76.0\pm 0.8\pm 2.0\pm 2.6\rm\,pb$
where the first uncertainty is statistical, the second is systematic and the
third is the uncertainty in the luminosity. The measurement is performed as a
function of Z rapidity and as a function of an angular variable which is
closely related to the Z transverse momentum. The results are compared with
previous LHCb measurements and with theoretical predictions from QCD.
Submitted to Journal of High Energy Physics
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, A. Bates48, Th. Bauer38, A. Bay36, J.
Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-
Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A.
Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E.
Bowen37, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N. Chiapolini37, M.
Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M.
Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins35, A. Comerma-Montells33, A. Contu15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua51, M. De Cian37, J.M. De Miranda1, L. De
Paula2, W. De Silva57, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F.
Dettori39, A. Di Canto11, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, M.
Dogaru26, F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34,
D. Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, S. Eisenhardt47, U. Eitschberger9, R. Ekelhof9, L. Eklund48, I. El
Rifai5, Ch. Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G.
Fardell47, C. Farinelli38, S. Farry12, V. Fave36, D. Ferguson47, V. Fernandez
Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C.
Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M.
Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, E. Furfaro21, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J. Garofoli53,
P. Garosi51, J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E.
Gersabeck11, M. Gersabeck51, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V.
Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O.
Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-
Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, C.
Hombach51, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52,
D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, E. Jans38, F. Jansen38, P. Jaton36, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji20, O. Kochebina7, I. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G.
Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36, H.
Lu3, J. Luisier36, H. Luo47, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, S. Malde52, G. Manca15,d, G.
Mancinelli6, N. Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G.
Martellotti22, A. Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38,
D. Martinez Santos34, D. Martinez Santos39, D. Martins Tostes2, A.
Massafferri1, R. Matev35, Z. Mathe35, C. Matteuzzi20, M. Matveev27, E.
Maurice6, A. Mazurov16,30,35,e, J. McCarthy42, R. McNulty12, B. Meadows57,52,
F. Meier9, M. Meissner11, M. Merk38, D.A. Milanes13, M.-N. Minard4, J. Molina
Rodriguez54, S. Monteil5, D. Moran51, P. Morawski23, R. Mountain53, I. Mous38,
F. Muheim47, K. Müller37, R. Muresan26, B. Muryn24, B. Muster36, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, T.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin29, T. Nikodem11, S. Nisar56, A. Nomerotski52, A. Novoselov32, A.
Oblakowska-Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41,
R. Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K.
Pal53, A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M.
Pappagallo48, C. Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M.
Patel50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, K. Petridis50, A. Petrolini19,i,
A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45,
D. Pinci22, S. Playfer47, M. Plo Casasus34, F. Polci8, G. Polok23, A.
Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A.
Powell52, J. Prisciandaro36, V. Pugatch41, A. Puig Navarro36, W. Qian4, J.H.
Rademacker43, B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, N.
Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1, S.
Ricciardi46, A. Richards50, K. Rinnert49, V. Rives Molina33, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35,
V. Romanovsky32, A. Romero Vidal34, J. Rouvinet36, T. Ruf35, H. Ruiz33, G.
Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d,
C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios34, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P. Schaack50, M.
Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M. Schmelling10, B.
Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B.
Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I.
Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35,
P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L. Shekhtman31, O.
Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K. Sobczak5, M.D.
Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci37, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, V. Syropoulos39, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, M. Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14,
G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona33,n, A. Vollhardt37, D. Volyanskyy10, D. Voong43,
A. Vorobyev27, V. Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12,
S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D.
Websdale50, M. Whitehead45, J. Wicht35, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50,p, F.F. Wilson46, J. Wishahi9,
M. Witek23, W. Witzeling35, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35,
Y. Xie47,35, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O.
Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov28, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The measurement of vector boson production permits a number of tests of
electroweak physics and of quantum chromodynamics (QCD) to be performed. In
particular, the angular acceptance of LHCb, roughly $2<\eta<5$ in the case of
the main tracking system where $\eta$ denotes pseudorapidity, complements that
of the general purpose detectors ATLAS and CMS. LHCb measurements provide
sensitivity to the proton structure functions at very low Bjorken $x$ values
where the parton distribution functions (PDFs) are not particularly well
constrained by previous data from HERA (see for example Ref. [1]).
The most straightforward decay modes in which the $\mathrm{W}^{\pm}$ and Z
bosons can be studied using the LHCb data are the muonic channels, $\rm
Z\rightarrow\upmu^{+}\upmu^{-}$ and
$\mathrm{W}^{+}\rightarrow\upmu^{+}\upnu_{\mu}$. Measurements of $\rm
Z\rightarrow\upmu^{+}\upmu^{-}$ and of $\rm Z\rightarrow\uptau^{+}\uptau^{-}$
using the LHCb data at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ have already
been presented [2, 3]. To complement these studies, the electron channels $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ and
$\mathrm{W}^{+}\rightarrow\mathrm{e}^{+}\upnu_{e}$, which offer statistically
independent samples with different sources of systematic uncertainties, are
examined.
The main difficulty with electron111The term “electron” is used generically to
refer to either $\mathrm{e}^{+}$ or $\mathrm{e}^{-}$. reconstruction in LHCb
is the energy measurement. A significant amount of material is traversed by
the electrons before they reach the momentum analysing magnet, and their
measured momenta are therefore liable to be reduced by bremsstrahlung. For low
energy electrons, the bremsstrahlung photons can frequently be identified in
the electromagnetic calorimeter and their energies added to the measured
momentum of the electron. However, in the case of $\mathrm{W}^{\pm}$ and Z
decays, the electrons are of high momentum and transverse momentum ($p_{\rm
T}$), so that the bremsstrahlung photons often overlap with the electrons. The
LHCb calorimeters were designed so as to optimise the the measurement of
photons and $\uppi^{0}$s from heavy flavour decays, whose transverse energy
($E_{\rm T}$) values are generally well below 10 GeV. As a consequence,
individual calorimeter cells saturate at $E_{\rm T}$ around
10$\mathrm{\,Ge\kern-1.00006ptV}$, so it is not possible to substitute the
calorimeter energy for the momentum measured using the spectrometer. We
therefore have a situation in which the electron directions are well
determined, but their energies are underestimated by a variable amount,
typically around 25%. Nevertheless, the available information can be used to
study certain interesting variables.
In this paper, we present a measurement of the cross-section for
$\mathrm{p}\mathrm{p}\rightarrow\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$
using the data recorded by LHCb in 2011 at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. Throughout this paper we use $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ to refer to the process $\rm
Z/\gamma^{*}\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ where either a virtual
photon or a $\rm Z$ boson is produced and decays to
$\mathrm{e}^{+}\mathrm{e}^{-}$. For consistency, the measurement is presented
in the same kinematic region as the recent measurement of $\rm
Z\rightarrow\upmu^{+}\upmu^{-}$ using the 2010 LHCb data at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ [2]: $2<\eta<4.5$ and $\mbox{$p_{\rm
T}$}>20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for the leptons and
$60<M<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the dileptons where
$M$ is the invariant mass. Since the rapidity of the Z boson can be determined
to a precision of $\sim$0.05, the rapidity distribution will be presented.
However, the $p_{\rm T}$ of the Z boson is poorly determined and its
distribution will not be discussed. A similar problem was encountered by the
D0 collaboration [4], who employed a new variable proposed in Ref. [5]
depending only on track angles
$\mbox{$\phi^{*}$}\equiv\tan\left(\frac{\phi_{\mathrm{acop}}}{2}\right)\left/\cosh\left(\frac{\Delta\eta}{2}\right)\approx\frac{\mbox{$p_{\rm
T}$}}{Mc}\,,\right.$ (1)
where $M$ and $p_{\rm T}$ refer to the lepton pair, $\Delta\eta$ and
$\Delta\phi$ are the differences in pseudorapidity and azimuthal angles
respectively between the leptons, and the acoplanarity angle is
$\phi_{\mathrm{acop}}=\pi-|\Delta\phi|$. The $p_{\rm T}$ of the Z boson is
correlated with $\phi^{*}$, and the resolution on $\phi^{*}$ is excellent,
with a precision better than 0.001. The measurement of $\phi^{*}$ presented
here therefore largely accesses the same physics as a measurement of the Z
$p_{\rm T}$ distribution. The measurement of the distribution of Z rapidity
(denoted $y_{\rm Z}$) is expected to show sensitivity to the choice of PDFs,
while $\phi^{*}$ is likely to be more sensitive to higher order effects in the
QCD modelling.
After a brief description of the detector, Sect. 3 describes the event
selection, and Sect. 4 outlines the determination of the cross-section. The
results are given in Sect. 5 followed by a short summary.
## 2 LHCb detector
The LHCb detector [6] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed primarily for the study of particles
containing $\mathrm{b}$ or $\mathrm{c}$ quarks. The detector includes a high
precision tracking system consisting of a silicon-strip vertex detector
surrounding the pp interaction region, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for hadrons and muons, and an impact
parameter resolution of 20$\,\upmu\rm m$ for tracks with high transverse
momentum. Charged hadrons are identified using two ring-imaging Cherenkov
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad (SPD) and preshower (PRS)
detectors, an electromagnetic calorimeter (ECAL) and a hadronic calorimeter
(HCAL). The acceptance of the calorimeter system is roughly $1.8<\eta<4.3$.
Muons are identified by a system composed of alternating layers of iron and
multiwire proportional chambers.
The trigger [7] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies full
event reconstruction. A significant improvement to the trigger was implemented
during August 2011 which affected the trigger efficiency for $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$. The data samples before and after
this change are treated separately and will be referred to as data sample I
and data sample II. These correspond to integrated luminosities of $581\pm
20$$\mbox{\,pb}^{-1}$ and $364\pm 13$$\mbox{\,pb}^{-1}$ respectively, yielding
a total of $945\pm 33$$\mbox{\,pb}^{-1}$.
## 3 Event selection
The $\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ sample is initially
selected by single-electron triggers, which require electrons to have an
$E_{\rm T}$ above a given threshold between 10 and
15$\mathrm{\,Ge\kern-1.00006ptV}$ depending on the data-taking period and
specific trigger. The $\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ selection
starts from a sample of $\mathrm{e}^{+}\mathrm{e}^{-}$ candidates with high
invariant mass, which is refined by requiring the following selection
criteria:
* •
At least one of the candidate electrons must be selected by a high-$E_{\rm T}$
electron trigger.
* •
The electrons are both required to have $\mbox{$p_{\rm
T}$}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and pseudorapidity in the range
$2.0<\eta<4.5$. The invariant mass of the $\mathrm{e}^{+}\mathrm{e}^{-}$ pair
should be greater than 40${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
* •
Requirements on calorimeter information are imposed to provide particle
identification (PID) of electrons. The particle must satisfy
$E_{\mathrm{ECAL}}/pc>0.1$, where $p$ is the particle momentum, with
bremsstrahlung correction if available, and $E_{\mathrm{ECAL}}$ is the ECAL
energy associated with the particle. The particle is required to lie within
the HCAL acceptance and to satisfy $E_{\mathrm{HCAL}}/pc<0.05$, where
$E_{\mathrm{HCAL}}$ is the HCAL energy associated with the particle. The
energy in the preshower detector associated with the particle is required to
satisfy $E_{\mathrm{PRS}}>50$$\mathrm{\,Me\kern-1.00006ptV}$. These
requirements impose an electromagnetic shower profile, while being loose
enough to maintain a high electron efficiency despite the effects of
calorimeter saturation and bremsstrahlung.
* •
If more than one $\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ candidate
satisfies the above requirements in an event, just one candidate is used,
chosen at random. This only affects around 0.5% of cases, and in all instances
the multiple candidates share one daughter.
A sample of same-sign $\mathrm{e}^{\pm}\mathrm{e}^{\pm}$ combinations, subject
to the same selection criteria, is used to provide a data-based estimate of
background. The main background is expected to arise from hadrons that shower
early in the ECAL and consequently fake the signature of an electron. These
will contribute approximately equally to same-sign and opposite-sign pairs.
The contribution from semileptonic heavy flavour decays should be similar to
the small level ($\sim 0.2\%$) estimated for the $\rm
Z\rightarrow\upmu^{+}\upmu^{-}$ channel [2]; in any case, subtracting the
same-sign contribution should account for most of this effect.
Simulated event samples of $\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ with
$M(\mathrm{e}^{+}\mathrm{e}^{-})>40$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
are also used to assess some efficiencies as discussed below. Simulated
samples of $\rm Z\rightarrow\uptau^{+}\uptau^{-}$ and of
$\mathrm{t}\overline{}\mathrm{t}$ are used to assess possible background
contributions. For the simulation, pp collisions are generated using Pythia
6.4 [8] with a specific LHCb configuration [9] and the CTEQ6L1 PDF set [10].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [11, *Agostinelli:2002hh] as
described in Ref. [13]. Simulated samples based on different versions of GEANT
and of the detector model are employed, which allows the reliability of the
simulation to be assessed. The simulated events are then reconstructed in the
same way as the data, including simulation of the relevant trigger conditions.
The invariant mass distribution of the selected candidates is shown in Fig. 1.
The distribution falls off abruptly above the Z mass and is spread to lower
masses by bremsstrahlung. Good agreement in shape is observed between data and
the simulation sample used in the data correction; this will be further
discussed below. The background estimated from same-sign events amounts to
4.5% of the total number of $\mathrm{e}^{+}\mathrm{e}^{-}$ candidates. The
backgrounds from $\uptau^{+}\uptau^{-}$ and $\mathrm{t}\overline{}\mathrm{t}$
events are estimated to be around 0.1% and are neglected.
Figure 1: Invariant mass distribution of $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ candidates. The data are shown as
points with error bars, the background obtained from same-sign data is shown
in red (dark shading), to which the expectation from signal simulation is
added in yellow (light shading). The $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ simulated distribution has been
normalised to the (background-subtracted) data.
## 4 Cross-section determination
In a given bin of Z rapidity or $\phi^{*}$, the cross-section is calculated
using
$\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-})=\frac{N(\mathrm{e}^{+}\mathrm{e}^{-})-N(\mathrm{e}^{\pm}\mathrm{e}^{\pm})}{\epsilon_{\mathrm{GEC}}\cdot\epsilon_{\mathrm{trig}}\cdot\epsilon_{\mathrm{track}}\cdot\epsilon_{\mathrm{kin}}\cdot\epsilon_{\mathrm{PID}}\cdot\int\mathcal{L}\mathrm{d}t}\cdot
f_{\mathrm{FSR}}\cdot f_{\mathrm{MZ}}\;\;,$ (2)
where $N(\mathrm{e}^{+}\mathrm{e}^{-})$ is the number of Z candidates selected
in data, $N(\mathrm{e}^{\pm}\mathrm{e}^{\pm})$ is the background estimated
from the number of same-sign candidates and $\int\mathcal{L}\mathrm{d}t$ is
the integrated luminosity. The cross-section
$\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-})$ denotes the product of the
inclusive production cross-section for the $\rm Z$ or $\gamma^{*}$ and the
branching ratio to $\mathrm{e}^{+}\mathrm{e}^{-}$. The meaning and estimation
of the other factors are described below. The values obtained for each,
averaged over the acceptance, are summarised in Table 1.
Table 1: Quantities entering into the cross-section determination, averaged over the range of Z rapidity used. | Data sample I | Data sample II
---|---|---
$\int\mathcal{L}\mathrm{d}t\;[\mbox{\,pb}^{-1}]$ | $581\pm 20$ | $364\pm 13$
$\epsilon_{\mathrm{GEC}}$ | $0.947\pm 0.004$
$\epsilon_{\mathrm{trig}}$ | $0.715\pm 0.021$ | $0.899\pm 0.003$
$\epsilon_{\mathrm{track}}$ | $0.913\pm 0.015$
$\epsilon_{\mathrm{kin}}$ | $0.500\pm 0.007$
$\epsilon_{\mathrm{PID}}$ | $0.844\pm 0.011$
$f_{\mathrm{FSR}}$ | $1.049\pm 0.005$
$f_{\mathrm{MZ}}$ | $0.967\pm 0.001$
The luminosity is determined as described in Ref. [14] and has an uncertainty
of $3.5\%$. The factor $f_{\mathrm{FSR}}$ accounts for the effects of final-
state electromagnetic radiation, correcting the measurement to the Born level.
As in the $\rm Z\rightarrow\upmu^{+}\upmu^{-}$ analysis [2] it is determined
using Photos [15] interfaced to Pythia [8], with Horace [16] used as a cross-
check. An overall systematic uncertainty of 0.5% is assigned to this
correction [17]. The factor $f_{\mathrm{MZ}}$ corrects for
$\mathrm{e}^{+}\mathrm{e}^{-}$ events outside the mass range
$60<M(\mathrm{e}^{+}\mathrm{e}^{-})<120$$\mathrm{\,Ge\kern-1.00006ptV}$ which
pass the event selection, and is estimated from simulation by examining the
true mass for selected events.
The probability for a $\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ event to
satisfy the trigger and selection requirements is given by the product of the
efficiency factors, $\epsilon$, as described below.
* •
Global event cuts (GEC) are applied in the trigger in order to prevent very
large events from dominating the processing time. Their efficiency for
selecting signal events is given by $\epsilon_{\mathrm{GEC}}$. In the $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ case, the most important requirement
is on the multiplicity of SPD hits, $N_{\mathrm{SPD}}\leq 600$. This is
strongly correlated with the number of primary vertices reconstructed in the
event. The inefficiency is assessed by comparing with $\rm
Z\rightarrow\upmu^{+}\upmu^{-}$ candidates recorded in the same running period
using a dimuon trigger for which a less stringent requirement of 900 hits is
imposed. A correction is made for the small difference in the numbers of SPD
hits associated with the electrons and muons themselves. This procedure is
adopted for each number of reconstructed primary vertices and the results are
combined to obtain the overall efficiency.
* •
The trigger efficiency for events passing the final selection,
$\epsilon_{\mathrm{trig}}$, is determined from data. A sample of events
triggered independently of the $\mathrm{e}^{+}$ is identified and used to
determine the efficiency for triggering the $\mathrm{e}^{+}$, and likewise for
the $\mathrm{e}^{-}$. Using the total numbers of candidates for which the
single electron trigger is satisfied at each stage by the $\mathrm{e}^{+}$
($N^{+}$), by the $\mathrm{e}^{-}$ ($N^{-}$) and by both ($N^{+-}$), the
efficiency for triggering the $\mathrm{e}^{+}$ is given by
$\varepsilon^{+}=N^{+-}/N^{-}$. The overall efficiency is then taken to be
$\varepsilon^{-}+\varepsilon^{+}-\varepsilon^{-}\varepsilon^{+}$ assuming that
the $\mathrm{e}^{+}$ and $\mathrm{e}^{-}$ are triggered independently. The
procedure is validated on simulated events. The determination is performed
separately in each bin of Z rapidity and $\phi^{*}$. In all cases, the
statistical uncertainty on the efficiency is taken as a contribution to the
systematic uncertainty on the measurement.
* •
The track-finding efficiency, $\epsilon_{\mathrm{track}}$, represents the
probability that both of the electrons are successfully reconstructed. The
simulation is used to determine the track-finding efficiency, in bins of Z
rapidity and $\phi^{*}$, by calculating the probability that, in a $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ event whose generated electrons lie
within the kinematic acceptance, both of the electrons are associated with
reconstructed tracks that satisfy the track quality requirements, but not
necessarily the kinematic requirements. Its statistical precision is
propagated as a contribution to the systematic uncertainty.
This efficiency is checked in data using a tag-and-probe approach. One
electron is tagged using the standard requirements, and a search is made for
an accompanying cluster of electromagnetic energy having a high $E_{\rm T}$
and forming a high invariant mass with the tag electron. If such a cluster has
no associated track it provides evidence of a failure to reconstruct the other
electron. This sample contains significant background, which can be
discriminated by examining the $p_{\rm T}$ distribution of the tag electron
for cases where the photon candidate is and is not isolated. The $p_{\rm T}$
distribution of the electrons in signal events in data displays a clear
shoulder extending to $\sim$ 45${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ while
that for background falls monotonically, as shown in Fig. 2. The number of
signal-like events in which a cluster is not associated with a track can be
used to estimate a tracking efficiency, and the ratio of efficiencies between
data and simulation is applied as a correction to the tracking efficiency. The
precision of the test is taken to define a systematic uncertainty, assumed to
be fully correlated between bins of rapidity and $\phi^{*}$.
Figure 2: Distribution of $p_{\rm T}$ for the “tag” electron in cases where
an isolated cluster of energy of high $E_{\rm T}$ is found in the
electromagnetic calorimeter. This is fitted with two components obtained from
data, the $\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ signal whose shape is
taken from those candidates where the cluster is associated with an identified
electron track, and background whose shape is obtained from candidates where
the cluster is not isolated.
* •
The kinematic efficiency, $\epsilon_{\mathrm{kin}}$, represents the
probability that, in a $\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ event
whose generated electrons lie within the kinematic acceptance and are
associated with reconstructed tracks, both tracks pass the kinematic selection
requirements $2<\eta<4.5$ and $\mbox{$p_{\rm
T}$}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The efficiency is estimated
from simulation, with its statistical precision being treated as a
contribution to the systematic uncertainty.
This determination relies on a correct simulation, which can be tested using
data. For example, underestimation of the amount of material in the simulation
would cause a discrepancy between data and simulation in the $p_{\rm T}$
distributions of the electrons or the reconstructed mass spectrum shown in
Fig. 1. By comparing the shapes of the reconstructed mass spectrum and other
kinematic distributions in data with different simulation samples, a
systematic uncertainty on the momentum scale and hence on the kinematic
efficiency is assigned. This is combined with the statistical uncertainty
mentioned above, with the systematic contribution taken to be fully correlated
between bins of rapidity and $\phi^{*}$.
* •
The PID efficiency, $\epsilon_{\mathrm{PID}}$, represents the probability
that, in a $\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ event with
reconstructed electron tracks satisfying the kinematic requirements, both
tracks fulfil the calorimeter energy requirements for identified electrons.
This includes the probability that the tracks are within the calorimeter
acceptance and have been successfully associated with calorimeter information.
Because of the acceptance contribution, the efficiency has a strong dependence
on the Z rapidity. This dependence is taken from simulation, while the overall
normalisation of the PID efficiency is estimated directly from data, using a
tag-and-probe method.
Starting from a sample which requires just one high $p_{\rm T}$ electron,
events are selected by applying the usual criteria except that only one of the
$\mathrm{e}^{+}$ and $\mathrm{e}^{-}$ (the “tag”) is required to pass the
calorimeter-based electron identification requirements. The other track is
used as a “probe” to test the PID efficiency. The requirement of only one
identified electron admits a significant level of background, which is
assessed similarly to the tracking efficiency by examining the $p_{\rm T}$
distribution of the tag or alternatively the $p_{\rm T}$ of the probe electron
or the invariant mass of the two particles. The size of the signal component
can be used to define the number of Z events which fail the PID, and hence to
determine the PID efficiency and its uncertainty.
A systematic uncertainty is also assigned to the same-sign background
subtraction. The assumption that same-sign $\mathrm{e}^{\pm}\mathrm{e}^{\pm}$
combinations model background in $\mathrm{e}^{+}\mathrm{e}^{-}$ events is
tested by selecting events which satisfy all criteria except that one of the
particles fails the calorimeter energy requirements. This sample should be
dominated by background, and shows an excess of $\sim$8% of opposite-sign
events over same-sign events. Accordingly a systematic uncertainty amounting
to 8% of the number of same-sign events is assigned to the measurements.
## 5 Results
Using the efficiencies described above, the event yields detailed in Table 2
and Eq. (2) separate cross-section measurements for the two data-taking
periods are obtained. Since these are in good agreement, the results are
combined using a weighted average, and assuming their uncertainties are fully
correlated apart from the statistical contribution and the uncertainty in the
trigger efficiency. Data sample II has a smaller integrated luminosity but a
higher and more precisely estimated trigger efficiency. The weighting of the
two samples is chosen to minimise the total uncertainty on the cross-section
integrated over Z rapidity. The values of the differential cross-sections
obtained are given in Table 2. Correlation matrices may be found in the
Appendix. The bin $4.25<y_{\rm Z}<4.5$ is empty in data, and is expected to
have close to zero detection efficiency since the calorimeter acceptance
extends only slightly beyond 4.25. Hence no measurement is possible. However,
the QCD calculations discussed below predict a cross-section below
$\sim$0.01$\rm\,pb$ in this bin, which is negligibly small, so comparisons
with the Z$\rightarrow$ $\upmu^{+}\upmu^{-}$ results or with theoretical
calculations in the range $2<y_{\rm Z}<4.5$ are still meaningful.
The cross-section integrated over Z rapidity is obtained by summing the cross-
sections of all bins of $y_{\rm Z}$, taking the uncertainties associated with
the GEC and the luminosity to be fully correlated between bins, along with
parts of the tracking, kinematic and PID efficiencies, and treating the other
contributions as uncorrelated. The cross-section is measured to be
$\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-})=76.0\pm 0.8\,(\mathrm{stat.})\pm
2.0\,(\mathrm{syst.})\pm 2.6\,(\mathrm{lumi.})\pm 0.4\,(\mathrm{FSR})\rm\,pb,$
where the first uncertainty is statistical, the second is the experimental
systematic uncertainty, the third is the luminosity uncertainty and the last
represents the uncertainty in the FSR correction. Since the results have been
corrected to the Born level using the factor $f_{\mathrm{FSR}}$, it is
possible to compare this measurement with that found in the Z$\rightarrow$
$\upmu^{+}\upmu^{-}$ analysis [2] using 37$\mbox{\,pb}^{-1}$ of data, namely
$76.7\pm 1.7\,(\mathrm{stat.})\pm 3.3\,(\mathrm{syst.})\pm
2.7\,(\mathrm{lumi.})$ pb. Accounting for correlated uncertainties, the ratio
of cross-sections is
$\frac{\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-})}{\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\upmu^{+}\upmu^{-})}=0.990\pm 0.024\,(\mathrm{stat.})\pm
0.044\,(\mathrm{syst.}).$
This may be regarded as a cross-check of the analyses. Assuming lepton
universality, the two cross-sections can be combined in a weighted average so
as to minimise the total uncertainty, yielding
$\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\ell^{+}\ell^{-})=76.1\pm 0.7\,(\mathrm{stat.})\pm
1.8\,(\mathrm{syst.})\pm 2.7\,(\mathrm{lumi.})\pm 0.4\,(\mathrm{FSR})\rm\,pb.$
A recent measurement in $\rm Z\rightarrow\uptau^{+}\uptau^{-}$ decays which
has a larger statistical uncertainty [3] can also be combined with the
electron and muon channels, yielding
$\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\ell^{+}\ell^{-})=75.4\pm 0.8\,(\mathrm{stat.})\pm
1.7\,(\mathrm{syst.})\pm 2.6\,(\mathrm{lumi.})\pm 0.4\,(\mathrm{FSR})\rm\,pb.$
Table 2: Event yields and measurements for the differential cross-section of
$\mathrm{p}\mathrm{p}\rightarrow\rm Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$
at $\sqrt{s}=$7 TeV as a function of Z rapidity, $y_{\rm Z}$, and of
$\phi^{*}$. The first uncertainty is statistical, the second and third are the
uncorrelated and correlated experimental systematic uncertainties
respectively, and the fourth is the uncertainty in the FSR correction. The
common luminosity uncertainty of 3.5% is not explicitly included here. The
results are given for the combined data sample. The right-hand column gives
the values used for the FSR correction factor.
$y_{\rm Z}$ | $N(\mathrm{e}^{+}\mathrm{e}^{-})$ | $N(\mathrm{e}^{\pm}\mathrm{e}^{\pm})$ | d$\sigma$/d$y_{\rm Z}$ [pb] | $f_{\mathrm{FSR}}$
---|---|---|---|---
2.00– | 2.25 | 988 | 40 | $13.6\pm 0.7\pm 0.4\pm 0.3\pm 0.1$ | $1.049\pm 0.004$
2.25– | 2.50 | 3064 | 121 | $39.4\pm 1.0\pm 0.6\pm 0.8\pm 0.2$ | $1.046\pm 0.002$
2.50– | 2.75 | 4582 | 202 | $56.7\pm 1.2\pm 0.7\pm 1.3\pm 0.3$ | $1.050\pm 0.002$
2.75– | 3.00 | 5076 | 214 | $63.2\pm 1.3\pm 0.8\pm 1.5\pm 0.3$ | $1.049\pm 0.002$
3.00– | 3.25 | 4223 | 181 | $59.9\pm 1.4\pm 0.8\pm 1.6\pm 0.3$ | $1.056\pm 0.002$
3.25– | 3.50 | 2429 | 135 | $43.8\pm 1.3\pm 0.8\pm 1.1\pm 0.2$ | $1.054\pm 0.003$
3.50– | 3.75 | 906 | 61 | $20.5\pm 1.0\pm 0.7\pm 0.6\pm 0.1$ | $1.030\pm 0.006$
3.75– | 4.00 | 143 | 18 | $5.9\pm 0.8\pm 0.5\pm 0.3\pm 0.1$ | $1.074\pm 0.029$
4.00– | 4.25 | 9 | 2 | $0.66\pm 0.44\pm 0.30\pm 0.04\pm 0.02$ | $1.074\pm 0.029$
4.25– | 4.50 | 0 | 0 | — |
$\phi^{*}$ | $N(\mathrm{e}^{+}\mathrm{e}^{-})$ | $N(\mathrm{e}^{\pm}\mathrm{e}^{\pm})$ | d$\sigma$/d$\phi^{*}$ [pb] | $f_{\mathrm{FSR}}$
---|---|---|---|---
0.00– | 0.05 | 9696 | 363 | $693\pm 10\pm 6\pm 17\pm 3$ | $1.059\pm 0.001$
0.05– | 0.10 | 4787 | 219 | $326\pm 7\pm 4\pm 8\pm 2$ | $1.047\pm 0.002$
0.10– | 0.15 | 2382 | 115 | $164\pm 5\pm 3\pm 4\pm 1$ | $1.039\pm 0.002$
0.15– | 0.20 | 1384 | 80 | $99.1\pm 4.0\pm 2.0\pm 2.2\pm 0.5$ | $1.043\pm 0.003$
0.20– | 0.30 | 1434 | 82 | $49.6\pm 2.0\pm 1.1\pm 1.0\pm 0.3$ | $1.042\pm 0.003$
0.30– | 0.40 | 707 | 39 | $25.5\pm 1.4\pm 0.8\pm 0.6\pm 0.1$ | $1.049\pm 0.004$
0.40– | 0.60 | 583 | 41 | $10.8\pm 0.7\pm 0.4\pm 0.3\pm 0.1$ | $1.052\pm 0.005$
0.60– | 0.80 | 217 | 13 | $4.05\pm 0.38\pm 0.20\pm 0.09\pm 0.03$ | $1.054\pm 0.005$
0.80– | 1.00 | 91 | 9 | $1.41\pm 0.23\pm 0.11\pm 0.03\pm 0.02$ | $1.051\pm 0.009$
1.00– | 2.00 | 119 | 9 | $0.41\pm 0.06\pm 0.03\pm 0.01\pm 0.02$ | $1.035\pm 0.011$
The results may be compared with theoretical calculations similar to those
used in the interpretation of the $\rm Z\rightarrow\upmu^{+}\upmu^{-}$
analysis [2]. These calculations are performed at NNLO (${\cal
O}(\alpha_{\scriptscriptstyle S}^{2})$) with the program FEWZ [18] version
2.1.1 and using the NNLO PDF sets of MSTW08 [19], NNPDF21 [20] or CTEQ (CT10
NNLO) [21, *Nadolsky:2012ia]. In Fig. 3 we present the measured cross-section
and in Fig. 4(a) the measurements of the Z rapidity distribution, compared in
each case with the three calculations. The uncertainties in the predictions
include the effect of varying the renormalisation and factorisation scales by
factors of two around the nominal value, which is set to the Z mass, combined
in quadrature with the PDF uncertainties at 68% confidence level. The data
agree with expectations within the uncertainties.
Figure 3: Cross-section for $\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ measured in LHCb, shown as the
yellow band. The inner (darker) band represents the statistical uncertainty
and the outer the total uncertainty. The measurement corresponds to the
kinematic acceptance, $\mbox{$p_{\rm
T}$}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2<\eta<4.5$ for the
leptons and $60<M<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the
dilepton. The points show the various theoretical predictions with their
uncertainties as described in the text.
The differential cross-section as a function of $\phi^{*}$ is shown in Fig.
4(b), compared with the predictions of QCD to NNLO. Figure 5(a) displays the
ratios of these predictions to the measurements. The NNLO calculations tend to
overestimate the data at low $\phi^{*}$ and to underestimate the data at high
$\phi^{*}$. It is expected that the $\phi^{*}$ distribution, like that of
$p_{\rm T}$, is significantly affected by multiple soft gluon emissions, which
are not sufficiently accounted for in fixed order calculations. A QCD
calculation which takes this into account through resummation is provided by
Resbos [23, *Balazs:1997xd, *Landry:2002ix].222The P branch of Resbos is used
with grids for LHC at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ based on
CTEQ6.6. Another resummed calculation [26] has been compared with ATLAS data
[27] in the central region of rapidity, but is not yet available for the LHCb
acceptance. Alternatively, Powheg [28, *Alioli:2010qp] provides a framework
whereby a NLO QCD (${\cal O}(\alpha_{\scriptscriptstyle S})$) calculation can
be interfaced to a parton shower model such as Pythia which can approximate
higher order effects. Comparisons with these models, and with the LHCb version
[9] of Pythia [8] are shown in Fig. 5(b). The Resbos and Powheg distributions
are normalised to their own cross-section predictions, while the Pythia
distribution is normalised to the cross-section measured in data. It is seen
that Resbos gives a reasonable description of the $\phi^{*}$ distribution.
Powheg shows that the combination of a parton shower with the ${\cal
O}(\alpha_{\scriptscriptstyle S})$ QCD prediction significantly improves the
description of data in the low $\phi^{*}$ region, while in the high $\phi^{*}$
region the data are still underestimated. Pythia models the data reasonably
well. Overall, Resbos and Pythia seem to be the more successful of the
calculation schemes considered here.
Figure 4: Differential cross-section for $\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ as a function of (a) Z rapidity and
(b) $\phi^{*}$. The measurements based on the
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ LHCb data are shown as the yellow
bands where the inner (darker) band represents the statistical uncertainty and
the outer the total uncertainty. NNLO QCD predictions are shown as points with
error bars reflecting their uncertainties as described in the text.
Figure 5: Ratios of various QCD calculations to data for the differential
cross-section for $\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ as a function of $\phi^{*}$. The
measurements based on the $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ LHCb data
are shown as the yellow band centred at unity where the inner (darker) band
represents the statistical uncertainty and the outer the total uncertainty.
(a) NNLO QCD predictions shown as points with error bars reflecting their
uncertainties as described in the text. Small lateral displacements of the
theory points are made to improve clarity. (b) Ratios of the predictions of
Pythia, Resbos and Powheg to the data shown as points, with error bars that
reflect the statistical uncertainties in the predictions. For most points,
these errors are so small that they are not visible.
## 6 Summary
A measurement of the $\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ cross-section in pp collisions at
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ using 0.94$\mbox{\,fb}^{-1}$ of
data recorded by LHCb is presented. Although the characteristics of the LHCb
detector prevent a sharp mass peak from being seen, a clean sample of events
is identified with less than 5% background. Within the kinematic acceptance,
$\mbox{$p_{\rm T}$}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2<\eta<4.5$
for the leptons and $60<M<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for
the dielectron, the cross-section is measured to be
$\sigma(\mathrm{p}\mathrm{p}\rightarrow\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-})=76.0\pm 0.8\,(\mathrm{stat.})\pm
2.0\,(\mathrm{syst.})\pm 2.6\,(\mathrm{lumi.})\pm 0.4\,(\mathrm{FSR})\rm\,pb.$
The cross-section is also measured in bins of the rapidity of the Z and of the
angular variable $\phi^{*}$. The measurements of the rapidity distribution and
of the integrated cross-sections are consistent with previous measurements
using Z decays to $\upmu^{+}\upmu^{-}$ and $\uptau^{+}\uptau^{-}$ and show
good agreement with the expectations from NNLO QCD calculations. The
$\phi^{*}$ distribution, related to the Z $p_{\rm T}$ distribution, is better
modelled by calculations which approximately include the effects of higher
orders.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
## Appendix
Table A.1: Correlation coefficients for the differential cross-section of $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ at 7 TeV between bins of Z rapidity,
$y_{\rm Z}$. Both statistical and systematic contributions are included.
$y_{\rm Z}$ bin | 2.–2.25 | 2.25–2.5 | 2.5–2.75 | 2.75–3. | 3.–3.25 | 3.25–3.5 | 3.5–3.75 | 3.75–4. | 4.–4.25
---|---|---|---|---|---|---|---|---|---
2.00– | 2.25 | 1 | | | | | | | |
2.25– | 2.50 | 0.47 | 1 | | | | | | |
2.50– | 2.75 | 0.50 | 0.70 | 1 | | | | | |
2.75– | 3.00 | 0.51 | 0.70 | 0.75 | 1 | | | | |
3.00– | 3.25 | 0.50 | 0.69 | 0.74 | 0.75 | 1 | | | |
3.25– | 3.50 | 0.45 | 0.62 | 0.66 | 0.67 | 0.66 | 1 | | |
3.50– | 3.75 | 0.35 | 0.49 | 0.52 | 0.52 | 0.51 | 0.46 | 1 | |
3.75– | 4.00 | 0.20 | 0.27 | 0.29 | 0.29 | 0.29 | 0.26 | 0.20 | 1 |
4.00– | 4.25 | 0.05 | 0.07 | 0.08 | 0.08 | 0.08 | 0.07 | 0.06 | 0.03 | 1
Table A.2: Correlation coefficients for the differential cross-section of $\rm
Z\rightarrow\mathrm{e}^{+}\mathrm{e}^{-}$ at 7 TeV between bins of $\phi^{*}$.
Both statistical and systematic contributions are included.
$\phi^{*}$ bin | 0.–0.05 | 0.05–0.1 | 0.1–0.15 | 0.15–0.2 | 0.2–0.3 | 0.3–0.4 | 0.4–0.6 | 0.6–0.8 | 0.8–1. | 1.–2.
---|---|---|---|---|---|---|---|---|---|---
0.00– | 0.05 | 1 | | | | | | | | |
0.05– | 0.10 | 0.80 | 1 | | | | | | | |
0.10– | 0.15 | 0.73 | 0.67 | 1 | | | | | | |
0.15– | 0.20 | 0.63 | 0.58 | 0.53 | 1 | | | | | |
0.20– | 0.30 | 0.62 | 0.58 | 0.53 | 0.45 | 1 | | | | |
0.30– | 0.40 | 0.51 | 0.48 | 0.43 | 0.38 | 0.38 | 1 | | | |
0.40– | 0.60 | 0.46 | 0.43 | 0.39 | 0.34 | 0.34 | 0.28 | 1 | | |
0.60– | 0.80 | 0.34 | 0.31 | 0.29 | 0.25 | 0.25 | 0.20 | 0.18 | 1 | |
0.80– | 1.00 | 0.21 | 0.20 | 0.18 | 0.16 | 0.15 | 0.13 | 0.11 | 0.08 | 1 |
1.00– | 2.00 | 0.23 | 0.21 | 0.19 | 0.17 | 0.17 | 0.14 | 0.12 | 0.09 | 0.06 | 1
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|
arxiv-papers
| 2012-12-19T10:48:52 |
2024-09-04T02:49:39.430914
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R. B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J. J. Back, C. Baesso, V. Balagura, W. Baldini, R. J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J.\n Beddow, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M.\n Benayoun, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A. B\\\"uchler-Germann,\n I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M.\n Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic,\n H. V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier,\n G. A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P. N.\n Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De\n Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, H.\n Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori,\n A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz Batista, M. Dogaru, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, A. Jaeger, E. Jans, F. Jansen, P. Jaton, F. Jing, M.\n John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson,\n T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, I. Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, H. Luo, A.\n Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martinez Santos, D. Martins Tostes, A. Massafferri, R.\n Matev, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J.\n McCarthy, R. McNulty, B. Meadows, F. Meier, M. Meissner, M. Merk, D. A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P.\n Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A. D. Nguyen, T. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, S. Nisar, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J. M. Otalora Goicochea, P. Owen, B. K. Pal, A. Palano,\n M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C. J.\n Parkinson, G. Passaleva, G. D. Patel, M. Patel, G. N. Patrick, C. Patrignani,\n C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D. L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet,\n T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E.\n Smith, M. Smith, K. Sobczak, M. D. Sokoloff, F. J. P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert,\n C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M. T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, R.\n Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G.\n Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi, M. Witek,\n W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "David Ward",
"url": "https://arxiv.org/abs/1212.4620"
}
|
1212.4724
|
# Event Identification in 3He Proportional Counters Using Risetime
Discrimination
T. J. Langford Department of Physics, University of Maryland, College Park
MD, 20742 USA Institute for Research in Electronics and Applied Physics,
University of Maryland, College Park MD, 20742 USA C. D. Bass111Current
Address: Thomas Jefferson National Accelerator Facility, Newport News VA,
23606 USA National Institute of Standards and Technology, Gaithersburg MD,
20899 USA E. J. Beise H. Breuer D. K. Erwin C. R. Heimbach J. S. Nico
###### Abstract
We present a straightforward method for particle identification and background
rejection in 3He proportional counters for use in neutron detection. By
measuring the risetime and pulse height of the preamplifier signals, one may
define a region in the risetime versus pulse height space where the events are
predominately from neutron interactions. For six proportional counters
surveyed in a low-background environment, we demonstrate the ability to reject
alpha-particle events with an efficiency of 99%. By applying the same method,
we also show an effective rejection of microdischarge noise events that, when
passed through a shaping amplifier, are indistinguishable from physical events
in the counters. The primary application of this method is in measurements
where the signal-to-background for counting neutrons is very low, such as in
underground laboratories.
###### keywords:
Helium proportional counters , Neutron detectors , Thermal neutrons , Pulse
shape discrimination , Particle identification , Microdischarges
††journal: Nuclear Instruments and Methods A
## 1 Introduction
Helium-3 proportional counters are used as neutron detectors in a variety of
fields in physics such as neutron scattering and nuclear and particle physics,
particularly those experiments that operate underground in a low-background
environment. They also have a commercial application as neutron detectors in
health physics, medical physics, and nuclear safeguards. 3He has the
advantages of a large thermal neutron capture cross section, easily detected
charged reaction products, and low gamma-ray sensitivity. Although the basic
properties of 3He proportional counters have been known for decades [1], more
recent applications in counting low rates of neutrons has spurred work in
better understanding their backgrounds. Methods for improving background
rejection in the counters [2, 3] can be a significant benefit for those
experiments or applications where the neutron signal-to-background ratio is
particularly poor.
Specifically, alpha-particle decays are a pernicious background for 3He
detectors used in low counting rate applications. The rate of alpha-particle
decays originating from the walls of the counter can be comparable to, or even
exceed, the rate of neutron captures. Alpha particles arise primarily from
daughters in the uranium and thorium decay chains. The amount of energy
deposited in the detector can be from essentially zero to several MeV,
spanning the region of the neutron capture products. Thus, energy
discrimination alone is not sufficient to reject background events from alpha
particles. The method of risetime discrimination provides a straightforward
method to reject a large fraction of these events.
For experiments operating in deep underground environments, cosmic-ray induced
neutron events can be a particularly difficult background because one cannot
turn off the source of the background [4, 5]. The experiments can only reduce
the ambient neutron background through shielding and precisely quantifying the
remaining neutron flux [6]. Solar neutrino experiments have used 3He
proportional counters as triggers for neutral current interactions and
monitors of muon-induced neutron backgrounds [7]. Dark matter experiments are
particularly susceptible to neutron backgrounds, due to neutron recoil events
that can mimic WIMP dark matter interactions. To this end, many experiments
use 3He proportional counters to monitor and characterize the neutron flux at
their experimental site [8, 9].
In one notable example, the Sudbury Neutrino Observatory (SNO) collaboration
took considerable effort in constructing ultra-low background 3He proportional
counters containing low concentration of alpha emitters. That effort involved
manufacturing proportional counters from materials with low concentrations of
uranium and thorium, as well as working in a radon-free environment to prevent
surface contamination [10, 11]. For experiments that rely upon neutron
counters as their triggers, these alpha-particle interactions in the
proportional counter can be a leading source of background. Therefore,
experiments such as SNO have distinguished alpha-particle background from
neutron signals either through risetime analysis [12, 13, 14] or chi-squared
fitting of the entire pulse waveforms [15].
More generally, fast neutron spectroscopy can be done directly using a 3He
proportional counter [16, 17], and it is also routinely done using Bonner
spheres, where fast neutrons are moderated in a hydrogenous material and a
fraction of those thermalized neutrons may be captured by a central 3He
proportional counter [18, 19]. Another technique for fast neutron detection
using 3He proportional counters is capture-gated spectroscopy [20]. The fast
neutrons recoil from protons in an active hydrogenous moderator; the moderated
neutron diffuses in the medium and may be subsequently captured on 3He. The
time-delayed capture signifies that the preceding interaction was a fully
thermalized neutron. This allows for an accurate measurement of the incident
particles energy, as well as a highly efficient method for background
reduction.
Work in nuclear safeguards has made helium-based neutron detectors the focus
of detecting special nuclear material. Most neutron-detecting portal monitors
use a 3He proportional counter surrounded by a passive neutron moderator [21].
Because the monitors are searching for very low levels of neutron activity,
the technical requirements for these detectors and those operating in an
underground environment are very similar. In this case, background events are
false positives, and hence alpha particles and noise in the detectors are a
detriment to the effectiveness of the portal monitors [22].
Improving the particle identification and rejection of events in 3He
proportional counters that are not neutrons has clear relevance to these
applications. More specifically, this collaboration is constructing a fast
neutron spectrometer [23] that employs a large number of 3He counters, and
hence it is critical to understand all the background contributions in each
counter that may be part of the spectrometer. Approximately 75 3He
proportional counters of the same model were surveyed as part of this study.
In this paper, we discuss how a combination of risetime and energy analysis
can significantly improve the identification of neutron capture events in a
3He proportional counter. Section 2 briefly reviews the operation of 3He
proportional counters. In Section 3 we describe the experimental setup and
analysis method for testing 3He proportional counters. Section 4 gives the
approach for identifying the origin of various waveforms that may occur in a
3He proportional counters. We apply that method to quantify the rejection of
non-neutron events for a specific counter in Section 5. Section 6 describes an
auxiliary measurement to quantify alpha-particle rates from the proportional
counter body, and Section 7 summarizes the results.
## 2 3He proportional counters
### 2.1 Principles of operation
3He provides an excellent medium for detecting thermal neutrons due to the
high capture cross section ($5330\times 10^{-24}$ cm2 ) and a final state that
consists of charged particles, as opposed to gamma rays. A neutron is captured
by a 3He nucleus, resulting in a proton and a triton, which share 764 keV of
kinetic energy
$n\;+\;^{3}\textrm{He}\rightarrow p\;+\;t\;+\;764\textrm{\;keV.}$ (1)
These heavy charged particles ionize the gas in the proportional counter,
creating electron-ion pairs that are proportional to the energy deposited. For
cylindrical proportional counters, the electrons drift towards a central anode
wire, and when the electric field gradient reaches a critical level, begin to
avalanche. Thus, the detected current is created from the drifted charge.
Because the neutron capture on 3He has a two body final state, the charged
particles will be emitted in opposite directions. If the ionization tracks are
parallel to the central anode wire, all of the charge will be collected at
approximately the same time, leading to a short risetime of the detected
signal. However, if the particles are emitted perpendicular to the anode wire,
one particle will be moving towards the wire while the other is moving away.
The collected charge will be spread out in time due to the radial variation
caused by the track geometry [24].
Typically, a 3He proportional counter is biased through a preamplifier, and
the resulting signals are sent through a shaping amplifier to be analyzed with
a multi-channel analyzer or peak-sensing analog to digital converter. However,
shaping the preamplifier signal smoothes out the slight deviations in signal
shape caused by the track geometry. With high-speed waveform digitizers, it is
possible to study the shape of preamplifier signals directly and apply digital
signal processing to identify the type of incident radiation.
### 2.2 Specific energy loss and particle range
The specific energy loss of charged particles with charge $Z$, mass $m$, and
kinetic energy $E_{kin}$, is approximately proportional to
$-\frac{dE}{dx}\propto\frac{Z^{2}m}{E_{kin}}.$ (2)
Thus, the track length for a given $E_{kin}$ decreases with increasing mass
and charge of the projectile. For neutron capture on 3He, the proton is
emitted with 573 keV and the triton with 191 keV. At these energies and using
the method of Anderson et al. [25], the proton has a range of 1.88 mg/cm2 and
the triton has a range of 0.71 mg/cm2 in our proportional counters. Because
the proton and triton are back-to-back, this yields a total track length of
2.6 mg/cm2, compared to a 764 keV alpha particle, for example, with a range of
1.0 mg/cm2.
Beta emitters, electrons from $\gamma$-interactions, and cosmic rays leave
long tracks and deposit little energy resulting in small signals with a wider
range of risetimes. Alpha particles with much higher specific energy loss
leave shorter tracks and deposit more of their energy, which yield large
signals with a relatively fast risetime. These features can be exploited in
analysis to help identify the original radiation. Details about the energy
loss of charged particles and the resulting pulse shapes in proportional
counters can be found elsewhere [25, 26, 24, 27, 28].
## 3 Experimental setup
Figure 1: A schematic showing the setup for a single channel of our
measurements. Particles in the 3He proportional counters generate current
signals that go to the preamplifier. The waveform digitizer captures the
resulting long-tailed signals and records them for off-line analysis.
The experimental setup is shown in Fig. 1. We used 2.54 cm outer diameter,
46.3 cm active length, aluminum-bodied, cylindrical 3He proportional counters
manufactured by GE-Reuter Stokes222Certain trade names and company products
are mentioned in the text or identified in an illustration in order to
adequately specify the experimental procedure and equipment used. In no case
does such identification imply recommendation or endorsement by the National
Institute of Standards and Technology, nor does it imply that the products are
necessarily the best available for the purpose.. The 3He partial pressure in
the counters was 404 kPa (4 atm) with a buffer gas consisting of 111 kPa (1.1
atm) of krypton. Helium does not have a high stopping power for heavy charged
particles. Therefore, manufacturers tend to add in a buffer gas with higher
stopping power to ensure full energy deposition from the capture products. The
krypton increases the stopping power of the gas for charged particles but has
little effect on neutron capture.
Figure 2: Typical energy spectrum of neutron capture events in one of the 3He
proportional counters used in this study. The dominant feature is the full-
deposition peak at 764 keV. The two edges in the spectrum near 200 keV and 600
keV are related to partial energy deposition in the gas when either the proton
or the triton interacts in the wall.
The proportional counters were biased through a charge sensitive preamplifier,
and the traces were recorded for offline analysis by a 125 MSamples/s 12-bit
GaGe waveform digitizer or a 250 MSamples/s 12-bit CAEN waveform digitizer.
Data were taken at different voltage levels to test the variability of the
counters’ response. For these specific counters, 2000 V was a reasonable
operating voltage. An exploration of the effect of different counter bias
voltage indicated that there was little variation of the risetime
distribution.
Figure 2 shows a typical energy spectrum measured by one of the proportional
counters used in this work. Note that although the neutron capture reaction is
monoenergetic, there are still features in the energy spectrum related to when
one of the particles interacts in the counter wall, leading to reduced energy
deposition. This well-known wall effect is clearly seen in the two edges
around 200 keV and 600 keV in Fig. 2. Events below 200 keV are largely betas
liberated from the counter body by gammas from the 252Cf source.
### 3.1 Analysis method
The digitized waveforms were analyzed, and both the amplitude and risetime of
the signals were extracted. Figure 3 shows sample traces corresponding to
three different sources: a neutron event, an alpha-particle event, and a
microdischarge from the proportional counter. Microdischarges are electronic
artifacts originating from current leakage in the bias voltage breakdown
applied to the counter [29, 30, 31]. To determine the pulse height of the
waveform, one first removes the baseline offset by subtracting off the average
of the first few hundred samples of the trace, well before the pulse onset. To
remove high-frequency noise, a Gaussian smoothing routine was applied to the
waveform, and the amplitude was determined from the resulting wave. Because
the signals were sent though a charge-sensitive preamplifier, the energy of
the particle is proportional to the amplitude of the recorded signal. The fall
time of the signals were a property of the AC coupling and decay time of the
preamplifier. The detectors were calibrated using the thermal neutron capture
peak to set the energy scale.
For this work, the risetime was defined as the time difference between the
signal reaching 10% and 50% of its maximum amplitude. This is the same
interval used in Ref. [12]. This choice was based upon a reasonable separation
of the three signal types. If the interval was too small (less than 20%),
there was not adequate separation between the microdischarges and the real
signals. However, the interval could not be made too large, otherwise the slow
component of the risetime, due to the drift of positive ions [27], decreases
the separation of alpha-particle and neutron events. For both the alpha
particles and neutron traces shown in Fig. 3, this begins at about 75% of the
maximum signal amplitude.
Figure 3: Example of the raw preamplifier traces showing the risetimes from
three different sources. Shown here, in sequence of increasing risetime, are a
microdischarge (red), an alpha-particle trace (gray) and a neutron trace
(black).
## 4 Event identification through risetime analysis
### 4.1 Physical signals from source radiation
Events in the counters arise from other radiation sources or bias voltage
breakdown in the counter. Data were collected under a few different conditions
to characterize the 3He proportional counters’ response to different sources
of radiation. By displaying the risetime versus the pulse height of each
signal, it is possible to identify regions of this two-dimensional space where
events from different source radiation will occur. Figure 4 shows the
proportional counter response to thermal neutron, gamma, beta, alpha, and
microdischarge events.
(a)
(b)
(c)
(d)
(e)
Figure 4: Scatter plots showing the risetime versus pulse height from
different sources of radiation and microdischarges in 3He proportional
counters: a) neutron source data; b) gamma source data; c) beta source data;
d) alpha-particle background data; and e) microdischarges. The solid black
lines illustrate regions where the indicated events occur. Further description
is found in the text of Section 4.
Thermal neutrons were generated from neutrons from 252Cf that were moderated
by a 5 cm thick piece of polyethylene placed between the source and
proportional counter. Figure 4a illustrates the region in both risetime and
pulse height where the counter detects the neutron-capture reaction products.
The large concentration of events around 764 keV correspond to the full energy
deposition. The risetime varies depending upon the orientation of the decay
products in the counter with respect to the central anode wire.
To test the counter’s response to beta and gamma radiation, 90Sr and 60Co
sources were placed directly above the counter in separate measurements.
Figures 4b and 4c show that these interactions are comparatively low energy
that can be easily discriminated from neutron events. Note that there are
still neutron events from the ambient background in the laboratory where the
measurements were performed.
Figure 4d illustrates why alpha particles are the most difficult background to
address. Many events overlap with neutrons in both energy and risetime. For
this measurement, the source of the alpha particles was the trace amount of
uranium and thorium contamination in the wall of the proportional counter
body. To emphasize the alpha-particle events, the counter was wrapped in
boron-loaded silicone rubber to absorb most of the external ambient background
of thermal neutrons. As the typical alpha-particle detection rates for the
counters in this work are quite low, data were taken over the course of a few
days to acquire sufficient statistics. Note that with such a long run, there
is still some leakage of thermal neutron events into the spectrum.
We characterized the events from about 75 3He proportional counters for this
risetime study. We note that they had a wide range of alpha activity as well
as variations in spectral shape, even though they all came from the same
manufacturer. Some counters had rates consistent with or higher than the raw
aluminum expectation, discussed in Section 6, while some were considerably
lower. The counter with the lowest alpha-particle rate had approximately
$6\times 10^{-4}$ s-1, and the highest alpha-particle rate counter was
approximately $3\times 10^{-2}$ s-1. Certain counters exhibited a broad
spectrum of alphas with no peaks while others displayed peaks at energies
consistent with isotopes in the decay chains of uranium and thorium. During
construction of the counters, a thin layer of nickel was deposited onto the
inner surface of the cylinders. This was done by the manufacturer to minimize
the contribution of alpha particles to the background [32]. We speculate that
deviations in the thickness of this coating could cause the observed
variation.
In counters with high alpha-particle rates, events were observed with pileup
of a large alpha-like signal on top of a small gamma-like signal. These events
are possibly from correlated decays, where a nucleus alpha-decays to an
excited state, which subsequently emits a gamma. An example trace of this
process is shown in Fig. 5. The low-amplitude gamma signal will appear first
because its track is extended and closer the anode. The alpha particles
predominately originate from the counter walls, the furthest distance from the
anode, and therefore arrive after the gamma event. As a result, these signals
have longer risetimes, which could lead to misidentification as neutron-like
signals with this analysis. Other techniques, such as waveform fitting, should
be able to identify such events.
Figure 5: A sample trace attributed to an alpha-particle and gamma-ray event
in the proportional counter. These are possibly from a correlated decay, where
a nucleus alpha decays to an excited state, which then de-excites by emitting
a gamma.
### 4.2 Microdischarges
Figure 6: Risetime versus pulse height data from a 3He proportional counter
placed underground at KURF. The black lines and labels serve to indicate the
predominate type of event in the region. The low neutron flux facilitates the
study of the internal alpha-particle contamination of the counters. This
counter has a broad alpha spectrum and a low microdischarge rate, as shown in
the labeled regions.
In the course of the measurements, a few counters were found to have spurious
signals with very rapid risetimes. After studying the counters in question,
the source was determined to be microdischarges from the high voltage
feedthrough to the grounded case of the counter. An example of such a signal
is shown in Fig. 3 along with neutron-capture and alpha-particle signals. This
is a known effect in 3He proportional counters, and a thorough treatment of
the origin of these signals can be found in Ref. [33]. The microdischarge is
seen by the preamp as a current pulse and is treated as a normal signal. When
sent through a shaping amplifier, there is no way to distinguish a discharge
from a signal generated by an incident particle. By using the preamplifier
signal, it is possible to measure the fast risetime of these spurious signals
and discriminate against them. Figure 4e shows an example of a detector with a
clear band of fast ($<100$ ns) risetime events that span a wide range of pulse
height. The rate of such events may vary significantly among proportional
counters, even for counters made by the same manufacturer.
## 5 Non-neutron event rejection using energy and risetime discrimination
Utilizing the energy and risetime of these signals provides a straightforward
and fast method to characterize the shape and origin of events in a counter,
as well as a rationale to define cuts that minimize backgrounds specific to a
given application. Figure 6 is a risetime versus pulse height plot from one of
the 3He proportional counters placed underground. The black lines indicate
where one may apply cuts in an analysis to isolate the different events. Two
methods of performing cuts on the data proved useful in this work and are
described in this section. First, one may place restrictive cuts that define
an essentially neutron-only region, This is particularly important for the
applications where the rate from the neutron signal is expected to be very
low. Second, one may place less restrictive cuts that eliminate
microdischarges and gamma and beta backgrounds but accept the contribution
from alpha-particle contamination.
### 5.1 Neutron-only cut region
In the risetime versus pulse height space, one can define a region essentially
free of alpha and beta/gamma backgrounds, as well as microdischarge noise.
This region is clearly shown in Fig. 6. Three cuts are required to isolate
neutron events: two diagonal cuts and one in pulse height only. The first
diagonal cut is defined by events above the upper edge of the alpha region.
The second diagonal cut is defined as events below the upper edge of the
distribution from neutron source data. Finally, an upper energy threshold is
made to reject events with energy above 0.8 MeV, which is the approximate end
of the neutron capture peak. These cuts define a triangular region which is
essentially free from alpha-particle and beta/gamma backgrounds and
microdischarge events.
To test leakage of alpha-particle events into the neutron-only region, we
operated several of the proportional counters in a low-neutron environment at
the Kimballton Underground Research Facility (KURF) in Blacksburg, VA. The
facility is located in an active limestone mine at a depth of 1450 meters
water equivalent and provides a good low-radioactivity counting environment
[34]. Six helium detectors were counted for approximately 16 hours. These
detectors showed a variety of alpha and microdischarge rates; the risetime
versus pulse height spectra from one of the counters is shown in Fig. 6.
It is possible to estimate the leakage of alpha events into the signal region
by looking at events above the neutron capture peak in Fig. 6. There should be
no neutrons in this region, but the alpha events should have the same
probability to spill into the cut-region. For the six detectors tested at
KURF, we see an average of 1 % of alpha events which are not cut by this
method. When these cuts are applied to neutron source data, we find that
approximately 55 % of the neutrons are rejected. The location of the cuts can
be refined to match distributions present in specific counters, as well as the
desired signal-to-background ratio.
### 5.2 Elimination of microdischarge noise only
After surveying the 75 proportional counters using the risetime method, we
selected counters that contained the lowest alpha-particle backgrounds to
minimize the contamination of the neutron signal region. This permits one to
focus on the microdischarge backgrounds. Because the microdischarge region is
well-separated from the neutron region, we can use this cut to eliminate the
microdischarge events while preserving the entire neutron region.
Without utilizing the information in the pulse risetime, there is nothing to
distinguish microdischarge events from alpha-particle events. Thus, more
proportional counters would have been categorized as high-background that are
low in alpha contamination but high in microdischarges. Counters that would
otherwise have been unsuitable for low background experiments can still be
used, as long as the microdischarge events are rejected.
## 6 Direct measurement of alpha-particle rates from 3He counter walls
To verify that the background attributed to alpha particles was being
correctly identified, we performed a separate measurement of the alpha
activity at the surface of a 3He proportional counter using a silicon
detector. These measured rates were compared with existing measurements [35,
36] and the internal alpha rates from the counters themselves. The comparison
will not exact because some parameters among the counters cannot be
controlled, such as the manufacturing process and knowledge of an absorbing
layer deposited on the inner surface.
Figure 7: Energy spectra recorded with a silicon surface barrier detector. The
red trace shows the spectrum from the outside surface of a 3He proportional
counter that had been scrubbed clean to remove surface contamination. The blue
trace shows a background spectrum obtained when the 3He counter was removed.
The broad spectrum of alpha-particle energies is consistent with bulk
contamination of the counter body material.
To measure the alpha-particle rate from the aluminum surface, a 3He
proportional counter was placed inside a vacuum chamber, and a silicon surface
barrier detector was placed on the surface. The chamber was lined with thin
plastic and evacuated to minimize events originating from the chamber and
radon in the atmosphere, respectively. The silicon detector signal was
processed through a preamp and shaping amplifier, and the peak heights were
registered on a multichannel analyzer. Data were collected under a number of
different conditions in order to identify sources of background. It is
important to differentiate between alphas that originate from the metallic
surface of the 3He counter and those that may come from the chamber, detector
mounting hardware, or the silicon detector itself. Contamination on the
surface the 3He counter body is a particular concern, and care was taken to
clean the surface with a mild abrasive and detergent before measurements were
performed.
Figure 7 shows the energy spectrum of particles from a 3He counter along with
a background spectrum when the counter was removed. To compare with values in
the literature, we summed the counts between 2 MeV and 10 MeV after background
subtraction when the counter was removed. Taking into account the detector
solid angle and scaling up to the full size of the proportional counter, a
typical alpha rate was $3.9\times 10^{-2}$ s-1. Using the internal surface
area of 370 cm2 for a counter, one obtains a count rate per area of $1.1\times
10^{-4}$ cm-2s-1. In Ref. [36], researchers tabulated alpha count rates per
area ranging from $1.4\times 10^{-5}$ to $1.2\times 10^{-4}$ cm-2s-1 from
commercial aluminum. Rates will vary widely due to several factors, such as
the supplier and surface contamination [37], but the values obtained with this
proportional counter are consistent with both the literature and our own
internal measurements. This gives additional confidence that the alpha
particle in the proportional counters are being correctly identified.
## 7 Summary
We have detailed a straightforward method to identify neutron, alpha, gamma
and beta, and microdischarge events from 3He proportional counter preamplifier
signals collected with waveform digitizers. By using the risetime and energy
information of each signal, one can isolate neutron capture events and
eliminate contamination of the neutron signal from the gamma and beta
backgrounds and microdischarges. We have verified that the dominant background
in the neutron risetime and energy window arises from alpha decays from the
counter body itself. We have identified a set of selection criteria to improve
significantly the neutron signal-to-background ratio in that region. If
counting statistics are sufficient for given experiment, or if one desires the
highest signal-to-background ratio, one may choose to sacrifice approximately
half of the neutron events in order to eliminate nearly all of the alpha-
particle events. In many low event rate applications, that may not be the
preferred option; in which case, this method will provide an important tool to
identify and mitigate the contribution of alpha backgrounds in 3He
proportional counters.
A fast neutron spectrometer utilizing six 3He proportional counters [23] was
placed underground at KURF. Using the risetime method to identify and reduce
backgrounds has led to a significant improvement of the experiment. In future
work, we will investigate the improvement in the neutron signal-to-background
from data acquired in a low-background environment.
## 8 Acknowledgments
We thank Vladimir Gavrin and Johnrid Abdurashitov of the Institute for Nuclear
Research - Russian Academy of Sciences for useful discussions. We acknowledge
the NIST Center for Neutron Research for the loan of the 3He proportional
counters used in this study. We also acknowledge KURF and Lhoist North
America, especially Mark Luxbacher, for providing us access to the underground
site and logistical support. The research has been partially supported by NSF
grant 0809696. T. Langford acknowledges support under the National Institute
for Standards and Technology American Recovery and Reinvestment Act
Measurement Science and Engineering Fellowship Program Award 70NANB10H026
through the University of Maryland.
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|
arxiv-papers
| 2012-12-19T16:22:52 |
2024-09-04T02:49:39.442802
|
{
"license": "Public Domain",
"authors": "T. J. Langford, C. D. Bass, E. J. Beise, H. Breuer, D. K. Erwin, C. R.\n Heimbach, J. S. Nico",
"submitter": "Thomas Langford",
"url": "https://arxiv.org/abs/1212.4724"
}
|
1212.4728
|
# Network Multibaker Maps, Information Theory and Stochastic Thermodynamics
Bernhard Altaner Jürgen Vollmer Max-Planck-Institut für Dynamik und
Selbstorganisation (MPIDS), 37077 Göttingen, Germany Fakultät für Physik,
Universität Göttingen, 37077 Göttingen, Germany
###### Abstract
Stochastic thermodynamics (ST) summarizes recent advances in nonequilibrium
statistical mechanics of systems featuring local thermodynamic equilibrium.
However, the mathematical description of ST is largely independent of this
thermodynamic context. Here, we introduce novel network multibaker maps as a
model system for reversible, chaotic dynamics. As a result, the mathematical
formalism of ST emerges from information theory, and is shown to be consistent
with previous work.
###### pacs:
05.45.-a, 05.70.Ln,05.40.-a, 89.70.Cf
## Introduction
_Stochastic thermodynamics_ (ST) is a recent approach to the statistical
mechanics of nonequilibrium systems Seifert (2008). Using both random
variables and ensemble properties it extends the notions of entropy and
entropy production to stochastic trajectories Maes (2004); Seifert (2005).
Using the notion of time-reversal, ST provides elegant derivations Seifert
(2012) of stochastic fluctuation Kurchan (1998); Lebowitz and Spohn (1999) and
work relations Jarzynski (1997); Crooks (1999). It also yields universal
results on thermodynamic efficiency of small machines Esposito _et al._
(2009) and the conversion of information to work Horowitz and Parrondo (2011);
Sagawa and Ueda (2012). Moreover, advanced experimental and simulation
techniques allow direct testing of these theoretical results Toyabe _et al._
(2010); Tietz _et al._ (2006); Ciliberto _et al._ (2010). ST relies on a
thermodynamically consistent interpretation of the stochastic process within a
framework of an assumed thermodynamic _local equilibrium_. Surprisingly, the
mathematical formulation of ST seems not to depend on the specific physical
context Seifert (2011); Esposito (2012). What is needed is the _Markov
property_ of the dynamics, though for certain relations even this assumptions
can be relaxed Speck and Seifert (2007).
In this work, we introduce the notion of _network multibaker maps_ (NMBM) and
take a dynamical system’s perspective on ST. Multibaker maps serve as a
paradigm for studies on the relation of microscopic dynamics, statistical
physics and transport phenomena Vollmer (2002); Colangeli _et al._ (2011).
They fulfill many of the generic features of chaotic dynamics: Reversibility,
hyberbolicity, homo- and heteroclinic orbits, a natural Markov partition and
ergodicity Hopf (1948); Gaspard (2005). Yet, their simple structure admits an
explicit calculation of the evolution of ensembles which are represented by
non-stationary phase-space densities Vollmer _et al._ (1997); Vollmer (2002).
Here, we explicitly calculate the full evolution of information-theoretical
quantities for NMBM. We distinguish three fundamental contributions, which are
in one-to-one correspondance to the entropy expressions used in ST. Further,
we show that our approach is consistent with previous work based on phase-
space contraction Rondoni _et al._ (2000); Evans and Searles (2002).
## Network multibaker maps
Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a connected graph with $N$
vertices $i\in\mathcal{V}:=[1,2,\ldots,N]$ and edges
$e\in\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$. Vertices represent the
fundamental states of some effective (for instance, thermodynamic)
description. Edges indicate possible transitions between these states. We
assume bi-directional edges, i.e.
$e:=(i,j)\in\mathcal{E}\Leftrightarrow\overline{e}:=(j,i)\in\mathcal{E}$, and
that each vertex is connected to itself, i.e. $(i,i)\in\mathcal{E},\,\forall
i\in\mathcal{V}$. Let
$\mathcal{V}_{i}:={\\{j\in\mathcal{V}:(i,j)\in\mathcal{E}\\}}$ be the set of
vertices that are connected to a state $i$. Denote by
$k_{i}:=\left|\mathcal{V}_{i}\right|$ the degree of vertex $i$. Associate with
each vertex $i$ a rectangular _cell_
$\mathcal{C}_{i}:=[0,1]\times\Pi_{i}\cdot[0,1]$ with area (Lebesgue measure)
$\Pi_{i}$ consisting of the set of phase-space points $x\in\mathcal{C}_{i}$
which belong to state $i$. The overall _phase space_ $\Gamma$ of the system is
the disjoint union $\Gamma:=\cup_{i=1}^{N}\mathcal{C}_{i}$.
Figure 1: Dynamics of the network multibaker map. For each of the $k_{i}$
neighbours vertices $j\in\mathcal{V}_{j}$ of vertex $i$, a horizontal strip
$C^{i}_{j}\subset\mathcal{C}_{i}$ of relative height $s^{i}_{j}$ is affine-
linearly mapped to a vertical strip
$\tilde{}\mathcal{C}^{i}_{j}\subset\mathcal{C}_{j}$ of width
$\tilde{s}^{i}_{j}$.
A NMBM, $\Phi:\Gamma\to\Gamma$, deterministically maps phase-space points to
adjacent cells. It is specified geometrically (cf. Fig. 1) by dividing each
cell $C_{i}$ into $k_{i}$ horizontal strips of finite relative height
$s^{i}_{j}>0$. More precisely, with the offset
$b^{i}_{j}=\sum_{k<j}s^{i}_{k}$:
$\mathcal{C}^{i}_{j}:=\begin{cases}[0,1]\times\Pi_{i}\cdot[b^{i}_{j},b^{i}_{j}+s^{i}_{j})&i\in\mathcal{V},j\in\mathcal{V}_{i},\\\
\emptyset&\text{else}.\end{cases}$ (1)
The dynamics maps each horizontal strip
$\mathcal{C}^{i}_{j}\subset\mathcal{C}_{i}$ to a vertical strip
$\tilde{}\mathcal{C}^{i}_{j}:=\Phi\mathcal{C}^{i}_{j}\subset\mathcal{C}_{j}$
in an affine-linear way:
$\tilde{}\mathcal{C}^{i}_{j}:=[\tilde{b}^{i}_{j},\tilde{b}^{i}_{j}+\tilde{s}^{i}_{j})\times\Pi_{j}\cdot[0,1]$
(2)
Analogously, $\tilde{s}^{i}_{j}$ denote the relative width of the vertical
strips and the offsets read $\tilde{b}^{i}_{j}=\sum_{k<i}\tilde{s}^{i}_{j}$.
Points in any strip are mapped affine-linearly, such that the horizontal
direction is contracted by a factor $\tilde{s}^{i}_{j}<1$ whereas the vertical
direction is expanded by a factor $(s^{i}_{j})^{-1}>1$. The multibaker map is
fully defined by specifying $\mathcal{G}$ and the numbers $\Pi_{i}$,
$s^{i}_{j}$ and $\tilde{s}^{i}_{j}$.
In deriving fluctuation relations in both stochastic and deterministic
dynamics, one applies a natural notion of time-reversal Lebowitz and Spohn
(1999); Rondoni _et al._ (2000); Maes and Netočný (2003); Seifert (2005)
which has its origins in the time-reversibility of fundamental physical laws.
Following Ref. Tel and Vollmer (2000) we are interested in dynamics with a
transformation $\mathcal{I}$ on $\Gamma$ which a) is an involution, b)
preserves phase-space volume $\left|\cdot\right|$ and c) facilitates time-
reversal:
$\displaystyle\mathcal{I}\circ\mathcal{I}$ $\displaystyle=\text{id},$ (3a)
$\displaystyle\left|\mathcal{I}^{-1}A\right|$
$\displaystyle=\left|A\right|,~{}\forall A\subset\Gamma,$ (3b)
$\displaystyle\Phi^{-1}x$
$\displaystyle=\left(\mathcal{I}\circ\Phi\circ\mathcal{I}\right)x,~{}\forall
x\in\Gamma.$ (3c)
_Properly thermostatted multibaker maps_ Tel and Vollmer (2000) obey
$\displaystyle\tilde{s}^{i}_{j}=s^{j}_{i},\,\forall i,j,$ (4)
such that Eq. (3) is fulfilled by the involution Vollmer (2002)
$\displaystyle\mathcal{I}:(x,y)$
$\displaystyle\mapsto(1-\Pi_{i}^{-1}y,\Pi_{i}(1-x)),\,\forall(x,y)\in\mathcal{C}_{i},$
(5)
which additionally is local each cell (cf. Fig. 2), i.e.
$\mathcal{I}\mathcal{C}_{i}=\mathcal{C}_{i},\,\forall i$ (6)
With the general definition
$\tilde{}\mathcal{C}^{i}_{j}:=\Phi\mathcal{C}_{i}\cap\mathcal{C}_{j}$, Eq. (4)
follows solely from Eqs. (3) and (6), which ensure that
$\tilde{}\mathcal{C}^{i}_{j}$ and $\mathcal{C}^{j}_{i}$ are interchanged by
the involution (cf. Fig. 2).
Figure 2: The multibaker dynamics feature a volume preserving involution
$\mathcal{I}$, cf. Eq. (5), which interchanges horizontal and vertical
stripes in one cell, i.e.
$\mathcal{I}\tilde{}\mathcal{C}^{i}_{j}=\mathcal{C}^{j}_{i}$ The dynamics
$\Phi$ is reversible, but generally not volume preserving.
## Densities and trajectories
In statistical physics, the probability of finding a system at some point
$x\in\Gamma$ is given by a time-dependent ensemble measure. We assume that
this ensemble is absolutely continuous with respect to the Lebesgue measure,
i.e. it is characterized by a phase-space density
$\varrho^{\left(\nu\right)}(x)$, where the superscript (ν) indicates discrete
time. This phase-space density evolves according to $\Phi$ and generically
develops an intricate structure as the number of iterations increases.
Practically, preparation and resolution of phase-space densities is restricted
to a coarse-grained level, which is represented by the vertices of
$\mathcal{G}$. Hence, we define a coarse-grained density
$\rho^{\left(\nu\right)}(x)$ to take the average value
$\rho^{\left(\nu\right)}_{i}$ of $\varrho^{\left(\nu\right)}$ on cell
$\mathcal{C}_{i}$:
$\displaystyle\rho_{i}^{\left(\nu\right)}$
$\displaystyle:=\Pi_{i}^{-1}\int_{\mathcal{C}_{i}}\varrho^{\left(\nu\right)}(x)\,\mathrm{d}x,$
(7a) $\displaystyle\rho^{\left(\nu\right)}(x)$
$\displaystyle:=\sum_{i}\chi_{i}(x)\rho^{\left(\nu\right)}_{i},$ (7b)
where $\chi_{i}$ is the characteristic or indicator function of
$\mathcal{C}_{i}$.
Due to the preparation procedure, fine- and coarse-grained ensembles agree
initially, i.e. $\varrho^{(0)}(x)\equiv\rho^{(0)}(x)$. To follow the evolution
of $\varrho^{(0)}$ to $\varrho^{(\nu)}$ we introduce trajectories
$\underline{\omega}$ as $(\nu+1)$-tuples with entries
$\omega_{k}\in\mathcal{V}$:
$\underline{\omega}:=\left(\omega_{0},\omega_{1},\ldots,\omega_{\nu}\right).$
(8)
The entry $\omega_{k}$ denotes the vertex visited at time $k$. Hence, a
trajectory is a finite string of a the symbolic sequence associated with the
(natural) Markov partition Beck and Schögl (1995).
Let the set $\tilde{}\mathcal{C}\left[\underline{\omega}\right]$ denote all
phase-space points that move together along a fixed trajectory
$\underline{\omega}$ and hence reside in cell $\mathcal{C}_{\omega_{\nu}}$ at
time $\nu$. Denoting by $\underline{\omega}^{(k)}$ the first $k\leq\nu$ steps
of $\underline{\omega}$, we define recursively:
$\displaystyle\tilde{}\mathcal{C}[\underline{\omega}^{(k)}]:=\Phi(\tilde{}\mathcal{C}[\underline{\omega}^{(k-1)}]\cap\mathcal{C}^{\omega_{k-1}}_{\omega_{k}}),\quad\forall\,0<k\leq\nu,$
(9)
where
$\tilde{}\mathcal{C}\left[\underline{\omega}^{(0)}\right]:=\mathcal{C}_{\omega_{0}}$.
One can easily verify that $\tilde{}\mathcal{C}[\underline{\omega}^{(k-1)}]$
is a vertical strip. Intersecting with the horizontal strip
$\mathcal{C}^{\omega_{k-1}}_{\omega_{k}}$ reduces its volume by a factor
$s^{\omega_{k-1}}_{\omega_{k}}<1$. Through contraction and expansion in
horizontal and vertical directions, respectively, $\Phi$ changes the volume by
another factor
$\tilde{s}^{\omega_{k-1}}_{\omega_{k}}/s^{\omega_{k-1}}_{\omega_{k}}$. Hence,
$\left|\mathcal{C}\left[\underline{\omega}^{(k)}\right]\right|=\tilde{s}^{\omega_{k-1}}_{\omega_{k}}\left|\mathcal{C}\left[\underline{\omega}^{(k-1)}\right]\right|$
and after iteration to $k=\nu$,
$\displaystyle\left|\mathcal{C}[\underline{\omega}^{(\nu)}]\right|=\Pi_{\omega_{\nu}}\prod_{k=1}^{\nu}\tilde{s}^{\omega_{k-1}}_{\omega_{k}}=\Pi_{\omega_{\nu}}\prod_{k=1}^{\nu}s^{\omega_{k}}_{\omega_{k-1}},$
(10)
where for the last equality we used Eq. (4).
Because the initial microscopic density $\varrho^{(0)}$ is uniform on each
cell, so is the density on $\tilde{}\mathcal{C}[\underline{\omega}]$ for any
$\underline{\omega}$. Denote by $\varrho[\underline{\omega};k]$ the density on
the $(\nu-k)$th pre-image
$\Phi^{\nu-k}\tilde{}\mathcal{C}[\underline{\omega}]$. Probability
conservation and uniform contraction imply
$\displaystyle\varrho[\underline{\omega};k]=(\eta^{\omega_{k-1}}_{\omega_{k}})^{-1}\varrho[\underline{\omega};k-1]$
(11)
where we introduced the contraction factor
$\eta^{\omega_{k-1}}_{\omega_{k}}:=\frac{\left|\tilde{}\mathcal{C}^{\omega_{k-1}}_{\omega_{k}}\right|}{\left|\mathcal{C}^{\omega_{k-1}}_{\omega_{k}}\right|}\equiv\frac{\Pi_{\omega_{k}}\tilde{s}^{\omega_{k-1}}_{\omega_{k}}}{\Pi_{\omega_{k-1}}s^{\omega_{k-1}}_{\omega_{k}}}.$
(12)
Iterating this condition, we find with Eq. (4) that the density
$\varrho^{(\nu)}(x)$ at a point $x$ with history $\underline{\omega}(x)$
obeys:
$\varrho^{(\nu)}(x)=\varrho\left[\underline{\omega}(x);\nu\right]=\rho_{\omega_{0}}\prod_{k=1}^{\nu}\left[\frac{s^{\omega_{k-1}}_{\omega_{k}}}{s^{\omega_{k}}_{\omega_{k-1}}}\right]\frac{\Pi_{\omega_{0}}}{\Pi_{\omega_{\nu}}}.$
(13)
Integrating $\varrho(x)$ over a cell $\Pi_{i}$ yields the probability
$\displaystyle
p_{i}^{(k)}:=\int_{\mathcal{C}_{i}}\varrho^{\left(k\right)}(x)\,\,\mathrm{d}x\,=\Pi_{i}\rho_{i}^{(k)},$
(14)
which evolves according to the _Master equation_ Beck and Schögl (1995)
$\displaystyle p_{i}^{(k)}=\sum_{j}\left[p_{j}^{(k-1)}s^{j}_{i}\right].$ (15)
This establishes the connection to _Markov chains_ and hence the (discrete)
formulation of ST Seifert (2012).
## Information and entropy
The Shannon entropy Shannon (1948) of a probability density $\varrho(x)$ on
$\Gamma$ is defined as
$\displaystyle\mathcal{S}\left[\varrho\right]:=-\int_{\Gamma}\varrho\,\log\varrho\,\,\mathrm{d}x.$
(16)
With the results (10) and (13) we can calculate the fine-grained entropy of
$\varrho^{\left(\nu\right)}(x)$ as a sum over all trajectories
$\underline{\omega}^{\left(\nu\right)}$ of length $\nu$ weighted with
$\left|\mathcal{C}[\underline{\omega}^{\left(\nu\right)}]\right|$
$\displaystyle
S_{\mathrm{fg}}^{(\nu)}:=\mathcal{S}\left[\varrho^{(\nu)}\right]=-\sum_{\underline{\omega}^{\left(\nu\right)}}\left[\left|\mathcal{C}[\underline{\omega}^{\left(\nu\right)}]\right|\times\left(\varrho[\underline{\omega};\nu]\log\varrho[\underline{\omega};\nu]\right)\right]$
$\displaystyle=\sum_{\underline{\omega}^{\left(\nu\right)}}\left[\mathbb{P}[\underline{\omega}]\left(\log\Pi_{\omega_{\nu}}-\log
p^{(0)}_{\omega_{0}}-\sum_{k=1}^{\nu}B^{\omega_{k-1}}_{\omega_{k}}\right)\right]$
(17)
where
$\displaystyle\mathbb{P}[\underline{\omega}^{\left(\nu\right)}]$
$\displaystyle\equiv\left|\mathcal{C}[\underline{\omega}^{\left(\nu\right)}]\right|\varrho[\underline{\omega};\nu]=p^{\left(0\right)}_{\omega_{0}}\prod_{k=1}^{\nu}s^{\omega_{k-1}}_{\omega_{k}},$
(18) $\displaystyle B^{i}_{j}$
$\displaystyle:=\log\,\left({s^{i}_{j}}/{s^{j}_{i}}\right).$ (19)
Similarly, we find
$\displaystyle S_{\mathrm{cg}}^{(\nu)}$
$\displaystyle:=\mathcal{S}\left[\varrho^{(\nu)}\right]=\sum_{\underline{\omega}^{\left(\nu\right)}}\left[\mathbb{P}[\underline{\omega}^{\left(\nu\right)}]\left(\log\Pi_{\omega_{\nu}}-\log
p^{(\nu)}_{\omega_{\nu}}\right)\right].$ (20)
## Trajectory functionals
The forms of Eqs. (17) and (20) suggest to write them as weighted averages of
trajectory functionals $s[\underline{\omega};\nu]$,i.e. :
$\displaystyle\left\langle\\!\left\langle
s\right\rangle\\!\right\rangle^{(\nu)}=\sum_{\underline{\omega}^{\left(\nu\right)}}\left[\mathbb{P}[\underline{\omega}]\,s[\underline{\omega};\nu]\right]$
(21)
With that we can write $S_{\mathrm{fg}}^{(\nu)}=\left\langle\\!\left\langle
s_{\mathrm{fg}}\right\rangle\\!\right\rangle^{\left(\nu\right)}$ and
$S_{\mathrm{cg}}^{(\nu)}=\left\langle\\!\left\langle
s_{\mathrm{cg}}\right\rangle\\!\right\rangle^{\left(\nu\right)}$, where the
functionals
$\displaystyle s_{\mathrm{fg}}[\underline{\omega},\nu]$
$\displaystyle=s_{\mathrm{int}}[\underline{\omega},\nu]+s_{\mathrm{vis}}[\underline{\omega},0]-s_{\mathrm{mot}}[\underline{\omega},\nu],$
(22a) $\displaystyle s_{\mathrm{cg}}[\underline{\omega},\nu]$
$\displaystyle=s_{\mathrm{int}}[\underline{\omega},\nu]+s_{\mathrm{vis}}[\underline{\omega},\nu],$
(22b)
are linear combinations of three fundamental functionals:
$\displaystyle s_{\mathrm{int}}[\underline{\omega},\nu]$
$\displaystyle=s_{\mathrm{int}}(\omega_{\nu})=\log\Pi_{\omega_{\nu}},$ (23a)
$\displaystyle s_{\mathrm{vis}}[\underline{\omega},\nu]$
$\displaystyle=s_{\mathrm{vis}}(\omega_{\nu},\nu)=-\log
p^{\left(\nu\right)}_{\omega_{\nu}},$ (23b) $\displaystyle
s_{\mathrm{mot}}[\underline{\omega},\nu]$
$\displaystyle=s_{\mathrm{mot}}[\underline{\omega}]=\sum_{k=1}^{\nu}B^{\omega_{k-1}}_{\omega_{k}}.$
(23c)
Note that the functionals $s_{\mathrm{int}}$ and $s_{\mathrm{vis}}$ depend on
the final state $\omega_{\nu}$ of the trajectory only, i.e.
$s[\underline{\omega};\nu]=s(\omega_{\nu};\nu)$. Hence, we say they have a
_local form_. Trajectory averages of local forms reduce to ensemble averages
via Eq. (15):
$\displaystyle\left\langle\\!\left\langle
s\right\rangle\\!\right\rangle^{(\nu)}=\left\langle
s\right\rangle^{\left(\nu\right)}:=\sum_{i}\left[p_{i}^{\left(\nu\right)}s(i;\nu)\right].$
(24)
Henceforth, we regard NMBM as a model for thermostatted reversible dynamics
Jepps and Rondoni (2010). The Markovian description (15) for the coarse-
grained ensemble on the discrete state establishes the connection to ST.
The _intrinsic entropy_ $s_{\mathrm{int}}$ accounts for the entropy of the
unobservable microscopic configurations. In the present dynamical context it
is calculated à la Boltzmann as the logarithm of a phase-space volume (23a).
In the thermodynamic context, it is the entropy of the local equilibrium
ensemble which reflects the state of the environment Seifert (2011). Hence,
the intrinsic entropy can be considered a “state variable” and in this respect
is closest to standard thermodynamics.
The entropy that is operationally accesible in experiments is the _visible
entropy_ $s_{\mathrm{vis}}$. It only depends on the present ensemble, i.e.
the $p_{i}^{(\nu)}$, which can be sampled by performing many measurements on
identical systems. Though it has local form and averages can be calculated
with Eq. (24), it depends on the ensemble and hence is not a state variable.
The conceptually most difficult term is the third functional,
$s_{\mathrm{mot}}$ related to the quantity $B^{i}_{j}$. In analogy to the
electromotance for electrical circuits Altaner _et al._ (2012), we call
$B^{i}_{j}$ the _motance_ of a transition $i\to j$. Physically, a finite
motance arises from a system-environment interaction. Unlike the affinity
$A^{i}_{j}=\log({p_{i}s^{i}_{j}}/({p_{j}s^{j}_{i}}))$, the motance $B^{i}_{j}$
only depends on the dynamics and not on the ensemble Schnakenberg (1976).
Consequently, we call $s_{\mathrm{mot}}$ the _accumulated motance_.
## Entropy variation and production
For each fundamental functional $s[\underline{\omega},\nu]$ there is an
associated fundamental variation
$\sigma[\underline{\omega},\nu]:=s[\underline{\omega},\nu]-s[\underline{\omega}^{(\nu-1)},\nu-1]$.
The fundamental variations
$\displaystyle\sigma_{\mathrm{int}}[\underline{\omega},\nu]$
$\displaystyle=\log{\Pi_{\omega_{\nu}}}-\log{\Pi_{\omega_{\nu-1}}},$ (25a)
$\displaystyle\sigma_{\mathrm{vis}}[\underline{\omega},\nu]$
$\displaystyle=-\log{p_{\omega_{\nu}}^{(\nu)}}+\log{p_{\omega_{\nu-1}}^{(\nu-1)}},$
(25b) $\displaystyle\sigma_{\mathrm{mot}}[\underline{\omega},\nu]$
$\displaystyle=B^{\omega_{\nu-1}}_{\omega_{\nu}}$ (25c)
can be used to construct the previously known fundamental forms of entropy
variation Schnakenberg (1976):
$\displaystyle\sigma_{\mathrm{sys}}[\underline{\omega},\nu]$
$\displaystyle=\sigma_{\mathrm{int}}[\underline{\omega},\nu]+\sigma_{\mathrm{vis}}[\underline{\omega},\nu],$
(26a) $\displaystyle\sigma_{\mathrm{tot}}[\underline{\omega},\nu]$
$\displaystyle=\sigma_{\mathrm{vis}}[\underline{\omega},\nu]+\sigma_{\mathrm{mot}}[\underline{\omega},\nu],$
(26b) $\displaystyle\sigma_{\mathrm{med}}[\underline{\omega},\nu]$
$\displaystyle=\sigma_{\mathrm{mot}}[\underline{\omega},\nu]-\sigma_{\mathrm{int}}[\underline{\omega},\nu].$
(26c)
The system entropy variation $\sigma_{\mathrm{sys}}$ is equivalent to the
change of coarse-grained entropy. It consists of the visible entropy of the
ensemble and the intrinsic entropy of the current state. Often, the latter is
forgotten Seifert (2011).
We can write the average of the _total entropy production_
$\sigma_{\mathrm{tot}}$ as a Kullback-Leiber divergence,
$\displaystyle\left\langle\\!\left\langle\sigma_{\mathrm{tot}}\right\rangle\\!\right\rangle^{(\nu)}=$
$\displaystyle\sum_{i,j}\left[p_{i}^{(\nu)}s^{i}_{j}\log\frac{p_{i}^{(\nu)}s^{i}_{j}}{p_{j}^{(\nu+1)}s^{j}_{i}}\right]\geq
0,$
which is always positive. It is the temporal variation of the difference
$\left\langle\\!\left\langle
s_{\mathrm{cg}}-s_{\mathrm{fg}}\right\rangle\\!\right\rangle$, which itself is
a Kullback-Leibler divergence. It characterizes the information gain of the
microscopic over the coarse-grained density. Its positive variation (i.e. its
monotonous growth) reflects the fact that one cannot resolve the ever-refining
features of the microscopic density.
The _entropy change in the medium_ $\sigma_{\mathrm{med}}$ consists of a
motance and an intrinsic part. It has been understood that the latter is
important for a consistent thermodynamic interpretation Seifert (2011, 2012).
From Eqs. (19) and (12) we see that it is directly related to phase-space
contraction:
$\sigma_{\mathrm{med}}[\underline{\omega}]=\log\frac{s^{\omega_{\nu-1}}_{\omega_{\nu}}}{s^{\omega_{\nu}}_{\omega_{\nu-1}}}-\log\frac{\Pi_{\omega_{\nu}}}{\Pi_{\omega_{\nu-1}}}=\log\eta^{\omega_{\nu-1}}_{\omega_{\nu}}$
(27)
For maps, the phase-space contraction per unit time, $\Lambda=\log J$ Ruelle
(1999); Rondoni and Mejia-Monasterio (2007), is given by the logarithm of the
Jacobian determinant
$J=\left|\frac{\,\mathrm{d}\Phi\,}{\,\mathrm{d}x\,}\right|$ of the dynamics.
In the current affine-linear case, $J|_{x}=\eta^{\omega_{k-1}}_{\omega_{k}}$
for $x\in\mathcal{C}^{\omega_{k-1}}_{\omega_{k}}$.
## Connection to the continuous description
For us, NMBM serve as a model for a _stroboscopic observation_ of physical
dynamics with a time lag $\tau_{\mathrm{obs}}$. However, stochastic
thermodynamics is usually formulated using _continuous-time Markov processes_
rather than Markov chains. In the continuous time limit, the Master equation
(15) becomes a differential equation
$\dot{p}_{i}(t)=\sum_{j}W^{j}_{i}p_{j}(t)$. Its infinitesimal generator
$W^{j}_{i}$ consists of transition rates $w^{i}_{j}$, which can be thought of
as the $\tau_{\mathrm{obs}}\to 0$ limit of transition probability divided by
$\tau_{\mathrm{obs}}$. The continuous-time versions of the fundamental entropy
and variation functionals have been identified previously Seifert (2005). They
resemble the discrete-time functionals with $s^{i}_{j}$ substituted by
$w^{i}_{j}$ and finite-time variations substituted by derivatives. However,
there are some subtleties to be taken care of. Firstly, in the continuous-time
description, the measure $\mathbb{P}[\underline{\omega}]$ of a finite
trajectory is zero and summing over trajectories involves integration over
stochastic jump times. Secondly, in the present case, the average value
$\left\langle\\!\left\langle\sigma\right\rangle\\!\right\rangle^{(\nu)}\equiv\left\langle\\!\left\langle
s[\underline{\omega},\nu]-s[\underline{\omega}^{(\nu-1)},\nu-1]\right\rangle\\!\right\rangle^{(\nu)}$
of a variation is the same as the change in one time-step of the averaged
entropies, $\left\langle\\!\left\langle
s\right\rangle\\!\right\rangle^{(\nu)}-\left\langle\\!\left\langle
s\right\rangle\\!\right\rangle^{\nu-1}$. This does not apply to the
continuous-time case, because averaging and taking the time-derivative do not
commute for non-stationary probabilities. Hence, temporal boundary-terms need
to be attributed for. Thirdly, we can rewrite the average total entropy
production in the following form:
$\displaystyle\left\langle\\!\left\langle\sigma_{\mathrm{tot}}\right\rangle\\!\right\rangle^{(\nu)}=$
$\displaystyle\sum_{i,j}\left[p_{i}^{(\nu)}s^{i}_{j}\log\frac{p_{i}^{(\nu)}s^{i}_{j}}{p_{j}^{(\nu)}s^{j}_{i}}\right]+\sum_{i}\left[p_{i}^{(\nu+1)}\log\frac{p_{i}^{(\nu)}}{p_{i}^{(\nu+1)}}\right]$
Substituting $s^{i}_{j}$ by $w^{i}_{j}$ in the first term, one obtains the
familiar term used in the continuous-time case, which is already a Kullback-
Leibler divergence on its own. However, for the discrete-time case the second
term is needed to account for the change of the ensemble between two jumps. It
vanishes for $\tau_{\mathrm{obs}}\to 0$ because $p_{i}^{(\nu)}\to
p_{i}^{(\nu+1)}$.
## Phase-space preserving systems
Finally, we demonstrate the consistency of our approach in the context of
phase-space preserving maps. From the _dynamical systems perspective_ , we
consider such maps as stroboscopic pictures of a Hamiltonian dynamics.
_Thermodynamically_ , a Hamiltonian system is an isolated system which does
not exchange energy, particles or entropy with its environment at any time.
Its steady-state distribution is _microcanonical_ , i.e. uniform on phase
space. In _stochastic terms_ , the steady state of an isolated system features
_detailed balance_ Schnakenberg (1976) for the steady-state distribution
$p^{\infty}_{i}$:
$p_{i}^{\infty}s^{i}_{j}=p_{j}^{\infty}s^{j}_{i}.$ (28)
Henceforth, we assume Eq. (28) and reversibility, Eq. (4), to show that the
interpretations are consistent. Definition (14) directly yields
$s^{i}_{j}=c_{ij}\Pi_{j}$ with some constant $c_{ij}=c_{ji}$. Because
$\sum_{j}s^{i}_{j}=1,\,\forall i$, it follows that $c_{ji}=1$, $\forall i,j$.
With $p_{i}^{\prime}$ denoting the iterated values of the probabilities
$p_{i}$ in Eq. (15), we hence find
$\displaystyle\left\langle\\!\left\langle\sigma_{\mathrm{med}}\right\rangle\\!\right\rangle$
$\displaystyle=\sum_{i,j}\left[p_{i}s^{i}_{j}\log\frac{s^{i}_{j}}{s^{j}_{i}}\right]-\sum_{i}\left[\left(p^{\prime}_{i}-p_{i}\right)\log\Pi_{i}\right]$
$\displaystyle=\sum_{i,j}\left[p_{i}\Pi_{j}\log\Pi_{j}-p_{i}\Pi_{j}\log\Pi_{i}-p_{j}\Pi_{i}\log\Pi_{i}\right]$
$\displaystyle\quad+\sum_{i}\left[p_{i}\log\Pi_{i}\right]=\sum_{i}\left[\left(-p_{i}+p_{i})\log\Pi_{i}\right)\right]=0.$
Hence, the average entropy change in the medium—which equals the average
phase-space contraction—identically vanishes _for all times and independent of
the initial ensemble_ , as is expected.
## Summary and outlook
We have shown how for NMBM the functionals of stochastic thermodynamics emerge
from information theory without relying on thermodynamic assumptions. We
presented interpretations of the different notions of entropy and entropy
variation, which are consistent both with information theory and work on non-
phase-space preserving “thermostatted” dynamics.
In analogy to the role of Smale’s horseshoe Smale (1967) as a hallmark of
chaos, NMBM could hence help understanding salient features of reversible
chaotic dynamics and their connection to thermodynamics. A possible approach
in that direction is to treat the NMBM vertices as linearizations of
fundamental structures in such dynamical systems. Then, a coarse-graining
procedure under the assumption of separation of scales, in the spirit of Ref.
Esposito (2012) could yield a better dynamical picture of local equilibrium.
###### Acknowledgements.
The authors are indebted to A. Wachtel, J. Horowitz and L. Rondoni for
illuminating discussions.
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|
arxiv-papers
| 2012-12-19T16:32:26 |
2024-09-04T02:49:39.450687
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bernhard Altaner, J\\\"urgen Vollmer",
"submitter": "Bernhard Altaner",
"url": "https://arxiv.org/abs/1212.4728"
}
|
1212.4755
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-346 LHCb-PAPER-2012-034 March 22, 2013
Measurement of the forward energy flow in $pp$ collisions at $\sqrt{s}=7$ TeV
The LHCb collaboration111Authors are listed on the following pages.
The energy flow created in $pp$ collisions at $\sqrt{s}=7$ TeV is studied
within the pseudorapidity range $1.9<\eta<4.9$ with data collected by the LHCb
experiment. The measurements are performed for inclusive minimum-bias
interactions, hard scattering processes and events with an enhanced or
suppressed diffractive contribution. The results are compared to predictions
given by Pythia-based and cosmic-ray event generators, which provide different
models of soft hadronic interactions.
Submitted to the European Physical Journal C
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B.
Appleby51, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov32, M. Artuso53,
E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, V.
Balagura28, W. Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W.
Barter44, A. Bates48, Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S.
Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R. Bernet37, M.-O.
Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T. Bird51, A.
Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50, J.
Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz23, K. Ciba35, X. Cid
Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D.C. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua51, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, M. Dogaru26, F. Domingo Bonal33,n, S.
Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F.
Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46,35, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, U.
Eitschberger9, R. Ekelhof9, L. Eklund48,35, I. El Rifai5, Ch. Elsasser37, D.
Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S.
Farry12, V. Fave36, D. Ferguson47, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, M. Fiore16, C. Fitzpatrick35, M.
Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C.
Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M.
Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, P. Garosi51,
J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M.
Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C.
Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M.
Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33,
G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B.
Gui53, E. Gushchin30, Yu. Guz32,35, T. Gys35, C. Hadjivasiliou53, G.
Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-
Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P.
Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, D.
Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F. Jansen38, P.
Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35, I.R.
Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20, Y.M. Kim47, O.
Kochebina7, I. Komarov36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V. Kudryavtsev31, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18,35, C.
Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29, J. Lefrançois7, O. Leroy6, Y. Li3, L.
Li Gioi5, M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von
Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J.
Luisier36, H. Luo47, A. Mac Raighne48, F. Machefert7, I.V. Machikhiliyan4,28,
F. Maciuc26, O. Maev27,35, S. Malde52, G. Manca15,d, G. Mancinelli6, N.
Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38, D. Martinez
Santos39, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z. Mathe35, C.
Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J. McCarthy42,
G. McGregor51, R. McNulty12, F. Meier9, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, T.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin29, T. Nikodem11, A. Nomerotski52, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S.
Playfer47, M. Plo Casasus34, F. Polci8, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro36, W. Qian4, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, N. Rauschmayr35, G.
Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A.
Richards50, K. Rinnert49, V. Rives Molina33, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35, V.
Romanovsky32, A. Romero Vidal34, J. Rouvinet36, T. Ruf35, H. Ruiz33, G.
Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d,
C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P.
Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval35,40, P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L.
Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva
Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, F. Soomro18, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci37, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, V. Syropoulos39, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, M. Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14,
G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C. Voß55, H.
Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44,
N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov28, L.
Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
In Quantum Chromodynamics (QCD), the final state of an inelastic hadron-hadron
collision can be described by contributions from hard and soft scattering
occurring between the constituents of the hadrons, initial- and final-state
(gluon) radiation and the fragmentation of the initially coloured partonic
final state into colour-neutral hadrons. The soft component of a collision is
called the underlying event. Its precise theoretical description remains a
challenge, while the dynamics of hard scattering processes is well described
by perturbative QCD. One source of the underlying event activity is multi-
parton interactions (MPI). These arise mainly in the region of a very low
parton momentum fraction, where parton densities are high so that the
probability of more than a single parton-parton interaction per hadron-hadron
collision is large. MPI effects become increasingly important at LHC collision
energies, where inelastic interactions between very soft partons are
sufficiently energetic to contribute to final state particle production [1].
MPI phenomena can be probed by measuring in the centre-of-mass system the
amount of energy created in inelastic hadron-hadron interactions at large
values of the pseudorapidity $\eta=-\ln[\tan(\theta/2)]$, with $\theta$ being
the polar angle of particles with respect to the beam axis. The energy flow is
expected to be directly sensitive to the amount of parton radiation and MPI
[2]. For a particular pseudorapidity interval with width $\Delta\eta$, the
total energy flow, which is normalised to the number of inelastic $pp$
interactions $N_{\rm int}$, is defined as
$\frac{1}{N_{\rm int}}\frac{dE_{\rm
total}}{d\eta}=\frac{1}{\Delta\eta}\left(\frac{1}{N_{\rm
int}}\sum_{i=1}^{N_{\rm part,\eta}}E_{i,\eta}\right)\;,$ (1)
where $N_{\rm part,\eta}$ is the total number of stable particles and
$E_{i,\eta}$ is the energy of the individual particles.
In this study, the energy flow is measured in $pp$ collisions at $\sqrt{s}$ =
7$\mathrm{\,Te\kern-1.00006ptV}$ within the pseudorapidity range
$1.9<\eta<4.9$. This extends the previous measurements that have been made in
$p\bar{p}$ [3] and $ep$ collisions [4] to larger pseudorapidity values and
higher centre-of-mass energies, and complements the studies performed by the
CMS and ATLAS collaborations [5, 6]. Experimental results are compared to
predictions given by Pythia-based [7, 8] and cosmic-ray event generators [9,
10], which model the underlying event activity in different ways. In order to
probe various aspects of multi-particle production in high-energy hadron-
hadron collisions, the measurements are performed for the following four
classes of events: inclusive minimum-bias, hard scattering, diffractive, and
non-diffractive enriched interactions.
## 2 The LHCb detector
The LHCb detector [11] is a single-arm forward spectrometer with an angular
coverage from 10$\rm\,mrad$ to 300 (250)$\rm\,mrad$ in the bending (non-
bending) plane, designed for the study of $b$\- and $c$-hadrons. The detector
includes a high precision tracking system consisting of a silicon-strip vertex
detector (VELO) surrounding the $pp$ interaction region, a large-area silicon-
strip detector located upstream of a dipole magnet with a bending power of
about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw
drift tubes placed downstream. The VELO has a larger angular acceptance than
the rest of the spectrometer, including partial coverage of the backward
region. It allows reconstruction of charged particle tracks in the
pseudorapidity ranges $1.5<\eta<5.0$ and $-4<\eta<-1.5$. The combined tracking
system has a momentum resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum, $p_{\rm T}$.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are distinguished by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter (ECAL) and a hadronic calorimeter (HCAL). The
calorimeters have an energy resolution of $\sigma(E)/E=10\%/\sqrt{E}\oplus
1\%$ and $\sigma(E)/E=69\%/\sqrt{E}\oplus 9\%$ (with $E$ in GeV),
respectively. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. For the minimum-bias data used in this analysis,
the hardware trigger was accepting all beam-beam crossings, while the presence
of at least one reconstructed track was required in the software stage to
record an event.
## 3 Data analysis
### 3.1 Data and Monte Carlo samples
The analysis is performed using a sample of minimum-bias data collected in
$pp$ collisions at $\sqrt{s}$ = 7$\mathrm{\,Te\kern-1.00006ptV}$ during the
initial running period of the LHC with low interaction rate. The fraction of
bunch crossings with two or more collisions (“pile-up events”) is estimated to
be approximately $5\%$. The total number of events available in the sample is
$5.8\times 10^{6}$, corresponding to an integrated luminosity of about
0.1$\mbox{\,nb}^{-1}$.
Fully simulated minimum-bias $pp$ events at $\sqrt{s}$ =
7$\mathrm{\,Te\kern-1.00006ptV}$ were generated using the LHCb tune [12] of
the Pythia 6.4 event generator [7]. Here, decays of hadronic particles are
described by EvtGen [13] in which final state QED radiation is generated using
Photos [14]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [15, 16] as described in
Ref. [17]. Additional Monte Carlo (MC) samples with fully simulated minimum-
bias $pp$ interactions at $\sqrt{s}$ = 7$\mathrm{\,Te\kern-1.00006ptV}$ were
generated using the Perugia 0 and Perugia NOCR [18] tunes of Pythia 6.4. These
models along with the LHCb tune use different values for the MPI energy
scaling parameter and MPI $p_{\rm T}$ cut-off, which entails a sizeable
deviation in the amount of MPI predicted by these tunes.
The LHCb tune utilises the CTEQ6L parton density functions (PDFs) [19], while
both Perugia tunes use the CTEQ5L PDFs [20]. Colour reconnection effects are
not included in the Perugia NOCR tune. In the MC samples generated with the
Perugia 0 and Perugia NOCR tunes, diffractive $pp$ interactions are not
included, whereas the sample generated with the LHCb tune contains the
contributions from both single and double diffractive processes. A sample of
fully simulated diffractive events generated with Pythia 8.130 [8], which
utilises the CTEQ5L PDFs, is used in addition. This event generator gives a
more accurate description of diffractive $pp$ interactions than Pythia 6,
especially at high-$p_{\rm T}$, as it includes the contribution from hard
diffractive processes, which is absent in Pythia 6 [21].
In addition to the models above, experimental results are compared to
generator level predictions given by the Pythia 8.135 model with default
parameters. Furthermore, the measurements are compared with predictions given
by the cosmic-ray interaction models Epos 1.99 [22], Qgsjet01, QgsjetII-03
[23], and Sibyll 2.1 [24], which are widely used in extensive air shower
simulations and are not tuned to LHC data. These generate inelastic $pp$
interactions taking into account the contributions from both soft and hard
scattering processes. Soft contributions are described with Gribov’s Reggeon
field theory [25] via exchanges between virtual quasi-particle states
(Pomerons), while hard processes are described by perturbative QCD via
exchanges of hard or semi-hard Pomerons. The predictions given by these models
diverge mainly because of different treatments of non-linear interaction
effects related to parton saturation [26] and shadowing [27]. The Qgsjet01
model describes hadronic multiple scattering processes as multiple exchanges
of Pomerons without specific treatment of saturation effects. A distinct
feature of the QgsjetII model is the treatment of non-linear parton effects
via Pomeron interactions taking into account all order re-summation of the
corresponding Reggeon field theory diagrams. Based on the dual parton model
[28], Sibyll utilises the Lund string model [29] for hadronisation and
describes soft and hard processes using the Pomeron formalism and the minijet
model [30], correspondingly. The treatment of non-linear effects in this model
is based on a simple geometrical approach of parton saturation. The Epos model
takes into account energy-momentum correlations between multiple re-
scatterings and describes non-linear effects using an effective treatment of
lowest order Pomeron-Pomeron interaction graphs. It also accounts for the
final state interaction of the produced particles.
### 3.2 Analysis strategy
The energy flow, as defined in Eq. 1, is the energy-weighted pseudorapidity
distribution of particles, normalised to the number of inelastic interactions
and the $\eta$-bin size. The measurements are performed in ten equidistant
pseudorapidity bins of width $\Delta\eta=0.3$ over the range $1.9<\eta<4.9$.
The primary measurement is the energy flow carried by charged particles
(charged energy flow). It is performed with reconstructed tracks which contain
hits in the VELO and downstream tracking stations and have momentum in the
range $2<p<1000$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Particle
identification is not required in this analysis, as the energy is taken from
the momentum, neglecting particle masses. In order to be able to compare the
results of the measurements with generator level predictions, the
reconstructed charged energy flow is corrected for detector effects. The total
energy flow is determined by using a data-constrained MC estimate of the
neutral component based on information from the ECAL, while the HCAL is not
used. Details of the procedure are discussed below.
### 3.3 Event classes
The event classes studied in this analysis are defined as follows. Inclusive
minimum-bias events are selected by requesting the presence of at least one
track originating from the luminous region in order to suppress pollution from
beam-gas interactions and beam halo related background. Events with two or
more reconstructed primary vertices are rejected to suppress pile-up
contamination. To minimise biases on the track multiplicity of the event, the
information on the primary vertex is not used. The selected inclusive minimum-
bias interactions are further classified as hard scattering, diffractive and
non-diffractive enriched events using the following criteria:
* •
Hard scattering events: at least one track with $\mbox{$p_{\rm
T}$}>3$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $1.9<\eta<4.9$.
* •
Diffractive enriched events: no tracks reconstructed with $-3.5<\eta<-1.5$.
* •
Non-diffractive enriched events: at least one track reconstructed with
$-3.5<\eta<-1.5$.
The selection requirements applied for the last two event classes are
motivated by the fact that a sizeable rapidity gap is an experimental
signature of diffractive processes [31]. The level of enrichment of the
diffractive and non-diffractive samples was studied in simulation, by
retrieving the Pythia process type of the $pp$ interaction for every selected
diffractive and non-diffractive candidate. In the case of the LHCb tune of
Pythia 6.4, the purities of the selected diffractive and non-diffractive
enriched samples are found to be about $70\%$ and $90\%$, respectively.
Although the actual percentages are only meaningful within the specific model,
the study shows that the applied selection criteria indeed lead to sizeable
enhancement of the respective event classes.
To minimise the experimental corrections, the definition of the event classes
at generator level is similar to that at detector level. Inclusive minimum-
bias events at generator level are selected by requiring the presence of at
least one outgoing final-state charged particle (lifetime $\tau>10^{-8}$ s) in
the pseudorapidity range $1.9<\eta<4.9$, but without imposing any condition on
its energy. The sample of hard scattering events is selected by requesting at
least one final-state charged particle with $p_{\rm T}$
$>3$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $1.9<\eta<4.9$. The absence or
presence of at least one final-state charged particle in $-3.5<\eta<-1.5$ is
used as criterion to select diffractive and non-diffractive enriched events
among inclusive minimum-bias interactions, respectively. For the selected
events, the energy flow at generator level is determined using the outgoing
final-state charged and neutral particles111These include $\pi^{\pm}$,
$K^{\pm}$, $e^{\pm}$, $\mu^{\pm}$, $p$, $\overline{}p$, $\gamma$, $n$,
$\overline{}n$ and $K^{0}_{\rm\scriptscriptstyle L}$. which are either prompt,
originating directly from the fragmentation, or the decay products of unstable
particles. Since neutrinos are not reconstructed by the LHCb spectrometer the
energy carried by these particles is not taken into account. Only MC events
simulated with exactly one inelastic $pp$ interaction are considered in this
study.
### 3.4 Corrections
The reconstructed charged energy flow measured with data is corrected for
detector effects using bin-by-bin correction factors, which are estimated as
the ratio of the charged energy flow at generator and detector level in
simulation for each $\eta$ region and event class under consideration. The
overall correction factor for each bin is taken as the average of the
correction factors obtained with different MC models used in this analysis.
For inclusive, hard scattering and non-diffractive enriched events, the
average and standard deviation of the correction factors, which is included in
the model-dependent systematic uncertainty, are determined from the LHCb,
Perugia 0 and Perugia NOCR tunes of Pythia 6.4. In the case of the diffractive
enriched event class, only the LHCb tune and the Pythia 8 diffractive
simulation are used. Except for the lowest $\eta$ bin, which suffers from
reduced acceptance for low-$p_{\rm T}$ particles and thus exhibits large
corrections and a sizeable model dependence, the correction factors are found
to be stable among the models with a slight rise towards the edges of the
detector acceptance. The majority of the factors are well below two,
indicating that most of the energy is measured by the detector. In the case of
diffractive enriched events, the correction factors obtained with the LHCb
tune are slightly smaller than unity for some of the bins, i.e. the energy
flow at detector level is found to be larger than at generator level. This is
due to detection inefficiency for charged particles over the pseudorapidity
range $-3.5<\eta<-1.5$. As a result, some of the events containing backward
going charged particles migrate into the diffractive sample, which leads to
enhanced energy flow at detector level.
For the measurement of the total energy flow, the neutral component $F_{\rm
neut,\eta}$ is estimated in the following way. To first order $F_{\rm
neut,\eta}$ is assumed to be proportional to the corrected charged energy flow
$F_{\rm char,\eta}$ with a factor $R_{\rm gen,\eta}$, which is the average
ratio of the neutral energy flow to the charged energy flow obtained at
generator level for each $\eta$ bin and event class with different Pythia
tunes. This ratio is found to be rather stable over the entire pseudorapidity
range of the measurements with only small variations between the Pythia tunes.
This reflects the usage of the same hadronisation mechanism governed in the
Pythia generator by the Lund string model [7, 8]. The latter successfully
describes the hadronisation of quarks and gluons emerging from high energy
interactions and was rigorously tested for high-$p_{\rm T}$ processes [32, 33,
34]. The $R_{\rm gen,\eta}$ ratio is found to be around $0.6$ for all event
types except the hard scattering interactions. For the latter, it is about
$15\%$ smaller for all $\eta$ bins. This feature is found to be a consequence
of the requirement of a high-$p_{\rm T}$ charged particle in the definition of
this event class.
Under the assumption outlined above, the total energy flow for a particular
event class and pseudorapidity bin $F_{\rm total,\eta}$ can be written as
$F_{\rm total,\eta}=F_{\rm char,\eta}+F_{\rm neut,\eta}=F_{\rm
char,\eta}\times\left(1+R_{\rm gen,\eta}\right).$ (2)
In order to constrain this initially purely model-based estimate of the
neutral energy flow to data, the total energy flow is further multiplied by an
additional correction factor $k_{\eta}$. It accounts for differences between
simulation and data being defined for every $\eta$ bin as
$k_{\eta}=\frac{1+R_{\rm data,\eta}}{1+R_{\rm mc,\eta}}\;.$ (3)
Here, $R_{\rm data,\eta}$ and $R_{\rm mc,\eta}$ are ratios of the uncorrected
neutral to charged energy flow measured in data and simulation, respectively.
The $R_{\rm mc,\eta}$ ratio is obtained with different Pythia tunes and its
average is taken for the estimation of the $k_{\eta}$ factors. The neutral
component of $R_{\rm data,\eta}$ and $R_{\rm mc,\eta}$ is measured using
reconstructed photon candidates which are selected from neutral clusters in
the ECAL with an energy greater than 2$\mathrm{\,Ge\kern-1.00006ptV}$ and
$p_{\rm T}$ $>0.2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Since the polar
angular coverage of the ECAL begins at about 30$\rm\,mrad$, there are no
measurements of the neutral energy for the last two $\eta$ bins ($\eta>4.3$).
The $k_{\eta}$ factors for this pseudorapidity region are estimated using a
linear extrapolation of the $k_{\eta}$ factors obtained for the pseudorapidity
interval $3.1<\eta<4.3$. The bins with $\eta<3.1$ are not considered for the
extrapolation, as these are affected by the detection inefficiency for
low-$p_{\rm T}$ charged particles. The latter have a low average momentum in
this $\eta$ region and thus are unlikely to reach downstream tracking
stations. Except for the lowest $\eta$ bin, which suffers most from the
detection inefficiency especially in the case of the diffractive enriched
event class, the $k_{\eta}$ factors are found to be rather close to unity,
reflecting the fact that the ratio of the neutral to charged energy flow is
well simulated at detector level.
## 4 Systematic uncertainties
The total uncertainties on the results are dominated by systematic effects, as
the statistical uncertainties are found to be negligible for all $\eta$ bins
and event classes. The various contributions to the systematic uncertainties
are summarised in Table 1.
Table 1: Relative systematic uncertainties (in percent) affecting the energy flow measurements for all event classes. The total uncertainties are obtained by adding the individual sources in quadrature. The ranges indicate the variation of the uncertainty as a function of $\eta$. Source of | Inclusive | Hard | Diffractive | Non-diffractive
---|---|---|---|---
uncertainty | minbias | scattering | enriched | enriched
Model uncertainty on | $0.6-9.2$ | $0.7-4.1$ | $16-43$ | $0.7-8.6$
correction factors | | | |
Selection cuts | $1.0-4.9$ | $2.7-8.8$ | $0.9-2.8$ | $1.1-5.0$
Tracking efficiency | $3$ | $3$ | $3$ | $3$
Multiple tracks | $1$ | $1$ | $1$ | $1$
Spurious tracks | $0.3-1.2$ | $0.4-1.7$ | $0.2-0.7$ | $0.3-1.2$
Magnet polarity | — | — | $2.6-7.7$ | —
Residual pile-up | $1.7$ | $1.7$ | $1.7$ | $1.7$
Total on $F_{\rm char,\eta}$ | $3.9-11$ | $4.9-10$ | $16-43$ | $4.0-11$
Variation of $R_{\rm gen,\eta}$ | $0.8-6.1$ | $0.7-2.9$ | $1.5-23$ | $0.9-5.5$
and $k_{\eta}$ factors | | | |
Photon efficiency | $1.4-1.6$ | $1.2-1.3$ | $1.3-2.3$ | $1.3-1.6$
ECAL miscalibration | $<1$ | $<1$ | $<1$ | $<1$
Total on $F_{\rm total,\eta}$ | $4.4-13$ | $5.4-11$ | $17-49$ | $4.4-12$
For all event types except hard scattering interactions, the largest
uncertainty on the charged energy flow arises from the model dependence of the
bin-by-bin correction factors, which is estimated as the standard deviation of
the correction factors obtained with different Pythia tunes. Here, the largest
impact is at low $\eta$, reaching $9\%$ for inclusive and non-diffractive
enriched events, $4\%$ for hard scattering interactions and up to $43\%$ for
diffractive enriched events. At large $\eta$ this effect generally drops to
about 1–2$\%$ for all event classes except diffractive enriched interactions
for which it stays above $15\%$.
Systematic uncertainties related to the track selection requirements are
estimated by comparing the fraction of the energy flow from tracks which are
rejected by the selection cuts in data and simulation. For the majority of the
bins the resulting systematic uncertainty is found to be less than $4\%$. Only
for hard scattering events this uncertainty approaches $9\%$ at low $\eta$.
To account for differences between the true tracking efficiency and that
estimated using simulation, a global $3\%$ systematic uncertainty is assigned
across the entire $\eta$ range following the analysis presented in Ref. [35].
This applies for all event classes under consideration.
The other tracking related factors having an influence on the charged energy
flow measurements are contaminations from multiply reconstructed tracks and
tracks created from random combinations of hits (spurious tracks). The impact
of the former is estimated by removing from the measurement all tracks found
within the same event with similar momentum vectors. It is observed that the
charged energy flow drops by less than $1\%$ for all $\eta$ bins and event
classes in case of both data and simulation. For the final results, a global
$1\%$ systematic uncertainty for multiply reconstructed tracks is
conservatively assigned. The effect of spurious tracks is estimated in
simulation by determining the energy flow carried by reconstructed tracks
which cannot be associated with particles at generator level and accounting
for the difference between the rate of spurious tracks in data and simulation.
The corresponding systematic uncertainty is found to vary between $0.2\%$ and
$2\%$.
It has been checked that reversing the LHCb magnet polarity has only an
influence on the measurements of the charged energy flow for the diffractive
enriched event class, which mainly consists of low-multiplicity events. Here,
the corresponding effect is assigned as a systematic uncertainty.
Table 2: Charged energy flow for all event classes and $\eta$ bins with the corresponding systematic uncertainties. The statistical uncertainties are insignificant and not listed. All values are in GeV per unit pseudorapidity interval. Pseudorapidity | Inclusive | Hard | Diffractive | Non-diffractive
---|---|---|---|---
range | minbias | scattering | enriched | enriched
$1.9<\eta<2.2$ | $12\pm 1$ | $37\pm 4$ | $4\pm 2$ | $13\pm 1$
$2.2<\eta<2.5$ | $16\pm 1$ | $50\pm 4$ | $5\pm 2$ | $17\pm 1$
$2.5<\eta<2.8$ | $21\pm 1$ | $64\pm 4$ | $6\pm 2$ | $22\pm 1$
$2.8<\eta<3.1$ | $27\pm 1$ | $83\pm 5$ | $9\pm 3$ | $29\pm 1$
$3.1<\eta<3.4$ | $35\pm 2$ | $105\pm 6$ | $12\pm 3$ | $38\pm 2$
$3.4<\eta<3.7$ | $46\pm 2$ | $132\pm 6$ | $17\pm 4$ | $49\pm 2$
$3.7<\eta<4.0$ | $58\pm 2$ | $161\pm 8$ | $22\pm 5$ | $61\pm 2$
$4.0<\eta<4.3$ | $73\pm 3$ | $194\pm 10$ | $31\pm 7$ | $77\pm 3$
$4.3<\eta<4.6$ | $88\pm 4$ | $219\pm 12$ | $41\pm 7$ | $93\pm 4$
$4.6<\eta<4.9$ | $112\pm 5$ | $256\pm 13$ | $57\pm 9$ | $118\pm 6$
Table 3: Total energy flow for all event classes and $\eta$ bins with the corresponding systematic uncertainties. The statistical uncertainties are insignificant and not listed. All values are in GeV per unit pseudorapidity interval. Pseudorapidity | Inclusive | Hard | Diffractive | Non-diffractive
---|---|---|---|---
range | minbias | scattering | enriched | enriched
$1.9<\eta<2.2$ | $18\pm 2$ | $55\pm 6$ | $4\pm 2$ | $19\pm 2$
$2.2<\eta<2.5$ | $26\pm 2$ | $77\pm 7$ | $6\pm 2$ | $28\pm 2$
$2.5<\eta<2.8$ | $36\pm 2$ | $102\pm 7$ | $10\pm 3$ | $38\pm 2$
$2.8<\eta<3.1$ | $48\pm 3$ | $133\pm 8$ | $15\pm 5$ | $51\pm 3$
$3.1<\eta<3.4$ | $60\pm 3$ | $164\pm 9$ | $20\pm 5$ | $64\pm 3$
$3.4<\eta<3.7$ | $75\pm 3$ | $203\pm 11$ | $27\pm 6$ | $80\pm 4$
$3.7<\eta<4.0$ | $95\pm 4$ | $246\pm 15$ | $37\pm 9$ | $100\pm 4$
$4.0<\eta<4.3$ | $118\pm 5$ | $296\pm 17$ | $50\pm 11$ | $125\pm 6$
$4.3<\eta<4.6$ | $144\pm 7$ | $329\pm 20$ | $65\pm 11$ | $151\pm 7$
$4.6<\eta<4.9$ | $182\pm 9$ | $380\pm 21$ | $89\pm 15$ | $191\pm 10$
Events with more than one reconstructed primary vertex are vetoed in the
analysis in order to suppress pile-up contamination. Its residual effect is
estimated to be $1.7\%$ by taking the efficiencies to accept pile-up events
from simulation. This factor is included in the normalisation of the energy
flow and conservatively taken as the systematic uncertainty.
The total energy flow acquires an additional uncertainty from the variation of
the $R_{\rm gen,\eta}$ and $k_{\eta}$ factors between the Pythia tunes and the
extrapolation procedure used for the $k_{\eta}$ factors in two highest $\eta$
bins. No systematic uncertainty is assigned to account for inaccuracies of the
Lund string model in describing the ratio of the neutral energy flow to the
charged energy flow. The uncertainties associated with the ECAL energy
calibration and photon reconstruction efficiency also affect the accuracy of
the total energy flow. To account for a possible difference in the photon
reconstruction efficiency between simulation and data, a global $3.7\%$
systematic uncertainty is assigned following the analysis presented in Ref.
[36]. The energy calibration of the ECAL has been studied by measuring the
invariant masses of diphoton resonances ($\pi^{0}\rightarrow\gamma\gamma$ and
$\eta\rightarrow\gamma\gamma$) and has been assigned a global systematic
uncertainty of $1.5\%$. Both uncertainties are scaled with a factor $F_{\rm
neut,\eta}/F_{\rm total,\eta}$ and are listed in Table 1.
Other potential sources of systematic uncertainties such as momentum- and
$\eta$-smearing, effect of the beam crossing angle, neglecting the masses of
charged particles, pollution from elastic scattering and beam-gas interactions
have been studied as well. Their impacts on the accuracy of the measurements
are found to be negligible.
The total systematic uncertainties on the corrected charged and total energy
flow are listed in Tables 2 and 3 for all event classes and $\eta$ bins. It
should be noted that the uncertainties are strongly correlated between the
bins.
## 5 Results
The fully-corrected measurements for the charged energy flow are shown in Fig.
1 for each event class together with the generator level predictions given by
the Pythia tunes and the corresponding systematic uncertainties. By comparing
experimental results obtained for different event classes one can clearly see
that the amount of energy flow strongly correlates with the momentum transfer
in an underlying $pp$ inelastic interaction. The charged energy flow rises
more steeply with pseudorapidity in data than predicted by the majority of the
Pythia tunes. As a consequence, the discrepancy between the measurements and
generator level predictions increases towards large $\eta$ rising to $20\%$ in
the last $\eta$ bin. At lower $\eta$ the data are reasonably well described by
the Pythia tunes. This is the case for all event classes except the
diffractive enriched one. For the latter, the measurements are well described
by the Pythia 8.135 generator with default parameters. However, this model
overestimates the charged energy flow in the case of hard scattering events
over the entire pseudorapidity range of the measurements.
Figure 2 illustrates the charged energy flow along with the predictions given
by the cosmic-ray interaction models. It is interesting to note that the
measurements performed with inclusive minimum-bias and non-diffractive
enriched events are well described by the Epos 1.99 and Sibyll 2.1 models,
while the Qgsjet01 and QgsjetII-03 generators overestimate the charged energy
flow for these event classes. The latter also occurs at large $\eta$ in the
case of hard scattering interactions for all cosmic-ray interaction models
except the QgsjetII-03. The diffractive enriched charged energy flow is
underestimated by all cosmic-ray interaction models.
Figure 1: Charged energy flow as a function of $\eta$ for all event classes as
indicated in the figures. The corrected measurements are given by points with
error bars, while the predictions by the Pythia tunes are shown as histograms.
The error bars represent the systematic uncertainties, which are highly
correlated between the bins. The statistical uncertainties are negligible. The
ratios of MC predictions to data are shown in addition.
Figure 2: Charged energy flow as a function of $\eta$ for all event classes as
indicated in the figures. The corrected measurements are given by points with
error bars, while the predictions by the cosmic-ray interaction models are
shown as histograms. The error bars represent the systematic uncertainties,
which are highly correlated between the bins. The statistical uncertainties
are negligible. The ratios of MC predictions to data are shown in addition.
Figure 3: Total energy flow as a function of $\eta$ for all event classes as
indicated in the figures. The corrected measurements are given by points with
error bars, while the predictions by the Pythia tunes are shown as histograms.
The data obtained with extrapolated $k_{\eta}$ factors are shown in grey. The
error bars represent the systematic uncertainties, which are highly correlated
between the bins. The statistical uncertainties are negligible. The ratios of
MC predictions to data are shown in addition.
Figure 4: Total energy flow as a function of $\eta$ for all event classes as
indicated in the figures. The corrected measurements are given by points with
error bars, while the predictions by the cosmic-ray interaction models are
shown as histograms. The data obtained with extrapolated $k_{\eta}$ factors
are shown in grey. The error bars represent the systematic uncertainties,
which are highly correlated between the bins. The statistical uncertainties
are negligible. The ratios of MC predictions to data are shown in addition.
The total energy flow is shown for each event class in Fig. 3 along with the
generator level predictions given by the Pythia tunes and the corresponding
systematic uncertainties. It can be clearly seen that all Pythia 6 tunes
underestimate the amount of energy flow at large pseudorapidity for all event
classes. The Pythia 8.135 generator gives the best description of the
measurements performed with inclusive minimum-bias, diffractive and non-
diffractive enriched events among the Pythia tunes, except for the
pseudorapidity range $1.9<\eta<2.5$. None of these models provide an accurate
description of the energy flow measured with hard scattering events. The
predictions given by the LHCb and Perugia NOCR tunes for non-diffractive
enriched and hard scattering events are rather similar, while the Perugia 0
tune significantly underestimates the energy flow for all event classes. For
diffractive enriched events, the inconsistency between the data and the
prediction given by the LHCb tune is found to be rather large throughout the
entire pseudorapidity range $1.9<\eta<4.9$, while the Pythia 8.135 generator
with default parameters gives a good description of the corresponding energy
flow at large $\eta$.
Figure 4 illustrates the total energy flow together with the predictions given
by the cosmic-ray interaction models. It is observed that the Sibyll 2.1
generator gives the best description of the energy flow measured with
inclusive minimum-bias and non-diffractive enriched events at large $\eta$.
The predictions given by the Epos 1.99 generator for these event classes also
describe the measurements reasonably well. In the case of hard scattering
interactions, the best description of the data at large $\eta$ is given by the
QgsjetII-03 generator. The total energy flow measured with diffractive
enriched events is underestimated at large $\eta$ by all cosmic-ray
interaction models used in this study. The measurements of the charged and
total energy flow are summarised in Tables 2 and 3 for all event classes and
$\eta$ bins.
The results obtained in this study cannot be directly compared with the
measurements performed by the CMS collaboration [5], since different event
selection criteria are applied in the analyses. Nevertheless, both
measurements demonstrate that the energy flow is underestimated by Pythia 6
tunes at large pseudorapidity, while the results of the ATLAS collaboration
indicate that the amount of transverse energy is also underestimated by
various Pythia tunes at large $\eta$ [6].
## 6 Conclusions
The energy flow is measured in the pseudorapidity range $1.9<\eta<4.9$ with
data collected by the LHCb experiment in $pp$ collisions at $\sqrt{s}$ =
7$\mathrm{\,Te\kern-1.00006ptV}$ for inclusive minimum-bias interactions, hard
scattering processes and events with enhanced or suppressed diffractive
contribution. The primary measurement is the energy flow carried by charged
particles. For the measurement of the total energy flow, a data-constrained MC
estimate of the neutral component is used. The energy flow is found to
increase with the momentum transfer in an underlying $pp$ inelastic
interaction. The evolution of the energy flow as a function of pseudorapidity
is reasonably well reproduced by the MC models. Nevertheless, the majority of
the Pythia tunes underestimate the measurements at large pseudorapidity, while
most of the cosmic-ray interaction models overestimate them, except for
diffractive enriched interactions. For inclusive and non-diffractive enriched
events, the best description of the data at large $\eta$ is given by the
Sibyll 2.1 and Pythia 8.135 generators. The latter also provides a good
description of the energy flow measured with diffractive enriched events,
especially at large $\eta$. The comparison shows that the absence of hard
diffractive processes moderates the amount of the forward energy flow meaning
that their inclusion is vital for a more precise description of partonic
interactions. It also demonstrates that higher-order QCD effects as contained
in the Pomeron phenomenology play an important role in the forward region.
None of the event generators used in this analysis are able to describe the
energy flow measurements for all event classes that have been studied.
## Acknowledgements
We are thankful to Colin Baus and Ralf Ulrich from the Karlsruhe Institute of
Technology for providing the predictions of the cosmic-ray Monte Carlo
generators. We express our gratitude to our colleagues in the CERN accelerator
departments for the excellent performance of the LHC. We thank the technical
and administrative staff at the LHCb institutes. We acknowledge support from
CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2012-12-19T17:36:51 |
2024-09-04T02:49:39.458875
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche,\n J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S.\n Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, K.\n De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, P. De\n Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P.\n Diniz Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M.\n Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier,\n J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu.\n Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, P.F.\n Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A.\n Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y.M. Kim, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, H. Luo, A.\n Mac Raighne, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R.\n McNulty, F. Meier, M. Meissner, M. Merk, J. Merkel, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R.\n Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K. Pal, A. Palano,\n M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J.\n Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C. Patrignani,\n C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V.\n Rives Molina, D.A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez,\n G.J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf,\n H. Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta,\n C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina\n Rios, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A.\n Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, S.\n Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, K. Sobczak, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De\n Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp,\n S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah,\n S. Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I. Videau, D.\n Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Dmytro Volyanskyy",
"url": "https://arxiv.org/abs/1212.4755"
}
|
1212.4796
|
# Relativistic rotation curve for cosmological structures
Mohammadhosein Razbin Georg-August-Universität Göttingen, Institut für
Theoretische Physik, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany, EU and
Max-Planck-Institut für Dynamik und Selbstorganisation, Am Fassberg 17, 37077
Göttingen, Germany, EU [email protected] Javad T.
Firouzjaee School of Physics and School of Astronomy, Institute for Research
in Fundamental Sciences (IPM), Tehran, Iran [email protected] Reza
Mansouri Department of Physics, Sharif University of Technology, Tehran, Iran
and
School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran, Iran [email protected]
###### Abstract
Using a general relativistic exact model for spherical structures in a
cosmological background, we have put forward an algorithm to calculate the
test particle geodesics within such cosmological structures in order to obtain
the velocity profile of stars or galaxies. The rotation curve thus obtained is
based on a density profile and is independent of any mass definition which is
not unique in general relativity. It is then shown that this general
relativistic rotation curves for a toy model and a NFW density profile are
almost identical to the corresponding Newtonian one, although the general
relativistic masses may be quite different.
## I Introduction
There has been recently many attempts to understand fully the consistency of
the linearized Einstein equations; in other words how far we are right in
using Newtonian versus general relativistic gravity in astrophysical and
cosmological applications (see for example relativity-newtonian and wald ).
We face this controversy in this paper by calculating the general relativistic
rotation curve for a spherical structure within a FRW universe. The exact
solution gives us the possibility to see if and in which sense the Newtonian
approximation is valid in the case of the weak gravity.
The relativistic rotation curve has been the subject of some recent papers
(see Cooperstock1 and Cooperstock2 ). The model discussed in these papers,
however, is based on the simple Schwarzschild metric, dust collapse, or a
simple axially symmetric space-time with a singularity at the central plane,
not taking into account the full dynamics of a cosmological structure. We,
however, look for an exact solution of Einstein equations representing a
cosmological structure as an overdensity region within a cosmological
background assumed to be asymptotically FRW. There are not much viable exact
models representing such a structure. We will use an analytical model proposed
recently based on an inhomogeneous cosmological LTB solution javad1 . Its
time-like geodesics are then integrated numerically to obtain the rotation
curve of the cosmological structure. Obviously we need a specified density
profile for our cosmological structure to obtain the rotation curve, without a
need of specifying a mass. In the Newtonian case this is identical to have a
unique mass of the structure. In general relativity, however, there is no
unique mass definition for astrophysical objects given a specific density
profilejavad2 . Therefore, relying on a density profile which is an
observational quantity, the non-uniqueness of the concept of mass does not
disturb our conclusion about the comparison of the Newtonian versus general
relativistic rotation curve.
Section II is a brief introduction to the LTB metric, followed by definitions
of some quasi-local masses. In section III we introduce an algorithm how to
calculate the rotation curve for a test particle motion around a cosmological
structure within a FRW universe. Section IV is devoted to the calculation of
the general relativistic rotation curve for a cosmological structure using
this algorithm and to compare the results with the Newtonian case. Some
general relativistic quasi-local masses are also calculated and compared to
the corresponding Newtonian one for the same density profile. We then conclude
in section V. Throughout the paper we assume $8\pi G=c=1$.
## II LTB model of a structure
The LTB metric in synchronous coordinates is written as
$ds^{2}=-dt^{2}+\frac{R^{\prime 2}}{1+f(r)}dr^{2}+R(r,t)^{2}d\Omega^{2},$ (1)
representing a pressure-less perfect fluid satisfying
$\displaystyle\rho(r,t)=\frac{2M^{\prime}(r)}{R^{2}R^{\prime}},\hskip
22.76228pt\dot{R}^{2}=f+\frac{2M}{R}.$ (2)
Here dot and prime denote partial derivatives with respect to the parameters
$t$ and $r$ respectively. The angular distance $R$, depending on the value of
$f$, is given by
$\displaystyle R=-\frac{M}{f}(1-\cos\eta(r,t)),$ $\displaystyle\hskip
22.76228pt\eta-\sin\eta=\frac{(-f)^{3/2}}{M}(t-t_{b}(r)),$ (3)
for $f<0$, and
$R=(\frac{9}{2}M)^{\frac{1}{3}}(t-t_{b})^{\frac{2}{3}},$ (4)
for $f=0$, and
$\displaystyle R=\frac{M}{f}(\cosh\eta(r,t)-1),$ $\displaystyle\hskip
22.76228pt\sinh\eta-\eta=\frac{f^{3/2}}{M}(t-t_{b}(r)),$ (5)
for $f>0$.
The metric is covariant under the rescaling $r\rightarrow\tilde{r}(r)$.
Therefore, one can fix one of the three free functions of the metric, i.e.
$t_{b}(r)$, $f(r)$, or $M(r)$. The function $M(r)$ corresponds to the Misner-
Sharp mass in general relativity (mis-sha , see also javad2 ). The $r$
dependence of the bang time $t_{b}(r)$ corresponds to a non-simultaneous big-
bang or big-crunch singularity.
There are two generic singularities of this metric, where the Kretschmann and
Ricci scalars become infinite: the shell focusing singularity at $R(t,r)=0$,
and the shell crossing one at $R^{\prime}(t,r)=0$. However, there may occur
that in the case of $R(t,r)=0$ the density
$\rho=\frac{M^{\prime}}{R^{2}R^{\prime}}$ and the term $\frac{M}{R^{3}}$
remain finite. In this case the Kretschmann scalar remains finite and there is
no shell focusing singularity. Similarly, if in the case of vanishing
$R^{\prime}$ the term $\frac{M^{\prime}}{R^{\prime}}$ is finite, then the
density remains finite and there is no shell crossing singularity either (see
javad1 for more detail).
For our model structure, depending on our model calculation, we arrive in the
case of a toy model at a specific density profile, or specify a density
profile like the NFW one from the beginning. In each case there is a unique
Newtonian mass. However, in the relativistic case we may allocate different
masses without any preference. Although this does not change our aim of a
comparison between the relativistic rotation curve and the Newtonian one, we
prefer to show how different these masses may be. Note that the observational
quantity is the density profile and not the total mass which is a derived
quantity. In fact more than 10 general relativistic mass definitions have
already been proposed in the literature. The difference between some of the
proposed quasi-local mass definitions has been studied in javad2 , where it
has been shown that in the case of spherically symmetric structures the
Hawking quasi-local mass is equal to the Misner-sharp one. The Brown-York mass
for the 2-boundary specified by $r=constant$ and $t=constant$ ($M_{BY.r}$) in
an asymptotically FRW solution is given by
$\displaystyle M_{BY.r}=-R\sqrt{1+f}+(R\sqrt{1+f})\mid_{FRW}.$ (6)
If we specify the 2-boundary by $R(r,t)=costant$ and $t=constant$ with $R$
being the areal radius, then the Brown-York mass ($M_{BY.R}$) is given by
$\displaystyle
M_{BY.R}=-R\sqrt{1+\frac{2M}{R}}+(R\sqrt{1+\frac{2M}{R}})\mid_{FRW}.$ (7)
The Liu-Ya and Epp masses are equal in our spherically symmetric case and are
given by
$\displaystyle
M_{LY}=M_{Epp}=-R\sqrt{1+\frac{2M}{R}}+(R\sqrt{1+\frac{2M}{R}})\mid_{FRW}.$
(8)
We therefore concentrate on three mass definitions: the Misner-Sharp one which
is equal to that of Hawking; The Epp mass being equal to Liu-Yau and the
Brown-York mass at either the constant areal radius ($R=constant$) or the
constant comoving radius ($r=constant$).
## III Constructing the model
The model we are going to construct and study should describe a simple model
of a spherically symmetric mass condensation within a FRW universe as an exact
solution of Einstein equations with a pressure-less ideal fluid. Therefore we
will choose a density profile reflecting an overdensity at the center and
almost constant density far from the center as expected for a FRW universe.
Within such a model structure we then study timelike geodesics to extract
information about the rotation curve in such a dynamical setting. This is
achieved by specifying the three LTB functions $t_{b}(r)$, $f(r)$, and $M(r)$.
Assuming an expanding universe, this cosmic LTB model structure starts
expanding with the universe before its expansion decouple from the universe
and a collapsing phase starts. There are different ways to specify the LTB
functions depending on our needs or our methodology. Once the solution is
fixed we may study the timelike geodesics to extract the rotation curve within
the structure. Now, the geodesic equations for the LTB metric are given by
$\frac{d^{2}r}{d{\lambda}^{2}}=\frac{R(1+f)}{R^{\prime}}[\sin(\theta)^{2}({\frac{d\phi}{d{\lambda}}})^{2}+({\frac{d\theta}{d{\lambda}}})^{2}]-\frac{[R^{\prime}R^{\prime\prime}-\frac{{R^{\prime}}^{2}f^{\prime}}{2(1+f)}]}{{R^{\prime}}^{2}}{({\frac{dr}{d{\lambda}}})^{2}}-2\frac{\dot{R^{\prime}}}{R^{\prime}}{({\frac{dt}{d{\lambda}}})({\frac{dr}{d{\lambda}}})},$
(9)
${\frac{d^{2}\theta}{d{\lambda}^{2}}=\sin(\theta)\cos(\theta)({\frac{d\phi}{d{\lambda}}})^{2}-2\frac{R^{\prime}}{R}({\frac{dr}{d{\lambda}}})({\frac{d\theta}{d{\lambda}}})-2\frac{R^{\prime}}{R}({\frac{dt}{d{\lambda}}})({\frac{d\theta}{d{\lambda}}})},$
(10)
$\frac{d^{2}\phi}{d{\lambda}^{2}}=-2\frac{\cos(\theta)}{\sin(\theta)}(\frac{d\theta}{d{\lambda}})(\frac{d\phi}{d{\lambda}})-2\frac{R^{\prime}}{R}(\frac{dr}{d{\lambda}})(\frac{d\phi}{d{\lambda}})-2\frac{R^{\prime}}{R}(\frac{dt}{d{\lambda}})(\frac{d\phi}{d{\lambda}})$
(11)
and
$\frac{d^{2}t}{d{\lambda}^{2}}=-{R{\dot{R}}{\sin(\theta)}^{2}({\frac{d\phi}{d{\lambda}}})^{2}}-R\dot{R}({\frac{d\theta}{d{\lambda}}})^{2}-\frac{R^{\prime}\dot{R^{\prime}}}{1+f}({\frac{dr}{d{\lambda}}})^{2}.$
(12)
These equations can be simplified by choosing the particle rotation plane in
the $\theta=\pi/2$ and by the assumption of the geodesic being time like:
$-1=-(\frac{dt}{d\lambda})^{2}+\frac{R^{\prime
2}}{1+f}(\frac{dr}{d\lambda})^{2}+R^{2}(\frac{d\theta}{d\lambda})^{2}+R^{2}\sin^{2}(\theta)(\frac{d\phi}{d\lambda})^{2}.$
Using these geodesic equations, we are able to find dynamical properties of a
test particle within this structure. Given that our cosmic structure is
dynamic we choose the initial conditions such that the structure is in its
late dynamic phase, where the time scale of the evolution of the structure is
greater than the time of revolution of the circular path of a test particle.
Therefore, we may expect to have quasi-circular geodesics. These are defined
either as those having a vanishing radial velocity and acceleration respect to
areal radius at the initial conditions, or have an almost constant radius
within the numerical precision for a finite angular displacement. We have
checked both procedures leading to the same numerical result. Although we have
calculated the particle path, its velocity and the rotation curve at distances
far from the center using general relativity, it is obvious that these local
results should be well within the Newtonian approximation. Now, using the
following Newtonian relation, we may define the Newtonian mass corresponding
to the structure leading to the circular velocity reflected in the geodesic
equations:
$M_{N}(R)={R}^{2}[{R}{(\frac{d\phi}{d\lambda})}^{2}-\frac{{d}^{2}R}{d{\lambda}^{2}}].$
(13)
where $R$ is the areal radius with $\lambda$ its affine parameter, and the
acceleration given by
$\frac{{d}^{2}R}{d{\lambda}^{2}}=R^{\prime\prime}(\frac{dr}{d\lambda})^{2}+2{\dot{R^{\prime}}}(\frac{dr}{d\lambda})(\frac{dt}{d\lambda})+R^{\prime}({\frac{d^{2}r}{d{\lambda}^{2}}})+{\ddot{R}}(\frac{dt}{d\lambda})^{2}+\dot{R}(\frac{{d}^{2}t}{d{\lambda}^{2}})$
(14)
The terms ${\frac{dt}{d\lambda}},\frac{dr}{d\lambda},\frac{d\phi}{d\lambda}$
may be calculated from the geodesic equations.
In addition, we may use any general relativistic mass definition to calculate
the relativistic quasi-local mass of the structure. Understanding the
difference between these masses and the Newtonian one allocated to the
structure and its rotation curve will be the subject of our discussion.
## IV Newtonian versus general relativistic rotation curve
We report the results in three steps. To justify our algorithm and the code,
we first apply it to the vacuum Schwarzschild case. We then go over to a toy
model structure, and finally apply the algorithm to a more realistic case with
a NFW density profile.
### IV.1 Rotation curve in the vacuum case of Schwarzschild
Noting that the Schwarzschild space-time is a particular case of LTB space-
time, we choose the three LTB functions in the following way:
$f(r)=0,$ (15) $M(r)=m,$ (16) $t_{b}(r)=r.$ (17)
The resulting Schwarzschild metric is then given by
$ds^{2}=-dt^{2}+\frac{1}{(\frac{3}{4}m(r-t))^{\frac{2}{3}}}dr^{2}+(2m)^{\frac{2}{3}}(\frac{3}{2}(r-t))^{\frac{4}{3}}({d\theta}^{2}+{sin(\theta)}^{2}{d\phi}^{2}).$
(18)
This is the metric of the Schwarzschild space-time in a synchronous frame. By
studying circular geodesics having vanishing radial acceleration and velocity,
we obtain the rotation curve as given in the Fig.(1). Here the mass is the
unique Schwarzschild one, which is the same as the Misner-Sharp mass. It is
well known that in the case of Schwarzschild space-time at distances far from
the center, i.e. $\frac{2M}{R}<<1$, the Misner Sharp mass is equal to all the
other general relativistic masses defined so far. Now, to obtain the Newtonian
rotation curve we take the Newtonian mass to be equal to the Schwarzschild one
and then obtain the corresponding rotation curve. It is seen from Fig.(1) that
the relativistic and the Newtonian rotation curves coincide as expected.
Figure 1: General relativistic and the Newtonian rotation curves for the
Schwarzschild metric
### IV.2 A toy model
Now we go over to study the rotation curve for a non-trivial but simple toy
model. It is constructed as an exact solution of the Einstein equations to
represent a cosmic structure within a FRW universe. The three LTB functions
are fixed in the following wayjavad1 :
$\displaystyle M(r)=\frac{{r}^{3/2}\left(1+{r}^{3/2}\right)}{a},\hskip
22.76228pt\ t_{b}(r)=0,\hskip 22.76228pt\ f(r)=-(\frac{r{{\rm e}^{-r}}}{b}).$
(19)
The model represents a typical galaxy as a cosmological structure showing the
formation of a central black hole with distinct event- and apparent horizon.
As expected for a cosmological structure, the density profile shows a void
before entering the asymptotic FRW region. Fig.(2) shows the density profile
of the toy model and its corresponding relativistic and Newtonian rotation
curve. The Newtonian rotation curve is calculated using the mass related to
the density profile at a specific radius. As we see from the figure, both
rotation curves are almost identical.
We note by passing that the density profile is obtained without fixing the
mass which is not uniquely defined in general relativity in contrast to the
Newtonian case. Therefore, as our algorithm shows, the rotation curve may be
uniquely defined once we have fixed the density profile for a specific
solution of Einstein equations. In other words, any rotation curve within a
cosmological structure corresponds to a unique density profile but not a
unique mass, in contrast to the Newtonian case. This is shown in Fig. (3).
Figure 2: Density profile for the toy model of a typical galaxy and the
corresponding general relativistic and Newtonian rotation curve
Figure 3: Quasi-local masses for a toy model of a galaxy corresponding to the
density profile of the toy model
### IV.3 NFW model
Now, we try a model structure with a NFW density profile nfw . This density
profile is used to find the areal radius through an algorithm formulated in
krasinki , see also javad2 . To use this algorithm, the density profile has to
be specified at two different initial and final times, $t_{i},t_{f}$ as a
function of the coordinate $r$. Now, the algorithm is based on the choice of
$r$-coordinate such that $M(r)=r$. This is due to the fact that $M(r)$ is an
increasing function of $r$. Therefore, $E$ and $t_{b}$ become functions of
$M$. The LTB functions $E(M)$ and $t_{b}(M)$ may then be extracted from the
algorithm. For the initial time we choose the time of the last scattering
surface: $t_{i}\simeq 3.77\times 10^{5}yr$. The initial density profile should
show a small over-density near the center imitating otherwise a FRW universe.
Therefore, we add a Gaussian peak to the FRW background density. We know
already that having an over-density in an otherwise homogeneous universe needs
a void to compensate for the extra mass within the over-density region.
Therefore, to compensate this mass we subtract a wider gaussian peak. These
density profiles may then be written as
$\rho_{i}(r)=\rho_{crit}(t_{i})((\delta_{i}e^{-(\frac{r}{R_{i1}})^{2}}-b_{i})e^{-(\frac{r}{R_{i2}})^{2}}+{1}),$
(20)
$\rho_{f}(r)=\rho_{crit}(t_{f})((\frac{\delta_{c}}{(\frac{r}{r_{s}})(1+\frac{r}{r_{s}})^{2}}-b_{f})e^{-(\frac{r}{R_{f}})^{2}}+1),$
(21)
where $\delta_{i}$ is the density contrast of the Gaussian peak, $R_{i1}$ is
the width of the Gaussian peak and $R_{i2}$ is the width of the negative
Gaussian profile at the initial time. The two constants $b_{i}$ and $b_{f}$
are then obtained by the mass compensation condition. The results for the NFW
density profile of a typical galaxy and the corresponding relativistic and
Newtonian rotation curves are given in Fig.(4). We see again that both
Newtonian and relativistic rotation curves coincide.
We have also calculated different general relativistic masses to see how
different they are, although this does not change our results as far as the
rotation curve for a specific density profile is concerned. The results are
shown in Fig.(5). As in the case of the toy model, we realize that the Misner-
Sharp mass is almost identical to the Newtonian one. There is however
substantial difference to other general relativistic masses.
Figure 4: Density profile for the NFW model of a galaxy and the corresponding
Newtonian and relativistic rotation curve
Figure 5: Ratio of GR masses to the Newtonian mass for the NFW density
profile of a galaxy
## V discussion and conclusion
We have tried for the first time to answer the question of how a relativistic
rotation curve for a general relativistic structure within an otherwise
expanding universe looks like based on the Einstein equations. To achieve
this, we have modeled our so-called cosmological structure using a LTB
cosmological solution tending to FRW universe at infinity and having an
overdense structure at the center. After the original expansion, the overdense
region starts collapsing leading to a dense structure before going over to a
FRW universe. The mass in-fall to the structure reduces at late times leading
to an almost static structure. This late time behavior gives us the
possibility of defining quasi-circular orbits for particles revolving around
it.
We have then defined an algorithm how to obtain the relativistic rotation
curve for a specific density profile, without relying on a mass definition for
the structure which is not unique in general relativity. The numerical
procedure is tested first for a Schwarzschild metric where we have shown that
it is equivalent to the Newtonian rotation curve. We know already that in this
case all the general relativistic masses are equivalent to the Newtonian mass.
In the case of a general relativistic cosmological structure we choose first a
toy model and then a NFW density profile.
Now, for our cosmological structure which is embedded in a dynamical FRW
background, it turns out that the rotation curve at the galactic scale and at
far distances from the center of the structure (where the gravity is weak), is
almost identical to the Newtonian one, which is the main result of our work.
We have also calculated the general relativistic mass definitions for our
models to see how different they are, despite having a unique rotation curve
due to the specific density profile.
## VI ACKNOWLEDGMENT
We would like to thank Mojahed Parsi Mood for helpful discussions and
providing us his NFW model data.
## References
* (1) S. F. Flender and D. J. Schwarz, Phys. Rev. D 86, 063527 (2012) [arXiv:1207.2035 [astro-ph.CO]]; J. -c. Hwang and H. Noh, Gen. Rel. Grav. 38, 703 (2006) [astro-ph/0512636]; L. Lopez-Honorez, O. Mena and S. Rigolin, Phys. Rev. D 85, 023511 (2012) [arXiv:1109.5117 [astro-ph.CO]].
* (2) Stephen R. Green, Robert M. Wald, Phys. Rev. D 83: 084020, (2011); Stephen R. Green, Robert M. Wald, Phys. Rev. D 85: 063512, (2012).
* (3) J. D. Carrick, F. I. Cooperstock, [arXiv:1101.3224 [astro-ph]].
* (4) F. I. Cooperstock, S. Tieu, Mod. Phys. Lett. A23, 1745-1755 (2008). [arXiv:0712.0019 [astro-ph]].
* (5) J.T. Firouzjaee, Reza Mansouri, Gen. Relativ. Gravit. 42, 2431 (2010), [arXiv:0812.5108]; J.T. Firouzjaee, Int. J. Mod. Phys. D, 21, 1250039 (2012) [arXiv:1102.1062]; Krasinski A. and Hellaby C., Phys. Rev. D69, 043502 (2004); Rahim Moradi, J. T. Firouzjaee, Reza Mansouri, [arXiv:1301.1480].
* (6) J. T. Firouzjaee, M. Parsi Mood and Reza Mansouri, Gen. Relativ. Gravit. 44, 639 (2012) [arXiv:1010.3971]
* (7) Arnowitt R, Deser S and Misner C W 1962 in Gravitation, an Introduction to Current Research ed: Witten L (New York: Wiley)
* (8) Bondi H, van der Burg M G J and Metzner A W K 1962 Proc. Roy. Soc. Lond. A269 21; Sachs R K 1962 Proc. Roy. Soc. Lond. A270 103
* (9) Misner C W and Sharp D H 1964 Phys. Rev. 136 B571.
* (10) J.D. Brown and J.W. York, Jr., Phys. Rev. D 47, 1407- 1419 (1993).
* (11) Hawking S W 1968 J. Math. Phys. 9 598.
* (12) R.J. Epp, Phys. Rev. D 62, 124018 (2000)
* (13) C-C.M. Liu and S-T. Yau, Phys. Rev. Lett. 90, 231102 (2003).
* (14) Classical Dynamics of Particles and Systems,Stephen T. Thornton, Jerry B. Marion,3th Edition,1988
* (15) Hawking S W, J. Math. Phys. 9 598 (1968).
* (16) J.F. Navarro, C.S. Frenk, S.D.M. White, ApJ, 462, 563 (1996).
* (17) A. Krasiński, C. Hellaby, Phys. Rev. D 65, 023501 (2001).
|
arxiv-papers
| 2012-12-19T19:17:50 |
2024-09-04T02:49:39.470048
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mohammadhosein Razbin, Javad T. Firouzjaee and Reza Mansouri",
"submitter": "Javad Taghizadeh Firouzjaee",
"url": "https://arxiv.org/abs/1212.4796"
}
|
1212.5142
|
# Maxallent: Maximizers of all Entropies and Uncertainty of Uncertainty
A. N. Gorban [email protected] Department of Mathematics, University of
Leicester, Leicester, LE1 7RH, UK
###### Abstract
The entropy maximum approach (Maxent) was developed as a minimization of the
subjective uncertainty measured by the Boltzmann–Gibbs–Shannon entropy. Many
new entropies have been invented in the second half of the 20th century. Now
there exists a rich choice of entropies for fitting needs. This diversity of
entropies gave rise to a Maxent “anarchism”. The Maxent approach is now the
conditional maximization of an appropriate entropy for the evaluation of the
probability distribution when our information is partial and incomplete. The
rich choice of non-classical entropies causes a new problem: which entropy is
better for a given class of applications? We understand entropy as a measure
of uncertainty which increases in Markov processes. In this work, we describe
the most general ordering of the distribution space, with respect to which all
continuous-time Markov processes are monotonic (the Markov order). For
inference, this approach results in a set of conditionally “most random”
distributions. Each distribution from this set is a maximizer of its own
entropy. This “uncertainty of uncertainty” is unavoidable in the analysis of
non-equilibrium systems. Surprisingly, the constructive description of this
set of maximizers is possible. Two decomposition theorems for Markov processes
provide a tool for this description.
###### keywords:
uncertainty; Markov process; Lyapunov function; entropy; Maxent; inference
###### PACS:
05.45.-a , 82.40.Qt , 82.20.-w , 82.60.Hc
## 1 Introduction
Entropy was born in the 19th century as a daughter of energy:
${\mathrm{d}}S=\delta Q/T$. Clausius [1], Boltzmann [2] and Gibbs [3] (and
others) had developed the physical notion of entropy. At the same time, the
famous Boltzmann’s formula $S=k\log W$ had opened the informational
interpretation of entropy. In the 20th century, Hartley [4] and Shannon [5]
introduced a logarithmic measure of information in electronic communication in
order “to eliminate the psychological factors involved and to establish a
measure of information in terms of purely physical quantities” ([4], p. 536).
Information theory is focused on entropy as a measure of uncertainty of
subjective choice. This understanding of entropy was returned from information
theory to statistical mechanics by Jaynes [6] as a basis of “subjective”
statistical mechanics: “Information theory provides a constructive criterion
for setting up probability distributions on the basis of partial knowledge,
and leads to a type of statistical inference which is called the maximum
entropy estimate. It is least biased estimate possible on the given
information; i.e., it is maximally noncommittal with regard to missing
information. That is to say, when characterizing some unknown events with a
statistical model, we should always choose the one that has Maximum Entropy.”
This is the brief manifesto of the Maxent (maximum of entropy) methodology.
Entropy is used for measurement of uncertainty in a probability distribution.
The Maxent method finds the maximally uncertain distribution under given
values of some moments. After Jaynes, this approach became very popular in
physics [7, 8], statistics [9, 10], econometrics [11, 12] and other
disciplines.
The non-classical entropies were invented by Rényi [13] in the middle of the
20th century, simultaneously with the expansion of the Maxent approach. This
invention introduced additional uncertainty in the uncertainty evaluation.
Maximization of different entropies produces different probability
distributions under the same conditions. Now, one has to select the proper
entropy functional to use in the Maxent approach. This choice may be non-
obvious. The beautiful and transparent understanding of the Maxent
distribution as a unique “least biased estimate possible on the given
information” is now destroyed by the non-classical entropies. If we consider
the non-classical entropies seriously then we have to select the proper
entropy for each problem.
If we do not find solid reasons for the entropy selection then we have to
accept this “Uncertainty of Uncertainty” (UoU) as the nature of things. In
this case, the set of all the Maxent distributions for different entropies
will evaluate the unknown “maximally uncertain” distribution under given
conditions. We call this method of handling the UoU the “maximization of all
entropies” or Maxallent. If there are some reasons for selection of a class of
entropy function then we have to select the conditional maximizer of the
entropies from this class.
The widest class of entropies we use in this paper are the Csiszár–Morimoto
conditional entropies ($f$-divergencies). They were introduced by Rényi in his
famous work [13] where he proposed also the “Rényi entropy”. The
$f$-divergencies were studied further by Csiszar [14] and T. Morimoto [15].
For a discrete probability distribution $P=(p_{i})$ and the positive
“equilibrium distribution” $P^{*}=(p^{*}_{i})$, $p^{*}_{i}>0$ the general form
of the $f$-divergence is
$\boxed{H_{h}(P\|P^{*})=\sum_{i}p^{*}_{i}h\left(\frac{p_{i}}{p_{i}^{*}}\right),}$
(1)
where $h(x)$ is a convex function defined on the open ($x>0$) or closed
($x\geq 0$) semi-axis. We use here the notation $H_{h}(P\|P^{*})$ to stress
the dependence of $H_{h}$ both on $p_{i}$ and $p^{*}_{i}$.
In some practical problems, it is convenient to use a convex function $h(x)$
with singularity at $x=0$, for example, $h(x)=-\ln x$ (the Burg relative
entropy [16]). Therefore, we assume that the function $H_{h}(P\|P^{*})$ is
defined for positive $P$ and $P^{*}$. Convexity of $h(x)$ implies convexity of
$H_{h}(P\|P^{*})$ as a function of $P$. It achieves its minimal value on the
equilibrium probability, $P=P^{*}$ (under conditions $\sum_{i}p_{i}=1$, and
$p_{i}>0$). If $h(x)$ is strictly convex then $H_{h}(P\|P^{*})$ is also
strictly convex and this minimizer (the equilibrium) is unique.
### 1.1 Maxallent, approach #1: parametrization by monotonic function of one
variable
The standard settings for the Maxent approach are: an event space $\Omega$, a
divergency $H_{h}(P\|P^{*})$ and a set of moments $M_{r}(P)$ ($r=1,\ldots,k$)
are given. Here, $P$ is a probability distribution, $P^{*}$ is the “maximally
disordered” probability distribution (“equilibrium”) and $H_{h}(P\|P^{*})$
measures the deviation of $P$ from $P^{*}$. Of course, for general probability
spaces we have to assume that $P$ is absolutely continuous with respect to
$P^{*}$ and that it is possible to compute the divergence $H_{h}(P\|P^{*})$.
The Maxent problem is: for given values of the moments $M_{r}(P)$
($r=1,\ldots,k$) find the minimizers of $H_{h}(P\|P^{*})$. That is, on the set
of probability distributions with given values of $M_{r}(P)$ ($r=1,\ldots,k$)
find the distributions that are the closest to the equilibrium $P^{*}$ if we
measure the deviation by $H_{h}(P\|P^{*})$. The terminological mess (Maxent
and minimizers) appears due to historical reasons. Divergences measure the
differences between distributions and we always look for minimizers of them.
To avoid the irrelevant technicalities we consider discrete distributions. Let
$\Omega=\\{A_{1},A_{2},\ldots,A_{n}\\}$ be a finite event space with
probability distributions $P=(p_{i})$. The set of probability distribution is
the standard simplex $\Delta^{n-1}$ in $\mathbb{R}^{n}$. The set of positive
distribution ($p_{i}>0$) is $\Delta^{n-1}_{+}$, the relative interior of the
standard simplex.
The Maxent problem for $H_{h}(P\|P^{*})$ and given values of moments
$\sum_{j}m_{rj}p_{j}=M_{r}$ ($r=1,\ldots,k$) reads: find $P\in\Delta^{n-1}$
such that
$H_{h}(P\|P^{*})\to\min,\;\mbox{ subject to }\;\sum_{j}m_{rj}p_{j}=M_{r}.$
The total probability condition gives $\sum_{j}m_{0j}p_{j}=1$ ($m_{0j}=1$,
$M_{0}=1$). Assume that $k+1<n$ and
${\rm rank}(m_{rj})=k+1\;\;(r=0,1,\ldots,k;\,j=1,2,\ldots,n).$
If ${\rm rank}(m_{rj})<k+1$ then just exclude some moments.
The method of Lagrange multipliers gives for $P\in\Delta^{n-1}_{+}$
$h^{\prime}\left(\frac{p_{j}}{p_{j}^{*}}\right)=\sum_{r=0}^{k}\lambda_{r}m_{rj}\;\;(j=1,\ldots,n).$
(2)
The derivative $h^{\prime}$ is a monotonic function. Let $h$ be strictly
convex. Then the inverse function $g(y)$ exists, $g(h^{\prime}(x))=x$ (for
positive $x$). We can apply the function $g$ to both sides of (2) and write
the expression of $P$ and the equations for the Lagrange multipliers
$\lambda_{r}$ that are just the moment conditions $\sum_{j}m_{rj}p_{j}=M_{r}$:
$\boxed{\begin{gathered}p_{j}=p_{j}^{*}g\left(\sum_{r=0}^{k}\lambda_{r}m_{rj}\right)\;\;(j=1,\ldots,n);\\\
\sum_{j}m_{\rho
j}p_{j}^{*}g\left(\sum_{r=0}^{k}\lambda_{r}m_{rj}\right)=M_{\rho}\;\;(\rho=0,\ldots,k)\,.\end{gathered}}$
(3)
Therefore, for the class of the strictly convex functions $h$ all the positive
solutions of the Maxent problem for all $f$-divergencies are parameterized (3)
by the monotonic function $g$.
The function $g$ should be defined on a real interval
$(a,b)=h^{\prime}((0,\infty))$ (it might be that $a=-\infty$ or $b=\infty$).
The image of $g$ should be the real semi-axis $(0,\infty)$ because $p/p^{*}$
may be any positive number. Therefore, $\lim_{y\to a}g(y)=0$ and for finite
$a$ the function $g$ is defined on $[a,b)$. For each monotonically increasing
function $g$ on a real interval $(a,b)$ with ${\rm im}\,g=(0,\infty)$, the
corresponding solution of the Maxent problem is given by the distribution (3),
where $\lambda_{i}$ are the solutions of the corresponding equation. This
solution of the Maxent problem is the conditional minimizer of
$H_{h}(P\|P^{*})$ with $h(x)=\int h^{\prime}(x){\mathrm{d}}x$, where
$h^{\prime}(x)$ is the inverse function of $g(y)$, i.e. $h^{\prime}(x)=y$,
where $y$ is the solution to the equation $g(y)=x$. The additive constant in
$\int h^{\prime}(x){\mathrm{d}}x$ does not affect the solution of any Maxent
problems and may be chosen arbitrarily. Thus, we present the parametric
description of the minimizers of all strictly convex divergences
$H_{h}(P\|P^{*})$. A monotonic function $g$ with the values range $(0,\infty)$
serves as a parameter in this description.
For the existence of a positive distribution $P$ which satisfies (3) the
moment conditions $\sum_{j}m_{rj}p_{j}=M_{r}$ ($\rho=0,\ldots,k$) should be
compatible with the positivity of $p_{i}$. Of course, for arbitrary $g$ this
may be not sufficient for the existence of such a positive distribution. To
guarantee the existence of a positive Maxent distribution it is sufficient to
add to the function $h(x)$ a term $\varepsilon x\ln x$ with arbitrarily small
positive $\varepsilon$. This term creates a logarithmic singularity of
$h^{\prime}(x)$ at zero. It is easy to check that this singularity guarantees
the existence of a positive solution of (3) if the moment conditions are
compatible with the positivity of $p_{i}$. For some applied purposes an
additional term $-\varepsilon\ln x$ may be even more convenient [17] because
it guarantees the logarithmic singularity of entropy and $h^{\prime}(x)$ has
the singularity $\sim-1/x$ at zero.
In this paper, the question about existence of the positive Maxent
distribution is not important. We need only the conditions (3) which are
necessary and sufficient for a positive distribution $P=(p_{i})$ to provide a
minimizer of the given $f$-divergency under moment conditions.
### 1.2 Maxallent, approach #2: the Markov order
Any Markov process with equilibrium $P^{*}$ increases disorder. The classical
Boltzmann–Gibbs–Shannon entropy grows in Markov processes. This theorem (the
“data processing lemma”) was proved in the first paper of Shannon [5] but of
course the entropy growth in kinetics was known before (Boltzmann’s
$H$-theorem [2] and its generalization for the systems without detailed
balance [18]).
A. Rényi proved in the first paper about the non-classical entropies [13] that
all $f$-divergencies (1) decrease in Markov processes with equilibrium
$P^{*}$. Later on, it was demonstrated that this property characterizes
$f$-divergencies among all functions which can be presented in the form of the
sums over states (the “trace form”) [21, 22, 23].
The generalized data processing lemma was proven [24, 25]: For every two
positive probability distributions $P,Q$ the divergence $H_{h}(P\|Q)$
decreases under action of a stochastic matrix $A=(a_{ij})$
$H_{h}(AP\|AQ)\leq\overline{\alpha}(A)H_{h}(P\|Q),$
where
$\overline{\alpha}(A)=\frac{1}{2}\max_{i,k}\left\\{\sum_{j}|a_{ij}-a_{kj}|\right\\}$
is the ergodicity contraction coefficient, $0\leq\overline{\alpha}(A)\leq 1$.
A second method of handling the UoU is based on a simple remark: “uncertainty
of a probability distribution should increase in Markov processes”. More
precisely, let the most uncertain distribution $P^{*}$ be given (the
equilibrium). If a distribution $P^{\prime}$ can be obtained from a
distribution $P$ in a Markov process with equilibrium $P^{*}$ then we can
assume:
$\mbox{uncertainty of }P\leq\mbox{uncertainty of }P^{\prime}.$
Thus, we do not care about the values of the uncertainty measure, we just
compare the uncertainty of distributions: $P^{\prime}$ is more uncertain than
$P$ under given equilibrium $P^{*}$ (in this sense, the values vanish but the
(pre)order appears [23]).
In the Maxent approach, the entropy is used as a (pre)order in the
distribution space, not as a function, and the values are not important
because any monotonically increasing transformation of the entropy does not
change the solution of the Maxent problem. Of course, in some other
applications the values of entropy are important: in coding theory (bits per
symbol) and in thermodynamics (${\mathrm{d}}U=T{\mathrm{d}}S$) the values of
the entropy have a specific important sense. Nevertheless, when we discuss the
entropy as a measure of uncertainty and work with the huge population of non-
classical entropies, these entropies are, in their essence, (pre)orders on the
space of distributions.
We consider the continuous time Markov processes with a given equilibrium
distribution $P^{*}$. By definition, the equilibrium is the unconditionally
maximally uncertain distribution. To add the moment conditions we define a
linear manifold in the space of distributions. For every non-equilibrium
distribution $P$ each Markov process with the equilibrium distribution $P^{*}$
determines the direction of $P$ evolution, ${\mathrm{d}}P/{\mathrm{d}}t$. In
this direction, the distribution becomes more uncertain. Let us take this
property as a definition of the uncertainty. Instead of an entropy functional
we use the transitive closure of this relation, define an order on the space
of distributions and call it the “Markov order” [23].
Let $\mathbf{Q}(P,P^{*})$ be a cone of possible time derivatives
${\mathrm{d}}P/{\mathrm{d}}t$ for a given probability distribution $P$, the
equilibrium $P^{*}$, and all Markov processes with equilibrium $P^{*}$.
For fixed values of moments, $M_{r}$, the conditionally linear manifold $L$ in
the space of the probability distributions is given by equations
$\sum_{j}m_{rj}p_{j}=M_{r}$ ($r=0,\ldots,k$). We can consider $P^{0}\in L$ as
a possibly extremely disordered distribution on $L$, if for any Markov process
with equilibrium $P^{*}$ the solution $P(t)$ of the Kolmogorov equation with
initial condition $P(0)=P^{0}$ has no points on the conditionally linear
manifold $L$ for $t>0$ (we assume that $P^{0}$ is not a steady state for this
process). Instead of this global condition, we consider the local condition
(Fig. 1).
###### Definition 1.
The distribution $P^{0}\in L\cap\Delta^{n-1}_{+}$ is a local minimum of the
Markov order on $L\cap\Delta^{n-1}_{+}$ if
$\boxed{(P^{0}+\mathbf{Q}(P^{0},P^{*}))\cap L=\\{P^{0}\\}.}$ (4)
Further, for short, we can omit $\Delta^{n-1}_{+}$ and call $P^{0}$ “a local
minimum of the Markov order on $L$”. In this definition, we substitute the
trajectories $P(t)$ by their tangent directions at point $P^{0}$,
${\mathrm{d}}P(t)/{\mathrm{d}}t\in\mathbf{Q}(P^{0},P^{*})$. In Sec. 2 we
justify this substitution and prove that the local condition (4) holds if and
only if for every Markov process with equilibrium $P^{*}$ the solution $P(t)$
of the Kolmogorov equation with initial condition $P(0)=P^{0}$ has no points
on the condition linear manifold $L$ for $t>0$ (if $P^{0}$ is not a steady
state for the process).
Figure 1: The local condition (4): $P^{0}$ may be an extremely disordered
distribution on the condition linear manifold $L$ if the set
$P^{0}+\mathbf{Q}(P^{0},P^{*})$ intersects the linear manifold of conditions
at the only point $P^{0}$; (a)$(P^{0}+\mathbf{Q}(P,P^{*}))\cap L=\\{P^{0}\\}$
and $P^{0}$ may be an extremely disordered distribution on $L$; (b)
$(P^{0}+\mathbf{Q}(P,P^{*}))\cap L\varsupsetneqq\\{P^{0}\\}$ and $P^{0}$ has
no chance to be an extremely disordered distribution on $L$. In case (b),
there are more disordered distributions on $L$ achievable by the Markov
processes from the initial distribution $P^{0}$.
For applications, we need the local minima condition formalized by Definition
1 and the local order generated by the cone $\mathbf{Q}(P^{0},P^{*}))$ only.
The general notion of (global) Markov order appears later, in Section 3, where
we prove equivalence of the Maxima of all entropies and the Markov order
approaches. Surprisingly, the set of the conditional minimizers of all
$f$-divergencies and the set of the conditionally minimal elements of the
Markov order coincide for the same conditions (Sec. 3). These sets include all
reasonable hypotheses about conditionally most uncertain distributions. Let us
call the problem of description of all the conditional minima of the Markov
order the Maxallent problem.
### 1.3 Main tool: decomposition theorems
The main tools for constructive work with the Markov orders are the
decomposition theorems for Markov chains. The first decomposition theorem
states that every Markov chain with a positive equilibrium distribution is a
convex combination of the simple directed cyclic Markov chains with the same
equilibrium. The coefficients in this decomposition do not depend on the
current probability distribution: the vector field
${\mathrm{d}}P/{\mathrm{d}}t$ for a general Markov chain is a convex
combination of these vector fields for simple cyclic Markov chains with the
same positive equilibrium.
The second decomposition theorem states that for every Markov chain with a
positive equilibrium distribution and for any non-equilibrium distribution $P$
the velocity vector ${\mathrm{d}}P/{\mathrm{d}}t$ is a convex combination of
the velocity vectors for the simple cyclic Markov chains of the length two
with the same equilibrium (i.e. of the reversible transitions between two
states, $A_{i}\rightleftharpoons A_{j}$). The coefficients in this
decomposition typically depend on the current probability distribution.
The idea of the first decomposition theorem was used by Boltzmann in 1882 [18]
in his proof of the $H$-theorem for systems without detailed balance. (This
was his answer to the Lorentz objections [19].) He did not formulate this
theorem separately but efficiently used the cycle decomposition for
generalization of detailed balance. Later on, his extension of the detailed
balance conditions were analyzed by many authors under different names as
“cyclic balance”, “semi-detailed balance” or “complex balance” (see, for
example, the review [20]). Now, the theory of the cycle decomposition is a
well developed area of the theory and applications of the random processes
[26].
The second decomposition theorem is less known. We found this theorem in the
analysis of the Markov order [23]. This decomposition means that for the
general first-order kinetics and an arbitrary non-equilibrium probability
distribution $P$ there exists a system with detailed balance and the same
equilibrium that has the same velocity $dP/dt$ at point $P$ [27]: the classes
of the general Markov processes and the Markov processes with detailed balance
are pointwise equivalent.
The decomposition theorems are discussed in Appendix B in more detail.
## 2 Local minima of Markov order
Let us consider continuous time Markov chains with $n$ states
$A_{1},\ldots,A_{n}$. The Kolmogorov equation (or master equation) for the
probability distribution $P=(p_{i})$ is
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}(q_{ij}p_{j}-q_{ji}p_{i})\;\;(i=1,\ldots,n),$ (5)
where $q_{ij}$ ($i,j=1,\ldots,n$, $i\neq j$) are non-negative.
In this notation, $q_{ij}$ is the rate constant for the transition $A_{j}\to
A_{i}$. Any non-negative values of the coefficients $q_{ij}$ ($i\neq j$)
correspond to a master equation. Therefore, the set of all the Kolmogorov
equations (5) may be considered as the positive orthant
$\mathbb{R}_{+}^{n(n-1)}$ in $\mathbb{R}^{n(n-1)}$ with coordinates $q_{ij}$
($i\neq j$).
Now, let us restrict our consideration to the set of the Markov chains with
the given positive equilibrium distribution $P^{*}$ ($p^{*}_{i}>0$).
$\sum_{j,\,j\neq i}q_{ij}p^{*}_{j}=\left(\sum_{j,\,j\neq
i}q_{ji}\right)p^{*}_{i}\;\mbox{ for all }i=1,\ldots,n.$ (6)
This system of uniform linear equations define a cone of the $q_{ij}$
($i,j=1,\ldots,n$, $i\neq j$) in $\mathbb{R}_{+}^{n(n-1)}$.
Under the balance condition (6), the Kolmogorov equations (5) may be rewritten
in a convenient equivalent form:
$\boxed{\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}q_{ij}p^{*}_{j}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\;\;(i=1,\ldots,n).}$
(7)
We use below one of the $f$-divergencies (1) with $h(x)=(x-1)^{2}$. It is a
quadratic divergence, the weighted $l_{2}$ distance between $P$ and $P^{*}$:
$H_{2}(P\|P^{*})=\sum_{i}\frac{(p_{i}-p^{*}_{i})^{2}}{p^{*}_{i}}\,.$
With the master equation in the form (7), it is straightforward to calculate
the time derivative of $H_{2}(P\|P^{*})$
$\frac{{\mathrm{d}}H_{2}(P\|P^{*})}{{\mathrm{d}}t}=-\sum_{i,j,\,j\neq
i}q_{ij}p^{*}_{j}\left(\frac{p_{i}}{p^{*}_{i}}-\frac{p_{j}}{p^{*}_{j}}\right)^{2}\leq
0\,.$ (8)
Each term in the sum is non-negative. The time derivative (8) is strictly
negative if for a transition $A_{j}\to A_{i}$ the rate constant is positive,
$q_{ij}>0$, and $\frac{p_{i}}{p^{*}_{i}}\neq\frac{p_{j}}{p^{*}_{j}}$. Hence,
if the state $P$ is not an equilibrium (i.e., the right hand side in (7) is
not zero) then $\frac{{\mathrm{d}}H_{2}(P\|P^{*})}{{\mathrm{d}}t}<0$.
An important class of the Markov chains is formed by reversible chains with
detailed balance. The detailed balance condition reads:
$q_{ij}p^{*}_{j}=q_{ji}p^{*}_{i}\;\;\mbox{ for all }i,j=1,\ldots,n.$ (9)
Under this condition, there are only $\frac{n(n-1)}{2}$ independent
coefficients among $n(n-1)$ numbers $q_{ij}$. For example, we can arbitrarily
select $q_{ij}\geq 0$ for $i>j$ and then take
$q_{ij}=q_{ji}\frac{p_{i}^{*}}{p^{*}_{j}}$ for $i<j$. So, for given $P^{*}$,
the cone of the detailed balance systems (9) is a positive orthant in
$\mathbb{R}^{\frac{n(n-1)}{2}}$ embedded in $\mathbb{R}_{+}^{n(n-1)}$. The
equilibrium fluxes
$w_{ij}^{*}=q_{ij}p^{*}_{j}=q_{ji}p^{*}_{i}\;\;(i>j)$
are the convenient coordinates in $\mathbb{R}^{\frac{n(n-1)}{2}}$ for a
description of the systems with detailed balance.
Let $\mathbf{Q}(P,P^{*})$ be the set of all possible velocities
${\mathrm{d}}P/{\mathrm{d}}t$ at a non-equilibrium distribution $P$ for all
Markov chains which obey a given positive equilibrium $P^{*}$. According to
the second decomposition theorem, the set of all possible velocities
${\mathrm{d}}P/{\mathrm{d}}t$ for the chains with detailed balance and the
same equilibrium is the same cone $\mathbf{Q}(P,P^{*})$. Therefore,
$\mathbf{Q}(P,P^{*})$ is a convex polyhedral cone and its extreme rays consist
of the velocity vectors for two-state Markov chains $A_{i}\rightleftharpoons
A_{j}$ with rate constants $q_{ji}=\kappa/p_{j}^{*}$,
$q_{ji}=\kappa/p_{i}^{*}$ ($\kappa>0$).
The construction of the cones of possible velocities was proposed in 1979 [28]
for systems with detailed balance in the general setting, for nonlinear
chemical kinetics. These systems are represented by stoichiometric equations
of the elementary reactions coupled with the reverse reactions:
$\alpha_{\rho 1}A_{1}+\ldots+\alpha_{\rho n}A_{n}\rightleftharpoons\beta_{\rho
1}A_{1}+\ldots+\beta_{\rho n}A_{n}\,,$ (10)
where $\alpha_{\rho i},\,\beta_{\rho i}\geq 0$ are the stoichiometric
coefficient, $\rho$ is the reaction number ($\rho=1,\ldots,m$). The
stoichiometric vector of the $\rho$th reaction is an $n$ dimensional vector
$\gamma_{\rho}$ with coordinates $\gamma_{\rho i}=\beta_{\rho i}-\alpha_{\rho
i}$. The reaction rate is $w_{\rho}=w_{\rho}^{+}-w_{\rho}^{-}$, where
$w_{\rho}^{+}$ is the rate of the direct elementary reaction and
$w_{\rho}^{-}$ is the rate of the reverse reaction
The equilibria of the $\rho$th pair of reactions (10) form a hypersurface in
the space of concentrations. The intersection of these surfaces for all $\rho$
is the equilibrium (with detailed balance). Each surface of the equilibria of
a pair of elementary reactions (10) divides the non-negative orthant of
concentrations into three sets: (i) $w_{\rho}>0$, (ii) $w_{\rho}=0$ (the
surface of the equilibria) and (iii) $w_{\rho}<0$. All the surfaces of
equilibria ($w_{\rho}=0$) divide the non-negative orthant of concentrations
into compartments. In each compartment, the dominant direction of each
reaction (10) is fixed and, hence, the cone of possible velocities is also
constant. It is a piecewise constant function of concentrations:
$\mathbf{Q}={\rm cone}\\{\gamma_{\rho}{\rm
sign}(w_{\rho})\,|\,\rho=1,\ldots,m\\}\,,$
where “cone” stands for the conic hull, that is the set of all linear
combinations with non-negative coefficients. Here and below we use the three-
valued sign function (with values $\pm 1$ and 0).
Let us apply this construction to Markov chains with detailed balance. Let us
join the transitions $A_{i}\rightleftharpoons A_{j}$ in pairs (say, $i>j$) and
introduce the stoichiometric vectors $\gamma^{ji}$ with coordinates:
$\gamma^{ji}_{k}=\left\\{\begin{array}[]{ll}-1&\mbox{ if }k=j,\\\ 1&\mbox{ if
}k=i,\\\ 0&\mbox{ otherwise}.\end{array}\right.$ (11)
Let us rewrite the Kolmogorov equation for the Markov process with detailed
balance (9) in the quasichemical form:
$\frac{{\mathrm{d}}P}{{\mathrm{d}}t}=\sum_{i>j}w_{ij}^{*}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\gamma^{ji}\,.$
(12)
Here, $w_{ij}^{*}=q_{ij}p^{*}_{j}=q_{ji}p^{*}_{i}$ is the equilibrium flux
from $A_{i}$ to $A_{j}$ and back.
The cone of possible velocities for (12) is
$\boxed{\mathbf{Q}(P,P^{*})={\rm cone}\left\\{\gamma^{ji}{\rm
sign}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\,\Big{|}\,i>j\right\\}\,.}$
(13)
The standard simplex of distributions $P$ is divided by linear manifolds
$\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}$ into compartments. They are
the polyhedra where the cone of the local Markov order $\mathbf{Q}{(P,P^{*})}$
is constant. The compartments for the Markov chains with the positive
equilibrium $P^{*}$ correspond to various partial orders on the finite set
$\\{p_{i}/p_{i}^{*}\\}$ ($i=1,\ldots,n$).
Let us describe the compartments and cones in more detail following [23]. For
every natural number $k\leq n-1$ the $k$-dimensional compartments are
enumerated by surjective functions
$\sigma:\\{1,2,\ldots,n\\}\to\\{1,2,\ldots,k+1\\}$. Such a function defines
the partial ordering of quantities $\frac{p_{j}}{p_{j}^{*}}$ inside the
compartment:
$\frac{p_{i}}{p_{i}^{*}}>\frac{p_{j}}{p_{j}^{*}}\;\;{\rm
if}\;\;\sigma(i)<\sigma(j);\;\;\;\frac{p_{i}}{p_{i}^{*}}=\frac{p_{j}}{p_{j}^{*}}\;\;{\rm
if}\;\;\sigma(i)=\sigma(j)\,.$ (14)
Let $\mathcal{C}_{\sigma}$ be the corresponding compartment and $Q_{\sigma}$
be the corresponding local Markov order cone ($\mathbf{Q}(P,P^{*})=Q_{\sigma}$
if $P\in\mathcal{C}_{\sigma}$).
For a given surjection $\sigma$ the compartment $\mathcal{C}_{\sigma}$ and the
cone $Q_{\sigma}$ have the following description:
$\boxed{\begin{gathered}\mathcal{C}_{\sigma}=\left\\{P\ \Big{|}\
\frac{p_{i}}{p^{*}_{i}}=\frac{p_{j}}{p^{*}_{j}}\;\;{\rm
for}\;\;\sigma(i)=\sigma(j)\;\;{\rm
and}\;\;\frac{p_{i}}{p^{*}_{i}}>\frac{p_{j}}{p^{*}_{j}}\;\;{\rm
for}\;\;\sigma(j)=\sigma(i)+1\right\\}\,;\\\ Q_{\sigma}={\rm
cone}\\{\gamma^{ij}\ |\ \sigma(j)=\sigma(i)+1\\}.\end{gathered}}$ (15)
In Fig. 2, the partition of the standard distribution simplex into
compartments, and the cones (angles) of possible velocities are presented for
the Markov chains with three states. In the construction of this cone,
reversible chains with detailed balance are used. Due to the second
decomposition theorem, this construction of the cone of possible velocities is
valid for the class of general Markov chains (and not only for reversible
chains) with the same equilibrium. It seems quite surprising that the Markov
order for general Markov chains is generated by the reversible Markov chains
which satisfy the detailed balance principle.
Figure 2: The cones of possible velocities $\mathbf{Q}$ for all Markov chains
with three states and equilibrium equidistribution ($p_{i}^{*}=1/3$). The
triangle of the probability distributions $(p_{1},p_{2},p_{3})$ ($p_{i}\geq
0$, $p_{1}+p_{2}+p_{3}=1$) has the vertices $A_{i}$, where $p_{i}=1$ and other
probabilities are zeros. Equilibrium is the centre of the triangle. This
triangle is divided by three lines of partial equilibria
($A_{i}\rightleftharpoons A_{i}$) into 12 compartments and the equilibrium
point. Six compartments are triangles and six other compartments are segments.
For all compartments the cones (here the angles) of possible velocities are
shown. Each cone is connected with the corresponding compartment by a dashed
line. In each cone, the vectors $\gamma^{ji}{\rm
sign}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)$ are
presented. For the 2D (triangle) compartments all three vectors are non-zero.
For the 1D compartments (segments) only two these vectors are non-zero. The
vectors $\gamma^{ji}$ are presented separately, in the top left corner.
Let $L$ be a linear manifold in the probability distribution space. Due to
Definition 1, $P^{0}\in L\cap\Delta^{n-1}_{+}$ is a local minimum of the
Markov order on $L\cap\Delta^{n-1}_{+}$ if the condition (4) holds.
In Fig. 3 the sets of conditional minimizers are presented for the Markov
order on the straight line $L$ for the Markov chain with three states and
symmetric equilibrium ($p_{i}^{*}=1/3$). Two general positions of $L$ in the
probability triangle are used (Fig. 3a,b). If $L$ is parallel to one side of
the triangle (Fig. 3c) then the moments are just some of the $p_{i}$ and the
extreme points of the Markov order on $L\cap\Delta^{n-1}_{+}$ coincide with
the partial equilibria.
Figure 3: a) and b) Extreme points of the Markov order for the Markov chain
with three states and different positions of the condition line $L$. c)
Extreme points of the Markov order coincide with the partial equilibria, when
the moments are just some of $p_{i}$
Let $J$ be a set of pairs of indexes $(i,j)$ ($i>j$) and $\mathcal{K}_{J}$ be
the class of kinetic equations (12) with $w_{ij}^{*}=0$ for $(i,j)\notin J$
and $w_{ij}^{*}\geq 0$ for $(i,j)\in J$ ($i\neq j$). We define
$\Phi_{J}(P^{0})$ for an initial distribution $P^{0}$ as a set of all values
$P(t)$ ($t>0$) for solutions $P(t)$ of all equations from the class
$\mathcal{K}_{J}$ with initial value $P(0)=P^{0}$.
Consider a cone of possible velocities for the set of transitions
$A_{i}\rightleftharpoons A_{j}$, $(i,j)\in J$:
$\mathbf{Q}_{J}(P,P^{*})={\rm cone}\left\\{\gamma^{ji}{\rm
sign}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\,\Big{|}\,(i,j)\in
J\right\\}\,.$
The following proposition states that in a vicinity of the distribution
$P^{0}$ the sets $\Phi_{J}(P^{0})$ and $P^{0}+\mathbf{Q}_{J}(P^{0},P^{*})$
coincide. This gives a justification of the use of the cone of the tangent
directions $\mathbf{Q}_{J}(P^{0},P^{*})$ in the definition of the local minima
of the Markov order (4).
###### Proposition 1.
Let $\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\neq 0$ ($(i,j)\in J$) for
a distribution $P=P^{0}$. There exists a vicinity $U$ of $P^{0}$ where
$P^{0}+\mathbf{Q}_{J}(P^{0},P^{*})$ coincides with $\Phi_{J}(P^{0})$:
$(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))\cap U=\Phi_{J}(P^{0})\cap U\,.$
###### Proof.
There exists a Euclidean ball $B_{r}$ around $P^{0}$ where
$\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\neq 0$ ($(i,j)\in J$). Due to
(8), inside $B_{r}$, the divergence $H_{2}(P\|P^{*})$ strictly decreases with
$\lambda$ increasing along any ray $P^{0}+\lambda e$,
$e\in\mathbf{Q}_{J}(P,P^{*})$ ($\lambda>0$). For each ray, we can find the
minimum of $H_{2}(P\|P^{*})$ in $B_{r}$. Let the maximum of these minima be
$h_{r}(P^{0})$:
$h_{r}(P^{0})=\max_{e\in\mathbf{Q}_{J}(P,P^{*})}\left\\{\min_{\lambda>0}\\{H_{2}(P^{0}+\lambda
e\|P^{*})\,|\,P^{0}+\lambda e\in B_{r}\\}\right\\}\,.$
By construction, $H_{2}(P^{0}\|P^{*})>h_{r}(P^{0})$. The set
$U=\\{P\in B_{r}\,|\,H_{2}(P\|P^{*})>h_{r}(P^{0})\\}$
is a vicinity of $P^{0}$. The intersection
$(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))\cap U$ is
$(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))\cap
U=\\{P\in(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))\,|\,H_{2}(P\|P^{*})>h_{r}(P^{0})\\}\,.$
For any system from $\mathcal{K}_{J}$ on $B_{r}$ and for any distribution
$P\in(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))$ the velocity vector
${\mathrm{d}}P/{\mathrm{d}}t$ belongs to $\mathbf{Q}_{J}(P^{0},P^{*})$.
Obviously,
$(P+\mathbf{Q}_{J}(P^{0},P^{*}))\subset(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))$.
Therefore, the solution of this system with the initial condition $P(0)=P^{0}$
may leave the intersection $(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))\cap U$ through
the level surface $H_{2}(P^{0}\|P^{*})=h_{r}(P^{0})$ only. After that, the
solution cannot return to $U$ because in $U$ the value of
$H_{2}(P^{0}\|P^{*})$ are bigger, $H_{2}(P^{0}\|P^{*})>h_{r}(P^{0})$, and
$H_{2}(P(t)\|P^{*})$ should decrease in time along every solution of any
system from $\mathcal{K}_{J}$. Thus, one inclusion is proven,
$(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))\cap U\supseteq\Phi_{J}(P^{0})\cap U\,.$
To prove the second inclusion, $(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))\cap
U\subseteq\Phi_{J}(P^{0})\cap U$, we have to demonstrate that the solutions
$P(t)$ ($P(0)=P^{0}$, $t\geq 0$) of the equations from $\mathcal{K}_{J}$ cover
$(P^{0}+\mathbf{Q}_{J}(P^{0},P^{*}))$ in some vicinity of $P^{0}$.
The polyhedral cone $\mathbf{Q}_{J}(P^{0},P^{*})$ is covered by the simplicial
cones spanned by the sets of linearly independent vectors $\gamma^{ji}{\rm
sign}(\frac{p_{j}^{0}}{p_{j}^{*}}-\frac{p_{i}^{0}}{p_{i}^{*}})$. Therefore, it
is sufficient to prove the second inclusion for the simplicial cones
$\mathbf{Q}_{J}(P^{0},P^{*})$.
Let the vectors $\\{\gamma^{ji}|(i,j)\in J\\}$ be linearly independent. For
the simplicity of notation, let us enumerate the states in the order of the
values of ${p_{j}^{0}}/{p_{j}^{*}}$:
$\frac{p_{1}^{0}}{p_{1}^{*}}\geq\frac{p_{2}^{0}}{p_{2}^{*}}\geq\ldots\geq\frac{p_{n}^{0}}{p_{n}^{*}}\,.$
In these notations, ${\rm
sign}(\frac{p_{j}^{0}}{p_{j}^{*}}-\frac{p_{i}^{0}}{p_{i}^{*}})=1$ for all
$(i,j)\in J$ because $i>j$ and
$\frac{p_{j}^{0}}{p_{j}^{*}}-\frac{p_{i}^{0}}{p_{i}^{*}}\neq 0$ for $(i,j)\in
J$.
Consider a subset of the cone $\mathbf{Q}_{J}(P^{0},P^{*})$ (a “pyramid”)
$\mathcal{Q}_{J}(P^{0},P^{*})=\left\\{\sum_{(i,j)\in
J}\theta_{ij}\gamma^{ji}\,\Big{|}\,\theta_{ij}\geq 0,\,\sum_{(i,j)\in
J}\theta_{ij}<1\right\\}\,.$ (16)
The “base” of this pyramid is a simplex
$\mathcal{B}_{J}(P^{0},P^{*})=\left\\{\sum_{(i,j)\in
J}\theta_{ij}\gamma^{ji}\,\Big{|}\,\theta_{ij}\geq 0,\,\sum_{(i,j)\in
J}\theta_{ij}=1\right\\}\,.$
Let $\alpha>0$ be sufficiently small and, therefore,
$\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\neq 0$ ($(i,j)\in J$) in
$P^{0}+\alpha\mathcal{Q}_{J}(P^{0},P^{*})$. For this $\alpha$, a solution
$P(t)$ ($t\geq 0$) of an equation from the class $\mathcal{K}_{J}$ with
initial data $P(0)=P^{0}$ may leave $P^{0}+\alpha\mathcal{Q}_{J}(P^{0},P^{*})$
only through its base, $P^{0}+\alpha\mathcal{B}_{J}(P,P^{*})$.
Let us prove that if $\alpha$ is sufficiently small then for each point
$y\in\mathcal{B}_{J}(P,P^{*})$ there exists a system in $\mathcal{K}_{J}$
whose solution $P(t)$ ($t>0$, $P(0)=P^{0}$) leaves
$P^{0}+a\mathcal{Q}_{J}(P^{0},P^{*})$ through the point $P^{0}+\alpha y$. This
means that $P(t_{1})=P^{0}+\alpha x$ for some $t_{1}>0$ and
$P(t)\in\mathcal{Q}_{J}(P,P^{*})$ for $0<t<t_{1}$.
Each vector $x\in\mathcal{B}_{J}(P,P^{*})$ can be expanded into a linear
combination of $\gamma^{ji}$ ($(i,j)\in J$):
$x=\sum_{(i,j)\in J}\theta_{ij}\gamma^{ji},\;\theta_{ij}\geq 0\mbox{ and
}\sum_{(i,j)\in J}\theta_{ij}=1.$ (17)
With this expansion we define the system $K_{x}\in\mathcal{K}_{J}$ by the
condition $\frac{{\mathrm{d}}P}{{\mathrm{d}}t}\big{|}_{P=P^{0}}=x$:
$\frac{{\mathrm{d}}P}{{\mathrm{d}}t}=\sum_{(i,j)\in
J}\theta_{ij}\left(\frac{p_{j}^{0}}{p_{j}^{*}}-\frac{p_{i}^{0}}{p_{i}^{*}}\right)^{-1}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\gamma^{ji}\,.$
(18)
(Just take
$w_{ij}^{*}=\theta_{ij}\left(\frac{p_{j}^{0}}{p_{j}^{*}}-\frac{p_{i}^{0}}{p_{i}^{*}}\right)$
for $(i,j)\in J$ in (12).) A solution $P(t)$ ($P(0)=P^{0}$) of this equation
(18) can be also expanded into a linear combination of $\gamma^{ji}$
($(i,j)\in J$) ($x\in\mathcal{B}_{J}(P,P^{*})$):
$P(t)=P^{0}+tx+\frac{t^{2}}{2}f(t,x)=P^{0}+\sum_{\theta_{ij}>0}\left(t\theta_{ij}+\frac{t^{2}}{2}\nu_{ij}(t,\\{\theta_{lm}\\})\right)\gamma^{ji},$
(19)
where $\nu_{ij}(t,\\{\theta_{lm}\\})$ are analytic functions. If $x$ belongs
to a face $F$ of the cone $\mathbf{Q}_{J}(P^{0},P^{*})$ then $P(t)\in P^{0}+F$
for sufficiently small $t$.
The moment $t=t(\alpha,x)$ when the solution $P(t)$ (19) leaves
$P^{0}+\alpha\mathcal{Q}_{J}(P^{0},P^{*})$ is a root of equation
$t+\frac{t^{2}}{2}\sum_{(i,j)\in J}\nu_{ij}(t,\\{\theta_{lm}\\})=\alpha.$
Due to the standard inverse function theorems this root exists and the
function $t(\alpha,x)$ is smooth for sufficiently small $\alpha$ for all
$x\in\mathcal{B}_{J}(P,P^{*})$, and $t(\alpha,x)=\alpha+o(\alpha)$. The
solution $P(t)$ (19) of the system $K_{x}$ (18) leaves
$P^{0}+\alpha\mathcal{Q}_{J}(P^{0},P^{*})$ at the point
$P(t(\alpha,x))=P^{0}+\alpha y(x)$, where $y(x)\in\mathcal{B}_{J}(P,P^{*})$.
To prove that $y(\bullet):\mathcal{B}_{J}(P,P^{*})\to\mathcal{B}_{J}(P,P^{*})$
is a homeomorphism of the simplex $\mathcal{B}_{J}(P,P^{*})$ onto itself, let
us notice that the map $x\mapsto y(x)$ leaves the faces of the simplex
$\mathcal{B}_{J}(P,P^{*})$ invariant: vertices transform into themselves, the
same for edges, etc.
We use the following topological lemma, the multidimensional intermediate
value theorem. Consider a continuous map $\Psi:\Delta_{n}\to\Delta_{n}$ of the
$n$-dimensional standard simplex into itself. Let each face
$F\subset\Delta_{n}$ be $\Psi$-invariant, i.e. $\Psi(F)\subset F$. Then $\Psi$
is surjective. The proof is possible by induction in $n$: for $n=0$ it is
obvious, for $n=1$ this is just a 1D intermediate value theorem. In all
dimensions, it can be proved on the basis of the “no-retraction theorem” [29]
and simple inductive topological reasoning, which reduces the general case to
the situation when all the faces $F\subset\Delta_{n}$ consist of fixed points
of the map $\Psi$.
Therefore, for sufficiently small $\alpha$ the solutions $P(t)$ ($P(0)=P^{0}$,
$t\geq 0$) of the equations from $\mathcal{K}_{J}$ cover
$(P^{0}+a\mathcal{Q}_{J}(P^{0},P^{*}))$ in some vicinity of $P^{0}$. The
second inclusion is proven. Let us combine the inclusions and reduce the
vicinities, if necessary. ∎
If $\frac{p^{0}_{i}}{p^{*}_{i}}\neq\frac{p^{0}_{j}}{p^{*}_{j}}$ for all $i,j$
($i\neq j$) then
$\mathbf{Q}(P^{0},P^{*})=\mathbf{Q}(P,P^{*})$
for $P$ in some vicinity of $P^{0}$. If for some pairs $i,j$ ($i\neq j$)
$\frac{p^{0}_{i}}{p^{*}_{i}}=\frac{p^{0}_{j}}{p^{*}_{j}}$ (see Fig. 3c) then
for some $P\in P^{0}+\mathbf{Q}(P^{0},P^{*})$ the cone $\mathbf{Q}(P,P^{*})$
may be bigger than $\mathbf{Q}(P^{0},P^{*})$ even in a small vicinity of
$P^{0}$. Nevertheless, the set of trajectories $P(t)$ ($t>0$, $P(0)=P^{0}$)
remains in $P^{0}+\mathbf{Q}(P^{0},P^{*})$ for sufficiently small $t$. Let us
prove this statement.
Let $\mathcal{K}$ be the class of all master equations with detailed balance
with the positive equilibrium $P^{*}$ (12) with $w_{ij}^{*}\geq 0$ for all
$(i,j)$ ($i>j$). We define $\Phi(P^{0})$ for an initial distribution $P^{0}$
as a set of all values $P(t)$ ($t>0$) for solutions $P(t)$ of all equations
from the class $\mathcal{K}$ with initial value $P(0)=P^{0}$.
###### Proposition 2.
For every probability distribution $P^{0}$ there exists a vicinity $U$ of
$P^{0}$ where $P^{0}+\mathbf{Q}(P^{0},P^{*})$ coincides with $\Phi(P^{0})$:
$(P^{0}+\mathbf{Q}(P^{0},P^{*}))\cap U=\Phi(P^{0})\cap U\,.$
###### Proof.
The inclusion $(P^{0}+\mathbf{Q}(P^{0},P^{*}))\cap U\subset\Phi(P^{0})\cap U$
is proven in the second part of the proof of Proposition 1 because
$\mathcal{K}_{J}\subset\mathcal{K}$. We have to prove the inclusion
$(P^{0}+\mathbf{Q}(P^{0},P^{*}))\cap U\supset\Phi(P^{0})\cap U$.
Let us use the combinatorial description of compartments and cones (15). We
assume that $P^{0}\in\mathcal{C}_{\sigma}$ for a surjection
$\sigma:\\{1,2,\ldots,n\\}\to\\{1,2,\ldots,k+1\\}$. Let us recall that
$k=\dim\mathcal{C}_{\sigma}$. If $k=n-1$ then $\mathcal{C}_{\sigma}$ is an
open subset of the distribution space and the preimage of every
$l=1,2,\ldots,n$ consists of one point. For every $P\in\mathcal{C}_{\sigma}$
the cone $\mathbf{Q}(P,P^{*})$ coincides with $\mathbf{Q}(P^{0},P^{*})$ and
due to Proposition 1 there exists a vicinity $U$ of $P^{0}$ where
$P^{0}+\mathbf{Q}(P^{0},P^{*})$ coincides with $\Phi(P^{0})$.
Let $k<n-1$. Then for some $i=1,\ldots,k+1$ the preimage of $i$ includes more
than 1 point, $|\sigma^{-1}(i)|>1$. Let $I$ be the set of such $i$ and
$S_{i}=\sigma^{-1}(i)$ is the preimage of $i$. Due to (15),
$\mathbf{Q}(P^{0},P^{*})={\rm cone}\\{\gamma^{ij}\ |\
\sigma(j)=\sigma(i)+1\\}.$
For a sufficiently small ball $U_{r}$ with the centre $P^{0}$ and
$P\in(P^{0}+\mathbf{Q}(P^{0},P^{*}))\cap U$ the cone $\mathbf{Q}(P,P^{*})$ may
include also some $\gamma^{ij}$ with $\sigma(i)=\sigma(j)$ but
$\mathbf{Q}(P,P^{*})\subset{\rm cone}\\{\gamma^{ij}\ |\
\sigma(j)=\sigma(i)+1\mbox{ or }\sigma(j)=\sigma(i)\\}.$ (20)
Let us prove that for any Markov chain with equilibrium $P^{*}$ for
sufficiently small time $\tau>0$ and a ball $U_{r/2}$ with the centre $P^{0}$
the solutions of the Kolmogorov equations $P(t)$ do not leave
$P^{0}+\mathbf{Q}(P^{0},P^{*})$ during the time interval $[0,\tau]$ if
$P(0)\in(P^{0}+\mathbf{Q}(P^{0},P^{*}))\cap U_{r/2}$.
A set $V$ is positively invariant with respect to a dynamical system if every
motion that starts in $V$ at $t=0$ remains there for $t>0$. Let a convex set
$V$ be positively invariant with respect to several dynamical system given by
Lipschitz vector fields $\mathbf{w}_{1},\ldots,\mathbf{w}_{r}$. Then $V$ is
positively invariant with respect to any combination
$\mathbf{w}=\sum_{j}f_{j}\mathbf{w}_{j}$, where $f_{j}$ are non-negative
functions and $\mathbf{w}$ is a Lipschitz vector field. Therefore, the problem
of positive invariance of a convex set with respect to such combinations of
vector fields can be “split” into problems of the positive invariance of $V$
with respect to summands $\mathbf{w}_{j}$. Due to the second decomposition
theorem, we can always assume that the vector field of the Kolmogorov equation
for the Markov kinetics is a linear combination of the vector fields of the
pairs of elementary transitions $A_{i}\rightleftharpoons A_{j}$ with the same
equilibrium. The coefficients in these combinations are non-negative
functions.
The motion $P(t)$ with $P(0)\in(P^{0}+\mathbf{Q}(P^{0},P^{*}))$ does not leave
$(P^{0}+\mathbf{Q}(P^{0},P^{*}))$ in time $t\in[0,\tau]$ if
${\mathrm{d}}P(t)/{\mathrm{d}}t\in\mathbf{Q}(P^{0},P^{*})$ on $[0,\tau]$.
The cone $\mathbf{Q}(P^{0},P^{*})$ is generated by vectors $\gamma^{ij}$ with
$\sigma(j)=\sigma(i)+1$. To generate a cone $\mathbf{Q}(P,P^{*})$ for a point
$P\in U_{r}$ we have to add to the set of $\gamma^{ij}$
($\sigma(j)=\sigma(i)+1$) some of $\gamma^{ij}$ with $\sigma(j)=\sigma(i)$.
Let us consider the pyramid (compare to (16))
$\mathcal{Q}(P^{0})=\left\\{\sum_{\sigma(j)=\sigma(i)+1}\theta_{ij}\gamma^{ji}\,\Big{|}\,\theta_{ij}\geq
0,\,\sum_{(i,j)\in J}\theta_{ij}<1\right\\}\,.$
We will prove that the set $P^{0}+a\mathcal{Q}(P^{0})$ is positively invariant
with respect to any first order kinetics with transitions
$A_{i}\rightleftharpoons A_{j}$ ($i,j\in S_{l}$) and equilibrium $P^{*}$ for
any $l=1,\ldots,k+1$.
It is sufficient to consider dynamics in projections on the coordinate
subspace $\mathbf{R}_{S_{l}}$ with coordinates $p_{i}$, $i\in S_{l}$ for every
$l\in I$ separately. In this space, vectors $\gamma^{ij}$ ($i,j\in S_{l}$)
correspond to the standard first order kinetics like (12) with the reduced
vector $P\in\mathbf{R}_{S_{l}}$ but without compulsory unit balance
($\sum_{i\in S_{l}}p_{i}=const$ with any $const>0$). A projection of $P^{0}$
on $\mathbf{R}_{S_{l}}$, $P^{0}_{S_{l}}$ is an equilibrium for this first
order kinetics with the balance $\sum_{i\in S_{l}}p_{i}=\sum_{i\in
S_{l}}p^{0}_{i}$ because
$\frac{p^{0}_{i}}{p^{*}_{i}}=\frac{p^{0}_{j}}{p^{*}_{j}}$ for $i,j\in S_{l}$.
The vectors $\gamma^{ij}$ that generate $\mathbf{Q}(P^{0},P^{*})$
($\sigma(j)=\sigma(i)+1$) (20) have non-zero projections on
$\mathbf{R}_{S_{l}}$ if and only if either $l=\sigma(j)=\sigma(i)+1$ or
$\sigma(j)=\sigma(i)+1=l+1$. In the first case, $l=\sigma(j)=\sigma(i)+1$,
vector $\gamma^{ij}$ is the standard basis vector $e_{j}$ in
$\mathbf{R}_{S_{l}}$. In the second case, $\sigma(j)=\sigma(i)+1=l+1$, we have
$\gamma^{ij}=-e_{i}$. If $l=1$ then only the second case is possible, and if
$l=k+1$ then only the first case can take place.
Let $V_{l}=\\{P\in\mathbf{R}_{S_{l}}\ |\ p_{i}\geq 0,\sum_{i\in
S_{l}}p_{i}<1\\}$. The projection of the pyramid $\mathcal{Q}(P^{0})$ onto
$\mathbf{R}_{S_{l}}$ is $\mbox{conv}(V_{l}-V_{l}))$ if $1<l<k+1$; it is
$V_{l}$ if $l=k+1$ and $-V_{l}$ if $l=1$. (For sets $X,Y$, the sum $X+Y$ is
the set of all sums $x+y$ ($x\in X$, $y\in Y$), the difference $X-Y$ is the
set of all differences $x-y$, therefore $V-V$ is not $\\{0\\}$ if $V$ includes
more than one element.)
The set $V_{l}$ is positively invariant with respect to the first order
kinetics in $\mathbf{R}_{S_{l}}$. Therefore, the following sets are also
positively invariant with respect to the first order kinetics in
$\mathbf{R}_{S_{l}}$ with equilibrium $P^{0}_{S_{l}}$ for every $a>0$:
$P^{0}_{S_{l}}+aV_{l},\,(P^{0}_{S_{l}}+aV_{l})-aV_{l},\,\mbox{conv}((P^{0}_{S_{l}}+aV_{l})-aV_{l})\,.$
Thus, the set $P^{0}+a\mathcal{Q}(P^{0})$ is positively invariant with respect
to any first order kinetics with transitions $A_{i}\rightleftharpoons A_{j}$
($i,j\in S_{l}$) and equilibrium $P^{*}$ for any $l=1,\ldots,k+1$. A
combination of these statements for all $l=1,\ldots,k+1$ finalizes the proof.
∎
This proposition finalizes the justification of the use of the cone of the
tangent directions $\mathbf{Q}(P^{0},P^{*})$ in the definition of the local
minimum of the Markov order (4).
## 3 Equivalence of the maxima of all entropies and the Markov order
approaches
The cone $\mathbf{Q}(P^{0},P^{*})$ is a piecewise constant function of
$P^{0}$: it is the same for all $P^{0}$ from one compartment
$\mathcal{C}_{\sigma}$ and, hence, depends on $\sigma$ only. Therefore, if the
condition of the local minimum (4) holds for one $P^{0}\in
L\cap\mathcal{C}_{\sigma}$ then it holds also for all elements of
$L\cap\mathcal{C}_{\sigma}$. There is a finite number of compartments
$\mathcal{C}_{\sigma}$.
Let the linear manifold of conditions $L$ be given by the values of moments
$\sum_{i}m_{ri}p_{i}=M_{r}$, $L^{0}=\ker m$ and
$L\cap\Delta^{n-1}_{+}\neq\emptyset$. The set of all conditional local minima
of the Markov order on the linear manifold of conditions $L$ is
$\boxed{\bigcup\left\\{L\cap\mathcal{C}_{\sigma}\ \big{|}\
L\cap\mathcal{C}_{\sigma}\neq\emptyset\;\mbox{ and }\;L^{0}\cap
Q_{\sigma}=\\{0\\}\right\\}\,,}$ (21)
where $\mathcal{C}_{\sigma}$ and $Q_{\sigma}$ are defined by (15). It is
sufficient to find all $\sigma$ such that
$L\cap\mathcal{C}_{\sigma}\neq\emptyset$ and $L^{0}\cap Q_{\sigma}=\\{0\\}$
and then describe the union of the compartments $\mathcal{C}_{\sigma}$ for
these $\sigma$.
The approach based on the minimization of all $f$-divergencies seems to be
very different. For all monotonically increasing functions $g$ we have to
solve the equations for the Lagrange multipliers and represent the probability
distribution in the form (3). Nevertheless, these approaches are equivalent
and describe the same set of the “conditionally maximally disordered
distributions”.
###### Theorem 1.
A positive distribution $P^{0}\in L$ satisfies the local conditional minimum
conditions of the Markov order (4) if and only if there exists a strictly
monotonic function $g$ on $\mathbb{R}$ with ${\rm im}\,g=(0,\infty)$ such that
the conditions (3) hold for some Lagrange multipliers and for
$p_{i}^{0}=p_{i}$.
This means that every conditionally minimal distribution of the Markov order
on the linear manifold $L\cap\Delta^{n-1}_{+}$ is a conditional minimum on
$L\cap\Delta^{n-1}_{+}$ of a strictly convex $f$-divergence (1).
###### Proof.
Due to the classical theorems about separation of convex sets and linear
spaces by linear functionals [30], a distribution $P^{0}$ satisfies the
condition of the local minimum (4) if and only if there exists a linear
functional $\psi(P)=\sum_{i}\psi_{i}p_{i}$ such that
$\psi|_{L}=\psi(P^{0})=const$ and $\psi(P)>\psi(P^{0})$ for every $P\in
P^{0}+\mathbf{Q}(P^{0},P^{*})$ if $P\neq P^{0}$. In other words,
$\psi|_{L^{0}}\equiv 0$ and
$(\psi_{i}-\psi_{j})\left(\frac{p_{i}^{0}}{p_{i}^{*}}-\frac{p_{j}^{0}}{p_{j}^{*}}\right)>0\;\mbox{
if }\;\frac{p_{i}^{0}}{p_{i}^{*}}\neq\frac{p_{j}^{0}}{p_{j}^{*}}\,.$ (22)
according to the definition of $\mathbf{Q}(P,P^{*})$ (13). Condition
$\psi|_{L^{0}}\equiv 0$ is equivalent to the existence of the coefficients
$\lambda_{r}$ such that for all $i$
$\psi_{i}=\sum_{r}\lambda_{r}m_{ri}\,.$
Condition (22) is equivalent to the existence of a strictly monotonic function
$\eta(x)$ defined for $x\geq 0$ such that
$\psi_{i}=\eta\left(\frac{p_{i}^{0}}{p_{i}^{*}}\right)\,.$
To find such a function $\eta(x)$ we can take the known values $\psi_{i}$ for
$x={p_{i}^{0}}/{p_{i}^{*}}$ and then use, for example, linear interpolation
$\eta(x)$ between ${p_{i}^{0}}/{p_{i}^{*}}$. To extrapolate $\eta(x)$ from
$\max\\{{p_{i}^{0}}/{p_{i}^{*}}\\}$ to $+\infty$ we can use an increasing
linear function. To extrapolate $\eta(x)$ on the interval
$(0,\min\\{{p_{i}^{0}}/{p_{i}^{*}}\\})$ we can use $\varepsilon\log x+const$.
Finally, we can take $h^{\prime}(x)=\eta(x)$,
$h(x)=\int\eta(\xi)\,{\mathrm{d}}\xi$; and $g(y)$ is the inverse function:
$g(\eta(x))=x$ for $x\geq 0$. The distribution $P^{0}$ is the local minimum of
$H_{h}(P\|P^{*})$ on $L$.
Conversely, if $P^{0}$ is a minimum of a strictly convex Lyapunov function $H$
on $L$ and ${\mathrm{d}}H/{\mathrm{d}}t|_{P^{0}}<0$ for every Markov chain
with equilibrium $P^{*}$ for which $P^{0}$ is a non-equilibrium distribution
then we can take
$\psi_{i}=-\left.\frac{\partial H}{\partial p_{i}}\right|_{P^{0}}\,.$
This choice of $\psi_{i}$ provides (22) (because $H$ is strictly decreasing in
time Lyapunov function) and $\psi|_{L^{0}}\equiv 0$ because grad$H$ is
orthogonal to $L$ (the condition of local minimum). ∎
This equivalence of two definitions of the maximally uncertain distribution
under given conditions has several important consequences.
Let us introduce the notion of the (global) Markov order [23].
* 1.
If for distributions $P^{0}$ and $P^{1}$ there exists such a Markov process
with equilibrium $P^{*}$ that for the solution of the Kolmogorov equation with
$P(0)=P^{0}$ we have $P(1)=P^{1}$ then we say that $P^{0}$ and $P^{1}$ are
connected by the Markov preorder [23] with equilibrium $P^{*}$ and use
notation $P^{0}\succ^{0}_{P^{*}}P^{1}$.
* 2.
The (global) Markov order is the closed transitive closure of the Markov
preorder. For the Markov order with equilibrium $P^{*}$ we use notation
$P^{0}\succ_{P^{*}}P^{1}$.
The local Markov order at point $P^{0}$ is just a vector order generated by
the tangent cone $\mathbf{Q}(P^{0},P^{*})$ [23]. We use for this local order
the notation $>_{P^{0},P*}$:
$P>_{P^{0},P^{*}}P^{\prime}\;\mbox{ if
}\;P^{\prime}-P\in\mathbf{Q}(P^{0},P^{*}).$
The proofs of Propositions 1 and 2 give us the possibility to use the relation
$P^{0}>_{P^{0},P^{*}}P^{1}$ instead of the Markov preorder for the definition
of the Markov order minimizers on linear manifolds. The relation
$P^{0}>_{P^{0},P^{*}}P^{1}$ is defined by the local Markov order in a vicinity
of $P^{0}$:
$P^{1}-P^{0}\in\mathbf{Q}(P^{0},P^{*}).$
The cone $\mathbf{Q}(P^{0},P^{*})$ depends on $P^{0}$, therefore, the relation
$P^{0}>_{P^{0},P^{*}}P^{1}$ is antisymmetric locally, in a vicinity of
$P^{0}$.
###### Remark 1.
It is possible to generate the Markov order by the relation
$P^{0}>_{P^{0},P^{*}}P^{1}$. Let us specify the vicinity of $P^{0}$ where this
relation is defined and introduce a new relation: $P^{0}>_{P^{*}}^{0}P^{1}$ if
$P^{0}>_{P^{0},P^{*}}P^{1}$ for all $i,j=1,\ldots,n$ and
$\left(\frac{p^{0}_{i}}{p^{*}_{i}}-\frac{p^{0}_{j}}{p^{*}_{j}}\right)\left(\frac{p^{1}_{i}}{p^{*}_{i}}-\frac{p^{1}_{j}}{p^{*}_{j}}\right)\geq
0\,.$
This condition means that the pairs of numbers
$(\frac{p^{0}_{i}}{p^{*}_{i}},\frac{p^{0}_{j}}{p^{*}_{j}})$ and
$(\frac{p^{1}_{i}}{p^{*}_{i}},\frac{p^{1}_{j}}{p^{*}_{j}})$ cannot have an
opposite order on the real line. The closed transitive closure of the relation
$P^{0}>_{P^{*}}^{0}P^{1}$ is the Markov order $P^{0}\succ_{P^{*}}P^{1}$
Let $L$ be a linear manifold in the space of distributions. By definition,
$P^{0}\in L$ is a minimal point on $L\cap\Delta^{n-1}_{+}$ with respect to the
order $\succ_{P^{*}}$ if and only if there is no point $P^{1}\in
L\cap\Delta^{n-1}_{+}$, $P^{1}\neq P^{0}$ such that $P^{0}\succ_{P^{*}}P^{1}$.
###### Corollary 1.
$P^{0}\in L\cap\Delta^{n-1}_{+}$ is a minimal point on $L\cap\Delta^{n-1}_{+}$
with respect to the (global) Markov order if and only if it satisfies the
local minimum condition (4).
###### Proof.
If $P^{0}\in L\cap\Delta^{n-1}_{+}$ is a minimal point on
$L\cap\Delta^{n-1}_{+}$ with respect to the (global) Markov order then it
satisfies the condition (4) due to the definition of the Markov order through
the transitive closure of the relation $P^{0}\succ^{0}_{P^{*}}P^{1}$ and
Propositions 1 and 2.
Let $P^{0}$ satisfy the local minimum condition (4). Then there exists a
divergence $H_{h}(P\|P^{*})$ with strictly convex $h(x)$ ($x\geq 0$) such that
$P^{0}$ is a local minimum of $H_{h}(P\|P^{*})$ on $L$. Because of strong
convexity, this local minimum is a global one. $H_{h}(P\|P^{*})$ is a Lyapunov
function for all Markov chains with equilibrium $P^{*}$. Therefore, a broken
line, which is combined from solutions of the Kolmogorov equations for such
Markov chains and starts at $P^{0}$, leaves a small vicinity of $P^{0}$
(Propositions 1 and 2) and never returns in a sufficiently small vicinity of
$L$. Thus, for the closed transitive closure of the relation
$P\succ^{0}_{P^{*}}P^{\prime}$, point $P^{0}$ is a minimal point on $L$. ∎
Of course, there may be infinitely many minimal points of the Markov order on
$L$ and each of them corresponds to a different Lyapunov functions
$H_{h}(P\|P^{*})$.
Another remarkable order on the space of distributions is
$P^{0}>_{H,P^{*}}P^{1}$ if for all strictly convex functions $h(x)$ ($x\geq
0$)
$H_{h}(P^{0}\|P^{*})>H_{h}(P^{1}\|P^{*})\,,$
that is, $P^{1}$ is closer to equilibrium than $P^{0}$ with respect to all
divergencies $H_{h}(P\|P^{*})$.
###### Corollary 2.
For any linear manifold $L$ in the distribution space the minimal elements of
the Markov order $\succ_{P^{*}}$ on $L\cap\Delta^{n-1}_{+}$ coincide with the
minimal elements of the order $>_{H,P^{*}}$ on $L\cap\Delta^{n-1}_{+}$.
###### Proof.
We just have to combine Theorem 1 with Corollary 1. ∎
Thus, the minimal elements of the orders $\succ_{P^{*}}$ and $>_{H,P^{*}}$ on
the linear manifolds coincide. Nevertheless, it is necessary to mention the
difference between these orders. Let $P^{0}$ be a distribution. For
$>_{H,P^{*}}$ the set of distributions $\\{P\ |\ P^{0}>_{H,P^{*}}P\\}$ is
convex as an intersection of convex sets $\\{P\ |\
H_{h}(P^{0}\|P^{*})>H_{h}(P\|P^{*})\\}$ for various strictly convex $h$. This
is not the case for the Markov order. The set of distributions $\\{P\ |\
P^{0}\succ_{P^{*}}P\\}$ may be non-convex. The examples may be extracted from
the papers [28, 31] (see Fig. 4).
Figure 4: The set $\\{P\ |\ P^{0}>_{H,P^{*}}\\}$ for different $P^{0}$ and for
the Markov chains with three states and equilibrium ($p_{i}^{*}=1/3$): (a)
$\\{P\ |\ P^{0}>_{H,P^{*}}\\}$ is convex, (b) it is not convex. The border of
the set $\\{P\ |\ P^{0}>_{H,P^{*}}\\}$ is highlighted by bold lines. The
arrows on these lines correspond to the directions of the extreme rays of the
cones $\mathbf{Q}(P,P^{*})$ (i.e. the angles represented in Fig. 2).
###### Corollary 3.
Let $P^{0}\succ_{P^{*}}P^{1}$. Then $P^{1}\in P^{0}+\mathbf{Q}(P^{0},P^{*})$.
###### Proof.
Let us apply Corollary 1 to all support hyperplanes $L$ of the convex set
$(P^{0}+\mathbf{Q}(P^{0},P^{*}))$ for which
$(P^{0}+\mathbf{Q}(P^{0},P^{*}))\cap L=\\{P^{0}\\}$. ∎
## 4 Example: generalization of the normal distribution
In this section, we discuss distributions $p(x)$ on a continuous space of
states, the non-negative real semi-axis, $\mathbb{R}_{+}=\\{x\ |\ x\geq 0\\}$.
We have in mind two classical examples of distributions of the quantity
bounded from below: energy (physics) and wealth (economics and
microeconomics).
Let two moments be fixed, the total probability
$M_{0}=\int_{0}^{\infty}p(x)\,{\mathrm{d}}x$ and the average quantity
$M_{1}=\int_{0}^{\infty}xp(x)\,{\mathrm{d}}x$. The conditional maximization of
the classical Boltzmann–Gibbs–Shannon entropy gives:
$\int p(x)(\ln p(x)-1)\,{\mathrm{d}}x\to\min\;\mbox{ for given
}\;\int_{0}^{\infty}p(x)=M_{0},\;\int_{0}^{\infty}p(x)\,{\mathrm{d}}x=M_{1}\,;$
$\ln
p(x)=\lambda_{0}+\lambda_{1}x,\;p(x)=\exp(\lambda_{0}+\lambda_{1}x),\;\exp\lambda_{0}=-\lambda_{1}\;\exp\lambda_{0}=M_{1}\lambda_{1}^{2}\,;$
$\boxed{p^{*}(x)=\frac{1}{M_{1}}\exp\left(-\frac{x}{M_{1}}\right)\,.}$ (23)
This Boltzmann distribution appears always as a first candidate for the
equilibrium distribution of an additive conserved quantity bounded from below.
Khinchin (1943) clearly explained this law as a version of the limit theorem
[32].
Technically, it is not difficult to involve the higher moments and obtain the
distribution of the form
$p(x)=\exp(\lambda_{0}+\lambda_{1}x+\lambda_{2}x^{2}+\ldots+\lambda_{r}x^{r})\,.$
(24)
One can expect that this extension of the set of moments may improve the
description. This is a traditional belief in Extended Irreversible
Thermodynamics (EIT) [7].
There may be many different approaches to evaluation of the quality of the
approximation (24) but at least one important property of these functions is
wrong: the asymptotic behavior at large $x$ is $p(x)\asymp\exp(-const\times
x^{r})$. These “super-light” tails of the distribution $p(x)$ change
qualitatively with the change of the order $r$ in (24).
If we use, for example, the “regularizing” forth moment in the moment chain
for the Boltzmann equation [33] then we corrupt the ${\rm e}^{-const\times
v^{2}}$ tails of the Maxwell distribution. Therefore, other approaches which
do not modify the tails of the distribution qualitatively (like [8]) may be
more appreciated.
The asymptotic behavior of the distribution’s tails was thoroughly studied in
many cases. Very often, the tails of the distributions are, without a doubt,
heavier than normal ${\rm e}^{-const\times x^{2}}$ and definitely are not cut
as ${\rm e}^{-const\times x^{4}}$. For example, it is demonstrated that the
distribution of money between people has the exponential tail with a possible
transformation into a heavier power tail for very rich people [34].
The general solution (3) with the Boltzmann equilibrium (23) gives the
following expression instead of (24)
$p(x)=g(\lambda_{0}+\lambda_{1}x+\lambda_{2}x^{2}+\ldots+\lambda_{r}x^{r})\frac{1}{M_{1}}\exp\left(-\frac{x}{M_{1}}\right)\,,$
where $g$ is a monotonically increasing function. In particular, for the
moments $M_{0}$, $M_{1}$ and $M_{2}$ we obtain
$p(x)=g(\lambda_{0}+\lambda_{1}x+\lambda_{2}x^{2})\frac{1}{M_{1}}\exp\left(-\frac{x}{M_{1}}\right)\,.$
(25)
There are four qualitatively different cases of (25). Let $\lambda_{2}\neq 0$
and $\mu=-\frac{\lambda_{1}}{2\lambda_{2}}$. Then
$\boxed{p(x)=\frac{f(x)}{M_{1}}\exp\left(-\frac{x}{M_{1}}\right)\,,}$ (26)
and
1. 1.
if $\mu\leq 0$ and $\lambda_{2}>0$ then $f(x)$ is a monotonically increasing
function on $[0,\infty)$;
2. 2.
if $\mu\leq 0$ and $\lambda_{2}<0$ then $f(x)$ is a monotonically decreasing
function on $[0,\infty)$;
3. 3.
if $\mu>0$ and $\lambda_{2}>0$ then $f(x)$ is a monotonically increasing
function on $[\mu,\infty)$ and $f(x)=f(2\mu-x)$ for $x\in[0,\mu]$;
4. 4.
if $\mu>0$ and $\lambda_{2}<0$ then $f(x)$ is a monotonically decreasing
function on $[\mu,\infty)$ and $f(x)=f(2\mu-x)$ for $x\in[0,\mu]$.
Each of these “generalized normal distributions” (26) is a minimizer of the
corresponding $f$-divergence. For the construction of such a divergence in
general case, it is convenient to define the convex functions $h$ in (1) with
values on an extended real line with additional possible value $+\infty$. This
is a natural general definition of convex functions [30]. In case 1 ($\mu\leq
0$, $\lambda_{2}>0$, and $f$ increases), we can take in (25), (26) without
loss of generality $\mu=0$, $f(x)=g(x^{2})$, and $g(y)=f(\sqrt{y})$. The
monotonically increasing function $g(y)$ is, therefore, defined on
$[0,\infty)$ with the set of values $[\underline{g},\overline{g})$, where
$\underline{g}=f(0)\geq 0$, $\overline{g}=\lim_{x\to\infty}f(x)>0$ and the
upper limit may be finite or infinite. The inverse function $\xi(z)$ is
defined for $z\in[\underline{g},\overline{g})$ with the interval of values
$[0,\infty)$. Let us take
$h^{\prime}(z)=\left\\{\begin{array}[]{ll}0&\mbox{ if }z<\underline{g};\\\
\xi(z)&\mbox{ if }z\in[\underline{g},\overline{g});\\\ \infty&\mbox{ if
}z\geq\overline{g};\end{array}\right.\;\;h(z)=\left\\{\begin{array}[]{ll}0&\mbox{
if }z<\underline{g};\\\
\int_{\underline{g}}^{z}\xi(\varsigma){\mathrm{d}}\varsigma&\mbox{ if
}z\in[\underline{g},\overline{g}];\\\ \infty&\mbox{ if
}z>\overline{g}.\end{array}\right.$ (27)
The improper integral
$\int_{\underline{g}}^{\overline{g}}\xi(\varsigma){\mathrm{d}}\varsigma$ may
take finite or infinite values.
Similarly, in case 2 ($\mu\leq 0$ and $\lambda_{2}<0$, $f$ decreases) we
define $g(y)=f(\sqrt{-y})$ for $y\in(-\infty,0]$. The function $g(y)$
monotonically increases and takes values on $(\underline{g},\overline{g}]$,
where $\underline{g}=\lim_{x\to\infty}f(x)$ and $\overline{g}=f(0)$. The
inverse function $\xi(z)$ is defined for $z\in(\underline{g},\overline{g}]$
with the interval of values $(-\infty,0]$. In this case, we can take
$h^{\prime}(z)=\left\\{\begin{array}[]{ll}-\infty&\mbox{ if
}z<\underline{g};\\\ \xi(z)&\mbox{ if }z\in(\underline{g},\overline{g}];\\\
0&\mbox{ if
}z>\overline{g};\end{array}\right.\;\;h(z)=\left\\{\begin{array}[]{ll}\infty&\mbox{
if }z\leq\underline{g};\\\
-\int^{\overline{g}}_{z}\xi(\varsigma){\mathrm{d}}\varsigma&\mbox{ if
}z\in[\underline{g},\overline{g}];\\\ 0&\mbox{ if
}z>\overline{g}.\end{array}\right.$ (28)
In case 3, the construction is almost the same as for the case 1 but
$f(x)=g((x-\mu)^{2})$ and $g(y)=f(\sqrt{y}+\mu)$. In this case, $g(y)$ is a
monotonically increasing function defined on the interval $[0,\infty)$ with
the set of values $[\underline{g},\overline{g})$, where $\underline{g}=f(\mu)$
and $\overline{g}=\lim_{x\to\infty}f(x)$. Similarly, for case 4, the
construction of $h(z)$ is almost the same as in case 2.
Thus, for every distribution in the form (26) we can find a $f$-divergence
$H_{h}(P\|P^{*})$, which conditional minimization produces this distribution.
For example, if in (26) $f(x)=ax^{\beta}$ then we can take $h$ in the form
$h(z)=\frac{\beta}{\beta+2}(z/a)^{1+2/\beta}$.
## 5 Conclusion
The Maxallent approach aims to bring some order to the modern anarchy of the
measures of disorder. If there is no clear idea which entropy is better then
we have to use all of them together.
The Markov order approach was also proposed as an alternative to the entropic
anarchism. It is based on the idea that the disorder has to increase in random
processes with given equilibrium distribution, which is considered as the
maximally disordered state. Here, we have proved that these two approaches
produce the same conditional minimizers on the planes of given values of
moments (Theorem 1).
In this paper, we have considered several relations between positive
distributions:
1. 1.
$P^{0}\succ^{0}_{P^{*}}P^{1}$ if there exists a Markov chain with equilibrium
$P^{*}$ such that for the solution of the Kolmogorov equation $P(t)$ with
$P(0)=P^{0}$ we have $P(1)=P^{1}$;
2. 2.
$P^{0}\succ_{P^{*}}P^{1}$ if there exist integrable bounded functions
$q_{ij}(t)$ ($i,j=1,\ldots,n$, $i\neq j$, $t\geq 0$) such that $q_{ij}(t)$
satisfy the balance condition (6) for given $P^{*}$ ($p^{*}_{i}>0$) (for all
$t\geq 0$), and $P(1)=P^{1}$ for solution $P(t)$ of the equations
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}(q_{ij}(t)p_{j}-q_{ji}(t)p_{i})\;\;(i=1,\ldots,n)$
with $P(0)=P^{0}$ (that is, $\succ_{P^{*}}$ is the transitive closure of
$\succ^{0}_{P^{*}}$);
3. 3.
$P^{0}>_{H,P^{*}}P^{1}$ if $H_{h}(P^{0}\|P^{*})>H_{h}(P^{1}\|P^{*})$ for all
strictly convex functions $h(x)$ on a semi-axis $x\geq 0$.
4. 4.
$P^{0}>_{P^{0},P^{*}}P^{1}$ if $P^{1}-P^{0}\in\mathbf{Q}(P^{0},P^{*})$, where
$\mathbf{Q}(P^{0},P^{*})$ is the cone of possible velocities
${\mathrm{d}}P/{\mathrm{d}}t$ (13) at point $P^{0}$ for all Markov chains with
equilibrium $P^{*}$.
All these relations are different. Three of them are antisymmetric, and one,
$P^{0}>_{P^{0},P^{*}}P^{1}$, is locally antisymmetric, in a vicinity of
$P^{0}$. Their interrelations are described by the follows implications:
$(P^{0}\succ^{0}_{P^{*}}P^{1})\Rightarrow(P^{0}\succ_{P^{*}}P^{1})\Rightarrow(P^{0}>_{H,P^{*}}P^{1})\Rightarrow(P^{0}>_{P^{0},P^{*}}P^{1})\,.$
The local Markov order $P^{0}>_{P^{*}}P^{1}$ is the weakest and the connection
by a solution of the Kolmogorov equation $P^{0}\succ^{0}_{P^{*}}P^{1}$ is the
strongest of these relations. Nevertheless, locally, in a small vicinity of a
positive non-equilibrium distribution $P^{0}$, these relations coincide and
they define the same set of locally minimal distributions on a linear manifold
of conditions $L$ (Propositions 1, 2, Theorem 1, Corollaries 1, 2 and 3).
Of course, there is the other, the classical way to reduce the variability of
the measures of disorder. The divergences $H(P\|P^{*})$ can be defined by
their main properties. This is an axiomatic approach: we postulate some
“natural properties” of the divergence, then find the divergences with these
properties, evaluate the result and decide whether we have to change the
system of axiom or not. The axiomatic approach to definition of entropy was
used by Shannon [5] and elaborated in detail by Khinchin [35].
Two distinguished additivity properties are important for the Maxent
reasoning:
* 1.
Additivity on the algebra of states: $H(P\|P^{*})$ is a sum in states
$H(P\|P^{*})=\sum_{i}\eta(p_{i},p^{*}_{i})\,.$
* 2.
Additivity with respect to the joining of independent subsystems. This means
that if $P$ and $P^{*}$ are products of distributions then $H(P\|P^{*})$ is
the sum of the corresponding entropies: if $P=(p_{jl})=(q_{j}r_{l})$ and
$P^{*}=(p^{*}_{jl})=(q^{*}_{j}r^{*}_{l})$ then
$H(P\|P^{*})=H(Q\|Q^{*})+H(R\|R^{*})$.
If we join the first additivity property with the requirement that the
divergence should be a Lyapunov function for all Markov chains with
equilibrium $P^{*}$ then we get $H_{h}(P\|P^{*})$ of the form (1) [21, 22,
23]. If we add the second additivity property and require continuity of
$H_{h}(P\|P^{*})$ for all values of $P$ (including vectors with some
$p_{i}=0$) then the classical Boltzmann–Gibbs–Shannon relative entropy will be
the only possibility (that is, $H_{h}(P\|P^{*})$ with $h(x)=x\ln x$ up to
unimportant constant factors and summand). If we relax the requirement of the
continuity to the set of strictly positive distributions then we will get the
one-parametric family $H_{h}(P\|P^{*})$ with $h(x)=\beta x\ln x-(1-\beta)\ln
x$ [21, 23].
Let us accept the point of view that the divergency is an order. Then the
values are not important and all the divergencies connected by a monotonic
transformation of a scale, $H=f(H^{\prime})$ (with a monotonically increasing
$f$), are equivalent. If the first additivity property is valid in one scale,
and the second may be valid in another one, then one more one-parametric
family appear, the Cressie–Read divergences (see Appendix A) [21, 23]. The
Tsallis entropy is a particular case of them. The Boltzmann–Gibbs–Shannon
relative entropy (or the Kullback–Leibler entropy, which is the same), the
convex combination of $H_{h}(P\|P^{*})$ and $H_{h}(P^{*}\|P)$ for $h(x)=x\ln
x$, and the Cressie–Read divergences (including the Tsallis relative entropy)
form the “entropic aristocracy” distinguished mostly by the additivity
properties.
If we accept the additivity on the algebra of states (i.e., the trace form)
and the additivity with respect to joining of independent subsystems, both,
then we have to use some of these functions. If additivity with respect to
joining of independent subsystems seems to be too restrictive then we have to
take the wider class of divergencies, for example, $H_{h}(P\|P^{*})$ of the
form (1). If we reject the requirement of the trace form then the variety of
the admissible divergences becomes even richer. This uncertainty in the choice
of divergence forces us to use the Maxallent approach.
The Maxallent approach produces a set of conditionally maximally disordered
distributions instead of a single distribution that maximizes a selected
distinguished entropy in the usual Maxent method. These Maxallent sets of
distributions may be considered as probabilistic analogues of the type-2 fuzzy
sets introduced by L. Zadeh [36] to capture the uncertainty of the fuzzy
systems. The Maxallent approach is invented to manage the uncertainty of the
measures of uncertainty. If there is no uncertainty of uncertainty then the
set of distributions reduces to a single distribution.
The decomposition theorems for Markov chains provide us with tools for the
efficient calculation of the Markov order. Following [27], we compare the
general Markov chains and the reversible chains with detailed balance. For any
general chain there is a reversible chain with the same velocity vector at a
given point. The classes of general and reversible chains locally coincide
because they have the same cone of possible velocities at every non-
equilibrium distribution (the second decomposition theorem, Appendix B). This
theorem gives us the possibility to describe the set of the conditionally
maximally uncertain distributions combinatorially, in the finite form (21).
For the classical Boltzmann–Gibbs–Shannon entropy the distribution on
$\mathbb{R}_{+}$ with two given moments has the Gaussian form
$a\exp(-b(x-c)^{2})$. The class of the Maxallent distributions on
$\mathbb{R}_{+}$ with two given moments is also simple (26) but much richer.
It can be produced by multiplication of the Boltzmann distribution (23) by a
monotonic function or unimodal function (with one local maximum) or by a
function with one local minimum.
There exists an attractive possibility: if a distribution can be obtained in
the Maxallent approach then it is a conditional minimum of a divergence. If we
find or guess a distribution of the Maxallent type for an empirical system
then we can restore the divergence and then use it in the standard Maxent
reasoning.
The Maxallent approach is, surprisingly, efficient enough to analyze some
practical problems. It gives an answer that does not depend on the subjective
choice and, therefore, returns us to the “mission” of information theory: “to
eliminate the psychological factors involved…” [4]. At the same time, it has a
solid basis in the theory of Lyapunov functions for the Kolmogorov equations.
Now, essential mathematical work on the basic notion of entropy is needed.
Gromov suggests that the natural mathematical language for this work will
involve nonstandard analysis and category theory [37]. These abstract
languages seem to be closer to the basic intuition than the set theory of
Cantor and the $\varepsilon-\delta$ reasoning of the classical analysis.
Nevertheless, the basic idea of Maxallent is so simple and natural, that it
should persist in the future advanced theory of entropy: order is something
that decreases in Markov processes.
## Appendix A. The most popular examples of $H_{h}(P\|P^{*})$
The most popular examples of $H_{h}(P\|P^{*})$ are [23]:
1. 1.
Let $h(x)$ be the step function, $h(x)=0$ if $x=0$ and $h(x)=-1$ if $x>0$. In
this case,
$H_{h}(P\|P^{*})=-\sum_{i,\ p_{i}>0}1\,.$ (29)
The quantity $-H_{h}$ is the number of non-zero probabilities $p_{i}$ and does
not depend on $P^{*}$. Sometimes it is called the Hartley entropy.
2. 2.
$h=|x-1|$,
$H_{h}(P\|P^{*})=\sum_{i}|p_{i}-p_{i}^{*}|\,;$
this is the $l_{1}$-distance between $P$ and $P^{*}$.
3. 3.
$h=x\ln x$,
$H_{h}(P\|P^{*})=\sum_{i}p_{i}\ln\left(\frac{p_{i}}{p_{i}^{*}}\right)=D_{\mathrm{KL}}(P\|P^{*})\,;$
(30)
this is the usual Kullback–Leibler divergence or the relative
Boltzmann–Gibbs–Shannon (BGS) entropy;
4. 4.
$h=-\ln x$,
$H_{h}(P\|P^{*})=-\sum_{i}p^{*}_{i}\ln\left(\frac{p_{i}}{p_{i}^{*}}\right)=D_{\mathrm{KL}}(P^{*}\|P)\,;$
(31)
this is the relative Burg entropy. It is obvious that this is again the
Kullback–Leibler divergence, but for another order of arguments.
5. 5.
Convex combinations of $h=x\ln x$ and $h=-\ln x$ also produces a remarkable
family of divergences: $h=\beta x\ln x-(1-\beta)\ln x$ ($\beta\in[0,1]$),
$H_{h}(P\|P^{*})=\beta
D_{\mathrm{KL}}(P\|P^{*})+(1-\beta)D_{\mathrm{KL}}(P^{*}\|P)\,;$ (32)
this convex combination of divergences was used by Gorban in the early 1980s
[38] and studied further by Gorban and Karlin [39]. It becomes a symmetric
functional of $(P,P^{*})$ for $\beta=1/2$. There exists a special name for
this case, “Jeffreys’ entropy”.
6. 6.
$h=\frac{(x-1)^{2}}{2}$,
$H_{h}(P\|P^{*})=\frac{1}{2}\sum_{i}\frac{(p_{i}-p_{i}^{*})^{2}}{p_{i}^{*}}=H_{2}(P\|P^{*})\,;$
(33)
this is the quadratic term in the Taylor expansion of the relative
Boltzmann–Gibbs-Shannon entropy, $D_{\mathrm{KL}}(P\|P^{*})$, near
equilibrium. We have used its time derivative in (8).
7. 7.
$h=\frac{x(x^{\lambda}-1)}{\lambda(\lambda+1)}$,
$H_{h}(P\|P^{*})=\frac{1}{\lambda(\lambda+1)}\sum_{i}p_{i}\left[\left(\frac{p_{i}}{p_{i}^{*}}\right)^{\lambda}-1\right]\,;$
(34)
this is the Cressie–Read (CR) family of power divergences [40] (the modern
exposition of the history, properties and applications of these entropies is
presented in [41]). For this family we use the notation $H_{\rm CR\ \lambda}$.
If $\lambda\to 0$ then $H_{\rm CR\ \lambda}\to D_{\mathrm{KL}}(P\|P^{*})$,
this is the classical BGS relative entropy; if $\lambda\to-1$ then $H_{\rm CR\
\lambda}\to D_{\mathrm{KL}}(P^{*}\|P)$, this is the relative Burg entropy.
8. 8.
For the CR family in the limits $\lambda\to\pm\infty$ only the maximal terms
“survive”. Exactly as we get the limit $l^{\infty}$ of $l^{p}$ norms for
$p\to\infty$, we can use $({\lambda(\lambda+1)}H_{\rm CR\
\lambda})^{1/|\lambda|}$ for $\lambda\to\pm\infty$ and write in these limits:
$H_{\rm CR\
\infty}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}}{p_{i}^{*}}\right\\}-1\,;$ (35)
$H_{\rm CR\
-\infty}(P\|P^{*})=\max_{i}\left\\{\frac{p_{i}^{*}}{p_{i}}\right\\}-1\,.$ (36)
The existence of two limiting divergences $H_{{\rm CR\ \pm\infty}}$ seems very
natural: there may be two types of extremely non-equilibrium states: with a
high excess of current probability $p_{i}$ above $p_{i}^{*}$ and, inversely,
with an extremely small current probability $p_{i}$ with respect to
$p_{i}^{*}$.
9. 9.
The Tsallis relative entropy [42]: $h=\frac{(x^{\alpha}-x)}{\alpha-1}$,
$\alpha>0$,
$H_{h}(P\|P^{*})=\frac{1}{\alpha-1}\sum_{i}p_{i}\left[\left(\frac{p_{i}}{p_{i}^{*}}\right)^{\alpha-1}-1\right]\,.$
(37)
For this family we use notation $H_{\rm Ts\ \alpha}$.
## Appendix B. The decomposition theorems
###### The first decomposition theorem.
Every Markov chain with a positive equilibrium is a conic combination of
simple cycles with the same equilibrium.
###### Proof.
If a non-zero Markov chain has a positive equilibrium then it cannot be
acyclic: there exists at least one oriented cycle of transitions with nonzero
rate constants. The length of this cycle can vary from 2 to $n$. The set of
all Markov chains with a positive equilibrium $P^{*}$ is an intersection of a
linear subspace given by the balance equations (6) with the positive orthant
$\mathbb{R}_{+}^{n(n-1)}$. This is a polyhedral cone which does not include a
whole straight line. It is well known in convex geometry that every such
polyhedral cone is a convex hull of a finite number of its extreme rays [30].
A ray $l$ with direction vector $x\neq 0$ is a set $l=\\{\kappa x\\}$
($\kappa\geq 0$). By definition, it is an extreme ray of a cone $\mathbf{Q}$
if for any $u\in l$ and any $x,y\in\mathbf{Q}$, whenever $u=(x+y)/2$, we must
have $x,y\in l$.
Any extreme ray of the cone of Markov chains with equilibrium $P^{*}$ is a
simple cycle $A_{i_{1}}\to\ldots\to A_{i_{k}}\to A_{i_{1}}$ with rate
constants $q_{i_{j+1}i_{j}}=\kappa/p_{j}^{*}$. Indeed, let a non-zero Markov
chain $Q$ with coefficients $q_{ij}$ belong to an extreme ray of this cone.
This chain includes a simple cycle with non-zero coefficients,
$A_{i_{1}}\to\ldots\to A_{i_{k}}\to A_{i_{1}}$ ($k\leq n$, all the numbers
$i_{1},\ldots,i_{k}$ are different, $q_{i_{j+1}\,i_{j}}>0$ for $j=1,\ldots,k$,
and $i_{k+1}=i_{1}$). For sufficiently small $\kappa$ ($0<\kappa<\kappa_{0}$),
$q_{i_{j+1}\,i_{j}}-\frac{\kappa}{p^{*}_{i_{j}}}>0$ ($j=1,\ldots,k$). Let
$Q_{\kappa}$ be the same simple cycle with the rate constants
$q_{i_{j+1}i_{j}}=\kappa/p_{j}^{*}$. Then for $0<\kappa<\kappa_{0}$ vectors
$Q\pm Q_{\kappa}$ also represent Markov chains with the equilibrium $P^{*}$.
Obviously, $Q=\frac{(Q+Q_{\kappa})+(Q-Q_{\kappa})}{2}$, hence, $Q$ should be
proportional to $Q_{\kappa}$, by the definition of extreme rays.
So, any Markov chain with a positive equilibrium $P^{*}$ is a linear
combination with positive coefficients of the cycles with the same
equilibrium. This decomposition is global, it does not depend on the current
distribution $P$. ∎
###### The second decomposition theorem.
For every Markov chain with a positive equilibrium $P^{*}$ and any probability
distribution $P^{0}$ the vector ${\mathrm{d}}P/{\mathrm{d}}t|_{P^{0}}$ is a
conic combination of the vectors ${\mathrm{d}}P/{\mathrm{d}}t|_{P^{0}}$ for
the simple cycles of length two $A_{i}\rightleftharpoons A_{j}$ with the same
equilibrium.
###### Proof.
Let us start from a simple cycle $A_{1}\to A_{2}\to\ldots\to A_{n}\to A_{1}$
with the constants $q_{i+1i}=1/p_{i}^{*}$, where $p_{i}^{*}>0$ is the
equilibrium. At a non-equilibrium distribution $P$ the right hand side of
equation (5) is the vector ${\mathrm{d}}P/{\mathrm{d}}t=\mathbf{v}_{n}$ with
coordinates
$\frac{{\mathrm{d}}p_{j}}{{\mathrm{d}}t}=(\mathbf{v}_{n})_{j}=\frac{p_{j-1}}{p^{*}_{j-1}}-\frac{p_{j}}{p^{*}_{j}}\,.$
(38)
The flux $A_{j}\to A_{j+1}$ is ${p_{j}}/{p^{*}_{j}}$. Let us find $A_{j}$ with
the minimum value of this flux and, for convenience, let us put this $A_{j}$
in the first position by a cyclic permutation. We will represent the right
hand side vector $\mathbf{v}_{n}$ in the form
$\mathbf{v}_{n}=\mathbf{v}_{n-1}+\kappa\mathbf{v}_{2}\,,$
where $\mathbf{v}_{n-1}$ corresponds to the cycle of the length $n-1$,
$A_{2}\to\ldots A_{n}\to A_{2}$, with the rate constants
$q_{i+1i}=1/p_{i}^{*}$ (and the cyclic convention $n+1=2$), $\mathbf{v}_{2}$
corresponds to the cycle of the length 2, $A_{1}\rightleftharpoons A_{2}$,
with the rate constants $q_{21}=1/{p^{*}_{1}}$, $q_{12}=1/{p^{*}_{2}}$, and
$\kappa\geq 0$. Both velocities $\mathbf{v}_{n-1}$ and $\mathbf{v}_{2}$ should
be calculated for the same distribution $P$.
We find the constant $\kappa$ from the conditions:
$\mathbf{v}_{n}=\mathbf{v}_{n-1}+\kappa\mathbf{v}_{2}$ at the point $P$,
hence, the two following reaction schemes, (a) and (b), should have the same
velocities, ${\mathrm{d}}P/{\mathrm{d}}t$:
$\mbox{(a)
}A_{n}{\overset{1/p_{n}^{*}}{\rightarrow}}A_{1}{\overset{1/p_{1}^{*}}{\rightarrow}}A_{2}\mbox{
and (b)
}A_{n}{\overset{1/p_{n}^{*}}{\rightarrow}}A_{2}\,;A_{1}\underset{\kappa/p_{2}^{*}}{\overset{\kappa/p_{1}^{*}}{\rightleftharpoons}}A_{2}\,.$
From this condition,
$\kappa=\left(\frac{p_{n}}{p_{n}^{*}}-\frac{p_{1}}{p_{1}^{*}}\right)\left(\frac{p_{2}}{p_{2}^{*}}-\frac{p_{1}}{p_{1}^{*}}\right)^{-1}\,.$
The inequality $\kappa\geq 0$ holds because ${p_{1}}/{p^{*}_{1}}$ is the
minimal value of the flux ${p_{j}}/{p^{*}_{j}}$.
We just delete the vertex with the smallest outgoing flux from the initial
cycle of length $n$ and add a cycle of the length 2 with the same equilibrium.
Let us repeat this operation for the remaining cycle of the length $n-1$, and
so on. At the end, the left hand side vector $\mathbf{v}_{n}$ will be
represented as the combination with positive coefficients the vectors
${\mathrm{d}}P/{\mathrm{d}}t$ for the cycles of the length 2,
$A_{i}\rightleftharpoons A_{j}$ with the same equilibrium. This is the system
with detailed balance. We have to stress here that the set of these
transitions and the coefficients $\kappa$ depend on the current distribution
$P$.
For every distribution $P$, the velocity ${\mathrm{d}}P/{\mathrm{d}}t$ of
every cycle with equilibrium $P^{*}$ is a combination with positive
coefficients of the velocities for some cycles of the length two
$A_{i}\rightleftharpoons A_{j}$ with the same equilibrium. Therefore, the
right hand side of the Kolmogorov equation for any Markov chain with
equilibrium $P^{*}$ also allows such a decomposition.
It is necessary to stress that the decomposition of the right hand side of the
Kolmogorov equation (5) into a conic combination of cycles of length 2 depends
on the ordering of the ratios $p_{i}/p_{i}^{*}$ and cannot be performed for
all values of $P$ simultaneously. ∎
For more details and further references see [27].
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|
arxiv-papers
| 2012-12-20T16:59:50 |
2024-09-04T02:49:39.487386
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.N. Gorban",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1212.5142"
}
|
1212.5153
|
§ INTRODUCTION
Let $X: = (X_t)_{t\geq 0}$ be a one-dimensional Lévy process,
starting from zero,
with law $\stP$. The Lévy–Khintchine formula
states that for all
$\theta\in \RR$,
the characteristic exponent
$\Psi(\theta) : = %t^{-1}
-\log \stE (e^{\iu\theta X_1})$ satisfies
\[
\Psi(\theta)
= \iu a\theta
+ \frac{1}{2}\sigma^2\theta^2
+ \int_{\RR} (1 - e^{\iu\theta x} + \iu\theta x\Indic{|x|\leq 1})\Pi(\dd x)
+ q,
\]
where $a\in\mathbb{R}$, $\sigma\geq 0$ and $\Pi$ is a measure
(the Lévy measure) concentrated on
such that $\int_{\RR}(1\wedge x^2)\Pi(\dd x)<\infty$. The
parameter $q$ is the killing rate. When $q=0$, we say that
the process $X$ is unkilled, and it remains in $\RR$
for all time; when $q > 0$, the process $X$ is sent to
a cemetery state at a random time, independent of the
path of $X$ and with an exponential distribution of rate $q$.
The process $(X,\stP)$ is
said to be a (strictly) $\alpha$-stable process
if it is an unkilled Lévy process which
also satisfies the scaling property: under $\stP$,
for every $c > 0$,
the process $(cX_{t c^{-\alpha}})_{t \ge 0}$
has the same law as $X$.
It is known that $\alpha \in (0,2]$, and the case $\alpha = 2$
corresponds to Brownian motion, which we exclude.
In fact, we will assume $\alpha \in (1,2)$.
The Lévy-Khintchine representation of such a process
is as follows:
$\sigma = 0$, and
$\Pi$ is absolutely continuous with density given by
\[
c_+ x^{-(\alpha+1)} \Indic{x > 0} + c_- \abs{x}^{-(\alpha+1)} \Indic{x < 0},
\for x \in \RR,
\]
where $c_+,\, c_- \ge 0$, and $a = (c_+-c_-)/(\alpha-1)$.
The process $X$ has the
characteristic exponent
\begin{equation}\label{e:stable CE}
\Psi(\theta) =
(1 - \iu\beta\tan\tfrac{\pi\alpha}{2}\sgn\theta), \for \theta\in\RR,
\end{equation}
where $\beta = (c_+- c_-)/(c_+ + c_-)$ and
$c = - (c_++c_-)\Gamma(-\alpha)\cos (\pi\alpha/2)$. For more details,
see <cit.>.
For consistency with the literature that we shall appeal to in this article,
we shall always parametrise our $\alpha$-stable process such that
\[ c_+ = \Gamma(\alpha+1) \frac{\sin(\pi \alpha \rho)}{\pi} \quad \text{and} \quad
c_- = \Gamma(\alpha+1) \frac{\sin(\pi \alpha \rhoh)}{\pi },
\]
$\rho = \stP(X_t \ge 0)$ is the positivity parameter,
and $\rhoh = 1-\rho$. In that case, the constant $c$ simplifies to just $c = \cos (\pi\alpha(\rho - 1/2))$.
We take the point of view that the class of stable processes,
with this normalisation, is
parametrised by $\alpha$ and $\rho$; the reader will note that
all the quantities above can be written in terms of these parameters.
We shall restrict ourselves a little further within this class
by excluding the possibility of having only one-sided jumps.
This gives
us the following set of admissible parameters (see <cit.>):
{ (α,ρ) : α∈(1,2), ρ∈(1-1/α, 1/α) }.
Let us write $\stP_x$ for the law of the shifted process $X+x$ under $\stP$.
We are interested in computing the distribution of
\[ T_0 = \inf\{ t \ge 0 : X_t = 0 \} ,\]
the first hitting time of zero for $X$, under $\stP_x$, for $x \ne 0$.
When $\alpha > 1$, this random variable is a.s. finite, while
when $\alpha \le 1$, points are
polar for the stable process, so $T_0 = \infty$ a.s. (see
this explains our exclusion of such processes.
This paper consists of two parts: the first deals with the case
where $X$ is symmetric. Here, we may identify a positive, self-similar Markov process
$R$ and make use of the Lamperti transform to write down the
Mellin transform of $T_0$.
The second part concerns the general case where $X$ may be asymmetric.
Here we present instead
a method making use of the generalised Lamperti transform
and Markov additive processes.
It should be noted that the
symmetric case can be deduced from the general case, and so
in principle we need not go into details when $X$ is symmetric; however,
this case provides familiar ground on which to rehearse the arguments
which appear in a more complicated situation in the general case.
Let us also note here that, in the symmetric case, the distribution of
$T_0$ has been characterised in <cit.>, and
the Mellin transform appears
in <cit.>; however, these authors proceed
via a different method.
For the spectrally one-sided case, which our range of parameters omits,
representations of law of $T_0$ have been given by Peskir, 2008 and Simon, 2011.
This justifies
our exclusion of the one-sided case.
Nonetheless, as we explain in Remark <ref>, our methodology can also
be used in this case.
We now give a short outline of the coming material.
In section <ref>, we suppose that the stable process $X$ is symmetric,
that is $\rho = 1/2$,
and we define a process $R$ by
\[ R_t = \abs{X_t} \Indic{t < T_0}, \for t \ge 0, \]
the radial part of $X$.
The process $R$ satisfies the $\alpha$-scaling property,
and indeed is a positive, self-similar Markov process, whose
Lamperti representation, say $\xi$, is a Lévy process;
see section <ref> for definitions.
It is then known that $T_0$ has the same distribution as
the random variable
\[ I(\alpha\xi) := \int_0^\infty \exp(\alpha \xi_t) \, \dd t, \]
the so-called
exponential functional of $\alpha\xi$.
In order to find the distribution of $T_0$, we compute the
Mellin transform, $\LevE[I(\alpha\xi)^{s-1}]$, for a suitable range of $s$.
The result is given in Proposition <ref>.
This then characterises the
distribution, and the transform here can be analytically inverted.
In section <ref> we consider
the general case, where $X$ may not be symmetric, and our reasoning
is along very similar lines. The process $R$ still satisfies the scaling
but, since its dynamics depend on the sign of $X$, it is no longer a Markov process,
and the method above breaks down. However,
due to the recent work of Chaumont et al., 2013, there is still a type of Lamperti representation for $X$,
not in terms of a Lévy process, but in terms of a so-called Markov additive process, say $\xi$.
Again, the distribution of $T_0$ is equal to that of $I(\alpha\xi)$
(but now with the role of $\xi$
taken by the Markov additive process), and we develop techniques
to compute a vector-valued Mellin transform for the exponential function of
this Markov additive process.
Further, we invert the Mellin transform of $I(\alpha\xi)$ in order to deduce
explicit series representations for the law of $T_0$.
After the present article was submitted, the preprint of Letemplier and Simon, 2013 appeared,
in which the authors begin from the classical potential theoretic formula
\[ \LevE_x [e^{-q T_0}] = \frac{u^q(-x)}{u^q(0)} , \for q > 0, \; x \in \RR, \]
where $u^q$ is the $q$-potential of the stable process. Manipulating this
formula, they derive the Mellin transform of $T_0$. Their proof is rather shorter
than ours, but it appears to us that the Markov
additive point of view offers a good insight into the structure of
real self-similar Markov processes in general, and, for example,
will be central to the development of a theory of entrance laws of recurrent extensions
of rssMps.
In certain scenarios the distribution of $T_0$ is a very convenient quantity to have,
and we consider some applications in section <ref>:
for example, we give an alternative description of the stable process
conditioned to avoid zero, and we give some identities in law similar
to the result of Bertoin and Yor, 2002 for the entrance law
of a pssMp started at zero.
§ THE SYMMETRIC CASE
In this section, we give a brief derivation of the Mellin transform
of $T_0$ for a symmetric stable process. As we have said, we do this by
considering the Lamperti transform of the radial part of the process;
let us therefore recall some relevant definitions and results.
A positive self-similar Markov process (pssMp) with
self-similarity index $\alpha > 0$ is a standard Markov process
$R = (R_t)_{t\geq 0}$ with associated filtration $\FFt$ and probability laws
$(\stP_x)_{x > 0}$, on $[0,\infty)$, which has $0$ as an absorbing state and
which satisfies the scaling property, that for every $x, c > 0$,
\[
\label{scaling prop}%
\text{ the law of } (cR_{t c^{-\alpha}})_{t \ge 0}
\text{ under } \stP_x \text{ is } \stP_{cx} \text{.}
\]
Here, we mean “standard” in the sense of [Blumenthal and Getoor, 1968],
which is to say, $\FFt$ is a complete, right-continuous filtration,
and $R$ has càdlàg paths and is strong Markov
and quasi-left-continuous.
In the seminal paper [Lamperti, 1972], Lamperti describes a one-to-one correspondence
between pssMps and Lévy processes, which we now outline.
It may be worth noting that we have presented a slightly
different definition of pssMp from Lamperti; for the connection, see
$S(t) = \int_0^t (R_u)^{-\alpha}\, \dd u .$
This process is continuous and strictly increasing until $R$ reaches zero.
Let $(T(s))_{s \ge 0}$ be its inverse, and define
\[ \eta_s = \log R_{T(s)} \qquad s\geq 0.
\]
Then $\eta : = (\eta_s)_{s\geq 0}$ is a Lévy process started
at $\log x$, possibly killed at an independent
exponential time; the law of the Lévy process and the
rate of killing do not depend
on the value of $x$. The real-valued process
$\eta$ with probability laws
$(\LevP_y)_{y \in \RR}$ is called the
Lévy process associated to $R$,
or the Lamperti transform of $R$.
An equivalent definition of $S$ and $T$, in terms of $\eta$ instead
of $R$, is given by taking
$T(s) = \int_0^s \exp(\alpha \eta_u)\, \dd u$
and $S$ as its inverse. Then,
\begin{equation*}
\label{Lamp repr}
R_t = \exp(\eta_{S(t)})
\end{equation*}
for all $t\geq 0$, and this shows that the Lamperti transform is a bijection.
Let $X$ be the symmetric stable process, that is,
the process defined in the introduction with $\rho = 1/2$.
We note that this process is such that
$\stE(e^{\iu\theta X_1}) = \exp(-\abs{\theta}^{\alpha})$,
and remark briefly that it has Lévy measure
$\Pi(\dd x) = k \abs{x}^{-\alpha-1}\,\dd x$,
\[ k = \Gamma(\alpha+1)\frac{\sin(\pi\alpha/2)}{\pi}
= \frac{\Gamma(\alpha+1)}{\Gamma(\alpha/2)\Gamma(1-\alpha/2)} .\]
Connected to $X$ is an important example where the Lamperti transform
can be computed explicitly. In Caballero and Chaumont, 2006, the authors
compute Lamperti transforms of killed
and conditioned stable processes; the simplest of their examples,
given in <cit.>, is as follows.
\[ \tau_0^- = \inf\{ t > 0 : X_t \le 0 \} , \]
and denote by $\LSabs$ the Lamperti transform of the
pssMp $\stproca{X_t \Indic{t < \tau_0^-}}$.
Then $\LSabs$ has Lévy density
\begin{equation*}
%\label{LSabs density}
k e^x\abs{e^x-1}^{-(\alpha+1)}, \for x \in \RR,
\end{equation*}
and is killed at rate
Using this, we can analyse a pssMp which will give us the information we seek. Let
\[ R_t = \abs{X_t} \Indic{t < T_0}, \for t \ge 0, \]
be the radial part of the symmetric stable process. It is simple to see
that this is a pssMp. Let us denote its Lamperti transform by $\xi$.
In Caballero et al., 2011, the authors study the Lévy process $\xi$;
they find its characteristic function, and
Wiener-Hopf factorisation, when
$\alpha < 1$, as well as a decomposition into two simpler processes
when $\alpha \le 1$. We will now demonstrate that their expression for the
characteristic function is also valid when $\alpha > 1$, by showing
that their decomposition has meaning in terms of the Lamperti transform.
The Lévy process $\xi$ is the sum of two independent Lévy processes,
$\xiLS$ and $\xiCPP$, such that
* The Lévy process $\xiLS$ has characteristic exponent
\[ \Psi^*(\theta) - k/\alpha, \for \theta \in \RR , \]
where $\Psi^*$ is the characteristic exponent of the process
$\xi^*$, which is the Lamperti transform of the stable process killed
upon first passage below zero.
That is, $\xiLS$ is formed by removing the independent
killing from $\xi^*$.
* The process $\xiCPP$ is a compound Poisson process whose jumps occur
at rate $k/\alpha$, whose Lévy density is
\begin{equation}\label{e:ldcpp}
\LDCPP(y) = k \frac{e^y}{(1+e^y)^{\alpha+1}} , \for y \in \RR.
\end{equation}
Precisely the same argument as in <cit.>
gives the decomposition into $\xiLS$ and $\xiCPP$,
and the process $\xiLS$ is exactly as in that case.
The expression (<ref>), which determines the law of $\xiCPP$,
follows from <cit.>,
once one observes that the computation in that article does not require
any restriction on $\alpha$.
We now compute the characteristic exponent of $\xi$.
As we have mentioned, when $\alpha < 1$,
this has already been computed in <cit.>,
but whereas in that paper the authors were concerned with computing
the Wiener-Hopf factorisation, and the characteristic function
was extracted as a consequence of this, here we
provide a proof directly from the above decomposition.
The characteristic exponent of the Lévy process $\xi$
is given by
\begin{equation}
\CE(\theta) = 2^{\alpha}
\frac{\Gamma(\alpha/2-\iu\theta/2)}{\Gamma(-\iu\theta/2)}
% \times
\frac{\Gamma(1/2+\iu\theta/2)}{\Gamma((1-\alpha)/2+\iu\theta/2)},
\for \theta \in \RR.
\label{e:WHF low index}
\end{equation}
We consider separately the two Lévy processes
in Proposition <ref>.
By <cit.>,
\[ \CELS(\theta) =
\frac{\Gamma(\alpha-\iu\theta) \Gamma(1+\iu\theta)}
- \frac{\Gamma(\alpha)}{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}, \]
and via the beta integral,
= k ∫_-∞^∞(1-e^θy) π^C( y)y
= Γ(α+1)/Γ(α/2)Γ(1-α/2)
[ 1/α
- Γ(1+θ)Γ(α-θ)/Γ(α+1)
] .
Summing these, then using product-sum identities and <cit.>,
= Γ(α-θ)Γ(1+θ)
[ sin(π(α/2-θ))/π
- sin(πα/2)/π ]
= 2/π Γ(α-θ)Γ(1+θ)
cosπ(α-θ)/2 sin-θπ/2
= 2πΓ(α-θ)/Γ(1/2 + (α-θ)/2)
Γ(1+θ)/Γ(1/2 + (1+θ)/2)
Now, applying the Legendre–Gauss duplication formula <cit.> for the gamma function,
we obtain the expression in the theorem.
We now characterise the law of the exponential functional
\[ I(\alpha \xi) = \int_0^\infty e^{\alpha \xi_t} \, \dd t \]
of the process $\alpha\xi$, which we do via the Mellin transform,
\[ \MT(s) = \LevE[ I(\alpha\xi)^{s - 1} ] , \]
for $s \in \CC$ whose real part lies in some open interval which we will specify.
To begin with, we observe that the Laplace exponent $\psi $ of
the process $-\alpha \xi$, that is, the function such that
$\LevE e^{-z \alpha \xi_1} = e^{\psi (z)}$,
is given by
\[ \psi (z)
= - 2^\alpha\frac{ \Gamma(1/2-\alpha z /2) }{ \Gamma(1/2-\alpha(1+z)/2) }
\frac{\Gamma(\alpha(1+z)/2)}{\Gamma(\alpha z/2)},
\for \Re z \in (-1,1/\alpha).
\]
We will now proceed via the `verification result' <cit.>:
essentially, this result says that
we must find a candidate for $\MT$
which satisfies the functional equation
\begin{equation}
\MT(s+1) = -\frac{s}{\psi (-s)} \MT(s),
\label{e:verif}
\end{equation}
for certain $s$, together with some additional conditions. Let us now state our result.
The Mellin transform of $T_0$ satisfies
\begin{equation}\label{e:MT symm}
\stE_1[T_0^{s-1}]
= \LevE_0[I(\alpha\xi)^{s-1}]
= \sin(\pi/\alpha)
\frac{\cos\bigl(\frac{\pi\alpha}{2}(s-1)\bigr)}
\frac{\Gamma(1+\alpha-\alpha s)}{\Gamma(2-s)},
\end{equation}
for $\Re s \in \bigl(-\tfrac{1}{\alpha}, 2-\tfrac{1}{\alpha}\bigr)$.
Denote the right-hand side of (<ref>) by $f(s)$.
We begin by noting that $-\alpha\xi$ satisfies the Cramér condition
$\psi(1/\alpha-1)=0$; the verification result
<cit.> therefore allows us to prove the proposition
for $\Re s \in (0,2-1/\alpha)$ once we verify some conditions on
this domain.
There are three conditions to check.
For the first, we require $f(s)$ is analytic and zero-free in
the strip $\Re s \in (0, 2-1/\alpha)$; this is straightforward. The second requires us to verify that
$f$ satisfies (<ref>). To this end, we expand $\cos$ and $\sin$
in gamma functions (via reflection formulas <cit.>) and apply the Legendre duplication formula
<cit.> twice.
Finally, there is an asymptotic property which is needed. More precisely, we need
to investigate the asymptotics of $f(s)$ as $\im s \to \infty$. To do this,
we will use the following formula
lim_|y|→∞ |Γ(x+y)|/|y|^x-1/2e^-π/2|y|=√(2π).
This can be derived (see <cit.>)
from Stirling's asymptotic formula:
as $ z\to \infty$ and $\abs{\arg(z)}<\pi-\delta$, for fixed $\delta > 0$.
Since Stirling's asymptotic formula is uniform in any sector $|\arg(z)|<\pi-\delta$,
it is easy to see that the
convergence in (<ref>) is also uniform in $x$ belonging to a compact subset of $\r$.
Using formula \eqref{Gamma_asymptotics} we check that $|1/f(s)|=O(exp(πs))$ as $s →∞$,
uniformly in the strip $s ∈(0,2-1/α)$.
This is the asymptotic result that we require.
The conditions of \cite[Proposition 2]{KP-HG} are therefore
satisfied, and it follows that the formula in the proposition
holds for $s ∈(0,2-1/α)$. Since $f(s)$ is analytic in the wider strip
$s ∈(-1/α, 2-1/α)$, we conclude the proof by
analytic extension.
%% The fact that this analytic extension can be made is part of the folklore.
%% The article of Lukacs clarifies the situation a little.
% we conclude that $\LevE_0[I(\alpha\xi)^{s-1}]$ can be analytically
% extended to this wider strip and that identity \eqref{e:MT symm} holds
% for all $s\in \c$ with $\re s \in (-1/\alpha, 2-1/\alpha)$.
%(see \cite[Theorem 2]{Lukacs}).
\end{proof}
\end{proposition}
We note that the expression of \citet[equation (1.36)]{Cor-thesis} can be derived
from this result via the duplication formula for the gamma function; furthermore,
it is not difficult to deduce it from \cite[Theorem 5.3]{YYY-laws}.
Now, this Mellin transform completely characterises the law of $T_0$,
and we could at this point
invert the Mellin transform to find a series expansion for the
density of $T_0$.
However, as we will shortly perform precisely this calculation in section
\ref{s:asymm} for the general case,
we shall leave the Mellin transform as it is and proceed
to consider what happens when $X$ may not be symmetric.
\section{The asymmetric case}
\label{s:asymm}
With the symmetric case as our model, we will now tackle the general case
where $X$ may be asymmetric. The ideas here are much the same, but the
possibility of asymmetry leads us to introduce more complicated
objects: our positive self-similar Markov processes become
real self-similar Markov processes; our L\'evy processes
become Markov additive processes; and our functional equation for
the Mellin transform (\ref{e:verif}) becomes vector-valued.
The section is laid out as follows. We
devote the first two subsections to a discussion of Markov
additive processes and their exponential functionals, and then discuss
real self-similar Markov processes and the generalised Lamperti
representation. Finally, in the last
subsection, we apply the theory which we have developed to the
problem of determining the law of $T_0$ for a general two-sided jumping stable process with $α∈(1,2)$.
\subsection{Markov additive processes}
\label{ss:MAP}
Let $E$ be a finite state space and $$ a standard
A c\`adl\`ag process $(ξ,J)$ in $×E$
with law $$ is called a
\define{Markov additive process (MAP)} with respect to $$
if $(J(t))_t ≥0$ is a continuous-time Markov chain in $E$, and
the following property is satisfied,
for any $i ∈E$, $s,t ≥0$:
\begin{multline}\label{e:MAP}
\text{given $\{J(t) = i\}$,
the pair $(\xi(t+s)-\xi(t), J(t+s))$ is independent of
$\GG_t$,} \\ \text{and has the same distribution as $(\xi(s)-\xi(0), J(s))$
given $\{J(0) = i\}$.}
\end{multline}
Aspects of the theory of Markov additive processes
are covered in a number of texts, among them
[Asmussen, 2000] and [Asmussen, 2003].
We will mainly use the notation of
[Ivanovs, 2011],
which principally works under the assumption that
$ξ$ is spectrally negative; the results which
we quote are valid without this hypothesis, however.
Let us introduce some notation.
We write $_i = ( ·|ξ(0) = 0, J(0) = i)$;
and if $μ$ is a probability distribution on $E$, we write
$_μ= ( ·|ξ(0) = 0, J(0) ∼μ)
= ∑_i ∈E μ(i) _i$. % \mu_i?
% It will also be convenient to denote by $\MAPP(A)$ the column vector
% with $i$th element $\MAPP_i(A)$,
% and by $\MAPP(A; J(t))$ the matrix whose $(i,j)$th entry
% is $\MAPP_i(A, J(t) = j)$.
We adopt a similar convention
for expectations.
It is well-known that a Markov additive process $(ξ,J)$ also satisfies
\eqref{e:MAP} with $t$ replaced by a stopping time. Furthermore, it
has the structure given by the following proposition;
see \cite[\S XI.2a]{Asm-apq2} and \cite[Proposition 2.5]{Iva-thesis}.
\begin{proposition}
The pair $(\xi,J)$ is a Markov additive process if and only if, for each $i,j\in E$,
there exist a sequence of iid L\'evy processes
$(\xi_i^n)_{n \ge 0}$ and a sequence of iid random variables
$(U_{ij}^n)_{n\ge 0}$, independent
of the chain $J$, such that if $T_0 = 0$
and $(T_n)_{n \ge 1}$ are the
jump times of $J$, the process $\xi$ has the representation
\[ \xi(t) = \Indic{n > 0}( \xi(T_n -) + U_{J(T_n-), J(T_n)}^n) + \xi_{J(T_n)}^n(t-T_n),
\for t \in [T_n, T_{n+1}),\, n \ge 0. \]
\end{proposition}
For each $i ∈E$, it will be convenient to define,
on the same probability space, $ξ_i$ as a L\'evy process whose distribution is the common
law of the $ξ_i^n$ processes in the above representation; and similarly, for each $i,j ∈E$, define $U_ij$ to
be a random variable having the common law of the $U_ij^n$ variables.
Let us now fix the following setup. Firstly, we confine ourselves
to irreducible Markov chains $J$.
Let the state space $E$ be the finite set ${1 N}$, for some $N ∈$.
Denote the transition rate matrix of the chain $J$ by
$Q = (q_ij)_i,j ∈E$.
For each $i ∈E$, the Laplace exponent of the L\'evy process $ξ_i$
will be written $ψ_i$, in the sense that $e^ψ_i(z) = (e^z ξ_i(1))$, for all $z ∈$ for which the right-hand side exists.
For each pair of $i,j ∈E$,
the Laplace transform $G_ij(z) = (e^z U_ij)$
of the jump
distribution $U_ij$,
where this exists; write $G(z)$ for the $N ×N$ matrix
whose $(i,j)$th element is $G_ij(z)$.%\footnote{Here, we understand the Laplace transform of a random variable $W$ with law $P$ (and associated expectation operator $E$) to mean $E(\exp\{z W\})$, for all $z\in\mathbb{C}$ such that the expectation is finite. Moreover, we define the associated Laplace exponent as $\log E(\exp\{z W\}).$}
We will adopt the convention that $U_ij = 0$ if
$q_ij = 0$, $i j$, and also set $U_ii = 0$ for each $i ∈E$.
A multidimensional analogue of the Laplace exponent of a L\'evy process is
provided by the matrix-valued function
\begin{equation}\label{e:MAP F}
F(z) = \diag( \psi_1(z) \cdotsc \psi_N(z))
+ Q \Had G(z),
\end{equation}
for all $z ∈$ where the elements on the right are defined,
where $$ indicates elementwise multiplication, also called
Hadamard multiplication.
It is then known
%\cite[Lemma 2.1]{AK-mg}
\[ \MAPE_i( e^{z \xi(t)} ; J(t)=j) = \bigl(e^{F(z) t}\bigr)_{ij} , \for i,\,j \in E, \]
for all $z ∈$ where one side of the equality is defined.
For this reason, $F$ is called the \define{matrix exponent} of
the MAP $ξ$.
We now describe the existence of the \define{leading eigenvalue}
of the matrix $F$, which will play a key role in our analysis
of MAPs. This is sometimes also called the
\define{Perron–Frobenius eigenvalue};
see \cite[\S XI.2c]{Asm-apq2} and \cite[Proposition 2.12]{Iva-thesis}.
\begin{proposition}
Suppose that $z \in \CC$ is such that $F(z)$ is defined.
Then, the matrix $F(z)$ has a real simple eigenvalue $\kappa(z)$,
which is larger than the real part of all its other eigenvalues.
Furthermore, the
right-eigenvector $v(z)$ may be chosen so that
$v_i(z) > 0$ for every $i \in E$,
and normalised such that
\begin{eqnarr}
\pi v(z) &=& 1 \label{e:h norm}
\end{eqnarr}
where $\pi$ is the equilibrium distribution of
the chain $J$.
\end{proposition}
This leading eigenvalue features in the following probabilistic
result, which identifies a martingale (also known as the Wald martingale) and change of measure
analogous to the exponential martingale and Esscher transformation
of a L\'evy process; cf.\ \cite[Proposition XI.2.4, Theorem XIII.8.1]{Asm-apq2}.
\begin{proposition}
\label{p:mg and com}
\[ M(t,\gamma) = e^{\gamma \xi(t) - \kappa(\gamma)t}
\frac{v_{J(t)}(\gamma)}{v_{J(0)}(\gamma)} ,
\for t \ge 0, \]
for some $\gamma$ such that the right-hand side is defined.
% \begin{enumerate}[(i)]
% \item
$M(\cdot,\gamma)$ is a unit-mean martingale with respect to $\GGt$,
under any initial distribution of $(\xi(0),J(0))$.
% \item
% Define the change of measure
% \[ \left.\frac{\dd \MAPPc{\gamma}}{\dd \MAPP}\right\rvert_{\GG_t}
% = \Mn(t,\gamma) . \]
% Under $\MAPPc{\gamma}$, the process $\xi$ is still a Markov additive
% process, and it has the following characteristics, for each $i,j \in E$:
% \begin{itemize}
% \item
% $ \MAPPc{\gamma}(U_{ij} \in \dd x)
% = \dfrac{e^{\gamma x}}{G_{ij}(\gamma)}\MAPP(U_{ij} \in \dd x), $
% and hence
% $ G_{ij}^{(\gamma)}(z)
% = \dfrac{G_{ij}(z + \gamma)}{G_{ij}(\gamma)}, $
% \item
% $ q_{ij}^{(\gamma)} = \dfrac{v_j(\gamma)}{v_i(\gamma)}
% q_{ij} G_{ij}(\gamma) \text{ and} $
% \item
% $ \psi_i^{(\gamma)}(z) = \psi_i(z+\gamma) - \psi_i(\gamma) . $
% \end{itemize}
% Furthermore,
% \[ F^{(\gamma)}(z)
% = (\diag(v_i(\gamma),\, i \in E))^{-1}
% [F(z+\gamma) - \kappa(\gamma)\!\Id]%XXX spacing
% \diag(v_i(\gamma),\, i \in E) , \]
% and hence,
% \[ \kappa^{(\gamma)}(z) = \kappa(z+\gamma) - \kappa(\gamma) . \]
% \end{enumerate}
% %\begin{proof}
% %For (i), see \cite[Proposition XI.2.4]{Asm-apq2}; and for (ii), see
% %\cite[Theorem XIII.8.1]{Asm-apq2}.
% %\end{proof}
\end{proposition}
% Making use of this, the following proposition can be obtained; the properties
% of $\kappa$ given here are often used in the literature, but for convenience,
% we also provide a short proof.
The following properties of $κ$ will also prove useful.
\begin{proposition}
Suppose that $F$ is defined in some open interval $D$ of $\RR$.
Then, the leading eigenvalue $\kappa$ of $F$
is smooth and convex on $D$.
\begin{proof}
Smoothness follows from results on the perturbation of eigenvalues; see
\cite[Proposition 2.13]{Iva-thesis} for a full proof. The convexity of
$\kappa$ is a consequence of the convexity properties of the entries
of $F$. The proof follows simply from \cite[Corollary 9]{BN-Perron};
see also [Kingman, 1961, Miller, 1961].
%% This proof is incorrect since it assumes some perturbation properties
%% of v which may not hold:
% To prove that $\kappa$ is convex, we begin by proving that for
% $\lambda \in (0,1)$ such that $z, \, \lambda z \in D$,
% \begin{equation}\label{e:cvx 1}
% \kappa(\lambda z) \le \lambda \kappa(z) .
% \end{equation}
% Taking expectations in the martingale from Proposition \ref{p:mg and com},
% and using the normalisation condition \eqref{e:h norm}, we see that,
% starting $J$ in equilibrium,
% \[ \MAPE_\pi[ e^{z \xi(1)} v_{J(1)}(z)] = e^{\kappa(z)} . \]
% Now let $p = 1/\lambda > 1$, and let $q$ be its conjugate exponent,
% that is, $\frac{1}{p} + \frac{1}{q} = 1$. We apply H\"older's inequality
% as follows.
% \begin{eqnarr*}
% e^{\kappa(z/p)} =
% \MAPE_\pi[ e^{(z/p) \xi(1)} v_{J(1)}(z)]
% &\le& \bigl(\MAPE_\pi [ e^{(zp/p) \xi(1)} v_{J(1)}^{p/p}(z) ] \bigr)^{1/p}
% \bigl( \MAPE_\pi [ v_{J(1)}^{q/q}(z) ] \bigr)^{1/q} \\
% &=& \bigl( \MAPE_\pi [ e^{z \xi(1)} v_{J(1)}(z) ] \bigr)^{1/p}
% (\pi v(z))^{1/q} \\
% &=& \bigl( \MAPE_\pi [ e^{z \xi(1)} v_{J(1)}(z) ] \bigr)^{1/p}
% = e^{\kappa(z)/p}
% \end{eqnarr*}
% Replacing $1/p$ by $\lambda$, we obtain \eqref{e:cvx 1}.
% Now let $u,v \in D$ and $\lambda \in (0,1)$; recall from Proposition
% \ref{p:mg and com} that for the leading eigenvalue under the change of measure,
% $\kappa^{(u)}(z) = \kappa(z+u) - \kappa(u)$. We then calculate:
% \begin{eqnarr*}
% \kappa((1-\lambda)u + \lambda v) - \kappa(u)
% &=& \kappa^{(u)}(\lambda(v-u)) \\
% &\le& \lambda \kappa^{(u)}(v-u) \\
% &=& \lambda( \kappa(v)-\kappa(u))
% \end{eqnarr*}
% and so
% \[ \kappa((1-\lambda)u + \lambda v) \le (1-\lambda)\kappa(u) + \lambda \kappa(v) , \]
% which completes the proof.
\end{proof}
\end{proposition}
\subsection{The Mellin transform of the exponential functional}
In section \ref{s:symm}, we studied the exponential functional
of a certain L\'evy process associated to the radial part of the
stable process; now we are interested in obtaining some results
which will assist us in computing the law of an integrated exponential functional
associated to Markov additive processes.
For a MAP $ξ$, let
\[ I(-\xi) = \int_0^\infty \exp(-\xi(t)) \, \dd t. \]
One way to characterise the law of $I(-ξ)$ is via its
Mellin transform, which
we write as $(s)$. This is the vector in $^N$
whose $i$th element is given by
\[ \MT_i(s) = \MAPE_i[I(-\xi)^{s-1}] , \for i \in E. \]
We will shortly obtain a functional equation for $$,
analogous to the functional equation \eqref{e:verif} which we saw
in section \ref{s:symm}. For L\'evy processes, proofs of the
result can be found in \citet[Proposition 3.1]{CPY97},
\citet[Lemma 2.1]{MZ-ef} and \citet[Lemma 2]{Riv-RE2};
our proof follows the
latter, making changes to account for the Markov additive
We make the following assumption, which is analogous to the Cram\'er
condition for a L\'evy process; recall that $κ$ is the leading
eigenvalue of the matrix $F$, as discussed in section \ref{ss:MAP}.
\begin{assumption}[Cram\'er condition for a Markov additive process]
\label{a:Cramer}
There exists $z_0 < 0$ such that $F(s)$ exists on $(z_0,0)$,
and some $\theta \in (0,-z_0)$, called the \define{Cram\'er number},
such that $\kappa(-\theta) = 0$.
\end{assumption}
Since the leading eigenvalue $κ$ is smooth and convex
where it is
defined, it follows also that $κ(-s) < 0$ for $s ∈(0,θ)$.
In particular, this renders the matrix $F(-s)$ negative definite, and hence invertible.
Furthermore, it follows that $κ^'(0-) > 0$, and hence
(see \cite[Corollary XI.2.7]{Asm-apq2} and \cite[Lemma 2.14]{Iva-thesis}) that $ξ$
drifts to $+∞$ independently of its initial state. This
implies that $I(-ξ)$ is an a.s.\ finite random variable.
\begin{proposition}\label{p:MT func eq}
Suppose that $\xi$ satisfies the Cram\'er condition
(Assumption \ref{a:Cramer})
with Cram\'er number $\theta\in(0,1)$.
Then, $\MT(s)$ is finite and analytic
when $\Re s \in (0,1+\theta)$, and we have the
following vector-valued functional equation:
\[ \MT(s+1) = - s (F(-s))^{-1} \MT(s) , \for s \in (0,\theta). \]
\begin{proof}
At the end of the proof, we shall require the existence of certain
moments of the random variable
\[ Q_t = \int_0^t e^{- \xi(u)} \, \dd u , \]
and so we shall begin by establishing this.
Suppose that $s \in (0,\theta]$, and let $p > 1$.
Then, by the Cram\'er condition, it follows that $\kappa(-s/p) < 0$,
and hence for any $u \ge 0$, $e^{-u \kappa(-s/p)} \ge 1$.
Recall that the process
\[ M(u,z) = e^{z \xi(u) - \kappa(z)u}\frac{ v_{J(u)}(z)}{v_{J(0)}(z)} ,
\for u \ge 0 \]
is a martingale (the Wald martingale)
under any initial distribution $(\xi(0),J(0))$, and set
\[ V(z) = \min_{j \in E} v_j(z) > 0, \]
so that for each $j \in E$, $v_j(z)/V(z) \ge 1$.
We now have everything in place to make the following calculation,
which uses the Doob maximal inequality in connection with the Wald martingale
in the third line, and the Cram\'er condition in the fourth.
\begin{eqnarr*}
\MAPE_i [ Q_t^s ]
% &\le& t^s \MAPE_i\bigl[ \sup_{u \le t} e^{-s \xi(u)}\bigr] \\
&\le& t^s \MAPE_i \left[ \sup_{u \le t}\bigl[ e^{-s \xi(u)/p}\bigr]^p \right] \\
&\le& t^s \MAPE_i \left[ \sup_{u \le t}\bigl[ M(u,-s/p) v_i(-s/p)(V(-s/p))^{-1}\bigr]^p \right] \\
&\le& t^s v_i(-s/p)^pV(-s/p)^{-p} \biggl(\frac{p}{p-1}\biggr)^p
\MAPE_i \bigl[ M(t, -s/p)^p \bigr] \\
&\le& t^s V(-s/p)^{-p} \biggl(\frac{p}{p-1}\biggr)^p
e^{-tp \kappa(-s/p)} \max_{j \in J} v_j(-s/p)^p
\MAPE_i\bigl[ e^{-s \xi(t)} \bigr]
< \infty.
\end{eqnarr*}
Now, it is simple to show that for all $s > 0$, $t \ge 0$,
\[ \biggl( \int_0^\infty e^{-\xi(u)} \, \dd u \biggr)^s
- \biggl( \int_t^\infty e^{-\xi(u)} \, \dd u \biggr)^s
= s \int_0^t e^{-s \xi(u)}
\biggl( \int_0^\infty e^{-(\xi(u+v) - \xi(u))} \, \dd v \biggr)^{s-1}
\, \dd u . \]
For each $i \in E$, we take expectations and apply the Markov additive
\begin{eqnarr*}
\eqnarrLHS{\MAPE_i \biggl[
\biggl( \int_0^\infty e^{-\xi(u)} \, \dd u \biggr)^s
- \biggl( \int_t^\infty e^{-\xi(u)} \, \dd u \biggr)^s \biggr]}
&=& s \sum_{j \in E} \int_0^t
\MAPE_i\Bigl[ e^{-s \xi(u)} ; J(u) = j \Bigr]
\MAPE_j\biggl[ \int_0^\infty e^{-\xi(v)} \, \dd v \biggr]^{s-1}
\, \dd u \\
&=& s \int_0^t \sum_{j \in E} \Bigl(e^{F(-s)u} \Bigr)_{ij}
\MAPE_j\bigl[I(-\xi)^{s-1}\bigr] \, \dd u .
\end{eqnarr*}
% This may be expressed using our vector notation as
% \[
% \MAPE \biggl[
% \biggl( \int_0^\infty e^{-\xi(u)} \, \dd u \biggr)^s
% - \biggl( \int_t^\infty e^{-\xi(u)} \, \dd u \biggr)^s \biggr]
% = s \int_0^t e^{F(-s)u}
% \MAPE\bigl[I(-\xi)^{s-1}\bigr] \, \dd u.
% \]
Since $0<s<\theta<1$, it follows that
$\bigabs{\abs{x}^s -\abs{y}^s} \leq \abs{x-y}^s$
for any $x, y\in\mathbb{R}$, and so we see that
for each $i \in E$, the left-hand side of the above equation
is bounded by
$\MAPE_i(Q_t^s) < \infty$. Since $\bigl( e^{F(-s)u} \bigr)_{ii} \ne 0$,
it follows that $\MAPE_i[I(-\xi)^{s-1}] < \infty$ also.
If we now take $t \to \infty$, the left-hand side of the previous
equality is monotone increasing,
while on the right, the Cram\'er condition ensures that $F(-s)$
is negative definite, which is a sufficient condition for
convergence, giving the limit:
\[ \MT(s+1) = -s(F(-s))^{-1} \MT(s), \for s \in (0,\theta). \]
Furthermore, as we know the right-hand side is finite, this
functional equation allows us to conclude that $\MT(s) < \infty$
for all $s \in (0,1+\theta)$.
It then follows from the general properties of Mellin transforms that
$\MT(s)$ is finite and analytic for all $s \in \CC$ such that
$\Re s \in (0,1+\theta)$.
\end{proof}
\end{proposition}
\subsection{Real self-similar Markov processes}
\newcommand{\CPRE}{\mathcal{E}}
\newcommand{\mT}{\mathcal{T}} % my notation for the T_n etc. in CPR
In section \ref{s:symm}, we studied a L\'evy process which was
associated through the Lamperti representation to a positive, self-similar
Markov process. Here we see that Markov additive processes also admit
an interpretation as Lamperti-type representations of \emph{real}
self-similar Markov processes.
The structure of real self-similar Markov processes has been investigated
by Chybiryakov, 2006 in the symmetric case, and Chaumont et~al., 2013 in general.
Here, we give an interpretation of these authors' results
in terms of a two-state Markov additive process. We begin with some
relevant definitions, and state some of the results of these authors.
A \define{real self-similar Markov process} with \define{self-%
similarity index} $α> 0$ is a standard (in the sense of [Blumenthal and Getoor, 1968])
Markov process $X = X$ with probability laws
$(_x)_x ∈∖{0}$ which satisfies the
\define{scaling property}, that for all $x ∈∖{0}$
and $c > 0$,
\[ \text{the law of }(c X_{t c^{-\alpha}})_{t \ge 0}
\text{ under } \stP_x \text{ is } \stP_{cx} . \]
In [Chaumont et~al., 2013] the authors confine their attention to processes
in `class \textbf{C.4}'. An rssMp $X$ is in this class if, for
all $x 0$, $_x( ∃t > 0: X_t X_t - < 0 ) = 1$;
that is, with probability one, the process $X$ changes
sign infinitely often.
As with the stable process, define
\[ T_0 = \inf\{t \ge 0: X_t = 0 \} .\]
% For such processes, $X$ may be identified up to the time $T_0$ as the time-changed
% exponential of a certain complex-valued process $\CPRE$,
% which we call the \define{Lamperti--Kiu representation} of $X$.
% The following result is taken from
% [Chaumont et~al., 2013].
% \def\CPRthref{\cite[Theorem 6]{CPR}}
% \begin{prop}[\CPRthref] \label{p:CPR main}
% Let $X$ be a rssMp in class \textbf{C.4}, and let $x \ne 0$.
% It is possible to define independent sequences
% $(\xi^{\pm,k})_{k \ge 0}, \, (\zeta^{\pm,k})_{k \ge 0}, \, (U^{\pm,k})_{k \ge 0}$
% of iid random objects with the following proprties:
% \begin{enumerate}
% \item The elements of these sequences are distributed such that:
% the $\xi^{\pm}$ are real-valued
% L\'evy processes; $\zeta^\pm$ are exponential random variables
% with parameters $q^\pm$; and $U^\pm$ are real-valued random variables.
% \item For each $x \ne 0$, define the following objects:
% \[ (\xi^{(k)}, \zeta^{(k)}, U^{(k)})
% = \begin{cases}
% ( \xi^{+,k}, \zeta^{+,k}, U^{+,k} ),
% & \text{if } \sgn(x) (-1)^{k} = 1 \\
% ( \xi^{-,k}, \zeta^{-,k}, U^{-,k} ),
% & \text{if } \sgn(x) (-1)^{k} = -1, \\
% \end{cases} \]
% \[ \mT_0 = 0, \quad \mT_n = \sum_{k = 0}^{n-1} \zeta^{(k)} , \]
% \[ N_t = \max\{ n \ge 0 : \mT_n \le t \}, \]
% \[ \sigma_t = t - \mT_{N_t}, \]
% \[ \CPRE_t = \xi_{\sigma_t}^{(N_t)}
% + \sum_{k = 0}^{N_t - 1}
% ( \xi_{\zeta^{(k)}}^{(k)} + U^{(k)})
% + \iu\pi N_t , \for t \ge 0, \]
% \begin{equation*}
% \tau(t) = \inf \biggl\{ s > 0 : \int_0^s \abs{\exp(\alpha \CPRE_u)}
% \, \dd u > t \abs{x}^{-\alpha} \biggr\}, \for t < T_0.
% \end{equation*}
% Then, the process $X$ under the measure $\stP_x$ has the representation
% \[ X_t = x \exp(\CPRE_{\tau(t)}) , \for 0 \le t < T_0. \]
% \end{enumerate}
% \end{prop}
% The abundance of notation necessary to be precise in this context
% may obscure the fundamental idea, which is as follows.
% At any given time, the process $\CPRE$ evolves as a L\'evy process
% $\xi^\pm$, moving along a line $\Im z = \pi N$, up until an exponential
% `clock' $\zeta^\pm$ (corresponding to the process $X$ changing sign)
% rings. At this point the imaginary part of $\CPRE$
% is incremented by $\pi$, the real part jumps by $U^\pm$, and the
% process begins to evolve as the other L\'evy process, $\xi^\mp$.
% Particularly in light of the above discussion, our interpretation
% is simple to state.
% \begin{prop}
% Let $X$ be an rssMp, with Lamperti--Kiu representation $\CPRE$.
% Define furthermore
% \[ [n] = \begin{cases}
% 1, & \text{if } n \text{ is odd} \\
% 2, & \text{if } n \text{ is even}.
% \end{cases}
% \]
% Then, for each $x \ne 0$, the process
% \[ (\xi(t), J(t))
% = \bigl( \Re \CPRE_t,
% \bigl[ \Im \CPRE_t/\pi + \Indic{x > 0} \bigr] \bigr)
% \]
% defined in Proposition \ref{p:CPR main}
% is a Markov additive process with state space $E = \{1, 2\}$,
% and $X$ under $\stP_x$ has the representation
% \[ X_t = x \exp\bigl(
% \xi( \tau(t))
% + \iu \pi (J( \tau(t)) + 1 )\bigr) , \for 0 \le t < T_0, \]
% where we note that $(\xi(0),J(0))$ is equal to $(0,1)$ if $x > 0$,
% or $(0,2)$ if $x < 0$.
% Furthermore, the time-change $\tau$ has the representation
% \begin{equation}\label{e:Lamp time change}
% \tau(t) = \inf \biggl\{ s > 0 : \int_0^s \exp(\alpha \xi(u))
% \, \dd u > t \abs{x}^{-\alpha} \biggr\}, \for t < T_0,
% \end{equation}
% in terms of the real-valued process $\xi$.
% \end{prop}
Such a process may be identified, under a deformation of space and
time, with a Markov additive process which we call the
\define{Lamperti--Kiu representation} of $X$. The following
result is a simple corollary of \cite[Theorem 6]{CPR}.
\begin{proposition}
Let $X$ be an rssMp in class \textbf{C.4} and fix $x \ne 0$.
Define the symbol
\[ [y] = \begin{cases}
1, & y > 0, \\
2, & y < 0.
\end{cases}
\]
Then there exists a time-change $\sigma$, adapted to the filtration of $X$,
such that, under the law $\stP_x$, the process
\[ (\xi(t),J(t)) = (\log\abs{X_{\sigma(t)}}, [X_t]) , \qquad t \ge 0, \]
is a Markov additive process with state space $E = \{1,2\}$
under the law $\MAPP_{[x]}$.
Furthermore, the process $X$ under $\stP_x$ has the representation
\[ X_t = x \exp\bigl( \xi( \tau(t))
+ \iu \pi (J( \tau(t)) + 1 )\bigr) , \for 0 \le t < T_0, \]
where $\tau$ is the inverse of the time-change $\sigma$,
and may be given by
\begin{equation}\label{e:Lamp time change}
\tau(t) = \inf \biggl\{ s > 0 : \int_0^s \exp(\alpha \xi(u))
\, \dd u > t \abs{x}^{-\alpha} \biggr\}, \for t < T_0,
\end{equation}
\end{proposition}
% Note that the MAP $(\xi,J)$ under $\MAPP_1$ corresponds to the rssMp
% $X$ started at a point $x > 0$, and the MAP under $\MAPP_2$
% corresponds to the rssMp started at a point $x < 0$.
We observe from the expression \eqref{e:Lamp time change}
for the time-change $τ$ that under $_x$, for any $x 0$,
the following identity holds for $T_0$, the hitting time of zero:
\[ \abs{x}^{\alpha} T_0 = \int_0^\infty e^{\alpha \xi(u)} \, \dd u. \]
% where either $T_0$ is under $\stP_1$ and $\xi$ is under $\MAPPp$,
% or $T_0$ is under $\stP_{-1}$ and $\xi$ is under $\MAPPm$.
Implicit in this statement is that the MAP on the right-hand side
has law $ℙ_1$ if $x > 0$, and law $ℙ_2$ if
$x < 0$.
This observation will be exploited in the coming
section, in which we put together the theory we have outlined
so far.
\subsection{The hitting time of zero}
We now return to the central problem of this paper: computing
the distribution of $T_0$ for a stable process. We already have in
hand the representation of $T_0$ for an rssMp as the exponential
functional of a MAP, as well as a functional equation for this
quantity which will assist us in the computation.
\medskip \noindent
Let $X$ be the stable process with parameters $(α,ρ) ∈$,
defined in the introduction. We will restrict our attention for now
to $X$ under the measures $_±1$; the results for other initial
values can be derived via scaling.
Since $X$ is an rssMp, it has a representation in terms of a MAP
$(ξ,J)$; futhermore, under $_±1$,
\[ T_0 = \int_0^\infty e^{\alpha \xi(s)} \, \dd s
= I(\alpha \xi);
\]
to be precise, under $_1$ the process $ξ$ is under $_1$,
while under $_-1$ it is under $_2$.
In \cite[\S 4.1]{CPR}, the authors calculate the characteristics
of the Lamperti--Kiu representation for $X$, that is, the processes
$ξ_i$, and the jump distributions $U_ij$ and rates $q_ij$.
Using this information, and the representation \eqref{e:MAP F},
one sees that the MAP $(-αξ,J)$ has matrix exponent
\[%\label{def_Fz}
F(z) =
% F(-\alpha z) =
\begin{pmatrix}
- \dfrac{\Gamma(\alpha(1+z))\Gamma(1-\alpha z)}
{\Gamma(\alpha\rhoh+\alpha z)\Gamma(1-\alpha\rhoh- \alpha z)}
& \dfrac{\Gamma(\alpha(1+z))\Gamma(1-\alpha z)}
\\
\dfrac{\Gamma(\alpha(1+z))\Gamma(1-\alpha z)}
& - \dfrac{\Gamma(\alpha(1+z))\Gamma(1-\alpha z)}
{\Gamma(\alpha\rho+\alpha z)\Gamma(1-\alpha\rho-\alpha z)}
\end{pmatrix} ,
\]
for $z ∈(-1,1/α)$.
\begin{remark}\label{r:spec os}
It is well-known that, when $X$ does not have one-sided jumps,
it changes sign infinitely often; that is, the rssMp $X$ is in [Chaumont et~al., 2013]'s
class \textbf{C.4}. When the stable process has only one-sided jumps, which
corresponds to the parameter values $\rho = 1-1/\alpha,\,1/\alpha$,
then it jumps over $0$ at most once before hitting it; the
rssMp is therefore in class \textbf{C.1} or \textbf{C.2} according to the
classification of [Chaumont et~al., 2013].
The Markov chain component of
the corresponding MAP then has one absorbing state,
and hence is no longer irreducible.
Although it seems plain that our calculations can be carried
out in this case, we omit it for the sake of simplicity.
As we remarked in the introduction, it is considered in
[Peskir, 2008, Simon, 2011].
\end{remark}
\smallskip \noindent
We now analyse $F$ in order to deduce the Mellin transform of $T_0$.
The equation $F(z) = 0$ is equivalent to
\[
\sin(\pi(\alpha\rho+\alpha z))\sin(\pi(\alpha\hat\rho+\alpha z))
- \sin(\pi \alpha \rho) \sin(\pi\alpha\hat\rho)
= 0,
\]
% By considering
% the equation $\det F(z) = 0$ with $z = 1/\alpha - 1$, it is not difficult to deduce
% that
and considering the solutions of this, it is not difficult to deduce that
$κ(1/α-1) = 0$; that is, $-αξ$ satisfies the
Cram\'er condition (Assumption \ref{a:Cramer}) with
Cram\'er number $θ= 1-1/α$.
\[
f_1(s) := \stE_1[ T_0^{s-1} ] = \MAPE_1[ I(\alpha\xi)^{s-1}],
\quad
f_2(s) := \stE_{-1} [ T_0^{s-1}] = \MAPE_2[ I(\alpha\xi)^{s-1}],
\]
which by Proposition \ref{p:MT func eq} are defined when $s ∈(0,2-1/α)$.
This proposition also implies that
\beq\label{matrix_equation2}
{\mathbf B}(s)\times \left[ \begin{array}{c} f_1(s+1) \\ f_2(s+1) \end{array} \right]=
\left[ \begin{array}{c} f_1(s) \\ f_2(s) \end{array} \right],
\for s \in (0,1-1/\alpha),
\eeq
where $𝐁(s):=-F(-s)/s$. Using the reflection formula for the gamma function we find that
\beqq
{\mathbf B}(s)=\frac{\alpha}{\pi} \Gamma(\alpha-\alpha s) \Gamma(\alpha s)
\left[ \begin{array}{cc} \sin(\pi \alpha (\hat \rho-s)) & -\sin(\pi \alpha \hat \rho) \\
-\sin(\pi \alpha \rho) & \sin(\pi \alpha ( \rho-s)) \end{array} \right],
\eeqq
% Using the above formula and Ptolemy's identity
% \beq\label{Ptolemy_theorem}
% \sin(\delta_1+\delta_2)\sin(\delta_2+\delta_3)=\sin(\delta_1)\sin(\delta_3)+\sin(\delta_1+\delta_2+\delta_3)
% \sin(\delta_2)
% \eeq
% with $\delta_1=\pi\alpha \rho$, $\delta_2=-\pi \alpha s$ and $\delta_3=\pi \alpha \rhoh$, we conclude that
for $s ∈(-1/α,1), s 0$, and
\beq\label{det_Bs}
{\textnormal{det}}({\mathbf B}(s))=-\alpha^2 \frac{\Gamma(\alpha-\alpha s) \Gamma(\alpha s)}
{\Gamma(1-\alpha+\alpha s) \Gamma(1-\alpha s)}, \for \Re s \in (-1/\alpha,1), \, s \ne 0.
\eeq
Therefore, if we define $𝐀(s) = (𝐁(s))^-1$, we have
\[%\label{def_As}
{\mathbf A}(s)
= -\frac{1}{\pi \alpha} \Gamma(1-\alpha+\alpha s) \Gamma(1-\alpha s)
\begin{bmatrix} \sin(\pi \alpha (\rho-s)) & \sin(\pi \alpha \hat \rho) \\
\sin(\pi \alpha \rho) & \sin(\pi \alpha (\hat \rho-s)) \end{bmatrix}
\]
for $s ∈(1-2/α,1-1/α)$, and may rewrite \eqref{matrix_equation2} in the form
\beq\label{matrix_equation}
\left[ \begin{array}{c} f_1(s+1) \\ f_2(s+1) \end{array} \right]={\mathbf A}(s) \times
\left[ \begin{array}{c} f_1(s) \\ f_2(s) \end{array} \right],
\for s \in (0,1-1/\alpha).
\eeq
The following theorem is our main result.
\begin{theorem}\label{thm_Mellin_transform}
For $-1/\alpha<\re(s)<2-1/\alpha$ we have
\beq\label{Mellin_transform}
\stE_1[T_0^{s-1}]
= \frac{\sin\left(\frac{\pi}{\alpha} \right)}{\sin(\pi \hat \rho)}
\frac{\sin\left( \pi \hat \rho (1-\alpha+\alpha s)\right)}
{\sin\left(\frac{\pi}{\alpha} (1-\alpha+\alpha s) \right)}
\frac{\Gamma(1+\alpha-\alpha s)}{\Gamma(2-s)}.
\eeq
% this function has:
% poles at k+1-1/alpha, k \in \ZZ,
% 1+ (k+1)/\alpha, k \ge 0.
% zeroes at 1-1/\alpha+k/(\alpha\rhoh), k \in \ZZ,
% 2+k, k \ge 0.
The corresponding expression for $\stE_{-1}[T_0^{s-1}]$ can be obtained from \eqref{Mellin_transform}
by changing $\hat \rho \mapsto \rho$.
\end{theorem}
Let us denote the function in the right-hand side of \eqref{Mellin_transform} by $h_1(s)$,
and by $h_2(s)$
the function obtained from $h_1(s)$ by replacing $ρ̂↦ρ$.
Before we are able to prove Theorem
\ref{thm_Mellin_transform}, we need to establish several properties of these functions.
\begin{lemma}\label{lemma1}
\mbox{}
\begin{itemize}
\item[(i)] There exists $\epsilon \in (0,1-1/\alpha)$ such that the functions $h_1(s)$, $h_2(s)$ are analytic and zero-free in the vertical strip $0 \le \re(s) \le 1+\epsilon$.
\item[(ii)] For any $-\infty< a<b < +\infty$ there exists $C>0$ such that
\beqq
e^{-\pi |\im(s)|}<|h_i(s)|< e^{-\frac{\pi}{2}(\alpha-1) |\im(s)|}, \;\;\; i=1,2
\eeqq
for all $s$ in the vertical strip $a\le \re(s) \le b$ satisfying $|\im(s)|>C$.
\end{itemize}
\end{lemma}
\begin{proof}
It is clear from the definition of $h_1(s)$ that it is a meromorphic function.
Its zeroes are
contained in the set
\beqq
\{2,3,4,\dots\}\cup \{1-1/\alpha+n/(\alpha \hat \rho) : \; n \in {\mathbb Z}, n\ne 0 \}
\eeqq
and its poles are contained in the set
\beqq
\{1+n/\alpha : \; n\ge 1\} \cup \{n-1/\alpha : \; n \in {\mathbb Z}, n \ne 1\}.
\eeqq
In particular, $h_1(s)$ possesses neither zeroes nor poles in the strip $0 \le \re(s) \le 1$.
The same is clearly true for $h_2(s)$, which proves part (i).
We now make use of Stirling's formula, as we did in section \ref{s:symm}.
Applying \eqref{Gamma_asymptotics} to $h_1(s)$ we find
that as $s \to \infty$ (uniformly in the strip $a\le \re(s) \le b$) we have
\beqq
\ln (|h_1(s)|)=- \frac{\pi}{2} (1+\alpha - 2\alpha \hat \rho) |\im(s)|+O(\ln(|\im(s)|)).
\eeqq
Since for $\alpha>1$, the admissible parameters $\alpha$, $\rho$ must satisfy $\alpha-1<\alpha \hat \rho<1$, this shows that
\beqq
\alpha-1<1+\alpha - 2\alpha \hat \rho<3-\alpha<2,
\eeqq
and completes the proof of part (ii).
\end{proof}
\begin{lemma}\label{lemma2}
The functions $h_1(s)$, $h_2(s)$ satisfy the system of equations \eqref{matrix_equation}.
\end{lemma}
\begin{proof}
Denote the elements of the matrix $\mathbf{A}(s)$ by $A_{ij}(s)$.
Multiplying the first row
of $\mathbf{A}(s)$ by the column vector $[h_1(s), h_2(s)]^T$, and using identity $\sin(\pi \rho)=\sin(\pi \hat \rho)$, we obtain
\begin{eqnarr*}
\eqnarrLHS{A_{11}(s)h_1(s)+A_{12}(s)h_2(s)}
-\frac{1}{\pi \alpha} \frac{\sin\left(\frac{\pi}{\alpha} \right)}{\sin(\pi \hat \rho)}
\frac{\Gamma(1-\alpha s)}{\sin\left(\frac{\pi}{\alpha} (1-\alpha+\alpha s) \right)}
\left[\frac{\Gamma(1+\alpha-\alpha s)}{\Gamma(2-s)}
\Gamma(1-\alpha+\alpha s) \right] \\
&& {} \times
\big\{ \sin(\pi \alpha (\rho-s)) \sin\left( \pi \hat \rho (1-\alpha+\alpha s)\right)
+\sin(\pi \alpha \hat \rho) \sin\left( \pi \rho (1-\alpha+\alpha s)\right)\big\}.
\end{eqnarr*}
Applying identity $\Gamma(z+1)=z \Gamma(z)$ and reflection
formula for the gamma function, we rewrite the expression in the square brackets as follows:
\beqq
\left[\frac{\Gamma(1+\alpha-\alpha s)}{\Gamma(2-s)}
\Gamma(1-\alpha+\alpha s) \right] =
\frac{\alpha \Gamma(\alpha-\alpha s)}{\Gamma(1-s)}
\Gamma(1-\alpha+\alpha s)=\frac{\pi \alpha}{\sin(\pi \alpha(1-s)) \Gamma(1-s)}.
\eeqq
% Using Ptolemy's identity \eqref{Ptolemy_theorem} with $\delta_1=\pi \alpha(1-s)$,
% $\delta_2=\pi \alpha (s-\rho)$ and $\delta_3=\pi (\rho-\alpha s + \alpha \rho s)$,
% we obtain
Applying certain trigonometric identities, we obtain
\begin{multline*}
\sin(\pi \alpha (\rho-s)) \sin\left( \pi \hat \rho (1-\alpha+\alpha s)\right)+
\sin(\pi \alpha \hat \rho) \sin\left( \pi \rho (1-\alpha+\alpha s)\right) \\
=\sin(\pi \alpha (1-s)) \sin(\pi \hat \rho (1+\alpha s)).
\end{multline*}
Combining the above three formulas we conclude
\beqq
-\frac{\sin\left(\frac{\pi}{\alpha} \right)}{\sin(\pi \hat \rho)}
\frac{\sin\left( \pi \hat \rho (1+\alpha s)\right)}
{\sin\left(\frac{\pi}{\alpha} (1-\alpha+\alpha s) \right)}
\frac{\Gamma(1-\alpha s)}{\Gamma(1-s)}=h_1(s+1).
\eeqq
The derivation of the identity $A_{21}(s)h_1(s)+A_{22}(s)h_2(s)=h_2(s+1)$ is identical.
We have now established that two functions $h_i(s)$ satisfy
the system of equations \eqref{matrix_equation}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm_Mellin_transform}]
Our goal now is to establish the uniqueness of solutions to the system
\eqref{matrix_equation} in a certain class of meromorphic functions,
which contains both $h_i(s)$ and $f_i(s)$.
This will imply $h_i(s) \equiv f_i(s)$.
Our argument is similar in spirit to the proof of Proposition 2 in [Kuznetsov and Pardo, 2013].
First of all, we check that there exists $\epsilon\in (0,1/2-1/(2\alpha))$, such that
the functions $f_1(s)$, $f_2(s)$ are analytic and bounded in the open strip
\beqq
{\mathcal S}_{\epsilon}=\{s\in \c: \epsilon < \Re(s) < 1+2\epsilon\}
\eeqq
This follows from Proposition \ref{p:MT func eq} and the estimate
\[
\abs{f_1(s)}=\abs{\stE_1[ T_0^{s-1} ]}\le \stE_1[ \abs{T_0^{s-1}} ]=\stE_1[ T_0^{\re(s)-1} ]=f_1(\Re(s)).
\]
The same applies to $f_2$. Given results of Lemma \ref{lemma1},
we can also assume that $\epsilon$ is small enough, so that the functions $h_i(s)$
are analytic, zero-free and bounded in the strip ${\mathcal S}_{\epsilon}$.
Let us define $D(s):=f_1(s)h_2(s) - f_2(s) h_1(s)$ for $s\in {\mathcal S}_{\epsilon}$.
From the above properties of $f_i(s)$ and $h_i(s)$ we conclude that $D(s)$ is analytic and bounded in ${\mathcal S}_{\epsilon}$. Our first goal is to show that $D(s)\equiv 0$.
If both $s$ and $s+1$ belong to ${\mathcal S}_{\epsilon}$, then the function $D(s)$ satisfies the equation
\beq\label{eqn_Fs}
D(s+1)=-\frac{1}{\alpha^2} \frac{\Gamma(1-\alpha+\alpha s) \Gamma(1-\alpha s)}{\Gamma(\alpha-\alpha s) \Gamma(\alpha s)}D(s),
\eeq
as is easily established by taking determinants of the matrix equation
\beqq
\left[ \begin{array}{cc} f_1(s+1) & h_1(s+1) \\ f_2(s+1) & h_2(s+1) \end{array} \right]={\mathbf A}(s) \times
\left[ \begin{array}{cc} f_1(s) & h_1(s) \\ f_2(s) & h_2(s) \end{array} \right],
\eeqq
and using \eqref{det_Bs} and the identity $\mathbf{A}(s) = \mathbf{B}(s)^{-1}$.
Define also
\[%\label{def_G}
G(s):=\frac{\Gamma(s-1)\Gamma(\alpha-\alpha s)}{\Gamma(1-s)\Gamma(-\alpha + \alpha s)}
\sin\left( \pi \left(s+\frac{1}{\alpha} \right) \right).
\]
It is simple to check that:
\begin{enumerate}[(i)]
\item\label{G1} $G$ satisfies the functional equation \eqref{eqn_Fs};
\item\label{G2} $G$ is analytic and zero-free in the strip ${\mathcal S}_{\epsilon}$;
\item\label{G3} $|G(s)|\to \infty$ as $\im(s)\to \infty$, uniformly in the strip ${\mathcal S}_{\epsilon}$
(use \eqref{Gamma_asymptotics} and $\alpha > 1$).
\end{enumerate}
We will now take the ratio of $D$ and $G$ in order to obtain a periodic function,
borrowing a technique from the theory of functional equations
(for a similar idea applied to the gamma function, see \cite[\S 6]{Art-gamma}.)
We thus define $H(s):=D(s)/G(s)$ for $s\in {\mathcal S}_{\epsilon}$.
% NB. H may be defined in (0,2-1/\alpha).
The property \pref{G2} guarantees that $H$ is analytic in the strip $ {\mathcal S}_{\epsilon}$,
while property \pref{G1} and \eqref{eqn_Fs} show that\linebreak % final version
if both $s$ and $s+1$ belong to $ {\mathcal S}_{\epsilon}$.
Therefore, we can extend $H(s)$ to an entire function satisfying
$H(s+1)=H(s)$ for all $s\in \c$.
Using the periodicity of $H(s)$, property \pref{G3} of the function $G(s)$ and the fact that the function $D(s)$ is bounded in the strip ${\mathcal S}_{\epsilon}$, we conclude that
$H(s)$ is bounded on $\CC$ and $H(s)\to 0$ as $\im(s)\to \infty$.
Since $H$ is entire, it follows that $H \equiv 0$.
So far, we have proved that for all $s\in {\mathcal S}_{\epsilon}$ we have $f_1(s)h_2(s) = f_2(s) h_1(s)$. Let us define
$w(s):=f_1(s)/h_1(s)=f_2(s)/h_2(s)$. Since both $f_i(s)$ and $h_i(s)$ satisfy the same functional equation
\eqref{matrix_equation}, if $s$ and $s+1$ belong to ${\mathcal S}_{\epsilon}$ we have
\begin{eqnarr*}
&=& f_1(s+1) \\
&=& A_{11}(s)f_1(s)+A_{12}(s)f_2(s) \\
&=& w(s)[A_{11}(s)h_1(s)+A_{12}(s)h_2(s)],
\end{eqnarr*}
and therefore $w(s+1)=w(s)$.
Using again the fact that
$f_i$ and $h_i$ are analytic in this strip and $h_i$ is also zero free there,
we conclude that $w(s)$ is analytic in ${\mathcal S}_{\epsilon}$,
and the periodicity of $w$ implies that it may be extended to an entire periodic function.
Lemma \ref{lemma1}(ii) together with the uniform boundedness of $f_i(s)$ in ${\mathcal S}_{\epsilon}$ imply
that there exists a constant $C>0$ such that for all $s\in {\mathcal S}_{\epsilon}$,
\beqq
|w(s)|< C e^{\pi|\im(s)|}.
\eeqq
By periodicity of $w$, we conclude that the above bound holds for all $s\in \c$.
Since $w$ is periodic with period one, this bound implies that
$w$ is a constant function (this follows from the Fourier series representation
of periodic analytic functions; see the proof of Proposition 2 in [Kuznetsov and Pardo, 2013]).
Finally, we know that $f_i(1)=h_i(1)=1$, and so we conclude that
$w(s)\equiv 1$. Hence, $f_i(s)\equiv h_i(s)$ for all $s\in {\mathcal S}_{\epsilon}$.
Since $h_i(s)$ are analytic in the wider strip\linebreak % final
$-1/\alpha<\re(s)<2-1/\alpha$, by analytic continuation
%% See comments on earlier citation of this result
% (see \cite[Theorem 2]{Lukacs})
we conclude that \eqref{Mellin_transform} holds for all $s$ in $-1/\alpha<\re(s)<2-1/\alpha$.
\end{proof}
\begin{remark}
Since the proof of Theorem \ref{thm_Mellin_transform} is based on a verification technique,
it does not reveal how we derived the formula
on the right-hand side of \eqref{Mellin_transform}, for which we took a trial and error approach.
The expression
in \eqref{Mellin_transform} is already known, or may be easily computed, for
the spectrally positive case ($\rho=1-1/\alpha$; in this case
$T_0$ is the time of first passage below the level zero, and indeed
has a positive $1/\alpha$-stable law, as may be seen from \cite[Theorem 46.3]{Sato}),
the spectrally negative case ($\rho=1/\alpha$; due to \cite[Corollary 1]{Sim-hit})
and for the symmetric case ($\rho=1/2$; due to [Yano et~al., 2009, Cordero, 2010]),
we sought a function which interpolated these three cases and satisfied
the functional equation \eqref{matrix_equation}.
After a candidate was found, we verified that this was indeed the Mellin transform,
using the argument above.
\end{remark}
\newcommand*{\Lnorm}[1]{\lVert #1 \rVert}
\medskip\noindent
We turn our attention to computing the density of $T_0$.
Let us define $x=min_n ∈x-n$, and
\beqq
{\mathcal L}=\r \setminus (\q \cup \{x \in \r : \lim_{n\to \infty} \tfrac{1}{n}\ln\Lnorm{nx}=0\}).
\eeqq
This set was introduced in [Hubalek and Kuznetsov, 2011], where it was shown that $ℒ$ is
a subset of the Liouville numbers %,which implies
and that
$x∈ℒ$ if and only if the coefficients
of the continued fraction representation of $x$ grow extremely fast.
It is known that $ℒ$ is dense, yet it is a rather small set:
it has Hausdorff dimension zero, and therefore its Lebesgue measure is also zero.
For $α∈$̊ we also define
𝒦(α)={N∈ℕ : (N-12)α>exp(-α-12 (N-2)ln(N-2)) }.
Assume that $\alpha \notin \q$.
* The set $ {\mathcal K}(\alpha)$ is unbounded and has density equal to one:
lim_n→∞ # { 𝒦(α) ∩[1,n] }/n =1.
* If $\alpha \notin {\mathcal L}$, the set ${\mathbb N}\setminus {\mathcal K}(\alpha)$ is finite.
Part i:prop_K_alpha:1 follows, after some short manipulation,
from the well-known fact that for any irrational
$\alpha$ the sequence $\Lnorm{(N-\tfrac12)\alpha}$
is uniformly distributed on the interval $(0,1/2)$.
To prove part i:prop_K_alpha:2, first assume that $\alpha \notin {\mathcal L}$.
Since $\lim_{n\to +\infty} \tfrac{1}{n}\ln\Lnorm{n \alpha}=0$, there exists $C>0$ such that
for all $n$ we have $\Lnorm{n \alpha}>C2^{-n}$. Then for all $N$ we have
(N-12)α≥1/2 (2N-1)α>C2^-2N.
Since for all $N$ large enough it is true that
C2^-2N>exp(-α-12 (N-2)ln(N-2)),
we conclude that all $N$ large enough will be in the set ${\mathcal K}(\alpha)$, therefore
the set ${\mathbb N}\setminus {\mathcal K}(\alpha)$ is finite.
Let $p$ be the density of $T_0$ under $\stP_1$.
If $\alpha \notin \q$ then for all $t>0$ we have
= lim_N∈𝒦(α)
sin(π/α )/πsin(πρ̂)
∑_1≤k < α(N-1/2)-1
sin(π/α k )/sin( π/α(k+1) )
×Γ(k/α +1) /k!
(-1)^k-1 t^-1-k/α
2l -
sin(π/α )^2/πsin(πρ̂)
∑_1≤k < N
Γ(k-1/α)/ Γ(αk -1)
The above limit is uniform for $t\in [\epsilon,\infty)$ and any $\epsilon>0$.
If $\alpha=m/n$ (where $m$ and $n$ are coprime natural numbers) then for all $t>0$ we have
sin(π/α )/πsin(πρ̂)
k≠-1 (mod m)
sin(π/α k )/sin( π/α(k+1) )
Γ(k/α +1) /k!
(-1)^k-1 t^-1-k/α
sin(π/α )^2/πsin(πρ̂)
k≠0 (mod n)
Γ(k-1/α)/ Γ(αk -1)
sin(π/α )^2/π^2 αsin(πρ̂)
R_k (t)
R_k (t) := παρ̂cos(πρ̂km)
- sin(πρ̂km)
- ψ(kn - 1α )
+ αψ(km - 1)
+ ln(t) ]
and $\psi$ is the digamma function.
The three series in (<ref>) converge uniformly for $t\in [\epsilon,\infty)$ and any $\epsilon>0$.
For all values of $\alpha$ and any $c>0$, the following asymptotic expansion holds as $t\downto 0$:
\[
p(t) =
\frac{\alpha\sin\left(\frac{\pi}{\alpha} \right)}{\pi\sin(\pi \hat \rho)}
\sum\limits_{1\le n <1+c }
\sin(\pi \alpha \hat \rho n)
\frac{\Gamma(\alpha n+1)}{ \Gamma\left(n+\frac{1}{\alpha} \right)}
(-1)^{n-1} t^{n-1+\frac{1}{\alpha}}+
\]
Recall that $h_1(s)=\stE_1[T_0^{s-1}]$ denotes the function in (<ref>). According to
Lemma <ref>(ii), for any $x\in \r$, $h_1(x+y)$ decreases to zero exponentially
fast as $y→∞$. This implies that the density of $T_0$ exists and is a smooth function. It also implies that $p(t)$ can be written as the inverse Mellin transform,
\beq\label{px_inverse_Mellin}
p(t)=\frac{1}{2\pi \iu} \int\limits_{1+\iu \r} h_1(s) t^{-s} \d s.
\eeq
The function $h_1(s)$ is meromorphic, and it has poles at points
\beqq
\{s^{(1)}_n:=1+n/\alpha\; : \; n\ge 1\} \cup \{s^{(2)}_n:=n-1/\alpha \; : \; n\ge 2\}
\cup\{s^{(3)}_n:=-n-1/\alpha \;:\; n\ge 0\}
\eeqq
If $α∉$, all these points are distinct and $h_1(s)$ has only simple poles. When $α∈$,
some of $s^(1)_n$ and $s^(2)_m$ will coincide, and $h_1(s)$ will have double poles at these points.
Let us first consider the case $α∉$, so that all poles are simple.
Let $N∈𝒦(α)$ and define $c=c(N)=N+12-1α$.
Lemma \ref{lemma1}(ii) tells us that $h_1(s)$ decreases exponentially to zero as $(s)→∞$,
uniformly in any finite vertical strip. Therefore, we can shift the contour of integration in
\eqref{px_inverse_Mellin} and obtain, by Cauchy's residue theorem,
\begin{eqnarr}
\nonumber
- \sum_n%\limits_{\substack{n\ge 1 \\ s^{(1)}_n<c(N)}}
\Res_{s=s^{(1)}_n}(h_1(s)t^{-s})
- \sum_m%\limits_{\substack{m\ge 2 \\ s^{(2)}_m<c(N)}}
\Res_{s=s^{(2)}_m}(h_1(s)t^{-s}) \\
&& {} + \frac{1}{2\pi \iu} \int\limits_{c(N)+\iu \r} h_1(s) t^{-s} \d s,
\label{px_inverse_Mellin2}
\end{eqnarr}
where $∑_n$ and $∑_m$ indicate
summation over $n ≥1$ such that $s_n^(1) < c(N)$
and over $m ≥2$ such that $s_m^(2) < c(N)$, respectively.
Computing the residues we obtain the two sums in the right-hand side of \eqref{e:T0 density}.
Now our goal is to show that the integral term
\beqq
I_N(t):=\frac{1}{2\pi \iu} \int\limits_{c(N)+\iu \r} h_1(s) t^{-s} \d s
\eeqq
converges to zero as $N→+∞$, $N∈𝒦(α)$. We use the reflection formula
for the gamma function and the inequalities
\begin{IEEEeqnarray*}{l"l}
\abs{\sin(\pi x)} > \Lnorm{x}, & x \in \RR, \\
\abs{\sin(x)}\cosh(y) \le \abs{\sin(x+\iu y)} = \sqrt{\cosh^2(y)-\cos^2(x)} \le \cosh(y),
& x,y\in \r,
\end{IEEEeqnarray*}
to estimate $h_1(s)$, $s=c(N)+u$, as follows
\beq\label{estimate1}
\abs{h_1(s)}&=&
\frac{\sin\left(\frac{\pi}{\alpha} \right)}{\sin(\pi \hat \rho)}
\biggabs{ \frac{\sin\left( \pi \hat \rho (1-\alpha+\alpha s)\right)}
{\sin\left(\frac{\pi}{\alpha} (1-\alpha+\alpha s) \right)}
\frac{\sin(\pi s)}{\sin(\pi \alpha (s-1))}
\frac{\Gamma(s-1)}{\Gamma(\alpha (s-1))} } \nonumber \\
& \le& \frac{C_1}{\Lnorm{\alpha (N-\frac{1}{2})}}
\frac{ \cosh(\pi \alpha \hat \rho u) }{\cosh(\pi \alpha u)}
\biggabs{
\frac{\Gamma(s-1)}{\Gamma(\alpha (s-1))} } .
\eeq
Using Stirling's formula (\ref{Stirlings_formula}),
% \cite[8.237]{GR},
we find that
\beq\label{e:ratio of gammas}
\frac{\Gamma(s)}{\Gamma(\alpha s)}=\sqrt{\alpha}
e^{-s \left((\alpha-1)\ln(s)+A\right)+O(s^{-1})}, \for s\to \infty, \; \re(s)>0,
\eeq
where $A:=1-α+αln(α)>0$.
Therefore, there exists a constant $C_2>0$ such that for $s >0$ we can estimate
\begin{eqnarr*}
\biggabs{ \frac{\Gamma(s)}{\Gamma(\alpha s)}}
% &<& C_2 \bigabs{ e^{-s \left((\alpha-1)\ln(s)+A\right)) } } \\
% &=& C_2 e^{- (\alpha-1) \re (s \ln(s)) - A\Re(s))} \\
% &=& C_2 e^{- (\alpha-1) \re (s) \ln|s| - A\Re(s)+(\alpha-1) \Im(s)\arg(s) } \\
&<& C_2 e^{- (\alpha-1) \re (s) \ln(\Re(s)) +(\alpha-1) \abs{\Im(s)} \frac{\pi}{2} }.
\end{eqnarr*}
Combining the above estimate with \eqref{estimate1} and using the fact that $N∈𝒦(α)$ we find that
\begin{eqnarr*}
\abs{h_1(c(N)+\iu u)}
&<& \frac{C_1C_2}{\Lnorm{\alpha (N-\frac{1}{2})}}
\frac{ \cosh(\pi \alpha \hat \rho u) }{\cosh(\pi \alpha u)}
e^{- (\alpha-1) (c(N)-1) \ln(c(N)-1) +(\alpha-1) \abs{u} \frac{\pi}{2} } \\
&<& C_1C_2 e^{-\frac{\alpha-1}{2} (N-2)\ln(N-2)}
\frac{ \cosh(\pi \alpha \hat \rho u) }{\cosh(\pi \alpha u)} e^{(\alpha-1) \abs{u} \frac{\pi}{2} } .
\end{eqnarr*}
Note that the function in the right-hand side of the above inequality decreases to zero exponentially fast
as $u→∞$ (since $αρ̂+ 12 (α-1)-α<0$),
and hence in particular is integrable on $$.
Thus we can estimate
\begin{multline*}
\abs{I_N(t)}
= \frac{t^{-c(N)}}{2\pi}
\left\lvert\int_\r h_1(c(N)+\iu u)) t^{-\iu u} \, \d u \right\rvert
< \frac{t^{-c(N)}}{2\pi} \int_\r |h_1(c(N)+\iu u))| \, \d u \\
{} < \frac{t^{-c(N)}}{2\pi}C_1C_2 e^{-\frac{\alpha-1}{2} (N-2)\ln(N-2)}
\int_\r \frac{ \cosh(\pi \alpha \hat \rho u) }{\cosh(\pi \alpha u)} e^{(\alpha-1) \abs{u} \frac{\pi}{2} } \, \d u .
\end{multline*}
When $N→∞$, the quantity in the right-hand side of the above inequality
converges to zero for every $t>0$. This ends the proof of part (i).
The proof of part (ii) is very similar, and we offer only a sketch.
It also begins with \eqref{px_inverse_Mellin2}
and uses the above estimate for $h_1(s)$. The only difference is that when $α∈$
some of $s^(1)_n$ and $s^(2)_m$ will coincide, and $h_1(s)$ will have
double poles at these points.
The terms with double poles give rise to the third series in \eqref{e:T0 density2}. In this case all
three series are convergent, and we can express the limit of partial sums as
a series in the usual sense.
The proof of part (iii) is much simpler: we need to shift the contour of integration in
\eqref{px_inverse_Mellin2} in the opposite direction ($c<0$). The proof is identical to the
proof of Theorem 9 in [Kuznetsov, 2011].
\end{proof}
\end{theorem}
\begin{remark}
We offer some remarks on the asymptotic expansion.
When $\rhohat = 1/\alpha$, all of its terms are equal to zero. This is the spectrally
positive case, in which $T_0$ has the law of a positive $1/\alpha$-stable
random variable, and it is known
that its density is exponentially small at zero; see \cite[equation (14.35)]{Sato}
for a more precise result.
Otherwise, the series given by including all the terms in (iii) is divergent for all $t > 0$.
This may be seen from the fact that the terms do not approach $0$;
we sketch this now. When $\alpha\rhoh \in \QQ$, some terms are zero,
and in the rest the sine term is bounded away from zero;
when $\alpha\rhoh \notin \QQ$, it follows that
$\limsup_{n \to \infty} \abs{\sin(\pi\alpha\rhoh n)} = 1$.
One then bounds the
ratio of gamma functions from below by $\Gamma(\alpha n)/\Gamma((1+\epsilon)n)$,
for some small enough $\epsilon > 0$ and large $n$.
This grows superexponentially due to \eqref{e:ratio of gammas},
so $t^n\Gamma(\alpha n+1)/\Gamma(n+1/\alpha)$ is unbounded as $n \to \infty$.
\end{remark}
The next corollary shows that, for almost all irrational $α$, the expression
\eqref{e:T0 density} can be written in a simpler form.
\begin{corollary}\label{cor_alpha_notin_L}
If $\alpha \notin {\mathcal L} \cup \q$ then
\begin{eqnarr}
p(t)&=&\frac{\sin\left(\frac{\pi}{\alpha} \right)}{\pi\sin(\pi \hat \rho)}
\sum\limits_{k\ge 1}
\sin(\pi \hat \rho(k+1))
\frac{\sin\left(\frac{\pi}{\alpha} k \right)}{\sin\left( \frac{\pi}{\alpha}(k+1) \right)}
\frac{\Gamma\left(\frac{k}{\alpha} +1\right) }{k! }
(-1)^{k-1} t^{-1-\frac{k}{\alpha}} \nonumber \\
&& {} -
\frac{\sin\left(\frac{\pi}{\alpha} \right)^2}{\pi\sin(\pi \hat \rho)}
\sum\limits_{k\ge 1}
\frac{\sin(\pi \alpha \hat \rho k)}{\sin(\pi \alpha k)}
\frac{\Gamma\left(k-\frac{1}{\alpha}\right)}{ \Gamma\left(\alpha k -1\right)}
\label{e:T0 density4}
\end{eqnarr}
The two series in the right-hand side of the above formula converge uniformly for
$t\in [\epsilon,\infty)$ and any $\epsilon>0$.
\end{corollary}
\begin{proof}
As we have shown in Proposition \ref{prop_K_alpha}, if $\alpha \notin {\mathcal L} \cup \q$ then
the set ${\mathbb N} \setminus{\mathcal K}(\alpha)$ is finite. Therefore we can remove the restriction
$N \in {\mathcal K}(\alpha)$ in \eqref{e:T0 density}, and need only show that
both series in \eqref{e:T0 density4} converge.
% The remaining part of the proof if based on the following simple result:
% If the two series
% $\sum a_n$ and $\sum b_n$ converge, then for any increasing, unbounded sequences $c_n$ and $d_n$
% \beqq
% \lim_{N\to +\infty} \left[\sum_{n=1}^{c_N} a_n+ \sum_{n=1}^{d_N}b_n \right]=\sum_{n\ge 1} a_n +\sum_{n\ge 1} b_n.
% \eeqq
% Therefore,
In \cite[Proposition 1]{ECP1601} it was shown that $x \in {\mathcal L}$ iff $x^{-1} \in {\mathcal L}$. Therefore,
according to our assumption,
both $\alpha$ and $1/\alpha$ are not in the set ${\mathcal L}$.
From the definition of ${\mathcal L}$ we see that there exists $C>0$ such that
$\Lnorm{\alpha n}>C2^{-n}$ and $\Lnorm{\alpha^{-1} n}>C2^{-n}$ for all integers $n$.
Using the estimate $|\sin(\pi x)|\ge \Lnorm{x}$ and
Stirling's formula \eqref{Stirlings_formula}, it is easy to see that both series in
in \eqref{e:T0 density4} converge (uniformly for $t\in [\epsilon,\infty)$ and any $\epsilon>0$),
which ends the proof of the corollary.
\end{proof}
\begin{remark}
Note that formula \eqref{e:T0 density4} may not be true if $\alpha \in {\mathcal L}$, as the series
may fail to converge. An example where this occurs is given after Theorem 5 in [Kuznetsov, 2011].
\end{remark}
\section{Applications}
\label{s:app}
\subsection{Conditioning to avoid zero}
In \cite[\S 4.2]{CPR}, Chaumont, Pant{\'{\i}} and Rivero
discuss a harmonic transform
of a stable process with $α> 1$
which results in \define{conditioning to avoid zero}. The
results quoted in that paper are a special case of the
notion of
conditioning a L\'evy process to avoid zero, which is explored
in Pant{\'{\i}}, 2013.
In these works, in terms of the parameters used in the introduction, the authors define
\begin{equation}
\label{e:h}
h(x) = %K(\alpha)
\frac{\Gamma(2-\alpha) \sin(\pi\alpha/2)}
{c \pi (\alpha-1) (1+\beta^2 \tan^2(\pi\alpha/2))}
(1-\beta \sgn(x)) \abs{x}^{\alpha-1},
\for x \in \RR.
\end{equation}
%where the constant
%\[ K(\alpha) =
% \frac{\Gamma(2-\alpha) \sin(\pi\alpha/2)}
% {c \pi (\alpha-1) (1+\beta^2 \tan^2(\pi\alpha/2))}
%is defined in terms of
%\[ c
% = - (c_+ + c_-)\Gamma(-\alpha) \cos(\pi\alpha/2)
% = \cos (\pi\alpha(\rho - 1/2)) \]
%\[ \beta = \frac{c_+ - c_-}{c_+ + c_-}
% = \frac{ \tan(\pi\alpha(\rho-1/2))}{\tan(\pi\alpha/2)},
%which are the quantities appearing in \eqref{e:stable CE}.
If we write the function $h$ in terms of the $(α,ρ)$
parameterisation which we prefer, this gives
\[ h(x) =
- \Gamma(1-\alpha)
\frac{\sin (\pi\alpha\rhoh)}{\pi}
\abs{x}^{\alpha-1} , \for x > 0, \]
and the same expression with $$ replaced by $ρ$ when $x < 0$.
In [Pant{\'{\i}}, 2013], Pant{\'\i} proves the following
proposition for all L\'evy processes, and $x ∈$,
with a suitable
definition of $h$. Here we quote only the result for stable
processes and $x 0$.
Hereafter, $$ is the standard filtration associated with $X$,
$n$ refers to the excursion measure of the stable process
away from zero, normalised
(see \cite[(7)]{Pan-cond} and \cite[(4.11)]{FG-bridges}) such that
\[ n(1-e^{-q\zeta}) = 1/u^q(0), \]
where $ζ$ is the excursion length and $u^q$ is the $q$-potential
density of the stable process.
\def\tempPprop{\cite[Theorem 2, Theorem 6]{Pan-cond}}
\begin{proposition}[\tempPprop]\label{p:Panti}
Let $X$ be a stable process, and $h$ the function in \eqref{e:h}.
\begin{enumerate}[(i)]
\item \label{p:Panti:1}
The function $h$ is invariant for the stable process
killed on hitting $0$, that is,
\begin{equation}\label{e:h invariant}
\stE_x[ h(X_t), t < T_0] = h(x), \for t > 0, \, x \neq 0.
\end{equation}
Therefore, we may define a family of measures $\stPaz_x$ by
\[ \stPaz_x(\Lambda)
= \frac{1}{h(x)} \stE_x[ h(X_t) \Ind_\Lambda, t < T_0 ],
\for x \ne 0, \, \Lambda \in \FF_t, \]
for any $t \ge 0$.
\item \label{p:Panti:2}
The function $h$ can be represented as
\begin{equation*}%\label{e:h lim}
h(x) = \lim_{q \downto 0} \frac{\stP_x(T_0 > \ee_q)}{n(\zeta > \ee_q)},
\for x \ne 0,
\end{equation*}
where $\ee_q$ is an independent exponentially distributed random
variable with parameter $q$. Furthermore, for any stopping time $T$
and $\Lambda \in \FF_T$, and any $x \ne 0$,
\[ \lim_{q \downto 0} \stP_x(\Lambda, T < \ee_q \vert T_0 > \ee_q) = \stPaz_x(\Lambda) . \]
\end{enumerate}
\end{proposition}
This justifies the name `the stable process conditioned to avoid zero'
for the canonical process associated
with the measures $(_x)_x 0$.
We will denote this process by $$.
\medskip\noindent
Our aim in this section is to prove the following variation
of Proposition
\ref{p:Panti}(ii), making use of our expression for the density of $T_0$.
Our presentation here owes much to
\citet[\S 4.3]{YYY-pen}.
\begin{proposition}\label{p:lim P} Let $X$ be a stable process adapted to the filtration $\FFt$,
and $h \colon \RR \to \RR$ as in \eqref{e:h}.
\begin{enumerate}[(i)]
\item \label{p:lim P:1}
Define the function
\begin{equation*} Y(s,x) = \frac{\stP_x(T_0 > s)}{h(x) n(\zeta > s)}
\for s > 0, \, x \ne 0.
\end{equation*}
Then, for any $x \ne 0$,
\begin{equation}
\lim_{s \to \infty} Y(s,x) = 1,
\label{YYY}
\end{equation}
and furthermore, $Y$ is bounded away from $0$ and $\infty$ on its whole domain.
\item \label{p:lim P:2}
For any $x \ne 0$,
stopping time $T$ such that $\LevE_x[T] < \infty$, and
$\Lambda \in \FF_T$,
% $\stP_x$-a.s.\ finite stopping time $T$ and $\Lambda \in \FF_T$
% such that there exists $\epsilon > 0$ for which
% $\LevE_x[\Ind_{\Lambda} T^{1-1/\alpha+\epsilon}] < \infty$,
\[ \stPaz_x(\Lambda)
= \lim_{s \to \infty} \stP_x (\Lambda \vert T_0 > T + s) . \]
\end{enumerate}
\begin{proof}
We begin by proving
\begin{equation}\label{e:h lim s}
h(x) = \lim_{s \to \infty} \frac{\stP_x(T_0 > s)}{n(\zeta > s)},
\end{equation}
for $x > 0$, noting that when $x < 0$, we may deduce the same limit by duality.
Let us denote the density of the measure $\stP_x(T_0 \in \cdot)$
by $p(x,\cdot)$. A straightforward application of scaling
shows that
%\[ p(x,t) = x^{\alpha-1} p(1,t) \for x > 0,\, t \ge 0 , \]%WRONG
\[ \stP_x(T_0 > t) = \stP_1(T_0 > x^{-\alpha} t), \for x > 0, \, t \ge 0, \]
and so we may focus our attention on $p(1,t)$, which
is the quantity given as $p(t)$ in Theorem \ref{t:T0 densities}.
In particular, we have
\[
p(1,t) = - \frac{ \sin^2 (\pi/\alpha)}{\pi \sin(\pi \rhoh)}
\frac{\sin(\pi\alpha\rhoh)}{\sin(\pi\alpha)}
\frac{\Gamma(1-1/\alpha)}{\Gamma(\alpha-1)}
% + O(t^{1/\alpha-3})
+ O(t^{-1/\alpha-1})
\]
Denote the coefficient of $t^{1/\alpha-2}$ in the
first term of this expression by $P$.
%% Removed per referee
% Note that all limits in Theorem 3.15 exist uniformly in
% $t \in [\epsilon,\infty)$, so we can integrate in $t$ on any interval $[t_0,\infty)$ for $t_0 > 0$.
% This allows us
% to compute $\stP_x(T_0 > t)$ in the asymptotic sense
% from the expansions in Theorem \ref{t:T0 densities} by
% integrating term-by-term.
To obtain an expression for $n(\zeta > t)$, we turn to
Fitzsimmons and Getoor, 1995, in which the authors compute
explicitly the density
of $n(\zeta \in \cdot)$ for a stable process; see p.~84
in that work, where $n$ is denoted $P^*$ and $\zeta$
is denoted $R$.
The authors work with
a different normalisation of the stable process; they have
$c = 1$. In our context, their result says
\begin{equation}\label{e:n density}
n(\zeta \in \dd t)
= \frac{\alpha-1}{\Gamma(1/\alpha)}
\frac{\sin(\pi/\alpha)}{\cos(\pi(\rho-1/2))}
t^{1/\alpha-2} \, \dd t, \qquad t\geq 0.
\end{equation}
Denote the coefficient in the above power law by $W$.
We can now compute $h$. We will use elementary
properties of
trigonometric functions and the reflection identity for the gamma
function. For $x > 0$,
\begin{eqnarr*}
\frac{\stP_x(T_0 > t)}{n(\zeta > t)}
&=& \frac{P}{W} x^{\alpha-1}
+ O(t^{1-2/\alpha}) \\
- \frac{\cos(\pi(\rho-1/2)) \sin(\pi\alpha\rhoh)}
{\Gamma(\alpha) \sin(\pi\rhoh) \sin(\pi\alpha)} x^{\alpha-1}
+ o(1) \\
- \frac{1}{\Gamma(\alpha)}
\frac{\sin(\pi\alpha\rhoh)}{\sin(\pi\alpha)} x^{\alpha-1}
+ o(1) \\
&=& - \Gamma(1-\alpha) \frac{\sin(\pi\alpha\rhoh)}{\pi} x^{\alpha-1}
+ o(1).
\end{eqnarr*}
This proves \eqref{e:h lim s} for $x > 0$, and it is simple to deduce
via duality that this limit holds for $x \ne 0$.
We now turn our attention to the slightly more delicate result about $Y$.
It is clear that the limit in \eqref{YYY} holds, so we only need to prove that $Y$ is bounded.
We begin by noting that, for fixed $x \ne 0$, $Y(t,x)$ is bounded
away from $0$ and $\infty$ in $t$
since the function is continuous and converges to $1$ as $t \to \infty$.
Now, due to the expression \eqref{e:n density} and the scaling property of
$X$, we have the relation $Y(t,x) = Y(\abs{x}^{-\alpha}t,\sgn x)$.
This then shows that $Y$ is bounded as a function of two variables.
With this in hand, we move on to the calculation of the limiting measure.
This proceeds along familiar lines, using the strong Markov property:
\begin{eqnarr*}
\stP_x( \Lambda \vert T_0 > T+s)
&=& \stE_x \biggl[ \frac{ \stP_x(\Ind_\Lambda, T_0 > T+s \vert \FF_T) }
{\stP_x(T_0 > T+s)} \biggr] \\
&=& \stE_x\biggl[ \Ind_\Lambda \Indic{T_0 > T}
\frac{\stP_{X_T}(T_0 > s)}{\stP_x(T_0 > T+s)} \biggr] \\
&=& \stE_x\biggl[ \Ind_\Lambda \Indic{T_0 > T} h(X_T)
Y(s, X_T)
\frac{n(\zeta > s)}{n(\zeta > T+s)}
\frac{1}{h(x) Y(s+T,x)}
\biggr].
\end{eqnarr*}
Now, as $h$ is invariant for the stable process killed at zero,
\eqref{e:h invariant} also holds at $T$, and in particular
the random variable $h(X_T)\Indic{T_0 > T}$ is integrable; meanwhile, $Y$ is bounded
away from zero and $\infty$.
\newcommand\notfrac[2]{#1/#2}
Finally, $\notfrac{n(\zeta > s)}{n(\zeta>T+s)} = \bigl(1+\frac{T}{s}\bigr)^{1-1/\alpha}$;
the moment condition in the statement of the theorem permits us to
apply the dominated convergence theorem, and this gives the result.
\end{proof}
\end{proposition}
\medskip \noindent
We offer a brief comparison to conditioning a L\'evy process to stay
positive. In this case, \citet[Remark 1]{Cha-cond} observes that
the analogue of
Proposition \refpref{p:lim P}{p:lim P:2}
holds under Spitzer's condition, and in particular for a stable process.
However, it appears that in general, a key role is played by
the exponential random variable
analogous to that
appearing in Proposition \refpref{p:Panti}{p:Panti:2}.
\subsection{The radial part of the stable process conditioned to avoid zero}
\label{ss:Raz}
For this section, consider $X$ to be the symmetric stable process,
as in section \ref{s:symm}. There we computed the Lamperti transform $ξ$
of the pssMp
\[ R_t = \abs{X_t}\Indic{t < T_0}, \for t \ge 0, \]
and gave its characteristic exponent $Ψ$ in \eqref{e:WHF low index}.
Consider now the process
\[ \Raz_t = \bigabs{\Xaz_t}, \for t \ge 0.\]
This is also
a pssMp, and we may consider its Lamperti transform, which we will
denote by $$. The characteristics of the Lamperti--Kiu
representation of $$ have been computed explicitly in [Chaumont et~al., 2013],
and the characteristic exponent, $$, of $$ could be
computed from this information; however, the harmonic transform
in Proposition \refpref{p:Panti}{p:Panti:1}
gives us the following straightforward relationship
between characteristic exponents:
\[ \CEaz(\theta) = \CE(\theta - \iu(\alpha-1)) . \]
This allows us to calculate
\[ \CE^\updownarrow(\theta)
= 2^\alpha \frac{\Gamma(1/2-\iu\theta/2)}{\Gamma((1-\alpha)/2-\iu\theta/2)}
\frac{\Gamma(\alpha/2 + \iu\theta/2)}{\Gamma(\iu\theta/2)} , \for \theta \in \RR.
\]
It is immediately apparent that $$ is the dual L\'evy process
to $ξ$. It then follows that $R$ is a time-reversal
of $$, in the sense of \cite[\S 2]{CP-lower}: roughly speaking,
if one fixes $x > 0$,
starts the process $$ at zero (as in [Caballero and Chaumont, 2006], say)
and runs it backward from its last moment below some level $y$, where $y>x$,
simultaneously conditioning on the position of the left limit at this time taking value $x$,
then one obtains the law of $R$ under $_x$.
We remark that this relationship is already known for Brownian
motion, where $$ is a Bessel process of dimension $3$.
However, it seems unlikely that any such time-reversal
property will hold for a general L\'evy process conditioned to avoid
\subsection{The entrance law of the excursion measure}
It is known, from a more general result \cite[(2.8)]{CFY-ext}
on Markov processes in weak duality, that for any Borel function
$f$, the equality
\[
\int_0^\infty e^{-qt} n(f(X_t)) \, \dd t
= \int_{\RR} f(x) \stEh_x\bigl[ e^{-qT_0} \bigr] \, \dd x \]
holds, where $n$ is the excursion measure of $X$ from zero.
(This formulation is from \cite[(3.9)]{YYY-pen}.) In terms of
densities, this may be written
\begin{eqnarr*}
n(X_t \in \dd x) \, \dd t
&=& \stPh_x(T_0 \in \dd t) \, \dd x \\
&=& \abs{x}^{-\alpha} \stP_{\sgn(-x)}(T_0 \in \abs{x}^{-\alpha}\dd t) \, \dd x
\end{eqnarr*}
Therefore, our expressions in Theorem \ref{t:T0 densities}
for the density of $T_0$ yield expressions
for the density of the entrance law of the excursions of the stable process
from zero.
% lots of clauses here...
\subsection{Identities in law using the exponential functional}
In a series of papers (Bertoin and Yor, 2002, Caballero and Chaumont, 2006, Chaumont et~al., 2012)
it is proved that under certain conditions, the laws $(_x)_x > 0$ of an
$α$-pssMp $X$ admit a weak limit $_0$ as $x 0$, in the Skorokhod space
of c\`adl\`ag paths.
If $ξ$ is the Lamperti
transform of $X$ under $_1$,
then provided that $ξ_1 < ∞$
and $m := ξ_1 > 0$, it is known that the entrance law of $_0$
\[ \stE_0( f(X_t^\alpha)) = \frac{1}{\alpha m} \LevE( I(-\alpha\xi)^{-1} f(t/I(-\alpha\xi))) , \]
for any $t > 0$ and Borel function $f$.
Similar expressions are available under
less restrictive conditions on $ξ$.
It is tempting to speculate that any rssMp may admit a weak limit
$_0$ along similar lines, but we do not propose any results
in this direction; instead, we demonstrate
similar formulae for the entrance law $n(X_t ∈·)$ of the
stable process, and the corresponding measure $_0$ for the stable
process conditioned to avoid zero.
\medskip \noindent
Let $X$ be a stable process, possibly asymmetric.
From the previous subsection, we have that
\[ n(f(X_t))
= \int_{-\infty}^\infty
\abs{x}^{-\alpha} p(\sgn(-x), \abs{x}^{-\alpha}t) f(x) \, \dd x.
\]
Substituting in the integral, and recalling that the law of $T_0$
for the stable process is equal to the law of the exponential
functional $I(αξ)$ of the Markov additive process associated with it,
we obtain
\begin{eqnarr*}
&=& \frac{1}{\alpha}\int_0^\infty p(1,u)
f( -(u/t)^{-1/\alpha}) u^{-1/\alpha} t^{1/\alpha-1} \, \dd u \\
&& {} + \frac{1}{\alpha}\int_0^\infty p(-1,u)
f( (u/t)^{-1/\alpha}) u^{-1/\alpha} t^{1/\alpha-1} \, \dd u \\
&=& \frac{1}{\alpha}\MAPE_1\bigl[ f(-(t/I(\alpha\xi))^{1/\alpha})
I(\alpha\xi)^{-1/\alpha} t^{1/\alpha-1} \bigr] \\
&& {} + \frac{1}{\alpha}\MAPE_2\bigl[ f((t/I(\alpha\xi))^{1/\alpha})
I(\alpha\xi)^{-1/\alpha} t^{1/\alpha-1} \bigr].
\end{eqnarr*}
Recall from [Pant{\'{\i}}, 2013] that the
law $_0$ of the stable process conditioned to avoid
zero is given by the following harmonic transform of the
stable excursion measure $n$:
\[ \stPaz_0(\Lambda) = n( \Ind_\Lambda h(X_t), t < \zeta),
\for t \ge 0, \, \Lambda \in \FF_t , \]
with $h$ as in \eqref{e:h}. Therefore, applying the above result to the Borel function $hf$, we
\begin{eqnarr*}
\stEaz_0(f(X_t))
&=& n( h(X_t) f(X_t) ) \\
\Gamma(-\alpha)\frac{\sin(\pi\alpha\rho)}{\pi}
\MAPE_1\bigl[I(\alpha\xi)^{-1} f(-(t/I(\alpha\xi))^{1/\alpha})
\bigr] \\
&&{} + \Gamma(-\alpha)\frac{\sin(\pi\alpha\rhoh)}{\pi}
\MAPE_2\bigl[ I(\alpha\xi)^{-1} f((t/I(\alpha\xi))^{1/\alpha})
\bigr] ,
\end{eqnarr*}
where we emphasise that $I(αξ)$ (under $_i$)
is the exponential functional of the
Markov additive process associated to $X$.
%% %%
%% You may add acknowledgments (optional). %%
%% %%
\bigskip
\ACKNO{%
Some of this research was carried out whilst AEK and ARW were visiting
ETH Z\"urich and CIMAT. Both authors would like to offer thanks to both
institutions for their hospitality. AK acknowledges the support by the
Natural Sciences and Engineering Research Council of Canada. JCP acknowledges
the support by CONACYT (grant 128896).
All authors would like to thank the referee for their detailed comments that led
to an improved version of this paper.%
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\end{document}
|
arxiv-papers
| 2012-12-20T17:29:14 |
2024-09-04T02:49:39.502518
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexey Kuznetsov, Andreas E. Kyprianou, Juan Carlos Pardo, Alexander\n R. Watson",
"submitter": "Managing Editor",
"url": "https://arxiv.org/abs/1212.5153"
}
|
1212.5205
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-362 LHCb-PAPER-2012-042 December 18, 2012
Measurement of $C\\!P$ observables in $B^{0}\rightarrow DK^{*0}$ with
$D\rightarrow K^{+}K^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
The decay $B^{0}\rightarrow DK^{*0}$ and the charge conjugate mode are studied
using 1.0$\mbox{\,fb}^{-1}$ of $pp$ collision data collected by the LHCb
experiment at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ in 2011. The $C\\!P$
asymmetry between the $B^{0}\rightarrow DK^{*0}$ and the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay rates, with the neutral $D$
meson in the $C\\!P$-even final state $K^{+}K^{-}$, is found to be
${\cal A}_{d}^{KK}=-0.45\pm 0.23\pm 0.02,$
where the first uncertainty is statistical and the second is systematic. In
addition, favoured $B^{0}\rightarrow DK^{*0}$ decays are reconstructed with
the $D$ meson in the non-$CP$ eigenstate $K^{+}\pi^{-}$. The ratio of the
$B$-flavour averaged decay rates in $D$ decays to $CP$ and non-$CP$
eigenstates is measured to be
${\cal R}_{d}^{KK}=1.36\,^{+\,0.37}_{-\,0.32}\pm 0.07,$
where the ratio of the branching fractions of $D^{0}\rightarrow K^{-}\pi^{+}$
to $D^{0}\rightarrow K^{+}K^{-}$ decays is included as multiplicative factor.
The $C\\!P$ asymmetries measured with two control channels, the favoured
$B^{0}\rightarrow DK^{*0}$ decay with $D\rightarrow K^{+}\pi^{-}$ and the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow DK^{*0}$
decay with $D\rightarrow K^{+}K^{-}$, are also reported.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves
Jr22,35, S. Amato2, Y. Amhis7, L. Anderlini17,f, J. Anderson37, R.
Andreassen57, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J.
Back45, C. Baesso54, V. Balagura28, W. Baldini16, R.J. Barlow51, C.
Barschel35, S. Barsuk7, W. Barter44, A. Bates48, Th. Bauer38, A. Bay36, J.
Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-
Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A.
Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi51, A. Borgia53, T.J.V. Bowcock49, E.
Bowen37, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N. Chiapolini37, M.
Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M.
Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins35, A. Comerma-Montells33, A. Contu15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua51, M. De Cian37, J.M. De Miranda1, L. De
Paula2, W. De Silva57, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
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Dzyuba27, S. Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
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Rifai5, Ch. Elsasser37, D. Elsby42, A. Falabella14,e, C. Färber11, G.
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Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C.
Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M.
Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, E. Furfaro21, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J. Garofoli53,
P. Garosi51, J. Garra Tico44, L. Garrido33, C. Gaspar35, R. Gauld52, E.
Gersabeck11, M. Gersabeck51, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V.
Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara5, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O.
Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-
Menzemer11, N. Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
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Hombach51, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52,
D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, E. Jans38, F. Jansen38, P. Jaton36, F. Jing3, M.
John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M.
Karacson35, T.M. Karbach35, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
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Lu3, J. Luisier36, H. Luo47, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, S. Malde52, G. Manca15,d, G.
Mancinelli6, N. Mangiafave44, U. Marconi14, R. Märki36, J. Marks11, G.
Martellotti22, A. Martens8, L. Martin52, A. Martín Sánchez7, M. Martinelli38,
D. Martinez Santos39, D. Martins Tostes2, A. Massafferri1, R. Matev35, Z.
Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J.
McCarthy42, R. McNulty12, B. Meadows57,52, F. Meier9, M. Meissner11, M.
Merk38, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D.
Moran51, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, P. Naik43, T. Nakada36, R. Nandakumar46,
I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, T.D. Nguyen36, C.
Nguyen-Mau36,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin29, T. Nikodem11, S.
Nisar56, A. Nomerotski52, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C.
Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G.
Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33,
B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F.
Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian4, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues51, P. Rodriguez Perez34,
G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J. Rouvinet36,
T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P.
Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28,29, P.
Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49,35, L.
Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva
Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, M.D. Sokoloff57, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53, B. Storaci37, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, V. Syropoulos39,
M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S. T’Jampens4, M.
Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D. Tonelli35, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, M.
Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g,
G. Veneziano36, M. Vesterinen35, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona33,n, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V.
Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, J. Wicht35, J. Wiechczynski23, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50,p, F.F. Wilson46, J. Wishahi9,
M. Witek23, W. Witzeling35, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35,
Y. Xie47,35, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O.
Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov28, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated
to 53
57University of Cincinnati, Cincinnati, OH, United States, associated to 53
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
Direct $C\\!P$ violation can arise in $B^{0}\rightarrow DK^{*0}$ decays111Here
and in the following, $D$ represents a neutral meson that is an admixture of
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$. Inclusion of
charge conjugate modes is implied unless specified otherwise. from the
interference between two colour-suppressed transitions:
$\bar{}b\rightarrow\bar{}c$ (Cabibbo favoured) and $\bar{}b\rightarrow\bar{}u$
(Cabibbo suppressed). The corresponding Feynman diagrams are shown in Fig. 1;
interference occurs if the $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons decay to a common final
state. The magnitude of the $C\\!P$-violating asymmetry that arises from this
interference is related to the value of the phase
$\gamma=\arg\left[-(V_{ud}V_{ub}^{*})/(V_{cd}V_{cb}^{*})\right]$, the least-
well determined angle of the Unitarity Triangle. A method to determine
$\gamma$ from hadronic $B$-decay rates was originally proposed by Gronau,
London and Wyler (GLW) in Ref. [1, *bib:GW] for various charged and neutral
$B\rightarrow DK$ decay modes and can be applied to the decay mode
$B^{0}\rightarrow DK^{*0}$. In this mode, the charge of the kaon from the
$K^{*0}\rightarrow K^{+}\pi^{-}$ decay unambiguously tags the flavour of the
decaying $B$ meson [3], hence no time-dependent tagged analysis is required.
\begin{overpic}[width=223.0721pt]{B2DbarKstar.pdf} \put(10.0,60.0){(a)}
\end{overpic}\begin{overpic}[width=223.0721pt]{B2DKstar.pdf}
\put(10.0,60.0){(b)} \end{overpic}
Figure 1: Feynman diagrams for (a) $B^{0}\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{*0}$ and (b) $B^{0}\rightarrow
D^{0}K^{*0}$.
The use of these neutral $B$ decays is particularly interesting because the
magnitude of the ratio of the suppressed over the favoured amplitude, which
controls the size of the interference, is expected to be relatively large
(naively a factor three larger than the analogous ratio for $B^{+}\rightarrow
DK^{+}$ decays), hence the system can exhibit large $C\\!P$-violating effects,
depending on the $D$ decay. Among the modes used in the GLW method, which are
studied in this paper, large $C\\!P$ asymmetries can be expected when the $D$
meson is reconstructed in a $C\\!P$ eigenstate. Contributions from $B^{0}$
decays to the non-resonant $DK^{+}\pi^{-}$ final state, which can pollute the
$DK^{*0}$ reconstructed signal candidates due to the large natural width of
the $K^{*0}$, can be treated in a model-independent way, as shown in Ref. [4].
Studies with simulated events have shown that the $B^{0}\rightarrow DK^{*0}$
mode is one of the most promising channels to provide a precise measurement of
$\gamma$ at LHCb [5]. Results with this channel will therefore complement
those from $B^{+}\rightarrow DK^{+}$, which have recently been used by LHCb to
constrain $\gamma$ [6, *Aaij:1483187].
This paper presents the measurement of the $B^{0}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ partial width asymmetry using $D$
decays into the $C\\!P$ eigenstate $K^{+}K^{-}$,
${\cal A}_{d}^{KK}=\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{[K^{+}K^{-}]}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})-\Gamma(B^{0}\rightarrow
D_{[K^{+}K^{-}]}K^{*0})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{[K^{+}K^{-}]}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})+\Gamma(B^{0}\rightarrow
D_{[K^{+}K^{-}]}K^{*0})},$ (1)
together with the measurement of the ratio of the average of the $B^{0}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ partial widths with
$D\rightarrow K^{+}K^{-}$, to the average partial width with $D\rightarrow
K^{+}\pi^{-}$ (where the sign of the kaon charge from the $D$ decay is the
same as that of the kaon from the $K^{*0}$ decay),
${\cal R}_{d}^{KK}=\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{[K^{+}K^{-}]}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})+\Gamma(B^{0}\rightarrow
D_{[K^{+}K^{-}]}K^{*0})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{[K^{-}\pi^{+}]}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})+\Gamma(B^{0}\rightarrow
D_{[K^{+}\pi^{-}]}K^{*0})}.$ (2)
These quantities can be used together with other inputs to determine the value
of $\gamma$. Note that the suppressed decay mode $B^{0}\rightarrow
D_{[K^{-}\pi^{+}]}K^{*0}$, where the sign of the kaon charge from the $D$
decay is opposite to that of the kaon from the $K^{*0}$ decay, is not included
in this analysis. This decay mode can exhibit large $C\\!P$-violating effects
and can be studied with a larger dataset. The measured asymmetry in the
favoured decay $B^{0}\rightarrow D_{[K^{+}\pi^{-}]}K^{*0}$,
${\cal A}_{d}^{\rm fav}=\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{[K^{-}\pi^{+}]}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})-\Gamma(B^{0}\rightarrow
D_{[K^{+}\pi^{-}]}K^{*0})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{[K^{-}\pi^{+}]}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})+\Gamma(B^{0}\rightarrow
D_{[K^{+}\pi^{-}]}K^{*0})}$ (3)
is a useful cross-check since it is expected to be compatible with zero given
the size of the current dataset.
In $pp$ collisions, $B^{0}_{s}$ mesons are produced and can decay to the same
final state, $B^{0}_{s}\rightarrow D\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ [8]. In these $B^{0}_{s}$ decay
modes, the interference between the two contributing amplitudes is expected to
be small, since the relative magnitude of the suppressed to the favoured
amplitude is small compared to the $B^{0}$ case. Therefore, these modes are
valuable control channels, and the asymmetry
${\cal A}^{KK}_{s}=\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{[K^{+}K^{-}]}K^{*0})-\Gamma(B^{0}_{s}\rightarrow D_{[K^{+}K^{-}]}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{[K^{+}K^{-}]}K^{*0})+\Gamma(B^{0}_{s}\rightarrow D_{[K^{+}K^{-}]}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})},$ (4)
similar to that defined in Eq. (1), is also obtained in this analysis. Since
the favoured (suppressed) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ ($B^{0}$) decay gives kaons
with opposite charges from $D$ and $K^{*0}$ decays, ${\cal A}_{s}^{\rm fav}$
is not used as a control measurement in the analysis, to avoid biasing a
potential future measurement of ${\cal A}_{d}^{\rm sup}$.
## 2 The LHCb detector, dataset and event selection
The study reported here is based on a data sample collected at the Large
Hadron Collider (LHC) with the LHCb detector at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ during the year 2011, corresponding to an
integrated luminosity of 1.0$\mbox{\,fb}^{-1}$. The LHCb detector [9] is a
single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$,
designed for the study of particles containing $b$ or $c$ quarks. The detector
includes a high precision tracking system consisting of a silicon-strip vertex
detector surrounding the $pp$ interaction region, a large-area silicon-strip
detector located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger [10] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction.
This analysis uses events selected by the hardware level trigger either when
one of the charged particles of the signal decay gives a large enough energy
deposit in the calorimeter system (hadron trigger), or when one of the
particles in the event, not coming from the signal decay, fulfills the trigger
requirements (i.e. mainly events triggered by one particle coming from the
decay of the other $B$ in the event). The software trigger requires a two-,
three- or four-track secondary vertex with a high scalar sum of the $p_{\rm
T}$ of the tracks and a significant displacement from the primary $pp$
interaction vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and an impact parameter (IP)
$\chi^{2}$ with respect to the PV greater than 16. The IP $\chi^{2}$ is
defined as the difference between the $\chi^{2}$ of the PV reconstructed with
and without the considered track. A multivariate algorithm is used for the
identification of secondary vertices consistent with the decay of a $b$
hadron.
Candidates are selected from combinations of charged particles. $D$ mesons are
reconstructed in the decay modes $D\rightarrow K^{+}\pi^{-}$ and $K^{+}K^{-}$.
The $p_{\rm T}$ of the daughters is required to be larger than
400${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Particle identification (PID) is
used to distinguish between charged pions and kaons. The difference between
the log-likelihoods of the kaon and pion hypotheses ($\mathrm{DLL}_{K\pi}$) is
required to be larger than 0 for kaons and smaller than 4 for pions. This aids
the reduction of cross-feed between the signal $D$ decay modes to a negligible
level. A fit is applied to the two-track vertex, requiring that the
corresponding $\chi^{2}$ per degree of freedom is less than 5. In order to
separate $D$ mesons coming from a $B$ decay from those produced at the PV, the
$D$ candidates are required to have an IP $\chi^{2}$ greater than 4 with
respect to any PV. To suppress background from $B$ decays without an
intermediate $D$ meson ($B^{0}\rightarrow K^{*0}K^{+}K^{-}$ for example), for
which all four charged hadrons are produced at the $B$-decay vertex, a
condition on the $D$ flight distance with respect to the $B$ vertex is
applied, requiring that it is larger than 0 by at least 2.5 standard
deviations. Finally, $D$ candidates with an invariant mass within $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $D^{0}$ mass are
retained.
$K^{*0}$ mesons are reconstructed in the mode $K^{*0}\rightarrow
K^{+}\pi^{-}$. The $p_{\rm T}$ of the $K^{+}$ and $\pi^{-}$ mesons must be
larger than 300${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. PID is also used,
requiring that $\mathrm{DLL}_{K\pi}$ is larger than 3 for the kaon and lower
than 3 for the pion, reducing the cross-feed from $B^{0}\rightarrow D\rho^{0}$
to a manageable level and rejecting non-resonant $B^{0}\rightarrow
DK^{+}K^{-}$ [11]. Possible contamination from protons in the kaon sample,
e.g. from $\mathchar 28931\relax^{0}_{b}\rightarrow Dp\pi^{-}$ decays, is
reduced by removing kaon candidates with a difference between the log-
likelihoods of the proton and kaon hypotheses (${\rm DLL}_{pK}$) of less than
10. The IP $\chi^{2}$ of the $K^{*0}$ mesons must be larger than 25, to select
those coming from a $B$ decay, and their invariant mass within $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal mass.
${\hbox{}\\!\mathop{B}\limits^{\hbox{\tiny(---)}}\\!\hbox{}}{{}^{0}_{\mskip-3.0mu(s)}}$
meson candidates are formed by combining $D$ and $K^{*0}$ candidates selected
with the above requirements. A fit to a common vertex is performed, keeping
only combinations with $\chi^{2}$ per degree of freedom lower than 4, and a
kinematic fit is performed to constrain the invariant mass of the
reconstructed $D$ to the nominal $D^{0}$ mass [12]. Since $B$ mesons are
produced at the PV, only candidates with IP $\chi^{2}$ lower than 9 are
retained. In case several PVs are reconstructed, the one for which the
$B$-candidate IP $\chi^{2}$ is the smallest is taken as reference.
Additionally, the momentum of the reconstructed $B$ candidate is required to
point back to the PV, by requiring that the angle between the $B$ momentum
direction and its direction of flight from the PV is smaller than
10$\rm\,mrad$. Furthermore, the sum of the square roots of the IP $\chi^{2}$
of the four charged particles must be larger than 32. The absolute value of
the cosine of the $K^{*0}$ helicity angle is required to be larger than 0.4.
This angle is defined as the angle between the kaon-daughter momentum
direction in the $K^{*0}$ rest frame, and the $K^{*0}$ direction in the $B$
rest frame.
Specific peaking backgrounds from $B^{0}_{\mskip-3.0mu(s)}\rightarrow
D_{\mskip-3.0mu(s)}^{\mp}h^{\pm}$ decays, where $h$ is a $\pi$ or a $K$ meson,
are eliminated by vetoing candidates for which the invariant mass of
$K^{+}K^{-}\pi^{+}$($K^{-}\pi^{+}\pi^{+}$ and $K^{+}K^{-}\pi^{+}$) is within
$\pm 15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal mass of a
$D^{+}_{s}$ ($D^{+}$) meson.
Where possible, data-driven methods are used to determine selection
efficiencies and invariant mass distribution shapes. Otherwise, they are
determined from fully simulated events. The $pp$ collisions are generated
using Pythia 6.4 [13] with a specific LHCb configuration [14] where, in
particular, decays of hadronic particles are described by EvtGen [15]. The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [16, *Agostinelli:2002hh] as described in
Ref. [18].
## 3 Determination of signal yields
The numbers of reconstructed signal $B^{0}$ and $B^{0}_{s}$ candidates are
determined from an unbinned maximum likelihood fit to their mass
distributions. Candidates are split into four categories, which are fitted
simultaneously: $D(K^{+}K^{-})K^{*0}$, $D(K^{+}K^{-})\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, $D(K^{+}\pi^{-})K^{*0}$, and
$D(K^{-}\pi^{+})\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. The mass
distribution of each category is fitted with a sum of probability density
functions (PDF) modelling the different contributing components:
1. 1.
the $B^{0}$ and $B^{0}_{s}$ signals are described by double Gaussian
functions;
2. 2.
the combinatorial background is described by an exponential function;
3. 3.
the cross-feed from $B^{0}\rightarrow D\rho^{0}$ decays, where one pion from
the $\rho^{0}\rightarrow\pi^{+}\pi^{-}$ decay is misidentified as a kaon, is
described by a non-parametric PDF [19] determined from fully simulated and
selected events;
4. 4.
the partially reconstructed $B^{0}\rightarrow D^{*}K^{*0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{*}K^{*0}$ decays,
where the $D^{*}$ is a $D^{*0}$ or a $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}$ and the $\pi^{0}$ or photon from
the $D^{*}$ decay is not reconstructed, are modelled by a non-parametric PDF
determined from fully simulated and selected events.
There are 23 free parameters in the fit. These include the $B^{0}$ PDF peak
position, the core Gaussian resolution for the $B^{0}$ and the $B^{0}_{s}$ and
the slope of the combinatorial background, all of which are common to the four
fit categories. The remaining free parameters are yields for each fit
component within each category. Yields for $B^{0}_{\mskip-3.0mu(s)}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{\mskip-3.0mu(s)}$ are
constrained to be identical for the background components where $C\\!P$
violation effects can be excluded or are expected to be compatible with zero
with the current data sample size.
A separate fit to $B^{0}\rightarrow D(K^{+}\pi^{-})\rho^{0}$ candidates in the
same data sample is performed. The yield of such candidates and the
probability to reconstruct them as $B^{0}\rightarrow D(K^{+}\pi^{-})K^{*0}$ is
used to constrain the number of cross-feed events in the
$D(K^{+}\pi^{-})K^{*0}$ category. The number of cross-feed candidates from
$B^{0}\rightarrow D(K^{+}K^{-})\rho^{0}$ in the $D(K^{+}K^{-})K^{*0}$ category
is derived from the $D(K^{+}\pi^{-})K^{*0}$ category using the relative $D$
branching fractions and $B$ selection efficiencies. As no flavour asymmetry is
expected for this background, the numbers of cross-feed events in the $D\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ categories are constrained to be
identical to those of the corresponding $DK^{*0}$ categories.
The partially reconstructed background component accumulates at masses lower
than the nominal $B^{0}$ mass. Its shape depends on the unknown fraction of
transverse polarisation in the
${\hbox{}\\!\mathop{B}\limits^{\hbox{\tiny(---)}}\\!\hbox{}}{{}^{0}_{\mskip-3.0mu(s)}}\rightarrow
D^{*}K^{*0}$ decays. In order to model the
${\hbox{}\\!\mathop{B}\limits^{\hbox{\tiny(---)}}\\!\hbox{}}{{}^{0}_{\mskip-3.0mu(s)}}\rightarrow
D^{*}K^{*0}$ contribution, a PDF is built from a linear combination of three
non-parametric functions corresponding to the three orthogonal helicity
eigenstates. The functions are derived from simulated
${\hbox{}\\!\mathop{B}\limits^{\hbox{\tiny(---)}}\\!\hbox{}}{{}^{0}_{\mskip-3.0mu(s)}}\rightarrow
D^{*}K^{*0}$ events reconstructed as $B^{0}\rightarrow DK^{*0}$. Each function
corresponds to the weighted sum of the $D^{*}\rightarrow D\gamma$ and
$D^{*}\rightarrow D\pi^{0}$ contributions for a defined helicity eigenstate,
where the weights take into account the relative $D^{*}$ decay branching
fractions and the corresponding reconstruction efficiencies.
The invariant mass distributions together with the function resulting from the
fit are shown in Fig. 2. Note that the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D(K^{+}\pi^{-})K^{*0}$ is not observed since the charge combination of the
kaons in the final state corresponds to the suppressed decay. The signal yield
in each category is summarized in Table 1. The significance of the
$B^{0}\rightarrow DK^{*0}$ signal for $D\rightarrow K^{+}K^{-}$ decays,
summing $B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and
including both statistical and systematic uncertainties, is found to be equal
to $5.1\,\sigma$, by comparing the maximum of the likelihood of the nominal
fit and the maximum with the yield of the $B^{0}\rightarrow
D(K^{+}K^{-})K^{*0}$ category set to zero.
The yields determined from the simultaneous mass fit are corrected for
selection efficiencies in order to evaluate the asymmetries and ratios
described in the introduction. The selection efficiencies account for the
geometrical acceptance of the detector, the reconstruction, the PID, and the
trigger efficiencies. All efficiencies are computed from fully simulated
events, except for the PID and trigger efficiencies, which are obtained
directly from data using clean calibration samples of $D^{0}\rightarrow
K^{-}\pi^{+}$ from $D^{*+}$ decays.
Table 1: Signal yields with their statistical uncertainties. Category | Signal yield | Category | Signal yield
---|---|---|---
$B^{0}\rightarrow D_{[K^{+}K^{-}]}K^{*0}$ | $21\,$ | ${}_{-\,5}^{+\,6}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{[K^{+}K^{-}]}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | $8\,$ | $\pm\,4$
$B^{0}_{s}\rightarrow D_{[K^{+}K^{-}]}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | $23\,$ | ${}_{-\,5}^{+\,6}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{[K^{+}K^{-}]}K^{*0}$ | $24\,$ | ${}_{\,-\,\,\,5}^{\,+\,\,\,6}$
$B^{0}\rightarrow D_{[K^{+}\pi^{-}]}K^{*0}$ | $\phantom{11}108\,$ | ${}_{-\,11}^{+\,12}$ | $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D_{[K^{-}\pi^{+}]}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | $\phantom{111}94\,$ | $\pm\,11$
Figure 2: Invariant mass distributions of (a) $D_{[K^{+}K^{-}]}\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$, (b) $D_{[K^{+}K^{-}]}K^{*0}$,
(c) $D_{[K^{-}\pi^{+}]}\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ and
(d) $D_{[K^{+}\pi^{-}]}K^{*0}$ candidates. The $D\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ distributions correspond to
$\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}$ and $B^{0}_{s}$ decays
whereas the $DK^{*0}$ distributions correspond to $B^{0}$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ decays. The fit functions are
superimposed; the different $B$ decays and combinatorial background components
are detailed in the legends.
## 4 Systematic uncertainties
Several sources of systematic uncertainty are considered, affecting either the
determination of the signal yields or the computation of the efficiencies.
They are summarized in Table 2. In order to take into account the measured
difference in the production rate between $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{0}$, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ yields are multiplied by a
correction factor,
$a_{\rm prod}^{d}=\frac{1-\kappa A_{\rm prod}}{1+\kappa A_{\rm prod}},$ (5)
where $A_{\rm prod}=0.010\pm 0.013$ [20] is the asymmetry between $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{0}$ at production in $pp$
collisions, and $\kappa$ is a decay-dependent factor,
$\kappa=\frac{\int_{0}^{+\infty}e^{-\Gamma t}\cos(\Delta
mt)~{}\epsilon(B^{0}\rightarrow DK^{*0},t)~{}{\rm
d}t}{\int_{0}^{+\infty}e^{-\Gamma t}~{}\epsilon(B^{0}\rightarrow
DK^{*0},t)~{}{\rm d}t}$, which takes into account dilution effects due to the
$B^{0}-\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ oscillation
frequency, $\Delta m$, and includes the acceptance as a function of the decay
time for the reconstructed decay, $\epsilon(B^{0}\rightarrow DK^{*0},t)$. The
value of $\kappa$ is found to be $0.46\pm 0.01$ using fully simulated events
and PID efficiencies from calibration samples. The uncertainty on $a_{\rm
prod}^{d}$ is propagated to the measured observables to estimate the
systematic uncertainty from the production asymmetry. Owing to the large
$B^{0}_{s}$ oscillation frequency, the potential production asymmetry of
$B^{0}_{s}$ mesons does not significantly affect the measurement presented
here and is neglected.
The PID calibration introduces a systematic uncertainty on the calculated PID
efficiencies, which propagates to the final results. All PID correction
factors are compatible with unity within their uncertainties which are of the
order of 1%.
The systematic uncertainty associated to the trigger is estimated by varying
in the simulation the fraction of events triggered by the hadron trigger with
respect to the fraction of events triggered by the other $b$-hadron in the
event. Other selection efficiencies cancel in the ratio of yields, except for
the efficiencies of the $p_{\rm T}$ cuts on the $D$ daughters, which are
different between different $D$ decay modes. ${\cal{R}}_{d}^{KK}$ has to be
corrected by a multiplicative factor $0.94\pm 0.04$, where the statistical
uncertainty on the correction, which arises from finite simulated sample size,
is assigned as systematic uncertainty due to the relative selection
efficiencies.
The fit procedure is validated with simulated experiments. A bias of
statistical nature, owing to the small number of events in the
$B^{0}\rightarrow D(K^{+}K^{-})K^{*0}$ channel, is found to be 5% for $B^{0}$
and 8% for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$. The signal
yields are corrected for this bias before computing the asymmetries and
ratios. A systematic uncertainty equal to half the size of the correction has
been assigned.
Simulated experiments are also used to determine the systematic uncertainties
due to the low-mass background, the $B^{0}\rightarrow D\rho^{0}$ cross-feed,
and the signal shape. Samples are generated with different values of the
polarisation parameters, the cross-feed fraction and the fixed signal
parameters. The corresponding systematic uncertainty is estimated from the
bias in the results obtained by performing the fit described in the previous
section to these samples.
Table 2: Summary of the absolute systematic uncertainties on the measured observables. Source | ${\cal{A}}^{KK}_{d}$ | ${\cal{A}}_{d}^{\rm fav}$ | ${\cal{A}}^{KK}_{s}$ | ${\cal{R}}^{KK}_{d}$
---|---|---|---|---
Production asymmetry | 0.005 | 0.006 | $-$ | 0.003
PID efficiency | 0.004 | 0.008 | 0.005 | 0.014
Trigger efficiency | 0.004 | 0.001 | 0.005 | 0.022
Selection efficiency | $-$ | $-$ | $-$ | 0.040
Bias correction | 0.004 | $-$ | 0.001 | 0.013
Low-mass background | 0.017 | 0.001 | 0.004 | 0.042
$B^{0}\rightarrow D\rho^{0}$ cross-feed | 0.001 | $-$ | 0.002 | 0.008
Signal description | 0.001 | 0.001 | 0.001 | 0.005
$D$ branching fractions | $-$ | $-$ | $-$ | 0.022
Total | 0.019 | 0.010 | 0.008 | 0.069
## 5 Results and summary
This paper reports the analysis of $B^{0}\rightarrow DK^{*0}$ decays using 1.0
$\mbox{\,fb}^{-1}$ of $pp$ collision data. Potential contributions to the
decay amplitudes from the non-resonant $B^{0}\rightarrow DK^{+}\pi^{-}$ mode
are reduced by requiring that the $K^{*0}$ reconstructed mass is within $\pm
50$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal mass and the
absolute value of the cosine of the $K^{*0}$ helicity angle is greater than
0.4. The results for the $C\\!P$-violating observables are
$\displaystyle{\cal A}_{d}^{KK}$ $\displaystyle\,=$ $\displaystyle-0.45$
$\displaystyle\,\pm\,0.23$ $\displaystyle\ ({\rm stat})\pm 0.02\ ({\rm
syst}),$ $\displaystyle{\cal A}_{d}^{\rm fav}$ $\displaystyle\,=$
$\displaystyle-0.08$ $\displaystyle\,\pm\,0.08$ $\displaystyle\ ({\rm
stat})\pm 0.01\ ({\rm syst}),$ $\displaystyle{\cal A}_{s}^{KK}$
$\displaystyle\,=$ $\displaystyle 0.04$ $\displaystyle\,\pm\,0.16$
$\displaystyle\ ({\rm stat})\pm 0.01\ ({\rm syst}),$ $\displaystyle{\cal
R}_{d}^{KK}$ $\displaystyle\,=$ $\displaystyle 1.36$
${}_{-\phantom{1}0.32\phantom{1}}^{+\phantom{1}0.37}$ $\displaystyle\ ({\rm
stat})\pm 0.07\ ({\rm syst}).$
The value of ${\cal R}_{d}^{KK}$ takes into account the ratio of the branching
fractions of $D^{0}\rightarrow K^{+}K^{-}$ to $D^{0}\rightarrow K^{-}\pi^{+}$
decays [12]. The correlation between ${\cal A}_{d}^{KK}$ and ${\cal
R}_{d}^{KK}$ is equal to 0.16 and the correlations between the other
observables are negligible.
These are the first measurements of $C\\!P$ asymmetries in $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to $DK^{*0}$ decays with the
neutral $D$ meson decaying into a $C\\!P$-even final state. Triggering,
reconstructing and selecting a pure sample of these fully hadronic $B$ decays
is challenging in a high rate and high track-multiplicity environment,
especially in the forward direction of LHCb. The present statistical
limitations are due to a combination of several factors, the most important
one being the trigger. In order to keep the output rate below its maximum of 1
MHz, the current hardware trigger imposes relatively restrictive criteria on
the minimum transverse momentum of hadrons, which affect the efficiency for
fully-hadronic modes. This limitation is overcome in the proposed LHCb upgrade
[21, *CERN-LHCC-2012-007] by reading out the detector at the maximum LHC
bunch-crossing frequency of 40 MHz. With more data, improved measurements of
these and other quantities in $B^{0}\rightarrow DK^{*0}$ decays will result in
important constraints on the angle $\gamma$ of the Unitarity Triangle.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2012-12-20T19:42:47 |
2024-09-04T02:49:39.518725
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J.\n Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J.\n Beddow, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M.\n Benayoun, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V.\n Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D.\n Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I.\n Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo\n Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic,\n H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier,\n G.A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y.\n David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L.\n De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi,\n L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto,\n J. Dickens, H. Dijkstra, P. Diniz Batista, M. Dogaru, F. Domingo Bonal, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell,\n C. Farinelli, S. Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao,\n J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu.\n Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, P.F.\n Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A.\n Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A.\n Jaeger, E. Jans, F. Jansen, P. Jaton, F. Jing, M. John, D. Johnson, C.R.\n Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach, I.R.\n Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, Y. Li, L. Li Gioi, M. Liles, R.\n Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez Asamar,\n N. Lopez-March, H. Lu, J. Luisier, H. Luo, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, N.\n Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L.\n Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins\n Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, A. Mazurov, J. McCarthy, R. McNulty, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller,\n R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M.\n Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, S. Nisar, A. Nomerotski, A.\n Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D.\n Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, K. Sobczak, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D.\n Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert,\n C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi,\n R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D.\n Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G.\n Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W.\n Witzeling, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Edmund Smith",
"url": "https://arxiv.org/abs/1212.5205"
}
|
1212.5303
|
Relational Foundations For Functorial Data Migration
David I. Spivak
[Spivak acknowledges support from ONR grants N000141010841 and N000141310260.]
Massachusetts Institute of Technology
[email protected] Ryan Wisnesky
Harvard University
We study the data transformation capabilities associated with schemas that are presented by labeled directed multi-graphs and path equivalence constraints. Unlike most approaches which treat graph-based schemas as abbreviations for relational schemas, we treat graph-based schemas as categories. A morphism $M$ between schemas $S$ and $T$, which can be generated from a visual mapping between graphs, induces three adjoint data migration functors, $\Sigma_M\taking S \to T$, $\Pi_M\taking S \to T$, and $\Delta_M\taking T \to S$. We present an algebraic query language FQL based on these functors, investigate its metatheory, and present SQL query generation algorithms for FQL.
§ INTRODUCTION
In this paper we describe how to implement functorial data migration [18] using relational algebra, and vice versa. In the functorial data model [18], a database schema is a finitely presented category (a directed multigraph with path equations [3]), and a database instance on a schema $S$ is a functor from $S$ to the category of sets. The database instances on a schema $S$ constitute a category, denoted $S\inst$, and a functor $M$ between schemas $S$ and $T$ induces three adjoint data migration functors, $\Sigma_M\taking S\inst \to T\inst$, $\Pi_M\taking S\inst \to T\inst$, and $\Delta_M\taking T\inst \to S\inst$. We define a functorial query language, FQL, in which every query denotes a data migration functor, prove that FQL is closed under composition, and define a translation from FQL to the select-project-product-union (SPCU) relational algebra extended with an operation for creating globally unique IDs (GUIDs)[An operation to create IDs is required because functorial data migration can create constants not contained in input instances.]. In addition, we describe how to implement the SPCU relational algebra with FQL.
Our primary motivation for this work is to use SQL as a practical deployment platform for the functorial data model. Using the results of this paper we have implemented a SQL compiler for FQL as part of a visual schema mapping tool available at wisnesky.net/fql.html. We are interested in the functorial data model because we believe its category-theoretic underpinnings provide a better mathematical foundation for data migration than the relational model. For example, many practical “relational” systems (including SQL-based RDBMSs) are not actually relational, but are based on bags and unique row IDs – like the functorial data model. More advantages of the functorial data model are described in [18] and [8].
§.§ Related Work
Although category presentations – directed mutli-graphs with path equivalence constraints – are a very common notation for schemas [5], there has been very little work to treat such schemas categorically [8]. Instead, most work treats such schemas as abbreviations for relational schemas. For example, in Clio [10], users draw lines connecting related elements between two schemas-as-graphs and Clio generates a relational query (semantically similar to our $\Sigma$) that implements the user's intended data transformation. Behind the scenes, the user's correspondence is translated into a formula in the relational language of second-order tuple generating dependencies, from which a query is generated [7].
In many ways, our work is an extension and improvement of Rosebrugh et al's initial work on understanding category presentations as database schemas [8]. In that work, the authors identify the $\Sigma$ and $\Pi$ data migration functors, but they do not identify $\Delta$ as their adjoint. Moreover, they remark that they were unable to implement $\Sigma$ and $\Pi$ using relational algebra, they do not formalize a query language or investigate the behavior of the $\Sigma$ and $\Pi$ with respect to composition, and they do not consider “attributes”. Our mathematical development diverges from their subsequent work on “sketches” [13].
Category-theoretic techniques were instrumental in the development of nested relational data model [19]. However, at this time we do not believe the functorial data model and the nested relational model to be closely connected. The functorial data model is more closely related to work on categorical logic and type theory, where operations similar to $\Sigma, \Pi$, and $\Delta$ often appear under the slogan that “quantification is adjoint to substitution” [12].
§.§ Contributions and Outline
We make the following contributions:
* The naive functorial data model [18], as sketched above, is not quite appropriate for many practical information management tasks. Intuitively, this is because every instance contains only meaningless GUIDs. So, we extend the functorial data model with “attributes” to capture meaningful concrete data such as strings and integers. The practical result is that our schemas become special kinds of entity-relationship (ER) diagrams (those in “categorical normal form”), and our instances can be represented as relational tables that conform to such diagrams. (Sections 2 and 3)
* We define an algebraic query language FQL where every query denotes a data migration functor. We show that FQL queries are closed under composition, and how every query in FQL can be described as a triplet of graph correspondences (similar to schema mappings [4]). Determining whether a triplet of arbitrary graph correspondences is an FQL query is semi-decidable. (Section 4)
* We provide a translation of FQL into the SPCU relational algebra of selection, projection, cartesian product, and union extended with a globally unique ID generator that constructs $N+1$-ary tables from $N$-ary tables by adding a globally unique ID to each row. This allows us to easily generate SQL programs that implement FQL. An immediate corollary is that materializing result instances of FQL queries has polynomial time data complexity. (Section 5)
* We show that every SPCU query under bag semantics is expressible in FQL, and how to extend FQL to capture every SPCU query under set semantics. (Section 6)
§ CATEGORICAL DATA
In this section we define the original signatures and instances of [18], as well as “typed signatures” and “typed instances”, which are our extension of [18] to attributes. The basic idea is that signatures are stylized ER diagrams that denote categories, and our database instances can be represented as instances of such ER diagrams, and vice versa (up to natural isomorphism). From this point on, we will distinguish between a signature, which is a presentation of a category, and a schema, which is a category.
§.§ Signatures
The functorial data model [18] uses directed labeled multi-graphs and path equalities for signatures. A path $p$ is defined inductively as:
p ::= node \ | \ p.edge
A signature $S$ is a finite presentation of a category. That is, a signature $S$ is a triple $S := (N,E,C)$ where $N$ is a finite set of nodes, $E$ is a finite set of labeled directed edges , and $C$ a finite set of path equivalences. E.g,
\begin{align}\label{dia:maincat}
\xymatrix@=10pt{&\LTO{Emp}\ar@<.5ex>[rrrrr]^{\tn{worksIn}}\ar@(l,u)[]+<0pt,10pt>^{\tn{manager}}&&&&&\LTO{Dept}\ar@<.5ex>[lllll]^{\tn{secretary}} }
\end{align}
\begin{eqnarray*}
& {\tt Emp}.{\tt manager}.{\tt worksIn} = {\tt Emp}.{\tt worksIn} & \\
& {\tt Dept}.{\tt secretary}.{\tt worksIn} = {\tt Dept} &
\end{eqnarray*}
Here we see a signature $S$ with two vertices and three arrows, and two path equivalence statements. This information generates a category $\church{S}$: the free category on the graph, modulo the equivalence relation induced by the path equivalences. The category $\church{S}$ is the schema for $S$, and database instances over $\church{S}$ are functors $\church{S}\to\Set$. Every path $p\taking X \to Y$ in a signature $S$ denotes a morphism $\church{p}\taking \church{X} \to \church{Y}$ in $\church{S}$. Given two paths $p_1, p_2$ in a signature $S$, we say that they are equivalent, written $p_1 \cong p_2$ if $\church{p_1}$ and $\church{p_2}$ are the same morphism in $\church{S}$. Two signatures $S$ and $T$ are isomorphic, written $S \cong T$, if they denote isomorphic schema, i.e. if the categories they generate are isomorphic.
§.§ Cyclic Signatures
If a signature contains a loop, it may or may not denote a category with infinitely many morphisms. Hence, some constructions over signatures may not be computable. Testing if two paths in a signature are equivalent is known as the word problem for categories. The word problem can be semi-decided using the “completion without failure” extension [2] of the Knuth-Bendix algorithm. This algorithm first attempts to construct a strongly normalizing re-write system based on the path equalities; if it succeeds, it yields a linear time decision procedure for the word problem [11]. If a signature denotes a finite category, the Carmody-Walters algorithm [6] will compute its denotation. The algorithm computes left Kan extensions and can be used for many other purposes in computational category theory [8]. In fact, every $\Sigma$ functor arises as a left Kan extension, and vice versa.
§.§ Instances
Let $S$ be a signature. A $\llbracket S \rrbracket$-instance is a functor from $\llbracket S \rrbracket$ to the category of sets. We will represent instances as relational tables using the following binary format:
* To each node $N$ corresponds an “identity” or “entity” table named $N$, a reflexive table with tuples of the form $(x,x)$. We can specify this using first-order logic:
\begin{align}\label{dia:entity FBR}
\forall x y. N(x,y) \Rightarrow x = y.
\end{align}
The entries in these tables are called IDs or keys, and for the purposes of this paper we require them to be globally unique. We call this the globally unique key assumption.
* To each edge $e\taking N_1 \to N_2$ corresponds a “link” table $e$ between identity tables $N_1$ and $N_2$. The axioms below merely say that every edge $e\taking N_1 \to N_2$ designates a total function $N_1 \to N_2$:
\begin{align}\label{dia:link FBR}
\begin{array}{l}
\forall x y. \ e(x,y) \Rightarrow N_1(x,x) \\
\forall x y. \ e(x,y) \Rightarrow N_2(y,y) \\
\forall x y z. \ e(x,y) \wedge e(x,z) \Rightarrow y = z \\
\forall x. \ N_1(x,x) \Rightarrow \exists y. e(x,y)
\end{array}
\end{align}
An example instance of our employees schema is:
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt Emp}}\\\bbhline
{\bf Emp}&{\bf Emp}\\\hline
101& 101 \\\hline
102& 102 \\\hline
103& 103 \\\bhline
\end{tabular}
\ \ \ \ \ \
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt Dept}}\\\bbhline
{\bf Dept}&{\bf Dept}\\\hline
q10& q10\\\hline
x02& x02 \\\bhline
\end{tabular}
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt manager}}\\\bbhline
{\bf Emp}&{\bf Emp}\\\hline
103&103 \\\bhline
\end{tabular}
\ \ \ \ \ \
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt worksIn}}\\\bbhline
{\bf Emp}&{\bf Dept}\\\hline
\end{tabular}
\ \ \ \ \ \ \
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt secretary}}\\\bbhline
{\bf Dept}&{\bf Emp}\\\hline
\end{tabular}
To save space, we will sometimes present instances in a “joined” format:
\begin{tabular}{| c | c | c |}\bhline
\multicolumn{3}{| c |}{{\tt Emp}}\\\bbhline
{\bf Emp}&{\bf manager }&{\bf worksIn}\\\hline
101& 103 & q10 \\\hline
102& 102 & q10 \\\hline
103& 103 & x02 \\\bhline
\end{tabular}
\ \ \ \ \ \
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt Dept}}\\\bbhline
{\bf Dept}&{\bf secretary}\\\hline
q10& 102\\\hline
x02& 101 \\\bhline
\end{tabular}
The natural notion of equality of instances is isomorphism. In particular, the actual constants in the above tables should be considered meaningless IDs.
§.§ Attributes
Signatures and instances, as defined above, do not have quite enough structure to be useful in practice. At a practical level, we usually need fixed atomic domains like String and Nat to store actual data. Hence, in this section we extend the functorial data model with attributes.
Let $S$ be a signature. A typing $\Gamma$ for $S$ is a mapping from each node $N$ of $S$ to a (possibly empty) set of attribute names and associated base types (Nat, String, etc), written $\Gamma(N)$. We call a pairing of a signature and a typing a typed signature. Borrowing from ER-diagram notation, we will write attributes as open circles. For example, we might enrich our previous signature with a typing as follows:
\begin{align}
\xymatrix@=10pt{&\LTO{Emp}\ar@<.5ex>[rrrrr]^{\tn{worksIn}}\ar@(l,u)[]+<0pt,10pt>^{\tn{manager}}\ar@{-}[dddl]\ar@{-}[dddr]&&&&&\LTO{Dept}\ar@<.5ex>[lllll]^{\tn{secretary}}\ar@{-}[ddd]\\\\\\\LTOO{FName}&&\LTOO{LName}&~&~&~&\LTOO{DName} }
\end{align}
{\tt Emp}.{\tt manager}.{\tt worksIn} = {\tt Emp}.{\tt worksIn}
{\tt Dept}.{\tt secretary}.{\tt worksIn} = {\tt Dept}
Importantly, path expressions may not refer to attributes; they may only refer to nodes and directed edges. The meaning of $\Gamma $ is a triple $\llbracket \Gamma \rrbracket := (A, i, \gamma)$, where $A$ is a discrete category (category containing only objects and identity morphisms) consisting of the attributes of $\Gamma$, $i$ is a functor from $A$ to $\llbracket S \rrbracket$ mapping each attribute to its corresponding node, and $\gamma$ is a functor from $A$ to $Set$, mapping each attribute to its domain (e.g., the set of strings, the set of natural numbers):
A\ar[rr]^i\ar[dr]_\gamma&&\llbracket S \rrbracket \\
§.§ Typed Instances
Let $S$ be a signature and $\Gamma$ a typing such that $\llbracket \Gamma \rrbracket = (A,i,\gamma)$. A typed instance $I$ is a pair $(I^\prime,\delta)$ where $I'\taking\mc \llbracket S \rrbracket \to\Set$ is an (untyped) instance together with a natural transformation $\delta\taking I^\prime \circ i\Rightarrow\gamma$. Intuitively, $\delta$ associates an appropriately typed constant (e.g., a string) to each globally unique ID in $I^\prime$:
A\ar[rr]^i\ar[dr]_\gamma&\ar@{}[d]|(.4){\stackrel{\delta}{\Leftarrow}}&\llbracket S \rrbracket \ar[dl]^{I^\prime}\\
We represent the $\delta$-part of a typed instance as a set of binary tables as follows:
* To each node table $N$ and attribute $A$ with type $t$ in $\Gamma(N)$ corresponds a binary “attribute” table mapping the domain of $N$ to values of type $t$.
In our employees example, we might add the following:
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt FName}}\\\bbhline
{\bf Emp}&{\bf String}\\\hline
\end{tabular}
\ \
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt LName}}\\\bbhline
{\bf Emp}&{\bf String}\\\hline
\end{tabular}
\ \
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{{\tt DName}}\\\bbhline
{\bf Dept}&{\bf Str}\\\hline
\end{tabular}
Typed instances form a category, and two instances are equivalent, written $\cong$, when they are isomorphic objects in this category. Isomorphism of typed instances captures our expected notion of equality on typed instances, where the “structure parts” are compared for isomorphism and the “attribute parts” are compared for equality under such an isomorphism.
§ FUNCTORIAL DATA MIGRATION
In this section we define the original signature morphisms and data migration functors of [18], as well as “typed signature morphisms” and “typed data migration functors”, which are our extension of [18] to attributes. The basic idea is that associated with a mapping between signatures $F : S \to T$ is a data transformation $\Delta_F : T$-Inst $\to S$-Inst and left and right adjoints $\Sigma_F, \Pi_F : S$-Inst $\to T$-Inst.
§.§ Signature Morphism
Let $\mcC$ and $\mcD$ be signatures. A signature morphism $F\taking\mcC\to\mcD$ is a mapping (indeed, a functor) that takes vertices in $\mcC$ to vertices in $\mcD$ and arrows in $\mcC$ to paths in $\mcD$; in so doing, it must respect arrow sources, arrow targets, and path equivalences. In other words, if $p_1 \cong p_2$ is a path equivalence in $\mcC$, then $F(p_1) \cong F(p_2)$ is a path equivalence in $\mcD$. Each signature morphism $F\taking\mcC\to\mcD$ determines a unique schema morphism $\church{F}\taking\church{\mcC}\to\church{\mcD}$ in the obvious way. Two signature morphisms $F_1\taking \mcC \to \mcD$ and $F_2\taking \mcC \to \mcD$ are equivalent, written $F_1 \cong F_2$, if they denote isomorphic functors. Below is an example of a signature morphism.
\begin{align*}\mcC:=\parbox{1in}{\fbox{\xymatrix@=10pt{&\\&\LTO{A1}\\\color{red}{\LTO{N1}}\ar[ur]\ar[dr]&&\color{red}{\LTO{N2}}\ar[ul]\ar[dl]\\&\LTO{A2}\\&}}}\Too{F}\parbox{.8in}{\fbox{\xymatrix@=10pt{&\\&\LTO{B1}\\\color{red}{\LTO{N}}\ar[ur]\ar[dr]\\&\LTO{B2}\\&}}}=:\mcD\end{align*}
In the above example, the nodes N1 and N2 are mapped to N, the two morphisms to A1 are mapped to the morphism to B1, and the two morphisms to A2 are mapped to the morphism to B2.
§.§ Typed Signature Morphisms
Intuitively, signature morphisms are extended to typed signatures by providing an additional mapping between attributes. For example, we might have Name and Age attributes in our source and target typings:
\begin{align*}\parbox{1.15in}{\fbox{\xymatrix@=10pt{&\LTOO{Name}\\&\LTO{A1}\\\color{red}{\LTO{N1}}\ar@{-}[uur]\ar[ur]\ar[dr]&&\color{red}{\LTO{N2}}\ar[ul]\ar[dl]\ar@{-}[ddl]\\&\LTO{A2}\\&\LTOO{Age}}}}\Too{ }\parbox{.8in}{\fbox{\xymatrix@=10pt{&\LTOO{Name}\\&\LTO{B1}\\\color{red}{\LTO{N}}\ar@{-}[uur]\ar[ur]\ar[dr]\ar@{-}[ddr]\\&\LTO{B2}\\&\LTOO{Age}}}}\end{align*}
In the above, we map Name to Name and Age to Age. More complicated mappings of attributes are also possible, as we will see in the next section.
Formally, let $S$ be a signature and $\llbracket \Gamma \rrbracket = (A, i, \gamma)$ a typing for S. Let $S'$ be a signature and $\llbracket \Gamma' \rrbracket = (A', i', \gamma')$ a typing for $S'$. A typed signature morphism from (S,$\Gamma$) to (S',$\Gamma'$) consists of a signature morphism $F : S \to S'$ and a functor $G : A \to A'$ such that the following diagram commutes:
A\ar[rr]^i\ar[dr]_\gamma\ar[dd]_{G}&&\llbracket S \rrbracket\ar[dd]^{\llbracket F \rrbracket}\\
A'\ar[ur]^{\gamma'}\ar[rr]_{i'}&&\llbracket S' \rrbracket
§.§ Data Migration Functors
Each signature morphism $F\taking\mcC\to\mcD$ is associated with three data migration functors, $\Delta_F, \Sigma_F$, and $\Pi_F$, which migrate instance data on $\mcD$ to instance data on $\mcC$ and vice versa. A summary is given is Figure 1.
Data migration functors induced by a translation $F\taking\mcC\to\mcD$
5| c |
Name Symbol Type Idea of definition Relational partner
Pullback $\Delta_F$ $\Delta_F\taking\mcD\inst\to\mcC\inst$ Composition with $F$ Project
Right Pushforward $\Pi_F$ $\Pi_F\taking\mcC\inst\to\mcD\inst$ Right adjoint to $\Delta_F$ Join
Left Pushforward $\Sigma_F$ $\Sigma_F\taking\mcC\inst\to\mcD\inst$ Left adjoint to $\Delta_F$ Union
Let $F\taking \mcC\to \mcD$ be a functor. We will define a functor $\Delta_F\taking \mcD\inst\to \mcC\inst$; that is, given a $\mcD$-instance $I\taking \mcD\to\Set$ we will construct a $\mcC$-instance $\Delta_F(I)$. This is obtained simply by composing the functors $I$ and $F$ to get
$$\parbox{1.6in}{$\Delta_F(I):=I\circ F\taking \mcC\to\Set$}
\parbox{1.5in}{\xymatrix{\mcC\ar[r]^F\ar@/_1pc/[rr]_{\Delta_FI}&\mcD\ar[r]^I&\Set}}
Then, $\Sigma_F \taking \mcC\inst\to \mcD\inst$ is defined as the left adjoint to $\Delta_F$, and $\Pi_F \taking \mcC\inst\to \mcD\inst $ is defined as the right adjoint to $\Delta_F$.
Data migration functors extend to typed data migration functors over typed instances in a natural way. We will use typed signatures and instances as we examine each data migration functor in turn below.
§.§ $\Delta$
Consider the following signature morphism $F : C \to D$:
\begin{align*} C := \parbox{1.15in}{\fbox{\xymatrix@=10pt{&\LTOO{Name}\\&\LTOO{Salary}\\\color{red}{\LTO{N1}}\ar@{-}[uur]\ar@{-}[ur]&&\color{red}{\LTO{N2}}\ar@{-}[dl]\\&\LTOO{Age}}}}\Too{F}\parbox{.8in}{\fbox{\xymatrix@=10pt{&\LTOO{Name}\\&\LTOO{Salary}\\\color{red}{\LTO{N}}\ar@{-}[uur]\ar@{-}[ur]\ar@{-}[dr]\\&\LTOO{Age}}}} =: D \end{align*}
Even though our translation $F$ points forwards (from $C$ to $D$), our migration functor $\Delta_F$ points “backwards" (from $D$-instances to $C$-instances). Consider the instance $J$, on schema $D$, defined by
\begin{align*}
J :=
\begin{tabular}{| c || c | c | c | }\bhline\multicolumn{4}{| c |}{{\tt N}}\\\bhline {\bf ID}&{\bf Name}&{\bf Age}&{\bf Salary}\\\bbhline
\end{tabular}
\end{align*}
$\Delta_F(J)$ splits up the columns of table ${\tt N}$ according to the translation $F$, resulting in
\begin{align*}
I :=
\begin{tabular}{| c || c | c | }\bhline
\multicolumn{3}{| c |}{{\tt N1}}\\\bhline {\bf ID}&{\bf Name}&{\bf Salary}\\\bbhline
\end{tabular}
\hspace{.25in}
\begin{tabular}{| c || c | }\bhline
\multicolumn{2}{| c |}{{\tt N2}}\\\bhline {\bf ID}&{\bf Age}\\\bbhline
\end{tabular}
%\begin{tabular}{| l || l | l | l | l |}\bhline\multicolumn{5}{| c |}{{\tt N2}}\\\bhline
%{\bf ID}&{\bf SSN}&{\bf Name}&{\bf Age}&{\bf Salary}\\\bbhline
\end{align*}
Because of the globally unique ID requirement, the IDs of the two tables N1 and N2 must be disjoint.
Note that $\Delta$ never changes the number of IDs associated with a node. For example, table N1 is ostensibly the “projection” of Age, yet has two rows with age 20.
§.§ $\Pi$
Consider the morphism $F : C \to D$ and $C$-instance $I$ defined in the previous section. Our migration functor $\Pi_F$ points in the same direction as $F$: it takes $C$-instances to $D$-instances. In general, $\Pi$ will take the join of tables. We can $\Pi$ along any typed signature morphism whose attribute mapping is a surjection[We require this restriction so that we never need to create NULLs]. Continuing with our example, we find that $\Pi_F(I)$ will take the cartesian product of N1 and N2:
\begin{align*}
\begin{tabular}{| c || c | c | c | }\bhline\multicolumn{4}{| c |}{{\tt N}}\\\bhline {\bf ID}&{\bf Name}&{\bf Age}&{\bf Salary}\\\bbhline
\end{tabular}
\end{align*}
This example illustrates that adjoints are not, in general, inverses. Intuitively, the above instance is a product rather than a join because in there is no path between N1 and N2.
Remark. When the target schema is infinite, on finite inputs $\Pi$ may create uncountably infinite result instances. Consider the unique signature morphism
Here $\church{\mcD}$ has arrows $\{f^n\|n\in\NN\}$ so it is infinite. Given the two-element instance $I\taking\mcC\to\Set$ with $I(s)=\{\tn{Alice},\tn{Bob}\}$, the rowset in the right pushforward $\Pi_F(I)$ is the (uncountable) set of infinite streams in $\{\tn{Alice},\tn{Bob}\}$, i.e. $\Pi_F(I)(s)=I(s)^\NN.$
§.§ $\Sigma$
For the purposes of this paper we will only define $\Sigma_F$ for functors $F$ that are discrete op-fibrations. Inasmuch as $\Sigma$ can be thought of as computing unions, functors that are discrete op-fibrations intuitively express the idea that all such unions are over “union compatible” tables.
A functor $F\taking\mcC\to\mcD$ is called a discrete op-fibration if, for every object $c\in\Ob(\mcC)$ and every arrow $g\taking d\to d'$ in $\mcD$ with $F(c)=d$, there exists a unique arrow $\bar{g}\taking c\to c'$ in $\mcC$ such that $F(\bar{g})=g$.
Consider the following discrete op-fibration, which maps $a$s to $A$, $b$s to $B$, $c$s to $C$, $g$s to $G$, and $h$s to $H$:
\LMO{b_1}&\hspace{.3in}&\LMO{a_1}\ar[rr]^{h_1}\ar[ll]_{g_1}&\hspace{.3in}&\LMO{c_1}\\
\LMO{b_2}&&\LMO{a_2}\ar[ll]_{g_2}\ar[rr]^{h_2}&&\LMO{c_2}\\
\hspace{.4in}\xymatrix{~\ar[d]^F\\~}
\LMO{B}&\hspace{.3in}&\LMO{A}\ar[rr]^H\ar[ll]_G&\hspace{.3in}&\LMO{C}
Intuitively, $F\taking\mcC\to\mcD$ is a discrete op-fibration if “the columns in each table $T$ of $D$ are exactly matched by the columns in each table in $C$ mapping to $T$.” Since $a_1,a_2,a_3\mapsto A$, they must have the same column structure and they do: each has two non-ID columns. Similarly each of the $b_i$ and each of the $c_i$ have no non-ID columns, just like their images in $D$.
To explain the action of $\Sigma_F$, consider the following instance:
\begin{tabular}{| c || c | c | }\bhline
\multicolumn{3}{| c |}{{\tt a1}}\\\bhline {\bf ID}&{\bf g1}&{\bf h1}\\\bbhline
\end{tabular}
\hspace{.1in}
\begin{tabular}{| c || c | c | }\bhline
\multicolumn{3}{| c |}{{\tt a2}}\\\bhline {\bf ID}&{\bf g2}&{\bf h2}\\\bbhline
16 & 9 & 3 \\\hline
15 & 10 & 4 \\\hline
\end{tabular}
\hspace{.1in}
\begin{tabular}{| c || c | c | }\bhline
\multicolumn{3}{| c |}{{\tt a3}}\\\bhline {\bf ID}&{\bf g3}&{\bf h3}\\\bbhline
13 & 10 & 17 \\\hline
12 & 9 & 18 \\\bhline
\end{tabular}
\begin{tabular}{| c || }\bhline
\multicolumn{1}{| c |}{{\tt b1}}\\\bhline {\bf ID}\\\bbhline
\end{tabular}
\hspace{.1in}
\begin{tabular}{| c || }\bhline
\multicolumn{1}{| c |}{{\tt b2}}\\\bhline {\bf ID}\\\bbhline
\end{tabular}
\hspace{.25in}
\begin{tabular}{| c || }\bhline
\multicolumn{1}{| c |}{{\tt c1}}\\\bhline {\bf ID}\\\bbhline
\end{tabular}
\hspace{.1in}
\begin{tabular}{| c || }\bhline
\multicolumn{1}{| c |}{{\tt c2}}\\\bhline {\bf ID}\\\bbhline
\end{tabular}
\hspace{.1in}
\begin{tabular}{| c || }\bhline
\multicolumn{1}{| c |}{{\tt c3}}\\\bhline {\bf ID}\\\bbhline
\end{tabular}
\hspace{.1in}
\begin{tabular}{| c || }\bhline
\multicolumn{1}{| c |}{{\tt c4}}\\\bhline {\bf ID}\\\bbhline
\end{tabular}
The result of $\Sigma_F$ is:
\begin{tabular}{| c || c | c | }\bhline
\multicolumn{3}{| c |}{{\tt A}}\\\bhline {\bf ID}&{\bf G}&{\bf H}\\\bbhline
16 & 9 & 3 \\\hline
15 & 10 & 4 \\\hline
14 & 8 & 4 \\\hline
13 & 10 & 17 \\\hline
12 & 9 & 18 \\\hline
11 & 7 & 1 \\\bhline
\end{tabular}
\hspace{.25in}
\begin{tabular}{| c || }\bhline
\multicolumn{1}{| c |}{{\tt B}}\\\bhline {\bf ID}\\\bbhline
\end{tabular}
\hspace{.25in}
\begin{tabular}{| c || }\bhline
\multicolumn{1}{| c |}{{\tt C}}\\\bhline {\bf ID}\\\bbhline
\end{tabular}
By counting the number of rows it is easy to see that $\Sigma$ computes union: ${\tt A}$ has $6 = 1 + 3 + 2$ rows, B has $5 = 3 + 2$ rows, C has $7 = 2 + 2 + 1 + 2$ rows.
We can $\Sigma$ along any typed signature morphism for which the attribute mapping is also “union compatible in the sense of Codd”: string attributes must map to string attributes, integer attributes must map to integer attributes, and so forth. Intuitively, the attribute data will be unioned together in exactly the same way as ID data.
Technically, $\Sigma$ is a disjoint union. However, by requiring our IDs to be globally unique, we can use regular union to implement disjoint union: the globally unique ID assumption ensures that for all distinct tables $X,Y$ in a functorial instance, $|X \cup Y| = |X| + |Y|$
Remark. In this paper we require that signature morphisms used with $\Sigma$ be discrete op-fibrations. However, it is possible to define $\Sigma$ for arbitrary, un-restricted signature morphisms, at the following cost:
* Unrestricted $\Sigma$s may not exist for typed instances.
* An unrestricted variant of FQL will probably not be closed under composition.
* To implement unrestricted $\Sigma$ we may be required to synthesize new IDs, and termination of this process is semi-decidable.
§ FQL
The goal of this section is to define and study an algebraic query language where every query denotes a composition of data migration functors. Our syntax for queries is designed to build-in the syntactic restrictions discussion in the previous section, and to provide a convenient normal form.
A FQL query $Q$ from $S$ to $T$, denoted $Q\taking S\queryto T$ is a triple of typed signature morphisms $(F,G,H)$:
such that
* $\llbracket S \rrbracket, \llbracket S' \rrbracket, \llbracket S'' \rrbracket$, and $\llbracket T \rrbracket$ are finite
* $G$'s attribute mapping is a surjection
* $H$ is a discrete op-fibration with a union compatible attribute mapping
Semantically, the query $Q\taking S \queryto T$ corresponds to a functor $\church{Q}\taking\church{S}\inst\to\church{T}\inst$ given as follows:
$$\church{Q}:=\Sigma_{\church{H}}\Pi_{\church{G}}\Delta_{\church{F}}\taking \church{S}\inst\to \church{T}\inst$$
By choosing two of $F$, $G$, and $H$ to be the identity mapping, we can recover $\Delta$, $\Sigma$, and $\Pi$. However, grouping $\Delta, \Sigma$, and $\Pi$ together like this formalizes a query as a disjoint union of a join of a projection. (Interestingly, in the SPCU relational algebra, the order of join and projection are swapped: the normal form is that of unions of projections of joins.)
§.§ Composition
Given any FQL queries $f\taking S \queryto X$ and $g\taking X \queryto T$, we can compute an FQL query $g \cdot f\taking S\queryto T$ such that
$$\church{g\cdot f} \cong \church{g}\circ\church{f}$$
Following the proof in <cit.>, it suffices to show the Beck-Chevalley conditions hold for $\Delta$ and its adjoints, and that dependent sums distribute over dependent products when the former are indexed by discrete op-fibrations. The proof and algorithm are given in the appendix as Theorem <ref> and Corollary <ref>.
Detailing the algorithm involves a number of constructions we have not defined, so we only give a brief sketch here. To compose $\Sigma_t \Pi_f \Delta_s$ with $\Sigma_v \Pi_g \Delta_u$ we must construct the following diagram:
\begin{align*}
\xymatrix{
\end{align*}
First we form the pullback $(i)$. Then we form the typed distributivity diagram $(ii)$ as in Proposition <ref>. Then we form the typed comma categories $(iii)$ and $(iv)$ as in Proposition <ref>. The result then follows from Proposition <ref>, <ref>, and Corollary <ref>. The composed query is $\Sigma_{w;v}\Pi_{p;q}\Delta_{n;m;s}$.
§ SQL GENERATION
In this section we define SQL generation algorithms for $\Delta$, $\Sigma$, and $\Pi$. Let $F\taking S \to T$ be a signature morphism.
§.§ $\Delta$
We can compute a SQL program $[F]_\Delta\taking T \to S$ such that for every $T$-instance $I\in T\inst$, we have $\Delta_F(I) \cong [F]_\Delta(I)$.
See Construction <ref>.
We sketch the algorithm as follows. We are given a $T$-instance $I$, presented as a set of binary functions, and are tasked with creating the $S$-instance $\Delta_F(I)$. We describe the result of $\Delta_F(I)$ on each table in the result instance by examining the schema $S$:
* for each node $N$ in $S$, the binary table $\Delta_F(N)$ is the binary table $I_{F(N)}$
* for each attribute $A$ in $S$, the binary table $\Delta_F(A)$ is the binary table $I_{F(A)}$
* Each edge $E : X \to Y$ in $S$ maps to a path $F(E) : FX \to FY$ in $T$. We compose the binary edges tables making up the path $F(E)$, and that becomes the binary table $\Delta_F(E)$.
The SQL generation algorithm for $\Delta$ sketched above does not maintain the globally unique ID requirement. For example, $\Delta$ can copy tables. Hence we must also generate SQL to restore this invariant.
§.§ $\Sigma$
Suppose $F$ is a discrete op-fibration and has a union compatible attribute mapping. Then we can compute a SQL program $[F]_\Sigma\taking S \to T$ such that for every $S$-instance $I\in S\inst$, we have $\Sigma_F(I) \cong [F]_\Sigma(I)$.
See Construction <ref>.
We sketch the algorithm as follows. We are given a $S$-instance $I$, presented as a set of binary functions, and are tasked with creating the $T$-instance $\Sigma_F(I)$. We describe the result of $\Sigma_F(I)$ on each table in the result instance by examining the schema $T$:
* for each node $N$ in $T$, the binary table $\Delta_F(N)$ is the union of the binary node tables in $I$ that map to $N$ via $F$.
* for each attribute $A$ in $T$, the binary table $\Delta_F(A)$ is the union of the binary attribute tables in $I$ that map to $A$ via $F$.
* Let $E : X \to Y$ be an edge in $T$. We know that for each $c \in F^{-1}(X)$ there is at least one path $p_c$ in $C$ such that $F(p_c) \cong e$. Compose $p_c$ to a single binary table, and define $\Sigma_F(E)$ to be the union over all such $c$. The choice of $p_c$ will not matter.
§.§ $\Pi$
Suppose $\llbracket S \rrbracket$ and $\llbracket T \rrbracket$ are finite, and $F$ has a bijective attribute mapping. Then we can compute a SQL program $[F]_\Pi\taking S \to T$ such that for every $S$-instance $I\in S\inst$, we have $\Pi_F(I) \cong [F]_\Pi(I)$.
See Construction <ref>.
The algorithm for $\Pi_F$ is more complicated than for $\Delta_F$ and $\Sigma_F$. In particular, its construction makes use of comma categories, which we have not yet defined, as well as “limit tables”, which are a sort of “join all”. We define these now.
Let $B$ be a typed signature and $H$ a typed $B$-instance. The limit table $lim_B$ is computed as follows. First, take the cartesian product of every node table in $B$, and naturally join the attribute tables of $B$. Then, for each edge $e : n_1 \to n_2$ filter the table by $n_1 = n_2$. This filtered table is the limit table $lim_B$.
Let $S : A \to C$ and $T : B \to C$ be functors. The comma category $(S \downarrow T)$ has for objects triples $(\alpha, \beta, f)$, with $\alpha$ an object in $A$, $\beta$ and object in $B$, and $f : S(\alpha) \to T(\beta)$ a morphism in $C$. The morphisms from $(\alpha, \beta, f)$ to $(\alpha', \beta', f')$ are all the pairs $(g,h)$ where $g : \alpha \to \alpha'$ and $h : B \to B'$ are morphisms in $A$ and $B$ respectively such that $T(h) \circ f = f' \circ S(g)$.
The algorithm for $\Pi_F$ proceeds as follows. First, for every object $d \in T$ we consider the comma category $B_d := (d \downarrow F)$ and its projection functor $q_d : (d \downarrow F) \to C$. (Here we treat $d$ as a functor from the empty category). Let $H_d := I \circ q_d : B_d \to {\bf Set}$, constructed by generating SQL for $\Delta(I)$. We say that the limit table for $d$ is $lim_{B_d} H_d$, as described above. Now we can describe the target tables in $T$:
* for each node $N$ in $T$, generate globally unique IDs for each row in the limit table for $N$. These GUIDs are $\Pi_F(N)$.
* for each attribute $A : X \to type$ in $T$, $\Pi_F(A)$ will be a projection from the limit table for $X$.
* for each edge $E : X \to Y$ in $T$, $\Pi_F(E)$ will be a map from $X$ to $Y$ obtained by joining the limit tables for $X$ and $Y$ on columns which “factor through” $E$.
Remark. Our SQL generation algorithms for $\Delta$ and $\Sigma$ work even when $\llbracket S \rrbracket$ and $\llbracket T \rrbracket$ are infinite, but this is not the case for $\Pi$. But if we are willing to take on infinite SQL queries, we have $\Pi$ in general; see Construction <ref>. Alternatively, it is possible to target a category besides ${\bf Set}$, provided that category is complete and co-complete. For example, it is possible to store our data not as relations, but as programs in the Turing-complete language PCF [18] [16].
§ SQL IN FQL
Because FQL operates over functorial instances, it is not possible to implement many relational/SQL operations directly in FQL. However, we can always “encode” arbitrary relational databases as functorial instances using an explicit active domain construction.
Relational signatures are encoded as “pointed” signatures with a single attribute that can intuitively be thought of as the active domain. For example, the signature for a relational schema with two relations $R(c_1, \ldots, c_n)$ and $R^\prime(c^\prime_1, \ldots, c^\prime_{n^\prime})$ has the following form:
\fbox{\parbox{1in}{\xymatrix{\LMO{R}\ar@/_1pc/[dr]_{c_1}\ar@{}[dr]|{\cdots}\ar@/^1pc/[dr]^(.3){c_n}&&\LMO{R'}\ar@/_1pc/[dl]_(.3){c'_{n'}}\ar@{}[dl]|{\cdots}\ar@/^1pc/[dl]^{c'_1}\\&\LMO{D} \\ & \stackrel{A}{\circ} \ar@{-}[u]}}}
We might expect that the $c_1, \ldots, c_n$ would be attributes of node $R$, and hence there would be no node $D$, but that doesn't work: attributes may not be joined on. Instead, we must think of each column of $R$ as a mapping from $R$'s domain to IDs in $D$, and $A$ as a mapping from IDs in $D$ to constants. We will write $[R]$ for the encoding of a relational schema $R$ and $[I]$ for the encoding of a relational $R$-instance $I$.
§.§ Conjunctive queries (Bags)
FQL can implement relational conjunctive (SPC/select-from-where) queries under bag semantics directly using the above encoding. In what follows we will omit the attribute $A$ from the diagrams. For simplicity, we will assume the minimal number of tables required to illustrate the construction. We may express the (bag) operations $\pi, \sigma, \times$ as follows:
* Let $R$ be a table. We can express $\pi_{i_1, \ldots, i_k} R$ using the pullback $\Delta_F$, where $F$ is the following functor
\fbox{\parbox{1in}{\xymatrix{\LMO{\pi R}\ar@/_1pc/[d]_{i_1}\ar@{}[d]|{\cdots}\ar@/^1pc/[d]^{i_k}\\\LMO{D}}}}
\Too{F}
\fbox{\parbox{1in}{\xymatrix{\LMO{R}\ar@/_2pc/[d]_{c_1}\ar@/_1pc/[d]_{c_2}\ar@{}[d]|{\cdots}\ar@/^1pc/[d]^{c_n}\\\LMO{D}}}}
This construction is only appropriate for bag semantics because $\pi R$ will will have the same number of rows as $R$.
* Let $R$ be a table. We can express $\sigma_{i=j} R$ using the pullback of
the right pushforward $\Delta_F\Pi_F$, where $F$ is the following functor:
\fbox{\parbox{1in}{\xymatrix{\LMO{R}\ar@/_3pc/[d]_{c_1}\ar@/_1pc/[d]_{\cdots\;\; c_i}\ar@{}[d]|{\cdots}\ar@/^1pc/[d]^{c_j\;\;\cdots}\ar@/^3pc/[d]^{c_n}\\\LMO{D}}}}
\Too{F}
\fbox{\parbox{1in}{\xymatrix{\LMO{\sigma R}\ar@/_2pc/[d]_{s}\ar@/_1pc/[d]_{c_1}\ar@{}[d]|\cdots\ar@/^1pc/[d]^{c_n}\\\LMO{D}}}}
Here $F(c_i)=F(c_j)=s$.
Remark. In fact, pulling back along $F$ creates the duplicated column. If we wanted the more economical query in which the column is not duplicated, we would use $\Pi_F$ instead of $\Delta_F\Pi_F$.
* Let $R$ and $R'$ be tables. We can express $R \times R'$ as the right pushforward $\Pi_F$, where $F$ is the following functor:
\fbox{\parbox{1in}{\xymatrix{\LMO{R}\ar@/_1pc/[dr]_{c_1}\ar@{}[dr]|{\cdots}\ar@/^1pc/[dr]^(.3){c_n}&&\LMO{R'}\ar@/_1pc/[dl]_(.3){c'_{n'}}\ar@{}[dl]|{\cdots}\ar@/^1pc/[dl]^{c'_1}\\&\LMO{D}}}}
\Too{F}
\fbox{\parbox{1in}{\xymatrix{\LMO{R \times R'}\ar@/_3pc/[d]_{c_1}\ar@/_1pc/[d]_{\cdots\;\; c_n}\ar@{}[d]|{\cdots}\ar@/^1pc/[d]^{c'_1\;\;\cdots}\ar@/^3pc/[d]^{c'_{n'}}\\\LMO{D}}}}
Let $R$ be a relational schema, and $I$ an $R$-instance. For every conjunctive (SPC) query $q$ under bag semantics we can compute a FQL program $[q]$ such that $[q(I)] \cong [q]([I])$.
See Proposition <ref>.
§.§ Conjunctive Queries (Sets)
The encoding strategy described above fails for relational conjunctive queries under their typical set-theoretic semantics. For example, consider a simple two column relational table $R$, its encoded FQL instance $[R]$, and an attempt to project off col1 using $\Delta$:
\begin{align*}
\begin{tabular}{| l | l |}\bhline
\multicolumn{2}{| c |}{R}\\\bbhline
{\bf col1}&{\bf col2}\\\hline
\end{tabular}\hspace{.1in}
\begin{tabular}{| c || c | c |}\bhline
\multicolumn{3}{| c |}{$[R]$}\\\bbhline
{\bf ID} & {\bf col1}&{\bf col2}\\\hline
0 & x&y\\\hline
1 & x&z\\\bhline
\end{tabular}\hspace{.1in}
\begin{tabular}{| c || c |}\bhline
\multicolumn{2}{| c |}{$\Delta [R]$}\\\bbhline
{\bf ID} & {\bf col1}\\\hline
0& x \\\hline
1 & x\\\bhline
\end{tabular}
\end{align*}
This answer is incorrect under projection's set-theoretic semantics, because it has the wrong number of rows. However, it is possible to extend FQL with an operation, $relationalize_T : T$-Inst $\to T$-Inst, such that FQL+$relationalize$ can implement every relational conjunctive query under normal set-theoretic semantics. Intuitively, $relationalize_T$ converts a $T$-instance to a smaller $T$-instance by equating IDs that cannot be distinguished. The relationalization of the above instance would be
\begin{align*}
\begin{tabular}{| c || c |}\bhline
\multicolumn{2}{| c |}{r$elationalize(\Delta [R])$}\\\bbhline
{\bf ID} & {\bf col1}\\\hline
0& x \\\hline
\end{tabular}
\end{align*}
which is correct for the set-theoretic semantics. We have the the following theorem:
Let $R$ be a relational schema, and $I$ an $R$-instance. For every conjunctive (SPC) query $q$ under set semantics we can compute a FQL program $[q]$ such that $[q(I)] \cong relationalize_T([q]([I]))$.
See Proposition <ref>
Provided $T$ is obtained from a relational schema (i.e., has the pointed form described in this section), $relationalize_T$ can easily be implemented using the same SPCU+GUIDgen operations as $\Delta,\Sigma,\Pi$.
§.§ Union
The bag union of two relational instances $R_1$ and $R_2$ can be expressed using disjoint union as $R_1 + R_2$; similarly, the set-union can be expressed as $relationalize(R_1 + R_2)$. Hence, given the results of the previous sections, to translate the SPCU relational algebra to FQL it is sufficient to implement disjoint union using FQL. However, because FQL queries are unary and union is binary, implementing disjoint union using FQL requires encoding two $C$-instances as an instance on the co-product schema $C+C$.
For any two signatures $S, T$, the co-product signature $S+T$ is formed by taking the disjoint union of $S, T$'s nodes, attributes, edges, and equations. An $S$ instance can be injected into $(S+T)\inst$ using $\Sigma_F : S\inst \to (S+T)\inst$, where $F : S \to S+T$ is the canonical inclusion map. We have the following theorem:
Let $C$ be a signature and $I:C\inst$ and $J:C\inst$ be instances. Then the co-product instance $I+J : C\inst$ is expressible as $\Sigma_F(K) : C\inst$, where $F : C+C \to C$ is the “fold” signature morphism taking each copy of $C$ in the co-product schema $C+C$ to $C$, and $K:(C+C)\inst$ is formed by injecting $I:C\inst$ and $J:C\inst$ into $(C+C)\inst$.
Let $D=C+C$ denote the coproduct signature of $C$ with itself, let $L,R\taking C\to D$ be the inclusions, and let $F=\id_C+\id_C\taking D\to C$ denote the above “fold" functor. Each of $L,R,F$ are discrete op-fibrations. We have $I\taking C\to\Set$ and $J\taking C\to\Set$. It is easy to see that we are defining $K=\Sigma_L(I)+\Sigma_R(J)$. The result follows from the fact that $F\circ L=\id_C\taking C\to C$ and $F\circ R=\id_C\taking C\to C$. Indeed, since $\Sigma_G$ is a left adjoint functor (for any $G$), and left adjoints preserve colimits, we have isomorphisms
\begin{align*}
\Sigma_F(K)&\iso\Sigma_F(\Sigma_L(I) + \Sigma_R(J))\\
&\iso\Sigma_{F\circ L}(I)+\Sigma_{F\circ R}(J)\iso I + J.
\end{align*}
§ CONCLUSION
We are working to extend FQL with additional operations such as difference, selection by a constant, and aggregation. In addition, we are studying the systems aspects of FQL, such as FQL's equational theory and the data structures and algorithms that would be appropriate for a native, non-SQL implementation of FQL. But we are most excited about FQL's potential as a schema mapping and information integration system in the spirit of Clio [10] and Rondo <cit.>. Such systems, which are defined abstractly as “schema transformation frameworks” <cit.> and “generic model managers” <cit.>, allow users to integrate information from disparate sources while providing guarantees about the semantics of the integrated data. Despite a decade of work, building expressive and performant systems remains a challenging problem [4]. We have proved that FQL is a schema transformation framework in the sense of <cit.>, and we believe FQL's categorical underpinnings make it a quite powerful one. For example, FQL's $\Sigma$ operation, when not restricted to discrete op-fibrations, has a similar semantics to Clio's primary data migration operation, yet is more expressive: Clio's operation cannot take into account path-equality constraints on the target schema, but $\Sigma$ can. As another example, there is as yet no consensus about the “correct” way to invert a data migration operation, and notions such as “quasi-inverses” <cit.> are continually proposed. In FQL, $\Sigma_F$ and $\Delta_F$ are adjoint, and indeed when $F$ is a fully faithful functor $\Sigma_F$ and $\Delta_F$ are quasi-inverses in the sense of <cit.>. To fully understand FQL as an information integration platform requires first understanding its relational foundations, which we have done in this paper.
Acknowledgements. The authors would like to thank Lucian Popa and Scott Morrison.
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§ CATEGORICAL CONSTRUCTIONS FOR DATA MIGRATION
Here we give many standard definitions from category theory and a few less-than-standard (or original) results. The proofs often assume more knowledge than the definitions do. Let us also note our use of the term essential, which basically means “up to isomorphism". Thus an object $X$ having a certain property is essentially unique if every other object having that property is isomorphic to $X$; an object $X$ is in the essential image of some functor if it is isomorphic to an object in the image of that functor; etc.
§.§ Basic constructions
Suppose given the diagram of sets and functions as to the left in (<ref>).
\begin{align}\label{dia:fp sets}
\xymatrix{&B\ar[d]^g&&A\cross_CB\ar[r]^{f'}\ar[dr]^h\ar[d]_{g'}&B\ar[d]^g\\
\end{align}
Its fiber product is the commutative diagram to the right in (<ref>), where we define
and $f', g'$, and $h=g\circ f'=f\circ g'$ are the obvious projections (e.g. $f'(a,c,b)=b$). We sometimes refer to just $(A\cross_CB,f',g', h)$ or even to $A\cross_CB$ as the fiber product.
[Chain signatures]
Let $n\in\NN$ be a natural number. The chain signature on $n$ arrows, denoted $\vect{n}$, [The chain signature $\vect{n}$ is often denoted $[n]$ in the literature, but we did not want to further overload the bracket [-] notation] is the graph
with no path equivalences. One can check that $\vect{n}$ has ${n+2}\choose{2}$-many paths.
The chain signature $\vect{0}$ is the terminal object in the category of signatures.
A signature morphism $c\taking \vect{0}\to C$ can be identified with an object in $C$. A signature morphism $f\taking\vect{1}\to C$ can be identified with a morphism in $C$. It is determined up to path equivalence by a pair $(n,p)$ where $n\in\NN$ is a natural number and $p\taking\vect{n}\to C$ is a morphism of graphs.
Suppose given the diagram of signatures and signatures mappings as to the left in (<ref>).
\begin{align}\label{dia:fiber prod of sigs}
\xymatrix{&B\ar[d]^g&&A\cross_CB\ar[r]^{f'}\ar[dr]^h\ar[d]_{g'}&B\ar[d]^g\\
\end{align}
On objects and arrows we have the following diagrams of sets:
\begin{align}\label{dia:fiber prod on objects and arrows}
\parbox{1in}{
\xymatrix{&\Ob(B)\ar[d]^{\Ob(g)}\\
\Ob(A)\ar[r]_{\Ob(f)}&\Ob(C)}
\parbox{1in}{
\xymatrix{&\Arr(B)\ar[d]^{\Arr(g)}\\
\Arr(A)\ar[r]_{\Arr(f)}&\Mor(C)}
\end{align}
We define the fiber product of the left-hand diagram in (<ref>) to be the right-hand commutative diagram in (<ref>), where we the objects and arrows of $A\cross_CB$ are given by the fiber product of the left and right diagram of sets in (<ref>), respectively. A pair of paths in $A\cross_CB$ are equivalent if they map to equivalent paths in both $A$ and $B$ (and therefore in $C$). Note that we have given $A\cross_CB$ as a signature, i.e. we have provided a finite presentation of it.
[Pre-image of an object or morphism]
Let $F\taking C\to D$ be a functor. Given an object $d\in \Ob(D)$ (respectively a morphism $d\in\Mor(D)$), we can regard it as a functor $d\taking\vect{0}\to D$ (respectively a functor $d\taking \vect{1}\to D$), as in Example <ref>. The pre-image of $d$ is the subcategory of $C$ that maps entirely into $d$. This is the fiber product of the diagram $C\To{F}D\From{d}\vect{0}$ (respectively the diagram $C\To{F}D\From{d}\vect{1}$).
A signature is assumed to have finitely many arrows, but because it may have loops, $C$ may have infinitely many non-equivalent paths. We make the following definitions to deal with this.
A signature $C$ is called finite if there is a finite set $S$ of paths such that every path in $C$ is equivalent to some $s\in S$. An equivalent condition is that the schema $\church{C}$ has finitely many morphisms. (Note that if $C$ is finite then $C$ has finitely many objects too, since each object has an associated trivial path.)
Recall (from the top of Section <ref>) that every signature is assumed to have a finite set of objects and arrows. The above notion is a semantic one: a signature $C$ is called finite if $\church{C}$ is a finite category.
A signature $C$ is called acyclic if it has no non-trivial loops. In other words, for all objects $c\in\Ob(C)$ every path $p\taking c\to c$ is a trivial path.
If $C$ is an acyclic signature then it is finite.
All of our signatures are assumed to include only finitely many objects and arrows. If $C$ is acyclic then no edge can appear twice in the same path. Suppose that $C$ has $n$ arrows. Then the number of paths in $C$ is bounded by the number of lists in $n$ letters of length at most $n$. This is finite.
Given a diagram $A\To{F}C\From{G}B$ of signatures such that each is acyclic (respectively, finite), the fiber product signature $A\cross_CB$ is acyclic (respectively, finite).
The fiber product of finite sets is clearly finite, and the fiber product of categories is computed by twice taking the fiber products of sets (see Definition <ref>). We now show that acyclicity is preserved.
Let $\Loop$ denote the signature
\begin{align}\label{dia:loop}
\Loop:=\LoopSchema
\end{align}
A category $C$ is acyclic if and only if every functor $L\taking\Loop\to C$ factors through the unique functor $\Loop\to\vect{0}$, i.e. if and only if every loop is a trivial path. The result follows by the universal property for fiber products.
To a diagram of categories $A\To{f}C\From{g}B$ we can associate the comma category of $f$ over $g$, denoted $(f\down g)$, defined as follows.
$$\Ob(f\down g):=\{(a,b,\gamma)\|a\in\Ob(A),b\in\Ob(B),\gamma\taking f(a)\to g(b)\text{ in }C\}$$
$$\Hom_{(f\down g)}((a,b,\gamma),(a',b',\gamma')):=\{(m,n)\|m\taking a\to a'\text{ in } A,n\taking b\to b'\text{ in } B, \gamma'\circ f(m)=g(n)\circ\gamma\}$$
There are obvious projection functors $(f\down g)\To{p} A$ and $(f\down g)\To{q} B$ and a natural transformation $\alpha\taking f\circ p\to g\circ q$, and we summarize this in a canonical natural transformation diagram
\begin{align}\label{dia:nt for comma}
\xymatrix{(f\down g)\ar[r]^q\ar[d]_p\ar@{}[dr]|{\stackrel{\alpha}{\Nearrow}}&B\ar[d]^g\\A\ar[r]_f&C}
\end{align}
Below is a heuristic picture that may be helpful. The outside square consists of categories and functors (note that it does not commute!). The inside square represents a morphism in $(f\down g)$ from object $(a,b,\gamma)$ to object $(a',b',\gamma')$; this diagram is required to commute in $C$.
&&&(f\down g)\ar@/_2pc/[lllddd]_p\ar@/^2pc/[rrrddd]^q\\\\
\\\color{red}{A}\ar@/_2pc/[dddrrr]_f&\color{red}{a}\ar@[red][d]_{\color{red}{m}}&\color{blue}{f(a)}\ar@[blue][rr]^{\color{blue}{\gamma}}\ar@[blue][d]_{\color{blue}{f(m)}}&&\color{blue}{g(b)}\ar@[blue][d]^{\color{blue}{g(n)}}&\color{ForestGreen}{b}\ar@[ForestGreen][d]^{\color{ForestGreen}{n}}&\color{ForestGreen}{B}\ar@/^2pc/[dddlll]^g\\
[Slice category and coslice category]
Let $f\taking A\to B$ be a functor, and let $b\in\Ob(B)$ be an object, which we represent as a functor $b\taking\vect{0}\to B$. The slice category of $f$ over $b$ is defined as $(f\down b)$. The coslice category of $b$ over $F$ is defined as $(b\down f)$.
For example let $\singleton$ denote a one element set, and let $\id_\Set$ be the identity functor on $\Set$. Then the coslice category $(\singleton\down\id_\Set)$ is the category of pointed sets, which we denote by $\Set_*$. The slice category $(\id_\Set\down\singleton)$ is isomorphic to $\Set$.
Given a diagram $A\To{f}C\From{g}B$, the comma category $(f\down g)$ can be obtained as the fiber product in the diagram
\xymatrix{(f\down g)\ar[r]^{\alpha}\ar[d]_{(p,q)}&C^{\vect{1}}\ar[d]^{(dom,cod)}\\A\cross B\ar[r]_{(f,g)}&C\cross C}
where $C^{\vect{1}}$ is the category of functors $\vect{1}\to C$.
Let $A, B$, and $C$ be acyclic (resp. finite) categories, and suppose we have functors $A\To{f}C\From{g}B$. Then the comma category $(f\down g)$ is acyclic (resp. finite).
Since the fiber product of finite sets is finite, Lemma <ref> implies that the formation of comma categories preserves finiteness. Suppose that $A$ and $B$ are acyclic (in fact this is enough to imply that $(f\down g)$ is acyclic). Given a functor $L\taking\Loop\to(f\down g)$, this implies that composing with the projections to $A$ and $B$ yields trivial loops. In other words $L$ consists of a diagram in $C$ of the form
\xymatrix{f(a)\ar[r]^\gamma\ar@{=}[d]&g(b)\ar@{=}[d]\\f(a)\ar[r]_{\gamma'}&g(b)}
But this implies that $\gamma=\gamma'$, so $L$ is a trivial loop too, competing the proof.
§.§ Data migration functors
Let $F\taking C\to D$ be a functor. We will define a functor $\Delta_F\taking D\inst\to C\inst$; that is, given a $D$-instance $I\taking D\to\Set$ we will naturally construct a $C$-instance $\Delta_F(I)$. Mathematically, this is obtained simply by composing the functors to get
$$\parbox{1.6in}{$\Delta_F(I):=I\circ F\taking C\to\Set$}
\hspace{1in}
\parbox{1.5in}{\xymatrix{C\ar[r]^F\ar@/_1pc/[rr]_{\Delta_FI}&D\ar[r]^I&\Set}}
To understand this on the level of binary tables, we take $F$ and $I$ as given and proceed to define the $C$-instance $J:=\Delta_FI$ as follows. Given an object $c\in\Ob(C)$, it is sent to an object $d:=F(c)\in\Ob(D)$, and $I(d)$ is an entity table. Assign $J(c)=I(d)$. Given an arrow $g\taking c\to c'$ in $C$, it is sent to a path $p=F(g)$ in $D$. We can compose this to a binary table $[p]$. Assign $J(g)=[p]$. It is easy to check that these assignments preserve the axioms.
Suppose that $C$ and $D$ are finite categories and $F\taking C\to D$ is a functor. If $I\taking D\to\Set$ is a finite $D$-instance then $\Delta_FI$ is a finite $C$-instance.
This follows by construction: every object and arrow in $C$ is assigned some finite join of the tables associated to objects and arrows in $D$, and finite joins of finite tables are finite.
Recall the definition of discrete op-fibrations from Definition <ref>.
Let $F\taking C\to D$ be a signature morphism. The following are conditions on $F$ are equivalent:
* The functor $[F]\taking[C]\to[D]$ is a discrete op-fibration.
* For every choice of object $c_0$ in $C$ and path $q\taking d_0\to d_n$ in $D$ with $F(c_0)=d_0$, there exists a path $p\in C$ such that $F(p)\sim q$, and $p$ is unique up to path equivalence.
* For every choice of object $c_0$ in $C$ and edge $q\taking d_0\to d_1$ in $D$ with $F(c_0)=d_0$, there exists a path $p\in C$ such that $F(p)\sim q$, and $p$ is unique up to path equivalence.
It is clear that condition (2) implies condition (3), and the converse is true because concatenations of equivalent paths are equivalent. It suffices to show that (1) and (2) are equivalent.
It is shown in [17] that a functor $\pi\taking X\to S$ between categories $X$ and $S$ is a discrete op-fibration if and only if for each solid-arrow square of the form
\xymatrix@=30pt{\vect{0}\ar[r]^{c_0}\ar[d]_{i_0}&X\ar[d]^\pi\\\vect{1}\ar@{-->}[ur]^{\exists!p}\ar[r]_q&S,}
where $i_0\taking \vect{0}\to\vect{1}$ sends the unique object of $\vect{0}$ to the initial object of $\vect{1}$, there exists a unique functor $\ell$ such that both triangles commute. We now translate this statement to the language of signatures with $\church{C}=X$ and $\church{D}=S$. A functor $\vect{1}\to\church{C}$ corresponds to a path in $C$, and the uniqueness of lift $p$ corresponds to uniqueness of equivalence class. This completes the proof.
[A discrete op-fibration]
Let $C, D$ be signatures, and let $F\taking C\to D$ be as suggested by the picture below.
\LMO{b_1}&\hspace{.3in}&\LMO{a_1}\ar[rr]^{h_1}\ar[ll]_{g_1}&\hspace{.3in}&\LMO{c_1}\\
\LMO{b_2}&&\LMO{a_2}\ar[ll]_{g_2}\ar[rr]^{h_2}&&\LMO{c_2}\\
\hspace{.4in}\xymatrix{~\ar[d]^F\\~}
\LMO{B}&\hspace{.3in}&\LMO{A}\ar[rr]^H\ar[ll]_G&\hspace{.3in}&\LMO{C}
This is a discrete op-fibration.
Let $F\taking C\to D$ be a discrete op-fibration and let $d\in\Ob(D)$ be an object. Then the pre-image $F^\m1(d)$ is a discrete signature (i.e. every path in $F^\m1(d)$ is equivalent to a trivial path).
For any object $d\in\Ob(D)$ the pre-image $F^\m1(d)$ is either empty or not. If it is empty, then it is discrete. If $F^\m1(d)$ is non-empty, let $e\taking c_0\to c_1$ be an edge in it. Then $F(e)=d=F(c_0)$ so we have an equivalence $e\sim c_0$ by Lemma <ref>.
Let $F\taking A\to B$ be a functor. For any object $b\in B$, considered as a functor $b\taking\vect{0}\to B$, the induced functor $(b\down F)\to B$ is a discrete op-fibration. Indeed, given an object $b\To{g}F(a)$ in $(b\down F)$ and a morphism $h\taking a\to a'$ in $A$, there is a unique map $b\To{g}F(a)\To{F(h)}F(a')$ over it.
A signature $C$ is said to have non-redundant edges if it satisfies the following condition for every edge $e$ and path $p$ in $C$: If $e\sim p$, then $p$ has length 1 and $e=p$.
Every acyclic signature is equivalent to a signature with non-redundant edges.
The important observation is that if $C$ is acyclic then for any edge $e$ and path $p$ in $C$, if $e$ is an edge in $p$ and $e\sim p$ then $e=p$. Thus, we know that if $C$ is acyclic and $e\sim p$ then $e$ is not in $p$. So enumerate the edges of $C$ as $E_C:=\{e_1,\ldots,e_n\}$. If $e_1$ is equivalent to a path in $E_C-\{e_1\}$ then there is an equivalence of signatures $C\To{\iso}C-\{e_1\}=:C_1$; if not, let $C_1:=C$. Proceed by induction to remove each $e_i$ that is equivalent to a path in $C_{i-1}$, and at the end no edge will be redundant.
If $C$ and $D$ have non-redundant edges and $F\taking C\to D$ is a discrete op-fibration, then for every choice of object $c_0$ in $C$ and path $q=d_0.e'_1.e'_2.\ldots.e'_n$ of length $n$ in $D$ with $F(c_0)=d_0$, there exists a unique path $p=c_0.e_1.e_2.\ldots.e_n$ of length $n$ in $C$ such that $F(e_i)=e'_i$ for each $1\leq i\leq n$, so in particular $F(p)=q$.
Let $c_0, d_0$, and $q$ be as in the hypothesis. We proceed by induction on $n$, the length of $q$. In the base case $n=0$ then Corollary <ref> implies that every edge in $F^\m1(d_0)$ is equivalent to the trivial path $c_0$, so the result follows by the non-redundancy of edges in $C$. Suppose the result holds for some $n\in\NN$. To prove the result for $n+1$ it suffices to consider the final edge of $q$, i.e. we assume that $q=e'$ is simply an edge. By Lemma <ref> there exists a path $p\in C$ such that $F(p)\sim e'$, so by non-redundancy in $D$ we know that $F(p)$ has length 1. This implies that for precisely one edge $e_i$ in $p$ we have $F(e_i)=e'$, and for all other edges $e_j$ in $p$ we have $F(e_j)$ is a trivial path. But by the base case this implies that $p$ has length 1, completing the proof.
[$\Sigma$ for discrete opfibrations]
Suppose that $F\taking C\to D$ is a discrete op-fibration. We can succinctly define $\Sigma_F\taking C\inst\to D\inst$ to be the left adjoint to $\Delta_F$, however the formula has a simple description which we give now. Suppose we are given $F$ and an instance $I\taking C\to\Set$, considered as a collection of binary tables, one for each object and each edge in $C$. We are tasked with finding a $D$-instance, $J:=\Sigma_FI\taking D\to\Set$.
We first define $J$ on an arbitrary object $d\in\Ob(D)$. By Corollary <ref>, the pre-image $F^\m1(d)$ is discrete in $C$; that is, it is equivalent to a finite collection of object tables. We define $J(d)$ to be the disjoint union
$$J(d):=\coprod_{c\in F^\m1(d)}I(c).$$
Similarly, let $e\taking d\to d'$ be an arbitrary arrow in $D$. By Lemma <ref> we know that for each $c\in F^\m1(d)$ there is a unique equivalence class of paths $p_c$ in $C$ such that $F(p_c)\sim e$. Choose one, and compose it to a single binary table — all other choices will result in the same result. Then define $J(e)$ to be the disjoint union
$$J(e):=\coprod_{c\in F^\m1(d)}I(p_c).$$
Suppose that $C$ and $D$ are finite categories and $F\taking C\to D$ is a functor. If $I\taking C\to\Set$ is a finite $C$-instance then $\Sigma_FI$ is a finite $D$ instance.
For each object or arrow $d$ in $D$, the table $\Sigma_FI(d)$ is a finite disjoint union of composition-joins of tables in $C$. The finite join of finite tables is finite, and the finite union of finite tables is finite.
For any functor $F\taking C\to D$ the functor $\Delta_F\taking D\inst\to C\inst$ has a left adjoint, which we can denote by $\Sigma_F\taking C\inst\to D\inst$ because it agrees with the $\Sigma_F$ constructed in <ref> in the case that $F$ is a discrete op-fibration. Certain queries are possible if we can use $\Sigma_F$ in this more general case—namely, quotienting by equivalence relations and the introduction of labeled nulls (a.k.a. Skolem variables). We do not consider it much in this paper for a few reasons. First, quotients and skolem variables are not part of the relational algebra. Second, the set of queries that include such quotients and skolem variables are not obviously closed under composition (see Section <ref>).
[Limit as a kind of “join all"]
Let $B$ be a signature and let $H$ be a $B$-instance. The functor $[H]\taking [B]\to\Set$ has a limit $\lim_BH\in\Ob(\Set)$, which can be computed as follows. Forgetting the path equivalence relations, axioms (<ref>) and (<ref>) imply that $H$ consists of a set $\{N_1,\ldots,N_m\}$ of node tables, a set $\{e_1,\ldots,e_n\}$ of edge tables, and functions $s,t\taking\{1,\ldots,n\}\to\{1,\ldots,m\}$ such that for each $1\leq i\leq n$ the table $e_i$ constitutes a function $e_i\taking N_{s(i)}\to N_{t(i)}$. For each $i$ define $X_i$ as follows:
\begin{align*}
X_0&:=\pi_{2,4,\ldots,2m}(N_1\cross\cdots\cross N_m)\\
X_1&:=\pi_{1,2,\ldots m}\sigma_{s(1)=m+1}\sigma_{t(1)=m+2}(X_0\cross e_1)\\
X_i&:=\pi_{1,2,\ldots m}\sigma_{s(i)=m+1}\sigma_{t(i)=m+2}(X_{i-1}\cross e_i)\\
X_n&:=\pi_{1,2,\ldots m}\sigma_{s(n)=m+1}\sigma_{t(n)=m+2}(X_{n-1}\cross e_n)
\end{align*}
Then $\ol{X}:=X_n$ is a table with $m$ columns, and its set $|\ol{X}|$ of rows (which can be constructed if one wishes by concatenating the fields in each row) is the limit, $|\ol{X}|\iso\lim_BH$.
Let $F\taking C\to D$ be a signature morphism. We can succinctly define $\Pi_F\taking C\inst\to D\inst$ to be the right adjoint to $\Delta_F$, however the formula has an algorithmic description which we give now. Suppose we are given $F$ and an instance $I\taking C\to\Set$, considered as a collection of binary tables, one for each object and each edge in $C$. We are tasked with finding a $D$-instance, $J:=\Pi_FI\taking D\to\Set$.
We first define $J$ on an arbitrary object $d\taking\vect{0}\to D$ (see Example <ref>). Consider the comma category $B_d:=(d\down F)$ and its projection $q_d\taking (d\down F)\to C$. Note that in case $C$ and $D$ are finite, so is $B_d$. Let $H_d:=I\circ q_d\taking B_d\to\Set$. We define $J(d):=\lim_{B_d}H_d$ as in Construction <ref>.
Now let $e\taking d\to d'$ be an arbitrary arrow in $D$. For ease of notation, rewrite
$$B:=B_d,\;\; B':=B_{d'},\;\; q:=q_d,\;\; q':=q_{d'},\;\; H:=H_d, \text{ and } H':=H_{d'}.$$
We have the following diagram of categories
\xymatrix@=10pt{
(d'\down F)\ar[rr]^{(e\down F)}\ar[ddr]_{q'}\ar@/_1pc/[ddddr]_{H'}&&(d\down F)\ar[ddl]^{q}\ar@/^1pc/[ddddl]^{H}\\\\
A unique natural map $J(d)=\lim_BH\to\lim_BH'=J(d')$ is determined by the universal property for limits, but we give an idea of its construction. There is a map from the set of nodes in $B'$ to the set of nodes in $B$ and for each node $N$ in $B'$ the corresponding node tables in $H$ and $H'$ agree. Let $X_i$ and $X_i'$ be defined respectively for $(B,H)$ and $(B',H')$ as in Construction <ref>. It follows that $X'_0$ is a projection (or column duplication) on $X_0$. The set of edges in $B'$ also map to the set of edges in $B$ and for each edge $e$ in $B'$ the corresponding edge tables in $H$ and $H'$ agree. Thus the select statements done to obtain $J(d')=\ol{X'}$ contains the set of select statements performed to obtain $J(d')=\ol{X}$. Therefore, the map from $J(d)$ to $J(d')$ is given as the inclusion of a subset followed by a projection.
If $C$ or $D$ is not finite, then the right pushforward $\Pi_F$ of a finite instance $I\in C\set$ may have infinite, even uncountable results.
Consider the unique signature morphism
Here $\church{\mcD}$ has arrows $\{f^n\|n\in\NN\}$ so it is infinite. In this case $(d\down F)$ is the discrete category with a countably infinite set of objects $\Ob(d\down F)\iso\NN$.
Given the two-element instance $I\taking\mcC\to\Set$ with $I(s)=\{\tn{Alice},\tn{Bob}\}$, the rowset in the right pushforward $\Pi_F(I)$ is the (uncountable) set of infinite streams in $\{\tn{Alice},\tn{Bob}\}$, i.e.
Let $H\taking B\to\Set$ be a functor, and let $q\taking B\to\vect{0}$ be the terminal functor. Noting that there is an isomorphism of categories $\vect{0}\set\iso\Set$, we have a bijection $\lim_BH\iso\Pi_qH.$
Obvious by construction of $\Pi$.
Let $C$ and $D$ be categories, let $F,G\taking C\to D$ be functors, and let $\alpha\taking F\to G$ be a natural transformation as depicted in the following diagram:
Then $\alpha$ induces natural transformations
For any instance $J\taking D\to\Set$ and any object $c\in\Ob(C)$ we have $\alpha_c\taking J\circ F(c)\to J\circ G(c)$, and the naturality of $\alpha$ implies that we can gather these into a natural transformation $\Delta_\alpha(J)\taking\Delta_F(J)\to\Delta_G(J)$. One checks easily that this assignment is natural in $J$, so we have $\Delta_\alpha\taking\Delta_F\to\Delta_G$ as desired.
Now suppose that $I\taking C\to\Set$ is an instance on $C$. Then for any $J\in D\set$ we have natural maps
so by the Yoneda lemma, we have a natural map $\Pi_G\to\Pi_F$, as desired. We also have natural maps
so by the Yoneda lemma, we have a natural map $\Sigma_G\to\Sigma_F$, as desired.
§.§ Relation to SQL queries
Suppose we are working in a domain $DOM\in\Ob(\Set)$.
A set-theoretic SQL query $q$ is an expression of the form
S̄ELECT DISTINCT $̄P$\\
\>FROM \>$(c_1,1,…,c_1,k_1),…,(c_n,1,…,c_n,k_n)$\\
\>WHERE\>$W$
\end{tabbing}
where $n,k_1,…,k_n∈$ are natural numbers, $C$ is a set of the form $C=∐_i{c_i,1,…,c_i,k_i}$, $W$ is a set of pairs $WßC×C$, and $PP_0→C$ is a function, for some set $P_0$.
A {\em bag-theoretic SQL query} $q'$ is an expression of the form
\begin{tabbing}
\hsp\=SELECT \;\;\=$P$\\
\>FROM \>$(c_{1,1},\ldots,c_{1,k_1}),\ldots,(c_{n,1},\ldots,c_{n,k_n})$\\
\>WHERE\>$W$
\end{tabbing}
where $n,k_1,\ldots,k_n, C, W, P, P_0$ are as above. We call $(n,k_1,\ldots,k_n, C, W, P, P_0)$ an {\em agnostic SQL query}.
Suppose that for each $1\leq i\leq n$ we are given a relation $R_i\ss \prod_{1\leq j\leq k_i}DOM$.
The {\em evaluation of $q$} (respectively, the {\em evaluation of $q'$}) on $R_1,\ldots,R_n$ is a relation $Q(R_1,\ldots,R_n)\ss \prod_{i\in P_0}R_i$ (respectively, a function $Q'(R_1,\ldots,R_n)\taking B'\to \prod_{i\in P_0}R_i$ for some set $B'$), defined as follows. Let $$B'=\{r\in R_1\times\cdots\times R_n\|r.c_1=r.c_2\tn{ for all }(c_1,c_2)\in W\}.$$ We compose an inclusion with a projection to get a function
$$B'\ss R_1\times\cdots\times R_n\To{P} \prod_{i\in P_0}R_i.$$
and define $Q'(R_1,\ldots,R_n)$ to be this function. Its image is the desired relation $Q(R_1,\ldots,R_n)\ss\prod_{i\in P_0}R_i$.
\end{definition}
\begin{definition}
Let $q=(n,k_1,\ldots,k_n,C,W,P,P_0)$ be an agnostic SQL query as in Definition \ref{def:SQL bag set}. The set $W$ generates an equivalence relation $\sim$ on $C$; let $C':=C/\sim$ be the quotient and $\ell\taking C\to C'$ the induced function. Suppose that $C'$ has $r=|C'|$ many elements and that $P_0$ has $s=|P_0|$ many elements.
We define the {\em categorical setup for $q$}, denoted $CS(q)=\mcS\To{F}\mcT\From{G}\mcU$ as follows.
\parbox{2.2in}{\begin{center}$\mcS:=$\end{center}\fbox{\xymatrix{
\LMO{1}\ar@/_1pc/[ddrr]_{c_{1,1}}\ar@/^1pc/[ddrr]^{c_{1,k_1}}\ar@{}[ddrr]|\cdots&&\cdots&&\LMO{n}\ar@/_1pc/[ddll]_{c_{n,1}}\ar@/^1pc/[dlld]^{c_{n,k_n}}\\\\
\parbox{.45in}{\xymatrix{~\ar[r]^F&~}}
\parbox{.9in}{\begin{center}$\mcT:=$\end{center}\fbox{\xymatrix{
\LMO{B'}\ar@/_1.5pc/[dd]_{c'_1}\ar@/_.5pc/[dd]_{c'_2}\ar@{}[dd]^\cdots\ar@/^1.5pc/[dd]^{c'_r}\\\\
\LMO{dom}
\parbox{.45in}{\xymatrix{~&~\ar[l]_G}}
\parbox{1in}{\begin{center}$\mcU:=$\end{center}\fbox{\xymatrix{
\LMO{B'}\ar@/_1.5pc/[dd]_{p_1}\ar@/_.5pc/[dd]_{p_2}\ar@{}[dd]^\cdots\ar@/^1.5pc/[dd]^{p_s}\\\\
\LMO{dom}
Here, the functor $F$ sends $dom$ to $dom$, sends the other objects to $B$, and acts on morphisms by $\ell\taking C\to C'$; the functor $G$ sends $dom$ to $dom$, $B'$ to $B'$, and $P_0\to C'$ by the composite $P_0\To{P}C\To{\ell}C'$.
We define the {\em categorical analogue of $q$}, denoted $CA(q)$, to be the FQL query $\Delta_G\Pi_F$.\footnote{The functor $\Delta_G\Pi_F$ is technically not in the correct $\Sigma\Pi\Delta$ order, but it is equivalent to a query in that order by Theorem \ref{thm:query comp}.}
\end{definition}
\begin{proposition}\label{prop:conjunctive RA bag}
Let $q'=(n,k_1,\ldots,k_n,C,W,P,P_0)$ be an agnostic SQL query, let $Q(R_1,\ldots,R_n)$ be its set-theoretic evaluation, and let $Q'(R_1,\ldots,R_n)$ be its bag-theoretic evaluation as in Definition \ref{def:SQL bag set}. Let $\mcS\To{F}\mcT\From{G}\mcU$ be the categorical setup for $q$. Suppose that for each $1\leq i\leq n$ we are given a relation $R_i\ss \prod_{1\leq j\leq k_i}DOM$. Let $I\taking\mcS\to\Set$ be the corresponding functor (i.e., for each $1\leq i\leq n$ let $I(i)=R_i$, let $I(dom)=DOM$, and let $c_{i,j}\taking R_i\to DOM$ be the projection).
Then the categorical analogue $CA(q)(I)=\Delta_G\Pi_F(I)$ is equal to the bag-theoretic evaluation $Q'(R_1,\ldots,R_n)$, and the relationalization $REL(CA(q)(I))$ (see Definiton \ref{def:relationalize}) is equal to the set-theoretic evaluation $Q(R_1,\ldots,R_n)$.
\end{proposition}
\begin{proof}
The first claim follows from Construction \ref{const:Pi}. The second claim follows directly, since both sides are simply images.
\end{proof}
\subsection{Query composition}
In this section and those that follow we were greatly inspired by, and follow closely, the work of [9]. While their work does not apply directly, the adaptation to our context is fairly straightforward.
\begin{lemma}[Unit and counit for $\Sigma$ are Cartesian]\label{lemma:cartesian}
Let $F\taking C\to D$ be a discrete op-fibration, and let
$$\eta\taking\id_{C\inst}\to\Delta_F\Sigma_F\hsp\text{and}\hsp \epsilon\taking\Sigma_F\Delta_F\to\id_{D\inst}$$
be (respectively) the unit and counit of the $(\Sigma_F,\Delta_F)$ adjunction. Each is a Cartesian natural transformation. In other words, for any morphism $a\taking I\to I'$ of $C$-instances and for any morphism $b\taking J\to J'$ of $D$-instances, each of the induced naturality squares (left below for the unit and right below for the counit)
\parbox{1in}{
\xymatrix@=30pt{I\ar[r]^{a}\ar[d]_{\eta_I}\ullimit&I'\ar[d]^{\eta_{I'}}\\
\Delta_F\Sigma_F(I)\ar[r]_{\Delta_F\Sigma_F(a)}&\Delta_F\Sigma_F(I')}
\hspace{1in}
\parbox{1in}{
\xymatrix@=30pt{
\Sigma_F\Delta_F(J)\ar[r]^{\Sigma_F\Delta_F(b)}\ar[d]_{\epsilon_J}\ullimit&\Sigma_F\Delta_F(J')\ar[d]^{\epsilon_{J'}}\\
is a pullback in $C\inst$ and $D\inst$ respectively.
\end{lemma}
\begin{proof}
It suffices to check that for an arbitrary object $c\in\Ob(C)$ and $d\in\Ob(D)$ respectively, the induced commutative diagram in $\Set$
\parbox{1in}{
\xymatrix@=35pt{I(c)\ar[r]^{a_c}\ar[d]_{(\eta_I)_c}&I'(c)\ar[d]^{(\eta_{I'})_c}\\
\Delta_F\Sigma_F(I)(c)\ar[r]_{\Delta_F\Sigma_F(a)_c}&\Delta_F\Sigma_F(I')(c)}
\hspace{.8in}
\parbox{1in}{
\xymatrix@=35pt{
\Sigma_F\Delta_F(J)(d)\ar[r]^{\Sigma_F\Delta_F(b)_d}\ar[d]_{(\epsilon_J)_d}\ullimit&\Sigma_F\Delta_F(J')(d)\ar[d]^{(\epsilon_{J'})_d}\\
is a pullback. Unpacking definitions, these are:
\parbox{2.7in}{
\xymatrix@=30pt{I(c)\ar[r]^{a_c}\ar[d]_{c'=c\;\;}&I'(c)\ar[d]^{\;\;c'=c}\\
\parbox{.7in}{\vspace{-.1in}$$\coprod_{\{c'\|F(c')=F(c)\}}$$\vspace{-.2in}}I(c')\ar[r]_{\amalg a_{c'}}&\parbox{.7in}{\vspace{-.1in}$$\coprod_{\{c'\|F(c')=F(c)\}}$$\vspace{-.2in}}I'(c')}
\hspace{.8in}
\parbox{2.2in}{
\xymatrix@=30pt{
\parbox{.6in}{$$\coprod_{\{c\|F(c)=d\}}$$\vspace{-.1in}}J(d)\ar[r]^{\amalg b_d}\ar[d]&\parbox{.6in}{$$\coprod_{\{c\|F(c)=d\}}$$\vspace{-.1in}}J'(d)\ar[d]\\
Roughly, the fact that these are pullbacks squares follows from the way coproducts work (e.g. the disjointness property) in $\Set$, or more precisely it follows from the fact that $\Set$ is a positive coherent category \cite[p. 34]{Johnstone:MR1953060}.
\end{proof}
\begin{definition}[Grothendieck construction]
Let $C$ be a signature and $I\taking C\to\Set$ an instance. The {\em Grothendieck category of elements for $I$ over $C$} consists of a pair $(\int_CI,\pi_I)$, where $\int_CI$ is a signature and $\pi_I\taking\int_CI\to C$ is a signature morphism, constructed as follows. The set of nodes in $\int_CI$ is $\{(c,x)\|c\in\Ob(C), x\in I(c)\}$. The set of edges in $\int_CI$ from node $(c,x)$ to node $(c',x')$ is $\{e\taking c \to c'\|I(e)(x)=x'\}$. The signature morphism $\pi_I\taking\int_CI\to C$ is obvious: send $(c,x)$ to $c$ and send $e$ to $e$. Two paths are equivalent in $\int_CI$ if and only if their images under $\pi_I$ are equivalent. We sometimes denote $\int_C$ simply as $\int$.
\end{definition}
\begin{lemma}\label{lemma:grothendieck is dop}
Let $I\taking C\to\Set$ be an instance. Then $\pi_I\taking\int I\to C$ is a discrete op-fibration.
\end{lemma}
\begin{proof}
This follows by Lemma \ref{lemma:discrete op-fibrations}.
\end{proof}
\begin{lemma}[Acyclicity and finiteness are preserved under the Grothendieck construction]
Suppose that $C$ is an acyclic (respectively, a finite) signature and that $I\taking C\to\Set$ is a finite instance. Then $\int I$ is an acyclic (respectively, a finite) signature.
\end{lemma}
\begin{proof}
Both are obvious by construction.
\end{proof}
\begin{definition}[DeGrothendieckification]
Let $\pi\taking X\to C$ be a discrete op-fibration. Let $\singleton^X$ denote a terminal object in $X\inst$, i.e. any instance in which every node table and edge table consists of precisely one row. Define {\em the deGrothendieckification of $\pi$}, denoted $\partial\pi\taking C\to\Set$ to be $\partial\pi:=\Sigma_\pi\left(\singleton^X\right)\in C\inst$.
\end{definition}
One checks that for a discrete op-fibration $\pi\taking X\to C$ and object $c\in\Ob(C)$ we have the formula $$\partial\pi(c)=\pi^\m1(c),$$ so we can say that deGrothendieckification is given by pre-image.
\begin{lemma}[Finiteness is preserved under DeGrothendiekification]
If $X$ is a finite signature and $\pi\taking X\to C$ is any discrete op-fibration, then $\partial\pi$ is a finite $C$-instance.
\end{lemma}
\begin{proof}
\end{proof}
\begin{proposition}\label{prop:on dopf}
Given a signature $C$, let $Dopf_C\ss\Cat_{/C}$ denote the full category spanned by the discrete op-fibrations over $C$. Then $\int\taking C\inst\to Dopf_C$ and $\partial\taking Dopf_C\to C\inst$ are functorial, $\partial$ is left adjoint to $\int$, and they are mutually inverse equivalences of categories.
\end{proposition}
\begin{proof}
See \cite[Lemma 2.3.4, Proposition 3.2.5]{Spivak:1202.2591}.
\end{proof}
\begin{corollary}
Suppose that $C$ and $D$ are categories, that $F,G\taking C\to D$ are discrete op-fibrations, and $\alpha\taking F\to G$ is a natural transformation. Then there exists a functor $p\taking C\to C$ such that $F\circ p=G$, and $\Sigma_\alpha=\partial(p)$, where $\Sigma_\alpha\taking\Sigma_G\to\Sigma_F$ is the natural transformation given in Proposition \ref{prop:dmf under nt}.
\end{corollary}
\begin{proof}
By Proposition \ref{prop:dmf under nt} the natural transformation $\alpha\taking F\to G$ induces a natural transformation $\Sigma_G\to\Sigma_F$, and applying deGrothendieckification supplies a map $\partial G\to\partial F$. By Proposition \ref{prop:on dopf} this induces a map $p\taking C\to C$ over $D$ with the above properties.
\end{proof}
\begin{proposition}\label{prop:functor between dopfs is dopf}
Given a commutative diagram
\xymatrix@=10pt{A\ar[rr]^f\ar[rdd]_h&&B\ar[ldd]^g\\\\&C}
in which $g$ and $h$ are discrete op-fibrations, it follows that $f$ is also a discrete op-fibration.
\end{proposition}
\begin{proof}
Consider the diagram to the left below
\xymatrix{\vect{0}\ar[r]^a\ar[d]_i&A\ar[d]^f\\\vect{1}\ar[r]_q\ar@{=}[d]&B\ar[d]^g\\\vect{1}\ar[r]_{gq}&C}
\hspace{.8in}
\xymatrix{\vect{0}\ar[r]^a\ar[d]_i&A\ar[d]^h\\\vect{1}\ar[ur]^{\ell}\ar[r]_{gq}&C}
\hspace{.8in}
\xymatrix@=30pt{\vect{0}\ar[r]^{fa}\ar[d]_i&B\ar[d]^g\\\vect{1}\ar@<.5ex>[ur]^q\ar@<-.5ex>[ur]_{f\ell}\ar[r]_{gq}&C}
Then since $h$ is a discrete op-fibration there exists a unique $\ell\taking \vect{1}\to A$ making the middle diagram commute. But now we have two lifts, $q$ and $f\ell$, and since $g$ is a discrete op-fibration, it follows that $q=f\ell$, completing the proof.
\end{proof}
\begin{proposition}[Discrete op-fibrations are stable under pullback]\label{prop:dop and pullback}
Let $F\taking C'\to C$ be a functor and $\pi\taking X\to C$ be a discrete op-fibration. Then given the pullback square
\xymatrix{C'\cross_CX\ar[r]\ar[d]_{\pi'}\ullimit&X\ar[d]^\pi\\C'\ar[r]_F&C}
the map $\pi'$ is a discrete op-fibration, and there is a natural isomorphism $$\pi'\iso\int_{C'}\Delta_F(\partial\pi).$$
\end{proposition}
\begin{proof}
A functor $p\taking Y\to D$ is a discrete op-fibration if and only there exists a functor $d\taking D\to\Set$ such that the diagram
\xymatrix{Y\ar[r]\ar[d]_{p}\ullimit&\Set_*\ar[d]\\D\ar[r]_d&\Set}
is a pullback square, where $\Set_*$ is the category of pointed sets as in Example \ref{ex:slice}. The result follows from the pasting lemma for fiber products, and Lemma \ref{lemma:grothendieck is dop}.
\end{proof}
\begin{lemma}[Comparison morphisms for squares]\label{lemma:comparison morphism}
Suppose given the following diagram of categories:
Then there are natural transformations of functors $S\inst\to T\inst$:
If $hf=ge$ and $\alpha=\id$ (i.e. if the diagram commutes) then, by symmetry, there are natural transformations of functors $T\inst\to S\inst$:
\end{lemma}
\begin{proof}
These arise from units and counits, together with Proposition \ref{prop:dmf under nt}.
\begin{align*}
\Sigma_f\Delta_e\To{\eta_g}\Sigma_f\Delta_e\Delta_g\Sigma_g\To{\Delta_\alpha}\Sigma_f\Delta_f\Delta_h\Sigma_g\To{\epsilon_f}\Delta_h\Sigma_g\\
\Delta_g\Pi_h\To{\eta_e}\Pi_e\Delta_e\Delta_g\Pi_h\To{\Delta_\alpha}\Pi_e\Delta_f\Delta_h\Pi_h\To{\epsilon_h}\Pi_e\Delta_f.
\end{align*}
The second claim is symmetric to the first if $\alpha=\id$.
\end{proof}
\begin{definition}
Suppose given a diagram of the form
We say it is {\em exact} if the comparison morphism $\Sigma_f\Delta_e\to\Delta_h\Sigma_g$ is an isomorphism. Note that this is the case if and only if the comparison morphism $\Delta_g\Pi_h\to\Pi_e\Delta_f$ is an isomorphism.
\end{definition}
%\begin{proposition}\label{prop:complexity of dopf}
%The complexity of checking that a functor is a discrete op-fibration is a combination of the complexity of checking equivalence of categories and the complexity of the word problem for categories (which is r.e.).
%Any equivalence of categories is a discrete op-fibration, so one containment is trivial. Suppose that $F\taking C\to D$ is a signature morphism. Let $C'$ be the signature with the same objects and arrows as $C$ and with all the path equivalences in $C$, plus a path equivalence $p\sim p'$ whenever $F(p)\sim F(p')$ in $D$. Then $F$ factors as a composition $$\xymatrix{C\ar[r]^{i}\ar@/_1pc/[rr]_F&C'\ar[r]^{F'}&D}$$ and $F$ is a discrete op-fibration if and only if $i\taking C\to C'$ is an equivalence of categories. %Thus the complexity of checking that $F$ is a discrete op-fibration is the sum of the complexity of checking equivalence of categories and the complexity of the word problem, which is r.e.
\begin{proposition}[Comma Beck-Chevalley for $\Sigma,\Delta$]\label{prop:BC Sigma}
Let $F\taking C\to D$ and $G\taking E\to D$ be functors, and consider the canonical natural transformation diagram (see Definition \ref{def:comma})
\begin{align}\label{dia:comma sigma delta exact}
\xymatrix{(F\down G)\ar[r]^q\ar[d]_p\ar@{}[dr]|{\stackrel{\alpha}{\Nearrow}}&E\ar[d]^G\\C\ar[r]_F&D}
\end{align}
Let $\Sigma_F$ be the generalized left push-forward as defined in Remark \ref{rmk:generalized sigma}. Then the comparison morphism (Lemma \ref{lemma:comparison morphism}) is an isomorphism
of functors $C\inst\to E\inst$. In other words, (\ref{dia:comma sigma delta exact}) is exact.
\end{proposition}
\begin{proof}
The map is given by the composition
\xymatrix{\Sigma_q\Delta_p\ar[r]^-{\eta_F}&\Sigma_q\Delta_p\Delta_F\Sigma_F\ar[r]^{\alpha}&\Sigma_q\Delta_q\Delta_G\Sigma_F\ar[r]^-{\epsilon_q}&\Delta_G\Sigma_F}
To prove it is an isomorphism, one checks it on each object $e\in\Ob(E)$ by proving that the diagonal map $(F\down Ge)\to(q\down e)$ in the diagram
\xymatrix{
(F\down Ge)\ar@/^1pc/[rrd]\ar[rd]\ar@/_1pc/[rdd]\\
&(q\down e)\ar[r]\ar[d]\ar@{}[rd]|{\Nearrow}&[0]\ar[d]^e\\
&(F\down G)\ar@{}[dr]|{\Nearrow}\ar[d]_p\ar[r]^q&E\ar[d]^G\\
is a final functor, and this is easily checked.
\end{proof}
\begin{corollary}[Comma Beck-Chevalley for $\Delta,\Pi$]\label{cor:BC Pi}
Let $F\taking C\to D$ and $G\taking E\to D$ be functors, and consider the canonical natural transformation diagram (\ref{dia:nt for comma})
\xymatrix{(F\down G)\ar[r]^q\ar[d]_p\ar@{}[dr]|{\stackrel{\alpha}{\Nearrow}}&E\ar[d]^G\\C\ar[r]_F&D}
Then the comparison morphism (Lemma \ref{lemma:comparison morphism}) is an isomorphism $$\Delta_F\Pi_G\To{\iso}\Pi_p\Delta_q$$ of functors $E\inst\to C\inst$.
\end{corollary}
\begin{proof}
This follows from Proposition \ref{prop:BC Sigma} by adjointness.
\end{proof}
Sometimes when we push forward an instance $I$ along a functor $B\To{G}C$, the resulting instance $\Sigma_GI$ agrees with $I$, at least on some subcategory $A\ss B$. This is very useful to know, because it gives an easy way to calculate row sets and name rows in the pushforward. The following lemma roughly says that this occurs if for all $b\in\Ob(B)$ and $a\in\Ob(A)$, the sense to which $b$ is to the left of $a$ in $C$, is the same as the sense to which $b$ is to the left of $a$ in $B$.
\begin{lemma}
Suppose that we are given a diagram of the form
\xymatrix{A\ar[r]^q\ar@{=}[d]&B\ar[d]^G\\A\ar[r]_F&C}
Then the map $\beta\taking\Delta_q\to\Delta_F\Sigma_G$ is an isomorphism if, for all objects $a\in\Ob(A)$ the functor
$$\ol{G}\taking(\id_B\down qa)\to (G\down Fa)$$
is final.
\end{lemma}
\begin{proof}
For typographical convenience, let $[0]$ denote the terminal category, usually denoted $\vect{0}$. To check that $\beta$ is an isomorphism, we may choose an arbitrary object $a\taking[0]\to A$ and check that $\Delta_a\beta$ is an isomorphism. Consider the diagram
\xymatrix{
(G\down Fa)\ar@/^1pc/[rrrd]^v\ar@/_.5pc/[rdd]_u\\
&(\id_A\down a)\ar[r]^-p\ar[d]_t\ar[ul]_{G'}\ar@{}[dr]|{\Swarrow}&A\ar@{=}[d]\ar[r]^(.4)q&B\ar[d]^G\\
The map $\beta$ is the composite
and this is equivalent to showing that $\beta'$ is an isomorphism. It suffices to show that $\Sigma_t\to\Sigma_u\Delta_v$ is an isomorphism, but this is equivalent to the assertion that $G'$ is a final functor. One can factor $G'$ as
(\id_A\down a)\To{\ol{q}}(\id_B\down qa)\To{\ol{G}}(G\down Fa)
and the first of these is easily checked to be final. Thus the composite is final if and only if the latter map is final, as desired.
\end{proof}
\begin{proposition}[Pullback Beck-Chevalley for $\Sigma,\Delta$ when $\Sigma$ is along a discrete op-fibration]\label{prop:pb beck sigma}
Suppose that $F\taking D\to C$ is a functor and $p\taking X\to C$ is a discrete op-fibration, and form the pullback square
\begin{align}\label{dia:pullback sigma delta exact}
\xymatrix{Y\ullimit\ar[r]^G\ar[d]_{q}&X\ar[d]^p\\D\ar[r]_F&C}
\end{align}
The comparison morphism (Lemma \ref{lemma:comparison morphism}) is a natural isomorphism $$\Sigma_q\Delta_G\To{\iso}\Delta_F\Sigma_p$$ of functors $X\inst\to D\inst$. In other words, (\ref{dia:pullback sigma delta exact}) is exact.
\end{proposition}
\begin{proof}
Consider the morphism given in Lemma \ref{lemma:comparison morphism}.
Since $\eta_p$ and $\epsilon_q$ are Cartesian (by Lemma \ref{lemma:cartesian}), it suffices to check the claim on the terminal object of $X\inst$, where it follows by Proposition \ref{prop:dop and pullback}.
\end{proof}
\begin{corollary}[Pullback Beck-Chevalley for $\Delta,\Pi$ when $\Delta$ is along a discrete op-fibration]\label{cor:pb bc DP dop}
Suppose that $F\taking D\to C$ is a functor and $p\taking X\to C$ is a discrete op-fibration, and form the pullback square
\begin{align}\label{dia:pullback delta pi exact}
\xymatrix{Y\ullimit\ar[r]^G\ar[d]_{q}&X\ar[d]^p\\D\ar[r]_F&C}
\end{align}
The comparison morphism (Lemma \ref{lemma:comparison morphism}) is an isomorphism $$\Delta_p\Pi_F\To{\iso}\Pi_G\Delta_q$$ of functors $D\inst\to X\inst$. In other words, (\ref{dia:pullback delta pi exact}) is exact.
\end{corollary}
\begin{proof}
This follows from Proposition \ref{prop:pb beck sigma} by adjointness.
\end{proof}
\begin{proposition}[Distributive law]\label{prop:distributive}
Let $u\taking C\to B$ be a discrete op-fibration and $f\taking B\to A$ any functor. Construct the {\em distributivity diagram}
\begin{align}\label{dia:dd}
\xymatrix@=15pt{&N\ullimit\ar[ld]_e\ar[rr]^g\ar[dd]^w&&M\ar[dd]^v\\
\end{align}
as follows. Form $v\taking M\to A$ by $v:=\int_A\Pi_f(\partial u)$, form $w\taking N\to B$ by $w:=\int_B\Delta_f\Pi_f\partial u$. Finally let $e\taking N\to C$ be given by $\int_B\epsilon_f(\partial u)$, where $\epsilon_f\taking\Delta_f\Pi_f\to\id_B$ is the counit. Note that $N\iso B\cross_AM$ by Proposition \ref{prop:dop and pullback} and that one has $w=u\circ e$ by construction.
Then there is a natural isomorphism
\begin{align}\label{dia:Pi past Sigma}
\Sigma_v\Pi_g\Delta_e\To{\iso}\Pi_f\Sigma_u
\end{align}
of functors $C\inst\to A\inst$.
\end{proposition}
\begin{proof}
We have that $u,v,$ and $w$ are discrete op-fibrations. By Corollary \ref{cor:pb bc DP dop} we have the following chain of natural transformations
\begin{align*}
\Sigma_v\Pi_g\Delta_e\To{\eta_u}\Sigma_v\Pi_g\Delta_e\Delta_u\Sigma_u=\Sigma_v\Pi_g\Delta_w\Sigma_u\To{\iso}\Sigma_v\Delta_v\Pi_f\Sigma_u\To{\epsilon_v}\Pi_f\Sigma_u.
\end{align*}
Every natural transformation in the chain is Cartesian, so it suffices to check that the composite is an isomorphism when applied to the terminal object $\singleton^C$ in $C\inst$. But there the composition is simply the identity transformation on $\Pi_f(\partial u)$, proving the result.
\end{proof}
\begin{remark}\label{rem:interpretation of distributive law}
The distributive law above takes on a much simpler form when we realize that any discrete opfibration over $B$ is the category of elements of some functor $I\taking B\to\Set$. We have an equivalence of categories $\int(I)\set\iso B\set_{/I}$. Thinking about the distributive law in these terms is quite helpful.
Let $f\taking B\to A$ be a functor and consider the functor
\footnote{The quotes around $``\Pi_f"$ indicate that this is not actually a right Kan extension, but that $``\Pi_f"$ is a suggestive name for the functor we indicate.}$$
given by $(J\to I)\mapsto(\Pi_fJ\to\Pi_fI)$. Let $u\taking\int I\to B$ and $v\taking\int(\Pi_fI)\to A$ be the canonical projections. Form the distributivity diagram
\int\Delta_f\Pi_fI\ullimit\ar[d]_e\ar[r]^g\ar[d]_e&\int\Pi_fI\ar[dd]^v\\
\int I\ar[d]_u&\\
Then the following diagram of categories commutes:
(\int I)\set\ar[r]_{\Pi_g\Delta_e}\ar[d]_{\Sigma_u}&(\int\Pi_fI)\set\ar[d]^{\Sigma_v}\\
where the vertical composites are the ``forgetful" functors. In this diagram, the big rectangle clearly commutes. The distributive law precisely says that the bottom square commutes. Deducing that the top square commutes, we realize that the rather opaque looking $\Pi_g\Delta_e$ is quite a simple functor.
\end{remark}
\begin{lemma}\label{lemma:lots of pullbacks}
Suppose given a pullback square.
For any functor $\Gamma\taking T\to\Set$, every square in the following diagram is a pullback:
\int\Delta_g\Pi_g\Gamma\ar[dd]\ar[rr]&&\int\Pi_g\Gamma\ar[dddd]\\
\int\Gamma\ar[dd]\\
\end{lemma}
\begin{proof}
While the format of this diagram contains a couple copies of the distributivity diagram, this result has nothing to do with that one. Indeed it follows from Proposition \ref{prop:dop and pullback} and Corollary \ref{cor:pb bc DP dop}. The bottom square is a pullback by hypothesis. The lower left-hand square, the front square, and the back square are each pullbacks by the proposition. It now follows that the top square is a pullback. The right-hand square is a pullback by a combination of the proposition and the corollary. It now follows that the upper left-hand square is a pullback.
\end{proof}
\begin{example}[Why $\Sigma$-restrictedness appears to be necessary for composability of queries]\label{ex:why dist needs dopf}
Without the condition that $u$ be a discrete op-fibration, Proposition \ref{prop:distributive} does not hold. Indeed, let $C=\Loop$ as in (\ref{dia:loop}) and let $B=A=\vect{0}$ be the terminal category. One finds that $N=M=\vect{0}$ too in (\ref{dia:dd}), and $e$ is the unique functor. If one considers $C$-sets as special kinds of graphs (discrete dynamical systems) then we can say that $\Pi_f\Sigma_u$ will extract the set of connected components for a $C$-set, whereas $\Sigma_v\Pi_g\Delta_e$ will extract its set of vertices.
We may still wish to ask ``whether a $\Pi$ can be pushed past a $\Sigma$", i.e. whether for any $C\To{u}B\To{f}A$, some appropriate $v,g,e$ exist such that an isomorphism as in (\ref{dia:Pi past Sigma}) holds. The following example may help give intuition. Let $C\To{u}B\To{f}A$ be given as follows
\parbox{.75in}{
\fbox{\xymatrix{&\LMO{e}\ar@/_.5pc/[d]\ar@/^.5pc/[d]\\\LMO{r}\ar@/^.5pc/[r]\ar@/_.5pc/[r]&\LMO{v}}}}
\xymatrix{~\ar[r]^u&~}
\parbox{.75in}{
\fbox{\xymatrix{&\color{white}{\LMO{e}}\\\LMO{r}\ar@/^.5pc/[r]\ar@/_.5pc/[r]&\LMO{v}}}}
\xymatrix{~\ar[r]^f&~}
\parbox{.75in}{
\fbox{\xymatrix{&\color{white}{\LMO{e}}\\\color{white}{\LMO{r}}&\LMO{v}}}}
where $u(e)=v$ and $f(r)=v$. The goal is to find some $C\From{e}N\To{g}M\To{v}A$ such that isomorphism (\ref{dia:Pi past Sigma}) holds. This does not appear possible if $M$ and $N$ are assumed finitely presentable.
\end{example}
\begin{theorem}[Query composition]\label{thm:query comp}
Suppose that one has $\Sigma$-restricted data migration queries $Q\taking S\queryto T$ and $Q'\taking T\queryto U$ as follows:
\begin{align}\label{dia:composable queries}
\xymatrix{&B\ar[r]^f\ar[dl]_s&A\ar[dr]^t&&&D\ar[r]^g\ar[dl]_u&C\ar[dr]^v\\
\end{align}
Then there exists a $\Sigma$-restricted data migration query $Q''\taking S\queryto U$ such that $\church{Q''}\iso\church{Q'}\circ\church{Q}$.
\end{theorem}
\begin{proof}
We follow the proof in [9], as we have been throughout this section. We will form $Q''$ by constructing the following diagram, which we will explain step by step:
\begin{align}\label{dia:query comp}
\xymatrix{
\end{align}
First, form (i) by taking the pullback; note that $k$ is a discrete op-fibration by Proposition \ref{prop:dop and pullback}. Second, form (ii) as a distributivity diagram (see Proposition \ref{prop:distributive}) and note that $w$ is a discrete op-fibration. Third, form (iii) and (iv) with $B'=(f\down h)$ and $N=(r\down e)$.
By Proposition \ref{prop:pb beck sigma} we have $\Delta_u\Sigma_t\iso\Sigma_k\Delta_h$. By Proposition \ref{prop:distributive} we have $\Pi_g\Sigma_k\iso\Sigma_w\Pi_q\Delta_e$. By Corollary \ref{cor:BC Pi} we have both $\Delta_h\Pi_f\iso\Pi_r\Delta_m$ and $\Delta_e\Pi_r\iso\Pi_p\Delta_n$. Pulling this all together, we have an isomorphism
\begin{align*}
\Sigma_v\Pi_g\Delta_u\Sigma_t\Pi_f\Delta_s
\iso\Sigma_v\Pi_g\Sigma_k\Delta_h\Pi_f\Delta_s
\end{align*}
which proves that $\church{Q''}\iso\church{Q'}\circ\church{Q}$, where $Q''$ is the data migration functor given by the triple of morphisms $Q':=(s\circ m\circ n, q\circ p, v\circ w)$, i.e. $\church{Q}=\Sigma_{vw}\Pi_{qp}\Delta_{smn}.$ This completes the proof.
\end{proof}
\begin{corollary}\label{cor:query comp for Delta-restricted}
Suppose that $Q\taking S\queryto T$ and $Q'\taking T\queryto U$ are queries as in (\ref{dia:composable queries}) and that both are $(\Delta,\Sigma)$-restricted. Then there exists a $(\Delta,\Sigma)$-restricted data migration query $Q''\taking S\queryto U$ such that $\church{Q''}\iso\church{Q'}\circ\church{Q}$.
\end{corollary}
\begin{proof}
We form a diagram similar to (\ref{dia:query comp}), as follows. First, form (i) by taking the pullback; note that $k$ is a discrete op-fibration by Proposition \ref{prop:dop and pullback}. Second, form (ii) as a distributivity diagram (see Proposition \ref{prop:distributive}) and note that $w$ is a discrete op-fibration. Third, form (iii) and (iv) as pullbacks. Note that $h$, $m$, and $n$ will be discrete opfibrations. Following the proof of Theorem \ref{thm:query comp}, the result follows by Corollary \ref{cor:pb bc DP dop} in place of Corollary \ref{cor:BC Pi}.
\end{proof}
\subsection{Syntactic characterization of query morphism and query equivalence}
The present section is not used in the main part of the paper. In an older version of this paper, this section contained a theorem, which we later realized is incorrect. We record the theorem as Remark \ref{rem:syntactic query morphism}, which explains the mistake and the counterexample. The ideas in this section were adapted from [9]; however, the notion of tensorial strength used throughout [9] does not seem to have a useful analogue in our setting.
\begin{definition}[Morphism of data migration queries]\label{def:morphism of dmqs}
A {\em morphism of $(\Delta,\Sigma)$-restricted data migration queries from $P=(u,g,v)$ to $Q=(s,f,t)$}, denoted $(c,a,\beta)\taking P\to Q$, consists of two functors $c,a$ and a natural transformation $\beta$, fitting into a diagram of the form
\begin{align}\label{dia:morphism of polys}
\xymatrix@=35pt{
\end{align}
where blue arrows $(u,v,s,t,c)$ are required to be discrete op-fibrations and every square commutes except for the top middle square in which $\beta\taking f'\to ga$ is a natural transformation.
\end{definition}
Note that for any diagram of the form (\ref{dia:morphism of polys}), the maps $a$ and $b$ will automatically be discrete op-fibrations too.
\begin{definition}[Composition of data migration queries]
Suppose given data migration queries $P,Q,R\taking S\queryto T$ and morphisms $P\to Q$ and $Q\to R$ as in Definition \ref{dia:morphism of polys}. Then they can be composed to give a query morphism $P\to R$, as follows.
We begin the following diagram
\xymatrix{
By Proposition \ref{prop:dmf under nt} there is a canonical map $f'\cross_Ce\to (ga)\cross_Ce$, and this allows us to construct a composite
\xymatrix{
\end{definition}
\begin{remark}
Note that this composition is associative {\em up to isomorphism}, but not on the nose. Thus we cannot speak of the category of data migration queries $P\to Q$, but only the $(\infty,1)$-category of such.
\end{remark}
The following definition is slightly abbreviated, i.e. not fully spelled out, but hopefully it is clear to anyone who is following so far.
\begin{definition}[Category of data migration queries]\label{def:cat of dmq}
We define the {\em category of $(\Delta,\Sigma)$-restricted data migration queries from $S$ to $T$}, denoted $\RQry(S,T)$, to be the category whose objects are $(\Delta,\Sigma)$-restricted data migration queries $S\queryto T$ and whose morphisms are equivalence classes of diagrams as in (\ref{dia:morphism of polys}), where two such diagrams are equivalent if there are equivalences of categories between corresponding objects in their middle rows and the functors commute appropriately.
\end{definition}
\begin{lemma}
Given a morphism of $(\Delta,\Sigma)$-restricted data migration queries $P\to Q$, there is an induced natural transformation $\church{P}\to\church{Q}$.
\end{lemma}
\begin{proof}
By Proposition \ref{prop:dmf under nt} and Corollary \ref{cor:pb bc DP dop} we have the natural natural transformations below:
\begin{align}\label{dia:poly morphism to nt}
\Sigma_v\Pi_g\Delta_u=\Sigma_t\Sigma_c\Pi_g\Delta_u\To{\eta_a}\;\;&\Sigma_t\Sigma_c\Pi_g\Pi_a\Delta_a\Delta_u\\
\nonumber\To{\;\beta\;}\;\;&\Sigma_t\Sigma_c\Pi_{f'}\Delta_a\Delta_u\\
\nonumber=\;\;\;&\Sigma_t\Sigma_c\Pi_{f'}\Delta_b\Delta_s\\
\nonumber\iso\;\;\;&\Sigma_t\Sigma_c\Delta_c\Pi_f\Delta_s\To{e_c}\Sigma_t\Pi_f\Delta_s
\end{align}
\end{proof}
\begin{lemma}\label{lemma:cartesian over poly is poly}
Let $Q\taking S\queryto T$ be a $(\Delta,\Sigma)$-restricted data migration query, and let $P\taking S\inst\to T\inst$ be any functor. If there exists a Cartesian natural transformation $\phi\taking P\to\church{Q}$, then there is an essentially unique $(\Delta,\Sigma)$-restricted data migration query isomorphic to $P$.
\end{lemma}
\begin{proof}
Suppose that $Q$ is represented by the bottom row in the diagram below
\begin{align}\label{dia:cartesian map of polys}
\xymatrix{Q':&S\ar@{=}[d]&B\cross_AC\ar[l]_{s'}\ar[r]^{f'}\ar[d]_b\ullimit&C\ar[r]^{t'}\ar[d]_c&T\ar@{=}[d]\\
\end{align}
We form the rest of the diagram as follows. Let $\ast$ be the terminal object in $S\inst$, so in particular $\int_TQ(\ast)\iso t$. Let $t'=\int_TP(\ast)$ and define $c\taking t'\to t$ over $T$ to be $\int_T\phi(\ast)$. Let $b$ be the pullback of $c$ and $s'=s\circ b$. The Diagram (\ref{dia:cartesian map of polys}) now formed, let $Q'\taking S\queryto T$ be the top row. Note that $c$, and therefore $b$ and $s$, are discrete op-fibrations, so in particular $Q'$ is a $(\Delta,\Sigma)$-restricted data migration query. Note also that there was essentially no choice in the definition of $Q'$.
The map $\church{Q'}\to\church{Q}$ is given by
\Sigma_{t'}\Pi_{f'}\Delta_{s'}=\Sigma_{t}\Sigma_{c}\Pi_{f'}\Delta_b\Delta_s\To{\iso}\Sigma_t\Sigma_c\Delta_c\Pi_f\Delta_s\To{\eta_c}\Sigma_t\Pi_f\Delta_s,
so it is Cartesian by Lemma \ref{lemma:cartesian}. But if both $P$ and $Q'$ are Cartesian over $Q$ and if they agree on the terminal object (as they do by construction), then they are isomorphic. Thus $P\iso Q'$ is a $(\Delta,\Sigma)$-restricted data migration query.
\end{proof}
\begin{lemma}[Yoneda Functorialization]\label{lemma:yoneda}
Let $C$ be a category and let $b\taking C\to\Set$ be a functor with $s\taking B\to C$ its Grothendieck category of elements. Then there is a natural isomorphism of functors
where $t\taking I\to\vect{0}$ is the unique functor. Moreover, for any $b'\taking C\to\Set$ with $s'\taking B'\to C$ its category of elements, there is a bijection
\end{lemma}
\begin{proof}
The second claim follows from the first by Proposition \ref{prop:on dopf} and the usual Yoneda-style argument. For the first, we have \begin{align*}
\Pi_u\Pi_s\Delta_s(-)\iso\Hom_\Set(\ast,\Pi_u\Pi_s\Delta_s(-))\iso\Hom_{B\set}(\ast^B,\Delta_s(-))&\iso\Hom_{C\set}(\Sigma_s(\ast^B),-)\\&\iso\Hom_{C\set}(b,-).
\end{align*}
\end{proof}
\begin{lemma}\label{lemma:crucial}
Suppose given the diagram to the left, where $u$ and $u'$ are discrete op-fibrations:
\xymatrix@=30pt{
\hspace{1in}
\xymatrix@=30pt{
and a natural transformation $j\taking\Pi_g\Delta_u\to\Pi_{g'}\Delta_{u'}$. Then there exists an essentially unique diagram as to the right, such that $j$ is the composition (see Proposition \ref{prop:dmf under nt}),
\begin{align}\label{dia:composite for j}
\Pi_g\Delta_u\To{\eta_a}\Pi_g\Pi_a\Delta_a\Delta_u\To{\beta}\Pi_{g'}\Delta_a\Delta_u\iso\Pi_{g'}\Delta_{u'}.
\end{align}
\end{lemma}
\begin{proof}
Choose an object $c\taking\vect{0}\to C$ in $C$. Form the diagram
\xymatrix{
&D\ar[ld]_u\ar[dr]^g&&(c\down g)\ar[ll]_p\ar[rd]^q\\
&X\ar[ul]^{u'}\ar[ur]_{g'}&&(c\down g')\ar[ll]^{p'}\ar[ur]_{q'}}
Let $s=u\circ p$ and $s'=u'\circ p'$, and let $t\taking S\to\vect{0}$ denote the unique functor. It is easy to show that $p$ and $p'$, and hence $s$ and $s'$ are discrete op-fibrations. We have a natural transformation
\begin{align*}
\Pi_t\Pi_s\Delta_s=\Pi_q\Delta_s=\Pi_q\Delta_p\Delta_u\;\iso\;&\Delta_c\Pi_g\Delta_u\\
\To{j}&\Delta_c\Pi_{g'}\Delta_{u'}\iso\Pi_{q'}\Delta_{p'}\Delta_{u'}=\Pi_{q'}\Delta_{s'}=\Pi_t\Pi_{s'}\Delta_{s'}.
\end{align*}
By Lemma \ref{lemma:yoneda} this is equivalent to giving a map $j_c\taking(c\down g')\to (c\down g)$ over $S$. Moreover, given a map $f\taking c_0\to c_1$ in $C$, the natural transformation $\Delta_f\taking\Delta_{c_0}\to\Delta_{c_1}$ (see Proposition \ref{prop:dmf under nt}) induces a commutative diagram of functors $S\to\Set$, as to the left
\xymatrix{\Delta_{c_1}\Pi_{g'}\Delta_{u'}&\Delta_{c_1}\Pi_g\Delta_u\ar[l]\\\Delta_{c_0}\Pi_{g'}\Delta_{u'}\ar[u]&\Delta_{c_0}\Pi_g\Delta_u\ar[u]\ar[l]}
\hspace{1in}
\xymatrix{(c_1\down g')\ar[r]^{j_{c_1}}\ar[d]_f&(c_1\down g)\ar[d]^f\\(c_0\down g')\ar[r]_{j_{c_0}}&(c_0\down g)}
which induces a commutative diagram of categories over $S$ as to the right.
We are now in a position to construct a unique functor $a\taking X\to D$ with $u'=u\circ a$ and natural transformation $\beta\taking g'\to g\circ a$ agreeing with each $j_c$. Given an object $x\in\Ob(X)$, we apply the map $j_{g'x}\taking (g'x\down g')\to(g'x\down g)$ to $(x,\id_{g'x})$ to get some $(d,g'x\To{b} gd)$. We assign $a(x):=d$ and $\beta_x:=b$, and note that $u'(x)=u(d)$ as above. A similar argument applies to morphisms, so $a$ and $\beta$ are constructed. To see that $j$ is the composite given in (\ref{dia:composite for j}), one checks it on objects and arrows in $C$ using the formula from Construction \ref{const:Pi} in conjunction with Proposition \ref{prop:limits are pushforwards}.
\end{proof}
\begin{remark}\label{rem:syntactic query morphism}
A previous version of this paper contained the following statement as a theorem, which we later realized was incorrect:
\begin{quote}
Suppose that $P,Q\taking S\queryto T$ are $(\Delta,\Sigma)$-restricted data migration queries, and $X\taking\church{P}\to\church{Q}$ is a natural transformation of functors. Then there exists an essentially unique morphism of $(\Delta,\Sigma)$-restricted data migration queries $x\taking P\to Q$ (i.e. a diagram of the form (\ref{dia:morphism of polys})) such that $\church{x}\iso X$. In other words, the functor $\RQry(S,T)\to\Fun(S,T)$ is fully faithful.
\end{quote}
The proof was as follows:
\begin{proof}
Let $*$ denote the terminal object in $S\inst$. We define a functor $\ol{P}\taking S\inst\to T\inst$ as follows. For $I\taking S\to\Set$, define $\ol{P}(I)$ as the fiber product
\xymatrix{\ol{P}(I)\ar[r]\ar[d]\ullimit&\church{Q}(I)\ar[d]\\\church{P}(*)\ar[r]_{X(\ast)}&\church{Q}(*).}
There is a induced natural transformation $i\taking\church{P}\to\ol{P}$ and an induced Cartesian natural transformation $X'\taking \ol{P}\to\church{Q}$, such that $X=X'\circ i$.
Let $v\taking C\to T$ and $t\taking A\to T$ denote the category of elements $v:=\int_T\church{P}(*)=\int_T\ol{P}(*)$ and $t:=\int_T\church{Q}(*)$ respectively, and let $c:=\int_TX(*)\taking v\to t$ over $T$. If $P$ and $Q$ are the top and bottom rows in the diagram below, then Lemma \ref{lemma:cartesian over poly is poly} and in particular Diagram (\ref{dia:cartesian map of polys}) implies that there is an essentially unique way to fill out the middle row, and its maps down to the bottom row, as follows:
with $s'=s\circ b$ and $\church{P'}=\ol{P}$.
{\color{red}Lemma \ref{lemma:cartesian} implies that $i\taking\Sigma_v\Pi_g\Delta_u\to\Sigma_v\Pi_{f'}\Delta_{s'}$ induces a natural transformation $j\taking\Pi_g\Delta_u\to\Pi_{f'}\Delta_{s'}$ with $\Sigma_vj=i$.} The result follows by Lemma \ref{lemma:crucial}.
\end{proof}
\noindent This proof is incorrect in its application (second-to-last sentence) of Lemma \ref{lemma:cartesian}.
A counterexample to the theorem statement can be found for which every category involved is discrete. Let $P$ and $Q$ be given by the top and bottom line below:
Here $P$ represents the constant functor $\church{P}(x,y)=\ul{2}$ and $Q$ represents the polynomial $\church{Q}(x,y)=1+y$, where $x,y\in\Ob(\Set)$. There is a natural transformation $\church{P}\to\church{Q}$ given by $\ul{2}\To{!}\ul{1}\To{inl}1+y$. However, it cannot be realized by a morphism of polynomials in the sense of Definition \ref{def:morphism of dmqs} because there is no functor $\ul{1}\to\ul{0}$.
\end{remark}
%\begin{corollary}\label{cor:query iso}
%Let $f, g\taking S \queryto T$ be $(\Delta,\Sigma)$-restricted data migration queries. Then the complexity of $\church{f}\iso\church{g}$ is at most the complexity of testing equivalence of categories.
%By Remark \ref{rem:syntactic query morphism} proving that $\church{f}\iso\church{g}$ is equivalent to proving that $f\iso g$, and by Definition \ref{def:cat of dmq}, we need only check that certain functors are equivalences of categories.
%\subsection{Finite approximation}
%Recall the schema $$\Loop:=\LoopSchema$$ and for any $n\in\NN$, let $\Loop_n$ denote the result of quotienting $\Loop$ by $f^{n+1}=f^n$. Then we have an infinite chain $$\cdots\To{p_{n+1}}\Loop_{n+1}\To{p_n}\Loop_n\To{p_{n-1}}\cdots\To{p_2}\Loop_2\To{p_1}\Loop_1\To{p_0}\Loop_0,$$ and $\Loop_0\iso\fbox{$\bullet$}.$ In fact $\Loop$ is the limit of this diagram of categories. Let $F_n\taking\Loop\to\Loop_n$ so $p_{n}\circ F_{n+1}=F_n$ for all $n\in\NN$.
%Given a finitely presented category $\mcD$, consider the set $\Hom_\Cat(\Loop,\mcD)$. There is an evaluation map $$ev\taking\Hom_\Cat(\Loop,\mcD)\cross\Loop\too\mcD.$$ For $N\in\NN$, let $D_N$ denote the pushout in $\Cat$,
%$$\xymatrix{\Hom_\Cat(\Loop,\mcD)\cross\Loop\ar[r]^-{ev}\ar[d]_{F_N}&\mcD\ar[d]^{G_N}\\\Hom_\Cat(\Loop,\mcD)\cross\Loop_N\ar[r]&\mcD_N}$$ There are maps $q_N\taking D_{N+1}\to D_N$, defined in the obvious way.
%\begin{corollary}[Decidability of Morphism]
%Let $f, g\taking S \queryto T$ be data migration queries. Then we can decide if there is a natural transformation $f\to g$. The complexity of this is equivalent to the complexity of finding a functor and a natural transformation.
%This again follows from Remark \ref{rem:syntactic query morphism}.
\section{Typed signatures}
\subsection{Definitions and data migration}
\begin{definition}
A {\em typed signature} $\ol{\mcC}$ is a sequence $\ol{\mcC}:=(\mcC,\mcC_0,i,\Gamma)$ where $\mcC$ is a signature, $\mcC_0$ is a discrete category,
\footnote{In fact, one does not need to assume that $\mcC_0$ is a discrete category for the following results to hold. Still, it is conceptually simpler. To get a feeling for what could be done if $\mcC_0$ were not discrete, consider the possibility of a table having two data columns, an attribute $A$ of type string and an attribute $B$ of type integer, where the integer in $B$ was the {\em length} of the string in $A$.}
$i\taking\mcC_0\to\mcC$ is a functor, and $\Gamma\taking\mcC_0\to\Set$ is a functor,
\mcC_0\ar[rr]^i\ar[dr]_\Gamma&&\mcC\\
The signature $\mcC$ is called the {\em structure part} of $\ol{\mcC}$, and the rest is called the {\em typing setup}.
Suppose $(\mcC',\mcC_0,'i',\Gamma')$ is another typed signature. A {\em typed signature morphism from $\ol{\mcC}$ to $\ol{\mcC'}$}, denoted $\ol{F}=(F,F_0)\taking\ol{\mcC}\to\ol{\mcC'}$, consists of a functor $F\taking\mcC\to\mcC'$ and a functor $F_0\taking\mcC_0\to\mcC'_0$ such that the following diagram commutes:
\mcC_0\ar[rr]^i\ar[dr]_\Gamma\ar[dd]_{F_0}&&\mcC\ar[dd]^F\\
\mcC_0'\ar[ur]^{\Gamma'}\ar[rr]_{i'}&&\mcC'
The category of typed signatures is denoted $\TSig$.
A {\em $\ol{\mcC}$-instance} $\ol{I}$ is a pair $\ol{I}:=(I,\delta)$ where $I\taking\mcC\to\Set$ is a functor, called the {\em structure part of $\ol{I}$} together with a natural transformation $\delta\taking I\circ i\to\Gamma$, called the {\em data part of $\ol{I}$}.
\mcC_0\ar[rr]^i\ar[dr]_\Gamma&\ar@{}[d]|(.4){\stackrel{\delta}{\Leftarrow}}&\mcC\ar[dl]^I\\
Suppose $\ol{I'}:=(I',\delta')$ is another $\ol{\mcC}$-instance. An {\em instance morphism from $\ol{I}$ to $\ol{I'}$}, denoted $\alpha\taking\ol{I}\to\ol{I'}$, is a natural transformation $\alpha\taking I\to I'$ such that $\delta'\circ\alpha=\delta$.
\mcC_0\ar[rr]^i\ar[dr]_\Gamma&\ar@{}[d]|(.4){\stackrel{\delta'}{\Leftarrow}}&\mcC\ar@/^.5pc/[dl]^{I}\ar@{}[dl]|{\stackrel{\alpha}{\Leftarrow}}\ar@/_.5pc/[dl]_{I'}\\
The category of $\ol{\mcC}$-instances is denoted $\ol{\mcC}\inst$.
\end{definition}
\begin{proposition}
Let $\ol{\mcC}$ be a typed signature. Then the category $\ol{\mcC}\inst$ is a topos.
\end{proposition}
\begin{proof}
$\ol{\mcC}\inst$ is equivalent to the slice topos $\mcC\set/\Pi_i\Gamma$.
\end{proof}
\begin{construction}
Let $\ol{\mcC}:=(\mcC,\mcC_0,i,\Gamma)$ be a typed signature. We set up the tables for it as follows. For every arrow in $\mcC$ we make a binary table; we call these {\em arrow tables}. For every object $c\in\Ob(\mcC)$, let $N=i^\m1(c)\ss\mcC_0$. We make an $1+|N|$ column table, where $|N|$ is the cardinality of $N$; we call these {\em node tables}. For each node table, one column is the primary key column and the other $N$ columns are called {\em data columns}. The {\em data type} for each $n\in N$ is the set $\Gamma(n)$.
Let $(I,\delta)$ be an instance of $\ol{\mcC}$. For each object $c\in\Ob(\mcC)$ we fill in the primary key column of the node table with the set $I(c)\in\Ob(\Set)$. For each data column $n\in N$ we have a function $\delta_n\taking I(c)\to\Gamma(n)$, which we use to fill in the data in that column. For every arrow $f\taking c\to c'$ in $\mcC$ we fill in the primary key column of the arrow table with $I(c)$, and we fill in the other column with the function $I(f)\taking I(c)\to I(c')$.
\end{construction}
\begin{example}
If $\mcC_0=\emptyset$ then for any category presentation $\mcC$ there is a unique functor $i\taking\mcC_0\to\mcC$ and a unique functor $\Gamma\taking\mcC_0\to\Set$, so there is a unique schema $\ol{\mcC}=(\mcC,\emptyset,i,\Gamma)$ with structure $\mcC$ and empty domain setup.
Given a functor $I\taking\mcC\to\Set$, there is a unique instance $\ol{I}:=(I,!)\in\ol{\mcC}\inst$ with that structure part. The data part of $\ol{I}$ is empty.
\end{example}
\begin{example}
Let $\mcC=\fbox{$\bullet^X$}$ be a terminal category, and let $\mcC_0=\{\tn{First},\tn{Last}\}$. There is a unique $i\taking\mcC_0\to\mcC$. Let
Thus we have our schema $\ol{\mcC}$.
An instance on $\ol{\mcC}$ consists of a functor $I\taking\mcC\to\Set$ and a natural transformation $\delta\taking I\circ i\to\Gamma$. Let $I(X)=\{1,2\}$, let $\delta_{\tn{First}}(1)=\tt{David}$, let $\delta_{\tn{First}}(2)=\tt{Ryan}$, let $\delta_{\tn{Last}}(1)=\tt{Spivak}$, let $\delta_{\tn{Last}}(2)=\tt{Wisnesky}.$ We display this as
$$\begin{tabular}{| l || l | l |}
\bhline
\multicolumn{3}{|c|}{X}\\\bhline
{\bf ID}&{\bf First}&{\bf Last}\\\bbhline
\bhline
\end{tabular}
\end{example}
\begin{definition}\label{def:relationalize}
Suppose given a typed signature $\ol{\mcC}$ and a typed instance $\ol{I}$,
\mcC_0\ar[rr]^i\ar[dr]_\Gamma&\ar@{}[d]|(.4){\stackrel{\delta}{\Leftarrow}}&\mcC\ar[dl]^I\\
We have a morphism $\delta\taking I\to\Pi_i\Gamma$. We say our instance $\ol{I}$ is {\em relational} if $\delta$ is a monomorphism. We define the {\em relationalization of $\ol{I}$} denoted $REL(\ol{I})$ to be the image $\tn{im}(\delta)\ss\Pi_i\Gamma$.
\end{definition}
\begin{proposition}\label{prop:typed Delta}
Let $\ol{\mcC}=(\mcC,\mcC_0,i,\Gamma)$ and $\ol{\mcC'}=(\mcC',\mcC_0',i',\Gamma')$ be typed signatures, and let $\ol{F}=(F,F_0)\taking\ol{\mcC}\to\ol{\mcC'}$ be a typed signature morphism. Then the pullback functor $\Delta_F\taking\mcC'\inst\to\mcC\inst$ of untyped instances extends to a functor of typed instances
\end{proposition}
\begin{proof}
An object $(I',\delta')\in\Ob(\ol{\mcC'}\inst)$ is drawn to the left; simply compose with $F$ to get the diagram on the right
\xymatrix@=25pt{
\mcC_0\ar[rr]^i\ar[dr]_\Gamma\ar[dd]_{F_0}&&\mcC\ar[dd]^F\\
\mcC_0'\ar[ur]^{\Gamma'}\ar[rr]_{i'}&\ar@{}[u]|{\stackrel{\delta'}{\Leftarrow}}&\mcC'\ar[ul]_{I'}
\hsp
\xymatrix@=25pt{
\mcC_0\ar[rr]^i\ar[dr]_\Gamma\ar[dd]_{F_0}&&\mcC\ar[dd]^F\ar[dl]^{I'\circ F}\\
\mcC_0'\ar[ur]^{\Gamma'}\ar[rr]_{i'}&\ar@{}[u]|(.4){\stackrel{\delta'}{\Leftarrow}}&\mcC'\ar[ul]_{I'}
More formally, by Lemma \ref{lemma:comparison morphism} we have a natural transformation $\Delta_F\Pi_{i'}\to\Pi_i\Delta_{F_0}$, so we set $I=\Delta_FI'$ and we set $\delta'$ to be the composite
\Delta_iI=\Delta_i\Delta_FI'=\Delta_{F_0}\Delta_{i'}I'\To{\Delta_{F_0}\delta'}\Delta_{F_0}\Gamma'=\Gamma
\end{proof}
\begin{remark}\label{rem:interpretation of typed delta}
Suppose given the following diagram:
\begin{align}\label{dia:typed delta helper}
\xymatrix@=25pt{
\mcC_0\ar[rr]^i\ar[dr]_\Gamma\ar[dd]_{F_0}&&\mcC\ar[dd]^F\\
\mcC_0'\ar[ur]^{\Gamma'}\ar[rr]_{i'}&&\mcC'
\end{align}
A $\ol{\mcC'}$-instance is a functor $\int\Pi_{i'}\Gamma'\to\Set$, whereas a $\ol{\mcC}$ instance is a functor $\int\Pi_{i}\Gamma\to\Set$. We can reinterpret the typed-$\Delta$ functor using the following commutative diagram
\int\Pi_i\Gamma\ar[dr]&\int\Delta_F\Pi_{i'}\Gamma'\ar[l]_q\ar[d]\ar[r]^r\ullimit&\int\Pi_{i'}\Gamma'\ar[d]\\
The functor $\Delta_{\ol{F}}$ is given by the composition $\Sigma_q\Delta_r$.
Note that if (the exterior square of) Diagram \ref{dia:typed delta helper} is a pullback square, then $q$ is an isomorphism.
\end{remark}
\begin{definition}
Let $\ol{\mcC}=(\mcC,\mcC_0,i,\Gamma)$ and $\ol{\mcC'}=(\mcC',\mcC_0,i',\Gamma)$ be typed signatures, and let $\ol{F}=(F,F_0)\taking\ol{\mcC}\to\ol{\mcC'}$ be a typed signature morphism. We say that $\ol{F}$ is {\em $\Pi$-ready} if $F_0=\id_{\mcC_0}$.
\end{definition}
\begin{proposition}\label{prop:typed pi}
Let $\ol{\mcC}=(\mcC,\mcC_0,i,\Gamma)$ and $\ol{\mcC'}=(\mcC',\mcC_0,i',\Gamma)$ be typed signatures, and let $\ol{F}=(F,F_0)\taking\ol{\mcC}\to\ol{\mcC'}$ be a typed signature morphism that is $\Pi$-ready. Then the right pushforward functor $\Pi_F\taking\mcC\inst\to\mcC'\inst$ of untyped instances extends to a functor of typed instances
\end{proposition}
\begin{proof}
Given the diagram to the left, we form $\Pi_F(I)\taking\mcC'\to\Set$ to get the diagram on the right:
\xymatrix@=25pt{
\mcC_0\ar[rr]^i\ar[dr]_\Gamma\ar@{=}[dd]&\ar@{}[d]|(.4){\stackrel{\delta}{\Leftarrow}}&\mcC\ar[dd]^F\ar[dl]^I\\
\mcC_0\ar[ur]^{\Gamma}\ar[rr]_{i'}&&\mcC'
\hsp
\xymatrix@=25pt{
\mcC_0\ar[rr]^i\ar[dr]_\Gamma\ar@{=}[dd]&\ar@{}[d]|(.4){\stackrel{\delta}{\Leftarrow}}&\mcC\ar[dd]^F\ar[dl]^I\\
\mcC_0\ar[ur]^{\Gamma}\ar[rr]_{i'}&&\mcC'\ar[ul]_{\Pi_FI}
The morphism $\epsilon\taking\Delta_F\Pi_FI\to I$ is the counit map. We set $\delta'\taking\Delta_{i'}\Pi_FI\to \Gamma$ to be the composite
\end{proof}
\begin{remark}\label{rem:interpretation of typed pi}
A $\ol{\mcC}$-instance is a functor $\int\Pi_i\Gamma\to\Set$, whereas a $\ol{\mcC'}$ instance is a functor $\int\Pi_{i'}\Gamma=\int\Pi_F\Pi_i\Gamma\to\Set$. We can reinterpret the typed-$\Pi$ functor using the following diagram (see Remark \ref{rem:interpretation of distributive law}).
\int\Delta_F\Pi_F\Pi_i\Gamma\ullimit\ar[d]_e\ar[r]^g\ar[d]_e&\int\Pi_F\Pi_i\Gamma\ar[dd]^v\\
\int\Pi_i\Gamma\ar[d]_u&\\
\mcC\ar[r]_F&\mcC'
The functor $\Pi_{\ol{F}}$ is given by
\end{remark}
\begin{definition}
Let $\ol{\mcC}=(\mcC,\mcC_0,i,\Gamma)$ and $\ol{\mcC'}=(\mcC',\mcC_0,i',\Gamma)$ be typed signatures, and let $\ol{F}=(F,F_0)\taking\ol{\mcC}\to\ol{\mcC'}$ be a typed signature morphism. We say that $\ol{F}$ is {\em $\Sigma$-ready} if
\begin{itemize}
\item $p$ is a discrete opfibration, and
\item $C_0=p^{\m1}(C_0')$, i.e. the following is a fiber product of categories,
\end{itemize}
\end{definition}
\begin{proposition}\label{prop:typed sigma}
Let $\ol{\mcC}=(\mcC,\mcC_0,i,\Gamma)$ and $\ol{\mcC'}=(\mcC',\mcC_0',i',\Gamma')$ be typed signatures, and let $\ol{p}=(p,p_0)\taking\ol{\mcC}\to\ol{\mcC'}$ be a typed signature morphism that is $\Sigma$-ready. Then the left pushforward functor $\Sigma_p\taking\mcC\inst\to\mcC'\inst$ of untyped instances extends to a functor of typed instances
\end{proposition}
\begin{proof}
An instance $\delta\taking\Delta_iI\to\Gamma$ is drawn below
\xymatrix@=25pt{
\mcC_0\ar[rr]^i\ar[dr]_\Gamma\ar[dd]_{p_0}&\ar@{}[d]|(.4){\stackrel{\delta}{\Leftarrow}}&\mcC\ar[dd]^p\ar[dl]^I\\
\mcC_0'\ar[ur]^{\Gamma'}\ar[rr]_{i'}&&\mcC'
Let $I'=\Sigma_pI$; we need a morphism $\Delta_{i'}I'\to\Gamma'$. Since $\Gamma\iso\Delta_{p_0}\Gamma'$, we indeed have by Proposition \ref{prop:pb beck sigma}
\Delta_{i'}\Sigma_pI\To{\iso}\Sigma_{p_0}\Delta_iI\To{\Sigma_{p_0}\delta}\Sigma_{p_0}\Delta_{p_0}\Gamma'\To{\epsilon}\Gamma'.
\end{proof}
\begin{remark}\label{rem:interpretation of typed sigma}
A $\ol{\mcC}$-instance is a functor $\int\Pi_i\Gamma\to\Set$, whereas a $\ol{\mcC'}$ instance is a functor $\int\Pi_{i'}\Gamma'\to\Set$. We can reinterpret the typed-$\Sigma$ functor using the following diagram.
\begin{align}\label{dia:typed sigma helper}
\xymatrix{
\int\Pi_i\Gamma\ullimit\ar[d]\ar[r]^{p'}&\int\Pi_{i'}\Gamma'\ar[d]\\
\mcC\ar[r]_p&\mcC'
\end{align}
The above diagram is a fiber product diagram because of the isomorphism (see Corollary \ref{cor:pb bc DP dop}):
\Pi_i\Gamma=\Pi_i\Delta_{p_0}\Gamma'\iso\Delta_p\Pi_{i'}\Gamma'.
At this point we can interpret our typed pushforward using the natural isomorphism
\end{remark}
\begin{definition}\label{def:typed FQL query}
Let $\ol{C}=(S,S_0,i_S,\Gamma_S)$ and $\ol{T}=(T,T_0,i_T,\Gamma_T)$ be typed signatures. A {\em typed FQL query} $Q$ from $\ol{S}$ to $\ol{T}$, denoted $Q\taking\ol{S}\queryto\ol{T}$ is a triple of typed signature morphisms $(\ol{F},\ol{G},\ol{H})$:
\ol{S}\From{\ol{F}}\ol{S'}\To{\ol{G}}\ol{S''}\To{\ol{H}}\ol{T}
such that $\ol{G}$ is $\Pi$-ready and $\ol{H}$ is $\Sigma$-ready.
\end{definition}
\subsection{Typed query composition}
\begin{proposition}[Comparison morphism for typed $\Delta,\Pi$]\label{prop:typed comparison Delta Pi}
Suppose given the following diagram of typed signatures,
\begin{align}\label{dia:compare delta pi}
\xymatrix{
\end{align}
such that all diagrams commute, except that the front square for which we have a natural transformation $\alpha\taking Fq\to pG$:
\xymatrix{Y\ar[r]^-G\ar[d]_q\ar@{}[dr]|{\stackrel{\alpha}{\Nearrow}}&X\ar[d]^p\\D\ar[r]_F&C
Then the comparison transformation for untyped instances $\Delta_{F}\Pi_{p}\to\Pi_{q}\Delta_{G}$, from Lemma \ref{lemma:comparison morphism}, extends to a {\em typed comparison morphism} of typed queries $\ol{X}\inst\to\ol{D}\inst$,
\end{proposition}
\begin{proof}
Suppose given an $\ol{X}$-instance $(I,\delta\taking\Delta_{i_X}I\to\Gamma_C$. The formulas in Propositions \ref{prop:typed Delta} and \ref{prop:typed pi} become
\begin{align}\label{dia:delta-pi 1}
\Delta_{i_D}\Delta_F\Pi_pI=\Delta_{F_0}\Delta_{i_C}\Pi_pI
\To{\epsilon_p}\Delta_{F_0}\Delta_{i_X}I
\To{\delta}\Delta_{F_0}\Gamma_C=\Gamma_D\\\label{dia:delta-pi 2}
\Delta_{i_D}\Pi_q\Delta_GI=\Delta_{i_Y}\Delta_q\Pi_q\Delta_GI
\To{\epsilon_q}\Delta_{i_Y}\Delta_GI
\To{\delta}\Delta_{F_0}\Gamma_C=\Gamma_D
\end{align}
We need to show that the comparison morphism above commutes with these maps to $\Gamma_D$. To see this, consider the following commutative diagram:
\color{blue}{\Delta_{i_D}\Delta_F\Pi_p}\ar@{=}[d]\\
\Delta_{i_Y}\Delta_q\Delta_F\Pi_p\ar[d]_{\alpha}\ar[r]^{\eta_q}&\Delta_{i_Y}\Delta_q\Pi_q\Delta_q\Delta_F\Pi_p\ar[d]^\alpha\\
\Delta_{i_Y}\Delta_G\Delta_p\Pi_p\ar@{=}[d]&\Delta_{i_Y}\Delta_q\Pi_q\Delta_G\Delta_p\Pi_p\ar[l]_{\epsilon_q}\ar[d]^{\epsilon_p}\\
\Delta_{F_0}\Delta_{i_X}\Delta_p\Pi_p\ar[d]_{\epsilon_p}&\Delta_{i_Y}\Delta_q\Pi_q\Delta_G\ar@{=}[r]\ar[d]^{\epsilon_q}&\color{ForestGreen}{\Delta_{i_D}\Pi_q\Delta_G}\\
\color{red}{\Delta_{F_0}\Delta_{i_X}}\ar@{=}[r]&\Delta_{i_Y}\Delta_G
The left-hand composite $\color{blue}{\Delta_{i_D}\Delta_F\Pi_p}\to\color{red}{\Delta_{F_0}\Delta_{i_X}}$ is the first part of (\ref{dia:delta-pi 1}), the lower-right zig-zag $\color{ForestGreen}{\Delta_{i_D}\Pi_q\Delta_G}\to\color{red}{\Delta_{F_0}\Delta_{i_X}}$ is the first part of (\ref{dia:delta-pi 2}), and the map from top to lower-right $\color{blue}{\Delta_{i_D}\Delta_F\Pi_p}\to\color{ForestGreen}{\Delta_{i_D}\Pi_q\Delta_G}$ is $\Delta_{i_D}$ applied to the comparison morphism. The commutativity of the diagram (which follows by the triangle identities and the associativity law) is what we were trying to prove.
\end{proof}
\begin{corollary}[Pullback Beck-Chevalley for typed $\Delta,\Pi$]\label{cor:pullback BC for typed Delta Pi}
Suppose given Diagram (\ref{dia:compare delta pi}) such that the front square is a pullback:
\xymatrix{Y\ar[r]^{G}\ar[d]_{q}\ullimit&X\ar[d]^p\\D\ar[r]_{F}&C
and $p$ is a discrete op-fibration. Then the typed comparison morphism is an isomorphism:
\end{corollary}
\begin{proof}
By Corollary \ref{cor:pb bc DP dop}, the comparison morphism is an isomorphism,
The result follows from Proposition \ref{prop:typed comparison Delta Pi}, where $i_Y\taking D_0\to Y$ is induced by the fact that $Y$ is a fiber product.
\end{proof}
\begin{corollary}[Comma Beck-Chevalley for typed $\Delta, \Pi$]\label{cor:comma BC for typed Delta Pi}
Suppose given Diagram (\ref{dia:compare delta pi}) such that the front square is a comma category:
\xymatrix{Y=(F\down p)\ar[r]^-G\ar[d]_q\ar@{}[dr]|{\stackrel{\alpha}{\Nearrow}}&X\ar[d]^p\\D\ar[r]_F&C}
Then the typed comparison morphism is an isomorphism:
\end{corollary}
\begin{proof}
By Corollary \ref{cor:BC Pi}, the comparison morphism is an isomorphism,
The result follows from Proposition \ref{prop:typed comparison Delta Pi}, where $i_Y\taking D_0\to Y$ is the induced map factoring through the canonical morphism $F\times_Cp\to (F\down p)$.
\end{proof}
\begin{proposition}[Pullback Beck-Chevalley for typed $\Sigma,\Delta$]\label{prop:pullback BC for typed Delta Sigma}
Suppose given the following commutative diagram of typed signatures,
in which $p\taking X\to C$ is a discrete op-fibration, and the four side squares
\xymatrix{Y_0\ar[r]^{i_Y}\ar[d]_{q_0}\ullimit&Y\ar[d]^q\\D_0\ar[r]_{i_D}&D}\hsp
\xymatrix{Y_0\ar[r]^{G_0}\ar[d]_{q_0}\ullimit&X_0\ar[d]^{p_0}\\D_0\ar[r]_{F_0}&C_0}\hsp
\xymatrix{Y\ar[r]^{G}\ar[d]_{q}\ullimit&X\ar[d]^p\\D\ar[r]_{F}&C}\hsp
\xymatrix{X_0\ar[r]^{i_X}\ar[d]_{p_0}\ullimit&X\ar[d]^p\\C_0\ar[r]_{i_C}&C}
are pullbacks. Then there is an isomorphism
of functors $\ol{X}\inst\to\ol{D}\inst$.
\end{proposition}
\begin{proof}
Note that the functors $q,p_0$, and $q_0$ are discrete opfibrations too. Suppose given an $\ol{X}$-instance $(I,\delta\taking\Delta_{i_X}I\to\Gamma_X$. By Proposition \ref{prop:pb beck sigma}, we have a Beck-Chevalley isomorphism $\Sigma_{q}\Delta_{G}\To{\iso}\Delta_{F}\Sigma_{p}$. The formulas in Propositions \ref{prop:typed Delta} and \ref{prop:typed sigma} become
\begin{align}\label{dia:sigma delta thing1}
\Delta_{i_D}\Sigma_q\Delta_GI\To{\iso}\Sigma_{q_0}\Delta_{i_Y}\Delta_GI
&=\Sigma_{q_0}\Delta_{q_0}\Gamma_D\To{\epsilon_{q_0}}\Gamma_D\\\label{dia:sigma delta thing2}
\Delta_{i_D}\Delta_F\Sigma_pI=\Delta_{F_0}\Delta_{i_C}\Sigma_pI
\end{align}
We need to show that the Beck-Chevalley isomorphism commutes with these maps to $\Gamma_D$. It suffices to show that that the following diagram commutes:
\color{red}{\Sigma_{q_0}\Delta_{G_0}\Delta_{i_X}}\ar[r]^-{\eta_{p_0}}\ar@{=}[d]&
\Sigma_{q_0}\Delta_{G_0}\Delta_{p_0}\Sigma_{p_0}\Delta_{i_X}\ar@{=}[r]&
\Sigma_{q_0}\Delta_{q_0}\Delta_{F_0}\Sigma_{p_0}\Delta_{i_X}\ar[r]^{\epsilon_{q_0}}&
\color{blue}{\Delta_{F_0}\Sigma_{p_0}\Delta_{i_X}}\ar[d]^{\eta_p}\\
\Sigma_{q_0}\Delta_{i_Y}\Delta_G\ar[d]_{\eta_q}&&&\Delta_{F_0}\Sigma_{p_0}\Delta_{i_X}\Delta_p\Sigma_p\ar@{=}[d]\\
\Sigma_{q_0}\Delta_{i_Y}\Delta_q\Sigma_q\Delta_G\ar@{=}[d]&&&\Delta_{F_0}\Sigma_{p_0}\Delta_{p_0}\Delta_{i_C}\Sigma_p\ar[d]^-{\epsilon_{p_0}}\\
\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Sigma_q\Delta_G\ar[d]_{\epsilon_{q_0}}&&&\Delta_{F_0}\Delta_{i_C}\Sigma_p\ar@{=}[d]\\
\color{ForestGreen}{\Delta_{i_D}\Sigma_q\Delta_G}\ar[r]_-{\eta_p}&
\Delta_{i_D}\Sigma_q\Delta_G\Delta_p\Sigma_p\ar@{=}[r]&
\Delta_{i_D}\Sigma_q\Delta_q\Delta_F\Sigma_p\ar[r]_-{\epsilon_q}&
\color{orange}{\Delta_{i_D}\Delta_F\Sigma_p}
Indeed, the left-hand composite is the inverse to (\ref{dia:sigma delta thing1}), the right-hand composite is the inverse to (\ref{dia:sigma delta thing2}), the bottom map is the Beck-Chevalley isomorphism, and the top map relates $\Sigma_{q_0}\Delta_{G_0}$ to $\Delta_{F_0}\Sigma_{p_0}$, which completes the comparison between the two maps to $\Gamma_D$ above.
Proving that the above diagram commutes may be much easier than what follows, which is quite unenlightening. It is mainly an exercise in finding something analogous to a least common denominator in the square above. We include it for completeness.
\color{blue}{\Delta_{F_0}\Sigma_{p_0}\Delta_{i_X}}\ar[r]^{\eta_p}&\Delta_{F_0}\Sigma_{p_0}\Delta_{i_X}\Delta_p\Sigma_p\ar@{=}[r]&\Delta_{F_0}\Sigma_{p_0}\Delta_{p_0}\Delta_{i_C}\Sigma_p\ar@/^1pc/[ddr]^{\epsilon_{p_0}}\\
\Sigma_{q_0}\Delta_{q_0}\Delta_{F_0}\Sigma_{p_0}\Delta_{i_X}\ar[u]^{\epsilon_{q_0}}\ar[r]^{\eta_p}&\Sigma_{q_0}\Delta_{q_0}\Delta_{F_0}\Sigma_{p_0}\Delta_{i_X}\Delta_p\Sigma_p\ar[u]_{\epsilon_{q_0}}\ar@{=}[r]&\Sigma_{q_0}\Delta_{q_0}\Delta_{F_0}\Sigma_{p_0}\Delta_{p_0}\Delta_{i_C}\Sigma_p\ar[d]^{\epsilon_{p_0}}\ar[u]_{\epsilon_{q_0}}\\
\color{red}{\Sigma_{q_0}\Delta_{G_0}\Delta_{i_X}}\ar[u]^{\eta_{p_0}}\ar[r]^{\eta_p}\ar@{=}[d]&\Sigma_{q_0}\Delta_{G_0}\Delta_{i_X}\Delta_p\Sigma_p\ar[u]_{\eta_{p_0}}\ar@{=}[d]\ar@{=}[r]&\Sigma_{q_0}\Delta_{q_0}\Delta_{F_0}\Delta_{i_C}\Sigma_p\ar@{=}[d]\ar[r]^{\epsilon_{q_0}}&\Delta_{F_0}\Delta_{i_C}\Sigma_p\ar@{=}[d]\\
\Sigma_{q_0}\Delta_{i_Y}\Delta_G\ar[d]_{\eta_q}\ar[r]^{\eta_p}&\Sigma_{q_0}\Delta_{i_Y}\Delta_G\Delta_p\Sigma_p\ar[d]^{\eta_q}\ar@{=}[r]&\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Delta_F\Sigma_p\ar[r]^{\epsilon_{q_0}}&\color{orange}{\Delta_{i_D}\Delta_F\Sigma_p}\\
\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Sigma_q\Delta_G\ar[d]_{\epsilon_{q_0}}\ar[r]^{\eta_p}&\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Sigma_q\Delta_G\Delta_p\Sigma_p\ar[d]^{\epsilon_{q_0}}\ar@{=}[r]&\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Sigma_q\Delta_q\Delta_F\Sigma_p\ar[u]_{\epsilon_q}\ar[d]^{\epsilon_{q_0}}\\
\color{ForestGreen}{\Delta_{i_D}\Sigma_q\Delta_G}\ar[r]^{\eta_p}&\Delta_{i_D}\Sigma_q\Delta_G\Delta_p\Sigma_p\ar@{=}[r]&\Delta_{i_D}\Sigma_q\Delta_q\Delta_F\Sigma_p\ar@/_1pc/[uur]_{\epsilon_q}
Every square in the above diagram clearly commutes, completing the proof.
\end{proof}
\begin{proposition}[Typed distributivity]\label{prop:typed distributivity}
Suppose given the diagram of typed signatures
\Set&C_0\ar[l]_{\Gamma_C}\ar[r]^{i_C}\ar[d]_{u_0}\ullimit&C\ar[d]^u\\
such that $u$ is a discrete op-fibration and the square is a pullback. Note that $\ol{u}=(u,u_0)$ is $\Sigma$-ready and that $\ol{f}=(f,\id_{B_0})$ is $\Pi$-ready. Construct the {\em typed distributivity diagram}
as follows. First form the distributivity diagram as in Proposition \ref{prop:distributive}, which is the big rectangle to the right. Now take $N_0$ to be the pullback of either the top-left square, the big-left rectangle, or the big outer square --- all are equivalent. The functors $v$ and $e$ are discrete opfibrations. Note that $\ol{g}=(g,\id_{N_0})$ is $\Pi$-ready and $\ol{v}=(v,u_0\circ e_0)$ is $\Sigma$-ready.
Then there is an isomorphism
of functors $\ol{C}\inst\to\ol{A}\inst$.
\end{proposition}
\begin{proof}
By Proposition \ref{prop:distributive} we have a distributivity isomorphism
\begin{align}\label{dia:dist for typed}
\Sigma_v\Pi_g\Delta_e\To{\iso}\Pi_f\Sigma_u
\end{align}
of functors $C\inst\to A\inst$, but we want an isomorphism of functors $\ol{C}\inst\to\ol{A}\inst$.
Suppose given a $\ol{C}$-instance $(I,\delta\taking\Delta_{i_C}I\to\Gamma_C)$. This is equivalent to a natural transformation $\delta\taking I\to\Pi_{i_C}\Gamma_C$ or equivalently a functor $\delta\taking\int\Pi_{i_C}\Gamma_C\to\Set$. The two sides of isomorphism (\ref{dia:dist for typed}) give rise to the same instance $A\to\Set$, but not a priori the same typed instance. A typed $A$-instance is a functor $\int\Pi_f\Pi_{i_B}\Gamma_B\to\Set$. It may be useful to find $\int\Pi_{i_C}\Gamma_C$ (middle vertex of left-hand square) and $\int\Pi_f\Pi_{i_B}\Gamma_B$ (bottom right back vertex) in the diagram below, which we will presently describe:
\begin{align}\label{dia:typed distributivity helper}
\xymatrix@=15pt{
\int\Delta_f\Pi_f\partial u\ar[rrr]^(.6)g\ar[dd]_e&&&\int\Pi_f\partial u\ar[dddd]^(.3)v\\
\end{align}
The front square is the (untyped) distributivity diagram of Proposition \ref{prop:distributive}; note that it is a pullback. The bottom square is our interpretation of $\Pi_{\ol{f}}$ from Remark \ref{rem:interpretation of typed pi}; note that it is a pullback. In the left-hand square we begin by form the little square in the front portion (the part that includes $q,u,u', u''$) as a pullback; see Remark \ref{rem:interpretation of typed sigma}. Form the other little square as a pullback along $e$. Form the diagonal square as another distributivity diagram for $\Pi_f$ of $\partial u'$. Note that we can complete the right-hand square by the functoriality of $\Pi_f$ and that it is a pullback because $\Pi_f$ preserves limits. Finally, form the back square as a pullback. It is now easy to see that all six sides are pullbacks.
Our only task is to show that the top square is a distributivity square. It is already a pullback, so our interest is in the top right corner: we need to show that there is a natural isomorphism
$$\Pi_f\Pi_{i_C}\Gamma_C=\Pi_f\partial u'\iso^?\Pi_g\partial r=\Pi_g\Delta_e\Pi_{i_C}\Gamma_C.$$
The adjunction isomorphism for $\Pi_g\partial r$ says that for any discrete opfibration $X\to\int\Pi_{f}\partial u$ there is an isomorphism
$$\Hom_{/\int\Delta_f\Pi_{f}\partial u}(g^\m1X,\dispInt\Delta_e\Pi_{i_C}\Gamma_C)\iso\Hom_{/\int\Pi_f\partial u}(X,\dispInt\Pi_g\Delta_e\Pi_{i_C}\Gamma_C).$$
We thus need to show that there is a one-to-one correspondence between functors $X\to\dispInt\Pi_f\Pi_{i_C}\Gamma_C)$ over $\int\Pi_f\partial u$ and dotted arrows in the diagram
\int\Delta_e\Pi_{i_C}\Gamma_C\ar[r]_r&\int\Delta_f\Pi_f\partial u\ar[r]_g&\int\Pi_f\partial u
Given a map $X\to\dispInt\Pi_f\Pi_{i_C}\Gamma_C)$, we can pull it back along $g$ to get a map $g^\m1X\to\int\Delta_f\Pi_f\Pi_{i_C}\Gamma_C$ because the top square in (\ref{dia:typed distributivity helper}) is a pullback; this induces the required dotted arrow. Conversely, given a dotted arrow, we obtain the diagram
\xymatrix{
\int\Delta_e\Pi_{i_C}\Gamma_C\ar[r]_r\ar[d]&\int\Delta_f\Pi_f\partial u\ar[r]_g\ar[d]&\int\Pi_f\partial u\ar[d]\\
\int\Pi_{i_C}\Gamma_C\ar[r]_{u'}&B\ar[r]_f&A
and because the diagonal diagram in (\ref{dia:typed distributivity helper}) was constructed as a distributivity diagram, this induces a map $X\to\int\Pi_f\Pi_{i_C}\Gamma_C$ as desired.
Now that we have completed the aforementioned task, we are ready to prove the result. Our goal is as follows. We begin with an instance in the middle of the left square, a functor $\delta\taking\int\Pi_{i_C}\Gamma_C\to\Set$. We are interested in $\Pi_{\ol{f}}\Sigma_{\ol{u}}\delta$ and $\Sigma_{\ol{v}}\Pi_{\ol{g}}\Delta_{\ol{e}}\delta$; we consider them in this order.
By Remark \ref{rem:interpretation of typed sigma}, we have a natural isomorphism
Because $e$ is a discrete opfibration, Remarks \ref{rem:interpretation of typed delta}, \ref{rem:interpretation of typed pi}, and \ref{rem:interpretation of typed sigma}, we have a natural isomorphism
The result now follows from the usual distributive law, Proposition \ref{prop:distributive}.
\end{proof}
\begin{theorem}
Suppose that one has typed FQL queries $Q\taking\ol{S}\queryto\ol{T}$ and $Q'\taking\ol{T}\queryto \ol{U}$ as follows:
\begin{align}
\xymatrix{&\ol{B}\ar[r]^f\ar[dl]_s&\ol{A}\ar[dr]^t&&&\ol{D}\ar[r]^g\ar[dl]_u&\ol{C}\ar[dr]^v\\
\ol{S}\ar@{}[rrr]|Q&&&\ol{T}&\ol{T}\ar@{}[rrr]|{Q'}&&&\ol{U}}
\end{align}
Then there exists a typed FQL query $Q''\taking\ol{S}\queryto\ol{U}$ such that $\church{Q''}\iso\church{Q'}\circ\church{Q}$.
\end{theorem}
\begin{proof}
We will form $Q''$ by constructing the following diagram, which we will explain step by step:
\begin{align}
\xymatrix{
\ol{S}&&&&\ol{T}&&&\ol{U}}
\end{align}
We form the pullback $(i)$; note that since $\ol{t}$ was $\Sigma$-ready, so is $\ol{k}$. Now form the typed distributivity diagram $(ii)$ as in Proposition \ref{prop:typed distributivity}; note that $\ol{w}$ is $\Sigma$-ready and $\ol{q}$ is $\Pi$-ready. Now form the typed comma categories $(iii)$ and $(iv)$ as in Proposition \ref{prop:typed comparison Delta Pi}. Note that $\ol{r}$ and $\ol{p}$ are $\Pi$-ready. The result now follows from Proposition \ref{prop:pullback BC for typed Delta Sigma}, \ref{prop:typed distributivity}, and Corollary \ref{cor:comma BC for typed Delta Pi},
\end{proof}
\end{document}
|
arxiv-papers
| 2012-12-20T23:58:26 |
2024-09-04T02:49:39.533193
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "David I. Spivak and Ryan Wisnesky",
"submitter": "Ryan Wisnesky",
"url": "https://arxiv.org/abs/1212.5303"
}
|
1212.5324
|
# Hypercontractive inequalities via SOS,
with an application to Vertex-Cover
Manuel Kauers Research Institute for Symbolic Computation, Johannes Kepler
Universität. Supported by FWF grant Y464-N18. Ryan O’Donnell Department of
Computer Science, Carnegie Mellon University. Supported by NSF grants
CCF-0747250 and CCF-1116594, a Sloan fellowship, and a grant from the MSR–CMU
Center for Computational Thinking. Li-Yang Tan Department of Computer Science,
Columbia University. Research done while visiting CMU. Yuan Zhou† Also
supported by a grant from the Simons Foundation (Award Number 252545).
###### Abstract
Our main result is a formulation and proof of the reverse hypercontractive
inequality in the sum-of-squares (SOS) proof system. As a consequence, we show
that for any constant $\gamma>0$, the $O(1/\gamma)$-round SOS/Lasserre SDP
hierarchy certifies the statement
“$\mathsf{Min}\text{-}\mathsf{Vertex\text{-}Cover}(G^{n}_{\gamma})\geq(1-o_{n}(1))|V|$”,
where $G^{n}_{\gamma}=(V,E)$ is the “Frankl–Rödl graph” with $V=\\{0,1\\}^{n}$
and $(x,y)\in E$ whenever $\Delta(x,y)=(1-\gamma)n$. This is despite the fact
that $k$ rounds of various LP and SDP hierarchies fail to certify the
statement
“$\mathsf{Min}\text{-}\mathsf{Vertex\text{-}Cover}(G^{n}_{\gamma})\geq(\tfrac{1}{2}+\epsilon)|V|$”
once $\gamma=\gamma(k,\epsilon)>0$ is small enough. Finally, we also give an
SOS proof of (a generalization of) the sharp $(2,q)$-hypercontractive
inequality for any even integer $q$.
## 1 Introduction
Hypercontractive inequalities play an important role in analysis of Boolean
functions. They are concerned with the _noise operator_ $T_{\rho}$ which acts
on functions $f:\\{-1,1\\}^{n}\to\mathbbm{R}$ via $T_{\rho}f(x)=\mathop{\bf
E\/}[f(\boldsymbol{y})]$, where $\boldsymbol{y}$ is a “$\rho$-correlated copy”
of $x$. Equivalently,
$T_{\rho}f=\sum_{S\subseteq[n]}\rho^{|S|}\widehat{f}(S)\chi_{S}$, where the
numbers $\widehat{f}(S)$ are the Fourier coefficients of $f$. The standard
hypercontractivity inequality was first proved by Bonami [Bon70] and the
reverse hypercontractivity inequality was first proved by Borell [Bor82]. We
state both, recalling the notation $\|f\|_{p}=\mathop{\bf
E\/}_{{\boldsymbol{x}}\sim\\{-1,1\\}^{n}}[|f({\boldsymbol{x}})|^{p}]^{1/p}$.
###### Hypercontractive Inequality.
Let $f:\\{-1,1\\}^{n}\to\mathbbm{R}$, let $1\leq p\leq q\leq\infty$, and let
$0\leq\rho\leq\sqrt{(p-1)/(q-1)}$. Then $\|T_{\rho}f\|_{q}\leq\|f\|_{p}$.
###### Reverse Hypercontractive Inequality.
Let $f:\\{-1,1\\}^{n}\to\mathbbm{R}^{\geq 0}$, let $-\infty\leq q\leq p\leq
1$, and let $0\leq\rho\leq\sqrt{(1-p)/(1-q)}$. Then
$\|T_{\rho}f\|_{q}\geq\|f\|_{p}$.
The hypercontractive inequality is almost always used with either $p=2$ or
$q=2$. The $(2,4)$-hypercontractivity inequality — i.e., the case $q=4$,
$p=2$, $\rho=1/\sqrt{3}$ — is a particularly useful case, as is the following
easy corollary:
###### Theorem 1.1.
For $k\in\mathbbm{N}$, let $\mathcal{P}^{\leq k}$ be the projection operator
which maps $f:\\{-1,1\\}^{n}\to\mathbbm{R}$ to its low-degree part
$\mathcal{P}^{\leq k}f=\sum_{|S|\leq k}\widehat{f}(S)\chi_{S}$. Then the
$2\rightarrow 4$ operator norm of $\mathcal{P}^{\leq k}$ is at most $3^{k/2}$.
I.e., $\|\mathcal{P}^{\leq k}f\|_{4}\leq 3^{k/2}\|f\|_{2}$.
Theorem 1.1 is known to have a proof which is noticeably simpler than that of
the general hypercontractive inequality [MOO10]. Theorem 1.1 can be used to
prove, e.g., the KKL Theorem [KKL88], the sharp small-set expansion statement
for the $1/3$-noisy hypercube, and the Invariance Principle of [MOO10]. More
generally, the hypercontractivity inequality has the following corollary:
###### Theorem 1.2.
For any $q\geq 2$ and $f:\\{-1,1\\}^{n}\to\mathbbm{R}$ we have
$\|\mathcal{P}^{\leq k}f\|_{q}\leq(q-1)^{k/2}\|f\|_{2}$.
This corollary is often use to control the behavior of low-degree polynomials
of random bits.
Reverse hypercontractivity is perhaps most often used to show that if
$A,B\subseteq\\{-1,1\\}^{n}$ are large sets and
$({\boldsymbol{x}},\boldsymbol{y})$ is a $\rho$-correlated pair of random
strings then there is a substantial chance that ${\boldsymbol{x}}\in A$ and
$\boldsymbol{y}\in B$. This was first deduced in [MOR+06] by deriving the
following consequence of reverse hypercontractivity:
###### Theorem 1.3.
Let $f,g:\\{-1,1\\}^{n}\to\mathbbm{R}^{\geq 0}$, let $0\leq q\leq 1$, and let
$0\leq\rho\leq 1-q$. Then $\mathop{\bf
E\/}[f({\boldsymbol{x}})g(\boldsymbol{y})]\geq\|f\|_{q}\|g\|_{q}$ when
$({\boldsymbol{x}},\boldsymbol{y})$ is a pair of $\rho$-correlated random
strings.
The reverse hypercontractive inequality has been used, e.g., in problems
related to approximability and hardness of approximation [FKO07, She09,
BHM12], and problems in quantitative social choice [MOO10, Mos12a, MOS12b,
Kel12, MR12].
### 1.1 Sum-of-squares proofs of hypercontractive inequalities
The present work is concerned with proving hypercontractive inequalities via
“sums of squares” (SOS); i.e., in the Positivstellensatz proof system
introduced by Grigoriev and Vorobjov [GV01]. A recent work of Barak et al.
[BBH+12] showed that the Khot–Vishnoi [KV05] SDP integrality gap instances for
Unique-Games are actually well-solved by the “$4$-round Lasserre SDP
hierarchy”; equivalently, the “degree-$8$ SOS hierarchy”. This is despite the
fact that they are strong gap instances for superconstantly many rounds of
other weaker SDP hierarchies such as Lovász–Schrijver+ and Sherali–Adams+
[RS09, KS09]. The key to analyzing the optimum value of the Khot–Vishnoi
instances is the hypercontractive inequality, and perhaps the key technical
component of the Barak et al. result is showing that Theorem 1.1 has a
degree-$4$ “SOS proof”. That is, if we treat the each $f(x)$ as a formal
“indeterminate”, then $9^{k}\|f\|_{2}^{4}-\|\mathcal{P}^{\leq k}f\|_{4}^{4}$
is a degree-$4$ polynomial in the $2^{n}$ indeterminates, and Barak et al.
showed that it is a sum of squared polynomials (hence always nonnegative).
The connection between SOS proofs and SDP relaxations for optimization
problems was made independently by Lasserre [Las00] and Parrilo [Par00].
Roughly speaking, if a system of $n$-variate polynomial inequalities can be
refuted within the degree-$d$ SOS proof system of Grigoriev and Vorobjov
[GV01], then this refutation can also be found efficiently by solving a
semidefinite program of size $n^{O(d)}$. (For more details, see e.g. [OZ13].)
The associated “degree-$d$ SOS hierarchy” for approximating optimization
problems is known to be at least as strong as the Lovász–Schrijver+ and
Sherali–Adams+ SDP hierarchies, and the [BBH+12] result shows that it can be
noticeably stronger for the notorious Unique Games problem.
Later, [OZ13] showed that the degree-$4$ SOS hierarchy correctly analyzes the
value of the [DKSV06] instances of Balanced-Separator, which are known to be
superconstant-factor integrality gap instances for superconstantly many rounds
of the “LH SDP hierarchy” [RS09]. It was also shown in [OZ13] that the
degree-$O(1)$ SOS hierarchy certifies the value of the [KV05] instances of
Max-Cut to within factor $.952$, whereas superconstantly many rounds of the
Sherali–Adams+ hierarchy are still off by a factor of $.878$ [RS09, KS09]. The
key to the former result was an SOS proof of the KKL Theorem (relying on
[BBH+12]’s SOS proof of Theorem 1.1); the key to the latter was an SOS proof
of an Invariance Principle variant, which in turned needed an SOS proof
Theorem 1.2. The work [OZ13] was unable to actually obtain Theorem 1.2 with an
SOS proof, but instead obtained a weaker version which sufficed for their
purposes.
Still, the full power of the SOS hierarchy is far from well-understood.
Analyzing what can and cannot be proved with low-degree SOS proofs is
evidently very important; for example, it’s consistent with our current
knowledge that the degree-$4$ SOS hierarchy refutes the Unique-Games
Conjecture, gives a $1.01$-approximation for Uniform Sparsest-Cut, a
$.94$-approximation for Max-Cut, and a $1.4$-approximation for Vertex-Cover.
In particular, hypercontractive inequalities have played a key role in many of
the sophisticated SDP integrality gap instances. Thus it is natural to ask:
Can a sharp version of the hypercontractive inequality be proved in the SOS
proof system? Can any version of the reverse hypercontractive inequality be
proved? As we will see, the latter question is particularly relevant for the
known SDP integrality instances of the Vertex-Cover problem, a basic
optimization task not yet analyzed via SOS.
### 1.2 Our results
The main result in this paper is an SOS proof of the reverse
hypercontractivity Theorem 1.3 for all $q$ equal to the reciprocal of an even
integer. As an application of this, we show that the $O(1)$-degree SOS
hierarchy correctly analyzes the Frankl–Rödl SDP integrality gap instances for
Vertex-Cover for a large range of parameters. Finally, we also give an SOS
proof of the sharp $(2,q)$-hypercontractive inequality for all even integers
$q$; in fact, a version with relaxed moment conditions. We find it interesting
to see that the two powerful hypercontractive inequalities admit proofs as
“elementary” as sum-of-squares proofs. On the other hand, to obtain these
proofs we had to use somewhat elaborate methods, including computer algebra
techniques.
##### The hypercontractive inequality for even integer norms.
As mentioned, Barak et al. [BBH+12] gave an SOS proof of Theorem 1.1, that
$\|\mathcal{P}^{\leq k}f\|_{4}^{4}\leq 9^{k}\|f\|_{2}^{4}$. Although there is
a very easy proof of this theorem “in ZFC” [MOO10], that proof uses the
Cauchy–Schwarz inequality, whose square-roots do not obviously translate into
SOS statements. The SOS proof in [BBH+12] gets around this by proving the
_generalized_ statement $\mathop{\bf E\/}[(\mathcal{P}^{\leq
k}f)^{2}(\mathcal{P}^{\leq k^{\prime}}g)^{2}]\leq 3^{k+k^{\prime}}\mathop{\bf
E\/}[f^{2}]\mathop{\bf E\/}[g^{2}]$, allowing them to replace Cauchy–Schwarz
with $XY\leq\tfrac{1}{2}X^{2}+\tfrac{1}{2}Y^{2}$. In [OZ13] this SOS proof was
very slightly generalized to cover the $(2,4)$-hypercontractive inequality,
$\mathop{\bf E\/}[(T_{\rho}f)^{2}(T_{\rho}g)^{2}]\leq\mathop{\bf
E\/}[f^{2}]\mathop{\bf E\/}[g^{2}]$ for $\rho=1/\sqrt{3}$. That work also gave
an SOS proof of a weakened version of Theorem 1.2 for all even integers $q$,
namely $\|\mathcal{P}^{\leq k}f\|_{q}^{q}\leq q^{O(qk/2)}\|f\|_{2}^{q}$.
(Attention is restricted to even integers $q$ because the
$(2,q)$-hypercontractive inequality cannot even be stated as a polynomial
inequality otherwise.)
In Section 3 we prove the full $(2,q)$-hypercontractive inequality for all
even integers $q$. Our strategy is as follows. First, we give a simple proof
(“in ZFC”) of $(2,q)$-hypercontractivity for all even integers $q$; our proof
works not just for random $\pm 1$ bits but for any random variables satisfying
fairly liberal moment bounds. Indeed, we are not aware of any previous work
showing that such moment bounds are sufficient for hypercontractivity. However
this proof relies on the well-known fact that the hypercontractivity
inequality tensorizes [KS88], which in turn uses the triangle inequality for
the $(q/2)$-norm, an inequality that cannot even be stated in SOS. For our SOS
extension of this result we move to a $(q/2)$-function version of the
statement as in [BBH+12]; this requires a little more work and a very mild use
of computer algebra at the end to prove a binomial sum identity. Our final
theorem is as follows:
###### Theorem 1.4.
(Informal.) Let $s\in\mathbbm{N}^{+}$ and write $q=2s$. Let
$0\leq\rho\leq\frac{1}{\sqrt{q-1}}$. Let
${\boldsymbol{x}}=({\boldsymbol{x}}_{1},\dots,{\boldsymbol{x}}_{n})$ be a
sequence of independent real random variables, with each
${\boldsymbol{x}}_{i}$ satisfying
$\mathop{\bf E\/}[{\boldsymbol{x}}_{i}^{2j-1}]=0,\quad\mathop{\bf
E\/}[{\boldsymbol{x}}_{i}^{2j}]\leq(2s-1)^{j}\tfrac{\tbinom{s}{j}}{\tbinom{2s}{2j}}\quad\text{for
all integers $1\leq j\leq s$};$
further assume that $\mathop{\bf E\/}[{\boldsymbol{x}}_{i}^{2}]=1$ for each
$i$. (Rademachers and standard Gaussians qualify.) Then for functions
$f_{1},\dots,f_{s}:\\{-1,1\\}^{n}\to\mathbbm{R}$ there is an SOS proof of
$\mathop{\bf
E\/}\left[\mathop{{\textstyle\prod}}_{i=1}^{s}(T_{\rho}f_{i}({\boldsymbol{x}}))^{2}\right]\leq\prod_{i=1}^{s}\mathop{\bf
E\/}[f_{i}({\boldsymbol{x}})^{2}].$
As corollaries we have SOS proofs of $\|T_{\rho}f\|_{q}^{q}\leq\|f\|_{2}^{q}$
and $\|\mathcal{P}^{\leq k}f\|_{q}^{q}\leq(q-1)^{qk/2}\|f\|_{2}^{q}$.
##### The reverse hypercontractive inequality.
Giving an SOS proof of this theorem proved to be significantly more difficult;
it is our main result and the source of our application to Vertex-Cover
integrality gaps. The theorem cannot even be stated in the SOS proof system
directly since the $p$-“norms” are not polynomials in the values $f(x)$ for
$p<1$. We turn to the $2$-function version from [MOR+06], Theorem 1.3; if
$q=\frac{1}{2k}$ for some $k\in\mathbbm{N}^{+}$ and if we replace $f$ and $g$
by $f^{2k}$ and $g^{2k}$ then we get a polynomial statement (and we can even
drop the hypothesis that $f$ and $g$ are nonnegative). The resulting theorem
is:
###### Theorem 1.5.
Let $k\in\mathbbm{N}^{+}$ and let $0\leq\rho\leq 1-\frac{1}{2k}$. Then for
functions $f,g:\\{-1,1\\}^{n}\to\mathbbm{R}$ there is an SOS proof of
$\mathop{\bf E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\text{$\rho$-corr'd}\end{subarray}}[f({\boldsymbol{x}})^{2k}g(\boldsymbol{y})^{2k}]\geq\mathop{\bf
E\/}[f]^{2k}\mathop{\bf E\/}[g]^{2k}.$
We prove this result in Section 4. An induction on $n$ easily reduces the
problem to the $n=1$ case; for each $k$, this is an inequality in four real
indeterminates. Then by homogeneity we can further reduce to an inequality in
just two indeterminates. Nevertheless, giving an SOS-proof of this “two-point
inequality” for all $k$ seems to be surprisingly tricky. As an example of the
problem we need to solve (the $k=3$ case), the reader is invited to try the
following puzzle:
“Show that
${\tfrac{11}{24}}\left(1+a\right)^{6}\left(1+b\right)^{6}+{\tfrac{11}{24}}\left(1-a\right)^{6}\left(1-b\right)^{6}+\tfrac{1}{24}\left(1+a\right)^{6}\left(1-b\right)^{6}+\tfrac{1}{24}\left(1-a\right)^{6}\left(1+b\right)^{6}-1$
is a sum of squared polynomials in $a$ and $b$.”
Our solution is presented in Section 4.1. Our high level approach is to employ
a change of variables which reduces the task to proving a sequences of one-
variable real inequalities. This is helpful because every nonnegative
univariate polynomial is SOS; hence we can use any mathematical technique to
verify the one-variable inequalities. We establish the one-variable
inequalities using techniques from computer algebra. Peculiarly, this approach
only works for the specific choice $\rho=1-\frac{1}{2k}$; however the proof
for general $0\leq\rho\leq 1-\frac{1}{2k}$ can be deduced since the two-point
inequality is linear in $\rho$.
#### 1.2.1 Application to integrality gap instances for minimum vertex cover
Along the lines of [BBH+12, OZ13], our SOS proof of the reverse
hypercontractive inequality also has application to integrality gap instances;
specifically, for the Minimum Vertex-Cover problem. Let us briefly recall the
relevant work on Vertex-Cover, and then state our result.
##### Previous work on approximating Vertex-Cover.
We are interested in the problem of finding a $C$-approximate minimum vertex
cover in a graph; i.e., one whose cardinality is at most $C$ times that of the
minimum vertex cover. For $C=2$ the problem is easily solved in linear time
[GJ79, Gavril 1974]; for $C=1.36$ the problem is known to be
$\mathsf{NP}$-hard [DS05]. But for $C=1.5$, say, we know neither polynomial-
time solvability, nor $\mathsf{NP}$-hardness. Based on the nearly $30$-year
lack of progress on the algorithms side, it is reasonable to suspect that
there is no efficient $(2-\epsilon)$-approximation algorithm. Indeed, this is
known to be true assuming the Unique-Games Conjecture [KR08]. However there is
reasonable doubt about the Unique-Games Conjecture [ABS10] and one may seek
alternative evidence of hardness. One fairly strong form of evidence is
showing that various linear programming-based and/or semidefinite programming-
based generic polynomial-time optimization algorithms fail to even
$(2-\epsilon)$-certify minimum vertex covers. Here we are using the following
definition:
###### Definition 1.6.
We say that an algorithm _$C$ -certifies_ the minimum vertex cover in a graph
if, on input a graph $G$, it outputs a number $\alpha$ guaranteed to satisfy
$\alpha\leq\mathsf{Min}\text{-}\mathsf{Vertex\text{-}Cover}(G)\leq C\alpha$.
(This is a strictly easier problem than actually _finding_ $C$-approximate
minimum vertex covers.)
There is a long line of work of this type [KG98, Cha02, ABL02, ABLT06, Tou06,
FO06, STT07a, STT07b, GMT08, GM08, Sch08, CMM09, Tul09, GMPT10, GM10, BCGM11];
see Georgiou’s thesis [Geo10] for a recent survey. Briefly, it is known that
the following algorithms fail to provide $(2-\epsilon)$-certifications for the
minimum vertex cover: $N^{\Omega(1)}$ levels of the Sherali–Adams LP hierarchy
[CMM09] (where $N$ is the number of vertices); $\Omega(\sqrt{\frac{\log
N}{\log\log N}})$ levels of the Lovász–Schrijver+ SDP hierarchy; and, $6$
levels of the Sherali–Adams+ SDP hierarchy [BCGM11]. Further, [BCGM11]
conjectures (based on numerical evidence) that their $6$-level result can be
extended to any constant number of levels. Since the Sherali–Adams+ hierarchy
is stronger than the Sherali–Adams and Lovász–Schrijver+ hierarchies, this
conjecture would subsume the other two mentioned results, at least with
regards to ruling out polynomial-time (constant-level) algorithms.
The _Frankl–Rödl graphs_ , defined as follows, are used to show failure of the
Sherali–Adams+ hierarchy to $(2-\epsilon)$-certify the minimum vertex cover.
As far as we are aware, these are the only known “hard instances” even for the
basic SDP relaxation (for factor $(2-\epsilon)$).
###### Definition 1.7.
Let $n\in\mathbbm{N}$ and let $0\leq\gamma\leq 1$ be such that $(1-\gamma)n$
is an even integer. The _Frankl–Rödl_ graph $G^{n}_{\gamma}$ is the undirected
graph with vertex set $\\{-1,1\\}^{n}$ and edge set
$\\{(x,y):\Delta(x,y)=(1-\gamma)n\\}$, where $\Delta(\cdot,\cdot)$ denotes
Hamming distance.
The following theorem is essentially due to Frankl and Rödl [FR87], though a
few details of the proof are only worked out in [GMPT10].
###### Theorem 1.8.
There exists $\gamma_{0}>0$ such that for all $\gamma\leq\gamma_{0}$ it holds
that
$\mathsf{Min}\text{-}\mathsf{Vertex\text{-}Cover}(G^{n}_{\gamma})\geq(1-n(1-\gamma^{2}/64)^{n})N$,
where $N=2^{n}$ is the number of vertices in the graph. In particular,
$\mathsf{Min}\text{-}\mathsf{Vertex\text{-}Cover}(G^{n}_{\gamma})\geq(1-o_{N}(1))N$
for $.1\sqrt{\frac{\log n}{n}}\leq\gamma\leq\gamma_{0}$.
On the other hand, Benabbas, Chan, Georgiou, and Magen [BCGM11] showed that
for all $\epsilon>0$ there exists $\gamma>0$ such that the level-$6$
Sherali–Adams+ algorithm can only certify that
$\mathsf{Min}\text{-}\mathsf{Vertex\text{-}Cover}(G^{n}_{\gamma})\geq(1/2+\epsilon)N$
(for $n$ sufficiently large). They further conjectured that this remains true
for level-$k$ Sherali–Adams+ assuming $\gamma=\gamma(\epsilon,k)$ is small
enough. Specifically, they suggest fixing $\gamma=\Theta(\sqrt{\frac{\log
n}{n}})$.
##### Our result.
As an application of our SOS proof for the reverse hypercontractive
inequality, we are able to show that the SOS/Lasserre hierarchy _is_ an
effective certification algorithm for the Frankl–Rödl graphs for any constant
$\gamma>0$. Specifically, we show that the $O(1/\gamma)$-degree SOS hierarchy
provides a $(1+o_{n}(1))$-certification for the minimum vertex cover in the
Frankl–Rödl graph $G^{n}_{\gamma}$. In fact, our analysis works even for any
$\gamma>\omega(\frac{1}{\log n})$; whether it continues to hold for $\gamma$
as small as $\Theta(\sqrt{\frac{\log n}{n}})=\Theta(\sqrt{\frac{\log\log
N}{\log N}})$ (the parameter setting suggested in [GMPT10, BCGM11]) remains
open.
To accomplish our goal, we need to show an SOS proof for the Frankl–Rödl
Theorem. A key ingredient in Frankl and Rödl’s original proof is the vertex
isoperimetric inequality on $\\{-1,1\\}^{n}$, due to Harper. The standard
proof of this inequality involves a “shifting” argument which we do not see
how to carry out with SOS. However, it is known that inequalities of this type
can also be proved using the reverse hypercontractive inequality studied in
this paper. In particular, Benabbas, Hatami, and Magen [BHM12] have very
recently proven a “density” variation of the Frankl–Rödl Theorem using the
reverse hypercontractive inequality. We obtain the SOS proof for the
Frankl–Rödl Theorem by combining our SOS proof for the reverse
hypercontractive inequality and an SOS version of the Benabbas–Hatami–Magen
proof; see Section 5.
## 2 Preliminaries
##### The SOS proof system.
We describe the SOS (Positivstellensatz) proof system of Grigoriev and
Vorobjov [GV01] using the notation from the work [OZ13]; for more details,
please see that paper.
###### Definition 2.1.
Let $X=(X_{1},\dots,X_{n})$ be indeterminates, let
$q_{1},\dots,q_{m},r_{1},\dots,r_{m^{\prime}}\in\mathbbm{R}[X]$, and let
$A=\\{q_{1}\geq 0,\dots,q_{m}\geq
0\\}\cup\\{r_{1}=0,\dots,r_{m^{\prime}}=0\\}.$
Given $p\in\mathbbm{R}[X]$ we say that $A$ _SOS-proves_ $p\geq 0$ _with
degree_ $k$, written
$A\quad\vdash_{{k}}\quad p\geq 0,$
whenever
$\exists v_{1},\dots,v_{m^{\prime}}\text{ and SOS
}u_{0},u_{1},\dots,u_{m}\text{ such that }\\\
p=u_{0}+\sum_{i=1}^{m}u_{i}q_{i}+\sum_{j=1}^{m^{\prime}}v_{j}r_{j},\qquad\text{with
}\deg(u_{0}),\deg(u_{i}q_{i}),\deg(v_{j}r_{j})\leq k\ \forall
i\in[m],j\in[m^{\prime}].$
Here we use the abbreviation “$w\in\mathbbm{R}[X]$ is SOS” to mean
$w=s_{1}^{2}+\cdots+s_{t}^{2}$ for some $s_{i}\in\mathbbm{R}[X]$. We say that
$A$ has a _degree- $k$ SOS refutation_ if
$A\quad\vdash_{{k}}\quad-1\geq 0.$
Finally, when $A=\emptyset$ we will sometimes use the shorthand
$\vdash_{{k}}\quad p\geq 0,$
which simply means that $p$ is SOS and $\deg(p)\leq k$.
##### Analysis of boolean functions.
Let us recall some standard notation from the field. We write
${\boldsymbol{x}}\sim\\{-1,1\\}^{n}$ to denote that the string
${\boldsymbol{x}}$ is drawn uniformly at random from $\\{-1,1\\}^{n}$. Given
$f:\\{-1,1\\}^{n}\to\mathbbm{R}$ we sometimes use abbreviations like
$\mathop{\bf E\/}[f]$ for $\mathop{\bf
E\/}_{{\boldsymbol{x}}\sim\\{-1,1\\}^{n}}[f({\boldsymbol{x}})]$. For
$f,g:\\{-1,1\\}^{n}\to\mathbbm{R}$ we also write $\langle
f,g\rangle=\mathop{\bf E\/}[fg]=\mathop{\bf
E\/}_{{\boldsymbol{x}}\sim\\{-1,1\\}^{n}}[f({\boldsymbol{x}})g({\boldsymbol{x}})]$.
For $-1\leq\rho\leq 1$ we say that
$({\boldsymbol{x}},\boldsymbol{y})\sim\\{-1,1\\}^{n}\times\\{-1,1\\}^{n}$ is a
pair of _$\rho$ -correlated random strings_ if the pairs
$({\boldsymbol{x}}_{i},\boldsymbol{y}_{i})$ are independent for $i\in[n]$ and
satisfy $\mathop{\bf E\/}[{\boldsymbol{x}}_{i}]=\mathop{\bf
E\/}[\boldsymbol{y}_{i}]=0$ and $\mathop{\bf
E\/}[{\boldsymbol{x}}_{i}\boldsymbol{y}_{i}]=\rho$. The operator $T_{\rho}$
acts on functions $f:\\{-1,1\\}^{n}\to\mathbbm{R}$ via
$T_{\rho}f(x)=\mathop{\bf E\/}[f(\boldsymbol{y})\mid{\boldsymbol{x}}=x]$,
where $({\boldsymbol{x}},\boldsymbol{y})$ is a pair of $\rho$-correlated
random strings.
##### Simple SOS facts and lemmas.
We will use the following facts and lemmas in our SOS proofs. The first one,
in particular, we use throughout without comment.
###### Lemma 2.2.
$\displaystyle\text{If}\quad A\ \ \vdash_{{k}}\ \ p\geq 0,\quad A^{\prime}\ \
\vdash_{{k^{\prime}}}\ \ p^{\prime}\geq 0,$ $\displaystyle\text{then}\quad
A\cup A^{\prime}\ \ \vdash_{{\max(k,k^{\prime})}}\ \ p+p^{\prime}\geq 0.$
The following fact is a well-known consequence of the Fundamental Theorem of
Algebra.
###### Fact 2.3.
A univariate polynomial $p(x)$ is SOS if it is nonnegative. In other words, we
have
$\vdash_{{\deg(p)}}p(x)\geq 0,$
when $p(x)\geq 0$ for all $x\in\mathbbm{R}$.
It is also well known that for homogeneous polynomials, one can reduce the
number of variables by $1$ by “dehomogenizing” the polynomial, getting an SOS
representation (if there is one), and rehomogenizing it to get an SOS
representation of the original polynomial. Applying this trick to Fact 2.3, we
get:
###### Fact 2.4.
A homogeneous bivariate polynomial $p(x,y)$ is SOS if it is nonnegative.
Here are some additional lemmas:
###### Lemma 2.5.
Let $c\geq 0$ be a constant and $X$ an indeterminate. Then for any
$k\in\mathbbm{N}^{+}$,
$X\geq c\quad\vdash_{{k}}\quad X^{k}\geq c^{k}.$
###### Proof.
This follows because
$X^{k}-c^{k}=(X-c+c)^{k}-c^{k}=\sum_{i=1}^{k}\tbinom{k}{i}c^{k-i}(X-c)^{i}$
and each power $(X-c)^{i}$ is either a square or $(X-c)$ times a square. ∎
###### Lemma 2.6.
For any $k\in\mathbbm{N}^{+}$ we have
$\vdash_{{2k}}\quad\left(\tfrac{X+Y}{2}\right)^{2k}\leq\tfrac{X^{2k}+Y^{2k}}{2}.$
###### Proof.
Since $\tfrac{X^{2k}+Y^{2k}}{2}-\left(\tfrac{X+Y}{2}\right)^{2k}$ is a
degree-$2k$ homogeneous polynomial, the claim follows from Fact 2.4: the
inequality is indeed true by convexity of $t\mapsto t^{k}$. ∎
## 3 The hypercontractive inequality in SOS
As a warmup, we give a simple proof (“in ZFC”) of the $(2,q)$-hypercontractive
inequality $\|T_{\rho}f\|_{q}\leq\|f\|_{q}$ for all even integers $q$, which
implies Theorem 1.2 for all even integers $q$. As mentioned, we do this under
a significantly weakened moment condition:
##### “$\boldsymbol{s}$-Moment Conditions.”
_For a real random variable ${\boldsymbol{x}}_{i}$, the condition is that
$\mathop{\bf E\/}[{\boldsymbol{x}}_{i}^{2}]=1$ and_
$\mathop{\bf E\/}[{\boldsymbol{x}}_{i}^{2j-1}]=0,\quad\mathop{\bf
E\/}[{\boldsymbol{x}}_{i}^{2j}]\leq(2s-1)^{j}\tfrac{\tbinom{s}{j}}{\tbinom{2s}{2j}}\quad\textit{for
all integers $1\leq j\leq s$.}$
Our proof will show that these moment conditions are sharp; none of them can
be relaxed.
###### Remark 3.1.
By converting to factorials and expanding, one verifies that
$(2s-1)^{j}\tfrac{\tbinom{s}{j}}{\tbinom{2s}{2j}}=(2j-1)!!\cdot\prod_{i=1}^{j-1}\frac{2s-1}{2s-(2i+1)}.$
It follows that for each fixed $j\in\mathbbm{N}^{+}$, the quantity decreases
as a function of $s$ (for $s\geq j$) to the limit $(2j-1)!!$, which is the
$(2j)$th moment of a standard Gaussian. This shows that a standard Gaussian
and a uniformly random $\pm 1$ bit both satisfy all of the above moment
conditions.
###### Theorem 3.2.
Let ${\boldsymbol{x}}=({\boldsymbol{x}}_{1},\dots,{\boldsymbol{x}}_{n})$ be a
sequence independent real random variables satisfying the $s$-Moment
Conditions. Let $f:\\{-1,1\\}^{n}\to\mathbbm{R}$, $s\in\mathbbm{N}^{+}$, and
$0\leq\rho\leq\sqrt{1/(2s-1)}$. Then
$\|T_{\rho}f({\boldsymbol{x}})\|_{2s}\leq\|f({\boldsymbol{x}})\|_{2}$.
###### Proof.
It is well-known that the hypercontractive inequality tensorizes [KS88] and so
it suffices to treat the case $n=1$. By homogeneity we may also assume
$\mathop{\bf E\/}[f]=1$; we thus write
$f({\boldsymbol{x}}_{1})=1+\epsilon{\boldsymbol{x}}_{1}$ for some
$\epsilon\in\mathbbm{R}$. Raising both sides of the inequality to the
$(2s)^{\text{th}}$ power and using the odd moment conditions ($\mathop{\bf
E\/}[{\boldsymbol{x}}_{1}^{2j-1}]=0$ for all integers $1\leq j\leq s$), we
have
$\displaystyle\|T_{\rho}f({\boldsymbol{x}}_{1})\|_{2s}^{2s}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{s}{2s\choose{2j}}\rho^{2j}\epsilon^{2j}\mathop{\bf
E\/}[x_{1}^{2j}]$ (1) $\displaystyle\|f({\boldsymbol{x}}_{1})\|_{2}^{2s}$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{s}{s\choose j}\epsilon^{2j}.$ (2)
By the even moment conditions $\mathop{\bf
E\/}[{\boldsymbol{x}}_{1}^{2j}]\leq(2s-1)^{j}{s\choose j}/{{2s}\choose{2j}}$,
each summand in (1) is at most the corresponding term in (2) and the proof is
complete. ∎
By considering $\epsilon\to 0$ in (1) and (2) it is easy to see for each
$j=1,2,\dots,s$ in turn that the associated $s$-moment condition cannot be
further relaxed.
Our SOS extension of this result requires the following lemma:
###### Lemma 3.3.
Let $v$ be an even positive integer and let
$G_{1},\dots,G_{v},H_{1},\dots,H_{v}$ be indeterminates. Then
$\vdash_{{2v}}\quad\prod_{i=1}^{v}G_{i}H_{i}\leq\frac{1}{\tbinom{v}{v/2}}\sum_{\begin{subarray}{c}T\subset[v]\\\
|T|=v/2\end{subarray}}\prod_{i\in T}G_{i}^{2}\prod_{i\in[v]\setminus
T}H_{i}^{2}.$
###### Proof.
The non-SOS proof would be to just apply the AM-GM inequality. For the SOS
proof we first trivially write
$\prod_{i\in[v]}G_{i}H_{i}=\frac{1}{\tbinom{v}{v/2}}\sum_{\begin{subarray}{c}T\subseteq
V\\\ |T|=v/2\end{subarray}}\left(\prod_{i\in T}G_{i}\prod_{i\in[v]\setminus
T}H_{i}\right)\left(\prod_{i\in[v]\setminus T}G_{i}\prod_{i\in
T}H_{i}\right).$
We then apply the fact that
$\vdash_{{2}}XY\leq\tfrac{1}{2}X^{2}+\tfrac{1}{2}Y^{2}$ to each summand to
deduce
$\displaystyle\vdash_{{2v}}\quad\prod_{i=1}^{v}G_{i}H_{i}$
$\displaystyle\leq\frac{1}{2\tbinom{v}{v/2}}\sum_{\begin{subarray}{c}T\subseteq[v]\\\
|T|=v/2\end{subarray}}\prod_{i\in T}G_{i}^{2}\prod_{i\in[t]\setminus
T}H_{i}^{2}+\frac{1}{2\tbinom{v}{v/2}}\sum_{\begin{subarray}{c}T\subseteq[v]\\\
|T|=v/2\end{subarray}}\prod_{i\in[v]\setminus T}G_{i}^{2}\prod_{i\in
T}H_{i}^{2}$
$\displaystyle=\frac{1}{\tbinom{v}{v/2}}\sum_{\begin{subarray}{c}T\subset[v]\\\
|T|=v/2\end{subarray}}\prod_{i\in T}G_{i}^{2}\prod_{i\in[v]\setminus
T}H_{i}^{2}.\qed$
We are now ready to state and prove the full version of Theorem 1.4.
###### Theorem 3.4.
Fix $s\in\mathbbm{N}^{+}$ and write $q=2s$. Let
$0\leq\rho\leq\frac{1}{\sqrt{q-1}}$. Let $n\in\mathbbm{N}$ and for each $1\leq
i\leq s$ and each $S\subseteq[n]$, introduce an indeterminate
$\widehat{f_{i}}(S)$. For each $x=(x_{1},\dots,x_{n})\in\mathbbm{R}^{n}$ we
write
$f_{i}(x)=\sum_{S\subseteq[n]}\widehat{f_{i}}(S)\prod_{j\in S}x_{i},\quad
T_{\rho}f_{i}(x)=\sum_{S\subseteq[n]}\rho^{|S|}\widehat{f_{i}}(S)\prod_{j\in
S}x_{i}.$
Let ${\boldsymbol{x}}=({\boldsymbol{x}}_{1},\dots,{\boldsymbol{x}}_{n})$ be a
sequence independent real random variables satisfying the $s$-Moment
Conditions. Then
$\vdash_{{q}}\quad\mathop{\bf
E\/}\left[\mathop{{\textstyle\prod}}_{i=1}^{s}(T_{\rho}f_{i}({\boldsymbol{x}}))^{2}\right]\leq\prod_{i=1}^{s}\mathop{\bf
E\/}[f_{i}({\boldsymbol{x}})^{2}].$ (3)
###### Proof.
We prove (3) by induction on $n$. The base case, $n=0$, is trivial. For
general $n\geq 1$, we can decompose each $f_{i}(x)$ as
$f_{i}(x_{1},\dots,x_{n})=x_{n}g_{i}(x_{1},\dots,x_{n-1})+h_{i}(x_{1},\dots,x_{n-1}).$
Formally, this means introducing the shorthand
$h_{i}(x_{1},\dots,x_{n-1})=\sum_{S\not\ni n}\widehat{f_{i}}(S)\prod_{j\in
S}x_{i}$, and similarly for $g_{i}$. We also introduce the notation
$\boldsymbol{F}_{i}=f_{i}({\boldsymbol{x}})$,
$\widetilde{\boldsymbol{F}}_{i}=T_{\rho}f_{i}({\boldsymbol{x}})$ for each $i$,
and similarly
$\boldsymbol{G}_{i},\widetilde{\boldsymbol{G}}_{i},\boldsymbol{H}_{i},\widetilde{\boldsymbol{H}}_{i}$.
Note that these latter four do not depend on ${\boldsymbol{x}}_{n}$. By
definition we have
$\widetilde{\boldsymbol{F}}_{i}=\rho{\boldsymbol{x}}_{n}\widetilde{\boldsymbol{G}}_{i}+\widetilde{\boldsymbol{H}}_{i}$.
Using the fact that ${\boldsymbol{x}}_{n}$ is independent of all
$\boldsymbol{G}_{i}$, $\boldsymbol{H}_{i}$ and has zero odd moments, the left-
hand side of (3) can be written as follows:
$\displaystyle\phantom{=}\ \mathop{\bf
E\/}\left[\mathop{{\textstyle\prod}}_{i=1}^{s}\left(\rho^{2}{\boldsymbol{x}}_{n}^{2}\widetilde{\boldsymbol{G}}_{i}^{2}+2\rho{\boldsymbol{x}}_{n}\widetilde{\boldsymbol{G}}_{i}\widetilde{\boldsymbol{H}}_{i}+\widetilde{\boldsymbol{H}}_{i}^{2}\right)\right]$
$\displaystyle=\sum_{\begin{subarray}{c}\text{partitions }\\\ (U,V,W)\text{ of
}[s]\end{subarray}}\rho^{2|U|+|V|}2^{|V|}\mathop{\bf
E\/}\left[{\boldsymbol{x}}_{n}^{2|U|+|V|}\right]\mathop{\bf
E\/}\left[\mathop{{\textstyle\prod}}_{i\in
U}\widetilde{\boldsymbol{G}}_{i}^{2}\mathop{{\textstyle\prod}}_{i\in
V}\widetilde{\boldsymbol{G}}_{i}\widetilde{\boldsymbol{H}}_{i}\mathop{{\textstyle\prod}}_{i\in
W}\widetilde{\boldsymbol{H}}_{i}^{2}\right]$
$\displaystyle=\sum_{u=0}^{s}\sum_{\begin{subarray}{c}v=0\\\ v\text{
even}\end{subarray}}^{s-u}\rho^{2u+v}2^{v}\mathop{\bf
E\/}[{\boldsymbol{x}}_{n}^{2u+v}]\sum_{\begin{subarray}{c}(U,V,W)\\\ |U|=u\\\
|V|=v\end{subarray}}\mathop{\bf E\/}\left[\mathop{{\textstyle\prod}}_{i\in
U}\widetilde{\boldsymbol{G}}_{i}^{2}\mathop{{\textstyle\prod}}_{i\in
W}\widetilde{\boldsymbol{H}}_{i}^{2}\mathop{{\textstyle\prod}}_{i\in
V}\widetilde{\boldsymbol{G}}_{i}\widetilde{\boldsymbol{H}}_{i}\right].$ (4)
We apply Lemma 3.3 to each $\mathop{{\textstyle\prod}}_{i\in
V}\widetilde{\boldsymbol{G}}_{i}\widetilde{\boldsymbol{H}}_{i}$ (notice that
each is multiplied against an SOS polynomial) to obtain
$\displaystyle\vdash_{{q}}\quad\eqref{eqn:insanity}$
$\displaystyle\leq\sum_{u=0}^{s}\sum_{\begin{subarray}{c}v=0\\\ v\text{
even}\end{subarray}}^{s-u}\frac{\rho^{2u+v}2^{v}}{\tbinom{v}{v/2}}\mathop{\bf
E\/}[{\boldsymbol{x}}_{n}^{2u+v}]\sum_{\begin{subarray}{c}(U,V,W)\\\ |U|=u\\\
|V|=v\end{subarray}}\sum_{\begin{subarray}{c}T\subseteq V\\\
|T|=v/2\end{subarray}}\mathop{\bf E\/}\left[\mathop{{\textstyle\prod}}_{i\in
U\cup T}\widetilde{\boldsymbol{G}}_{i}^{2}\mathop{{\textstyle\prod}}_{i\in
W\cup(V\setminus T)}\widetilde{\boldsymbol{H}}_{i}^{2}\right]$
$\displaystyle\leq\sum_{u=0}^{s}\sum_{\begin{subarray}{c}v=0\\\ v\text{
even}\end{subarray}}^{s-u}\frac{2^{v}}{\tbinom{v}{v/2}}\frac{\tbinom{s}{u+v/2}}{\tbinom{2s}{2u+v}}\sum_{\begin{subarray}{c}(U,V,W)\\\
|U|=u\\\ |V|=v\end{subarray}}\sum_{\begin{subarray}{c}T\subseteq V\\\
|T|=v/2\end{subarray}}\mathop{\bf E\/}\left[\mathop{{\textstyle\prod}}_{i\in
U\cup T}\widetilde{\boldsymbol{G}}_{i}^{2}\mathop{{\textstyle\prod}}_{i\in
W\cup(V\setminus T)}\widetilde{\boldsymbol{H}}_{i}^{2}\right]$
$\displaystyle\leq\sum_{u=0}^{s}\sum_{\begin{subarray}{c}v=0\\\ v\text{
even}\end{subarray}}^{s-u}\frac{2^{v}}{\tbinom{v}{v/2}}\frac{\tbinom{s}{u+v/2}}{\tbinom{2s}{2u+v}}\sum_{\begin{subarray}{c}(U,V,W)\\\
|U|=u\\\ |V|=v\end{subarray}}\sum_{\begin{subarray}{c}T\subseteq V\\\
|T|=v/2\end{subarray}}\prod_{i\in U\cup T}\mathop{\bf
E\/}[\boldsymbol{G}_{i}^{2}]\prod_{i\in W\cup(V\setminus T)}\mathop{\bf
E\/}[\boldsymbol{H}_{i}^{2}],$ (5)
where the second inequality uses the $s$-Moments Condition and the bound on
$\rho$, and the third inequality uses the induction hypothesis. (Again, note
that each inequality is multiplied against an SOS polynomial.) It is easy to
check that $\mathop{\bf E\/}[\boldsymbol{F}_{i}^{2}]=\mathop{\bf
E\/}[\boldsymbol{G}_{i}^{2}]+\mathop{\bf E\/}[\boldsymbol{H}_{i}^{2}]$ and so
the right-hand side of (3) is simply
$\sum_{R\subseteq[s]}\prod_{i\in R}\mathop{\bf
E\/}[\boldsymbol{G}_{i}^{2}]\prod_{i\in[s]\setminus R}\mathop{\bf
E\/}[\boldsymbol{H}_{i}^{2}].$
Thus to complete the inductive proof, it suffices to show that for each
$R\subseteq[s]$, the coefficient on $\prod_{i\in R}\mathop{\bf
E\/}[\boldsymbol{G}_{i}^{2}]\prod_{i\in[s]\setminus R}\mathop{\bf
E\/}[\boldsymbol{H}_{i}^{2}]$ in (5) is equal to $1$. By symmetry, and taking
the sum over $v$ first in (5), it suffices to check that for each
$r=|R|=|U\cup T|\in\\{0,1,\dots,s\\}$ we have
$\sum_{v^{\prime}=0}^{r}\frac{2^{2v^{\prime}}}{\tbinom{2v^{\prime}}{v^{\prime}}}\frac{\tbinom{s}{r}}{\tbinom{2s}{2r}}\tbinom{r}{v^{\prime}}\tbinom{s-r}{v^{\prime}}=1\quad\Leftrightarrow\quad\sum_{v^{\prime}=0}^{r}\frac{2^{2v^{\prime}}}{\tbinom{2v^{\prime}}{v^{\prime}}}\tbinom{r}{v^{\prime}}\tbinom{s-r}{v^{\prime}}=\frac{\tbinom{2s}{2r}}{\tbinom{s}{r}}.$
This identity can be proved computationally using Zeilberger’s algorithm
[Zei90, PWZ97]. Denoting the sum on the left side of the left equation by
$T(r)$, the algorithm delivers the recurrence equation $T(r+1)-T(r)=0$, which
together with the initial value $T(0)=1$ yields that $T(r)=1$ for all $r$. ∎
## 4 The reverse hypercontractive inequality in SOS
This section is devoted to providing a proof Theorem 1.5, the reverse
hypercontractivity in the SOS proof system. More precisely:
###### Theorem 4.1.
Let $k,n\in\mathbbm{N}^{+}$, let $0\leq\rho\leq 1-\frac{1}{2k}$, and let
$f(x)$, $g(x)$ be indeterminates for each $x\in\\{-1,1\\}^{n}$. Then
$\vdash_{{4k}}\quad\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\text{$\rho$-corr'd}\end{subarray}}[f({\boldsymbol{x}})^{2k}g(\boldsymbol{y})^{2k}]\geq\mathop{\bf
E\/}[f]^{2k}\mathop{\bf E\/}[g]^{2k}.$
For each fixed $k$, we prove Theorem 4.1 by induction on $n$. The $n=1$ base
case of the induction is the following $4$-variable inequality:
###### Theorem 4.2.
Let $k\in\mathbbm{N}^{+}$ and let $0\leq\rho\leq 1-\frac{1}{2k}$. Let
$F_{0},F_{1},G_{0},G_{1}$ be real indeterminates. Then
$\vdash_{{4k}}\quad(\tfrac{1}{4}+\tfrac{1}{4}\rho)\Bigl{(}F_{0}^{2k}G_{0}^{2k}+F_{1}^{2k}G_{1}^{2k}\Bigr{)}+(\tfrac{1}{4}-\tfrac{1}{4}\rho)\Bigl{(}F_{0}^{2k}G_{1}^{2k}+F_{1}^{2k}G_{0}^{2k}\Bigr{)}\geq\Bigl{(}\tfrac{F_{0}+F_{1}}{2}\Bigr{)}^{2k}\Bigl{(}\tfrac{G_{0}+G_{1}}{2}\Bigr{)}^{2k}.$
Proving this base case will be the key challenge; for now, we give the
induction which proves Theorem 4.1.
###### Proof of Theorem 4.1.
Let $n>1$. Given indeterminates $f(x)$, $g(x)$ for $x\in\\{-1,1\\}^{n}$, let
$f_{0}(x)$ be shorthand for $f(x_{1},\dots,x_{n-1},1)$, let $f_{1}(x)$ be
shorthand for $f(x_{1},\dots,x_{n-1},-1)$, and similarly define shorthands
$g_{0}$, $g_{1}$. Now
$\displaystyle\smash{\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\text{$\rho$-corr'd}\end{subarray}}}[f({\boldsymbol{x}})^{2k}g(\boldsymbol{y})^{2k}]=\
$ $\displaystyle(\tfrac{1}{4}+\tfrac{1}{4}\rho)\mathop{\bf
E\/}[f_{0}({\boldsymbol{x}})^{2k}g_{0}(\boldsymbol{y})^{2k}]$ $\displaystyle+\
$ $\displaystyle(\tfrac{1}{4}+\tfrac{1}{4}\rho)\mathop{\bf
E\/}[f_{1}({\boldsymbol{x}})^{2k}g_{1}(\boldsymbol{y})^{2k}]$ $\displaystyle+\
$ $\displaystyle(\tfrac{1}{4}-\tfrac{1}{4}\rho)\mathop{\bf
E\/}[f_{0}({\boldsymbol{x}})^{2k}g_{1}(\boldsymbol{y})^{2k}]$ $\displaystyle+\
$ $\displaystyle(\tfrac{1}{4}-\tfrac{1}{4}\rho)\mathop{\bf
E\/}[f_{1}({\boldsymbol{x}})^{2k}g_{0}(\boldsymbol{y})^{2k}].$
By four applications of induction, we deduce
$\displaystyle\vdash_{{4k}}\quad\smash{\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\text{$\rho$-corr'd}\end{subarray}}}[f({\boldsymbol{x}})^{2k}g(\boldsymbol{y})^{2k}]\geq\
$ $\displaystyle(\tfrac{1}{4}+\tfrac{1}{4}\rho)\mathop{\bf
E\/}[f_{0}({\boldsymbol{x}})]^{2k}\mathop{\bf
E\/}[g_{0}(\boldsymbol{y})]^{2k}$ $\displaystyle+\ $
$\displaystyle(\tfrac{1}{4}+\tfrac{1}{4}\rho)\mathop{\bf
E\/}[f_{1}({\boldsymbol{x}})]^{2k}\mathop{\bf
E\/}[g_{1}(\boldsymbol{y})]^{2k}$ $\displaystyle+\ $
$\displaystyle(\tfrac{1}{4}-\tfrac{1}{4}\rho)\mathop{\bf
E\/}[f_{0}({\boldsymbol{x}})]^{2k}\mathop{\bf
E\/}[g_{1}(\boldsymbol{y})]^{2k}$ $\displaystyle+\ $
$\displaystyle(\tfrac{1}{4}-\tfrac{1}{4}\rho)\mathop{\bf
E\/}[f_{1}({\boldsymbol{x}})]^{2k}\mathop{\bf
E\/}[g_{0}(\boldsymbol{y})]^{2k}.$
Now applying the $n=1$ base case of the induction (Theorem 4.2) to the right-
hand side of the above we conclude that
$\vdash_{{4k}}\quad\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\text{$\rho$-corr'd}\end{subarray}}[f({\boldsymbol{x}})^{2k}g(\boldsymbol{y})^{2k}]\geq\left(\tfrac{\mathop{\bf
E\/}[f_{0}({\boldsymbol{x}})]+\mathop{\bf
E\/}[f_{1}({\boldsymbol{x}})]}{2}\right)^{2k}\left(\tfrac{\mathop{\bf
E\/}[g_{0}(\boldsymbol{y})]+\mathop{\bf
E\/}[g_{1}(\boldsymbol{y})]}{2}\right)^{2k}=\mathop{\bf
E\/}[f]^{2k}\mathop{\bf E\/}[g]^{2k}.\qed$
Our remaining task is to prove the $4$-variable base case, Theorem 4.2. Let us
make a few simplifications. First, we claim it suffices to prove it in the
case $\rho=\rho^{*}=1-\frac{1}{2k}$. To see this, note that
$(\tfrac{1}{4}+\tfrac{1}{4}\rho)\Bigl{(}F_{0}^{2k}G_{0}^{2k}+F_{1}^{2k}G_{1}^{2k}\Bigr{)}+(\tfrac{1}{4}-\tfrac{1}{4}\rho)\Bigl{(}F_{0}^{2k}G_{1}^{2k}+F_{1}^{2k}G_{0}^{2k}\Bigr{)}-\Bigl{(}\tfrac{F_{0}+F_{1}}{2}\Bigr{)}^{2k}\Bigl{(}\tfrac{G_{0}+G_{1}}{2}\Bigr{)}^{2k}$
is linear in $\rho$. Thus if we can show it is SOS for both $\rho=0$ and
$\rho=\rho^{*}$, it follows easily that it is SOS for all $0<\rho<\rho^{*}$.
And for $\rho=0$ the task is easy:
$\vdash_{{4k}}\quad\tfrac{1}{4}\left(F_{0}^{2k}G_{0}^{2k}+F_{1}^{2k}G_{1}^{2k}\right)+\tfrac{1}{4}\left(F_{0}^{2k}G_{1}^{2k}+F_{1}^{2k}G_{0}^{2k}\right)=\left(\tfrac{F_{0}^{2k}+F_{1}^{2k}}{2}\right)\left(\tfrac{G_{0}^{2k}+G_{1}^{2k}}{2}\right)\geq\left(\tfrac{F_{0}+F_{1}}{2}\right)^{2k}\left(\tfrac{G_{0}+G_{1}}{2}\right)^{2k}$
by Lemma 2.6. Next, for clarity we make a change of variables; our task
becomes showing that for real indeterminates $\mu,\nu,\alpha,\beta$,
$\displaystyle\vdash_{{4k}}\quad$
$\displaystyle(\tfrac{1}{4}+\tfrac{1}{4}\rho^{*})\Bigl{(}(\mu+\alpha)^{2k}(\nu+\beta)^{2k}+(\mu-\alpha)^{2k}(\nu-\beta)^{2k}\Bigr{)}$
$\displaystyle+\ $
$\displaystyle(\tfrac{1}{4}-\tfrac{1}{4}\rho^{*})\Bigl{(}(\mu+\alpha)^{2k}(\nu-\beta)^{2k}+(\mu-\alpha)^{2k}(\nu+\beta)^{2k}\Bigr{)}-\mu^{2k}\nu^{2k}\geq
0.$ (6)
Finally, by homogeneity we can reduce the above to proving the following “two-
point inequality”:
###### Two-Point Inequality.
Let $k\in\mathbbm{N}^{+}$ and let $\rho^{*}=1-\frac{1}{2k}$. Then
$\displaystyle\vdash_{{4k}}\quad P_{k}(a,b)\coloneqq\ $
$\displaystyle(\tfrac{1}{4}+\tfrac{1}{4}\rho^{*})\Bigl{(}(1+a)^{2k}(1+b)^{2k}+(1-a)^{2k}(1-b)^{2k}\Bigr{)}$
$\displaystyle+\ $
$\displaystyle(\tfrac{1}{4}-\tfrac{1}{4}\rho^{*})\Bigl{(}(1+a)^{2k}(1-b)^{2k}+(1-a)^{2k}(1+b)^{2k}\Bigr{)}-1\geq
0.$
###### Proof that (6) follows from the Two-Point Inequality.
Suppose we show that $P_{k}(a,b)$ is equal to a sum of squares, say
$\sum_{i=1}^{m}R_{i}(a,b)^{2}$ where each $R_{i}(a,b)$ is a bivariate
polynomial. Viewing this as an SOS identity in $a$ only, we deduce that
$\deg_{a}(R_{i})\leq k$ for each $i$; similarly, $\deg_{b}(R_{i})\leq k$ for
each $i$. Then
$\sum_{i=1}^{m}(\mu^{k}\nu^{k}R_{i}(\tfrac{\alpha}{\mu},\tfrac{\beta}{\nu}))^{2}=\mu^{2k}\nu^{2k}\sum_{i=1}^{m}R_{i}(\tfrac{\alpha}{\mu},\tfrac{\beta}{\nu})^{2}=\mathrm{LHS}\eqref{eqn:rev-
goal1},$
and in the summation each expression
$\mu^{k}\nu^{k}R_{i}(\tfrac{\alpha}{\mu},\tfrac{\beta}{\nu})$ is a polynomial
in $\mu,\nu,\alpha,\beta$. ∎
It remains to establish the Two-Point Inequality via an SOS proof.
###### Remark 4.3.
We remind the reader that there is of course a “ZFC” proof of the Two-Point
Inequality, since it follows as a special case of the reverse hypercontractive
inequality.
### 4.1 The Two-Point Inequality in SOS
This section is devoted to proving the Two-Point Inequality; i.e., showing
$P_{k}(a,b)$ is SOS. The crucial idea turns out to be rewriting it under the
following substitutions:
$\displaystyle r=a-b,\qquad s=a+b,\qquad t=ab.$
We may then express
$\displaystyle P_{k}(a,b)$
$\displaystyle=-1+\left(\tfrac{1}{4}+\tfrac{1}{4}\rho^{*}\right)\bigl{(}(1+t+s)^{2k}+(1+t-s)^{2k}\bigr{)}+\left(\tfrac{1}{4}-\tfrac{1}{4}\rho^{*}\right)\bigl{(}(1-t+r)^{2k}+(1-t-r)^{2k}\bigr{)}$
$\displaystyle=-1+\left(\tfrac{1}{2}+\tfrac{1}{2}\rho^{*}\right)\sum_{i=0}^{k}\tbinom{2k}{2i}(1+t)^{2k-2i}s^{2j}+\left(\tfrac{1}{2}-\tfrac{1}{2}\rho^{*}\right)\sum_{j=0}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}r^{2j},$
where we used the identity
$\tfrac{1}{2}\bigl{(}(c+d)^{2k}+(c-d)^{2k}\bigr{)}=\sum_{i=0}^{k}\tbinom{2k}{2i}c^{2k-2i}d^{2i}.$
Next we use $r^{2}=s^{2}-4t$ to eliminate $r$, obtaining
$P_{k}(a,b)=-1+\left(\tfrac{1}{2}+\tfrac{1}{2}\rho^{*}\right)\sum_{i=0}^{k}\tbinom{2k}{2i}(1+t)^{2k-2i}s^{2i}+\left(\tfrac{1}{2}-\tfrac{1}{2}\rho^{*}\right)\sum_{j=0}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}(s^{2}-4t)^{j}.$
Now we expand $(s^{2}-4t)^{j}$ in the latter sum so that we can write it as an
even polynomial in $s$. We get
$\displaystyle\sum_{j=0}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}(s^{2}-4t)^{j}$
$\displaystyle=\sum_{j=0}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}\sum_{i=0}^{j}\tbinom{j}{i}s^{2i}(-4t)^{j-i}$
$\displaystyle=\sum_{i=0}^{k}s^{2i}\sum_{j=i}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}\tbinom{j}{i}(-4t)^{j-i}.$
Thus we have
$\displaystyle P_{k}(a,b)$
$\displaystyle=-1+\sum_{i=0}^{k}\left(\left(\tfrac{1}{2}+\tfrac{1}{2}\rho^{*}\right)\tbinom{2k}{2i}(1+t)^{2k-2i}+\left(\tfrac{1}{2}-\tfrac{1}{2}\rho^{*}\right)\sum_{j=i}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}\tbinom{j}{i}(-4t)^{j-i}\right)s^{2i}$
$\displaystyle=Q_{k,0}(t)+Q_{k,1}(t)s^{2}+Q_{k,2}(t)s^{4}+\cdots+Q_{k,k}(t)s^{2k},$
(7)
where
$\displaystyle
Q_{k,0}(t)=-1+\left(\tfrac{1}{2}+\tfrac{1}{2}\rho^{*}\right)(1+t)^{2k}+\left(\tfrac{1}{2}-\tfrac{1}{2}\rho^{*}\right)\sum_{j=0}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}(-4t)^{j},$
$\displaystyle
Q_{k,i}(t)=\left(\tfrac{1}{2}+\tfrac{1}{2}\rho^{*}\right)\tbinom{2k}{2i}(1+t)^{2k-2i}+\left(\tfrac{1}{2}-\tfrac{1}{2}\rho^{*}\right)\sum_{j=i}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}\tbinom{j}{i}(-4t)^{j-i},\qquad
i=1\dots k.$
Suppose we could show that $Q_{k,0}(t)$ and also $Q_{k,1}(t),\dots,Q_{k,k}(t)$
are nonnegative for all $t\in\mathbbm{R}$. Then by Fact 2.3 they are also SOS,
and hence $P_{k}(a,b)$ is SOS in light of (7). This would complete the proof
of the Two-Point Inequality.
In fact that is precisely what we show below, using some computer algebra
assistance. We remark, though, that is not a priori clear that this strategy
should work; i.e., that $Q_{k,0}(t),\dots,Q_{k,k}(t)$ should be nonnegative.
It does not follow from the truth of the Two-Point Inequality. To see this,
observe that whereas the Two-Point Inequality is known to hold for any
$0\leq\rho\leq\rho^{*}$, it is _not_ true that $Q_{k,0}(t)\geq 0$ for all
$0\leq\rho\leq\rho^{*}$. In fact, for $k=1$ we have
$Q_{1,0}(t)=t^{2}-(2-4\rho^{*})t$ (8)
which is nonnegative for all $t$ _only_ for the specific choice
$\rho^{*}=1-\frac{1}{2k}=\frac{1}{2}$.
Nevertheless, we now complete the proof of the Two-Point Inequality by showing
that $Q_{k,0}(t),\dots,Q_{k,k}(t)$ are all nonnegative.
###### Proposition 4.4.
For each $k\in\mathbbm{N}^{+}$ (with $\rho^{*}=1-\frac{1}{2k}$), the
polynomial $Q_{k,0}(t)$ is nonnegative.
###### Proof.
For $k=1$ we have $Q_{1,0}(t)=t^{2}$ (as noted in (8)); henceforth we may
assume $k\geq 2$. For $t<0$ we substitute $a=\sqrt{-t}$, $b=-\sqrt{-t}$ into
(7); since $s=a+b=0$ we get $P_{k}(\sqrt{-t},-\sqrt{-t})=Q_{k,0}(t)$. By
Remark 4.3 we have $P_{k}(\sqrt{-t},-\sqrt{-t})\geq 0$ and hence
$Q_{k,0}(t)\geq 0$ for all $t<0$.
For $t\geq 0$ we first rewrite
$Q_{k,0}(t)={-1}+(1+t)^{2k}\left(1-\frac{1}{4k}+\frac{1}{4k}\sum_{j=0}^{k}\tbinom{2k}{2j}\frac{(1-t)^{2k-2j}}{(1+t)^{2k\phantom{-2j}}}(-4t)^{j}\right).$
Denoting the sum in this expression by $S_{k}(t)$, Zeilberger’s algorithm
[Zei90, PWZ97] finds the recurrence equation
$(t+1)^{2}S_{k+2}(t)-2(t^{2}-6t+1)S_{k+1}(t)+(t+1)^{2}S_{k}(t)=0,$
valid for all $k\geq 0$. Since the coefficients in this recurrence do not
depend on $k$ but only on $t$, the recurrence can be solved in closed form.
Together with the initial values $S(0)=1$ and
$S(1)=\tfrac{t^{2}-6t+1}{(t+1)^{2}}$, it follows that
$S(t)=\cos(4k\arctan(\sqrt{t}))$. (Not every computer algebra system may
deliver the solution in this form; however, for the correctness of the proof
it is sufficient to check that $\cos(4k\arctan(\sqrt{t}))$ is indeed a
solution of the recurrence. This is easy to verify.) Hence,
$\displaystyle Q_{k,0}(t)$
$\displaystyle={-1}+(1+t)^{2k}\left(1-\tfrac{1}{4k}+\tfrac{1}{4k}\cos(4k\arctan(\sqrt{t}))\right)$
$\displaystyle\geq{-1}+(1+2kt)\left(1-\tfrac{1}{4k}+\tfrac{1}{4k}\cos(4k\arctan(\sqrt{t}))\right),$
using the fact that the parenthesized expression is clearly nonnegative. We
now split into two cases.
##### Case 1:
$t\geq\frac{1}{2k(2k-1)}$. In this case we simply use that
$\cos(4k\arctan(\sqrt{t}))\geq-1$ to obtain
$Q_{k,0}(t)\geq{-1}+(1+2kt)\left(1-\tfrac{1}{2k}\right)=-\tfrac{1}{2k}+(2k-1)t,$
which is indeed nonnegative when $t\geq\frac{1}{2k(2k-1)}$.
##### Case 2:
$0\leq t\leq\frac{1}{2k(2k-1)}$. In this case we use the following estimates:
$\arctan(\sqrt{t})\leq\sqrt{t}\quad\forall t\geq
0,\qquad\cos(x)\geq\kappa(x)\coloneqq
1-\tfrac{1}{2}x^{2}+\tfrac{1}{24}x^{4}-\tfrac{1}{720}x^{6}\quad\forall
x\in\mathbbm{R}.$
Note that $4k\arctan(\sqrt{t})\leq 4k\sqrt{t}\leq
4k\sqrt{\frac{1}{2k(2k-1)}}$, and the latter quantity is at most $\sqrt{2}$
for all $k\geq 2$. Since $\cos(x)$ is decreasing for $x\in[0,\sqrt{2}]$ we
have
$\cos(4k\arctan(\sqrt{t}))\geq\cos(4k\sqrt{t})\geq\kappa(4k\sqrt{t})\\\
\begin{aligned} \Rightarrow\quad
Q_{k,0}(t)&\geq{-1}+(1+2kt)\left(1-\tfrac{1}{4k}+\tfrac{1}{4k}\kappa(4k\sqrt{t})\right)\\\
&=-1+(1+2kt)\left(1-\tfrac{1}{4k}+\tfrac{1}{4k}\left(1-8k^{2}t+\tfrac{32}{3}k^{4}t^{2}-\tfrac{256}{45}k^{6}t^{3}\right)\right)\\\
&=q(t)t^{2},\qquad\text{where
}q(t)=-\tfrac{128}{45}k^{6}t^{2}-\left(\tfrac{64}{45}k-\tfrac{16}{3}\right)k^{4}t+\left(\tfrac{8}{3}k-4\right)k^{2}.\end{aligned}$
It remains to show that $q(t)\geq 0$ for $0\leq t\leq\frac{1}{2k(2k-1)}$.
Since $q(t)$ is a quadratic polynomial with negative leading coefficient, we
only need to check that $q(0),~{}q(\frac{1}{2k(2k-1)})\geq 0$. We have
$q(0)=\left(\tfrac{8}{3}k-4\right)k^{2}$, which is clearly nonnegative for
$k\geq 2$. Finally, one may check that
$\displaystyle q\left(\tfrac{1}{2k(2k-1)}\right)$
$\displaystyle=\tfrac{4k^{2}}{45(2k-1)^{2}}\left(19+136(k-2)\left((k-\tfrac{13}{34})^{2}+\tfrac{103}{1156}\right)\right),$
which is evidently nonnegative for $k\geq 2$. ∎
###### Proposition 4.5.
For all $1\leq i\leq k\in\mathbbm{N}^{+}$, the polynomial $Q_{k,i}(t)$ is
nonnegative (with $\rho^{*}=1-\frac{1}{2k}$).
###### Proof.
In fact, we will prove the stronger claim that each $Q_{k,i}(t)$ is
nonnegative even when $\rho^{*}$ is set to $0$. I.e., we will show that
$\widetilde{Q}_{k,i}(t)\coloneqq\tfrac{1}{2}\tbinom{2k}{2i}(1+t)^{2k-2i}+\tfrac{1}{2}\sum_{j=i}^{k}\tbinom{2k}{2j}(1-t)^{2k-2j}\tbinom{j}{i}(-4t)^{j-i}$
is nonnegative. To see that this is indeed stronger, simply note that
$Q_{k,i}(t)$ and $\widetilde{Q}_{k,i}(t)$ are convex combinations of the same
two main quantities, but $\widetilde{Q}_{k,i}(t)$ has less of its “weight” on
the first quantity $\tbinom{2k}{2i}(1+t)^{2k-2i}$, which is clearly
nonnegative. We will furthermore show that even $\widetilde{Q}_{k,0}(t)\geq
0$.
One may check that the following recurrence holds for all integers $0\leq
i\leq k$:
$(1+i)(1+k)\widetilde{Q}_{k+2,i+1}(t)=(1+i)(2+k)(1+t)^{2}\widetilde{Q}_{k+1,i+1}(t)+(2+k)(2+2k-i)\widetilde{Q}_{k+1,i}(t).$
(This was found by guessing the form of a polynomial recurrence and then
solving via computer.) Therefore we only need to prove $\tilde{Q}_{k,i}(t)\geq
0$ for the cases that $k=i$ and $i=0$; the nonnegativity of
$\widetilde{Q}_{k,i}(t)$ for general $k$ and $i$ then follows by induction.
For $k=i$ we have $\widetilde{Q}_{k,k}(t)=1\geq 0$. For $i=0$ the proof of
nonnegativity is similar to, but easier than, that of Proposition 4.4. For
$t<0$ it’s obvious from its definition that $\tilde{Q}_{k,0}(t)$ is
nonnegative. For $t\geq 0$, the proof of Proposition 4.4 gives
$\widetilde{Q}_{k,0}(t)=\tfrac{1}{2}(1+t)^{2k}\left(1+\cos(4k\arctan(\sqrt{t}))\right)\geq
0.\qed$
## 5 Vertex-Cover and the Frankl-Rödl Theorem in SOS
It will be slightly more convenient to work with the maximum independent set
($\mathsf{Max}\text{-}\mathsf{IS}$) problem, rather than the minimum vertex
cover problem; there is an equivalence between the problems because the
complement of a vertex cover is an independent set, and vice versa. We will
consider the fractional size of independent sets.
###### Definition 5.1.
Given a graph $G=(V,E)$ we define
$\mathsf{Max}\text{-}\mathsf{IS}(G)=\max\\{|S|/|V|:S\subseteq V\text{ is an
independent set}\\}\in[0,1],$
where $S$ is said to be an independent set if $E\cap(S\times S)=\emptyset$.
We repeat the Frankl–Rödl Theorem with this notation:
###### Theorem 5.2.
There exists $\gamma_{0}>0$ such that for all $\gamma\leq\gamma_{0}$ it holds
that $\mathsf{Max}\text{-}\mathsf{IS}(G^{n}_{\gamma})\leq
n(1-\gamma^{2}/64)^{n}$. In particular,
$\mathsf{Max}\text{-}\mathsf{IS}(G^{n}_{\gamma})\leq o_{n}(1)$ for
$.1\sqrt{\frac{\log n}{n}}\leq\gamma\leq\gamma_{0}$.
The main goal of this section is to give a low-degree proof of
“$\mathsf{Max}\text{-}\mathsf{IS}(G^{n}_{\gamma})\leq o_{n}(1)$” (for all
constant $\gamma>0$) in the SOS proof system. We state our theorem as follows.
###### Theorem 5.3.
Let $n\in\mathbbm{N}^{+}$ and let $\frac{1}{\log n}\leq\gamma\leq\frac{1}{2}$
such that $(1-\gamma)n$ is an even integer. Given the Frankl–Rödl graph
$G^{n}_{\gamma}=(V,E)$, for each $x\in V=\\{-1,1\\}^{n}$ let $f(x)$ be an
indeterminate. Then there is a degree-$O(1/\gamma)$ SOS refutation of the
system expressing the statement that $\mathsf{Max}\text{-}\mathsf{IS}(G)\geq
O(n^{-\gamma/10})$; i.e.,
$\\{f(x)^{2}=f(x)\ \forall x\in V,\ \ f(x)f(y)=0\ \forall(x,y)\in E,\ \
\tfrac{1}{|V|}\mathop{{\textstyle\sum}}_{x\in V}f(x)\geq
Cn^{-\gamma/10}\\}\vdash_{{O(1/\gamma)}}\ \ -1\geq 0$
for a universal constant $C$.
This theorem implies that the $N^{O(1/\gamma)}$-time SOS/Lasserre SDP
hierarchy algorithm certifies that
$\mathsf{Max}\text{-}\mathsf{IS}(G^{n}_{\gamma})\leq O(n^{-\gamma/10})$; in
other words, that
$\mathsf{Min}\text{-}\mathsf{Vertex\text{-}Cover}(G^{n}_{\gamma})\geq(1-O(n^{-\gamma/10}))N=(1-O(n^{-\gamma/10}))2^{n}$.
Note that our bound is nontrivial only for $\gamma\gg\frac{1}{\log n}$.
We prove Theorem 5.3 by “SOS-izing” the Benabbas–Hatami–Magen Fourier-
theoretic proof [BHM12] of the following “density” version of the Frankl–Rödl
Theorem:
###### Theorem 5.4.
([BHM12]) Fix $0<\gamma<1/2$ and $0<\alpha\leq 1$. In the graph
$G^{n}_{\gamma}=(V,E)$, if $S\subseteq V$ has $|S|/2^{n}\geq\alpha$ then
$\mathop{\bf Pr\/}_{({\boldsymbol{x}},\boldsymbol{y})\sim
E}[{\boldsymbol{x}}\in S,\boldsymbol{y}\in S]\geq
2(\alpha/2)^{1/\gamma}-o_{n}(1).$
Here the $o_{n}(1)$ goes to $0$ rather slowly in $n$, which means that the
Benabbas–Hatami–Magen proof only recovers the Frankl–Rödl Theorem for
$\gamma>\omega(\frac{1}{\log n})$.
### 5.1 The Benabbas–Hatami–Magen argument in SOS
Benabbas, Hatami, and Magen [BHM12] introduce the following operator:
###### Definition 5.5.
For integer $0\leq d\leq n$ the operator $S_{d}$ is defined on functions
$f:\\{-1,1\\}^{n}\to\mathbbm{R}$ by $S_{d}f(x)=\mathop{\bf
E\/}_{\boldsymbol{y}}[f(\boldsymbol{y})]$, where $\boldsymbol{y}$ is a chosen
uniformly at random subject to $\Delta(x,\boldsymbol{y})=d$.
The key technical contribution of [BHM12] is showing how to pass between the
$S_{d}$ operators (which are relevant for Frankl–Rödl analysis) and the
$T_{\rho}$ operators (for which we have reverse hypercontractivity).
Intuitively, the operators $S_{d}$ and $T_{1-2d/n}$ should be similar (at
least if $d/n$ is bounded away from $0$ and $1$). However there is one caveat:
“parity” issues with $S_{d}$. For example, if $f:\\{-1,1\\}^{n}\to\\{0,1\\}$
is the indicator of the strings of even Hamming weight, then
$\langle f,S_{d}f\rangle=\begin{cases}0&\text{if $d$ is odd,}\\\
\tfrac{1}{2}&\text{if $d$ is even;}\end{cases}\qquad\text{but, }\langle
f,T_{1-2d/n}f\rangle\approx\tfrac{1}{4}\text{ for $d$ odd or even}.$
Benabbas, Hatami, and Magen evade this parity issue by considering the
operator $\tfrac{1}{2}S_{d}+\tfrac{1}{2}S_{d+1}$.
###### Definition 5.6.
For integer $0\leq d<n$, we define the operator
$S_{d}^{\prime}=\tfrac{1}{2}S_{d}+\tfrac{1}{2}S_{d+1}$.
The crucial theorem in [BHM12]’s work is the following:
###### Theorem 5.7.
(Follows from Lemma 3.4 in [BHM12].) Let $f:\\{-1,1\\}^{n}\to\mathbbm{R}$. Let
$d=n-c$ for some integer $e^{2}\sqrt{n}\leq c\leq n/2$. Then for
$\rho=1-2d/n$,
$\langle f,S^{\prime}_{d}f\rangle-\langle
f,T_{\rho}f\rangle=\sum_{U\subseteq[n]}\widehat{f}(U)^{2}\cdot\delta(U),$
where each real number $\delta(U)$ satisfies
$|\delta(U)|\leq
O(\max\\{n^{-1/5},\tfrac{n}{c^{2}}\log^{2}(\tfrac{c^{2}}{n})\\}).$
Given this Theorem 5.7, Benabbas, Hatami, and Magen are able to deduce their
main Theorem 5.4 from the reverse hypercontractivity result Theorem 1.3
without too much trouble. We now show that this deduction can also be carried
out in the SOS proof system. Specifically, we give here the proof of our
Theorem 5.3, relying on the SOS proof of hypercontractivity (Theorem 4.1) from
Section 4.
###### Proof of Theorem 5.3.
Write $d=(1-\gamma)n$ (where $\frac{1}{\log n}\leq\gamma\leq\tfrac{1}{2}$) and
write $\rho^{\prime}=1-2d/n=-(1-2\gamma)$. For $i=0,1$ let us denote
$f_{i}(x)=\begin{cases}f(x)&\text{if $x$'s Hamming weight equals $i$ mod
2,}\\\ 0&\text{else.}\end{cases}$
We have
$\\{f(x)f(y)=0\ \forall\Delta(x,y)=d\\}\quad\vdash_{{2}}\quad\langle
f_{0},S^{\prime}_{d}f_{0}\rangle+\langle f_{1},S^{\prime}_{d}f_{1}\rangle=0$
because if $x$’s Hamming weight has the same parity as $y$’s then their
distance can only be $d$ (an even integer) not $d+1$ (an odd one). Using
Theorem 5.7 it follows that
$\\{f(x)f(y)=0\ \forall\Delta(x,y)=d\\}\quad\vdash_{{2}}\\\ \langle
f_{0},T_{\rho^{\prime}}f_{0}\rangle+\langle
f_{1},T_{\rho^{\prime}}f_{1}\rangle\leq\delta\mathop{{\textstyle\sum}}_{U}\widehat{f_{0}}(U)^{2}+\delta\mathop{{\textstyle\sum}}_{U}\widehat{f_{1}}(U)^{2}=\delta(\mathop{\bf
E\/}[f_{0}^{2}]+\mathop{\bf E\/}[f_{1}^{2}])=\delta\mathop{\bf E\/}[f^{2}].$
where
$\delta=O(\max\\{n^{-1/5},\tfrac{1}{\gamma^{2}n}\log^{2}(\gamma^{2}n)\\})=O(n^{-1/5})$
(with the second bound using $\gamma\geq\frac{1}{\log n}$.) We now write
$g_{i}(x)$ to denote $f_{i}(-x)$ and also $\rho=-\rho^{\prime}=1-2\gamma$;
then
$\langle f_{i},T_{\rho^{\prime}}f_{i}\rangle=\langle
f_{i},T_{\rho}g_{i}\rangle=\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\rho\text{-corr'd}\end{subarray}}[f_{i}({\boldsymbol{x}})g_{i}(\boldsymbol{y})]$
so we conclude
$\\{f(x)f(y)=0\ \forall\Delta(x,y)=d\\}\quad\vdash_{{2}}\quad\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\rho\text{-corr'd}\end{subarray}}[f_{0}({\boldsymbol{x}})g_{0}(\boldsymbol{y})]+\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\rho\text{-corr'd}\end{subarray}}[f_{1}({\boldsymbol{x}})g_{1}(\boldsymbol{y})]\leq\delta\mathop{\bf
E\/}[f^{2}].$
Next, let $k=\lceil\frac{1}{4\gamma}\rceil$. We have
$f(x)^{2}=f(x)\vdash_{{2k}}f(x)^{2k}=f(x)$, from which we may easily deduce
$\\{f(x)^{2}=f(x)\ \forall x,\quad f(x)f(y)=0\ \forall\Delta(x,y)=d\\}\\\
\vdash_{{2k}}\quad\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\rho\text{-corr'd}\end{subarray}}[f_{0}({\boldsymbol{x}})^{2k}g_{0}(\boldsymbol{y})^{2k}]+\mathop{\bf
E\/}_{\begin{subarray}{c}({\boldsymbol{x}},\boldsymbol{y})\\\
\rho\text{-corr'd}\end{subarray}}[f_{1}({\boldsymbol{x}})^{2k}g_{1}(\boldsymbol{y})^{2k}]\leq\delta\mathop{\bf
E\/}[f].$
Since $\rho=1-2\gamma\leq 1-\frac{1}{2k}$ we may apply our reverse
hypercontractivity result Theorem 4.1 (in Section 4) to deduce
$\\{f(x)^{2}=f(x)\ \forall x,\quad f(x)f(y)=0\
\forall\Delta(x,y)=d\\}\quad\vdash_{{2k}}\quad\mathop{\bf
E\/}[f_{0}]^{2k}\mathop{\bf E\/}[g_{0}]^{2k}+\mathop{\bf
E\/}[f_{1}]^{2k}\mathop{\bf E\/}[g_{1}]^{2k}\leq\delta\mathop{\bf E\/}[f].$
We’re now almost done. First, $\mathop{\bf E\/}[f_{i}]=\mathop{\bf
E\/}[g_{i}]$ formally for each $i=0,1$. Second, for simplicity we use the
bound
$\\{f(x)^{2}=f(x)\ \forall x\\}\quad\vdash_{{2}}{\quad}\delta\mathop{\bf
E\/}[f]=\delta\mathop{\bf E\/}[2f-f^{2}]=\delta\mathop{\bf
E\/}[1-(1-f)^{2}]\leq\delta.$
Thus we have
$\displaystyle\\{f(x)^{2}=f(x)\ \forall x,\quad f(x)f(y)=0\
\forall\Delta(x,y)=d\\}\quad\vdash_{{2k}}\quad\delta$
$\displaystyle\geq\mathop{\bf E\/}[f_{0}]^{4k}+\mathop{\bf E\/}[f_{1}]^{4k}$
$\displaystyle\geq 2\left(\frac{\mathop{\bf E\/}[f_{0}]+\mathop{\bf
E\/}[f_{1}]}{2}\right)^{4k}$ $\displaystyle=2(\mathop{\bf E\/}[f]/2)^{4k}$
$\displaystyle\Rightarrow\quad$ $\displaystyle 2^{4k-1}\delta\geq\mathop{\bf
E\/}[f]^{4k},$
where the second inequality is Lemma 2.6. Finally, from Lemma 2.5 we may
deduce
$\mathop{\bf E\/}[f]\geq Cn^{-\gamma/10}\quad\vdash_{{4k}}\quad\mathop{\bf
E\/}[f]^{4k}\geq C^{4k}n^{-4k\gamma/10}\geq C^{4k}n^{-\gamma/5}\geq
2^{4k}\delta$
for $C$ sufficiently large. By combining the previous two statements we can
get
$\\{f(x)^{2}=f(x)\ \forall x,\quad f(x)f(y)=0\
\forall\Delta(x,y)=d,\quad\mathop{\bf E\/}[f]\geq
Cn^{-\gamma/10}\\}\quad\vdash_{{4k}}\quad-1\geq 0,$
as required. ∎
## 6 Conclusions
We describe here a few questions left open by our work. Regarding reverse
hypercontractivity, it seems we may not have given the Book proof of the SOS
Two-Point Inequality. We would be happy to see a more elegant “human proof”,
but even more interesting would be a computer algebra technique that could
automatically prove SOS-ness, symbolically for all $k$.
An additional open question regarding the Frankl–Rödl Theorem is whether the
Benabbas–Hatami–Magen proof can be improved to work for $\gamma$ as small as
$\sqrt{\frac{\log n}{n}}$. However even if this is possible, the resulting SOS
proof would (presumably) be of degree $\Omega(\sqrt{\frac{n}{\log
n}})=\Omega(\sqrt{\frac{\log N}{\log\log N}})$, slightly superconstant. Could
there be an $O(1)$-degree SOS of the Frankl–Rödl Theorem? An interesting toy
version of this question is the following: The vertex isoperimetric inequality
for the hypercube immediately implies that if $A,B\subseteq\\{-1,1\\}^{n}$
satisfy $\mathrm{dist}(A,B)\geq\sqrt{n\log n}$ then
$\frac{|A|}{2^{n}}\frac{|B|}{2^{n}}=o_{n}(1)$. Does this have an $O(1)$-degree
SOS proof?
### Acknowledgments
The authors would like to thank Siavosh Benabbas and Hamed Hatami for the
advance copy of [BHM12]. Proposition 4.5 was independently proven by Fedor
Nazarov; we are very grateful to him for sharing his proof with us. Thanks
also to Toni Pitassi and Doron Zeilberger for helpful discussions.
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## Appendix A Solution to the puzzle
${a}^{6}{b}^{6}+15\left(a+b\right)^{2}\left(1+ba\right)^{4}+10\left(a+b\right)^{4}\left(1+ba\right)^{2}+5\left({a}^{3}b+{b}^{2}+{a}^{2}+a{b}^{3}\right)^{2}\\\
+\
35{a}^{4}{b}^{4}+\left({a}^{3}+{b}^{3}\right)^{2}+{\tfrac{17}{3}}\left({a}^{2}b+a{b}^{2}\right)^{2}+35{a}^{2}{b}^{2}+{\tfrac{148}{3}}{a}^{2}{b}^{4}+{\tfrac{148}{3}}{a}^{4}{b}^{2}.$
|
arxiv-papers
| 2012-12-21T03:28:28 |
2024-09-04T02:49:39.554275
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Manuel Kauers, Ryan O'Donnell, Li-Yang Tan, Yuan Zhou",
"submitter": "Li-Yang Tan",
"url": "https://arxiv.org/abs/1212.5324"
}
|
1212.5515
|
# The Curve shortening Flow with Parallel 1-form
Hengyu Zhou The Graduate Center,
The City University of New York,
365 fifth Ave.,
New York, NY 10016, USA [email protected]
###### Abstract.
Let $M$ be a closed Riemannian manifold with a parallel 1-form $\Omega$. We
prove two theorems about the curve shortening flow in $M$. One is that the
curve shortening flow $\mathcal{C}_{t}$ in $M$ exists for all $t$ in
$[0,\infty)$, if it satisfies $\Omega(T)\geq 0$ on the initial curve
$\mathcal{C}_{0}$. Here $T$ is the unit tangent vector on $\mathcal{C}_{0}$.
The other one is about the convergence. It says that in a closed Riemannian
manifold $\tilde{M}$, assume the curve shortening flow $\mathcal{C}_{t}$
exists for all $t\in[0,\infty)$ and its length converges to a positive limit,
then
$\lim\limits_{t\rightarrow\infty}max_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}=0$ for
all $m=0,1,\cdots$. Here $A$ denotes the second fundamental form of
$\mathcal{C}_{t}$ in $\tilde{M}$.
###### Key words and phrases:
Curve shortening flow, mean curvature flow, parallel form.
###### 2010 Mathematics Subject Classification:
Primary 53C44, Secondary 58J35
## Introduction
We use $\nabla$ denote the Levi-Civita connection of a Riemannian manifold . A
1-form $\Omega$ is called parallel if $\nabla\Omega=0$. In this paper we
consider the long time existence and the convergence of the curve shortening
flow in the Riemannian manifold with arbitrary codimension. More precisely,
the curve shortening flow $\mathcal{C}_{t}$ is given as the solution of the
follow equation:
(0.1) $\left\\{\begin{aligned} &\frac{\partial\mathcal{C}_{t}}{\partial
t}=\vec{H}\\\ &\mathcal{C}(x,0)=\mathcal{C}_{0}(x);\end{aligned}\right.$
Here $\vec{H}=\nabla_{T}T$ is the mean curvature vector of $\mathcal{C}_{t}$
in the ambient Riemannian manifold.
The evolution of closed curves under (0.1) has received considerable study.
For example, Gage ([Gag84]), Gage-Hamilton ([GH86]) and Grayson ([Gra87]),
considered the evolution of the convex curve flowed by the mean curvature in
the plane. Furthermore, Grayson ([Gra89]) generalized the results into the
case on the embedded closed curve of Riemannian surfaces. Huisken ([Hui98])
and Andrews-Bryan ([AB11]) used the technique of distance comparison to
investigate the curve shortening flow on the plane. For the mean curvature
flow of the submanifolds with codimension $k\geq 2$, Andrews-Baker ([AB10])
proposed a machinery about the evolution of the second fundamental form $A$
with arbitrary codimension. They also obtained the convergence of the mean
curvature flow for the submanifold pinched to sphere in Euclidean space. More
recently, with an integral curvature condition ([KFLZ10]) and a pinched
condition in hyperbolic space form ([KFLZ11]) Liu, Xu, Ye and Zhao also proved
two extension theorems for the mean curvature flow of the submanifold with
high codimension. Smoczyk’s survey ([Smo12]) is a good reference.
In a different direction, Wang ([Wan02]) initialized a way to use parallel
n-forms $\Omega$, $n\geq 2$, to investigate the graphic mean curvature flow
for arbitrary codimension ($\geq 2$) in product manifolds. With the evolution
equation of $*\Omega$, ($*$ is the Hodge dual), Wang obtained its long-time
existence and an estimate of its second fundamental form along the mean
curvature flow. Moreover, Wang and his coauthors generalized the results of
graphic mean curvature flow into other settings
([MW11],[TW04],[Wan01],[SW02],). Both of ([AB10]) and ([Wan02]) did not
consider the case of the curve shortening flow, which is the topic of this
paper.
There are two special properties about the curve shortening flow whose
codimension in its ambient manifold is greater than 1. First the embeddedness
of the evolving curve is not necessarily preserved ([Alt91]). Therefore, we
only consider the case of the immersed curve. On the other hand,
$|\vec{H}|^{2}=|A|^{2}$. Based on this fact, we can overcome the barrier that
the curve shortening flow has arbitrary codimension. This makes many
computations possible, especially in Section 2.
Now let us state two main theorems:
###### Theorem 0.1.
Let $M$ be a closed Riemannian manifold, and $\Omega$ be a parallel 1-form in
$M$. If $\mathcal{C}_{0}$ is a closed curve satisfying $\Omega(T)>0$ on
$\mathcal{C}_{0}$. Here $T$ is the unit tangent vector on $\mathcal{C}_{0}$.
Then the curve shortening flow $\mathcal{C}_{t}$ in (0.1) exists for all $t$
in $[0,\infty)$.
###### Corollary 1.
Let $(N,g)$ be a closed manifold, $du^{2}$ is the canonical metric on $S^{1}$.
Assume $M=S^{1}\times N$ is the product manifold with the product metric
$\bar{g}=g+du^{2}$. Let $\Omega=du$, which is a parallel 1-form in $M$.
$\mathcal{C}_{0}$ is a closed curve in $M$. If the unit tangent vector $T$ of
$\mathcal{C}_{0}$ satisfies $\Omega(T)>0$, the curve shortening flow
$\mathcal{C}_{t}$ in (0.1) exists for all $t$ in $[0,\infty)$.
###### Remark 0.1.
Corollary 1 can be viewed the 1-dimension version of Theorem A in ([Wan02]).
The existence result here with $\Omega(T)>0$ is stronger than that of
graphical mean curvature flow with $*\Omega>\frac{1}{2}$ in ([Wan02]). It also
can be compared with the results of Tsui-Wang ([TW04]) if we choose $N=S^{n}$.
Now we consider the convergence of the curve shortening flow in any closed
Riemannian manifold. Notice that we do not require there is a parallel 1-form
in the ambient manifold. The convergence of the curve shortening flow in the
$C^{\infty}$ sense can be stated as follows.
###### Theorem 0.2.
Let $\tilde{M}$ be a closed manifold. If the curve shortening flow
$\mathcal{C}_{t}$ in $\tilde{M}$ exists for all $t\in[0,\infty)$ and the
length of $\mathcal{C}_{t}$ converges to a positive number,
$\lim\limits_{t\rightarrow\infty}max_{\mathcal{C}_{t}}|\nabla^{m}A|=0$ for all
$m=0,1,\cdots$.
###### Remark 0.2.
The definition of $\nabla^{m}A$ is very subtle. In ([Hui84],[Hui86]),
Huisken’s definitions about $\nabla^{m}A$ only work in the hypersurface’s
case. Andrews-Bakers ([AB10]) gave a rigorous definition of $\nabla^{m}A$ for
any mean curvature flow with arbitrary codimension in Riemannian manifold. The
basic idea is that $|\nabla^{m}A|$ can contain all information about the mean
curvature flow. Namely, when $|\nabla^{m}A|$ goes to 0 when $t$ goes to
infinity, the mean curvature flow $F_{t}$ will converge to a geodesic
submanifold in some natural sense. For example, one of these cases is that
$F_{t}$ has long time existence and always stay in a compact region of its
ambient Riemannian manifold. We will state the definition of $\nabla^{m}A$ in
Section 2.
###### Remark 0.3.
The condition that $M$ and $\tilde{M}$ are closed is not essential. We take
them only for the sake of the exposition. In Theorem 0.1, what we really need
is that for all finite time $t$, the curve shortening flow $\mathcal{C}_{t}$
is in a compact region in $M$, which can depend on time $t$. In Theorem 0.2,
we can only require that all covariant derivatives of the Riemann tensor in
$\tilde{M}$ are uniformly bounded.
### Outline of this paper
In section 1, we give two examples to illustrate our motivations of using
parallel form to investigate the curve shortening flow. In Section 2, we give
the definition of $\nabla^{m}A$. Then we derive the evolution equations
related to the second fundamental form $A$ (lemma 2.2 and lemma 2.3). With
those equations, we derive the evolution equation of $\Omega(T)$ for any
1-form $\Omega$ along the curve shortening flow (lemma 2.4). In particular,
when $\Omega$ is a parallel 1-form, this evolution equation states that the
lower positive bound of $\Omega(T)$ can be preserved along the curve
shortening flow. In section 3, we prove Theorem 0.1 by using the results from
section 2. In section 4, with Grayson’s idea ([Gra89]), we prove Theorem 0.2.
### Notation
We collect some geometric quantities and facts used later.
1. (1)
$\mathcal{C}(u,t)$ denotes the solution of (0.1). $\frac{\partial}{\partial
t}$ and $\frac{\partial}{\partial u}$ are the abbreviations of
$\mathcal{C}_{*}(\frac{\partial}{\partial t})$ and
$\mathcal{C}_{*}(\frac{\partial}{\partial u})$ respectively.
2. (2)
$\nabla$ is the Levi-Civita connection, $du^{2}$ is the canonical metric on
$S^{1}$.
3. (3)
Let $R$ denote Riemann curvature tensor, given by
$R(X,Y)=\nabla_{Y}\nabla_{X}-\nabla_{X}\nabla_{Y}+\nabla_{[X,Y]}$.
4. (4)
Let $s$ be the arc-length parameter of a curve $\mathcal{C}$. Its unit tangent
vector $T$ is equal to $\frac{\partial}{\partial
s}=\frac{\mathcal{C}_{*}(\frac{\partial}{\partial
u})}{<\mathcal{C}_{*}(\frac{\partial}{\partial
u}),\mathcal{C}_{*}(\frac{\partial}{\partial u})>^{\frac{1}{2}}}.$
5. (5)
$\vec{H}=\frac{\partial}{\partial t}=A(T,T)=\nabla_{T}T$.
$|A|^{2}=|\vec{H}|^{2}$.
6. (6)
For a curve $\mathcal{C}$, $|\vec{H}|^{2}=|A|^{2}=<\nabla_{T}T,\nabla_{T}T>$.
All curves in this paper are closed.
### Acknowledgements
The author would like to thank Professor Zeno Huang for suggesting the
problem, and for many helpful discussions and encouragement, and Professor
Yunping Jiang for his interest and suggestions. This work is completed while
the author is partially supported, as a student associate, by a CIRG award
$\\#1861$. The author is also indebted to Xiangwen Zhang, Longzhi Lin and Andy
Huang. These discussions with them greatly help to simplify the computations
in this paper.
## 1\. The Examples about Parallel form
In this section we use two examples to illustrate the motivations that we
choose parallel 1-form $\Omega$ to investigate the curve shortening flow. The
first one is the graphic mean curvature flow in the product manifolds
([Wan02]). Based on this, we can use Wang’s idea ([Wan02]) about parallel form
to propose a problem about the open curve shortening flow in $R^{1+p}$.
###### Example 1 (graphic mean curvature flow).
In ([Wan02]), Mu-Tao Wang initialized a way to use parallel form to deform a
map along the mean curvature flow in product manifolds. The basic setting is
as followed: Let $N_{1}$, $N_{2}$ be closed manifolds with constant sectional
curvatures, dimension $\geq 2$. $f$ is a smooth map from $N_{1}$ to $N_{2}$.
Let $F(p,t):N_{1}\times[0,q)\rightarrow N_{1}\times N_{2}$ satisfies the
following equation.
(1.1) $\left\\{\begin{aligned} \frac{\partial F_{t}}{\partial t}&=\vec{H}\\\
F(p,0)&=(p,f(p));\end{aligned}\right.$
Let $\Omega$ be the volume form of $N_{1}$, which is parallel in the product
manifold $N_{1}\times N_{2}$. Together with some technical conditions about
the sectional curvature of $N_{1}$ and $N_{2}$, Wang ([Wan02]) obtained that
if $*\Omega>\frac{1}{2}$ at the initial manifold, the evolution equations of
$*\Omega$ indicates the long time existence of the solution in (1.1).
###### Example 2 (open curve shortening flow).
In this example, we propose a question about open curve shortening flow in
$R^{1+p}$. $\mathcal{C}_{0}$ in $R^{1+p}$ is defined by
(1.2) $\mathcal{C}_{0}:R^{1}\rightarrow
R^{1+p}\quad\mbox{and}\qquad\mathcal{C}_{0}(x)=(x,f_{1}(x),\cdots,f_{p}(x));$
Let $|Df|^{2}=\sum_{i=1,\cdots,p}(f_{i}^{{}^{\prime}}(x))^{2}$. Then,
$T=\left(\frac{\partial}{\partial
x^{1}}+\sum_{i=1,\cdots,p}f_{i}^{{}^{\prime}}(x)\frac{\partial}{\partial
x^{i+1}}\right)\frac{1}{\sqrt{1+|Df|^{2}}};$
Let $\Omega=dx_{1}$. Again $\Omega$ is a parallel 1-form. On
$\mathcal{C}_{0}$, $\Omega(T)=\frac{1}{\sqrt{1+|Df|^{2}}}$. From lemma 2.6, if
$\mathcal{C}_{t}$ exists for $t\in[0,q)$, then $\Omega(T)$ on
$\mathcal{C}_{t}$ satisfies
$\frac{\partial}{\partial
t}\Omega(T)=\triangle^{\mathcal{C}_{t}}\Omega(T)+|A|^{2}\Omega(T)$
Here $A$ is its second fundamental form. Recall that with
$*\Omega>\frac{1}{2}$ in the initial data, Wang ([Wan02]) answered the long
time existence of graphic mean curvature flow positively. Naturally if we
assume that $\mathcal{C}_{0}$ satisfies $\Omega(T)\geq\delta>0$, does the
curve shortening flow $\mathcal{C}_{t}$ in (0.1) exist for all
$t\in[0,\infty)$? If it exists, can we give any description about the
convergence of this curve shortening flow in $R^{1+p}$?
###### Remark 1.1.
The above examples reflect our basic perspectives about the parallel form and
the mean curvature flow. (2.6) and (3.3) are some possible evidences to
support these connections between them. (2.6) can be viewed as a
generalization of Wang’s graphical mean curvature flow ([Wan02]) in the
curve’s case. (3.3) indicates that $\mu=\frac{1}{\Omega(T)}$ be thought as the
gradient function ([EH89]) in the case of the curve shortening flow. There are
no any oblivious connections between ([Wan02]) and ([EH89]). For the curve
shortening flow, however, (3.3) and (2.6) are two different forms of the same
evolution equation.
## 2\. Evolution Equations
We derive the evolution equations of the curve shortening flow. Generally, the
forms of those equations are very complicated for the mean curvature flow
whose dimension and codimension are both $>1$. ([AB10], [Wan02])
### 2.1. Definition of $\nabla^{m}A$
We briefly state the definition of $\nabla^{m}A$ which can be found in
([AB10]). Assume $M$ is a Riemannian manifold with the Levi-Civita connection
$\bar{\nabla}$, $N$ is an immersed submanifold of $M$. Suppose $I$ is a real
interval, then the tangent space $T(N\times I)$ splits into a direct product
$\mathcal{H}\otimes R\partial t$, where $\mathcal{H}=\\{u\in T(N\times
I);dt(u)=0\\}$ is the ”spatial” tangent bundle.
We consider a smooth map $F:N\times I\rightarrow M$ which is a time-dependent
immersion, i.e.,for each fixed $t\in I$, $F(\,,t):N\rightarrow M$ is an
immersion. $F^{*}TM$ is a vector bundle over $N\times I$. We can define a
metric $g_{F}$ and the connection ${}^{F}\nabla$ on $F^{*}TM$ by the pull-back
from $(\bar{\nabla},\bar{g})$ on $M$. Let $\mathcal{N}$ be the orthogonal
complement of $\mathcal{H}$ in $F^{*}TM$. We denote the $\pi,\pi^{\bot}$ be
the orthogonal projections from $F^{*}TM$ onto $\mathcal{H}$ and $\mathcal{N}$
respectively. The connections ${}^{\mathcal{H}}\nabla$ and ${}^{N}\nabla$ are
given by.
$\displaystyle{}^{\mathcal{H}}\nabla$ $\displaystyle=\pi\circ^{F}\nabla\circ
F_{*};$ $\displaystyle{}^{\mathcal{N}}\nabla$
$\displaystyle=\pi^{\bot}\circ^{F}\nabla\circ F_{*};$
For a tensor $K\in\Gamma(\mathcal{H}^{*}\otimes\mathcal{H}^{*}\otimes N)$. We
have to define $\nabla^{m}K\in\Gamma(\otimes^{m+2}\mathcal{H}^{*}\otimes N)$
by induction on $m$. Let $u_{0},u_{i},i=1,2,3,\cdots,m+1$, then $\nabla^{m}K$
and $\nabla_{\frac{\partial}{\partial t}}\nabla^{m}K$ are given by
$\displaystyle\nabla_{u_{0}}$
$\displaystyle(\nabla^{m-1}K)(u_{1},\cdots,u_{m+1})=^{\mathcal{N}}\nabla_{u_{0}}(\nabla^{m-1}K(u_{1},\cdots,u_{m+1}))-\sum\limits_{i=1}^{m+1}\nabla^{m-1}K(\cdots,^{\mathcal{H}}\nabla_{u_{0}}u_{i},\cdots);$
$\displaystyle\nabla_{\frac{\partial}{\partial t}}$
$\displaystyle(\nabla^{m}K)(u_{0},u_{1},\cdots,u_{m+1})=^{\mathcal{N}}\nabla_{\frac{\partial}{\partial
t}}(\nabla^{m}K(u_{0},\cdots,u_{m+1}))-\sum\limits_{k=0}^{m+1}\nabla^{m}K(\cdots,^{\mathcal{H}}\nabla_{\frac{\partial}{\partial
t}}u_{k},\cdots);$
### 2.2. The Evolution Equations for the Curve Shortening Flow
For the curve shortening flow $\mathcal{C}_{t}$ in (0.1), we define a new
tensor
$\tilde{R}\in\Gamma(\mathcal{H}^{*}\otimes\mathcal{H}^{*}\otimes\mathcal{N})$
by $\tilde{R}(T,T)=R(T,\vec{H})T$. We also have the definition of
$\nabla^{m}\tilde{R}$. Since
$A\in\Gamma(\mathcal{H}^{*}\otimes\mathcal{H}^{*}\otimes\mathcal{N})$, with
the definitions in the previous subsection, we can obtain a theorem about the
evolution equation of $A$ from equation (18) in ([AB10]).
###### Theorem 2.1 (Andrews-Baker).
For the curve shortening flow $\mathcal{C}_{t}$, the evolution equation of
$A(T,T)$ is given by
(2.1) $\nabla_{\frac{\partial}{\partial
t}}A(T,T)=\nabla_{T}\nabla_{T}A(T,T)+|A|^{2}\vec{H}+\tilde{R}(T,T);$
Recall that $S*T$ ([Hui84]) means any linear combination of tensors formed by
contraction on $S$ and $T$. (2.1) can be rewritten as
$\nabla_{\frac{\partial}{\partial t}}A=\nabla^{2}A+\sum A*A*A+\tilde{R}(T,T)$.
This form can be generalized into any $m$-th covariant derivative $A$ for the
curve shortening flow $\mathcal{C}_{t}$.
###### Lemma 2.2.
The evolution equation of $m$-th covariant derivative $A$ is of the form.
(2.2) $\nabla_{\frac{\partial}{\partial
t}}\nabla^{m}A=\nabla^{m+2}A+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+\nabla^{m}\tilde{R}(T,T);$
###### Proof.
We prove the lemma through the induction on $m$. The case $m=0$ is given by
(2.1). Now suppose the results hold up to $k\leq m-1$. In ([Hui84]), the time
derivative of the Christoffel symbols $\Gamma_{jk}^{i}$ has the form $A*\nabla
A$. Therefore we obtain,
$\displaystyle\nabla_{\frac{\partial}{\partial t}}\nabla^{m}A$
$\displaystyle=\nabla(\nabla_{\frac{\partial}{\partial
t}}\nabla^{m-1}A)+\nabla^{m-1}A*A*\nabla A;$
$\displaystyle=\nabla\\{\nabla^{m+1}A)+\sum\limits_{i+j+k=m-1}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+\nabla^{m-1}\tilde{R}(T,T)\\}$
$\displaystyle+\nabla^{m-1}A*A*\nabla A;$
$\displaystyle=\nabla^{m+2}A+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+\nabla^{m}\tilde{R}(T,T);$
∎
Moreover, we obtain the evolution equations of
$|\nabla^{m}A|^{2},m=0,1,\cdots,$.
###### Lemma 2.3.
The evolution equations of $|\nabla^{m}A|^{2}$ are given by
(2.3) $\displaystyle\frac{\partial}{\partial
t}|A|^{2}=\triangle^{\mathcal{C}_{t}}|A|^{2}-2|\nabla
A|^{2}+2|A|^{4}+2R(T,\vec{H},T,\vec{H});$ (2.4)
$\displaystyle\begin{split}\frac{\partial}{\partial
t}|\nabla^{m}A|^{2}&=\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A*\nabla^{m}A\\\
&+2<\nabla^{m}\tilde{R}(T,T),\nabla^{m}A>\quad\text{for}\quad m\geq
1;\end{split}$
###### Proof.
We differentiate $|\nabla^{m}A|^{2}$ with respect to $t$. For $m=0$,
$\displaystyle\frac{\partial}{\partial t}|A|^{2}$
$\displaystyle=2<\nabla_{\frac{\partial}{\partial t}}A(T,T),A(T,T)>;$
$\displaystyle=2<\nabla^{2}A(T,T)+|A|^{2}\vec{H}+R(T,\vec{H})T,A(T,T)>;$ Since
$A(T,T)=\vec{H}$ $\displaystyle=\triangle^{\mathcal{C}_{t}}|A|^{2}-2|\nabla
A|^{2}+2|A|^{4}+2R(T,\vec{H},T,\vec{H});$
For $m\geq 1$.
$\displaystyle\frac{\partial}{\partial t}|\nabla^{m}A|^{2}$
$\displaystyle=2<\nabla_{\frac{\partial}{\partial
t}}\nabla^{m}A,\nabla^{m}A>;$
$\displaystyle=2<\nabla^{m+2}A+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+\nabla^{m}\tilde{R}(T,T),\nabla^{m}A>;$
$\displaystyle=\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A*\nabla^{m}A+$
$\displaystyle+2<\nabla^{m}\tilde{R}(T,T),\nabla^{m}A>;$
∎
Now we derive the evolution equation of $\Omega(T)$ along the curve shortening
flow $\mathcal{C}_{t}$.
###### Lemma 2.4.
Let $\Omega$ be a 1-form in a Riemannian manifold $\tilde{M}$, $s$ be the arc-
length parameter of $\mathcal{C}_{t}$. Let
$\triangle^{\mathcal{C}_{t}}=\frac{\partial^{2}}{\partial s^{2}}$ be the
Laplace operator on $\mathcal{C}_{t}$, and $T=\frac{\partial}{\partial s}$ be
the unit tangent vector. Thus we have
(2.5) $\frac{\partial}{\partial
t}\Omega(T)=\triangle^{\mathcal{C}_{t}}\Omega(T)+|A|^{2}\Omega(T)-\nabla^{2}\Omega(T,T,T)-2\nabla\Omega(T,\nabla_{T}T);$
In particular, if $\Omega$ is parallel in $\tilde{M}$, we obtain
(2.6) $\frac{\partial}{\partial
t}\Omega(T)=\triangle^{\mathcal{C}_{t}}\Omega(T)+|A|^{2}\Omega(T);$
###### Proof.
Recall $T=\frac{C_{*}(\frac{\partial}{\partial
u})}{<C_{*}(\frac{\partial}{\partial u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{1}{2}}}$ and $\frac{\partial}{\partial t}=\vec{H}$. And
(2.7) $\begin{split}[\frac{\partial}{\partial
t},T]&=\frac{[C_{*}(\frac{\partial}{\partial
t}),C_{*}(\frac{\partial}{\partial u})]}{<C_{*}(\frac{\partial}{\partial
u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{1}{2}}}+C_{*}(\frac{\partial}{\partial u})\frac{\partial}{\partial
t}(\frac{1}{<C_{*}(\frac{\partial}{\partial u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{1}{2}}});\\\ &=-C_{*}(\frac{\partial}{\partial
u})\frac{<\nabla_{C_{*}(\frac{\partial}{\partial
t})}(C_{*}(\frac{\partial}{\partial u})),C_{*}(\frac{\partial}{\partial
u})>}{<C_{*}(\frac{\partial}{\partial u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{3}{2}}};\\\ &=-C_{*}(\frac{\partial}{\partial
u})\frac{<\nabla_{C_{*}(\frac{\partial}{\partial
u})}(\vec{H}),C_{*}(\frac{\partial}{\partial
u})>}{<C_{*}(\frac{\partial}{\partial u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{3}{2}}};\\\ &=|A|^{2}T;\end{split}$
Therefore, the relation of $\nabla_{\frac{\partial}{\partial t}}T$ and
$\nabla_{T}\frac{\partial}{\partial t}$ is given by
(2.8) $\nabla_{\frac{\partial}{\partial
t}}T-\nabla_{T}\frac{\partial}{\partial t}=|A|^{2}T;$
(2.9)
$\begin{split}\triangle^{\mathcal{C}_{t}}\Omega(T)&=T\big{(}\Omega(\nabla_{T}T)+\nabla\Omega(T,T)\big{)};\\\
&=\Omega(\nabla_{T}\frac{\partial}{\partial
t})+\nabla\Omega(\frac{\partial}{\partial
t},T)+\nabla^{2}\Omega(T,T,T)+2\nabla\Omega(T,\nabla_{T}T);\end{split}$
The $t$-derivative of $\Omega(T)$
(2.10)
$\begin{split}\frac{d}{dt}\Omega(T)&=\Omega(\nabla_{\frac{\partial}{\partial
t}}T)+\nabla\Omega(\frac{\partial}{\partial t},T);\\\
&=\Omega(\nabla_{T}\frac{\partial}{\partial
t})+|A|^{2}\Omega(T)+\nabla\Omega(\frac{\partial}{\partial
t},T);\quad\end{split}$
Therefore, (2.9) and (2.10) lead to the lemma. ∎
###### Remark 2.1.
It’s easily checked $min_{\mathcal{C}_{t}}\Omega$ is a Lipschitz function of
$t$, and $min_{\mathcal{C}_{t}}\Omega(T)$ is differentiable with respect to
$t$ almost everywhere. When $\Omega$ ia a parallel 1-form in $\tilde{M}$,
$\Omega(T)>0$ for $t\in[0,q_{0})$. For
$\triangle^{\mathcal{C}_{t}}\Omega(T)\geq 0$ for the points which attain the
minimal value of $\Omega(T)$, for a.e $t\in[0,q_{0})$ we have the following:
$\frac{d}{dt}min_{\mathcal{C}_{t}}\Omega(T)\geq|A|^{2}min_{\mathcal{C}_{t}}\Omega(T)\geq
0;$
This implies that $min_{\mathcal{C}_{t}}\Omega(T)$ is nondecreasing along the
flow. Then $\Omega(T)\geq\delta_{0}$ will be preserved whenever the curve
shortening flow exists.
## 3\. Long time existence
With the evolution results in the previous section, we prove Theorem 0.1. The
short time existence of $\mathcal{C}_{t}$ is from the short time existence of
solution of the quasilinear parabolic equation for closed initial data. It’s
well known that if the mean curvature flow $F_{t}$ of the hypersurface in
Euclidean space exists for only finite time interval $[0,q)$,
$max_{F_{t}}|A|\rightarrow\infty$ when $t$ approach to $q$ ([Hui84]). For the
submanifold in Euclidean space with high codimension, such kind of lemma is
proved by Andrews-Baker ([AB10]). For the curve shortening flow’s case, we
give the proof of the following lemma only for the sake of completeness.
###### Lemma 3.1.
Let $\tilde{M}$ be a Riemannian manifold. If the curve shortening flow
$\mathcal{C}_{t}$ in $\tilde{M}$ exists for $t$ in the maximal finite interval
$[0,q)$, then $max_{\mathcal{C}_{t}}|A|^{2}\rightarrow\infty\ \text{as}\
t\rightarrow q$.
###### Proof.
If the lemma is false, there exists a constant $C_{4}<\infty$ such that
$max_{\mathcal{C}_{t}}|A|\leq C_{4}$ for $t\in[0,q)$. It follows that for all
$u\in S^{1}$ and $t_{1},t_{2}\in[0,q)$.
$\displaystyle dist(\mathcal{C}_{t_{1}}(u),\mathcal{C}_{t_{2}}(u))$
$\displaystyle\leq|\int\limits_{t_{1}}^{t_{2}}\vec{H}dt|\leq
C_{4}|t_{1}-t_{2}|;$ $\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}ds_{t}$
$\displaystyle\geq-C_{4}\int_{\mathcal{C}_{t}}ds_{t};$
Therefore, $\mathcal{C}_{t}$ converges uniformly to some continuous limit
$\mathcal{C}_{q}(u)$ and the length $l_{\mathcal{C}_{t}}\geq\delta>0$ for
$t\in[0,q)$. We want to show that $\mathcal{C}_{q}(u)$ actually represents a
smooth limit curve. Then we can extend the flow $\mathcal{C}_{t}$ over time
$q$. It’s a contradiction to the maximal finite interval in the assumption.
By (2.4), we are led to
(3.1) $\begin{split}\frac{\partial}{\partial
t}|\nabla^{m}A|^{2}&\leq\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}+C_{m}\sum_{i+j+k=m}|\nabla^{i}A||\nabla^{j}A||\nabla^{k}A||\nabla^{m}A|\\\
&+D_{m}\sum_{j\leq
m}|\nabla^{j}A||\nabla^{m}A|+\tilde{C}_{m}|\nabla^{m}A|;\end{split}$
By Cauchy inequality, we have the follows.
(3.2) $\displaystyle|\nabla^{m}A|^{2}$ $\displaystyle\leq
min_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}+\int_{\mathcal{C}_{t}}\frac{\partial}{\partial
s}|\nabla^{m}A|^{2}ds_{t};$
$\displaystyle\leq\frac{1}{l_{\mathcal{C}_{t}}}\int|\nabla^{m}A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}<\nabla^{m}A,\nabla^{m+1}A>ds_{t};$
$\displaystyle\leq(\frac{1}{\delta}+2)\int|\nabla^{m}A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$
If $\mathcal{C}_{t}$ satisfies $|\nabla^{m}A|\leq C_{m}$ for all $t<q$, then
$\mathcal{C}_{q}$ is a smooth curve. From our definition about $\nabla^{m}A$,
the proof of this fact is classical but a little tedious. Therefore we omit it
here. For the Euclidean case, please refer to the proof of Theorem 3 in
([AB10]).
By (3.2), we have to prove that
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq C_{m}$ for all $m$. We
argue $|\int_{\mathcal{C}_{t}}\nabla^{m}Ads_{t}|\leq C_{m}$ by induction on
$m$. If it holds up for $k\leq m-1$. Then by (3.2) for $k\leq m-2$,
$max_{\mathcal{C}_{t}}|\nabla^{k}A|\leq C_{k}$ for $t\in[0,q)$. (3.1) and
(3.2) imply that
$\frac{\partial}{\partial
t}\int_{ct}|\nabla^{m}A|^{2}ds_{t}\leq\tilde{C}(m)\big{(}\int_{ct}|\nabla^{m}A|^{2}ds_{t}+(\int_{ct}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}\big{)};$
With the above inequality, we obtain
$(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}+1\leq
e^{\frac{\tilde{C}(m)}{2}t}((\int_{\mathcal{C}_{0}}|\nabla^{m}A|^{2}ds_{0})^{\frac{1}{2}}+1);$
Then we conclude that for all
$m=1,\cdots,max_{\mathcal{C}_{t}}|\nabla^{m}A|\leq C_{m}$. $C_{q}(u)$ is the
smooth limit curve of $\mathcal{C}_{t}$ for $t<q$. This will lead to the
contradiction, since we suppose $[0,q)$ is the maximal time interval for the
existence of $\mathcal{C}_{t}$. ∎
Now let $\mu=\frac{1}{\Omega(T)}$. The following lemma tells us $\mu$ can be
viewed as the gradient function $\nu$ in ([EH89]). They have the same form of
evolution equation.
###### Lemma 3.2.
Let $M$ be a closed manifold, $\Omega$ be a parallel 1-form in $M$, $s$ be the
arc-length parameter of $\mathcal{C}_{t}$. If the curve shortening flow
$\mathcal{C}_{t}$ exists for $t\in[0,q)$ and $\mathcal{C}_{0}$ satisfies
$\Omega(T)>0$, then $\mu$ satisfies
(3.3)
$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})\mu=-|A|^{2}\mu-2\mu^{-1}|\frac{\partial\mu}{\partial
s}|^{2};$
###### Proof.
$\displaystyle(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})\mu$
$\displaystyle=-\frac{1}{\Omega(T)^{2}}(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})\Omega(T)-2\Omega(T)|\frac{\partial\Omega(T)}{\partial
s}|;$
$\displaystyle=-\frac{1}{\Omega(T)^{2}}(|A|^{2}\Omega(T))-2\Omega(T)|\frac{\partial}{\partial
s}\frac{1}{\Omega(T)}|^{2};$
$\displaystyle=-|A|^{2}\mu-2\mu^{-1}|\frac{\partial\mu}{\partial s}|^{2};$
∎
###### Lemma 3.3.
Use the assumption in lemma 3.2. Let $C_{0}=max_{x\in M}R$. If the curve
shortening flow $\mathcal{C}_{t}$ exists for $t\in[0,q)$, then
(3.4)
$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})|A|^{2}\mu^{2}\leq-2\mu^{-1}\frac{\partial\mu}{\partial
s}\frac{\partial|A|^{2}\mu^{2}}{\partial s}+2C_{0}|A|^{2}\mu^{2};$
###### Proof.
$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})|A|^{2}=-2|\nabla
A|^{2}+2|A|^{4}+2R(T,\vec{H},T,\vec{H});$
and together with the identity
$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})\mu^{2}=-2|A|^{2}\mu^{2}-6|\frac{\partial\mu}{\partial
s}|^{2}$
yields
(3.5) $(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})|A|^{2}\mu^{2}\leq-2|\nabla
A|^{2}\mu^{2}-6|\frac{\partial\mu}{\partial
s}|^{2}|A|^{2}-2\frac{\partial|A|^{2}}{\partial
s}\frac{\partial\mu^{2}}{\partial s}+2R(T,\vec{H},T,\vec{H})\mu^{2};$
and
(3.6) $\displaystyle-2\frac{\partial|A|^{2}}{\partial
s}.\frac{\partial\mu^{2}}{\partial s}$
$\displaystyle\leq-\frac{\partial|A|^{2}}{\partial
s}\frac{\partial\mu^{2}}{\partial s}-4\mu|A|\frac{\partial|A|}{\partial
s}\frac{\partial\mu}{\partial s};$
$\displaystyle\leq-\mu^{-2}\frac{\partial\mu^{2}}{\partial
s}\frac{\partial|A|^{2}\mu^{2}}{\partial s}+4|\frac{\partial\mu}{\partial
s}|^{2}|A|^{2}-4\mu|A|\frac{\partial|A|}{\partial
s}\frac{\partial\mu}{\partial s}$
$\displaystyle\leq-2\mu^{-1}\frac{\partial\mu}{\partial
s}\frac{\partial|A|^{2}\mu^{2}}{\partial s}+2|\frac{\partial|A|}{\partial
s}|^{2}\mu^{2}+6|\frac{\partial\mu}{\partial s}|^{2}|A|^{2};$
Here we use the fact $\frac{\partial|A|}{\partial s}\leq|\nabla A|$. For $M$
is closed, $|\vec{H}|^{2}=|A|^{2}$, $|R(T,\vec{H},T,\vec{H})|\leq
C_{0}|A|^{2}$. By (3.5) and (3.6), we obtain
(3.7)
$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})|A|^{2}\mu^{2}\leq-2\mu^{-1}\frac{\partial\mu}{\partial
s}\frac{\partial|A|^{2}\mu^{2}}{\partial s}+2C_{0}|A|^{2}\mu^{2};$
∎
###### Corollary 2.
Under the assumption of Theorem 0.1. Then $max_{\mathcal{C}_{t}}|A|\leq
C(q,C_{0})$ when $t<q$. Here $C(q,C_{0})$ is a constant only dependent on $q$
and $C_{0}$.
###### Proof.
Because $M$ is closed, for any unit vector field $S$ in $TM$, $|\Omega(S)|\leq
C_{M}$.
$\mu^{-1}\frac{\partial\mu}{\partial s}=\Omega(T)\frac{\partial}{\partial
s}\frac{1}{\Omega(T)}=-\mu\Omega(\vec{H});$
Since $|\mu\Omega(\vec{H})|$ is continuous for $(u,t)\in S^{1}\times[0,q)$,
the maximal principle of parabolic equation in (3.7) gives ,
(3.8) $\frac{\partial}{\partial t}max_{\mathcal{C}_{t}}|A|^{2}\mu^{2}\leq
2C_{0}max_{\mathcal{C}_{t}}|A|^{2}\mu^{2};$
From $\mu\geq\frac{1}{C_{M}}$ and (3.8), we conclude the corollary. ∎
#### The proof of Theorem 1
Now we prepare everything which we need to prove Theorem 0.1. From corollary
2, for all finite time interval $[0,q)$, $max_{\mathcal{C}_{t}}|A|$ is always
uniformly bounded (depending on $q$). If $\mathcal{C}_{t}$ exists only for a
finite interval, Lemma 3.1 will lead to a contraction. Then Theorem 1 is
followed.
## 4\. The Convergence
This section is devoted to prove Theorem 0.2. Recall Theorem 0.2 concludes
that the limit of $max_{\mathcal{C}_{t}}|\nabla^{m}A|$ is $0$ for all
$m=0,1,\cdots$ if the curve shortening flow $\mathcal{C}_{t}$ in the closed
manifold $\tilde{M}$ exists for all $t\in[0,\infty)$ and the length of
$\mathcal{C}_{t}$ converges to a positive number. The idea of its proof
originated from Section 7 in ([Gra89]). We suppose the length of
$\mathcal{C}_{t}$ satisfies $l_{\mathcal{C}_{t}}\geq l_{\infty}>0$. First,
let’s prove a technique lemma.
###### Lemma 4.1.
For all $m\geq 0$,
(4.1) $|\nabla^{m}A|^{2}\leq
a\int|\nabla^{m}A|^{2}ds_{t}+2\int|\nabla^{m+1}A|^{2}ds_{t};$
Here $a=\frac{1}{l_{\infty}}+2$. For $m\geq 1$,
(4.2)
$(\int|\nabla^{m}A|^{2}ds_{t})^{2}\leq\int|\nabla^{m-1}A|^{2}ds_{t}\int|\nabla^{m+1}A|^{2}ds_{t};$
###### Proof.
For $|\nabla^{m}A|^{2}$,
$\displaystyle|\nabla^{m}A|^{2}$ $\displaystyle\leq
min_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}+\int_{\mathcal{C}_{t}}\frac{\partial}{\partial
s}|\nabla^{m}A|^{2}ds_{t};$
$\displaystyle\leq\frac{1}{l_{\mathcal{C}_{t}}}\int|\nabla^{m}A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}<\nabla^{m}A,\nabla^{m+1}A>ds_{t};$
$\displaystyle\leq(\frac{1}{l_{\mathcal{C}_{t}}}+2)\int|\nabla^{m}A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$
Integrating $\int|\nabla^{m}A|^{2}ds_{t}$ by parts,
$\int|\nabla^{m}A|^{2}ds_{t}=\int\frac{\partial}{\partial
s}<\nabla^{m-1}A,\nabla^{m}A>ds_{t}-\int<\nabla^{m-1}A,\nabla^{m+1}A>ds_{t};$
The first term is 0. Using the Cauchy inequality we obtain (4.2). ∎
#### The proof of Theorem 0.2
Our strategy is to prove for all $m=0,1,\cdots$, $\int|\nabla^{m}A|^{2}ds_{t}$
converges to 0 as $t\rightarrow\infty$. First, we prove $m=0$ case, then argue
by induction.
From the defintion of the curve shortening flow, we get the following two
facts.
(4.3)
$\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}ds_{t}=-\int|\vec{H}|^{2}ds_{t}=-\int_{\mathcal{C}_{t}}|A|^{2}ds_{t};$
(4.4) $\displaystyle
l_{\mathcal{C}_{t}}-l_{\mathcal{C}_{0}}=\int\limits_{0}^{t}\frac{d}{dt}\int_{\mathcal{C}_{t}}ds_{t}dt=-\int\limits_{0}^{t}\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}dt<\infty;$
There exists a measure zero set $E\,\text{in}\,[0,\infty)$. Let
$E_{n}=[n,\infty)\cap E^{c}$. We get
(4.5) $max_{t\in E_{n}}\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}\rightarrow
0\quad\text{as}\quad n\rightarrow\infty;$
From (2.3), we differentiate $\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}$ with
respect to $t$.
(4.6) $\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}$
$\displaystyle\leq-2\int|\nabla
A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}|A|^{4}ds_{t}+D_{0}\int|A|^{2}ds_{t};$
$\displaystyle\leq-2\int|\nabla
A|^{2}ds_{t}+(a\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}|\nabla
A|^{2}ds_{t})\int|A|^{2}ds_{t}$ $\displaystyle+D_{0}\int|A|^{2}ds_{t};$
$\displaystyle\leq-(2-2\int_{\mathcal{C}_{t}}|A|^{2}ds_{t})\int_{\mathcal{C}_{t}}|\nabla
A|^{2}ds_{t}+\int|A|^{2}ds_{t}(D_{0}+a\int|A|^{2}ds_{t});$
Here we use (4.1). Let $n$ be a sufficiently large number. We can assume
$(2-2\int_{\mathcal{C}_{t}}|A|^{2}ds_{t})\geq 1$ and
$D_{0}+a\int|A|^{2}ds_{t}\leq D_{0}+1$ for all $t\in E_{n}$. Furthermore,
(4.7)
$\frac{d}{dt}\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}\leq(D_{0}+1)\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}$
For both sides of (4.7) are continuous with respect to $t$, (4.7) holds up for
$t\in[n,\infty)$. $\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}$ increases at most with
exponential growth when $n$ is sufficiently large. By (4.5), we obtain
$\lim_{t\rightarrow\infty}\int|A|^{2}ds_{t}=0$.
We notice that $\tilde{M}$ is closed, all $|\nabla^{m}R|\leq C_{m}\,\text{for
all}\,m$. Together with (2.4), we have the following inequality for
$m=1,\cdots$.
(4.8) $\frac{\partial}{\partial
t}|\nabla^{m}A|^{2}\leq\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}\\\
+\tilde{C}_{m}\sum_{i+j+k=m}|\nabla^{i}A||\nabla^{j}A||\nabla^{k}A||\nabla^{m}A|+D_{m}\sum_{j\leq
m}|\nabla^{j}A||\nabla^{m}A|+\tilde{C}_{m}|\nabla^{m}A|;$
Here all constants in this proof are only depending on $C_{m}$.
When $m\geq 1$, we argue by the induction on $m$. Suppose
$\lim\limits_{t\rightarrow\infty}\int_{\mathcal{C}_{t}}|\nabla^{k}A|^{2}ds_{t}=0$
for $k\leq m-1$. Then by (4.1), for $k\leq m-2$,
$\lim\limits_{t\rightarrow\infty}max_{\mathcal{C}_{t}}|\nabla^{k}A|^{2}=0$.
For any $\epsilon>0$, let $t_{0}$ be a positive number sufficiently large,
such that for $t\geq t_{0}$ and $k\leq m-2$,
$\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}\leq\epsilon$ and
$max_{\mathcal{C}_{t}}|\nabla^{k}A|^{2}\leq\epsilon$. By (4.8), we obtain the
following estimates.
$\displaystyle\frac{\partial}{\partial t}|\nabla^{m}A|^{2}$
$\displaystyle\leq\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}+\tilde{C}_{m}(\epsilon)(|\nabla^{m}A|^{2}$
$\displaystyle+|\nabla^{m}A||\nabla^{m-1}A|)+\tilde{D}_{m}|\nabla^{m}A|;$
$\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$
$\displaystyle\leq-2\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t}+\tilde{C}_{m}(\epsilon)(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$
$\displaystyle+\int_{\mathcal{C}_{t}}|\nabla^{m}A||\nabla^{m-1}A|ds_{t})+\tilde{D}_{m}\int_{\mathcal{C}_{t}}|\nabla^{m}A|ds_{t};$
Here $\tilde{C}_{m}(\epsilon)$ goes to 0 as $\epsilon$ converges to 0.
Let $C>3$ be a constant determined later. If
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\geq
C\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}$, the following estimates are
given by (4.2).
(4.9) $\displaystyle\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$
$\displaystyle\leq\frac{1}{C}\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$
$\displaystyle\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}$
$\displaystyle\leq\frac{1}{C^{2}}\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$
As a result,
(4.10) $\displaystyle\int_{\mathcal{C}_{t}}|\nabla^{m}A||\nabla^{m-1}A|ds_{t}$
$\displaystyle\leq\frac{1}{\sqrt{C^{3}}}\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$
$\displaystyle\int_{\mathcal{C}_{t}}|\nabla^{m}A|ds_{t}$
$\displaystyle\leq\frac{\sqrt{l_{\mathcal{C}_{0}}}}{\sqrt{C}}(\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}};$
The $t$-derivative of $\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$ can be
written as followed:
(4.11)
$\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$
$\displaystyle\leq-(2-\tilde{C}_{m}(\epsilon)(\frac{1}{\sqrt{C^{3}}}+\frac{1}{C}))\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t}$
$\displaystyle+\frac{\tilde{D}_{m}}{\sqrt{C}}(\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}};$
Let $C$ be sufficiently large such that
$(2-\tilde{C}_{m}(\epsilon)(\frac{1}{\sqrt{C^{3}}}+\frac{1}{C}))\geq 1$ and
$\frac{\tilde{D}_{m}}{\sqrt{C}}\leq 2\epsilon$. (4.11) becomes
(4.12)
$\frac{d}{dt}\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq-\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}}(\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}}-2\epsilon);$
Now we can conclude Theorem 0.2 as follows.
1. (1)
If $\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\geq
C\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}$,
1. (a)
If $(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}\geq
3\epsilon$, we have
$(\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}}\geq
3\epsilon$. From (4.12), then
$\frac{d}{dt}\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq-2\epsilon^{2}$.
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$ will decrease until
$(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})\leq(3\epsilon)^{2}$
If we can find a $t_{0}$ such that
$(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}\leq 3\epsilon;$
then for all $t>t_{0}$,
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq(3\epsilon)^{2}$.
2. (b)
If $(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}\leq
3\epsilon$, the conclusion of theorem 0.2 is obviously true.
2. (2)
If
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}<C\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}$,
we have $\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq C\epsilon$.
In a word, finally $\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$ converges
to 0 when $t\rightarrow\infty$. By (4.1), $max_{\mathcal{C}_{t}}|\nabla^{m}A|$
converges to 0 for all $m$ as $t$ goes to $\infty$.
## References
* [AB10] Ben Andrews and Charles Baker. Mean curvature flow of pinched submanifolds to spheres. J. Differential Geom., 85(3):357–395, 2010.
* [AB11] Ben Andrews and Paul Bryan. A comparison theorem for the isoperimetric profile under curve-shortening flow. Comm. Anal. Geom., 19(3):503–539, 2011.
* [Alt91] Steven J. Altschuler. Singularities of the curve shrinking flow for space curves. J. Differential Geom., 34(2):491–514, 1991.
* [EH89] Klaus Ecker and Gerhard Huisken. Mean curvature evolution of entire graphs. Ann. of Math. (2), 130(3), 1989.
* [Gag84] M. E. Gage. Curve shortening makes convex curves circular. Invent. Math., 76(2):357–364, 1984.
* [GH86] M. Gage and R. S. Hamilton. The heat equation shrinking convex plane curves. J. Differential Geom., 23(1):69–96, 1986.
* [Gra87] Matthew A. Grayson. The heat equation shrinks embedded plane curves to round points. J. Differential Geom., 26(2):285–314, 1987.
* [Gra89] Matthew A. Grayson. Shortening embedded curves. Ann. of Math. (2), 129(1):71–111, 1989.
* [Hui84] Gerhard Huisken. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom., 20(1):237–266, 1984.
* [Hui86] Gerhard Huisken. Contracting convex hypersurfaces in riemannian manifolds by their mean curvature. Invent. Math., 84(3):463–480, 1986.
* [Hui98] Gerhard Huisken. A distance comparison principle for evolving curves. Asian J. Math., 2(1):127–133, 1998.
* [KFLZ10] F. Ye K. F. Liu, H.W. Xu and E. T. Zhao. The extension and convergence of mean curvature flow in higher codimension, 2010. arXiv:1104.0971v1 [math.DG].
* [KFLZ11] F. Ye K. F. Liu, H. W. Xu and E. T. Zhao. Mean curvature flow of higher codimension in hyperbolic spaces, 2011. arXiv:1105.5686v1 [math.DG].
* [MW11] Ivana Medoš and Mu-Tao Wang. Deforming symplectomorphisms of complex projective spaces by the mean curvature flow. J. Differential Geom., 87(2):309–341, 2011.
* [Smo12] Knut Smoczyk. Mean Curvature Flow in Higher Codimension Introduction and Survey, volume 17 of Springer Proceedings in Mathematics, pages 231–274. Springer Berlin Heidelberg, 2012.
* [SW02] Knut Smoczyk and Mu-Tao Wang. Mean curvature flows of Lagrangians submanifolds with convex potentials. J. Differential Geom., 62(2):243–257, 2002.
* [TW04] Mao-Pei Tsui and Mu-Tao Wang. Mean curvature flows and isotopy of maps between spheres. Comm. Pure Appl. Math., 57(8):1110–1126, 2004.
* [Wan01] Mu-Tao Wang. Mean curvature flow of surfaces in Einstein four-manifolds. J. Differential Geom., 57(2):301–338, 2001.
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|
arxiv-papers
| 2012-12-21T16:39:34 |
2024-09-04T02:49:39.569150
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hengyu Zhou",
"submitter": "Hengyu Zhou",
"url": "https://arxiv.org/abs/1212.5515"
}
|
1212.5568
|
# Symmetry breaking in dipolar matter-wave solitons in dual-core couplers
Yongyao Li1,2,3, Jingfeng Liu2, Wei Pang4, and Boris A. Malomed1
[email protected] 1Department of Physical Electronics, School of
Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv
69978, Israel
2Department of Applied Physics, South China Agricultural University, Guangzhou
510642, China
3Modern Educational Technology Center, South China Agricultural University,
Guangzhou 510642, China
4 Department of Experiment Teaching, Guangdong University of Technology,
Guangzhou 510006, China.
###### Abstract
We study effects of the spontaneous symmetry-breaking (SSB) in solitons built
of the dipolar Bose-Einstein condensate (BEC), trapped in a dual-core system
with the dipole-dipole interactions (DDIs) and hopping between the cores. Two
realizations of such a matter-wave coupler are introduced, weakly- and
strongly-coupled. The former one in based on two parallel pipe-shaped traps,
while the latter one is represented by a single pipe sliced by an external
field into parallel layers. The dipoles are oriented along axes of the pipes.
In these systems, the dual-core solitons feature the SSB of the supercritical
type and subcritical types, respectively. Stability regions are identified for
symmetric and asymmetric solitons, and, in addition, for non-bifurcating
antisymmetric ones, as well as for symmetric flat states, which may also be
stable in the strongly-coupled system, due to competition between the
attractive and repulsive intra- and inter-core DDIs. Effects of the contact
interactions are considered too. Collisions between moving asymmetric solitons
in the weakly-symmetric system feature elastic rebound, merger into a single
breather, and passage accompanied by excitation of intrinsic vibrations of the
solitons, for small, intermediate, and large collision velocities,
respectively. A $\mathcal{PT}$-symmetric version of the weakly-coupled system
is briefly considered too, which may be relevant for matter-wave lasers.
Stability boundaries for $\mathcal{PT}$-symmetric and antisymmetric solitons
are identified.
###### pacs:
42.65.Tg; 03.75.Lm; 47.20.Ky; 05.45.Yv
## I Introduction
Studies of Bose-Einstein condensates (BECs) made of dipolar atoms or molecules
had produced a great deal of fascinating experimental and theoretical results,
which were summarized in recent reviews manual ; Lahaye and Baranov ; Lahaye
, respectively. The continuation of the work in this direction has yielded new
remarkable findings, such as the prediction of various pattern-formation
mechanisms Patterns ; roton-layered (which share some features with the
formation of patterns in ferrofluids Richter ), analysis of the stability of
the dipolar BEC trapped in optical-lattice (OL) potentials OL and of the
roton instability roton-layered ; roton , the possibility of the Einstein - de
Haas effect Haas , etc. Important experimental achievements, which offer new
perspectives for studies of dipole-dipole interactions (DDIs) in atomic
condensates, are the creation of BEC in dysprosium Dy and erbium Er .
Parallel to that, essential results have been obtained for degenerate quantum
gases of dipolar fermions Baranov ; Fermi .
In addition to their own physical significance, dipolar condensates may also
be used as quantum simulators simulator ; Lewenstein representing other
physical media where nonlocal nonlinearities play a fundamental role. These
include the heating and ionization of plasmas Litvak , nonlinear optics of
nematic liquid crystals Peccianti , of waveguides sensitive to temperature
variations Krolik , and of semiconductor cavities Falk , BECs with long-range
interactions induced by laser illumination Gershon , and others settings.
An interesting ramification of the study of collective nonlinear modes in the
dipolar BEC is the prediction of solitons (which have not yet been reported in
experimental works). In effectively one-dimensional (1D) traps, solitons were
analyzed in both continual 1D-cont and discrete 1D-discr settings, the
latter one corresponding to the fragmentation of the BEC by a deep OL. In a
similar form, 1D solitons supported by the attractive DDIs were predicted in
the Tonks-Girardeau gas of dipolar hard-core bosons TG . Taking into account
the 3D structure of the quasi-1D cigar-shaped traps, the solitons, including
ones with embedded vorticity (cf. similar modes introduced earlier in the
context of BEC with local interactions Luca ) were further studied in Ref.
SKA1 , and gap solitons in a similar setting, but including an OL potential,
were considered too SKA2 . In the 2D system, discrete fundamental solitons and
solitary vortices with long-range DDIs between sites of the lattice can be
constructed easily 2D-discr . In the continual 2D model, fundamental Pedri
and vortical vortex solitons were constructed in the isotropic setting,
assuming that the sign of the DDI could be reversed from repulsion to
attraction by means of a rapidly oscillating ac field inversion . 2D
fundamental and vortex solitons supported by a trapping potential were
introduced in Ref. Lashkin .
Without reversing the DDI sign, stable 2D anisotropic solitons, corresponding
to the in-plane polarization of dipoles, were constructed in Ref. Tikhonenkov
, by means of the variational approximation and systematical numerical
simulations. The variational approximation for 2D solitons supported by the
DDI was analyzed in Ref. Wunner-VA , and a rigorous proof of the existence of
such solitons was provided too rigorous . Also studied were more complex
situations, such as the formation of a multi-soliton patterns as a result of
the development of the modulational instability of an extended state MI-
formation .
The long-range character of the DDI makes it possible to consider interactions
between condensate layers trapped in parallel planar waveguides. The DDI
couples them by nonlinear forces even in the absence of hopping (tunneling) of
atoms across gaps separating the layers roton-layered (the isolation of
parallel layers can be provided by a strong OL field whose axis is
perpendicular to the layers OL-first ). This nonlocal interaction gives rise
to “indirect” scattering of 2D solitons moving in the separated layers stack-
scattering , and the formation of bound states of such solitons stack-molecule
. The creation of multi-soliton filaments and checkerboard crystals in multi-
layered stack was predicted too crystal-in-stack .
The model of nonlocal DDIs between parallel layers considered in Refs. stack-
scattering ; roton-layered ; crystal-in-stack ; stack-molecule did not take
into regard the hopping (tunneling of atoms, alias linear coupling) between
the layers. On the other hand, models of dual-core couplers, with intrinsic
local nonlinearity acting in both cores, were studied in detail in terms of
optics and matter waves, starting from the analysis of the spontaneous
symmetry breaking (SSB) of CW (continuous-wave, i.e., uniform) states in dual-
core optical fibers with the cubic and more general forms of the intra-core
nonlinearity. In that system, the linear coupling is caused by the overlap of
the evanescent field, originating from each core, with the parallel one. The
SSB happens in the dual-core fiber, as a result of the interplay of the linear
coupling and intrinsic nonlinearity, with the increase of the total power of
the CW beam. The analysis of the SSB was extended, in full detail, to temporal
and spatial solitons fibers ; fibers2 , and to optical domain walls DW . The
SSB effects were also studied for solitons in dual-core fiber Bragg gratings
BraggSSB , in two-tier waveguiding arrays (for discrete solitons) Amherst , in
parallel-coupled waveguides with the quadratic (second-harmonic-generating)
chi2 and cubic-quintic (CQ) CQ nonlinearities, as well as for dissipative
solitons in linearly-coupled CQ complex Ginzburg-Landau equations CGL .
Recently, a similar analysis was developed for the SSB of solitons in
$\mathcal{PT}$-symmetric couplers, with mutually balanced loss and gain (and
identical cubic nonlinearities) acting in the two cores PT ; PT2 . Unlike the
above-mentioned settings, in the latter case the SSB destroys symmetric
solitons, rather than replacing them by stable asymmetric ones. The linear
coupling in optics may also represent the mutual interconversion of two
polarizations of light in twisted fibers (which, in particular, are used in
the so-called rocking filters) twist , twisted photonic-crystal fibers PCF-
twist , twisted fiber gratings grating-twist , or the interconversion of two
waves with different carrier frequencies, caused by the electromagnetically-
induced transparency yongyao .
Similar dual-core (alias double-well) settings, approximated by linearly-
coupled Gross-Pitaevskii equations (GPEs), were introduced for the mean-field
wave functions describing trapped BEC with local interactions Arik . A similar
linear coupling accounts for the mutual interconversion in a mixture of two
different atomic states, induced by a resonant electromagnetic wave Ballagh .
The linearly coupled GPEs were used to predict the shift of the miscibility-
immiscibility transition in BEC or fermionic mixtures of two states connected
by the linear interconversion Merhasin and the stabilization of 2D solitons
against the collapse in a linearly-coupled binary system with attractive and
repulsive intra-component interactions Ueda . The SSB of matter-wave solitons
trapped in 1D and 2D linearly-coupled cores was analyzed in Ref. Arik . In the
1D situation, a more accurate description, which, nevertheless, yields similar
results for the solitons’ SSB, is provided by the two-dimensional GPE, which,
instead of postulating two 1D wave functions in the two parallel cores with
the linear exchange between them, introduces a single 2D wave function
comprising both cores Marek . This includes the case when the cores are
defined by nonlinear pseudopotentials (rather than by a linear trapping
potential), i.e., local modulation of the self-attraction coefficient Hung .
Although the previous works analyzed many aspects of the SSB in dual-core CW
and solitonic states, those works were dealing solely with local intrinsic
nonlinearities. Only in a very recent paper Fangwei , a shift of the SSB
transition of solitons in the coupler with nonlocal nonlinearity of the
thermal type, typical to optical systems Krolik , was considered. The analysis
was performed for two opposite limit cases, viz., the weak nonlocality
characterized by a small correlation radius, which may be approximated by the
first two terms of the expansion of the nonlocal cubic term, and the opposite
limit of the infinite correlation radius, which corresponds to the two-
component quasi-linear model of “accessible solitons” accessible .
The aim of the present work is to consider the SSB of solitons in the
effectively 1D dual-core coupler filled by the dipolar condensate, which
exhibits the interplay of the long-range DDIs and linear hopping between the
quasi-1D cores. In particular, the DDIs act both inside the cores and between
them, while the thermal nonlocality considered in Ref. Fangwei could not act
across the gap separating the parallel waveguides. It is relevant to stress
that, in the absence of the longitudinal dimension, the double-well setting is
not sufficient to exhibit the nonlocal character of the interactions in the
dipolar condensate, the minimum necessary configuration being based on a set
of three potential wells triple .
Two coupler configurations are considered here, as shown in Fig. 1. The first
setting, presented in Fig. 1, is based on two identical condensate-trapping
pipes of diameter $b$, separated by distance $a$. In this case, $b<a$ is
implied, hence the system may be naturally called a weakly-coupled one. The
other setting is shown in Fig. 2, with the condensate loaded into the single
pipe of diameter $b$, which is sliced into two parallel layers by a thin
potential barrier of small thickness $a$. The barrier can be induced by a
repulsive light sheet (blue-shifted one, with respect to the atoms) sheet .
The latter setting implies $b>a$, and it will be named, accordingly, a
strongly-coupled system. In either case, the dipoles are polarized along the
pipes’ axes, hence the DDIs are attractive inside the cores, which makes it
possible to form solitons in each one 1D-cont .
Figure 1: (Color online) (a) In the weakly-coupled system, the dipolar
condensates are trapped in parallel pipes of diameter $b$, separated by
distance $a$. The arrows represent the orientation of the dipoles. (b) The
strongly-coupled system is shown by means of the cross-section image of the
condensate trapped in the single pipe of diameter $b$, which is sliced by a
repelling laser sheet into two layers, with effective separation $a$ between
them. Symbol $\otimes$ represents the orientation of the dipoles, which are
perpendicular to the figure’s plane.
Aiming to study the SSB, alias the symmetry-breaking phase transition, in
these settings, it is relevant to recall that there are two types of the SSB
bifurcation, namely, the subcritical and supercritical ones Joseph , which are
tantamount to the phase transitions of the first and second kinds,
respectively. In the subcritical situation, branches representing asymmetric
modes emerge as unstable states at the bifurcation point, then go backward in
the bifurcation diagram, and get stabilized after turning forward. In the
supercritical setting, the asymmetric branches emerge as stable ones at the
bifurcation point and immediately continue forward.
The rest of the paper is organized as follows. In Sec. II, we construct
symmetric solitons in the two settings presented in Fig. 1, and then identify
the SSB transitions to asymmetric solitons. Antisymmetric solitons and their
stability are considered too (these solitons do not undergo any bifurcation).
Basic results for the stability of different modes are presented (including a
flat state, which may also be stable in the strongly-coupled system). Effects
of the local (contact) nonlinearity on the SSB are considered too. In Sec.
III, we study collisions between moving asymmetric solitons. In Sec. IV, a
$\mathcal{PT}$-symmetric Bender extension of this system is briefly
considered, which includes gain and loss applied to the two cores (similar to
the $\mathcal{PT}$-symmetric coupler with the local nonlinearity introduced in
Refs. PT ; PT2 ). The paper is concluded by section V.
## II The symmetry-breaking bifurcation and stability of solitons
### II.1 The coupled Gross-Pitaevskii equations
In the usual mean-field approximation Lahaye ; Baranov , both dual-core
settings introduced in Fig. 1 are described by the system of 1D linearly-
coupled GPEs for the wave functions in the two cores, $\psi_{1}$ and
$\psi_{2}$. In the scaled form, the equations are
$\displaystyle i{\frac{\partial\psi_{n}}{\partial
t}}=-{\frac{1}{2}}{\frac{\partial^{2}\psi_{n}}{\partial
x^{2}}}+g|\psi_{n}|^{2}\psi_{n}-\kappa\psi_{3-n}$ $\displaystyle-
G_{\mathrm{DD}}\psi_{n}(x)\int_{-\infty}^{+\infty}\left[{\frac{|\psi_{n}(x^{\prime})|^{2}}{(b^{2}+|x-x^{\prime}|^{2})^{3/2}}}-\frac{1}{2}{\frac{(1-3\cos^{2}\theta)|\psi_{3-n}(x^{\prime})|^{2}}{2(a^{2}+|x-x^{\prime}|^{2})^{3/2}}}\right]dx^{\prime},$
(1)
where $n=1,2$ and
$\cos\theta=|x-x^{\prime}|/(a^{2}+|x-x^{\prime}|^{2})^{1/2}$, see Fig. 1(a).
In this notation, $\kappa$ is the coupling parameter (hopping coefficient),
$g$ represent the local interaction (repulsive in the case of $g>0$), and the
orientation of the dipoles in Fig. 1 corresponds to $G_{\mathrm{DD}}>0$, which
implies the attraction between dipoles in the given core and repulsion between
the cores for $\cos^{2}\theta<1/3$. The first term in the integrand, that
accounts for the intra-core DDIs, is regularized by the transverse diameter,
$b$, which is an approximation sufficient for producing 1D solitons 1D-cont .
The solitons will be characterized by the total norm (proportional to the
number of atoms in the condensate),
$P\equiv
P_{1}+P_{2}=\int_{-\infty}^{+\infty}\left(|\psi_{1}|^{2}+|\psi_{2}|^{2}\right)dx.$
To focus on SSB effects dominated by the DDI, we will first drop the local
nonlinearity, setting $g=0$ (in the experiment, this can be done by means of
the Feshbach resonance Feshbach ); effects of the contact interactions will be
considered afterwards. Then, we scale the units to fix $\kappa\equiv 1$ and
$G_{\mathrm{DD}}\equiv 1$, the remaining free parameters being $a$, $b$, and
$P$ (and $g$ too, in the end).
Stationary solutions to Eq. (1) with chemical potential $\mu$ are looked for
in the usual form, $\psi_{1,2}(x,t)=\exp\left(-i\mu t\right)\phi_{1,2}(x)$,
with real functions $\phi_{1,2}(x)$. In particular,
$\phi_{1,2}(x)\equiv\phi(x)$ for symmetric solutions obeys the stationary
equation,
$\displaystyle\left(\mu+1\right)\phi=-{\frac{1}{2}}{\frac{d^{2}\phi}{dx^{2}}}+g\phi^{3}$
$\displaystyle-\phi(x)\int_{-\infty}^{+\infty}\left[{\frac{1}{(b^{2}+|x-x^{\prime}|^{2})^{3/2}}}-{\frac{1-3\cos^{2}\theta}{2(a^{2}+|x-x^{\prime}|^{2})^{3/2}}}\right]\phi^{2}(x^{\prime})dx^{\prime},$
(2)
where $\kappa=G_{\mathrm{DD}}=1$ is fixed, as said above. Equations (1) and
(2) are solved below by means of numerical methods. In particular, stationary
solutions were found below by means of the imaginary-time propagation method
imaginary with periodic boundary condition, while the stability of the
solutions was tested by means of the integration in real time.
### II.2 The weakly-coupled system ($a>b$)
Typical examples of stable symmetric, asymmetric, and antisymmetric solitons
found in the system represented by Fig. 1 are displayed in Fig. 2. Naturally,
the asymmetric solitons, morphed by the stronger nonlinearity, are narrower
and taller than their symmetric counterparts.
It is relevant to mention that, unlike the standard model of the coupler with
the local cubic nonlinearity, where symmetric and antisymmetric solitons are
available in an obvious exact form, and asymmetric ones can be effectively
described by means of the variational approximation fibers2 ; Progress , in
the present strongly nonlocal system such an approximation is not practically
possible. Nevertheless, some results can be obtained in an approximate
analytical form for the present system too, see Eqs. (5) and (6) below.
Before proceeding to the consideration of the transition between the symmetric
and asymmetric solitons, we display the stability area of the antisymmetric
ones (which do not undergo any antisymmetry-breaking bifurcation), in the
plane of $\left(b,P\right)$, in Fig. 2. The stability boundary may be fitted
to curve
$P=7b^{3}.$ (3)
Unstable antisymmetric solitons gradually decay into radiation (not shown here
in detail).
Figure 2: (Color online) Examples of stable symmetric (a), asymmetric (b), and
antisymmetric (c) solitons found in the weekly-coupled system [see Fig. 1(a)]
for $a=1,$ $b=0.4$, and total norm $P=0.4$ (a) and $P=0.6$ (b,c) The
antisymmetric solitons are stable below the boundary, $P\approx 7b^{3}$, shown
in the plane of $\left(b,P\right)$ in panel (d), at different values of $a$.
The SSB for solitons is summarized in Fig. 2 by means of the bifurcation
diagrams, which clearly show that the symmetry-breaking bifurcation, driven by
the nonlocal attractive DDIs, is supercritical, on the contrary to the
commonly known subcritical bifurcation in the coupler with the local self-
focusing nonlinearity fibers ; fibers2 . A similar trend to the change of the
character of the SSB bifurcation for solitons from sub- to supercritical, with
the increase of the degree of the nonlocality, was recently demonstrated in
the model of the optical coupler with the weakly nonlocal thermal nonlinearity
Fangwei .
Figure 3: (Color online) (a) The bifurcation diagram for solitons in the
weakly-coupled system: The soliton’s asymmetry, measured by the deviation of
the share of the total power in one core ($P_{1}/P$) from $0.5$, versus total
norm $P$. (b) The bifurcation diagram as a function of the pipes’ diameter,
$b$. The circles located along the solid and dashed lines represent stable and
unstable solutions, respectively. (c) The critical value of the total norm at
the symmetry-breaking point as a function of $b$, for fixed $a=1$. Stable
asymmetric solitons exist above the curve, while the symmetric solitons are
stable below it. The curve is well fitted by $P_{\mathrm{cr}}=3b^{2}$.
As seen from the structure of the first term in the integrand of Eq. (1), the
decrease of diameter $b$ of the parallel pipes implies effective enhancement
of the nonlinearity. This, as well as the direct strengthening of the
nonlinearity due to the increase of the total norm ($P$), leads to the
symmetry breaking, as seen in Figs. 3 and 3. Further, Fig. 3 demonstrates the
related effect of the decrease of the critical value, $P_{\mathrm{cr}}$, of
the total norm at the bifurcation point with the decrease of $b$. The latter
dependence may be fitted to formula $P_{\mathrm{cr}}=3b^{2}$, which is
explained by the fact that, at small $b$, the nearly diverging first integral
term in Eq. (2) may be estimated as $A^{2}/b$, where $A$ is the soliton’s
amplitude. Its balance with other terms in the equation leads to estimates for
scalings of the amplitude and width:
$A\sim P/b,W\sim b^{2}/P.$ (4)
Then, as the SSB point is determined by the competition between the nonlinear
intra-core terms and linear inter-core-coupling ones in Eq. (1) Snyder -fibers
, the corresponding scaling for the value of $P$ at the critical point indeed
takes the form of
$P_{\mathrm{cr}}\sim\sqrt{\kappa}b^{2}$ (5)
(here, $\kappa$ is kept for clarity, although it was actually scaled to be
$\kappa\equiv 1$).
Note that, according to Eq. (3), the stability region for antisymmetric
solitons is much smaller at small $b$. This agrees with the general trend of
the antisymmetric solitons in nonlinear couplers to be more fragile modes than
their symmetric counterparts fibers ; PT ; PT2 , due to the obvious fact that
they correspond to a larger coupling energy. In fact, the cubic scaling in Eq.
(3) may be qualitatively explained too, although in a more vague form than Eq.
(5). Indeed, the antisymmetric soliton with amplitude $A$ is subject to an
oscillatory instability characterized by complex growth rates (eigenvalues),
$\lambda=\pm i\kappa+\mathrm{Re}\left(\lambda\right)$, where, in the generic
case, an estimate $\mathrm{Re}\left(\lambda\right)\sim A$ is valid (see, e.g.
Ref. PT2 ). Because the instability is oscillatory, the corresponding
perturbations tend to escape from the region of width $W$ , occupied by the
soliton [see Eq. (4)], within time $\tau\sim W/V_{\mathrm{gr}}$, where the
group velocity is determined by the characteristic wavenumber of the
perturbation mode, $k\sim W^{-1}$, i.e., $\tau\sim W^{2}$. The instability
accounted for by the escaping perturbations seems as convective one, that will
have enough time to destroy the antisymmetric soliton under condition
$\mathrm{Re}\left(\lambda\right)\tau\sim 1$, i.e., according to the above
estimates,
$AW^{2}\sim b^{3}/P\sim 1,$ (6)
which qualitatively explains the numerically found fit (3).
It is also worthy to note that additional analysis demonstrates that, as is
strongly suggested by Figs. 2 and 3, the general picture of the SSB, being
sensitive to the value of $b$, shows little dependence on the separation
between the cores, $a$ (in the case of $a>b$, which is considered here). In
other words, the SSB in the weakly-coupled system, in accordance with its
name, is weakly sensitive to the DDI between the parallel cores, which renders
the picture relatively simple.
### II.3 The strongly-coupled system ($a<b$)
In the setting displayed in Fig. 1, the small separation between the effective
cores makes effects of the inter-core DDIs essentially stronger, in comparison
with the weakly-coupled system. In fact, the strongly-coupled system realizes
an example of competing interactions, namely, intra-core attraction and inter-
core repulsion. A somewhat similar example is the discrete Salerno model with
competing signs of the onsite and intersite cubic nonlinearities, which was
studied in 1D and 2D settings Zaragoza .
The numerical solution demonstrates that, in addition to symmetric and
asymmetric solitons, stable flat states with unbroken symmetry between the
cores also exist in the strongly-coupled system, being stabilized by the
strong dipolar repulsion between the cores. A typical example of such stable
states is displayed in Fig. 4.
Figure 4: (Color online) Examples of stable modes found in the strongly-
coupled system. (a) A symmetric soliton , with parameters
$(a,b,P)=(0.3,0.5,0.2)$. (b) A symmetric flat state with
$(a,b,P)=(0.1,0.5,0.2)$. (c) An asymmetric soliton, with
$(a,b,P)=(0.2,0.5,1)$.
In comparison to their counterparts in the weakly coupled system (cf. Fig. 2),
the symmetric solitons are wider in the case of the strong coupling. This is
also an effect of the repulsive DDI between the cores, which partly cancels
the intra-core attraction, which forms the symmetric modes. The transition to
the asymmetric soliton occurs, as before, with the increase of the total norm,
the difference from the weakly coupled system being that the smaller component
of the asymmetric mode demonstrates a split-peak structure in Fig. 4(b). This
feature originates from the anisotropy [i.e., the $\theta$-dependence in the
second term in the integrand of Eq. (1)] of the DDI between the cores.
Results of the systematic analysis of the strongly-coupled system are
summarized in Fig. 5 and 5, in the form of stability diagrams for all the
three types of the modes, in the plane of $\left(a,P\right)$, for two
different fixed values of $b$. There are two bistability areas, where the
stable flat state and asymmetric solitons coexist [the large yellow triangular
and trapezoidal regions in the left parts of Figs. 5 and 5, respectively], or
the symmetric and asymmetric solitons are simultaneously stable (small orange
regions in the right parts if the same figures). The presence of the latter
bistability area implies that the SSB transition between the symmetric and
asymmetric solitons is subcritical in the strongly-coupled system, as
explicitly shown in the bifurcation diagram in Fig. 5 [cf. Fig. 3]. It is thus
concluded that the repulsive DDI between the cores plays an increasingly more
important role with the decrease of the separation between them, $a,$ which
becomes the dominant parameter of the system at $a\lesssim(2/3)b$. In
particular, only in this region the symmetric flat state may be stable. On the
other hand, the effect of parameter $a$ is inconspicuous at $a>(2/3)b$, like
in the weakly-coupled system, see the previous subsection. Accordingly, at
$a\rightarrow b$, the area of the bistability between the symmetric and
asymmetric solitons (the small orange area in Figs. 5 and 5) shrinks to zero,
which demonstrates that the SSB changes to the supercritical type, like in the
weekly-couple case. Finally, antisymmetric solitons and flat states can be
also found as stationary solutions of the strongly-coupled system, but both
these species of the modes turn out to be completely unstable.
Figure 5: (Color online) (a), (b) Stability diagrams in the plane of the total
norm ($P)$ and separation $a$ between the cores of the strongly-coupled
system, for two different values of the overall diameter, $b$. Symmetric
solitons are stable in the bottom right areas colored by red, flat states in
the bottom left areas, shown in blue, and asymmetric solitons—in the green top
areas on the left and right sides. The middle-left yellow areas are regions of
the bistability area of flat state and asymmetric solitons, while the small
orange areas at the right center harbor the bistability of the symmetric and
asymmetric solitons. (c) The bifurcation diagram for solitons in the strongly-
coupled system: The soliton’s asymmetry, measured by the deviation of the
share of the total power in one core ($P_{1}/P$) from $0.5$, versus total norm
$P$. The type of the symmetry-breaking bifurcation is subcritical here.The
dashed segments, which link the stable asymmetric branches and the stable
symmetric one, designate the actually missing unstable branches, which, as
usual Marek , could not be found by means of the imaginary-time-integration
method.
### II.4 Effects of the local nonlinearity
The inclusion of the local self-attractive ($g<0$) or self-repulsive ($g>0$)
nonlinearity into the coupled GPEs, Eq. (1), shifts the point of the SSB
bifurcation (the critical value of the total power, $P_{\mathrm{cr}}$), but it
does not change the supercritical character of the bifurcation in the weakly-
coupled system. The bifurcation diagrams for the system of this type, in the
presence of the local nonlinearity of either sign, are presented in Figs. 6(a)
and (b), cf. Fig. 3. In addition, Fig. 6(c) displays the dependence of the
bifurcation point, $P_{\mathrm{cr}}$, on strength $g$ of the local
nonlinearity. Naturally, the latter dependence is weaker for smaller values of
the pipes’ diameter, $b$, as the DDI is stronger at smaller $b$, suppressing
the effect of the local nonlinearity. It is natural too that the self-
attractive local nonlinearity ($g<0$) makes $P_{\mathrm{cr}}$ smaller, while
the local self-repulsion ($g>0$) makes it larger. Similarly, the addition of
the contact nonlinearity does not change the type of SSB bifurcation in the
strongly-coupled system either (not shown here in detail).
Figure 6: (Color online) The bifurcation diagrams in the weakly-coupled
system, with $(a,b)=(1,0.4)$, in the presence of the self-attractive (a),
$g=-2$, or self-repulsive (b), $g=+2$, local nonlinearity. (c) The total norm
at the bifurcation point as a function of strength $g$ of the additional
contact nonlinearity, at different fixed values of $b$.
## III Mobility and collisions between solitons
Collisions between solitons represent an important aspect of the dynamics of
integrable and nonintegrable models Yang . In dual-core systems, one can study
collisions between symmetric solitons, and, which is most interesting, between
asymmetric ones with equal or opposite polarities, i.e., with larger
components belonging to the same or different cores polarity . These three
cases are schematically defined in Fig. 7.
Figure 7: Three types of collisions between solitons in the dual-core system:
(a) “Unipolar” asymmetric solitons, with the larger components belonging to
the same core. (b) Asymmetric solitons with opposite “polarities”. (c) The
collision between symmetric solitons.
We here report numerical results for collisions between solitons in the
weakly-coupled system, taking the initial state at $t=0$ as a pair of far-
separated kicked solitons,
$\psi_{1,2}^{(0)}=U_{1,2}^{(1)}(x+x_{0},P)e^{i\eta(x+x_{0})}+U_{1,2}^{(2)}(x-x_{0},P)e^{-i\eta(x-x_{0})},$
(7)
where $U_{1,2}^{(1,2)}(x\pm x_{0},P)$ are the two-component soliton solutions
with total powers $P$, which are centered at $x=\mp x_{0}$, with sufficiently
large initial separation $2x_{0}$, and $\eta$ is the kick (momentum imparted
to the soliton). Solutions $U^{(1)}$ and $U^{(2)}$ are either identical ones,
or, in the above-mentioned case of opposite polarities, these are two
asymmetric solitons with swapped components, see Fig. 7(b).
In Figs. 8 and 9, we display typical results of collisions for asymmetric
soliton pairs in the case which is close to the border between the weakly- and
strongly-bound systems, $(a,b,P)=(1,0.4,0.5)$. In panels 8(a,b) and 9(a,b), it
is observed that slowly moving solitons bounce back from each other
elastically. When the kick and ensuing velocities are larger, the collision
becomes inelastic, leading to the merger of the solitons into a single
asymmetric breather. Panels 8(c,d) and 8(e,f) demonstrate, severally, that the
merger of the colliding unipolar solitons may switch the polarity of the
emerging breather, in comparison with the original solitons, or produce a
breather which keeps the original polarity. The merger of the solitons with
opposite polarities gives rise to an asymmetric breather, whose polarity is
established spontaneously, as seen in Figs. 9(c,d). Finally, at still larger
values of the kick, the moving solitons pass through each other, reappearing
in an excited form (each one as a moving breather), see Figs. 8(g,h) and
9(e,f). The norms of the outgoing vibrating solitons are equal, as they were
before the collision.
Figure 8: (Color online) Examples of collisions between unipolar asymmetric
solitons: (a,b) for the slow solitons, with $\eta=0.1$; (c,d) for the
intermediate velocity, $\eta=0.4$; (e,f) for a larger intermediate velocity,
$\eta=0.8$; (g,h) for the fast solitons, $\eta=1.6$.
Figure 9: (Color online) Examples of collisions between asymmetric solitons
with opposite polarities: (a,b) for the slow solitons, with $\eta=0.1$; (c,d)
for the intermediate velocity, $\eta=0.4$; (e,f) for the fast solitons,
$\eta=1.6$.
The same sequence of outcomes of the collisions—rebound, merger into a
breather, and passage in the form of moving breathers—is observed, with the
increase of the kick ($\eta$), at other values of the parameters. Collisions
between symmetric solitons seem simpler (not shown here in detail): Rebound at
small values of $\eta$, and passage, without conspicuous excitation of
intrinsic vibrations, at larger $\eta$.
## IV The PT-symmetric version of the weakly coupled system
It was proposed to realize $\mathcal{PT}$-symmetric systems in BEC by linearly
coupling two traps (cores, in the present terms) with the loss of atoms in one
of them, and compensating supply in the other, which may be provided by a
matter-wave laser PT-BEC . The objective of the analysis was to provide for a
direct realization of the $\mathcal{PT}$ symmetry in quantum media, after it
was proposed PT-optics and implemented Guo in classical optics, see also
review doulides .
Accordingly, the $\mathcal{PT}$-balanced version of linearly-coupled GPEs (1)
is
$\displaystyle i{\frac{\partial\psi_{1}}{\partial
t}}=-{\frac{1}{2}}{\frac{\partial^{2}\psi_{1}}{\partial
x^{2}}}-\psi_{2}+i\gamma\psi_{1}$
$\displaystyle-\psi_{1}\int_{-\infty}^{+\infty}\left[{\frac{|\psi_{1}(x^{\prime})|^{2}}{(b^{2}+|x-x^{\prime}|^{2})^{3/2}}}-\frac{1}{2}(1-3\cos^{2}\theta){\frac{|\psi_{2}(x^{\prime})|^{2}}{(a^{2}+|x-x^{\prime}|^{2})^{3/2}}}\right]dx^{\prime},$
$\displaystyle i{\frac{\partial\psi_{2}}{\partial
t}}=-{\frac{1}{2}}{\frac{\partial^{2}\psi_{2}}{\partial
x^{2}}}-\psi_{1}-i\gamma\psi_{2}$
$\displaystyle-\psi_{2}\int_{-\infty}^{+\infty}\left[{\frac{|\psi_{2}(x^{\prime})|^{2}}{(b^{2}+|x-x^{\prime}|^{2})^{3/2}}}-\frac{1}{2}(1-3\cos^{2}\theta){\frac{|\psi_{1}(x^{\prime})|^{2}}{(a^{2}+|x-x^{\prime}|^{2})^{3/2}}}\right]dx^{\prime},$
(8)
where $\gamma>0$ is the coefficient of the gain and loss in the first and
second cores, respectively, $\kappa=G_{\mathrm{DD}}\equiv 1$ is fixed, as
above, and the local nonlinearity is dropped ($g=0$). As well as in the
recently studied model of the $\mathcal{PT}$-symmetric coupler with the cubic
nonlinearity PT ; PT2 , stationary $\mathcal{PT}$-symmetric and antisymmetric
solutions to Eq. (8) can be found as
$\displaystyle\psi_{1,2}^{\mathrm{(symm)}}(x,t)$ $\displaystyle=$
$\displaystyle e^{-i\mu
t}\phi(x)\exp\left(\pm\frac{1}{2}i\arcsin\gamma\right)\equiv e^{-i\mu
t}\phi_{1,2}(x),$ (9) $\displaystyle\psi_{1,2}^{\mathrm{(anti)}}(x,t)$
$\displaystyle=$ $\displaystyle\pm ie^{-i\mu
t}\phi(x)\exp\left(\mp\frac{1}{2}i\arcsin\gamma\right),$ (10)
where the upper and lower signs correspond to subscripts $1$ and $2$,
respectively, $\mu$ is a real chemical potential, and real function $\phi(x)$
is the solution of stationary equation (2) for symmetric solitons in the
system without the $\mathcal{PT}$ terms, and with the same value of $\mu$.
Obviously, the symmetric and antisymmetric solitons exist for $\gamma<1$, and
they form continuous families parameterized by $\mu$, like their counterparts
in the conservative system.
Here we focus on the analysis of the stability of the $\mathcal{PT}$-symmetric
and antisymmetric solitons (9), which is a nontrivial problem in the present
context. Typical examples of stable symmetric solitons are displayed in Fig.
10, along with their counterpart in the system without the $\mathcal{PT}$
terms ($\gamma=0$).
Figure 10: (Color online) Examples of stable $\mathcal{PT}$-symmetric
solitons, defined as per Eq. (9), with $(a,b,P)=(1,0.4,0.3)$, for $\gamma=0$,
$0.3$, and $0.6$. (a) The profile of $\left|\psi_{1,2}(x)\right|$, which is
common for the three solitons. (b,c) Real and imaginary part of the solitons.
The $\mathcal{PT}$-symmetric and antisymmetric solitons remain stable up to a
certain critical value, $\gamma_{\mathrm{cr}}$, of the gain-loss coefficient,
and are unstable in the interval of $\gamma_{\mathrm{cr}}<\gamma<1$, which is
a situation typical for solitons in $\mathcal{PT}$-symmetric systems PT-
soliton ; PT ; PT2 . The unstable solitons suffer a blowup of the pumped
component and decay of the damped one, which is a typical scenario too (not
shown here in detail). The most essential results are presented in Fig. 11 for
the weakly-coupled system, in the form of the dependence of
$\gamma_{\mathrm{cr}}$ on the total norm for different diameters of the
parallel-coupled pipes, $b$ (recall $b$ was demonstrated above to be the most
essential coefficient for the weakly-coupled system). The figure demonstrates
that the instability region expands as the nonlinear interactions get
stronger, which is caused either by the increase of the total norm ($P$), or
decrease of $b$.
Figure 11: (Color online) Critical value $\gamma_{\mathrm{cr}}$ (the stability
boundary for the $\mathcal{PT}$-symmetric solitons) versus the total norm,
$P$, at different fixed values of the pipe’s parameter, $b$, in the weakly-
coupled system. (a) For the symmetric solitons (9); (b) for antisymmetric ones
(10). The continuous and dashed curves are guides for eye.
## V Conclusion
The objective of this work is to extend the study of the symmetry breaking of
solitons in dual-core systems with the cubic nonlinearity and linear coupling
between the cores, that has been previously analyzed in full detail for local
interactions, to dipolar BEC with the long-range interactions, which act both
inside each core and between them. Two versions of the system were introduced,
weakly- and strongly-coupled ones, depending on the relation between the
diameter of the pipe-shaped traps and the distance between them. In either
case, the linear coupling accounts for hopping of atoms between the cores. The
symmetry-breaking bifurcation and stability regions for symmetric and
asymmetric solitons, as well as for non-bifurcating antisymmetric ones, have
been identified in both systems. In addition to the solitons, the strongly-
coupled system supports a stability region for flat states with the unbroken
symmetry between the cores, due to the competition between the attractive and
repulsive intra- and inter-core dipole-dipole interactions. Collisions between
kicked asymmetric solitons in the weakly-coupled system were systematically
studied too, showing bouncing back at small velocities, merger into an
asymmetric breather in the intermediate range, and reappearance of vibrating
fast solitons after the collision. Finally, a $\mathcal{PT}$-symmetric
generalization of the weakly coupled system was introduced as a more “exotic”
extension of the system, and a stability boundary for $\mathcal{PT}$-symmetric
and antisymmetric solitons was found.
A challenging generalization of the analysis may be its application to 2D
dual-core systems, where, in particular, the symmetry breaking may occur not
only for fundamental solitons, but also for solitary vortices.
###### Acknowledgements.
We appreciate a valuable discussion with L. Santos. This work was supported by
Chinese agency CNNSF (grants No. 11104083, 11204089, 11205063), by the
Guangdong Provincial Science and technology projects (2011B090400325), by the
German-Israel Foundation through grant No. I-1024-2.7/2009, and by the Tel
Aviv University in the framework of the “matching” scheme for a postdoctoral
fellowship of Y.L.
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|
arxiv-papers
| 2012-12-21T19:30:40 |
2024-09-04T02:49:39.578045
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yongyao Li, Jingfeng Liu, Wei Pang, and Boris A. Malomed",
"submitter": "Yongyao Li",
"url": "https://arxiv.org/abs/1212.5568"
}
|
1212.5585
|
# The presheaf of Quantum Realities and constructions of the space-time from
the Space of Ultimation
Innocenti V. Maresin
###### Abstract
The paper presents several quantum models constructed with the formalism of
algebraic geometry. The Universe is presented as a presheaf on the “space of
ultimation” (i.e. the branching future) with values in certain category which
enhances the category opposite to C*-algebras. Given models show the
possibility to derive ordinary space-times (of both Special and General
Relativity) from ultrametric spaces.
This work is supported in part by the Russian grant NSh-2928.2012.1.
This work is supported in part by Russian Foundation for Basic Research,
grant 10-01-00178.
### 0.1 Introduction
The paper actually presents two things. First, it is an interpretation of
Quantum Mechanics with multiple possible futures. Its central concept is a
Quantum Reality, which consists of an algebra of observables and one state on
it, and actually is a variant of the Heisenberg’s picture. This contrasts to
the Schrödinger’s picture, where the state depends on the time coordinate.
Though, Quantum Realities postulate certain non-unitary “causal” evolution of
the state, together with the algebra.
This is a non-local theory, but not in the same sense as hidden parameters
theories are. Quantum Realities admit that any reality, in some sense, depends
on its futures, and is not related to causally independent realities (except
through their common past).
Second, an appropriate mathematical formalism is presented, which is also used
to derive the relativistic space-time, from an underlying structure. This
formulation can be used for theories where solutions branch in future
directions. If any reality depends on all its possible futures, then it is
handy to parameterize realities by sets of their futures. This leads to the
concept of the ultimate future, the topological “space of ultimation” and to
application of the presheaf formalism, where “restriction morphisms” to
subsets are, essentially, possible outcomes of a quantum measurement.
## 1 Basic concepts
### 1.1 C*-algebras
C*-algebra ${\mathcal{A}}$ is a complex Banach algebra with an antilinear
involution, which reverses the order of multiplication: ${(a\hskip
1.25ptb)}^{*}={b}^{*}\hskip 1.25pt{a}^{*}$. The norm must satisfy:
$||a||^{2}=||{a}^{*}\hskip 1.25pta||$
The theory of C*-algebras was presented in details, for example, in [1].
${{\mathcal{A}}_{herm}}$ is a real linear subspace of self-conjugate
(Hermitian) elements. By definition, the set of “positive” elements
${\mathcal{A}}^{+}=\\{a\hskip 1.25pt{a}^{*}\,|\,a\in{\mathcal{A}}\,\\}$;111 An
estabilshed, but confusing terminology. Actually, it is a generalization for
non-negative real numbers $[0,+\infty)$. it is a cone in
${{\mathcal{A}}_{herm}}$.
A *-homomorphism does not have to keep the norm. Though, one can prove that a
*-homomorphism cannot increase the norm of an element:
$||f({a}^{*}\hskip 1.25pta)||=||f(a)||^{2},\ ||f({a}^{*}\hskip 1.25pta\hskip
1.25pt{a}^{*}\hskip 1.25pta)||=||f(a)||^{4},\ ||f(({a}^{*}\hskip
1.25pta)^{4})||=||f(a)||^{8},\ \cdots$
The standard definition of a C*-algebra does not require existence of the
unity. Henceforth, we’ll assume that any of our algebras has the “1” element,
although it’s possible that ${\mathbf{1}}={\mathbf{0}}$ (in the {0} algebra,
which consists of the only “0” element). It always exists in a finite-
dimensional C*-algebra. If “1” does not exist in an infinite-dimensional
algebra, then it can always be created by adding exactly one dimension.
Note that we does not require a *-homomorphism to preserve “1”. Obviously,
$f({\bf 1})={f({\bf 1})}^{*}\hskip 1.25ptf({\bf 1})$ is “positive”.
### 1.2 Categories and presheaves
Category is a mathematical concept which formalizes relations between
mathematical “objects” (of the same structure) via morphisms from one object
to another; refer to [5] for detailed theory. It usually, but not always, is
used to represent some additional structure on sets. Most well-known
categories (such as sets, groups and other algebraic structures, topological
spaces…) are so named concrete categories,[6] which means that objects and
morphisms can be encoded as sets and functions between them. Not all
categories are concrete, though. In other words, an object of the category
does not necessarily have the structure of a set.
The concept of a topological space is assumed to be well-known. A presheaf $F$
on a topological space $X$ with values in a category C
($F\in\operatorname{{\mathrm{Preshv}}}(X,\mathrm{C})$) maps any open subset
$U$ of $X$ to an object $F(U)$ of C. Also, for any two open subsets $V\subset
U\subset X$ a restriction morphism $F(U)\to F(V)$ must be defined, which
equals to the identity morphism if $V=U$.
$F(W)\hskip 25.83325pt\to\hskip 25.83325ptF(U)$ $\to\ F(V)\ \to$ The last
condition on $F$ to be a presheaf is for any open $W\subset V\subset U\subset
X$ the diagram of three restriction morphisms must commute. must commute. This
means that the restriction morphism from $F(U)$ to $F(W)$ must be equal to the
composition of ones from $F(V)$ to $F(W)$ and from $F(U)$ to $F(V)$.
Usually, C is a concrete category; then, elements of objects from a presheaf
are referred to as germs. Though, Quantum Realities need the general,
categorical definition of a presheaf, because their morphisms are not
functions.
### 1.3 Metric spaces, ultrametrics and their generalizations
A metric space has a real-valued two-arguments metric function, or distance,
such that:
$d(\cdot,\cdot)\geq 0,\ \ \forall\epsilon,\zeta:\
d(\epsilon,\zeta)=d(\zeta,\epsilon),\ \ \forall\epsilon,\zeta:\
(d(\epsilon,\zeta)=0)\Leftrightarrow(\epsilon=\zeta),$
$\forall\epsilon,\zeta,\eta:d(\epsilon,\zeta)\leq
d(\epsilon,\eta)+d(\eta,\zeta)$
The topology of a metric space can be defined through the family of open
balls,
${\mathrm{B}_{r}}(\epsilon)=\\{\zeta\,|\,d(\epsilon,\zeta)<r\\}$
Also, if
$\forall\epsilon,\zeta,\eta:d(\epsilon,\zeta)\leq\max(d(\epsilon,\eta),d(\eta,\zeta))\,,$
then $d$ is referred to as an ultrametric.[7] One can easily realize that
$\varphi\operatorname{\circ}d$, where $\varphi:[0,+\infty)\to[0,+\infty)$
monotonically increases and $\varphi(0)=0$, is also an ultrametric. This
suggests that the codomain of an ultrametric may be any bounded upper
semilattice.[8] “Bounded” means that is has “0”, the minimal element. 222 For
upper semilattices, we’ll use notations of calculus (“0” and “max”), not the
symbol $\vee$ used in Wikipedia, which is ubiquitous in mathematical logic and
algebra. Though, it is not clear how to make such a space a topological
space, because there is no strict inequality “$<$” on a lattice, which is
necessary to define open balls.
For models of the space-time, where the codomain of a (generalized)
ultrametric will be referred to as the Locale of Time, we will assume the
following properties of it:
* •
It is a bounded upper semilattice with the partial order “$\leq$”, the minimal
element “0” and the join operation “max”.
* •
It is a non-Hausdorff topological space where for any open $R$:
$\forall\sigma\in R\ \forall\tau\leq\sigma:\ \tau\in R\,.$
* •
The topology of the Locale of Time is defined by a base, which generalizes
intervals $[0,r)$ of a real-valued ultrametric. So, the topology of a
(generalized) ultrametric space can be defined through the family of open
balls
${\mathrm{B}_{r}}(\epsilon)=\\{\zeta\,|\,d(\epsilon,\zeta)\in r\\}\,,$
where $r$ belongs to the aforementioned base, which implies that it is an open
subset (non-Hausdorff) of the Locale of Time.
* •
The intersection of any two subsets in the aforementioned base also belongs to
it.
First two requirements and the ultrametric inequality provide us with the fact
that any point of an open ball can be its center.
For a real-values ultrametric (or, conceptually speaking, if its codomain is
totally ordered) there are no non-trivial intersections of balls: any two
balls either do not intersect at all, or one includes another. For a
generalized ultrametric defined above it is not generally true. The purpose of
the last requirement is to ensure that the intersection of two balls is always
a ball. Since we actually use only balls, but never the value of the metric
itself, the Locale of Time may be even a pointless topological space (also
known as locale),[9] where open sets form a special case of a lattice, known
as frame.
We’ll use almost nothing of a (ultra)metric geometry beyond these two
properties.
## 2 Quantum Realities and measurements
### 2.1 Categories of quantum realities
A state $\rho$ on a C*-algebra ${\mathcal{A}}$ is such bounded linear
functional that:
$\forall a\in{\mathcal{A}}^{+}:\rho(a)\geq 0\,,\ \ \ ({\bf
1}>0)\Rightarrow(\rho({\bf 1})=1)$
Remind that ${\bf 1}=0$ in {0}. Such functionals corresponds to average values
of an observable in quantum mechanics. If there exist two C*-algebras
${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}\neq\\{0\\}$, a *-homomorphism
$f:{\mathcal{A}}_{2}\to{\mathcal{A}}_{1}$ and a state $\rho_{1}$ on
${\mathcal{A}}_{1}\,$, then $\rho_{1}(f({\bf 1}))$ is a non-negative real
number. If is is positive, then
$\rho_{2}:=\frac{\rho_{1}\operatorname{\circ}f}{\rho_{1}(f({\bf 1}))}\mbox{ is
a state on }{\mathcal{A}}_{2}\,.$
Let’s define the category QR-1 of Quantum Realities. An object of QR-1 is a
C*-algebra ${\mathcal{A}}$ and a state $\rho$ on it.
A morphism from an object $({\mathcal{A}}_{1},\rho_{1})$ to an object
$({\mathcal{A}}_{2},\rho_{2})$ is such *-homomorphism
$f:{\mathcal{A}}_{2}\to{\mathcal{A}}_{1}$ that
$\exists c\in[0,1]\ \forall a\in{\mathcal{A}}_{2}:\rho_{1}(f(a))=c\hskip
1.25pt\rho_{2}(a).$
We call the factor $c$ the Born’s factor. Obviously, $c=\rho_{1}(f({\bf 1}))$
if ${\mathcal{A}}_{2}\neq\\{0\\}$ and is indeterminate if
${\mathcal{A}}_{2}=\\{0\\}$.
The physical sense of a morphism is a transition (by quantum measurement) from
the reality 1 to the reality 2. Unless the Born’s factor equals to zero, the
state functional $\rho_{2}$ is determined uniquely, with aforementioned
equation, by the state $\rho_{1}$ and *-homomorphism $f$. So, unlike
observables, states are transformed in the same direction as morphisms in the
category QR-1, from past to future.
We’ll call the nihil 333 “Nothing” (Latin). Cognates with “annihilation”.
reality a reality based on the {0} algebra, and we’ll denote it as (), the
0-tuple. It is, obviously, the zero object[10] of corresponding categories;
its existence is crucial for the presheaf formalism. It also implies existence
of the zero morphism between any pair of realities, which map all observables
to 0 and have zero Born’s factor.
It is possible to consider also a full subcategory of QR-1, which includes the
nihil object. For example, the category where all algebras are full matrix
algebras (including $Mat(0\times 0)$).
Another possibility to define a Quantum Reality category is stateless
realities. The C*algopis the category opposite to the category of C*-algebras.
We’ll refer to both QR-1 (and its full subcategories which include “()”) and
C*algop(and its full subcategories which include {0}) to as Quantum Reality
categories (QR).
The singleton (reality) is the algebra ${\mathbb{C}}$ with its only possible
state, the identity functional. We’ll denote it as (1).
Other objects in a category of Quantum Realities based on full matrix algebras
and pure states444 The state is called pure if it cannot be represented as a
non-trivial convex combination of other states. For finite-dimensional
C*-algebras, any such state has the form
$\rho(\cdot)=\langle{\psi}^{*}|\cdot|\psi\rangle$, where $\psi$ is a ket-
vector, $|\psi|=1$, in the space of the representation of the algebra. In full
matrix algebras all states $\langle{\psi}^{*}|\cdot|\psi\rangle$ are pure.
may be denoted as $(z_{0}:z_{1}:\ldots:z_{n-1})$, where “:” refers to
projectivization, because pure states are actually parameterized by
${\mathbb{C}}\mathrm{P}^{n-1}$.
### 2.2 Relationship to quantum measurements
The most usual type of quantum measurements, a Von Neumann’s measurement,
relies on the *-homomorphism (and monomorphism)
${\mathcal{B}}({\mathcal{H}}_{l})\to{\mathcal{B}}(\mathop{\oplus}\limits_{k}{\mathcal{H}}_{k})\,$
which maps the element 1 to a projector and transforms a pure state
$\psi=\sum\limits_{k}\psi_{k}\mbox{\ to \ }\frac{\psi_{l}}{|\psi_{l}|^{2}}\
(\mbox{if }\psi_{l}\neq 0)$
with Born’s factor $|\psi_{l}|^{2}$ (it follows from the fact that “1” in
${\mathcal{B}}({\mathcal{H}}_{l})$ maps to the projector to
${\mathcal{H}}_{l}$). Its simplest example is two homomorphisms from the
singleton algebra ${\mathbb{C}}$ to $Mat(2\times 2)$:
${\mathfrak{p}}_{\uparrow}:a\mapsto\begin{pmatrix}a&0\\\ 0&0\end{pmatrix};\
{\mathfrak{p}}_{\downarrow}:a\mapsto\begin{pmatrix}0&0\\\ 0&a\end{pmatrix}.$
Another type of measurement represented in the QR formalism correspond to the
natural *-monomorphism of algebras of bounded operators
$\mathop{\oplus}\limits_{k}{\mathcal{B}}({\mathcal{H}}_{k})\to{\mathcal{B}}(\mathop{\oplus}\limits_{k}{\mathcal{H}}_{k})$
Note that, since this homomorphism preserves “1”, its Born’s factor must be
equal to 1. Several physicists refer to such case as a non-selective
measurement; see e.g. [3] 5.3.2 or [4].
Its “counterpart” in a complete Von Neumann’s measurement is a “selection”
homomorphism
${\mathcal{A}}_{l}\to\mathop{\oplus}\limits_{k}{\mathcal{A}}_{k}\,,$
the splitting of a mixed reality. Like Von Neumann’s measurement, it also maps
the element 1 to a projector. Its Born’s factor equals to
$\rho({\mathbf{1}}_{l}\oplus 0_{\mathrm{others}})$.
Note that Quantum Realities always map observables from the future to the
past. Such things as the “partial trace” has nothing to do with morphisms of
Realities.
### 2.3 Presheaf formalism and physical restrictions
The idea behind this interpretation of Quantum Mechanics is that any reality,
in some sense, includes all its possible futures. When we advance into the
future, the set of available futures shrinks. Hence, the formalism is based on
certain “space of ultimation” ${\mathcal{E}}$, which assumed to have a
topological structure. Its open subsets are called ultimation sets, and its
point are “points of ultimation”, or just “ultimations”.
The model of the Universe is described as:
$F\in\operatorname{{\mathrm{Preshv}}}({\mathcal{E}},\mathrm{QR}),\
F(\emptyset)=()$
where $\mathrm{QR}$ is a category of quantum realities (usually, QR-1).
From here forth, we will think about realities only as ultimation sets, and
causal relation will be expressed in set-theoretical language. Namely,
$V\subset U$ or $U\supset V$ means “$V$ is a future of $U$” or, the same, “$V$
is a consequence of the cause $U$”. It is handy to use inclusion signs to
denote the causal relation instead of arrows (for morphisms in QR), because it
helps to avoid confusion with *-homomorphisms, whose direction is opposite.
The mathematical requirement for $F$ to be a presheaf is, itself, quite weak.
To construct a physical theory some extra conditions must be satisfied. The
following three requirements, ordered from weakest to strongest, can be easily
guessed from philosophical and quantum-mechanical considerations.
Weak ex nihilo nihil fit 555 “Nothing comes from nothing.” (Latin) principle:
no restriction morphisms from the nihil reality
to a non-nihil reality. In other words, a presheaf must be nihil on any subset
of a nihil ultimation set. If we annihilate a presheaf on the union of all
nihil ultimation sets (i.e. put $F=()$ for any subset of the aforementioned
union), then it will be satisfied.
Strong ex nihilo nihil fit principle: any restriction morphism,
but a morphism to the nihil reality, may not be zero. It is stronger, because
zero restriction morphisms between non-nihil realities are possible in a
presheaf.
Positive Born’s factor principle: any restriction morphism,
but a morphism to the nihil reality, must have a positive Born’s factor. It is
stronger than the previous one, because even a non-zero morphism can map all
observables of the future reality to a subalgebra vanished by the state
functional of the past reality, and hence to have the zero Born’s factor. Note
that, for a presheaf of stateful quantum realities satisfying this principle,
the state in each reality is uniquely determined by the state corresponding to
the entire Space of Ultrimation (which is also an open set, the Absolute
Past), given algebras and their *-homomorphisms only.
Although this paper does not discuss probabilistic implications, it should be
noted that the positive Born’s factor principle does not preclude “zero-
probability” points of ultimation. In fact, it precludes zero-probability open
sets of ultimation, which appears to accord with such concept as the spectral
decomposition.[12] It is one of arguments in favor of the presheaf formalism.
Henceforth we assume that, in a physical theory of the presheaf on the space
of ultimation, the positive Born’s factor principle holds. Other physically
motivated restrictions apparently exist. For example, from quantum-mechanical
theory of measurement follows that ${\bf 1}$ of $F(U)$ must have a
representation as the sum (possibly, infinite; see e.g. [1] B29 for
explanations) of images of observables from any covering of $U$ by its
ultimation subsets. Such restrictions are not considered in this paper and
will be investigated later.
### 2.4 One-dimensional branching time model
Suppose that ${\mathcal{E}}$ is an ultrametric space, with an ordinary, real-
valued ultrametric. If $U={\mathrm{B}_{r}}(\epsilon)$ for certain
$\epsilon\in{\mathcal{E}}$ and $r>0$, then there exists a possibility that
different $r$ may fit to the same open set $U$. Namely, let
$r_{\mathrm{inf}}:=\sup\limits_{\zeta\in U}d(\epsilon,\zeta);\
r_{\mathrm{max}}:=\inf\limits_{\zeta\in{\mathcal{E}}\setminus
U}d(\epsilon,\zeta);$
and if $r_{\mathrm{max}}>r_{\mathrm{inf}}$ then any $r_{\mathrm{max}}\geq
r>r_{\mathrm{inf}}$ parametrizes the same ball $U$. In such a case, let us
think that $U$ corresponds to certain reality whose duration is the time
interval $[-\log r_{\mathrm{max}},-\log r_{\mathrm{inf}})$. If
$r_{\mathrm{max}}=r=r_{\mathrm{inf}}$, then the reality $F(U)$ is point-time
with $t=-\log r\,$. If the ball $V$ is a subset of $U$, then it’s easy to
realize that its $r_{\mathrm{max}}$ is not greater than $r_{\mathrm{inf}}$ of
$U$. Physical interpretation of the restriction morphism of $F(U)$ to $F(V)$
is a transition from $U$ to one of its futures. Possible future realities or,
more strictly, the covering of $U$ with smaller balls,666 In $p$-adic numbers,
and some other types of ultrametric spaces, the radius of balls in the
covering can be set as $r_{\mathrm{inf}}$. If values of the ultrametric do not
form a discrete set even locally and $r_{\mathrm{max}}=r_{\mathrm{inf}}$, then
smaller radii must be chosen for balls of a covering. can form a Von
Neumann’s quantum measurement, or other type of it.
Note that we have many different realities for any time moment $t$. They form
a branching structure, which is equivalent to a directed tree if values of the
ultrametric are (locally) discrete and, hence,
$r_{\mathrm{max}}>r_{\mathrm{inf}}$ for any ball.
## 3 Constructions based on the Locale of Time
### 3.1 1+1-dimensional Minkowski space model
Suppose that ${\mathcal{E}}$ is an semilattice-valued ultrametric space, where
the ultrametric takes values in $[0,+\infty)\times[0,+\infty)$, the Locale of
Time of this particular model.
Two real components of this ultrametric will be referred as $d_{u}$ and
$d_{v}$, and open balls (see 1.3.) are defined with:
$(d_{u}<r_{u})\wedge(d_{v}<r_{v});\ r_{u},r_{v}\in(0,+\infty)$ It is,
obviously, a semilattice with the componentwise $\max$ operation and
$0=(0,0)$. Geometrically, it corresponds to the projective 1+1-dimensional
Minkowski space, more strictly, to the conformal compactification of the 1+1d
Minkowski space. Namely, the (temporal) infinity $\Omega$ maps to 0, and two
light infinities map to $\\{0\\}\times(0,+\infty)$ and
$(0,+\infty)\times\\{0\\}$.
On the other hand, the affine part (the Minkowski space proper) map to
$(0,+\infty)\times(0,+\infty)$.
The main difference of this model from the model of 2.4. is that this Locale
of Time and radii of balls defined with it are only partially ordered. In one-
dimensional time, all ultimation balls between some large (past) ball and some
smaller (future) ball are totally ordered by their radii. A semilattice-valued
ultrametric makes a more complicated causal structure, which, in this case,
resembles the graph product of two directed trees, a directed acyclic graph,
but not a tree itself.
### 3.2 The EPR paradox
This subsection gives an interpretation of the quantum entanglement and
Einstein–Podolsky–Rosen paradox within the model constructed in the previous
subsection. It does not give an explicit construction of the space of
ultimation and the presheaf, but concrete examples may be created from this
description.
Suppose we have an ultrametric ball $B$ where the algebra of observables of
$F(B)$ has the form $Mat(4\times 4)\otimes\mathrm{Environment}$ and the state
has the form $\psi\otimes\rho_{\mathrm{Environment}}$, where $\psi$ is some
pure state
$(\psi_{\uparrow\uparrow}:\psi_{\uparrow\downarrow}:\psi_{\downarrow\uparrow}:\psi_{\downarrow\downarrow})$.
Define four *-homomorphisms from $Mat(2\times 2)$ to $Mat(4\times 4)$ and four
*-homomorphisms from ${\mathbb{C}}$ to $Mat(4\times 4)$ (see 2.2.):
${\mathfrak{p}}_{\uparrow{\mathchar 45\relax}}:\begin{pmatrix}a_{1}&a_{2}\\\
a_{3}&a_{4}\end{pmatrix}\mapsto\begin{pmatrix}a_{1}&0&a_{2}&0\\\ 0&0&0&0\\\
a_{3}&0&a_{4}&0\\\ 0&0&0&0\end{pmatrix};\ {\mathfrak{p}}_{\downarrow{\mathchar
45\relax}}:\begin{pmatrix}a_{1}&a_{2}\\\
a_{3}&a_{4}\end{pmatrix}\mapsto\begin{pmatrix}0&0&0&0\\\ 0&a_{1}&0&a_{2}\\\
0&0&0&0\\\ 0&a_{3}&0&a_{4}\end{pmatrix};$ ${\mathfrak{p}}_{{\mathchar
45\relax}\uparrow}:{\mathbf{a}}\mapsto\begin{pmatrix}{\mathbf{a}}&0\\\
0&0\end{pmatrix};\ {\mathfrak{p}}_{{\mathchar
45\relax}\downarrow}:{\mathbf{a}}\mapsto\begin{pmatrix}0&0\\\
0&{\mathbf{a}}\end{pmatrix};\
{\mathfrak{p}}_{\alpha\beta}:={\mathfrak{p}}_{{\mathchar
45\relax}\beta}\operatorname{\circ}{\mathfrak{p}}_{\alpha}={\mathfrak{p}}_{\alpha{\mathchar
45\relax}}\operatorname{\circ}{\mathfrak{p}}_{\beta}\,.$
Let an “advance by $u$” to denote a transition to a smaller ball with smaller
$d_{u}$ (the first component of the two-component ultrametric) and the same
$d_{v}\,$, and, similarly, an “advance by $v$” to denote a transition to a
smaller ball with smaller $d_{v}\,$ and the same $d_{u}\,$. Suppose that a
sequence of one on more advances by $v$ from $B$ changed only the
“Environment”. It means that the algebra ${\mathcal{A}}^{\prime}_{v}$ of a
$v$-advanced reality $F(B^{\prime}_{v})$ has the same structure $Mat(4\times
4)\otimes\mathrm{Environment}^{\prime}_{v}\,$, the morphism from $F(B)$ to
$F(B^{\prime}_{v})$ correspond to a *-homomorphism from
${\mathcal{A}}^{\prime}_{v}$ to ${\mathcal{A}}$ of the form
$\mathrm{id}_{4\times 4}\otimes E^{\prime}_{v}\,$. On the next step, let the
ball split to such two $v$-advanced balls $B_{{\mathchar 45\relax}\uparrow}$
and $B_{{\mathchar 45\relax}\downarrow}$, where ${\mathcal{A}}_{\,{\mathchar
45\relax}\beta}=Mat(2\times 2)\otimes\mathrm{Environment}_{\,{\mathchar
45\relax}\beta}\,$, that *-homomorphisms from ${\mathcal{A}}_{\,{\mathchar
45\relax}\beta}$ to ${\mathcal{A}}$ have the form ${\mathfrak{p}}_{{\mathchar
45\relax}\beta}\otimes E_{{\mathchar 45\relax}\beta}\,$.
Similarly, suppose the same picture on $u$ and the split to two $u$-advanced
balls $B_{\uparrow{\mathchar 45\relax}}$ and $B_{\downarrow{\mathchar
45\relax}}$, that *-homomorphisms from ${\mathcal{A}}_{\alpha{\mathchar
45\relax}}$ to ${\mathcal{A}}$ have the form ${\mathfrak{p}}_{\alpha{\mathchar
45\relax}}\otimes E_{\alpha{\mathchar 45\relax}}\,$.
What can we say about ultimation intersections
$B_{\alpha\beta}:=B_{\alpha{\mathchar 45\relax}}\cap B_{{\mathchar
45\relax}\beta}\,$, namely,
$B_{\uparrow\uparrow},B_{\uparrow\downarrow},B_{\downarrow\uparrow}$, and
$B_{\downarrow\downarrow}$? First of all, some intersections may be empty. We
supposed that ${\mathcal{E}}$ has a structure like the product of two
ultrametric spaces, but it is not necessarily such a product, and the
intersection of a $v$-ball with $u$-ball can be empty.
But what corresponding algebras of observables appear to be? Because morphisms
between Realities commute, their images
$\operatorname{{\mathrm{im}}}{\mathcal{A}}_{\alpha\beta}$ in ${\mathcal{A}}$
must lie in the intersection
$\operatorname{{\mathrm{im}}}{\mathcal{A}}_{\alpha{\mathchar
45\relax}}\cap\operatorname{{\mathrm{im}}}{\mathcal{A}}_{\,{\mathchar
45\relax}\beta}$ of images of observables from corresponding $u$-advanced and
$v$-advanced realities. These intersections have the form
$\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}\otimes$
some subalgebra of Environment, for $\uparrow\uparrow$, and so on. If for the
original $\psi:\psi_{\uparrow\downarrow}=\psi_{\downarrow\uparrow}=0$ (with
$\psi_{\uparrow\uparrow}=\psi_{\downarrow\downarrow}$ it gives an EPR pair),
then balls $B_{\uparrow\downarrow}$ and $B_{\downarrow\uparrow}$ must be
nihil, because corresponding Born’s factors must be 0. A nihil intersection
can be either the empty set, or a non-empty set annihilated by the presheaf.
$\scriptstyle\begin{pmatrix}\psi_{\uparrow\uparrow}\\\ \cdot\\\ \cdot\\\
\cdot\end{pmatrix}$ $\displaystyle\psi_{\uparrow\uparrow}$ !
$\displaystyle\psi_{\uparrow\downarrow}$ $\scriptstyle\begin{pmatrix}\cdot\\\
\psi_{\uparrow\downarrow}\\\ \cdot\\\ \cdot\end{pmatrix}$ \- - - \- - -
$\scriptstyle\begin{pmatrix}\cdot\\\ \cdot\\\ \psi_{\downarrow\uparrow}\\\
\cdot\end{pmatrix}$ $\displaystyle\psi_{\downarrow\uparrow}$ !
$\displaystyle\psi_{\downarrow\downarrow}$
$\scriptstyle\begin{pmatrix}\cdot\\\ \cdot\\\ \cdot\\\
\psi_{\downarrow\downarrow}\end{pmatrix}$
This example demonstrates the approach of Quantum Realities to the problem of
locality. Although $v$-advanced realities do not depend on $u$-advanced
realities and vice versa, they not always can combine to a joint future, which
makes and illusion that a measurement in a $v$-advanced reality can change
something in a $u$-advanced reality.
### 3.3 Locale of Time in more dimensions
The main obstacle to apply this construction straightforwardly to
1+$d$-dimensional Minkowski space ${{\mathbb{M}}^{1+d}}$ is that pseudo-
Riemannian manifolds for $d>1$ are not (semi)lattices. It means that the
Locale of Time cannot be made from ${{\mathbb{M}}^{1+d}}$ proper, with only
addition of some infinite points. The Locale of Time can be constructed,
although a detailed construction (for ${{\mathbb{M}}^{1+d}}$ and some other
globally hyperbolic manifolds, as well for space-times with different
geometries in different futures) will be presented in a separate paper. A
possible way to generalize the Locale of Time construction correctly is to
consider the conformal compactification of ${{\mathbb{M}}^{1+d}}$ and its
infinite part ${\mathrm{P}\mathbb{N}}$, referred to as the space of “null
projective twistors” for $d=3$.[11]
## 4 Miscellaneous considerations
### 4.1 The choice of direction of morphisms and direction of time
Yes, both are firmly based on physical grounds. A (pre)sheaf of C*-algebras
cannot encode a quantum measurement in such a way that possible results are
parts of the original reality. There exist the *-homomorphism from “measured”
to “original”, but an opposite direction is impossible, al least for full
matrix algebras, because “measured” has less dimensions than “original” (if we
understand a measurement as a decomposition).
The direction of time is determined, first, by semantics of the “measurement”.
Also, a morphism of Quantum Realities easily maps an original pure state to a
future mixed state, but not versa. It’s this which appears to be compatible
with entropy considerations.
### 4.2 Relationship to algebraic quantum field theory
The algebraic quantum field theory[13] also uses sheaves and C*-algebras, but
there are two major differences between it and the Quantum Realities. First,
in algebraic quantum field theory the direction of morphisms is not reversed.
This allows it to use products (see the next subsection), but restriction
morphisms in Algebraic QFT are not measurements.
Second, algebraic quantum field theory postulates the space-time, not derives
it.
### 4.3 Why not sheaves?
The definition of a sheaf[14] requires the product of objects, but categories
of Quantum Realities with states have no such operation. The category
C*algophas the product – it is the coproduct of C*-algebras, i.e. the free
product completed in the appropriate norm, an infinite-dimensional algebra for
any two non-nihil factors. Though, it unlikely has a physical sense.
Let us prove that any Quantum Reality category with states cannot have the
product operation. By contradiction, suppose that there exists the product
$(1)\times(1)$ of two singletons. By definition,[15] for an object $S$ any
pair of morphisms $m_{1}:S\to(1),\ m_{2}:S\to(1)$ such morphism
$m:S\to(1)\times(1)$ must exist that:
$m_{k}=\pi_{k}\operatorname{\circ}m\,,\mbox{ where }\pi_{k}\mbox{ are
canonical projections.}$
Due to symmetry (and a universal property of the product) $\pi_{k}$ must have
the same Born’s factor and it is some definite number, because $(1)\times(1)$
cannot be nihil. This implies that $m_{1}$ and $m_{2}$ always have equal
Born’s factors, which contradicts to the requirement that they are any pair of
morphisms.
### 4.4 Relationship to categorical quantum mechanics
Another application of formalism of the theory of categories to quantum
physics is categorical quantum mechanics.[16] But that formalism describes
quantum systems; there are operations on systems such as the composite system
(tensor product). The formalism of Realities can consider (sub)systems only
inside an algebra of observables (as we demonstrated in 3.2.), but not on the
categorical level. There is no such operation as composite system in the
Quantum Realities formalism. Such operation cannot be used with the presheaf
formalism because a composite system of two systems having the common past
would violate the no-cloning principle.
In short, Quantum Realities are not Categorical QM in any way.
### 4.5 Origin of symmetries
Ultrametric constructions do not explain the origin of physical symmetries,
such as translations, Lorentz transforms and gauge groups. It’s plausible that
symmetries do not rely on the space of ultimation, but are properties of
algebras of observables, where corresponding (families of) vector fields are
preserved by homomorphisms between realities. Some insights can be found in
the paper [2] of the same author. If such vector fields generate semigroups,
then homomorphisms should be intertwiners of corresponding endomorphism
semigroups in different realities.
### 4.6 Is the space of ultimation actually ultrametric?
First, I think that it is a physical object. Second, I do not think that it
has not, in fact, a disjoint topological structure, because it makes all
branches “sharp”, which is not accepted well by the physical intuition.
Possibly, the space of ultimation is not even a Hausdorff space. Ultrametric
spaces are, indeed, a good approximation of the actual structure of the space
of ultimation. Ultrametric spaces were used for many years in researches of
conformational spaces of large, complicated molecules (such as proteins), but
without realizing its theoretical value in application to the process of
branching of the time. Certainly, a conformational space is not an ultrametric
space. But it should be approximated with an ultrametric space, because its
terrible complexity hinders the use of more accurate models. Actually, the
conformational space of a molecule which occupied certain state in a given
moment of the past, is a projection of the corresponding ultimation set.
## References
* [1] Dixmier, Jaques. Les C*-algèbres et leurs représentations. Paris, 1969 (in French)
* [2] Maresin, Innocenti. Vector fields on C*-algebras, semigroups of endomorphisms and gauge groups. arXiv:1212.0500.
* [3] Ivanov, Mikhail G. Kak ponimat’ kvantovuyu mekhaniku? (English: How to understand the Quantum Mechanics?) Moscow, 2012. http://mezhpr.fizteh.ru/biblio/q-ivanov.html (in Russian)
* [4] Pechen, Alexander et al. Quantum control by von Neumann measurements. arXiv:quant-ph/0606187
* [5] http://en.wikipedia.org/wiki/Category_%28mathematics%29
* [6] http://en.wikipedia.org/wiki/Concrete_category
* [7] http://en.wikipedia.org/wiki/Ultrametric_space
* [8] http://en.wikipedia.org/wiki/Semilattice
* [9] http://en.wikipedia.org/wiki/Pointless_topology
* [10] http://en.wikipedia.org/wiki/Zero_object
, http://en.wikipedia.org/wiki/Zero_object_%28algebra%29
* [11] http://en.wikipedia.org/wiki/Twistor_theory
* [12] http://en.wikipedia.org/wiki/Spectral_decomposition
* [13] http://en.wikipedia.org/wiki/Algebraic_quantum_field_theory
* [14] http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29
* [15] http://en.wikipedia.org/wiki/Product_%28category_theory%29
* [16] http://en.wikipedia.org/wiki/Categorical_quantum_mechanics
|
arxiv-papers
| 2012-12-21T20:50:47 |
2024-09-04T02:49:39.588320
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Innocenti Maresin",
"submitter": "Innocenti Maresin",
"url": "https://arxiv.org/abs/1212.5585"
}
|
1212.5639
|
# A Comparison Between Nonlinear Force-Free Field and Potential Field Models
Using Full-Disk SDO/HMI Magnetogram
Tilaye Tadesse1 1 Department of Physics, Drexel University, Philadelphia, PA
19104-2875, U.S.A, email: [email protected],email:
[email protected]
3 Max Planck Institut für Sonnensystemforschung, Max-Planck Str. 2, D–37191
Katlenburg-Lindau, Germany, email: [email protected], email:
[email protected]
4 NASA, Goddard Space Flight Center, Code 674, Greenbelt, MD 20771, U.S.A.
email: [email protected]
5 National Solar Observatory, Sunspot, NM 88349, U.S.A. email:
[email protected]
T. Wiegelmann3 1 Department of Physics, Drexel University, Philadelphia, PA
19104-2875, U.S.A, email: [email protected],email:
[email protected]
3 Max Planck Institut für Sonnensystemforschung, Max-Planck Str. 2, D–37191
Katlenburg-Lindau, Germany, email: [email protected], email:
[email protected]
4 NASA, Goddard Space Flight Center, Code 674, Greenbelt, MD 20771, U.S.A.
email: [email protected]
5 National Solar Observatory, Sunspot, NM 88349, U.S.A. email:
[email protected]
B. Inhester3 1 Department of Physics, Drexel University, Philadelphia, PA
19104-2875, U.S.A, email: [email protected],email:
[email protected]
3 Max Planck Institut für Sonnensystemforschung, Max-Planck Str. 2, D–37191
Katlenburg-Lindau, Germany, email: [email protected], email:
[email protected]
4 NASA, Goddard Space Flight Center, Code 674, Greenbelt, MD 20771, U.S.A.
email: [email protected]
5 National Solar Observatory, Sunspot, NM 88349, U.S.A. email:
[email protected]
K. Olson1 1 Department of Physics, Drexel University, Philadelphia, PA
19104-2875, U.S.A, email: [email protected],email:
[email protected]
3 Max Planck Institut für Sonnensystemforschung, Max-Planck Str. 2, D–37191
Katlenburg-Lindau, Germany, email: [email protected], email:
[email protected]
4 NASA, Goddard Space Flight Center, Code 674, Greenbelt, MD 20771, U.S.A.
email: [email protected]
5 National Solar Observatory, Sunspot, NM 88349, U.S.A. email:
[email protected]
P. J. MacNeice4 1 Department of Physics, Drexel University, Philadelphia, PA
19104-2875, U.S.A, email: [email protected],email:
[email protected]
3 Max Planck Institut für Sonnensystemforschung, Max-Planck Str. 2, D–37191
Katlenburg-Lindau, Germany, email: [email protected], email:
[email protected]
4 NASA, Goddard Space Flight Center, Code 674, Greenbelt, MD 20771, U.S.A.
email: [email protected]
5 National Solar Observatory, Sunspot, NM 88349, U.S.A. email:
[email protected]
A. Pevtsov5 1 Department of Physics, Drexel University, Philadelphia, PA
19104-2875, U.S.A, email: [email protected],email:
[email protected]
3 Max Planck Institut für Sonnensystemforschung, Max-Planck Str. 2, D–37191
Katlenburg-Lindau, Germany, email: [email protected], email:
[email protected]
4 NASA, Goddard Space Flight Center, Code 674, Greenbelt, MD 20771, U.S.A.
email: [email protected]
5 National Solar Observatory, Sunspot, NM 88349, U.S.A. email:
[email protected]
###### Abstract
Measurements of magnetic fields and electric currents in the pre-eruptive
corona are crucial to study solar eruptive phenomena, like flare and coronal
mass ejections(CMEs). However, spectro-polarimetric measurements of certain
photospheric lines permit a determination of the vector magnetic field at the
photosphere. Thus, substantial collection of magnetograms relate to the
photospheric surface field only. Therefore, there is considerable interest in
accurate modeling of the solar coronal magnetic field using photospheric
vector magnetograms as boundary data. This numerical modeling is carried out
by applying state-of-the-art nonlinear force-free field (NLFFF)
reconstruction. Cartesian nonlinear force-free field (NLFFF) codes are not
well suited for larger domains, since the spherical nature of the solar
surface cannot be neglected when the field of view is large. One of the most
significant results of Solar Dynamic Observatory (SDO) mission to date has
been repeated observations of large, almost global scale events in which large
scale connection between active regions may play fundamental role. Therefore,
it appears prudent to implement a NLFFF procedure in spherical geometry for
use when large scale boundary data are available, such as from the
Helioseismic and Magnetic Imager (HMI) on board SDO. In this work, we model
the coronal magnetic field above multiple active regions with the help of a
potential field and a NLFFF extrapolation codes in a full-disk using HMI data
as a boundary conditions. We compare projections of the resulting magnetic
field lines solutions with full-disk coronal images from the Atmospheric
Imaging Assembly (SDO/AIA) for both models. This study has found that the
NLFFF model reconstructs the magnetic configuration better than the potential
field model. We have concluded that much of trans-equatorial loops connecting
the two solar hemispheres are current-free.
###### keywords:
Active Regions, Magnetic Fields; Active Regions, Models; Magnetic fields,
Corona; Magnetic fields, Photosphere; Magnetic fields, Models
## 1 Introduction
Magnetic fields are believed to play a dominant role for active phenomena
carried out in the solar corona. The study of solar eruptive phenomena
requires that we understand how magnetic energy is stored in the pre-eruptive
corona. One of the key questions is: What is the three dimensional (3D)
structure of magnetic fields and electric currents in the pre-eruptive corona,
and how much free energy is stored in the field? [Schrijver and Title (2005),
Jiang and Feng (2012)]. To answer this question we must measure the magnetic
field in the coronal volume. However, spectro-polarimetric measurements of
certain photospheric lines permit a determination of the vector magnetic field
at the photosphere. Thus, substantial collection of magnetograms relate to the
photospheric surface field only. Although measurement of magnetic fields in
the chromosphere and the corona has considerably improved in recent decades
[Lin, Penn, and Tomczyk (2000), Liu and Lin (2008)], further developments are
needed before accurate data are routinely available. The problem of measuring
the coronal field and its embedded electrical currents thus leads us to use
numerical modeling to infer the field strength in the higher layers of the
solar atmosphere from the measured photospheric field. Due to the low value of
the plasma $\beta$ (the ratio of gas pressure to magnetic pressure), the solar
corona is magnetically dominated [Gary (2001)]. To describe the equilibrium
structure of the static coronal magnetic field when non-magnetic forces are
negligible, the force-free assumption is appropriate:
$(\nabla\times\textbf{B})\times\textbf{B}=0\ilabel{one}$ (1)
$\nabla\cdot\textbf{B}=0\ilabel{two}$ (2)
subject to the boundary condition
$\textbf{B}=\textbf{B}_{\textrm{obs}}\quad\mbox{on photosphere}\ilabel{three}$
(3)
where B is the magnetic field and $\textbf{B}_{\textrm{obs}}$ is measured
vector field on the photosphere. Equation (one) states that the Lorentz force
vanishes (as a consequence of $\textbf{J}\parallel\textbf{B}$, where J is the
electric current density) and Equation (two) describes the absence of magnetic
monopoles.
As an alternative to real measurements, force-free extrapolation of
photosphericmagnetic fields is currently being used as a the primary tool for
the modeling of coronal magnetic fields [Inhester and Wiegelmann (2006),
Valori, Kliem, and Keppens (2005), Wiegelmann (2004), Wheatland (2004),
Wheatland and Régnier (2009), Tadesse, Wiegelmann, and Inhester (2009),
Wheatland and Leka (2011), Amari and Aly (2010), Wiegelmann, T. and Thalmann,
J. K. and Inhester, B. and Tadesse, T. et al. (2012), Jiang and Feng (2012),
Jiang, Feng, and Xiang (2012), Aschwanden and Malanushenko (2012), Wiegelmann
and Sakurai (2012)].
Cartesian nonlinear force-free field (NLFFF) codes are not well suited for
larger domains, since the spherical nature of the solar surface cannot be
neglected when the field of view is large [Wiegelmann (2007), Tadesse,
Wiegelmann, and Inhester (2009), Tadesse et al. (2012a), Guo et al. (2012)].
One of the most significant results of Solar Dynamic Observatory (SDO) mission
to date has been repeated observations of large, almost global scale events in
which large scale connection between active regions may play fundamental role.
Therefore, it appears prudent to implement a NLFFF procedure in spherical
geometry for use when large scale boundary data are available, such as from
the Helioseismic and Magnetic Imager (HMI) on board SDO. DeRosa has studied
that different NLFFF models have markedly different field line configurations
and provided widely varying estimates of the magnetic free energy in the
coronal volume. The main reasons for that problem are (1) the forces acting on
the field within the photosphere, (2) the uncertainties on vector-field
measurements, particularly on the transverse component, and (3) the large
domain that needs to be modeled to capture the connections of an active region
to its surroundings [Wiegelmann (2007), Tadesse et al. (2011), Tadesse et al.
(2012b), Tadesse et al. (2012c)]. In this study, we have considered those
three points explicitly into account for modeling coronal magnetic field in a
full-disk.
In this work, we apply our spherical NLFFF procedure to a group of active
regions observed on November 09 2011 around 17:45UT by SDO/HMI. During this
observation, there were four active regions (ARs 11338, 11339, 11341 and
11342) along with other smaller sunspots spreading on the disk. We compare the
extrapolated (both potential and NLFFF) magnetic loops with extreme
ultraviolet (EUV) observations by the Atmospheric Imaging Assembly (AIA)
[Lemen, J. R. and Title, A. M. and Akin, D. J. et al. (2012)] on board SDO.
During comparison, we check whether the NLFFF model reconstructs the magnetic
configuration better than the potential field model. This comparison can be
used to evaluate how well the model field lines approximate the observed
coronal loops.
## 2 Instrumentation and data set
The Helioseismic and Magnetic Imager (HMI) [Schou, J. and Scherrer, P. H. and
Bush, R. I. et al. (2012)] is part of the Solar Dynamics Observatory (SDO) and
observes the full Sun at six wavelengths and full Stokes profile in the Fe
slowromancapi@ 617.3 nm spectral line. It consists of a refracting telescope,
a polarization selector, an image stabilization system, a narrow band tunable
filter and two 4096 pixel CCD cameras with mechanical shutters and control
electronics.
The transverse components of vector magnetic fields suffer from the so-called
$180^{\circ}$ ambiguity. The $180^{\circ}$ ambiguity for the HMI data in this
study has been resolved by an improved version of the minimum energy method
[Metcalf (1994), Metcalf, T. R. and Leka, K. D. and Barnes, G. et al. (2006),
Leka, Barnes, and Crouch (2009)]. As described in Leka:2009, in weak-field
areas, the minimization may not return a good solution due to large noise. The
noise level is $\approx$ 10G and $\approx$ 100G for the longitudinal and
transverse magnetic field, respectively. Therefore, in order to get a
spatially smooth solution in weak-field areas, we divide the magnetic field
into two regions, i.e., strong-field region and weak-field region, which is
defined to be where the field strength is below 200 G at the disk center, and
400 G on the limb. The values vary linearly with distance from the center to
the limb. The ambiguity solution in the strong-field area is derived by the
annealing in the minimization method and released in the data series
“hmi.B_720s_e15w1332_cutout“. Magnetic field azimuths in the weak-field area
are finally determined by the potential-field acute-angle method after the
annealing. This is different from the original minimum energy method, which
uses a neighboring-pixel acute-angle algorithm to revisit the weak field area.
Figure fig3a) shows the vector magnetic fields observed by HMI on November 09
2011 around 17:45UT.
## 3 Optimization principle and preprocessing of HMI data
Molodenskii69 and Aly89 pointed out that vector magnetic fields on a closed
surface that fully encloses any force-free domain have to satisfy the force-
free and torque-free conditions. To serve as suitable lower boundary condition
for a force-free modeling, on average a net tangential force acting on the
boundary and shear stresses along axes lying on the boundary have to reduce to
zero [Molodensky (1969), Aly (1989), Wiegelmann, Inhester, and Sakurai (2006),
Tadesse, Wiegelmann, and Inhester (2009)]. Preprocessing method as implemented
in Wiegelmann06sak has to be used to fulfill those conditions. The
preprocessing scheme of tilaye09 involves minimizing a two-dimensional
functional of quadratic form in spherical geometry similar to
$\textbf{B}=\emph{argmin}(L_{p}),$
$L_{p}=\mu_{1}L_{1}+\mu_{2}L_{2}+\mu_{3}L_{3}+\mu_{4}L_{4},\ilabel{3}$ (4)
where B is the preprocessed surface magnetic field from the input observed
field $\textbf{B}_{obs}$. Each of the constraints $L_{n}$ is weighted by an as
yet undetermined factor $\mu_{n}$. The first term $(n=1)$ corresponds to the
force-balance condition, the next $(n=2)$ to the torque-free condition, and
the last term $(n=4)$ controls the smoothing. The explicit form of $L_{1}$,
$L_{2}$, and $L_{4}$ can be found in tilaye09. The term $(n=3)$ controls the
difference between measured and preprocessed vector fields. After
preprocessing the HMI data, we solve the force-free equations (one) and (two)
using optimization principle [Wheatland, Sturrock, and Roumeliotis (2000),
Wiegelmann (2004)] in spherical geometry [Wiegelmann (2007), Tadesse,
Wiegelmann, and Inhester (2009), Tadesse et al. (2012c)].
## 4 Potential field model
The simplest way to model the coronal field is to assume that it is potential,
i.e. that it carries no electric current. Solutions for this model in plane
geometry have been obtained by schmidt, for the case where the vertical
component of the field is specified at the photospheric boundary. Potential
field model has now led to an almost routine type reconstruction, used for
observational purposes [Sakurai (1989)], but also for building initial
conditions for dynamical MHD numerical simulations [Amari et al. (1996)]. This
assumption has proven to be adequate for many quiescent, old active regions
and even for the non-eruptive global coronal-heliospheric interface [Wang and
Sheeley (1990), Schrijver and De Rosa (2003)].
Studies of the coupling of the coronal field into the heliosphere suggest that
the global coronal magnetic field is often largely potential. For the
practical calculation of the global field, the so-called source-surface model
has been introduced [Schatten, Wilcox, and Ness (1969)], in which the
influence of the solar wind is artificially taken account of by the
requirement that the field be radial at some exterior spherical (source)
surface typically at $R_{s}=2.5R_{\odot}$ from the sun’s center. The
potential-field source surface (PFSS) model, uses this concept to extrapolate
the line-of-sight surface magnetic field through the corona with the boundary
assumed to be at the source surface.
The potential field approximation has often been used to deduce the magnetic
structure of the solar corona from measurements of the photospheric field. It
obeys the equation $\nabla\times\textbf{B}=0$, so that the magnetic field can
be expressed as the gradient of a scalar potential, i.e.
$\textbf{B}=-\nabla\Phi$. Since the magnetic field is divergence-free
($\nabla\cdot\textbf{B}=0$), the scalar potential obeys the Laplace equation
$\nabla^{2}\Phi=0$. Together with the photospheric boundary condition, which
is traditionally provided by a map of the radial magnetic field, the Laplace
equation can be solved in the spherical coordinate system $(r,\theta,\phi)$,
where $\theta$ stands for colatitude. Using the radial magnetic field observed
at $R_{\odot}$ and the source surface assumption at $R_{s}$ as the boundary
condition, together with the spherical harmonic expansion in the domain
$R_{\odot}<r<R_{s}$, we finally obtain the potential field model.
## 5 Results
In this study, we have used potential field model and our spherical NLFFF
optimization procedure to a group of active regions observed on November 09
2011 around 17:45UT by SDO/HMI instrument. During this observation there were
four active regions (ARs 11338, 11339, 11341 and 11342, see Fig. fig2a) along
with other smaller sunspots spreading on the disk. In order to accommodate the
connectivity between those ARs and their surroundings, and to study trans-
equatorial loops (loops connecting north and south hemisphere), we adopt a non
uniform spherical grid $r$, $\theta$, $\phi$ with $n_{r}=225$,
$n_{\theta}=375$, $n_{\phi}=425$ grid points in the direction of radius,
latitude, and longitude, respectively, with the field of view of
$[r_{\mbox{min}}=1R_{\odot}:r_{\mbox{max}}=2R_{\odot}]\times[\theta_{\mbox{min}}=-50^{\circ}:\theta_{\mbox{max}}=50^{\circ}]\times[\phi_{\mbox{min}}=90^{\circ}:\phi_{\mbox{max}}=270^{\circ}]$.
The main purpose of this work is to study the magnetic and electric current
connectivities of the northern and southern solar hemispheres and to compare
which of the two models (potential or NLFFF model) is best suited to fully
describe those connectivities in full-disk environment.
We use our preprocessing routine in spherical geometry to derive suitable
boundary conditions for force-free modeling from the measured photospheric
data [Wiegelmann, Inhester, and Sakurai (2006), Tadesse, Wiegelmann, and
Inhester (2009)]. We used a standard preprocessing parameter set
$\mu_{1}=\mu_{2}=1$, $\mu_{3}=0.001$ and $\mu_{4}=0.05$, which are similar to
the values calculated from vector data used in previous studies [Wiegelmann,
T. and Thalmann, J. K. and Inhester, B. and Tadesse, T. et al. (2012), Tadesse
et al. (2012c)] for HMI data. The preprocessing influences the structure of
the magnetic vector data. It does not only smooths $\textbf{B}_{t}$
(transverse field) but also alters its values in order to reduce the net force
and torque. The change in $\textbf{B}_{t}$ is more pronounced than the radial
component $\textbf{B}_{r}$ (radial field) since $\textbf{B}_{t}$ is measured
with lower accuracy than the longitudinal magnetic field.
Figure 1.: Magnetic vector maps of HMI data on part of the lower boundary. The
color coding shows $B_{r}$ on the photosphere and the black arrow indicates
the transverse components of the field. The vertical and horizontal axes show
latitude, $\theta$ and longitude, $\phi$ on the photosphere respectively.fig1
For computing the potential field, we use the preprocessed radial component
$\textbf{B}_{r}$ of the HMI-data using spherical harmonic expansion. This
potential field solution has been used to initialize our NLFFF code. For
nonlinear force-free fields we minimize the functional Eq. (4). We implement
the new term $L_{photo}$ in Eq. (4) to work with boundary data of different
noise levels and qualities [Wiegelmann and Inhester (2010), Tadesse et al.
(2011)]. For those pixels, for which $\textbf{B}_{obs}$ was successfully
inverted, we allow deviations between the model field B and the input fields
observed $\textbf{B}_{obs}$ surface field using Eq. (4), so that the model
field can be iterated closer to a force-free solution even if the observations
are inconsistent. In order to control the speed with which the lower boundary
is injected during the NLFFF extrapolation, we have used the Langrangian
multiplier of $\nu=0.001$ as suggested by Tadesse:2013. For more details of
the method used in this work we direct the readers to the study by
Tilaye:2010.
We plot the surface potential field and NLFFF solutions in Figure fig1 for
selected region excluding the polar regions. In this figure, one can see that
there are substantial differences between the surface radial and transverse
field components of the two models. To quantify the degree of disagreement
between the respective vector components of potential and NLFFF model
solutions on the bottom surface, we calculate their pixel-wise correlations.
The correlation were calculated[Schrijver, C. J. and Derosa, M. L. and
Metcalf, T. R. et al. (2006), Metcalf, T. R. and De Rosa, M. L. and Schrijver,
C. J. et al. (2008)] from
$C_{\mathrm{vec}}=\frac{\sum_{i}\textbf{v}_{i}\cdot\textbf{u}_{i}}{\Big{(}\sum_{i}|\textbf{v}_{i}|^{2}\sum_{i}|\textbf{u}_{i}|^{2}\Big{)}^{1/2}},\ilabel{6}$
(5)
where $\textbf{v}_{i}$ and $\textbf{u}_{i}$ are the vectors at each grid point
$i$ on the bottom surface. If the vector fields are identical, then
$C_{vec}=1$; if $\textbf{v}_{i}\perp\textbf{u}_{i}$ , then $C_{vec}=0$. Table
table1 shows the correlation ($C_{vec}$) of the 2D surface magnetic field
vectors of potential and NLFFF models for the radial and transverse
components. The vector correlation between $\textbf{B}_{t}$ of the potential
and NLFFF surface vector maps is much more less than that of respective
$\textbf{B}_{r}$ components. This is due to the fact that the transverse
components ($\textbf{B}_{t}$) of the potential field model are computed using
spherical harmonics expansion from the radial components of the magnetic field
used as a boundary condition for NLFFF model. The average vector correlations
between the radial and transverse components of the two models in Table table1
indicates that the NLFFF solutions are somewhat far from potential as they
carry electric currents.
Table 1.: The correlations between the components of surface fields from potential and NLFFF models.table1 v | u | $C_{\mathrm{vec}}$
---|---|---
$(\textbf{B}_{pot})_{r}$ | $(\textbf{B}_{NLFFF})_{r}$ | $0.937$
$(\textbf{B}_{pot})_{t}$ | $(\textbf{B}_{NLFFF})_{t}$ | $0.813$
Figure 2.: a) Full-disk SDO/HMI magnetogram, the rectangles figure outline the
main active regions during this observation and the yellow line indicates the
solar equator ($\theta=0^{\circ}$), b) Full-disk AIA 171 Å image. Both data
sets were obtained on November 09, 2011 around 17:45UT, c) Field lines of the
potential field model at 17:45 UT overlaid on AIA 193Å image and d) Field
lines of the NLFFF model at 17:45 UT overlaid on AIA 193Å image. Green and red
lines represent open and closed magnetic field lines.fig2
In order to compare our reconstructions with observation, we plot the selected
fieldlines of the potential and NLFFF models in Figure fig2c and d. We overlay
the field lines with AIA 193 Å image. For observation time in Figure fig2c and
d, the field lines of the potential and NLFFF models are reconstructed from
the same footpoints. The potential field lines in Figure fig2(c) have an
obvious deviation from the observed EUV loops, since the projection of the
field lines at the bottom of AR 11339 leans toward the east, while the loops
toward the west. The loops connecting positive and negative polarities of AR
11341 are best overlaid by NLFFF lines than potential ones. The spatial
correspondence between the overall shape of the NLFFF field lines and the EUV
loops is much improved as shown in Figure fig2(d). Therefore, the qualitative
comparison between the model magnetic field lines and the observed EUV loops
indicates that the NLFFF model provides a more consistent field for full-disk
magnetic field reconstruction. In the absence of a more reliable quantitative
comparison, it remains the best option even if the qualitative nature is not
ideal.
In addition to the above comparison to quantify the degree of disagreement
between vector field solutions of the two models in the computational volume
that are specified on identical sets of grid points, we use five metrics that
compare either local characteristics (e.g., vector magnitudes and directions
at each point) or the global energy content in addition to the force and
divergence integrals as defined in Schrijver06. The vector correlation
($C_{vec}$) metric of Equation (6) is also used analogous to the standard
correlation coefficient for scalar functions. The second metric, $C_{CS}$ is
based on the Cauchy-Schwarz
inequality($|\textbf{a}\cdot\textbf{b}\leq|\textbf{a}||\textbf{b}|$ for any
vector a and b)
$C_{\rm CS}=\frac{1}{N}\sum_{i}\frac{{\bf B_{i}}\cdot{\bf b_{i}}}{|{\bf
B_{i}}||{\bf b_{i}}|},\ilabel{7}$ (6)
where $\textbf{B}=\textbf{B}_{\mbox{pot}}$,
$\textbf{b}=\textbf{B}_{\mbox{NLFFF}}$, and $N$ is the number of vectors in
the field. This metric is mostly a measure of the angular differences of the
vector fields: $C_{CS}=1$ when B and b are parallel and $C_{CS}=-1$ if they
are anti-parallel; $C_{CS}=0$ if $\textbf{B}_{i}\perp\textbf{b}_{i}$ at each
point. Next, we introduce two measures for the vector errors, one normalized
to the average vector norm, one averaging over relative differences. The
normalized vector error $E_{N}$ is defined as
$E_{\rm N}=\sum_{i}|{\bf b_{i}}-{\bf B_{i}}|/\sum_{i}|{\bf B_{i}}|,\ilabel{8}$
(7)
The mean vector error $E_{M}$ is defined as
$E_{\rm M}=\frac{1}{N}\sum_{i}\frac{|{\bf b_{i}}-{\bf B_{i}}|}{|{\bf
B_{i}}|}.\ilabel{9}$ (8)
Unlike the first two metrics, perfect disagreement of the two vector fields
results in $1-E_{M}=1-E_{N}=0$. As we are also interested in determining how
well the models estimate the energy contained in the field, we use the total
magnetic energy in NLFFF model field normalized to the total magnetic energy
in potential field as a global measure of the quality of the fit:
$\epsilon=\frac{\sum_{i}|{\bf b_{i}}|^{2}}{\sum_{i}|{\bf
B_{i}}|^{2}}.\ilabel{10}$ (9)
$\epsilon=1$, if there is no difference between the potential field and the
nonlinear force-free model solutions.
Table 2.: Comparision of figure of merits between potential field and NLFFF models. We have used pherical grids of $225\times 375\times 425$.table2 Model | $L_{f}$ | $L_{d}$ | $C_{\rm vec}$ | $C_{\rm CS}$ | $1-E_{N}$ | $1-E_{M}$ | $\epsilon$ | Time
---|---|---|---|---|---|---|---|---
Potential | $0.000$ | $0.002$ | $1$ | $1$ | $0$ | $0$ | $1$ | 30min
NLFFF | $0.591$ | $0.997$ | $0.878$ | $0.833$ | $0.735$ | $0.876$ | $1.220$ | 4h:39min
The degree of convergence towards a force-free and divergence-free model
solution can be quantified by the integral measures of the Lorentz force and
divergence terms in the minimization functional in Equation (4), computed over
the entire model volume V. $L_{f}$ and $L_{d}$ of Equation (4) measure how
well the force-free and divergence-free conditions are fulfilled,
respectively. In Table table2, we provide some quantitative measures to rate
the quality (figure of merit) of our reconstruction. Column $1$ names the
corresponding models. Columns $2-3$ show how well the force and solenoidal
condition are fulfilled for both models. The next five columns of Table table2
contain different measures which compare our NLFFF solution with potential
field. Those figures of merit indicate that there is a clear difference
between potential field and NLFFF models solutions for full-disk. The last
column shows the computing time on $1$ processor. The figures of merit show
that the potential field is far away from the true solutions and contains only
$81.97\%$ of the total magnetic energy.
We estimate the free magnetic energy to be the difference between the
extrapolated NLFFF and the potential field with the same normal boundary
conditions in the photosphere[Régnier and Priest (2007), Thalmann, Wiegelmann,
and Raouafi (2008)]. We therefore estimate the upper limit to the free
magnetic energy associated with coronal currents of the form
$E_{\mathrm{free}}=\frac{1}{8\pi}\int_{V}\Big{(}B_{nlff}^{2}-B_{pot}^{2}\Big{)}r^{2}sin\theta
drd\theta d\phi,\ilabel{ten}$ (10)
Our result for the estimation of free-magnetic energy in Table table3 shows
that the NLFFF model has $18.83\%$ more energy than the corresponding
potential field model. Therefore, whenever we try to analysis magnetic field
configuration and its magnetic field contents in full-disk, it is inaccurate
to use potential field model as it undermines the real features of the field
in the corona.
Table 3.: The magnetic energy associated with extrapolated potential and NLFFF field configurations from full disk SDO/HMI data.table3 Model | $E_{\mbox{total}}(10^{33}{\mbox{erg}})$ | $E_{\mbox{free}}(10^{33}{\mbox{erg}})$
---|---|---
Potential | $7.306$ | $0$
NLFFF | $8.913$ | $1.607$
In our previous work [Tadesse et al. (2012a)], we have studied the
connectivity between three neighbouring active regions. In this work, we study
the magnetic and electric connectivities between active regions in northern
and southern solar hemispheres using full-disk HMI data. During this
observation, there were three active regions in the northern hemisphere and
one active region with surrounding sunspots in the southern hemisphere. As a
result the total unsigned magnetic field flux in the northern hemisphere is
larger than that of the southern one. In order to quantify these
connectivities, we have calculated the magnetic flux and the electric currents
shared between those active regions. For the magnetic flux, e.g., we use
$\Phi_{\alpha\beta}=\sum_{i}|\textbf{B}_{i}\cdot\hat{r}|R^{2}_{\odot}\textrm{sin}(\theta_{i})\Delta\theta_{i}\Delta\phi_{i}\ilabel{ten}$
(11)
where the summation is over all pixels of $\mbox{AR}_{\alpha}$ from which the
field line ends in $\mbox{AR}_{\beta}$ or
$i\in\mbox{AR}_{\alpha}\|\,\mbox{conjugate footpoint}(i)\in\mbox{AR}_{\beta}$.
The indices $\alpha$ and $\beta$ denote the active region. For the electric
current we replace the magnetic field, $\textbf{B}_{i}\cdot\hat{r}$, by the
vertical current density $\textbf{J}_{i}\cdot\hat{r}$ in Equation (ten). Table
table4 shows the percentage of the total magnetic flux and electric current
shared between the ARs 11339 and 11338 and between AR 11341 and sunspots in
the southern hemisphere (see Fig. fig3). We have calculated total an unsigned
flux for each active regions 11339 and 11341 (both being in northern
hemisphere) and the flux due to those field lines ending in the AR 11338 and
magnetic patches in the southern hemisphere, respectively. The first column of
Table table4 shows that $7.31\%(6.17\%)$ of positive/negative polarity of AR
11339 (11341) in the northern hemisphere is connected to positive/negative
polarity of AR 11338 (magnetic patches) in southern hemisphere for potential
field configuration. The percentage share between those ARs are
$7.57\%(6.91\%)$ for NLFFF configuration. In Table table4, the percentages in
the bracket show the share in electric current between the corresponding
active regions and small magnetic patches. Figure fig3 display the trans-
equatorial loops connecting those active regions. In the figure, we rotate
those active regions to the limb to show loops connecting the two hemispheres.
The surface contour plots in Figures fig4 and fig5 show the magnetic flux and
electric current density flux of field lines crossing equatorial plane,
respectively. From those figures we can see that NLFFF model has more magnetic
and electric current density fluxes compared to the potential field. Figure
fig5 shows that there are more electric currents crossing the plane carried by
trans-equatorial loops connecting AR 11341 in northern hemisphere to sunspots
in the south than those currents connecting ARs 11339 and 11338. This indicate
that much of the trans-equatorial loops connecting ARs 11339 and 11338 are
potential. In general the two hemispheres are more magnetically connected than
electric current. However, one has to use NLFFF model over potential model to
study magnetic field structure in any phenomena involving the field.
Figure 3.: a) Selected potential field lines connecting ARs 11338 and 11339 rotated to limb. b) The corresponding field lines for NLFFF in a), c) Selected potential field lines connecting AR 11341 and Sun spots in southern hemisphere rotated to limb d) The corresponding field lines for NLFFF in c).fig3 Figure 4.: Surface contour plot of perpendicular component of magnetic fields crossing a plane of the solar equator ($\theta=0^{\circ}$). The color coding shows $B_{\perp}$ crossing the $\theta=0^{\circ}$ equatorial plane. The vertical and horizontal axes show height, $r$(in solar radius) and longitude, $\phi$(in degree) on the plane.fig4 Figure 5.: Surface contour plot of perpendicular component of electric current density crossing a plane of the solar equator ($\theta=0^{\circ}$). The color coding shows $J_{\perp}$ crossing the $\theta=0^{\circ}$ equatorial plane. The vertical and horizontal axes show height, $r$(in solar radius) and longitude, $\phi$(in degree) on the plane.fig5 Table 4.: The percentage of the total magnetic flux and electric current density shared between the ARs 11339 and 11338 and between AR 11341 and sunspots in the southern hemisphere ( see Fig. fig3). The percentage in brackets denote that of electric currents.table4 Model | $\mbox{AR 11339}\longmapsto\mbox{AR 11338}$ | $\mbox{AR 11341}\longmapsto\mbox{Patches in South}$
---|---|---
Potential | $7.31\,(0.0)$ | $6.17\,(0.01)$
NLFFF | $7.57\,(0.13)$ | $6.91\,(3.35)$
## 6 Conclusion and outlook
Both potential and nonlinear force-free field (NLFFF) codes in Cartesian
geometry are not well suited for larger domains, since the spherical nature of
the solar surface cannot be neglected when the field of view is large. One of
the most significant results of Solar Dynamic Obseratory (SDO) mission to date
has been repeated observations of large, almost global scale events in which
large scale connection between active regions may play fundamental role.
Therefore, it appears prudent to implement a NLFFF procedure in spherical
geometry for use when large scale boundary data are available, such as from
the Helioseismic and Magnetic Imager (HMI) on board SDO.
In this study, we have investigated the coronal magnetic field associated with
full solar disk on November 09 2011 by analysing SDO/HMI data using potential
and NLFFF models. During this particular observation, there were three active
regions in the northern hemisphere and one active region surrounded by
sunspots in the south. We have used our spherical NLFFF and potential codes to
compute the magnetic field solutions in full-disk. For computing the potential
field, we use the preprocessed radial component $\textbf{B}_{r}$ of the HMI-
data using spherical harmonic expansion. We implement our NLFFF code
initialized by the this potential field solution (except the lower observed
bottom boundary) during relaxation towards force-freeness state in the
computational volume.
We have compared the magnetic field solutions from both models. The
qualitative comparison between the model magnetic field lines and the observed
EUV loops indicates that the NLFFF model provides a more consistent field for
full-disk magnetic field reconstruction. In addition to this, the figures of
merits have been used to quantify the disagreements between the two models
field lines. The figures of merit indicate that the magnetic field lines
structures are not potential in full-disk field-of-views. However, much of the
trans-equatorial field lines connecting the active regions in the northern and
southern hemispheres are potential. This indicates that the two solar
hemispheres are more magnetically connected than electric current. The
magnetic field lines obtained from nonlinear force-free extrapolation bear
larger total magnetic energy that of the potential field model. Therefore, one
has to use NLFFF model over potential model to study magnetic field structure
in any phenomena involving the field in full-disk.
## Acknowledgements
Data are courtesy of NASA/SDO and the AIA and HMI science teams. SOLIS/VSM
vector magnetograms are produced cooperatively by NSF/NSO and NASA/LWS. The
National Solar Observatory (NSO) is operated by the Association of
Universities for Research in Astronomy, Inc., under cooperative agreement with
the National Science Foundation. This work was supported by NASA grant
NNX07AU64G and the work of T. Wiegelmann was supported by DLR-grant $50$ OC
$453$ $0501$.
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|
arxiv-papers
| 2012-12-22T00:29:40 |
2024-09-04T02:49:39.597436
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tilaye Tadesse, T. Wiegelmann, B.Inhester, K.Olson, P. J. MacNeice, A.\n Pevtsov",
"submitter": "Tilaye Tadesse",
"url": "https://arxiv.org/abs/1212.5639"
}
|
1212.5668
|
section subsection subsubsection part partnumber partentrypagenumber title
descriptionlabel
# Initial Semantics for Reduction Rules
Benedikt Ahrens
###### Abstract
We give an algebraic characterization of the syntax and operational semantics
of a class of simply–typed languages, such as the language $\mathsf{PCF}$: we
characterize simply–typed syntax with variable binding and equipped with
reduction rules via a universal property, namely as the initial object of some
category of models. For this purpose, we employ techniques developed in two
previous works: [2] models syntactic translations between languages _over
different sets of types_ as initial morphisms in a category of models. [1]
characterizes untyped syntax _with reduction rules_ as initial object in a
category of models. In the present work, we show that those techniques are
modular enough to be combined: we thus characterize simply–typed syntax with
reduction rules as initial object in a category. The universal property yields
an operator which allows to specify translations — that are semantically
faithful by construction — between languages over possibly different sets of
types.
We specify a language by a _2–signature_ , that is, a signature on two levels:
the _syntactic_ level specifies the types and terms of the language, and
associates a type to each term. The _semantic_ level specifies, through
_inequations_ , reduction rules on the terms of the language. To any given
2–signature we associate a category of models. We prove that this category has
an initial object, which integrates the types and terms freely generated by
the 2–signature, and the reduction relation on those terms generated by its
inequations. We call this object the _(programming) language generated by the
2–signature_.
This paper is an extended version of an article published in the proceedings
of WoLLIC 2012.
## 1 Introduction
We give an algebraic characterization, via a universal property, of the
programming language generated by a _signature_. More precisely, we define a
notion of _2–signature_ which allows the specification of the _types and
terms_ of a programming language — via a 1–signature, say, $\Sigma$ — as well
as its _semantics_ in form of reduction rules, specified through a set $A$ of
inequations over $\Sigma$. To any 1–signature $\Sigma$ we associate a category
of _models of $\Sigma$_. Given a 2–signature $(\Sigma,A)$, the inequations of
$A$ give rise to a _satisfaction_ predicate on the models of $\Sigma$, and
thus specify a full subcategory of models of $\Sigma$ which satisfy the
inequations of $A$. We call this subcategory the _category of models of
$(\Sigma,A)$_. Our main theorem states that this category has an initial
object — the programming language associated to $(\Sigma,A)$ —, which
integrates the types and terms generated by $\Sigma$, equipped with the
reduction relation generated by the inequations of $A$.
As an example, we specify a translation from $\mathsf{PCF}$ to the untyped
lambda calculus $\operatorname{\mathsf{ULC}}$ using the category–theoretic
iteration operator. This translation is by construction faithful with respect
to reduction in $\mathsf{PCF}$ and $\operatorname{\mathsf{ULC}}$. This example
is verified formally in the proof assistant Coq [9]. The Coq files as well as
documentation are available online at
http://math.unice.fr/laboratoire/logiciels.
The present work is an extended version of another work by the author [3]. In
that previous work, the main theorem [3, Thm. 44] is stated, but no proof is
given. In the present work, we review the definitions given in the earlier
work and present a proof of the main theorem. Afterwards, we explain in detail
the formal verification in the proof assistant Coq [9] of an instance of this
theorem, for the simply–typed programming language $\mathsf{PCF}$. Finally, we
illustrate the iteration operator coming from initiality by specifying an
executable certified translation in Coq from $\mathsf{PCF}$ to the untyped
lambda calculus.
### 1.1 Summary
We define a notion of 2–signature in order to specify the types and terms and
reduction rules of functional programming languages. Given any 2–signature, we
characterize its associated programming language as initial object in some
category. This characterization of syntax with reduction rules is given in two
steps:
1. 1.
At first _pure syntax_ is characterized as initial object in some category.
Here we use the term “pure” to express the fact that no semantic aspects such
as reductions on terms are considered. As will be explained in subsubsection
1.1.1, this characterization is actually a consequence of an earlier result
[2].
2. 2.
Afterwards we consider _inequations_ specifying _reduction rules_. Given a set
of reduction rules for terms, we build up on the preceding result to give an
algebraic characterization of syntax _with reduction_. Inequations for
_untyped_ syntax are considered in earlier work [1]; in the present work, the
main result of that earlier work is carried over to _simply–typed_ syntax.
In summary, the merit of this work is to give an algebraic characterization of
simply–typed syntax _with reduction rules_ , building up on such a
characterization for _pure_ syntax given earlier [2]. Our approach is based on
relative monads as defined by [5] from the category $\mathsf{Set}$ of sets to
the category $\mathsf{Pre}$ of preorders. Compared to traditional monads,
relative monads allow for different categories as domain and codomain.
We now explain the above two points in more detail:
#### 1.1.1 Pure Syntax
A 1–signature $(S,\Sigma)$ is a pair which specifies the types and terms of a
language, respectively. Furthermore, it associates a type to any term. To any
1–signature $(S,\Sigma)$ we associate a category
$\operatorname{Rep}^{\Delta}(S,\Sigma)$ of _representations_ , or “models”, of
$\Sigma$, where a model of $(S,\Sigma)$ is built from a model $T$ of the types
specified by $S$ and a relative monad on the functor
${\Delta}^{T}:{\mathsf{Set}}^{T}\to{\mathsf{Pre}}^{T}$.
This category has an initial object (cf. Sect. 3.2), which integrates the
types and terms freely generated by $(S,\Sigma)$. We call this object the
_(pure) syntax associated to $(S,\Sigma)$_. As mentioned above, we use the
term “pure” to distinguish this initial object from the initial object
associated to a _2–signature_ , which gives an analogous characterization of
syntax _with reduction rules_ (cf. below).
Initiality for pure syntax is actually a consequence of a related initiality
theorem proved in another work [2]: in that work, we associate, to any
signature $(S,\Sigma)$, a category $\operatorname{Rep}(S,\Sigma)$ of models of
$(S,\Sigma)$, where a model is built from a (traditional) monad over
${\mathsf{Set}}^{T}$ instead of a relative monad as above. We connect the
corresponding categories by exhibiting a pair of adjoint functors (cf. Sect.
3.2) between our category $\operatorname{Rep}^{\Delta}(S,\Sigma)$ of
representations of $(S,\Sigma)$ and that of [2],
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We thus obtain an initial object in our category
$\operatorname{Rep}^{\Delta}(S,\Sigma)$ using the fact that left adjoints are
cocontinuous: the image under the functor
$\Delta_{*}:\operatorname{Rep}(S,\Sigma)\to\operatorname{Rep}^{\Delta}(S,\Sigma)$
of the initial object in the category $\operatorname{Rep}(S,\Sigma)$ is
initial in $\operatorname{Rep}^{\Delta}(S,\Sigma)$.
#### 1.1.2 Syntax with Reduction Rules
Given a 1–signature $(S,\Sigma)$, an _$(S,\Sigma)$ –inequation_
$E=(\alpha,\gamma)$ associates a pair $(\alpha^{R},\gamma^{R})$ of _parallel_
morphisms in a suitable category to any representation $R$ of $(S,\Sigma)$. In
a sense made precise later, we can ask whether
$\alpha^{R}\enspace\leq\enspace\gamma^{R}\enspace,$
due to our use of relative monads towards families of _preordered_ sets. If
this is the case, we say that $R$ _satisfies_ the inequation $E$. A
_2–signature_ is a pair $((S,\Sigma),A)$ consisting of a 1–signature
$(S,\Sigma)$, which specifies the types and terms of a language, together with
a set $A$ of _$(S,\Sigma)$ –inequations_, which specifies reduction rules on
those terms. Given a 2–signature $((S,\Sigma),A)$, we call _representation of
$((S,\Sigma),A)$_ any representation of $(S,\Sigma)$ that satisfies each
inequation of $A$. The _category of representations of $((S,\Sigma),A)$_ is
defined to be the full subcategory of representations of $(S,\Sigma)$ whose
objects are representations of $((S,\Sigma),A)$.
We would like to exhibit an initial object in the category of representations
of $((S,\Sigma),A)$, and thus must rule out inequations which are never
satisfied. We call _classic_ $(S,\Sigma)$–inequation any
$(S,\Sigma)$–inequation whose codomain is of a particular form. Our main
result states that for any set $A$ of classic $(S,\Sigma)$–inequations the
category of representations of $((S,\Sigma),A)$ has an initial object. The
class of classic inequations is large enough to account for the fundamental
reduction rules; in particular, beta and eta reductions are given by classic
inequations.
Our definitions ensure that any reduction rule between terms that is expressed
by an inequation $E\in A$ is automatically propagated into subterms. The set
$A$ of inequations hence only needs to contain some “generating” inequations,
a fact that is well illustrated by the example 2–signature $\Lambda\beta$ of
the untyped lambda calculus with beta reduction [1]: This signature has only
one inequation $\beta$ which expresses beta reduction at the root of a term,
$\lambda x.M(N)\enspace\leadsto\enspace M[x:=N]\enspace.$
The initial representation of $\Lambda\beta$ is given by the untyped lambda
calculus, equipped with the reflexive and transitive beta reduction relation
$\twoheadrightarrow_{\beta}$ as presented by [6].
### 1.2 Related Work
Initial Semantics results for syntax with variable binding were first
presented on the LICS’99 conference. Those results are concerned only with the
_syntactic aspect_ of languages: they characterize the _set of terms_ of a
language as an initial object in some category, while not taking into account
reductions on terms. In lack of a better name, we refer to this kind of
initiality results as _purely syntactic_.
Some of these initiality theorems have been extended to also incorporate
semantic aspects, e.g., in form of equivalence relations between terms. These
extensions are reviewed in the second paragraph.
##### Purely syntactic results
Initial Semantics for “pure” syntax — i.e. without considering semantic
aspects — with variable binding were presented by several people
independently, differing in the modelling of variable binding:
The _nominal approach_ by [16] (see also [15, 27]) uses a set theory enriched
with _atoms_ to establish an initiality result. Their approach models lambda
abstraction as a constructor which takes a pair of a variable name and a term
as arguments. In contrast to the other techniques mentioned in this list, in
the nominal approach syntactic equality is different from
$\alpha$–equivalence. [22] proves an initiality result modelling variable
binding in a Higher–Order Abstract Syntax (HOAS) style. [13] (also [11, 12])
model variable binding through nested datatypes as introduced by [7]. [13]’s
approach [13] is extended to _simply–typed syntax_ by [26]. [30] generalize
and subsume those three approaches to a general category of contexts. An
overview of this work and references to more technical papers is given by
[29]. [18] prove an initiality result for untyped syntax based on the notion
of _module over a monad_. Their work has been extended to simply–typed syntax
by [31].
##### Incorporating Semantics
Rewriting in nominal settings has been examined by [10]. [17] present
rewriting for algebraic theories without variable binding; they characterize
equational theories (with a _symmetry_ rule) resp. rewrite systems (with
_reflexivity_ and _transitivity_ rule, but without _symmetry_) as
_coequalizers_ resp. _coinserters_ in a category of monads on the categories
$\mathsf{Set}$ resp. $\mathsf{Pre}$. [14] have extended Fiore’s work to
integrate semantic aspects into initiality results. In particular, Hur’s
thesis [23] is dedicated to _equational_ systems for syntax with variable
binding. In a “Further research” section [23, Chap. 9.3], Hur suggests the use
of preorders, or more generally, arbitrary relations to model _in_ equational
systems. [18] prove initiality of the set of lambda terms modulo beta and eta
conversion in a category of _exponential monads_. In an unpublished paper,
[19] define a notion of _half–equation_ and _equation_ to express congruence
between terms. We adopt their definition in this paper, but interpret a pair
of half–equations as _in_ equation rather than equation. This emphasizes the
_dynamic_ viewpoint of reductions as _directed_ equalities rather than the
_static_ , mathematical viewpoint one obtains by considering symmetric
relations. In a “Future Work” section, [20, Sect. 8] mention the idea of using
preorders as an approach to model semantics, and they suggest interpreting the
untyped lambda calculus with beta and eta reduction rule as a monad over the
category $\mathsf{Pre}$ of preordered sets. The present work gives an
alternative viewpoint to their suggestion by considering the lambda calculus
with beta reduction — and a class of programming languages in general — as a
preorder–valued _relative_ monad on the functor
$\Delta:\mathsf{Set}\to\mathsf{Pre}$. The rationale underlying our use of
relative monads from sets to preorders is that we consider _contexts_ to be
given by unstructured sets, whereas _terms_ of a language carry structure in
form of a reduction relation. In this view it is reasonable to suppose
variables and terms to live in _different_ categories, which is possible
through the use of relative monads on the functor
$\Delta:\mathsf{Set}\to\mathsf{Pre}$ (cf. Def. 2.6) instead of traditional
monads (cf. also [1]). Relative monads were introduced by [5]. In that work,
the authors characterize the untyped lambda calculus as a relative monad over
the inclusion functor from finite sets to sets. Their point of view can be
combined with ours, leading to considering monads on the functor
${\Delta}\circ{i}:\mathsf{Fin}\to\mathsf{Pre}$, cf. Sect. 2.1. [21] , taking
the viewpoint of Categorical Semantics, defines a category Sig of 2–signatures
for _simply–typed_ syntax with reduction rules, and constructs an adjunction
between Sig and the category $\mathsf{2CCCat}$ of small cartesian closed
2–categories. He thus associates to any signature a 2–category of types, terms
and reductions satisfying a universal property. More precisely, terms are
given by morphisms in this category, and reductions are expressed by the
existence of 2–cells between terms. His approach differs from ours in the way
in which variable binding is modelled: Hirschowitz encodes binding in a
Higher–Order Abstract Syntax (HOAS) style through exponentials.
### 1.3 Synopsis
In the second section we review the definition of relative monads and modules
over such monads as well as their morphisms. Some constructions on monads and
modules are given, which will be of importance in what follows.
In the third section we define arities, half–equations and inequations, as
well as their representations. Afterwards we prove our main result.
In the fourth section we describe the formalization in the proof assistant Coq
of an instance of our main result, for the particular case of the language
$\mathsf{PCF}$.
## 2 Relative Monads and Modules
The functor underlying a monad is necessarily endo — this is enforced by the
type of monadic multiplication. _Relative monads_ were introduced by [5] to
overcome this restriction. One of their motivations was to consider the
untyped lambda calculus over _finite_ contexts as a monad–like structure —
similar to the monad structure on the lambda calculus over arbitrary contexts
exhibited by [4].
We review the definition of relative monads and define suitable _colax
morphisms of relative monads_. Afterwards we define _modules over relative
monads_ and port the constructions on modules over monads defined by [18] to
modules over _relative_ monads.
### 2.1 Definitions
We review the definition of relative monad as given by [5] and define suitable
morphisms for them. As an example we consider the lambda calculus with beta
reduction as a relative monad from sets to preorders, on the functor
$\Delta:\mathsf{Set}\to\mathsf{Pre}$ (cf. Def. 2.6). Afterwards we define
_modules over relative monads_ and carry over the constructions on modules
over regular monads of [18] to modules over relative monads.
The definition of relative monads is analogous to that of monads _in Kleisli
form_ , except that the underlying map of objects is between _different_
categories. Thus, for the operations to remain well–typed, one needs an
additional “mediating” functor, in the following usually called $F$, which is
inserted wherever necessary:
###### 2.1 Definition (Relative Monad, [5]):
Given categories ${\mathcal{C}}$ and $\mathcal{D}$ and a functor
$F:{\mathcal{C}}\to\mathcal{D}$, a _relative monad
$P:{\mathcal{C}}\stackrel{{\scriptstyle F}}{{\to}}\mathcal{D}$ on $F$_ is
given by the following data:
* •
a map $P\colon{\mathcal{C}}\to\mathcal{D}$ on the objects of ${\mathcal{C}}$,
* •
for each object $c$ of ${\mathcal{C}}$, a morphism
$\eta_{c}\in\mathcal{D}(Fc,Pc)$ and
* •
for each two objects $c,d$ of ${\mathcal{C}}$, a _substitution_ map (whose
subscripts we usually omit)
$\sigma_{c,d}\colon\mathcal{D}(Fc,Pd)\to\mathcal{D}(Pc,Pd)$
such that the following diagrams commute for all suitable morphisms $f$ and
$g$:
$\textstyle{Fc\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{c}}$$\scriptstyle{f}$$\textstyle{Pc\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma\left({f}\right)}$$\textstyle{Pc\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma\left({\eta_{c}}\right)}$$\scriptstyle{\operatorname{id}}$$\textstyle{Pc\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma\left({f}\right)}$$\scriptstyle{\sigma\left({{\sigma\left({g}\right)}\circ{f}}\right)}$$\textstyle{Pd\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma\left({g}\right)}$$\textstyle{Pd\enspace,}$$\textstyle{Pc\enspace,}$$\textstyle{Pe\enspace.}$
###### 2.2 Remark __:
Relative monads on the identity functor
$\operatorname{Id}:{\mathcal{C}}\to{\mathcal{C}}$ precisely correspond to
monads.
Various examples of relative monads are given by [5]. They give one example
related to syntax and substitution:
###### 2.3 Example (Lambda Calculus over Finite Contexts):
[5] consider the untyped lambda calculus as a relative monad on the functor
$J:\mathsf{Fin}_{\mathrm{skel}}\to\mathsf{Set}$. Here the category
$\mathsf{Fin}_{\mathrm{skel}}$ is the category of finite cardinals, i.e. the
skeleton of the category $\mathsf{Fin}$ of _finite_ sets and maps between
finite sets. The category $\mathsf{Set}$ is the category of sets, cf. Def.
2.4.
We will give another example (cf. Sect. 2.1) of how to view syntax _with
reduction rules_ as a relative monad. For this, we first fix some definitions.
###### 2.4 Definition:
The category $\mathsf{Set}$ is the category of sets and total maps between
them, together with the usual composition of maps.
###### 2.5 Definition:
The category $\mathsf{Pre}$ of preorders has, as objects, sets equipped with a
preorder, and, as morphisms between any two preordered sets $A$ and $B$, the
monotone functions from $A$ to $B$. We consider $\mathsf{Pre}$ as a category
enriched over itself as follows: given $f,g\in\mathsf{Pre}(A,B)$, we say that
$f\leq g$ iff for any $a\in A$, $f(a)\leq g(a)$ in $B$.
###### 2.6 Definition (Functor $\Delta:\mathsf{Set}\to\mathsf{Pre}$ and
Forgetful Functor):
We call $\Delta:\mathsf{Set}\to\mathsf{Pre}$ the left adjoint of the forgetful
functor $U:\mathsf{Pre}\to\mathsf{Set}$,
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The functor $\Delta$ associates, to each set $X$, the set itself together with
the smallest preorder, i.e. the diagonal of $X$,
$\Delta(X):=(X,\delta_{X})\enspace.$
In other words, for any $x,y\in X$ we have $x\delta_{X}y$ if and only if
$x=y$. The functor $\Delta:\mathsf{Set}\to\mathsf{Pre}$ is a _full embedding_
, i.e. it is fully faithful and injective on objects. We have
${U}\circ{\Delta}=\operatorname{Id}_{\mathsf{Set}}$. Altogether, the embedding
$\Delta:\mathsf{Set}\to\mathsf{Pre}$ is a coreflection. We denote by $\varphi$
the family of isomorphisms
$\varphi_{X,Y}:\mathsf{Pre}(\Delta X,Y)\cong\mathsf{Set}(X,UY)\enspace.$
We omit the indices of $\varphi$ whenever they can be deduced from the
context.
###### 2.7 Definition (Category of Families):
Let ${\mathcal{C}}$ be a category and $T$ be a set, i.e. a discrete category.
We denote by ${{\mathcal{C}}}^{T}$ the functor category, an object of which is
a $T$–indexed family of objects of ${\mathcal{C}}$. Given two families $V$ and
$W$, a morphism $f:V\to W$ is a family of morphisms in ${\mathcal{C}}$,
$f:t\mapsto f(t):V(t)\to W(t)\enspace.$
We write $V_{t}:=V(t)$ for objects and morphisms. Given another category
$\mathcal{D}$ and a functor $F:{\mathcal{C}}\to\mathcal{D}$, we denote by
${F}^{T}$ the functor defined on objects and morphisms as
${F}^{T}:{{\mathcal{C}}}^{T}\to{\mathcal{D}}^{T},\quad
f\mapsto\bigl{(}t\mapsto F(f_{t})\bigr{)}\enspace.$
###### 2.8 Remark __:
Given a set $T$, the adjunction of Def. 2.6 induces an adjunction
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3.0pt\raise-3.075pt\hbox{$\textstyle{\scriptstyle{\Delta}^{T}}$}}}}}\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
65.20688pt\raise 4.56238pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
29.0712pt\raise-17.5pt\hbox{\hbox{\kern
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4.66019pt\raise-3.00433pt\hbox{\hbox{\kern 0.0pt\raise
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3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathsf{Pre}}^{T}}$}}}}}}}\ignorespaces\ignorespaces}}}}\enspace.$
###### 2.9 Example (Simply–Typed Lambda Calculus as Relative Monad on
$\Delta^{T}$):
Let
$T:=T_{\operatorname{\mathsf{TLC}}}::=\leavevmode\nobreak\
*\leavevmode\nobreak\ \mid\leavevmode\nobreak\
T_{\operatorname{\mathsf{TLC}}}\leadsto T_{\operatorname{\mathsf{TLC}}}$
be the set of types of the simply–typed lambda calculus. Consider the set
family of simply–typed lambda terms over $T_{\operatorname{\mathsf{TLC}}}$,
indexed by typed contexts:
⬇
Inductive TLC (V : T -> Type) : T -> Type :=
| Var : forall t, V t -> TLC V t
| Abs : forall s t TLC (V + s) t -> TLC V (s $\sim$> t)
| App : forall s t, TLC V (s $\sim$> t) -> TLC V s -> TLC V t.
Here the context V + s is the context V extended by a fresh variable of type s
— the variable that is bound by the constructor Abs (cf. also Sect. 2.3). We
leave the object type arguments implicit and write $\lambda M$ and $M(N)$ for
$\operatorname{Abs}M$ and $\operatorname{App}MN$, respectively. We equip each
set $\operatorname{\mathsf{TLC}}(V)(t)$ of lambda terms over context $V$ of
object type $t$ with a preorder taken as the reflexive–transitive closure of
the relation generated by the rule
$\quad\lambda M(N)\leavevmode\nobreak\ \leq\leavevmode\nobreak\ M[*:=N]$
and its propagation into subterms. This defines a monad
$\operatorname{\mathsf{TLC}}_{\beta}$ from families of sets to families of
preorders over the functor $\Delta^{T}$,
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}:{\mathsf{Set}}^{T}\stackrel{{\scriptstyle\Delta^{T}}}{{\longrightarrow}}{\mathsf{Pre}}^{T}.$
The family $\eta^{\operatorname{\mathsf{TLC}}}$ is given by the constructor
$\operatorname{Var}$, and the substitution map
$\sigma_{X,Y}:{\mathsf{Pre}}^{T}\bigl{(}\Delta^{T}(X),\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(Y)\bigr{)}\to{\mathsf{Pre}}^{T}\bigl{(}\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(X),\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(Y)\bigr{)}$
((1))
is given by capture–avoiding simultaneous substitution. Via the adjunction of
Sect. 2.1 the substitution can also be read as
$\sigma_{X,Y}:{\mathsf{Set}}^{T}\bigl{(}X,\operatorname{\mathsf{TLC}}(Y)\bigr{)}\to{\mathsf{Pre}}^{T}\bigl{(}\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(X),\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(Y)\bigr{)}\enspace.$
In the previous example, the substitution of the lambda calculus satisfies an
additional monotonicity property: the map $\sigma_{X,Y}$ in Disp. (1) is
monotone for the preorders on hom–sets defined in Def. 2.5 and its propagation
in products. This motivates the following definition:
###### 2.10 Definition:
Given a monad $P$ on $\Delta^{T}$ for some set $T$. We say that $P$ is _a
reduction monad_ if for any $X$ and $Y$ the substition $\sigma_{X,Y}$ is
_monotone_ for the preorders on ${\mathsf{Pre}}^{T}(\Delta^{T}X,PY)$ and
${\mathsf{Pre}}^{T}(PX,PY)$.
The monad $\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$ is thus a reduction
monad. It will be clear from Def. 2.40 why we are interested in reduction
monads.
###### 2.11 Remark _Relative Monads are functorial_ :
Given a monad $P$ over $F:{\mathcal{C}}\to\mathcal{D}$, a functorial action
(rlift) for $P$ is defined by setting, for any morphism $f:c\to d$ in
${\mathcal{C}}$,
$P(f):=\operatorname{lift}_{P}(f):=\sigma\left({{\eta}\circ{Ff}}\right)\enspace.$
The functor axioms are easily proved from the monadic axioms.
The substitution $\sigma=(\sigma_{c,d})_{c,d\in|{\mathcal{C}}|}$ of a relative
monad $P$ is binatural:
###### 2.12 Remark _Naturality of Substitution_ :
Given a relative monad $P$ over $F:{\mathcal{C}}\to\mathcal{D}$, then its
substitution $\sigma$ is natural in $c$ and $d$. We write
$f^{*}(h):={h}\circ{f}$. For naturality in $c$ we check that the diagram
$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathcal{D}(Fc,Pd)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{c,d}}$$\textstyle{\mathcal{D}(Pc,Pd)}$$\textstyle{c^{\prime}}$$\textstyle{\mathcal{D}(Fc^{\prime},Pd)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{c^{\prime},d}}$$\scriptstyle{(Ff)^{*}}$$\textstyle{\mathcal{D}(Pc^{\prime},Pd)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(Pf)^{*}}$
commutes. Given $g\in\mathcal{D}(Fc^{\prime},Pd)$, we have
$\displaystyle{\sigma(g)}\circ{Pf}$
$\displaystyle={\sigma(g)}\circ{\sigma({\eta_{c^{\prime}}}\circ{Ff})}$
$\displaystyle\stackrel{{\scriptstyle
3}}{{=}}\sigma({\sigma(g)}\circ{{\eta_{c^{\prime}}}\circ{Ff}})$
$\displaystyle\stackrel{{\scriptstyle 1}}{{=}}\sigma({g}\circ{Ff})\enspace,$
where the numbers correspond to the diagrams of Def. 2.1 used to rewrite in
the respective step. Similarly we check naturality in $d$. Writing
$h_{*}(g):={h}\circ{g}$, the diagram
$\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\mathcal{D}(Fc,Pd)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{c,d}}$$\scriptstyle{(Ph)_{*}}$$\textstyle{\mathcal{D}(Pc,Pd)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(Ph)_{*}}$$\textstyle{d^{\prime}}$$\textstyle{\mathcal{D}(Fc^{\prime},Pd)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{c^{\prime},d}}$$\textstyle{\mathcal{D}(Pc^{\prime},Pd)}$
commutes: given $g\in\mathcal{D}(Fc,Pd)$, we have
$\displaystyle{Ph}\circ{\sigma(g)}$
$\displaystyle={\sigma({\eta_{d^{\prime}}}\circ{Fh})}\circ{\sigma(g)}$
$\displaystyle\stackrel{{\scriptstyle
3}}{{=}}\sigma({\sigma({\eta_{d^{\prime}}}\circ{Fh})}\circ{g})$
$\displaystyle=\sigma({Ph}\circ{g})\enspace.$
If $(T_{i})_{i\in I}$ is a family of sets and $f:I\to J$ a map of sets, then
we obtain a family of sets $(T^{\prime}_{j})_{j\in J}$ by setting
$T^{\prime}_{j}:=\coprod_{\\{i|f(i)=j\\}}T_{i}$. The following construction
generalizes this reparametrization:
###### 2.13 Definition (Retyping Functor):
Let $T$ and $T^{\prime}$ be sets and $g:T\to T^{\prime}$ be a map. Let
${\mathcal{C}}$ be a cocomplete category. The map $g$ induces a functor
$g^{*}:{{\mathcal{C}}}^{T^{\prime}}\to{{\mathcal{C}}}^{T}\enspace,\quad
W\mapsto{W}\circ{g}\enspace.$
The _retyping functor associated to $g:T\to T^{\prime}$_,
$\vec{g}:{{\mathcal{C}}}^{T}\to{{\mathcal{C}}}^{T^{\prime}}\enspace,$
is defined as the left Kan extension operation along $g$, that is, we have an
adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.89917pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-9.89917pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathcal{C}}}^{T}\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
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26.58716pt\raise 17.5pt\hbox{\hbox{\kern
3.0pt\raise-3.57222pt\hbox{$\textstyle{\scriptstyle\vec{g}}$}}}}}\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
58.12418pt\raise 5.10718pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
27.06131pt\raise-17.5pt\hbox{\hbox{\kern
3.0pt\raise-1.29167pt\hbox{$\textstyle{\scriptstyle
g^{*}}$}}}}}\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
4.21939pt\raise-3.00433pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
27.13402pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise-2.43056pt\hbox{$\textstyle{\scriptstyle\bot}$}}}}}\ignorespaces{\hbox{\kern
58.11697pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathcal{C}}}^{T^{\prime}}}$}}}}}}}\ignorespaces\ignorespaces}}}}\enspace.$
((2))
Put differently, the map $g:T\to T^{\prime}$ induces an endofunctor $\bar{g}$
on ${{\mathcal{C}}}^{T}$ with object map
$\bar{g}(V):={\vec{g}(V)}\circ{g}$
and we have a natural transformation ctype — the unit of the adjunction of
Disp. (2),
${\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{ctype}}}}}}:\operatorname{Id}\Rightarrow\bar{g}:{{\mathcal{C}}}^{T}\to{{\mathcal{C}}}^{T}\enspace.$
###### 2.14 Definition (Pointed index sets):
Given a category ${\mathcal{C}}$, a set $T$ and a natural number $n$, we
denote by ${{\mathcal{C}}}^{T}_{n}$ the category with, as objects, diagrams of
the form
$n\stackrel{{\scriptstyle\mathbf{t}}}{{\to}}T\stackrel{{\scriptstyle
V}}{{\to}}{\mathcal{C}}\enspace,$
written $(V,t_{1},\ldots,t_{n})$ with $t_{i}:=\mathbf{t}(i)$. A morphism $h$
to another such $(W,\mathbf{t})$ with the same pointing map $\mathbf{t}$ is
given by a morphism $h:V\to W$ in ${{\mathcal{C}}}^{T}$. Any functor
$F:{{\mathcal{C}}}^{T}\to{\mathcal{D}}^{T}$ extends to
$F_{n}:{{\mathcal{C}}}^{T}_{n}\to{\mathcal{D}}^{T}_{n}$ via
$F_{n}(V,t_{1},\ldots,t_{n}):=(FV,t_{1},\ldots,t_{n})\enspace.$
###### 2.15 Remark __:
The category ${{\mathcal{C}}}^{T}_{n}$ consists of $T^{n}$ copies of
${{\mathcal{C}}}^{T}$, which do not interact. Due to the “markers”
$(t_{1},\ldots,t_{n})$ we can act differently on each copy, cf., e.g., Def.
2.35 and 2.36.
Retyping functors generalize to categories with pointed indexing sets; when
changing types according to a map of types $g:T\to T^{\prime}$, the markers
must be adapted as well:
###### 2.16 Definition:
Given a map of sets $g:T\to T^{\prime}$, by postcomposing the pointing map
with $g$, the retyping functor generalizes to the functor
$\vec{g}(n):{{\mathcal{C}}}^{T}_{n}\to{{\mathcal{C}}}^{T^{\prime}}_{n}\enspace,\quad(V,\mathbf{t})\mapsto\bigl{(}\vec{g}V,g_{*}(\mathbf{t})\bigr{)}\enspace,$
where $g_{*}(\mathbf{t}):={g}\circ{\mathbf{t}}:n\to T^{\prime}$.
We are interested in monads on the category ${\mathsf{Set}}^{T}$ of families
of sets indexed by $T$ and relative monads on
$\Delta^{T}:{\mathsf{Set}}^{T}\to{\mathsf{Pre}}^{T}$ as well as their
relationship:
###### 2.17 Lemma (Relative Monads on $\Delta^{T}$ and Monads on
${\mathsf{Set}}^{T}$):
Let $P$ be a relative monad on
$\Delta^{:}{\mathsf{Set}}^{T}\to{\mathsf{Pre}}^{T}$. By postcomposing with the
forgetful functor ${U}^{T}:{\mathsf{Pre}}^{T}\to{\mathsf{Set}}^{T}$ we obtain
a monad
$UP:{\mathsf{Set}}^{T}\to{\mathsf{Set}}^{T}\enspace.$
The substitution is defined, for $m:X\to UPY$ by setting
$U\sigma:m\mapsto U\left(\sigma\left({\varphi^{-1}m}\right)\right)\enspace,$
making use of the adjunction $\varphi$ of Sect. 2.1. Conversely, to any monad
$P$ over ${\mathsf{Set}}^{T}$, we associate a relative monad $\Delta P$ over
$\Delta^{T}$ by postcomposing with $\Delta^{T}$.
We generalize the definition of colax monad morphisms [25] to relative monads:
###### 2.18 Definition (Colax Morphism of Relative Monads):
Suppose given two relative monads $P:{\mathcal{C}}\stackrel{{\scriptstyle
F}}{{\to}}\mathcal{D}$ and $Q:{\mathcal{C}}^{\prime}\stackrel{{\scriptstyle
F^{\prime}}}{{\to}}\mathcal{D}^{\prime}$. A _colax morphism of relative
monads_ from $P$ to $Q$ is a quadruple $h=(G,G^{\prime},N,\tau)$ of a functor
$G\colon{\mathcal{C}}\to{\mathcal{C}}^{\prime}$, a functor
$G^{\prime}:\mathcal{D}\to\mathcal{D}^{\prime}$ as well as a natural
transformation $N:F^{\prime}G\to G^{\prime}F$ and a natural transformation
$\tau:PG^{\prime}\to GQ$ such that the following diagrams commute for any
objects $c,d$ and any suitable morphism $f$:
$\textstyle{G^{\prime}Pc\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G^{\prime}\sigma^{P}\left({f}\right)}$$\scriptstyle{\tau_{c}}$$\textstyle{G^{\prime}Pd\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{d}}$$\textstyle{QGc\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{Q}\left({{{\tau_{d}}\circ{G^{\prime}f}}\circ{Nc}}\right)}$$\textstyle{QGd}$
$\textstyle{F^{\prime}Gc\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Nc}$$\scriptstyle{\eta^{Q}_{Gc}}$$\textstyle{G^{\prime}Fc\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G^{\prime}\eta^{P}_{c}}$$\textstyle{G^{\prime}Pc\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{c}}$$\textstyle{QGc.}$
Naturality of $\tau$ in the preceding definition is actually a consequence of
the commutative diagrams of Def. 2.18, cf. Lemma colax_RMonad_Hom_NatTrans in
the Coq library.
###### 2.19 Remark __:
In Sect. 3 we are going to use the following instance of the preceding
definition: the categories ${\mathcal{C}}$ and ${\mathcal{C}}^{\prime}$ are
instantiated by ${\mathsf{Set}}^{T}$ and ${\mathsf{Set}}^{T^{\prime}}$,
respectively, for sets $T$ and $T^{\prime}$. The functor $G$ is the retyping
functor (cf. Def. 2.13) associated to some translation of types $g:T\to
T^{\prime}$. Similarly, the categories $\mathcal{D}$ and
$\mathcal{D}^{\prime}$ are instantiated by ${\mathsf{Pre}}^{T}$ and
${\mathsf{Pre}}^{T^{\prime}}$, and the functor $F$ by
$F:=\Delta^{T}:{\mathsf{Set}}^{T}\to{\mathsf{Pre}}^{T}\enspace,$
and similar for $F^{\prime}$:
$\textstyle{{\mathsf{Set}}^{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta^{T}}$$\scriptstyle{\vec{g}}$$\textstyle{{\mathsf{Pre}}^{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vec{g}}$$\textstyle{{\mathsf{Set}}^{T^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta^{T^{\prime}}}$$\textstyle{{\mathsf{Pre}}^{T^{\prime}}.\ignorespaces\ignorespaces}$$\textstyle{\scriptstyle\operatorname{Id}}$
Given a monad $P$ on $F:{\mathcal{C}}\to\mathcal{D}$, the notion of _module
over $P$_ generalizes the notion of monadic substitution:
###### 2.20 Definition (Module over a Relative Monad):
Let $P\colon{\mathcal{C}}\stackrel{{\scriptstyle F}}{{\to}}\mathcal{D}$ be a
relative monad and let $\mathcal{E}$ be a category. A _module $M$ over $P$
with codomain $\mathcal{E}$_ is given by
* •
a map $M:{\mathcal{C}}\to\mathcal{E}$ on the objects of the categories
involved and
* •
for all objects $c,d$ of ${\mathcal{C}}$, a map
$\varsigma_{c,d}:\mathcal{D}(Fc,Pd)\to\mathcal{E}(Mc,Md)$
such that the following diagrams commute for all suitable morphisms $f$ and
$g$:
$\textstyle{Mc\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma({f})}$$\scriptstyle{\varsigma({{\sigma\left({g}\right)}\circ{f}})}$$\textstyle{Md\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma({g})}$$\textstyle{Mc\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma({\eta_{c}})}$$\scriptstyle{\operatorname{id}}$$\textstyle{Me}$$\textstyle{Mc.}$
A functoriality (rmlift) for such a module $M$ is then defined similarly to
that for relative monads: for any morphism $f:c\to d$ in ${\mathcal{C}}$ we
set
$M(f):=\operatorname{rmlift}_{M}(f):=\varsigma({{\eta}\circ{Ff}})\enspace.$
The following examples of modules are instances of constructions explained in
the next section:
###### 2.21 Example (Sect. 2.1 cont.):
The map
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}:V\mapsto\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(V)$
yields a module over the relative monad
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$, the _tautological
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$–module_
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$.
###### 2.22 Example:
Given $V\in{\mathsf{Set}}^{T}$ and $s\in T$, we denote by $V+s$ the context
$V$ enriched by an additional variable of type $s$. The map
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}^{s}:V\mapsto\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(V^{s})$
inherits the structure of a
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$–module from the tautological
module $\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$ (cf. Sect. 2.1). We call
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}^{s}$ the _derived module with
respect to $s\in T$_ of the module
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$; cf. also Sect. 2.2.
###### 2.23 Example:
Given $t\in T$, the map
$V\mapsto\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(V)(t):{\mathsf{Set}}^{T}\to\mathsf{Pre}$
inherits a structure of a
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$–module, the _fibre module
$[{\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}}]_{t}$ with respect to $t\in
T$_.
###### 2.24 Example:
Given $s,t\in T$, the map
$V\mapsto\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(V)(s\leadsto
t)\times\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}(V)(s)$ inherits a
structure of an $\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$–module.
A _module morphism_ is a family of morphisms that is compatible with module
substitution in the source and target modules:
###### 2.25 Definition (Morphism of Relative Modules):
Let $M$ and $N$ be two relative modules over
$P\colon{\mathcal{C}}\stackrel{{\scriptstyle F}}{{\to}}\mathcal{D}$ with
codomain $\mathcal{E}$. A _morphism of relative $P$–modules_ from $M$ to $N$
is given by a collection of morphisms $\rho_{c}\in\mathcal{E}(Mc,Nc)$ such
that for all morphisms $f\in\mathcal{D}(Fc,Pd)$ the following diagram
commutes:
$\textstyle{Mc\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma^{M}({f})}$$\scriptstyle{\rho_{c}}$$\textstyle{Md\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{d}}$$\textstyle{Nc\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma^{N}({f})}$$\textstyle{Nd.}$
The modules over $P$ with codomain $\mathcal{E}$ and morphisms between them
form a category called $\operatorname{\mathsf{RMod}}({P},{\mathcal{E}})$ (in
the digital library: RMOD P E). Composition and identity morphisms of modules
are defined by pointwise composition and identity, similarly to the category
of monads.
###### 2.26 Example (Sect. 2.1, 2.1, Sect. 2.1 cont.):
Abstraction and application are morphisms of
$\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}$–modules:
$\displaystyle\operatorname{Abs}_{s,t}$
$\displaystyle:[{\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}^{s}}]_{t}\to[{\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}}]_{s\leadsto
t}\enspace,$ $\displaystyle\operatorname{App}_{s,t}$
$\displaystyle:[{\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}}]_{s\leadsto
t}\times[{\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}}]_{s}\to[{\operatorname{\mathsf{TLC}}_{\mathsf{\beta}}}]_{t}\enspace.$
### 2.2 Constructions on Relative Monads and Modules
The following constructions are analogous to those used by [18].
Any relative monad $P$ comes with the _tautological_ module over $P$ itself:
###### 2.27 Definition (Tautological Module):
Every relative monad $P$ on $F:{\mathcal{C}}\to\mathcal{D}$ yields a module
$(P,\sigma^{P})$ — also denoted by $P$ — over itself, i.e. an object in the
category $\operatorname{\mathsf{RMod}}({P},{\mathcal{D}})$.
###### 2.28 Definition (Constant and Terminal Module):
Let $P$ be a relative monad on $F:{\mathcal{C}}\to\mathcal{D}$. For any object
$e\in\mathcal{E}$ the constant map $T_{e}\colon{\mathcal{C}}\to\mathcal{E}$,
$c\mapsto e$ for all $c\in{\mathcal{C}}$, is equipped with the structure of a
$P$–module by setting $\varsigma_{c,d}(f)=\operatorname{id}_{e}$. In
particular, if $\mathcal{E}$ has a terminal object $1_{\mathcal{E}}$, then the
constant module $T_{1_{\mathcal{E}}}:c\mapsto 1_{\mathcal{E}}$ is terminal in
$\operatorname{\mathsf{RMod}}({P},{\mathcal{E}})$.
###### 2.29 Definition (Postcomposition with a functor):
Let $P$ be a relative monad on $F:{\mathcal{C}}\to\mathcal{D}$, and let $M$ be
a $P$–module with codomain $\mathcal{E}$. Let $G:\mathcal{E}\to\mathcal{X}$ be
a functor. Then the object map ${G}\circ{M}:{\mathcal{C}}\to\mathcal{X}$
defined by $c\mapsto G(M(c))$ is equipped with a $P$–module structure by
setting, for $c,d\in{\mathcal{C}}$ and $f\in\mathcal{D}(Fc,Pd)$,
$\varsigma^{{G}\circ{M}}(f):=G(\varsigma^{M}(f))\enspace.$
For $M:=P$ (considered as tautological module over itself) and $G$ a constant
functor mapping to an object $x\in\mathcal{X}$ and its identity morphism
$\operatorname{id}_{x}$, we obtain the constant module
$(T_{x},\operatorname{id})$ as in the preceding definition.
Given a module $N$ over a relative monad $Q$ and a monad morphism $\tau:P\to
Q$ into $Q$, we can rebase or “pull back” the module $N$ along $\tau$:
###### 2.30 Definition (Pullback Module):
Suppose given two relative monads $P$ and $Q$ and a morphism $\tau:P\to Q$ as
in Def. 2.18. Let $N$ a $Q$-module with codomain $\mathcal{E}$. We define a
$P$-module $h^{*}M$ to $\mathcal{E}$ with object map
$c\mapsto M(Gc)$
by defining the substitution map, for $f:Fc\to Pd$, as
$\varsigma^{h^{*}M}({f}):=\varsigma^{M}({{{h_{d}}\circ{G^{\prime}f}}\circ{N_{c}}})\enspace.$
The module thus defined is called the _pullback module of $N$ along $h$_. The
pullback extends to module morphisms and is functorial.
###### 2.31 Definition (Induced Module Morphism):
With the same notation as before, the monad morphism $h$ induces a morphism of
$P$–modules $h:G^{\prime}P\to h^{*}Q$. Note that the domain module is the
module obtained by postcomposing (the tautological module of) $P$ with
$G^{\prime}$, whereas for (traditional) monads the domain module was just the
tautological module of the domain monad [18].
One big difference between monads — both traditional and relative ones — and
modules over them is that for the latter we know how to take _products_ :
###### 2.32 Definition (Product):
Suppose the category $\mathcal{E}$ has products. Let $M$ and $N$ be
$P$–modules with codomain $\mathcal{E}$. Then the map
$M\times N:{\mathcal{C}}\to\mathcal{E},\quad c\mapsto Mc\times Nc$
is canonically equipped with a substitution and thus constitutes a module
called the _product of $M$ and $N$_. This construction extends to a product on
$\operatorname{\mathsf{RMod}}({P},{\mathcal{E}})$.
### 2.3 Derivation & Fibre
We are particularly interested in relative monads on the functor
$\Delta^{T}:{\mathsf{Set}}^{T}\to{\mathsf{Pre}}^{T}$ for some set $T$, and
modules over such monads. _Derivation_ and _fibre_ , two important
constructions of [20] on modules over monads on families of sets, carry over
to modules over relative monads on ${\Delta}^{T}$.
Given $u\in T$, we denote by $D(u)\in{\mathsf{Set}}^{T}$ the context with
$D(u)(u)=\\{*\\}$ and $D(u)(t)=\emptyset$ for $u\neq t$. For a context
$V\in{\mathsf{Set}}^{T}$ we set $V^{*u}:=V+D(u)$.
###### 2.33 Definition:
Given a monad $P$ over $\Delta^{T}$ and a $P$–module $M$ with codomain
$\mathcal{E}$, we define the derived module of $M$ with respect to $u\in T$ by
setting
$M^{u}(V):=M(V^{*u})\enspace.$
The module substitution is defined, for
$f\in{\mathsf{Pre}}^{T}(\Delta^{T}V,PW)$, by
$\varsigma^{M^{u}}({f}):=\varsigma^{M}({{}_{u}{f}})\enspace.$
Here the “shifted” map
${}_{u}{f}\in{\mathsf{Pre}}^{T}\bigl{(}\Delta^{T}(V^{*u}),P(W^{*u})\bigr{)}$
is the adjunct under the adjunction of Sect. 2.1 of the coproduct map
$\varphi({}_{u}{f}):=[{P(\mathrm{inl})}\circ{f},\eta(\mathrm{inr}(*))]:V^{*u}\to
UP(W^{*u})\enspace,$
where $[\mathrm{inl},\mathrm{inr}]=\operatorname{id}:W^{*u}\to W^{*u}$.
Derivation is an endofunctor on the category of $P$–modules with codomain
$\mathcal{E}$.
###### 2.34 Notation __:
In case the set $T$ of types is $T=\\{*\\}$ the singleton set of types, i.e.
when talking about untyped syntax, we denote by $M^{\prime}$ the derived
module of $M$. Given a natural number $n$, we denote by $M^{n}$ the module
obtained by deriving $n$ times the module $M$.
Analogously to [2], we derive more generally with respect to a natural
transformation $\tau:1\to\mathcal{T}U_{n}$:
###### 2.35 Definition (Derived Module):
Let $\tau:1\to\mathcal{T}U_{n}$ be a natural transformation. Let $T$ be a set
and $P$ be a relative monad on $\Delta^{T}_{n}$. Given any $P$–module $M$, we
call _derivation of $M$ with respect to $\tau$_ the module with object map
$M^{\tau}(V):=M\left(V^{\tau(V)}\right)$.
###### 2.36 Definition:
Let $P$ be a relative monad over $F$, and $M$ a $P$–module with codomain
$\mathcal{E}^{T}$ for some category $\mathcal{E}$. The _fibre module
$[{M}]_{t}$ of $M$ with respect to $t\in T$_ has object map
$c\mapsto M(c)(t)=M(c)_{t}$
and substitution map
$\varsigma^{[{M}]_{t}}({f}):=\bigl{(}\varsigma^{M}({f})\bigr{)}_{t}\enspace.$
This definition generalizes to fibres with respect to a natural transformation
as in Def. 2.35.
The pullback operation commutes with products, derivations and fibres :
###### 2.37 Lemma:
Let ${\mathcal{C}}$ and $\mathcal{D}$ be categories and $\mathcal{E}$ be a
category with products. Let $P\colon{\mathcal{C}}\to\mathcal{D}$ and
$Q\colon{\mathcal{C}}\to D$ be monads over $F:{\mathcal{C}}\to\mathcal{D}$ and
$F^{\prime}:{\mathcal{C}}^{\prime}\to\mathcal{D}^{\prime}$, resp., and
$\rho:P\to Q$ a monad morphism. Let $M$ and $N$ be $P$–modules with codomain
$\mathcal{E}$. The pullback functor is cartesian:
$\rho^{*}(M\times N)\cong\rho^{*}M\times\rho^{*}N\enspace.$
###### 2.38 Lemma:
Consider the setting as in the preceding lemma, with $F=\Delta^{T}$, and $t\in
T$. Then we have
$\rho^{*}(M^{t})\cong(\rho^{*}M)^{t}\enspace.$
###### 2.39 Lemma:
Suppose $N$ is a $Q$–module with codomain ${\mathcal{E}}^{T}$, and $t\in T$.
Then
$\rho^{*}[{M}]_{t}\cong[{\rho^{*}M}]_{t}\enspace.$
###### 2.40 Definition (Substitution of _one_ Variable):
Let $T$ be a (nonempty) set and let $P$ be a _reduction monad_ (cf. Def. 2.10)
over ${\Delta}^{T}$. For any $s,t\in T$ and $X\in{\mathsf{Set}}^{T}$ we define
a binary substitution operation
$\displaystyle\operatorname{subst}_{s,t}(X):P(X^{*s})_{t}\times P(X)_{s}$
$\displaystyle\to P(X)_{t},$ $\displaystyle(y,z)$ $\displaystyle\mapsto
y[*:=z]:=\sigma\left({[\eta_{X},\lambda x.z]}\right)(y)\enspace.$
For any pair $(s,t)\in T^{2}$, we thus obtain a morphism of $P$–modules
$\operatorname{subst}^{P}_{s,t}:[{{P}^{s}}]_{t}\times[{{P}}]_{s}\to[{{P}}]_{t}\enspace.$
Observe that this substitution operation is monotone in both arguments:
monotonicity in the first argument is a consequence of the monadic axioms.
Monotonicity in the second argument is enforced by considering reduction
monads (Def. 2.10).
## 3 Signatures, Representations, Initiality
We combine the techniques of [2] and [1] in order to obtain an initiality
result for simple type systems with reductions on the term level. As an
example, we specify, via the iteration principle coming from the universal
property, a semantically faithful translation from $\mathsf{PCF}$ with its
usual reduction relation to the untyped lambda calculus with beta reduction.
More precisely, in this section we define a notion of _signature_ and suitable
_representations_ for such signatures, such that the types and terms generated
by the signature, equipped with reductions according to the inequations
specified by the signature, form the _initial representation_. Analogously to
[1], we define a notion of _2–signature_ with two levels: a _syntactic_ level
specifying types and terms of a language, and, on top of that, a _semantic_
level specifying reduction rules on the terms.
### 3.1 1–Signatures
A _1–signature_ specifies types and terms over these types. We give two
presentations of 1–signatures, a _syntactic_ one (cf. Def. 3.7) and a
_semantic_ one (cf. Def. 3.18). The syntactic presentation is the same as in
[2]. However, the semantic presentation is here adapted to our use of
_relative_ monads or, to be more precise, _reduction monads_.
#### 3.1.1 Signatures for Types
We present _algebraic signatures_ , which later are used to specify the
_object types_ of the languages we consider. Algebraic signatures and their
models were first considered by [8].
###### 3.1 Definition (Algebraic Signature):
An _algebraic signature $S$_ is a family of natural numbers, i.e. a set
$J_{S}$ and a map (carrying the same name as the signature)
$S:J_{S}\to\mathbb{N}$. For $j\in J_{S}$ and $n\in\mathbb{N}$, we also write
$j:n$ instead of $j\mapsto n$. An element of $J$ resp. its image under $S$ is
called an _arity_ of $S$.
###### 3.2 Example (Algebraic Signature of $T_{\operatorname{\mathsf{TLC}}}$,
Sect. 2.1):
The algebraic signature of the types of the simply–typed lambda calculus is
given by
$S_{\operatorname{\mathsf{TLC}}}:=\\{*:0\enspace,\quad(\leadsto):2\\}\enspace.$
To any algebraic signature we associate a category of _representations_. We
call _representation of $S$_ any set $U$ equipped with operations according to
the signature $S$. A _morphism of representations_ is a map between the
underlying sets that is compatible with the operations on either side in a
suitable sense. Representations and their morphisms form a category. We give
the formal definitions:
###### 3.3 Definition (Representation of an Algebraic Signature):
A representation $R$ of an algebraic signature $S$ is given by
* •
a set $X$ and
* •
for each $j\in J_{S}$, an operation $j^{R}:X^{S(j)}\to X$.
In the following, given a representation $R$, we write $R$ also for its
underlying set.
###### 3.4 Definition (Morphisms of Representations):
Given two representations $T$ and $U$ of the algebraic signature $S$, a
_morphism_ from $T$ to $U$ is a map $f:T\to U$ such that, for any arity
$n=S(j)$ of $S$, we have
${f}\circ{j^{T}}={j^{U}}\circ{(\underbrace{f\times\ldots\times f}_{n\text{
times}})}\enspace.$
###### 3.5 Example:
The language $\mathsf{PCF}$ [28, 24] is a simply–typed lambda calculus with a
fixed point operator and arithmetic constants. Let
$J:=\\{\iota,o,(\Rightarrow)\\}$. The signature of the types of $\mathsf{PCF}$
is given by the arities
$S_{\mathsf{PCF}}:=\\{\iota:0\enspace,\quad
o:0\enspace,\quad(\Rightarrow):2\\}\enspace.$
A representation $T$ of $S_{\mathsf{PCF}}$ is given by a set $T$ and three
operations,
$\iota^{T}:T\enspace,\quad o^{T}:T\enspace,\quad(\Rightarrow)^{T}:T\times T\to
T\enspace.$
Given two representations $T$ and $U$ of $S_{\mathsf{PCF}}$, a morphism from
$T$ to $U$ is a map $f:T\to U$ between the underlying sets such that, for any
$s,t\in T$,
$\displaystyle f(\iota^{T})$ $\displaystyle=\iota^{U}\enspace,$ $\displaystyle
f(o^{T})$ $\displaystyle=o^{U}\quad\text{ and}$ $\displaystyle
f(s\Rightarrow^{T}t)$ $\displaystyle=f(s)\Rightarrow^{U}f(t)\enspace.$
#### 3.1.2 Signatures for Terms
Consider the example of the simply–typed lambda calculus over a set
$T_{\operatorname{\mathsf{TLC}}}$ of types. Its signature for terms may be
given as follows:
$\\{\operatorname{abs}_{s,t}:=\bigl{[}([s],t)\bigr{]}\to(s\leadsto
t)\enspace,\quad\operatorname{app}_{s,t}:=\bigl{[}([],s\leadsto
t),([],s)\bigr{]}\to t\\}_{s,t\in T_{\operatorname{\mathsf{TLC}}}}\enspace.$
((3))
The parameters $s$ and $t$ range over the set
$T_{\operatorname{\mathsf{TLC}}}$ of types, the initial representation of the
signature for types from Sect. 3.1.1. Our goal is to consider representations
of the simply–typed lambda calculus in monads over categories of the form
${\mathsf{Set}}^{T}$ for _any_ set $T$ — provided that $T$ is equipped with a
representation of the signature $S_{\operatorname{\mathsf{TLC}}}$. It thus is
more suitable to specify the signature of the simply–typed lambda calculus as
follows:
$\\{\operatorname{abs}:=\bigl{[}([1],2)\bigr{]}\to(1\leadsto
2)\enspace,\quad\operatorname{app}:=\bigl{[}([],1\leadsto 2),([],1)\bigr{]}\to
2\\}\enspace.$ ((4))
For any representation $T$ of $S_{\operatorname{\mathsf{TLC}}}$, the variables
$1$ and $2$ range over elements of $T$. In this way the number of abstractions
and applications depends on the representation $T$ of
$S_{\operatorname{\mathsf{TLC}}}$: intuitively, a representation of the above
signature of Disp. (4) over a representation $T$ of
$T_{\operatorname{\mathsf{TLC}}}$ has $T^{2}$ abstractions and $T^{2}$
applications — one for each pair of elements of $T$.
###### 3.6 Definition (Type of Degree $n$):
For $n\geq 1$, we call _types of $S$ of degree $n$_ the elements of the set
$S(n)$ of types associated to the signature $S$ with free variables in the set
$\\{1,\ldots,n\\}$. We set $S(0):=\hat{S}$. Formally, the set $S(n)$ may be
obtained as the initial representation of the signature $S$ enriched by $n$
nullary arities.
Types of degree $n$ are used to form classic arities of degree $n$:
###### 3.7 Definition (Classic Arity of Degree $n$):
A classic arity for terms over the signature $S$ for types of degree $n$ is of
the form
$\bigl{[}([t_{1,1},\ldots,t_{1,m_{1}}],t_{1}),\ldots,([t_{k,1},\ldots,t_{k,m_{k}}],t_{k})\bigr{]}\to
t_{0}\enspace,$ ((5))
where $t_{i,j},t_{i}\in S(n)$. More formally, a classic arity of degree $n$
over $S$ is a pair consisting of an element $t_{0}\in S(n)$ and a list of
pairs. where each pair itself consists of a list
$[t_{i,1},\ldots,t_{i,m_{i}}]$ of elements of $S(n)$ and an element $t_{i}$ of
$S(n)$.
A classic arity of the form given in Disp. (5) denotes a constructor — or a
family of constructors, for $n\geq 1$ — whose output type is $t_{0}$, and
whose $k$ inputs are terms of type $t_{i}$, respectively, in each of which
variables of type according to the list $[t_{i,1},\ldots,t_{i,m_{i}}]$ are
bound by the constructor.
We have to adapt the _semantic_ definition of signatures for terms, however,
since we now work with reduction monads on $\Delta^{T}$ for some set $T$
instead of monads over families of sets. The following definition is the
analogue of earlier work [2], adapted to the use of _relative_ monads:
###### 3.8 Definition (Relative $S$–Monad):
Given an algebraic signature $S$, the _category ${S}\text{-}\mathsf{RMnd}$ of
relative $S$–monads_ is defined as the category whose objects are pairs
$(T,P)$ of a representation $T$ of $S$ and a reduction monad
$P:{\mathsf{Set}}^{T}\stackrel{{\scriptstyle\Delta^{T}}}{{\longrightarrow}}{\mathsf{Pre}}^{T}\enspace.$
A morphism from $(T,P)$ to $(T^{\prime},P^{\prime})$ is a pair $(g,f)$ of a
morphism of $S$–representations $g:T\to T^{\prime}$ and a morphism of relative
monads $f:P\to P^{\prime}$ over the retyping functor $\vec{g}$ as in Sect.
2.1.
Given $n\in\mathbb{N}$, we write ${S}\text{-}\mathsf{RMnd}_{n}$ for the
category whose objects are pairs $(T,P)$ of a representation $T$ of $S$ and a
reduction monad $P$ over $\Delta^{T}_{n}$. A morphism from $(T,P)$ to
$(T^{\prime},P^{\prime})$ is a pair $(g,f)$ of a morphism of
$S$–representations $g:T\to T^{\prime}$ and a monad morphism $f:P\to
P^{\prime}$ over the retyping functor $\vec{g}_{n}$ defined in Def. 2.16.
Similarly, we have a large category of modules over relative monads:
###### 3.9 Definition (Large Category
$\operatorname{\mathsf{LRMod}}_{n}({S},{\mathcal{D}})$ of Modules):
Given a natural number $n\in\mathbb{N}$, an algebraic signature $S$ and a
category $\mathcal{D}$, we call
$\operatorname{\mathsf{LRMod}}_{n}({S},{\mathcal{D}})$ the category an object
of which is a pair $(P,M)$ of a relative $S$–monad
$P\in{S}\text{-}\mathsf{RMnd}_{n}$ and a $P$–module with codomain
$\mathcal{D}$. A morphism to another such $(Q,N)$ is a pair $(f,h)$ of a
morphism of relative $S$–monads $f:P\to Q$ in ${S}\text{-}\mathsf{RMnd}_{n}$
and a morphism of relative modules $h:M\to f^{*}N$.
As before, we sometimes just write the module — i.e. the second — component of
an object or morphism of the large category of modules. Given
$M\in\operatorname{\mathsf{LRMod}}_{n}({S},{\mathcal{D}})$, we thus write
$M(V)$ or $M_{V}$ for the value of the module on the object $V$.
A _half–arity over $S$ of degree $n$_ is a functor from relative $S$–monads to
the category of large modules of degree $n$:
###### 3.10 Definition (Half–Arity over $S$ (of degree $n$)):
Given an algebraic signature $S$ and $n\in\mathbb{N}$, we call _half–arity
over $S$ of degree $n$_ a functor
$\alpha:{S}\text{-}\mathsf{RMnd}\to\operatorname{\mathsf{LRMod}}_{n}({S},{\mathsf{Pre}})\enspace.$
which is pre–inverse to the forgetful functor.
As before we restrict ourselves to a class of such functors. Again, we start
with the _tautological_ module:
###### 3.11 Definition (Tautological Module of Degree $n$):
Given $n\in\mathbb{N}$, any relative monad $R$ over $\Delta^{T}$ induces a
monad $R_{n}$ over $\Delta^{T}_{n}$ with object map
$(V,t_{1},\ldots,t_{n})\mapsto(RV,t_{1},\ldots,t_{n})$. To any relative
$S$–monad $R$ we associate the tautological module of $R_{n}$,
$\Theta_{n}(R):=(R_{n},R_{n})\in\operatorname{\mathsf{LRMod}}_{n}({S},{{\mathsf{Pre}}^{T}_{n}})\enspace.$
Furthermore, we use _canonical natural transformations_ (cf. Def. 3.13) to
build _classic_ half–arities; these transformations specify context extension
(derivation) and selection of specific object types (fibre):
###### 3.12 Definition ($S{\mathcal{C}}_{n}$):
Given a category ${\mathcal{C}}$ — think of it as the category $\mathsf{Set}$
of sets — we define the category $S{\mathcal{C}}_{n}$ to be the category an
object of which is a triple $(T,V,\mathbf{t})$ where $T$ is a representation
of $S$, the object $V\in{{\mathcal{C}}}^{T}$ is a $T$–indexed family of
objects of ${\mathcal{C}}$ and $\mathbf{t}$ is a vector of elements of $T$ of
length $n$. We denote by $SU_{n}:S{\mathcal{C}}_{n}\to\mathsf{Set}$ the
functor mapping an object $(T,V,\mathbf{t})$ to the underlying set $T$.
We have a forgetful functor
$S{\mathcal{C}}_{n}\to\mathcal{T}{\mathcal{C}}_{n}$ which forgets the
representation structure. On the other hand, any representation $T$ of $S$ in
a set $T$ gives rise to a functor ${{\mathcal{C}}}^{T}_{n}\to
S{\mathcal{C}}_{n}$, which “attaches” the representation structure.
The meaning of a term $s\in S(n)$ as a natural transformation
$s:1\Rightarrow SU_{n}:S{\mathcal{C}}_{n}\to\mathsf{Set}$
is now given by recursion on the structure of $s$:
###### 3.13 Definition (Canonical Natural Transformation):
Let $s\in S(n)$ be a type of degree $n$. Then $s$ denotes a natural
transformation
$s:1\Rightarrow SU_{n}:S{\mathcal{C}}_{n}\to\mathsf{Set}\enspace$
defined recursively on the structure of $s$ as follows: for
$s=\alpha(a_{1},\ldots,a_{k})$ the image of a constructor $\alpha\in S$ we set
$s(T,V,\mathbf{t})=\alpha(a_{1}(T,V,\mathbf{t}),\ldots,a_{k}(T,V,\mathbf{t}))$
and for $s=m$ with $1\leq m\leq n$ we define
$s(T,V,\mathbf{t})=\mathbf{t}(m)\enspace.$
We call a natural transformation of the form $s\in S(n)$ _canonical_.
###### 3.14 Definition (Classic Half–Arity):
As with monads (cf. [2]), we restrict our attention to _classic_ half–arities,
which we define analogously to [2] as constructed using derivations and
products, starting from the fibres of the tautological module and the constant
singleton module. We omit the precise statement of this definition.
The following clauses define an inductive set of _classic_ half–arities, to
which we restrict our attention:
* •
The constant functor $*:R\mapsto 1$ is a classic half–arity.
* •
Given any canonical natural transformation $\tau:1\to SU_{n}$ (cf. Def. 3.13),
the point-wise fibre module with respect to $\tau$ (cf. Def. 2.36) of the
tautological module $\Theta_{n}:R\mapsto(R_{n},R_{n})$ (cf. Def. 3.11) is a
classic half–arity of degree $n$,
$[{\Theta_{n}}]_{\tau}:{S}\text{-}\mathsf{RMnd}\to\operatorname{\mathsf{LRMod}}_{n}({S},{\mathsf{Set}})\enspace,\quad
R\mapsto[{R_{n}}]_{\tau}\enspace.$
* •
Given any (classic) half–arity
$M:{S}\text{-}\mathsf{Mnd}\to\operatorname{\mathsf{LMod}}_{n}({S},{\mathsf{Set}})$
of degree $n$ and a canonical natural transformation $\tau:1\to SU_{n}$, the
point-wise derivation of $M$ with respect to $\tau$ is a (classic) half–arity
of degree $n$,
$M^{\tau}:{S}\text{-}\mathsf{RMnd}\to\operatorname{\mathsf{LRMod}}_{n}({S},{\mathsf{Set}})\enspace,\quad
R\mapsto\bigl{(}M(R)\bigr{)}^{\tau}\enspace.$
Here $\bigl{(}M(R)\bigr{)}^{\tau}$ really means derivation of the module, i.e.
derivation in the second component of $M(R)$.
* •
For a half–arity $M$, let $M_{i}:R\mapsto\pi_{i}M(R)$ denote the $i$–th
projection. Given two (polynomial) half–arities $M$ and $N$ of degree $n$,
which coincide pointwise on the first component, i.e. such that $M_{1}=N_{1}$.
Then their product $M\times N$ is again a (polynomial) half–arity of degree
$n$. Here the product is really the pointwise product in the second component,
i.e.
$M\times N:R\mapsto\bigl{(}M_{1}(R),M_{2}(R)\times N_{2}(R)\bigr{)}\enspace.$
A half–arity of degree $n$ thus associates, to any relative $S$–monad $P$ over
a set of types $T$, a _family of $P$–modules_ indexed by $T^{n}$:
###### 3.15 Remark _Module of Higher Degree corresponds to a Family of
Modules_ :
Let $T$ be a set and let $R$ be a monad on the functor ${\Delta}^{T}$. Then a
module $M$ over the monad $R_{n}$ corresponds precisely to a family of
$R$–modules $(M_{\mathbf{t}})_{\mathbf{t}\in T^{n}}$ by (un)currying.
Similarly, a morphism $\alpha:M\to N$ of modules of degree $n$ is equivalent
to a family $(\alpha_{\mathbf{t}})_{\mathbf{t}\in T^{n}}$ of morphisms of
modules of degree zero with $\alpha_{\mathbf{t}}:M_{\mathbf{t}}\to
N_{\mathbf{t}}$.
An arity of degree $n\in\mathbb{N}$ for terms over an algebraic signature $S$
is defined to be a pair of functors from relative $S$–monads to modules in
$\operatorname{\mathsf{LRMod}}_{n}({S},{\mathsf{Pre}})$. The degree $n$
corresponds to the number of object type indices of its associated
constructor. As an example, the arities of $\operatorname{Abs}$ and
$\operatorname{App}$ of Disp. (4) are of degree $2$.
###### 3.16 Definition (Weighted Set):
A weighted set is a set $J$ together with a map $d:J\to\mathbb{N}$.
###### 3.17 Definition (Term–Arity, Signature over $S$):
A _classic arity $\alpha$ over $S$ of degree $n$_ is a pair
$s=\bigl{(}\operatorname{dom}(\alpha),\operatorname{cod}(\alpha)\bigr{)}$
of half–arities over $S$ of degree $n$ such that
* •
$\operatorname{dom}(\alpha)$ is classic and
* •
$\operatorname{cod}(\alpha)$ is of the form $[{\Theta_{n}}]_{\tau}$ for some
canonical natural transformation $\tau$ as in Def. 3.13.
Any classic arity is thus _syntactically_ of the form given in Def. 3.7. We
write $\operatorname{dom}(\alpha)\to\operatorname{cod}(\alpha)$ for the arity
$\alpha$, and $\operatorname{dom}(\alpha,R):=\operatorname{dom}(\alpha)(R)$
and similar for the codomain and morphisms of relative $S$–monads. Given a
weighted set $(J,d)$ as in Def. 3.16, a _term–signature_ $\Sigma$ over $S$
indexed by $(J,d)$ is a $J$-family $\Sigma$ of algebraic arities over $S$, the
arity $\Sigma(j)$ being of degree $d(j)$ for any $j\in J$.
###### 3.18 Definition (Typed Signature):
A _typed signature_ is a pair $(S,\Sigma)$ consisting of an algebraic
signature $S$ for sorts and a term–signature $\Sigma$ (indexed by some
weighted set) over $S$.
###### 3.19 Example:
The terms of the simply typed lambda calculus over the type signature of Sect.
3.1.1 are given by the arities
$\displaystyle\operatorname{abs}$
$\displaystyle:[{\Theta^{1}}]_{2}\to[{\Theta}]_{1\leadsto 2}\enspace,$
$\displaystyle\operatorname{app}$ $\displaystyle:[{\Theta}]_{1\leadsto
2}\times[{\Theta}]_{1}\to[{\Theta}]_{2}\quad,$
both of which are of degree $2$ — we leave the degree implicit. The outer
lower index and the exponent are to be interpreted as de Bruijn variables,
ranging over types. They indicate the fibre (cf. Def. 2.36) and derivation
(cf. Def. 2.35), respectively, in the special case where the corresponding
natural transformation is given by a natural number as in Def. 3.13. In
particular, contrast that to the signature for the simply–typed lambda
calculus we gave in Disp. (3). The difference is that now “similar” arities
which differ only in an object type parameter, are grouped together, whereas
this is not the case in Disp. (3).
Those two arities, $\operatorname{abs}$ and $\operatorname{app}$, can in fact
be considered over any algebraic signature $S$ with an arrow constructor, in
particular over the signature $S_{\mathsf{PCF}}$ (cf. Sect. 3.1.2).
###### 3.20 Example (Sect. 3.1.1 continued):
We continue considering $\mathsf{PCF}$. The signature $S_{\mathsf{PCF}}$ for
its types is given in Sect. 3.1.1. The term–signature of $\mathsf{PCF}$ is
given in Fig. 1: it consists of an arity for abstraction and an arity for
application, each of degree 2, an arity (of degree 1) for the fixed point
operator, and one arity of degree 0 for each logic and arithmetic constant —
some of which we omit:
$\displaystyle\operatorname{abs}$
$\displaystyle:[{\Theta^{1}}]_{2}\to[{\Theta}]_{1\Rightarrow 2}\enspace,$
$\displaystyle\operatorname{app}$ $\displaystyle:[{\Theta}]_{1\Rightarrow
2}\times[{\Theta}]_{1}\to[{\Theta}]_{2}\enspace,$ $\displaystyle\mathbf{Fix}$
$\displaystyle:[{\Theta}]_{1\Rightarrow 1}\to[{\Theta}]_{1}\enspace,$
$\displaystyle\mathbf{n}$ $\displaystyle:*\to[{\Theta}]_{\iota}\qquad\text{for
$n\in\mathbb{N}$}$ $\displaystyle\mathbf{Succ}$
$\displaystyle:*\to[{\Theta}]_{\iota\Rightarrow\iota}$
$\displaystyle\mathbf{Pred}$
$\displaystyle:*\to[{\Theta}]_{\iota\Rightarrow\iota}$
$\displaystyle\mathbf{Zero?}$ $\displaystyle:*\to[{\Theta}]_{\iota\Rightarrow
o}$ $\displaystyle\mathbf{cond_{\iota}}$
$\displaystyle:*\to[{\Theta}]_{o\Rightarrow\iota\Rightarrow\iota\Rightarrow\iota}$
$\displaystyle\mathbf{T},\mathbf{F}$ $\displaystyle:*\to[{\Theta}]_{o}$
$\displaystyle\vdots$
Figure 1: Term Signature of $\mathsf{PCF}$
Our presentation of $\mathsf{PCF}$ is inspired by [24], who — similarly to
[28] — consider, e.g., the successor as a constant of arrow type. As an
alternative, one might consider the successor as a constructor expecting a
term of type $\iota$ as argument, yielding a term of type $\iota$. For our
purpose, those two points of view are equivalent.
### 3.2 Representations of 1–Signatures
###### 3.21 Definition (Representation of an Arity, a Signature over $S$):
A representation of an arity $\alpha$ over $S$ in an $S$–monad $R$ is a
morphism of relative modules
$\operatorname{dom}(\alpha,R)\to\operatorname{cod}(\alpha,R)\enspace.$
A representation $R$ of a signature over $S$ is a given by a relative
$S$–monad — called $R$ as well — and a representation $\alpha^{R}$ of each
arity $\alpha$ of $S$ in $R$.
Representations of $(S,\Sigma)$ are the objects of a category
$\operatorname{Rep}^{\Delta}(S,\Sigma)$, whose morphisms are defined as
follows:
###### 3.22 Definition (Morphism of Representations):
Given representations $P$ and $R$ of a typed signature $(S,\Sigma)$, a
morphism of representations $f:P\to R$ is given by a morphism of relative
$S$–monads $f:P\to R$, such that for any arity $\alpha$ of $S$ the following
diagram of module morphisms commutes:
$\textstyle{\operatorname{dom}(\alpha,P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{dom}(\alpha,f)}$$\scriptstyle{\alpha^{P}}$$\textstyle{\operatorname{cod}(\alpha,P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cod}(\alpha,f)}$$\textstyle{\operatorname{dom}(\alpha,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{R}}$$\textstyle{\operatorname{cod}(\alpha,R).}$
###### 3.23 Lemma:
For any typed signature $(S,\Sigma)$, the category of representations of
$(S,\Sigma)$ has an initial object.
###### Proof.
The initial object is obtained, analogously to the untyped case (cf. [1]), via
an adjunction $\Delta_{*}\dashv U_{*}$ between the categories of
representations of $(S,\Sigma)$ in relative monads and those in monads as in
[2].
In more detail, to any relative $S$–monad $(T,P)\in{S}\text{-}\mathsf{RMnd}$
we associate the $S$–monad $U(T,P):=(T,UP)$ where $U_{*}P$ is the monad
obtained by postcomposing with the forgetful functor
${U}^{T}:{\mathsf{Pre}}^{T}\to{\mathsf{Set}}^{T}$. Substitution for $U_{*}P$
is defined, in each fibre, as in Sect. 2.1. For any arity $s\in\Sigma$ we have
that
$U_{*}\operatorname{dom}(s,P)\cong\operatorname{dom}(s,U_{*}P)\enspace,$
and similar for the codomain. The postcomposed representation morphism
$U_{*}s(P)$ hence represents $s$ in $U_{*}P$ in the sense of [2]. This defines
the functor
$U_{*}:\operatorname{Rep}^{\Delta}(S,\Sigma)\to\operatorname{Rep}(S,\Sigma)$.
Conversely, to any $S$–monad we can associate a relative $S$–monad by
postcomposing with ${\Delta}^{T}:{\mathsf{Set}}^{T}\to{\mathsf{Pre}}^{T}$,
analogous to the untyped case in [1], yielding
$\Delta_{*}:\operatorname{Rep}(S,\Sigma)\to\operatorname{Rep}^{\Delta}(S,\Sigma)$.
In summary, the natural isomorphism
$\varphi_{R,P}:\bigl{(}\operatorname{Rep}^{\Delta}(S,\Sigma)\bigr{)}(\Delta_{*}R,P)\cong\bigl{(}\operatorname{Rep}(S,\Sigma)\bigr{)}(R,U_{*}P)$
is given by postcomposition with the forgetful functor (from left to right)
resp. the functor $\Delta$ (from right to left).
∎
### 3.3 Inequations
Analogously to the untyped case (cf. [1]), an inequation associates, to any
representation of $(S,\Sigma)$ in a relative monad $P$, two parallel morphisms
of $P$–modules. However, similarly to arities, an inequation may now be, more
precisely, a _family of inequations_ , indexed by object types. Consider the
simply–typed lambda calculus, which was defined with _typed_ abstraction and
application. Similarly, we have a _typed substitution_ operation for
$\operatorname{\mathsf{TLC}}$, which substitutes a term of type $s\in
T_{\operatorname{\mathsf{TLC}}}$ for a free variable of type $s$ in a term of
type $t\in T_{\operatorname{\mathsf{TLC}}}$, yielding again a term of type
$t$. For $s,t\in T_{\operatorname{\mathsf{TLC}}}$ and
$M\in\operatorname{\mathsf{TLC}}(V^{*s})_{t}$ and
$N\in\operatorname{\mathsf{TLC}}(V)_{s}$, beta reduction is specified by
$\lambda_{s,t}M(N)\leadsto M[*:=N]\enspace,$
where our notation hides the fact that not only abstraction, but also
application and substitution are typed operations. More formally, such a
reduction rule might read as a family of inequations between morphisms of
modules
${\operatorname{app}_{s,t}}\circ{(\operatorname{abs}_{s,t}\times\operatorname{id})}\enspace\leq\enspace\\_[*^{s}:=_{t}\\_]\enspace,$
where $s,t\in T_{\operatorname{\mathsf{TLC}}}$ range over types of the
simply–typed lambda calculus. Analogously to Sect. 3.1.2, we want to specify
the beta rule without referring to the set $T_{\operatorname{\mathsf{TLC}}}$,
but instead express it for an arbitrary representation $R$ of the typed
signature
$(S_{\operatorname{\mathsf{TLC}}},\Sigma_{\operatorname{\mathsf{TLC}}})$ (cf.
Sect. 3.1.1, 3.1.2), as in
${\operatorname{app}^{R}}\circ{(\operatorname{abs}^{R}\times\operatorname{id})}\enspace\leq\enspace\\_[*:=\\_]\enspace,$
where both the left and the right side of the inequation are given by suitable
$R$–module morphisms of degree 2. Source and target of a _half–equation_
accordingly are given by functors from representations of a typed signature
$(S,\Sigma)$ to a suitable category of modules. A half–equation then is a
natural transformation between its source and target functor:
###### 3.24 Definition (Category of Half–Equations):
Let $(S,\Sigma)$ be a signature. An _$(S,\Sigma)$ –module_ $U$ of degree
$n\in\mathbb{N}$ is a functor from the category of representations of
$(S,\Sigma)$ as defined in Sect. 3.2 to the category
$\operatorname{\mathsf{LRMod}}_{n}({S},{\mathsf{Pre}})$ (cf. Def. 3.9)
commuting with the forgetful functor to the category of relative monads. We
define a morphism of $(S,\Sigma)$–modules to be a natural transformation which
becomes the identity when composed with the forgetful functor. We call these
morphisms _half–equations_ (of degree $n$). We write $U^{R}:=U(R)$ for the
image of the representation $R$ under the $S$–module $U$, and similar for
morphisms.
###### 3.25 Definition (Substitution as Half–Equation):
Given a relative monad on ${\Delta}^{T}$, its associated
substitution–of–one–variable operation (cf. Def. 2.40) yields a family of
module morphisms, indexed by pairs $(s,t)\in T$. By Sect. 3.1.2 this family is
equivalent to a module morphism of degree 2. The assignment
$\operatorname{subst}:R\mapsto\operatorname{subst}^{R}:[{{R}_{2}^{1}}]_{2}\times[{{R}_{2}}]_{1}\to[{{R}_{2}}]_{2}$
thus yields a half–equation of degree $2$ over any signature $S$. Its domain
and codomain are classic.
###### 3.26 Example (Sect. 3.1.2 continued):
The map
${\operatorname{app}}\circ{(\operatorname{abs}\times\operatorname{id})}:R\mapsto{\operatorname{app}^{R}}\circ{(\operatorname{abs}^{R},\operatorname{id}^{R})}:[{{R}_{2}^{1}}]_{2}\times[{{R}_{2}}]_{1}\to[{{R}_{2}}]_{2}$
is a half–equation over the signature $\operatorname{\mathsf{TLC}}$, as well
as over the signature of $\mathsf{PCF}$.
###### 3.27 Definition:
Any classic arity of degree $n$,
$s=[{\Theta_{n}^{{{\tau_{1}}}}}]_{\sigma_{1}}\times\ldots\times[{\Theta_{n}^{{\tau_{m}}}}]_{\sigma_{m}}\to[{\Theta_{n}}]_{\sigma}\enspace,$
defines a classic $S$–module
$\operatorname{dom}(s):R\mapsto[{R^{\tau_{1}}_{n}}]_{\sigma_{1}}\times\ldots\times[{R^{\tau_{m}}_{n}}]_{\sigma_{m}}\enspace.$
###### 3.28 Definition (Inequation):
Given a signature $(S,\Sigma)$, an _inequation over $(S,\Sigma)$_, or
_$(S,\Sigma)$ –inequation_, of degree $n\in\mathbb{N}$ is a pair of parallel
half–equations between $(S,\Sigma)$–modules of degree $n$. We write
$\alpha\leq\gamma$ for the inequation $(\alpha,\gamma)$.
###### 3.29 Example (Beta Reduction):
For any suitable 1–signature — i.e. for any 1–signature that has an arity for
abstraction and an arity for application — we specify beta reduction using the
parallel half–equations of Defs. 3.25 and Sect. 3.3:
${\operatorname{app}}\circ{(\operatorname{abs}\times\operatorname{id})}\leq\operatorname{subst}:[{\Theta_{2}^{1}}]_{2}\times[{\Theta_{2}}]_{1}\to[{\Theta_{2}}]_{2}\enspace.$
###### 3.30 Example (Fixpoints and Arithmetics of $\mathsf{PCF}$):
The reduction rules of $\mathsf{PCF}$ are specified by the inequations — over
the 1–signature of $\mathsf{PCF}$ as given in Sect. 3.1.2 — of Fig. 2.
$\displaystyle\mathbf{Fix}$
$\displaystyle\leq{\operatorname{app}}\circ{(\operatorname{id},\mathbf{Fix})}:[{\Theta_{1}}]_{1\Rightarrow
1}\to[{\Theta_{1}}]_{1}$
$\displaystyle{\operatorname{app}}\circ{(\mathbf{Succ},\mathbf{n})}$
$\displaystyle\leq\mathbf{n+1}:*\to[{\Theta}]_{\iota}$
$\displaystyle{\operatorname{app}}\circ{(\mathbf{Pred},\mathbf{0})}$
$\displaystyle\leq\mathbf{0}:*\to[{\Theta}]_{\iota}$
$\displaystyle{\operatorname{app}}\circ{\left(\mathbf{Pred},{\operatorname{app}}\circ{({\mathbf{Succ}},{\mathbf{n}})}\right)}$
$\displaystyle\leq\mathbf{n}:*\to[{\Theta}]_{\iota}$
$\displaystyle{\operatorname{app}}\circ{(\mathbf{Zero?},\mathbf{0})}$
$\displaystyle\leq\mathbf{T}:*\to[{\Theta}]_{o}$
$\displaystyle{\operatorname{app}}\circ{\left(\mathbf{Zero?},{\operatorname{app}}\circ{(\mathbf{Succ},\mathbf{n})}\right)}$
$\displaystyle\leq\mathbf{F}:*\to[{\Theta}]_{o}$ $\displaystyle\vdots$
Figure 2: Reduction Rules of $\mathsf{PCF}$
###### 3.31 Definition (Representation of Inequations):
A _representation of an $(S,\Sigma)$–inequation $\alpha\leq\gamma:U\to V$_ (of
degree $n$) is any representation $R$ over a set of types $T$ of $(S,\Sigma)$
such that $\alpha^{R}\leq\gamma^{R}$ pointwise, i.e. if for any pointed
context $(X,\mathbf{t})\in{\mathsf{Set}}^{T}\times T^{n}$, any $t\in T$ and
any $y\in U^{R}_{(X,\mathbf{t})}(t)$,
$\alpha^{R}(y)\enspace\leq\enspace\gamma^{R}(y)\enspace,$ ((6))
where we omit the sort argument $t$ as well as the context $(X,\mathbf{t})$
from $\alpha$ and $\gamma$. We say that such a representation $R$ _satisfies_
the inequation $\alpha\leq\gamma$.
For a set $A$ of $(S,\Sigma)$–inequations, we call _representation of
$((S,\Sigma),A)$_ any representation of $(S,\Sigma)$ that satisfies each
inequation of $A$. We define the category of representations of the
2–signature $((S,\Sigma),A)$ to be the full subcategory of the category of
representations of $S$ whose objects are representations of $((S,\Sigma),A)$.
We also write $(\Sigma,A)$ for $((S,\Sigma),A)$.
According to Sect. 3.1.2, the inequation of Disps. (6) is equivalent to ask
whether, for any $\mathbf{t}\in T^{n}$, any $t\in T$ and any $y\in
U_{\mathbf{t}}^{R}(X)(t)$,
$\alpha_{\mathbf{t}}^{R}(y)\enspace\leq\enspace\gamma_{\mathbf{t}}^{R}(y)\enspace.$
### 3.4 Initiality for 2–Signatures
We are ready to state and prove an initiality result for typed signatures with
inequations:
###### 3.32 Theorem:
For any set of classic $(S,\Sigma)$–inequations $A$, the category of
representations of $((S,\Sigma),A)$ has an initial object.
###### Proof.
The proof is analogous to that of the untyped case [1]. The fact that we now
consider _typed_ syntax introduces a minor complication, on the presentation
of which we put the emphasis during the proof. The basic ingredients for
building the initial representation are given by the initial representation
$(\hat{S},\hat{\Sigma})$ — or just $\hat{\Sigma}$ for short — in the category
$\operatorname{Rep}(S,\Sigma)$ of representations in monads on set families
[2]. Equivalently, the ingredients come from the initial object
$(\hat{S},\Delta_{*}\hat{\Sigma})$ — or just $\Delta_{*}\hat{\Sigma}$ for
short — of representations _without inequations_ in the category
$\operatorname{Rep}^{\Delta}(S,\Sigma)$ (cf. Sect. 3.2). We call
$\hat{\Sigma}$ resp. $\Delta_{*}\hat{\Sigma}$ the monad resp. relative monad
underlying the initial representation
The proof consists of 3 steps: at first, we define a preorder $\leq_{A}$ on
the terms of $\hat{\Sigma}$, induced by the set $A$ of inequations. Afterwards
we show that the data of the representation $\hat{\Sigma}$ — substitution,
representation morphisms etc. — is compatible with the preorder $\leq_{A}$ in
a suitable sense. This will yield a representation $\hat{\Sigma}_{A}$ of
$(\Sigma,A)$. Finally we show that $\hat{\Sigma}_{A}$ is the initial such
representation.
_— The monad underlying the initial representation:_
For any context $X\in{\mathsf{Set}}^{\hat{S}}$ and $t\in\hat{S}$, we equip
$\hat{\Sigma}X(t)$ with a preorder $A$ by setting — _morally_ , cf. below —,
for $x,y\in\hat{\Sigma}X(t)$,
$x\leq_{A}y\quad:\Leftrightarrow\quad\forall
R:\operatorname{Rep}(\Sigma,A),\quad i_{R}(x)\leq_{R}i_{R}(y)\enspace,$ ((7))
where $i_{R}:\Delta_{*}\hat{\Sigma}\to R$ is the initial morphism of
representations of $(S,\Sigma)$, cf. Sect. 3.2. Note that the above definition
in Disp. (7) is ill–typed: we have $x\in\hat{\Sigma}X(t)$, which cannot be
applied to (a fibre of) $i_{R}(X):\vec{g}(\hat{\Sigma}X)\to R(\vec{g}X)$. We
denote by $\varphi=\varphi_{R}$ the natural isomorphism induced by the
adjunction of Def. 2.13 obtained by retyping — along the initial morphism of
types $g:\hat{S}\to T=T_{R}$ — towards the set $T$ of “types” of $R$,
$\varphi_{X,Y}:{\mathsf{Pre}}^{T}\left(\vec{g}(\hat{\Sigma}X),R(\vec{g}X)\right)\cong{\mathsf{Pre}}^{\hat{S}}\left(\hat{\Sigma}X,{R(\vec{g}X)}\circ{g}\right)\enspace.$
Instead of the above definition in Disp. (7), we should really write
$x\leq_{A}y\quad:\Leftrightarrow\quad\forall
R:\operatorname{Rep}(\Sigma,A),\quad\left(\varphi(i_{R,X})\right)(x)\leq_{R}\left(\varphi(i_{R,X})\right)(y)\enspace,$
((8))
where we omit the subscript “$R$” from $\varphi$. We have to show that the map
$X\mapsto\hat{\Sigma}_{A}X:=(\hat{\Sigma}X,\leq_{A})$
yields a relative monad on $\Delta^{\hat{S}}$. The missing fact to prove is
that the substitution with a morphism
$f\in{\mathsf{Pre}}^{\hat{S}}(\Delta
X,\hat{\Sigma}_{A}Y)\cong{\mathsf{Set}}^{\hat{S}}(X,\hat{\Sigma}Y)$
is compatible with the order $\leq_{A}$: given any
$f\in{\mathsf{Pre}}^{\hat{S}}(\Delta X,\hat{\Sigma}_{A}Y)$ we show that
$\sigma^{\hat{\Sigma}}(f)\in{\mathsf{Set}}^{\hat{S}}(\hat{\Sigma}X,\hat{\Sigma}Y)$
is monotone with respect to $\leq_{A}$ and hence (the carrier of) a morphism
$\sigma^{\hat{\Sigma}_{A}}(f)\in{\mathsf{Pre}}^{\hat{S}}(\hat{\Sigma}_{A}X,\hat{\Sigma}_{A}Y)\enspace.$
We overload the infix symbol ${}\gg={}$ to denote monadic substitution. Note
that this notation now hides an implicit argument giving the sort of the term
in which we substitute. Suppose $x,y\in\hat{\Sigma}X(t)$ with $x\leq_{A}y$, we
show
${x}\gg={f}\enspace\leq_{A}\enspace{y}\gg={f}\enspace.$
Using the definition of $\leq_{A}$, we must show, for a given representation
$R$ of $(\Sigma,A)$,
$\left(\varphi(i_{R})\right)({x}\gg={f})\enspace\leq_{R}\enspace\left(\varphi(i_{R})\right)({y}\gg={f})\enspace.$
((9))
Let $g$ be the initial morphism of types towards the types of $R$. Since
$i:=i_{R}$ is a morphism of representations — and thus in particular a _monad_
morphism, it is compatible with the substitution of $\hat{\Sigma}$ and $R$; we
have
$\textstyle{\vec{g}(\hat{\Sigma}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vec{g}(\sigma(f))}$$\scriptstyle{i_{X}}$$\textstyle{\vec{g}(\hat{\Sigma}Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{Y}}$$\textstyle{R(\vec{g}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{R}({i_{Y}}\circ{\vec{g}f})}$$\textstyle{R(\vec{g}Y).}$
((10))
By applying the isomorphism $\varphi$ on the diagram of Disp. (10), we obtain
$\displaystyle{\varphi(i_{Y})}\circ{\sigma(f)}$
$\displaystyle=\varphi\left({i_{Y}}\circ{\vec{g}(\sigma(f))}\right)$
$\displaystyle=\varphi\left({\sigma({i_{Y}}\circ{\vec{g}f})}\circ{i_{X}}\right)$
$\displaystyle={g^{*}\left(\sigma^{R}({i_{Y}}\circ{\vec{g}f})\right)}\circ{\varphi(i_{X})}\enspace.$
((11))
Rewriting the equality of Disp. (11) twice in the goal Disp. (9) yields the
goal
$g^{*}\left(\sigma^{R}({i_{Y}}\circ{\vec{g}f})\right)\left((\varphi(i_{X}))(x)\right)=g^{*}\left(\sigma^{R}({i_{Y}}\circ{\vec{g}f})\right)\left((\varphi(i_{X}))(y)\right)\enspace,$
which is true since $g^{*}\left(\sigma^{R}({i_{Y}}\circ{\vec{g}f})\right)$ is
monotone and $(\varphi(i_{X}))(x)\leq_{R}(\varphi(i_{X}))(y)$ by hypothesis.
We hence have defined a monad $\hat{\Sigma}_{A}$ over $\Delta^{\hat{S}}$.
It remains to show that this is a _reduction_ monad: for $f\leq f^{\prime}$,
we must prove that $\sigma(f)\leq\sigma(f^{\prime})$. By Disp. (11), it
suffices to show that
${g^{*}\left(\sigma^{R}({i_{Y}}\circ{\vec{g}f})\right)}\leq{g^{*}\left(\sigma^{R}({i_{Y}}\circ{\vec{g}f^{\prime}})\right)}$
which follows from the fact that $R$ is a reduction category.
###### 3.33 Lemma:
Given a classic $S$–module
$V:\operatorname{Rep}^{\Delta}(S,\Sigma)\to\operatorname{\mathsf{LRMod}}({S},{\mathsf{Pre}})$
from the category of representations of $(S,\Sigma)$ in $S$–monads to the
large category of modules over $S$–monads and $x,y\in V(\hat{\Sigma})(X)(t)$,
we have
$x\leq_{A}y\in V^{\hat{\Sigma}}_{X}(t)\quad\Leftrightarrow\quad\forall
R:\operatorname{Rep}(S,A),\quad
V(i_{R})(x)\leq_{V^{R}_{X}}V(i_{R})(y)\enspace,$
where now and later we omit the arguments $X$ and $(i_{R}(t))$, e.g., in
$V(i_{R})(X)(i_{R}(t))(x)$.
###### Proof of Sects. 3.4.
The proof is done by induction on the derivation of “$V$ classic”. The only
interesting case is where $V=M\times N$ is a product:
$\displaystyle(x_{1},y_{1})\leq(x_{2},y_{2})$ $\displaystyle\Leftrightarrow
x_{1}\leq x_{2}\wedge y_{1}\leq y_{2}$ $\displaystyle\Leftrightarrow\forall
R,M(i_{R})(x_{1})\leq M(i_{R})(x_{2})\wedge\forall R,N(i_{R})(y_{1})\leq
N(i_{R})(y_{2})$ $\displaystyle\Leftrightarrow\forall R,M(i_{R})(x_{1})\leq
M(i_{R})(x_{2})\wedge N(i_{R})(y_{1})\leq N(i_{R})(y_{2})$
$\displaystyle\Leftrightarrow\forall R,V(i_{R})(x_{1},y_{1})\leq
V(i_{R})(x_{2},y_{2})\enspace.$
∎
_— Representing $\Sigma$ in $\hat{\Sigma}_{A}$:_
Any arity $s\in\Sigma$ should be represented by the module morphism
$s^{\hat{\Sigma}}$, i.e. by the representation of $s$ in $\hat{\Sigma}$. We
have to show that those representations are compatible with the preorder $A$.
Given $x\leq_{A}y$ in $\operatorname{dom}(s,\hat{\Sigma})(X)$, we show
(omitting the argument $X$ in $s^{\hat{\Sigma}}(X)(x)$)
$s^{\hat{\Sigma}}(x)\quad\leq_{A}\quad s^{\hat{\Sigma}}(y)\enspace.$
By definition, we have to show that, for any representation $R$ with initial
morphism $i=i_{R}:\hat{\Sigma}\to R$ as before,
$\varphi(i_{X})(s^{\hat{\Sigma}}(x))\quad\leq_{R}\quad\varphi(i_{X})(s^{\hat{\Sigma}}(y))\enspace.$
But these two sides are precisely the images of $x$ and $y$ under the
upper–right composition of the diagram of Def. 3.22 for the morphism of
representations $i_{R}$. By rewriting with this diagram we obtain the goal
$s^{R}\Bigl{(}\bigl{(}\operatorname{dom}(s)(i_{R})\bigr{)}(x)\Bigr{)}\quad\leq_{R}\quad
s^{R}\Bigl{(}\bigl{(}\operatorname{dom}(s)(i_{R})\bigr{)}(y)\Bigr{)}\enspace.$
We know that $s^{R}$ is monotone, thus it is sufficient to show
$\bigl{(}\operatorname{dom}(s)(i_{R})\bigr{)}(x)\quad\leq_{R}\quad\bigl{(}\operatorname{dom}(s)(i_{R})\bigr{)}(y)\enspace.$
This goal follows from Sects. 3.4 (instantiated for the classic $S$–module
$\operatorname{dom}(s)$, cf. Defs. 3.27) and the hypothesis $x\leq_{A}y$. We
hence have established a representation — which we call $\hat{\Sigma}_{A}$ —
of $S$ in $\hat{\Sigma}_{A}$.
_— $\hat{\Sigma}_{A}$ satisfies $A$:_
The next step is to show that the representation $\hat{\Sigma}_{A}$ satisfies
$A$. Given an inequation
$\alpha\leq\gamma:U\to V$
of $A$ with a classic $S$–module $V$, we must show that for any context
$X\in{\mathsf{Set}}^{\hat{S}}$, any $t\in\hat{S}$ and any $x\in
U(\hat{\Sigma}_{A})(X)_{t}$ in the domain of $\alpha$ we have
$\alpha^{\hat{\Sigma}_{A}}(x)\quad\leq_{A}\quad\gamma^{\hat{\Sigma}_{A}}(x)\enspace,$
where here and later we omit the context argument $X$ and the sort argument
$t$. By Sects. 3.4 the goal is equivalent to
$\forall R:\operatorname{Rep}(\Sigma,A),\quad
V(i_{R})(\alpha^{\hat{\Sigma}_{A}}(x))\quad\leq_{V^{R}_{X}}\quad
V(i_{R})(\gamma^{\hat{\Sigma}_{A}}(x))\enspace.$ ((12))
Let $R$ be a representation of $(\Sigma,A)$. We continue by proving Disp. (12)
for $R$. The half–equations $\alpha$ and $\gamma$ are natural transformations.
The fact that $i_{R}$ is the carrier of a morphism of
$(S,\Sigma)$–representations from $\Delta\hat{\Sigma}$ to $R$ allows to
rewrite the goal as
$\alpha^{R}\bigl{(}U(i_{R})(x)\bigr{)}\quad\leq_{V^{R}_{X}}\quad\gamma^{R}\bigr{(}U(i_{R})(x)\bigr{)}\enspace,$
which is true since $R$ satisfies $A$.
_— Initiality of $\hat{\Sigma}_{A}$:_
Given any representation $R$ of $(\Sigma,A)$, the morphism $i_{R}$ is monotone
with respect to the orders on $\hat{\Sigma}_{A}$ and $R$ by construction of
$\leq_{A}$. It is hence a morphism of representations from $\hat{\Sigma}_{A}$
to $R$. Unicity of the morphisms $i_{R}$ follows from its unicity in the
category of representations of $(S,\Sigma)$, i.e. without inequations. Hence
$(\hat{S},\hat{\Sigma}_{A})$ is the initial object in the category of
representations of $((S,\Sigma),A)$.
∎
###### 3.34 Remark _Iteration Principle by Initiality_ :
The universal property of the language generated by a 2–signature yields an
_iteration principle_ to define maps — translations — on this language, which
are certified to be compatible with substitution and reduction in the source
and target languages. How does this iteration principle work? More precisely,
what data (and proof) needs to be specified in order to define such a
translation via initiality from a language, say, $(\hat{S},\hat{\Sigma}_{A})$
to another language $(\hat{S}^{\prime},\hat{\Sigma}^{\prime}_{A^{\prime}})$,
generated by signatures $(S,\Sigma,A)$ and
$(S^{\prime},\Sigma^{\prime},A^{\prime})$, respectively? The translation is a
morphism — an initial one — in the category of representations of the
signature $(S,\Sigma,A)$ of the source language. It is obtained by equipping
the relative monad $\hat{\Sigma}^{\prime}_{A^{\prime}}$ underlying the target
language with a representation of the signature $(S,\Sigma,A)$. In more
detail:
1. 1.
we give a representation of the type signature $S$ in the set
$\hat{S}^{\prime}$. By initiality of $\hat{S}$, this yields a translation
$\hat{S}\to\hat{S}^{\prime}$ of sorts.
2. 2.
Afterwards, we specify a representation of the term signature $\Sigma$ in the
monad $\hat{\Sigma}^{\prime}_{A^{\prime}}$ by defining suitable (families) of
morphisms of $\hat{\Sigma}^{\prime}_{A^{\prime}}$–modules. This yields a
representation $R$ of $(S,\Sigma)$ in the monad
$\hat{\Sigma}^{\prime}_{A^{\prime}}$.
By initiality, we obtain a morphism $f:(\hat{S},\hat{\Sigma})\to R$ of
representations of $(S,\Sigma)$, that is, we obtain a translation from
$(\hat{S},\hat{\Sigma})$ to $(\hat{S}^{\prime},\hat{\Sigma}^{\prime})$ as the
colax monad morphism underlying the morphism $f$. However, we have not yet
ensured that the translation $f$ is compatible with the respective reduction
preorders in the source and target languages.
1. 3.
Finally, we verify that the representation $R$ of $(S,\Sigma)$ satisfies the
inequations of $A$, that is, we check whether, for each $\alpha\leq\gamma:U\to
V\in A$, and for each context $V$, each $t\in\hat{S}$ and $x\in U^{R}_{V}(t)$,
$\alpha^{R}(x)\enspace\leq\enspace\gamma^{R}(x)\enspace.$
After verifying that $R$ satisfies the inequations of $A$, the representation
$R$ is in fact a representation of $(S,\Sigma,A)$. The initial morphism $f$
thus yields a faithful translation from $(\hat{S},\hat{\Sigma}_{A})$ to
$(\hat{S}^{\prime},\hat{\Sigma}^{\prime}_{A^{\prime}})$.
###### 3.35 Example (Translation from $\mathsf{PCF}$ to
$\operatorname{\mathsf{ULC}}$, Sects. 4):
We use the above explained iteration principle to specify a translation from
$\mathsf{PCF}$ to the untyped lambda calculus, that is semantically faithful
with respect to the usual reduction relation of $\mathsf{PCF}$ — generated by
the inequations of Sect. 3.3 — and beta reduction of the lambda calculus.
For the translation of $\mathsf{PCF}$ to the lambda calculus mapping the
fixedpoint operator of $\mathsf{PCF}$ to the Turing fixedpoint combinator, we
have formalized its specification via initiality in the proof assistant Coq
[9]. After constructing the category of representations of $\mathsf{PCF}$, we
equip the untyped lambda calculus with a representations of $\mathsf{PCF}$,
representing the arity $\mathbf{Fix}$ by the Turing operator ${\Theta}$. The
Coq source files as well as documentation is available on
http://math.unice.fr/laboratoire/logiciels.
Note that the translation is given by a Coq function and hence executable.
## 4 A Translation from $\mathsf{PCF}$ to $\operatorname{\mathsf{ULC}}$ via
Initiality, in Coq
In this section we describe the implementation of the category of
representations of $\mathsf{PCF}$, equipped with reduction rules, as well as
of its initial object. This yields an instance of Sects. 3.4. However, for the
implementation in Coq of this instance we make several simplifications
compared to the general theorem:
* •
we do not define a notion of 2–signature, but specify directly a Coq type of
representations of semantic $\mathsf{PCF}$;
* •
we use dependent Coq types to formalize arities of higher degree (cf. Def.
3.10), instead of relying on modules on pointed categories. A representation
of an arity of degree $n$ is thus given by a family of module morphisms (of
degree zero), indexed $n$ times over the respective object type as described
in Sect. 3.1.2;
* •
the relation on the initial object is not defined via the formula of Disp.
(7), but directly through an inductive type, cf. Sect. 4.4, and various
closures, cf. Sect. 4.4.
### 4.1 Representations of $\mathsf{PCF}$
In this section we explain the formalization of representations of
$\mathsf{PCF}$ with reduction rules (cf. Fig. 1 and Fig. 2). According to Def.
3.21 and Defs. 3.31, such a representation consists of
1. 1.
a representation of the types of $\mathsf{PCF}$ (in a Coq type U), cf. Sect.
3.1.1,
2. 2.
a reduction monad P over the functor ${\Delta}^{U}$ (in the formalization:
IDelta U) and
3. 3.
representations of the arities of $\mathsf{PCF}$ (cf. Sect. 3.1.2), i.e.
morphisms of $P$–modules with suitable source and target modules such that
4. 4.
the inequations defining the reduction rules of $\mathsf{PCF}$ are satisfied.
A representation of $\mathsf{PCF}$ should be a “bundle”, i.e. a record type,
whose components — or “fields” — are these 4 items. We first define what a
representation of the term signature of $\mathsf{PCF}$ in a monad $P$ is, in
the presence of an $S_{\mathsf{PCF}}$–monad (cf. Def. 3.8). Unfolding the
definitions, we suppose given a type Sorts, a relative monad P over IDelta
Sorts and three operations on Sorts: a binary function Arrow — denoted by an
infixed “$\sim$$\sim$>” — and two constants Bool and Nat.
⬇
Variable Sorts : Type.
Variable P : RMonad (IDelta Sorts).
Variable Arrow : Sorts -> Sorts -> Sorts.
Variable Bool : Sorts.
Variable Nat : Sorts.
Notation "a $\sim$$\sim$> b" := (Arrow a b) (at level 60, right
associativity).
In this context, a representation of $\mathsf{PCF}$ is given by a bunch of
module morphisms satisfying some conditions. We split the definition into
smaller pieces, cf. Sect. 4.1 – 4.1 . Note that M[t] denotes the fibre module
of module M with respect to t, and d M // u denotes derivation of module M
with respect to u. The module denoted by a star * is the terminal module,
which is the constant singleton module.
4.1 Code (1–Signature of $\mathsf{PCF}$):
⬇
Class PCFPO_rep_struct := {
app : forall u v, (P[u $\sim$$\sim$> v]) x (P[u]) —> P[v]
where "A @ B" := (app _ _ _ (A,B));
abs : forall u v, (d P // u)[v] —> P[u $\sim$$\sim$> v];
rec : forall t, P[t $\sim$$\sim$> t] —> P[t];
tttt : * —> P[Bool];
ffff : * —> P[Bool];
nats : forall m:nat, * —> P[Nat];
Succ : * —> P[Nat $\sim$$\sim$> Nat];
Pred : * —> P[Nat $\sim$$\sim$> Nat];
Zero : * —> P[Nat $\sim$$\sim$> Bool];
CondN: * —> P[Bool $\sim$$\sim$> Nat $\sim$$\sim$> Nat $\sim$$\sim$> Nat];
CondB: * —> P[Bool $\sim$$\sim$> Bool $\sim$$\sim$> Bool $\sim$$\sim$> Bool];
bottom: forall t, * —> P[t];
…
These module morphisms are subject to some inequations specifying the
reduction rules of $\mathsf{PCF}$. The beta rule reads as
4.2 Code (Beta Rule for Representations of $\mathsf{PCF}$):
⬇
beta_red : forall r s V y z, abs r s V y @ z << y[*:= z] ;
…
where y[*:= z] is the substitution of the freshest variable (cf. Def. 2.40) as
a special case of simultaneous monadic substitution. The rule for the fixed
point operator says that $\mathbf{Y}(f)\leadsto f\left(\mathbf{Y}(f)\right)$:
4.3 Code (Inequation for Fixedpoint Operator):
⬇
Rec_A: forall V t g, rec t V g << g @ rec _ _ g
…
The other inequations concern the arithmetic and logical constants of
$\mathsf{PCF}$. Firstly, we have that the conditionals reduce according to the
truth value they are applied to:
4.4 Code (Logic Inequations of $\mathsf{PCF}$ Representations):
⬇
CondN_t: forall V n m, CondN V tt @ tttt _ tt @ n @ m << n ;
CondN_f: forall V n m, CondN V tt @ ffff _ tt @ n @ m << m ;
CondB_t: forall V n m, CondB V tt @ tttt _ tt @ n @ m << n ;
CondB_f: forall V n m, CondB V tt @ ffff _ tt @ n @ m << m ;
…
Furthermore, we have that $\operatorname{succ}(n)$ reduces to $n+1$ (which in
Coq is written S n), reduction of the $\operatorname{zero}?$ predicate
according to whether its argument is zero or not, and that the predecessor is
post–inverse to the successor function:
4.5 Code (Arithmetic Inequations of $\mathsf{PCF}$ Representations):
⬇
Succ_red: forall V n, Succ V tt @ nats n _ tt << nats (S n) _ tt ;
Zero_t: forall V, Zero V tt @ nats 0 _ tt << tttt _ tt ;
Zero_f: forall V n, Zero V tt @ nats (S n) _ tt << ffff _ tt ;
Pred_Succ: forall V n, Pred V tt @ (Succ V tt @ nats n _ tt) << nats n _ tt;
Pred_Z: forall V, Pred V tt @ nats 0 _ tt << nats 0 _ tt }.
After abstracting over the section variables we package all of this into a
record type:
⬇
Record PCFPO_rep := {
Sorts : Type;
Arrow : Sorts -> Sorts -> Sorts;
Bool : Sorts ;
Nat : Sorts ;
pcf_rep_monad :> RMonad (IDelta Sorts);
pcf_rep_struct :> PCFPO_rep_struct pcf_rep_monad Arrow Bool Nat }.
Notation "a $\sim$$\sim$> b" := (Arrow a b) (at level 60, right
associativity).
The type PCFPO_rep constitutes the type of objects of the category of
representations of $\mathsf{PCF}$ with reduction rules.
### 4.2 Morphisms of Representations
A morphism of representations (cf. Def. 3.22) is built from a morphism $g$ of
_type_ representations and a colax monad morphism over the retyping functor
associated to the map $g$. In the particular case of $\mathsf{PCF}$, a
morphism of representations from $P$ to $R$ consists of a morphism of
representations of the types of $\mathsf{PCF}$ — with underlying map Sorts_map
— and a colax morphism of relative monads which makes commute the diagrams of
the form given in Def. 3.22. We first define the diagrams we expect to
commute, before packaging everything into a record type of morphisms. The
context is given by the following declarations:
⬇
Variables P R : PCFPO_rep.
Variable Sorts_map : Sorts P -> Sorts R.
Hypothesis HArrow : forall u v, Sorts_map (u $\sim$$\sim$> v) = Sorts_map u
$\sim$$\sim$> Sorts_map v.
Hypothesis HBool : Sorts_map (Bool _ ) = Bool _ .
Hypothesis HNat : Sorts_map (Nat _ ) = Nat _ .
Variable f : colax_RMonad_Hom P R
(RETYPE (fun t => Sorts_map t))
(RETYPE_PO (fun t => Sorts_map t))
(RT_NT (fun t => Sorts_map t)).
We explain the commutative diagrams of Def. 3.22 for some of the arities. For
the successor arity we ask the following diagram to commute:
4.6 Code (Commutative Diagram for Successor Arity):
⬇
Program Definition Succ_hom’ :=
Succ ;; f [(Nat $\sim$$\sim$> Nat)] ;; Fib_eq_RMod _ _ ;; IsoPF == *—>* ;; f
** Succ.
Here the morphism Succ refers to the representation of the successor arity
either of P (the first appearance) or R (the second appearance) — Coq is able
to figure this out itself. The domain of the successor is given by the
terminal module $*$. Accordingly, we have that
$\operatorname{dom}(\operatorname{Succ},f)$ is the trivial module morphism
with domain and codomain given by the terminal module. We denote this module
morphism by *—>*. The codomain is given as the fibre of $f$ of type
$\iota\Rightarrow\iota$. The two remaining module morphisms are isomorphisms
which do not appear in the informal description. The isomorphism IsoPF is
needed to permute fibre with pullback (cf. Sect. 2.3). The morphism
Fib_eq_RMod M H takes a module M and a proof H of equality of two object types
as arguments, say, H : u = v. Its output is an isomorphism M[u] —> M[v]. Here
the proof is of type
⬇
Sorts_map (Nat $\sim$$\sim$> Nat) = Sorts_map Nat $\sim$$\sim$> Sorts_map Nat
and Coq is able to figure out the proof itself. The diagram for application
uses the product of module morphisms, denoted by an infixed X:
4.7 Code (Commutative Diagram for Application Arity):
⬇
Program Definition app_hom’ := forall u v,
app u v;; f [( _ )] ;; IsoPF ==
(f [(u $\sim$$\sim$> v)] ;; Fib_eq_RMod _ (HArrow _ _);; IsoPF ) X (f [(u)] ;;
IsoPF ) ;;
IsoXP ;; f ** (app _ _ ).
In addition to the already encountered isomorphism IsoPF we have to insert an
isomorphism IsoXP which permutes pullback and product (cf. Sect. 2.3). As a
last example, we present the property for the abstraction:
4.8 Code (Commutative Diagram for Abstraction Arity):
⬇
Program Definition abs_hom’ := forall u v,
abs u v ;; f [( _ )] ==
DerFib_RMod_Hom _ _ _ ;; IsoPF ;; f ** (abs (_ u) (_ v)) ;; IsoFP ;;
Fib_eq_RMod _ (eq_sym (HArrow _ _ )) .
Here the module morphism DerFib_RMod_Hom f u v corresponds to the morphism
$\operatorname{dom}(\operatorname{Abs}(u,v),f)=[{f^{u}}]_{v}$, and IsoFP
permutes fibre with pullback, just like its sibling IsoPF, but the other way
round.
We bundle all those properties into a type class:
⬇
Class PCFPO_rep_Hom_struct := {
CondB_hom : CondB_hom’ ;
CondN_hom : CondN_hom’ ;
Pred_hom : Pred_hom’ ;
Zero_hom : Zero_hom’ ;
Succ_hom : Succ_hom’ ;
fff_hom : fff_hom’ ;
ttt_hom : ttt_hom’ ;
bottom_hom : bottom_hom’ ;
nats_hom : nats_hom’ ;
app_hom : app_hom’ ;
rec_hom : rec_hom’ ;
abs_hom : abs_hom’ }.
Similarly to what we did for representations, we abstract over the section
variables and define a record type of morphisms of representations from P to R
:
⬇
Record PCFPO_rep_Hom := {
Sorts_map : Sorts P -> Sorts R ;
HArrow : forall u v, Sorts_map (u $\sim$$\sim$> v) = Sorts_map u $\sim$$\sim$>
Sorts_map v;
HNat : Sorts_map (Nat _ ) = Nat R ;
HBool : Sorts_map (Bool _ ) = Bool R ;
rep_Hom_monad :> colax_RMonad_Hom P R (RT_NT Sorts_map);
rep_colax_Hom_monad_struct :> PCFPO_rep_Hom_struct
HArrow HBool HNat rep_Hom_monad }.
### 4.3 Equality of Morphisms, Category of Representations
We have already seen how some definitions that are trivial in informal
mathematics, turn into something awful in intensional type theory. Equality of
morphisms of representations is another such definition. Informally, two such
morphisms $a,c:P\to R$ of representations are equal if
1. 1.
their map of object types $f_{a}$ and $f_{c}$ (Sorts_map) are equal and
2. 2.
their underlying colax morphism of monads — also called $a$ and $c$ — are
equal.
In our formalization, the second condition is not even directly expressable,
since these monad morphisms do not have the same type: we have, for a context
$V\in{\mathsf{Set}}^{P}$,
$a_{V}:\vec{f_{a}}(PV)\to R(\vec{f_{a}}V)$
and
$c_{V}:\vec{f_{c}}(PV)\to R(\vec{f_{c}}V)\enspace.$
where ${\mathsf{Set}}^{P}$ is a notation for contexts typed over the set of
object types the representation $P$ comes with, formally the type Sorts P. We
can only compare $a_{V}$ to $c_{V}$ by composing each of them with a suitable
transport transp again, yielding morphisms
${{R(\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{transp}}}}}})}\circ{a_{V}}:\vec{f_{a}}(PV)\to
R(\vec{f_{a}}V)\to R(\vec{f_{c}}V)$
and
${{c_{V}}\circ{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{transp}}\textquoteright}}}}}:\vec{f_{a}}(PV)\to\vec{f_{c}}(PV)\to
R(\vec{f_{c}}V)\enspace.$
As before, for equal fibres $[{M}]_{u}$ and $[{M}]_{t}$ with $u=t$, the
carriers of those transports transp and transp’ are terms of the form eq_rect
_ _ _ H, where H is a proof term which depends on the proof of
⬇
forall x : Sorts P, Sorts_map c x = Sorts_map a x
of the first condition. Altogether, the definition of equality of morphisms of
representations is given by the following inductive proposition:
⬇
Inductive eq_Rep (P R : PCFPO_rep) : relation (PCFPO_rep_Hom P R) :=
| eq_rep : forall (a c : PCFPO_rep_Hom P R),
forall H : (forall t, Sorts_map c t = Sorts_map a t),
(forall V, a V ;; rlift R (Transp H V)
==
Transp_ord H (P V) ;; c V ) -> eq_Rep a c.
The formal proof that the relation thus defined is an equivalence is
inadequately long when compared to its mathematical complexity, due to the
transport elimination.
Composition of representations is done by composing the underlying maps of
sorts, as well as composing the underlying monad morphisms pointwise. Again,
this operation, which is trivial from a mathematical point of view, yields a
difficulty in the formalization, due to the fact that in the formalization
$\vec{g}(\vec{f}V)\not\equiv\vec{({g}\circ{f})}V\enspace.$
More precisely, suppose given two morphisms of representations $a:P\to Q$ and
$b:Q\to R$, given by families of morphisms indexed by $V$ resp. $W$,
$\displaystyle a_{V}$ $\displaystyle:\widetilde{PV}^{a}\to
Q(\widetilde{V}^{a})\quad\text{and}$ $\displaystyle b_{W}$
$\displaystyle:\widetilde{QW}^{b}\to R(\widetilde{W}^{b})\enspace,$
where we write $\widetilde{V}^{a}$ for $\vec{f_{a}}V$. The monad morphism
underlying the composite morphism of representations is given by the following
definition:
$\textstyle{\widetilde{PV}^{{b}\circ{a}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{b}\circ{a}_{V}}$match$\textstyle{R(\widetilde{V}^{{b}\circ{a}})}$$\textstyle{PV\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ctype$\textstyle{R\left(\widetilde{\widetilde{V}^{a}}^{b}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R(\cong)}$$\textstyle{\widetilde{PV}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{V}}$$\textstyle{Q(\widetilde{V}^{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ctype$\textstyle{\widetilde{Q(\widetilde{V}^{a})}^{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{\widetilde{V}^{a}}}$
or, in Coq code,
⬇
Definition comp_rep_car : (forall c : ITYPE U,
RETYPE (fun t => f’ (f t)) (P c) —>
R ((RETYPE (fun t => f’ (f t))) c)) :=
fun (V : ITYPE U) t (y : retype (fun t => f’ (f t)) (P V) t) =>
match y with ctype _ z =>
lift (M:=R) (double_retype_1 (f:=f) (f’:=f’) (V:=V)) _
(b _ _ (ctype (fun t => f’ t)
(a _ _ (ctype (fun t => f t) z ))))
end.
where double_retype_1 denotes the isomorphism in the upper right corner. The
proof of the commutative diagrams for the composite monad morphism is lengthy
due to the number of arities of the signature of $\mathsf{PCF}$. Definition of
the identity morphisms is routine, and in the end we define the category of
representations of semantic $\mathsf{PCF}$:
⬇
Program Instance REP_s :
Cat_struct (obj := PCFPO_rep) (PCFPO_rep_Hom) := {
mor_oid P R := eq_Rep_oid P R ;
id R := Rep_id R ;
comp a b c f g := Rep_comp f g }.
### 4.4 One Particular Representation
We define a particular representation, which we later prove to be initial.
First of all, the set of object types of $\mathsf{PCF}$ is given as follows:
⬇
Inductive Sorts :=
| Nat : Sorts
| Bool : Sorts
| Arrow : Sorts -> Sorts -> Sorts.
For this section we introduce some notations:
⬇
Notation "’TY’" := PCF.Sorts.
Notation "’Bool’" := PCF.Bool.
Notation "’Nat’" := PCF.Nat.
Notation "’IT’" := (ITYPE TY).
Notation "a ’$\sim$>’ b" := (PCF.Arrow a b) (at level 69, right
associativity).
We specify the set of $\mathsf{PCF}$ constants through the following inductive
type, indexed by the sorts of $\mathsf{PCF}$:
⬇
Inductive Consts : TY -> Type :=
| Nats : nat -> Consts Nat
| ttt : Consts Bool
| fff : Consts Bool
| succ : Consts (Nat $\sim$> Nat)
| preds : Consts (Nat $\sim$> Nat)
| zero : Consts (Nat $\sim$> Bool)
| condN: Consts (Bool $\sim$> Nat $\sim$> Nat $\sim$> Nat)
| condB: Consts (Bool $\sim$> Bool $\sim$> Bool $\sim$> Bool).
The set family of terms of $\mathsf{PCF}$ is given by an inductive family,
parametrized by a context V and indexed by object types:
⬇
Inductive PCF (V: TY -> Type) : TY -> Type:=
| Bottom: forall t, PCF V t
| Const : forall t, Consts t -> PCF V t
| Var : forall t, V t -> PCF V t
| App : forall t s, PCF V (s $\sim$> t) -> PCF V s -> PCF V t
| Lam : forall t s, PCF (opt t V) s -> PCF V (t $\sim$> s)
| Rec : forall t, PCF V (t $\sim$> t) -> PCF V t.
Notation "a @ b" := (App a b)(at level 43, left associativity).
Notation "M ’" := (Const _ M) (at level 15).
Monadic substitution is defined recursively on terms:
⬇
Fixpoint subst (V W: TY -> Type)(f: forall t, V t -> PCF W t)
(t : TY)(v : PCF V t) : PCF W t :=
match v with
| Bottom t => Bottom W t
| c ’ => c ’
| Var t v => f t v
| u @ v => u >>= f @ v >>= f
| Lam t s u => Lam (u >>= shift f)
| Rec t u => Rec (u >>= f)
end
where "y >>= f" := (@subst _ _ f _ y).
Here shift f is the substitution map f extended to account for an extended
context under the binder Lam. It is equal to the shifted map of Def. 2.33.
Finally, we define a relation on the terms of type PCF via the inductive
definition
4.9 Code (Reduction Rules for $\mathsf{PCF}$):
⬇
Inductive eval (V : IT): forall t, relation (PCF V t) :=
| app_abs : forall (s t:TY) (M: PCF (opt s V) t) N,
eval (Lam M @ N) (M [*:= N])
| condN_t: forall n m, eval (condN ’ @ ttt ’ @ n @ m) n
| condN_f: forall n m, eval (condN ’ @ fff ’ @ n @ m) m
| condB_t: forall u v, eval (condB ’ @ ttt ’ @ u @ v) u
| condB_f: forall u v, eval (condB ’ @ fff ’ @ u @ v) v
| succ_red: forall n, eval (succ ’ @ Nats n ’) (Nats (S n) ’)
| zero_t: eval ( zero ’ @ Nats 0 ’) (ttt ’)
| zero_f: forall n, eval (zero ’ @ Nats (S n)’) (fff ’)
| pred_Succ: forall n, eval (preds ’ @ (succ ’ @ Nats n ’)) (Nats n ’)
| pred_z: eval (preds ’ @ Nats 0 ’) (Nats 0 ’)
| rec_a : forall t g, eval (Rec g) (g @ (Rec (t:=t) g)).
which we then propagate into subterms (cf. Sect. 4.4) and close with respect
to transitivity and reflexivity:
4.10 Code (Propagation of Reductions into Subterms):
⬇
Reserved Notation "x :> y" (at level 70).
Variable rel : forall (V:IT) t, relation (PCF V t).
Inductive propag (V: IT) : forall t, relation (PCF V t) :=
| relorig : forall t (v v’: PCF V t), rel v v’ -> v :> v’
| relApp1: forall s t (M M’ : PCF V (s $\sim$> t)) N, M :> M’ -> M @ N :> M’ @
N
| relApp2: forall s t (M : PCF V (s $\sim$> t)) N N’, N :> N’ -> M @ N :> M @
N’
| relLam: forall s t (M M’:PCF (opt s V) t), M :> M’ -> Lam M :> Lam M’
| relRec: forall t (M M’ : PCF V (t $\sim$> t)), M :> M’ -> Rec M :> Rec M’
where "x :> y" := (@propag _ _ x y).
The data thus defined constitutes a relative monad PCFEM on the functor
$\Delta^{T_{\mathsf{PCF}}}$ (IDelta TY). We omit the details.
Now we need to define a suitable morphism (resp. family of morphisms) of
PCFEM–modules for any arity (of higher degree). Let $\alpha$ be any such
arity, for instance the arity $\operatorname{App}$. We need to verify two
things:
1. 1.
we show that the constructor of PCF which corresponds to $\alpha$ is monotone
with respect to the order on PCFEM. For instance, we show that for any two
terms r s:TY and any V : IDelta TY, the function
⬇
fun y => App (fst y) (snd y): PCFEM V (r$\sim$>s) x PCFEM V r -> PCFEM V s
is monotone.
2. 2.
We show that the monadic substitution defined above distributes over the
constructor, i.e. we prove that the constructor is the carrier of a _module_
morphism.
All of these are very straightforward proofs, resulting in a representation
PCFE_rep of semantic $\mathsf{PCF}$:
⬇
Program Instance PCFE_rep_struct :
PCFPO_rep_struct PCFEM PCF.arrow PCF.Bool PCF.Nat := {
app r s := PCFApp r s;
abs r s := PCFAbs r s;
rec t := PCFRec t ;
tttt := PCFconsts ttt ;
ffff := PCFconsts fff;
Succ := PCFconsts succ;
Pred := PCFconsts preds;
CondN := PCFconsts condN;
CondB := PCFconsts condB;
Zero := PCFconsts zero ;
nats m := PCFconsts (Nats m);
bottom t := PCFbottom t }.
Definition PCFE_rep : PCFPO_rep := Build_PCFPO_rep PCFE_rep_struct.
Note that in the instance declaration PCFE_rep_struct, the Program framework
proves automatically the properties of Sect. 4.1, 4.1, 4.1 and 4.1.
### 4.5 Initiality
In this section we define a morphism of representations from PCFE_rep to any
representation R : PCFPO_rep. At first we need to define a map between the
underlying sorts, that is, a map Sorts PCFE_rep -> Sorts R. In short, each
$\mathsf{PCF}$ type goes to its representation in R:
⬇
Fixpoint Init_Sorts_map (t : Sorts PCFE_rep) : Sorts R :=
match t with
| PCF.Nat => Nat R
| PCF.Bool => Bool R
| u $\sim$> v => (Init_Sorts_map u) $\sim$$\sim$> (Init_Sorts_map v)
end.
The function init is the carrier of what will later be proved to be the
initial morphism to the representation R. It maps each constructor of
$\mathsf{PCF}$ recursively to its counterpart in the representation R:
⬇
Fixpoint init V t (v : PCF V t) :
R (retype (fun t0 => Init_Sorts_map t0) V) (Init_Sorts_map t) :=
match v with
| Var t v => rweta R _ _ (ctype _ v)
| u @ v => app _ _ _ (init u, init v)
| Lam _ _ v => abs _ _ _ (rlift R
(@der_comm TY (Sorts R) (fun t => Init_Sorts_map t) _ V ) _ (init v))
| Rec _ v => rec _ _ (init v)
| Bottom _ => bottom _ _ tt
| y ’ => match y in Consts t1 return
R (retype (fun t2 => Init_Sorts_map t2) V) (Init_Sorts_map t1) with
| Nats m => nats m _ tt
| succ => Succ _ tt
| condN => CondN _ tt
| condB => CondB _ tt
| zero => Zero _ tt
| ttt => tttt _ tt
| fff => ffff _ tt
| preds => Pred _ tt
end
end.
We write $i_{V}$ for init V and $g$ for Init_Sorts_map. Note that
$i_{V}:\mathsf{PCF}(V)\to g^{*}\left(R(\vec{g}V)\right)$ really is _the image
of the initial morphism under the adjunction $\varphi$_ of Def. 2.13.
Intuitively, passing from init V$=i_{V}$ to its adjunct $\varphi^{-1}(i_{V})$
is done by precomposing with pattern matching on the constructor ctype. We
informally denote $\varphi^{-1}(i_{V})$ by
${{{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{init}}{\@listingGroup{ltx_lst_space}{
}}{\@listingGroup{ltx_lst_identifier}{V}}}}}}}\circ{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{match}}}}}}}$.
The map init is compatible with renaming and substitution in PCF and R,
respectively, in a sense made precise by the following two lemmas. The first
lemma states that, for any morphism $f:V\to W$ in
${\mathsf{Set}}^{T_{\mathsf{PCF}}}$, the following diagram commutes:
$\textstyle{\mathsf{PCF}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$init
V$\scriptstyle{\mathsf{PCF}(f)}$$\textstyle{\mathsf{PCF}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$init
W$\textstyle{g^{*}R(\vec{g}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{*}R(g^{*}f)}$$\textstyle{g^{*}R(\vec{g}W).}$
⬇
Lemma init_lift (V : IT) t (y : PCF V t) W (f : V —> W) :
init (y //- f) = rlift R (retype_map f) _ (init y).
The next commutative diagram concerns substitution; for any
$f:V\to\mathsf{PCF}(W)$, the diagram obtained by applying $\varphi$ to the
diagram given in Disp. (10) — i.e. the diagram corresponding to Disp. (11) —,
commutes:
$\textstyle{\mathsf{PCF}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$init
V$\scriptstyle{\sigma^{\mathsf{PCF}}\left({f}\right)}$$\textstyle{\mathsf{PCF}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$init
W$\textstyle{g^{*}R(\tilde{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$${\scriptstyle{g^{*}\sigma^{R}\left({{\varphi^{-1}(\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{init}}{\@listingGroup{ltx_lst_space}{
}}{\@listingGroup{ltx_lst_identifier}{W}}}}}})}\circ{(g^{*}f)}}\right)}$$\textstyle{g^{*}R(\tilde{W}).}$
In Coq the lemma init_subst proves commutativity of this latter diagram:
⬇
Lemma init_subst V t (y : PCF V t) W (f : IDelta _ V —> PCFE W):
init (y >>= f) =
rkleisli R (SM_ind (fun t v => match v with ctype t p => init (f t p) end))
_ (init y).
This latter lemma establishes almost the commutative diagram for the family
$\varphi^{-1}(i_{V})$ to constitute a (colax) _monad_ morphism, which reads as
follows:
$\textstyle{\vec{g}\left(\mathsf{PCF}(V)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$${{\scriptstyle{{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{init}}{\@listingGroup{ltx_lst_space}{
}}{\@listingGroup{ltx_lst_identifier}{V}}}}} }}\circ{\text{
\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{match}}}}}}}}$$\scriptstyle{\vec{g}\left(\sigma^{\mathsf{PCF}}\left({f}\right)\right)}$$\textstyle{\vec{g}\left({\mathsf{PCF}(W)}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$${{\scriptstyle{{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{init}}{\@listingGroup{ltx_lst_space}{
}}{\@listingGroup{ltx_lst_identifier}{W}}}}} }}\circ{\text{
\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{match}}}}}}}}$$\textstyle{R(\vec{g}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$${{\scriptstyle{\sigma^{R}\left({{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{init}}}}}
}}\circ{{\text{
\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{match}}}}}
}}\circ{(\vec{g}f)}}}\right)}$$\textstyle{R(\vec{g}{W}).}$ ((13))
Before we can actually build a monad morphism with carrier map
${{{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{init}}{\@listingGroup{ltx_lst_space}{
}}{\@listingGroup{ltx_lst_identifier}{V}}}}}}}\circ{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{match}}}}}}}$,
we need to verify that init — and thus its adjunct — is monotone. We do this
in 3 steps, corresponding to the 3 steps in which we built up the preorder on
the terms of $\mathsf{PCF}$:
1. 1.
init monotone with respect to the relation eval (cf. Sect. 4.4):
⬇
Lemma init_eval V t (v v’ : PCF V t) : eval v v’ -> init v <<< init v’.
2. 2.
init monotone with respect to the propagation into subterms of eval;
⬇
Lemma init_eval_star V t (y z : PCF V t) : eval_star y z -> init y <<< init z.
3. 3.
init monotone with respect to reflexive and transitive closure of above
relation.
⬇
Lemma init_mono c t (y z : PCFE c t) : y <<< z -> init y <<< init z.
We now have all the ingredients to define the initial morphism from
$\mathsf{PCF}$ to R. As already indicated by the diagram Disp. (13), its
carrier is not given by just the map init, since this map does not have the
right type: its domain is given, for any context
$V\in{\mathsf{Set}}^{T_{\mathsf{PCF}}}$, by $\mathsf{PCF}(V)$ and not, as
needed, by $\vec{g}\left(\mathsf{PCF}(V)\right)$. We thus precompose with
pattern matching in order to pass to its adjunct: for any context $V$, the
carrier of the initial morphism is given by
⬇
fun t y => match y with
| ctype _ p => init p
end
: retype _ (PCF V) —> R (retype _ W)
We recall that the constructor ctype is the carrier of the natural
transformation of the same name of Def. 2.13, and that precomposing with
pattern matching corresponds to specifying maps on a coproduct via its
universal property.
Putting the pieces together, we obtain a morphism of representions of semantic
$\mathsf{PCF}$:
⬇
Definition initR : PCFPO_rep_Hom PCFE_rep R :=
Build_PCFPO_rep_Hom initR_s.
Uniqueness is proved in the following lemma:
⬇
Lemma initR_unique : forall g : PCFE_rep —> R, g == initR.
The proof consists of two steps: first, one has to show that the translation
of _sorts_ coincide. Since the source of this translation is an inductive type
— the initial representation of the signature of Sect. 3.1.1 — this proof is
done by induction. Afterwards the translations of terms are proved to be
equal. The proof is done by induction on terms of $\mathsf{PCF}$. It makes
essentially use of the commutative diagrams (cf. Def. 3.22) which we
exemplarily presented for the arities of successor (Sect. 4.2), application
(Sect. 4.2) and abstraction (Sect. 4.2). Finally we can declare an instance of
Initial for the category REP of representations:
⬇
Instance PCF_initial : Initial REP := {
Init := PCFE_rep ;
InitMor R := initR R ;
InitMorUnique R := @initR_unique R }.
Checking the axioms used for the proof of initiality (and its dependencies)
yields the use of non–dependent functional extensionality (applied to the
translations of sorts) and uniqueness of identity proofs, which in the Coq
standard library is implemented as a consequence of another — logically
equivalent — axiom eq_rect_eq:
⬇
Print Assumptions PCF_initial.
Axioms:
CatSem.AXIOMS.functional_extensionality.functional_extensionality :
forall (A B : Type) (f g : A -> B),
(forall x : A, f x = g x) -> f = g
Eq_rect_eq.eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type)
(x : Q p) (h : p = p), x = eq_rect p Q x p h
### 4.6 A Representation of $\mathsf{PCF}$ in the Untyped Lambda Calculus
We use the iteration principle explained in Sects. 3.4 in order to specify a
translation from $\mathsf{PCF}$ to the untyped lambda calculus which is
compatible with reduction in the source and target. According to the
principle, it is sufficient to define a representation of $\mathsf{PCF}$ in
the relative monad of the lambda calculus (cf. Sect. 2.1) and to verify that
this representation satisfies the $\mathsf{PCF}$ inequations, formalized in
the Coq code snippets 4.1, 4.1, 4.1 and 4.1. The first task, specifying a
representation of the types of $\mathsf{PCF}$, in the singleton set of types
of $\operatorname{\mathsf{ULC}}$, is trivial. We furthermore specify
representations of the term arities of $\mathsf{PCF}$, presented in Sect. 4.1,
by giving an instance of the corresponding type class.
⬇
Program Instance PCF_ULC_rep_s :
PCFPO_rep_struct (Sorts:=unit) ULCBETAM (fun _ _ => tt) tt tt := {
app r s := ulc_app r s;
abs r s := ulc_abs r s;
rec t := ulc_rec t ;
tttt := ulc_ttt ;
ffff := ulc_fff ;
nats m := ulc_N m ;
Succ := ulc_succ ;
CondB := ulc_condb ;
CondN := ulc_condn ;
bottom t := ulc_bottom t ;
Zero := ulc_zero ;
Pred := ulc_pred }.
Before taking a closer look at the module morphisms we specify in order to
represent the arities of $\mathsf{PCF}$, we note that in the above instance
declaration, we have not given the proofs corresponding to code snippets 4.1
to 4.1. In the terms of Sects. 3.4, we have not completed the third task, the
verification that the given representation satisfies the inequations. The
Program feature we use during the above instance declaration is able to detect
that the fields called beta_red, rec_A, etc., are missing, and enters into
interactive proof mode to allow us to fill in each of the missing fields.
We now take a look at some of the lambda terms representing arities of
$\mathsf{PCF}$. The carrier of the representations ulc_app is the application
of lambda calculus, of course, and similar for ulc_abs. Here the parameters r
and s vary over terms of type unit, the type of sorts underlying this
representation. We use an infixed application and a de Bruijn notation instead
of the more abstract notation of nested data types:
⬇
Notation "a @ b" := (App a b) (at level 42, left associativity).
Notation "’1’" := (Var None) (at level 33).
Notation "’2’" := (Var (Some None)) (at level 24).
The truth values $\mathbf{T}$ and $\mathbf{F}$ are represented by
⬇
Eval compute in ULC_True.
= Abs (Abs 2)
Eval compute in ULC_False.
= Abs (Abs 1)
Natural numbers are given in Church style, the successor function is given by
the term $\lambda nfx.f(n\leavevmode\nobreak\ f\leavevmode\nobreak\ x)$. The
predecessor is represented by the constant
$\lambda nfx.n\leavevmode\nobreak\ (\lambda gh.h(g\leavevmode\nobreak\
f))(\lambda u.x)(\lambda u.u),$
and the test for zero is represented by $\lambda n.n(\lambda x.F)T$, where $F$
and $T$ are the lambda terms representing $\mathbf{F}$ and $\mathbf{T}$,
respectively.
⬇
Eval compute in ULC_Nat 0\.
= Abs (Abs 1)
Eval compute in ULC_Nat 2\.
= Abs (Abs (2 @ (Abs (Abs (2 @ (Abs (Abs 1) @ 2 @ 1))) @ 2 @ 1)))
Eval compute in ULC_succ.
= Abs (Abs (Abs (2 @ (3 @ 2 @ 1))))
Eval compute in ULC_pred.
= Abs (Abs (Abs (3 @ Abs (Abs (1 @ (2 @ 4))) @ Abs 2 @ Abs 1)))
Eval compute in ULC_zero.
= Abs (1 @ Abs (Abs (Abs 1)) @ Abs (Abs 2))
The conditional is represented by the lambda term $\lambda
pab.p\leavevmode\nobreak\ a\leavevmode\nobreak\ b$:
⬇
Eval compute in ULC_cond.
= Abs (Abs (Abs (3 @ 2 @ 1)))
The constant arity $\bot_{A}$ is represented by $\Omega$:
⬇
Eval compute in ULC_omega.
= Abs (1 @ 1) @ Abs (1 @ 1)
The fixed point operator $\mathbf{Fix}$ (rec) is represented by the _Turing_
fixed–point combinator, that is, the lambda term
⬇
Eval compute in ULC_theta.
= Abs (Abs (1 @ (2 @ 2 @ 1))) @ Abs (Abs (1 @ (2 @ 2 @ 1)))
The reason why we use the Turing operator instead of, say, the combinator
$\mathbf{Y}$,
⬇
Eval compute in ULC_Y.
= Abs (Abs (2 @ (1 @ 1)) @ Abs (2 @ (1 @ 1)))
is that the latter does not have a property that is crucial for us: It is
$\Theta(f)\rightsquigarrow^{*}f\left(\Theta(f)\right)$
but only
$\mathbf{Y}(f)\stackrel{{\scriptstyle*}}{{\leftrightsquigarrow}}f\left(\mathbf{Y}(f)\right)$
via a common reduct. Thus if we would attempt to represent the arity rec by
the fixed–point combinator $\mathbf{Y}$, we would not be able to prove the
condition expressed in Sect. 4.1. A way to allow for the use of $\mathbf{Y}$
as representation of rec would by to consider _symmetric_ relations on terms,
e.g., relative monads into a category of setoids.
As a final remark, we emphasize that while reduction is given as a relation in
our formalization, and as such is not computable, the obtained translation
from $\mathsf{PCF}$ to the untyped lambda calculus is executable in Coq. For
instance, we can translate the $\mathsf{PCF}$ term negating boolean terms as
follows:
4.11 Code:
⬇
Eval compute in
(PCF_ULC_c ((fun t => False)) tt (ctype _
(Lam (condB ’ @@ x_bool @@ fff ’ @@ ttt ’)))).
= Abs (Abs (Abs (Abs (3 @ 2 @ 1))) @ 1 @ Abs (Abs 1) @ Abs (Abs 2))
Here we use infixed “@@” to denote application of $\mathsf{PCF}$, and x_bool
is simply a notation for a de Bruijn variable of type Bool of the lowest
level, i.e. a variable that is bound by the Lam binder of $\mathsf{PCF}$ in
above term.
## References
* [1] Benedikt Ahrens “Modules over relative monads for syntax and semantics” To be published in Math. Struct. in Comp. Science, arXiv:1107.5252, 2011
* [2] Benedikt Ahrens “Extended Initiality for Typed Abstract Syntax” In _Logical Methods in Computer Science_ 8.2, 2012, pp. 1 –35 DOI: 10.2168/LMCS-8(2:1)2012
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* [4] Thorsten Altenkirch and Bernhard Reus “Monadic Presentations of Lambda Terms Using Generalized Inductive Types” In _CSL_ 1683, Lecture Notes in Computer Science Springer, 1999, pp. 453–468
* [5] Thorsten Altenkirch, James Chapman and Tarmo Uustalu “Monads Need Not Be Endofunctors” In _FOSSACS_ 6014, Lecture Notes in Computer Science Springer, 2010, pp. 297–311
* [6] Henk Barendregt and Erik Barendsen “Introduction to Lambda Calculus” revised 2000, ftp://ftp.cs.ru.nl/pub/CompMath.Found/lambda.pdf, 1994
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* [8] Garrett Birkhoff “On the Structure of Abstract Algebras” In _Proc. Cambridge Phil. Soc._ 31, 1935, pp. 433–454
* [9] Coq “The Coq Proof Assistant”, http://coq.inria.fr, 2010 URL: http://coq.inria.fr
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* [11] Marcelo Fiore “Semantic analysis of normalisation by evaluation for typed lambda calculus” In _Proceedings of the 4th ACM SIGPLAN international conference on Principles and practice of declarative programming_ , PPDP ’02 New York, NY, USA: ACM, 2002, pp. 26–37 DOI: http://doi.acm.org/10.1145/571157.571161
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* [24] J. M. E. Hyland and C.-H. Ong “On full abstraction for PCF: I. Models, observables and the full abstraction problem II. Dialogue games and innocent strategies III. A fully abstract and universal game model” In _Information and Computation_ 163, 2000, pp. 285–408
* [25] Tom Leinster “Higher Operads, Higher Categories”, London Mathematical Society Lecture Note Series 298 Cambridge: Cambridge University Press, 2004
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* [29] John Power “Abstract Syntax: Substitution and Binders” In _Electron. Notes Theor. Comput. Sci._ 173 Amsterdam, The Netherlands, The Netherlands: Elsevier Science Publishers B. V., 2007, pp. 3–16 DOI: 10.1016/j.entcs.2007.02.024
* [30] Miki Tanaka and John Power “A unified category-theoretic formulation of typed binding signatures” In _Proceedings of the 3rd ACM SIGPLAN workshop on Mechanized reasoning about languages with variable binding_ , MERLIN ’05 Tallinn, Estonia: ACM, 2005, pp. 13–24 DOI: http://doi.acm.org/10.1145/1088454.1088457
* [31] Julianna Zsidó “Typed Abstract Syntax” http://tel.archives-ouvertes.fr/tel-00535944/, http://tel.archives-ouvertes.fr/tel-00535944/, 2010
|
arxiv-papers
| 2012-12-22T08:48:22 |
2024-09-04T02:49:39.611469
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Benedikt Ahrens",
"submitter": "Thorsten Wissmann",
"url": "https://arxiv.org/abs/1212.5668"
}
|
1212.5695
|
# Higgs to diphoton decay rate and the antisymmetric tensor unparticle
mediation
E. O. Iltan
Physics Department, Middle East Technical University
Ankara, Turkey
E-mail address: [email protected]
###### Abstract
We study the contribution of the antisymmetric tensor unparticle mediation to
the diphoton production rate of the Higgs boson and try to explain the
discrepancy between the measured value of the decay width of the discovered
new resonance and that of the standard model Higgs boson. We observe that tree
level contribution of the antisymmetric unparticle mediation is a possible
candidate to explain the measured value of the diphoton decay rate.
The standard model (SM) electroweak symmetry breaking mechanism is based on
the existence of a scalar particle, the Higgs boson $H_{0}$, which is crucial
for productions of the masses of fundamental particles. Recently, the ATLAS
and CMS collaborations [1, 2] discovered a resonance with the invariant mass
$125-126\,(GeV)$. At this stage one needs a conformation that the properties
of the discovered resonance coincide with that of the SM Higgs boson. Current
data shows that there is no significant deviation in the decay widths of the
processes $H_{0}\rightarrow W\,W^{*}$ and $H_{0}\rightarrow Z\,Z^{*}$,
however, in the $H_{0}\rightarrow\gamma\gamma$ channel, there is a deviation
from the SM result, namely, the diphoton production rate reaches $1.5$ to $2$
times that of the SM prediction [1, 2, 3, 4]. Even if there needs more data in
order to check whether the excess is based on the statistical fluctuations or
not, a possible attempt to explain this excess from the theoretical side would
be worthwhile and it has been studied in various models beyond the SM
[5]-[38].
In the present work, we consider the antisymmetric tensor unparticle mediation
in order to explain the excess in the diphoton production and we restrict the
free parameters existing in the scenario. Unparticles, being massless, having
non integral scaling dimension $d_{U}$, around the scale $\Lambda_{U}\sim
1.0\,TeV$, are proposed by [39, 40]. They are new degrees of freedom arising
from a hypothetical scale invariant high energy ultraviolet sector with non-
trivial infrared fixed point. In the low energy level the effective
interaction of the SM-unparticle sector reads (see for example [41])
${\cal{L}}_{eff}=\frac{\eta}{\Lambda_{U}^{d_{U}+d_{SM}-n}}\,O_{SM}\,O_{U}\,,$
(1)
where $O_{U}$ ($O_{SM}$) is the unparticle (the SM) operator, $\Lambda_{U}$ is
the energy scale, $n$ is the space-time dimension and $\eta$ is the effective
coefficient [39, 40, 42]. The antisymmetric tensor unparticle propagator which
drives one of the outgoing photon in diphoton production is obtained by the
two point function arising from the scale invariance and it becomes
$\displaystyle\int\,d^{4}x\,e^{ipx}\,<0|T\Big{(}O^{\mu\nu}_{U}(x)\,O^{\alpha\beta}_{U}(0)\Big{)}0>=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,\Pi^{\mu\nu\alpha\beta}(-p^{2}-i\epsilon)^{d_{U}-2}\,,$
(2)
with
$\displaystyle
A_{d_{U}}=\frac{16\,\pi^{5/2}}{(2\,\pi)^{2\,d_{U}}}\,\frac{\Gamma(d_{U}+\frac{1}{2})}{\Gamma(d_{U}-1)\,\Gamma(2\,d_{U})}\,,$
(3)
and the projection operator
$\displaystyle\Pi_{\mu\nu\alpha\beta}=\frac{1}{2}(g_{\mu\alpha}\,g_{\nu\beta}-g_{\nu\alpha}\,g_{\mu\beta})\,.$
(4)
Notice that the projection operator has the transverse and the longitudinal
parts, namely,
$\displaystyle\Pi^{T}_{\mu\nu\alpha\beta}=\frac{1}{2}(P^{T}_{\mu\alpha}\,P^{T}_{\nu\beta}-P^{T}_{\nu\alpha}\,P^{T}_{\mu\beta})\,,\,\,\,\,\,\,\Pi^{L}_{\mu\nu\alpha\beta}=\Pi_{\mu\nu\alpha\beta}-\Pi^{T}_{\mu\nu\alpha\beta}\,,$
(5)
where $P^{T}_{\mu\nu}=g_{\mu\nu}-p_{\mu}\,p_{\nu}/{p^{2}}$ (see [43] and
references therein). At this stage we consider that the scale invariance is
broken at some scale $\mu_{U}$ and we take the antisymmetric tensor unparticle
propagator as
$\displaystyle\int\,d^{4}x\,e^{ipx}\,<0|T\Big{(}O^{\mu\nu}_{U}(x)\,O^{\alpha\beta}_{U}(0)\Big{)}0>=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,\Pi^{\mu\nu\alpha\beta}(-(p^{2}-\mu_{U}^{2})-i\epsilon)^{d_{U}-2}\,,$
(6)
by considering a simple model [44, 45] which provides a rough connection
between the unparticle sector and the particle sector.
Now, we are ready to present the low energy effective Lagrangian which drives
the new contribution to the diphoton production (see [43]):
$\displaystyle{\cal{L}}_{eff}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime}\,\lambda_{B}}{\Lambda_{U}^{d_{U}-2}}\,B_{\mu\nu}\,O^{\mu\nu}_{U}+\frac{g\,\lambda_{W}}{\Lambda_{U}^{d_{U}}}\,(H^{\dagger}\,\tau_{a}\,H)\,W^{a}_{\mu\nu}\,O^{\mu\nu}_{U}\,,$
(7)
where $H$ is the Higgs doublet, $g$ and $g^{\prime}$ are weak couplings,
$\lambda_{B}$ and $\lambda_{W}$ are unparticle-field tensor couplings,
$B_{\mu\nu}$ is the field strength tensor of the $U(1)_{Y}$ gauge boson
$B_{\mu}=c_{W}\,A_{\mu}+s_{W}\,Z_{\mu}$ and $W^{a}_{\mu\nu}$, $a=1,2,3$, are
the field strength tensors of the $SU(2)_{L}$ gauge bosons with
$W^{3}_{\mu}=s_{W}\,A_{\mu}-c_{W}\,Z_{\mu}$ where $A_{\mu}$ and $Z_{\mu}$ are
photon and Z boson fields respectively. The gauge invariant amplitude of the
$H_{0}\rightarrow\gamma\gamma$ decay is
$M=C_{eff}\,(k_{1}.k_{2}\,g^{\mu\nu}-k_{1}^{\nu}\,k_{2}^{\mu})\,\epsilon_{1\mu}\,\epsilon_{2\nu}\,,$
(8)
with the effective coefficient $C_{eff}$ and $i^{th}$ photon polarization
(momentum) four vector $\epsilon_{i\alpha}$ ($k_{i\beta}$). In the framework
of the SM this decay appears at least at the one loop level [46, 47] (see
Appendix for details). On the other hand the antisymmetric tensor unparticle
mediation results in the contribution to the decay in the tree level, with the
transition $H_{0}\rightarrow\gamma O_{U}\rightarrow\gamma\,\gamma$ (see
Fig.1). Here $H_{0}\rightarrow\gamma O_{U}$ transition is carried by the
vertex
$\displaystyle
i\,\frac{e\,v\,\lambda_{W}}{\Lambda_{U}^{d_{U}}}\,k_{1\mu}\,\epsilon_{1\nu}\,O^{\mu\nu}_{U}\,H_{0},$
which arises from the second term in eq.(7). The $O_{U}\rightarrow\gamma$
transition appears with the vertex
$\displaystyle
2\,i\,e\,\Bigg{(}\frac{\lambda_{B}}{\Lambda_{U}^{d_{U}-2}}-\frac{v^{2}\,\lambda_{W}}{4\,\Lambda_{U}^{d_{U}}}\Bigg{)}\,k_{2\mu}\,\epsilon_{2\nu}\,O^{\mu\nu}_{U}\,\,,$
which is coming from the first term and the second term in eq.(7). Here $v$ is
the vacuum expectation value of the SM Higgs $H_{0}$ and $a=3$ is taken in
both vertices. Finally the effective coefficient $C_{eff}$ reads
$\displaystyle C_{eff}=C_{SM}+C_{U}\,,$ (9)
where
$\displaystyle
C_{U}=\frac{-i\,e^{2}\,\lambda_{W}\,v\,\mu_{U}^{2\,(d_{U}-2)}\,A_{d_{U}}}{2\,sin\,(d_{U}\pi)\,\Lambda_{U}^{2\,d_{U}}}\,\Bigg{(}\lambda_{B}\,\Lambda_{U}^{2}-\frac{v^{2}\,\lambda_{W}}{4}\Bigg{)}$
(10)
(see appendix for $C_{SM}$).
Discussion
In this section we study the discrepancy between the measured value of the
decay width of the discovered new resonance, interpreted as the Higgs boson,
and that of the SM one, i.e.,
$\frac{\Gamma(H_{0}\rightarrow\gamma\gamma)_{Measured}}{\Gamma(H_{0}\rightarrow\gamma\gamma)_{SM}}\sim
1.5$. We see that the intermediate antisymmetric tensor unparticle mediation
(see Fig.1) can explain the deviation of diphoton production rate from the SM
prediction. In the present scenario the couplings $\lambda_{B}$,
$\lambda_{W}$, the scale $\Lambda_{U}$, the scaling dimension $d_{U}$ of the
antisymmetric tensor unparticle operator and the scale $\mu_{U}$, which drives
the transition from the unparticle sector to the particle one, are the free
parameters. We take $\lambda_{B}$ and $\lambda_{W}$ as universal and choose
$\lambda_{B}=\lambda_{W}=1$. For the antisymmetric tensor unparticle scale
dimension $d_{U}$ one needs a restriction $d_{U}>2$ not to violate the
unitarity (see [48]). However, since we consider that the scale invariance is
broken at some scale $\mu_{U}$ we relax the restriction and we choose the
range $1<d_{U}<2$ for the scaling dimension $d_{U}$. Here we switched on the
scale invariance breaking by following the simple model [44, 45] which is
based on the redefinition of the unparticle propagator (see eq. (6)). Notice
that the unparticle sector flows to particle sector when $d_{U}$ converges to
one and the range of $d_{U}$ we consider above is appropriate to establish the
connection between these two sectors. For the scale $\mu_{U}$ where the scale
invariance is broken we choose different values $\mu_{U}=1-20\,GeV$ and we
take the scale $\Lambda_{U}$ at the order of the magnitude of $10^{4}\,GeV$.
In our numerical calculations we also consider constraints coming from the
Peskin-Takeuchi parameter, S, which is used to restrict the new physics
contribution to the gauge boson self energy ((see [43]). Here the operator
$(H^{\dagger}\,\tau^{a}\,H)\,W^{a}_{\mu\nu}\,B^{\mu\nu}$ which is induced by
the tensor unparticle exchange results in the S parameter
$\displaystyle S=\frac{A_{d_{U}}}{sin(d_{U}\pi)}\,\frac{g^{2}\,g^{\prime
2}\,\lambda_{B}\,\lambda_{W}\,c_{W}\,v^{2}\,\mu^{2\,(d_{U}-2)}}{s_{W}\,\Lambda_{U}^{2\,d_{U}-2}}\,.$
(11)
The parameter $S$ from the electroweak precision data reads
$S=0.00^{+0.11}_{-0.10}$ [49] which is due to new physics only. Since the
unparticle contribution is negative in our choice of parameters (we choose
$\lambda_{B}=\lambda_{W}=1$), we take the lower bound $S_{LB}=-0.10$ and we
plot the contour diagram of the S parameter with $1\,\sigma$ bound in
$\Lambda_{U},\mu_{U}$ plane for different values of $d_{U}$, as shown in
figure 2. In this figure the pair of parameters $\Lambda_{U},\mu_{U}$ under
the curves are excluded.
Now, we start to study the discrepancy between the measured value of the decay
width of the discovered new resonance and the SM Higgs boson by considering
the restriction region coming from the S parameter. Fig.3 represents $d_{U}$
dependence of the ratio
$r=\frac{\Gamma(H_{0}\rightarrow\gamma\gamma)_{SM+U}}{\Gamma(H_{0}\rightarrow\gamma\gamma)_{SM}}$
for different values of the scales $\Lambda_{U}$ and $\mu_{U}$. Here111The
solid straight line represents the ratio
$\frac{\Gamma(H_{0}\rightarrow\gamma\gamma)_{Measured}}{\Gamma(H_{0}\rightarrow\gamma\gamma)_{SM}}\sim
1.5$ in each figure. the upper-intermediate-lower solid (dashed) line
represents $r$ for $\mu_{U}=1-10-20\ (GeV)$,
$\Lambda_{U}=5000\,(10000)\,(GeV)$. The ratio is strongly sensitive to the
scale dimension $d_{U}$ for the values far from $1.9$ and the decrease in the
scale $d_{U}$ results in the increase in the unparticle contribution which
makes it possible to overcome the the discrepancy between the measured value
and the SM result. We observe that the measured value is reached if the
scaling dimension is in the range $1.48<d_{U}<1.68$ for the given numerical
values of $\Lambda_{U}$ and $\mu_{U}$. In the case of
$\Lambda_{U}=10000\,(5000)\,(GeV)$ and $\mu_{U}=1\,(GeV)$, the measured decay
rate is obtained for $d_{U}\sim 1.63\,(1.68)$. For $\mu_{U}=20\,(GeV)$ the
measured value is reached for $d_{U}\sim(1,49)\,1.54$.
Fig.4 is devoted to $\mu_{U}$ dependence of the ratio $r$ for
$\Lambda_{U}=10000\,(GeV)$ and different values $d_{U}$. Here the solid (long
dashed, dashed, dotted) line represents $r$ for
$d_{U}=1.4\,(1.5,\,1.6,\,1.7)$. The ratio is sensitive to the scale $\mu_{U}$
and increases with its decreasing value. One can reach the measured decay rate
for $2.3\,(GeV)<\mu_{U}<16\,(GeV)$ if $d_{U}$ is in the range $d_{U}\sim
1.50-1.60$.
In Fig.5, we present $\Lambda_{U}$ dependence of the ratio $r$ for different
values of $d_{U}$ and $\mu_{U}$. Here the solid (long dashed, dashed, dotted)
line represents $r$ for $d_{U}=1.5;\,\mu_{U}=1.0\,(GeV)$
$(d_{U}=1.5;\,\mu_{U}=10\,(GeV),\,d_{U}=1.6;\,\mu_{U}=1.0\,(GeV),\,d_{U}=1.6;\,\mu_{U}=10\,(GeV))$.
The measured decay rate is reached for $d_{U}=1.6;\,\mu_{U}=1.0\,(GeV)$ if the
energy scale reads $\Lambda_{U}\sim 17000\,(GeV)$.
As a summary, we show that the intermediate antisymmetric tensor unparticle
mediation is a possible candidate to overcome the deviation of diphoton
production rate from the SM prediction. We study the ratio
$r=\frac{\Gamma(H_{0}\rightarrow\gamma\gamma)_{SM+U}}{\Gamma(H_{0}\rightarrow\gamma\gamma)_{SM}}$
and see that $r\sim 1.5$ is reached if the scaling dimension is almost in the
range $1.48<d_{U}<1.68$ for the given numerical values of
$5000\,(GeV)<\Lambda_{U}<17000\,(GeV)$ and $1.0\,(GeV)<\mu_{U}<20\,(GeV)$.
This result makes it possible to explain the discrepancy between the measured
value of the decay width of the discovered new resonance and that of the SM
Higgs boson. In addition, it also gives an opportunity to understand the role
and the type of the unparticle scenario and to determine the existing free
parameters.
Appendix
In the framework of the SM, the $H_{0}\rightarrow\gamma\gamma$ decay appears
at least in the loop level with the internal W boson and fermions where the
top quark gives the main contribution. The gauge invariant amplitude reads
$\displaystyle
M=C_{SM}\,(k_{1}.k_{2}\,g^{\mu\nu}-k_{1}^{\nu}\,k_{2}^{\mu})\,\epsilon_{1\mu}\,\epsilon_{2\nu}\,,$
(12)
where $C_{SM}=\frac{\alpha_{EM}\,g}{4\,\pi\,m_{W}}\,F(x_{W},x_{f})$ and
$\displaystyle
F(x_{W},x_{f})=F_{1}(x_{W})+\sum_{f}N_{C}\,Q_{f}^{2}\,F_{2}(x_{f})\,,$ (13)
with
$\displaystyle F_{1}(x)=2+3\,x+3\,x\,(2-x)\,g(x)\,,$ $\displaystyle
F_{2}(x)=-2\,\Big{(}1+(1-x)\,g(x)\Big{)}\,.$ (14)
Here
$\displaystyle
g(x)=\left\\{\begin{array}[]{ll}arcsin^{2}(x^{-1/2}),&\hbox{$x\geq 1$;}\\\
\frac{-1}{4}\Bigg{(}ln\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}-i\pi\Bigg{)}^{2},&\hbox{$x<1$
,}\\\ \end{array}\right.$ (17)
and $Q_{f}$ is the charge of the fermion $f$, $N_{C}=1\,(3)$ for lepton
(quark), $x_{i}=\frac{4\,m_{i}^{2}}{m_{H_{0}}^{2}}$, $i=W,f$. Finally the
decay width $\Gamma(H_{0}\rightarrow\gamma\gamma)$ is obtained as
$\displaystyle\Gamma(H_{0}\rightarrow\gamma\gamma)=\frac{m^{3}_{H_{0}}\,|C_{SM}|^{2}}{64\,\pi}\,.$
(18)
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Figure 1: Tree level diagram contributing to diphoton decay due to the
antisymmetric tensor unparticle mediation. Solid (wavy, double dashed) line
represents the Higgs (electromagnetic, antisymmetric tensor unparticle) field.
Figure 2: $\Lambda_{U}$ with respect to $\mu_{U}$ within $1\,\sigma$ bound.
Here the solid (long dashed, dashed, dotted) line represents $\Lambda_{U}$ for
$d_{U}=1.4\,(1.5,1.6,1.7)$. Figure 3: $r$ with respect to $d_{U}$. Here the
upper-intermediate-lower solid (dashed) line represents $r$ for
$\mu_{U}=1-10-20\ (GeV)$, $\Lambda_{U}=5000\,(10000)\,(GeV)$. Figure 4: $r$
with respect to $\mu_{U}$ for $\Lambda_{U}=10000\,(GeV)$. Here the solid (long
dashed, dashed, dotted) line represents $r$ for
$d_{U}=1.4\,(1.5,\,1.6,\,1.7)$. Figure 5: $r$ with respect to $\Lambda_{U}$.
Here the solid (long dashed, dashed, dotted) line represents $r$ for
$d_{U}=1.5;\,\mu_{U}=1.0\,(GeV)$
$(d_{U}=1.5;\,\mu_{U}=10\,(GeV),\,d_{U}=1.6;\,\mu_{U}=1.0\,(GeV),\,d_{U}=1.6;\,\mu_{U}=10\,(GeV))$.
|
arxiv-papers
| 2012-12-22T14:49:00 |
2024-09-04T02:49:39.628475
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. O. Iltan",
"submitter": "Erhan Iltan",
"url": "https://arxiv.org/abs/1212.5695"
}
|
1212.5837
|
# A note on the modified $q$-Dedekind sums
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY [email protected];
[email protected]; [email protected] , Erdoğan Şen Department of
Mathematics, Faculty of Science and Letters, Namık Kemal University, 59030
Tekirdağ, TURKEY [email protected] and Mehmet Acikgoz University of
Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310
Gaziantep, TURKEY [email protected]
###### Abstract.
In the present paper, the fundamental aim is to consider a $p$-adic continuous
function for an odd prime to inside a $p$-adic $q$-analogue of the higher
order Extended Dedekind-type sums related to $q$-Genocchi polynomials with
weight $\alpha$ by using fermionic $p$-adic invariant $q$-integral on
$\mathbb{Z}_{p}$.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. Dedekind Sums, $q$-Dedekind-type Sums, $p$-adic
$q$-integral, $q$-Genocchi polynomials with weight $\alpha.$
## 1\. Introduction
Imagine that $p$ be a fixed odd prime number. We now start with definition of
the following notations. Let $\mathbb{Q}_{p}$ be the field $p$-adic rational
numbers and let $\mathbb{C}_{p}$ be the completion of algebraic closure of
$\mathbb{Q}_{p}$.
Thus,
$\boldsymbol{\mathbb{Q}}_{p}=\left\\{x=\sum_{n=-k}^{\infty}a_{n}p^{n}:0\leq
a_{n}<p\right\\}\text{.}$
Then $\mathbb{Z}_{p}$ is integral domain, which is defined by
$\boldsymbol{\mathbb{Z}}_{p}=\left\\{x=\sum_{n=0}^{\infty}a_{n}p^{n}:0\leq
a_{n}<p\right\\}$
or
$\boldsymbol{\mathbb{Z}}_{p}=\left\\{x\in\mathbb{Q}_{p}:\left|x\right|_{p}\leq
1\right\\}\text{.}$
We assume that $q\in\mathbb{C}_{p}$ with $\left|1-q\right|_{p}<1$ as an
indeterminate. The $p$-adic absolute value $\left|.\right|_{p}$, is normally
defined by
$\left|x\right|_{p}=\frac{1}{p^{n}}$
where $x=p^{n}\frac{s}{t}$ with
$\left(p,s\right)=\left(p,t\right)=\left(s,t\right)=1$ and $n\in\mathbb{Q}$
(for details, see [1-19]).
The $p$-adic $q$-Haar distribution is defined by Kim as follows: for any
postive integer $n$,
$\mu_{q}\left(a+p^{n}\mathbb{Z}_{p}\right)=\left(-q\right)^{a}\frac{\left(1+q\right)}{1+q^{p^{n}}}$
for $0\leq a<p^{n}$ and this can be extended to a measure on $\mathbb{Z}_{p}$
(for details, see [12], [14], [17]).
In [7], the $q$-Genocchi polynomials are defined by Araci et al. as follows:
(1)
$\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x\right)=n\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}\right)^{n-1}d\mu_{q}\left(\xi\right)$
for $n\in\mathbb{Z}_{+}:=\left\\{0,1,2,3,\cdots\right\\}$. We easily see that
$\lim_{q\rightarrow
1}\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x\right)=G_{n}\left(x\right)$
where $G_{n}\left(x\right)$ are Genocchi polynomials, which are given in the
form:
$\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!}=e^{tx}\frac{2t}{e^{t}+1},\text{
}\left|t\right|<\pi$
(for details, see [7]). Taking $x=0$ into (1), then we have
$\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(0\right):=\widetilde{G}_{n,q}^{\left(\alpha\right)}$
are called $q$-Genocchi numbers with weight $\alpha$.
The $q$-Genocchi numbers and polynomials have the following identities:
(2) $\displaystyle\widetilde{G}_{n+1,q}^{\left(\alpha\right)}$
$\displaystyle=$
$\displaystyle\left(n+1\right)\frac{1+q}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{1}{1+q^{\alpha
l+1}}\text{,}$ (3)
$\displaystyle\widetilde{G}_{n+1,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\left(n+1\right)\frac{1+q}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{q^{\alpha
lx}}{1+q^{\alpha l+1}}\text{,}$ (4)
$\displaystyle\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x\right)$
$\displaystyle=$ $\displaystyle\sum_{l=0}^{n}\binom{n}{l}q^{\alpha
lx}\widetilde{G}_{l,q}^{\left(\alpha\right)}\left(\frac{1-q^{\alpha
x}}{1-q^{\alpha}}\right)^{n-l}\text{.}$
Additionally, for $d$ odd natural number, we have
(5)
$\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(dx\right)=\left(\frac{1+q}{1+q^{d}}\right)\left(\frac{1-q^{\alpha
d}}{1-q^{\alpha}}\right)^{n-1}\sum_{a=0}^{d-1}q^{a}\left(-1\right)^{a}\widetilde{G}_{n,q}^{\left(\alpha\right)}\left(x+\frac{a}{d}\right)\text{,}$
(for details about this subject, see [7]).
For any positive integer $h,k$ and $m$, Dedekind-type DC sums are given by Kim
in [9], [10] and [11] as follows:
$S_{m}\left(h,k\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\frac{M}{k}\overline{E}_{m}\left(\frac{hM}{k}\right)$
where $\overline{E}_{m}\left(x\right)$ are the $m$-th periodic Euler function.
In 2011, Taekyun Kim added a weight to $q$-Bernoulli polynomials in [16]. He
derived not only new but also ineteresting properties for weighted
$q$-Bernoulli polynomials. After, many mathematicians, by utilizing from Kim’s
paper [16], have introduced a new concept in Analytic numbers theory as
weighted $q$-Bernoulli, weighted $q$-Euler, weighted $q$-Genocchi polynomials
in [17], [6], [7], [1], [3] and [5]. Also, the generating function of weighted
$q$-Genocchi polynomials was introduced by Araci et al. in [7]. They also
derived several arithmetic properties for weighted $q$-Genocchi polynomials.
Kim has given some interesting properties for Dedekind-type DC sums. He
firstly considered a $p$-adic continuous function for an odd prime number to
contain a $p$-adic $q$-analogue of the higher order Dedekind-type DC sums
$k^{m}S_{m+1}\left(h,k\right)$ in [10].
By the same motivation, we, by using $p$-adic invariant $q$-integral on
$\mathbb{Z}_{p}$, shall get weighted $p$-adic $q$-analogue of the higher order
Dedekind-type DC sums $k^{m}S_{m+1}\left(h,k\right)$.
## 2\. Extended $q$-Dedekind-type Sums in connection with $q$-Genocchi
polynomials with weight $\alpha$
If $x$ is a $p$-adic integer, then $w\left(x\right)$ is the unique solution of
$w\left(x\right)=w\left(x\right)^{p}$ that is congruent to
$x\mathop{\mathrm{m}od}p$. It can also be defined by
$w\left(x\right)=\lim_{n\rightarrow\infty}x^{p^{n}}\text{.}$
The multiplicative group of $p$-adic units is a product of the finite group of
roots of unity, and a group isomorphic to the $p$-adic integers. The finite
group is cylic of order $p-1$ or $2$, as $p$ is odd or even, respectively, and
so it is isomorphic. Actually, the teichmüller character gives a canonical
isomorphism between these two groups.
Let $w$ be the $Teichm\ddot{u}ller$ character ($\mathop{\mathrm{m}od}p$). For
$x\in\mathbb{Z}_{p}^{\ast}$ $:=\mathbb{Z}_{p}/p\mathbb{Z}_{p}$, set
$\left\langle
x:q\right\rangle=w^{-1}\left(x\right)\left(\frac{1-q^{x}}{1-q}\right)\text{.}$
Let $a$ and $N$ be positive integers with $\left(p,a\right)=1$ and $p\mid N$.
We now introduce the following
$\widetilde{E}_{q}^{\left(\alpha\right)}\left(s,a,N:q^{N}\right)=w^{-1}\left(a\right)\left\langle
x:q^{\alpha}\right\rangle^{s}\sum_{j=0}^{\infty}\binom{s}{j}q^{\alpha
aj}\left(\frac{1-q^{\alpha N}}{1-q^{\alpha
a}}\right)^{j}\widetilde{G}_{j,q^{N}}^{\left(\alpha\right)}\text{.}$
In particular, if $m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$, then we have
$\displaystyle\widetilde{E}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)$
$\displaystyle=$ $\displaystyle\left(\frac{1-q^{\alpha
a}}{1-q^{\alpha}}\right)^{m}\sum_{j=0}^{m}\binom{m}{j}q^{\alpha
aj}\widetilde{G}_{j,q^{N}}^{\left(\alpha\right)}\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha a}}\right)^{j}$ $\displaystyle=$
$\displaystyle\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)\text{.}$
Then, $\widetilde{E}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)$ is a
continuous $p$-adic extension of
$\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\frac{\widetilde{G}_{m+1,q^{N}}^{\left(\alpha\right)}\left(\frac{a}{N}\right)}{m+1}\text{.}$
Suppose that $\left[.\right]$ be the Gauss’ symbol and let
$\left\\{x\right\\}=x-\left[x\right]$. Thus, we are now ready to treat
$q$-extension of the higher order Dedekind-type DC sums
$\widetilde{S}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{l}\right)$ in the form:
$\widetilde{S}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{l}\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha
k}}\right)\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha\left(l\xi+l\left\\{\frac{hM}{k}\right\\}\right)}}{1-q^{\alpha
l}}\right)^{m}d\mu_{q^{l}}\left(\xi\right)\text{.}$
If $m+1\equiv 0\left(\mathop{\mathrm{m}od}p-1\right)$
$\displaystyle\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha k}}\right)\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$ $\displaystyle=$
$\displaystyle\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$
where $p\mid k$, $\left(hM,p\right)=1$ for each $M$. Via the equation (1), we
easily state the following
(6) $\displaystyle\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{S}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)$
$\displaystyle=\sum_{M=1}^{k-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m}\left(-1\right)^{M-1}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
k\left(\xi+\frac{hM}{k}\right)}}{1-q^{\alpha
k}}\right)^{m}d\mu_{q^{k}}\left(\xi\right)$
$\displaystyle=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{E}_{q}^{\left(\alpha\right)}\left(m,\left(hM\right)_{k}:q^{k}\right)$
where $(hM)_{k}$ denotes the integer $x$ such that $0\leq x<n$ and
$x\equiv\alpha\left(\mathop{\mathrm{m}od}k\right)$.
It is simple to show the following:
(7)
$\displaystyle\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}\right)^{k}d\mu_{q}\left(\xi\right)$
$\displaystyle=\left(\frac{1-q^{\alpha
m}}{1-q^{\alpha}}\right)^{k}\frac{1+q}{1+q^{m}}\sum_{i=0}^{m-1}\left(-1\right)^{i}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
m\left(\xi+\frac{x+i}{m}\right)}}{1-q^{\alpha
m}}\right)^{k}d\mu_{q^{m}}\left(\xi\right)\text{.}$
Due to (6) and (7), we easily obtain
(8) $\displaystyle\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)$
$\displaystyle=\frac{1+q^{N}}{1+q^{Np}}\sum_{i=0}^{p-1}\left(-1\right)^{i}\left(\frac{1-q^{\alpha
Np}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
pN\left(\xi+\frac{a+iN}{pN}\right)}}{1-q^{\alpha
pN}}\right)^{m}d\mu_{q^{pN}}\left(\xi\right)\text{.}$
Thanks to (6), (7) and (8), we discover the following $p$-adic integration:
$\widetilde{E}_{q}^{\left(\alpha\right)}\left(s,a,N:q^{N}\right)=\frac{1+q^{N}}{1+q^{Np}}\sum_{\underset{a+iN\neq
0(\mathop{\mathrm{m}od}p)}{0\leq i\leq
p-1}}\left(-1\right)^{i}\widetilde{E}_{q}^{\left(\alpha\right)}\left(s,\left(a+iN\right)_{pN},p^{N}:q^{pN}\right)\text{.}$
On the other hand,
$\displaystyle\widetilde{E}_{q}^{\left(\alpha\right)}\left(m,a,N:q^{N}\right)=\left(\frac{1-q^{\alpha
N}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
N\left(\xi+\frac{a}{N}\right)}}{1-q^{\alpha
N}}\right)^{m}d\mu_{q^{N}}\left(\xi\right)$
$\displaystyle-\left(\frac{1-q^{\alpha
Np}}{1-q^{\alpha}}\right)^{m}\int_{\mathbb{Z}_{p}}\left(\frac{1-q^{\alpha
pN\left(\xi+\frac{a+iN}{pN}\right)}}{1-q^{\alpha
pN}}\right)^{m}d\mu_{q^{pN}}\left(\xi\right)$
where $\left(p^{-1}a\right)_{N}$ denotes the integer $x$ with $0\leq x<N$,
$px\equiv a\left(\mathop{\mathrm{m}od}N\right)$ and $m$ is integer with
$m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$. Therefore, we can state the following
$\displaystyle\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{E}_{q}^{\left(\alpha\right)}\left(m,hM,k:q^{k}\right)$
$\displaystyle=\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{S}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)-\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\left(\frac{1-q^{\alpha kp}}{1-q^{\alpha
k}}\right)\widetilde{S}_{m,q}^{\left(\alpha\right)}\left(\left(p^{-1}h\right),k:q^{pk}\right)$
where $p\nmid k$ and $p\nmid hm$ for each $M$. Thus, we obtain the following
definition, which seems interesting for further studying in theory of Dedekind
sums.
###### Definition 1.
Let $h,k$ be positive integer with $\left(h,k\right)=1$, $p\nmid k$. For
$s\in\mathbb{Z}_{p},$ we define $p$-adic Dedekind-type DC sums as follows:
$\widetilde{S}_{p,q}^{\left(\alpha\right)}\left(s:h,k:q^{k}\right)=\sum_{M=1}^{k-1}\left(-1\right)^{M-1}\left(\frac{1-q^{\alpha
M}}{1-q^{\alpha}}\right)\widetilde{E}_{q}^{\left(\alpha\right)}\left(m,hM,k:q^{k}\right)\text{.}$
As a result of the above definition, we derive the following theorem.
###### Theorem 2.1.
For $m+1\equiv 0(\mathop{\mathrm{m}od}p-1)$ and $\left(p^{-1}a\right)_{N}$
denotes the integer $x$ with $0\leq x<N$, $px\equiv
a\left(\mathop{\mathrm{m}od}N\right)$, then, we have
$\displaystyle\widetilde{S}_{p,q}^{\left(\alpha\right)}\left(s:h,k:q^{k}\right)=\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\widetilde{S}_{m,q}^{\left(\alpha\right)}\left(h,k:q^{k}\right)$
$\displaystyle-\left(\frac{1-q^{\alpha
k}}{1-q^{\alpha}}\right)^{m+1}\left(\frac{1-q^{\alpha kp}}{1-q^{\alpha
k}}\right)\widetilde{S}_{m,q}^{\left(\alpha\right)}\left(\left(p^{-1}h\right),k:q^{pk}\right)\text{.}$
In the special case $\alpha=1$, our applications in theory of Dedekind sums
resemble Kim’s results in [10]. These results seem to be interesting for
further studies in [9], [11] and [18].
## References
* [1] S. Araci, D. Erdal and J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [2] S. Araci, M. Acikgoz and J. J. Seo, Explicit formulas involving $q$-Euler numbers and polynomials, Abstract and Applied Analysis, Volume 2012, Article ID 298531, 11 pages.
* [3] S. Araci, J. J. Seo and D. Erdal, New construction weighted ($h,q$)-Genocchi numbers and polynomials related to Zeta-type functions, Discrete Dynamics in Nature and Society, Volume 2011 (2011), Article ID 487490, 7 pages.
* [4] S. Araci, M. Acikgoz, K. H. Park and H. Jolany, On the unification of two families of multiple twisted type polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, accepted in Bulletin of the Malaysian Mathematical Sciences and Society.
* [5] S. Araci, M. Acikgoz, H. Jolany, J. J. Seo, A unified generating function of the $q$-Genocchi polynomials with their interpolation functions, Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 227–233.
* [6] S. Araci, M. Acikgoz and K. H. Park, A note on the $q$-analogue of Kim’s $p$-adic $\log$ gamma type functions associated with $q$-extension of Genocchi and Euler numbers with weight $\alpha$, accepted in Bulletin of the Korean Mathematical Society.
* [7] S. Araci, M. Acikgoz and J. J. Seo, A study on the weighted $q$-Genocchi numbers and polynomials with their interpolation function, Honam Mathematical J. 34 (2012), No. 1, pp. 11-18.
* [8] M. Acikgoz, S. Araci and I. N. Cangul, A note on the modified $q$-Bernstein polynomials for functions of several variables and their reflections on $q$-Volkenborn integration, Applied Mathematics and Computation, vol. 218, no. 3, pp. 707–712, 2011.
* [9] T. Kim, A note on $p$-adic $q$-Dedekind sums, C. R. Acad. Bulgare Sci. 54 (2001), 37–42.
* [10] T. Kim, Note on $q$-Dedekind-type sums related to $q$-Euler polynomials, Glasgow Math. J. 54 (2012), 121-125.
* [11] T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contemp. Math. 18 (2009), 249–260.
* [12] T. Kim, The modified $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 16 (2008), 161–170.
* [13] T. Kim, $q$-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288–299.
* [14] T. Kim, On $p$-adic interpolating function for $q$-Euler numbers and its derivatives, J. Math. Anal. Appl. 339 (2008), 598–608.
* [15] T. Kim, On a $q$-analogue of the $p$-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320-329.
* [16] T. Kim, On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.
* [17] T. Kim and J. Choi, On the $q$-Euler Numbers and Polynomials with Weight $0$, Abstract and Applied Analysis, vol. 2012, Article ID 795304, 7 pages.
* [18] Y. Simsek, $q$-Dedekind type sums related to $q$-zeta function and basic $L$-series, J. Math. Anal. Appl. 318 (2006), 333-351.
* [19] M. Acikgoz, Y. Simsek, On multiple interpolation function of the Nörlund-type $q$-Euler polynomials, Abst. Appl. Anal. 2009 (2009), Article ID 382574, 14 pages.
|
arxiv-papers
| 2012-12-23T21:30:37 |
2024-09-04T02:49:39.636731
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Erdo\\u{g}an \\c{S}en, and Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1212.5837"
}
|
1212.5854
|
The Reasonable Effectiveness of Mathematics in the Natural Sciences
Alex Harveya)
Visiting Scholar
New York University
New York, NY 10003
###### Abstract
Mathematics and its relation to the physical universe have been the topic of
speculation since the days of Pythagoras. Several different views of the
nature of mathematics have been considered: Realism \- mathematics exists and
is discovered; Logicism \- all mathematics may be deduced through pure logic;
Formalism: mathematics is just the manipulation of formulas and rules invented
for the purpose; Intuitionism: mathematics comprises mental constructs
governed by self evident rules. The debate among the several schools has major
importance in understanding what Eugene Wigner called, “The Unreasonable
Effectiveness of Mathematics in the Natural Sciences.” In return, this
‘Unreasonable Effectiveness’ suggests a possible resolution of the debate in
favor of Realism. The crucial element is the extraordinary predictive capacity
of mathematical structures descriptive of physical theories.111It is a
pleasure to dedicate this paper to Josh Goldberg, long a valued member of the
community of Generl Relativists and a good friend.
PACS: 0170w *43.10.10Mq
## 1 Introduction
In an essay which appeared in his collection, Symmetries and Reflections,
Wigner [1] explored the connection between mathematics and science. He
wondered that,
> “$\dots$the enormous usefulness of mathematics in the natural sciences is
> something bordering on the mysterious and that there is no rational
> explanation for this.”
And, later,
> “$\ldots$ the mathematical formulation of the physicist’s often crude
> experience leads in an uncanny number of cases to an amazingly accurate
> description of a large class of phenomena.”
A similar view was expressed by Bochner222It is noteworthy that Wigner was a
Nobel Laureate in physics and Bochner, a distinguished mathematician. The
talent for the two skills does not seem to be the same. Only a few gifted
individuals such as Archimedes, Newton, or Gauss had supreme talents in both.
This difference in talents can be observed in college upperclassmen. Expert
knowledge and superior skill in the use of mathematics is the hallmark of the
gifted physicist. [2]
> “What makes mathematics so effective when it enters science is a mystery of
> mysteries and the present book wants to achieve no more than explicate how
> deep this mystery is.”
And again, in Chapter 22 of the London Philosophical Study Guide [3].
> “Mathematical reality is in itself mysterious: how can it be highly abstract
> and yet applicable to the physical world? How can mathematical theorems be
> necessary truths about an unchanging realm of abstract entities and at the
> same time so useful in dealing with the contingent, variable and inexact
> happenings evident to the senses?”
None mention the central mystery, which is the ability of an appropriate
formulation to predict phenomena yet undiscovered. Indeed, as often noted, the
principal problem in progress in physics is to find this formulation. It is
our purpose to explore this “mysterious” relation between physical theories
and their mathematical formulation and suggest an answer. Firstly an
understanding of mathematics is essential.
## 2 Mathematics
Mathematics arose from that most fundamental of mathematical activities -
counting objects. In the most ancient of times, a person might enumerate
objects of the same kind - say goats or sheaves of wheat and if he wanted to
record the results, would have to invent a system of symbols to identify and
differentiate two quantites. We can be sure that he would use the same symbol
system to record different quantities of apples. He would quickly learn to add
two collections of objects. This could well have happened anywhere groups of
people progressed beyond being hunter-gatherers.
This simple skill evolved to the sophisticated arithmetic of the Babylonians
[4]. They exhibited great ingenuity in their ability to use tables of squares
to solve various arithmetic problems and could solve quadratic and cubic
equations. Similarly, a direct descent for spatial relations can be discerned.
The Pharaonic Egyptians devised an elementary geometry for purposes of
resurveying fields after the recession of the annual Nile floodwaters. This
skill was elaborated to the extent that construction of temples and pyramids
became possible. To accomplish all this, a unit of length was defined. In this
way geometry was conflated with arithmetic. Despite their achievements,
neither the Babylonians nor the Egyptians considered their mathematics to be
more than a computational scheme. Their mathematics was devoted primarily to
mensuration, computing taxes, keeping accounts and making astronomical
calculations. They had no concepts of theorems or proofs. This was left to the
Greeks.
The genius of the Greeks, beginning with the Pythagorean school was to discern
an abstract coherent structure beneath the arithmetic of the Babylonians and
Egyptians. They introduced the concept of theorem and proof. In this way, from
the simple act of counting and the drawing of geometric figures, an elaborate
system of mathematics was constructed in Classical Greece.
The Pythagoreans [5] left no written record of their accomplishments. These
are known largely through the writings of later Greek philosophers [6]. They
were besotted with number per se effectively founding number theory. But, it
was their belief in the esoteric character of mathematics and its relation to
the physical universe which is of interest here. They believed that all that
could be known of the physical universe would be known through number. Of the
Pythagoreans, Aristotle commented [7],
> “Contemporaneously with these philosophers and before them, the so-called
> Pythagoreans, who were the first to take up mathematics, not only advanced
> this study, but also having been brought up in it they thought its
> principles were the principles of all things. Since of these principles
> numbers are by nature the first, and in numbers they seemed to see many
> resemblances to the things that be modeled on numbers, and numbers seemed to
> be the first things in the whole of nature, they supposed the elements of
> numbers to be the elements of all things, and they exist and come into being
> - more than in fire and earth and water $\ldots$ all other things seemed in
> their whole nature to hold heaven to be a musical scale and a number.”
Many centuries later, Galileo [8] put it more precisely (and famously).
> “Philosophy is written in this grand book–I mean the universe–which stands
> continually open to our gaze, but it cannot be understood unless one first
> learns to comprehend the language and interpret the characters in which it
> is written. It is written in the language of mathematics, and its characters
> are triangles, circles, and other geometrical figures, without which it is
> humanly impossible to understand a single word of it; without these, one is
> wandering around in a dark labyrinth.”
Whether his view is Pythagorean, that is, does he imply an identity between
the mode of description and the object described or that the mode and object
are distinct, is not clear.
A similar sentiment, with clearer distinction, was expressed some three
centuries later by Weyl [9] in connection with the “language” appropriate for
application to gravitation theory.
> “As everyone has to work hard learning language and writing before he can
> use them freely for expressing his thoughts, so here too the only way to
> shift the load of formulas from one’s own shoulders is to be well acquainted
> with tensor analysis so that one can turn unimpeded by formalities to the
> true problems: Insight into the nature of space, time and matter insofar as
> they contribute to the construction of the objective reality.”
## 3 Nature of Mathematics
For the scientist, mathematics is a tool; for the mathematician, it is an end
in itself; for the philosopher, the philosophy of mathematics is, as Körner
[10] puts it,
> “… not mathematics. It is a reflection on mathematics.”
The nature of mathematics has been debated by philosophers333For a concise
introduction and useful bibliography, see the Wikipedia article, “The
Philosophy of Mathematics”. For a detailed introduction and discussion,
consult Körner’s [11]“The Philosophy of Mathematics”. beginning with the
Pythagoreans. They considered numbers to be inextricably connected with the
physical world. Plato refined Pythagorean ideas to a scheme of ideal abstract
structures mirrored in the immediate world as discovered through human
activity. Subsequent modifications led to Realism in which the severe concept
of ideals is modified. Simply put, mathematical structures have an independent
existence and are uncovered by observation and ratiocination. This view has
been held by many mathematicians such as Paul Erdös who often referred to an
imaginary “Book” in which God had written down all the “beautiful”
mathematical proofs. This was expressed more picturesquely by Everett [12]
> “In the pure mathematics we contemplate absolute truths which existed in the
> divine mind before the morning stars sang together, and which will continue
> to exist there when the last of their host shall have fallen from heaven.”
A more temperate exposition of the same view was expressed by the
distinguished British mathematician, Hardy [13],
> “I have myself always thought of a mathematician in the first instance as an
> observer, who gazes at a distant range of mountains and writes down his
> observations.”
In the late 19th Century, a radically new approach by Frege [14], subsequently
termed Logicism claimed that all mathematics was just a logical structure.
> “Arithmetic thus becomes simply a development of logic, and every
> proposition of arithmetic a law of logic albeit a derivative one. To apply
> arithmetic in the physical sciences is to bring logic to bear on observed
> facts; calculation becomes deduction. The laws of number will not, $\ldots$,
> need to stand up to practical tests if they are to be applicable to the
> external world; for in the external world, in the whole of space and all
> that therein is, there are no concepts, no properties of concepts, no
> numbers. The laws of number, therefore, are not really applicable to
> external things; they are not laws of nature.”
A modified view was earlier given by Kronecker [15].
> “God created the numbers, all the rest is the work of man.”
This avoids the extreme difficulty of establishing the system of numbers by
purely logical arguments encountered later by Whitehead and Russell in their
Principia Mathematica [16]. They do not establish that $1+1=2$ until page 362
of Volume 1.
The concept of Formalism is due to Hilbert who described it simply as444Though
widely quoted, the attribution of this comment to Hilbert is apocryphal. [17].
> “Mathematics is a game played according to certain simple rules with
> meaningless marks on paper.”
Wigner, early in his essay, in the section titled What is Mathematics?,
provides the same view of mathematics.
> “$\ldots$ mathematics is the science of skillful operations with concepts
> and rules invented for just this purpose.”
Intuitionism, created by the Dutch mathematician Brouwer [18], proposes that
mathematical objects are mental constructions governed by self-evident laws.
In a class by itself is the school of thought that considers the brain to have
been hard-wired with the number concept at birth555For an example of this
approach see Dahaene [19].
The various schools of thought have enabled intense debates among philosophers
of science and mathematics. However, a conclusive argument strongly in favor
of Realism can be constructed out the interaction of mathematics and physics,
This will be described below.
## 4 The Practice of Physics
The physicist attempts to understand physical universe. The endeavor is
enormously facilitated through employment of Galileo’s “language of
mathematics”. The use of mathematics entails representing each physical entity
of interest with a mathematical object. This is done by mapping these entities
uniquely onto a mathematical concept. A velocity is a polar vector; a moment
is an axial vector; time is a one-dimensional manifold; flat space-time is a
four-dimensional manifold with a Minkowski metric. The process is a species of
conceptual isomorphism. It can’t be a homeomorphism if ambiguity is to be
avoided. Once this mapping has been completed, the mathematics takes over
completely; it has a life of its own. The rules of the particular mathematical
structure govern. There is no longer any physics per se. At the end of the
calculation, the results are mapped back onto the physical universe.
At intermediate steps it should be possible to map the mathematics back onto
the physics. That is, it should be possible to interpret every mathematical
symbol with a physical construct. Dirac [20] expressed it this way,
> “The new scheme [the representation of quantum states and variables] becomes
> a precise physical theory when all the axioms and rules of manipulation
> governing the mathematical quantities are specified and when in addition
> certain laws are laid down connecting physical facts with the mathematical
> formalism, so that from any given physical conditions equations between the
> mathematical quantities may be inferred and vice versa. In an application of
> the theory one would be given certain physical information, which one would
> proceed to express by equations between the mathematical quantities. One
> would then deduce new equations with the help of the axioms and rules of
> manipulation and would conclude by interpreting these new equations as
> physical conditions.”
The “Unreasonable Effectiveness” is two-fold. Firstly, there is the incredible
richness of phenomena which can be accurately described by a mathematical
structure. Maxwell’s equations furnish the means to accurately describe
electromagnetic phenomena from power transformers to radio transmission.
Einstein’s equations of general relativity provide the basis for calculating
gravitational phenomena from remarkably accurate solar system ephemerides to
the gravitational radiation from pulsar PSR1913+16. The wealth of accurate
description of atomic phenomena by the Schrödinger equation needs no
elaboration. This brief list could be extended dramatically.
Secondly, and of decisive importance, the mathematics may describe phenomena
completely unforeseen when the problem was originally formulated. think of the
Dirac equation for the electron. What Dirac [21] said about it later is
precisely to the point.
> “It was found that this equation gave the particle a spin of half a quantum.
> And also gave it a magnetic moment. It gave just the properties that one
> needed for an electron. That was really an unexpected bonus for me,
> completely unexpected.”
[Emphasis added] What he did not know at the time he published the equation
[1927] was that, quite remarkably, it also described the positron which had
yet to be discovered [1932].
The Schwarzschild singularity in the solution of the Einstein equations for an
isolated point mass is another example. So was the yet to be observed radio
waves in Maxwell’s equations. Physics is full of these surprises. A less well
known far more simple example is that of the exact solution of the simple
pendulum. This is an elliptic integral [22], the real value of which
corresponds to the given initial conditions. The imaginary period corresponds
to a reversed gravitational field and an initial displacement of the bob which
is supplementary to that in the original problem. The latter is an example of
the principle that every prediction of a mathematical formulation of a
physical has a “real” significance.
## 5 Reflections
The practice of physics is based on the concept that the physical universe,
locally and globally, evolves in accordance with fixed rules. This is attested
to by the consistent ability of independent observers to replicate each others
results. It is this regularity which admits the use of mathematics to describe
these results.
All of mathematics starts from a set of integers used for counting. The
historical record shows that the ability to count evolved in different areas
independently. The Babylonian and Mayan civilizations, among others, developed
this skill. Contrarily, some cultures developed counting no further than ”one,
two, many.” This suggests an objective independence of the integers and that
they were discovered rather than created or, as suggested, that the number
concept is hard-wired in the brain [19]. Further, evidence provided by
astrophysical observations implies that the laws of physics are invariant with
respect to location and independent of era. This implies that the mathematical
models created by extraterrestrials (if such exist) are identical, modulo
symbology, to those here on earth. In turn, these imply the universal common
existence of mathematics.
It was noted earlier that the relation between mathematics and physical
phenomena is a species of isomorphism. The phenomena is mapped uniquely onto a
mathematical structure and then mapped in the reverse direction. The extent to
which the initial choice of mathematical structure is an accurate mapping is
determined by how well the evolving physical phenomena remains isomorphic to
the mathematics The relationship between mathematics and phenomena is
reciprocal; each is a representation of the other. It is in this sense that
mathematics has an objective reality. The problem of the physicist is to find
that representation.
The identification of mathematics as objectively existent entails an
interesting problem. Hawking [23] has noted that mathematical structures are
subject to the incompleteness theorems of Gödel:
> “… we and our models are both part of the universe we are describing. Thus a
> physical theory is self referencing, … . One might therefore expect it to be
> either inconsistent or incomplete. … The theories we have so far are both
> inconsistent and incomplete.”
This is certainly correct but only of theoretical interest. There has yet to
constructed mathematical models which are precisely descriptive of the
phenomena we do observe. Despite an incredible accuracy of ten parts in a
billion, the present formulation of quantum electrodynamics is notoriously
lacking in rigor [24]; the “standard model”, however accurate its predictions,
is substantially more a protocol than a theory; a quantized version of the
theory of general relativity has yet to be found. In brief, there is nothing
yet in the mathematical models of observed physical phenomena to which Gödel’s
theorems might be applied. Moreover, our theories are certainly incomplete and
will always remain so. With respect to the cosmos this is obvious. The
strictures imposed by the principle of special relativity limits what
information we can collect to the interior of the past light-cone. That is
finite in extent, some 13+ billion years at the present time. The best that a
physicist may hope for is a model which is both a) accurate in description b)
possessed of predictive capability and c) mathematically rigorous. While the
success of (a) and (b) have been substantial, this is not the case for (c).
The approach to a final, mathematically rigorous “theory of everything” is at
best, asymptotic. Thus, for the finite future, Gödel’s theorem is not a
problem.
Physicists do not (with rare exception such as Newton’s fluxions, the
aforementioned Dirac delta function, or Heaviside’s operational calculus)
create any of the mathematics they utilize. The elaboration of existing and
discovery of new mathematics is mostly the province of mathematicians. We thus
have a very large apparatus of which only part is presently used by
physicists. This suggested the question [25], “Does any piece of mathematics
exist for which there is no application whatsoever in physics?” The question
has been addressed several times, most recently in 2001 [26] [27]. The answers
were uniformly no. This, though suggestive, is not conclusive.
If either Formalism or Constructivism were accurate characterizations of
mathematics then every mathematician could construct his own universe. There
could be as many different universes as there were mathematicians engaged in
the enterprise. The consequences of this are not easily described. The
decisive evidence that Realism is the proper characterization is the
predictive capability of the mathematical structures in which physical
theories are couched. This strongly infers that mathematics and the reality it
describes are both part of the objective universe and await discovery.
## 6 Acknowledgments
Thanks: to (the late) Professor Arthur Komar for stimulating conversations; to
Professor Hilail Gildin for useful information on the philosophy of
mathematics and comments on the manuscript; and especially to Professor
Engelbert Schucking who not only provided the translation of the quotation of
Weyl, page 4, above but also read the manuscript critically.
a)Professor Emeritus, Queens College, City University of New York,
[email protected]
## References
* [1] Wigner, E., Symmetries and Reflections, Indiana University Press, Bloomington, IN (1967).
* [2] Bochner, S., The Role of Mathematics in the Rise of Science, $4^{th}$ ed., Princeton University Press, Princeton, NJ (1981). See p.v (Preface). The first edition was written in the mid sixties.
* [3] London Philosophy Study Guide, University of London,
http://www.ucl.ac.uk/philosophy/LPSG/Ch22.pdf. NB Do not google the acronym
LPSG to find the site.
* [4] Neugebauer, O.,The Exact Sciences in Antiquity, 2nd ed., Dover Publications, New York (1969).
* [5] Allen, D., Pythagoras and the Pythagoreans,
http://www.math.tamu.edu/d̃allen/history/pythag/pythag.html (1997).
* [6] Klein, J., Greek Mathematical Thought and the Origin of Algebra, pp. 63-68, Dover Publications, Inc., New York (1992).
* [7] Ross, W. D., Aristotle, Metaphysics, Book 1-Section 5, Clarendon Press, Oxford (1924).
* [8] Galileo Galilei, Il Saggiatore (The Assayer) (1623), Stillman Drake , and C. D. O’Malley, translators, 1960, The Controversy on the Comets of 1618 University of Pennsylvania Press, Philadelphia, PA (1960).
* [9] Weyl, H., Raum - Zeit - Materie, 5th ed., Springer-Verlag, Berlin (1923). See page 136. Translation furnished by E. Schucking, New York University.
* [10] Körner, S., THE PHILOSOPHY OF MATHEMATICS - An Introductory Essay, Dover Publications, Inc., New York (1986). See the first paragraph of the Introduction.
* [11] Körner, S., loc cit, It contains a complete account of the various philosophical approaches from an historical perspective in lucid style. Chapters VIII, on the relationship between pure and applied mathematics, contains the material most closely relevant to this paper.
* [12] Everett, E., This observation was uttered in an address at the inauguration of Washington University of St. Louis in 1947.
* [13] Hardy, G. H., Mind, new series, 38, 1-25 (1929).
* [14] Frege, G., Foundations of Arithmetic, revised $2^{nd}$ edition, p. 99, Northwestern University Press, Evanston, IL, (1986).
* [15] Weber, H., in a eulogy, (Jahresberichte Deutscher Matematiker-Vereinigung, 2, 5-31, (1893)), cites a lecture by Kronecker before a session of the Berliner Naturforschung-Sammelung in 1886, in which Kronecker stated, Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk..
* [16] Whitehead, A. N. & Russell, B., Principia Mathematica, vol 1, Cambridge University Press, Cambridge (1910).
* [17] Quoted in N. Rose, Mathematical Maxims and Minims, Rome Press Inc., Raleigh, NC (1988).
* [18] Kleene, S., Introduction to Metamathematics, , North-Holland, Amsterdam (1952 with corrections 1971, 10th reprint 1991), See in particular Chapter III: “Critique of Mathematical Reasoning”, sect. 13 “Intuitionism”.
* [19] Dahaene, S., The Number Sense, Oxford University Press, Oxford (1999).
* [20] Dirac, P. A. M., The Principles of Quantum Mechanics , 4th ed. revised, p. 15, Oxford University Press, Oxford (1958).
* [21] Dirac, P. A. M., The Relativistic Wave Electron, European Conference on Particle Physics, Budapest (1977).
* [22] Appell, P., “Sur une interprétation des valeurs imaginaires du temps en M canique”, Comptes Rendus Hebdomadaires des Sc ances de l’Acad mie des Sciences, 87, No. 1, July, 1878.
* [23] Hawking, S. W., “Gödel and the end of Physics”, http://www.hawking.org.uk/index.php/lectures/publiclectures (2002).
* [24] Valenti, M. B., “The renormalization of charge and temporality in quantum electrodynamics”, philsci-archive.pitt.edu (2008).
* [25] Neuenschwander, D. E., “Does any piece of mathematics exist for which is no application whatsoever in physics?”, Amer. J. Phys., 63, 1065 (1996).
* [26] de la Torre, A. C. & Zamora, R., “Answer to Question #31”, Amer. J. Phys., 69, 103 (2001).
* [27] H. P. W. Gottlieb, H. P. W., “Answer to Question #31”, Amer. J. Phys., 69, 103 (2001).
|
arxiv-papers
| 2012-12-24T01:35:22 |
2024-09-04T02:49:39.642480
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alex Harvey",
"submitter": "Alex Harvey",
"url": "https://arxiv.org/abs/1212.5854"
}
|
1212.5932
|
## Title page
PREPRINT; SUBMITTED FOR REVIEW, December 2012
Title Fully scalable online-preprocessing algorithm for short oligonucleotide
microarray atlases
Running title Online-RPA
Authors Leo Lahti, Department of Veterinary Bioscience, University of
Helsinki, Finland and Laboratory of Microbiology, Wageningen University,
Netherlands; Aurora Torrente, European Bioinformatics Institute, Wellcome
Trust Genome Campus, Hinxton, Cambridge, UK and Department of Material Science
and Engineering, Universidad Carlos III de Madrid, Leganés, Spain; Laura L.
Elo, Department of Mathematics and Statistics, University of Turku, Turku,
Finland and Turku Centre for Biotechnology, Turku, Finland; Alvis Brazma,
European Bioinformatics Institute, Wellcome Trust Genome Campus, Hinxton,
Cambridge, UK; Johan Rung, European Bioinformatics Institute, Wellcome Trust
Genome Campus, Hinxton, Cambridge, UK
Corresponding author Leo Lahti, Department of Veterinary Bioscience,
University of Helsinki, PO Box 66, FI-00014 University of Helsinki, Finland;
Tel: +31 6 1673 7991; Email: [email protected]; Fax: +31 317 483829
Keywords: microarrays, online-learning, preprocessing, probabilistic analysis,
robust probabilistic averaging
###### Abstract
Accumulation of standardized data collections is opening up novel
opportunities for holistic characterization of genome function. The limited
scalability of current preprocessing techniques has, however, formed a
bottleneck for full utilization of contemporary microarray collections. While
short oligonucleotide arrays constitute a major source of genome-wide
profiling data, scalable probe-level preprocessing algorithms have been
available only for few measurement platforms based on pre-calculated model
parameters from restricted reference training sets. To overcome these key
limitations, we introduce a fully scalable online-learning algorithm that
provides tools to process large microarray atlases including tens of thousands
of arrays. Unlike the alternatives, the proposed algorithm scales up in linear
time with respect to sample size and is readily applicable to all short
oligonucleotide platforms. This is the only available preprocessing algorithm
that can learn probe-level parameters based on sequential hyperparameter
updates at small, consecutive batches of data, thus circumventing the
extensive memory requirements of the standard approaches and opening up novel
opportunities to take full advantage of contemporary microarray data
collections. Moreover, using the most comprehensive data collections to
estimate probe-level effects can assist in pinpointing individual probes
affected by various biases and provide new tools to guide array design and
quality control. The implementation is freely available in R/Bioconductor at
http://www.bioconductor.org/packages/devel/bioc/html/RPA.html.
## 1 Introduction
Accumulation of research data in in-house and public repositories, such as the
ArrayExpress (Parkinson et al., 2010) and Gene Expression Omnibus (Barrett et
al., 2005) has created data collections that include tens of thousands of
microarray experiments originating from standardized measurement platforms
(Parkinson et al., 2010), helping to overcome some of the inherent biases in
meta-analyses involving multiple measurement platforms (Kilpinen et al., 2008;
Rudy and Valafar, 2011). By combining data from hundreds of individual studies
and thousands of commensurable microarray experiments it is possible to obtain
a more holistic picture of genome function and carry out detailed
investigations of targeted model systems and diseases that can benefit from
large sample sizes of the new data collections (Lukk et al., 2010). A major
portion of contemporary microarray collections originates from short
oligonucleotide microarrays (Lockhart et al., 1996), whose applications range
from gene expression profiling (Parkinson et al., 2010) to alternative
splicing and microbial community analysis (Brodie et al., 2007; Rajilić-
Stojanović et al., 2009; Nikkilä and de Vos, 2010). With nearly a million
arrays in the ArrayExpress database, being able to combine and analyse very
large sets of arrays together is a key challenge with a variety of
applications (Kohane and Valtchinov, 2012; Lukk et al., 2010; Schmid et al.,
2012; Zheng-Bradley et al., 2010).
Reliable preprocessing of the data is central for investigations. While multi-
array preprocessing techniques that combine information across multiple arrays
to quantify probe effects have led to improved preprocessing performance
(Bolstad et al., 2003), the applicability of the standard multi-array
techniques, such as RMA (Irizarry et al., 2003), GC-RMA (Wu and Irizarry,
2004), MBEI (Li and Wong, 2001) and PLIER (Affymetrix Inc., 2005), has been
limited to a few thousand arrays due to the considerable memory requirements
associated with processing up to a million probe-level features per a single
array. This has formed a bottleneck for comprehensive meta-analysis of
contemporary microarray data collections. Scalable variants of the standard
preprocessing algorithms have been recently developed to tackle these
shortcomings. The scalable refRMA (Katz et al., 2006) and fRMA (McCall et al.,
2010) algorithms rely on pre-calculated probe effect terms that are estimated
from restricted reference training sets and then extrapolated to preprocess
further microarray data. The applicability of these methods has, however, been
limited to few specific microarray platforms with pre-calculated probe effect
terms; while the ArrayExpress database lists 203 distinct Affymetrix
platforms, fRMA is currently available only for a single mouse platform and 2
human platforms (McCall and Irizarry, 2011). No platform-independent
preprocessing methods have been available to date to extract and utilize
complete probe-level information in large-scale sample collections in a
scalable manner.
We introduce a fully scalable probe-level model for multi-array preprocessing
based on Bayesian online-learning to overcome the key limitations of the
current approaches. We have extended the Robust Probabilistic Averaging
framework introduced in (Lahti et al., 2011) to estimate probe affinities and
to incorporate prior information of the probes in the analysis. This provides
the basis for scalable online-learning based on sequential hyperparameter
updates, allowing rigorous preprocessing of large microarray atlases on an
ordinary desktop computer in small consecutive batches with minimal memory
requirements and in linear time with respect to sample size. The proposed
Online-RPA algorithm (RPA) provides the means to extract probe-level
information across arbitrarily large data collections and it is readily
applicable to all short oligonucleotide microarray platforms. Moreover, the
analysis of probe performance can now be based on the most comprehensive
collections of short oligonucleotide array data, which provides useful
information also for microarray design and quality control. To our knowledge
this is the only probe-level preprocessing method which is both fully scalable
and platform-independent, providing new tools to take full advantage of
contemporary microarray data collections.
## 2 Results
### 2.1 Scalable preprocessing with online-learning: an overview
The standard steps in microarray preprocessing include background correction,
normalization, and probe summarization. Probe-level procedures that combine
information across multiple arrays have been found to improve preprocessing
performance (Bolstad et al., 2003) but their applicability to large sample
collections has been limited due to huge memory requirements associated with
increasing sample sizes. The available solutions have been based on learning
and extrapolation of probe-level effects from smaller reference training sets
(Katz et al., 2006; McCall et al., 2010). We propose an alternative online-
learning procedure that can extract probe-level information across the
complete microarray collection with minimal memory requirements and in linear
time with respect to sample size based on Bayesian hyperparameter updates.
Assuming that appropriate microarray quality controls have been applied prior
to the analysis, the standard steps of background correction, normalization,
and probe summarization are applied to consecutive batches of the data in four
sweeps over the complete data collection:
Step 1: Background correction In the first step, each individual array is
background-corrected with the standard RMA background correction (Irizarry et
al., 2003). The processed data can be stored temporarily on hard disk to speed
up preprocessing in the following steps that operate on background-corrected
data.
Step 2: Quantile basis estimation The base distribution for quantile
normalization is obtained as the average over sorted probe-level signals from
background-corrected data (Bolstad et al., 2003). We have implemented a
scalable version of standard quantile normalization where the initial base
distribution is calculated from the first batch and updated at each new batch
by taking weighted average between the current base distribution and the one
from the new batch, weighted by their corresponding sample sizes. The final
base distribution is identical to the one which would be obtained by jointly
normalizing all arrays in a single batch. Similar approach is used in parallel
implementations of RMA where the quantile basis from individual batches are
calculated and applied to the complete data collection after averaging over
the individual basis distributions (Schmidberger et al., 2009).
Step 3: Hyperparameter estimation The key novelty of our approach is in
introducing the online-learning model for consecutive probe-level
hyperparameter updates. The background-corrected batches are normalised with
the quantile base distribution, log-transformed, and finally used to update
the model parameters. At the first batch, the model can be initialized by
giving equal priors for the probes if no probe-specific prior information is
available. The probe-level hyperparameters are then updated at each new batch,
providing priors for the next batch. The final probe-level parameters are
obtained after scanning over all batches in the data collection. Ideally, this
is expected to yield an identical result with a single batch approach.
Step 4: Probe summarization In the last step, the final probe-level parameters
obtained from the third step are applied to summarize the probes in each
batch, yielding the final preprocessed data matrix.
#### 2.1.1 The probe-level model
Let us summarize the probe-level model for a fixed probeset with $J$ probes
measured at $T+1$ arrays. The model is based on background-corrected,
normalized, and log-transformed probe-level data. We assume a Gaussian model
for probe effects, where the signal $s_{ij}$ of probe $j\in\\{1,\dots,J\\}$ in
sample $i\in\\{1,\dots,T+1\\}$ is modeled as a sum of the underlying target
signal $a_{i}$ and Gaussian mean and variance parameters $\mathbf{\mu}_{j}$,
$\mathbf{\tau}_{j}^{2}$ that are directly interpretable as constant affinity
$\mathbf{\mu}_{j}$ and stochastic noise $\mathbf{\varepsilon}_{ij}\sim
N(0,\mathbf{\tau}_{j}^{2})$, respectively:
$s_{ij}=a_{i}+\mathbf{\mu}_{j}+\mathbf{\varepsilon}_{ij}.$ (1)
In the following, let us describe the estimation procedure for the model
parameters $\mathbf{a}=[a_{1},\dots,a_{T+1}]$,
$\boldsymbol{\mu}=[\mu_{1},\dots,\mu_{J}]$,
$\boldsymbol{\tau}^{2}=[\tau_{1}^{2},\dots,\tau_{J}^{2}]$, and the additional
modeling assumptions that are necessary to circumvent unidentifiability of the
probe affinity terms.
#### 2.1.2 Incorporating prior information of the probes
We start the analysis by estimating the variance parameters
$\boldsymbol{\tau}^{2}$, following the procedure in (Lahti et al., 2011). In
summary, the analysis is based on probe-level differential expression signal
between each sample $t=[1,\dots,T]$ and a randomly selected reference sample
$r$. Then, given Eq. 1, the unidentifiable affinity parameters
$\mathbf{\mu}_{j}$ cancel out, yielding
$s_{tj}-s_{rj}=(a_{t}-a_{r})+(\varepsilon_{tj}-\varepsilon_{rj})$. Denoting
$m_{t}=s_{t}-s_{r}$, $d_{t}=a_{t}-a_{r}$, and applying the vector notation
$\mathbf{m}=[m_{1},\dots,m_{T}]$, $\mathbf{d}=[d_{1},\dots,d_{T}]$, the full
posterior density for the model parameters $\mathbf{d},\boldsymbol{\tau}^{2}$
is obtained with the Bayes’ rule as
$P(\mathbf{d},\boldsymbol{\tau}^{2}|\mathbf{m})\sim
P(\mathbf{m}|\mathbf{d},\boldsymbol{\tau}^{2})P(\mathbf{d},\boldsymbol{\tau}^{2}).$
(2)
Assuming independent observations $\mathbf{m}_{j}$, given the model
parameters, and marginalizing over $\varepsilon_{rj}$, the likelihood term in
Eq. 2 is then given (Lahti et al., 2011) by
$\begin{split}P(\mathbf{m}|\mathbf{d},\boldsymbol{\tau}^{2})=\prod_{tj}\int
N(m_{tj}|d_{t}-\varepsilon_{rj},\mathbf{\tau}_{j}^{2})N(\varepsilon_{rj}|0,\mathbf{\tau}_{j}^{2})d\varepsilon_{rj}\\\
\sim\prod_{j}(2\pi\mathbf{\tau}_{j}^{2})^{-\frac{T}{2}}exp(-\frac{\sum_{t}(m_{tj}-d_{t})^{2}-\frac{[\sum_{t}(m_{tj}-d_{t})]^{2}}{T+1}}{2\mathbf{\tau}_{j}^{2}}).\end{split}$
(3)
With non-informative priors for $P(\mathbf{d},\boldsymbol{\tau}^{2})$ the
posterior of Eq. 2 would reduce to maximum-likelihood-estimation of Eq. 3 as
in (Lahti et al., 2011). The Bayesian online-learning version, developed and
validated in this paper, takes full advantage of the prior term. This forms
the basis for sequential updates of the posterior in Eq. 2. Assuming
independent prior terms, a non-informative prior $P(\mathbf{d})\sim 1$, and
inverse Gamma conjugate priors for the variances with probe-specific
hyperparameters $\alpha_{j}$ and $\beta_{j}$ (Gelman et al., 2003), the prior
takes the form
$P(\mathbf{d},\boldsymbol{\tau}^{2})=P(\mathbf{d})P(\boldsymbol{\tau}^{2})\sim\prod_{j}\Gamma^{-1}(\mathbf{\tau}_{j}^{2};\alpha_{j},\beta_{j}).$
(4)
The posterior in Eq. 2 is now fully specified given the likelihood (Eq. 3),
the prior (Eq. 4), and the hyperparameters
$\boldsymbol{\alpha}=[\alpha_{1},\dots,\alpha_{J}]$,
$\boldsymbol{\beta}=[\beta_{1},\dots,\beta_{J}]$.
Our primary interest is in estimating the probe-specific variances
$\boldsymbol{\tau}^{2}$, while $\mathbf{d}$ is a nuisance parameter that could
be marginalized out from the model to obtain more robust estimates of
$\boldsymbol{\tau}^{2}$. Since no analytical solution is available and
sampling-based marginalization approaches would slow down computation, we find
a single point estimate for the joint posterior in Eq. 2 as a fast
approximation. Point estimates are found by iterative optimization of
$\mathbf{d}$ and $\boldsymbol{\tau}^{2}$. A mode for $\mathbf{d}$, given
$\boldsymbol{\tau}^{2}$, is searched for by a standard quasi-Newton
optimization method (Goldfarb, 1970). Then, given $\mathbf{d}$, the priors
($\boldsymbol{\alpha},\boldsymbol{\beta}$) and sample size $T+1$, the variance
$\mathbf{\tau}_{j}^{2}$ follows inverse Gamma distribution with
hyperparameters $\hat{\alpha}_{j}=\alpha_{j}+\frac{T}{2}$ and
$\hat{\beta}_{j}=\beta_{j}+\frac{1}{2}(\sum_{t}(m_{tj}-d_{t})^{2}-\frac{(\sum_{t}(m_{tj}-d_{t}))^{2}}{T+1}))$,
yielding
$P(\mathbf{\tau}_{j}^{2}|\mathbf{m},\mathbf{d})\sim\Gamma^{-1}(\mathbf{\tau}_{j}^{2}|\hat{\alpha}_{j},\hat{\beta}_{j}).$
(5)
The point estimate for $\mathbf{\tau}_{j}^{2}$ in this model is readily given
by the mode at $\mathbf{\tau}_{j}^{2}=\hat{\beta}_{j}/(\hat{\alpha}_{j}+1)$.
We give equal weight for all probes by initializing the process with
$\mathbf{\tau}_{j}^{2}=\beta/(\alpha+1)$ with $\alpha=\beta=1$ for all $j$ if
specific prior information is not available. The parameters $\mathbf{d}$ and
$\boldsymbol{\tau}^{2}$ are then iteratively updated until convergence
($<0.01$ change in parameter values in our experiments). The inverse Gamma
hyperparameters corresponding to the final $\boldsymbol{\tau}^{2}$ can then be
retrieved as $\hat{\alpha}_{j}=\alpha_{j}+\frac{T}{2}$ and
$\hat{\beta}_{j}=\mathbf{\tau}_{j}^{2}(\hat{\alpha}_{j}+1)$.
#### 2.1.3 Online-learning of variance hyperparameters
The above formulation allows incorporation of prior information of the probes
in the analysis and sequential updates where the updated hyperparameters
$\hat{\boldsymbol{\alpha}}$, $\hat{\boldsymbol{\beta}}$ from the previous
batch provide priors for the next batch through Eq. 2 and the prior in Eq. 5.
In the absence of prior information we shall give equal weight for all probes
$j$ at the first batch by setting $\alpha_{j}=1$; $\beta_{j}=1$ for all $j$.
The hyperparameters $\hat{\boldsymbol{\alpha}}$, $\hat{\boldsymbol{\beta}}$
are then updated with new observations at each batch until the complete data
collection has been screened through, yielding the final probe-level
hyperparameters.
#### 2.1.4 Affinity estimation and probe summarization
The remaining task after learning the probe-specific variance hyperparameters
is to estimate the probeset-level signal $\mathbf{a}$ and probe affinities
$\boldsymbol{\mu}$ in Eq. 1. Unidentifiability of probe affinities is a well-
recognized issue in microarray preprocessing (Irizarry et al., 2003), and
further assumptions are necessary to formulate an identifiable model. A
standard approach, used in the widely used RMA algorithm (Irizarry et al.,
2003) is to assume that the probes capture the underlying signal correctly on
average and the probe affinities sum to zero: $\Sigma_{j}\mathbf{\mu}_{j}=0$.
We propose a more flexible probabilistic approach where this hard constraint
is replaced by soft priors that keep the expected probe affinities at zero and
allow higher deviations for the more noisy probes with a higher variance
$\mathbf{\tau}_{j}^{2}$. To implent this we apply a Gaussian prior
$\mathbf{\mu}_{j}\sim N(0,\mathbf{\tau}_{j}^{2})$ for the affinities in Eq. 1.
This allows higher fluctuations for the more noisy probes, yielding a better
fit between the probeset-level signal estimate $\mathbf{a}$ and the less noisy
probes with smaller $\mathbf{\tau}_{j}^{2}$. The prior expectation of the sum
of probe affinities, $\Sigma_{j}\mathbf{\mu}_{j}$, will remain at zero but in
contrast to RMA, deviations from this are allowed if supported by the data.
Alternatively, the affinity priors could be determined based on known probe-
specific factors, such as GC-content which is a key element in probe affinity
estimation in the GC-RMA algorithm (Wu and Irizarry, 2004). As probe
performance is affected by a number of factors, however, we prefer the data-
driven approach which can accommodate various, potentially unknown probe-level
noise sources. Point estimates for the probe affinities are identified based
on the following procedure. From Eq. 1 we have
$a_{i}=s_{ij}-\mathbf{\mu}_{j}-\mathbf{\varepsilon}_{ij}\sim
N(s_{ij},2\mathbf{\tau}_{j}^{2})$. A maximum-likelihood estimate for $a_{i}$
is then obtained as a weighted sum of $s_{ij}$ over the probes $j$, weighted
by the inverse variances:
$a_{i}=\frac{1}{\Sigma_{j}\frac{1}{2\mathbf{\tau}_{j}^{2}}}\Sigma_{j}\frac{1}{2\mathbf{\tau}_{j}^{2}}(s_{ij})$.
Given the estimated $a_{i}$, the corresponding maximum-likelihood estimate for
each $\mathbf{\mu}_{j}$ is then given by
$\mathbf{\mu}_{j}^{(i)}=s_{ij}-a_{i}$. Robust estimates of $\mathbf{\mu}_{j}$
are obtained by taking an average over affinity estimates across multiple
samples, yielding the final estimates
$\boldsymbol{\mu}=[\mu_{1},\dots,\mu_{J}]$. This completes the specification
of the model and allows probe summarization according to Eq. 1. Given
$\boldsymbol{\mu}$, $\boldsymbol{\tau}^{2}$, the final estimate for the
probeset-level signal $a_{i}$ is now readily obtained by Eq. 1 as the weighted
sum of $s_{ij}-\mathbf{\mu}_{j}$ over the probes $j$, weighted by the inverse
variances:
$a_{i}=\frac{1}{\Sigma_{j}\frac{1}{\mathbf{\tau}_{j}^{2}}}\Sigma_{j}\frac{1}{\mathbf{\tau}_{j}^{2}}(s_{ij}-\mathbf{\mu}_{j})$.
### 2.2 Validation
In the following, let us investigate the scalability and parameter convergence
of the new online-learning algorithm. The model performance is further
validated by comparisons to alternative preprocessing methods with standard
benchmarking procedures based on the AffyComp spike-in data sets (Cope et al.,
2004) and a large-scale human gene expression atlas (Lukk et al., 2010).
#### 2.2.1 Scalability and parameter convergence
Ideally, the online-learning procedure is expected to yield identical results
with the standard single-batch algorithm. We confirmed this by comparing the
results obtained with the regular single-batch and online-learning versions of
the RPA algorithm for a moderately sized data set of 300 randomly selected
samples, which can be preprocessed with both models. The probeset-level signal
estimates ($\mathbf{a}$) correlated to a high degree (Pearson correlation
$r>0.995;p<10^{-6}$) between the single-batch and online-learning versions.
The selected batch size may affect the running time depeding on the available
memory resources but the results were robust to varying batch sizes of 10, 50,
100 and 300 samples (data not shown); we have used a batch size of 50 samples
in the experiments. The high correspondence between the single-batch and
online-learning models confirms the technical validity of the online version.
Parameter convergence depends on the versatility of the data collection and
overall probe-level noise, which can vary between probesets (McCall et al.,
2010). For certain probesets, the probe parameters start to convergence before
2000-3000 samples or later, indicating that the sample sizes of $\sim 1000$
arrays typically applied with fRMA (McCall et al., 2010; McCall and Irizarry,
2011) may in some cases be be too low to ensure parameter convergence
(Supplementary Fig. 1).
The scalability of Online-RPA was investigated by preprocessing up to 20,000
HG-U133A CEL files from ArrayExpress. The model scales up linearly with
respect to sample size (Supplementary Fig. 2), with 10 hours preprocessing
time for 20,000 CEL files on a standard desktop (Z400 with four 3.06 GHz
processor cores).
#### 2.2.2 Comparison with alternatives
The currently available approaches for scalable preprocessing can be
classified in (i) traditional single-array preprocessing methods and (ii)
frozen multi-array techniques. In single-array algorithms, the preprocessing
is performed separately for each array. Such approaches are fully scalable but
cannot combine probe-level information across multiple arrays, limiting their
accuracy compared to multi-array procedures (Bolstad et al., 2003). We include
the MAS5 algorithm (Affymetrix Inc., 2005) as the reference as this is one of
the most well-known single-array preprocessing techniques. The frozen multi-
array techniques include the refRMA (Katz et al., 2006) and fRMA (McCall et
al., 2010) algorithms. To our knowledge, fRMA (McCall et al., 2010), which
incorporates ideas from refRMA (Katz et al., 2006), is the only available
multi-array preprocessing algorithm for scalable preprocessing of large-scale
microarray collections. The fRMA is based on a standardized database of pre-
calculated probe effect terms, which are then applied to preprocess new arrays
in the database. The estimation procedure for probe effects is not scalable,
however, and applicability of fRMA is currently limited to three Affymetrix
platforms. In addition to the standard “single-array” fRMA model (fRMA), we
consider a second variant which includes an additional model for batch effects
(fRMA-batch). This incorporates additional experiment-specific information to
the analysis which cannot be utilized by the other methods. While this can
improve preprocessing performance, batch information is not necessarily
available for heterogeneous microarray collections, and sets further
requirements for the application of the fRMA-batch procedure. Finally, we
include in the comparisons the widely used RMA algorithm (Irizarry et al.,
2003). In contrast to the other approaches RMA is not fully scalable, but the
RMA preprocessed version of the Lukk et al. (2010) data set is readily
available, and as one of the most widely used standard preprocessing
algorithms this is a relevant reference.
#### 2.2.3 Spike-in experiments
The AffyComp website (http://affycomp.biostat.jhsph.edu; Cope et al. (2004))
provides standard benchmarking tests for microarray preprocessing based on
spike-in experiments on the Affymetrix HG-U95av2 and HG-U133A_tag platforms.
The tests are designed to quantify the relative sensitivity and accuracy of
the alternative preprocessing algorithms based on known target transcript
concentrations. We focus here on the scalable MAS5, fRMA, and RPA algorithms
since most other methods at the AffyComp website have been designed for
moderately sized data sets and have therefore a different scope than the
scalable approaches. However, fRMA is not applicable to the spike-in data sets
as pre-calculated probe-effect vectors are not available for these microarray
platforms. Therefore we have included in the comparisons the widely used RMA
algorithm which has a closely related probe-level model and is the key
preprocessing method in the field. All arrays were preprocessed in a single
batch with RPA. The Figure 1 summarizes the 14 AffyComp benchmarking tests for
MAS5, RMA, and RPA. The complete comparison results for various preprocessing
algorithms are available the AffyComp
website111http://affycomp.biostat.jhsph.edu/AFFY3/comp_form.html.
RPA outperformed RMA in 13 and 11 tests (out of 14) on the HG-U95Av2 and
HG-U133A_tag data sets, respectively (Figure 1). In particular, RPA
outperformed RMA with respect to bias (AffyComp tests 5-10; Supplementary Fig.
3) and true positive-false positive rate, quantified by ROC/AUC analysis
(AffyComp tests 11-14); the differences between RPA and the other methods were
particularly salient with low concentration targets (Fig. 2). In the other
benchmarking tests, RPA and RMA had comparable performance. Interestingly,
MAS5 had the smallest bias although RPA and RMA in general outperformed this
method, and in certain tests such as ROC/AUC analysis MAS5 failed to
distinguish the spike-in transcripts from noise. Certain methods in general
outperform RPA at the AffyComp benchmarking tests, including the FARMS
(Hochreiter et al., 2006) and GCRMA (Wu et al., 2003) algorithms. These
methods have a more limited scalability than RPA, however, and hence a
different scope.
In summary, the fully scalable RPA had a similar or improved preprocessing
performance in spike-in data sets compared to the standard RMA and MAS5
algorithms, and a wider scope than the only scalable probe-level alternative,
fRMA, which is not available for the spike-in platforms (McCall and Irizarry,
2011).
#### 2.2.4 Classification performance
We investigated the classification performance of the comparison methods based
on the (Lukk et al., 2010) data collection of 5,372 samples with 369 cell and
tissue types, disease states and cell lines (Fig. 3). A random forest
classifier (Breiman, 2001) was trained to distinguish between these classes
based on 1000 randomly selected probe sets and 500 trees at 10 cross-
validation folds, where the data was split into training (90%) and test (10%)
sets. The singleton classes (150 samples) were excluded. The comparisons were
performed with both the standard Affymetrix probe sets and alternative
probesets based on the Ensembl Gene (ENSG) identifiers. RPA outperformed RMA
and MAS5 ($p<0.05$; paired Wilcoxon test). Differences between RPA and fRMA
were not significant, and the fRMA with batch effect model (fRMA-batch)
outperformed the other methods ($p<0.05$). It is important to note, however,
that fRMA and fRMA-batch algorithms have a considerably more limited scope
than Online-RPA, as detailed in Discussion. Further details on the data, class
and batch definitions are provided in Materials and Methods.
#### 2.2.5 Correlation between technical gene replicates
The standard Affymetrix arrays contain multiple probesets for certain
transcripts. As the final benchmarking test, we compared the probeset-level
summaries for such technical gene replicates on the (Lukk et al., 2010) data
set, following Elo et al. (2005); Lahti et al. (2011). Pearson correlation was
calculated for each Affymetrix probeset pair sharing the same EnsemblID
(Bioconductor package hgu133a.db). The average correlations over all pairs
were as follows: MAS5 0.46, RMA 0.53, fRMA 0.51, fRMA.batch 0.55 and RPA 0.54.
The differences between the methods were significant (paired Wilcoxon test
$p<0.01$). In this comparison, RPA outperformed MAS5, RMA and fRMA
(Supplementary Fig. 4).
## 3 Discussion
High memory requirements of the standard preprocessing techniques for short
oligonucleotide arrays have limited their applicability to moderately sized
data sets. Scalable probe-level preprocessing techniques, the refRMA (Katz et
al., 2006) and fRMA (McCall et al., 2010), have been available only for few
microarray platforms based on precalculated probe effects from restricted
reference training sets. The lack of scalable, general-purpose preprocessing
algorithms is hence forming a bottleneck for large-scale meta-analyses and
full utilization of contemporary microarray collections. Our proposed method
provides novel tools for taking advantage of the full information content in
these data collections. With nearly a million arrays in the ArrayExpress
database, being able to combine and analyse very large sets of arrays together
is a key challenge with a variety of applications ranging from gene expression
profiling (Lukk et al., 2010; Kohane and Valtchinov, 2012; Parkinson et al.,
2010; Schmid et al., 2012; Zheng-Bradley et al., 2010) to alternative splicing
and microbial community analysis (Brodie et al., 2007; Rajilić-Stojanović et
al., 2009; Nikkilä and de Vos, 2010).
We have introduced the first fully scalable online-learning algorithm that can
extract and utilize individual probe effects across arbitrarily large
microarray collections, overcoming the key scalability limitations of the
current preprocessing techniques. The algorithm learns probe-level parameters
by sequential hyperparameter updates, processing the data in small consecutive
batches. This makes the model readily applicable to contemporary microarray
collections with tens of thousands of samples by extending the probabilistic
framework introduced in Lahti et al. (2011). In contrast to the alternatives
our model is fully scalable and readily applicable to all microarray platforms
as its application does not depend on precalculated probe effect terms. The
model scales up in linear time with respect to sample size, with 10 hours
running time for 20,000 CEL files in our experiments. The running time could
be further accelerated by optimizing the implementation, using more efficient
processors and parallelizing with multiple cores (Schmidberger et al., 2009).
Interestingly, we noticed that averaging of the probes, weighted by the probe-
specific variance estimates provided by RPA, yielded similar results with the
full model that additionally includes probe affinities. The differences
between the two models were not significant, indicating that affinity analysis
could be omitted to further speed up computation without compromising
preprocessing performance. In analogy to fRMA, our model also allows the
utilization of estimated model parameters as priors to preprocess further data
sets. This can provide the advantages of single-chip methods and fRMA of not
having to recompute the whole preprocessing procedure when new arrays are
included in the data collection. Providing frozen parameter estimates can not
only speed up computations but also allow reproducible analysis of single
arrays for diagnostic and other purposes, as suggested in (McCall et al.,
2010).
Probe performance is affected by RNA degradation, non-specific hybridization,
GC- and SNP-content, annotation errors and other, potentially unknown factors.
While modeling of the probe effects have been shown to yield improved
estimates of the target signal (Irizarry et al., 2003; Li and Wong, 2001), the
various sources of the probe-level noise remain poorly understood. While our
model can be applied to learn the probe-level parameters based on subsets of
the data, with the fully scalable extension the analysis of probe performance
can now be based on the most comprehensive collections of short
oligonucleotide array data that are available in the public repositories. As
such, RPA can assist in nailing down individual probes affected by various
biases, giving tools to guide microarray preprocessing and probe design in
future studies and industry standards (Katz et al., 2006).
The widely used RMA algorithm is a special case of our single-batch model,
assuming that the affinities sum to zero and all probes within a probeset have
identical variances. The recent scalable extension, fRMA (McCall et al.,
2010), has a more detailed model for probe-specific variances. While RPA was
comparable to or outperformed the standard fRMA algorithm in our experiments,
the fRMA-batch which utilizes additional sample metadata outperformed RPA. It
is important to note, however, that the modeling of batch effects in fRMA is
only possible when sufficient sample metadata is available which is not always
the case with large and heterogeneous microarray collections, which are the
primary target of scalable preprocessing algorithms. Moreover, batch effects
could be modeled in a separate step for instance based on linear models (Chen
et al., 2011; Leek et al., 2012). Comparison of the various batch effect
modeling techniques is out of the scope of this paper, however. Finally, the
applicability of the fRMA and fRMA-batch algorithms is currently limited to
only three Affymetrix platforms: HG-U133Plus2.0, HG-U133A, and MG-430 2.0
(McCall and Irizarry, 2011), and could therefore not be included in the
standard AffyComp spike-in benchmarking comparisons which rely on other
microarray platforms. Therefore fRMA and fRMA-batch have a more limited scope
than RPA. While certain methods, such as FARMS (Hochreiter et al., 2006) and
GCRMA (Wu et al., 2003) with more detailed probe-level models outperform RPA
in spike-in experiments, their scalability and hence the scope are more
limited.
The introduced online-learning approach therefore remains currently the only
fully scalable preprocessing algorithm that is readily applicable on all short
oligonucleotide platforms, outperforms the standard RMA algorithm, and can be
used for scalable probe-level analysis and preprocessing of gene expression
atlases involving tens of thousands of samples. This provides new tools to
scale up investigations and to take advantage of the full information content
of the rapidly expanding microarray data collections.
## 4 Methods and Data Access
### 4.1 Data
We have selected for comparisons a reasonably large, well-annotated and
quality assessed microarray data set including 5372 human samples from a
versatile collection of 369 cell and tissue types, disease states and cell
lines from 206 public experiments and 162 laboratories, measured with the
Affymetrix HG-U133A microarray (Lukk et al., 2010). The biological groups are
of varying sizes, and include 150 classes with only one sample (singleton
classes); the annotations describing the group of each sample in the data set
can be retrieved from the ArrayExpress archive (accession number:
E-MTAB-62)222http://www.ebi.ac.uk/gxa/experimentDesign/E-MTAB-62. This data
set is ideal for benchmarking of scalable preprocessing methods since the
alternative fRMA preprocessing model depends on the availability of
precalculated probe effect terms, which are available for this microarray.
Moreover, sufficient sample metadata is available for this data set to include
batch effects in the fRMA model. In addition, despite of the heterogeneous
origin of the data set, which made unfeasible to obtain ‘batches’ in strictly
the same manner as defined in (McCall et al., 2010), we could approximate them
with the following approach. For each array within each experiment in the data
set we retrieved the creation date of the CEL file from its HEADER section,
under the DatHeader TAG, and assigned to the same batch those arrays from the
same experiment (and laboratory) that were scanned on the same day. Thus, it
was possible to assess the two available versions of fRMA, specifically the
“single-array” and “batch-of-arrays”. Moreover, the sample size allows
comparisons with the standard RMA preprocessing method (Irizarry et al.,
2003). Finally, in addition to the standard Affymetrix probe sets we have
included in the comparisons alternative probe sets based on updated Ensembl
gene mappings available through the hgu133ahsensgcdf (14.1.0) annotation
package (Dai et al., 2005). The reference probe effects for fRMA and the
alternative mapping of probes to genes were built using the Bioconductor
frmaTools (McCall and Irizarry, 2011).
### 4.2 AffyComp spike-in experiments
The score components in Figure 1 are as follows: (1) median SD across
replicates; (2) Inter-quartile range of the log-fold-changes from genes that
should not change; (3) 99.9% percentile of the log-fold-changes if from the
genes that should not change; (4) $R^{2}$ obtained from regressing expression
values on nominal concentrations in the spike-in data; (5) slope obtained from
regressing expression values on nominal concentrations in the spike-in data;
(6) slope from regression of observed log concentration versus nominal log
concentration for genes with low intensities; (7) same for genes with medium
intensities; (8) same for genes with high intensities; (9) slope obtained from
regressing observed log-fold-changes against nominal log-fold-changes; (10)
slope obtained from regressing observed log-fold-changes against nominal log-
fold-changes for genes with nominal concentrations less than or equal to 2;
(11) area under the ROC curve (AUC; up to 100 false positives) for genes with
low intensity standardized so that the optimum is 1; (12) AUC for genes with
medium intensities; (13) AUC for genes with high intensities; (14) a weighted
average of the ROC curves 11-13 with weights related to amount of data in each
class. For full details, see Cope et al. (2004).
## 5 Acknowledgements
LL received funding from the Finnish Alfred Cordelin foundation and Academy of
Finland (decision 256950). AT was supported by the Ramón Areces Foundation.
LLE was supported by the Academy of Finland (decisions 127575, 218591). JR was
supported through funds from The European Community’s Seventh Framework
Programme (FP7/2007-2013), ENGAGE Consortium, grant agreement
HEALTH-F4-2007-201413. All authors have reviewed the manuscript.
## 6 Disclosure declaration
The authors do not have competing interests.
## 7 Figure Legends
Figure 1: The benchmarking statistics for the AffyCompIII spike-in data for
RPA, RMA, and MAS5 for the HG-U95Av2 (top) and HG-U133A_tag (bottom)
platforms. RPA and MAS5 represent fully scalable algorithms, and the standard
RMA algorithm has been included as a benchmark since its fully scalable
extension, fRMA, is not available for the spike-in platforms. For clarity of
presentation, we have transformed the scores 1-3 with $1-x$; so that the score
value of 1 corresponds here to ideal performance at all 14 scores. For a full
description of the 14 benchmarking components, see Materials and Methods.
Figure 2: Average ROC curves for low-abundance targets with nominal
concentrations at most 4 picoMolar and nominal fold changes at most 2 in the
AffyCompIII spike-in data for MAS5 (solid line), RMA (dashed line), and RPA
(dotted line) on the HG-U95Av2 (left) and HG-U133A_tag (right) platforms. The
Figure has been adapted from AffyCompIII test 5C. For details, see Cope et al.
(2004).
Figure 3: Classification performance in Lukk et al. data set (Lukk et al.,
2010) for the comparison algorithms. The 5372 samples were classified into 369
cell and tissue types, and after excluding the singleton classes the
classification performance was quantified by random forest classifier based on
10-fold cross-validation as described in the Results. Online-RPA outperforms
RMA and MAS5 ($p<0.05$). Differences between RPA-online and fRMA are not
significant, and fRMA-batch outperforms the other methods ($p<0.05$).
## 8 Figures
---
Figure 1:
---
Figure 2:
Figure 3:
## 9 References
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|
arxiv-papers
| 2012-12-24T16:41:08 |
2024-09-04T02:49:39.651233
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leo Lahti, Aurora Torrente, Laura L. Elo, Alvis Brazma, Johan Rung",
"submitter": "Leo Lahti",
"url": "https://arxiv.org/abs/1212.5932"
}
|
1212.5983
|
# An ideal multi-secret sharing scheme based on minimal privileged coalitions
Yun Song , Zhihui Li ∗ College of Mathematics and Information Science, Shaanxi
Normal University, Xi’an, 710062, P. R. China
###### Abstract
How to construct an ideal multi-secret sharing scheme for general access
structures is difficult. In this paper, we solve an open problem proposed by
Spiez et al.recently [Finite Fields and Their Application, 2011(17) 329-342],
namely to design an algorithm of privileged coalitions of any length if such
coalitions exist. Furthermore, in terms of privileged coalitions, we show that
most of the existing multi-secret sharing schemes based on Shamir threshold
secret sharing are not perfect by analyzing Yang et al.’s scheme and Pang et
al.’s scheme. Finally, based on the algorithm mentioned above, we devise an
ideal multi-secret sharing scheme for families of access structures, which
possesses more vivid authorized sets than that of the threshold scheme.
###### keywords:
$(t,n)$ threshold, track, access structure, minimal privileged coalitions,
multi-secret sharing
## 1 Introduction
_Single-secret sharing schemes(SSSS)._ A secret sharing scheme is a method to
distribute a secret among a set $P$ of participants, which includes a pair of
efficient algorithms:a distribution algorithm and a reconstruction algorithm,
implemented by a dealer and some participants. The distribution algorithm
allows a dealer to split a secret s into different pieces, called shares, and
distribute them to participants. The reconstruction algorithm is executed by
the authorized subsets of parties who are able to reconstruct the secret by
using their respective shares. The collection of these authorized sets of
participants is called the access structure, $\Gamma\subset 2^{P}$. A group of
participants is called a minimal authorized subsets if they can recover the
secret with their shares, and any of its proper subgroups cannot do so. Then,
the access structure is determined by the family of minimal authorized
subsets, $(\Gamma)_{min}$. The notion of secret sharing was introduced by
Shamir [1] and Blakley [2] , who considered the only schemes with a
$(t,n)$-threshold access structure formed by the set of authorized subsets of
participants is the set of all subsets of size at least $t$, for some integer
$t$. In 1987, Ito et al.[3] proved that there exists a secret sharing scheme
for any access structure which is more general than the threshold ones.
Afterwards, secret sharing schemes have been widely employed in the
construction of more elaborate cryptographic primitives and several types of
cryptographic protocols(see [4-8]).
A secret sharing scheme is called perfect if any non-authorized subset of
participants have no information about the secret, and ideal if the shares are
of the same size as that of the secret. An ideal secret sharing scheme is
well-known to be the best efficiency that one can achieve with lowest storage
complexity and the communication complexity[9].
_Multi-secret sharing schemes(MSSS)._ Multi-secret sharing can be seen as a
natural generalization of single secret sharing schemes. In 1994, Blundo et
al. [10] studied the more general case in which the set of participants share
more than one secret and different secret is related to different access
structure. Let $\textbf{$\Gamma$}=(\Gamma_{0},\ldots,\Gamma_{m-1})$ be the
m-tuple of access structures on $P$ and let $S_{0}\times
S_{1}\times\cdots\times S_{m-1}$be the set from which the secrets are chosen,
where for any $0\leq j\leq m-1$, each secret $s_{j}$ to be shared is chosen in
$S_{j}$. In the definition of a perfect multi-secret sharing scheme, an
m-tuple of secrets $(s_{0},\ldots,s_{m-1})\in S_{0}\times\cdots\times S_{m-1}$
is shared in an m-tuple $(\Gamma_{0},\ldots,\Gamma_{m-1})$ of access
structures on $P$ in such a way that, for each $0\leq j\leq m-1$, the access
structure $\Gamma_{j}$ is the set of all subsets of $P$ that can recover
secret $s_{j}.$ A perfect multi-secret sharing scheme is defined in [10] such
that the following requirements are satisfied.
###### Definition 1.1
Let $\textbf{$\Gamma$}=\\{\Gamma_{0},\ldots,\Gamma_{m-1}\\}$ be an m-tuple of
access structures on the set of participants $P=\\{P_{1},\ldots,P_{n}\\}$. A
multi-secret sharing scheme for
$\textbf{$\Gamma$}=\\{\Gamma_{0},\ldots,\Gamma_{m-1}\\}$ is a sharing of the
secrets $(s_{0},\ldots,s_{m-1})\in S_{0}\times\cdots\times S_{m-1}$ in such a
way that, for $0\leq j\leq m-1$,
(1) Correctness requirement: Any subset $A\subseteq P$ of participants enabled
to recover
$s_{j}$ can compute $s_{j}$. Formally, for all $A\in\Gamma_{j}$, it holds
$H(S_{j}\ |\ A)=0,$
(2) Security requirement: Any subset $A\subseteq P$ of participants not
enabled to recover
$s_{j}$, even knowing some of the other secrets, has no more information on
$s_{j}$
than that already conveyed by the known secrets. Formally, for all
$A\not\in\Gamma_{j}$ and
$T\subseteq\\{S_{0},\cdots,S_{m-1}\\}\backslash\\{S_{j}\\}$, it holds
$H(S_{j}\ |\ AT)=H(S_{j}\ |\ T).$
So far, Multi-secret sharing are widely applied not only in the field of
information security but also the theories and models of secret sharing
schemes[11-19].
_Our results._ Although fruitful results for the multi-secret sharing have
been obtained, it is still difficult to devise ideal multi-secret sharing
schemes for general access structures. In this paper, by using the theory of
privileged coalitions[20,21], we point out that several multi-secret sharing
schemes (see[13-18])based on Shamir’s threshold scheme are not perfect, and
thus are not ideal. In[20] Spiez et al. obtained an algorithm to construct the
privileged coalitions of maximal length and put forward how to design an
algorithm of privileged coalitions of any length if such coalitions exist.
Motivated by these concers, we solve this open problem and devise an ideal
multi-secret sharing scheme(IMSSS) for families of access structures based on
the algorithm mentioned above. Finally, we compare our scheme with two
additional schemes[13,14] according to their performance analysis.
The rest of the paper is organized as follows: In Section 2, the basic
definitions of secret Shamir’s secret sharing scheme and privileged coalitions
are reviewed, an algorithm of privileged coalitions of any length is designed
as well. In Section 3 and 4, based on theories of privileged coalitions and
corresponding algorithm, we shall present our ideal multi-secret sharing
scheme and make some discussions. Finally, some remarks are given in the
conclusion Section.
## 2 Preliminaries
### 2.1 Shamir’s secret sharing scheme
In a $(t,n)$ threshold secret sharing scheme(a scheme with $n$ participants
and threshold $t$), where $2\leq t\leq n$, a dealer does not disclose a secret
data to the participants but only distributes $n$ shares amongest them in such
a way that at least $t$ or more participants can collectively efficiently
reconstruct the secret but no coalition of less than $t$ participants can
obtain noting about the secret.
In the paper, we consider Shamir’s secret sharing schemes[20] with the secret
placed as a coefficient $a_{i}$ of the scheme polynomial
$f(x)=a_{0}+a_{1}x+\cdots+a_{t-1}x^{t-1}$, where
$\textbf{a}=(a_{0},\ldots,a_{t-1})\in F^{t}_{q}$ ($q$ is a prime power). For a
fix $f(x)$ and an $j$, such scheme is uniquely defined by a sequence
$\textbf{L}=(l_{1},l_{2}\ldots,l_{n})\in F_{q^{*}}^{n}$ of pairwise different
public identities, allocated to participants, called in [20] a track. The
shares $y_{i}=f(l_{i})$ $(1\leq i\leq n)$ assigned to participants are secret.
Remark 1. Shamir’s $(t,n)$ threshold secret sharing and Shamir’s secret
sharing are different. A Shamir’s $(t,n)$ threshold secret sharing must be a
Shamir’s secret sharing, but a Shamir’s secret sharing may not be a Shamir’s
$(t,n)$ threshold secret sharing.
In fact, the public identities L define the $n\times t$ matrix
$\textbf{A(L)}=(l^{\nu}_{\mu})_{1\leq\mu\leq n\atop{0\leq\nu\leq t-1}}$ over
$F_{q}$ which gives the shares by $\textbf{A(L)a}^{T}=\textbf{y}^{T}$. Since L
is a track any coalition of $t$ participants determines a $t\times t$ non-
singular Vandermonde submatrix of the matrix A(L) consisting of the
corresponding rows of the matrix A(L), i.e.,
(i) all $t\times t$ submatrices of A(L) are non-singular.
A Shamir’s secret sharing is a Shamir’s $(t,n)$ threshold secret sharing if
and only if
(ii) all $(t-1)\times(t-1)$ submatrices of the matrix obtained from the matrix
A(L)
by removing its $j$-th column are non-singular.
The track L corresponding to such matrix A(L) satisfying two above conditions
is called $(t,j)-$admissible[21].
###### Definition 2.1
Let $0\leq j\leq t-1$. If the track $\textbf{L}\in F^{n}_{q}$, where $n\geq
k$, defines a Shamir’s $(t,n)$ threshold secret sharing with the secret placed
as a coefficient $a_{j}$ of the scheme polynomial
$f(x)=a_{0}+a_{1}x+\cdots+a_{t-1}x^{t-1}$, then L is called a
$(t,j)-$admissible track.
Note that if the track $\textbf{L}\in F^{n}_{q}$ is not $(t,j)-$admissible,
then it contains a subtrack consisting of less than $t$ participants which can
reconstruct the secret by themselves, forming a privileged coalition[20].
### 2.2 Minimal privileged coalition
###### Definition 2.2
[20] Let $r<t$ and fix $j$, $0<j<t-1.$ A coalition of $r$ participants
$\textbf{L}=(l_{1},l_{2},\cdots,l_{r})\in F^{r}_{q}$ is said to be a
$(t,j)-$privileged coalition, if they can reconstruct the secret, placed as
the coefficient $a_{j}$ of the scheme polynomial
$f(x)=a_{0}+a_{1}x+\cdots+a_{t-1}x^{t-1}$.
###### Definition 2.3
Let $r<t$ and fix $j$, $0<j<t-1.$ A coalition of $r$ participants
$\textbf{L}=(l_{1},l_{2},\cdots,l_{r})\in F^{r}_{q}$ is said to be a
$(t,j)-$minimal privileged coalition, if L is a privileged coalition without
any subtracks that can reconstruct the secret.
Note that the tracks of length $t$, which can be extended by privileged
coalitions of length $r$, can reconstruct the secrets placed as any
coefficients of the scheme polynomial. Then we would have
###### Definition 2.4
Fix $j$, $0<j<t-1.$ A coalition of $t$ participants
$\textbf{L}=(l_{1},l_{2},\cdots,l_{t})\in F^{t}_{q}$ is said to be a
$(t,j)-$unextended track, if L can not be extended by a $(t,j)-$privileged
coalition.
By Definition 2.3, the minimal authorized subsets of Shamir’s secret sharing
schemes are determined by minimal privileged coalitions and unextended tracks.
Therefore, constructing minimal privileged coalitions become the key to the
determination of the access structures. In order to devise an algorithm to
obtain privileged coalitions, we will need the following theorem about the
characterization of $(t,j)-$privileged coalitions whose proof can be found
in[20].
###### Theorem 2.5
Assume that $0<j<t-1$, $j<r\leq t-1$. Let
$\textbf{L}=(l_{1},l_{2},\cdots,l_{r})\in F^{r}_{q}$ be a track, and let
$t\leq q$. Then L is a $(t,j)-$privileged coalition if and only if
$\tau_{\omega}(\textbf{L})=0,\ \ \ for\ all\ \omega\in\\{r-j,\cdots,t-1-j\\},$
where $\tau_{\omega}(\textbf{L})$ denotes the elementary symmetric polynomial
of total degree $\omega$.
### 2.3 Privileged coalitions of any length
In this section, we solve an open problem proposed in[20] recently, namely to
design an algorithm of privileged coalitions of any length if such coalitions
exist. For simplicity, we will focus on the tracks whose value of each
component is clamped to the range of $\\{1,2,\cdots,N\\}$, and we call these
coalitions $(t,j)-$privileged coalitions with respect to $N$. Using Algorithm
1 we can obtain $(t,j)-$privileged coalitions with respect to $N$ of length
$r$ over $F_{p}$ ($p$ is a odd prime). By Definition 2.3, for the given $t,r$
with $\frac{t+1}{2}\leq r\leq t-1$, we can obtain $(t,j)-$minimal privileged
coalitions with respect to $N$ of length $r$ by detaching ${r-r_{min}\choose
p-r_{min}}N_{min}$ non-minimal privileged coalitions from $(t,j)-$privileged
coalitions of length $r$, where $N_{min}$ denotes the number of privileged
coalitions of the shortest length $r_{min}$.
As an illustration, we investigate the Shamir’s secret scheme with the number
of participants $n=13$ and threshold $t=7$. In the appendix, we then present
two tables of $(7,j)-$minimal privileged coalitions with respect to $N=13$ of
any length if such coalitions exist.
Algorithm 1
Input positive integers $t,r,j$ with $t\geq 3$, $\frac{t+1}{2}\leq r\leq t-1$,
$t-r\leq j\leq r-1$.
Output all of the $(t,j)-$privileged coalitions of length $r$ with respect to
$N$ over $F_{p}$.
1\. Compute the range of values for $j$, and set $a\leftarrow r-j$,
$b\leftarrow t-1-j$. Take
$J$ is a set of all integers from $a$ to $b$.
2\. Obtain all of the tracks of length $r$ with respect to $N$ over $F_{p}$.
2.1. $B=(1,2,\cdots,n)$, for $i=1$ to $N$
2.1.1 For $j=i+1$ to $N$
If $B(j)>B(i)$ ($B(j)$ denotes the elements of the $j-$th position
of the array $B$),
2.1.1.1 For $k=j+1$ to $N$
If $B(k)>B(j)$
⋮
$r-1$ Nested loops are carried out in turn, then
$\underbrace{(\cdots,B(k),B(j),B(i))}_{r}$
is a track.
3\. Find $(t,j)-$privileged coalitions from the tracks obtained in Step 2 by
means
of the principle about Vieta theorem of high power equation.
3.1. For $i=1$ to $b-a+1$
3.1.1. Set $p\leftarrow J(i)$, and go through tracks that obtained in Step 2.
Let $C=1$. For $j=1$ to $r$
Set $C\leftarrow C\times(x+\textbf{L}(j))$
3.1.2. Set$f\leftarrow C$, and select m as a vector whose components are
coefficients of the expansion of $f(x)$ in ascending power of $x$.
Set $s\leftarrow\textbf{m}(r-p+1)$, then set $v_{p}\leftarrow mod(s,p)$.
3.2. If $v_{p}=0$, then return $(\textbf{L})$.
## 3 An ideal multi-secret sharing scheme
When the probability distributions over the secrets and shares are uniform, a
secret sharing scheme is said to be ideal if all secrets and shares are the
same size[19]. In this section we firstly define a (t-1)-tuple
$\textbf{$\Gamma$}=(\Gamma_{0},\ldots,\Gamma_{t-2})$ of access structures and
then we devise an IMSSS which realizes such a (t-1)-tuple
$\textbf{$\Gamma$}=(\Gamma_{0},\ldots,\Gamma_{t-2})$ of access structures.
### 3.1 Definition of the access structures
Let $P=\\{P_{1},\ldots,P_{n}\\}$ be the set of participants, we firstly define
such an (t-1)-tuple $\textbf{$\Gamma$}=(\Gamma_{0},\ldots,\Gamma_{t-2})$ as
follows:
(1) $(\Gamma_{0})_{min}=\\{A\subseteq P\mid|A|=t\\}.$
(2) $(\Gamma_{j})_{min}=\\{A\subseteq P\mid\textbf{L}_{A}\ is\ either\ a\
(t,j)-unextended\ track\ or\ a\ (t,j)-$
$privileged\ coalition,(1\leq j\leq t-2)\\}$.
As each component of a track can be used as the identity of the participants,
we can determine the minimal authorized subsets of $(\Gamma_{j})_{min}$ for
the secret placed as a coefficient $a_{j}$ of the scheme polynomial
$f(x)=a_{0}+a_{1}x+\cdots+a_{t-1}x^{t-1}$ by getting both (t,j)-unextended
track and (t,j)-privileged coalition using the Algorithm 1.
Obviously, a subset $A\subseteq P$ is likely to reconstruct more than one
secret. For example, if $A\in\Gamma_{i}$ and $A\in\Gamma_{j}$, then $A$ can
reconstruct not only $s_{i}$ but also $s_{j}$, where $0\leq i,j\leq t-2$ and
$i\neq j$.
Example 3.1 Let $P=\\{P_{1},P_{2},P_{3},P_{4},P_{5},P_{6}\\},t=5,q=7.$ Without
loss of generality, D distributes $i$ to participant $P_{i}$ for $1\leq i\leq
6.$ It follows from algorithm 1 that (5,1)-privileged coalition over $F_{7}$
is $(1,2,5,6),\ (1,3,4,6)$ and $(2,3,4,5)$; (5,2)-privileged coalition is
$(1,2,4),(3,5,6)$; (5,3)-privileged coalition is $(1,2,5,6),\ (1,3,4,6)$ and
$(2,3,4,5)$, but without (t,j)-unextended track. Hence $(t-1)$-tuple
$\textbf{$\Gamma$}=(\Gamma_{0},\ldots,\Gamma_{m-2})$ can be defined as
follows:
$(\Gamma_{0})_{min}=\\{\\{P_{1},P_{2},P_{3},P_{4},P_{5}\\},\\{P_{1},P_{2},P_{3},P_{4},P_{6}\\}\\{P_{1},P_{2},P_{4},P_{5},P_{6}\\},$
$\\{P_{1},P_{2},P_{3},P_{5},P_{6}\\},\\{P_{1},P_{3},P_{4},P_{5},P_{6}\\},\\{P_{2},P_{3},P_{4},P_{5},P_{6}\\}\\},$
$(\Gamma_{1})_{min}=(\Gamma_{3})_{min}=\\{\\{P_{1},P_{3},P_{4},P_{6}\\},\\{P_{1},P_{2},P_{5},P_{6}\\},\\{P_{2},P_{3},P_{4},P_{5}\\}\\},$
$(\Gamma_{2})_{min}=\\{\\{P_{1},P_{2},P_{4}\\},\\{P_{3},P_{5},P_{6}\\}\\}.$
### 3.2 Construction of the IMSSS
#### 3.2.1 Initialization phase
Note that $s_{0},\ldots,s_{t-2}$ denote t-1 secrets to be shared, where
$(s_{0},\ldots,s_{t-2})\in S_{0}\times\cdots\times S_{t-2}.$ The dealer D
randomly chooses $n$ pairwise different $l_{i}$ and assigns them to every
participant $P_{i}$ as their public identity, where $l_{i}\in F_{q}^{*}$ for
$1\leq i\leq n.$ Then D computes the $(t,j)$-unextended track and
$(t,j)$-privileged coalition by means of algorithm 1 for $0\leq j\leq t-2.$
#### 3.2.2 Distribute phase
The dealer D performs the following steps:
(1) Choose an integer $a_{t-1}$ from $F_{q}^{*}$ and construct $(t-1)$th
degree polynomial
$f(x)$ mod $q$, where $0<s_{0},\ldots,s_{t-2},a_{t-1}<q$ as
follows:$f(x)=s_{0}+\cdots+$
$s_{t-2}x^{t-2}+a_{t-1}x^{t-1}\mod q.$
(2) Compute $y_{i}=f(l_{i})\mod q$ for $i=1,2\ldots,n$ and distribute them to
every
participant $P_{i}$ as secret shares by a secret channel.
#### 3.2.3 Recovery phase
Assume that $(0\leq j\leq t-2).$ for any $A\in(\Gamma_{j})_{min}$, if $|A|=t$,
then any subset $A\in(\Gamma_{j})_{min}$ can reconstruct the secret $s_{j}$ by
solving equation system $\textbf{A}(\textbf{L}_{A})\
\textbf{a}^{T}=\textbf{y}^{T}$, where $\textbf{L}_{A}$ is corresponding track
of $A$; if $|A|<t$ and $\textbf{L}=(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{r}})\in
F^{r}_{q}$ is a $(t,j)$-privileged coalition of $A$, let
$\textbf{L}^{{}^{\prime}}_{A}=(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{r}},l_{i_{r+1}},\ldots,l_{i_{t}})\in
F^{t}_{q}$, then the participants in $A$ can reconstruct the secret $s_{j}$ by
solving equation system $\textbf{A}(\textbf{L}^{{}^{\prime}}_{A})\
\textbf{a}^{T}=\textbf{y}^{T},$ where
$\textbf{a}=(s_{0},\ldots,s_{t-2},a_{t-1})\in F^{t}_{q},$
$\textbf{y}=(y_{i_{1}},\ldots,y_{i_{t}})$, and $\textbf{A}(\textbf{L}_{A})$,
$\textbf{A}(\textbf{L}^{{}^{\prime}}_{A})$ are all $t\times t$ matrix which
can be specifically written as $(l^{\nu}_{\mu})_{1\leq\mu\leq
n\atop{0\leq\nu\leq t-1}}.$
## 4 Correctness and security proof
In order to prove that our scheme is perfect, we need to introduce some
results on the generalized the Vandermonde determinants. As usual, for a
$k-$tuple of indeterminates $\textbf{x}=(x_{1},\ldots,x_{k})$ and a $k-$tuple
of increasing non-negative integers $\textbf{c}=(c_{1},\ldots,c_{k})$ we call
$V_{\textbf{c}}(\textbf{x})=\textrm{det}((x^{c_{\nu}}_{\mu})_{1\leq\nu,\mu\leq
k})$ generalized Vandermonde determinant. Write
$\textbf{e}_{k}=(0,\ldots,k-1).$ If $\textbf{c}=\textbf{e}_{k}$ then
$V_{\textbf{c}}(\textbf{x})$ equals the classical Vandermonde determinant
$V(\textbf{x})=\prod_{1\leq i<j\leq k}(x_{j}-x_{i})$.
###### Theorem 4.1
The scheme presented in Section 3 is a perfect multi-secret sharing scheme.
###### Proof 1
When $j=0$ due to the perfect property of the $(t,n)$ secret sharing schemes,
our scheme satisfies the two conditions of Definition 1.1. Now we consider
that $1\leq j\leq t-2.$
(1) If $|A|=t$, then by solving equation system $\textbf{A}(\textbf{L}_{A})\
\textbf{a}^{T}=\textbf{y}^{T},$ the participants in $A$ can obtain the unique
solution ${s_{j}}$ in terms of Cramer rule. If $|A|<t$ and
$\textbf{L}=(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{r}})\in F^{r}_{q}$ is a
$(t,j)$-privileged coalition of $A$, by Theorem 2.5,
$\tau_{\omega}(\textbf{L})=0,$ for all $\omega\in\\{r-j,\cdots,t-1-j\\}$. By
Lemma 2[20], $\tau_{\omega}(\textbf{L})=0$ for all
$\omega\in\\{r-j,\cdots,t-1-j\\}$ if and only if
$\quad\quad\quad\quad\quad\quad\tau_{t-1-j}(\textbf{L}\ ||\
\hat{\textbf{u}}_{m})=0\ \ \ for\ all\ m,\ 1\leq m\leq
t-r,\quad\quad\quad\quad(1^{{}^{\prime}})$
let $\textbf{u}=(u_{1},\ldots,u_{t-r})\in F_{q}^{t-r}$ be a track disjoint
with L, and $\hat{\textbf{u}}_{m}$ denotes the sequence obtained from u by
removing the term $u_{m}.$ Let
$y_{k}=\left\\{\begin{array}[]{ll}f(l_{k})&\textrm{ \
$k\in\\{1,\ldots,r\\}$}\\\ f(u_{k-r})&\textrm{ \ $k\in\\{r+1,\ldots,t\\}$}\\\
\end{array},\right.$
and let
$\textbf{L}^{{}^{\prime}}_{A}=(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{r}},u_{1},\ldots,u_{t-r})\in
F^{t}_{q}$, then $s_{j}$ can be obtained by solving equation system
$\textbf{A}(\textbf{L}^{{}^{\prime}}_{A})\ \textbf{a}^{T}=\textbf{y}^{T}$
$\displaystyle s_{j}$ $\displaystyle=$ $\displaystyle\frac{1}{V(\textbf{L}\
||\ \textbf{u})}(\sum^{r}_{k=1}(-1)^{k+j+1}V(\hat{\textbf{L}}_{k}\ ||\
\textbf{u})\tau_{t-1-j}(\hat{L}_{k}\ ||\ \textbf{u})y_{k}+$
$\displaystyle\sum^{t}_{k=r+1}(-1)^{k+j+1}V(\textbf{L}\ ||\
\hat{\textbf{u}}_{k-r})\tau_{t-1-j}(\textbf{L}\ ||\
\hat{\textbf{u}}_{k-r})y_{k}.\quad(2^{{}^{\prime}})$
By $(1^{{}^{\prime}})$ and $(2^{{}^{\prime}}),$ $(t,j)$-privileged coalition
of $A$ can compute the secret $s_{j}$. Hence, it holds that for all
$A\in\Gamma_{j},H(S_{j}\ |\ A)=0.$
(2) If $A\not\in\Gamma_{j}$, then $|A|<t$ and $\textbf{L}_{A}$ is not a
$(t,j)-$ privileged coalition. In view of $(1^{{}^{\prime}})$, there exists a
$m^{{}^{\prime}}$, where $1\leq m^{{}^{\prime}}\leq t-r$ such that
$\tau_{t-1-j}(\textbf{L}\ ||\ \hat{\textbf{u}}_{m^{{}^{\prime}}})\neq 0$.
Thus, the share $y_{m^{{}^{\prime}}+r}$ of $u_{m^{{}^{\prime}}}$ is needed. By
$(2^{{}^{\prime}}$) we can obtain that the participants in $A$ have no
information on $s_{j}$, even knowing some of the other secrets. Hence, it
holds that $H(S_{j}\ |\ AT)=H(S_{j}\ |\ T),$ where $T$ denotes the secrets
that $A$ can compute.
Therefore, according to Definition 1.1, the scheme is a perfect multi-secret
sharing scheme.
As a consequence, our scheme is an ideal and perfect linear multi-secret
sharing scheme. In 2004 and 2005, Yang et al.[13] and Pang et al.[14] proposed
multi-secret sharing schemes based on Shamir’s threshold secret sharing,
respectively, which are relatively efficient with lower cost of computing due
to the Lagrange interpolation operation that is employed in the process of
schemes construction. However, the existence of the privileged coalitions and
the Lagrange interpolation polynomial participants use will result in a fact
that the union of less than $t$ participants may compute the coefficients of
the polynomial corresponding to secrets by solving equation system, thereby
obtaining further information about the secret. Consequently, both of the two
schemes are not perfect, i.e., there is information leakages. Table 1 is for
the comparison among three schemes.
Likewise, the multi-secret sharing schemes[15-18] are not perfect, and thus
are not ideal.
Remark 2. The validity of the shares can be verified in a verifiable secret
sharing scheme, thus participants are not able to cheat. Based on our scheme,
we can further construct an ideal verifiable multi-secret sharing scheme by
adding the existing verifiability methods where the intractability of discrete
logarithm problem is frequently employed (see[15-18]).
Table 1 The comparison of performance among three schemes
Capability | Our scheme | Yang’s scheme. | Pang’s scheme
---|---|---|---
Multi-secret | Yes | Yes | Yes
Each participant holds only one share | Yes | Yes | Yes
Recover multi-secrets by Lagrange interpolating polynomials | No | Yes | Yes
Access structures corresponding to each secret is the same | No | Yes | Yes
Access structures possess more vivid authorized sets | Yes | No | No
The scheme is perfect | Yes | No | No
The scheme is ideal | Yes | No | No
## 5 Conclusions
In this paper, we consider an ideal multi-secret sharing scheme based on the
theories of minimal privileged coalitions and Shamir’s secret sharing, where
for a set of participants $P=\\{P_{1},\ldots,P_{n}\\}$, each subset of
$\Gamma_{j}$ carries different target secret $s_{j}$ for $0\leq j\leq t-2$. In
particular, in order to obtain privileged coalitions, we devise an algorithm
of privileged coalitions of any length if such coalitions exist. In real
terms, we can integrate data into a database which obtained from the
experiment for different value $t$ in accordance with our needs, and then we
can extract the results as identities of participants applied to the
construction of the scheme for practical applications.
## 6 Acknowledgements
This work was supported by the National Natural Science Foundation of China
(Grant No. 60873119).
## 7 References
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Table 2 The number of $(7,j)$-minimal privileged coalitions in the track
$(1,\ldots,13)$ over $F_{p}$ for $1\leq j\leq 5$
p | j=1 | j=2 | j=3 | j=4 | j=5 | p | j=1 | j=2 | j=3 | j=4 | j=5
---|---|---|---|---|---|---|---|---|---|---|---
13 | 72 | 114 | 71 | 93 | 132 | 137 | 15 | 10 | 11 | 18 | 0
17 | 101 | 69 | 76 | 72 | 98 | 139 | 18 | 12 | 10 | 13 | 0
19 | 90 | 53 | 72 | 57 | 92 | 149 | 13 | 13 | 8 | 8 | 0
23 | 72 | 61 | 58 | 63 | 83 | 151 | 12 | 9 | 13 | 13 | 0
29 | 60 | 43 | 51 | 46 | 26 | 157 | 14 | 10 | 13 | 10 | 0
31 | 58 | 51 | 40 | 39 | 32 | 163 | 13 | 16 | 9 | 7 | 0
37 | 51 | 49 | 38 | 27 | 76 | 167 | 13 | 12 | 9 | 11 | 0
41 | 44 | 36 | 38 | 27 | 94 | 173 | 8 | 13 | 11 | 10 | 0
43 | 41 | 40 | 35 | 34 | 94 | 179 | 7 | 11 | 9 | 10 | 0
47 | 41 | 32 | 36 | 39 | 76 | 181 | 7 | 8 | 12 | 9 | 0
53 | 26 | 35 | 35 | 27 | 31 | 191 | 5 | 4 | 11 | 10 | 0
59 | 32 | 25 | 28 | 24 | 5 | 193 | 12 | 8 | 7 | 8 | 0
61 | 27 | 32 | 31 | 28 | 2 | 197 | 10 | 10 | 11 | 9 | 0
67 | 21 | 32 | 25 | 22 | 0 | 199 | 7 | 13 | 9 | 8 | 0
71 | 26 | 22 | 22 | 23 | 0 | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
73 | 23 | 24 | 21 | 31 | 0 | 809 | 1 | 2 | 3 | 0 | 0
79 | 24 | 20 | 22 | 15 | 0 | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
83 | 23 | 21 | 24 | 25 | 0 | 5231 | 0 | 0 | 1 | 0 | 0
89 | 18 | 12 | 15 | 18 | 0 | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
97 | 18 | 24 | 15 | 17 | 0 | 31601 | 0 | 1 | 0 | 0 | 0
101 | 19 | 10 | 21 | 17 | 0 | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
103 | 18 | 20 | 14 | 20 | 0 | 199999 | 0 | 0 | 0 | 0 | 0
107 | 21 | 14 | 12 | 18 | 0 | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
109 | 18 | 18 | 9 | 16 | 0 | 499253 | 0 | 0 | 0 | 0 | 0
113 | 12 | 14 | 16 | 16 | 0 | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
127 | 12 | 12 | 16 | 19 | 0 | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
131 | 13 | 7 | 11 | 12 | 0 | $\geq 725597$ | 0 | 0 | 0 | 0 | 0
Table 3 The $(7,3)$-privileged coalitions with the shortest length in the
track $(1,\ldots,13)$ over $F_{p}$
p | Nr. | $(7,3)$-privileged coalitions with the shortest length
---|---|---
13 | 3 | {12, 8, 5, 1} {11, 10, 3, 2} {9, 7, 6, 4}
17 | 1 | {11, 10, 7, 6}
19 | 3 | {13, 5, 4, 3, 1} {13, 9, 8, 5, 2} {10, 8, 7, 6, 2}
23 | 2 | {10, 9, 8, 6, 1} {13, 11, 8, 7, 1}
29 | 3 | {11, 4, 3, 2, 1} {11, 9, 7, 5, 1} {12, 9, 6, 4, 3}
31 | 1 | {9, 8, 5, 3, 1}
37 | 2 | {12, 10, 8, 5, 1} {7, 6, 4, 3, 2}
41 | 1 | {11, 9, 7, 6, 4}
| | {10, 5, 4, 3, 2, 1}{12, 6, 5, 3, 2, 1} {11, 10, 9, 5, 3, 1} {13, 7, 5, 4, 2, 1}{12, 11, 6, 4, 2, 1}
| | {13, 11, 9, 5, 2, 1} {11, 10, 7, 6, 2, 1}{11, 8, 6, 3, 2, 1} {12, 10, 8, 6, 5, 4}{13, 12, 11, 6, 5, 4}
43 | 35 | {11, 9, 8, 5, 4, 1}{13, 9, 8, 6, 4, 1}{11, 10, 8, 7, 4, 1}{12, 11, 10, 9, 4, 1}{12, 11, 8, 6, 5, 1}
| | {13, 11, 10, 8, 6, 1}{13, 9, 5, 4, 3, 2}{12, 10, 8, 6, 3, 2} {10, 9, 8, 7, 3, 2}{13, 12, 10, 5, 4, 2}
| | {12, 10, 8, 7, 4, 2}{12, 9, 8, 6, 5, 2} {13, 10, 8, 6, 5, 2, }{11, 9, 8, 7, 6, 2,}{8, 7, 5, 4, 3, 1}
| | {12, 11, 10, 6, 5, 3}{13, 12, 11, 8, 6, 3}{11, 10, 9, 8, 7, 5}{11, 7, 6, 5, 4, 3}{12, 11, 8, 4, 3, 2}
| | {11, 9, 7, 4, 2, 1}{13, 10, 8, 7, 6, 4}{13, 11, 9, 7, 6, 1}{11, 10, 9, 8, 4, 3}{13, 11, 9, 6, 4, 2, }
47 | 1 | {13, 11, 9, 8, 2}
| | {11, 6, 5, 3, 2, 1} {12, 10, 7, 3, 2, 1}{7, 6, 5, 4, 2, 1}{12, 11, 8, 4, 2, 1}{12, 8, 6, 5, 2, 1}
| | {11, 9, 7, 5, 2, 1}{11, 10, 9, 6, 2, 1} {12, 11, 5, 4, 3, 1}{12, 9, 7, 5, 3, 1}{13, 10, 8, 6, 3, 1}
| | {13, 12, 11, 10, 3, 1}{9, 8, 7, 6, 4, 1}{13, 10, 9, 6, 4, 1}{13, 12, 10, 6, 5, 1} {13, 12, 9, 8, 6, 1}
53 | 35 | {9, 8, 6, 5, 3, 2}{13, 9, 8, 7, 3, 2}{13, 11, 8, 7, 4, 2} {13, 10, 8, 7, 5, 3}{13, 11, 8, 7, 6, 3}
| | {10, 8, 7, 5, 4, 2}{13, 12, 7, 5, 4, 2}{11, 10, 9, 5, 4, 2}{13, 11, 10, 9, 8, 2} {12, 11, 9, 7, 6, 2}
| | {11, 8, 6, 5, 4, 3}{12, 10, 9, 5, 4, 3}{11, 10, 8, 7, 4, 3}{11, 9, 7, 6, 5, 3}{13, 12, 11, 9, 2, 1}
| | {3, 9, 6, 5, 2, 1} {10, 9, 8, 7, 3, 1}{13, 12, 10, 4, 3, 2}{12, 11, 10, 7, 6, 3}{13, 12, 10, 8, 6, 2}
| | {10, 7, 6, 3, 2, 1}{11 8, 5, 4, 2, 1}{12, 10, 5, 4, 2, 1} {10, 8, 7, 4, 2, 1}{11, 7, 6, 5, 2, 1}
| | {12, 9, 8, 7, 6, 5} {12, 9, 8, 6, 3, 1}{12, 8, 7, 5, 4, 1}{13, 10, 9, 7, 4, 1}{13, 12, 8, 6, 5, 1}
59 | 28 | {11, 10, 7, 4, 3, 2}{13, 12, 11, 10, 6, 5} {13, 11, 8, 4, 3, 2}{13, 11, 10, 6, 3, 2}{13, 12, 10, 8, 3, 2}
| | {13, 11, 10, 8, 5, 2} {13, 12, 11, 9, 8, 2}{12, 9, 7, 5, 4, 3}{13, 12, 11, 7, 4, 3}{11, 10, 9, 8, 4, 3}
| | {12, 11, 10, 9, 8, 7}{11, 10, 8, 6, 5, 4}{13, 8, 7, 6, 3, 1}{10, 9, 8, 7, 5, 3}{13, 6, 5, 4, 3, 1}
| | {12, 10, 9, 5, 3, 2}{13, 10, 7, 6, 5, 2}{12, 11, 9, 8, 5, 1}
| | {10, 9, 5, 3, 2, 1}{11, 10, 6, 3, 2, 1}{12, 10, 7, 4, 2, 1}{12, 9, 6, 5, 2, 1}{13, 11, 7, 5, 2, 1}
| | {13, 8, 6, 4, 3, 1}{12, 11, 6, 5, 3, 1}{11, 9, 8, 5, 3, 1}{12, 11, 10, 5, 4, 1}{10, 8, 7, 6, 5, 1}
| | {13, 12, 11, 8, 7, 1}{12, 6, 5, 4, 3, 2}{10, 9, 6, 4, 3, 2}{12, 10, 8, 5, 3, 2}{13, 12, 10, 7, 6, 1}
61 | 31 | {13, 11, 10, 5, 3, 2}{13, 9, 7, 6, 3, 2}{12, 11, 8, 7, 3, 2}{13, 12, 9, 7, 4, 2}{10, 9, 7, 6, 5, 2}
| | {12, 10, 7, 5, 4, 3}{13, 11, 9, 6, 4, 3}{13, 12, 10, 8, 4, 3}{13, 11, 9, 4, 2, 1}{13, 12, 10, 9, 5, 1}
| | {11, 10, 7, 6, 5, 3} {13, 12, 8, 6, 2, 1}{13, 12, 11, 9, 6, 5} {13, 12, 8, 6, 5, 4}{13, 12, 11, 10, 9, 2}
| | {13, 12, 9, 7, 5, 3}
| | {12, 10, 6, 3, 2, 1}{10, 8, 7, 3, 2, 1}{8, 6, 5, 4, 2, 1}{11, 9, 8, 5, 2, 1}{12, 11, 9, 6, 2, 1}
| | {13, 12, 11, 6, 4, 1}{10, 8, 7, 6, 5, 1}{12, 9, 8, 7, 6, 1}{12, 11, 10, 8, 6, 1}{9, 7, 6, 5, 3, 2}
67 | 25 | {12, 9, 6, 5, 4, 2}{13, 9, 8, 6, 5, 2}{13, 9, 7, 5, 4, 3}{12, 11, 8, 5, 4, 3}{9, 8, 7, 6, 4, 3}
| | {12, 11, 9, 6, 5, 3}{13, 11, 10, 8, 7, 3}{13, 10, 8, 7, 5, 4}{13, 11, 9, 8, 5, 4}{11, 9, 7, 5, 4, 1}
| | {13, 12, 10, 9, 8, 6}{13, 12, 8, 5, 3, 1}{13, 12, 9, 5, 3, 2}{11, 10, 7, 6, 4, 3}{12, 11, 9, 8, 4, 1}
| | {13, 12, 4, 3, 2, 1}{13, 10, 9, 3, 2, 1}{11, 10, 6, 4, 2, 1}{7, 6, 5, 4, 3, 1}{13, 8, 5, 4, 3, 1}
| | {9, 8, 5, 4, 2, 1}{11, 9, 8, 4, 3, 1}{12, 11, 9, 7, 4, 1}{10, 9, 7, 6, 5, 1}{12, 11, 10, 9, 6, 1}
71 | 22 | {10, 9, 7, 4, 3, 2}{12, 10, 8, 5, 4, 2}{13, 12, 11, 6, 4, 2}{13, 11, 10, 7, 4, 2}{13, 12, 9, 8, 5, 2}
| | {10, 9, 8, 6, 4, 3}{12, 11, 8, 7, 4, 3}{13, 12, 10, 9, 7, 3}{13, 12, 11, 10, 9, 5}{13, 9, 7, 4, 3, 1}
| | {10, 6, 5, 4, 3, 2}{13, 9, 8, 7, 6, 2}
| | {13, 8, 4, 3, 2, 1}{10, 8, 5, 3, 2, 1}{12, 10, 9, 5, 2, 1}{12, 11, 9, 6, 2, 1}{10, 9, 7, 4, 3, 1}
| | {13, 10, 7, 5, 3, 1}{13, 9, 8, 7, 3, 1}{12, 11, 8, 7, 6, 1}{12, 11, 10, 8, 3, 2}{11, 9, 8, 5, 4, 2}
73 | 21 | {13, 11, 7, 6, 5, 2}{11, 10, 9, 6, 5, 2}{13, 12, 8, 7, 5, 2}{13, 11, 10, 9, 7, 2}{11, 10, 8, 6, 4, 3}
| | {12, 10, 9, 8, 6, 4}{13, 12, 9, 8, 5, 4}{12, 11, 10, 7, 5, 3}{13, 11, 10, 8, 4, 2}{13, 12, 9, 4, 3, 1}
| | {12, 8, 7, 6, 5, 3}
| | {9, 7, 6, 3, 2, 1}{13, 11, 10, 3, 2, 1}{13, 10, 5, 4, 2, 1}{13, 7, 6, 5, 2, 1}{10, 9, 6, 5, 2, 1}
| | {12, 11, 7, 5, 3, 1}{10, 9, 7, 5, 4, 1}{12, 11, 8, 5, 4, 1}{9, 6, 5, 4, 3, 2}{11, 9, 7, 5, 3, 2}
| | {13, 10, 8, 6, 5, 2}{11, 8, 6, 5, 4, 3}{13, 12, 10, 5, 4, 3}{13, 9, 8, 7, 4, 3}{12, 8, 7, 6, 5, 3}
79 | 22 | {13, 11, 9, 8, 5, 4}{12, 11, 10, 7, 6, 4}{13, 11, 10, 9, 6, 5}{13, 11, 10, 8, 7, 6}{13, 12, 10, 8, 2, 1}
| | {12, 8, 7, 6, 4, 2}{13, 12, 9, 7, 5, 3}
| | {12, 5, 4, 3, 2, 1}{13, 7, 4, 3, 2, 1}{8, 7, 6, 4, 2, 1}{12, 11, 7, 4, 2, 1}{13, 10, 5, 4, 3, 1}
| | {13, 10, 7, 6, 3, 1}{13, 12, 9, 8, 4, 1}{12, 9, 8, 6, 5, 1}{11, 10, 9, 8, 5, 1}{13, 11, 8, 7, 6, 1}
83 | 24 | {12, 9, 8, 6, 3, 2}{11, 10, 9, 6, 3, 2}{13, 9, 7, 6, 4, 2}{12, 10, 9, 6, 4, 2}{12, 11, 10, 8, 4, 2}
| | {13, 12, 10, 7, 5, 2}{9, 8, 7, 5, 4, 3}{13, 12, 10, 8, 4, 3}{12, 10, 9, 7, 5, 4}{11, 10, 9, 8, 7, 4}
| | {10, 8, 6, 5, 3, 1}{12, 11, 7, 5, 3, 2}{12, 11, 10, 6, 5, 2} {13, 12, 11, 10, 9, 7}
| | {12, 9, 5, 4, 2, 1}{9, 8, 6, 5, 2, 1}{13, 11, 5, 4, 3, 1}{12, 11, 8, 5, 3, 1}{11, 8, 6, 5, 4, 1}
---|---|---
89 | 15 | {13, 7, 5, 4, 3, 2}{13, 11, 10, 8, 7, 3} {13, 11, 10, 4, 3, 2}{12, 10, 9, 5, 3, 2}{11, 10, 9, 6, 5, 2}
| | {12, 8, 6, 5, 4, 3}{13, 12, 7, 6, 5, 3}{13, 12, 11, 10, 6, 3}{13, 12, 8, 5, 4, 1}{12, 11, 10, 7, 6, 2}
97 | 1 | {12, 11, 10, 5, 4}
| | {12, 10, 4, 3, 2, 1}{11, 10, 7, 3, 2, 1}{13, 9, 8, 3, 2, 1}{12, 10, 7, 5, 2, 1}{13, 10, 7, 6, 2, 1}
| | {13, 11, 9, 8, 6, 3} {13, 12, 11, 7, 2, 1}{10, 7, 5, 4, 3, 1}{13, 11, 8, 4, 3, 1}{9, 7, 6, 5, 4, 1}
101 | 21 | {11, 10, 9, 7, 6, 1}{13, 11, 7, 6, 5, 4} {12, 11, 8, 7, 3, 2}{13, 9, 6, 5, 4, 2}{13, 10, 9, 8, 6, 2}
| | {12, 11, 10, 8, 4, 3}{12, 10, 9, 6, 5, 3}{13, 11, 10, 9, 8, 5}{11, 9, 8, 7, 2, 1}{12, 10, 9, 6, 4, 1}
| | {13, 10, 8, 6, 4, 3}
| | {10, 7, 5, 4, 2, 1}{12, 10, 6, 4, 2, 1}{12, 11, 9, 4, 2, 1}{12, 11, 8, 6, 2, 1}{12, 7, 6, 4, 3, 1}
103 | 14 | {12, 11, 10, 7, 5, 4} {13, 10, 9, 5, 3, 2}{11, 10, 9, 5, 4, 2}{13, 12, 11, 9, 5, 2}{10, 9, 6, 5, 4, 3}
| | {12, 9, 7, 6, 5, 3}{13, 11, 10, 8, 7, 4}{9, 8, 5, 4, 3, 2}{13, 12, 10, 5, 4, 3}
| | {13, 12, 10, 3, 2, 1}{10, 8, 6, 4, 2, 1}{12, 9, 8, 4, 2, 1}{12, 11, 7, 6, 2, 1}{13, 12, 7, 5, 4, 1}
107 | 12 | {11, 10, 9, 8, 6, 1}{11, 9, 8, 5, 3, 2}{13, 10, 7, 6, 5, 2}{13, 12, 11, 9, 8, 3}{12, 10, 9, 7, 5, 4}
| | {13, 12, 11, 10, 4, 1}{13, 9, 8, 7, 6, 5}
| | {13, 12, 10, 3, 2, 1}{10, 8, 6, 4, 2, 1}{12, 9, 8, 4, 2, 1}{12, 11, 7, 6, 2, 1}{13, 12, 7, 5, 4, 1}
107 | 12 | {11, 10, 9, 8, 6, 1}{11, 9, 8, 5, 3, 2}{13, 10, 7, 6, 5, 2}{13, 12, 11, 9, 8, 3}{12, 10, 9, 7, 5, 4}
| | {13, 12, 11, 10, 4, 1}{13, 9, 8, 7, 6, 5}
| | {10, 9, 5, 4, 3, 1}{13, 12, 7, 4, 3, 1}{11, 10, 9, 7, 5, 1}{13, 11, 10, 8, 7, 1}{13, 12, 9, 7, 3, 2}
109 | 9 | {10, 8, 6, 5, 4, 3}{13, 12, 9, 8, 6, 4}{10, 9, 8, 7, 6, 5}{13, 7, 6, 5, 4, 3}
113 | 1 | {13, 12, 7, 5, 2}
| | {9, 6, 5, 3, 2, 1}{13, 11, 9, 3, 2, 1}{13, 12, 5, 4, 2, 1}{11, 10, 8, 5, 3, 1}{13, 11, 8, 7, 3, 1}
127 | 16 | {13, 10, 9, 8, 7, 1}{13, 12, 8, 4, 3, 2}{13, 9, 8, 5, 3, 2}{12, 11, 8, 5, 3, 2}{11, 10, 8, 5, 4, 2}
| | {10, 9, 8, 5, 4, 3}{13, 8, 7, 6, 5, 3}{13, 11, 10, 9, 5, 4}{13, 12, 10, 8, 6, 4}{9, 8, 7, 6, 5, 1}
| | {13, 12, 10, 9, 7, 2}
| | {11, 6, 5, 4, 2, 1}{13, 8, 6, 4, 2, 1}{12, 11, 9, 5, 2, 1}{9, 8, 6, 4, 3, 2}{13, 12, 11, 10, 8, 2}
131 | 11 | {10, 8, 7, 6, 5, 3} {13, 11, 10, 7, 5, 3}{12, 11, 10, 9, 5, 3}{13, 10, 8, 7, 5, 4}{13, 11, 10, 8, 6, 4}
| | {13, 12, 8, 7, 6, 5}
| | {13, 12, 10, 7, 2, 1}{13, 7, 6, 4, 3, 1}{11, 10, 8, 5, 4, 1}{13, 11, 10, 9, 4, 1{10, 8, 7, 5, 3, 2}
137 | 11 | {12, 11, 10, 9, 7, 2}{10, 9, 7, 6, 5, 3}{13, 11, 7, 6, 5, 3}{13, 9, 8, 7, 6, 4}{12, 11, 9, 8, 6, 5}
| | {11, 10, 9, 8, 5, 2}
| | {13, 12, 8, 3, 2, 1}{9, 7, 6, 4, 2, 1}{13, 7, 6, 5, 3, 1}{12, 10, 9, 7, 3, 1}{11, 10, 9, 6, 5, 1}
139 | 10 | {11, 6, 5, 4, 2}{10, 9, 8, 5, 4, 2}{13, 12, 9, 7, 4, 3}{13, 12, 11, 7, 5, 3}{8, 7, 6, 5, 4, 2}
149 | 1 | {13, 10, 6, 5, 1}
| | {10, 9, 8, 7, 2, 1}{11, 9, 8, 7, 3, 1}{13, 12, 10, 7, 6, 1}{12, 11, 9, 8, 6, 1}{11, 8, 6, 4, 3, 2}
151 | 13 | {11, 10, 7, 6, 4, 2} {12, 9, 7, 6, 5, 2}{12, 11, 9, 8, 7, 2}{12, 10, 7, 6, 4, 3}{12, 11, 10, 9, 8, 3}
| | {13, 12, 8, 7, 6, 4}{13, 10, 9, 8, 7, 5}{10, 9, 8, 4, 3, 2}
| | {12, 9, 6, 3, 2, 1}{11, 7, 5, 4, 2, 1}{13, 8, 5, 4, 2, 1}{10, 7, 6, 5, 3, 1}{13, 11, 9, 7, 3, 1}
157 | 13 | {13, 10, 7, 5, 4, 1}{12, 9, 8, 5, 3, 2}{12, 11, 8, 7, 5, 2}{13, 9, 8, 6, 4, 3}{12, 10, 9, 8, 6, 3}
| | {13, 10, 8, 6, 5, 4}{13, 12, 11, 10, 6, 3}{11, 10, 9, 8, 3, 1}
| | {11, 10, 7, 5, 3, 2}{13, 12, 11, 5, 3, 2}{9, 8, 6, 5, 4, 2}{11, 8, 7, 5, 4, 2}{13, 12, 7, 6, 4, 2}
163 | 9 | {11, 10, 9, 8, 6, 2}{11, 9, 7, 5, 4, 3}{13, 12, 11, 10, 9, 6} {13, 10, 8, 7, 5, 2}
| | {12, 9, 6, 4, 2, 1}{13, 11, 6, 5, 2, 1}{12, 9, 7, 4, 3, 1}{12, 11, 10, 7, 3, 1}{13, 12, 7, 6, 4, 1}
167 | 9 | {12, 11, 10, 9, 6, 2}{11, 9, 7, 5, 4, 3}{13, 12, 11, 10, 9, 6} {12, 10, 7, 5, 3, 2}
173 | 1 | {13, 12, 9, 7, 6}
| | {12, 11, 9, 3, 2, 1}{12, 10, 6, 5, 2, 1}{13, 8, 7, 5, 2, 1}{8, 7, 6, 5, 3, 1}{10, 9, 7, 6, 3, 1}
179 | 9 | {12, 8, 7, 5, 4, 3}{13, 11, 10, 7, 4, 3}{13, 11, 10, 7, 6, 5}{12, 11, 7, 6, 4, 1}
| | {8, 5, 4, 3, 2, 1}{13, 11, 7, 4, 2, 1}{9, 8, 7, 6, 2, 1}{13, 11, 10, 8, 3, 1}{13, 12, 10, 6, 4, 1}
181 | 12 | {11, 9, 6, 5, 3, 2}{12, 11, 10, 7, 3, 2}{13, 9, 8, 7, 5, 2}{13, 12, 10, 8, 7, 2}{13, 11, 10, 9, 7, 3}
| | {12, 10, 8, 7, 5, 4}{9, 7, 5, 4, 3, 2}
| | {10, 7, 5, 3, 2, 1}{11, 7, 6, 4, 2, 1}{13, 10, 7, 4, 3, 1}{13, 11, 7, 6, 3, 1}{13, 12, 11, 9, 4, 1}
191 | 11 | {12, 10, 9, 6, 3, 2}{13, 8, 7, 6, 5, 4}{11, 10, 9, 7, 5, 4}{13, 12, 10, 8, 5, 4}{12, 11, 10, 8, 6, 4}
| | {13, 12, 7, 6, 3, 2}
| | {12, 11, 10, 8, 2, 1}{11, 10, 8, 4, 3, 1}{13, 11, 5, 4, 3, 2}{10, 8, 7, 4, 3, 2}{11, 10, 8, 6, 3, 2}
193 | 7 | {12, 10, 9, 8, 7, 2}{13, 11, 8, 7, 6, 5}
| | {11, 9, 5, 3, 2, 1}{12, 9, 8, 6, 2, 1}{12, 11, 10, 7, 2, 1}{13, 9, 7, 5, 3, 1}{11, 8, 7, 6, 3, 1}
197 | 11 | {12, 8, 7, 5, 4, 2}{13, 9, 7, 6, 5, 2}{12, 11, 7, 5, 4, 3}{12, 11, 10, 6, 4, 3}{13, 11, 10, 8, 5, 4}
| | {13, 11, 8, 5, 3, 2}
| | {13, 9, 4, 3, 2, 1}{13, 11, 8, 5, 2, 1}{13, 10, 8, 6, 4, 1}{11, 10, 9, 8, 7, 1}{12, 8, 6, 5, 4, 2}
199 | 9 | {12, 10, 7, 6, 5, 2} {10, 9, 8, 7, 5, 2}{12, 11, 10, 6, 5, 3}
| | {12, 11, 10, 6, 4, 2}
⋮ | | ⋮
$\geq 22787$ | | no privileged coalitions (proven)
|
arxiv-papers
| 2012-12-25T01:56:29 |
2024-09-04T02:49:39.663324
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yun Song, Zhihui Li",
"submitter": "Yun Song",
"url": "https://arxiv.org/abs/1212.5983"
}
|
1212.6212
|
# Investigating polaron transitions with polar molecules
Felipe Herrera Department of Chemistry, University of British Columbia,
Vancouver, B.C., V6T 1Z1, Canada Department of Chemistry, Purdue University,
West Lafayette, IN 47907, USA Department of Chemistry and Chemical Biology,
Harvard University, 12 Oxford St., Cambridge, MA 02138, USA Kirk W. Madison
Department of Physics and Astronomy, University of British Columbia,
Vancouver, B.C., V6T 1Z1, Canada Roman V. Krems Department of Chemistry,
University of British Columbia, Vancouver, B.C., V6T 1Z1, Canada Mona Berciu
Department of Physics and Astronomy, University of British Columbia,
Vancouver, B.C., V6T 1Z1, Canada
###### Abstract
We determine the phase diagram of a polaron model with mixed breathing-mode
and Su-Schrieffer-Heeger couplings and show that it has two sharp transitions,
in contrast to pure models which exhibit one (for Su-Schrieffer-Heeger
coupling) or no (for breathing-mode coupling) transition. Our results indicate
that the physics of realistic mixed polaron models is much richer than that of
simplified models. We then show that ultracold molecules trapped in optical
lattices can be used to study precisely this mixed Hamiltonian, and that the
relative contributions of the two couplings can be tuned with external
electric fields. The parameters of current experimental set-ups place them in
the region where one of the transitions occurs. We propose a scheme to measure
the polaron dispersion using stimulated Raman spectroscopy.
###### pacs:
34.50.Cx, 67.85.-d, 37.10.Gh, 37.10.De, 34.20.Cf, 52.55.Jd, 52.55.Lf, 37.10.Jk
Introduction: Polarons, which are the low-energy dressed quasiparticles
appearing in the spectrum of particles coupled to bosonic fields, have been of
broad interest in physics ever since their first study by Landau Landau . The
generic Hamiltonian for a single polaron problem:
${\cal
H}=\sum_{k}\epsilon_{k}c_{k}^{\dagger}c_{k}+\sum_{q}\hbar\Omega_{q}b^{\dagger}_{q}b_{q}+\sum_{k,q}g_{k,q}c^{\dagger}_{k+q}c_{k}\left(b^{\dagger}_{-q}+b_{q}\right)$
(1)
includes the kinetic energy of the bare particle (first term), the energy of
the bosonic mode (second term) and their interaction (third term), where the
particle scatters by absorbing or emitting bosons.
There are two mechanisms for the particle-boson coupling, as the presence of
bosons can change (i) the potential or (ii) the kinetic energy of the bare
particle. The former always leads to a vertex $g_{q}$ independent of the
particle’s momentum $k$, while in the latter case the vertex $g_{k,q}$ depends
explicitly on both momenta. As an example, consider interactions between
electrons and phonons. Vibrations of nearby atoms modulate the potential
energy of an electron. Much studied examples of such type (i) interactions are
the Holstein Holstein:1959 and Fröhlich Frohlich models. At the same time,
by modulating the distance between lattice sites, lattice vibrations also
affect the hopping integrals. Such effects are described by type (ii) models
like the Su-Schrieffer-Heeger (SSH) model SSH:1988 .
Most of the early polaron studies focused on type (i) models, in particular on
the search for a self-trapping transition whereby at strong coupling the
bosons create a potential well so deep that it traps the polaron. In the
absence of impurities, however, this idea has proved to be wrong: type (i)
polarons have a finite effective mass for any finite coupling strength. The
absence of transitions for type (i) models has been demonstrated analytically
Gerlach:1991 . These models have a smooth crossover from light, highly mobile
polarons at weak coupling to heavy, small polarons at strong coupling.
This standard view of the polaron as a quasiparticle that becomes heavier with
increased coupling is now strongly challenged by results for type (ii) models.
Recent work has shown that in such models, the polaron can be lighter than the
bare particle, since the bosons affect the particle’s hopping so it may move
more easily in their presence Holger ; Berciu:2010 ; Marchand:2010 . The
boson-mediated dispersion can be quite different from that of the bare
particle, for example it may have a ground state with a different momentum
Berciu:2010 . If this happens, a sharp transition should occur when the boson-
mediated contribution to the dispersion becomes dominant. Indeed, such a
transition was recently predicted for the single SSH polaron Stojanovic:2008 ;
Marchand:2010 .
These results are interesting not just for challenging a long-established
paradigm, but also for raising the question of what happens in realistic
systems, where a mix of both types of coupling is generally expected. For
example, what is the fate of the transition in a mixed model if the coupling
is varied smoothly from $g_{k,q}$ to $g_{q}$? This also makes it highly
desirable to find systems described by such mixed Hamiltonians but where,
unlike in solid-state systems, the various parameters can be tuned
continuously so that the resulting polaron behaviour can be systematically
investigated.
In this Letter we elucidate the evolution of this transition as the coupling
interpolates between type (ii) SSH and type (i) breathing mode (BM)
Goodvin:2008 . Surprisingly, we find that the phase diagram has two sharp
transitions, and that these may occur even when the type (i) coupling is
dominant. This shows that polaron physics is much richer than generally
assumed. We then show that this mixed Hamiltonian describes polar molecules
trapped in an optical lattice, and moreover, that the parameters of current
experimental set-ups place them in the region where a transition is expected
to occur. Thus, experimental confirmation of these transitions is within
reach. Furthermore, we propose a detection scheme equivalent to Angle-Resolved
Photoemission Spectroscopy (ARPES) in solid-state systems Andrea , which
directly measures the polaron dispersion and can therefore pinpoint the
transition. Finally, as we discuss in conclusion, such experiments hold the
promise not only to clarify many other effects of interplay between type (i)
and type (ii) couplings on single polarons, but also to investigate the finite
concentration regime to look for quantum phase transitions.
Phase diagram: Consider the particle-boson coupling
$g_{k,q}=\frac{2i}{\sqrt{N}}\left\\{\alpha\left[\sin(k+q)-\sin(k)\right]+\beta\sin(q)\right\\},$
(2)
where $N$ is the number of lattice sites in a one-dimensional chain. The
$(k,q)$-dependent part, with energy scale $\alpha$, describes SSH coupling
Marchand:2010 while the $k$-independent part, with energy scale $\beta$,
describes BM type modulations of the site energy by bosons Goodvin:2008 . We
assume Einstein bosons $\Omega_{q}=\Omega$ and a free particle dispersion
$\epsilon_{k}=+2t\cos(k)$ with $t>0$ note2 . Following Ref. Marchand:2010 , we
define the effective SSH coupling strength
$\lambda=2\alpha^{2}/(t\hbar\Omega)$ and the adiabaticity ratio
$A=\hbar\Omega/t$. In addition, we introduce $R\equiv\beta/\alpha$ to
characterize the relative strength of the two types of coupling.
Figure 1: (color online). Phase diagram $\lambda$ vs. $A$ at fixed $R$.
Symbols show the location of the transition if $R=0$ (blue circles) and
$R=-0.5$ (red circles). The three shaded regions represent molecules with
large (a, LiCs), intermediate (b, RbCs) and small (c, KRb) dipoles, and
dressing schemes 1 ($R=-0.5$ for a) and 2 ($R=0$ for b and c), respectively.
For each region, the three lines correspond to lattice constants of 256 nm
(lower bound), 532 nm (dashed line) and 775 nm (upper bound, not shown for
region (c) because it is off-scale). For each curve $\hbar\Omega$ varies
between 1 and 100 kHz.
The SSH polaron ($R=0$) was predicted to undergo a sharp transition at a value
$\lambda^{*}$ Marchand:2010 . Its physical origin is simple to understand in
the anti-adiabatic limit $A\gg 1$ where the SSH coupling leads to an effective
next-nearest neighbor hopping $i\leftrightarrow i+2$ of the particle, by first
creating and then removing a boson at site $i+1$ Marchand:2010 . Its amplitude
is $t_{2}=-\alpha^{2}/(\hbar\Omega)=-\lambda t/2<0$, so its contribution
$-2t_{2}\cos(2k)$ to the total dispersion has a minimum at $\pi/2$, unlike the
bare dispersion which has a minimum at $\pi$. If $4|t_{2}|=t$, corresponding
to $\lambda^{*}={1\over 2}$ in the limit of $A\gg 1$, a sharp transition marks
the switch from a non-degenerate ground state with momentum $k_{gs}=\pi$ (for
$\lambda<\lambda^{*}$) to a doubly-degenerate one with
$|k_{gs}|\rightarrow{\pi\over 2}$ (for $\lambda>\lambda^{*}$). As $A$
decreases the number of phonons in the polaron cloud increases. This
renormalizes both hoppings $t\rightarrow t^{*}$, $t_{2}\rightarrow t_{2}^{*}$,
so $\lambda^{*}$ changes smoothly with $A$ as shown by the blue circles in
Fig. 1 (see also Fig. 4 of Ref. Marchand:2010 ). We also plot $\lambda^{*}$
for $R=-0.5$ (red circles), showing that the sharp polaron transition persists
for coexisting type (i) and type (ii) couplings. These results were generated
with the Momentum Average (MA) approximation, specifically its variational
flavor where the polaron cloud is allowed to extend over any three consecutive
sites Berciu:2010 ; Marchand:2010 . For $A\geq 0.3$, MA was shown to be very
accurate for both SSH and BM couplings Marchand:2010 ; Goodvin:2008 .
Figure 2: (color online). Phase diagram $\lambda$ vs. $R$ at fixed $A$,
showing two sharp transitions: one from a non-degenerate ground state with
$k_{\text{gs}}=\pi$ to a doubly-degenerate ground state with
$0<|k_{\text{gs}}|<\pi$, and the second back to a non-degenerate ground state
with $k_{\text{gs}}=0$. The results are for $A=\hbar\Omega/t\rightarrow\infty$
(red lines) and for $A=5$ (blue circles).
To understand the evolution of $\lambda^{*}$ with $R$, consider again the
limit $A\gg 1$. In addition to the second-nearest neighbor hopping $t_{2}$,
there is now also a dynamically generated nearest-neighbor hopping
$t_{1}=2\alpha\beta/(\hbar\Omega)=R\lambda t$. This describes processes where
the particle hops from site $i$ to $i+1$ leaving behind a boson at $i$ (SSH
coupling) followed by absorption of the boson while the particle stays at
$i+1$ (BM coupling); or vice versa, hence the factor of 2. The total nearest-
neighbor hopping is thus $t^{*}=t-t_{1}$, and the transition now occurs when
$4|t_{2}|=t^{*}\rightarrow\lambda^{*}=1/(2+R)$. Thus, for $R<0$, the
interference between the SSH and the BM couplings results in a larger
effective $t^{*}$ leading to a larger $\lambda^{*}$. In particular,
$\lambda^{*}\rightarrow\infty$ for $R\leq-2$, i.e. no transition occurs here.
The lack of a transition is not surprising when $R\rightarrow-\infty$, since
here the BM coupling is dominant and pure $g_{q}$ models do not have
transitions Gerlach:1991 . Our results show that for mixed SSH+BM coupling,
the switch from having to not having a transition occurs abruptly at $R=-2$ if
$A\gg 1$. This value must change continuously with $A$, therefore we expect
this switch to always occur at a finite $R$.
This is confirmed in Fig. 2, where we plot $\lambda^{*}$ vs. $R$ for
$A\rightarrow\infty$ and $A=5$. Surprisingly, we find not just the transition
at $\lambda^{*}\sim 1/(R+2)$, but also a second one which marks the crossing
to a ground state with $k_{gs}=0$. Its origin is also easy to understand in
the anti-adiabatic limit: if $R\lambda>1$, $t^{*}$ is negative and favors a
ground state at $k_{\text{gs}}=0$ instead of $k_{\text{gs}}=\pi$. For $A\gg 1$
this second transition is at $\lambda^{*}=1/(R-2)$ if $R>2$. At finite $A$, it
moves towards smaller $(R,\lambda)$ values, see Fig. 2.
Interestingly, this shows that for $R\rightarrow+\infty$ there are two nearby
transitions for the shift $k_{gs}=\pi$ to $k_{\text{gs}}=0$. This seems to
contradict the proof that a type-(i) Hamiltonian cannot have transitions
Gerlach:1991 , however, even for $R\rightarrow\infty$ this is a mixed
Hamiltonian if $\lambda\neq 0$. The transition is indeed absent if $\alpha=0$.
This is an example of the rare occurrence where a perturbatively small term
has a large effect on the behavior of the system.
Cold molecule implementation: Polar molecules in optical lattices can be used
to implement Hamiltonian (1) in a wide region of the parameter space.
Specifically, we consider molecules prepared in the ro-vibrational ground
state of the spinless electronic state ${}^{1}\Sigma$ and trapped on an
optical lattice in the Mott insulator phase, as recently demonstrated
experimentally Ospelkaus:2006 ; Chotia:2012 . We assume that there is at most
one molecule per lattice site.
The dipole-dipole interaction between molecules in different sites can be
modified by applying a DC electric field
$\mathbf{E}=E_{\text{DC}}\hat{\mathbf{z}}$ Micheli:2006 ; Micheli:2007 ;
Rabl:2007 ; Herrera:2010 ; Jesus:2010 ; Gorshkov:2011 . Here we consider two
schemes for dressing the rotational states of molecules with electric fields
that are relevant for polaron observation, scheme 1 involving a DC electric
field only, and scheme 2 involving combined optical and DC electric fields.
For the former, we define the two-state subspace
$|g\rangle=|\tilde{0},0\rangle$ and $|e\rangle=|\tilde{1},0\rangle$, where
$|\tilde{N},M_{N}\rangle$ denotes the field-dressed state that correlates
adiabatically with the field-free rotational state $|N,M_{N}\rangle$. $N$ is
the rotational angular momentum and $M_{N}$ is the projection of $N$ along the
electric field vector. In this basis we define the pseudospin operator
$\hat{c}^{\dagger}_{i}\equiv|e_{i}\rangle\langle g_{i}|$ that creates a
rotational excitation at site $i$. This excitation (the “bare particle”) can
be transferred between molecules in different lattice sites with an amplitude
$t_{ij}=\gamma\,U_{ij}\,(1-3\cos^{2}\Theta)$, where
$U_{ij}=d^{2}/|\mathbf{r}_{i}-\mathbf{r}_{j}|^{3}$, $d$ is the permanent
dipole moment, $\mathbf{r}_{i}$ is the position of molecule in site $i$,
$\Theta$ is the polar angle of the intermolecular separation vector,
$\gamma=\mu_{eg}^{2}/d^{2}\leq 1$ is the dimensionless transition dipole
moment that depends on the strength of the DC electric field. The excitation
hopping amplitude is finite even for vanishing field strengths. The field-
induced dipole-dipole interaction shifts the energy of the state
$|e_{i}\rangle$ by $D_{i}=\sum_{j}D_{ij}$. Here
$D_{ij}=-\kappa\,U_{ij}\,(1-3\cos^{2}\Theta)$, where
$\kappa=|\mu_{g}(\mu_{e}-\mu_{g})|/d^{2}$ and $\mu_{g}$($\mu_{e}$) is the
induced dipole of the ground(excited) state. Dipolar couplings outside this
two-level subspace are suppressed when the electric field separates state
$|e\rangle$ from other excited states.
The free quasiparticle dispersion is $\epsilon_{k}=\varepsilon_{0}+2t\cos(k)$
where the site energy is $\varepsilon_{0}=\hbar\omega_{eg}+D_{0}$, with the
single-molecule rotational excitation energy $\hbar\omega_{eg}\sim 10$ GHz and
$t\equiv t_{12}$. The center-of-mass vibration of molecules in the optical
lattice potential is coupled to their internal rotation through the dependence
of $U_{ij}$ on $\mathbf{r}_{i}-\mathbf{r}_{j}$. For harmonic vibrations with
linear coupling between internal and external degrees of freedom Rabl:2007 ;
Herrera:2011 , the boson term in Eq. (1) describes lattice phonons whose
spectrum depends on the trapping laser intensity and the DC electric field
Herrera:2011 . Here we consider weak DC fields and moderate trapping
frequencies which give a gapped and nearly dispersionless (Einstein) phonon
spectrum with frequency $\Omega$.
With these definitions, the system is described by SSH and BM-like couplings
with the energy scales $\alpha=-3(t_{12}/a_{L})\sqrt{\hbar/2m\Omega}$ and
$\beta=-3(D_{12}/a_{L})\sqrt{\hbar/2m\Omega}$, respectively, where $m$ is the
mass of the molecule and $a_{L}$ is the lattice constant. The ratio
$R=\beta/\alpha=\mu_{g}(\mu_{e}-\mu_{g})/\mu_{eg}^{2}$ is independent of the
intensity of the trapping laser or of the orientation of the array with
respect to the DC field. In the field dressing scheme 1 we have $|R|<1/2$ for
$dE_{\text{DC}}/B_{e}\leq 1$. In the combined AC-DC dressing scheme 2, the
same DC field strength and orientation is used as above, but an additional
two-color Raman coupling redefines the two-level subspace as
$|g\rangle=\sqrt{a}\;|\tilde{0},0\rangle+\sqrt{1-a}\;|\tilde{2},0\rangle$ and
$|e\rangle=\sqrt{b}\;|\tilde{1},0\rangle+\sqrt{1-b}\;|\tilde{3},0\rangle$
(details in the Supplementary Information). This dressing scheme can be used
to effectively enhance the hopping amplitude by a factor $f>1$ yielding
$\alpha\rightarrow f\alpha$, without changing the value of $\beta$, nor the
phonon dispersion. When using this dressing scheme, any point in the phase
diagram transforms as $\lambda\rightarrow f\lambda$ and $A\rightarrow A/f$,
thus shifting the system towards stronger SSH couplings.
The frequency of lattice phonons in a 1D array is
$\Omega=(2/\hbar)\sqrt{V_{0}E_{R}}$ where $V_{0}$ is the lattice depth and
$E_{R}=\hbar^{2}\pi^{2}/2ma_{L}^{2}$ is the recoil energy. The particle-boson
coupling can thus be written as $\lambda=18(E_{R}/t)(\pi A)^{-2}$. The shaded
regions in Fig. 1 show accessible points in the polaron phase diagram
$(\lambda,A)$ for LiCs, RbCs and KRb molecules, illustrating the flexibility
in varying the Hamiltonian parameters when using the two field dressing
scenarios and different experimental settings. Figure 1 shows that the
transition characterized by the shift from a non-degenerate ground state
$k_{\text{gs}}=\pi$ to a degenerate ground state $0<|k_{\text{gs}}|<\pi$ can
be studied using molecular species with moderate dipole moments such as RbCs,
in lattices with a site separation $a_{L}\approx 500$ nm. However, the
transition is easier to observe for molecules with large dipole moments such
as LiRb and LiCs. For weakly dipolar molecules such as KRb, the strong
coupling region can be achieved using $a_{L}<500$ nm.
The most direct way to detect the transition is to measure the polaron
dispersion. We propose the stimulated Raman spectroscopic scheme illustrated
in Fig. 3 to achieve this goal. We consider a one-dimensional array initially
prepared in the absolute ground state
$|g\rangle=|g_{1},\ldots,g_{N}\rangle|\\{0\\}\rangle$, where $|\\{0\\}\rangle$
is the phonon vacuum. We consider two linearly-polarized laser beams with
wavevectors arranged such that $\mathbf{k}_{1}-\mathbf{k}_{2}$ is parallel to
the molecular array. If the laser beams are far-detuned from any vibronic
resonance the effective light-matter interaction operator can be written as
$\hat{V}(t)=-g_{N}[\hat{c}^{\dagger}_{q}\text{e}^{-i\omega
t}+\hat{c}_{q}\text{e}^{i\omega t}]$, where $g_{N}$ is a size-dependent
coupling energy proportional to the amplitudes of both laser beams,
$q=|\mathbf{k}_{1}-\mathbf{k}_{2}|$ and $\omega=\omega_{1}-\omega_{2}$ are the
net momentum and energy transferred from the fields to the molecules. For
short interaction times (linear response), the system is excited from
$|g\rangle$ into the one-particle sector with a probability proportional to
the spectral function $\mathcal{A}(q,\omega)=-\text{Im}[G(q,\omega)]/\pi$
where $G(q,\omega)=\langle g|\hat{c}_{q}(\hbar\omega-{\cal
H}+i\eta)^{-1}\hat{c}^{\dagger}_{q}|g\rangle$ is the one-particle Green’s
function. As a result, for any $q$ the polaron energy $E_{q}$ equals the
energy $\hbar\omega$ of the lowest-energy peak in $\mathcal{A}(q,\omega)$, in
analogy with ARPES measurements Andrea .
To measure $\mathcal{A}(q,\omega)$, the stimulated Raman excitation rate can
be determined using state selective resonance enhanced multi-photon ionization
(REMPI), see Fig. 3(b). With some probability this converts the rotational
excitation (the “particle”) into a molecular ion which can be extracted from
the chain and detected by a multi-channel plate ion detector. Ionization and
subsequent detection efficiencies for a 2 step REMPI processes can easily
exceed 20% Stwalley:2011 , and a properly gated integrator can resolve the
arrival of a single molecular ion. Using a 3D lattice, a set of uncoupled
parallel 1D arrays can be realized and excited simultaneously, increasing the
signal to noise ratio of the detection step.
Figure 3: (Color online) (a) A two-photon stimulated Raman transition creates
a polaron state with a well defined momentum $q$ and energy
$\omega=\omega_{1}-\omega_{2}$. (b) The presence of the quasiparticle is
subsequently detected using resonantly-enhanced multi-photon ionization
(REMPI).
In summary, we presented the first (to our knowledge) phase diagram for a
mixed type polaron Hamitonian note3 , which showed that polaron physics is
much richer than previously thought and that sharp transitions may occur even
for dominantly type-(i) Hamiltonians ($R\gg 1$). We showed that polar
molecules trapped in optical lattices can be used to study this physics, and
proposed an ARPES-like detection scheme to directly measure the polaron
dispersion and thus identify the transitions expected to occur in such
systems.
Polarons with type (i) coupling have been observed for a single atomic
impurity immersed in a Fermi gas Schirotzek:2009 ; Nascimbene:2009 ; Will:2011
, and proposed for realization in lattice setups using atom-molecule systems
Ortner:2009 , self-assembled crystals in strong DC fields Rabl:2007 , and
recently trapped ions Stojanovic:2012 . Realization of type (ii) coupling has
been considered using Rydberg atoms Hague:2012 . Using these systems it would
be possible to explore the region of the phase diagram with $k_{\text{gs}}=0$
for $t<0$ (or equivalently $k_{\text{gs}}=\pi$ for $t>0$). Here we showed that
using trapped molecules in weak DC electric fields it is possible to explore
transitions into the phase with $0<|k_{\text{gs}}|<\pi$, where novel polaron
physics is expected to occur Marchand:2010 ; Stojanovic:2008 .
Many other aspects of single polaron physics can be investigated with trapped
polar molecules. Examples include investigating the effects of dispersive
phonons (most theoretical work assumes Einstein bosons), or novel effects
resulting from quadratic particle-boson coupling in the strong coupling
regime. Studying polaron phase diagrams in higher dimensions is easily
achieved with the same experimental scheme. Generalizations to studies of bi-
polarons are also of significant interest, to understand the pairing mechanism
for dominantly type-(ii) models (most bi-polaron studies are for type-(i)
Holstein and Fröhlich models). Finally, one may also be able to adapt the
polar molecules systems to study finite polaron concentrations and search for
quantum phase transitions QPT .
Acknowledgements: Work supported by NSERC and CIFAR. FH would also like to
thank NSF CCI center “Quantum Information for Quantum Chemistry (QIQC)”, Award
number CHE-1037992.
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## Supplementary Material
### Rotational state dressing schemes
Static field dressing.- This dressing scheme involves an homogeneous DC
electric field acting on a one-dimensional array of $\mathcal{N}$ polar
molecules. The field can have an arbitrary orientation with respect to the
array axis, described by the polar angle $\Theta$. We consider ${}^{1}\Sigma$
molecules each prepared in its lower vibrational and electronic state. The
rotational state of the molecules is described by the pendular state
$|\tilde{N},M\rangle$, which is an eigenstate of $\hat{N}_{z}$ and
$\hat{H}_{\text{R}}=B_{e}\hat{N}^{2}-dE_{\text{DC}}\cos\theta$, where
$\hat{N}$ is the rotational angular momentum operator, $B_{e}$ is the
rotational constant, $d$ is magnitude of the permanent dipole moment, and
$\theta$ is the polar angle of the internuclear axis Carrington:2003. We use
the two-level subspace $|g\rangle=|\tilde{0},0\rangle$ and
$|e\rangle=|\tilde{1},0\rangle$ to define the particle creation operator
$\hat{c}^{\dagger}_{{}_{i}}=|e_{i}\rangle\langle g_{i}|$. If the state
$|\tilde{1},0\rangle$ is energetically separated from the degenerate states
$|\tilde{1},\pm 1\rangle$ by an amount larger than the dipole-dipole
interaction energy between neighbouring molecules in an array, the two-level
approximation is valid and the dipole-dipole interaction operator reduces to
$\hat{V}_{\text{dd}}=|\mathbf{r}_{ij}|^{-3}\left(1-3\cos^{2}\Theta\right)\hat{d}_{0}(i)\hat{d}_{0}(j)$,
where $\mathbf{r}_{ij}$ is the intermolecular separation vector, $\Theta$ its
polar angle with respect to the dc field, and $\hat{d}_{p}$ is the
$p$-component of the space-fixed dipole operator in spherical coordinates.
This simplification of the dipole operator is accurate for electric field
strengths $E_{\text{DC}}>1$ V/cm, regardless of its orientation with respect
to the molecular array. The dipolar energies $t_{12}=\langle
e_{i}g_{j}|\hat{V}_{\text{dd}}|g_{i}e_{j}\rangle$ and $D_{12}=\langle
e_{1}g_{2}|\hat{V}_{\text{dd}}|e_{1}g_{2}\rangle-\langle
g_{1}g_{2}|\hat{V}_{\text{dd}}|g_{1}g_{2}\rangle$ are evaluated by
diagonalizing $\hat{H}_{\text{R}}$ numerically and using the eigenvectors to
compute the matrix elements of $\hat{V}_{\text{dd}}$ for a given value of the
field strength $E_{\text{DC}}$.
Combined static and infrared dressing.- This schme involves a DC electric
field as before and an additional infrared laser fields that induce Raman
couplings between rotational states of the same parity. The Raman coupling
scheme is illustrated in Fig. 1. It consists of two uncoupled lambda systems
$\Lambda_{1}=\\{|^{1}\Sigma,v=0\rangle|\tilde{0},0\rangle,|^{1}\Sigma,v=1\rangle|\tilde{1},0\rangle,|^{1}\Sigma,v=0\rangle|\tilde{2},0\rangle\\}$
and
$\Lambda_{2}=\\{|^{1}\Sigma,v=0\rangle|\tilde{1},0\rangle,|^{1}\Sigma,v=1\rangle|\tilde{2},0\rangle,|^{1}\Sigma,v=0\rangle|\tilde{3},0\rangle\\}$.
Here $|^{1}\Sigma,v\rangle$ denotes the electronic and vibrational quantum
numbers. Four linearly polarized infrared fields are needed to implement the
scheme. The corresponding transitions are far detuned by $\delta\omega>10^{2}$
MHz from other rotational lines in the presence of a DC field
$E_{\text{DC}}\sim 1$ kV/cm.
Let $\Omega=|\Omega|\text{e}^{i\phi}$ be the Rabi frequency associated with a
dipole-allowed transition. For each lambda system the adiabatic state with
zero eigenvalue (dark state) can be written as [2]
$|\lambda_{0}\rangle=\cos\alpha(t)|\epsilon_{1}\rangle-\sin\alpha(t)\exp(i\beta)|\epsilon_{3}\rangle,$
(3)
where the mixing angle $\alpha(t)$ is defined by
$\tan\alpha(t)=|\Omega_{P}(t)/\Omega_{S}(t)|$, and the relative phase of the
fields is $\beta=\phi_{P}-\phi_{S}$. The RWA holds when
$\text{max}\left\\{\Delta,\Omega_{P},\Omega_{S}\right\\}\ll B_{e}$. The two
lambda systems in Fig. 1 are uncoupled due to the anharmonicity of the
rotational spectrum for small detunings $\Delta_{1}$ and $\Delta_{2}$.
Selection rules restrict the coupling between rovibrational states with
different angular momentum projection $M$.
Figure 4: (a) Infrared dressing of rotational states in the presence of a dc
electric field. Two independent lambda systems are used to construct the dark
state superpositions
$|g\rangle=\sqrt{a}|\tilde{0},0\rangle+\sqrt{1-a}|\tilde{2},0\rangle$ and
$|e\rangle=\sqrt{b}|\tilde{1},0\rangle+\sqrt{1-b}|\tilde{3},0\rangle$. (b)
Dipole-dipole energy ratio $R=D_{12}/t_{12}$ as a function of the DC fields
strength $E_{\text{DC}}$ for DC field rotational dressing ($a=b=1$) and
combined AC-DC dressing with $a=b=1/2$.
We want to use the dark states of the lambda systems $\Lambda_{1}$ and
$\Lambda_{2}$ to define the two-level subspace $|g\rangle$ and $|e\rangle$,
respectively. It is in this field-dressed space where excitation hopping
processes take place in a molecular array. Let the molecules be initially
prepared in the rovibrational ground state
$|g\rangle=|^{1}\Sigma,v=0\rangle|\tilde{0},0\rangle$. We assume that only the
Stokes fields $\Omega_{S_{1}}(t)$ and $\Omega_{S_{2}}(t)$ are present. In this
case the mixing angles $\alpha_{1}(t)$ and $\alpha_{2}(t)$ vanish and the dark
states are the same as the DC dressing scheme
$|g\rangle=|^{1}\Sigma,v=0\rangle|\tilde{0},0\rangle$ and
$|e\rangle=|^{1}\Sigma,v=0\rangle|\tilde{1},0\rangle$. By slowly turning on
the pump fields $\Omega_{P_{1}}(t)$ and $\Omega_{P_{2}}(t)$ the dark states
become
$\displaystyle|g\rangle$ $\displaystyle=$
$\displaystyle\cos\alpha_{1}(t)|^{1}\Sigma,v=0\rangle|\tilde{0},0\rangle+\sin\alpha_{1}(t)|^{1}\Sigma,v=0\rangle|\tilde{2},0\rangle,$
$\displaystyle|e\rangle$ $\displaystyle=$
$\displaystyle\cos\alpha_{2}(t)|^{1}\Sigma,v=0\rangle|\tilde{1},0\rangle+\sin\alpha_{2}(t)|^{1}\Sigma,v=0\rangle|\tilde{3},0\rangle.$
The Raman lasers have the relative phases $\beta_{1}=\beta_{2}=\pi$, which can
be achieved experimentally. The states $|g\rangle$ and $|e\rangle$ can be
manipulated independently by adiabatically tuning the pump laser intensities
relative to the Stokes beams. We consider the specific situation where
$\alpha_{1}=\alpha_{2}=\pi/2$. Omitting the labels $v$ and $M$, the excitation
hopping amplitude $t_{ij}$ is then given by
$\displaystyle t_{ij}$ $\displaystyle\approx$
$\displaystyle\frac{1-3\cos^{2}\Theta}{|\mathbf{r}_{ij}|^{3}}\left(\frac{1}{4}\right)\left\\{d_{10}^{2}+d_{21}^{2}+d_{32}^{2}+2d_{21}d_{10}+2d_{32}d_{10}+2d_{21}d_{32}\right\\}$
(4)
where
$d_{NN^{\prime}}=\langle\tilde{N}|\hat{d}_{0}|\tilde{N}^{\prime}\rangle=d_{N^{\prime}N}$.
We have ignored couplings with $\Delta\tilde{N}>1$, which are suppressed in
the presence of weak dc electric fields $E_{\text{DC}}\leq B_{e}/d$. Equation
4 becomes exact in the limit $E_{\text{DC}}\rightarrow 0$ where the transition
dipole moments can be evaluated analytically using angular momentum algebra
[1].The transition dipole moments in Eq. (4) have the same sign and therefore
contribute to the enhancement of the hopping amplitude
$t_{12}^{\prime}=f\,t_{12}$ with respect to the pure dc dressing scheme. The
diagonal energy $D_{12}$ has a similar behaviour as in the type-I coupling
only for weak dc electric fields $dE_{\text{DC}}/B_{e}\ll 1$. For non-
perturbative fields, contributions from higher rotational states
$\tilde{N}\geq 2$ modify significantly its static field dependence. In Fig.
1(b) we show the achievable values of the ratio $R=D_{12}/t_{12}$ using the
combined AC-DC dressing scheme described above, as function of the DC field
strength. The hopping amplitude is enhanced by a factor $f\approx 2$ with
respect to the static scheme. The ratio $R$ has the same sign as the static
dressing case ($R<0$) for weak fields $E_{\text{DC}}\leq 2B_{e}/d$, but is up
to an order of magnitude smaller. The ratio changes sign at
$E_{\text{DC}}\approx 2.8B_{e}/d$ and has a local maximum
$R_{\text{max}}\approx 0.32$ at $E_{\text{DC}}=7B_{e}/d$.
Phonon dispersion.- In this work the dispersion of the phonon spectrum is
ignored due to the current limitations of the Momentum Average (MA)
approximation we use in the main text to solve the polaron problem [3]. For a
1D array the phonon frequency can be written as
$\omega_{q}=\omega_{0}\sqrt{1+12\rho\gamma(q)}$, where
$\rho=V^{gg}_{12}/V_{0}$ is the ratio between the ground state dipolar energy
$V_{12}^{gg}=\langle g_{1}g_{2}|\hat{V}_{\text{dd}}|g_{1}g_{2}\rangle$ and the
optical lattice depth $V_{0}$, and $\gamma(q)=2\sin^{2}(q)$ [4]. The
dispersion is determined by the parameter $\rho$, which can be kept small for
shallow lattices $V_{0}\approx 18-20E_{R}$ in the static dressing scheme only
for weak dc fields $E_{\text{DC}}\leq B_{e}/d$. For higher fields we have
$|V_{12}|^{gg}>|t_{12}|$, thus the phonon dispersion becomes an important
energy scale. For combined AC-DC dressing, the phonon dispersion can be safely
ignored since $|V^{gg}_{12}|\ll|t_{12}|$ over the entire range of electric
fields in Fig. 1.
1. (1)
J. Brown and A. Carrington, Rotational Spectroscopy of Diatomic Molecules,
(Cambridge University Press, 2003).
2. (2)
K. Bergmann, H. Theuer and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998).
3. (3)
G. L. Goodvin and M. Berciu, Phys. Rev. B 23, 235120 (2008).
4. (4)
F. Herrera and R. V. Krems, Phys Rev. A. 84, 051401(R) (2011).
|
arxiv-papers
| 2012-12-26T16:58:47 |
2024-09-04T02:49:39.675931
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Felipe Herrera, Kirk W. Madison, Roman V. Krems and Mona Berciu",
"submitter": "Felipe Herrera",
"url": "https://arxiv.org/abs/1212.6212"
}
|
1212.6341
|
# $D^{0}$ MIXING AND CP VIOLATION IN $D$ DECAYS
Érica Polycarpo Universidade Federal do Rio de Janeiro
CP 68528, 20511-010, Rio de Janeiro, RJ, Brasil
E-mail: [email protected]
###### Abstract
We present a brief review of CPV and mixing measurements in the charm sector,
with emphasys in results published or presented since the previous edition of
the Physics in Collision Symposia.
## 1 Introduction
Neutral mesons mixing occur in nature due to the fact that the mass
eigenstates are not weak interaction eigenstates. An initially pure beam of
$D^{0}$ or $\bar{D}^{0}$ mesons evolves in time to a mixture of $D^{0}$ and
$\bar{D}^{0}$ states with rates determined by the solution of the Schröedinger
equation
$i\frac{d}{dt}\begin{pmatrix}D^{0}\\\ \bar{D}^{0}\\\ \end{pmatrix}(t)={\bf
H_{eff}}\begin{pmatrix}D^{0}\\\ \bar{D}^{0}\\\ \end{pmatrix}(t),$ (1)
where the diagonal elements of the effective Hamiltonian $\bf H_{eff}\equiv
M-\frac{i}{2}\Gamma$ and of the hermitian matrices $M$ and $\Gamma$ are
related to the flavour conserving $D^{0}(\bar{D}^{0})\rightarrow
D^{0}(\bar{D}^{0})$ transitions, while the non-diagonal ones are related to
the flavour changing $D^{0}\leftrightarrow\bar{D}^{0}$ transitions. The
solution of Eq.1 are the eigenstates $|D_{1,2}\rangle=p|D^{0}\rangle\pm
q|\bar{D}^{0}\rangle$, with complex coefficients $p$ e $q$ satisfying
$\sqrt{|p|^{2}+|q|^{2}}=1$. The eigenstates have well defined masses and
widths $m_{1},\Gamma_{1}$ and $m_{2},\Gamma_{2}$. The parameters which govern
the mixing are $x\equiv\frac{m1-m2}{\Gamma}$ and
$y\equiv\frac{\Gamma_{1}-\Gamma_{2}}{2\Gamma}$, where
$\Gamma\equiv\frac{\Gamma_{1}+\Gamma_{2}}{2}$ is the mean decay width. For
charm mesons, both $x$ and $y$ are small. There are essentially two approaches
used to predict their values in the Standard Model (SM), none of them leading
to fully reliable results (see, for example, [1, 2, 3] and references
therein). The transitions leading to the $D^{0}-\bar{D}^{0}$ oscillations are
those represented in Fig. 1. Short distance contributions are illustrated by
the box diagram on the left and long distance transitions, involving
intermediate states accessible by both $D^{0}$ and $\bar{D}^{0}$, on the
right. As seen from these diagrams, charm mixing gives us access to
transitions involving intermediate states with down-type quarks. In fact, this
is the only system which offers this probe, since the top quark is too heavy
to hadronize and the $\pi^{0}$ is its own anti-particle.
While $x$ receives contributions from both the short distance and long
distance processes, $y$ only receives contributions from the long distance
transitions. Since $x$ is the only one to receive contributions from virtual
intermediate states, new physics introducing off-shell particles in the box
diagrams would only affect $x$ and therefore it is a sort of consensus that an
unambiguous sign of NP would be given by a $x$ value well above $y$. The first
evidences of mixing were obtained in 2007 by BaBar [4], Belle [5] and CDF [6].
At present, both $x$ and $y$ are constrained by several measurements at the
values $x=(0.65^{+.018}_{-0.19})$% and $y=(0.73\pm 0.12)$% and the no mixing
hypothesis is excluded with a statistical significance above 10 standard
deviations ($\sigma$) [7], under the assumption of no CP violation. Although
theoretical predictions are not precise, the experimental values of the mixing
parameters are now consistent with SM expectations [8], even if still
considered at the upper level of possible values [9].
|
---|---
Figure 1: Short and long distance contributions to $D^{0}-\bar{D}^{0}$ mixing.
The CP symmetry can be violated in the mixing, if $|q|\neq|p|$; in the decay,
if the magnitudes of the instantaneous decay amplitudes ${\cal A}_{f}=\langle
f|{\cal H}|D^{0}\rangle$ and $\bar{\cal A}_{\bar{f}}=\langle\bar{f}|{\cal
H}|\bar{D}^{0}\rangle$ for CP conjugate processes are not the same; and, for
decays reachable by both the $D^{0}$ and $\bar{D}^{0}$, in the interference
between mixing and decay. This kind of mixing induced CPV can occur even in
the absence of CPV in the mixing and in the decay, if there is a non null
phase difference $\phi$ between the mixing and the decay amplitudes.
In the SM, CP violation (CPV) is suppressed by the small values of the CKM
parameters $V_{ub}$ and $V_{cb}$. In the mixing, it is expected to be at the
level of 10-4 [10], independently of the final state. Direct CPV can only
occur in singly Cabibbo suppressed (SCS) decays, since their amplitudes can
receive contributions from processes with virtual b-quarks, involving the
complex elements of the CKM matrix. Existing predictions for SCS modes in the
SM were, at least until recently, of the order of $10^{-3}$ or less. The
interest in CPV in charm decays is, therefore, related to searches for new
physics effects in a SM suppressed environment. Experimentally, the only
existing evidence of CPV in the charm sector is the measurement of the
difference between the time integrated asymmetries of the decays $D^{0}\to
K^{+}K^{-}$ and $D^{0}\to\pi^{+}\pi^{-}$ [11], that will be further discussed
in this document.
A very good review on CPV and mixing on the charm sector is given in [3]. Here
we report on some recent measurements performed by BaBar, Belle, CDF and LHCb
since the previous edition of the Physics in Collision conference series.
## 2 Time dependent measurements of $D^{0}-\bar{D}^{0}$ mixing and CP
violation
There are several ways to access the mixing parameters $x$ and $y$
experimentally, using decays to various final states. Each method provide
different sensitivities to these parameters and, by combining these
measurements, we can improve our knowledge about them. The HFAG group [12]
provides world averages and best fit values to these parameters using results
provided by several experiments at different machines around the world. A good
overview of the most up to date averages are given in [7].
One of the most sensitive ways of determining $x$ and $y$ is via a time
dependent amplitude analysis of the decays $D^{0}\to K_{s}\pi^{+}\pi^{-}$ and
$D^{0}\to K_{s}K^{+}K^{-}$. The two final states are CP eigenstates reachable
by both the $D^{0}$ and $\bar{D}^{0}$. The time dependent rate for these
decays is proportional to terms arising from the interference between the
direct decay and the decay via mixing:
${\mathcal{R}}(t)\propto{|A_{1}|^{2}e^{-yt}+|A_{2}|^{2}e^{-yt}+2Re[A_{1}A^{*}_{2}]\cos(xt)+2Im[A_{1}A^{*}_{2}]\sin(xt)}$
(2)
where $A_{1,2}=\frac{{\cal A}\pm\bar{\cal A}}{2}$ and ${\cal A}$ and
$\bar{\cal A}$ are the instantaneous decay amplitudes for $D^{0}$ and
$\bar{D}^{0}$, written as a function of the 2-body invariant masses $M^{+}$
($K^{0}_{S}K^{+}$) and $M^{-}$ ($K^{0}_{s}K^{-}$). In order to determine the
flavour of the meson at production, it is required to form a vertex with a low
momentum pion $\pi_{s}$, consistent with the decay vertex of a
$D^{*+}\to\pi^{+}_{s}D^{0}$ or $D^{*-}\to\pi^{-}_{s}\bar{D}^{0}$. The charge
of the $\pi_{s}$ identifies the $D^{0}$ flavour. This analysis was first
performed by CLEO [13] and subsequently by Belle [14] and BaBar [15], with
much larger samples. An update of the $D^{0}\to K_{s}\pi^{+}\pi^{-}$ analysis
by Belle was presented this summer, using their full data set of 920 fb-1 and
an improved tracking, which provides a signal yield of 1.23 M events with a
purity of 95.6% [16]. They parametrize the amplitudes ${\cal A}$ and
$\bar{\cal A}$ as a sum of 12 quasi-two-body amplitudes describing the P and D
waves and two components to describe the $K\pi$ (LASS model) and the $\pi\pi$
(K-matrix model) s-waves. From a maximum likelihood fit to the data in the
signal region, the mixing parameters $x$ and $y$, the average $D^{0}$-meson
lifetime $\tau_{D^{0}}$ and the magnitudes and phases of the resonances are
directly extracted. The preliminary results shown are $x=(0.56\pm
0.19^{+0.03+0.06}_{0.09-0.09})$% and $y=(0.30\pm
0.15^{+0.04+0.03}_{-0.05-0.06})$%, where the first uncertainty is statistical,
the second is due to experimental systematic effects and the third
systematical due to the model used to fit the Dalitz plot. At the time of the
conference, this was the most precise single measurement of the mixing
parameters.
Another measurement which is sensitive to mixing is the deviation from unity
of the ratio of the effective lifetimes measured in decays to the CP
eigenstates $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ with respect to the CP mixed
state $K^{-}\pi^{+}$. Since mixing is very slow, the decay time distributions
are, to a good approximation, simply exponential. The fits to the decay time
distributions of the different samples use a model where signal events are
described by a single exponential and provide the effective lifetimes used to
build the observable
$y_{CP}\equiv\frac{\tau(D^{0}\to h^{-}h^{+})}{\tau(D^{0}\to
K^{-}\pi^{+})}-1\approx\frac{1}{2}\left[\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}+\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)y\cos\phi-\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}-\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)x\sin\phi\right]$
(3)
where $h^{-}h^{+}$ corresponds to $\pi^{-}\pi^{+}$ or $K^{-}K^{+}$, the phase
$\phi$ is considered universal and a contribution from CPV in the decay is
neglected. In the absence of CPV, $|q/p|=1$ and $\phi=0$, hence $y_{CP}=y$.
This measurement was previously performed by several collaborations and
provided the first evidence for mixing obtained by the BELLE experiment in
2007 [5]. In April 2011, both BaBar and Belle presented new results using
their full data samples, with integrated luminosities of 468 fb-1 [17] and 976
fb-1 [18], respectively. LHCb also published a first result for $y_{CP}$ using
29 pb-1 of data collected in 2010 [19]. In addition to the $D^{*\pm}$ tagged
samples, BaBar used also untagged samples of $D^{0}(\bar{D}^{0})\to
K^{\pm}\pi^{\mp}$ and $D^{0}(\bar{D}^{0})\to K^{-}K^{+}$ decays. These results
are presented in Table 1, together with the size of the samples used in each
measurement.
The effective lifetimes obtained by fitting the decay time distributions of
the tagged samples separately for the CP conjugate decays provide a second
quantity, $A_{\Gamma}$, sensitive to CPV:
$\small A_{\Gamma}(h^{-}h^{+})\equiv\frac{\tau(\bar{D}^{0}\to
h^{-}h^{+})-\tau(D^{0}\to h^{-}h^{+})}{\tau(\bar{D}^{0}\to
h^{-}h^{+})+\tau(D^{0}\to
h^{-}h^{+})}\approx\frac{1}{2}\left[\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}-\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)y\cos\phi-\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}+\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)x\sin\phi\right],$
(4)
considering as valid the same approximations used for $y_{CP}$.
The three results for $y_{CP}$ and $A_{\Gamma}$ are consistent with each other
and with previous measurements and also compatible with the hypothesis of no
CPV. They bring down the world average for $y_{CP}$ from $(1.064\pm 0.209)$%
to $(0.866\pm 0.155)$% [7], relaxing the previously existing tendency for
$y_{CP}>y$ [3]. The new world average for $A_{\Gamma}$ is $(-0.022\pm 0.161)$%
[7].
Table 1: Recent experimental results for the quantities $y_{CP}$ and $A_{\Gamma}$. The $h^{-}h^{+}$ yield is given for $K^{-}K^{+}$ and inside the parentheses for $\pi^{-}\pi^{+}$. First uncertainty is statistical, second is systematical. BaBar measures, instead of $A_{\Gamma}$, $\Delta Y=(1+y_{CP})A_{\Gamma}$. Experiment | $K^{-}\pi^{+}$ | $h^{-}h^{+}$ | $y_{CP}$ (%) | $A_{\Gamma}$ ($\Delta Y$) (%)
---|---|---|---|---
| yield ($\times 10^{6}$) | yield ($\times 10^{3}$) | |
LHCb | 0.225 | 30 | $5.5\pm 6.3\pm 4.1$ | $-5.9\pm 5.9\pm 2.1$
BaBar | 7.312 | 633 (65) | $0.72\pm 0.18\pm 0.12$ | $0.09\pm 0.26\pm 0.09$
Belle | 2.61 | 242 (114) | $1.11\pm 0.22\pm 0.11$ | $-0.03\pm 0.20\pm 0.08$
## 3 Time integrated CP asymmetries
### 3.1 Two-body decays
One of the most discussed results from the LHC in the past year was the first
evidence of CPV in the charm sector, provided by the difference of the time
integrated CP asymmetries measured in the decays of the $D^{0}$ meson to
$K^{-}K^{+}$ and $\pi^{-}\pi^{+}$, presented by LHCb [11] and supported by new
results by CDF [20] and Belle [21]. To first order of approximation, the raw
asymmetry can be written as a sum of small terms
$A_{raw}(f)=A_{CP}(f)+A_{det}(f)+A_{det}(\pi_{s})+A_{prod}(D^{*+}),$ (5)
where $A_{CP}(f)$ is the physical CP asymmetry of interest, $A_{det}(f)$ is
the asymmetry in the detection efficiency for the decay to final state $f$,
$A_{det}(\pi_{s})$ is the reconstruction efficiency for the slow pion from the
$D^{*+}\to\pi^{+}_{s}D^{0}$ tagging chain and $A_{prod}(D^{*+})$ is a physical
asymmetry for $D^{*+}$ production. This asymmetry vanishes in CP conserving
strong collisions, as the $p\bar{p}$ collisions at the Tevatron, if the events
are reconstructed symmetrically in the pseudo-rapidity $\eta$. However, it is
not expected to be null in the $pp$ collisions at the LHC. In $e^{+}e^{-}$
colliders, there is a forward-backward asymmetry in the production of charm
mesons arising from $\gamma Z^{0}$ interference and higher order QED effects
[22].
When dealing with two-body decays of a spin 0 particle to self-conjugate final
states $f$, the detection efficiency $A_{det}(f)$ is null. Since the remaining
terms, apart from $A_{CP}(f)$, are independent of the final state, they cancel
out in the difference of asymmetries $\Delta
A_{raw}=A_{raw}(K^{+}K^{-})-A_{raw}(\pi^{+}\pi^{-})$, which thus becomes a
robust measurement of the difference between the physical CP asymmetries:
$\Delta A_{CP}\equiv A_{CP}(K^{+}K^{-})-A_{CP}(\pi^{+}\pi^{-})=\Delta
A_{raw}.$ (6)
Experimentally, the raw asymmetries are obtained from
$A_{raw}(f)\equiv\frac{N(D^{0}\to f)-N(\bar{D}^{0}\to f)}{N(D^{0}\to
f)+N(\bar{D}^{0}\to f)}$ (7)
where $N(X)$ corresponds to the number of reconstructed decays $X$ after
background subtraction.
LHCb performed this measurement using 0.62 fb-1 of data collected in 2011.
Maximum likelihood fits to the $\delta m=m(\pi_{s}h^{-}h^{+})-m(h^{-}h^{+})$
spectra provide the number of signal events used to calculate the raw
asymmetries in different kinematic bins, in order to guarantee that the
detection and production asymmetries are cancelled. The kinematic bins are
defined in intervals of the $D^{*\pm}$ candidate transverse momentum $p_{T}$
and $\eta$ and the final result, shown in Table 2, is obtained from a weighted
average over this kinematic space. This result deviates from zero with a
significance of 3.5 $\sigma$.
CDF also performed a new $\Delta A_{CP}$ measurement, using the full data
sample collected at Tevatron Run II, which corresponds to 9.7 fb-1 [20].
Compared to the previously published measurement [23], their new result relies
on a larger data sample and an optimized selection procedure. In order to take
into account differences in the instrumental asymmetries due to phase space
variations among the two final states, a reweighting of the samples is
performed, so as to equalize the $\pi_{s}$ impact parameter, $p_{T}$ and
$\eta$ distributions. A simultaneous $\chi^{2}$ fit of the resulting
$D^{0}\pi^{+}$ and $\bar{D}^{0}\pi^{-}$ mass distributions provides the number
of signal events used in the raw asymmetries. The result, also shown in Table
2, is 2.7$\sigma$ away from zero, with a precision level very close to the
obtained by LHCb. The individual asymmetries, measured with a sample of 6.0
fb-1, are $A_{CP}(K^{-}K^{+})=-0.24\pm 0.22\pm 0.09$% and
$A_{CP}(\pi^{-}\pi^{+})=(+0.22\pm 0.24\pm 0.11)$% [23].
In July 2012, Belle presented a preliminary update of the individual
asymmetries $A_{CP}(K^{-}K^{+})$ and $A_{CP}(\pi^{-}\pi^{+})$, reporting also
a $\Delta A_{CP}$ measurement, using a data sample corresponding to 976 fb-1.
Instrumental asymmetries are obtained using control samples of CP conserving
decays and the production asymmetry is eliminated by making the measurement in
bins of $cos\theta^{*}$, where $\theta^{*}$ is the polar angle of the $D^{*+}$
production in the center of mass system. The results are
$A_{CP}(K^{-}K^{+})=-0.3\pm 0.21\pm 0.09$% and
$A_{CP}(\pi^{-}\pi^{+})=(+0.55\pm 0.36\pm 0.09)$%. The $\Delta A_{CP}$ value,
given in Table 2, deviates from zero with a 2.1$\sigma$ significance.
Table 2: Recent experimental results for $\Delta A_{CP}$. The last column shows the difference between the average proper time of the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ samples used to extract $\Delta A_{CP}$. Experiment | $K^{-}K^{+}$ | $\pi^{-}\pi^{+}$ | $\delta A_{CP}$ | $\Delta\langle t\rangle$
---|---|---|---|---
| yield ($\times 10^{6}$) | yield ($\times 10^{6}$) | |
LHCb | 1.44 | 0.38 | ($−0.82\pm 0.21\pm 0.11$)% | $\sim 0.098\tau_{D^{0}}$
CDF | 0.55 | 1.2 | ($-0.62\pm 0.21\pm 0.10$)% | $\sim 0.26\tau_{D^{0}}$
Belle | 0.282 | 0.123 | ($-0.87\pm 0.41\pm 0.06$)% | 0
The asymmetry difference can be written, to first order of approximation, in
terms of the direct and indirect components as [23]
$\Delta A_{CP}\approx\Delta a^{dir}_{CP}+\frac{\Delta\langle
t\rangle}{\tau_{D^{0}}}a^{ind},$ (8)
where $\langle t\rangle$ is the average reconstructed decay time,
$a^{dir}_{CP}(f)=(|{\cal A}_{f}|^{2}-|\bar{\cal A}_{f}|^{2})/(|{\cal
A}_{f}|^{2}+|\bar{\cal A}_{f}|^{2})$ and
$a^{ind}_{CP}=-\frac{1}{2}\left[\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}-\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)y\cos\phi-\left(\begin{vmatrix}\frac{q}{p}\end{vmatrix}+\begin{vmatrix}\frac{p}{q}\end{vmatrix}\right)x\sin\phi\right]$
and the phase $\phi$ is assumed to be universal. From this equation and the
$\Delta\langle t\rangle/\tau_{D^{0}}$ values given in Tab. 2 for the different
experiments, it becomes clear that $\Delta A_{CP}$ is mostly a measurement of
direct CPV.
Including higher order corrections, the first term in the right side of Eq. 8
[24] changes to $\Delta a^{dir}_{CP}[1+y_{CP}(\langle
t(K^{-}K^{+})\rangle+\langle t(K^{-}K^{+})\rangle)/(2\tau_{D^{0}})]$. Assuming
that the current precision of $A_{\Gamma}$ is such that it is not sensitive to
$\Delta a^{dir}_{CP}$, HFAG fits the contributions of direct and indirect CPV
to the most recent results of $y_{CP}$, $A_{\Gamma}$ and $\Delta A_{CP}$ by
Babar, Belle, CDF and LHCb [12], as shown in Fig. 2. This fit yields $\Delta
a^{dir}_{CP}=(-6.78\pm 1.47)\times 10^{-3}$ and $a^{ind}_{CP}=(0.27\pm
1.63)\times 10^{-3}$, again favouring a direct CPV effect over an indirect,
and is consistent with CP conservation with a confidence level of only 0.002%.
Figure 2: Combined measurements of $\Delta A_{CP}$ and $A_{\Gamma}$, where the
bands represent $\pm 1\sigma$ intervals. The point of no CP violation $(0,0)$
is shown as a filled circle, and two-dimensional 68% CL, 95% CL, and 99.7% CL
regions are plotted as ellipses with the best fit value as a cross indicating
the one-dimensional uncertainties in their center. Reproduced from [3, 12].
BaBar and Belle searched for CPV in the decays of the $D^{+}$ and $D^{+}_{s}$
mesons to two-body final states containing a $K^{0}_{S}$ using data samples
corresponding to 469 fb-1 [25, 27] and 673 fb-1 [26], respectively. The
results are presented in Table 3. The first two decays are SCS decays, while
the latter result from the coherent sum of Cabibbo favoured and doubly Cabibbo
suppressed transitions and are not expected to exhibit CPV in the charm
dynamics. For all those decays, however, the SM predicts a small CP asymmetry
due to the $K^{0}-\bar{K}^{0}$ mixing. This contribution is measured to be
$(+0.332\pm 0.006)$% for the $D^{+}_{s}\to K^{0}_{S}\pi^{+}$ mode and
$(-0.332\pm 0.006)$% for the other modes [29]. The Belle result for the
$D^{+}\to K^{0}_{s}\pi^{+}$ asymmetry was performed with a larger sample,
corresponding to the full data set of 977 fb-1, and deviates from zero with a
significance of 3.2$\sigma$. In this work, they estimate a correction to the
$K^{0}-\bar{K}^{0}$ mixing induced asymmetry following [30] and, after
subtracting this contribution, they estimate a CP asymmetry due to the charm
dynamics alone of $(-0.024\pm 0.094\pm 0.067)$%, which is the most precise
measurement of $A_{CP}$ in charm decays to date. The $A_{CP}$ values obtained
for the other channels are all consistent with zero and with the SM
expectation. In particular the Babar measurements presented in May 2011 are
the most precise [27] determination of these asymmetries.
Table 3: Most recent experimental results for CP asymmetries in the decays $D^{+}\to K^{0}_{S}K^{+}$, $D^{+}_{s}\to K^{0}_{S}\pi^{+}$, $D^{+}\to K^{0}_{S}\pi^{+}$ and $D^{+}_{s}\to K^{0}_{S}K^{+}$. Decay | BaBar $A_{CP}$(%) | Belle $A_{CP}$(%)
---|---|---
$D^{+}\to K^{0}_{S}K^{+}$ | $+0.13\pm 0.36\pm 0.25$[27] | $-0.16\pm 0.58\pm 0.25$[26]
$D^{+}_{s}\to K^{0}_{S}\pi^{+}$ | $+0.6\pm 2.0\pm 0.3$[27] | $+5.45\pm 2.50\pm 0.33$[26]
$D^{+}_{s}\to K^{0}_{S}K^{+}$ | $-0.05\pm 0.23\pm 0.24$[27] | $+0.12\pm 0.36\pm 0.22$[26]
$D^{+}\to K^{0}_{S}\pi^{+}$ | $-0.44\pm 0.13\pm 0.10$[25] | $-0.363\pm 0.094\pm 0.067$[28]
### 3.2 Multi-body decays
In the decays of D mesons to final states containing more than 2 hadrons, the
rich resonant structure of the intermediate state provide the strong phases
necessary to the observation of CPV, under the hypothesis of existing weak CPV
phases. The interference between the resonances can lead to CPV asymmetries
which vary across the phase space. These local asymmetries can be large
compared to the overall asymmetry or can even happen in the absence of phase
space integrated asymmetry.
LHCb used a model independent approach to look for a CPV signal in the SCS
decay $D^{+}\to K^{-}K^{+}\pi^{+}$ using a high purity sample ($\sim$90%)
containing about 370$\times 10^{3}$ decays selected from about 35 pb-1 of data
collected in 2010 [31]. Within this technique [32, 33], the Dalitz plot is
divided in bins and for each bin the statistical significance of the asymmetry
$S^{i}_{CP}=\frac{N^{i}_{+}-\alpha
N^{i}_{-}}{\sqrt{N^{i}_{+}+\alpha^{2}N^{i}_{-}}}$ (9)
is calculated, where $N^{i}_{\pm}$ is the number of $D^{\pm}$ decays in each
bin $i$ and $\alpha$ is the ratio between the total number of $D^{+}$ and
$D^{-}$ events and is used as a correction due to a global production
asymmetry. The distribution of $S^{i}_{CP}$ is normal under the hypothesis of
CP conservation. A $\chi^{2}$ test using $\chi^{2}=\sum^{N}_{i=1}S^{i}_{CP}$
provides a numerical evaluation for the degree of confidence for the
assumption that the differences between the $D^{+}$ and $D^{-}$ Dalitz plots
are driven only by statistical fluctuations. Four different binning schemes
are considered, two of them uniform and two adaptative which account for the
resonant structure of the decay. Higher statistics control samples consisting
of the CF decays $D^{+}_{s}\to K^{-}K^{+}\pi^{+}$ and $D^{+}\to
K^{-}\pi^{+}\pi^{+}$ are used to demonstrate that possible instrumental
asymmetries varying over the Dalitz plot are under the sensitivity of the
method. The method is also applied to the events in the side bands of the
$D^{+}$ mass distribution to prove that no effect can be introduced by the
background. The p-values obtained with the $\chi^{2}$ tests for the
$S^{i}_{CP}$ distributions using the four different binning choices are 12.7%,
10.6%, 82.1% and 60.5%. The widths and means obtained from the fits to the
distributions are all consistent with 1 and 0. No evidence for CPV is found in
any of the binning schemes.
BaBar has also performed a similar analysis of $D^{+}\to K^{-}K^{+}\pi^{+}$
events [34], using a sample of 476 fb-1 of data with $\sim 228\times 10^{3}$
events (92% pure). They divide the Dalitz plane into 100 equally populated
bins and find a 72% probability for the consistency with the assumption of no
CPV. The measured integrated asymmetry is $A_{CP}=(0.35\pm 0.30\pm 0.15)$%. A
full amplitude analysis provides CP-violating differences for the phases and
magnitudes of the intermediate resonances, all consistent with zero.
CDF searched for CPV in the resonant substructure of $D^{*+}$ tagged $D^{0}\to
K^{0}_{S}\pi^{-}\pi^{+}$ decays using a sample corresponding to 6 fb-1
collected at Run II [35]. The SM predictions for this channel are of the order
of 10-6, dominated by the $K^{0}_{S}$ mixing contribution. The selected sample
is about 90% pure and contains approximately $350\times 10^{3}$ events. A
simultaneous maximum likelihood fit to the $D^{0}$ and $\bar{D}^{0}$ Dalitz
plots, taking into account the effect of efficiency over the Dalitz space,
provide resonances parameters and CP-violating fractions, amplitudes and
phases. Differences between the $p_{T}$ distributions of the slow pions of
opposite charges are used to reweight the $\bar{D}^{0}$ distribution. All the
asymmetries are measured to be consistent with zero, as is the overall
integrated asymmetry, $A_{CP}=(−0.05\pm 0.57\pm 0.54)$%. A model independent
analysis similar to the one described above for the $D^{+}\to
K^{-}K^{+}\pi^{+}$ decay also provides null CPV signal.
For the decay $D^{0}\to\pi^{-}\pi^{+}\pi^{-}\pi^{+}$, the phase space becomes
5-dimensional. LHCb performed the first model independent CPV analysis of this
decay, using about $180\times 10^{3}$ $D^{*+}$ tagged candidates, selected
from a sample of 1 fb-1 collected in 2011 [36]. The decay $D^{0}\to
K^{-}\pi^{+}\pi^{-}\pi^{+}$ is used as control mode and events in the $D^{0}$
sample are rejected in a way such to ensure that the $\eta$ and $p_{T}$
distributions of the $D^{0}$ decays match those of the $\bar{D}^{0}$ decays.
The distribution of the asymmetry significances across the phase space is
shown to be compatible with the hypothesis of CP conservation.
## 4 Summary and Conclusions
Thanks for the pioneering work of the charm factories and the excellent work
done by Babar, Belle and CDF, mixing in the $D^{0}-\bar{D}^{0}$ system is well
established, though no single measurement with a significance above 5 standard
deviations was available until the beginning of the preparation of these
proceedings. In November 2012, LHCb has submitted a paper reporting the
observation of oscillations in the time dependent rate of $D^{0}\to
K^{+}\pi^{-}$ relative to $D^{0}\to K^{-}\pi^{+}$, which excludes the no
mixing hypothesis with a significance of 9.1 $\sigma$ [37].
The recent results obtained for $\Delta A_{CP}$ raised the interest about CPV
in charm decays. A lot of effort was put, from the theory side, to understand
on whether this can be, if confirmed, due to a new physics effect or to an
enhancement of SM penguin contributions [38]. Until now, no clear picture
emerged yet. More precise measurements of the difference of asymmetries and of
the individual asymmetries $A_{CP}(K^{-}K^{+})$ and $A_{CP}(\pi^{-}\pi^{+})$
are needed. Advances in the theoretical understanding of charm decays can also
help to disentangle this puzzle, as well as precise measurements in additional
neutral and charged decay modes. CDF, BaBar and Belle ended operation and are
finishing to analyse their samples. LHCb has shown the ability to reconstruct
charm decays with purities comparable to the experiments at $e^{+}e^{-}$
machines and has already collected the largest charm sample of the world,
corresponding to a luminosity of 1 fb-1 at $\sqrt{s}=7$ TeV and 2 fb-1 at
$\sqrt{s}=8$ TeV. Precisions comparable or below the SM predictions will only
be achievable at the LHCb upgrade [39] and the super B factories [40, 41].
## Acknowledgements
I am grateful to the organizers of PIC2012 and to my colleagues at the LHCb
charm working group for useful discussions. My participation at this
conference was supported by CNPq and FINEP.
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|
arxiv-papers
| 2012-12-27T10:55:28 |
2024-09-04T02:49:39.685713
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Erica Polycarpo",
"submitter": "\\'Erica Polycarpo",
"url": "https://arxiv.org/abs/1212.6341"
}
|
1212.6374
|
# New physics from flavour
Physics Dept., 201 Physics Building, Syracuse University, Syracuse, NY, USA,
13244-1130
E-mail Work supported by U.S. National Science Foundation
###### Abstract:
I report on investigations of physics beyond the Standard Model using
experiments designed to investigate $b$ quark and $c$ quark interactions.
Generalized searches involving loop diagrams using heavy meson mixing and $CP$
violation experiments imply limits on New Physics (NP) at large scales from
100-100,000 TeV. There are ways of avoiding these limits, but satisfying their
requirements puts severe restrictions on NP properties. Specific constraints
on NP models are discussed from individual measurements including $b\to
s\gamma$, $\overline{B}_{s}^{0}\to\mu^{+}\mu^{-}$,
$\overline{B}^{0}\to\overline{K}^{*}\ell^{+}\ell^{-}$, and $CP$ violation in
the $B^{0}$ and $B_{s}^{0}$ systems. Hints of deviations from Standard Model
predictions from other decays including $B^{-}\to\tau^{-}\overline{\nu}$, and
$\overline{B}\to D^{(*)}\tau^{-}\overline{\nu}$ are investigated. Finally,
searches for dark matter and Majorana neutrinos are reviewed.
## 1 Introduction: Reasons for physics beyond the Standard Model
Although the Standard Model (SM) of particle physics provides an excellent
description of electroweak and strong interactions, there are many reasons
that we expect to observe new forces giving rise to new particles at larger
masses than the known fermions or bosons. One oft noted source of this belief
is the observation of dark matter in the cosmos as evidenced by galactic
angular velocity distributions [1], gravitational lensing [2], and galactic
collisions [3]. The existence of dark energy, believed to cause the
accelerating expansion of the Universe, is another source of mystery [4]. The
fine tuning of quantum corrections needed to keep, for example, the Higgs
boson mass at the electroweak scale rather than near the Planck scale is
another reason habitually mentioned for new physics (NP) and is usually called
“the hierarchy problem” [5].
It is interesting to note that the above cited reasons are all tied in one way
or another to gravity. Dark matter may or may not have purely gravitational
interactions, dark energy may be explained by a cosmological constant or at
least be a purely general relativistic phenomena, and the Planck scale is
defined by gravity; other scales may exist at much lower energies, so the
quantum corrections could be much smaller. There are, however, many
observations that are not explained by the SM, and have nothing to do with
gravity, as far as we know. Consider the size of the quark mixing matrix (CKM)
elements [6] and also the neutrino mixing matrix (PMNS) elements [7]. These
are shown pictorially in Fig. 1. We do not understand the relative sizes of
these values or nor the relationship between quarks and neutrinos.
Figure 1: (left) Sizes of the the CKM matrix elements for quark mixing, and
(right) the PMNS matrix elements for neutrino mixing. The area of the squares
represents the square of the matrix elements. Figure 2: Lepton and quark
masses.
We also do not understand the masses of the fundamental matter constituents,
the quarks and leptons. Not only are they not predicted, but also the
relationships among them are not understood. These masses, shown in Fig. 2,
span 12 orders of magnitude [7]. There may be a connections between the mass
values and the values of the mixing matrix elements, but thus far no
connection besides simple numerology exists.
What we are seeking is a new theoretical explanation of the above mentioned
facts. Of course, any new model must explain all the data, so that any one
measurement could confound a model. It is not a good plan, however, to try and
find only one discrepancy; experiment must determine a consistent pattern of
deviations to restrict possible theoretical explanations, and to be sure the
new phenomena is real.
## 2 Use of flavour physics as a new physics discovery tool
While measurements of CKM parameters and masses are important, as they are
fundamental constants of nature, the main purpose of flavour physics is to
find and/or define the properties of physics beyond the SM. Flavour physics
probes large mass scales via virtual quantum loops. Examples of the importance
of such loops are changes in the $W$ boson mass ($M_{W}$) from the existence
of the $t$ quark of mass $m_{t}$, $dM_{W}\propto m_{t}^{2}$, and changes due
to the existence of the Higgs boson of mass $M_{H}$, $dM_{W}\propto{\rm
ln}(M_{H})$.
Strong constraints on NP are provided by individual processes. Each process
provides a different constraint. Consider for example the inclusive decay
$b\to s\gamma$. The SM diagrams are shown in Fig. 3(a). The SM calculation
considers either a $B^{-}$ or $\overline{B}^{0}$ meson and then sums over all
decays where a hard photon emerges. The Feynman diagrams corresponding to a NP
process with a virtual charged Higgs boson are shown in Fig. 3(b).
Figure 3: (a) SM diagrams for the quark level process $b\to s\gamma$ and (b)
NP diagrams mediated by a charged Higgs boson.
My average of measured branching fractions, including a recent result at this
conference from BaBar [8], is
${\cal{B}}\left(b\to s\gamma\right)=(3.37\pm 0.23)\times 10^{-4}~{}.$ (1)
The SM prediction is $(3.15\pm 0.23)\times 10^{-4}$ [9, 10]; thus the ratio of
the measured to theoretical prediction is 1.07$\pm$0.10, which serves to
severely limit many, if not most, NP models. An example of such a constraint
for the two-Higgs-double model (2HDM), with the ratio of the vacuum
expectation values of the two Higgs doublets having a value of $\tan\beta=2$,
is given in Fig. 4. The intersection of the lower edge of the NP prediction
with the upper line of the experiment gives a limit at the approximately two
standard deviation $(\sigma)$ level of $m_{H^{+}}>385$ GeV [11].
Figure 4: ${\cal{B}}\left(b\to s\gamma\right)$ as a function of the charged
Higgs boson mass. There are three sets of curves including the experimental
average shown in dotted (red), the SM prediction in dashed (blue) and the 2HDM
model in solid (black). The middle curve of each set gives the central value
and higher and lower ones the $\pm 1\sigma$ bands [9].
It is of course true that experiments measure the sum of SM and NP, so to set
limits or see signals from NP we have to be confident in understanding the SM
contribution. This was clear in the above example of $b\to s\gamma$. In other
processes SM calculations are not possible, or are not unambiguous. An often
used paradigm is that “tree-level” diagrams are dominated by SM processes, but
loop level diagrams can be strongly influenced by NP. Examples of tree
diagrams are given in Fig. 5(a), while both SM and loop diagrams that could
have new as yet undiscovered particles in the loop are shown in Fig. 5(b);
here both diagrams would add in amplitude and thus interfere allowing for
either increases or decrease in rates and asymmetries.
Figure 5: (a) Examples of SM tree-level diagrams. (b) Example of a SM loop
diagram for $B_{s}$ mixing interfering that can interfere with (c) a NP
diagram.
NP can also be limited via a general study of loop processes, again assuming
that it doesn’t contribute to tree-level processes. One can write a general
local Lagrangian which includes an infinite tower of operators with dimension
$d>4$, constructed in terms of SM fields, suppressed by inverse powers of an
effective scale $\Lambda>M_{W}$ as
${\cal{L}}_{\rm eff}={\cal{L}}_{\rm
SM}+\sum{\frac{c_{i}^{(d)}}{\Lambda^{(d-4)}}{\cal{O}}^{(d)}_{i}},$ (2)
where the ${\cal{O}}$’s represent the SM fields [12]. Lower bounds on
$\Lambda$ are shown in Table 1, taking the $c_{i}$ coefficients equal to one
for different operators.
Operator | $\Lambda$ lower bound (TeV) | Observable
---|---|---
Re$[(\overline{s}_{L}\gamma^{\mu}d_{L})^{2}]$ | $9.8\times 10^{2}$ | $K_{L}-K_{S}$ mass difference, $\Delta m_{K}$
Im$[(\overline{s}_{L}\gamma^{\mu}d_{L})^{2}]$ | $1.6\times 10^{4}$ | $CP$ violation in $K_{L}$ decay, $\epsilon_{K}$
Re$[(\overline{s}_{R}d_{L})(\overline{s}_{L}d_{R})]$ | $1.8\times 10^{4}$ | $K_{L}-K_{S}$ mass difference, $\Delta m_{K}$
Im$[(\overline{s}_{R}d_{L})(\overline{s}_{L}d_{R})]$ | $3.2\times 10^{5}$ | $CP$ violation in $K_{L}$ decay, $\epsilon_{K}$
Re$[(\overline{b}_{L}\gamma^{\mu}d_{L})^{2}]$ | $9.8\times 10^{2}$ | $B_{L}-B_{S}$ mass difference, $\Delta m_{B}$
Im$[(\overline{b}_{L}\gamma^{\mu}d_{L})^{2}]$ | $1.6\times 10^{4}$ | $CP$ violation in $B^{0}\to J/\psi K_{S}$ decay
Re$[(\overline{b}_{R}d_{L})(\overline{s}_{L}d_{R})]$ | $1.8\times 10^{4}$ | $B_{L}-B_{S}$ mass difference, $\Delta m_{B}$
Im$[(\overline{b}_{R}d_{L})(\overline{s}_{L}d_{R})]$ | $3.2\times 10^{5}$ | $CP$ violation in $B^{0}\to J/\psi K_{S}$ decay
Table 1: Lower limits on the size of representative dimension six loop diagram
operators ($\Lambda$), taking the coefficients $c_{i}$ equal to one. Adopted
from Ref. [12].
These limits are impressive ranging from approximately 100 to 10,000 TeV, well
beyond the direct collider limit search possibilities. Of course it is
possible that the $c_{i}$ are smaller than one, but they have to be in range
from $10^{-6}$ to $10^{-9}$ to bring these limits to the TeV scale. There are
ways, however, that NP can be consistent with these limits. One way is for the
new particles in the loop diagrams to be degenerate in mass and have opposite
phase, so they cancel in the loops. Another is for the quark mixing angles to
be the same in the NP sector as in the SM. This defacto prescription, called
“Minimum Flavor Violation,” is well described by Buras [13]. Of course very
heavy new particles are also a possibility. Overall, these measurements put
very severe constraints on NP models.
## 3 Mixing and $CP$ violation
Let us consider in more detail the mixing diagrams which allow transformation
of neutral mesons into their anti-particles. Fig. 6 shows the second order
weak interaction diagrams for $D^{0}$, $B^{0}$ and $B_{s}^{0}$ mesons. The
amplitude strength depends on the values of the CKM elements at the four
vertices and approximately as the magnitude of the mass-squared of the virtual
quarks permitted. $B^{0}$ and $B_{s}^{0}$ mixing transitions are far larger
than that of the $D^{0}$, because the $t$ quark is allowed in the $B$ cases,
but for the $D^{0}$ the heaviest allowed exchange quark is the $b$. Hence
$D^{0}$ mixing is quite small, but finite [28], while the $B^{0}$ is quite
significant and the $B_{s}^{0}$ huge, the difference in the two $B$ cases is
due entirely to relative sizes of the CKM elements.
Figure 6: (top) Diagrams for $D^{0}$, $B^{0}$ and $B_{s}^{0}$ mixing. (bottom)
Sketches of the time dependence for mesons to decay unmixed and mixed.
(Adapted from Van-kooton [15].)
The Schrödinger equation is used to describe time dependent transformations
between meson and anti-meson states for diagrams similar to the one in Fig.
5(b) in terms of a mass matrix $M$ and a decay matrix $\Gamma$:
$i\frac{d}{dt}\left(\begin{array}[]{c}B^{0}_{s}\\\
\overline{B}^{0}_{s}\end{array}\right)=\left(\begin{array}[]{cc}M_{11}-i\Gamma_{11}/2&M_{12}-i\Gamma_{12}/2\\\
M_{12}^{*}-i\Gamma_{12}^{*}/2&M_{22}-i\Gamma_{22}/2\end{array}\right)\left(\begin{array}[]{c}B^{0}_{s}\\\
\overline{B}^{0}_{s}\end{array}\right)$ (3)
Diagonalizing we find
$\ket{M_{L}}=p\ket{B^{0}_{s}}+q\ket{\overline{B}^{0}_{s}}$,
$\ket{M_{H}}=p\ket{B^{0}_{s}}-q\ket{\overline{B}^{0}_{s}}$, where $p$ and $q$
are complex numbers that are equal to $1/\sqrt{2}$ in the absence of $CP$
violation. Also, assuming $CPT$ invariance, the mass
$m(B_{s})=M_{11}=M_{22}=(M_{H}+M_{L})/2$, and the mass difference is defined
as $\Delta M=M_{H}-M_{L}$, while for the decay widths,
$\Gamma=1/\tau(B_{s})=(\Gamma_{H}+\Gamma_{L})/2=\Gamma_{11}=\Gamma_{22}$, and
$\Delta\Gamma=\Gamma_{L}-\Gamma_{H}$. For convenience the variable
$y\equiv\Delta\Gamma/2\Gamma$ is defined.
Time dependent $CP$ violation depends on the process in many ways. If the
final state $f$ is a $CP$ eigenstate then $f=\bar{f}$ and the $CP$ violating
asymmetry is defined as [16]
$a(t)=\frac{\Gamma\left(\overline{M}\to f\right)-\Gamma\left({M}\to
f\right)}{\Gamma\left(\overline{M}\to f\right)+\Gamma\left({M}\to
f\right)}~{}.$ (4)
The amplitudes are defined as $A_{f}\equiv A(M\to f)$,
$\bar{A}_{f}\equiv\bar{A}(M\to f)$. The magnitude of the $CP$ violating effect
for each channel depends on the CKM elements present in the decay, and is
given by
$\lambda_{f}=\frac{p}{q}\frac{\bar{A}_{f}}{A_{f}}~{}.$ (5)
The decay rates, when $|\lambda_{f}|=1$, are given by
$\displaystyle\Gamma\left(\overline{M}\to f\right)$ $\displaystyle=$
$\displaystyle N_{f}|A_{f}|^{2}e^{-\Gamma t}\left(\cosh\frac{\Delta\Gamma
t}{2}-{\rm Re}\lambda_{f}\sinh\frac{\Delta\Gamma t}{2}+{\rm
Im}\lambda_{f}\sin\left(\Delta Mt\right)\right)$
$\displaystyle\Gamma\left({M}\to f\right)$ $\displaystyle=$ $\displaystyle
N_{f}|A_{f}|^{2}e^{-\Gamma t}\left(\cosh\frac{\Delta\Gamma t}{2}-{\rm
Re}\lambda_{f}\sinh\frac{\Delta\Gamma t}{2}-{\rm
Im}\lambda_{f}\sin\left(\Delta Mt\right)\right)~{}.$ (6)
Thus the $CP$ violating asymmetry is given by
$a(t)=\frac{{\rm Im}\lambda_{f}\sin\left(\Delta
Mt\right)}{\cosh\frac{\Delta\Gamma t}{2}-{\rm
Re}\lambda_{f}\sinh\frac{\Delta\Gamma t}{2}}~{}.$ (7)
### 3.1 $CP$ violation in $\overline{B}^{0}_{s}$ decays
Let us consider the decays $\overline{B}^{0}_{s}\to J/\psi\phi$, where
$\phi\to K^{+}K^{-}$ and $J/\psi\pi^{+}\pi^{-}$. The tree-level decay diagram
is shown in Fig. 7. Here we use the interference between the direct decay and
the one that proceeds via mixing: $\overline{B}^{0}_{s}\to B^{0}_{s}\to
J/\psi\phi$, for example.
Figure 7: Tree level diagram for $\overline{B}^{0}_{s}\to
J/\psi\pi^{+}\pi^{-}$ or $K^{+}K^{-}$ decays.
The $CP$ violating asymmetry $a(t)$ can be expressed in terms of the phase
angle $\phi_{s}$, and the $CP$ parity of the final state, $\eta_{\rm CP}$. In
the SM ${\rm Im}(\lambda_{f})=-\eta_{\rm CP}\sin\phi_{s}=\eta_{\rm
CP}\arg\left(\frac{V_{ts}V^{*}_{tb}}{V^{*}_{ts}V_{tb}}\frac{V^{*}_{cs}V_{cb}}{V_{cs}V^{*}_{cb}}\right)=-0.04$
rad for $CP$ odd states [17].
Based on a conjecture by Stone and Zhang [18], the LHCb collaboration found
the decay mode $\overline{B_{s}^{0}}\to J/\psi f_{0}(980)$ where
$f_{0}(980)\to\pi^{+}\pi^{-}$, close to the predicted level [19], and was soon
confirmed by others [20]. As the final state consists of a scalar $f_{0}(980)$
and a vector $J/\psi$, the relative angular momentum between the $f_{0}(980)$
and the $J/\psi$ is a P-wave and the final state is pure $CP$ odd. LHCb then
examined the entire $\pi^{+}\pi^{-}$ mass spectrum in the
$\overline{B}^{0}_{s}\to J/\psi\pi^{+}\pi^{-}$ channel, shown in Fig. 8. The
region between the arrows has a large signal to background ratio. In this
region a full Dalitiz like analysis was performed that identified the
individual components, and more importantly showed that the final state is
$>97.7$% CP-odd [21]. Measurement of $CP$ violation based on Eq. 7 gives a
value $\phi_{s}=-0.019^{+0.173+0.004}_{-0.174-0.003}$ rad [22].
Figure 8: The $\pi^{+}\pi^{-}$ invariant mass in $\overline{B}^{0}_{s}\to
J/\psi\pi^{+}\pi^{-}$ decays. The dashed (red) curve shows the background, and
the arrows indicated the region chosen for $CP$ studies.
The $J/\psi\phi$ state being composed of two vector particles is not a $CP$
eigenstate. Such final states still can be used but an angular analysis is
necessary to separate the $CP$-even and $CP$-odd components [23], which
requires fitting time dependent angular distributions. The three angles are
shown in Fig. 9.
Figure 9: Pictoral description of the decay angles. On the left $\theta$ and
$\phi$ defined in the $J/\psi$ rest frame and on the right $\psi$ defined in
the $\phi$ rest frame. (From T. Kuhr [24].).
In addition to the three P-wave amplitudes in the decay width, an additional
amplitude due to possible $K^{+}K^{-}$ S-wave also must be considered [18].
The complexity of the analysis scheme is shown by writing the distribution of
the signal decay time and angles, described by a sum of ten terms,
corresponding to the four polarization amplitudes and their interference
terms. Each of these is the product of a time-dependent function and an
angular function [23]
$\frac{\mathrm{d}^{4}\Gamma({B}_{s}^{0})}{\mathrm{d}t\;\mathrm{d}\theta\mathrm{d}\psi\mathrm{d}\phi}\;\propto\;\sum^{10}_{k=1}\>h_{k}(t)\>f_{k}(\theta,\psi,\phi)\,.$
(8)
The time-dependent functions $h_{k}(t)$ can be written as
$h_{k}(t)\;=\;N_{k}e^{-\Gamma_{s}t}\>\left[c_{k}\cos(\Delta
m_{s}t)\,+d_{k}\sin(\Delta
m_{s}t)\,+a_{k}\cosh\left(\tfrac{1}{2}\Delta\Gamma_{s}t\right)+b_{k}\sinh\left(\tfrac{1}{2}\Delta\Gamma_{s}t\right)\right].$
(9)
where $\Delta m_{s}{}$ is the $B^{0}_{s}$ oscillation frequency. The
coefficients $N_{k}$ and $a_{k},\ldots,d_{k}$ can be expressed in terms of
$\phi_{s}$ and four complex transversity amplitudes $A_{i}$ at $t=0$. The
label $i$ takes the values $\\{\perp,\parallel,0\\}$ for the three P-wave
amplitudes and S for the S-wave amplitude. For a particle produced in a
$B^{0}_{s}$ flavour eigenstate the coefficients in Eq. 9 and the angular
functions $f_{k}(\theta,\psi,\phi)$ are then, given in Table 2 [26], where
$\delta_{0}=0$ is chosen arbitrarily
${\small\begin{array}[]{c|c|c|c|c|c|c}\hline\cr\hline\cr
k&f_{k}(\theta,\psi,\phi)&N_{k}&a_{k}&b_{k}&c_{k}&d_{k}\\\ \hline\cr
1&2\,\cos^{2}\psi\left(1-\sin^{2}\theta\cos^{2}\phi\right)&|A_{0}(0)|^{2}&1&-\cos\phi_{s}&0&\sin\phi_{s}\\\
2&\sin^{2}\psi\left(1-\sin^{2}\theta\sin^{2}\phi\right)&|A_{\|}(0)|^{2}&1&-\cos\phi_{s}&0&\sin\phi_{s}\\\
3&\sin^{2}\psi\sin^{2}\theta&|A_{\perp}(0)|^{2}&1&\cos\phi_{s}&0&-\sin\phi_{s}\\\
4&-\sin^{2}\psi\sin
2\theta\sin\phi&|A_{\|}(0)A_{\perp}(0)|&0&-\cos(\delta_{\perp{\|}})\sin\phi_{s}&\sin(\delta_{\perp{\|}})&-\cos(\delta_{\perp{\|}})\cos\phi_{s}\\\
5&\tfrac{1}{2}\sqrt{2}\sin 2\psi\sin^{2}\theta\sin
2\phi&|A_{0}(0)A_{\|}(0)|&\cos(\delta_{\|0})&-\cos(\delta_{\|0})\cos\phi_{s}&0&\cos(\delta_{\|0})\sin\phi_{s}\\\
6&\tfrac{1}{2}\sqrt{2}\sin 2\psi\sin
2\theta\cos\phi&|A_{0}(0)A_{\perp}(0)|&0&-\cos(\delta_{\perp
0})\sin\phi_{s}&\sin(\delta_{\perp 0})&-\cos(\delta_{\perp 0})\cos\phi_{s}\\\
7&\tfrac{2}{3}(1-\sin^{2}\theta\cos^{2}\phi)&|A_{\mathrm{S}}(0)|^{2}&1&\cos\phi_{s}&0&-\sin\phi_{s}\\\
8&\tfrac{1}{3}\sqrt{6}\sin\psi\sin^{2}\theta\sin
2\phi&|A_{\mathrm{S}}(0)A_{\|}(0)|&0&-\sin(\delta_{\|\mathrm{S}})\sin\phi_{s}&\cos(\delta_{\|\mathrm{S}})&-\sin(\delta_{\|\mathrm{S}})\cos\phi_{s}\\\
9&\tfrac{1}{3}\sqrt{6}\sin\psi\sin
2\theta\cos\phi&|A_{\mathrm{S}}(0)A_{\perp}(0)|&\sin(\delta_{\perp\mathrm{S}})&\sin(\delta_{\perp\mathrm{S}})\cos\phi_{s}&0&-\sin(\delta_{\perp\mathrm{S}})\sin\phi_{s}\\\
10&\tfrac{4}{3}\sqrt{3}\cos\psi(1-\sin^{2}\theta\cos^{2}\phi)&|A_{\mathrm{S}}(0)A_{0}(0)|&0&-\sin(\delta_{0\mathrm{S}})\sin\phi_{s}&\cos(\delta_{0\mathrm{S}})&-\sin(\delta_{0\mathrm{S}})\cos\phi_{s}\\\
\hline\cr\hline\cr\end{array}}$ Table 2: The components of the decay width
contributing to Eqs. 3.6 and 3.7, where
$\delta_{ij}\equiv\delta_{i}-\delta_{j}$, e.g. $\delta_{\perp
0}=\delta_{\perp}-\delta_{0}$.
The differential decay rates for a $\overline{B}^{0}_{s}$ meson produced at
time $t=0$ are obtained by changing the signs of $\phi_{s}$, $A_{\perp}(0)$
and $A_{\mathrm{S}}(0)$, or, equivalently, the signs of $c_{k}$ and
$d_{k}$.111The decay width is invariant under the transformation
$(\phi_{s},\Delta\Gamma_{s},\delta_{\|},\delta_{\perp},\delta_{\mathrm{S}})\mapsto(\pi-\phi_{s},-\Delta\Gamma_{s},-\delta_{\|},\pi-\delta_{\perp},-\delta_{\mathrm{S}})$
which gives rise to a two-fold ambiguity in the results. This ambiguity has
been removed by analyzing the interference between the S and P waves [27].
Note, that for the $J/\psi\pi^{+}\pi^{-}$ final state, only line #7 in Table 2
contributes.
Results of the analyses are presented in the $\Delta\Gamma_{s}-\phi_{s}$
plane, since earlier low statistics data had significant correlations. The
current results are shown in Fig. 10 for several experiments. The LHCb result,
$\phi_{s}=0.001\pm 0.101\pm 0.027$ rad is the most accurate and dominates the
average. Combining with the $J/\psi\pi^{+}\pi^{-}$ result LHCb finds [25]
$\phi_{s}=-0.002\pm 0.083\pm 0.027~{}{\rm rad,}$ (10)
which is consistent with the SM prediction.
Figure 10: Measurements of $\Delta\Gamma$ versus $\phi_{s}$ as compiled by
HFAG [28] .
I choose to determine the $\overline{B}_{s}^{0}$ lifetime and
$\Delta\Gamma_{s}$ based on measurements of only fully reconstructed decays
where all the final state particles are seen. Fig. 11 shows the data from four
separate determinations using $J/\psi\phi$, two using $J/\psi f_{0}(980)$, one
using $D_{s}^{+}\pi^{-}$ at fully mixed $CP$ state, and one using
$K^{+}K^{-}$.The last mode is a $CP$-even eigenstate where the effects of $CP$
violation are taken to be negligible. The results to an overall fit are shown
in the black oval covering 39% confidence level (CL). They are
$\displaystyle\tau_{s}$ $\displaystyle=$ $\displaystyle 1.509\pm 0.010~{}{\rm
ps},$ $\displaystyle\Delta\Gamma_{s}$ $\displaystyle=$ $\displaystyle 0.092\pm
0.011~{}{\rm ps}^{-1},$ $\displaystyle y_{s}$ $\displaystyle=$
$\displaystyle\frac{\Delta\Gamma_{s}}{2\Gamma_{s}}=0.07\pm 0.01~{}.$ (11)
These result agree very well with the theoretical predictions.
Figure 11: Measurements of $\Delta\Gamma_{s}$ versus the effective lifetime in
each mode labeled as $1/\Gamma_{s}$, and the average as shown in the black
ellipse as determined by A. Phan [29]. The ovals show 39% CL contours, while
the bands are at 68% CL.
There is another way $CP$ violation can show itself in $\overline{B}^{0}$
decays. Consider the asymmetry defined as
$a(t)=\frac{\Gamma\left(\overline{M}\to
f\right)-\Gamma\left({M}\to\overline{f}\right)}{\Gamma\left(\overline{M}\to
f\right)+\Gamma\left({M}\to\overline{f}\right)}~{}.$ (12)
The difference here with Eq. 4 is that the final state $f$ is flavour
specific, $f\neq\overline{f}$, and at zero decay time both
$\Gamma\left(\overline{M}\to f\right)=0$ and
$\Gamma\left({M}\to\overline{f}\right)=0$. For example, in the case of
$\overline{B}^{0}_{s}$ decays the asymmetry between the decay rates for
$\overline{B}^{0}_{s}\to D^{+}_{s}\mu^{-}\overline{\nu}$ and ${B}^{0}_{s}\to
D^{-}_{s}\mu^{+}{\nu}$ can be measured. Another method is to use events where
both $b$-flavoured hadrons decay semileptonically, i.e. to $X\mu\nu$. When one
$b$ mixes and the other decays these events produce like-sign muons, and the
difference between $\mu^{+}\mu^{+}$ and $\mu^{-}\mu^{-}$ events can be
examined.
In the SM the asymmetry is related to decay width as
$a_{sl}=\left(\Delta\Gamma/\Delta M\right)\tan\phi_{12}$, where
$\tan\phi_{12}=-\arg\left(-\Gamma_{12}/M_{12}\right)$. The asymmetries are
expected to be very small in the SM, $-4.1\times 10^{-4}$, and $1.9\times
10^{-5}$ for $B^{0}$ ($a_{sl}^{d}$) and $B_{s}^{0}$ ($a_{sl}^{s}$),
respectively [30]. The D0 collaboration, however, reported an anomalously
large value for the dimuon asymmetry of $A^{b}_{sl}=(-0.787\pm 0.172\pm
0.093)$% [31]. As they are summing over $B^{0}$ and $B_{s}^{0}$ decays, they
graph their results as the purple band shown in Fig. 12. As the width of the
band corresponds to $\pm 1\sigma$ uncertainty, their result is 3.9$\sigma$
from the SM value. If confirmed, this would be a clear indication of NP.
Figure 12: Measurements of the semileptonic asymmetry using dimuon events from
D0. The purple band ($A^{b}_{sl}$) is their base result, while the bands have
impact parameter (IP) cuts applied that change the relative $B^{0}$ and
$B_{s}^{0}$ yields [32].
Subsequently D0 measured the individual semleptonic asymmetries $a_{sl}^{s}$
[33] and $a_{sl}^{d}$ [34] using semileptonic $B$ decays, where a single muon
is detected in conjunction with a charmed meson. The invariant
$K^{+}K^{-}\pi^{\pm}$ mass distribution, where the $K^{+}K^{-}$ is required to
be consistent with the $\phi$ meson mass, for putative ${B}_{s}^{0}\to
D_{s}^{\pm}\mu^{\mp}X$ decays is shown in Fig. 13, where the $X$ indicates a
missing neutrino plus possibly additional mesons. The events in the $D_{s}$
signal peak come predominantly from $B_{s}^{0}$ decays. The measurement
results from counting the number of $D_{s}^{+}\mu^{-}$ versus
$D_{s}^{-}\mu^{+}$ events and correcting for any residual detector
asymmetries. From these data D0 finds $a_{sl}^{s}=(-1.08\pm 0.72\pm 0.17)$%,
and from studying $D^{\pm}\mu^{\mp}X$ decays, with $D^{\pm}\to
K^{\mp}\pi^{+}\pi^{-}$, $a_{sl}^{d}=(0.68\pm 0.45\pm 0.14)\%$.
Figure 13: The $\phi\pi^{\pm}$ mass spectrum for events where the tracks form
a common vertex detached from the primary. The peaks are at the $D^{+}$ and
$D_{s}^{+}$ masses. (Note the zero suppressed vertical scale.)
Combining their di-muon results, including some separation into $a_{sl}^{s}$
and $a_{sl}^{d}$ values based on muon impact parameter, with the semileptonic
measurements, Bertram compares the D0 measurements with the SM in Fig. 14
[35].
Figure 14: Combination D0 semileptonic decay asymmetry measurements with
measurements using di-muons separated by into two impact parameter categories
by a cut of 120 $\mu$m. The error bands represent $1\sigma$ uncertainties on
each individual measurement. The ellipses represent the 1, 2, 3, and 4$\sigma$
two-dimensional C.L. regions, respectively. The SM point is shown with a black
dot.
There is a differing world view on these asymmetries. The B-factories have
produced measurements of $a_{sl}^{d}$ that average to $0.0002\pm 0.0031$ [28].
LHCb recently measured the asymmetry in semileptonic $B_{s}^{0}$ decays, in a
similar manner as D0, using a 1.0 fb-1 data sample. The $K^{+}K^{-}\pi^{\pm}$
mass spectra is shown in Fig. 15 for candidate decays with a muon of opposite
charge to the candidate $D_{s}$ meson. LHCb periodically reverses the polarity
of their spectrometer magnet, as did D0. This data is for one configuration
only (magnetic field down). The excellent mass resolution allows the $D^{+}$
and $D_{s}^{+}$ signals to be completely separated.
Figure 15: Invariant mass distributions for (a) $K^{+}K^{-}\pi^{+}$ candidates
and (b) $K^{+}K^{-}\pi^{-}$ candidates for magnet down data, requiring that
$m(K^{+}K^{-}$) be within $\pm$20 MeV of the $\phi$ meson mass. The vertical
dashed lines indicate the signal region.
LHCb has different detection efficiencies, in principle, for $\pi^{+}$ versus
$\pi^{-}$ and $\mu^{+}$ versus $\mu^{-}$. In each magnet setting the tracking
efficiency differences mostly cancel considering the binary pairs
$\pi^{+}\mu^{-}$ versus $\pi^{-}\mu^{+}$. Most of the remaining differences
are cancelled by averaging the two opposite magnet polarities.The different
muon triggering efficiencies are evaluated using a sample of about one million
$J/\psi\to\mu^{+}\mu^{-}$ found in events that were triggered independently of
the $J/\psi$. LHCb finds $a_{sl}^{s}=(-0.24\pm 0.54\pm 0.33)$% [36].
Measurements from the B-factories and LHCb are plotted in Fig. 16. The data
are completely consistent with the SM, and differ somewhat from the D0
results.
Figure 16: The LHCb measurement of $a_{sl}^{s}$, and the B-factory measurement
of $a_{sl}^{d}$. The band edges are the central values $\pm 1\sigma$
uncertainties.
### 3.2 Mixing and $CP$ violation in Charm
LHCb recently measured mixing in the $D^{0}-\overline{D}^{0}$ system at more
than 5$\sigma$ significance [37], consistent with the average of several other
less significant results [28]. Measurement of charm mixing at the levels seen
is expected in the SM. Measurement of $CP$ violation at the $\sim 1$% level,
however, was not expected. An interesting paper discussed the possibility of
seeing NP in charm $CP$ violation specifically by looking at singly Cabibbo
suppressed decays [38].
Using the same definition for asymmetry as in Eq. 4, the Cabibbo suppressed
final states $D^{0}\to K^{+}K^{-}$ and $D^{0}\to\pi^{+}\pi^{-}$ have been
investigated. A flavor tag is provided by using $D^{0}$ that result from
$D^{*\pm}\to\pi^{\pm}D^{0}$ decays. In order to cancel the effects of
$D^{*\pm}$ production asymmetries and $\pi^{+}$ versus $\pi^{-}$ detection
asymmetries, the difference between time-integrated asymmetries $\Delta
A_{CP}=a_{CP}(K^{+}K^{-})-a_{CP}(\pi^{+}\pi^{-})$ is determined. LHCb first
published a 3.5 standard deviation result [39]. When averaging data from CDF
[40] and Belle [41], I find $\Delta A_{CP}=(-0.74\pm 0.15)$%. However, none of
the experiments by itself has a result of more than 3.5$\sigma$ significance,
and more data are needed to see if the result is real. Meanwhile the theory
community continues to argue if this asymmetry difference can be explained by
the SM [42]. I choose to treat this as a limit on NP of 1%$>-\Delta
A_{CP}>0$%.
## 4 Exclusive rare decays
We have seen that the inclusive “rare” decay $b\to s\gamma$ has a relatively
accurate SM prediction (see Sec. 2). Other well determined predictions exist
for exclusive rare decays. Since these decays have suppressed branching
fractions the possibility that NP can make large contributions is enhanced.
### 4.1 $\overline{B}_{s}^{0}\to\mu^{+}\mu^{-}$
Heavily suppressed in the SM partially due to helicity conservation, the decay
$\overline{B}_{s}^{0}\to\mu^{+}\mu^{-}$ is predicted by Buras [43] to have a
branching fraction, after a correction due to the relatively large size of
$\Delta\Gamma_{s}$, of $(3.5\pm 0.2)\times 10^{-9}$ [44]. The SM branching
fraction for the corresponding $\overline{B}^{0}$ decay is even smaller due to
the smallness of the CKM element $|V_{td}|$ compared with $|V_{ts}|$. The SM
diagrams are shown in Fig. 17, along with possible interfering diagrams from
NP processes. We note that many NP models are possible not just the one
example of super-symmetry that is shown.
Figure 17: Decay diagrams for $\overline{B}_{s}^{0}\to\mu^{+}\mu^{-}$ in the
SM (left) and in a particular NP model, that of the MSSM (right).
While upper limits on this process have been established by several
experiments, as shown in Fig. 18, a new LHCb result sees evidence of a signal
[45].222Most LHCb results reviewed here use a 1.0 fb-1 data sample. This
result uses 2.1 fb-1.
Figure 18: Upper limits for $\overline{B}_{s}^{0}\to\mu^{+}\mu^{-}$ from
different experiments, as of July 4, 2012. The “LHC combination,” includes
individual ATLAS, CMS and LHCb contributions shown in the figure.
LHCb has unique capabilities at the LHC to study purely hadronic decays of
$b$-flavoured hadrons. In the case of this analysis such decays are used to
study selection criteria, backgrounds and to measure the transverse momentum,
$p_{T}$, dependence of the ratio of $B_{s}$ to $B^{0}$ production, called
$f_{s}/f_{d}$. Multivariate discriminants are trained using several variables,
including track impact parameters, $B$ lifetime, $B$ $p_{T}$, $B$ isolation,
muon isolation, minimum impact parameter of muons, etc… The signal properties
are established using fully reconstructed $B$ decays into two oppositely
charged hadrons. The signals for four different final states are shown in Fig.
19.
Figure 19: Invariant mass spectra of the two final state hadrons for
$\overline{B}_{(s)}^{0}\to h^{+}h^{\prime-}$ from different final states. Fits
are shown to signal and background shapes. Signals exist for both
$\overline{B}^{0}$ and $\overline{B}_{s}^{0}$ decays in some cases.
The analysis is done applying two different multivariate discriminates in
sequence. The last one is a Boosted Decision Tree (BDT), which is constructed
to have the signal probability flat in BDT, and the background peaked around
zero. The data show a 3.5 standard deviation significance signal from the
final fit. The projection for the final boosted decision tree variable,
BDT$>$0.7 is shown in Fig. 20.
Figure 20: Invariant $\mu^{+}\mu^{-}$ mass distribution of the selected
candidates (black dots) with ${\rm BDT}>0.7$. The result of the fit is
overlaid (blue solid line) and the different components detailed:
$\overline{B}_{s}^{0}$ (red long dashed), $\overline{B}^{0}$ (green medium
dashed), $\overline{B}_{s}^{0}\to h^{+}h^{\prime-}$ (pink dotted),
$\overline{B}^{0}\to\pi^{-}\mu^{+}\nu$ (black short dashed) and
$\overline{B}\to\pi\mu^{+}\mu^{-}$ (light blue dot dashed), and the
combinatorial background (blue medium dashed).
To find a branching fraction one needs to determine not only the number of
signal events and their average detection efficiency, but also the total
number of $\overline{B}_{s}^{0}$ events. This number is inferred by measuring
the number of events in the similar decay channels $B^{-}\to J/\psi K^{-}$ and
$\overline{B}^{0}\to K^{-}\pi^{+}$, using their known branching fractions, and
knowledge of the production ratio for $\overline{B}_{s}^{0}/\overline{B}^{0}$
decay, $f_{s}/f_{d}$ (assuming that $B^{-}$ and $\overline{B}^{0}$ production
are equal.) LHCb determines this ratio from two sources. The most precise is
semileptonic $b$ decays [46], and the second is specific hadronic channels
using a theoretical model. The average of the two methods gives
$f_{s}/f_{d}=0.256\pm 0.020$ [47].333The hadronic channels used are
$\overline{B}^{0}\to D^{+}K^{-}$, $\overline{B}^{0}\to D^{+}\pi^{-}$ and
$\overline{B}_{s}^{0}\to D_{s}^{+}K^{-}$. The hadronic channels also allows an
exploration of the pseudo-rapidity, $\eta$, and $p_{T}$ dependence. It is
found to be independent of $\eta$, but in $p_{T}$ the
$\overline{B}_{s}^{0}/\overline{B}^{0}$ yield decreases as shown in Fig. 21.
For LHCb the $p_{T}$ distribution of the signal is quite similar so little
systematic uncertainty is introduced. Other LHC experiments have in the past
used the LHCb value for $f_{s}/f_{d}$; they will have to take into account
this $p_{T}$ dependency in the future.
Figure 21: The variation of $f_{s}/f_{d}$ as a function of $p_{T}$ measured by
LHCb using fully reconstructed hadronic decays.
The most important implication of this measurement is that it is consistent
with the SM prediction. It rules out many SUSY models especially those with
large values of the parameter $\tan\beta$ [48]. Many NP models are eliminated
or severally constrained when looked combined with other measurements. For
example, the concurrently available upper limit on
${\cal{B}}(\overline{B}^{0}\to\mu^{+}\mu^{-})<0.94\times 10^{-9}$ at 95% CL is
plotted versus ${\cal{B}}(\overline{B}_{s}^{0}\to\mu^{+}\mu^{-})$ for several
models including the SM in Fig. 22(a). Here the space taken by any particular
model is caused by a “reasonable” variation of its parameters. In Fig. 22(b)
the models are plotted in $\phi_{s}$ versus
${\cal{B}}(\overline{B}_{s}^{0}\to\mu^{+}\mu^{-})$ plane. Many models are gone
or mostly eliminated. SM4, a fourth-generation model, is eliminated by the
Higgs discovery. Here the 4th generation quarks would cause the Higgs
production cross-section to be nine times larger and suppress the decay into
$\gamma\gamma$ [49]. Other models with multiple Higgs doublets are allowed for
heavy fourth-generation quarks with masses larger than 400 GeV [50].
Figure 22: (a) Correlation between
${\cal{B}}(\overline{B}_{s}^{0}\to\mu^{+}\mu^{-})$ and
${\cal{B}}(\overline{B}^{0}\to\mu^{+}\mu^{-})$ in minimum flavor violation
(MFV) model, the four-generation SM4 model, and four SUSY models. Correlation
between ${\cal{B}}(\overline{B}_{s}^{0}\to\mu^{+}\mu^{-})$ and the mixing-
induced CP asymmetry $\phi_{s}$ in SM4, the two-Higgs doublet model with
flavour blind phases, and three SUSY models. In both cases, the SM point is
marked by a star. The gray area is ruled out experimentally. In both (a) and
(b) the blue dashed region outlines the allowed region within one standard
deviation of their measured values. Adapted from D. Straub [51].
### 4.2 $\overline{B}\to K^{*}\ell^{+}\ell^{-}$
The process $\overline{B}\to\overline{K}^{*}\ell^{+}\ell^{-}$ is well worth
studying as it proceeds via two SM model loop diagrams, thus allowing for
substantial NP contributions, and it is well predicted in the SM [52]. The
diagram for the specific decay channel
$\overline{B}^{0}\to\overline{K}^{*0}\mu^{+}\mu^{-}$ is shown in Fig. 23.
Although this is an exclusive hadronic decay as contrasted with corresponding
inclusive decay $b\to s\ell^{+}\ell^{-}$, there is sufficient precision in the
theoretical predictions to make this an exceeding interestly mode to study
[52], especially useful as the exclusive decay is much more experimentally
accessible. Indeed, there are many suggestions for variables to use to uncover
NP in this decay. Here I will use just $q^{2}$ the four-momentum transfer
between the $B$ and the $K^{*}$, and the forward-backward asymmetry, $A_{\rm
FB}$, defined as the angle of the $\mu^{+}$ with respect to the $B$ direction
in the di-muon rest frame. An essentially identical diagram can be drawn for
the $\overline{K}\mu^{+}\mu^{-}$, but the spin-0 $K$ provides a less
interesting final state in terms of decay amplitudes.
Figure 23: Decay diagram for
$\overline{B}^{0}\to\overline{K}^{*0}\mu^{+}\mu^{-}$. The isospin related
diagram for $B^{-}$ decays is almost identical, as is the one for $e^{+}e^{-}$
in the final state.
The decay rates as a function of $q^{2}$ from four experiments BaBar [53],
Belle [54], CDF [55]. and LHCb [56] are shown in Fig. 24 and compared with a
theoretical prediction [57]. All four experiments agree with the prediction;
LHCb has the most precise measurements.
Figure 24: Measured decay rates for
$\frac{d{\cal{B}}}{dq^{2}}\left(\overline{B}^{0}\to\overline{K}^{*0}\mu^{+}\mu^{-}\right)$
compared to a theoretical calculation from Bobeth et al. [57].
$A_{\rm FB}$ is more sensitive to NP. The LHCb results and an average of the
data from all four experiments shown in Fig. 25, however, are quite consistent
with the SM prediction. LHCb also measure the point at which $A_{\rm FB}$
crosses zero as $(4.9^{+1.1}_{-1.3})$ GeV2, where the SM predictions range
from 4.0 to 4.3 GeV2 [58].
Figure 25: Data and theory comparison of the forward-backward asymmetry,
$A_{\rm FB}$ for $\overline{B}^{0}\to\overline{K}^{*0}\mu^{+}\mu^{-}$ decay.
The data are from LHCb, and the theory curves are from Bobeth et al. [57]. The
SM prediction is shown shaded with the band covering $\pm 1$ standard
deviation uncertainty estimates. The non-SM curves are labeled with $C_{i}$,
the meaning of which will be discussed in the next section. Also shown is the
data averaged over all four experiments, where the horizontal axis is shifted
somewhat with respect to the LHCb data.
### 4.3 Other rare $b\to s$ processes
Other processes dominated by the $b\to s$ transition can be combined with the
previously discussed ones in a common analysis framework. The formalism
developed starts with writing the effective Hamiltonian as [59]
${\cal H}_{\rm
eff}=-\frac{4\,G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\frac{e^{2}}{16\pi^{2}}\sum_{i}(C_{i}O_{i}+C^{\prime}_{i}O^{\prime}_{i})+\text{h.c.}~{},$
(13)
where the $C_{i}$ and $O_{i}$ represent SM coefficients and operators, and the
$C^{\prime}_{i}$ and $O^{\prime}_{i}$ represent NP coefficients and operators.
The operators most sensitive to NP effects are
$\displaystyle O_{7}$
$\displaystyle=\frac{m_{b}}{e}(\bar{s}\sigma_{\mu\nu}P_{R}b)F^{\mu\nu},$
$\displaystyle O_{8}$
$\displaystyle=\frac{gm_{b}}{e^{2}}(\bar{s}\sigma_{\mu\nu}T^{a}P_{R}b)G^{\mu\nu\,a},$
$\displaystyle O_{9}$
$\displaystyle=(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\ell)\,,$
$\displaystyle O_{10}$
$\displaystyle=(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell)\,,$
$\displaystyle O_{S}$ $\displaystyle=m_{b}(\bar{s}P_{R}b)(\bar{\ell}\ell)\,,$
$\displaystyle O_{P}$
$\displaystyle=m_{b}(\bar{s}P_{R}b)(\bar{\ell}\gamma_{5}\ell)\,,$ (14)
where $m_{b}$ denotes the running $b$ quark mass in the $\overline{\rm MS}$
scheme, $P_{L}=(1-\gamma_{5})/2$, and $P_{R}=(1+\gamma_{5})/2$. The NP
operators $O^{\prime}$ are found by switching $P_{R}$ for $P_{L}$ and vice-
versa [60].
There are other rare $b\to s$ processes that contribute to the NP search. One
important one is the exclusive decay $\overline{B}^{0}\to K^{*0}\gamma$,
$K^{*0}\to K_{s}\pi^{0}$. Time dependent CP violation for this process is
given by
$\frac{\Gamma\left(\overline{B}^{0}(t)\to
K^{*0}\gamma\right)-\Gamma\left({B}^{0}(t)\to
K^{*0}\gamma\right)}{\Gamma\left(\overline{B}^{0}(t)\to
K^{*0}\gamma\right)+\Gamma\left({B}^{0}(t)\to
K^{*0}\gamma\right)}=S_{K^{*}\gamma}\sin\left(\Delta
m_{d}t\right)-C_{K^{*}\gamma}\cos\left(\Delta m_{d}t\right)~{}.$ (15)
Including generic NP operators, the expected asymmetry is parameterized as
$S_{K^{*}\gamma}=\frac{2}{\left|C_{7}\right|^{2}+\left|C^{\prime}_{7}\right|^{2}}{\rm
Im}\left(e^{-2i\beta}C_{7}C^{\prime}_{7}\right).$ (16)
The SM prediction for $S_{K^{*}\gamma}$, where $C^{\prime}_{7}=0$, is
$(-0.023\pm 0.016)$, while the measurement from BaBar and Belle [61] is
$-0.16\pm 0.22$. Clearly reducing the error would be useful, but the
measurement, as we will see, is already useful.
Another useful process is the inclusive decay $b\to s\ell^{+}\ell^{-}$. Here
two four-momentum transfer, $q^{2}$, regions are defined, one at low values
$1<q^{2}<6$ GeV2 which again probes $C^{\prime}_{7}$ and the other with
$q^{2}>14.4$ GeV2 which examines both $C^{\prime}_{9}$ and $C^{\prime}_{10}$.
This is similar to the coefficients probed in the exclusive
$\overline{B}^{0}\to\overline{K}^{*0}\ell^{+}\ell^{-}$ channel.
Putting all of the channels together Altmannshofer and Straub provide
constraints on the real and imaginary parts of the NP parameters shown in Fig.
26. The SM points are at (0,0), consistent with the analysis within $2\sigma$
in all cases.
Figure 26: Individual two standard deviation constraints on the primed Wilson
coefficients as well as combined one and two standard deviation constraints
shown in red (dark and light) from $B\to X_{s}\ell^{+}\ell^{-}$ (brown),
${\cal{B}}(B\to X_{s}\gamma$) (yellow), $A_{\text{CP}}(b\to s\gamma)$
(orange), $B\to K^{*}\gamma$ (purple), $B\to K^{*}\mu^{+}\mu^{-}$ (green),
$B\to K\mu^{+}\mu^{-}$ (blue) and $\overline{B}_{s}\to\mu^{+}\mu^{-}$ (gray)
from [60] .
## 5 Tree level decays with “abnormal” behavior
### 5.1 $B^{-}\to\tau^{-}\overline{\nu}$
Here we depart from the paradigm of considering NP via loop diagrams and look
instead at a the suppressed diagram shown in Fig. 27. The branching fraction
in the SM is given by
${\cal{B}}(B^{-}\to\tau^{-}\overline{\nu})=\frac{G_{F}^{2}m_{B^{-}}\tau_{B^{-}}}{8\pi}m_{\tau}^{2}\left(1-\frac{m_{\tau}^{2}}{m_{B^{-}}^{2}}\right)^{2}f_{B}^{2}\left|V_{ub}\right|^{2},$
(17)
where $V_{ub}$ is a small CKM element and $f_{B}$ is the $B$ meson decay
constant. Helicity suppression, caused by having the spin-0 $B^{-}$ decay into
a left-handed lepton and a right-handed anti-lepton whose helicities align for
massless leptons and thus tend to form a spin-1 system, is partially broken
due to the relative large value of $m_{\tau}$ with respect to $m_{B^{-}}$. New
physics could enter here by having another charged boson, e.g. a virtual
$H^{-}$ also participate in the decay instead of the $W^{-}$ [69].
Figure 27: Feynman diagram for ${\cal{B}}(B^{-}\to\tau^{-}\overline{\nu})$.
The measurements of the branching fraction have been made by BaBar and Belle.
Both collaborations have separate determinations where they either fully
reconstruct a hadronic decay or a semi-leptonic decay tag, and look for
evidence for a $\tau^{-}$ decay on the other side requiring little extra
energy in the event after accounting for the tag and the $\tau^{-}$ decay
products. The experimental measurements have been compiled by Rosner and Stone
[64] and listed in Table 3.
| Experiment | Tag | ${\cal{B}}$ (units of $10^{-4}$)
---|---|---|---
| Belle [65] | Hadronic | $1.79\,^{+0.56\,+0.46}_{-0.49\,-0.51}$
| Belle [66] | Semileptonic | $1.54\,^{+0.38\,+0.29}_{-0.37\,-0.31}$
| Belle | Our average | $1.62\pm 0.40$
| BaBar [67] | Hadronic | $1.80\,^{+0.57}_{-0.54}\pm 0.26$
| BaBar [68] | Semileptonic | $1.7\pm 0.8\pm 0.2$
| BaBar | Average [67] | $1.76\pm 0.49$
| | Our average | $1.68\pm 0.31$
Table 3: Experimental results for ${\cal{B}}(B^{-}\to\tau^{-}\overline{\nu})$
prior to this conference from Rosner and Stone [64].
The Belle average was constructed assuming that the systematic uncertainties
in the two measurements are uncorrelated, while BaBar supplies their average.
The measurements are all very consistent, perhaps too consistent, considering
the quoted uncertainties.
Prior to this conference one inconsistency stood out that could be the first
sign of NP. The CKM fitter [62] and UTfit [63] groups had explored the
consistency of all processes, by relating the the CKM constants $A$,
$\lambda$, $\rho$ and $\eta$ to individual measurements, and deriving values
for these numbers. While the overall consistency was at the $\sim\\!2\sigma$
level, delving into the individual measurements showed a looming difference
between the value of the $CP$ violating asymmetry in $\overline{B}^{0}\to
J/\psi K_{S}^{0}$ decays, $\sin 2\beta$, and the value of
${\cal{B}}\left(B^{-}\to\tau^{-}\overline{\nu}\right)$. The CKM fitter group
showed this discrepancy nicely by predicting the values of $\sin 2\beta$ and
${\cal{B}}\left(B^{-}\to\tau^{-}\overline{\nu}\right)$ without using either
measurement and comparing the prediction with actual measurements; see Fig.
28. The experimental average for $\sin 2\beta$ is consistent with the
predicted value, while the pre-conference value of
${\cal{B}}\left(B^{-}\to\tau^{-}\overline{\nu}\right)$ is higher by a bit more
than $2\sigma$. This problem was taken quite seriously by some: one paper
appeared entitled “Demise of CKM and its aftermath” [70]. As we shall see
shortly, this conclusion may have been a bit premature.
Figure 28: Allowed correlated values of
${\cal{B}}\left(B^{-}\to\tau^{-}\overline{\nu}\right)$ and $\sin 2\beta$,
shown in the banana like shape, compared with the experimental averages, shown
as the small black circle, prior to this conference. The pink x shows the
newly measured value from Belle for hadronic tags only. In all cases the
errors are a combination of statistical and systematic uncertainties.
Figure 29: Distribution of ${\rm E}_{\rm ECL}$ (left) combined for all the
$\tau^{-}$ decays from Belle. The M${}^{2}_{\rm miss}$, distribution (right)
is shown for ${\rm E}_{\rm ECL}<0.2~{}{\rm GeV}$. The solid circles with error
bars are data, while the the solid histograms show the projections of the
fits, with the dashed and dotted histograms indicating the signal and
background components, respectively.
At this conference Belle presented a new measurement using hadronic tags [71].
In doing so, they made several significant advances. First of all they
improved their low momentum tracking efficiency thus obtaining a factor of 2.2
improvement in the number of hadronic tags. Secondly, they increased the data
sample by a factor of 1.7, they also changed their analysis technique
obtaining an improved signal efficiency by a factor of 1.8, and using besides
extra energy, ${\rm E}_{\rm ECL}$, they include information from the missing
mass squared, M${}^{2}_{\rm miss}$, calculated from the hadronic tag and the
$\tau^{-}$ decay products. These distributions are shown in Fig. 29.
The ${\rm E}_{\rm ECL}$ signal distribution peaks at zero but is widened by
detector and initial state radiation effects. The M${}^{2}_{\rm miss}$ signal
distribution is easy to predict from well understood $\tau^{-}$ decays and is
flatter than the background distribution adding to the discrimination power of
the analysis. With these improvements the result is only 40% of its previous
value:
${\cal{B}}\left(B^{-}\to\tau^{-}\overline{\nu}\right)=\left(0.72^{+0.27}_{-0.25}\pm
0.11\right)\times 10^{-4}~{},$ (18)
being only about $3\sigma$ significant. This branching fraction is also shown
in Fig. 28 and is completely consistent with the $\sin 2\beta$ measurement. We
await updates using semileptonic tags from Belle as well as BaBar updates.
### 5.2 $\overline{B}\to D^{(*)}\tau^{-}\overline{\nu}$
The semileptonic decay $\overline{B}\to D^{(*)}\tau^{-}\overline{\nu}$
proceeds in the SM via the tree-level diagram shown in Fig. 5(a), where the
$c$ quark and light spectator quark form a $D$ or $D^{*}$ meson. As in the
previous case a new charged boson could also modify this diagram especially if
its coupling to the $\tau^{-}$ were enhanced [72, 73].
Figure 30: BaBar data for M${}^{2}_{\rm miss}$ and $|p^{*}_{\ell}|$ (see
insets) for M${}^{2}_{\rm miss}>1$ GeV2 for the $D^{+}\ell^{-}$ and
$D^{*+}\ell^{-}$ channels. (Data exist but are not shown for the
$D^{0}\ell^{-}$ and $D^{*0}\ell^{-}$ channels.) In the background component
the region above the dashed line is from charge cross-feed, while the region
below contains continuum and $B\overline{B}$.
BaBar has recently shown a new analysis of these decays [74]. This analysis
also requires tagging, but here only fully hadronic decay tags are used. The
$\tau^{-}$ is observed only in its fully leptonic decays to $\mu^{-}$ or
$e^{-}$, so three neutrinos are missing. The main variables used to
discriminate signal from background are the M${}^{2}_{\rm miss}$ calculated
here using the tag plus the lepton, and the magnitude of the momentum of the
lepton in the $B$ rest frame, $|p^{*}_{\ell}|$. The data are shown in Fig. 30.
The large shaded (blue) peaks correspond to either the $\overline{B}\to
D\ell^{-}\overline{\nu}$ or the $\overline{B}\to D^{*}\ell^{-}\overline{\nu}$
channel and thus in this case constitute a background. The solid black regions
correspond to $D^{**}$ production and also are a background that has a similar
shape to the signal. Only the non-shaded (red) region in the top plot and the
non-shaded (green) region in the bottom plot constitute signal. While it is
difficult to separate the $D^{**}$ from the signal components, BaBar has used
events with an extra $\pi^{0}$ to estimate this contribution directly from the
data.
Results are reported in terms of ratios ${\cal R}(D^{(*)})={\cal
B}(\overline{B}\rightarrow D^{(*)}\tau^{-}\overline{\nu})/{\cal
B}(\overline{B}\rightarrow D^{(*)}\ell^{-}\overline{\nu})$. These quantities
are well predicted in the SM [73]. The comparison is shown in Table 4.
| SM theory | BaBar value | Difference
---|---|---|---
${\cal R}(D)$ | $0.297\pm 0.017$ | $0.440\pm 0.058\pm 0.042$ | +2.0$\sigma$
${\cal R}(D^{*})$ | $0.252\pm 0.003$ | $0.332\pm 0.024\pm 0.018$ | +2.7$\sigma$
Table 4: BaBar results for the ratios ${\cal R}(D^{(*)})$ compared to the SM
prediction.
Taken together, the results disagree with these expectations at the
$3.4\sigma$ level.444The two results are inconsistent the prediction of the
Type II two Higgs doublet model, because they imply different values for the
model parameters.
Belle had previously measured these ratios [75] and found excesses beyond the
SM predictions; their results are listed in Table 5.
| SM theory | Belle value | Difference
---|---|---|---
${\cal R}(D^{0})$ | $0.297\pm 0.017$ | $0.70^{+0.19+0.11}_{-0.18-0.09}$ | +2.0$\sigma$
${\cal R}(D^{+})$ | $0.297\pm 0.017$ | $0.48^{+0.22+0.06}_{-0.19-0.05}$ | +0.9$\sigma$
${\cal R}(D^{*0})$ | $0.252\pm 0.003$ | $0.47^{+0.11+0.06}_{-0.10-0.07}$ | +1.8$\sigma$
${\cal R}(D^{*+})$ | $0.252\pm 0.003$ | $0.48^{+0.14+0.06}_{-0.12-0.04}$ | +1.8$\sigma$
Table 5: Belle results for the ratios ${\cal R}(D^{(*)})$ compared to the SM
prediction.
Note however that the Belle results are not published, even though they were
disseminated in 2009, and have rather large uncertainties compared to BaBar.
It would be really interesting to have an updated analysis using the new
tagging techniques that Belle uses in their most recent
$B^{-}\to\tau^{-}\overline{\nu}$ paper to see if NP in these channels can be
established.
## 6 Other searches: the dark sector and Majorana neutrinos
### 6.1 The dark sector
Could it be that there are 3 classes of matter? One being well known SM
particles with charges [SU(3)xSU(2)xU(1)], the second being dark matter
particles with “dark” charges, and the third being a form of matter that has
both charges, called “mediators.”
In one such proposal [76] the mediator is a particle of the U(1) gauge group
and is nicknamed a “dark photon,” playing on the oxymoronic nature of the
name. In one experimental study BaBar looked for such mediators coupling to
$b$ quarks [77]. The idea was to see if there were any events where the
$\Upsilon(1S)$ decayed into a real photon plus no visible energy. No such
events were found and limits on branching fraction between $\approx
10^{-6}-10^{-4}$ are set, depending on the kinematical distributions of the
photon.
The coupling between the dark sector and the quark sector is specified by
$\epsilon$, and the dark photon mass is labeled as mA′. Several other searches
have been made and reviewed by Echenard [78]. They are summarized in Fig. 31.
There will be more experimental tests of these ideas, although these searches
have already produced significant limits.
Figure 31: Constraints on the mixing parameters, $\epsilon$, as a function of
the hidden photon mass derived from searches in $\Upsilon(2S,3S)$ decays at
BaBar (orange shading) and from other experiments [76, 79, 80] (gray shading).
The red line shows the value of the coupling required to explain the
discrepancy between the calculated and measured anomalous magnetic moment of
the muon [81]. From Echenard [78].
### 6.2 Majorana neutrino production in $B^{-}$ decays
A crucial question in formulating theories beyond the SM is whether or not
neutrinos are normal Dirac spin 1/2 fermions, or if they are their own anti-
particles as suggested by Majorana [82]. If neutrinos are indeed Majorana then
they allow a process called neutrinoless double-$\beta$ decay, shown in Fig.
32(a) [83]. This process can proceed for any value of the Majorana neutrino
mass as the particle is virtual in this diagram. A similar process for $B^{-}$
decays is shown in Fig. 32(b). Each $\mu^{-}$ here can be replaced by an
$e^{-}$ or a $\tau^{-}$.
Figure 32: (a) Feynman diagram for nuclear double-$\beta$ decay. (b) A related
diagram for $B^{-}\to D^{+(*)}\mu^{-}\mu^{-}$ decays.
Belle [84] and LHCb [88] have searched for these decays without success. Upper
limits are given in Table 6. The upper limit in the $e^{-}e^{-}$ mode is not
competitive with nuclear double-$\beta$ decay. The other modes, however, are
unique since the measure coupling of the Majorana neutrino to muons.
Mode | Experiment | Upper limit $\times 10^{-6}$
---|---|---
$B^{-}\to D^{+}e^{-}e^{-}$ | Belle | $<2.6$
$B^{-}\to D^{+}e^{-}\mu^{-}$ | Belle | $<1.8$
$B^{-}\to D^{+}\mu^{-}\mu^{-}$ | Belle | $<1.0$
$B^{-}\to D^{+}\mu^{-}\mu^{-}$ | LHCb | $<0.69$
$B^{-}\to D^{*+}\mu^{-}\mu^{-}$ | LHCb | $<3.6$
Table 6: Upper limits at 90% CL on $B^{-}$ decays to like-sign dilepton final
states.
Majorana neutrinos with mass below that of the $B$ can be searched for
directly. Consider the annihilation diagram shown in Fig. 33. The process is
similar to the one for $B^{-}\to\tau^{-}\overline{\nu}$ shown in Fig. 27, but
here the neutrino being Majorana and on shell can transform into a $\mu^{-}$
and a virtual $W^{+}$, which can materialize into hadrons. In particular the
decay rate into $\pi^{+}$ and $D_{s}^{+}$ has been predicted as a function of
the coupling between the heavy Majorana neutrino and the lepton, $|V_{\mu 4}|$
[86]. 555Similar calculations have been performed for semi-leptonic decays
[87], but these turn out to be less sensitive.
Figure 33: (Feynman diagram for nuclear double-$\beta$ decay. (b) A related
diagram for $B^{-}\to D^{+(*)}\mu^{-}\mu^{-}$ decays.
LHCb has searched for these decays. They did not find any signals but have set
the best experimental limits on $|V_{\mu 4}|$ as a function of the Majorana
neutrino mass, for values just above the pion mass to just below the $B^{-}$
mass, shown in Fig. 34 [88]. Other searches have been carried out using like-
sign dileptons at higher masses [89].
Figure 34: Upper limits on $|V_{\mu 4}|$ at 95% CL as a function of the
Majorana neutrino mass derived from the $B^{-}\to\mu^{-}\mu^{-}\pi^{+}$
channel.
## 7 Conclusions
Although there is no compelling evidence yet for NP, Flavour Physics is very
sensitive to potential effects at high mass scales. All NP theories must
satisfy stringent experimental constraints. LHCb has been very effective at
dispelling effects with marginal statistical significance, although a few
remain. Will some be established when precision increases? Improving
measurements in such decays $\overline{B}\to\overline{K}^{*}\mu^{+}\mu^{-}$,
$\overline{B}_{s}^{0}\to\mu^{+}\mu^{-}$, $B^{-}\to\tau^{-}\overline{\nu}$,
$\overline{B}\to D^{(*)}\tau^{-}\overline{\nu}$, $CP$ violation in
$\overline{B}_{s}^{0}$ decays may show NP effects with increasing precision,
and need to be aggressively pursued. Flavour based experiments also are
looking for evidence that neutrinos are Majorana, and direct links to dark
matter. We are looking forward to defining the next theory beyond the SM
either with our current and near term data or with new future facilities
including Belle-II [90], and the proposed LHCb upgrade [91].
## Acknowledgements
I thank the U. S. National Science Foundation for support. My LHCb colleagues
have provided a great intellectual environment conducive to thinking about
this physics. I want to especially thank Marina Artuso and Liming Zhang for
their help. My scientific secretary, Antonio Limosani, was very helpful in
preparing my conference talk. I also thank the organizers, especially Geoffrey
Taylor, Raymond Volkas, and Elisabetta Barberio who brought us to wonderful
Melbourne.
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|
arxiv-papers
| 2012-12-27T14:10:56 |
2024-09-04T02:49:39.693995
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sheldon Stone",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1212.6374"
}
|
1212.6381
|
# Heavy flavour production in $pp$ collisions and intrinsic quark components
in proton
Joint Institute for Nuclear Research
E-mail A. A. Grinyuk
Joint Institute for Nuclear Research
E-mail [email protected] I. V. Bednyakov
Joint Institute for Nuclear Research
E-mail: [email protected]
###### Abstract:
The LHC data on the forward heavy flavour hadron production can be a new
unique source for estimation of intrinsic heavy quark contributions to the
proton. We discuss in detail the $D$-meson production in $pp$ collisions at
the LHC including the intrinsic charm in the proton. We present also some
predictions for the $K$-meson production in the $pp$ collision at the initial
energies 158 GeV and 7 TeV made within the perturbative QCD including the
intrinsic strangeness in the proton that can be verified in the NA61
experiment and at LHC.
## 1 Intrinsic heavy flavours in the proton
The Large Hadron Collider (LHC) at CERN can be a useful laboratory for
investigation of the unique structure of the proton, in particular for the
study of the parton distribution functions (PDFs) with high accuracy. It is
well known that the precise knowledge of these PDFs is very important for
verification of the Standard Model and search for New Physics.
By definition, the PDF $f_{a}(x,\mu)$ is a function of the proton momentum
fraction $x$ carried by parton $a$ (quark $q$ or gluon $g$) at the QCD
momentum transfer scale $\mu$. For small values of $\mu$, corresponding to the
long distance scales less than $1/\mu_{0}$, the PDF cannot be calculated from
the first principles of QCD (although some progress in this direction has been
recently achieved within the lattice methods [1]). The PDF $f_{a}(x,\mu)$ at
$\mu>\mu_{0}$ can be calculated by solving the perturbative QCD evolution
equations (DGLAP) [2]. The unknown (input for the evolution) functions
$f_{a}(x,\mu_{0})$ can usually be found empirically from some “QCD global
analysis” [3, 4] of a large variety of data, typically at $\mu>\mu_{0}$.
In general, almost all $pp$ processes that took place at the LHC energies,
including the Higgs boson production, are sensitive to the charm
$f_{c}(x,\mu)$ or bottom $f_{b}(x,\mu)$ PDFs. Nevertheless, within the global
analysis the charm content of the proton at $\mu\sim\mu_{c}$ and the bottom
one at $\mu\sim\mu_{b}$ are both assumed to be negligible. Here $\mu_{c}$ and
$\mu_{b}$ are typical energy scales relevant to the $c$\- and $b$-quark QCD
excitation in the proton. These heavy quark components arise in the proton
only perturbatively with increasing $Q^{2}$-scale through the gluon splitting
in the DGLAP $Q^{2}$ evolution [2]. Direct measurement of the open charm and
open bottom production in the deep inelastic processes (DIS) confirms the
perturbative origin of heavy quark flavours [5]. However, the description of
these experimental data is not sensitive to the heavy quark distributions at
relatively large $x$ ($x>0.1$).
As was assumed by Brodsky with coauthors in [6, 7], there are extrinsic and
intrinsic contributions to the quark-gluon structure of the proton. Extrinsic
(or ordinary) quarks and gluons are generated on a short time scale associated
with a large-transverse-momentum processes. Their distribution functions
satisfy the standard QCD evolution equations. Intrinsic quarks and gluons
exist over a time scale which is independent of any probe momentum transfer.
They can be associated with bound-state (zero-momentum transfer regime) hadron
dynamics and are believed to be of nonperturarbative origin. Figure 1 gives a
schematic view of a nucleon, which consists of three valence quarks qv, quark-
antiquark q${\bar{\rm q}}$ and gluon sea, and, for example, pairs of the
intrinsic charm (q${}_{\rm in}^{\rm c}{\bar{\rm q}_{\rm in}^{\rm c}}$) and
intrinsic bottom quarks (q${}_{\rm in}^{\rm b}{\bar{\rm q}_{\rm in}^{\rm
b}}$).
Figure 1: Schematic presentation of a nucleon consisting of three valence
quarks qv, quark-antiquark q${\bar{\rm q}}$ and gluon sea, and pairs of the
intrinsic charm (q${}_{\rm in}^{\rm c}{\bar{\rm q}_{\rm in}^{\rm c}}$) and
intrinsic bottom quarks (q${}_{\rm in}^{\rm b}{\bar{\rm q}_{\rm in}^{\rm
b}}$).
It was shown in [7] that the existence of intrinsic heavy quark pairs
$c{\bar{c}}$ and $b{\bar{b}}$ within the proton state could be due to the
virtue of gluon-exchange and vacuum-polarization graphs. On this basis, within
the MIT bag model [8], the probability to find the five-quark component
$|uudc{\bar{c}}\rangle$ bound within the nucleon bag was estimated to be about
1–2%.
Initially in [6, 7] S.Brodsky with coauthors have proposed existence of the
5-quark state $|uudc{\bar{c}}\rangle$ in the proton (Fig. 1). Later some other
models were developed. One of them considered a quasi-two-body state
${\bar{D}}^{0}(u{\bar{c}})\,{\bar{\Lambda}}_{c}^{+}(udc)$ in the proton [9].
In [9]–[11] the probability to find the intrinsic charm (IC) in the proton
(the weight of the relevant Fock state in the proton) was assumed to be
1–3.5%. The probability of the intrinsic bottom (IB) in the proton is
suppressed by the factor $m^{2}_{c}/m^{2}_{b}\simeq 0.1$ [12], where $m_{c}$
and $m_{b}$ are the masses of the charmed and bottom quarks. Nevertheless, it
was shown that the IC could result in a sizable contribution to the forward
charmed meson production [13]. Furthermore the IC “signal” can constitute
almost 100% of the inclusive spectrum of $D$-mesons produced at high
pseudorapidities $\eta$ and large transverse momenta $p_{T}$ in $pp$
collisions at LHC energies [14].
If the distributions of the intrinsic charm or bottom in the proton are hard
enough and are similar in the shape to the valence quark distributions (have
the valence-like form), then the production of the charmed (bottom) mesons or
charmed (bottom) baryons in the fragmentation region should be similar to the
production of pions or nucleons. However, the yield of this production depends
on the probability to find the intrinsic charm or bottom in the proton, but
this yield looks too small. The PDF which included the IC contribution in the
proton have already been used in the perturbative QCD calculations in
[9]-[11].
The probability distribution for the 5-quark state ($uudc{\bar{c}}$) in the
light-cone description of the proton was first calculated in [6]. Assuming
that the light quark ($u,d$) masses and the proton mass are smaller than the
$c$-quark mass one can get the following form for this probability [10] at
$Q^{2}=m_{c}^{2}$ ($m_{c}=1.69$ is the c-quark mass):
$\displaystyle\frac{dP}{dx}=f_{c}(x)=f_{\bar{c}}(x)={\cal
N}x^{2}\,\left\\{(1-x)(1+10x+x^{2})+6x(1+x)\ln(x)\right\\}~{},$ (1)
where the normalization constant ${\cal N}$ determines some probability
$w_{\rm IC}$ to find the Fock state $|uudc{\bar{c}}\rangle$ in the proton. The
solid line in Fig. 3 shows the intrinsic charm PDF $xf_{c}(x)$ as a function
of $x$ at $Q^{2}=m_{c}^{2}$ when the probability $w_{\rm IC}=3.5$%. The dashed
curve in Fig. 3 is the density distribution of the (ordinary) sea charm in the
proton. One can see from Fig. 3 that the IC distribution (with $w_{\rm
IC}=3.5$% [11]) given by Eq. (1) has rather visible enhancement at $x\sim
0.2-0.3$ and it is much larger (a few orders of magnitude) than the sea
(ordinary) charm density distribution in the proton.
---
Figure 2: The distributions of charmed quarks in the proton; the dashed line
is the sea charmed quarks $c(x)$, the solid curve is the sum of the intrinsic
charm ans the sea one $c(x)+c_{\rm in}(x)$
.
---
Figure 3: The distributions of the valence ($u,d$) quarks and sea quarks in
the proton as a function of the Bjorken variable $x$.
The valence quark density distributions ($xf_{u.d}$), the distributions of the
sea charm ($c_{\rm sea}$) and the intrinsic charm ($xf_{\rm IC}(x)$) at
$Q^{2}=100$ GeV/$c^{2}$ are presented in Fig. 3. One can see from Fig. 3. that
the sea charm and the intrinsic charm distributions are much “smaller” than
the valence ones in the whole region of $x$. However, in the hard $pp$
collisions the gluons and sea quarks make the main contribution to the
inclusive hadron spectra. Therefore, the inclusion of the intrinsic heavy
quark components in the proton makes sense.
Due to the nonperturbative intrinsic heavy quark components one can expect
some excess of the heavy quark PDFs over the ordinary sea quark PDFs at
$x>0.1$. The “signal” of these components can be visible in the observables of
the heavy flavour production in semi-inclusive $ep$ DIS and inclusive $pp$
collisions at high energies. For example, it was recently shown that rather
good description of the HERMES data on the
$xf_{s}(x,Q^{2})+xf_{\bar{s}}(x,Q^{2})$ at $x>0.1$ and $Q^{2}=2.5$ GeV$/c^{2}$
[15] could be achieved due to existence of intrinsic strangeness in the
proton. Similarly, possible existence of the intrinsic charm in the proton can
lead to some enhancement in the inclusive spectra of the open charm hadrons,
in particular $D$-mesons, produced at the LHC in $pp$-collisions at high
pseudorapidities $\eta$ and large transverse momenta $p_{T}$ [14].
## 2 Intrinsic heavy quarks at the LHC
$\bullet~{}$Intrinsic charm
It is known that in the open charm/beauty $pp$-production at large momentum
transfer the hard QCD interactions of two sea quarks, two gluons and a gluon
with a sea quark play the main role. According to the model of hard scattering
[21]–[29] the relativistic invariant inclusive spectrum of the hard process
$p+p\rightarrow h+X$ can be related to the elastic parton-parton subprocess
$i+j\rightarrow i^{\prime}+j^{\prime}$, where $i,j$ are the partons (quarks
and gluons). This spectrum can be presented in the following general form
[25]–[27] (see also [30, 31]):
$\displaystyle
E\frac{d\sigma}{d^{3}p}=\sum_{i,j}\\!\int\\!d^{2}k_{iT}\\!\int\\!d^{2}k_{jT}\\!\int_{x_{i}^{\min}}^{1}dx_{i}\\!\int_{x_{j}^{\min}}^{1}dx_{j}f_{i}(x_{i},k_{iT})f_{j}(x_{j},k_{jT})\frac{d\sigma_{ij}({\hat{s}},{\hat{t}})}{d{\hat{t}}}\frac{D_{i,j}^{h}(z_{h})}{\pi
z_{h}}.$ (2)
Here $k_{i,j}$ and $k_{i,j}^{\prime}$ are the four-momenta of the partons $i$
or $j$ before and after the elastic parton-parton scattering, respectively;
$k_{iT},k_{jT}$ are the transverse momenta of the partons $i$ and $j$; $z$ is
the fraction of the hadron momentum from the parton momentum; $f_{i,j}$ is the
PDF; and $D_{i,j}$ is the fragmentation function (FF) of the parton $i$ or $j$
into a hadron $h$.
When the transverse momenta of the partons are neglected in comparison with
the longitudinal momenta, the variables ${\hat{s}}$, ${\hat{t}}$, ${\hat{u}}$
and $z_{h}$ can be presented in the following forms [25]:
$\displaystyle{\hat{s}}=x_{i}x_{j}s,\quad{\hat{t}}=x_{i}\frac{t}{z_{h}},\quad{\hat{u}}=x_{j}\frac{u}{z_{h}},\quad
z_{h}=\frac{x_{1}}{x_{i}}+\frac{x_{2}}{x_{j}},$ (3)
where
$\displaystyle x_{1}=-\frac{u}{s}=\frac{x_{T}}{2}\cot({\theta}/{2}),\quad
x_{2}=-\frac{t}{s}=\frac{x_{T}}{2}\tan({\theta}/{2}),\quad
x_{T}=2\sqrt{tu}/s=2p_{T}/\sqrt{s}.$ (4)
Here as usual, $s=(p_{1}+p_{2})^{2}$, $t=(p_{1}-p_{1}^{\prime})^{2}$,
$u=(p_{2}-p_{1}^{\prime})^{2}$, and $p_{1}$, $p_{2}$, $p_{1}^{\prime}$ are the
4-momenta of the colliding protons and the produced hadron $h$, respectively;
$\theta$ is the scattering angle for the hadron $h$ in the $pp$ c.m.s. The
lower limits of the integration in (2) are
$\displaystyle
x_{i}^{\min}=\frac{x_{T}\cot(\frac{\theta}{2})}{2-x_{T}\tan(\frac{\theta}{2})},\qquad
x_{j}^{\min}=\frac{x_{i}x_{T}\tan(\frac{\theta}{2})}{2x_{i}-x_{T}\cot(\frac{\theta}{2})}.$
(5)
Actually, the parton distribution functions $f_{i}(x_{i},k_{iT})$ also depend
on the four-momentum transfer squared $Q^{2}$ that is related to the
Mandelstam variables ${\hat{s}},{\hat{t}},{\hat{u}}$ for the elastic parton-
parton scattering [27]
$\displaystyle
Q^{2}~{}=~{}\frac{2{\hat{s}}{\hat{t}}{\hat{u}}}{{\hat{s}}^{2}+{\hat{t}}^{2}+{\hat{u}}^{2}}$
(6)
Calculating spectra by Eq.(2) we used the PDF which includes the IS (and does
not include it) [11], the FF of the type AKK08 [29] and
$d\sigma_{ij}({\hat{s}},{\hat{t}})/d{\hat{t}}$ calculated within the LO QCD
and presented, for example, in [28].
In particular, Fig. 4 shows our estimation of the inclusive yield of single
$D^{0}$-mesons in $pp$-production at $\sqrt{s}=7$ TeV and 10 GeV$/c<p_{T}<25$
GeV$/c$ as a function of the pseudorapidity $\eta$.
Figure 4: The $D+{\bar{D}}_{0}$ distributions (with and without intrinsic
charm contribution) over the pseudorapidity $\eta$ for
$pp\rightarrow(D_{0}+{\bar{D}}_{0})+X$ at $\sqrt{s}=7$ TeV and $10\leq
p_{T}\leq 25$ GeV$/c$ [14].
This estimation is obtained within PYTHIA8, where the PDF CTEQ66 (without the
IC, given by the dashed blue curve in Fig. 4) and CTEQ66c (with the IC for
$w_{\rm IC}\simeq 3.5$%, given by the solid red curve) were used at
$Q^{2}=m_{c}^{2}=$(1.69 GeV)2 [11].
One can see from Fig. 4 that there is some enhancement due to the IC
contribution at large $\eta$, which is due to the above-mentioned enhancement
in the IC PDF at $x>0.1$, given in Fig. 3. Its amount increases with growing
$p_{T}$. For example, due to the IC the spectrum increases by a factor of 2 at
$\eta=4.5$. A similar effect was predicted in [32].
One can see that the Feynman variable $x_{F}$ of the produced hadron, for
example, the $D^{0}$-meson, can be expressed via the variables $p_{T}$ and
$\eta$, or $\theta$ the hadron scattering angle in the $pp$ c.m.s,
$\displaystyle
x_{F}\equiv\frac{2p_{z}}{\sqrt{s}}=\frac{2p_{T}}{\sqrt{s}}\frac{1}{\tan\theta}=\frac{2p_{T}}{\sqrt{s}}\sinh(\eta).$
(7)
At small scattering angles of the produced hadron this formula becomes
$\displaystyle x_{F}\sim\frac{2p_{T}}{\sqrt{s}}\frac{1}{\theta}.$ (8)
It is clear that for fixed $p_{T}$ an outgoing hadron must possess a very
small $\theta$ or very large $\eta$ in order to have large $x_{F}$ (to follow
forward, or backward direction).
In the fragmentation region (of large $x_{F}$) the Feynman variable $x_{F}$ of
the produced hadron is related to the variable $x$ of the intrinsic charm
quark in the proton, and according to the longitudinal momentum conservation
law, the $x_{F}\simeq x$ (and $x_{F}<x$). Therefore, the visible excess of the
solid (red) histogram over the dashed (blue) one in Fig. 4 at $\eta>3.5$ is
due to the enhancement of the IC distribution (see Fig. 3) at $x>$ 0.1.
One expects similar enhancement in the experimental distributions of the open
bottom production due to the (hidden) intrinsic bottom (IB) in the proton,
which could have the PDF very similar to the one given in (1). However, the
probability $w_{\rm IB}$ to find in the proton the Fock state with the IB
contribution $|uudb{\bar{b}}\rangle$ is about 10 times lower than the IC
probability $w_{\rm IC}$ due to the relation $w_{\rm IB}/w_{\rm IC}\sim
m_{c}^{2}/m_{b}^{2}$, where $m_{b}$ is the bottom quark mass [7, 12].
$\bullet~{}$Intrinsic strangeness
Let us analyze now how the possible existence of the intrinsic strangeness in
the proton can be visible in $pp$ collisions. For example, consider the
$K^{-}$-meson production in the process $pp\rightarrow K^{-}+X$. Considering
the intrinsic strangeness in the proton [15] we calculated the inclusive
spectrum $ED\sigma/d^{3}p$ of such mesons within the hard scattering model
(Eq.(2)), which describes satisfactorily the HERA and HERMES data on the DIS.
The FF and the parton cross sections were taken from [29, 28], respectively,
as mentioned above.
Figure 5: The $K^{-}$-meson distributions (with and without intrinsic
strangeness contribution) over the transverse momentum $p_{t}$ for
$pp\rightarrow K^{-}+X$ at the initial energy $E=$ 158 GeV, the rapidity
$y=$1.3 and $p_{t}\geq$ 0.8 GeV$/$c. Figure 6: The $K^{-}$-meson
distributions (with and without intrinsic strangeness contribution) over the
transverse momentum $p_{t}$ for $pp\rightarrow K^{-}+X$ at the initial energy
$E_{p}=$ 158 GeV, the rapidity $y=$1.7 and $p_{t}\geq$ 0.8 GeV$/$c.
In Figs. (5,6) the inclusive $p_{t}$-spectra of $K^{-}$-mesons produced in
$pp$ collision at the initial energy $E_{p}=$158 GeV are presented at the
rapidity $y=$1.3 (Fig 5) and $y=$1.7 (Fig 6). The solid lines in Figs. (5,6)
correspond to our calculation ignoring the intrinsic strangeness (IS) in the
proton and the dashed curves correspond to the calculation including the IS
with the probability about 2.5$\%$, according to [15]. The crosses show the
ratio of our calculation with the IS and without the IS minus 1. One can see
from Figs. (5,6, right axis) that the IS signal can be above 200 $\%$ at $y=$
1.3, $p_{t}=$ 3.6-3.7 Gev$/$c and slightly smaller, than 200 $\%$ at $y=$ 1.7,
$p_{t}\simeq$ 2.5 Gev$/$c. Actually, this is our prediction for the NA61
experiment that is now under way at CERN.
We also calculated the inclusive spectra of $K^{-}$ and $K^{+}$ mesons
produced in the $pp$ collision at LHC energies. Note that the spectra of
$K^{-}$-mesons, which consist of $s$\- and ${\bar{u}}$\- quarks, can give us
information on the IS at large $x_{F}$ ($x_{F}>$0.1) or at the certain region
of $p_{T}$ and $\eta$, according to Eq.(7) and Fig. (3), because they are
produced mainly from the fragmentation of the strange sea ($s_{sea}$) and the
intrinsic strange ($s_{in}$) quarks in the proton. One can see from Fig. (7,
right axis) that the IS signal at $\sqrt{s}=$7 TeV, $y=$4.5 can be about 400
$\%$ at $p_{T}\simeq$ 32-33 GeV$/$c, the measuring of which at LHC can be
difficult. Anyway, at $p_{T}>$15 GeV$/$c and $y=$4.5 the IS signal is about
200 $\%$ and more, as the crosses show in Fig. (7,right axis). However, the
spectra of $K^{+}$-mesons almost do not give us such information because
$K^{+}$-meson consists of the $u$\- and ${\bar{s}}$-quarks and its production
at large $x_{F}$ is due to the fragmentation of the valence $u$-quark by the
$pp$ collision. It illustrates Fig. (8), the notations are the same as in
Figs. (5-7). The similar effect but for the $IC$ signal can be visible in the
inclusive spectra of $D^{-}$\- and $D^{+}$-mesons produced in $pp$ collision
at LHC energies.
Let us also note that we calculated the inclusive spectra of $K$\- and
$D$-mesons within the LO QCD. The NLO calculation, in principle, can change
the inclusive spectra. However, the ratio of the spectrum of $K^{-}$ (or
$D^{-}$) to the spectrum of $K^{+}$ (or $D^{+}$) can not be very sensitive to
the NLO corrections.
Figure 7: The $K^{-}$-meson distributions (with and without intrinsic
strangeness contribution) over the transverse momentum $p_{t}$ for
$pp\rightarrow K^{-}+X$ at $\sqrt{s}=$ 7 TeV, the rapidity $y=$4.5 and
$p_{t}\geq$ 10 GeV$/$c. Figure 8: The $K^{+}$-meson distributions (with and
without intrinsic strangeness contribution) over the transverse momentum
$p_{t}$ for $pp\rightarrow K^{+}+X$ at $\sqrt{s}=$ 7 TeV, the rapidity $y=$4.5
and $p_{t}\geq$ 10 GeV$/$c.
## 3 Conclusion
In this paper we have shown that possible existence of the intrinsic heavy
flavour quark components in the proton can be seen by the forward production
of the open heavy flavour in $pp$-collisions.
Our calculations of the charmed meson production in $pp$ collisions within the
hard scattering model using the PYTHIA8 MC generator [14] and the PDF
including the intrinsic charm [11] showed the following. We found that the
contribution of the intrinsic charm in the proton could be studied in the
production of $D$-mesons in $pp$ collisions at the LHC. The IC contribution
for the single $D^{0}$-meson production can be large, about 100 $\%$ at high
rapidities 3 $\leq y\leq$ 4.5 and large transverse momenta 10 $\leq p_{t}\leq$
25 GeV$/$c. For the double $D^{0}$ production this contribution is not larger
than 30 $\%$ at $p_{t}\geq$ 5 GeV$/$c and 3 $\leq y\leq$ 4.5. These IC
contributions for the single and double $D$-meson production [14] were
obtained with the probability of the intrinsic charm taken to be
$w_{c{\bar{c}}}=$3.5 $\%$ [11], and they will decrease by a factor of 3 when
$w_{c{\bar{c}}}\simeq$ 1 $\%$. Therefore, this value can be verified
experimentally at the LHCb.
The presented predictions could stimulate measurement of the single and double
D-meson production in $pp$ collisions at the CERN LHCb experiment in the
kinematic region mentioned above to observe a possible signal for the
intrinsic charm. The intrinsic beauty in the proton is suppressed by a factor
of 10, therefore its signal in the inclusive spectra of $B$-mesons will
probably be very weak.
We also analyzed the inclusive $K^{-}$-meson production in $pp$ collision at
the initial energy $E_{p}=$158 GeV and gave some predictions for the NA61
experiment going on at CERN, and made some predictions for the inclusive
spectra of $K^{\pm}$-mesons produced in $pp$ at the LHC energy $\sqrt{s}=$ 7
TeV. We showed that in the inclusive spectrum of $K^{-}$-mesons as a function
of $p_{t}$ at the initial energy $E_{p}=$158 GeV and some values of their
rapidities the signal of the intrinsic strangeness can be visible and reach
about 200$\%$ and more at large momentum transfer we took. The signal of the
intrinsic strangeness in the inclusive spectrum of $K^{-}$-mesons produced in
$pp$ collision at $\sqrt{s}=$7 TeV can be about 200$\%$ and even 400$\%$ at
$p_{t}>$15 GeV$/$c, $y=$4.5. The probability of the intrinsic strangeness to
be about 2.5$\%$, as was found from the best description of the HERA and
HERMES data on the DIS, see [15] and references therein.
## 4 Acknowledgements
We are very grateful to A.F Pikelner for his help with the MC calculations. We
thank S.J.Brodsky, M. Gazdzicki, S.M. Pulawski and A.Rustamov for extremely
helpful discussions and recommendations for the predictions on the search for
the possible intrinsic heavy flavour components in $pp$ collisions at high
energies. We are also grateful to V.A.Bednyakov, I.Belyaev, V.Gligorov,
H.Jung, B.Kniehl, B.Z.Kopeliovich, A.Likhoded, P.Spradlin, M. Poghosyan, V.V
Uzhinsky and N.P.Zotov for very helpful discussions. This work was supported
in part by the Russian Foundation for Basic Research, grant No: 11-02-01538-a.
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* [30] V.A. Bednyakov, A.A. Grinyuk, G.I. Lykasov, M. Poghosyan, Nucl.Phys. B 219-220 [Proc.Suppl.] (2011) 225; arXiv:11040532 [hep-ph].
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|
arxiv-papers
| 2012-12-27T14:59:30 |
2024-09-04T02:49:39.705191
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. I. Lykasov, A. A. Grinyuk, I. V. Bednyakov",
"submitter": "Gennady Lykasov I",
"url": "https://arxiv.org/abs/1212.6381"
}
|
1212.6390
|
# Non-backtracking random walk
(Preprint)
Robert Fitzner Department of Mathematics and Computer Science, Eindhoven
University of Technology, 5600 MB Eindhoven, The Netherlands.
[email protected],[email protected] Remco van der Hofstad∗
(December 11, 2012)
###### Abstract
We consider non-backtracking random walk (NBW) in the nearest-neighbor setting
on the ${\mathbb{Z}}^{d}$-lattice and on tori. We evaluate the eigensystem of
the $m\times m$-dimensional transition matrix of NBW where $m$ denote the
degree of the graph. We use its eigensystem to show a functional central limit
theorem for NBW on ${\mathbb{Z}}^{d}$ and to obtain estimates on the
convergence towards the stationary distribution for NBW on the torus.
## 1 Introduction
The non-backtracking walk (NBW) is a simple random walk that is conditioned
not to jump back along the edge it has just traversed. NBW can be viewed as a
Markov chain on the set of directed edges, where the edge has the
interpretation of being the last edge traversed by the NBW, but we will not
rely on this interpretation. We study NBWs on ${\mathbb{Z}}^{d}$ in the
nearest-neighbor setting, and NBWs on tori in various settings, derive the
transition matrix for the NBW and analyse its eigensystem in Fourier space. We
use this to study asymptotic properties of the NBW, such as its Green’s
function and a functional central limit theorem (CLT) on ${\mathbb{Z}}^{d}$,
and its convergence towards the stationary distribution on the torus. In
particular, our analysis allows us to give an explicit formula for the Fourier
transform of the number of $n$-step NBWs traversing fixed positions at given
times. We use this to prove that the finite-dimensional distributions of the
NBW displacements, after diffusive rescaling, converge to those of Brownian
motion. By an appropriate tightness argument, this proves a functional CLT. We
further evaluate the Fourier transform of the NBW $n$-step transition
probabilities on the torus to identify when the NBW transition probabilities
are close to the stationary distribution. Our paper is inspired by the study
of various high-dimensional statistical mechanical models. For example, our
derivation allows us to give detailed estimates of the probability that NBW on
a torus is at a given vertex, a fact used in the analysis of hypercube
percolation in [6].
NBWs have been investigated on finite graphs in [3], in particular expanders,
where it was shown that NBWs mix faster than ordinary random walks. See also
[4], where a Poisson limit was proved for the number of visits of an $n$-step
NBW to a vertex in an $r$-regular graph of size $n$. In the nearest-neighbor
setting on ${\mathbb{Z}}^{d}$, NBWs were investigated in [9, Section 5.3],
where an explicit expression of its Green’s function and many related
properties are derived (see also [13, Exercise 3.8]). Noonan [10] investigates
the generating function of NBWs, and his results also apply to walks that
avoid their last 7 previous positions (i.e., with memory up to 8), and were
used to improve the known upper bounds on the SAW connective constant. See
[12] for an extension up to memory 22 for $d=2$, further improving the upper
bound on the SAW connective constant. The methods in [10, 12] allow to compute
the generating function of the number of memory-$k$ SAWs for appropriate
values of $k$, but do not investigate the number of memory-$k$ SAWs ending in
a particular position in ${\mathbb{Z}}^{d}$. Finally, [11] studies relations
between the exponential growth of the transition probabilities of NBW and non-
amenability of the underlying graph. For interesting connections to zeta
functions on graphs, which allow one to study the number of NBWs of arbitrary
length ending in the starting point, we refer the reader to [7].
This paper is oganized as follows. In Section 2 we investigate NBW on
${\mathbb{Z}}^{d}$ and in Section 3 we study NBW on tori.
## 2 Non-backtracking random walk on ${\mathbb{Z}}^{d}$
### 2.1 The setting
An _$n$ -step nearest-neighbor simple random walk_ (SRW) on ${\mathbb{Z}}^{d}$
is an ordered $(n+1)$-tuple
$\omega=(\omega_{0},\omega_{1},\omega_{2},\dots,\omega_{n})$, with
$\omega_{i}\in{\mathbb{Z}}$ and $\|\omega_{i}-\omega_{i+1}\|_{1}=1$, where
$\|x\|_{1}=\sum_{i=1}^{d}|x_{i}|$. We always take $\omega_{0}=(0,0,\dots,0)$.
The step distribution of SRW is given by
$\displaystyle
D(x)=\frac{1}{2d}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\|x\|_{1}=1\\}},$
(2.1)
where
${\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{A}$
is the indicator of the event $A$. If an $n$-step SRW $\omega$ additionally
satisfies that $\omega_{i}\not=\omega_{i+2}$, then we call the walk a _non-
backtracking walk_ (NBW). As the problem of NBW is trivial for $d=1$, we
always assume that $d\geq 2$. For the NBW we also count walks conditioned not
to take their first step in a certain direction $\iota$. We exclusively use
the Greek letters $\iota$ and $\kappa$ for values in
$\\{-d,-d+1,\dots,-1,1,2,\dots,d\\}$ and denote by
${e}_{\iota}\in{\mathbb{Z}}^{d}$ the unit vector in direction $\iota$, i.e.
$({e}_{\iota})_{\kappa}=\text{sign}(\iota)\delta_{|\iota|,\kappa}$ (beware of
the minus sign when $\iota$ is negative, which is somewhat different from the
usual choice of a unit vector). Let $b_{n}(x)$ be the number of $n$-step NBWs
with $\omega_{n}=x$, and $b^{\iota}_{n}(x)$ the number of $n$-step NBWs
$\omega$ with $\omega_{n}=x$ and $\omega_{1}\not={e}_{\iota}$. For $n\geq 1$,
the following relations between these objects hold:
$\displaystyle b_{n}(x)$ $\displaystyle=$ $\displaystyle\sum_{\iota\in\\{\pm
1,\dots,\pm d\\}}b^{-\iota}_{n-1}(x-{e}_{\iota}),$ (2.2) $\displaystyle
b_{n}(x)$ $\displaystyle=$ $\displaystyle
b^{\iota}_{n}(x)+b^{-\iota}_{n-1}(x-{e}_{\iota})\quad\forall\iota,$ (2.3)
$\displaystyle b^{\iota}_{n}(x)$ $\displaystyle=$
$\displaystyle\sum_{\kappa\in\\{\pm 1,\dots,\pm
d\\}\setminus\\{\iota\\}}b^{-\kappa}_{n-1}(x-{e}_{\kappa}).$ (2.4)
We analyse $b_{n}$ and $b^{\iota}_{n}$ using _Fourier theory_. For an
absolutely summable function $f\colon{\mathbb{Z}}^{d}\mapsto{\mathbb{C}}$, we
define its Fourier transform by
$\displaystyle\hat{f}(k)=\sum_{x\in{\mathbb{Z}}}f(x){\mathrm{e}}^{{\mathrm{i}}k\cdot
x}\qquad(k\in[-\pi,\pi]^{d}),$ (2.5)
where $k\cdot x=\sum^{d}_{s=1}k_{s}x_{s}$ and ${\mathrm{i}}$ denotes the
imaginary unit. We use $k$ exclusively to denote values in the Fourier dual
space $[-\pi,\pi]^{d}$. For $f,g\colon{\mathbb{Z}}^{d}\mapsto{\mathbb{C}}$ we
denote their convolution by $f\star g$, i.e.,
$\displaystyle(f\star g)(x)=\sum_{y\in{\mathbb{Z}}}f(y)g(x-y),$ (2.6)
and note that the Fourier transform of $f\star g$ is given by
$\hat{f}\hat{g}$. Applying the Fourier transformation to (2.1)-(2.4) yields
$\displaystyle\hat{D}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{d}\sum_{i=1}^{d}\cos(k_{\iota}),$ (2.7)
$\displaystyle\hat{b}_{n}(k)$ $\displaystyle=$
$\displaystyle\sum_{\iota\in\\{\pm 1,\dots,\pm
d\\}}\hat{b}^{-\iota}_{n-1}(k){\mathrm{e}}^{{\mathrm{i}}k_{\iota}},$ (2.8)
$\displaystyle\hat{b}_{n}(k)$ $\displaystyle=$
$\displaystyle\hat{b}^{\iota}_{n}(k)+\hat{b}^{-\iota}_{n-1}(k){\mathrm{e}}^{{\mathrm{i}}k_{\iota}}\quad\forall\iota,$
(2.9) $\displaystyle\hat{b}^{\iota}_{n}(k)$ $\displaystyle=$
$\displaystyle\sum_{\kappa\in\\{\pm 1,\dots,\pm
d\\}\setminus\\{\iota\\}}\hat{b}^{-\kappa}_{n-1}(x){\mathrm{e}}^{{\mathrm{i}}k_{\kappa}}.$
(2.10)
In our further analysis, we use ${\mathbb{C}}^{2d}$-valued and
${\mathbb{C}}^{2d}\times{\mathbb{C}}^{2d}$-valued functions. For a clear
distinction between scalar-, vector- and matrix-valued quantities, we always
write ${\mathbb{C}}^{2d}$-valued functions with a vector arrow (e.g.
$\vec{v}$) and matrix-valued functions with bold capital letters (e.g. ${\bf
M}$). We do not use $\\{1,2,\dots,2d\\}$ for the indices of the elements of a
vector or a matrix, but use $\\{-d,-d+1,\dots,-1,1,2,\dots,d\\}$ instead.
Here, for a negative index $\iota\in\\{-d,-d+1,\dots,-1\\}$ and a vector
$k\in[-\pi,\pi]^{d}$, we define $k_{\iota}:=-k_{|\iota|}$.
We denote the identity matrix by ${\bf I}\in{\mathbb{C}}^{2d\times 2d}$ and
the all-one vector by ${\vec{1}}=(1,1,\dots,1)^{T}\in{\mathbb{C}}^{2d}$.
Moreover we define the matrices ${\bf C},{\bf J}\in{\mathbb{C}}^{2d\times 2d}$
by $({\bf C})_{\iota,\kappa}=1$ and $({\bf
J})_{\iota,\kappa}=\delta_{\iota,-\kappa}$. To characterize the displacement
of a step in a direction $\iota$, we define the diagonal matrix
${\bf\hat{D}}(k)$ with the entries
$({\bf\hat{D}}(k))_{\iota,\iota}={\mathrm{e}}^{{\mathrm{i}}k_{\iota}}$. We
define the vector ${\vec{b}}_{n}(k)$ with entries
$\left({\vec{b}}_{n}(k)\right)_{\iota}=\hat{b}^{\iota}_{n}(k)$. Then, we can
rewrite (2.8)-(2.10) as
$\displaystyle\hat{b}_{n}(k)$ $\displaystyle=$
$\displaystyle{\vec{1}}^{T}{\bf\hat{D}}(-k){\vec{b}}_{n-1}(k),$ (2.11)
$\displaystyle\hat{b}_{n}(k){\vec{1}}$ $\displaystyle=$
$\displaystyle{\vec{b}}_{n}(k)+{\bf\hat{D}}(k){\bf J}{\vec{b}}_{n-1}(k),$
(2.12) $\displaystyle{\vec{b}}_{n}(k)$ $\displaystyle=$ $\displaystyle({\bf
C}-{\bf J}){\bf\hat{D}}(-k){\vec{b}}_{n-1}(k)=\left(({\bf C}-{\bf
J}){\bf\hat{D}}(-k)\right)^{n}{\vec{1}}.$ (2.13)
We define the transition matrix
${\bf\hat{A}}(k)=({\bf C}-{\bf J}){\bf\hat{D}}(-k),$ (2.14)
so that
$({\bf\hat{A}}(k))_{\iota,\kappa}={\mathrm{e}}^{-ik_{\iota}}(1-\delta_{\iota,-\kappa})$.
With this notation in hand, we are ready to identify the NBW Green’s function.
### 2.2 The Green’s function
We start by deriving a formula for the NBW Green’s function using the
relations in (2.11)-(2.11). While these results are not novel, the analysis
presented here is efficient and simple. We define the NBW Green’s function as
the generating function of $\hat{b}_{n}$ and $\hat{b}^{\iota}_{n}$:
$\displaystyle\hat{B}_{z}(x)$ $\displaystyle:=$
$\displaystyle\sum_{n=0}^{\infty}\hat{b}_{n}(k)z^{n},\qquad\qquad\hat{B}^{\iota}_{z}(x):=\sum_{n=0}^{\infty}\hat{b}^{\iota}_{n}(k)z^{n}$
(2.15) $\displaystyle{\vec{B}}_{z}(k)^{T}$ $\displaystyle:=$
$\displaystyle(\hat{B}^{1}_{z}(k),\hat{B}^{-1}_{z}(k),\hat{B}^{2}_{z}(k),\dots,\hat{B}^{-d}_{z}(k)),$
(2.16)
where $\vec{y}^{T}$ denotes the transpose of the vector
$\vec{y}\in{\mathbb{R}}^{d}$. By (2.11)-(2.12),
$\displaystyle\hat{B}_{z}(k)$ $\displaystyle=$ $\displaystyle
1+z{\vec{1}}^{T}{\bf\hat{D}}(-k)\vec{B}_{z}(k),$ (2.17)
$\displaystyle\hat{B}_{z}(k){\vec{1}}$ $\displaystyle=$
$\displaystyle\vec{B}_{z}(k)+z{\bf\hat{D}}(k){\bf J}\vec{B}_{z}(k)\
\Rightarrow\ \vec{B}_{z}(k)=\left[{\bf I}+z{\bf\hat{D}}(k){\bf
J}\right]^{-1}{\vec{1}}\hat{B}_{z}(k),$ (2.18)
Using ${\bf\hat{D}}(k){\bf J}{\bf\hat{D}}(k){\bf J}={\bf I}$, it is easy to
check that
$\displaystyle\left[{\bf I}+z{\bf\hat{D}}(k){\bf J}\right]^{-1}$
$\displaystyle=$ $\displaystyle\frac{1}{1-z^{2}}\left({\bf
I}-z{\bf\hat{D}}(k){\bf J}\right),$ (2.19)
and we use (2.17)-(2.19) to obtain
$\displaystyle\hat{B}_{z}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{1-z{\vec{1}}^{T}{\bf\hat{D}}(-k)\left[{\bf
I}+z{\bf\hat{D}}(k){\bf
J}\right]^{-1}{\vec{1}}}=\frac{1-z^{2}}{1+(2d-1)z^{2}-2dz\hat{D}(k)}.$ (2.20)
Note in particular that
$\hat{B}_{z}(k)=\frac{1-z^{2}}{1+(2d-1)z^{2}}\hat{C}_{\mu_{z}}(k),$ (2.21)
where $\mu_{z}=2dz/(1+(2d-1)z^{2})$ and $\hat{C}_{\mu}(k)=1/[1-\mu\hat{D}(k)]$
is the SRW Green’s function. By (2.18),
$\displaystyle\vec{B}_{z}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{1-z^{2}}\left[{\bf I}-z{\bf\hat{D}}(k){\bf
J}\right]{\vec{1}}\hat{B}_{z}(k)=\frac{\left[{\bf
I}-z{\bf\hat{D}}(k)\right]{\vec{1}}}{1+(2d-1)z^{2}-2dz\hat{D}(k)},$
so that
$\displaystyle\hat{B}^{\iota}_{z}(k)$ $\displaystyle=$
$\displaystyle\frac{1-z{\mathrm{e}}^{{\mathrm{i}}k_{\iota}}}{1+(2d-1)z^{2}-2dz\hat{D}(k)}.$
(2.22)
### 2.3 The transition matrix
#### The eigensystem.
We start by evaluating the transition matrix (2.14) by characterizing its
eigenvalues and eigenvectors:
###### Lemma 2.1 (Dominant eigenvalues).
For $d\geq 2$ and $k\in(-\pi,\pi)^{d}$, let ${\bf\hat{A}}(k)$ be the matrix
given in (2.14). Then
$\hat{\lambda}_{\pm 1}=\hat{\lambda}_{\pm
1}(k)=d\hat{D}(k)\pm\sqrt{(d\hat{D}(k))^{2}-(2d-1)}$ (2.23)
are eigenvalues of ${\bf\hat{A}}(k)$. For $k\not=(0,0,\dots,0)^{T}$, the right
eigenvectors ${\vec{v}}^{\scriptscriptstyle(\pm 1)}$ to the eigenvalue
$\hat{\lambda}_{\pm 1}$ are given by
$\displaystyle{\vec{v}}^{\scriptscriptstyle(\pm 1)}=\hat{\lambda}_{\pm
1}{\vec{1}}\pm{\bf\hat{D}}(k){\vec{1}}.$ (2.24)
For $k=(0,0,\dots,0)^{T}$, the eigenvectors are given by
${\vec{v}}^{\scriptscriptstyle(1)}(0)=(2d-2){{\vec{1}}}$ and
${\vec{v}}^{\scriptscriptstyle(-1)}:=(1,-1,0,0,0,\dots,0)\in\mathbb{Z}^{2d}$.
As we will see below, Lemma 2.1 yields the two most important eigenvalues.
When $\hat{\lambda}_{+}=\hat{\lambda}_{-}$, which occurs when
$(d\hat{D}(k))^{2}-(2d-1)=0$, it turns out that
${\vec{v}}^{\scriptscriptstyle(1)}$ has geometric multiplicity $1$, and that
${\vec{1}}$ is a generalized eigenvector satisfying
${\bf\hat{A}}(k){\vec{1}}={\vec{v}}^{\scriptscriptstyle(1)}+\hat{\lambda}_{+}{\vec{1}}$.
We continue by computing the remaining eigenvalues and -vectors:
###### Lemma 2.2 (Simple eigenvalues).
For $d\geq 2$, $k\in(-\pi,\pi)^{d}$, let ${\bf\hat{A}}(k)$ be the matrix given
in (2.14) and ${\vec{e}}_{\iota}\in{\mathbb{C}}^{2d}$ the $\iota$th unit
vector, i.e., $({\vec{e}}_{\iota})_{\kappa}=\delta_{\iota,\kappa}$ for
$\kappa\in\\{\pm 1,\dots,\pm d\\}$.
For $\iota\in\\{2,3,\dots,d\\}$, let
$\displaystyle{\vec{v}}^{\scriptscriptstyle(\iota)}$ $\displaystyle=$
$\displaystyle(1+{\mathrm{e}}^{{\mathrm{i}}k_{\iota}})({\mathrm{e}}^{{\mathrm{i}}k_{\iota}}{\vec{e}}_{1}+{\vec{e}}_{-1})-(1+{\mathrm{e}}^{{\mathrm{i}}k_{1}})({\mathrm{e}}^{{\mathrm{i}}k_{\iota}}{\vec{e}}_{\iota}+{\vec{e}}_{-\iota})\qquad\
\forall k\in[-\pi,\pi]^{d}$
$\displaystyle{\vec{v}}^{\scriptscriptstyle(-\iota)}$ $\displaystyle=$
$\displaystyle(1-{\mathrm{e}}^{{\mathrm{i}}k_{\iota}})({\mathrm{e}}^{{\mathrm{i}}k_{1}}{\vec{e}}_{1}-{\vec{e}}_{-1})-(1-{\mathrm{e}}^{{\mathrm{i}}k_{1}})({\mathrm{e}}^{{\mathrm{i}}k_{\iota}}{\vec{e}}_{\iota}-{\vec{e}}_{-\iota})\qquad\forall
k\in[-\pi,\pi]^{d}\text{ if }{\mathrm{e}}^{{\mathrm{i}}k_{1}}\not=1,$
Then, ${\vec{v}}^{\scriptscriptstyle(\pm\iota)}$ is an eigenvector of
${\bf\hat{A}}(k)$ to the eigenvalue $\mp 1$. Both eigenvalues have a
geometrical multiplicity of $d-1$ for all $k$.
Lemmas 2.1–2.2 identify a collection of $2d$ independent eigenvectors, and
thus the complete eigensystem, of ${\bf\hat{A}}(k)$. Now we prove these two
lemmas:
###### Proof of Lemma 2.1.
Let $\hat{\lambda}\in\\{\hat{\lambda}_{1},\hat{\lambda}_{-1}\\}$ and
${\vec{v}}=\hat{\lambda}{\bf 1}-{\bf\hat{D}}(k){\bf 1}$. The values
$\hat{\lambda}_{1}$ and $\hat{\lambda}_{-1}$ are the solutions of the
quadratic equation
$\displaystyle\hat{\lambda}^{2}=2d\hat{\lambda}\hat{D}(k)-(2d-1).$ (2.25)
Using ${\bf C}{\bf\hat{D}}(-k){\vec{1}}=2d\hat{D}(k){\vec{1}}$ and ${\bf
J}{\bf\hat{D}}(-k)={\bf\hat{D}}(k){\bf J}$, we compute
$\displaystyle{\bf\hat{A}}(k){\vec{v}}$ $\displaystyle=$ $\displaystyle({\bf
C}-{\bf J}){\bf\hat{D}}(-k)(\hat{\lambda}{\bf
I}-{\bf\hat{D}}(k)){\vec{1}}=\left(2d\hat{\lambda}\hat{D}(k){\bf
I}-\hat{\lambda}{\bf\hat{D}}(k)-(2d-1){\bf I}\right){\vec{1}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{e:quadend}}}{{=}}$
$\displaystyle\left(\hat{\lambda}^{2}{\bf
I}-\hat{\lambda}{\bf\hat{D}}(k)\right){\vec{1}}=\hat{\lambda}\vec{v}.$
This proves that $\vec{v}$ is a eigenvector of ${\bf\hat{A}}(k)$ corresponding
to the eigenvalue $\hat{\lambda}$ for all $k\neq 0$ and also for the case of
$\iota=1$ for $k=0$. For $k=(0,\dots,0)$ we note that
$\hat{\lambda}_{-1}(0)=1$ and see that
$\displaystyle{\bf\hat{A}}(0){\vec{v}}^{\scriptscriptstyle(-1)}(0)=({\bf
C}-{\bf J}){\vec{v}}^{\scriptscriptstyle(-1)}(0)=-{\bf
J}{\vec{v}}^{\scriptscriptstyle(-1)}(0)={\vec{v}}^{\scriptscriptstyle(-1)}(0).$
∎
###### Proof of Lemma 2.2.
For $\iota\in\\{1,2,\dots,d\\}$, the vectors
$\displaystyle\vec{u}^{\scriptscriptstyle(\iota)}={\mathrm{e}}^{{\mathrm{i}}k_{\iota}}{\vec{e}}_{\iota}+{\vec{e}}_{-\iota}\qquad\text{and}\qquad\vec{u}^{\scriptscriptstyle(-\iota)}$
$\displaystyle=$
$\displaystyle{\mathrm{e}}^{{\mathrm{i}}k_{\iota}}{\vec{e}}_{\iota}-{\vec{e}}_{-\iota}$
are eigenvectors of ${\bf J}{\bf\hat{D}}(-k)$, where
$\vec{u}^{\scriptscriptstyle(\pm\iota)}$ is associated to the eigenvalue $\pm
1$. For $\iota\in\\{2,3,\dots,d\\}$, we define
${\vec{v}}^{\scriptscriptstyle(\iota)}=\vec{u}^{\scriptscriptstyle(1)}\sum_{\kappa}\vec{u}^{\scriptscriptstyle(\iota)}_{\kappa}-\vec{u}^{\scriptscriptstyle(\iota)}\sum_{\kappa}\vec{u}^{\scriptscriptstyle(1)}_{\kappa},\qquad\qquad{\vec{v}}^{\scriptscriptstyle(-\iota)}=\vec{u}^{\scriptscriptstyle(-1)}\sum_{\kappa}\vec{u}^{\scriptscriptstyle(-\iota)}_{\kappa}-\vec{u}^{\scriptscriptstyle(-\iota)}\sum_{\kappa}\vec{u}^{\scriptscriptstyle(-1)}_{\kappa}.$
(2.26)
By construction, ${\vec{v}}^{\scriptscriptstyle(\iota)}$ and
${\vec{v}}^{\scriptscriptstyle(-\iota)}$ are also eigenvalues of ${\bf
J}{\bf\hat{D}}(-k)$. For ${\bf C}{\bf\hat{D}}(-k)$, we compute that
${\bf C}{\bf\hat{D}}(-k)\vec{u}^{\scriptscriptstyle(\iota)}={\bf
C}({\vec{e}}_{\iota}+{\mathrm{e}}^{{\mathrm{i}}k_{\iota}}{\vec{e}}_{-\iota})=\sum_{\kappa}\vec{u}^{\scriptscriptstyle(\iota)}_{\kappa}{\vec{1}},\qquad\qquad{\bf
C}{\bf\hat{D}}(-k)\vec{u}^{\scriptscriptstyle(-\iota)}={\bf
C}({\vec{e}}_{\iota}-{\mathrm{e}}^{{\mathrm{i}}k_{\iota}}{\vec{e}}_{-\iota})=-\sum_{\kappa}\vec{u}^{\scriptscriptstyle(-\iota)}_{\kappa}{\vec{1}},$
so that
$\displaystyle{\bf C}{\bf\hat{D}}(-k)\vec{v}^{\scriptscriptstyle(\iota)}={\bf
C}{\bf\hat{D}}(-k)\vec{v}^{\scriptscriptstyle(-\iota)}=0.$
Knowing this, it follows that
$\displaystyle{\bf\hat{A}}(k)\vec{v}^{\scriptscriptstyle(\iota)}=({\bf C}-{\bf
J}){\bf\hat{D}}(-k)\vec{v}^{\scriptscriptstyle(\iota)}=-\vec{v}^{\scriptscriptstyle(\iota)},$
(2.27) $\displaystyle{\bf\hat{A}}(k)\vec{v}^{\scriptscriptstyle(-\iota)}=({\bf
C}-{\bf
J}){\bf\hat{D}}(-k)\vec{v}^{\scriptscriptstyle(\iota)}=\vec{v}^{\scriptscriptstyle(-\iota)}.$
(2.28)
By (2.27), $\vec{v}^{\scriptscriptstyle(\iota)}$ is an eigenvector for all
$k$. Since the set of vectors
$(\vec{v}^{\scriptscriptstyle(\iota)})_{\iota=2,3,\dots,d}$ is linearly
independent we know that the eigenvalue $-1$ has geometric multiplicity $d-1$.
From (2.28), we conclude the existence of $d-1$ linear independent eigenvalue
for $1$ only when ${\mathrm{e}}^{{\mathrm{i}}k_{\iota}}\neq 1$ for all
$\iota\in\\{1,2,\dots,d\\}$. To prove that the eigenvalue $1$ has geometric
multiplicity $d-1$ for all $k$, we show how to choose $d-1$ linear independent
eigenvectors when ${\mathrm{e}}^{{\mathrm{i}}k_{\kappa}}=1$ for a
$\kappa\in\\{1,2,\dots,d\\}$. For this, let $S_{1}$ be the set of all
$\kappa\in\\{1,2,\dots,d\\}$ with $k_{\kappa}=0$ and $S_{2}$ the set of all
$\kappa\in\\{1,2,\dots,d\\}$ with $k_{\kappa}\neq 0$. Let $s_{1}$ and $s_{2}$
be the number of elements in $S_{1}$ and $S_{2}$. Then $s_{1}+s_{2}=d$. For
all $\iota\in S_{1}$, we define
${\vec{v}}^{\scriptscriptstyle(-\iota)}={e}_{\iota}-{e}_{-\iota}$.
If $s_{1}=d$, then $k=0$. In Lemma 2.1, we define for this case
$\hat{\lambda}_{2}=1$ and
${\vec{v}}^{\scriptscriptstyle(-1)}={e}_{1}-{e}_{-1}$. Then
$\\{\vec{v}^{\scriptscriptstyle(-1)},\vec{v}^{\scriptscriptstyle(-2)},\dots,\vec{v}^{\scriptscriptstyle(-d)}\\}$
is a set of independent eigenvectors of ${\bf\hat{A}}(k)$ to the eigenvalue
$1$.
If $s_{1}<d$, then let $\rho$ be the smallest number in $S_{2}$ and define
$\displaystyle{\vec{v}}^{\scriptscriptstyle(-\iota)}$ $\displaystyle=$
$\displaystyle\vec{u}^{\scriptscriptstyle(-\rho)}\sum_{\kappa}\vec{u}^{\scriptscriptstyle(-\iota)}_{\kappa}-\vec{u}^{\scriptscriptstyle(-\iota)}\sum_{\kappa}\vec{u}^{\scriptscriptstyle(-\rho)}_{\kappa}.$
for all $\iota\in S_{2}\setminus\\{\kappa\\}$. Then it is easy to verify that
the vectors $(\vec{v}^{\scriptscriptstyle(-\iota)})_{\iota=2,3,\dots,d}$ are
linearly independent and are eigenvectors of ${\bf\hat{A}}(k)$ with eigenvalue
$1$. ∎
We use the eigensystem of the matrix ${\bf\hat{A}}(k)$ to identify
$\hat{b}_{n}(k)$ and $\vec{b}_{n}(k)$:
###### Lemma 2.3 (NBW characterization).
Let $d\geq 2$, $n\geq 1$ and $k\in(-\pi,\pi^{d})$ such that
$\hat{\lambda}_{1}(k)\neq\hat{\lambda}_{-1}(k)$. Then,
$\displaystyle\hat{b}_{n}(k)$ $\displaystyle=$ $\displaystyle
2d\frac{\hat{D}(k)(\hat{\lambda}^{n}_{1}(k)-\hat{\lambda}^{n}_{-1}(k))+\hat{\lambda}^{n-1}_{-1}(k)-\hat{\lambda}^{n-1}_{1}(k)}{\hat{\lambda}_{1}(k)-\hat{\lambda}_{-1}(k)},$
(2.29) $\displaystyle\vec{b}_{n}(k)$ $\displaystyle=$
$\displaystyle\frac{\hat{\lambda}^{n}_{1}(k)-\hat{\lambda}^{n}_{-1}(k)}{\hat{\lambda}_{1}(k)-\hat{\lambda}_{-1}(k)}{\bf\hat{D}}(k){\vec{1}}-\frac{\hat{\lambda}^{n-1}_{1}(k)-\hat{\lambda}^{n-1}_{-1}(k)}{\hat{\lambda}_{1}(k)-\hat{\lambda}_{-1}(k)}{\vec{1}}.$
(2.30)
When $\hat{\lambda}_{1}(k)=\hat{\lambda}_{-1}(k)$,
$\hat{b}_{n}(k)=2d[(n-1)\hat{\lambda}_{1}(k)^{n-2}+\hat{D}(k)n\hat{\lambda}_{1}(k)^{n-1}],\qquad\vec{b}_{n}(k)=[(n+1)\hat{\lambda}_{1}(k)^{n}+n\hat{\lambda}_{1}(k)^{n-1}{\bf\hat{D}}(k)]{\vec{1}}.$
(2.31)
Clealry, one can reprove (2.20) and (2.22) using Lemma 2.3.
###### Proof.
For $\hat{\lambda}_{1}(k)\neq\hat{\lambda}_{-1}(k)$, we define
$\displaystyle\alpha(k)$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{\lambda}_{1}(k)-\hat{\lambda}_{-1}(k)}=\frac{1}{2\sqrt{(d\hat{D}(k))^{2}-(2d-1)}}.$
We can write
$\displaystyle{\vec{1}}$ $\displaystyle=$
$\displaystyle\alpha(k)\left(\hat{\lambda}_{1}(k){\bf
I}-\hat{\lambda}_{-1}(k){\bf
I}+{\bf\hat{D}}(k)-{\bf\hat{D}}(k)\right){\vec{1}}=\alpha(k)\vec{v}^{\scriptscriptstyle(1)}(k)-\alpha(k)\vec{v}^{\scriptscriptstyle(-1)}(k).$
Using (2.13) and the fact that $\vec{v}^{\scriptscriptstyle(\pm 1)}(k)$ are
eigenvectors of ${\bf\hat{A}}(k)$ with eigenvalue $\hat{\lambda}_{\pm}(k)$, we
obtain
$\displaystyle\vec{b}_{n}(k)$ $\displaystyle=$
$\displaystyle{\bf\hat{A}}(k)^{n}{\vec{1}}=\alpha(k)\left(\hat{\lambda}^{n}_{1}(k)\vec{v}^{\scriptscriptstyle(1)}(k)-\hat{\lambda}^{n}_{-1}(k)\vec{v}^{\scriptscriptstyle(-1)}(k)\right),$
(2.32)
which proves (2.30). Combining (2.11) and (2.32) gives
$\displaystyle\hat{b}_{n}(k)$ $\displaystyle=$
$\displaystyle\alpha(k){\vec{1}}^{T}{\bf\hat{D}}(-k)\left(\hat{\lambda}^{n}_{1}(k)\vec{v}^{\scriptscriptstyle(1)}(k)-\hat{\lambda}^{n}_{-1}(k)\vec{v}^{\scriptscriptstyle(-1)}(k)\right).$
(2.33)
Inserting the definition of $\vec{v}^{\scriptscriptstyle(\pm 1)}(k)$ gives
(2.29). The proof of (2.31) is similar, now using that
${\bf\hat{A}}(k){\vec{1}}={\vec{v}}^{\scriptscriptstyle(1)}+\hat{\lambda}_{+}{\vec{1}}$,
which implies that
${\bf\hat{A}}(k)^{n}{\vec{1}}=n\hat{\lambda}_{+}(k)^{n-1}{\vec{v}}^{\scriptscriptstyle(1)}+\hat{\lambda}_{+}(k)^{n}{\vec{1}}$.
∎
### 2.4 Central limit theorem
This section is devoted to the proof of a functional central limit theorem for
the NBW. The uniform measures on $n$-step NBWs form a consistent family of
measures, so that there is a unique law that describes them as a process. We
let $\omega=(\omega_{r})_{r\geq 0}$ be distributed according to this law. For
a NBW $\omega$ and $t\geq 0$, we define
$\displaystyle X_{n}(t)=\frac{\omega_{\lfloor nt\rfloor}}{\sqrt{n}},$ (2.34)
where $\lfloor x\rfloor$ denotes the integer part of $x\in{\mathbb{R}}$.
###### Theorem 2.4 (Functional central limit theorem).
The processes $(X_{n}(t))_{t\geq 0}$ converge weakly to $(B(t))_{t\geq 0}$,
where $(B(t))_{t\geq 0}$ is a Brownian motion with covariance matrix ${\bf
I}/(d-1)$.
Our proof of Theorem 2.4 is organized as follows. We use Lemma 2.3 to prove a
CLT for the endpoint in Lemma 2.5. We then prove the convergence of the
finite-dimensional distributions to the Gaussian distribution (see Lemma 2.6),
followed by a proof of tightness (see Lemma 2.8). This implies Theorem 2.4,
see e.g. [5, Theorem 15.6]. We now fill in the details.
###### Lemma 2.5 (Central limit theorem).
Let $d\geq 2$ and $k\in[-\pi,\pi]^{d}$. Then,
$\displaystyle\lim_{n\rightarrow\infty}{\mathbb{E}}[{\mathrm{e}}^{{\mathrm{i}}k\cdot\omega_{n}/\sqrt{n}}]$
$\displaystyle=$ $\displaystyle{\mathrm{e}}^{-\|k\|^{2}_{2}/(2d-2)},$ (2.35)
where $\|k\|_{2}^{2}=\sum_{i=1}^{d}k_{i}^{2}$ is the Euclidean norm of $k$.
Lemma 2.5 implies that the distribution of the endpoint of an $n$-step NBW
converges in distribution to a normal distribution with mean zero and
covariance matrix $(d-1)^{-1}{\mathbf{I}}$.
###### Proof.
We can rewrite the expectation as
$\displaystyle{\mathbb{E}}[{\mathrm{e}}^{{\mathrm{i}}k\cdot\omega_{n}/\sqrt{n}}]$
$\displaystyle=$
$\displaystyle\sum_{x\in{\mathbb{Z}}}\frac{b_{n}(x)}{\hat{b}_{n}(0)}{\mathrm{e}}^{{\mathrm{i}}k\cdot
x/\sqrt{n}}=\frac{\hat{b}_{n}(k/\sqrt{n})}{2d(2d-1)^{n}}.$ (2.36)
As $n\rightarrow\infty$, we can assume without loss of generality that $k$ is
small and therefore that $\hat{\lambda}_{1}(k)>\hat{\lambda}_{-1}(k)$. Then we
can use (2.29) to compute the limit of (2.36). For $\iota={1,-1}$, we compute
$\displaystyle\lim_{n\rightarrow\infty}\frac{\hat{\lambda}^{n-1}_{\iota}(k/\sqrt{n})}{(2d-1)^{n-1}}\alpha(k/\sqrt{n})\frac{{\vec{1}}^{T}{\bf\hat{D}}(-k/\sqrt{n}){\vec{v}}^{\scriptscriptstyle(\iota)}(k/\sqrt{n})}{2d}.$
(2.37)
We first consider the case $\iota=1$. The coefficient $\alpha(k)$ as well as
$\vec{v}^{\scriptscriptstyle(1)}(k)$ are continuous in a neighborhood of
$\vec{0}$, so we can directly compute
$\displaystyle\lim_{n\rightarrow\infty}\alpha(k)\frac{{\vec{1}}^{T}{\bf\hat{D}}(-k/\sqrt{n}){\vec{v}}^{\scriptscriptstyle(1)}(k/\sqrt{n})}{2d}=\alpha(0)\frac{{\vec{1}}^{T}{\bf\hat{D}}(0){\vec{v}}^{\scriptscriptstyle(1)}(0)}{2d}=1.$
Further, $\hat{\lambda}_{1}(k)$ is differentiable in $k$, so we can Taylor
expand $\hat{\lambda}_{1}(k)$ at $0$ to obtain
$\displaystyle\hat{\lambda}_{1}(k)$ $\displaystyle=$ $\displaystyle
2d-1-\frac{2d-1}{2d-2}\|k\|^{2}_{2}+O(\|k\|_{2}^{4}).$
Therefore,
$\displaystyle\lim_{n\rightarrow\infty}\frac{\hat{\lambda}^{n-1}_{1}(k/\sqrt{n})}{(2d-1)^{n-1}}$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left(1-\frac{1}{2d-2}\frac{\|k\|^{2}_{2}}{n}+O\left(\frac{\|k\|_{4}^{4}}{n^{2}}\right)\right)^{n-1}={\mathrm{e}}^{-\|k\|^{2}_{2}/(2d-2)},$
(2.38)
so that the limit (2.37) for $\iota=1$ is given by
${\mathrm{e}}^{-\frac{1}{2d-2}\|k\|^{2}_{2}}$.
We next consider the case $\iota=-1$, for which we use that
$k\mapsto\hat{\lambda}_{-1}(k)$ is continuous and $\hat{\lambda}_{-1}(0)=1$,
Therefore, for $d\geq 2$,
$\displaystyle\lim_{n\rightarrow\infty}\frac{\hat{\lambda}^{n-1}_{-1}(k/\sqrt{n})}{(2d-1)^{n-1}}=0.$
(2.39)
The second factor in (2.37) can easily be bounded uniformly for small $k$, so
that for $\iota=-1$ the limit of (2.37) is zero. ∎
###### Lemma 2.6 (Convergence of finite-dimensional distributions).
For $d\geq 2$ and $N>0$, let $0=t_{0}<t_{1}<t_{2}<\dots<t_{N}=1$ and
$k^{\scriptscriptstyle(r)}\in(-\pi,\pi]^{d}$ for $r=1,\dots,N$. Then,
$\displaystyle\lim_{n\rightarrow\infty}{\mathbb{E}}[{\mathrm{e}}^{{\mathrm{i}}(\sum_{r=1}^{N}k^{\scriptscriptstyle(r)}(\omega_{\lfloor
t_{r}n\rfloor}-\omega_{\lfloor t_{r-1}n\rfloor})/\sqrt{n})}]$ $\displaystyle=$
$\displaystyle{\mathrm{e}}^{-\sum_{r=1}^{N}\|k^{\scriptscriptstyle(r)}\|_{2}^{2}(t_{r}-t_{r-1})/(2d-2)}.$
(2.40)
###### Proof.
As we take the limit $n\rightarrow\infty$, without loss of generality we can
assume that $\eta_{r}(n):=\lfloor t_{r}n\rfloor-\lfloor t_{r-1}n\rfloor\geq
1$, $t_{r}>t_{r-1}$ and each
$k^{\scriptscriptstyle(r)}_{n}:=k^{\scriptscriptstyle(r)}/\sqrt{n}$ is so
small that $\hat{\lambda}_{1}(k)>\hat{\lambda}_{-1}(k)$ for $r=1,\dots,N$ and
$n\in{\mathbb{N}}$. Let $\mathcal{W}_{n}$ be the set of all $n$-step NBW. For
any function $f\colon\mathbb{Z}^{d\times N}\mapsto\mathbb{C}$, we know that
$\displaystyle{\mathbb{E}}[f(\omega_{\lfloor
t_{1}n\rfloor},\dots,\omega_{\lfloor t_{N}n\rfloor})]$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{b}_{n}(0)}\sum_{\omega\in\mathcal{W}_{n}}f(\omega_{\lfloor
t_{1}n\rfloor},\dots,\lfloor\omega_{t_{N}n\rfloor})$ (2.41) $\displaystyle=$
$\displaystyle\frac{1}{\hat{b}_{n}(0)}\
\sum_{\omega\in\mathcal{W}_{n}}\sum_{x_{1},\dots,x_{N}\in{\mathbb{Z}}}f(x_{1},\dots,x_{N})\prod_{i=1,\dots,N}\delta_{x_{i},\lfloor\omega_{t_{i}n}\rfloor}.$
Let $b_{n}^{\iota,\kappa}(x)$ be the number of $n$-step NBW $\omega$ with
$\omega_{1}\neq{e}_{\iota}$, $\omega_{n-1}=x+{e}_{\kappa}$ and $\omega_{n}=x$.
We define the matrix $\hat{\bf B}_{n}(k)$ with entries
$({\bf\hat{B}}_{n}(k))_{\iota,\kappa}=\hat{b}_{n}^{\iota,\kappa}(k)$. By a
relation similar to (2.10), we conclude that
${\bf\hat{B}}_{n}(k)={\bf\hat{A}}(k)^{n}$. We fix $N$ points
$x_{1},\dots,x_{N}\in{\mathbb{Z}}$, then the number of NBW $\omega$ with
$\omega_{\lfloor t_{i}n\rfloor}=x_{i}$ for all $i=1,2,\dots,N$ is given by
$\displaystyle\sum_{\stackrel{{\scriptstyle\iota_{1},\dots,\iota_{N}}}{{\text{in
}\\{\pm 1,\dots,\pm
d\\}}}}b_{\eta_{1}(n)-1}^{\iota_{0},\iota_{1}}(x_{1}+{e}_{\iota_{0}})\prod_{r=2,\dots,N}b_{\eta_{r}(n)}^{\iota_{r-1},\iota_{r}}(x_{r}-x_{r-1}).$
(2.42)
We insert
$f(x_{1},\dots,x_{N})={\mathrm{e}}^{{\mathrm{i}}\sum_{r}k^{\scriptscriptstyle(r)}_{n}(x_{r}-x_{r-1})}$
and obtain
$\displaystyle{\mathbb{E}}[{\mathrm{e}}^{{\mathrm{i}}\sum_{r}k^{\scriptscriptstyle(r)}_{n}(\omega_{n_{r}}-\omega_{n_{r-1}})}]$
$\displaystyle=$
$\displaystyle\frac{1}{\hat{b}_{n}(0)}\sum_{\stackrel{{\scriptstyle\iota_{1},\dots,\iota_{N}}}{{\text{in
}\\{\pm 1,\dots,\pm
d\\}}}}\hat{b}_{\eta_{1}(n)-1}^{\iota_{1}}(k^{\scriptscriptstyle(1)}_{n}){\mathrm{e}}^{-{\mathrm{i}}k_{\iota_{1}}^{\scriptscriptstyle(1)}/\sqrt{n}}\prod_{r=2}^{N}\hat{b}_{\eta_{r}(n)}^{\iota_{r-1},\iota_{r}}(k^{\scriptscriptstyle(r)}_{n})$
$\displaystyle=$
$\displaystyle\frac{1}{\hat{b}_{n}(0)}{\vec{1}}^{T}{\bf\hat{D}}(-k^{\scriptscriptstyle(1)}_{n}){\bf\hat{A}}(k^{\scriptscriptstyle(1)}_{n})^{\eta_{1}(n)-1}\prod_{r=2}^{N}{\bf\hat{B}}_{\eta_{r}(n)}(k^{\scriptscriptstyle(r)}_{n}){\vec{1}}$
$\displaystyle=$
$\displaystyle\frac{1}{\hat{b}_{n}(0)}{\vec{1}}^{T}{\bf\hat{D}}(-k^{\scriptscriptstyle(1)}_{n}){\bf\hat{A}}(k^{\scriptscriptstyle(1)}_{n})^{\eta_{1}(n)-1}\prod_{r=2}^{N}{\bf\hat{A}}(k^{\scriptscriptstyle(r)}_{n})^{\eta_{r}(n)}{\vec{1}}.$
(2.43)
To proceed, we define, for $\tau\in\\{1,2,\dots,N\\}$,
$\displaystyle\vec{h}_{n}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{b}_{\lfloor
t_{\tau}n\rfloor}(0)}{\vec{1}}^{T}{\bf\hat{D}}(-k^{\scriptscriptstyle(1)}_{n}){\bf\hat{A}}(k^{\scriptscriptstyle(1)}_{n})^{\eta_{1}(n)-1}\cdots{\bf\hat{A}}(k^{\scriptscriptstyle(\tau)}_{n})^{\eta_{\tau}(n)},\qquad\qquad\vec{h}_{n}(0)=\frac{1}{2d}{\vec{1}}^{T}{\bf\hat{D}}(-k^{\scriptscriptstyle(1)}_{n}).$
As $\sum^{\tau}_{r=1}\eta_{r}(n)\leq t_{\tau}n$, by construction of
$\vec{h}_{n}(\tau)$, for all ${\vec{v}}\in\mathbb{C}^{2d}$
$\displaystyle|\vec{h}_{n}(\tau){\vec{v}}|$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\left[{\mathrm{e}}^{{\mathrm{i}}(\sum_{r=1}^{\tau}k^{\scriptscriptstyle(r)}(\omega_{\lfloor
t_{r}n\rfloor}-\omega_{\lfloor
t_{r-1}n\rfloor})/\sqrt{n})}\sum_{\iota}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\omega_{t_{\tau}n}-\omega_{t_{\tau}n-1}={e}_{\iota}\\}}|\vec{v}_{\iota}|\right]$
(2.44) $\displaystyle\leq$
$\displaystyle{\mathbb{E}}[\sum_{\iota}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\omega_{t_{\tau}n}-\omega_{t_{\tau}n-1}={e}_{\iota}\\}}|\vec{v}_{\iota}|]\leq\|\vec{v}\|_{\infty},$
where we write $\|v\|_{\infty}:=\max_{\iota}|v_{\iota}|$. Therefore, we can
rewrite
$\displaystyle\eqref{e:FCLTchar2}$ $\displaystyle=$
$\displaystyle\vec{h}_{n}(N){\vec{1}}=\vec{h}_{n}(N-1)\left(\frac{{\bf\hat{A}}(k^{\scriptscriptstyle(N)}_{n})}{2d-1}\right)^{\eta_{N}(n)}\frac{\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(N)}_{n})}{2d-2}+\vec{h}_{n}(N)\left({\vec{1}}-\frac{\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(N)}_{n})}{2d-2}\right)$
(2.47) $\displaystyle=$
$\displaystyle\vec{h}_{n}(N-1)\left(\frac{\hat{\lambda}_{1}(k^{\scriptscriptstyle(N)}_{n})}{2d-1}\right)^{\eta_{N}(n)}\frac{\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(N)}_{n})}{2d-2}+\vec{h}_{n}(N)({\vec{1}}-\frac{\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(N)}_{n})}{2d-2})$
$\displaystyle=$
$\displaystyle\vec{h}_{n}(0)\frac{\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(1)}_{n})}{2d-2}\left(\frac{2d-1}{\hat{\lambda}_{1}(k^{\scriptscriptstyle(1)}_{n})}\right)\prod_{r=1}^{N}\left(\frac{\hat{\lambda}_{1}(k^{\scriptscriptstyle(r)}_{n})}{2d-1}\right)^{\eta_{r}(n)}$
$\displaystyle+\sum_{r=1}^{N-1}\frac{\vec{h}_{n}(r)}{2d-2}\left(\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(r+1)}_{n})-\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(r)}_{n})\right)\prod_{i=r+1}^{N}\left(\frac{\hat{\lambda}_{1}(k^{\scriptscriptstyle(i)}_{n})}{2d-1}\right)^{\eta_{i}(n)}$
$\displaystyle+\vec{h}_{n}(N)\left({\vec{1}}-\frac{\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(N)}_{n})}{2d-2}\right).$
To compute the limit of (2.43), we show that (2.47) and (2.47) converge to
zero and identify the limit of (2.47). From (2.44), it follows that we can
bound (2.47) by
${\vec{1}}-\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(N)}_{n})/(2d-2)$
and we know that
$\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(N)}_{n})\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\rightarrow}}(2d-2){\vec{1}}$. Thereby we conclude that
(2.47) converges to $0$.
To compute the limit of (2.47) we note that $\hat{\lambda}_{1}(k)\leq 2d-1$
and that
$\vec{v}^{\scriptscriptstyle(r+1)}(k^{\scriptscriptstyle(r+1)}_{n})-\vec{v}^{\scriptscriptstyle(r)}(k^{\scriptscriptstyle(r)}_{n})\stackrel{{\scriptstyle
n\rightarrow\infty}}{{\rightarrow}}\vec{0}$, so that
$\displaystyle\lim_{n\rightarrow\infty}|\eqref{e:FCLTchar3b}|$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow\infty}\Big{|}\sum_{r=1}^{N-1}\frac{\vec{h}_{n}(r)}{2d-2}\left(\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(1)}_{n})-\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(1)}_{n})\right)\Big{|}$
$\displaystyle\stackrel{{\scriptstyle\eqref{e:matrixprodBounded}}}{{\leq}}$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{2d-2}\sum_{r=1}^{N-1}\|\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(1)}_{n})-\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(1)}_{n})\|_{\infty}=0.$
To compute the limit of (2.47), we use that
$\displaystyle\lim_{n\rightarrow\infty}\vec{h}_{n}(0)\frac{\vec{v}^{\scriptscriptstyle(1)}(k^{\scriptscriptstyle(1)}_{n})}{2d-2}\frac{2d-1}{\hat{\lambda}_{1}(k^{\scriptscriptstyle(1)}_{n})}$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{{\vec{1}}^{T}{\bf\hat{D}}(-k)(\hat{\lambda}_{1}(k^{\scriptscriptstyle(1)}_{n}){\vec{1}}+{\bf\hat{D}}(k^{\scriptscriptstyle(1)}_{n}){\vec{1}})}{2d(2d-2)}\frac{2d-1}{\hat{\lambda}_{1}(k^{\scriptscriptstyle(1)}_{n})}=1,$
and, for each $r\in\\{1,\ldots,N\\}$,
$\displaystyle\lim_{n\rightarrow\infty}\left(\frac{\hat{\lambda}_{1}(k^{\scriptscriptstyle(r)}_{n})}{2d-1}\right)^{\eta_{r}(n)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{e:lampart}}}{{=}}$
$\displaystyle{\mathrm{e}}^{-\|k^{\scriptscriptstyle(i)}\|^{2}_{2}(t_{i}-t_{i-1})/(2d-2)}.$
Combining this yields that the limit of (2.47) is the right-hand side of
(2.40), which completes the proof of Lemma 2.6. ∎
To prove tightness we make use of the following lemma, which computes the
second moment of the end-point of NBW. The leading order in this result was
alternatively proved in [9, (5.3.11)] using residues.
###### Lemma 2.7 (The second moment).
For $d\geq 2$ and $n\in{\mathbb{N}}$,
$\displaystyle{\mathbb{E}}[\|\omega_{n}\|_{2}^{2}]=\frac{d}{d-1}n+\frac{4d-1}{2(d-1)^{2}}+\frac{d}{2(d-1)^{2}(2d-1)^{n-2}}.$
###### Proof.
We define the differential operator
$\nabla^{2}:=\sum_{i=1}^{d}\left(\frac{\partial}{\partial k_{i}}\right)^{2}$
and see that
$\displaystyle{\mathbb{E}}[\|\omega_{n}\|_{2}^{2}]=\frac{1}{\hat{b}_{n}(0)}\sum_{x\in{\mathbb{Z}}}b_{n}(x)\|x\|_{2}^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{\hat{b}_{n}(0)}\nabla^{2}\hat{b}_{n}(0).$
To compute the second derivative in a neighborhood of the origin, we recall
(2.29). By Lemma 2.3, for $k$ such that
$\hat{\lambda}_{+}(k)\neq\hat{\lambda}_{-}(k)$,
$\displaystyle\hat{b}_{n}(k)$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\\{-1,1\\}}d\iota\frac{\hat{\lambda}^{n}_{\sigma}(k)\hat{D}(k)-\hat{\lambda}^{n-1}_{\sigma}(k)}{\sqrt{(d\hat{D}(k))^{2}-(2d-1)}}\equiv\sum_{\sigma\in\\{-1,1\\}}\hat{g}_{\sigma}(k).$
(2.48)
Since, $\hat{\lambda}_{+}(k)\neq\hat{\lambda}_{-}(k)$ for $k$ small, (2.48)
holds in particular for $k$ small. A straightforward computation gives
$\displaystyle\nabla^{2}g_{\sigma}(k)\Big{|}_{k=0}$ $\displaystyle=$
$\displaystyle
d\sigma\left(\nabla^{2}\hat{\lambda}_{\sigma}(0)\right)\frac{n\hat{\lambda}^{n-1}_{\sigma}(0)\hat{D}(0)-(n-1)\hat{\lambda}^{n-2}_{\sigma}(0)}{\sqrt{(d\hat{D}(0))^{2}-(2d-1)}}$
$\displaystyle+d\sigma\left(\nabla^{2}\hat{D}(0)\right)\Big{(}\frac{\hat{\lambda}^{n}_{\sigma}(0)}{\sqrt{(d\hat{D}(0))^{2}-(2d-1)}}-d^{2}\hat{D}(0)\frac{\hat{\lambda}^{n}_{\sigma}(0)\hat{D}(0)-\hat{\lambda}^{n-1}_{\sigma}(0)}{\left(\sqrt{(d\hat{D}(0))^{2}-(2d-1)}\right)^{3}}\Big{)}$
$\displaystyle=$
$\displaystyle-d\sigma\frac{d((1+\sigma)d-1)}{(d-1)}\frac{n\hat{\lambda}^{n-1}_{\sigma}(0)-(n-1)\hat{\lambda}^{n-2}_{\sigma}(0)}{d-1}$
$\displaystyle+\frac{d\sigma}{d-1}\hat{\lambda}^{n-1}_{\sigma}(0)\left(\hat{\lambda}_{\sigma}(0)-d^{2}\frac{\hat{\lambda}_{\sigma}(0)-1}{(d-1)^{2}}\right),$
so that
$\displaystyle\nabla^{2}g_{1}(k)\Big{|}_{k=0}$ $\displaystyle=$
$\displaystyle-\frac{d}{d-1}\frac{(2d-1)^{n-1}}{d-1}\left[2d(d-1)n+(4d-1)\right]$
$\displaystyle\nabla^{2}g_{-1}(k)\Big{|}_{k=0}$ $\displaystyle=$
$\displaystyle\frac{d}{d-1}\left[\frac{d(-1)}{(d-1)}-1\right]=-d\frac{2d-1}{(d-1)^{2}}.$
We arrive at
$\displaystyle-\frac{1}{\hat{b}_{n}(0)}\nabla^{2}\hat{b}_{n}(0)$
$\displaystyle=$
$\displaystyle\frac{1}{2d(2d-1)^{n-1}}\sum_{\sigma\in\\{-1,1\\}}-\nabla^{2}g_{\sigma}(k)\Big{|}_{k=0}$
$\displaystyle=$
$\displaystyle\frac{d}{d-1}n+\frac{4d-1}{2(d-1)^{2}}+\frac{d}{2(d-1)^{2}(2d-1)^{n-2}}.$
∎
###### Lemma 2.8 (Tightness).
For $d\geq 2$ and $0\leq t_{1}<t_{2}<t_{3}\leq 1$, there exists a $K>0$, such
that, for all $n\geq 1$,
$\displaystyle{\mathbb{E}}[\|X_{n}(t_{2})-X_{n}(t_{1})\|^{2}_{2}\|X_{n}(t_{3})-X_{n}(t_{2})\|^{2}_{2}]\leq
K(t_{2}-t_{1})(t_{3}-t_{2}).$
###### Proof.
We use the same notation as in the proof of Lemma 2.6. In (2.42), we have seen
how to describe the number of NBWs that visit a number of fixed points. This
time we forget about the non-backtracking constraint between two subsequent
NBWs to upper bound
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\mathbb{E}}[\|\omega_{\lfloor
t_{2}n\rfloor}-\omega_{\lfloor t_{1}n\rfloor}\|^{2}_{2}\|\omega_{\lfloor
t_{3}n\rfloor}-\omega_{\lfloor t_{2}n\rfloor}\|^{2}_{2}]$ $\displaystyle\leq$
$\displaystyle\frac{1}{\hat{b}_{n}(0)}\sum_{x_{1},x_{2},x_{3},x_{4}\in{\mathbb{Z}}^{d}}\|x_{2}-x_{1}\|_{2}^{2}\|x_{3}-x_{2}\|_{2}^{2}b_{\eta_{1}(n)}(x_{1})\prod_{r=2}^{4}b_{\eta_{r}(n)}(x_{r}-x_{r-1})$
$\displaystyle=$
$\displaystyle\frac{1}{\hat{b}_{n}(0)}\sum_{x_{1},x_{2},x_{3},x_{4}\in{\mathbb{Z}}^{d}}\|x_{2}\|_{2}^{2}\|x_{3}\|_{2}^{2}\prod_{r=1}^{4}b_{\eta_{r}(n)}(x_{r})$
$\displaystyle=$
$\displaystyle\frac{\hat{b}_{\eta_{1}(n)}\hat{b}_{\eta_{4}(n)}}{b_{n}}\sum_{x_{2}\in{\mathbb{Z}}}b_{\eta_{2}(n)}(x_{2})\|x_{2}\|_{2}^{2}\sum_{x_{3}\in{\mathbb{Z}}}b_{\eta_{3}(n)}(x_{3})\|x_{3}\|_{2}^{2}$
$\displaystyle=$
$\displaystyle\left(\frac{2d-1}{2d}\right)^{3}{\mathbb{E}}[\|\omega_{\eta_{2}(n)}\|_{2}^{2}]{\mathbb{E}}[\|\omega_{\eta_{3}(n)}\|_{2}^{2}].$
Applying Lemma 2.7 completes the proof. ∎
### 2.5 Extension to non-nearest-neighbor setting
In this section, we extend the analysis of NBW on ${\mathbb{Z}}^{d}$ to other
bond sets. We start by introducing the bond sets that we consider. We let
$\mathbb{B}\subset{\mathbb{Z}}^{d}\times{\mathbb{Z}}^{d}$ be a translation
invariant collection of bonds. Let
$\mathbb{V}_{0}=\\{x\colon\\{0,x\\}\in\mathbb{B}\\}$ denote the set of
endpoints of bonds containing the origin and write $m=|\mathbb{V}_{0}|$. We
assume that $0\not\in\mathbb{V}_{0}$, and that $\mathbb{V}_{0}$ is symmetric,
i.e., $-x\in\mathbb{V}_{0}$ for every $x\in\mathbb{V}_{0}$. Thus, $m$ is even.
We define the simple random walk step distribution by
$D(x)=\frac{1}{m}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{x\in\mathbb{V}_{0}\\}}.$
(2.49)
Define the matrices ${\bf C},{\bf J}\in{\mathbb{C}}^{m\times m}$ by $({\bf
C})_{x,y}=1$ and $({\bf J})_{x,y}=\delta_{x,-y}$, and let the diagonal matrix
${\bf\hat{D}}(k)$ have entries
$({\bf\hat{D}}(k))_{x,x}={\mathrm{e}}^{{\mathrm{i}}k\cdot x},$ where
$x,y\in{\mathbb{B}}_{0}$. Then, we define the matrix ${\bf\hat{A}}(k)$ of size
$m\times m$ by
${\bf\hat{A}}(k)=({\bf C}-{\bf J}){\bf\hat{D}}(-k).$ (2.50)
With this definition at hand, we see that (2.11)–(2.13) remain to hold. As a
result, also Lemmas 2.1–2.2, whose proof only depends on (2.11)–(2.13),
continue to hold when we replace each occurrence of $2d$ by $m$. Since the
proof of Lemma 2.3, in turn, only depends on Lemmas 2.1–2.2, also it extends
to this setting, so that, for example
$\hat{\lambda}_{\pm}(k)=F_{\pm}(\hat{D}(k);m),\qquad\text{where}\qquad
F_{\pm}(x;m)=\frac{1}{2}\left(mx\pm\sqrt{(mx)^{2}-4(m-1)}\right),$ (2.51)
and, when $\hat{\lambda}_{1}(k)\neq\hat{\lambda}_{-1}(k)$,
$\hat{b}_{n}(k)=\frac{m}{2}\frac{\hat{D}(k)(\hat{\lambda}^{n}_{1}(k)-\hat{\lambda}^{n}_{-1}(k))+(\hat{\lambda}^{n-1}_{-1}(k)-\hat{\lambda}^{n-1}_{1}(k))}{\hat{\lambda}_{1}(k)-\hat{\lambda}_{-1}(k)}.$
(2.52)
Naturally, Theorem 2.4 needs to be adapted, and now reads that the processes
$(X_{n}(t))_{t\geq 0}$ converge weakly to a Brownian motion with covariance
matrix ${{\mathbf{M}}}$ of size $d\times d$, where, for $\iota,\kappa\in\\{\pm
1,\ldots,\pm d\\}$, we define
${\mathbf{M}}_{\iota,\kappa}=\frac{\partial^{2}\hat{\lambda}_{1}(k)}{\partial
k_{\iota}\partial k_{\kappa}}\Big{|}_{k=0}\hat{\lambda}_{1}(0)^{-1}.$ (2.53)
We next compute ${{\mathbf{M}}}$ explicitly in terms of the covariance matrix
of the transition kernel $D$. We compute that $F_{+}(1;m)=m/2+(m-2)/2=m-1,$
and
$F_{+}^{\prime}(x;m)=m/2+\frac{m^{2}x}{2\sqrt{(mx)^{2}-4(m-1)}},\qquad\text{so
that}\qquad F_{+}^{\prime}(1;m)=m(m-1)/(m-2).$ (2.54)
By symmetry, the odd derivatives of $\hat{D}(k)$ are zero, so that a Taylor
expansion yields
$\hat{D}(k)=1-\tfrac{1}{2}k^{T}\mathbf{H}k+O(\|k\|_{2}^{4}),$ (2.55)
where, for $\iota,\kappa\in\\{1,\ldots,d\\}$,
$\mathbf{H}_{\iota,\kappa}=\sum_{x}x_{\iota}x_{\kappa}D(x)$ (2.56)
denotes the covariance matrix of SRW. As a result,
${\mathbf{M}}=\mathbf{H}\frac{F_{+}^{\prime}(1;m)}{F_{+}(1;m)}=\mathbf{H}m/(m-2).$
(2.57)
In the nearest-neighbor case, $m=2d$ and $\mathbf{H}={\bf I}/d$, so that we
retrieve the result in Theorem 2.4.
## 3 NBW on tori
In this section, we extend the results in Section 2 to NBWs on tori. In
Section 3.1 and 3.2, we investigate NBW on a torus of width $r\geq 2$, and in
Section 3.3 we investigate NBW on the hypercube, for which $r=2$. The study of
random walks on various finite transitive graphs has attracted considerable
attention. See e.g., [8] for a recent book on the subject, and [2] for a book
in preparation. Here we restrict ourselves to NBWs on tori.
### 3.1 Setting
For $d\geq 2$ and $r\geq 3$, we denote by
${\mathbb{T}}={\mathbb{T}}_{r,d}=({\mathbb{Z}}/r{\mathbb{Z}})^{d}$ the
discrete $d$-dimensional torus with side length $r$. The torus has periodic
boundaries, i.e., we identify two points $x,y\in{\mathbb{T}}_{r,d}$ if
$x_{i}\text{ mod }r=y_{i}\text{ mod }r$ for all $i=1,\dots,d$ where mod denote
the modulus. We define the Fourier dual torus of ${\mathbb{T}}$ as
$\displaystyle{\mathbb{T}}^{*}_{r,d}:=\frac{2\pi}{r}\left\\{-\left\lfloor\frac{r-1}{2}\right\rfloor,\dots,\left\lceil\frac{r-1}{2}\right\rceil\right\\}^{d},$
(3.1)
so that each component of $k\in{\mathbb{T}}^{*}_{r,d}$ is between $-\pi$ and
$\pi$. The Fourier transform of
$f\colon{\mathbb{T}}_{r,d}\rightarrow{\mathbb{C}}$ is defined by
$\displaystyle\hat{f}(k)=\sum_{x\in{\mathbb{T}}_{r,n}}f(x){\mathrm{e}}^{{\mathrm{i}}k\cdot
x},\qquad\qquad k\in{\mathbb{T}}^{*}_{r,d}.$
As in Section 2, we define an _$n$ -step random walk_ on ${\mathbb{T}}_{r,d}$
to be an ordered tuple $\omega=(\omega_{0},\dots,\omega_{n})$, with
$\omega_{i}\in{\mathbb{T}}_{r,d}$ and
$\omega_{i}-\omega_{i+1}\in\mathbb{V}_{0}$, where we recall that
$\mathbb{V}_{0}=\\{x\colon\\{0,x\\}\in{\mathbb{B}}\\}$, and ${\mathbb{B}}$ is
the translationally invariant bond set on which our random walks moves. We
always assume that $0\not\in\mathbb{V}_{0}$, and that $\mathbb{V}_{0}$ is
symmetric, i.e., if $x\in\mathbb{V}_{0}$, then also $-x\in\mathbb{V}_{0}$.
Further, we always assume that $\omega_{0}=(0,\dots,0)$. The simple random
walk step distribution is given by
$\displaystyle
D(x)=\frac{1}{m}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{x\in\mathbb{V}_{0}\\}}\qquad\text{
and
}\qquad\hat{D}(k)=\frac{1}{m}\sum_{x\in{\mathbb{T}}_{r,d}}{\mathrm{e}}^{{\mathrm{i}}k\cdot
x}.$ (3.2)
If an $n$-step random walk on ${\mathbb{T}}_{r,d}$ additionally satisfies
$\omega_{i}\neq\omega_{i-2}$, then we call the walk a non-backtracking walk
(NBW) on ${\mathbb{T}}_{r,d}$. Let $b_{n}(x)$ be the number of $n$-step NBWs
with $\omega_{n}=x$. Further, let $b^{\iota}_{n}(x)$ be the number of $n$-step
NBWs $\omega$ with $\omega_{n}=x$ and $\omega_{1}\neq{e}_{\iota}$.
In this setting, we can express $b_{n}(x)$ for NBW on ${\mathbb{T}}_{r,d}$ in
terms of NBW on ${\mathbb{Z}}^{d}$. Indeed, identify ${\mathbb{T}}_{r,d}$ with
$\\{0,\ldots,r-1\\}^{d}\subset{\mathbb{Z}}^{d}$, and also identify
$\mathbb{V}_{0}=\\{x\colon\\{0,x\\}\in{\mathbb{B}}\\}$ as a subset of
$\\{0,\ldots,r-1\\}^{d}\subset{\mathbb{Z}}^{d}$. Define
$\mathbb{V}_{0}^{\scriptscriptstyle{\mathbb{Z}}}=\mathbb{V}_{0}\cup(-\mathbb{V}_{0})$
(which, by construction, are disjoint subsets of ${\mathbb{Z}}^{d}$), and
define the random walk step distribution
$D_{\scriptscriptstyle{\mathbb{Z}}}(x)$ by the uniform distribution on
$\mathbb{V}_{0}^{\scriptscriptstyle{\mathbb{Z}}}$. Then, for
$x\in{\mathbb{T}}_{r,d}$,
$b_{n}(x)=\sum_{y\colon y\stackrel{{\scriptstyle
r}}{{\sim}}x}b_{n}^{\scriptscriptstyle{\mathbb{Z}}}(y),$ (3.3)
where, for $y\in{\mathbb{Z}}^{d}$ and $x\in{\mathbb{T}}_{r,d}$, we say that
$x\stackrel{{\scriptstyle r}}{{\sim}}y$ when $x=y\mod r$, and
$b_{n}^{\scriptscriptstyle Z}(y)$ denotes the number of $n$-step NBWs on
${\mathbb{Z}}^{d}$ with step distribution
$D_{\scriptscriptstyle{\mathbb{Z}}}(x)$. As a result, $\hat{b}_{n}(k)$ for NBW
on ${\mathbb{T}}_{r,d}$ is equal to
$\hat{b}_{n}^{\scriptscriptstyle{\mathbb{Z}}}(k)$ for every
$k\in{\mathbb{T}}^{*}_{r,d}$ as defined in (3.1). Therefore, we can use most
results for NBW on ${\mathbb{Z}}^{d}$ to study NBW on ${\mathbb{T}}_{r,d}.$ We
define the probability mass function of the endpoint of an $n$-step NBW by
$p_{n}(x)=\frac{b_{n}(x)}{\sum_{y}b_{n}(y)}=\frac{b_{n}(x)}{m(m-1)^{n-1}}.$
(3.4)
In order to study the asymptotic behavior of NBW, we investigate
$\hat{\lambda}_{\pm}(k)$ for $k\neq 0$. Our main result in this section is the
following theorem:
###### Theorem 3.1 (Pointwise bound on $\hat{b}_{n}(k)$).
Let $\mathbb{V}_{0}$ be symmetric and satisfy $0\not\in\mathbb{V}_{0}$. Then,
for $n\in{\mathbb{N}}$, NBW with steps in $\mathbb{V}_{0}$ satisfies
$|\hat{p}_{n}(k)|\leq\big{(}1/\sqrt{m-1}\vee|\hat{D}(k)|\big{)}^{n-1}.$ (3.5)
To prove Theorem 3.1, we start by investigating $\hat{\lambda}_{\pm}(k)$ for
$k\neq 0$. For this, we use Lemma 2.1 to note that
$\hat{\lambda}_{\pm}(k)=F_{\pm}(\hat{D}(k);m),\qquad\text{where}\qquad
F_{\pm}(x;m)=\frac{1}{2}\left(mx\pm\sqrt{(mx)^{2}-4(m-1)}\right),$ (3.6)
and where $m$ denotes the degree of our graph. We bound
$\hat{\lambda}_{\pm}(k)$ in the following lemma:
###### Lemma 3.2 (Bounds on $\hat{\lambda}_{\pm}(k)$).
For any $k\in{\mathbb{T}}^{*}_{r,d}$,
$|\hat{\lambda}_{+}(k)|\begin{cases}=\sqrt{m-1}&\text{when
}(m\hat{D}(k))^{2}-4(m-1)\leq 0;\\\
\leq(m-1)[1-(1-\hat{D}(k))m/(m-2)]&\text{when }\hat{D}(k)\geq
0,(m\hat{D}(k))^{2}-4(m-1)>0;\\\ \leq 1&\text{when }\hat{D}(k)\leq
0,(m\hat{D}(k))^{2}-4(m-1)>0.\end{cases}$ (3.7)
###### Proof.
The function $x\mapsto F_{\pm}(x;m)$ is real when $(mx)^{2}-4(m-1)\geq 0$, and
complex when $(mx)^{2}-4(m-1)<0$. When $(mx)^{2}-4(m-1)<0$,
$|F_{\pm}(x;m)|^{2}=m-1,$ (3.8)
so that $|F_{\pm}(x;m)|=\sqrt{m-1}$.
When $(mx)^{2}-4(m-1)\geq 0$, by the symmetry $F_{+}(x;m)=F_{-}(-x;m)$, we
only need to investigate $x\in[0,1]$. We start with $F_{-}(x;m)$, which
clearly satisfies $F_{-}(x;m)\geq 0$. Further, we can compute that
$F_{-}(1;m)=1$, and
$F_{-}^{\prime}(x;m)=m/2-\frac{m^{2}x}{2\sqrt{(mx)^{2}-4(m-1)}}=\frac{m}{2}\Big{[}1-\frac{mx}{\sqrt{(mx)^{2}-4(m-1)}}\Big{]}<0,$
(3.9)
so that $F_{-}(x;m)\leq 1$ for all $x\in[0,1]$ for which $(mx)^{2}-4(m-1)>0$.
To bound $F_{+}(x;m)$, we use (2.54) as well as
$\displaystyle F_{+}^{\prime\prime}(x;m)$
$\displaystyle=\frac{m^{2}}{2\sqrt{(mx)^{2}-4(m-1)}}-\frac{m^{4}x^{2}}{2((mx)^{2}-4(m-1))^{3/2}}$
(3.10)
$\displaystyle=\frac{m^{2}}{2((mx)^{2}-4(m-1))^{3/2}}\big{\\{}((mx)^{2}-4(m-1))-(mx)^{2}\big{\\}}$
$\displaystyle=-\frac{2m^{2}(m-1)}{((mx)^{2}-4(m-1))^{3/2}}<0.$
As a result, a Taylor expansion yields
$\displaystyle F_{+}(x;m)$ $\displaystyle\leq
F_{+}(1;m)+(x-1)F_{+}^{\prime}(1;m)=(m-1)+(x-1)m(m-1)/(m-2)$ (3.11)
$\displaystyle=(m-1)[1-(1-x)m/(m-2)].$
∎
Proof of Theorem 3.1. By (2.11) and (2.13),
$|\hat{b}_{n}(k)|\leq\|{\bf\hat{D}}(-k){\vec{1}}\|_{2}\|{\vec{b}}_{n-1}(k)\|_{2}\leq\|{\vec{1}}\|_{2}\|{\bf\hat{A}}(k)\|^{n-1}_{\rm\scriptscriptstyle
OP}\|{\vec{1}}\|_{2},$ (3.12)
where we write $\|{\mathbf{M}}\|_{\rm\scriptscriptstyle
OP}=\sup\|{\mathbf{M}}x\|_{2}/\|x\|_{2}$ for the operator norm of the matrix
${\mathbf{M}}$. We next use that ${\bf\hat{A}}(k)$ has eigenvalues
$\hat{\lambda}_{+}(k),\hat{\lambda}_{-}(k)$ and $\pm 1$ by Lemmas 2.1-2.2, so
that
$\|{\bf\hat{A}}(k)\|_{\rm\scriptscriptstyle
OP}=|\hat{\lambda}_{+}(k)|\vee|\hat{\lambda}_{-}(k)|\vee 1,$ (3.13)
where we use that for finite-dimensional matrices, the operator norm is equal
to the maximal eigenvalue, and for $x,y\in{\mathbb{R}}$, we write $(x\vee
y)=\max\\{x,y\\}$. Thus, we arrive at
$|\hat{b}_{n}(k)|\leq
m\big{(}|\hat{\lambda}_{+}(k)|\vee|\hat{\lambda}_{-}(k)|\vee 1\big{)}^{n-1}.$
(3.14)
By Lemma 3.2, and since $m\geq 2$ so that $\sqrt{m-1}\geq 1$,
$\frac{|\hat{\lambda}_{\pm}(k)|}{m-1}\leq(m-1)^{-1/2}\vee\big{(}1-[1-\hat{D}(k)]\frac{m}{m-2}\big{)}\leq(m-1)^{-1/2}\vee|\hat{D}(k)|.$
(3.15)
Substitution into (3.12) yields the claim. ∎
### 3.2 Asymptotics for NBW on the torus
In this section, we study the convergence towards equilibrium of NBW on tori
of width $r\geq 3$. We focus on two different examples. The first is random
walk on products of complete graphs, where
$\mathbb{V}_{0}=\\{x\colon\exists!i\in\\{1,\ldots,d\\}\text{ such that
}x_{i}\neq 0\\}.$ (3.16)
Our second example is NBW on the nearest-neighbor torus. The reason why we
study these cases separately is that NBW on products of complete graphs is
aperiodic, while nearest-neighbor NBW is periodic. Therefore, the stationary
distribution for NBW on products of complete graphs equals the uniform
distribution on the torus, while for nearest-neighbor NBW, the parity of the
position after $n$ steps always equals that of $n$. In Section 3.3, we further
study random walk on the $m$-dimensional hypercube.
For any small $\xi>0,$ we write $T_{{\rm mix}}(\xi)$ for the $\xi$-uniform
mixing time of NBW, that is,
$T_{{\rm
mix}}(\xi)=\min\Big{\\{}n\colon\max_{x,y}\,\,\frac{p_{n}(x,y)+p_{n+1}(x,y)}{2}\leq(1+\xi)V^{-1}\Big{\\}},$
(3.17)
where $V=r^{d}$ is the volume of the torus. We start to investigate NBW on
products of complete graphs:
#### NBW on products of complete graphs.
Our main result is as follows:
###### Lemma 3.3.
For every $d\geq 1$, $r\geq 3$,
$n>\frac{d(r-1)}{r}\log{\big{(}(r-1)/[(1+\xi)^{1/d}-1]\big{)}}$, NBW on
products of complete graphs satisfies that
$\max_{x\in{\mathbb{T}}_{r,d}}\big{|}p_{n}(x)-r^{-d}\big{|}\leq(m-1)^{-(n-1)/2}+\xi
r^{-d}.$ (3.18)
In particular, for every $\varepsilon>0$, there exists $V_{0}$ such that when
$r^{d}\geq V_{0}$,
$T_{{\rm
mix}}(\xi)\leq(1+\varepsilon)\frac{d(r-1)}{r}\log{\big{(}(r-1)/[(1+\xi)^{1/d}-1]\big{)}}.$
When $\xi$ is quite small, we obtain that $T_{{\rm
mix}}(\xi)\leq(1+2\varepsilon)\frac{r}{r-1}\log{(d(r-1)/\xi)}$.
###### Proof.
The inverse Fourier transform on ${\mathbb{T}}_{r,d}$ is given by
$\displaystyle
f(x)=\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}}\hat{f}(k){\mathrm{e}}^{{\mathrm{i}}k\cdot
x},$
so that
$b_{n}(x)=\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}}\hat{b}_{n}(k){\mathrm{e}}^{{\mathrm{i}}k\cdot
x}=r^{-d}\hat{b}_{n}(0)+\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}\colon
k\neq\vec{0}}\hat{b}_{n}(k){\mathrm{e}}^{{\mathrm{i}}k\cdot x}.$ (3.19)
Therefore,
$\big{|}p_{n}(x)-r^{-d}\big{|}\leq\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}\colon
k\neq\vec{0}}\frac{|\hat{b}_{n}(k)|}{\hat{b}_{n}(0)}.$ (3.20)
To bound $|\hat{b}_{n}(k)|$, we rely on Theorem 3.1, and start by computing
$\hat{D}(k)=\frac{1}{d(r-1)}\sum_{i=1}^{d}\sum_{j=1}^{r-1}{\mathrm{e}}^{{\mathrm{i}}k_{i}j}.$
(3.21)
Since
$k_{i}\in\frac{2\pi}{r}\left\\{-\left\lfloor\frac{r-1}{2}\right\rfloor,\dots,\left\lceil\frac{r-1}{2}\right\rceil\right\\}$,
we have that $\sum_{j=0}^{r-1}{\mathrm{e}}^{{\mathrm{i}}k_{i}j}=0$ for
$k_{i}\neq 0$. Therefore,
$\hat{D}(k)=\frac{1}{d(r-1)}\sum_{i=1}^{d}\sum_{j=1}^{r-1}{\mathrm{e}}^{{\mathrm{i}}k_{i}j}=\frac{1}{d(r-1)}\sum_{i=1}^{d}(r{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{k_{i}=0\\}}-1)=1-\frac{r}{d(r-1)}a(k),$
(3.22)
where
$a(k)=\sum_{i=1}^{d}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{k_{i}\neq
0\\}}$ denotes the number of non-zero coordinates of $k$. By this observation,
$\hat{D}(k(j))=1-\frac{rj}{d(r-1)}$ for any $k(j)\in{\mathbb{T}}^{*}_{r,d}$
for which $a(k(j))=j$. Then, by Theorem 3.1 and the fact that there are
${{d}\choose{j}}(r-1)^{j}$ values of $k\in{\mathbb{T}}^{*}_{r,d}$ for which
$a(k(j))=j$,
$\displaystyle\big{|}p_{n}(x)-r^{-d}\big{|}$ $\displaystyle=$ $\displaystyle
r^{-d}\sum_{j=1}^{d}(1/\sqrt{m-1}\vee|\hat{D}(k(j))|)^{n-1}{{d}\choose{j}}(r-1)^{j}$
$\displaystyle\leq$ $\displaystyle
r^{-d}\sum_{j=1}^{d}{{d}\choose{j}}(r-1)^{j}\Big{[}\big{|}1-\frac{rj}{d(r-1)}\big{|}^{n-1}+(m-1)^{-(n-1)/2}\Big{]}$
$\displaystyle\leq$
$\displaystyle(m-1)^{-(n-1)/2}+r^{-d}\sum_{j=1}^{d}{{d}\choose{j}}(r-1)^{j}{\mathrm{e}}^{-rj(n-1)/[d(r-1)]},$
where we use that $|1-\frac{rj}{d(r-1)}|\leq{\mathrm{e}}^{-rj(n-1)/[d(r-1)]}$
for any $j=1,\ldots,d$. Thus,
$\displaystyle\big{|}p_{n}(x)-r^{-d}\big{|}$
$\displaystyle\leq(m-1)^{-(n-1)/2}+r^{-d}\big{[}(1+(r-1){\mathrm{e}}^{-r(n-1)/[d(r-1)]})^{d}-1\big{]}$
$\displaystyle\leq(m-1)^{-(n-1)/2}+\xi r^{-d},$ (3.24)
when $n>\frac{d(r-1)}{r}\log{\big{(}(r-1)/[(1+\xi)^{1/d}-1]\big{)}}$. The
result on $T_{{\rm mix}}(\xi)$ follows immediately. ∎
#### NBW on the nearest-neighbor torus.
Our main result is as follows:
###### Lemma 3.4.
For every $d\geq 1$, $r\geq 3$ and
$n>\log{(2/[(1+\xi/2)^{1/d}-1]})/[1-\cos(2\pi/r)]$, NBW on the $d$-dimensional
nearest-neighbor torus satisfies that
$\max_{x\in{\mathbb{T}}_{r,d}}\big{|}p_{n}(x)-[1+(-1)^{\|x\|_{1}+n}]r^{-d}\big{|}\leq(2d-1)^{-n/2}+\xi
r^{-d}.$ (3.25)
In particular, for every $\varepsilon>0$, there exists $V_{0}$ such that
whenever $r^{d}\geq V_{0}$,
$T_{{\rm
mix}}(\xi)\leq(1+\varepsilon)\log{(2/[(1+\xi/2)^{1/d}-1]})/[1-\cos(2\pi/r)].$
When $r^{d}$ is large and $\xi$ small, the above implies that $T_{{\rm
mix}}(\xi)\leq(1+2\varepsilon)(r^{2}/(2\pi^{2}))(\log{(2d/\xi)})$.
###### Proof.
We adapt the proof of Lemma 3.3 to this setting. We have
$b_{n}(x)=\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}}\hat{b}_{n}(k){\mathrm{e}}^{{\mathrm{i}}k\cdot
x}=r^{-d}[\hat{b}_{n}(0)+(-1)^{\|x\|_{1}}\hat{b}_{n}(\vec{\pi})]+\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}\colon
k\neq\vec{0},\vec{\pi}}\hat{b}_{n}(k){\mathrm{e}}^{{\mathrm{i}}k\cdot x}.$
(3.26)
Further, $\hat{D}(-\vec{\pi})=-1$, so that, by (2.30)
$\hat{b}_{n}(\vec{\pi})=\hat{b}_{n}(0)(-1)^{n}.$ Therefore,
$\displaystyle\big{|}p_{n}(x)-[1+(-1)^{\|x\|_{1}+n}]r^{-d}\big{|}$
$\displaystyle=\Big{|}\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}\colon
k\neq\vec{0},\vec{\pi}}\frac{\hat{b}_{n}(k)}{\hat{b}_{n}(0)}{\mathrm{e}}^{{\mathrm{i}}k\cdot
x}\Big{|}\leq\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}\colon
k\neq\vec{0},\vec{\pi}}\frac{|\hat{b}_{n}(k)|}{\hat{b}_{n}(0)}$
$\displaystyle\leq(m-1)^{(n-1)/2}+\frac{1}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}\colon
k\neq\vec{0},\vec{\pi}}|\hat{D}(k)|^{n-1}.$
In the nearest-neighbor case, $\hat{D}(k-\vec{\pi})=-\hat{D}(k)$, so we may
restrict to $k$ for which $\hat{D}(k)\geq 0$. Let
${\mathbb{T}}^{*}_{r,d,+}=\\{k\in{\mathbb{T}}^{*}_{r,d}\colon\hat{D}(k)\geq
0\\}$ (3.27)
denote the set of $k$’s for which $\hat{D}(k)\geq 0$. Then,
$\displaystyle\big{|}p_{n}(x)-[1+(-1)^{\|x\|_{1}+n}]r^{-d}\big{|}$
$\displaystyle\leq(m-1)^{(n-1)/2}+\frac{2}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d,+}\colon
k\neq\vec{0}}\hat{D}(k)^{n-1}.$ (3.28)
We use that
$\hat{D}(k)=1-[1-\hat{D}(k)]\leq{\mathrm{e}}^{-[1-\hat{D}(k)]},$ (3.29)
so that
$\displaystyle\big{|}p_{n}(x)-[1+(-1)^{\|x\|_{1}+n}]r^{-d}\big{|}$
$\displaystyle\leq(m-1)^{(n-1)/2}+\frac{2}{r^{d}}\sum_{k\in{\mathbb{T}}^{*}_{r,d}\colon
k\neq\vec{0}}{\mathrm{e}}^{-(n-1)[1-\hat{D}(k)]}$ (3.30)
$\displaystyle=(m-1)^{(n-1)/2}+\frac{2}{r^{d}}\Big{[}\sum_{k\in{\mathbb{T}}^{*}_{r,d}}{\mathrm{e}}^{-(n-1)[1-\hat{D}(k)]}-1\Big{]}$
$\displaystyle=(m-1)^{(n-1)/2}+\frac{2}{r^{d}}\Big{[}\Big{(}\sum_{k\in{\mathbb{T}}^{*}_{r,1}}{\mathrm{e}}^{-(n-1)[1-\cos(k)]/d}\Big{)}^{d}-1\Big{]}.$
We use that the dominant contributions to the sum over
$k\in{\mathbb{T}}^{*}_{r,1}$ comes from $k=0$ and $k=\pm 2\pi/r$, so that
$\sum_{k\in{\mathbb{T}}^{*}_{r,1}}{\mathrm{e}}^{-(n-1)[1-\cos(k)]/d}=1+2{\mathrm{e}}^{-(n-1)[1-\cos(2\pi/r)]/d)}(1+o(1)),$
(3.31)
so that
$\displaystyle\big{|}p_{n}(x)-[1+(-1)^{\|x\|_{1}+n}]r^{-d}\big{|}$
$\displaystyle\leq(m-1)^{(n-1)/2}+\frac{2}{r^{d}}\Big{[}\Big{(}1+2{\mathrm{e}}^{-(n-1)[1-\cos(2\pi/r)]/d)}(1+o(1))\Big{)}^{d}-1\Big{]}$
$\displaystyle\leq(2d-1)^{-(n-1)/2}+\xi r^{-d}$ (3.32)
when $n>(\log{(2/[(1+\xi/2)^{1/d}-1]})/[1-\cos(2\pi/r)]$. ∎
### 3.3 NBW on the hypercube
In this section we specialize the results of Sections 2 and Sections 3.1–3.2
to the hypercube ${\mathbb{T}}_{2,m}=\\{0,1\\}^{m}$. The results in this
section are an important ingredient to the analysis of percolation on the
hypercube in [6]. We start with some notation. It will be convenient to let
the Fourier dual space of $\\{0,1\\}^{m}=\\{0,1\\}^{m}$ be
${\mathbb{Q}}^{*}_{m}=\\{0,1\\}^{m}$, so that the Fourier transform of a
summable function $f\colon\\{0,1\\}^{m}\rightarrow{\mathbb{C}}$ is given by
$\displaystyle\hat{f}(k)=\sum_{x\in\\{0,1\\}^{m}}f(x){\mathrm{e}}^{{\mathrm{i}}\pi
k\cdot x}=\sum_{x\in\\{0,1\\}^{m}}f(x)(-1)^{k\cdot
x},\qquad(k\in\\{0,1\\}^{m}).$
For $k\in\\{0,1\\}^{m}$, let $a(k)$ be its number of non-zero entries. Then,
the SRW step distribution on $\\{0,1\\}^{m}$ satisfies
$\displaystyle\hat{D}(k)=\frac{1}{m}\sum_{j=1}^{m}(-1)^{k_{j}}=1-2a(k)/m.$
(3.33)
The main result of this section is as follows:
###### Theorem 3.5 (NBW on hypercube).
For NBW on the hypercube $\\{0,1\\}^{m}$,
$\hat{p}_{n}(k)\leq\Big{(}|\hat{D}(k)|\vee 1/\sqrt{m-1}\Big{)}^{n-1}.$ (3.34)
Consequently, for every $\varepsilon>0$, there exists $m_{0}$ such that for
all $m\geq m_{0}$,
$T_{{\rm mix}}(\xi)\leq\frac{m(1+\varepsilon)}{2}\log{(2m/\xi)}.$ (3.35)
Random walks on hypercubes have attracted considerable attention, In fact, the
bound on the uniform mixing time for NBW in discrete time closely matches the
one for SRW in continuous time (see [1, Lemma 2.5(a)]).
###### Proof.
The bound in (3.34) follows directly from Theorem 3.1 . We continue to
investigate the convergence of the NBW transition probabilities $x\mapsto
p_{n}(x)$ to its quasi-stationary distribution, which is $2\times 2^{-m}$ when
$n$ and $x$ have the same parity and $0$ otherwise. Here we say that $n$ and
$x$ have the same parity when there exists an $n$-step path from $0$ to $x$.
###### Lemma 3.6 (Convergence to equilibrium for NBW on hypercube).
For NBW on $\\{0,1\\}^{m}$ and every $n>d(\log{d}+\log{\xi})/2$,
$\max_{x\in\\{0,1\\}^{m}}\big{|}p_{n}(x)-2^{-m}[1+(-1)^{\|x\|_{1}+n}]\big{|}\leq(m-1)^{-(n-1)/2}+\xi
2^{-m}.$ (3.36)
Consequently, for every $\varepsilon>0$, there exists $m_{0}$ such that for
every $m\geq m_{0}$,
$T_{{\rm mix}}(\xi)\leq-\frac{m(1+\varepsilon)}{2}\log{([1+\xi/2)]^{1/m}-1)}.$
###### Proof.
The inverse Fourier transform on the $m$-dimensional hypercube is given by
$\displaystyle f(x)=2^{-m}\sum_{k\in\\{0,1\\}^{m}}\hat{f}(k)(-1)^{k\cdot x},$
so that, using $\hat{b}_{n}({\vec{1}})=(-1)^{n}\hat{b}_{n}(0)$,
$b_{n}(x)=2^{-m}\sum_{k\in\\{0,1\\}^{m}}\hat{b}_{n}(k)(-1)^{k\cdot
x}=2^{-m}[1+(-1)^{\|x\|_{1}+n}]\hat{b}_{n}(0)+\sum_{k\in\\{0,1\\}^{m}\colon
k\neq{\vec{1}},\vec{0}}\hat{b}_{n}(k)(-1)^{k\cdot x}.$ (3.37)
Substituting the bound (3.34) in (3.37) leads to
$\displaystyle\big{|}p_{n}(x)-2^{-m}[1+(-1)^{\|x\|_{1}+n}]\big{|}$
$\displaystyle\leq 2\cdot
2^{-m}\sum_{j=1}^{m/2}{{m}\choose{j}}\big{[}(1-2j/m)^{n-1}+(m-1)^{-n/2}\big{]}$
(3.38) $\displaystyle\leq(m-1)^{-n/2}+2\cdot
2^{-m}\sum_{j=1}^{m/2}{{m}\choose{j}}{\mathrm{e}}^{-2j(n-1)/m}$
$\displaystyle\leq(m-1)^{-n/2}+2\cdot
2^{-m}\big{[}(1+{\mathrm{e}}^{-2(n-1)/m})^{m}-1\big{]}$
$\displaystyle\leq(m-1)^{-n/2}+\xi 2^{-m},$
when $n>-\frac{m}{2}\log{([1+\xi/2]^{1/m}-1)}$. ∎
#### Acknowledgements.
This work was supported in part by the Netherlands Organization for Scientific
Research (NWO). RvdH thanks Asaf Nachmias for valuable comments on an early
version of the paper, and Gordon Slade for pointing us at the relevant
literature.
## References
* [1] D. Aldous. Minimization algorithms and random walk on the $d$-cube. Ann. Probab., 11(2):403–413, (1983).
* [2] D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs (monograph in preparation.), (2002).
* [3] N. Alon, I. Benjamini, E. Lubetzky, and S. Sodin. Non-backtracking random walks mix faster. Commun. Contemp. Math., 9(4):585–603, (2007).
* [4] N. Alon and E. Lubetzky. Poisson approximation for non-backtracking random walks. Israel J. Math., 174:227–252, (2009).
* [5] P. Billingsley. Probability and Measure. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, third edition, (1995). A Wiley-Interscience Publication.
* [6] R. van der Hofstad and A. Nachmias. Hypercube percolation. Preprint (2012).
* [7] M. Kotani and T. Sunada. Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo, 7(1):7–25, (2000).
* [8] D. Levin, Y. Peres, and E. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, (2009). With a chapter by James G. Propp and David B. Wilson.
* [9] N. Madras and G. Slade. The Self-Avoiding Walk. Birkhäuser, Boston, (1993).
* [10] J. Noonan. New upper bounds for the connective constants of self-avoiding walks. J. Statist. Phys., 91(5-6):871–888, (1998).
* [11] R. Ortner and W. Woess. Non-backtracking random walks and cogrowth of graphs. Canad. J. Math., 59(4):828–844, (2007).
* [12] A. Pönitz and P. Tittmann. Improved upper bounds for self-avoiding walks in ${\bf Z}^{d}$. Electron. J. Combin., 7:Research Paper 21, 10 pp. (electronic), (2000).
* [13] G. Slade. The lace expansion and its applications, volume 1879 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, (2006). Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, Edited and with a foreword by Jean Picard.
|
arxiv-papers
| 2012-12-27T16:23:01 |
2024-09-04T02:49:39.712327
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Robert Fitzner, Remco van der Hofstad",
"submitter": "Robert Fitzner",
"url": "https://arxiv.org/abs/1212.6390"
}
|
1212.6417
|
# Interaction of Diatomic Molecules with Photon Angular Momentum
Thomas B. Bahder Aviation and Missile Research, Development, and Engineering
Center,
US Army RDECOM, Redstone Arsenal, AL 35898, U.S.A.
###### Abstract
The interaction of a diatomic molecule with photons carrying well-defined
angular momentum and parity is investigated to determine whether photon
absorption can induce molecular rotational transitions between states having
angular momentum $\Delta J>1$. A transformation from laboratory coordinates to
coordinates with origin at the center-of-mass of the nuclei is used to obtain
the interaction between the photons and the molecule’s center-of-mass,
electronic, and rotational degrees of freedom. For molecules making
transitions between rotational levels, there is a small parameter, $ka\ll 1$,
where $k$ is the photon wave vector and $a$ is the size of the molecule, which
enters into the $Ej$ and $Mj$ photon absorption probabilities. For electric
photons having arbitrary angular momentum $j\hbar$, the probability of
absorbing an $E(j+1)$ photon divided by the probability of absorbing an $Ej$
photon, scales as $(ka)^{2}/(2j+1)^{2}$. The probability of absorbing an $Mj$
photon, divided by the probability of absorbing and $Ej$ photon scales
according to the same factor.
## I Introduction
A molecule is significantly more complicated than an atom because, in addition
to center of mass motion, a molecule has vibrational and rotational degrees of
freedom. Work on molecular spectra has a long history. A significant milestone
was made in 1927 by Born and Oppenheimer Born and Oppenheimer (1927), who
assumed that the lighter electrons in a molecule adjust adiabatically to the
slower motion of the heavier nuclei, which remain at all times in their
instantaneous ground state. Further progress in understanding molecular
spectra was made by treating a rotating molecule as a symmetric top or
asymmetric top Wang (1929), sometimes called a rigid rotor Landau and Lifshitz
(1984); Davydov (1965), see Ref. Van Vleck (1929) and references contained
therein. In the rigid rotor treatment of molecules, vibrational and center-of-
mass motion are essentially neglected.
Solving the complete quantum mechanical many body problem consisting of
electrons and nuclei for a given type of molecule will in-principle provide
the energy spectrum of the molecule. In practice, this is a complicated task
which usually results in heavy numerical work. For this reason, much analytic
work has been done on simple diatomic molecules Landau and Lifshitz (1984);
Davydov (1965). For modern reviews on the theory of the molecular
Hamiltonians, see Meyer Meyer (2002), and for an overview about diatomic
molecules, see Brown and Carrington Brown and Carrington (2003).
While the spectrum of a molecule can in-principle be obtained by solving the
corresponding quantum mechanical many body problem, in experiments we usually
probe molecules with electromagnetic fields. In order to interpret
experiments, we must calculate selection rules and probabilities of transition
between molecular quantum states. This involves investigating the coupling of
a molecule to the electromagnetic field. The fundamental theory of a second
quantized electromagnetic radiation field coupled to charges is discussed in
Ref. Akheizer and Berestetskii (1965); Berestetskii et al. (1982). Detailed
treatments dealing with radiation coupling to molecules are given by Atkins
and Wooley Atkins and Woolley (1970), Woolley Woolley (1971, 1975), Craig and
Thirunamachandran Craig and Thirunamachandran (1998) and a recent detailed
treatment is given by Sindelka Sindelka and Moiseyev (2006).
In the past two decades, a completely different line of investigation has
dealt with the angular momentum carried by the electromagnetic field. Beth
Beth (1936) made the first experimental observation that light carries angular
momentum. Beth demonstrated that if a beam of right-hand circularly polarized
light passed through a birefringent plate, and the beam was converted into a
left-hand polarized beam, the plate experienced a torque of $2\hbar$ for each
photon in the beam. This experiment is cited as a demonstration that a light
beam carries spin angular momentum. Years later, in 1992, Allen et al. showed
that laser light with a Laguerre-Gaussian amplitude distribution has well-
defined orbital angular momentum Allen et al. (1992). Many theoretical and
experimental investigations followed, see Ref. Torres and Torner (2011). For
paraxial beams, i.e., in the paraxial approximation, it has been demonstrated
that the total angular momentum can be separated into a sum of spin angular
momentum and orbital angular momentum Allen et al. (1999). However, for a
general electromagnetic radiation field, spin and orbital angular momentum
cannot be separately defined, and only the total angular momentum is a well-
defined quantity Akheizer and Berestetskii (1965); Davydov (1965);
Berestetskii et al. (1982). The interaction of atoms with light carrying
angular momentum were explored theoretically in a number of works Huang
(1994); Jáuregui (2004); Grinter (2008).
Within the rigid rotor model of a molecule, one can easily see that an
electromagnetic radiation field can induce (dipole, quadrupole, etc…)
rotational transitions in a diatomic molecule through its multipole moments,
see for example p. 601 in Section 136 of DavydovDavydov (1965). However, the
rigid rotor treatment neglects the molecule center-of-mass degree of freedom.
In the absence of a radiation field, the molecular center-of-mass motion
decouples from the other degrees of freedom and there is no issue to consider
further, see Eqs. (30) and (38). However, in the presence of a radiation
field, the molecular center of mass plays a key role in the molecule-field
interaction, see Eq. (42) and Ref. Thomas (1970a); Thomas and Joy (1970);
Thomas (1970b, 1971); Sindelka and Moiseyev (2006). References Thomas (1970a);
Thomas and Joy (1970); Thomas (1970b, 1971) discuss the non-trivial role of
the center of molecular mass when a classical plane polarized electromagnetic
wave (corresponding to photons having $j=\hbar$) interacts with a molecule. In
order to investigate in detail molecular absorption of photons having
arbitrary angular momentum ($j>n\hbar)$, where $n>1$, one has to revisit the
interaction Hamiltonian between the molecule and radiation field. As already
remarked, this is complicated because a molecule has several degrees of
freedom: center-of-mass motion, electronic motion, vibrational motion, and
rotational motion.
In this manuscript, I consider a simple diatomic molecule interacting with
photons carrying arbitrary angular momentum, $j\hbar$. I investigate the
mechanism that leads to transitions between molecular rotational levels, with
emphasis on the role of the molecular center-of-mass. Of particular interest
are probabilities of transitions between rotational levels whose angular
momentum differs by more than one unit, $\Delta J>1$. I write down a
Hamiltonian for a diatomic molecule, for the radiation field, and for their
interaction. By making successive transformations from the laboratory
coordinates into the center-of-mass coordinates, and then to center-of-
molecule coordinates, I obtain interaction terms that show the coupling of a
diatomic molecule to the radiation field. For the molecule, there is a small
parameter, $ka\ll 1$, where $k$ is the wave vector and $a$ is the size of the
molecule. I find how the probability of the molecule absorbing an $Ej$ and
$Mj$ multipole photon scales with $ka$. In this treatment, I neglect all
relativistic interactions containing electron and nuclear spins.
## II Molecular and Field Hamiltonian
I consider a closed system consisting of a diatomic molecule interacting with
a quantized radiation field. The Hamiltonian can be described as a sum of
three terms Davydov (1965); Berestetskii et al. (1982); Akheizer and
Berestetskii (1965); Atkins and Woolley (1970); Woolley (1971, 1975):
${{\hat{H}}}={\hat{H}}_{molecule}+{{\hat{H}}_{field}}+{\hat{H}}_{\mathop{\rm
int}}(t)$ (1)
where ${\hat{H}}_{molecule}$ is the Hamiltonian for the molecule alone (with
no field present), ${{\hat{H}}_{field}}$ is the Hamiltonian for the field
(with no molecule present) and ${\hat{H}}_{\mathop{\rm int}}$ is the
interaction Hamiltonian between the radiation field and the molecule. I assume
the diatomic molecule consists of $n$ electrons and two nuclei. I use a
laboratory system of Cartesian coordinates, ${\bf R}_{i}=(X_{i},Y_{i},Z_{i})$,
$i=1,2,\ldots,n$, for the electrons and ${\bf
R}_{\alpha}=(X_{\alpha},Y_{\alpha},Z_{\alpha})$, $\alpha=1,2$, for the nuclei.
I take the basis vectors of the Cartesian coordinate system to be $({\bf
e}_{x},{\bf e}_{y},{\bf e}_{z})\equiv({\bf e}_{1},{\bf e}_{2},{\bf e}_{3})$.
The position of the origin of the laboratory system of coordinates is
arbitrary. In what follows, I distinguish between coordinates, e.g.,
coordinates of the $i$th electron, ${\bf R}_{i}=(X_{i},Y_{i},Z_{i})$, and the
geometric object, $(X_{i}{\bf e}_{x},Y_{i}{\bf e}_{y},Z_{i}{\bf e}_{z},)$,
which represents the position of the $i$th electron and is independent of the
coordinate system. Note that these two quantities, ${\bf
R}_{i}=(X_{i},Y_{i},Z_{i})$ and $(X_{i}{\bf e}_{x},Y_{i}{\bf e}_{y},Z_{i}{\bf
e}_{z},)$, transform differently under change of coordinate systems. The first
one transforms as coordinate components and the second is a geometric quantity
that does not transform.
The Hamiltonian for the diatomic molecule can be written as Akheizer and
Berestetskii (1965); Meyer (2002)
${{\hat{H}}_{molecule}}=\hat{T}+{{\hat{V}}_{e-e}}+{{\hat{V}}_{n-n}}+{{\hat{V}}_{e-n}}$
(2)
The kinetic energy operator for electrons and nuclei is given by
$\hat{T}=\sum\limits_{i=1}^{n}{\frac{{{\bf{\hat{P}}}_{i}^{2}}}{{2m}}}+\sum\limits_{\alpha=1,2}{\frac{{{\bf{\hat{P}}}_{\alpha}^{2}}}{{2{M_{\alpha}}}}}$
(3)
where $m$ is the mass of an electron and $M_{1}$ and $M_{2}$ are the masses of
the nuclei, and $n$ is the number of electrons. Note that I have not assumed
that the molecule is electrically neutral. The electron and nuclear momentum
operators, ${\bf{\hat{P}}}_{i}=\frac{\partial}{{\partial{{\bf{R}}_{i}}}}$ and
${\bf{\hat{P}}}_{\alpha}=\frac{\partial}{{\partial{{\bf{R}}_{\alpha}}}}$, are
conjugate to the coordinates, ${\bf R}_{i}=(X_{i},Y_{i},Z_{i})$ and ${\bf
R}_{\alpha}=(X_{\alpha},Y_{\alpha},Z_{\alpha})$. The Coulomb interaction terms
for electrons, nuclei, and electrons and nuclei, respectively, are given by
$\begin{array}[]{l}{{\hat{V}}_{e-e}}=\frac{{\,{e^{2}}}}{{4\pi{\varepsilon_{0}}}}\sum\limits_{i<j}{\frac{1}{{\left|{{{\bf{R}}_{i}}-{{\bf{R}}_{j}}}\right|}}}\\\
{{\hat{V}}_{n-n}}=\frac{{{Z_{1}}\,{Z_{2}}\,{e^{2}}}}{{4\pi{\varepsilon_{0}}}}\frac{1}{{\left|{{{\bf{R}}_{2}}-{{\bf{R}}_{1}}}\right|}}\\\
{{\hat{V}}_{e-n}}=-\frac{{\,{e^{2}}}}{{4\pi{\varepsilon_{0}}}}\sum\limits_{\alpha=1,2}{\sum\limits_{i=1}^{n}{\frac{{{Z_{\alpha}}}}{{\left|{{{\bf{R}}_{i}}-{{\bf{R}}_{\alpha}}}\right|}}}}\\\
\end{array}$ (4)
where $e$ is the electronic charge, $\varepsilon_{0}$ is the permitivity of
vacuum, $Z_{\alpha}$ is the atomic number and $M_{\alpha}$ is the mass of
nucleus $\alpha$.
## III Photons with Well-Defined Angular Momentum
For the the radiation field, I can choose Laguerre-Gaussian modes Allen et al.
(1999); Romero et al. (2002), which are solutions of the paraxial wave
equation, and in which a separation can be made between spin angular momentum
and orbital angular momentum. These types of photons carry angular momentum
$\hbar l$. Such modes are convenient when dealing with cylindrical symmetry,
however, they are only approximate solutions of Maxwell equations.
Alternatively, we can choose vector spherical harmonic modes Davydov (1965);
Berestetskii et al. (1982), which are exact solutions of Maxwell’s equations,
but then spin angular momentum and orbital angular momentum cannot be
separated. This is not a flaw of vector spherical harmonics, rather it is a
general feature of electromagnetic radiation fields. The vector spherical
harmonic modes are a complete set of vector eigenfunctions, so any field can
be expressed in terms of these modes. For these reasons, I take the radiation
field to be described by vector spherical harmonic modes. The Hamiltonian for
the radiation field, see Eq. (1), is given by
${{\hat{H}}_{field}}=\sum\limits_{\lambda,k,j,m}{\hbar{\omega_{k}}\left({{{\hat{a}}^{\dagger}}_{\lambda
kjm}\,{{\hat{a}}_{\lambda kjm}}+\frac{1}{2}}\right)\,}$ (5)
where ${{{\hat{a}}^{\dagger}}_{\lambda kjm}}$ and ${{{\hat{a}}}_{\lambda
kjm}}$ are the creation and destruction operators for electric or magnetic
multipole photons, labeled by $\lambda=(E,M)$, respectively, having wave
vector magnitude $k=\omega_{k}/c$, and carrying angular momentum $j\hbar$ and
angular momentum projection $m\hbar$ on z-axis . The creation and annihilation
operators satisfy the commutation relations Davydov (1965)
$\left[{{{\hat{a}}_{\lambda
kjm}},{{\hat{a}}^{\dagger}}_{\lambda^{\prime}{\kern 1.0pt}k^{\prime}{\kern
1.0pt}j^{\prime}{\kern 1.0pt}m^{\prime}}}\right]={\delta_{\lambda{\kern
1.0pt}\lambda^{\prime}}}{\delta_{k{\kern 1.0pt}k^{\prime}}}\,{\delta_{j{\kern
1.0pt}j^{\prime}}}\,{\delta_{m{\kern 1.0pt}m^{\prime}}}$ (6)
Each field state with well-defined photon number has well-defined energy. Each
photon in a state labeled by quantum numbers ($\lambda,k,j,m$) has wave vector
$k$, square of angular momentum $j(j+1)\hbar^{2}$, z-component of angular
momentum $m\hbar$, and parity $(-1)^{j}$ for $\lambda=E$ and parity
$(-1)^{j+1}$ for $\lambda=M$. Photons are of two types: an $Ej$ photon is an
electric photon and a $Mj$ photon is a magnetic photon, each carrying $j\hbar$
units of angular momentum Davydov (1965); Akheizer and Berestetskii (1965);
Berestetskii et al. (1982). The vector potential operator is given by
${\bf{\hat{A}}}({\bf{r}},t)=\sum\limits_{\lambda,k,j,m}{\left[{{{\hat{a}}_{\lambda
kjm}}\,{\bf{A}}_{kjm}^{\lambda}({\bf{r}},t)+{{\hat{a}}^{\dagger}}_{\lambda
kjm}\,{\bf{A}}_{kjm}^{\lambda\;*}({\bf{r}},t)}\right]}$ (7)
where ${{\bf{A}}_{kjm}^{E}({\bf{r}},t)}$ is the photon wave function for
electric 2j pole radiation and ${{\bf{A}}_{kjm}^{M}({\bf{r}},t)}$ is the
photon wave function for magnetic 2j pole radiation Davydov (1965); Akheizer
and Berestetskii (1965); Berestetskii et al. (1982). In a large sphere of
radius $\cal{R}$, the magnetic and electric photon wave functions are given by
Davydov (1965)
${\bf{A}}_{kjm}^{M}({\bf{r}},t)=\frac{1}{{\sqrt{V}}}{f_{j}}(kr)\,{\bf{Y}}_{jjm}(\theta,\phi)\,{e^{-i{\omega_{k}}t}}$
(8)
and
$\displaystyle{\bf{A}}_{kjm}^{E}({\bf{r}},t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{V}\,\sqrt{2j+1}}\left[\sqrt{j}\,f_{j+1}(kr){\bf{Y}}_{j,j+1,m}(\theta,\phi)\right.$
(9)
$\displaystyle\left.-\sqrt{j+1}\,{f_{j-1}}(kr){\bf{Y}}_{j,j-1,m}(\theta,\phi)\right]\,{e^{-i{\omega_{k}}t}}$
where photon frequency is $\omega_{k}=ck$ and the spherical quantization
volume is $V$. The radial functions, $f_{j}(kr)$, are given in terms of Bessel
functions of the first kind Davydov (1965), $J_{\nu+\frac{1}{2}}(x)$
${f_{l}}(k\,r)=\frac{{{{\left({2V}\right)}^{1/2}}}}{R}\frac{1}{{{J_{l+\frac{3}{2}}}\left({{x_{l+\frac{1}{2},n}}}\right)}}\frac{{{J_{l+\frac{1}{2}}}\left({k\,r}\right)}}{{{r^{1/2}}}}$
(10)
and the allowed set of discrete wave vectors, $k_{ln}$, where $l$ and $n$ are
positive integers, are given by
${J_{l+\frac{1}{2}}}\left({{k_{l{\kern 1.0pt}n}}\,\cal{R}}\right)=0$ (11)
These discrete wave vectors determine the photon energies,
$\hbar\omega_{k}=\hbar ck_{l\,n}$, in a sphere of radius $R$. For a given
value of $l$, the spectrum of allowed wave vectors, $\\{k_{l\,n}\\}$ is the
same for electric and magnetic photons.
The three vector wave functions (vector spherical harmonics), ${\bf
Y}_{j,l,m}(\theta,\phi)$, $j=l+1,l,|l-1|$, are defined as direct products of
two different irreducible representations of the three-dimensional rotation
group, ${\bf e}_{\mu}$, and $Y_{lm}(\theta,\phi)$, coupled by Clebsch-Gordon
coeficients:
${{\bf{Y}}_{j,l,m}}(\theta,\phi)=\sum\limits_{\mu}{\left\langle{{1,l,\mu,m-\mu}}\mathrel{\left|{\vphantom{{1,l,\mu,m-\mu}{jm}}}\right.\kern-1.2pt}{{jm}}\right\rangle}\,{{\bf{e}}_{\mu}}\,{Y_{l,m-\mu}}$
(12)
Here ${\bf e}_{\mu}$ are three spin $S=1$ eigenfunctions,
$\chi_{\mu}(\sigma)$, whose components are ${\bf
e}_{\mu}=\\{\chi_{\mu}(+1),\chi_{\mu}(0),\chi_{\mu}(-1)\\}$, and
$\sigma={+1,0,-1}$ is the spin variable Davydov (1965); Landau and Lifshitz
(1984). The ${\bf e}_{\mu}$, $\mu=\\{-1,0,+1\\}$, are the spherical basis
vectors, which are simultaneous eigenfunctions of the spin angular momentum
operators $\hat{{\bf S}}^{2}$ and $\hat{S}_{z}$:
$\displaystyle\hat{{\bf S}}^{2}\,{\bf e}_{\mu}$ $\displaystyle=$
$\displaystyle 2{\kern 1.0pt}{\bf e}_{\mu}$ $\displaystyle\hat{S}_{z}\,{\bf
e}_{\mu}$ $\displaystyle=$ $\displaystyle\mu\,{\bf e}_{\mu}$ (13)
The spherical basis vectors are related to the Cartesian basis vectors,
$\\{{\bf e}_{x},{\bf e}_{y},{\bf e}_{z}\\}$, by
$\displaystyle{{\bf{e}}_{+}}$ $\displaystyle=$
$\displaystyle-\frac{1}{{\sqrt{2}}}\left({{{\bf{e}}_{x}}+i{{\bf{e}}_{y}}}\right)$
$\displaystyle{{\bf{e}}_{-}}$ $\displaystyle=$
$\displaystyle\frac{1}{{\sqrt{2}}}\left({{{\bf{e}}_{x}}-i{{\bf{e}}_{y}}}\right)$
$\displaystyle{{\bf{e}}_{0}}$ $\displaystyle=$ $\displaystyle{{\bf{e}}_{z}}$
(14)
In terms of the Cartesian basis vectors, $\\{{\bf e}_{x},{\bf e}_{y},{\bf
e}_{z}\\}$, the action of the spin operator is given by
$\hat{S}_{i}{\bf e}_{k}=i\sum_{l=1,2,3}\,e_{ikl}\,{\bf e}_{l}$ (15)
where $e_{ikl}$ is the totally antisymmetric Levi-Civita tensor, where
$e_{123}=+1$.
The $Y_{l,m}$ in Eq. (12) are the usual scalar spherical Harmonics that are
eigenfunctions of the orbital angular momentum operators $\hat{{\bf L}}^{2}$
and $\hat{L}_{z}$:
$\displaystyle\hat{{\bf L}}^{2}\,Y_{lm}$ $\displaystyle=$ $\displaystyle
l(l+1)Y_{lm}$ $\displaystyle\hat{L}_{z}\,Y_{lm}$ $\displaystyle=$
$\displaystyle mY_{lm}$ (16)
According to the definitions in Eq.(12)–(16), the product functions,
${{\bf{e}}_{\mu}}\,{Y_{l,m}}$, are simultaneous eigenfunctions of $\hat{{\bf
L}}^{2}$, $\hat{L}_{z}$, $\hat{{\bf S}}^{2}$ and $\hat{S}_{z}$, with
eigenvalues $l(l+1)$, $m$, $2$, and $\mu$, respectively.
The spherical vector harmonics, ${{\bf{Y}}_{j,l,m}}(\theta,\phi)$, form an
irreducible representaion of the three-dimensional rotation group on the unit
sphere, and are simultaneous eigenfunctions of $\hat{J}^{2}$, $\hat{J}_{z}$,
$\hat{{\bf L}}^{2}$, and $\hat{{\bf S}}^{2}$:
$\displaystyle{{\hat{J}}_{z}}{{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right)$
$\displaystyle=$ $\displaystyle m{{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right)$
(17)
$\displaystyle{{\hat{J}}^{2}}{{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right)$
$\displaystyle=$ $\displaystyle
j(j+1){{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right)$ (18)
$\displaystyle{{\hat{L}}^{2}}{{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right)$
$\displaystyle=$ $\displaystyle
l(l+1){{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right)$ (19)
$\displaystyle{{\hat{S}}^{2}}{{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right)$
$\displaystyle=$ $\displaystyle 2{{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right)$
(20)
The vector spherical harmonics, ${{\bf{Y}}_{j,l,m}}(\theta,\phi)$, satisfy the
orthogonality relation
$\int{{d^{2}}\Omega}\,{\bf{Y}}_{jlm}^{*}\left({\theta,\phi}\right)\cdot{\bf{Y}}_{j^{\prime}l^{\prime}m^{\prime}}\left({\theta,\phi}\right)={\delta_{j\,j^{\prime}}}\,{\delta_{l\,l^{\prime}}}\,{\delta_{m\,m^{\prime}}}$
(21)
where the integration is over the unit sphere, and completeness relation
$\sum\limits_{j\,l\,m}{{{\bf{Y}}_{j,l,m}}\left({\theta,\phi}\right){{\bf{Y}}_{j,l,m}}\left({\theta^{\prime},\phi^{\prime}}\right)}={\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over{I}}}}\,{\delta^{2}}\left({\Omega-\Omega^{\prime}}\right)$
(22)
Under the parity operator (spatial inversion), the basis vectors ${\bf
e}_{\mu}$ change sign, and we take the scalar spherical harmonics, $Y_{lm}$,
to be defined so that they are multiplied by $(-1)^{l}$, and therefore the
vector spherical harmonics ${\bf Y}_{j,l,m}(\theta,\phi)$ are eigenfunctions
of parity with eigenvalues $(-1)^{l+1}$.
From Eq. (8) we see that magnetic photons have well-defined spin and orbital
angular momentum, since ${\bf{A}}_{kjm}^{M}({\bf{r}},t)$ is an eigenfunction
of spin and orbital angular momentum. However, from Eq. (9) we see that
electric photons with quantum numbers $(j,m)$ are a superposition of orbital
angular momentum $l=j-1$ and $l=j+1$, and do not have well-defined orbital
angular momentum.
The number operator for photons of type $\lambda$ is given by
$\hat{{n}}_{\lambda}=\sum\limits_{k,j,m}{{{\hat{a}}^{\dagger}}_{\lambda
kjm}\,{{\hat{a}}_{\lambda kjm}}\,}$ (23)
## IV Molecule-Field Interaction
The electromagnetic field couples to the charges in the molecule, which
consist of electrons and nuclei. The interaction Hamiltonian describing the
coupling of the radiation field to the molecule is given by Atkins and Woolley
(1970); Woolley (1971, 1975); Berestetskii et al. (1982); Meyer (2002);
Sindelka and Moiseyev (2006)
${{\hat{H}}_{{\mathop{\rm
int}}}}(t)=\frac{e}{m}\sum\limits_{i=1}^{n}{{\bf{\hat{A}}}({{\bf{R}}_{i}},t)\cdot{{{\bf{\hat{P}}}}_{i}}}-e\sum\limits_{\alpha=1}^{2}{\frac{{{Z_{\alpha}}}}{{{M_{\alpha}}}}{\bf{\hat{A}}}({{\bf{R}}_{\alpha}},t)\cdot{{{\bf{\hat{P}}}}_{\alpha}}}$
(24)
where the vector potential operator is in laboratory coordinates, and is given
in Eq. (7). In Eq. (24), the first term is the coupling of the field to
electrons and the second term is the coupling of the field to the nuclei. The
radiation field, ${\bf{\hat{A}}}({\bf r},t)$, is quantized in the laboratory
coordinates, ${\bf R}$. The electron and nuclear momentum operators in Eq.
(24), ${\bf P}_{i}$ and ${\bf P}_{\alpha}$, are also in laboratory
coordinates. I have dropped terms that are quadratic in the vector potential
because they are responsible for photon scattering and are small for photon
absorption and emission. Also, I have dropped small relativistic terms that
couple the electron and nuclear spins to the field Landau and Lifshitz (1984);
Davydov (1965); Berestetskii et al. (1982). Furthermore, in Eq. (24), I have
assumed the Coulomb gauge for the vector potential, so that ${\rm
div}\,\,{\bf{\hat{A}}}({\bf r},t)=0$ and the scalar potential is zero.
In many experiments, the wavelength of the electromagnetic radiation field is
long compared to the dimensions of the physical system, such as the molecule.
We can make a Taylor series expansion of the vector potential terms
${\bf{\hat{A}}}({\bf R}_{i},t)$ and ${\bf{\hat{A}}}({\bf R}_{\alpha},t)$ about
the center of mass of the molecule ${\bf R}_{o}$. Keeping only the first term,
I make the approximations
${\bf{\hat{A}}}({{\bf{R}}_{i}},t)\approx{\bf{\hat{A}}}({{\bf{R}}_{\alpha}},t)\approx{\bf{\hat{A}}}({{\bf{R}}_{o}},t)$
(25)
## V Transformation to Center of Mass of the Molecule
In order to explore the transfer of angular momentum from photons to
molecules, we must look at degrees of freedom that can ”absorb” angular
momentum. To this end, I transform the molecular Hamiltonian in Eq. (2) from
the laboratory coordinates to center-of-mass-of-molecule coordinates. I take
the Cartesian basis vectors in laboratory coordinates to be collinear with the
basis vectors in center-of-mass-of-molecule coordinates. I follow the notation
used by Brown and Carrington Brown and Carrington (2003). The position of the
center of mass of the molecule, in laboratory coordinates, is given by
${{\bf{R}}_{o}}=\frac{m}{M}\sum\limits_{i=1}^{n}{{{\bf{R}}_{i}}+\frac{1}{M}\sum\limits_{\alpha=1}^{2}{{M_{\alpha}}{{\bf{R}}_{\alpha}}}}$
(26)
where the total molecular mass is given by
$M=\sum\limits_{i=1}^{n}{m_{i}}+\sum\limits_{\alpha=1}^{2}{{M_{\alpha}}}=nm+M_{1}+M_{2}$
(27)
where $n$ is the number of electrons in the molecule. The transformation of
electron and nuclear coordinates, from laboratory coordinates,
${{\bf{R}}_{i}}$ and ${{\bf{R}}_{\alpha}}$, to the center of mass of molecule
coordinates, ${{{\bf{R^{\prime}}}}_{i}}$ and ${{{\bf{R^{\prime}}}}_{\alpha}}$,
is given by
$\displaystyle{{{\bf{R^{\prime}}}}_{i}}$ $\displaystyle=$
$\displaystyle{{\bf{R}}_{i}}-{{\bf{R}}_{o}}$
$\displaystyle{{{\bf{R^{\prime}}}}_{\alpha}}$ $\displaystyle=$
$\displaystyle{{\bf{R}}_{\alpha}}-{{\bf{R}}_{o}}$ (28)
Note that the transformations in Eq. (28) are transformations of vector
components, see comment above Eq. (2). I introduce the inter-nuclear vector,
${\bf R}$, in terms of the coordinates of the two nuclei (in laboratory
coordinates)
${\bf R}={\bf R}_{2}-{\bf R}_{1}$ (29)
Using the above definitions of the transformation to the center-of-mass-of-
molecule coordinates, the transformation of the kinetic energy of the
molecule, given in Eq. (3), to the center-of-mass-of-molecule coordinates,
gives Brown and Carrington (2003)
$T=\frac{1}{{2M}}{\bf{P}}_{o}^{2}+\frac{1}{{2\mu}}{\bf{P}}_{\bf{R}}^{2}+\frac{1}{{2m}}\sum\limits_{i=1}^{n}{{\bf{P^{\prime}}}_{i}^{2}-}\frac{1}{{2M}}\sum\limits_{j,k=1}^{n}{{{{\bf{P^{\prime}}}}_{i}}\cdot}{{{\bf{P^{\prime}}}}_{k}}$
(30)
where the momentum operators
${{\bf{P}}_{o}}=\frac{\hbar}{i}\frac{\partial}{{\partial{{\bf{R}}_{o}}}}$ (31)
${{\bf{P}}_{\bf{R}}}=\frac{\hbar}{i}\frac{\partial}{{\partial{\bf{R}}}}$ (32)
${{\bf{P}}_{j}^{\prime}}=\frac{\hbar}{i}\frac{\partial}{{\partial{{\bf{R}}_{j}^{\prime}}}}$
(33)
are conjugate to the coordinates, ${\bf R}_{o}$, ${\bf R}$ and ${\bf
R}_{j}^{\prime}$, and where $\mu$ is the reduced nuclear mass
$\frac{1}{\mu}=\frac{1}{{{M_{1}}}}+\frac{1}{{{M_{2}}}}$ (34)
The first term in Eq. (30) is the center of mass kinetic energy of the
molecule, expressed in laboratory coordinates. The second term in Eq. (30) is
the relative kinetic energy of the nuclei, expressed in terms of the inter-
nuclear vector ${\bf R}$. The third term in Eq. (30) is the kinetic energy of
the electrons. The last term in Eq. (30) is the so-called mass polarization
terms and results from fluctuations of electron position about the massive
nuclei.
The transformation of the Coulomb interaction terms, given in Eq. (4), from
the laboratory coordinates to center-of-mass-of-molecule coordinates, is done
similarly.
## VI Transformation to Center of Mass of the Nuclei
In order to discuss rotation of the molecule about the line joining the
nuclei, I define the position of the center of mass of the nuclei (in center-
of-mass-of-molecule coordinates):
${{{\bf{R^{\prime}}}}_{N}}=\frac{1}{{{M_{1}}+{M_{2}}}}\left({{M_{1}}{{{\bf{R^{\prime}}}}_{1}}+{M_{2}}{{{\bf{R^{\prime}}}}_{2}}}\right)$
(35)
The positions of the electrons in the center-of-mass-of-nuceli coordinates are
given by
$\displaystyle{{{\bf{R^{\prime\prime}}}}_{i}}$ $\displaystyle=$
$\displaystyle{{{\bf{R^{\prime}}}}_{i}}-{{{\bf{R^{\prime}}}}_{N}}$ (36)
$\displaystyle=$
$\displaystyle{{{\bf{R^{\prime}}}}_{i}}+\frac{1}{{{M_{1}}+{M_{2}}}}\sum\limits_{j=1}^{n}{{m_{j}}}{{{\bf{R^{\prime}}}}_{j}}$
In the center-of-mass-of-nuceli coordinates, the momentum operator conjugate
to the electron position, ${{{\bf{R^{\prime\prime}}}}_{i}}$, is given by
${{{\bf{P^{\prime\prime}}}}_{j}}=\frac{\hbar}{i}\frac{\partial}{{\partial{{{\bf{R^{\prime\prime}}}}_{j}}}}$
(37)
The transformation of nuclear coordinates, ${\bf R}_{\alpha}$, to center-of-
mass-of-nuceli coordinates is similar.
The total kinetic energy of the molecule, given in equation Eq. (30),
transformed to center-of-nuclei coordinates, is given by Judd (1975); Brown
and Carrington (2003)
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2M}{\bf{P}}_{o}^{2}+\frac{1}{{2\mu}}{\bf{P}}_{\bf{R}}^{2}+\frac{1}{{2m}}\sum\limits_{i=1}^{n}{\bf{P^{\prime\prime}}}_{i}^{2}$
(38) $\displaystyle+$
$\displaystyle\frac{1}{{2\left({{M_{1}}+{M_{2}}}\right)}}\sum\limits_{j,k=1}^{n}{{{{\bf{P^{\prime\prime}}}}_{j}}\cdot}{{{\bf{P^{\prime\prime}}}}_{k}}$
Transforming the Coulomb terms in Eq. (2) and (4) to the center-of-mass-of-
the-nuclei coordinates (which are designated by two primed superscripts),
these terms become
$\begin{array}[]{l}{{\hat{V}}_{e-e}}=\frac{{\,{e^{2}}}}{{4\pi{\varepsilon_{0}}}}\sum\limits_{i<j}{\frac{1}{{\left|{{{{\bf{R^{\prime\prime}}}}_{i}}_{j}}\right|}}}\\\
{{\hat{V}}_{n-n}}=\frac{{{Z_{1}}\,{Z_{2}}\,{e^{2}}}}{{4\pi{\varepsilon_{0}}}}\frac{1}{R}\\\
{{\hat{V}}_{e-n}}=-\frac{{\,{e^{2}}}}{{4\pi{\varepsilon_{0}}}}\sum\limits_{\alpha=1,2}{\sum\limits_{i=1}^{n}{\frac{{{Z_{\alpha}}}}{{\left|{{{{\bf{R^{\prime\prime}}}}_{i\alpha}}}\right|}}}}\\\
\end{array}$ (39)
where $|{{{\bf R^{\prime\prime}}}}_{ij}|=\left|{{{\bf
R^{\prime\prime}}}}_{i}-{{{\bf R^{\prime\prime}}}}_{j}\right|$ is the distance
between electrons, $|{{{\bf R^{\prime\prime}}}}_{i\alpha}|=\left|{{{{\bf
R^{\prime\prime}}}}_{i}}-{{{\bf R^{\prime\prime}}}}_{\alpha}\right|$ is the
distance between the $i$th electron and $\alpha$th nucleus, and ${\bf R}$ is
the inter-nuclear vector defined in Eq. (29), with magnitude $R={\bf R}$. Note
that the ${{{\bf R^{\prime\prime}}}}_{i\alpha}$ are expressible in terms of
the electron coordinates, ${{{\bf R^{\prime\prime}}}}_{i}$, and the inter-
nuclear vector, ${\bf R}$, so the molecular wavefunction depends on the center
of mass coordinate, ${\bf R}_{o}$, the inter-nuclear coordinate, ${\bf R}$,
and the electron coordinates ${{{\bf R^{\prime\prime}}}}_{i}$, for
$i=1,2,\cdots,n$.
In summary, when the molecule Hamiltonian in laboratory coordinates, given in
Eq. (2), is transformed into the center-of-mass-of-nuceli coordinates, it is
given by the sum of Eq. (38) and (39). Note that in the absence of the
electromagnetic field, the center-of-mass motion of the molecule decouples
from the internal motion of the molecule, i.e., the molecular Hamiltonian does
not contain mixed terms with center-of-mass and internal degrees of freedom.
So, in the absence of the radiation field, the total molecular wavefunction is
the product:
${\psi_{QLM}}\left({{{\bf{R}}_{o}}}\right)\,\phi\left({{\bf{R}},{{{\bf{R^{\prime\prime}}}}_{1}},{{{\bf{R^{\prime\prime}}}}_{2}},\cdots,{{{\bf{R^{\prime\prime}}}}_{n}},}\right)$
(40)
where the center-of-mass wavefunction is of the form
${\psi_{QLM}}\left({{{\bf{R}}_{o}}}\right)={R_{QL}}\left({Q{R_{o}}}\right)\,{Y_{LM}}\left({{\theta_{o}},{\phi_{o}}}\right)$
(41)
and
$\phi\left({{\bf{R}},{{{\bf{R^{\prime\prime}}}}_{1}},{{{\bf{R^{\prime\prime}}}}_{2}},\cdots,{{{\bf{R^{\prime\prime}}}}_{n}},}\right)$
is the wave function for internal degrees of freedom.
## VII Interaction Hamiltonian in CM of Nuclei Coordinates
As previously stated, in the absence of the radiation field, the molecule is
described by a Hamiltonian that is the sum of Eq. (38) and (39). In this case,
the center of mass motion, described by coordinate ${\bf R}_{o}$, decouples
from the internal motion, which is described by coordinate ${\bf R}$. However,
when an external radiation field is present, see the interaction Hamiltonian
given in Eq. (42), the radiation field introduces a coupling between the
center of mass motion of the molecule and its internal motions Sindelka and
Moiseyev (2006).
Next, I assume that the molecule interacts with radiation, according to the
interaction Hamiltonian in Eq. (24). It is possible to re-write the molecule-
field interaction in a form that allows more insight than that given by the
interaction Hamiltonian in Eq. (24). In order to do this, I make the long
wavelength approximation, given in Eq. (25), in the interaction Hamiltonian in
Eq. (24). Then, I transform the electron and nuclear momentum operators to
center-of-mass-of-nuclei coordinates. The molecule-field interaction
Hamiltonian ${\hat{H}_{{\mathop{\rm int}}}}(t)$ in Eq. (24) then becomes
$\displaystyle{\hat{H}_{{\mathop{\rm int}}}}(t)$ $\displaystyle=$
$\displaystyle\frac{e}{M}(n-{Z_{1}}-{Z_{2}})\,{\bf{\hat{A}}}({{\bf{R}}_{o}},t)\cdot\,{{{\bf{\hat{P}}}}_{o}}+$
(42) $\displaystyle+$ $\displaystyle{\kern
1.0pt}e\left({\frac{1}{m}+\frac{{{Z_{1}}+{Z_{2}}}}{{{M_{1}}+{M_{2}}}}}\right){\kern
1.0pt}{\bf{\hat{A}}}({{\bf{R}}_{o}},t)\cdot\,\sum\limits_{i=1}^{n}{{{{\bf{P^{\prime\prime}}}}_{i}}}$
$\displaystyle+$ $\displaystyle{\kern
1.0pt}e\left({\frac{{{Z_{1}}}}{{{M_{1}}}}-\frac{{{Z_{2}}}}{{{M_{2}}}}}\right){\kern
1.0pt}{\bf{\hat{A}}}({{\bf{R}}_{o}},t)\cdot\,{{{\bf{\hat{P}}}}_{R}}$
where ${{{\bf{\hat{P}}}}_{o}}$ is the relative momentum operator for the
center of mass of the molecule (in laboratory coordinates) that is defined in
Eq. (31), ${{{{\bf{P^{\prime\prime}}}}_{i}}}$ is the momentum operator of the
$i$th electron (in center-of-mass-of-nuclei coordinates) given in Eq. (37),
and ${{{\bf{\hat{P}}}}_{R}}$ is the momentum operator that is conjugate to the
inter-nuclear vector ${\bf R}$ and is given by Eq. (29).
The total Hamiltonian for the molecule, the field, and their interaction is
given by the sum of Eq. (38), (39) and (42). The coordinates in the total
Hamiltonian are the center-of-mass position (in laboratory coordinates), ${\bf
R}_{o}$, the inter-nuclear position (nuclear relative coordinate) ${\bf R}$,
and the electron positions in the center-of-mass-of-nuclei coordinates,
${{\bf{R^{\prime\prime}}}_{i}}$. The basis states for the combined system of
molecule and field are of the form
$\Psi\left({{{\bf{R}}_{o}},{\bf{R}},{{{\bf{R^{\prime\prime}}}}_{1}},\ldots,{{{\bf{R^{\prime\prime}}}}_{n}}}\right)\,\left|{{{({n_{1}})}_{{\alpha_{1}}}}\,,{{({n_{2}})}_{{\alpha_{2}}}}\,,\cdots}\right\rangle$
(43)
where
$\Psi\left({{{\bf{R}}_{o}},{\bf{R}},{{{\bf{R^{\prime\prime}}}}_{1}},\ldots,{{{\bf{R^{\prime\prime}}}}_{n}}}\right)$
are the molecular states that are eigenstates of the sum of Eq. (38) and (39),
and
$\left|{{{({n_{1}})}_{{\alpha_{1}}}}\,,{{({n_{2}})}_{{\alpha_{2}}}}\,,\cdots}\right\rangle$
are the photon states that are eigenstates of Eq. (5), where there are $n_{i}$
photons in mode $\alpha_{i}$ and I use the short-hand notation
$\alpha=\\{\lambda,k,j,m\\}$.
The first term in the interaction Hamiltonian in Eq. (42) is an ionic term,
and is proportional to the pre-factor $n-{Z_{1}}-{Z_{2}}$. This pre-factor is
zero for neutral molecules, whose electron number is equal to the sum of the
atomic numbers of the two nuclei, $n={Z_{1}}+{Z_{2}}$. For the case of a
molecule that is not neutral (an ion) this first term couples the momentum of
the center of mass of the molecule to the radiation field. This term leads to
center of mass motion of the (charged) molecule as a whole.
The second term in the interaction Hamiltonian in Eq. (42) couples the sum of
electron momenta (in center-of-mass-of-nuclei coordinates) to the radiation
field. This term allows a transfer of angular momentum from the radiation
field to the electrons. The pre-factor for this second term is never zero.
The third term in the interaction Hamiltonian in Eq. (42) couples the
momentum,
${{\bf{P}}_{\bf{R}}}=-i\hbar\left[{\frac{\partial}{{\partial{R_{X}}}}{{\bf{e}}_{x}}+\frac{\partial}{{\partial{R_{Y}}}}{{\bf{e}}_{y}}+\frac{\partial}{{\partial{R_{Z}}}}{{\bf{e}}_{z}}}\right]$
(44)
which is conjugate to the inter-nuclear vector,
${\bf{R}}=\left({{R_{X}},{R_{Y}},{R_{Z}}}\right)$, to the electromagnetic
radiation field, ${\bf{\hat{A}}}({{\bf{R}}_{o}},t)$. Here, $({\bf e}_{x},{\bf
e}_{y},{\bf e}_{z})$ are the basis vectors for the space-fixed coordinates of
${\bf{R}}=\left({{R_{X}},{R_{Y}},{R_{Z}}}\right)$. The pre-factor for this
term is zero for homo-nuclear diatomic molecules Sindelka and Moiseyev (2006),
which have $M_{1}=M_{2}$ and $Z_{1}=Z_{2}$. Homo-nuclear diatomic molecules
have an inversion symmetry about the nuclear center of mass. Molecules which
have $Z_{1}=Z_{2}$ but $M_{1}\neq M_{2}$, (e.g., different isotopes, such as
HD), lack this inversion symmetry, and then this third term in Eq. (42) is
generally non-zero. This third term contains a coupling of the radiation field
to the rotation of the molecule about the line joining the nuclei. This can be
seen by considering the unitary transformation, $\cal{M}$, from space-fixed
coordinates with origin at the center-of-nuclear mass,
${\bf{X}}=\left({{X},{Y},{Z}}\right)$, to molecule-fixed coordinates,
${\bf{x}}=\left({{x},{y},{z}}\right)$, where the $z$-axis is pointing along
the inter-molecular vector ${\bf R}_{2}-{\bf R}_{1}$:
$\left({\begin{array}[]{*{20}{c}}X\\\ Y\\\ Z\\\
\end{array}}\right)=\cal{M}\left({\phi,\theta,\chi}\right)\cdot\left[{\begin{array}[]{*{20}{c}}x\\\
y\\\ z\\\ \end{array}}\right]$ (45)
where $\left({\phi,\theta,\chi}\right)$ are the Euler angles between
coordinates ${\bf{X}}=\left({{X},{Y},{Z}}\right)$ and
${\bf{x}}=\left({{x},{y},{z}}\right)$. I take the sequence of rotations
through angles $\\{\phi,\theta,\chi\\}$ to be along the $z$, $y$, $z$ axes, so
the rotation matrix is given by Judd (1975); Zare (1988); Brown and Carrington
(2003)
$\cal{M}\left({\phi,\theta,\chi}\right)=\left({\begin{array}[]{*{20}{c}}{\cos\phi\cos\theta\cos\chi-\sin\phi\sin\chi}&{-\sin\phi\cos\chi-\cos\phi\cos\theta\sin\chi}&{\cos\phi\sin\theta}\\\
{\sin\phi\cos\theta\cos\chi+\cos\phi\sin\chi}&{\cos\phi\cos\chi-\sin\phi\cos\theta\sin\chi}&{\sin\phi\sin\theta}\\\
{-\sin\theta\cos\chi}&{\sin\theta\sin\chi}&{\cos\theta}\\\
\end{array}}\right)$ (46)
For internuclear vector ${\bf R}=\left({{R_{X}},{R_{Y}},{R_{Z}}}\right)$, the
transformation from space-fixed coordinates to molecule-fixed coordinates,
$\left({{R_{X}},{R_{Y}},{R_{Z}}}\right)\rightarrow\left({R,\theta,\phi,\chi}\right)$,
is given by
$\left({\begin{array}[]{*{20}{c}}{{R_{X}}}\\\ {{R_{Y}}}\\\ {{R_{Z}}}\\\
\end{array}}\right)=\cal{M}\left({\phi,\theta,\chi}\right)\cdot\left[{\begin{array}[]{*{20}{c}}0\\\
0\\\ R\\\ \end{array}}\right]$ (47)
For a diatomic molecule, since the vector
${\bf{R}}=\left({{R_{X}},{R_{Y}},{R_{Z}}}\right)$ is parallel to the molecule-
fixed $z$-axis, the components $\left({{R_{X}},{R_{Y}},{R_{Z}}}\right)$ do not
depend on $\chi$, and we are free to choose its value for convenience Judd
(1975); Zare (1988); Brown and Carrington (2003). Using the transformation in
Eq. (47), the relative momemtum in Eq. (44) becomes Brown and Carrington
(2003):
${{\bf{P}}_{\bf{R}}}=\hbar\left[{-\frac{1}{R}{\bf{n}}\times{\bf{N}}-i{\bf{n}}\frac{\partial}{{\partial
R}}}\right]$ (48)
where ${\bf n}$ is the inter-nuclear unit vector, given in space-fixed
coordinates by
${\bf{n}}=\frac{{\bf R}_{2}-{\bf R}_{1}}{|{\bf R}_{2}-{\bf
R}_{1}|}=\sin\theta\,\cos\phi\,{{\bf{e}}_{x}}+\sin\theta\,\sin\phi\,{{\bf{e}}_{y}}+\cos\theta\,\,{{\bf{e}}_{z}}$
(49)
and ${\bf N}$ is the angular momentum operator in space-fixed coordinates with
origin at center-of-mass of the nuclei
$\displaystyle{\bf{N}}$ $\displaystyle=$ $\displaystyle
i\left[{\cot\theta\cos\phi\frac{\partial}{{\partial\phi}}+\sin\phi\frac{\partial}{{\partial\theta}}-{\mathop{\rm
c}\nolimits}\sec\theta\cos\phi\frac{\partial}{{\partial\chi}}}\right]{{\bf{e}}_{x}}$
(50) $\displaystyle+$ $\displaystyle
i\left[{\cot\theta\sin\phi\frac{\partial}{{\partial\phi}}-\cos\phi\frac{\partial}{{\partial\theta}}-{\mathop{\rm
c}\nolimits}\sec\theta\sin\phi\frac{\partial}{{\partial\chi}}}\right]{{\bf{e}}_{y}}$
$\displaystyle-$ $\displaystyle
i\frac{\partial}{{\partial\phi}}{{\bf{e}}_{z}}$
Using the form of ${\bf P}_{\bf R}$ in Eq. (48) in the third term in Eq. (42),
gives
${\bf{\hat{A}}}({{\bf{R}}_{o}})\cdot\,{{{\bf{\hat{P}}}}_{R},t}=-{\bf{\hat{A}}}({{\bf{R}}_{o}},t)\cdot\left[{\frac{1}{R}{\bf{n}}\times{\bf{N}}+i{\bf{n}}\frac{\partial}{{\partial
R}}}\right]$ (51)
which shows that the radiation field couples to both molecular rotations and
molecular vibrations. The first term in Eq. (51) is a coupling of the
radiation field to molecular rotations, through the molecular angular momentum
${\bf N}$. The second term in Eq. (51) is a coupling of the radiation field to
molecular vibrations through the length of the inter-nuclear distance $R$.
Babiker et al. investigated the interaction of a hydrogenic bound system
consisting of a nucleus and a bound electron, interacting with a linearly
polarized radiation field carrying orbital angular momentum Babiker et al.
(2002). They found that an exchange of orbital angular momentum in an electric
dipole transition occurs only between light and the center of mass motion of
the atom—they concluded that the internal electronic-type motion does not
participate in any exchange of orbital angular momentum in a dipole
transition. The presence of the second term in Eq. (42) suggests that this is
not the case for a diatomic molecule.
Alexandrescu et al. studied a charged diatomic molecule (an ion, with two
nuclei) interacting with a Laguerre-Gaussian beam carrying orbital angular
momentum Alexandrescu et al. (2006). They seem to disagree with the
conclusions of Babiker et al., stating that the electronic motion and the
electromagnetic field can exchange one unit of OAM within the electronic
dipole interaction.
I note that the interaction Hamiltonian in Eq. (42) is not of the form $-{\bf
d}\cdot{\bf E}$, which is a dipole coupling to the radiation field ${\bf E}$.
The interaction Hamiltonian in Eq. (42) contains all multipole couplings with
the radiation field. The interaction Hamiltonian in Eq. (42) allows
investigation of transfer of angular momentum from the radiation field to the
molecule, including transfer of angular momentum to the rotation of the
nuclei. Regarding the dipole moment, consider the dipole moment of the
molecule, expressed in terms of variables ${\bf R^{\prime\prime}}_{i}$ and
${\bf R^{\prime\prime}}_{\alpha}$ in the center-of-mass-of-nuclei coordinates:
${\bf{d}}=-{\bf{e}}\sum\limits_{i=1}^{n}{{{{\bf{R^{\prime\prime}}}}_{i}}}+e\sum\limits_{\alpha=1}^{2}{{Z_{\alpha}}{{{\bf{R^{\prime\prime}}}}_{\alpha}}}$
(52)
When the dipole moment is transformed to the set of coordinates $\\{{\bf
R}_{o},{\bf R},{\bf R^{\prime\prime}}_{1},\ldots{\bf
R^{\prime\prime}}_{n}\\}$, it becomes
$\displaystyle{\bf{d}}$ $\displaystyle=$
$\displaystyle-{\bf{e}}\left[{1+\frac{m}{M}\left({{Z_{1}}+{Z_{2}}}\right)}\right]\sum\limits_{i=1}^{n}{{{{\bf{R^{\prime\prime}}}}_{i}}}$
(53) $\displaystyle+$ $\displaystyle
e\left({{Z_{1}}+{Z_{2}}}\right){{\bf{R}}_{o}}-e\left({\frac{{{Z_{1}}}}{{{M_{1}}}}-\frac{{{Z_{2}}}}{{{M_{2}}}}}\right)\mu{\kern
1.0pt}{\bf{R}}$
In order to isolate the rotational part of the Hamiltonian, contained in ${\bf
P}_{\bf R}$, it was necessary to transform to coordinates $\\{{\bf R}_{o},{\bf
R},{\bf R^{\prime\prime}}_{1},\ldots{\bf R^{\prime\prime}}_{n}\\}$. However,
after performing his transformation, the Hamiltonian in Eq. (42) is not of the
form of a dipole coupling to the radiation field, $-{\bf d}\cdot{\bf E}$. This
is not a result of the transformation itself, rather it is due to the added
rotational degrees of freedom of a molecule as compared to an atom.
## VIII Transition Probabilities
The form of the molecule-radiation interaction in Eq. (42) shows that there is
a clear coupling of the radiation field to molecular rotations. The center of
mass coordinate, ${\bf R}_{o}$, plays a central role in this coupling. As
stated earlier, in order to determine the effect of this coupling on
observable quantities, transition matrix elements and probabilities for photon
absorption must be calculated. In order to estimate these matrix elements, we
must have knowledge of the form of the wavefunctions for the diatomic
molecule, which are schematically shown in Eq. (43). We must also assume some
form for the modes of the radiation field, for example, vector spherical
harmonics or Laguerre-Gaussian modes. Group theoretic methods based on tensor
operators can then be used to determine ratios of transition matrix elements
Judd (1975); Zare (1988); Brown and Carrington (2003). This work is in
progress. However, some statements can be made about parity selection rules
and how the transition matrix elements scale.
Exact selection rules based on conservation of parity can be written down.
Assume that under inversion of coordinates (under the parity operator), the
parity of the initial and final molecular wavefunctions are $P_{i}$ and
$P_{f}$, respectively, and the parity of the emitted photon is $P_{\gamma}$,
where parity eigenvalues takes values $\pm 1$. If the initial state of the
whole system has one photon, and the final state has no photons, then parity
conservation requires $P_{i}\,P_{\gamma}=P_{f}$, or, since each parity
eigenvalue is $\pm 1$, parity conservation requires Berestetskii et al. (1982)
$P_{i}\,P_{f}=P_{\gamma}$ (54)
Electric $Ej$ photons have parity $P^{Ej}_{\gamma}=(-1)^{j}$ and magnetic $Mj$
photons have parity $P^{Mj}_{\gamma}=(-1)^{j+1}$. For absorbing one photon of
a given type, parity conservation requires
$P_{i}\,P_{f}=(-1)^{j}\qquad{\rm electric\,\,\,{\it Ej}\,\,\,photon}$ (55)
or
$P_{i}\,P_{f}=(-1)^{j+1}\qquad{\rm magnetic\,\,\,{\it Mj}\,\,\,photon}$ (56)
An estimate of the scaling of transition probabilities can be made by
considering the molecule as a system of charges interacting with a radiation
field. Consider a molecule in an initial state with energy $E_{i}$, absorbing
a single photon of energy $\hbar\omega_{k}=\hbar ck$, and making a transition
to a final state with energy $E_{f}$. The transition matrix element contains
the quantity $kr$, which enters in the vector potential in Eqs. (8) and (9).
The quantity $r$ has a characteristic scale of a diatomic molecule, on the
order of $a\sim 10^{-10}$ m. For transitions between rotational states, I take
the frequency $f\sim 300$ GHz. Therefore, there is a small dimensionless
quantity, $kr\sim ka\sim 2\pi a/\lambda\sim 6\times 10^{-7}$ in the transition
matrix elements. The vector potentials in Eqs. (8) and (9) depend on the
function $f_{l}(kr)$, which in turn depend on the Bessel function
$J_{l+\frac{1}{2}}(kr)$. Using the small argument, $kr\ll 1$, expansion of the
Bessel function, in Eq. (10), the radial function, $f_{l}(kr)$, scales as (up
to a constant):
${f_{l}}\left({k\,r}\right)\sim{\left({k{\kern
1.0pt}r}\right)^{l+\frac{1}{2}}}\frac{1}{{1\cdot 3\cdot 5\cdots(2l+1)}}$ (57)
The vector potential for $Ej$ photons scales as
${\bf{A}}_{kjm}^{E}\left({\bf{r}}\right)\sim{f_{j-1}}\left({k\,r}\right)$, and
the vector potential for $Mj$ photons scales as
${\bf{A}}_{kjm}^{M}\left({\bf{r}}\right)\sim{f_{j}}\left({k\,r}\right)$. The
probability of absorbing a photon scales as the square of the vector potential
(square of the matrix element). Therefore, the ratio of the probability for
absorbing an $Mj$ photon, $P(Mj)$, divided by the probability of absorbing an
$Ej$ photon, $P(Ej)$, scales as
$\frac{{P\left({Mj}\right)}}{{P\left({Ej}\right)}}\sim\frac{{{{\left({k{\kern
1.0pt}a}\right)}^{2}}}}{{{{\left({2j+1}\right)}^{2}}}}\sim\frac{{{{4\times
10}^{-13}}}}{{{{\left({2j+1}\right)}^{2}}}}$ (58)
For a given angular momentum $j$, the probability of absorption of a magnetic
photon is much less than absorption of an electric photon.
The ratio of the probability for absorbing an $E(j+1)$ photon, $P(E(j+1)$,
divided by the probability of absorbing an $Ej$ photon, $P(Ej)$, also scales
as
$\frac{{P\left({E(j+1)}\right)}}{{P\left({Ej}\right)}}\sim\frac{{{{\left({k{\kern
1.0pt}a}\right)}^{2}}}}{{{{\left({2j+1}\right)}^{2}}}}\sim\frac{{{{4\times
10}^{-13}}}}{{{{\left({2j+1}\right)}^{2}}}}$ (59)
Form Eq. (59), we see that the probability of absorbing higher angular
momentum photons rapidly decreases with increasing $j$. As the simplest
example, the ratio of probability of absorbing an $E2$ photon to the
probability of absorbing an $E1$ photon is:
$\frac{{P\left({E2}\right)}}{{P\left({E1}\right)}}\sim{4\times 10^{-14}}$ (60)
The scaling of the absorption probabilities for molecules, given in Eqs.
(58)–(60), makes it a challenging proposition to observe absorption of photons
having angular momentum $j>1$.
## IX Summary
I have investigated the interaction between a diatomic molecule and photons
with well-defined angular momentum. I have exploited the transformation from
laboratory coordinates to coordinates with origin at the center of mass of the
nuclei to show the nuclear rotational degrees of freedom explicitly (through
the vector ${\bf R}$). For the photon modes, I have assumed spherical vector
harmonics in a large sphere of radius $R$. These modes are exact solutions of
Maxwell equations, and consequently, any other modes, such as the Laguerre-
Gaussian modes, can be expanded in terms of them. However, in general, spin
and orbital angular momentum are not separable in vector spherical harmonic
modes. I used the standard minimal coupling between elementary charges and
radiation field to couple the radiation field to the molecule, neglecting
quadratic terms in the vector potential and neglecting relativistic
interaction with spins. For molecules, which absorb at millimeter wavelengths,
there is a small parameter, $k\,a$ where $k$ is a wave vector and $a$ is the
size of the molecule. This small parameter was used to determine the scaling
of absorption probabilities of $Ej$ and $Mj$ type photons. For transitions
between rotational levels of the molecule having $\Delta J>1$, the probability
of absorption of photons with angular momentum $j>1$ is small, because the
factor $ka\ll 1$, where wave vector $k=2\pi/\lambda$ corresponds to photons in
the millimeter wavelength region, see Section VIII. Note that the small
parameter, $k\,a$, also exists for atoms Huang (1994); Jáuregui (2004);
Grinter (2008), however, it is larger for atoms because $k$ corresponds to
visible wavelengths, whereas for rotational transitions in molecules $k$
corresponds to millimeter wavelengths.
For electric photons having arbitrary angular momentum $j\hbar$, the
probability of rotational transition by absorbing an electric $E(j+1)$ photon
divided by the probability of absorbing an electric $Ej$ photon, scales as
$(ka)^{2}/(2j+1)^{2}$. The probability of absorbing a magnetic $Mj$ photon is
much smaller than the probability of absorbing an electric $Ej$ photon. The
probability of absorbing a magnetic $Mj$ photon, divided by the probability of
absorbing an electric $Ej$ photon, also scales as $(ka)^{2}/(2j+1)^{2}$.
###### Acknowledgements.
The author acknowledges interesting discussions with Henry Everitt and Matt
Goodman.
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|
arxiv-papers
| 2012-12-27T19:11:22 |
2024-09-04T02:49:39.723070
|
{
"license": "Public Domain",
"authors": "Thomas B. Bahder",
"submitter": "Thomas B. Bahder",
"url": "https://arxiv.org/abs/1212.6417"
}
|
1212.6434
|
December 27, 2012
Time-dependent Dalitz-plot formalism for
$B_{q}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}h^{-}$
Liming Zhang and Sheldon Stone
Physics Department Syracuse University, Syracuse, NY, USA 13244-1130
A formalism for measuring time-dependent $C\\!P$ violation in
$B^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}h^{-}$
decays with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ is developed for the general case where there
can be many $h^{+}h^{-}$ final states of different angular momentum present.
Here $h$ refers to any spinless meson. The decay amplitude is derived using
similar considerations as those in a Dalitz like analysis of three-body
spinless mesons taking into account the fact that the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is spin-1, and the various
interferences allowed between different final states. Implementation of this
procedure can, in principle, lead to the use of a larger number of final
states for $C\\!P$ violation studies.
## 1 Introduction
Measurement of $C\\!P$ violation in the $B^{0}$ and $B_{s}^{0}$ systems is
important for testing the Standard Model, as new particles can appear in
mixing diagrams. Previous measurements have been made in many modes [1]. To
measure the phase in $B_{s}^{0}$ decays the final states
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
for $K^{+}K^{-}$ masses close to that of the $\phi$ meson has been used [2, 3,
4, *Abazov:2011ry, *:2012fu], as well as
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ [7, *LHCb:2011ab]. In the latter case the final state is
$C\\!P$ odd [9] over most of the $\pi^{+}\pi^{-}$ mass range, while in the
case of $K^{+}K^{-}$ the final state even in the mass region near the $\phi$
meson has both $C\\!P$ odd and even components, that can be resolved using
time-dependent angular analysis [10, *Dighe:1998vk]. In this paper we present
a formalism that allows the entire $K^{+}K^{-}$ mass region to be used in
$C\\!P$ violation measurements regardless of the final state angular momentum.
This formalism can also be applied to $B^{0}$ decays, e.g.
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$.
The basic concept here is to couple a three-body Dalitz like analysis [12] to
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}h^{-}$ final state,
where the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ and concurrently measure the time-dependent
$C\\!P$ violation by splitting the final state into odd and even $C\\!P$
components.
## 2 Time-dependent decay rates
The time evolution of the $B^{0}_{q}$-$\overline{B}{}_{q}^{0}$ system is
described by the Schrödinger equation
$i\frac{\partial}{\partial t}\left(\begin{array}[]{c}{|B^{0}_{q}}(t)\rangle\\\
{|\overline{B}{}_{q}^{0}}(t)\rangle\end{array}\right)=\left(\mbox{\boldmath$\rm
M$}-\frac{i}{2}\mbox{\boldmath$\rm\Gamma$}\right)\left(\begin{array}[]{c}{|B^{0}_{q}}(t)\rangle\\\
{|\overline{B}{}_{q}^{0}}(t)\rangle\end{array}\right),$ (1)
where the $\rm M$ and $\rm\Gamma$ matrices are Hermitian, and $CPT$ invariance
implies that $M_{11}=M_{22}$ and $\Gamma_{11}=\Gamma_{22}$. The off-diagonal
elements, $M_{12}$ and $\Gamma_{12}$, of these matrices describe the off-shell
(dispersive) and on-shell (absorptive) contributions to
$B^{0}_{q}$-$\overline{B}{}_{q}^{0}$ mixing, respectively.
The mass eigenstates $|B_{H}\rangle$ and $|B_{L}\rangle$ of the effective
Hamiltonian matrix are given by
$\displaystyle|B_{L}\rangle$ $\displaystyle=$ $\displaystyle
p|B_{q}^{0}\rangle+q|\overline{B}{}_{q}^{0}\rangle,$
$\displaystyle|B_{H}\rangle$ $\displaystyle=$ $\displaystyle
p|B_{q}^{0}\rangle-q|\overline{B}{}_{q}^{0}\rangle,$ (2)
with $|p|^{2}+|q|^{2}=1$. The decay amplitudes for $B_{q}^{0}$ and
$\overline{B}{}_{q}^{0}$ into a self-charge-conjugated final state $f$, where
for this paper $f={J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}h^{-}$, are
defined as
$A_{f}\equiv\langle
f|S|B_{q}^{0}\rangle,\quad\quad\quad\overline{A}_{f}\equiv\langle
f|S|\overline{B}{}_{q}^{0}\rangle.$ (3)
With the additional definitions
$A\equiv A_{f},\quad{\rm
and}\quad\overline{A}\equiv\frac{q}{p}\overline{A}_{f},$ (4)
the time dependent decay rates can be written as [13, *Bigi:2000yz]
$\displaystyle\Gamma(t)=\quad\quad{\cal N}e^{-\Gamma
t}\left\\{\frac{|A|^{2}+|\overline{A}|^{2}}{2}\cosh\frac{\Delta\Gamma
t}{2}+\frac{|A|^{2}-|\overline{A}|^{2}}{2}\cos(\Delta
mt)\right.\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle-\left.\mathcal{R}e(A^{*}\overline{A})\sinh\frac{\Delta\Gamma
t}{2}-\mathcal{I}m(A^{*}\overline{A})\sin(\Delta mt)\right\\},$ (5)
$\displaystyle\overline{\Gamma}(t)=\left|\frac{p}{q}\right|^{2}{\cal
N}e^{-\Gamma
t}\left\\{\frac{|A|^{2}+|\overline{A}|^{2}}{2}\cosh\frac{\Delta\Gamma
t}{2}-\frac{|A|^{2}-|\overline{A}|^{2}}{2}\cos(\Delta
mt)\right.\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle-\left.\mathcal{R}e(A^{*}\overline{A})\sinh\frac{\Delta\Gamma
t}{2}+\mathcal{I}m(A^{*}\overline{A})\sin(\Delta mt)\right\\},$ (6)
where $\cal N$ is a normalization constant, $\Delta m=m_{H}-m_{L}$,
$\Delta\Gamma=\Gamma_{L}-\Gamma_{H}$, and $\Gamma=(\Gamma_{L}+\Gamma_{H})/2$.
## 3 Angular dependent formulas
### 3.1 Definition of helicity angles
We express the angular dependence of the decay in terms of “helicity” angles
defined as (i) $\theta_{\ell}$, the angle between the $\mu^{+}$ direction in
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame with respect to
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ direction in the
$B_{q}^{0}$ rest frame; (ii) $\theta_{h}$ the angle between the $h^{+}$
direction in the $h^{+}h^{-}$ rest frame with respect to the $h^{+}h^{-}$
direction in the $B_{q}^{0}$ rest frame, and (iii) $\chi$ the angle between
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $h^{+}h^{-}$ decay
planes in the $B_{q}^{0}$ rest frame. These angles are shown pictorially in
Fig. 1. (These definitions are the same for $B_{q}^{0}$ and
$\overline{B}{}_{q}^{0}$, namely, using $\mu^{+}$ and $h^{+}$ to define the
angles for both $B_{q}^{0}$ and $\overline{B}{}_{q}^{0}$ decays.)
Figure 1: Definition of helicity angles. For details see text.
### 3.2 Time-independent part of the rate for $B^{0}_{q}$ decays
For the decays of $B_{q}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}h^{+}h^{-}$ with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ the decay rate is found by summing over the
unobserved lepton polarizations. The time-independent part of the rate is111In
$d_{-\lambda,0}^{J}(\theta_{h})$, $-\lambda$ is used instead of $\lambda$ in
order to be consistent with the convention used in [3].
$|A_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)|^{2}=\sum_{\alpha=\pm
1}\left|\sum^{|\lambda|\leq
J}_{\lambda,J}\sqrt{\frac{2J+1}{4\pi}}H_{\lambda}^{J}(m_{hh})e^{i\lambda\chi}d_{\lambda,\alpha}^{1}(\theta_{\ell})d_{-\lambda,0}^{J}(\theta_{h})\right|^{2},$
(7)
where $\lambda=0,\pm 1$ is the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
helicity, $\alpha=\pm 1$ is the helicity difference between the two muons, $J$
is the spin of the $h^{+}h^{-}$ intermediate state, and
$H_{\lambda}^{J}(m_{hh})$ is a helicity amplitude depending on $m_{hh}$ that
can be expressed using a formalism similar to that in a Dalitz-plot analyses.
We define the term which contains the sum over spin-$J$ as
${\cal
H}_{\lambda}(m_{hh},\theta_{h})=\sum_{J}\sqrt{\frac{2J+1}{4\pi}}H_{\lambda}^{J}(m_{hh})d_{-\lambda,0}^{J}(\theta_{h}).$
(8)
Then Eq. (7) becomes
$\displaystyle|A_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)|^{2}=\sum_{\alpha=\pm
1}\left|\sum_{\lambda}e^{i\lambda\chi}d_{\lambda,\alpha}^{1}(\theta_{\ell}){\cal
H}_{\lambda}(m_{hh},\theta_{h})\right|^{2}$ $\displaystyle\hskip
14.22636pt=\sum_{\alpha=\pm
1}\left[\left(\sum_{\lambda^{\prime}}e^{i\lambda^{\prime}\chi}d_{\lambda^{\prime},\alpha}^{1}(\theta_{\ell}){\cal
H}_{\lambda^{\prime}}(m_{hh},\theta_{h})\right)^{*}\left(\sum_{\lambda}e^{i\lambda\chi}d_{\lambda,\alpha}^{1}(\theta_{\ell}){\cal
H}_{\lambda}(m_{hh},\theta_{h})\right)\right]$ $\displaystyle\hskip
14.22636pt=\sum_{\lambda^{\prime},\lambda}\left(\sum_{\alpha=\pm
1}d_{\lambda^{\prime},\alpha}^{1}(\theta_{\ell})d_{\lambda,\alpha}^{1}(\theta_{\ell})\right){\cal
H}_{\lambda^{\prime}}^{*}(m_{hh},\theta_{h}){\cal
H}_{\lambda}(m_{hh},\theta_{h})\;e^{i(\lambda-\lambda^{\prime})\chi}.$ (9)
Defining
$\Theta_{\lambda^{\prime}\lambda}(\theta_{\ell})\equiv\displaystyle\sum_{\alpha=\pm
1}d_{\lambda^{\prime},\alpha}^{1}(\theta_{\ell})d_{\lambda,\alpha}^{1}(\theta_{\ell}),$
(10)
results in
$|A_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)|^{2}=\sum_{\lambda^{\prime},\lambda}{\cal
H}_{\lambda}(m_{hh},\theta_{h}){\cal
H}_{\lambda^{\prime}}^{*}(m_{hh},\theta_{h})\;e^{i(\lambda-\lambda^{\prime})\chi}\Theta_{\lambda^{\prime}\lambda}(\theta_{\ell}).$
(11)
Table 1 lists the functions $\Theta_{\lambda^{\prime}\lambda}(\theta_{\ell})$.
They are invariant under the interchange of $\lambda$ and $\lambda^{\prime}$,
i.e.
$\Theta_{\lambda^{\prime}\lambda}(\theta_{\ell})=\Theta_{\lambda\lambda^{\prime}}(\theta_{\ell})$,
and transform with respect to a change of the sign of both $\lambda$ and
$\lambda^{\prime}$ as
$\Theta_{\lambda\lambda^{\prime}}(\theta_{\ell})=(-1)^{\lambda-\lambda^{\prime}}\Theta_{-\lambda^{\prime}-\lambda}(\theta_{\ell})$.
Inserting the explicit functional forms in Eq. (11) allows us to express the
amplitude as
$\displaystyle|A_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)|^{2}=$
$\displaystyle|{\cal
H}_{0}(m_{hh},\theta_{h})|^{2}\sin^{2}\theta_{\ell}+\frac{1}{2}\left(|{\cal
H}_{+}(m_{hh},\theta_{h})|^{2}+|{\cal H}_{-}(m_{hh},\theta_{h})|^{2}\right)$
$\displaystyle\times(1+\cos^{2}\theta_{\ell})+\mathcal{R}e\left[{\cal
H}_{+}(m_{hh},\theta_{h}){\cal
H}_{-}^{*}(m_{hh},\theta_{h})e^{2i\chi}\right]\sin^{2}\theta_{\ell}$
$\displaystyle+\sqrt{2}\mathcal{R}e\left[\left({\cal
H}_{0}(m_{hh},\theta_{h}){\cal H}_{+}^{*}(m_{hh},\theta_{h})-{\cal
H}^{*}_{0}(m_{hh},\theta_{h}){\cal
H}_{-}(m_{hh},\theta_{h})\right)e^{-i\chi}\right]$
$\displaystyle\times\sin\theta_{\ell}\cos\theta_{\ell},$ (12)
where we denote ${\cal H}_{\lambda}$ by 0, +, and $-$, rather than 0, $+1$ and
$-1$.
Table 1: Functional forms of $\Theta_{\lambda^{\prime}\lambda}(\theta)$ defined in Eq. (10) for different values of $\lambda$ and $\lambda^{\prime}$. $\lambda$ | $\lambda^{\prime}$ | $\Theta_{\lambda^{\prime}\lambda}(\theta)$
---|---|---
0 | 0 | $\sin^{2}\theta$
0 | 1 | $\frac{1}{\sqrt{2}}\sin\theta\cos\theta$
0 | $-$1 | $-\frac{1}{\sqrt{2}}\sin\theta\cos\theta$
1 | 0 | $\frac{1}{\sqrt{2}}\sin\theta\cos\theta$
1 | 1 | $\frac{1}{2}(1+\cos^{2}\theta)$
1 | $-$1 | $\frac{1}{2}\sin^{2}\theta$
$-$1 | 0 | $-\frac{1}{\sqrt{2}}\sin\theta\cos\theta$
$-$1 | 1 | $\frac{1}{2}\sin^{2}\theta$
$-$1 | $-$1 | $\frac{1}{2}(1+\cos^{2}\theta)$
### 3.3 Time-independent part of the rate for $\overline{B}^{0}_{q}$ decays
For $\overline{B}^{0}_{q}$ decays, the expression for
$|\overline{A}_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)|^{2}$, results from
replacing ${\cal H}_{\lambda}(m_{hh},\theta_{h})$ in Eq. (3.2) by
${\cal\overline{H}}_{\lambda}(m_{hh},\theta_{h})$, which contains the helicity
amplitudes for $\overline{B}{}_{q}^{0}$ decays. ${\cal
H}_{\lambda}(m_{hh},\theta_{h})$ and
${\cal\overline{H}}_{\lambda}(m_{hh},\theta_{h})$ are related by transversity
$C\\!P$ eigenstates [15], that are discussed in Section 4. Using these we find
$\displaystyle|\overline{A}_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)|^{2}=$
$\displaystyle|{\cal\overline{H}}_{0}(m_{hh},\theta_{h})|^{2}\sin^{2}\theta_{\ell}+\frac{1}{2}\left(|{\cal\overline{H}}_{+}(m_{hh},\theta_{h})|^{2}+|{\cal\overline{H}}_{-}(m_{hh},\theta_{h})|^{2}\right)$
$\displaystyle\times(1+\cos^{2}\theta_{\ell})+\mathcal{R}e\left[{\cal\overline{H}}_{+}(m_{hh},\theta_{h}){\cal\overline{H}}_{-}^{*}(m_{hh},\theta_{h})e^{2i\chi}\right]\sin^{2}\theta_{\ell}$
$\displaystyle+\sqrt{2}\mathcal{R}e\left[\left({\cal\overline{H}}_{0}(m_{hh},\theta_{h}){\cal\overline{H}}_{+}^{*}(m_{hh},\theta_{h})-{\cal\overline{H}}^{*}_{0}(m_{hh},\theta_{h}){\cal\overline{H}}_{-}(m_{hh},\theta_{h})\right)e^{-i\chi}\right]$
$\displaystyle\times\sin\theta_{\ell}\cos\theta_{\ell}~{}.$ (13)
### 3.4 The interference term
Next we calculate the complex term
$A_{f}^{*}(m_{hh},\theta_{h},\theta_{\ell},\chi)\overline{A}_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)$.
We have
$\displaystyle
A_{f}^{*}(m_{hh},\theta_{h},\theta_{\ell},\chi)\overline{A}_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)=$
$\displaystyle\hskip 28.45274pt\sum_{\alpha=\pm
1}\left[\left(\sum_{\lambda^{\prime}}e^{i\lambda^{\prime}\chi}d_{\lambda^{\prime},\alpha}^{1}(\theta_{\ell}){\cal
H}_{\lambda^{\prime}}(m_{hh},\theta_{h})\right)^{*}\left(\sum_{\lambda}e^{i\lambda\chi}d_{\lambda,\alpha}^{1}(\theta_{\ell}){\cal\overline{H}}_{\lambda}(m_{hh},\theta_{h})\right)\right]$
$\displaystyle\hskip
17.07164pt=\sum_{\lambda^{\prime},\lambda}{\cal\overline{H}}_{\lambda}(m_{hh},\theta_{h}){\cal
H}_{\lambda^{\prime}}^{*}(m_{hh},\theta_{h})\;e^{i(\lambda-\lambda^{\prime})\chi}\Theta_{\lambda^{\prime}\lambda}(\theta_{\ell}).$
(14)
Replacing the explicit terms leads to
$\displaystyle
A_{f}^{*}(m_{hh},\theta_{h},\theta_{\ell},\chi)\overline{A}_{f}(m_{hh},\theta_{h},\theta_{\ell},\chi)={\cal\overline{H}}_{0}(m_{hh},\theta_{h}){\cal
H}_{0}^{*}(m_{hh},\theta_{h})\sin^{2}\theta_{\ell}$
$\displaystyle+\frac{1}{2}\left({\cal\overline{H}}_{+}(m_{hh},\theta_{h}){\cal
H}_{+}^{*}(m_{hh},\theta_{h})+{\cal\overline{H}}_{-}(m_{hh},\theta_{h}){\cal
H}_{-}^{*}(m_{hh},\theta_{h})\right)(1+\cos^{2}\theta_{\ell})$
$\displaystyle+\frac{1}{2}\left({\cal\overline{H}}_{+}(m_{hh},\theta_{h}){\cal
H}_{-}^{*}(m_{hh},\theta_{h})e^{2i\chi}+{\cal\overline{H}}_{-}(m_{hh},\theta_{h}){\cal
H}_{+}^{*}(m_{hh},\theta_{h})e^{-2i\chi}\right)\sin^{2}\theta_{\ell}$
$\displaystyle+\frac{1}{\sqrt{2}}\left({\cal\overline{H}}_{0}(m_{hh},\theta_{h}){\cal
H}_{+}^{*}(m_{hh},\theta_{h})e^{-i\chi}-{\cal\overline{H}}_{0}(m_{hh},\theta_{h}){\cal
H}_{-}^{*}(m_{hh},\theta_{h})e^{i\chi}\right.$
$\displaystyle\left.+{\cal\overline{H}}_{+}(m_{hh},\theta_{h}){\cal
H}_{0}^{*}(m_{hh},\theta_{h})e^{i\chi}-{\cal\overline{H}}_{-}(m_{hh},\theta_{h}){\cal
H}_{0}^{*}(m_{hh},\theta_{h})e^{-i\chi}\right)\sin\theta_{\ell}\cos\theta_{\ell}.$
(15)
## 4 Time-dependent Dalitz-plot formalism
Here we discuss the general formalism which includes S, P, D or higher waves
of the $h^{+}h^{-}$ intermediate states.
Apart from the proper decay-time $t$, the decay of
$B_{q}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}h^{-}$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ can be
described by four variables, we choose to use $m_{hh}$ and three helicity
angles ($\theta_{\ell},\theta_{h},\chi$), where ($m_{hh},\cos\theta_{h}$)
space is used instead of the usual variables in a Dalitz-plot analysis:
$m^{2}_{hh},m^{2}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}}$; the
advantage is the former has an rectangle phase space which is easier for
calculating the normalization.
Assuming $|p/q|=1$, the differential decay rates in Eqs. (2) and (6) can be
written in terms of the five variables
$t,m_{hh},\theta_{\ell},\theta_{h},\chi$ as
$\displaystyle\frac{{\rm d}^{5}\Gamma}{{\rm d}t{\rm d}m_{hh}{\rm
d}\cos\theta_{\ell}{\rm d}\cos\theta_{h}{\rm d}\chi}\propto e^{-\Gamma
t}\left\\{\frac{|A_{f}|^{2}+|\overline{A}_{f}|^{2}}{2}\cosh\frac{\Delta\Gamma
t}{2}+\frac{|A_{f}|^{2}-|\overline{A}_{f}|^{2}}{2}\cos(\Delta mt)\right.$
$\displaystyle-\left.\mathcal{R}e\left(\frac{q}{p}A_{f}^{*}\overline{A}_{f}\right)\sinh\frac{\Delta\Gamma
t}{2}-\mathcal{I}m\left(\frac{q}{p}A_{f}^{*}\overline{A}_{f}\right)\sin(\Delta
mt)\right\\},$ (16) $\displaystyle\frac{{\rm d}^{5}\overline{\Gamma}}{{\rm
d}t{\rm d}m_{hh}{\rm d}\cos\theta_{\ell}{\rm d}\cos\theta_{h}{\rm
d}\chi}\propto e^{-\Gamma
t}\left\\{\frac{|A_{f}|^{2}+|\overline{A}_{f}|^{2}}{2}\cosh\frac{\Delta\Gamma
t}{2}-\frac{|A_{f}|^{2}-|\overline{A}_{f}|^{2}}{2}\cos(\Delta mt)\right.$
$\displaystyle-\left.\mathcal{R}e\left(\frac{q}{p}A_{f}^{*}\overline{A}_{f}\right)\sinh\frac{\Delta\Gamma
t}{2}+\mathcal{I}m\left(\frac{q}{p}A_{f}^{*}\overline{A}_{f}\right)\sin(\Delta
mt)\right\\}.$ (17)
The functions $|A_{f}|^{2}$, $|\overline{A}_{f}|^{2}$ and
$A_{f}^{*}\overline{A}_{f}$ are defined in Eqs. (3.2), (3.3) and (3.4)
respectively. We now substitute in Eq. (8) explicit variables for
$H_{\lambda}^{J}(m_{hh})$ in terms of our chosen Dalitz-plot variables
$m_{hh}$ and $\theta_{h}$, resulting in
${\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{\lambda}(m_{hh},\theta_{h})=\sum_{R}\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\bf
h}_{\lambda}^{R}\sqrt{2J_{R}+1}\sqrt{P_{R}P_{B}}F_{B}^{(L_{B})}F_{R}^{(L_{R})}A_{R}(m_{hh})\left(\frac{P_{B}}{m_{B}}\right)^{L_{B}}\left(\frac{P_{R}}{m_{hh}}\right)^{L_{R}}d_{-\lambda,0}^{J_{R}}(\theta_{h}),$
(18)
where the function $A_{R}(m_{hh})$ describes the mass squared shape of the
resonance $R$, that in most cases is a Breit-Wigner function, $P_{B}$ is the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum in the
$\overline{B}^{0}_{q}$ rest frame, $P_{R}$ is the momentum of either of the
two hadrons in the dihadron rest frame, $m_{B}$ is the $\overline{B}^{0}_{q}$
mass, $L_{B}$ is the orbital angular momentum between the $J/\psi$ and
$h^{+}h^{-}$ system, and $L_{R}$ the orbital angular momentum in the
$h^{+}h^{-}$ decay, and thus is the same as the spin of the $h^{+}h^{-}$
resonance. $F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$ are the Blatt-Weisskopf
barrier factors for $\overline{B}^{0}_{q}$ and $R$ resonance respectively [9].
The factor $\sqrt{P_{R}P_{B}}$ results from converting the phase space of the
natural Dalitz-plot variables $m^{2}_{hh}$ and
$m^{2}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}}$ to that of
$m_{hh}$ and $\cos\theta_{h}$.
${\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{\lambda}$
is summed over all $h^{+}h^{-}$ intermediate states ($R$) with different
spins, denoted as $J_{R}$. The function defined in Eq. (18) is based on
previous Dalitz plot analyses [16, 9], but here all allowed values of $L_{B}$
and $L_{R}$ are included.
In order to use $C\\!P$ relations, it is convenient to replace the helicity
complex coefficients
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{\lambda}^{R}$
by the transversity complex coefficients
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{i}^{R}$
using their relations
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{0}^{R}$
$\displaystyle=$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{0}^{R},$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\parallel}^{R}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{+}^{R}+\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{-}^{R}),$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\perp}^{R}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{+}^{R}-\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{-}^{R}).$
(19)
Here
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{0}^{R}$
corresponds to longitudinal polarization of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, and the other two
coefficients correspond to polarizations of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson and $h^{+}h^{-}$ system
transverse to the decay axis:
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\parallel}^{R}$
for parallel polarization of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $h^{+}h^{-}$ and
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\perp}^{R}$
for their perpendicular polarization.
In the SM, if we assume that only one diagram contributes to the decay and
there is no direct $C\\!P$ violation, then the CKM weak phase only appears as
$\frac{q}{p}=e^{-i\phi_{s}}$ for the $B_{s}^{0}$ decays and
$\frac{q}{p}=e^{-2i\beta}$ for the $B^{0}$ decays. The
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{i}$
amplitudes only contain strong phases, so
$\bar{\textbf{a}}^{R}_{i}=\eta^{R}_{i}\textbf{a}^{R}_{i}$, where
$\eta^{R}_{i}$ is $C\\!P$ eigenvalue of the $i$th transversity component for
the intermediate state $R$. (Here $i=0,~{}\parallel,~{}\perp$.) Note that for
the $h^{+}h^{-}$ system both $C$ and $P$ are given by $(-1)^{L_{R}}$, so the
$C\\!P$ of the $h^{+}h^{-}$ system is always even. The total CP of the final
state is $(-1)^{L_{B}}$, since the $C\\!P$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is also even. The final state
$C\\!P$ parities for S, P, and D-waves are shown in Table 2.
Table 2: $C\\!P$ parity for different spin resonances. Note that spin-0 only has one transversity component-$0$. Spin | $\eta_{0}$ | $\eta_{\parallel}$ | $\eta_{\perp}$
---|---|---|---
0 | $-1$ | – | –
1 | 1 | 1 | $-1$
2 | $-1$ | $-1$ | 1
Direct $C\\!P$ violation can also be considered, i.e.
$\bar{\textbf{a}}^{R}_{i}\neq\eta^{R}_{i}\textbf{a}^{R}_{i}$. The complex
coefficients can be parameterized as
${\bf
a}^{R}_{i}=c^{R}_{i}(1+b^{R}_{i})e^{i(\delta^{R}_{i}+\phi^{R}_{i})},\quad\bar{\textbf{a}}^{R}_{i}=\eta^{R}_{i}c^{R}_{i}(1-b^{R}_{i})e^{i(\delta^{R}_{i}-\phi^{R}_{i})},$
(20)
where $c^{R}_{i}$, $b^{R}_{i}$, $\delta^{R}_{i}$ and $\phi^{R}_{i}$ are real
numbers that can be determined in the experiment. Note that $b^{R}_{i}$ and
$\phi^{R}_{i}$ are $C\\!P$ violating, while $c^{R}_{i}$ and $\delta^{R}_{i}$
are $C\\!P$ conserving. The direct $C\\!P$ asymmetry for a particular
intermediate state $R$ with the transversity $i$ component is
$A_{C\\!P}(R)_{i}\equiv\frac{|\bar{\textbf{a}}^{R}_{i}|^{2}-|{\bf
a}^{R}_{i}|^{2}}{|\bar{\textbf{a}}^{R}_{i}|^{2}+|{\bf
a}^{R}_{i}|^{2}}=\frac{-2b_{i}^{R}}{1+(b^{R}_{i})^{2}}.$ (21)
In the case that direct $C\\!P$ violation is present, the experiment measures
an “effective” phase that is the sum of the $C\\!P$ violation due to the
interference between mixing the decay and direct $C\\!P$ violation, given by
$\phi_{s}^{\rm eff}(R)_{i}=\phi_{s}+2\phi_{i}^{R},{~{}\rm or~{}}2\beta^{\rm
eff}(R)_{i}=2\beta+2\phi_{i}^{R}.$ (22)
To implement this procedure data need to be fit with the probability density
functions (PDFs) given in Eqs. (16) and (17). The normalization can be
computed by first integrating over $t$, $\theta_{\ell}$ and $\chi$
analytically, then by using numerical integration for the remaining variables;
the terms containing variable $\chi$ in Eqs. (3.2), (3.3) and (3.4) are zero
when integrating over $\chi\in[-\pi,\pi]$. The data can be either flavour
tagged or not [17, *Abazov:2006qp]. In the latter case the two PDFs are
averaged.
Without considering $m_{hh}$ dependence, time-dependent angular analysis for
$\phi_{s}$ determination in
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ decay
[2, 3, 4, *Abazov:2011ry, *:2012fu] cannot distinguish between two ambiguous
solutions, one that is ($\phi_{s}$, $\Delta\Gamma)$ and the other being
$(\pi-\phi_{s}$,$-\Delta\Gamma$), because the time-dependent differential
decay rates are invariant under this transformation together with a similar
transformation for the strong phases. This ambiguity has been resolved by the
LHCb collaboration [19] using the P-wave $\phi$ interference with the
$K^{+}K^{-}$ S-wave [20] as a function of dikaon invariant mass as suggested
in [21]. Our Dalitz-plot formalism automatically takes the strong phases as a
function of $m_{hh}$ into account in the complex function $A_{R}(m_{hh})$ in
Eq. (18), and thus provides only one solution for ($\phi_{s}$, $\Gamma_{s}$)
without any ambiguity.
## 5 Conclusions
We have presented a method that can be used to extract the $C\\!P$ violating
phase for neutral $B$ meson decays into a spin-1 resonance that decays to a
dilepton pair and a $\pi^{+}\pi^{-}$ or $K^{+}K^{-}$ pair, using the full set
of mass and angular variables. Thus $C\\!P$ violation can be measured using a
much larger set of final states. For example, the $K^{+}K^{-}$ mass range in
$\overline{B}{}_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ can be used including higher mass states such as the
$f_{2}^{\prime}(1525)$ [22, *Abazov:2012dz].
## Acknowledgments
We thank Peter Clarke, Greig Cowan, Jeroen van Leerdam, Stephanie Hansmann-
Menzemer and Yuehong Xie for useful discussions. We are grateful for the
support we have received from the U. S. National Science Foundation.
## Appendix: Application to S- and P-waves in
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
Time-dependent angular analysis [3, 4, *Abazov:2011ry, *:2012fu] has been
applied to $B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ considering both a P-wave resonance, $\phi\rightarrow K^{+}K^{-}$,
and S-wave contamination [20]. Here we show that our formulation reduces to
previously used expressions [21, 24] by considering only the $\phi$ mass
region in
$\overline{B}{}_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ decays. With this simplification, it is necessary to
consider only S- and P-waves in the $K^{+}K^{-}$ system. Eq. (8) can be
rewritten as
$\displaystyle{\cal H}_{0}$ $\displaystyle=$
$\displaystyle\frac{H_{S}}{\sqrt{3}}d_{0,0}^{0}(\theta_{h})+H_{0}d_{0,0}^{1}(\theta_{h})=\frac{H_{S}}{\sqrt{3}}+H_{0}\cos\theta_{h},$
$\displaystyle{\cal H}_{+}$ $\displaystyle=$ $\displaystyle
H_{+}d_{-1,0}^{1}(\theta_{h})=H_{+}\frac{\sin\theta_{h}}{\sqrt{2}},$ (23)
$\displaystyle{\cal H}_{-}$ $\displaystyle=$ $\displaystyle
H_{-}d_{1,0}^{1}(\theta_{h})=-H_{-}\frac{\sin\theta_{h}}{\sqrt{2}},$
where $H_{S}$ is the helicity amplitude for S-wave, and $H_{0,\pm}$ are the
helicity amplitudes for P-wave ($\lambda=0,\pm 1$). Then Eq. (3.2) can be
expressed as
$|A_{f}|^{2}=\displaystyle\sum_{k=1}^{10}p_{k}G_{k}(\bf\Omega_{\rm hel})$ (24)
in terms of the amplitudes $H$, where ${\bf\Omega}_{\rm hel}$ is short hand
for the three angular variables $(\theta_{h},\theta_{\ell},\chi)$. The
individual terms for $p_{k}$ and $G_{k}(\bf\Omega_{\rm hel})$ for $k$=1–10 are
listed in Table 3.
Table 3: Definition of the functions $p_{k}$ and $G_{k}(\bf\Omega_{\rm hel})$ of Eq. (24). $k$ | $p_{k}$ | $G_{k}(\bf\Omega_{\rm hel})$
---|---|---
1 | $|\frac{H_{S}}{\sqrt{3}}|^{2}$ | $\sin^{2}\theta_{\ell}$
2 | $|H_{0}|^{2}$ | $\sin^{2}\theta_{\ell}\cos^{2}\theta_{h}$
3 | $|H_{+}|^{2}+|H_{-}|^{2}$ | $\frac{1}{4}(1+\cos^{2}\theta_{\ell})\sin^{2}\theta_{h}$
4 | $\mathcal{R}e(\frac{H_{S}}{\sqrt{3}}H_{0}^{*})$ | $2\sin^{2}\theta_{\ell}\cos\theta_{h}$
5 | $\mathcal{R}e(H_{+}H_{-}^{*})$ | $-\frac{1}{2}\sin^{2}\theta_{\ell}\sin^{2}\theta_{h}\cos 2\chi$
6 | $\mathcal{I}m(H_{+}H_{-}^{*})$ | $\frac{1}{2}\sin^{2}\theta_{\ell}\sin^{2}\theta_{h}\sin 2\chi$
7 | $\mathcal{R}e[\frac{H_{S}}{\sqrt{3}}(H_{+}^{*}+H_{-}^{*})]$ | $\frac{1}{2}\sin 2\theta_{\ell}\sin\theta_{h}\cos\chi$
8 | $\mathcal{I}m(\frac{H_{S}}{\sqrt{3}}(H_{+}^{*}-H_{-}^{*}))$ | $\frac{1}{2}\sin 2\theta_{\ell}\sin\theta_{h}\sin\chi$
9 | $\mathcal{R}e(H_{0}(H_{+}^{*}+H_{-}^{*}))$ | $\frac{1}{4}\sin 2\theta_{\ell}\sin 2\theta_{h}\cos\chi$
10 | $\mathcal{I}m(H_{0}(H_{+}^{*}-H_{-}^{*}))$ | $-\frac{1}{4}\sin 2\theta_{\ell}\sin 2\theta_{h}\sin\chi$
The functions can be expressed using transversity amplitudes by using the
relations between helicity and transversity amplitudes (A) [25], and the
relations between helicity $(\theta_{h},\theta_{\ell},\chi)$,
${\bf\Omega}_{\rm hel}$, and transversity angles $(\psi_{\rm tr},\theta_{\rm
tr},\phi_{\rm tr})$, ${\bf\Omega}_{\rm tr}$. The amplitudes relations are
$\displaystyle A_{S}$ $\displaystyle=$ $\displaystyle H_{S},$ $\displaystyle
A_{0}$ $\displaystyle=$ $\displaystyle H_{0},$ $\displaystyle A_{\parallel}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(H_{+}+H_{-}),$
$\displaystyle A_{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(H_{+}-H_{-}),$ (25)
and the angular relationships are
$\displaystyle\cos\psi_{\rm tr}$ $\displaystyle=$
$\displaystyle\cos\theta_{h},$ $\displaystyle\sin\theta_{\rm tr}\cos\phi_{\rm
tr}$ $\displaystyle=$ $\displaystyle-\cos\theta_{\ell},$
$\displaystyle\sin\theta_{\rm tr}\sin\phi_{\rm tr}$ $\displaystyle=$
$\displaystyle-\sin\theta_{\ell}\cos\chi,$ $\displaystyle\ \cos\theta_{\rm
tr}$ $\displaystyle=$ $\displaystyle\sin\theta_{\ell}\sin\chi.$ (26)
The amplitudes $\overline{A}_{i}$ are related to $A_{i}$ as
$\frac{q}{p}\frac{\overline{A}_{i}}{A_{i}}=\eta_{i}e^{-i\phi_{s}},$ (27)
where $\eta_{i}$ is $C\\!P$ eigenvalue of the $i$ component; $\eta_{S}$ and
$\eta_{\perp}=-1$, and $\eta_{0}$ and $\eta_{\parallel}=1$. We express the
amplitutes as functions of either the helicity distributions or transversity
distributions as the sums
$|\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{A}_{f}|^{2}=\displaystyle\sum_{k=1}^{10}\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{q}_{k}g_{k}({\bf\Omega}_{\rm
hel})=\sum_{k=1}^{10}\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{q}_{k}g_{k}({\bf\Omega}_{\rm
tr}).$ (28)
From Eqs. (3.4) and (27), we compute the interference terms as
$\frac{q}{p}A_{f}^{*}\overline{A}_{f}=e^{-i\phi_{s}}\left(\sum_{k=1}^{10}r_{k}g_{k}(\bf\Omega_{\rm
tr})\right).$ (29)
Each term is listed in Table 4.
Table 4: Definition of the functions used in Eqs. (28) and (29). When two signs appear, the upper one corresponds to $q_{k}$ and the lower to $\bar{q}_{k}$. $\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{q}_{k}$ | $r_{k}$ | $g_{k}(\bf\Omega_{\rm hel})$ | $g_{k}(\bf\Omega_{\rm tr})$
---|---|---|---
$|A_{0}|^{2}$ | $|A_{0}|^{2}$ | $\sin^{2}\theta_{\ell}\cos^{2}\theta_{h}$ | $\cos\psi_{\rm tr}(1-\sin^{2}\theta_{\rm tr}\cos^{2}\phi_{\rm tr})$
$|A_{\parallel}|^{2}$ | $|A_{\parallel}|^{2}$ | $\frac{1}{2}(1-\sin^{2}\theta_{\ell}\cos^{2}\chi)\sin^{2}\theta_{h}$ | $\frac{1}{2}\sin^{2}\psi_{\rm tr}(1-\sin^{2}\theta_{\rm tr}\sin^{2}\phi_{\rm tr})$
$|A_{\perp}|^{2}$ | $-|A_{\perp}|^{2}$ | $\frac{1}{2}(1-\sin^{2}\theta_{\ell}\sin^{2}\chi)\sin^{2}\theta_{h}$ | $\frac{1}{2}\sin^{2}\psi_{\rm tr}\sin^{2}\theta_{\rm tr}$
$|\frac{A_{S}}{\sqrt{3}}|^{2}$ | $-|\frac{A_{S}}{\sqrt{3}}|^{2}$ | $\sin^{2}\theta_{\ell}$ | $1-\sin^{2}\theta_{\rm tr}\cos^{2}\phi_{\rm tr}$
$\mathcal{R}e(A_{0}^{*}A_{\parallel})$ | $\mathcal{R}e(A_{0}^{*}A_{\parallel})$ | $\frac{1}{2\sqrt{2}}\sin 2\theta_{\ell}\sin 2\theta_{h}\cos\chi$ | $\frac{1}{2\sqrt{2}}\sin 2\psi_{\rm tr}\sin^{2}\theta_{\rm tr}\sin 2\phi_{\rm tr}$
$\pm\mathcal{I}m(A_{0}^{*}A_{\perp})$ | $i\mathcal{R}e(A_{0}^{*}A_{\perp})$ | $-\frac{1}{2\sqrt{2}}\sin 2\theta_{\ell}\sin 2\theta_{h}\sin\chi$ | $\frac{1}{2\sqrt{2}}\sin 2\psi_{\rm tr}\sin 2\theta_{\rm tr}\cos\phi_{\rm tr}$
$\pm\mathcal{R}e(A_{0}^{*}\frac{A_{S}}{\sqrt{3}})$ | $-i\mathcal{I}m(A_{0}^{*}\frac{A_{S}}{\sqrt{3}})$ | $2\sin^{2}\theta_{\ell}\cos\theta_{h}$ | $2\cos\psi_{\rm tr}(1-\sin^{2}\theta_{\rm tr}\cos^{2}\phi_{\rm tr})$
$\pm\mathcal{I}m(A_{\parallel}^{*}A_{\perp})$ | $i\mathcal{R}e(A_{\parallel}^{*}A_{\perp})$ | $\frac{1}{2}\sin^{2}\theta_{\ell}\sin^{2}\theta_{h}\sin 2\chi$ | $-\frac{1}{2}\sin\psi_{\rm tr}\sin 2\theta_{\rm tr}\sin\phi_{\rm tr}$
$\pm\mathcal{R}e(A_{\parallel}^{*}\frac{A_{S}}{\sqrt{3}})$ | $-i\mathcal{I}m(A_{\parallel}^{*}\frac{A_{S}}{\sqrt{3}})$ | $\frac{1}{\sqrt{2}}\sin 2\theta_{\ell}\sin\theta_{h}\cos\chi$ | $\frac{1}{\sqrt{2}}\sin\psi_{\rm tr}\sin^{2}\theta_{\rm tr}\sin 2\phi_{\rm tr}$
$\mathcal{I}m(A_{\perp}^{*}\frac{A_{S}}{\sqrt{3}})$ | $-\mathcal{I}m(A_{\perp}^{*}\frac{A_{S}}{\sqrt{3}})$ | $\frac{1}{\sqrt{2}}\sin 2\theta_{\ell}\sin\theta_{h}\sin\chi$ | $-\frac{1}{\sqrt{2}}\sin\psi_{\rm tr}\sin 2\theta_{\rm tr}\cos\phi_{\rm tr}$
In Ref. [3] the time-dependent and angular-dependent rate for
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ is
written as
$\frac{{\rm d}^{4}\Gamma}{{\rm d}t{\rm d}{\bf\Omega_{\rm
tr}}}\propto\sum_{k=1}^{10}h_{k}(t)f_{k}({\bf\Omega_{\rm tr}}),$ (30)
where the time-dependent function
$h_{k}(t)=N_{k}e^{-\Gamma t}[a_{k}\cosh\frac{\Delta\Gamma
t}{2}+c_{k}\cos(\Delta mt)+b_{k}\sinh\frac{\Delta\Gamma t}{2}+d_{k}\sin(\Delta
mt)],$ (31)
and the $f_{k}({\bf\Omega_{\rm tr}})$ represent angular-dependent functions.
Comparing with Eq. (2) it can be seen that $a_{k}$ corresponds to
$\displaystyle\frac{|A_{f}|^{2}+|\overline{A}_{f}|^{2}}{2}$, $c_{k}$ to
$\displaystyle\frac{|A_{f}|^{2}-|\overline{A}_{f}|^{2}}{2}$, $b_{k}$ to
$-\mathcal{R}e(\frac{q}{p}A_{f}^{*}\overline{A}_{f})$ and $d_{k}$ to
$-\mathcal{I}m(\frac{q}{p}A_{f}^{*}\overline{A}_{f})$. Using Table 4, we find
the same equations as shown in Ref. [3].
## References
* [1] Particle Data Group, J. Beringer et al., Review of Particle Physics (RPP), Phys. Rev. D86 (2012) 010001
* [2] LHCb collaboration, R. Aaij et al., Tagged time-dependent angular analysis of $B_{s}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ decays at LHCb, LHCb-CONF-2012-002
* [3] LHCb collaboration, R. Aaij et al., Measurement of the CP-violating phase $\phi_{s}$ in the decay $B_{s}^{0}\rightarrow J/\psi\phi$, Phys. Rev. Lett. 108 (2012) 101803, arXiv:1112.3183
* [4] CDF collaboration, T. Aaltonen et al., Measurement of the CP-violating phase $\beta_{s}^{J/\Psi\phi}$ in $B^{0}_{s}\rightarrow J/\Psi\phi$ decays with the CDF II Detector, Phys. Rev. D85 (2012) 072002, arXiv:1112.1726
* [5] D0 collaboration, V. M. Abazov et al., Measurement of the CP-violating phase $\phi_{s}^{J/\psi\phi}$ using the flavor-tagged decay $B_{s}^{0}\rightarrow J/\psi\phi$ in 8 fb-1 of $p\overline{p}$ collisions, Phys. Rev. D85 (2012) 032006, arXiv:1109.3166
* [6] ATLAS collaboration, G. Aad et al., Time-dependent angular analysis of the decay $B_{s}^{0}\rightarrow J/\psi\phi$ and extraction of $\Delta\Gamma_{s}$ and the $CP$-violating weak phase $\phi_{s}$ by ATLAS, arXiv:1208.0572
* [7] LHCb collaboration, R. Aaij et al., Measurement of the CP-violating phase $\phi_{s}$ in $\overline{B}{}_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays, Phys. Lett. B713 (2012) 378, arXiv:1204.5675
* [8] LHCb collaboration, R. Aaij et al., Measurement of the $C\\!P$ violating phase $\phi_{s}$ in $\overline{B}{}_{s}^{0}\rightarrow J/\psi f_{0}(980)$, Phys. Lett. B707 (2012) 497, arXiv:1112.3056
* [9] LHCb collaboration, R. Aaij et al., Analysis of the resonant components in $\overline{B}{}_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ , Phys. Rev. D86 (2012) 052006, arXiv:1204.5643
* [10] A. S. Dighe, I. Dunietz, H. J. Lipkin, and J. L. Rosner, Angular distributions and lifetime differences in $B_{s}\rightarrow J/\psi\phi$ decays, Phys. Lett. B369 (1996) 144, arXiv:hep-ph/9511363
* [11] A. S. Dighe, I. Dunietz, and R. Fleischer, Extracting CKM phases and $B_{s}-\bar{B}_{s}$ mixing parameters from angular distributions of nonleptonic $B$ decays, Eur. Phys. J. C6 (1999) 647, arXiv:hep-ph/9804253
* [12] R. Dalitz, On the analysis of $\tau$-meson data and the nature of the $\tau$-meson, Phil. Mag. 44 (1953) 1068
* [13] U. Nierste, Three lectures on Meson mixing and CKM phenomenology, arXiv:0904.1869
* [14] I. I. Bigi and A. Sanda, CP violation, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 9 (2000) 1
* [15] I. Dunietz et al., How to extract CP violating asymmetries from angular correlations, Phys. Rev. D43 (1991) 2193
* [16] Belle collaboration, R. Mizuk et al., Observation of two resonance-like structures in the $\pi^{+}\chi_{c1}$ mass distribution in exclusive $\overline{B}^{0}\rightarrow K^{-}\pi^{+}\chi_{c1}$ decays, Phys. Rev. D78 (2008) 072004, arXiv:0806.4098
* [17] LHCb Collaboration, R. Aaij et al., Opposite-side flavour tagging of $B$ mesons at the LHCb experiment, Eur. Phys. J. C72 (2012) 2022, arXiv:1202.4979
* [18] D0 Collaboration, V. Abazov et al., Measurement of $B_{d}$ mixing using opposite-side flavor tagging, Phys. Rev. D74 (2006) 112002, arXiv:hep-ex/0609034
* [19] LHCb Collaboration, R. Aaij et al., Determination of the sign of the decay width difference in the $B_{s}$ system, Phys. Rev. Lett. 108 (2012) 241801, arXiv:1202.4717
* [20] S. Stone and L. Zhang, S-waves and the measurement of CP violating phases in $B_{s}$ decays, Phys. Rev. D79 (2009) 074024, arXiv:0812.2832
* [21] Y. Xie, P. Clarke, G. Cowan, and F. Muheim, Determination of 2$\beta_{s}$ in $B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ Decays in the Presence of a $K^{+}K^{-}$ S-Wave Contribution, JHEP 0909 (2009) 074, arXiv:0908.3627
* [22] LHCb Collaboration, R. Aaij et al., Observation of $B_{s}\rightarrow J/\psi f^{\prime}_{2}(1525)$ in $J/\psi K^{+}K^{-}$ final states, Phys. Rev. Lett. 108 (2012) 151801, arXiv:1112.4695
* [23] D0 Collaboration, V. M. Abazov et al., Study of the decay $B_{s}^{0}\rightarrow J/\psi f_{2}^{\prime}(1525)$ in $\mu^{+}\mu^{-}K^{+}K^{-}$ final states, Phys. Rev. D (2012) arXiv:1204.5723
* [24] F. Azfar et al., Formulae for the Analysis of the Flavor-Tagged Decay $B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$, JHEP 1011 (2010) 158, arXiv:1008.4283
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|
arxiv-papers
| 2012-12-27T21:58:07 |
2024-09-04T02:49:39.732774
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liming Zhang and Sheldon Stone",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1212.6434"
}
|
1212.6439
|
# Conformal infinity in Robinson–Trautman spacetimes with cosmological
constant
Otakar Svítek
###### Abstract
In past, the future asymptotic behavior (with respect to initial data on null
hypersurface) of Robinson–Trautman spacetime was examined and its past horizon
characterized. Therefore, only the investigation of conformal infinity is
missing from the picture. We would like to present some initial results
concerning conformal infinity when negative cosmological constant is present
motivated by the AdS/CFT correspondence.
###### Keywords:
conformal infinity, cosmological constant, boundary stress energy tensor,
Hamilton-Jacobi theory, renormalization
###### :
04.20.Jb, 04.20.Gz, 11.25.Tq
## 1 Introduction
Various aspects of an important Robinson–Trautman family of expanding,
shearfree and twistfree spacetimes [1, 2] were studied during the last 50
years. The existence, asymptotic behavior, possible extensions beyond future
horizon and other specific properties of _cosmological and/or pure radiation_
solutions of algebraic type II with spherical topology were investigated, see
[3, 4, 5, 6, 7, 8, 9] for more recent expositions. Also the past horizon
properties were extensively covered [10, 11, 12]. Several results were also
generalized to higher dimensional version of this family [13, 14].
Having global causal structure in mind, one is naturally interested in
learning something about conformal infinity. Here, we will show initial
results concerning Robinson–Trautman spacetimes with negative cosmological
constant.
Inspired by the fact that the selected subfamily is locally asymptotically AdS
at timelike conformal infinity we will use the view provided by AdS/CFT
correspondence [15, 16]. Namely, we will determine the associated boundary
stress energy tensor with the possible identification of dual field theory in
mind. For black hole spacetimes the favored interpretation is in terms of
strongly coupled fluid [17].
Robinson–Trautman spacetimes of Petrov type II (with a cosmological constant
$\Lambda$ and pure radiation with density $w$) can be described by the
following metric
${\rm d}s^{2}=-2H\,{\rm d}u^{2}-\,2\,{\rm d}u\,{\rm
d}r+2\,\frac{r^{2}}{P^{2}}\,{\rm d}\zeta\,{\rm d}\bar{\zeta}$ (1)
where ${2H=\Delta(\,\ln P)-2r(\,\ln P)_{,u}-{2m(u)/r}-(\Lambda/3)r^{2}}$.
Using the Laplacian $\Delta\equiv
2P^{2}\partial_{\zeta}\partial_{\bar{\zeta}}$ the field equations reduce to
the well-known Robinson–Trautman equation
$\Delta\Delta(\,\ln P)+12\,m(\,\ln P)_{,u}-4\,m_{,u}=2\kappa\ w^{2}$ (2)
(a) (b)
Figure 1: (a) Robinson–Trautman-AdS (b) Foliation
These spacetimes have well defined evolution into the future of a null
hypersurface ($u=u_{0}$, see Figure 1a) with arbitrary smooth initial data.
Asymptotically (as $u\to\infty$) they approach Schwarzschild–((anti-) de
Sitter) or Vaidya–((anti-) de Sitter) when pure radiation is present. Past
black hole horizon can be localized quasilocally using any of the following
definitions: apparent horizon, trapping horizon, dynamical horizon.
## 2 Boundary stress energy tensor for asymptotically AdS spacetimes
We will try to compute the boundary stress energy tensor (SET) associated with
the gravitating asymptotically AdS spacetime. The result can be generally
interpreted as an expectation value of the stress energy tensor in a dual
conformal field theory (AdS/CFT).
To compute the SET we will use a method developed by J.D. Brown and J.W. York
[18](so called Brown-York SET). The derivation is based on Hamilton-Jacobi
theory. They consider a spacetime region $M=\Sigma\times R$ with metric $g$
and denote $B^{2}$ the two-dimensional boundary of $\Sigma$ (equipped with
metric $h$). Further, $B^{3}$ (with metric $\gamma$) is the timelike part of
$\partial M$ which is sliced by two-dimensional surfaces $B^{2}$ (see Figure
1b).
Now, consider the following gravitational action with negative cosmological
constant, boundary terms and matter
$S=\frac{1}{2\kappa}\int_{M}d^{4}x\sqrt{-g}\left(R+\frac{6}{\ell^{2}}\right)+\frac{1}{\kappa}\int_{t_{i}}^{t_{f}}d^{3}x\sqrt{h}\,K-\frac{1}{\kappa}\int_{B^{3}}\sqrt{-\gamma}\
\Theta+S_{\mathrm{matter}}$ (3)
where $\Lambda=-\frac{3}{\ell^{2}}$. In a standard way, arbitrary variation
gives terms corresponding to equations of motion and relevant boundary terms.
However, when the action $S$ is evaluated at classical solution ($S$ $\to$
$S_{\mathrm{cl}}$) it is a functional of boundary data:
$\gamma,h(t_{i}),h(t_{f})$. So restricting only to variations among classical
solutions we determine the dependence of $S_{\mathrm{cl}}$ on them (omitting
indices for now)
$\delta S_{\mathrm{cl}}=\\{\mathrm{variations\ of\ matter\
fields}\\}+\int_{t_{i}}^{t_{f}}d^{3}x\ P_{\mathrm{cl}}\,\delta
h+\int_{B^{3}}d^{3}x\ \pi_{\mathrm{cl}}\,\delta\gamma$ (4)
where $\pi_{\mathrm{cl}}$ is the gravitational momentum conjugate to $\gamma$.
Quasilocal boundary SET is then defined in analogy with mechanical systems as
$T=\frac{2}{\sqrt{-\gamma}}\frac{\delta
S_{\mathrm{cl}}}{\delta\gamma}=\frac{2}{\sqrt{-\gamma}}\pi_{\mathrm{cl}}$ (5)
However, the prescription diverges as the boundary is taken to infinity and
therefore some form of regularization is needed. We will select the approach
by Balasubramanian and Kraus [19] who proposed a renormalization procedure
consisting of subtracting suitable counterterms. These counterterms consist
only from boundary terms and so they do not influence bulk equations of
motion.
To arrive at the formula we start with ADM-like decomposition (foliation is
timelike)
${\rm d}s^{2}=N^{2}{\rm d}r^{2}+\gamma_{\mu\nu}({\rm d}x^{\mu}+N^{\mu}{\rm
d}r)({\rm d}x^{\nu}+N^{\nu}{\rm d}r)$ (6)
Since the boundary metric $\gamma$ acquires an infinite Weyl factor
asymptotically we will formally consider conformal class of boundaries. We
assume the action (from now we are only interested in dependence on boundary
metric $\gamma$) in the form
$S_{\gamma}=S_{\mathrm{cl}}[\gamma]+\frac{1}{\kappa}S_{\mathrm{ct}}[\gamma]$
and introduce the extrinsic curvature
$\Theta^{\mu\nu}=-\nabla^{(\mu}n^{\nu)}$, where $\mathbf{n}$ is an outward
pointing normal to $B^{3}$.
Then, $S_{\mathrm{ct}}$ is constructed by requiring cancellation of all
divergences using technique resembling tree unitarity approach in QFT.
Specifically, using prescription (5) with $S_{\gamma}$ in four dimensions [19]
arrive at
$T^{\mu\nu}=\frac{1}{\kappa}\left[\Theta^{\mu\nu}-\Theta\,\gamma^{\mu\nu}-\frac{2}{\ell}\gamma^{\mu\nu}-\ell\,G^{\mu\nu}(\gamma)\right]$
(7)
Taking an additional ADM foliation (now spacelike) of boundary one can compute
interesting physical quantities by projecting $T^{\mu\nu}$ on the vector
generating time flow on $B^{3}$.
## 3 Robinson–Trautman-AdS boundary SET
After introducing new coordinate $l=r^{-1}$ and taking the conformal factor to
be ${\Omega=l}$, the Robinson–Trautman metric becomes ${\rm
d}s^{2}=\Omega^{-2}[({\textstyle\frac{1}{3}}\Lambda+2\,l(\,\ln
P)_{,u}-l^{2}\Delta\ln P+2m\,l^{3}){\rm d}u^{2}+2\,{\rm d}u\,{\rm
d}l+2\,P^{-2}\,{\rm d}\zeta\,{\rm d}\bar{\zeta}\,]$ so it is locally
asymptotically AdS for negative cosmological constant.
Before applying the previously derived results we have to account for the
additional boundary segment corresponding to hypersurface $u=u_{0}$. Since
this boundary is equipped with initial data determining the bulk evolution we
have to assume that its boundary metric is subject to variations corresponding
to changes in the bulk solution. However, we will not investigate canonical
pair for this boundary explicitly because it is not influencing the
calculation of SET for conformal infinity.
Using a simple transformation $u=t-r$ on the metric (1) we arrive at
${\rm d}s^{2}=-2H{\rm d}t^{2}-2(1-2H)\,{\rm d}t{\rm d}r+2(1-H)\,{\rm
d}r^{2}+\frac{r^{2}}{P^{2}}\,{\rm d}\zeta\,{\rm d}\bar{\zeta}$ (8)
which is suitable for reading of the necessary quantities:
$\gamma_{\mu\nu}=\mathrm{diag}(-2H,{\textstyle\frac{r^{2}}{P^{2}}},{\textstyle\frac{r^{2}}{P^{2}}}),\,N=(2H)^{-1/2},\,N^{\mu}=(N^{2}-1,0,0),\,\mathbf{n}=N{\rm
d}r$.
From (7) we compute that the leading term of SET is $T_{00}=\frac{2m(u)}{\ell
r}+\ \cdots$ which generalizes the previous results for Schwarzschild–AdS
spacetime. Using the above mentioned additional ADM splitting on the boundary
with parameters
$\sigma_{ab}=\mathrm{diag}({\textstyle\frac{r^{2}}{P^{2}},\frac{r^{2}}{P^{2}}}),\
N_{B^{2}}=\sqrt{2H},\ \mathbf{v}=\sqrt{2H}{\rm d}t$ we can compute the proper
energy density on the boundary
$\int_{B^{2}}d\zeta
d\bar{\zeta}\,\sqrt{\sigma}N_{B^{2}}T_{\mu\nu}v^{\mu}v^{\nu}=\int_{B^{2}}d\zeta
d\bar{\zeta}\,\frac{4m(u)^{2}}{P^{2}\ell}$ (9)
which however differs from corresponding expression given by Robinson and
Trautman that involved term $m^{2/3}$ instead of $m^{2}$.
## 4 Conclusions and outlook
The presented results are so far only preliminary and have to be analyzed and
extended. Specifically we would like to understand the correspondence with
strongly coupled fluid in this context. Having the causal structure in mind
one is also interested in explicitly connecting the behavior at conformal
infinity with the dynamics of past horizon.
This work was supported by grant GACR 202/09/0772.
## References
* [1] I. Robinson and A. Trautman, _Phys. Rev. Lett._ 4, 431 (1960).
* [2] I. Robinson and A. Trautman, _Proc. Roy. Soc. Lond._ A265, 463 (1962).
* [3] P. T. Chruściel, _Commun. Math. Phys._ 137, 289 (1991).
* [4] P. T. Chruściel, _Proc. Roy. Soc. Lond._ A436, 299 (1992).
* [5] P. T. Chruściel and D. B. Singleton, _Commun. Math. Phys._ 147, 137 (1992).
* [6] J. Bičák and J. Podolský, _Phys. Rev. D_ 52, 887 (1995).
* [7] J. Bičák and J. Podolský, _Phys. Rev. D_ 55, 1985 (1997).
* [8] J. Podolský and O. Svítek, _Phys. Rev. D_ 71, 124001 (2005).
* [9] L. Rezzolla, R. P. Macedo and J. L. Jaramillo, _Phys. Rev. Lett._ 104, 221101 (2010).
* [10] E. W. M. Chow and A. W. C. Lun, _J. Austr. Math. Soc. B_ 41, 217 (1999).
* [11] W. Natorf and J. Tafel, _Class. Quantum Grav._ 25, 195012 (2008).
* [12] J. Podolský and O. Svítek, _Phys. Rev. D_ 80, 124042 (2009).
* [13] J. Podolský and M. Ortaggio, _Class. Quant. Grav._ 23, 5785 (2006).
* [14] O. Svítek, _Phys. Rev. D_ 84, 044027 (2011).
* [15] J. M. Maldacena, _Adv. Theor. Math. Phys._ 2, 231 (1998).
* [16] E. Witten, _Adv. Theor. Math. Phys._ 2, 253 (1998).
* [17] S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, _JHEP_ 0802, 045 (2008).
* [18] J. D. Brown and J. W. York, _Phys. Rev. D_ 47, 1407 (1993).
* [19] V. Balasubramanian and P. Kraus, _Commun. Math. Phys._ 208, 413 (1999)
|
arxiv-papers
| 2012-12-28T00:08:00 |
2024-09-04T02:49:39.739526
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Otakar Svitek",
"submitter": "Otakar Svitek",
"url": "https://arxiv.org/abs/1212.6439"
}
|
1212.6546
|
# Numerical Approximation of Probability Mass Functions Via the Inverse
Discrete Fourier Transform
Richard L. Warr111Department of Mathematics and Statistics, Air Force
Institute of Technology, Wright-Patterson Air Force Base, Ohio, USA. The views
expressed in this article are those of the authors and do not reflect the
official policy or position of the United States Air Force, Department of
Defense, or the U.S. Government.
###### Abstract
First passage distributions of semi-Markov processes are of interest in fields
such as reliability, survival analysis, and many others. The problem of
finding or computing first passage distributions is, in general, quite
challenging. We take the approach of using characteristic functions (or
Fourier transforms) and inverting them, to numerically calculate the first
passage distribution. Numerical inversion of characteristic functions can be
numerically unstable for a general probability measure, however, we show for
lattice distributions they can be quickly calculated using the inverse
discrete Fourier transform. Using the fast Fourier transform algorithm these
computations can be extremely fast. In addition to the speed of this approach,
we are able to prove a few useful bounds for the numerical inversion error of
the characteristic functions. These error bounds rely on the existence of a
first or second moment of the distribution, or on an eventual monotonicity
condition. We demonstrate these techniques in an example and include R-code.
KEY WORDS: characteristic function, discrete Weibull, first passage
distribution, fast Fourier transform, semi-Markov process, statistical
flowgraph
## 1 Introduction
Statistical literature abounds with proofs using characteristic functions
(CFs). Asymptotic results such as the central limit theorem rely heavily on
the properties of CFs. However, when it comes to applied statistics the
reverse is true. In general statisticians seem very uncomfortable using the CF
or numeric approximations of it, even though they routinely use numerical
approximations and calculations for various other procedures. This could be
partly due to the fact that CFs are complex functions, but this is a deterrent
based on fear of the unknown, not difficulty.
This is not the case in many applied sciences. Engineers make heavy use of the
Fourier transform and are quite comfortable using it in practice. The Fourier
transform and the characteristic function are the same function with a change
of variables. To use one over the other is a matter of preference, the curves,
as drawn, are identical. There are some areas in applied statistics where use
of CFs are helpful. One such area is semi-Markov processes (SMPs),
specifically finding first passage distributions from one state to another, as
in statistical flowgraph models Huzurbazar (2005). Using characteristic
functions is a natural way to find first passage distributions in SMPs.
For example consider the graphical representation of a SMP in Figure 1, where
the nodes on the graph represent a state, and the branches represent waiting
time distributions before transition to the next state. Each branch is labeled
with a characteristic function $\varphi_{{}_{ij}}(s)$ and a probability of
taking that branch. The characteristic function of the first passage
distribution is:
$\varphi_{{}_{13*}}(s)=\frac{(1-p)\varphi_{{}_{12}}(s)\varphi_{{}_{23}}(s)}{1-p\varphi_{{}_{12}}(s)\varphi_{{}_{21}}(s)},$
(1)
the “*” denotes a first passage. For details on finding the first passage CF
see Huzurbazar (2005) or for a more theoretical development refer to Pyke
(1961). Once the CFs for the branches are known, it is trivial to calculate
the CF of the first passage from state $1$ to $3$. The challenge is to convert
the CF to a distribution. In most cases this must be done numerically.
Figure 1: An example semi-Markov process.
Work has been done to show that the discrete Fourier transform (DFT) can
accurately invert characteristic functions given sufficiently fast decay of
both the density and the characteristic function (see Hughett (1998)). One
class of distributions that has not been well studied for inversion using the
DFT is discrete distributions. Specifically we focus on distributions with
support $n\Delta t$ for $n=0,\pm 1,\pm 2,\ldots$ and $\Delta
t\in\mathbb{R}^{+}$. This type of distribution has been termed a lattice or an
arithmetic distribution, these include most of the well studied discrete
distributions such as the binomial, Poisson, negative binomial and many
others.
The primary goal of this paper is to promote the use of the inverse discrete
Fourier transform (iDFT) for the inversion of CFs to probability mass
functions (PMFs). A secondary benefit of using the DFT is easy calculation of
CFs when they do not have a closed form.
This paper is outlined as follows. The next section details some properties of
the DFT, in particular how the DFT is applied when dealing with lattice
distributions. Next, the DFT is shown in action, in an application. We include
code using the R software package (see R Development Core Team (2009)).
Finally, we conclude with a few comments.
## 2 The discrete Fourier transform
For a random variable $T$, with distribution $F$, its characteristic function
is
$\varphi_{{}_{T}}(s)\equiv\int_{-\infty}^{\infty}e^{ist}\,dF.$ (2)
Let $\hat{f}(\omega)$ be the Fourier transform of $T$, then the relationship
between the characteristic function and the Fourier transform is
$\varphi_{{}_{T}}(-2\pi\omega)=\hat{f}(\omega).$ (3)
In the reminder of the paper we use Fourier transforms, with the understanding
that the CF is easily obtained from the FT. We do this primarily for ease in
applying DFT methods, specifically using the fast Fourier transform (FFT)
algorithm.
The DFT can be used to approximate the Fourier transform and its inverse.
Under certain conditions, the error of this approximation can be controlled.
An excellent reference on the DFT is Briggs and Henson (1995). To use the DFT,
the support of the RV must be discretized at evenly spaced points, where
$\Delta t$ is the space between adjacent points. We term these points the
support of the DFT samples. Similarly the support of the iDFT samples the FT
at evenly spaced intervals of width $\Delta\omega$ in the transform domain.
These sampling widths are related by the reciprocity relation
$\Delta t\Delta\omega=\frac{1}{N}.$ (4)
With this information we apply the DFT and iDFT to lattice distributions.
### 2.1 DFT of lattice distributions
In this section error bounds for lattice distributions with nonnegative
support are derived. If $f(t_{n})$ is the PMF of a lattice distribution, then
the DFT of that distribution is:
$F_{k}\equiv\sum_{n=0}^{N-1}f(t_{n})e^{-2\pi i\frac{nk}{N}},\text{ for
}k=0,1,\ldots,N-1.$ (5)
Clearly the utility of the DFT as an approximation of the FT is dependent on
the size of $N$. The major benefit of using the DFT with discrete random
variables is that unlike RVs with PDFs, when sampling from lattice
distributions we are able to capture essentially all the information about the
random variable. In this paper we use the word “sampling” loosely to mean the
PMF values of the support that are included in the DFT. When sampling a RV
with a PDF the measure of our finite number of samples is zero, whereas when
sampling from a certain class of lattice distributions our “sample” has
measure that can approach one. In essence, we are losing little or no
information when sampling from a DRV, or in other words, with a finite number
of points we obtain more of a “census” than a “sample” of where the
probability is located in a distribution. This allows the DFT to accurately
approximate the FT at specified points given $N$ sufficiently large.
We confine our focus to nonnegative lattice RVs $T$ where, given any
$\varepsilon>0$, there exists an $N$ such that
$\sum_{i=0}^{N-1}P(T=t_{i})>1-\varepsilon$. Nearly all properly defined
nonnegative lattice RVs fall into this class. There may be a few pathological
cases which do not, such as the counting measure on all the integers, however,
in many cases one would argue that these are not properly defined
distributions. The assumption that $T$ is nonnegative is only slightly less
general than the unrestricted case, and derivations for lattice RVs with
unbounded support would be very similar.
To find the error bound for the DFT on lattice distributions one only needs to
look at the definition of the Fourier transform. Let $f(t_{n})\equiv
P(T=n\,\Delta t)$, then the definition of the Fourier transform is:
$\hat{f}(\omega)\equiv\sum_{n=0}^{\infty}f(t_{n})e^{-2\pi i\omega n\Delta
t}=\sum_{n=0}^{\infty}P(T=n\Delta t)e^{-2\pi i\omega n\Delta t}$ (6)
Now our goal is to bound the error $|\hat{f}(\omega_{k})-F_{k}|$ for all index
$k=0,1,\ldots,N-1$. Therefore we must find an optimal $\omega_{k}$ that
matches well with the DFT’s $F_{k}$. To do this we simply set the exponents of
Equations 5 and 6 equal to each other and solve for $\omega_{k}$. Therefore
$-2\pi i\omega_{k}n\Delta t=-2\pi i\frac{nk}{N},$ (7)
implies that
$\omega_{k}=\frac{k}{\Delta tN}.$ (8)
Now using the constraint that
$\sum_{n=0}^{N-1}P(T=n\,\Delta t)=\sum_{n=0}^{N-1}f(t_{n})\geq 1-\varepsilon,$
we show that the error can be controlled.
###### Lemma 2.1
For a nonnegative lattice random variable, defined on the support $n\,\Delta
t$ for $n=0,1,\ldots,\infty$, the pointwise error of the forward DFT is less
than or equal to $P(T\geq N\,\Delta t)$.
* Proof
$\displaystyle\left|\hat{f}(\omega_{k})-F_{k}\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{n=0}^{\infty}P(T=n\Delta t)e^{-2\pi
i\omega_{k}n\Delta t}-\sum_{n=0}^{N-1}P(T=n\Delta t)e^{-2\pi
i\frac{nk}{N}}\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{n=0}^{\infty}P(T=n\Delta t)e^{-2\pi
i\frac{nk}{N}}-\sum_{n=0}^{N-1}P(T=n\Delta t)e^{-2\pi i\frac{nk}{N}}\right|$
$\displaystyle=$ $\displaystyle\left|\sum_{n=N}^{\infty}P(T=n\Delta t)e^{-2\pi
i\frac{nk}{N}}\right|$ $\displaystyle\leq$
$\displaystyle\sum_{n=N}^{\infty}P(T=n\Delta t)\left|e^{-2\pi
i\frac{nk}{N}}\right|\ =\ \sum_{n=N}^{\infty}P(T=n\,\Delta t)$
$\displaystyle=$ $\displaystyle\sum_{n=N}^{\infty}f(t_{n})\ =P(T\geq N\Delta
t).\nobreak\hskip 0.0pt\hskip 15.00002pt minus 5.0pt\nobreak\vrule
height=7.5pt,width=5.0pt,depth=2.5pt$
To give an example, consider a Poisson random variable $U$, with rate
parameter $\lambda=5$. The FT of $U$ is $\hat{f}(\omega)=\exp\\{5e^{-2\pi
i\omega}-5\\}$. If we choose $N=16$, where $\Delta t=1$, then
$\omega_{k}=k/16$ for $k=0,1,\ldots,15$. We set the error
$\varepsilon=P(U>15)\approx 0.000069$. The greatest error occurs when
computing $|\hat{f}(0/16)-F_{0}|\approx|1-0.999931|=0.000069$. Therefore we
can see in this example that
$\left|\hat{f}(\omega_{k})-F_{k}\right|\leq\varepsilon$.
Now that we have shown that the DFT can accurately approximate the Fourier
transform of a DRV, at specified points, we focus on the inverse DFT. This is
the more difficult of the two problems, but is in general more useful. Given
the Fourier transform of a DRV we now demonstrate how to accurately calculate
the PMF.
Given that $\hat{f}(\omega)$ is a known function, we show that the inverse DFT
has the same error bound as the forward DFT. The inverse DFT is defined as:
$f_{k}\equiv\frac{1}{N}\sum_{n=0}^{N-1}\hat{f}(\omega_{n})e^{2\pi
i\omega_{n}k\Delta t}.$ (9)
###### Lemma 2.2
Given the Fourier transform of a nonnegative lattice RV, $T$, evaluated at
points $k/(N\,\Delta t$), for $k=0,1,\ldots N-1$, the pointwise error of the
inverse DFT is less than or equal to $P(T\geq N\,\Delta t)$.
We introduce a couple of notational items to help with the following proof.
Let “mod” be the modulo operator, so for the positive integer $b$, and non-
negative integers $a$, $n$ and $r$; then $r=a\text{ mod }b$, where $n$ is the
largest non-negative integer that satisfies the equation $a=r+nb$. This
implies that $r<b$. Also, let
$I_{(\text{condition})}=\left\\{\begin{array}[]{ll}1&\mbox{if ``condition" is
true}\\\ 0&\mbox{otherwise}\end{array}\right.$ (10)
We’ll call $I_{(\text{condition})}$ the indicator function.
* Proof
Using Equations 6 and 9 we can write the iDFT as:
$\displaystyle f_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\sum_{j=0}^{\infty}P(T=j\Delta
t)e^{-2\pi i\omega_{n}j\Delta t}e^{2\pi i\omega_{n}k\Delta t}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\sum_{j=0}^{\infty}P(T=j\Delta
t)e^{-2\pi i\frac{nj}{N}}e^{2\pi i\frac{nk}{N}}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{j=0}^{\infty}P(T=j\Delta
t)\sum_{n=0}^{N-1}e^{-2\pi i\frac{nj}{N}}e^{2\pi i\frac{nk}{N}}$
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{j=0}^{\infty}P(T=j\Delta
t)\,N\,\text{I}_{(k=j\text{ mod }N)}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}P(T=j\Delta t)\,\text{I}_{(k=j\text{ mod
}N)}+\sum_{j=N}^{\infty}P(T=j\Delta t)\,\text{I}_{(k=j\text{ mod }N)}$
$\displaystyle=$ $\displaystyle P(T=k\Delta t)+\sum_{j=N}^{\infty}P(T=j\Delta
t)\,\text{I}_{(k=j\text{ mod }N)}.$
Therefore we can now express the error as:
$\displaystyle\left|f_{k}-P(T=k\Delta t)\right|$ $\displaystyle=$
$\displaystyle\left|P(T=k\Delta t)+\sum_{j=N}^{\infty}P(T=j\Delta
t)\,\text{I}_{(k=j\text{ mod }N)}-P(T=k\Delta t)\right|$ $\displaystyle\leq$
$\displaystyle P(T\geq N\Delta t).\nobreak\hskip 0.0pt\hskip 15.00002pt minus
5.0pt\nobreak\vrule height=7.5pt,width=5.0pt,depth=2.5pt$
This result is not surprising, because of the close relationship between the
forward and inverse DFT.
The next result assumes the the FT is unknown and must be approximated by the
DFT, following which, the iDFT is used to invert the approximated FT to a PMF.
This theorem shows this error is also bounded.
###### Theorem 2.3
Given that the FT of a nonnegative lattice random variable is known up to a
fixed error bound, then the error bound of the iDFT of this FT can be bounded
by $2P(T\geq N\,\Delta t)$.
* Proof
Mimicking the proof of Lemma 2.2 we replace $\hat{f}(\omega_{n})$ with
$\hat{f}(\omega_{n})+\delta_{n}$, where $\varepsilon=P(T\geq N\,\Delta t)$ and
$|\delta_{n}|\leq\varepsilon$. This gives:
$\displaystyle f_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\left[\hat{f}(\omega_{n})+\delta_{n}\right]e^{2\pi
i\omega_{n}k\Delta t}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\hat{f}(\omega_{n})e^{2\pi
i\frac{nk}{N}}+\frac{1}{N}\sum_{n=0}^{N-1}\delta_{n}e^{2\pi i\frac{nk}{N}}$
$\displaystyle=$ $\displaystyle P(T=k\Delta t)+\sum_{j=N}^{\infty}P(T=j\Delta
t)\,\text{I}_{(k=j\text{ mod }N)}+\frac{1}{N}\sum_{n=0}^{N-1}\delta_{n}e^{2\pi
i\frac{nk}{N}}$
Therefore if we only have an approximation for $\hat{f}(\omega_{n})$ then,
$\displaystyle\left|f_{k}-P(T=k\Delta t)\right|$ $\displaystyle=$
$\displaystyle\left|\sum_{j=N}^{\infty}P(T=j\Delta t)\,\text{I}_{(k=j\text{
mod }N)}+\frac{1}{N}\sum_{n=0}^{N-1}\delta_{n}e^{2\pi i\frac{nk}{N}}\right|$
$\displaystyle\leq$ $\displaystyle\left|\sum_{j=N}^{\infty}P(T=j\Delta
t)\,\text{I}_{(k=j\text{ mod
}N)}\right|+\left|\frac{1}{N}\sum_{n=0}^{N-1}\delta_{n}e^{2\pi
i\frac{nk}{N}}\right|$ $\displaystyle\leq$
$\displaystyle\varepsilon+\frac{1}{N}\sum_{n=0}^{N-1}\left|\delta_{n}\right|\left|e^{2\pi
i\frac{nk}{N}}\right|$ $\displaystyle\leq$
$\displaystyle\varepsilon+\frac{\varepsilon}{N}\sum_{n=0}^{N-1}1\ =\
2\varepsilon.\nobreak\hskip 0.0pt\hskip 15.00002pt minus 5.0pt\nobreak\vrule
height=7.5pt,width=5.0pt,depth=2.5pt$
What we have not addressed, thus far, is the error introduced when
manipulating FTs in the Fourier domain. The primary reason we use CFs and FTs
is for the convenient theoretical properties but when $\hat{f}(\omega_{n})$ is
estimated, then when mathematical manipulation such as multiplication or
division occurs the error is magnified. We do not specifically address this
problem, but warn the practitioner of this added source of error.
Additionally, machine precision error also plays a part in the overall error
bound.
### 2.2 Finding the inversion error bounds
The primary problem when using the inverse DFT is that one may not know a
priori $N$ such that
$\sum_{n=0}^{N-1}P(T=n\Delta t)\geq 1-\varepsilon,$
for a given $\varepsilon$. So although the error is bounded, this bound is not
obvious without the exact PMF, which we do not usually have. There are a few
approaches to overcome this difficulty. One solution is to use an inequality
to bound $P(T>n\Delta t)$. One bound is Markov’s inequality:
$P(T\geq a)\leq\frac{E(T)}{a},\ \ \text{ for }a>0.$
Therefore if the first moment exists for the RV $T$, and we either know or can
estimate it, this method provides a conservative error bound. The primary
issue is finding $E(T)$. In our case the random variable $T$’s distribution is
a first passage distribution in a semi-Markov process. It turns out if all the
expectations to the individual transitions in the SMP are known, then finding
the first passage expectation is the solution to a set of linear equations.
Literature exists that explains how one can find the moments of first passage
distributions, for example see Zhang and Hou (2012) and Yao (1985).
Another inequality that can assist in bounding the error is Cantelli’s
inequality. For a random variable $T$ with $\mu=E(T)$, $\sigma^{2}=Var(T)$,
and $k>0$ we have
$P(T\geq k\sigma+\mu)\leq\frac{1}{1+k^{2}}.$
Therefore if $Var(T)<\infty$, and given $E(T)$ and $Var(T)$, Cantelli’s
inequality can be useful in bounding the DFT error. Where $k>\sigma/\mu$,
Cantelli’s inequality provides a smaller bound and should be used over
Markov’s inequality. Finding the variance of a first passage random variable
is similar to the expectation, again see Yao (1985) for details.
If the two inequalities prove to be impractical, then another approach can be
taken if certain assumptions can be made about the distribution of $T$.
###### Theorem 2.4
If for some index $M$ where the following inequalities are satisfied
$P(T=n\Delta t)\leq P(T=m\Delta t)$ and $M<m<n$, then the error for the iDFT
can be expressed as: Choose an even $N$ (the number of DFT samples) such that
$N>2M$, then for all $n=0,1,\ldots,(N/2-1)$
$\left|f_{n}-P(T=n\Delta t)\right|\leq f_{N/2+n}.$
Essentially what this is saying is that after a certain point, if the first
passage PMF, $f(t)$, is monotonically nonincreasing, then we can bound the
error using that fact alone. This technique is certainly not universally
applicable, but it does provide a working error bound if monotonicity
eventually occurs. To our knowledge PMFs used for modeling processes will
often meet this assumption.
* Proof
From the proof of Lemma 2.2 we know
$f_{n}=\sum_{j=0}^{\infty}P(T=j\Delta t)\,\text{I}_{(n=j\text{ mod }N)},$
Which can be rewritten as
$f_{n}=\sum_{j=0}^{\infty}P(T=(n+jN)\Delta t).$
Which implies
$f_{N/2+n}=\sum_{j=0}^{\infty}P(T=(N/2+n+jN)\Delta t)$
therefore
$\left|f_{n}-P(T=n\Delta t)\right|=\sum_{j=1}^{\infty}P(T=(n+jN)\Delta t)$
but by monotonicity we know
$P\left(T=\left(N/2+n+jN\right)\Delta t\right)\geq
P\left(T=\left(n+(j+1)N\right)\Delta t\right)\text{ for all }j=0,1,2\ldots$
This implies
$f_{N/2+n}\geq\sum_{j=1}^{\infty}P(T=(n+jN)\Delta t).$
Therefore
$\left|f_{n}-P(T=n\Delta t)\right|\leq f_{N/2+n}.\nobreak\hskip 0.0pt\hskip
15.00002pt minus 5.0pt\nobreak\vrule height=7.5pt,width=5.0pt,depth=2.5pt$
This bound can be the more effective then the previous bounds, if the
condition exists for it to be applied. However, it is difficult to prove the
monotonicity condition in a particular situation, and if sufficient doubt
exists, another error bounding method should be applied.
Before we move to the application section, we suggest an even broader
condition than eventual monotonicity that will provide the same error bound.
This condition is:
###### Theorem 2.5
If there exists some nonnegative integers $n$ and $N/2$ and a positive integer
$k$, such that, if $n<N/2$, and $P\left(T=\left(n+kN/2\right)\Delta
t\right)\geq P\left(T=\left(n+(k+1)N/2\right)\Delta t\right)$ then, for a DFT
with $N$ samples,
$\left|f_{n}-P(T=n\Delta t)\right|\leq f_{n+N/2}.$
These types of distributions might be termed eventually periodic
nonincreasing. This condition is very broad and can be applied in nearly all
first passage distributions, although we are sure there exist some
pathological cases that do not fit in this class of distributions. The proof
of Theorem 2.5 follows the same argument as Theorem 2.4.
In the next section we demonstrate using the DFT to calculate FTs, their
inversion, and also it’s error bounds.
## 3 An application of discrete time semi-Markov processes
Consider the problem described in Barbu and Liminios (2008), the waste
treatment for a textile factory. If the treatment facility is working then the
waste can be discarded; if the unit fails the untreated waste must be stored
in a holding tank until the treatment facility is again functioning. However,
if the repair to the treatment facility takes too long the holding tank
becomes full, and the textile factory must halt production until the repair is
complete. Figure 2 is a graphical depiction of the process.
Figure 2: A graphical depiction of the textile waste treatment process.
We use the model parameterization of Barbu and Liminios (2008), which is:
$\displaystyle T_{12}$ $\displaystyle\sim$
$\displaystyle\text{geometric}(0.8),$ $\displaystyle T_{21}$
$\displaystyle\sim$ $\displaystyle\text{discrete Weibull}(0.3,0.5),$
$\displaystyle T_{23}$ $\displaystyle\sim$ $\displaystyle\text{discrete
Weibull}(0.5,0.7),\text{ and}$ $\displaystyle T_{31}$ $\displaystyle\sim$
$\displaystyle\text{discrete Weibull}(0.6,0.9).$
The discrete Weibull as defined in Nakagawa and Osaki (1975) (slightly
modified) is
$f(t|q,b)=\begin{cases}q^{(t-1)^{b}}-q^{t^{b}}&\mbox{if
}t\in\\{1,2,\ldots\\}\\\ 0&\text{otherwise.}\end{cases}$
The SMP parameter $p$ (see Figure 2) is set at 0.95, or in words, on average
95% of the repairs to the treatment facility are completed before the tank
becomes full.
A few of the quantities that may be of interest from this process are the
first passage PMF from state $1$ to state $3$, the expected amount of time
spent in state $3$ at some time $t$, and the probability the factory is
shutdown due to waste treatment failure at some time $t$. The formulas for
these quantities can be found in Warr and Collins (2011). For this paper we
focus on finding the first passage PMF from state $1$ to state $3$.
The first passage CF is given in Equation 1, where we replace CFs with FTs in
all instances. So to calculate the first passage FT from state 1 to state 3,
we need the FT of $T_{12}$, $T_{21}$, and $T_{23}$. The FT of geometric RV
$T_{12}$ is known in closed form, however, the FT for the discrete Weibull is
not. So in our computations we use the exact FT for $T_{12}$ and use the DFT
approximation for the other two.
We first want to calculate the mean and variance of $T_{12}$, $T_{21}$, and
$T_{23}$ which will allow us to use the Cantelli inequality to bound our
error. Table 1 shows the expectations and variances for these RVs.
Table 1: The mean and variance of waiting time random variables in the textile waste treatment process. | Mean | Variance
---|---|---
$T_{12}$ | 1.2500 | 0.3125
$T_{21}$ | 2.0769 | 9.0765
$T_{23}$ | 2.7300 | 9.4618
$T_{13*}$ | 67.191 | 4394.1
These quantities allow us to calculate the number $N$ we need to use to obtain
a desired error bound. We choose $\varepsilon=10^{-6}$ which implies $k\geq
66356$. Therefore if $N=2^{17}$ the pointwise error for each probability will
be smaller than 2$\varepsilon$. The computation time of calculating the first
passage PMF is less than one second (on a Windows 7 PC with an AMD AthlonTM
7750 2.7 GHz dual-core processor).
The plot in Figure 3 shows what the first passage PMF from state $1$ to state
$3$ looks like for the first one hundred points. Using the developed theory in
this paper, we are assured the pointwise error is less than or equal to
2$\varepsilon$. Again this ignores the error introduced by multiplying and
dividing the FTs (as in Equation 1) and the machine round-off error.
Figure 3: The PMF of the first passage from state 1, “Working”, to state 3,
“Treatment failed, tank full”.
Figure 3 indicates that the PMF of this first passage distribution is
monotonically decreasing after roughly time $20$. Therefore we can attempt to
apply Theorem 2.4. Since we estimated that $P(T_{13*}=609)<\varepsilon$
Therefore if we choose $N=1218$ we only need to calculate roughly $1\%$ of the
values in the FFT and can still claim the pointwise error is less than or
equal to 2$\varepsilon$. This example shows the potential computation savings
of using Theorem 2.4 for first passage random variables. Also, Theorem 2.4
does not assume a finite mean or variance, but a monotonically decreasing PMF
after a selected point on the support. In this example we worked our way, by
trial and error, from a large $N$ down to a smaller $N$, however, we are also
able to work our way up from a small $N$ to a larger $N$. This can reduce the
computation time to find the desired error bound.
## 4 Discussion
It has been our goal to show for lattice random variables the DFT/FFT computes
the desired first passage PMF extremely quickly and to a prescribed accuracy.
It is often very desirable to handle probability problems in the transformed
CF domain, but computational methods to invert them back into the time or
spatial domain have been the largest deterrent. The DFT is a excellent
solution for the inversion of characteristic functions to PMFs. The DFT is
very fast using the FFT algorithm and it also provides error bounds for
estimates.
## 5 Acknowledgments
The authors would like to thank Patrick Chapin, David Collins and Aparna
Huzurbazar for their advice, assistance, and encouragement.
## 6 Appendix – R Code
##Discrete Weibull PMF
dweibulldisc<-function(x,q,b){
temp1=q^((x-1)^b)-q^(x^b); temp2=x-floor(x)
temp3=temp2<1e-15 ; temp4<-x==0; temp1[temp4]<-0
return(temp1*temp3)
}
##p is the probability of the treatment facility will be fixed
## before the holding tank becomes full
p = 0.95
N = 2^17
support = 0:(N-1)
## The DFT samples of the two of the transition distributions
T21 = dweibulldisc(support,0.3,0.5)
T23 = dweibulldisc(support,0.5,0.7)
## The Fourier transforms of the 3 transition distributions
F12 = p*exp(-2*pi*1i*support/N)/(1-(1-p)*exp(-2*pi*1i*support/N))
F21 = fft(T21); F23 = fft(T23)
## The Fourier transform of the first passage distribution from
## state 1 to state 3
F_1st_Pas_13 = ((1-p)*F12*F23)/(1-p*F12*F21)
## The first passage PMF from state 1 to state 3
T_1st_Pas_13 = 1/N*Re( fft(F_1st_Pas_13,inverse=T) )
## References
* (1)
* Barbu and Liminios (2008) Barbu, V. S. and Liminios, N. (2008). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications: Their Use in Reliability and DNA Analysis, Springer, New York.
* Briggs and Henson (1995) Briggs, W. L. and Henson, V. E. (1995). The DFT: An Owner’s Manual for the Discrete Fourier Transform, SIAM, Philadelphia.
* Hughett (1998) Hughett, P. (1998). Error bounds for numerical inversion of a probability characteristic function, SIAM Journal on Numerical Analysis 35(4): 1368–1392.
* Huzurbazar (2005) Huzurbazar, A. V. (2005). Flowgraph Models for Multistate Time-to-Event Data, Wiley, New York.
* Nakagawa and Osaki (1975) Nakagawa, T. and Osaki, S. (1975). The discrete Weibull distribution, IEEE Transactions on Reliability R-24(5): 300–301.
* Pyke (1961) Pyke, R. (1961). Markov renewal processes with finitely many states, The Annals of Mathematical Statistics 32(4): 1243–1259.
* R Development Core Team (2009) R Development Core Team (2009). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.
http://www.R-project.org
* Warr and Collins (2011) Warr, R. L. and Collins, D. H. (2011). A comprehensive method for solving finite-state semi-Markov processes, Technical Report LA-UR-11-01596, Los Alamos National Laboratory, Los Alamos, NM.
* Yao (1985) Yao, D. D. (1985). First-passage-time moments of Markov processes, Journal of Applied Probability 22(4): 939–945.
* Zhang and Hou (2012) Zhang, X. and Hou, Z. (2012). The first-passage times of phase semi-Markov processes, Statistics & Probability Letters 82(1): 40 – 48.
|
arxiv-papers
| 2012-12-28T18:01:26 |
2024-09-04T02:49:39.746915
|
{
"license": "Public Domain",
"authors": "Richard L. Warr",
"submitter": "Richard Warr",
"url": "https://arxiv.org/abs/1212.6546"
}
|
1212.6560
|
# Spinor Frenet equations in three dimensional Lie groups
O. Zeki Okuyucu Department of Mathematics, Bilecik Şeyh Edebali University,
Turkey [email protected] , Ö. Gökmen Yıldız Department of
Mathematics, Bilecik Şeyh Edebali University, Turkey
[email protected] and Murat Tosun Department of Mathematics,
Sakarya University, Turkey [email protected]
###### Abstract.
In this paper, we study spinor Frenet equations in three dimensional Lie
Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for
special cases of three dimensional Lie groups.
###### Key words and phrases:
Spinor, Frenet equations, Lie groups.
###### 2010 Mathematics Subject Classification:
Primary 11A66; Secondary 22E15.
## 1\. Introduction
In differential geometry, we think that curves are geometric set of points of
loci. Curves theory is a important workframe in the differential geometry
studies. Geometric properties of a curve are heavily studied for a long time
and are still studied in Euclidean space and other spaces. One of the most
important tools used to analyze a curve is the Frenet frame, a moving frame
that provides an orthonormal coordinate system at each point of the curve. We
can show that this orthonormal system with $\left\\{T,N,B\right\\}$ in
Euclidean $3$-space, which are called tangent vector field, principal normal
vector field and binormal vector field, respectively. And for a curve, we can
calculate curvature and torsion along the curve with the help of Frenet
vectors. They are so useful for characterizations of special curves like that
general helices, Slant helices, Mannheim curves, Bertrand curves etc.
Characterizations of these curves are studied in different spaces and we can
see the applications of special curves in nature, mechanic tools, computer
aided design and computer graphics etc.
Recently, Castillo and Barrales studied spinor formulation of the Frenet
equations of the curve and they obtained spinor equivalent of the Frenet
equations for a curve [5]. Moreover Kişi et.al. and Ünal et. al. defined
spinor formulations of the Frenet equations of a curve according to Darboux
Frame and Bishop Frame, respectively (see [6] and [7]).
In this paper, we define spinor formulation of the Frenet equations of the
curve in three dimensional Lie Groups. Also we give this formulation for
special cases of three dimensional Lie groups. We believe that this study will
be useful for curves theory in Lie groups.
## 2\. Preliminaries
Let $G$ be a Lie group with a bi-invariant metric $\left\langle\text{
},\right\rangle$ and $D$ be the Levi-Civita connection of Lie group $G.$ If
$\mathfrak{g}$ denotes the Lie algebra of $G$ then we know that $\mathfrak{g}$
is issomorphic to $T_{e}G$ where $e$ is neutral element of $G.$ If
$\left\langle\text{ },\right\rangle$ is a bi-invariant metric on $G$ then we
have
$\left\langle
X,\left[Y,Z\right]\right\rangle=\left\langle\left[X,Y\right],Z\right\rangle$
(2.1)
and
$D_{X}Y=\frac{1}{2}\left[X,Y\right]$ (2.2)
for all $X,Y$ and $Z\in\mathfrak{g}.$
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted curve and
$\left\\{X_{1},X_{2,}...,X_{n}\right\\}$ be an orthonormal basis of
$\mathfrak{g}.$ In this case, we write that any two vector fields $W$ and $Z$
along the curve $\alpha\ $as $W=\sum_{i=1}^{n}w_{i}X_{i}$ and
$Z=\sum_{i=1}^{n}z_{i}X_{i}$ where $w_{i}:I\rightarrow\mathbb{R}$ and
$z_{i}:I\rightarrow\mathbb{R}$ are smooth functions. Also the Lie bracket of
two vector fields $W$ and $Z$ is given
$\left[W,Z\right]=\sum_{i=1}^{n}w_{i}z_{i}\left[X_{i},X_{j}\right]$
and the covariant derivative of $W$ along the curve $\alpha$ with the notation
$D_{\alpha^{\shortmid}}W$ is given as follows
$D_{\alpha^{\shortmid}}W=\overset{\cdot}{W}+\frac{1}{2}\left[T,W\right]$ (2.3)
where $T=\alpha^{\prime}$ and
$\overset{\cdot}{W}=\sum_{i=1}^{n}\overset{\cdot}{w_{i}}X_{i}$ or
$\overset{\cdot}{W}=\sum_{i=1}^{n}\frac{dw}{dt}X_{i}.$ Note that if $W$ is the
left-invariant vector field to the curve $\alpha$ then $\overset{\cdot}{W}=0$
(see [3] for details).
Let $G$ be a three dimensional Lie group and
$\left(T,N,B,\varkappa,\tau\right)$ denote the Frenet apparatus of the curve
$\alpha$, and calculate $\varkappa=\overset{\cdot}{\left\|T\right\|}.$
###### Definition 2.1.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$ then
$\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ (2.4)
or
$\tau_{G}=\frac{1}{2\varkappa^{2}\tau}\overset{\cdot\cdot\text{ \ \ \ \ \ \ \
\ }\cdot}{\left\langle
T,\left[T,T\right]\right\rangle}+\frac{1}{4\varkappa^{2}\tau}\overset{\text{ \
\ }\cdot}{\left\|\left[T,T\right]\right\|^{2}}$
(see [2]).
###### Proposition 2.2.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B\right\\}$. Then the following
equalities
$\displaystyle\left[T,N\right]$
$\displaystyle=\left\langle\left[T,N\right],B\right\rangle B=2\tau_{G}B$
$\displaystyle\left[T,B\right]$
$\displaystyle=\left\langle\left[T,B\right],N\right\rangle N=-2\tau_{G}N$
hold (see [8]).
In the following sentences we give some properties about spinors:
We can represent a spinor
$\phi=\left(\begin{array}[c]{c}\phi_{1}\\\ \phi_{2}\end{array}\right)$
by way of three vector $a,b,c\in\mathbb{R}^{3}$ so that
$a+ib=\phi^{t}\sigma\phi,\text{ \ \ }c=-\widehat{\phi^{t}}\sigma\phi,$ (2.5)
where the superscript $t$ denotes transposition and $\widehat{\phi}$ is the
mate ( or conjugate [1])
$\widehat{\phi}\equiv-\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\text{,
}\overline{\phi}=\left(\begin{array}[c]{c}-\overline{\phi}_{2}\\\
\overline{\phi}_{1}\end{array}\right)$
where the bar indicates complex conjugation. In addition,
$\sigma=\left(\sigma_{1},\sigma_{2},\sigma_{3}\right)$ is a vector whose
cartesian components are the complex symmetrics $2\times 2$ matrices
$\sigma_{1}=\left(\begin{array}[c]{cc}1&0\\\ 0&-1\end{array}\right),\text{
}\sigma_{2}=\left(\begin{array}[c]{cc}i&0\\\ 0&i\end{array}\right),\text{
}\sigma_{3}=\left(\begin{array}[c]{cc}0&-1\\\ -1&0\end{array}\right).$ (2.6)
Then, the vector $a,b$ and $c$ are clearly given by
$\displaystyle a+ib$
$\displaystyle=\left(\phi_{1}^{2}-\phi_{2}^{2},i(\phi_{1}^{2}+\phi_{2}^{2}),-2\phi_{1}\phi_{2}\right)$
$\displaystyle c$
$\displaystyle=\left(\phi_{1}\overline{\phi}_{2}+\overline{\phi}_{1}\phi_{2},i\left(\overline{\phi}_{2}\phi_{1}-\overline{\phi}_{1}\phi_{2}\right),\left|\phi_{1}\right|^{2}-\left|\phi_{2}\right|^{2}\right)$
Since the vector $a+ib$ is a isotropic vector, by way of an explict
computation one finds that $a,b$ and $c$ are one another orthogonal and
$\left|a\right|=\left|b\right|=\left|c\right|=\overline{\phi}^{t}\phi$.
Moreover $\left\langle a\times b,c\right\rangle=\det(a,b,c)>0.$
Conversely, if the vector $a,b$ and $c$ one another orthogonal vectors of same
magnitude $\left(\left\langle a\times b,c\right\rangle>0\right),$ then there
is a spinor defined up to sign such that the Eq. (2.5) holds.
As it is mentioned the above, for two arbitrary $\psi$ and $\phi$, there exist
following equalities
$\displaystyle\overline{\psi^{t}\sigma\phi}$
$\displaystyle=-\widehat{\psi^{t}}\sigma\widehat{\phi}$
$\displaystyle\widehat{a\psi+b\phi}$
$\displaystyle=\overline{a}\widehat{\psi}+\overline{b}\widehat{\phi}$
$\displaystyle\widehat{\widehat{\phi}}$ $\displaystyle=-\phi$
where $a$ and $b$ are complex numbers [1] and [5].
The relations between spinors and orthonormal bases given in Eq. (2.5) is two
to on; the spinors $\phi$ and $-\phi$ correspond to the same ordered
orthonormal bases $\left\\{a,b,c\right\\}$, with
$\left|a\right|=\left|b\right|=\left|c\right|$ and $\left\langle a\times
b,c\right\rangle>0.$ Moreover we can define different spinors by using the
ordered triads $\left\\{a,b,c\right\\},\left\\{b,c,a\right\\}$ and
$\left\\{c,a,b\right\\}.$
The equation $\psi^{t}\sigma\phi=\phi^{t}\sigma\psi$ is satisfied for any pair
of $\psi$ and $\phi$, because the matrices $\sigma$ given by Eq. (2.6) are
symmetric. For any spinor $\phi$ and its conjugate $\widehat{\phi}$ are
linearly independent, where $\phi\neq 0.$
## 3\. Spinor Frenet equations in a three dimensional Lie groups
In this section we define the spinor Frenet equations for a curve in a three
dimensional Lie group $G$ with a bi-invariant metric $\left\langle\text{
},\right\rangle$. Also we give the spinor Frenet equations in the special
cases of $G$.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a curve with arc-length
parameter $s$. Then we can wirte with the help of the Eq. (2.3) and
Proposition 2.2 the following equations for Frenet vectors of the curve
$\alpha$
$\begin{bmatrix}\frac{dT}{ds}\\\ \frac{dN}{ds}\\\
\frac{dB}{ds}\end{bmatrix}=\begin{bmatrix}0&\varkappa&0\\\
-\varkappa&0&\left(\tau-\tau_{G}\right)\\\
0&-\left(\tau-\tau_{G}\right)&0\end{bmatrix}\begin{bmatrix}T\\\ N\\\
B\end{bmatrix}$ (3.1)
where $\left\\{T,N,B\right\\}$ is Frenet frame,
$\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ and
$\varkappa,$ $\tau$ are curvatures of the curve $\alpha$ in three dimensional
Lie group $G,$ respectively.
According to the results presented in section $2$, there exists a spinor
$\phi,$ such that
$N+iB=\phi^{t}\sigma\phi,\text{ \ \ \ \ \ \ \ \ \
}T=\widehat{\phi^{t}}\sigma\phi$ (3.2)
with $\overline{\phi^{t}}\phi=1$. Therefore the spinor $\phi$ denotes the
triad $\left\\{N,B,T\right\\}.$ And the Frenet equations for each point of the
curve $\alpha$ must correspond to some expression for $\frac{d\phi}{ds}.$
As $\left\\{\phi,\widehat{\phi}\right\\}$ is a basis for the two component
spinors $\left(\phi\neq 0\right)$, there are two functions $g$ and $h$, such
that
$\frac{d\phi}{ds}=g\phi+h\widehat{\phi},$ (3.3)
where the functions $g$ and $h$ are possibly complex-valued functions.
Diferentiating the first equation in the Eq. (3.2) with respect to $s$ and by
using the Frenet equations, we get
$\displaystyle\frac{dN}{ds}+i\frac{dB}{ds}$
$\displaystyle=\frac{d}{ds}\left(\phi^{t}\sigma\phi\right)$
$\displaystyle-\varkappa T-i\left(\tau-\tau_{G}\right)\left\\{N+iB\right\\}$
$\displaystyle=\left(\frac{d\phi}{ds}\right)^{t}\sigma\phi+\phi^{t}\sigma\left(\frac{d\phi}{ds}\right).$
(3.4)
If we think together the Eq. (3.3) and Eq. (3.4), using again the Eq. (3.2),
we get
$-\varkappa
T-i\left(\tau-\tau_{G}\right)\left\\{N+iB\right\\}=2g(N+iB)-2h(T).$
From the last equation we can write,
$g=-i\frac{\left(\tau-\tau_{G}\right)}{2}\text{ and }h=\frac{\varkappa}{2}.$
Thus, we have proved the following theorem.
###### Theorem 3.1.
Let $\alpha$ be an arc-lenghted regular curve with the Frenet vector fields
$\left\\{T,N,B\right\\}$ in the Lie group $G$. If its two-component spinors
$\phi$ represents the triad $\left\\{N,B,T\right\\}$, then the Frenet
equations are equivalent to the single spinor equation
$\frac{d\phi}{ds}=-i\frac{\left(\tau-\tau_{G}\right)}{2}\phi+\frac{\varkappa}{2}\widehat{\phi},$
where $\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ and
$\varkappa$, $\tau$ are curvatures of the curve $\alpha.$
Now, we introduce this equation for special cases of three dimensional Lie
group. In the following remark, we note that three dimensional Lie groups
admitting bi-invariant metrics are $S^{3},$ $SO^{3}$ and Abelian Lie groups
using the same notation as in [2] and [4] as follows:
###### Remark 3.2.
Let $G$ be a Lie group with a bi-invariant metric
$\left\langle,\right\rangle$. Then the following equalities can be given in
different Lie groups:
i) If $G$ is abelian group then $\tau_{G}=0.$
ii) If $G$ is $SO^{3}$ then $\tau_{G}=\frac{1}{2}.$
iii) If $G$ is $S^{3}$ then $\tau_{G}=1$
(see for details [2] and [4]).
###### Corollary 3.3.
Let $\alpha$ be an arc-lenghted regular curve with the Frenet vector fields
$\left\\{T,N,B\right\\}$ in the Abelian Lie group $G$. If its two-component
spinors $\phi$ represents the triad $\left\\{N,B,T\right\\}$, then the Frenet
equations are equivalent to the single spinor equation
$\frac{d\phi}{ds}=-i\frac{\tau}{2}\phi+\frac{\varkappa}{2}\widehat{\phi},$
where $\varkappa$ and $\tau$ are curvatures of the curve $\alpha.$
###### Proof.
If $G$ is Abellian Lie group then using the above Remark and the Theorem 3.1
we have the result. ∎
From the above corollary we can see easily this study is a generalization of
the spinor equivalent of the Frenet equations for a curve defined by Del
Castillo and Barrales [5] in Euclidean 3-space. Moreover, with a similar
proof, we have the following two corollaries.
###### Corollary 3.4.
Let $\alpha$ be an arc-lenghted regular curve with the Frenet vector fields
$\left\\{T,N,B\right\\}$ in the Lie group $SO^{3}$. If its two-component
spinors $\phi$ represents the triad $\left\\{N,B,T\right\\}$, then the Frenet
equations are equivalent to the single spinor equation
$\frac{d\phi}{ds}=-i\frac{\left(\tau-\frac{1}{2}\right)}{2}\phi+\frac{\varkappa}{2}\widehat{\phi},$
where $\varkappa$ and $\tau$ are curvatures of the curve $\alpha.$
###### Corollary 3.5.
Let $\alpha$ be an arc-lenghted regular curve with the Frenet vector fields
$\left\\{T,N,B\right\\}$ in the Lie group $S^{3}$. If its two-component
spinors $\phi$ represents the triad $\left\\{N,B,T\right\\}$, then the Frenet
equations are equivalent to the single spinor equation
$\frac{d\phi}{ds}=-i\frac{\left(\tau-1\right)}{2}\phi+\frac{\varkappa}{2}\widehat{\phi},$
where $\varkappa$ and $\tau$ are curvatures of the curve $\alpha.$
## References
* [1] E. Cartan, The theory of spinors, Hermann, Paris, 1966. (Dover, New York, reprinted 1981).
* [2] Ü. Çiftçi, A generalization of Lancert’s theorem, J. Geom. Phys. 59 (2009) 1597-1603.
* [3] P. Crouch, F. Silva Leite, The dynamic interpolation problem: on Riemannian manifoldsi Lie groups and symmetric spaces, J. Dyn. Control Syst. 1 (2) (1995) 177-202.
* [4] N. do Espírito-Santo, S. Fornari, K. Frensel, J. Ripoll, Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math. 111 (4) (2003) 459 470.
* [5] T. del Castillo, G. F., Barrales, G.S., Spinor formulation of the differential geometry of curve, Revista Colombiana de Matematicas, Vol 38, (2004), 27-34.
* [6] İ., Kişi and M., Tosun, Spinor Darboux Equations of Curves in Euclidean 3-Space, arXiv:1210.4185v1 [math.GM].
* [7] D., Ünal, İ., Kişi and M., Tosun. Spinor Bishop Equations of Curves in Euclidean 3-Space, arXiv:1210.0727v1 [math.DG].
* [8] O., Z., Okuyucu, I., Gok, Y., Yayl , and N., Ekmekci, Slant Helices in three Dimensional Lie Groups, arXiv:1203.1146v2 [math.DG].
|
arxiv-papers
| 2012-12-28T20:24:29 |
2024-09-04T02:49:39.753990
|
{
"license": "Public Domain",
"authors": "O. Zeki Okuyucu, \\\"O. G\\\"okmen Y{\\i}ld{\\i}z, Murat Tosun",
"submitter": "Osman Zeki Okuyucu",
"url": "https://arxiv.org/abs/1212.6560"
}
|
1212.6632
|
11institutetext: The Blavatnik School of Computer Science, Tel Aviv
University, Israel
# The Complexity of
Infinitely Repeated Alternating Move Games
Yaron Velner
###### Abstract
We consider infinite duration alternating move games. These games were
previously studied by Roth, Balcan, Kalai and Mansour [10]. They presented an
FPTAS for computing an approximate equilibrium, and conjectured that there is
a polynomial algorithm for finding an exact equilibrium [9]. We extend their
study in two directions: (1) We show that finding an exact equilibrium, even
for two-player zero-sum games, is polynomial time equivalent to finding a
winning strategy for a (two-player) mean-payoff game on graphs. The existence
of a polynomial algorithm for the latter is a long standing open question in
computer science. Our hardness result for two-player games suggests that two-
player alternating move games are harder to solve than two-player simultaneous
move games, while the work of Roth et al., suggests that for $k\geq 3$,
$k$-player games are easier to analyze in the alternating move setting. (2) We
show that optimal equilibria (with respect to the social welfare metric) can
be obtained by pure strategies, and we present an FPTAS for computing a pure
approximated equilibrium that is $\delta$-optimal with respect to the social
welfare metric. This result extends the previous work by presenting an FPTAS
that finds a much more desirable approximated equilibrium. We also show that
if there is a polynomial algorithm for mean-payoff games on graphs, then there
is a polynomial algorithm that computes an optimal exact equilibrium, and
hence, (two-player) mean-payoff games on graphs are inter-reducible with
$k$-player alternating move games, for any $k\geq 2$.
## 1 Introduction
In this work, we investigate infinitely repeated games in which players
_alternate_ making moves. This framework can model, for example, five
telecommunication providers competing for customers: each company can observe
the price that is set by the others, and it can update the price at any time.
In the short term, each company can benefit from undercutting its opponents
price, but since the game is repeated indefinitely, in some settings, it might
be better to coordinate prices with the other companies. Such examples
motivate us to study equilibria in alternating move games.
In this work, we study infinitely repeated $k$-player $n$-action games. In
such games, in every _round_ , a player chooses an action, and the utility of
each player (for the current round) is determined according to the $k$-tuple
of actions of the players. Each player goal is to maximize his own _long-run
average_ utility as the number of rounds tends to infinity.
These games were studied by Roth et al. in [10], and they showed an FPTAS for
computing an $\epsilon$-equilibrium. Their result provided a theoretical
separation between the alternating move model and the simultaneous move model,
since for the latter, it is known that an FPTAS for computing approximate
equilibria does not exists for games with $k\geq 3$ players unless P=PPAD.
Their result was obtained by a simple reduction to mean-payoff games on
graphs. These games were presented in [5], and they play an important rule in
automata theory and in economics. The computational complexity of finding an
exact equilibrium for such games is a long standing open problem, and despite
many efforts [6, 12, 1, 7, 2, 3], there is no known polynomial solution for
this problem.
We extend the work in [10] by investigating the complexity of an exact
equilibrium (which was stated as an open question in [10]), and by
investigating the computational complexity of finding an _$\delta$ -optimal_
approximated equilibrium with respect to the _social welfare_ metric. Our main
technical results are as follows:
* •
We show a reduction from mean-payoff games on graphs to two-player zero-sum
alternating move games, and thus we prove that $k$-player alternating move
games are computationally equivalent to mean-payoff games on graphs for any
$k\geq 2$.
* •
We show that optimal equilibrium can be obtained by pure strategies, and we
show an FPTAS for computing an $\delta$-optimal $\epsilon$-equilibrium. In
addition, we show that computing an exact optimal equilibrium is polynomial
time equivalent to solving mean-payoff games on graphs.
We note that the first result may suggest that two-player alternating move
games are harder than two-player simultaneous move games, since a polynomial
time algorithm to solve the latter is known [8]. Hence, along with the result
of [10], we get that simultaneous move games are easier to solve with
comparison to alternating move games for two-player games, and are harder to
solve for $k\geq 3$ player games.
This paper is organized as following. In the next section we bring formal
definitions for alternating move games and mean-payoff games on graphs. In
Section 3, we show that alternating move games are at least as hard as mean-
payoff games on graphs. In Section 4 we investigate the properties of optimal
equilibria, and we present an FPTAS for computing an $\delta$-optimal
$\epsilon$-equilibrium. Due to lack of space, some of the proofs were omitted,
and the full proofs are given in the appendix.
## 2 Definitions
In this section we bring the formal definitions for alternating move repeated
games and mean-payoff games on graphs. Alternating move games are presented in
Subsection 2.1, and mean-payoff games are presented in Subsection 2.2.
### 2.1 Alternating move repeated games
#### Actions, plays and utility function.
A $k$-player $n$-action game is defined by an action set $A_{i}$ for every
player $i$, and by $k$ utility functions, one for each player,
$u_{i}:A_{1}\times\dots A_{k}\to[-1,1]$. W.l.o.g we assume that the size of
all action set is the same, and we denote it by $n$. We note that any game can
be rescaled so its utilities are bounded in $[-1,1]$, however, the FPTAS that
was presented in [10], and our results in Subsection 4.4 crucially rely on the
assumption that the utilities are in the interval $[-1,1]$.
An alternating move game is played for infinitely many rounds. In round $t$
player $j=1+(t\mod k)$ plays action $a_{j}^{t}$, and a _vector of actions_
$a^{t}=(a_{1},\dots,a_{k})$ is produced, where $a_{i}\in A_{i}$ is the last
action of player $i$. In every round $t$, player $i$ receives a utility
$u_{i}(a^{t})$, which depends only in the last action of each of the $k$
players (W.l.o.g the utility in the first $k$ rounds is zero for all players).
A sequence of infinite rounds forms a play, and we characterize a play either
by an infinite sequence of actions or by the corresponding sequence of vectors
of actions. The utility of player $i$ in a play $a^{1}a^{2}\dots
a^{t}a^{t+1}\dots$ is the _limit average payoff_ , namely,
$\lim_{n\to\infty}\frac{1}{n}\sum_{t=1}^{n}u_{i}(a^{t})$. When this limit does
not exist, we define the utility of the play for player $i$ to be
$\liminf_{n\to\infty}\frac{1}{n}\sum_{t=1}^{n}u_{i}(a^{t})$. We note that in
the frame work of [10], the utility of a play was undefined when the limit
does not exist. The results we present in this paper for the $\liminf$ metric
holds also for the framework of [10]. On the other hand, if we would take the
$\limsup$ value instead, then the problem is much easier. An optimal
equilibrium is obtained when all players join forces and maximize player
$1+(i\pmod{k})$ utility for $2^{i}$ rounds (for $i=1,2,\dots,\infty$). Hence,
we can easily produce a polynomial algorithm to solve these games.
#### Strategies.
A _strategy_ is a recipe for player’s next action, based on the entire
_history_ of previous actions. Formally, a (mixed) strategy for player $i$ is
a function $\sigma_{i}:(A_{1}\times\dots\times A_{k})^{*}\times
A_{1}\times\dots\times A_{i-1}\to\Delta(A_{i})$, where $\Delta(S)$ denotes the
set probability distribution over any finite set $S$. We say that $\sigma_{i}$
is a _pure strategy_ if $\Delta$ is a degenerated distribution. A _strategy
profile_ is a vector $\sigma=(\sigma_{1},\dots,\sigma_{k})$ that defines a
strategy for every player. A profile of pure strategies uniquely determines
the action vector in every round and yields a utility vector for the players.
A profile of mixed strategies determines, for every round $t$ in the play, a
distribution of sequences of action vectors, and the _average payoff_ in round
$t$ is the expected average payoff over the distribution of action vectors.
Formally, for a strategy profile $\sigma$ we denote the average payoff of
player $i$ in round $t$ by
$P_{i,t}(\sigma)=E_{a^{1}a^{2}\dots
a^{t}\sim\sigma}[\frac{u_{i}(a^{1})+\dots+u_{i}(a^{t})}{t}]$
and the utility of player $i$ is $\liminf_{t\to\infty}P_{i,t}$.
#### Equilibria, $\epsilon$ equilibria and optimal equilibria
A strategy profile forms an _equilibrium_ if none of the players can strictly
improve his utility (that is induced by the profile) by unilaterally deviating
from his strategy (that is defined by the profile). For every $\epsilon>0$, we
say that a strategy profile forms an _$\epsilon$ -equilibrium_ if none of the
players can improve his utility by more than $\epsilon$ by unilaterally
deviating from his strategy.
The _social welfare_ of a strategy profile is the sum of the utilities of all
players. An equilibrium (resp. an $\epsilon$-equilibrium) is called an
_optimal equilibrium_ (optimal $\epsilon$-equilibrium) if its social welfare
is not smaller than the social welfare of any other equilibrium
($\epsilon$-equilibrium). For $\delta>0$, an equilibrium (resp. an
$\epsilon$-equilibrium) is called an _$\delta$ -optimal equilibrium_ if its
social welfare is not smaller by more than $\delta$ with comparison to the
social welfare of any other equilibrium ($\epsilon$-equilibrium).
### 2.2 Mean-payoff games on graphs
#### Plays and payoffs.
A _mean-payoff game_ on a graph is defined by a weighted directed bipartite
graph $G=(V=V_{1}\cup V_{2},E,w:E\to\mathbb{Q})$ and an initial vertex
$v_{0}\in V$. The game consists of two players, namely, maximizer (who owns
$V_{1}$) and minimizer (who owns $V_{2}$). Initially, a pebble is place on the
initial vertex, and in every round, the player who owns the vertex in which
the pebble resides, advance the pebble into an adjacent vertex. This process
is repeated forever and forms a _play_. A play is characterized by a sequence
of edges, and the _average payoff_ of a play $\rho=e_{1}\dots e_{t}$ up to
round $t$ is denoted by $P_{t}=\frac{1}{t}\sum_{i=1}w(e_{i})$. The _value_ of
a play is the limit average payoff (mean-payoff), namely,
$\liminf_{t\to\infty}P_{t}$. (We note that for games on graphs, the $\limsup$
metric gives the same complexity results.) The objective of the maximizer is
to maximize the mean-payoff of a play, and the minimizer aims to minimize the
mean-payoff.
#### Strategies, memoryless strategies, optimal strategies and winning
strategies.
In this work, we consider only pure strategies for games on graphs, and it is
well-known that randomization does not give better strategies for mean-payoff
games. A _strategy_ for maximizer is a function $\sigma:(V_{1}\times
V_{2})^{*}\times V_{1}\to E$ that decides the next move, and similarly, for
the minimizer a strategy is a function $\tau:(V_{1}\times V_{2})^{*}\to E$. A
strategy is called _memoryless_ if it depends only on the current position of
the pebble. Formally, a memoryless strategy for the maximizer is a function
$\sigma:V_{1}\to E$ and similarly a memoryless strategy for the minimizer is a
function $\tau:V_{2}\to E$.
A profile of strategies $(\sigma,\tau)$ uniquely determines the mean-payoff
value of a game. We say that a play $\pi=e_{1}e_{2}\dots e_{n}\dots$ is
_consistent_ with a maximizer strategy $\sigma$ if there exists a minimizer
strategy $\tau$ such that $\pi$ is formed by $(\sigma,\tau)$. We say that the
_value_ of a maximizer strategy is $p$ if it can assure a value of at least
$p$ against any minimizer strategy. Analogously, we say that the value of a
minimizer strategy is $p$ if it can assure a value of at most $p$ against any
maximizer strategy.
We say that a maximizer strategy is _optimal_ if its value is maximal (with
respect to all possible maximizer strategies). Analogously, a minimizer
strategy is optimal if its value is minimal. For a given threshold, we say
that a maximizer strategy is a _winning strategy_ if it assures mean-payoff
value that is greater or equal to the given threshold, and a minimizer
strategy is winning if it assures value that is strictly smaller than the
given threshold.
#### One-player games, and games according to memoryless strategies
A special (and easier) case of games on graphs is when the out-degree is one
for all the vertices that are owned by a certain player. In this case, all the
_choices_ are done by one player. For a two-player game on graph $G$, and a
player-1 strategy $\sigma$, we define the one-player game graph $G^{\sigma}$
to be the game graph that is formed by removing, for every player-1 vertex
$v$, the out-edges that are not equal to $\sigma(v)$.
#### Classical results on mean-payoff games.
Mean-payoff games were introduced in ’79 by Ehrenfeucht and Mycielski [5], and
their main result was that optimal strategies (for both players) exist, and
moreover, the optimal value can be obtained by a memoryless strategy. The
decision problem for mean-payoff games is to determine if the maximizer has a
winning strategy with respect to a given threshold. The existence of optimal
memoryless strategies almost immediately proves that the decision problem for
mean-payoff games is in NP$\cap$coNP, and thus it is unlikely to be NP-hard
(or coNP-hard). Zwick and Paterson [12] introduced the first pseudo-polynomial
algorithm, which runs in polynomial time when the weights of the edges are
encoded in unary. They also provided a polynomial algorithm for the special
case of one-player mean-payoff games. A randomized sub-exponential algorithm
for mean-payoff games is also known [2], but despite many efforts, the
existence of a polynomial algorithm to solve mean-payoff games remains an open
question, and it is one of the rare problems in computer science that is known
to be in NP$\cap$coNP but no polynomial algorithm is known.
We summarize the known results on mean-payoff games in the next theorems. The
first theorem states that optimal strategies exist and moreover, there exist
optimal strategies that are memoryless.
###### Theorem 1 ([5])
For every mean-payoff game there exists a maximizer memoryless strategy
$\sigma$ and a minimizer memoryless strategy $\tau$ such that $\sigma$ is
optimal for the maximizer and $\tau$ is optimal for the minimizer.
The next theorem shows that there is a polynomial algorithm that computes
optimal strategies if and only if there is a polynomial algorithm for the
mean-payoff games decision problem.
###### Theorem 2 ([12])
The following problems are polynomial time inter-reducible: (i) Compute
maximizer optimal memoryless strategy. (ii) Compute the optimal value that
maximizer can assure. (iii) Determine whether maximizer optimal value is at
least zero. (iv) Determine whether maximizer optimal value is greater than
zero.
## 3 Two-Player Zero-Sum (Alternating Move) Games are Inter-Reducible with
Mean-Payoff Games
In this section we prove that there is a polynomial algorithm that computes an
exact equilibrium for two-player zero-sum (alternating move) games if and only
if there exists a polynomial algorithm that solves mean-payoff games.
The reduction from two-player zero-sum games to mean-payoff games is trivial,
for a two-player zero-sum game with actions $A_{1}$, $A_{2}$ and utility
functions $u_{1}:A_{1}\times A_{2}\to\mathbb{Q}$ and $u_{2}=-u_{1}$, we
construct a complete bipartite game graph $G=(V=A_{1}\cup A_{2},E=(A_{1}\times
A_{2})\cup(A_{2}\times A_{1}),w:E\to\mathbb{Q})$ such that the weight of the
transition from $a_{1}\in A_{1}$ to $a_{2}\in A_{2}$ is simply
$u_{1}(a_{1},a_{2})$, and the weight of the transition from $a_{2}$ to $a_{1}$
is also $u_{1}(a_{1},a_{2})$. It is a simple observation that a pair of
optimal strategies (for the maximizer and minimizer) in the mean-payoff game
induces an equilibrium strategy profile in the two-player zero-sum game and
vice versa.
The reduction for the converse direction is more complicated. For this purpose
we bring the notion of _undirected game graph_. A mean-payoff game graph is
said to be _undirected_ if its edge relation is symmetric, and
$w(v_{1},v_{2})=w(v_{2},v_{1})$ for every edge $(v_{1},v_{2})$. (Basically, it
is a game on an undirected graph.) The next simple lemma shows a reduction
from mean-payoff games on a complete bipartite undirected graphs to two-player
zero-sum game.
###### Lemma 1
There is a polynomial reduction from mean-payoff games on complete bipartite
undirected graphs to two-player zero-sum games.
###### Proof
The proof is straight forward. Let $V_{1}$ and $V_{2}$ be the maximizer and
minimizer (resp.) vertices in the mean-payoff game. We construct a two-player
zero-sum game in the following way. The set of action of player 1 is
$A_{1}=V_{2}$ and the set of action for player 2 is $A_{2}=V_{1}$. We denote
by $W$ the least value for which all the weights in the undirected graphs are
in $[-W,+W]$, and the utility function of player 1 is
$u_{1}(a_{1},a_{2})=\frac{w(a_{1},a_{2})}{W}=\frac{w(a_{2},a_{1})}{W}$, and
$u_{2}=-u_{1}$. It is trivial to observe that an equilibrium profile induces a
pair of optimal strategies for the mean-payoff game, and the proof follows. ∎
Due to Lemma 1, all that is left is to prove that mean-payoff games on
complete bipartite undirected graphs are equivalent to mean-payoff games. A
recent result by Chatterjee, Henzinger, Krinninger and Nanongkai [4] gives us
the first step towards such proof.
###### Theorem 3 (Corollary 24 in [4])
Solving mean-payoff games on complete bipartite (directed) graphs is as hard
as solving mean-payoff games on arbitrary graphs.
We use the above result as a black box and extend it to complete bipartite
undirected graphs. We note that the main difference between directed and
undirected graphs is that for undirected graphs the weight function is
symmetric. In the rest of this section we will describe a process that for a
given complete bipartite directed graph, generates a suitable symmetric weight
function, and the winner in the generated graph is the same as in the original
graph.
We say that a directed game graph has a _normalized weight function_ if it
assigns a positive weight to every out-edge of maximizer vertex, and a
negative weight for every out-edge of minimizer vertex. The next lemma shows
that we may assume w.l.o.g that a directed game graph has a normalized weight
function.
###### Lemma 2
Solving mean-payoff games on (directed) bipartite graphs is polynomial time
inter-reducible to solving mean-payoff games on (directed) bipartite graphs
with normalized weights.
###### Proof
Let $G$ be a non-normalized graph and let us denote by $W$ the heaviest weight
(in absolute value) that is assigned by its weight function $w$. We construct
a normalized graph $G^{\prime}$ from $G$ by defining a weight function
$w^{\prime}$ as:
$w^{\prime}(u,v)=\left\\{\begin{array}[]{rl}w(u,v)+(W+1)&\mbox{ if $u$ is
owned by maximizer}\\\ w(u,v)-(W+1)&\mbox{ otherwise (if $u$ is owned by the
minimizer)}\end{array}\right.$
Clearly, $G^{\prime}$ is a normalized graph, and since $G^{\prime}$ and $G$
are bipartite, it is straight forward to observe that for any finite path in
$\pi$ we have that $|w(\pi)-w^{\prime}(\pi)|\leq W+1$, and thus, for every
infinite path $\rho$, we have that that mean-payoff value of $\rho$ according
to $w$ is identical to the mean-payoff of $\rho$ according to $w^{\prime}$.
Therefore, a maximizer winning (resp. optimal) strategy in graph $G$ is a
winning (optimal) strategy also in $G^{\prime}$ and vice versa, and the proof
of the lemma follows. ∎
In the next lemma we show that mean-payoff games on direct normalized
bipartite complete graphs are as hard as mean-payoff games on undirected
normalized bipartite complete graphs.
###### Lemma 3
The problem of determining whether maximizer has a winning strategy for a
threshold $0$ for a mean-payoff games on a directed normalized bipartite
complete graph is as hard as the corresponding problem for mean-payoff games
on an undirected normalized bipartite complete graph
To conclude, by Lemma 1 we get that a polynomial algorithm for alternating
move two-player zero-sum games exists if and only if there exists a polynomial
algorithm for solving mean-payoff games on a complete bipartite undirected
graph, and by Lemmas 2, 1 and 3 and by Theorem 3 we get that the latter exists
if and only if there exists a polynomial algorithm for solving mean-payoff
games on arbitrary (directed) graphs. Hence, the main result of this section
follows.
###### Theorem 4
There exists a polynomial time algorithm for computing exact equilibrium for
two-player zero-sum (alternating move) games if and only if there exists a
polynomial time algorithm for solving mean-payoff games on graphs.
## 4 Complexity of Computing Optimal Equilibrium
In this section, we investigate the complexity of computing an optimal
equilibrium. Our main results are summarized in the next theorem:
###### Theorem 5
1. 1.
Optimal equilibrium can be obtained by a profile of pure strategies.
2. 2.
If mean-payoff games are in P, then there is a polynomial algorithm for
computing an exact optimal equilibrium.
3. 3.
If mean-payoff games are not in P, then there is no FPTAS that approximate the
social welfare of the optimal equilibrium.
4. 4.
There is an FPTAS to compute an $\epsilon$-equilibrium that is
$\delta$-optimal. (Note that it does not necessarily approximate the value of
an exact optimal equilibrium.)
We will prove Theorem 5 in the next four subsections: In Subsection 4.1 we
show the naive algorithm for computing an equilibrium that is based on Folk
Theorem, and we prove basic properties of equilibria in alternating move
games. In Subsection 4.2 we prove Theorem 5(1). In Subsection 4.3 we
investigate the complexity of computing the social welfare of the optimal
equilibrium, and prove Theorem 5(2) and Theorem 5(3). Finally, in Subsection
4.4 we prove Theorem 5(4) which is the main result of this section.
In this section, we will model $n$-action $k$-player alternating move games by
a multi-weighted graph, according to the following conventions: The vertices
of the graph are the vertices in the set $V=(A_{1}\times
A_{2}\times\dots\times A_{k})\times\\{1,\dots,k\\}$, and we say that player
$i$ owns the vertex set $V_{i}=(A_{1}\times A_{2}\times\dots\times
A_{k})\times\\{i\\}$. Intuitively, a vertex is characterized by an action
vector and by a player that owns it. The pair $(u,v)$ is in the edge relation
if $u$ is owned by player $i$, $v$ is owned by player $i+1$ (where player
$k+1$ is player 1), and there is at most one difference in the action vector
of $u$ and $v$ and it is in position $i$. The weight of every edge is a vector
of size $k$ that corresponds to the utility vector of the actions. Formally,
if $u=(\overline{a_{1}},i)$ and $v=(\overline{a_{2}},i+1)$ then
$w(u,v)=(u_{1}(\overline{a_{2}}),u_{2}(\overline{a_{2}}),\dots,u_{k}(\overline{a_{2}}))$.
For an infinite path in the multi-weighted graph we define the dimension $i$
of _mean-payoff vector_ of the path to be the mean-payoff value of the path
according to dimension $i$. It is an easy observation that every infinite path
in the graph corresponds to a play and its mean-payoff vector corresponds to
the utility vector of the play. We note that the size of the graph is
$k^{2}\cdot n^{k}$ which is polynomial in the size of the encoding of the
utility functions (which is $k\cdot n^{k}$), hence this graph can be
constructed in polynomial time.
### 4.1 Basic properties of equilibria
The Folk Theorem gives a conceptually simple (but inefficient) technique to
construct an equilibrium. Intuitively, an equilibrium is obtained when each of
the players play as if the goal of all the other players is to minimize its
utility, and if one of the players deviates from this strategy, then all the
other players switch to playing according to a strategy that will minimize the
utility of the rebellious player. Formally, let $G$ be the corresponding
$k$-player game graph that models the alternating move game. For every player
$i$, we consider a zero-sum two-player mean-payoff game graph $G^{i}$ in which
the maximizer owns player $i$ vertices and the minimizer owns the other
vertices. Let $\sigma_{i}$ be an arbitrary optimal strategy for the maximizer
in graph $G^{i}$, let $\overline{\sigma_{i}}$ be an arbitrary optimal strategy
for the minimizer in $G^{i}$, and let $\nu_{i}$ be the value that is obtained
by the strategy profile $(\sigma_{i},\overline{\sigma_{i}})$. Then if every
player $i$ plays according to the strategy:
> If player $j\neq i$ deviated from $\sigma_{j}$, then play according to
> $\overline{\sigma_{j}}$ forever, and otherwise play according to
> $\sigma_{i}$
an equilibrium is formed (since by definition, playing according to
$\sigma_{i}$ assures utility at least $\nu_{i}$, and deviating from
$\sigma_{i}$ assures utility at most $\nu_{i}$).
In the next lemma, we extend the basic principle of Folk Theorem and get a
characterization of all the equilibria that are obtained by a profile of pure
strategies.
###### Lemma 4
Let $(t_{1},t_{2},\dots,t_{k})$ be a utility vector such that
$t_{i}\geq\nu_{i}$ (for every player $i$), then there exists a pure
equilibrium with utility exactly $t_{i}$ for every player $i$ if and only if
there exists an infinite path $\pi$ in the graph $G$ with mean-payoff vector
$(t_{1},t_{2},\dots,t_{k})$.
### 4.2 Optimal equilibrium can be obtained by pure strategies
In this subsection, we extend Lemma 4 also for the case of mixed strategies,
and as a consequence we get that optimal equilibrium can be obtained by pure
strategies. Intuitively, we wish to show that if a profile of (mixed)
strategies yields a utility vector $(t_{1},\dots,t_{k})$, then there exists an
infinite path in the graph with mean-payoff vector that is greater or equal
(in every dimension) to $(t_{1},\dots,t_{k})$. Then we get that if a utility
vector is obtained by a profile of mixed strategies, and then by Lemma 4 it is
also obtained by a profile of pure strategies.
We formally prove the above by the next two lemmas.
###### Lemma 5
Let $G$ be a multi-weighted graph that is strongly connected, and let
$(t_{1},\dots,t_{k})$ be a vector. Then if for every $\alpha>0$ there exists a
(finite) cyclic path with average weight at least $t_{i}-\alpha$ in every
dimension, then there exists an infinite path with mean-payoff vector at least
$(t_{1},\dots,t_{k})$.
###### Lemma 6
Let $\sigma$ be a profile of (mixed) strategies with utility vector
$(t_{1},\dots,t_{k})$. Then for every $\alpha>0$ there exists a cyclic path in
the game graph with average weight at least $t_{i}-\alpha$ in every dimension.
We are now ready to prove that the utility vector of a mixed equilibrium can
be obtained by a pure equilibrium.
###### Proposition 1
Let $\sigma$ be a profile of mixed strategies that induces a utility vector
$(t_{1},\dots,t_{k})$. Then there exists a profile $\sigma^{\prime}$ of pure
strategies that induces exactly the same utility vector. Moreover, if $\sigma$
is an equilibrium, then so is $\sigma^{\prime}$.
###### Proof
By Lemma 6 we get that for every $\alpha>0$ there is a cyclic path with
average weight at least $t_{i}-\alpha$ in every dimension. Therefore, by Lemma
5 we get that there is an infinite path in $G$ with mean-payoff vector at
least $(t_{1},\dots,t_{k})$, and by Lemma 4 we get that there is a profile of
pure strategies that has utility at least $(t_{1},\dots,t_{k})$. If $\sigma$
is an equilibrium we get that $t_{i}\geq\nu_{i}$ (since otherwise, player $i$
would deviate to strategy $\sigma_{i}$), and thus, by Lemma 4, we get that
there is a pure equilibrium that gives the same utility vector. ∎
The next corollary immediately follows from Proposition 1.
###### Corollary 1 (Theorem 5(1))
An optimal equilibrium can be obtained by a profile of pure strategies.
### 4.3 The complexity of computing the social welfare of the optimal (exact)
equilibrium
In this section we show that if there is a polynomial algorithm for mean-
payoff games, then there is a polynomial algorithm to compute an optimal
equilibrium in a $k$-player alternating move games. We also prove the converse
direction, that is, we show that if there is a polynomial algorithm that
computes the social welfare of the optimal equilibrium, then there is a
polynomial algorithm that solves mean-payoff games. We prove these two
assertions in the next two lemmas.
###### Lemma 7
Suppose that mean-payoff games are in P, then there is a polynomial algorithm
that computes the social welfare of an optimal equilibrium.
###### Proof
Due to Corollary 1, it is enough to consider only pure strategies, and due to
Lemma 4 a vector of utilities $(t_{1},\dots,t_{k})$ is obtained by a pure
equilibrium if and only if $t_{i}\geq\nu_{i}$ (for $i=1,\dots,k$) and there is
an infinite path in the game graph with mean-payoff $(t_{1},\dots,t_{k})$.
Since we assume that there is a polynomial algorithm for computing $\nu_{i}$,
our problem boils down to
> Find the maximal value of $\sum_{i=1}^{k}t_{i}$ subject to
>
> * •
>
> $t_{i}\geq\nu_{i}$; and
>
> * •
>
> there exists an infinite path with mean-payoff vector at least
> $(t_{1},\dots,t_{k})$
>
>
It was shown in [11] (in the proof of Theorem 18) that the problem of deciding
whether there exists an infinite path with mean-payoff vector at least
$(t_{1},\dots,t_{k})$ can be reduced (in polynomial time) to a set of linear
constraints. Moreover, the generated set of constraints remain linear even
when $t_{i}$ is a variable. Hence, we can find a feasible threshold vector
$(t_{1},\dots,t_{k})$ (that is, a vector that is realizable by an infinite
path in the graph) that maximizes $\sum_{i=1}^{k}t_{i}$ by linear programming.
Therefore, if we have a polynomial algorithm that computes $\nu_{i}$, then we
can find the social welfare of the optimal equilibrium in polynomial time. ∎
Lemma 7 proves Theorem 5(2) and gives an upper bound to the complexity of
computing optimal equilibrium. In the next lemma we show that this bound is
tight, and that the social welfare of the optimal equilibrium cannot be
approximated, unless mean-payoff games are in P.
###### Lemma 8 (Theorem 5(3))
There is no FPTAS that approximates the social welfare of an optimal
equilibrium, unless mean-payoff games are in P.
### 4.4 An FPTAS to compute an $\epsilon$-equilibrium that is
$\delta$-optimal
In this subsection, we assume that the utilities of the players are scaled to
rationals in $[-1,1]$, and we will describe an algorithm that computes an
$\epsilon$-equilibrium that is $\delta$-optimal (with respect to all
$\epsilon$-equilibria) and runs in time complexity that is polynomial in the
input size and in $\frac{1}{\epsilon}$ and $\frac{1}{\delta}$. Subsection 4.3
suggests that in order to compute a $\delta$-optimal $\epsilon$-equilibrium we
should approximate (by some value) the values of $\nu_{1},\dots,\nu_{k}$ and
then compute the optimal infinite path (with respect to the sum of utilities)
that has utility for player $i$ that is greater than the approximation of
$\nu_{i}$. However, this approach would not work, since the optimal social
welfare is not a continuous function with respect to the values
$\nu_{1},\dots,\nu_{k}$.
We denote by $\mathit{OPT}_{\epsilon}$ the social welfare of the optimal
$\epsilon$-equilibrium. We base our solution on the next lemma, which gives
two key properties of $\mathit{OPT}_{\epsilon}$.
###### Lemma 9
1. 1.
If $\epsilon_{1}\geq\epsilon_{2}$, then
$\mathit{OPT}_{\epsilon_{1}}\geq\mathit{OPT}_{\epsilon_{2}}$.
2. 2.
For every $\alpha\in[0,1]$ and $\epsilon_{1},\epsilon_{2}>0$, let
$\epsilon=\alpha\epsilon_{1}+(1-\alpha)\epsilon_{2}$, then there exists an
$\epsilon$-equilibrium with social welfare
$\alpha\mathit{OPT}_{\epsilon_{1}}+(1-\alpha)\mathit{OPT}_{\epsilon_{2}}$.
###### Proof
The first item of the lemma is a trivial observation. In order to prove the
second item, we observe that by Lemma 4 (and since by Proposition 1 it is
enough to consider only pure equilibria) it is enough to prove that there is
an infinite path $\pi$ with utility at least $\nu_{i}-\epsilon$ in every
dimension and with social welfare
$\alpha\mathit{OPT}_{\epsilon_{1}}+(1-\alpha)\mathit{OPT}_{\epsilon_{2}}$. By
Proposition 1, it is enough to show that there is a profile $\sigma$ of mixed
strategies (that need not be an equilibrium) that has a utility at least
$\nu-\epsilon$ in every dimension and has a social welfare at least
$\alpha\mathit{OPT}_{\epsilon_{1}}+(1-\alpha)\mathit{OPT}_{\epsilon_{2}}$. The
construction of $\sigma$ is trivial. For $i=1,2$, let $\sigma_{\epsilon_{i}}$
be a profile of strategies that induces an $\epsilon_{i}$-optimal equilibrium,
then we construct $\sigma$ by playing according to $\sigma_{\epsilon_{1}}$
with probability $\alpha$ and playing according to $\sigma_{\epsilon_{2}}$
with probability $1-\alpha$. ∎
###### Corollary 2
For every $\zeta\leq\frac{\epsilon\delta}{4k}$ we have
$\mathit{OPT}_{\epsilon+\zeta}-\frac{\delta}{2}\leq\mathit{OPT}_{\epsilon}\leq\mathit{OPT}_{\epsilon+\zeta}$
By the above corollary, to approximate $\mathit{OPT}_{\epsilon}$, it is enough
to approximate by $\frac{\delta}{2}$ the value of
$\mathit{OPT}_{\epsilon+\zeta}$ for some $\zeta\leq\frac{\epsilon\delta}{4k}$.
For this purpose, we extend the notion of $\epsilon$-equilibrium also for
$k$-dimensional vectors, and we say that a profile of strategies is a
$\overline{\beta}$-equilibrium if player $i$ cannot improve its utility by at
least $\beta_{i}$. Let us denote by $\min\overline{\beta}$ and by
$\max\overline{\beta}$ the minimal and maximal element of $\beta$
(respectively). Then by definition,
$\mathit{OPT}_{\min\overline{\beta}}\leq\mathit{OPT}_{\overline{\beta}}\leq\mathit{OPT}_{\max\overline{\beta}}$,
and by Corollary 2 we get that
$\mathit{OPT}_{\overline{\beta}}\leq\mathit{OPT}_{\max\overline{\beta}}\leq\mathit{OPT}_{\overline{\beta}}+(\max\overline{\beta}-\min\overline{\beta})\cdot\frac{4k}{\epsilon}$.
We are now ready to present an FPTAS that computes a $\delta$-approximation
for $\mathit{OPT}_{\epsilon}$: (1) Set $\zeta=\frac{\epsilon\delta}{4k}$, and
compute a $\zeta$ approximation of $\nu_{i}$ for every player $i$, and denote
it by $r_{i}$. (2) Compute the optimal path $\pi$ (with respect to social
welfare) that has utility at least $r_{i}-(\epsilon-\zeta)$ for every player,
and return its social welfare.
We note that we can execute the first step of the algorithm in polynomial time
due to [10](Observation 3.1), and we can execute the second step in polynomial
time by solving the linear programming problem that we described in the proof
of Lemma 7. The next lemma proves the correctness of our approximation
algorithm.
###### Lemma 10
Let $S(\pi)$ be the social welfare of $\pi$. Then
$S(\pi)-\delta\leq\mathit{OPT}_{\epsilon}\leq S(\pi)$
Lemma 10 along with the complexity analysis that we provided, proves that
there is an FPTAS to compute an $\epsilon$-equilibrium that is
$\delta$-optimal, and Theorem 5(4) follows. We also note that our proof for
Theorem 5(4) gives a constructive (and polynomial) algorithm that computes a
description of an actual $\epsilon$-equilibrium that is $\delta$-optimal.
## Acknowledgement
This work was carried out in partial fulfillment of the requirements for the
course Computational Game Theory that was given by Prof. Amos Fiat in Tel Aviv
University.
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## Appendix
## Appendix 0.A Proof of Lemma 3
###### Proof
We show that for a given directed normalized bipartite complete graph
$G=(V,E,w)$ we can construct (in polynomial time) an undirected normalized
bipartite complete graph $G^{\prime}=(V^{\prime},E^{\prime},w^{\prime})$ such
that maximizer has a winning strategy in $G$ (for threshold $0$) if and only
if he has a winning strategy in $G^{\prime}$ (again for threshold $0$).
Informally, we build $G^{\prime}$ from $G$ by taking the same set of vertices
and by assigning a weight $w^{\prime}(u,v)$ that matches either to $w(u,v)$ or
to $w(v,u)$.
To formally construct $w^{\prime}$ we present the notion of _surely loosing
edges_. Recall that $V_{1}$ are maximizer vertices and $V_{2}$ are minimizer
vertices. We say that a maximizer vertex out-edge $(v_{1},v_{2})$ is _surely
loosing for maximizer_ if $w(v_{1},v_{2})+w(v_{2},v_{1})<0$. We claim that if
maximizer has a winning strategy, then all its memoryless winning strategies
do not contain surely loosing edges. That is, for every memoryless winning
strategy $\sigma$ and a surely loosing edge $(v_{1},v_{2})$ we have
$\sigma(v_{1})\neq(v_{1},v_{2})$. Indeed, towards contradiction let us assume
that $\sigma(v_{1})=(v_{1},v_{2})$, and recall that $G$ is a complete
bipartite graph, than for the minimizer strategy $\tau$ that leads from every
minimizer vertex to $v_{1}$ we have that the mean-payoff value of
$(\sigma,\tau)$ is $\frac{w(v_{1},v_{2})+w(v_{2},v_{1})}{2}<0$, in
contradiction to the assumption that $\sigma$ is a winning strategy.
Analogously, $(v_{2},v_{1})$ is a surely loosing edge for minimizer if
$w(v_{1},v_{2})+w(v_{2},v_{1})\geq 0$ and by the same arguments there is no
memoryless winning strategy for minimizer that contains a surely loosing edge.
We are now ready to formally define
$G^{\prime}=(V^{\prime},E^{\prime},w^{\prime})$. We define $V^{\prime}=V$, and
since $G^{\prime}$ is complete bipartite the definition of $E^{\prime}$
follows immediately. For an edge $\\{v_{1},v_{2}\\}\in E^{\prime}$, where
$v_{i}\in V_{i}$, we define
$w^{\prime}(v_{1},v_{2})=\left\\{\begin{array}[]{rl}w(v_{1},v_{2})&\mbox{ if
$w(v_{1},v_{2})+w(v_{2},v_{1})\geq 0$}\\\ w(v_{2},v_{1})&\mbox{ otherwise (if
$w(v_{1},v_{2})+w(v_{2},v_{1})<0$)}\end{array}\right.$
We first prove that if $\sigma$ is a memoryless winning strategy for maximizer
in $G$, then it is also a winning strategy in $G^{\prime}$. Indeed, let $\rho$
be a path (in $G$ and in $G^{\prime}$) that is consistent with $\sigma$. We
claim that for every path $\pi$ that is a finite prefix of $\rho$ we have
$w(\pi)\leq w^{\prime}(\pi)$ (that is, its weight in $G^{\prime}$ is not less
than its weight in $G$), and we prove it by a simple induction on the length
of $\pi$. For $|\pi|=0$ the claim is trivial. For $|\pi|>0$ let $(u,v)$ be the
last edge in $\pi$. If $u\in V_{1}$, then we get that $\sigma(u)=(u,v)$ and
therefore $(u,v)$ is not a surely loosing edge. Therefore, by definition,
$w(u,v)=w^{\prime}(u,v)$ and the claim follows by the induction hypothesis
(since it holds for the prefix of length $|\pi|-1$). If $u\in V_{2}$, then
since $G$ is normalized we have that $w(u,v)<0$ and $w(v,u)>0$. Therefore, by
definition, $w^{\prime}(u,v)\geq w(u,v)$, and by the induction hypothesis the
claim follows. Since $\sigma$ is a winning strategy we get that the mean-
payoff value of $\rho$ is non-negative according to $w$, and by the last claim
we get that the mean-payoff of $\rho$ according to $w^{\prime}$ is greater or
equal to the mean-payoff value according to $w$. Hence $\sigma$ is a winning
strategy for maximizer also in $G^{\prime}$.
By the same arguments we get that if minimizer has a memoryless winning
strategy in $G$, then the same strategy is winning for minimizer also in
$G^{\prime}$.
Finally, due to Theorem 1, we get that maximizer has a winning strategy in $G$
if and only if he has a memoryless winning strategy in $G$ if and only if he
has a memoryless winning strategy in $G^{\prime}$ if and only if it has a
winning strategy in $G^{\prime}$ and the proof of the lemma follows. ∎
## Appendix 0.B Proof of Lemma 4
###### Proof
The direction from left to right is trivial, since a profile of pure
strategies has a (unique) infinite path in the graph that is consistent with
the strategies. To prove the converse direction we introduce the notion of a
_path strategy_ and the notion of _path equilibrium_. For a path $\pi$ we
define the _path strategy_ for player $i$ to be: at round $j$ (which is a
player $i$ round), play according to the $j$-th edge in $\pi$. And we define
the strategy $\sigma_{i}^{\pi}$ to be: If player $j\neq i$ deviated from the
path strategy of $\pi$, then play forever according to
$\overline{\sigma_{j}}$, and otherwise play according to the path strategy of
$\pi$. The _path equilibrium_ of the infinite path $\pi$ is the profile
$(\sigma_{1}^{\pi},\dots,\sigma_{k}^{\pi})$. The profile is an equilibrium
since it assures a utility $t_{i}\geq\nu_{i}$ for every player, and if player
$i$ deviates from the strategy $\sigma_{i}^{\pi}$ he will end up with a
utility $\nu_{i}$. ∎
## Appendix 0.C Proof of Lemma 5
###### Proof
We assume that for every $\alpha>0$ there exists a finite cyclic path
$C_{\alpha}$ with average weight at least $t_{i}-\alpha$ in every dimension,
and we let $v_{\alpha}$ be an arbitrary vertex in the cycle $C_{\alpha}$. For
every two vertices $u$ and $v$, we denote by $\pi_{u,v}$ the shortest path
from $u$ to $v$ (recall that $G$ is strongly connected), and we denote by $W$
the size of the biggest weight (in absolute value) in $G$. Intuitively, we
obtain an infinite path with mean-payoff vector at least $(t_{1},\dots,t_{n})$
by following the cycle $C_{\alpha}$ for $\alpha=1$, and then we follow the
path $\pi_{v_{\alpha},v_{\frac{\alpha}{2}}}$ and we follow the cycle
$C_{\frac{\alpha}{2}}$ twice, and then we follow $C_{\frac{\alpha}{3}}$ three
times and so on. However, we have to make sure that the average payoff does
not decrease too much when following a cycle for the first time.
Formally, for every $i\in\mathbb{N}$, we denote by $L_{i}$ the length of the
cycle $C_{\frac{1}{i}}$, and by $m_{i}=iWL_{i+1}$. We define $\rho_{0}$ to be
the empty path, and for every $i>0$ we define
$\rho_{i}=\rho_{i-1}\pi_{v_{\frac{1}{i-1}},v_{\frac{1}{i}}}C_{\frac{1}{i}}^{m_{i}}$,
and we define $\rho$ to be the infinite path that is the limit of the sequence
$\rho_{0},\rho_{1},\dots,\rho_{i},\dots$. Due to the fact that the length of
$\pi_{v_{\frac{1}{i-1}},v_{\frac{1}{i}}}$ is bounded (by the size of the
graph), and since the maximal weight is at most $W$ (and the minimal weight is
at least $-W$), we get, by a simple algebra, that the mean-payoff vector of
$\rho$ is at least $(t_{1},\dots,t_{k})$. ∎
## Appendix 0.D Proof of Lemma 6
###### Proof
By definition of the utility function, for every $\delta>0$, there exists a
round $j$ such that the expected average utility, in every round after $j$, is
at least $t_{i}-\delta$ in every dimension. We denote by $\Pi_{j}$ the
(finite) set of paths with length $j$ that have non-zero probability according
to the strategy profile $\sigma$, and w.l.o.g we assume that all the paths
have the same probability (and a path may occur more than once in $\Pi_{j}$).
For a path $\pi\in\Pi_{j}$, we denote by $C_{\pi}$ the longest cyclic path
that is a sub-path of $\pi$. We note that since $G$ is a finite graph, we get
that $|C_{\pi}|\geq|\pi|-2|G|$ (where $|G|$ denotes the number of vertices in
$G$), and in every dimension: $w(C_{\pi})\geq w(\pi)-2|G|W$ and
$\mathit{Avg}(C_{\pi})\geq\mathit{Avg}(\pi)-\frac{2|G|W}{|\pi|-2|G|W}=\mathit{Avg}(\pi)-\frac{2|G|W}{j-2|G|W}$.
We partition $\Pi_{j}$ to $|G|$ sets (some of them may be empty) namely
$\Pi_{j}^{v}$ for every $v\in G$, such that for every path $\pi\in\Pi_{j}^{v}$
we have that $v$ is in the cyclic path $C_{\pi}$. For a set
$\Pi_{j}^{v}=\\{\pi_{1},\dots,\pi_{m}\\}$, we denote
$C(\Pi_{j}^{v})=C_{\pi_{1}}C_{\pi_{2}}\dots C_{\pi_{m}}$ (note that
$C(\Pi_{j}^{v})$ is a path). For every two vertices $u,v\in G$, we denote by
$\pi_{u,v}$ the shortest path between $u$ and $v$, and we note that
$|\pi_{u,v}|\leq|G|$ and $w(\pi_{u,v})\geq-|G|W$ in every dimension. Finally,
we assume that the vertex set of $G$ is $V=\\{v_{1},\dots,v_{m}\\}$ and we
define the cyclic path
$\pi=C(\Pi_{j}^{v_{1}})\pi_{v_{1},v_{2}}C(\Pi_{j}^{v_{2}})\pi_{v_{2},v_{3}}\dots\pi_{v_{m-1},v_{m}}C(\Pi_{j}^{v_{m}})\pi_{v_{m},v_{1}}$
The average weight of $\pi$ in every dimension is at least
$\mathit{Avg}(\Pi_{j})-\frac{3|G|^{2}W}{j}$
and since in every dimension $\mathit{Avg}(\Pi_{j})\geq t_{i}-\delta$, then
for $j\geq\frac{3|G|^{2}W}{\delta}$ we get that in every dimension
$\mathit{Avg}(\pi)\geq t_{i}-2\delta$
and for $\delta=\frac{\alpha}{2}$ we get that $\pi$ is a cyclic path with
average weight at least $t_{i}-\alpha$ in every dimension, and the proof of
the lemma follows. ∎
## Appendix 0.E Proof of Lemma 8
###### Proof
Due to Theorem 2 and Theorem 4 it is enough to show that if we could
approximate the social welfare of the optimal equilibrium in a three-player
game, then we would be able to determine whether in a two-player zero-sum
game, player 1 has a strategy that assures a value that is strictly greater
than $0$. The proof is straight forward. Let $(A_{1},A_{2})$ be the actions of
a zero-sum two-player game with utility functions $u_{1}:A_{1}\times
A_{2}\to[-1,1]$ and $u_{2}=-u_{1}$. We construct a three-player game
$(A_{1}^{\prime},A_{2}^{\prime},A_{3}^{\prime},u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})$
in the following way:
* •
$A_{1}^{\prime}=A_{1}\cup\\{\$\\}$, $A_{2}^{\prime}=A_{2}\cup\\{\$\\}$ and
$A_{3}^{\prime}=\\{\$\\}$ (where $\$$ is a fresh action).
* •
Let $a_{2}^{*}$ be an arbitrary action in $A_{2}^{\prime}-\\{\$\\}$. We define
the utility function $u_{1}^{\prime}$ to be
$u_{1}^{\prime}(a_{1},a_{2},a_{3})=\left\\{\begin{array}[]{rl}u_{1}(a_{1},a_{2})&\mbox{
if $a_{1}\neq\$$ and $a_{2}\neq\$$}\\\ u_{1}(a_{1},a_{2}^{*})&\mbox{ if
$a_{1}\neq\$$ and $a_{2}=\$$}\\\ 0&\mbox{ otherwise (if
$a_{1}=\$$)}\end{array}\right.$
We define $u_{2}^{\prime}=-u_{1}^{\prime}$, and
$u_{3}^{\prime}(a_{1},a_{2},a_{3})=\left\\{\begin{array}[]{rl}1&\mbox{ if
$a_{1}=\$$}\\\ -1&\mbox{ otherwise (if $a_{1}\neq\$$)}\end{array}\right.$
The reader can verify that if player 1 has a strategy to assure utility
greater than $0$ in the zero-sum game, then in any profile of equilibrium in
the three-player game he will play $\$$ only for a negligible number of
rounds, and the social welfare of the equilibrium will be $-1$. On the other
hand, if player 1 cannot assure utility at least $0$, then a profile of
strategies in which all three players play $\$$ forever is an equilibrium and
its social welfare is $1$. Hence, even for $\epsilon=1$ we cannot approximate
the social welfare of the optimal equilibrium, unless mean-payoff games are in
P. ∎
## Appendix 0.F Proof of Corollary 2
###### Proof
The fact that $\mathit{OPT}_{\epsilon}\leq\mathit{OPT}_{\epsilon+\zeta}$
follows immediately from Lemma 9(1). To prove that
$\mathit{OPT}_{\epsilon+\zeta}-\delta\leq\mathit{OPT}_{\epsilon}$, we set
$\alpha=\frac{\zeta}{\epsilon}$ and by Lemma 9(2), and since
$\alpha\zeta+(1-\alpha)(\epsilon+\zeta)=\epsilon$ we get
$\alpha\mathit{OPT}_{\zeta}+(1-\alpha)\mathit{OPT}_{\epsilon+\zeta}\leq\mathit{OPT}_{\epsilon}$
since the utility function is scaled to $[-1,1]$ we get that
$\mathit{OPT}_{\zeta}\geq-k$ and $\mathit{OPT}_{\epsilon+\zeta}\leq k$, and
thus we have
$-2k\alpha+\mathit{OPT}_{\epsilon+\zeta}\leq\mathit{OPT}_{\epsilon}$
since $\alpha=\frac{\zeta}{\epsilon}$ we get
$\frac{-2k\zeta}{\epsilon}+\mathit{OPT}_{\epsilon+\zeta}\leq\mathit{OPT}_{\epsilon}$
and since $\zeta\leq\frac{\epsilon\delta}{4k}$ we have
$\mathit{OPT}_{\epsilon+\zeta}-\frac{\delta}{2}\leq\mathit{OPT}_{\epsilon}$
∎
## Appendix 0.G Proof of Lemma 10
###### Proof
We denote player-$i$ utility according to $\pi$ by $u_{i}(\pi)$ and we
construct the vector $\overline{\beta}$ by defining
$\beta_{i}=\left\\{\begin{array}[]{rl}\nu_{i}-u_{i}(\pi)&\mbox{ if
$\nu_{i}-u_{i}(\pi)\geq(\epsilon-\frac{\zeta}{2})$}\\\
\epsilon-\frac{\zeta}{2}&\mbox{ otherwise}\end{array}\right.$
and we claim that $\pi$ is an optimal $\overline{\beta}$-equilibrium. The
claim holds because for every $\overline{\beta}$-equilibrium, the utility of
player $i$ is at least $\nu_{i}-(\epsilon-\zeta)$, and $\pi$ is the optimal
path with utility at least $\nu_{i}-(\epsilon-\zeta)$ for every player. In
addition, we claim that $\max\overline{\beta}-\min\overline{\beta}$ is at most
$\frac{\zeta}{2}$. The claim holds because by the construction of $\beta$ we
have that $\min\overline{\beta}\geq\epsilon-\frac{\zeta}{2}$, and
$\nu_{i}-u_{i}(\pi)$ is at most $\epsilon-\zeta$ (since $\pi$ is an
$(\epsilon-\zeta)$-equilibrium). Hence
$\max\overline{\beta}-\min\overline{\beta}\leq\frac{\zeta}{2}$. Therefore,
since $\max\overline{\beta}\leq\epsilon$ and since
$\mathit{OPT}_{\overline{\beta}}\leq\mathit{OPT}_{\max\overline{\beta}}\leq\mathit{OPT}_{\overline{\beta}}+(\max\overline{\beta}-\min\overline{\beta})\cdot\frac{4k}{\epsilon}$
we get that
$S(\pi)\leq\mathit{OPT}_{\epsilon}\leq
S(\pi)+\frac{\zeta}{2}\cdot\frac{4k}{\epsilon\delta}$
and since $\zeta=\frac{\epsilon\delta}{4k}$ we get that
$S(\pi)\leq\mathit{OPT}_{\epsilon}\leq S(\pi)+\frac{\delta}{2}$
and the assertion of the lemma follows. ∎
|
arxiv-papers
| 2012-12-29T14:14:34 |
2024-09-04T02:49:39.761545
|
{
"license": "Public Domain",
"authors": "Yaron Velner",
"submitter": "Yaron Velner",
"url": "https://arxiv.org/abs/1212.6632"
}
|
1212.6727
|
# Strategies in a Symmetric Quantum Kolkata Restaurant Problem
Puya Sharif and Hoshang Heydari
###### Abstract
The Quantum Kolkata restaurant problem is a multiple-choice version of the
quantum minority game, where a set of $n$ non-communicating players have to
chose between one of $m$ choices. A payoff is granted to the players that make
a unique choice. It has previously been shown that shared entanglement and
quantum operations can aid the players to coordinate their actions and acquire
higher payoffs than is possible with classical randomization. In this paper
the initial quantum state is expanded to a family of GHZ-type states and
strategies are discussed in terms of possible final outcomes. It is shown that
the players individually seek outcomes that maximize the collective good.
###### Keywords:
quantum game theory, quantum games, qutrit, local quantum operations, kolkata
restaurant problem
###### :
03.67.Ac
## 1 Introduction
### 1.1 Games and framework
Game theory is the study of systematic and strategic decision-making in
interactive situations of conflict and cooperation. The models are widely used
in economics, political science, biology and computer science to capture the
behavior of individual participants in terms of responses to strategies of the
rest. The field attempts to describe how decision makers do and should
interact within a well-defined system of rules to maximize their satisfaction
with the outcome GT-Critical ; GT-fudenberg ; Course in GT . A game is a model
of the strategies of these decision makers or players as we will call them, in
terms of choices made by each of them. They are assumed to all have individual
preference profiles
$\sigma_{x_{1}}\succeq\sigma_{x_{2}}\succeq\cdots\succeq\sigma_{x_{m}}$ over a
set of $m$ outcomes $\\{\sigma_{x_{j}}\\}$, where
${}^{\prime}\succeq\,^{\prime}$ should be interpreted as ”preferred by”, and
the $x_{j}$’s as indices for possible outcomes. An outcome or strategy profile
$\sigma\in S_{n}\times S_{n-1}\times\cdots\times S_{1}=S$ is equivalent to the
combination of the strategies $s^{i}_{j}\in S_{i}$ of the participants, where
$s^{i}_{j}$ is the $j$’th strategy of player $i$, $S_{i}$ the set of
strategies or choices available to that player and $S$ the set of all possible
strategy profiles. In order to evaluate the profit or satisfaction of player
$i$ with regards to a strategy profile we need to define for each player a
payoff function $\$_{i}$ that takes a strategy profile $\sigma$ as input and
outputs a real numerical value as a measure of desirability. We have
$\$:S\rightarrow\mathbb{R}$ and
$\$_{i}(\sigma_{k})\geq\$_{i}(\sigma_{l})\Leftrightarrow\sigma_{k}\succeq_{i}\sigma_{l}$.
The question to answer is, what should rational players choose to do given
that they have partial or complete information of the content of $S$ and the
payoff functions $\$_{i}$? The main approach is to find a _solution concept_ ,
with the most famous one being the _Nash Equilibrium_ , where all players
simply make the choice $s^{i}_{j}$ that is the best possible response to any
configuration in $S/S_{i}$, i.e to any combination of strategies by their
counter-parties. In situations where such equilibrium does not exist one needs
to extend the game to allow for mixed (probabilistic) strategies where the
players extend the sets $S_{i}$ to $\Delta(S_{i})$, i.e the set of convex
combinations of the $s^{i}_{j}$’s to acquire one. Just as classical
probability distributions extend pure strategy games to mixed ones, quantum
probabilities, operations and entanglement can extend the framework to
outperform any classical setup.
### 1.2 Quantum games
A quantum game is defined by a set $\Gamma$ of objects and the relationships
between them:
$\Gamma=\\{\rho_{\mathcal{Q}},\mathcal{H}_{\mathcal{Q}},n,S_{i},\$_{i}\\}\;\;\textrm{for}\;\;i=1,\cdots,n$
(1)
where $\mathcal{H}_{\mathcal{Q}}$ is the Hilbert space of the composite
quantum system, $\rho_{\mathcal{Q}}$ is the initial state of the game defined
on $\mathcal{H}_{\mathcal{Q}}$, $n$ is the number of players, $S_{i}$ the set
of available strategies of player $i$ and $\$_{i}$ the payoffs available to
player $i$ for each game outcome. In our quantum game protocol the $m_{i}$
different pure strategies available to a player $i$ will be encoded in the
basis states of an $m_{i}$-level quantum system
$\rho_{\mathcal{Q}_{i}}\in\mathcal{H}_{{\mathcal{Q}_{i}}}$. With $n$ players
we’ll end up needing a initial quantum state
$\rho_{\mathcal{Q}}\in\mathcal{H}_{\mathcal{Q}}=\mathcal{H}_{\mathcal{Q}_{n}}\otimes\mathcal{H}_{\mathcal{Q}_{n-1}}\cdots\otimes\mathcal{H}_{\mathcal{Q}_{1}}$
with
$\textrm{dim}(\mathcal{H}_{\mathcal{Q}})=\prod_{i=1}^{n}\textrm{dim}(\mathcal{H}_{\mathcal{Q}_{i}})$
to accommodate for all possible game outcomes review .
The strategies are chosen and played by each player trough the application of
a unitary operator $U_{i}\in S_{i}=S(m_{i})$ on their own sub-systems, where
the set of allowed quantum operations $S(m_{i})$ is some subset of the special
unitary group $\textrm{SU}(m_{i})$. The general procedure of a quantum game
consists of a transformation of the composite initial state trough local
unitary operations by the players: $U_{n}\otimes U_{n-1}\otimes\cdots\otimes
U_{1}:\mathcal{H}_{\mathcal{Q}}\rightarrow\mathcal{H}_{\mathcal{Q}}$. Followed
by a measurement outcome, or in terms of pre-measurement reasoning, an
expectation value: $\$:\mathcal{H}_{\mathcal{Q}}\rightarrow\mathbb{R}$.
## 2 Quantum Kolkata restaurant problem
This is a general form of a minority game kolkata1 ; Hayden ; Chen , where $n$
non-communicating agents (players), have to choose between $m$ choices. A
payoff of $\$=1$ is payed out to the players that make _unique_ choices.
Players making the same choice receive $\$=0$. The challenge is to come up
with a strategy profile that maximizes the expected payoffs $E_{i}(\$)$ of all
players $i$, and has the property of being a Nash equilibrium. In the absence
of communication, in a classical framework, there is nothing else to do, but
to randomize.
### 2.1 Collective aim in the quantum case
It has been shown for the case of three players and three choices in the
quantum setting, starting with a GHZ-type state
$|\psi_{in}\rangle=\frac{1}{\sqrt{3}}\left(|000\rangle+|111\rangle+|222\rangle\right)$,
that shared entanglement and local SU(3) operations (by the players on their
own subsystems) will lead to a expected payoff $E(\$)=\frac{2}{3}$. This is a
50% increase compared to the classical payoff of $\frac{4}{9}$ reachable
trough randomization. Although the details of the protocol can be found in
puya , it is instructive to jump back a couple steps. Since we have three
players with three allowed pure choices the Hilbert space we are dealing with
is the space of three qutrit states; with a basis
$B=\\{|ijk\rangle\\};\;i,j,k\in\\{0,1,2\\}$, each of which representing a
post-measurement outcome of the game, where $i,j,k$ denotes the final choices
of players $1,2,3$ respectively. We have
$\textrm{span}(B)=\\{\sum_{i,j,k=0}^{2}a_{ijk}|ijk\rangle:i,j,k=0,1,2\;\textrm{and}\;a_{ijk}\in\mathbb{C}\\}$
which with a normalization condition gives us the complete Hilbert space of
the game. We can divide $B$ into subsets that are interesting from the point
of view of the possible outcomes:
$\displaystyle L\,$ $\displaystyle=$
$\displaystyle\\{|000\rangle,|111\rangle,|222\rangle\\},$ (2) $\displaystyle
G\,$ $\displaystyle=$
$\displaystyle\\{|012\rangle,|120\rangle,|201\rangle,|021\rangle,|102\rangle,|210\rangle\\},$
(3) $\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle\\{|011\rangle,|022\rangle,|100\rangle,|122\rangle,|200\rangle,|211\rangle\\},$
(4) $\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle\\{|101\rangle,|202\rangle,|010\rangle,|212\rangle,|020\rangle,|121\rangle\\},$
(5) $\displaystyle D_{3}$ $\displaystyle=$
$\displaystyle\\{|110\rangle,|220\rangle,|001\rangle,|221\rangle,|002\rangle,|112\rangle\\},$
(6)
where $L$ contains all states for which none the players 1,2,3 receive any
payoff. It is thus a collective objective to avoid these states. $G$ contains
all those states that returns a payoff $\$=1$ to the three of them and the
sets $D_{i}$ contains the post-measurement states leads to a payoff $\$=1$ for
player $i$ and $\$=0$ to players $\neq i$. Thus the general goal of each
player $i$ is to maximize the probability of the post-measurement outcome to
be a state in $G_{i}=G\cup D_{i}$. Starting with an initial state
$|\psi_{in}\rangle$, each player $i\in{1,2,3}$ applies an operator from its
set of allowed strategies $S_{i}\subseteq\textrm{SU}(3)$, transforming it to
its final state $|\psi_{fin}\rangle=U_{1}\otimes U_{2}\otimes
U_{3}|\psi_{in}\rangle$. The expected payoff $E(\$_{i})$ of player $i$ is the
probability of the post-measurement outcome to be a state in $G_{i}$:
$E(\$_{i})=\sum_{|\xi\rangle\in
G_{i}}\left|\langle\psi_{fin}|\xi\rangle\right|^{2}.$ (7)
Given that we have an initial state in $\textrm{span}(L)$ containing all
states of the form
$|\psi_{in}\rangle=\alpha|000\rangle+\beta|111\rangle+\gamma|222\rangle$ with
$\alpha,\beta,\gamma\in\mathbb{C}$ (We will assume
$0\leq\alpha,\beta,\gamma\in\mathbb{R}$ later for simplicity), what is the
rational aim of player $i$? First, note that all states in $\textrm{span}(L)$
are unbiased with regards of change in player positions. We can assume that
they don’t even know which qutrit they control since that knowledge doesn’t
add any useful information for the choice of $U_{i}\in S_{i}$. Second, any
choice of $U_{i}$ aimed at increasing the probability of post-measurement
state to end up being in $D_{i}$ must due to the symmetry of the setup
increase the probability of the state being in $D_{j\neq i\vee k}\cup D_{k\neq
i\vee j}$ with 2:1. Third, an outcome in $G=G_{1}\cap G_{2}\cap G_{3}$ is as
favorable as any outcome in $D_{i}$ for player $i$. It follows therefore that
the players should aim for producing a state in $\textrm{span}(G)$ to the
extent this is possible. Although it was shown in puya that they fail to
fully depart from $\textrm{span}(L)$, whereby they reach a maximum payoff of
$E(\$)=\frac{2}{3}$ rater than $E(\$)=1$ for
$\alpha=\beta=\gamma=\frac{1}{\sqrt{3}},$ with a final state of the following
form:
$\mid\psi_{fin}\rangle=\frac{1}{3}\left(|000\rangle+|012\rangle+|021\rangle+|102\rangle\right.+\\\
\left.|111\rangle+|120\rangle+|201\rangle+|210\rangle+|222\rangle\right).$ (8)
### 2.2 Expected payoffs for initial states in span(_L_)
Due to the symmetries mentioned in the previous section the three players will
have to chose a unitary operator $U$ that takes states from $\textrm{span}(L)$
to $\textrm{span}(L\cup G)$, without the possibility to favor any subset of
$G$ (even if that possibility existed, that would only put a roof for the
individual payoffs and any choice of subset other than the whole would
decrease the expected payoff, due to the lack of coordination the choice of
subset). This leads to the conclusion that there exists a $U$ (fixed up to a
global phase) for any initial state
$|\psi_{in}\rangle=\alpha|000\rangle+\beta|111\rangle+\gamma|222\rangle$, that
maximizes the individual expected payoffs and is a Nash equilibrium solution,
since any departure from this strategy will lead to a lower payoff. We have:
$E_{max}(\$)=\sum_{|\xi\rangle\in G}\left|\langle\psi_{fin}|U^{\dagger}\otimes
U^{\dagger}\otimes U^{\dagger}|\xi\rangle\right|^{2},$ (9)
simultaneously for all three of them. Figure 1 shows numerically calculated
expected payoffs $E(\$)$ for
$\alpha=\sin\vartheta\cos\varphi;\,\beta=\sin\vartheta\sin\varphi;\,\gamma=\cos\vartheta$
where $\varphi=\frac{\pi}{40}M,\vartheta=\frac{\pi}{40}N$ and
$M,N=1,2,\cdots,20$. A total of 400 optimizations with equally many different
associated operators $U_{MN}$. We see that the expected payoff is maximized
for $\varphi=\frac{\pi}{4}$ and $\vartheta=\cos^{-1}\frac{1}{\sqrt{3}}$, where
the initial state is maximally entangled, and falls off towards the classical
expected payoff as the entanglement decreases.
Figure 1: Payoffs associated with optimal strategies in a three player game
with a variable initial state. The expected payoff $E(\$)$ decreases as the
level of entanglement decreases.
## 3 Conclusions
The ambition of self-maximization in the studied Kolkata restaurant problem
leads individual players to act in such way that the collective good is
maximized. The game is symmetric with regards to permutations of player
positions which guides the participants to aim for a set of outcomes that
favors them all. The expected payoff reachable trough local operations changes
with the level of entanglement in the initial state.
### Acknowledgements:
The work was supported by the Swedish Research Council (VR).
## References
* (1) S. Hargreaves Heap, Y. Varoufakis, Game theory - A critical introduction, (Routledge, 2004).
* (2) D. Fudenberg, J. Tirole, Game Theory, (MIT Press, 1991).
* (3) M. J. Osbourne, A Rubinstein, A course in game theory, (MIT Press, 1994).
* (4) P. Sharif, H. Heydari, Econophysics of Systemic Risk and Network Dynamics, New Economic Windows 2013, pp 217-236
* (5) A. S. Chakrabarti, B. K. Chakrabarti, A. Chatterjee, M. Mitra, ”The Kolkata Paise Restaurant problem and resource utilization”, Physica A 388,(2009) 2420-2426.
* (6) S. C. Benjamin, P.M. Hayden, Phys. Rev. Lett 87, 069801 (2001).
* (7) Q. Chen, Y. Wang, N-player quantum minority game, Physics Letters A, A 327, 98,102, (2004).
* (8) P. Sharif, H. Heydari, ”Quantum solution to a three player Kolkata restaurant problem using entangled qutrits”, Preprint:http://arxiv.org/abs/1111.1962
|
arxiv-papers
| 2012-12-30T14:35:18 |
2024-09-04T02:49:39.772065
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Puya Sharif and Hoshang Heydari",
"submitter": "Puya Sharif",
"url": "https://arxiv.org/abs/1212.6727"
}
|
1212.6767
|
###### Abstract
$H$-theorem states that the entropy production is nonnegative and, therefore,
the entropy of a closed system should monotonically change in time. In
information processing, the entropy production is positive for random
transformation of signals (the information processing lemma). Originally, the
$H$-theorem and the information processing lemma were proved for the classical
Boltzmann-Gibbs-Shannon entropy and for the correspondent divergence (the
relative entropy). Many new entropies and divergences have been proposed
during last decades and for all of them the $H$-theorem is needed. This note
proposes a simple and general criterion to check whether the $H$-theorem is
valid for a convex divergence $H$ and demonstrates that some of the popular
divergences obey no $H$-theorem. We consider systems with $n$ states $A_{i}$
that obey first order kinetics (master equation). A convex function $H$ is a
Lyapunov function for all master equations with given equilibrium if and only
if its conditional minima properly describe the equilibria of pair transitions
$A_{i}\rightleftharpoons A_{j}$. This theorem does not depend on the principle
of detailed balance and is valid for general Markov kinetics. Elementary
analysis of pair equilibria demonstrates that the popular Bregman divergences
like Euclidean distance or Itakura-Saito distance in the space of distribution
cannot be the universal Lyapunov functions for the first-order kinetics and
can increase in Markov processes. Therefore, they violate the second law and
the information processing lemma. In particular, for these measures of
information (divergences) random manipulation with data may add information to
data. The main results are extended to nonlinear generalized mass action law
kinetic equations. In Appendix, a new family of the universal Lyapunov
functions for the generalized mass action law kinetics is described.
###### keywords:
Markov process; Lyapunov function; non-classical entropy; information
processing; quasiconvexity; directional convexity; Schur convexity
10.3390/e16052408 16 Received: 9 March 2014; in revised form: 15 April 2014 /
Accepted: 24 April 2014 / Published: 29 April 2014 / Postprint with new
Appendix 03 June 2014 General $H$-theorem and Entropies that Violate the
Second Law Alexander N. Gorban .
## 1 The Problem
The first non-classical entropy was proposed by Rényi in 1960 Renyi1961 . In
the same paper he discovered the very general class of divergences, the so-
called $f$-divergences (or Csiszár-Morimoto divergences because of the works
of Csiszár Csiszar1963 and Morimoto Morimoto1963 published simultaneously in
1963):
$H_{h}(p)=H_{h}(P\|P^{*})=\sum_{i}p^{*}_{i}h\left(\frac{p_{i}}{p_{i}^{*}}\right)$
(1)
where $P=(p_{i})$ is a probability distribution, $P^{*}$ is an equilibrium
distribution, $h(x)$ is a convex function defined on the open ($x>0$) or
closed $x\geq 0$ semi-axis. We use here the notation $H_{h}(P\|P^{*})$ to
stress the dependence of $H_{h}$ on both $p_{i}$ and $p^{*}_{i}$.
These divergences have the form of the relative entropy or, in the
thermodynamic terminology, the (negative) free entropy, the Massieu-Planck
functions Callen1985 , or $F/RT$ where $F$ is the free energy. They measure
the deviation of the current distribution $P$ from the equilibrium $P^{*}$.
After 1961, many new entropies and divergences were invented and applied to
real problems, including Burg entropy Burg1972 , Cressie-Red family of power
divergences CR1984 , Tsallis entropy Tsa1988 ; Abe , families of $\alpha$-,
$\beta$\- and $\gamma$-divergences CichockiAmari2010 and many others (see the
review papers EstMor1995 ; GorGorJudge2010 ). Many of them have the
$f$-divergence form, but some of them do not. For example, the squared
Euclidean distance from $P$ to $P^{*}$ is not, in general, a $f$-divergence
unless all $p^{*}_{i}$ are equal (equidistribution). Another example gives the
Itakura-Saito distance:
$\sum_{i}\left(\frac{p_{i}}{p_{i}^{*}}-\ln\frac{p_{i}}{p_{i}^{*}}-1\right)$
(2)
The idea of Bregman divergences Bregman1967 provides a new general source of
divergences different from the $f$-divergences. Any strictly convex function
$F$ in an closed convex set $V$ satisfies the Jensen inequality
$D_{F}(p,q)=F(p)-F(q)-(\nabla_{q}F(q),p-q)>0$ (3)
if $p\neq q$, $p,q\in V$. This positive quantity $D_{F}(p,q)$ is the Bregman
divergence associated with $F$. For example, for a positive quadratic form
$F(x)$ the Bregman distance is just $D_{F}(p,q)=F(p-q)$. In particular, if $F$
is the squared Euclidean length of $x$ then $D_{F}(p,q)$ is the squared
Euclidean distance. If $F$ is the Burg entropy, $F(x)=-\sum_{i}\ln p_{i}$,
then $D_{F}(p,q)$ is the Itakura-Saito distance. The Bregman divergences have
many attractive properties. For example, the mean vector minimizes the
expected Bregman divergence from the random vector Banerjee2005 . The Bregman
divergences are convenient for numerical optimization because generalized
Pythagorean identity CsizarMatus2012 . Nevertheless, for information
processing and for many physical applications one more property is crucially
important.
The divergence between the current distribution and equilibrium should
monotonically decrease in Markov processes. It is the ultimate requirement for
use of the divergence in information processing and in non-equilibrium
thermodynamics and kinetics. In physics, the first result of this type was
Boltzmann’s $H$-theorem proven for nonlinear kinetic equation. In information
theory, Shannon Shannon1948 proved this theorem for the entropy (“the data
processing lemma”) and Markov chains.
In his well-known paper Renyi1961 , Rényi also proved that $H_{h}(P\|P^{*})$
monotonically decreases in Markov processes (he gave the detailed proof for
the classical relative entropy and then mentioned that for the $f$-divergences
it is the same). This result, elaborated further by Csiszár Csiszar1963 and
Morimoto Morimoto1963 , embraces many later particular $H$-theorems for
various entropies including the Tsallis entropy and the Rényi entropy (because
it can be transformed into the form (1) by a monotonic function, see for
example GorGorJudge2010 ). The generalized data processing lemma was proven
Cohen1993 ; CohenIwasa1993 : for every two positive probability distributions
$P,Q$ the divergence $H_{h}(P\|Q)$ decreases under action of a stochastic
matrix $A=(a_{ij})$
$H_{h}(AP\|AQ)\leq\overline{\alpha}(A)H_{h}(P\|Q)$ (4)
where
$\overline{\alpha}(A)=\frac{1}{2}\max_{i,k}\left\\{\sum_{j}|a_{ij}-a_{kj}|\right\\}$
(5)
is the ergodicity contraction coefficient, $0\leq\overline{\alpha}(A)\leq 1$.
Here, neither $Q$ nor $P$ must be the equilibrium distribution: divergence
between any two distributions decreases in Markov processes.
Under some additional conditions, the property to decrease in Markov processes
characterizes the $f$-divergences ENTR3 ; Amari2009 . For example, if a
divergence decreases in all Markov processes, does not change under
permutation of states and can be represented as a sum over states (has the
trace form), then it is the $f$-divergence ENTR3 ; GorGorJudge2010 .
The dynamics of distributions in the continuous time Markov processes is
described by the master equation. Thus, the $f$-divergences are the Lyapunov
functions for the master equation. The important property of the divergences
$H_{h}(P\|P^{*})$ is that they are universal Lyapunov functions. That is, they
depend on the current distribution $P$ and on the equilibrium $P^{*}$ but do
not depend on the transition probabilities directly.
For each new divergence we have to analyze its behavior in Markov processes
and to prove or refute the $H$-theorem. For this purpose, we need a simple and
general criterion. It is desirable to avoid any additional requirements like
the trace form or symmetry. In this paper we develop this criterion.
It is obvious that the equilibrium $P^{*}$ is a global minimum of any
universal Lyapunov function $H(P)$ in the simplex of distributions (see the
model equation below). In brief, the general $H$-theorem states that a convex
function $H(P)$ is a universal Lyapunov function for the master equation if
and only if its conditional minima correctly describe the partial equilibria
for pairs of transitions $A_{i}\rightleftharpoons A_{j}$. These partial
equilibria are given by proportions $p_{i}/p_{i}^{*}=p_{j}/p_{j}^{*}$. They
should be solutions to the problem
$\begin{split}&H(P)\rightarrow\min\;\mbox{ subject to }p_{k}\geq
0\,(k=1,\ldots,n),\\\ &\sum_{k=1}^{n}p_{k}=1,\;\mbox{ and given values of
}p_{l}\,(l\neq i,j)\end{split}$ (6)
These solutions are minima of $H(P)$ on segments $p_{i}+p_{j}=1-\sum_{l\neq
i,j}p_{l}$, $p_{i,j}\geq 0$. They depend on $n-2$ parameters $p_{l}\geq 0$
($l\neq i,j$, $\sum_{l\neq i,j}p_{l}<1$).
Using this general $H$-theorem we analyze several Bregman divergences that are
not $f$-divergences and demonstrate that they do not allow the $H$-theorem
even for systems with three states. We present also the generalizations of the
main results for Generalized Mass Action Law (GMAL) kinetics.
## 2 Three Forms of Master Equation and the Decomposition Theorem
We consider continuous time Markov chains with $n$ states
$A_{1},\ldots,A_{n}$. The Kolmogorov equation or master equation for the
probability distribution $P$ with the coordinates $p_{i}$ (we can consider $P$
as a vector-column $P=[p_{1},\ldots,p_{n}]^{T}$) is
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}(q_{ij}p_{j}-q_{ji}p_{i})\;\;(i=1,\ldots,n)$ (7)
where $q_{ij}$ ($i,j=1,\ldots,n$, $i\neq j$) are nonnegative. In this
notation, $q_{ij}$ is the rate constant for the transition $A_{j}\to A_{i}$.
Any set of nonnegative coefficients $q_{ij}$ ($i\neq j$) corresponds to a
master equation. Therefore, the class of the master equations can be
represented as a nonnegative orthant in $\mathbb{R}^{n(n-1)}$ with coordinates
$q_{ij}$ ($i\neq j$). Equations of the same class describe any first order
kinetics in perfect mixtures. The only difference between the general first
order kinetics and master equation for the probability distribution is in the
balance conditions: the sum of probabilities should be 1, whereas the sum of
variables (concentrations) for the general first order kinetics may be any
positive number.
It is useful to mention that the model equation with equilibrium $P^{*}$ and
relaxation time $\tau$
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\frac{1}{\tau}(p_{i}^{*}-p_{i})\;\;(i=1,\ldots,n)$
(8)
is a particular case of master equation for normalized variables $p_{i}$
($p_{i}\geq 0$, $\sum_{i}p_{i}=1$). Indeed, let us take in Equation (7)
$q_{ij}=\frac{1}{\tau}p_{i}^{*}$.
The graph of transitions for a Markov chain is a directed graph. Its vertices
correspond to the states $A_{i}$ and the edges correspond to the transitions
$A_{j}\to A_{i}$ with the positive transition coefficients, $q_{ij}>0$. The
digraph of transitions is strongly connected if there exists an oriented path
from any vertex $A_{i}$ to every other vertex $A_{j}$ ($i\neq j$). The
continuous-time Markov chain is ergodic if there exists a unique strictly
positive equilibrium distribution $P^{*}$ ($p^{*}_{i}>0$,
$\sum_{i}p^{*}_{i}=1$) for master equation (7) MeynNets2007 ; MeynMarkCh2009 .
Strong connectivity of the graph of transitions is necessary and sufficient
for ergodicity of the corresponding Markov chain.
A digraph is weakly connected if the underlying undirected graph obtained by
replacing directed edges by undirected ones is connected. The maximal weakly
connected components of a digraph are called connected (or weakly connected)
components. The maximal strongly connected subgraphs are called strong
components. The necessary and sufficient condition for the existence of a
strongly positive equilibrium for master equation (7) is: the weakly connected
components of the transition graph are its strong components. An equivalent
form of this condition is: if there exists a directed path from $A_{i}$ to
$A_{j}$, then there exists a directed path from $A_{j}$ to $A_{i}$. In
chemical kinetics this condition is sometimes called the “weak reversibility”
condition Feinberg1974 ; SzederkHang2012 . This implies that the digraph is
the union of disjoint strongly connected digraphs. For each strong component
of the transition digraph the normalized equilibrium is unique and the
equilibrium for the whole graph is a convex combination of positive normalized
equilibria for its strong components. If $m$ is the number of these components
then the set of normalized positive equilibria of master equation ($P^{*}$:
$p^{*}_{i}>0$, $\sum_{i}p^{*}_{i}=1$) is a relative interior of a
$m-1$-dimensional polyhedron in the unit simplex $\Delta_{n}$. The set of non-
normalized positive equilibria ($P^{*}$: $p^{*}_{i}>0$) is a relative interior
of a $m$-dimensional cone in the positive orthant $\mathbb{R}_{+}^{n}$.
We reserve notation $\mathbb{R}_{+}^{n}$ for the positive orthant and for the
nonnegative orthant we use $\overline{\mathbb{R}_{+}^{n}}$ (the closure of
$\mathbb{R}_{+}^{n}$)
The Markov chain in Equation (7) is weakly ergodic if it allows the only
conservation law: the sum of coordinates, $\sum_{i}p_{i}\equiv const$. Such a
system forgets its initial condition: the distance between any two
trajectories with the same value of the conservation law tends to zero when
time goes to infinity. Among all possible norms, the $l_{1}$ distance
($\|P-Q\|_{l_{1}}=\sum_{i}|p_{i}-q_{i}|$) plays a special role: it does not
increase in time for any first order kinetic system in master equation (7) and
strongly monotonically decreases to zero for normalized probability
distributions ($\sum_{i}p_{i}=\sum_{i}q_{i}=1$) and weakly ergodic chains. The
difference between weakly ergodic and ergodic systems is in the obligatory
existence of a strictly positive equilibrium for an ergodic system. A Markov
chain is weakly ergodic if and only if for each two vertices
$A_{i},\>A_{j}\>(i\neq j)$ we can find such a vertex $A_{k}$ that is reachable
by oriented paths both from $A_{i}$ and from $A_{j}$. This means that the
following structure exists GorbanPathSumm2011 :
$A_{i}\to\ldots\to A_{k}\leftarrow\ldots\leftarrow A_{j}\ .$ (9)
One of the paths can be degenerated: it may be $i=k$ or $j=k$.
Now, let us restrict our consideration to the set of the Markov chains with
the given positive equilibrium distribution $P^{*}$ ($p^{*}_{i}>0$). We do not
assume that this distribution is compulsory unique. The transition graph
should be the union of disjoint strongly connected digraphs (in particular, it
may be strongly connected). Using the known positive equilibrium $P^{*}$ we
can rewrite master equation (7) in the following form
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}q_{ij}p^{*}_{j}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\
$ (10)
where $p_{i}^{*}$ and $q_{ij}$ are connected by the balance equation
$\sum_{j,\,j\neq i}q_{ij}p^{*}_{j}=\left(\sum_{j,\,j\neq
i}q_{ji}\right)p^{*}_{i}\;\mbox{ for all }i=1,\ldots,n$ (11)
For the next transformation of master equation we join the mutually reverse
transitions in pairs $A_{i}\rightleftharpoons A_{j}$ in pairs (say, $i>j$) and
introduce the stoichiometric vectors $\gamma^{ji}$ with coordinates:
$\gamma^{ji}_{k}=\left\\{\begin{array}[]{ll}-1&\mbox{ if }k=j,\\\ 1&\mbox{ if
}k=i,\\\ 0&\mbox{ otherwise}\end{array}\right.$ (12)
Let us rewrite the master equation (7) in the quasichemical form:
$\frac{{\mathrm{d}}P}{{\mathrm{d}}t}=\sum_{i>j}(w^{+}_{ij}-w^{-}_{ij})\gamma^{ji}$
(13)
where $w_{ij}^{+}=q_{ij}{p_{j}^{*}}\frac{p_{j}}{p_{j}^{*}}$ is the rate of the
transitions $A_{j}\to A_{i}$ and
$w_{ij}^{-}=q_{ji}{p_{i}^{*}}\frac{p_{i}}{p_{i}^{*}}$ is the rate of the
reverse process $A_{j}\leftarrow A_{i}$ ($i>j$).
The reversible systems with detailed balance form an important class of first
order kinetics. The detailed balance condition reads VanKampen : at
equilibrium, $w^{+}_{ij}=w^{-}_{ij}$, i.e.,
$q_{ij}p^{*}_{j}=q_{ji}p^{*}_{i}\,(=w_{ij}^{*})\;\;i,j=1,\ldots,n$ (14)
Here, $w_{ij}^{*}$ is the equilibrium flux from $A_{i}$ to $A_{j}$ and back.
For the systems with detailed balance the quasichemical form of the master
equation is especially simple:
$\frac{{\mathrm{d}}P}{{\mathrm{d}}t}=\sum_{i>j}w_{ij}^{*}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\gamma^{ji}$
(15)
It is important that any set of nonnegative equilibrium fluxes $w_{ij}^{*}$
($i>j$) defines by Equation (15) a system with detailed balance with a given
positive equilibrium $P^{*}$. Therefore, the set of all systems with detailed
balance presented by Equation (15) and a given equilibrium may be represented
as a nonnegative orthant in $\mathbb{R}^{\frac{n(n-1)}{2}}$ with coordinates
$w_{ij}^{*}$ ($i>j$).
The decomposition theorem GorbanEq2012arX ; GorbanCAMWA2013 states that for
any given positive equilibrium $P^{*}$ and any positive distribution $P$ the
set of possible values ${{\mathrm{d}}P}/{{\mathrm{d}}t}$ for Equations (13)
under the balance condition (11) coincides with the set of possible values
${{\mathrm{d}}P}/{{\mathrm{d}}t}$ for Equations (15) under detailed balance
condition (14).
In other words, for every general system of the form (13) with positive
equilibrium $P^{*}$ and any given non-equilibrium distribution $P$ there
exists a system with detailed balance of the form (15) with the same
equilibrium and the same value of the velocity vector
${{\mathrm{d}}P}/{{\mathrm{d}}t}$ at point $P$. Therefore, the sets of the
universal Lyapunov function for the general master equations and for the
master equations with detailed balance coincide.
## 3 General $H$-Theorem
Let $H(P)$ be a convex function on the space of distributions. It is a
Lyapunov function for a master equations with the positive equilibrium $P^{*}$
if ${\mathrm{d}}H(P(t))/{\mathrm{d}}t\leq 0$ for any positive normalized
solution $P(t)$. For a system with detailed balance given by Equation (15)
$\frac{{\mathrm{d}}H(P(t))}{{\mathrm{d}}t}=-\sum_{i>j}w_{ij}^{*}\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\left(\frac{\partial
H(P)}{\partial p_{j}}-\frac{\partial H(P)}{\partial p_{i}}\right)$ (16)
The inequality ${\mathrm{d}}H(P(t))/{\mathrm{d}}t\leq 0$ is true for all
nonnegative values of $w_{ij}^{*}$ if and only is it holds for any term in
Equation (16) separately. That is, for any pair $i,j$ ($i>j$) the convex
function $H(P)$ is a Lyapunov function for the system (15) where one and only
one $w_{ij}^{*}$ is not zero.
A convex function on a straight line is a Lyapunov function for a one-
dimensional system with single equilibrium if and only if the equilibrium is a
minimizer of this function. This elementary fact together with the previous
observation gives us the criterion for universal Lyapunov functions for
systems with detailed balance. Let us introduce the partial equilibria
criterion:
###### Definition 1 (Partial equilibria criterion).
A convex function $H(P)$ on the simplex $\Delta_{n}$ of probability
distributions satisfies the partial equilibria criterion with a positive
equilibrium $P^{*}$ if the proportion $p_{i}/p_{i}^{*}=p_{j}/p_{j}^{*}$ give
the minimizers in the problem (6).
###### Proposition 1.
A convex function $H(P)$ on the simplex $\Delta_{n}$ of probability
distributions is a Lyapunov function for all master equations with the given
equilibrium $P^{*}$ that obey the principle of detailed balance if and only if
it satisfies the partial equilibria criterion with the equilibrium $P^{*}$.
Combination of this Proposition with the decomposition theorem GorbanEq2012arX
gives the same criterion for general master equations without hypothesis about
detailed balance
###### Proposition 2.
A convex function $H(P)$ on the simplex $\Delta_{n}$ of probability
distributions is a Lyapunov function for all master equations with the given
equilibrium $P^{*}$ if and only if it satisfies the partial equilibria
criterion with the equilibrium $P^{*}$.
These two propositions together form the general $H$-theorem.
###### Theorem 1.
The partial equilibria criterion with a positive equilibrium $P^{*}$ is a
necessary condition for a convex function to be the universal Lyapunov
function for all master equations with detailed balance and equilibrium
$P^{*}$ and a sufficient condition for this function to be the universal
Lyapunov function for all master equations with equilibrium $P^{*}$.
Let us stress that here the partial equilibria criterion provides a necessary
condition for systems with detailed balance (and, therefore, for the general
systems without detailed balance assumption) and a sufficient condition for
the general systems (and, therefore, for the systems with detailed balance
too).
Figure 1: The triangle of distributions for the system with three states
$A_{1}$, $A_{2}$, $A_{3}$ and the equilibrium $p_{1}^{*}=\frac{4}{7}$,
$p_{2}^{*}=\frac{2}{7}$, $p_{3}^{*}=\frac{1}{7}$. The lines of partial
equilibria $A_{i}\rightleftharpoons A_{j}$ given by the proportions
$p_{i}/p_{i}^{*}=p_{j}/p_{j}^{*}$ are shown, for $A_{1}\rightleftharpoons
A_{2}$ by solid straight lines (with one end at the vertex $A_{3}$), for
$A_{2}\rightleftharpoons A_{3}$ and for $A_{1}\rightleftharpoons A_{3}$ by
dashed lines. The lines of conditional minima of $H(P)$ in problem (6) are
presented for the partial equilibrium $A_{1}\rightleftharpoons A_{2}$ (a) for
the squared Euclidean distance (a circle here is an example of the $H(P)$
level set) and (b) for the Itakura-Saito distance. Between these lines and the
line of partial equilibria the “no $H$-theorem zone” is situated. In this
zone, $H(P)$ increases in time for some master equations with equilibrium
$P^{*}$. Similar zones (not shown) exist near other partial equilibrium lines
too. Outside these zones, $H(P)$ monotonically decreases in time for any
master equation with equilibrium $P^{*}$.
## 4 Examples
The simplest Bregman divergence is the squared Euclidean distance between $P$
and $P^{*}$, $\sum_{i}(p_{i}-p_{i}^{*})^{2}$. The solution to the problem (6)
is: $p_{i}-p_{i}^{*}=p_{j}-p_{j}^{*}$. Obviously, it differs from the
proportion required by the partial equilibria criterion
$\frac{p_{i}}{p_{j}}=\frac{p_{i}^{*}}{p_{j}^{*}}$ (Figure 1a).
For the Itakura-Saito distance (2) the solution to the problem (6) is:
$\frac{1}{p_{i}}-\frac{1}{p_{i}^{*}}=\frac{1}{p_{j}}-\frac{1}{p_{j}^{*}}$. It
also differs from the proportion required (Figure 1b).
If the single equilibrium in 1D system is not a minimizer of a convex function
$H$ then ${\mathrm{d}}H/{\mathrm{d}}t>0$ on the interval between the
equilibrium and minimizer of $H$ (or minimizers if it is not unique).
Therefore, if $H(P)$ does not satisfy the partial equilibria criterion then in
the simplex of distributions there exists an area bordered by the partial
equilibria surface for $A_{i}\rightleftharpoons A_{j}$ and by the minimizers
for the problem (6), where for some master equations
${\mathrm{d}}H/{\mathrm{d}}t>0$ (Figure 1). In particular, in such an area
${\mathrm{d}}H/{\mathrm{d}}t>0$ for the simple system with two mutually
reverse transitions, $A_{i}\rightleftharpoons A_{j}$, and the same
equilibrium.
If $H$ satisfies the partial equilibria criterion, then the minimizers for the
problem (6) coincide with the partial equilibria surface for
$A_{i}\rightleftharpoons A_{j}$, and the “no $H$-theorem zone” vanishes.
The partial equilibria criterion allows a simple geometric interpretation. Let
us consider a sublevel set of $H(P)$ in the simplex $\Delta_{n}$:
$U_{h}=\\{P\in\Delta_{n}\ |\ H(P)\leq h\\}$. Let the level set be
$L_{h}=\\{P\in\Delta_{n}\ |\ H(P)=h\\}$. For the partial equilibrium
$A_{i}\rightleftharpoons A_{j}$ we use the notation $E_{ij}$. It is given by
the equation $p_{i}/p_{i}^{*}=p_{j}/p_{j}^{*}$. The geometric equivalent of
the partial equilibrium condition is: for all $i,j$ ($i\neq j$) and every
$P\in L_{h}\cap E_{ij}$ the straight line $P+\lambda\gamma_{ij}$
($\lambda\in\mathbb{R}$) is a supporting line of $U_{h}$. This means that this
line does not intersect the interior of $U_{h}$.
We illustrate this condition on the plane for three states in Figure 2. The
level set of $H$ is represented by the dot-dash line. It intersects the lines
of partial equilibria (dashed lines) at points $B_{1,2,3}$ and $C_{1,2,3}$.
For each point $P$ from these six intersections ($P=B_{i}$ and $P=C_{i}$) the
line $P+\lambda\gamma_{jk}$ ($\lambda\in\mathbb{R}$) should be a supporting
line of the sublevel set (the region bounded by the dot-dash line). Here,
$i,j,k$ should all be different numbers. Segments of these lines form a
hexagon circumscribed around the level set (Figure 2b).
The points of intersection $B_{1,2,3}$ and $C_{1,2,3}$ cannot be selected
arbitrarily on the lines of partial equilibria. First of all, they should be
the vertices of a convex hexagon with the equilibrium $P^{*}$ inside.
Secondly, due to the partial equilibria criterion, the intersections of the
straight line $P+\lambda\gamma_{ij}$ with the partial equilibria $E_{ij}$ are
the conditional minimizers of $H$ on this line, and therefore should belong to
the sublevel set $U_{H(P)}$. If we apply this statement to $P=B_{i}$ and
$P=C_{i}$, then we will get two projections of this point onto partial
equilibria $E_{ij}$ parallel to $\gamma_{ij}$ (Figure 2a). These projections
should belong to the hexagon with the vertices $B_{1,2,3}$ and $C_{1,2,3}$.
They produce a six-ray star that should be inscribed into the level set.
In Figure 2 we present the following characterization of the level set of a
Lyapunov function for the Markov chains with three states. This convex set
should be circumscribed around the six-ray star (Figure 2a) and inscribed in
the hexagon of the supporting lines (Figure 2b).
Figure 2: Geometry of the Lyapunov function level set. The triangle of
distributions for the system with three states $A_{1}$, $A_{2}$, $A_{3}$ and
the equilibrium $p_{1}^{*}=\frac{4}{7}$, $p_{2}^{*}=\frac{2}{7}$,
$p_{3}^{*}=\frac{1}{7}$. The lines of partial equilibria
$A_{i}\rightleftharpoons A_{j}$ given by the proportions
$p_{i}/p_{i}^{*}=p_{j}/p_{j}^{*}$ are shown by dashed lines. The dash-dot line
is the level set of a Lyapunov function $H$. It intersects the lines of
partial equilibria at points $B_{1,2,3}$ and $C_{1,2,3}$. (The points $B_{i}$
are close to the vertices $A_{i}$, the points $C_{i}$ belong to the same
partial equilibrium but on another side of the equilibrium $P^{*}$.) For each
point $B_{i}$, $C_{i}$ the corresponding partial equilibria of two transitions
$A_{i}\rightleftharpoons A_{j}$ ($j\neq i$) are presented (a). These partial
equilibria should belong to the sublevel set of $H$. They are the projections
of $B_{i}$, $C_{i}$ onto the lines of partial equilibria
$A_{i}\rightleftharpoons A_{j}$ ($j\neq i$) with projecting rays parallel to
the sides $[A_{i},A_{j}]$ of the triangle (i.e., to the stoichiometric vectors
$\gamma^{ji}$ (12)). The six-ray star with vertices $B_{i}$, $C_{i}$ should be
inside the dash-dot contour (a). Therefore, the projection of $B_{i}$ onto the
partial equilibrium $A_{i}\rightleftharpoons A_{j}$ should belong to the
segment $[C_{k},P^{*}]$ and the projection of $C_{i}$ onto the partial
equilibrium $A_{i}\rightleftharpoons A_{j}$ should belong to the segment
$[B_{k},P^{*}]$ (a). The lines parallel to the sides $A_{j},A_{k}$ of the
triangle should be supporting lines of the level set of $H$ at points $B_{i}$,
$C_{i}$ ($i,j,k$ are different numbers) (b). Segments of these lines form a
circumscribed hexagon around the level set (b).
All the $f$-divergences given by Equation (1) satisfy the partial equilibria
criterion and are the universal Lyapunov functions but the reverse is not
true: the class of universal Lyapunov functions is much wider than the set of
the $f$-divergences. Let us consider the set “$PEC$” of convex functions
$H(P\|P^{*})$, which satisfy the partial equilibria criterion. It is closed
with respect to the following operations
* •
Conic combination: if $H_{j}(P\|P^{*})\in PEC$ then
$\sum_{j}\alpha_{j}H_{j}(P\|P^{*})\in PEC$ for nonnegative coefficients
$\alpha_{j}\geq 0$.
* •
Convex monotonic transformation of scale: if $H(P\|P^{*})\in PEC$ then
$F(H(P\|P^{*}))\in PEC$ for any convex monotonically increasing function of
one variable $F$.
Using these operations we can construct new universal Lyapunov functions from
a given set. For example,
$\frac{1}{2}\sum_{i}\frac{(p_{i}-p_{i}^{*})^{2}}{p_{i}^{*}}+\prod_{j}\exp\frac{(p_{j}-p_{j}^{*})^{2}}{2p_{j}^{*}}$
is a universal Lyapunov function that does not have the $f$-divergence form
because the first sum is an $f$-divergence given by Equation (1) with
$h(x)=\frac{1}{2}(x-1)^{2}$ and the product is the exponent of this
$f$-divergence (exp is convex and monotonically increasing function).
The following function satisfies the partial equilibria criterion for every
$\varepsilon>0$.
$\frac{1}{2}\sum_{i}\frac{(p_{i}-p_{i}^{*})^{2}}{p_{i}^{*}}+\frac{\varepsilon}{4n^{2}}\prod_{i,j,i\neq
j}{({p_{i}}{p_{j}^{*}}-{p_{j}}{p_{i}^{*}})^{2}}$ (17)
It is convex for $0<\varepsilon<1$. (Just apply the Gershgorin theorem Golub
to the Hessian and use that all $p_{i},p_{i}^{*}\leq 1$.) Therefore, it is a
universal Lyapunov function for master equation in $\Delta_{n}$ if
$0<\varepsilon<1$. The partial equilibria criterion together with the
convexity condition allows us to construct many such examples.
## 5 General $H$-Theorem for Nonlinear Kinetics
### 5.1 Generalized Mass Action Law
Several formalisms are developed in chemical kinetics and non-equilibrium
thermodynamics for the construction of general kinetic equations with a given
“thermodynamic Lyapunov functional”. The motivation of this approach “from
thermodynamics to kinetics” is simple G1 ; YBGE : (i) the thermodynamic data
are usually more reliable than data about kinetics and we know the
thermodynamic functions better than the details of kinetic equations, and (ii)
positivity of entropy production is a fundamental law and we prefer to respect
it “from scratch”, by the structure of kinetic equations.
GMAL is a method for the construction of dissipative kinetic equations for a
given thermodynamic potential $H$. Other general thermodynamic approaches
GENERIC ; Grmela2012 ; GiovangigliMatus2012 give similar results for a given
stoichiometric algebra. Below we introduce GMAL following G1 ; YBGE ;
GorbanShahzad2011 .
The list of components is a finite set of symbols $A_{1},\ldots,A_{n}$.
A reaction mechanism is a finite set of the stoichiometric equations of
elementary reactions:
$\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho i}A_{i}\,$ (18)
where $\rho=1,\ldots,m$ is the reaction number and the stoichiometric
coefficients $\alpha_{\rho i}$, $\beta_{\rho i}$ are nonnegative numbers.
Usually, these numbers are assumed to be integer but in some applications the
construction should be more flexible and admit real nonnegative values. Let
$\alpha_{\rho}$, $\beta_{\rho}$ be the vectors with coordinates $\alpha_{\rho
i}$, $\beta_{\rho i}$ correspondingly.
A stoichiometric vector $\gamma_{\rho}$ of the reaction in Equation (18) is a
$n$-dimensional vector $\gamma_{\rho}=\beta_{\rho}-\alpha_{\rho}$ with
coordinates
$\gamma_{\rho i}=\beta_{\rho i}-\alpha_{\rho i}\,$ (19)
that is, “gain minus loss” in the $\rho$th elementary reaction. We assume
$\alpha_{\rho}\neq\beta_{\rho}$ to avoid trivial reactions with zero
$\gamma_{\rho}$.
One of the standard assumptions is existence of a strictly positive
stoichiometric conservation law, a vector $b=(b_{i})$, $b_{i}>0$ such that
$\sum_{i}b_{i}\gamma_{\rho i}=0$ for all $\rho$. This may be the conservation
of mass, of the total probability, or of the total number of atoms, for
example.
A nonnegative extensive variable $N_{i}$, the amount of $A_{i}$, corresponds
to each component. We call the vector $N$ with coordinates $N_{i}$ “the
composition vector”. The concentration of $A_{i}$ is an intensive variable
$c_{i}=N_{i}/V$, where $V>0$ is the volume. The vector $c=N/V$ with
coordinates $c_{i}$ is the vector of concentrations.
Let us consider a domain $U$ in $n$-dimensional real vector space
$\mathbb{R}^{n}$ with coordinates $N_{1},\ldots,N_{n}\geq 0$
($U\subset\overline{\mathbb{R}_{+}^{n}}$). For each $N_{i}$, a dimensionless
entropy (or free entropy, for example, Massieu, Planck, or Massieu-Planck
potential that corresponds to the selected conditions Callen1985 ) $S(N)$ is
defined in $U$. “Dimensionless” means that we use $S/R$ instead of physical
$S$. This choice of units corresponds to the informational entropy ($p\log p$
instead of $k_{\rm B}p\ln p$).
The dual variables, potentials, are defined as the partial derivatives of
$H=-S$:
$\check{\mu}_{i}=\frac{\partial H}{\partial
N_{i}},\;\;\check{\mu}=\nabla_{N}H$ (20)
This definition differs from the chemical potentials Callen1985 by the factor
${1}/{RT}$. We keep the same sign as for the chemical potentials, and this
differs from the standard Legendre transform for $S$. (It is the Legendre
transform for the function $H=-S$.) The standard condition for the
reversibility of the Legendre transform is strong positive definiteness of the
Hessian of $H$.
For each reaction, a nonnegative quantity, reaction rate $r_{\rho}$ is
defined. We assume that this quantity has the following structure (compare
with Equations (4), (7), and (14) in Grmela2012 and Equation (4.10) in
GiovangigliMatus2012 ):
$r_{\rho}=\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})$ (21)
where $(\alpha_{\rho},\check{\mu})=\sum_{i}\alpha_{\rho i}\check{\mu}_{i}$ is
the standard inner product. Here and below, $\exp(\;\ ,\;)$ is the exponent of
the standard inner product. The kinetic factor $\varphi_{\rho}\geq$ is an
intensive quantity and the expression $\exp(\alpha_{\rho},\check{\mu})$ is the
Boltzmann factor of the $\rho$th elementary reaction.
In the standard formalism of chemical kinetics the reaction rates are
intensive variables and in kinetic equations for $N$ an additional factor—the
volume—appears. For heterogeneous systems, there may be several “volumes”
(including interphase surfaces).
A nonnegative extensive variable $N_{i}$, the amount of $A_{i}$, corresponds
to each component. We call the vector $N$ with coordinates $N_{i}$ “the
composition vector”. $N\in\overline{\mathbb{R}_{+}^{n}}$. The concentration of
$A_{i}$ is an intensive variable $c_{i}=N_{i}/V$, where $V>0$ is the volume.
If the system is heterogeneous then there are several “volumes” (volumes,
surfaces, etc.), and in each volume there are the composition vector and the
vector of concentrations YBGE ; GorbanShahzad2011 . Here we will consider
homogeneous systems.
The kinetic equations for a homogeneous system in the absence of external
fluxes are
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=V\sum_{\rho}r_{\rho}\gamma_{\rho}=V\sum_{\rho}\gamma_{\rho}\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})$
(22)
If the volume is not constant then the equations for concentrations include
$\dot{V}$ and have different form (this is typical for combustion reactions,
for example).
The classical Mass Action Law gives us an important particular case of GMAL
given by Equation (21). Let us take the perfect free entropy
$S=-\sum_{i}N_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{*}}\right)-1\right)$ (23)
where $c_{i}=N_{i}/V\geq 0$ are concentrations and $c_{i}^{*}>0$ are the
standard equilibrium concentrations.
For the perfect entropy function presented in Equation (23)
$\check{\mu}_{i}=\ln\left(\frac{c_{i}}{c_{i}^{*}}\right)\,,\;\exp(\alpha_{\rho},\check{\mu})=\prod_{i}\left(\frac{c_{i}}{c_{i}^{*}}\right)^{\alpha_{\rho
i}}$ (24)
and for the GMAL reaction rate function given by (21) we get
$r_{\rho}=\varphi_{\rho}\prod_{i}\left(\frac{c_{i}}{c_{i}^{*}}\right)^{\alpha_{\rho
i}}$ (25)
The standard assumption for the Mass Action Law in physics and chemistry is
that $\varphi$ and $c^{*}$ are functions of temperature:
$\varphi_{\rho}=\varphi_{\rho}(T)$ and $c^{*}_{i}=c^{*}_{i}(T)$. To return to
the kinetic constants notation and in particular to first order kinetics in
the quasichemical form presented in Equation (13), we should write:
$\frac{\varphi_{\rho}}{\prod_{i}{c_{i}^{*}}^{\alpha_{\rho i}}}=k_{\rho}$ (26)
### 5.2 General Entropy Production Formula
Thus, the following entities are given: the set of components $A_{i}$
($i=1,\ldots,n$), the set of $m$ elementary reactions presented by
stoichiometric equations (18), the thermodynamic Lyapunov function
$H(N,V,\ldots)$ Callen1985 ; YBGE ; Hangos2010 , where dots (marks of
omission) stand for the quantities that do not change in time under given
conditions, for example, temperature for isothermal processes or energy for
isolated systems. The GMAL presents the reaction rate $r_{\rho}$ in Equation
(21) as a product of two factors: the Boltzmann factor and the kinetic factor.
Simple algebra gives for the time derivative of $H$:
$\begin{split}\frac{{\mathrm{d}}H}{{\mathrm{d}}t}&=\sum_{i}\frac{\partial
H}{\partial N_{i}}\frac{{\mathrm{d}}N_{i}}{{\mathrm{d}}t}\\\
&=\sum_{i}\check{\mu}_{i}V\sum_{\rho}\gamma_{\rho
i}\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})\\\
&=V\sum_{\rho}(\gamma_{\rho},\check{\mu})\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})\end{split}$
(27)
An auxiliary function $\theta(\lambda)$ of one variable $\lambda\in[0,1]$ is
convenient for analysis of ${\mathrm{d}}S/{\mathrm{d}}t$ (see G1 ;
GorbanShahzad2011 ; OrlovRozonoer1984 ):
$\theta(\lambda)=\sum_{\rho}\varphi_{\rho}\exp[(\check{\mu},(\lambda\alpha_{\rho}+(1-\lambda)\beta_{\rho}))]$
(28)
With this function, $\dot{H}$ defined by Equation (27) has a very simple form:
$\frac{{\mathrm{d}}H}{{\mathrm{d}}t}=-V\left.\frac{{\mathrm{d}}\theta(\lambda)}{{\mathrm{d}}\lambda}\right|_{\lambda=1}$
(29)
The auxiliary function $\theta(\lambda)$ allows the following interpretation.
Let us introduce the deformed stoichiometric mechanism with the stoichiometric
vectors,
$\alpha_{\rho}(\lambda)=\lambda\alpha_{\rho}+(1-\lambda)\beta_{\rho}\,,\;\beta_{\rho}(\lambda)=\lambda\beta_{\rho}+(1-\lambda)\alpha_{\rho}$
(30)
which is the initial mechanism when $\lambda=1$, the reverted mechanism with
interchange of $\alpha$ and $\beta$ when $\lambda=0$, and the trivial
mechanism (the left and right hand sides of the stoichiometric equations
coincide) when $\lambda=1/2$. Let the deformed reaction rate be
$r_{\rho}(\lambda)=\varphi_{\rho}\exp(\alpha_{\rho}(\lambda),\check{\mu})$
(the genuine kinetic factor is combined with the deformed Boltzmann factor).
Then $\theta(\lambda)=\sum_{\rho}r_{\rho}(\lambda)$.
It is easy to check that $\theta^{\prime\prime}(\lambda)\geq 0$ and,
therefore, $\theta(\lambda)$ is a convex function.
The inequality
$\theta^{\prime}(1)\geq 0$ (31)
is necessary and sufficient for accordance between kinetics and thermodynamics
(decrease of free energy or positivity of entropy production). This inequality
is a condition on the kinetic factors. Together with the positivity condition
$\varphi_{\rho}\geq 0$, it defines a convex cone in the space of vectors of
kinetic factors $\varphi_{\rho}$ ($\rho=1,\ldots,m$). There exist two less
general and more restrictive sufficient conditions: detailed balance and
complex balance (known also as semidetailed or cyclic balance).
### 5.3 Detailed Balance
The detailed balance condition consists of two assumptions: (i) for each
elementary reaction $\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho
i}A_{i}$ in the mechanism (18) there exists a reverse reaction
$\sum_{i}\alpha_{\rho i}A_{i}\leftarrow\sum_{i}\beta_{\rho i}A_{i}$. Let us
join these reactions in pairs
$\sum_{i}\alpha_{\rho i}A_{i}\rightleftharpoons\sum_{i}\beta_{\rho i}A_{i}$
(32)
After this joining, the total number of stoichiometric equations decreases. We
distinguish the reaction rates and kinetic factors for direct and inverse
reactions by the upper plus or minus:
$r_{\rho}^{+}=\varphi_{\rho}^{+}\exp(\alpha_{\rho},\check{\mu})\,,\;r_{\rho}^{-}=\varphi_{\rho}^{-}\exp(\beta_{\rho},\check{\mu})$
(33)
The kinetic equations take the form
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=V\sum_{\rho}(r^{+}_{\rho}-r^{-}_{\rho})\gamma_{\rho}$
(34)
The condition of detailed balance in GMAL is simple and elegant:
$\varphi_{\rho}^{+}=\varphi_{\rho}^{-}$ (35)
For the systems with detailed balance we can take
$\varphi_{\rho}=\varphi_{\rho}^{+}=\varphi_{\rho}^{-}$ and write for the
reaction rate:
$r_{\rho}=r^{+}_{\rho}-r^{-}_{\rho}=\varphi_{\rho}(\exp(\alpha_{\rho},\check{\mu})-\exp(\beta_{\rho},\check{\mu}))$
(36)
M. Feinberg called this kinetic law the “Marselin-De Donder” kinetics
Feinberg1972_a .
Under the detailed balance conditions, the auxiliary function
$\theta(\lambda)$ is symmetric with respect to change
$\lambda\mapsto(1-\lambda)$. Therefore, $\theta(1)=\theta(0)$ and, because of
convexity of $\theta(\lambda)$, the inequality holds: $\theta^{\prime}(1)\geq
0$. Therefore, $\dot{H}\leq 0$ and kinetic equations obey the second law of
thermodynamics.
The explicit formula for $\dot{H}\leq 0$ has the well known form since
Boltzmann proved his $H$-theorem in 1872:
$\frac{{\mathrm{d}}H}{{\mathrm{d}}t}=-V\sum_{\rho}(\ln r_{\rho}^{+}-\ln
r_{\rho}^{-})(r_{\rho}^{+}-r_{\rho}^{-})\leq 0$ (37)
A convenient equivalent form of $\dot{H}\leq 0$ is proposed in GorMirYab2013 :
$\frac{{\mathrm{d}}H}{{\mathrm{d}}t}=-V\sum_{\rho}(r_{\rho}^{+}+r_{\rho}^{-})\mathbb{A}_{\rho}\tanh\frac{\mathbb{A}_{\rho}}{2}\leq
0$ (38)
where
$\mathbb{A}_{\rho}=-(\gamma_{\rho},\check{\mu})\;(=-{(\gamma_{\rho},\mu)}/{RT},\mbox{
where }\mu\mbox{ is the chemical potential})$
is a normalized affinity. In this formula, the kinetic information is
collected in the nonnegative factors, the sums of reaction rates
$(r_{\rho}^{+}+r_{\rho}^{-})$. The purely thermodynamic multiplier
$\mathbb{A}\tanh({\mathbb{A}}/{2})\geq 0$ is positive for non-zero
$\mathbb{A}$. For small $|\mathbb{A}|$, the expression
$\mathbb{A}\tanh({\mathbb{A}}/{2})$ behaves like $\mathbb{A}^{2}/2$ and for
large $|\mathbb{A}|$ it behaves like the absolute value, $|\mathbb{A}|$.
The detailed balance condition reflects “microreversibility”, that is, time-
reversibility of the dynamic microscopic description and was first introduced
by Boltzmann in 1872 as a consequence of the reversibility of collisions in
Newtonian mechanics.
### 5.4 Complex Balance
The complex balance condition was invented by Boltzmann in 1887 for the
Boltzmann equation Boltzmann1887 as an answer to the Lorentz objections
Lorentz1887 against Boltzmann’s proof of the $H$-theorem. Stueckelberg
demonstrated in 1952 that this condition follows from the Markovian
microkinetics of fast intermediates if their concentrations are small
Stueckelberg1952 . Under this asymptotic assumption this condition is just the
probability balance condition for the underlying Markov process. (Stueckelberg
considered this property as a consequence of “unitarity” in the $S$-matrix
terminology.) It was known as the semidetailed or cyclic balance condition.
This condition was rediscovered in the framework of chemical kinetics by Horn
and Jackson in 1972 HornJackson1972 and called the complex balance condition.
Now it is used for chemical reaction networks in chemical engineering
SzederkHangos2011 . Detailed analysis of the backgrounds of the complex
balance condition is given in GorbanShahzad2011 .
Formally, the complex balance condition means that $\theta(1)\equiv\theta(0)$
for all values of $\check{\mu}$. We start from the initial stoichiometric
equations (18) without joining the direct and reverse reactions. The equality
$\theta(1)\equiv\theta(0)$ reads
$\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\alpha_{\rho})=\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\beta_{\rho})$
(39)
Let us consider the family of vectors $\\{\alpha_{\rho},\beta_{\rho}\\}$
($\rho=1,\ldots,m$). Usually, some of these vectors coincide. Assume that
there are $q$ different vectors among them. Let $y_{1},\ldots,y_{q}$ be these
vectors. For each $j=1,\ldots,q$ we take
$R_{j}^{+}=\\{\rho\,|\,\alpha_{\rho}=y_{j}\\}\,,\;R_{j}^{-}=\\{\rho\,|\,\beta_{\rho}=y_{j}\\}$
(40)
We can rewrite Equation (39) in the form
$\sum_{j=1}^{q}\exp(\check{\mu},y_{j})\left[\sum_{\rho\in
R_{j}^{+}}\varphi_{\rho}-\sum_{\rho\in R_{j}^{-}}\varphi_{\rho}\right]=0$ (41)
The Boltzmann factors $\exp(\check{\mu},y_{j})$ are linearly independent
functions. Therefore, the natural way to meet these conditions is: for any
$j=1,\ldots,q$
$\sum_{\rho\in R_{j}^{+}}\varphi_{\rho}-\sum_{\rho\in
R_{j}^{-}}\varphi_{\rho}=0$ (42)
This is the general complex balance condition. This condition is sufficient
for the inequality $\dot{H}=\theta^{\prime}(1)\leq 0$, because it provides the
equality $\theta(1)=\theta(0)$ and $\theta(\lambda)$ is a convex function.
It is easy to check that for the first order kinetics given by Equation (10)
(or Equation (13)) with positive equilibrium, the complex balance condition is
just the balance equation (11) and always holds.
### 5.5 Cyclic Decomposition of the Systems with Complex Balance
The complex balance conditions defined by Equation (42) allow a simple
geometric interpretation. Let us introduce the digraph of transformation of
complexes. The vertices of this digraph correspond to the formal sums $(y,A)$
(“complexes”), where $A$ is the vector of components, and
$y\in\\{y_{1},\ldots,y_{q}\\}$ are vectors $\alpha_{\rho}$ or $\beta_{\rho}$
from the stoichiometric equations of the elementary reactions (18). The edges
of the digraph correspond to the elementary reactions with non-zero kinetic
factor.
Let us assign to each edge $(\alpha_{\rho},A)\to(\beta_{\rho},A)$ the
auxiliary current—the kinetic factor $\varphi_{\rho}$. For these currents, the
complex balance condition presented by Equation (42) is just Kirchhoff’s first
rule: the sum of the input currents is equal to the sum of the output currents
for each vertex. (We have to stress that these auxiliary currents are not the
actual rates of transformations.)
Let us use for the vertices the notation $\Theta_{j}$: $\Theta_{j}=(y_{j},A)$,
($j=1,\ldots,q$) and denote $\varphi_{lj}$ the fluxes for the edge
$\Theta_{j}\to\Theta_{l}$.
The simple cycle is the digraph
$\Theta_{i_{1}}\to\Theta_{i_{2}}\to\ldots\to\Theta_{i_{k}}\to\Theta_{i_{1}}$,
where all the complexes $\Theta_{i_{l}}$ ($l=1,\ldots,k$) are different. We
say that the simple cycle is normalized if all the corresponding auxiliary
fluxes are unit: $\varphi_{i_{j+1}\,i_{j}}=\varphi_{i_{1}\,i_{k}}=1$.
The graph of the transformation of complexes cannot be arbitrary if the system
satisfies the complex balance condition Feinberg1974 .
###### Proposition 3.
If the system satisfies the complex balance condition (i.e. Equation (42)
holds) then every edge of the digraph of transformation of complexes is
included into a simple cycle.
###### Proof.
First of all, let us formulate Kirchhoff’s first rule (42) for subsets: if the
digraph of transformation of complexes satisfies Equation (42), then for any
set of complexes $\Omega$
$\sum_{\Theta\in\Omega,\Phi\notin\Omega}\varphi_{\Theta\Phi}=\sum_{\Theta\in\Omega,\Phi\notin\Omega}\varphi_{\Phi\Theta}$
(43)
where $\varphi_{\Phi\Theta}$ is the positive kinetic factor for the reaction
$\Theta\to\Phi$ if it belongs to the reaction mechanism (i.e., the edge
$\Theta\to\Phi$ belongs to the digraph of transformations) and
$\varphi_{\Phi\Theta}=0$ if it does not. Equation (43) is just the result of
summation of Equations (42) for all $(y_{j},A)=\Theta\in\Omega$.
We say that a state $\Theta_{j}$ is reachable from a state $\Theta_{k}$ if
$k=i$ or there exists a non-empty chain of transitions with non-zero
coefficients that starts at $\Theta_{k}$ and ends at $\Theta_{j}$:
$\Theta_{k}\to\ldots\to\Theta_{j}$. Let $\Theta_{i\downarrow}$ be the set of
states reachable from $\Theta_{i}$. The set $\Theta_{i\downarrow}$ has no
output edges.
Assume that the edge $\Theta_{j}\to\Theta_{i}$ is not included in a simple
cycle, which means $\Theta_{j}\notin\Theta_{i\downarrow}$. Therefore, the set
$\Omega=\Theta_{i\downarrow}$ has the input edge ($\Theta_{j}\to\Theta_{i}$)
but no output edges and cannot satisfy Equation (43). This contradiction
proves the proposition.∎
This property (every edge is included in a simple cycle) is equivalent to the
so-called “weak reversibility” or to the property that every weakly connected
component of the digraph is its strong component.
For every graph with the system of fluxes, which obey Kirchhoff’s first rule,
the cycle decomposition theorem holds. It can be extracted from many books and
papers MeynNets2007 ; GorbanEq2012arX ; Kalpazidou2006 . Let us recall the
notion of extreme ray. A ray with direction vector $x\neq 0$ is a set
$\\{\lambda x\\}$ ($\lambda\geq 0$). A ray $l$ is an extreme ray of a cone
$\mathbf{Q}$ if for any $u\in l$ and any $x,y\in\mathbf{Q}$, whenever
$u=(x+y)/2$, we must have $x,y\in l$. If a closed convex cone does not include
a whole straight line then it is the convex hull of its extreme rays
Rockafellar1970 .
Let us consider a digraph $Q$ with vertices $\Theta_{i}$, the set of edges $E$
and the system of auxiliary fluxes along the edges $\varphi_{ij}\geq 0$
($(j,i)\in E$). The set of all nonnegative functions on $E$,
$\varphi:(j,i)\mapsto\varphi_{ij}$, is a nonnegative orthant
$\mathbb{R}^{|E|}_{+}$. Kirchhoff’s first rule (Equation (42)) together with
nonnegativity of the kinetic factors define a cone of the systems with complex
balance $\mathcal{Q}\subset\mathbb{R}^{|E|}_{+}$.
###### Proposition 4 (Cycle decomposition of systems with complex balance).
Every extreme ray of $\mathcal{Q}$ has a direction vector that corresponds to
a simple normalized cycle
$\Theta_{i_{1}}\to\Theta_{i_{2}}\to\ldots\to\Theta_{i_{k}}\to\Theta_{i_{1}}$,
where all the complexes $\Theta_{i_{l}}$ ($l=1,\ldots,k$) are different, all
the corresponding fluxes are unit,
$\varphi_{i_{j+1}\,i_{j}}=\varphi_{i_{1}\,i_{k}}=1$, and other fluxes are
zeros.
###### Proof.
Let a function $\phi:E\to\mathbb{R}_{+}$ be an extreme ray of $\mathcal{Q}$
and $\mbox{supp}\phi=\\{(j,i)\in E\,|\,\phi_{ij}>0\\}$. Due to Proposition 3
each edge from $\mbox{supp}\phi$ is included in a simple cycle formed by edges
from $\mbox{supp}\phi$. Let us take one this cycle
$\Theta_{i_{1}}\to\Theta_{i_{2}}\to\ldots\to\Theta_{i_{k}}\to\Theta_{i_{1}}$.
Denote the fluxes of the corresponding simple normalized cycle by $\psi$. It
is a function on $E$: $\psi_{i_{j+1}\,i_{j}}=\psi_{i_{1}\,i_{k}}=1$ and
$\psi_{ij}=0$ if $(i,j)\in E$ but $(i,j)\neq(i_{j+1},i_{j})$ and
$(i,j)\neq(i_{1},i_{k})$ ($i,j=1,\ldots,k$, $i\neq j$).
Assume that $\mbox{supp}\phi$ includes at least one edge that does not belong
to the cycle
$\Theta_{i_{1}}\to\Theta_{i_{2}}\to\ldots\to\Theta_{i_{k}}\to\Theta_{i_{1}}$.
Then, for sufficiently small $\kappa>0$, $\phi\pm\kappa\psi\in\mathcal{Q}$ and
the vector $\phi\pm\kappa\psi$ is not proportional to $\phi$. This
contradiction proves the proposition.∎
This decomposition theorem explains why the complex balance condition was
often called the “cyclic balance condition”.
### 5.6 Local Equivalence of Systems with Detailed and Complex Balance
The class of systems with detailed balance is the proper subset of the class
of systems with complex balance. A simple (irreversible) cycle of the length
$k>2$ gives a simplest and famous example of the complex balance system
without detailed balance condition.
For Markov chains, the complex balance systems are all the systems that have a
positive equilibrium distribution presented by Equation (11), whereas the
systems with detailed balance form the proper subclass of the Markov chains,
the so-called reversible chains.
In nonlinear kinetics, the systems with complex balance provide the natural
generalization of the Markov processes. They deserve the term “nonlinear
Markov processes”, though it is occupied by a much wider notion Kolokoltzov .
The systems with detailed balance form the proper subset of this class.
Nevertheless, in some special sense the classes of systems with detailed
balance and with the complex balance are equivalent. Let us consider a
thermodynamic state given by the vector of potentials $\check{\mu}$ defined by
Equation (20). Let all the reactions in the reaction mechanism be reversible
(i.e., for every transition $\Theta_{i}\to\Theta_{j}$ the reverse transition
$\Theta_{i}\leftarrow\Theta_{j}$ is allowed and the corresponding edge belongs
to the digraph of complex transformations). Calculate the right hand side of
the kinetic equations (34) with the detailed balance condition given by
Equation (35) for a given value of $\check{\mu}$ and all possible values of
$\varphi_{\rho}^{+}=\varphi_{\rho}^{-}$. The set of these values of $\dot{N}$
is a convex cone. Denote this cone $\mathbf{Q}_{\rm DB}(\check{\mu})$. For the
same transition graph, calculate the right hand side of the kinetic equation
(22) under the complex balance condition (42). The set of these values of
$\dot{N}$ is also a convex cone. Denote it $\mathbf{Q}_{\rm CB}(\check{\mu})$.
It is obvious that $\mathbf{Q}_{\rm DB}(\check{\mu})\subseteq\mathbf{Q}_{\rm
CB}(\check{\mu})$. Surprisingly, these cones coincide. In GorbanEq2012arX we
proved this fact on the basis of the Michaelis-Menten-Stueckelberg theorem
GorbanShahzad2011 about connection of the macroscopic GMAL kinetics and the
complex balance condition with the Markov microscopic description and under
some asymptotic assumptions. Below a direct proof is presented.
###### Theorem 2 (Local equivalence of detailed and complex balance).
$\mathbf{Q}_{\rm DB}(\check{\mu})=\mathbf{Q}_{\rm CB}(\check{\mu})$ (44)
###### Proof.
Because of the cycle decomposition (Proposition 4) it is sufficient to prove
this theorem for simple normalized cycles. Let us use induction on the cycle
length $k$. For $k=2$ the transition graph is
$\Theta_{1}\rightleftharpoons\Theta_{2}$ and the detailed balance condition
(35) coincides with the complex balance condition (42). Assume that for the
cycles of the length below $k$ the theorem is proved. Consider a normalized
simple cycle $\Theta_{1}\to\Theta_{2}\to\ldots\Theta_{k}\to\Theta_{1}$,
$\Theta_{i}=(y_{i},A)$. The corresponding kinetic equations are
$\begin{split}\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=&(y_{2}-y_{1})\exp(\check{\mu},y_{1})+(y_{3}-y_{2})\exp(\check{\mu},y_{2})+\ldots\\\
&+(y_{k-1}-y_{k})\exp(\check{\mu},y_{k-1})+(y_{k}-y_{1})\exp(\check{\mu},y_{k})\end{split}$
(45)
At the equilibrium, all systems with detailed balance or with complex balance
give $\dot{N}=0$. Assume that the state $\check{\mu}$ is non-equilibrium and
therefore not all the Boltzmann factors $\exp(\check{\mu},y_{i})$ are equal.
Select $i$ such that the Boltzmann factor $\exp(\check{\mu},y_{i})$ has
minimal value, while for the next position in the cycle this factor becomes
bigger. We can use a cyclic permutation and assume that the factor
$\exp(\check{\mu},y_{1})$ is the minimal one and
$\exp(\check{\mu},y_{2})>\exp(\check{\mu},y_{1})$.
Let us find a kinetic factor $\varphi$ such that the reaction system
consisting of two cycles, a cycle of the length 2 with detailed balance
$\Theta_{1}\underset{\varphi}{\overset{\varphi}{\rightleftharpoons}}\Theta_{2}$
(here the kinetic factors are shown above and below the arrows) and a simple
normalized cycle of the length $k-1$,
$\Theta_{2}\to\ldots\Theta_{k}\to\Theta_{2}$, gives the same $\dot{N}$ at the
state $\check{\mu}$ as the initial scheme. We obtain from Equation (45) the
following necessary and sufficient condition
$(y_{1}-y_{k})\exp(\check{\mu},y_{k})+(y_{2}-y_{1})\exp(\check{\mu},y_{1})=(y_{2}-y_{k})\exp(\check{\mu},y_{k})+\varphi(y_{2}-y_{1})(\exp(\check{\mu},y_{1})-\exp(\check{\mu},y_{2}))$
. It is sufficient to equate here the coefficients at every $y_{i}$
($i=1,2,k$). The result is
$\varphi=\frac{\exp(\check{\mu},y_{k})-\exp(\check{\mu},y_{1})}{\exp(\check{\mu},y_{2})-\exp(\check{\mu},y_{1})}$
By the induction assumption we proved that theorem for the cycles of arbitrary
length and, therefore, it is valid for all reaction schemes with complex
balance. ∎
The cone $\mathbf{Q}_{\rm DB}(\check{\mu})$ of the possible values of
$\dot{N}$ in Equation (34) is a polyhedral cone with finite set of extreme
rays at any non-equilibrium state $\check{\mu}$ for the systems with detailed
balance. Each of its extreme rays has the direction vector of the form
$\gamma_{\rho}\mbox{sign}(\exp(\check{\mu},\alpha_{\rho})-\exp(\check{\mu},\beta_{\rho}))$
(46)
This follows from the form of the reaction rate presented by Equation (36) for
the kinetic equations (34). Following Theorem 2, the cone of the possible
values of $\dot{N}$ for systems with complex balance has the same set of
extreme rays. Each extreme ray corresponds to a single reversible elementary
reaction with the detailed balance condition (35).
### 5.7 General $H$-Theorem for GMAL
Consider GMAL kinetics with the given reaction mechanism presented by
stoichiometric equations (32) and the detailed balance condition (35). The
reaction rates of the elementary reaction for the kinetic equations (34) are
proportional to the nonnegative parameter $\varphi_{\rho}$ in Equation (36).
These $m$ nonnegative numbers $\varphi_{\rho}$ ($\rho=1,\ldots,m$) are
independent in the following sense: for any set of values $\varphi_{\rho}\geq
0$ the kinetic equations (34) satisfy the $H$-theorem in the form of Equation
(38):
$\frac{{\mathrm{d}}H}{{\mathrm{d}}t}=-V\sum_{\rho}\varphi_{\rho}(\exp(\alpha_{\rho},\check{\mu})+\exp(\beta_{\rho},\check{\mu}))\mathbb{A}_{\rho}\tanh\frac{\mathbb{A}_{\rho}}{2}\leq
0$ (47)
Therefore, nonnegativity is the only a priori restriction on the values of
$\varphi_{\rho}$ ($\rho=1,\ldots,m$).
One Lyapunov function for the GMAL kinetics with the given reaction mechanism
and the detailed balance condition obviously exists. This is the thermodynamic
Lyapunov function $H$ used in GMAL construction. For ideal systems (in
particular, for master equation) $H$ has the standard form
$\sum_{i}N_{i}(\ln(c_{i}/c_{i}^{*})-1)$ given by Equation (23). Usually, $H$
is assumed to be convex and some singularities (like $c\ln c$) near zeros of
$c$ may be required for positivity preservation in kinetics ($\dot{N}_{i}\geq
0$ if $c_{i}=0$). The choice of the thermodynamic Lyapunov function for GMAL
construction is wide. We consider kinetic equations in a compact convex set
$U$ and assume $H$ to be convex and continuous in $U$ and differentiable in
the relative interior of $U$ with derivatives continued by continuity to $U$.
Assume that we select the thermodynamic Lyapunov function $H$ and the reaction
mechanism in the form (32). Are there other universal Lyapunov functions for
GMAL kinetics with detailed balance and given mechanism? “Universal” here
means “independent of the choice of the nonnegative kinetic factors”.
For a given reaction mechanism we introduce the partial equilibria criterion
by analogy to Definition 1. Roughly speaking, a convex function $F$ satisfies
this criterion if its conditional minima correctly describe the partial
equilibria of elementary reactions.
For each elementary reaction $\sum_{i}\alpha_{\rho
i}A_{i}\rightleftharpoons\sum_{i}\beta_{\rho i}A_{i}$ from the reaction
mechanism given by the stoichiometric equations (32) and any $X\in U$ we
define an interval of a straight line
$I_{X,\rho}=\\{X+\lambda\gamma_{\rho}\,|\,\lambda\in\mathbb{R}\\}\cap U.$ (48)
###### Definition 2 (Partial equilibria criterion for GMAL).
A convex function $F(N)$ on $U$ satisfies the partial equilibria criterion
with a given thermodynamic Lyapunov function $H$ and reversible reaction
mechanism given by stoichiometric equations (32) if
$\underset{{N\in
I_{X,\rho}}}{\operatorname{argmin}}H(N)\subseteq\underset{{N\in
I_{X,\rho}}}{\operatorname{argmin}}F(N)$ (49)
for all $X\in U$, $\rho=1,\ldots,m$.
###### Theorem 3.
A convex function $F(N)$ on $U$ is a Lyapunov function for all kinetic
equations (34) with the given thermodynamic Lyapunov function $H$ and reaction
rates presented by Equation (36) (detailed balance) if and only if it
satisfies the partial equilibria criterion (Definition 2).
###### Proof.
The partial equilibria criterion is necessary because $F(N)$ should be a
Lyapunov function for a reaction mechanism that consists of any single
reversible reaction from the reaction mechanism (32). It is also sufficient
because for the whole reaction mechanism the kinetic equations (34) are the
conic combinations of the kinetic equations for single reversible reactions
from the reaction mechanism (32). ∎
For the general reaction systems with complex balance we can use the theorem
about local equivalence (Theorem 2). Consider a GMAL reaction system with the
mechanism (18) and the complex balance condition.
###### Theorem 4.
A convex function $F(N)$ on U is a Lyapunov function for all kinetic equations
(22) with the given thermodynamic Lyapunov function $H$ and the complex
balance condition (42) if it satisfies the partial equilibria criterion
(Definition 2).
###### Proof.
The theorem follows immediately from Theorem 3 about Lyapunov functions for
systems with detailed balance and the theorem about local equivalence between
systems with local and complex balance (Theorem 2). ∎
The general $H$-theorems for GMAL is similar to Theorem 1 for Markov chains.
Nevertheless, many non-classical universal Lyapunov functions are known for
master equations, for example, the $f$-divergences given by Equation (1),
while for a nonlinear reaction mechanism it is difficult to present a single
example different from the thermodynamic Lyapunov function or its monotonic
transformations. The following family of example generalizes Equation (17).
$F(N)=H(N)+\varepsilon
f(N)\prod_{\rho}(\exp(\alpha_{\rho},\check{\mu})-\exp(\beta_{\rho},\check{\mu}))^{2}$
(50)
where $f(N)$ is a non-negative differentiable function and $\varepsilon>0$ is
a sufficiently small number. This function satisfies the conditional
equilibria criterion. For continuous $H(N)$ on compact $U$ with the spectrum
of the Hessian uniformly separated from zero, this $F(N)$ is convex for
sufficiently small $\varepsilon>0$.
## 6 Generalization: Weakened Convexity Condition, Directional Convexity and
Quasiconvexity
In all versions of the general $H$-theorems we use convexity of the Lyapunov
functions. Strong convexity of the thermodynamic Lyapunov functions $H$ (or
even positive definiteness of its Hessian) is needed, indeed, to provide
reversibility of the Legendre transform $N\leftrightarrow\nabla H$.
Figure 3: Monotonicity on both sides of the minimizer $\lambda^{*}$ for convex
(a) and non-convex but quasiconvex (b) functions
For the kinetic Lyapunov functions that satisfy the partial equilibria
criterion, we use, actually, a rather weak consequence of convexity in
restrictions on the straight lines $X+\lambda\gamma$ (where
$\lambda\in\mathbb{R}$ is a coordinate on the real line, $\gamma$ is a
stoichiometric vector of an elementary reaction): if
$\lambda^{*}=\underset{\lambda\in\mathbb{R}}{\operatorname{argmin}}F(X+\lambda\gamma)$
then on the half-lines (rays) $\lambda\geq\lambda^{*}$ and
$\lambda\leq\lambda^{*}$ function $F(X+\lambda\gamma)$ is monotonic. It does
not decrease for $\lambda\geq\lambda^{*}$ and does not increase for
$\lambda\leq\lambda^{*}$. Of course, convexity is sufficient (Figure 3a) but a
much weaker property is needed (Figure 3b).
A function $F$ on a convex set $U$ is quasiconvex Greenberg1971 if all its
sublevel sets are convex. It means that for every $X,Y\in U$
$F(\lambda X+(1-\lambda)Y)\leq\max\\{F(X),F(Y)\\}\mbox{ for all
}\lambda\in[0,1]$ (51)
In particular, a function $F$ on a segment is quasiconvex if all its sublevel
sets are segments.
Among many other types of convexity and quasiconvexity (see, for example
Ponstein1967 ) two are important for the general $H$-theorem. We do not need
convexity of functions along all straight lines in $U$. It is sufficient that
the function is convex on the straight lines $X+\mathbb{R}\gamma_{\rho}$,
where $\gamma_{\rho}$ are the stoichiometric (direction) vectors of the
elementary reactions.
Let $D$ be a set of vectors. A function $F$ is $D$-convex if its restriction
to each line parallel to a nonzero $v\in D$ is convex Matousek2001 . In our
case, $D$ is the set of stoichiometric vectors of the transitions,
$D=\\{\gamma_{\rho}\,|\,\rho=1,\ldots,m\\}$. We can use this directional
convexity instead of convexity in Propositions 1, 2 and Theorems 1, 3, 4.
Finally, we can relax the convexity conditions even more and postulate
directional quasiconvexity Hwang1996 for the set of directions
$D=\\{\gamma_{\rho}\,|\,\rho=1,\ldots,m\\}$. Propositions 1, 2 and Theorems 1,
3, 4 will be still true if the functions are continuous, quasiconvex in
restrictions on all lines $X+\mathbb{R}\gamma_{\rho}$ and satisfy the partial
equilibria criterion.
Relations between these types of convexity are schematically illustrated in
Figure 4.
Figure 4: Relations between different types of convexity
## 7 Discussion
Many non-classical entropies are invented and applied to various problems in
physics and data analysis. In this paper, the general necessary and sufficient
criterion for the existence of $H$-theorem is proved. It has a simple and
physically transparent form: the convex divergence (relative entropy) should
properly describe the partial equilibria for transitions
$A_{i}\rightleftharpoons A_{j}$. It is straightforward to check this partial
equilibria criterion. The applicability of this criterion does not depend on
the detailed balance condition and it is valid both for the class of the
systems with detailed balance and for the general first order kinetics without
this assumption.
If an entropy has no $H$-theorem (that is, it violates the second law and the
data processing lemma) then there should be unprecedentedly strong reasons for
its use. Without such strong reasons we cannot employ it. Now, I cannot find
an example of sufficiently strong reasons but people use these entropies in
data analysis and we have to presume that they may have some hidden reasons
and that these reasons may be sufficiently strong. We demonstrate that this
problem arises even for such popular divergences like Euclidean distance or
Itakura-Saito distance.
The general $H$-theorem is simply a reduction of a dynamical question
(Lyapunov functionals) to a static one (partial equilibria). It is not
surprising that it can be also proved for nonlinear Generalized Mass Action
Law kinetics. Here kinetic systems with complex balance play the role of the
general Markov chains, whereas the systems with detailed balance correspond to
the reversible Markov chains. The requirement of convexity of Lyapunov
functions can be relaxed to the directional convexity (in the directions of
reactions) or even directional quasiconvexity.
For the reversible Markov chains presented by Equations (15) with the
classical entropy production formula (16), every universal Lyapunov function
$H$ should satisfy inequalities
$\left(\frac{p_{j}}{p_{j}^{*}}-\frac{p_{i}}{p_{i}^{*}}\right)\left(\frac{\partial
H(P)}{\partial p_{j}}-\frac{\partial H(P)}{\partial p_{i}}\right)\leq 0\mbox{
for all }i,j,i\neq j$ (52)
These inequalities are closely related to another generalization of convexity,
the Schur convexity MarshallOlkin2011 . They turn into the definition of the
Schur convexity when equilibrium is the equidistribution with $p_{i}^{*}=1/n$
for all $i$. Universal Lyapunov functions for nonlinear kinetics give one more
generalization of the Schur convexity.
Introduction of many non-classical entropies leads to the “uncertainty of
uncertainty” phenomenon: we measure uncertainty by entropy but we have
uncertainty in the entropy choice GorbanCAMWA2013 . The selection of the
appropriate entropy and introduction of new entropies are essentially
connected with the class of kinetics. $H$-theorems in physics are
formalizations of the second law of thermodynamics: entropy of isolated
systems should increase in relaxation to equilibrium. If we know the class of
kinetic equations (for example, the Markov kinetics given by master equations)
then the $H$ theorem states that it is possible to use this entropy with the
given kinetics. If we know the entropy and are looking for kinetic equations
then such a statement turns into the thermodynamic restriction on the
thermodynamically admissible kinetic equations. For information processing,
the class of kinetic equations describes possible manipulations with data. In
this case, the $H$-theorems mean that under given class of manipulation the
information does not increase. It is not possible to compare different
entropies without any relation to kinetics. It is useful to specify the class
of kinetic equations, for which they are the Lyapunov functionals. For the
GMAL equations, we can introduce the dynamic equivalence between divergences
(free entropies or conditional entropies). Two functionals $H(N)$ and $F(N)$
in a convex set $U$ are dynamically consistent with respect to the set of
stoichiometric vectors $\\{\gamma_{\rho}\\}$ ($\rho=1,\ldots,m$) if
1. (1)
$F$ and $H$ are directionally quasiconvex functions in directions
$\\{\gamma_{\rho}\\}$ ($\rho=1,\ldots,m$)
2. (2)
For all $\rho=1,\ldots,m$ and $N\in U$
$(\nabla_{N}F(N),\gamma_{\rho})(\nabla_{N}H(N),\gamma_{\rho})\geq 0$
For the Markov kinetics, the partial equilibria criterion is sufficient for a
convex function $H(P)$ to be dynamically consistent with the relative entropy
$\sum_{i}p_{i}(\ln(p_{i}/p_{i}^{*})-1)$ in the unit simplex $\Delta_{n}$. For
GMAL, any convex function $H(N)$ defines a class of kinetic equations. Every
reaction mechanism defines a family of kinetic equations from this class and a
class of Lyapunov functions $F$, which are dynamically consistent with $H$.
The main message of this paper is that it is necessary to discuss the choice
of the non-classical entropies in the context of kinetic equations.
## Appendix: Quasiequilibrium entropies and forward–invariant peeling
### A1. Maximum of quasiequilibrium entropies – a new family
of universal Lyapunov functions for generalized mass action law
The general $H$ theorems for the Generalized Mass Action Law (GMAL) and for
its linear version, master equation, look very similar. For the linear systems
many Lyapunov functionals are known in the explicit form: for every convex
function $h$ on the positive ray $\mathbb{R}_{+}$ we have such a functional
(1). On the contrary, for the nonlinear systems we, typically, know the only
Lyapunov function $H$, it is the thermodynamic potential which is used for the
system construction. The situation looks rather intriguing and challenging:
for every finite reaction mechanism there should be many Lyapunov functionals,
but we cannot construct them. (There is no chance to find many Lyapunov
functions for all nonlinear mechanisms together under given thermodynamics
because in this case the cone of the possible velocities $\dot{N}$ is a half-
space and locally there is the only divergence with a given tangent
hyperplane. Globally, such a divergence can be given by an arbitrary monotonic
function on the thermodynamic tree G11984 ; GorbanSIADS2013 ).
In this Appendix, we present a general procedure for the construction of a
family of new Lyapunov functionals from $H$ for nonlinear GMAL kinetics and a
given reaction mechanism. We will use two auxiliary construction, the
quasiequilibrium entropies (or divergences) and the forward–invariant peeling.
Let us consider isochoric systems (constant volume $V$). For them,
concentrations $c_{i}$ (intensive variables) and amounts $N_{i}$ (extensive
variables are proportional with a constant extensive factor $V$ and we take
$N_{i}=c_{i}$ in a standard unit volume without loss of generality.
We assume that $H$ is strongly convex in the second approximation in
$\mathbb{R}_{+}^{n}$. This means that it is twice differentiable and the
Hessian $\partial^{2}H/\partial N_{i}\partial N_{j}$ is positively definite in
$\mathbb{R}_{+}^{n}$. In addition, we assume logarithmic singularities of the
partial derivatives of $H$ near zeros of concentrations:
$H(N)=\sum_{i}N_{i}(\ln c_{i}-1+\mu_{0i}(c))\,,$ (53)
where the functions $\mu_{0i}(c)$ are bounded continuously differentiable
functions in a vicinity of the non-negative orthant. This assumption
corresponds to the physical hypothesis about the logarithmic singularity of
the chemical potentials, $\mu_{i}=RT\ln c_{i}+\ldots$ where $\ldots$ stands
for a continuous function of $c,T$, and to the supposition about the classical
mass action law for small concentrations. Assume also that all the described
properties of $H$ hold for its restrictions on the faces of
$\overline{\mathbb{R}_{+}^{n}}$: these restrictions are strictly convex,
differentiable in the relative interior, etc.
For every linear subspace $E\subset\mathbb{R}^{n}$ and a given composition
vector $N^{0}\in\mathbb{R}_{+}^{n}$ the quasiequilibrium composition is the
partial equilibrium
$N^{*}_{E}(N^{0})=\underset{{N\in(N^{0}+E)\cap\mathbb{R}_{+}^{n}}}{\operatorname{argmin}}H(N)$
The quasiequilibrium divergence is the value of $H$ at the partial
equilibrium:
$H^{*}_{E}(N^{0})=\min_{N\in(N^{0}+E)\cap\mathbb{R}_{+}^{n}}H(N)$
Due to the assumption about strong convexity of $H$ and logarithmic
singularity (53), for a positive vector $N^{0}\in\mathbb{R}_{+}^{n}$ and a
subspace $E\subset\mathbb{R}^{n}$ the quasiequilibrium composition
$N^{*}_{E}(N^{0})$ is also positive.
Such quasiequilibrium “entropies” are discussed by Jaynes Jaynes1965 . He
considered the quasiequilibrium $H$-function as the Boltzmann $H$-function
$H_{\rm B}$ in contrast to the original Gibbs $H$-function, $H_{\rm G}$. The
Gibbs $H$-function is defined for the distributions on the phase space of the
mechanical systems. The Boltzmann function is a conditional minimum of the
Gibbs function, therefore the inequality holds $H_{\rm B}\leq H_{\rm G}$
Jaynes1965 . Analogously,
$H^{*}_{E}(N^{0})\leq H(N^{0})$
and this inequality turns into the equality if and only if $N^{0}$ is the
quasiequilibrium state for the subspace $E$: $N^{0}=N^{*}_{E}(N^{0})$. After
Jaynes, these functions are intensively used in the discussion of time arrow
Ocherki1986 ; Lebowith1993 ; GoldsteinLebow2004 . In the theory of
information, quasiequilibrium was studied in detail under the name information
projection (or I-projection) CsiszarMatus2003 .
Analysis of partial equilibria is useful in chemical engineering in the
presence of uncertainty: when the reaction rate constants are unknown then the
chains of partial equilibria together with information about the
thermodynamically preferable directions of reactions may give some important
information about the process GorbanKagan2006 .
Let us prove several elementary properties of $H^{*}_{E}(N)$. Let $E$ and $L$
be subspaces of $\mathbb{R}^{n}$.
###### Proposition 5.
1. 1.
The function $H^{*}_{E}(N)$ is convex.
2. 2.
If $E$ is a proper subspace of $R^{n}$ then the function $H^{*}_{E}(N)$ is not
strictly convex: for each $N\in\mathbb{R}_{+}^{n}$ the level set
$\\{N^{\prime}\,|\,H^{*}_{E}(N^{\prime})=H^{*}_{E}(N)\\}$ includes faces
$(N+E)\cap\mathbb{R}_{+}^{n}$.
3. 3.
The function $H^{*}_{E}([N])$ is strictly convex on the quotient space
$\mathbb{R}_{+}^{n}/E$ (here, $[N]\in\mathbb{R}_{+}^{n}/E$ is the equivalence
class, $[N]=(N+E)\cap\mathbb{R}_{+}^{n}$).
4. 4.
If $E\subseteq L$ then $H^{*}_{E}(N)\geq H^{*}_{L}(N)$ and this inequality
turns into the equality if and only if the corresponding quasiequlibria
coincide: $N^{*}_{E}(N)=N^{*}_{L}(N)$ (this is a generalization of the Jaynes
inequality $H_{\rm B}\leq H_{\rm G}$).
5. 5.
If $N=N^{*}_{E}(N)$ then $H^{*}_{E}(N)\geq H^{*}_{L}(N)$ for all $L$ and this
inequality turns into the equality if and only if $N=N^{*}_{(E+L)}(N)$.
###### Proof.
1. 1.
Convexity of $H^{*}_{E}(N)$ means that for every positive $N^{1}$ and $N^{2}$
and a number $\lambda\in[0,1]$ the inequality holds:
$H^{*}_{E}(\lambda N^{1}+(1-\lambda)N^{2})\leq\lambda
H^{*}_{E}(N^{1})+(1-\lambda)H^{*}_{E}(N^{2})$
Let us prove this inequality. First,
$H(\lambda N^{*}_{E}(N^{1})+(1-\lambda)N^{*}_{E}(N^{2})\leq\lambda
H(N^{*}_{E}(N^{1}))+(1-\lambda)H(N^{*}_{E}(N^{2}))$
because convexity $H$. Secondly, $H(N^{*}_{E}(N^{1,2}))=H^{*}_{E}(N^{1,2})$ by
definition and the last inequality reads
$H(\lambda N^{*}_{E}(N^{1})+(1-\lambda)N^{*}_{E}(N^{2})\leq\lambda
H^{*}_{E}(N^{1})+(1-\lambda)H^{*}_{E}(N^{2})$
Finally, $N^{*}_{E}(N^{1,2})\in N^{1,2}+E$, hence,
$\lambda N^{*}_{E}(N^{1})+(1-\lambda)N^{*}_{E}(N^{2})\in\lambda
N^{1}+(1-\lambda)N^{2}+E$
and $H(\lambda N^{*}_{E}(N^{1})+(1-\lambda)N^{*}_{E}(N^{2}))\geq
H^{*}_{E}(\lambda N^{1}+(1-\lambda)N^{2})$ because the last value is the
minimum of $H$ on the linear manifold $\lambda N^{1}+(1-\lambda)N^{2}+E$.
Inequality is proven.
2. 2.
Indeed, the function $H^{*}_{E}$ is constant on the set
$(N+E)\cap\mathbb{R}_{+}^{n}$, by construction.
3. 3.
In the proof of item 1 the inequality $H(\lambda
N^{*}_{E}(N^{1})+(1-\lambda)N^{*}_{E}(N^{2}))>H^{*}_{E}(\lambda
N^{1}+(1-\lambda)N^{2})$ is strong for $\lambda\neq 0,1$ and
$N^{1}-N^{2}\notin E$. Therefore, under these conditions the convexity
inequality is strong.
4. 4.
If $E\subseteq L$ then $N+E\subset N+L$ and $H^{*}_{E}(N)\geq H^{*}_{L}(N)$ by
definition of $H^{*}$ as a conditional minimum. This inequality turns into the
equality if and only if the corresponding quasiequlibria coincide because of
strong convexity of $H$.
5. 5.
This follows directly from the definitions of $H^{*}_{E}(N)$ as a conditional
minimum and $N^{*}_{E}(N)$ as the corresponding minimizer of $H$ on
$N+E\cap\mathbb{R}^{n}_{+}$
∎
Consider the reversible reaction mechanism (32) with the set of the
stoichiometric vectors $\Upsilon$. For each $\Gamma\subset\Upsilon$ we can
take $E={\rm Span}(\Gamma)$ and define the quasiequilibrium. The subspace
${\rm Span}(\Gamma)$ may coincide for different $\Gamma$ and the
quasiequilibrium depends on the subspace $E$ only, therefore, it is useful to
introduce the set of these subspaces for a given reaction mechanism (32). Let
$\mathcal{E}_{\Upsilon}$ be the set of all subspaces of the form $E={\rm
Span}(\Gamma)$ ($\Gamma\subset\Upsilon$). For each dimension $k$ we denote
$\mathcal{E}^{k}_{\Upsilon}$ the set of $k$-dimensional subspaces from
$\mathcal{E}_{\Upsilon}$.
For each dimension $k=0,\ldots,{\rm rank}(\Upsilon)$ we define the function
$H^{k,\max}_{\Upsilon}$: $H^{0,\max}_{\Upsilon}=H$, and for $0<k\leq{\rm
rank}(\Upsilon)$
$H^{k,\max}_{\Upsilon}(N)=\max_{E\in\mathcal{E}^{k}_{\Upsilon}}H^{*}_{E}(N)$
(54)
Immediate consequence of the definition of the quasiequilibrium divergence and
Theorem 3 is:
###### Proposition 6.
$H^{1,\max}_{\Upsilon}(N)$ is a Lyapunov function in $\mathbb{R}_{+}^{n}$ for
all kinetic equations (34) with the given thermodynamic Lyapunov function $H$,
reaction rates presented by Equation (36) (detailed balance) and the
reversible reaction mechanism with the set of stoichiometric vectors
$\Gamma\subseteq\Upsilon$.
###### Proof.
$H^{1,\max}_{\Upsilon}(N)$ is a convex function as the maximum of several
convex functions. Let us consider a restriction of this function onto an
interval of the straight line
$I=(N^{0}+\mathbb{R}\gamma)\cap\mathbb{R}^{n}_{+}$ for a stoichiometric vector
$\gamma\in\Upsilon$. The partial equilibrium
$N^{**}=N^{*}_{\\{\mathbb{R}\gamma\\}}(N^{0})$ is the minimizer of $H(N)$ on
$I$. Assume that this partial equilibrium is not a partial equilibrium for
other 1D subspaces $E\in\mathcal{E}^{1}_{\Upsilon}$. Then for all
$E\in\mathcal{E}^{1}_{\Upsilon}$ ($E\neq\mathbb{R}\gamma$)
$H^{*}_{E}(N^{**})<H^{*}_{\\{\mathbb{R}\gamma\\}}(N^{0})$ and
$H^{1,\max}_{\Upsilon}(N)=H^{*}_{\\{\mathbb{R}\gamma\\}}(N)$
in some vicinity of $N^{**}$. This function is constant on $I$.
If a convex function $h$ on an interval $I$ is constant on an subinterval
$J=(a,b)\subset I$ ($a\neq b$) then the value $h(J)$ is the minimum of $h$ on
$I$. Therefore, $N^{*}_{\\{\mathbb{R}\gamma\\}}(N^{0})$ is a minimizer of the
convex function $H^{1,\max}_{\Upsilon}(N)$ on $I$ in the case, when $N^{**}$
is not a partial equilibrium for other 1D subspaces
$E\in\mathcal{E}^{1}_{\Upsilon}$ ($E\neq\mathbb{R}\gamma$).
Let us assume now that the partial equilibrium
$N^{**}=N^{*}_{\\{\mathbb{R}\gamma\\}}(N^{0})$ is, at the same time, the
partial equilibrium for several other $E\in\mathcal{E}^{1}_{\Upsilon}$. Let
$\mathcal{B}$ be the set of subspaces $E\in\mathcal{E}^{1}_{\Upsilon}$ for
which $N^{**}$ is a partial equilibrium, i.e. $H(N^{**})=H^{*}_{E}(N^{**})$.
In this case, for all $E\notin\mathcal{B}$ ($E\in\mathcal{E}^{1}_{\Upsilon}$)
$H^{*}_{E}(N^{**})<H(N^{**})$. Therefore, in a sufficiently small vicinity of
$N^{**}$ the function $H^{1,\max}_{\Upsilon}(N)$ can be defined as
$H^{1,\max}_{\Upsilon}(N)=\max_{E\in\mathcal{B}}H^{*}_{E}(N)$
Point $N^{**}$ is a minimizer of $H$ on a linear manifold
$N^{**}+(\bigoplus_{E\in B}E)$. It is also a minimizer of convex function
$H^{1,\max}_{\Upsilon}(N)$ on this linear manifold, particularly, it is a
minimizer of this convex function on the interval $I$. (Convexity plays a
crucial role in this reasoning because for convex functions the local minima
are the global ones.)
We proved that the function $H^{1,\max}_{\Upsilon}(N)$ satisfies the partial
equilibria criterion and, hence, it is a Lyapunov function in
$\mathbb{R}_{+}^{n}$ for all kinetic equations (34) with the given
thermodynamic Lyapunov function $H$, reaction rates presented by Equation (36)
(detailed balance) and the reversible reaction mechanism with the set of
stoichiometric vectors $\Gamma\subseteq\Upsilon$. ∎
Let a positive vector $N^{**}$ be a minimizer of $H$ on
$(N^{**}+E)\cap\mathbb{R}^{n}$, where $E$ is a linear subspace of
$\mathbb{R}^{n}$. It may be useful to represent the structure of the Lyapunov
function $H^{1,\max}_{\Upsilon}(N)$ near $N^{**}$ in the quadratic
approximation. Assume that $H(N)$ is $m$ times continuously differentiable in
$\mathbb{R}^{n}$ for sufficiently large $m$. In the vicinity of $N^{**}$
$H(N)-H(N^{**})=(DH)_{**}(\Delta)+\frac{1}{2}\langle\Delta,\Delta\rangle_{**}+o(\|\Delta\|^{2})$
where $\Delta=N-N^{**}$, $(DH)_{**}=(DH)\left|{}_{N^{**}}\right.$ is the
differential of $H$ at $N^{**}$, and
$\langle\bullet,\bullet\rangle_{**}=(\bullet,(D^{2}H)_{**}\bullet)$ is the
entropic inner product, with the positive symmetric operator
$(D^{2}H)_{**}=(D^{2}H)\left|{}_{N^{**}}\right.$ (the second differential of
$H$ at $N^{**}$). The entropic inner product is widely used in kinetics and
nonequilibrium thermodynamics, see, for example G11984 ; Ocherki1986 ; UNIMOLD
; GorGorKar2004 ; InChLANL ; GorKarLNP2005 .
Let us split $\mathbb{R}^{n}$ into the orthogonal sum: $\mathbb{R}^{n}=E\oplus
E^{\bot}$, $E^{\bot}$ is the orthogonal supplement to $E$ in the entropic
inner product $\langle\bullet,\bullet\rangle_{**}$. Each vector
$\Delta\in\mathbb{R}^{n}$ is represented in the form
$\Delta=\Delta^{\|}\oplus\Delta^{\bot}$, where $\Delta^{\|}\in E$ and
$\Delta^{\bot}\in E^{\bot}$. By the definition of the partial equilibrium as a
conditional minimizer of $H$, $(DH)_{**}(\Delta^{\|})=0$, and we have the
following representation of $H$
$H(N)-H(N^{**})=(DH)_{**}(\Delta^{\bot})+\frac{1}{2}\langle\Delta^{\|},\Delta^{\|}\rangle_{**}+\frac{1}{2}\langle\Delta^{\bot},\Delta^{\bot}\rangle_{**}+o(\|\Delta\|^{2})$
In particular, when $\Delta\in E$ ($\Delta^{\bot}=0$), this formula gives
$H(N)-H(N^{**})=\frac{1}{2}\langle\Delta,\Delta\rangle_{**}+o(\|\Delta\|^{2})$
From these formulas, we easily get the approximations of $N^{*}_{E}(N)$ and
$H^{*}_{E}(N)$ in a vicinity of $N^{**}$. Let
$N-N^{**}=\Delta=\Delta^{\|}\oplus\Delta^{\bot}$. Then
$N^{*}_{E}(N)-N^{**}=\Delta^{\bot}+o(\|\Delta\|)$ (55)
in particular, $(N^{*}_{E}(N)-N^{**})^{\bot}=\Delta^{\bot}$ (exactly) and
$(N^{*}_{E}(N)-N^{**})^{\|}=o(\|\Delta\|)$. Therefore,
$H^{*}_{E}(N)-H(N^{**})=H(N^{*}_{E}(N))-H(N^{**})=(DH)_{**}(\Delta^{\bot})+\frac{1}{2}\langle\Delta^{\bot},\Delta^{\bot}\rangle_{**}+o(\|\Delta\|^{2})$
(56)
If $E$ is a 1D subspace with the directional vector $\gamma$
($E=\mathbb{R}\gamma$) then
$\Delta^{\|}=\frac{\gamma\langle\gamma,\Delta\rangle_{**}}{\langle\gamma,\gamma\rangle_{**}},\;\;\Delta^{\bot}=\Delta-\frac{\gamma\langle\gamma,\Delta\rangle_{**}}{\langle\gamma,\gamma\rangle_{**}}$
here, $\frac{\gamma\langle\gamma|}{\langle\gamma,\gamma\rangle}$ is the
orthogonal projector onto $E$ and
$1-\frac{\gamma\langle\gamma|}{\langle\gamma,\gamma\rangle}$ is the orthogonal
projector onto $E^{\bot}$, the orthogonal complement to $E$.
Let us use the normalized vectors $\gamma$. In this case,
$N^{*}_{E}(N)-N^{**}=\Delta-\gamma\langle\gamma,\Delta\rangle_{**}+o(\|\Delta\|)$
$H^{*}_{E}(N)-H(N^{**})=(DH)_{**}(\Delta-\gamma\langle\gamma,\Delta\rangle_{**})+\frac{1}{2}\langle\Delta-\gamma\langle\gamma,\Delta\rangle_{**},\Delta-\gamma\langle\gamma,\Delta\rangle_{**}\rangle_{**}+o(\|\Delta\|^{2})$
(57)
Let us assume now that $N^{**}$ is the partial equilibrium for several (two or
more) different 1D subspaces $E$ and $\mathcal{B}$ is a finite set of these
subspaces. The set $\mathcal{B}$ includes two or more different subspaces $E$.
Select a normalized directional vector $\gamma_{E}$ for each
$E\in\mathcal{B}$. Let $E_{\mathcal{B}}={\rm
Span}\\{\gamma_{E}\,|\,E\in\mathcal{B}\\}$. $N^{**}$ is a critical point of
$H$ on $(N^{**}+E_{\mathcal{B}})\cap\mathbb{R}^{n}$ because
$\gamma_{E}\in\ker(DH)_{**}$ for all $E\in\mathcal{B}$ and, therefore,
$E_{\mathcal{B}}\subset\ker(DH)_{**}$.
Consider a function $H^{1,\max}(N)=\max_{E\in\mathcal{B}}H^{*}_{E}(N)$. This
function is strictly convex on $(N^{**}+E_{\mathcal{B}})\cap\mathbb{R}^{n}$
and $N^{**}$ is its minimizer on this set. Indeed, in a vicinity of $N^{**}$
in $(N^{**}+E_{\mathcal{B}})\cap\mathbb{R}^{n}$ for every $E\in\mathcal{B}$
Equation (57) holds. Consider a direct sum of $k=|\mathcal{B}|$ copies of
$E_{\mathcal{B}}$, $E_{1}\oplus E_{2}\oplus\ldots\oplus E_{k}$, where all
$E_{i}$ are the copies of $E_{\mathcal{B}}$ equipped by the entropic inner
product $\langle\bullet,\bullet\rangle_{**}$ and the corresponding Euclidean
norm, and the norm of the sum is the maximum of the norm in the summands:
$\|x_{1}\oplus\ldots\oplus x_{k}\|=\max\\{\|x_{1}\|,\ldots,\|x_{k}\|\\}$. The
following linear map $\psi$ is a surjection $\psi:E_{\mathcal{B}}\to
E_{1}\oplus E_{2}\oplus\ldots\oplus E_{k}$ because $E_{\mathcal{B}}={\rm
Span}\\{\gamma_{E}\,|\,E\in\mathcal{B}\\}$ and if all the summands are zero
for $\Delta\in E_{\mathcal{B}}$ then $\Delta=0$:
$\psi:\Delta\mapsto\bigoplus_{E\in\mathcal{B}}(\Delta-\gamma_{E}\langle\gamma_{E},\Delta\rangle_{**})$
Let us mention that on $E_{\mathcal{B}}$
$H^{1,\max}(N)-H(N^{**})=\frac{1}{2}\|\psi(\Delta)\|^{2}+o(\|\Delta\|^{2})$
and, therefore, $N^{**}$ is a unique minimizer of $H^{1,\max}(N)$ on
$(N^{**}+E_{\mathcal{B}})\cap\mathbb{R}^{n}$
Because of the local equivalence of the systems with detailed and complex
balance and Theorem 4 we also get the following proposition.
###### Proposition 7.
$H^{1,\max}_{\Upsilon}(N)$ is a Lyapunov function in $\mathbb{R}_{+}^{n}$ for
all kinetic equations (34) with the given thermodynamic Lyapunov function $H$,
the complex balance condition (42) and the reaction mechanism (18) with the
set of stoichiometric vectors $\Gamma\subseteq\Upsilon\cup-\Upsilon$.
Here, we use the set $\Upsilon\cup-\Upsilon$ instead of just $\Upsilon$ in
Proposition 6 because the direct and reverse reactions are included in the
stoichiometric equations (18) separately.
### A2. Forward–invariant peeling
We use the quasiequilibrium functions and their various combinations for
construction of new Lyapunov functions from the known thermodynamic Lyapunov
functions, $H$, and an arbitrary convex function $F$. In this procedure, we
delete some parts from the sublevel sets of $F$ to make the rest
positively–invariant with respect to GMAL kinetics with the given reaction
mechanism and detailed or complex balance. We call these procedures the
forward–invariant peeling.
Figure 5: Peeling of convex sets (2D): (a) The reaction mechanism
$A_{1}\rightleftharpoons A_{2}$, $A_{2}\rightleftharpoons A_{3}$,
$2A_{1}\rightleftharpoons A_{2}+A_{3}$. The concentration triangle
$c_{1}+c_{2}+c_{3}=b$ is split by the partial equilibria lines into six
compartments. In each compartment, the cone of possible directions (the angle)
$\mathbf{Q}_{\rm DB}$ is presented. The positively invariant set which
includes the $A_{1}$ vertex is outlined by bold. (b) The set $U$ is outlined
by the dashed line, the level set $H=h-\varepsilon$ is shown by the dotted
line. The levels of $H^{*}_{\gamma}$ are the straight lines $\|\gamma$. The
boundary of the peeled set
$U^{\varepsilon}_{\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}}$ is shown by red.
(c) The sets $U$, $U^{\varepsilon}_{\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}}$
and the level set $H=h-\varepsilon$ without auxiliary lines are presented.
$None$ $None$
$None$
Figure 6: Peeling of convex sets (3D): The unpeeled potato corresponds to the
convex set $U$. The partial equilibria $(\nabla H,\gamma_{i})=0$ ($i=1,2$) are
presented with the corresponding stoichiometric vectors $\gamma_{i}$. (a) Near
the intersection of the partial equilibrium surfaces the peeling
$\|\gamma_{1}$ is separated from the peeling $\|\gamma_{2}$ by the cross of
the dashed lines. (b) After deformation of the partial equilibria (red dashed
lines) the peeled set remains forward–invariants if the deformed partial
equilibria for individual reactions do not leave the corresponding peeled 1D
faces and, in particular, the intersection of the partial equilibria does not
change. (c) The additional peeling $\|{\rm Span}\\{\gamma_{1},\gamma_{2}\\}$
makes the peeled set forward–invariant with respect to the set of systems with
interval reaction rate constants. The partial equilibria for any combination
of reactions can move in the limits of the corresponding faces (red dashed
lines and their intersection in the Figure, panel c).
Let $U\subset\overline{\mathbb{R}_{+}^{*}}$ be a convex compact set of non-
negative $n$-dimensional vectors $N$ and for some $\eta>\min H$ the
$\eta$-sublevel set of $H$ belongs to $U$: $\\{N\,|\,H(N)\leq\eta\\}\subset
U$. Let $h>\min H$ be the maximal value of such $\eta$. Select a thickness of
peel $\varepsilon>0$.
We define the peeled set $U$ as
$U^{\varepsilon}_{\Upsilon}=U\cap\\{N\in\overline{R^{n}_{+}}\,|\,H^{1,max}_{\Upsilon}(N)\leq
h-\varepsilon\\}$
For sufficiently small $\varepsilon>0$ ($\varepsilon<h-\min H$) this set is
non-empty and forward–invariant.
###### Proposition 8.
For sufficiently small $\varepsilon>0$ ($\varepsilon<h-\min H$) the peeled set
$U^{\varepsilon}_{\Upsilon}$ is non-empty. If it is non-empty then it is
forward–invariant with respect to kinetic equations (34) with the
thermodynamic Lyapunov function $H$, reaction rates presented by Equation (36)
(detailed balance) and the reversible reaction mechanism (32) with the set of
stoichiometric vectors $\Gamma\subseteq\Upsilon$.
###### Proposition 9.
For sufficiently small $\varepsilon>0$ ($\varepsilon<h-\min H$) the peeled set
$U^{\varepsilon}_{\Upsilon}$ is non-empty. If it is non-empty then it is
forward–invariant with respect to all kinetic equations (34) with the given
thermodynamic Lyapunov function $H$, the complex balance condition (42) and
the reaction mechanism (18) with any set of stoichiometric vectors
$\Gamma\subseteq\Upsilon\cup-\Upsilon$.
The forward–invariant peeling for a 2D nonlinear kinetic scheme is
demonstrated in Figure 5. A 3D example is presented in Figure 6. It is worth
to mention that the peeled froward–invariant sets have 1D faces near the
partial equilibria. These faces are parallel to the stiochiometric vectors of
the equilibrating reactions.
Let $F(N)$ be a continuous strictly convex function with bounded level sets on
the non-negative orthant. For each level of $H$, $h\in{\rm im}H$, we define
the level
$f(h)=\max_{H(N)=h}F(N)$
. Let $f^{*}(h)\geq f(h)$ be any strictly increasing function. In particular,
we can take $f^{*}(h)=f(h)+\varepsilon$ ($\varepsilon>0$). Introduce the
peeled function
$F^{f^{*}(h)}_{\Upsilon}(N)=\max\\{F(N),f^{*}(H^{1,\max}_{\Upsilon}(N))\\}$
Applying Proposition 8 to sublevel sets of $F^{f^{*}(h)}_{\Upsilon}$ we obtain
the following propositions.
###### Proposition 10.
$F^{f^{*}(h)}_{\Upsilon}(N)$ is a Lyapunov function in $\mathbb{R}_{+}^{n}$
for all kinetic equations (34) with the thermodynamic Lyapunov function $H$,
reaction rates presented by Equation (36) (detailed balance) and the
reversible reaction mechanism (32) with any set of stoichiometric vectors
$\Gamma\subseteq\Upsilon$.
###### Proposition 11.
$F^{f^{*}(h)}_{\Upsilon}(N)$ is a Lyapunov function in $\mathbb{R}_{+}^{n}$
for all kinetic equations (34) with the given thermodynamic Lyapunov function
$H$, the complex balance condition (42) and the reaction mechanism (18) with
the set of stoichiometric vectors $\Gamma\subseteq\Upsilon\cup-\Upsilon$.
The level sets of $F^{f^{*}(h)}_{\Upsilon}(N)$ have 1D faces parallel to the
stoichiometric vectors $\gamma\in\Upsilon$ near the corresponding partial
equilibria outside a vicinity of the intersections of these partial equilibria
hypersurfaces. At the intersections of two partial equilibria there is a
singularity and the size of both faces tends to zero (Figure 6 a).
Let us consider kinetic systems with perturbed thermodynamic potentials
$H^{\prime}=H+\Delta H$, where $\Delta H$ is uniformly small with it second
derivatives. For such the perturbed systems in a bounded set all the partial
equilibria are close to the partial equilibria of the original system. (An
important case is the perturbation of $H$ by a linear functional $\Delta H$.)
We will modify the peeling procedure to create forward–invariant sets for
sufficiently small perturbations.
Let us look on the forward–invariant peeled set on Figure 6 a. If we slightly
deform the partial equilibria surfaces for each reaction (Figure 6 b, red
dashed lines) but keep their intersection unchanged then the peeled set may
remain forward–invariant. It is sufficient that the intersections of the
partial equilibria with the border of $U$ in $\mathbb{R}^{n}$ do not leave the
corresponding 1D faces (Figure 6 b). If we perform additional peeling near the
intersections of the partial equilibria (Figure 6 c), then the peeled set may
be positively invariant with respect to the kinetic equations with perturbed
thermodynamic potentials (or, even a bit stronger, with respect to kinetic
equations with interval coefficients for sufficiently small intervals).
We consider the reversible reaction mechanism (32) with the set of the
stoichiometric vectors $\Upsilon$. $U\subset\overline{R^{n}_{+}}$ is a convex
compact set, and for some $\eta>\min H$ the $\eta$-sublevel set of $H$ belongs
to $U$: $\\{N\in\overline{R^{n}_{+}}\,|\,H(N)\leq\eta\\}\subset U$. Let
$h>\min H$ be the maximal value of such $\eta$.
Select a sequence of thicknesses of peels
$\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}>0$, where $k={\rm
rank}\Upsilon$. Let us use for the peeling the functions
$H^{i,\max}_{\Upsilon}(N)$ (54) ($i=1,\ldots{\rm rank}\Upsilon$).
For each $i$ we consider the sublevel set
$U_{i}=\\{N\in\overline{\mathbb{R}_{+}^{n}}\,|\,H^{i,\max}_{\Upsilon}(N)\leq
h-\sum_{j=1}^{i}\varepsilon_{j}\\}$
for sufficiently small numbers $\varepsilon_{j}>0$ all these sets are non-
empty. The peeled $U$ for this sequence of thicknesses is defined as
$U^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon}=\bigcap_{i=0}^{k}U_{i}$
(58)
where we take $U_{0}=U$.
Definition of $U_{k}$ ($k={\rm rank}\Upsilon$) requires some comments. If
${\rm rank}\Upsilon=n$ then ${\rm Span}(\Upsilon)=\mathbb{R}^{n}$ and the
corresponding quasiequilibrium $N^{*}_{\mathbb{R}^{n}}$ is the global
equilibrium, i.e.
$H^{*}_{\mathbb{R}^{n}}=\min_{N\in\mathbb{R}^{n}_{+}}H(N)\,(=\min
H),\;\;N^{*}_{\mathbb{R}^{n}}=\underset{{N\in\mathbb{R}^{n}_{+}}}{\operatorname{argmin}}H(N)$
In this case, $H^{k,\max}_{\Upsilon}(N)\equiv\min H$ and either $U_{k}$ is the
nonnegative orthant (if $h-\sum_{j=1}^{i}\varepsilon_{j}\geq\min H$) or it is
empty (if $h-\sum_{j=1}^{i}\varepsilon_{j}<\min H$). Therefore, in this case
the term $U_{k}$ is not needed in Equation (58).
If $k={\rm rank}\Upsilon<n$ then the term $U_{k}$ is necessary. In this case,
$H^{k,\max}_{\Upsilon}(N)=H^{*}_{{\rm Span}(\Upsilon)}$ and $U_{k}$ defines
non-trivial peeling.
###### Proposition 12.
1. 1.
For sufficiently small thicknesses $\varepsilon_{1},\ldots,\varepsilon_{k}>0$
the peeled set $U^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon}$ is non-
empty and forward–invariant with respect to kinetic equations (34) with the
thermodynamic Lyapunov function $H$, reaction rates presented by Equation (36)
(detailed balance) and the reversible reaction mechanism with the set of
stoichiometric vectors $\Gamma\subseteq\Upsilon$.
2. 2.
For these thicknesses $\varepsilon_{1},\ldots,\varepsilon_{k}>0$ the peeled
set $U^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon}$ is
forward–invariant with respect to kinetic equations (34) with the perturbed
thermodynamic Lyapunov function $H+\Delta H$, reaction rates presented by
Equation (36) (detailed balance) and the reversible reaction mechanism with
the set of stoichiometric vectors $\Gamma\subseteq\Upsilon$ if the
perturbation $\Delta H$ is sufficiently uniformly small with its second
derivatives.
The similar proposition is valid for the systems with complex balance (because
of the local equivalence theorem). Peeling of the sublevel sets of a convex
function will produce a Lyapunov function similarly to Proposition 10.
Essential difference of Proposition 12 from Proposition 8 is in the ultimate
positive–invariance of $U^{\varepsilon}_{\Upsilon}$ if it is non-empty. To
provide the forward–invariance of
$U^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon}$ we need an additional
property.
Let $E\in\mathcal{E}_{\Upsilon}$ be a subspace of the form $E={\rm
Span}(\Gamma)$, $\Gamma\subset\Upsilon$. The quasiequilibrium surface
$\Phi_{E}\subset\mathbb{R}^{n}_{+}$ is a set of all quasiequilibria
$N^{*}_{E}(N)$ ($N\in\mathbb{R}^{n}_{+}$). The Legendre transform of
$\Phi_{E}$ (its image in the space of potentials $\check{\mu}$) is the
orthogonal supplement to $E$, for every $\check{\mu}$ from this image
$(\check{\mu},\gamma)=0$, and this is an equivalent definition of $\Phi_{E}$.
For every $E\in\mathcal{E}_{\Upsilon}$ we define the $E$-faces of
$U^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon}$ as follows. Let $\dim
E=i$. Consider the generalized cylindrical surface with the given value
$H^{*}_{E}(N)=q$
$S_{E}^{q}=\\{N\in R^{n}_{+}\,|\,H^{*}_{E}(N)=q\\}$
The $E$-faces of $U^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon}$
belong to the intersection
$\Psi^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon,\,E}=S_{E}^{h-(\varepsilon_{1}+\ldots+\varepsilon_{i})}\cap
U^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon}$
###### Proposition 13.
Let $h-\sum_{j=1}^{k}\varepsilon_{j}>\min H$ and for every
$E,L\in\mathcal{E}_{\Upsilon}$ the following property holds:
$\Psi^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon,\,E}\cap\Phi_{L}=\emptyset$
if $L\nsubseteq E$. Then the peeled set
$U^{\varepsilon_{1},\ldots,\varepsilon_{k}}_{\Upsilon}$ is non-empty and
forward–invariant with respect to kinetic equations (34) with the perturbed
thermodynamic Lyapunov function $H+\Delta H$, reaction rates presented by
Equation (36) (detailed balance) and the reversible reaction mechanism with
the set of stoichiometric vectors $\Gamma\subseteq\Upsilon$ if the
perturbation $\Delta H$ is sufficiently uniformly small with its second
derivatives.
The proof is an application of the general $H$-theorem based on the partial
equilibrium criterion (for illustration see Figure 6 c). The condition of
Proposition 13 means that the result of peeling in higher dimensions does not
destroy the main property of the 1D peeling (Proposition 8): if a positive
boundary point $N$ of the bodily peeled set is a partial equilibrium in
direction $\gamma\in\Upsilon$ then the stoichiometric vector $\gamma$ belongs
to a supporting hyperplane of this peeled set at $N$.
### A3. Greedy peeling
The goal of the forward–invariant peeling is to create a forward–invariant
convex set from an initial convex set $U$ by deletion (peeling) of its non-
necessary parts. The resulting (peeled) set should be forward–invariant with
respect to any GMAL kinetics with a given reaction mechanism and the
thermodynamic Lyapunov function $H$ (“free energy”). In more general but
practically useful settings, we consider not a single system but a family of
systems with the given reaction mechanism but for a set of Lyapunov functions
$H(N)=H_{0}(N)+(l,N)$ where $l\in Q$ and $Q\subset\mathbb{R}^{n}$ is a convex
compact set. We would like to produce a set that is forward–invariant with
respect to all these systems (and, therefore, with respect to the differential
inclusion (compare to (22))
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}\in
V\sum_{\rho}\gamma_{\rho}\varphi_{\rho}\left[\exp(\alpha_{\rho},\check{\mu})-\exp(\beta_{\rho},\check{\mu})\right]$
(59)
where
$\check{\mu}-\nabla H_{0}(N)=l\in Q$
and $\varphi_{\rho}\in\mathbb{R}_{+}$.
Further on, we consider systems with fixed volume, therefore we omit the
factor $V$ and make no difference between the amounts $N_{i}$ and the
concentrations $c_{i}$.
The peeling procedure proposed in the previous subsection works but it is
often too extensive and produces not the maximal possible forward–invariant
set. We would like to produce the maximal forward–invariant subset of $U$ and,
therefore, have to minimize peeling. Here we meet a slightly unexpected
obstacle. The union (and the closure) of forward–invariant sets is also
forward–invariant, whereas the union of convex sets may be non-convex.
Therefore, there exists the unique maximal forward–invariant subset of $U$ but
it may be non-convex and the maximal convex forward–invariant subset may be
non-unique. If we would like to find the maximal forward–invariant subset then
we have to relax the requirement of convexity.
A set $U$ is directionally convex with respect to a set of vectors $\Gamma$ if
for every $x\in U$ and $\gamma\in\Gamma$ the intersection
$(x+\mathbb{R}\gamma)\cap U$ is a segment of a straight line:
$(x+\mathbb{R}\gamma)\cap
U=(x+[a,b]\gamma),\;\mbox{or}\;(x+]a,b]\gamma),\;\mbox{or}\;(x+[a,b[\gamma),\mbox{or}\;(x+]a,b[\gamma)$
The minimal forward–invariant non-convex (but directionally convex) sets were
introduced in Gorban1979 and studied for chemical kinetics in G11984 and for
Markov chains (master equation) in Zylka1985 .
Let $U\subset\mathbb{R}^{n}_{+}$ be a compact subset. The greedy peeling of
$U$ is constructed for an inclusion (59) as a sequence of peeling operations
$\Pi_{\gamma}$, where $\gamma$ is a stoichiometric vector of an elementary
reaction. A point $x\in U$ belongs to $\Pi_{\gamma}(U)$ if and only if there
exists such a segment $[a,b]\subset\mathbb{R}$ that
* •
$0\in[a,b]$;
* •
$x+[a,b]\gamma\subset U$;
* •
if $y\in(x+\mathbb{R}\gamma)\cap\mathbb{R}_{+}^{n}$ and
$(\nabla_{N}H_{0}(N)\left.\right|_{N=y}+l,\gamma)=0$ for some $l\in Q$ then
$y\in x+[a,b]\gamma$.
We call the set
$S_{\gamma}=\\{N\in\mathbb{R}_{+}^{n}\,|\,(\nabla
H_{0}\left.\right|_{N}+l,\gamma)=0\mbox{ for some }l\in Q\\}$
the equilibrium strip for the elementary reaction with the stoichiometric
vector $\gamma$.
Another equivalent description of the operation $\Pi_{\gamma}$ may be useful.
Find the orthogonal projection of $U\cap S_{\gamma}$ onto the orthogonal
complement to $\gamma$, the hyperplane $\gamma^{\bot}\subset\mathbb{R}^{n}$.
Let $\pi_{\gamma}^{\bot}$ be the orthogonal projector onto this hyperplane.
Find all such $z\in\pi_{\gamma}^{\bot}(U\cap S_{\gamma})$ that
$((\pi_{\gamma}^{\bot})^{-1}z)\cap S_{\gamma}=U\cap S_{\gamma}$
This set is the base of $\Pi_{\gamma}(U)$, i.e. it is
$B_{\gamma}(U)=\pi_{\gamma}^{\bot}(\Pi_{\gamma}(U))$
For each $z\in B_{\gamma}(U)$ consider the straight line
$(\pi_{\gamma}^{\bot})^{-1}z$. This line is parallel to $\gamma$ and its
orthogonal projection onto ${\gamma}^{\bot}$ is one point $z$. The
intersection $S_{\gamma}\cap((\pi_{\gamma}^{\bot})^{-1}z)$ is a segment. Find
the maximal connected part of $U\cap((\pi_{\gamma}^{\bot})^{-1}z)$ that
includes this segment. This is also a segment (a fiber). Let us call it
$F_{z,\gamma}(U)$. We define
$\Pi_{\gamma}(U)=\bigcup_{z\in B_{\gamma}(U)}F_{z,\gamma}(U)$
The set $\Pi_{\gamma}(U)$ is forward–invariant with respect to the
differential inclusion (59) if the reaction mechanism consists of one reaction
with the stoichiometric vector $\gamma$. It is directionally convex in the
direction $\gamma$. Of course, if we apply the operation
$\Pi_{\gamma^{\prime}}$ with a different stoichiometric vector
$\gamma^{\prime}$ to $\Pi_{\gamma}(U)$ then the forward-invariance with
respect to the differential inclusion (59) for one reaction with the previous
stoichiometric vector $\gamma$ may be destroyed. Nevertheless, if we apply an
infinite sequence of operations $\Pi_{\gamma_{\rho}}$ (${\rho}=1,\ldots,m$)
where all the stoichiometric vectors $\gamma_{\rho}$ from the reaction
mechanism appear infinitely many times then the sequence converges to the
maximal forward–invariant subset of $U$ because of monotonicity (in
particular, this limit may be empty if there is no positively invariant subset
in $U$). The limit set is directionally convex in directions ${\gamma_{\rho}}$
(${\rho}=1,\ldots,m$) and is the same for all such sequences.
### A4. A toy example
Let us consider a reaction mechanism
$A_{1}{\overset{k_{1}}{\rightarrow}}A_{2}{\overset{k_{2}}{\rightarrow}}A_{3}{\overset{k_{3}}{\rightarrow}}A_{1},\;\;\;2A_{1}\underset{k_{-4}}{\overset{k_{4}}{\rightleftharpoons}}3A_{2}$
(60)
with the classical mass action law and interval constants $0<k_{i\,\min}\leq
k_{i}\leq k_{i\,\max}<\infty$. Consider the kinetic equations with such
interval constants and classical mass action law.
The stoichiometric vectors of the reactions are
$\gamma_{1}=\left(\begin{array}[]{c}-1\\\ 1\\\
0\end{array}\right);\;\gamma_{2}=\left(\begin{array}[]{c}0\\\ -1\\\
1\end{array}\right);\;\gamma_{3}=\left(\begin{array}[]{c}1\\\ 0\\\
-1\end{array}\right);\;\gamma_{4}=\left(\begin{array}[]{c}-2\\\ 3\\\
0\end{array}\right);\;$ (61)
We will demonstrate how to use peeling for solving of the following problem
for the system (60): is it possible that the solution of the differential
inclusion with these interval constants starting from a positive vector will
go to zero when $t\to\infty$? (This question for this system was considered
recently as an unsolved problem GopaShiu2013 .)
Let us use the local equivalence of systems with complex and detailed balance
and represent this system as a particular case of differential inclusion (59)
(with possible extension of the interval of constants).
The equilibrium concentrations $c_{i}^{*}$ in the irreversible cycle satisfy
the following identities:
$k_{1}c_{1}^{*}=k_{2}c_{2}^{*}=k_{3}c_{3}^{*},\;\;\frac{c_{i}^{*}}{c^{*}_{j}}=\frac{k_{j}}{k_{i}}$
Instead of the irreversible cycle of linear reactions we will take the
reversible cycle
$A_{1}\underset{\kappa_{-1}}{\overset{\kappa_{1}}{\rightleftharpoons}}A_{2}\underset{\kappa_{-2}}{\overset{\kappa_{2}}{\rightleftharpoons}}A_{3}\underset{\kappa_{-3}}{\overset{\kappa_{3}}{\rightleftharpoons}}A_{1}$
(62)
with the interval restrictions on the equilibrium constants (the ratios of the
reaction rate constants $\kappa_{j}/\kappa_{-j}$)
$\frac{\min k_{2}}{\max k_{1}}\leq\frac{\kappa_{1}}{\kappa_{-1}}\leq\frac{\max
k_{2}}{\min{k_{1}}},\;\;\frac{\min k_{3}}{\max
k_{2}}\leq\frac{\kappa_{2}}{\kappa_{-2}}\leq\frac{\max
k_{3}}{\min{k_{2}}},\;\;\frac{\min k_{1}}{\max
k_{3}}\leq\frac{\kappa_{3}}{\kappa_{-3}}\leq\frac{\max k_{1}}{\min{k_{3}}}$
(63)
The detailed balance condition should also hold for the constants $\kappa_{\pm
j}$:
$\kappa_{1}\kappa_{2}\kappa_{3}=\kappa_{-1}\kappa_{-2}\kappa_{-3}$ (64)
The equilibria for this cycle satisfy the conditions
$\kappa_{1}c_{1}^{*}=\kappa_{-1}c_{2}^{*},\;\kappa_{2}c_{2}^{*}=\kappa_{-2}c_{3}^{*},\;\kappa_{3}c_{3}^{*}=\kappa_{-3}c_{1}^{*}$
.
Figure 7: Partial equilibria of the reversible cycle (62) with the interval
restrictions on the equilibrium constants. The triangle is split by the lines
of partial equilibria $A_{i}\rightleftharpoons A_{j}$ into several
compartments. The borders of these compartments are combined from the segments
of the dashed lines. These dashed lines correspond to the minima and maxima of
the equilibrium constants $\kappa_{j}/\kappa_{-j}$. In each compartment, the
cone (the angle) of possible directions of $\dot{c}$ is given. This is a
proper cone (an angle that is less than $\pi$) outside the equilibrium strips,
a halfplane in an equilibrium strip of a single reaction, and a whole plane in
the intersection of two such strips. The area of the possible equilibria
(where the angle of possible directions of $\dot{c}$ is the whole plane) is
outlined by bold line and colored in green.
These conditions provide the same range of equilibrium concentrations for the
reversible and irreversible cycles. Therefore, the possible value of $\dot{c}$
for the irreversible cycle in the given interval of reaction rate constants
always belongs to the cone of possible values of $\dot{c}$ of the reversible
cycle under the given restrictions (63) and the detailed balance condition
(64).
For the reversible cycle the reaction rates are
$r_{1}=\kappa_{1}c_{1}-\kappa_{-1}c_{2},\;r_{2}=\kappa_{2}c_{2}-\kappa_{-2}c_{3},\;r_{3}=\kappa_{3}c_{3}-\kappa_{-3}c_{1}$
The reaction rate of the reaction $2A_{1}\rightleftharpoons 3A_{2}$ is
$r_{4}=k_{4}c_{1}^{2}-k_{-4}c_{3}^{3}$.
The time derivatives of the concentrations are
$\dot{c}_{1}=-r_{1}+r_{3}-2r_{4},\;\dot{c}_{2}=r_{1}-r_{2}+3r_{4},\;\dot{c}_{3}=r_{2}-r_{3}$
(65)
The differential inclusion for the reversible linear cycle (62) is represented
in Fig. 7. There are three types of areas: (i) area where the equilibria may
be located and the direction of $\dot{c}$ may coincide with any vector of the
linear subspace $\sum_{i}\dot{c}_{i}=0$, (ii) areas where direction of one
reaction is indefinite but the signs of two other reactions rates are fixed,
and (iii) areas where the signs of all reaction rates are fixed. The cones
(angles) of possible vectors $\dot{c}$ are drawn in Fig. 7
For the linear system the scheme presented in Figure 7 does not depend on the
positive value of the balance $\sum_{i}c_{i}=\varepsilon$. We can just rescale
$c_{i}\leftarrow c_{i}/\varepsilon$ and return to the unit triangle with the
unit sum of $c_{i}$. The situation is different for the nonlinear reaction
$2A_{1}\rightleftharpoons 3A_{2}$. Consider the “equilibrium strip” where the
reaction rate $r_{4}=k_{4}c_{1}^{2}-k_{-4}c_{2}^{3}$ may be zero for the
admissible reaction rate constants:
$\frac{\min k_{-4}}{\max k_{4}}\leq\frac{c_{1}^{2}}{c_{2}^{3}}\leq\frac{\max
k_{-4}}{\min k_{4}}$
Let us take this strip on the plane $\sum_{i}c_{i}=\varepsilon$ and return it
to the unit triangle by rescaling ($c_{i}\leftarrow c_{i}/\varepsilon$). For
small $\varepsilon$ this strip approaches the $[A_{2},A_{3}]$ edge of the
triangle. It is situated between the line
${c_{1}}=\sqrt{\varepsilon}\sqrt{\frac{\max k_{-4}}{\min
k_{4}}}(1-{c_{3}})^{3/2}$
and the segment $[A_{2},A_{3}]$. Further we use the notation $\vartheta$ for
the coefficient in this formula:
$\vartheta=\sqrt{\varepsilon}\sqrt{\frac{\max k_{-4}}{\min k_{4}}}$
The line
$c_{1}=\vartheta(1-c_{3})^{3/2}$ (66)
separates the equilibrium strip of the reaction $2A_{1}\rightleftharpoons
3A_{2}$ (where $r_{4}=0$ for some admissible combinations of the reaction rate
constants) from the area where $r_{4}>0$ (i.e. $k_{4}({\varepsilon
c_{1}})^{2}-k_{-4}(\varepsilon c_{2})^{3}>0$ for all admissible
$k_{4},k_{-4}$. (We use the rescaling from the triangle with $\sum
c_{i}=\varepsilon$ to the unit triangle without further comments.)
We will study intersection of the equilibrium strip for the reaction
$2A_{1}\rightleftharpoons 3A_{2}$ with different planes and then scale the
result to the balance plane $\sum_{i}c_{i}=1$. The projection of the strip
from all planes $\sum_{i}c_{i}=a\varepsilon$ onto the unit triangle for
$a\in[\min a,\max a]>0$ belong to the projection of the strip from the plane
$\sum_{i}c_{i}=\varepsilon$ with the extended range of the equilibrium
constants:
$\min a\frac{\min k_{-4}}{\max k_{4}}\geq\frac{k_{-4}}{k_{4}}\leq\max
a\frac{\max k_{-4}}{\min k_{4}}$ (67)
This rescaling does not cause any difficulty but requires additional check at
the end of construction: does the set of the constructed faces (“peels”) has
the bounded ratio
$\frac{\max{\sum_{i}c_{i}}}{\min{\sum_{i}c_{i}}}$
with the upper estimate does not dependent on the values of
$\frac{k_{-4}}{k_{4}}$.
This line is tangent to the segment at the vertex $A_{3}$ (Fig. 8). On the
other side of the line the time derivative of $\sum_{i}c_{i}$ is positive:
$\sum_{i}\dot{c}_{i}=r_{4}>0$
Figure 8: The equilibrium strip of the reaction $2A_{1}\rightleftharpoons
3A_{2}$ (yellow) and the area where $\sum_{i}\dot{c}_{i}>0$ (blue) rescaled
from the triangle with $\sum c_{i}=\varepsilon$ to the unit triangle Figure
9: Faces of the peeled invariant set in the central projection onto unit
triangle. The borders between faces are highlighted by bold.
Let us describe first the structure of the peeled set. Select for peeling the
set $U=\\{c\,|\,\sum_{i}c_{i}\geq\varepsilon,\,c_{i}\geq 0\\}$. The structure
of peeling scaled to $c_{1}+c_{2}+c_{3}=1$ is presented in Fig. 9. It appears
that the piecewise linear peeling is sufficient. There are five faces
different from the coordinate planes. The face F0 is a polygon on the plane
$\sum c_{i}=1$. The face F1 is situated at the $A_{2}$ corner. It is produced
by the peeling parallel to ${\rm Span}\\{\gamma_{3},\gamma_{4}\\}$. The plane
of F1 is given by the equation $3c_{1}+2c_{2}+3c_{3}=const$. The face F2 is
presented by a parallelogram at the middle of the edge $[A_{2},A_{3}]$ (Fig.
9). It covers the intersection of the equilibrium strips of the reactions
$2A_{1}\rightleftharpoons 3A_{2}$ and the reaction $A_{2}\rightleftharpoons
A_{3}$. F2 is produced by the peeling parallel to ${\rm
Span}\\{\gamma_{2},\gamma_{4}\\}$. The plane is given by the equation
$3c_{1}+2c_{2}+2c_{3}=const$. Its intersection with the plane
$c_{1}+c_{2}+c_{3}=1$ is a straight line
$c_{1}=c_{1}^{\circ},c_{2}+c_{3}=1-c_{1}^{\circ}$ for a sufficiently small
$c_{1}^{\circ}>0$.
The final fragment of peeling is situated near the vertex $A_{3}$ (Fig. 9). It
consists of two triangles. The first (F3) is a fragment of a plane
$c_{1}+c_{2}+vc_{3}=const$ ($0<v<1$). Parameter $v$ is defined from the
condition of positive invariance below.
The second triangle (F4) situated near the vertex $A_{3}$ is parallel to
$\gamma_{4}$ and has the common edge with F3. The general plane parallel to
$\gamma_{4}$ is given by the equation $3c_{1}+2c_{2}+lc_{3}=D$. We will define
the parameters $l$ and $D$ using the vertices of the face F3, V34 and V0234
(see Fig. 9).
Let us define the parameters of this peeling. At the $A_{2}$ corner the
peeling is parallel to ${\rm Span}\\{\gamma_{3},\gamma_{4}\\}$. The plane can
be given by the equation $3c_{1}+2c_{2}+3c_{3}=const$. The edge between this
face and the face $\sum c_{i}=1$ belongs to the straight line
$c_{2}=c_{2}^{\circ}$, $c_{1}+c_{3}=1-c_{2}^{\circ}$. The level
$c_{2}^{\circ}$ should be selected above all the equilibria of the linear
reactions (Fig. 7) but below the intersection of the curve (66) with the right
border of the equilibrium strip of the reaction $A_{1}\rightleftharpoons
A_{3}$ given by the equation
$c_{3}=c_{1}\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}$. For the
intersection we have
$c_{3}=\vartheta\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}(1-c_{3})^{3/2}$
Therefore, at this point
$c_{3}<\vartheta\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}$
and $c_{1}<\vartheta$ on the line (66). Therefore, we can select
$c_{2}^{\circ}=1-\vartheta\left(\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}+1\right)$
This $c_{2}^{\circ}$ is smaller than the value of $c_{2}$ at the intersection,
and for sufficiently small $\vartheta$ the line $c_{2}=c_{2}^{\circ}$ is close
to the vertex $A_{2}$ and does not intersect the area of possible equilibria
of linear reactions (the area colored in green in Fig. 7).
Consider intersection of the straight line $c_{2}=c_{2}^{\circ}$,
$c_{1}+c_{3}=1-c_{2}^{\circ}$ with the curve (66) and evaluate the value of
$c_{1}$ at this intersection from above:
$c_{1}=\vartheta(c_{2}^{\circ}+c_{1})^{3/2}$, $c_{1}<\vartheta$, hence,
$c_{1}<c_{1}^{\circ}=\vartheta(c_{2}^{\circ}+\vartheta)^{3/2}$.
Thus, the vertex V012 at the intersection of three faces, F0, F1, and F2 is
selected as $(c_{1}^{\circ},c_{2}^{\circ},c_{3}^{\circ})$, where
$c_{1}^{\circ}=\vartheta\left(1-\vartheta\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}\right)^{3/2}$
$c_{2}^{\circ}=1-\vartheta\left(\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}+1\right)$
$c_{3}^{\circ}=1-c_{1}^{\circ}-c_{2}^{\circ}=\vartheta\left(1+\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}-\left(1-\vartheta\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}\right)^{3/2}\right)$
To check that this point is outside the equilibrium strip of the reaction
$A_{1}\rightleftharpoons A_{3}$, we calculate
$\frac{c_{3}^{\circ}}{c_{1}^{\circ}}=\frac{1+\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}}{\left(1-\vartheta\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}\right)^{3/2}}-1>\max\left\\{\frac{\kappa_{-3}}{\kappa_{3}}\right\\}$
The next group of parameters we have to identify are the coordinates of the
vertex V0234 $(c_{1}^{\prime},c_{2}^{\prime},c_{3}^{\prime})$ at the
intersection of four faces F0, F2, F3, and F4. We will define it as the
intersection of F0, F2, and F3 and then use its coordinates for defining the
parameters of F4. One coordinate, $c_{1}^{\prime}$ is, obviously,
$c_{1}^{\prime}=c_{1}^{\circ}$ because the intersection of F2 and F0 is
parallel to $\gamma_{2}$, i.e. it is parallel to the edge $[A_{2},A_{3}]$ of
the unit triangle and $c_{1}$ is constant on this edge. Another coordinate,
$c_{3}^{\prime}$ can be easily determined from the condition that the line
$c_{3}=c_{3}^{\prime}$ in the unit triangle should not intersect the strips of
equilibria for the reactions $A_{2}\rightleftharpoons A_{3}$ and
$A_{1}\rightleftharpoons A_{3}$. Immediately, these condition give the
inequalities that should hold for all admissible reaction rate constants:
$c_{3}^{\prime}>\frac{\kappa_{-3}}{\kappa_{3}+\kappa_{-3}},\;\;c_{3}^{\prime}>\frac{\kappa_{2}}{\kappa_{2}+\kappa_{-2}}$
Finally,
$c_{3}^{\prime}>\max\left\\{\frac{1}{\min\left\\{\frac{\kappa_{3}}{\kappa_{-3}}\right\\}+1},\;\frac{1}{1+\min\left\\{\frac{\kappa_{-2}}{\kappa_{2}}\right\\}}\right\\}$
We can take $c_{3}^{\prime}$ between this maximum and 1: for example, we
propose
$c_{3}^{\prime}=\frac{1}{2}+\frac{1}{2}\max\left\\{\frac{1}{\min\left\\{\frac{\kappa_{3}}{\kappa_{-3}}\right\\}+1},\;\frac{1}{1+\min\left\\{\frac{\kappa_{-2}}{\kappa_{2}}\right\\}}\right\\}$
For sufficiently small $\vartheta$, the inequality
$c_{3}^{\prime}+c_{1}^{\circ}<1$ holds, and we can take
$c_{2}^{\prime}=1-c_{3}^{\prime}-c_{1}^{\circ}>0$.
If we know $c_{3}^{\prime}$ and $v$ then we know the equation of the plane F4:
$c_{1}+c_{2}+vc_{3}=1-(1-v)c_{3}^{\prime}$
We also find immediately the coordinates of the vertex V34, the intersection
of F3 (and F4) with the coordinate axis $A_{3}$. This vertex is
$(0,0,\frac{1}{v}(1-c_{3}^{\prime})+c_{3}^{\prime})$.
Let us define the parameters $l$ and $D$ for the face F4. This face should
include the vertices V0234 $(c_{1}^{\circ},c_{2}^{\prime},c_{3}^{\prime})$ and
V34 $(0,0,\frac{1}{v}(1-c_{3}^{\prime})+c_{3}^{\prime})$. Therefore,
$l=v\left(2+\frac{c_{1}^{\circ}}{c_{1}^{\circ}+c_{2}^{\prime}}\right),\;D=3c_{1}^{\circ}+2c_{2}^{\prime}+lc_{3}^{\prime}$
To demonstrate the positive invariance of the peeled set we have to evaluate
the sign of the inner product of $\dot{c}$ onto the inner normals to the faces
on the faces.
The signs of some reaction rates are unambiguously defined on the faces:
* •
On F0 $r_{4}>0$;
* •
On F1 $r_{1}<0$, and $r_{2}>0$;
* •
On F2 $r_{1}<0$, and $r_{3}>0$;
* •
On F3 $r_{2}<0$, $r_{3}>0$, and $r_{4}>0$;
* •
On F4 $r_{1}<0$, $r_{2}<0$, and $r_{3}>0$.
The inner products of $\dot{c}$ (65) onto the inner normals to the faces are:
* •
On F0 $\frac{d}{dt}(c_{1}+c_{2}+c_{3})=r_{4}>0$;
* •
On F1 $\frac{d}{dt}(3c_{1}+2c_{2}+3c_{3})=-r_{1}+r_{2}>0$;
* •
On F2 $\frac{d}{dt}(3c_{1}+2c_{2}+2c_{3})=-r_{1}+r_{3}>0$;
* •
On F3 $\frac{d}{dt}(c_{1}+c_{2}+vc_{3})=(1-v)(-r_{2}+r_{3})+r_{4}>0$
($0<v<1$);
* •
On F4 $\frac{d}{dt}(3c_{1}+2c_{2}+lc_{3})=-r_{1}-(2-l)r_{2}+(3-l)r_{3}<0$ if
$0<l<2$.
Thus, the peeled set is positively invariant if $0<l<2$. This means
$0<v<\frac{1}{1+\frac{c_{1}^{\circ}}{2(c_{1}^{\circ}+c_{2}^{\prime})}}$
It is sufficient to take $0<v\leq\frac{2}{3}$ (for example, $v=\frac{2}{3}$)
because of the obvious inequality,
$\frac{c_{1}^{\circ}}{2(c_{1}^{\circ}+c_{2}^{\prime})}<\frac{1}{2}$.
We see that the peeled faces are located between the planes
$\sum_{i}c_{i}=\varepsilon$ and $\sum_{i}c_{i}=\frac{3}{2}\varepsilon$ (for
$v=2/3$). Therefore, it is sufficient to take in the rescaling (67) the
constants $\max a=\frac{3}{2}$, $\min a=1$ which do not depend on the
equilibrium constant.
We have demonstrated that for any given range of positive kinetic constants
any positive solution of the kinetic inclusion for the system (60) cannot
approach the origin when $t\to\infty$.
We have started from a system (60) with interval rate constants and have
embedded the corresponding differential inclusion into a differential
inclusion for a reversible system with detailed balance (64) and interval
restrictions onto equilibrium constants (63).
We have constructed a piecewise-linear surface that isolated the
$\varepsilon$-vicinity of the origin from the outside for sufficiently small
$\varepsilon>0$. This surface cannot be intersected by the solutions of the
kinetic inclusion in the motion from the outside to the origin.
The peeling procedure used in this toy-example differs from the universal
greedy peeling. (It is the simplified version of the greedy peeling.) We have
guessed the structure of the corner near $A_{3}$ and build two plain faces, F3
and F4, instead of a sequence of the curvilinear “cylindric” faces. This
piecewise peeling is not minimal but is simpler for drawing.
### Acknowledgement
I am very grateful to Dr Anne Shiu from the Department of Mathematics at the
University of Chicago. She ensured me that my results annotated in 1979
GorbanRKCL1980 may be still of interest for the chemical dynamics community.
In this Appendix, I explain one of the methods (forward–invariant peeling)
used in this work, the further details will follow.
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|
arxiv-papers
| 2012-12-30T19:37:08 |
2024-09-04T02:49:39.780889
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexander N. Gorban",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1212.6767"
}
|
1212.6784
|
# On Quantization, the Generalized Schrödinger Equation and Classical
Mechanics111Author’s note: This is the original text of a pre-print that was
lost for some twenty years. Over the years, people have asked me for a copy
but I had none. This work is not refereed and was never published (since it
was lost). Given what I know now, I would write it differently, but place it
here under Creative Commons 3.0 - Attribution License under the condition that
this footnote is displayed. It is old, but may be of interest for the ideas it
explores. Opinions expressed may have been superseded by later works.
Kingsley R. W. Jones School of Physics, University of Melbourne, Parkville
3052,
Melbourne, Australia
(Original Preprint: June 1991)
###### Abstract
Using a new state-dependent, $\lambda$-deformable, linear functional operator,
${\cal Q}_{\psi}^{\lambda}$, which presents a natural $C^{\infty}$ deformation
of quantization, we obtain a uniquely selected non–linear,
integro–differential Generalized Schrödinger equation. The case ${\cal
Q}_{\psi}^{1}$ reproduces linear quantum mechanics, whereas ${\cal
Q}_{\psi}^{0}$ admits an exact dynamic, energetic and measurement theoretic
reproduction of classical mechanics. All solutions to the resulting classical
wave equation are given and we show that functionally chaotic dynamics exists.
###### pacs:
0230, 0365, 0545
††preprint: UM-P-91/47
This paper reports some new and quite elementary results that we believe must
lie at the very heart of the connection between classical and quantum
mechanics. In the interests of rapid communication we shall not describe how
we obtained them. This omission will be rectified in subsequent publications.
Our work appears in the spirit of de Brogliebrog (1). It is inspired by the
early work of Diracdirac1 (2, 3), but evolved purely from a single geometric
picturejones1 (4) that grew out of our parallel measurement theoretic
researchjones2 (5), coupled with thinking stimulated by a single prescient
paper by Klauderklau (6). The work potentially opens up a vast new territory
of quantal dynamical systems research.
Perhaps the most topical result we shall demonstrate is that quantum chaos
certainly exists within a dynamical system that contains standard quantum
mechanics. Moreover, the nonlinear dynamics described appears to fit within
the recently elaborated formal scheme of Weinbergwein (7). However, this work
does more than that. It both solves an old problem and generates a new
dynamical structure of obvious physical application. This structure should
prove useful both for phenomenological model building, and as a theoretical
guide to some potentially exciting new physics at the boundary of the quantal
and classical domains.
A brief outline is in order. We first define quantization, then deformed
quantization. From the latter we derive the Generalized Schrödinger equation.
This includes both the usual Schrödinger equation and a new Classical
Schrödinger equation as particular values of $\lambda$. We exhibit the entire
infinite family of solutions of the latter equation in parametric form, and
then show that it gives an exact energetic, dynamic and measurement theoretic
reproduction of classical mechanics via a travelling wave double solution.
Consider any classical phase space function $f_{c}({\bf q},{\bf p})$ of
$n$–degrees of freedom. Quantization of this phase space function amounts to
application of the following $\psi$-dependent linear functional operator:
${\cal
Q}_{\psi}\equiv\exp\left\\{\sum_{k=1}^{n}(\hat{q}_{k}-\langle\hat{q}_{k}\rangle_{\psi})\frac{\partial}{\partial
q_{k}}+(\hat{p}_{k}-\langle\hat{p}_{k}\rangle_{\psi})\frac{\partial}{\partial
p_{k}}\right\\}.$ (1)
The domain of this map is any region of phase space where $f_{c}({\bf q},{\bf
p})$ is analytic, its range the set of quantal operators. The domain of the
resulting quantal operator proves to be precisely the set of all $\psi$ such
that the numerical values of the expectation values
$\langle\hat{q}_{k}\rangle_{\psi}$ and $\langle\hat{p}_{k}\rangle_{\psi}$ lie
in an analytic domain of the initial classical function $f_{c}({\bf q},{\bf
p})$.
The linear operator (1) is to be understood as a formal power series. It
generates a natural, uniquely defined, Weyl–ordered operator Taylor series
whose action upon any classical phase space function is: ${\cal Q}_{\psi}\circ
f_{c}({\bf q},{\bf p})=\hat{f}_{q}(\hat{\bf q},\hat{\bf p}),$ where $q_{k}$
and $p_{k}$ are general classical canonical coordinates satisfying the rules:
$\\{q_{j},q_{k}\\}=0$, $\\{p_{j},p_{k}\\}=0$ and
$\\{q_{j},p_{k}\\}=\delta_{ij},$ whereas $\hat{q}_{k}$ and $\hat{p}_{k}$ are
general quantal canonical operators satisfying the rules
$[\hat{q}_{j},\hat{q}_{k}]=0$, $[\hat{p}_{j},\hat{p}_{k}]=0$ and
$[\hat{q}_{j},\hat{p}_{k}]=i\hbar\delta_{ij}\hat{1}$. The classical
differential operators $\partial q_{k}$ and $\partial p_{k}$ are taken to
commute with $\hat{q}_{k}$ and $\hat{p}_{k}$ and merely evaluate derivatives
of the classical function $f({\bf q},{\bf p})$ in a $\psi$-dependent way at
the quantal expectation values $\langle\hat{q}_{k}\rangle_{\psi}$ and
$\langle\hat{p}_{k}\rangle_{\psi}$. However, note that the $c$–number
derivatives need not be taken to commute with one another so that non–analytic
behaviour such as $\partial^{2}_{qp}f(q,p)\neq\partial^{2}_{pq}f(q,p)$ is also
catered for.
Because the $\hat{q}_{j}$ and $\hat{p}_{j}$ do not commute, the generalized
Taylor series implict in (1) should not have like order terms collected. This
is the source of its important Weyl–ordering property. An example will
suffice. Consider the one dimensional generalized harmonic oscillator:
$H_{c}(q,p)=\frac{1}{2}(ap^{2}+bpq+cq^{2})$. Applying (1) we obtain
$\hat{H}_{q}(\hat{q},\hat{p})\equiv{\cal Q}_{\psi}\circ
H_{c}(q,p)=\frac{1}{2}(a\hat{p}^{2}+b/2[\hat{p}\hat{q}+\hat{q}\hat{p}]+c\hat{q}^{2}).$
The $\psi$-dependence of the result has disappeared. This may be understood
via the deep observation of Weierstrassnote (8) that analytic functions can be
considered as a family of power series whose totality comprises the analytic
function and whose consistency of definition requires only that the individual
members of that family should share continuously connected overlapping domains
of agreement. The construction of the full quantal operator therefore amounts
to a kind of analytic continuation in the wave function $\psi$, with reference
to the analytic domain of the starting classical function via the expectation
values of each $\psi$ so considered.
The above we believe to be a new result. It systematizes a significant body of
previously known resultsmess (9), resolves operator ordering ambiguities, and
suggests new results as welljones1 (4).
Let us now introduce the $\lambda$–deformed operator:
${\cal
Q}^{\lambda}_{\psi}\equiv\exp\left\\{\sum_{k=1}^{n}\lambda(\hat{q}_{k}-\langle\hat{q}_{k}\rangle_{\psi})\frac{\partial}{\partial
q_{k}}+\lambda(\hat{p}_{k}-\langle\hat{p}_{k}\rangle_{\psi})\frac{\partial}{\partial
p_{k}}\right\\}.$ (2)
This is to be understood in exactly the same manner as (1). However, for
$\lambda\neq 1$ the properties of (1) as a Taylor series are modified. In
general, (2) produces a $\psi$-dependent quantization prescription:
${\cal Q}_{\psi}^{\lambda}:f_{c}({\bf q},{\bf
p})\mapsto\hat{f}_{q}(\psi;\lambda)=\hat{f}_{q}^{\lambda}(\langle\hat{\bf
q}\rangle_{\psi}+\lambda(\hat{\bf q}-\langle\hat{\bf
q}\rangle_{\psi}),\langle\hat{\bf p}\rangle_{\psi}+\lambda(\hat{\bf
p}-\langle\hat{\bf p}\rangle_{\psi})).$
Moreover, it is clear that this happens in a smooth fashion and yields an
entirely natural and inifinitely differentiable deformation of the dynamical
structure of quantum mechanics.
Application of the deformed rule (2) now produces a new family of non–linear
integro–differential equations, which family we elect to call The Generalized
Schrödinger equation.
To $H_{c}({\bf q},{\bf p})$ we apply ${\cal Q}_{\psi}^{\lambda}$ to obtain
$\hat{H}_{q}(\psi;\lambda)\equiv{\cal Q}_{\psi}^{\lambda}\circ H_{c}({\bf
q},{\bf p}),$ This yields its Generalized Schrödinger equation as
$i\hbar\frac{d}{dt}|\psi\rangle=\hat{H}_{q}(\psi;\lambda)|\psi\rangle.$ (3)
A precursorial formal structure into which (3) appears to fall has been given
recently by Weinbergwein (7).
Significantly, $\lambda$ is a mathematically free parameter, though if this is
new physics then its value in any physical situation probably includes $\hbar$
in dimensionless combination with other parameters.
The family (3) incorporates the following physically plausible features: a
state dependent energy behaviour in the non–linear regime ($\lambda\neq 1$);
unitary but non distance preserving dynamics; true quantum chaos and
representation independent physical predictions. As noted by numerous
authorswein (7, 10) the final condition is vital to the survival of symmetry
as a physical principle.
Note that the $\psi$–dependence of $\hat{H}_{q}(\psi;\lambda)$ enters in this
case purely via the representation independent quantities:
$\langle{\bf\hat{q}}\rangle_{\psi}$ and $\langle{\bf\hat{p}}\rangle_{\psi}$.
This, coupled with the fact that the dynamical generator is Hermitian, ensures
that we have a $\psi$–dynamics that is at all times unitary and norm
preserving, but one that need not preserve the natural quantal metric
distanceprov (11): $d(\psi_{1},\psi_{2})\equiv
1-|\langle\psi_{1}|\psi_{2}\rangle|^{2}$. It follows that the superposition
principle is no longer valid in the regime $\lambda\neq 1$. This may well
require careful consideration when attempting to construct a dynamics for
non–separable systemsbial (10).
True quantum chaos is now indicated as the sensitive dependence of
wavefunction trajectories upon wavefunction initial conditionsnote3 (12). We
now exhibit such chaotic wavefunction dynamics for ${\cal Q}_{\psi}^{0}$
quantised classically chaotic dynamics. We call the $\lambda=0$ case the
Classical Schrödinger equation.
In coordinate representation the Classical Schrödinger equation corresponds to
a nonlinear first order integro–differential equation. For the one dimensional
classical Hamiltonian:
$H_{c}(q,p)=\frac{p^{2}}{2m}+V(q),$
It has the explicit form
$i\hbar\left(\frac{\partial}{\partial t}+\frac{{\langle
p\rangle}}{m}\frac{\partial}{\partial q}\right)\psi(q,t)=\left(V(\langle
q\rangle)+\frac{\partial V}{\partial q}(\langle q\rangle)(q-\langle
q\rangle)-\frac{\langle p\rangle^{2}}{2m}\right)\psi(q,t),$ (4)
where of course $\langle q\rangle$ and $\langle p\rangle$ must at all times
satisfy the relations
$\langle
q\rangle=\int_{-\infty}^{\infty}\\!q\psi^{*}(q,t)\psi(q,t)\,dq\;\;\mbox{and}\;\;\langle
p\rangle=\int_{-\infty}^{\infty}\\!-i\hbar\psi^{*}(q,t)\frac{\partial}{\partial
q}\psi(q,t)\,dq.$
Such an equation is rather difficult to solve in general. Here are all of its
solutions:
$\psi(q,t)=e^{i\phi(t)}\exp\left\\{-\frac{i}{2\hbar}\langle p\rangle\langle
q\rangle\right\\}\exp\left\\{+\frac{i}{\hbar}\langle p\rangle
q\right\\}\psi_{0}(q-\langle q\rangle),$ (5)
where $\psi_{0}(q)$ belongs to the infinite family of square integrable and
differentiable wavefunctions whose expectation values for both position and
momentum operators are equal to zero. The phase factor is
$\phi(t)=\frac{1}{\hbar}\int_{0}^{t}\left[\frac{1}{2}\left(\langle
p\rangle\frac{d}{dt}\langle q\rangle-\langle q\rangle\frac{d}{dt}\langle
p\rangle\right)-\frac{\langle p\rangle^{2}}{2m}-V(\langle
q\rangle)\right]\,d\tau.$ (6)
To explain how (5) provides an infinite family of solutions we must point out
that the terms $\langle q\rangle$ and $\langle p\rangle$ are time dependent
parameters that enforce the functional evolution of $\psi(q,t)$. The required
values of these parameters are obtained at all times by solving the purely
classical equations:
$\frac{d\langle q\rangle}{dt}=\frac{\partial H_{c}}{\partial p}(\langle
q\rangle,\langle p\rangle),\;\;\mbox{and}\;\;\frac{d\langle
p\rangle}{dt}=-\frac{\partial H_{c}}{\partial q}(\langle q\rangle,\langle
p\rangle).$ (7)
In addition, due to the special form of (5), these parameters will always
correspond to the required expectation values. In this way (4) realizes Louis
de Broglie’s dream of the double solutionbrog (1).
In order to prove the above one simply substitutes (5) into (4). Taking real
and imaginary parts of the resulting equation, one can then derive (7) as an
essential consistency requirement. Many $\psi_{0}(q)$ will do, so that (4)
actually admits an infinite family of non–dispersive travelling wave solutions
whose expectation values follow precisely the trajectories of the classical
Hamiltonian evaluated upon each unique trajectory associated with each one of
an inifinite family of equivalent initial conditions (choice of
$\psi_{0}(q)$). This is the highly desirable version of Ehrenfest’s theorem
which one cannot derive from linear quantal mechanicsmess (9, 13). The proof
that (5) generates all of the solutions we shall present elsewherejones1 (4).
We remark that the ${\cal Q}_{\psi}^{1}$ harmonic oscillator has a zero point
energy, whereas the ${\cal Q}_{\psi}^{0}$ version does not, and $\lambda$
interpolates between the two. The existence of two possible quantizations, in
this case, was commented upon a long time ago by Klauderklau (6), who appears
to have been the first to encounter a manifestation of deformed quantization.
Note that the phase (6) includes an Aharanov–Anandanahar (14) contribution
such that the total value upon closed classical trajectories is numerically
equal to the classical actiondirac2 (3). Moreover, the geometric component of
the ${\cal Q}_{\psi}^{0}$ phase corresponds precisely to the abbreviated
action upon closed classical trajectories. This injects a natural quantal
geometric phase into classical mechanics and suggests that these phases are
the natural action variables of integrable quantal dynamicsjones1 (4).
Indeed for closed circuits $\Gamma$ of a ${\cal Q}_{\psi}^{0}$ quantised
$n$–degree of freedom classical system this geometric phase can be expressed
in terms of the first Poincaré integral invariant of classical mechanicsarn
(15) by the simple expression
$\gamma(\Gamma)=\frac{1}{\hbar}\oint\sum_{j=1}^{n}p_{j}\,dq_{j}.$ (8)
One now understands Einstein’s versionein (16) of the old Bohr–Sommerfeld
quantization condition to be a constraint upon the geometric phase
($2\pi\times$integer). To the ${\cal Q}_{\psi}^{0}$ quantised integrable
$n$–degree of freedom classical system there are $n$ geometric phase actions
$\gamma_{j}$ associated with the natural wave function evolution about a
functional torus parametrised by $\langle q\rangle$ and $\langle p\rangle$.
The values of these phases are obtained from (8) via integration upon the $n$
irreducible contours of the classical $n$–torus, which the functional
parameters $\langle q\rangle$ and $\langle p\rangle$ explorearn (15).
Note that (4) will exhibit regions of chaotic wavefunction dynamics for any
classically chaotic Hamiltonian system.
Turning now to energetic and measurement theoretic considerations, one can
readily verify that ${\cal Q}_{\psi}^{0}$ quantized classical mechanics yields
precisely the correct classical energy at all times where this is encoded in
the expectation value form: $H_{c}(\langle q\rangle,\langle
p\rangle,t)=\langle\psi(t)|\hat{H}_{q}(\psi(t);0)|\psi(t)\rangle$. This
quantal model of classical mechanics therefore amounts to a conservative
theory, just as one would expect, albeit one with $\psi$–dependent energy
dispersion.
Changing $\hbar$ does not alter the dynamical behaviour of the equation (4).
It simply rescales phase space. However, the quantum measurement theory is
changed under such rescaling. For instance, a simple argument based upon the
Heisenberg Uncertainty Principle $(\Delta x)^{2}(\Delta p)^{2}\geq\hbar^{2}/4$
shows that we might only expect to follow features of a wavefunction
trajectory that are large in comparison to a confidence region $\Delta$ of
area: $\int_{\Delta}dp\wedge dq\sim\hbar.$ More exotic arguments based upon
modern quantum measurement theory yield essentially the same conclusionart
(17, 18).
To conclude, ${\cal Q}_{\psi}^{\lambda}$ has the characteristic aesthetic
appeal of a most natural mathematical formalism. There is clearly considerable
scope for its extension (renormalized, deformed Taylor series may have
properties of general interest). The resulting Generalized Schrödinger
equation provides a sharp spur towards the general development of the theory
of integro–differential equations and clearly has great potential physical
application. Perturbative studies in $\lambda$ and the investigation of
physically coupled dynamics in $\lambda$ are indicatednote4 (19). Studies of
this kind offer a natural route to the exploration of functional attractors,
which would be likely to have important physical repercussions.
In this, a physical paper, we are more directly and vitally concerned with the
interpretation of $\lambda$. In view of the very natural appearance of this
dimensionless number we suggest that it encapsulates the degree to which a
quantal system may back react upon its environment. It is then entirely
plausible that the energy should be $\psi$–dependent (self–energy).
An empirical test that $\lambda$ need not be unity is needed. The author can
think of two possibilities. Classical mechanics has enjoyed a rather curious
success in explaining certain features of the microwave ionization
spectroscopy of Rydberg states in atomic hydrogenryd1 (20, 21). One might
envisage nonlinearity entering into this phenomenon due to the very high order
photon processes involved. Another possibility is the observed nonlinear
polarisation dynamics of optical beams traversing media of significant third-
order nonlinear susceptibilityshen (22, 23). It is suggested that $\lambda$
may enter here as an appropriate ratio of the linear and nonlinear
susceptibilities. A general search for such quantal two state nonlinear
dynamics is indicated.
Measurement theoretic tools for the detection of nonlinear quantal dynamics
are reported injones2 (5). On this note we might add that the measurement
theoretic consequences of a workable quantal nonlinearity are profound and may
lead to some unexpected results when coupled with current research into
nonlinear dynamics.
I am grateful to many people for their help, advice or comments at different
stages: Profs. Bruce McKellar and Ian Percival and Drs. Serdar Kuyucak, Andrew
Davies, Ray Volkas, Carmelo Pisani and Zwi Barnea.
## References
* (1) L. de Broglie, Non-Linear Wave Mechanics–A Causal Interpretation (Elsevier, Amsterdam, 1950).
* (2) P.A.M. Dirac, Quantum Mechanics (Oxford, London, 1958).
* (3) P.A.M. Dirac, Phys. Zeit. der. Sowjet. 3, 64 (1933).
* (4) K.R.W. Jones, paper in preparation (University of Melbourne, UM-P-91/45, 1991).
* (5) K.R.W. Jones, Ann. Phys. (N.Y.) 207, 140 (1991).
* (6) J.R. Klauder, J. Math. Phys. 4, 1058 (1963).
* (7) S. Weinberg, Ann. Phys. (N.Y.) 194, 336 (1989).
* (8) See any good textbook on complex analysis.
* (9) A. Messiah, Quantum Mechanics Vol I (North Holland, Amsterdam, 1961).
* (10) I. Biálynicki-Birula and J. Mycielski, Ann. Phys. (N.Y.) 100, 62 (1976).
* (11) J.P. Provost and G. Vallee, Commun. Math. Phys. 76, 289 (1980).
* (12) A possibility that is generally discounted in the contemporary literature. For a discussion of the reasons why see, for example; G.M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, London, 1985).
* (13) K. Gottfried, Quantum Mechanics Vol. I: Fundamentals (Benjamin, New York, 1966).
* (14) Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987).
* (15) V.I. Arn’old, Mathematical Methods of Classical Mechanics 2nd ed (Springer, Berlin, 1989).
* (16) A. Einstein, Verhandl. Dtsch. Phys. Ges. 19, 82 (1917).
* (17) E. Arthurs and J.L. Kelly Jr, Bell. Syst. Tech. J. 44, 725 (1965).
* (18) C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
* (19) The application of such ideas to quantum gravity is an enticing prospect, particularly since that is a natural nonlinear theory incorporating geometric properties that $\hat{H}(\psi)$ dynamics may share.
* (20) K.A.H. van Leeuwen et al., Phys. Rev. Lett. 55, 2231 (1985).
* (21) J.G. Leopold and I.C. Percival, Phys. Rev. Lett. 41, 944 (1978).
* (22) Y.R. Shen, The Principles of Nonlinear Optics (Wiley Interscience, New York, 1984).
* (23) D. David, D.D. Holm and M.V. Tratnik, Phys. Lett. A 138, 29 (1989).
|
arxiv-papers
| 2012-12-30T22:13:18 |
2024-09-04T02:49:39.798427
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. R. W. Jones",
"submitter": "Kingsley Jones",
"url": "https://arxiv.org/abs/1212.6784"
}
|
1212.6809
|
# About the bound of the $\text{C}^{*}$ exponential length
Qingfei Pan and Kun Wang Qingfei Pan: Department of Applied Mathematics
Sanming University
Sanming, Fujian
China 365004 [email protected] Kun Wang:Department of Mathematics
University of Puerto Rico, Rio Piedras Campus, P.O.Box 70377
San Juan, Puerto Rico, USA 00931 [email protected]
###### Abstract.
Let $X$ be a compact Hausdorff space. In this paper, we give an example to
show that there is $u\in\text{C}(X)\otimes\text{M}_{n}$ with $\det(u(x))=1$
for all $x\in X$ and $u\sim_{h}1$ such that the $\text{C}^{*}$ exponential
length of $u$ (denoted by $cel(u)$) can not be controlled by $\pi$. This
example answers a question of N.C. Phillips (see [21]). Moreover, in simple
inductive limit $\text{C}^{*}$-algebras, similar examples also exist.
###### 2000 Mathematics Subject Classification:
46L05
## 1\. Introduction
Exponential rank was introduced by Phillips and Ringrose [24]; and,
subsequently, exponential length was introduced by Ringrose [25]. These
invariants have been fundamental in the structure and classification of
$\text{C}^{*}$-algebras. Among other things, they have played important roles
in factorization and approximation properties for $\text{C}^{*}$-algebras
e.g., the weak FU property [17], Weyl-von Neumann Theorems [14], [15] (which
in turn have been important in various generalizations of BDF Theory beyond
the Calkin algebra case), and the uniqueness theorems of classification theory
[5], [16].
The $\text{C}^{*}$ exponential length and rank have been extensively studied
(see [25], [24], [17], [22], [30], [32], [31], [6], [21], [19], [11], [23],
[27], [12], etc. -an incomplete list). In [21] page 851, the paragraphs before
Proposition 7.9, N.C. Phillips mentioned that ”We believe that suitable
modifications of Lemma 5.2 and 5.3 will show that if $u\sim_{h}1$ and
$\det(u)=1$, then $cel(u)\leq\pi$ (even though, for general $u$, $cel(u)$ can
be arbitrarily large).” However, in fact for any $\varepsilon>0$, we can find
a unitary $u$ in a $\text{C}^{*}$-algebra with $\det(u)=1$, $u\sim_{h}1$ and
$cel(u)\geq 2\pi-\varepsilon$. In this paper, we provide a method for
constructing such examples. In simple inductive limit $\text{C}^{*}$-algebras
(simple AH algebras), we also get such examples. A recent paper of H. Lin’s
also proves similar results independently (not completely same) (see [12]) but
with different method. Note that for unital real rank zero
$\text{C}^{*}$-algebras, $\pi$ is an upper bound for the $\text{C}^{*}$
exponential length (see [11]). In a forthcoming paper, the second author will
show that $2\pi$ is an upper bound for the $\text{C}^{*}$ exponential length
of $\text{M}_{n}(C(X))$ provided the matrix size $n$ is large. Consequently,
for the simple AH algebras $A$ (with no dimension growth) classified by
Elliott-Gong-Li, the $\text{C}^{*}$ exponential length of elements in
$CU(A)$—the closure of the commutator subgroup of $U(A)$, is at most $2\pi.$
Acknowledgments The authors would like to thank the referee for giving many
valuable and constructive suggestions.
## 2\. Preliminaries
For convenience of the reader, we recall some definitions and lemmas (see [9],
[25] for more details).
###### Definition 2.1.
Let $X$ be a compact space and $B=\text{C}(X)\otimes\text{M}_{n}$, for $u\in
U(B)$(unitary group of $B$), let $\det(u)$ be a function from $X$ to $S^{1}$
whose value at $x$ is $\det(u(x))$.
###### Definition 2.2.
Let $A$ be a unital $\text{C}^{*}$-algebra and $u$ be a unitary which lies in
the connected component of the identity 1 in $A$. Define the $\text{C}^{*}$
exponential length of $u$ (denoted by $cel(u)$) as follows:
$cel(u)=\inf\\{\sum_{i=1}^{k}\|h_{i}\|:u=\text{exp}(ih_{1})\text{exp}(ih_{2})...\text{exp}(ih_{k})\\}.$
###### Remark 2.3.
For a unital $\text{C}^{*}$-algebra $A$, let $U_{0}(A)$ be the connected
component of $U(A)$ containing the identity $1$. Recall from [25] that if
$u\in U_{0}(A)$, then the $\text{C}^{*}$ exponential length $cel(u)$ is equal
to the infimum of the lengths of rectifiable paths from $u$ to $1$ in $U(A)$.
The following lemma is an easy example for calculating the $\text{C}^{*}$
exponential length.
###### Lemma 2.4.
Let $\alpha\in\mathbb{R}$ and $u\in\text{C}[0,1]$ be defined by
$u(t)=\exp{(it\alpha)}$, then
$cel(u)=\min_{k\in\mathbb{Z}}\max_{t\in[0,1]}|\alpha t-2k\pi|.$
Moreover, if $|\alpha|\leq 2\pi$, then $cel(u)=|\alpha|$.
###### Proof.
Since $cel(u)=\inf\\{\text{length}(u_{s}):u_{s}\mbox{ is a path in
}U(\text{C}[0,1])\mbox{ from }u\mbox{ to }1\\}$, let $v_{s}(t)$ be any path
from $1$ to $u$, that is, $v_{0}(t)=1,v_{1}(t)=u(t).$ Without loss of
generality, we can assume $v_{s}$ is piecewise smooth. Then
$\text{length}(v_{s})=\int_{0}^{1}\|\frac{dv}{ds}\|ds.$ Since $v_{s}(t)$ can
be considered as a map from $[0,1]\times[0,1]$ to $S^{1}$ and $\mathbb{R}$ is
a covering space of $S^{1},$ there exists a unique map $\tilde{v}_{s}(t)$ from
$[0,1]\times[0,1]$ to $\mathbb{R}$ such that:
$v_{s}(t)=\pi(\tilde{v}_{s}(t))~{}~{}\mbox{and}~{}~{}\widetilde{v}_{0}(0)=0,\qquad(*)$
where $\pi(x)=e^{ix}.$ Therefore
$\frac{dv}{ds}=\pi^{\prime}(\tilde{v}_{s}(t))\cdot\frac{d\widetilde{v}}{ds},$
which implies $\|\frac{dv}{ds}\|=\|\frac{d\tilde{v}}{ds}\|.$
By $(*)$, $\pi(\tilde{v}_{0}(t))=v_{0}(t)=1$, hence $\tilde{v}_{0}(t)\in
2\pi\mathbb{Z}$ for all $t\in[0,1]$. Since $\tilde{v}_{0}(0)=0$ and
$\tilde{v}_{0}(t)$ is continuous, $\tilde{v}_{0}(t)=0$ for all $t\in[0,1]$. In
addition, by $(*)$, we can also get
$\pi(\tilde{v}_{1}(t))=v_{1}(t)=\exp(it\alpha)$, thus $\tilde{v}_{1}(t)-\alpha
t\in 2\pi\mathbb{Z}$ for all $t$. By continuity of $\tilde{v}_{1}(t)-\alpha
t$, there exists some integer $k$ such that $\tilde{v}_{1}(t)-\alpha t=2k\pi$
for all $t\in[0,1]$. Therefore,
$\int_{0}^{1}\|\frac{d\tilde{v}}{ds}\|ds\geq\|\int_{0}^{1}\frac{d\tilde{v}}{ds}ds\|=\max_{t\in[0,1]}|\tilde{v}_{1}(t)-\tilde{v}_{0}(t)|\geq\min_{k\in\mathbb{Z}}\max_{t\in[0,1]}|\alpha
t-2k\pi|.$
Let $L=\min\limits_{k\in\mathbb{Z}}\max\limits_{t\in[0,1]}|\alpha t-2k\pi|$
and $k_{0}\in\mathbb{Z}$ be such that $L=\max\limits_{t\in[0,1]}|\alpha
t-2k_{0}\pi|$. Fix
$v_{s}(t)=\exp\\{is(\alpha t-2k_{0}\pi)\\},$
then $v_{0}(t)=\exp\\{0\\}=1$ and $v_{1}(t)=\exp\\{i\alpha t-2k_{0}\pi
i\\}=\exp\\{i\alpha t\\}$ and
$\int_{0}^{1}||\frac{d{v}}{ds}||ds=\int_{0}^{1}||\alpha
t-2k_{0}\pi||ds=\int_{0}^{1}\max\limits_{t\in[0,1]}|\alpha
t-2k_{0}\pi|ds=\int_{0}^{1}Lds=L.$
Thus $v_{s}(t)$ is a path in $U(C[0,1])$ connecting 1 and $u(t)$ with length
$L$. Therefore, $cel(u)=L.$
Let us assume $\alpha\leq 2\pi$. For $k=0,$
$\max\limits_{t\in[0,1]}|\alpha t-2k\pi|=\max\limits_{t\in[0,1]}|\alpha
t-0|=|\alpha|.$
For $k\neq 0$,
$\max\limits_{t\in[0,1]}|\alpha t-2k\pi|\geq|0-2k\pi|=2|k|\pi\geq|\alpha|.$
Hence, $\min\limits_{k\in\mathbb{Z}}\max\limits_{t\in[0,1]}|\alpha
t-2k\pi|=|\alpha|.$ That is $cel(u)=|\alpha|$. ∎
## 3\. Counter-examples
###### Lemma 3.1.
Let $f(s,t):X\triangleq[0,1]\times[0,1]\rightarrow
U(\text{M}_{n}(\mathbb{C}))$ be a smooth map. For any $\delta>0$, there is a
smooth map $g(s,t):[0,1]\times[0,1]\rightarrow U(\text{M}_{n}(\mathbb{C}))$
such that:
(1) $\|f-g\|<\delta$, $\|\frac{\partial f}{\partial s}(s,t)-\frac{\partial
g}{\partial s}(s,t)\|<\delta$, $\|\frac{\partial f}{\partial
t}(s,t)-\frac{\partial g}{\partial t}(s,t)\|<\delta$;
(2) $g(s,t)$ has no repeated eigenvalues for all $(s,t)\in[0,1]\times[0,1]$.
###### Proof.
This is the standard transversal theorem. Even though in the original
statement in [8], it does not assert that the derivatives are also close. But
the proof really shows that (see pages 70-71 of [8]). For convenience of the
reader, we repeat the construction here for our special case.
By smoothly extending $f$ to an open neighborhood of $[0,1]\times[0,1]$, we
can assume $f$ is defined on an open manifold without boundary. Let $Z$ be a
subspace of $U(\text{M}_{n}(\mathbb{C}))$ defined by
$Z=\\{u\in U(\text{M}_{n}(\mathbb{C})):~{}u\mbox{ has repeated
eigenvalues}\\}.$
Since $U(\text{M}_{n}(\mathbb{C}))$ is a subspace of
$\text{M}_{n}(\mathbb{C})$ and the latter can be identified with
$\mathbb{R}^{2n^{2}}$ as a topological space, $f$ is a smooth map from $X$ to
$\mathbb{R}^{2n^{2}}$. Let $B$ be the open unit ball of $\mathbb{R}^{2n^{2}}$
(with Euclidean metric), then $B$ corresponds to some open ball (contained in
the unit ball of $\text{M}_{n}(\mathbb{C})$) in $\text{M}_{n}(\mathbb{C})$
with the matrix norm, for which we still use the notation $B$. Let
$0<\varepsilon<1/2$, for $x\in X$, $r\in B$, define
$F(x,r)=\pi[f(x)+\varepsilon r],$
where $\pi:Gl_{n}(\mathbb{C})\rightarrow U(\text{M}_{n}(\mathbb{C}))$ is
defined by the Polar decomposition, which serves as the map $\pi$ in [8] from
the tubular neighborhood of $U(\text{M}_{n}(\mathbb{C}))$ (which is $Y$ in the
notation of [8] page 70) to $U(\text{M}_{n}(\mathbb{C}))$. By the definition,
$\pi$ is a smooth map. Define $f_{r}:X\rightarrow U(\text{M}_{n}(\mathbb{C}))$
by
$f_{r}(x)=F(x,r).$
Since $\pi$ restricts to the identity on $U(\text{M}_{n}(\mathbb{C}))$,
$f_{0}(x)=F(x,0)=\pi(f(x))=f(x).$
For fixed $x$, $r\rightarrow f(x)+\varepsilon r$ is certainly a submersion of
$B\rightarrow\text{M}_{n}(\mathbb{C})$. As the composition of two submersions
is another, $r\rightarrow F(x,r)$ is a submersion. Therefore, $F$ is
transversal to $Z$. Then the Transversality Theorem (see page 68 of [8])
implies that $f_{r}$ is transversal to $Z$ for almost all $r\in B$.
By a result of M. Choi and G. Elliott (see the second paragraph on page 77 of
[2]), $Z$ is a finite union of embedded submanifolds of codimension at least
three. Thus
$\dim(X)+\dim(Z)<\dim(U(\text{M}_{n}(\mathbb{C}))).$
Therefore, $f_{r}$ transversal to $Z$ implies $\text{Im}f_{r}\bigcap
Z=\emptyset.$
Since $f+\varepsilon r$ is in a neighborhood of $f$ and $\pi$ restricts to the
identity on $U(\text{M}_{n}(\mathbb{C}))$, there exist positive real numbers
$\varepsilon_{1}=\varepsilon_{1}(\varepsilon,r),~{}\varepsilon_{2}=\varepsilon_{2}(\varepsilon,r)$
such that:
$\|\frac{\partial f_{r}}{\partial s}\|\leq(1+\varepsilon_{1})\|\frac{\partial
f}{\partial s}\|,\quad\|\frac{\partial f_{r}}{\partial
t}\|\leq(1+\varepsilon_{2})\|\frac{\partial f}{\partial t}\|.$
Therefore, by taking $r$ appropriately, we can get $f_{r}$ satisfies the
properties (1) and (2). Finally, let $g=f_{r}$ and this completes the proof.
∎
###### Remark 3.2.
In [17] Lemma 2.5, N. C. Phillips proves a similar result except for the
property of derivatives.
###### Corollary 3.3.
Let $\widetilde{F}_{s}$ be a path in $U(\text{M}_{k}(C[0,1]))$. For any
$\varepsilon>0$, there exists a path $F_{s}$ in $U(\text{M}_{k}(C[0,1]))$ such
that:
(1) $\|F-\widetilde{F}\|<\varepsilon$;
(2) $F_{s}(t)$ has no repeated eigenvalues for all $(s,t)\in[0,1]\times[0,1]$;
(3) $|\text{length}(\widetilde{F})-\text{length}(F)|<\varepsilon$.
Moreover, if for each $t\in[0,1]$, $\widetilde{F}_{1}(t)$ has no repeated
eigenvalues, then $F$ can be chosen to be such that
$F_{1}(t)=\widetilde{F}_{1}(t)$ for all $t\in[0,1]$.
###### Proof.
Let $\varepsilon_{1}$ be a small number to be determined later. Let
$\delta_{0}$ be such that $|1-e^{i\theta}|\leq\delta_{0}$ implies
$|\theta|\leq(1+\varepsilon_{1})|1-e^{i\theta}|$ for $\theta\in\mathbb{R}$.
For $\varepsilon>0$, let $\delta=\min\\{\delta_{0},\varepsilon/6,1/2\\}$. By
the definition of the length, there exist $0=s_{0}<s_{1}<s_{2}<\cdots<s_{n}=1$
such that
$\|\widetilde{F}_{s_{j+1}}-\widetilde{F}_{s_{j}}\|<\delta/2,\mbox{ for
}j=0,1,\cdots,n-1,$
and
$\sum_{j=0}^{n-1}\|\widetilde{F}_{s_{j+1}}-\widetilde{F}_{s_{j}}\|\leq
length(\widetilde{F}_{s})\leq\sum_{j=0}^{n-1}\|\widetilde{F}_{s_{j+1}}-\widetilde{F}_{s_{j}}\|+\varepsilon/4.$
Note that for each $j$, $\widetilde{F}_{s_{j}}(t)$ is a continuous map from
$[0,1]$ to $U(\text{M}_{k}(\mathbb{C}))$. There exist smooth maps
$G_{s_{j}}(t):[0,1]\rightarrow U(\text{M}_{k}(\mathbb{C}))$, such that
$\|G_{s_{j}}-\widetilde{F}_{s_{j}}\|=\sup_{t\in[0,1]}\|G_{s_{j}}(t)-\widetilde{F}_{s_{j}}(t)\|<\frac{\delta}{4n},~{}~{}~{}\quad~{}~{}j=0,1,\cdots,n.$
Then
$\displaystyle\|G_{s_{j+1}}-G_{s_{j}}\|$
$\displaystyle=\|G_{s_{j+1}}-\widetilde{F}_{s_{j+1}}+\widetilde{F}_{s_{j+1}}-\widetilde{F}_{s_{j}}+\widetilde{F}_{s_{j}}-G_{s_{j}}\|$
$\displaystyle\leq\|\widetilde{F}_{s_{j+1}}-\widetilde{F}_{s_{j}}\|+\frac{\delta}{2n}\leq\delta.$
And by the first equality, we can also get
$\|G_{s_{j+1}}-G_{s_{j}}\|\geq\|\widetilde{F}_{s_{j+1}}-\widetilde{F}_{s_{j}}\|-\frac{\delta}{2n}.$
Therefore,
$|\|G_{s_{j+1}}-G_{s_{j}}\|-\|\widetilde{F}_{s_{j+1}}-\widetilde{F}_{s_{j}}\||\leq\frac{\delta}{2n},$
$|\sum_{j=0}^{n-1}\|G_{s_{j+1}}-G_{s_{j}}\|-\sum_{j=0}^{n-1}\|\widetilde{F}_{s_{j+1}}-\widetilde{F}_{s_{j}}\||\leq\frac{\delta}{2}.$
Thus
$\sum_{j=0}^{n-1}\|G_{s_{j+1}}-G_{s_{j}}\|-\frac{\delta}{2}\leq\mbox{length}(\widetilde{F}_{s})\leq\sum_{j=0}^{n-1}\|G_{s_{j+1}}-G_{s_{j}}\|+\frac{\delta}{2}+\varepsilon/4,$
$\sum_{j=0}^{n-1}\|G_{s_{j+1}}-G_{s_{j}}\|-\frac{\varepsilon}{8}\leq\mbox{length}(\widetilde{F}_{s})\leq\sum_{j=0}^{n-1}\|G_{s_{j+1}}-G_{s_{j}}\|+\varepsilon/2.\quad(*)$
Now we want to define a smooth function
$\widetilde{G}_{s}(t):[0,1]\times[0,1]\rightarrow U(\text{M}_{k}(\mathbb{C}))$
such that $\widetilde{G}_{s_{j}}(t)=G_{s_{j}}(t)\mbox{ for
}j=0,1,\cdots,n,~{}t\in[0,1].$ We will define it piece by piece on each
subinterval $[s_{j},s_{j+1}]$ ($j=0,1,\cdots,n-1$).
Suppose
$G^{*}_{s_{j}}G_{s_{j+1}}=U_{j}\begin{pmatrix}e^{i\alpha_{1}(t)}&0&\cdots&0\\\
0&e^{i\alpha_{2}(t)}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{i\alpha_{k}(t)}\end{pmatrix}U_{j}^{*}.$
Since $\|G^{*}_{s_{j}}G_{s_{j+1}}-I\|=\|G_{s_{j}}-G_{s_{j+1}}\|\leq\delta<1$,
there exists a self-adjoint element $H_{j}(t)$ in $\text{M}_{k}(C[0,1])$ such
that $G^{*}_{s_{j}}G_{s_{j+1}}(t)=e^{iH_{j}(t)}$, (here
$H_{j}(t)=-i\log[G_{s_{j}}^{*}(t)G_{s_{j+1}}(t)]$ which is a smooth function).
Define
$\widetilde{G}_{s}(t)=G_{s_{j}}(t)e^{i\frac{s-s_{j}}{s_{j+1}-s_{j}}H_{j}}\mbox{
for }s_{j}\leq s\leq s_{j+1},~{}t\in[0,1],~{}j=0,1,\cdots,n-1.$
Then $\widetilde{G}_{s}(t)~{}(s_{j}\leq s\leq s_{j+1})$ is a path in
$U(\text{M}_{k}(C[0,1]))$ from $G_{s_{j}}$ to $G_{s_{j+1}}$ and
$\widetilde{G}_{s}(t)$ is smooth for $(s,t)\in[s_{j},s_{j+1}]\times[0,1]$.
Moreover,
$\displaystyle\mbox{length}(\widetilde{G}_{s}|_{s_{j}\leq s\leq s_{j+1}})$
$\displaystyle=$
$\displaystyle\int_{s_{j}}^{s_{j+1}}\|\frac{\partial\widetilde{G}_{s}}{\partial
s}\|ds$ $\displaystyle\leq$
$\displaystyle\int_{s_{j}}^{s_{j+1}}\frac{1}{s_{j+1}-s_{j}}\|G_{s_{j}}(t)H_{j}(t)\|ds$
$\displaystyle\leq$
$\displaystyle(1+\varepsilon_{1})\|G_{s_{j}}U_{j}\begin{pmatrix}1-e^{i\alpha_{1}(t)}&0&\cdots&0\\\
0&1-e^{i\alpha_{2}(t)}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&1-e^{i\alpha_{k}(t)}\end{pmatrix}U_{j}^{*}\|$ $\displaystyle=$
$\displaystyle(1+\varepsilon_{1})\|G_{s_{j}}[I-U_{j}\begin{pmatrix}e^{i\alpha_{1}(t)}&0&\cdots&0\\\
0&e^{i\alpha_{2}(t)}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{i\alpha_{k}(t)}\end{pmatrix}U_{j}^{*}]\|$ $\displaystyle=$
$\displaystyle(1+\varepsilon_{1})\|G_{s_{j}}[I-G_{s_{j}}^{*}{G}_{s_{j+1}}]\|$
$\displaystyle=$
$\displaystyle(1+\varepsilon_{1})\|G_{s_{j}}-{G}_{s_{j+1}}\|.$
Therefore, $\widetilde{G}_{s}$ $(0\leq s\leq 1)$ is a piecewise smooth path in
$U(\text{M}_{k}(C[0,1]))$ and
$\sum_{j=0}^{n-1}\|G_{s_{j+1}}-G_{s_{j}}\|\leq\mbox{length}(\widetilde{G}_{s})\leq(1+\varepsilon_{1})\sum_{j=0}^{n-1}\|G_{s_{j+1}}-G_{s_{j}}\|.$
Thus by $(*)$, we have
$\mbox{length}(\widetilde{G}_{s})(1+\varepsilon_{1})^{-1}-\frac{\varepsilon}{8}\leq\mbox{length}(\widetilde{F}_{s})\leq\mbox{length}(\widetilde{G}_{s})+\varepsilon/2.$
Finally, pick any smooth monotone function $\xi:[0,1]\rightarrow[0,1]$ with
$\xi(0)=0,~{}\xi(1)=1,~{}\frac{d^{n}\xi}{ds^{n}}\mid_{s=0}=0,~{}\frac{d^{n}\xi}{ds^{n}}\mid_{s=1}=0~{}\mbox{
for all }n\geq 1.$
Let
$\widetilde{G}^{\prime}_{s}(t)=G_{s_{j}}(t)e^{i\xi(\frac{s-s_{j}}{s_{j+1}-s_{j}})H_{j}}\mbox{
for }s_{j}\leq s\leq s_{j+1},~{}t\in[0,1],~{}j=0,1,\cdots,n-1.$
Then $\widetilde{G}^{\prime}_{s}(t)$ is smooth for all
$(s,t)\in[0,1]\times[0,1]$ (since
$\frac{\partial\widetilde{G}^{\prime}_{s}(t)}{\partial s}|_{s=s_{j}}=0$ from
both left and right for all $j=1,2,\cdots,n-1$) and
$\mbox{length}(\widetilde{G}^{\prime}_{s})=\mbox{length}(\widetilde{G}_{s}).$
And for each $(s,t)\in[0,1]\times[0,1]$,
$\displaystyle\|\widetilde{G}^{\prime}_{s}(t)-\widetilde{F}_{s}(t)\|=$
$\displaystyle\|\widetilde{G}^{\prime}_{s}(t)-\widetilde{G}^{\prime}_{s_{j}}(t)+\widetilde{G}^{\prime}_{s_{j}}(t)-\widetilde{F}_{s_{j}}(t)+\widetilde{F}_{s_{j}}(t)-\widetilde{F}_{s}(t)\|$
$\displaystyle\leq$
$\displaystyle\|\widetilde{G}_{s_{j+1}}(t)-\widetilde{G}_{s_{j}}(t)\|+\|\widetilde{G}_{s_{j}}(t)-\widetilde{F}_{s_{j}}(t)\|+\|\widetilde{F}_{s_{j}}(t)-\widetilde{F}_{s}(t)\|$
$\displaystyle\leq$
$\displaystyle\delta+\frac{\delta}{4n}+\frac{\varepsilon}{4}\leq\varepsilon/2,$
where $s_{j}$ satisfies $s_{j}\leq s\leq s_{j+1}$.
Thus, by choosing $\varepsilon_{1}$ appropriately, we have
$|\mbox{length}(\widetilde{G}^{\prime}_{s})-\mbox{length}(\widetilde{F}_{s})|<\varepsilon/2\mbox{
and }\|\widetilde{G}^{\prime}-\widetilde{F}\|<\varepsilon/2.$
Since $\widetilde{G}^{\prime}_{s}$ can be seen as a smooth map from
$[0,1]\times[0,1]$ to $U(\text{M}_{k}(\mathbb{C}))$, by Lemma 3.1, there
exists $F$ such that $\|F-\widetilde{G}^{\prime}\|<\varepsilon/2$ and
$F_{s}(t)$ has no repeated eigenvalues for all $(s,t)\in[0,1]\times[0,1]$.
Moreover,
$|\text{length}(F_{s})-\text{length}(\widetilde{G}^{\prime}_{s})|=|\int_{0}^{1}\|\frac{\partial
F}{\partial
s}\|ds-\int_{0}^{1}\|\frac{\partial\widetilde{G}^{\prime}}{\partial
s}\|ds|<\varepsilon/2.$
Thus $F$ satisfies properties (1)-(3), which is what we want.
Moreover, if $\widetilde{F}_{1}(t)$ has no repeated eigenvalues for all
$t\in[0,1]$, then there exists $\eta>0$ such that
$\|u(t)-\widetilde{F}_{1}(t)\|<\eta$ implies $u(t)$ has no repeated
eigenvalues for all $t\in[0,1]$. For $\varepsilon=\eta/2$, by the first part
of the statement we can find a path $F_{s}$ satisfying properties (1), (2),
(3). Let $s_{0}\in[s_{n-1},1)$ be such that
$\|F_{s_{0}}(t)-F_{1}(t)\|\leq\eta/4$, (where $s_{n-1}$ is a point of the
partition of $[0,1]$ for which we mentioned in the proof of the first part of
the statement). Then
$\|F_{s_{0}}(t)-\widetilde{F}_{1}(t)\|=\|F_{s_{0}}(t)-F_{1}(t)+F_{1}(t)-\widetilde{F}_{1}(t)\|\leq
3\eta/4.$
Now let us redefine $F_{s}(t)$ on the subinterval $[s_{0},1]$ (still use the
notation $F_{s}(t)$) by a similar way as above:
$F_{s}(t)=F_{s_{0}}(t)e^{i\frac{s-s_{0}}{1-s_{0}}H_{n}(t)},~{}~{}\mbox{ for
}s_{0}\leq s\leq 1,$
where $H_{n}(t)=-i\log[F_{s_{0}}^{*}(t)\widetilde{F}_{1}(t)]$. Since this
newly defined path $F_{s}$ lies in the $\eta$ neighborhood of $F_{1}$,
$F_{s}(t)$ has no repeated eigenvalues for all $(s,t)\in[0,1]\times[0,1]$.
Thus this $F_{s}$ is what we want.
∎
###### Definition 3.4.
For a metric space $(Y,d)$, let
$P^{k}Y:=\\{(y_{1},y_{2},...,y_{k}):y_{i}\in Y\\}/\sim,$
where
$(y_{1},y_{2},...,y_{k})\sim(\widetilde{y}_{1},\widetilde{y}_{2},...,\widetilde{y}_{k})$
if $\exists\sigma\in S_{k}$ such that $y_{\sigma(i)}=\tilde{y}_{i}\mbox{ for
all }1\leq i\leq k$. Let $[y_{1},y_{2},...,y_{k}]$ denote the equivalent class
of $(y_{1},y_{2},\cdots,y_{k})$ in $P^{k}Y$. Define also the metric of
$P^{k}Y$ as:
$\text{dist}([y_{1},y_{2},...,y_{k}],[\widetilde{y}_{1},\widetilde{y}_{2},...,\widetilde{y}_{k}])=\min_{\sigma\in
S_{k}}\max_{1\leq i\leq k}d(y_{i},\widetilde{y}_{\sigma(i)}).$
The proof of the following lemma is straightforward.
###### Lemma 3.5.
Let $(Y,d)$ be a metric space and $\pi:\underbrace{Y\times Y\times...\times
Y}_{k}\longrightarrow P^{k}Y$ be the quotient map. Let
$X\subset\underbrace{Y\times Y\times...\times Y}_{k}$ be the set consisting of
those elements $(y_{1},y_{2},...,y_{k})$ with $y_{i}\neq y_{j}$ if $i\neq j$.
Then the restriction of $\pi$ to $X$ is a covering map.
We need the following easy lemma.
###### Lemma 3.6.
Let $F:[0,1]\times[0,1]\rightarrow P^{k}S^{1}$ be a continuous function,
suppose
$F(s,t)=[x_{1}(s,t),x_{2}(s,t),...,x_{k}(s,t)],$
and for all $(s,t)\in[0,1]\times[0,1]$, $x_{i}(s,t)\neq x_{j}(s,t)$ if $i\neq
j$. Then there are continuous functions
$f_{1},f_{2},...,f_{k}:[0,1]\times[0,1]\rightarrow S^{1}$ such that:
$F(s,t)=[f_{1}(s,t),f_{2}(s,t),...,f_{k}(s,t)].$
###### Proof.
Let $\pi:\underbrace{S^{1}\times S^{1}\times...\times S^{1}}_{k}\rightarrow
P^{k}S^{1}$ denote the quotient map, and let $X\subset\underbrace{S^{1}\times
S^{1}\times...\times S^{1}}_{k}$ be the set consisting of those elements
$(x_{1},x_{2},\cdots,x_{k})$ with $x_{i}\neq x_{j}$ if $i\neq j$. Then by
Lemma 3.5 $\pi|_{X}$ is a covering map from $X$ to $\pi(X)$ (which is a subset
of $P^{k}S^{1}$).
Note from the assumption of the Lemma, the image of $F$ is contained in
$\pi(X)$. Since $[0,1]\times[0,1]$ is simply connected, by the standard
lifting theorem for covering spaces, the map
$F:[0,1]\times[0,1]\to\pi(X)\subset P^{k}S^{1}$ can be lifted to a map
$F_{1}:[0,1]\times[0,1]\to X(\subset\underbrace{S^{1}\times
S^{1}\times...\times S^{1}}_{k})$.
Let $\pi_{j}:S^{1}\times S^{1}\times...\times S^{1}\rightarrow S^{1}$ be the
projection onto the $j$th coordinate. For $1\leq j\leq k,$ define functions
$f_{j}:[0,1]\times[0,1]\rightarrow S^{1}$ by
$f_{j}(s,t)=\pi_{j}(F_{1}(s,t)).$
Then it is easy to see that $f_{j}$’s satisfy the requirements. ∎
###### Remark 3.7.
Let $F_{s}$ be a path in $U(M_{k}(C[0,1]))$ such that $F_{s}(t)$ has no
repeated eigenvalues for all $(s,t)\in[0,1]\times[0,1]$. Let
$\Lambda:[0,1]\times[0,1]\rightarrow P_{k}S^{1}$ be the eigenvalue map of
$F_{s}(t)$, i.e. $\Lambda(s,t)=[x_{1}(s,t),x_{2}(s,t),\cdots,x_{k}(s,t)],$
where $\\{x_{i}(s,t)\\}_{i=1}^{k}$ are eigenvalues of the matrix $F_{s}(t)$.
By Lemma 3.6, there are continuous functions
$f_{1},f_{2},...,f_{k}:[0,1]\times[0,1]\rightarrow S^{1}$ such that:
$\Lambda(s,t)=[f_{1}(s,t),f_{2}(s,t),...,f_{k}(s,t)].$
For each fixed $(s,t)\in[0,1]\times[0,1]$, there is a unitary $U_{s}(t)$ such
that
$F_{s}(t)=U_{s}(t)diag[f_{1}(s,t),f_{2}(s,t),...,f_{k}(s,t)]U_{s}(t)^{*}.$
Note that $U_{s}(t)$ can be chosen to be continuous, but in this paper we
don’t need this property.
###### Proposition 3.8.
(see line 13-line 18 of page 71 of [1]) If $U,V\in M_{n}(\mathbb{C})$ are
unitaries with eigenvalues $u_{1},u_{2},\cdots,u_{n}$ and
$v_{1},v_{2},\cdots,v_{n}$ respectively, then
$\min\limits_{\sigma\in
S_{n}}\max\limits_{i}|u_{i}-v_{\sigma(i)}|\leq\|U-V\|.$
The same result for a pair of Hermitian matrices is due to H. Weyl (called
Weyl’s Inequality see [29]).
###### Lemma 3.9.
Let $F_{s}$ be a path in $U(\text{M}_{n}(\text{C}[0,1]))$ and
$f^{1}_{s}(t),f^{2}_{s}(t),...,f^{n}_{s}(t)$ be continuous functions such that
$F_{s}(t)=U_{s}(t)\text{diag}[f^{1}_{s}(t),f^{2}_{s}(t),...,f^{n}_{s}(t)]U_{s}(t)^{*},$
where $U_{s}(t)$ are unitaries. Suppose for any $(s,t)\in[0,1]\times[0,1]$,
$f_{s}^{i}(t)\neq f_{s}^{j}(t)$ if $i\neq j$, then
$\text{length}(F_{s})\geq\max_{1\leq j\leq n}\\{\text{length}(f_{s}^{j})\\}.$
(In this lemma, we assume that $F_{s}(t)$ is continuous, but we do not assume
$U_{s}(t)$ is continuous.)
###### Proof.
Let
$\varepsilon=\min\\{|(f_{s}^{i}(t)-f_{s}^{j}(t))|:~{}i\neq j,~{}1\leq i,j\leq
n,~{}s\in[0,1],~{}t\in[0,1]\\}.$
Since for each $j$, $f_{s}^{j}(t)$ is continuous with respect to $s$, there
exists $\delta>0$ such that: for any partition
$\mathcal{P}=\\{s_{1},s_{2},\cdots,s_{\lambda}\\}$ with
$|\mathcal{P}|<\delta$,
$\|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{j}(t)\|<\varepsilon/2~{}~{}\mbox{ for all
}2\leq i\leq\lambda,~{}~{}1\leq j\leq n.$
Then by Proposition 3.8,
$\displaystyle\text{length}(F_{s})\geq$
$\displaystyle\sum\limits_{i=2}^{\lambda}\|F_{s_{i}}-F_{s_{i-1}}\|=\sum\limits_{i=2}^{\lambda}\sup_{t\in[0,1]}\|F_{s_{i}}(t)-F_{s_{i-1}}(t)\|$
$\displaystyle\geq$
$\displaystyle\sum\limits_{i=2}^{\lambda}\sup_{t\in[0,1]}[\min\limits_{\sigma\in
S_{n}}\max\limits_{1\leq j\leq
n}|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{\sigma(j)}(t)|].$
If $\sigma(j)\neq j$, then
$\displaystyle|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{\sigma(j)}(t)|$
$\displaystyle\geq$
$\displaystyle|f_{s_{i}}^{j}(t)-f_{s_{i}}^{\sigma(j)}(t)|-|f_{s_{i}}^{\sigma(j)}(t)-f_{s_{i-1}}^{\sigma(j)}(t)|$
$\displaystyle>$ $\displaystyle\varepsilon-\varepsilon/2=\varepsilon/2.$
If $\sigma(j)=j$, then $|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{j}(t)|<\varepsilon/2$.
Therefore,
$\min\limits_{\sigma\in S_{n}}\max\limits_{1\leq j\leq
n}|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{\sigma(j)}(t)|=\max\limits_{1\leq j\leq
n}|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{j}(t)|.$ $\displaystyle\text{length}(F_{s})$
$\displaystyle\geq\sum\limits_{i=2}^{\lambda}\sup_{t\in[0,1]}[\min\limits_{\sigma\in
S_{n}}\max\limits_{1\leq j\leq
n}|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{\sigma(j)}(t)|]$
$\displaystyle\geq\sum\limits_{i=2}^{\lambda}\sup_{t\in[0,1]}\max\limits_{1\leq
j\leq n}|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{j}(t)|$
$\displaystyle=\sum\limits_{i=2}^{\lambda}\max\limits_{1\leq j\leq
n}\|f_{s_{i}}^{j}(t)-f_{s_{i-1}}^{j}(t)\|.~{}~{}~{}~{}(*)$
Since $(*)$ holds for any partition $\mathcal{P}$ with $|\mathcal{P}|<\delta$,
we have
$\text{length}(F_{s})\geq\max\limits_{1\leq j\leq n}\text{length}(f_{s}^{j}).$
∎
###### Example 3.10.
Let $A=\text{M}_{10}(\text{C}[0,1])$. Define
$u(t)=\begin{pmatrix}e^{-2\pi it\frac{9}{10}}&0&\cdots&0\\\ 0&e^{2\pi
it\frac{1}{10}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&e^{2\pi
it\frac{1}{10}}\end{pmatrix}_{10\times 10}.$
Then $u$ is a unitary in $A$ with $\det(u)=1$ and $u\sim_{h}1.$
###### Theorem 3.11.
Let $u\in\text{M}_{10}(\text{C}[0,1])$ be defined as in 3.10, then
$cel(u)\geq 2\pi\cdot\frac{9}{10}.$
###### Proof.
Let $\widetilde{F}_{s}$ be a path in $U(\text{M}_{10}(\text{C}[0,1]))$ with
$\widetilde{F}_{0}=1\in\text{M}_{10}(\text{C}[0,1])$ and
$\widetilde{F}_{1}(t)=u(t).$ By Corollary 3.3, there is a path $F_{s}$ in
$U(\text{M}_{10}(\text{C}[0,1]))$ such that (1)
$\|F-\widetilde{F}\|<\varepsilon/2$, (2) $F_{s}(t)$ has no repeated
eigenvalues for all $(s,t)\in[0,1]\times[0,1]$, and
(3)$|\text{length}(\widetilde{F})-\text{length}(F)|<\varepsilon/2$. By Lemma
3.6 and Remark 3.7, there are continuous maps
$f^{1},f^{2},...,f^{10}:[0,1]\times[0,1]\rightarrow S^{1}$ and unitaries
$U_{s}(t)$ such that
$F_{s}(t)=U_{s}(t)\text{diag}[f^{1}_{s}(t),f^{2}_{s}(t),...,f^{10}_{s}(t)]U_{s}(t)^{*}$.
By Lemma 3.9,
$\text{length}(F_{s})\geq\max_{1\leq i\leq 10}\\{\text{length}(f_{s}^{i})\\}.$
Since $\|F-\widetilde{F}\|<\varepsilon/2,$
$\|f^{j}_{0}-1\|<\varepsilon/2\mbox{ for all }1\leq j\leq 10$. For each fixed
$t\in(\varepsilon,1-\varepsilon)$, there exists one and only one $j_{0}$ such
that
$\|f^{j_{0}}_{1}(t)-e^{-2\pi it\frac{9}{10}}\|<\varepsilon/2.~{}~{}$
For other $j\neq j_{0}$
$\|f^{j}_{1}(t)-e^{2\pi it\frac{1}{10}}\|<\varepsilon/2.$
(We use the fact that $\|e^{2\pi it\frac{1}{10}}-e^{-2\pi
it\frac{9}{10}}\|\geq\varepsilon$ for $t\in(\varepsilon,1-\varepsilon)$ .)
Since all $f^{j}$’s are continuous, the index $j_{0}$ should be the same for
all $t\in(\varepsilon,1-\varepsilon)$.
Thus $f^{j_{0}}_{s}$ is a path in $U(\text{C}[0,1])$ connecting a point near 1
and a point near $e^{-2\pi it\frac{9}{10}}$. By Lemma 2.4,
$\text{length}(f^{j_{0}}_{s})\geq\frac{9}{10}\cdot 2\pi-\varepsilon.$
Therefore,
$\text{length}(\tilde{F}_{s})\geq\frac{9}{10}\cdot
2\pi-\varepsilon/2-\varepsilon\geq\frac{9}{10}\cdot 2\pi-2\varepsilon.$
Since $\varepsilon$ is arbitrary, we have $cel(u)\geq\frac{9}{10}\cdot 2\pi$,
which completes the proof. ∎
###### Example 3.12.
Examples in some simple inductive limit $\text{C}^{*}$-algebras.
Let $\\{x_{1},x_{2},...\\}$ be a countable distinct dense subset of $[0,1]$
and let $\\{k_{n}\\}_{n=2}^{\infty}$ be a sequence of integers satisfying
$\prod_{n}\frac{10^{k_{n}}-1}{10^{k_{n}}}>\frac{11}{12}.$
Let
$A_{1}=\text{M}_{10}(\text{C}[0,1]),~{}A_{2}=\text{M}_{10^{k_{2}}}(A_{1}),~{}...,~{}A_{n}=\text{M}_{10^{k_{n}}}(A_{n-1}),....$
Define $\varphi_{n,n+1}:A_{n}\rightarrow A_{n+1}$ by
$\varphi_{n,n+1}(f)=\begin{pmatrix}f&0&\cdots&0&0\\\ 0&f&\cdots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\cdots&f&0\\\
0&0&\cdots&0&f(x_{n})\\\ \end{pmatrix}_{10^{k_{n}}\times 10^{k_{n}}}$
and $A=\underrightarrow{\lim}(A_{i},\varphi_{i,i+1})$ be the inductive limit
$\text{C}^{*}$-algebra. Then $A$ is simple.
Let $u(t)\in A_{1}$ be defined as in Example 3.10, then (see Theorem 3.14 and
Corollary 3.15)
$cel(\varphi_{1,\infty}(u))\geq\frac{9}{10}\cdot 2\pi.$
Inductive limit of such form was studied by Goodearl [7] and its exponential
rank was calculated by Gong and Lin [6].
###### Lemma 3.13.
Let $\theta:P^{L}\mathbb{R}\rightarrow(\mathbb{R}^{L},d_{max})$ be defined by
$\theta[x_{1},x_{2},\cdots,x_{L}]=(y_{1},y_{2},\cdots,y_{L})$
iff $[x_{1},x_{2},\cdots,x_{L}]=[y_{1},y_{2},\cdots,y_{L}]$ and $y_{1}\leq
y_{2}\leq\cdots\leq y_{L}$. Then $\theta$ is an isometry.
###### Proof.
Let $a=[a_{1},a_{2},\cdots,a_{L}],~{}b=[b_{1},b_{2},\cdots,b_{L}]$ be any two
elements in $P^{L}\mathbb{R}$. Without loss of generality, we can assume that
$a_{1}\leq a_{2}\leq\cdots\leq a_{L}$ and $b_{1}\leq b_{2}\leq\cdots\leq
b_{L}$. Thus
$d_{max}(\theta(a),\theta(b))=\max_{1\leq i\leq L}|a_{i}-b_{i}|.$
If $dist(a,b)\neq\max\limits_{1\leq i\leq L}|a_{i}-b_{i}|$, then there exist a
permutation $\sigma\in S_{L}$ such that
$l\triangleq\max_{1\leq i\leq L}|a_{i}-b_{\sigma(i)}|<\max_{1\leq i\leq
L}|a_{i}-b_{i}|.$
Since $l<\max\limits_{1\leq i\leq L}|a_{i}-b_{i}|$, there exists $k$ such that
$|a_{k}-b_{k}|>l.$
If $a_{k}<b_{k}$, then $|a_{i}-b_{j}|>l$ for any $i\leq k,j\geq k.$ Since the
cardinality of the set $\\{\sigma(1),\sigma(2),\cdots,\sigma(k)\\}$ is $k$,
there is at least one element $i_{0}\in\\{1,2,\cdots,k\\}$ with
$\sigma(i_{0})\geq k$. Then $|a_{i_{0}}-b_{\sigma(i_{0})}|>l$. Therefore,
$\max_{1\leq i\leq k}|a_{i}-b_{\sigma(i)}|>l.$
Similarly, if $a_{k}>b_{k}$, one can prove
$\max_{k\leq i\leq L}|a_{i}-b_{\sigma(i)}|>l.$
In either case, it contradicts $l=\max\limits_{1\leq i\leq
L}|a_{i}-b_{\sigma(i)}|$. Therefore,
$dist(a,b)=\max\limits_{1\leq i\leq
L}|a_{i}-b_{i}|=d_{max}(\theta(a),\theta(b)),$
which means $\theta$ is an isometry.
∎
###### Theorem 3.14.
Let $A_{i},~{}\varphi_{i,i+1},~{}(i\in\mathbb{N})$ be defined as in 3.12, for
any $\varepsilon\in(0,\frac{1}{100})$, let $u_{\varepsilon}\in A_{1}$ be
defined by:
$u_{\varepsilon}(t)=\begin{pmatrix}e^{-2\pi
it(\frac{9}{10}-\varepsilon)}&0&\cdots&0\\\ 0&e^{2\pi
it(\frac{1}{10}-\varepsilon)}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{2\pi it(\frac{1}{10}-\varepsilon)}\end{pmatrix}_{10\times 10},$
then
$cel(\varphi_{1,n}(u_{\varepsilon}))\geq
2\pi(\frac{9}{10}-\varepsilon)-5\varepsilon~{}~{}\mbox{for all
}n\in\mathbb{N}.$
###### Proof.
By an easy calculation, we know that
$\varphi_{1,n}(u_{\varepsilon})\in M_{L}(C[0,1]),~{}~{}\text{ where
}L=10\prod_{i=2}^{n}10^{k_{i}}.$
On the diagonal of $\varphi_{1,n}(u_{\varepsilon})$, there are
$\prod\limits_{i=2}^{n}(10^{k_{i}}-1)$ terms equal to $e^{-2\pi
it(\frac{9}{10}-\varepsilon)}$, $9\cdot\prod\limits_{i=2}^{n}(10^{k_{i}}-1)$
terms equal to $e^{2\pi it(\frac{1}{10}-\varepsilon)}$ and the rest are
constants. Let $\alpha$, $\beta$ and $\gamma$ denote, respectively, the
numbers of terms of the forms $e^{-2\pi it(\frac{9}{10}-\varepsilon)}$,
$e^{2\pi it(\frac{1}{10}-\varepsilon)}$ and constants on the diagonal of
$\varphi_{1,n}(u_{\varepsilon})$ (i.e.
$\alpha=\prod\limits_{i=2}^{n}(10^{k_{i}}-1)$,
$\beta=9\cdot\prod\limits_{i=2}^{n}(10^{k_{i}}-1)$ and
$\gamma=L-\alpha-\beta$). Therefore
$\frac{\alpha+\beta}{L}=\prod\limits_{i=2}^{n}\frac{10^{k_{i}}-1}{10^{k_{i}}}>\frac{11}{12}$,
which implies $\frac{\alpha}{\gamma}>\frac{11}{10}$.
For each $t\in[0,1]$, let $E(t)$ be the set consisting of all eigenvalues of
$\varphi_{1,n}(u_{\varepsilon})(t)$ (counting multiplicities). Define
continuous functions $\bar{\bar{y}}_{k}(t)~{}(1\leq k\leq L)$ from $[0,1]$ to
$[-\frac{9}{10}+\varepsilon,\frac{1}{10}-\varepsilon]$ as follows:
$\displaystyle\bar{\bar{y}}_{k}(t)=-(\frac{9}{10}-\varepsilon)t,~{}~{}~{}~{}\mbox{if
}1\leq k\leq\alpha,$
$\displaystyle\bar{\bar{y}}_{k}(t)=(\frac{1}{10}-\varepsilon)t,~{}~{}~{}~{}~{}~{}\mbox{if
}\alpha+1\leq k\leq\alpha+\beta,$
$\displaystyle\bar{\bar{y}}_{k}(t)=-(\frac{9}{10}-\varepsilon)x_{\bullet}\mbox{
or }(\frac{1}{10}-\varepsilon)x_{\bullet},~{}~{}~{}~{}\mbox{if
}\alpha+\beta+1\leq k\leq L,$
for $x_{\bullet}\in\\{x_{1},x_{2},\cdots,x_{n}\\}$ such that $\\{\exp\\{2\pi
i\bar{\bar{y}}_{k}(t)\\}:1\leq k\leq L\\}=E(t)$ for $\forall t\in[0,1]$.
Let $\theta$ be the map defined in Lemma 3.13 and let $p_{k}$ $(1\leq k\leq
L)$ be projections from $\mathbb{R}^{L}$ to $\mathbb{R}$ with respect to the
$k$-th coordinate. Define functions
$\bar{y}_{k}(t):[0,1]\rightarrow[-\frac{9}{10}+\varepsilon,\frac{1}{10}-\varepsilon]$
for $1\leq k\leq L$ as follows:
$\bar{y}_{k}(t)=p_{k}\theta[\bar{\bar{y}}_{1}(t),\bar{\bar{y}}_{2}(t),\cdots,\bar{\bar{y}}_{L}(t)].$
Then $\bar{y}_{k}(t)$ $(1\leq k\leq L)$ are continuous functions with
$\bar{y}_{1}(t)\leq\bar{y}_{2}(t)\leq\cdots\leq\bar{y}_{L}(t)$ for all
$t\in[0,1]$ and $\\{\bar{y}_{1}(t),\bar{y}_{2}(t),\cdots,\bar{y}_{L}(t)\\}$
(counting multiplicities) is equal to
$\\{\bar{\bar{y}}_{1}(t),\bar{\bar{y}}_{2}(t),\cdots,\bar{\bar{y}}_{L}(t)\\}$.
For $t\in(0,1)$, there are at most $\gamma$ terms (the constants referred to
above) of $\\{\bar{y}_{k}(t)\\}_{k=1}^{L}$ which could be less than
$-(\frac{9}{10}-\varepsilon)t$, therefore,
$\bar{y}_{k}(t)\geq-(\frac{9}{10}-\varepsilon)t,~{}~{}~{}\mbox{for
}k\geq\gamma+1.$
Similarly, for $t\in(0,1)$ there are at most $\gamma+\beta$ terms (the
constants or terms of the form $(\frac{1}{10}-\varepsilon)t$) of
$\\{\bar{y}_{k}(t)\\}_{k=1}^{L}$ which could be greater than
$-(\frac{9}{10}-\varepsilon)t$, so
$\bar{y}_{k}(t)\leq-(\frac{9}{10}-\varepsilon)t,~{}~{}~{}\mbox{for }k\leq
L-\gamma-\beta=\alpha.$
Since $\alpha>\gamma$,
$\bar{y}_{k}(t)=-(\frac{9}{10}-\varepsilon)t,~{}~{}\mbox{ for }\gamma+1\leq
k\leq\alpha.$
Let $y_{k}(t)=\exp\\{2\pi i{\bar{y}_{k}(t)}\\}$ for $1\leq k\leq L$. Then we
have
$y_{k}(t)=e^{-2\pi it(\frac{9}{10}-\varepsilon)},~{}~{}\mbox{ for
}\gamma+1\leq k\leq\alpha,$
and
$\\{y_{k}(t):1\leq k\leq L\\}=E(t),~{}~{}\mbox{ for all }t\in[0,1].$
Let
$W(t)=\begin{pmatrix}y_{1}(t)&0&\cdots&0\\\ 0&y_{2}(t)&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&y_{L}(t)\end{pmatrix}.$
Then for all $t\in[0,1]$, $W(t)$ and $\varphi_{1,n}(u_{\varepsilon})(t)$ have
exactly same eigenvalues (counting multiplicities). By a result of Thomsen
(see [26] Theorem 1.5), $\exists\Lambda(t)\in U(M_{L}(C[0,1]))$ such that
$\|\Lambda(t)W(t)\Lambda(t)^{*}-\varphi_{1,n}(u_{\varepsilon})(t)\|<\varepsilon,$
for all $t\in[0,1].$ Therefore, $cel(\varphi_{1,n}(u_{\varepsilon}))\geq
cel(W)-\varepsilon\pi/2>cel(W)-2\varepsilon$. (Here we use two facts: for
unitaries $a,b$, (1) $\|a-b\|<\varepsilon<1$ implies
$|cel(a)-cel(b)|<\varepsilon\pi/2$ and (2) $cel(uau^{*})=cel(a)$ where $u$ is
a unitary).
Let
$-\varepsilon<\varepsilon_{1}<\varepsilon_{2}<\cdots<\varepsilon_{\alpha}=0<\varepsilon_{\alpha+1}<\varepsilon_{\alpha+2}<\cdots<\varepsilon_{L}<\varepsilon$
be chosen, and $\tilde{y}_{k}(t)=\exp\\{2\pi
i(\bar{y}_{k}(t)+\varepsilon_{k})\\}$ and let
$\widetilde{W}(t)=\begin{pmatrix}\widetilde{y}_{1}(t)&0&\cdots&0\\\
0&\widetilde{y}_{2}(t)&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\widetilde{y}_{L}(t)\end{pmatrix}\in U(M_{L}(C[0,1])).$
Then $\widetilde{W}(t)$ has no repeated eigenvalues for all $t\in[0,1]$ and
$\|\widetilde{W}(t)-W(t)\|<\varepsilon,$
$\widetilde{y}_{i}(t)\neq\widetilde{y}_{j}(t)\mbox{ for }i\neq j\mbox{ and
}\widetilde{y}_{{\alpha}}(t)=e^{-2\pi it(\frac{9}{10}-\varepsilon)}.$
Let $\widetilde{F}_{s}(t)$ be a path in $U(M_{L}(C[0,1]))$ from $1$ to
$\widetilde{W}(t)$ with $\widetilde{F}_{0}(t)=1$ and
$\widetilde{F}_{1}(t)=\widetilde{W}(t).$ By Corollary 3.3, there is a smooth
path $F_{s}$ in $U(M_{L}(C[0,1]))$ such that $\|F-\widetilde{F}\|<\varepsilon$
and $F_{1}(t)=\widetilde{F}_{1}(t)=\widetilde{W}(t)$ and $F_{s}(t)$ has no
repeated eigenvalues for all $(s,t)\in[0,1]\times[0,1]$ and
$|\text{length}(\widetilde{F})-\text{length}(F)|\leq\varepsilon$. By Lemma 3.6
and Remark 3.7, there exist continuous maps
$f^{1},f^{2},...,f^{L}:[0,1]\times[0,1]\longrightarrow S^{1}$ and unitaries
$U_{s}(t)$ such that
$F_{s}(t)=U_{s}(t)\text{diag}(f_{s}^{1}(t),f_{s}^{2}(t),...,f_{s}^{L}(t))U_{s}(t)^{*}.$
Since $F_{1}(t)=\widetilde{W}(t)$ and $\widetilde{y}_{\alpha}=e^{-2\pi
it(\frac{9}{10}-\varepsilon)}$, we can assume $f_{1}^{\alpha}=e^{-2\pi
it(\frac{9}{10}-\varepsilon)}.$ Therefore, $f_{s}^{\alpha}$ is a path in
$U(C[0,1])$ from a point near $1$ to $e^{-2\pi it(\frac{9}{10}-\varepsilon)}.$
By Lemma 2.4,
$\text{length}(f_{s}^{\alpha})\geq
2\pi(\frac{9}{10}-\varepsilon)-\varepsilon.$
Therefore,
$\text{length}(\tilde{F}_{s})\geq
2\pi(\frac{9}{10}-\varepsilon)-\varepsilon-\varepsilon=2\pi(\frac{9}{10}-\varepsilon)-2\varepsilon,$
and
$cel(\varphi_{1,n}(u_{\varepsilon}))\geq
2\pi(\frac{9}{10}-\varepsilon)-5\varepsilon.$
∎
###### Corollary 3.15.
Let $A=\lim A_{i}$ and $u\in A_{1}$ be defined as in 3.12, then
$\varphi_{1,\infty}(u)\in CU(A)$ and
$cel(\varphi_{1,\infty}(u))\geq 2\pi\cdot\frac{9}{10}.$
Note that $CU(A)$ is the closure of the commutator subgroup of $U(A)$ (see
[13], [3], [28]). For $A=\text{M}_{n}(C[0,1])$ ($n\in\mathbb{N}$), $x\in
CU(A)$ if and only if $\det(x(t))=1$ for each $t\in[0,1]$.
###### Proof.
For any $\varepsilon\in(0,\frac{1}{100})$, let ${u_{\varepsilon}}\in A_{1}$ be
defined as in Theorem 3.14, then
$cel(\varphi_{1,n}(u_{\varepsilon}))\geq
2\pi(\frac{9}{10}-\varepsilon)-5\varepsilon~{}~{}\mbox{for all
}n\in\mathbb{N}.$
Since $\|\varphi_{1,n}(u)-\varphi_{1,n}(u_{\varepsilon})\|\leq
2\pi\varepsilon$ for all $n\in\mathbb{N}$,
$cel(\varphi_{1,n}(u))\geq
cel(\varphi_{1,n}(u_{\varepsilon}))-2\pi\varepsilon\geq
2\pi(\frac{9}{10}-2\varepsilon)-5\varepsilon.$
Since $\varepsilon$ is arbitrary, we have $cel(\varphi_{1,n}(u))\geq
2\pi\cdot\frac{9}{10}$, for all $n\in\mathbb{N}$. Therefore,
$cel(\varphi_{1,\infty}(u))\geq 2\pi\cdot\frac{9}{10}.$
∎
###### Remark 3.16.
Let $A_{i},~{}\varphi_{i,i+1},~{}(i\in\mathbb{N})$ be defined as in 3.12, for
any $\varepsilon>0$, there exists $i$ such that
$\frac{10^{k_{i}}-1}{10^{k_{i}}}\geq 1-\frac{\varepsilon}{2\pi}$. Let $u\in
A_{i}$ be defined by
$u(t)=\begin{pmatrix}e^{-2\pi it\frac{10^{k_{i}}-1}{10^{k_{i}}}}&0&\cdots&0\\\
0&e^{2\pi it\frac{1}{10^{k_{i}}}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{2\pi it\frac{1}{10^{k_{i}}}}\end{pmatrix}_{10^{k_{i}}\times
10^{k_{i}}}.$
Then $\varphi_{i,\infty}(u)\in CU(A)$ and
$cel(\varphi_{i,\infty}(u))\geq 2\pi\frac{10^{k_{i}}-1}{10^{k_{i}}}\geq
2\pi(1-\frac{\varepsilon}{2\pi})=2\pi-\varepsilon.$
###### Remark 3.17.
Our paper with above results was first posted on arxiv on Dec. 2012. Later H.
Lin posted a paper on Feb. 2013 (see [12]). In his paper, he provides examples
with $cel_{CU}(A)>\pi$ for unital simple $AH$-algebras $A$ with tracial rank
one, whose $\text{K}_{0}$-group can realize all possible weakly unperforated
Riesz group (see 5.11, 5.12 of [12]). He also obtained our theorem 3.11 with
different method. But our example shows examples with $cel_{CU}(A)\geq 2\pi$
for some simple $AH$-algebras (see 3.12, 3.15, 3.16). Of course, his paper
contains many interesting results in the other directions.
## References
* [1] R. Bhatia and C. Davis, _A bound for the spectral variation of a unitary operator_ , Linear and Multi-linear Algebra 15(1984), 71-76.
* [2] M.-D. Choi and G. A. Elliott, _Density of the self-adjoint elements with finite spectrum in an irrational rotation $\text{C}^{*}$-algebra_, Math. Scand. 67(1990), 73-86.
* [3] de la Harpe, P. and Skandalis, G., _D $\acute{e}$terminant associ$\acute{e}$ $\grave{a}$ une trace sur une alg$\grave{e}$bre de Banach_, Ann. Inst. Fourier, Grenoble, 34-1 (1984), 169-202.
* [4] G. A. Elliott, _On the classification of $\text{C}^{*}$-algebras of real rank zero_, J. Reine Angew. Math. 443(1993), 179-219.
* [5] G. A. Elliott, G. Gong and L. Li, _On the classification of simple inductive limit $\text{C}^{*}$-algebras, II, the isomorphism theorem_, Invent. Math., 168 (2007) no. 2, 249-320.
* [6] G. Gong and H. Lin, _The exponential rank of inductive limit $\text{C}^{*}$-algebras_, Math. Scand., 71(1992), 301-319.
* [7] K. R. Goodearl, _Notes on a class of simple $\text{C}^{*}$-algebras with real rank zero_, Publicacions Matem$\grave{a}$tiques, Vol 36 (1992), 637-654.
* [8] V. Guillemin and A. Pollack, _Differential Topology_.
* [9] R. V. Kadison and J. R. Ringrose, _Fundamentals of the theory of operator algebras_ , vol. 1, Elementary theory (New York: Academic Press, 1983).
* [10] C. Jiang and K. Wang, _A complete classification of limits of splitting interval algebras with the ideal property_ , J. Ramanujan Math. Soc. 27, No.3 (2012) 305-354.
* [11] H. Lin, _Exponential rank of $\text{C}^{*}$-algebras with real rank zero and Brown-Pedersen conjectures_, J. Funct. Anal., 114(1993) no. 1, 1-11.
* [12] H. Lin, _Exponentials in simple $\mathcal{Z}$-stable $\text{C}^{*}$-algebras_, J. Funct. Anal., 266 (2014), no. 2, 754-791.
* [13] H. Lin, _An Introduction to the classification of amenable $C^{*}$-algebras_, World Scientific.
* [14] H. Lin, _Generalized Weyl-von Neumann Theorems_ , Int. J. Math., 02, 725 (1991).
* [15] H. Lin, _Generalized Weyl-von Neumann Theorems II_ , Math. Scand. 77 (1995), 129-147.
* [16] H. Lin, _Simple nuclear $\text{C}^{*}$-algebras of tracial topological rank one_, J. Funct. Anal., 251 (2007), no. 2, 601-679.
* [17] N. C. Phillips, _Simple $\text{C}^{*}$-algebras with the property weak (FU)_, Math. Scand. 69(1991), 127-151.
* [18] N. C. Phillips, _The rectifiable metric on the space of projections in a $\text{C}^{*}$-algebra_, Internat. J. Math. 3(1992), 679-698.
* [19] N. C. Phillips, _How many exponentials?_ , Amer. J. Math., Vol. 116, No. 6 (Dec., 1994), 1513-1543.
* [20] N. C. Phillips, _The $\text{C}^{*}$ projective length of $n$-homogeneous $\text{C}^{*}$-algebras_, J. Operator Theory, 31(1994), 253-276.
* [21] N. C. Phillips, _Reduction of exponential rank in direct limits of $\text{C}^{*}$-algebras_, Can. J. Math. Vol. 46(4), 1994, 818-853.
* [22] N. C. Phillips, _Approximation by unitaries with finite spectrum in purely infinite $\text{C}^{*}$-algebras_, J. Funct. Anal., 120 (1994), 98-106.
* [23] N. C. Phillips, _Exponential length and traces_ , Proc. Roy. Soc. Edinburgh Sect. A, Vol. 125, January 1995, 13-29.
* [24] N. C. Phillips and J. R. Ringrose, _Exponential rank in operator algebras, Current topics in operator algebras_ , (Nara, 1990), 395-413, World Sci. Publ., River Edge, NJ, 1991.
* [25] J. R. Ringrose, _Exponential length and exponential rank in $\text{C}^{*}$-algebras_, Proc. London Math. Soc. (3) 46(1983), 301-333.
* [26] K. Thomsen, _Homomorphisms between finite direct sums of circle algebra_ , Linear and Multi-linear Algebra 1992, Vol.32, 33-50.
* [27] K. Thomsen, _On the reduced $\text{C}^{*}$-exponential length_, Operator algebras and quantum field theory (Rome, 1996), 59-64, Int. Press, Cambridge, MA, 1997.
* [28] K. Thomsen, _Traces, Unitary Characters and Crossed Products by $\mathbb{Z}$_, Publ. RIMS, Kyoto Univ. 31 (1995), 1011-1029.
* [29] H. Weyl, _Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller differentialgleichungen_ , Math. Ann., 71 (1912), 441-479.
* [30] S. Zhang, _On the exponential rank and exponential length of $\text{C}^{*}$-algebras_, J. Operator Theory, 28 (1992), 337-355.
* [31] S. Zhang, _Exponential rank and exponential length of operators on Hilbert $\text{C}^{*}$-algebras_, Ann. of Math. 137 (1993), 121-144.
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|
arxiv-papers
| 2012-12-31T02:33:11 |
2024-09-04T02:49:39.804686
|
{
"license": "Public Domain",
"authors": "Qingfei Pan and Kun Wang",
"submitter": "Kun Wang",
"url": "https://arxiv.org/abs/1212.6809"
}
|
1212.6816
|
# The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions
Jason Murphy Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA
[email protected]
###### Abstract.
We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger
equation in dimensions $d\geq 5.$ In the spirit of Kenig and Merle [10], we
combine a concentration-compactness approach with the Lin–Strauss Morawetz
inequality to prove that if a solution $u$ is bounded in $\dot{H}^{1/2}$
throughout its lifespan, then $u$ is global and scatters.
## 1\. Introduction
We consider the initial-value problem for the defocusing
$\dot{H}_{x}^{1/2}$-critical nonlinear Schrödinger equation in dimensions
$d\geq 5$:
$\left\\{\begin{array}[]{ll}(i\partial_{t}+\Delta)u=|u|^{\frac{4}{d-1}}u\\\
u(0)=u_{0},\end{array}\right.$ (1.1)
with $u:\mathbb{R}_{t}\times\mathbb{R}_{x}^{d}\to\mathbb{C}$. This equation is
deemed $\dot{H}_{x}^{1/2}$-critical because the rescaling that preserves the
class of solutions to (1.1), that is,
$u(t,x)\mapsto\lambda^{\frac{d-1}{2}}u(\lambda^{2}t,\lambda x),$ leaves
invariant the $\dot{H}_{x}^{1/2}$-norm of the initial data.
In [10], Kenig and Merle considered (1.1) with $d=3$. They proved that if a
solution $u$ stays bounded in $\dot{H}_{x}^{1/2}$ throughout its lifespan,
then $u$ must be global and scatter. The same statement for $d=4$ was proven
as a special case of the results of [20]. In this short note, we establish
this result for all $d\geq 5$.
We begin with some definitions.
###### Definition 1.1 (Solution).
A function $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ on a time interval $I\ni 0$
is a _solution_ to (1.1) if it belongs to
$C_{t}\dot{H}_{x}^{1/2}(K\times\mathbb{R}^{d})\cap
L_{t,x}^{\frac{2(d+2)}{d-1}}(K\times\mathbb{R}^{d})$ for every compact
$K\subset I$ and obeys the Duhamel formula
$u(t)=e^{it\Delta}u_{0}-i\int_{0}^{t}e^{i(t-s)\Delta}(|u|^{\frac{4}{d-1}}u)(s)\,ds$
for all $t\in I$. We call $I$ the _lifespan_ of $u$; we say $u$ is a _maximal-
lifespan solution_ if it cannot be extended to any strictly larger interval.
If $I=\mathbb{R}$, we say $u$ is _global_.
###### Definition 1.2 (Scattering size and blowup).
We define the _scattering size_ of a solution $u$ to (1.1) on a time interval
$I$ by
$S_{I}(u):=\int_{I}\int_{\mathbb{R}^{d}}|u(t,x)|^{\frac{2(d+2)}{d-1}}\,dx\,dt.$
(1.2)
If there exists $t\in I$ such that $S_{[t,\sup I)}(u)=\infty$, then we say $u$
_blows up forward in time_. Similarly, if there exists $t\in I$ such that
$S_{(\inf I,t]}(u)=\infty$, then we say $u$ _blows up backward in time_.
On the other hand, if $u$ is global with $S_{\mathbb{R}}(u)<\infty$, then
standard arguments show that $u$ _scatters_ , that is, there exist unique
$u_{\pm}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$ such that
$\lim_{t\to\pm\infty}\|u(t)-e^{it\Delta}u_{\pm}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}=0.$
Our main result is the following
###### Theorem 1.3.
Let $d\geq 5$ and let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-
lifespan solution to (1.1) such that $u\in
L_{t}^{\infty}\dot{H}_{x}^{1/2}(I\times\mathbb{R}^{d}).$ Then $u$ is global
and scatters, with
$S_{\mathbb{R}}(u)\leq
C\big{(}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}(\mathbb{R}\times\mathbb{R}^{d})}\big{)}\quad\text{for
some function}\quad C:[0,\infty)\to[0,\infty).$
Following the approach of Kenig and Merle [10], we will establish Theorem 1.3
by combining a concentration-compactness argument with the Lin–Strauss
Morawetz inequality of [18]. This estimate is very useful in the study of
(1.1), as it has critical scaling for this problem. In fact, it is the
concentration-compactness component that comprises most of this note; once we
have reduced the problem to the study of almost periodic solutions, we can
quickly bring the argument to a conclusion.
We first need a good local-in-time theory. Building off arguments of Cazenave
and Weissler [3], we can prove the following local well-posedness result (see
Remark 3.8).
###### Theorem 1.4 (Local well-posedness).
Let $d\geq 5$ and $u_{0}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$. Then there
exists a unique maximal-lifespan solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) such that:
$\bullet$ $($Local existence$)$ $I$ is an open neighborhood of $0$.
$\bullet$ $($Blowup criterion$)$ If $\sup I<\infty$, then $u$ blows up forward
in time. If $|\inf I|<\infty$, then $u$ blows up backward in time.
$\bullet$ $($Existence of wave operators$)$ For any
$u_{+}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$, there is a unique solution $u$ to
(1.1) such that $u$ scatters to $u_{+}$, that is,
$\lim_{t\to\infty}\|u(t)-e^{it\Delta}u_{+}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}=0.$
A similar statement holds backward in time.
$\bullet$ $($Small-data global existence$)$ If
$\|u_{0}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}$ is sufficiently small
depending on $d$, then $u$ is global and scatters, with
$S_{\mathbb{R}}(u)\lesssim\|u_{0}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}^{\frac{2(d+2)}{d-1}}.$
### 1.1. Outline of the proof of Theorem 1.3
We argue by contradiction and suppose that Theorem 1.3 fails. Recalling from
Theorem 1.4 that Theorem 1.3 holds for sufficiently small initial data, we
deduce the existence of a threshold size, below which Theorem 1.3 holds, but
above which we can find (almost) counterexamples. We then use a limiting
argument to find blowup solutions _at_ this threshold, and show that such
minimal blowup solutions must possess strong concentration properties.
Finally, in Sections 5 and 6, we show that solutions to (1.1) with such
properties cannot exist.
The main property of these solutions is that of almost periodicity:
###### Definition 1.5 (Almost periodic solutions).
A solution $u$ to (1.1) with lifespan $I$ is said to be _almost periodic_
$($_modulo symmetries $)$_ if $u\in
L_{t}^{\infty}\dot{H}_{x}^{1/2}(I\times\mathbb{R}^{d})$ and there exist
functions $N:I\to\mathbb{R}^{+},$ $x:I\to\mathbb{R}^{d}$, and
$C:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
$\int_{|x-x(t)|\geq\frac{C(\eta)}{N(t)}}\big{|}|\nabla|^{1/2}u(t,x)\big{|}^{2}\,dx+\int_{|\xi|\geq
C(\eta)N(t)}|\xi|\,|\widehat{u}(t,\xi)|^{2}\,d\xi\leq\eta$
for all $t\in I$ and $\eta>0$. We call $N$ the _frequency scale function_ ,
$x$ the _spatial center function_ , and $C$ the _compactness modulus
function_.
###### Remark 1.6.
Using Arzelà–Ascoli and Sobolev embedding, one can derive the following: for a
nonzero almost periodic solution $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to
(1.1), there exists $C(u)>0$ such that
$\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{\frac{2d}{d-1}}\,dx\gtrsim_{u}1\quad\text{uniformly
for }t\in I.$
We can now describe the first major step in the proof of Theorem 1.3.
###### Theorem 1.7 (Reduction to almost periodic solutions).
If Theorem 1.3 fails, then there exists a maximal-lifespan solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) such that $u$ is almost
periodic and blows up in both time directions.
The reduction to almost periodic solutions has become a well-known and widely
used technique in the study of dispersive equations at critical regularity.
The existence of such solutions was first established by Keraani [12] in the
context of the mass-critical NLS, while Kenig and Merle [9] were the first to
use them to prove a global well-posedness result (in the energy-critical
setting). These techniques have since been adapted to a variety of settings
(see [7, 10, 13, 14, 15, 16, 20, 23, 24] for some examples in the case of
NLS), and the general approach is well-understood.
The argument, which we will carry out in Section 4, requires three
ingredients: (i) a profile decomposition for the linear propagator, (ii) a
stability result for the nonlinear equation, and (iii) a decoupling statement
for nonlinear profiles. The first profile decompositions established for
$e^{it\Delta}$ were adapted to the mass- and energy-critical settings (see [1,
2, 11, 19]); the case of non-conserved critical regularity was addressed in
[21]. We will be able to import the profile decomposition we need directly
from [21] (see Section 2.4).
Ingredients (ii) and (iii) are closely related, in that the decoupling must be
established in a space that is dictated by the stability result. Stability
results most often require errors to be small in a space with the scaling-
critical number of derivatives (say $s_{c}$). In [11], Keraani showed how to
establish the decoupling in such a space for the energy-critical problem (that
is, $s_{c}=1$). The argument relies on pointwise estimates and hence is also
applicable to the mass-critical problem ($s_{c}=0$). For
$s_{c}\notin\\{0,1\\}$, however, the nonlocal nature of $|\nabla|^{s_{c}}$
prevents the direct use of this argument.
In certain cases for which $s_{c}\notin\\{0,1\\}$ it has nonetheless been
possible to adapt the arguments of [11] to establish the decoupling in a space
with $s_{c}$ derivatives. Kenig and Merle [10] were able to succeed in the
case $s_{c}=1/2$, $d=3$ (for which the nonlinearity is cubic) by exploiting
the polynomial nature of the nonlinearity and making use of a paraproduct
estimate. Killip and Vişan [16] handled some cases for which $s_{c}>1$ by
utilizing a square function of Strichartz that shares estimates with
$|\nabla|^{s_{c}}$. In [20], some cases were treated for which $s_{c}\in(0,1)$
(and the nonlinearity is non-polynomial) by making use of the Littlewood–Paley
square function and working at the level of individual frequencies.
In this paper, we take a simpler approach to (ii) and (iii), inspired by the
work of Holmer and Roudenko [7] on the focusing $\dot{H}_{x}^{1/2}$-critical
NLS in $d=3$. It relies on the observation that for $s_{c}=1/2$, one can
develop a stability theory for NLS that only requires errors to be small in a
space without derivatives. In Section 3, we do exactly this (see Theorem 3.6).
To prove the decoupling in a space without derivatives, we can then rely
simply on pointwise estimates and apply the arguments of [11] directly (see
Lemma 4.3). By proving a more refined stability result, we are thus able to
avoid entirely the technical issues related to fractional differentiation
described above. In this way, we can greatly simplify the analysis needed to
carry out the reduction to almost periodic solutions in our setting.
Continuing from Theorem 1.7, we can make some further reductions to the class
of solutions that we consider. In particular, we can prove the following
###### Theorem 1.8.
If Theorem 1.3 fails, then there exists an almost periodic solution
$u:[0,T_{max})\to\mathbb{C}$ to (1.1) with the following properties:
$(i)$ $u$ blows up forward in time,
$(ii)$ $\inf_{t\in[0,T_{max})}N(t)\geq 1,$
$(iii)$ $|x(t)|\lesssim_{u}\int_{0}^{t}N(s)\,ds$ for all $t\in[0,T_{max}).$
Let us briefly sketch the proof of Theorem 1.8. Beginning with an almost
periodic solution as in Theorem 1.7 and using a rescaling argument (as in [24,
Theorem 3.3], for example), one can deduce the existence of an almost periodic
blowup solution that does not escape to artibrarily low frequencies on at
least half of its maximal lifespan, say $[0,T_{max})$. In this way, we may
find an almost periodic solution such that $(i)$ and $(ii)$ hold.
It remains to see how one can modify an almost periodic solution
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ so that $(iii)$ holds. We may
always translate $u$ so that $x(0)=0$; thus, it suffices to show that we can
modify the modulation parameters of $u$ so that $|\dot{x}(t)|\sim_{u}N(t)$ for
a.e. $t\in[0,T_{max})$. This will follow from the following local constancy
property for the modulation parameters of almost periodic solutions (see [14,
Lemma 5.18], for example):
###### Lemma 1.9 (Local constancy).
Let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan almost
periodic solution to (1.1). Then there exists $\delta=\delta(u)>0$ such that
if $t_{0}\in I$, then
$[t_{0}-\delta N(t_{0})^{-2},t_{0}+\delta N(t_{0})^{-2}]\subset I,$
with
$N(t)\sim_{u}N(t_{0}),\quad|x(t)-x(t_{0})|\lesssim_{u}N(t_{0})^{-1}\quad\text{for}\quad|t-t_{0}|\leq\delta
N(t_{0})^{-2}.$
Using Lemma 1.9, we may subdivide $[0,T_{max})$ into _characteristic
subintervals_ $I_{k}$ and set $N(t)$ to be constant and equal to some $N_{k}$
on each $I_{k}$. Note that $|I_{k}|\sim_{u}N_{k}^{-2}$ and that this requires
us to modify the compactness modulus function by a (time-independent)
multiplicative factor. We may then modify $x(t)$ by $O(N(t)^{-1})$ so that
$x(t)$ becomes piecewise linear on each $I_{k}$, with
$|\dot{x}(t)|\sim_{u}N(t)$ for $t\in I_{k}^{\circ}$. Thus, we get
$|\dot{x}(t)|\sim_{u}N(t)$ for a.e. $t\in[0,T_{max}),$ as desired.
To complete the proof of Theorem 1.3, it therefore suffices to rule out the
existence of the almost periodic solutions described in Theorem 1.8.
In Section 5, we preclude the possibility of finite time blowup (i.e.
$T_{max}<\infty$). To do this, we make use of the following ‘reduced’ Duhamel
formula for almost periodic solutions, which one can prove by adapting the
argument in [14, Proposition 5.23]:
###### Proposition 1.10 (Reduced Duhamel formula).
Let $u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ be an almost periodic
solution to (1.1). Then for all $t\in[0,T_{max})$, we have
$u(t)=i\lim_{T\to
T_{max}}\int_{t}^{T}e^{i(t-s)\Delta}(|u|^{\frac{4}{d-1}}u)(s)\,ds,$
where the limits are taken in the weak $\dot{H}_{x}^{1/2}$ topology.
Using Proposition 1.10, Strichartz estimates, and conservation of mass, we can
show that an almost periodic solution that blows up in finite time must have
zero mass, contradicting the fact that the solution blows up in the first
place.
In Section 6, we use the Lin–Strauss Morawetz inequality to rule out the
remaining case, $T_{max}=\infty$. This estimate tells us that solutions to NLS
that are bounded in $\dot{H}_{x}^{1/2}$ cannot remain concentrated near the
origin for too long. However, almost periodic solutions to (1.1) as in Theorem
1.8 with $T_{max}=\infty$ do essentially this; thus we can reach a
contradiction in this case.
### Acknowledgements
I am grateful to my advisors Rowan Killip and Monica Vişan for helpful
discussions and for a careful reading of the manuscript. This work was
supported in part by NSF grant DMS-1001531 (P.I. Rowan Killip).
## 2\. Notation and useful lemmas
### 2.1. Some notation
We write $X\lesssim Y$ or $Y\gtrsim X$ whenever $X\leq CY$ for some
$C=C(d)>0$. If $X\lesssim Y\lesssim X$, we write $X\sim Y$. Dependence on
additional parameters will be indicated with subscripts, for example,
$X\lesssim_{u}Y$.
For a spacetime slab $I\times\mathbb{R}^{d}$, we define
$\|u\|_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}:=\|\,\|u(t)\|_{L_{x}^{r}(\mathbb{R}^{d})}\,\|_{L_{t}^{q}(I)}.$
If $q=r$, we write $L_{t}^{q}L_{x}^{q}=L_{t,x}^{q}.$ We also sometimes write
$\|f\|_{L_{x}^{r}(\mathbb{R}^{d})}=\|f\|_{L_{x}^{r}}$.
We define the Fourier transform on $\mathbb{R}^{d}$ by
$\widehat{f}(\xi):=(2\pi)^{-d/2}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)\,dx.$
For $s>-d/2$, we then define the fractional differentiation operator
$|\nabla|^{s}$ and the homogeneous Sobolev norm via
$\widehat{|\nabla|^{s}f}(\xi):=|\xi|^{s}\widehat{f}(\xi)$ and
$\|f\|_{\dot{H}_{x}^{s}(\mathbb{R}^{d})}:=\||\nabla|^{s}f\|_{L_{x}^{2}(\mathbb{R}^{d})}.$
### 2.2. Basic harmonic analysis
Let $\varphi$ be a radial bump function supported in the ball
$\\{\xi\in\mathbb{R}^{d}:|\xi|\leq\tfrac{11}{10}\\}$ and equal to 1 on the
ball $\\{\xi\in\mathbb{R}^{d}:|\xi|\leq 1\\}.$ For $N\in 2^{\mathbb{Z}}$, we
define the Littlewood–Paley projection operators via
$\displaystyle\widehat{P_{\leq
N}f}(\xi):=\varphi(\tfrac{\xi}{N})\widehat{f}(\xi),\quad\widehat{P_{N}f}(\xi):=\big{(}\varphi(\tfrac{\xi}{N})-\varphi(\tfrac{2\xi}{N})\big{)}\widehat{f}(\xi),\quad
P_{>N}:=\text{Id}-P_{\leq N}.$
We note that these operators commute with $e^{it\Delta}$ and all differential
operators, as they are Fourier multiplier operators. They also obey the
following
###### Lemma 2.1 (Bernstein estimates).
For $1\leq r\leq q\leq\infty$ and $s\geq 0$,
$\displaystyle\|P_{>N}f\|_{L_{x}^{r}(\mathbb{R}^{d})}\lesssim
N^{-s}\||\nabla|^{s}f\|_{L_{x}^{r}(\mathbb{R}^{d})},\quad\|P_{N}f\|_{L_{x}^{q}(\mathbb{R}^{d})}\lesssim
N^{\frac{d}{r}-\frac{d}{q}}\|f\|_{L_{x}^{r}(\mathbb{R}^{d})}.$
We will also need some fractional calculus estimates.
###### Lemma 2.2 (Fractional chain rule, [5]).
Suppose $G\in C^{1}(\mathbb{C})$ and $s\in(0,1]$. Let $1<r,r_{1}<\infty,$
$1<r_{2}\leq\infty$ be such that
$\tfrac{1}{r}=\tfrac{1}{r_{1}}+\tfrac{1}{r_{2}}.$ Then
$\||\nabla|^{s}G(u)\|_{L_{x}^{r}}\lesssim\|G^{\prime}(u)\|_{L_{x}^{r_{1}}}\||\nabla|^{s}u\|_{L_{x}^{r_{2}}}.$
###### Lemma 2.3 (Derivatives of differences, [17]).
Let $F(u)=|u|^{p}u$ for some $p>0$ and let $0<s<1$. For
$1<r,r_{1},r_{2}<\infty$ such that
$\tfrac{1}{r}=\tfrac{1}{r_{1}}+\tfrac{p}{r_{2}},$ we have
$\||\nabla|^{s}[F(u+v)-F(u)]\|_{L_{x}^{r}}\lesssim\||\nabla|^{s}u\|_{L_{x}^{r_{1}}}\|v\|_{L_{x}^{r_{2}}}^{p}+\||\nabla|^{s}v\|_{L_{x}^{r_{1}}}\|u+v\|_{L_{x}^{r_{2}}}^{p}.$
### 2.3. Strichartz estimates
Let $e^{it\Delta}$ be the free Schrödinger propagator:
$[e^{it\Delta}f](x)=(4\pi
it)^{-d/2}\int_{\mathbb{R}^{d}}e^{i|x-y|^{2}/4t}f(y)\,dy\quad\text{for }t\neq
0.$
For $d\geq 3$, we call a pair of exponents $(q,r)$ _Schrödinger admissible_ if
$2\leq q,r\leq\infty$ and $\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}$. For a time
interval $I$ and $s\geq 0$, we define the Strichartz space $\dot{S}^{s}(I)$
via the norm
$\|u\|_{\dot{S}^{s}(I)}=\sup\big{\\{}\||\nabla|^{s}u\|_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}:(q,r)\text{
Schr\"{o}dinger admissible}\big{\\}}.$
We will make frequent use of the following standard estimates for
$e^{it\Delta}$:
###### Lemma 2.4 (Strichartz estimates, [6, 8, 22]).
Let $d\geq 3$, $s\geq 0$ and let $I$ be a compact time interval. Let
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a solution to the forced Schrödinger
equation $(i\partial_{t}+\Delta)u=F$. Then for any $t_{0}\in I$, we have
$\|u\|_{\dot{S}^{s}(I)}\lesssim\|u(t_{0})\|_{\dot{H}_{x}^{s}(\mathbb{R}^{d})}+\min\bigg{\\{}\||\nabla|^{s}F\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}(I\times\mathbb{R}^{d})},\||\nabla|^{s}F\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}(I\times\mathbb{R}^{d})}\bigg{\\}}.$
### 2.4. Concentration-compactness
We record here the linear profile decomposition that we will use in Section 4.
We begin with the following
###### Definition 2.5 (Symmetry group).
For any position $x_{0}\in\mathbb{R}^{d}$ and scaling parameter $\lambda>0$,
we define a unitary transformation
$g_{x_{0},\lambda}:\dot{H}_{x}^{1/2}(\mathbb{R}^{d})\to\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$
by
$[g_{x_{0},\lambda}f](x):=\lambda^{-\frac{d-1}{2}}f(\lambda^{-1}(x-x_{0})).$
We let $G$ denote the group of such transformations.
We now state the linear profile decomposition. For the mass-critical NLS, this
result was originally proven in [1, 2, 19], while for the energy-critical NLS,
it was established in [11]. In the generality we need, a proof can be found in
[21].
###### Lemma 2.6 (Linear profile decomposition, [21]).
Let $\\{u_{n}\\}_{n\geq 1}$ be a bounded sequence in
$\dot{H}_{x}^{1/2}(\mathbb{R}^{d}).$ After passing to a subsequence if
necessary, there exist functions $\\{\phi^{j}\\}_{j\geq
1}\subset\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$, group elements $g_{n}^{j}\in G$
$($with parameters $x_{n}^{j}$ and $\lambda_{n}^{j}$$)$, and times
$t_{n}^{j}\in\mathbb{R}$ such that for all $J\geq 1$, we have the following
decomposition:
$u_{n}=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+w_{n}^{J}.$
This decomposition satisfies the following properties:
$\bullet$ For each $j$, either $t_{n}^{j}\equiv 0$ or $t_{n}^{j}\to\pm\infty$
as $n\to\infty.$
$\bullet$ For $J\geq 1$, we have the following decoupling:
$\lim_{n\to\infty}\bigg{[}\|u_{n}\|_{\dot{H}_{x}^{1/2}}^{2}-\sum_{j=1}^{J}\|\phi^{j}\|_{\dot{H}_{x}^{1/2}}^{2}-\|w_{n}^{J}\|_{\dot{H}_{x}^{1/2}}^{2}\bigg{]}=0.$
(2.1)
$\bullet$ For any $j\neq k$, we have the following asymptotic orthogonality
condition:
$\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}+\frac{\lambda_{n}^{k}}{\lambda_{n}^{j}}+\frac{|x_{n}^{j}-x_{n}^{k}|^{2}}{\lambda_{n}^{j}\lambda_{n}^{k}}+\frac{|t_{n}^{j}(\lambda_{n}^{j})^{2}-t_{n}^{k}(\lambda_{n}^{k})^{2}|}{\lambda_{n}^{j}\lambda_{n}^{k}}\to\infty\quad\text{as
}n\to\infty.$ (2.2)
$\bullet$ For all $n$ and all $J\geq 1$, we have
$w_{n}^{J}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$, with
$\lim_{J\to\infty}\limsup_{n\to\infty}\|e^{it\Delta}w_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}(\mathbb{R}\times\mathbb{R}^{d})}=0.$
(2.3)
## 3\. Stability theory
In this section, we develop a stability theory for (1.1). The main result of
this section is Theorem 3.6, which will play a key role in the reduction to
almost periodic solutions carried out in Section 4. Throughout this section,
we will denote the nonlinearity $|u|^{\frac{4}{d-1}}u$ by $F(u)$.
We begin by recording a local well-posedness result of Cazenave and Weissler
[3]. This result requires the data to belong to the inhomogeneous Sobolev
space, so that a contraction mapping argument may be run in mass-critical
spaces.
###### Theorem 3.1 (Standard local well-posedness [3]).
Let $d\geq 5$ and $u_{0}\in{H}_{x}^{1/2}(\mathbb{R}^{d})$. If $I\ni 0$ is a
time interval such that
$\||\nabla|^{1/2}e^{it\Delta}u_{0}\|_{L_{t}^{\frac{2(d+1)}{d-1}}L_{x}^{\frac{2d(d+1)}{d^{2}-d+2}}(I\times\mathbb{R}^{d})}$
is sufficiently small, then we may find a unique solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1).
Next, we turn to the stability results. We will make use of function spaces
that are critical with respect to scaling, but do not involve any derivatives.
In particular, for a time interval $I$, we define the following norms:
$\displaystyle\|u\|_{X(I)}:=\|u\|_{L_{t}^{\frac{4(d+1)}{d-1}}L_{x}^{\frac{2(d+1)}{d-1}}(I\times\mathbb{R}^{d})},\quad\|F\|_{Y(I)}:=\|F\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}(I\times\mathbb{R}^{d})}.$
We first relate the $X$-norm to the usual Strichartz norms. By Sobolev
embedding, we get $\|u\|_{X(I)}\lesssim\|u\|_{\dot{S}^{1/2}(I)},$ while Hölder
and Sobolev embedding together imply
$\|u\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}(I\times\mathbb{R}^{d})}\lesssim\|u\|_{X(I)}^{c}\|u\|_{\dot{S}^{1/2}(I)}^{1-c}\quad\text{for
some}\quad 0<c(d)<1.$ (3.1)
Next, we record a Strichartz estimate, which one can prove via the standard
approach (namely, by applying the dispersive estimate and
Hardy–Littlewood–Sobolev).
###### Lemma 3.2.
Let $I$ be a compact time interval and $t_{0}\in I$. Then for all $t\in I$,
$\bigg{\|}\int_{t_{0}}^{t}e^{i(t-s)\Delta}F(s)\,ds\bigg{\|}_{X(I)}\lesssim\|F\|_{Y(I)}.$
(3.2)
Finally, we collect some estimates that will allow us to control the
nonlinearity.
###### Lemma 3.3.
Fix $d\geq 5$. Then, with spacetime norms over $I\times\mathbb{R}^{d}$, we
have
$\displaystyle\|F(u)\|_{Y(I)}$
$\displaystyle\lesssim\|u\|_{X(I)}^{\frac{d+3}{d-1}}$ (3.3)
$\displaystyle\|F(u)-F(\tilde{u})\|_{Y(I)}$
$\displaystyle\lesssim\big{\\{}\|u\|_{X(I)}^{\frac{4}{d-1}}+\|\tilde{u}\|_{X(I)}^{\frac{4}{d-1}}\big{\\}}\|u-\tilde{u}\|_{X(I)}$
(3.4) $\displaystyle\||\nabla|^{1/2}F(u)\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\|u\|_{X(I)}^{\frac{4}{d-1}}\|u\|_{\dot{S}^{1/2}(I)}$
(3.5)
$\displaystyle\||\nabla|^{1/2}[F(u)-F(\tilde{u})]\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\|u-\tilde{u}\|_{X(I)}^{\frac{4}{d-1}}\|\tilde{u}\|_{\dot{S}^{1/2}(I)}+\|u\|_{X(I)}^{\frac{4}{d-1}}\|u-\tilde{u}\|_{\dot{S}^{1/2}(I)}.$
(3.6)
###### Proof.
We first note that (3.3) follows from Hölder, while (3.4) follows from the
fundamental theorem of calculus followed by Hölder.
Next, we see that (3.5) follows from Hölder and the fractional chain rule.
Indeed,
$\||\nabla|^{1/2}F(u)\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}\lesssim\|u\|_{L_{t}^{\frac{4(d+1)}{d-1}}L_{x}^{\frac{2(d+1)}{d-1}}}^{\frac{4}{d-1}}\||\nabla|^{1/2}u\|_{L_{t}^{\frac{2(d+1)}{d-1}}L_{x}^{\frac{2d(d+1)}{d^{2}-d+2}}}.$
Using these same exponents with Lemma 2.3, we deduce (3.6). ∎
We may now state our first stability result.
###### Lemma 3.4 (Short-time perturbations).
Let $d\geq 5$ and let $I$ be a compact time interval, with $t_{0}\in I$. Let
$\tilde{u}:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a solution to
$(i\partial_{t}+\Delta)\tilde{u}=F(\tilde{u})+e$ with
$\tilde{u}(t_{0})=\tilde{u}_{0}\in\dot{H}_{x}^{1/2}$. Suppose
$\|\tilde{u}\|_{\dot{S}^{1/2}(I)}\leq
E\quad\text{and}\quad\||\nabla|^{1/2}e\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}(I\times\mathbb{R}^{d})}\leq
E$ (3.7)
for some $E>0$. Let $u_{0}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$ satisfy
$\displaystyle\|u_{0}-\tilde{u}_{0}\|_{\dot{H}_{x}^{1/2}}$ $\displaystyle\leq
E,$ (3.8)
and suppose that we have the smallness conditions
$\displaystyle\|\tilde{u}\|_{X(I)}$ $\displaystyle\leq\delta,$ (3.9)
$\displaystyle\|e^{i(t-t_{0})\Delta}(u_{0}-\tilde{u}_{0})\|_{X(I)}+\|e\|_{Y(I)}$
$\displaystyle\leq\varepsilon,$ (3.10)
for some small $0<\delta=\delta(E)$ and $0<\varepsilon<\varepsilon_{0}(E).$
Then there exists $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ solving (1.1) with
$u(t_{0})=u_{0}$ such that
$\displaystyle\|u-\tilde{u}\|_{X(I)}+\|F(u)-F(\tilde{u})\|_{Y(I)}$
$\displaystyle\lesssim\varepsilon,$ (3.11)
$\displaystyle\|u-\tilde{u}\|_{\dot{S}^{1/2}(I)}+\||\nabla|^{1/2}[F(u)-F(\tilde{u})]\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim_{E}1.$ (3.12)
###### Proof.
We first suppose $u_{0}\in L_{x}^{2}$, so that Theorem 3.1 provides the
solution $u$. We will then prove (3.11) and (3.12) as _a priori_ estimates.
After the lemma is proven for $u_{0}\in H_{x}^{1/2}$, we can use approximation
by $H_{x}^{1/2}$-functions to see that the lemma holds for
$u_{0}\in\dot{H}_{x}^{1/2}$. Throughout the proof, spacetime norms will be
over $I\times\mathbb{R}^{d}$.
We will first show
$\|u\|_{X(I)}\lesssim\delta.$ (3.13)
By the triangle inequality, (3.2), (3.3), (3.9), and (3.10), we get
$\displaystyle\|e^{i(t-t_{0})\Delta}\tilde{u}_{0}\|_{X(I)}\lesssim\|\tilde{u}\|_{X(I)}+\|F(\tilde{u})\|_{Y(I)}+\|e\|_{Y(I)}\lesssim\delta+\delta^{\frac{d+3}{d-1}}+\varepsilon.$
Combining this estimate with (3.10) and using the triangle inequality then
gives
$\|e^{i(t-t_{0})\Delta}u_{0}\|_{X(I)}\lesssim\delta$
for $\delta$ and $\varepsilon\lesssim\delta$ sufficiently small. Thus, by
(3.2) and (3.3), we get
$\displaystyle\|u\|_{X(I)}\lesssim\delta+\|F(u)\|_{Y(I)}\lesssim\delta+\|u\|_{X(I)}^{\frac{d+3}{d-1}},$
which (taking $\delta$ sufficiently small) implies (3.13).
We now turn to proving the desired estimates for $w:=u-\tilde{u}$. Note first
that $w$ is a solution to $(i\partial_{t}+\Delta)w=F(u)-F(\tilde{u})-e$, with
$w(t_{0})=u_{0}-\tilde{u}_{0}$; thus, we can use (3.2), (3.4), (3.9), (3.10),
and (3.13) to see
$\displaystyle\|w\|_{X(I)}$
$\displaystyle\lesssim\|e^{i(t-t_{0})\Delta}(u_{0}-\tilde{u}_{0})\|_{X(I)}+\|e\|_{Y(I)}+\|F(u)-F(\tilde{u})\|_{Y(I)}$
$\displaystyle\lesssim\varepsilon+\big{\\{}\|u\|_{X(I)}^{\frac{4}{d-1}}+\|\tilde{u}\|_{X(I)}^{\frac{4}{d-1}}\big{\\}}\|w\|_{X(I)}$
$\displaystyle\lesssim\varepsilon+\delta^{\frac{4}{d-1}}\|w\|_{X(I)}.$
Taking $\delta$ sufficiently small, we see that the first estimate in (3.11)
holds. Using the first estimate in (3.11), along with (3.4), (3.9), and
(3.13), we see that the remaining estimate in (3.11) holds, as well.
Next, by Strichartz, (3.6), (3.7), (3.8), (3.9), (3.11), and (3.13), we get
$\displaystyle\|w\|_{\dot{S}^{1/2}(I)}$
$\displaystyle\lesssim\|u_{0}-\tilde{u}_{0}\|_{\dot{H}_{x}^{1/2}}+\||\nabla|^{1/2}e\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}+\||\nabla|^{1/2}[F(u)-F(\tilde{u})]\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim_{E}1+\|\tilde{u}\|_{\dot{S}^{1/2}(I)}\|w\|_{X(I)}^{\frac{4}{d-1}}+\|w\|_{\dot{S}^{1/2}(I)}\|u\|_{X(I)}^{\frac{4}{d-1}}$
$\displaystyle\lesssim_{E}1+\delta^{\frac{4}{d-1}}\|w\|_{\dot{S}^{1/2}(I)}.$
Taking $\delta=\delta(E)$ sufficiently small then gives the first estimate in
(3.12). We get the remaining estimate in (3.12) by using (3.6) with (3.7),
(3.11), (3.13), and the first estimate in (3.12). ∎
###### Remark 3.5.
As mentioned in the introduction, the error $e$ is only required to be small
in a space without derivatives (see (3.10)); it merely needs to be _bounded_
in a space with derivatives (see (3.7)). This will also be the case in Theorem
3.6 below (see (3.14) and (3.15)). We will see the benefit of this refinement
when we carry out the proof of Theorem 1.7 in Section 4 (see Remark 4.4).
We continue to the main result of this section:
###### Theorem 3.6 (Stability).
Let $d\geq 5$, and let $I$ be a compact time interval, with $t_{0}\in I$.
Suppose $\tilde{u}$ is a solution to
$(i\partial_{t}+\Delta)\tilde{u}=F(\tilde{u})+e$, with
$\tilde{u}(t_{0})=\tilde{u}_{0}$. Suppose
$\displaystyle\|\tilde{u}\|_{\dot{S}^{1/2}(I)}\leq
E\quad\text{and}\quad\||\nabla|^{1/2}e\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}(I\times\mathbb{R}^{d})}$
$\displaystyle\leq E$ (3.14)
for some $E>0$. Let $u_{0}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$, and suppose
we have the smallness conditions
$\displaystyle\|u_{0}-\tilde{u}_{0}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}+\|e\|_{Y(I)}$
$\displaystyle\leq\varepsilon$ (3.15)
for some small $0<\varepsilon<\varepsilon_{1}(E)$. Then, there exists
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ solving (1.1) with $u(t_{0})=u_{0}$,
and there exists $0<c(d)<1$ such that
$\displaystyle\|u-\tilde{u}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim_{E}\varepsilon^{c}.$ (3.16)
One can derive Theorem 3.6 from Lemma 3.4 in the standard fashion, namely, by
applying Lemma 3.4 inductively (see [14], for example). We omit these details,
but pause to point out the following: this induction will actually yield the
bounds
$\|u-\tilde{u}\|_{X(I)}\lesssim\varepsilon\quad\text{and}\quad\|u-\tilde{u}\|_{\dot{S}^{1/2}(I)}\lesssim_{E}1.$
With these bounds in hand, we then use (3.1) to see that (3.16) holds.
###### Remark 3.7.
The smallness condition on $u_{0}-\tilde{u}_{0}$ appearing in (3.15) may
actually be relaxed to the condition appearing in (3.10). In our setting, it
will not be difficult to prove the stronger condition (see Lemma 4.3).
###### Remark 3.8.
Using arguments from [4, 5], one can establish Theorem 1.4 for data in the
imhomogeneous Sobolev space $H_{x}^{1/2}$. Using Theorem 3.6, one can then
remove the assumption $u_{0}\in L_{x}^{2}$ _a posteriori_ (by approximating
$u_{0}\in\dot{H}_{x}^{1/2}$ by ${H}_{x}^{1/2}$\- functions). We omit the
standard details.
## 4\. Reduction to almost periodic solutions
In this section, we sketch a proof of Theorem 1.7. As described in the
introduction, the key ideas come from [11, 12] and are well-known. Thus, we
will merely outline the argument, providing full details only when our
approach deviates from the usual one. We model our presentation primarily
after [15, Section 3]. Throughout this section, we denote the nonlinearity
$|u|^{\frac{4}{d-1}}u$ by $F(u)$.
We suppose that Theorem 1.3 fails. We then define $L:[0,\infty)\to[0,\infty]$
by
$L(E):=\sup\big{\\{}S_{I}(u)\ \big{|}\
u:I\times\mathbb{R}^{d}\to\mathbb{C}\text{ solving }\eqref{nls}\text{ with
}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}(I\times\mathbb{R}^{d})}^{2}\leq
E\big{\\}}.$
We note that $L$ is non-decreasing, with $L(E)\lesssim E^{\frac{d+2}{d-1}}$
for $E$ sufficiently small (cf. Theorem 1.4). Thus, there exists a unique
critical threshold $E_{c}\in(0,\infty]$ such that $L(E)<\infty$ for $E<E_{c}$
and $L(E)=\infty$ for $E>E_{c}$. The failure of Theorem 1.3 implies that
$0<E_{c}<\infty.$
The key to proving Theorem 1.7 is the following convergence result. With this
result in hand, establishing Theorem 1.7 is a straightforward exercise (see
[15, Section 3.2]).
###### Proposition 4.1 (Palais–Smale condition modulo symmetries).
Let $d\geq 5$ and let $u_{n}:I_{n}\times\mathbb{R}^{d}\to\mathbb{C}$ be a
sequence of solutions to (1.1) such that
$\limsup_{n\to\infty}\|u_{n}\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}(I_{n}\times\mathbb{R}^{d})}^{2}=E_{c}.$
Suppose $t_{n}\in I_{n}$ are such that
$\lim_{n\to\infty}S_{[t_{n},\sup I_{n})}(u_{n})=\lim_{n\to\infty}S_{(\inf
I_{n},t_{n}]}(u_{n})=\infty.$ (4.1)
Then, $\\{u_{n}(t_{n})\\}$ converges along a subsequence in
$\dot{H}_{x}^{1/2}(\mathbb{R}^{d})/G$.
###### Proof.
We first translate so that each $t_{n}=0$ and apply Lemma 2.6 to write
$u_{n}(0)=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+w_{n}^{J}$ (4.2)
along some subsequence. Recall that for each $j$, either $t_{n}^{j}\equiv 0$
or $t_{n}^{j}\to\pm\infty$. To prove Proposition 4.1, we need to show that
there is exactly one profile $\phi^{1}$, with $t_{n}^{1}\equiv 0$ and
$\|w_{n}^{1}\|_{\dot{H}_{x}^{1/2}}\to 0$.
First, using Theorem 1.4, for each $j$ we define
$v^{j}:I^{j}\times\mathbb{R}^{d}\to\mathbb{C}$ to be the maximal-lifespan
solution to (1.1) such that
$\left\\{\begin{array}[]{ll}v^{j}(0)=\phi^{j}&\text{if }t_{n}^{j}\equiv 0,\\\
v^{j}\text{ scatters to }\phi^{j}\text{ as }t\to\pm\infty&\text{if
}t_{n}^{j}\to\pm\infty.\end{array}\right.$
Next, we define nonlinear profiles
$v_{n}^{j}:I_{n}^{j}\times\mathbb{R}^{d}\to\mathbb{C}$ by
$v_{n}^{j}(t)=g_{n}^{j}v^{j}\big{(}(\lambda_{n}^{j})^{-2}t+t_{n}^{j}\big{)},\quad\text{where}\quad
I_{n}^{j}=\\{t:(\lambda_{n}^{j})^{-2}t+t_{n}^{j}\in I^{j}\\}.$
To complete the proof, we need the following three claims:
(i) There is at least one ‘bad’ profile $\phi^{j}$, in the sense that
$\limsup_{n\to\infty}S_{[0,\sup I_{n}^{j})}(v_{n}^{j})=\infty.$ (4.3)
(ii) There can then be at _most_ one profile (which we label $\phi^{1}$), and
$\|w_{n}^{1}\|_{\dot{H}_{x}^{1/2}}\to 0$.
(iii) We have $t_{n}^{1}\equiv 0$.
We will provide a proof of (i) below. The proofs of (ii) and (iii) require
only small variations of the analysis given for (i), so we will merely outline
the arguments here. For (ii), one can adapt the argument of [15, Lemma 3.3] to
show that the decoupling (2.1) persists in time (this is not obvious, as the
$\dot{H}_{x}^{1/2}$-norm is not a conserved quantity for (1.1)). The critical
nature of $E_{c}$ may then be used to preclude the possibility of multiple
profiles (and to show $\|w_{n}^{1}\|_{\dot{H}_{x}^{1/2}}\to 0$). For (iii), we
only need to rule out the cases $t_{n}^{1}\to\pm\infty.$ To do this, one can
argue by contradiction: if $t_{n}^{1}\to\pm\infty$, one can use the stability
result Theorem 3.6 (comparing $u_{n}$ to $e^{it\Delta}u_{n}(0)$) to contradict
(4.1). See [15, p. 391] for more details.
We now turn to the proof of (i). We first note that the decoupling (2.1)
implies that the $v_{n}^{j}$ are global and scatter for $j$ sufficiently
large, say for $j\geq J_{0}$; indeed, for $j$ sufficiently large, the
$\dot{H}_{x}^{1/2}$-norm of $\phi^{j}$ must be below the small-data threshold
described in Theorem 1.4. Thus, we need to show that there is at least one bad
profile $\phi^{j}$ (in the sense of (4.3)) in the range $1\leq j<J_{0}$.
Suppose towards a contradiction that there are no bad profiles. By the blowup
criterion of Theorem 1.4, this immediately implies that $\sup
I_{n}^{j}=\infty$ for all $j$ and for all $n$ sufficiently large. In fact, we
claim that we have the following:
$\limsup_{J\to\infty}\limsup_{n\to\infty}\sum_{j=1}^{J}\|v_{n}^{j}\|_{\dot{S}^{1/2}([0,\infty))}^{2}\lesssim_{E_{c}}1.$
(4.4)
Indeed, for $\eta>0$, the decoupling (2.1) implies the existence of
$J_{1}=J_{1}(\eta)$ such that
$\sum_{j>J_{1}}\|\phi^{j}\|_{\dot{H}_{x}^{1/2}}^{2}\lesssim\eta.$
Thus, choosing $\eta$ smaller than the small-data threshold, Strichartz and a
standard bootstrap argument give
$\sum_{j>J_{1}}\|v_{n}^{j}\|_{\dot{S}^{1/2}([0,\infty))}^{2}\lesssim\sum_{j>J_{1}}\|\phi^{j}\|_{\dot{H}_{x}^{1/2}}^{2}\lesssim\eta.$
As the $v_{n}^{j}$ satisfy $S_{[0,\infty)}(v_{n}^{j})\lesssim 1$ for $n$
large, we may use Strichartz and another bootstrap argument to see
$\|v_{n}^{j}\|_{\dot{S}^{1/2}}\lesssim 1$ for $1\leq j\leq J_{1}$ and $n$
large. Thus, we conclude that (4.4) holds.
Next, using the fact that there are no bad profiles, together with the
orthogonality condition (2.2), one can use the arguments of [11] to arrive at
the following
###### Lemma 4.2 (Orthogonality).
For $j\neq k$, we have
$\displaystyle\bigg{[}\|v_{n}^{j}v_{n}^{k}$
$\displaystyle\|_{L_{t,x}^{\frac{d+2}{d-1}}}+\|v_{n}^{j}v_{n}^{k}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2d(d+1)}{2d^{2}-d-5}}}+\|(|\nabla|^{1/2}v_{n}^{j})(|\nabla|^{1/2}v_{n}^{k})\|_{L_{t,x}^{\frac{d+2}{d}}}$
$\displaystyle+\|\big{(}|\nabla|^{1/2}F(v_{n}^{j})\big{)}\big{(}|\nabla|^{1/2}F(v_{n}^{k})\big{)}^{\frac{d}{d+4}}\|_{L_{t,x}^{1}}\bigg{]}\to
0\quad\text{as}\quad n\to\infty,$ (4.5)
where all spacetime norms are taken over $[0,\infty)\times\mathbb{R}^{d}$.
We now wish to use (4.4) and (4.5), together with Theorem 3.6, to deduce a
bound on the scattering size of the $u_{n}$, thus contradicting (4.1). To this
end, we define approximate solutions to (1.1) and collect the information we
need about them in the following
###### Lemma 4.3.
The approximate solutions
$u_{n}^{J}(t):=\sum_{j=1}^{J}v_{n}^{j}(t)+e^{it\Delta}w_{n}^{J}$ satisfy
$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}\|u_{n}(0)-u_{n}^{J}(0)\|_{\dot{H}_{x}^{1/2}}=0,$
(4.6)
$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}S_{[0,\infty)}(u_{n}^{J})\lesssim_{E_{c}}1,$
(4.7)
$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}\|u_{n}^{J}\|_{\dot{S}^{1/2}([0,\infty))}\lesssim_{E_{c}}1.$
(4.8)
The errors
$e_{n}^{J}:=(i\partial_{t}+\Delta)u_{n}^{J}-F(u_{n}^{J})=\sum_{j=1}^{J}F(v_{n}^{j})-F(u_{n}^{J})$
satisfy
$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}\||\nabla|^{1/2}e_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}([0,\infty)\times\mathbb{R}^{d})}\lesssim_{E_{c}}1,$
(4.9)
$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}\|e_{n}^{J}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}([0,\infty)\times\mathbb{R}^{d})}=0.$
(4.10)
###### Remark 4.4.
It is here that we see the benefit of the refined stability result Theorem
3.6. In particular, to apply Theorem 3.6 we only need to exhibit smallness of
the $e_{n}^{J}$ in the space appearing in (4.10). As this space contains no
derivatives, we can achieve this simply by relying on pointwise estimates.
###### Proof.
We first note that (4.6) follows from the construction of the $v^{j}$.
Next, we turn to (4.7). To begin, we notice that by Sobolev embedding and the
fact that $\frac{2(d+2)}{d-1}\geq 2$, we may deduce from (4.4) that
$\sum_{j\geq 1}S_{[0,\infty)}(v_{n}^{j})\lesssim_{E_{c}}1.$
Thus, recalling (2.3), we see that to prove (4.7) it will suffice to show
$\limsup_{J\to\infty}\limsup_{n\to\infty}\bigg{|}S_{[0,\infty)}\bigg{(}\sum_{j=1}^{J}v_{n}^{j}\bigg{)}-\sum_{j=1}^{J}S_{[0,\infty)}(v_{n}^{j})\bigg{|}=0.$
(4.11)
To this end, we fix $J$ and use Hölder’s inequality, (4.4), and (4.5) to see
$\displaystyle\bigg{|}S_{[0,\infty)}\bigg{(}\sum_{j=1}^{J}v_{n}^{j}\bigg{)}-\sum_{j=1}^{J}S_{[0,\infty)}(v_{n}^{j})\bigg{|}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\|v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}}^{\frac{6}{d-1}}\|v_{n}^{j}v_{n}^{k}\|_{L_{t,x}^{\frac{d+2}{d-1}}}\to
0$
as $n\to\infty$. This establishes (4.11) and completes the proof of (4.7).
Let us next turn to (4.9) (which we will later use in the proof of (4.8)). To
begin, we will derive the bound
$\limsup_{J\to\infty}\limsup_{n\to\infty}\||\nabla|^{1/2}u_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}\lesssim_{E_{c}}1.$
(4.12)
As $w_{n}^{J}\in\dot{H}_{x}^{1/2},$ it will suffice to show
$\limsup_{J\to\infty}\limsup_{n\to\infty}\|\sum_{j=1}^{J}|\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}\lesssim_{E_{c}}1.$
(4.13)
To this end, we first note that as $\frac{2(d+2)}{d}\geq 2$, we may use (4.4)
to see
$\limsup_{J\to\infty}\limsup_{n\to\infty}\sum_{j=1}^{J}\||\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}\lesssim_{E_{c}}1.$
(4.14)
On the other hand, for fixed $J$, we can use (4.4) and (4.5) to see
$\displaystyle\bigg{|}\|$
$\displaystyle\sum_{j=1}^{J}|\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}-\sum_{j=1}^{J}\||\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}\bigg{|}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\||\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{4}{d}}\|(|\nabla|^{1/2}v_{n}^{j})(|\nabla|^{1/2}v_{n}^{k})\|_{L_{t,x}^{\frac{d+2}{d}}}\to
0\quad\text{as}\quad n\to\infty.$
Then (4.14) implies (4.13), which in turn gives (4.12).
Next, by the fractional chain rule, (4.7), and (4.12), we get
$\displaystyle\||\nabla|^{1/2}F(u_{n}^{J})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}\lesssim\|u_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}}^{\frac{4}{d-1}}\||\nabla|^{1/2}u_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}\lesssim_{E_{c}}1$
(4.15)
as $n,J\to\infty$, which handles one of the terms appearing in (4.9).
To complete the proof of (4.9), it remains to show
$\limsup_{J\to\infty}\limsup_{n\to\infty}\|\sum_{j=1}^{J}|\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}\lesssim_{E_{c}}1.$
We claim it will suffice to establish
$\lim_{J\to\infty}\limsup_{n\to\infty}\sum_{j=1}^{J}\||\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}\lesssim_{E_{c}}1.$
(4.16)
Indeed, for fixed $J$, we have by (4.5)
$\displaystyle\bigg{|}\|\sum_{j=1}^{J}$
$\displaystyle|\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}-\sum_{j=1}^{J}\||\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}\bigg{|}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\||\nabla|^{1/2}F(v_{n}^{j})\,||\nabla|^{1/2}F(v_{n}^{k})|^{\frac{d}{d+4}}\|_{L_{t,x}^{1}}\to
0\quad\text{as }n\to\infty.$
To establish (4.16) and thereby complete the proof of (4.9), we use the
fractional chain rule and Sobolev embedding to see
$\displaystyle\sum_{j=1}^{J}\||\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}$
$\displaystyle\lesssim\sum_{j=1}^{J}\big{(}\|v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}}^{\frac{4}{d-1}}\||\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{)}^{\frac{2(d+2)}{d+4}}$
$\displaystyle\lesssim\sum_{j=1}^{J}\|v_{n}^{j}\|_{\dot{S}^{1/2}}^{\frac{2(d+2)(d+3)}{(d+4)(d-1)}}.$
Then (4.16) follows from (4.4) and the fact that
$\frac{2(d+2)(d+3)}{(d+4)(d-1)}\geq 2$.
Now (4.8) follows from an application of Strichartz, (4.9) and (4.15).
It remains to establish (4.10). We begin by rewriting
$e_{n}^{J}=\bigg{[}\sum_{j=1}^{J}F(v_{n}^{j})-F\bigg{(}\sum_{j=1}^{J}v_{n}^{j}\bigg{)}\bigg{]}+\big{[}F(u_{n}^{J}-e^{it\Delta}w_{n}^{J})-F(u_{n}^{J})\big{]}=:(e_{n}^{J})_{1}+(e_{n}^{J})_{2}.$
We first fix $J$ and use Hölder, Sobolev embedding, (4.4), and (4.5) to see
$\displaystyle\|(e_{n}^{J})_{1}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\|\,|v_{n}^{j}v_{n}^{k}|^{\frac{4}{d-1}}|v_{n}^{j}|^{\frac{d-5}{d-1}}\,\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\|v_{n}^{j}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2d(d+1)}{d^{2}-d-4}}}^{\frac{d-5}{d-1}}\|v_{n}^{j}v_{n}^{k}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2d(d+1)}{2d^{2}-d-5}}}^{\frac{4}{d-1}}\to
0$
as $n\to\infty$. Next, we note that we have the pointwise estimate
$|(e_{n}^{J})_{2}|\lesssim|e^{it\Delta}w_{n}^{J}||f_{n}^{J}|^{\frac{4}{d-1}},$
where $f_{n}^{J}:=u_{n}^{J}+e^{it\Delta}w_{n}^{J}$ satisfies
$\|f_{n}^{J}\|_{\dot{S}^{1/2}}\lesssim_{E_{c}}1$ as $n,J\to\infty$ (cf. (4.8)
and the fact that $w_{n}^{J}\in\dot{H}_{x}^{1/2}$). Thus, we can use Hölder,
Strichartz, Sobolev embedding, $w_{n}^{J}\in\dot{H}_{x}^{1/2}$, and (2.3) to
see
$\displaystyle\|(e_{n}^{J})_{2}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}}$
$\displaystyle\lesssim\|e^{it\Delta}w_{n}^{J}\|_{L_{t}^{\frac{4(d+1)}{d-1}}L_{x}^{\frac{2(d+1)}{d-1}}}\|f_{n}^{J}\|_{L_{t}^{\frac{4(d+1)}{d-1}}L_{x}^{\frac{2(d+1)}{d-1}}}^{\frac{4}{d-1}}$
$\displaystyle\lesssim\|e^{it\Delta}w_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}}^{\frac{d+2}{2(d+1)}}\|w_{n}^{J}\|_{\dot{H}_{x}^{1/2}}^{\frac{d}{2(d+1)}}\|f_{n}^{J}\|_{\dot{S}^{1/2}}^{\frac{4}{d-1}}\to
0\quad\text{as }n,J\to\infty.$
Combining the estimates for $(e_{n}^{J})_{1}$ and $(e_{n}^{J})_{2}$, we
conclude that (4.10) holds. ∎
Using Lemma 4.3, we may apply Theorem 3.6 to deduce that
$S_{[0,\infty)}(u_{n})\lesssim_{E_{c}}1$ for $n$ large, contradicting (4.1).
We conclude that there is at least one bad profile, that is, claim (i) holds.
This completes the proof of Proposition 4.1 and Theorem 1.7. ∎
## 5\. Finite time blowup
In this section, we use Proposition 1.10, Strichartz estimates, and
conservation of mass to preclude the existence of almost periodic solutions as
in Theorem 1.8 with $T_{max}<\infty$.
###### Theorem 5.1 (No finite time blowup).
Let $d\geq 5$. There are no almost periodic solutions
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) with
$T_{max}<\infty$ and $S_{[0,T_{max})}=\infty.$
###### Proof.
Suppose that $u$ were such a solution. Then, for $t\in[0,T_{max})$ and $N>0$,
Proposition 1.10, Strichartz, Hölder, Bernstein, and Sobolev embedding give
$\displaystyle\|P_{N}u(t)\|_{L_{x}^{2}}$
$\displaystyle\lesssim\|P_{N}(|u|^{\frac{4}{d-1}}u)\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}([t,T_{max})\times\mathbb{R}^{d})}$
$\displaystyle\lesssim(T_{max}-t)^{1/2}N^{1/2}\||u|^{\frac{4}{d-1}}u\|_{L_{t}^{\infty}L_{x}^{\frac{2d}{d+3}}}$
$\displaystyle\lesssim(T_{max}-t)^{1/2}N^{1/2}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}}^{\frac{d+3}{d-1}}.$
As $u\in L_{t}^{\infty}\dot{H}_{x}^{1/2}$, we deduce
$\displaystyle\|P_{\leq
N}u(t)\|_{L_{x}^{2}}\lesssim_{u}(T_{max}-t)^{1/2}N^{1/2}\quad\text{for all
}t\in I\text{ and }N>0.$ (5.1)
On the other hand, an application of Bernstein gives
$\displaystyle\|P_{>N}u\|_{L_{t}^{\infty}L_{x}^{2}}\lesssim
N^{-1/2}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}}\lesssim_{u}N^{-1/2}\quad\text{for
all }N>0.$ (5.2)
We now let $\eta>0$. We choose $N$ large enough that $N^{-1/2}<\eta$, and
subsequently choose $t$ close enough to $T_{max}$ that
$(T_{max}-t)^{1/2}N^{1/2}<\eta.$ Combining (5.1) and (5.2), we then get
$\|u(t)\|_{L_{x}^{2}}\lesssim_{u}\eta.$
As $\eta$ was arbitrary and mass is conserved, we conclude
$\|u(t)\|_{L_{x}^{2}}=0$ for all $t\in[0,T_{max})$. Thus $u\equiv 0$, which
contradicts the fact that $u$ blows up. ∎
## 6\. The Lin–Strauss Morawetz inequality
In this section, we use the Lin–Strauss Morawetz inequality to preclude the
existence of almost periodic solutions as in Theorem 1.8 such that
$T_{max}=\infty.$
###### Proposition 6.1 (Lin–Strauss Morawetz inequality, [18]).
Let $d\geq 3$ and let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a solution to
$(i\partial_{t}+\Delta)u=|u|^{p}u$. Then
$\int_{I}\int_{\mathbb{R}^{d}}\frac{|u(t,x)|^{p+2}}{|x|}\,dx\,dt\lesssim\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}(I\times\mathbb{R}^{d})}^{2}.$
(6.1)
As in [10], we will use this estimate to establish the following
###### Theorem 6.2.
Let $d\geq 5$. There are no almost periodic solutions
$u:[0,\infty)\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) such that $u$ blows
up forward in time, $\inf_{t\in[0,\infty)}N(t)\geq 1$, and
$|x(t)|\lesssim_{u}\int_{0}^{t}N(s)\,ds$ for all $t\geq 0$.
###### Proof.
Suppose $u$ were such a solution. In particular $u$ is nonzero, so that by
Remark 1.6 we may find $C(u)>0$ such that
$\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{\frac{2d}{d-1}}\,dx\gtrsim_{u}1\quad\text{uniformly
for }t\in[0,\infty).$
Applying Hölder and rearranging, this implies
$\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{\frac{2(d+1)}{d-1}}\,dx\gtrsim_{u}N(t)\quad\text{uniformly
for }t\in[0,\infty).$ (6.2)
We now let $T>1$ and use $u\in L_{t}^{\infty}\dot{H}_{x}^{1/2}$, (6.1), and
(6.2) to see
$\displaystyle
1\gtrsim_{u}\int_{1}^{T}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}\frac{|u(t,x)|^{\frac{2(d+1)}{d-1}}}{|x|}\,dx\,dt\gtrsim_{u}\int_{1}^{T}\frac{N(t)}{|x(t)|+N(t)^{-1}}\,dt.$
As $\inf_{t\in[1,\infty)}N(t)\geq 1$, to derive a contradiction it will
suffice to show that
$\lim_{T\to\infty}\int_{1}^{T}\frac{N(t)}{1+|x(t)|}\,dt=\infty.$ (6.3)
Recalling that $|x(t)|\lesssim_{u}\int_{0}^{t}N(s)\,ds$ for all $t\geq 0$, we
get
$\displaystyle\int_{1}^{T}\frac{N(t)}{1+|x(t)|}\,dt\gtrsim_{u}\int_{1}^{T}\frac{d}{dt}\log\bigg{(}1+\int_{0}^{t}N(s)\,ds\bigg{)}\,dt\gtrsim_{u}\log\bigg{(}\frac{1+\int_{0}^{T}N(s)\,ds}{1+\int_{0}^{1}N(s)\,ds}\bigg{)}.$
As $\inf_{t\in[1,\infty)}N(t)\geq 1$, we conclude that (6.3) holds, as needed.
∎
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|
arxiv-papers
| 2012-12-31T04:33:12 |
2024-09-04T02:49:39.812823
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jason Murphy",
"submitter": "Jason Murphy",
"url": "https://arxiv.org/abs/1212.6816"
}
|
1212.6831
|
11institutetext: School of Computer Science and Engineering, University of
Electronic Science and Technology of China, China, 11email: [email protected]
22institutetext: Department of Applied Mathematics and Physics, Graduate
School of Informatics, Kyoto University, Japan, 22email:
[email protected]
# An Exact Algorithm for TSP in Degree-$3$ Graphs via Circuit Procedure and
Amortization on Connectivity Structure ††thanks: Supported in part by Grant
60903007 of NSFC, China.
Mingyu Xiao 11 Hiroshi Nagamochi 22
###### Abstract
The paper presents an $O^{*}(1.2312^{n})$-time and polynomial-space algorithm
for the traveling salesman problem in an $n$-vertex graph with maximum degree
$3$. This improves the previous time bounds of $O^{*}(1.251^{n})$ by Iwama and
Nakashima and $O^{*}(1.260^{n})$ by Eppstein. Our algorithm is a simple
branch-and-search algorithm. The only branch rule is designed on a cut-circuit
structure of a graph induced by unprocessed edges. To improve a time bound by
a simple analysis on measure and conquer, we introduce an amortization scheme
over the cut-circuit structure by defining the measure of an instance to be
the sum of not only weights of vertices but also weights of connected
components of the induced graph.
Key words. Traveling Salesman Problem, Exact Exponential Algorithm, Graph
Algorithm, Connectivity, Measure and Conquer
## 1 Introduction
The traveling salesman problem (TSP) is one of the most famous and intensively
studied problems in computational mathematics. Many algorithmic methods have
been investigated to beat this challenge of finding the shortest route
visiting each member of a collection of $n$ locations and returning to the
starting point. The first $O^{*}(2^{n})$-time dynamic programming algorithm
for TSP is back to early 1960s. However, in the last half of a century no one
can break the barrier of $2$ in the base of the running time. To make steps
toward the long-standing and major open problem in exact exponential
algorithms, TSP in special classes of graphs, especially degree bounded
graphs, have also been intensively studied. Eppstein [5] showed that TSP in
degree-$3$ graphs (a graph with maximum degree $i$ is called a degree-$i$
graph) can be solved in $O^{*}(1.260^{n})$ time and polynomial space, and TSP
in degree-$4$ graphs can be solved in $O^{*}(1.890^{n})$ time and polynomial
space. Iwama and Nakashima [9] refined Eppstein’s algorithm for degree-$3$
graphs and improved the result to $O^{*}(1.251^{n})$ by showing that the worst
case in Eppstein’s algorithm will not always happen. Gebauer [8] designed an
$O^{*}(1.733^{n})$-time exponential-space algorithm for TSP in degree-$4$
graphs, which is improved to $O^{*}(1.716^{n})$ time and polynomial space by
Xiao and Nagamochi [13]. Bjorklund _et al._ [2] also showed TSP in degree
bounded graph can be solved in $O^{*}((2-\varepsilon)^{n})$ time, where
$\varepsilon>0$ depends on the degree bound only. There is a Monte Carlo
algorithm to decide a graph is Hamiltonian or not in $O^{*}(1.657^{n})$ time
[1]. For planar TSP and Euclidean TSP, there are sub-exponential algorithms
based on small separators [3].
In this paper, we present an improved deterministic algorithm for TSP in
degree-$3$ graphs, which runs in
$O^{*}(2^{{{3}\over{10}}n})=O^{*}(1.2312^{n})$ time and polynomial space. The
algorithm is simple and contains only one branch rule that is designed on a
cut-circuit structure of a graph induced by unprocessed edges. We will apply
the measure and conquer method to analyze the running time. Note that our
algorithm for TSP in degree-4 graphs in [13] is obtained by successfully
applying the measure and conquer method to TSP for the first time. However,
direct application of measure and conquer to TSP in degree-3 graphs may only
lead to an $O^{*}(1.260^{n})$-time algorithm. To effectively analyze our
algorithm, we use an amortization scheme over the cut-circuit structures by
setting weights to both vertices and connected components of the induced
graph.
## 2 Preliminaries
In this paper, a graph means an undirected edge-weighted graph with maximum
degree 3, which possibly has multiple edges, but no self-loops. Let $G=(V,E)$
be a graph with an edge weight. For a subset $V^{\prime}\subseteq V$ of
vertices and a subset $E^{\prime}\subseteq E$ of edges, the subgraphs induced
by $V^{\prime}$ and $E^{\prime}$ are denoted by $G[V^{\prime}]$ and
$G[E^{\prime}]$ respectively. We also use $\mathrm{cost}(E^{\prime})$ to
denote the total weight of edges in $E^{\prime}$. For any graph $G^{\prime}$,
the sets of vertices and edges in $G^{\prime}$ are denoted as $V(G^{\prime})$
and $E(G^{\prime})$ respectively. A graph consisting of a single vertex is
called trivial. A _cycle_ of length $l$ (also denoted as $l$-cycle) is a graph
with $l$ vertices $v_{i}$ and $l$ edges $v_{i}v_{i+1}$ ($i\in\\{1,2\dots,l\\}$
and $v_{l+1}=v_{1}$). An edge $v_{i}v_{j}$ ($|i-j|\geq 2$) between two
vertices in the cycle but different from the $l$ edges in it is called a chord
of the cycle. Two vertices in a graph are $k$-edge-connected if there are
$k$-edge-disjoint paths between them. A graph is $k$-edge-connected if every
pair of vertices in it are $k$-edge-connected. We treat a trivial graph as a
$k$-edge-connected graph for any $k\geq 1$. A Hamiltonian cycle is a cycle
through every vertex. Given a graph with an edge weight, the _traveling
salesman problem_ (TSP) is to find a Hamiltonian cycle of minimum total weight
in the edges.
### 2.1 Forced TSP
In some branch-and-search algorithms for TSP, we may branch on an edge in the
graph by including it to the solution or excluding it from the solution. In
this way, we need to maintain a set of edges that must be used in the
solution. We introduce the _forced traveling salesman problem_ as follows. An
instance is a pair $(G,F)$ of an edge-weighted undirected graph $G=(V,E)$ and
a subset $F\subseteq E$ of edges, called forced edges. A Hamiltonian cycle of
$G$ is called a tour if it passes though all the forced edges in $F$. The
objective of the problem is to compute a tour of minimum weight in the given
instance $(G,F)$. An instance is called infeasible if no tour exists. A vertex
is called _forced_ if there is a forced edge incident on it. For convenience,
we say that the _sign_ of an edge $e$ is 1 if $e$ is a forced edge and $0$ if
$e$ is an unforced edge. We use $\mathrm{sign}(e)$ to denote the sign of $e$.
### 2.2 $U$-graphs and $U$-components
We consider an instance $(G,F)$. Let $U=E(G)-F$ denote the set of unforced
edges. A subgraph $H$ of $G$ is called a _$U$ -graph_ if $H$ is a trivial
graph or $H$ is induced by a subset $U^{\prime}\subseteq U$ of unforced edges
(i.e., $H=G[U^{\prime}]$). A maximal connected $U$-graph is called a _$U$
-component_. Note that each connected component in the graph $(V(G),U)$ is a
$U$-component.
For a vertex subset $X$ (or a subgraph $X$) of $G$, let $\mathrm{cut}(X)$
denote the set of edges in $E=F\cup U$ between $X$ and $V(G)-X$, and denote
$\mathrm{cut}_{F}(X)=\mathrm{cut}(X)\cap F$ and
$\mathrm{cut}_{U}(X)=\mathrm{cut}(X)\cap U$. Edge set $\mathrm{cut}(X)$ is
also called a _cut_ of the graph. We say that an edge is _incident_ on $X$ if
the edge is in $\mathrm{cut}(X)$. The degree $d(v)$ of a vertex $v$ is defined
to be $|\mathrm{cut}(\\{v\\})|$. We also denote
$d_{F}(v)=|\mathrm{cut}_{F}(\\{v\\})|$ and
$d_{U}(v)=|\mathrm{cut}_{U}(\\{v\\})|$. A $U$-graph $H$ is $k$-pendent if
$|\mathrm{cut}_{U}(H)|=k$. A $U$-graph $H$ is called even (resp., odd) if
$|\mathrm{cut}_{F}(H)|$ is even (resp., odd). A $U$-component is 0-pendent.
In this paper, we will always keep every $U$-component 2-edge-connected. For
simplicity, we may regard a maximal path of forced edges between two vertices
$u$ and $v$ as a single forced edge $uv$ in an instance $(G,F)$, since we can
assume that $d_{F}(v)=2$ always implies $d(v)=2$ for any vertex $v$.
### 2.3 Circuits and blocks
We consider a nontrivial $U$-component $H$ in an instance $(G,F)$. A circuit
${\cal C}$ in $H$ is a maximal sequence $e_{1},e_{2},\ldots,e_{p}$ of edges
$e_{i}=u_{i}v_{i}\in E(H)$ $(1\leq i\leq p)$ such that for each $e_{i}\in{\cal
C}$ $(i\neq p)$, the next edge $e_{i+1}\in{\cal C}$ is given by a subgraph
$B_{i}$ of $H$ such that $\mathrm{cut}_{U}(B_{i})=\\{e_{i},e_{i+1}\\}$. See
Fig. 1 for an illustration. We say that each subgraph $B_{i}$ is a _block_
along ${\cal C}$ and vertices $v_{i}$ and $u_{i+1}$ are the endpoints of block
$B_{i}$. By the maximality of ${\cal C}$, we know that any two vertices in
each block $B_{i}$ are 2-edge-connected in the induced subgraph $G[B_{i}]$. It
is possible that a circuit in a 2-edge-connected graph $H$ may contain only
one edge $e=u_{1}v_{1}$. For this case, vertices $u_{1}$ and $v_{1}$ are
connected by three edge-disjoint paths in $H$ and the circuit is called
_trivial_ , where the unique block is the $U$-component $H$. Each nontrivial
circuit contains at least two blocks, each of which is a 2-pendent subgraph of
$H$. In our algorithm, we will consider only nontrivial circuits ${\cal C}$.
When $H$ is 2-edge-connected, there are $p\geq 2$ different blocks along a
nontrivial circuit ${\cal C}$, where $u_{1}$ and $v_{p}$ are in the same block
$B_{p}$ and $\mathrm{cut}_{U}(B_{p})=\\{e_{p},e_{1}\\}$. A block $B_{i}$ is
called trivial if $|V(B_{i})|=1$ and $d_{F}(v)=1$ for the only vertex $v$ in
it ($v$ is of degree 3 in $G$). A block $B_{i}$ is called reducible if
$|V(B_{i})|=1$ and $d_{F}(v)=0$ for the only vertex $v$ in it ($v$ is of
degree 2 in $G$). A block $B_{i}$ with $V(B_{i})=\\{v_{i}=u_{i+1}\\}$ is
either trivial or reducible in a 2-edge-connected graph.
Figure 1: A circuit in a 2-edge-connected graph $H$
We state more properties on circuits and blocks.
###### Lemma 1
In a degree-$3$ graph, let $H$ be a 2-edge-connected $U$-component and ${\cal
C}$ be any circuit in it. For each block $B_{i}$ of ${\cal C}$, $B_{i}$ is not
trivial or reducible if and only if the two endpoints $v_{i}$ and $u_{i+1}$ of
it are two different vertices of degree $3$ in $H$.
###### Lemma 2
Each edge in a $2$-edge-connected $U$-component $H$ of a degree-$3$ graph is
contained in exactly one circuit. A partition of $E(H)$ into circuits can be
obtained in linear time.
_Proof._ It is known that the set of all minimum cuts (a set of $k$ edges in a
$k$-edge-connected multigraph is called a minimum cut if the graph becomes
disconnected by removing the $k$ edges) can be represented by a cactus
structure (cf. [11]). In particular, when the size of a minimum cut is two,
the cactus structure of minimum cuts can be obtained in linear time by
contracting each 3-edge-connected component (a maximal set of vertices every
two of which are 3-connected in the given graph) into a single vertex, and for
each cycle $C$ in the resulting graph, a pair of any two edges in $C$
corresponds to a minimum cut in the original graph [10]. In a 2-edge-connected
$U$-component $H$, (i) an edge $e\in E(H)$ forms a circuit ${\cal C}$ having
only one block if and only if $e$ is not in any minimum cut of $H$; and (ii) A
circuit ${\cal C}$ with at least two edges in $H$ corresponds to a cycle $C$
in the cactus structure. Based on the cactus structure, we can obtain a
partition of edge sets into circuits in linear time.
### 2.4 Branch-and-search algorithms
Our algorithm is a branch-and-search algorithm: we search the solution by
iteratively branching on the current instance into several smaller instances
until the current instance becomes trivial (or polynomially solvable). In this
paradigm, we will get a search tree. In each leaf of the search tree, we can
solve the problem directly. The size of the search tree is the exponential
part of the running time of the search algorithm. Let $\mu$ be a measure of
the instance (for graph problems, the measure can be the number of vertices or
edges in the graph and so on). Let $C(\mu)$ denote the maximum number of
leaves in the search tree generated by the algorithm for any instance with
measure $\mu$. We shall determine an upper bound on $C(\mu)$ by evaluating all
the branches. When we branch on an instance $(G,F)$ with $k$ branches such
that the $i$-th branch decreases the measure $\mu$ of $(G,F)$ by at least
$a_{i}$, we obtain the following recurrence
$C(\mu)\leq C(\mu-a_{1})+C(\mu-a_{2})+\cdots+C(\mu-a_{k}).$
Solving this recurrence, we get
$C(\mu)=[\alpha(a_{1},a_{2},\ldots,a_{k})]^{\mu}$, where
$\alpha(a_{1},a_{2},\ldots,a_{k})$ is the largest root of the function
$f(x)=1-\sum_{i=1}^{k}x^{-a_{i}}$. In this paper, we represent the above
recurrence by a vector $(a_{1};a_{2};\cdots;a_{k})$ of measure decreases,
called a branch vector (cf. [7]). In particular, when
$a_{i}=a_{i+1}=\cdots=a_{j}$ for some $i\leq j$, it may be written as
$(a_{1};a_{2};\cdots a_{i-1};[a_{i}]_{j-i+1};a_{j+1};\cdots;a_{k})$, and a
vector $([a]_{k})$ is simply written as $[a]_{k}$. When we compare two branch
vectors $\mathbf{b}=(a_{1};a_{2})$ $(a_{1}\leq a_{2})$ and
$\mathbf{b}^{\prime}=(a^{\prime}_{1};a^{\prime}_{2})$ such that “$a_{i}\leq
a^{\prime}_{i}$ $(i=1,2)$” or “$a^{\prime}_{1}=a_{1}-\varepsilon$ and
$a^{\prime}_{2}=a_{2}+\varepsilon$ for some $0\leq\varepsilon\leq
a_{2}-a_{1}$,” we only consider branch vector $\mathbf{b}$ in analysis, since
a solution $\alpha$ from $\mathbf{b}$ is not smaller than that from
$\mathbf{b}^{\prime}$ (cf. [7]). We say that $\mathbf{b}$ covers
$\mathbf{b}^{\prime}$ in this case.
## 3 Reductions based on small cuts
For some special cases, we can reduce the instance directly without branching.
Most of out reduction rules are based on the structures of small cuts in the
graph. In fact, we will deal with cuts of size $1,2,3$ and $4$.
### 3.1 Sufficient conditions for infeasibility
The parity condition on an instance is: (i) every $U$-component is even; and
(ii) the number of odd blocks along every circuit is even.
###### Lemma 3
An instance $(G,F)$ is infeasible if $G$ is not 2-edge-connected or it
violates the parity condition.
_Proof._ Since any tour is a 2-edge-connected spanning graph of $G$, it cannot
exist when $G$ is not 2-edge-connected. Since any tour is an Eulerian graph,
it cannot exist in any instance with an odd $U$-component. For a circuit which
has an odd number of odd blocks, we see that at least one odd block will be an
odd $U$-component in any way of including/deleting edges in the circuit.
### 3.2 Eliminable, reducible and parallel edges
The unique unforced edge incident on a 1-pendent $U$-graph is _eliminable_.
From parity condition (i), we can decide whether each eliminable edge need to
be included to $F$ or deleted from the graph just by depending on the parity
of $|\mathrm{cut}_{F}(H)|$.
For any subgraph $H$ of $G$ with $|\mathrm{cut}(H)|=2$, we call the unforced
edges in $\mathrm{cut}(H)$ reducible. From the connectivity condition and
parity condition (i), we see that all reducible edges need to be included to
$F$. In particular, any edge $uv$ incident to a vertex $v$ with $d(v)=2$ (or
with a neighbor $v^{\prime}$ with multiple edges of $vv^{\prime}\in F$ and
$vv^{\prime}\in U$) is reducible, since $uv\in\mathrm{cut}(X)$ and
$|\mathrm{cut}(X)|=2$ for $X=\\{v\\}$ (or $X=\\{v,v^{\prime}\\}$).
If there are multiple edges with the same endpoints $u$ and $v$, we can reduce
the instance in the following way preserving the optimality: If the graph has
only two vertices $u$ and $v$, solve the problem directly; else if there are
forced edges between $u$ and $v$, the problem is infeasible; and otherwise
remove all unforced edges between $u$ and $v$ except one with the smallest
weight.
### 3.3 Reductions based on 3-cuts and 4-cuts
###### Lemma 4
Let $(G,F)$ be an instance where $G$ is a graph with maximum degree 3. For any
subgraph $X$ with $|\mathrm{cut}(X)|=3$, we can replace $X$ with a single
vertex $x$ and update the three edges incident on $x$ preserving the
optimality of the instance.
_Proof._ Denote $\mathrm{cut}(X)$ by $\\{y_{1}x_{1},y_{2}x_{2},y_{3}x_{3}\\}$
with $x_{i}\in V(X)$ and $y_{i}\in V-V(X)$. We will replace $X$ and
$\mathrm{cut}(X)$ with a single vertex $x$ and three new edges $xy_{1},xy_{2}$
and $xy_{3}$. Let $G^{\prime}$ denote the new graph. We only need to decide
the weights and signs of edges $xy_{1},xy_{2}$ and $xy_{3}$ in $G^{\prime}$ to
satisfy the lemma. Let $I_{i}$ ($i=1,2,3$) denote the problem of finding a
path $P$ from $x_{i_{1}}$ to $x_{i_{2}}$ ($\\{i,i_{1},i_{2}\\}=\\{1,2,3\\}$)
of minimum total cost in $X$ that passes through all vertices and forced edges
in $X$. We say that $I_{i}$ _infeasible_ if it has no such path. We consider
the three problems $I_{i}$ ($i=1,2,3$). There are four possible cases. Case 1.
None of the three problems is feasible: We can see that the original instance
$(G,F)$ is also infeasible. In $G^{\prime}$, we let
$\mathrm{sign}(xy_{1})=\mathrm{sign}(xy_{2})=\mathrm{sign}(xy_{3})=1$. Since
the trivial $U$-component $\\{x\\}$ is odd, the new instance is infeasible by
Lemma 3. Case 2. Only one of the three problems, say $I_{j_{1}}$
($\\{j_{1},j_{2},j_{3}\\}=\\{1,2,3\\}$), is feasible: Let $S_{j_{1}}$ be an
optimal solution to $I_{j_{1}}$. Then there is a solution $S$ to $(G,F)$ such
that $S\cap E(X)=E(S_{j_{1}})$ (if $(G,F)$ is feasible). Therefore, in
$G^{\prime}$, we let $\mathrm{sign}(xy_{j_{2}})=\mathrm{sign}(xy_{j_{3}})=1$,
$\mathrm{sign}(xy_{j_{1}})=\mathrm{sign}(x_{j_{1}}y_{j_{1}})$,
$\mathrm{cost}(xy_{j_{2}})=\mathrm{cost}(x_{j_{2}}y_{j_{2}})+\mathrm{cost}(S_{j_{1}})$,
$\mathrm{cost}(xy_{j_{3}})=\mathrm{cost}(x_{j_{3}}y_{j_{3}})$ and
$\mathrm{cost}(xy_{j_{1}})=\mathrm{cost}(x_{j_{1}}y_{j_{1}})$. Case 3. Exactly
two of the three problems, say $I_{j_{1}}$ and $I_{j_{2}}$
($\\{j_{1},j_{2},j_{3}\\}=\\{1,2,3\\}$), are feasible: Let $S_{j_{1}}$ and
$S_{j_{2}}$ be an optimal solution to $I_{j_{1}}$ and $I_{j_{2}}$
respectively. Then there is a solution $S$ to $(G,F)$ such that either $S\cap
E(X)=E(S_{j_{1}})$ or $S\cap E(X)=E(S_{j_{2}})$. Therefore, in $G^{\prime}$,
we let $\mathrm{sign}(xy_{j_{3}})=1$,
$\mathrm{sign}(xy_{j_{1}})=\mathrm{sign}(x_{j_{1}}y_{j_{1}})$,
$\mathrm{sign}(xy_{j_{2}})=\mathrm{sign}(x_{j_{2}}y_{j_{2}})$,
$\mathrm{cost}(xy_{j_{3}})=\mathrm{cost}(x_{j_{3}}y_{j_{3}})$,
$\mathrm{cost}(xy_{j_{1}})=\mathrm{cost}(x_{j_{1}}y_{j_{1}})+\mathrm{cost}(S_{j_{2}})$
and
$\mathrm{cost}(xy_{j_{2}})=\mathrm{cost}(x_{j_{2}}y_{j_{2}})+\mathrm{cost}(S_{j_{1}})$.
Case 4. All of the three problems are feasible: Let $S_{1}$, $S_{2}$ and
$S_{3}$ be an optimal solution to $I_{1}$, $I_{2}$ and $I_{3}$ respectively.
In $G^{\prime}$, we let $\mathrm{sign}(xy_{i})=\mathrm{sign}(x_{i}y_{i})$ and
$\mathrm{cost}(xy_{i})=\mathrm{cost}(x_{i}y_{i})+{1\over
2}{\sum_{j=1}^{3}\mathrm{cost}(S_{j})}-\mathrm{cost}(S_{i})$ ($i=1,2,3$).
Straightforward computation can verify that with these setting $G^{\prime}$
will preserve the optimality.
Similar to Lemma 4, we can simplify a subgraph $X$ with $|\mathrm{cut}(X)|=4$.
However, there are many cases needed to consider. In fact, in our algorithms,
we only need to consider a special case.
We consider a subgraph $X$ with $|\mathrm{cut}_{F}(X)|=4$ and
$|\mathrm{cut}_{U}(X)|=0$. We want to reduce $X$. Denote $\mathrm{cut}(X)$ by
$\\{y_{1}x_{1},y_{2}x_{2},y_{3}x_{3},y_{4}x_{4}\\}$ with $x_{i}\in V(X)$ and
$y_{i}\in V-V(X)$, where $x_{i}\neq x_{j}$ $(1\leq i<j\leq 4)$. We define
$I_{i}$ ($i=1,2,3$) to be instances of the problem of finding two disjoint
paths $P$ and $P^{\prime}$ of minimum total cost in $X$ such that all vertices
and forced edges in $X$ appear in exactly one of the two paths, and one of the
two paths is from $x_{i}$ to $x_{4}$ and the other one is from $x_{j_{1}}$ to
$x_{j_{2}}$ ($\\{j_{1},j_{2}\\}=\\{1,2,3\\}-\\{i\\}$). We say that $I_{i}$
_infeasible_ if it has no solution.
A subgraph $X$ is _$4$ -cut reducible_ if $|\mathrm{cut}_{F}(X)|=4$,
$|\mathrm{cut}_{U}(X)|=0$, and at least one of the three problems
$I_{1},I_{2}$ and $I_{3}$ defined above is infeasible. We have the following
lemma to reduce the $4$-cut reducible subgraph.
###### Lemma 5
Let $(G,F)$ be an instance where $G$ is a graph with maximum degree 3. A
$4$-cut reducible subgraph $X$ can be replaced with one of the following
subgraphs $X^{\prime}$ with four vertices and
$|\mathrm{cut}_{F}(X^{\prime})|=4$ so that the optimality of the instance is
preserved:
(i) four single vertices $($i.e., there is no solution to this instance$)$;
(ii) a pair of forced edges; and
(iii) a 4-cycle with four unforced edges.
_Proof._ We consider the three problems $I_{i}$ ($i=1,2,3$). Since at least
one of them is infeasible, there are three possible cases. Case 1. None of the
three problems is feasible: We can see that the original instance $(G,F)$ is
also infeasible. Then we can replace $X$ with a graph containing only four
vertices $\\{x_{1},x_{2},x_{3},x_{4}\\}$ and no edge. Now $x_{1}$ becomes a
degree-1 vertex in the new graph and the new instance is infeasible. Case 2.
Only one of the three problems, say $I_{i_{0}}$ ($i_{0}\in\\{1,2,3\\}$), is
feasible: Let $S_{i_{0}}$ be an optimal solution to $I_{i_{0}}$. Then there is
a solution $S$ to $(G,F)$ such that $S\cap E(X)=E(S_{i_{0}})$. Therefore, we
can replace $X$ with a graph of four vertices $\\{x_{1},x_{2},x_{3},x_{4}\\}$
and two edges $x_{i}x_{4}$ and $x_{j_{1}}x_{j_{2}}$ in $G$ preserving the
optimality of the instance, where the costs of $x_{i_{0}}x_{4}$ and
$x_{j_{1}}x_{j_{2}}$ are the costs of the two paths in $S_{i_{0}}$. Note that
in the new instance after the replacement, the four vertices
$\\{x_{1},x_{2},x_{3},x_{4}\\}$ become degree-2 vertices and the two new edges
$x_{i}x_{4}$ and $x_{j_{1}}x_{j_{2}}$ should be included into $F$. Case 3. Two
of the three problems, say $I_{i_{1}}$ and $I_{i_{2}}$
($i_{1},i_{2}\in\\{1,2,3\\}$), are feasible: Let $S_{i_{1}}$ and $S_{i_{2}}$
be optimal solutions to $I_{i_{1}}$ and $I_{i_{2}}$ respectively. Then there
is a solution $S$ to $(G,F)$ such that $S\cap E(X)=E(S_{i_{1}})$ or $S\cap
E(X)=E(S_{i_{2}})$. Therefore, we can replace $X$ with a $4$-cycle
$x_{i_{1}}x_{4}x_{i_{2}}x_{j}$ preserving the optimality of the instance,
where $\\{j\\}=\\{1,2,3\\}-\\{i_{1},i_{2}\\}$, the costs of $x_{i_{1}}x_{4}$
and $x_{i_{2}}x_{j}$ are the costs of the two paths in $S_{i_{1}}$, and the
costs of $x_{4}x_{i_{2}}$ and $x_{j}x_{i_{1}}$ are the costs of the two paths
in $S_{i_{2}}$.
###### Lemma 6
Let $X$ be an induced subgraph of a degree-$3$ graph $G$ such that $X$
contains at most eight vertices of degree 3 in $G$. Then $X$ is $4$-cut
reducible if $|\mathrm{cut}_{F}(X)|=4$, $|\mathrm{cut}_{U}(X)|=0$, and $X$
contains at most two unforced vertices.
_Proof._ We only need to show that at least one of the three problem instances
$I_{1},I_{2}$ and $I_{3}$ (defined before Lemma 5) is infeasible. Assume that
no two edges in $\mathrm{cut}_{F}(X)$ meet at a same vertex in $X$, since
otherwise only one of $I_{1},I_{2}$ and $I_{3}$ is feasible. Also assume that
$X$ has no multiple edges or induced triangles, since otherwise $X$ can be
reduced to a smaller graph preserving its optimality. Note that $X$ contains
an even number $k$ of degree 3 vertices in $G$. Since $k=4$ (i.e., $|V(X)|=4$)
implies the lemma, we consider the case of $k=6,8$. When $k=6$, $X$ is either
a 6-cycle with a chord or a graph obtained from a 5-cycle with a chord by
subdividing the chord with a new vertex. In any case, we see that one of
$I_{1},I_{2}$ and $I_{3}$ is infeasible. Let $k=8$. In this case, there are
four vertices $u_{i}$ $(i=1,2,3,4)$ which are not incident to any of the four
edges in $\mathrm{cut}(B)=\mathrm{cut}_{F}(B)$, and two of them, say $u_{1}$
and $u_{2}$ are joined by a forced edge $u_{1}u_{2}\in F$ by the assumption on
the number of unforced vertices in $X$. Then we see that there are three
possible configurations for such an induced graph $X$ with no induced
triangles, and a straightforward inspection shows that none of them admits a
set of three feasible instances $I_{1},I_{2}$ and $I_{3}$.
Lemma 4 and Lemma 5 imply a way of simplifying some local structures of an
instance. However, it is not easy to find solutions to problems $I_{i}$ in the
above two lemmas. In our algorithm, we only do this replacement for $X$
containing no more than 10 vertices and then the corresponding problems
$I_{i}$ can be solved in constant time by a brute force search.
We define the operation of _$3$ /$4$-cut reduction_: If there a subgraph $X$
of $G$ with $|V(X)|\leq 10$ such that $|\mathrm{cut}(X)|=3$ or $X$ is $4$-cut
reducible, then we simplify the graph by replacing $X$ with a graph according
to Lemma 4 or Lemma 5. Note that a $3$/$4$-cut can be found in polynomial time
if it exists and then this reduction operation can be implemented in
polynomial time.
### 3.4 A solvable case and reduced graphs
A 3/4-cut reduction reduce the subgraph $X$ to a trivial graph except for the
last case of Lemma 5 where $X$ will become a 4-cycle. Eppstein has identified
a polynomially solvable case of forced TSP [5], which can deal with
$U$-components of 4-cycles.
###### Lemma 7
[5] If every $U$-component is a component of a 4-cycle, then a minimum cost
tour of the instance can be found in polynomial time.
Based on this lemma, we do not need to deal with $U$-components of 4-cycles in
our algorithms.
All above reduction rules can be applied in polynomial time. An instance
$(G,F)$ is called a _reduced_ instance if $G$ is 2-edge-connected, $(G,F)$
satisfies the parity condition, and has none of reducible edges, eliminable
edges and multiple edges, and the 3/4-cut reduction cannot be applied on it
anymore. Note that a reduced instance has no triangle, otherwise 3-cut
reduction would be applicable. An instance is called 2-edge-connected if every
$U$-component in it is 2-edge-connected. The initial instance
$(G,F=\emptyset)$ is assumed to be 2-edge-connected, otherwise it is
infeasible by Lemma 3. In our algorithm, we will guarantee that the input
instance is always 2-edge-connected, and we branch on a reduced graph to
search a solution.
## 4 The circuit procedure
The _circuit procedure_ is one of the most important operations in our
algorithm. The procedure will determinate each edge in a circuit to be
included into $F$ or to be deleted from the graph. It will be widely used as
the only branching operation in our algorithm.
#### Processing circuits:
_Determining_ an unforced edge means either including it to $F$ or deleting it
from the graph. When an edge is determined, the other edges in the same
circuit containing this edge can also be determined directly by reducing
eliminable edges. We call the series of procedures applied to all edges in a
circuit together as a _circuit procedure_. Thus, in the circuit procedure,
after we start to process a circuit ${\cal C}$ either by including an edge
$e_{1}\in{\cal C}$ to $F$ or by deleting $e_{1}$ from the graph, the next edge
$e_{i+1}$ of $e_{i}$ becomes an eliminable edge and we continue to determine
$e_{i+1}$ either by deleting it from the graph if block $B_{i}$ is odd and
$e_{i}=u_{i}v_{i}$ is included to $F$ (or $B_{i}$ is even and $e_{i}$ is
deleted); or by including it to $F$ otherwise. Circuit procedure is a
fundamental operation to build up our proposed algorithm. Note that a circuit
procedure determines only the edges in the circuit. During the procedure, some
unforced edges outside the circuit may become reducible and so on, but we do
not determine them in this execution.
###### Lemma 8
Let $H$ be a 2-edge-connected $U$-component in an instance $(G,F)$ and ${\cal
C}$ be a circuit in $H$. Let $(G^{\prime},F^{\prime})$ be the resulting
instance after applying circuit procedure on ${\cal C}$. Then
(i) each block $B_{i}$ of ${\cal C}$ becomes a 2-edge-connected $U$-component
in $(G^{\prime},F^{\prime})$; and
(ii) any other $U$-component $H^{\prime}$ than $H$ in $(G,F)$ remains
unchanged in $(G^{\prime},F^{\prime})$.
_Proof._ Since $H$ is 2-edge-connected, we know that each block $B_{i}$
induces a 2-edge-connected subgraph from $H$ according to the definition of
circuits. Hence $B_{i}$ will be a 2-edge-connected $U$-component in
$(G^{\prime},F^{\prime})$. Then we get (i). Since $H$ and $H^{\prime}$ are
vertex-disjoint and only edges in $H$ are determined, we see that (ii) holds.
We call a circuit _reducible_ if it contains at least one reducible edge. We
can apply the circuit procedure on a reducible circuit directly starting by
including a reducible edge to $F$. In our algorithm, we will deal with
reducible edges by processing a reducible circuit. When the instance becomes a
reduced instance, we may not be able to reduce the instance directly. Then we
search the solution by “branching on a circuit.” _Branching on a circuit
${\cal C}$ at edge $e\in{\cal C}$_ means branching on the current instance to
generate two instances by applying the circuit procedure to ${\cal C}$ after
including $e$ to $F$ and deleting $e$ from the graph respectively. Branching
on a circuit is the only branching operation used in our algorithm.
## 5 A simple algorithm based on circuit procedures
We first introduce a simple algorithm for forced TSP to show the effectiveness
of the circuit procedures. Improved algorithms are given in the next sections.
The simple algorithm contains only two steps: First reduce the instance until
it becomes a reduced one; and then select a $U$-component $H$ that is neither
trivial nor a 4-cycle and branch on a circuit ${\cal C}$ in $H$ such that at
least one block along ${\cal C}$ is trivial. Note that there is always a
circuit having a trivial block as long as the forced edge set $F$ is not
empty.
Here we use a traditional method to analyze the simple algorithm. It is
natural to consider how many edges can be added to $F$ in each operation of
the algorithm. The size of $F$ will not decrease by applying the reduction
rules. Let $r=n-|F|$ and $C(r)$ denote the maximum number of leaves in the
search tree generated by the algorithm for any instance with measure $r$. We
only need to consider the branching operation in the second step.
For convenience, we call a maximal sequence $P=\\{e_{1},e_{2},\ldots,e_{p}\\}$
of edges $e_{i}=u_{i}u_{i+1}\in E(H)$ $(1\leq i\leq p-1)$ a chain if all
vertices $u_{j}$ ($j=2,3,\ldots,p-1$) are forced vertices. In the definition
of the chain, we allow $u_{1}=u_{p}$. Observe that each chain is contained in
the same circuit. Since the selected circuit ${\cal C}$ has some trivial
block, we know that ${\cal C}$ contains at least one chain $P$ of size $\geq
2$. We distinguish two cases according to the size of $P$ being even or odd.
Case 1. $|P|$ is even: If all the blocks of ${\cal C}$ are trivial, then the
$U$-component $H$ containing ${\cal C}$ is a cycle of even length. Since $H$
cannot be a $4$-cycle, the length of cycle $H$ is at least $6$ (see Fig. 2(a)
for an illustration). In each branch, after processing the circuit ${\cal C}$,
we can include at least $3$ edges to $F$. This gives us branch vector
$\displaystyle(3;3).$ (1)
Next, we assume that ${\cal C}$ has a nontrivial block. We look at the worst
case where $P$ is of size 2 and ${\cal C}$ has only one nontrivial block (see
Fig. 2(b) for an illustration). Now ${\cal C}=P$. Let ${\cal
C}=\\{u_{1}v_{1},u_{2}v_{2}\\}$, where $v_{1}=u_{2}$. We branch on the circuit
at edge $u_{1}v_{1}$. In the branch of deleting $u_{1}v_{1}$, we include
$u_{2}v_{2}$ into $F$ . Furthermore, we can include the remaining two edges
incident on $u_{1}$ into $F$ by simply applying reduction rules. In the other
branch of including $u_{1}v_{1}$ into $F$, we will delete $u_{2}v_{2}$ and
include the other two edges incident on $v_{2}$ into $F$. We still can get
branch vector (1). Note that when ${\cal C}$ is not of the worst case, it is
not hard to verify that we may include more edges to $F$ and we can get a
branch vector covered by (1). We omit the details here, since the detailed
proof can also be derived from the analysis of the improved algorithm in the
next sections.
Figure 2: Three bottleneck cases in branching on a cirucuit ${\cal C}$: (a)
${\cal C}$ is a 6-cycle; (b) ${\cal C}$ is a chain $P$ of length 2; (c) ${\cal
C}$ is a chain $P$ of length 3.
Case 2. $|P|$ is odd: Now circuit ${\cal C}$ must contain at least one
nontrivial block, otherwise the instance violates the parity condition. We
also look at the worst case where $P$ is of size 3 (since $|P|\geq 2$) and
${\cal C}$ has only one nontrivial block (see Fig. 2(c) for an illustration).
Let ${\cal C}=\\{u_{1}v_{1},u_{2}v_{2},u_{3}v_{3}\\}$, where $v_{1}=u_{2}$ and
$v_{2}=u_{3}$. We branch on the circuit at edge $u_{1}v_{1}$. In the branch of
deleting $u_{1}v_{1}$, we will also delete $u_{3}v_{3}$ and include the
following five edges to $F$: $v_{2}u_{2}$, the remaining two edges incident on
$u_{1}$ and the remaining two edges incident on $v_{3}$ (note that since the
graph is reduced, $u_{1}$ and $v_{3}$ are not adjacent and then the five edges
are different to each other). In the other branch of including $u_{1}v_{1}$ to
$F$, we will delete $u_{2}v_{2}$ and include $u_{3}v_{3}$ to $F$. We can get
branch vector
$\displaystyle(5;2).$ (2)
When ${\cal C}$ is not of the worst case, we can reduce more edges and get (2)
at least.
Since $C(r)=1.260^{r}$ satisfies the two recurrences corresponding to (1) and
(2), we know that the simple algorithm can solve the TSP problem in an
$n$-vertex degree-3 graph in $O^{*}(1.260^{n})$ time, which achieves the same
running time bound of Eppstein’s algorithm for TSP3 in [5].
## 6 The measure and conquer method
The measure and conquer method, first introduced by Fomin, Grandoni and
Kratsch [6], is one of the most powerful tools to analyze exact algorithms. It
can obtain improved running time bound for many branching-and-search
algorithms without making any modification to the algorithms. Currently, many
best exact algorithms for NP-hard problems are based on this method. In the
measure and conquer method, we may set a weight of vertices in the instance
and use the sum $w$ of the total weight in the graph as the measure to
evaluate the running time. In the algorithm, the measure $w$ should satisfy
the _measure condition_ : (i) when $w\leq 0$ the instance can be solved in
polynomial time; (ii) the measure $w$ will never increase in each operation in
the algorithm; and (iii) the measure will decrease in each of the subinstances
generated by applying a branching rule. With these constraints, we may build
recurrences for the branching operations. Next, we introduce a way of applying
the measure and conquer method to the above simple algorithm.
The graph has three different vertex-weight. For each vertex $v$, we set its
vertex-weight $w(v)$ to be
$w(v)=\left\\{\begin{array}[]{cl}w_{3}=1&\mbox{if $d_{U}(v)=3$ }\\\
w_{3^{\prime}}&\mbox{if $d_{U}(v)=2$ and $d_{F}(v)=1$}\\\
0&\mbox{otherwise.}\end{array}\right.$
We will determine the best value of $w_{3^{\prime}}$ such that the worst
recurrence in our algorithm is best. Let $\Delta_{3}=w_{3}-w_{3^{\prime}}$.
For a subset of vertices (or a subgraph) $X$, we also use $w(X)$ to denote the
total vertex-weight in $X$.
Now we analyze the simple algorithm presented in Section 5 by using this
vertex-weight setting. Note that since we require that $w_{3}=1$, the total
vertex weight $w$ in the graph is not greater than the number $n$ of vertices
in the graph. We can get a running time bound related to $n$ if we get a
running time bound related to $w$. Here we only examine the three bottleneck
cases in Fig. 2.
When we branch on a circuit of 6-cycle in Fig. 2(a), in each branch, all the
six forced vertices will be reduced and then we can reduce $w$ by
$6w_{3^{\prime}}$. We get the following branching vector:
$\displaystyle[6w_{3^{\prime}}]_{2}.$ (3)
When we branch on a circuit of chain of length 2 in Fig. 2(b), in the branch
where $u_{1}v_{1}$ is included to $F$, $u_{2}v_{2}$ is deleted, and
$v_{2}v^{\prime}_{2}$ and $v_{2}v^{\prime\prime}_{2}$ are also included to $F$
by reduction rules, where $v^{\prime}_{2}$ and $v^{\prime\prime}_{2}$ are the
two neighbors of $v_{2}$ other than $u_{2}$. Then we can reduce $w$ by
$w_{3^{\prime}}$ from $v_{1}$, $\Delta_{3}$ from $u_{1}$, $w_{3}$ from
$v_{2}$, and $2\delta_{1}$ from $v^{\prime}_{2}$ and $v^{\prime\prime}_{2}$,
where $\delta_{1}\geq\min\\{\Delta_{3},w_{3^{\prime}}\\}$. The second branch
can be analyzed in a similar manner. We get
$\displaystyle[w_{3^{\prime}}+\Delta_{3}+w_{3}+2\delta_{1}]_{2}=[2+2\delta_{1}]_{2}.$
(4)
When we branch on a circuit of chain of length 3 in Fig. 2(c), in the branch
of including $u_{1}v_{1}$ to $F$, we reduce $w$ by $2\Delta_{3}$ from $u_{1}$
and $v_{3}$ and $2w_{3^{\prime}}$ from $v_{1}$ and $v_{2}$. In the other
branch, we can reduce $w$ by $2w_{3}$ from $u_{1}$ and $v_{3}$,
$2w_{3^{\prime}}$ from $v_{1}$ and $v_{2}$, and $4\delta_{1}$ from
$\\{u^{\prime}_{1},u^{\prime\prime}_{1},v^{\prime}_{3},v^{\prime\prime}_{3}\\}$,
where $u^{\prime}_{1}$ and $u^{\prime\prime}_{1}$ are the two neighbors of
$u_{1}$ other than $v_{1}$ and $v^{\prime}_{3}$ and $v^{\prime\prime}_{3}$ are
the two neighbors of $v_{3}$ other than $u_{3}$. We get
$\displaystyle(2;2w_{3}+2w_{3^{\prime}}+4\delta_{1}).$ (5)
We can verify that under that above three constraints, the best value of
$w_{3^{\prime}}$ is ${\frac{1}{2}}w_{3}=\frac{1}{2}$. With this setting, we
can see that (3) and (4) become (1), and (5) becomes (2). This also tells us
that the measure and conquer method cannot directly derive a better running
time bound of the simple algorithm. In the next section, we present a new
technique and show an improvement by combining the new technique with the
traditional measure and conquer method.
## 7 Amortization on connectivity structures
To improve the time bound by the above simple analysis, we need to use more
structural properties of the graph. Note that for the bottleneck case of Fig.
2(a), the above algorithm reduces all the vertices in this $U$-component. It
is impossible to improve by reducing more vertices (or edges) and so on. But
we also reduce the number of $U$-components by 1. This observation gives us an
idea of an amortization scheme over the cut-circuit structure by setting a
weight on each $U$-component in the graph.
In this method, each vertex in the graph receives a nonnegative weight as
shown in Section 6. We also set a weight (which is possibly negative, but
bounded from by a constant $c\geq 0$) to each $U$-component. Let $\mu$ be the
sum of all vertex weight and $U$-component weight. We will use $\mu$ to
measure the size of the search tree generated by our algorithm. The measure
$\mu$ will also satisfy the measure condition. Initially there is only one
$U$-component and $\mu<n+c$ holds. If we get a running time bound related to
$\mu$ for our algorithm, then we get a running time bound related to $n$.
A simple idea is to set the same weight to each nontrivial $U$-component. It
is possible to improve the previous best result by using this simple idea.
However, to get further improvement, in this paper, we set several different
component-weights. Our purpose is to distinguish some “bad” $U$-components,
which will be characterized as “critical” $U$-components.
An extension of a 6-cycle is obtained from a 6-cycle
$v_{1}v_{2}v_{3}v_{4}v_{5}v_{6}$ and a 2-clique $ab$ by joining them with two
independent edges $av_{i}$ and $bv_{j}$ ($i\neq j$). An extension of a 6-cycle
always has exactly eight vertices. A chord of an extension of a 6-cycle is an
edge between two vertices in it but different from the eight edges
$v_{1}v_{2},v_{2}v_{3},v_{3}v_{4},v_{4}v_{5},v_{5}v_{6},v_{6}v_{1},av_{i}$,
$bv_{j}$ and $ab$.
A subgraph $H$ of a $U$-component in an instance $(G,F)$ is _$k$ -pendent
critical_, if it is a 6-cycle or an extension of a 6-cycle with
$|\mathrm{cut}_{U}(H)|=k$ and $|\mathrm{cut}_{F}(H)|=6-k$ (i.e., $H$ has no
chord of unforced/forced edge). A $0$-pendent critical $U$-component is also
simply called a _critical graph_ or _critical $U$-component_. Fig. 3
illustrates two examples of critical graphs of extensions of a $6$-cycle.
Branching on a critical $U$-component may lead to a bottleneck recurrence in
our algorithm. So we set a different component-weight to this kind of
components to get improvement.
Figure 3: Extensions of a 6-cycle
For each $U$-component $H$, we set its component-weight $c(H)$ to be
$c(H)=\left\\{\begin{array}[]{cl}0&\mbox{if $H$ is trivial}\\\
-4w_{3^{\prime}}&\mbox{if $H$ is a 4-cycle}\\\ \gamma&\mbox{if $H$ is a
critical $U$-component}\\\ \delta&\mbox{otherwise,}\end{array}\right.$
where we set $c(H)=-4w_{3^{\prime}}$ so that $c(H)+w(H)=0$ holds for every
4-cycle $U$-component $H$.
We also require that the vertex-weight and component-weight satisfy the
following requirements
$\displaystyle 2\Delta_{3}\geq\gamma\geq\delta\geq\Delta_{3}\geq{1\over
2}w_{3},~{}w_{3^{\prime}}\geq{1\over
5}w_{3}~{}~{}\mathrm{and}~{}~{}\gamma-\delta\leq w_{3^{\prime}}.$ (6)
Under these constraints, we still need to decide the values of
$w_{3^{\prime}},\gamma$ and $\delta$ such that the time bound derived by the
worst recurrences in our algorithm will be minimized. In (6),
$2\Delta_{3}\geq\gamma$ is important, because it will be used to satisfy the
measure condition (ii). The other constraints in (6) are mainly used to
simplify some arguments and they will not become the bottleneck in our
analysis. Next, we first describe our simple algorithm.
## 8 The algorithm
A block is called a _normal block_ if it is none of trivial, reducible and
2-pendent critical. A normal block is _minimal_ if no subgraph of it is a
normal block along any circuit. Note that when $F$ is not empty, each
$U$-component has at least one nontrivial circuit. Our recursive algorithm for
forced TSP only contains two main steps:
1. First apply the reduction rules to a given instance until it becomes a reduced one; and
2. Then take any $U$-component $H$ that is neither trivial nor a 4-cycle (if no such $U$-component $H$, then the instance is polynomially solvable by Lemma 7), and branch on a nontrivial circuit ${\cal C}$ in $H$, where ${\cal C}$ is chosen so that
(1) no normal block appears along ${\cal C}$ (i.e., ${\cal C}$ has only
trivial and 2-pendent critical blocks) if this kind of circuit exist; and
(2) a minimal normal block $B_{1}$ in $H$ appears along ${\cal C}$ otherwise.
We use $\mu$ as the measure to analyze the size of the search tree in our
algorithm. It is easy to see that after applying the reduction rules on a
2-edge-connected instance, the resulting instance remains 2-edge-connected. By
this observation and Lemma 8, we can guarantee that an input instance is
always 2-edge-connected. Our analysis is based on this.
### 8.1 Basic properties of the measure
Before analyzing the time bound on our algorithm, we first give basic
properties of the measure $\mu$.
We show that the measure will not increase after applying any reduction
operation in an 2-edge-connected instance. Since an input instance is 2-edge-
connected, there is no eliminable edge. In fact, we always deal with
eliminable edges in circuit procedures. For reducible edges, we deal with them
during a process of a reducible circuit (including the reducible edges to $F$
and dealing with the resulting eliminable edges). We will show that $\mu$
never increases after processing a circuit. The measure $\mu$ will not
increase after deleting any unforced parallel edge. The following lemma also
shows that applying the $3$/$4$-cut reduction does not increase $\mu$.
###### Lemma 9
For a given instance,
(i) applying the $3$-cut reduction does not increase the measure $\mu$; and
(ii) applying the $4$-cut reduction on a $U$-component $X$ decreases the
measure $\mu$ by $w(X)+c(X)$.
_Proof._ It is easy to observe (i). Next we prove (ii). In the case where the
resulting graph consists of four single vertices or a pair of forced edges
after applying the $4$-cut reduction, the whole component $X$ is eliminated,
decreasing $\mu$ by $w(X)+c(X)$ and then the lemma holds. Otherwise the
resulting component, say $X^{\prime}$ is a $4$-cycle, where
$w(X^{\prime})+c(X^{\prime})=0$ according to our setting on the component-
weight of $4$-cycles, and $\mu$ again decreases by $w(X)+c(X)$.
Next we consider how much amount of measure decreases by processing a circuit.
We consider that the measure $\mu$ becomes zero whenever we find an instance
infeasible by Lemma 3. After processing a circuit ${\cal
C}=\\{e_{i}=u_{i}v_{i}\mid 1\leq i\leq p\\}$ in a $U$-component $H$, each
block $B_{i}$ along ${\cal C}$ becomes a new $U$-component, which we denote by
$\bar{B_{i}}$. We define the direct benefit $\beta^{\prime}(B_{i})$ from
$B_{i}$ to be the decrease in vertex-weight of the endpoints $v_{i}$ and
$u_{i+1}$ of $B_{i}$ minus the component-weight $c(\bar{B_{i}})$ in the new
instance after the circuit procedure. Immediately after the procedure, the
measure $\mu$ decreases by
$w(H)+c(H)-\sum_{i}(w(\bar{B_{i}})+c(\bar{B_{i}}))=c(H)+\sum_{i}\beta^{\prime}(B_{i})$.
After the circuit procedure, we see that the vertex-weights of endpoints of
each non-reducible and nontrivial block $B_{i}$ decreases by $\Delta_{3}$ and
$\Delta_{3}$ (or $w_{3}$ and $w_{3}$) respectively if $B_{i}$ is even, and by
$\Delta_{3}$ and $w_{3}$ (or $w_{3}$ and $\Delta_{3}$) respectively if $B_{i}$
is odd. Summarizing these, the direct benefit $\beta^{\prime}(B)$ from a block
$B$ is given by
$\beta^{\prime}(B)=\left\\{\begin{array}[]{cl}0&\mbox{if $B$ is reducible},\\\
w_{3^{\prime}}&\mbox{if $B$ is trivial},\\\
w_{3}\\!+\\!\Delta_{3}\\!-\\!\delta&\mbox{if $B$ is odd and nontrivial},\\\
\mbox{$2w_{3}-\delta$}&\mbox{if $B$ is even and non-reducible, and
$\mathrm{cut}_{U}(B)$ is deleted},\\\ \mbox{$2\Delta_{3}-\gamma$}&\mbox{if $B$
is $2$-pendent critical, and $\mathrm{cut}_{U}(B)$ is included in $F$},\\\
\mbox{$w(B)$}&\mbox{if $B$ is a $2$-pendent 4-cycle, and $\mathrm{cut}_{U}(B)$
is included in $F$},\\\ \mbox{$2\Delta_{3}-\delta$}&\mbox{otherwise (i.e., $B$
is even, non-reducible but not }\\\ &\mbox{~{} a $2$-pendent critical
$U$-graph or a $2$-pendent 4-cycle,}\\\ &\mbox{~{} and $\mathrm{cut}_{U}(B)$
is included to $F$).}\\\ \end{array}\right.$ (7)
By (6), we have that $\beta^{\prime}(B_{i})\geq 0$ for any type of block
$B_{i}$, which implies that the decrease
$c(H)+\sum_{i}\beta^{\prime}(B_{i})\geq c(H)\geq 0$ (where $H$ is not a
4-cycle) is in fact nonnegative, i.e., the measure $\mu$ never increases by
processing a circuit.
After processing a circuit ${\cal C}$, a reduction operation may be applicable
to some $U$-components $\bar{B}_{i}$ and we can decrease $\mu$ more by
reducing them. The _indirect benefit_ $\beta^{\prime\prime}(B)$ from a block
$B$ is defined as the amount of $\mu$ decreased by applying reduction rules on
the $U$-component $\bar{B}$ after processing the circuit. Since we have shown
that $\mu$ never increases by applying reduction rules, we know that
$\beta^{\prime\prime}(B)$ is always nonnegative. The _total benefit_
(_benefit_ , for short) from a block $B$ is
$\beta(B)=\beta^{\prime}(B)+\beta^{\prime\prime}(B).$
###### Lemma 10
After processing a circuit ${\cal C}$ in a $2$-edge-connected $U$-component
$H$ $($not necessary being reduced$)$ and applying reduction rules until the
instance becomes a reduced one, the measure $\mu$ decreases by
$c(H)+\sum_{i}\beta(B_{i}),$
where $B_{i}$ are the blocks along circuit ${\cal C}$.
The indirect benefit from a block depends on the structure of the block. In
our algorithm, we hope that the indirect benefit is as large as possible. Here
we prove some lower bounds on it for some special cases.
###### Lemma 11
Let $H$ be a $U$-component containing no induced triangle and ${\cal
C}^{\prime}$ be a reducible circuit in it such that there is exactly one
reducible block along ${\cal C}^{\prime}$. The measure $\mu$ decreases by at
least $2\Delta_{3}$ by processing the reducible circuit ${\cal C}^{\prime}$
and applying reduction rules.
_Proof._ By assumption, the reducible circuit has at least two blocks, one
reducible block $B_{1}$ and one non-reducible block $B_{2}$. (1) Assume that
every other block than $B_{1}$ along ${\cal C}^{\prime}$ is trivial. Then $H$
should be a cycle of length $k\geq 5$ (if $H$ is a 4-cycle, then there are
three odd blocks along ${\cal C}^{\prime}$ and we find the instance infeasible
by Lemma 3). There are $k-1\geq 4$ trivial blocks along ${\cal C}^{\prime}$.
After processing the circuit, the whole component $H$ will be eliminated
decreasing $\mu$ by at least $c(H)+w(H)\geq\delta+4w_{3^{\prime}}\geq
2\Delta_{3}$ (by (6)). (2) Otherwise, i.e., there is a block $B_{2}$ of more
than one vertex ($B_{2}$ is not trivial or reducible): (2-i)
$\mathrm{cut}_{U}(B_{2})$ is deleted in the circuit procedure: Then $\mu$
decreases by at least
$c(H)+\beta^{\prime}(B_{2})=\delta+(2w_{3}-\delta)=2w_{3}\geq 2\Delta_{3}$ by
(7). Hence assume that $\mathrm{cut}_{U}(B_{2})$ is included to $F$ in (2).
(2-ii) $B_{2}$ is not 2-pendent critical: Then the measure $\mu$ decreases by
at least $c(H)+\beta^{\prime}(B_{2})=\delta+(2\Delta_{3}-\delta)=2\Delta_{3}$.
The remaining case is that $B_{2}$ is 2-pendent critical and
$\mathrm{cut}_{U}(B_{2})$ is included to $F$. (2-iii) there are only two
blocks $B_{1}$ and $B_{2}$ along ${\cal C}^{\prime}$: By processing the
circuit ${\cal C}^{\prime}$, the two edges in
$\\{xz,yz\\}=\mathrm{cut}_{U}(B_{2})$ become forced edges. Since they are
incident on the single vertex $z$ in $B_{1}$, we can replace $xz$ and $yz$
with a single forced edge $xy$ and then $B_{2}$ becomes a 0-pendent 6-cycle or
extension of a 6-cycle with only one forced chord $xy$. After the circuit
procedure of ${\cal C}^{\prime}$, we can apply 4-cut reduction to the
0-pendent 6-cycle $B_{2}$ (by Lemma 6) and then this decreases $\mu$ by
$c(H)+\beta(B_{2})=\delta+w(B_{2})>2\Delta_{3}$ (by Lemma 9(ii)). (2-iv) there
is a pair of two trivial blocks $B_{3}$ and $B_{4}$ or a nontrivial and non-
reducible block $B_{5}$ along ${\cal C}^{\prime}$: Now we can decrease $\mu$
by at least $c(H)+\beta(B_{2})+\sum_{i\neq
2}\beta(B_{i})\geq\delta+(2\Delta_{3}-\gamma)+\min\\{2w_{3^{\prime}},(2\Delta_{3}-\delta)\\}\geq
2\Delta_{3}$ (by (6)).
In any case, the measure $\mu$ decreases by at least $2\Delta_{3}$.
###### Lemma 12
In the circuit procedure for a circuit ${\cal C}$ in a reduced instance, the
indirect benefit from a block $B$ along ${\cal C}$ satisfies
$\beta^{\prime\prime}(B)\geq\left\\{\begin{array}[]{clc}2\Delta_{3}&\mbox{if
$B$ is odd and nontrivial},&{\rm(i)}\\\ w(B)-\beta^{\prime}(B)&\mbox{if $B$ is
a $2$-pendent cycle or critical graph},&\\\ &\mbox{~{}~{}and
$\mathrm{cut}_{U}(B)$ is deleted},&{\rm(ii)}\\\ \delta&\mbox{if $B$ is even
but not reducible or a $2$-pendent}&\\\ &\mbox{~{}~{} cycle, and
$\mathrm{cut}_{U}(B)$ is deleted},&{\rm(iii)}\\\
0&\mbox{otherwise}.&{\rm(iv)}\\\ \end{array}\right.$
_Proof._ We will use $\bar{B}$ to denote the $U$-component resulting from $B$
after the circuit procession. Case (i): Since $B$ is odd, only one edge in
$\mathrm{cut}_{U}(B)$ is deleted (the other one is included to $F$) in the
circuit procession. Then there is exactly one vertex of degree 2 in $\bar{B}$,
which is the only reducible block along a circuit ${\cal C}^{\prime}$ in
$\bar{B}$. Since the original instance is reduced and contains no triangle, we
know that circuit ${\cal C}^{\prime}$ satisfies the condition in Lemma 11. By
processing ${\cal C}^{\prime}$ in $\bar{B}$, we can decrease $\mu$ by at least
$2\Delta_{3}$ by Lemma 11. Then we get $\beta^{\prime\prime}(B)\geq
2\Delta_{3}$. For case (ii), if $B$ is a $2$-pendent cycle, we can reduce the
whole $U$-component $\bar{B}$ after the circuit procedure, and then we have
$\beta^{\prime\prime}(B)=c(\bar{B})+w(\bar{B})$. Otherwise $B$ in (ii) is a
$2$-pendent critical graph and the 4-cut reduction can be applied to $\bar{B}$
by Lemma 6 since the two end vertices of edges in $\mathrm{cut}_{U}(B)$ will
be of degree 2 in $\bar{B}$ after $\mathrm{cut}_{U}(B)$ is deleted. Then we
still have $\beta^{\prime\prime}(B)=c(\bar{B})+w(\bar{B})$ by Lemma 9(ii).
Note that $\beta^{\prime}(B)=w(B)-w(\bar{B})-c(\bar{B})$ (by the definition of
$\beta^{\prime}(B)$). We get $\beta^{\prime\prime}(B)=w(B)-\beta^{\prime}(B)$.
For Case (iii), we will get at least one reducible circuit ${\cal C}^{\prime}$
in $\bar{B}$. Note that $\bar{B}$ has some vertex of degree 2 and cannot be a
critical graph. Hence $c(\bar{B})=\delta$. By processing the reducible circuit
${\cal C}^{\prime}$, we can decrease $\mu$ by at least
$c(\bar{B})+\sum_{i}\beta(B^{\prime}_{i})\geq c(\bar{B})=\delta$, where
$B^{\prime}_{i}$ are the blocks along ${\cal C}^{\prime}$. The inequality in
(iv) holds since we have proved that $\mu$ will never increase after applying
reduction rules.
## 9 The Analysis
Now we are ready to analyze our algorithm. In the algorithm, branching on a
circuit generates two instances $(G_{1},F_{1})$ and $(G_{2},F_{2})$. By Lemma
10, we get branch vector
$(c(H)+\sum_{i}\beta_{1}(B_{i});c(H)+\sum_{i}\beta_{2}(B_{i})),$
where $\beta_{j}(B)$, $\beta^{\prime}_{j}(B)$ and
$\beta^{\prime\prime}_{j}(B)$ denote the functions $\beta(B)$,
$\beta^{\prime}(B)$ and $\beta^{\prime\prime}(B)$ evaluated in
$(G_{j},F_{j})$, $j=1,2$ for clarifying how branch vectors are derived in the
subsequent analysis. We consider this branch vector for different cases. First
we analyze the easy case where the chosen circuit ${\cal C}$ has no normal
block. Then we analyze the somewhat complicated case where there is a minimal
normal block along ${\cal C}$.
### 9.1 Circuits with only trivial and $2$-pendent critical blocks
In this subsection, we assume that the chosen circuit ${\cal C}$ in a
$U$-component $H$ (not a 4-cycle) has only trivial and $2$-pendent critical
blocks. We consider the following three cases.
Case 1. All blocks along ${\cal C}$ are trivial blocks: Now $H$ should be a
cycle of even length $l\geq 6$. By (7) and Lemma 10, we know that in each
branch we can decrease $\mu$ by at least
$c(H)+\sum_{i}\beta(B_{i})\geq\left\\{\begin{array}[]{cl}\gamma+6w_{3^{\prime}}&\mbox{if
$l=6$ }\\\ \delta+8w_{3^{\prime}}&\mbox{if $l\geq 8$.}\end{array}\right.$
Then we get branch vectors
$\displaystyle[6w_{3^{\prime}}+\gamma]_{2}$ (8)
for $l=6$ and $[8w_{3^{\prime}}+\delta]_{2}$ for $l\geq 8$, which is covered
by (8). Note that (8) is tight for a circuit of 6-cycle in Fig. 2(a).
Case 2. There is only one 2-pendent critical block $B_{1}$ along ${\cal C}$:
Since $B_{1}$ is even, the number $r>0$ of trivial blocks (odd blocks) along
${\cal C}$ is also even by the parity condition. Note that if $r=2$ and
$B_{1}$ is 2-pendent 6-cycle then $H$ is a critical graph (an extension of a
6-cycle) and $c(H)=\gamma$. Otherwise $H$ is not critical and $c(H)=\delta$.
Then in the branch where $\mathrm{cut}_{U}(B_{1})$ is deleted, the decrease
$c(H)+\sum_{i}\beta(B_{i})$ of $\mu$ is at least
$c(H)+w(B_{2})+rw_{3^{\prime}}\geq\min\\{\gamma+(2w_{3}+4w_{3^{\prime}})+2w_{3^{\prime}},\delta+(4w_{3}+4w_{3^{\prime}})+2w_{3^{\prime}},\delta+(2w_{3}+4w_{3^{\prime}})+4w_{3^{\prime}}\\}=$
$\gamma+2w_{3}+6w_{3^{\prime}}.$
In the branch where $\mathrm{cut}_{U}(B_{1})$ is included to $F$, the decrease
$c(H)+\sum_{i}\beta(B_{i})$ of $\mu$ is at least
$\min\\{\gamma,\delta\\}+(2\Delta_{3}-\gamma)+2w_{3^{\prime}}=\delta+2w_{3}-\gamma.$
This gives branch vector
$\displaystyle(\gamma+2w_{3}+6w_{3^{\prime}};\delta+2w_{3}-\gamma).$ (9)
Case 3. There are at least two 2-pendent critical blocks $B_{1}$ and $B_{2}$
along ${\cal C}$: If $\mathrm{cut}_{U}(B_{2})$ is included to $F$ in the
branch where $\mathrm{cut}_{U}(B_{1})$ is deleted, then we can get branch
vector
$[c(H)+\beta(B_{1})+\beta(B_{2})]_{2}=[\delta+(2w_{3}+4w_{3^{\prime}})+(2\Delta_{3}-\gamma)]_{2}=[\delta+4w_{3}+2w_{3^{\prime}}-\gamma]_{2}$,
which is covered by (8). On the other hand, if $\mathrm{cut}_{U}(B_{2})$ is
also deleted in the branch where $\mathrm{cut}_{U}(B_{1})$ is deleted, we get
branch vector
$\displaystyle(\delta+2(2\Delta_{3}-\gamma);\delta+2(2w_{3}+4w_{3^{\prime}})).$
(10)
### 9.2 Circuits with a minimal normal block
In this subsection, we assume that the chosen circuit ${\cal C}$ in a
$U$-component $H$ has a minimal normal block $B_{1}$. Now $c(H)=\delta$ always
holds. We distinguish several cases to analyze the branch vectors.
Case 1. Block $B_{1}$ is odd: Circuit ${\cal C}$ has another odd block
$B_{2}$. By (7) we have
$\beta(B_{2})\geq\beta^{\prime}(B_{2})=\min\\{w_{3^{\prime}},w_{3}\\!+\\!\Delta_{3}\\!-\\!\delta\\}=w_{3^{\prime}}$
in each branch. For $B_{1}$, we have that
$\beta^{\prime}(B_{1})=w_{3}+\Delta_{3}-\delta$ and
$\beta^{\prime\prime}(B_{1})\geq 2\Delta_{3}$ by Lemma 12. In each branch,
$\mu$ decreases by at least
$c(H)+\sum_{i}\beta(B_{i})\geq\delta+\beta^{\prime}(B_{1})+\beta^{\prime\prime}(B_{1})+\beta(B_{2})\geq\delta+(w_{3}+\Delta_{3}-\delta)+2\Delta_{3}+w_{3^{\prime}}=4w_{3}-2w_{3^{\prime}}$.
Therefore, we can get branch vector
$\displaystyle[4w_{3}-2w_{3^{\prime}}]_{2}.$ (11)
Note that (11) is tight for a circuit of chain of length 2 in Fig. 2(b).
Next, we assume that $B_{1}$ is even. In the branch where
$\mathrm{cut}_{U}(B_{1})$ is included to $F$, we have that
$\beta^{\prime}_{1}(B_{1})=2\Delta_{3}-\delta$. In the other branch, we have
that $\beta^{\prime}_{2}(B_{1})=2w_{3}-\delta$. By branching on ${\cal C}$, we
get branch vector
$(\delta+\beta^{\prime}_{1}(B_{1})+\beta^{\prime\prime}_{1}(B_{1})+\xi_{1};\delta+\beta^{\prime}_{2}(B_{1})+\beta^{\prime\prime}_{2}(B_{1})+\xi_{2}),$
where $\xi_{j}=\sum_{i\neq 1}\beta_{j}(B_{i})\geq 0$ ($j\in\\{1,2\\}$),
$\beta^{\prime}_{1}(B_{1})=2\Delta_{3}-\delta$ and
$\beta^{\prime}_{2}(B_{1})=2w_{3}-\delta$. We here evaluate $\xi_{1}$ and
$\xi_{2}$. If all blocks other than $B_{1}$ along ${\cal C}$ are 2-pendent
critical, then we have that $\xi_{1}\geq 2\Delta_{3}-\gamma$ and
$\xi_{2}=\sum_{i\neq 1}w(B_{i})\geq w(B_{2})\geq 2w_{3}+4w_{3^{\prime}}$.
Otherwise,
$\xi_{1}\geq\min\\{2w_{3^{\prime}},w_{3}+\Delta_{3}-\delta,2\Delta_{3}-\delta\\}=2w_{3^{\prime}}$
and
$\xi_{2}\geq\min\\{2w_{3^{\prime}},w_{3}+\Delta_{3}-\delta,2w_{3}-\delta\\}=2w_{3^{\prime}}$.
We have the following two choices for $(\xi_{1},\xi_{2})$:
$\displaystyle(\xi_{1},\xi_{2})=(2\Delta_{3}-\gamma,2w_{3}+4w_{3^{\prime}})\ \
\ {\rm and}\ \ \ (2w_{3^{\prime}},2w_{3^{\prime}}).$ (12)
In what follows, we derive some lower bounds on
$\beta^{\prime\prime}_{1}(B_{1})$ and $\beta^{\prime\prime}_{2}(B_{1})$ by
examining the structure of $B_{1}$.
Let $\mathrm{cut}_{U}(B_{1})=\\{xv,yu\\}$, where $x$ and $y$ are in $B_{1}$.
Let $(G_{1},F_{1})$ and $(G_{2},F_{2})$ be the two resulting instances after
branching and processing the circuit ${\cal C}$, where $(G_{1},F_{1})$
corresponds to the branch where $\mathrm{cut}_{U}(B_{1})$ is included to $F$
and $(G_{2},F_{2})$ corresponds to the branch where $\mathrm{cut}_{U}(B_{1})$
is deleted. Let $x_{1}x$ and $x_{2}x$ (resp., $y_{1}y$ and $y_{2}y$) be the
two unforced edges incident on $x$ (resp., $y$) in $(G_{1},F_{1})$. Note that
$x_{1}x$ and $x_{2}x$ (resp., $y_{1}y$ and $y_{2}y$) will be in the same
circuit ${\cal C}_{x}$ (resp., ${\cal C}_{y}$) in $(G_{1},F_{1})$, since $x$
(resp., $y$) is a forced vertex now. See Fig. 4 for illustrations of the
structure of the edges incident to $x$ and $y$.
Figure 4: Illustrations of block $B_{1}$: (a) ${\cal C}_{x}\neq{\cal C}_{y}$;
(b) ${\cal C}_{x}={\cal C}_{y}$ and $x$ and $y$ are adjacent; (c) ${\cal
C}_{x}={\cal C}_{y}$ and $x$ and $y$ are not adjacent.
Case 2. Block $B_{1}$ is even and ${\cal C}_{x}$ and ${\cal C}_{y}$ are two
different circuits in $(G_{1},F_{1})$ (see Fig. 4(a)): Now ${\cal C}_{x}$ and
${\cal C}_{y}$ are two different circuits also in $(G_{2},F_{2})$. In the
branch where $\mathrm{cut}_{U}(B_{1})$ is deleted, we have
$\beta^{\prime\prime}_{2}(B_{1})\geq 4\Delta_{3}$ (by applying Lemma 11 to
${\cal C}_{x}$ and ${\cal C}_{y}$). Next we consider the other blocks along
${\cal C}$.
Case 2.1. There are at least two odd blocks $B_{2}$ and $B_{3}$ along ${\cal
C}$: By (7), $\beta_{1}(B_{i})\geq w_{3^{\prime}}$ and $\beta_{2}(B_{i})\geq
w_{3^{\prime}}$ ($i\in\\{2,3\\}$). Since
$c(H)+\sum_{i}\beta(B_{i})\geq\delta+\beta(B_{1})+\beta(B_{2})+\beta(B_{3})$,
we get branch vector
$\displaystyle(\delta+(2\Delta_{3}-\delta)+2w_{3^{\prime}};\delta+(2w_{3}-\delta)+4\Delta_{3}+2w_{3^{\prime}})=(2w_{3};6w_{3}-2w_{3^{\prime}}).$
(13)
Case 2.2. There is no odd block along ${\cal C}$: Let $B_{2}$ be the block
along ${\cal C}$ containing vertex $v$. Then $B_{2}$ is an even block such
that $\mathrm{cut}_{U}(B_{2})$ is also included to $F$ in the branch where
$\mathrm{cut}_{U}(B_{1})$ is included to $F$. Then
$c(H)+\sum_{i}\beta(B_{i})\geq\delta+\beta(B_{1})+\beta(B_{2})$. If $B_{2}$ is
not a 2-pendent critical block, then
$\beta^{\prime}_{1}(B_{2})=2\Delta_{3}-\delta$,
$\beta^{\prime}_{2}(B_{2})=2w_{3}-\delta$ and
$\beta^{\prime\prime}_{2}(B_{2})\geq\delta$ (by Lemma 12), and we get branch
vector
$\displaystyle(\delta+2(2\Delta_{3}-\delta);\delta+(2w_{3}-\delta)+4\Delta_{3}+2w_{3})=(4\Delta_{3}-\delta;4w_{3}+4\Delta_{3}).$
(14)
Otherwise $B_{2}$ is a 2-pendent critical block. Then
$\beta^{\prime}_{1}(B_{2})=2\Delta_{3}-\gamma$ and
$\beta_{2}(B_{2})=\beta^{\prime}_{2}(B_{2})+\beta^{\prime\prime}_{2}(B_{2})=w(B_{2})\geq
2w_{3}+4w_{3^{\prime}}$ (by (7) and Lemma 12). We get branch vector
$(\delta+(2\Delta_{3}-\delta)+(2\Delta_{3}-\gamma);\delta+(2w_{3}-\delta)+4\Delta_{3}+(2w_{3}+4w_{3^{\prime}}))$,
i.e.,
$\displaystyle(4\Delta_{3}-\gamma;8w_{3}).$ (15)
Case 3. Block $B_{1}$ is even and ${\cal C}_{x}={\cal C}_{y}$ in
$(G_{1},F_{1})$ (see Fig. 4(b),(c)): First of all, we consider
$\beta^{\prime\prime}_{1}(B_{1})$, $\beta^{\prime\prime}_{2}(B_{1})$ and
others. We look at the circuit ${\cal C}_{x}$ in $(G_{1},F_{1})$. Except
blocks $\\{x\\}$ and $\\{y\\}$, there are some other blocks along ${\cal
C}_{x}$. Note that each block $B^{\prime}$ along ${\cal C}_{x}$ should be a
trivial or 2-pendent critical block since $B_{1}$ is a minimal normal block.
We distinguish three cases by considering the number of critical blocks along
${\cal C}_{x}$.
Case 3.1. $B_{1}$ is a 2-pendent cycle of length $\ell$ (all blocks along
${\cal C}_{x}$ are trivial in $(G_{1},F_{1})$): Then $\ell$ is an even integer
with $\ell=4$ or $\ell\geq 8$ (since $B_{1}$ is even and non-critical). When
$\ell=4$, in both branches, we have
$\beta(B_{1})=w(B_{1})=2w_{3}+2w_{3^{\prime}}$. Then we can branch with a
branch vector $[c(H)+\beta(B_{1})]_{2}=[\delta+2w_{3}+2w_{3^{\prime}}]_{2}$
covered by (8). Next we assume that $\ell\geq 8$. Now we may get only
$\beta^{\prime}_{1}(B_{1})=2\Delta_{3}-\delta$ and
$\beta^{\prime\prime}_{1}(B_{1})\geq 0$ instead of
$\beta_{1}(B_{1})=w(B_{1})$. But it holds that
$\beta_{2}(B_{1})=w(B_{1})=2w_{3}+(\ell-2)w_{3^{\prime}}\geq
2w_{3}+6w_{3^{\prime}}$. We get branch vector
$\displaystyle(\delta+(2\Delta_{3}-\delta)+\xi_{1};\delta+(2w_{3}+6w_{3^{\prime}})+\xi_{2}).$
(16)
Case 3.2. There are only three blocks $\\{x\\}$, $\\{y\\}$ and $B^{\prime}$
along ${\cal C}_{x}$ in $(G_{1},F_{1})$, where $B^{\prime}$ is a 2-pendent
critical block: Now $x$ and $y$ are adjacent (see Fig. 4(b)). We assume that
$x_{2}=y$, $y_{2}=x$, and $x_{1},y_{1}\in B_{1}$. We look at $(G_{2},F_{2})$
wherein $x$ and $y$ are degree-2 vertices. After including edges $xy$,
$xx_{1}$ and $yy_{1}$ to $F$, $B^{\prime}$ becomes a $4$-cut reducible graph
by Lemma 6. Then $\mu$ decreases by $w(B^{\prime})$ after applying the 4-cut
reduction. Then we know that
$\beta_{2}(B_{1})=w(B_{1})=w(x)+w(y)+w(B^{\prime})\geq
4w_{3}+4w_{3^{\prime}}$. Therefore, we get branch vector
$(\delta+(2\Delta_{3}-\delta)+\xi_{1};\delta+(4w_{3}+4w_{3^{\prime}})+\xi_{2}),$
which is covered by (16).
Case 3.3. $B_{1}$ is not a 2-pendent cycle and there are more than three
blocks along ${\cal C}_{x}$ in $(G_{1},F_{1})$: Then there is a nontrivial and
nonreducible block $B^{\prime}_{1}$ along ${\cal C}_{x}$ in $(G_{1},F_{1})$,
where $B^{\prime}_{1}$ is a 2-pendent critical block since $B_{1}$ is a
minimal normal block. For this case, we only get the following branch vector
by branching on ${\cal C}$:
$(\delta+(2\Delta_{3}-\delta)+\xi_{1};\delta+(2w_{3}-\delta)+\beta^{\prime\prime}_{2}(B_{1})+\xi_{2})=$
$\displaystyle(2\Delta_{3}+\xi_{1};2w_{3}+\beta^{\prime\prime}_{2}(B_{1})+\xi_{2}).$
(17)
In fact, the branch vector (17) in Case 3.3 can be the bottleneck in the
analysis of our algorithm. However, in $(G_{1},F_{1})$, circuit ${\cal C}_{x}$
is a circuit with only trivial and 2-pendent critical blocks. In our
algorithm, circuit ${\cal C}_{x}$ will be one of the circuits for the next
branching, and it will never be destroyed until we branch on it. In fact,
branching on ${\cal C}_{x}$ proves a branch vector better than (17). For the
purpose of analysis, we derive a branch vector for the three branches, i.e.,
the branching on ${\cal C}$ followed by the branching on ${\cal C}_{x}$ in
$(G_{1},F_{1})$ in the branch of including $\mathrm{cut}_{U}(B_{1})$ into $F$.
In Case 3.3, we see that: either ($a$) ${\cal C}_{x}$ has at least two trivial
blocks $B^{\prime}_{2}$ and $B^{\prime}_{3}$ different from $\\{x\\}$ and
$\\{y\\}$ (since the number of odd blocks is even); or ($b$) all blocks other
than $\\{x\\}$ and $\\{y\\}$ are 2-pendent critical.
Case ($a$): There is also a 2-pendent critical block $B^{\prime}_{1}$ along
${\cal C}_{x}$ (since $B_{1}$ is not a 2-pendent cycle). In $(G_{2},F_{2})$,
we can see that $\beta^{\prime\prime}_{2}(B_{1})\geq
c(B_{1})+w(B^{\prime}_{2})+w(B^{\prime}_{3})=\delta+2w_{3^{\prime}}$ (since
$B^{\prime}_{2}$ and $B^{\prime}_{3}$ are trivial blocks). In $(G_{1},F_{1})$,
by branching on ${\cal C}_{x}$, we can get branch vector
$(c(B_{1})+\beta_{1}(\\{x\\})+\beta_{1}(\\{y\\})+\sum_{i=1}^{3}\beta_{1}(B^{\prime}_{i});c(B_{1})+\beta_{2}(\\{x\\})+\beta_{2}(\\{y\\})+\sum_{i=1}^{3}\beta_{2}(B^{\prime}_{i}))=(\delta+4w_{3^{\prime}}+(2\Delta_{3}-\gamma);\delta+4w_{3^{\prime}}+(2w_{3}+4w_{3^{\prime}}))=$
$(\delta+2w_{3}+2w_{3^{\prime}}-\gamma;\delta+2w_{3}+8w_{3^{\prime}}).$
By combining it with (17) and taking
$\beta^{\prime\prime}_{2}(B_{1})=\delta+2w_{3^{\prime}}$, we get branch vector
$\displaystyle(2\Delta_{3}\\!+\\!\xi_{1}\\!\\!+\\!(\delta\\!+\\!2w_{3}\\!+\\!2w_{3^{\prime}}\\!-\\!\gamma);2\Delta_{3}\\!+\\!\xi_{1}\\!\\!+\\!(\delta\\!+\\!2w_{3}\\!+\\!8w_{3^{\prime}}\\!);2w_{3}\\!+\\!\delta\\!+\\!2w_{3^{\prime}}\\!+\\!\xi_{2}).~{}~{}~{}~{}$
(18)
Case ($b$): There are at least two 2-pendent critical blocks $B^{\prime}_{1}$
and $B^{\prime}_{2}$ along ${\cal C}_{x}$ (since there are at least four
blocks among ${\cal C}_{x}$). In $(G_{2},F_{2})$, we may get only
$\beta^{\prime\prime}_{2}(B_{1})\geq\delta$. In $(G_{1},F_{1})$, by branching
on ${\cal C}_{x}$, at least we can get branch vector
$(\delta+2w_{3^{\prime}}+2(2\Delta_{3}-\gamma);\delta+2w_{3^{\prime}}+2(2w_{3}+4w_{3^{\prime}}).$
By combining it with (17) and taking $\beta^{\prime\prime}_{2}(B_{1})=\delta$,
we get branch vector
$\displaystyle(2\Delta_{3}\\!+\\!\xi_{1}\\!+\\!(\delta\\!+\\!4w_{3}\\!-\\!2w_{3^{\prime}}\\!-\\!2\gamma);2\Delta_{3}\\!+\\!\xi_{1}\\!+\\!(\delta\\!+\\!4w_{3}\\!+\\!10w_{3^{\prime}});2w_{3}\\!+\\!\delta\\!+\\!\xi_{2}).$
(19)
Finally, by replacing $\xi_{1}$ and $\xi_{2}$ in (16), (18) and (19)
respectively with the bounds in (12), we get the following six branch vectors
$\displaystyle(4\Delta_{3}-\gamma;\delta+4w_{3}+10w_{3^{\prime}}),$ (20)
$\displaystyle(2w_{3};\delta+2w_{3}+8w_{3^{\prime}}),$ (21)
$\displaystyle(\delta+6w_{3}-2w_{3^{\prime}}-2\gamma;\delta+6w_{3}+4w_{3^{\prime}}-\gamma;\delta+4w_{3}+6w_{3^{\prime}}),$
(22)
$\displaystyle(\delta+4w_{3}+2w_{3^{\prime}}-\gamma;\delta+4w_{3}+8w_{3^{\prime}};\delta+2w_{3}+4w_{3^{\prime}}),$
(23)
$\displaystyle(\delta+8w_{3}-6w_{3^{\prime}}-3\gamma;\delta+8w_{3}+6w_{3^{\prime}}-\gamma;\delta+4w_{3}+4w_{3^{\prime}}),$
(24)
and
$\displaystyle(\delta+6w_{3}-2w_{3^{\prime}}-2\gamma;\delta+6w_{3}+10w_{3^{\prime}};\delta+2w_{3}+3w_{3^{\prime}}).$
(25)
### 9.3 Overall analysis
A quasiconvex program is obtained from (6) and 13 branch vectors (from (8) to
(15) and from (20) to (25)) in our analysis. There is a general method to
solve quasiconvex programs [4]. For our quasiconvex program, we observe a
simple way to solve it. We look at (8) and (11). Note that
$\min\\{6w_{3^{\prime}}+\gamma,4w_{3}-2w_{3^{\prime}}\\}$ under the constraint
$2\Delta_{3}\geq\gamma$ gets the maximum value at the time when
$6w_{3^{\prime}}+\gamma=4w_{3}-2w_{3^{\prime}}$ and $2\Delta_{3}=\gamma$. We
get $w_{3^{\prime}}={1\over 3}$ and $\gamma={4\over 3}$. With this setting, we
can verify that when $\delta\in[1.2584,1.2832]$, all branch vectors other than
(8) and (11) in our quasiconvex program will not be the bottleneck. We get a
time bound $O^{*}(\alpha^{\mu})$ with $\alpha=2^{3\over 10}<1.2312$ by setting
$w_{3^{\prime}}={1\over 3}$,$\gamma={4\over 3}$ and $\delta\in[1.2584,1.2832]$
for our problem. The bottlenecks in the analysis are (8), (11) and
$2\Delta_{3}\geq\gamma$ in (6).
###### Theorem 9.1
TSP in an $n$-vertex graph $G$ with maximum degree 3 can be solved in
$O^{*}(1.2312^{n})$ time and polynomial space.
## 10 Concluding Remarks
In this paper, we have presented an improved exact algorithm for TSP in
degree-3 graphs. The basic operation in the algorithm is to process the edges
in a circuit by either including an edge in the circuit to the solution or
excluding it from the solution. The algorithm is analyzed by using the measure
and conquer method and an amortization scheme over the cut-circuit structure
of graphs, wherein we introduce not only weights of vertices but also weights
of $U$-components to define the measure of an instance.
The idea of amortization schemes introducing weights on components may yield
better bounds for other exact algorithms for graph problems if how
reduction/branching procedures change the system of components is successfully
analyzed.
## References
* [1] Bjorklund, A.: Determinant sums for undirected Hamiltonicity. In: Proc. 51st Annual IEEE Symp. on Foundations of Computer Science (2010) 173-182
* [2] Bjorklund, A., Husfeldt, T., Kasaki, P. and Koivisto, M.: The travelling salesman problem in bounded degree graphs. In: ICALP 2008, LNCS 5125 (2008) 198-209
* [3] Dorn, F., Penninkx, E., Bodlaender, H. L., and Fomin, F. V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica 58(3) (2010) 790-810
* [4] Eppstein, D.: Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms. ACM Trans. on Algorithms 2(4) (2006) 492-509
* [5] Eppstein, D.: The traveling salesman problem for cubic graphs. J. Graph Algorithms and Applications 11(1) (2007) 61-81
* [6] Fomin, F., Grandoni, F., Kratsch, D. Measure and Conquer: Domination - A Case Study, ICALP Springer-Verlag LNCS 3580 (2005), pp. 191–203.
* [7] Fomin, F. V., Kratsch, D.: Exact Exponential Algorithms, Springer (2010)
* [8] Gebauer, H.: Finding and enumerating Hamilton cycles in 4-regular graphs. Theoretical Computer Science 412(35) (2011) 4579-4591
* [9] Iwama, K. and Nakashima, T.: An improved exact algorithm for cubic graph TSP. In: COCOON 2007. LNCS 4598 (2007) 108-117
* [10] Nagamochi, H., Ibaraki, T.: A linear time algorithm for computing 3-edge-connected components in multigraphs, J. of Japan Society for Industrial and Applied Mathematics, 9(2) (1992) 163-180
* [11] Nagamochi, H., Ibaraki, T.: Algorithmic Aspects of Graph Connectivities, Encyclopedia of Mathematics and Its Applications, Cambridge University Press (2008)
* [12] Woeginger, G. J.: Exact algorithms for NP-hard problems: A survey. In: Combinatorial Optimization. LNCS 2570 (2003) 185-207
* [13] Xiao, M. and Nagamochi, H.: An improved exact algorithm for TSP in degree-$4$ graphs. In: COCOON 2012. LNCS 7434 (2012) 74-85
|
arxiv-papers
| 2012-12-31T07:28:15 |
2024-09-04T02:49:39.822786
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mingyu Xiao and Hiroshi Nagamochi",
"submitter": "Mingyu Xiao",
"url": "https://arxiv.org/abs/1212.6831"
}
|
1301.0175
|
Holomorphic Lagrangian fibrations on hypercomplex manifolds
Andrey Soldatenkov111Andrey Soldatenkov is partially supported by AG
Laboratory NRU-HSE, RF government grant, ag. 11.G34.31.0023, Misha
Verbitsky222Misha Verbitsky is partially supported by RFBR grant
10-01-93113-NCNIL-a, RFBR grant 09-01-00242-a, Science Foundation of the SU-
HSE award No. 10-09-0015 and AG Laboratory NRU-HSE, RF government grant, ag.
11.G34.31.0023.
Abstract
A hypercomplex manifold is a manifold equipped with a triple of complex
structures satisfying the quaternionic relations. A holomorphic Lagrangian
variety on a hypercomplex manifold with trivial canonical bundle is a
holomorphic subvariety which is calibrated by a form associated with the
holomorphic volume form; this notion is a generalization of the usual
holomorphic Lagrangian subvarieties known in hyperkähler geometry. An HKT
(hyperkähler with torsion) metric on a hypercomplex manifold is a metric
determined by a local potential, in a similar way to the Kähler metric. We
prove that a base of a holomorphic Lagrangian fibration is always Kähler, if
its total space is HKT. This is used to construct new examples of hypercomplex
manifolds which do not admit an HKT structure.
###### Contents
1. 1 Introduction
1. 1.1 HKT metrics and $SL(n,{\mathbb{H}})$-structures
2. 1.2 Calibrations on manifolds and Lagrangian fibrations
2. 2 Preliminaries
1. 2.1 Hypercomplex manifolds and quaternionic Dolbeault complex
2. 2.2 $SL(n,{\mathbb{H}})$-manifolds and holomorphic Lagrangian calibration
3. 3 Holomorphic Lagrangian fibrations on $SL(n,{\mathbb{H}})$-manifolds
1. 3.1 HKT structures and Lagrangian fibrations
2. 3.2 Examples of Lagrangian fibrations
## 1 Introduction
### 1.1 HKT metrics and $SL(n,{\mathbb{H}})$-structures
Let $I,J,K$ be complex structures on a manifold $M$ satisfying the
quaternionic relation $I\circ J=-J\circ I=K$. Then $(M,I,J,K)$ is called a
hypercomplex manifold. Hypercomplex manifolds are quaternionic analogues of
complex manifolds, in the same way as hyperkähler manifolds are quaternionic
analogues of Kähler manifolds. However, the theory of hypercomplex manifolds
is much richer in examples and (unlike that of hyperkähler manifolds) still
not well understood.
For more details on hypercomplex and hyperkähler structures, please see
Subsection 2.1.
The term hypercomplex is due to C. P. Boyer, [Bo], who classified compact
hypercomplex manifolds in $\dim_{\mathbb{H}}=1$. These are: K3 surfaces,
compact 2-dimensional complex tori, and Hopf surfaces. However, in dimension
$>1$ there is no classification apparent.
The notion of hypercomplex structure was considered as early as in 1955 by
[Ob], who proved existence and uniqueness of a torsion-free connection
preserving a hypercomplex structure. However, the first non-trivial examples
were obtained only in 1988 by physicists Ph. Spindel, A. Sevrin, W. Troost, A.
Van Proeyen ([SSTV]), and the intensive study of hypercomplex structures began
with the paper [J] by D. Joyce, who classified hypercomplex structures on
homogeneous spaces, and constructed one on each compact Lie group multiplied
by a compact torus of appropriate dimension.
The main geometric tool allowing one to work with the hypercomplex manifolds
is a notion of an HKT (“hyperkähler with torsion”) metric, due to physicists
Howe and Papadopoulos ([HP]). For a definition and a discussion of HKT
structure, please see Subsection 2.1. HKT structures were much used in physics
since mid-1990-ies ([GP1], [GP2], [GPS]), and then mathematicians started
using HKT structures to study the hypercomplex manifolds.
In [V1], the second named author developed Hodge theory for HKT manifolds, and
it turned out to be quite useful. To illustrate the usability of HKT metrics,
let us quote the following result, obtained in [V4]. Let $(M,I,J,K)$ be a
hypercomplex manifold with $(M,I)$ a complex manifold of Kähler type. Then
$(M,I)$ is in fact hyperkähler (that it, admits a hyperkähler structure). The
idea of the proof is that any Kähler metric on $(M,I)$ gives an HKT metric on
$(M,I,J,K)$, and this metric can be used to apply Hodge theory. The interplay
between Kähler geometry and HKT geometry is quite extensive, and can be used
to study the geometry of hypercomplex manifolds extensively.
In the present paper, we obtain a result in a similar vein, constructing a
Kähler metric on a base of holomorphic Lagrangian fibration on an HKT
manifold.
Of course, the notion of “holomorphic Lagrangian fibration” is itself highly
non-trivial, because an HKT manifold is not necessarily holomorphically
symplectic. It was developed in [GV], using the theory of calibrations and the
holonomy of the Obata connection.
The Obata connection is an immensely powerful tool, known since 1950-ies
([Ob]), but even the most primitive invariants of Obata connection, such as
its holonomy, are hard to compute, and still unknown in most cases. When
$(M,I,J,K)$ admits a hyperkähler metric, Obata connection is equal to the
Levi-Civita connection of the hyperkähler metric and its holonomy is
$\operatorname{Sp}(n)$. The converse is also true: whenever holonomy of a
torsion-free connection is $Sp(n)$, it is a Levi-Civita connection of a
hyperkähler manifold.
In a general situation, it’s hard to say anything definite about the holonomy
$\operatorname{Hol}(\nabla)$ of the Obata connection. It is clear that
$\operatorname{Hol}(\nabla)\subset GL({\mathbb{H}},n)$, but the equality is
rarely realized. In fact, before the publication of the paper [Sol] in 2011,
there was no single example of a hypercomplex manifold with
$\operatorname{Hol}(\nabla)=GL({\mathbb{H}},n)$. In [Sol], the first named
author provided an example of such a manifold, by proving that
$\operatorname{Hol}(\nabla)=GL({\mathbb{H}},2)$ for the homogeneous
hypercomplex structure on $SU(3)$, constructed by D. Joyce in [J].
However, there are many examples of hypercomplex manifolds with
$\operatorname{Hol}(\nabla)$ $\subsetneq GL({\mathbb{H}},n)$. Two biggest
known families of hypercomplex manifolds are the Joyce’s homogeneous manifolds
from [J], and hypercomplex nilmanifolds ([BD]). As shown in [BDV], for all
nilmanifolds, the holonomy of the Obata connection lies in
$SL({\mathbb{H}},n)=[GL({\mathbb{H}},n),GL({\mathbb{H}},n)]\subset
GL({\mathbb{H}},n).$
A hypercomplex manifold with holonomy in $SL(n,{\mathbb{H}})$ is called an
$SL(n,{\mathbb{H}})$-manifold. Such manifold might have holonomy group which
is strictly less than $SL(n,{\mathbb{H}})$; in fact, no example of a manifold
with $\operatorname{Hol}(\nabla)=SL({\mathbb{H}},n)$ is known so far.
For any $SL(n,{\mathbb{H}})$-manifold, the Obata connection on the canonical
bundle $K(M,I)=\Lambda^{2n,0}(M,I)$ of $(M,I)$ preserves a non-trivial
section. This implies that $K(M,I)$ is holomorphically trivial (see [V3]). In
the presence of an HKT metric, the converse is also true: any compact
hypercomplex manifold with holomorphically trivial canonical bundle $K(M,I)$
and an HKT metric satisfies $\operatorname{Hol}(\nabla)\subset
SL({\mathbb{H}},n)$. This result is obtained in [V3] using the Hodge theory
for HKT manifolds developed in [V1].
The Hodge-theoretic constructions of [V1] work for
$SL(n,{\mathbb{H}})$-manifolds with HKT-structure especially well. For such
manifolds, one obtains Hodge-type decomposition on the holomorphic cohomology
bundle $H^{*}({\cal O}_{(M,I)})$.
### 1.2 Calibrations on manifolds and Lagrangian fibrations
Let $M$ be a Riemannian manifold. A calibration on $M$ is a closed $k$-form
$\eta$, such that $\eta(x_{1},...,x_{k})\leqslant 1$ for each orthonormal
$k$-tuple $x_{1},...,x_{k}\in TM$. A $k$-dimensional oriented subspace
$V\subset T_{x}M$ is called calibrated if the Riemannian volume of $V$ is
equal to $\eta{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$, and a subvariety
$Z\subset M$ of dimension $k$ is called calibrated if its singularities are of
Hausdorff codimension $\geqslant 1$, and all smooth tangent planes
$T_{z}Z\subset T_{z}M$ are calibrated.
The theory of calibrations has a long and distinguished history, starting from
the work of Harvey and Lawson [HL]. In [GV], several families of calibrations
were constructed on hyperkähler manifolds, using the quaternionic linear
algebra. Let $(M,I,J,K)$ be a hyperkähler manifold, and
$\omega_{I},\omega_{J},\omega_{K}$ its Kähler forms. These forms generate an
interesting commutative subalgebra in $\Lambda^{*}(M)$ (see [GV], and [V0] for
structure results about this algebra). In [GV], Grantcharov and Verbitsky
discovered several new calibrations which are expressed as polynomials of
$\omega_{I},\omega_{J},\omega_{K}$. One of these calibrations, denoted as
$\Psi$ in the sequel (see (2.2)), is called holomorphic Lagrangian
calibration. On a hyperkähler manifold, it calibrates holomorphic Lagrangian
subvarieties in $(M,I)$.
It is surprising (and quite wonderful) that this form remains closed, even
when the metric is not hyperkähler, provided that the Obata connection of
$(M,I,J,K)$ belongs to $SL(n,{\mathbb{H}})$. The hyperkähler condition can be
weakened drastically: it suffices to assume that the metric $g$ is
quaternionic Hermitian, and the Obata-parallel section of $K(M,I)$ has
constant length with respect to $g$. In this case $\Psi$ is closed. If
rescaled properly, this form is a calibration, calibrating a new class of
complex subvarieties of $(M,I)$ called holomorphic Lagrangian (see [GV] and
3.1 for more details).
In [SV] it was shown that holomorphic Lagrangian subvarieties of
$SL(n,{\mathbb{H}})$-manifolds exist only in a countable number of complex
structure of the form $L=aI+bJ+cK$ (such complex structures are called induced
by a hypercomplex structure, see Subsection 2.1). In the present paper, we
study holomorphic Lagrangian fibrations on hypercomplex manifolds. Such
fibrations are often present in examples ([GV]; see also Subsection 3.2).
The holomorphic Lagrangian fibrations are of significant interest for
mathematicians and physicists; for an early survey of the subject, please see
the paper [Saw], and for applications, see e.g. [KRS]. The non-Kähler version
of this geometry defined above should be even more useful, because the number
of examples is much greater.
In the present paper, we consider HKT metrics on
$SL(n,{\mathbb{H}})$-manifolds admitting holomorphic Lagrangian fibrations. We
prove that a base of such fibration is always Kähler, if the fibration is
smooth (3.1). This allows us to construct new examples of hypercomplex
manifolds admitting no HKT metrics.
Originally, it was conjectured that any compact hypercomplex manifold admits
an HKT metric. Examples of hypercomplex nilmanifolds not admitting HKT metrics
were constructed by Fino and Grantcharov ([FG]). Since then, hypercomplex
nilmanifolds admitting HKT metrics were classified completely in [BDV]. It was
found that in fact most of hypercomplex nilmanifolds are not HKT. However, no
other hypercomplex manifolds not admitting an HKT metrics were known before
now.
Acknowledgements. The work on this paper began during the visit to the
University of Science and Technology of China. The authors would like to
express their gratitude to USTC for the hospitality an to Prof. Xiuxiong Chen
for the invitation.
## 2 Preliminaries
### 2.1 Hypercomplex manifolds
and quaternionic Dolbeault complex
A $C^{\infty}$-manifold $M$ is called hypercomplex if $M$ is equipped with
complex structures $I$, $J$, $K$, that satisfy the quaternionic relations
$IJ=-JI=K.$
It was shown in [Ob] that any hypercomplex manifold admits a unique torsion-
free connection that preserves $I,J$ and $K$. This connection is called the
Obata connection. Note that the holonomy of the Obata connection is a subgroup
of $GL(n,{\mathbb{H}})$.
Any almost-complex structure which is preserved by a torsion-free connection
is integrable (this follows from Newlander-Nirenberg theorem). Therefore, for
any $a,b,c\in{\mathbb{R}}$ with $a^{2}+b^{2}+c^{2}=1$, the almost-complex
structure $L=aI+bJ+cK$ is integrable. We denote by $(M,L)$ the corresponding
complex manifold. Such complex structures are called induced by quaternions.
Let $(M,I,J,K)$ be a hypercomplex manifold of real dimension $4n$. The
hypercomplex structure induces the action of $SU(2)$ on all tensor bundles
over $M$. Recall that any irreducible complex representation of $SU(2)$ is of
the form $S^{k}U$, where $U$ is the standard two-dimensional representation
and $S^{k}$ denotes the symmetric power. We will refer to any representation
of the form $\left(S^{k}(U)\right)^{\oplus m}$ (for arbitrary $m$) as a weight
k representation of $SU(2)$.
We make the following observation about the weight decomposition of the
exterior algebra
$\Lambda_{\mathbb{C}}^{*}M=\Lambda^{*}M\otimes_{\mathbb{R}}{\mathbb{C}}$.
First, note that $\Lambda^{1}_{\mathbb{C}}M$ is an $SU(2)$ representation of
weight one. It follows from the Clebsch-Gordan formula that
$\Lambda^{k}_{\mathbb{C}}M$ is a sum of representations of weight $\leq k$. By
duality, the same is true about $\Lambda^{4n-k}_{\mathbb{C}}M$. Denote by
$\Lambda^{k}_{+}M$ the maximal subrepresentation of weight $k$ for $k\leq 2n$
and of weight $4n-k$ for $k>2n$ inside $\Lambda^{k}_{\mathbb{C}}M$. Denote by
$\Pi_{+}\colon\Lambda^{*}M\to\Lambda^{*}_{+}M$ the equivariant projection onto
the component of maximal weight.
Note that the Hodge decomposition with respect to an arbitrary complex
structure (we can pick $I$ without loss of generality) is compatible with the
weight decomposition:
$\Lambda^{k}_{+}M=\bigoplus_{p+q=k}\Lambda^{p,q}_{I,+}M,$
where $\Lambda^{p,q}_{I,+}M=\Lambda^{p,q}_{I}M\cap\Lambda^{k}_{+}M$. This is
actually a weight decomposition of an $SU(2)$-representation for a special
choice of Cartan subalgebra corresponding to $I$. Therefore,
$\Lambda^{k}_{+}M$ is generated by $\Lambda^{k,0}_{I,+}M=\Lambda^{k,0}_{I}M$
as an $SU(2)$-module. It follows that the summands $\Lambda^{p,q}_{I,+}M$ are
of the same dimension and actually isomorphic to $\Lambda^{p+q,0}_{I}M$. We
will denote these isomorphisms by
$\mathcal{R}_{p,q}\colon\Lambda^{p+q,0}_{I}M\tilde{\to}\Lambda^{p,q}_{I,+}M,$
see [V2] and formula (2.3) below.
Given a hyperhermitian metric $g$ on $M$, we can consider the corresponding
Kähler forms $\omega_{I}$, $\omega_{J}$ and $\omega_{K}$. It is easy to check
that the form
$\Omega_{I}=\omega_{J}+\sqrt{-1}\omega_{K}$
lies in $\Lambda^{2,0}_{I}M$. If $d\Omega_{I}=0$, then the manifold $M$ is
called hyperkähler. This condition implies that $g$ is Kähler with respect to
any induced complex structure. If $\Omega_{I}$ satisfies a strictly weaker
condition $\partial\Omega_{I}=0$, then $M$ is called an HKT manifold, and $g$
is called an HKT metric (HKT stands for HyperKähler with Torsion; see [GP] for
a general introduction to HKT geometry).
We can reformulate the HKT condition as follows. Denote by
$d_{+}\colon\Lambda^{k}_{+}M\to\Lambda^{k+1}_{+}M$ the composition
$\Pi_{+}\circ d$ of the de Rham differential and the projection onto the
component of maximal weight. Note that $\omega_{I}\in\Lambda^{1,1}_{I,+}M$.
Then by Theorem 5.7 in [V1], the condition $\partial\Omega_{I}=0$ is
equivalent to
$d_{+}\omega_{I}=0.$ (2.1)
That is, for an HKT metric $g$ the exterior differential of its Kähler form
$d\omega_{I}$ has weight one. This observation will be important for the proof
of the main theorem.
### 2.2 $SL(n,{\mathbb{H}})$-manifolds
and holomorphic Lagrangian calibration
As it was mentioned above, the holonomy of the Obata connection on a
hypercomplex manifold is a subgroup of $GL(n,{\mathbb{H}})$. Recall that this
group can be defined as follows. Consider a vector space $V$ of real dimension
$4n$ with a quaternionic structure $I,J,K$. Then $GL(n,{\mathbb{H}})$ consists
of those linear transformations of $V$ that commute with $I$, $J$ and $K$.
Consider the Hodge decomposition
$V_{\mathbb{C}}=V\otimes{\mathbb{C}}=V^{1,0}_{I}\oplus V^{0,1}_{I}$. The
action of $GL(n,{\mathbb{H}})$ preserves this decomposition, hence it induces
an action on all exterior powers of $V^{1,0}_{I}$. Denote by
$SL(n,{\mathbb{H}})$ the subgroup of $GL(n,{\mathbb{H}})$ consisting of those
elements that act identically on $V^{2n,0}_{I}$.
Consider a hypercomplex manifold $(M,I,J,K)$. If the holonomy of the Obata
connection is contained in $SL(n,{\mathbb{H}})$, then we call $M$ an
$SL(n,{\mathbb{H}})$-manifold. It follows almost immediately from the
definition (see [V3], Claim 1.1), that on an $SL(n,{\mathbb{H}})$-manifold the
canonical bundle $\Lambda^{2n,0}_{I}M$ is flat with respect to the Obata
connection. It also follows ([V3], Claim 1.2), that any parallel section of
$\Lambda^{2n,0}_{I}M$ is holomorphic. Moreover, if the manifold $(M,I,J,K)$ is
HKT and compact, the condition $\operatorname{Hol}(M)\subset
SL(n,{\mathbb{H}})$ is equivalent to holomorphic triviality of the canonical
bundle ([V3], Theorem 2.3).
Denote by $\Phi_{I}\in\Lambda^{2n,0}_{I}M$ a parallel section which we call a
holomorphic volume form. Note that the operator $J$ defines a real structure
(that is, a complex-antilinear involution) on $\Lambda^{2n,0}_{I}M$ by
$\eta\mapsto\overline{J\eta}$. We can always assume that $\Phi_{I}$ is real
with respect to this structure.
Given an $SL(n,{\mathbb{H}})$-manifold $M$ with a hyperhermitian metric $g$,
denote by $\Phi_{I}$ the holomorphic volume form and by
$\Omega_{I}=\omega_{J}+\sqrt{-1}\omega_{K}$ the $(2,0)$-form associated with
the metric. In [GV], a number of calibrations (in the sense of [HL]) on
$SL(n,{\mathbb{H}})$-manifolds were constructed. In particular, it was shown
that the form
$\Psi=(-\sqrt{-1})^{n}\mathcal{R}_{n,n}(\Phi_{I})\in\Lambda^{n,n}_{I,+}M$
(2.2)
is a calibration with respect to a properly rescaled metric. It was shown that
this form calibrates $\Omega_{I}$-Lagrangian subvarieties of $M$. Recall, that
a complex (with respect to the complex structure $I$) subvariety $N\subset M$
of complex dimension $n$ is called $\Omega_{I}$-Lagrangian if the restriction
of $\Omega_{I}$ to the smooth part of $N$ vanishes. This condition is
equivalent to the following: $T_{x}N$ is orthogonal to $J(T_{x}N)$ with
respect to the metric $g$ at any smooth point $x\in N$. To see this, observe
that for any vector fields $X,Y\in T^{1,0}_{I}M$ we have
$\Omega_{I}(X,Y)=2g(JX,Y)$.
We will not need the construction from [GV] in its full generality, but only
some basic properties of the form $\Psi$. For the convenience of the reader we
will formulate these properties in the following lemma and give a short proof,
independent from [GV].
Lemma 2.1: Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold of real
dimension $4n$, $\Phi_{I}\in\Lambda^{2n,0}_{I}M$ a holomorphic volume form and
$\Psi=(-\sqrt{-1})^{n}\mathcal{R}_{n,n}(\Phi_{I})\in\Lambda^{n,n}_{I,+}M$. Let
$N\subset M$ be an $I$-complex subvariety, $\dim_{\mathbb{C}}N=n$, with
$T_{x}N\cap J(T_{x}N)=0$ at all smooth points $x\in N$. Then:
1. 1.
$d\Psi=0$,
2. 2.
$\Psi|_{N}$ is a strictly positive volume form on the smooth part of $N$.
Proof: 1\. As it was mentioned above (see also formula (2.3) below), the
operator $\mathcal{R}_{n,n}$ is induced by the $SU(2)$-action on the exterior
algebra of $M$. Since the Obata connection $\nabla$ preserves the hypercomplex
structure, it commutes with the $SU(2)$-action and with the operator
$\mathcal{R}_{n,n}$. It was explained above that $\nabla\Phi_{I}=0$, thus
$\nabla\Psi=0$ also. The Obata connection is torsion-free, hence
$\nabla\Psi=0$ implies $d\Psi=0$.
2\. Let $x\in N$ be a smooth point. By the assumptions of the lemma, we can
choose a basis $e_{1},\ldots,e_{n}$ of $T^{1,0}_{I,x}N$, such that
$e_{1},\ldots,e_{n},J\overline{e}_{1},\ldots,J\overline{e}_{n}$ will form a
basis of $T^{1,0}_{I,x}M$. To show that $\Psi|_{N}$ is strictly positive, we
have to evaluate the form $\Psi$ on the polyvector
$\xi=(-\sqrt{-1})^{n}e_{1}\wedge\overline{e}_{1}\wedge\cdots\wedge
e_{n}\wedge\overline{e}_{n}\in\Lambda^{n,n}_{I}(T_{x}M)$.
We will use the following explicit description of the operator
$\mathcal{R}_{n,n}$. Consider the operators: $\mathcal{H}=-\sqrt{-1}I$,
$\mathcal{X}=\frac{1}{2}(\sqrt{-1}K-J)$ and
$\mathcal{Y}=\frac{1}{2}(\sqrt{-1}K+J)$ acting on $\Lambda^{1}_{\mathbb{C}}M$.
It is straightforward to check that these operators satisfy the standard
$\mathfrak{sl}_{2}({\mathbb{C}})$ commutator relations:
$[\mathcal{X},\mathcal{Y}]=\mathcal{H}$,
$[\mathcal{H},\mathcal{X}]=2\mathcal{X}$,
$[\mathcal{H},\mathcal{Y}]=-2\mathcal{Y}$. Also observe that
$\mathcal{H}|_{\Lambda^{1,0}_{I}M}=1,\hskip
10.0pt\mathcal{X}|_{\Lambda^{1,0}_{I}M}=0,\hskip
10.0pt\mathcal{Y}|_{\Lambda^{1,0}_{I}M}=J,$
$\mathcal{H}|_{\Lambda^{0,1}_{I}M}=-1,\hskip
10.0pt\mathcal{X}|_{\Lambda^{0,1}_{I}M}=-J,\hskip
10.0pt\mathcal{Y}|_{\Lambda^{0,1}_{I}M}=0.$
We can extend $\mathcal{H}$, $\mathcal{X}$, $\mathcal{Y}$ as derivations to
the whole exterior algebra. Then we have
$\mathcal{R}_{n,n}=\mathcal{Y}^{n},$ (2.3)
and
$\langle\Psi,\xi\rangle=(-\sqrt{-1})^{n}\langle\mathcal{Y}^{n}\Phi_{I},\xi\rangle=(\sqrt{-1})^{n}\langle\Phi_{I},\mathcal{Y}^{n}\xi\rangle.$
Observe, that $\mathcal{Y}|_{T^{1,0}_{I}M}=0$,
$\mathcal{Y}|_{T^{0,1}_{I}M}=J$, and since $\mathcal{Y}$ acts as a derivation,
we have
$\mathcal{Y}^{n}\xi=n!(-\sqrt{-1})^{n}e_{1}\wedge
J\overline{e}_{1}\wedge\cdots\wedge e_{n}\wedge J\overline{e}_{n}.$
The volume form is given by $\Phi_{I}=a\,e^{*}_{1}\wedge
J\overline{e^{*}_{1}}\wedge\cdots\wedge e^{*}_{n}\wedge J\overline{e^{*}_{n}}$
for some positive real number $a$, and we see that $\langle\Psi,\xi\rangle>0$.
## 3 Holomorphic Lagrangian fibrations
on $SL(n,{\mathbb{H}})$-manifolds
### 3.1 HKT structures and Lagrangian fibrations
Definition 3.1: A holomorphic Lagrangian subvariety of an
$SL(n,{\mathbb{H}})$-manifold is a subvariety calibrated by the form $\Psi$ of
2.2.
Definition 3.2: Let $M$ be an $SL(n,{\mathbb{H}})$-manifold, and
$\varphi\colon(M,I)\to X$ a smooth holomorphic fibration. It is called a
holomorphic Lagrangian fibration if all its fibers are holomorphic Lagrangian
subvarieties. We will say that the fibration $\varphi\colon(M,I)\to X$ is
smooth if $\varphi$ is a submersion (hence all the fibers are smooth).
Remark 3.3: Suppose that we are given an arbitrary holomorphic fibration
$\varphi\colon(M,I)\to X$ with $M$ compact and $\dim_{\mathbb{C}}X={1\over
2}\dim_{\mathbb{C}}M$. Denote by $F_{x}$ the fiber of $\varphi$ over $x\in X$.
Consider the set $U=\\{x\in X\colon J(TF_{x})\cap TF_{x}=0\\}$. It is clear
that $U$ is open and all the fibers over $U$ can be made Lagrangian with
respect to some hyperhermitian metric. Namely, the condition $J(TF_{x})\cap
TF_{x}=0$ implies that each fiber of the vector bundle $J(TF_{x})$ projects
onto $T_{x}X$ by $\varphi$, so that we can lift any Hermitian metric from $U$
to $J(TF_{x})$. Then we can uniquely extend it to a hyperhermitian metric on
$\varphi^{-1}(U)$ with $J(TF_{x})$ orthogonal to $TF_{x}$. This means that the
set of Lagrangian fibers of $\varphi$ is open in $M$.
The main result of this paper is the following theorem.
Theorem 3.4: Let $M$ be a compact $SL(n,{\mathbb{H}})$-manifold, and
$\varphi\colon(M,I)\to X$ a smooth holomorphic Lagrangian fibration. Assume
that $M$ admits an HKT-structure. Then $X$ is Kähler.
Proof: Let $\omega_{I}\in\Lambda^{1,1}_{I}M$ be a Kähler form, and $\Psi$ the
holomorphic Lagrangian calibration defined by (2.2). Consider the form
$\Theta=\Psi\wedge\omega_{I}\in\Lambda^{n+1,n+1}_{I}M.$
By 2.2, $\Psi$ is closed, so we have $d\Theta=\Psi\wedge
d\omega_{I}\in\Lambda^{2n+3}M$. It follows from (2.1) that $d\omega_{I}$ has
weight one, hence Clebsch-Gordan formula implies that $d\Theta$ is a sum of
weight $2n+1$ and weight $2n-1$ components. But the weight decomposition of
$\Lambda^{2n+3}M$ has components only up to weight $2n-3$, so $d\Theta$ has to
vanish.
We will obtain a Kähler metric on $X$ as a push-forward $\pi_{*}\Theta$, which
is a $(1,1)$-current on $X$. We have to show that this current is strictly
positive and smooth.
Consider any Hermitian metric on $X$ and denote by $\eta$ the corresponding
$(1,1)$-form. It is always possible to rescale $\eta$ so that it satisfies the
condition $\pi^{*}\eta\leq\omega_{I}$. In this case we have
$\Theta=\Psi\wedge\omega_{I}\geq\Psi\wedge\pi^{*}\eta$. The inequality is
preserved under taking push-forwards, so we have
$\pi_{*}\Theta\geq\pi_{*}(\Psi\wedge\pi^{*}\eta)$.
For any test-form $\alpha\in\Lambda^{n-1,n-1}_{I}X$ we have by definition
$\displaystyle\int_{X}\pi_{*}(\Psi\wedge\pi^{*}\eta)\wedge\alpha=\int_{M}\Psi\wedge\pi^{*}\eta\wedge\pi^{*}\alpha$
$\displaystyle=\int_{M}\Psi\wedge\pi^{*}(\eta\wedge\alpha)=\int_{X}\pi_{*}\Psi\wedge\eta\wedge\alpha.$
Thus, we have the inequality $\pi_{*}\Theta\geq\pi_{*}(\Psi)\wedge\eta$ where
$\pi_{*}(\Psi)$ is a closed $0$-current on $X$, that is a constant. To see
that this constant is positive, take $\alpha=\eta^{n-1}$ and observe that
$\Psi\wedge\pi^{*}(\eta^{n})$ is a smooth positive form of top degree on $M$.
It suffices to check that this form is non-zero at some point of $M$. Pick any
non-critical point $x$ of $\pi$ and decompose the tangent space
$T_{x}M=V_{0}\oplus V_{1}$, where $V_{0}$ is tangent to the fiber and $V_{1}$
is any complementary subspace. Since $\pi^{*}\eta^{n}|_{V_{0}}=0$ and
$\pi^{*}\eta^{n}|_{V_{1}}$ is a volume form on $V_{1}$, we should check that
$\Psi|_{V_{0}}$ is non-zero. But this follows from the second statement in
2.2, since the fiber is Lagrangian.
It remains to check that $\pi_{*}\Theta$ is smooth. This follows from
smoothness of the fibration $\varphi\colon(M,I)\to X$. Since $\varphi$ is a
submersion, we can choose a splitting $TM=\mathcal{V}\oplus\mathcal{H}$, where
$\mathcal{V}$ is a subbundle, tangent to the fibers of $\varphi$. Using this
splitting we can lift vector field from $X$ to $M$. Denote by $F_{x}$ the
fiber $\varphi^{-1}(x)$ over a point $x\in X$. Consider a $(1,1)$-form
$\theta$ on $X$, which is given by
$\theta(\xi,\overline{\eta})_{x}=\int_{F_{x}}(\pi^{*}\xi)\lrcorner\,(\pi^{*}\overline{\eta})\lrcorner\,\Theta,$
where $\xi,\eta\in T^{1,0}X$ and $\pi^{*}\xi$, $\pi^{*}\eta$ denote the the
lifts of these vector fields to $\mathcal{H}$. Note, that $\theta$ does not
actually depend on the choice of $\mathcal{H}$, since convolution of $\Theta$
with a vertical vector field from $\mathcal{V}$ gives a differential form that
restricts trivially to the fiber. The form $\theta$ coincides with
$\pi_{*}\Theta$ as a current, thanks to Fubini theorem. But it is clear from
the definition that $\theta$ is smooth. This completes the proof.
Remark 3.5: Note that for a non-smooth holomorphic Lagrangian fibration
$\varphi\colon(M,I)\to X$ the above proof shows that
$\pi_{*}(\Theta\wedge\omega_{I})$ is a Kähler current. Therefore, $X$ is in
Fujiki class $\mathcal{C}$, whenever $M$ admits an HKT-structure (see [DP]).
### 3.2 Examples of Lagrangian fibrations
In this subsection we construct a class of hypercomplex
$SL(n,{\mathbb{H}})$-manifolds that do not admit an HKT metric. The idea,
which was also used in [KS], is to consider a torus fibration over an affine
base.
Let $(X,I)$ be a complex manifold, $\dim_{\mathbb{C}}X=n$. We call $X$ an
affine complex manifold if there exists a torsion-free connection $D\colon
TX\to TX\otimes\Lambda^{1}X$ which is flat and preserves the complex
structure: $DI=0$.
Fix a point $x\in X$ and consider the holonomy group
$\mathrm{Hol}_{x}(D)\subset GL(T_{x}X)$. We call $X$ an affine manifold with
integer monodromy if there exists a lattice $\Lambda_{x}\subset T_{x}X$ which
is preserved by holonomy: $\mathrm{Hol}_{x}(D)\Lambda_{x}=\Lambda_{x}$. In
this case, we can construct a subset $\Lambda\subset TX$ in the total space of
the tangent bundle, obtained as parallel translation of $\Lambda_{x}$. For any
point $y\in X$ the intersection $\Lambda\cap T_{y}X$ is a lattice in $T_{y}X$.
Let $M=TX/\Lambda$, that is, a manifold obtained as a fiberwise quotient of
$TX$. We will introduce a pair of anticommuting complex structures on $TX$
that will descend to $M$. The connection $D$ defines a splitting
$T(TX)=\mathcal{V}\oplus\mathcal{H}$ (3.1)
into a direct sum of the vertical subbundle $\mathcal{V}$ which is tangent to
the fibers of the projection $TX\to X$, and a horizontal complement
$\mathcal{H}$. Note that for any point $(x,v)\in TX$ with $v\in T_{x}X$ we
have natural isomorphisms $\mathcal{H}_{(x,v)}\simeq
T_{x}X\simeq\mathcal{V}_{(x,v)}$, thus we can identify the components of the
splitting (3.1). This also shows that the complex structure $I$ acts naturally
on both $\mathcal{H}$ and $\mathcal{V}$. With these observations in mind,
define a pair of operators from $\operatorname{End}(T(TM))$:
$\mathcal{I}=\begin{pmatrix}-I&0\\\
0&I\end{pmatrix},\quad\mathcal{J}=\begin{pmatrix}0&\operatorname{Id}\\\
-\operatorname{Id}&0\end{pmatrix},$
where the block-matrix form corresponds to the decomposition (3.1). It is
clear that these operators define a pair of anticommuting almost-complex
structures. We need to check that these structures are integrable. This can be
done locally.
Using the fact that the connection $D$ is flat and $I$ is parallel, we can
choose a parallel local frame in $TX$ of the form $e_{1},Ie_{1},\ldots
e_{n},Ie_{n}$. Since $D$ is torsion-free, these vector fields commute, hence
they define a local coordinate system on $X$. Since the sections $e_{i}$ and
$Ie_{i}$ are flat, they are tangent to the subbundle $\mathcal{H}$. Therefore,
they also define a local coordinate system in the total space of $TX$ in which
the complex structures $\mathcal{I}$ and $\mathcal{J}$ act as standard complex
structures of ${\mathbb{H}}^{n}$. Thus, they are integrable.
This construction gives a hypercomplex structure on the total space of $TX$.
Observe that this structure is invariant with respect to translations along
the fiber. This implies that the hypercomplex structure descends to $M$. We
call the manifold $M$ constructed this way quaternionic double of an affine
complex manifold $X$.
The hypercomplex structure on the quaternionic double $M$ is locally modeled
on ${\mathbb{H}}^{n}$ by construction. Thus, the Obata connection on this
manifold is flat (and it is actually induced from the flat connection $D$ on
$X$). Since the connection $D$ preserves the lattice in the tangent bundle, it
also preserves the holomorphic volume form. To see this, we can again choose a
local frame $e_{1},Ie_{1},\ldots e_{n},Ie_{n}$ as above. We can assume that
the elements of this frame form a local basis for the lattice $\Lambda$
preserved by $D$. Then the volume form
$(e_{1}^{*}-\sqrt{-1}Ie_{1}^{*})\wedge\cdots\wedge(e_{n}^{*}-\sqrt{-1}Ie_{n}^{*})$
is well-defined globally and parallel with respect to $D$, hence holomorphic.
It follows that the Obata connection on $M$ also preserves an $I$-holomorphic
volume form. We conclude that the manifold $M$ is an
$SL(n,{\mathbb{H}})$-manifold.
Theorem 3.6: Let $M$ be a quaternionic double of an affine complex manifold
$X$. If $X$ is not Kähler, then $M$ does not admit an HKT metric.
Proof: We can choose a Hermitian metric $g$ on $X$. Then we can define a
hyperhermitian metric $h=g\oplus g$ on $M$, using the decomposition (3.1).
With respect to this metric the fibers of the projection $M\to X$ are
Lagrangian, hence we can apply 3.1.
We conclude this section by some examples of affine complex manifolds with
integer monodromy. We remark that the manifolds in the examples will be non-
Kähler (actually any complex affine manifold admitting a Kähler metric is a
quotient of a torus, as follows from the Calabi-Yau theorem and the
Bieberbach’s solution of Hilbert’s 18-th problem; see e.g. [Bes]). Therefore,
the corresponding quaternionic doubles possess no HKT metrics.
Example 3.7: Let $N$ be a three-dimensional real Heisenberg group, that is the
group of upper-triangular unipotent $3\times 3$ matrices with real elements.
Consider the nilpotent Lie group $G=N\times{\mathbb{R}}$. Then $G$ is
diffeomorphic to ${\mathbb{C}}^{2}$ and one can check (see [Ha], Example 2)
that the structure of the Lie group on ${\mathbb{C}}^{2}$ can be explicitly
given by
$(w_{1},w_{2})\cdot(z_{1},z_{2})=(w_{1}+z_{1},w_{2}-\sqrt{-1}\overline{w_{1}}z_{1}+z_{2}).$
It is clear from this formula, that left translations are holomorphic affine
transformations of $C^{2}$, so $G$ has a left-invariant complex structure
which is preserved by a flat torsion-free connection. We can choose a lattice
in ${\mathbb{C}}^{2}$, say $\Lambda={\mathbb{Z}}[\sqrt{-1}]^{2}$. Then the
manifold $X=\Lambda\backslash G$ (which is called a primary Kodaira surface)
will be an affine complex manifold with integer monodromy.
Example 3.8: Let $G$ be a three-dimensional complex Heisenberg group. It can
be described (see [Ha]) as ${\mathbb{C}}^{3}$ with multiplication given by
$(w_{1},w_{2},w_{3})\cdot(z_{1},z_{2},z_{3})=(w_{1}+z_{1},w_{2}+z_{2},w_{3}+z_{3}+w_{1}z_{2}).$
We can choose a lattice in $G$, for example
$\Lambda={\mathbb{Z}}[\sqrt{-1}]^{3}$ and take a quotient $X=\Lambda\backslash
G$. This is an affine complex manifold with integer monodromy (which is called
an Iwasawa manifold).
## References
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* [BDV] Maria L. Barberis, Isabel G. Dotti, Misha Verbitsky, Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry, arXiv:0712.3863, 22 pages.
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* [FG] Fino, A., Grantcharov, G., On some properties of the manifolds with skew-symmetric torsion and holonomy SU(n) and Sp(n), math.DG/0302358, Adv. Math. 189 (2004), no. 2, 439–450.
* [GP] Grantcharov, G., Poon, Y. S., Geometry of hyper-Kähler connections with torsion, math.DG/9908015, Comm. Math. Phys. 213 (2000), no. 1, 19–37.
* [GV] Grantcharov, G., Verbitsky, M., Calibrations in hyperkähler geometry, arXiv:1009.1178, 32 pages.
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* [GP2] Gutowski, J., Papadopoulos, G. The Moduli Spaces of Worldvolume Brane Solitons, Phys. Lett. B432 (1998) 97-102.
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* [V1] Verbitsky, M., Hyperkähler manifolds with torsion, supersymmetry and Hodge theory, math.AG/0112215, Asian J. Math. Vol. 6, No. 4, pp. 679-712 (2002).
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Misha Verbitsky
Laboratory of Algebraic Geometry,
National Research University HSE,
7 Vavilova Str. Moscow, Russia, 117312
[email protected], [email protected]
Andrey Soldatenkov
Laboratory of Algebraic Geometry,
National Research University HSE,
7 Vavilova Str. Moscow, Russia, 117312
[email protected]
|
arxiv-papers
| 2013-01-02T06:57:50 |
2024-09-04T02:49:39.842480
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrey Soldatenkov, Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1301.0175"
}
|
1301.0216
|
# Applying Strategic Multiagent Planning to Real-World Travel Sharing Problems
Jan Hrnčíř
Michael Rovatsos
Agent Technology Center Faculty of Electrical Engineering Czech Technical
University in Prague 121 35 Prague, Czech Republic
[email protected] School of Informatics The University of
Edinburgh Edinburgh EH8 9AB, United Kingdom [email protected]
###### Abstract
Travel sharing, i.e., the problem of finding parts of routes which can be
shared by several travellers with different points of departure and
destinations, is a complex multiagent problem that requires taking into
account individual agents’ preferences to come up with mutually acceptable
joint plans. In this paper, we apply state-of-the-art planning techniques to
real-world public transportation data to evaluate the feasibility of
multiagent planning techniques in this domain. The potential of improving
travel sharing technology has great application value due to its ability to
reduce the environmental impact of travelling while providing benefits to
travellers at the same time.
We propose a three-phase algorithm that utilises performant single-agent
planners to find individual plans in a simplified domain first and then merges
them using a best-response planner which ensures resulting solutions are
individually rational. Finally, it maps the resulting plan onto the full
temporal planning domain to schedule actual journeys.
The evaluation of our algorithm on real-world, multi-modal public
transportation data for the United Kingdom shows linear scalability both in
the scenario size and in the number of agents, where trade-offs have to be
made between total cost improvement, the percentage of feasible timetables
identified for journeys, and the prolongation of these journeys. Our system
constitutes the first implementation of strategic multiagent planning
algorithms in large-scale domains and provides insights into the engineering
process of translating general domain-independent multiagent planning
algorithms to real-world applications.
###### category:
I.2.11 Artificial Intelligence Distributed Artificial Intelligence –
Multiagent systems
###### keywords:
multiagent planning, real-world application, travel sharing
††terms: Algorithms, Design, Experimentation
## 1 Introduction
Travelling is an important and frequent activity, yet people willing to travel
have to face problems with rising fuel prices, carbon footprint and traffic
jams. These problems can be ameliorated by travel sharing, i.e., groups of
people travel together in one vehicle for parts of the journey. Participants
in such schemes can benefit from travel sharing in several ways: sharing parts
of a journey may reduce cost (e.g., through group tickets), carbon footprint
(e.g., when sharing a private car, or through better capacity utilisation of
public means of transport), and travellers can enjoy the company of others on
a long journey. In more advanced scenarios one could even imagine this being
combined with working together while travelling, holding meetings on the road,
etc.
Today, there exist various commercial online services for car111E.g.,
www.liftshare.com or www.citycarclub.co.uk., bike, and walk sharing as well as
services which assist users in negotiating shared journeys222E.g.,
www.companions2travel.co.uk, www.travbuddy.com., and, of course, plenty of
travel planning services333E.g., in the United Kingdom: www.nationalrail.co.uk
for trains, www.traveline.info and www.google.com/transit for multi-modal
transportation. that automate individual travel planning for one or several
means of transport. On the research side, there is previous work that deals
with the ridesharing and car-pooling problem [1, 8, 14]. However, no work has
been done that attempts to compute joint travel plans based on public
transportation timetable data and geographical stop locations, let alone in a
way that takes into account the strategic nature of the problem, which comes
about through the different (and potentially conflicting) preferences of
individuals who might be able to benefit from travelling together. From the
point of view of (multiagent) planning, this presents itself as a very complex
application scenario: Firstly, even if one restricted oneself to centralised
(non-strategic) planning, the domain is huge – public transportation data for
the UK alone currently involves $240,590$ timetable connections for trains and
coaches (even excluding local city buses), which would have to be translated
to a quarter of a million planning actions, at least in a naive formalisation
of the domain. Secondly, planning for multiple self-interested agents that are
willing to cooperate only if it is beneficial for them is known to be
exponentially harder than planning for each agent individually [2]. Yet any
automated service that proposes joint journeys would have to guarantee such
strategic properties in order to be acceptable for human users (who could then
even leave it to the service to negotiate trips on their behalf).
In this paper, we present an implementation of best-response planning (BRP)
[13] within a three-phase algorithm that is capable of solving strategic
travel sharing problems for several agents based on real-world transportation
data. Based on a simplified version of the domain that ignores timetabling
information, the algorithm first builds individual travel plans using state-
of-the-art single-agent planners that are available off the shelf. It then
merges these individual plans and computes a multiagent plan that is a Nash
equilibrium and guarantees individual rationality of solutions, as well as
stability in the sense that no single agent has an incentive to deviate from
the joint travel route. This is done using BRP as the underlying planner, as
it is the only available planner that can solve strategic multiagent planning
problems of such scale, and is proven to converge in domains that comply with
certain assumptions, as is the case for our travel sharing domain. In a third
and final step, the resulting multiagent plan is mapped onto the full temporal
planning domain to schedule actual journeys. This scheduling task is not
guaranteed to always find a feasible solution, as the previous simplification
ignores a potential lack of suitable connections. However, we show through an
extensive empirical evaluation that our method finds useful solutions in a
large number of cases despite its theoretical incompleteness.
The contribution of our work is threefold: Firstly, we show that current
multiagent planning technology can be used in important planning domains such
as travel sharing by presenting its application to a practical problem that
cannot be solved with other existing techniques. In the process, we describe
the engineering steps that are necessary to deal with the challenges of real-
world large-scale data and propose suitable solutions. Secondly, we present an
algorithm that combines different techniques in a practically-oriented way,
and which is largely based on domain-independent off-the-shelf heuristic
problem solvers. In fact, only data preprocessing and timetable mapping use
domain-specific knowledge, and much of the process of incorporating this
knowledge could be replicated for similar other domains (such as logistics,
manufacturing, and network communications). Finally, we provide a potential
solution to the hard computational problem of travel sharing that could be
exploited for automating important tasks in a future real-world application to
the benefits of users, who normally have to plan such routes manually and
would be overwhelmed by the choices in a domain full of different
transportation options which is inhabited by many potential co-travellers.
We start off by describing the problem domain in section 2 and specifying the
planning problem formally in section 3, following the model used in [13].
Section 4 introduces our three-phase algorithm for strategic planning in
travel sharing domains and we present an extensive experimental evaluation of
the algorithm in section 5. Section 6 presents a discussion of our results and
section 7 concludes.
## 2 The travel sharing domain
The real-world travel domain used in this paper is based on the public
transport network in the United Kingdom, a very large and complex domain which
contains $4,055$ railway and coach stops supplemented by timetable
information. An agent representing a passenger is able to use different means
of transport during its journey: walking, trains, and coaches. The aim of each
agent is to get from its starting location to its final destination at the
lowest possible cost, where the cost of the journey is based on the duration
and the price of the journey. Since we assume that all agents are travelling
on the same day and that all journeys must be completed within 24 hours, in
what follows below we consider only travel data for Tuesdays (this is an
arbitrary choice that could be changed without any problem). For the purposes
of this paper, we will make the assumption that sharing a part of a journey
with other agents is cheaper than travelling alone. While this may not
currently hold in many public transportation systems, defining hypothetical
cost functions that reflect this would help assess the potential benefit of
introducing such pricing schemes.
### 2.1 Source data
The travel sharing domain uses the National Public Transport Data Repository
(NPTDR)444data.gov.uk/dataset/nptdr which is publicly available from the
Department for Transport of the British Government. It contains a snapshot of
route and timetable data that has been gathered in the first or second
complete week of October since 2004. For the evaluation of the algorithm in
section 5, we used data from
2010555www.nptdr.org.uk/snapshot/2010/nptdr2010txc.zip, which is provided in
TransXChange XML666An XML-based UK standard for interchange of route and
timetable data..
National Public Transport Access Nodes (NaPTAN)777data.gov.uk/dataset/naptan
is a UK national system for uniquely identifying all the points of access to
public transport. Every point of access (bus stop, rail station, etc.) is
identified by an ATCO code888A unique identifier for all points of access to
public transport in the United Kingdom., e.g., 9100HAYMRKT for Haymarket Rail
Station in Edinburgh. Each stop in NaPTAN XML data is also supplemented by
common name, latitude, longitude, address and other pieces of information.
This data also contains information about how the stops are grouped together
(e.g., several bus bays that are located at the same bus station).
Figure 1: Overview of the data transformation and processing
To be able to use this domain data with modern AI planning systems, it has to
be converted to the Planning Domain Definition Language (PDDL). We transformed
the data in three subsequent stages, cf. Figure 1. First, we transformed the
NPTDR and NaPTAN XML data to a spatially-enabled PostgreSQL database. Second,
we automatically processed and optimised the data in the database. The data
processing by SQL functions in the procedural PL/pgSQL language included the
following steps: merging bus bays at bus stations and parts of train stations,
introducing walking connections to enable multi-modal journeys, and
eliminating duplicates from the timetable. Finally, we created a script for
generating PDDL specifications based on the data in the database. More details
about the data processing and PDDL specifications can be found in [11].
Figure 2: An example of the relaxed domain (e.g., it takes 50 minutes to
travel from the stop $\boldsymbol{A}$ to $\boldsymbol{B}$)
### 2.2 Planning domain definitions
Since the full travel planning domain is too large for any current state-of-
the-art planner to deal with, we distinguish the full domain from a relaxed
domain, which we will use to come up with an initial plan before mapping it to
the full timetable information in our algorithm below.
The relaxed domain is a single-agent planning domain represented as a directed
graph where the nodes are the stops and the edges are the connections provided
by a service. The graph must be directed because there exist stops that are
used in one direction only. There is an edge from $A$ to $B$ if there is at
least one connection from $A$ to $B$ in the timetable. The cost of this edge
is the minimal time needed for travelling from $A$ to $B$. A plan $P_{i}$
found in the relaxed domain for the agent $i$ is a sequence of connections to
travel from its origin to its destination. The relaxed domain does not contain
any information about the traveller’s departure time. This could be
problematic in a scenario where people are travelling at different times of
day. This issue could be solved by clustering of user requests, cf. chapter 7.
A small example of the relaxed domain is shown in Figure 2. An example plan
for an agent travelling from $C$ to $F$ is $P_{1}=\langle C\rightarrow
D,D\rightarrow E,E\rightarrow F\rangle$. To illustrate the difference between
the relaxed domain and the full timetable, there are $8,688$ connections in
the relaxed domain for trains and coaches in the UK compared to $240,590$
timetable connections.
Direct trains that do not stop at every stop are filtered out from the relaxed
domain for the following reason: Assume that in Figure 2, there is only one
agent travelling from $C$ to $F$ and that its plan in the relaxed domain is to
use a direct train from $C$ to $F$. In this case, it is only possible to match
its plan to direct train connections from $C$ to $F$, and not to trains that
stop at $C$, $D$, $E$, and $F$. Therefore, the agent’s plan cannot be matched
against all possible trains between $C$ and $F$ which is problematic
especially in the case where the majority of trains stop at every stop and
only a few trains are direct. On the other hand, it is possible to match a
plan with a train stopping in every stop to a direct train, as it is explained
later in section 4.3.
The full domain is a multiagent planning domain based on the joint plan $P$.
Assume that there are $N$ agents in the full domain (each agent $i$ has the
plan $P_{i}$ from the relaxed domain). Then, the joint plan $P$ is a merge of
single-agent plans defined by formula
$P=\bigcup_{i=1}^{N}P_{i}$
where we interpret $\bigcup$ as the union of graphs that would result from
interpreting each plan as a set of edges connecting stops. More specifically,
given a set of single-agent plans, the plan merging operator $\bigcup$
computes its result in three steps: First, it transforms every single-agent
plan $P_{i}$ to a directed graph $G_{i}$ where the nodes are the stops from
the single-agent plan $P_{i}$ and the edges represent the actions of $P_{i}$
(for instance, a plan $P_{1}=\langle C\rightarrow D,D\rightarrow
E,E\rightarrow F\rangle$ is transformed to a directed graph
$G_{1}=\\{C\rightarrow D\rightarrow E\rightarrow F\\}$). Second, it performs a
graph union operation over the directed graphs $G_{i}$ and labels every edge
in the joint plan with the numbers of agents that are using the edge (we don’t
introduce any formal notation for these labels here, and simply slightly abuse
the standard notation of sets of edges to describe the resulting graph).
As an example, the joint plan for agent 1 travelling from $C$ to $F$ and
sharing a journey from $D$ to $E$ with agent 2 would be computed as
$\langle C\rightarrow D,D\rightarrow E,E\rightarrow F\rangle\>\cup\>\langle
D\rightarrow E\rangle=\\\
\\{C\xrightarrow{(1)}D\xrightarrow{(1,2)}E\xrightarrow{(1)}F\\}$
With this, the full domain is represented as a directed multigraph where the
nodes are the stops that are present in the joint plan of the relaxed domain.
Edges of the multigraph are the service journeys from the timetable. Every
service is identified by a unique service name and is assigned a departure
time from each stop and the duration of its journey between two stops. In the
example of the full domain in Figure 3, the agents can travel using some
subset of five different services `S1` to `S5`. In order to travel from $C$ to
$D$ using service `S1`, an agent must be present at stop $C$ before its
departure.
Figure 3: An example of the full domain with stops $\boldsymbol{C}$,
$\boldsymbol{D}$, $\boldsymbol{E}$ and $\boldsymbol{F}$ for the joint plan
$\boldsymbol{P}\boldsymbol{=}\boldsymbol{\\{}\boldsymbol{C}\xrightarrow{(1)}\boldsymbol{D}\xrightarrow{(1,2)}\boldsymbol{E}\xrightarrow{(1)}\boldsymbol{F}\boldsymbol{\\}}$
## 3 The planning problem
Automated planning technology [9] has developed a variety of scalable
heuristic algorithms for tackling hard planning problems, where plans, i.e.,
sequences of actions that achieve a given goal from a given initial state, are
calculated by domain-independent problem solvers. To model the travel sharing
problem, we use a multiagent planning formalism which is based on MA-STRIPS
[2] and coalition-planning games [3]. States are represented by sets of ground
fluents, actions are tuples $a=\langle\mathit{pre}(a),\mathit{eff}(a)\rangle$.
After the execution of action $a$, positive fluents $p$ from $\mathit{eff}(a)$
are added to the state and negative fluents $\neg p$ are deleted from the
state. Each agent has individual goals and actions with associated costs.
There is no extra reward for achieving the goal, the total utility received by
an agent is simply the inverse of the cost incurred by the plan executed to
achieve the goal.
More formally, following the notation of [13], a multiagent planning problem
(MAP) is a tuple
$\Pi=\langle
N,F,I,\\{G_{i}\\}_{i=1}^{n},\\{A_{i}\\}_{i=1}^{n},\Psi,\\{c_{i}\\}_{i=1}^{n}\rangle$
where
* •
$N=\\{1,\ldots,n\\}$ is the set of agents,
* •
$F$ is the set of fluents,
* •
$I\subseteq F$ is the initial state,
* •
$G_{i}\subseteq F$ is agent $i$’s goal,
* •
$A_{i}$ is agent $i$’s action set,
* •
$\Psi:A\rightarrow\\{0,1\\}$ is an admissibility function,
* •
$c_{i}:\times_{i=1}^{n}A_{i}\rightarrow\mathbb{R}$ is the cost function of
agent $i$.
$A=A_{1}\times\ldots\times A_{n}$ is the joint action set assuming a
concurrent, synchronous execution model, and $G=\wedge_{i}G_{i}$ is the
conjunction of all agents’ individual goals. A MAP typically imposes
concurrency constraints regarding actions that cannot or have to be performed
concurrently by different agents to succeed which the authors of [13] encode
using an admissibility function $\Psi$, with $\Psi(a)=1$ if the joint action
$a$ is executable, and $\Psi(a)=0$ otherwise.
A plan $\pi=\langle a^{1},\ldots,a^{k}\rangle$ is a sequence of joint actions
$a^{j}\in A$ such that $a^{1}$ is applicable in the initial state $I$ (i.e.,
$\mathit{pre}(a^{1})\subseteq I$), and $a^{j}$ is applicable following the
application of $a^{1},\ldots,$ $a^{j-1}$. We say that $\pi$ solves the MAP
$\Pi$ if the goal state $G$ is satisfied following the application of all
actions in $\pi$ in sequence. The cost of a plan $\pi$ to agent $i$ is given
by $C_{i}(\pi)=\sum_{j=1}^{k}c_{i}(a^{j})$. Each agent’s contribution to a
plan $\pi$ is denoted by $\pi_{i}$ (a sequence of $a_{i}\in A_{i}$).
### 3.1 Best-response planning
The best-response planning (BRP) algorithm proposed in [13] is an algorithm
which, given a solution $\pi^{k}$ to a MAP $\Pi$, finds a solution $\pi^{k+1}$
to a transformed planning problem $\Pi_{i}$ with minimum cost
$C_{i}(\pi^{k+1})$ among all possible solutions:
$\pi^{k+1}=\arg\min\\{C_{i}(\pi)|\pi\textrm{ identical to }\pi^{k}\textrm{ for
all }j\neq i\\}$
The transformed planning problem $\Pi_{i}$ is obtained by rewriting the
original problem $\Pi$ so that all other agents’ actions are fixed, and agent
$i$ can only choose its own actions in such a way that all other agents still
can perform their original actions. Since $\Pi_{i}$ is a single-agent planning
problem, any cost-optimal planner can be used as a best-response planner.
In [13], the authors show how for a class of congestion planning problems,
where all fluents are private, the transformation they propose allows the
algorithm to converge to a Nash equilibrium if agents iteratively perform
best-response steps using an optimal planner. This requires that every agent
can perform its actions without requiring another agent, and hence can achieve
its goal in principle on its own, and conversely, that no agent can invalidate
other agents’ plans. Assuming infinite capacity of vehicles, the relaxed
domain is an instance of a congestion planning problem999 Following the
definition of a congestion planning problem in [13], all actions are private,
as every agent can use transportation means on their own and the other agents’
concurrently taken actions only affect action cost. A part of the cost
function defined in section 4.4 depends only on the action choice of
individual agent..
The BRP planner works in two phases: In the first phase, an initial plan for
each agent is computed (e.g., each agent plans independently or a centralised
multi-agent planner is used). In the second phase, the planner solves simpler
best-response planning problems from the point of view of each individual
agent. The goal of the planner in a BRP problem is to minimise the cost of an
agent’s plan without changing the plans of others. Consequently, it optimises
a plan of each agent with respect to the current joint plan.
This approach has several advantages. It supports full concurrency of actions
and the BRP phase avoids the exponential blowup in the action space resulting
in much improved scalability. For the class of potential games [16], it
guarantees to converge to a Nash equilibrium. On the other hand, it does not
guarantee the optimality of a solution, i.e., the quality of the equilibrium
in terms of overall efficiency is not guaranteed (it depends on which initial
plan the agents start off with). However, experiments have proven that it can
be successfully used for improving general multiagent plans [13]. Such non-
strategic plans can be computed using a centralised multiagent planner, i.e.,
a single-agent planner (for instance Metric-FF [10]) which tries to optimise
the value of the joint cost function (in our case the sum of the values of the
cost functions of agents in the environment) while trying to achieve all
agents’ goals. Centralised multiagent planners have no notion of self-
interested agents, i.e., they ignore the individual preferences of agents.
## 4 A three-phase strategic travel sharing algorithm
The main problem when planning for multiple agents with a centralised
multiagent planner is the exponential blowup in the action space which is
caused by using concurrent, independent actions [13]. Using a naive PDDL
translation has proven that a direct application of a centralised multiagent
planner to this problem does not scale well. For example, a simple scenario
with two agents, ferries to Orkney Islands and trains in the area between
Edinburgh and Aberdeen resulted in a one-day computation time.
As mentioned above, we tackle the complexity of the domain by breaking down
the planning process into different phases that avoid dealing with the full
fine-grained timetable data from the outset. Our algorithm, which is shown in
Figure 4, is designed to work in three phases.
### 4.1 The initial phase
First, in the initial phase, an initial journey is found for each agent using
the relaxed domain. A journey for each agent is calculated independently of
other agents in the scenario using a single-agent planner. As a result, each
agent is assigned a single-agent plan which will be further optimised in the
next phase. This approach makes sense in our domain because the agents do not
need each other to achieve their goals and they cannot invalidate each other’s
plans.
Input
* •
a relaxed domain
* •
a set of $N$ agents $A=\\{a_{1},\dots,a_{N}\\}$
* •
an origin and a destination for each agent
1\. The initial phase
For $i=1,\dots,N$ do
Find an initial journey for agent $a_{i}$ using
a single-agent planner.
2\. The BR phase
Do until no change in the cost of the joint plan
For $i=1,\dots,N$ do
1. 1.
Create a simpler best-response planning (BRP)
problem from the point of view of agent $a_{i}$.
2. 2.
Minimise the cost of $a_{i}$’s plan without changing
the plans of others.
End
3\. The timetabling phase
Identify independent groups of agents $G=\\{g_{1},\dots,g_{M}\\}$.
For $i=1,\dots,M$ do
1. 1.
Find the relevant timetable for group $g_{i}$.
2. 2.
Match the joint plan of $g_{i}$ to timetable using a temporal single-agent
planner in the full domain with the relevant timetable.
End
Figure 4: Three-phase algorithm for finding shared journeys for agents
### 4.2 The BR phase
Second, in the BR phase (best-response phase), which is also based on the
relaxed domain, the algorithm uses the BRP algorithm as described above. It
iteratively creates and solves simpler best-response planning problems from
the point of view of each individual agent. In the case of the relaxed domain,
the BRP problem looks almost the same as a problem of finding a single-agent
initial journey. The difference is that the cost of travelling is smaller when
an agent uses a connection which is used by one or more other agents, as will
be explained below, cf. equation (1).
Iterations over agents continue until there is no change in the cost of the
joint plan between two successive iterations. This means that the joint plan
cannot be further improved using the best-response approach. The output of the
BR phase is the joint plan $P$ in the relaxed domain (defined in section 2.2)
that specifies which connections the agents use for their journeys and which
segments of their journeys are shared. The joint plan $P$ will be matched to
the timetable in the final phase of the algorithm.
### 4.3 The timetabling phase
In the final timetabling phase, the optimised shared journeys are matched
against timetables using a temporal single-agent planner which assumes the
full domain. For this, as a first step, independent groups of agents with
respect to journey sharing are identified. An independent group of agents is
defined as an edge disjoint subgraph of the joint plan $P$. This means that
actions of independent groups do not affect each other so it is possible to
find a timetable for each independent group separately.
Then, for every independent group, parts of the group journey are identified.
A part of the group journey is defined as a maximal continuous segment of the
group journey which is performed by the same set of agents. As an example,
there is a group of two agents that share a segment of their journeys in
Figure 5: Agent 1 travels from $A$ to $G$ while agent 2 travels from $B$ to
$H$. Their group journey has five parts, with the shared part (part 3) of
their journey occurring between stops $C$ and $F$.
Figure 5: Parts of the group journey of two agents Figure 6: The full domain with services from the relevant timetable. There are five different trains T1 to T5, and train T1 is a direct train. Table 1: Parameters of the testing scenarios Scenario code | S1 | S2 | S3 | S4 | S5
---|---|---|---|---|---
Number of stops | 344 | 721 | $1\>044$ | $1\>670$ | $2\>176$
Relaxed domain connections | 744 | $1\>520$ | $2\>275$ | $4\>001$ | $4\>794$
Timetabled connections | $23\>994$ | $26\>702$ | $68\>597$ | $72\>937$ | $203\>590$
Means of transport | trains | trains, coaches | trains | trains, coaches | trains
In order to use both direct and stopping trains when the group journey is
matched to the timetable, the relevant timetable for a group journey is
composed in the following way: for every part of the group journey, return all
timetable services in the direction of agents’ journeys which connect the
stops in that part. An example of the relevant timetable for a group of agents
from the previous example is shown in Figure 6. Now, the agents can travel
using the direct train `T1` or using train `T2` with intermediate stops.
The relevant timetable for the group journey is used with the aim to cut down
the amount of data that will be given to a temporal single-agent planner. For
instance, there are $23\,994$ timetabled connections in Scotland. For an
example journey of two agents, there are only 885 services in the relevant
timetable which is approximately 4 % of the data. As a result, the temporal
single-agent planner gets only the necessary amount of data as input, to
prevent the time-consuming exploration of irrelevant regions of the state
space.
### 4.4 Cost functions
The timetable data used in this paper (cf. section 2.1) contains neither
information about ticket prices nor distances between adjacent stops, only
durations of journeys from one stop to another. This significantly restricts
the design of cost functions used for the planning problems. Therefore, the
cost functions used in the three phases of the algorithm are based solely on
the duration of journeys.
In the initial phase, every agent tries to get to its destination in the
shortest possible time. The cost of travelling between adjacent stops $A$ and
$B$ is simply the duration of the journey between stops $A$ and $B$. In the BR
phase, we design the cost function in such a way that it favours shared
journeys. The cost $c_{i,n}$ for agent $i$ travelling from $A$ to $B$ in a
group of $n$ agents is then defined by equation (1):
$c_{i,n}=\left(\frac{1}{n}\,0.8+0.2\right)c_{i}$ (1)
where $c_{i}$ is the individual cost of the single action to $i$ when
travelling alone. In our experiments below, we take this to be equal to the
duration of the trip from $A$ to $B$.
This is designed to approximately model the discount for the passengers if
they buy a group ticket: The more agents travel together, the cheaper the
shared (leg of a) journey becomes for each agent. Also, an agent cannot travel
any cheaper than 20 % of the single-agent cost. In reality, pricing for group
tickets could vary, and while our experimental results assume this specific
setup, the actual price calculation could be easily replaced by any
alternative model.
In the timetabling phase, every agent in a group of agents tries to spend the
shortest possible time on its journey. When matching the plan to the
timetable, the temporal planner tries to minimise the sum of durations of
agents’ journeys including waiting times between services.
## 5 Evaluation
We have evaluated our algorithm on public transportation data for the United
Kingdom, using various off-the-shelf planners for the three phases described
above, and a number of different scenarios. These are described together with
the results obtained from extensive experiments below.
### 5.1 Planners
All three single-agent planners used for the evaluation were taken from recent
International Planning Competitions (IPC) from 2008 and 2011. We use LAMA [18]
in the initial and the BR phase, a sequential satisficing (as opposed to cost-
optimal) planner which searches for any plan that solves a given problem and
does not guarantee optimality of the plans computed. LAMA is a propositional
planning system based on heuristic state-space search. Its core feature is the
usage of landmarks [17], i.e., propositions that must be true in every
solution of a planning problem.
SGPlan6 [12] and POPF2 [7] are temporal satisficing planners used in the
timetabling phase. Such temporal planners take the duration of actions into
account and try to minimise makespan (i.e., total duration) of a plan but do
not guarantee optimality. The two planners use different search strategies and
usually produce different results. This allows us to run them in sequence on
every problem and to pick the plan with the shortest duration. It is not
strictly necessary to run both planners, one could save computation effort by
trusting one of them.
SGPlan6 consists of three inter-related steps: parallel decomposition,
constraint resolution and subproblem solution [4, 10, 15, 19]. POPF2 is a
temporal forward-chaining partial-order planner with a specific extended
grounded search strategy described in [5, 6]. It is not known beforehand which
of the two planners will return a better plan on a particular problem
instance.
### 5.2 Scenarios
To test the performance of our algorithm, we generated five different
scenarios of increasing complexity, whose parameters are shown in Table 1.
They are based on different regions of the United Kingdom (Scotland for S1 and
S2, central UK for S3 and S4, central and southern UK for S5). Each scenario
assumes trains or trains and coaches as available means of transportation.
In order to observe the behaviour of the algorithm with different numbers of
agents, we ran our algorithm on every scenario with $2,4,6,\dots,14$ agents in
it. To ensure a reasonable likelihood of travel sharing to occur, all agents
in the scenarios travel in the same direction. This imitates a preprocessing
step where the agents’ origins and destinations are clustered according to
their direction of travel. For simplicity reasons, we have chosen directions
based on cardinal points (N–S, S–N, W–E, E–W). For every scenario and number
of agents, we generated 40 different experiments (10 experiments for each
direction of travel), resulting in a total of $1,400$ experiments. All
experiments are generated partially randomly as defined below.
To explain how each experiment is set up, assume we have selected a scenario
from S1 to S5, a specific number of agents, and a direction of travel, say
north–south. To compute the origin–destination pairs to be used by the agents,
we place two axes $x$ and $y$ over the region, dividing the stops in the
scenario into four quadrants I, II, III and IV. Then, the set $O$ of possible
origin–destination pairs is computed as
$O:=\\{(A,B)|(\left(A\in\mathbf{I}\wedge
B\in\mathbf{IV}\right)\vee\left(A\in\mathbf{II}\wedge
B\in\mathbf{III}\right))\\\ \wedge|AB|\in[20,160]\\}$
This means that each agent travels from $A$ to $B$ either from quadrant I to
IV or from quadrant II to III. The straight-line distance $\left|AB\right|$
between the origin and the destination is taken from the interval 20–160 km
(when using roads or rail tracks, this interval stretches approximately to a
real distance of 30–250 km). This interval is chosen to prevent journeys that
could be hard to complete within 24 hours. We sample the actual origin-
destination pairs from the elements of $O$, assuming a uniform distribution,
and repeat the process for all other directions of travel.
### 5.3 Experimental results
We evaluate the performance of the algorithm in terms of three different
metrics: the amount of time the algorithm needs to compute shared journeys for
all agents in a given scenario, the success rate of finding a plan for any
given agent and the quality of the plans computed. Unless stated otherwise,
the values in graphs are averaged over 40 experiments that were performed for
each scenario and each number of agents. The results obtained are based on
running the algorithm on a Linux desktop computer with 2.66 GHz Intel Core 2
Duo processor and 4 GB of memory. The data, source codes and scenarios in PDDL
are archived online101010
agents.fel.cvut.cz/download/hrncir/journey_sharing.tgz.
#### 5.3.1 Scalability
To assess the scalability of the algorithm, we measure the amount of time
needed to plan shared journeys for all agents in a scenario.
In many of the experiments, the SGPlan6 and POPF2 used in the timetabling
phase returned some plans in the first few minutes but then they continued
exploration of the search space without returning any better plan. To account
for this, we imposed a time limit for each planner in the temporal planning
stage to 5 minutes for a group of up to 5 agents, 10 minutes for a group of up
to 10 agents, and 15 minutes otherwise.
Figure 7 shows the computation times of the algorithm. The graph indicates
that overall computation time grows roughly linearly with increasing number of
agents, which confirms that the algorithm avoids the exponential blowup in the
action space characteristic for centralised multiagent planning. Computation
time also increases linearly with growing scenario size. Figure 8 shows
computation times for 4, 8 and 12 agents against the different scenarios.
While the overall computation times are considerable (up to one hour for 14
agents in the largest scenario), we should emphasise that the algorithm is
effectively computing equilibrium solutions in multi-player games with
hundreds of thousands of states. Considering this, the linear growth hints at
having achieved a level of scalability based on the structure of the domain
that is far above naive approaches to plan jointly in such state spaces.
Moreover, it implies that the runtimes could be easily reduced by using more
processing power.
Figure 7: Computation time against number of agents Figure 8: Computation time
against scenario size
#### 5.3.2 Success rate
To assess the value of the algorithm, we also need to look at how many agents
end up having a valid travel plan. Planning in the relaxed domain in the
initial and the BR phase of the algorithm is very successful. After the BR
phase, 99.4 % of agents have a journey plan. The remaining 0.6 % of all agents
does not have a single-agent plan because of the irregularities in the relaxed
domain caused by splitting the public transportation network into regions. The
agents without a single-agent plan are not matched to timetable connections in
the timetabling phase.
Figure 9: Percentage of groups for which a timetable was found as a function
of group size.
The timetabling phase is of course much more problematic. Figure 9 shows the
percentage of groups for which a timetable was found, as a function of group
size. In order to create this graph, number of groups with assigned timetable
and total number of groups identified was counted for every size of the group.
There are several things to point out here.
Naturally, the bigger a group is, the harder it is to find a feasible
timetable, as the problem quickly becomes overconstrained in terms of travel
times and actually available transportation services. When a group of agents
sharing parts of their journeys is big (5 or more agents), the percentage of
groups for which we can find a timetable drops below 50 %. With a group of 8
agents, almost no timetable can be found. Basically what happens here is that
the initial and BR phases find suitable ways of travelling together in
principle, but that it becomes impossible to find appropriate connections that
satisfy every traveller’s requirements, or do not add up to a total duration
of less than 24 hours.
We can also observe that the success rate is higher in scenarios that use only
trains than in those that combine trains and coaches. On closer inspection, we
can observe that this is mainly caused by different service densities in the
rail and coach networks, i.e., the ratios of timetabled connections over
connections in the relaxed domain. For example, the service density is 33
train services a day compared to only 4 coach services in Scotland. As a
consequence, it is much harder to find a timetable in a scenario with both
trains and coaches because the timetable of coaches is much less regular than
the timetable of trains. However, this does not mean that there is less
sharing if coaches are included. Instead, it just reflects the fact that due
to low service density, many of the envisioned shared journeys do not turn out
to be feasible using coaches. The fact that this cannot be anticipated in the
initial and BR phases is a weakness of our method, and is discussed further in
section 7.
Figure 10: Average cost improvement Figure 11: Percentage of groups with less
than 30 % journey prolongation
#### 5.3.3 Plan quality
Finally, we want to assess the quality of the plans obtained with respect to
improvement in cost of agents’ journeys and their prolongation, to evaluate
the net benefit of using our method in the travel sharing domain. We should
mention that the algorithm does not explicitly optimises the solutions with
respect to these metrics. To calculate cost improvement, recalling that
$C_{i}(\pi)=\sum_{j}c_{i}(a^{j})$ for a plan is the cost of a plan
$\pi=\langle a^{1},\ldots,a^{k}\rangle$ to agent $i$, assume $n(a^{j})$
returns the number of agents with whom the $j$th step of the plan is shared.
We can define a cost of a shared travel plan
$C_{i}^{{}^{\prime}}(\pi)=\sum_{j}c_{i,n(a^{j})}(a^{j})$ using equation (1).
With this we can calculate the improvement $\Delta C$ as follows:
$\Delta C=\frac{\sum_{i\in N}C_{i}(\pi_{i})-\sum_{i\in
N}C_{i}^{{}^{\prime}}(\pi_{N})}{\sum_{i\in N}C_{i}(\pi_{i})}$ (2)
where $N$ is the set of all agents, $\pi_{i}$ is the single-agent plan
initially computed for agent $i$, and $\pi_{N}$ is the final joint plan of all
agents after completion of the algorithm (which is interpreted as the plan of
the “grand coalition” $N$ and reflects how subgroups within $N$ share parts of
the individual journeys).
The average cost improvement obtained in our experiments is shown in Figure
10, and it shows that the more agents there are in the scenario, the higher
the improvement. However, there is a trade-off between the improvement in cost
and the percentage of groups that we manage to find a suitable timetable for,
cf. Figure 9.
On the one hand, travel sharing is beneficial in terms of cost. On the other
hand, a shared journey has a longer duration than a single-agent journey in
most cases. In order to evaluate this trade-off, we also measure the journey
prolongation. Assume that $T_{\mathrm{i}}(\pi)$ is the total duration of a
plan to agent $i$ in plan $\pi$, and, as above, $\pi_{i}$/$\pi_{N}$ denote the
initial single-agent plans and the shared joint plan at the end of the
timetabling phase, respectively. Then, the prolongation $\Delta T$ of a
journey is defined as follows:
$\Delta T=\frac{\sum_{i\in N}T_{i}(\pi_{N})-\sum_{i\in
N}T_{i}(\pi_{i})}{\sum_{i\in N}T_{i}(\pi_{i})}$ (3)
Journey prolongation can be calculated only when a group is assigned a
timetable and each member of the group is assigned a single-agent timetable.
For this purpose, in every experiment, we also calculate single-agent
timetables in the timetabling phase of the algorithm.
A graph of the percentage of groups that have a timetable with prolongation
less than 30 % as a function of group size is shown in Figure 11. The graph
shows which groups benefit from travel sharing, i.e., groups whose journeys
are not prolonged excessively by travelling together. Approximately 15 % of
groups with 3–4 agents are assigned a timetable that leads to a prolongation
of less than 30 %. Such a low percentage of groups can be explained by the
algorithm trying to optimise the price of the journey by sharing in the BR
phase. However, there is a trade-off between the price and the duration of the
journey. The more agents are sharing a journey, the longer the journey
duration is likely to be.
These results were obtained based on the specific cost function (1) we have
introduced to favour travel sharing, and which would have to be adapted to the
specific cost structure that is present in a given transportation system.
Also, the extent to which longer journey times are acceptable for the
traveller depends on their preferences, but these could be easily adapted by
using different cost functions.
## 6 Discussion
The computation of single-agent plans in the initial phase involves solving a
set of completely independent planning problems. This means that the planning
process could be speeded up significantly by using parallel computation on
multiple CPUs. The same is true for matching different independent groups of
agents to timetabled connections in the timetabling phase. As an example,
assume that there are $N$ agents in the scenario and $t_{1},\dots,t_{N}$ are
the computation times for respective single-agent initial plans. If computed
concurrently, this would reduce the computation time from
$t=\sum_{i=1}^{N}t_{i}$ to $t^{\prime}=\max_{i=1}^{N}(t_{i})$. Similar
optimisations could be performed for the timetabling phase of the algorithm.
In the experiments with 10 agents, for example, this would lead to a runtime
reduced by 48 % in scenario S1 and by 44 % in scenario S5.
A major problem of our method is the inability to find appropriate connections
in the timetabling phase for larger groups. There are several reasons for
this. Firstly, the relaxed domain is overly simplified, and many journeys
found in it do not correspond to journeys that would be found if we were
planning in the full domain. Secondly, there are too many temporal constraints
in bigger groups (5 or more agents), so the timetable matching problem becomes
unsolvable given the 24-hour timetable. However, it should also be pointed out
that such larger groups would be very hard to identify and schedule even using
human planning. Thirdly, some parts of public transportation network have very
irregular timetables.
Our method clearly improves the cost of agents’ journeys by sharing parts of
the journeys, even though there is a trade-off between the amount of
improvement, the percentage of found timetables and the prolongation of
journeys. On the one hand, the bigger the group, the better the improvement.
On the other hand, the more agents share a journey, the harder it is to match
their joint plan to timetable. Also, the prolongation is likely to be higher
with more agents travelling together, and will most likely lead to results
that are not acceptable for users in larger groups.
Regarding the domain-independence of the algorithm, we should point out that
its initial and BR phases are completely domain-independent so they could
easily be used in other problem domains such as logistics, network routing or
service allocation. In the traffic domain, the algorithm can be used to plan
routes that avoid traffic jams or to control traffic lights. What is more,
additional constraints such as staying at one city for some time or travelling
together with a specific person can be easily added. On the other hand, the
timetabling phase of the algorithm is domain-specific, providing an example of
the specific design choices that have to be made from an engineering point of
view.
To assess the practical value of our contribution, it is worth discussing how
it could be used in practice as a part of a travel planning system for real
passengers. In such a system, every user would submit origin, destination and
travel times. Different users could submit their preferences at different
times, with the system continuously computing shared journeys for them based
on information about all users’ preferences. Users would need to agree on a
shared journey in time to arrange meeting points and to purchase tickets,
subject to any restrictions on advance tickets etc. Because of this lead time,
it would be entirely sufficient if the users got an e-mail with a planned
journey one hour after the last member of the travel group submits his or her
journey details, which implies that even with our current implementation of
the algorithm, the runtimes would be acceptable.
From our experimental evaluation, we conclude that reasonable group sizes
range from two to four persons. Apart from the fact that such groups can be
relatively easily coordinated, with the price model used in this paper, cf.
formula (1), every member of a three-person group could save up to 53 % of the
single-agent price. The success rate of the timetabling phase of the algorithm
for three-person groups in the scenario S3 (trains in the central UK) is 70 %.
## 7 Conclusion
We have presented a multiagent planning algorithm which is able to plan
meaningful shared routes in a real-world travel domain. The algorithm has been
implemented and evaluated on five scenarios based on real-world UK public
transport data. The algorithm exhibits very good scalability, since it scales
linearly both with the scenario size and the number of agents. The average
computation time for 12 agents in the scenario with 90 % of trains in the UK
is less than one hour. Experiments indicate that the algorithm avoids the
exponential blowup in the action space characteristic for a centralised
multiagent planner.
To deal with thousands of users that could be in a real-world travel planning
system, a preprocessing step would be needed: The agents would have to be
divided into smaller groups by clustering them according to departure time,
direction of travel, origin, destination, length of journey and preferences
(e.g., travel by train only, find cheapest journey). Then, the algorithm could
be used to find a shared travel plan with a timetable. To prevent too large
groups of agents which are unlikely to be matched to the timetable, a limit
can be imposed on the size of the group. If a group plan cannot be mapped to a
timetable, the group can be split into smaller sub-groups which are more
likely to identify a suitable timetable.
Finally, the price of travel and flexibility of travel sharing can be
significantly improved by sharing a private car. In the future, we would like
to explore the problem of planning shared journeys when public transport is
combined with ride sharing. Then, in order to have a feasible number of nodes
in the travel domain, train and bus stops can be used as meeting points where
it is possible to change from a car to public transport or vice versa.
## 8 Acknowledgments
Partly supported by the Ministry of Education, Youth and Sports of Czech
Republic (grant No. LD12044) and European Commission FP7 (grant agreement No.
289067).
## References
* [1] S. Abdel-Naby, S. Fante, and P. Giorgini. Auctions Negotiation for Mobile Rideshare Service. In Procs. ICPCA 2007, pages 225–230, 2007.
* [2] R. I. Brafman and C. Domshlak. From One to Many: Planning for Loosely Coupled Multi-Agent Systems. In Procs. ICAPS 2008, pages 28–35. AAAI Press, 2008.
* [3] R. I. Brafman, C. Domshlak, Y. Engel, and M. Tennenholtz. Planning Games. In Procs. IJCAI 2009, pages 73–78, July 2009.
* [4] Y. Chen, B. W. Wah, and C.-W. Hsu. Temporal planning using subgoal partitioning and resolution in SGPlan. Journal of Artificial Intelligence Research, 26:323–369, Aug. 2006\.
* [5] A. J. Coles, A. I. Coles, A. Clark, and S. T. Gilmore. Cost-Sensitive Concurrent Planning under Duration Uncertainty for Service Level Agreements. In Procs. ICAPS 2011, pages 34–41. AAAI Press, June 2011.
* [6] A. J. Coles, A. I. Coles, M. Fox, and D. Long. Forward-Chaining Partial-Order Planning. In Procs. ICAPS 2010, pages 42–49. AAAI Press, May 2010.
* [7] A. J. Coles, A. I. Coles, M. Fox, and D. Long. POPF2: a Forward-Chaining Partial Order Planner. In Procs. IPC-7, 2011.
* [8] E. Ferrari, R. Manzini, A. Pareschi, A. Persona, and A. Regattieri. The car pooling problem: Heuristic algorithms based on savings functions. Journal of Advanced Transportation, 37(3):243–272, 2003.
* [9] M. Ghallab, D. Nau, and P. Traverso. Automated Planning: Theory and Practice. Morgan Kaufmann Publishers Inc., 2004.
* [10] J. Hoffmann and B. Nebel. The FF planning system: Fast plan generation through heuristic search. Journal of Artificial Intelligence Research, 14:253–302, 2001.
* [11] J. Hrnčíř. Improving a Collaborative Travel Planning Application. Master’s thesis, The University of Edinburgh, Aug. 2011.
* [12] C.-W. Hsu and B. W. Wah. The SGPlan Planning System in IPC-6. In Procs. IPC-6, 2008.
* [13] A. Jonsson and M. Rovatsos. Scaling Up Multiagent Planning: A Best-Response Approach. In Procs. ICAPS 2011, pages 114–121. AAAI Press, June 2011.
* [14] P. Lalos, A. Korres, C. K. Datsikas, G. S. Tombras, and K. Peppas. A Framework for Dynamic Car and Taxi Pools with the Use of Positioning Systems. In Computation World: Future Computing, Service Computation, Cognitive, Adaptive, Content, Patterns, pages 385–391, Nov. 2009.
* [15] N. Meuleau, M. Hauskrecht, K.-E. Kim, L. Peshkin, L. P. Kaelbling, T. Dean, and C. Boutilier. Solving very large weakly coupled Markov decision processes. In Procs. AAAI 1998, pages 165–172. AAAI Press, 1998.
* [16] D. Monderer and L. S. Shapley. Potential Games. Games and Economic Behavior, 14(1):124–143, 1996.
* [17] S. Richter, M. Helmert, and M. Westphal. Landmarks Revisited. In Procs. AAAI 2008, pages 975–982. AAAI Press, July 2008.
* [18] S. Richter and M. Westphal. The LAMA planner. Using landmark counting in heuristic search. In Procs. IPC-6, 2008.
* [19] B. W. Wah and Y. Chen. Constraint partitioning in penalty formulations for solving temporal planning problems. Artificial Intelligence, 170:187–231, Mar. 2006.
|
arxiv-papers
| 2013-01-02T12:06:59 |
2024-09-04T02:49:39.851190
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Hrn\\v{c}\\'i\\v{r} and Michael Rovatsos",
"submitter": "Jan Hrncir",
"url": "https://arxiv.org/abs/1301.0216"
}
|
1301.0302
|
3
# MANCaLog: A Logic for Multi-Attribute Network Cascades (Technical Report)
Paulo Shakarian
Network Science Center and
Dept. of Electrical Engineering
and Computer Science
U.S. Military Academy
West Point, NY 10996
Gerardo I. Simari
Dept. of Computer Science
University of Oxford
Wolfson Building, Parks Road
Oxford OX1 3QD, UK
Robert Schroeder
CORE Lab
Defense Analysis Dept.
Naval Postgraduate School
Monterey, CA 93943
[email protected] [email protected] [email protected]
###### Abstract
The modeling of cascade processes in multi-agent systems in the form of
complex networks has in recent years become an important topic of study due to
its many applications: the adoption of commercial products, spread of disease,
the diffusion of an idea, etc. In this paper, we begin by identifying a
desiderata of seven properties that a framework for modeling such processes
should satisfy: the ability to represent attributes of both nodes and edges,
an explicit representation of time, the ability to represent non-Markovian
temporal relationships, representation of uncertain information, the ability
to represent competing cascades, allowance of non-monotonic diffusion, and
computational tractability. We then present the MANCaLog language, a formalism
based on logic programming that satisfies all these desiderata, and focus on
algorithms for finding minimal models (from which the outcome of cascades can
be obtained) as well as how this formalism can be applied in real world
scenarios. We are not aware of any other formalism in the literature that
meets all of the above requirements.
###### Key Words.:
Complex Networks, Cascades, Logic Programming
I.2.4Artificial IntelligenceKnowledge Representation Formalisms and
Methods[Representation Languages]
Languages, Algorithms
## 1 Introduction and Related Work
An epidemic working through a population, cascading electrical power failures,
product adoption, and the spread of a mutant gene are all examples of
diffusion processes that can happen in multi-agent systems structured as
complex networks. These network processes have been studied in a variety of
disciplines, including computer science kleinberg , biology
lieberman_evolutionary_2005 , sociology Gran78 , economics schelling , and
physics sood08 . Much existing work in this area is based on pre-existing
models in sociology and economics – in particular the work of Gran78 ;
schelling . However, recent examinations of social networks – both analysis of
large data sets and experimental – have indicated that there may be additional
factors to consider that are not taken into account by these models. These
include the attributes of nodes and edges, competing diffusion processes, and
time. In this paper, we outline seven design criteria (Section 1.1) for such a
framework and introduce MANCaLog (Section 2), which is to the best of our
knowledge the first logical language for modeling diffusion in complex
networks that meets these criteria. MANCaLog is a rule-based framework
(inspired by logic programming) that can richly express how agents adopt or
fail to adopt certain behaviors, and how these behaviors cascade through a
network. We also introduce fixed-point based algorithms that allow for the
calculation of the result of the diffusion process in Section 3. Note that
these algorithms are proven not only to be correct, but also to run in
polynomial time. Hence, our approach can not only better express many aspects
of cascades in complex networks, but it can do so in a reasonable amount of
time. We conclude by discussing applications of MANCaLog in Section 4.
Proofs of all results stated in this paper can be found in the appendix.
### 1.1 Desiderata of Properties
We begin by identifying a set of criteria that we believe a framework for
reasoning about cascades in complex networks should satisfy.
1\. Multiply labeled and weighted nodes and edges. Many existing frameworks
for studying diffusion in complex networks assume that there is only one type
of vertex that may become “active” kleinberg or may “mutate”
lieberman_evolutionary_2005 ; sood08 and only one possible relationship
between nodes. In reality, nodes and edges often have different properties.
For instance, labels on edges can be used to differentiate between strong and
weak ties (edge types) – a concept that is well studied granovetter1973 .
Recently, such attributes of nodes have been shown to impact influence in a
network aral12 .
2\. Explicit Representation of Time. Most work in the literature assumes
static models, with the exception of the recent developments in
goyal2008discovering ; goyal2010learning ; goyal2011data , which assume the
existence of a timestamped log referring to actions taken in the network in
order to learn how nodes influence each other. Though goyal2008discovering
tackles the problem of predicting the time at which a certain node will take
an action, the authors make several simplifying assumptions such as
monotonicity of probability functions, probabilistic independence, sub-
modularity and, most importantly for this criterion, a modeling of time solely
based on temporal decay of influence. We seek a richer model of temporal
relationships between conditions in the network structure, the current state
of the cascades in process, and how influence propagates.
3\. Non-Markovian Temporal Relationships. Apart from time being explicitly
represented, the temporal dependencies should be able to span multiple units
of time. Hence, the “memoryless” mode of a standard Markov process, where only
the information of the current state is required, is insufficient. Here, we
strive to create a framework where dependencies can be from other earlier time
steps. This issue has been previously studied with respect to more general
logic programming frameworks such as apt1 , but to our knowledge has not been
applied to social networks.
4\. Representation of Uncertainty. As in practice it is not always possible to
judge the attributes of all individuals in a network, an element of
uncertainty must be included. However, in connection with point 7, this should
not be at the expense of tractability. For instance, the probabilistic models
of kleinberg are normally addressed with simulation (and hence do not scale
well) as the computation of the expected number of activated nodes is a
$\\#P$-hard problem chen10 .
5\. Competing Cascades. Often, in real-world situations there will be
competing cascading processes. For example, in evolutionary graph theory
lieberman_evolutionary_2005 , “mutants” and “residents” compete for nodes in
the network – the success of one hinges on the failure of the other.
6\. Non-Monotonic Cascades. In much existing work on cascades in complex
networks, the number of nodes attaining a certain property at each time step
can only increase. However, if we allow for competing cascades in the same
model, we cannot have such a strong restriction as the success of one cascade
may come at the expense of another.
7\. Tractability. The social networks of interest in today’s data mining
problems often have millions of nodes. It is reasonable to expect that soon
billion-node networks will be commonplace. Any framework for dealing with
these problems must be solvable in a reasonable amount of time and offer areas
for practical improvement for further scalability.
### 1.2 Related Work
The above criteria can be summarized as the desire to design the most
expressive language for network cascades possible while still allowing
computation of the outcome of a diffusion process to be completed in a
tractable amount of time. As a comparison, let us briefly describe some
relevant related work. Perhaps the best known general model for representing
diffusion in complex networks is the independent cascade/linear threshold
(IC/LT) model of kleinberg . However, although this framework was shown to be
capable of expressing a wide variety of sociological models, it assumes the
Markov property and does not allow for the representation of multiple
attributes on vertices and edges. A more recent framework, social network
optimization problems (SNOPs) snops uses logic programming to allow for the
representation of attributes, but this framework does not allow for competing
processes or non-monotonic cascades. A related logic programming framework,
competitive diffusion (CD) bss10 allows for competitive diffusion and non-
monotonic processes but does not explicitly represent time and also makes
Markovian assumptions. Further, we also note that the semantics of CD yields a
“most probable interpretation” that is not a unique solution. Hence, a given
model in that framework can lead to multiple and possibly contradictory,
outcomes to a cascade (this problem is avoided in MANCaLog). Another popular
class of models is Evolutionary Graph Theory (EGT) lieberman_evolutionary_2005
, which is highly related to the voter model (VM) sood08 . Although this
framework allows for competing processes and non-monotonic diffusion, it also
makes Markovian assumptions while not explicitly representing time. Further,
determining the outcome of a cascade in those models is NP-hard, while
determining the outcome in MANCaLog can be accomplished in polynomial time.
Table 1 lists how these models compare to MANCaLog when considering our design
criteria.
Table 1: Comparison with other models Criterion | MANCaLog | IC/LT kleinberg | SNOP snops | CD bss10 | EGT/VM lieberman_evolutionary_2005
---|---|---|---|---|---
1\. Labels | Yes | No | Yes | Yes | No
2\. Explicit Representation of Time | Yes | No | Yes | No | Yes
3\. Non-Markovian Temporal Relationships | Yes | No | No | No | No
4\. Uncertainty | Yes | Yes | Yes | Yes | Yes
5\. Competing Cascades | Yes | No | No | Yes | Yes
6\. Non-monotonic Cascades | Yes | No | No | Yes | Yes
7\. Tractablity | PTIME | $\\#$P-hard | PTIME | PTIME | NP-hard
## 2 Framework
### 2.1 Syntax and Semantics
In this paper we assume that agents are arranged in a directed graph (or
network) $G=(V,E)$, where the set of nodes corresponds to the agents, and the
edges model the relationships between them. We also assume a set of labels
$\mathcal{L}$, which is partitioned into two sets: fluent labels
$\mathcal{L}_{f}$ (labels that can change over time) and non-fluent labels
$\mathcal{L}_{nf}$ (labels that do not); labels can be applied to both the
nodes and edges of the network. We will use the notation $\mathcal{G}=V\cup E$
to be the set of all components (nodes and edges) in the network. Thus,
$c\in\mathcal{G}$ could be either a node or an edge.
###### Example 2.0
We will use the sample online social network $G_{soc}$ shown in Figure 1 as
the running example; $\mathcal{G}_{soc}$ is used to denote the set of
components of $G_{soc}$. Here we have
$\mathcal{L}_{nf}=\\{\textit{male},\textit{fem},$
$\textit{strTie},\textit{wkTie}\\}$ representing male, female, strong ties and
weak ties, respectively. Additionally, we have
$\mathcal{L}_{f}=\\{\textit{visPgA},\textit{visPgB}\\}$ representing visiting
webpage A and visiting webpage B, respectively. $\blacksquare$
Figure 1: Simple online social network $G_{soc}$. Solid edges are labeled with
strTie, while dashed edges are labeled with wkTie. White nodes are labeled
with male, while gray nodes are labeled with fem. Arrows represent the
direction of the edge; double-headed edges represent two edges with the same
label.
In this paper, we present a logical language where we use atoms, referring to
labels and weights, to describe properties of the nodes and edges. Though
labels themselves could be modeled as atoms instead of predicates (to model
non-ground labelings that allow for greater expressibility), for simplicity of
presentation we leave this to future work. The first piece of the syntax is
the network atom.
###### Definition 2.0 (Network Atom)
Given label $L\in\mathcal{L}$ and weight interval
$\mathit{bnd}\subseteq[0,1]$, then $\langle L,\mathit{bnd}\rangle$ is a
network atom. A network atom is fluent (resp., non-fluent) if
$L\in\mathcal{L}_{f}$ (resp., $L\in\mathcal{L}_{\mathit{nf}}$). We use $NA$ to
denote the set of all possible network atoms.
Network atoms describe properties of nodes and edges. The definition is
intuitive: $L$ represents a property of the vertex or edge, and associated
with this property is some weight that may have associated uncertainty – hence
represented as an interval $\mathit{bnd}$, which can be open or closed. An
invalid bound is represented by $\emptyset$, which is equivalent to all other
invalid bounds.
###### Definition 2.0 (World)
A world $W$ is a set of network atoms such that for each $L\in\mathcal{L}$
there is no more than one network atom of the form $\langle
L,\mathit{bnd}\rangle$ in $W$.
A network formula over $NA$ is defined using conjunction, disjunction, and
negation in the usual way. If a formula contains only non-fluent (resp.,
fluent) atoms, it is a non-fluent (resp., fluent) formula.
###### Definition 2.0 (Satisfaction of Worlds)
Given a world $W$ and network formula $f$, satisfaction of $W$ by $f$ (denoted
$W\models f$) is defined:
* •
If $f=\langle L,[0,1]\rangle$ then $W\models f$.
* •
If $f=\langle L,\emptyset\rangle$ then $W\not\models f$.
* •
If $f=\langle L,\mathit{bnd}\rangle$, with $\mathit{bnd}\neq\emptyset$ and
$\mathit{bnd}\neq[0,1]$, then $W\models f$ iff there exists $\langle
L,\mathit{bnd}^{\prime}\rangle\in W$ s.t.
$\mathit{bnd}^{\prime}\subseteq\mathit{bnd}$.
* •
If $f=\neg f^{\prime}$ then $W\models f$ iff $W\not\models f^{\prime}$.
* •
If $f=f_{1}\wedge f_{2}$ then $W\models f$ iff $W\models f_{1}$ and $W\models
f_{2}$.
* •
If $f=f_{1}\vee f_{2}$ then $W\models f$ iff $W\models f_{1}$ or $W\models
f_{2}$.
For some arbitrary label $L\in\mathcal{L}$, we will use the notation
$\textsf{Tr}=\langle L,[0,1]\rangle$ and $\textsf{F}=\langle
L,\emptyset\rangle$ to represent a tautology and contradiction, respectively.
For ease of notation (and without loss of generality), we say that if there
does not exist some $\mathit{bnd}$ s.t. $\langle L,\mathit{bnd}\rangle\in W$,
then this implies that $\langle L,[0,1]\rangle\in W$.
###### Example 2.0
Following from Example 2.1, the network atom $\langle female,[1,1]\rangle$ can
be used to identify a node as a woman. Likewise, the world $W=$
$\Big{\\{}\langle\textit{fem},[1,1]\rangle,\langle\textit{male},[0,0]\rangle,\langle\textit{visPgA},[1,1]\rangle,\langle\textit{visPgB},[0,0]\rangle\Big{\\}}$
might be used to identify a woman who visits webpage A. Clearly, we have that
$W\models$
$\langle\textit{fem},[1,1]\rangle\wedge\neg\langle\textit{visPgA},[0.5,0.9]\rangle\wedge\neg\langle\textit{visPgB},[0.1,0.7]\rangle$
Note that the network atoms formed with strTie and wkTie are not present; this
could be due to the fact that such a world is used to describe a node and not
an edge, and hence there is no information about those two labels. As such is
the case,
$W\models\langle\textit{strTie},[0,1]\rangle\wedge\langle\textit{wkTie},[0,1]\rangle$.
$\blacksquare$
The idea is to use MANCaLog to describe how properties (specified by labels)
of the nodes in the network change over time. We assume that there is some
natural number $t_{\mathit{max}}$ that specifies the total amount of time we
are considering, and we use $\tau=\\{t\;|\;t\in[0,t_{\mathit{max}}]\\}$ to
denote the set of all time points. How well a certain property can be
attributed to a node is based on a weight (to which the $\mathit{bnd}$ bound
in the network atom refers). As time progresses, a weight can either
increase/decrease and/or become more/less certain. We now introduce the
MANCaLog fact, which states that some network atom is true for a node or edge
during certain times.
###### Definition 2.0 (MANCaLog Fact)
If $[t_{1},t_{2}]\subseteq[0,t_{\mathit{max}}]$, $c\in\mathcal{G}$, and $a\in
NA$, then $(a,c):[t_{1},t_{2}]$ is a MANCaLog fact. A fact is fluent (resp.,
non-fluent) if atom $a$ is fluent (resp., non-fluent). All non-fluent facts
must be of the form $(a,c):[0,t_{\mathit{max}}]$. Let $\mathcal{F}$ be the set
of all facts and $\mathcal{F}_{nf},\mathcal{F}_{f}$ be the set of all non-
fluent and fluent facts, respectively.
###### Example 2.0
Following from Example 2.5, the following facts are based on Figure 1:
$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle(\langle\textit{male},[1,1]\rangle,1):[0,t_{\mathit{max}}]$
$\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle(\langle\textit{fem},[1,1]\rangle,1):[0,t_{\mathit{max}}]$
$\displaystyle F_{3}$ $\displaystyle=$
$\displaystyle(\langle\textit{male},[1,1]\rangle,3):[0,t_{\mathit{max}}]$
$\displaystyle F_{4}$ $\displaystyle=$
$\displaystyle(\langle\textit{strTie},[1,1]\rangle,(1,2)):[0,t_{\mathit{max}}]$
$\displaystyle F_{5}$ $\displaystyle=$
$\displaystyle(\langle\textit{strTie},[1,1]\rangle,(2,1)):[0,t_{\mathit{max}}]$
$\displaystyle F_{6}$ $\displaystyle=$
$\displaystyle(\langle\textit{wkTie},[1,1]\rangle,(2,3)):[0,t_{\mathit{max}}]$
$\displaystyle F_{7}$ $\displaystyle=$
$\displaystyle(\langle\textit{visPgA},[0.8,1.0]\rangle,1):[0,t_{\mathit{max}}]$
$\displaystyle F_{8}$ $\displaystyle=$
$\displaystyle(\langle\textit{visPgA},[0.5,1.0]\rangle,2):[0,t_{\mathit{max}}]$
For instance, agent 1 is male, and has a strong tie to agent 2, who is female.
$\blacksquare$
Next, we introduce integrity constraints (ICs).
###### Definition 2.0
Given fluent network atom $a$ and conjunction of network atoms $b$, an
integrity constraint is of the form $a\hookleftarrow b$.
Intuitively, integrity constraint $\langle L,\mathit{bnd}\rangle\hookleftarrow
b$ means that if at a certain time point a component (vertex or edge) of the
network has a set of properties specified by conjunction $b$, then at that
same time the component’s weight for label $L$ must be in interval
$\mathit{bnd}$.
###### Example 2.0
Following from the previous examples, the integrity constraint
$\langle\textit{male},[0,0]\rangle\hookleftarrow\langle\textit{fem},[1,1]\rangle$
would require any node designated as a female to not be male. $\blacksquare$
We now define MANCaLog rules. The idea behind rules is simple: an agent that
meets some criteria is influenced by the set of its neighbors who possess
certain properties. The amount of influence exerted on an agent by its
neighbors is specified by an influence function, whose precise effects will be
described later on when we discuss the semantics. As a result, a rule consists
of four major parts: (i) an influence function, (ii) neighbor criteria, (iii)
target criteria, and (iv) a target. Intuitively, (i) specifies how the
neighbors influence the agent in question, (ii) specifies which of the
neighbors can influence the agent, (iii) specifies the criteria that cause the
agent to be influenced, and (iv) is the property of the agent that changes as
a result of the influence.
We will discuss each of these parts in turn, and then define rules in terms of
these elements. First, we define influence functions and neighbor criteria.
###### Definition 2.0 (Influence Function)
An influence function is a function
$\mathit{ifl}:\textbf{N}\times\textbf{N}\rightarrow[0,1]\times[0,1]$ that
satisfies the following two axioms:
1\. $\mathit{ifl}$ can be computed in constant ($O(1)$) time.
2\. For $x^{\prime}>x$ we have
$\mathit{ifl}(x^{\prime},y)\subseteq\mathit{ifl}(x,y)$.
We use $\mathit{IFL}$ to denote the set of all influence functions.
Intuitively, an influence function takes the number of qualifying influencers
and the number of eligible influencers and returns a bound on the new value
for the weight of the property of the target node that changes. In practice,
we expect the time complexity of such a function to be a polynomial in terms
of the two arguments. However, as both arguments are naturals bounded by the
maximum degree of a node in the network, this value will be much smaller than
the size of the network – we thus treat it as a constant here.
###### Example 2.0
The well-known “tipping model” originally introduced in Gran78 ; schelling
states that an agent adopts a behavior when a certain fraction of his incoming
neighbors do so. A common tipping function is the majority threshold where at
least half of the agent’s neighbors must previously adopt the behavior. We can
represent this using the following influence function:
$\mathit{tip}(x,y)=\begin{cases}[1.0,1.0]&\textit{if }x/y\geq 0.5\\\
[0.0,1.0]&\text{otherwise}\end{cases}$
This function says that an agent adopts a certain behavior if at least half of
his incoming neighbors have some property (strong ties, weak ties, meet some
requirement of gender, income, etc.) and that we have no information
otherwise. In our framework, we can leverage the bounds associated with the
influence function to create a “soft” tipping function:
${\mathit{sftTp}}(x,y)=\begin{cases}[0.7,1.0]&\textit{if }x/y\geq 0.5\\\
[0.0,1.0]&\text{otherwise}\end{cases}$
Intuitively, the above function says that an agent adopts a behavior with a
weight of at least $0.7$ if half of the incoming neighbors that have some
attribute and meet some criteria, and we have no information otherwise.
Another possibility is to have an influence function that may reduce the
weight that an agent adopts a certain behavior:
${\mathit{ngTp}}(x,y)=\begin{cases}[0.0,0.2]&\textit{if }x=y\\\
[0.0,1.0]&\text{otherwise}\end{cases}$
The ${\mathit{ngTp}}$ function says that an agent will adopt a behavior with a
weight no greater than $0.2$ if all of the incoming neighbors possessing some
property meet some criteria, and that we have no information otherwise.
$\blacksquare$
###### Definition 2.0 (Neighbor Criterion)
If $g_{edge},g_{node}$ are non-fluent network formulas (formed over edges and
nodes, respectively), $h$ is a conjunction of network atoms, and
$\mathit{ifl}$ is an influence function, then
$(g_{edge},g_{node},h)_{\mathit{ifl}}$ is a neighbor criterion.
Formulas $g_{node}$ and $h$ in a neighbor criterion specify the (non-fluent
and fluent, respectively) criteria on a given neighbor, while formula
$g_{edge}$ specifies the non-fluent criteria on the directed edge from that
neighbor to the node in question.
The next component is the “target criteria”, which are the criteria that an
agent must satisfy in order to be influenced by its neighbors. Ideas such as
“susceptibility” aral12 can be integrated into our framework via this
component. We represent these criteria with a formula of non-fluent network
atoms. The final component, the “target”, is simply the label of the target
agent that is influenced by its neighbors. Hence, we now have all the pieces
to define a rule.
###### Definition 2.0 (Rule)
Given fluent label $L$, natural number $\Delta t$, target criteria $f$ and
neighbor criteria
$(g_{edge},g_{node},h)_{\mathit{ifl}}$, a MANCaLog Rule is of the form:
$\displaystyle r=L$ $\displaystyle\stackrel{{\scriptstyle\Delta
t}}{{\leftarrow}}$ $\displaystyle f,(g_{edge},g_{node},h)_{\mathit{ifl}}$
We will use the notation $head(r)$ to denote $L$.
Note that the target (also referred to as the head) of the rule is a single
label; essentially, the body of the rule characterizes a set of nodes, and
this label is the one that is modified for each node in this set. More
specifically, the rule is essentially saying that when certain conditions for
an agent and its neighbors are met, the $\mathit{bnd}$ bound for the network
atom formed with label $L$ on that agent changes. Later, in the semantics, we
introduce network interpretations, which map components (nodes and edges) of
the network to worlds at a given point in time. The rule dictates how this
mapping changes in the next time step.
###### Definition 2.0 (MANCaLog Program)
A program $P$ is a set of rules, facts, and integrity constraints s.t. each
non-fluent fact $F\in\mathcal{F}_{nf}$ appears no more than once in the
program. Let $\mathbf{P}$ be the set of all programs.
###### Example 2.0
Following from the previous examples, we can have a MANCaLog program that
leverage the ${\mathit{sftTp}}$ and ${\mathit{ngTp}}$ influence functions in
rules that are more expressive than previous models. Consider the following
rules:
$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\textit{visPgA}\stackrel{{\scriptstyle 2}}{{\leftarrow}}$
$\displaystyle\langle\textit{fem},[1,1]\rangle,(\langle\textit{strTie},[0.9,1]\rangle,\textsf{Tr},\langle\textit{visPgA},[0.9,1.0]\rangle)_{\mathit{sftTp}}$
$\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\textit{visPgB}\stackrel{{\scriptstyle 1}}{{\leftarrow}}$
$\displaystyle\langle\textit{male},[1,1]\rangle,(\textsf{Tr},\textsf{Tr},\langle\textit{visPgB},[0.8,1.0]\rangle)_{\mathit{sftTp}}$
$\displaystyle R_{3}$ $\displaystyle=$
$\displaystyle\textit{visPgA}\stackrel{{\scriptstyle 3}}{{\leftarrow}}$
$\displaystyle\langle\textit{male},[1,1]\rangle,(\textsf{Tr},\langle\textit{fem},[1,1]\rangle,\neg\langle\textit{visPgA},[0.7,1.0]\rangle)_{\mathit{ngTp}}$
Rule $R_{1}$ says that a female agent in the network visits page A with a
weight of at least $0.7$ (this is specified in the $\mathit{sftTp}$ influence
function) if at least half of her strong ties (with weight of at least $0.9$)
visited the page (with a weight of at least $0.9$) two days ago. The rest of
the rules can be read analogously. $\blacksquare$
We now introduce our first semantic structure: the network interpretation.
###### Definition 2.0 (Network Interpretation)
A network interpretation is a mapping of network components to sets of network
atoms, $NI:\mathcal{G}\rightarrow 2^{NA}$. We will use $\mathbf{NI}$ to denote
the set of all network interpretations.
We note that not all labels will necessarily apply to all nodes and edges in
the network. For instance, certain labels may describe a relationship while
others may only describe a property of an individual in the network. If a
given label $L$ does not describe a certain component $c$ of the network, then
in a valid network interpretation $NI$, $\langle L,[0,1]\rangle\in NI(c)$.
###### Example 2.0
Consider $G_{soc}^{\prime}$, the induced subgraph of $G_{soc}$ that has only
nodes $\\{1,2,3,4,5\\}$. Table 2 shows the contents of $NI_{1}$, an example
network interpretation. $\blacksquare$
Table 2: Example network interpretation, $NI_{1}$. Comp. | male | fem | strTie | wkTie | visPgA | visPgB
---|---|---|---|---|---|---
1 | $[1,1]$ | $[0,0]$ | - | - | $[0.9,1.0]$ | $[0.8,1.0]$
2 | $[0,0]$ | $[1,1]$ | - | - | $[0.0,0.3]$ | $[0.0,0.2]$
3 | $[1,1]$ | $[0,0]$ | - | - | $[0.6,1.0]$ | $[0.0,0.2]$
4 | $[0,0]$ | $[1,1]$ | - | - | $[0.0,0.2]$ | $[0.9,1.0]$
5 | $[1,1]$ | $[0,0]$ | - | - | $[0.0,0.2]$ | $[0.7,1.0]$
(1,2) | - | - | $[1,1]$ | $[0,0]$ | - | -
(2,1) | - | - | $[1,1]$ | $[0,0]$ | - | -
(1,3) | - | - | $[0,0]$ | $[1,1]$ | - | -
(2,3) | - | - | $[0,0]$ | $[1,1]$ | - | -
(3,4) | - | - | $[1,1]$ | $[0,0]$ | - | -
(4,3) | - | - | $[1,1]$ | $[0,0]$ | - | -
(4,5) | - | - | $[1,1]$ | $[0,0]$ | - | -
We define a MANCaLog interpretation as follows.
###### Definition 2.0 (Interpretation)
A MANCaLog interpretation $I$ is a mapping of natural numbers in the interval
$[0,t_{\mathit{max}}]$ to network interpretations, i.e.,
$I:\textbf{N}\rightarrow\mathbf{NI}$. Let $\mathcal{I}$ be the set of all
possible interpretations.
### 2.2 Satisfaction
First, we define what it means for an interpretation to satisfy a fact and a
rule.
###### Definition 2.0 (Fact Satisfaction)
An interpretation $I$ satisfies MANCaLog fact $(a,c):[t_{1},t_{2}]$, written
$I\models(a,c):[t_{1},t_{2}]$, iff $\forall t\in[t_{1},t_{2}]$,
$I(t)(c)\models a$.
###### Example 2.0
Consider interpretation $I_{1}$, where $I_{1}(0)=NI_{1}$ (from Example 2.17),
and MANCaLog facts $F_{7}$ and $F_{8}$ from Example 2.7. In this case,
$I_{1}\models F_{7}$ and $I_{1}\not\models F_{8}$. $\blacksquare$
For non-fluent facts, we introduce the notion of strict satisfaction, which
enforces the bound in the interpretation to be set to exactly what the fact
dictates.
###### Definition 2.0 (Strict Fact Satisfaction)
Interpretation $I$ strictly satisfies MANCaLog fact $(c,a):[t_{1},t_{2}]$ iff
$\forall t\in[t_{1},t_{2}]$, $a\in I(t)(c)$.
Next, we define what it means for an interpretation to satisfy an integrity
constraint.
###### Definition 2.0 (IC Satisfaction)
An interpretation $I$ satisfies integrity constraint $a\hookleftarrow b$ iff
for all $t\in\tau$ and $c\in\mathcal{G}$, $I(t)(c)\models\neg b\vee a$.
Before we define what it means for an interpretation to satisfy a rule, we
require two auxiliary definitions that are used to define the bound enforced
on a label by a given rule, and the set of time points that are affected by a
rule.
###### Definition 2.0 ($\mathit{Bound}$ function)
For a given rule $r=L\stackrel{{\scriptstyle\Delta
t}}{{\leftarrow}}f,(g_{edge},g_{node},h)_{\mathit{ifl}}$, node $v$, and
network interpretation $NI$, $\mathit{Bound}(r,v,NI)=$
$\mathit{ifl}\Big{(}\Big{|}\mathit{Qual}\big{(}v,g_{edge},g_{node},h,NI\big{)}\Big{|},\Big{|}\mathit{Elig}\big{(}v,g_{edge},g_{node},NI\big{)}\Big{|}\Big{)},$
where $\mathit{Elig}(v,g_{edge},g_{node},NI)=$
$\Big{\\{}v^{\prime}\in V\;|\;NI(v^{\prime})\models
g_{node}\wedge(v^{\prime},v)\in E\wedge NI\big{(}(v^{\prime},v)\big{)}\models
g_{edge}\Big{\\}}$
and $\mathit{Qual}(v,g_{edge},g_{node},h,NI)=$
$\Big{\\{}v^{\prime}\in\mathit{Elig}(v,g_{edge},g_{node},NI)\;|\;NI(v^{\prime})\models
h\Big{\\}}$
Intuitively, the bound returned by the function depends on the influence
function and the number of qualifying and eligible nodes that influence it.
###### Definition 2.0 (Target Time Set)
For interpretation $I$, node $v$, and rule $r=L\stackrel{{\scriptstyle\Delta
t}}{{\leftarrow}}f,(g_{edge},g_{node},h)_{\mathit{ifl}}$, the target time set
of $I,v,r$ is defined as follows:
$\mathit{TTS}(I,v,r)=\Big{\\{}t\in[0,t_{\mathit{max}}]\;|\;I(t-\Delta
t)(v)\models f\Big{\\}}$
We also extend this definition to a program $P$, for a given $c\in\mathcal{G}$
and $L\in\mathcal{L}$, as follows; $\mathit{TTS}(I,c,L,P)=$
$\bigcup_{r\in
P,\mathit{head}(r)=L}\mathit{TTS}(I,c,r)\cup\big{\\{}t\in[t_{1},t_{2}]\;|\;(\langle
L,\mathit{bnd}\rangle,c):[t_{1},t_{2}]\in P\big{\\}}$
$\cup\;\Big{\\{}t\;|\;\langle L,\mathit{bnd}\rangle\hookleftarrow b\in P\wedge
I(t)(c)\models b\Big{\\}}$
We can now define satisfaction of a rule by an interpretation.
###### Definition 2.0
An interpretation $I$ satisfies a rule $r=L\stackrel{{\scriptstyle\Delta
t}}{{\leftarrow}}f,(g_{edge},g_{node},h)_{\mathit{ifl}}$ iff for all $v\in V$
and $t\in\mathit{TTS}(I,v,r)$ it holds that
$I(t)(v)\models\Big{\langle}L,\mathit{Bound}\big{(}r,v,I(t-\Delta
t)\big{)}\Big{\rangle}.$
###### Example 2.0
Let $I_{1}$ be the interpretation from Example 2.20. Suppose that
$\langle\textit{visPgB},[0.8,1.0]\rangle\in I(1)(5)$. In this case,
$I_{1}\models R_{2}$. Let $I_{2}$ be equivalent to $I_{1}$ except that we have
$\langle\textit{visPgB},[0.0,0.5]\rangle\in I_{2}(1)(3)$. In this case,
$I_{2}\not\models R_{2}$. $\blacksquare$
We now define satisfaction of programs, and introduce canonical
interpretations, in which time points that are not “targets” retain
information from the last time step.
###### Definition 2.0
For interpretation $I$ and program $P$:
$I$ is a model for $P$ iff it satisfies all rules, integrity constraints, and
fluent facts in that program, strictly satisfies all non-fluent facts in the
program, and for all $L\in\mathcal{L},$ $c\in\mathcal{G}$ and
$t\notin\mathit{TTS}(I,c,L,P)$, $\langle L,[0,1]\rangle\in I(c)(t)$.
$I$ is a canonical model for $P$ iff it satisfies all rules, integrity
constraints, and fluent facts in $P$, strictly satisfies all non-fluent facts
in $P$, and for all $L\in\mathcal{L},$ $c\in\mathcal{G},$ and
$t\notin\mathit{TTS}(I,c,L,P)$, $\langle L,[0,1]\rangle\in I(c)(t)$ when $t=0$
and $\langle L,bnd\rangle\in I(t)(c)$ where $\langle L,bnd\rangle\in
I(t-1)(c)$, otherwise.
###### Example 2.0
Following from previous examples, if we consider interpretation $I_{1}$ and
program $P=\\{F_{7},R_{2}\\}$, we have that
$\langle\textit{visPgB},[0.0,0.2]\rangle$ must be in $I_{1}(1)(2)$ in order
for $I_{1}$ to be canonical. $\blacksquare$
### 2.3 Consistency and Entailment
In this section we discuss consistency and entailment in MANCaLog programs,
and explore the use of minimal models towards computing answers to these
problems.
###### Definition 2.0 (Consistency)
A MANCaLog program $P$ is (canonically) consistent iff there exists a
(canonical) model $I$ of $P$.
###### Definition 2.0 (Entailment)
A MANCaLog program $P$ (canonically) entails MANCaLog fact $F$ iff for all
(canonical) models $I$ of $P$, it holds that $I\models F$.
Now we define an ordering over models and define the concept of minimal model.
We then show that if we can find a minimal model then we can answer
consistency, entailment, and tight entailment queries. To do so, we first
define a pre-order over interpretations.
###### Definition 2.0 (Preorder over Interpretations)
Given interpretations $I,I^{\prime}$ we say $I\sqsubseteq^{pre}I^{\prime}$ if
and only if for all $t,v,L$ if there exists $\langle L,\mathit{bnd}\rangle\in
I(t)(v)$ then there must exist $\langle L,\mathit{bnd}^{\prime}\rangle\in
I^{\prime}(t)(v)$ s.t. $\mathit{bnd}^{\prime}\subseteq\mathit{bnd}$.
Next, we define an equivalence relation for interpretations denoted with
$\sim$; we will use the notation $[I]$ for the set of all interpretations
equivalent to $I$ w.r.t. $\sim$. This allows us to define a partial ordering.
###### Definition 2.0
Two interpretations $I,I^{\prime}$ are equivalent (written $I\sim I^{\prime}$)
iff for all $P\in\mathbf{P}$, $I\models P$ iff $I^{\prime}\models P$.
###### Definition 2.0 (Partial Ordering)
Given classes of interpretations $[I],[I^{\prime}]$ that are equivalent w.r.t.
$\sim$, we say that $[I]$ precedes $[I^{\prime}]$, written
$[I]\sqsubseteq[I^{\prime}]$, iff $I\sqsubseteq^{pre}I^{\prime}$.
The partial ordering is clearly reflexive, antisymmetric, and transitive. Note
that we will often use $I\sqsubseteq I^{\prime}$ as shorthand for
$[I]\sqsubseteq[I^{\prime}]$. We define two special interpretations, $\bot$
and $\top$, such that $\forall t\in\tau,c\in\mathcal{G}$,
$\bot(t)(c)=\emptyset$ and there exists network atom $\langle
L,\emptyset\rangle\in\top(t)(c)$. Clearly, no other interpretation can be
below $\bot$ as the $[\ell,u]$ bound on all network atoms for each time step
and each component is $[0,1]$; similarly, no other interpretation is above
$\top$, since for any interpretation $I$ for which there exists $\langle
L,\mathit{bnd}\rangle\in I(t)(c)$ where $\mathit{bnd}\neq\emptyset$, we have
$\emptyset\subseteq\mathit{bnd}$. We can prove (see the full version of the
paper for details) that with $\top$ and $\bot$,
$\langle\mathcal{I},\sqsubseteq\rangle$ is a complete lattice. We can now
arrive at the notion of minimal model for a MANCaLog program.
###### Definition 2.0 (Minimal Model)
Given program $P$, the minimal model of $P$ is a (canonical) interpretation
$I$ s.t. $I\models P$ and for all (canonical) interpretation $I^{\prime}$ s.t.
$I^{\prime}\models P$, we have that $I\sqsubseteq I^{\prime}$.
Suppose we have some algorithm $A$ that takes as input a program $P$ and
returns an interpretation $I$ (where $I$ does not necessarily satisfy $P$)
s.t. for all $I^{\prime}$ where $I^{\prime}\models P$, $I\sqsubseteq
I^{\prime}$. It is easy to show that if $A(P)\models P$ then $P$ is
consistent. Likewise, if $A(P)=\top$ then $P$ is inconsistent, as all models
must then have a tighter weight bound for the network atoms than an invalid
interpretation (hence, making such an interpretation invalid as well).
Clearly, any such algorithm $A$ would provide a sound and complete answer to
the consistency problem. Likewise, if we consider the entailment problem, it
is easy to show that for fact $F=(\langle
L,\mathit{bnd}\rangle,c):[t_{1},t_{2}]$, $P$ (canonically) entails $F$ iff the
minimal model of $P$ (canonically) satisfies $F$. This is because for minimal
model $A(P)$ of $P$, for any time $t\in[t_{1},t_{2}]$, if
$A(P)(t)(c)\models\langle L,\mathit{bnd}\rangle$ then there is network atom
$\langle L,\mathit{bnd}^{\prime}\rangle\in A(P)(t)(c)$ s.t.
$\mathit{bnd}^{\prime}\subseteq\mathit{bnd}$. We note that for any other
interpretation $I$ of $P$ with $\langle
L,\mathit{bnd}^{\prime\prime}\rangle\in I(t)(c)$ we have that
$\mathit{bnd}^{\prime}\supseteq\mathit{bnd}^{\prime\prime}$. Hence, having a
minimal model allows us to solve any entailment query. We can think of a
minimal model of a MANCaLog program as the outcome of a diffusion process in a
multi-agent system modeled as a complex network. Hence, a question such as
“how many agents will adopt the product with a weight of at least $0.9$ in two
months?” can be easily answered once the minimal model is obtained.
## 3 Fixed Point Model Computation
In this section we introduce a fixed-point operator that produces the non-
canonical minimal model of a MANCaLog program in polynomial time. This is
followed by an algorithm to find a canonical minimal model also in polynomial
time. First, we introduce three preliminary definitions.
###### Definition 3.0
For a given MANCaLog program $P$, $c\in\mathcal{G}$, $L\in\mathcal{L}$, and
$t\in\tau$ we define function $\mathit{FBnd}(P,c,t,L)=$
$\bigcap_{(\langle L,\mathit{bnd}\rangle,c):[t_{1},t_{2}]\in P\textit{ s.t.
}t\in[t_{1},t_{2}]}\mathit{bnd}$
###### Definition 3.0
For a given MANCaLog program $P$, $c\in\mathcal{G}$, $L\in\mathcal{L}$, and
$t\in\tau$ we define function $\mathit{IBnd}(P,c,t,L)=$
$\bigcap_{\langle L,\mathit{bnd}\rangle\hookleftarrow a\in P\textit{ s.t.
}I(t)(c)\models a}\mathit{bnd}$
###### Definition 3.0
Given MANCaLog program $P$, interpretation $I$, $v\in V$, $L\in\mathcal{L}$,
and $t\in\tau$, we define $\mathit{RBnd}(P,I,v,t,L)=$
$\bigcap_{r\in P\textit{ s.t.
}t\in\mathit{TTS}(I,v,L,P)\cap\mathit{TTS}(I,v,r)}\mathit{Bound}(r,v,I(t-\Delta
t))$
We can now introduce the operator.
###### Definition 3.0 ($\Gamma$ Operator)
For a given MANCaLog program $P$, we define the operator
$\Gamma_{P}:\mathcal{I}\rightarrow\mathcal{I}$ as follows: For a given $I$,
for each $t\in\tau$, $c\in\mathcal{G}$, and $L\in\mathcal{L}$, add
$\langle\mathcal{L},\mathit{bnd}\rangle$ to $\Gamma_{P}(I)(t)(c)$ where
$\mathit{bnd}$ is defined as:
$\displaystyle\mathit{bnd}$ $\displaystyle=$
$\displaystyle\mathit{bnd}_{prv}\cap\mathit{FBnd}(P,c,t,L)\cap$
$\displaystyle\mathit{IBnd}(P,I,c,t,L)\cap\mathit{RBnd}(P,I,c,t,L)$
where $\langle L,\mathit{bnd}_{prv}\rangle\in I(t)(c)$.
It is easy to show that $\Gamma$ can be computed in polynomial time (the proof
is in the full version). Next, we introduce notation for repeated applications
of $\Gamma$.
###### Definition 3.0 (Iterated Applications of $\Gamma$)
Given natural number $i>0$, interpretation $I$, and program $P$, we define
$\Gamma_{P}^{i}(I)$, the multiple applications of $\Gamma$, as follows:
$\Gamma^{i}_{P}(I)=\begin{cases}\Gamma_{P}(I)&\text{if $i=1$}\\\
\Gamma_{P}(\Gamma^{i-1}_{P}(I))&\text{otherwise}\end{cases}$
We can prove that the iterated $\Gamma$ operator converges after a polynomial
number of applications:
###### Theorem 3.6
Given interpretation $I$ and program $P$, there exists a natural number $k$
s.t. $\Gamma_{P}^{k}(I)=\Gamma_{P}^{k+1}(I)$, and
$k\in O\Big{(}|P|\cdot{d_{*}^{in}}\cdot t_{\mathit{max}}\cdot|E|\Big{)}$
where ${d_{*}^{in}}$ is the maximum in-degree in the network.
###### Proof ((sketch))
For a given vertex $i\in V$, we will use the notation $d_{i}^{in}$ to denote
the number of incoming neighbors (of any edge type). First note that for a
given $t\in\tau,i\in V,$ and $L\in\mathcal{L}$, a given rule $r$ can tighten
the bound on a network atom formed with $L$ no more than
$(d_{i}^{in}+1)\cdot({d_{*}^{in}}+1)$ times. At each application of $\Gamma$,
at least one network atom must tighten. Hence, as there are only
$O\Big{(}|P|\cdot{d_{*}^{in}}\cdot t_{\mathit{max}}\cdot|E|\Big{)}$
tightenings possible, this is also the bound on the number of applications of
$\Gamma$.
In the following, we will use the notation $\Gamma^{*}_{P}$ to denote the
iterated application of $\Gamma$ after a number of steps sufficient for
convergence; Theorem 3.6 means that we can efficiently compute
$\Gamma^{*}_{P}$. We also note that as a single application of $\Gamma$ can be
computed in polynomial time, this implies that we can find a minimal model of
a MANCaLog program in polynomial time. We now prove the correctness of the
operator. We do this first by proving a key lemma that, when combined with a
claim showing that for consistent program $P$, $\Gamma^{*}_{P}$ is a model of
$P$, tells us that $\Gamma^{*}_{P}$ is a minimal model for $P$. Following
directly from this, we have that $P$ is inconsistent iff
$\Gamma^{*}_{P}=\top$.
###### Lemma 3.7
If $I\models P$ and $I^{\prime}\sqsubseteq I$ then
$\Gamma(I^{\prime})\sqsubseteq I$.
###### Theorem 3.8
If program $P$ is consistent then $\Gamma^{*}_{P}$ is a minimal model for $P$.
These results, when taken together, prove that tight entailment and
consistency problems for MANCaLog can be solved in polynomial time, which is
precisely what we set out to accomplish as part of our desiderata described in
Section 1.1. Next, we develop an algorithm for the canonical versions of
consistency and tight entailment, and show that we can bound the running time
of the algorithm with a polynomial. We also note that subsequent runs of the
convergence of $\Gamma$ will likely complete quicker in practice, as the
initial interpretation is the last interpretation calculated (cf. line 11). We
also show that the interpretation produced by the algorithm is a canonical
minimal model. Following from that, a program is inconsistent iff the
algorithm returns $\top$.
Algorithm 1 CANON_PROC
0: Program $P$
0: Interpretation $I$
1: $cur\\_interp=\Gamma^{*}_{P}(\bot)$;
2: Initialize matrix array $cur\\_free[\cdot][\cdot]$ where for $v\in V,$ and
$L\in\mathcal{L}$,
$cur\\_free[v][L]=\tau-\mathit{TTS}(cur\\_interp,v,L,P)-\\{0\\}$;
3: Initialize array $vl\\_pr[\cdot]$ where for each
$t\in[1,t_{\mathit{max}}]$, $vl\\_pr[t]=\\{(v,L)\;|\;t\in
cur\\_free[v][L]\\}$;
4: for $t=1,\ldots,t_{\mathit{max}}$ do
5: if $vl\\_pr[t]\neq\emptyset$ then
6: for $(v,L)\in vl\\_pr[t]$ do
7: Remove $\langle L,bnd\rangle$ from $I(t)(v)$;
8: Let $a$ be the atom in $I(t-1)(v)$ of the form $\langle
L,bnd^{\prime}\rangle$;
9: Add $a$ to $I(t)(v)$;
10: end for
11: Set $cur\\_interp=\Gamma^{*}_{P}(cur\\_interp)$;
12: end if
13: For $v\in V$, and $L\in\mathcal{L}$,
$cur\\_free[v][L]=\tau-\mathit{TTS}(cur\\_interp,v,L,P)-\\{0,\ldots,t\\}$
14: For each $t\in[t+1,t_{\mathit{max}}]$, $vl\\_pr[t]=\\{(v,L)\;|\;t\in
cur\\_free[v][L]\\}$
15: end for
16: return $I$
###### Proposition 3.0
Algorithm CANON_PROC performs no more than
$1+t_{\mathit{max}}\cdot\min(|\mathcal{L}|,|P|)\cdot|V|$ calculations of the
convergence of $\Gamma$.
###### Theorem 3.10
If $P$ is consistent, then $\textsf{CANON\\_PROC}(P)$ is the minimal canonical
model of $P$.
## 4 Applications
In this section, we will briefly discuss work in progress on how MANCaLog can
be applied in real world settings.
It is widely acknowledged that modeling influence in multi-agent systems (most
usefully modeled as complex networks) is highly desirable for many practical
problems as varied as viral marketing, prevention of drug use, vaccination,
and power plant failure. Though MANCaLog programs are a rich model to work
with, the acquisition of rules is the principal hurdle to overcome; this is
mainly due to this richness of representation, since for each rule we must
provide a set of conditions on the agents being influenced, conditions on
their neighbors and their ties to their neighbors, and how capable these
neighbors are of influencing them. A domain expert is likely able to provide
important insights into these components, but the best way to obtain these
rules is undoubtedly to leverage the presence of large amounts of data in
domains like Twitter (with about 340M messages sent per day, available through
public APIs), Facebook (over 950M users with more complex information; not
publicly available, but data can be requested through apps), and blogging and
photo hosting sites such as Blogger and Flickr (which have millions of users
as well).
Concretely, we have begun working towards this goal by extracting several
time-series, multi-attribute network data sets on which to apply MANCaLog. For
instance, to study the proliferation of research on different topics, we
looked at research on “niacin” indexed by Thomson Reuters Web of Knowledge
(http://wokinfo.com). This topic was chosen due to its interest to a variety
of disciplines, such as medicine, biology, and chemistry; this gives the data
more variety compared with more discipline-specific topics. We extracted an
author-paper bipartite network consisting of $3,790$ papers with $10,465$
authors and $16,722$ edges (cf. Figure 3); from this data we can easily focus
on various kinds of networks (co-author, citation, etc.). We have also
collected attribute and time-series data for this network, as well as the
subjects of the papers; the propagation of these subjects is a good starting
point to test methods for the acquisition of MANCaLog rules. We are harvesting
larger datasets from various online social networks. Further details can be
found in the full version of the paper.
A proposed learning architecture. We are currently developing a MANCaLog
learning architecture (depicted in Fig. 2) based on the use of state-of-the-
art data analysis, clustering, and influence learning techniques as building
blocks for the acquisition of MANCaLog rules from data sets. The key question
is not just the identification of the best techniques to adopt, but how to
adapt them and combine them in such a way as to produce meaningful and useful
outputs.
Consider the diagram in Fig. 2: the data first flows from raw data sources to
the cluster identification component, which has the goal of identifying sets
of agents behaving as groups (for instance, teens influencing other teens of
the same sex in the consumption of music, or scientists of a certain field
influencing the research topics of others in a related field)
warren2005clustering ; jain2010data ; the main output here is a set of
conditions on nodes and edges that characterize groups of nodes. Once clusters
are identified, the influence recognition component will make use of both the
clusters and the data sources to recognize what kind of influence is present
in the system aral12 ; goyal2010learning ; goyal2011data ; the main output of
this component is the influence function to be used in the MANCaLog rules. The
rule generation component then takes the output of the cluster identification
and influence recognition components, along with the raw data (e.g., to
analyze time stamps) and produces MANCaLog rules; the output of this component
is involved in a refinement cycle with experts who can provide feedback on the
rules being produced (such as possible combinations of rules, identification
of cases of overfitting, etc.).
Figure 2: An architecture for obtaining MANCaLog programs from available data
sources. Figure 3: (Left) Visualization of a multi-attribute time-series
author-paper network from 1952 to 2012. (Top-Right) Close-up of the data
inside the small box in the main figure. (Bottom-Right) Close-up showing node
attributes. In all cases, authors are colored green and papers are colored
red. Data extracted from Thomson Reuters Web of Knowledge.
## 5 Conclusion
In this paper, we presented the MANCaLog language for modeling cascades in
multi-agent systems organized in the form of complex networks. We started by
establishing seven criteria in the form of desiderata for such a formalism,
and proved that MANCaLog meets all of them; to the best of our knowledge, this
has not been accomplished by any previous model in the literature. We also
note that MANCaLog is the first language of its kind to consider network
structure in the semantics, potentially opening the door for algorithms that
leverage features of network topology in more efficiently answering queries.
Our current work involves implementing the algorithms described in this paper,
as well as the real-world applications described in Section 4; though our
algorithms have polynomial time complexity, it is likely that further
optimizations will be needed in practice to ensure scalability for very large
data sets.
In the near future, we shall also explore various types of queries that have
been studied in the literature, such as finding agents of maximum influence,
identifying agents that cause a cascade to spread more quickly, and
identifying agents that can be influenced in order to halt a cascade.
## 6 Acknowledgments
P.S. is supported by the Army Research Office (project 2GDATXR042). G.I.S. is
supported under (UK) EPSRC grant EP/J008346/1 – PrOQAW. The opinions in this
paper are those of the authors and do not necessarily reflect the opinions of
the funders, the U.S. Military Academy, or the U.S. Army.
## References
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* (2) M. Broecheler, P. Shakarian, and V. Subrahmanian. A scalable framework for modeling competitive diffusion in social networks. In Proc. of SocialCom. IEEE, 2010.
* (3) W. Chen, C. Wang, and Y. Wang. Scalable influence maximization for prevalent viral marketing in large-scale social networks. In Proc. of KDD ’10, pages 1029–1038. ACM, 2010.
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* (5) A. Goyal, F. Bonchi, and L. Lakshmanan. Learning influence probabilities in social networks. In Proc. of WSDM, pages 241–250. ACM, 2010.
* (6) A. Goyal, F. Bonchi, and L. Lakshmanan. A data-based approach to social influence maximization. Proc. of VLDB, 5(1):73–84, 2011.
* (7) M. Granovetter. The Strength of Weak Ties. The American Journal of Sociology, 78(6):1360–1380, 1973.
* (8) M. Granovetter. Threshold models of collective behavior. The American Journal of Sociology, 83(6):1420–1443, 1978.
* (9) A. Jain. Data clustering: 50 years beyond K-means. Pattern Recognition Letters, 31(8):651–666, 2010.
* (10) D. Kempe, J. Kleinberg, and E. Tardos. Maximizing the spread of influence through a social network. In Proc. of KDD ’03, pages 137–146. ACM, 2003.
* (11) E. Lieberman, C. Hauert, and M. A. Nowak. Evolutionary dynamics on graphs. Nature, 433(7023):312–316, 2005.
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## 7 Appendix
### 7.1 Set of interpretations form a complete lattice
With top interpretation $\top$ and bottom interpretation $\bot$,
$\langle\mathcal{I},\sqsubseteq\rangle$ is a complete lattice.
###### Proof
Let $\mathcal{I}^{\prime}$ be a subset of $\mathcal{I}$. We can create
$\mathit{inf}(\mathcal{I}^{\prime})$ as follows. We build interpretation
$I^{\prime}$. For each $t\in\tau,c\in\mathcal{G},L\in\mathcal{L}$, let
$\ell_{1}$ be the least of the set
$\cup_{I\in\mathcal{I}^{\prime}}\\{\ell|\langle L,[\ell,u]\rangle\in
I(t)(c),\langle L,[\ell,u)\rangle\in I(t)(c)\\}$ and $\ell_{2}$ be the least
of the set $\cup_{I\in\mathcal{I}^{\prime}}\\{\ell|\langle
L,(\ell,u]\rangle\in I(t)(c),\langle L,(\ell,u)\rangle\in I(t)(c)\\}$. Then,
for each $t\in\tau,c\in\mathcal{G},L\in\mathcal{L}$ let $u_{1}$ be the
greatest element of the set $\cup_{I\in\mathcal{I}^{\prime}}\\{u|\langle
L,[\ell,u]\rangle\in I(t)(c),\langle L,(\ell,u]\rangle\in I(t)(c)\\}$
and $u_{2}$ be the greatest of the set
$\cup_{I\in\mathcal{I}^{\prime}}\\{u|\langle L,[\ell,u)\rangle\in
I(t)(c),\langle L,(\ell,u)\rangle\in I(t)(c)\\}$. If there is any
interpretation $I$ in $\mathcal{I}$ where there is not some $\mathit{bnd}$
s.t. $\langle L,\mathit{bnd}\rangle\in I(t)(c)$ then add $\langle
L,[0,1]\rangle$ to $I^{\prime}(t)(c)$. If $\ell_{2}\leq\ell_{1}$ and
$u_{1}\geq u_{2}$ then add $\langle L,(\ell_{2},u_{1}]\rangle$ to
$I^{\prime}(t)(c)$. If $\ell_{2}\leq\ell_{1}$ and $u_{2}>u_{1}$ then add
$\langle L,(\ell_{2},u_{2})\rangle$ to $I^{\prime}(t)(c)$. If
$\ell_{2}>\ell_{1}$ and $u_{2}>u_{1}$ then add $\langle
L,[\ell_{1},u_{2})\rangle$ to $I^{\prime}(t)(c)$. Finally, if
$\ell_{2}>\ell_{1}$ and $u_{1}\geq u_{2}$ then add $\langle
L,[\ell_{1},u_{1}]\rangle$ to $I^{\prime}(t)(c)$. Clearly,
$I^{\prime}=\mathit{inf}(\mathcal{I}^{\prime})$.
In the next part of the proof, we show we can create
$\mathit{sup}(\mathcal{I}^{\prime})$ as follows. We build interpretation
$I^{\prime}$. For each $t\in\tau,c\in\mathcal{G},L\in\mathcal{L}$ let
$\ell_{1}$ be the greatest of the set
$\cup_{I\in\mathcal{I}^{\prime}}\\{\ell|\langle L,[\ell,u]\rangle\in
I(t)(c),\langle L,[\ell,u)\rangle\in I(t)(c)\\}$ and $\ell_{2}$ be the
greatest of the set
$\cup_{I\in\mathcal{I}^{\prime}}\\{\ell|\langle L,(\ell,u]\rangle\in
I(t)(c),\langle L,(\ell,u)\rangle\in I(t)(c)\\}$. Then, for each
$t\in\tau,c\in\mathcal{G},L\in\mathcal{L}$ let $u_{1}$ be the least element of
the set $\cup_{I\in\mathcal{I}^{\prime}}\\{u|\langle L,[\ell,u]\rangle\in
I(t)(c),\langle L,(\ell,u]\rangle\in I(t)(c)\\}$ and $u_{2}$ be the least of
the set $\cup_{I\in\mathcal{I}^{\prime}}\\{u|\langle L,[\ell,u)\rangle\in
I(t)(c),\langle L,(\ell,u)\rangle\in I(t)(c)\\}$. If
$\max(\ell_{1},\ell_{2})>\min(u_{1},u_{2})$ or
$(\ell_{2}>\ell_{1})\wedge(u_{2}<u_{1})\wedge(\ell_{2}=u_{2})$ then add
$\langle L,\emptyset\rangle$ to $I^{\prime}(t)(c)$. If $\ell_{2}>\ell_{1}$ and
$u_{1}\leq u_{2}$ then add $\langle L,(\ell_{2},u_{1}]\rangle$ to
$I^{\prime}(t)(c)$. If $\ell_{2}>\ell_{1}$ and $u_{2}<u_{1}$ then add $\langle
L,(\ell_{2},u_{2})\rangle$ to $I^{\prime}(t)(c)$. If $\ell_{2}\leq\ell_{1}$
and $u_{2}<u_{1}$ then add $\langle L,[\ell_{1},u_{2})\rangle$ to
$I^{\prime}(t)(c)$. Finally, if $\ell_{2}\leq\ell_{1}$ and $u_{1}\leq u_{2}$
then add $\langle L,[\ell_{1},u_{1}]\rangle$ to $I^{\prime}(t)(c)$. Clearly,
$I^{\prime}=\mathit{sup}(\mathcal{I}^{\prime})$.
As both $\mathit{inf}(\mathcal{I}^{\prime})$ and
$\mathit{sup}(\mathcal{I}^{\prime})$ exist and are clearly in $\mathcal{I}$
then the statement follows.
### 7.2 A single application of $\Gamma$ can be computed in polynomial time
For interpretation $I$, $\Gamma(I)$ can be computed by conducting
$O(|P|\cdot|V|\cdot t_{\mathit{max}}\cdot{d_{*}^{in}})$ satisfaction checks
where ${d_{*}^{in}}$ is the maximum in-degree of a node in the network. (This
combined with the assumption that the influence function is computed in
constant time results in polynomial time computation for a single application
of $\Gamma$.)
###### Proof
We note that a given rule will require the most satisfaction checks, as a rule
will potentially affect a network atom of a certain label for each vertex-time
point pair. By the definition of $\mathit{RBnd}$, a given rule clearly causes
no more than $O({d_{*}^{in}})$ satisfaction checks. As the number of rules is
no more than $|P|$, the statement follows.
### 7.3 Proof of Theorem 3.6
Given interpretation $I$ and program $P$, there exists a natural number $k$
s.t. $\Gamma_{P}^{k}(I)=\Gamma_{P}^{k+1}(I)$, and
$k\in O\Big{(}|P|\cdot{d_{*}^{in}}\cdot t_{\mathit{max}}\cdot|E|\Big{)}$
where ${d_{*}^{in}}$ is the maximum in-degree in the network.
###### Proof
For a given vertex $i\in V$, we will use the notation $d_{i}^{in}$ to denote
the number of incoming neighbors (of any edge type) and
${d_{*}^{in}}=\max_{i}d_{i}^{in}$. First we show that for a given
$t\in\tau,i\in V,$ and $L\in\mathcal{L}$, a given rule $r$ can tighten the
bound on a network atom formed with $L$ no more than
$(d_{i}^{in}+1)\cdot({d_{*}^{in}}+1)$ times. This is because a given rule
adjusts the bound on a network atom based on the number of eligible and
qualifying neighbors, which are bounded by $d_{i}^{in},{d_{*}^{in}}$
respectively. At each application of $\Gamma$, at least one network atom must
tighten. Hence, as there are only $O\Big{(}|P|\cdot{d_{*}^{in}}\cdot
t_{\mathit{max}}\cdot\sum_{i}d_{i}^{in}\Big{)}=$$O\Big{(}|P|\cdot{d_{*}^{in}}\cdot
t_{\mathit{max}}\cdot|E|\Big{)}$ tightenings possible, this is also the bound
on the number of applications of $\Gamma$.
### 7.4 Proof of Lemma 3.7
If $I\models P$ and $I^{\prime}\sqsubseteq I$ then
$\Gamma(I^{\prime})\sqsubseteq I$.
###### Proof
Suppose, BWOC, that $\Gamma(I^{\prime})\sqsupset I$. Then, there exists some
$L\in\mathcal{L}$, $t\in\tau$ and $c\in\mathcal{G}$ s.t. $\langle
L,\mathit{bnd}\rangle\in I(t)(c)$, $\langle L,\mathit{bnd}^{\prime}\rangle\in
I^{\prime}(t)(c)$, and $\langle
L,\mathit{bnd}^{\prime\prime}\rangle\in\Gamma(I^{\prime})(t)(c)$ s.t.
$\mathit{bnd}\supset\mathit{bnd}^{\prime\prime}$ and
$\mathit{bnd}^{\prime}\supseteq\mathit{bnd}^{\prime\prime}$. There are four
things that affect $\mathit{bnd}^{\prime\prime}$: facts, rules, integrity
constraints and $\mathit{bnd}^{\prime}$. Clearly, we need not consider the
effect that either facts or $\mathit{bnd}^{\prime}$ have on
$\mathit{bnd}^{\prime\prime}$, as $I$ satisfies all facts and
$I^{\prime}\sqsubseteq I$. We also note that a given integrity constraint
imposed by Definition 3.2 can tighten $\mathit{bnd}^{\prime\prime}$ no more
than the associated bound in any model. Hence, there must be some rule
$r=L\stackrel{{\scriptstyle\Delta
t}}{{\leftarrow}}f,(g_{edge},g_{node},h)_{\mathit{ifl}}$ that causes
$\mathit{bnd}^{\prime\prime}$ to become less than $\mathit{bnd}$. As
$\mathit{bnd}^{\prime\prime}\neq\mathit{bnd}^{\prime}$, we know that
$t\in\mathit{TTS}(\Gamma(I^{\prime}),c,r)\cap\mathit{TTS}(I^{\prime},c,r)$. As
a result, we have $\Gamma(I^{\prime})(t-\Delta t)(c)\models f$ and
$I^{\prime}(t-\Delta t)(c)\models f$. Further, as $I\models
P,I^{\prime}\sqsubseteq I,$ and no rule can modify a non-fluent atom, we have
> $|\mathit{Elig}(v,g_{edge},g_{node},\Gamma(I^{\prime})(t-\Delta t)|=$
> $|\mathit{Elig}(v,g_{edge},g_{node},I^{\prime}(t-\Delta t)|=$
> $|\mathit{Elig}(v,g_{edge},g_{node},\Gamma(I^{\prime})(t-\Delta t)|.$
Further, we know that as $I^{\prime}\sqsubseteq I$, it must be the case that
$|\mathit{Qual}(v,g_{edge},g_{node},h,I(t-\Delta t))|\geq$
$|\mathit{Qual}(v,g_{edge},g_{node},h,I^{\prime}(t-\Delta t))|.$
This implies, by Axiom 2 that, $\mathit{Bound}(r,v,I(t-\Delta
t))\subseteq\mathit{Bound}(r,v,I^{\prime}(t-\Delta t))$. This then implies
that $\mathit{bnd}\subseteq\mathit{bnd}^{\prime\prime}$, which is a
contradiction.
### 7.5 Proof of Theorem 3.8
$\Gamma^{*}_{P}$ is a minimal model for $P$.
###### Proof
Claim: If program $P$ is consistent then $\Gamma^{*}_{P}$ is a model of $P$.
Suppose, BWOC, that there is a fact in $P$ that $\Gamma^{*}_{P}$ does not
satisfy. However, by the definition of $\Gamma$ and the definition of a fact,
$\Gamma^{*}_{P}$ must satisfy all facts as the bound on the weight associated
with each fact is included in the intersection. Further, we can also see by
the definition of $\Gamma$ that $\Gamma^{*}_{P}$ strictly satisfies all non-
fluent facts in $P$. We also note that the final application of the $\Gamma$
operator ensures that all integrity constraints are satisfied by
$\Gamma^{*}_{P}$. Now, Suppose, BWOC, that there is a rule in $P$ that
$\Gamma^{*}_{P}$ does not satisfy. However, with each application of $\Gamma$,
for each rule, we include the bound on the weight returned by the
$\mathit{Bound}$ function for each time step in the target time step
associated with that rule. As $\Gamma$ is applied to convergence, and new
bounds are intersected with each application, then we know that all time
points in any associated target time set are considered in the intersection.
Proof of Theorem: The above claim tells us that $\Gamma^{*}_{P}\models P$. Now
consider interpretation $I$ s.t. $I\models P$. As $\bot\sqsubseteq I$,
multiple applications of Lemma 3.7 tell us that $\Gamma^{*}_{P}\sqsubseteq I$.
Hence, the statement follows.
### 7.6 Proof of Theorem 3.10
If $P$ is consistent, then $\textsf{CANON\\_PROC}(P)$ is the minimal canonical
model of $P$.
###### Proof
CLAIM 1: If $P$ is consistent, then $\textsf{CANON\\_PROC}(P)$ is a canonical
model of $P$.
Clearly, $I=\textsf{CANON\\_PROC}(P)$ satisfies all facts and integrity
constraints in $P$. Hence, we shall consider programs that only consist of
rules in this proof. We say $I$ $L$-canonically satisfies $P$ iff $I$
canonically satisfies $\\{r\in P\;|\;\textit{head}(r)=L\\}$. Clearly, $I$
canonically satisfies $P$ if for all $L\in\mathcal{L}$, $P$ $L$-canonically
satisfies by $I$. We say that $I$ is an $(L,c,q)$-canonically consistent
interpretation if for $c\in\mathcal{G}$, for the first
$t\in\tau-\mathit{TTS}(I,c,L,P)-\\{0\\}$, $I(t)(c)\models\langle L,bnd\rangle$
where $\langle L,bnd\rangle\in I(t-1)(c)$. Consider some $L\in\mathcal{L}$ and
$c\in\mathcal{G}$. Clearly, $I$ is an $(L,c,0)$-model for $P$. Let us assume,
for some value $q$, that $I$ is an $(L,c,q-1)$ model for $P$. Let time point
$t$ be the $q$-th element of $\tau-\mathit{TTS}(I,c,L,P)-\\{0\\}$. Consider
the time step before time $t$ is considered in the for-loop at line 4 of
CANON_PROC, which causes the condition at line 5 to be true. By line 13,
$\tau-\mathit{TTS}(I,c,L,P)-\\{0\\}\subseteq cur\\_free[c][L]$. This means
that $t$ is a member of both. Hence, when $t$ is considered at line 4, the
condition at line 5 is true, causing $\langle L,bnd\rangle\in I(t)(c)\cap
I(t-1)(c)$ and as the element $\langle L,bnd\rangle\in I(t-1)(c)$ is not
changed here, we have shown the $I$ is an $(L,c,q)$-model for $P$. By the for-
loop at line 6, for all $L\in\mathcal{L}$ and $c\in\mathcal{G}$, $I$ is an
$(L,c,q)$-model for $P$. Hence, at the for-loop at line 4, we can be assured
that for $L\in\mathcal{L}$ and $c\in\mathcal{G}$ that $I$
$(L,c,|\tau-\mathit{TTS}(I,c,L,P)-\\{0\\}|)$ satisfies $P$ – which means that
$I$ canonically satisfies $P$
CLAIM 2: If $I$ is a canonical model for $P$,
$cur\\_interp\sqsubseteq I$ is an interpretation that also strictly satisfies
all non-fluent facts in $P$, and $cur\\_interp^{\prime}$ is $cur\\_interp$
after being manipulated in lines 6-10 of CANON_PROC, then
$cur\\_interp^{\prime}\sqsubseteq I$.
We note that by the definition of satisfaction of a non-fluent fact, and the
fact that both $cur\\_interp$ and $I$ must strictly satisfy all non-fluent
facts in $P$, we know that for all $c\in\mathcal{G}$ and $L\in\mathcal{L}$
that:
$\displaystyle TTS(I,c,L,P)$ $\displaystyle=$ $\displaystyle
TTS(cur\\_interp,c,L,P)$ $\displaystyle=$ $\displaystyle
TTS(cur\\_interp^{\prime},c,L,P)$
Let us assume that lines 6-10 of the algorithm are changing $cur\\_interp$
when the outer loop is considering time $t$ and that the condition at line 5
is true. Clearly,
$\displaystyle t\in\tau-TTS(I,v^{\prime},L^{\prime},P)-\\{0\\}$
As a result, for any $(v,L)$ pair considered at this point by the algorithm,
if $\langle L,\mathit{bnd}\rangle\in I(t)(v)$ and $\langle
L,\mathit{bnd}^{\prime}\rangle\in I(t-1)(v)$ then we have
$\mathit{bnd}=\mathit{bnd}^{\prime}$. By the algorithm, if we have $\langle
L,\mathit{bnd}^{*}\rangle\in cur\\_interp^{\prime}(t)(v)$ and $\langle
L,\mathit{bnd}^{**}\rangle\in cur\\_interp^{\prime}(t-1)(v)$ we have that
$\mathit{bnd}^{*}=\mathit{bnd}^{**}$. As
$\langle L,\mathit{bnd}^{**}\rangle\in cur\\_interp(t-1)(v)$, we know that
$\mathit{bnd}^{\prime}\subseteq\mathit{bnd}^{**}$. As a result, we have
$cur\\_interp^{\prime}\sqsubseteq I$, completing the claim.
Proof of theorem: As initially $cur\\_interp=\Gamma^{*}_{P}$ and
$\Gamma^{*}_{P}\sqsubseteq I$ by Theorem 3.8, we note that the algorithm
changes
$cur\\_interp$ either by applying $\Gamma$ or manipulating it in lines 6-10,
which tells us (by claim 2) that for all models $I$ of $P$ that
$\textsf{CANON\\_PROC}(P)\sqsubseteq I$. Since by claim 1 we know that
$\textsf{CANON\\_PROC}(P)\models P$, the statement of the theorem follows.
### 7.7 Details on the Extracted Dataset
One way in which MANCaLog can be used is looking at proliferation of research
on different topics. We look at research conducted on niacin, an organic
compound commonly used for increasing levels of high density lipoproteins
(HDL). Using Thomson Reuters Web of Knowledge (http://wokinfo.com) we were
able to extract information on $4,202$ articles about niacin. This information
was then processed using the Science of Science (Sci2) Tool
(http://sci2.cns.iu.edu) to extract numerous different networks such as author
by paper networks, citation networks, and paper by subject networks. Each
paper has attributes about when it was published, what journal it was
published in, and what subjects the paper was about. During the first time
period there is a total of $508$ papers with $856$ different authors and
$1,231$ connections based on an author being cited as an author of a given
paper. During the second time period, there is a total of $3,790$ papers with
$10,465$ different authors and $16,772$ connections.
|
arxiv-papers
| 2013-01-02T20:01:04 |
2024-09-04T02:49:39.867663
|
{
"license": "Public Domain",
"authors": "Paulo Shakarian, Gerardo I. Simari, Robert Schroeder",
"submitter": "Paulo Shakarian",
"url": "https://arxiv.org/abs/1301.0302"
}
|
1301.0398
|
# Energy evolution of the large-$t$ elastic scattering and its correlation
with multiparticle production
S.M. Troshin
Institute for High Energy Physics,
Protvino, Moscow Region, 142280, Russia
It is emphasized that the collective dynamics associated with color
confinement is dominating over a point-like mechanism related to a scattering
of the proton constituents at the currently available values of the momentum
transferred in proton elastic scattering at the LHC. Deep–elastic scattering
and its role in the dissimilation of the absorptive and reflective asymptotic
scattering mechanisms are discussed with emphasis on the experimental
signatures associated with the multiparticle production processes.
## Introduction
Studies of elastic hadron scattering where initial particles are keeping their
identity can lead to a new knowledge on the nonperturbative dynamics of
hadronic interactions, mechanism of confinement and asymptotic regime of
strong interactions.
Concerning relation to the color confinement phenomena, it should be noted
that according to the superselection rules (SSR) colored quarks and gluons
live in the coherent Hilbert subspaces, and those are different from the
physical Hilbert subspace populated by the white hadrons. No self-adjont
operator (related to the observable quantity) describing transition between
colored and bleached Hilbert subspaces can exist. It means that the color
degrees of freedom can never be observed. It is the result of SSR for the
color degrees of freedom which is combined with the non-abelian nature of QCD
[1]. But it is not a proof of confinement yet — according to it, color should
be confined inside hadron. There is no known dynamical mechanism providing
this nowadays. Indeed, what is known is that such mechanism should be based on
the collective dynamics of quarks and gluons and, as it was demonstrated in
[2], the unitarity might be a consequence of the confinement.
We discuss here possible manifestations of the collective effects in hadron
elastic scattering keeping in mind the connection of these effects with
phenomena of color confinement.
## 1 Coherence in the elastic scattering
In this Section the large-$t$ elastic scattering discussed. It should be noted
that in the region of the transferred momenta beyond the second maximum in the
differential cross-sections, additional dips and bumps are absent. This smooth
decrease can be considered as a manifestation of the composite hadron
structure, then a power-like dependence can be used as a relevant function
reproducing the experimental data behavior. On the other hand, the exponential
function can also be applied. Those dependencies are based on the different
dynamical mechanisms, namely, power–like behavior corresponds to the composite
scattering dynamics where coherence is absent and point–like constituents
being independent, while the exponential form should be associated with
coherent collective interactions.
The power-like parametrization $d\sigma/dt\sim|t|^{-7.8}$ has been applied for
the description of the differential cross-section in the region between 1.5
(GeV)2 and 2.0 (GeV)2 in the paper [3]. This dependence is depicted on the
Fig. 1 (red line).
Figure 1: Dependence of the large-$t$ elastic scattering differential cross-
section.
At the LHC energy $\sqrt{s}=7$ TeV the power-like dependence allows to fit
data in the rather narrow region of the transferred momenta. At the same time
the Orear dependence of the form [4]
$d\sigma/dt\sim\exp(-c_{\mbox{o}}\sqrt{-t})$ (1)
can describe the experimental data better with $c_{\mbox{o}}\simeq 12$
(GeV)-1, cf. Fig. 1 (solid line). The slope parameter is about twice as much
bigger compared to the value of $c_{\mbox{o}}$ at the CERN ISR and at lower
energies [5]. It is evident that the exponential dependence on $\sqrt{-t}$
describes experimental data in the wider region of $-t$-values and use of the
power-like dependence for the data analysis seems to be premature and
misleading.
Different various dynamical mechanisms can provide the Orear dependence and
all of them are associted with the non-perturbative dynamics of white hadron
interactions. For the first time such dependence has been obtained in the
multiperipheral model [6], it has also been interpreted as a result of
scattering into a classically prohibited region in [7] and as one originating
from the contribution of the branching point in the complex angular momentum
plane in [8]111I am indebted to S.S. Gershtein and L.N. Lipatov for bringing
the papers [7] and [8], respectively, to my attention.. The presence of poles
in the complex impact parameter plane which can result [9] from the rational
form of the scattering amplitude unitarization leads to such dependence of the
scattering amplitude too. For the case of pure imaginary scattering amplitude
the poles in the impact parameter plane provide the additional oscillating
factors in front of the Orear exponent in the amplitude. Such oscillations are
common for the picture of diffractive scattering. The absence of the
oscillations at lower energies in the region of large $-t$ can be explained by
the significant role of the phase but this explanation could stop working at
the LHC energies.
Alternatively, the smooth dependence of the differential cross–section
observed at lower energies can be associated with the presence of the
essential double helicity-flip amplitude contribution. It has been shown that
the double helicity–flip amplitudes $F_{2}$ and $F_{4}$ are important at large
values of $-t$ and compensates oscillations of the helicity non-flip
amplitudes [10].
If the spin effects can be neglected at the LHC energies, i.e. any helicity-
flip amplitudes would not survive at such high energies, it would result in
appearance of the oscillations at higher $-t$-values. Thus, a possible
appearance of the above oscillations in the differential cross-section at
higher values of $-t$ can be interpreted in this case as an observation of the
$s$-channel helicity conservation in $pp$-scattering at the LHC energies.
## 2 Asymptotics: reflective vs black disk
The existing experimental accelerator and cosmic rays data set for the total,
elastic and total inelastic cross–sections cannot lead to the definite
conclusion on the possible asymptotic hadron scattering mechanism. Therefore
one should try to search for the independent experimental manifestations of
the possible asymptotic mechanism. In this connection it is instrumental to
consider a deep–elastic scattering. The notion of deep–elastic scattering
introduced in the paper [11] uses an analogy with the deep-inelastic
scattering and refers to the elastic scattering with the large transferred
momenta $-t>4$ (GeV/c)2.
With the elastic scattering amplitude being a purely imaginary function,
($f\to if$), the function $S(s,b)$ becomes real ($S=1+2if$) and can be
interpreted as a survival amplitude of the prompt elastic channel. The
relevant expressions for the survival amplitude $S(s,b)$ are the following
$S(s,b)=\pm\sqrt{1-4h_{inel}(s,b)},$ (2)
i.e. the probability of absorptive (destructive) collisions is
$1-S^{2}(s,b)=4h_{inel}(s,b)$ ($h_{inel}(s,b)\leq 1/4$). Simultaneous
vanishing of elastic and inelastic scattering amplitudes at $b\to\infty$
should always take place and therefore only one root in Eq. (2) (with plus
sign) being usually taken into account, while another one (with minus sign) is
omitted as a rule. This is a well known shadow approach to elastic scattering.
This is only valid in the case when $h_{inel}(s,b)$ is a monotonically
decreasing function of the impact parameter and reaches its maximum value at
$b=0$. Thus, the inelastic overlap function has a central impact parameter
profile and approaches its maximum value $1/4$, i.e. $h_{inel}(s,b=0)\to 1/4$
at $s\to\infty$. The survival amplitude described above, vanishes in the high
energy limit in central hadron collisions, $S(s,b=0)\to 0$. However, the self-
damping of inelastic channels at very high energies would lead to a peripheral
dependence on the impact parameter of the inelastic overlap function
$h_{inel}(s,b)$, it is vanishing at $b=0$ in the high energy limit
$s\to\infty$. In this limit the inelastic overlap function $h_{inel}(s,b)$
reaches its maximum value at nonzero values of impact parameter $b=R(s)$ [12].
This conclusion results from the unitarity saturation by the elastic amplitude
when $f(s,b)\to 1$ at $s\to\infty$ and $b=0$. This saturation can be realized
in the framework of the rational form of unitarization (cf. e.g. [12]). Thus,
we should take $S(s,b)=-\sqrt{1-4h_{inel}(s,b)}$ when $1/2<f(s,b)<1$. The
scattering dynamics starts to be reflective in the region where very high
energies combined with small and moderate values of $b$ (it means that hard
core appears) and approaches asymptotically to the completely reflecting limit
($S=-1$) at $b=0$ and $s\to\infty$ since $h_{inel}(s,b=0)\to 0$. The
probability of reflective scattering at $b<R(s)$ is to be determined then by
the magnitude of $S^{2}(s,b)$.
Thus, the deep–elastic scattering (DES) is associated with reflective
scattering at very high energies where the colliding hadrons do not suffer
from absorption anymore. The DES dominates over multiparticle production (at
small impact parameter values $h_{inel}(s,b=0)\to 0$). This ensures favorable
conditions for the experimental measurements, since the peripheral profile of
$h_{inel}(s,b)$, associated with reflective scattering, suppresses the
probability of the inelastic collisions in the region of small impact
parameters. The main contribution to the mean multiplicity is due to the
peripheral region of $b\sim R(s)$. DES in this case is correlated with
inelastic events of low cross–sections, i.e. it has a small background due to
production and high experimental visibility. The reflective mechanism
associated with the complete unitarity saturation will asymptotically decouple
from particle production asymptotically and at finite energies it corresponds
to observation of the DES with decreasing correlations with particle
production. Contrary, the saturation of the black disk limit implies strong
correlation of DES with multiparticle production processes [13].
## Acknowledgement
I am pleased to express the gratitude to IHEP, Protvino and Organizers of
Diffraction 2012 in Puerto del Carmen, Lanzarote, for the support of my
participation to the Conference. The talk is based in part on the papers
prepared in collaboration with N.E. Tyurin. I am grateful to him for the many
useful comments and discussions. I would also like to thank L.L. Jenkovszky,
U. Maor, V.A. Petrov and N.P. Zotov for the interesting remarks and
discussions.
## References
* [1] S. Tanimura, arXiv: 1112.5701.
* [2] S.M. Troshin, N.E. Tyurin, _Mod. Phys. Lett. A_ 25, 3363 (2010).
* [3] G. Antchev et al. (The TOTEM Collaboration), _Europhys. Lett._ 95, 41001 (2011) .
* [4] J. Orear et al., _Phys. Rev._ 152, 1162 (1966).
* [5] J.L. Hartmann et al., _Phys. Rev. Lett._ 39, 975 (1977).
* [6] D. Amati, M. Cini, A. Stanghellini, _Nuov. Cim._ 30, 193 (1963).
* [7] S.P. Alliluev, S.S. Gershtein, A.A. Logunov, _Phys. Lett._ 18, 195 (1965).
* [8] A.A. Anselm, I.T. Dyatlov, _Phys. Lett._ 24, 479 (1967).
* [9] V.F. Edneral, S.M. Troshin, N.E. Tyurin, _Theor. Math. Phys._ 44, 652 (1980).
* [10] V.F. Edneral, S.M. Troshin, N.E. Tyurin, _JETP Lett._ 30, 330 (1979)
* [11] M.M. Islam, J. Kaspar, R.J. Luddy, _Mod. Phys. Lett._ A24, 485 (2009).
* [12] S.M. Troshin, N.E. Tyurin, _Int. J. Mod. Phys._ A22, 4437 (2007).
* [13] S.M. Troshin, N.E. Tyurin, _Phys. Lett. B_ 707, 558 (2012).
|
arxiv-papers
| 2013-01-03T09:06:22 |
2024-09-04T02:49:39.878906
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. M. Troshin",
"submitter": "Sergey Troshin",
"url": "https://arxiv.org/abs/1301.0398"
}
|
1301.0457
|
# A New Result for Second Order BSDEs
with Quadratic Growth and its applications
Yiqing LIN Institut de Recherche Mathématique de Rennes,
Université de Rennes 1,
35042 Rennes Cedex, France [email protected]
(Date: November 30th, 2012)
###### Abstract.
In this paper, we study a class of second order backward stochastic
differential equations (2BSDEs) with quadratic growth in coefficients. We
first establish solvability for such 2BSDEs and then give their applications
to robust utility maximization problems.
###### Key words and phrases:
second order BSDEs, quadratic growth, robust utility maximization
###### 2000 Mathematics Subject Classification:
60H10, 60H30
## 1\. Introduction
Typically, nonlinear backward stochastic differential equations (BSDEs) are
defined on a Wiener probability space $(\Omega,\mathcal{F},\mathbb{P})$ and of
the following type:
(1.1) $Y_{t}=\xi+\int^{T}_{t}g(s,Y_{s},Z_{s})ds-\int^{T}_{t}Z_{s}dB_{s},\
0\leq t\leq T,$
where $B$ is a Brownian motion, $\mathcal{F}$ is the $\mathbb{P}$-augmented
natural filtration generated by $B$, $g$ is a nonlinear generator, $T$ is the
terminal time and $\xi\in\mathcal{F}_{T}$ is the terminal value. A solution to
BSDE (1.1) is a couple of processes $(Y,Z)$ adapted to the filtration
$\mathcal{F}$.
Under a Lipschitz condition on the generator $g$, Pardoux and Peng [19] first
provided the wellposedness of (1.1). Since then, the theory of nonlinear BSDEs
has been extensively studied in the past twenty years. Among all the
contributions, we only quote the results which is highly related to our
present work.
A weaker assumption on the generator is that $g$ has a quadratic growth in
$z$. This kind of BSDEs with bounded terminal value condition was first
examined by Kobylanski [13], who used a weak convergence technique borrowed
from PDE literatures to prove the existence and also obtained the uniqueness
result under some additional condition on $g$. With the help of contraction
mapping principle, Tevzadze [28] re-considered this type of BSDEs when the
terminal value $\xi$ is small enough in norm. The advantage of the method
adopted by Tevzadze [28] is its applicability to not only one-dimensional
quadratic BSDEs but also to multidimensional ones. Particularly, the
restriction on $\xi$ can be loosen when $g$ satisfies some restrictive
condition on regularity. Briand and Hu [1, 2] extended the existence result
for (1.1) to the case that $\xi$ is not uniformly bounded and provided the
uniqueness result when $g$ is convex. Besides, Morlais [16] considered some
similar type of BSDEs driven by continuous martingales.
Motivated by expected utility theory, Peng [20] defined a so-called
$g$-expectation $\mathcal{E}^{g}[\xi]:=Y_{0}$ on $\mathcal{F}_{T}$ via
nonlinear BSDEs with Lipschitz generator. Also, a conditional expectation can
be consistently defined: $\mathcal{E}^{g}[\xi|\mathcal{F}_{t}]:=Y_{t}$, under
which the solution $Y$ of the BSDE with the generator $g$ is a $g$-martingale.
As the counterparts in the classical framework under a linear expectation,
Peng [21] gave the notion of $g$-supermartingle ($g$-submartingle) and
established the nonlinear Doob-Meyer type decomposition theorem. Subsequently,
Chen and Peng [3] proved the downcrossing inequality for $g$-martingales. For
the case that $g$ is allowed to have a quadratic growth in $z$, similar
results can be found in Ma and Yao [14].
Recently, Soner et al. [26] established a framework of “quasi-sure” stochastic
analysis under a non-dominated class of probability measures. This provided a
new approach for Soner et al. [24, 25] to re-consider the wellposedness of
second order BSDEs (2BSDEs) introduced by Cheridito et al. [4]. The key idea
in Soner et al. [25] is to reinforce a condition that the following 2BSDE
holds true $\mathcal{P}_{H}$-quasi-surely, i.e., $\mathbb{P}$-a.s. for all
$\mathbb{P}\in\mathcal{P}_{H}$, which is a class of mutually singular
probability measures (cf. Definition 2.1):
(1.2)
$Y_{t}=\xi+\int^{1}_{t}\hat{F}_{s}(Y_{s},Z_{s})ds-\int^{1}_{t}Z_{s}dB_{s}+K_{1}-K_{t},\
0\leq t\leq 1.$
Under a uniformly Lipschitz condition on the generator $\hat{F}$, Soner et al.
[25] provided a complete wellposedness result for the 2BSDE (1.2). In this
pioneering work, a representation theorem of the solution $Y$ is established
and thus, the uniqueness is a straightforward corollary. For the existence, a
process $Y$ is pathwisely constructed and verified as a
$\hat{F}$-supermartingale under each $\mathbb{P}\in\mathcal{P}_{H}$. Applying
the nonlinear Doob-Meyer decomposition theorem, the right-hand side of (1.2)
comes out, where $K$ is a (family of) non-decreasing process(es) that
satisfies the minimum condition (cf. Definition 2.11). Moreover, both
Cheridito et al. [4] and Soner et al. [25] explained the connection between
the Markov 2BSDEs and a large class of fully nonlinear PDEs, which was one of
the motivations initiate this 2BSDEs topic.
Meanwhile, Peng [22, 23] independently introduced another framework (so-called
$G$-framework) of a time consistent nonlinear expectation
$\mathbb{E}_{G}[\cdot]$, in which a new type of Brownian motion was
constructed and the related Itô type stochastic calculus was established. By
explicit constructions, Denis et al. [5] showed that $G$-expectation is in
fact an upper expectation related to a non-dominated family $\mathcal{P}_{G}$
that consists of some probability measures similar to the elements in
$\mathcal{P}_{H}$. In this regards, the $G$-framework is highly related to the
2BSDE one. Adopted the idea in Denis and Martini [7], Denis et al. [5] defined
a Choquet capacity $\bar{C}(\cdot)$ on $(\Omega,\mathcal{B}(\Omega))$ as
follows:
$\bar{C}(A):=\sup_{\mathbb{P}\in\mathcal{P}_{G}}\mathbb{P}(A),\
A\subset\mathcal{B}(\Omega),$
and then they gave the the notion of “quasi-surely” in a standard capacity-
related vocabulary: a property holds true quasi-surely if and only if it holds
outside a polar set, i.e., outside a set $A\subset\Omega$ that satisfies
$\bar{C}(A)=0$. We notice that this notion is a little bit stronger than the
corresponding one in the 2BSDE framework, so that it yields another type of
“quasi-sure” stochastic analysis. In this $G$-framework, Hu et al. [10] have
worked on nonlinear BSDEs driven by $G$-brownian motion (GBSDEs), which is of
the same form as (1.2) but holds in the stronger “quasi-sure” sense. In that
paper, the solution is an aggregated triple $(Y,Z,K)$ which quasi-surely
solves (1.2), where $-K$ is a decreasing $G$-martingale that comes from the
$G$-martingale decomposition. To ensure that (1.2) is well defined in
$G$-framework, an additional condition to the Lipschitz one is imposed on the
regularity of the generator (cf. (H1) in Hu et al. [10]). This cost is
intelligible since the definition of $G$-stochastic integrals is under a
stronger norm induced by $\mathbb{E}_{G}[\cdot]$ and it makes the space of
admissible integrands smaller than the classical one.
Following the works of Soner et al. [24, 25, 26], Possamai and Zhou [18]
generalized the existence and uniqueness results for the 2BSDE whose generator
has a quadratic growth. Based on the previous one of Tevzadze [28] for
quadratic BSDEs, this work requires some additional condition, either on the
terminal value or on the regularity of the generator. Our aim of the present
paper is to remove these conditions, that is, to redo the job of Possamai and
Zhou [18] under some weaker assumptions of the type similar to that in
Kobylanski [13] and Morlais [16].
In the classical framework, the quadratic BSDE is a powerful technique to deal
with the utility maximization problems. El Karoui and Rouge [8] computed the
value function of an exponential utility maximization problem when the
strategies are confined to a convex cone, and they found that its dual problem
is related to a quadratic BSDE. In contrast to this, Hu et al. [9] and Morlais
[16] directly treated the primal problem rather than the dual one and obtained
an similar result without the convex condition on the constrain set. The value
function was characterized by also a solution of a quadratic BSDE.
Corresponding to thses works above, Matoussi et al. [15] found that a robust
utility maximization problem with non-dominated models can be solved via the
2BSDE technique. This kind of problem was first consider by Denis and Kervarec
[6] under a weakly compact class of probability measures. Just because of this
weakly compact assumption, one can find a least favorable probability in this
class and work under this probability to find an optimal strategy similarly to
how we solve the classical problem under a single probability. With the help
of 2BSDEs, Matoussi et al. [15] solved this problem globally and characterized
the value function by using a solution of a 2BSDE. This method does not
require that the class is weakly compact. However, the result in Matoussi et
al. [15] has some limitations: for example, when the utility function is
exponential, they are able to solve only the case that $\xi$ is small enough
or the border of the constraint domain satisfies an extra regularity
condition. This limitations is derived from the theory of quadratic 2BSDEs in
Possamai and Zhou [18]. Since we shall remove these extra conditions adopted
by Possamai and Zhou [18], we can have a better result on solving this robust
utility maximization problem.
This paper is organized as follows: Section 2 includes preliminaries for
2BSDEs theory. Section 3 introduce a priori estimates, a representation
theorem and the uniqueness result for 2BSDEs with quadratic growth. Section 4
studies the existence of solutions while Section 5 is the applications of
quadratic 2BSDEs to robust maximization problems.
## 2\. Preliminaries
The aim of this section is to list some basic definitions for 2BSDEs
introduced by Soner et al. [24, 25, 26] and Possamai and Zhou [18]. The reader
interested in a more detailed description of these notation is referred to
these papers listed above.
### 2.1. The class of probability measures
Let
$\Omega:=\\{\omega:\omega\in\mathcal{C}([0,1],\mathbb{R}^{d}),\omega_{0}=0\\}$
be the canonical space equipped with the uniform norm
$||\omega||^{\infty}_{1}:=\sup_{0\leq t\leq 1}|\omega_{t}|$, $B$ the canonical
process, $\mathcal{F}$ the filtration generated by $B$, $\mathcal{F}^{+}$ the
right limit of $\mathcal{F}$.
We call $\mathbb{P}$ a local martingale measure if under which the canonical
process $B$ is a local martingale. By Karandikar [11], the quadratic variation
process of $B$ and its density can be defined universally, such that under
each local martingale measure $\mathbb{P}$:
$\langle B\rangle:=B^{2}_{t}-2\int^{t}_{0}B_{s}dB_{s}\ {\rm and}\
\hat{a}_{t}:=\overline{\lim_{\varepsilon\downarrow{0}}}\frac{1}{\varepsilon}(\langle
B\rangle_{t}-\langle B\rangle_{t-\varepsilon}),\ 0\leq t\leq 1,\
\mathbb{P}-a.s..$
Adapting to Soner et al. [26], we denote $\overline{\mathcal{P}}_{W}$ the
collection of all local martingale measures $\mathbb{P}$ such that $\langle
B\rangle_{t}$ is absolutely continuous in $t$ and $\hat{a}$ takes values in
$\mathbb{S}_{d}^{>0}$, $\mathbb{P}$-a.s.. It is easy to verify that the
following stochastic integral defines a $\mathbb{P}$-Brownian motion:
$W^{\mathbb{P}}_{t}:=\int^{t}_{0}\hat{a}_{s}^{{-1/2}}dB_{s},\ 0\leq t\leq 1.$
We define a subclass of $\overline{\mathcal{P}}_{W}$ that consists of the
probability measures induced by the strong formulation (cf. Lemma 8.1 in Soner
et al. [26]):
$\overline{\mathcal{P}}_{S}:=\\{\mathbb{P}\in\overline{\mathcal{P}}_{W}:\overline{\mathcal{F}^{W^{\mathbb{P}}}}^{\mathbb{P}}=\overline{\mathcal{F}}^{\mathbb{P}}\\},$
where $\overline{\mathcal{F}}^{\mathbb{P}}$
($\overline{\mathcal{F}^{W^{\mathbb{P}}}}^{\mathbb{P}}$, respectively) is the
$\mathbb{P}$-augmentation of the filtration generated by $B$
($W^{\mathbb{P}}$, respectively).
### 2.2. The nonlinear generator
We consider a mapping
$H_{t}(\omega,y,z,\eta):[0,1]\times\Omega\times\mathbb{R}\times\mathbb{R}^{d}\times
D_{H}\rightarrow\mathbb{R}$ and its Fenchel-Legendre conjugate with respect to
$\eta$:
$F_{t}(\omega,y,z,a):=\sup_{\eta\in D_{H}}\bigg{\\{}\frac{1}{2}{\rm
tr}(a\eta)-H_{t}(\omega,y,z,\eta)\bigg{\\}},\ a\in\mathbb{S}_{d}^{>0}.$
where $D_{H}\subset\mathbb{R}^{d\times d}$ a given subset that contains $0$.
For simplicity of notation, we note
$\hat{F}_{t}(y,z):=F_{t}(y,z,\hat{a}_{t})\ {\rm and}\
F^{0}_{t}:=\hat{F}_{t}(0,0),$
and we denote by $D_{F_{t}(y,z)}$ the domain of $F$ in $a$ for a fixed
$(t,\omega,y,z)$. In accordance with the settings previous literatures, we
assume the following assumptions on $F$, which is needed for the “quasi-sure”
technique:
(A1) $D_{F_{t}(y,z)}=D_{F_{t}}$ is independent of $(\omega,y,z)$;
(A2) $F$ is $\mathcal{F}$-progressively measurable and uniformly continuous in
$\omega$.
### 2.3. The spaces and the norms
For the wellposedness of 2BSDEs, we consider a restrictive subclass
$\mathcal{P}_{H}\subset\mathcal{P}_{S}$ defined as follows:
###### Definition 2.1.
Let $\mathcal{P}_{H}$ denote the collection of all those
$\mathbb{P}\in\overline{\mathcal{P}}_{S}$ such that
$\underline{a}^{\mathbb{P}}\leq\hat{a}_{t}\leq\overline{a}^{\mathbb{P}}\
{(usual\ partial\ ordering\ on}\ \mathbb{S}^{>0}_{d}{)\ and}\ \hat{a}_{t}\in
D_{F_{t}},\ \lambda\times\mathbb{P}-a.e.,$
for some $\underline{a}^{\mathbb{P}}$,
$\overline{a}^{\mathbb{P}}\in\mathbb{S}_{d}^{>0}$ and all
$(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$.
###### Remark 2.2.
Soner et al. [25] mentioned that the bounds $\underline{a}^{\mathbb{P}}$ and
$\overline{a}^{\mathbb{P}}$ may vary in $\mathbb{P}$. Thanks to the quadratic
growth assumption on $F$, i.e., (A3) in the sequel, $\hat{F}^{0}_{t}$ is
bounded so that $\mathcal{P}_{H}$ is not empty in our case (cf. Remark 2.5 in
Possamai and Zhou [18]).
###### Definition 2.3.
We say that a property holds $\mathcal{P}_{H}$-quasi-surely
($\mathcal{P}_{H}$-q.s.) if it holds $\mathbb{P}$-a.s. for all
$\mathbb{P}\in\mathcal{P}_{H}$.
For each $p\geq 1$, $L^{p}_{H}$ denotes the space of all
$\mathcal{F}_{1}$-measurable scalar random variable $\xi$ that satisfies
$||\xi||_{L^{p}_{H}}:=\sup_{\mathbb{P}\in\mathcal{P}_{H}}\mathbb{E}^{\mathbb{P}}[|\xi|^{p}]<+\infty.$
Letting $p\rightarrow+\infty$, we denote by $L^{\infty}_{H}$ the space of all
$\mathbb{P}_{H}$-q.s. bounded random variable $\xi$ with
$||\xi||_{L^{\infty}_{H}}:=\sup_{\mathbb{P}\in\mathcal{P}_{H}}||\xi||_{L^{\infty}(\mathbb{P})}<+\infty.$
Let $\mathbb{D}^{\infty}_{H}$ denote the space of all $\mathbb{R}$-valued
$\mathcal{F}^{+}$-progressively measurable process $Y$ that satisfies
$\mathcal{P}_{H}-q.s.\ c\grave{a}dl\grave{a}g\ {\rm and}\
||Y||_{D^{\infty}_{H}}:=\sup_{0\leq t\leq
1}||Y_{t}||_{L^{\infty}_{H}}<+\infty,$
and $\mathbb{H}^{2}_{H}$ denotes the space of all $\mathbb{R}^{d}$-valued
$\mathcal{F}^{+}$-progressively measurable process $Z$ that satisfies
$||Z||^{2}_{\mathbb{H}^{2}_{H}}:=\sup_{\mathbb{P}\in\mathcal{P}_{H}}\mathbb{E}^{\mathbb{P}}\bigg{[}\int^{1}_{0}|\hat{a}^{1/2}_{t}Z_{t}|^{2}dt\bigg{]}<+\infty.$
###### Remark 2.4.
We emphasize that the monotone convergence theorem no long holds true on each
space listed above in this framework, i.e. that the monotone
$\mathcal{P}_{H}$-q.s. convergence yields the convergence in norm may fail. As
stated in section 4 of Possamai and Zhou [18], this is one of the main
difficulties to prove the existence of quadratic 2BSDEs by global
approximation.
With a little abuse of notation, we introduce the notion of
$BMO(\mathcal{P}_{H})$-martingale and its generator, which is an extension of
the classical one. For the convenience of notation, $H$ can refer to either a
single process or a family of non-aggregated processes
$\\{H^{\mathbb{P}}\\}_{\mathbb{P}\in\mathcal{P}_{H}}$ in the definition and
lemmas below.
###### Definition 2.5.
We call $H$ a $BMO(\mathcal{P}_{H})$-martingale if for each
$\mathbb{P}\in\mathcal{P}_{H}$, $H^{\mathbb{P}}$ is a $\mathbb{P}$-square
integrable martingale and
$||H||^{2}_{BMO_{2}(\mathcal{P}_{H})}:=\sup_{\mathbb{P}\in\mathcal{P}_{H}}\sup_{\tau\in\mathcal{T}^{1}_{0}}||\mathbb{E}^{\mathbb{P}}_{\tau}[\langle
H^{\mathbb{P}}\rangle_{1}-\langle
H^{\mathbb{P}}\rangle_{\tau}]||_{L^{\infty}(\mathbb{P})}<+\infty,$
where $\mathcal{T}^{1}_{0}$ is the collection of all $\mathcal{F}$-stopping
times $\tau$ that take values in $[0,1]$.
From the definition above, for a fixed $BMO(\mathcal{P}_{H})$-martingale $H$,
there exists a uniform bound constant $M_{H}>0$, such that for all
$\mathbb{P}\in\mathcal{P}_{H}$ and $\sigma\in\mathcal{T}^{1}_{0}$,
$||H_{\cdot\wedge\sigma}||^{2}_{BMO_{2}(\mathbb{P})}\leq||H||^{2}_{BMO_{2}(\mathbb{P})}\leq
M_{H}.$
Applying Theorem 2.4 and Theorem 3.1 in Kazamaki [12] under each
$\mathbb{P}\in\mathcal{P}_{H}$, we have the following lemmas:
###### Lemma 2.6.
Suppose $H$ is a $BMO(\mathcal{P}_{H})$-martingale, then there exist two
constants $r>1$ and $C>0$, such that
$\sup_{\mathbb{P}\in\mathcal{P}_{H}}\sup_{0\leq t\leq
1}\mathbb{E}^{\mathbb{P}}[|\mathcal{E}(H^{\mathbb{P}})_{t}|^{r}]\leq C,$
and for some $q>1$, the following reverse Hölder’s inequality holds under each
$\mathbb{P}\in\mathcal{P}_{H}$ with a uniform constant $C_{RH}$: for each
$0\leq t_{1}\leq t_{2}\leq 1$,
$\mathbb{E}^{\mathbb{P}}_{t_{1}}[\mathcal{E}(H^{\mathbb{P}})^{q}_{t_{2}}]\leq
C_{RH}\mathcal{E}(H^{\mathbb{P}})^{q}_{t_{1}},\ \mathbb{P}-a.s.,$
where $r$, $C$, $q$ and $C_{RH}$ simply depend on $M_{H}$.
###### Lemma 2.7.
Suppose $H$ is a $BMO(\mathcal{P}_{H})$-martingale, then there exist a $p>1$
and a $C_{E}>0$ that simply depend on $M_{H}$, such that for each $t\in[0,1]$,
$\sup_{\mathbb{P}\in\mathcal{P}_{H}}\sup_{\tau\in{\mathcal{T}^{t}_{0}}}\bigg{|}\bigg{|}\mathbb{E}^{\mathbb{P}}_{\tau}\bigg{[}\bigg{(}\frac{\mathcal{E}(H^{\mathbb{P}})_{\tau}}{\mathcal{E}(H^{\mathbb{P}})_{t}}\bigg{)}^{\frac{1}{p-1}}\bigg{]}\bigg{|}\bigg{|}_{L^{\infty}(\mathbb{P})}\leq
C_{E}.$
###### Definition 2.8.
We call $Z\in\mathbb{H}^{2}_{H}$ a $BMO(\mathcal{P}_{H})$-martingale generator
if
$\displaystyle||Z||^{2}_{\mathbb{H}^{2}_{BMO(\mathcal{P}_{H})}}:$
$\displaystyle=\sup_{\mathbb{P}\in\mathcal{P}_{H}}\bigg{|}\bigg{|}\int^{\cdot}_{0}Z_{t}dB_{t}\bigg{|}\bigg{|}^{2}_{BMO_{2}(\mathbb{P})}$
$\displaystyle=\sup_{\mathbb{P}\in\mathcal{P}_{H}}\sup_{\tau\in\mathcal{T}^{1}_{0}}\bigg{|}\bigg{|}\mathbb{E}^{\mathbb{P}}_{\tau}\bigg{[}\int^{1}_{\tau}|\hat{a}^{1/2}_{t}Z_{t}|^{2}dt\bigg{]}\bigg{|}\bigg{|}_{L^{\infty}(\mathbb{P})}<+\infty.$
It is evident that if $Z$ is a $BMO(\mathcal{P}_{H})$-martingale generator,
defining for each $\mathbb{P}\in\mathcal{P}_{H}$,
$H^{\mathbb{P}}_{t}:=\int^{t}_{0}Z_{s}dB_{s},\ 0\leq t\leq 1,$
then $H$ is a $BMO(\mathcal{P}_{H})$-martingale. We denote by
$\mathbb{H}^{2}_{BMO(\mathcal{P}_{H})}$ the space of all
$BMO(\mathcal{P}_{H})$-martingale generators.
Applying energy inequality under each $\mathbb{P}\in\mathcal{P}_{H}$, we have
the following lemma:
###### Lemma 2.9.
Suppose $Z\in\mathbb{H}^{2}_{BMO(\mathcal{P}_{H})}$, for each $p\geq 1$,
$\mathbb{P}\in\mathcal{P}_{H}$ and all $\tau\in\mathcal{T}^{1}_{0}$,
$\mathbb{E}^{\mathbb{P}}_{\tau}\bigg{[}\bigg{(}\int^{1}_{\tau}|\hat{a}^{1/2}_{t}Z_{t}|^{2}dt\bigg{)}^{p}\bigg{]}\leq
C_{p}||Z||^{2p}_{\mathbb{H}^{2}_{BMO}},\ \mathbb{P}-a.s..$
Finally, we denote by $UC_{b}(\Omega)$ the collection of all bounded and
uniformly continuous maps $\xi:\Omega\rightarrow\mathbb{R}$ and denote by
$\mathcal{L}^{\infty}_{H}$ the closure of $UC_{b}(\Omega)$ under the
norm$||\cdot||_{L^{\infty}_{H}}$.
### 2.4. Formulation to quadratic 2BSDEs
We shall consider the 2BSDE of the following form, which is first introduce in
Soner et al. [25]:
(2.1)
$Y_{t}=\xi+\int^{1}_{t}\hat{F}_{s}(Y_{s},Z_{s})ds-\int^{1}_{t}Z_{s}dB_{s}+K_{1}-K_{t},\
0\leq t\leq 1,\ \mathcal{P}_{H}-q.s..$
In addition to (A1)-(A2), we assume the following conditions on the generator
$F$:
(A3) $F$ is continuous in $(y,z)$ and has a quadratic growth, i.e. there
exists a triple
$(\alpha,\beta,\gamma)\in\mathbb{R}^{+}\times\mathbb{R}^{+}\times\mathbb{R}^{+}$,
such that for all
$(\omega,t,y,z,a)\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}\times
D_{F_{t}}$,
(2.2)
$|F_{t}(\omega,y,z,a)|\leq\alpha+\beta|y|+\frac{\gamma}{2}|a^{1/2}z|^{2};$
(A4) $F$ is uniform Lipschitz in $y$, i.e. there exists a $\mu>0$, such that
for all $(\omega,t,y,y^{\prime},z,\\\
a)\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{d}\times
D_{F_{t}}$,
$|F_{t}(\omega,y,z,a)-F_{t}(\omega,y^{\prime},z,a)|\leq\mu|y-y^{\prime}|;$
(A5) $F$ is local Lipschitz in $z$, i.e. for each
$(\omega,t,y,z,z^{\prime}a)\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times
D_{F_{t}}$,
$|F_{t}(\omega,y,z,a)-F_{t}(\omega,y,z^{\prime},a)|\leq
C(1+|a^{1/2}z|+|a^{1/2}z^{\prime}|)|a^{1/2}(z-z^{\prime})|.$
###### Remark 2.10.
We have some comments on these conditions above: (A3) is a quadratic growth
condition for the proof of existence and similar ones for quadratic BSDEs can
be found in Kobylanski [13] and Marlais [16]; (A4) and (A5) are necessary for
the proof of uniqueness and analogous conditions were adopted by Hu et al. [9]
and Morlais [16] for quadratic BSDEs. All these conditions above could be
slightly weakened and further discussion will be made in Remark 5.11.
###### Definition 2.11.
We say that $(Y,Z)\in\mathbb{D}^{\infty}_{H}\times\mathbb{H}^{2}_{H}$ is a
solution of 2BSDE (2.1) if:
\- $Y_{T}=\xi$, $\mathcal{P}_{H}$-q.s.;
\- The process $K^{\mathbb{P}}$ defined as below: for each
$\mathbb{P}\in\mathcal{P}_{H}$,
(2.3)
$K^{\mathbb{P}}_{t}:=Y_{0}-Y_{t}-\int^{t}_{0}\hat{F}_{s}(Y_{s},Z_{s})ds+\int^{t}_{0}Z_{s}dB_{s},\
0\leq t\leq 1,\ \mathbb{P}-a.s.,$
has non-decreasing paths $\mathbb{P}$-a.s.;
\- The family $\\{K^{\mathbb{P}}\\}_{\mathbb{P}\in\mathcal{P}_{H}}$ satisfies
the minimum condition: for each $\mathbb{P}\in\mathcal{P}_{H}$,
(2.4)
$K^{\mathbb{P}}_{t}=\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,inf}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+},\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{T}],\
0\leq t\leq 1,\ \mathbb{P}-a.s..$
Moreover, if the family $\\{K^{\mathbb{P}}\\}_{\mathbb{P}\in\mathcal{P}_{H}}$
can be aggregated into a universal process $K$, we call $(Y,Z,K)$ a solution
of 2BSDE (2.1).
In the sequel, positive constants $C$ and $M$ vary from line to line.
## 3\. Representation and uniqueness of solutions to 2BSDEs
In this section, we give a representation theorem of solutions to the 2BSDE
(2.1) under (A1)-(A5), which is similar to those in Soner et al. [25] and
Possamai and Zhou [18]. The representation theorem shows the relationship
between the solution to the 2BSDE (2.1) and those to quadratic BSDEs with the
generator $\hat{F}$ under each $\mathbb{P}\in\mathcal{P}_{H}$. Also, some a
priori estimates to solutions is given which are useful to the proof of the
existence.
### 3.1. Representation theorem
Before proceeding the argument, we first introduce a lemma (cf. Lemma 3.1 in
Possamai and Zhou [18]), the parallel version of which for quadratic BSDEs
plays a very important role to show the connection between the boundness of
$Y$ and the $BMO$ property of the martingale part
$\int^{T}_{\cdot}Z_{t}dB_{t}$.
###### Lemma 3.1.
We assume (A1)-(A3) and $\xi\in L^{\infty}_{H}$. If
$(Y,Z)\in\mathbb{D}^{\infty}_{H}\times\mathbb{H}^{2}_{H}$ is a solution to the
2BSDE (2.1), then $Z\in\mathbb{H}^{2}_{BMO(\mathcal{P}_{H})}$ and
(3.1)
$||Z||^{2}_{\mathbb{H}^{2}_{BMO(\mathcal{P}_{H})}}\leq\frac{1}{\gamma^{2}}e^{4\gamma||Y||_{\mathbb{D}^{\infty}_{H}}}(1+2\gamma(\alpha+\beta||Y||_{\mathbb{D}^{\infty}_{H}})).$
Consider the following quadratic BSDE under each
$\mathbb{P}\in\mathcal{P}_{H}$:
(3.2)
$y^{\mathbb{P}}_{s}=\eta+\int^{t}_{s}\hat{F}_{u}(y^{\mathbb{P}}_{u},z^{\mathbb{P}}_{u})du-\int^{t}_{s}z^{\mathbb{P}}_{u}dB_{u},\
0\leq s\leq t,\ \mathbb{P}-a.s.,$
where $t\in[0,1]$ and $\eta$ is a $\mathcal{F}_{t}$-measurable random variable
in $L^{\infty}(\mathbb{P})$. Under (A1)-(A5), the BSDE (3.2) admits a unique
solution $(y^{\mathbb{P}}(t,\eta),z^{\mathbb{P}}(t,\eta))$ according to
Kobylanski [13] and Morlais [16].
Then, we have the following representation theorem for the solution of the
2BSDE (2.1):
###### Theorem 3.2.
Let (A1)-(A5) hold. Assume that $\xi\in L^{\infty}_{H}$ and
$(Y,Z)\in\mathbb{D}^{\infty}_{H}\times\mathbb{H}^{2}_{H}$ is a solution of the
2BSDE (2.1). Then, for each $\mathbb{P}\in\mathcal{P}_{H}$ and all $0\leq
t_{1}\leq t_{2}\leq 1$,
(3.3)
$Y_{t_{1}}=\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})}y^{\mathbb{P}^{\prime}}_{t_{1}}(t_{2},Y_{t_{2}}),\
\mathbb{P}-a.s.,$
where
$\mathcal{P}_{H}(t^{+}_{1},\mathbb{P}):=\\{\mathbb{P}^{\prime}\in{\mathcal{P}_{H}}:\mathbb{P}^{\prime}|_{\mathcal{F}^{+}_{t_{1}}}=\mathbb{P}|_{\mathcal{F}^{+}_{t_{1}}}\\}.$
###### Remark 3.3.
Applying Theorem 2.7 (comparison principle) in Morlais [16], the theorem above
also implies a comparison principle for quadratic 2BSDEs.
Proof: First of all, Lemma 3.1 shows that $Z$ is a
$BMO(\mathcal{P}_{H})$-martingale generator, then we deduce by the BDG type
inequalities, Lemma 2.9 and (3.1) that for each $p\geq 1$,
$\mathbb{P}\in\mathcal{P}_{H}$ and all $0\leq t_{1}\leq t_{2}\leq 1$,
(3.4)
$\mathbb{E}^{\mathbb{P}}_{t_{1}}[(K^{\mathbb{P}}_{t_{2}}-K^{\mathbb{P}}_{t_{1}})^{p}]\leq
C_{p}:=Ce^{4p\gamma||Y||_{\mathbb{D}^{\infty}_{H}}}(1+||Y||^{p}_{\mathbb{D}^{\infty}_{H}}),\
\mathbb{P}-a.s.,$
Since $\mathbb{P}$ is arbitrary in (3.4), we have
$\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t_{1}}[(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})^{p}]<C_{p},\
\mathbb{P}-a.s..$
We are now ready to prove that for a fixed $\mathbb{P}\in\mathcal{P}_{H}$ and
all $0\leq t_{1}\leq t_{2}\leq 1$,
(3.5)
$Y_{t_{1}}\leq\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{{\mathbb{P}^{\prime}}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})}y^{\mathbb{P}^{\prime}}_{t_{1}}(t_{2},Y_{t_{2}}),\
\mathbb{P}-a.s..$
Fixing $t_{2}\in[0,1]$, for each
$\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})$, we note
$\delta Y^{\mathbb{P}^{\prime}}:=Y-y^{\mathbb{P}^{\prime}}(t_{2},Y_{t_{2}})\
{\rm and}\ \delta
Z^{\mathbb{P}^{\prime}}:=Z-z^{\mathbb{P}^{\prime}}(t_{2},Y_{t_{2}}),$
then, for each $t\in[0,t_{2}]$,
$\displaystyle\delta
Y^{\mathbb{P}^{\prime}}_{t}=\int^{t_{2}}_{t}\lambda_{s}\delta
Y_{s}ds-\int^{t_{2}}_{t}\delta
Z_{s}\hat{a}^{1/2}_{s}(-\kappa^{\mathbb{P}^{\prime}}_{s}ds+dW^{\mathbb{P}^{\prime}}_{s})+K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t},\
\mathbb{P}^{\prime}-a.s.,$
where $\lambda$ is a scalar valued process and $\kappa$ is an
$\mathbb{R}^{d}$-valued process defined by
$\kappa^{\mathbb{P}^{\prime}}_{t}=\left\\{\begin{array}[]{l@{\quad,
\quad}l}\frac{(\hat{F}_{t}(y^{\mathbb{P}^{\prime}}_{t}(t_{2},Y_{t_{2}}),z^{\mathbb{P}^{\prime}}_{t}(t_{2},Y_{t_{2}}))-\hat{F}_{t}(y^{\mathbb{P}^{\prime}}_{t}(t_{2},Y_{t_{2}}),Z_{t}))\hat{a}^{1/2}_{t}\delta
Z_{t}}{|\hat{a}^{1/2}_{t}\delta Z_{t}|^{2}}&|\hat{a}^{1/2}_{t}\delta
Z_{t}|\neq 0;\\\ 0&otherwise.\end{array}\right.$
By (A4) and (A5), we have
$||\lambda||_{D^{\infty}(\mathbb{P}^{\prime})}\leq\mu$ and $\kappa$ satisfies
$|\kappa^{\mathbb{P}^{\prime}}_{t}|\leq
1+|\hat{a}^{1/2}_{t}z^{\mathbb{P}^{\prime}}_{t}(t_{2},Y_{t_{2}})|+|\hat{a}^{1/2}_{t}Z_{t}|.$
Defining
$H^{\mathbb{P}^{\prime}}_{t}:=\int^{t}_{0}\kappa^{\mathbb{P}^{\prime}}_{s}dW^{\mathbb{P}^{\prime}}_{s},\
0\leq t\leq t_{2},$
we have
(3.6) $||H^{\mathbb{P}^{\prime}}||^{2}_{BMO(\mathbb{P}^{\prime})}\leq
C(1+||\hat{a}^{1/2}_{t}z^{\mathbb{P}^{\prime}}_{t}(t_{2},Y_{t_{2}})||^{2}_{\mathbb{H}^{2}_{BMO(\mathbb{P}^{\prime})}}+||\hat{a}^{1/2}_{t}Z_{t}||^{2}_{\mathbb{H}^{2}_{BMO_{2}(\mathbb{P}^{\prime})}}).$
Applying a priori estimates for quadratic BSDEs (cf. Lemma 3.1 in Morlais
[16]), it is readily observed that
(3.7)
$||\hat{a}^{1/2}z^{\mathbb{P}^{\prime}}(t_{2},Y_{t_{2}})||^{2}_{H^{2}_{BMO(\mathbb{P}^{\prime})}}\leq
Ce^{4\gamma||Y||_{\mathbb{D}^{\infty}_{H}}}(1+||Y||_{\mathbb{D}^{\infty}_{H}}).$
Putting (3.1) and (3.7) into (3.6), we deduce the following estimate uniformly
in $\mathbb{P}^{\prime}$:
(3.8) $||H^{\mathbb{P}^{\prime}}||^{2}_{BMO(\mathbb{P}^{\prime})}\leq M_{H},$
where $M_{H}$ depends simply on $||Y||_{\mathbb{D}^{\infty}_{H}}$. This
implies that $H$ is a $BMO(\mathcal{P}_{H})$-martingale.
Define a probability measure $\mathbb{Q}^{\prime}\ll\mathbb{P}^{\prime}$ by
$\frac{d\mathbb{Q}^{\prime}}{d\mathbb{P}^{\prime}}|_{\mathcal{F}_{t}}=\mathcal{E}(\int^{\cdot}_{0}\kappa^{\mathbb{P}^{\prime}}_{s}dW^{\mathbb{P}^{\prime}}_{s})_{t}$
and a process $M_{t}:=\exp(\int^{t}_{t_{1}}\lambda_{s}ds),\ t_{1}\leq t\leq
t_{2}.$ Applying Itô’s formula to $M\delta Y$ under $\mathbb{Q}^{\prime}$, we
have
(3.9) $\displaystyle\delta
Y^{\mathbb{P}^{\prime}}_{t_{1}}=\mathbb{E}^{\mathbb{Q}^{\prime}}_{t_{1}}\bigg{[}\int^{t_{2}}_{t_{1}}M_{t}dK^{\mathbb{P}^{\prime}}_{t}\bigg{]}$
$\displaystyle\leq\mathbb{E}^{\mathbb{Q}^{\prime}}_{t_{1}}[\sup_{t_{1}\leq
t\leq
t_{2}}(M_{t})(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})]$
$\displaystyle\leq
e^{\mu}\mathbb{E}^{\mathbb{P}^{\prime}}_{t_{1}}\bigg{[}\frac{\mathcal{E}(H^{\mathbb{P}^{\prime}})_{t_{2}}}{\mathcal{E}(H^{\mathbb{P}^{\prime}})_{t_{1}}}(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})\bigg{]},\
\mathbb{P}-a.s..$
Thanks to (3.8) and Lemma 2.6, we can find uniformly for all
$\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})$ two constants
$q>1$ and $C_{RH}>0$ such that
$\displaystyle\delta Y^{\mathbb{P}^{\prime}}_{t_{1}}$ $\displaystyle\leq
C_{RH}^{{1}/{q}}e^{\mu}\mathbb{E}^{\mathbb{P}^{\prime}}_{t_{1}}[(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})^{p}]^{{1}/{p}}$
$\displaystyle\leq
C^{{1}/{q}}_{RH}e^{\mu}(\mathbb{E}^{\mathbb{P}^{\prime}}_{t_{1}}[(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})^{2p-1}])^{1/2p}\mathbb{E}^{\mathbb{P}^{\prime}}_{t_{1}}[(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})]^{{1}/{2p}}$
$\displaystyle\leq
C^{{1}/{q}}_{RH}e^{\mu}(\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t_{1}}[(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})^{2p-1}])^{1/2p}\mathbb{E}^{\mathbb{P}^{\prime}}_{t_{1}}[(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})]^{{1}/{2p}}$
$\displaystyle\leq
C^{{1}/{q}}_{RH}C_{2p-1}^{{1}/{2p}}e^{\mu}\mathbb{E}^{\mathbb{P}^{\prime}}_{t_{1}}[(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})]^{{1}/{2p}},\
\mathbb{P}-a.s.,$
where $1/p+1/q=1$. Since $C_{RH}$ and $C_{2p-1}$ are independent of
$\mathbb{P}^{\prime}$, we can take essential infimum over all
$\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})$ on the left-hand
side of the above inequality and deduce by the minimum condition (2.4) that
$\displaystyle Y_{t_{1}}$
$\displaystyle-\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})}y^{\mathbb{P}^{\prime}}_{t_{1}}(t_{2},Y_{t_{2}})$
$\displaystyle\leq
C^{{1}/{q}}_{RH}C_{2p-1}^{{1}/{2p}}e^{\mu}\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,inf}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})}[(K^{\mathbb{P}^{\prime}}_{t_{2}}-K^{\mathbb{P}^{\prime}}_{t_{1}})]^{{1}/{2p}}=0,\
\mathbb{P}-a.s..$
From (3.9), it is easily observed that $\delta
Y^{\mathbb{P}^{\prime}}_{t_{1}}\geq 0$, $\mathbb{Q}^{\prime}$-a.s. and thus,
$\mathbb{P}-a.s.$, for all
$\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+}_{1},\mathbb{P})$, which directly
yields the reverse inequality of (3.5). The proof of (3.3) is complete.
$\square$
### 3.2. A priori estimates
We now give some a priori estimates for quadratic 2BSDEs:
###### Lemma 3.4.
Let (A1)-(A5) hold. Assume that $\xi\in L^{\infty}_{H}$ and that
$(Y,Z)\in\mathbb{D}^{\infty}_{H}\times\mathbb{H}^{2}_{H}$ is a solution to
2BSDE (2.1). Then, there exists a $C>0$ such that
(3.10) $||Y||_{\mathbb{D}^{\infty}_{H}}\leq C(1+||\xi||_{L^{\infty}_{H}})\
and\ ||Z||^{2}_{\mathbb{H}^{2}_{BMO(\mathcal{P}_{H})}}\leq
Ce^{4\gamma||\xi||_{L^{\infty}_{H}}}(1+||\xi||_{L^{\infty}_{H}}).$
Proof: From (3.4) and a priori estimates for quadratic BSDEs, we deduce the
left-hand side of (3.10), whereas the right-hand side comes after (3.1).
$\square$
###### Lemma 3.5.
Let (A1)-(A5) hold. Assume that $\xi^{i}\in L^{\infty}_{H}$ and that
$(Y^{i},Z^{i})\in\mathbb{D}^{\infty}_{H}\times\mathbb{H}^{2}_{H}$, $i=1,2$,
are two solution to 2BSDE (2.1). Denote
$\displaystyle\delta\xi:=\xi^{1}-\xi^{2},\ \delta Y:=Y^{1}-Y^{2},\ \delta
Z:=Z^{1}-Z^{2},$ $\displaystyle\delta
K^{\mathbb{P}}:=(K^{1})^{\mathbb{P}}-(K^{2}){{}^{\mathbb{P}}}\ and\
\Delta\delta Y_{t}=\delta Y_{t}-\delta Y_{t^{-}},$
then we have the following estimates
$||\delta Y||_{\mathbb{D}^{\infty}_{H}}\leq C||\delta\xi||_{L^{\infty}_{H}};$
$\mathbb{E}^{\mathbb{P}}_{\tau}\bigg{[}\int^{1}_{\tau}|\hat{a}^{1/2}_{t}\delta
Z_{t}|^{2}dt\bigg{]}\leq
C||\delta\xi||^{2}_{L^{\infty}_{H}}\sum^{2}_{i=1}(1+e^{4\gamma||\xi^{i}||_{L^{\infty}_{H}}})(1+||\xi^{i}||_{L^{\infty}_{H}}).$
Proof: Similar to (3.9), we can easily obtain the first inequality. For the
second one, we apply Itô’s formula to $\delta Y^{2}$, then we have for a fixed
$\mathbb{P}\in\mathcal{P}_{H}$ and a $\tau\in\mathcal{T}^{1}_{0}$,
$\displaystyle\delta Y^{2}_{\tau}+\int^{1}_{\tau}|\hat{a}^{1/2}_{t}\delta
Z_{t}|^{2}dt$ $\displaystyle\leq\delta\xi^{2}+2\int^{1}_{\tau}\delta
Y_{t}(\hat{F}_{t}(Y^{1}_{t},Z^{1}_{t})-\hat{F}_{t}(Y^{2}_{t},Z^{2}_{t}))dt$
$\displaystyle-$ $\displaystyle 2\int^{1}_{\tau}\delta Y_{t}\delta
Z_{t}dB_{t}+2\int^{1}_{\tau}\delta Y_{t^{-}}\delta
dK^{\mathbb{P}}_{t}-\sum_{\tau<t\leq 1}(\Delta\delta Y_{t})^{2},\
\mathbb{P}-a.s..$
Taking expectation on the left-hand side and by (A3) and (3.4), we deduce
$\displaystyle\mathbb{E}^{\mathbb{P}}_{\tau}\bigg{[}\int^{1}_{\tau}|\hat{a}^{1/2}_{t}\delta
Z_{t}|^{2}dt\bigg{]}$
$\displaystyle\leq||\delta\xi||^{2}_{L^{\infty}_{H}}+2||\delta
Y||_{\mathbb{D}^{\infty}_{H}}\bigg{(}2\alpha+\beta\sum^{2}_{i=1}||Y^{i}||_{\mathbb{D}^{\infty}_{H}}+\frac{\gamma}{2}\sum^{2}_{i=1}||Z^{i}||_{H^{2}_{BMO(\mathcal{P}_{H})}}\bigg{)}$
$\displaystyle+2||\delta
Y||_{\mathbb{D}^{\infty}_{H}}(\mathbb{E}^{\mathbb{P}}_{\tau}[(K^{1})^{\mathbb{P}}_{1}-(K^{1})^{\mathbb{P}}_{\tau}]+\mathbb{E}^{\mathbb{P}}_{\tau}[(K^{2})^{\mathbb{P}}_{1}-(K^{2})^{\mathbb{P}}_{\tau}])$
$\displaystyle\leq
C||\delta\xi||_{L^{\infty}_{H}}\sum^{2}_{i=1}(1+e^{4\gamma||\xi^{i}||_{L^{\infty}_{H}}})(1+||\xi^{i}||_{L^{\infty}_{H}}).$
We complete the proof. $\square$
By either Theorem 3.2 or Lemma 3.5, we deduce immediately the uniqueness of
$Y$. We observe that $d\langle Y,B\rangle_{t}=Z_{t}d\langle B\rangle_{t}$,
$0\leq t\leq 1$, $\mathcal{P}_{H}$-q.s., which implies the uniqueness of $Z$.
## 4\. Existence of solutions to 2BSDEs
In this section, we provide the existence result for the 2BSDE (2.1) under
(A1)-(A5) by a pathwise construction introduced in Soner et al. [24] and [25]
with the technique so-called regular conditional probability distribution
(r.p.c.d.), which can be find in Stroock and Varadhan [27].
### 4.1. Regular conditional probability measures
For the convenience of the reader, we recall some notations of r.p.c.d. in
Soner et al. [24].
* •
For each $t\in[0,1]$, let
$\Omega^{t}:=\\{\tilde{\omega}\in\mathcal{C}([t,1],\mathbb{R}^{d}),\tilde{\omega}(t)=0\\}$
be the shifted space, $B^{t}$ the shifted canonical process, $\mathcal{F}^{t}$
the shifted filtration generated by $B^{t}$.
* •
For each $0\leq s\leq t\leq 1$ and $\omega\in\Omega^{s}$, we define the
shifted path $\tilde{\omega}\in\Omega^{t}$ by
$\tilde{\omega}_{u}:=\omega_{u}-\omega_{t},\ u\in[t,1].$
* •
For each $0\leq s\leq t\leq 1$, $\omega\in\Omega^{s}$ and
$\tilde{\omega}\in\Omega^{t}$, we define the concatenation path
$\omega\otimes_{t}\tilde{\omega}\in\Omega^{s}$ by
$(\omega\otimes_{t}\tilde{\omega})_{u}:=\omega_{u}\textnormal{{1}}_{[s,t)}(u)+(\omega_{t}+\tilde{\omega}_{u})\textnormal{{1}}_{[t,1]}(u),\
u\in[s,1].$
* •
For each $0\leq s\leq t\leq 1$, $\omega\in\Omega^{s}$ and an
$\mathcal{F}^{s}_{1}$-measurable random variable $\xi$ on $\Omega^{s}$, we
define the shifted $\mathcal{F}^{t}_{1}$-measurable random variable
$\xi^{t,\omega}$ on $\Omega^{t}$ by
$\xi^{t,\omega}(\tilde{\omega}):=\xi(\omega\otimes_{t}\tilde{\omega}),\tilde{\omega}\in\Omega^{t}.$
* •
For each $0\leq s\leq t\leq 1$, the shifted process $X^{t,\omega}$ of an
$\mathcal{F}^{s}$-progressively measurable $X$ is
$\mathcal{F}^{t}$-progressively measurable.
* •
For each $t\in[0,1]$ and $\omega\in\Omega$, we define our shifted generator by
$\hat{F}^{t,\omega}_{s}(\tilde{\omega},y,z):=F_{s}(\omega\otimes_{t}\tilde{\omega},y,z,\hat{a}^{t}_{s}(\tilde{\omega})),\
(s,\tilde{\omega})\in[t,1]\times\Omega^{t}.$
* •
For each $t\in[0,1]$, $\mathcal{P}^{t}_{H}$ denotes the collection of all
those $\mathbb{P}\in\overline{\mathcal{P}}^{t}_{S}$ such that
$\underline{a}^{\mathbb{P}}\leq\hat{a}^{t}_{s}\leq\overline{a}^{\mathbb{P}}\
{\rm and}\ \hat{a}^{t}_{s}\in D_{F_{s}},\ \lambda\times\mathbb{P}-a.e.,$
for some $\underline{a}^{\mathbb{P}}$,
$\overline{a}^{\mathbb{P}}\in\mathbb{S}_{d}^{>0}$ and all
$(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$.
* •
For $t\in[0,1]$ and $\mathbb{P}\in\mathcal{P}_{H}$, the r.p.c.d
$\mathbb{P}^{\omega}_{t}$ of $\mathbb{P}$ induces naturally a probability
measure $\mathbb{P}^{t,\omega}$ on $(\Omega^{t},\mathcal{F}^{t}_{1})$ which
satisfies that for each bounded and $\mathcal{F}_{1}$-measurable random
variable $\xi$,
$\mathbb{E}^{\mathbb{P}^{\omega}_{t}}[\xi]=\mathbb{E}^{\mathbb{P}^{t,\omega}}[\xi^{t,\omega}].$
* •
By Lemma 4.1 in [24], $\mathbb{P}^{t,\omega}$ is an elements in
$\mathcal{P}^{t}_{H}$ and for each $t\in[0,1]$ and
$\mathbb{P}\in\mathcal{P}_{H}$, it holds for $\mathbb{P}$-a.s.
$\omega\in\Omega$,
$\displaystyle
F_{s}(\omega\otimes_{t}\tilde{\omega},y,z,\hat{a}^{t}_{s}(\tilde{\omega}))=F_{s}(\omega\otimes_{t}\tilde{\omega},y,z$
$\displaystyle,\hat{a}_{s}(\omega\otimes_{t}\tilde{\omega})),\
\lambda\times\mathbb{P}^{t,\omega}-a.e..$
### 4.2. Existence result
For some fixed $\omega\in\Omega$ and $0\leq t_{1}\leq t_{2}\leq 1$, we
consider a quadratic BSDE of the following type on the shifted space
$\Omega^{t_{1}}$ under each $\mathbb{P}^{t_{1}}\in\mathcal{P}^{t_{1}}_{H}$:
(4.1) $\displaystyle
y^{\mathbb{P}^{t_{1}},t_{1},\omega}_{s}=\eta^{t_{1},\omega}$
$\displaystyle+\int^{t_{2}}_{s}\hat{F}^{t_{1},\omega}_{u}(y^{\mathbb{P}^{t_{1}},t_{1},\omega}_{u},z^{\mathbb{P}^{t_{1}},t_{1},\omega}_{u})du$
$\displaystyle-\int^{t_{2}}_{s}z^{\mathbb{P}^{t_{1}},t_{1},\omega}_{u}dB^{t_{1}}_{u},\
t_{1}\leq s\leq t_{2},\ \mathbb{P}^{t_{1}}-a.s.,$
where $\eta\in L^{\infty}_{H}$ is a $\mathcal{F}_{t_{2}}$-measurable random
variable. It is well known that (4.1) admits a unique solution
$(y^{\mathbb{P}^{t_{1}},t_{1},\omega}(t_{2},\eta),z^{\mathbb{P}^{t_{1}},t_{1},\omega}(t_{2},\eta))$
under (A1)-(A5). In view of the Blumenthal zero-one law,
$y^{\mathbb{P}^{t_{1}},t_{1},\omega}_{t_{1}}(t_{2},\eta)$ is a deterministic
constant $\mathbb{P}^{t_{1}}$-a.s. for any given $\eta$ and
$\mathbb{P}^{t_{1}}$.
The following lemma describes the relationship between
$y_{t}^{\mathbb{P}}(1,\xi)$ and
$y^{\mathbb{P}^{t,\omega},t,\omega}_{t}(1,\xi)$, where the first one is the
solution to (3.2) with the parameters $(1,\xi)$ under a fixed
$\mathbb{P}\in\mathcal{P}_{H}$ and the second one is the solution to (4.1)
when $t_{1}$ is a fixed $t$, $\mathbb{P}^{t}$ is in fact the r.p.c.d
$\mathbb{P}^{t,\omega}$ of $\mathbb{P}$ and $(t_{2},\eta)=(1,\xi)$.
###### Lemma 4.1.
Assume (A1)-(A5) hold. For a given $\xi\in L^{\infty}_{H}$ and a fixed
$\mathbb{P}\in\mathcal{P}_{H}$, we have, for each $t\in[0,1]$ and
$\mathbb{P}$-a.s. $\omega\in\Omega$,
(4.2)
$y^{\mathbb{P}}_{t}(1,\xi)(\omega)=y^{\mathbb{P}^{t,\omega},t,\omega}_{t}(1,\xi).$
Proof: Similar to the proof of Lemma 4.1 in Soner et al. [24], we have the
following conclusion: because $\xi\in L^{\infty}(\mathbb{P})$, for
$\mathbb{P}$-a.s. $\omega\in\Omega$,
$|\xi^{t,\omega}|\leq||\xi||_{L^{\infty}(\mathbb{P})}$,
$\mathbb{P}^{t,\omega}$-a.s.. Thus, (4.1) is well defined under our setting
and the right-hand side of (4.2) is the unique solution of (4.1).
We emphasize that the wellposedness of both (3.2) and (4.1) as well as the
estimates of the solutions are already provided by Kobylanski [13] and Morlais
[16]. Our job here is only to redo the construction of two sequences formed by
the solutions of Lipschitz BSDEs, which approximate the solutions on both
sides of (4.2).
By Lemma3.1 in Morlais [16], we can find an
$M:=e^{\beta}(\alpha+||\xi(\omega)||_{L^{\infty}(\mathbb{P})}),$ which is the
bound of both side of (4.2). Then, we choose a
$\mathcal{C}^{\infty}(\mathbb{R})$ function which takes value in $[0,1]$ and
satisfies that
$\phi(u)=\left\\{\begin{array}[]{l@{\quad, \quad}l}1&u\in[e^{-\gamma
M},e^{\gamma M}];\\\
0&u\in(-\infty,e^{-\gamma(M+1)}]\cup[e^{\gamma(M+1)},+\infty).\end{array}\right.\\\
$
We can verify that for each $t\in[0,1]$,
(4.3) $\mathcal{Y}^{\mathbb{P}}_{t}(1,e^{\gamma\xi},\hat{G}):=\exp(\gamma
y^{\mathbb{P}}_{t}(1,\xi)),$
solves a quadratic BSDE with the parameters $(1,e^{\gamma\xi})$ and the
generator $\hat{G}$ of the following form: for each
$(\omega,t,\mathcal{Y},\mathcal{Z})\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}$,
(4.4)
$\hat{G}_{t}(\omega,\mathcal{Y},\mathcal{Z}):=\phi(\mathcal{Y})\bigg{(}\gamma\mathcal{Y}\hat{F}_{t}\bigg{(}\omega,\frac{\ln(\mathcal{Y})}{\gamma},\frac{\mathcal{Z}}{\gamma\mathcal{Y}}\bigg{)}-\frac{1}{2\mathcal{Y}}|\hat{a}^{1/2}_{t}(\omega)\mathcal{Z}|^{2}\bigg{)}.$
On the other hand, fixing $(\omega,t)\in\Omega\times[0,1]$,
(4.5)
$\mathcal{Y}^{\mathbb{P}^{t,\omega},t,\omega}_{s}(1,e^{\gamma\xi},\hat{G}^{t,\omega})(\tilde{\omega}):=\exp(\gamma
y^{\mathbb{P}^{t,\omega},t,\omega}_{s}(1,\xi)(\tilde{\omega})),\ t\leq s\leq
1,$
defines a solution that solves a quadratic BSDE under $\mathbb{P}^{t,\omega}$
with the parameters $(1,e^{\gamma\xi})$ and the generator $\hat{G}^{t,\omega}$
of the following form: for each
$(\tilde{\omega},s,\mathcal{Y},\mathcal{Z})\in\Omega^{t}\times[t,1]\times\mathbb{R}\times\mathbb{R}^{d}$,
$\hat{G}^{t,\omega}_{s}(\tilde{\omega},\mathcal{Y},\mathcal{Z}):=\phi(\mathcal{Y})\bigg{(}\gamma\mathcal{Y}\hat{F}_{s}^{t,\omega}\bigg{(}\tilde{\omega},\frac{\ln(\mathcal{Y})}{\gamma},\frac{\mathcal{Z}}{\gamma\mathcal{Y}}\bigg{)}-\frac{1}{2\mathcal{Y}}|(\hat{a}^{t}_{s})^{1/2}(\tilde{\omega})\mathcal{Z}|^{2}\bigg{)}.$
Now, our main aim is changed into that for each $t\in[0,1]$ and
$\mathbb{P}$-a.s. $\omega\in\Omega$,
$\mathcal{Y}^{\mathbb{P}}_{t}(1,e^{\gamma\xi},\hat{G})(\omega)=\mathcal{Y}^{\mathbb{P}^{t,\omega},t,\omega}_{t}(1,e^{\gamma\xi},\hat{G}^{t,\omega}).$
For each
$(\omega,t,\mathcal{Y},\mathcal{Z})\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}$,
we set
(4.6)
$\hat{G}^{n}_{t}(\omega,\mathcal{Y},\mathcal{Z}):=\sup_{(p,q)\in\mathbb{Q}\times\mathbb{Q}^{d}}\\{\hat{G}_{t}(\omega,p,q)-n|(p-\mathcal{Y})|-n|\hat{a}^{1/2}_{t}(\omega)(q-\mathcal{Z})|\\},\
n\in\mathbb{N}$
and also for fixed $(\omega,t)\in\Omega\times[0,1]$ and each
$(\tilde{\omega},s,\mathcal{Y},\mathcal{Z})\in\Omega^{t}\times[t,1]\times\mathbb{R}\times\mathbb{R}^{d}$,
we define
$\displaystyle((\hat{G}^{t,\omega})^{n}_{s}$
$\displaystyle(\tilde{\omega},\mathcal{Y},\mathcal{Z})$
$\displaystyle:=\sup_{(p,q)\in\mathbb{Q}\times\mathbb{Q}^{d}}\\{\hat{G}^{t,\omega}_{s}(\tilde{\omega},p,q)-n|(p-\mathcal{Y})|-n|(\hat{a}^{t}_{s})^{1/2}(\tilde{\omega})(q-\mathcal{Z})|\\},\
n\in\mathbb{N}.$
By Lemma 4.1 in Soner et al. [24], for each $t\in[0,1]$, $\mathbb{P}$-a.s.
$\omega\in\Omega$ and each $(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$,
(4.7) $\displaystyle\hat{F}^{t,\omega}_{s}(\tilde{\omega},y,z)$
$\displaystyle=F^{t,\omega}_{s}(\tilde{\omega},y,z,\hat{a}^{t}_{s}(\tilde{\omega}))=F_{s}(\omega\otimes_{t}\tilde{\omega},y,z,\hat{a}_{s}(\omega\otimes_{t}\tilde{\omega}))$
$\displaystyle=(\hat{F}(\cdot,\cdot))^{t,\omega}_{s}(\tilde{\omega},y,z),\
{\rm and}\
\hat{a}^{t}_{s}(\tilde{\omega})=\hat{a}^{t,\omega}_{s}(\tilde{\omega}),\
\lambda\times\mathbb{P}^{t,\omega}-a.e..$
We call $(\hat{F}(\cdot,\cdot))^{t,\omega}$ the globally shifted generator of
$\hat{F}$. From (4.7), we can deduce that for each $t\in[0,1]$,
$\mathbb{P}$-a.s. $\omega\in\Omega$ and each
$(\mathcal{Y},\mathcal{Z})\in\mathbb{R}\times\mathbb{R}^{d}$,
$\hat{G}^{t,\omega}_{s}(\tilde{\omega},\mathcal{Y},\mathcal{Z})=(\hat{G}(\cdot,\cdot))^{t,\omega}_{s}(\tilde{\omega},\mathcal{Y},\mathcal{Z}),\
\lambda\times\mathbb{P}^{t,\omega}-a.e.,$
and furthermore that for each $n\in\mathbb{N}$,
(4.8)
$(\hat{G}^{t,\omega})^{n}_{s}(\tilde{\omega},\mathcal{Y},\mathcal{Z})=(\hat{G}^{n}(\cdot,\cdot))^{t,\omega}_{s}(\tilde{\omega},\mathcal{Y},\mathcal{Z}),\
\lambda\times\mathbb{P}^{t,\omega}-a.e..$
Moreover, it is easy to verify that for each
$(\omega,t,\mathcal{Y},\mathcal{Z})\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}$,
$\displaystyle-e^{M+1}(\alpha\gamma+\beta(M+1))$
$\displaystyle-e^{M+1}|\hat{a}_{t}^{1/2}(\tilde{\omega})\mathcal{Z}|^{2}\leq\hat{G}_{t}(\omega,\mathcal{Y},\mathcal{Z})$
$\displaystyle\leq\hat{G}^{n+1}_{t}(\omega,\mathcal{Y},\mathcal{Z})\leq\hat{G}^{n}_{t}(\omega,\mathcal{Y},\mathcal{Z})\leq
e^{M+1}(\alpha\gamma+\beta(M+1)),$
and
$\hat{G}^{n}_{t}(\omega,\mathcal{Y},\mathcal{Z})\downarrow\hat{G}_{t}(\omega,\mathcal{Y},\mathcal{Z})$
uniformly on compact sets in $[0,1]\times\mathbb{R}\times\mathbb{R}^{d}$.
Similarly, for fixed $(\omega,t)\in\Omega\times[0,1]$ and each
$(\tilde{\omega},s,\mathcal{Y},\mathcal{Z})\in\Omega^{t}\times[t,1]\times\mathbb{R}\times\mathbb{R}^{d}$,
$\displaystyle-e^{M+1}(\alpha\gamma+\beta(M+1))$
$\displaystyle-e^{M+1}|(\hat{a}^{t}_{s})^{1/2}(\tilde{\omega})\mathcal{Z}|^{2}\leq\hat{G}_{t}(\omega,\mathcal{Y},\mathcal{Z})$
$\displaystyle\leq(\hat{G}^{t,\omega})^{n+1}_{t}(\omega,\mathcal{Y},\mathcal{Z})\leq(\hat{G}^{t,\omega})^{n}_{t}(\omega,\mathcal{Y},\mathcal{Z})\leq
e^{M+1}(\alpha\gamma+\beta(M+1)),$
and
$(\hat{G}^{t,\omega})^{n}_{t}(\omega,\mathcal{Y},\mathcal{Z})\downarrow(\hat{G}^{t,\omega})_{t}(\omega,\mathcal{Y},\mathcal{Z})$
uniformly on compact sets in $[t,1]\times\mathbb{R}\times\mathbb{R}^{d}$. By
Lemma 3.3 (monotone stability) in Morlais [16], we have, for each $t\in[0,1]$
and $\mathbb{P}$-a.s. $\omega\in\Omega$,
(4.9)
$\mathcal{Y}_{t}^{\mathbb{P}}(1,e^{\gamma\xi},\hat{G}^{n})(\omega)\downarrow\mathcal{Y}_{t}^{\mathbb{P}}(1,e^{\gamma\xi},\hat{G})(\omega),\
{\rm as}\ n\rightarrow+\infty,$
and for fixed $(\omega,t)\in\Omega\times[0,1]$,
(4.10)
$\mathcal{Y}_{t}^{\mathbb{P}^{t,\omega},t,\omega}(1,e^{\gamma\xi},(\hat{G}^{t,\omega})^{n})\downarrow\mathcal{Y}_{t}^{\mathbb{P}^{t,\omega},t,\omega}(1,e^{\gamma\xi},\hat{G}^{t,\omega}),\
{\rm as}\ n\rightarrow+\infty.$
To obtain the desired result, it is suffice to prove that for each
$n\in\mathbb{N}$, a fixed $t\in[0,1]$ and $\mathbb{P}$-a.s. $\omega\in\Omega$,
(4.11)
$\mathcal{Y}^{\mathbb{P}}_{t}(1,e^{\gamma\xi},\hat{G}^{n})(\omega)=\mathcal{Y}^{\mathbb{P}^{t,\omega},t,\omega}_{t}(1,e^{\gamma\xi},(\hat{G}^{t,\omega})^{n}).$
We notice that the generators of both sides of (4.11) satisfy the following
uniform Lipschitz conditions: for each $n\in\mathbb{N}$ and
$(\omega,t,\mathcal{Y}^{1},\mathcal{Y}^{2},\mathcal{Z}^{1},\mathcal{Z}^{2})\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$,
$|\hat{G}^{n}_{t}(\omega,\mathcal{Y}^{1},\mathcal{Z}^{1})-\hat{G}^{n}_{t}(\omega,\mathcal{Y}^{2},\mathcal{Z}^{2})|\leq
n|\mathcal{Y}^{1}-\mathcal{Y}^{2}|+n|\hat{a}^{1/2}_{t}(\omega)(\mathcal{Z}^{1}-\mathcal{Z}^{2})|;$
and for fixed $(\omega,t)\in\Omega\times[0,1]$, each $n\in\mathbb{N}$ and
$(\tilde{\omega},s,\mathcal{Y}^{1},\mathcal{Y}^{2},\mathcal{Z}^{1},\mathcal{Z}^{2})\in\Omega^{t}\times[t,1]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$,
$|(\hat{G}^{t,\omega})^{n}_{s}(\tilde{\omega},\mathcal{Y}^{1},\mathcal{Z}^{1})-(\hat{G}^{t,\omega})^{n}_{s}(\tilde{\omega},\mathcal{Y}^{2},\mathcal{Z}^{2})|\leq
n|\mathcal{Y}^{1}-\mathcal{Y}^{2}|+n|(\hat{a}^{t}_{s})^{1/2}(\tilde{\omega})(\mathcal{Z}^{1}-\mathcal{Z}^{2})|.$
Since the solutions of these Lipschitz BSDEs can be constructed via Picard
iteration, from (4.8), we can obtain (4.11) (cf. (i) in the proof of
Proposition 4.7 in Soner et al. [24] and (i) in the proof of Proposition 5.1
in Possamai and Zhou [18]). Then, (4.9) and (4.10) give the desired result.
$\square$
###### Remark 4.2.
The lemma above is the key point of this paper, which ensure us to prove the
following proposition under the Kobylanski [13] type condition (A3) instead of
Tevzadze [28] type one, which is adopted by Possamai and Zhou [18] to make
sure that the solutions of quadratic BSDEs on both original and shifted spaces
can be constructed via Picard iteration, so that the statement corresponding
to (4.2) in Soner et al. [24] for Lipschitz BSDEs still holds. In the present
paper, the lemma above is proved by a monotonic convergence technique for
classical BSDEs under a fixed $\mathbb{P}$, but it is still difficult to
obtain a globally monotonic convergence theorem for quadratic 2BSDEs, as
Possamai and Zhou [18] has already stated.
Similar to the one in Soner et al. [24], we define the following value process
$V_{t}$ pathwisely: for each $(\omega,t)\in\Omega\times[0,1]$,
(4.12)
$V_{t}(\omega):=\sup_{\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}}y^{\mathbb{P}^{t},t,\omega}_{t}(1,\xi).$
For the rest part of the proof of the existence, we assume moreover that the
terminal value $\xi$ is an element in $UC_{b}(\Omega)$. Therefore, it is
readily observed that for all $(\omega,t)\in\Omega\times[0,1]$,
(4.13) $V_{t}(\omega)\leq C(1+\sup_{\omega\in\Omega}|\xi(\omega)|),$
and there exists a modulus of continuity $\rho$, such that for each
$t\in[0,1]$ and
$(\omega,\omega^{\prime},\tilde{\omega})\in\Omega\times\Omega\times\Omega^{t}$,
$|\xi^{t,\omega}(\tilde{\omega})-\xi^{t,\omega^{\prime}}(\tilde{\omega})|\leq\rho(||\omega-\omega^{\prime}||^{\infty}_{t}).$
Recalling the uniform continuity of $F$ in $\omega$, we have moreover that for
each $0\leq t\leq s\leq 1$,
$(\omega,\omega^{\prime},\tilde{\omega},y,z)\in\Omega\times\Omega\times\Omega^{t}\times\mathbb{R}\times\mathbb{R}^{d}$,
$|\hat{F}^{t,\omega}_{s}(\tilde{\omega},y,z)-\hat{F}^{t,\omega^{\prime}}_{s}(\tilde{\omega},y,z)|\leq\rho(||\omega-\omega^{\prime}||^{\infty}_{t}).$
We define
$\delta
y:=y^{\mathbb{P}^{t},t,\omega}(1,\xi)-y^{\mathbb{P}^{t},t,\omega^{\prime}}(1,\xi),\
\delta
z:=z^{\mathbb{P}^{t},t,\omega}(1,\xi)-z^{\mathbb{P}^{t},t,\omega^{\prime}}(1,\xi),$
$\delta\xi:=\xi^{t,\omega}-\xi^{t,\omega^{\prime}},\
\delta\hat{F}(y,z):=\hat{F}^{t,\omega}(y,z)-\hat{F}^{t,\omega^{\prime}}(y,z).$
Proceeding the same in the proof of Theorem 3.2, for each
$(\omega,\omega^{\prime},t)\in\Omega\times\Omega\times[0,1]$ and a fixed
$\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$, we can find a
$\mathbb{Q}^{t}\ll\mathbb{P}^{t}$ and a bounded process $M$, such that
(4.14) $|\delta
y_{t}|=\mathbb{E}^{\mathbb{Q}^{t}}\bigg{[}M_{1}\delta\xi+\int^{1}_{t}M_{s}\delta\hat{F}_{s}(y_{s}^{\mathbb{P}^{t},t,\omega}(1,\xi),z_{s}^{\mathbb{P}^{t},t,\omega}(1,\xi))ds\bigg{]}\leq
C\rho(||\omega-\omega^{\prime}||^{\infty}_{t}).$
By the arbitrariness of $\mathbb{P}^{t}$, it follows that
(4.15) $|V_{t}(\omega)-V_{t}(\omega^{\prime})|\leq
C\rho(||\omega-\omega^{\prime}||^{\infty}_{t}),$
from which we can deduce that $V_{t}\in\mathcal{F}_{t}$.
Parallel to Proposition 4.7 in Soner et al. [24] and Proposition 5.1 in
Possamai and Zhou [18], we give the following dynamic programming principle:
###### Proposition 4.3.
Under (A1)-(A5) and for a given $\xi\in UC_{b}$, we have, for each $0\leq
t_{1}\leq t_{2}\leq 1$ and $\omega\in\Omega$,
(4.16)
$V_{t_{1}}(\omega)=\sup_{\mathbb{P}^{t_{1}}\in\mathcal{P}^{t_{1}}_{H}}y^{\mathbb{P}^{t_{1}},t_{1},\omega}_{t_{1}}(t_{2},V_{t_{2}}).$
Proof: Without loss of generality, we only need to prove the case when
$t_{1}=0$ and $t_{2}=t$, i.e.,
$V_{0}=\sup_{\mathbb{P}\in\mathcal{P}_{H}}y^{\mathbb{P}}_{0}(t,V_{t}).$
Fixing $\mathbb{P}\in\mathcal{P}_{H}$, for each $\omega\in\Omega$ and
$t\in[0,1]$, $\mathbb{P}^{t,\omega}\in\mathcal{P}^{t}_{H}$. By Lemma 4.1 and
from (4.12), we have, for each $t\in[0,1]$ and $\mathbb{P}$-a.s.
$\omega\in\Omega$,
$y^{\mathbb{P}}_{t}(1,\xi)(\omega)=y^{\mathbb{P}^{t,\omega},t,\omega}_{t}(1,\xi)\leq\sup_{\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}}y^{\mathbb{P}^{t},t,\omega}_{t}(1,\xi)=V_{t}(\omega).$
Applying Theorem 2.7 (comparison principle) in Morlais [16], it follows that
$V_{0}\leq\sup_{\mathbb{P}\in\mathcal{P}_{H}}y^{\mathbb{P}}_{0}(t,V_{t})$.
We omit the rest part of the proof, i.e., the proof of the reverse inequality
of (4.16) via the r.c.p.d. technique, since it goes in a exact same way as the
one in Soner et al. [24] and Possamai and Zhou [18]. $\square$
We shall head to the Doob-Meyer type decomposition of $V$ based on some
results for quadratic $g$-supermartingales in Ma and Yao [14]. These results
were obtained under the assumptions for the proof of uniqueness in Kobylanski
[13], since these assumptions ensure that the wellposedness of the
corresponding quadratic BSDEs, so that the $g$-expectation can be well
defined. However, Morlais [16] also provided the wellposedness of quadratic
BSDEs under (A3)-(A5) for the BSDEs of the form (3.2), then the applicability
of these arguments to such type of $\hat{F}$-supermartingales will not alter
under each $\mathbb{P}\in\mathcal{P}_{H}$.
For a fixed $\mathbb{P}\in\mathcal{P}_{H}$, from (4.16) and by Lemma 4.1, we
have for each $0\leq t_{1}\leq t_{2}\leq 1$,
(4.17) $V_{t_{1}}\geq y^{\mathbb{P}}_{t_{1}}(t_{2},V_{t_{2}}),\
\mathbb{P}-a.s..$
Thus, by Definition 5.1 in Ma and Yao [14], $V$ is an
$\hat{F}$-supermartingale under $\mathbb{P}$. Then, for each
$(\omega,t)\in\Omega\times[0,1]$, we define
$V^{+}_{t}(\omega):=\limsup_{\mathbb{Q}\cap(t,1]\ni r\downarrow
t}V_{r}(\omega).$
Applying corollary 5.6 (downcrossing inequality) in Ma and Yao [14] , one can
see that for $\mathbb{P}$-a.s. $\omega\in\Omega$,
$\lim_{\mathbb{Q}\cap(t,1]\ni r\downarrow t}V_{r}$ exists for all $t\in[0,1]$.
Therefore, we have
(4.18) $V^{+}_{t}=\lim_{\mathbb{Q}\cap(t,1]\ni r\downarrow t}V_{r},\ 0\leq
t\leq 1,\ \mathcal{P}_{H}-q.s.,$
which implies that $V^{+}$ has $\mathcal{P}_{H}$-q.s. càdlàg paths.
The following proposition (corresponding to Proposition 4.10 and 4.11 in Soner
et al. [24] and Proposition 5.2 in Possamai and Zhou [18]) demonstrates the
relationship between $V$ and $V^{+}$, from the second part of which, we can
deduce that $V$ is a càdlàg $\hat{F}$-supermartingale under each
$\mathbb{P}\in\mathcal{P}_{H}$, then we apply the Doob-Meyer type
decomposition theorem (cf. Theorem 5.8 in Ma and Yao [14]) directly to $V$.
###### Proposition 4.4.
Assume (A1)-(A5) hold. For a given $\xi\in UC_{b}(\Omega)$ and a fixed
$\mathbb{P}\in\mathcal{P}_{H}$, we define
(4.19)
$V^{\mathbb{P}}_{t}:=\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t,\mathbb{P})}y^{\mathbb{P}^{\prime}}_{t}(1,\xi)\
{and}\
V^{\mathbb{P},+}_{t}:=\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+},\mathbb{P})}y^{\mathbb{P}^{\prime}}_{t}(1,\xi).$
Then, we have
$V_{t}=V^{\mathbb{P}}_{t}\ {and}\ V^{+}_{t}=V^{\mathbb{P},+}_{t},\ 0\leq t\leq
1,\ \mathbb{P}-a.s..$
Moreover,
(4.20) $V_{t}=V^{+}_{t},\ 0\leq t\leq 1,\ \mathcal{P}_{H}-q.s..$
Proof: For the proof of the first equality in (4.19) and that $V^{+}_{t}\geq
V^{\mathbb{P},+}_{t}$, we can proceed the same steps in the proof of
Proposition 4.10 in Soner et al. [24]. Here, we would like only to prove that
for a fixed $\mathbb{P}\in\mathcal{P}_{H}$,
$V^{+}_{t}\leq V^{\mathbb{P},+}_{t},\ 0\leq t\leq 1,\ \mathbb{P}-a.s.,$
since the technique will be a little different.
Fixing a $\mathbb{P}\in\mathcal{P}_{H}$ , a $t\in[0,1]$ and an
$r\in\mathbb{Q}\cap(t,1]$, from the first equality, we have
$V^{\mathbb{P}}_{r}:=\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t,\mathbb{P})}y^{\mathbb{P}^{\prime}}_{r}(1,\xi).$
Following step 3 in the proof of Theorem 4.3 in Soner et al. [25], we could
find a sequence of probability measures such that
$\\{\mathbb{P}_{n}\\}_{n\in\mathbb{N}}\subset\mathcal{P}_{H}(r,\mathbb{P})\subset\mathcal{P}_{H}(t^{+},\mathbb{P})$
and $y^{\mathbb{P}_{n}}_{r}(1,\xi)\uparrow V_{r}$, $\mathbb{P}$-a.s.. We
consider the following BSDE with the parameters $(r,V_{r})$ and the generator
$\hat{F}$:
$y^{\mathbb{P}}_{s}=V_{r}+\int^{r}_{s}\hat{F}_{u}(y^{\mathbb{P}}_{u},z^{\mathbb{P}}_{u})du-\int^{r}_{s}z^{\mathbb{P}}_{u}dB_{u},\
0\leq s\leq r,\ \mathbb{P}-a.s..$
and denote by $(y^{\mathbb{P}}(r,V_{r}),z^{\mathbb{P}}(r,V_{r})))$ its
solution. Then, it follows by Lemma 3.3 (monotone stability) in Morlais [16]
that
$y^{\mathbb{P}}_{t}(r,V_{r})=y^{\mathbb{P}}_{t}(r,\lim_{n\rightarrow+\infty}y^{\mathbb{P}_{n}}_{r}(1,\xi))=\lim_{n\rightarrow+\infty}y^{\mathbb{P}^{n}}_{t}(1,\xi)\leq
V^{\mathbb{P},+}_{t},\ \mathbb{P}-a.s..$
Now, our aim is to find a sequence $\\{r_{m}\\}_{m\in\mathbb{N}}\subset(t,1]$
such that $r_{m}\downarrow t$ and
(4.21)
$\lim_{m\rightarrow+\infty}y^{\mathbb{P}}_{t}(r_{m},V_{r_{m}})=V^{+}_{t},\
\mathbb{P}-a.s..$
Noticing that the generator $\hat{F}$ is no longer Lipschitz in
$|\hat{a}^{1/2}Z|$, in general, the statement above (4.21) is not
straightforward if only the conditions that $V$ is uniformly bounded on
$(t,1]$ and that $V_{r}\rightarrow V^{+}_{t}$, $\mathbb{P}$-a.s. are given. We
define a sequence of BSDEs under $\mathbb{P}$ with the parameters
$(r,e^{\gamma V_{r}})$ and the generator $\hat{G}$ in the form of (4.4) and
denote by $(\mathcal{Y}^{\mathbb{P}}(r,e^{\gamma
V_{r}},\hat{G}),\mathcal{Z}^{\mathbb{P}}(r,e^{\gamma V_{r}},\hat{G}))$ their
solution. From the relationship that
$\mathcal{Y}^{\mathbb{P}}_{t}(r,e^{\gamma V_{r}},\hat{G})=e^{\gamma
y^{\mathbb{P}}_{t}(r,V_{r})},\ 0\leq t\leq r,\ \mathbb{P}-a.s.,$
the statement (4.21) is equivalent to
(4.22) $\lim_{m\rightarrow+\infty}\mathcal{Y}^{\mathbb{P}}_{t}(r_{m},e^{\gamma
V_{r_{m}}},\hat{G})=e^{\gamma V^{+}_{t}},\ \mathbb{P}-a.s..$
Proceeding the same in the proof of Lemma 4.1, for each $n$, we consider the
solutions $\mathcal{Y}^{\mathbb{P}}(r,e^{\gamma V_{r}},\hat{G}^{n})$ of the
BSDE with parameters $(r,e^{\gamma V_{r}})$ and the generator $\hat{G}^{n}$ in
the form of (4.6). Note $M:=C(1+\sup_{\omega\in\Omega}e^{\xi(\omega)})$ that
is the uniform bound for all $\mathcal{Y}^{\mathbb{P}}(r,e^{\gamma
V_{r}},\hat{G}^{n})$, we have
$\mathbb{E}^{\mathbb{P}}[|\mathcal{Y}^{\mathbb{P}}(r,e^{\gamma
V_{r}},\hat{G}^{n})-e^{\gamma V_{r}}|^{2}]\leq C(1+n+\alpha_{M})(r-t),$
where $n$ is the Lipschitz constant of $\hat{G}^{n}$ and $\alpha_{M}>0$
depends simply on $M$. Therefore, for a fixed $n\in\mathbb{N}$, there exists a
sequence $\\{r^{n}_{m}\\}_{m\in\mathbb{N}}\subset(t,1]$ such that
$r^{n}_{m}\downarrow t$ and
$\lim_{m\rightarrow+\infty}|\mathcal{Y}^{\mathbb{P}}_{t}(r^{n}_{m},e^{\gamma
V_{r^{n}_{m}}},\hat{G}^{n})-e^{\gamma V_{r_{m}}}|=0,\ \mathbb{P}-a.s.,$
which implies
$\displaystyle\lim_{m\rightarrow+\infty}$
$\displaystyle|\mathcal{Y}^{\mathbb{P}}_{t}(r^{n}_{m},e^{\gamma
V_{r^{n}_{m}}},\hat{G}^{n})-e^{\gamma V^{+}_{t}}|$
$\displaystyle\leq\lim_{m\rightarrow+\infty}|\mathcal{Y}^{\mathbb{P}}_{t}(r^{n}_{m},e^{\gamma
V_{r^{n}_{m}}},\hat{G}^{n})-e^{\gamma
V_{r^{n}_{m}}}|+\lim_{m\rightarrow+\infty}|e^{\gamma V_{r^{n}_{m}}}-e^{\gamma
V^{+}_{t}}|=0,\ \mathbb{P}-a.s..$
By the diagonal argument, we could find a universal sequence
$\\{\tilde{r}_{m}\\}_{m\in\mathbb{N}}\subset(t,1]$ such that
$\tilde{r}_{m}\downarrow t$ and for each $n\in\mathbb{N}$,
$\lim_{m\rightarrow+\infty}|\mathcal{Y}^{\mathbb{P}}_{t}(\tilde{r}_{m},e^{\gamma
V_{\tilde{r}_{m}}},\hat{G}^{n})-e^{\gamma V^{+}_{t}}|=0,\ \mathbb{P}-a.s..$
For each $n$, $m\in\mathbb{N}$,
$|\mathcal{Y}^{\mathbb{P}}_{t}(\tilde{r}_{m},e^{\gamma
V_{\tilde{r}_{m}}},\hat{G}^{n})|\leq M,$
and Lemma 3.3 (monotone stability) in Morlais [16] shows that for each
$m\in\mathbb{N}$, the following statement holds true $\mathbb{P}$-a.s.:
$\mathcal{Y}^{\mathbb{P},n}_{t}(\tilde{r}_{m},e^{\gamma
V_{\tilde{r}_{m}}},\hat{G}^{n})\downarrow\mathcal{Y}^{\mathbb{P}}_{t}(\tilde{r}_{m},e^{\gamma
V_{\tilde{r}_{m}}}),\ as\ n\rightarrow+\infty.$
Thus,
$\displaystyle\lim_{m\rightarrow+\infty}\mathcal{Y}^{\mathbb{P}}_{t}(\tilde{r}_{m},e^{\gamma
V_{\tilde{r}_{m}}})$
$\displaystyle=\lim_{m\rightarrow+\infty}\lim_{n\rightarrow+\infty}\mathcal{Y}^{\mathbb{P}}_{t}(\tilde{r}_{m},e^{\gamma
V_{\tilde{r}_{m}}},\hat{G}^{n})$
$\displaystyle=\lim_{n\rightarrow+\infty}\lim_{m\rightarrow+\infty}\mathcal{Y}^{\mathbb{P}}_{t}(\tilde{r}_{m},e^{\gamma
V_{\tilde{r}_{m}}},\hat{G}^{n})=\lim_{n\rightarrow+\infty}e^{\gamma
V^{+}_{t}}=e^{\gamma V^{+}_{t}},$
which ends the proof of (4.19). Subsequently, the statement (4.20) could be
proved in a similar way as Proposition 4.11 in Soner et al. [24]. $\square$
###### Theorem 4.5.
Under (A1)-(A5) and for a given $\xi\in UC_{b}(\Omega)$, the 2BSDE (2.1) has a
unique solution $(Y,Z)\in\mathbb{D}^{\infty}_{H}\times\mathbb{H}^{2}_{H}$.
Proof: From (4.17) and (4.20), we know that $V$ is a càdlàg
$\hat{F}$-supermartingale. Applying the Doob-Meyer type decomposition (cf.
Theorem 5.8 in Ma and Yao [14]) under each $\mathbb{P}\in\mathcal{P}_{H}$,
$V_{t}=V_{1}+\int^{1}_{t}\hat{F}_{s}(V_{s},Z^{\mathbb{P}}_{s})ds-\int^{1}_{t}Z^{\mathbb{P}}_{s}dBs+K^{\mathbb{P}}_{1}-K^{\mathbb{P}}_{t},\
0\leq t\leq 1,\ \mathbb{P}-a.s.,$
where $K^{\mathbb{P}}$ is a non-decreasing process null at $0$. As shown in
the proof of Theorem 4.5 in Soner et al. [24] one can find a universal $Z$
such that for each $\mathbb{P}\in\mathcal{P}_{H}$,
$Z_{t}=Z^{\mathbb{P}}_{t},\ 0\leq t\leq 1,\ \mathbb{P}-a.s..$
Defining $Y=V$, from (4.13), $Y\in\mathbb{D}^{\infty}_{H}$. Similar to Lemma
3.1 in Possamai and Zhou [18], we deduce that $Z\in\mathbb{H}^{2}_{H}$. Then,
it suffices to verify that the family of non-decreasing processes
$\\{K^{\mathbb{P}}\\}_{\mathbb{P}\in\mathcal{P}_{H}}$ satisfies the minimum
condition (2.4). For a fixed $\mathbb{P}\in\mathcal{P}_{H}$, $t\in[0,1]$ and
each $\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+},\mathbb{P})$, using the
notations from the proof of Theorem 3.2, we have
$V_{t}-y^{\mathbb{P}^{\prime}}_{t}(1,\xi)=\mathbb{E}^{\mathbb{Q}^{\prime}}_{t}\bigg{[}\int^{1}_{t}M_{s}dK^{\mathbb{P}^{\prime}}_{s}\bigg{]}\geq
e^{-\mu}\mathbb{E}^{\mathbb{Q}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}],$
where
$\frac{d\mathbb{Q}^{\prime}}{d\mathbb{P}^{\prime}}\bigg{|}_{\mathcal{F}_{t}}=H^{\mathbb{P}^{\prime}}_{t}:=\int^{t}_{0}\kappa^{\mathbb{P}^{\prime}}_{s}dW^{\mathbb{P}^{\prime}}_{s},\
0\leq t\leq 1,\ \mathbb{P}^{\prime}-a.s.,$
By Definition 2.5,
$H:=\\{H^{\mathbb{P^{\prime}}}\\}_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+},\mathbb{P})}$
is a $BMO(\mathcal{P}_{H}(t^{+},\mathbb{P}))$-martingale. Applying Lemma 2.7
to the family $\mathcal{E}(H)$, there exists a $p>1$ such that
$\sup_{\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t^{+},\mathbb{P})}\sup_{0\leq
t\leq
1}\bigg{|}\bigg{|}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}\bigg{[}\bigg{(}\frac{\mathcal{E}(H)_{t}}{\mathcal{E}(H)_{1}}\bigg{)}^{\frac{1}{p-1}}\bigg{]}\bigg{|}\bigg{|}_{L^{\infty}(\mathbb{P}^{\prime})}\leq
C_{E},$
then
$\displaystyle\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}]$
$\displaystyle\leq\mathbb{E}^{\mathbb{P}^{\prime}}_{t}\bigg{[}\frac{\mathcal{E}(H^{\mathbb{P}^{\prime}})_{1}}{\mathcal{E}(H^{\mathbb{P}^{\prime}})_{t}}(K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t})\bigg{]}^{\frac{1}{2p-1}}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}\bigg{[}\bigg{(}\frac{\mathcal{E}(H^{\mathbb{P}^{\prime}})_{1}}{\mathcal{E}(H^{\mathbb{P}^{\prime}})_{t}}\bigg{)}^{-\frac{1}{2p-2}}(K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t})\bigg{]}^{\frac{2p-2}{2p-1}}$
$\displaystyle\leq\mathbb{E}^{\mathbb{Q}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}]^{\frac{1}{2p-1}}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}\bigg{[}\bigg{(}\frac{\mathcal{E}(H^{\mathbb{P}^{\prime}})_{1}}{\mathcal{E}(H^{\mathbb{P}^{\prime}})_{t}}\bigg{)}^{-\frac{1}{p-1}}\bigg{]}^{\frac{p-1}{2p-1}}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[(K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t})^{2}]^{\frac{p-1}{2p-1}}$
$\displaystyle\leq
C_{E}^{\frac{p-1}{2p-1}}C_{2}^{\frac{p-1}{2p-1}}e^{\frac{\mu}{2p-1}}(V_{t}-y^{\mathbb{P}^{\prime}}_{t}(1,\xi))^{\frac{1}{2p-1}}.$
From (4.19) and (4.20), we obtain
$\displaystyle
0\leq\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,inf}}}_{\mathbb{P}\in\mathcal{P}_{H}(t^{+},\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}]$
$\displaystyle\leq
C(V_{t}-\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}\in\mathcal{P}_{H}(t^{+},\mathbb{P})}y^{\mathbb{P}^{\prime}}_{t}(1,\xi))^{\frac{1}{2p-1}}$
$\displaystyle=C(V^{+}_{t}-\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}\in\mathcal{P}_{H}(t^{+},\mathbb{P})}y^{\mathbb{P}^{\prime}}_{t}(1,\xi))^{\frac{1}{2p-1}}=0,\
\mathbb{P}-a.s.,$
which is the desired result. $\square$
For each $\xi\in\mathcal{L}^{\infty}_{H}$, one can find a sequence
$\\{\xi^{n}\\}_{n\in\mathbb{N}}\subset UC_{b}(\Omega)$, such that
$||\xi^{n}-\xi||_{L^{\infty}_{H}}\rightarrow 0$. Thanks to a prior estimates,
we have the following main result of the section.
###### Theorem 4.6.
Under (A1)-(A5) and for a given $\xi\in\mathcal{L}^{\infty}_{H}$, the 2BSDE
(2.1) has a unique solution
$(Y,Z)\in\mathbb{D}^{\infty}_{H}\times\mathbb{H}^{2}_{H}$.
###### Remark 4.7.
Recently, working under the Zermelo-Fraenkel set theory with axiom of choice
(ZFC) and the continuum hypothesis (CH), Nutz [17] has developed a way to
define pathwisely a universal process that $\mathbb{P}$-a.s. coincides with
the Itô type stochastic integral of a predicable process $H$ with respect to a
process $X$ that is a semimartingale under each $\mathbb{P}$ (for a continuous
integrator $X$, $H$ needs only to be progressively measurable). We notice that
for each $\mathbb{P}\in\mathcal{P}_{H}$, the canonical process $B$ satisfies
Assumption 2.1 in Nutz [17], then the stochastic integral
$\int^{T}_{t}Z_{s}dB_{s}$ could be defined universally if we add ZFC and CH
into our framework, but the aggregated process is only
$\mathcal{F}^{*}$-adapted, where
$\mathcal{F}^{*}_{t}:=\bigcap_{\mathbb{P}\in\mathcal{P}_{H}}\mathcal{F}^{+}_{t}\vee\mathcal{N}^{\mathbb{P}}.$
On this occasion, $K$ could be a universal process in (2.1).
## 5\. Application to finance
In this section, we re-solve some robust utility maximization problems
introduced by Matoussi et al. [15].
### 5.1. Statement of the problem
The problem under consideration in Matoussi et al. [15] is to maximize in a
robust way the expected utility of the terminal value of a portfolio on a
financial market with some uncertainty on the objective probability and to
choose an optimal trading strategy to attain this optimal goal under some
restrictions.
This problem can be formulated into
(5.1) $\displaystyle
V(x):=\sup_{\pi\in\tilde{\mathcal{A}}}\inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}^{\mathbb{P}}[U(X^{\pi}_{T}-\xi)],$
where $X^{\pi}_{T}$ is the terminal value of the wealth process associated
with a strategy $\pi$ from a given set $\tilde{\mathcal{A}}$ of all admissible
trading strategies, $\xi$ is a liability that matures at time $T$, $U$ denotes
the utility function and $\mathcal{P}$ is a set of all possible probability
measures. Without loss of generality, we always assume that $T=1$ in the
sequel.
In the present paper, we study the problem consists of non-dominated models
i.e., the probability measures from the collection $\mathcal{P}$ could not be
dominated by a finite measure. Consistent with the setting for 2BSDE theory,
we assume that $\mathcal{P}$ is a subset of the class $\bar{\mathcal{P}}_{S}$
(cf. Definition 2.1), in which all the probability measures are mutually
singular.
###### Definition 5.1.
In (5.1), let $\mathcal{P}=\tilde{\mathcal{P}}_{H}$ denote the collection of
all those $\mathbb{P}\in\overline{\mathcal{P}}_{S}$ such that
$\underline{a}\leq\hat{a}_{t}\leq\overline{a}\ {\rm and}\ \hat{a}_{t}\in
D_{F_{t}},\ \lambda\times\mathbb{P}-a.e.,$
for some $\underline{a}$, $\overline{a}\in\mathbb{S}_{d}^{>0}$ and each
$(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$.
Adapted to this setting of $\tilde{\mathcal{P}}_{H}$, we shall change a little
our settings for quadratic 2BSDEs, that is, (A3) and (A5) will be replaced by
the following (A3’) and (A5’):
(A3’) $F$ is continuous in $(y,z)$ and has a quadratic growth, i.e., for each
$(\omega,t,y,z,a)\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}\times
D_{F_{t}}$,
$|F_{t}(\omega,y,z,a)|\leq\alpha(a)+\beta(a)|y|+\frac{\gamma}{2}|a^{1/2}z|^{2},$
where $\gamma$ is a strictly positive constant and $\alpha$, $\beta$ are non-
negative deterministic functions satisfy that for some strictly positive
constants $\overline{\alpha}$ and $\overline{\beta}$,
$\alpha(\hat{a}_{t})\leq\overline{\alpha},\ {\rm and}\
\beta(\hat{a}_{t})\leq\overline{\beta},\ \ 0\leq t\leq 1,\
\tilde{\mathcal{P}}_{H}-q.s..$
(A5’) $F$ is local Lipschitz in $z$, i.e., for each
$(\omega,t,y,z,z^{\prime},a)\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times
D_{F_{t}}$,
$|F_{t}(\omega,y,z,a)-F_{t}(\omega,y,z^{\prime},a)|\leq
C(|a^{1/2}\phi(a)|+|a^{1/2}z|+|a^{1/2}z^{\prime}|)|a^{1/2}(z-z^{\prime})|,$
where $C$ is a strictly positive constant and $\bar{\phi}(a)=a^{1/2}\phi(a)$
satisfies that for some strictly positive constant $\overline{\gamma}$,
$|\bar{\phi}(\hat{a}_{t})|=|\hat{a}^{1/2}_{t}\phi(\hat{a}_{t})|\leq\overline{\gamma},\
0\leq t\leq 1,\ \tilde{\mathcal{P}}_{H}-q.s..$
Repeating all the proof for the wellposedness of quadratic 2BSDEs in the last
section, we can have the following theorem:
###### Theorem 5.2.
Under (A1)- (A2), (A3’), (A4) and (A5’) and for a given
$\xi\in\mathcal{L}^{\infty}_{\tilde{H}}$, the 2BSDE (2.1) has a unique
solution $(Y,Z)\in\mathbb{D}^{\infty}_{H}\times\mathbb{H}^{2}_{H}$.
We will have a detailed discussion for this kind of settings later in
Subsection 5.4.
The financial market consists of one bond with zero interest rate and $d$
stocks. The price process of the stocks is give by the following stochastic
differential equations:
$dS^{i}_{t}=S^{i}_{t}(b^{i}_{t}dt+dB^{i}_{t}),\ 0\leq t\leq 1,\ i=1,\ldots,d,\
\tilde{\mathcal{P}}_{H}-q.s.,$
where $B$ is a $d$-dimensinal canonical process, $b^{i}$ is an
$\mathbb{R}$-valued process that is uniformly bounded by a constant $M>0$ and
is uniformly continuous in $\omega$ under the uniform norm
$||\cdot||^{\infty}_{1}$, $i=1,2,\ldots,d$. By the definition of
$\tilde{\mathcal{P}}_{H}$, for each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$,
$B_{t}=\int^{t}_{0}\hat{a}^{1/2}_{s}dW^{\mathbb{P}}_{s},\ 0\leq t\leq 1,\
\mathbb{P}-a.s.,$
where $\hat{a}^{1/2}$ plays in fact the role of volatility in (1) of Hu et al.
[9]. Thus, the difference of $\hat{a}^{1/2}$ under each
$\mathbb{P}\in\tilde{\mathcal{P}}_{H}$ allows us to model the volatility
uncertainty.
In the following subsections, we study the problem (5.1) for two kinds of
utility functions, the exponential and the power ones.
### 5.2. Robust exponential utility maximization
In this subsection, we consider the robust utility maximization problem (5.1)
with an exponential utility function:
$U(x):=-\exp(-cx),\ c>0,\ x\in\mathbb{R}.$
In this case, we denote $\pi=\\{\pi_{t}\\}_{0\leq t\leq 1}$ the trading
strategy, which is a $d$-dimensional $\mathcal{F}$-progressive measurable
process. The $i$th component $\pi^{i}_{t}$ describes the amount of money
invested in stock $i$ at time $t$, $i=1,\ldots,d$, then, for a given trading
strategy $\pi$, the wealth process $X^{\pi}$ can be written as
$X^{\pi}_{t}=x+\sum^{d}_{i=1}\int^{t}_{0}\frac{\pi^{i}_{s}}{S^{i}_{s}}dS^{i}_{s}=x+\int^{t}_{0}\pi_{s}(dB_{s}+b_{s}ds),\
0\leq t\leq 1,\ \tilde{\mathcal{P}}_{H}-q.s..$
We now give the definition of the admissible trading strategies.
###### Definition 5.3.
Let $\tilde{C}$ be a closed set in $\mathbb{R}^{d}$. The set of admissible
trading strategies $\tilde{\mathcal{A}}$ consists of all $d$-dimensional
progressively measurable processes $\pi=\\{\pi_{t}\\}_{0\leq t\leq 1}$ that
take values in $\tilde{C}$, $\lambda\otimes\mathcal{P}_{H}$-q.s., such that
for each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$,
$\int^{1}_{0}|\hat{a}^{1/2}_{t}\pi_{t}|^{2}dt<+\infty$, $\mathbb{P}$-a.s. and
$\\{\exp(-cX^{\pi}_{\tau})\\}_{\tau\in\mathcal{T}^{1}_{0}}$ is a
$\mathbb{P}$-uniformly integrable family.
Then, the utility maximization problem is equivalent to
(5.2)
$V(x):=\sup_{\pi\in\tilde{\mathcal{A}}}\inf_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}\mathbb{E}^{\mathbb{P}}\bigg{[}-\exp\bigg{(}-c\bigg{(}x+\int^{1}_{0}\pi_{t}(dB_{t}+b_{t}dt)-\xi\bigg{)}\bigg{)}\bigg{]}.$
We can also consider a reduced utility maximization problem under each
$\mathbb{P}\in\tilde{\mathcal{P}}_{H}$, which is introduced by Theorem 7 in Hu
et al. [9] and Theorem 4.1 in Morlais [16]. Following these well known
results, one can find a
${\pi^{\mathbb{P}}}^{*}\in\tilde{\mathcal{A}}^{\mathbb{P}}$ that solves the
reduced utility maximization problem:
(5.3)
$V^{\mathbb{P}}(x):=\sup_{\pi\in\tilde{\mathcal{A}^{\mathbb{P}}}}\mathbb{E}^{\mathbb{P}}\bigg{[}-\exp\bigg{(}-c\bigg{(}x+\int^{1}_{0}\pi_{t}(dB_{t}+b_{t}dt)-\xi\bigg{)}\bigg{)}\bigg{]},$
where $\tilde{\mathcal{A}}^{\mathbb{P}}$ is the collection of all admissible
trading strategies given by Definition 1 in Hu et al. [9] under $\mathbb{P}$
and thus, $\tilde{\mathcal{A}}\subset\tilde{\mathcal{A}}^{\mathbb{P}}$. It is
evident that
$V(x)\leq\inf_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}V^{\mathbb{P}}(x).$
Therefore, the robust utility maximization problem (5.2) is solved if one can
find an optimal strategy $\pi^{*}$ such that
$V(x)=\inf_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}\mathbb{E}^{\mathbb{P}}\bigg{[}-\exp\bigg{(}-c\bigg{(}x+\int^{1}_{0}\pi^{*}_{t}(dB_{t}+b_{t}dt)-\xi\bigg{)}\bigg{)}\bigg{]}=\inf_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}V^{\mathbb{P}}(x).$
In what follows, we give the theorem similar to Theorem 4.1 in Matoussi et al.
[15] but without some additional condition on $\xi$ or on the border of
$\tilde{C}$.
###### Theorem 5.4.
Assume that $\xi\in\mathcal{L}^{\infty}_{\tilde{H}}$. The value function of
the utility maximization problem (5.2) is given by
$V(x)=-\exp(-c(x-Y_{0})),$
where $Y_{0}$ is defined by the unique solution
$(Y,Z)\in\tilde{\mathbb{D}}^{\infty}_{H}\times\tilde{\mathbb{H}}^{2}_{H}$ of
the following 2BSDE:
(5.4)
$Y_{t}=\xi+\int^{1}_{t}\hat{F}_{s}(Z_{s})ds-\int^{1}_{t}Z_{s}dB_{s}+K_{1}-K_{t},\
0\leq t\leq 1,\ \tilde{\mathcal{P}}_{H}-q.s.,$
where for each
$(\omega,t,z,a)\in\Omega\times[0,1]\times\mathbb{R}^{d}\times\mathbb{S}^{>0}_{d}$,
(5.5)
$F_{t}(\omega,z,a):=\frac{c}{2}\sideset{}{{}^{2}}{\operatorname*{\textnormal{dist}}}\bigg{(}a^{1/2}z+\frac{1}{c}a^{-1/2}b_{t}(\omega),a^{1/2}\tilde{C}\bigg{)}-z^{\textnormal{{Tr}}}b_{t}(\omega)-\frac{1}{2c}|a^{-1/2}b_{t}(\omega)|^{2}.$
Moreover, there exists an optimal trading strategy
$\pi^{*}\in\tilde{\mathcal{A}}$ with
(5.6)
$\hat{a}^{1/2}_{t}\pi^{*}_{t}\in\Pi_{\hat{a}^{1/2}_{t}\tilde{C}}\bigg{(}\hat{a}^{1/2}_{t}Z_{t}+\frac{1}{c}\hat{a}^{-1/2}_{t}b_{t}\bigg{)},\
\lambda\otimes\tilde{\mathcal{P}}_{H}-q.s.,$
where $\Pi_{A}(r)$ denotes the collection of the elements in the closed set
$A$ that realize the minimal distance to the point $r$.
###### Remark 5.5.
Some of the assumptions adopted by Theorem 4.1 in Matoussi et al. [15] are
removed: our assumptions for the wellposedness of quadratic 2BSDEs does not
concern the size of $\xi$, so we do not need to assume in addition that the
liability $\xi$ is small enough in norm; on the other hand, we do not have any
requirement on the regularity of the derivatives of the generator $\hat{F}$
and thus, the border of $\tilde{C}$ is no longer assumed to be a
$\mathcal{C}^{2}$ curve. It is evident that these two additional assumptions
have limitations in real financial market: the one on $\xi$ is not practical;
the other one on the border of $\tilde{C}$ is often difficult to verify.
Sketch of the proof: We prove this theorem by following procedures adopted by
Matoussi et al. [15] but with some modifications, and we only give the sketch.
Step 1: In this step, we show that the 2BSDE (5.4) has a unique solution by
verifying that the generator $F$ satisfies (A1)-(A2), (A3’) and (A5’). Then,
Theorem 5.2 states that the 2BSDE (5.4) admits a unique solution
$(Y,Z)\in\tilde{\mathbb{D}}^{\infty}_{H}\times\tilde{\mathbb{H}}^{2}_{H}$.
* •
From that $b$ is uniform bounded and that $\tilde{C}$ is closed, we have, for
each $(\omega,z)\in\Omega\times\mathbb{R}^{d}$,
$D_{F_{t}(\omega,z)}=\mathbb{S}^{>0}_{d}$, which implies that (A1) is
satisfied.
* •
Since $b$ is $\mathcal{F}$-progressive measurable and uniformly continuous in
$\omega$ under the uniform norm, for each
$(z,a)\in\mathbb{R}^{d}\times\mathbb{S}^{>0}_{d}$, $F(z,a)$ is
$\mathcal{F}$-progressive measurable and uniformly continuous in $\omega$.
* •
For each $a\in\mathbb{S}^{+}_{d}$ that satisfies $\underline{a}\leq
a\leq\overline{a}$, there exist a $\overline{K}>0$ that depends only on
$\overline{a}$ and $\tilde{C}$ such that
$\inf\\{|r|:r\in a^{1/2}\tilde{C}\\}\leq\overline{K},$
and another $\underline{K}>0$ that depends only on $\underline{a}$ and $M$,
such that for each $\omega\in\Omega$,
$|a^{-1/2}b_{t}(\omega)|^{2}\leq{\rm tr}(a^{-1})M^{2}=\underline{K}^{2}.$
Then, for each $(t,z)\in[0,1]\times\mathbb{R}^{d}$,
(5.7)
$\sideset{}{{}^{2}}{\operatorname*{\textnormal{dist}}}\bigg{(}a^{1/2}z+\frac{1}{c}a^{-1/2}b_{t},a^{1/2}\tilde{C}\bigg{)}\leq
2|a^{1/2}z|^{2}+2\bigg{(}\frac{1}{c}|a^{-1/2}b_{t}|+\overline{K}\bigg{)}^{2},$
from which we deduce
$|F_{t}(\omega,z,a)|\leq\bigg{(}2c\overline{K}^{2}+\frac{5+c}{2c}\underline{K}^{2}\bigg{)}+\bigg{(}\frac{1}{2}+c\bigg{)}|a^{1/2}z|^{2}.$
That is to say (A3’) is satisfied.
* •
For each $(t,z^{1},z^{2})\in[0,1]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ and
$a\in\mathbb{S}^{>0}_{d}$ that satisfies $\underline{a}\leq
a\leq\overline{a}$,
$\displaystyle F_{t}(\omega,z^{1},a)-F_{t}(\omega,z^{2},a)=$ $\displaystyle\
\frac{c}{2}\bigg{(}\sideset{}{{}^{2}}{\operatorname*{\textnormal{dist}}}\bigg{(}a^{1/2}z^{1}+\frac{1}{c}a^{-1/2}b_{t},a^{1/2}\tilde{C}\bigg{)}$
$\displaystyle-$ $\displaystyle\
\sideset{}{{}^{2}}{\operatorname*{\textnormal{dist}}}\bigg{(}a^{1/2}z^{2}+\frac{1}{c}a^{-1/2}b_{t},a^{1/2}\tilde{C}\bigg{)}\bigg{)}-(z^{1}-z^{2})^{\textnormal{{Tr}}}b_{t}.$
By the Lipschitz property of the distance function with respect to a closed
set, we obtain the following inequality:
$\displaystyle|F_{t}(\omega,z^{1},a)-F_{t}(\omega,z^{2},a)|$
$\displaystyle\leq\frac{c}{2}\bigg{(}\bigg{(}2\overline{K}+\frac{4}{c}\underline{K}\bigg{)}+|a^{1/2}z^{1}|+|a^{1/2}z^{2}|\bigg{)}|a^{1/2}(z^{1}-z^{2})|,$
from which (A5’) is satisfied.
Step 2: We define, for each $\pi\in\tilde{\mathcal{A}}$,
(5.8) $R^{\pi}_{t}=-\exp(-c(X^{\pi}_{t}-Y_{t})),\ 0\leq t\leq 1,$
where $Y$ is the solution to 2BSDE (5.4). Then, we decompose $R^{\pi}$ into a
product of two processes, i.e., $R^{\pi}=M^{\pi}A^{\pi}$, where for each
$\mathbb{P}\in\tilde{\mathcal{P}}_{H}$,
$\displaystyle M^{\pi}_{t}:=e^{-c(x-Y_{0})}\exp\bigg{(}$
$\displaystyle-\int^{t}_{0}c(\pi_{s}-Z_{s})dB_{s}$
$\displaystyle-\frac{1}{2}\int^{t}_{0}c^{2}|\hat{a}^{1/2}_{s}(\pi_{s}-Z_{s})|^{2}ds-
cK^{\mathbb{P}}_{t}\bigg{)},\ 0\leq t\leq 1,\ \mathbb{P}-a.s.,$
and
$A^{\pi}_{t}:=-\exp\bigg{(}-\int^{t}_{0}\bigg{(}c\pi^{\textnormal{{Tr}}}_{s}b_{s}+c\hat{F}_{s}(Z_{s})-\frac{1}{2}c^{2}|\hat{a}^{1/2}_{s}(\pi_{s}-Z_{s})|^{2}\bigg{)}ds\bigg{)},\
0\leq t\leq 1,\ \mathbb{P}-a.s..$
We rewrite $A^{\pi}$ into the following form,
$\displaystyle
A^{\pi}_{t}=-\exp\bigg{(}-\int^{t}_{0}\bigg{(}\frac{c^{2}}{2}\bigg{|}\hat{a}^{1/2}_{s}\pi_{s}-\bigg{(}\hat{a}^{1/2}_{s}Z_{s}$
$\displaystyle+\frac{1}{c}\hat{a}^{-1/2}_{s}b_{s}\bigg{)}\bigg{|}^{2}$
$\displaystyle-
cZ^{\textnormal{{Tr}}}_{s}b_{s}-\frac{1}{2}|\hat{a}^{-1/2}_{s}b_{s}|^{2}-c\hat{F}_{s}(Z_{s})\bigg{)}ds\bigg{)}.$
It is readily to observe that if $\pi=\pi^{*}$ that satisfies (5.6), then
$A^{\pi^{*}}_{t}\equiv-1,\ 0\leq t\leq 1,\ \mathbb{P}-a.s..$
Moreover, Lemma 11 in Hu et al. [9] says that one can define such a $\pi^{*}$
that is $\mathcal{F}$-progressively measurable if $Z$ is
$\mathcal{F}$-progressively measurable.
In the previous section, we have already proved that
$Z\in\tilde{\mathbb{H}}^{2}_{BMO(\tilde{\mathcal{P}}_{H})}$. To show that
$\pi^{*}-Z\in\tilde{\mathbb{H}}^{2}_{BMO(\tilde{\mathcal{P}}_{H})}$, it
suffices to verify that $\pi^{*}$ is also in
$\tilde{\mathbb{H}}^{2}_{BMO(\tilde{\mathcal{P}}_{H})}$. Applying triangle
inequality to $|\hat{a}^{1/2}_{t}\pi^{*}_{t}|$ and recalling (5.7), we have,
for each $t\in[0,1]$,
$\displaystyle|\hat{a}^{1/2}_{t}\pi^{*}_{t}|$
$\displaystyle\leq\bigg{|}\hat{a}^{1/2}_{t}Z_{t}+\frac{1}{c}\hat{a}^{-1/2}_{t}b_{t}\bigg{|}+\bigg{|}\hat{a}^{1/2}_{t}\pi^{*}_{t}-\bigg{(}\hat{a}^{1/2}_{t}Z_{t}+\frac{1}{c}\hat{a}^{-1/2}_{t}b_{t}\bigg{)}\bigg{|}$
(5.9)
$\displaystyle\leq|\hat{a}^{1/2}_{t}Z_{t}|+\frac{1}{c}|\hat{a}^{-1/2}_{t}b_{t}|+\operatorname*{\textnormal{dist}}\bigg{(}a^{1/2}_{t}Z_{t}+\frac{1}{c}\hat{a}^{-1/2}_{t}b_{t},\hat{a}^{1/2}\tilde{C}\bigg{)}$
$\displaystyle\leq
2|\hat{a}^{1/2}_{t}Z_{t}|+\frac{2}{c}\underline{K}+2\overline{K},\
\mathcal{P}_{H}-q.s.,$
which implies that $\pi^{*}$ is an element in
$\tilde{\mathbb{H}}^{2}_{BMO(\tilde{\mathcal{P}}_{H})}$.
As $\pi^{*}\in\tilde{\mathbb{H}}^{2}_{BMO(\tilde{\mathcal{P}}_{H})}$, for each
$\mathbb{P}\in\tilde{\mathcal{P}}_{H}$, $\hat{a}^{1/2}\pi^{*}$ is a
$BMO(\mathbb{P})$-martingale generator. By Remark 8 in Hu et al. [9],
$\\{\exp{-cX^{\pi}_{\tau}}\\}_{\tau\in\mathcal{T}^{1}_{0}}$ is a
$\mathbb{P}$-uniformly integrable family and it is easy to verify that
$\mathbb{E}^{\mathbb{P}}[\int^{1}_{0}|\hat{a}^{1/2}_{t}\pi^{*}_{t}|^{2}dt]<+\infty$.
Thus, $\pi^{*}\in\tilde{\mathcal{A}}$.
Step 3: We now prove that for each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$,
(5.10)
$\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[M^{\pi^{*}}_{1}]=M^{\pi^{*}}_{t},\
0\leq t\leq 1,\ \mathbb{P}-a.s.,$
so that
(5.11)
$\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,inf}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[R^{\pi^{*}}_{1}]=R^{\pi^{*}}_{t},\
0\leq t\leq 1,\ \mathbb{P}-a.s..$
Since $-c(\pi^{*}-Z)$ is a $BMO(\tilde{\mathcal{P}}_{H})$-martingale
generator, under each
$\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})$,
$\mathcal{E}(-c\int^{\cdot}_{0}(\pi^{*}_{t}-Z_{t})dB_{t})$ is an exponential
martingale, and $M^{\pi^{*}}$ can be regard as a product of a martingale and a
positive non-increasing process. Thus, it is easy to show that for each
$\mathbb{P}\in\tilde{\mathcal{P}}_{H}$,
(5.12)
$\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[M^{\pi^{*}}_{1}]\leq
M^{\pi^{*}}_{t},\ 0\leq t\leq 1,\ \mathbb{P}-a.s..$
To get the desired result, it suffices to prove the reverse inequality.
Noticing that $M^{\pi^{*}}_{1}$ and $M^{\pi^{*}}_{t}$ are both positive, we
can consider the ratio $\frac{M^{\pi^{*}}_{1}}{M^{\pi^{*}}_{t}}$. We calculate
for each $t\in[0,1]$ and
$\mathbb{P}^{\prime}\in\mathcal{P}_{H}(t,\mathbb{P})$,
$\displaystyle\frac{M^{\pi^{*}}_{1}}{M^{\pi^{*}}_{t}}=\exp\bigg{(}$
$\displaystyle-\int^{1}_{t}c(\pi^{*}_{s}-Z_{s})dB_{s}$
$\displaystyle-\frac{1}{2}\int^{1}_{t}c^{2}|\hat{a}^{1/2}_{s}(\pi^{*}_{s}-Z_{s})|^{2}ds-c(K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t})\bigg{)},\
\mathbb{P}^{\prime}-a.s..$
Changing measure by
$\frac{d\mathbb{Q}^{\prime}}{d\mathbb{P}^{\prime}}\bigg{|}_{\mathcal{F}_{t}}=\mathcal{E}\bigg{(}-c\int^{\cdot}_{0}(\pi^{*}_{s}-Z_{s})\hat{a}^{1/2}_{s}dW^{\mathbb{P}^{\prime}}_{s}\bigg{)}_{t},$
we have
$\mathbb{E}^{\mathbb{P}^{\prime}}_{t}\bigg{[}\frac{M^{\pi^{*}}_{1}}{M^{\pi^{*}}_{t}}\bigg{]}=\mathbb{E}^{\mathbb{Q}^{\prime}}_{t}[\exp(-c(K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}))],\
\mathbb{P}^{\prime}-a.s..$
By Jensen’s inequality and the convexity of $\exp(-cx)$, we obtain
$\displaystyle\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}\bigg{[}\frac{M^{\pi^{*}}_{1}}{M^{\pi^{*}}_{t}}\bigg{]}$
$\displaystyle=\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{Q}^{\prime}}_{t}[\exp(-c(K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}))]$
$\displaystyle\geq\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\exp(-c\mathbb{E}^{\mathbb{Q}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}])$
$\displaystyle\geq\exp(-c\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,inf}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{Q}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}]).$
Similar to (3.9), we know, for some $p$, $q>1$ that satisfy $1/{p}+{1}/{q}=1$,
$\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,inf}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{Q}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}]\leq
C_{RH}^{1/q}C^{1/2p}_{2p-1}\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,inf}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[K^{\mathbb{P}^{\prime}}_{1}-K^{\mathbb{P}^{\prime}}_{t}]=0,$
where $C_{RH}$ is the constant in Lemma 2.6 and $C_{2p-1}$ is from (3.4). The
inequality above implies that
(5.13)
$\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,sup}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}\bigg{[}\frac{M^{\pi^{*}}_{1}}{M^{\pi^{*}}_{t}}\bigg{]}\geq
1.$
Then, (5.10) comes after (5.12) and (5.13).
Step 4: Under each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$, the canonical
process $B$ is a $\mathbb{P}$-martingale and $\hat{F}_{t}(z)$ is in fact (2.6)
in Morlais [16]. Thus, the value function of the reduced utility maximization
problem is given by
$V^{\mathbb{P}}(x)=-\exp(-c(x-Y^{\mathbb{P}}_{0})),$
where $Y^{\mathbb{P}}_{0}$ is defined by the unique solution
$(Y^{\mathbb{P}},Z^{\mathbb{P}})\in D^{\infty}({\mathbb{P}})\times
H^{2}(\mathbb{P})$ of the following BSDE:
(5.14)
$Y^{\mathbb{P}}_{t}=\xi+\int^{1}_{t}\hat{F}_{s}(Z_{s})ds-\int^{1}_{t}Z_{s}dB_{s},\
0\leq t\leq 1,\ \mathbb{P}-a.s..$
By Theorem 3.2, we have
$Y_{0}=\sup_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}Y^{\mathbb{P}}_{0}.$
From (5.11) and (5.14), it holds true that
$\displaystyle\inf_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}\mathbb{E}^{\mathbb{P}}[-\exp(-c(X^{\pi*}_{t}-\xi))]$
$\displaystyle=\inf_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}\mathbb{E}^{\mathbb{P}}[R^{\pi^{*}}_{1}]=R^{\pi^{*}}_{0}$
$\displaystyle=-\exp(-c(x-Y_{0}))=\inf_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}-\exp(-c(x-Y^{\mathbb{P}}_{0})),$
which implies that $\pi^{*}$ is the optimal strategy. We complete the proof.
$\square$
###### Remark 5.6.
In fact, we adopt a weaker assumption on the admissible strategy than the one
in Theorem 4.1 in Matoussi et al. [15]. We only assume that $\pi$ is an
admissible strategy defined by Hu et al. [9] and Morlais [16] under each
$\mathbb{P}\in\tilde{\mathcal{P}}_{H}$, i.e.,
$\tilde{\mathcal{A}}=\bigcap_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}\tilde{\mathcal{A}}^{\mathbb{P}},$
while Matoussi et al. [15] assumed that
$\pi\in\tilde{\mathbb{H}}^{2}_{BMO(\tilde{\mathcal{P}}_{H})}$. Under this
stronger assumption, all $R^{\pi}$ satisfies the minimal condition (5.11) and
they verified that $\pi^{*}$ is optimal only $A^{\pi}\leq A^{*}\equiv-1$, for
all $\pi$ is admissible. In our present paper, we justify that $\pi^{*}$ is
optimal for this larger set of admissible strategies by a min-max property as
we showed in Step 4, which is regardless of whether the admissible strategy
other than the optimal one is an $BMO(\tilde{\mathcal{P}}_{H})$-martingale
generator. Although we have still proved that
$\pi^{*}\in\tilde{\mathbb{H}}^{2}_{BMO(\tilde{\mathcal{P}}_{H})}$, this result
is more general.
### 5.3. Robust power utility maximization
In this subsection, we redo the problem (5.1) with a power utility function:
$U(x):=\frac{1}{\gamma}x^{\gamma},\ \gamma<1,\ x\in\mathbb{R}.$
In this case, a $d$-dimensional $\mathcal{F}$-progressively measuable process
$\\{\rho_{t}\\}_{0\leq t\leq 1}$ denotes the trading strategy, whose component
$\rho^{i}_{t}$ describes the proportion of money invested in stock $i$ at time
$t$, $0\leq t\leq 1$, $i=1,2,\ldots,d$, then, for a given trading strategy
$\rho$, the wealth process $X^{\rho}$ can be written as
(5.15)
$X^{\rho}_{t}=x+\sum^{d}_{i=1}\int^{t}_{0}\frac{X^{\rho}_{s}\rho^{i}_{s}}{S^{i}_{s}}dS^{i}_{s}=x+\int^{t}_{0}X^{\rho}_{s}\rho_{s}(dB_{s}+b_{s}ds),\
0\leq t\leq 1,\ \tilde{\mathcal{P}}_{H}-q.s.,$
where the initial capital $x$ is positive. One can find an $X^{\rho}$ defined
by
$X^{\rho}_{t}:=x\mathcal{E}\bigg{(}\int^{\cdot}_{0}\rho_{s}(dB_{s}+b_{s}ds)\bigg{)}_{t},\
0\leq t\leq 1,$
which is the unique solution of (5.15) under each
$\mathbb{P}\in\tilde{\mathcal{P}}_{H}$.
###### Definition 5.7.
Let $\tilde{C}$ be a closed set in $\mathbb{R}^{d}$. The set of admissible
trading strategies $\tilde{\mathcal{A}}$ consists of all $d$-dimensional
progressively measurable processes $\rho=\\{\rho_{t}\\}_{0\leq t\leq 1}$ that
take values in $\tilde{C}$, $\lambda\otimes\tilde{\mathcal{P}}_{H}$-q.s. and
for each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$,
$\int^{1}_{0}|\hat{a}^{1/2}_{t}\rho_{t}|^{2}dt<+\infty$, $\mathbb{P}$-a.s..
For each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$, we define a probability
measure $\mathbb{Q}\ll\mathbb{P}$ by
$\frac{d\mathbb{Q}}{d\mathbb{P}}\bigg{|}_{\mathcal{F}_{t}}=\mathcal{E}\bigg{(}-\int^{\cdot}_{0}b^{\textnormal{{Tr}}}_{s}\hat{a}^{1/2}_{s}dW^{\mathbb{P}}_{s}\bigg{)}_{t},$
then, by the definition above, for each $\rho\in\tilde{\mathcal{A}}$,
$X^{\rho}$ is a $\mathbb{Q}$-local martingale bounded from below. Thus,
$X^{\rho}$ is a $\mathbb{Q}$-supermartingale. Since $\mathbb{Q}\ll\mathbb{P}$,
the strategy $\rho$ is free of arbitrage under $\mathbb{P}$.
We suppose that the investor has no liability, i.e., $\xi=0$, then the
maximization problem is equivalent to
(5.16)
$V(x):=\frac{1}{\gamma}x^{\gamma}\sup_{\rho\in\tilde{\mathcal{A}}}\inf_{\mathbb{P}\in\tilde{\mathcal{P}}_{H}}\mathbb{E}^{\mathbb{P}}\bigg{[}\exp\bigg{(}\gamma\int^{1}_{0}\rho_{s}(dB_{s}+b_{s}ds)-\frac{\gamma}{2}\int^{1}_{0}|\hat{a}^{1/2}_{s}\rho_{s}|^{2}ds\bigg{)}\bigg{]}.$
Similar to that in the last subsection, we have the following theorem:
###### Theorem 5.8.
The value function of the utility maximization problem (5.16) is given by
$V(x)=\frac{1}{\gamma}x^{\gamma}\exp(Y_{0}),$
where $Y_{0}$ is defined by the unique solution
$(Y,Z)\in\tilde{\mathbb{D}}^{\infty}_{H}\times\tilde{\mathbb{H}}^{2}_{H}$ of
the following 2BSDE:
(5.17)
$Y_{t}=0+\int^{1}_{t}\hat{F}_{s}(Z_{s})ds-\int^{1}_{t}Z_{s}dB_{s}+K_{1}-K_{t},\
0\leq t\leq 1,\ \tilde{\mathcal{P}}_{H}-q.s.,$
where for each
$(\omega,t,z,a)\in\Omega\times[0,1]\times\mathbb{R}^{d}\times\mathbb{S}^{>0}_{d}$,
(5.18) $\displaystyle
F_{t}(\omega,z,a):=-\frac{\gamma(1-\gamma)}{2}\sideset{}{{}^{2}}{\operatorname*{\textnormal{dist}}}\bigg{(}\frac{1}{1-\gamma}(a^{1/2}z$
$\displaystyle+a^{-1/2}b_{t}(\omega)),a^{1/2}_{t}\tilde{C}\bigg{)}$
$\displaystyle+\frac{\gamma|a^{1/2}z+a^{-1/2}b_{t}(\omega)|^{2}}{2(1-\gamma)}+\frac{1}{2}|a^{1/2}z|^{2}.$
Moreover, there exists an optimal trading strategy
$\rho^{*}\in\tilde{\mathcal{A}}$ with
(5.19)
$\displaystyle\hat{a}^{1/2}_{t}\rho^{*}_{t}(\omega)\in\Pi_{\hat{a}^{1/2}_{t}\tilde{C}}\bigg{(}\frac{1}{1-\gamma}(a^{1/2}_{t}z+a^{-1/2}_{t}b_{t}(\omega))\bigg{)},\
0\leq t\leq 1,\ \tilde{\mathcal{P}}_{H}-q.s.,$
where $\Pi_{A}(r)$ denotes the collection of the elements in the closed set
$A$ that realize the minimal distance to the point $r$.
Sketch of the proof: Following similar procedures in the proof of Theorem 5.4,
we verify that the generator $F$ in 2BSDE (5.17) satisfies (A1)-(A2), (A3’)
and (A5’) and define a family of processes
$\\{R^{\rho}\\}_{\rho\in\tilde{\mathcal{A}}}$ by
$R^{\rho}_{t}:=\frac{1}{\gamma}x^{\gamma}\exp\bigg{(}\gamma\int^{t}_{0}\rho_{s}(dB_{s}+b_{s}ds)-\frac{\gamma}{2}\int^{t}_{0}|\hat{a}^{1/2}_{s}\rho_{s}|^{2}ds+Y_{t}\bigg{)},\
0\leq t\leq 1,$
such that for each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$,
* •
$R^{\rho}_{0}$ is a constant indepent of $\rho$;
* •
$R^{\rho}_{1}=\frac{1}{\gamma}(X^{\rho}_{1})^{\gamma}$, for each
$\rho\in\tilde{\mathcal{A}}$.
Then, we rewrite $R^{\rho}$ under each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$
as follows:
$R^{\rho}_{t}=\frac{1}{\gamma}x^{\gamma}\exp(Y_{0})\mathcal{E}\bigg{(}\int^{\cdot}_{0}(\gamma\rho_{s}+Z_{s})dB_{s}\bigg{)}_{t}e^{-K^{\mathbb{P}}_{t}}\exp\bigg{(}\int^{t}_{0}\nu_{s}ds\bigg{)},\
0\leq t\leq 1,\ \mathbb{P}-a.s.,$
where
$\displaystyle\nu_{t}:=-\frac{\gamma(1-\gamma)}{2}\bigg{|}\hat{a}^{1/2}_{t}\rho_{t}$
$\displaystyle-\frac{1}{1-\gamma}(\hat{a}^{1/2}_{t}Z_{t}+\hat{a}^{-1/2}_{t}b_{t})\bigg{|}^{2}$
$\displaystyle+\frac{\gamma|\hat{a}^{1/2}_{t}Z_{t}+\hat{a}^{-1/2}_{t}b_{t}|^{2}}{2(1-\gamma)}+\frac{1}{2}|\hat{a}^{1/2}_{t}Z_{t}|^{2}-\hat{F}_{t}(Z_{t}).$
Similar to (5.11), we could find an optimal strategy $\rho^{*}$ such that for
each $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$,
$\nu^{\rho^{*}}_{t}\equiv 0,\ 0\leq t\leq 1,\ \mathbb{P}-a.s.$
and thus,
(5.20)
$\sideset{}{{}^{\mathbb{P}}}{\operatorname*{\textnormal{ess\,inf}}}_{\mathbb{P}^{\prime}\in\tilde{\mathcal{P}}_{H}(t,\mathbb{P})}\mathbb{E}^{\mathbb{P}^{\prime}}_{t}[R^{\rho^{*}}_{1}]=R^{\rho^{*}}_{t},\
0\leq t\leq 1,\ \mathbb{P}-a.s..$
The desired result comes after (5.20) and the min-max property. $\square$
###### Remark 5.9.
In Matoussi et al. [15], only the case that $\gamma<0$ was considered.
According to their assumption that $\tilde{C}$ contains $0$, we calculate
$\hat{F}^{0}_{t}=-\frac{\gamma}{2(1-\gamma)}|a^{-1/2}_{t}b_{t}|^{2},$
where $-\frac{\gamma}{2(1-\gamma)}$ is dominated by $\frac{1}{2}$ when
$\gamma<0$ and so that they can give a uniform assumption on $b$, which is
regardless of $\gamma$, to make sure that $F^{0}$ is small enough.
### 5.4. Some remarks on the class of probability measures and the
assumptions
We have already seen that the 2BSDEs (5.4) and (5.17) are discussed under some
new settings, where $\mathcal{P}_{H}$ was changed into
$\tilde{\mathcal{P}}_{H}$; (A3) and (A5) were changed into (A3’) and (A5’). In
what follows, we would like to discuss more about these conditions and class
and probability measures.
Since these weakened conditions shall be related to some given series of
probability measure classes $\\{\mathcal{P}^{t}_{H}\\}_{t\in[0,1]}$ of
probability measures, we first give the following definition:
###### Definition 5.10.
We say a series of probability measure classes
$\\{\mathcal{P}^{t}_{H}\\}_{t\in[0,1]}$ is consistent if the following points
are satisfied (we note $\mathcal{P}^{0}_{H}=\mathcal{P}_{H}$.):
* •
For each $\mathbb{P}\in\mathcal{P}_{H}$, for $\mathbb{P}$-a.e.
$\omega\in\Omega$ and each $\tau\in\mathcal{T}^{1}_{0}$,
$\mathbb{P}^{\tau,\omega}\in\mathcal{P}^{\tau(\omega)}_{H}$;
* •
For each $\tau\in\mathcal{T}^{1}_{0}$, $A\in\mathcal{F}_{\tau}$,
$\mathbb{P}\in\mathcal{P}_{H}$ and
$\hat{\mathbb{P}}^{\tau}\in\mathcal{P}^{\tau}_{H}$,
$\mathbb{P}\otimes^{A}_{\tau}\hat{\mathbb{P}}^{\tau}\in\mathcal{P}_{H}$, where
for each $E\subset\Omega$,
$\mathbb{P}\otimes^{A}_{\tau}\hat{\mathbb{P}}^{\tau}(E):=\mathbb{E}^{\mathbb{P}}[\mathbb{E}^{\hat{\mathbb{P}}^{\tau}}[({\bf
1}_{E})^{\tau,\omega}]{\bf 1}_{A}]+\mathbb{P}(E\cap A^{c}).$
In the 2BSDE framework, the series of classes defined by Definition 2.1 is
consistent, since the first point is guaranteed by Lemma 4.1 in Soner et al.
[24] and the second one is in fact the reduced version ($n=1$) of the
statement (4.19) in Soner et al. [24]. These two properties play an important
role in our proof of the dynamic programming principle (cf. Proposition 4.3).
In what follows, we verify that the series of classes defined by Definition
5.1 is consistent. In this case, $\tilde{\mathcal{P}}^{t}_{H}$ consists of all
those $\mathbb{P}\in\overline{\mathcal{P}}^{t}_{S}$ such that
$\underline{a}\leq\hat{a}^{t}_{s}\leq\overline{a}\ {\rm and}\
\hat{a}^{t}_{s}\in D_{F_{s}},\ \lambda\times\mathbb{P}^{t}-a.e.,$
for some $\underline{a}$, $\overline{a}\in\mathbb{S}_{d}^{>0}$ and each
$(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$. Since
$\tilde{\mathcal{P}}_{H}\subset\overline{\mathcal{P}}_{S}$, by Lemma 4.1 in
Soner et al. [24], for a given $\mathbb{P}\in\tilde{\mathcal{P}}_{H}$ and
$\mathbb{P}$-a.e. $\omega\in\Omega$,
$\mathbb{P}^{\tau,\omega}\in\overline{\mathcal{P}}^{\tau(\omega)}_{S}$ and
$\underline{a}\leq\hat{a}^{\tau(\omega)}_{t}(\tilde{\omega})=\hat{a}^{\tau,\omega}_{t}(\tilde{\omega})=\hat{a}_{t}(\omega\otimes^{\tau}\tilde{\omega})\leq\overline{a},\
\lambda\times\mathbb{P}^{\tau,\omega}-a.e..$
On the other hand, the proof of statement (4.19) in Soner et al. [24] showed
that
$\mathbb{P}\otimes^{A}_{\tau}\hat{\mathbb{P}}^{\tau}\in\overline{\mathcal{P}}_{S}$.
Defining
$\tilde{\mathbb{P}}:=\mathbb{P}\otimes^{A}_{\tau}\hat{\mathbb{P}}^{\tau}$, it
suffices to verify that
(5.21) $\underline{a}\leq\hat{a}_{t}\leq\overline{a},\
\lambda\times\tilde{\mathbb{P}}-a.e..$
We calculate
$\displaystyle\int^{1}_{0}\mathbb{E}^{\tilde{\mathbb{P}}}[{\bf
1}_{\\{\hat{a}_{t}\notin[\underline{a},\overline{a}]\\}}]dt=\int^{1}_{0}(\mathbb{E}^{\mathbb{P}}[\mathbb{E}^{\hat{\mathbb{P}}^{\tau}}[({\bf
1}_{\\{\hat{a}_{t}\notin[\underline{a},\overline{a}]\\}})^{\tau,\omega}]{\bf
1}_{A}]+\mathbb{E}^{\mathbb{P}}[{\bf
1}_{\\{\\{\hat{a}_{t}\notin[\underline{a},\overline{a}]\\}\cap A^{c}\\}}])dt,$
where
$\mathbb{E}^{\hat{\mathbb{P}}^{\tau}}[({\bf
1}_{\\{\hat{a}_{t}\notin[\underline{a},\overline{a}]\\}})^{\tau,\omega}]{\bf
1}_{A}(\omega)=\left\\{\begin{array}[]{l@{\quad,
\quad}l}\mathbb{E}^{\hat{\mathbb{P}}^{\tau}}[{\bf
1}_{\\{\hat{a}^{\tau}_{t}\notin[\underline{a},\overline{a}]\\}}(\tilde{\omega})]=0&\omega\in
A,\ t\geq\tau(\omega);\\\\[3.0pt] {\bf
1}_{\\{\hat{a}_{t}\notin[\underline{a},\overline{a}]\\}}(\omega)&\omega\in A,\
t<\tau(\omega);\\\\[3.0pt] 0&otherwise.\end{array}\right.$
Thus,
$\int^{1}_{0}\mathbb{E}^{\tilde{\mathbb{P}}}[{\bf
1}_{\\{\hat{a}_{t}\notin[\underline{a},\overline{a}]\\}}]dt\leq\int^{1}_{0}\mathbb{E}^{{\mathbb{P}}}[{\bf
1}_{\\{\hat{a}_{t}\notin[\underline{a},\overline{a}]\\}}]dt=0,$
which implies (5.21).
###### Remark 5.11.
Suppose that a consistent series of probability measure classes
$\\{\mathcal{P}^{t}_{H}\\}_{t\in[0,1]}\subset\bar{\mathcal{P}}_{S}$ is given
(not limited to the form defined by Definition 2.1 and 5.1), then (A3) can be
even weakened to the following form, which is similar to (H1) in Morlais [16]
for quadratic BSDEs:
(A3”) $F$ is continuous in $(y,z)$ and has a quadratic growth in $z$, i.e.,
for each
$(\omega,t,y,z,a)\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}\times
D_{F_{t}}$,
(5.22)
$|F_{t}(\omega,y,z,a)|\leq\alpha_{t}(a)+\beta_{t}(a)|y|+\frac{\gamma}{2}|a^{1/2}z|^{2},$
where $\gamma$ is a strictly positive constant and $\alpha$, $\beta$ satisfy
that
* •
For each $a\in\mathbb{S}^{>0}_{d}$, $\alpha(a)$, $\beta(a)$ are nonnegative
$\mathcal{F}$-progressive measurable processes;
* •
For some $\overline{\alpha}$, $\overline{\beta}$ which are strictly positive
constants,
$\int^{1}_{0}\alpha_{t}(\hat{a}_{t})dt\leq\overline{\alpha}\ {\rm and}\
\int^{1}_{0}\beta_{t}(\hat{a}_{t})dt\leq\overline{\beta},\
\mathcal{P}_{H}-q.s.;$
* •
For each $(\omega,t)\in\Omega\times(0,1]$ and
$\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$,
$\int^{1}_{t}\alpha^{t,\omega}_{s}(\hat{a}^{t}_{s})ds\leq\overline{\alpha}\
{\rm and}\
\int^{1}_{t}\beta^{t,\omega}_{s}(\hat{a}^{t}_{s})ds\leq\overline{\beta},\
\mathbb{P}^{t}-a.s.,$
where $\overline{\alpha}$, $\overline{\beta}$ are the same as above.
We recall (4.12) that for each $(\omega,t)\in\Omega\times[0,1]$,
$V_{t}(\omega)$ concerns the solutions of the $(t,\omega)$-shifted quadratic
BSDEs under all $\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$. Therefore, for each
$t\in[0,1]$, at least $\mathcal{P}_{H}$-q.s. $\omega\in\Omega$, $(t,\omega)$
shifted generator should satisfy (H1) in Morlais [16] (or similar conditions
for quadratic BSDEs) under each $\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$ to
ensure the existence of these solutions. We notice that the orignal condition
(A3) is posed pathwisely, that is, it holds for all
$(\omega,t)\in\Omega\times[0,1]$, whereas (A3”) also involves pathwise
settings for each $(\omega,t)\in\Omega\times[0,1]$. Therefore, (4.12) can be
well defined under these two conditions.
A natural question arises: if (2.2) and (5.22) can be written in a
$\mathcal{P}_{H}$-q.s. version; if the third point of (A3”) can be removed?
We consider the first question: suppose that for all
$(t,y,z,a)\in[0,1]\times\mathbb{R}\times\mathbb{R}^{d}$,
$|\hat{F}_{t}(y,z)|\leq\alpha+\beta|y|+\frac{\gamma}{2}|\hat{a}^{1/2}_{t}z|^{2},\
\mathcal{P}_{H}-q.s..$
Fixing an $\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$, we can choose an arbitrage
$\mathbb{P}\in\mathcal{P}_{H}$ and construct a concatenation probability
$\hat{\mathbb{P}}:=\mathbb{P}\otimes^{\Omega}_{t}\mathbb{P}^{t}$. Since
$\\{\mathcal{P}^{t}_{H}\\}_{t\in[0,1]}$ is consistent,
$\hat{\mathbb{P}}\in\mathcal{P}_{H}$,
$\mathbb{P}|_{\mathcal{F}_{t}}=\hat{\mathbb{P}}|_{\mathcal{F}_{t}}$ and for
each $\omega\in\Omega$, $\hat{\mathbb{P}}^{t,\omega}=\mathbb{P}^{t}$. Thus, we
have for $\mathbb{P}$-a.s, $\omega\in\Omega$ and all
$(s,y,z,a)\in[t,1]\times\mathbb{R}\times\mathbb{R}^{d}$,
(5.23) $\displaystyle|\hat{F}^{t,\omega}_{s}(y,z)|$
$\displaystyle=|F_{s}(\omega\otimes_{t}\tilde{\omega},y,z,\hat{a}_{s}(\omega\otimes_{t}\tilde{\omega}))$
$\displaystyle\leq\alpha+\beta|y|+\frac{\gamma}{2}|\hat{a}^{1/2}_{s}(\omega\otimes_{t}\tilde{\omega})z|^{2}=\alpha+\beta|y|+\frac{\gamma}{2}|(\hat{a}^{t}_{s})^{1/2}(\tilde{\omega})z|^{2},\
\mathbb{P}^{t}-a.s..$
Since $\mathbb{P}$ is arbitrage, we can deduce that for $\mathcal{P}_{H}$-q.s.
$\omega\in\Omega$, (5.23) is satisfied. In other words, defining for each
$\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$ a set:
$E^{\mathbb{P}^{t}}:=\\{\omega:\ \hat{F}^{t,\omega}_{s}(y,z)\ satisfies\
(\ref{feret}),\ \mathbb{P}^{t}-a.s.\\},$
we have $\mathbb{P}(E^{\mathbb{P}^{t}})=1$ for all
$\mathbb{P}\in\mathcal{P}_{H}$. At the end of the day, we still have no idea
about
$\mathbb{P}(\cap_{\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}}E^{\mathbb{P}^{t}})$,
since it is a probability of an intersection of non-countable sets. Therefore,
the answer to the first question is negative.
For the same reason, the answer to the second question is negative either,
unless we could find an $\alpha$ such that for each $a\in\mathbb{S}^{>0}_{d}$,
$\alpha^{t,\omega}(a)$ is independent of $\omega$, i.e.,
$\alpha^{t,\omega}_{s}(a)\equiv\alpha^{t}_{s}(a)$. In such case, if we only
assume the second point and define
$E^{\mathbb{P}^{t}}:=\bigg{\\{}\omega:\int^{1}_{t}\alpha^{t,\omega}_{s}(\hat{a}^{t}_{s})ds\leq\overline{\alpha},\
\mathbb{P}^{t}-a.s.\bigg{\\}},$
then $E^{\mathbb{P}^{t}}=\Omega$ for all
${\mathbb{P}^{t}}\in\mathcal{P}^{t}_{H}$, which implies the third point in
(A3”). As we have shown in (A3’), a special case of such $\alpha$ is that for
each $a\in\mathbb{S}^{>0}_{d}$, $\alpha_{\cdot}(a)$ is a deterministic
function in $t$.
###### Remark 5.12.
Corresponding to (v) of Assumption 2.2 in Possamai and Zhou [18], (A5) can be
weakened to the following form:
(A5”) $F$ is local Lipschitz in $z$, i.e., for each
$(\omega,t,y,z,z^{\prime},a)\in\Omega\times[0,1]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times
D_{F_{t}}$,
$|F_{t}(\omega,y,z,a)-F_{t}(\omega,y,z^{\prime},a)|\leq
C(|a^{1/2}\phi_{t}(a)|+|a^{1/2}z|+|a^{1/2}z^{\prime}|)|a^{1/2}(z-z^{\prime})|,$
where $C$ is a strictly positive constant and $\phi$ satisfies that
* •
For each $a\in\mathbb{S}^{>0}_{d}$, $\phi(a)$ is an
$\mathcal{F}$-progressively measurable process;
* •
$\phi(\hat{a})$ is a $BMO(\mathcal{P}_{H})$-martingale generator;
* •
For each $(\omega,t)\in\Omega\times(0,1]$, $\phi^{t,\omega}(\hat{a}^{t})$ is a
$BMO(\mathbb{P}^{t})$-martingale generator under each
$\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$.
Based on the argument in remark 5.11, only having that $\phi$ is a
$BMO(\mathcal{P}_{H})$-martingale generator, we have no idea weather for some
$(\omega,t)\in\Omega\times[0,1]$, $\phi^{t,\omega}$ is a
$BMO(\mathbb{P}^{t})$-martingale generator under all
$\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$, unless for each
$a\in\mathbb{S}^{>0}_{d}$, $\phi(a)$ is independent of $\omega$. Thus, the
third point in (A5”) is necessary. We would like to point out that (v) in
Assumption 2.2 in Possamai and Zhou [18] is ambiguous, which may cause some
slight problems for their setting of $V_{t}(\omega)$ and for the proof of
Lemma 5.1 in that paper.
Taking the 2BSDE (5.4) as an example, we explain these settings ((A3’) and
(A5’) are special cases of (A3”) and (A5”), respectively). We observe that the
generator (5.5) satisfies the quadratic condition (A3”) for
$\alpha_{t}(a):=2c\inf\\{|r|^{2}:r\in a^{1/2}\tilde{C}\\}+\frac{5+c}{2c}{\rm
tr}(a^{-1})M^{2},\ a\in\mathbb{S}^{>0}_{d},$
in which $\alpha(a)$ is a deterministic function. In general,
$\inf\\{|r|^{2}:r\in a^{1/2}\tilde{C}\\}$ and $|a^{-1/2}b|^{2}$ could be
unbounded, so that (A3) is no longer satisfied. If we choose
$\overline{\alpha}=2c\overline{K}^{2}+\frac{5+c}{2c}\underline{K}^{2}$, then
(A3”) is satisfied.
Similarly, the generator (5.5) satisfies no longer and (A5). We define
$\phi_{t}(a):=2\inf\\{|r|:r\in a^{1/2}\tilde{C}\\}+\frac{4}{c}({\rm
tr}(a^{-1}))^{1/2}M,\ a\in\mathbb{S}^{>0}_{d},$
which is bounded by $2\overline{K}+\frac{4}{c}\underline{K}$ when $a$ is
replaced by $\hat{a}$ (or $\hat{a}^{t}$, respectively),
$\tilde{\mathcal{P}}_{H}$ (or $\tilde{\mathcal{P}}^{t}_{H}$,
respectively)-q.s.. By Definition 5.1, we know that a constant process is a
$BMO(\tilde{\mathcal{P}}_{H})$ (or $BMO(\tilde{\mathcal{P}}^{t}_{H})$,
respectively)-martingale generator. Then, (A5”) is satisfied.
The wellposedness of 2BSDEs will not alter under (A3”) and (A5”). First, the
statement (3.1) remains true if we change a little of its expression:
$\mathbb{E}^{\mathbb{P}}_{\tau}\bigg{[}\int^{1}_{\tau}|\hat{a}^{1/2}_{t}Z_{t}|^{2}\bigg{]}\leq\frac{1}{\gamma^{2}}e^{4\gamma||Y||_{\mathbb{D}^{\infty}_{H}}}(1+2\gamma(\overline{\alpha}+\overline{\beta}||Y||_{\mathbb{D}^{\infty}_{H}})),$
which yields that $Z$ is a $BMO(\mathcal{P}_{H})$-martingale generator if
$Y\in\mathbb{D}^{\infty}_{H}$. Lemma 2.6 and 2.7 ensure that the constants
that we need for the proof of the representation theorem and the last step of
the proof to the existence are uniform in $\mathbb{P}$. For the existence
result, we have already explained that $V_{t}(\omega)$ in (4.12) is well
defined and all the properties still hold since (A3”) and (A5”) provide
existence and uniqueness results as well as the estimates of solutions to
quadratic BSDEs with the parameters $(\xi^{t,\omega},\hat{F}^{t,\omega})$
under each $\mathbb{P}^{t}\in\mathcal{P}^{t}_{H}$.
###### Remark 5.13.
If we assume in addition that $0\in\tilde{C}$, then
$\overline{K}=\inf\\{|r|:r\in a^{1/2}\tilde{C}\\}=0$, so that the upper bound
of $\hat{a}$ is not necessary. Both Theorem 5.4 and 5.8 can hold true under a
larger class of probability measures $\hat{\mathcal{P}}_{H}$:
###### Definition 5.14.
We denote by $\hat{\mathcal{P}}_{H}$ the collection which consists of all
those $\mathbb{P}\in\overline{P}_{S}$ such that
$\overline{a}^{\mathbb{P}}\leq\hat{a}_{t}\leq\underline{a}^{\mathbb{P}},\ {\rm
tr}(\hat{a}_{t}^{-1})\leq\alpha_{t},\ {\rm and}\ \hat{a}_{t}\in D_{F_{t}},\
\lambda\times\mathbb{P}-a.e.,$
for some $\overline{a}^{\mathbb{P}}$,
$\underline{a}^{\mathbb{P}}\in\mathbb{S}^{>0}_{d}$, a strictly positive
$\alpha\in L^{1}([0,1])$ and each $(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$.
Correspondingly, we denote by $\hat{\mathcal{P}}^{t}_{H}$ the collection of
all those $\mathbb{P}^{t}\in\overline{P}^{t}_{S}$ such that
$\overline{a}^{\mathbb{P}^{t}}\leq\hat{a}^{t}_{s}\leq\underline{a}^{\mathbb{P}^{t}},\
{\rm tr}((\hat{a}^{t}_{s})^{-1})\leq\alpha_{s}\ {\rm and}\ \hat{a}^{t}_{s}\in
D_{F_{s}},\ \lambda\times\mathbb{P}^{t}-a.e.,$
for some $\overline{a}^{\mathbb{P}^{t}}$,
$\underline{a}^{\mathbb{P}^{t}}\in\mathbb{S}^{>0}_{d}$, the same $\alpha$ as
above and each $(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$.
We can verify that this series $\\{\mathcal{P}^{t}_{H}\\}_{t\in[0,1]}$ defined
by Definition 5.14 is consistent and they ensure that (5.5) and (5.18) satisfy
(A3”) and (A5”), respectively.
However, if we consider the same problems under an even larger class of
probability measures $\breve{\mathcal{P}}_{H}$:
###### Definition 5.15.
We denote by $\breve{\mathcal{P}}_{H}$ the collection which consists of all
$\mathbb{P}\in\overline{P}_{S}$ such that
$\overline{a}^{\mathbb{P}}\leq\hat{a}_{t}\leq\underline{a}^{\mathbb{P}},\
\int^{1}_{0}{\rm tr}(\hat{a}_{t}^{-1})dt\leq\overline{\alpha}\ {\rm and}\
\hat{a}_{t}\in D_{F_{t}},\ \lambda\times\mathbb{P}-a.e.,$
for some $\overline{a}^{\mathbb{P}}$,
$\underline{a}^{\mathbb{P}}\in\mathbb{S}^{>0}_{d}$, some strictly positive
constant $\overline{\alpha}$ and each
$(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$.
then the wellposedness of (5.4) and (5.17) will no longer hold true, since one
is difficult to find a series of class
$\\{\breve{\mathcal{P}}^{t}_{H}\\}_{t\in[0,1]}$ consistent with
$\breve{\mathcal{P}}_{H}$ defined by Definition 5.15. In another word, once
$\breve{\mathcal{P}}^{t}_{H}$ contains all the r.p.c.d.
$\mathbb{P}^{t,\omega}$ of $\mathbb{P}\in\breve{\mathcal{P}}_{H}$, the second
point in Definition 5.10 could not hold true.
Acknowledgement The author express special thanks to Prof. Hu, who provided
both the initial inspiration for the work and useful suggestions.
## References
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* [8] El Karoui, N., Rouge., R., Pricing via utility maximization and entropy, Mathematical Finance, 10-2: 259-276, 2000.
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* [10] Hu, M., Ji, S., Peng, S., Song, Y., Backward stochastic differential equations driven by $G$-Brownian Motion, arXiv:1206.5889v1.
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* [17] Nutz, M., Pathwise construction of stochastic integrals, Electron. Commun. Probab., 17: no. 24, 1–7, 2012.
* [18] Possamai, D., Zhou, C., Second order backward stochastic differential equations with quadratic growth, arXiv:1201.1050v3.
* [19] Pardoux, E., Peng, S., Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 14-1: 55-61, 1990.
* [20] Peng, S., Backward SDE and related $g$-expectation, Backward stochastic differential equations, Pitman Res. Notes Math. Ser., 364, Harlow: Longman, 1997, 141-159.
* [21] Peng, S., Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type, Probab. Theory Relat. Fields, 113-4: 473-499, 1999.
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|
arxiv-papers
| 2013-01-03T13:56:18 |
2024-09-04T02:49:39.886695
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yiqing Lin",
"submitter": "Yiqing Lin",
"url": "https://arxiv.org/abs/1301.0457"
}
|
1301.0693
|
# On the Inviscid Limit of the 3D Navier-Stokes Equations with Generalized
Navier-slip Boundary Conditions111This research is supported in part by NSFC
10971174, and Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research
Grants CUHK-4041/11P, CUHK-4042/08P, a Focus Area Grant from The Chinese
University of Hong Kong, and a grant from Croucher Foundation.
Yuelong Xiao222Institute for Computational and Applied mathematics, Xiangtan
University.$\ {}^{,{\ddagger}}$ and Zhouping Xin333The Institute of
Mathematical Sciences, The Chinese University of Hong Kong.
###### Abstract
In this paper, we investigate the vanishing viscosity limit problem for the
3-dimensional (3D) incompressible Navier-Stokes equations in a general bounded
smooth domain of $R^{3}$ with the generalized Navier-slip boundary conditions
(1.11). Some uniform estimates on rates of convergence in
$C([0,T],L^{2}(\Omega))$ and $C([0,T],H^{1}(\Omega))$ of the solutions to the
corresponding solutions of the idea Euler equations with the standard slip
boundary condition are obtained.
## 1 Introduction
Let $\Omega\subset R^{3}$ be a bounded smooth domain. We consider the Navier-
Stokes equations
$\displaystyle\partial_{t}u^{\varepsilon}-\varepsilon\Delta
u^{\varepsilon}+u^{\varepsilon}\cdot\nabla u^{\varepsilon}+\nabla
p^{\varepsilon}=0,\textrm{ in }\Omega$ (1.1) $\displaystyle\nabla\cdot
u^{\varepsilon}=0,\textrm{ in }\Omega$ (1.2)
and the Euler equations
$\displaystyle\partial_{t}u+u\cdot\nabla u+\nabla p=0,\textrm{ in }\Omega$
(1.4) $\displaystyle\nabla\cdot u=0,\textrm{ in }\Omega$
We are interested in the vanishing viscosity limit problem: do the solutions
of the Navier-Stokes equations (1.1),(1.2) converge to that of the Euler
equations (1.4) with the same initial data as $\varepsilon$ vanishes?.
The vanishing viscosity limit problem for the Navier-Stokes equations is a
classical issue and has been well studied when the domain has no boundaries,
various convergence results have been obtained, see for instances[13, 14, 16,
19, 20, 27, 34].
However, in the presence of a physical boundary, the situations is much more
complicated, and the problem becomes challenging due to the boundary layers.
This is so even in the case that the corresponding solution to the initial
boundary value problem of the Euler equations remains smooth.
Indeed, in presence of a boundary $\partial\Omega$, one of the most important
physical boundary conditions for the Euler equations (1.4) is the following
slip boundary condition, i.e.,
$u\cdot n=0\ {\rm on}\ \partial\Omega$ (1.5)
and the initial boundary value problem of the Euler equations (1.4) with the
slip boundary condition (1.5) has a smooth solution at least locally in time.
Corresponding to the slip boundary condition (1.5) for the Euler equations,
there are different choices of boundary conditions for the Navier-Stokes
equations, one is the mostly-used no-slip boundary condition, i.e.,
$u^{\varepsilon}=0,\ {\rm on}\ \Omega$ (1.6)
This Dirichlet type boundary condition was proposed by Stokes ([33]) assuming
that fluid particles are adherent to the boundary due to the positive
viscosity. Although the well-posedness of the smooth solution to the initial
boundary value problem of the Navier-Stokes equations with the no-slip
boundary condition can be established quiet easily (at least local in time),
the asymptotic convergence of the solution to the corresponding solution of
the Euler equations (1.4) with the boundary condition (1.5) as the viscosity
coefficient $\varepsilon$ tends to zero is one of major open problems except
special cases (see [15, 29, 31, 32]) due to the possible appearance of the
boundary layers. In general, only some sufficient conditions are obtained for
the convergence in $L^{2}(\Omega)$, see [21, 23, 38]. However, there are some
new results in 2D case announced recently in [46].
Another class of boundary conditions for the Navier-Stokes equations is the
Navier-slip boundary conditions, i.e,
$u^{\varepsilon}\cdot n=0,\ [2(S(u^{\varepsilon})n)+\gamma
u^{\varepsilon}]_{\tau}=0,\ {\rm on}\ \partial\Omega$ (1.7)
where $2S(u^{\varepsilon})=(\nabla u^{\varepsilon}+(\nabla
u^{\varepsilon})^{T})$ is the viscous stress tensor, and $\gamma$ is a smooth
given function. Such conditions were proposed by Navier in [30], which allow
the fluid to slip at the boundary, and have important applications for
problems with rough boundaries, prorated boundaries, and interfacial boundary
problems, see [6, 9, 35, 37]. The well-posedness of the Naver-Stokes equations
(1.1), (1.2) with the Navier-slip boundary conditions and related results have
been established, see [9, 43] and the references therein.
In a recent paper [25], the Navier-slip boundary condition (1.7) is written to
the following generalized one
$u^{\varepsilon}\cdot n=0,\
[2(S(u^{\varepsilon})n)+Au^{\varepsilon}]_{\tau}=0,\ {\rm on}\ \partial\Omega$
(1.8)
with $A$ is a smooth symmetric tensor. The well-posedness of the Naver-Stokes
equations (1.1), (1.2) with the generalized Navier-slip boundary condition is
indeed similar to the classical one.
On the other hand, the following vorticity-slip boundary condition
$u^{\varepsilon}\cdot n=0,\ n\times(\omega^{\varepsilon})=\beta
u^{\varepsilon}\ {\rm on}\ \partial\Omega$ (1.9)
with $\beta$ is a smooth function is introduced in [43], where
$\omega=\nabla\times u$ is the vorticity of the fluid.
It is noticed that
$(2(S(v)n)-(\nabla\times v)\times n)_{\tau}=GD(v)_{\tau}\ {\rm on}\
\partial\Omega$ (1.10)
where $GD(v)=-2S(n)v$ (see[43]). Hence the generalized slip condition (1.8) is
equivalent to the following generalized vorticity-slip condition
$u^{\varepsilon}\cdot n=0,\
n\times(\omega^{\varepsilon})=[Bu^{\varepsilon}]_{\tau}\ {\rm on}\
\partial\Omega$ (1.11)
with $B$ a given smooth symmetric tensor on the boundary. The equivalence can
also be seen for instance in [1, 2, 3, 7, 39, 41].
Compared to the case of no-slip boundary condition, the asymptotic behavior of
solutions to the Navier-Stokes equations with the Navier-slip boundary
conditions as $\varepsilon\rightarrow 0$ is relatively easy to establish. Some
strong convergence results in 2D (see [12, 26]) and various weak convergence
results in 3D (see [17, 24, 25]) are also established, where it has been shown
that
$\|u^{\varepsilon}-u\|^{2}+\varepsilon\int_{0}^{t}\|u^{\varepsilon}-u\|_{1}^{2}dt\leq
c\varepsilon^{\frac{3}{2}}$ (1.12)
and
$\|u^{\varepsilon}-u\|_{L^{\infty}([0,T]\times\Omega)}\leq
c\varepsilon^{\frac{3}{8}(1-s)}$ (1.13)
for some $s>0$, see [25] for details.
For the homogeneous case, i.e, $B=0$ in (1.11), better convergence results are
available. The $H^{3}(\Omega)$ convergence and $H^{2}(\Omega)$ estimate on the
rate of convergence are obtained in [41] for flat domains. These results are
improved to $W^{k,p}(\Omega)$ in [2, 3]. However, such a strong convergence
can not be expected for general domains, except the case that the initial
vorticity vanishes on the boundary [42]. Note that the convergence in
$H^{2}(\Omega)$ or, $W^{k,p}(\Omega)$ (for $k\geq 2$), implies that limiting
solution of the Euler equation satisfies the boundary condition (1.11) with
$B=0$.
However, it is shown in [4, 5] that the solution to the Euler equations with
the slip boundary condition (1.5) can not satisfy the extra condition
$n\times\omega=0$ on the boundary in general, and satisfy $n\times\omega=0\
\rm{on}\ \partial\Omega$ only if $\omega\cdot n=0\ \rm{on}\ \partial\Omega$.
This implies that the only possible case for convergence of solutions to the
Navier-Stokes equations in $H^{2}(\Omega)$ is the one obtained in [42], where
the following estimate on the convergence rate
$\|u^{\varepsilon}-u\|^{2}+\varepsilon\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\int_{0}^{t}(\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\|u^{\varepsilon}-u\|_{2}^{2})dt\leq
c\varepsilon^{2}$ (1.14)
is obtained. Furthermore, a $W^{s,p}(\Omega)$ convergence result can also be
obtained in this case, see [10].
For general initial data and domains in the homogeneous case, i.e, $B=0$ in
(1.11), the best estimate on the rate of convergence established so far is in
$W^{1,p}(\Omega)$ in [40], which implies in the particular case $p=2$ that
$\|u^{\varepsilon}-u\|^{2}+\varepsilon\|u^{\varepsilon}-u\|_{1}^{2}\leq
c\varepsilon^{2-s}$ (1.15)
with $s=\frac{1}{2}$, where $u^{\varepsilon}$ and $u$ are solutions to (1.1),
(1.2), (1.9) and (1.4), (1.5) respectively.
However, all the strong convergence results depend essentially the homogeneous
property, i.e, $n\times\omega=0$ on $\partial\Omega$. Indeed, for general
$B$(not identically equal to zero on $\partial\Omega$), there is no any
estimate on the rate of convergence in $H^{k}(\Omega)$ (or $W^{k,p}(\Omega)$)
for $k\geq 1$, as far as the authors are aware. It should be noted that the
co-normal uniform estimates have been established in [28] for general domains
and general Navier-slip boundary conditions, which grantee the convergence of
solutions to the Navier-Stokes equations to that of the Euler equations in
$W^{1,\infty}(\Omega)$. Yet, some additional efforts are still needed to
obtain an estimate on the rate of convergence as in (1.14).
In a recent paper [44], we proposed the following slip boundary condition
$u^{\varepsilon}\cdot n=0,\ \omega^{\varepsilon}\cdot n=0,n\times(\Delta
u^{\varepsilon})=0\ {\rm on}\ \partial\Omega$ (1.16)
for the Navier-Stokes equations (1.1), (1.2) and obtained the following
estimate on the rate of convergence
$\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\|u^{\varepsilon}-u\|_{2}^{2}+\varepsilon\int_{0}^{t}(\|u^{\varepsilon}-u\|_{2}^{2}+\varepsilon\|u^{\varepsilon}-u\|_{3}^{2})dt\leq
c\varepsilon^{2-s}$ (1.17)
for any $s>0$.
In this paper, we investigate the vanishing viscosity problem for the Navier-
Stokes equations (1.1), (1.2) on a general bounded smooth domain
$\Omega\subset R^{3}$ with the generalized vorticity-slip boundary condition
(1.11). Our main result is the following estimate on the rate of convergence
for the solutions:
Theorem 1.1. Let $u(t)$ be the smooth solution of the boundary value problem
of Euler equation (1.4), (1.4), (1.5) on $[0,T]$ with $u(0)=u_{0}\in
H^{3}(\Omega)$ which satisfies the corresponding assumptions on the initial
data in Theorem 1 of [28] (see also Lemma 2.1 in next section), and
$u^{\varepsilon}(t)$ be the solution of the boundary value problem of the
Navier-Stokes equation (1.1), (1.2), (1.11) with the same initial data
$u_{0}$. Then, there is $T_{0}>0$ such that
$\|u^{\varepsilon}-u\|^{2}+\varepsilon\int_{0}^{T_{1}}\|u^{\varepsilon}-u\|_{1}^{2}dt\leq
c\varepsilon^{2-s},\ {\rm on}\ [0,T_{0}]$ (1.18)
$\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\int_{0}^{T_{1}}\|u^{\varepsilon}-u\|_{2}^{2}dt\leq
c\varepsilon^{1-s},\ {\rm on}\ [0,T_{0}]$ (1.19)
for any $s>0$ and $\varepsilon$ small enough. Consequently,
$\|u^{\varepsilon}-u\|_{1,p}^{p}\leq c_{p}\varepsilon^{1-s},\ {\rm on}\
[0,T_{0}]$ (1.20)
for $2\leq p<\infty$, any $s>0$ and $\varepsilon$ small enough, and
$\|u^{\varepsilon}-u\|_{L^{\infty}([0,T]\times\Omega)}\leq
c\varepsilon^{\frac{2}{5}(1-s^{\prime})}$ (1.21)
for any $s^{\prime}>0$.
The estimate (1.19) yields the desired estimates on the rate of convergence of
the solutions of the Navier-Stokes equations to that of the Euler equations in
$C([0,T],H^{1}(\Omega))$ norm for general domains with general Navier-slip
boundary conditions. It should be noted that the estimates (1.18) and (1.21)
improve (1.12) and (1.13), and (1.19), (1.20) are even better than (1.15)
stated in [40] for the special case $B=0$. Indeed, The estimate (1.19) is
optimal in the sense that $s$ can not be taken to be $0$ due to boundary
layers in general, see Remark 3.1. below for details.
These estimates are motivated by our recent work [44], where the first order
derivative estimates for the solutions are obtained for the Navier-Stokes
equations with a new vorticity boundary condition (1.16), and the work of
Masmoudi and Rousset in [28], where uniform estimates on
$\|u^{\varepsilon}\|_{W^{1.\infty}}$ of the solutions of the Navier-Stokes
equation with the general Navier-slip boundary condition are obtained.
Our approach is an elementary energy estimate for the difference of the
solutions between the Navier-Stokes equations and the Euler equations. Some
suitable integrating by part formulas in terms of the vorticity are
successfully used to get the optimal rate of convergence. In comparison to the
asymptotic analysis method associate to the boundary layers (see [18, 24, 25,
40]), we should not need any correctors near the boundary. Meanwhile, the
uniform regularity obtained in [28] plays an essential role in our analysis.
The rest of the paper is organized as follows: In the next section, we give a
preliminary on the well-posedness of the initial boundary value problem of the
Navier-Stokes equations (1.1),(1.2) and (1.11), and the local uniform
regularity of the solutions. Then, the estimates on the asymptotic convergence
(1.18), (1.19), (1.20) and (1.21) are established in section 3.
## 2 Preliminaries
Let $\Omega\subset R^{3}$ be a general bounded smooth domain. We begin by
considering the following Stokes problem:
$\displaystyle\alpha u-\Delta u+\nabla q=f\textrm{ in }\Omega$ (2.1)
$\displaystyle\nabla\cdot u=0\textrm{ in }\Omega$ (2.2) $\displaystyle u\cdot
n=0,n\times(\nabla\times u)=[Bu]_{\tau}\ {\rm on}\ \partial\Omega$ (2.3)
where $B$ is smooth symmetric tensor. Set
$H=\\{u\in L^{2}(\Omega);\nabla\cdot u=0,{\rm in}\ \Omega;\ u\cdot n=0;\ {\rm
on}\ \partial\Omega\\}$ $V=H^{1}(\Omega)\cap H$ $W=\\{u\in H^{2}(\Omega);\
n\times(\nabla\times u)=[Bu]_{\tau}\ {\rm on},\partial\Omega\\}$
It is well known that for any $v\in V$, one has
$\displaystyle\|v\|_{1}\leq c\|\nabla\times v\|$ (2.4)
Note that
$[Bu]_{\tau}\cdot\phi=Bu\cdot\phi$
for any $\phi$ satisfying $\phi\cdot n=0$ on the boundary. We associate the
Stokes problem (2.1)-(2.3) with the following bilinear form
$\displaystyle
a_{\alpha}(u,\phi)=\alpha(u,\phi)+\int_{\partial\Omega}Bu\cdot\phi+\int_{\Omega}(\nabla\times
u)\cdot(\nabla\times\phi)$ (2.5)
Note that
$\displaystyle|\int_{\partial\Omega}Bu\cdot u|\leq c\|u\|\|u\|_{1}$ (2.6)
for all $u\in V$ and (2.4). It follows that $a_{\alpha}(u,\phi)$ is a positive
definite symmetric bilinear form if $\alpha$ large enough, and is closed with
the domain $\mathcal{D}(a_{\alpha})=V$. Similar to the discussions in [43],
one can define the self-adjoint operator associated with $a_{\alpha}$ as
$A=\alpha I-P\Delta$
with the domain $\mathcal{D}(A)=V\cap W$, which implies
$\displaystyle\|v\|_{2}\leq c\|v\|+\|P\Delta v\|,\ \forall v\in\mathcal{D}(A)$
(2.7)
Now, we turn to the boundary value problem of the Navier-Stokes equations
(1.1), (1.2) on $\Omega\subset R^{3}$ with the generalized vorticity-slip
boundary condition (1.11).
By using the Galerkin method based on the orthogonal eigenvectors of $A$,
noting the energy equation
$\frac{1}{2}\frac{d}{dt}\|u^{\varepsilon}\|^{2}+\varepsilon\|\nabla\times
u^{\varepsilon}\|^{2}+\varepsilon\int_{\partial\Omega}Bu^{\varepsilon}\cdot
u^{\varepsilon}=0$ (2.8)
valid for approximate solutions, and (2.4), (2.6), one can obtain the global
existence of weak solutions to the the boundary value problem of Navier-Stokes
equations (1.1), (1.2), (1.11) (see the corresponding definition in, for
instance, [43]).
Noting the energy equation
$\frac{1}{2}\frac{d}{dt}(\|\nabla\times
u^{\varepsilon}\|^{2}+\int_{\partial\Omega}Bu^{\varepsilon}\cdot
u^{\varepsilon})+\varepsilon\|P\Delta
u^{\varepsilon}\|^{2}+(u^{\varepsilon}\cdot\nabla u^{\varepsilon},P\Delta
u^{\varepsilon})=0$ (2.9)
for approximate solutions, (2.4), and (2.6)-(2.8), one can show that if
$u^{\varepsilon}_{0}\in V$, then there is a maximum time interval
$[0,T^{\varepsilon})$ such that the weak solutions are the unique strong one
on $[0,T^{\varepsilon})$ (see the corresponding definition in, for instance,
[43]).
It follows from Lemma 3.10 in [43] that
$(2(S(u)n)-\omega\times n)_{\tau}=GD(u)_{\tau}$ (2.10)
with $GD(u)=-2S(n)u$.
Then, the boundary conditions (1.7), (1.8) and (1.9) can be written to (1.11),
and (1.8) is equivalent to (1.11). Hence, one has the following uniform
regularity result for the solutions to the Navier-Stokes equations:
Lemma 2.1.(Masmoudi and Rousset [28]) Let $m>6$ be an integer and $\Omega$ be
a $C^{m+2}$ domain. Consider $u_{0}\in E^{m}\cap H$ such that $\nabla u_{0}\in
W^{1,\infty}_{co}$. Then there is $T_{m}>0$ such that for all sufficient small
$\varepsilon$, there is a unique solution $u^{\varepsilon}\in
C([0,T_{m}],E^{m})$ to the Navier-Stokes problem (1.1), (1.2), (1.11) with
$u^{\varepsilon}(0)=u_{0}$. Moreover, there is a constant $C$ such that
$\|u^{\varepsilon}\|_{H_{co}^{m}(\Omega)}+\|\nabla
u^{\varepsilon}\|_{H_{co}^{m-1}(\Omega)}+\|\nabla
u^{\varepsilon}\|_{W_{co}^{1,\infty}(\Omega)}+\varepsilon\int_{0}^{t}\|\nabla^{2}u^{\varepsilon}\|_{H_{co}^{m-1}(\Omega)}^{2}dt\leq
C$ (2.11)
on $[0,T_{m}]$.
Where $H_{co}^{m}(\Omega)$ and $W_{co}^{1,\infty}(\Omega)$ are co-norm vector-
spaces, $\|\cdot\|_{H_{co}^{m}(\Omega)}$ and
$\|\cdot\|_{W_{co}^{1,\infty}(\Omega)}$ denote the corresponding norms, and
$E^{m}=\\{u\in H_{co}^{m}(\Omega)|\nabla u\in H_{co}^{m-1}(\Omega)\\}$
Since the notations are rather complicate to be expressed, we omit it here and
refer to Masmoudi and Rousset [28] for the details. This uniform regularity
implies in particular the following uniform bound
$\|u^{\varepsilon}\|_{W^{1,\infty}}\leq C,\ {\rm on}\ [0,T_{m}],$
which plays an essential role in our estimates.
## 3 Convergence of the solutions
We now turn to the purpose of this paper to establish the convergence with a
rate for the solutions $u^{\varepsilon}$ to $u$. We start with the basic
$L^{2}-$estimate.
Theorem 3.1. Let $u_{0}\in H^{3}(\Omega)$ satisfy the assumptions stated in
Theorem 1.1. and $u(t)$ be the solution to the Euler equations (1.4), (1.4),
(1.5) on $[0,T]$ with $u(0)=u_{0}$, and $u(t)=u^{\varepsilon}(t)$ be the
solution to the Navier-Stokes problem (1.1), (1.2), (1.11) with
$u^{\varepsilon}(0)=u_{0}$. Then
$\|u^{\varepsilon}-u\|^{2}+\varepsilon\int_{0}^{T_{0}}\|u^{\varepsilon}-u\|_{1}^{2}dt\leq
c\varepsilon^{2-s}\ {\rm on}\ [0,T_{0}]$ (3.1)
for any $s>0$ and $\varepsilon$ small enough, where $T_{0}=\min\\{T,T_{m}\\}$.
Consequently,
$\|u^{\varepsilon}-u\|_{L^{\infty}([0,T]\times\Omega)}\leq
c\varepsilon^{\frac{2}{5}(1-s^{\prime})}$ (3.2)
for any $s^{\prime}>0$.
Proof: Note that $u^{\varepsilon}-u$ satisfies
$\displaystyle\partial_{t}(u^{\varepsilon}-u)-\varepsilon\Delta(u^{\varepsilon}-u)+\Phi+\nabla(p^{\varepsilon}-p)=\varepsilon\Delta
u,\textrm{ in }\Omega$ (3.3) $\displaystyle\nabla\cdot
u^{\varepsilon}=0,\textrm{ in }\Omega$ (3.4)
$\displaystyle(u^{\varepsilon}-u)\cdot n=0,\
n\times(\omega^{\varepsilon}-\omega)=[B(u^{\varepsilon}-u)+Bu]_{\tau}-n\times\omega\
{\rm on}\ \partial\Omega$ (3.5)
where $\omega=\nabla\times u$, $\omega^{\varepsilon}=\nabla\times
u^{\varepsilon}$,
$\Phi=u\cdot\nabla(u^{\varepsilon}-u)+(u^{\varepsilon}-u)\cdot\nabla
u+(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)$
Then, the following identity holds
$\frac{1}{2}\frac{d}{dt}\|u^{\varepsilon}-u\|^{2}+\varepsilon\|\nabla\times(u^{\varepsilon}-u)\|^{2}+\mathcal{B}_{0}+(\Phi,u^{\varepsilon}-u)=\varepsilon(\Delta
u,u^{\varepsilon}-u)$ (3.6)
where
$\mathcal{B}_{0}=\varepsilon\int_{\partial\Omega}n\times(\omega^{\varepsilon}-\omega)(u^{\varepsilon}-u)=\varepsilon\int_{\partial\Omega}(B(u^{\varepsilon}-u)+Bu-n\times\omega)(u^{\varepsilon}-u)$
Note that
$\int_{\partial\Omega}n\times(\omega^{\varepsilon}-\omega)(u^{\varepsilon}-u)=\int_{\partial\Omega}(B(u^{\varepsilon}-u)+Bu-n\times\omega)(u^{\varepsilon}-u)$
$\leq c\int_{\partial\Omega}(|(u^{\varepsilon}-u)|^{2}+|(u^{\varepsilon}-u)|)$
$\leq
c(\|u^{\varepsilon}-u\|\|\omega^{\varepsilon}-\omega\|+|u^{\varepsilon}-u|_{L(\partial\Omega)})$
It follows from the trace theorem that
$|u^{\varepsilon}-u|_{L(\partial\Omega)}\leq|u^{\varepsilon}-u|_{L^{q}(\partial\Omega)}\leq
c\|u^{\varepsilon}-u\|_{H^{s}(\Omega)}$ (3.7)
for any $q>1$ and then any $s>0$.
By interpolation (see [36]), we have
$\|u^{\varepsilon}-u\|_{H^{s}(\Omega)}\leq
c\|u^{\varepsilon}-u\|^{(1-s)}\|u^{\varepsilon}-u\|_{1}^{s}\leq
c\|u^{\varepsilon}-u\|^{(1-s)}\|\omega^{\varepsilon}-\omega\|^{s}$ (3.8)
Note that
$\varepsilon\|u^{\varepsilon}-u\|^{(1-s)}\|\omega^{\varepsilon}-\omega\|^{s}\leq\delta\varepsilon\|\omega^{\varepsilon}-\omega\|^{2}+c_{\delta}(\|u^{\varepsilon}-u\|^{2}+\varepsilon^{2-s})$
(3.9)
for any $s\in(0,1)$ and
$\varepsilon\|u^{\varepsilon}-u\|\|\omega^{\varepsilon}-\omega\|\leq\varepsilon^{2}\|\omega^{\varepsilon}-\omega\|^{2}+\|u^{\varepsilon}-u\|^{2}$
Then, it holds that
$\mathcal{B}_{0}\leq
2\delta\varepsilon\|\omega^{\varepsilon}-\omega\|^{2}+c_{\delta}\|u^{\varepsilon}-u\|^{2}+\varepsilon^{2-s}$
for any $s\in(0,1)$ and $\varepsilon$ small enough.
Note also that
$(\Phi,u^{\varepsilon}-u)=((u^{\varepsilon}-u)\cdot\nabla
u,u^{\varepsilon}-u)\leq c\|u^{\varepsilon}-u\|^{2},$
and
$\varepsilon(\Delta
u,u^{\varepsilon}-u)\leq\|u^{\varepsilon}-u\|^{2}+c\varepsilon^{2}$
It follows that
$\frac{d}{dt}\|u^{\varepsilon}-u\|^{2}+\varepsilon\|\nabla\times(u^{\varepsilon}-u)\|^{2}\leq
c(\|u^{\varepsilon}-u\|^{2}+\varepsilon^{2-s})$ (3.10)
for any $s\in(0,1)$, and then for any $s>0$ and $\varepsilon$ small enough.
Note that $u^{\varepsilon}(0)-u(0)=0$. Then, (3.1) follows from (3.10) and the
Gronwall lemma.
Consequently, by using the Gagliardo-Nirenberg interpolation inequality, one
has
$\|u^{\varepsilon}-u\|_{L^{\infty}(\Omega)}\leq
c\|u^{\varepsilon}-u\|^{\frac{2}{5}}\|u^{\varepsilon}-u\|_{W^{1,\infty}(\Omega)}^{\frac{3}{5}}\leq
c\varepsilon^{\frac{2}{5}(1-s^{\prime})}$ (3.11)
for any $s^{\prime}>0$. The theorem is proved.
Next, we prove the major estimate in this paper.
Theorem 3.2. Let $u_{0}\in H^{3}(\Omega)$ satisfy the assumptions stated in
Theorem 1.1., and $u(t)$ be the solution to the Euler equations (1.4), (1.4),
(1.5) on $[0,T]$ with $u(0)=u_{0}$, and $u(t)=u^{\varepsilon}(t)$ be the
solution to the Navier-Stokes problem (1.1), (1.2), (1.11) with
$u^{\varepsilon}(0)=u_{0}$. Then
$\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\int_{0}^{T}\|u^{\varepsilon}-u\|_{2}^{2}dt\leq
c\varepsilon^{1-s}\ {\rm on}\ [0,T_{0}]$ (3.12)
for any $s>0$ and $\varepsilon$ small enough, where $T_{0}=\min\\{T,T_{m}\\}$.
Proof: Let $s$ be the same as in Theorem 3.1.. Note the smoothness of the
solutions and
$\partial_{t}(u^{\varepsilon}-u)\cdot n=0\ \rm{on}\ \partial\Omega$
It follows from (3.3),(3.4) and (3.5) that
$\frac{1}{2}\frac{d}{dt}\|(\omega^{\varepsilon}-\omega)\|^{2}+\varepsilon\|P\Delta(u^{\varepsilon}-u)\|^{2}-(\Phi,P\Delta(u^{\varepsilon}-u))$
$=\int_{\partial\Omega}\partial_{t}(u^{\varepsilon}-u)\cdot(n\times(\omega^{\varepsilon}-\omega))-\varepsilon(\Delta
u,P\Delta(u^{\varepsilon}-u))$
where $P$ is the Lerray projection,
$\Phi=u\cdot\nabla(u^{\varepsilon}-u)+(u^{\varepsilon}-u)\cdot\nabla
u+(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)$
and that
$\int_{\partial\Omega}\partial_{t}(u^{\varepsilon}-u)\cdot(n\times(\omega^{\varepsilon}-\omega))=\int_{\partial\Omega}\partial_{t}(u^{\varepsilon}-u)\cdot(B(u^{\varepsilon}-u)+Bu-n\times\omega)$
$=\frac{1}{2}\frac{d}{dt}(\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot(u^{\varepsilon}-u)+2\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot(Bu-n\times\omega))-\mathcal{B}_{1}$
where
$\mathcal{B}_{1}=\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot\partial_{t}(Bu-n\times\omega)$
It follows from Theorem 3.1. and (3.9)
$|\mathcal{B}_{1}|\leq
c\int_{\partial\Omega}|(u^{\varepsilon}-u)|\leq\delta\|(\omega^{\varepsilon}-\omega)\|^{2}+c\varepsilon^{1-s}$
Note also that
$\varepsilon|(\Delta
u,P\Delta(u^{\varepsilon}-u))|\leq\frac{\varepsilon}{2}\|P\Delta(u^{\varepsilon}-u)\|^{2}+c\varepsilon$
and that
$-(\Phi,P\Delta(u^{\varepsilon}-u))=(P\Phi,-\Delta(u^{\varepsilon}-u))$
$=(\nabla\times\Phi,(\omega^{\varepsilon}-\omega))+\int_{\partial\Omega}n\times(\omega^{\varepsilon}-\omega)\cdot
P\Phi$
$=(\nabla\times\Phi,(\omega^{\varepsilon}-\omega))+\int_{\partial\Omega}(B(u^{\varepsilon}-u)+Bu-n\times\omega)\cdot
P\Phi$
Then, we get
$\frac{1}{2}\frac{d}{dt}E+\frac{\varepsilon}{2}\|P\Delta(u^{\varepsilon}-u)\|^{2}\leq
N+BN+BNL+c(\|(\omega^{\varepsilon}-\omega)\|^{2}+\varepsilon^{1-s})$ (3.13)
for $\varepsilon$ small enough, where
$E=\|(\omega^{\varepsilon}-\omega)\|^{2}-(\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot(u^{\varepsilon}-u)+2\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot(Bu-n\times\omega))$
$N=|(\nabla\times\Phi,(\omega^{\varepsilon}-\omega))|$
$BN=|\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot P\Phi|$
$BNL=|\int_{\partial\Omega}(Bu-n\times\omega)\cdot P\Phi|$
The term $N$ can be estimated easily by using the Sobolev inequalities and the
known uniform bounds for $\|u^{\varepsilon}\|_{\infty}$ and $\|\nabla
u^{\varepsilon}\|_{\infty}$. The term $BN$ can be estimated after integrating
by part properly. While, the estimate for the leading order term $BNL$ is
rather complicated due to the possible appearance of boundary layers. We now
carry out these estimates.
Estimates on $N$:
The term $N$ is estimated as follows:
Note that
$\nabla\times\Phi=\nabla\times(\omega\times(u^{\varepsilon}-u)+(\omega^{\varepsilon}-\omega)\times
u+(\omega^{\varepsilon}-\omega)\times(u^{\varepsilon}-u)$
$=[\omega,u^{\varepsilon}-u]+[\omega^{\varepsilon}-\omega,u]+[\omega^{\varepsilon}-\omega,\omega^{\varepsilon}-\omega]$
where
$[\varphi,\psi]=\psi\cdot\nabla\varphi-\varphi\cdot\nabla\psi$
And
$|((u^{\varepsilon}-u)\cdot\nabla\omega,\omega^{\varepsilon}-\omega)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
$|(\omega\cdot\nabla(u^{\varepsilon}-u),\omega^{\varepsilon}-\omega)|\leq
c\|u^{\varepsilon}-u\|_{1}\|\omega^{\varepsilon}-\omega\|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
It follows that
$|([\omega,u^{\varepsilon}-u],\omega^{\varepsilon}-\omega)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
On the other hand,
$(u\cdot\nabla(\omega^{\varepsilon}-\omega),\omega^{\varepsilon}-\omega)=0$
$((\omega^{\varepsilon}-\omega)\nabla u,\omega^{\varepsilon}-\omega)\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
It follows that
$|([\omega^{\varepsilon}-\omega,u],\omega^{\varepsilon}-\omega)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
Note that
$((u^{\varepsilon}-u)\cdot\nabla(\omega^{\varepsilon}-\omega),\omega^{\varepsilon}-\omega)=0$
$((\omega^{\varepsilon}-\omega)\cdot\nabla(u^{\varepsilon}-u),\omega^{\varepsilon}-\omega)$
$\leq\|\nabla(u^{\varepsilon}-u)\|_{\infty}\|\omega^{\varepsilon}-\omega\|^{2}\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
(here and below we will use the uniform regularity that $\|\nabla
u^{\varepsilon}\|_{\infty}\leq c$). It follows that
$|([\omega^{\varepsilon}-\omega,u^{\varepsilon}-u],\omega^{\varepsilon}-\omega)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
Hence
$N\leq c\|\omega^{\varepsilon}-\omega\|^{2}$ (3.14)
Estimates on $BN$:
Next, we estimate the term
$BN=|\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot P\Phi|$
Note that
$P\Phi=\Phi+\nabla p_{\Phi}$
involves a scalar function $p_{\Phi}$, which is difficult to estimate on the
boundary. We will transform it to an estimate on $\Omega$ by an integrating by
part formula.
To this end, we first extend $n,B$ to the interior of $\Omega$ as follows:
$n(x)=\varphi(r(x))\nabla(r(x))$ $B(x)=\varphi(r(x))B(\Pi x)$
where
$r(x)=\min_{y\in\partial\Omega}d(x,y)$ $\Pi x=y_{x}\in\partial\Omega$
such that
$r(x)=d(x,y_{x})$
which is uniquely defined on $\Omega_{\sigma}=\\{x\in\Omega,\ r(x)\leq
2\sigma\\}$ for some $\sigma>0$, and $\varphi(s)$ is smooth and compactly
supported in $[0,2\sigma)$ such that
$\varphi(0)=1,\ {\rm on}\ [0,\sigma]$
Then, we can deduce the estimate of the boundary term $BN$ to an interior
estimate on $\Omega$ by the Stokes formula
$\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot
P\Phi=\int_{\partial\Omega}(n\times B(u^{\varepsilon}-u)\cdot(n\times P\Phi)$
$=(n\times B(u^{\varepsilon}-u),\nabla\times\Phi)-(\nabla\times(n\times
B(u^{\varepsilon}-u)),P\Phi)$
since $P\Phi\cdot n=0$ on the boundary, and $\nabla\times
P\Phi=\nabla\times\Phi$.
Note that
$|(n\times B(u^{\varepsilon}-u),(u^{\varepsilon}-u)\cdot\nabla\omega)|\leq
c\|u^{\varepsilon}-u\|^{2}\leq c\|\omega^{\varepsilon}-\omega\|^{2}$
$|(n\times B(u^{\varepsilon}-u),\omega\cdot\nabla(u^{\varepsilon}-u))|\leq
c\|u^{\varepsilon}-u\|_{1}^{2}\leq c\|\omega^{\varepsilon}-\omega\|^{2}$
It follows that
$|(n\times B(u^{\varepsilon}-u),[\omega,u^{\varepsilon}-u])|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
Note that $u\cdot n=0$ on $\partial\Omega$, $\nabla\cdot u=0$. Then
$|(n\times B(u^{\varepsilon}-u),u\cdot\nabla(\omega^{\varepsilon}-\omega))|$
$=|(\omega^{\varepsilon}-\omega,u\cdot\nabla(n\times
B(u^{\varepsilon}-u)))|\leq c\|u^{\varepsilon}-u\|_{1}^{2}\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
and
$|(n\times B(u^{\varepsilon}-u),(\omega^{\varepsilon}-\omega)\cdot\nabla
u)|\leq c\|\omega^{\varepsilon}-\omega\|^{2}$
It follows that
$|(n\times B(u^{\varepsilon}-u),[\omega^{\varepsilon}-\omega,u])|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
Similarly
$|(n\times
B(u^{\varepsilon}-u),(u^{\varepsilon}-u)\cdot\nabla(\omega^{\varepsilon}-\omega))|=|(\omega^{\varepsilon}-\omega,(u^{\varepsilon}-u)\cdot\nabla(n\times
B(u^{\varepsilon}-u)))|$ $\leq c\|u^{\varepsilon}-u\|_{1}^{2}\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
and
$|(n\times
B(u^{\varepsilon}-u),(\omega^{\varepsilon}-\omega)\cdot\nabla(u^{\varepsilon}-u))|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
It follows that
$|(n\times
B(u^{\varepsilon}-u),[\omega^{\varepsilon}-\omega,u^{\varepsilon}-u])|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
Hence
$|(n\times B(u^{\varepsilon}-u),\nabla\times\Phi)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$ (3.15)
Note also that
$|(\nabla\times(n\times
B(u^{\varepsilon}-u)),P\Phi)|\leq\|\nabla\times(n\times
B(u^{\varepsilon}-u))\|\|\Phi\|$ $\leq c\|(u^{\varepsilon}-u))\|_{1}^{2}\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
Hence, we have
$BN\leq c\|\omega^{\varepsilon}-\omega\|^{2}$ (3.16)
Estimate on $\int_{\partial\Omega}(Bu-n\times\omega)\cdot P\Phi$:
Finally, we estimate the leading order term on the boundary
$BNL=|\int_{\partial\Omega}(Bu-n\times\omega)\cdot P\Phi|$
It should be noted that the estimate is trivial if
$[Bu]_{\tau}-n\times\omega=0$
which is that the Euler solution satisfies the same boundary condition (1.11)
just as the Navier-Stokes solution does so there is no strong boundary layer.
This applies to the cases treated in [41, 42]. Here,
$[Bu]_{\tau}-n\times\omega$ may be not equal zero, then boundary layers may
occur. Additional efforts are needed to overcome the new difficulties.
Similar to the above, we have
$|\int_{\partial\Omega}(Bu-n\times\omega)\cdot
P\Phi|=|\int_{\partial\Omega}n\times(Bu-n\times\omega)\cdot n\times P\Phi|$
$=|(n\times(Bu-n\times\omega),\nabla\times\Phi)-(\nabla\times(n\times(Bu-n\times\omega)),P\Phi)|$
The term $|(n\times(Bu-n\times\omega),\nabla\times\Phi)|$ is relatively easy
to be estimated since $\nabla\times\Phi$ does not involve the pressure
function.
Rewrite $\Phi$ as
$\Phi=\Phi_{1}+\Phi_{2}+\Phi_{3}$
where
$\Phi_{1}=(u^{\varepsilon}-u)\cdot\nabla u$
$\Phi_{2}=u\cdot\nabla(u^{\varepsilon}-u)$
$\Phi_{3}=(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)$
Note that
$|(n\times(Bu-n\times\omega),\nabla\times\Phi_{1})|=|(n\times(Bu-n\times\omega),\nabla\times((u^{\varepsilon}-u)\cdot\nabla
u))|$ $=|\int_{\partial\Omega}n\times((u^{\varepsilon}-u)\cdot\nabla
u)\cdot(n\times(Bu-n\times\omega))+(\nabla\times(n\times(Bu-n\times\omega)),(u^{\varepsilon}-u)\cdot\nabla
u)|$ $\leq
c(\int_{\partial\Omega}|u^{\varepsilon}-u|+\|u^{\varepsilon}-u\|)\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$
and
$|(n\times(Bu-n\times\omega),\nabla\times\Phi_{3})|=|(n\times(Bu-n\times\omega),\nabla\times((u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)))|$
$=|\int_{\partial\Omega}n\times((u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u))\cdot(n\times(Bu-n\times\omega))+(\nabla\times(n\times(Bu-n\times\omega)),(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u))|$
$\leq
c(\int_{\partial\Omega}|u^{\varepsilon}-u|+\|u^{\varepsilon}-u\|_{1}^{2})\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$
since $u^{\varepsilon}-u$ is smooth, so that
$|\nabla(u^{\varepsilon}-u)|\leq c\|\nabla(u^{\varepsilon}-u)\|_{\infty}\leq
c,\ {\rm on}\ \partial\Omega$
To estimate $(n\times(Bu-n\times\omega),\nabla\times\Phi_{2})$, we note
$\nabla\times(\psi\cdot\nabla\phi)=\psi\cdot\nabla(\nabla\times\phi)+\nabla\psi^{\perp}\cdot\nabla\phi$
where $\nabla\psi^{\perp}$ is expressed in component by
$(\nabla\psi^{\perp}\cdot\nabla\phi)_{j}=(-1)^{j+1}\partial_{j+1}\psi\cdot\nabla\phi_{j+1}+(-1)^{j+2}\partial_{j+2}\psi\cdot\nabla\phi_{j+2}$
(3.17)
with the index modulated by 3. Hence, we have
$|(n\times(Bu-n\times\omega),\nabla\times\Phi_{2})|=|(n\times(Bu-n\times\omega),\nabla\times(u\cdot\nabla(u^{\varepsilon}-u)))|$
$=|(n\times(Bu-n\times\omega),u\cdot\nabla(\omega^{\varepsilon}-\omega))+(n\times(Bu-n\times\omega),\nabla
u^{\perp}\cdot\nabla(u^{\varepsilon}-u))|$
Note that
$|(n\times(Bu-n\times\omega),u\cdot\nabla(\omega^{\varepsilon}-\omega))|=|(\omega^{\varepsilon}-\omega,u\cdot\nabla(n\times(Bu-n\times\omega)))|$
$=(\nabla\times(u^{\varepsilon}-u),u\cdot\nabla(n\times(Bu-n\times\omega)))$
$=|\int_{\partial\Omega}n\times(u^{\varepsilon}-u)\cdot(u\cdot\nabla(n\times(Bu-n\times\omega)))+(u^{\varepsilon}-u,\nabla\times(u\cdot\nabla(n\times(Bu-n\times\omega))))|$
$\leq c(\int_{\partial\Omega}|u^{\varepsilon}-u|+\|u^{\varepsilon}-u\|)\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$
Note also that
$(\partial_{j}u\cdot\nabla(u^{\varepsilon}-u)_{j},(n\times(Bu-n\times\omega))_{k})=$
$(\partial_{j}u,\nabla((u^{\varepsilon}-u)_{j}(n\times(Bu-n\times\omega))_{k})-(\partial_{j}u\cdot\nabla(n\times(Bu-n\times\omega))_{k},(u^{\varepsilon}-u)_{j}),$
and that
$|(\partial_{j}u,\nabla((u^{\varepsilon}-u)_{j}(n\times(Bu-n\times\omega))_{k})|=|\int_{\partial\Omega}(u^{\varepsilon}-u)_{j}(n\times(Bu-n\times\omega))_{k}\partial_{j}u\cdot
n|$ $\leq c\|u\|_{1,\infty}^{2}\int_{\partial\Omega}|u^{\varepsilon}-u|\leq
c\int_{\partial\Omega}|u^{\varepsilon}-u|\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$
since $\nabla\cdot\partial_{j}u=0$, and that
$|(\partial_{j}u\cdot\nabla(n\times(Bu-n\times\omega))_{k},(u^{\varepsilon}-u)_{j})|\leq
c\|u^{\varepsilon}-u\|\leq c\varepsilon^{1-s}$
it follows that
$|(n\times(Bu-n\times\omega),\nabla\times\Phi_{2})|\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$
Then, we get
$|(n\times(Bu-n\times\omega),\nabla\times\Phi)|\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$ (3.18)
It remains to estimate $|(\nabla\times(n\times(Bu-n\times\omega)),P\Phi)|$.
There is also a difficulty arising from $P\Phi$. To do it, it is noticed that
$P\Phi=u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla
u+P\tilde{\Phi}$
where
$\tilde{\Phi}=2(u^{\varepsilon}-u)\cdot\nabla
u+(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)$
with
$u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla
u=\nabla\times((u^{\varepsilon}-u)\times u)\in H$
since that $(u^{\varepsilon}-u)\cdot n=0$ and $u\cdot n=0$ on $\partial\Omega$
will imply
$(u^{\varepsilon}-u)\times u=\lambda n$
and
$H=\\{v=\nabla\times\phi;\ \phi\in H^{1}(\Omega),n\times\phi=0\\}$
Hence, we have
$P(u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla
u)=u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla u$
so that
$(\nabla\times(n\times(Bu-n\times\omega)),P\Phi)=$
$(\nabla\times(n\times(Bu-n\times\omega)),u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla
u)+(\nabla\times(n\times(Bu-n\times\omega)),P\tilde{\Phi})$
and
$\|P\tilde{\Phi}\|\leq c\|\tilde{\Phi}\|\leq
c(\|u\|_{1,\infty}+\|u^{\varepsilon}-u\|_{1,\infty})\|u^{\varepsilon}-u\|\leq
c\|u^{\varepsilon}-u\|\leq c\varepsilon^{1-s}$
These properties help us to complete the rest of estimates.
First, we have
$|(\nabla\times(n\times(Bu-n\times\omega)),P\tilde{\Phi})|\leq
c\|P\tilde{\Phi}\|\leq c\|\tilde{\Phi}\|\leq c\|u^{\varepsilon}-u\|\leq
c\varepsilon^{1-s}$
Next, note that
$|(\nabla\times(\times(Bu-n\times\omega)),u\cdot\nabla(u^{\varepsilon}-u))=$
$|(u^{\varepsilon}-u,u\cdot\nabla(\nabla\times(\times(Bu-n\times\omega))))|\leq
c\|u\|_{3}^{2}\|u^{\varepsilon}-u\|\leq c\varepsilon^{1-s}$
since $\nabla u=0$ in $\Omega$ and $u\cdot n=0$ on $\partial\Omega$.
Note also that
$|(\nabla\times(n\times(Bu-n\times\omega)),(u^{\varepsilon}-u)\nabla u)|\leq
c\|u\|_{2}^{2}\|u^{\varepsilon}-u\|\leq c\varepsilon^{1-s}$
Hence, we have
$|(\nabla\times(n\times(Bu-n\times\omega)),P\Phi)|\leq c\varepsilon^{1-s}$
(3.19)
Then, we conclude that
$BNL=|\int_{\partial\Omega}(Bu-n\times\omega)\cdot P\Phi|\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$ (3.20)
In conclusion, it follows from (3.13),(3.14),(3.16) and (3.20) that
$\frac{1}{2}\frac{d}{dt}E+\frac{\varepsilon}{2}\|P\Delta(u^{\varepsilon}-u)\|^{2}\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$ (3.21)
Recall that
$E=\|(\omega^{\varepsilon}-\omega)\|^{2}-(\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot(u^{\varepsilon}-u)+2\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot(Bu-n\times\omega))$
and
$|\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot(u^{\varepsilon}-u)|\leq
c\int_{\partial\Omega}|(u^{\varepsilon}-u)|^{2}\leq
c\|u^{\varepsilon}-u\|\|\omega^{\varepsilon}-\omega\|\leq\|\omega^{\varepsilon}-\omega\|^{2}+c\|u^{\varepsilon}-u\|^{2}$
and
$|\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot(Bu-n\times\omega)|\leq
c\int_{\partial\Omega}|(u^{\varepsilon}-u)|\leq\|\omega^{\varepsilon}-\omega\|^{2}+c\|u^{\varepsilon}-u\|^{2}$
It follows that
$\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon
c\int_{0}^{t}\|P\Delta(u^{\varepsilon}-u)\|^{2}\leq
c\int_{0}^{t}\|\omega^{\varepsilon}-\omega\|^{2}+c\varepsilon^{1-s}$ (3.22)
By using the Gronwall Lemma, one gets that
$\|\omega^{\varepsilon}-\omega\|^{2}\leq c\varepsilon^{1-s}$ (3.23)
on $[0,T_{0}]$ for any given $s>0$ and $\varepsilon$ for small enough. Then
$\|u^{\varepsilon}-u\|_{1}^{2}\leq c\varepsilon^{1-s}$ (3.24)
and
$\varepsilon\int_{0}^{t}\|P\Delta(u^{\varepsilon}-u)\|^{2}\leq
c\varepsilon^{1-s}$ (3.25)
Combining this with (2.7) and Theorem 3.1 implies that
$\varepsilon\int_{0}^{t}\|(u^{\varepsilon}-u)\|_{2}^{2}\leq
c\varepsilon^{1-s}$ (3.26)
Note that
$\|\nabla(u^{\varepsilon}-u)\|_{p}^{p}\leq
c\|\nabla(u^{\varepsilon}-u)\|_{\infty}^{p-2}\|\nabla(u^{\varepsilon}-u)\|^{2}$
It follows that
$\|\nabla(u^{\varepsilon}-u)\|_{p}^{p}\leq c\varepsilon^{1-s}$
The theorem is proved.
Remark 3.1. The estimate (1.19) is optimal in the sense that $s$ can not be
taken to be $0$. Since if $s=0$, then (1.19) will imply that
$u^{\varepsilon}-u$ is uniform bounded in $L^{2}(0,T;H^{2}(\Omega))$. Then,
there is a subsequence such that $u^{\varepsilon_{n}}-u$ weakly convergence in
$H^{2}(\Omega)$ for a.e. $t\in[0,T]$. Note that $u^{\varepsilon_{n}}-u$
strongly convergence to $0$ in $H^{1}(\Omega)$ for all $t\in[0,T]$. It follows
that
$u^{\varepsilon_{n}}\rightarrow u\ weakly\ in\ H^{2}(\Omega),\ a.e.\ t$
Note that $W$ is closed in $H^{2}(\Omega)$, then is weakly closed. Hence,
$u\in W$ for a.e. $t\in[0,T]$, i.e.,
$[Bu]_{\tau}-n\times\omega=0,\ \ a.e.\ t$
Since $u$ is smooth, then it holds for all $t\in[0,T]$. This is impossible in
general, see [4, 5].
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|
arxiv-papers
| 2013-01-04T09:53:15 |
2024-09-04T02:49:39.905959
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuelong Xiao and Zhouping Xin",
"submitter": "Yuelong Xiao",
"url": "https://arxiv.org/abs/1301.0693"
}
|
1301.0713
|
# 8Dim calculations of the third barrier in 232Th and a conflict between
theory and experiment on uranium nuclei.
P. Jachimowicz Institute of Physics, University of Zielona Góra, Szafrana 4a,
65516 Zielona Góra, Poland M. Kowal [email protected] National Centre for
Nuclear Research, Hoża 69, PL-00-681 Warsaw, Poland J. Skalski National
Centre for Nuclear Research, Hoża 69, PL-00-681 Warsaw, Poland
###### Abstract
We find the height of the third fission barrier $B_{III}$ and energy of the
third minimum $E_{III}$ in 232Th using the macroscopic - microscopic model,
very well tested in this region of nuclei. For the first time it is done on an
8-dimensional deformation hypercube. The dipole distortion is included among
the shape variables to assure that no important shapes are missed. The saddle
point is found on a lattice containing more than 50 million points by the
immersion water flow (IWF) method. The shallow third minimum,
$B_{III}-E_{III}\approx 0.36$ MeV, agrees with experimetal data of Blons et
al. This is in a sharp contrast with the status of the IIIrd minima in
232-236U: their experimental depth of $\geq 3$ MeV contradicts all realistic
theoretical predictions. We emphasize the importance of repeating the
experiment on 232Th, by a technique similar to that used in the uranium
nuclei, for settling the puzzle of the third minima in actinides.
###### pacs:
25.85.Ca,21.10.Gv,21.60.-n,27.90.+b
Search for a hyperdeformation (HD), i.e., extremely elongated shapes of atomic
nuclei (spheroid with a major to minor axis ratio of 3:1), is one of the most
difficult challenges of modern nuclear structure studies. There is still no
convincing evidence for discrete gamma transitions of HD rotational bands. The
only experimental evidence for hyperdeformation, except in light nuclei, has
been reported in the region of light actinides, in particular in uranium
isotopes 232U, 234 U and 236U, see Kra and references therein. In these
experiments, the fission probability was studied as a function of the
excitation energy via different reactions. Strong resonances were observed and
a fine structure of some interpreted as a signature of multiple hyperdeformed
bands. Measurements of the angular distribution of the fragments arising from
the induced fission support this interpretation. Moreover, by measuring the
angular distribution of the fission fragments one can obtain some information
about the spin and $K$ \- quantum number. Due to the predicted static mass-
asymmetry of the third minima, their intrinsic parity would be broken, so the
low-energy excitations should show a pattern of alternating parity (even
nuclei) or parity doublet (odd nuclei) bands, different from that in the
second minima. Since the third minima are predicted axially symmetric, the
excitations would have an approximate (up to a Coriolis mixing, much weaker
than in the first or second minimum) $K$ quantum number.
The current experimental status of the third barrier in 232Th is different
from that in uranium isotopes. No recent data exist, the only available are
those from the prior 30 years Blons1 ; Blons2 ; Blons3 ; Blons4 . The
experiment conducted by Blons et al pointed to a shallow third minimum (no
deeper than 0.5 MeV). Only a little more than 1 MeV deep third minimum in a
neighboring nucleus 232Pa was recently reported by the Munich-Debrecen group
Csige2012 . The technique used to resolve observed resonances was the same as
in the study of uranium nuclei by the same group. Deep third minima in uranium
( $3\div 4$ MeV) disagree with all calculations Mcdonell1 ; Mcdonell2 ; Bender
; Gogny ; HFBCS ; Delar ; MollerPRC2012 except those of S. Ćwiok et al,
within the Woods-Saxon model Stefan ; Stefan1 ; Stefan2 . Self-consitent
calculations based on the Skyrme SkM* interaction do not give a hyperdeformed
minimum in 232Th Mcdonell1 at all. With the help of a temperature effect, a
slight minimum appears, but does not exceed 500 keV. Moreover, with an
increasing excitation, the depth of this minimum decreases again. Situation is
very similar in 232U and 234U nuclei Mcdonell2 . Selfconsitent calculations
with the Gogny D1S interaction do not give third minima in those nuclei Gogny
; HFBCS ; Delar as well. Moreover, for 236U, a recent result of Möller et al
MollerPRC2012 within the macro-micro approach also does not support a deep
minimum reported in HyperU : one can read from the map (Fig. 8. in ref.
MollerPRC2012 ) that the minimum is not deeper than 300 keV, ten times smaller
than the experimental value.
In fact, hyperdeformed minima predicted in Stefan ; Stefan1 ; Stefan2 were
quite abundant: they not only appeared in nuclei where nobody saw them in
experiment, but were double, with different octupole deformations
$\beta_{3}\approx 0.3$ and 0.6. Using the same model, we have just showed in
kowskalIIImin that both types of the IIIrd minima in uranium nuclei disappear
completely after proper nuclear shapes are considered. The more mass-
asymmetric minima were unphysical from the beginning. Their quadrupole moment
is large, $Q_{20}\simeq 170$ b, which situates them behind the barrier, closer
to the scission point. The existence of minima with a smaller mass asymmetry
(and quadrupole moments) requires a more detailed study. Finding the barrier
in many-dimensional space requires hypercube calculations. This is the case of
232Th, where the less mass-asymmetric minimum is the deeper one.
Previously, we found in kowskalIIImin a tentative upper limit of 330 keV for
the IIIrd barrier in 232Th and 0 in even uranium isotopes 232-236U by 6Dim
hypercube calculations and by probing various trial 8Dim fission paths. The
main aim of the present study is to ascertain the prediction for 232Th,
showing that a third minimum is there. To accomplish this, an 8-dimensional
grid calculation has been done. All deformations were treated on the same
footing, without introducing any subdivision into relevant and irrelevant
subspaces. Let us emphasize that till now only 5-dimensional macroscopic-
microscopic calculations of fission saddles were available in the literature
MollerNATURE2001 ; MollerPRL2004 ; MollerPRC2012 .
There is one more important motivation for such studies. To distinguish
between resonances in the second and third minimum, the excitation energy
should be higher than the first barrier but simultaneously lower than the
second one. We show that our multidimensional calculations predict such a
possibility in 232Th, which is a natural candidate for the future experimental
study.
The energy is calculated within the microscopic-macroscopic method with a
deformed Woods-Saxon potential. Nuclear shapes are defined in terms of the
nuclear surface WS . Since we found that nonaxiality in the region of the
third barrier is less important than a proper treatment of the axially
symmetric shapes, we parameterize the surface as
$R(\theta)=c(\\{\beta\\})R_{0}(1+\sum_{\lambda=1}\beta_{\lambda 0}Y_{\lambda
0}(\theta)),$ (1)
where $c(\\{\beta\\})$ is the volume fixing factor. The macroscopic energy is
calculated using the Yukawa plus exponential model KN with parameters
specified in WSpar . All other parameters of the model are the same as in a
number of our previous studies, e.g. in KOW10 ; kowskal ; IIbarriers . A
hyperdeformed minimum was searched by exploring energy surfaces in a region of
deformations beyond the second minimum. We generate the grid in axially
symmetric deformations:
$\displaystyle\beta_{10}$ $\displaystyle=$ $\displaystyle\ \ -0.35\ (0.05)\
0.00;\,\,\,\beta_{20}=0.55\ (0.05)\ 1.50,$ $\displaystyle\beta_{30}$
$\displaystyle=$ $\displaystyle\ \ 0.00\ (0.05)\ 0.35;\,\,\,\beta_{40}=-0.10\
(0.05)\ 0.35,$ $\displaystyle\beta_{50}$ $\displaystyle=$ $\displaystyle\ \
-0.20\ (0.05)\ 0.20;\,\,\,\beta_{60}=-0.15\ (0.05)\ 0.15,$
$\displaystyle\beta_{70}$ $\displaystyle=$ $\displaystyle\ \ -0.20\ (0.05)\
0.20;\,\,\,\beta_{80}=-0.15\ (0.05)\ 0.15.$
Numbers in the parentheses specify the step with which the calculation is done
for a given variable. Thus, we finally have the values of energy at a total of
$50803200$ grid points. On a such giant 8-dimensional grid the IWF saddle
point searching method is used.
Let us emphasize that our model, without the $\lambda=1$ term in (1), very
well reproduces first KOW10 and second barriers IIbarriers and second minima
kowskal in actinide nuclei. One could hope that an extrapolation to more
elongated shapes would be still valid within such a parametrization. What then
causes troubles beyond the second peak? They result from the truncation of the
expansion (1) that restricts possible shapes for large deformations. Two
shapes close to the IIIrd saddle are shown in Fig. 1: one from the 8D
calculation (nonzero $\beta_{10}$-$\beta_{80}$, in red) and the other, from
the 7D calculation $\beta_{20}-\beta_{80}$ (in blue). Even if they seem not
much different, the difference in barrier is 4 MeV. It appears that the dipole
deformation $\beta_{10}$, usually associated only with a shift of the center
of mass, effectively makes up for truncated multipoles in very deformed, mass-
asymmetric configurations. We emphasize that our program keeps automatically
the center of mass at the coordinate origin, thus $\beta_{10}$ serves only
probing somewhat different shapes.
Figure 1: Shapes $R(\theta)/R_{0})$ near to the IIIrd saddle: red line - 8D
calculation ($\beta_{10}-\beta_{80}$), blue line \- 7D calculation (without
$\beta_{10}$). Figure 2: Potential energy surfaces
$E({\beta_{20},\beta_{40}})$ for 232Th from the 8D $\beta_{10}-\beta_{80}$
calculation, minimized over remaining degrees of freedom; upper panel -
$\beta_{10}=0$, bottom panel - $\beta_{1}=min$.
A modification of the energy landscape by including $\beta_{10}$ is shown in
Fig. 2. Since dipole is the first spherical harmonic (Eq.1), one can suspect
its pronounced effect. Indeed, that effect is significant. Not only the height
of the third saddle is clearly reduced, but the whole landscape changed. The
result indicates a large effect of new shapes, unattainable previously.
Figures show also examples of two fission paths. In the version of calculation
without the dipole (blue trajectory), after passing through the second saddle,
the nucleus falls into a deep third minimum. To be split, it needs to tunnel
through a more than 4 MeV barrier. In the case with included $\beta_{10}$ (red
trajectory), after passing the second barrier nucleus can easily split. One
can notice that $\beta_{10}$ deformation has almost no effect on the position
and height of the second saddle. Precise lowering of the third barrier by
$\beta_{10}$ may be read from Fig. 3 as 4.1 MeV. Beside the significant
reduction, one can also see a change in the barrier shape. This rather
dramatic change of the fission path in the multidimensional deformation space
has a direct effect on fission half-life.
Table 1: Calculated and experimental features of the fission barrier in 232Th and 232U. $M$ \- ground state mass excess, $B_{I}$ \- first barrier, $E_{II}$\- second minimum $B_{II}$ \- second barrier $E_{III}$\- third minimum $B_{III}$ \- third barrier. All quantities (in MeV) are shown relative to the ground state. Theory: | $M^{th}$ | $B^{th}_{I}$ | $E^{th}_{II}$ | $B^{th}_{II}$ | $E^{th}_{III}$ | $B^{th}_{III}$
---|---|---|---|---|---|---
Experiment: | $M^{exp}$ | $B^{exp}_{I}$ | $E^{exp}_{II}$ | $B^{exp}_{II}$ | $E^{exp}_{III}$ | $B^{exp}_{III}$
232Th | 35.33 | 4.4 | 2.2 | 6.1 | 4.1 | 4.4
RIPL3 | 35.45 | 5.8 | - | 6.7 (6.2) | - | -
Blons2 | - | 4.6 | - | 5.7 | 4.0 | 4.4
232U | 34.34 | 4.5 | 3.2 | 5.7 | 0.0 | 0.0
RIPL3 | 34.61 | 5.4 | - | 5.4 (5.3) | - | -
Hyper232U | - | 4.0 | 3.1 | 4.9 | 3.2 | 6.0
A detailed information about the structure of the fission barrier in 232Th is
collected in Table 1. The recent theoretical results and experimental data for
232U Hyper232U are also given. It is seen that the model calculation well
reproduces essential features of the fission barrier up to superdeformed
shapes in both nuclei. The inner fission barrier $B^{th}_{I}$ has been found
according to the prescription presented in KOW10 .
Here, a comment is needed. Usually the calculated first saddles are
substantially higher than experimental values but the opposite situation takes
place in the light Thorium isotopes what is called in the literature the
”Thorium anomaly”. In our case the nonaxiality reduces the first saddle for
light Thorium only non-significantly, to a nearly proper height (see column 3
of Table 1 column 3). Thus, the mentioned anomaly does not occur in our case
(we use the modern data of Hyper232U ).
In order to determine the second peak $B^{th}_{II}$, a mass asymmetry is
included, details can be found in IIbarriers . One can see that our calculated
second minimum and second barrier reasonably agree with experiment. As already
mentioned, the experimental situation (when it comes to the depth of
hyperdeformed minima) in two isobars: 232Th and 232U is completely different.
While in 232U, the IIIrd minimum is quite deep, in 232Th it is very shallow.
Our results support the existence of a shallow third minimum in 232Th as in
old experiments of Blons and thus agree with the majority of modern
theoretical models.
A designed experiment for 232Th, by a technique described in ref Kra ; Thirolf
, would be crucial for solving the mystery of the third minima in actinides.
Particularly promising are experiments employing highly monochromatic
$\gamma$-ray beams for photofission studies. With a high quality of photon and
spectral intensity, exceeding the performance of existing facilities by
several orders of magnitude seems to be possible in the near future Thirolfexp
. If the result of Blons et al is confirmed, we will have to understand _why
between 232Th and 232U, two beta decays away, the energy landscape changes so
dramatically_. On the other hand, if in the future experiment, a depth of the
IIIrd minimum is obtained similar as that in 232U, we will have a total
contradiction between theory and experiment. _Assuming that the existing
resonances cannot be interpreted otherwise, all current meaningful theoretical
models would have to be reconstructed anew to give deep hyperdeformed minima_.
One can expect that in hyperdeformed nuclei, some high particle states at
normal deformation become occupied with increasing deformation. Since these
are the orbits that are occupied in superheavy nuclei at moderate
deformations, the whole question may have some impact on the understanding of
superheavy nuclei.
Figure 3: Energy along a sequence of smoothly elongating shapes, beginning at
the superdeformed minimum. Appropriate shapes along trajectories are shown.
In summary:
(i) We have presented for the first time an 8D hypercube calculation for
232Th. To find the third barrier on a giant grid, the IWF method has been
applied. After a proper inclusion of the dipole deformation we found the depth
of the third minimum of about 0.36 MeV. This would rather exclude
spectroscopic studies in the IIIrd well.
(ii) Including a dipole distortion lowers the third saddle by more than 4 MeV.
It seems likely that, with the shape parametrization (1), the dipole
deformation is important everywhere, where large elongation and necking is
combined with a sizable mass asymmetry. For example, it may be the case of the
Jacobi shape transition at high spins in medium-heavy nuclei.
(iii) New experimental study dedicated to hyperdeformation in 232Th seems
essential for the understanding of the third minima in actinide nuclei.
Acknowledgments This work was partially supported by Narodowe Centrum Nauki
Grant No. 2011/01/B/ST2/05131. The support of the LEA COPIGAL fund is
gratefully acknowledged.
## References
* (1) A. Krasznahorkay, ,,_Tunneling Through Triple-Humped Fission barriers_ ”, Handbook of Nuclear Chemistry, Chapter 5, DOI10.1007/978-1-4419-0720-2-5, Springer Science+Business MediaB.V. (2011).
* (2) J. Blons, C. Mazur and D. Paya, Phys. Rev. Lett. 35, 1749 (1975).
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* (6) L. Csige et al., Phys. Rev. C 85 , 054306 (2012).
* (7) J.D. McDonnell, W. Nazarewicz, J.A. Sheikh, ” _Thermal Fission Pathways in 232Th_ ”, Proceedings of the 4th International Workshop on Fission and Fission Product Spectroscopy (2009).
* (8) J.D. McDonnell, PhD Thesis (unpublished) (2011).
* (9) M. Bender, PhD Thesis, Universitat Frankfurt (unpublished); M. Bender P-H. Heenen and P-G. Reinhard, Rev. Mod. Phys. 75, 121 180 (2003).
* (10) J. F. Beger, M. Girod and D. Gogny, Nucl. Phys. A, 502, 85c (1989).
* (11) L. Bonneau, P. Quentine and Samsoen, Eur. Phys. J. A, 21, (2004).
* (12) J.-P. Delaroche, M. Girod, H. Goutte and J. Libert, Nucl. Phys. A 771, 103 (2006).
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* (14) S. Ćwiok, W. Nazarewicz, J .X . Saladin, W. Płóciennik and A. Johnson, Phys. Lett. B, 322, 304 (1994).
* (15) W. Nazarewicz, S. Ćwiok, J. Dobaczewski, J .X . Saladin, Acta Phys. Pol. B, 26, (1996).
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* (18) L. Csige et al., Phys. Rev. C 80, 011301(R) (2009).
* (19) M. Kowal, J. Skalski Phys. Rev. C 85, 061302(R) (2012).
* (20) P. Möller, D. G. Madland, A. J. Sierk, and A. Iwamoto, Nature, Vol 409, 15 February 2001, p 485-490.
* (21) P. Möller, A. J. Sierk, and A. Iwamoto Phys. Rev. Lett., 92, 072501 (2004).
* (22) S. Ćwiok, J. Dudek, W. Nazarewicz, J. Skalski and T. Werner, Comput. Phys. Commun., 46, 379 (1987).
* (23) H. J. Krappe, J. R. Nix and A. J. Sierk, Phys. Rev. C 20, 992 (1979).
* (24) I. Muntian, Z. Patyk and A. Sobiczewski, Acta Phys. Pol. B 32, 691 (2001).
* (25) M. Kowal, P. Jachimowicz, A. Sobiczewski Phys. Rev. C 82, (2010).
* (26) P. Jachimowicz, M. Kowal, J. Skalski Phys. Rev. C, 85, 034305 (2012).
* (27) M. Kowal, J. Skalski Phys. Rev. C 82, 054303 (2010).
* (28) G. N. Smirenkin, IAEA Report INDC(CCP)-359, Vienna (1993), http://www-nds.iaea.org/RIPL-3/; A. Mamdouh, J.M. Pearson, M. Rayet, and F. Tondeur, Nucl. Phys. A 644, 389 (1998), http://www-nds.iaea.org/RIPL-2/.
* (29) P. G. Thirolf and D. Habs, Prog. Part. Nucl. Phys., 49, 325 (2002); P. G. Thirolf, D.Dc. Thesis, Ludwig-Maximilians-Universitat Munchen (2003).
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|
arxiv-papers
| 2013-01-04T13:14:01 |
2024-09-04T02:49:39.913979
|
{
"license": "Public Domain",
"authors": "P. Jachimowicz, M. Kowal, J. Skalski",
"submitter": "Michal Kowal",
"url": "https://arxiv.org/abs/1301.0713"
}
|
1301.0767
|
# Periodic Solutions for Circular Restricted 4-body Problems with Newtonian
Potentials ***Supported by National Natural Science Foundation of China.
Xiaoxiao Zhao, and Shiqing Zhang
Yangtze Center of Mathematics and College of Mathematics, Sichuan University,
Chengdu 610064, People’s Republic of China
> Abstract: We study the existence of non-collision periodic solutions with
> Newtonian potentials for the following planar restricted 4-body problems:
> Assume that the given positive masses $m_{1},m_{2},m_{3}$ in a Lagrange
> configuration move in circular obits around their center of masses, the
> sufficiently small mass moves around some body. Using variational minimizing
> methods, we prove the existence of minimizers for the Lagrangian action on
> anti-T/2 symmetric loop spaces. Moreover, we prove the minimizers are non-
> collision periodic solutions with some fixed wingding numbers.
>
> Keywords: Restricted 4-body problem; non-collision periodic solution;
> variational minimizer; wingding number.
>
> 2000 AMS Subject Classification 34C15, 34C25, 58F.
## 1 Introduction and Main Results
In this paper, we study the planar circular restricted 4-body problems with
Newtonian potentials. Suppose points of positive masses $m_{1},m_{2},m_{3}$
move in a plane of their circular orbits $q_{1}(t),q_{2}(t),q_{3}(t)$ and the
center of masses is at the origin; suppose the sufficiently small mass point
does not influence the motion of $m_{1},m_{2},m_{3}$, and moves in the plane
for the given masses $m_{1},m_{2},m_{3}$.
It is well-known that $q_{1}(t),q_{2}(t),q_{3}(t)$ satisfy the Newtonian
equations:
$m_{i}\ddot{q_{i}}=\frac{\partial U}{\partial q_{i}},\ \ \ \ i=1,2,3,$ (1.1)
where
$U=\sum\limits_{1\leq i<j\leq 3}\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}.$ (1.2)
Without loss of generality, we assume that there exists
$\theta_{1},\theta_{2},\theta_{3}\in[0,2\pi)$ such that the planar circular
orbits are
$q_{1}(t)=r_{1}e^{\sqrt{-1}\frac{2\pi}{T}t}e^{\sqrt{-1}\theta_{1}},\ \
q_{2}(t)=r_{2}e^{\sqrt{-1}\frac{2\pi}{T}t}e^{\sqrt{-1}\theta_{2}},\ \
q_{3}(t)=r_{3}e^{\sqrt{-1}\frac{2\pi}{T}t}e^{\sqrt{-1}\theta_{3}},$ (1.3)
where the radius $r_{1},r_{2},r_{3}$ are positive constants depending on
$m_{i}(i=1,2,3)$ and $T$ (see Lemma 2.6). We also assume that
$m_{1}q_{1}(t)+m_{2}q_{2}(t)+m_{3}q_{3}(t)=0$ (1.4)
and
$|q_{i}-q_{j}|=l,\ \ 1\leq i\neq j\leq 3,$ (1.5)
where the constant $l>0$ depends on $m_{i}(i=1,2,3)$ and $T$ (see Lemma 2.5).
The orbit $q(t)\in R^{2}$ for sufficiently small mass is governed by the
gravitational forces of $m_{1},m_{2},m_{3}$ and therefore it satisfies the
following equation
$\ddot{q}=\sum\limits_{i=1}^{3}\frac{m_{i}(q_{i}-q)}{|q_{i}-q|^{3}}.$ (1.6)
For $N$-body problems, there are many papers concerned with the periodic
solutions by using variational methods, see [1-9,13-16,18] and the references
therein. In [3], Chenciner-Montgomery proved the existence of the remarkable
figure-“8” type periodic solution for planar Newtonian 3-body problems with
equal masses. Marchal[6] studied the fixed end problem for Newtonian n-body
problems and proved the minimizer for the Lagrangian action has no interior
collision. Especially, in [8], Simó used computer to discover many new
periodic solutions for Newtonian n-body problems. Zhang-Zhou[13-15] decomposed
the Lagrangian action for n-body problems into some sum for two body problems
and [14,15] avoid collisions by comparing the lower bound for the Lagrangian
action on the symmetry collision orbits and the upper bound for the Lagrangian
action on test orbits in some cases.
Motivated by the above works, we use variational methods to study the circular
restricted 3+1-body problem with some fixed wingding numbers and some masses.
For the readers’ conveniences, we recall the definition of the winding number,
which can be found in many books on the classical differential geometry.
Definition 1.1 Let $\Gamma:x(t),t\in[a,b]$ be an given oriented continuous
closed curve, and $p$ be a point of the plane not on the curve. Then, the
mapping $\varphi:\Gamma\rightarrow S^{1}$, given by
$\varphi(x(t))=\frac{x(t)-p}{|x(t)-p|},\ \ \ t\in[a,b]$
is defined to be the position mapping of the curve $\Gamma$ relative to $p$,
when the point on $\Gamma$ goes around the curve once, its image point
$\varphi(x(t))$ will go around $S^{1}$ a number of times, this number is
called the winding number of the curve $\Gamma$ relative to $p$, and we denote
it by $deg(\Gamma,p)$. If $p$ is the origin, we write $deg\Gamma$.
Define
$W^{1,2}(R/TZ,R^{2})=\bigg{\\{}x(t)\Big{|}x(t),\dot{x}(t)\in L^{2}(R,R^{2}),\
x(t+T)=x(t)\bigg{\\}}.$
The norm of $W^{1,2}(R/TZ,R^{2})$ is
$\|x\|=\Big{[}\int_{0}^{T}|x|^{2}dt\Big{]}^{\frac{1}{2}}+\Big{[}\int_{0}^{T}|\dot{x}|^{2}dt\Big{]}^{\frac{1}{2}}.$
(1.7)
The functional corresponding to the equation (1.6) is
$f(q)=\int_{0}^{T}\Big{[}\frac{1}{2}|\dot{q}|^{2}+\sum\limits_{i=1}^{3}\frac{m_{i}}{|q-q_{i}|}\Big{]}dt,\
\ \ \ q\in\Lambda_{\pm},$ (1.8)
where
$\Lambda_{-}=\left\\{q\in
W^{1,2}(R/TZ,R^{2})\bigg{|}\begin{array}[]{c}q(t+\frac{T}{2})=-q(t),\
deg(q-q_{1})=-1,\\\ q(t)\neq q_{i}(t),\ \forall
t\in[0,T],i=1,2,3\end{array}\right\\}.$
and
$\Lambda_{+}=\left\\{q\in
W^{1,2}(R/TZ,R^{2})\bigg{|}\begin{array}[]{c}q(t+\frac{T}{2})=-q(t),\
deg(q-q_{1})=1,\\\ q(t)\neq q_{i}(t),\ \forall
t\in[0,T],i=1,2,3\end{array}\right\\}.$
Our main results are the following:
Theorem 1.1 Let $T=1$, for the values of $m_{1},m_{2},m_{3}$ given in Table 1
with $M=1$, the minimizer of $f(q)$ on the closure $\overline{\Lambda}_{-}$ of
$\Lambda_{-}$ is a non-collision 1-periodic solution of (1.6); for the values
of $m_{1},m_{2},m_{3}$ given in Table 2 with $m_{1}=m_{2}=m_{3}=1$, the
minimizer of $f(q)$ on $\overline{\Lambda}_{-}$ is a non-collision 1-periodic
solution of (1.6).
Remark 1 In proving Theorem 1, we need to use test functions. We find that if
the test functions are circular orbits, we can not get the desired results on
$\overline{\Lambda}_{-}$. Therefore, we select elliptic orbits as test
functions.
Theorem 1.2 Let $T=1$, for the values of $m_{1},m_{2},m_{3}$ given in Table 3
with $M=1$, the minimizer of $f(q)$ on the closure $\overline{\Lambda}_{+}$ of
$\Lambda_{+}$ is a non-collision 1-periodic solution of (1.6); for the values
of $m_{1},m_{2},m_{3}$ given in Table 4 with $m_{1}=m_{2}=m_{3}=1$, the
minimizer of $f(q)$ on $\overline{\Lambda}_{+}$ is a non-collision 1-periodic
solution of (1.6).
Remark 2 When we take elliptic orbits as test functions, we find that the
biggest symmetric space is the anti-T/2 symmetric loop space if the wingding
number $n$ is odd($n=\pm 1,\pm 3,\cdots$); we can not find suitable symmetric
space if the wingding number is even. When the wingding number $n\neq\pm 1$
and we take circular orbits as test functions, we find that the biggest
symmetric space is
$\Lambda=\left\\{q\in
W^{1,2}(R/TZ,R^{2})\bigg{|}\begin{array}[]{c}q(t+\frac{T}{|n-1|})=R(|n-1|)q(t),\
deg(q-q_{1})=n,\\\ q(t)\neq q_{i}(t),\ \forall
t\in[0,T],i=1,2,3\end{array}\right\\},$
where
$R(|n-1|)=\left(\begin{array}[]{ccc}cos\frac{2\pi}{|n-1|}&-sin\frac{2\pi}{|n-1|}\\\
sin\frac{2\pi}{|n-1|}&cos\frac{2\pi}{|n-1|}\end{array}\right)\in SO(2)$
is a counter-clockwise rotation of angle $\frac{2\pi}{|n-1|}$ in $R^{2}$. But
the Lagrangian actions on the circular test orbits are bigger than the lower
bound for the Lagrangian actions on collision symmetric orbits. Hence we
consider the anti-T/2 symmetric loop spaces $\Lambda_{\pm}$.
## 2 Preliminaries
In this section, we will list some basic Lemmas and inequality for proving our
Theorems 1.1 and 1.2.
Lemma 2.1(Tonelli[1],[11]) Let $X$ be a reflexive Banach space, $S$ be a
weakly closed subset of $X$, $f:S\rightarrow R\cup\\{+\infty\\}$. If
$f\not\equiv+\infty$ is weakly lower semi-continuous and
coercive($f(x)\rightarrow+\infty$ as $\|x\|\rightarrow+\infty$), then $f$
attains its infimum on $S$.
Lemma 2.2(Poincare-Wirtinger Inequality[10]) Let $q\in W^{1,2}(R/TZ,R^{K})$
and $\int_{0}^{T}q(t)dt=0$, then
$\int_{0}^{T}|q(t)|^{2}dt\leq\frac{T^{2}}{4\pi^{2}}\int_{0}^{T}|\dot{q}(t)|^{2}dt.$
Lemma 2.3(Palais’s Symmetry Principle([12])) Let $\sigma$ be an orthogonal
representation of a finite or compact group $G$, $H$ be a real Hilbert space,
$f:H\rightarrow R$ satisfies $f(\sigma\cdot x)=f(x),\forall\sigma\in G,\forall
x\in H$.
Set $F=\\{x\in H|\sigma\cdot x=x,\ \forall\sigma\in G\\}$. Then the critical
point of $f$ in $F$ is also a critical point of $f$ in $H$.
Remark 2.1 By Palais’s Symmetry Principle and the perturbation invariance for
wingding numbers, we know that the critical point of $f(q)$ in $\Lambda_{\pm}$
is a periodic solution of Newtonian equation (1.6).
Lemma 2.4
(1)(Gordon’s Theorem[17]) Let $x\in W^{1,2}([t_{1},t_{2}],R^{K})$ and
$x(t_{1})=x(t_{2})=0$. Then for any $a>0$, we have
$\int_{t_{1}}^{t_{2}}(\frac{1}{2}|\dot{x}|^{2}+\frac{a}{|x|})dt\geq\frac{3}{2}(2\pi)^{2/3}a^{2/3}(t_{2}-t_{1})^{1/3}.$
(2)(Long-Zhang[18]) Let $x\in W^{1,2}(R/TZ,R^{K}),\int_{0}^{T}xdt=0$, then for
any $a>0$, we have
$\int_{0}^{T}(\frac{1}{2}|\dot{x}|^{2}+\frac{a}{|x|})dt\geq\frac{3}{2}(2\pi)^{2/3}a^{2/3}T^{1/3}.$
Lemma 2.5 Let $M=m_{1}+m_{2}+m_{3}$, we have
$l=\sqrt[3]{\frac{MT^{2}}{4\pi^{2}}}$.
Proof. It follows from (1.1) and (1.2) that
$\ddot{q_{1}}=m_{2}\frac{q_{2}-q_{1}}{|q_{2}-q_{1}|^{3}}+m_{3}\frac{q_{3}-q_{1}}{|q_{3}-q_{1}|^{3}}.$
(2.1)
Then by (1.3)-(1.5), we obtain
$\displaystyle-\frac{4\pi^{2}}{T^{2}}q_{1}$
$\displaystyle=\frac{1}{l^{3}}(m_{2}q_{2}+m_{3}q_{3}-m_{2}q_{1}-m_{3}q_{1})$
(2.2) $\displaystyle=\frac{1}{l^{3}}(-m_{1}q_{1}-m_{2}q_{1}-m_{3}q_{1}),$
which implies
$l^{3}=\frac{MT^{2}}{4\pi^{2}},\ \ \ \ \ \ $ (2.3)
that is,
$l=\sqrt[3]{\frac{MT^{2}}{4\pi^{2}}}.\ \ \ \Box$ (2.4)
Lemma 2.6 The radius $r_{1},r_{2},r_{3}$ of the planar circular orbits for the
masses $m_{1},m_{2},m_{3}$ are
$r_{1}=\frac{\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ \ $
$r_{2}=\frac{\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ \ $
$r_{3}=\frac{\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}{M}l.\ \ \ \ \ \ $
Proof. Choose the geometrical center of the initial configuration
($q_{1}(0),q_{2}(0),q_{3}(0)$) as the origin of the coordinate (x,y). Without
loss of generality, by (1.5), we suppose the location coordinates of
$q_{1}(0),q_{2}(0),q_{3}(0)$ are
$A_{1}(\frac{\sqrt{3}l}{3},0),A_{2}(-\frac{\sqrt{3}l}{6},\frac{l}{2}),A_{3}(-\frac{\sqrt{3}l}{6},-\frac{l}{2})$.
Then we can get the coordinate of the center of masses $m_{1},m_{2},m_{3}$ is
$C(\frac{\frac{\sqrt{3}}{3}m_{1}l-\frac{\sqrt{3}}{6}m_{2}l-\frac{\sqrt{3}}{6}m_{3}l}{M},\frac{\frac{m_{2}}{2}l-\frac{m_{3}}{2}l}{M})$.
To make sure the Assumption (1.4) holds, we introduce the new coordinate
$\left\\{\begin{array}[]{ll}X=x-\frac{\frac{\sqrt{3}}{3}m_{1}l-\frac{\sqrt{3}}{6}m_{2}l-\frac{\sqrt{3}}{6}m_{3}l}{M},\\\
Y=y-\frac{\frac{m_{2}}{2}l-\frac{m_{3}}{2}l}{M}.\end{array}\right.$
Hence in the new coordinate (X,Y), the location coordinates of
$q_{1}(0),q_{2}(0),q_{3}(0)$ are
$A_{1}(\frac{\frac{\sqrt{3}}{2}m_{2}l+\frac{\sqrt{3}}{2}m_{3}l}{M},$
$\frac{-\frac{m_{2}}{2}l+\frac{m_{3}}{2}l}{M}),$
$A_{2}(-\frac{\frac{\sqrt{3}}{2}m_{1}l}{M},\frac{\frac{m_{1}}{2}l+m_{3}l}{M}),A_{3}(-\frac{\frac{\sqrt{3}}{2}m_{1}l}{M},-\frac{\frac{m_{1}}{2}l+m_{2}l}{M})$
and the center of masses $m_{1},m_{2},m_{3}$ is at the origin $O(0,0)$. Then
compared with (1.3), we have
$r_{1}=|A_{1}O|=\frac{\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \ \ \ $
(2.5) $r_{2}=|A_{2}O|=\frac{\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}}{M}l,\ \ \ \
\ \ $ (2.6) $r_{3}=|A_{3}O|=\frac{\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}{M}l,\
\ \ \ \ \ $ (2.7)
and
$\sin\theta_{1}=\frac{-m_{2}+m_{3}}{2\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}},\
\ \ \ \ \ \
\cos\theta_{1}=\frac{\sqrt{3}(m_{2}+m_{3})}{2\sqrt{m^{2}_{2}+m_{2}m_{3}+m^{2}_{3}}},\
\ \ \ \ \ \ $ (2.8)
$\sin\theta_{2}=\frac{m_{1}+2m_{3}}{2\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}},\
\ \ \ \ \ \
\cos\theta_{2}=-\frac{\sqrt{3}m_{1}}{2\sqrt{m^{2}_{1}+m_{1}m_{3}+m^{2}_{3}}},\
\ \ \ \ $ (2.9)
$\sin\theta_{3}=-\frac{m_{1}+2m_{2}}{2\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}},\
\ \ \ \
\cos\theta_{3}=-\frac{\sqrt{3}m_{1}}{2\sqrt{m^{2}_{1}+m_{1}m_{2}+m^{2}_{2}}}.\
\ \ \Box$ (2.10)
## 3 Proof of Theorems
In order to get Theorems, we need two steps to complete the proof.
Step 1: We will establish the existence of variational minimizers of $f(q)$ in
(1.8) on $\bar{\Lambda}_{\pm}$.
Lemma 3.1 $f(q)$ in (1.8) attains its infimum on $\bar{\Lambda}_{\pm}$.
Proof. By using Lemma 2.2, for $\forall q\in\Lambda_{\pm}$, we can get that
the equivalent norm of (1.7) in $\bar{\Lambda}_{\pm}$ is
$\|q\|\cong\Big{[}\int_{0}^{T}|\dot{q}|^{2}dt\Big{]}^{\frac{1}{2}}.$ (3.1)
Hence by the definition of $f(q)$, $f$ is coercive on $\bar{\Lambda}_{\pm}$.
Next, we claim that $f$ is weakly lower semi-continuous on
$\bar{\Lambda}_{\pm}$. In fact, for $\forall q^{k}\in\Lambda_{\pm}$, if
$q^{k}\rightharpoonup q$ weakly, by compact embedding theorem, we have the
uniformly convergence:
$\max\limits_{0\leq t\leq T}|q^{k}(t)-q(t)|\rightarrow 0,\ \ \ \
k\rightarrow\infty,$ (3.2)
which implies
$\int_{0}^{T}\sum\limits_{i=1}^{3}\frac{m_{i}}{|q^{k}-q_{i}|}dt\rightarrow\int_{0}^{T}\sum\limits_{i=1}^{3}\frac{m_{i}}{|q-q_{i}|}dt.$
(3.3)
It is well-known that the norm and its square are weakly lower semi-
continuous. Therefore, combined with (3.3), we obtain
$\liminf\limits_{k\rightarrow\infty}f(q^{k})\geq f(q),$ (3.4)
that is, $f$ is weakly lower semi-continuous on $\bar{\Lambda}_{\pm}$. By
Lemma 2.1, we can get that $f(q)$ in (1.8) attains its infimum on
$\bar{\Lambda}_{\pm}$. $\Box$
Step 2: We will prove the variational minimizers in Lemma 3.1 is the
noncollision T-period solution of (1.6).
For any collision generalized solution $q$, we can estimate the lower bound
for the value of Lagrangian action functional.
Lemma 3.2 For $\partial\Lambda_{\pm}=\\{q\in
W^{1,2}(R/TZ,R^{2})|q(t+\frac{T}{2})=-q(t),\ \exists 1\leq i^{\pm}_{0}\leq
3,t_{i^{\pm}_{0}}\in[0,T]\ s.t.\
q_{i^{\pm}_{0}}(t_{i^{\pm}_{0}})=q(t_{i^{\pm}_{0}})\\}$, we have
$\inf\limits_{q\in\partial\Lambda_{\pm}}f(q)\geq\frac{3}{2}(2\pi)^{2/3}CM^{-1/3}T^{1/3}\triangleq
d_{1},$
where
$C=\min\left\\{\begin{array}[]{c}2^{\frac{2}{3}}m_{1}+m_{2}+m_{3}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}),\\\
2^{\frac{2}{3}}m_{2}+m_{1}+m_{3}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}),\\\
2^{\frac{2}{3}}m_{3}+m_{1}+m_{2}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3})\end{array}\right\\}.$
Proof. It follows from (1.4) that
$\sum\limits_{i=1}^{3}m_{i}\dot{q}_{i}=0,$ (3.5)
which implies
$\displaystyle\sum\limits_{i=1}^{3}m_{i}|\dot{q}-\dot{q}_{i}|^{2}$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{3}m_{i}\Big{(}|\dot{q}|^{2}+|\dot{q}_{i}|^{2}-2\langle\dot{q},\dot{q}_{i}\rangle\Big{)}$
(3.6) $\displaystyle=$ $\displaystyle
M|\dot{q}|^{2}+\sum\limits_{i=1}^{3}m_{i}|\dot{q}_{i}|^{2}-2\Big{\langle}\dot{q},\sum\limits_{i=1}^{3}m_{i}\dot{q}_{i}\Big{\rangle}$
$\displaystyle=$ $\displaystyle
M|\dot{q}|^{2}+\sum\limits_{i=1}^{3}m_{i}|\dot{q}_{i}|^{2}.$
Therefore
$|\dot{q}|^{2}=\frac{1}{M}\sum\limits_{i=1}^{3}m_{i}\Big{(}|\dot{q}-\dot{q}_{i}|^{2}-|\dot{q}_{i}|^{2}\Big{)}.$
(3.7)
Hence
$\displaystyle f(q)$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\Big{[}\frac{1}{2}|\dot{q}|^{2}+\sum\limits_{i=1}^{3}\frac{m_{i}}{|q-q_{i}|}\Big{]}dt$
(3.8) $\displaystyle=$
$\displaystyle\frac{1}{M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\Big{[}\frac{1}{2}|\dot{q}-\dot{q}_{i}|^{2}+\frac{M}{|q-q_{i}|}\Big{]}dt-\frac{1}{2M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}|\dot{q}_{i}|^{2}dt.$
If $q\in\bar{\Lambda}_{-}$ is a collision generalized solution, then there
exists $t_{i^{-}_{0}}\in[0,T]$ and $1\leq i^{-}_{0}\leq 3$ such that
$q(t_{i^{-}_{0}})=q_{i^{-}_{0}}(t_{i^{-}_{0}})$. Since
$q_{i}(t+\frac{T}{2})=-q_{i}(t)$, we obtain
$q(t_{i^{-}_{0}}+\frac{kT}{2})=q_{i^{-}_{0}}(t_{i^{-}_{0}}+\frac{kT}{2}),\ \
\forall 0\leq k\leq 2$. So, by (1) of Lemma 2.4, we get
$\displaystyle\frac{1}{M}\int_{0}^{T}m_{i^{-}_{0}}\Big{[}\frac{1}{2}|\dot{q}-\dot{q}_{i^{-}_{0}}|^{2}+\frac{M}{|q-q_{i^{-}_{0}}|}\Big{]}dt$
$\displaystyle=$
$\displaystyle\frac{2}{M}m_{i^{-}_{0}}\int_{0}^{\frac{T}{2}}\Big{[}\frac{1}{2}|\dot{q}-\dot{q}_{i^{-}_{0}}|^{2}+\frac{M}{|q-q_{i^{-}_{0}}|}\Big{]}dt$
(3.9) $\displaystyle\geq$
$\displaystyle\frac{3}{2}(2\pi)^{2/3}2^{2/3}m_{i^{-}_{0}}M^{-1/3}T^{1/3}.$
For noncollision pair $q,q_{i}(i\neq i^{-}_{0})$, we have
$\int_{0}^{T}q(t)dt=0$, $\int_{0}^{T}q_{i}(t)dt=0$. Therefore
$\int_{0}^{T}\big{(}q(t)-q_{i}(t)\big{)}dt=0$. Hence by (2) of Lemma 2.4, we
can get
$\frac{1}{M}\int_{0}^{T}\sum\limits_{i\neq
i^{-}_{0}}m_{i}\Big{[}\frac{1}{2}|\dot{q}-\dot{q}_{i}|^{2}+\frac{M}{|q-q_{i}|}\Big{]}dt\geq\frac{3}{2}(2\pi)^{2/3}(M-m_{i^{-}_{0}})M^{-1/3}T^{1/3}.$
(3.10)
For the other term of $f$, using the expression for the orbits
$q_{1},q_{2},q_{3}$ as in (1.3), Lemma 2.5 and Lemma 2.6, we obtain
$-\frac{1}{2M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}|\dot{q}_{i}|^{2}dt=-\frac{1}{2}(2\pi)^{2/3}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3})M^{-4/3}T^{1/3}.$
(3.11)
Therefore, it follows from (3.9) - (3.11) that
$\inf\limits_{q\in\partial\Lambda_{-}}f(q)\geq\frac{3}{2}(2\pi)^{2/3}CM^{-1/3}T^{1/3}\triangleq
d_{1},$ (3.12)
where
$C=\min\left\\{\begin{array}[]{c}2^{\frac{2}{3}}m_{1}+m_{2}+m_{3}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}),\\\
2^{\frac{2}{3}}m_{2}+m_{1}+m_{3}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}),\\\
2^{\frac{2}{3}}m_{3}+m_{1}+m_{2}-\frac{1}{3M}(m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3})\end{array}\right\\}.$
Similarly, if $q\in\bar{\Lambda}_{+}$ is a collision generalized solution, we
have
$\inf\limits_{q\in\partial\Lambda_{+}}f(q)\geq\frac{3}{2}(2\pi)^{2/3}CM^{-1/3}T^{1/3}\triangleq
d_{1},\ \ \Box$ (3.13)
Proof of Theorem 1.1 In order to get Theorem 1.1, we are going to find a test
loop $\tilde{q}\in\Lambda_{-}$ such that $f(\tilde{q})\leq d_{2}$. Then the
minimizer of $f$ on $\bar{\Lambda}_{-}$ must be a noncollision solution if
$d_{2}<d_{1}$.
Let $a>0,b>0$, $\theta\in[0,2\pi)$ and
$\tilde{q}-q_{1}=\bigg{(}a\cos\Big{(}-\frac{2\pi}{T}t+\theta\Big{)},b\sin\Big{(}-\frac{2\pi}{T}t+\theta\Big{)}\bigg{)}^{T}.\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ $ (3.14)
Hence
$\displaystyle\tilde{q}-q_{2}$ $\displaystyle=$
$\displaystyle\tilde{q}-q_{1}+q_{1}-q_{2}$ (3.15) $\displaystyle=$
$\displaystyle(q_{1}-q_{2})+(\tilde{q}-q_{1})$ $\displaystyle=$
$\displaystyle\bigg{(}r_{1}\cos\Big{(}\frac{2\pi}{T}t+\theta_{1}\Big{)}-r_{2}\cos\Big{(}\frac{2\pi}{T}t+\theta_{2}\Big{)}+a\cos\Big{(}-\frac{2\pi}{T}t+\theta\Big{)},r_{1}\sin\Big{(}\frac{2\pi}{T}t+\theta_{1}\Big{)}\
\ \ \ \ \ \ $ $\displaystyle-
r_{2}\sin\Big{(}\frac{2\pi}{T}t+\theta_{2}\Big{)}+b\sin\Big{(}-\frac{2\pi}{T}t+\theta\Big{)}\bigg{)}^{T},$
$\displaystyle\tilde{q}-q_{3}$ $\displaystyle=$
$\displaystyle\bigg{(}r_{1}\cos\Big{(}\frac{2\pi}{T}t+\theta_{1}\Big{)}-r_{3}\cos\Big{(}\frac{2\pi}{T}t+\theta_{3}\Big{)}+a\cos\Big{(}-\frac{2\pi}{T}t+\theta\Big{)},r_{1}\sin\Big{(}\frac{2\pi}{T}t+\theta_{1}\Big{)}\
\ \ \ \ \ \ $ (3.16) $\displaystyle-
r_{3}\sin\Big{(}\frac{2\pi}{T}t+\theta_{3}\Big{)}+b\sin\Big{(}-\frac{2\pi}{T}t+\theta\Big{)}\bigg{)}^{T}.$
It is easy to see that $\tilde{q}\in\Lambda_{-}$ and
$\displaystyle|\dot{\tilde{q}}-\dot{q_{1}}|^{2}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{2\pi}{T}\Big{)}^{2}\bigg{[}\frac{a^{2}+b^{2}}{2}-\frac{a^{2}-b^{2}}{2}cos\Big{(}\frac{4\pi}{T}t-2\theta\Big{)}\bigg{]},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
(3.17)
$\displaystyle|\tilde{q}-q_{1}|=\sqrt{\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big{(}\frac{4\pi}{T}t-2\theta\Big{)}},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
(3.18) $\displaystyle|\dot{\tilde{q}}-\dot{q_{2}}|^{2}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{2\pi}{T}\Big{)}^{2}\bigg{\\{}\frac{a^{2}+b^{2}}{2}-\frac{a^{2}-b^{2}}{2}cos\Big{(}\frac{4\pi}{T}t-2\theta\Big{)}+r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})$
(3.19)
$\displaystyle-(a+b)\Big{[}r_{1}cos\Big{(}\frac{4\pi}{T}t+\theta_{1}-\theta\Big{)}-r_{2}cos\Big{(}\frac{4\pi}{T}t+\theta_{2}-\theta\Big{)}\Big{]}$
$\displaystyle+(a-b)\big{[}r_{1}cos(\theta_{1}+\theta)-r_{2}cos(\theta_{2}+\theta)\big{]}\bigg{\\}},$
$\displaystyle|\tilde{q}-q_{2}|$ $\displaystyle=$
$\displaystyle\bigg{\\{}\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big{(}\frac{4\pi}{T}t-2\theta\Big{)}+r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})\
\ \ \ \ \ \ \ $ (3.20)
$\displaystyle+(a+b)\Big{[}r_{1}cos\Big{(}\frac{4\pi}{T}t+\theta_{1}-\theta\Big{)}-r_{2}cos\Big{(}\frac{4\pi}{T}t+\theta_{2}-\theta\Big{)}\Big{]}$
$\displaystyle+(a-b)\big{[}r_{1}cos(\theta_{1}+\theta)-r_{2}cos(\theta_{2}+\theta)\big{]}\bigg{\\}}^{\frac{1}{2}},$
$\displaystyle|\dot{\tilde{q}}-\dot{q_{3}}|^{2}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{2\pi}{T}\Big{)}^{2}\bigg{\\{}\frac{a^{2}+b^{2}}{2}-\frac{a^{2}-b^{2}}{2}cos\Big{(}\frac{4\pi}{T}t-2\theta\Big{)}+r_{1}^{2}+r_{3}^{2}-2r_{1}r_{3}cos(\theta_{3}-\theta_{1})$
(3.21)
$\displaystyle-(a+b)\Big{[}r_{1}cos\Big{(}\frac{4\pi}{T}t+\theta_{1}-\theta\Big{)}-r_{3}cos\Big{(}\frac{4\pi}{T}t+\theta_{3}-\theta\Big{)}\Big{]}$
$\displaystyle+(a-b)\big{[}r_{1}cos(\theta_{1}+\theta)-r_{3}cos(\theta_{3}+\theta)\big{]}\bigg{\\}},$
$\displaystyle|\tilde{q}-q_{3}|$ $\displaystyle=$
$\displaystyle\bigg{\\{}\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big{(}\frac{4\pi}{T}t-2\theta\Big{)}+r_{1}^{2}+r_{3}^{2}-2r_{1}r_{3}cos(\theta_{2}-\theta_{1})\
\ \ \ \ \ \ \ $ (3.22)
$\displaystyle+(a+b)\Big{[}r_{1}cos\Big{(}\frac{4\pi}{T}t+\theta_{1}-\theta\Big{)}-r_{3}cos\Big{(}\frac{4\pi}{T}t+\theta_{3}-\theta\Big{)}\Big{]}$
$\displaystyle+(a-b)\big{[}r_{1}cos(\theta_{1}+\theta)-r_{3}cos(\theta_{3}+\theta)\big{]}\bigg{\\}}^{\frac{1}{2}},$
$|\dot{q_{1}}|^{2}=\Big{(}\frac{2\pi}{T}\Big{)}^{2}r_{1}^{2},\ \ \ \
|\dot{q_{2}}|^{2}=\Big{(}\frac{2\pi}{T}\Big{)}^{2}r_{2}^{2},\ \ \ \
|\dot{q_{3}}|^{2}=\Big{(}\frac{2\pi}{T}\Big{)}^{2}r_{3}^{2}.$ (3.23)
Therefore by (3.17)-(3.23), we get
$\displaystyle f(\tilde{q})$ $\displaystyle=$
$\displaystyle\frac{1}{M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\Big{[}\frac{1}{2}|\dot{\tilde{q}}-\dot{q}_{i}|^{2}+\frac{M}{|\tilde{q}-q_{i}|}\Big{]}dt-\frac{1}{2M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}|\dot{q}_{i}|^{2}dt$
(3.24) $\displaystyle=$
$\displaystyle\frac{2\pi^{2}}{T}\Big{\\{}\frac{a^{2}+b^{2}}{2}+\frac{m_{2}+m_{3}-m_{1}}{M}r_{1}^{2}-\frac{2m_{2}r_{2}cos(\theta_{2}-\theta_{1})+2m_{3}r_{3}cos(\theta_{3}-\theta_{1})}{M}r_{1}$
$\displaystyle+\frac{m_{2}(a-b)}{M}[r_{1}cos(\theta_{1}+\theta)-r_{2}cos(\theta_{2}+\theta)]+\frac{m_{3}(a-b)}{M}[r_{1}cos(\theta_{1}+\theta)$
$\displaystyle-
r_{3}cos(\theta_{3}+\theta)]\Big{\\}}+m_{1}\int_{0}^{T}\bigg{[}\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big{(}\frac{4\pi}{T}t-2\theta\Big{)}\bigg{]}^{-\frac{1}{2}}dt$
$\displaystyle+\sum\limits_{i=2}^{3}\int_{0}^{T}m_{i}\bigg{\\{}\frac{a^{2}+b^{2}}{2}+\frac{a^{2}-b^{2}}{2}cos\Big{(}\frac{4\pi}{T}t-2\theta\Big{)}+r_{1}^{2}+r_{i}^{2}-2r_{1}r_{i}cos(\theta_{i}-\theta_{1})$
$\displaystyle+(a+b)\Big{[}r_{1}cos\Big{(}\frac{4\pi}{T}t+\theta_{1}-\theta\Big{)}-r_{i}cos\Big{(}\frac{4\pi}{T}t+\theta_{i}-\theta\Big{)}\Big{]}$
$\displaystyle+(a-b)\big{[}r_{1}cos(\theta_{1}+\theta)-r_{i}cos(\theta_{i}+\theta)\big{]}\bigg{\\}}^{-\frac{1}{2}}dt$
$\displaystyle=$ $\displaystyle d_{2}(a,b,\theta).$
In order to estimate $d_{2}$, we have computed the numerical values of
$d_{2}=f(q)$ over some selected test loops. The computation of the integral
that appears in (3.24) has been done using the function $\\{quad\\}$ of
Mathematica 7.1 with an error less than $10^{-6}$. Let $T=1$, the results of
the numerical explorations are given in Table 1 with $M=1$ and Table 2 with
$m_{1}=m_{2}=m_{3}=1$.
Table 1: Parameters for test loops for Theoerm 1.1 a | b | $\theta$ | m1 | m2 | m3 | d1 | d2
---|---|---|---|---|---|---|---
0.13 | 0.49 | $\frac{\pi}{20}$ | 0.29 | 0.42 | 0.29 | 5.419669 | 5.417862
0.15 | 0.49 | $\frac{\pi}{20}$ | 0.29 | 0.41 | 0.30 | 5.417626 | 5.416591
0.15 | 0.49 | $\frac{\pi}{20}$ | 0.29 | 0.42 | 0.29 | 5.419669 | 5.413794
0.15 | 0.49 | $\frac{\pi}{20}$ | 0.30 | 0.35 | 0.35 | 5.441499 | 5.436767
0.15 | 0.51 | $\frac{\pi}{20}$ | 0.30 | 0.36 | 0.34 | 5.441669 | 5.437985
0.15 | 0.51 | $\frac{\pi}{20}$ | 0.30 | 0.37 | 0.33 | 5.442180 | 5.433615
0.15 | 0.51 | $\frac{\pi}{20}$ | 0.30 | 0.38 | 0.32 | 5.443031 | 5.429587
0.15 | 0.51 | $\frac{\pi}{20}$ | 0.30 | 0.39 | 0.31 | 5.444223 | 5.425898
0.15 | 0.51 | $\frac{\pi}{20}$ | 0.30 | 0.40 | 0.30 | 5.445755 | 5.422550
0.15 | 0.53 | $\frac{\pi}{20}$ | 0.31 | 0.35 | 0.34 | 5.470820 | 5.467576
0.15 | 0.53 | $\frac{\pi}{20}$ | 0.31 | 0.36 | 0.33 | 5.471160 | 5.462971
0.15 | 0.53 | $\frac{\pi}{20}$ | 0.31 | 0.37 | 0.32 | 5.471841 | 5.458707
0.15 | 0.53 | $\frac{\pi}{20}$ | 0.31 | 0.38 | 0.31 | 5.472863 | 5.454784
0.17 | 0.45 | $\frac{\pi}{20}$ | 0.32 | 0.32 | 0.36 | 5.500992 | 5.488608
0.17 | 0.47 | $\frac{\pi}{20}$ | 0.32 | 0.33 | 0.35 | 5.500481 | 5.454518
0.17 | 0.47 | $\frac{\pi}{20}$ | 0.32 | 0.34 | 0.34 | 5.500311 | 5.449987
0.17 | 0.47 | $\frac{\pi}{20}$ | 0.33 | 0.34 | 0.33 | 5.530142 | 5.444254
0.45 | 0.15 | $\pi$ | 0.33 | 0.31 | 0.36 | 5.471160 | 5.456006
0.45 | 0.15 | $\pi$ | 0.33 | 0.32 | 0.35 | 5.500481 | 5.455325
0.45 | 0.15 | $\pi$ | 0.33 | 0.33 | 0.34 | 5.530142 | 5.454984
0.47 | 0.13 | $\pi$ | 0.34 | 0.30 | 0.36 | 5.441669 | 5.439671
0.47 | 0.13 | $\pi$ | 0.34 | 0.31 | 0.35 | 5.470820 | 5.438820
0.47 | 0.13 | $\pi$ | 0.34 | 0.32 | 0.34 | 5.500311 | 5.438309
0.47 | 0.15 | $\pi$ | 0.35 | 0.30 | 0.35 | 5.441499 | 5.417900
0.49 | 0.15 | $\pi$ | 0.36 | 0.29 | 0.35 | 5.412519 | 5.411552
0.49 | 0.15 | $\pi$ | 0.36 | 0.32 | 0.32 | 5.500992 | 5.410020
0.49 | 0.15 | $\pi$ | 0.37 | 0.29 | 0.34 | 5.412859 | 5.411962
0.49 | 0.15 | $\pi$ | 0.37 | 0.30 | 0.33 | 5.442180 | 5.411281
0.49 | 0.15 | $\pi$ | 0.37 | 0.31 | 0.32 | 5.471841 | 5.410940
0.49 | 0.15 | $\pi$ | 0.38 | 0.29 | 0.33 | 5.413540 | 5.412712
0.49 | 0.15 | $\pi$ | 0.38 | 0.30 | 0.32 | 5.443031 | 5.412201
0.49 | 0.15 | $\pi$ | 0.38 | 0.31 | 0.31 | 5.472863 | 5.412031
0.49 | 0.15 | $\pi$ | 0.39 | 0.29 | 0.32 | 5.414562 | 5.413803
0.49 | 0.15 | $\pi$ | 0.39 | 0.30 | 0.31 | 5.444223 | 5.413462
0.49 | 0.17 | $\pi$ | 0.40 | 0.29 | 0.31 | 5.415924 | 5.415807
0.49 | 0.17 | $\pi$ | 0.40 | 0.30 | 0.30 | 5.445755 | 5.415637
0.49 | 0.17 | $\pi$ | 0.41 | 0.30 | 0.29 | 5.417626 | 5.416078
0.49 | 0.17 | $\pi$ | 0.42 | 0.29 | 0.29 | 5.419669 | 5.416689
Table 2: Parameters for test loops for Theoerm 1.1 a | b | $\theta$ | m1 | m2 | m3 | d1 | d2
---|---|---|---|---|---|---|---
0.15 | 0.67 | $\frac{\pi}{30}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.505860
0.15 | 0.67 | $\frac{\pi}{30}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.505860
0.15 | 0.69 | $\frac{\pi}{30}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.444212
0.17 | 0.65 | $\frac{\pi}{30}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.493238
0.17 | 0.67 | $\frac{\pi}{20}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.452135
0.17 | 0.69 | $\frac{\pi}{20}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.400124
0.19 | 0.63 | $\frac{\pi}{30}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.519350
0.19 | 0.65 | $\frac{\pi}{20}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.455969
0.19 | 0.67 | $\frac{\pi}{20}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.386608
0.19 | 0.69 | $\frac{\pi}{20}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.344747
0.61 | 0.23 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.516685
0.63 | 0.19 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.489791
0.63 | 0.21 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.436105
0.65 | 0.17 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.461786
0.65 | 0.19 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.392115
0.65 | 0.21 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.349366
0.67 | 0.15 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.472422
0.67 | 0.17 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.383978
0.67 | 0.19 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.324970
0.67 | 0.21 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.291915
0.69 | 0.13 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.522980
0.69 | 0.15 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.412094
0.69 | 0.17 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.334189
0.69 | 0.19 | $\pi$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.284714
For the parameters $a,b,\theta$ given in Table 1 and Table 2, we all have
$d_{2}<d_{1}$. This completes the Proof of Theorem 1.1. $\Box$
Proof of Theorem 1.2 To get Theorem 1.2, we are going to find a test loop
$\bar{q}\in\Lambda_{+}$ such that $f(\bar{q})\leq d_{3}$. Then the minimizer
of $f$ on $\bar{\Lambda}_{+}$ must be a noncollision solution if
$d_{3}<d_{1}$.
Let $a>0$, $\theta\in[0,2\pi)$ and
$\bar{q}-q_{1}=ae^{\sqrt{-1}(\frac{2\pi}{T}t+\theta)}.\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (3.25)
Hence
$\displaystyle\bar{q}-q_{2}$ $\displaystyle=$ $\displaystyle
q_{1}+ae^{\sqrt{-1}(\frac{2\pi}{T}t+\theta)}-q_{2}$ (3.26) $\displaystyle=$
$\displaystyle
r_{1}e^{\sqrt{-1}(\frac{2\pi}{T}t+\theta_{1})}-r_{2}e^{\sqrt{-1}(\frac{2\pi}{T}t+\theta_{2})}+ae^{\sqrt{-1}(\frac{2\pi}{T}t+\theta)},$
$\displaystyle\bar{q}-q_{3}$ $\displaystyle=$ $\displaystyle
q_{1}+ae^{\sqrt{-1}(\frac{2\pi}{T}t+\theta)}-q_{3}$ (3.27) $\displaystyle=$
$\displaystyle
r_{1}e^{\sqrt{-1}(\frac{2\pi}{T}t+\theta_{1})}-r_{3}e^{\sqrt{-1}(\frac{2\pi}{T}t+\theta_{3})}+ae^{\sqrt{-1}(\frac{2\pi}{T}t+\theta)}.$
It is easy to see that $\bar{q}\in\Lambda_{+}$ and
$\displaystyle|\dot{\bar{q}}-\dot{q_{1}}|^{2}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{2\pi}{T}\Big{)}^{2}a^{2},\ \ \ \
|\bar{q}-q_{1}|=a,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (3.28)
$\displaystyle|\dot{\bar{q}}-\dot{q_{2}}|^{2}=\Big{(}\frac{2\pi}{T}\Big{)}^{2}\big{[}a^{2}+r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta)-2ar_{2}cos(\theta_{2}-\theta)\big{]},$
(3.29)
$\displaystyle|\bar{q}-q_{2}|=\big{[}a^{2}+r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta)-2ar_{2}cos(\theta_{2}-\theta)\big{]}^{\frac{1}{2}},\
\ \ \ \ \ $ (3.30)
$\displaystyle|\dot{\bar{q}}-\dot{q_{3}}|^{2}=\Big{(}\frac{2\pi}{T}\Big{)}^{2}\big{[}a^{2}+r_{1}^{2}+r_{3}^{2}-2r_{1}r_{3}cos(\theta_{3}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta)-2ar_{3}cos(\theta_{3}-\theta)\big{]},$
(3.31)
$\displaystyle|\bar{q}-q_{3}|=\big{[}a^{2}+r_{1}^{2}+r_{3}^{2}-2r_{1}r_{3}cos(\theta_{3}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta)-2ar_{3}cos(\theta_{3}-\theta)\big{]}^{\frac{1}{2}},\
\ \ \ \ \ $ (3.32)
$|\dot{q_{1}}|^{2}=\Big{(}\frac{2\pi}{T}\Big{)}^{2}r_{1}^{2},\ \ \ \
|\dot{q_{2}}|^{2}=\Big{(}\frac{2\pi}{T}\Big{)}^{2}r_{2}^{2},\ \ \ \
|\dot{q_{3}}|^{2}=\Big{(}\frac{2\pi}{T}\Big{)}^{2}r_{3}^{2}.$ (3.33)
Therefore by (3.28)-(3.33), we get
$\displaystyle f(\bar{q})$ $\displaystyle=$
$\displaystyle\frac{1}{M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}\Big{[}\frac{1}{2}|\dot{\bar{q}}-\dot{q}_{i}|^{2}+\frac{M}{|\bar{q}-q_{i}|}\Big{]}dt-\frac{1}{2M}\int_{0}^{T}\sum\limits_{i=1}^{3}m_{i}|\dot{q}_{i}|^{2}dt$
(3.34) $\displaystyle=$
$\displaystyle\frac{2\pi^{2}}{T}\Big{[}a^{2}+\frac{m_{2}+m_{3}-m_{1}}{M}r_{1}^{2}-\frac{2m_{2}r_{2}cos(\theta_{2}-\theta_{1})+2m_{3}r_{3}cos(\theta_{3}-\theta_{1})}{M}r_{1}$
$\displaystyle+\frac{2(m_{2}+m_{3})}{M}ar_{1}cos(\theta_{1}-\theta)-\frac{2m_{2}r_{2}cos(\theta_{2}-\theta)+2m_{3}r_{3}cos(\theta_{3}-\theta)}{M}a\Big{]}$
$\displaystyle+\frac{m_{1}T}{a}+m_{2}\int_{0}^{T}\big{[}a^{2}+r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}cos(\theta_{2}-\theta_{1})+2ar_{1}cos(\theta_{1}-\theta)$
$\displaystyle-2ar_{2}cos(\theta_{2}-\theta)\big{]}^{-\frac{1}{2}}dt+m_{3}\int_{0}^{T}\big{[}a^{2}+r_{1}^{2}+r_{3}^{2}-2r_{1}r_{3}cos(\theta_{3}-\theta_{1})$
$\displaystyle+2ar_{1}cos(\theta_{1}-\theta)-2ar_{3}cos(\theta_{3}-\theta)\big{]}^{-\frac{1}{2}}dt$
$\displaystyle=$ $\displaystyle d_{3}(a,\theta).$
In order to estimate $d_{3}$, we have computed the numerical values of
$d_{3}=f(q)$ over some selected test loops. The computation of the integral
that appears in (3.34) has been done using the function $\\{quad\\}$ of
Mathematica 7.1 with an error less than $10^{-6}$. Let $T=1$, the results of
the numerical explorations are given in Table 3 with $M=1$ and Table 4 with
$m_{1}=m_{2}=m_{3}=1$.
Table 3: Parameters for test loops for Theoerm 1.2 a | $\theta$ | m1 | m2 | m3 | d1 | d3
---|---|---|---|---|---|---
0.17 | $\frac{\pi}{2}$ | 0.10 | 0.75 | 0.15 | 5.062791 | 5.060773
0.17 | $\frac{\pi}{2}$ | 0.10 | 0.77 | 0.13 | 5.083903 | 5.071551
0.17 | $\frac{\pi}{2}$ | 0.10 | 0.78 | 0.12 | 5.094969 | 5.077450
0.17 | $\frac{\pi}{2}$ | 0.10 | 0.80 | 0.10 | 5.118123 | 5.090270
0.17 | $\frac{\pi}{2}$ | 0.15 | 0.53 | 0.32 | 5.051742 | 5.050040
0.17 | $\frac{\pi}{2}$ | 0.15 | 0.57 | 0.28 | 5.068768 | 5.046398
0.17 | $\frac{\pi}{2}$ | 0.15 | 0.60 | 0.25 | 5.085112 | 5.047242
0.17 | $\frac{\pi}{2}$ | 0.15 | 0.65 | 0.20 | 5.119162 | 5.055458
0.17 | $\frac{\pi}{2}$ | 0.15 | 0.70 | 0.15 | 5.161725 | 5.072186
0.17 | $\frac{\pi}{2}$ | 0.15 | 0.72 | 0.13 | 5.121130 | 5.081261
0.17 | $\frac{\pi}{2}$ | 0.20 | 0.31 | 0.49 | 5.176554 | 5.175168
0.17 | $\frac{\pi}{2}$ | 0.20 | 0.35 | 0.45 | 5.167020 | 5.144967
0.17 | $\frac{\pi}{2}$ | 0.20 | 0.40 | 0.40 | 5.162763 | 5.114876
0.17 | $\frac{\pi}{2}$ | 0.20 | 0.50 | 0.30 | 5.179789 | 5.080232
0.17 | $\frac{\pi}{2}$ | 0.20 | 0.55 | 0.25 | 5.201070 | 5.075680
0.17 | $\frac{\pi}{2}$ | 0.20 | 0.60 | 0.20 | 5.230864 | 5.079639
0.19 | $\frac{\pi}{2}$ | 0.25 | 0.22 | 0.53 | 5.249837 | 5.237465
0.19 | $\frac{\pi}{2}$ | 0.25 | 0.25 | 0.50 | 5.325541 | 5.202291
0.19 | $\frac{\pi}{2}$ | 0.25 | 0.30 | 0.45 | 5.308516 | 5.150479
0.19 | $\frac{\pi}{2}$ | 0.25 | 0.35 | 0.40 | 5.300003 | 5.107178
0.19 | $\frac{\pi}{2}$ | 0.25 | 0.62 | 0.13 | 5.041112 | 5.020454
0.19 | $\frac{\pi}{2}$ | 0.30 | 0.22 | 0.48 | 5.230258 | 5.222385
0.19 | $\frac{\pi}{2}$ | 0.30 | 0.25 | 0.45 | 5.308516 | 5.189765
0.19 | $\frac{\pi}{2}$ | 0.30 | 0.30 | 0.40 | 5.445755 | 5.142208
0.19 | $\frac{\pi}{2}$ | 0.30 | 0.35 | 0.35 | 5.441499 | 5.103164
0.19 | $\frac{\pi}{2}$ | 0.30 | 0.56 | 0.14 | 5.036553 | 5.032137
0.21 | $\frac{\pi}{2}$ | 0.35 | 0.21 | 0.44 | 5.192935 | 5.184596
0.21 | $\frac{\pi}{2}$ | 0.35 | 0.29 | 0.36 | 5.412519 | 5.092092
0.21 | $\frac{\pi}{2}$ | 0.35 | 0.39 | 0.26 | 5.327621 | 5.007107
0.21 | $\frac{\pi}{2}$ | 0.35 | 0.48 | 0.17 | 5.091316 | 4.959734
0.21 | $\frac{\pi}{2}$ | 0.35 | 0.53 | 0.12 | 4.971952 | 4.945333
0.21 | $\frac{\pi}{3}$ | 0.40 | 0.28 | 0.32 | 5.386433 | 5.342981
0.21 | $\frac{\pi}{3}$ | 0.40 | 0.32 | 0.28 | 5.386433 | 5.287294
0.21 | $\frac{\pi}{3}$ | 0.40 | 0.36 | 0.24 | 5.271874 | 5.237055
0.21 | $\frac{\pi}{3}$ | 0.40 | 0.38 | 0.22 | 5.216638 | 5.213978
0.23 | $\frac{\pi}{2}$ | 0.45 | 0.19 | 0.36 | 5.139742 | 5.127834
0.23 | $\frac{\pi}{2}$ | 0.45 | 0.29 | 0.26 | 5.337836 | 5.003006
0.23 | $\frac{\pi}{2}$ | 0.45 | 0.37 | 0.18 | 5.112805 | 4.927660
0.23 | $\frac{\pi}{2}$ | 0.45 | 0.46 | 0.09 | 4.885693 | 4.868944
0.23 | $\frac{\pi}{2}$ | 0.50 | 0.18 | 0.32 | 5.123871 | 5.108878
0.23 | $\frac{\pi}{2}$ | 0.50 | 0.23 | 0.27 | 5.266218 | 5.044762
0.23 | $\frac{\pi}{2}$ | 0.50 | 0.29 | 0.21 | 5.208258 | 4.979058
0.23 | $\frac{\pi}{2}$ | 0.50 | 0.37 | 0.13 | 4.990036 | 4.910522
0.23 | $\frac{\pi}{2}$ | 0.50 | 0.41 | 0.09 | 4.889098 | 4.884426
Table 4: Parameters for test loops for Theoerm 1.2 a | $\theta$ | m1 | m2 | m3 | d1 | d3
---|---|---|---|---|---|---
0.21 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.327950
0.23 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.036769
0.23 | $\frac{\pi}{3}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.336568
0.25 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 10.821272
0.25 | $\frac{\pi}{3}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.187475
0.25 | $\frac{\pi}{4}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.453195
0.27 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 10.667031
0.27 | $\frac{\pi}{3}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.107374
0.27 | $\frac{\pi}{4}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.411685
0.29 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 10.563849
0.29 | $\frac{\pi}{3}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.085761
0.29 | $\frac{\pi}{4}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.430090
0.31 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 10.504424
0.31 | $\frac{\pi}{3}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.114860
0.31 | $\frac{\pi}{4}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.500414
0.33 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 10.483477
0.33 | $\frac{\pi}{3}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.188786
0.35 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 10.497161
0.35 | $\frac{\pi}{3}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.302997
0.37 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 10.542652
0.37 | $\frac{\pi}{3}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 11.453926
0.39 | $\frac{\pi}{2}$ | 1.00 | 1.00 | 1.00 | 11.523843 | 10.617860
For the parameters $a,\theta$ given in Table 3 and Table 4, we all have
$d_{3}<d_{1}$. This completes the Proof of Theorem 1.2. $\Box$
## References
* [1] A. Ambrosetti, V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkhäuser, Basel, 1993.
* [2] A. Bahri, P. Rabinowitz, Periodic solutions of Hamiltonian systems of three body type, Ann. IHP. nonlineaire, 8(6)(1991), 561-649.
* [3] A. Chenciner, R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152(3)(2000), 881-901.
* [4] A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: From homology to symmetry, ICM 2002, Vol. 3, 255-264.
* [5] A. Venturelli, Une caracterisation variationnelle des solutions de Lagrange du probleme plan des trois corps, C. R. Acad. Sci. Paris, 332(7)(2001), 641-644.
* [6] C. Marchal, How the method of minimization of action avoids singularities, Cel. Mech. and Dyn. Astronomy, 83(1-4)(2002), 325-353.
* [7] C. Moore, Braids in classical gravity, Physical Review Letters, 70(24)(1993), 3675-3679.
* [8] C. Simó, New families of solutions in N-body problems, Progress in Math., 201(2001), 101-115.
* [9] D. Ferrario, S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invention Math., 155(2)(2004), 305-362.
* [10] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989.
* [11] M. Struwe, Variational methods, Third Edition, Springer-Verlag, 1990.
* [12] R. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69(1)(1979), 19-30.
* [13] S. Q. Zhang, Q. Zhou, A minimizing property of Lagrangian solutions, Acta Math. Sinica, 17(3)(2001), 497-500.
* [14] S. Q. Zhang, Q. Zhou, Variational methods for the choreography solution to the three-body problem, Science in China, 45(5)(2002), 594-597.
* [15] S. Q. Zhang, Q. Zhou, Nonplanar and noncollision periodic solutions for N-body problems, DCDS-A, 10(3)(2004), 679-685.
* [16] U. Bessi, V. Coti Zelati, Symmetries and noncollision closed orbits for planar N-body-type problems, Nonlinear Anal. TMA, 16(1991), 587-598.
* [17] W. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99(5)(1977), 961-971.
* [18] Y. M. Long, S. Q. Zhang, Goemetric characterizations for variational minimization solutions of the 3-body problems, Acta Math. Sinica, 16(4)(2000), 579-592.
|
arxiv-papers
| 2013-01-04T16:42:42 |
2024-09-04T02:49:39.922931
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaoxiao Zhao and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1301.0767"
}
|
1301.0791
|
# Tower multiplexing and slow weak mixing
Terrence M. Adams Terrence M. Adams is with the Department of Defense, 9161
Sterling Drive Laurel, MD 20723 United States of America
###### Abstract
A technique is presented for multiplexing two ergodic measure preserving
transformations together to derive a third limiting transformation. This
technique is used to settle a question regarding rigidity sequences of weak
mixing transformations. Namely, given any rigidity sequence for an ergodic
measure preserving transformation, there exists a weak mixing transformation
which is rigid along the same sequence. This establishes a wide range of
rigidity sequences for weakly mixing dynamical systems.
## 1 Introduction
Fix a Lebesgue probability space. Endow the set of invertible measure
preserving transformations with the weak topology. It is well known that both
the properties of weak mixing and rigidity are generic properties in this
topological space [16]. This is interesting since the behaviors of these two
properties contrast greatly. Weak mixing occurs when a system equitably
spreads mass throughout the probability space for most times. Rigidity occurs
when a system evolves to resemble the identity map infinitely often. Since
both of these behaviors exist simultaneously in a large class of
transformations, it is natural to ask what types of rigidity sequences are
realizable by weak mixing transformations. Here we resolve this question to
show all rigidity sequences are realizable by the class of weak mixing
transformations.
###### Theorem 1.1.
Given any ergodic measure preserving transformation $R$ on a Lebesgue
probability space, and any rigid sequence $\rho_{n}$ for $R$, there exists a
weak mixing transformation $T$ on a Lebesgue probability space such that $T$
is rigid on $\rho_{n}$.
Prior to proving this main result, we present a new and direct method for
combining two invertible ergodic finite measure preserving transformations to
obtain a third limiting transformation. The technique iteratively utilizes the
Kakutani-Rokhlin lemma ([19],[24]). A measure preserving transformation $T$ on
a separable probability space $(X,\mathbb{B},\mu)$ is ergodic if any invariant
measurable set $A$ has measure 0 or 1. In particular, $TA=A$ implies
$\mu(A)=\mbox{0 or 1}$.
###### Lemma 1.2.
(Kakutani 1943, Rokhlin 1948) Let $T:X\to X$ be an ergodic measure preserving
transformation on a nonatomic probability space $(X,\mathcal{B},\mu)$, $h$ a
positive integer and $\epsilon>0$. There exists a measurable set $B\subset X$
such that $B,TB,\ldots,T^{h-1}B$ are pairwise disjoint and
$\mu(\bigcup_{i=0}^{h-1}T^{i}(B))>1-\epsilon$. The collection
$\\{B,TB,\ldots,T^{h-1}B\\}$ is referred to as a Rokhlin tower of height $h$
for transformation $T$.
Clearly, this lemma demonstrates that any ergodic measure preserving
transformation can be approximated arbitrarily well by periodic
transformations in an appropriate topology (i.e. uniform topology). See Halmos
[17], [16], Rokhlin [25], Katok and Stepin [22]. Much of the early work in
this regard focuses on the topological genericity of specific properties of
measure perserving transformations. In [22], results are presented on rates of
approximation by periodic transformations, and connections with dynamical
properties. Recent research of Kalikow demonstrates the utility of developing
a general theory of Rokhlin towers [20]. Also, it is clear from the Kakutani-
Rokhlin lemma that any ergodic measure preserving tranformation can be
approximated arbitrarily well by another ergodic measure preserving
transformation from any isomorphism class. This observation is utilized
repeatedly in this work.
Two input transformations $R$ and $S$ are multiplexed together to derive an
output transformation $T$ with prescribed properties. The multiplexing
operation is defined using an infinite chain of measure theoretic
isomorphisms. In the case where $R$ is ergodic and rigid, and $S$ weak mixing,
we present a method for unbalanced multiplexing of $R$ and $S$. Over time,
transformations isomorphic to $R$ are used on a higher proportion of the
measure space, as the action by $S$ dissipates over time. We refer to this
process informally as slow weak mixing.
A measure preserving transformation $T:X\to X$ is weak mixing, if for all
measurable sets $A$ and $B$,
$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}|\mu(T^{i}A\cap
B)-\mu(A)\mu(B)|=0.$
Clearly, if $T$ is weak mixing, then $T$ is ergodic. Also, $T$ is weak mixing,
if and only if $T$ has only 1 as an eigenvalue, and all eigenfunctions are
constants almost everywhere. An ergodic measure preserving transformation $R$
is rigid on a sequence $\rho_{n}\to\infty$, if for any measurable set $A$,
$\lim_{n\to\infty}\mu(T^{\rho_{n}}A\triangle A)=0.$
The sequence $\rho_{n}$ is called a rigidity sequence for $R$.
Several forms of rigidity have been studied in both ergodic theory and
topological dynamics. In the case of topological dynamics, both rigidity and
uniform rigidity are considered. Uniform rigidity was introduced in [14] and
given a specific generic characterization. In [18], it is shown the notion of
uniform rigidity is mutually exclusive from measurable weak mixing on a Cantor
set. In particular, every finite measure preserving weak mixing transformation
has a representation that is not uniformly rigid. Weak mixing and rigidity
have been studied for interval exchange transformations. See [7] and [3] for
recent results in this regard. Rigid, weak mixing transformations have been
studied in the setting of infinite measure preserving transformations, as well
as nonsingular transformations. Mildly mixing transformations are finite
measure preserving transformations that do not contain a rigid factor. These
are the transformations which yield ergodic products with any infinite ergodic
transformation [13]. See works [1], [2], [4] and the references therein for
results related to notions of weak mixing and rigidity for infinite measure
preserving or nonsingular transformations. The notion of IP-sequences was
introduced by Furstenberg and Weiss in connection with rigid transformations.
There has been recent research on IP-rigidity sequences (i.e. IP-sequences
which form a rigid sequence) for weak mixing transformations. See [4] and [15]
for results on IP-rigidity.
The notion of rigidity was extended to $\alpha$-rigidity by Friedman [10].
Transformations are constructed which are $\alpha$-rigid and
$(1-\alpha)$-partial mixing for any $0<\alpha<1$. See [12] and [9] for further
research on $\alpha$-rigid transformations. Many of these notions have been
studied for more general group actions. See [5] for a survey of weak mixing
group actions. Since our results depend mainly on the use of Lemma 1.2 which
extends to more general groups (i.e. amenable, abelian), there should exist an
extension of techniques provided in this work to a wider class of groups.
Since some of the principles provided in this work appear new, we focus
exclusively on the case of measure preserving $\mathbb{Z}$-actions on $[0,1)$
with Lebesgue measure.
For a recent comprehensive account on rigidity sequences, we recommend recent
publications [4] and [8]. Both of these works provide much detail on the
current understanding of rigidity for weak mixing transformations.
## 2 Towerplex Constructions
The main result is established constructively using Lemma 1.2. Given two
transformations $R$ and $S$, we define a third transformation $T$ which is
constructed as a blend of $R$ and $S$, such that $T$ acts more like $R$,
asymptotically. We will define a sequence of positive integers
$h_{n},n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, and a sequence of real numbers
$\epsilon_{n}>0$ such that $\sum_{n=1}^{\infty}{1}/{h_{n}}<\infty$ and
$\sum_{n=1}^{\infty}\epsilon_{n}<\infty$. Also, let $r_{n}$ and $s_{n}$ for
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ be sequences of real numbers
satisfying: $0\leq r_{n},s_{n}\leq 1$.
### 2.1 Initialization
Suppose $R$ and $S$ are ergodic measure preserving transformations defined on
a Lebesgue probability space $(X,\mu,\mathbb{B})$. Partition $X$ into two
equal sets $X_{1}$ and $Y_{1}$ (i.e. $\mu(X_{1})=\mu(Y_{1})=1/2$). Initialize
$R_{1}$ isomorphic to $R$ and $S_{1}$ isomorphic to $S$ to operate on $X_{1}$
and $Y_{1}$, respectively. Define $T_{1}(x)=R_{1}(x)$ for $x\in X_{1}$ and
$T_{1}(x)=S_{1}(x)$ for $x\in Y_{1}$. Produce Rohklin towers of height $h_{1}$
with residual less than $\epsilon_{1}/2$ for each of $R_{1}$ and $S_{1}$. In
particular, let $I_{1},J_{1}$ be the base of the $R_{1}$-tower and
$S_{1}$-tower such that
$\mu(\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}I_{1})>1/2(1-\epsilon_{1})$ and
$\mu(\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}J_{1})>1/2(1-\epsilon_{1})$. Let
$X_{1}^{*}=X_{1}\setminus\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}(I_{1})$ and
$Y_{1}^{*}=Y_{1}\setminus\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J_{1})$ be the
residuals for the $R_{1}$ and $S_{1}$ towers, respectively. Choose
$I^{\prime}_{1}\subset I_{1}$ and $J^{\prime}_{1}\subset J_{1}$ such that
$\mu(I^{\prime}_{1})=r_{1}\mu(I_{1})\mbox{ and
}\mu(J^{\prime}_{1})=s_{1}\mu(J_{1}).$
Set
$X_{2}=X_{1}\setminus[\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}(I_{1}^{\prime})]\cup[\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J_{1}^{\prime})]$
and
$Y_{2}=Y_{1}\setminus[\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J_{1}^{\prime})]\cup[\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}(I_{1}^{\prime})]$.
We will define second stage transformations $R_{2}:X_{2}\to X_{2}$ and
$S_{2}:Y_{2}\to Y_{2}$. First, it may be necessary to add or subtract measure
from the residuals so that $X_{2}$ is scaled properly to define $R_{2}$, and
$Y_{2}$ is scaled properly to define $S_{2}$.
### 2.2 Tower Rescaling
In the case where $\mu(I_{1}^{\prime})\neq\mu(J_{1}^{\prime})$, we give a
procedure for transferring measure between the towers and the residuals. This
is done in order to consistently define $R_{2}$ and $S_{2}$ on the new
inflated or deflated towers. Let
$a=\mu(\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}I_{1})$ and
$b=h_{1}(\mu(J^{\prime}_{1})-\mu(I^{\prime}_{1}))$. Let $c$ be the scaling
factor and $d$ the amount of measure transferred between
$\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J^{\prime}_{1})$ and $X_{1}^{*}$. Thus,
$a+b-d=ca$ and $1/2-a+d=c(1/2-a)$. The goal is to solve two unknowns $d$ and
$c$ in terms of the other values. Hence, $d=(1-2a)b$ and $c=1+2b$.
#### 2.2.1 $R$ Rescaling
If $d>0$, define $I_{1}^{*}\subset J^{\prime}_{1}$ such that
$\mu(I_{1}^{*})=d/h_{1}$. Let
$X_{1}^{\prime}=X_{1}^{*}\cup(\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}(I_{1}^{*}))$.
If $d=0$, set $X_{1}^{\prime}=X_{1}^{*}$. If $d<0$, transfer measure from
$X_{1}^{*}$ to the tower. Choose disjoint sets
$I_{1}^{*}(0),I_{1}^{*}(1),\ldots,I_{1}^{*}(h_{1}-1)$ contained in $X_{1}^{*}$
such that $\mu(I_{1}^{*}(i))=d/h_{1}$. Denote $I_{1}^{*}=I_{1}^{*}(0)$. Begin
by defining $\mu$ measure preserving map $\alpha_{1}$ such that
$I_{1}^{*}(i+1)=\alpha_{1}(I_{1}^{*}(i))$ for $i=0,1,\ldots,h_{1}-2$. In this
case, let
$X_{1}^{\prime}=X_{1}^{*}\setminus[\bigcup_{i=0}^{h_{1}-1}I_{1}^{*}(i)]$.
#### 2.2.2 $S$ Rescaling
The direction mass is transferred depends on the sign of $b$ above. If $d>0$,
then $\mu(J^{\prime}_{1})>\mu(I^{\prime}_{1})$ and mass is transferred from
the residual $Y_{1}^{*}$ to the $S_{1}$-tower. Choose disjoint sets
$J_{1}^{*}(0),J_{1}^{*}(1),\ldots,J_{1}^{*}(h_{1}-1)$ contained in $Y_{1}^{*}$
such that $\mu(J_{1}^{*}(i))=d/h_{1}$. Denote $J_{1}^{*}=J_{1}^{*}(0)$. Begin
by defining $\mu$ measure preserving map $\beta_{1}$ such that
$J_{1}^{*}(i+1)=\beta_{1}(J_{1}^{*}(i))$ for $i=0,1,\ldots,h_{1}-2$. In this
case, let
$Y_{1}^{\prime}=Y_{1}^{*}\setminus[\bigcup_{i=0}^{h_{1}-1}J_{1}^{*}(i)]$. If
$d=0$, set $Y_{1}^{\prime}=Y_{1}^{*}$. If $d<0$, transfer measure from the
$S_{1}$-tower to the residual $Y_{1}^{*}$. Define $J_{1}^{*}\subset
J_{1}\setminus J^{\prime}_{1}$ such that $\mu(J_{1}^{*})=d/h_{1}$. Let
$Y_{1}^{\prime}=Y_{1}^{*}\cup(\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J_{1}^{*}))$.
Note, if $d\neq 0$, then both $\epsilon_{1}$ and $\mu(X_{1}^{*})$ may be
chosen small enough (relative to $r_{1}$) to ensure the following solutions
lead to well-defined sets and mappings. For subsequent stages, assume
$\epsilon_{n}$ is chosen small enough to force well-defined rescaling
parameters, transfer sets and mappings $R_{n}$, $S_{n}$.
### 2.3 Stage 2 Construction
We have specified three cases: $d>0$, $d=0$ and $d<0$. The case $d=0$, can be
handled along with the case $d>0$. This gives two essential cases. Note the
case $d<0$ is analogous to the case $d>0$, with the roles of $R_{1}$ and
$S_{1}$ reversed. However, due to a key distinction in the handling of the
$R$-rescaling and the $S$-rescaling, it is important to clearly define $R_{2}$
and $S_{2}$ in both cases.
###### Case 2.1 ($d\geq 0$).
Define $\tau_{1}:X_{1}^{\prime}\to X_{1}^{*}$ as a measure preserving map
between normalized spaces $(X_{1}^{\prime},\mathbb{B}\cap
X_{1}^{\prime},\frac{\mu}{\mu(X_{1}^{\prime})})$ and
$(X_{1}^{*},\mathbb{B}\cap X_{1}^{*},\frac{\mu}{\mu(X_{1}^{*})})$. Extend
$\tau_{1}$ to the new tower base,
$\tau_{1}:[I_{1}\setminus I_{1}^{\prime}]\cup[J^{\prime}_{1}\setminus
I_{1}^{*}]\to I_{1}$
such that $\tau_{1}$ preserves normalized measure between
$\frac{\mu}{\mu([I_{1}\setminus I_{1}^{\prime}]\cup[J^{\prime}_{1}\setminus
I_{1}^{*}])}\mbox{ and }\frac{\mu}{\mu(I_{1})}.$
Define $\tau_{1}$ on the remainder of the tower consistently such that
$\displaystyle\tau_{1}(x)=\left\\{\begin{array}[]{ll}R_{1}^{i}\circ\tau_{1}\circ
R_{1}^{-i}(x)&\mbox{if $x\in R_{1}^{i}(I_{1}\setminus I_{1}^{\prime})$ for
$0\leq i<h_{1}$}\\\ R_{1}^{i}\circ\tau_{1}\circ S_{1}^{-i}(x)&\mbox{if $x\in
S_{1}^{i}(J^{\prime}_{1}\setminus I_{1}^{*})$ for $0\leq
i<h_{1}$}\end{array}\right.$
Define $R_{2}:X_{2}\to X_{2}$ as $R_{2}=\tau_{1}^{-1}\circ
R_{1}\circ\tau_{1}$. Note
$\displaystyle R_{2}(x)=\left\\{\begin{array}[]{ll}S_{1}(x)&\mbox{if $x\in
S_{1}^{i}(J^{\prime}_{1}\setminus I_{1}^{*})$ for $0\leq i<h_{1}-1$}\\\
R_{1}(x)&\mbox{if $x\in R_{1}^{i}(I_{1}\setminus I^{\prime}_{1})$ for $0\leq
i<h_{1}-1$}\end{array}\right.$
Clearly, $R_{2}$ is isomorphic to $R_{1}$ and $R$.
Define $\psi_{1}:Y_{1}^{\prime}\to Y_{1}^{*}$ as a measure preserving map
between normalized spaces $(Y_{1}^{\prime},\mathbb{B}\cap
Y_{1}^{\prime},\frac{\mu}{\mu(Y_{1}^{\prime})})$ and
$(Y_{1}^{*},\mathbb{B}\cap Y_{1}^{*},\frac{\mu}{\mu(Y_{1}^{*})})$. Extend
$\psi_{1}$ to the new tower base,
$\psi_{1}:[J_{1}\setminus J_{1}^{\prime}]\cup J_{1}^{*}\cup I^{\prime}_{1}\to
J_{1}$
such that $\psi_{1}$ preserves normalized measure between
$\frac{\mu}{\mu([J_{1}\setminus J_{1}^{\prime}]\cup J_{1}^{*}\cup
I^{\prime}_{1})}\mbox{ and }\frac{\mu}{\mu(J_{1})}.$
Define $\psi_{1}$ on the remainder of the tower consistently such that
$\displaystyle\psi_{1}(x)=\left\\{\begin{array}[]{ll}S_{1}^{i}\circ\psi_{1}\circ
S_{1}^{-i}(x)&\mbox{if $x\in S_{1}^{i}(J_{1}\setminus J_{1}^{\prime})$ for
$0\leq i<h_{1}$}\\\ S_{1}^{i}\circ\psi_{1}\circ R_{1}^{-i}(x)&\mbox{if $x\in
R_{1}^{i}(I^{\prime}_{1})$ for $0\leq i<h_{1}$}\\\
\beta_{1}^{i}\circ\psi_{1}\circ\beta_{1}^{-i}(x)&\mbox{if $x\in J_{1}^{*}(i)$
for $0\leq i<h_{1}$}\end{array}\right.$
In this case, define $S_{2}:Y_{2}\to Y_{2}$ such that
$S_{2}=\psi_{1}^{-1}\circ S_{1}\circ\psi_{1}$. Note
$\displaystyle S_{2}(x)=\left\\{\begin{array}[]{ll}R_{1}(x)&\mbox{if $x\in
R_{1}^{i}I^{\prime}_{1}$ for $0\leq i<h_{1}-1$}\\\ S_{1}(x)&\mbox{if $x\in
S_{1}^{i}(J_{1}\setminus J^{\prime}_{1})$ for $0\leq i<h_{1}-1$}\\\
\beta_{1}(x)&\mbox{if $x\in J_{1}^{*}(i)$ for $0\leq i<h_{1}-1$}\\\
\psi_{1}^{-1}\circ S_{1}\circ\psi_{1}(x)&\mbox{if $x\in Y_{1}^{\prime}\cup
S_{1}^{h_{1}-1}(J_{1}\setminus J_{1}^{\prime})\cup
R_{1}^{h_{1}-1}I^{\prime}_{1}\cup\beta_{1}^{h_{1}-1}J_{1}^{*}$}\end{array}\right.$
and $S_{2}$ is isomorphic to $S_{1}$ and $S$.
###### Case 2.2 ($d<0$).
Define $\tau_{1}:X_{1}^{\prime}\to X_{1}^{*}$ as a measure preserving map
between normalized spaces $(X_{1}^{\prime},\mathbb{B}\cap
X_{1}^{\prime},\frac{\mu}{\mu(X_{1}^{\prime})})$ and
$(X_{1}^{*},\mathbb{B}\cap X_{1}^{*},\frac{\mu}{\mu(X_{1}^{*})})$. Extend
$\tau_{1}$ to the new tower base,
$\tau_{1}:[I_{1}\setminus I_{1}^{\prime}]\cup I_{1}^{*}\cup J^{\prime}_{1}\to
I_{1}$
such that $\tau_{1}$ preserves normalized measure between
$\frac{\mu}{\mu([I_{1}\setminus I_{1}^{\prime}]\cup I_{1}^{*}\cup
J^{\prime}_{1})}\mbox{ and }\frac{\mu}{\mu(I_{1})}.$
Define $\tau_{1}$ on the remainder of the tower consistently such that
$\displaystyle\tau_{1}(x)=\left\\{\begin{array}[]{ll}R_{1}^{i}\circ\tau_{1}\circ
R_{1}^{-i}(x)&\mbox{if $x\in R_{1}^{i}(I_{1}\setminus I_{1}^{\prime})$ for
$0\leq i<h_{1}$}\\\ R_{1}^{i}\circ\tau_{1}\circ S_{1}^{-i}(x)&\mbox{if $x\in
S_{1}^{i}(J^{\prime}_{1})$ for $0\leq i<h_{1}$}\\\
\alpha_{1}^{i}\circ\tau_{1}\circ\alpha_{1}^{-i}(x)&\mbox{if $x\in
I_{1}^{*}(i)$ for $0\leq i<h_{1}$}\end{array}\right.$
In this case, define $R_{2}:X_{2}\to X_{2}$ such that
$\displaystyle R_{2}(x)=\left\\{\begin{array}[]{ll}S_{1}(x)&\mbox{if $x\in
S_{1}^{i}J^{\prime}_{1}$ for $0\leq i<h_{1}-1$}\\\ R_{1}(x)&\mbox{if $x\in
R_{1}^{i}(I_{1}\setminus I^{\prime}_{1})$ for $0\leq i<h_{1}-1$}\\\
\alpha_{1}(x)&\mbox{if $x\in I_{1}^{*}(i)$ for $0\leq i<h_{1}-1$}\\\
\tau_{1}^{-1}\circ R_{1}\circ\tau_{1}(x)&\mbox{if $x\in X_{1}^{\prime}\cup
R_{1}^{h_{1}-1}(I_{1}\setminus I_{1}^{\prime})\cup
S_{1}^{h_{1}-1}J^{\prime}_{1}\cup\alpha_{1}^{h_{1}-1}I_{1}^{*}$}\end{array}\right.$
Clearly, $R_{2}$ is isomorphic to $R_{1}$ and $R$.
Define $\psi_{1}:Y_{1}^{\prime}\to Y_{1}^{*}$ as a measure preserving map
between normalized spaces $(Y_{1}^{\prime},\mathbb{B}\cap
Y_{1}^{\prime},\frac{\mu}{\mu(Y_{1}^{\prime})})$ and
$(Y_{1}^{*},\mathbb{B}\cap Y_{1}^{*},\frac{\mu}{\mu(Y_{1}^{*})})$. Extend
$\psi_{1}$ to the new tower base,
$\psi_{1}:[J_{1}\setminus(J_{1}^{\prime}\cup J_{1}^{*})]\cup I^{\prime}_{1}\to
J_{1}$
such that $\psi_{1}$ preserves normalized measure between
$\frac{\mu}{\mu([J_{1}\setminus(J_{1}^{\prime}\cup J_{1}^{*})]\cup
I^{\prime}_{1})}\mbox{ and }\frac{\mu}{\mu(J_{1})}.$
Define $\psi_{1}$ on the remainder of the tower consistently such that
$\displaystyle\psi_{1}(x)=\left\\{\begin{array}[]{ll}S_{1}^{i}\circ\psi_{1}\circ
S_{1}^{-i}(x)&\mbox{if $x\in S_{1}^{i}(J_{1}\setminus[J_{1}^{\prime}\cup
J_{1}^{*}])$ for $0\leq i<h_{1}$}\\\ S_{1}^{i}\circ\psi_{1}\circ
R_{1}^{-i}(x)&\mbox{if $x\in R_{1}^{i}(I^{\prime}_{1})$ for $0\leq
i<h_{1}$}\end{array}\right.$
Define $S_{2}:Y_{2}\to Y_{2}$ such that $S_{2}=\psi_{1}^{-1}\circ
S_{1}\circ\psi_{1}$. Note
$\displaystyle S_{2}(x)=\left\\{\begin{array}[]{ll}R_{1}(x)&\mbox{if $x\in
R_{1}^{i}(I^{\prime}_{1})$ for $0\leq i<h_{1}-1$}\\\ S_{1}(x)&\mbox{if $x\in
S_{1}^{i}(J_{1}\setminus[J^{\prime}_{1}\cup J_{1}^{*}])$ for $0\leq
i<h_{1}-1$}\end{array}\right.$
Transformation $S_{2}$ is isomorphic to $S_{1}$ and $S$.
Define $T_{2}$ as
$\displaystyle T_{2}(x)=\left\\{\begin{array}[]{ll}R_{2}(x)&\mbox{if $x\in
X_{2}$}\\\ S_{2}(x)&\mbox{if $x\in Y_{2}$}\end{array}\right.$
Clearly, neither $T_{1}$ nor $T_{2}$ are ergodic. For $T_{1}$, $X_{1}$ and
$Y_{1}$ are ergodic components, and $X_{2}$, $Y_{2}$ are ergodic components
for $T_{2}$. See the appendix for a pictorial of the multiplexing operation
used to produce $R_{2}$ and $S_{2}$ from $R_{1}$, $S_{1}$ and the intermediary
maps defined in this section.
### 2.4 General Multiplexing Operation
For $n\geq 1$, suppose that $R_{n}$ and $S_{n}$ have been defined on $X_{n}$
and $Y_{n}$ respectively. Construct Rohklin towers of height $h_{n}$ for each
$R_{n}$ and $S_{n}$, and such that $I_{n}$ is the base of the $R_{n}$ tower,
$J_{n}$ is the base of the $S_{n}$ tower, and
$\mu(\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}I_{n})+\mu(\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}J_{n})>1-\epsilon_{n}$.
Let $I^{\prime}_{n}\subset I_{n}$ be such that
$\mu(I^{\prime}_{n})=r_{n}\mu(I_{n})$. Similarly, suppose
$J^{\prime}_{n}\subset J_{n}$ such that $\mu(J^{\prime}_{n})=s_{n}\mu(J_{n})$.
We define $R_{n+1}$ and $S_{n+1}$ by switching the subcolumns
$\\{I^{\prime}_{n},R_{n}(I^{\prime}_{n}),R_{n}^{2}(I^{\prime}_{n}),\ldots,R_{n}^{h_{n}-1}(I^{\prime}_{n})\\}$
and
$\\{J^{\prime}_{n},S_{n}(J^{\prime}_{n}),S_{n}^{2}(J^{\prime}_{n}),\ldots,S_{n}^{h_{n}-1}(J^{\prime}_{n})\\}.$
Let
$\displaystyle X_{n+1}$ $\displaystyle=$
$\displaystyle[\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}(I_{n}\setminus
I_{n}^{\prime})]\cup[\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}J_{n}^{\prime}]\cup[X_{n}\setminus\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}I_{n}]$
$\displaystyle Y_{n+1}$ $\displaystyle=$
$\displaystyle[\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}(J_{n}\setminus
J_{n}^{\prime})]\cup[\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}I_{n}^{\prime}]\cup[Y_{n}\setminus\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}J_{n}].$
As in the initial case, it may be necessary to transfer measure between each
column and its respective residual. We can follow the same algorithm as above,
and define maps $\tau_{n},\alpha_{n},\psi_{n}$ and $\beta_{n}$. Thus, we get
the following definitions:
###### Case 2.3 ($d\geq 0$).
$\displaystyle\tau_{n}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}R_{n}^{i}\circ\tau_{n}\circ
R_{n}^{-i}(x)&\mbox{if $x\in R_{n}^{i}(I_{n}\setminus I_{n}^{\prime})$ for
$0\leq i<h_{n}$}\\\ R_{n}^{i}\circ\tau_{n}\circ S_{n}^{-i}(x)&\mbox{if $x\in
S_{n}^{i}(J^{\prime}_{n}\setminus I_{1}^{*})$ for $0\leq
i<h_{n}$}\end{array}\right.$ $\displaystyle R_{n+1}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}S_{n}(x)&\mbox{if $x\in
S_{n}^{i}(J^{\prime}_{n}\setminus I_{n}^{*})$ for $0\leq i<h_{n}-1$}\\\
R_{n}(x)&\mbox{if $x\in R_{n}^{i}(I_{n}\setminus I^{\prime}_{n})$ for $0\leq
i<h_{n}-1$}\\\ \tau_{n}^{-1}\circ R_{n}\circ\tau_{n}(x)&\mbox{if $x\in
X_{n}^{\prime}\cup R_{n}^{h_{n}-1}(I_{n}\setminus I_{n}^{\prime})\cup
S_{n}^{h_{n}-1}(J^{\prime}_{n}\setminus I_{n}^{*})$}\end{array}\right.$
and $R_{n+1}=\tau_{n}^{-1}\circ R_{n}\circ\tau_{n}$.
$\displaystyle\psi_{n}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}S_{n}^{i}\circ\psi_{n}\circ
S_{n}^{-i}(x)&\mbox{if $x\in S_{n}^{i}(J_{n}\setminus J_{n}^{\prime})$ for
$0\leq i<h_{n}$}\\\ S_{n}^{i}\circ\psi_{n}\circ R_{n}^{-i}(x)&\mbox{if $x\in
R_{n}^{i}(I^{\prime}_{n})$ for $0\leq i<h_{n}$}\\\
\beta_{n}^{i}\circ\psi_{n}\circ\beta_{n}^{-i}(x)&\mbox{if $x\in J_{n}^{*}(i)$
for $0\leq i<h_{n}$}\end{array}\right.$ $\displaystyle S_{n+1}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}R_{n}(x)&\mbox{if $x\in
R_{n}^{i}I^{\prime}_{n}$ for $0\leq i<h_{n}-1$}\\\ S_{n}(x)&\mbox{if $x\in
S_{n}^{i}(J_{n}\setminus J^{\prime}_{n})$ for $0\leq i<h_{n}-1$}\\\
\beta_{n}(x)&\mbox{if $x\in J_{n}^{*}(i)$ for $0\leq i<h_{n}-1$}\\\
\psi_{n}^{-1}\circ S_{n}\circ\psi_{n}(x)&\mbox{if $x\in Y_{n}^{\prime}\cup
S_{n}^{h_{n}-1}(J_{n}\setminus J_{n}^{\prime})\cup
R_{n}^{h_{n}-1}I^{\prime}_{n}\cup\beta_{n}^{h_{n}-1}J_{n}^{*}$}\end{array}\right.$
and $S_{n+1}=\psi_{n}^{-1}\circ S_{n}\circ\psi_{n}$.
###### Case 2.4 ($d<0$).
$\displaystyle\tau_{n}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}R_{n}^{i}\circ\tau_{n}\circ
R_{n}^{-i}(x)&\mbox{if $x\in R_{n}^{i}(I_{n}\setminus I_{n}^{\prime})$ for
$0\leq i<h_{n}$}\\\ R_{n}^{i}\circ\tau_{n}\circ S_{n}^{-i}(x)&\mbox{if $x\in
S_{n}^{i}(J^{\prime}_{n})$ for $0\leq i<h_{n}$}\\\
\alpha_{n}^{i}\circ\tau_{n}\circ\alpha_{n}^{-i}(x)&\mbox{if $x\in
I_{n}^{*}(i)$ for $0\leq i<h_{n}$}\end{array}\right.$ $\displaystyle
R_{n+1}(x)=$ $\displaystyle\left\\{\begin{array}[]{ll}S_{n}(x)&\mbox{if $x\in
S_{n}^{i}J^{\prime}_{n}$ for $0\leq i<h_{n}-1$}\\\ R_{n}(x)&\mbox{if $x\in
R_{n}^{i}(I_{n}\setminus I^{\prime}_{n})$ for $0\leq i<h_{n}-1$}\\\
\alpha_{n}(x)&\mbox{if $x\in I_{n}^{*}(i)$ for $0\leq i<h_{n}-1$}\\\
\tau_{n}^{-1}\circ R_{n}\circ\tau_{n}(x)&\mbox{if $x\in X_{n}^{\prime}\cup
R_{n}^{h_{n}-1}(I_{n}\setminus I_{n}^{\prime})\cup
S_{n}^{h_{n}-1}J^{\prime}_{n}\cup\alpha_{n}^{h_{n}-1}I_{n}^{*}$}\end{array}\right.$
and $R_{n+1}=\tau_{n}^{-1}\circ R_{n}\circ\tau_{n}$.
$\displaystyle\psi_{n}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}S_{n}^{i}\circ\psi_{n}\circ
S_{n}^{-i}(x)&\mbox{if $x\in S_{n}^{i}(J_{n}\setminus[J_{n}^{\prime}\cup
J_{n}^{*}])$ for $0\leq i<h_{n}$}\\\ S_{n}^{i}\circ\psi_{n}\circ
R_{n}^{-i}(x)&\mbox{if $x\in R_{n}^{i}(I^{\prime}_{n})$ for $0\leq
i<h_{n}$}\end{array}\right.$ $\displaystyle S_{n+1}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}R_{n}(x)&\mbox{if $x\in
R_{n}^{i}(I^{\prime}_{n})$ for $0\leq i<h_{n}-1$}\\\ S_{n}(x)&\mbox{if $x\in
S_{n}^{i}(J_{n}\setminus[J^{\prime}_{n}\cup J_{n}^{*}])$ for $0\leq
i<h_{n}-1$}\\\ \psi_{n}^{-1}\circ S_{n}\circ\psi_{n}(x)&\mbox{if $x\in
Y_{n}^{\prime}\cup S_{n}^{h_{n}-1}(J_{n}\setminus[J_{n}^{\prime}\cup
J_{n}^{*}])\cup R_{n}^{h_{n}-1}(I^{\prime}_{n})$}\end{array}\right.$
and $S_{n+1}=\psi_{n}^{-1}\circ S_{n}\circ\psi_{n}$.
### 2.5 The Limiting Transformation
Define the transformation $T_{n+1}:X_{n+1}\cup Y_{n+1}\to X_{n+1}\cup Y_{n+1}$
such that
$\displaystyle T_{n+1}(x)=\left\\{\begin{array}[]{ll}R_{n+1}(x)&\mbox{if $x\in
X_{n+1}$}\\\ S_{n+1}(x)&\mbox{if $x\in Y_{n+1}$}\end{array}\right.$
The set where $T_{n+1}\neq T_{n}$ is determined by the top levels of the
Rokhlin towers, the residual and the transfer set. Note the transfer set has
measure $d$. Since this set is used to adjust the size of the residuals
between stages, it can be bounded below a constant multiple of $\epsilon_{n}$.
Thus, there is a fixed constant $\kappa$, independent of $n$, such that
$T_{n+1}(x)=T_{n}(x)$ except for $x$ in a set of measure less than
$\kappa(\epsilon_{n}+{1}/{h_{n}})$. Since
$\sum_{n=1}^{\infty}(\epsilon_{n}+{1}/{h_{n}})<\infty$,
$T(x)=\lim_{n\to\infty}T_{n}(x)$ exists almost everywhere, and preserves
normalized Lebesgue measure. Without loss of generality, we may assume
$\kappa$ and $h_{n}$ are chosen such that if
$E_{n}=\\{x\in X|T_{n+1}(x)\neq T_{n}(x)\\}$
then $\mu(E_{n})<\kappa\epsilon_{n}$ for
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. In the following section, additional
structure and conditions are implemented to ensure that $T$ inherits
properties from $R$ and $S$, and is also ergodic.
For the remainder of this paper, assume the parameters are chosen such that
1. 1.
$\lim_{n\to\infty}r_{n}=0$;
2. 2.
$\sum_{n=1}^{\infty}r_{n}=\sum_{n=1}^{\infty}s_{n}=\infty$;
3. 3.
$\lim_{n\to\infty}\mu(Y_{n})=0$;
4. 4.
$\sum_{n=1}^{\infty}\epsilon_{n}<\infty$.
### 2.6 Isomorphism Chain Consistency
In the following sections, rigidity and ergodicity will be established on sets
from a refining sequence of partitions. For
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, let $P_{n}$ be a refining sequence
of finite partitions which generates the sigma algebra. By refining $P_{n}$
further if necessary, assume $X_{n},Y_{n},X_{n}^{*},Y_{n}^{*}\in P_{n}$. Also,
assume $R_{n}^{i}(I^{\prime}_{n}),R_{n}^{i}(I_{n}\setminus
I^{\prime}_{n}),S_{n}^{i}(J^{\prime}_{n}),S_{n}^{i}(J_{n}\setminus
J^{\prime}_{n})$ are elements of $P_{n}$ for $0\leq i<h_{n}$. Finally, assume
for $0\leq i<h_{n}-1$, if $p\in P_{n}$ and $p\subset R_{n}^{i}(I_{n})$ then
$R_{n}(p)\in P_{n}$. Likewise, assume for $0\leq i<h_{n}-1$, if $p\in P_{n}$
and $p\subset S^{i}(J_{n})$ then $S_{n}(p)\in P_{n}$. Previously, we required
that $\tau_{n}$ map certain finite orbits from the $R_{n}$ and $S_{n}$ towers
to a corresponding orbit in the $R_{n+1}$ tower. In this section, further
regularity is imposed on $\tau_{n}$ relative to $P_{n}$ to ensure dynamical
properties of $R_{n}$ are inherited by $R_{n+1}$.
Let $P_{n}^{\prime}=\\{p\in
P_{n}|p\subset\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}(I_{n}\setminus
I_{n}^{\prime})\\}$. For each of the following three cases, impose the
corresponding restriction on $\tau_{n}$:
1. 1.
for $d_{R}=0$ and $p\in P_{n}^{\prime}$, $\tau_{n}$ is the identity map (i.e.
$\tau_{n}(p)=p$);
2. 2.
for $d_{R}>0$ and $p\in P_{n}^{\prime}$, $\tau_{n}(p)\subset p$;
3. 3.
for $d_{R}<0$ and $p\in P_{n}^{\prime}$, $p\subset\tau_{n}(p)$.
This can be accomplished by uniformly distributing the appropriate mass from
the sets $R_{n}^{i}(I_{n}^{*})$ using $\tau_{n}$. Note that $\tau_{n}$ either
preserves Lebesgue measure in the case $d_{R}=0$, or $\tau_{n}$ contracts sets
relative to Lebesgue measure in the case $d_{R}>0$, or it inflates measure in
the case $d_{R}<0$. In all three cases, for $p\in P_{n}^{\prime}$,
$\frac{\mu(p)}{\mu(\tau_{n}(p))}=\frac{\mu(X_{n+1})}{\mu(X_{n})}.$
It is straightforward to verify for any set $A$ measurable relative to
$P_{n}^{\prime}$,
$\mu(A\triangle\tau_{n}A)<|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|.$
The properties of $\tau_{n}$ allow approximation of $R_{n+1}$ by $R_{n}$
indefinitely over time. This is needed to establish our rigidity sequence for
the limiting transformation $T$. This lemma is not required for establishing
ergodicity, but for convenience we will reuse it to prove our limiting $T$ is
ergodic.
###### Lemma 2.5.
Suppose $\delta>0$ and $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ is chosen such
that
$\displaystyle|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|<\frac{\delta}{7},$
$\displaystyle r_{n}+\epsilon_{n}+\mu(Y_{n})<\frac{\delta}{7}.$
Then for $A,B\in P_{n}$ and $i\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, the
following holds:
1. 1.
$|\mu(R_{n+1}^{i}A\cap B)-\mu(A)\mu(B)|<|\mu(R_{n}^{i}A\cap
B)-\mu(A)\mu(B)|+\delta$;
2. 2.
$\mu(R_{n+1}^{i}A\triangle A)<\mu(R_{n}^{i}A\triangle A)+\delta$.
###### Proof.
For $A,B\in P_{n}$, let $A^{\prime}=\bigcup_{p\in P_{n}^{\prime}}p\cap A$ and
$B^{\prime}=\bigcup_{p\in P_{n}^{\prime}}p\cap B$. Since
$\mu(\bigcup_{j=0}^{h_{n}-1}R_{n}^{j}(I_{n}^{\prime}))=h_{n}\mu(I_{n}^{\prime})<r_{n}$
and $\mu(X_{n}^{*})<\epsilon_{n}$, then $\mu(A\triangle
A^{\prime})<r_{n}+\epsilon_{n}<\frac{\delta}{7}$. Likewise, $\mu(B\triangle
B^{\prime})<\frac{\delta}{7}$. Since
$|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|<\frac{\delta}{7}$, then
$\mu(A\triangle\tau_{n}A)<\frac{\delta}{7}$. By applying the triangle
inequality several times, we can get our approximations. Below is a sequence
of quantities to chain through such that consecutive values in the chain are
less than $\delta/7$ apart.
$\displaystyle\mu(R_{n+1}^{i}A\cap B)\rightarrow\mu(R_{n+1}^{i}A\cap
B^{\prime})\rightarrow\mu(R_{n+1}^{i}A^{\prime}\cap
B^{\prime})=\mu(\tau_{n}^{-1}R_{n}^{i}\tau_{n}A^{\prime}\cap B^{\prime})$
$\displaystyle\mu(\tau_{n}^{-1}R_{n}^{i}\tau_{n}A^{\prime}\cap
B^{\prime})\rightarrow\mu(R_{n}^{i}\tau_{n}A^{\prime}\cap\tau_{n}B^{\prime})\rightarrow\mu(R_{n}^{i}\tau_{n}A^{\prime}\cap
B^{\prime})$ $\displaystyle\rightarrow\mu(R_{n}^{i}A^{\prime}\cap
B^{\prime})\rightarrow\mu(R_{n}^{i}A^{\prime}\cap
B)\rightarrow\mu(R_{n}^{i}A\cap B)$
Each arrow in the chain signifies less than $\frac{\delta}{7}$ difference.
Hence,
$|\mu(R_{n+1}^{i}A\cap B)-\mu(R_{n}^{i}A\cap B)|<\delta$
which implies
$|\mu(R_{n+1}^{i}A\cap B)-\mu(A)\mu(B)|<|\mu(R_{n}^{i}A\cap
B)-\mu(A)\mu(B)|+\delta.$
The second part of the lemma can be proven in a similar fashion using the
triangle inequality, or chaining through the following six approximations.
$\displaystyle\mu(R_{n+1}^{i}A\triangle A)\rightarrow\mu(R_{n+1}^{i}A\triangle
A^{\prime})\rightarrow\mu(R_{n+1}^{i}A^{\prime}\triangle
A^{\prime})=\mu(\tau_{n}^{-1}R_{n}^{i}\tau_{n}A^{\prime}\triangle A^{\prime})$
$\displaystyle\mu(\tau_{n}^{-1}R_{n}^{i}\tau_{n}A^{\prime}\triangle
A^{\prime})\rightarrow\mu(R_{n}^{i}\tau_{n}A^{\prime}\triangle\tau_{n}A^{\prime})\rightarrow\mu(R_{n}^{i}\tau_{n}A^{\prime}\triangle
A^{\prime})$ $\displaystyle\rightarrow\mu(R_{n}^{i}A^{\prime}\triangle
A^{\prime})\rightarrow\mu(R_{n}^{i}A^{\prime}\triangle
A)\rightarrow\mu(R_{n}^{i}A\triangle A)$
Since each arrow indicates a difference less than $\frac{\delta}{7}$, then
$|\mu(R_{n+1}^{i}A\triangle A)-\mu(R_{n}^{i}A\triangle A)|<\delta.$
This completes the proof of the lemma. ∎
## 3 Establishing Rigidity
Suppose that $\rho_{n}$ is a rigidity sequence for $R$. In this section, we
define parameters such that $T$ is rigid on $\rho_{n}$.
### 3.1 Waiting for Rigidity
Let $\delta_{n}$ be a sequence of positive real numbers such that
$\lim_{n\to\infty}\delta_{n}=0$. Since $T_{n}|_{X_{n}}=R_{n}$ is rigid, choose
natural number $M_{n}^{1}>\max{\\{h_{n-1},M_{n-1}^{1}\\}}$ such that for
$N\geq M_{n}^{1}$, and $A\in P_{n-1}\cap X_{n}$,
$\mu(R_{n}^{\rho_{N}}A\triangle A)<\delta_{n}.$
Choose $\epsilon_{n}$ such that
$\displaystyle\frac{\epsilon_{n-1}}{M_{n}^{1}}<\epsilon_{n}.$ (19)
Also, without loss of generality, assume $h_{n}>M_{n}^{1}$. Below we show this
choice of $\epsilon_{n}$ is sufficient to produce
$T(x)=\lim_{n\to\infty}T_{n}(x)$ rigid on $\rho_{n}$. First, we provide a
diagram and heuristic description of our method for establishing rigidity on
$\rho_{n}$.
### 3.2 The Key Idea
$h_{n-1}$$M_{n}^{1}$$h_{n}$$M_{n+1}^{1}$$h_{n+1}$
To establish rigidity of $T$, we can focus on the asymptotic rigidity of $T$
on the intervals $(M_{n}^{1},M_{n+1}^{1}]$. We have chosen $M_{n}^{1}$
sufficiently large such that rigidity ”kicks in” for $R_{n}$ and
$\rho_{i}>M_{n}^{1}$. Lemma 2.5 allows us to approximate $R_{n}$ by $R_{n+1}$
as $\rho_{i}$ becomes closer to $M_{n+1}^{1}$. The fact that we can choose
$\epsilon_{n+1}$ arbitrarily small compared to ${1}/{M_{n+1}^{1}}$ allows us
to carry over the approximation to $T$. A precise proof is given below.
### 3.3 Rigidity Proof
If $E_{n+1}=\\{x\in X:T_{n+2}(x)\neq T_{n+1}(x)\\}$ and
$E_{n+1}^{1}=\bigcup_{i=0}^{M_{n+1}^{1}-1}[T_{n+2}^{-i}E_{n+1}\cup
T_{n+1}^{-i}E_{n+1}]$
then $\mu(E_{n+1}^{1})<2M_{n+1}^{1}\kappa\epsilon_{n+1}$. For $x\notin
E_{n+1}^{1}$, $T_{n+2}^{i}(x)=T_{n+1}^{i}(x)$ for $0\leq i\leq M_{n+1}^{1}$.
Let $\hat{E}_{n+1}^{1}=\bigcup_{k=n+1}^{\infty}E_{k}^{1}$. For
$x\notin\hat{E}_{n+1}^{1}$ and $0\leq i\leq M_{n+1}^{1}$,
$T^{i}(x)=T_{n+1}^{i}(x)$. Also,
$\mu(\hat{E}_{n+1}^{1})<\sum_{k=n+1}^{\infty}2M_{k}^{1}\kappa\epsilon_{k}<\sum_{k=n+1}^{\infty}2\kappa\epsilon_{k-1}\to
0$
as $n\to\infty$.
###### Proof of rigidity.
Let $A$ be a set in $P_{n_{1}}$ for some $n_{1}$, and let $\delta>0$. Choose
$n_{2}\geq n_{1}$ such that for $n\geq n_{2}$,
1. 1.
$|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|<\delta/28$;
2. 2.
$r_{n}+\epsilon_{n}+\mu(Y_{n})<\delta/28$;
3. 3.
$\delta_{n}<\delta/6$;
4. 4.
$\sum_{i=n_{2}}^{\infty}2\kappa\epsilon_{i}<\delta/12$.
For $n>n_{2}$, let $M_{n}^{1}<N\leq M_{n+1}^{1}$,
$A_{1}=A\setminus\hat{E}_{n+1}^{1}$ and $A_{2}=A\cap X_{n}$. Thus,
$\displaystyle\mu(T^{\rho_{N}}A\triangle A)$ $\displaystyle\leq$
$\displaystyle\mu(T^{\rho_{N}}A\triangle
T^{\rho_{N}}A_{1})+\mu(T^{\rho_{N}}A_{1}\triangle A)$ $\displaystyle=$
$\displaystyle\mu(A\triangle A_{1})+\mu(R_{n+1}^{\rho_{N}}A_{1}\triangle A)$
$\displaystyle<$
$\displaystyle\frac{\delta}{4}+\mu(R_{n+1}^{\rho_{N}}A_{1}\triangle
R_{n+1}^{\rho_{N}}A)+\mu(R_{n+1}^{\rho_{N}}A\triangle A)$ $\displaystyle<$
$\displaystyle\frac{\delta}{2}+\mu(R_{n+1}^{\rho_{N}}A\triangle A).$
By Lemma 2.5,
$\displaystyle\mu(T^{\rho_{N}}A\triangle A)$ $\displaystyle<$
$\displaystyle\frac{\delta}{2}+\mu(R_{n+1}^{\rho_{N}}A\triangle
A)<\frac{3\delta}{4}+\mu(R_{n}^{\rho_{N}}A\triangle A)$ $\displaystyle\leq$
$\displaystyle\frac{3\delta}{4}+\mu(R_{n}^{\rho_{N}}A\triangle
R_{n}^{\rho_{N}}A_{2})+\mu(R_{n}^{\rho_{N}}A_{2}\triangle
A_{2})+\mu(A_{2}\triangle A)$ $\displaystyle<$
$\displaystyle\frac{3\delta}{4}+2\mu(Y_{n})+\delta_{n}<\delta.$
Therefore, $\rho_{n}$ is a rigidity sequence for $T$. ∎
## 4 Ergodicity
A measure preserving transformation $T$ on a Lebesgue space is ergodic if any
invariant set has measure zero or one. It is well known this is equivalent to
the mean and pointwise ergodic theorem. For our purposes, we use the following
equivalent condition of ergodicity: for all measurable sets $A$ and $B$,
$\lim_{N\to\infty}\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A\cap B)=\mu(A)\mu(B).$
Let $P_{n},n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ be a sequence of finite
refining partitions as defined in the previous section. Using approximation,
$T$ is ergodic if the previous condition holds for all natural numbers $n$ and
sets $A$ and $B$ from $P_{n}$.
### 4.1 Ergodic Parameter Choice
Let $\delta_{n}$ be a sequence of positive real numbers such that
$\lim_{n\to\infty}\delta_{n}=0$. Since $T_{n}|_{X_{n}}=R_{n}$ is ergodic,
choose natural number $M_{n}=M_{n}^{2}$ such that for $N\geq M_{n}$, and sets
$A,B\in P_{n}\cap X_{n}$,
$|\frac{1}{N}\sum_{i=0}^{N-1}\frac{\mu(T_{n}^{i}A\cap
B)}{\mu(X_{n})}-\frac{\mu(A)\mu(B)}{\mu(X_{n})^{2}}|<\delta_{n}.$
Note that
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T_{n}^{i}A\cap B)$
$\displaystyle-$ $\displaystyle\mu(A)\mu(B)|$ $\displaystyle=$
$\displaystyle\mu(X_{n})|\frac{1}{N}\sum_{i=0}^{N-1}\frac{\mu(T_{n}^{i}A\cap
B)}{\mu(X_{n})}-\frac{\mu(A)\mu(B)}{\mu(X_{n})}|$ $\displaystyle\leq$
$\displaystyle\mu(X_{n})|\frac{1}{N}\sum_{i=0}^{N-1}\frac{\mu(T_{n}^{i}A\cap
B)}{\mu(X_{n})}-\frac{\mu(A)\mu(B)}{\mu(X_{n})^{2}}|$ $\displaystyle+$
$\displaystyle|\frac{\mu(A)\mu(B)}{\mu(X_{n})}-\mu(A)\mu(B)|$ $\displaystyle<$
$\displaystyle\delta_{n}+\frac{\mu(Y_{n})}{\mu(X_{n})}$
Choose $\epsilon_{n+1}$ such that
$\displaystyle\frac{\epsilon_{n}}{M_{n}}<\epsilon_{n+1}.$ (20)
### 4.2 Approximation
As previously, set $E_{n+1}=\\{x\in X:T_{n+2}(x)\neq T_{n+1}(x)\\}$. Let
$E_{n+1}^{2}=\bigcup_{i=0}^{M_{n+1}-1}[T_{n+2}^{-i}E_{n+1}\cup
T_{n+1}^{-i}E_{n+1}]$
Thus, $\mu(E_{n+1}^{2})<2M_{n+1}\kappa\epsilon_{n+1}$. For $x\notin
E_{n+1}^{2}$, $T_{n+2}^{i}(x)=T_{n+1}^{i}(x)$ for $0\leq i\leq M_{n+1}$. Let
$\hat{E}_{n+1}^{2}=\bigcup_{k=n+1}^{\infty}E_{k}^{2}$. For
$x\notin\hat{E}_{n+1}^{2}$ and $0\leq i\leq M_{n+1}$,
$T^{i}(x)=T_{n+1}^{i}(x)$. Also,
$\mu(\hat{E}_{n+1}^{2})<\sum_{k=n+1}^{\infty}2M_{k}\kappa\epsilon_{k}<\sum_{k=n+1}^{\infty}2\kappa\epsilon_{k-1}\to
0$
as $n\to\infty$.
###### Proof of ergodicity.
Let $A$ and $B$ be sets in $P_{n_{1}}$ for some $n_{1}$, and let $\delta>0$.
Choose $n_{2}\geq n_{1}$ such that for $n\geq n_{2}$,
1. 1.
$|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|<\delta/28$;
2. 2.
$r_{n}+\epsilon_{n}+\mu(Y_{n})<\delta/28$;
3. 3.
$\delta_{n}+\frac{\mu(Y_{n})}{\mu(X_{n})}<\delta/4$;
4. 4.
$\sum_{i=n_{2}}^{\infty}2\kappa\epsilon_{i}<\delta/12$.
For $n>n_{2}$, let $M_{n}<N\leq M_{n+1}$, $A_{1}=A\setminus\hat{E}_{n+1}^{2}$
and $B_{1}=B\setminus\hat{E}_{n+1}^{2}$.
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A\cap B)$ $\displaystyle-$
$\displaystyle\mu(A)\mu(B)|$ $\displaystyle\leq$
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A\cap
B)-\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A_{1}\cap B)|$ $\displaystyle+$
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A_{1}\cap B)-\mu(A)\mu(B)|$
$\displaystyle\leq$
$\displaystyle\frac{1}{N}\sum_{i=0}^{N-1}|\mu(T^{i}A)-\mu(T^{i}A_{1})|$
$\displaystyle+$
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(R_{n+1}^{i}A_{1}\cap
B)-\mu(A)\mu(B)|$ $\displaystyle<$
$\displaystyle\frac{\delta}{4}+|\frac{1}{N}\sum_{i=0}^{N-1}\mu(R_{n+1}^{i}A_{1}\cap
B)-\mu(R_{n+1}^{i}A\cap B)|$ $\displaystyle+$
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(R_{n+1}^{i}A\cap
B)-\mu(A)\mu(B)|$ $\displaystyle<$
$\displaystyle\frac{\delta}{2}+|\frac{1}{N}\sum_{i=0}^{N-1}\mu(R_{n+1}^{i}A\cap
B)-\mu(A)\mu(B)|.$
Since $A,B\in P_{n}$, then by Lemma 2.5,
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A\cap B)$ $\displaystyle-$
$\displaystyle\mu(A)\mu(B)|$ $\displaystyle<$
$\displaystyle\frac{\delta}{2}+\frac{1}{N}\sum_{i=0}^{N-1}|\mu(R_{n+1}^{i}A\cap
B)-\mu(A\cap B)|$ $\displaystyle<$
$\displaystyle\frac{3\delta}{4}+\frac{1}{N}\sum_{i=0}^{N-1}|\mu(R_{n}^{i}A\cap
B)-\mu(A\cap B)|<\delta.$
Since $\delta$ is chosen arbitrarily, and the above holds for any $n>n_{2}$
and $M_{n}<N\leq M_{n+1}$, then $T$ is ergodic. ∎
## 5 Weak Mixing
Since the weak mixing component is dissipative, and the resulting
transformation inherits its rigidity properties from $R$, we do not focus on
multiplexing with general weak mixing transformations. Instead, we set $S$
equal to the famous Chacon transformation. It is defined via cutting and
stacking, and considered the earliest construction demonstrated to be weak
mixing and not mixing. See [11] for a precise definition. For the remainder of
this paper, assume both $R$ and $S$ are defined on $([0,1),\mu,\mathbb{B})$
where $\mu$ is Lebesgue measure. In this section, we further specify $h_{n}$
and switching sets $C_{n}=\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}(I_{n}^{\prime})$
for $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. As in previous sections, all
conditions imposed are easily satisfied by choosing a faster growing sequence
of tower heights $h_{n}$. No upper bounds are imposed on the growth rate of
$h_{n}$.
### 5.1 Switching Set Definition
For each $k\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ and $n>k$, denote
$U_{k}^{n}=\bigcup_{j=k}^{n-1}C_{j}$, $V_{k}^{n}=(U_{k}^{n})^{c}$ and
$\dot{V}_{k}^{n}=V_{k}^{n}\cap X_{n}$. Since $R_{n}$ is ergodic on $X_{n}$,
$r_{n}$ is fixed, and $C_{n}$ predominantly represents long orbits of $R_{n}$,
then $h_{n}$ may be chosen sufficiently large such that $C_{n}$ is near
conditionally independent of $\dot{V}_{k}^{n}$ for each $k<n$.
Precisely, define $h_{n}$ and $C_{n}$ such that
$\displaystyle|\frac{\mu(C_{n}\cap\dot{V}_{k}^{n})}{\mu(X_{n})}-\frac{\mu(C_{n})\mu(\dot{V}_{k}^{n})}{\mu(X_{n})^{2}}|\leq\frac{1}{2}\mu(C_{n})\mu(\dot{V}_{k}^{n}).$
(21)
###### Lemma 5.1.
For each $k\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$,
$\lim_{n\to\infty}\mu(V_{k}^{n})=0$.
###### Proof.
Suppose the claim is not true, and there exists
$k_{0}\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that
$\lim_{n\to\infty}\mu(V_{k}^{n})>0.$
Since $\lim_{n\to\infty}\mu(Y_{n})=0$, we can choose $k_{1}>k_{0}$ such that
$\mu(Y_{j})<\frac{1}{2}\mu(V_{k_{1}}^{k_{1}+n})$ for $j\geq k_{1}$ and
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. Thus,
$\displaystyle\frac{\mu(C_{k_{1}+1}\cap\dot{V}_{k_{1}}^{k_{1}+1})}{\mu(X_{k_{1}+1})}$
$\displaystyle\geq$
$\displaystyle\frac{\mu(C_{k_{1}+1})\mu(\dot{V}_{k_{1}}^{k_{1}+1})}{\mu(X_{k_{1}+1})^{2}}-\frac{1}{2}\mu(C_{k_{1}+1})\mu(\dot{V}_{k_{1}}^{k_{1}+1})$
$\displaystyle\mu(C_{k_{1}+1}\cap\dot{V}_{k_{1}}^{k_{1}+1})$
$\displaystyle\geq$
$\displaystyle\mu(V_{k_{1}}^{k_{1}+1})\frac{\mu(V_{k_{1}}^{k_{1}+1}\cap
X_{k_{1}+1})}{\mu(V_{k_{1}}^{k_{1}+1})}\mu(C_{k_{1}+1})[\frac{1}{\mu(X_{k_{1}+1})}-\frac{\mu(X_{k_{1}+1})}{2}]$
$\displaystyle>$
$\displaystyle\frac{1}{4}\mu(C_{k_{1}+1})\mu(V_{k_{1}}^{k_{1}+1}).$
Hence,
$\displaystyle\mu(V_{k_{1}}^{k_{1}+2})$ $\displaystyle=$
$\displaystyle\mu(V_{k_{1}}^{k_{1}+1})-\mu(C_{k_{1}+1}\cap
V_{k_{1}}^{k_{1}+1})$ $\displaystyle<$
$\displaystyle\mu(V_{k_{1}}^{k_{1}+1})[1-\frac{1}{4}\mu(C_{k_{1}+1})]$
$\displaystyle<$
$\displaystyle(1-\frac{1}{4}\mu(C_{k_{1}}))(1-\frac{1}{4}\mu(C_{k_{1}+1})).$
Extending this inductively produces
$\mu(V_{k_{1}}^{k_{1}+n})<\prod_{i=0}^{n-1}(1-\frac{1}{4}\mu(C_{k_{1}+i})).$
Note that
$\mu(C_{n})=\mu(I_{n}^{\prime})h_{n}=\frac{\mu(I_{n}^{\prime})}{\mu(I_{n})}\mu(I_{n})h_{n}=r_{n}\mu(X_{n}).$
Since $\sum_{n=1}^{\infty}r_{n}=\infty$ and $\lim_{n\to\infty}\mu(X_{n})=1$,
then both $\sum_{n=1}^{\infty}\mu(C_{n})=\infty$ and
$\sum_{n=1}^{\infty}\frac{1}{4}\mu(C_{n})=\infty$. This is sufficient to force
$\lim_{n\to\infty}\prod_{i=0}^{n-1}(1-\frac{1}{4}\mu(C_{k_{1}+i}))=0$ which
proves our claim by contradiction. ∎
The previous claim establishes the following property that almost every point
falls in infinitely many sets $C_{n}$.
###### Property 5.2.
$\mu(\bigcap_{n=1}^{\infty}\bigcup_{i=n}^{\infty}C_{i})=1$.
### 5.2 Multiplexing Chacon’s Transformation
Chacon’s transformation $S$ is typically defined using cutting and stacking
[11]. Initialize $I_{1}^{0}=[0,2/3)$ and $\mathcal{C}_{1}=I_{1}^{0}$. Cut
$I_{1}$ into 3 pieces of equal width,
$I_{2}^{0}=[0,2/9),I_{2}^{1}=[2/9,4/9),I_{2}^{3}=[4/9,2/3)$, and add a single
spacer $I_{2}^{2}=[2/3,8/9)$ above interval $I_{2}^{1}$. Stack into a single
column $\mathcal{C}_{2}=<I_{2}^{0},I_{2}^{1},I_{2}^{2},I_{2}^{3}>$. Define $S$
as the linear map from $I_{2}^{i}$ to $I_{2}^{i+1}$ for $i=0,1,2$. Let
$H_{n}={(3^{n}-1)}/{2}$ be the height of column $\mathcal{C}_{n}$. Obtain
$\mathcal{C}_{n+1}$ by cutting $\mathcal{C}_{n}$ into 3 subcolumns of equal
width, $\mathcal{C}_{n}^{0},\mathcal{C}_{n}^{1},\mathcal{C}_{n}^{2}$, adding
one spacer above the second subcolumn and stacking left to right. Again, $S$
maps each level linearly to the level directly above it. Also, notice the
height of $\mathcal{C}_{n+1}$ equals $H_{n+1}=3H_{n}+1=\frac{3^{n+1}-1}{2}$.
The main property we utilize in this work is related to one of its limit
joinings.
###### Lemma 5.3.
Let $S$ be Chacon’s transformation. Given any two measurable sets, $A$ and
$B$,
$\lim_{n\to\infty}\mu(S^{H_{n}}A\cap B)=(\mu(A\cap B)+\mu(S^{-1}A\cap B))/2.$
###### Proof.
Each column $\mathcal{C}_{n}$, $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, has a
single level of spacer above precisely half the mass of the top level of
$\mathcal{C}_{n}$. This includes the spacers added when $\mathcal{C}_{n}$ is
cut into 3 subcolumns, as well as the infinitely many spacers added when
$\mathcal{C}_{n+1},\mathcal{C}_{n+2},\ldots$ are cut into 3 subcolumns and
stacked. Thus, $S^{H_{n}}$ maps half of each level to the same level, and maps
the other half to the level directly below itself. This establishes the lemma
for sets consisting of a finite union of levels. Since the levels of the
columns form a refining sequence of partitions which generate the sigma
algebra, the lemma follows by approximation. ∎
### 5.3 Weak Mixing Stage
Now we define $S_{n}$ inductively to ensure the final transformation $T$ is
weak mixing. Let $S_{1}$ be the Chacon transformation defined on $Y_{1}$.
Suppose $S_{n}\simeq S$ has been defined on $Y_{n}$. Now we specify the manner
in which $S_{n+1}$ should be defined.
#### 5.3.1 Local Approximation of Switching Sets
Choose natural number $k_{n}>n$ such that for each $i=0,1,\ldots,h_{n}-1$,
there exists a finite collection of indices $\hat{K}_{n}^{i}$ and dyadic
intervals $K_{n}^{i}(j)$, $j\in\hat{K}_{n}^{i}$, such that
$\mu(K_{n}^{i}(j))=\frac{1}{2^{k_{n}}}$ and
$K_{n}^{i}=\bigcup_{j\in\hat{K}_{n}^{i}}K_{n}^{i}(j)$ satisfies
$\mu(R_{n}^{i}I_{n}^{\prime}\triangle
K_{n}^{i})<(\frac{\epsilon_{n}}{h_{n}})^{2}\mu(I_{n}^{\prime})$. Let
$\hat{G}_{n}^{i}=\\{j\in\hat{K}_{n}^{i}:\mu(R_{n}^{i}I_{n}^{\prime}\cap
K_{n}^{i}(j))>(1-\frac{\epsilon_{n}}{h_{n}})\mu(K_{n}^{i}(j))\\}$. It is not
difficult to show
$\mu(\bigcup_{j\in\hat{G}_{n}^{i}}K_{n}^{i}(j))>(1-\frac{\epsilon_{n}}{h_{n}})\mu(I_{n}^{\prime})$.
Set $G_{n}^{i}=\bigcup_{j\in\hat{G}_{n}^{i}}K_{n}^{i}(j)$. For each
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, define
$D_{n}=\bigcup_{\ell=0}^{h_{n}-1}G_{n}^{\ell}.$
Note that
$\mu(C_{n}\setminus
D_{n})<\sum_{\ell=0}^{h_{n}-1}\frac{\epsilon_{n}}{h_{n}}=\epsilon_{n}.$
Next, we show almost every point falls in infinitely many $D_{n}$.
###### Property 5.4.
$\mu(\overline{D})=1$ where
$\overline{D}=\bigcap_{n=1}^{\infty}\bigcup_{i=n}^{\infty}D_{i}$.
###### Proof.
Given $\epsilon>0$, choose
$N=N(\epsilon)\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that
$\sum_{n=N}^{\infty}\epsilon_{n}<\epsilon$. Thus,
$\displaystyle\mu(\bigcup_{n=N}^{\infty}D_{n})$ $\displaystyle\geq$
$\displaystyle\mu(\bigcup_{n=N}^{\infty}C_{n})-\sum_{n=N}^{\infty}\mu(C_{n}\setminus
D_{n})$ $\displaystyle>$ $\displaystyle
1-\sum_{n=N}^{\infty}\epsilon_{n}>1-\epsilon.$
Since $\epsilon$ is arbitrarily small, then
$\mu(\bigcup_{n=N}^{\infty}D_{n})=1$, and Property 5.4 is establshed. ∎
#### 5.3.2 Weak Mixing Component
The main goal in this work is to demonstrate how properties of a given ergodic
transformation can be transferred to produce a tailored weak mixing
transformation. Since the weak mixing component will dissipate over time, we
do not focus on introducing general properties using $S$. Instead, we set $S$
to the Chacon transformation inside our towerplex construction. Thus, $S_{n}$
will be isomorphic to Chacon’s transformation. By Lemma 5.3, for each
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, there exists
$m_{n}\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that for each
$i=0,1,\ldots,h_{n}-1$, $j\in\hat{K}_{n}^{i}$ and $A=K_{n}^{i}(j)$,
$|\mu(S_{n+1}^{H_{m_{n}}}A\cap A)-\frac{1}{2}\mu(A)|<\epsilon_{n}\mu(A)$
and
$|\mu(S_{n+1}^{H_{m_{n}}}A\cap
S^{-1}(A)-\frac{1}{2}\mu(A)|<\epsilon_{n}\mu(A).$
Let $w_{n}=\min{\\{\mu(K_{\ell}^{i}(j))>0:1\leq\ell\leq n,0\leq i\leq
h_{n}-1,j\in\hat{K}_{\ell}^{i}\\}}$. Choose $h_{n+1}$ such that
$\displaystyle h_{n+1}>\frac{H_{m_{n}}}{\epsilon_{n}w_{n}}.$ (22)
## 6 Slow Weak Mixing Theorem
In this final section, we prove our main result using the towerplex
constructions. First, we give explicit parameters $r_{n}$ and $s_{n}$ that can
be used to generate our rigid weak mixing examples. Let
$r_{n}=\frac{\mu(I_{n}^{\prime})}{\mu(I_{n})}=\frac{1}{2(n+2)}$ and
$s_{n}=\frac{\mu(J_{n}^{\prime})}{\mu(J_{n})}=\frac{1}{2}$. Thus, each of the
switching sets have measure
$\mu(\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}(I_{n}^{\prime}))={(\mu(X_{n})-\mu(X_{n}^{*}))}/{2(n+2)}$
and
$\mu(\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}(J_{n}^{\prime}))={(\mu(Y_{n})-\mu(Y_{n}^{*}))}/2$
for $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. This implies
$\mu(Y_{n+1})=\frac{1}{2(n+2)}[(n+1)\mu(Y_{n})+1]+\kappa_{n}\epsilon_{n}$
where $|\kappa_{n}|$ is bounded for all
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. If all residuals had zero mass, then
$\kappa_{n}\epsilon_{n}=0$ and by induction:
$\mu(X_{n})=\frac{n}{n+1}\mbox{ and }\mu(Y_{n})=\frac{1}{n+1}.$
In the case the residuals are not null, the next lemma obtains
$\lim_{n\to\infty}\mu(X_{n})=1\mbox{, }\lim_{n\to\infty}\mu(Y_{n})=0.$
Parameters given here are called the canonical towerplex settings.
###### Lemma 6.1.
If real numbers $\epsilon_{n}>0$ are chosen sufficiently small for
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, then a canonical towerplex
construction, given by $r_{n}=\frac{1}{2(n+2)}$ and $s_{n}=\frac{1}{2}$ has
the property, for $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$,
$\displaystyle\frac{1}{n+2}<\mu(Y_{n})<\frac{1}{n}.$ (23)
###### Proof.
The function $f(y)=({1}/{2(n+2)})[(n+1)y+1]$ has a fixed point at
$y={1}/{(n+3)}$. If $y>{1}/{(n+3)}$, then $f(y)>{1}/{(n+3)}$. Thus, if
$\epsilon_{n}$ is sufficiently small, and $\mu(Y_{n})>{1}/{(n+2)}$, then
$\mu(Y_{n+1})>{1}/{(n+3)}$. This establishes the first inequality from (23).
To prove the second inequality, assume $y=\mu(Y_{n})<{1}/{n}$ for fixed
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. Thus,
$\displaystyle f(y)$ $\displaystyle<$
$\displaystyle\frac{1}{2(n+2)}[(n+1)(\frac{1}{n})+1]=\frac{1}{2(n+2)}[2+\frac{1}{n}]$
$\displaystyle=$
$\displaystyle\frac{1}{n+2}+\frac{1}{2n(n+2)}=\frac{1}{n+1}+\frac{1-n}{2n(n+1)(n+2)}\leq\frac{1}{n+1}.$
Therefore, if $\epsilon_{n}$ is sufficiently small,
$\mu(Y_{n+1})<\frac{1}{n+1}$. ∎
Now, we are ready to prove our main theorem.
###### Theorem 6.2.
Given an ergodic measure preserving transformation $R$ on a Lebesgue
probability space, and a rigid sequence $\rho_{n}$ for $R$, there exists a
weak mixing transformation $T$ on a Lebesgue probability space such that $T$
is rigid on $\rho_{n}$.
###### Proof.
Much of the details have been established in the previous sections. In
particular, the conditions imposed in each of the sections on ergodicity,
rigidity and weak mixing, are consistent. Essentially, $\epsilon_{n}\to 0$
arbitrarily fast which is possible since only the extra mass from successive
Rokhlin towers is bounded by $\epsilon_{n}$. Also, each section imposes a
lower bound on the growth rate of the tower heights $h_{n}$, but no upper
bound. Appendix B lists conditions that can be used to support the explicit
proofs. Below, we need to complete the argument that $T$ is weak mixing.
Suppose $f\neq 0$ is an eigenfunction for $T$ with eigenvalue $\lambda$. Since
we established that $T$ is ergodic, we may assume $|f|$ is a constant. Without
loss of generality, assume $|f|=|\lambda|=1$. Given $\delta>0$, there exists a
set $\Lambda_{\delta}$ of positive measure such that for
$x,y\in\Lambda_{\delta}$, $|f(x)-f(y)|<\delta$. Let
$\Lambda_{\delta}^{\prime}$ be the set of Lebesgue density points of
$\Lambda_{\delta}$. In particular, if
$\Lambda_{\delta}^{\prime}=\\{x\in\Lambda_{\delta}:\lim_{\eta\to
0}\frac{\mu(\Lambda_{\delta}\cap(x-\eta,x+\eta))}{2\eta}=1\\}$, then
$\mu(\Lambda_{\delta}^{\prime})=\mu(\Lambda_{\delta})>0$. Choose
$x\in\Lambda_{\delta}^{\prime}\cap\overline{D}$. Choose $\eta^{\prime}>0$ such
that for $\eta<\eta^{\prime}$,
$\frac{\mu(\Lambda_{\delta}\cap(x-\eta,x+\eta))}{2\eta}>1-\delta$. Choose
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that
$\frac{1}{2^{k_{n}}}<\eta^{\prime}$, $\sum_{i=n}^{\infty}\epsilon_{i}<\delta$
and $x\in D_{n}$. There exists $i=i(x)$ such that $x\in G_{n}^{i}$, and
subsequently $j=j(x)$ such that $x\in K_{n}^{i}(j)$. Let
$\eta_{x}=\max{\\{|y-x|:y\in K_{n}^{i}(j)\\}}$. Note $\eta_{x}<\eta^{\prime}$,
and
$\frac{\mu(\Lambda_{\delta}\cap(x-\eta_{x},x+\eta_{x}))}{2\eta_{x}}>1-\delta$.
Thus,
$\displaystyle\mu(\Lambda_{\delta}\cap K_{n}^{i}(j))$ $\displaystyle>$
$\displaystyle\mu(K_{n}^{i}(j))-2\eta_{x}\delta\mu(K_{n}^{i}(J))-2\delta\mu(K_{n}^{i}(j))$
(24) $\displaystyle\geq$ $\displaystyle(1-2\delta)\mu(K_{n}^{i}(j)).$ (25)
Hence,
$\displaystyle|\mu(S_{n+1}^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap
K_{n}^{i}(j)))-\frac{1}{2}\mu(\Lambda_{\delta}\cap K_{n}^{i}(j))|$
$\displaystyle\leq$
$\displaystyle|\mu(S_{n+1}^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap
K_{n}^{i}(j)))-\mu(S_{n+1}^{H_{m_{n}}}(K_{n}^{i}(j))\cap K_{n}^{i}(j))|$
$\displaystyle+$
$\displaystyle|\mu(S_{n+1}^{H_{m_{n}}}(K_{n}^{i}(j))\cap(K_{n}^{i}(j)))-\frac{1}{2}\mu(K_{n}^{i}(j))|$
$\displaystyle+$
$\displaystyle|\frac{1}{2}\mu(K_{n}^{i}(j))-\frac{1}{2}\mu(\Lambda_{\delta}\cap
K_{n}^{i}(j))|$ $\displaystyle<$ $\displaystyle
4\delta\mu(K_{n}^{i}(j))+\epsilon_{n}\mu(K_{n}^{i}(j))+\delta\mu(K_{n}^{i}(j))=(5\delta+\epsilon_{n})\mu(K_{n}^{i}(j)).$
We wish to establish that $T$ is weak mixing, and $T$ does not equal $S_{n+1}$
everywhere. In particular, $T$ may differ from $S_{n+1}$ on the top levels of
the towers of height $h_{n+1},h_{n+2},\ldots$, on the accompanying residuals,
and on the transfer sets. However, we have chosen the growth of the tower
heights sufficient to ensure the set where $T$ and $S_{n+1}$ may differ will
be small relative to interval, $K_{n}^{i}(j)$. Thus,
$\displaystyle\mu(\\{x\in Y_{n+1}:Tx\neq S_{n+1}x\\})$ $\displaystyle<$
$\displaystyle\sum_{i=n+1}^{\infty}[\frac{1}{h_{i}}+4\epsilon_{i}]$ (26)
$\displaystyle<$
$\displaystyle\sum_{i=n}^{\infty}[\frac{5\epsilon_{i}w_{n}}{H_{m_{i}}+1}].$
(27)
This implies
$\displaystyle\mu(\\{x\in Y_{n+1}$ $\displaystyle:$ $\displaystyle T^{i}x\neq
S_{n+1}^{i}x,i=1,2,\ldots,H_{m_{n}}+1\\})$ $\displaystyle<$ $\displaystyle
w_{n}(H_{m_{n}}+1)\sum_{i=n}^{\infty}\frac{5\epsilon_{i}}{H_{m_{i}}+1}<5w_{n}\sum_{i=n}^{\infty}\epsilon_{i}<5\delta
w_{n}.$
Hence,
$\displaystyle|\mu(T^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap
K_{n}^{i}(j)))-\frac{1}{2}\mu(\Lambda_{\delta}\cap K_{n}^{i}(j))|$
$\displaystyle\leq$ $\displaystyle|\mu(T^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap K_{n}^{i}(j)))$ $\displaystyle-$
$\displaystyle\mu(S_{n+1}^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap K_{n}^{i}(j)))|$ $\displaystyle+$
$\displaystyle|\mu(S_{n+1}^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap
K_{n}^{i}(j)))-\frac{1}{2}\mu(\Lambda_{\delta}\cap K_{n}^{i}(j))|$
$\displaystyle<$ $\displaystyle 5\delta
w_{n}+(5\delta+\epsilon_{n})\mu(K_{n}^{i}(j))\leq(10\delta+\epsilon_{n})\mu(K_{n}^{i}(j)).$
For $\delta$ and $\epsilon$ sufficiently small, there exists
$x_{1}\in\Lambda_{\delta}\cap K_{n}^{i}(j)$ such that
$T^{H_{m_{n}}}x_{1}\in\Lambda_{\delta}\cap K_{n}^{i}(j)$, and
$x_{2}\in\Lambda_{\delta}\cap K_{n}^{i}(j)$ such that
$T^{H_{m_{n}}+1}x_{2}\in\Lambda_{\delta}\cap K_{n}^{i}(j)$. Thus,
$\displaystyle|\lambda^{H_{m_{n}}}f(x_{1})-f(x_{1})|=|f(T^{H_{m_{n}}}x_{1})-f(x_{1})|<\delta,$
(28)
and
$\displaystyle|\lambda^{H_{m_{n}}+1}f(x_{2})-f(x_{2})|=|f(T^{H_{m_{n}}+1}x_{2})-f(x_{2})|<\delta.$
(29)
Hence,
$|\lambda^{H_{m_{n}}}-1|<\frac{\delta}{|f(x_{1})|}=\delta\mbox{ and
}|\lambda^{H_{m_{n}}+1}-1|<\frac{\delta}{|f(x_{2})|}=\delta.$
Therefore,
$|\lambda-1|=|\lambda^{H_{m_{n}}+1}-\lambda^{H_{m_{n}}}|\leq|\lambda^{H_{m_{n}}+1}-1|+|\lambda^{H_{m_{n}}}-1|<2\delta.$
Since $\delta>0$ may be chosen arbitrarily small, then $\lambda=1$. Since it
was established that $T$ is ergodic in an earlier section, then $f$ must be a
constant. Therefore, $T$ is weak mixing. ∎
Our theorem establishes the following corollaries which answer questions
raised in the ground-breaking works [4] and [8].
###### Corollary 6.3.
Given any ergodic measure preserving transformation $R$ on a Lebesgue
probability space with discrete spectrum, and a rigidity sequence $\rho_{n}$
for $R$, there exists a weak mixing transformation $T$ with rigidity sequence
$\rho_{n}$. In particular, for any
$k\in\hbox{\rm\hbox{I}\kern-1.62498ptN},k\geq 2$, there exists a weak mixing
transformation with $k^{n}$, $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, as a
rigidity sequence.
The next corollary gives an explicit characterization of ”large” rigid
sequences for weak mixing transformations. While this corollary appears known
in [2], our characterization gives a general concrete method for establishing
”large” rigidity sequences of weak mixing transformations. Given a sequence
$\mathcal{A}$, define the density function
$g_{\mathcal{A}}:\hbox{\rm\hbox{I}\kern-1.62498ptN}\to[0,1]$ such that
$g_{\mathcal{A}}(k)={\\#(\mathcal{A}\cap\\{1,2,\ldots k\\})}/{k}$.
###### Corollary 6.4.
Given any real-valued function
$f:\hbox{\rm\hbox{I}\kern-1.62498ptN}\to(0,\infty)$ such that
$\lim_{n\to\infty}f(n)=0,$
there exists a weak mixing transformation with rigidity sequence $\mathcal{A}$
such that
$\lim_{n\to\infty}\frac{f(n)}{g_{\mathcal{A}}(n)}=0.$
Also, there exist weak mixing transformations with rigidity sequences
$\rho_{n}$ satisfying
$\lim_{n\to\infty}\frac{\rho_{n+1}}{\rho_{n}}=1.$
###### Proof.
Let $\alpha$ be an irrational number and $R_{\alpha}$ the rotation by
$2\pi\alpha$ on the unit circle. Given $\epsilon>0$, define
$\mathcal{A}(\epsilon)=\\{j\in\hbox{\rm\hbox{I}\kern-1.62498ptN}:|\exp{(2\pi\alpha
j)}-1|<\epsilon\\}$, and for $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, define
$\mathcal{A}(\epsilon,n)=\mathcal{A}(\epsilon)\cap\\{1,2,\ldots n\\}$. For
$\bar{\epsilon}=\\{\epsilon_{1}<\epsilon_{2}<\ldots\\}$, let
$\mathcal{A}(\bar{\epsilon})=\bigcup_{n=1}^{\infty}\mathcal{A}(\epsilon_{n},n)$.
If $\lim_{n\to\infty}\epsilon_{n}=0$ and $\mathcal{A}(\bar{\epsilon})$ is
infinite, then $\mathcal{A}(\bar{\epsilon})$ forms a rigidity sequence for
$R_{\alpha}$. Let $f:\hbox{\rm\hbox{I}\kern-1.62498ptN}\to(0,\infty)$ be such
that $\lim_{n\to\infty}f(n)=0$. Since $\mathcal{A}(1/2^{i})$ has positive
density for $i\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, there exists
$j_{i}\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that
$\frac{|\mathcal{A}({1}/{2^{i}},j)|}{j}>f(j).$
for all $j\geq j_{i}$. For $k\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, choose
$i=i_{k}\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that $j_{i}+1\leq k\leq
j_{i+1}$. Set $\epsilon_{k}=\frac{1}{2^{i}}$ and let
$\mathcal{A}=\mathcal{A}(\bar{\epsilon})$. Thus,
$\displaystyle g_{\mathcal{A}}(k)$ $\displaystyle=$
$\displaystyle\frac{|\mathcal{A}\cap\\{1,2,\ldots,k\\}|}{k}$ (30)
$\displaystyle\geq$
$\displaystyle\frac{|\mathcal{A}(\epsilon_{k},k)|}{k}>2^{i}f(k).$ (31)
Hence,
$\displaystyle\frac{f(k)}{g_{\mathcal{A}}(k)}$ $\displaystyle>$
$\displaystyle\frac{1}{2^{i}}.$ (32)
This confirms that $\lim_{k\to\infty}{f(k)}/{g_{\mathcal{A}}(k)}=0$ for the
rigidity sequence $\mathcal{A}$. Therefore, by Theorem 6.2, $\mathcal{A}$ is a
rigidity sequence for a weak mixing transformation. The second assertion of
Corollary 6.4 can be established in a similar manner. Since ergodic rotations
on the unit circle have rigid sequences $\rho_{n}$ such that
$\lim_{n\to\infty}{\rho_{n+1}}/{\rho_{n}}=1$, then weak mixing transformations
admit these rigid sequences as well. ∎
Previously, it was established that denominators from convergents of continued
fractions serve as rigidity sequences for weak mixing transformations. A
partial result was provided in [8] for restricted convergents, and then a
general result was established in [4]. In this paper, we extend these results
to show that any rigidity sequence for an ergodic rotation on the unit circle
is also a rigidity sequence for a weak mixing transformation. This includes
sequences $q_{n}$ formed from the denominators of convergents
${p_{n}}/{q_{n}}$ of an irrational $\alpha$.
###### Corollary 6.5.
Let $\alpha\in(0,1)$ be any irrational number, and let $\rho_{n}$ be a
sequence of natural numbers satisfying
$\lim_{n\to\infty}|\exp{(2\pi i\alpha\rho_{n})}-1|=0.$
Then there exists a weak mixing transformation $T$ such that $\rho_{n}$ is a
rigidity sequence for $T$.
Acknowledgements
The author wishes to thank Joseph Rosenblatt, Andrew Parrish, Ayse Sahin, Karl
Petersen, Nathaniel Friedman, Cesar Silva, Keri Kornelson and Tatjana Eisner
for feedback on a previous version of this article.
## Appendix A Towerplex Pictorial
This appendix provides an illustration of towers for $R_{1}$, $S_{1}$, and the
multiplexing operation applied to obtain towers for $R_{2}$ and $S_{2}$. The
picture below represents only the case where $d_{R}>0$ and $d_{S}<0$. The
other cases are handled as described in the section on towerplex
constructions. Also, the general case of deriving $R_{n+1}$ and $S_{n+1}$ from
$R_{n}$ and $S_{n}$ is analogous to the initial multiplexing operation for
deriving $R_{2}$ and $S_{2}$.
$I_{1}$$R_{1}$$r_{1}$$I_{1}^{\prime}$$X_{1}^{*}$$X_{1}$$J_{1}$$S_{1}$$s_{1}$$J_{1}^{\prime}$$d_{R}/h_{1}$$\supset
I_{1}^{*}$$Y_{1}^{*}$$Y_{1}$
Transformations $R_{2}$ and $S_{2}$ are derived from $R_{1}$ and $S_{1}$ by
switching the red subcolumn with the green subcolumn. We refer to these sets
as the switching sets, and are the main multiplexing operation. In order to
preserve maps isomorphic to $R$ and $S$, and avoid redefining $R_{1}$ or
$S_{1}$ on most of the probability space, it may be necessary to transfer
measure between the towers and residuals. This is a rescaling operation, and
these sets are referred to as transfer sets. In the case where $d_{R}>0$, the
blue colored subcolumn $I_{1}^{*}$ from $J_{1}^{\prime}\subset Y_{1}$ is
absorbed into $X_{1}^{\prime}$. For $d_{S}<0$, mass is removed from
$Y_{1}^{*}$ and added as a blue subcolumn to define $S_{2}$.
$I_{1}$$R_{1}$$R_{2}$$s_{1}$$J_{1}^{\prime}\setminus{I_{1}^{*}}$$S_{1}$$X_{1}^{\prime}$$X_{2}$$d_{S}/h_{1}$$J_{1}^{*}$$J_{1}$$S_{1}$$S_{2}$$r_{1}$$I_{1}^{\prime}$$R_{1}$$Y_{1}^{\prime}$$Y_{2}$
## Appendix B Towerplex Conditions
Below is a list of explicit conditions that can be used to prove theorem 6.2.
1. 1.
$\lim_{n\to\infty}r_{n}=0$;
2. 2.
$\sum_{n=1}^{\infty}r_{n}=\sum_{n=1}^{\infty}s_{n}=\infty$;
3. 3.
$\lim_{n\to\infty}\mu(Y_{n})=0$;
4. 4.
${\epsilon_{n}}/{\max{\\{M_{n}^{1},M_{n}^{2}\\}}}<\epsilon_{n+1}$;
5. 5.
$h_{n-1}<M_{n}^{1},M_{n}^{2}<h_{n}$;
6. 6.
$h_{n}\mbox{ sufficiently large such that equation \ref{SwitchingSetInd}
holds}$;
7. 7.
$h_{n+1}\epsilon_{n}w_{n}>H_{m_{n}}+1$;
8. 8.
$\epsilon_{n+1}(H_{m_{n}}+1)<\epsilon_{n}w_{n}$;
9. 9.
$H_{m_{n+1}}\geq H_{m_{n}}$.
If $r_{n}={1}/{2(n+2)}$ and $s_{n}=1/2$, and $\epsilon_{n}$ is sufficiently
small such that Lemma 6.1 holds, then we have a canonical towerplex
construction.
## References
* [1] Aaronson, J., Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle, Israel Journal of Mathematics, 33:3 (1979), 181-197.
* [2] Aaronson, J., Hosseini, M. and Lemanczyk, M, IP-rigidity and eigenvalue groups, arXiv:1203.2257 (25 May 2012).
* [3] Avila, A. and Forni, G., Weak mixing for interval exchange transformations and translation flows, Annals of mathematics, 165:2 (2007), 637-664.
* [4] Bergelson, V., del Junco, A., Lemańczyk, M, Rosenblatt, J., Rigidity and non-recurrence along sequences, arXiv 1103.0905 (4 Mar 2011).
* [5] Bergelson, V. and Gorodnik, A., Weakly mixing group actions: a brief survey and an example, Modern dynamical systems and applications (2005), 3-25.
* [6] Chacon, R. V., Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), 559-562.
* [7] Chaika, J., Every ergodic transformation is disjoint from almost every interval exchange transformation, Annals of Mathematics, 175 (2012), 237-253.
* [8] Eisner, T. and Grivaux, S., Hilbertian Jamison sequences and rigid dynamical systems, J. Funct. Anal. 261:7 (2011), 2013-2052.
* [9] El Abdalaoui and El Houcein, On the spectrum of $\alpha$-rigid maps, Journal of dynamical and control systems, 15.4 (2009), 453-470.
* [10] Friedman, N., Partial mixing, partial rigidity, and factors, Contemp. Math, 94 (1989), 141-145.
* [11] Friedman, N. A., Introduction to ergodic theory. Vol. 29. Van Nostrand Reinhold, New York (1970).
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* [13] Furstenberg, H. and Weiss, B., The finite multipliers of infinite ergodic transformations, The structure of attractors in dynamical systems (1978), 127-132.
* [14] Glasner, S. and Maon, D., Rigidity in topological dynamics, Ergodic Theory Dynam. Systems, 9:2 (1989), 309-320.
* [15] Grivaux, S., IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products, arXiv:1209.2884 (13 Sep 2012).
* [16] Halmos, P., Lectures on Ergodic Theory, Math. Soc. Japan, (1956) MR0097489
* [17] Halmos, P., Approximation Theories for Measure Preserving Transformations, Transactions of the American Mathematical Society, (1944) MR0009703
* [18] James, Jennifer, Thomas Koberda, Kathryn Lindsey, Cesar E. Silva, and Peter Speh, On ergodic transformations that are both weakly mixing and uniformly rigid, New York J. Math, 15 (2009), 393-403.
* [19] Kakutani, S., Induced Measure Preserving Transformations, Proc. Japan Acad., 19 (1943), 65-41.
* [20] Kalikow, S., Infinite partitions and Rokhlin towers, Ergodic Theory and Dynami. Systems, 32:2 (2012), 707-738. doi:10.1017/S0143385711000381.
* [21] Katok, A. and Lemanczyk, M., Some new cases of realization of spectral multiplicity function for ergodic transformations, Fundamenta Mathematicae, 206 (2009), 185-215.
* [22] Katok, A.B. and Stepin, A.M. Approximations in Ergodic Theory, Russian Math. Surveys , 22:5 (1967), 77-102. MR0219697
* [23] Petersen, K. E. Ergodic theory. Vol. 2. Cambridge University Press (1989).
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|
arxiv-papers
| 2013-01-04T18:22:58 |
2024-09-04T02:49:39.930842
|
{
"license": "Public Domain",
"authors": "Terrence Adams",
"submitter": "Terrence Adams",
"url": "https://arxiv.org/abs/1301.0791"
}
|
1301.0899
|
# Structure of Spin Polarized Strange Quark Star in the Presence of Magnetic
Field at Finite Temperature
G. H. Bordbar1,2 111Corresponding author. E-mail: [email protected],
F. Kayanikhoo 1 and H. Bahri 1 Department of Physics, Shiraz University,
Shiraz 71454, Iran
and
Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box
55134-441, Maragha 55177-36698, Iran
###### Abstract
In this paper, we have calculated the thermodynamic properties of spin
polarized strange quark matter at finite temperature in the presence of a
strong magnetic field using MIT bag model. We have also computed the equation
of state of spin polarized strange quark matter in the presence of strong
magnetic field and finally, using this equation of states we have investigated
the structure of spin polarized strange quark star at different temperatures
and magnetic fields.
## I Introduction
One of the interesting features of the compact objects is the strange quark
star (SQS) composed of the strange quark matter (SQM). The composition of SQS
was first proposed by Itoh rk3 ; simultaneously with formulation of Quantum
chromo dynamics (QCD). Later, Bodmer discussed the fate of an astronomical
object collapsing to such a state of matter rk4 . The perturbative
computations of the equation of state of the SQM was developed after the
formation of QCD, but the region of validity of these calculations was
restricted to very high densities rk50 . The concept of SQS was also discussed
by witten. He proposed that SQM composed of light quarks, and is more stable
than the nuclei rk5 . Therefore the SQM can be the ground state of matter.
With this point of view, other authors proposed the concept of SQS. The SQM is
composed of 3-flavors of quarks (up, down and strange) and a small number of
electrons to ensure the charge neutrality rk6 ; rk6p . A typical electron
fraction is less than $10^{-3}$ $(fm^{-3})$ and it decreases from the surface
to the center of SQS.
The collapsing star of a supernova will turn into a neutron star only if its
mass is about $1.4-3$ $M_{\odot}$. After formation of the neutron star, if
density of core is high enough ($10^{15}\ gr/cm^{3}$) the nucleons dissolve to
their components, quarks, and a hybrid star (neutron star with a core of SQM)
is formed . But if after the explosion of supernova, in the stage of proto-
neutron, the density of the core is high enough ($10^{15}\ gr/cm^{3}$), the
pure SQS may be formed, directly. Making a quark-novae. If after collapse the
leftover is more than $3M_{\odot}$ mass the star will be transform to a black
hole. For a neutron star, the radius decrease with mass, or in other word
$M\propto\ 1/R$; while for an SQS, the mass-radius relation is as $M\propto\
R^{3}$. Some result of observations show that for a typical SQS, the mass is
more than $2.1\pm 0.28\ M_{\odot}$ and the radius is more than $13.8\pm 1.8\
km$ rk7 . Besides, recent observations indicate that objects RX J185635-3754
and 3C58 may be SQS rk51 .
The maximum magnetic field on the surface of magnetars is about $10^{15}\ G$,
however the highest possible magnetic field in the center of SQS that was
estimated by virial theorem is between $10^{18}-10^{20}\ G$ rk75 ; rk62 .
Therefore, investigating the effect of an strong magnetic field on the SQM
properties is important in astrophysics. Some studies that investigate the
effect of magnetic field on quark star and quark matter are available.
Chakrabarty rk63 studied the effects of an strong magnetic field on quark
matter. He showed that the equation of state of SQS changes significantly at
the presence of strong magnetic fields. Anand et.al. rk64 expressed that the
maximum mass, the radius and gravitational red-shift of the SQS are increasing
as a function of the magnetic field.
Recently, we have calculated the structure of polarized SQS at zero
temperature rk8 and unpolarized SQS at finite temperature rk9 . We have also
calculated the structure of the neutron star with the quark core at zero
temperature rk10 and at finite temperature rk11 . In this paper we calculate
some properties of polarized SQS at finite temperature in the presence of an
strong magnetic field. The plan of this work is as follows. In section II, we
calculate the energy and equation of state of SQM by the MIT bag model at
finite temperature in the presence of a strong magnetic field and in section
III, we present the results of our calculations for different temperatures and
magnetic fields.
## II Calculation of energy and equation of state of strange quark matter
The equation of state plays an important role in obtaining the structure of a
star. There are different statistical approaches for calculation of the
equation of state of quark matter (SQM) and all of them are based on the QCD,
for example the MIT bag model rk20 ; rk21 , NJL model rk22 ; rk23 and
perturbation QCD model rk24 ; rk25 . In this paper we calculate the equation
of state of SQM by MIT bag model with a density dependent bag constant. The
MIT bag model considers free particles confined to a bounded region by a bag
pressure $B$. This pressure depends on the quark-quark interaction rk55 . In
other words, the bag constant ($B$) is the difference between the energy
densities of the noninteracting quarks and the interacting ones. For the bag
constant in the MIT bag model, different values such as $55$ and $90\
MeV/fm^{3}$ are considered.
To obtain the equation of state of SQM, at first, we should calculate the
total energy of the system. We do this work using the MIT bag model with a
density dependent bag constant in the next section.
### II.1 Energy of spin polarized strange quark matter at finite temperature
in the presence of magnetic field
At densities high enough ($10^{15}\ gr/cm^{3}$) the hadronic phase is allowed
to undergo a phase transition to the strange quark matter phase. Strange quark
matter consists of $u$, $d$ and $s$ quarks, as well as electrons in weak
equilibrium;
$d\rightarrow u+e+\overline{\vartheta}_{e},$ (1) $u+e\rightarrow
d+\vartheta_{e},$ (2) $u+d\rightarrow u+s,$ (3) $s\rightarrow
u+e+\overline{\vartheta}_{e}.$ (4)
To calculate the energy of SQM, we need to know the quark densities in term of
the baryonic number density. To compute these densities, we use the beta
equilibrium and charge neutrality conditions in the above reactions. The beta
equilibrium and charge neutrality conditions lead to the following relations
for the chemical potentials and densities of relevant quarks,
$\mu_{d}=\mu_{u}+\mu_{e},$ (5) $\mu_{s}=\mu_{u}+\mu_{e},$ (6)
$\mu_{s}=\mu_{d},$ (7)
$\frac{2}{3}n_{u}-\frac{1}{3}n_{s}-\frac{1}{3}n_{d}-n_{e}=0,$ (8)
where $\mu_{i}$ and $n_{i}$ are the chemical potential and the number density
of particle $i$ respectively. It should be noted that because the neutrinos
escape from the interior matter of star, the chemical potential of neutrinos
vanishes ($\mu_{\vartheta_{e}}=\mu_{\overline{\vartheta}_{e}}=0$). Therefore,
we get the above relations between the chemical potentials rk6 ; rk6p . As it
was mentioned in the previous section, we consider the system as pure SQM and
ignore the electrons($n_{e}=0$) rk13 ; rk14 ; rk15 , therefore Eq. (8) becomes
as follows,
$n_{u}=\frac{1}{2}(n_{s}+n_{d}).$ (9)
Now, we want to calculate the energy of SQM in the presence of an strong
magnetic field. Therefore, we consider the spin polarized SQM composed of the
$u$, $d$ and $s$ quarks with up and down spins. We show the number density of
spin-up quarks by $n_{i}^{+}$ and spin-down quarks by $n_{i}^{-}$. The
parameter of polarization is defined by the following relation,
$\zeta_{i}=\frac{n_{i}^{+}-n_{i}^{-}}{n_{i}},$ (10)
where $0\leq\zeta_{i}\leq 1$ and $n_{i}=n_{i}^{+}+n_{i}^{-}$. The chemical
potential $\mu_{i}$ for any value of the temperature ($T$) and number density
($n_{i}$), using the following constraint,
$n_{i}=\sum_{P=\pm}\frac{g}{2\pi^{2}}\int_{0}^{\infty}f(n_{i},k^{(p)},T)k^{2}dk,$
(11)
where
$f(n_{i},k^{(p)},T)=\sum_{P=\pm}\frac{1}{exp\left(\beta((m_{i}^{2}c^{4}+\hbar^{2}(k^{(p)})^{2}c^{2})^{1/2}-\mu_{eff})\right)+1},$
(12)
is the Fermi-Dirac distribution function. In the above equation
$\beta=1/k_{B}T$ and $g$ is degeneracy number of the system. In this equation
$\mu_{eff}$ is the effective chemical potential in the presence of magnetic
field. The effective chemical potential is equal to $\mu_{i}\mp\mu_{s}B$ where
$\mu_{s}$ is magnetic moment and $B$ is the magnetic field. In other word, the
single particle energy is
$\varepsilon_{i}=\sqrt{m_{i}^{2}c^{4}+\hbar^{2}k^{p}}\pm\mu_{s}B$.
The energy of spin polarized SQM in the presence of the magnetic field within
the MIT bag model is as follows,
$\varepsilon_{tot}=\varepsilon_{u}+\varepsilon_{d}+\varepsilon_{s}+\varepsilon_{M}+B,$
(13)
where
$\varepsilon_{i}=\sum_{P=\pm}\frac{g}{2\pi^{2}}\int_{0}^{\infty}(m_{i}^{2}c^{4}+\hbar^{2}(k^{(P)})^{2}c^{2})^{1/2}f(n_{i},k^{(p)},T)k^{2}dk,$
(14)
here $k^{\pm}=(\pi^{2}n_{i})^{1/3}(1\pm\zeta_{i})^{1/3}$. In our calculations,
we suppose that $\zeta=\zeta_{u}=\zeta_{d}=\zeta_{s}$. In Eq. (13),
$\varepsilon_{M}=\frac{E_{M}}{V}$ is the magnetic energy density of SQM where
$E_{M}=-M.B$ is the magnetic energy. By considering a uniform magnetic field
along $z$ direction, the contribution of magnetic energy of the spin polarized
SQM is given by
$E_{M}=-\sum_{i=u,d,s}M_{z}^{(i)}B,$ (15)
where $M_{z}^{(i)}$ is the magnetization of the system corresponding to
particle $i$ which is given by
$M_{z}^{(i)}=N_{i}\mu_{i}\zeta_{i}.$ (16)
In the above equation $N_{i}$ and $\mu_{i}$ are the number density and
magnetic moment of particle $i$. Finally, the magnetic energy density of SQM
can be obtained by the following relation,
$\varepsilon_{M}=-\sum_{i}n_{i}\mu_{i}\zeta_{i}B.$ (17)
We obtain the thermodynamic properties of the system using the Helmholtz free
energy,
$F=\varepsilon_{tot}-TS_{tot},$ (18)
where $S_{tot}$ is the total entropy of SQM,
$S_{tot}=S_{u}+S_{d}+S_{s}.$ (19)
In Eq. (19), $S_{i}$ is entropy of particle $i$,
$\displaystyle S_{i}(n_{i},T)$ $\displaystyle=$
$\displaystyle-\frac{3}{\pi^{2}}k_{B}\int_{0}^{\infty}f(n_{i},k,T)ln(f(n_{i},k,T))$
$\displaystyle+$ $\displaystyle(1-f(n_{i},k,T))ln(1-f(n_{i},k,T))]k^{2}dk.$
### II.2 Equation of state of spin polarized strange quark matter
For deriving the equation of state of strange quark matter (SQM), the
following equation is used,
$P(n,T)=\sum_{i}(n_{i}\frac{\partial F_{i}}{\partial n_{i}}-F_{i}),$ (21)
where $P$ is the pressure of system, $F_{i}$ is the free energy of particle
$i$ and $n_{i}$ is the numerical density of particle $i$.
## III Results and discussion
### III.1 Thermodynamic properties of spin polarized strange quark matter
In Fig. 1, we present the total free energy per volume versus the polarization
parameter $\zeta$, for $B=5\times 10^{18}\ G$, at different densities and
temperatures. From Fig. 1, we can see that at each density and temperature,
the energy is minimum at a particular value of the polarization parameter.
This indicates that at each density and temperature, there is a meta stable
state. We have found that for a fixed temperature, the polarization parameter
corresponding to the meta stable state reaches zero by increasing the density.
Similarly, we can see that at a fixed density, in the meta stable state the
polarization parameter decreases by increasing the temperature. These results
agree with those of reference rk94 .
In Fig. 2, we have plotted the polarization parameter versus baryonic density
at different temperatures in the presence of magnetic field $B=5\times
10^{18}\ G$. We can see that the polarization parameter decreases by
increasing the baryonic density. Fig. 2 shows that at a fixed density, the
polarization parameter decreases by increasing the temperature. This is due to
the fact that contribution of magnetic energy is more significant at low
temperature. The polarization parameter has been also drawn versus baryonic
density at a fixed temperature ($T=30\ MeV$) for different magnetic fields in
Fig. 3. It is seen that for each density, the polarization parameter increases
by increasing the magnetic field. As it can be seen from Fig.3, at high
baryonic densities, for the magnetic fields lower than about $5\times 10^{18}\
G$ the polarization parameter nearly becomes zero. In other words, for lower
magnetic fields at higher densities, the strange quark matter (SQM) becomes
nearly unpolarized. In Fig. 4, we have shown the polarization parameter as a
function of the magnetic field for density $n=0.5\ fm^{-3}$ at different
temperatures. We can see that at lower values of the magnetic field, the
polarization parameter is nearly zero, and at the magnetic fields greater than
about $5\times 10^{17}\ G$, the polarization parameter increases as the
magnetic field increases. Also, this figure shows that for each value of the
magnetic field, the polarization parameter increases by decreasing the
temperature. This is because at a higher temperatures the contribution of the
kinetic energy is higher compared to that of the magnetic energy.
We have shown the total free energy per volume of the spin polarized SQM as a
function of the baryonic density in Fig. 5 for the magnetic field $B=5\times
10^{18}\ G$ at different temperatures. From this figure,we have found that the
free energy of the spin polarization SQM has positive values for all
temperatures and densities. For each density, it is shown that the free energy
decreases by increasing the temperature.
The pressure of spin polarized SQM at different temperatures in the presence
of magnetic field ($B=5\times 10^{18}\ G$) has been plotted in Fig. 6. This
figure shows that the pressure of SQM increases by increasing the density. We
can also see that the pressure increases by increasing temperature. These
results indicate that the equation of state of spin polarized SQM becomes
stiffer by increasing the temperature.
### III.2 Structure of spin polarized strange quark star
The structures of compact objects such as white dwarfs, neutron stars, pulsars
and strange quark stars (SQS), are determined from the Tolman-Oppenheimer-
Volkov equations (TOV) rk16 ,
$\frac{dP}{dr}=-\frac{G\left[\varepsilon(r)+\frac{P(r)}{c^{2}}\right]\left[m(r)+\frac{4\pi
r^{3}P(r)}{c^{2}}\right]}{r^{2}\left[1-\frac{2Gm(r)}{rc^{2}}\right]},$ (22)
$\frac{dm}{dr}=4\pi r^{2}\varepsilon(r).$ (23)
Using the equation of state which was obtained in the previous section, we
integrate the TOV equation to compute the structure of SQS.
In Fig. 7, we have plotted the gravitational mass of spin polarized strange
quark star (SQS) versus energy density at different temperatures for a
magnetic field $B=5\times 10^{18}\ G$. We can see that for all temperatures,
the gravitational mass increases rapidly by increasing the energy density and
finally reaches a limiting value (maximum gravitational mass). Fig. 7 shows
that this limiting value decreases by increasing the temperature. From this
result, we can conclude that the stiffer equation of state leads to the lower
values for the gravitational mass of SQS.
We have shown the gravitational mass of spin polarized SQS as a function of
the radius at different temperatures in Fig. 8. This figure shows that the
gravitational mass increases by increasing radius. We can see that the
increasing rate of gravitational mass versus radius increases by increasing
the temperature. According to Fig. 6, by increasing temperature, the equation
of state becomes stiffer leading to the conclusion that by increasing
temperature, mass and the radius of spin polarized SQS decrease. Fig. 9 shows
the gravitational mass of SQS as a function of the energy density at a fixed
temperature ($T=30\ MeV$) for different magnetic fields. It can be seen that
as the magnetic field increases, the rate of increasing of gravitational mass
versus central energy density decreases, getting to a lower value at the
limiting case. In Table 1, we have shown the maximum mass and the
corresponding radius of an spin polarized SQS at different temperatures for
$B=5\times 10^{18}\ G$. It is shown that by increasing the temperature, the
maximum mass and radius of spin polarized SQS decreases. We have also shown
the maximum mass and the corresponding radius of the spin polarized SQS for
different magnetic fields at a fixed temperature $T=30\ MeV$ in Table 2. We
see that the maximum mass and radius of the spin polarized SQS decreases by
increasing the magnetic field. One of the method to compare the mass and
radius in our calculation with the observational values is calculation of the
surface red-shift value. The maximum surface red-shift calculated in our work
is $z_{s}=0.38$ (for $T=30\ MeV$ $B=5\times 10^{19}\ G$ ) that is $55.34$
lower than the upper bound on the surface red-shift for sub-luminal equation
of states (i.e. $z_{s}^{CL}=0.8509$). Furthermore the minimum amount of red-
shift in our calculations is equal to $z_{s}=0.37$ that is almost near the
red-shift of RX J185635-3754 is $z_{s}=0.35\pm=0.37$.
## IV Summary and conclusions
In this paper, we investigated the thermodynamic properties of the strange
quark matter (SQM) composed of the spin-up and spin-down of the up, down and
strange quarks at finite temperature in the presence of an strong magnetic
field using the MIT bag model. We computed the total free energy of the spin
polarized SQM at finite temperature in the presence of an strong magnetic
field. We found that the free energy gets a minimum at a particular value of
the polarization parameter showing a meta-stable state. We have also shown
that the polarization parameter corresponding to the meta-stable state reaches
zero by increasing both density and temperature. In other words, by increasing
the density and the temperature, the spin polarized SQM becomes nearly
unpolarized. We calculated the equation of state of the spin polarized SQM at
different temperatures for $B=5\times 10^{18}\ G$. We showed that by
increasing temperature, the equation of state of spin polarized SQM becomes
stiffer. Finally, using the general relativistic TOV equation, we studied the
structure of the spin polarized strange quark star (SQS). We calculated the
maximum mass and the corresponding radius of spin polarized SQS at different
temperatures and magnetic fields. We have shown that the gravitational mass of
spin polarized SQS rapidly increases by increasing the central energy density.
We also observed that the mass and the radius of the spin polarized SQS
decreases by increasing both temperature and magnetic field.
###### Acknowledgements.
We wish to thank the Research Institute for Astronomy and Astrophysics of
Maragha and Shiraz University Research Council for financial support.
## References
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Table 1: Maximum mass and the corresponding radius of SQS for $B=5\times 10^{18}\ G$ at different temperatures. The results of $T=0\ MeV$ (Bordbar & Peyvand rk8 ) have been also given for comparison. $T\ (MeV)$ | $M_{max}\ (M_{\odot})$ | $R\ (km)$
---|---|---
$0$ | 1.33 | 7.44
$30$ | 1.17 | 7.37
$70$ | 1.16 | 7.21
Table 2: Maximum mass and the corresponding radius of SQS for different magnetic fields at $T=30\ MeV$. $B\ (G)$ | $M_{max}\ (M_{\odot})$ | $R\ (km)$
---|---|---
$0$ | 1.40 | 8.08
$5\times 10^{18}$ | 1.17 | 7.37
$5\times 10^{19}$ | 1.16 | 7.16
Figure 1: Total free energy per volume as a function of the polarization
parameter at different temperatures and densities for $B=5\times 10^{18}\ G$.
Figure 2: The polarization parameter versus baryonic density at different
temperatures for $B=5\times 10^{18}\ G$.
Figure 3: The polarization parameter versus baryonic density at $T=30\ MeV$
for different magnetic fields.
Figure 4: The polarization parameter as a function of magnetic field at
$n=0.5\ fm^{-3}$ for different temperatures.
Figure 5: The total free energy per volume of the spin polarized SQM as a
function of the baryonic density for $B=5\times 10^{18}\ G$ at different
temperatures.
Figure 6: The pressure of the spin polarized SQM for $B=5\times 10^{18}\ G$ at
different temperatures.
Figure 7: The gravitational mass of spin polarized SQS versus energy density
at different temperatures for $B=5\times 10^{18}\ G$. The results of $T=0\
MeV$ (Bordbar & Peyvand rk8 ) have been also given for comparison.
Figure 8: The gravitational mass of spin polarized SQS as a function of the
radius at different temperatures for $B=5\times 10^{18}\ G$. The results of
$T=0\ MeV$ (Bordbar & Peyvand rk8 ) have been also given for comparison.
Figure 9: The gravitational mass versus energy density at $T=30\ MeV$ for
different magnetic fields.
|
arxiv-papers
| 2013-01-05T12:52:02 |
2024-09-04T02:49:39.945718
|
{
"license": "Public Domain",
"authors": "G. H. Bordbar, F. Kayanikhoo and H. Bahri",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1301.0899"
}
|
1301.1015
|
# The concepts of depth of a pair of ideals $(I,J)$ on modules and
$(I,J)$-Cohen–Macaulay modules
M. Aghapournahr1 1 Department of Mathematics, Faculty of Science, Arak
University, Arak, 38156-8-8349, Iran. [email protected] , Kh. Ahmadi-
amoli2 2 Department of Mathematics, Payame Noor University, Tehran,
19395-3697, Iran. [email protected] and M. Y. Sadeghi3 3 Department of
Mathematics, Payame Noor University, Tehran, 19395-3697, Iran.
[email protected]
###### Abstract.
We introduce a generalization of the notion of depth of an ideal on a module
by applying the concept of local cohomology modules with respect to a pair of
ideals. Some vanishing theorems are given for this invariant. Vanishing of
these kind of local cohomology modules are related to the vanishing of local
cohomology modules with respect to an ideal. We show that local cohomology
with respect to an arbitrary pair of ideals $(I,J)$ are concerned with the
local cohomology with respect to a pair of ideals whose the first ideal is
generated by any $k$-regular sequence in $I$. We also introduce the concept of
$(I,J)$-Cohen–Macaulay modules as a generalization of the concept of
Cohen–Macaulay modules. These kind of modules are different from
Cohen–Macaulay modules, as an example shows. Also an artinian result for such
modules is given.
###### Key words and phrases:
local cohomology modules defined by a pair of ideals, local cohomology, Serre
subcategory, associated primes, ZD-modules, depth of a pair of ideals,
$(I,J)$-Cohen–Macaulay modules, Cohen–Macaulay modules.
###### 2010 Mathematics Subject Classification:
13D45, 13E05, 14B15.
## 1\. Introduction
Let $R$ be a commutative noetherian ring, $I$ , $J$ be two ideals of $R$, and
$M$ be an $R$-module. Many generalizations of the notion of depth of an ideal
on a module have been introduced for various purposes, such as depth,
$f$-depth, g depth, and $k$-depth $(k\geq-1)$ of an ideal introduced in [15],
[4], [12], and [14], respectively.
In this paper, we introduce a new depth related to a pair of ideals on a
module which is concerned with the local cohomology modules with respect to
that pair of ideals introduced by Takahashi, Yoshino, and Yoshizawa [17]. For
this purpose, in section 2, we first prove a main Theorem 2.6 on the vanishing
of both local cohomology with respect to a pair of ideals and the ordinary
one. This theorem plays a main role to obtain the other results in this paper.
For instance, for a non-negative integer $t$, vanishing of
$\operatorname{H}^{i}_{I,J}(M)$ for all $i<t$ is equivalent to the vanishing
of $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ for any
$\mathfrak{a}\in\tilde{W}(I,J)$ and all $i<t$ (see Corollary 2.7). In
Corollary 2.11, we obtain a general result on an arbitrary Serre class
$\mathcal{S}$. Applying this result for $\mathcal{S}=0$, we obtain a necessary
and sufficient condition for the vanishing and finiteness of
$\operatorname{H}^{t}_{I,J}(M)$ (see Corollary 2.12).
By using the concept of $k$-regular sequence [6], we show that local
cohomology with respect to an arbitrary pair of ideals is concerned with the
local cohomology with respect to a pair of ideals whose one of the ideals is
generated by any $k$-regular sequence. This generalizes [13, Lemma 3.4] to a
noetherian ring which is not necessary local (see Theorem 2.13).
In section 3, we introduce the concept of depth of a pair of ideals $(I,J)$ on
an $R$-module $M$, denoted by depth$(I,J,M)$. This invariant equals to
${\textmd{inf~{}}}\big{\\{}i\in\mathbb{N}_{0}\mid\operatorname{H}^{i}_{I,J}(M)\neq
0\big{\\}}$
(see Proposition 3.3). We prove some formulas and inequalities for this
invariant and examine how it behaves under various ideal theoretic operations
and short exact sequences (see Corollary 3.4–Theorem 3.6). Also, we show that
this is less than or equal to the ordinary depth of an ideal (see Corollary
3.4). The non-equality is shown in Example 4.3. For a finite $R$-module $M$
and any $\mathfrak{a}\in\tilde{W}(I,J)$ with $\mathfrak{a}M\neq M$, we present
some conditions under which depth$(\mathfrak{a},M)$ equals to depth$(I,J,M)$
(see Theorem 3.7). It is well-known that
$\textmd{depth}(I,M)={\textmd{inf~{}}}\big{\\{}i\in\mathbb{N}_{0}\mid\operatorname{H}^{i}_{I}(M)\neq
0\big{\\}}.$
We show that if $t=$ depth$(I,J,M)$ and $\operatorname{H}^{t}_{I}(M)\neq 0$,
then $t=$ depth$(I,M)$ (see Theorem 3.9).
In section 4, by applying the concept of depth of a pair of ideals, we
introduce the concept of $(I,J)$-Cohen–Macaulay modules over noetherian rings,
as a generalization of Cohen–Macaulay modules over local rings, for
$I=\mathfrak{m}$ and $J=0$. These two concepts are not the same, as it is
shown in Propositions 4.8 and 4.9 and Example 4.11 (iii). Indeed, for a local
integral domain $R$, and a faithful finite $R$-module $M$, which is
$(I,J)$-Cohen–Macaulay module (with $J\neq 0$), if $t=$ depth$(I,J,M)$ and
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{t}_{I,J}(M))\big{)}\neq
0$, then $M$ is not Cohen–Macaulay. In Example 4.4, we show that the
Grothendieck,s non-vanishing Theorem does not hold for local cohomology
modules with respect to a pair of ideals and in Examples 4.11, we show that if
the condition
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{t}_{I,J}(M))\big{)}\neq
0$ does not hold in Proposition 4.8, the assertions do not necessarily hold.
Finally, an artinian result for $M/JM$ is proved in Proposition 4.14, for a
finite $(I,J)$-torsion $R$-module $M$ which is $(I,J)$-Cohen–Macaulay module.
## 2\. Vanishing of $\operatorname{H}^{i}_{I,J}(M)$
In the next proposition, $\textmd{Mod}(R)$ is denoted the category of
$R$-modules and $R$-homomorphisms, and $\mathcal{S}$ is denoted a Serre
subcategory of $\textmd{Mod}(R)$.
###### Proposition 2.1.
Let $\mathfrak{a}\in\tilde{W}(I,J)$. Let $\emph{G :
Mod}(R)\rightarrow\emph{Mod}(R)$ be a left exact covariant functor such that
$\big{(}0:_{X}\mathfrak{a}\big{)}=\big{(}0:_{\emph{G}(X)}\mathfrak{a}\big{)}$
for any $R$-module $X$ and $\emph{G}(E)$ be an injective $R$-module for any
injective $R$-module $E$. For a non-negative integer $t$, consider the natural
homomorphism
$\psi:\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{a},M\big{)}\longrightarrow\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\emph{G}^{t}(M)\big{)}.$
1. (i)
If
$\operatorname{Ext}^{t-j}_{R}\big{(}R/\mathfrak{a},\emph{G}^{j}(M)\big{)}\in\mathcal{S}$
for all $j<t$, then _Ker_ $\psi\in\mathcal{S}$.
2. (ii)
If
$\operatorname{Ext}^{t+1-j}_{R}\big{(}R/\mathfrak{a},\emph{G}^{j}(M)\big{)}\in\mathcal{S}$
for all $j<t$, then _Coker_ $\psi\in\mathcal{S}$.
3. (iii)
If
$\operatorname{Ext}^{n-j}_{R}\big{(}R/\mathfrak{a},\emph{G}^{j}(M)\big{)}\in\mathcal{S}$
for $n=t,~{}t+1$ and all $j<t$, then _Ker_ $\psi$ and _Coker_ $\psi$ both
belong to $\mathcal{S}$. Thus
$\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{a},M\big{)}\in\mathcal{S}~{}$
iff
$~{}\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\emph{G}^{t}(M)\big{)}\in\mathcal{S}$.
###### Proof.
Let $\textmd{F}(-)=\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},-\big{)}$. Then
for any $R$-module $X$ we have, $\textmd{FG}(X)=\textmd{F}(X)$. Now, the
assertion follows from [2, Proposition 3.1]. ∎
###### Lemma 2.2.
Let $\mathfrak{a}\in\tilde{W}(I,J)$ and $X$ be an $R$-module. Then
$\big{(}0:_{X}\mathfrak{a}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{a}}(X)}\mathfrak{a}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{a},J}(X)}\mathfrak{a}\big{)}=\big{(}0:_{\Gamma_{I,J}(X)}\mathfrak{a}\big{)}.$
In particular, for any ideal $\mathfrak{b}\in\tilde{W}(I,0)$, we have
1. (i)
$\big{(}0:_{X}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{b}}(X)}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{b},J}(X)}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{I,J}(X)}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{I}(X)}\mathfrak{b}\big{)}$.
2. (ii)
$\big{(}0:_{X}\mathfrak{b}\big{)}=\big{(}0:_{\Gamma_{\mathfrak{b},J}(X)}\mathfrak{b}\big{)}\subseteq\big{(}0:_{\Gamma_{\mathfrak{b},J}(X)}I\big{)}\subseteq\big{(}0:_{\Gamma_{I,J}(X)}I\big{)}=\big{(}0:_{X}I\big{)}$.
###### Remark 2.3.
_Let $\big{(}\Lambda,\leq\big{)}$ be a (non-empty) directed partially ordered
set. By an inverse family of ideals of $R$ over $\Lambda$, we mean a family
$\big{(}{\mathfrak{b}}_{\alpha}\big{)}_{\alpha\in\Lambda}$ of ideals of $R$
such that, whenever $(\alpha,\beta)\in\Lambda\times\Lambda$ with
$\alpha\geq\beta$, we have
${\mathfrak{b}}_{\alpha}\subseteq{\mathfrak{b}}_{\beta}$. For the inverse
family of ideals
$\mathcal{B}=\big{(}{\mathfrak{b}}_{\alpha}\big{)}_{\alpha\in\Lambda}$, we
shall denote
$\operatorname{H}^{i}_{\mathcal{B}}(M):={\varinjlim_{\alpha\in\Lambda}}\operatorname{Ext}^{i}_{R}\big{(}R/{\mathfrak{b}_{\alpha}},M\big{)}$
(see [8, 1.2.10, 1.2.11, and 2.1.10])._
###### Proposition 2.4.
Let $\mathcal{B}=\big{(}{\mathfrak{b}}_{\alpha}\big{)}_{\alpha\in\Lambda}$ be
an inverse family of elements of $\tilde{W}(I,J)$ over $\Lambda$ and
$t\in\mathbb{N}_{0}$. Let
$\operatorname{Ext}^{n-j}_{R}\big{(}R/{\mathfrak{b}_{\alpha}},\operatorname{H}^{j}_{I,J}(M)\big{)}=0$
for any ${\mathfrak{b}_{\alpha}}\in\mathcal{B}$, $n=t,~{}t+1$, and all $j<t$.
Then
$\operatorname{H}^{t}_{\mathcal{B}}(M)\subseteq\operatorname{H}^{t}_{I,J}(M)$
and $\operatorname{H}^{i}_{\mathcal{B}}(M)=0$ for all $i<t$.
###### Proof.
By Proposition 2.1 and Lemma 2.2, for $\mathcal{S}=0$ and
$\textmd{G}(-)=\Gamma_{I,J}(-)$, we have
$\operatorname{Ext}^{t}_{R}\big{(}R/{\mathfrak{b}_{\alpha}},M\big{)}\cong\operatorname{Hom}_{R}\big{(}R/{\mathfrak{b}_{\alpha}},\operatorname{H}^{t}_{I,J}(M)\big{)}$
for any ${\mathfrak{b}_{\alpha}}\in\mathcal{B}$. Now, by [8, Theorem 1.2.11
and Remarks 1.3.7], we get
$\operatorname{H}^{t}_{\mathcal{B}}(M)\cong\Gamma_{\mathcal{B}}(\operatorname{H}^{t}_{I,J}(M))\subseteq\operatorname{H}^{t}_{I,J}(M)$,
as required. ∎
###### Corollary 2.5.
Let $~{}t\in\mathbb{N}_{0}$ be such that $\operatorname{H}^{i}_{I,J}(M)=0$ for
all $i<t$. Then
$\operatorname{H}^{i}_{\mathfrak{b}}(M)\subseteq\operatorname{H}^{i}_{I,J}(M)$
for any $\mathfrak{b}\in\tilde{W}(I,J)$ and all $i\leq t$ and so
$\operatorname{H}^{i}_{\mathfrak{b}}(M)=0$ for all $i<t$. In particular, when
$J=0$.
###### Proof.
Apply Proposition 2.4 for $\mathcal{B}=\\{\mathfrak{b}^{n}\\}_{n\geq 1}$ . ∎
The next theorem is one of the main results in this section which is used in
some results of this paper. Recall that an $R$-module $M$ is said to be ZD-
module, if for every submodule $N$ of $M$, the set of zero-divisors of $M/N$
is a union of finitely many prime ideals in Ass$(M/N)$.(See [10])
###### Theorem 2.6.
Let $~{}t\in\mathbb{N}_{0}$ be such that $\operatorname{H}^{i}_{I,J}(M)=0$ for
all $i<t$. Then the following statements hold for any
$\mathfrak{a}\in\tilde{W}(I,J)$.
1. (i)
${\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{\mathfrak{a}}(M)\big{)}}~{}\cong~{}\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{a},M\big{)}$
2. $~{}\cong~{}\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{\mathfrak{a},J}(M)\big{)}$
3. $~{}\cong~{}\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{I,J}(M)\big{)}$
4. (ii)
$\Gamma_{\mathfrak{a}}\big{(}\operatorname{H}^{t}_{\mathfrak{a}}(M)\big{)}\cong\operatorname{H}^{t}_{\mathfrak{a}}(M)\cong\Gamma_{\mathfrak{a}}\big{(}\operatorname{H}^{t}_{\mathfrak{a},J}(M)\big{)}\cong\Gamma_{\mathfrak{a}}\big{(}\operatorname{H}^{t}_{I,J}(M)\big{)}$
5. (iii)
$\operatorname{H}^{i}_{\mathfrak{a}}(M)\subseteq\operatorname{H}^{i}_{\mathfrak{a},J}(M)$
for all $i\leq t$.
6. (iv)
$\operatorname{H}^{i}_{\mathfrak{a},J}(M)\subseteq\operatorname{H}^{i}_{I,J}(M)$
for all $i\leq t$.
7. (v)
$\operatorname{H}^{i}_{\mathfrak{a}}(M)\subseteq\operatorname{H}^{i}_{I,J}(M)$
for all $i\leq t$.
8. (vi)
$\operatorname{H}^{i}_{\mathfrak{a},J}(M)=\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$
for all $i<t$.
9. (vii)
$\emph{Ass}\big{(}\operatorname{H}^{t}_{\mathfrak{a}}(M)\big{)}=\emph{Ass}\big{(}\operatorname{H}^{t}_{\mathfrak{a},J}(M)\big{)}\cap\emph{V}(\mathfrak{a})$
10. $=\emph{Ass}\big{(}\operatorname{H}^{t}_{I,J}(M)\big{)}\cap\emph{V}(\mathfrak{a})$.
11. (viii)
If $\mathfrak{a}\neq 0$ and $M$ is _ZD_ -module, then there exists a regular
$M$-sequence of length $~{}t$ contained in $\mathfrak{a}$.
###### Proof.
(i) Let $\textmd{G}_{1}(-)=\Gamma_{\mathfrak{a}}(-)$ ,
$\textmd{G}_{2}(-)=\Gamma_{\mathfrak{a},J}(-)$,
$\textmd{G}_{3}(-)=\Gamma_{I,J}(-)$, and $\mathcal{S}=0$. Now, the assertion
follows from Proposition 2.1 and Lemma 2.2.
(ii) Since $\mathfrak{a}\in\tilde{W}(I,J)$ implies that
${\mathfrak{a}^{n}}\in\tilde{W}(I,J)$ for any $n\in\mathbb{N}$, the result
follows from (i).
(iii) It is obvious by part (ii).
(iv) Apply Corollary 2.5 and [17, Theorem 3.2].
(v) It is obvious by parts (iii) , (iv).
(vi) It is a consequence of parts (iv) , (v).
(vii) It is obvious by part (i) and [9, Exercise 1.2.27].
(viii) First note that $\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for all
$i<t$, by (vi). By induction on $t$, we construct a regular $M$-sequence
$x_{1},x_{2},\cdots,x_{t}$ such that $x_{j}\in\mathfrak{a}$ for
$j=1,2,\cdots,t$. When $t=0$, there is nothing to prove. Let $t=1$. Then
$\Gamma_{\mathfrak{a}}(M)=0$. Since $M$ is ZD-module and
$\mathfrak{a}$-torsion free, by the prime Avoidance Theorem, $\mathfrak{a}$ is
not contained in Zd$(M)$. So, there exists an element $x_{1}\in\mathfrak{a}$
such that $x_{1}\not\in$ Zd$(M)$. This proves the case $t=1$. Now, let $t>1$
and the assertion be true for $t-1$. By the above observation, $\mathfrak{a}$
contains an element $x_{1}$ which is a non-zerodivisor on $M$. Considering the
exact sequence
$\operatorname{H}^{j}_{\mathfrak{a}}(M)\rightarrow\operatorname{H}^{j}_{\mathfrak{a}}(M/x_{1}M)\rightarrow\operatorname{H}^{j+1}_{\mathfrak{a}}(M),$
for all $j\in\mathbb{N}_{0}$, we obtain
$\operatorname{H}^{j}_{\mathfrak{a}}(M/x_{1}M)=0$ for all $j<t-1$. Now, by
inductive hypothesis, there is a regular $M/x_{1}M$-sequence
$x_{2},x_{3},\cdots,x_{t}$ such that $x_{j}\in\mathfrak{a}$ for all
$j=2,3,\cdots,t$. Therefore $x_{1},x_{2},\cdots,x_{t}$ is a regular
$M$-sequence. ∎
###### Corollary 2.7.
Let $t\in\mathbb{N}_{0}$. Then $\operatorname{H}^{i}_{I,J}(M)=0$ for all $i<t$
if and only if $\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for any
$\mathfrak{a}\in\tilde{W}(I,J)$ and all $i<t$.
###### Proof.
The assertion follows easily from Theorem 2.6 (vi) and [17, Theorem 3.2]. ∎
###### Corollary 2.8.
Let $M$ be a finite $R$-module and $t\in\mathbb{N}_{0}$ be such that
$\operatorname{H}^{i}_{I,J}(M)=0$ for all $i<t$. If
$\operatorname{H}^{t}_{I,J}(M)$ is finite, then
$\mathfrak{a}^{m}\operatorname{H}^{t}_{\mathfrak{a}}(M)=0$ for any
$\mathfrak{a}\in\tilde{W}(I,J)$ and some $m\in\mathbb{N}_{0}$.
###### Proof.
The assertion follows easily from Theorem 2.6 (v) and [8, Proposition 9.1.2].
∎
###### Corollary 2.9.
Let $~{}t\in\mathbb{N}_{0}$ be such that $\operatorname{H}^{i}_{I,J}(M)=0$ for
all $i<t$. Then for any $\mathfrak{a}\in\tilde{W}(I,J)$ and any
$\mathfrak{b}\in\tilde{W}(I,0)$, we have
1. (i)
$\operatorname{Hom}_{R}\big{(}R/I,\operatorname{H}^{t}_{\mathfrak{a},J}(M)\big{)}\subseteq\operatorname{Ext}^{t}_{R}\big{(}R/I,M\big{)}$.
2. (ii)
$\Gamma_{I}\big{(}\operatorname{H}^{t}_{\mathfrak{a},J}(M)\big{)}\subseteq\operatorname{H}^{t}_{I}(M)$.
3. (iii)
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{b},\operatorname{H}^{t}_{I}(M)\big{)}\cong\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{b},M\big{)}$.
4. (iv)
$\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{b},M\big{)}\subseteq\operatorname{Ext}^{t}_{R}\big{(}R/I^{m},M\big{)}$
for some $m\in\mathbb{N}$. In particular, when $\mathfrak{b}\supseteq I$ the
result holds for all $m\in\mathbb{N}$, and so
$\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{b},M\big{)}\subseteq\operatorname{H}^{t}_{I}(M)$.
5. (v)
$\Gamma_{\mathfrak{b}}\big{(}\operatorname{H}^{t}_{I}(M)\big{)}\cong\operatorname{H}^{t}_{\mathfrak{b}}(M)$
and so
$\operatorname{H}^{t}_{\mathfrak{b}}(M)\subseteq\operatorname{H}^{t}_{I}(M)$
and
$\emph{Ass}\big{(}\operatorname{H}^{t}_{\mathfrak{b}}(M)\big{)}\subseteq\emph{Ass}\big{(}\operatorname{H}^{t}_{I}(M)\big{)}$.
6. (vi)
$\emph{Ass}\big{(}\operatorname{H}^{t}_{\mathfrak{b}}(M)\big{)}=\emph{Ass}\big{(}\operatorname{H}^{t}_{I}(M)\big{)}\cap\emph{V}(\mathfrak{b})$.
###### Proof.
(i) By Theorem 2.6 (iv),
$\big{(}0:_{\operatorname{H}^{t}_{\mathfrak{a},J}(M)}{I}\big{)}\subseteq\big{(}0:_{\operatorname{H}^{t}_{I,J}(M)}{I}\big{)}$.
Now, again apply Theorem 2.6 (i) for $\mathfrak{a}=I$.
(ii) Use part (i).
(iii) Apply Lemma 2.2 and Proposition 2.1 for $\textmd{G}(-)=\Gamma_{I}(-)$
and $\mathcal{S}=0$.
(iv) Let $m\in\mathbb{N}$ be such that $I^{m}\subseteq\mathfrak{b}$. Then
since
$\big{(}0:_{\operatorname{H}^{t}_{I,J}(M)}{\mathfrak{b}}\big{)}\subseteq\big{(}0:_{\operatorname{H}^{t}_{I,J}(M)}{I^{m}}\big{)}$,
the assertion follows from Theorem 2.6 (i).
(v) It is clear by (iii), since ${\mathfrak{b}}^{n}\in\tilde{W}(I,0)$ for all
$n\in\mathbb{N}$.
(vi) Apply part (v) and [17, Proposition 1.10] for $J=0$. ∎
As an application of the above results, we obtain vanishing and finiteness
results for $\operatorname{H}^{t}_{I,J}(M)$ in Corollary 2.12. To achieve the
result, we need the following proposition which gets the same result as in [3,
Lemma 3.1] for an arbitrary Serre subcategory $\mathcal{S}$ of the category of
$R$-modules.
###### Proposition 2.10.
Let $\mathfrak{a}$ be an ideal of $R$. Then $\mathfrak{a}M\in\mathcal{S}$ if
and only if $M/(0:_{M}\mathfrak{a})\in\mathcal{S}$.
###### Proof.
Suppose that $\mathfrak{a}:=(a_{1},...,a_{n})$. $(\Rightarrow)$ Define
$f:M\rightarrow(\mathfrak{a}M)^{n}$ by $f(m)=\big{(}a_{i}m\big{)}_{i=1}^{n}$.
Then since $M/(0:_{M}\mathfrak{a})$ is isomorphic to a submodule of
$(\mathfrak{a}M)^{n}$ and $(\mathfrak{a}M)^{n}\in\mathcal{S}$, the assertion
follows.
$(\Leftarrow)$ Define $g:M^{n}\rightarrow\mathfrak{a}M$ by
$g\big{(}\big{(}m_{i})_{i=1}^{n})=\sum_{i=1}^{n}a_{i}m_{i}$. Then
$\mathfrak{a}M$ is a homomorphic image of
$\big{(}M/(0:_{M}\mathfrak{a})\big{)}^{n}$. Since
$\big{(}M/(0:_{M}\mathfrak{a})\big{)}^{n}\in\mathcal{S}$, the assertion
follows. ∎
###### Corollary 2.11.
Let $\mathfrak{a}\in\tilde{W}(I,J)$. Let$~{}t\in\mathbb{N}_{0}$ be such that
$\operatorname{Ext}^{t}_{R}(R/\mathfrak{a},M)$ and
$\operatorname{Ext}^{n-j}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{j}_{I,J}(M)\big{)}$
belong to $\mathcal{S}$ for $n=t,~{}t+1$, and all $j<t$. Then
$\operatorname{H}^{t}_{I,J}(M)\in\mathcal{S}$ if and only if
$\mathfrak{a}\operatorname{H}^{t}_{I,J}(M)\in\mathcal{S}$. In particular for
$\mathfrak{a}=I$.
###### Proof.
$(\Rightarrow)$ It is obvious.
$(\Leftarrow)$ The assertion follows from Proposition 2.10, [1, Theorem 2.9
(i)], and the short exact sequence
$0\rightarrow(0:_{\operatorname{H}^{t}_{I,J}(M)}\mathfrak{a})\rightarrow\operatorname{H}^{t}_{I,J}(M)\rightarrow\operatorname{H}^{t}_{I,J}(M)/(0:_{\operatorname{H}^{t}_{I,J}(M)}\mathfrak{a})\rightarrow
0.$
∎
###### Corollary 2.12.
Let $\mathfrak{a}\in\tilde{W}(I,J)$. Let$~{}t\in\mathbb{N}_{0}$ be such that
$\operatorname{Ext}^{t}_{R}(R/\mathfrak{a},M)$ and
$\operatorname{Ext}^{n-j}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{j}_{I,J}(M)\big{)}$
are finite for $n=t,~{}t+1$, and all $j<t$. Then
1. (i)
$\operatorname{H}^{t}_{I,J}(M)$ is finite if and only if
$\mathfrak{a}\operatorname{H}^{t}_{I,J}(M)$ is finite.
2. (ii)
$\operatorname{H}^{t}_{I,J}(M)=0$ if and only if
$\mathfrak{a}\operatorname{H}^{t}_{I,J}(M)=0$.
###### Proof.
Apply Corollary 2.11 for the classes of finite and Zero $R$-modules. ∎
In local case, it is shown that the local cohomology with respect to the
maximal ideal of $R$ is concerned with the local cohomology with respect to an
ideal generated by any filter regular sequence. (See [13, Lemma 3.4]). In
Theorem 2.13, as a final result of this section, we obtain this in a
noetherian (not necessary local) ring for a $k$-regular $M$-sequence
$(k\geq-1)$.
For a subset $T$ of Spec$(R)$ and an integer $i\geq-1$, we set
$(T)_{>i}:=\\{\mathfrak{p}\in T\mid\textmd{dim}(R/\mathfrak{p})>i\\},$
$(T)_{\leq i}:=\\{\mathfrak{p}\in T\mid\textmd{dim}(R/\mathfrak{p})\leq i\\}.$
Recall that, for an integer $k\geq-1$, a sequence $a_{1},\ldots,a_{n}$ of
elements of $R$ is called a poor $k$-regular $M$-sequence whenever
$a_{i}\notin\mathfrak{p}$ for all
$\mathfrak{p}\in\big{(}{\textmd{Ass}}(M/\sum_{j=1}^{i-1}a_{j}M)\big{)}_{>k}$
and all $i=1,\ldots,n.$ Moreover, if $\textmd{dim}(M/\sum^{n}_{i=1}a_{i}M)>k$,
then $a_{1},\ldots,a_{n}$ is called a $k$-regular $M$-sequence. It easy to see
that, any poor $k$-regular $M$-sequence is a poor $(k+1)$-regular $M$-sequence
and for an ideal $I$ of $R$ if
$\big{(}\operatorname{Supp}(M/IM)\big{)}_{>k+1}\neq\phi$, then any $k$-regular
$M$-sequence in $I$ is a $(k+1)$-regular $M$-sequence in $I$. ( To verify for
various $k$, see [6], [11], [16], [4], [14], and [7]).
###### Theorem 2.13.
Let $a_{1},a_{2},\ldots,a_{n}$ be a $k$-regular $M$-sequence in $I$ and set
$\mathfrak{a}:=(a_{1},\ldots,a_{n})$. If
$\big{(}\big{(}\emph{Supp}(M)\cap\emph{W}(\mathfrak{a},J)\big{)}\smallsetminus\emph{W}(I,J)\big{)}_{\leq{k}}=\emptyset$,
then $\ \operatorname{H}^{i}_{I,J}(M)\cong{\
\operatorname{H}^{i}_{\mathfrak{a},J}(M)}\ $ for all $i<n$. In particular for
$J=0$.
###### Proof.
Let
$0{\longrightarrow}{E^{0}}^{d^{0}\atop{\longrightarrow}}{E^{1}}^{d^{1}\atop{\longrightarrow}}\cdots\longrightarrow{E^{i}}^{d^{i}\atop{\longrightarrow}}\cdots$
be a minimal injective resolution for $M$. For all $i\in{\mathbb{N}_{0}}$ we
have
$E^{i}=\bigoplus_{\mathfrak{p}\in\textmd{Supp}(M)}\mu_{i}(\mathfrak{p},M)E(R/\mathfrak{p})$
in which $\mu_{i}(\mathfrak{p},M)$ is the $i-th$ Bass number of $M$ at the
prime ideal $\mathfrak{p}$ of $R$. Let $i<n$ and
$\mathfrak{p}\in{\big{(}\textmd{Supp}(M)\cap\textmd{W}(\mathfrak{a},J)\big{)}_{>k}}.$
By [6, Theorem 2.3 (ii)], since $a_{1}/1,\ldots,a_{n}/1$ is a regular
$M_{\mathfrak{p}}$-sequence, we have
(1) $\mu_{i}(\mathfrak{p},M)=0.$
Now, by [17, Proposition 1.11], we have
(2)
$\Gamma_{I,J}(E^{i})~{}~{}=\bigoplus_{\mathfrak{p}\in\textmd{Supp}(M)\cap\textmd{W}(I,J)}\mu_{i}(\mathfrak{p},M)E(R/\mathfrak{p}).$
for all $i<n$. Similarly, for the ideal $\mathfrak{a}$, we get
(3)
$\Gamma_{\mathfrak{a},J}(E^{i})~{}~{}=\bigoplus_{\mathfrak{p}\in\textmd{Supp}(M)\cap\textmd{W}(\mathfrak{a},J)}\mu_{i}(\mathfrak{p},M)E(R/\mathfrak{p}),$
for all $i<n$. On the other hand by [17, Proposition 1.6],
$\textmd{W}(I,J)\subseteq\textmd{W}(\mathfrak{a},J)$. So that by (1), (2), (3)
and our assumption, we get
$\Gamma_{I,J}(E^{i})=\Gamma_{\mathfrak{a},J}(E^{i}),$
for all $i<n$. It therefore follows that
$\operatorname{H}^{i}_{I,J}(M)=\operatorname{H}^{i}_{\mathfrak{a},J}(M),$
for all $i<n$, as required. ∎
## 3\. Depth of $(I,J)$ on modules
In this section, we introduce the concept of depth of a pair of ideals $(I,J)$
on a module $M$. By [8, Theorem 6.2.7], for an ideal $\mathfrak{a}$ of $R$ and
a finite $R$-module $M$ with $\mathfrak{a}M\neq M$, the
depth$(\mathfrak{a},M)$ is the least integer $i$ such that
$\operatorname{H}^{i}_{\mathfrak{a}}(M)\neq 0$. Having this in mind, we give
the following definition.
###### Definition 3.1.
_Let $I$ , $J$ be two ideals of $R$ and let $M$ be an $R$-module. We define
the depth of $(I,J)$ on $M$ by_
$\textmd{depth}(I,J,M)={\textmd{inf~{}}}\big{\\{}\textmd{depth}(\mathfrak{a},M)\mid\mathfrak{a}\in\tilde{W}(I,J)\big{\\}},$
_if this infimum exists, and $\infty$ otherwise_.
###### Remark 3.2.
_If $M$ is a finite $R$-module such that $\mathfrak{a}M=M$ for any
$\mathfrak{a}\in\tilde{W}(I,J)$, then by [8, Exercise 6.2.6] and [17, Theorem
3.2], we get $\operatorname{H}^{i}_{I,J}(M)=0$, for all $i\in\mathbb{N}_{0}$.
But if there exists $\mathfrak{a}\in\tilde{W}(I,J)$ such that
$\mathfrak{a}M\neq M$, then depth$(I,J,M)<\infty$. Also, it is clear by
definition that if $J=0$, then depth$(I,J,M)$ coincides with depth$(I,M)$._
###### Proposition 3.3.
For a finite $R$-module $M$, we have the following equalities.
1. $\emph{depth}(I,J,M)={\emph{inf~{}}}\big{\\{}i\in\mathbb{N}_{0}\mid\operatorname{H}^{i}_{I,J}(M)\neq 0\big{\\}}$.
2. $={\emph{inf~{}}}\big{\\{}\emph{depth}(\mathfrak{p},M)\mid\mathfrak{p}\in\emph{W}(I,J)\big{\\}}$.
3. $={\emph{inf~{}}}\big{\\{}\emph{depth}(M_{\mathfrak{p}})\mid\mathfrak{p}\in\emph{W}(I,J)\big{\\}}$.
###### Proof.
Let $s:=\textmd{depth}(I,J,M)$ and
$t:={\textmd{inf~{}}}\big{\\{}i\in\mathbb{N}_{0}\mid\operatorname{H}^{i}_{I,J}(M)\neq
0\big{\\}}$. Thus there exists $\mathfrak{b}\in\tilde{W}(I,J)$ such that
$s=\textmd{depth}(\mathfrak{b},M)$. Since
$\textmd{V}(\mathfrak{b})\subseteq\textmd{W}(I,J)$, so by [9, Proposition
1.2.10] and [17, Theorem 4.1], we have
1. $s=\textmd{depth}(\mathfrak{b},M)={\textmd{inf~{}}}\big{\\{}\textmd{depth}(M_{\mathfrak{p}})\mid\mathfrak{p}\in\textmd{V}(\mathfrak{b})\big{\\}}$
2. $\geq{\textmd{inf~{}}}\big{\\{}\textmd{depth}(M_{\mathfrak{p}})\mid\mathfrak{p}\in\textmd{W}(I,J)\big{\\}}=t$.
If $t<s$, then $\operatorname{H}^{t}_{\mathfrak{a}}(M)=0$ for all
$\mathfrak{a}\in\tilde{W}(I,J)$. Thus, by [17, Theorem 3.2], we get
$\operatorname{H}^{t}_{I,J}(M)=0$, which is a contradiction. Thus $t=s$. For
the second equality, since
$\textmd{depth}(\mathfrak{a},M)={\textmd{inf~{}}}\big{\\{}\textmd{depth}(\mathfrak{p},M)\mid\mathfrak{p}\in\textmd{V}(\mathfrak{a})\big{\\}}$,
the equality follows from definition. Finally, the third equality follows from
[17, Theorem 4.1]. ∎
Comparing the concept of depth of a pair of ideals with the ordinary depth of
an ideal, we have the following result.
###### Corollary 3.4.
For a finite $R$-module $M$ and all $\mathfrak{a}\in\tilde{W}(I,J)$, we have
$\emph{depth}(I,J,M)\leq\emph{depth}(\mathfrak{a},J,M)\leq\emph{depth}(\mathfrak{a},M),$
in particular for $\mathfrak{a}=I$.
###### Proof.
Apply Proposition 3.3 and Theorem 2.6 (iv) and (iii). ∎
There is an example which shows that this new invariant is different from the
ordinary depth (see Example 4.3).
In next two propositions, we prove formulas and inequalities for this
invariant. Also, we examine how it behaves under various ideal theoretic
operations and short exact sequences.
###### Proposition 3.5.
Let $I$, $J$, $\mathfrak{b}$, $\mathfrak{c}$ be ideals of $R$ and $M$ be a
finite $R$-module.
1. (i)
If $J^{n}\subseteq{\mathfrak{b}}^{m}$ for some $n,m\in\mathbb{N}$, then
$\emph{depth}(I,\mathfrak{b},M)\leq\emph{depth}(I,J,M)$.
2. (ii)
If $J^{n}\supseteq{\mathfrak{c}}^{m}$for some $n,m\in\mathbb{N}$, then
$\emph{depth}(I,J,M)=\emph{depth}(I+{\mathfrak{c}},J,M)$.
3. (iii)
If ${\sqrt{I}}={\sqrt{\mathfrak{b}}}$, then
$\emph{depth}(I,J,M)=\emph{depth}({\mathfrak{b}},J,M)$.
4. (iv)
If ${\sqrt{J}}={\sqrt{\mathfrak{c}}}$, then
$\emph{depth}(I,J,M)=\emph{depth}(I,\mathfrak{c},M)$.
5. (v)
$\emph{depth}(I,J,M)=\emph{depth}({\sqrt{I}},J,M)=\emph{depth}(I,{\sqrt{J}},M)=\emph{depth}({\sqrt{I}},{\sqrt{J}},M)$.
6. (vi)
$\emph{depth}(I,J\mathfrak{b},M)=\emph{depth}(I,J\cap\mathfrak{b},M)$.
###### Proof.
All these statements follow easily from Proposition 3.3 and [17, Proposition
1.4, 1.6]. As an illustration, we just prove statement (i). Since
$J^{n}\subseteq{\mathfrak{b}}^{m}$, we have
$\textmd{W}(I,J)\subseteq\textmd{W}(I,\mathfrak{b})$, by [17, Proposition
1.6]. Now, by Proposition 3.3, we get
1. $\textmd{depth}(I,\mathfrak{b},M)={\textmd{inf~{}}}\big{\\{}\textmd{depth}(\mathfrak{p},M)\mid\mathfrak{p}\in\textmd{W}(I,\mathfrak{b})\big{\\}}$
2. $\leq{\textmd{inf~{}}}\big{\\{}\textmd{depth}(\mathfrak{q},M)\mid\mathfrak{p}\in\textmd{W}(I,J)\big{\\}}=\textmd{depth}(I,J,M)$.
∎
###### Proposition 3.6.
Let $~{}0\rightarrow U\rightarrow M\rightarrow N\rightarrow 0$ be an exact
sequence of finite $R$-modules. Let $r:=\emph{depth}(I,J,U)$ ,
$t:=\emph{depth}(I,J,M)$ and $s:=\emph{depth}(I,J,N)$. Then
1. (i)
$t\geq\textmd{min}\\{r,s\\}$.
2. (ii)
$r\geq\textmd{min}\\{t,{s+1}\\}$.
3. (iii)
$s\geq\textmd{min}\\{{r-1},t\\}$.
4. (iv)
$One~{}of~{}the~{}following~{}equalities~{}holds:$
5. $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~{}t=r~{},~{}t=s~{},~{}t=r=s~{},~{}s={r-1}$.
###### Proof.
For (i),(ii),(iii) apply Corollary 3.4 and [9, Proposition 1.2.9].
To prove (iv), suppose that none of the equalities holds. Then only one of the
following cases happens:
1. $\textmd{(a)}~{}r<s<t\ \ \ \ \ \ \ \ \ \textmd{(b)}~{}s<r<t\ \ \ \ \ \ \ \ \ \ \textmd{(c)}~{}t<r<s$
2. $\textmd{(d)}~{}t<s<r\ \ \ \ \ \ \ \ \ \textmd{(e)}~{}s<t<r\ \ \ \ \ \ \ \ \ \ \textmd{(f)}~{}r<t<s$.
Let $r<s<t$. Then $s+1\leq t$ and by (ii), $s+1\leq r<s$, which is a
contradiction.
If $s<r<t$, then $s\leq r-1<t$. Thus by (iii), $r-1\leq s$. Therefore $s=r-1$,
which is a contradiction.
The same method can be applied to the other cases. ∎
Next, we give some conditions, for which the depth of a pair of ideals equals
to the ordinary depth.
###### Theorem 3.7.
Let $M$ be a finite $R$-module. Let $~{}t:=\textmd{depth}(I,J,M)$. Then for
any $\mathfrak{a}\in\tilde{W}(I,J)$ with $\mathfrak{a}M\neq M$, the following
statements are equivalent.
1. (i)
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{\mathfrak{a}}(M)\big{)}\neq
0$.
2. (ii)
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{\mathfrak{a},J}(M)\big{)}\neq
0$.
3. (iii)
$\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{a},M\big{)}\neq 0$
4. (iv)
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{a},\operatorname{H}^{t}_{I,J}(M)\big{)}\neq
0$
5. (v)
$\operatorname{depth}(\mathfrak{a},M)=t$.
###### Proof.
All the equivalence of parts (i)–(iv) are consequences of Theorem 2.6 (i).
To prove (iii)$\Leftrightarrow$(v) Apply Proposition 3.3, Theorem 2.6, and [9,
Proposition 1.2.10 (e)] ∎
###### Theorem 3.8.
Let $M$ be a finite $R$-module and $~{}t:=\operatorname{depth}(I,J,M)$. Then
for any ideal $\mathfrak{b}\in\tilde{W}(I,0)$ such that $\mathfrak{b}M\neq M$,
the following statements are equivalent.
1. (i)
$\operatorname{Hom}_{R}\big{(}R/\mathfrak{b},\operatorname{H}^{t}_{I}(M)\big{)}\neq
0$.
2. (ii)
$\operatorname{Ext}^{t}_{R}\big{(}R/\mathfrak{b},M\big{)}\neq 0$
3. (iii)
$\operatorname{depth}(\mathfrak{b},M)=t$.
###### Proof.
Apply Corollary 2.9 and Theorems 3.7 and 2.6 (vi). ∎
###### Theorem 3.9.
Let $M$ be a finite $R$-module such that $IM\neq M$. Let
$~{}t:=\operatorname{depth}(I,J,M)$. Then the following statements are
fulfilled.
1. (i)
$\operatorname{H}^{t}_{I}(M)\neq 0$ if and only if
$~{}t=\operatorname{depth}(I,M)$.
2. (ii)
If $\operatorname{H}^{t}_{I}(M)\neq 0$, then
$\operatorname{depth}(I,M)\leq\operatorname{depth}(\mathfrak{a},M)$ for all
$\mathfrak{a}\in\tilde{W}(I,J)$.
3. (iii)
$\operatorname{depth}(\mathfrak{p},M)=t$ for all
$\mathfrak{p}\in\operatorname{Ass}\big{(}\operatorname{H}^{t}_{I,J}(M)\big{)}$.
4. (iv)
$\operatorname{depth}(\mathfrak{p},M)\leq\operatorname{depth}(\mathfrak{a},M)$
for all
$\mathfrak{p}\in\operatorname{Ass}\big{(}\operatorname{H}^{t}_{I,J}(M)\big{)}$
and all $\mathfrak{a}\in\tilde{W}(I,J)$.
5. (v)
$\operatorname{Hom}_{R}\big{(}R/I,\operatorname{H}^{t}_{I,J}(M)\big{)}\cong\operatorname{Ext}^{t}_{R}\big{(}R/I,M\big{)}$
6. $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cong\operatorname{Hom}_{R}\big{(}R/I,\operatorname{H}^{t}_{I}(M)\big{)}$
7. $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cong\operatorname{Hom}_{R}\big{(}R/I,M/(x_{1},x_{2},\cdots,x_{t})M\big{)}$,
where $x_{1},x_{2},\cdots,x_{t}$ is a poor regular $M$-sequence in $I$.
###### Proof.
(i) The assertion follows from Theorem 2.6 (vi) and [8, Theorem 6.2.7].
(ii) This is an immediate consequence of part (i) and Corollary 3.4.
(iii) Apply Corollary 3.3 and [18, Theorem 3.6].
(iv) Apply part (iii) and Corollary 3.4.
(v) Apply Proposition 2.6 (i) and [9, Lemma 1.2.4]. ∎
## 4\. $(I,J)$-Cohen–Macaulay modules
As a generalization of the concept of Cohen–Macaulay modules, we introduce the
concept of $(I,J)$-Cohen–Macaulay modules by using the concept of depth of a
pair of ideals. The following lemma and proposition have a main role in some
results on this section.
###### Lemma 4.1.
Let $R$ be an integral domain and $\mathfrak{a}$ be an ideal of $R$. Let $M$
be a faithful finite $R$-module of finite Krull dimension $d$. Then $\dim
M/\mathfrak{a}M=d$ if and only if $\mathfrak{a}=0$.
###### Proof.
The assertion is obvious, since $(0)\in\operatorname{Supp}(M)$ and if
$\mathfrak{a}\neq 0$, then $(0)\not\in\operatorname{Supp}(M/\mathfrak{a}M)$. ∎
###### Proposition 4.2.
Let $(R,\mathfrak{m})$ be a local integral domain, $I+J$ is an
$\mathfrak{m}$-primary ideal, and $M$ be a faithful finite $R$-module of Krull
dimension $d$. Then the following statements are equivalent:
1. (i)
$\dim M/JM=d$;
2. (ii)
$J=0$;
3. (iii)
$\operatorname{H}^{d}_{I,J}(M)\neq 0$.
In particular for $M=R$.
###### Proof.
Apply Lemma 4.1 and [17, Theorem 4.5]. ∎
###### Corollary 4.3.
Let $(R,\mathfrak{m})$ be a local integral domain of dimension $n$ and $J$ be
a non-zero proper ideal of $R$. Then $\operatorname{H}^{n}_{I,J}(N)=0$, for
any $R$-module $N$, in particular when $R$ is a regular local ring and $N=R$.
###### Proof.
Since $J\neq 0$, then $\dim R/J<n$, by Lemma 4.1. Now, the assertion follows
from [17, Corollary 4.4]. ∎
###### Example 4.4.
_By[8, Corollary 6.2.9], if $(R,\mathfrak{m})$ is a regular local ring of
dimension $n$, then $n$ is the unique integer $i$ for which
$\operatorname{H}^{n}_{\mathfrak{m}}(R)\neq 0$. This result may not be true
for local cohomology modules with respect to a pair of ideals. To see this,
let $k$ be a field and $R:=k[[x,y]]$. Let $\mathfrak{m}:=(x,y)R$ and
$J:=(x)R$. Then $R$ is a regular local ring with $\dim R=2>\dim R/J=1$. So
that $\operatorname{H}^{2}_{\mathfrak{m},J}(R)=0$, by Corollary 4.3. Moreover
$\Gamma_{\mathfrak{m},J}(R)=0$. So, by Theorem 2.6 (vi) and Grothendieck,s
non-vanishing Theorem, we get $\operatorname{H}^{1}_{\mathfrak{m},J}(R)\neq
0$. Therefore, by Proposition 3.3,
$\operatorname{depth}(\mathfrak{m},J,R)=1=\dim R/J<\dim
R=\operatorname{depth}R$. This motivates us to give the following definition._
###### Definition 4.5.
_Let $R$ be a noetherian ring and $I$ , $J$ be two ideals of $R$. An
$R$-module $M$ is called an $(I,J)$-Cohen–Macaulay $R$-module if $M\neq 0$ and
$\operatorname{depth}(I,J,M)=\dim M/JM$ or if $M=0$. If $R$ itself is an
$(I,J)$-Cohen–Macaulay module we say that $R$ is an $(I,J)$-Cohen–Macaulay
ring._
By definition, it is obvious that if $M$ is an $(I,J)$-Cohen–Macaulay module,
then $J$ is a proper ideal. Also, if $(R,\mathfrak{m})$ is local,
$I=\mathfrak{m}$, and $J=0$, then the concept of $(I,J)$-Cohen–Macaulay finite
$R$-modules coincides with Cohen–Macaulay $R$-modules.
###### Remark 4.6.
_By Corollary 3.4 and [17, Theorem 4.3], if $(R,\mathfrak{m})$ is a local
ring, then any $(I,J)$-Cohen–Macaulay finite $R$-module $M$ is
$(\mathfrak{m},J)$-Cohen–Macaulay $R$-module. Moreover, by [17, Proposition
1.4 (6),(7)], if $I+J$ is $\mathfrak{m}$-primary, then $M$ is
$(I,J)$-Cohen–Macaulay if and only if $M$ is
$(\mathfrak{m},J)$-Cohen–Macaulay._
###### Proposition 4.7.
Let $M$ be a finite $R$-module. If $M$ is $(I,J)$-_Cohen–Macaulay_ , then $M$
is $(\mathfrak{a},J)$-_Cohen–Macaulay_ for any
$\mathfrak{a}\in\tilde{W}(I,J)$.
###### Proof.
The assertion follows easily from Corollary 3.4. ∎
In next two propositions, we show that the two concepts of
$(I,J)$-Cohen–Macaulay and Cohen–Macaulay modules are not the same.
###### Proposition 4.8.
Let $(R,\mathfrak{m})$ be a local integral domain and $J\neq 0$. Let $M$ be a
faithful finite $R$-module and let $~{}t=\operatorname{depth}(I,J,M)$ be such
that
$~{}\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{t}_{I,J}(M)\big{)}\neq
0$. If $M$ is $(I,J)$-Cohen–Macaulay, then $M$ is not Cohen–Macaulay.
###### Proof.
By Remark 4.6, $M$ is $(\mathfrak{m},J)$-Cohen–Macaulay. Now, the assertion
follows from definition, Theorem 3.7 for $\mathfrak{a}=\mathfrak{m}$, and
Lemma 4.1. ∎
###### Proposition 4.9.
Let $(R,\mathfrak{m})$ be a local integral domain and $J\neq 0$. Let $M$ be a
faithful finite $R$-module and let
$~{}t=\operatorname{depth}(\mathfrak{m},J,M)$ be such that
$~{}\operatorname{H}^{t}_{\mathfrak{m}}(M)\neq 0$. If $M$ is
$(\mathfrak{m},J)$-Cohen–Macaulay, then $M$ is not Cohen–Macaulay. In
particular for $M=R$.
###### Proof.
Apply Theorem 3.9 and Lemma 4.1. ∎
###### Proposition 4.10.
Let $(R,\mathfrak{m})$ be a local ring and $I,J$ be proper ideals of $R$. Let
$M\neq 0$ be a finite $(I,J)$-_Cohen–Macaulay_ $R$-module. If
$t=\operatorname{grade}(I,J,M)\geq 1$, then $\operatorname{H}^{t}_{I,J}(M)$ is
not finite.
###### Proof.
Since $M$ is $(I,J)$-Cohen–Macaulay, so $\dim M/JM=t$. Now, the assertion
follows from [1, Corollary 4.12 (ii)]. ∎
###### Remark and Examples 4.11.
_It is considerable that that if
$~{}\operatorname{Hom}_{R}\big{(}R/\mathfrak{m},\operatorname{H}^{t}_{I,J}(M)\big{)}=0$
in Proposition 4.8, and $~{}\operatorname{H}^{t}_{\mathfrak{m}}(M)=0$ in
Proposition 4.9, then the assertions do not necessarily hold. For example, let
$k$ be a field and $R:=k[[x,y]]$, $\mathfrak{m}:=(x,y)R$, and $J:=(x)R$._
1. (i)
$R$ is a Cohen–Macaulay ring and as we saw in Example 4.4,
$\operatorname{depth}(\mathfrak{m},J,R)=1$. So that $R$ is
$(\mathfrak{m},J)$-Cohen–Macaulay ring.
2. (ii)
By Proposition 4.10 and Example 4.4,
$\operatorname{H}^{1}_{\mathfrak{m},J}(R)$ is not finite.
3. (iii)
If $I:=(x^{2})R$, then $\Gamma_{I,J}(R)=R$ and so $R$ is not
$(I,J)$-Cohen–Macaulay.
The next result can be a generalization of [8, Corollary 6.2.8] which gives a
bound for non-vanishing of local cohomology modules in local case.
###### Proposition 4.12.
Let $M$ be a non-zero finite module over the local ring $(R,\mathfrak{m})$ and
$J\neq R$. Then any integer $i$ for which
$~{}\operatorname{H}^{i}_{\mathfrak{m},J}(M)\neq 0$ must satisfy
$\operatorname{depth}(\mathfrak{m},J,M)\leq i\leq\dim M/JM,$
while for $i$ at either extremity of this range we do have
$~{}\operatorname{H}^{i}_{\mathfrak{m},J}(M)\neq 0$.
###### Proof.
Apply Corollary 3.3 and [17, Theorem 4.5]. ∎
An immediate consequence of Proposition 4.12 is the following result.
###### Corollary 4.13.
Let $(R,\mathfrak{m})$ be a local ring and $M\neq 0$ be a finite $R$-module.
Suppose that $I+J$ is an $\mathfrak{m}$-primary ideal. Then there is exactly
one integer $i$ for which $~{}\operatorname{H}^{i}_{I,J}(M)\neq 0$ if and only
if $\operatorname{depth}(I,J,M)=\dim M/JM$, that is, if and only if $M$ is an
$(I,J)$-_Cohen–Macaulay_ $R$-module.
###### Proposition 4.14.
Let $M$ be a finite $(I,J)$-torsion $R$-module. If $M$ is
$(I,J)$-_Cohen–Macaulay_ , then $M/JM$ is artinian.
###### Proof.
It is enough to show that $\operatorname{depth}(I,J,M)=0$. Since
$\Gamma_{I,J}(M)=M$, so $\operatorname{H}^{i}_{I,J}(M)=0$, for all $i\geq 1$,
by [17, Corollary 1.13]. Thus $\operatorname{depth}(I,J,M)=0$, as required. ∎
## References
* [1] M. Aghapournahr, Kh. Ahmadi-Amoli, and M. Y. Sadeghi, Upper bounds, cofiniteness and artinianness of local cohomology modules defined by a pair of ideals, arXiv:1211.4204v1, [Math.AC] 18-Nov 2012.
* [2] M. Aghapournahr and L. Melkersson, A natural map in local cohomology, Ark. Mat.,48 (2010), 243–251.
* [3] M. Aghapournahr and L. Melkersson, Finiteness properties of minimax and coatomic local cohomology modules, Arch. Math., 94 (2010), 519–528.
* [4] Kh. Ahmadi-Amoli, Filter regular sequences, local cohomology modules and singular sets. Ph. D. Thesis, University for Teacher Education, Iran (1996).
* [5] Kh. Ahmadi-Amoli and M. Y. Sadeghi, On the local cohomology modules definde by a pair of ideals and Serre subcategories, arXiv:1208-5934v1, [Math.AC] 29-Aug 2012.
* [6] Kh. Ahmadi-Amoli and N. Sanaei, On the k-regular sequences and the generalization of F-modules, J. Korean Math. Soc. 49 (2012), (5), 1083–1096.
* [7] M. P. Brodmann and L. T. Nhan, A finitness result for associated primes of certain ext-modules, Comm. Algebra, 36, (2008), no. 4, 1527–1536.
* [8] M. P. Brodmann and R.Y. Sharp, Local cohomology : an algebraic introduction with geometric applications, Cambridge University Press, 1998.
* [9] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, revised ed., 1998.
* [10] K. Divaani-Aazar and M. A. Esmkhani, Artinianness of local cohomology modules of ZD-modules, Comm. Algebra, 33, (2005), 2857–2863.
* [11] R. L$\ddot{u}$ and Z. Tang, The f-depth of an ideal on a module, Proc. Amer. Math. Soc. 130 (2002), no. 7, 1905–1912.
* [12] L. Melkersson, Some applications of a criterion for artinianness of a module, J. Pure. Appl. Algebra. 101, (1995), no. 3, 291–303.
* [13] U. Nagal and P. Schenzel, Cohomological annihilators and Castelnuovo-Mumford regularity, Commutative Algebra: syzygies, multiplicities, and birational algebra, (South Hadley, MA, 1992), 307–328, Contemp. Math., 159, Amer. Math. Soc., Providence, RI, (1994),
* [14] L. T. Nhan, On generalized regular sequences and the finiteness for associated primes of local cohomology modules. Comm. Alg. 33 (2005), no. 3, 793–806.
* [15] D. Rees, The grade of an ideal or module. Proc. Camb. Philos. Soc. 53,(1957), 28–42.
* [16] P. Schenzel, N. V. Trung, and N. T. Coung Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85, (1978), 57–73.
* [17] R. Takahashi, Y. Yoshino, and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure. Appl. Algebra. 213, (2009), 582–600.
* [18] A. Tehranian and A. P. Talemi, Cofiniteness of local cohomology based on a nonclosed support defined by a pair of ideals, Bull. Iranian Math. Soc. 36 (2010), 145–155.
|
arxiv-papers
| 2013-01-06T15:49:13 |
2024-09-04T02:49:39.954061
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Aghapournahr, Kh. Ahmadi-amoli, M. Y. Sadeghi",
"submitter": "Moharram Aghapournahr",
"url": "https://arxiv.org/abs/1301.1015"
}
|
1301.1119
|
# The martingale representation in a progressive enlargement of a filtration
with jumps ††thanks: Supported by National Natural Science Foundation of
China(no.11171215) and Shanghai 085 Project for financial supports. The
authors thank Prof. Michael Kohlmann of University of Konstanz for his advice
and revising.
Kun Tian1, Dewen Xiong 1 and Zhongxing Ye 1,2
1\. Department of Mathematics , Shanghai Jiao Tong University, Shanghai
200240,
China
2\. Business Information Management School, Shanghai Institute of Foreign
Trade,
Shanghai 201620, China The corresponding author, his Email is
[email protected].
( )
###### Abstract
In this paper, we assume that the filtration $\mathbb{F}$ is generated by a
$d$-dimensional Brownian motion $W=(W_{1},\cdots,W_{d})^{\prime}$ as well as
an integer-valued random measure $\mu(du,dy)$. The random variable
${\widetilde{\tau}}$ is the default time and $L$ is the default loss. Let
$\mathbb{G}=\\{\mathscr{G}_{t};t\geq 0\\}$ be the progressive enlargement of
$\mathbb{F}$ by $({\widetilde{\tau}},L)$, i.e, $\mathbb{G}$ is the smallest
filtration including $\mathbb{F}$ such that ${\widetilde{\tau}}$ is a
$\mathbb{G}$-stopping time and $L$ is
$\mathscr{G}_{\widetilde{\tau}}$-measurable. Under the density hypothesis, we
consider the $\mathbb{G}$-decomposition of a $(P,\mathbb{F})$ martingale and
the representation of a $\mathbb{G}$-martingale. We characterize the
conditional density process by $p_{s}(s,l)$,
$\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and $\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$,
which allows us to describe the survival process $G$ explicitly. Then we give
the explicit $\mathbb{G}$-decomposition of a $\mathbb{F}$ martingale and
obtain the predictable representation theorems both for a
$(P,\mathbb{G})$-martingale and a $(P^{*},\mathbb{G})$-martingale, which are
different as shown in Callegaro, Jeanblanc and Zargari(2010)
${}^{\text{\cite[cite]{[\@@bibref{}{Callegaro-Jeanblanc-
Zargari(2010)}{}{}]}}}$.
Key words: default time and default loss, progressive enlargement of
filtration, conditional density, canonical decomposition, martingale
representation.
## 1 Introduction
In this paper, we assume that the filtration $\mathbb{F}$ is generated by a
$d$-dimensional Brownian motion $W=(W_{1},\cdots,W_{d})^{\prime}$ as well as
an integer-valued random measure $\mu(du,dy)$. Let ${\widetilde{\tau}}\geq 0$
be the default time and $L$ be the default loss, which are random variables.
Let $\mathbb{G}=\\{\mathscr{G}_{t};t\geq 0\\}$ be the progressive enlargement
of $\mathbb{F}$ by $({\widetilde{\tau}},L)$, i.e, $\mathbb{G}$ is the smallest
filtration including $\mathbb{F}$ such that ${\widetilde{\tau}}$ is a
$\mathbb{G}$-stopping time and $L$ is
$\mathscr{G}_{\widetilde{\tau}}$-measurable. We note here that the progressive
enlargement filtration in this paper is different from traditional progressive
enlargement of filtration in the literature, see e.g. in
Jeulin(1980)${}^{\text{\cite[cite]{[\@@bibref{}{Jeulin1980}{}{}]}}}$,
Jacod(1987)${}^{\text{\cite[cite]{[\@@bibref{}{Jacod1987}{}{}]}}}$, Jeanblanc,
Yor and Chesney(2009) ${}^{\text{\cite[cite]{[\@@bibref{}{Jeanblanc-Yor-
Chesney2009}{}{}]}}}$, El Karoui, Jeanblanc and
Jiao(2009)${}^{\text{\cite[cite]{[\@@bibref{}{El Karoui-Jeanblanc-
Jiao2009}{}{}]}}}$, Jeanblanc and Le
Cam(2009)${}^{\text{\cite[cite]{[\@@bibref{}{Jeanblanc-Cam2009}{}{}]}}}$,
Jeanblanc and Song(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Jeanblanc-
Song2010}{}{}]}}}$, Callegaro, Jeanblanc and
Zargari(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Callegaro-Jeanblanc-
Zargari(2010)}{}{}]}}}$ and Jeanblanc and
Song(2012)${}^{\text{\cite[cite]{[\@@bibref{}{Jeanblanc-Song2012}{}{}]}}}$,
among many others. In traditional progressive enlargement, the authors didn’t
consider the more practical situation that once the default happens, the
default loss is immediately generated. Hence the filtration $\mathbb{G}$ we
consider here including the default time and the default loss simultaneously
is more interesting and realistic.
It is well known that for a general enlargement of filtration, a
$(P,\mathbb{F})$-martingale might not be a $(P,\mathbb{G})$-semimartingale,
thus Jacod(1987)${}^{\text{\cite[cite]{[\@@bibref{}{Jacod1987}{}{}]}}}$ and El
Karoui, Jeanblanc and Jiao(2009)${}^{\text{\cite[cite]{[\@@bibref{}{El Karoui-
Jeanblanc-Jiao2009}{}{}]}}}$ proposed the concept of density hypothesis (H’),
which is adapted in the following:
###### Assumption 1.1.
i) the $\mathbb{F}$-regular conditional law of $({\widetilde{\tau}},L)$ is
equivalent to the law of $({\widetilde{\tau}},L)$, i.e
$P({\widetilde{\tau}}\in ds,L\in dl)\sim\eta(ds,dl),~{}for~{}every~{}t\geq 0;$
ii) $\eta(ds,dl)$ has no atoms.
From Assumption 1.1, one can see that there exits a so-called conditional
density $p_{t}(s,l)$ to describe $\mathbb{F}$-regular conditional law of
$({\widetilde{\tau}},L)$, such that
$P(({\widetilde{\tau}},L)\in
B|\mathscr{F}_{t})=\int\int_{B}p_{t}(s,l)\eta(ds,dl),\quad\text{for every
}t\geq 0,P\text{-a.s.}$
One can see that $\\{p_{t}(s,l);t\geq 0\\}$ is a $(P,\mathbb{F})$-martingale.
In this paper, we investigate $\\{p_{t}(s,l);t\geq 0\\}$ more deeply and find
that under Assumption 2.1, it is completely determined by $p_{s}(s,L)$,
$\theta_{1}(u;s,l)^{\prime}\mathbb{I}_{u>s}$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$ as
$\displaystyle p_{t}(s,l)$
$\displaystyle=E[p_{s}(s,l)\big{|}\mathscr{F}_{t}]+p_{s}(s,l)\mathbb{I}_{t\geq
s}+\int_{0}^{t}p_{u-}(s,l)\theta_{1}(u;s,l)^{\prime}\mathbb{I}_{u>s}dW(u)$
$\displaystyle+\int_{0}^{t}\int_{E}p_{u-}(s,l)\theta_{2}(u,y;s,l)\mathbb{I}_{u>{\widetilde{\tau}}}\\{\mu(du,dy)-\nu(du,dy)\\}.$
We show in Theorem 2.9 that $p_{s}(s,l)$,
$\theta_{1}(u;s,l)^{\prime}\mathbb{I}_{u>s}$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$ must satisfy the following condition
$\begin{array}[]{ll}\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)=1-\displaystyle\int_{0}^{\infty}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\Big{\\}}^{\prime}dW(u)\\\
\qquad-\displaystyle\int_{0}^{\infty}\displaystyle\int_{E}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\Big{\\}}\\{\mu(du,dy)-\nu(du,dy)\\},\end{array}$
(1.1)
where
$\begin{array}[]{ll}Z^{\theta_{1},\theta_{2}}_{s,l}(t)=\exp\bigg{\\{}\displaystyle\int_{s}^{t}\theta_{1}(u;s,l)^{\prime}dW(u)-\dfrac{1}{2}\displaystyle\int_{s}^{t}\|\theta_{1}(u;s,l)\|^{2}du\bigg{\\}}\\\
\qquad\times\exp\bigg{\\{}\displaystyle\int_{s}^{t}\displaystyle\int_{E}\big{\\{}\ln(1+\theta_{2}(u,y;s,l))\big{\\}}\mu(du,dy)-\displaystyle\int_{s}^{t}\displaystyle\int_{E}\theta_{2}(u,y;s,l)\nu(du,dy)\bigg{\\}}.\end{array}$
Then we prove that the corresponding survival process $G_{t}$ has the
following Doob-Meyer’s decomposition
$\begin{array}[]{ll}G_{t}&=1-\displaystyle\int_{0}^{t}\int_{\mathbb{R}}p_{s}(s,l)\eta(ds,dl)\\\
&\qquad-\displaystyle\int_{0}^{t}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\Big{\\}}^{\prime}dW(u)\\\
&\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{E}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\Big{\\}}\\{\mu(du,dy)-\nu(du,dy)\\},\end{array}$
(1.2)
which is completely determined by $p_{s}(s,l)$,
$\theta_{1}(u;s,l)^{\prime}\mathbb{I}_{u>s}$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$ , which appears to be quite interesting.
Similar to Callegaro, Jeanblanc and
Zargari(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Callegaro-Jeanblanc-
Zargari(2010)}{}{}]}}}$, we show that
$M_{t}:=M_{1}(t)\mathbb{I}_{t<{\widetilde{\tau}}}+M_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$
is a $(P,\mathbb{G})$-martingale if only if the following two conditions are
satisfied:
* (i)
for $\eta$-almost every $u\geq 0$ and $l\in\mathbb{R}$,
$\\{\widehat{y}_{t}(u;l)p_{t}(u;l);t\geq u\\}$ is a
$(P,\mathbb{F})$-martingale;
* (ii)
the process
$\\{\widetilde{y}_{t}G_{t}+\int_{0}^{t}\int_{\mathbb{R}}\widehat{y}_{u}(u,l)p_{u}(u,l)\eta(du,dl);t\geq
0\\}$ is a $(P,\mathbb{F})$-martingale.
Then we mainly study the $\mathbb{G}$-decomposition of a $(P,\mathbb{F})$
martingale and the representation of a $\mathbb{G}$-martingale. This
representation is very important also as it has many applications in
mathematical finance, see Duffie and
Huang(1986)${}^{\text{\cite[cite]{[\@@bibref{}{Duffie-Huang1986}{}{}]}}}$,
Karatzas and Pikovsky(1996)${}^{\text{\cite[cite]{[\@@bibref{}{Pikovsky-
Karatzas1996}{}{}]}}}$, Amendinger, Becherer and
Schweizer(2003)${}^{\text{\cite[cite]{[\@@bibref{}{Amendinger-Becherer-
Schweizer2003}{}{}]}}}$, Ankirchner, Dereichner and
Imkeller(2005)${}^{\text{\cite[cite]{[\@@bibref{}{Ankirchner-Dereichner-
Imkeller2005}{}{}]}}}$, Jiao and
Pham(2009)${}^{\text{\cite[cite]{[\@@bibref{}{Jiao-Pham2009}{}{}]}}}$ and
Eyraud-Loisel(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Eyraud-
Loisel2010}{}{}]}}}$, etc. Let $m$ be a càdlàg $(P,\mathbb{F})$-local
martingale of the following form
$m_{t}=m_{0}+\int_{0}^{t}\xi_{1}(u)^{\prime}dW(u)+\int_{0}^{t}\int_{E}\xi_{2}(u,y)\\{\mu(du,dy)-\nu(du,dy)\\},$
we show that
$\begin{array}[]{ll}X_{t}=m_{t}+\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{1}{G_{u-}}\bigg{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\Big{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;s,l)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\displaystyle\int_{E}\theta_{2}(u,y;s,l))\xi_{2}(u,y)F_{u}(dy)\Big{\\}}\eta(ds,dl)\bigg{\\}}du\\\
\qquad\qquad-\displaystyle\int_{\widetilde{\tau}}^{t}\bigg{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;{\widetilde{\tau}},L)+\displaystyle\int_{E}\theta_{2}(u,y;{\widetilde{\tau}},L)\xi_{2}(u,y)F_{u}(dy)\bigg{\\}}du\end{array}$
is a $(P,\mathbb{G})$-local martingale, from which one can see the
$\mathbb{G}$-decomposition of $m$. It is notable that the derived
$\mathbb{G}$-decomposition also only depends on $p_{s}(s,l)$,
$\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and $\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$.
In the end, we obtain the predictable representation theorems both for a
$(P,\mathbb{G})$-martingale and a $(P^{*},\mathbb{G})$-martingale. Although
recently Jeanblanc and
Song(2012)${}^{\text{\cite[cite]{[\@@bibref{}{Jeanblanc-Song2012}{}{}]}}}$
gave the more general discussion for martingale representation property in
traditional progressively enlarged filtration, it does not apply to our
filtration $\mathbb{G}$ which is enlarged by the default time and the default
loss. Furthermore, since we have the density hypothesis, our method is more
simple and more directly, and the results are also seen to be different.
The paper is organized as follows: In Section 2, we characterize the
conditional density process and obtain a more explicit form of the Doob-
Meyer’s decomposition of the survival process. Then we provide a necessary and
sufficient condition for a $\mathbb{G}$-adapted process to be a
$\mathbb{G}$-martingale in Section 3. In Section 4, we will first prove Lemma
4.1 and then explicitly describe the $\mathbb{G}$-decomposition of a
$(P,\mathbb{F})$-martingale. In the last section, we obtain the martingale
representation theorem for a $(P^{*},\mathbb{G})$-martingale and then give the
martingale representation theorem for a $(P,\mathbb{G})$-martingale.
## 2 The setup and notations
We assume that $\mathbb{F}=\\{\mathscr{F}_{t};t\geq 0\\}$ is a filtration on
$(\Omega,\mathscr{F},P)$ carrying an $d$-dimensional Brownian motion
$W=(W_{1},\cdots,W_{d})^{\prime}$ as well as an integer-valued random measure
$\mu(du,dy)$ on $\mathbb{R}_{+}\times E$, where $(E,\mathscr{E})$ is a
Blackwell space.
###### Assumption 2.1.
The filtration $\mathbb{F}=\\{\mathscr{F}_{t};t\geq 0\\}$ is the natural
filtration generated by $W$ and $\mu$, i.e.,
$\mathscr{F}_{t}=\sigma\big{\\{}W(s),\mu([0,s]\times A),B;\quad 0\leq s\leq
t,\;A\in\mathscr{B},\;B\in\mathscr{N}\big{\\}}$
where $\mathscr{N}$ is the collection of $P$-null sets from $\mathscr{F}$.
In the following, we let $\mathscr{P}(\mathbb{F})$ be the family of all
$\mathbb{F}$-predictable processes and
$\widetilde{\mathscr{P}}(\mathbb{F})=\mathscr{P}(\mathbb{F})\otimes\mathscr{E}$.
We assume that the compensator of $\mu(du,dy)$ is given by
$\nu(du,dy)=F_{u}(dy)du$, where $F_{u}(dy)$ is a transition kernel from
$(\Omega\times\mathbb{R}_{+},\mathscr{P}(\mathbb{F}))$ to into
$(E,\mathscr{E})$ with $\int_{E}F_{u}(dy)<\infty$, for each $u$. One can see
that any local $(P,\mathbb{F})$-martingale $M$ has the form
$M_{t}=M_{0}+\displaystyle\int_{0}^{t}f_{1}(u)^{\prime}dW(u)+\displaystyle\int_{0}^{t}\displaystyle\int_{E}f_{2}(u,y)(\mu(du,dy)-\nu(du,dy)),$
where $f_{1}$ is an $\mathbb{R}^{d}$-valued $\mathbb{F}$-predictable process
and $f_{2}$ is a $\widetilde{\mathscr{P}}(\mathbb{F})$-measurable function.
Remark. It is easy to see that there exists an $\mathbb{F}$-optional process
$\varphi=(\varphi_{t})$ and a sequence of stopping times $(\hat{\tau}_{k})$
such that for all positive $\widetilde{\mathscr{P}}(\mathbb{F})$-measurable
function $W(\omega,t,y)$,
$\int_{0}^{t}\displaystyle\int_{E}W(\omega,u,y)\mu(\omega;du,dy)=\sum_{(k)}W(\hat{\tau}_{k},\varphi_{\hat{\tau}_{k}})\mathbb{I}_{\hat{\tau}_{k}\leq
t}.$
Furthermore, as the compensator of $\mu(du,dy)$ is $\nu(du,dy)=F_{u}(dy)du$,
so the filtration $\mathbb{F}$ is quasi-left continuous.
Let $\widetilde{\tau}$ be a non-negative random variable and $L$ be a random
variable on $(\Omega,\mathscr{F})$, and let the
$\mathbb{G}=\\{\mathscr{G}_{t};t\geq 0\\}$ be the smallest progressive
enlargement filtration of $\mathbb{F}$ such that ${\widetilde{\tau}}$ is a
$\mathbb{G}$-stopping time and $L$ is a
$\mathscr{G}_{\widetilde{\tau}}$-measurable random variable. Let $\eta(ds,dl)$
be the law of $({\widetilde{\tau}},L)$, i.e.,
$\eta(ds,dl)=P({\widetilde{\tau}}\in ds,L\in dl)$. We need the following
assumptions
Assumption 1.1. i) the $\mathbb{F}$-regular conditional law of
$({\widetilde{\tau}},L)$ is equivalent to the law of $({\widetilde{\tau}},L)$,
i.e.,
$P({\widetilde{\tau}}\in ds,L\in
dl|\mathscr{F}_{t})\sim\eta(ds,dl),\quad\text{for every }t\geq 0;$
ii) $\eta(ds,dl)$ has no atoms.
###### Lemma 2.2.
$Y_{t}$ is a $\mathscr{G}_{t}$-measurable random variable (r.v.) if and only
if
$Y_{t}=y^{0}_{t}1_{t<{\widetilde{\tau}}}+y^{1}_{t}({\widetilde{\tau}},L)1_{{\widetilde{\tau}}\leq
t},$
where $y^{0}_{t}$ is a $\mathscr{F}_{t}$-measurable r.v. and $y^{1}_{t}(s,l)$
is an
$\mathscr{F}_{t}\otimes\mathscr{B}(\mathbb{R}^{+}\times\mathbb{R})$-measurable
function.
From Pham(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Pham-2010}{}{}]}}}$, we
have the following lemma
###### Lemma 2.3.
1) Any $\mathbb{G}$-predictable process $Y=(Y_{t})_{t\geq 0}$ is represented
as
$Y_{t}=Y^{0}_{t}1_{t\leq{\widetilde{\tau}}}+Y^{1}({\widetilde{\tau}},L)1_{t>{\widetilde{\tau}}},$
where $Y^{0}$ is a $\mathbb{F}$-predictable process and where $Y^{1}_{t}(s,l)$
is a
$\mathscr{P}(\mathbb{F})\otimes\mathscr{B}(R^{+})\otimes\mathscr{B}(R)$-measurable
function.
2) Any $\mathbb{G}$-optional process $Y=(Y_{t})_{t\geq 0}$ is represented as
$Y_{t}=Y^{0}_{t}1_{t<{\widetilde{\tau}}}+Y^{1}({\widetilde{\tau}},L)1_{t\geq{\widetilde{\tau}}},$
where $Y^{0}$ is a $\mathbb{F}$-optional process and where $Y^{1}_{t}(s,l)$ is
a
$\mathscr{O}(\mathbb{F})\otimes\mathscr{B}(R^{+})\otimes\mathscr{B}(R)$-measurable
function.
In this paper, we will consider the following basic questions:
1)
whether a $\mathbb{F}$-martingale is a $\mathbb{G}$-martingale and the
$\mathbb{G}$-decomposition of an $\mathbb{F}$-martingale;
2)
the representation of a $\mathbb{G}$-martingale.
From Assumption 1.1, one can see from
Jacod(1987)${}^{\text{\cite[cite]{[\@@bibref{}{Jacod1987}{}{}]}}}$ or
Amendinger(1999)${}^{\text{\cite[cite]{[\@@bibref{}{Amendinger(1999)}{}{}]}}}$
that there exists a strictly positive
$\mathscr{O}(\mathbb{F})\otimes\mathscr{B}(\mathbb{R}_{+}\times\mathbb{R})$-measurable
function $(t;\omega;s,l)\longrightarrow p_{t}(\omega;s,l)$, called the
$(P,\mathbb{F})$-conditional density of $({\widetilde{\tau}},L)$ with respect
to $\eta$, such that for every $(s,l)\in\mathbb{R}_{+}\times\mathbb{R}$,
$p(s,l)$ is a càdlàg $(P,\mathbb{F})$-martingale and for any
$B\in\mathscr{B}(\mathbb{R}_{+}\times\mathbb{R})$,
$P(({\widetilde{\tau}},L)\in
B|\mathscr{F}_{t})=\int\int_{B}p_{t}(s,l)\eta(ds,dl),\quad\text{for every
}t\geq 0,P\text{-a.s.}$
One can see that $p_{0}(s,l)=1$ for every $s,l$.
By the “change of probability measure” viewpoint of
Song(1987)${}^{\text{\cite[cite]{[\@@bibref{}{Song1987}{}{}]}}}$ and similar
to Callegaro, Jeanblanc and
Zargari(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Callegaro-Jeanblanc-
Zargari(2010)}{}{}]}}}$, we introduce the filtration
$\mathbb{G}^{{\widetilde{\tau}},L}=\\{\mathscr{G}^{{\widetilde{\tau}},L}_{t};t\geq
0\\}$ with
$\mathscr{G}^{{\widetilde{\tau}},L}_{t}=\mathscr{F}_{t}\bigvee\sigma({\widetilde{\tau}},L)$,
one can see that $\mathbb{G}^{{\widetilde{\tau}},L}$ is the initial
enlargement of the filtration $\mathbb{F}$ with ${\widetilde{\tau}}$ and $L$
and that
$\mathbb{F}\subset\mathbb{G}\subset\mathbb{G}^{{\widetilde{\tau}},L}$. Let
$Z_{t}=\dfrac{1}{p_{t}({\widetilde{\tau}},L)},$
similar to Grorud and Pontier(1998)${}^{\text{\cite[cite]{[\@@bibref{}{Grorud-
Pontier-1998}{}{}]}}}$ or
Amendinger(1999)${}^{\text{\cite[cite]{[\@@bibref{}{Amendinger(1999)}{}{}]}}}$,
one can see that $Z$ is a strictly positive
$(P,\mathbb{G}^{{\widetilde{\tau}},L})$-martingale with $E(Z_{t})=1$, for
every $t\geq 0$. Thus one can define a locally equivalent probability measure
$P^{*}$ by
$\dfrac{dP^{*}}{dP}|_{\mathscr{G}^{{\widetilde{\tau}},L}_{t}}=Z_{t}.$
Similar to Grorud-Pontier(1998)${}^{\text{\cite[cite]{[\@@bibref{}{Grorud-
Pontier-1998}{}{}]}}}$ or
Amendinger(1999)${}^{\text{\cite[cite]{[\@@bibref{}{Amendinger(1999)}{}{}]}}}$,
one can show that
1. 1.
under $P^{*}$, $({\widetilde{\tau}},L)$ is independent of $\mathscr{F}_{t}$
for every $t\geq 0$;
2. 2.
$P^{*}|_{\mathscr{F}_{t}}=P|_{\mathscr{F}_{t}}$;
3. 3.
$P^{*}|_{\sigma({\widetilde{\tau}},L)}=P|_{\sigma({\widetilde{\tau}},L)}$,
which implies $P^{*}({\widetilde{\tau}}\in ds,L\in
dl|\mathscr{F}_{t})=P({\widetilde{\tau}}\in ds,L\in dl)$. Similar to Lemma 1.4
in Callegaro, Jeanblanc and
Zargari(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Callegaro-Jeanblanc-
Zargari(2010)}{}{}]}}}$, we have the following lemma
###### Lemma 2.4.
1) Let $y_{t}({\widetilde{\tau}},L)$ be a
$\mathscr{G}^{{\widetilde{\tau}},L}_{t}$-measurable r.v., then for any $s\leq
t$,
$E_{P^{*}}\big{(}y_{t}({\widetilde{\tau}},L)\big{|}\mathscr{G}^{{\widetilde{\tau}},L}_{s}\big{)}=E_{P^{*}}\big{(}y_{t}(u,l)\big{|}\mathscr{F}_{s}\big{)}\big{|}_{u={\widetilde{\tau}},\>l=L}\>;$
2) if $y_{t}({\widetilde{\tau}},L)$ is $P$-integrable, then
$E\big{(}y_{t}({\widetilde{\tau}},L)\big{|}\mathscr{G}^{{\widetilde{\tau}},L}_{s}\big{)}=\dfrac{1}{p_{s}({\widetilde{\tau}},L)}E\big{(}y_{t}(u,l)p_{t}(u,l)\big{|}\mathscr{F}_{s}\big{)}\big{|}_{u={\widetilde{\tau}},\>l=L}\>.$
We have the following Corollaries
###### Corollary 2.5 (Characterization of
$(P,\mathbb{G}^{{\widetilde{\tau}},L})$-martingales in terms of
$(P,\mathbb{F})$-martingales).
A process $y_{t}({\widetilde{\tau}},L)$ is a
$(P,\mathbb{G}^{{\widetilde{\tau}},L})$-martingale if and only if
$\\{y_{t}(u,l)p_{t}(u,l);t\geq 0\\}$ is a $(P,\mathbb{F})$-martingale, for
almost every $u\geq 0,l\in\mathbb{R}$.
###### Corollary 2.6.
Let $M=\\{M_{t};t\geq 0\\}$ be a bounded $(P^{*},\mathbb{F})$-martingale, then
$M$ is a $(P^{*},\mathbb{G}^{{\widetilde{\tau}},L})$-martingale and hence a
$(P^{*},\mathbb{G})$-martingale.
###### Proof.
From part 1) of Lemma 2.4, one can see that for any $s\leq t$
$\begin{array}[]{ll}E_{P^{*}}\big{(}M_{t}\big{|}\mathscr{G}^{{\widetilde{\tau}},L}_{s}\big{)}&=E_{P^{*}}\big{(}M_{t}\big{|}\mathscr{F}_{s}\big{)}\big{|}_{u={\widetilde{\tau}},\>l=L}\\\
&=M_{s},\end{array}$
thus $M$ is a $(P^{*},\mathbb{G}^{{\widetilde{\tau}},L})$-martingale. Since
$M_{s}$ is $\mathscr{F}_{s}$-measurable, $M_{s}$ is
$\mathscr{G}_{s}$-measurable, thus
$\begin{array}[]{ll}E_{P^{*}}\big{(}M_{t}\big{|}\mathscr{G}_{s}\big{)}&=E_{P^{*}}\\{E_{P^{*}}\big{(}M_{t}\big{|}\mathscr{G}^{{\widetilde{\tau}},L}_{s}\big{)}|\mathscr{G}_{s}\\}\\\
&=E_{P^{*}}\\{M_{s}|\mathscr{G}_{s}\\}\\\ &=M_{s},\end{array}$
which completes the proof. ∎
Let
$\begin{array}[]{ll}G_{t}&:=P({\widetilde{\tau}}>t|\mathscr{F}_{t})=\displaystyle\int_{t}^{\infty}\int_{R}p_{t}(s,l)\eta(ds,dl)\quad\text{and}\\\
G^{*}_{t}&:=P^{*}({\widetilde{\tau}}>t|\mathscr{F}_{t})=P^{*}({\widetilde{\tau}}>t)=P({\widetilde{\tau}}>t)=\displaystyle\int_{0}^{t}\int_{R}\eta(ds,dl),\end{array}$
from Callegaro, Jeanblanc and
Zargari(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Callegaro-Jeanblanc-
Zargari(2010)}{}{}]}}}$, we find that $G$ is a
$(P,\mathbb{F})$-supermartingale and $G^{*}$ is a deterministic continuous and
decreasing function.
###### Theorem 2.7.
Let $y_{t}({\widetilde{\tau}},L)$ be a
$\mathscr{G}^{{\widetilde{\tau}},L}_{t}$-measurable $P$-integrable r.v., then
for $s\leq t$,
$E(y_{t}({\widetilde{\tau}},L)|\mathscr{G}_{s})=\widetilde{y}_{s}1_{s<{\widetilde{\tau}}}+\widehat{y}_{s}({\widetilde{\tau}},L)1_{{\widetilde{\tau}}\leq
s}$
with
$\begin{array}[]{ll}\widetilde{y}_{s}&=\dfrac{1}{G_{s}}E\Big{(}\displaystyle\int_{s}^{\infty}\int_{\mathbb{R}}y_{t}(u,l)p_{t}(u,l)\eta(du,dl)\Big{|}\mathscr{F}_{s}\Big{)}\\\
\widehat{y}_{s}(u,l)&=\dfrac{1}{p_{s}(u,l)}E\big{\\{}y_{t}(u,l)p_{t}(u,l)\big{|}\mathscr{F}_{s}\big{\\}}.\end{array}$
###### Proof.
The proof follows from the proof of Lemma 1.5 of Callegaro-Jeanblanc-
Zargari(2010)([5]). ∎
### 2.1 The $(P,\mathbb{F})$-density process
Since for any $s,l$, $p(s,l)=\\{p_{t}(s,l);t\geq 0\\}$ is a strictly positive
$(P,\mathbb{F})$-martingale, $\\{p(s,l);s\geq 0,l\in\mathbb{R}\\}$ is a
$(P,\mathbb{F})$-martingale system. Since $\\{p_{t}(s,l);t\geq 0\\}$ is a
strictly positive martingale, one can see that $p_{t}(s,l)$ can be represented
in the following form
$\begin{array}[]{ll}p_{t}(s,l)=E(p_{s}(s,l)|\mathscr{F}_{t})\mathbb{I}_{t<s}+p_{s}(s,l)\exp\bigg{\\{}\displaystyle\int_{s}^{t}\theta_{1}(u;s,l)^{\prime}dW(u)-\dfrac{1}{2}\displaystyle\int_{s}^{t}\|\theta_{1}(u;s,l)\|^{2}du\bigg{\\}}\\\
\qquad\times\exp\bigg{\\{}\displaystyle\int_{s}^{t}\displaystyle\int_{E}\big{\\{}\ln(1+\theta_{2}(u,y;s,l))\big{\\}}\mu(du,dy)-\displaystyle\int_{s}^{t}\displaystyle\int_{E}\theta_{2}(u,y;s,l)\nu(du,dy)\bigg{\\}}\mathbb{I}_{t\geq
s},\end{array}$ (2.1)
which implies that $p_{t}(s,l)$ is completely determined by $p_{s}(s,l)$,
$\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and $\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$.
Since $E(p_{s}(s,l)|\mathscr{F}_{t})$ can be written in the following form
$\begin{array}[]{ll}E(p_{s}(s,l)|\mathscr{F}_{t})&=1+\displaystyle\int_{0}^{t\wedge
s}p_{u-}(s,l)\theta^{*}_{1}(u;s,l)^{\prime}dW(u)\\\
&\qquad+\displaystyle\int_{0}^{t\wedge
s}\int_{E}p_{u-}(s,l)\theta^{*}_{2}(u,y;s,l)\\{\mu(du,dy)-\nu(du,dy)\\},\end{array}$
hence $p_{t}(s,l)$ can also be represented in the following way
$\begin{array}[]{ll}p_{t}(s,l)=1+\displaystyle\int_{0}^{t}p_{u-}(s,l)\\{\theta^{*}_{1}(u;s,l)\mathbb{I}_{u\leq
s}+\theta_{1}(u;s,l)\mathbb{I}_{u>s}\\}^{\prime}dW(u)\\\
\qquad+\displaystyle\int_{0}^{t}\int_{E}p_{u-}(s,l)\\{\theta^{*}_{2}(u,y;s,l)\mathbb{I}_{u\leq
s}+\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}\\}\\{\mu(du,dy)-\nu(du,dy)\\}.\end{array}$
(2.2)
First, we noted that the following equation by Fubini theorem.
$\displaystyle\int_{t}^{\infty}\displaystyle\int_{\mathbb{R}}E(p_{s}(s,l)|\mathscr{F}_{t})\eta(ds,dl)=E\Big{(}\displaystyle\int_{t}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)\Big{|}\mathscr{F}_{t}\Big{)}.$
(2.3)
We first give an examples before characterizing the conditional density
process.
###### Example 2.8.
Let $\theta_{1}(u;s,l)\mathbb{I}_{u>s}:=0$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}:=0$, then
$p_{t}(s,l)=E(p_{s}(s,l)|\mathscr{F}_{t})\mathbb{I}_{t<s}+p_{s}(s,l)\mathbb{I}_{t\geq
s}.$
Since $\int_{0}^{\infty}\int_{\mathbb{R}}p_{t}(s,l)\eta(ds,dl)\equiv 1$ for
each $t$, one can see from (2.3) that
$\begin{array}[]{ll}\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{t}(s,l)\eta(ds,dl)\\\
=E\Big{(}\displaystyle\int_{t}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)\Big{|}\mathscr{F}_{t}\Big{)}+\displaystyle\int_{0}^{t}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\eta(ds,dl)\\\
=E\Big{(}\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)\Big{|}\mathscr{F}_{t}\Big{)}=1,\end{array}$
for every $t\geq 0$. Let $t\rightarrow\infty$, one can see that $p_{s}(s,l)$
must satisfy the following condition
$\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)=1.$
More generally, we have the following theorem
###### Theorem 2.9.
For any given bounded $p_{s}(s,l)$, $\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$, let
$\begin{array}[]{ll}Z^{\theta_{1},\theta_{2}}_{s,l}(t)=\exp\bigg{\\{}\displaystyle\int_{s}^{t}\theta_{1}(u;s,l)^{\prime}dW(u)-\dfrac{1}{2}\displaystyle\int_{s}^{t}\|\theta_{1}(u;s,l)\|^{2}du\bigg{\\}}\\\
\qquad\times\exp\bigg{\\{}\displaystyle\int_{s}^{t}\displaystyle\int_{E}\big{\\{}\ln(1+\theta_{2}(u,y;s,l))\big{\\}}\mu(du,dy)-\displaystyle\int_{s}^{t}\displaystyle\int_{E}\theta_{2}(u,y;s,l)\nu(du,dy)\bigg{\\}}.\end{array}$
If
$\begin{array}[]{ll}p_{t}(s,l)=E(p_{s}(s,l)|\mathscr{F}_{t})\mathbb{I}_{t<s}+p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(t)\mathbb{I}_{t\geq
s}\end{array}$
is the density process of a pair $({\widetilde{\tau}},L)$ with respect to
$(P,\mathbb{F})$, then $p_{s}(s,l)$, $\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$ satisfies the following condition
$\begin{array}[]{ll}\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)=1-\displaystyle\int_{0}^{\infty}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\Big{\\}}^{\prime}dW(u)\\\
\qquad-\displaystyle\int_{0}^{\infty}\displaystyle\int_{E}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\Big{\\}}\\{\mu(du,dy)-\nu(du,dy)\\}.\end{array}$
(2.4)
###### Proof.
One can see that
$\begin{array}[]{ll}Z^{\theta_{1},\theta_{2}}_{s,l}(t)=1+\displaystyle\int_{s}^{t}Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)^{\prime}dW(u)\\\
\qquad\qquad\qquad+\displaystyle\int_{s}^{t}\displaystyle\int_{E}Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\\{\mu(du,dy)-\nu(du,dy)\\},\end{array}$
From
$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}p_{t}(s,l)\eta(ds,dl)\equiv 1$
for each $t$, one derives from (2.3) that
$\begin{array}[]{ll}1=\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{t}(s,l)\eta(ds,dl)\\\
=E\Big{(}\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)\Big{|}\mathscr{F}_{t}\Big{)}\\\
\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)+\displaystyle\int_{0}^{t}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(t)\eta(ds,dl)\\\
=E\Big{(}\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)\Big{|}\mathscr{F}_{t}\Big{)}\\\
\qquad+\displaystyle\int_{0}^{t}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\Big{\\{}\displaystyle\int_{s}^{t}Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)^{\prime}dW(u)\\\
\qquad\qquad\qquad+\displaystyle\int_{s}^{t}\displaystyle\int_{E}Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\\{\mu(du,dy)-\nu(du,dy)\\}\Big{\\}}\eta(ds,dl)\\\
=E\Big{(}\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)\mathbb{\eta}(ds,dl)\Big{|}\mathscr{F}_{t}\Big{)}\\\
\qquad+\displaystyle\int_{0}^{t}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\Big{\\}}^{\prime}dW(u)\\\
\qquad+\displaystyle\int_{0}^{t}\displaystyle\int_{E}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\Big{\\}}\\{\mu(du,dy)-\nu(du,dy)\\},\end{array}$
where the last equality comes from the stochastic Funini theorem for general
semimartingales, see Theorem 4.1.1 of Jeanblanc, Yor and Chesney(2009)
${}^{\text{\cite[cite]{[\@@bibref{}{Jeanblanc-Yor-Chesney2009}{}{}]}}}$.
∎
###### Corollary 2.10.
Under the conditions of Theorem 2.9, the survival process of
${\widetilde{\tau}}$ with respect to $(P,\mathbb{F})$ is given by
$\begin{array}[]{ll}G_{t}&=1-\displaystyle\int_{0}^{t}\int_{\mathbb{R}}p_{s}(s,l)\eta(ds,dl)\\\
&\qquad-\displaystyle\int_{0}^{t}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\Big{\\}}^{\prime}dW(u)\\\
&\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{E}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\Big{\\}}\\{\mu(du,dy)-\nu(du,dy)\\}.\end{array}$
(1.2)
###### Proof.
From the definition of $G_{t}$, one sees that
$\begin{array}[]{ll}G_{t}&=P({\widetilde{\tau}}>t|\mathscr{F}_{t})=\displaystyle\int_{t}^{\infty}\int_{\mathbb{R}}p_{t}(s,l)\eta(ds,dl)\\\
&=\displaystyle\int_{t}^{\infty}\int_{\mathbb{R}}E[p_{s}(s,l)|\mathscr{F}_{t}]\eta(ds,dl)\\\
&=E\Big{[}\displaystyle\int_{t}^{\infty}\int_{\mathbb{R}}p_{s}(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t}\Big{]}\\\
&=-\displaystyle\int_{0}^{t}\int_{\mathbb{R}}p_{s}(s,l)\eta(ds,dl)+E\Big{[}\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}p_{s}(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t}\Big{]}\\\
&=1-\displaystyle\int_{0}^{t}\int_{\mathbb{R}}p_{s}(s,l)\eta(ds,dl)\\\
&\qquad-\displaystyle\int_{0}^{t}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\Big{\\}}^{\prime}dW(u)\\\
&\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{E}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\Big{\\}}\\{\mu(du,dy)-\nu(du,dy)\\},\end{array}$
which completes the proof. ∎
Remark.We note here that (1.2) does not directly depend on
$\theta^{*}_{1}(u;s,l)$ and $\theta^{*}_{2}(u;s,l)$, that is to say, the
survival process $G$ is completely determined by $p_{s}(s,l)$,
$\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and $\theta_{2}(u;s,l)\mathbb{I}_{u>s}$.
Remark. From Corollary 2.14, one can this if and only if
$\theta_{1}=\theta_{2}=0$ and the survival process $G$ of ${\widetilde{\tau}}$
with respect to $(P,\mathbb{F})$ is a decreasing function, i,e, the
$\mathbb{F}$ is immersed in $\mathbb{G}$.
## 3 $\mathbb{G}$-martingales’ characterization
In this section, we will characterize $(P,\mathbb{G})$-martingales in terms of
$(P,\mathbb{F})$-martingales. Similar to Proposition 2.2 of [5], we have the
theorem
###### Theorem 3.1 (Characterization of $(P,\mathbb{G})$-martingales in terms
of $(P,\mathbb{F})$-martingales).
Let $y=\\{y_{t};t\geq 0\\}$, where
$y_{t}:=\widetilde{y}_{t}1_{t<{\widetilde{\tau}}}+\widehat{y}_{t}({\widetilde{\tau}},L)1_{t\geq{\widetilde{\tau}}}$,
be a $\mathbb{G}$-adapted process, then $y$ is a $(P,\mathbb{G})$-martingale
if and only if the following two conditions are satisfied:
* (i)
for $\eta$-almost every $u\geq 0$ and $l\in\mathbb{R}$,
$\\{\widehat{y}_{t}(u;l)p_{t}(u;l);t\geq u\\}$ is a
$(P,\mathbb{F})$-martingale;
* (ii)
the process
$\\{\widetilde{y}_{t}G_{t}+\int_{0}^{t}\int_{\mathbb{R}}\widehat{y}_{u}(u,l)p_{u}(u,l)\eta(du,dl);t\geq
0\\}$ is a $(P,\mathbb{F})$-martingale.
To give a proof, we need the following lemma
###### Lemma 3.2 (Projection).
Let $\mathbb{F}^{*}$ be a filtration larger than $\mathbb{F}$, i.e.,
$\mathbb{F}\subset\mathbb{F}$. If $x$ is a uniformly integrable (u.i.)
$\mathbb{F}$-martingale, then there exists an $\mathbb{F}^{*}$-martingale
$x^{*}$ such that $E(x^{*}_{t}|\mathscr{F}_{t})=x_{t}$.
###### Proof of Theorem 3.1.
Let $y$ is a u.i. $(P,\mathbb{G})$-martingale(otherwise consider the case of
stopped martingale as shown in [5]), since
$\mathbb{F}\subset\mathbb{G}\subset\mathbb{G}^{{\widetilde{\tau}},L}$, one can
see that there exits a $\mathbb{G}^{{\widetilde{\tau}},L}$-martingale
$y^{*}({\widetilde{\tau}},L)$ such that
$y_{t}=E(y^{*}_{t}({\widetilde{\tau}},L)|\mathscr{G}_{t})$, i.e., for
$\eta$-almost every $u\geq 0$ and $l\in\mathbb{R}$, the process
$\\{y^{*}_{t}(u,l)p_{t}(u,l);t\geq 0\\}$ is a u.i.
$(P,\mathbb{F})$-martingale. For any $t\geq 0$, since
$y^{*}_{t}({\widetilde{\tau}},L)1_{t\geq{\widetilde{\tau}}}$ is
$\mathscr{G}_{t}$-measurable and $1_{t\geq{\widetilde{\tau}}}$ is
$\mathscr{G}_{t}$-measurable, one can see that
$\begin{array}[]{ll}y^{*}_{t}({\widetilde{\tau}},L)1_{t\geq{\widetilde{\tau}}}=E[y^{*}_{t}({\widetilde{\tau}},L)|\mathscr{G}_{t}]1_{t\geq{\widetilde{\tau}}}=y_{t}1_{t\geq{\widetilde{\tau}}}=\widehat{y}_{t}({\widetilde{\tau}},L)1_{t\geq{\widetilde{\tau}}},\end{array}$
which implies that $y^{*}_{t}(u,l)1_{t\geq u}=\widehat{y}(u,l)1_{t\geq u}$,
$\eta$-a.s., for every $t\geq 0$, and hence
$\\{\widehat{y}_{t}(u;l)p_{t}(u;l);t\geq u\\}$ is a
$(P,\mathbb{F})$-martingale and (i) is proved. Furthermore, from Theorem 2.7,
one can see that
$E(y^{*}_{t}({\widetilde{\tau}},L)|\mathscr{G}_{t})=\mathbb{I}_{t<{\widetilde{\tau}}}\frac{1}{G_{t}}E\Big{(}\displaystyle\int_{t}^{\infty}\displaystyle\int_{\mathbb{R}}y_{t}^{*}(u,l)p_{t}(u,l)\eta(du,dl)\Big{|}\mathscr{F}_{t}\Big{)}+\widehat{y_{t}}({\widetilde{\tau}},L)\mathbb{I}_{{\widetilde{\tau}}\leq
t},$
thus
$\displaystyle\widetilde{y}_{t}\mathbb{I}_{t<{\widetilde{\tau}}}$
$\displaystyle=y_{t}\mathbb{I}_{t<{\widetilde{\tau}}}=E(y^{*}_{t}({\widetilde{\tau}},L)|\mathscr{G}_{t})\mathbb{I}_{t<{\widetilde{\tau}}}=\mathbb{I}_{t<{\widetilde{\tau}}}\frac{1}{G_{t}}E\Big{(}\int_{t}^{\infty}\int_{R}y_{t}^{*}(u,l)p_{t}(u,l)\eta(du,dl)\Big{|}\mathscr{F}_{t}\Big{)}$
$\displaystyle=\mathbb{I}_{t<{\widetilde{\tau}}}\frac{1}{G_{t}}E\big{(}\int_{t}^{\infty}\int_{R}y_{u}^{*}(u,l)p_{u}(u,l)\eta(du,dl)|\mathscr{F}_{t}\big{)},$
where the last equality results from the $(P,\mathbb{F})$-martingale property
of the process $y^{*}(u,l)p(u,l)$, for $\eta$-almost every $u\geq 0$, $l\geq
0$. We deduce
$\displaystyle\widetilde{y_{t}}G_{t}$
$\displaystyle=E\big{(}\int_{t}^{\infty}\int_{R}y_{u}^{*}(u,l)p_{u}(u,l)\eta(du,dl)|\mathscr{F}_{t}\big{)}$
$\displaystyle=E\big{(}\int_{0}^{\infty}\int_{R}y_{u}^{*}(u,l)p_{u}(u,l)\eta(du,dl)|\mathscr{F}_{t}\big{)}-\int_{0}^{t}\int_{R}y_{u}^{*}(u,l)p_{u}(u,l)\eta(du,dl),$
which implies that
$\Big{\\{}\widetilde{y}_{t}G_{t}+\displaystyle\int_{0}^{t}\displaystyle\int_{R}\widehat{y}_{u}(u,l)p_{u}(u,l)\eta(du,dl),~{}0\leq
t\leq\infty\Big{\\}}$ is a $(P,\mathbb{F})$-martingale (since
$y_{u}^{*}(u,l)=\widehat{y}_{u}(u,l)$) and (ii) immediately follows.
Conversely, assuming (i) and (ii), we verify $E(y_{t}|\mathscr{G}_{s})=y_{s}$
for $s\leq t$. Indeed,
$\displaystyle E(y_{t}|\mathscr{G}_{s})$
$\displaystyle=E(\mathbb{I}_{t<{\widetilde{\tau}}}\widetilde{y}_{t}+\mathbb{I}_{s<{\widetilde{\tau}}\leq
t}\widehat{y}_{t}({\widetilde{\tau}},L)|\mathscr{G}_{s})+E(\mathbb{I}_{{\widetilde{\tau}}\leq
s}\widehat{y}_{t}({\widetilde{\tau}},L)|\mathscr{G}_{s})$
$\displaystyle=\mathbb{I}_{s<{\widetilde{\tau}}}\frac{1}{G_{s}}E(\mathbb{I}_{t<{\widetilde{\tau}}}\widetilde{y}_{t}+\mathbb{I}_{s<{\widetilde{\tau}}\leq
t}\widehat{y}_{t}({\widetilde{\tau}},L)|\mathscr{F}_{s})+\frac{1}{p_{s}({\widetilde{\tau}},L)}\mathbb{I}_{{\widetilde{\tau}}\leq
s}E(\widehat{y}_{t}(u,l)p_{t}(u,l)|\mathscr{F}_{s})|_{u={\widetilde{\tau}},l=L}.$
where we also use Lemma 3.1.2 and Lemma 1.5 in Bielecki, Jeanblanc and
Rutkowski ${}^{\text{\cite[cite]{[\@@bibref{}{Bielecki-Jeanblanc-
Rutkowski2009}{}{}]}}}$ to obtain the last equality. Next, using condition
(i), it follows that
$\displaystyle E(y_{t}|\mathscr{G}_{s})$
$\displaystyle=\mathbb{I}_{s<{\widetilde{\tau}}}\frac{1}{G_{s}}E(\widetilde{y}_{t}G_{t}+\int_{s}^{t}\int_{R}\widehat{y}_{t}(u,l)p_{t}(u,l)\eta(du,dy)|\mathscr{F}_{s})+\mathbb{I}_{{\widetilde{\tau}}\leq
s}\frac{1}{p_{s}({\widetilde{\tau}},L)}\widehat{y}_{s}({\widetilde{\tau}},L)p_{s}({\widetilde{\tau}},L)$
$\displaystyle=\mathbb{I}_{s<{\widetilde{\tau}}}\frac{1}{G_{s}}E(\widetilde{y}_{t}G_{t}+\int_{0}^{t}\int_{R}\widehat{y}_{u}(u,l)p_{u}(u,l)\eta(du,dy)|\mathscr{F}_{s})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-\mathbb{I}_{s<{\widetilde{\tau}}}\frac{1}{G_{s}}\int_{0}^{s}\int_{R}\widehat{y}_{u}(u,l)p_{u}(u,l)\eta(du,dl)+\mathbb{I}_{{\widetilde{\tau}}\leq
s}\widehat{y}_{s}({\widetilde{\tau}},L)$
$\displaystyle=\mathbb{I}_{s<{\widetilde{\tau}}}\frac{1}{G_{s}}\widetilde{y}_{s}G_{s}+\mathbb{I}_{{\widetilde{\tau}}\leq
s}\widehat{y}_{s}({\widetilde{\tau}},L)$ $\displaystyle=y_{s},$
where we used condition (ii) to obtain the next-to-last identity. ∎
Similarly, we have the following corollary
###### Corollary 3.3.
Let $y=\\{y_{t};t\geq 0\\}$, where
$y_{t}:=\widetilde{y}_{t}1_{t<{\widetilde{\tau}}}+\widehat{y}_{t}({\widetilde{\tau}},L)1_{t\geq{\widetilde{\tau}}}$,
be a $\mathbb{G}$-adapted process, then $y$ is a
$(P^{*},\mathbb{G})$-martingale if and only if the following two conditions
are satisfied:
* (i)
for $\eta$-almost every $u\geq 0$ and $l\in\mathbb{R}$,
$\\{\widehat{y}_{t}(u;l);t\geq u\\}$ is a $(P^{*},\mathbb{F})$-martingale;
* (ii)
the process
$\\{\widetilde{y}_{t}G^{*}_{t}+\int_{0}^{t}\int_{\mathbb{R}}\widehat{y}_{u}(u,l)\eta(du,dl);t\geq
0\\}$ is a $(P^{*},\mathbb{F})$-martingale.
## 4 Canonical decomposition of a $(P,\mathbb{F})$-martingale in
$(P,\mathbb{G})$
We now consider the canonical decomposition of any $(P,\mathbb{F})$ martingale
$m$ in the enlarged filtration $\mathbb{G}$ respectively under Assumption 1.1.
Form Theorem 2.9, one can see that $p_{t}(s,l)$ can be determined by
$p_{s}(s,l)$, $\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$. We have
$p_{t}(s,l)=E[p_{s}(s,l)|\mathscr{F}_{t}]\mathbb{I}_{t<s}+p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(t)\mathbb{I}_{t\geq
s}$
and
$\begin{array}[]{ll}\\\ p_{t}(s,l)\mathbb{I}_{t\geq
s}=p_{s}(s,l)\mathbb{I}_{t\geq
s}+\displaystyle\int_{0}^{t}p_{u-}(s,l)\theta_{1}(u;s,l)^{\prime}\mathbb{I}_{u>s}dW(u)\\\
\qquad+\displaystyle\int_{0}^{t}\int_{E}p_{u-}(s,l)\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}\\{\mu(du,dy)-\nu(du,dy)\\},\end{array}$
thus
$\begin{array}[]{ll}p_{t}({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}=p_{\widetilde{\tau}}({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}+\displaystyle\int_{0}^{t}p_{u-}({\widetilde{\tau}},L)\theta_{1}(u;{\widetilde{\tau}},L)^{\prime}\mathbb{I}_{u>{\widetilde{\tau}}}dW(u)\\\
\qquad+\displaystyle\int_{0}^{t}\int_{E}p_{u-}({\widetilde{\tau}},L)\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\\{\mu(du,dy)-\nu(du,dy)\\}.\end{array}$
Recall that $G^{*}_{t}=P({\widetilde{\tau}}>t)$ is a deterministic continuous
function satisfying $0<G^{*}_{t}<1$ for each $t\in(0,\infty)$, there are no
atoms. To obtain the canonical decomposition of a $(P,\mathbb{F})$ martingale
in the filtration $\mathbb{G}$, we need the following lemma:
###### Lemma 4.1.
For any positive $\mathscr{O}(\mathbb{F})\times\mathscr{B}$-measurable
function $f(s,l)$ such that $E_{P^{*}}(|f({\widetilde{\tau}},L)|)<\infty$, let
$A^{f,*}_{t}:=\int_{0}^{t}\int_{\mathbb{R}}\dfrac{f(s,l)}{G^{*}_{s-}}\eta(ds,dl),$
then $A^{f,*}$ is a continuous increasing $\mathbb{F}$-adapted process and
$M^{f,*}_{t}=f({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}-A^{f,*}_{t\wedge{\widetilde{\tau}}}$
is a $(P^{*},\mathbb{G})$-martingale, i.e.,
$A^{f,*}_{\cdot\wedge{\widetilde{\tau}}}$ is the
$(P^{*},\mathbb{G})$-compensator of
$f({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$.
###### Proof.
For any $t_{1}<t_{2}$, we have
$\begin{array}[]{ll}E_{P^{*}}\big{[}f({\widetilde{\tau}},L)\mathbb{I}_{{\widetilde{\tau}}\leq
t}\big{|}\mathscr{F}_{t}\big{]}=\displaystyle\int_{0}^{t}\int_{\mathbb{R}}f(s,l)\eta(ds,dl).\end{array}$
(4.1)
In fact, one can see from the independence of $({\widetilde{\tau}},L)$ with
respect to $\mathscr{F}_{t}$ under $P^{*}$ that (4.1) holds for all positive
$\mathscr{B}(\mathbb{R}^{+}\times\mathbb{R})$-measurable functions $f(s,l)$.
In general, one gets from the independence lemma (see Lemma 2.3.4 of
Shreve(2003)${}^{\text{\cite[cite]{[\@@bibref{}{Shreve2003}{}{}]}}}$) and the
monotone class theorem that (4.1) still holds for any positive
$\mathscr{O}(\mathbb{F})\times\mathscr{B}$-measurable function $f(s,l)$. And
since there are no atoms, one can see that
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}f(s,l)\eta(ds,dl)$ is a continuous
increasing $\mathbb{F}$-adapted process, thus a $\mathbb{F}$-predictable
process.
For any $t_{1}<t_{2}$, we have
$\begin{array}[]{ll}E_{P^{*}}\big{[}f({\widetilde{\tau}},L)\mathbb{I}_{t_{1}<{\widetilde{\tau}}\leq
t_{2}}\big{|}\mathscr{F}_{t_{1}}\big{]}&=E_{P^{*}}\big{[}\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}f(s,l)\eta(ds,dl)\big{|}\mathscr{F}_{t_{1}}\big{]}\end{array}$
and that
$\begin{array}[]{ll}E_{P^{*}}\Big{[}\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}\dfrac{\mathbb{I}_{s\leq{\widetilde{\tau}}}}{G^{*}_{s}}f(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t_{1}}\Big{]}\\\
\qquad=E_{P^{*}}\Big{[}\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}\dfrac{E_{P^{*}}[\mathbb{I}_{s\leq{\widetilde{\tau}}}|\mathscr{F}_{s}]}{G^{*}_{s-}}f(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t_{1}}\Big{]}\\\
\qquad=E_{P^{*}}\Big{[}\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}f(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t_{1}}\Big{]},\end{array}$
hence
$\begin{array}[]{ll}E_{P^{*}}[M^{f,*}_{t_{2}}-M^{f,*}_{t_{1}}|\mathscr{G}_{t_{1}}]=E_{P^{*}}\Big{[}f({\widetilde{\tau}},L)\mathbb{I}_{t_{1}<{\widetilde{\tau}}\leq
t_{2}}-\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}\dfrac{\mathbb{I}_{s\leq{\widetilde{\tau}}}}{G^{*}_{s}}f(s,l)\eta(ds,dl)\Big{|}\mathscr{G}_{t_{1}}\Big{]}\\\
=\dfrac{1}{G^{*}_{t_{1}}}\Big{\\{}E_{P^{*}}\big{[}f({\widetilde{\tau}},L)\mathbb{I}_{t_{1}<{\widetilde{\tau}}\leq
t_{2}}\big{|}\mathscr{F}_{t_{1}}\big{]}\\\ \qquad\qquad-
E_{P^{*}}\Big{[}\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}\dfrac{\mathbb{I}_{s\leq{\widetilde{\tau}}}}{G^{*}_{s}}f(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t_{1}}\Big{]}\Big{\\}}\mathbb{I}_{t_{1}<{\widetilde{\tau}}}\\\
=0,\end{array}$
thus $E_{P^{*}}[M^{f,*}_{t_{2}}|\mathscr{G}_{t_{1}}]=M^{f,*}_{t_{1}}$ and
$M^{f,*}$ is a $(P^{*},\mathbb{G})$-martingale. ∎
We need the following assumption thereafter.
###### Assumption 4.2.
We assume that
$E_{P^{*}}(p_{{\widetilde{\tau}}}({\widetilde{\tau}},L))<\infty$ and
$E_{P^{*}}\Big{(}\dfrac{G_{{\widetilde{\tau}}}}{G^{*}_{\widetilde{\tau}}}\Big{)}<\infty$.
###### Corollary 4.3.
Suppose Assumption 4.2 holds. Let
$N_{1}(t):=p_{\widetilde{\tau}}({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\int_{\mathbb{R}}\dfrac{1}{G^{*}_{s}}p_{s}(s,l)\eta(ds,dl),$
then $\\{N_{1}(t);t\geq 0\\}$ is a uniformly integrable
$(P^{*},\mathbb{G})$-martingale.
###### Corollary 4.4.
Let Assumption 4.2 holds and let
$N_{2}(t):=\dfrac{G_{{\widetilde{\tau}}}}{G^{*}_{\widetilde{\tau}}}\mathbb{I}_{t\geq{\widetilde{\tau}}}+\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{G_{u}}{(G^{*}_{u})^{2}}dG^{*}_{u},$
then $\\{N_{1}(t);t\geq 0\\}$ is a uniformly integrable
$(P^{*},\mathbb{G})$-martingale.
###### Proof.
One can see that
$\begin{array}[]{ll}N_{2}(t)&=\dfrac{G_{{\widetilde{\tau}}}}{G^{*}_{\widetilde{\tau}}}\mathbb{I}_{t\geq{\widetilde{\tau}}}+\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{G_{u}}{(G^{*}_{u})^{2}}dG^{*}_{u}\\\
&=\dfrac{G_{{\widetilde{\tau}}}}{G^{*}_{\widetilde{\tau}}}\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{\mathbb{R}}\dfrac{1}{G^{*}_{u}}\dfrac{G_{u}}{G^{*}_{u}}\eta(du,dl),\end{array}$
since
$G^{*}_{u}=P^{*}({\widetilde{\tau}}>u)=\displaystyle\int_{u}^{\infty}\int_{\mathbb{R}}\eta(du,dl)$.
From Lemma 4.1, one can see that $N_{2}$ is a uniformly integrable
$(P^{*},\mathbb{G})$-martingale. ∎
###### Theorem 4.5.
Assume Assumption 4.2 holds. Let
$Z^{*}_{t}:=\dfrac{G_{t}}{G^{*}_{t}}\mathbb{I}_{t<{\widetilde{\tau}}}+p_{t}({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$,
then $Z^{*}$ is a uniformly integrable $(P^{*},\mathbb{G})$-martingale with
the following decomposition
$\begin{array}[]{ll}Z^{*}_{t}&=1-\displaystyle\int_{0}^{t}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
&\qquad\qquad\qquad-\theta_{1}(u;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}^{\prime}dW(u)\\\
&\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{E}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
&\qquad\qquad\qquad-\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}\\{\mu(du,dy)-\nu(du,dy)\\}\\\
&\qquad+N_{1}(t)-N_{2}(t).\end{array}$
###### Proof.
One can see that
$Z^{*}_{t}=\dfrac{1}{E\Big{[}\dfrac{1}{p_{t}({\widetilde{\tau}},L)}\Big{|}\mathscr{G}_{t}\Big{]}}$,
from which $Z^{*}$ is a uniformly integrable $(P^{*},\mathbb{G})$-martingale.
Since
$\begin{array}[]{ll}\dfrac{G_{t}}{G^{*}_{t}}&=1-\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\dfrac{1}{G^{*}_{s}}p_{s}(s,l)\eta(ds,dl)\\\
&\qquad-\displaystyle\int_{0}^{t}\dfrac{1}{G^{*}_{u}}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\Big{\\}}^{\prime}dW(u)\\\
&\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{E}\dfrac{1}{G^{*}_{u}}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\Big{\\}}\\{\mu(du,dy)-\nu(du,dy)\\}\\\
&\qquad-\displaystyle\int_{0}^{t}\dfrac{G_{u-}}{(G^{*}_{u})^{2}}dG^{*}_{u}\;,\end{array}$
one can see that
$\begin{array}[]{ll}Z^{*}_{t}&=1-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\int_{\mathbb{R}}\dfrac{1}{G^{*}_{s}}p_{s}(s,l)\eta(ds,dl)\\\
&\qquad-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{1}{G^{*}_{u}}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\Big{\\}}^{\prime}dW(u)\\\
&\qquad-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{E}\dfrac{1}{G^{*}_{u}}\Big{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\Big{\\}}\\{\mu(du,dy)-\nu(du,dy)\\}\\\
&\qquad-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{G_{u-}}{(G^{*}_{u})^{2}}dG^{*}_{u}-\dfrac{G_{{\widetilde{\tau}}-}}{G^{*}_{\widetilde{\tau}}}\mathbb{I}_{t\geq{\widetilde{\tau}}}\\\
\\\
&\qquad+p_{\widetilde{\tau}}({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}+\displaystyle\int_{0}^{t}p_{u-}({\widetilde{\tau}},L)\theta_{1}(u;{\widetilde{\tau}},L)^{\prime}\mathbb{I}_{u>{\widetilde{\tau}}}dW(u)\\\
&\qquad+\displaystyle\int_{0}^{t}\int_{E}p_{u-}({\widetilde{\tau}},L)\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\\{\mu(du,dy)-\nu(du,dy)\\}\end{array}$
$\begin{array}[]{ll}&=1-\displaystyle\int_{0}^{t}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
&\qquad\qquad\qquad-\theta_{1}(u;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}^{\prime}dW(u)\\\
&\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{E}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
&\qquad\qquad\qquad-\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}\\{\mu(du,dy)-\nu(du,dy)\\}\\\
\\\
&\qquad-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{G_{u-}}{(G^{*}_{u})^{2}}dG^{*}_{u}-\dfrac{G_{{\widetilde{\tau}}-}}{G^{*}_{\widetilde{\tau}}}\mathbb{I}_{t\geq{\widetilde{\tau}}}\\\
&\qquad+p_{\widetilde{\tau}}({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\int_{\mathbb{R}}\dfrac{1}{G^{*}_{s}}p_{s}(s,l)\eta(ds,dl),\end{array}$
which completes the proof. ∎
By Theorem 4.5, we give the $\mathbb{G}$-decomposition of a $(P,\mathbb{F})$
martingale as follows:
###### Theorem 4.6.
Let $p_{s}(s,l)$, $\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$ be given as in Theorem 2.9. If $m$ is a
càdlàg $(P,\mathbb{F})$-local martingale of the following form
$m_{t}=m_{0}+\int_{0}^{t}\xi_{1}(u)^{\prime}dW(u)+\int_{0}^{t}\int_{E}\xi_{2}(u,y)\\{\mu(du,dy)-\nu(du,dy)\\},$
then under Assumption 4.2,
$\begin{array}[]{ll}X_{t}:=m_{t}+\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{1}{G_{u-}}\bigg{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\Big{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;s,l)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\displaystyle\int_{E}\theta_{2}(u,y;s,l))\xi_{2}(u,y)F_{u}(dy)\Big{\\}}\eta(ds,dl)\bigg{\\}}du\\\
\qquad\qquad-\displaystyle\int_{\widetilde{\tau}}^{t}\bigg{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;{\widetilde{\tau}},L)+\displaystyle\int_{E}\theta_{2}(u,y;{\widetilde{\tau}},L)\xi_{2}(u,y)F_{u}(dy)\bigg{\\}}du,\end{array}$
is a $(P,\mathbb{G})$-local martingale.
Remark.
1)
Theorem 4.6 may be viewed as a corollary of Callegaro, Jeanblanc and
Zargari(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Callegaro-Jeanblanc-
Zargari(2010)}{}{}]}}}$ and El Karoui, Jeanblanc and
Jiao(2009)${}^{\text{\cite[cite]{[\@@bibref{}{El Karoui-Jeanblanc-
Jiao2009}{}{}]}}}$, the main difference is that the decomposition of
$\mathbb{F}$-local martingale in $\mathbb{G}$ we give here only depends on
$p_{s}(s,l)$, $\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and
$\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$, since in the integral
$\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\Big{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;s,l)+\displaystyle\int_{E}\theta_{2}(u,y;s,l))\xi_{2}(u,y)F_{u}(dy)\Big{\\}}\eta(ds,dl),$
$\theta_{1}(u;s,l)=\theta_{1}(u;s,l)\mathbb{I}_{u>s}$ and
$\theta_{1}(u,y;s,l)=\theta_{2}(u,y;s,l)\mathbb{I}_{u>s}$, which is quite
interesting.
2)
Furthermore, since a $\mathbb{G}^{{\widetilde{\tau}},L}$-stopping time might
not be a $\mathbb{G}$-stopping time and the optional projection of a
$(P,\mathbb{G}^{{\widetilde{\tau}},L})$-local martingale on $\mathbb{G}$ might
not be a $(P,\mathbb{G})$-local martingale, the proof of Proposition 3.3 in
Callegaro, Jeanblanc and Zargari(2010) is not strict. We can also use the
proof of Proposition 5.9 in El Karoui, Jeanblanc and Jiao(2009) to write
$X_{t}$ into the following form
$X_{t}=X_{1}(t)\mathbb{I}_{t<{\widetilde{\tau}}}+X_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$
where
$\begin{array}[]{ll}X_{1}(t)=m_{t}+\displaystyle\int_{0}^{t}\dfrac{1}{G_{u-}}\bigg{\\{}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\Big{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;s,l)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\displaystyle\int_{E}\theta_{2}(u,y;s,l))\xi_{2}(u,y)F_{u}(dy)\Big{\\}}\eta(ds,dl)\bigg{\\}}du\bigg{\\}}\\\
X_{2}(t;s,l)=\\{m_{t}-X_{1}(s)\\}\mathbb{I}_{t>s}-\displaystyle\int_{s}^{t}\bigg{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;s,l)+\displaystyle\int_{E}\theta_{2}(u,y;s,l)\xi_{2}(u,y)F_{u}(dy)\bigg{\\}}du,\end{array}$
and show that $\big{\\{}X_{2}(t;s,l)p_{t}(s,l);t\geq s\big{\\}}$ is a
$(P,\mathbb{F})$-local martingale and
$\big{\\{}X_{1}(t)G_{t}+\int_{0}^{t}\int_{\mathbb{R}}X_{2}(u;u,l)p_{u}(u,l)\eta(du,dl);t\geq
0\big{\\}}$ is a $(P,\mathbb{F})$-local martingale. However, we can not use
Theorem 3.1, since the sequence of the localization stopping times of
$\\{X_{2}(t;s,l)p_{t}(s,l);t\geq s\\}$ depends on $(s,l)$ which is uncountable
infinite, one can see that the proof of Proposition 5.9 in El Karoui-
Jeanblanc-Jiao(2009) is not strict. Here we would like to provide a strict
proof based on Proposition 4.5.
###### Proof of Theorem 4.6.
Let $m$ be a $(P,\mathbb{F})$-local martingale, then $m$ is a
$(P^{*},\mathbb{F})$-local martingale which is also a
$(P^{*},\mathbb{G}^{{\widetilde{\tau}},L})$-local martingale. Since
$\mathbb{F}\subset\mathbb{G}\subset\mathbb{G}^{{\widetilde{\tau}},L}$, thus
$m$ is a $(P^{*},\mathbb{G})$-local martingale. Furthermore, since
$\dfrac{1}{p_{t}({\widetilde{\tau}},L)}$ is the density process of $P^{*}$
with respect to $(P,\mathbb{G}^{{\widetilde{\tau}},L})$, one can see that the
density process of $P^{*}$ with respect to $(P,\mathbb{G})$ is given by
$\begin{array}[]{ll}L^{*}_{t}=E\Big{[}\dfrac{1}{p_{t}({\widetilde{\tau}},L)}\Big{|}\mathscr{G}_{t}\Big{]}\\\
\qquad=\dfrac{1}{G_{t}}E\Big{(}\displaystyle\int_{t}^{\infty}\int_{\mathbb{R}}\dfrac{1}{p_{t}(u,l)}p_{t}(u,l)\eta(du,dl)\Big{|}\mathscr{F}_{s}\Big{)}\mathbb{I}_{t<{\widetilde{\tau}}}+\dfrac{1}{p_{t}({\widetilde{\tau}},L)}\mathbb{I}_{t\geq{\widetilde{\tau}}}\\\
\qquad=\dfrac{1}{G_{t}}E\Big{(}\displaystyle\int_{t}^{\infty}\int_{\mathbb{R}}\eta(du,dl)\Big{|}\mathscr{F}_{s}\Big{)}\mathbb{I}_{t<{\widetilde{\tau}}}+\dfrac{1}{p_{t}({\widetilde{\tau}},L)}\mathbb{I}_{t\geq{\widetilde{\tau}}}\\\
\qquad=\dfrac{G^{*}_{t}}{G_{t}}\mathbb{I}_{t<{\widetilde{\tau}}}+\dfrac{1}{p_{t}({\widetilde{\tau}},L)}\mathbb{I}_{t\geq{\widetilde{\tau}}},\end{array}$
thus
$\begin{array}[]{ll}\dfrac{1}{L^{*}_{t}}=\dfrac{G_{t}}{G^{*}_{t}}\mathbb{I}_{t<{\widetilde{\tau}}}+p_{t}({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}=Z^{*}_{t}.\end{array}$
To prove $X$ is a $(P,\mathbb{G})$-local martingale, we need only to prove
that $X_{t}Z^{*}_{t}$ is a $(P^{*},\mathbb{G})$-local martingale. As a matter
of fact, one can see from Itô’s formula that
$\begin{array}[]{ll}X_{t}Z^{*}_{t}=m_{0}+\displaystyle\int_{0}^{t}Z^{*}_{u-}dm_{u}+\displaystyle\int_{0}^{t}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\Big{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;s,l)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\displaystyle\int_{E}\theta_{2}(u,y;s,l))\xi_{2}(u,y)F_{u}(dy)\Big{\\}}\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
\qquad\qquad-\Big{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;{\widetilde{\tau}},L)+\displaystyle\int_{E}\theta_{2}(u,y;{\widetilde{\tau}},L)\xi_{2}(u,y)F_{u}(dy)\Big{\\}}\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}du\\\
\qquad\qquad+\displaystyle\int_{0}^{t}X_{u-}dZ^{*}_{u}+[X,Z^{*}]_{t}\\\
=m_{0}+\displaystyle\int_{0}^{t}Z^{*}_{u-}dm_{u}+\displaystyle\int_{0}^{t}X_{u-}dZ^{*}_{u}\\\
\qquad\qquad+\displaystyle\int_{0}^{t}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\Big{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;s,l)\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\displaystyle\int_{E}\theta_{2}(u,y;s,l))\xi_{2}(u,y)F_{u}(dy)\Big{\\}}\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
\qquad\qquad\qquad\qquad-\Big{\\{}\xi_{1}(u)^{\prime}\theta_{1}(u;{\widetilde{\tau}},L)+\displaystyle\int_{E}\theta_{2}(u,y;{\widetilde{\tau}},L)\xi_{2}(u,y)F_{u}(dy)\Big{\\}}\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}du\\\
\qquad\qquad-\displaystyle\int_{0}^{t}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
\qquad\qquad\qquad-\theta_{1}(u;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}^{\prime}\xi_{1}(u)du\\\
\qquad\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{E}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
\qquad\qquad\qquad-\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}\xi_{2}(u,y)\mu(du,dy)\end{array}$
$\begin{array}[]{ll}=m_{0}+\displaystyle\int_{0}^{t}Z^{*}_{u-}dm_{u}+\displaystyle\int_{0}^{t}X_{u-}dZ^{*}_{u}\\\
\qquad\qquad-\displaystyle\int_{0}^{t}\displaystyle\int_{E}Z^{*}_{u-}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l))\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
\qquad\qquad\qquad-\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}\xi_{2}(u,y)\\{\mu(du,dy)-\nu(du,dy)\\}.\end{array}$
Since both $m$ and $Z^{*}$ are $(P^{*},\mathbb{G})$-local martingale, one can
see that $XZ^{*}$ is a $(P^{*},\mathbb{G})$-local martingale, which completes
the proof. ∎
###### Corollary 4.7.
Assume conditions of Theorem 4.6 hold, and let
$\begin{array}[]{lr}W^{\mathbb{G}}(t)&:=W(t)+\displaystyle\int_{0}^{t}\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{1}(u;s,l)\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
&-\theta_{1}(u;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}du,\end{array}$
then $W^{\mathbb{G}}$ is a $(P,\mathbb{G})$-Brownian motion.
###### Corollary 4.8.
Assume conditions of Theorem 4.6 hold, and let
$\begin{array}[]{lr}\nu^{\mathbb{G}}(du,dy)&:=\nu(du,dy)+\bigg{\\{}\dfrac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{\mathbb{R}}p_{s}(s,l)Z^{\theta_{1},\theta_{2}}_{s,l}(u-)\theta_{2}(u,y;s,l)\eta(ds,dl)\mathbb{I}_{u\leq{\widetilde{\tau}}}\\\
&-\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\bigg{\\}}F_{u}(dy)du,\end{array}$
then $\nu^{\mathbb{G}}(du,dy)$ is the compensator of $\mu(du,dy)$ with respect
to $(P,\mathbb{G})$.
Remark. From Theorem 4.6, we get the $\mathbb{G}$-decomposition of a
$\mathbb{F}$ martingale, the $(P,\mathbb{G})$-Brownian motion and the
compensator of $\mu(du,dy)$ with respect to $(P,\mathbb{G})$ explicitly.
## 5 The representation of a $(P,\mathbb{G})$-martingale
First, we have the following representation for a
$(P^{*},\mathbb{G})$-martingale:
###### Theorem 5.1.
Let
$M_{t}=M_{1}(t)\mathbb{I}_{t<{\widetilde{\tau}}}+M_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$
be a u.i. $(P^{*},\mathbb{G})$ martingale, then there exists a
$\mathbb{G}$-predictable process $\xi$ and a
$\widetilde{\mathscr{P}}(\mathbb{G})$-measurable function $\zeta(u,y)$ such
that
$\begin{array}[]{ll}M_{t}&=M_{0}+\displaystyle\int_{0}^{t}\xi(u)^{\prime}dW(u)+\displaystyle\int_{0}^{t}\int_{E}\zeta(u,y)\\{\mu(du,dy)-\nu(du,dy)\\}\\\
&\quad\qquad+M_{2}({\widetilde{\tau}};{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{\mathbb{R}}\dfrac{M_{2}(u;u,l)}{G^{*}_{u}}\eta(du,dl)\\\
&\quad\qquad-
M_{1}({\widetilde{\tau}}-)\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{M_{1}(u-)}{G^{*}_{u}}dG^{*}_{u},\end{array}$
(5.1)
###### Proof.
From Corollary 3.3, one can see that
$\\{M_{1}(t)G^{*}_{t}+\int_{0}^{t}\int_{\mathbb{R}}M_{2}(u;u,l)\eta(du,dl);t\geq
0\\}$ and $\\{M_{2}(t;s,l);t\geq s\\}$ are $(P^{*},\mathbb{F})$-martingales
for $\eta$-almost every $u\geq 0$ and $l\in\mathbb{R}$, thus there exist
$\mathbb{F}$-predictable processes $\xi_{1}$ and $\xi_{2}(s,l)$ and
$\widetilde{\mathscr{P}}(\mathbb{F})$-measurable functions $\zeta_{1}(u,y)$
and $\zeta_{2}(u,y;s,l)$ such that
$\begin{array}[]{ll}M_{1}(t)G^{*}_{t}+\displaystyle\int_{0}^{t}\int_{\mathbb{R}}M_{1}(u;u,l)\eta(du,dl)\\\
\qquad\qquad=M_{1}(0)+\displaystyle\int_{0}^{t}\xi_{1}(u)^{\prime}dW(u)+\displaystyle\int_{0}^{t}\displaystyle\int_{E}\zeta_{1}(u,y)\\{\mu(du,dy)-\nu(du,dy)\\}\text{
and}\\\
M_{2}(t;s,l)=M_{2}(s;s,l)+\displaystyle\int_{s}^{t}\xi_{2}(u;s,l)^{\prime}dW(u)+\displaystyle\int_{s}^{t}\displaystyle\int_{E}\zeta_{1}(u,y;s,l)\\{\mu(du,dy)-\nu(du,dy)\\}.\end{array}$
Thus
$\begin{array}[]{ll}M_{1}(t)G^{*}_{t}=M_{1}(0)+\displaystyle\int_{0}^{t}\xi_{1}(u)^{\prime}dW(u)+\displaystyle\int_{0}^{t}\displaystyle\int_{E}\zeta_{1}(u,y)\\{\mu(du,dy)-\nu(du,dy)\\}\\\
\qquad\qquad\qquad-\displaystyle\int_{0}^{t}\int_{\mathbb{R}}M_{1}(u;u,l)\eta(du,dl),\end{array}$
one can see from Itô’s formula that
$\begin{array}[]{ll}M_{1}(t)=M_{1}(t)G^{*}_{t}\dfrac{1}{G^{*}_{t}}\\\
=M_{1}(0)+\displaystyle\int_{0}^{t}\dfrac{\xi_{1}(u)^{\prime}}{G^{*}_{u-}}dW(u)+\displaystyle\int_{0}^{t}\displaystyle\int_{E}\dfrac{\zeta_{1}(u,y)}{G^{*}_{u-}}\\{\mu(du,dy)-\nu(du,dy)\\}\\\
\qquad\qquad\qquad-\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\dfrac{M_{1}(u;u,l)}{G^{*}_{u-}}\eta(du,dl)+\displaystyle\int_{0}^{t}M_{1}(u-)G^{*}_{u-}d\Big{(}\dfrac{1}{G^{*}_{u}}\Big{)}\\\
=M_{1}(0)+\displaystyle\int_{0}^{t}\dfrac{\xi_{1}(u)^{\prime}}{G^{*}_{u-}}dW(u)+\displaystyle\int_{0}^{t}\displaystyle\int_{E}\dfrac{\zeta_{1}(u,y)}{G^{*}_{u-}}\\{\mu(du,dy)-\nu(du,dy)\\}\\\
\qquad\qquad\qquad-\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\dfrac{M_{1}(u;u,l)}{G^{*}_{u-}}\eta(du,dl)-\displaystyle\int_{0}^{t}\dfrac{M_{1}(u-)}{G^{*}_{u}}dG^{*}_{u}\;.\end{array}$
Thus
$\begin{array}[]{ll}M_{t}=M_{1}(t)\mathbb{I}_{t<{\widetilde{\tau}}}+M_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}\\\
=M_{1}(t\wedge{\widetilde{\tau}})-M_{1}({\widetilde{\tau}})\mathbb{I}_{t\geq{\widetilde{\tau}}}+M_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}\\\
=M_{1}(0)+\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{\xi_{1}(u)^{\prime}}{G^{*}_{u-}}dW(u)+\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{E}\dfrac{\zeta_{1}(u,y)}{G^{*}_{u-}}\\{\mu(du,dy)-\nu(du,dy)\\}\\\
\qquad\qquad\qquad-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\int_{\mathbb{R}}\dfrac{M_{1}(u;u,l)}{G^{*}_{u-}}\eta(du,dl)-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{M_{1}(u-)}{G^{*}_{u}}dG^{*}_{u}\\\
\qquad\qquad\qquad-
M_{1}({\widetilde{\tau}})\mathbb{I}_{t\geq{\widetilde{\tau}}}+M_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}\\\
=M_{0}+\displaystyle\int_{0}^{t}\xi(u)^{\prime}dW(u)+\displaystyle\int_{0}^{t}\int_{E}\zeta(u,y)\\{\mu(du,dy)-\nu(du,dy)\\}\\\
\quad\qquad+M_{2}({\widetilde{\tau}};{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{\mathbb{R}}\dfrac{M_{2}(u;u,l)}{G^{*}_{u}}\eta(du,dl)\\\
\quad\qquad-
M_{1}({\widetilde{\tau}})\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\dfrac{M_{1}(u-)}{G^{*}_{u}}dG^{*}_{u}\;,\end{array}$
where
$\begin{array}[]{ll}\xi(u)=\dfrac{\xi_{1}(u)}{G^{*}_{u-}}\mathbb{I}_{u\leq{\widetilde{\tau}}}+\xi_{2}(u;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\;,\\\
\zeta(u,y)=\dfrac{\zeta_{1}(u,y)}{G^{*}_{u-}}\mathbb{I}_{u\leq{\widetilde{\tau}}}+\zeta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\;.\end{array}$
Since the $(P^{*},\mathbb{F})$-martingale
$\\{M_{1}(t)G^{*}_{t}+\int_{0}^{t}\int_{\mathbb{R}}M_{2}(u;u,l)\eta(du,dl);t\geq
0\\}$ has no jumps at ${\widetilde{\tau}}$ as a
$(P^{*},\mathbb{G})$-martingale and $G^{*}$ is continuous, one can see that
$M_{1}({\widetilde{\tau}})=M_{1}({\widetilde{\tau}}-)$, a.s. and (5.1)
follows. ∎
Now we turn to prove the predictable representation theorem for a
$(P,\mathbb{G})$-martingale. Similar to Lemma 4.1, we have the following lemma
###### Lemma 5.2.
For any positive $\mathscr{O}(\mathbb{F})\times\mathscr{B}$-measurable
function $f(s,l)$ such that $E_{P}(|f({\widetilde{\tau}},L)|)<\infty$, let
$A^{f}_{t}:=\int_{0}^{t}\int_{\mathbb{R}}\dfrac{f(s,l)}{G_{s-}}p_{s}(s,l)\eta(ds,dl),$
then $A^{f}$ is a continuous increasing process and
$M^{f}_{t}=f({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}-A^{f}_{t\wedge{\widetilde{\tau}}}$
is a $(P,\mathbb{G})$-martingale, i.e.,
$A^{f}_{\cdot\wedge{\widetilde{\tau}}}$ is the $(P,\mathbb{G})$-compensator of
$f({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$.
###### Proof.
Similar to the proof of Lemma 4.1, one can see that for any $t_{1}<t_{2}$,
$\begin{array}[]{ll}E\big{[}f({\widetilde{\tau}},L)\mathbb{I}_{t_{1}<{\widetilde{\tau}}\leq
t_{2}}\big{|}\mathscr{F}_{t_{1}}\big{]}&=\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}f(s,l)p_{t_{1}}(s,l)\eta(ds,dl)\\\
&=E\Big{[}\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}f(s,l)p_{s}(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t_{1}}\Big{]}\end{array}$
and that
$\begin{array}[]{ll}E\Big{[}\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}\dfrac{\mathbb{I}_{s\leq{\widetilde{\tau}}}}{G_{s-}}p_{s}(s,l)f(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t_{1}}\Big{]}\\\
=E\Big{[}\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}\dfrac{E[\mathbb{I}_{s\leq{\widetilde{\tau}}}|\mathscr{F}_{s}]}{G_{s-}}p_{s}(s,l)f(s,l)\eta(ds,dl)\Big{|}\mathscr{F}_{t_{1}}\Big{]}\\\
=\displaystyle\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}f(s,l)p_{s}(s,l)\eta(ds,dl),\end{array}$
and the rest is completely the same as the proof of Lemma 4.1. ∎
###### Theorem 5.3.
Let
$M_{t}:=M_{1}(t)\mathbb{I}_{t<{\widetilde{\tau}}}+M_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$
be a u.i. $(P,\mathbb{G})$-martingale, then there exits a
$\mathbb{F}$-predictable process $\xi$ and a
$\widetilde{\mathscr{P}}(\mathbb{F})$-measurable function $\zeta(u,y)$ such
that
$\begin{array}[]{ll}M_{t}&=M_{1}(0)+\displaystyle\int_{0}^{t}\xi(u)^{\prime}dW^{\mathbb{G}}(u)+\displaystyle\int_{0}^{t}\displaystyle\int_{E}\zeta(u,y)\\{\mu(du,dy)-\nu^{\mathbb{G}}(du,dy)\\}\\\
&~{}~{}~{}~{}~{}~{}~{}+M_{2}({\widetilde{\tau}};{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{R}\frac{M_{2}(u;u,l)}{G_{u-}}p_{u}(u,l)\eta(du,dl)\\\
&~{}~{}~{}~{}~{}~{}~{}-M_{1}({\widetilde{\tau}}-)\mathbb{I}_{t\geq{\widetilde{\tau}}}-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{R}\frac{M_{1}(u-)}{G_{u-}}p_{s}(s,l)\eta(ds,dl).\end{array}$
(5.2)
Remark. By Lemma 5.2, one can see that
$M_{2}({\widetilde{\tau}};{\widetilde{\tau}},L)-\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{R}\frac{M_{2}(u;u,l)}{G_{u-}}p_{u}(u,l)\eta(du,dl)$
and
$M_{1}({\widetilde{\tau}}-)\mathbb{I}_{t\geq{\widetilde{\tau}}}+\displaystyle\int_{0}^{t\wedge{\widetilde{\tau}}}\displaystyle\int_{R}\frac{M_{1}(u-)}{G_{u-}}p_{s}(s,l)\eta(ds,dl)$
are $(P,\mathbb{G})$-martingales. With the non-trivial conditional density
$p_{t}({\widetilde{\tau}},L)$ , Theorem 5.3 may be viewed as the more general
result of Theorem 5.2 and the representation of a $(P,\mathbb{G})$-martingale
is different from Callegaro, Jeanblanc and
Zargari(2010)${}^{\text{\cite[cite]{[\@@bibref{}{Callegaro-Jeanblanc-
Zargari(2010)}{}{}]}}}$.
###### Proof of Theorem 5.3.
From Theorem 3.1, one can see that $\Big{\\{}M_{2}(t;u,l)p_{t}(u,l)(t\geq
u)\Big{\\}}$ and
$\Big{\\{}M_{1}(t)G_{t}+\displaystyle\int_{0}^{t}\displaystyle\int_{R}M_{2}(u;u,l)p_{u}(u,l)\eta(du,dl)(t\geq
0)\Big{\\}}$ are $(P,\mathbb{F})$-martingales for $\eta$-almost every $u\geq
0$ and $l\in\mathbb{R}$, thus there exits $\mathbb{F}$-predictable processes
$\xi_{1}(u)$, $\xi_{2}(u;s,l)$ and
$\widetilde{\mathscr{P}}(\mathbb{F})$-measurable functions $\zeta_{1}(u,y)$,
$\zeta_{2}(u,y;s,l)$, such that
$\displaystyle
M_{1}(t)G_{t}+\int_{0}^{t}\int_{R}M_{2}(u;u,l)p_{t}(u,l)\eta(du,dl)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}=M_{1}(0)+\int_{0}^{t}\xi_{1}(u)dW(u)+\int_{0}^{t}\int_{E}\zeta_{1}(u,y)\\{\mu(du,dy)-\nu(du,dy)\\}\text{
and }$ $\displaystyle
M_{2}(t;s,l)p_{t}(s,l)=M_{2}(s;s,l)p_{s}(s,l)+\int_{s}^{t}\xi_{2}(u;s,l)^{\prime}dW(u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{s}^{t}\int_{E}\zeta_{2}(u,y;s,l)\\{\mu(du,dy)-\nu(du,dy)\\}\;.$
Let
$A_{u}^{\mathbb{G}}:=-\frac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{R}p_{s}(s,l)\eta(ds,dl),$
$\alpha_{1}(u):=\frac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{R}p_{s}(s,l)Z_{s,l}^{\theta_{1},\theta_{2}}(u-)\theta_{1}(u;s,l)\eta(ds,dl)$,
$\alpha_{2}(u,y):=\frac{1}{G_{u-}}\displaystyle\int_{0}^{u}\displaystyle\int_{R}p_{s}(s,l)Z_{s,l}^{\theta_{1},\theta_{2}}(u-)\theta_{2}(u;s,l)\eta(ds,dl)$,
then one can see that $G_{t}$ can be rewritten into the following form
$\displaystyle
G_{t}=1-\int_{0}^{t}G_{u-}dA_{u}^{\mathbb{G}}-\int_{0}^{t}G_{u-}\alpha_{1}(u)^{\prime}dW(u)-\int_{0}^{t}\int_{E}G_{u-}\alpha_{2}(u,y)\\{\mu(du,dy)-\nu(du,dy)\\},$
which implies that
$\displaystyle\frac{1}{G_{t}}$
$\displaystyle=1+\int_{0}^{t}\frac{1}{G_{u-}}dA_{u}^{\mathbb{G}}+\int_{0}^{t}\frac{1}{G_{u-}}\alpha_{1}(u)^{\prime}dW(u)-\int_{0}^{t}\int_{E}\frac{1}{G_{u-}}\alpha_{2}(u,y)\nu(du,dy)$
$\displaystyle\qquad+\int_{0}^{t}\frac{1}{G_{u-}}\parallel\alpha_{1}(u)\parallel^{2}du+\int_{0}^{t}\int_{E}\frac{1}{G_{u-}}\Big{\\{}\frac{1}{1-\alpha_{2}(u,y)}-1\Big{\\}}\mu(du,dy).$
Since
$\displaystyle p_{t}({\widetilde{\tau}},L)$
$\displaystyle=p_{\widetilde{\tau}}({\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}+\int_{0}^{t}p_{u-}({\widetilde{\tau}},L)\theta_{1}(u;{\widetilde{\tau}},L)^{\prime}\mathbb{I}_{u>{\widetilde{\tau}}}dW(u)$
$\displaystyle+\int_{0}^{t}\int_{E}p_{u-}({\widetilde{\tau}},L)\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\\{\mu(du,dy)-\nu(du,dy)\\},$
thus
$\displaystyle\frac{1}{p_{t}({\widetilde{\tau}},L)}$
$\displaystyle=\frac{1}{p_{\widetilde{\tau}}({\widetilde{\tau}},L)}\mathbb{I}_{t\geq{\widetilde{\tau}}}-\int_{0}^{t}\frac{1}{p_{u-}({\widetilde{\tau}},L)}\theta_{1}(u;{\widetilde{\tau}},L)^{\prime}\mathbb{I}_{u>{\widetilde{\tau}}}dW(u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{t}\int_{E}\frac{1}{p_{u-}({\widetilde{\tau}},L)}\theta_{2}(u,y;{\widetilde{\tau}},L)\mathbb{I}_{u>{\widetilde{\tau}}}\nu(du,dy)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{t}\frac{1}{p_{u-}({\widetilde{\tau}},L)}\|\theta_{1}(u;{\widetilde{\tau}},L)\|^{2}\mathbb{I}_{u>{\widetilde{\tau}}}du$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{t}\int_{E}\frac{1}{p_{u-}({\widetilde{\tau}},L)}\Big{\\{}\frac{1}{1+\theta_{2}(u,y;{\widetilde{\tau}},L)}-1\Big{\\}}\mathbb{I}_{u>{\widetilde{\tau}}}\mu(du,dy).$
From Itô’s formula, one has
$\displaystyle M_{1}(t)$ $\displaystyle=M_{1}(t)G_{t}\frac{1}{G_{t}}$
$\displaystyle=M_{1}(0)+\int_{0}^{t}\frac{\xi_{1}(u)^{\prime}}{G_{u-}}dW(u)+\int_{0}^{t}\int_{E}\frac{\zeta_{1}(u,y)}{G_{u-}}\\{\mu(du,dy)-\nu(du,dy)\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\int_{0}^{t}\int_{R}\frac{M_{2}(u;u,l)}{G_{u-}}p_{u}(u,l)\eta(du,dl)+\int_{0}^{t}M_{1}(u-)G_{u-}d(\frac{1}{G_{u}})+\int_{0}^{t}d[M_{1}G,\frac{1}{G}]_{u}$
$\displaystyle=M_{1}(0)+\int_{0}^{t}\frac{\xi_{1}(u)^{\prime}}{G_{u-}}dW(u)+\int_{0}^{t}\int_{E}\frac{\zeta_{1}(u,y)}{G_{u-}}\\{\mu(du,dy)-\nu(du,dy)\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\int_{0}^{t}\int_{R}\frac{M_{2}(u;u,l)}{G_{u-}}p_{u}(u,l)\eta(du,dl)$
$\displaystyle\qquad+\int_{0}^{t}M_{1}(u-)dA_{u}^{\mathbb{G}}+\int_{0}^{t}M_{1}(u-)\alpha_{1}(u)^{\prime}dW(u)-\int_{0}^{t}\int_{E}M_{1}(u-)\alpha_{2}(u,y)\nu(du,dy)$
$\displaystyle\qquad+\int_{0}^{t}M_{1}(u-)\parallel\alpha_{1}(u)\parallel^{2}du+\int_{0}^{t}\int_{E}M_{1}(u-)\Big{\\{}\frac{1}{1-\alpha_{2}(u,y)}-1\Big{\\}}\mu(du,dy)$
$\displaystyle\qquad+\int_{0}^{t}\frac{1}{G_{u-}}\xi_{1}(u)^{\prime}\alpha_{1}(u)du+\int_{0}^{t}\int_{E}\frac{1}{G_{u-}}\frac{\zeta_{1}(u,y)\alpha_{2}(u,y)}{1-\alpha_{2}(u,y)}\mu(du,dy),$
By calculation, we get
$\displaystyle M_{1}(t\wedge{\widetilde{\tau}})$
$\displaystyle=M_{1}(0)+\int_{0}^{t\wedge{\widetilde{\tau}}}M_{1}(u-)dA_{u}^{\mathbb{G}}-\int_{0}^{t\wedge{\widetilde{\tau}}}\int_{R}\frac{M_{2}(u;u,l)}{G_{u-}}p_{u}(u,l)\eta(du,dl)$
$\displaystyle+\int_{0}^{t\wedge{\widetilde{\tau}}}\Big{\\{}\frac{\xi_{1}(u)}{G_{u-}}+M_{1}(u-)\alpha_{1}(u)\Big{\\}}^{\prime}dW^{\mathbb{G}}(u)$
$\displaystyle+\int_{0}^{t\wedge{\widetilde{\tau}}}\int_{E}\frac{1}{1-\alpha_{2}(u,y)}\Big{\\{}\frac{\zeta_{1}(u,y)}{G_{u-}}+M_{1}(u-)\alpha_{2}(u,y)\Big{\\}}\\{\mu(du,dy)-\nu^{\mathbb{G}}(du,dy)\\}$
here we have used the following equalities
$W^{\mathbb{G}}(t)=W(t)+\displaystyle\int_{0}^{\widetilde{\tau}}\alpha_{1}(u)du-\displaystyle\int_{\widetilde{\tau}}^{t}\theta_{1}(u;{\widetilde{\tau}},L)du,$
$\nu^{\mathbb{G}}(du,dy)=\nu(du,dy)+\alpha_{2}(u,y)\nu(du,dy)\mathbb{I}_{u\leq{\widetilde{\tau}}}-\theta_{2}(u,y;{\widetilde{\tau}},L)\nu(du,dy)\mathbb{I}_{u>{\widetilde{\tau}}}.$
By Itô’s formula again, one can see that
$\displaystyle M_{2}(t;{\widetilde{\tau}},L)$
$\displaystyle=M_{2}(t;{\widetilde{\tau}},L)p_{t}({\widetilde{\tau}},L)\frac{1}{p_{t}({\widetilde{\tau}},L)}$
$\displaystyle=M_{2}({\widetilde{\tau}};{\widetilde{\tau}},L)+\int_{\widetilde{\tau}}^{t}\frac{\xi_{2}(u;{\widetilde{\tau}},L)^{\prime}}{p_{u-}({\widetilde{\tau}},L)}dW(u)+\int_{\widetilde{\tau}}^{t}\int_{E}\frac{\zeta_{2}(u,y;{\widetilde{\tau}},L)}{p_{u-}({\widetilde{\tau}},L)}\\{\mu(du,dy)-\nu(du,dy)\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\int_{\widetilde{\tau}}^{t}M_{2}(u-;{\widetilde{\tau}},L)p_{u-}({\widetilde{\tau}},L)d(\frac{1}{p_{u}({\widetilde{\tau}},L)})+\int_{\widetilde{\tau}}^{t}d[M_{2}(\cdot;{\widetilde{\tau}},L)p_{\cdot}({\widetilde{\tau}},L),\frac{1}{p_{\cdot}({\widetilde{\tau}},L)}]_{u}$
$\displaystyle=M_{2}({\widetilde{\tau}};{\widetilde{\tau}},L)+\int_{\widetilde{\tau}}^{t}\Big{\\{}\frac{\xi_{2}(u;{\widetilde{\tau}},L)^{\prime}}{p_{u-}({\widetilde{\tau}},L)}-M_{2}(u-;{\widetilde{\tau}},L)\theta_{1}(u;{\widetilde{\tau}},L)\Big{\\}}dW^{\mathbb{G}}(u)$
$\displaystyle+\int_{\widetilde{\tau}}^{t}\int_{E}\frac{1}{1+\theta_{2}(u,y;{\widetilde{\tau}},L)}\Big{\\{}\frac{\zeta_{2}(u,y;{\widetilde{\tau}},L)}{p_{u-}({\widetilde{\tau}},L)}-M_{2}(u-;{\widetilde{\tau}},L)\theta_{2}(u,y;{\widetilde{\tau}},L)\Big{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\\{\mu(du,dy)-\nu^{\mathbb{G}}(du,dy)\\}.$
Thus
$\displaystyle M_{t}$
$\displaystyle=M_{1}(t)\mathbb{I}_{t<{\widetilde{\tau}}}+M_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$
$\displaystyle=M_{1}({t\wedge{\widetilde{\tau}}})-M_{1}({\widetilde{\tau}})\mathbb{I}_{t\geq{\widetilde{\tau}}}+M_{2}(t;{\widetilde{\tau}},L)\mathbb{I}_{t\geq{\widetilde{\tau}}}$
$\displaystyle=M_{1}(0)+\int_{0}^{t}\xi(u)^{\prime}dW^{\mathbb{G}}(u)+\int_{0}^{t}\int_{E}\zeta(u,y)\\{\mu(du,dy)-\nu^{\mathbb{G}}(du,dy)\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+M_{2}({\widetilde{\tau}};{\widetilde{\tau}},L)-\int_{0}^{t\wedge{\widetilde{\tau}}}\int_{R}\frac{M_{2}(u;u,l)}{G_{u-}}p_{u}(u,l)\eta(du,dl)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-M_{1}({\widetilde{\tau}})\mathbb{I}_{t\geq{\widetilde{\tau}}}-\int_{0}^{t\wedge{\widetilde{\tau}}}\int_{R}\frac{M_{1}(u-)}{G_{u-}}p_{s}(s,l)\eta(ds,dl),$
where
$\xi(u)=\Big{\\{}\frac{\xi_{1}(u)}{G_{u-}}+M_{1}(u-)\alpha_{1}(u)\Big{\\}}+\Big{\\{}\frac{\xi_{2}(u;{\widetilde{\tau}},L)^{\prime}}{p_{u-}({\widetilde{\tau}},L)}-M_{2}(u-;{\widetilde{\tau}},L)\theta_{1}(u;{\widetilde{\tau}},L)\Big{\\}},$
$\displaystyle\zeta(u,y)$
$\displaystyle=\frac{1}{1-\alpha_{2}(u,y)}\Big{\\{}\frac{\zeta_{1}(u,y)}{G_{u-}}+M_{1}(u-)\alpha_{2}(u,y)\Big{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{1}{1+\theta_{2}(u,y;{\widetilde{\tau}},L)}\Big{\\{}\frac{\zeta_{2}(u,y;{\widetilde{\tau}},L)}{p_{u-}({\widetilde{\tau}},L)}-M_{2}(u-;{\widetilde{\tau}},L)\theta_{2}(u,y;{\widetilde{\tau}},L)\Big{\\}}.$
Since both
$\Big{\\{}M_{1}(t)G_{t}+\displaystyle\int_{0}^{t}\displaystyle\int_{R}M_{2}(u;u,l)p_{u}(u,l)\eta(du,dl)(t\geq
0)\Big{\\}}$ and $G$ have no jump at ${\widetilde{\tau}}$ as the
$(P,\mathbb{G})$ semimartingales, one can see that
$G_{\widetilde{\tau}}=G_{{\widetilde{\tau}}-}$, a.s. and
$M_{1}({\widetilde{\tau}})G_{\widetilde{\tau}}=M_{1}({\widetilde{\tau}}-)G_{{\widetilde{\tau}}-}$,
a.s., thus $M_{1}({\widetilde{\tau}})=M_{1}({\widetilde{\tau}}-)$, a.s., which
completes the proof. ∎
## 6 Conclusion
In this paper, we mainly study a new kind of progressive enlargement
filtration. We deeply characterize the conditional density process and give
the Doob-Meyer’s decomposition of the survival process. We also discuss the
necessary and sufficient conditions for a $\mathbb{G}$-martingale. By Lemma
4.1, we explicitly describe the $\mathbb{G}$-decomposition of a
$(P,\mathbb{F})$-martingale and prove the martingale representation theorems
which extend the traditional results.
## References
* [1] J. Amendinger(1999). _Initial Enlargement of Filtrations and Additional Information in Financial Markets_ , Phd thesis, Technischen Universität Berlin.
* [2] J. Amendinger, D. Becherer, and M. Schweizer(2003). _A monetary value for initial information in portfolio optimization._ Finance Stoch., 7(1):29-46.
* [3] S. Ankirchner, S. Dereich, and P. Imkeller(2005). _The Shannon information of filtrations and the additional logarithmic utility of insiders._ To appear in Annals of Probability.
* [4] T.R. Bielecki, M. Jeanblanc and M. Rutkowski(2009). _Credit Risk Modeling_ , CSFI lecture note series, Osaka University Press.
* [5] G. Callegaro, M. Jeanblanc and B. Zargari(2010). _Carthaginian Enlargement of Filtrations_. Preprint.
* [6] D. Duffie, and C. Huang(1986). _Multiperiod security markets with differential information: Martingales and resolution times._ J. Math. Econom. 15:283-303.
* [7] A. Eyraud-Loisel(2010) _An overview of some financial models using BSDE with enlarged filtrations_. Workshop “Enlargement of Filtrations and Applications to Finance and Insurance”, May 31-June 04, Jena(Germany).
* [8] N. El Karoui, M. Jeanblanc, and Y. Jiao (2009). _What happens after a default, the conditional density approach_. Stochastic Processes and their Applications, 120(7), 1011-1032.
* [9] A. Grorud and M. Pontier(1998). _Insider trading in a continuous time market model_ , IJTAF, 1(3), 331-347.
* [10] J. Jacod(1987). _Grossissement initial, Hypoth $\grave{e}$se (H) et théor$\grave{e}$me de Girsanov _, in Seminaire de Calcul Stochastique, (1982/1983), vol. 1118, Lecture Notes in Mathematics, Springer.
* [11] M. Jeanblanc, M. Yor, and M. Chesney(2009). _Mathematical Methods for Financial Markets_ , Springer-Verlag London.
* [12] M. Jeanblanc and Y. Le Cam(2009). _Progressive enlargement of filtration with initial times._ Stochastic Processes and their Applications, 119(8), 2523-2443.
* [13] M. Jeanblanc and S. Song (2010). _Explicit model of default time with given survival process_. Working paper.
* [14] M. Jeanblanc and S. Song (2012). _Martingale representation property in progressively enlarged filtrations_. Working paper.
* [15] T. Jeulin(1980). _Semi-martingales et Grossissement d’une Filtration._ Lecture Notes in Maths. 833. Berlin, Heidelberg, New York: Springe.
* [16] Y. Jiao and H. Pham (2009). _Optimal investment with counterparty risk: a defaultdensity approach._ To appear in Finance and Stochastics.
* [17] H. Pham(2010). _Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management_. _Preprint_.
* [18] I. Pikovsky, and I. Karatzas(1996). _Anticipative portfolio optimization._ Adv. in Appl. Probab. 28: 1095-1122.
* [19] Shreve, S.E.(2003). _Stochastic Calculus for Finance II: Continuous-Time Models_ , Springer.
* [20] S. Song(1987). _Grossissement defiltration et problemes connexes_ , Ph.D.thesis, Université Paris VI.
* [21] D. Xiong and Z. Ye(2005). _Optimal Utility with Some Additional Information_ , Stochastic Analysis and Applications, 23: 6, 1113-1140.
* [22] M.Yor(1985). _Grossissement de filtrations et absolue continuitiué de noyaux_ , Lecture Notes in Mathematics, 1118, Springer-Verlag.
|
arxiv-papers
| 2013-01-07T07:02:33 |
2024-09-04T02:49:39.964886
|
{
"license": "Public Domain",
"authors": "Kun Tian, Dewen Xiong and Zhongxing Ye",
"submitter": "Kun Tian",
"url": "https://arxiv.org/abs/1301.1119"
}
|
1301.1148
|
# On the fundamental tone of minimal submanifolds with controlled extrinsic
curvature
Vicent Gimeno Department of Mathematics, Universitat Jaume I, Castelló de la
Plana, Spain [email protected]
###### Abstract.
The aim of this paper is to obtain the fundamental tone for minimal
submanifolds of the Euclidean or hyperbolic space under certain restrictions
on the extrinsic curvature. We show some sufficient conditions on the norm of
the second fundamental form that allow us to obtain the same upper and lower
bound for the fundamental tone of minimal submanifolds in a Cartan-Hadamard
ambient manifold. As an intrinsic result, we obtain a sufficient condition on
the volume growth of a Cartan-Hadamard manifold to achieve the lowest bound
for the fundamental tone given by McKean.
###### Key words and phrases:
Fundamental tone and minimal submanifolds and hyperbolic space and Cartan-
Hadamard manifold
###### 1991 Mathematics Subject Classification:
53C40 and 53C42
## 1\. Introduction
Let $(M,g)$ be a Riemannian manifold, the _fundamental tone_
$\lambda^{\ast}(M)$ of $M$ is defined by
(1.1)
$\lambda^{\ast}(M)=\inf\\{\frac{\smallint_{M}|\operatorname{\nabla}f|^{2}}{\smallint_{M}f^{2}};\,f\in{L^{2}_{1,0}(M)\setminus\\{0\\}}\\}$
where $L^{2}_{1,\,0}(M)$ is the completion of $C^{\infty}_{0}(M)$ with respect
to the norm $\|\varphi\|^{2}=\int_{M}\varphi^{2}+\int_{M}|\nabla\varphi|^{2}$.
The fundamental tone is a powerful tool in the challenging area of geometric
analysis, being a classic feature its relation with the curvature of the
manifold. We refer to the book of I. Chavel [5] for a discussion of these and
related concepts.
For non-positively curved manifolds, lower bounds for the fundamental tone are
well-known. For these cases, McKean gave the following bounds for the
fundamental tone of a Cartan-Hadamard manifold.
###### Theorem 1.1 (See [14]).
Let $M^{n}$ be a complete, simply connected manifold with sectional curvature
bounded above by $\kappa<0$. Then
(1.2) $\lambda^{*}(M)\geq\frac{-(n-1)^{2}\kappa}{4}\quad.$
Similary, L.P. Cheung and L.F. Leung showed that minimal submanifolds of the
hyperbolic space retain this lower bound of the ambient manifold.
###### Theorem 1.2 (See [8]).
Let $M^{n}$ be a complete minimal submanifold in the hyperbolic space
$\mathbb{H}^{m}(\kappa)$ of constant sectional curvature $\kappa<0$. Then
(1.3) $\lambda^{*}(M)\geq\frac{-\left(n-1\right)^{2}\kappa}{4}\quad.$
Observe that inequality (1.2) and inequality (1.3) are sharp in the sense that
when $M$ is the hyperbolic space $\mathbb{H}^{n}(\kappa)$ (considered as a
manifold or considered as a totally geodesic submanifold of
$\mathbb{H}^{m}(\kappa)$) the fundamental tone $\lambda^{*}(M)$ satisfies
$\lambda^{*}(M)=\frac{-(n-1)^{2}\kappa}{4}$ (see [5] or [14]).
Moreover, it is known that other minimal submanifolds exist as well as the
hyperbolic space with that lowest fundamental tone. For example, all the
classical minimal catenoids in the hyperbolic space $\mathbb{H}^{3}(-1)$ given
by M. do Carmo and M. Dajczer in [4] achieve equality in (1.3) as shown by A.
Candel.
###### Theorem 1.3 (See [3]).
The fundamental tone of the minimal catenoids (given in [4]) in the hyperbolic
space $\mathbb{H}^{3}(-1)$ is
$\lambda^{*}(M)=\frac{1}{4}\quad.$
The purpose of this paper is to provide geometric conditions under which a
submanifold (or a manifold) attains the equality in the inequality (1.3) (or
inequality (1.2), respectively). Let us emphasize that the spherical catenoids
given in the previous example, as it was analyzed in [17], have finite
$L^{2}$-norm of the second fundamental form (denoted by $A$ throughout this
paper)
(1.4) $\int_{M}|A|^{2}d\mu<\infty\quad.$
This could suggest that the finiteness of the $L^{2}$-norm of the second
fundamental form for minimal surfaces in the hyperbolic space implies the
lowest fundamental tone. This is in fact true, and is the statement of our
first theorem.
###### Theorem A.
Let $M^{2}$ be a minimal surface immersed in the hyperbolic space
$\mathbb{H}^{m}(\kappa)$ of constant sectional curvature $\kappa<0$. Suppose
that $M^{2}$ has finite total extrinsic curvature, i.e,
$\int_{M}|A|^{2}d\mu<\infty$. Then, the fundamental tone satisfies
(1.5) $\lambda^{*}(M)=\frac{-\kappa}{4}\quad.$
Our approach to the problem makes use of our previous results about the
influence of the extrinsic curvature restrictions on the volume growth of the
submanifold (see [11],[10],[12] or §2), and also the relation between the
finite volume growth and the fundamental tone as a new tool (see the Main
Theorem).
For higher dimensional minimal submanifolds of the hyperbolic space, under an
appropriate decay of the norm of the second fundamental form on the extrinsic
distance (see §2 for a precise definition) we obtain
###### Theorem B.
Let $M^{n}$ be a minimal submanifold properly immersed in the hyperbolic space
$\mathbb{H}^{m}(\kappa)$ of constant sectional curvature $\kappa<0$. Suppose
moreover that $n>2$ and the submanifold is of faster than exponential decay of
its extrinsic curvature. Namely, there exists a point $p\in M$ such that
$|A|_{x}\leq\frac{\delta(r_{p}(x))}{e^{2\sqrt{-\kappa}r_{p}(x)}}\quad,$
where $\delta(r)$ is a function such that $\delta(r)\to 0$ when $r\to\infty$
and $r_{p}$ is the extrinsic distance function. Then, the fundamental tone
satisfies
(1.6) $\lambda^{*}(M)=\frac{-(n-1)^{2}\kappa}{4}\quad.$
We also prove for minimal submanifolds of the Euclidean space the following
well known result (see for example [2])
###### Theorem C.
Let $M^{n}$ be a minimal submanifold immersed in the Euclidean space
$\mathbb{R}^{m}$ with finite total scalar curvature, i.e,
$\int_{M}|A|^{n}d\mu<\infty$. Then
(1.7) $\lambda^{*}(M)=0\quad.$
###### Remark a.
As far as we know, the question 1.5 given in [2] about the existence (or not)
of complete minimal submanifolds properly immersed in the $\mathbb{R}^{m}$
with positive fundamental tone is still an open question. In any case, if such
kind of immersion exists, from our Main Theorem the immersion should be of
infinite volume growth, and from Theorem E the submanifold has no an extrinsic
doubling property.
Finally, as an intrinsic version of our Main Theorem one may obtain the
following theorem in the direction of the McKean’s Theorem 1.1
###### Theorem D.
Let $M^{n}$ be a complete, simply connected manifold with sectional curvature
bounded from above by $\kappa<0$. Suppose moreover that there exists a point
$p\in M$ such that
(1.8)
$\sup_{R>0}\frac{\operatorname{Vol}(B_{R}^{M}(p))}{\operatorname{Vol}(B_{R}^{\kappa})}<\infty\quad,$
where $B_{R}^{M}(p)$ is the geodesic ball in $M$ centered at $p$ of radius
$R$, and $B_{R}^{\kappa}$ is the geodesic ball in $\mathbb{H}^{n}(\kappa)$ of
the same radius $R$ .Then
(1.9) $\lambda^{*}(M)=\frac{-(n-1)^{2}\kappa}{4}\quad.$
###### Remark b.
In view of the intrinsic version of the results of [9] for complete, simply
connected, $2$-dimensional manifold $M^{2}$ with Gaussian curvature $K_{M}$
bounded from above by $K_{M}\leq\kappa<0$, the condition (1.8) can be achieved
provided the following
(1.10) $\int_{M}(\kappa-K_{M})d\mu<\infty\quad.$
In particular (in view of [9]), for every $2$-dimensional Cartan-Hadamard
manifold $M$ which is asymptotically locally $\kappa$-hyperbolic of order $2$
(see [18],[9]) and with sectional curvatures bounded from above by
$K_{M}\leq\kappa<0$, the fundamental tone satisfies
$\lambda^{*}(M)=\frac{-\kappa}{4}\quad.$
The proof of the three previous extrinsic theorems (Theorem A, Theorem B and
Theorem C) uses the fact that under the hypothesis of the theorems the
submanifold has finite volume growth
($\displaystyle\lim_{R\to\infty}\mathcal{Q}(R)<\infty$, see §2, and §2.1 for
precise definitions and see also [19] for an alternative definition in terms
of projective volume). Hence, the theorems will be proved using the following
Main Theorem:
###### Main Theorem.
Let $M^{n}$ be a $n-$dimensional minimal submanifold properly immersed in a
simply connected Cartan-Hadamard manifold $N$ of sectional curvature $K_{N}$
bounded from above by $K_{N}\leq\kappa\leq 0$. Suppose that
(1.11) $\lim_{R\to\infty}\mathcal{Q}(R)<\infty\quad.$
Then,
(1.12) $\lambda^{*}(M)=-\frac{(m-1)^{2}\kappa}{4}\quad.$
To obtain the Main Theorem we estimate upper bounds for the fundamental tone.
According to S.T. Yau ([20]) it is important to find upper bounds for the
fundamental tone. In the case of stable minimal submanifolds of the hyperbolic
space, A. Candel gave in 2007 the following upper bound for the fundamental
tone
###### Theorem 1.4 (See [3]).
Let $M$ be a complete simply connected stable minimal surface in the
$3-$dimensional hyperbolic space $\mathbb{H}^{3}(-1)$. Then the fundamental
tone of $M$ satisfies
(1.13) $\frac{1}{4}\leq\lambda^{*}(M)\leq\frac{3}{4}\quad.$
Using only as hypothesis the finiteness of the $L^{2}-$norm of the second
fundamental form, K. Seo has recently generalized the result of Theorem 1.4
without using the hypothesis about the simply connectedness
###### Theorem 1.5 (See [17]).
Let $M^{n}$ be a complete stable minimal hypersurface in
$\mathbb{H}^{n+1}(-1)$ with $\int_{M}|A|^{2}d\mu<\infty$. Then we have
(1.14) $\frac{\left(n-1\right)^{2}}{4}\leq\lambda^{*}(M)\leq n^{2}\quad.$
Another achievement of this paper is that using only the finiteness of the
$L^{2}-$norm of the second fundamental form (Theorem A), or an appropriate
decay of the norm of the second fundamental form, we can also remove the
hypothesis about the stability, and obtain not only an inequality but an
equality on the fundamental tone.
###### Remark c.
Note that the finiteness of the $L^{2}$-norm of the second fundamental form of
a minimal surface in the hyperbolic space does not imply the stability of the
surface. In fact, M. do Carmo and M. Dajczer also proved in [4] that there
exist some unstable spherical catenoids in $\mathbb{H}^{3}(-1)$. And observe,
moreover, that the codimension of the submanifold plays no role in our
theorems, in contrast to what happens in the theorems from A. Candel and K.
Seo where the codimension must be $1$.
The structure of the paper is as follows.
In §2 we recall the definition of the extrinsic distance function, the
extrinsic ball and the volume growth function $\mathcal{Q}$. Showing that
under the hypothesis of Theorems A, B, and C the submanifold has finite volume
growth
(1.15) $\sup_{R>0}\mathcal{Q}(R)<\infty\quad.$
In §3 we will prove the Main Theorem. Finally as a corollary from the proof of
the Main Theorem, supposing that the submanifold has an extrinsic doubling
property we state the Theorem E where is obtained lower and upper bounds for
the fundamental tone of a minimal submanifold properly immersed in a Cartan-
Hadamard ambient manifold.
## 2\. Preliminaries
The proof of Theorems A, B and C is based on upper and lower bounds for the
fundamental tone. The lower bounds are well known (see Theorem 1.2), but to
obtain upper bounds we use the so called _volume growth function_ and its
relation to the behavior of the extrinsic curvature. The volume growth
function is the quotient between the volume of an extrinsic ball of radius $R$
and the geodesic ball of the same radius $R$ in an appropriate real space
form.
Let $\mathbb{K}^{n}(\kappa)$ denote the $n-$dimensional simply connected real
space form of constant sectional curvature $\kappa\leq 0$, recall that the
volume of the geodesic sphere $S^{\kappa}_{R}$ and the geodesic ball
$B_{R}^{\kappa}$ of radius $R$ in $\mathbb{K}^{n}(\kappa)$ are (see [13])
$\operatorname{Vol}(S_{R}^{\kappa})=\omega_{n}\text{S}_{\kappa}(R)^{n-1}\quad\operatorname{Vol}(B_{R}^{\kappa})=\int_{0}^{R}\operatorname{Vol}(S_{s}^{\kappa})ds\quad,$
being $\omega_{n}$ the volume of the geodesic sphere of radius $1$ in
$\mathbb{R}^{n}$, where $\text{S}_{\kappa}$ is the usual function
$\text{S}_{\kappa}(t)=\begin{cases}t\quad\text{if}\quad\kappa=0,\\\
\frac{\sinh(\sqrt{-\kappa}t)}{\sqrt{-\kappa}}\quad\text{if}\quad\kappa<0\quad.\end{cases}$
And the mean curvature pointing inward of the geodesic spheres of radius $R$
is
$\text{Ct}_{\kappa}(t)=\frac{\text{S}_{\kappa}^{\prime}(t)}{\text{S}_{\kappa}(t)}\quad.$
On the other hand, given a submanifold $M$ immersed in a simply connected
Cartan-Hadamard manifold $N$ of sectional curvature $K_{N}$ bounded from a
above by $K_{N}\leq\kappa\leq 0$, the extrinsic ball $M_{p}^{R}$ centered at
$p\in M$ of radius $R$ is the sublevel set of the extrinsic distance function
$r_{p}$, where the extrinsic distance function is
###### Definition 2.1 (Extrinsic distance function).
Let $\varphi:M\to N$ be an immersion from the manifold $M$ to the simply
connected Cartan-Hadamard manifold of sectional curvature $K_{N}$ bounded from
above by $K_{N}\leq\kappa\leq 0$. Given a point $p\in M$, the extrinsic
distance function $r_{p}:M\to\mathbb{R}^{+}$ is defined by
$r_{p}(x)=\text{dist}^{N}(\varphi(p),\varphi(x))\quad,$
where $\text{dist}^{N}$ denotes the usual geodesic distance function in $N$.
Therefore the extrinsic ball $M_{p}^{R}$ centered at $p\in M$ of radius $R$ is
$M_{p}^{R}:=\left\\{x\in M\,:\,r_{p}(x)<R\right\\}\quad.$
With the extrinsic ball and the geodesic ball we can define the volume growth
function
###### Definition 2.2 (Volume growth function).
Let $\varphi:M^{n}\to N$ be an immersion from the $n-$dimensional manifold $M$
to the simply connected Cartan-Hadamard manifold of sectional curvature
$K_{N}$ bounded from above by $K_{N}\leq\kappa\leq 0$. The volume growth
function $\mathcal{Q}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is given by
$\mathcal{Q}(R):=\frac{\operatorname{Vol}(M_{p}^{R})}{\operatorname{Vol}(B_{R}^{\kappa})}\quad.$
Where $\operatorname{Vol}(B_{R}^{\kappa})$ is the geodesic ball of radius $R$
in $\mathbb{K}^{n}(\kappa)$.
### 2.1. Volume growth of minimal submanifolds of controlled extrinsic
curvature
Firstly to prove the Theorems A, B and C we have to study the behavior of the
volume comparison function for minimal submanifolds of a Cartan-Hadamard
manifold. Let us recall the following monotonicity formula
###### Proposition 2.3 (Monotonicity formula, see [15]).
Let $\varphi:M^{n}\to N$ be an immersion from the $n-$dimensional manifold $M$
to the simply connected Cartan-Hadamard manifold of sectional curvature
$K_{N}$ bounded from above by $K_{N}\leq\kappa\leq 0$. Then, the volume growth
function is a non-decreasing function.
Secondly to prove Theorems A, B, and C we need to check that an appropriate
control of their second fundamental form allow us to apply the Main Theorem
(§3). In order to apply the Main Theorem the submanifold should have finite
volume growth
(2.1) $\lim_{R\to\infty}\mathcal{Q}(R)<\infty\quad.$
But we can apply the Main Theorem under the assumptions of the Theorems A, B
and C because of the following three theorems that we recall here:
###### Theorem 2.4 (See [1],[6] and [11]).
Let $M^{n}$ be a minimal submanifold immersed in the Euclidean space
$\mathbb{R}^{m}$. If $M^{n}$ has finite total scalar curvature
$\int_{M}|A|^{n}d\mu<\infty\quad.$
Then
$\sup_{R>0}\mathcal{Q}(R)<\infty\quad.$
###### Theorem 2.5 (See [7], [11] and [16]).
Let $M^{2}$ be a minimal surface immersed in the hyperbolic space
$\mathbb{H}^{m}(\kappa)$ of constant sectional curvature $\kappa<0$ or in the
Euclidean space $\mathbb{R}^{m}$. If $M$ has finite total extrinsic curvature,
namely $\int_{M}|A|^{2}d\mu<\infty$, then $M$ has finite topological type, and
$\sup_{R>0}\mathcal{Q}(R)\leq\frac{1}{4}\int_{M}|A|^{2}d\mu+\chi(M)\quad,$
being $\chi(M)$ the Euler characteristic of $M$.
And
###### Theorem 2.6 (see [12]).
Let $M^{n}$ be a minimal $n-$dimensional submanifold properly immersed in the
hyperbolic space $\mathbb{H}^{m}(\kappa)$ of constant sectional curvature
$\kappa<0$. If $n>2$ and the submanifold is of faster than exponential decay
of its extrinsic curvature, namely, there exists a point $p\in M$ such that
$|A|_{x}\leq\frac{\delta(r_{p}(x))}{e^{2\sqrt{-\kappa}r_{p}(x)}}\quad,$
where $\delta(r)$ is a function such that $\delta(r)\to 0$ when $r\to\infty$.
Then the submanifold has finite topological type, and
$\sup_{R>0}\mathcal{Q}(R)\leq\mathcal{E}(M)\quad,$
being $\mathcal{E}(M)$ the (finite) number of ends of $M$.
## 3\. Proof of the Main Theorem: volume growth behavior and fundamental tone
The way to proof the equality (1.12) in the Main Theorem is to obtain the same
upper and lower bound for the fundamental tone. Hence, first of all, we need
lower bounds for the fundamental tone. But the lower bounds are well known,
and it is straight forward in a similar way to Theorem 1.2 that
$\lambda^{*}(M)\geq\frac{-(n-1)^{2}\kappa}{4}\quad.$
The above lower bound can also be proved using the Cheeger isoperimetric
constant of the submanifold. Taking into account that the Cheeger constant
$h(M)$ of a minimal submanifold $M^{n}$ properly immersed in a Cartan-Hadamard
manifold $N$ of sectional curvatures $K_{N}$ bounded from above by
$K_{N}\leq\kappa\leq 0$ is (see [10])
(3.1) $h(M)\geq(n-1)\sqrt{-\kappa}\quad,$
therefore,
(3.2)
$\lambda^{*}(M)\geq\frac{h(M)^{2}}{4}\geq\frac{-(n-1)^{2}\kappa}{4}\quad.$
To obtain the upper bounds for the fundamental tone, we use the Rayleigh
quotient definition (1.1) with an appropriate testing function. The first step
to obtain the testing function is to define the real function
$\phi:\mathbb{R}\to\mathbb{R}$, given by
(3.3) $\phi(t)=\begin{cases}f(t)\quad\text{ if }t\in[\frac{R}{2},R]\quad,\\\
0\quad\text{ otherwise}\end{cases}$
where the function $f:\mathbb{R}\to\mathbb{R}$ is
(3.4)
$f(s)=\frac{\sin(\frac{2\pi(s-\frac{R}{2})}{R})}{\operatorname{Vol}(S_{s}^{\kappa})^{1/2}}\quad.$
Now, the second step to take to construct the testing function $\Phi$, is to
transplant $\phi$ to $M$ using the extrinsic distance function by the
following definition:
$\Phi:M\to\mathbb{R};\quad\Phi(x)=\phi(r_{p}(x))\quad.$
By the Rayleigh quotient definition and the coarea formula
(3.5) $\displaystyle\lambda^{*}(M)\leq$
$\displaystyle\frac{\int_{M}\langle\nabla\Phi,\nabla\Phi\rangle
d\mu}{\int_{M}\Phi^{2}d\mu}=\frac{\int_{M}(\phi^{\prime})^{2}\langle\nabla
r_{p},\nabla r_{p}\rangle
d\mu}{\int_{M}\phi^{2}d\mu}\leq\frac{\int_{M}(\phi^{\prime})^{2}d\mu}{\int_{M}\phi^{2}d\mu}$
$\displaystyle=$ $\displaystyle\frac{\int_{0}^{R}\left[\int_{\partial
M_{p}^{s}}\frac{(\phi^{\prime})^{2}}{|\nabla
r|}\right]ds}{\int_{0}^{R}\left[\int_{\partial
M_{p}^{s}}\frac{\phi^{2}}{|\nabla
r|}\right]ds}=\frac{\int_{\frac{R}{2}}^{R}(\phi^{\prime}(s))^{2}\left[\int_{\partial
M_{p}^{s}}\frac{1}{|\nabla
r|}\right]ds}{\int_{\frac{R}{2}}^{R}\phi^{2}(s)\left[\int_{\partial
M_{p}^{s}}\frac{1}{|\nabla r|}\right]ds}$ $\displaystyle=$
$\displaystyle\frac{\int_{\frac{R}{2}}^{R}(\phi^{\prime}(s))^{2}\left(\operatorname{Vol}(M_{p}^{s})\right)^{\prime}ds}{\int_{\frac{R}{2}}^{R}\phi^{2}(s)\left(\operatorname{Vol}(M_{p}^{s})\right)^{\prime}ds}\quad.$
Using the following two lemmas
###### Lemma 3.1.
(3.6)
$\mathcal{Q}(s)\operatorname{Vol}(S_{s}^{\kappa})\leq\left(\operatorname{Vol}(M_{p}^{s})\right)^{\prime}=\left(\ln\mathcal{Q}(s)\right)^{\prime}\operatorname{Vol}(B_{s}^{\kappa})\mathcal{Q}(s)+\mathcal{Q}(s)\operatorname{Vol}(S_{s}^{\kappa})\quad.$
###### Proof.
From the definition of $\mathcal{Q}$ and taking into account that
$\mathcal{Q}$ is a non-decreasing function (by proposition 2.3)
(3.7)
$\left(\ln\mathcal{Q}(s)\right)^{\prime}=\frac{\left(\operatorname{Vol}M_{p}^{s}\right)^{\prime}}{\left(\operatorname{Vol}M_{p}^{s}\right)}-\frac{\operatorname{Vol}(S_{s}^{\kappa})}{\operatorname{Vol}(B_{s}^{\kappa})}\geq
0\quad.$
So,
(3.8)
$\mathcal{Q}(s)\operatorname{Vol}(S_{s}^{\kappa})\leq\left(\operatorname{Vol}(M_{p}^{s})\right)^{\prime}=\left(\ln\mathcal{Q}(s)\right)^{\prime}\operatorname{Vol}(B_{s}^{\kappa})\mathcal{Q}(s)+\mathcal{Q}(s)\operatorname{Vol}(S_{s}^{\kappa})\quad.$
$\hfill\Box$ ∎
###### Lemma 3.2.
There exists an upper bound function $\Lambda:\mathbb{R}^{+}\to\mathbb{R}^{+}$
such that:
(3.9)
$\frac{\int_{0}^{R}(\phi^{\prime})^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}{\int_{0}^{R}\phi^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}\leq\Lambda(R)\quad,$
and
(3.10) $\lim_{R\to\infty}\Lambda(R)=\frac{-(m-1)\kappa}{4}$
###### Proof.
(3.11)
$\displaystyle\frac{\int_{0}^{R}(\phi^{\prime}(s))^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}{\int_{0}^{R}\phi(s)^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}=\frac{\int_{\frac{R}{2}}^{R}(f^{\prime}(s))^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}{\int_{\frac{R}{2}}^{R}f(s)^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}\quad.$
But
(3.12) $\displaystyle\left(f^{\prime}(s)\right)^{2}=$
$\displaystyle\frac{\left(-\frac{m-1}{2}\text{Ct}_{\kappa}(s)\sin(\frac{2\pi(s-\frac{R}{2})}{R})+\frac{2\pi}{R}\cos(\frac{2\pi(s-\frac{R}{2})}{R})\right)^{2}}{\operatorname{Vol}(S_{s}^{\kappa})}$
$\displaystyle\leq$
$\displaystyle\frac{\frac{(m-1)^{2}}{4}\text{Ct}_{\kappa}(s)^{2}\sin(\frac{2\pi(s-\frac{R}{2})}{R})^{2}+\frac{4\pi^{2}}{R^{2}}+\frac{2(m-1)\pi}{R}\text{Ct}_{\kappa}(s)}{\operatorname{Vol}(S_{s}^{\kappa})}\quad.$
And, since $\text{Ct}_{\kappa}(t)$ is a non-increasing function and
$\int_{\frac{R}{2}}^{R}\sin(\frac{2\pi(s-\frac{R}{2})}{R})^{2}ds=\frac{R}{4}$
(3.13)
$\displaystyle\frac{\int_{0}^{R}(\phi^{\prime}(s))^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}{\int_{0}^{R}\phi(s)^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}\leq$
$\displaystyle\frac{(m-1)^{2}}{4}\text{Ct}_{\kappa}(R/2)^{2}+\frac{8\pi^{2}}{R^{2}}$
$\displaystyle+\frac{4\pi(m-1)}{R}\text{Ct}_{\kappa}(R/2)\quad.$
Then, letting
(3.14)
$\Lambda(R):=\frac{(m-1)^{2}}{4}\text{Ct}_{\kappa}(R/2)^{2}+\frac{8\pi^{2}}{R^{2}}+\frac{4\pi(m-1)}{R}\text{Ct}_{\kappa}(R/2)\quad,$
and taking into account that
(3.15) $\lim_{R\to\infty}\text{Ct}_{\kappa}(R/2)=\sqrt{-\kappa}\quad,$
the lemma is proven. $\hfill\Box$ ∎
Denoting now,
$F(R):=\left(\frac{(m-1)^{2}}{4}\text{Ct}_{\kappa}(R/2)^{2}+\frac{4\pi^{2}}{R^{2}}+\frac{2(m-1)\pi}{R}\text{Ct}_{\kappa}(R/2)\right)$
and
$\delta(R):=\int_{\frac{R}{2}}^{R}\left(\ln\mathcal{Q}(s)\right)^{\prime}ds$,
applying the lemma 3.1 and the lemma 3.2 to inequality (3.5) we get since
$\frac{\operatorname{Vol}(B_{s}^{\kappa})}{\operatorname{Vol}(S_{s}^{\kappa})}$
is a non-decreasing function
$\displaystyle\lambda^{*}(M)\leq$
$\displaystyle\frac{\mathcal{Q}(R)}{\mathcal{Q}(\frac{R}{2})}\frac{\int_{\frac{R}{2}}^{R}(\phi^{\prime}(s))^{2}\left(\ln\mathcal{Q}(s)\right)^{\prime}\operatorname{Vol}(B_{s}^{\kappa})ds+\int_{\frac{R}{2}}^{R}(\phi^{\prime}(s))^{2}\operatorname{Vol}(S_{s}^{\kappa})ds}{\int_{\frac{R}{2}}^{R}\phi^{2}(s)\operatorname{Vol}(S_{s}^{\kappa})ds}$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{Q}(R)}{\mathcal{Q}(\frac{R}{2})}\left(\frac{4}{R}\int_{\frac{R}{2}}^{R}(\phi^{\prime}(s))^{2}\left(\ln\mathcal{Q}(s)\right)^{\prime}\operatorname{Vol}(B_{s}^{\kappa})ds+\Lambda(R)\right)$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{Q}(R)}{\mathcal{Q}(\frac{R}{2})}\left[\frac{\operatorname{Vol}(B_{R}^{\kappa})}{\operatorname{Vol}(S_{R}^{\kappa})}\frac{4}{R}F(R)\delta(R)+\Lambda(R)\right]$
Letting $R$ tend to infinity and taking into account that
(3.16) $\displaystyle\lim_{R\to\infty}F(R)=$
$\displaystyle-\frac{(m-1)^{2}\kappa}{4}\quad,$
$\displaystyle\lim_{R\to\infty}\delta(R)=$ $\displaystyle 0\quad,$
$\displaystyle\lim_{R\to\infty}\frac{\operatorname{Vol}(B_{R}^{\kappa})}{\operatorname{Vol}(S_{R}^{\kappa})}\frac{4}{R}=$
$\displaystyle\begin{cases}\frac{4}{m-1}\text{ if }\kappa=0,\\\ 0\text{ if
}b<0.\end{cases}$
$\displaystyle\lim_{R\to\infty}\frac{\mathcal{Q}(R)}{\mathcal{Q}(\frac{R}{2})}=$
$\displaystyle 1\quad,$ $\displaystyle\lim_{R\to\infty}\Lambda(R)=$
$\displaystyle-\frac{(m-1)^{2}\kappa}{4}\quad.$
we conclude the proof of the theorem.
###### Remark d.
Observe that the finiteness of the volume growth function
(3.17) $\lim_{R\to\infty}\mathcal{Q}(R)<\infty\quad,$
is only used in the proof of the Main Theorem to achieve
(3.18)
$\lim_{R\to\infty}\frac{\mathcal{Q}(R)}{\mathcal{Q}(\frac{R}{2})}=1\quad.$
Therefore, we could use the slightly weaker assumption on the volume growth
given by the limit (3.18) instead the assumption on the finite volume growth
of the submanifold (limit (3.17)), or use an extrinsic doubling property to
obtain two sides estimates for the fundamental tone
###### Theorem E.
Let $M^{n}$ be a $n-$dimensional minimal submanifold properly immersed in a
simply connected Cartan-Hadamard manifold $N$ of sectional curvature $K_{N}$
bounded from above by $K_{N}\leq\kappa\leq 0$. Suppose that the immersion has
an extrinsic doubling property, namely
(3.19) $\frac{\mathcal{Q}(R)}{\mathcal{Q}(\frac{R}{2})}<C\quad.$
Then,
(3.20)
$-\frac{(m-1)^{2}\kappa}{4}\leq\lambda^{*}(M)\leq-\frac{C(m-1)^{2}\kappa}{4}\quad.$
## References
* [1] Anderson, M.T.: The compactification of a minimal submanifold in Euclidean space by the Gauss map (1984)
* [2] Bessa, G.P., Costa, M.S.: On submanifolds with tamed second fundamental form. Glasg. Math. J. 51(3), 669–680 (2009).
* [3] Candel, A.: Eigenvalue estimates for minimal surfaces in hyperbolic space. Trans. Amer. Math. Soc. 359(8), 3567–3575 (electronic) (2007). DOI 10.1090/S0002-9947-07-04104-9. URL http://dx.doi.org/10.1090/S0002-9947-07-04104-9
* [4] do Carmo, M., Dajczer, M.: Rotation hypersurfaces in spaces of constant curvature. Trans. Amer. Math. Soc. 277(2), 685–709 (1983).
* [5] Chavel, I.: Eigenvalues in Riemannian geometry, _Pure and Applied Mathematics_ , vol. 115. Academic Press Inc., Orlando, FL (1984). Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk
* [6] Chen, Q.: On the volume growth and the topology of complete minimal submanifolds of a Euclidean space. J. Math. Sci. Univ. Tokyo 2(3), 657–669 (1995)
* [7] Chern, S.s., Osserman, R.: Complete minimal surfaces in Euclidean $n$-space. J. Analyse Math. 19, 15–34 (1967)
* [8] Cheung, L.F., Leung, P.F.: Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236(3), 525–530 (2001).
* [9] Esteve, A., Palmer, V.: Chern-Osserman inequality for minimal surfaces in a cartan-hadamard manifold with strictly negative sectional curvatures (2011). Accepted in Arkiv för Matematik.
* [10] Gimeno, V., Palmer, V.: Volume growth of submanifolds and the Cheeger isoperimetric constant (2011) . To appear in Proceedings of the AMS.
* [11] Gimeno, V., Palmer, V.: Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces. Israel Journal of Mathematics 2(194), 539–553 (2013).
* [12] Gimeno, V., Palmer, V.: Volume growth, number of ends, and the topology of a complete submanifold. Journal of Geometric Analysis pp. 1–22 (2012). DOI 10.1007/s12220-012-9376-3. URL http://dx.doi.org/10.1007/s12220-012-9376-3
* [13] Gray, A.: Tubes, _Progress in Mathematics_ , vol. 221, second edn. Birkhäuser Verlag, Basel (2004). With a preface by Vicente Miquel
* [14] McKean, H.: An upper bound to the spectrum of $\Delta$ on a manifold of negative curvature. J. Differ. Geom. 4, 359–366 (1970)
* [15] Palmer, V.: Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications. J. London Math. Soc. (2) 60(2), 607–616 (1999).
* [16] Qing, C., Yi, C.: Chern-Osserman inequality for minimal surfaces in $\mathbb{H}^{n}$. Proc. Amer. Math Soc. 128, 2445–2450 (1999)
* [17] Seo, K.: Stable minimal hypersurfaces in the hyperbolic space. J. Korean Math. Soc. 48(2), 253–266 (2011).
* [18] Shi, Y., Tian, G.: Rigidity of asymptotically hyperbolic manifolds. Comm. Math. Phys. 259(3), 545–559 (2005).
* [19] Tkachev, V.G.: Finiteness of the number of ends of minimal submanifolds in Euclidean space. Manuscripta Math. 82(3-4), 313–330 (1994).
* [20] Yau, S.T.: Review of geometry and analysis. Asian J. Math. 4(1), 235–278 (2000). Kodaira’s issue
|
arxiv-papers
| 2013-01-07T10:33:48 |
2024-09-04T02:49:39.974760
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vicent Gimeno",
"submitter": "Vicent Gimeno",
"url": "https://arxiv.org/abs/1301.1148"
}
|
1301.1299
|
# Automated Variational Inference
in Probabilistic Programming
David Wingate, Theo Weber
###### Abstract
We present a new algorithm for approximate inference in probabilistic
programs, based on a stochastic gradient for variational programs. This method
is efficient without restrictions on the probabilistic program; it is
particularly practical for distributions which are not analytically tractable,
including highly structured distributions that arise in probabilistic
programs. We show how to automatically derive mean-field probabilistic
programs and optimize them, and demonstrate that our perspective improves
inference efficiency over other algorithms.
## 1 Introduction
### 1.1 Automated Variational Inference for Probabilistic Programming
Probabilistic programming languages simplify the development of probabilistic
models by allowing programmers to specify a stochastic process using syntax
that resembles modern programming languages. These languages allow programmers
to freely mix deterministic and stochastic elements, resulting in tremendous
modeling flexibility. The resulting programs define prior distributions:
running the (unconditional) program forward many times results in a
distribution over execution traces, with each trace generating a sample of
data from the prior. The goal of inference in such programs is to reason about
the posterior distribution over execution traces conditioned on a particular
program output. Examples include BLOG [1], PRISM [2], Bayesian Logic Programs
[3], Stochastic Logic Programs [4], Independent Choice Logic [5], IBAL [6],
Probabilistic Scheme [7], $\Lambda_{\circ}$ [8], Church [9], Stochastic Matlab
[10], and HANSEI [11].
It is easy to sample from the prior $p(x)$ defined by a probabilistic program:
simply run the program. But inference in such languages is hard: given a known
value of a subset $y$ of the variables, inference must essentially run the
program ‘backwards’ to sample from $p(x|y)$. Probabilistic programming
environments simplify inference by providing universal inference algorithms;
these are usually sample based (MCMC or Gibbs) [9, 1, 6], due to their
universality and ease of implementation.
Variational inference [12, 13, 14] offers a powerful, deterministic
approximation to exact Bayesian inference in complex distributions. The goal
is to approximate a complex distribution $p$ by a simpler parametric
distribution $q_{\theta}$; inference therefore becomes the task of finding the
$q_{\theta}$ closest to $p$, as measured by KL divergence. If $q_{\theta}$ is
an easy distribution, this optimization can often be done tractably; for
example, the mean-field approximation assumes that $q_{\theta}$ is a product
of marginal distributions, which is easy to work with.
Since variational inference techniques offer a compelling alternative to
sample based methods, it is of interest to automatically derive them,
especially for complex models. Unfortunately, this is intractable for most
programs. Even for models which have closed-form coordinate descent equations,
the derivations are often complex and cannot be done by a computer. However,
in this paper, we show that it _is_ tractable to construct a stochastic,
gradient-based variational inference algorithm automatically by leveraging
compositionality.
## 2 Automated Variational Inference
An unconditional probabilistic program $f$ is defined as a parameterless
function with an arbitrary mix of deterministic and stochastic elements.
Stochastic elements can either belong to a fixed set of known, atomic random
procedures called ERPs (for _elementary random procedures_), or be defined as
function of other stochastic elements.
The syntax and evaluation of a valid program, as well as the definition of the
library of ERPs, define the probabilistic programming language. As the program
$f$ runs, it will encounter a sequence of ERPs $x_{1},\cdots,x_{T}$, and
sample values for each. The set of sampled values is called the _trace_ of the
program. Let $x_{t}$ be the value taken by the $t^{\text{th}}$ ERP. The
probability of a trace is given by the probability of each ERP taking the
particular value observed in the trace:
$p(x)=\prod_{t=1}^{T}p_{t}(x_{t}\>|\>\psi_{t}(h_{t}))$ (1)
where $h_{t}$ is the history $(x_{1},\ldots,x_{t-1})$ of the program up to ERP
$t$, $p_{t}$ is the probability distribution of the $t^{\text{th}}$ ERP, with
history-dependent parameter $\psi_{t}(h_{t})$. Trace-based probabilistic
programs therefore define directed graphical models, but in a more general way
than many classes of models, since the language can allow complex programming
concepts such as flow control, recursion, external libraries, data structures,
etc.
### 2.1 KL Divergence
The goal of variational inference [13, 12, 14] is to approximate the complex
distribution $p(x|y)$ with a simpler distribution $p_{\theta}(x)$. This is
done by adjusting the parameters $\theta$ of $p_{\theta}(x)$ in order to
maximize a reward function $L(\theta)$, typically given by the KL divergence:
$\displaystyle KL(p_{\theta},p(x|y))$ $\displaystyle=$
$\displaystyle\int_{x}p_{\theta}(x)\log\left(\frac{p_{\theta}(x)}{p(x|y)}\right)$
(2) $\displaystyle=$
$\displaystyle\int_{x}p_{\theta}(x)\log\left(\frac{p_{\theta}(x)}{p(y|x)p(x)}\right)+\log
p(y)=-L(\theta)+\log p(y)$ where (3) $\displaystyle L(\theta)$
$\displaystyle\overset{\Delta}{=}$
$\displaystyle\int_{x}p_{\theta}(x)\log\left(\frac{p(y|x)p(x)}{p_{\theta}(x)}\right)$
(4)
Since the KL divergence is nonnegative, the reward function $L(\theta)$ is a
lower bound on the partition function $\log p(y)=\log\int_{x}p(y|x)p(x)$; the
approximation error is therefore minimized by maximizing the lower bound.
Different choices of $p_{\theta}(x)$ result in different kinds of
approximations. The popular mean-field approximation decomposes
$p_{\theta}(x)$ into a product of marginals as
$p_{\theta}(x)=\prod_{t=1}^{T}p_{\theta}(x_{t}|\theta_{t})$, where every
random choice ignores the history $h_{t}$ of the generative process.
### 2.2 Stochastic Gradient Optimization
Minimizing Eq. 2 is typically done by computing derivatives analytically,
setting them equal to zero, solving for a coupled set of nonlinear equations,
and deriving an iterative coordinate descent algorithm. However, this approach
only works for conjugate distributions, and fails for highly structured
distributions (such as those represented by, for example, probabilistic
programs) that are not analytically tractable.
One generic approach to solving this is (stochastic) gradient descent on
$L(\theta)$. We estimate the gradient according to the following computation:
$\displaystyle-\nabla_{\theta}\>L(\theta)$
$\displaystyle=\int_{x}\nabla_{\theta}\left(p_{\theta}(x)\log\left(\frac{p_{\theta}(x)}{p(y|x)p(x)}\right)\right)$
(5)
$\displaystyle=\int_{x}\nabla_{\theta}p_{\theta}(x)\left(\log\left(\frac{p_{\theta}(x)}{p(y|x)p(x)}\right)\right)+\int_{x}p_{\theta}(x)\left(\nabla_{\theta}\log(p_{\theta}(x))\right)$
(6)
$\displaystyle=\int_{x}\nabla_{\theta}p_{\theta}(x)\left(\log\left(\frac{p_{\theta}(x)}{p(y|x)p(x)}\right)\right)$
(7)
$\displaystyle=\int_{x}p_{\theta}(x)\nabla_{\theta}\log(p_{\theta}(x))\left(\log\left(\frac{p_{\theta}(x)}{p(y|x)p(x)}\right)\right)$
(8)
$\displaystyle=\int_{x}p_{\theta}(x)\nabla_{\theta}\log(p_{\theta}(x))\left(\log\left(\frac{p_{\theta}(x)}{p(y|x)p(x)}\right)+K\right)$
(9) $\displaystyle\approx\frac{1}{N}\sum_{x^{j}}\nabla_{\theta}\log
p_{\theta}(x^{j})\left(\log\left(\frac{p_{\theta}(x^{j})}{p(y|x^{j})p(x^{j})}\right)+K\right)$
(10)
with $x^{j}\sim p_{\theta}(x)$, $j=1\ldots N$ and $K$ is an arbitrary
constant. To obtain equations (7-9), we repeatedly use the fact that
$\nabla\log p_{\theta}(x)=\frac{\nabla_{\theta}p_{\theta}(x)}{p_{\theta}(x)}$.
Furthermore, for equations (7) and (9), we also use
$\int_{x}p_{\theta}(x)\nabla\log p_{\theta}(x)=0$, since
$\int_{x}p_{\theta}(x)\nabla\log
p_{\theta}(x)=\int_{x}\nabla_{\theta}p_{\theta}(x)=\nabla_{\theta}\int_{x}p_{\theta}(x)=\nabla_{\theta}1=0$.
The purpose of adding the constant $K$ is that it is possible to approximately
estimate a value of $K$ (optimal baseline), such that the variance of the
Monte-Carlo estimate (10) of expression (9) is minimized. As we will see,
choosing an appropriate value of $K$ will have drastic effects on the quality
of the gradient estimate.
### 2.3 Compositional Variational Inference
Consider a distribution $p(x)$ induced by an arbitrary, unconditional
probabilistic program. Our goal is to estimate marginals of the conditional
distribution $p(x|y)$, which we will call the _target program_. We introduce a
variational distribution $p_{\theta}(x)$, which is defined through another
probabilistic program, called the _variational program_. This distribution is
unconditional, so sampling from it is as easy as running it.
We derive the variational program from the target program. An easy way to do
this is to use a _partial mean-field approximation_ : the target probabilistic
program is run forward, and each time an ERP $x_{t}$ is encountered, a
variational parameter is used in place of whatever parameters would ordinarily
be passed to the ERP. That is, instead of sampling $x_{t}$ from
$p_{t}(x_{t}\>|\>\psi_{t}(h_{t}))$ as in Eq. 1, we instead sample from
$p_{t}(x_{t}\>|\>\theta_{t}(h_{t}))$, where $\theta_{t}(h_{t})$ is an
auxiliary variational parameter (and the true parameter $\psi_{t}(h_{t})$ is
ignored).
Fig. 1 illustrates this with pseudocode for a probabilistic program and its
variational equivalent: upon encountering the normal ERP on line 4, instead of
using parameter mu, the variational parameter $\theta_{3}$ is used instead
(normal is a Gaussian ERP which takes an optional argument for the mean, and
rand(a,b) is uniform over the set $[a,b]$, with $[0,1]$ as the default
argument). Note that a dependency between $X$ and $M$ exists through the
control logic, but not the parameterization. Thus, in general, stochastic
dependencies due to the parameters of a variable depending on the outcome of
another variable disappear, but dependencies due to control logic remain
(hence the term _partial mean-field approximation_).
Probabilistic program A
1: M = normal();
2: if M$>$1
3: mu = complex_deterministic_func( M );
4: X = normal( mu );
5: else
6: X = rand();
7: end;
Mean-Field variational program A
1: M = normal( $\theta_{1}$ );
2: if M$>$1
3: mu = complex_deterministic_func( M );
4: X = normal( $\theta_{3}$ );
5: else
6: X = rand($\theta_{4},\theta_{5}$);
7: end;
Figure 1: A probabilistic program and corresponding variational program
This idea can be extended to automatically compute the stochastic gradient of
the variational distribution: we run the forward target program normally, and
whenever a call to an ERP $x_{t}$ is made, we:
* $\bullet$
Sample $x_{t}$ according to $p_{\theta_{t}}(x_{t})$ (if this is the first time
the ERP is encountered, initialize $\theta_{t}$ to an arbitrary value, for
instance that given by $\psi_{t}(h_{t})$).
* $\bullet$
Compute the log-likelihood $\log p_{\theta_{t}}(x_{t})$ of $x_{t}$ according
to the mean-field distribution.
* $\bullet$
Compute the log-likelihood $\log p(x_{t}|h_{t})$ of $x_{t}$ according to the
target program.
* $\bullet$
Compute the reward $R_{t}=\log p(x_{t}|h_{t})-\log p_{\theta}(x_{t})$
* $\bullet$
Compute the local gradient $\psi_{t}=\nabla_{\theta_{t}}\log
p_{\theta_{t}}(x_{t})$.
When the program terminates, we simply compute $\log p(y|x)$, then compute the
gain $R=\sum R_{t}+\log p(y|x)+K$. The gradient estimate for the
$t^{\text{th}}$ ERP is given by $R\psi_{t}$, and can be averaged over many
sample traces $x$ for a more accurate estimate.
Thus, the only requirement on the probabilistic program is to be able to
compute the log likelihood of an ERP value, as well as its gradient with
respect to its parameters. Let us highlight that being able to compute the
gradient of the log-likelihood with respect to natural parameters is the only
additional requirement compared to an MCMC sampler.
Note that everything above holds: 1) regardless of conjugacy of distributions
in the stochastic program; 2) regardless of the control logic of the
stochastic program; and 3) regardless of the actual parametrization of
$p(x_{t};\theta_{t})$. In particular, we again emphasize that we do not need
the gradients of deterministic structures (for example, the function
complex_deterministic_func in Fig. 1).
### 2.4 Extensions
Here, we discuss three extensions of our core ideas.
Learning inference transfer. Assume we wish to run variational inference for
$N$ distinct datasets $y^{1},\ldots,y^{N}$. Ideally, one should solve a
distinct inference problem for each, yielding distinct
$\theta^{1},\cdots,\theta^{N}$. Unfortunately, finding
$\theta^{1},\cdots,\theta^{N}$ does not help to find $\theta^{N+1}$ for a new
dataset $y_{N+1}$. But perhaps our approach can be used to learn ‘approximate
samplers’: instead of $\theta$ depending on $y$ implicitly via the
optimization algorithm, suppose instead that $\theta_{t}$ depends on $y$
through some fixed functional form. For instance, we can assume
$\theta_{t}(y)=\sum_{j}\alpha_{i,j}f_{j}(y)$, where $f_{j}$ is a known
function, then find parameters $\theta_{t,j}$ such that for most observations
$y$, the variational distribution $p_{\theta(y)}(x)$ is a decent approximate
sampler to $p(x|y)$. Gradient estimates of $\alpha$ can be derived similarly
to Eq. 2 for arbitrary probabilistic programs.
Structured mean-field approximations. It is sometimes the case that a vanilla
mean-field distribution is a poor approximation of the posterior, in which
case more structured approximations should be used [15, 16, 17, 18]. Deriving
the variational update for structured mean-field is harder than vanilla mean-
field; however, from a probabilistic program point of view, a structured mean-
field approximation is simply a more complex (but still unconditional)
variational program that could be derived via program analysis (or perhaps
online via RL state-space estimation), with gradients computed as in the mean-
field case.
Online probabilistic programming One advantage of stochastic gradients in
probabilistic programs is simple parallelizability. This can also be done in
an online fashion, in a similar fashion to recent work for stochastic
variational inference by Blei et al. [19, 20, 21]. Suppose that the set of
variables and observations can be separated into a main set $X$, and a large
number $N$ of independent sets of latent variables $X_{i}$ and observations
$Y_{i}$ (where the $(X_{i},Y_{i})$ are only allowed to depend on $X$). For
instance, for LDA, $X$ represents the topic distributions, while the $X_{i}$
represent the document distribution over topics and $Y_{i}$ topic $i$. Recall
the gradient for the variational parameters of $X$ is given by $K\psi_{X}$,
with $K=R_{X}+\sum_{i}(R_{i}+\log P(Y_{i}|X_{i},X)$, where $R_{X}$ is the sum
of rewards for all ERPs in $X$, and $R_{i}$ is the sum of rewards for all ERPs
in $X_{i}$. $K$ can be rewritten as $X+NE[R_{v}+\log P(Y_{v}|X_{v},X)$, where
$v$ is a random integer in $\\{1,\ldots,N\\}$. The expectation can be
approximately computed in an online fashion, allowing the update of the
estimate of $X$ without manipulating the entire data set $Y$.
## 3 Experiments: LDA and QMR
(a) Results on QMR (b) Results on LDA (c) Steepest descent vs. conjugate
gradients
Figure 2: Numerical simulations of AVI
We tested automated variational inference on two common inference benchmarks:
the QMR-DT network (a binary bipartite graphical model with noisy-or directed
links) and LDA (a popular topic model). We compared three algorithms:
* $\bullet$
The first is vanilla stochastic gradient descent on Eq. 2, with the gradients
given by Eq. 5.
* $\bullet$
The Episodic Natural Actor Critic algorithm, a version of the algorithm
connecting variational inference to reinforcement learning – details are
reserved for a longer version of this paper. An important feature of ENAC is
optimizing over the baseline constant $K$.
* $\bullet$
A second-order gradient descent (SOGD) algorithm which estimates the Fisher
information matrix $F_{\theta}$ in the same way as the ENAC algorithm, and
uses it as curvature information.
For each algorithm, we set $M=10$ (i.e., far fewer roll-outs than parameters).
All three algorithms were given the same “budget” of samples; they used them
in different ways. All three algorithms estimated a gradient
$\hat{g}(\theta)$; these were used in a steepest descent optimizer:
$\theta=\theta+\alpha\hat{g}(\theta)$ with stepsize $\alpha$. All three
algorithms used the same stepsize; in addition, the gradients $g$ were scaled
to have unit norm. The experiment thus directly compares the quality of the
direction of the gradient estimate.
Fig. 2 shows the results. The ENAC algorithm shows faster convergence and
lower variance than steepest descent, while SOGD fares poorly (and even
diverges in the case of LDA). Fig. 2 also shows that the gradients from ENAC
can be used either with steepest descent or a conjugate gradient optimizer;
conjugate gradients converge faster.
Because both SOGD and ENAC estimate $F_{\theta}$ in the same way, we conclude
that the performance advantage of ENAC is _not_ due solely to its use of
second-order information: the additional step of estimating the baseline
improves performance significantly.
Once converged, the estimated variational program allows very fast approximate
sampling from the posterior, at a fraction of the cost of a sample obtained
using MCMC sampling. Samples from the variational program can also be used as
warm starts for MCMC sampling.
## 4 Related Work
Natural conjugate gradients for variational inference are investigated in
[25], but the analysis is mostly devoted to the case where the variational
approximation is Gaussian, and the resulting gradient equation involves an
integral which is not necessarily tractable.
The use of variational inference in probabilistic programming is explored in
[29]. The authors similarly note that it is easy to sample from the
variational program. However, they only use this observation to estimate the
free energy of the variational program, but they do not estimate the gradient
of that free energy. While they do highlight the need for optimizing the
parameters of the variational program, they do not offer a general algorithm
for doing so, instead suggesting rejection sampling or importance sampling.
Use of stochastic approximations for variational inference is also used by
Carbonetto [30]. Their approach is very different from ours: they use
Sequential Monte Carlo to refine gradient estimates, and require that the
family of variational distributions contains the target distribution. While
their approach is fairly general, it cannot be automatically generated for
arbitrarily complex probabilistic models.
Finally, stochastic gradient methods are also used in online variational
inference algorithms, in particular in the work of Blei et al. in stochastic
variational inference (for instance, online LDA [19], online HDP [20], and
more generally under conjugacy assumptions [21]), as a way to refine estimates
of latent variable distributions without processing all the observations.
However, this approach requires a manual derivation of the variational
equation for coordinate descent, which is only possible under conjugacy
assumptions which will in general not hold for arbitrary probabilistic
programs.
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|
arxiv-papers
| 2013-01-07T18:48:02 |
2024-09-04T02:49:39.984037
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David Wingate, Theophane Weber",
"submitter": "Th\\'eophane Weber",
"url": "https://arxiv.org/abs/1301.1299"
}
|
1301.1480
|
# The spherical perfect fluid collapse with pressure in the cosmological
background
Rahim Moradi Department of Physics, Shahid Beheshti University, G.C., Evin,
Tehran 19839, Iran Ra˙[email protected] Javad T. Firouzjaee School of Physics
and School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran, Iran [email protected] Reza Mansouri Department of Physics,
Sharif University of Technology, Tehran, Iran and
School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran, Iran [email protected]
###### Abstract
We have constructed a spherically symmetric structure model in a cosmological
background filled with perfect fluid with non-vanishing pressure and studied
its quasi-local characteristics. This is done by using the Lemaître solution
of the Einstein equations and suggesting an algorithm to integrate it
numerically. The result shows intriguing effects of the pressure inside the
structure. The evolution of the central black hole within the FRW universe,
its decoupling from the expanding parts of the model, the structure of its
space-like apparent horizon, the limiting case of the dynamical horizon
tending to a slowly evolving horizon, and the decreasing mass in-fall to the
black hole is also studied. The quasi-local features of this cosmological
black hole may not be inferred from the weak field approximation although the
gravity outside the structure is very weak.
###### pacs:
95.30.Sf,98.80.-k, 98.62.Js, 98.65.-r
## I introduction
The term cosmological black hole is used to describe a collapsing structure
within an otherwise expanding universe. Since the early beginning of the
discovery of the expansion of the universe people have been looking for models
describing an overdense region in a cosmological background (McVittie , see
also cosmological black hole ). The initial expansion of such an overdense
region will finally decouple from the background and collapses to a dynamical
black hole. The resulting structure and its central black hole, despite the
very weak gravitational field outside it, differ from the familiar
Schwarzschild one in being neither static no asymptotically flat man .
Therefore, such cosmological structures, if based on exact solutions of
general relativity and not produced by a cut-and-paste technology, are very
interesting laboratories to study not only general relativistic structures,
their quasi-local features such as mass and horizons man , but the validity of
the weak field approximation in the presence of the very weak gravitational
fields relevant to non-local concepts within general relativity mojahed .
After all, the universe is evolving and asymptotically not Minkowskian.
Therefore, one needs to have a dynamical model for a black hole to be compared
with the familiar results in the literature on black holes within a static and
asymptotically flat space-time waldbook where global concepts such as event
horizon are not defined. The need for a local definition of black holes and
their horizons has led us to concepts such as Hayward’s trapping horizon
Hayward94 , isolated horizon ashtekar99 , Ashtekar and Krishnan’s dynamical
horizon (DH) ashtekar02 , and Booth and Fairhurst’s slowly evolving horizon
booth04 .
Now, a widely used metric to describe the gravitational collapse of a
spherically symmetric dust cloud is the so-called Tolman-Bondi-Lemaître(LTB)
metric LTB . It was pointed out in man ; mangeneral that the model admits
cosmological black holes. These models may be extended to a perfect fluid with
a non zero pressure, the so-called Lemaître models hellaby . Our interest is
now using these cosmological solutions to construct dynamical black holes
within a FRW expanding universe, and study their characteristics. To do this,
we have to avoid any cut-and-paste method of finding the solution. Section II
is an introduction to these inhomogeneous prefect fluid cosmological models
and how numerically integrate the field equations leading to a dynamical
structure within an otherwise FRW universe. In section III the result of the
numerical integration is reported expressing the main characteristics of our
model assuming different pressure profiles. We will then discuss the result in
section IV. Throughout the paper we assume $8\pi G=c=1$.
## II General spherically symmetric solution
Consider a general inhomogeneous spherically symmetric spacetime hellaby
filled with a perfect fluid and a metric expressed in the comoving
coordinates, $x^{\mu}=(t,\,r,\,\theta,\,\phi)$:
$ds^{2}=-e^{2\sigma}\,dt^{2}+e^{2\lambda}\,dr^{2}+R^{2}\,d\Omega^{2}\;,$ (1)
where $\sigma=\sigma(t,r)$, $\lambda=\lambda(t,r)$ are functions to be
determined, $R=R(t,r)$ is the physical radius, and
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta\,d\phi^{2}$ is the metric of the unit
2-sphere. The energy momentum tensor of the perfect fluid is given by
$T^{\mu\nu}=(\rho+p)\,u^{\mu}\,u^{\nu}+g^{\mu\nu}p\;,$ (2)
where $\rho=\rho(t,r)$ is the mass-energy density, $p=p(t,r)$ is the pressure,
and $u^{\mu}=(e^{-\sigma},0,0,0)$ is the perfect fluid four-velocity.
### II.1 Field Equations
In addition to the Einstein field equations,
$G^{\mu\nu}=\kappa\,T^{\mu\nu}-g^{\mu\nu}\Lambda$, we will use the
conservation equations in the form
$\frac{2e^{2\sigma}}{(\rho+p)}\,\nabla_{\mu}T^{t\mu}=\dot{\lambda}+\frac{2\dot{\rho}}{(\rho+p)}+\frac{4\dot{R}}{R}\;=0$
(3)
$\frac{e^{\lambda}}{(\rho+p)}\,\nabla_{\mu}T^{r\mu}=\sigma^{\prime}+\frac{p^{\prime}}{p+\rho}=0,$
(4)
where the dot means the derivative with respect to $t$, and the prime means
the derivative with respect to $r$. The Einstein equations lead finally to
following equations
$\frac{\partial}{\partial r}\left[R+R\dot{R}^{2}e^{-2\sigma}-RR^{\prime
2}e^{-\lambda}-\frac{1}{3}\Lambda R^{3}\right]=\kappa\rho R^{2}R^{\prime},\;$
(5)
and
$\frac{\partial}{\partial t}\left[R+R\dot{R}^{2}e^{-2\sigma}-RR^{\prime
2}e^{-\lambda}-\frac{1}{3}\Lambda R^{3}\right]=-\kappa pR^{2}\dot{R}.\;$ (6)
The term in the brackets is related to the Misner-Sharp mass, $M,$ defined by
$\frac{2M}{R}=\dot{R}^{2}e^{-2\sigma}-R^{\prime
2}e^{-\lambda}+1-\frac{1}{3}\Lambda R^{2}\;.$ (7)
Eqs (5) and (6) may now be written as
$\kappa\rho=\frac{2M^{\prime}}{R^{2}R^{\prime}}~{},\kappa
p=-\frac{2\dot{M}}{R^{2}\dot{R}}~{}.$ (8)
We may write Eq (7) in the form of an evolution equation of the model:
$\dot{R}=\pm e^{\sigma}\sqrt{\frac{2M}{R}+f+\frac{\Lambda R^{2}}{3}}~{},\\\ $
(9)
where
$f(t,r)=R^{\prime 2}e^{-\lambda}-1~{}$ (10)
is the curvature term, or twice the total energy of test particle at $r$
(analogous to $f(r)$ in the LTB model). Note that $R(t,r)$ can not be directly
obtained from this equation because of the unknown functions $\lambda$,
$\sigma$, and $M$.
The metric functions $g_{tt}$ and $g_{rr}$ may be obtained by integrating (3)
and (4):
$\sigma=\ c(t)-\int_{r_{0}}^{r}\frac{p^{\prime}\,\
dr}{(\rho+p)}|_{t=const}=\sigma_{0}-\int_{\rho_{0}}^{\rho}\frac{(\frac{\partial
p}{\partial\rho})}{(\rho+p(\rho))}\,d\rho~{}|_{t=const},$ (11)
and
$\lambda=\lambda_{0}(r)-2\int_{\rho_{0}}^{\rho}\frac{d\rho}{(\rho+p(\rho))}-4\ln\left(\frac{R}{R_{0}}\right)~{}|_{t=const},$
(12)
where $c(t)$ and $\lambda_{0}(r)$ are arbitrary functions of integration (see
hellaby for more details). In the case of $c(t)$ it is easily seen that
requiring our coordinates to lead to the LTB synchronous ones for $p=0$ leads
to $c(t)=0$. We notice also that according to (10) and the LTB coordinate
conditions, the choice of $\lambda_{0}(r)$ is equivalent to the choice of
$f(t_{0},r)=f_{0}(r)$. One may prefer to choose $f_{0}(r)$ and then calculate
$\lambda_{0}(r)$ from $e^{\lambda_{0}}=R_{0}^{\prime 2}/(1+f_{0})$.
We have therefore 5 unknowns $p,\rho,\sigma,\lambda$, and $R$, four dynamical
equations $\dot{\rho}$, $\dot{M}$, $\dot{\lambda}$, and $\dot{R},$ in addition
to an equation of state $p=p(\rho)$, and the definition of the mass $M$ (7).
This defines a numerical algorithm to find solutions for the dynamics of our
spherical structure after assuming the initial conditions.
### II.2 Construction of the Lemaître Model
To generate a Lemaître model in the general case we need a numerical
procedure. We first specify the arbitrary initial functions,
$R_{0}(r)=R(t_{0},r)$, $\rho_{0}(r)=\rho(t_{0},r)$,
$\lambda_{0}(r)=\lambda(t_{0},r)$ , $\sigma_{0}(r)=\sigma(t_{0},r)$, and the
equation of state $p(\rho)$. We then integrate the dynamical equations for
constant $t$ or $r$ in the following order:
1. 1.
Choose an initial time $t_{0}$, and specify $R(t_{0},r)=R_{0}(r)$;
$R_{0}^{\prime}$ is then also known form the derivative of $R_{0}(r)$ with
respect to $r$ at $t=t_{0}$;
2. 2.
Specify $M(t_{0},r)=M_{0}(r)$ at $t=t_{0}$;
3. 3.
Once $M_{0}(r)$ and $R_{0}(r)$ are specified, $\rho_{0}(r)$ may be determined
from (8);
4. 4.
Select an equation of state, $p=p(\rho)$;
5. 5.
Choose $\lambda(t_{0},r)$, or choose first $f(t_{0},r)=f_{0}(r)$ and then
calculate $\lambda_{0}(r)$ from $e^{\lambda_{0}}=R_{0}^{\prime 2}/(1+f_{0})$;
6. 6.
$\sigma(t_{0},r)$ can then be obtained by integrating (11) along $t=t_{0}$.
We have now specified how to determine all the needed initial functions at the
time $t_{0}$. Therefore, their $r-$derivatives are also known. The time
evolution of the metric functions along the worldlines of constant $r$ may
then be calculated. This is done in the following way:
1. 1.
From equation (9) we obtain $\dot{R}$
2. 2.
Eq (8) then gives us $\dot{M}$:
$\dot{M}=\frac{-\kappa p\,\dot{R}R^{2}}{2}~{};$ (13)
3. 3.
Combining Eq (3) and $G_{01}$ allows us to eliminate $\sigma^{\prime}$. Then
substituting $\dot{\lambda}$ from (4), we arrive at $\dot{\rho}$:
$\dot{\rho}=-p^{\prime}\,\frac{\dot{R}}{R^{\prime}}-(\rho+p)\left[\frac{\dot{R}^{\prime}}{R^{\prime}}+\frac{2\dot{R}}{R}\right]~{};$
(14)
4. 4.
From the equation of state $p=p(\rho)$ we obtain $\dot{p}$
$\dot{p}=\frac{dp}{d\rho}\,\dot{\rho}~{};$ (15)
5. 5.
Eq (4) may be combined with (14) to give
$\dot{\lambda}=\frac{2}{R^{\prime}}\left(\frac{p^{\prime}\dot{R}}{(\rho+p)}+\dot{R}^{\prime}\right)~{};$
(16)
6. 6.
Using the initial values for $t=t_{0}$, we are then in a position to solve the
above 5 differential equations numerically to obtain $R(t,r)$, $M(t,r)$,
$\rho(t,r)$, $p(t,r)$ and $\lambda(t,r)$ for every $t$ and $r$. Note that in
each step the spatial derivatives $\dot{R}^{\prime}$, $p^{\prime}$ and
$M^{\prime}$ needs to be determined;
7. 7.
Finally, $\sigma(t,r)$ is obtained from Eq (11) by integrating along constant
$t$.
Notice that we have chosen 4 initial functions, $R_{0}(r)$, $\rho_{0}(r)$,
$\lambda_{0}(r)$ and $\sigma_{0}(t)$, as well as the equation of state
$p(\rho)$.
## III equation of state and the results
We are now ready to specify the equation of state and integrate the model to
see its characteristics. To have a comparative discussion of the results we
consider two types of equation of state: a perfect fluid with a constant state
function, $p=w\rho$, and a more general case with the equation of state
$p=ws(r)\rho$ matching our needs for a structure with pressure inside the
structure and a pressure-less matter dominated universe far from the
structure. We may then choose the function $s(r)$ in a way that the pressure
becomes zero at infinity, i.e. at $r>>r_{0}$. A suitable choice is
$s(r)=e^{-\frac{r}{r_{0}}}$ where the order of magnitiude $r_{0}$ is the
distance of void form the center (boundary of expanding and collapsing phase).
This is a more realistic model to describe a black hole collapse within the
FRW universe and to see the effect of the inside pressure while the universe
outside is matter dominated with no pressure.
The model we envisage starts from a small inhomogeneity within a FRW universe.
The density profile should be such that the metric outside the structure tends
to FRW independent of the time while the central overdensity region undergoes
a collapse after some initial expansion. At the initial conditions, where the
density contrast of the overdensity region is still too small, we may assume
that the metric is almost FRW or LTB; the density contrast and the pressure
does not play a significant role. The dynamics of Lemaître universe will give
us anyhow the expected structure at late times. To choose the initial
conditions at the time $t_{0}$, we will therefore use a LTB solution with a
negative curvature function. We have in fact tried both examples of LTB or FRW
initial data and received no significant difference between the final Lemaître
solutions.
Now, let us choose the the two initial functions $f(r)=f(t_{0},r)$ and
$M(r)=M(t_{0},r)$ in the following way to achieve an asymptotically FRW final
solution:
$f(r)=-\frac{1}{b}re^{-r},$ (17) $M(r)=\frac{1}{a}r^{3/2}(1+r^{3/2}).$ (18)
Far from the central overdensity region we have
$lim_{r\rightarrow\infty}f(r)=0,$ (19)
$lim_{r\rightarrow\infty}M(r)=\frac{r^{3}}{a},$ (20)
showing the asymptotically FRW behavior of the initial conditions. The
corresponding LTB solution of Einstein equations now gives us $R(r)=R_{0}(r)$
at the initial time $t_{0}$. Assuming an equation of state is now enough to
numerically calculate the necessary dynamical functions of the model.
Specifically, by looking at $\dot{R}(t,r)$ and $\rho(t,r)$ we may extract
informations of how the central region starts collapsing after the initial
expansion and how a black hole with distinct apparent and event horizons
develops while the outer region expands as a familiar FRW universe. We may
also find out the difference to the case of the pressure-less model. It will
also show if and how the very weak gravity outside the collapsed structure
affects the dynamic of the central structure in comparison to the familiar
Schwarzschild model. The results of the numerical calculation for both
equation of states are given in the following sections.
### III.1 The density behavior
The density profiles for both equation of states as a function or $t$ and $r$
are given in Figs.(1) and (2). A comparison of these figures shows the effect
of the pressure on the development of the central black hole. Obviously in
case of non-vanishing pressure outside the structure the collapse is more
highlighted with a more steep density profile. The over-density region in the
collapsing phase is always separated from the expanding under-density region
through a void not expressible in these figures. We will consider the deepest
place of the void as the boundary of the structure. This boundary is always
near by the boundary between the contracting and the expanding region of the
model structure.
Figure 1: Density evolution of our cosmological black hole for the perfect
fluid with the equation of state $p=w\rho$. Figure 2: Density evolution of
our cosmological black hole for the perfect fluid with the equation of state
$p=w\rho s(r)$. Note the less significant central density and the more flat
density profile near to the center of the structure.
### III.2 The pressure effect
Figs.(3(a)) and (4(a)) show the behavior of the collapsing and the expanding
regions for the equation of state $p=w\rho$ by depicting the corresponding
Lemaître Hubble parameter $\dot{R}/R$ versus the physical radius $R$. Figs.
(3(b)) and (4(b)) show similar data for the equation of state $p=ws(r)\rho$.
Note that the function $s(r)$, as defined to get matter dominated FRW universe
at far distances, has no significant effect, and the qualitative behavior of
the dynamics of the physical radius is independent of it. Therefore, as far as
we are interested in the qualitative features of the model, we will just use
the simple equation of state with $s(r)=1$.
The place of separation between the expanding and collapsing region defined by
$\dot{R}>0$ and $\dot{R}<0$ is almost coincident with the place of the void
where we have defined as the boundary of the structure. Now, from the figures
we realize that the effect of the pressure in different regions of the model
and its comparison to the homogeneous FRW model is an intriguing one. As we
know already from the Friedman equations in FRW models, the pressure adds up
to the density and has an attractive effect slowing down the expansion leading
to a more negative acceleration ($\frac{\ddot{a}}{a}=-\frac{1}{6}(\rho+3p)$).
This is evident from the figures at distances far from the center where our
model tends to an FRW one. Whereas within the structure where we have a
contracting overdensity region the behavior is counter-intuitive. Except for
the case of vanishing pressure, in all the other cases the pressure begins
somewhere to act classically like a repulsive force opposing the collapse of
the structure. To see this more clearly, we have also depicted the
acceleration in Fig.(5). As we approach distances near to the center, the
negative acceleration in the FRW limit and even inside the void, gradually
increases to positive values, meaning that somewhere within the structure the
contraction of the structure slows down due to the pressure like a classical
fluid. Therefore, the pressure effect begins somewhere within the structure to
act like a repulsive force in contrast to the outer regions where its
attractive nature dominates. Note that the central black hole and its horizon
has a much smaller radius than the region of the repulsive pressure effect we
are discussing.
(a) The overall scheme: $\dot{R}>0$ and $\dot{R}<0$ show the expanding and
collapsing regions, respectively. (b) A magnified scheme related to the
regions near to the center
Figure 3: The behavior of the Hubble parameter $\dot{R}/R$ in the case of
$p=w\rho$. Evidently the pressure slows down the collapse velocity near the
center of the structure
(a) The overall scheme: $\dot{R}>0$ and $\dot{R}<0$ show the expanding and
collapsing regions, respectively. (b) A magnified scheme related to the
regions near to the center.
Figure 4: The behavior of the Hubble parameter $\dot{R}/R$ in the case of
$p=w\rho s(r)$. The features are qualitatively as in Fig (3).
(a) The overall scheme: $\dot{R}>0$ and $\dot{R}<0$ show the expanding and
collapsing regions, respectively. (b) The overall scheme of the acceleration.
Figure 5: The behavior of the Hubble parameter $\dot{R}/R$ and acceleration
$\ddot{R}/R$ in the case of $p=w\rho$.
### III.3 The Apparent and Event Horizon
The boundary of a dynamical black hole, where the area law and the black hole
temperature are defined, is a non-trivial concept (see for example man and
manradiation ). Our model is again a good example to see the behavior of both
apparent and event horizon of a dynamical structure within an expanding
universe. It is easily seen that the apparent horizon for our cosmological
black hole is located at $R=2M$ mangeneral .
This apparent horizon is calculated in $t,r$ coordinates numerically. It is
always space-like tending to be light-like at late times. This can best be
seen by comparing the slope of the apparent horizon relative to the light cone
at every coordinate point of it. This is in contrast to the Schwarzschild
black hole horizon where it is always light-like. At the late times, however,
we expect the apparent horizon to become approximately light-like and
approaching the event horizon. This is reflected in the Figs.(6) and (7). It
is evident that $\frac{dt}{dr}|_{AH}<\frac{dt}{dr}|_{null}$ at all times on
the apparent horizon, the difference tending to zero at late times. Therefore,
the apparent horizon is always a space-like _dynamical horizon_ leading to a
_slowly varying horizon_ at late times ashtekar02 ; man . Note that the
qualitative result is independent of the equation of state.
Figure 6: The $p=w\rho$ case: $\frac{dt}{dr}|_{AH}<\frac{dt}{dr}|_{null}$ on
the apparent horizon. Therefore, the apparent horizon is always a space-like
dynamical horizon leading to a slowly varying horizon at late times. Figure 7:
The $p=ws(r)\rho$ case: $\frac{dt}{dr}|_{AH}<\frac{dt}{dr}|_{null}$ on the
apparent horizon. Qualitatively, there is no difference to the Fig…..
We now show how the dynamical horizon of our cosmological black hole becomes a
_slowly evolving horizon_ at late times. Let’s first define the evolution
parameter $c$ such that the tangent vector to the dynamical horizon, $V$, is
given by
$V^{\mu}=\ell^{\mu}-cn^{\mu},$ (21)
where the two vectors $\ell^{a}$ and $n^{a}$ are normal null vectors on a
space-like two surface $S$ in $(t,r)$ plane (see ashtekar02 ). We expect $c$
to go to zero at late times in order for our dynamical horizon to become a
_slowly evolving horizon_. In the case of our Lemaître model $c$ is calculated
to be
$\displaystyle
c=2\frac{M^{\prime}+wM^{\prime}}{M^{\prime}-wM^{\prime}-R^{\prime}}|_{AH}.$
(22)
The result of the numerical calculation for different equation of states and
different state functions is given in Figs.(8) and (9). The decreasing
behavior of the function $c$ in the course of time independent of the equation
of state is evident. We may then conclude that the dynamical horizon of the
cosmological black hole tends to a slowly evolving horizon.
Figure 8: The $p=w\rho$ case: the more pressure the sooner the dynamical
horizon becomes a slowly evolving horizon. Figure 9: The $p=w\rho s(r)$ case:
qualitatively, the same behavior as in Fig….
### III.4 Mass and matter flux
Due to the expanding background we expect the matter flux into the dynamical
black hole to be decreasing and the dynamical horizon to become a slowly
evolving horizon in the course of timemangeneral . We know already that there
is no unique concept of mass in general relativity corresponding to the
Newtonian concept. The question of what does general relativity tell us about
the mass of a cosmological structure in a dynamical setting was discussed
recently manmass . It was shown razbin that The Misner-Sharp quasi-local
mass, $M$, is very close to the Newtonian mass.
Let us then take the Misner-Sharp mass for this black hole and calculate the
corresponding matter flux into the black hole. In the case of Lemaîtremodel,
the matter flux is given by
$\displaystyle\frac{dM(r,t)}{dt}|_{AH}=\frac{\partial{M(r,t)}}{\partial{t}}|_{AH}+\frac{\partial{M(r,t)}}{\partial{r}}\frac{\partial{r}}{\partial{t}}|_{AH}=\dot{M}|_{AH}+M^{\prime}\frac{\partial{r}}{\partial{t}}|_{AH}$
(23)
The result of the numerical calculation is depicted in Figs.(10) and (11).
Note how the pressure decreases the rate of matter flux into the black hole.
Figure 10: The $p=w\rho$ case: the rate of matter flux into the black hole
decreases with the pressure. Figure 11: The $p=ws(r)\rho$ case: qualitatively
the same behavior as in Fig….
## IV DISCUSSION
We have studied the evolution of a structure made of perfect fluid with non-
vanishing pressure as an exact solution of Einstein equations within an
otherwise expanding FRW universe. The structure boundary is separated by a
void from the expanding part of the model which is very much like a FRW
universe already near by the void. We have noticed a counter-intuitive
pressure effect somewhere inside the structure where the existence of the
pressure slows down the collapse like a classical fluid in contrast to
distances far from the structure. The collapsed region develops to a dynamical
black hole with a space-like apparent horizon, in contrast to the
Schwarzschild black hole. This apparent horizon tends to a slowly evolving
horizon and becoming light-like at late times with a decreasing mater flux
into the black hole. We have, therefore, to conclude that the mere existence
of a cosmological matter, even dust, may have significant effect on the
central black hole differentiating it from a Schwarzschild one irrespective of
how small the density outside the structure is. Hence we may not be allowed to
speak about the Newtonian approximation because of the very weak gravity in
cases of non-local or quasi-local quantities such as the horizon and the mass.
## References
* (1) G.C. McVittie, Mon. Not. R. Astr. Soc. 93, 325 (1933); A. Einstein and E.G. Straus, Rev. Mod. Phys. 17, 120 (1945); 18, 148.
* (2) Wessel Valkenburg, Gen.Rel.Grav. 44 (2012) 2449-2476; Xuelei Chen, You-Gen Shen and Valerio Faraoni, Phys.Rev. D84 (2011) 104047; Krzysztof Bolejko, Marie-Noelle Celerier and Andrzej Krasinski, Class.Quant.Grav. 28 (2011) 164002.
* (3) J. T. Firouzjaee, Reza Mansouri, Gen. Relativity Gravitation. 42, 2431 (2010) [arXiv:0812.5108].
* (4) M. Parsi Mood, Javad T. Firouzjaee and Reza Mansouri, [arXiv:1304.5062]
* (5) Robert M. Wald, _General Relativity_ (University of Chicago Press, Chicago, 1984).
* (6) S. A. Hayward, Phys. Rev. D 49, 6467 (1994).
* (7) A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krishnan, J. Lewandowski and J. Wisniewski, Phys. Rev. Lett. 85, 3564-3567 (2000).
* (8) A. Ashtekar and B. Krishnan, Phys. Rev. Lett. 89, 261101 (2002); A. Ashtekar and B. Krishnan, Phys. Rev. D 68, 104030 (2003).
* (9) Booth I. and Fairhurst S., Phys. Rev. Lett. 92, 011102 (2004).
* (10) R. C. Tolman, Proc. Natl. Acad. Sci. U.S.A. 20, 410 (1934); G. Lemaître, Ann. Soc. Sci. Bruxelles I A53, 51 (1933); H. Bondi, Mon. Not. R. Astron. Soc. 107, 343 (1947).
* (11) J. T. Firouzjaee, Int. J. Mod. Phys. D, 21, 1250039 (2012) [arXiv:1102.1062].
* (12) Mohammadhosein Razbin, J. T. Firouzjaee and Reza Mansouri [arXiv:1212.4796].
* (13) J. T. Firouzjaee, M. Parsi Mood and Reza Mansouri, Gen. Relativ. Gravit. 44, 639 (2012) [arXiv:1010.3971].
* (14) Alnadhief A. H. Alfedeel, Charles Hellaby, Gen.Rel.Grav. 42, 1935 (2010) [arXiv:0906.2343].
* (15) J. T. Firouzjaee and Reza Mansouri, Europhys. Lett. 97, 29002 (2012) [arXiv:1104.0530].
|
arxiv-papers
| 2013-01-08T10:41:49 |
2024-09-04T02:49:39.994415
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rahim Moradi, Javad T. Firouzjaee and Reza Mansouri",
"submitter": "Javad Taghizadeh firouzjaee",
"url": "https://arxiv.org/abs/1301.1480"
}
|
1301.1710
|
# Evolutionary Exploration of the Finitely Repeated Prisoners’ Dilemma–The
Effect of Out-of-Equilibrium Play111Also appears as: _Lindgren, K.; Verendel,
V. Evolutionary Exploration of the Finitely Repeated Prisoners’ Dilemma–The
Effect of Out-of-Equilibrium Play. Games 2013, 4, 1-20._
Kristian Lindgren [email protected] Complex Systems Group,
Department of Energy and Environment, Chalmers University of Technology,
Göteborg, Sweden Vilhelm Verendel [email protected] Complex Systems Group,
Department of Energy and Environment, Chalmers University of Technology,
Göteborg, Sweden
###### Abstract
The finitely repeated Prisoners’ Dilemma is a good illustration of the
discrepancy between the strategic behaviour suggested by a game-theoretic
analysis and the behaviour often observed among human players, where
cooperation is maintained through most of the game. A game-theoretic reasoning
based on backward induction eliminates strategies step by step until defection
from the first round is the only remaining choice, reflecting the Nash
equilibrium of the game. We investigate the Nash equilibrium solution for two
different sets of strategies in an evolutionary context, using replicator-
mutation dynamics. The first set consists of conditional cooperators, up to a
certain round, while the second set in addition to these contains two strategy
types that react differently on the first round action: The ”Convincer”
strategies insist with two rounds of initial cooperation, trying to establish
more cooperative play in the game, while the ”Follower” strategies, although
being first round defectors, have the capability to respond to an invite in
the first round. For both of these strategy sets, iterated elimination of
strategies shows that the only Nash equilibria are given by defection from the
first round. We show that the evolutionary dynamics of the first set is always
characterised by a stable fixed point, corresponding to the Nash equilibrium,
if the mutation rate is sufficiently small (but still positive). The second
strategy set is numerically investigated, and we find that there are regions
of parameter space where fixed points become unstable and the dynamics
exhibits cycles of different strategy compositions. The results indicate that,
even in the limit of very small mutation rate, the replicator-mutation
dynamics does not necessarily bring the system with Convincers and Followers
to the fixed point corresponding to the Nash equilibrium of the game. We also
perform a detailed analysis of how the evolutionary behaviour depends on
payoffs, game length, and mutation rate.
## 1 Introduction
During the past two decades there has been a huge expansion in the development
and use of agent-based models for a variety of societal systems and economic
phenomena, ranging from markets of various types and societal activities such
as energy systems and land use, see e.g., [1, 2, 3, 4, 5] for a few
illustrative examples. A key issue in the construction of such models is the
design of the agents. To what extent are the agents rational? And what does it
mean that an agent is rational? This has divided scholars into different
camps. Herbert Simon [6] introduced the concept of ”bounded rationality”,
which can be implemented in a variety of ways in agent-based models, assuming
that there is some limitation on the reasoning capability of the agent. Game
theory provides useful methods for the analysis of interaction between agents
and their behaviour. But it is also well known, from experiments of human
behaviour in game-theoretic situations [7], that human subjects do not always
follow the behaviour predicted by game theory – and for good reasons. People
can in many cases establish cooperation for mutual benefit where game theory
would predict the opposite. This discrepancy between ”rational” game-theoretic
agents and human ones is often attributed to either limited rationality of
human reasoning or to social preferences. The latter can in some cases be
referred to as rule-based rationality [8] under which rules-of-thumb may have
developed over time in cultural evolution under positive selective feedback
from the benefits of cooperation.
In the modeling and construction of agents it is therefore of high importance
that the assumptions made on rationality and the reasoning process are made
explicit. Binmore discusses this in his classic papers ”Modeling rational
players” [9, 10]. He makes the distinction between eductive and evolutive
processes leading to an equilibrium in a game. The former refers to a process
internal to the agent representing a reasoning process, while the latter may
work with much simpler characterisation of the agents where evolutionary
processes lead to an equilibrium by mutation and selection on the population
level. In evolutionary game theory and agent-based modeling it is common to
use a combination of these, but one seldom designs agents who carefully reason
about possible actions and their consequences. And the fundamental question
still remains: What does it mean for an agent to be rational?
One of the major achievements in game theory is the establishment of the Nash
equilibrium concept and the existence proof that any finite game has at least
one such equilibrium [11]. The Nash equilibrium is a situation where no player
can gain by unilateral change of strategy, and in that sense this can be seen
as a rational equilibrium in providing the player a best response to other
players’ actions. The Nash equilibrium is thus often regarded as a result of
rational reasoning, reflecting the behaviour of rational players. Importantly,
for our discussion, this view of the Nash equilibrium has also been carried
over from single-shot to repeated games.
In several finitely repeated games, in which the number of rounds is known,
the solution of how players choose actions can be guided by the backward
induction procedure. This is often exemplified by the Prisoners’ Dilemma, for
which the single round game has a unique Nash equilibrium with both players
defecting, while the indefinitely repeated game has an uncountable infinity of
equilibria allowing for cooperation. However, when the exact number of rounds,
$n$, is known, a player can start with considering the last round, in which
the score is maximized by defecting. So, with both players being rational in
the sense that they want to maximize their own score, the outcome of the last
round is clear—mutual defection. But then the next-to-last round turns into
the last unresolved round, and the same reasoning applies again resulting in
mutual defection also for round $n-1$. The assumption needed is that each
player knows that the other one is rational. The procedure then repeats all
the way to the first round, showing that the Nash equilibrium is mutual
defection from the start of the game. Since the result of the backward
induction procedure does not seem to lead to an intuitively rational result it
is often referred to as the ”backward induction paradox” [12]. Note that
backward induction also applies to certain one-shot games, and the discrepancy
between theory and observation has been discussed also for several such
examples, e.g., the ”Beauty contest” [13] and the ”Traveler’s dilemma” [14].
There are at least two important objections against the generality of the
reasoning based on backward induction. The first objection is empirical, since
studies on how human players behave in the game show a substantial level of
cooperation, but with a transition to lower levels of cooperation towards the
end. Explanations are several, and this implies that several mechanisms are in
play. For example, it has been observed in the laboratory that subjects
cooperate initially but attempt to cheat each other by deviation in the end
[15, 7].
The second objection is conceptual and strongly connected to the notion of
rationality and what can be considered as a rational way of reasoning. The
only equilibrium that can exist in a given finite repetition is the Nash
equilibrium, but whether that is to be considered as rational is the question.
A critical point concerns what conclusion a player should draw if the opponent
deviates from what backward induction implies and instead cooperates _in the
first round_. It is then clear that the opponent is not playing according to
the Nash equilibrium, and there is a chance to get a higher score for some
time if cooperation can be established.
In this situation, the choice between (i) following backward induction and
defecting from start and (ii) deviating from backward induction by starting
with cooperation becomes a strategic decision. One can imagine ”rational”
players in both categories. In the first category, there are then two options,
either one just plays defect throughout the game whatever the opponent does,
as backward induction suggests, or one switches to cooperation if the opponent
cooperates. In the second category, both players are again faced with the
question of, provided the opponent is cooperating, when to switch to
defection. Obviously, there cannot exist a fixed procedure for deciding on
when it is optimal to switch from cooperation to defection, since such a
strategy would be dominated by the one that switches one round before.
However, the interaction and survival of different ways to handle first round
cooperation can be studied using evolutionary methods.
The purpose of this paper is to investigate in an evolutionary context the
performance of strategies representing the strategic choices discussed above
in the finitely repeated Prisoners’ Dilemma. In Binmore’s terms, we focus on
an evolutive process, in which each agent has a certain, relatively simple
strategy for the game, and the mix of strategies and their evolution is
investigated on the population level. Importantly, the chosen strategies can
all be seen as components in the reasoning processes discussed above: both (i)
the steps involved in the backward induction process, and (ii) the steps
initiating and responding to cooperation in the first round which then
reflects the possibility for strategies to deviate from equilibrium play. It
is well known that evolutionary drift or mutations, at least if sufficiently
strong, can drive the population away from a fixed point corresponding to the
Nash equilibrium. Under what circumstances does the evolutionary dynamics lead
to the same result as the backward induction process with a Nash equilibrium
as its fixed point, and when can deviation from Nash equilibrium play alter
that process? The answer, which is elaborated in this paper, depends on
choices of a number of critical model characteristics and parameters: selected
strategy space, mutation rate, payoff matrix, and the length of the game.
We prove that, for a simple set of strategies, i.e., conditional cooperators,
the replicator-mutation dynamics is always characterised by a stable fixed
point corresponding to the Nash equilibrium in the limit of zero (but
positive) mutation rate. Numerically we show that such a result does not hold
in general, even if the only Nash equilibrium is characterised by defection
from the start of the game. The strategy set that allows for different
reactions on the first round may lead to a path of actions different from what
is considered in the backward induction process. On the population level, this
turns out to destabilise the dynamics, and for a large part of parameter
space, the evolution does not bring the system to a stable fixed point, even
in the limit of zero (but positive) mutation rate. The dynamics is instead
characterised by oscillatory behaviour.
Most of the work related to evolutionary dynamics, backward induction, and the
finitely repeated Prisoners’ Dilemma concerns the replicator dynamics [16]
with no mutation, as this class has been examined analytically more thoroughly
than the replicator-mutation dynamics [17]. Nachbar [18] studies convergence
in the dynamics and shows that for 2-stage games all cooperation goes extinct
when starting from a mixed population. This result is extended by Cressman
[19, 20] to the finitely repeated Prisoners’ Dilemma of arbitrary length.
Several authors have investigated various types of evolutionary dynamics under
the effect of perturbations or mutations ([21, 22, 23, 24, 25]). Here the
focus has primarily been on other games than the finitely repeated Prisoners’
Dilemma. Gintis et al. [26] consider the replicator dynamics subject to
recurrent mutation when mutation rate goes to zero. For finite noncooperative
games, they show that the dynamics need not converge to the subgame perfect
equilibrium, and limiting equilibria can be far away from the this
equilibrium. They also show that in the $n$-player Centipede game, there
exists a unique limiting equilibrium as mutation rate goes to zero, which is
far from the subgame perfect equilibrium but equal in payoffs.
Ponti [27] studies replicator-mutation dynamics in the Centipede game [28].
Using simulations, he finds recurrent phases of cooperation in the
evolutionary dynamics, and for particular payoffs of the game, this is shown
to depend both on mutation rate as well as the length of the game. It is left
as an open question whether such behaviour would disappear in the long run, or
persist for negligible mutation rate. This result is most relevant to the
present paper in the context of the finitely repeated Prisoners’ Dilemma.
Our work is also related to the literature on the ”backward induction paradox”
[12], which has focused on deviation from equilibrium play in the first round
of extensive games. For the backward induction reasoning to be rational, it
has been assumed that each player has full knowledge about the rationality of
the other player, and that both players know this, et cetera, known as ”common
knowledge of rationality”. It has been proved by Aumann [29] that such common
knowledge implies backward induction in games with a unique subgame perfect
equilibrium. However, backward induction provides no firm basis to act when a
player deviates from equilibrium play [30, 31, 32]. Our study can be viewed as
an evolutionary study of a population in a setting where reacting to out-of-
equilibrium play in the first round is possible.
## 2 Evolutionary Dynamics
The evolution of strategies in the finitely repeated Prisoners’ Dilemma is
studied using replicator dynamics with a uniform mutation rate. This is a
model of an infinite population where all interact with all, and in which each
strategy $i$ occupies a certain fraction $x_{i}$ of the population. The
selection process gradually changes the population structure, based on a
comparison of the average score $s_{i}$ of strategy $i$ with the average score
$s$ in the population,
$\displaystyle s_{i}$ $\displaystyle=\sum_{j=1}^{n}u(i,j)x_{j}$ (1)
$\displaystyle s$ $\displaystyle=\sum_{i=1}^{n}s_{i}x_{i}\;$ (2)
where $u(i,j)$ is the score for strategy $i$ meeting strategy $j$ in the
$N$-round Prisoners’ Dilemma. Assuming a uniform mutation rate of $\epsilon$,
the replicator-mutation dynamics can be written,
$\displaystyle\frac{dx_{i}}{dt}=x_{i}\left(s_{i}-s-\epsilon\right)+\epsilon/n\;$
(3)
where $n$ is the number of different strategies. The dynamics conserve the
normalisation of $x$.
The score $u(i,j)$ between two strategies, $i$ and $j$, is calculated from $N$
rounds of the Prisoners’ Dilemma with a payoff matrix for the row player given
by
22 & $C$ $D$
$C$ $1$ $0$
$D$ $T$ $P$
This means that the reward $R$ for mutual cooperation (C, C) has been set to
1, and the ”suckers payoff” $S$ for cooperation against defection (C, D) is 0.
The temptation to defect against a cooperator (D, C) is associated with a
score $T$, and for mutual defection (D, D) the score is $P$. With the usual
constraints of $T>R>P>S$ and $2R>T+S$ it remains to study the parameter region
given by $T\in[1,2]$ and $P\in[0,1]$. There are in principle three independent
parameters in the game, but in combination with the replicator-mutation
dynamics, the third one is incorporated in the mutation rate $\epsilon$ (which
can be derived by subtraction and division by $S$ and $R-S$, respectively, in
the replicator-mutation dynamics equation, Equation (3)).
### 2.1 Selecting the Strategy Set
We investigate the evolutionary behaviour considering two sets of strategies.
The first one is a strategy set $\Gamma_{1}$ that represents various levels of
depth in applying the backward induction procedure to conditional cooperation.
A strategy in this set is denoted $S_{k}$ (with $k\in\\{0,1,...,N\\}$), which
means that the strategy is playing conditional cooperation up to a round $k$
and then defects throughout the game, see Figure 1. Conditional cooperation
means that if the opponent defects, one switches to unconditional defection
for the rest of the game. For example, $S_{N}$ is prepared to cooperate
through all rounds (or as long the opponent does), while $S_{0}$ defects from
the first round. It is then clear that strategy $S_{k}$ dominates $S_{k+1}$
(for $k\in\\{0,1,...,N-1\\}$), and backward induction leads us to the single
Nash Equilibrium in which both players choose strategy $S_{0}$.
$C$$S_{i\in\\{1..N\\}}$$D$$S_{0}$$D\lor(t\geq i)$$C\land(t<i)$
Figure 1: Finite state machine illustrating the first strategy set,
$\Gamma_{1}$, of the $S_{i}$ strategies. The action to perform is in the node,
and transitions are taken on the basis of the opponent’s action in the
previous round (or based on the previous round number $t$). All strategies
start in the left node (C), except $S_{0}$ that starts with defection in the
right node (D). $S_{i}$ cooperates conditionally for $i$ rounds after which it
starts to defect.
$C$$S_{i\in\\{1..N\\}}$$C$$CC_{i\in\\{2..N\\}}$$D$$DC_{i\in\\{2..N\\}}$$D$$S_{0}$$C$$D$$D\lor(t\geq
i)$$C\land(t<i)$
Figure 2: Finite state machine illustrating the extended strategy set
$\Gamma_{2}$ consisting of the strategies $S_{i},CC_{i}$, and $DC_{i}$.
$S_{i}$ are the conditional cooperators as described in Figure 1. The
Convincers are denoted $CC_{i}$ and the Followers $DC_{i}$. Strategies start
in the state at which the name is placed. The strategies $CC_{i}$ start with
at least two rounds of cooperation, which may trigger $DC_{i}$ to switch from
defection to cooperation. After that the latter two strategies act as the
conditional cooperators $S_{i}$.
Note that the entire strategy set $\Omega$ for the $N$-round Prisoners’
Dilemma is very large, as a strategy requires a specification how to react on
each possible history (involving the opponent’s actions) for every round of
the game. This results in $2^{2^{N}-1}$ possible strategies, e.g. for $N=10$
rounds this is in the order of $10^{308}$. The selection of strategies to
consider is critical. One can certainly introduce strategies so that other
Nash equilibria are introduced along with those characterized by always
defection. For example, selecting only three strategies, e.g.,
$\\{S_{0},S_{5},S_{10}\\}$ will lead to a game with three equilibria, one for
each strategy. But this is an artefact of the specific selection made. Here,
we do not want to create new Nash equilibria but we want to investigate how
and if the evolutionary dynamics brings the population to a fixed point
dominated by defect actions corresponding to the original Nash equilibrium.
One way to achieve this is to make sure that iterated elimination of weakly
dominated strategies can be applied to the constructed strategy set, in a way
so that only strategies that defect throughout the game remain, keeping the
non-cooperative characteristic of the Nash equilibrium.
The second set of strategies $\Gamma_{2}$ (in Figure 2) that we consider is
given by extending $\Gamma_{1}$ to include also strategies corresponding to
steps of reasoning in which one (i) tries to establish cooperation even if the
opponent defects in the first round and (ii) responds to such attempts by
switching to cooperation for a certain number of rounds. We refer to such
strategies as ”Convincers” and ”Followers”, respectively.
A Convincer strategy $CC_{k}$ starts with cooperation twice in a row,
regardless of the opponent’s first round action, and in this way it can be
seen as an attempt to establish cooperation even with first round defectors.
From round 3, the strategy plays as $S_{k}$, i.e., conditional cooperation up
to a certain round $k\in\\{2,3,...,N\\}$.
A Follower strategy $DC_{k}$ starts with defection, but can be triggered to
cooperation by a Convincer (or $S_{k}$ where $k>0$). So the Follower switches
to cooperation in the second round, after which it, like the Convincer, plays
as $S_{k}$, i.e., conditional cooperation up to round $k\in\\{2,3,...,N\\}$.
(A Follower strategy is also triggered to cooperation by an $S_{k}$ strategy,
but $S_{k}$ does not forgive the first round defect action and cooperation
cannot be established.)
For the extended strategy set, $\Gamma_{2}$, it is straightforward to see that
iterated elimination of weakly dominated strategies, starting with those
cooperating throughout the game, leads to a Nash equilibrium with only
defectors.
For the first strategy set, $\Gamma_{1}$, the Nash equilibrium of
$(S_{0},S_{0})$ is strict since any player deviating would score less. For the
second strategy set, $\Gamma_{2}$, this Nash equilibrium is no longer strict
as one of the players could switch to a Follower strategy, still defecting and
scoring the same. For the first strategy set, the NE is unique, but for the
second one that is not necessarily the case. Since backward induction still
applies in the second set, we know that any NE is characterized by defection
only, which can be represented by a pair $(S_{0},DC_{k})$. This is a NE only
if the $S_{0}$ player cannot gain by switching to $CC_{k-1}$ (or to $CC_{2}$
if $k=2$). This translates into $kP>S+(k-2)R+T$ (for $k>2$) or $T<kP-(k-2)$
for our parameter space, while for $k=2$, we have $2P>S+R$ or $P>1/2$.
Furthermore, if $(S_{0},DC_{k})$ is a NE, then also $(DC_{j},DC_{k})$, with
$j\leq k$, is a NE.
This means that in a part of the payoff parameter region, for $P<1/2$, there
is only one NE, while for larger $P$ values there are several NEs, with
increasing number the closer to $1$ the parameter $P$ is. Note, though, that
any additional NE here is characterized by defection from the first round,
corresponding to the unique subgame perfect Nash equilibrium $(S_{0},S_{0})$.
This illustrates the fact that in the NE one can switch from a pure defector
to a Follower without reducing the payoff. If this happens under genetic drift
in evolutionary dynamics, the situation may change so that Convincers may
benefit and cooperation can emerge.
## 3 Dynamic Behaviour and Stability Analysis
The dynamic behaviour and the stability properties of the fixed points are
investigated both analytically and numerically, for the two strategy sets
presented in Section 2.1. In each case, we investigate the dependence of the
payoff parameters, $T$ and $P$, as well as the mutation rate and game length,
$\epsilon$ and $N$, respectively. We are mainly interested in which influence
the presence of Convincers and Followers has on the stability of fixed points
and its impact on the dynamics. This will be illustrated in three different
ways as follows. First, we present and examine a few specific examples of the
evolutionary dynamics and discuss the qualitative difference between the
strategy sets. For the simple strategy set, we show analytically that the
fixed point associated with the Nash equilibrium remains stable for a
sufficiently small mutation rate. A numerical investigation is then performed
for the extended strategy set, characterising the fixed point stability.
Finally, for different lengths of the game, we study realisations of the
evolutionary dynamics from an initial condition of full cooperation ($x_{N}=1$
and $x_{i}=0$ for $i\neq N$, with $x_{k}$ denoting the fraction of strategy
$S_{k}$ in the population) to examine to which degree cooperation survives and
whether the dynamics is attracted to a fixed point or characterized by
oscillations. By studying variation of the dynamics over the different regions
in the parameter space and changing the payoff parameters $T$ and $P$ (by
adhering to the inequalities between $T,P,R,S$ as described in Section 2), it
is studied how the population evolves in various games over time.
### 3.1 Dynamics with the Simple and the Extended Strategy Set
$\begin{array}[]{cc}\noindent\includegraphics[width=227.62204pt]{StandardRPDScanPAPERPLOTS__N10__E2E_8.pdf}&\includegraphics[width=227.62204pt]{StandardRPDScanPAPERPLOTS__N10__E2E_12.pdf}\\\
\includegraphics[width=227.62204pt]{ExtendedScanPAPERPLOTS__N10__E2E_8.pdf}&\includegraphics[width=227.62204pt]{ExtendedScanPAPERPLOTS__N10__E2E_12.pdf}\end{array}$
Figure 3: Illustration of the dynamics for a particular game (P=0.2, T=1.33)
for 10-round Repeated Prisoners’ Dilemma ($N=10$). Each main plot displays how
the frequency of different strategies changes in the population as a function
of time. Smaller subplots show how the population average payoff changes with
time, with $NR$ and $NP$ denoting full cooperation and defection,
respectively. Top: simple strategy set ($S_{0},...,S_{10}$). Below: the
extended strategy set, which includes also Convincers ($CC_{2},...,CC_{10}$)
and Followers ($DC_{2},...,DC_{10}$). When lowering $\epsilon$, the extended
strategy set exhibits meta-stability with recurring cooperation, while for the
simple strategy set cooperation disappears.
Realisations of the evolutionary dynamics, Equation (3), for both the simple
and the extended strategy sets are shown in Figure 3. For particular payoff
values and a 10-round game, the mutation rate $\epsilon$ is varied to
illustrate how it affects the dynamics.
First, we consider the case with the simple strategies ($S_{0},...,S_{10}$) in
Figure 3. From an initial state of full cooperation, with a population
consisting only of fully cooperative $S_{10}$ players, the dynamics will, for
both levels of mutation rate, lead to a gradual unraveling of cooperation to a
point where $S_{0}$, full defection, dominates the population. The first step
of this unraveling occurs because $S_{9}$ defecting in the final round will
have higher payoff than $S_{10}$. At this stage $S_{0}$ is much worse off, but
the population goes through a series of transitions which reminds of a
backward induction process. This can also be seen in terms of average payoff,
as illustrated in Figure 3. When the population is in this non-cooperative
mode a positive mutation-rate $\epsilon$ may offset the situation. For the
higher mutation rate, cooperative strategies re-emerge after a period of
influx from other strategies. The mutations gradually introduce cooperative
behaviour to a critical point where some degree of cooperation has a selective
advantage over full defection, and we see a shift in the level of cooperation.
After a while, cooperative behaviour is again overtaken by full defection and
a cyclic behaviour becomes apparent. Comparing with the next realisation of
the simple strategies we see that this can happen only when the mutation rate
is high enough. As mutation rate becomes smaller, here illustrated with
$\epsilon=2^{-12}$, there is no re-appearance of cooperation. When the
mutation rate gets too low, strategies other than defection are kept on a
level that is too low to promote further cooperation. This demonstrates that
the mutation rate can affect whether cooperation re-appears or not.
Second, we consider the case with extended strategies (the $3N-1$ strategies
$S_{0},...S_{10}$, $CC_{2},...,CC_{10}$, $DC_{2},...,DC_{10}$) shown at the
bottom of Figure 3. We observe that adding Convincer and Follower strategies
changes the dynamics, but with a similar unraveling of cooperation. For both
mutation rates, the trajectories have periods with defection dominating in the
population, indicated by average payoff, but the population does not seem to
stabilize. Lowering the mutation rate increases the time between the outbreaks
of cooperation. Contrary to the simple strategy set, the system does not seem
to settle close to full defection. The explanation is due to the Follower
strategies being able to gradually enter the population by getting the same
payoff as the strategy of full defection. At a critical point, there are
sufficiently many Followers, which make mutations to Convincers successful and
cooperation re-enters the population for a period.
In the next section, we will investigate the dynamics and the stability
characteristics of both the simple and the extended strategy sets in detail,
varying the payoff parameters over the full ranges, and investigating the
behaviour in the limit of diminishing mutation rate.
### 3.2 Existence of Stable Fixed Points
We now turn to examine the existence of stable fixed points in the dynamics
for low mutation rates. For the simple strategy set the following proposition
holds.
#### Proposition 1:
For the simple set of strategies, $\Gamma_{1}$, the fixed point associated
with the Nash equilibrium, dominated by strategy $S_{0}$, is stable under the
replicator-mutation dynamics, if the mutation rate is sufficiently small (but
positive).
#### Proof 1:
See Appendix A.
Our results for the extended strategy set, $\Gamma_{2}$, are based on
numerical investigations: by using an eigenvalue analysis of the Jacobian of
the replicator-mutation dynamics, Equation (3), we determine for which
parameters $T,P$ and $\epsilon$ the dynamics is characterised by stable fixed
points. We are especially interested in the case of lowering $\epsilon$
towards 0 to examine whether fixed points become stable when mutation rate is
sufficiently small, like in the case of the simple strategy set.
This stability analysis over the parameter space and the results are presented
for different lengths of the game in Figure 4. Stable fixed points exist to
the _right_ of the line corresponding to a specific $\epsilon$, while to the
_left_ of the line no stable fixed points were found. When decreasing the
mutation rate $\epsilon$ towards 0, we see that an increasing fraction of
parameter space is characterised by a stable fixed point. Unlike the case for
the simple strategy set, the numerical investigation shows a convergence of
the delimiting line between stable and unstable fixed points, indicating that
there is a remaining region in parameter space (with low $P$ and low $T$) for
which fixed points are unstable in the limit of zero mutation rate.
Note from the discussion in Section 1 that while these results show the
existence of stable fixed points, it is left to consider whether the
population dynamics would actually converge to such states. Next, we turn to
consider population dynamics over the parameter space and its outcome.
$\begin{array}[]{cc}\includegraphics[width=227.62204pt]{FixpointExtendedN5.pdf}&\includegraphics[width=227.62204pt]{FixpointExtendedN6.pdf}\\\
\includegraphics[width=227.62204pt]{FixpointExtendedN7.pdf}&\includegraphics[width=227.62204pt]{PlotStableUnstableBoundary_N5_VaryEPS_SimpleAndExtended_Ver2.pdf}\end{array}$
Figure 4: Numerical analysis showing which games have stable fixed points.
Stable fixed points have been found to the right of a given line, at which
they disappear if $P$ is decreased further. Lowering the mutation rate
$\epsilon$ turns more evolutionary dynamics into having stable fixed points.
Runs for $N=5,6,7$ indicate a convergence, as the mutation rate is decreased,
towards a fraction of games being without stable fixed points, as seen in the
lower left of the parameter space for the different game lengths. To the
bottom right the boundary is shown for a particular $T=1.1$.
### 3.3 Recurring Phases of Cooperation
Motivated by the findings above in Section 3.1, we now turn to study recurrent
cooperative behaviour in the evolutionary dynamics. We investigate if and for
which games a population, starting from initial cooperation as before, settles
into a mode of oscillations that at some point in the cycle brings the system
back to a higher level of cooperation 222An instance of the dynamics was
counted as oscillating when the average payoff $A$ repeatedly returns to at
least $5\%$ above full defection, i.e. $A>1.05NP$. Frequently, it was the case
that the oscillations had phases of cooperation well above the $5\%$
threshold..
The interesting case is when mutation rates are small: higher mutation rates
introduce a background of all different strategies, which can be seen as
artificially keeping up cooperative behaviour in the population. To avoid this
effect, we investigate the dynamics with $0<\epsilon<<1$, and, in particular,
numerical analyses were made with a series of decreasing mutation rates. The
population is initialized as before, described in Section 3.1.
First, we characterise the simple and the extended strategy set for different
game lengths, varying mutation rates, and varying the parameters of $T$ and
$P$. Figure 5 displays the results for games of 5 and 10 rounds in the top and
the middle row. For a given $T$ a line denotes the border for which games to
the _left_ have recurring phases of cooperation, while games to the _right_
are characterised by a fixed point dominated by defection. For the simple
strategy set, we observe that by decreasing mutation rate, the fraction of
games where cooperation recurs becomes smaller and disappear. This suggests
that as we make the mutation rate sufficiently small, cooperation will die out
in the case of simple strategies, as is expected from Proposition 1 stating
that the fixed point dominated by the $S_{0}$ strategy becomes stable. On the
other hand, for the extended strategy set, a considerable fraction of games
seem to offer recurring phases of cooperation despite lowering the mutation.
For the 10-round game, the line describing the critical parameters is seen to
converge as the mutation rate decreases, i.e., there is a large part of the
parameter region for which the dynamics is not attracted to a fixed point. For
the $5$-round game, this occurs at least in part of the parameter space. The
bottom graphs in Figure 5 show that the longer the game, the larger is the
parameter region for $T$ and $P$ in which the fixed point is avoided and
recurring periods of cooperation are sustained. It should be noted, though,
that as the mutation rate decreases, the part of the cycle in which there is a
significant level of cooperation decreases towards zero. This is due to the
slow genetic drift that brings $DC_{k}$ strategies back into the population,
which eventually make it possible for the $CC_{k}$ strategies to re-establish
a significant level of cooperation.
$\begin{array}[]{cc}\includegraphics[width=227.62204pt]{ParseOscillationsPAPERPLOTS__N5Simple.pdf}&\includegraphics[width=227.62204pt]{ParseOscillationsPAPERPLOTS__N10Simple.pdf}\\\
\includegraphics[width=227.62204pt]{ParseOscillationsPAPERPLOTS__N5Extended.pdf}&\includegraphics[width=227.62204pt]{ParseOscillationsPAPERPLOTS__N10ExtendedAdditionalEps.pdf}\\\
\includegraphics[width=227.62204pt]{ParseOscillationsPAPERPLOTS__epsE2_8.pdf}&\includegraphics[width=227.62204pt]{ParseOscillationsPAPERPLOTS__epsE2_12.pdf}\\\
\end{array}$
Figure 5: Parameter diagram showing which games have recurrent cooperation in
the evolutionary dynamics from a starting point of initial cooperation. Top
row: simple strategy set. Middle row: extended strategy set with Convincers
and Followers. In the graphs, recurrent cooperation exists to the left of the
line and a fixed point with defectors characterizes the behaviour to the
right. For simple strategies, lowering $\epsilon$ steadily reduces the part of
parameter space dominated by recurrent cooperation. In the bottom row, the
delimiting line is shown for a variety of game lengths and for two levels of
mutation (left and right), illustrating the fact that the longer the game the
larger the parameter region supporting cooperative phases in the evolutionary
dynamics.
### 3.4 Co-existence between Fixed Point Existence and Recurring Cooperation
The discussion in Section 3.2 left the question of whether stable fixed points
are reached by population dynamics, and we found that it is not necessarily so
in Section 3.3. This was illustrated for the extended strategy set for low
$\epsilon$. Combining the different findings by considering their boundaries
in the parameter space, this suggests an additional property of the
evolutionary dynamics. By considering the joint results of our findings (in
Figures 4 and 5), one can note that there is a part of parameter space for
which there is co-existence between a stable fixed point of defect strategies
and a stable cycle with recurring cooperation.
## 4 Discussion and Conclusion
A key point in game theory is that a player’s strategic choice must consider
the strategic choice the opponent is making. For finitely repeated games,
backward induction as a solution concept has become established by assuming
player beliefs being based on common knowledge of rationality. However, this
assumption says nothing about how players would react to a deviation from full
defection—a deviation from the Nash equilibrium—since it a priori rules out
actions and reactions that exemplify other ways of reasoning.
Motivated by the general importance of backward induction and what has been
called its ”paradox”, we have introduced an evolutionary analysis of the
interaction in a population of strategies that react differently to out-of-
equilibrium play in the first round of the game. We have shown how extending a
strategy set for this possibility, in the special case of the repeated
Prisoners’ Dilemma, allows for stable limit cycles in which cooperative
players return after a period of defection. The introduction of Convincers and
Followers, representing both strategies that try to establish cooperation and
strategies that are capable of responding to that, are made in a way to
preserve the structure of the selected strategy set so that elimination of
weakly dominated strategies leads to full defection.
For the simple strategy set, as the mutation rate becomes sufficiently small,
the cyclic behaviour disappears and the system is attracted to a stable fixed
point. The stability of this fixed point was shown analytically for a
sufficiently small mutation rate $\epsilon$.
For the extended strategy set, for low levels of mutation, the numerical
investigation of fixed point stability and oscillatory modes indicates that,
for a certain part of payoff parameter space, the evolutionary dynamics does
not reach a stable fixed point but stays in an oscillatory mode, unlike the
case of simple strategy set. We characterise our results by a detailed
quantitative analysis of where this occurs: showing how the length of the
repeated game and the mutation rate affects the boundaries of this region.
One of the main results of the study is an affirmative answer to the question
whether different responses to out-of-equilibrium play in the first round can
make the dynamics avoid fixed points, and the corresponding Nash equilibrium.
Additionally, the fixed point analysis showed the co-existence of a stable
fixed point and stable oscillations with recurring phases of cooperation. This
means that a system with different responses to out-of-equilibrium play may be
found far from its possible stable fixed point. Taken together, this
illustrates that the Nash equilibrium play can be unstable at the population
level when mutations make explorations off the equilibrium path possible.
This paper contributes to the backward induction discussion in game theory,
but more broadly to the study of repeated social and economic interaction.
Many models, typically much larger and less transparent ones, of social and
economic systems involve agents. If solving these systems means finding the
Nash equilibria, then one may doubt whether that is a good representation of
rational behaviour except under certain conditions as we have discussed in the
paper. We have shown that strategies corresponding to the Nash equilibrium
cannot be taken for granted when they interact and compete with strategies
that act and respond differently to out-of-equilibrium play.
## Acknowledgements
Financial support from the Swedish Energy Agency is gratefully acknowledged.
We would also like to thank two anonymous reviewers for constructive criticism
and for inspiring us to prove Proposition 1. We also thank David Bryngelsson
for valuable comments on the introduction.
## Appendix A. Stability of fixed point in the simple strategy set for small
mutation rates
In order to show that the Nash equilibrium fixed point at zero mutation rate,
$\epsilon=0$, continuously translates into a stable fixed point as the
mutation rate becomes positive, $\epsilon>0$, we investigate the fixed point
more thoroughly. First, we note that the stability of the fixed point without
mutations ($\epsilon=0$), characterised by $x_{1}=1$ (strategy $S_{0}$
dominating), can be determined by the largest eigenvalue of the Jacobian
matrix ($\partial\dot{x}_{i}/\partial x_{j}$) derived from the dynamics,
Equation (3), where we use the notation $\dot{x}_{i}=dx_{i}/dt$. One finds
that the largest eigenvalue is given by $\lambda_{\text{max}}=-P<0$, which
shows that the fixed point is stable. This is also known from the stability
analysis of the finitely iterated game by Cressman [19].
We will now proceed with reformulating the fixed point condition, for positive
mutation rate $\epsilon>0$, which is a set of $n$ polynomial equations given
by $\dot{x}_{i}=0$ (with $i=1,...,n$), into an equation of only one variable
$x_{1}$. Based on this we show that the Nash equilibrium fixed point for zero
mutation rate $\epsilon=0$, characterised by $x_{1}=1$, continuously moves
into the interval $x_{1}\in[0,1[$, with retained stability.
We let index $i$ denote strategy $S_{i-1}$ in the simple strategy set
($i=1,...,n$), where the number of strategies is $n=N+1$. Due to the structure
of the repeated game for the simple strategy set, determining $u(i,j)$, we can
establish pair-wise relations between $x_{k}$ and $x_{k+1}$ for
$k=1,2,...,n-1$, at any fixed point. We use the slightly more compact notation
$s_{i,j}=u(i,j)$, for the score for a strategy $i$ against $j$.
For $x_{1}$ and $x_{2}$ (strategies $S_{0}$ and $S_{1}$), using Equation (3),
we have
$\displaystyle\frac{dx_{1}}{dt}=x_{1}\left(s_{1,1}x_{1}+s_{1,2}(1-x_{1})-\bar{s}-\epsilon\right)+\frac{\epsilon}{n}=0$
(4)
$\displaystyle\frac{dx_{2}}{dt}=x_{2}\left(s_{2,1}x_{1}+s_{2,2}x_{2}+s_{2,3}(1-x_{1}-x_{2})-\bar{s}-\epsilon\right)+\frac{\epsilon}{n}=0\;$
(5)
where we have used the fact that $s_{1,2}=s_{1,j}$ (for $j>2$), and
$s_{2,3}=s_{2,j}$ (for $j>3$). At a fixed point, for $\epsilon>0$, all
strategies are present, $x_{i}>0$, and we can divide these equations by
$x_{1}$ and $x_{2}$, respectively. Then, taking the difference between the
equations gives us an equation for the relation between $x_{1}$ and $x_{2}$,
$\displaystyle\frac{\epsilon}{n}\left(\frac{1}{x_{1}}-\frac{1}{x_{2}}\right)+x_{1}+(T-P)x_{2}-(1-P)=0\;$
(6)
where we have used that $s_{2,3}-s_{1,2}=1-P$,
$s_{1,1}-s_{2,1}-s_{1,2}+s_{2,3}=1$, and $s_{2,3}-s_{2,2}=T-P$. Solving this
quadratic equation gives $x_{2}$ as a function of $x_{1}$,
$\displaystyle x_{2}=f_{A}(x_{1})$ (7)
with the function $f_{A}(x)$ defined by
$\displaystyle
f_{A}(x)=\frac{1}{2(T-P)}\left(x-(1-P)+\frac{\epsilon^{\prime}}{x}\right)\left(-1+\sqrt{1+\frac{4(T-P)\epsilon^{\prime}}{\left(x-(1-P)+\frac{\epsilon^{\prime}}{x}\right)^{2}}}\;\right)\,$
(8)
Here we have introduced $\epsilon^{\prime}=\epsilon/n$. In the same way, for
$x_{k-1}$ and $x_{k}$ (for $k=3,...,N-1$), the fixed point implies that
$\displaystyle\frac{dx_{k-1}}{dt}=x_{k-1}\left(\sum_{j=1}^{k-2}s_{k-1,j}x_{j}+s_{k-1,k-1}x_{k-1}+s_{k-1,k}\left(1-\sum_{j=1}^{k-1}x_{j}\right)-\bar{s}-\epsilon\right)+\frac{\epsilon}{n}=0$
(9)
$\displaystyle\frac{dx_{k}}{dt}=x_{k}\left(\sum_{j=1}^{k-2}s_{k,j}x_{j}+s_{k,k-1}x_{k-1}+s_{k,k}x_{k}+s_{k,k+1}\left(1-\sum_{j=1}^{k}x_{j}\right)-\bar{s}-\epsilon\right)+\frac{\epsilon}{n}=0\;$
(10)
where we have used the fact that $s_{k,j}=s_{k,k+1}$ for $j>k$. Again, the
difference between these equations results in a relation between $x_{k}$ and
all $x_{j}$ with $j<k$ (for $k=3,...,N-1$). Since $s_{k-1,j}=s_{k,j}$ for
$j<k-1$, the difference can be written
$\displaystyle\frac{\epsilon}{n}\left(\frac{1}{x_{k-1}}-\frac{1}{x_{k}}\right)+Px_{k-1}+(T-P)x_{k}-(1-P)\left(1-\sum_{j=1}^{k-1}x_{j}\right)=0\;$
(11)
where the score differences, $s_{k-1,k-1}-s_{k,k-1}=P-S=P$,
$s_{k,k+1}-s_{k,k}=T-P$ and, $s_{k-1,k}-s_{k,k+1}=P-R=P-1$, have been used.
This results in an expression for $x_{k}$ in terms of all $x_{j}$ (with
$j<k$),
$\displaystyle x_{k}=f_{B}\left(x_{k-1},1-\sum_{j=1}^{k-1}x_{j}\right)$ (12)
with the function $f_{B}(x,w)$ defined by
$\displaystyle f_{B}(x,w)$
$\displaystyle=\frac{1}{2(T-P)}\left(Px-(1-P)w+\frac{\epsilon^{\prime}}{x}\right)\left(-1+\sqrt{1+\frac{4(T-P)\epsilon^{\prime}}{\left(Px-(1-P)w+\frac{\epsilon^{\prime}}{x}\right)^{2}}}\;\right)\,$
(13)
Finally, using the same approach for the last two variables, $x_{n-1}$ and
$x_{n}$, the equation for the relation between these can be written
$\displaystyle\frac{\epsilon}{n}\left(\frac{1}{x_{n-1}}-\frac{1}{x_{n}}\right)+Px_{n-1}+(T-1)x_{n}=0\;$
(14)
The last variable can then be expressed as
$\displaystyle x_{n}=f_{C}(x_{n-1})$ (15)
with the function $f_{C}(x)$ defined by
$\displaystyle
f_{C}(x)=\frac{1}{2(T-1)}\left(Px+\frac{\epsilon^{\prime}}{x}\right)\left(-1+\sqrt{1+\frac{4(T-1)\epsilon^{\prime}}{\left(Px+\frac{\epsilon^{\prime}}{x}\right)^{2}}}\;\right)\,$
(16)
This means that we can recursively express the fixed point abundancies $x_{k}$
of strategies $k=2,...,n$ in terms of $x_{1}$, using Eqs. (8), (13), and (16).
Together with the normalisation constraint, this results in an equation with
only one variable, $x_{1}$, that determines the fixed points. In other words:
Summation over all $x_{k}$ and subtraction of 1 gives us a function of
$x_{1}$, $F(x_{1})$, with zeroes at the fixed points,
$\displaystyle F(x_{1})=\sum_{k=1}^{n}x_{k}-1\;$ (17)
Note that we can extend the function $F$ also to the case $\epsilon=0$, since
all functions $f_{A},f_{B}$ and $f_{C}$ are identical to zero in that case and
we simply get $F(x_{1})=x_{1}-1$, capturing the Nash equilibrium fixed point
$x_{1}=1$. (Note, though, that in this extension none of the other fixed
points at $\epsilon=0$ are captured; any $x_{k}=1$ defines a fixed point for
zero mutation rate, but all except the first one are of no interest for us.)
All functions $f_{A},f_{B}$ and $f_{C}$ are continuous and bounded, since
$g(y)=y(1-\sqrt{1+1/y^{2}})$ is. We also see that $g(y)\rightarrow 0$ as
$|y|\rightarrow\infty$. Thus, $F(x)$ being composed of these functions is
continuous in the interval $[0,1]$.
As an example, Figure 6 illustrates $F(x)$ for $n=N+1=6,T=1.05,P=0.15$ with
four different choices of the mutation rate $\epsilon$. The graph shows that
as the mutation rate increases the fixed point at $x_{1}=1$ moves to the left
and also that new fixed points may emerge. Most importantly, though, the graph
indicates a discontinuous change of the function for $x$ below about $0.95$,
when going from $\epsilon=0$ to $\epsilon>0$. In order to show that the fixed
point always moves continuously from $x_{1}=1$, we need to be sure that any
such a discontinuity is bounded away from $x_{1}=1$.
Figure 6: The function $F(x)$ for the 5-round PD game with $T=1.05,P=0.15$,
and four different mutation rates $\epsilon$.
We already know that for $\epsilon=0$ there is a zero in $x=(1,0,...,0)$. We
want to show that as $\epsilon$ increases, this fixed point, characterised by
$F(x_{1})=0$ for $x_{1}=1$, continuously moves into the unit interval,
$0<x_{1}<1$. We accomplish this by performing a series expansion of $F(x)$
around $x=1$ and $\epsilon=0$, by showing that $F^{\prime}(x_{1})=1$ for
$x_{1}=1$, that $F$ is increasing with $\epsilon$ in the neighbourhood of
$x_{1}=1$, and that the coordinates $x_{k}$ ($k>1$) of the fixed point move
continuously with $x_{1}$ when sufficiently close to 1.
The first part is straightforward: At zero mutation rate, $\epsilon=0$, we
have already noted that $F(x_{1})=x_{1}-1$, and thus the derivative
$F^{\prime}(x_{1})=1$.
Next, we need to show that, at least sufficiently close to the fixed point and
for sufficiently small $\epsilon$, $F$ increases with $\epsilon$. We do this
term by term in the sum of $F$. First assume that the point $x_{1}$ is close
to fixed point value $1$ at zero mutation rate, i.e., $x_{1}=1-\delta$, and
that $\epsilon$ is small, so that $\epsilon<<P$ and $\delta<<P$. The first
term in $F$ is $x_{1}$ and does not depend on $\epsilon$. The second term,
$x_{1}$, is given by $f_{A}$,
$\displaystyle x_{2}$
$\displaystyle=f_{A}(1-\delta)=\frac{1}{2(T-P)}\left(P-\delta+\frac{\epsilon^{\prime}}{1-\delta}\right)\left(-1+\sqrt{1+\frac{4(T-P)\epsilon^{\prime}}{\left(P-\delta+\frac{\epsilon^{\prime}}{1-\delta}\right)^{2}}}\;\right)=$
$\displaystyle=\frac{\epsilon^{\prime}}{P}\left(1+\frac{\delta}{P}+O(\epsilon^{\prime})\right)+O(\epsilon^{\prime})^{2}+O(\delta)^{2}\;$
(18)
This term thus increases with $\epsilon^{\prime}$. For the next term $x_{3}$,
using Equation (13), we find that to first order in $\epsilon^{\prime}$ and
$\delta$,
$\displaystyle x_{3}$
$\displaystyle=f_{B}(x_{2},1-x_{1}-x_{2})=f_{B}\left(\frac{\epsilon^{\prime}}{P},\delta-\frac{\epsilon^{\prime}}{P}\left(1+\frac{\delta}{P}\right)\right)=$
$\displaystyle=\frac{\epsilon^{\prime}}{P}\left(1+\frac{2-P}{P}\delta+O(\delta)^{2}\right)+O(\epsilon^{\prime})^{2}+O(\delta)^{2}\;$
(19)
More generally, for $2<k<n$, repeatedly using Equation (13), we find
$\displaystyle x_{k}$
$\displaystyle=\frac{\epsilon^{\prime}}{P}\left(1+\frac{k-P(k-1)}{P}\delta+O(\delta)^{2}\right)+O(\epsilon^{\prime})^{2}+O(\delta)^{2}\;$
(20)
and finally, using Equation (16), that
$\displaystyle x_{n}$
$\displaystyle=\frac{\epsilon^{\prime}}{P}\left(1+\frac{(N-1)-P(N-2)}{P}\delta+O(\delta)^{2}\right)+O(\epsilon^{\prime})^{2}+O(\delta)^{2}\;$
(21)
From the linearisation we can conclude that when $x_{1}$ is sufficiently close
to 1, i.e., $\delta<<P$, the change in $F(x_{1})$ from an increase of
$\epsilon^{\prime}$ is given by the $n-1$ terms $x_{2},...,x_{n}$, or
$dF(x_{1})/d\epsilon^{\prime}\sim(n-1)/P>0$. Adding the fixed point
constraint, $F(x_{1})=0$, to the linearisation determines $x_{1}$ and thus
$\delta$ in terms of $\epsilon^{\prime}$. To first order in $\epsilon$ the
fixed point is given by
$\displaystyle x_{1}$ $\displaystyle\sim
1-(n-1)\frac{\epsilon^{\prime}}{P}+O(\epsilon)^{2}=1-(n-1)\frac{\epsilon}{nP}+O(\epsilon)^{2}\;,\text{and}$
(22) $\displaystyle x_{k}$
$\displaystyle\sim\frac{\epsilon}{nP}+O(\epsilon)^{2}\;\text{ for
}k=2,...,n\;$ (23)
where we have switched back to the original $\epsilon$. This analysis shows
that as $\epsilon$ increases from 0, the fixed point gradually moves into the
unit interval.
Since the fixed point is changed continuously as the mutation rate increases
from $\epsilon=0$, the eigenvalues of the Jacobian also change continuously.
The fixed point at $\epsilon=0$ is stable, with the largest eigenvalue being
$\lambda_{\text{max}}=-P<0$, and we can conclude that for sufficiently small
$\epsilon>0$, the real part of the largest eigenvalue remains negative. From
this we can conclude that _the fixed point associated with the Nash
equilibrium in the finitely repeated Prisoner’s Dilemma remains stable in the
simple strategy set if the mutation rate is sufficiently small_. This
concludes the proof of Proposition 1.
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This article is an open access article distributed under the terms and
conditions of the Creative Commons Attribution license
(http://creativecommons.org/licenses/by/3.0/).
|
arxiv-papers
| 2013-01-08T22:20:43 |
2024-09-04T02:49:40.006145
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kristian Lindgren, Vilhelm Verendel",
"submitter": "Vilhelm Verendel",
"url": "https://arxiv.org/abs/1301.1710"
}
|
1301.1723
|
# Study of dipole moments of LiSr and KRb molecules by quantum Monte Carlo
methods
Shi Guoa Michal Bajdich b Lubos Mitas a and Peter J. Reynoldsac ∗
a Department of Physics, North Carolina State University, Raleigh, NC 27695;
bJoint Center for Artificial Photosynthesis, Lawrence Berkeley Laboratory, and
Department of Chemical and Biomolecular Engineering, University of California,
Berkeley, CA 94720; cPhysics Division and Physical Sciences Directorate, Army
Research Office, Research Triangle Park, NC 27703
∗Corresponding author. Email: [email protected]
(final version received 15 March 2013)
###### Abstract
Heteronuclear dimers are of significant interest to experiments seeking to
exploit ultracold polar molecules in a number of novel ways including
precision measurement, quantum computing, and quantum simulation. We calculate
highly accurate Born-Oppenheimer total energies and electric dipole moments as
a function of internuclear separation for two such dimers, LiSr and KRb. We
apply fully-correlated, high-accuracy quantum Monte Carlo methods for
evaluating these molecular properties in a many-body framework. We use small-
core effective potentials combined with multi-reference Slater-Jastrow trial
wave functions to provide accurate nodes for the fixed-node diffusion Monte
Carlo method. For reference and comparison, we calculate the same properties
with Hartree-Fock and with restricted Configuration Interaction methods, and
carefully assess the impact of the recovered many-body correlations on the
calculated quantities. For LiSr we find a highly nonlinear dipole moment
curve, which may make this molecule’s dipole moment tunable through
vibrational state control.
###### keywords:
Ultracold polar molecules, LiSr, KRb, quantum Monte Carlo, fixed-node
approximation, electron correlation, quantum simulations.
## 1 Introduction
Motivated both by the desire for deeper understanding of basic physics and of
numerous potential applications, there has been a great deal of work over the
last few years devoted to cooling small molecules [1] and molecular ions [2].
This follows up on a few decades of exciting progress in cooling and trapping
of atoms and atomic ions. Molecules, of course, are far more challenging, due
to their complex internal structure, particularly their many internal degrees
of freedom, such as vibrational and rotational levels, in addition to fine and
hyperfine structure. A number of approaches have been pursued, with varying
degrees of success. While making molecules cold has been achieved in numerous
ways, reaching the ultra-cold regime, and particularly the quantum degenerate
regime, has been limited.
The most general methods of cooling molecules use buffer gases and supersonic
expansion [3, 4, 5], or velocity filtering [6] techniques. Another direct
approach is Stark decceleration [7]. On the other hand, direct laser cooling
of molecules, in analogy to the very successful approach for neutral atoms,
was long deemed impractical due to the complex level structures. The simplest
and most widespread method of direct laser cooling of atoms is Doppler
cooling, where radiative forces originate from momentum transfer to atoms from
a laser field, and subsequent spontaneous emission of slightly higher energy
photons into random directions. Repeating this optical cycle tens of thousands
of times cools neutral atoms very quickly to the Doppler limit (which is mass
dependent, but typically reaches sub-mK temperatures). For molecules, the
problem with the conventional scheme is that excited states can radiatively
decay out to a multitude of other states. This leads to decays that destroy
any closed cycling transitions. Exciting population from all these states back
to the starting point requires an impractically large number of lasers. Only
recently has laser cooling of a special class of molecules been demonstrated
[8, 9] based on insights from earlier work [10, 11].
Most successful at reaching into the ultra-cold regime are methods that create
the molecules from previously cooled atoms. However, the number of molecules
produced in this way is fairly small $(\approx 10^{4})$. Such methods include
photoassociation [12] and magnetoassociation (exploiting Feshbach resonances)
[13]. While the former has been a well-established technique to produce
homonuclear dimers from cold atoms, its application to produce heteronuclear
molecules from two different species of laser-cooled alkali atoms is more
recent. Such diatomic molecules can be polar, and this is of particular
interest (see, for example, reviews in [1, 4, 14, 15]).
The usefulness of polar molecules arises because they, unlike neutral atoms,
interact via the characteristic long-range, anisotropic dipolar interaction,
making them controllable by external electric fields. On the other hand, they
are less strongly coupled to the environment than ions, making them less prone
to decoherence [16]. This makes them attractive for quantum manipulation, such
as for quantum information processing [17, 18, 19, 20, 21, 22], and precision
measurements to test symmetries [23, 24, 25, 26] and the constancy of
fundamental “constants” [27, 28, 29, 30, 31, 32].
Additional interest lies in alkali-alkali earth dimers such as LiSr, one of
the molecules studied here. In contrast with alkali dimers, the unpaired spin
makes them able to be manipulated with both external electric and magnetic
fields. This provides an avenue for different physics to be explored, whether
in fundamental tests, as qubits, in optical lattices where the competing
interactions are important, or potentially even for use in atomic clocks where
atoms such as Sr are already showing great potential [33].
Cooling, trapping and quantum manipulation of polar molecules will have an
important impact on a diverse range of fields beyond just fundamental tests
and quantum computing. One such example is in condensed matter physics,
through quantum “emulation” or simulation of many-body quantum physics that is
inaccessible to even today’s (and any day’s) high-performance computers, let
alone to analytical solution (for example [34, 35, 36]). Cold polar molecules
trapped in optical lattices make an extremely interesting many-body system
(with long-range dipole-dipole interactions) promising a rich variety of novel
quantum phases, such as supersolids and topologically ordered states [35, 37].
One can imagine many other novel strongly-correlated quantum phases,
particularly in reduced dimensions [38, 39, 40]. Local control of the density
or orientation of polar molecules may allow one to create or simulate charge
density waves or (pseudo-) spin-density waves, or even a random, glassy
system.
Another and rapidly emerging area is ultra-cold chemistry [13, 14, 41, 42,
43]. In the regime where the de Broglie wavelength of the molecule becomes
comparable, or even orders of magnitude larger than the molecule itself, the
classical notion of a reaction coordinate is at odds with quantum mechanics.
Coherent population transfer methods (e.g., STIRAP) have already produced a
gas of fermionic 40K87Rb molecules with a temperature of a few hundred nano-
Kelvin at a phase-space density getting very close to that needed to achieve
degeneracy. In this regime one expects entirely new phenomena in looking at
chemical reactions at ultralow energies [44]. In addition to just potential
energy surfaces being relevant, the ultracold reaction rate is controlled by
quantum statistics and quantum coherence effects, including tunneling.
Moreover, quantum statistics only allows certain collisions, and tunneling
leads to threshold laws. In this regime, quantum control also can take on new
meaning, with resonance-mediated reactions and collective many-body effects
becoming prominent. Polar molecules also provide a handle for control through
application of relatively weak electric fields.
Among applications, cold molecules hold great promise for improving sensors of
all sorts. So far, all demonstrated matter-wave interferometric sensors
utilize atoms. Using dipolar molecules instead of atoms could lead to orders-
of-magnitude improvements in the sensitivity of such sensors [45]. The
advantage comes from the ability to guide the molecules with modest electric
field gradients, thereby creating steeper confining potentials. Steeper
potential gradients allow one to load waveguide structures more efficiently;
this leads to a larger number of particles and better statistical sensitivity.
Also because of the steeper potentials, molecules can be kept further away
from the wires and guiding surfaces, thus improving performance, as these are
major sources of decoherence. Moreover, because sensitivity (e.g., to
rotation) is proportional to the area enclosed by an interferometer, a large-
area storage ring (with electrostatic guiding) [46] becomes a possible
interferometer.
Bi-alkali molecules produced thus far include RbCs [12, 47], KRb [48, 49, 50,
51], NaCs [52] and LiCs [53, 54]. Photoassociation and magnetoassociation can
be combined with STIRAP to allow coherent population transfer into a low-
energy bound state, and significantly enhances the rate of molecular
production [48, 51, 55, 56].
The strength of the dipole moment in these diatomics is critical to their
possible uses, ranging from their potential as qubits, their use in precision
measurement experiments, the nature of their quantum phase transitions, as
well as simply their ability to be manipulated.
This motivates our study, which is focused on applying first principles
methods to two polar diatomics, one a bi-alkali, and one an alkali-alkali
earth. Specifically, we use quantum Monte Carlo in combination with basis set
methods to look carefully at KRb and LiSr, both in their ground states
($X^{1}\Sigma^{+}$ and $X^{2}\Sigma^{+}$, respectively).
## 2 Methods
Since the bonds in all these systems are weak, and consist of a mixture of
covalent and van der Waals bonding mechanisms, first principles calculations
tend to be quite involved and highly sensitive to basis sets, degree of
correlation included, nature of approximations, and other factors. Density
Functional Theory (DFT) for van der Waals systems is often less than
satisfactory, typically showing a varying degree of overbinding without any
obvious systematic trends. Several new DFT functionals with corrections for
dispersion interactions have been proposed very recently; see for example,
[57, 58]. However, thorough testing and benchmarking of these new DFT
approaches is necessary before they can be reliably applied across a variety
of systems. Dispersion interactions result from transient-induced
polarizations between the interacting constituents, and are therefore subtle
many-body effects which are difficult to capture in the functionals framework.
On the other hand, the powerful arsenal of basis-set correlated methods, such
as Configuration Interaction (CI), have their own challenges for systems such
as these. For example, the dipole moment can be very sensitive to the basis
set, and convergence for weakly bonded systems can be very slow [59]. In
addition, the dipole moment appears highly sensitive to the level of
correlation used, especially for problems which require multi-reference
treatment. This is well known, but a systematic study can be found in Ref.
[59]. Our CI results here also demonstrate this. It is therefore important to
explore other types of methods to understand the impact of many-body effects
more thoroughly. A highly accurate alternative approach is quantum Monte Carlo
(QMC) [60, 61]. QMC is very attractive since it is in principle exact; in
practice due to approximations it has a residual weak sensitivity to the size
of basis sets, and it captures the correlations at a level of 90-95% [60, 62,
63]. That is something that is quite difficult to achieve by correlated
methods based on expansions in basis sets.
In fact, there are previous studies of molecular dipole moments calculated by
QMC methods for a few molecular systems. Schautz and colleagues carried out
QMC study of the carbon monoxide molecule and obtained a very good estimate
for this non-trivial problem in which correlation reverses the sign of the
dipole obtained at the Hartree-Fock level [64]. The dipole moment of the
lithium hydride molecule was computed by fixed-node diffusion Monte Carlo
(DMC) in a good agreement with experiment [65]. Recently, several transition
metal monoxide molecules have also been calculated by QMC methods. Besides the
binding energies and equilibrium bond lengths, the dipole moments were also
calculated, all with reasonably good agreement to experiment [66]. In
addition, transition dipole moments such as for the Li atom were calculated by
QMC approaches some time ago [67].
### 2.1 Sampling
For our calculations here, we employ the two most common QMC methods,
variational and fixed-node diffusion Monte Carlo (VMC and DMC) [60, 61, 68,
62, 63, 69, 70]. We summarize just the main points of the algorithms. VMC is
based on the variational principle, and the Monte Carlo method of evaluation
of multi-variate integrals. Given a trial wavefunction $\Psi_{T}(R)$, we
sample a set of points in the 3$N$ dimensional space of electron
“configurations” according to the probability density $|\Psi_{T}(R)|^{2}$,
where $R$ denotes the set of spatial coordinates of all electrons. The
expectation value of an operator $A$ is evaluated over $M$ samples, and is
given by
$\langle
A\rangle_{VMC}=\frac{1}{M}\sum_{m=1}^{M}\Psi_{T}^{-1}(R_{m})[A\Psi_{T}(R_{m})]+\epsilon_{stat}$
where $m$ labels the samples (“random walkers”), and $\epsilon_{stat}$ is the
statistical error, which scales as $O(1/\sqrt{M})$.
Diffusion Monte Carlo in contrast is based on a stochastic process which
solves the imaginary-time many-body Schrodinger equation, projecting out the
ground state through iteration of a short-time Green’s function as
$f(R,t+\tau)=\int G(R\leftarrow R^{\prime},\tau)f(R^{\prime},t)dR^{\prime},$
where $f(R,t)=\Phi(R,t)\Psi_{T}(R)$. The wave function
$\Phi(R,t\rightarrow\infty)$ is the solution we are seeking, while
$\Psi_{T}(R)$ is a trial function, and $G(R\leftarrow R^{\prime},\tau)$ is the
Green’s function or propagator that evolves $f$ [61]. The fixed-node
approximation is imposed by preventing walkers from crossing the nodal
hypersurface of $\Psi_{T}$ (i.e., the zero locus of $\Psi_{T}(R)$). This
condition restricts $f(R,t)\geq 0$ so that the sampled probability density
remains non-negative everywhere. In this manner we can evade the well-known
fermion sign problem [60]. Since the nodes of the trial function are not
necessarily precisely those of the exact wave function, the fixed-node
condition introduces a bias. From a large body of work using this method, we
know that the typical size of the fixed-node bias is between 5 and 10% of the
correlation energy, essentially for all systems, including molecules, clusters
and solids [62].
### 2.2 Trial functions
The quality of the trial wave function plays two roles. First, highly accurate
trial functions lead to smaller statistical fluctutations and faster QMC
sampling and convergence. Second, better trial functions typically go hand-in-
hand with improvements of the nodal hypersurfaces, and hence result in smaller
fixed-node bias. Clearly, we want to use the best trial function we can.
Traditional methods, including correlated approaches, provide a convenient
starting point in this respect. In addition, however, quantum Monte Carlo can
employ explicitly correlated wave functions as well, without prohibitive
computational cost. This enables us to reach beyond the limits of traditional
approaches, even before beginning to project out the true (fixed node) ground
state.
In our calculations, we have used the Slater-Jastrow trial wave functions
$\Psi_{T}(R)=\Psi_{A}(R)e^{J(R)}$
which embody both the traditional computational chemistry starting point and
explicit correlation in the two factors. The antisymmetric part $\Psi_{A}(R)$
is often given by a single configuration state function which is a product of
Slater spin-up and spin-down determinants, or, more generally, by a linear
combination of such configurations. This linear combination of configurations
with, say, single and double excitations, becomes the Configuration
Interaction (CI) expansion
$\Psi_{A}=c_{0}\Psi_{0}+\sum_{ar}c_{a}^{r}\Psi_{a}^{r}+\sum_{a<b,r<s}c_{ab}^{rs}\Psi_{ab}^{rs}.$
The orbitals for our Slater determinats are obtained from Hartree-Fock
calculations, and our CI expansion is calculated using the GAMESS package.
The Jastrow factor contains electron-ion, electron-electron, and electron-
electron-ion correlation terms. The electron-electron correlation functions
build in the correct electron-electron cusps. The actual forms we use for the
Jastrow factor are described in detail in Ref [63]. The Jastrow variational
parameters and the CI expansion coefficients are optimized in VMC by
minimizing a linear combination of the variational energy and the variance of
the local energy, $[H\Psi_{T}]/\Psi_{T}$. The optimized trial function is then
employed in our fixed-node DMC runs.
### 2.3 Mixed and pure expectation values
In diffusion Monte Carlo, expectation values of quantities which commute with
the Hamiltonian are exact [60]; however, expectation values of non-commuting
operators, $\hat{O}$, are estimated by mixed estimators, i.e., with the
distribution $\Psi_{T}\Phi$, instead of $\Phi^{2}$. One can correct for this
in a number of ways [67, 71]. A commonly used approximation is
$<\Phi|\hat{O}|\Phi>\approx
2<\Phi|\hat{O}|\Psi_{T}>-<\Psi_{T}|\hat{O}|\Psi_{T}>.$
Exact sampling of $\Phi^{2}$, while computationally more complex and
significantly more costly, is also possible [72, 62, 71]. Accuracy gained from
exact sampling methods is limited by any approximations involved, both
fundamental and technical. These include, e.g., the fixed-node restriction and
the localization approximation in treating the nonlocal effective core
potentials (see below). Since the optimization of the wave function is done
stochastically on finite samples, it is difficult to eliminate a possible
optimization bias reflecting differences in the localization error with
different Jastrows. This is especially so in the case of weakly bonded systems
such as LiSr (with bonding of the order of 0.006 a.u.). Therefore, with any
likely improvements to be invisible, the results below are based on the mixed
estimator, which provides a good balance between accuracy and efficiency of
the calculations.
### 2.4 Effective core potentials
For heavy atoms we use effective core potentials (ECPs) to eliminate the core
electrons. In particular, we employ energy consistent, small-core ECPs [73,
74] for the atoms beyond the first row. For this work we explored ECPs with
different core sizes, and we found that the results appeared to converge with
the small-core ECPs. In fact, two different sets of small-core ECPs provided
essentially the same overall picture of the electronic structure as described
below. In the context of the QMC calculations, the ECPs are localized by a
projection onto the trial function, making the ECPs contributions to the local
energy formally similar to the other terms in the Hamiltonian, as described in
detail elsewhere [75, 76].
## 3 Calculational Details
The pseudopotential we ultimately used for the bulk of the calculations here
removes small cores for the elements beyond the first row, with the resulting
configurations of valence electrons in KRb given as K($3s^{2}3p^{6}4s^{1}$)
and Rb($4s^{2}4p^{6}5s^{1}$). Gaussian basis sets used in the calculations
were gradually increased in size until we observed saturation to about
$\approx$ 0.001 $E_{h}$ at the self-consistent level; basis sets of
25s22p13d/[4s4p3d] were used for both K and Rb. In the LiSr calculation, we
kept all Li atom electrons in the valence space, while for Sr we eliminated 18
core electrons, so that the valence configuration became
Sr$(4s^{2}4p^{6}5s^{2})$. Quite extensive basis sets for Sr and Li, with sizes
7s6p5d1f/[7s4p2d1f] and 27s7p1d/[7s5p1d], provided saturated HF energies at
the level of 0.001 $E_{h}$ or better.
A proper description of the dipole moments requires high-quality correlated
methods. This is true in particular for LiSr, which does not exhibit any
binding at the HF level. (KRb, on the other hand, has a single $\sigma$ bond,
so that using a single-reference wave function, and CI with single and double
excitations, was deemed to be appropriate in this case.) For LiSr we explored
several correlation levels and sizes of active occupied and virtual spaces. In
particular, we carried out singles and doubles (SD) with 5 active occupied and
70 virtual orbitals; SD and triples (SDT) with 5 active occupied and 55
virtual orbitals; SDT and quadruples (SDTQ) with 3 active occupied and 48
virtual orbitals. Excitations beyond doubles proved to be important for the
LiSr dipole moment. Even with quadruples included, it remained difficult to be
certain that the correlations were adequately described.
This is where QMC methods provide an important alternative, and a powerful
option to probe for correlations beyond what is practical with CI. For QMC,
the CI wave functions only serve as a starting point—as trial functions. They
are truncated to only their most significant terms by imposing a cutoff on the
weights of the configuration state functions. Different cutoff values from
0.05 to 0.01 were tested. It turned out that the optimal cutoff value is
around 0.03-0.05. (Values smaller than this no longer decreased the QMC energy
within the detectability of the error bars.) Using a cutoff of 0.05 for LiSr
and 0.03 for KRb, we were able to improve the nodal surfaces (evidenced by
lower QMC energy) with reasonable efficiency and statistical consistency.
## 4 Results and discussion
The energy that we obtain as a function of bond length is shown in Fig. 1, for
LiSr in Fig. 1a, and KRb in Fig. 1b. We compare our best CI calculation with
the diffusion QMC (DMC) results. For LiSr, we show the QMC results with two
different trial functions, DMC1 and DMC2. We see that the QMC correlation
energy for both molecules is considerably better than that found by the CI
method, and provides significant overall improvement of the potential energy
surfaces. We show only correlated methods, as HF gives no binding at all for
LiSr.
As previously discussed, the dipole moment of LiSr is particularly
sensitive—to pretty much everything (basis set size, level of correlation in
the theory, nodes in the QMC trial function, etc.). We show in Fig. 2 this
sensitivity with respect to amount of correlation in the theory. As is clear,
CI SDT, which is often sufficient, has not converged here. It would be
difficult to know if even CI SDTQ is sufficient were it not for our QMC
calculation. In Fig. 3a we compare the CI SDTQ result for the dipole moment
with our two QMC calculations with different trial functions. While the two
QMC results are not identical, they vary immensely less than the different CI
calculations. Moreover, the QMC with refined nodes from the multi-reference
trial function gives almost the same results as CI SDTQ. This provides
additional confidence in this result. This can be seen better in the expanded
Figure 4, where we compare the dipole moment of LiSr as computed from only
correlated methods, and compare our results against a recent result in the
literature as well.
|
---|---
Figure 1: The binding energies of (a) LiSr and (b) KRb as a function of the bond length with different methods. Two optimized Jastrow DMC trial wave functions are used for LiSr. DMC1 denotes a single Slater determinant, while DMC2 is multi-reference with a cutoff weight of 0.05. For KRb, the DMC trial wave function is multi-reference, with a cutoff weight of 0.03, and again an optimized Jastrow. Here and in subsequent figures, the statistical error bars of the QMC results are approximately of the symbol size. Additional deviations of the QMC results reflect the small errors discussed in Section 2. The CI curves correspond to CI/SDTQ for LiSr and CI/SD for KRb. Figure 2: The dipole moment of LiSr calculated by the CI method with SD, SDT and SDTQ levels of correlation, as described in the text. |
---|---
Figure 3: The dipole moment of (a) LiSr and (b) KRb as a function of
internuclear distance, as obtained in DMC, HF and CI methods (CI/SDTQ for LiSr
and CI/SD for KRb). The trial wave functions, DMC, DMC1 and DMC2 are the same
as described in Fig. 1. Figure 4: Dipole moment for LiSr from our best CI
and QMC calculations compared with results from Ref.[77]. The pronounced
nonlinearity allows switching of the value of $d$ by state selection to a
vibrational state near dissociation. See text.
Figure 3b shows the dipole moment results for KRb. The bonding behavior, as
well as the dipole moment of KRb, is rather straightforward to understand.
There is a single $\sigma$ bond formed from the K(4s) and Rb(5s) atomic
states. The dipole moment monotonically increases with the bond length,
corresponding to growth of the distance between the effective charges, and
also some possible growing enhancement of the electron density in the region
of the K atom. This trend is observed already at the HF level, indicating that
it is driven by the one-particle self-consistent balance of the energy
contributions. However, quantitatively, the HF dipole moment can be seen to be
too large. This is because electron correlation decreases the charge
polarization over the whole range of calculated interatomic distances. This
can be equivalently understood as a result of the well-known HF bias towards
larger ionicity of the bonds. The absence of correlation tends to make the
exchange more prominent, which drives the electronic structure towards
stronger charge polarization. It is interesting to observe that the dipole
moments computed from both of our correlated methods appear to be rather
close, although the degree of description of the many-body effects is very
different, with the CI recovering only about 30% of the correlation energy
compared to $\approx$ 90% in QMC.
On the other hand, LiSr has a more complicated electronic structure due to the
presence of the two $(5s^{2})$ electrons in the outermost shell of the Sr
atom, with the resulting open-shell $X^{2}\Sigma^{+}$ molecular ground state.
The last valence electron occupies an anti-bonding $\sigma$ level which is
formed by the combination of Li(2s) and Sr(5s) atomic states. The overall bond
is thus weaker, and this accounts for the lack of binding seen in the HF
method. Even after including $\approx 90$% of the correlation energy via QMC,
as we have done here, the binding is small, only of the order of $\approx$
0.25 eV. Even more interestingly, the dipole moment changes sign as a function
of separation! This behavior is visible in all three methods, implying that
one can trace its origin to changes at the single-particle level. Indeed, by
plotting the isosurfaces of the Hartree-Fock HOMO (antibonding $\sigma$) and
LUMO orbitals (see Figs. 5 and 6) we see that as the bond length increases,
the nature of the orbitals change very significantly. These changes lead to
sizeable restructuring of the electron density already at the HF level, with
the resulting sign change of the dipole moment as plotted in Figs. 2, 3a, and
4.
In addition, due to the reordering of the LUMO levels with varying bond length
as illustrated in Fig.6, the CI expansion coefficients change significantly as
well. The leading excited configuration for bond length $d=5$ a.u. is the
double excitation
$\sigma_{b}^{2}\sigma_{a}\to(\pi_{bx}^{2}+\pi_{by}^{2})\sigma_{a}$ where
$\sigma_{b}$ and $\sigma_{a}$ denote the highest bonding and anti-bonding
valence molecular orbitals, respectively. One of the corresponding $\pi_{b}$
orbitals is plotted in the upper part of Fig. 6. On the other hand, for bond
length of $d=7.5$ a.u., the lowest virtual state is the next bonding orbital
$2\sigma_{b}$, and the corresponding double excitation
$\sigma_{b}^{2}\sigma_{a}\to 2\sigma_{b}^{2}\sigma_{a}$ becomes dominant. This
restructuring, both in the one-particle (Fig. 6) and the many-body framework,
has a strong impact on the correlations. The dipole moment is affected, as is
seen both from smaller absolute values as compared to HF, as well as the shift
of the sign change towards longer bond lengths. Overall, our results indicate
that LiSr exhibits a sizeable dipole moment mainly for smaller interatomic
distances, while at larger bond lengths the dipole decreases significantly.
Figure 5: Isosufaces of the positive and negative lobes of the highest
occupied molecular orbital (HOMO) of LiSr, for interatomic distances 5 a.u.
(top) and 7.5 a.u. (bottom).
Figure 6: Isosurfaces of the positive and negative lobes of the lowest
unoccupied molecular orbital (LUMO) of LiSr, for interatomic distances 5 a.u.
(top) and 7.5 a.u. (bottom).
The quality of the trial wave function and its nodal surface has a
significantly more pronounced impact on the LiSr dipole moment result than on
the KRb moment, and one cannot rely on a single configuration for the former.
The inclusion of the most important configurations from the CI expansion
improves the accuracy of the DMC estimations quite significantly. It is
interesting, if somewhat unexpected, that our QMC results obtained with our
best trial function, apart from the statistical noise, agree remarkably well
with our most accurate CI calculations. This consistency is reassuring since
the employed methods are largely independent, and capture the electron
correlations in very different manners. The need to employ triples and
quadruples in CI indicates that correlations beyond multi-reference
configurations based on singles/doubles excitations are important. On the QMC
side, the single configuration trial function already captures the main
correction to the HF dipole moment, due to the correlations, as can be seen in
Fig 3a. However, for high accuracy results, the complicated orbital
restructuring of the LiSr molecule clearly requires a multi-reference
treatment which includes the dominant non-dynamical effects explicitly,
thereby improving the nodal surface. This accounts for the difference seen in
Fig 3a between DMC1 and DMC2. This observation is very much in line with
previous QMC calculations of other systems with significant multi-reference or
near-degeneracy effects.
The spectroscopic constants for these molecules have been computed, and are
shown in Tables 1 and 2, together with the most recently published
calculations [78, 77] which used much less fully converged coupled cluster and
CI methods. For LiSr we calculated the dipole moment also in the first
vibrationally excited state, and we found that it is basically the same as in
the ground state within the given error bar. This can be understood from the
fact that our results show the dipole being approximately linear with bond
length over the range $\approx$ 6.0 - 7.7 a.u., and this interval covers the
spatial range of both the ground and the first vibrational states.
Nevertheless, the nonlinearity of the LiSr dipole moment may be exploitable
for vibrational dependent control of $d$. In particular, it appears that the
highest bound vibrational states might have considerably smaller dipole
moments, making it possible to largely switch off (and on) the dipole moment
through tailored excitations.
The overall agreement between the calculational approaches, considering how
small the quantities are, is very reasonable. The differences reflect the
systematic biases of the approaches used, and clearly illustrate the challenge
of describing weakly-bonded systems with high accuracy. Since these
calculations reflect the state of the art of these computational
methodologies, this shows what is currently feasible. It is clearly desireable
to increase the accuracy further, in order to decrease the uncertainties. This
can be accomplished with further development of the methods; in particular,
for QMC approaches one would need much higher sensitivity in quantities which
do not commute with the Hamiltonian so that the optimization of much larger
multi-reference wave functions would be feasible. This is a promising
direction for future explorations.
Table 1: Spectroscopic constants for the LiSr molecule: bond length $R_{e}$, potential well depth $D_{e}$, harmonic constant $\omega_{e}$, and the ground state averaged dipole moment $\langle d\rangle$. The values in the row labeled DMC are calculated from the DMC2 data displayed in Figs. 1a and 3a. Numbers in parentheses give the statistical uncertainty in the last significant figure. | $R_{e}$ (a.u.) | $D_{e}$ ($cm^{-1}$) | $\omega_{e}$($cm^{-1}$) | $\langle d\rangle$ (D)
---|---|---|---|---
DMC | 6.80(5) | $2.7(3)\times 10^{3}$ | 167(7) | \- 0.14(2)
Ref [77] | 6.71 | $2.401\times 10^{3}$ | 184 | \- 0.244
Ref [78] | 6.57 | $2.587\times 10^{3}$ | 185 | \- 0.340
Table 2: Spectroscopic constants for the KRb molecule, defined as in Table 1. The values in the row labeled DMC correspond to the data displayed in Figs. 1b and 3b. | $R_{e}$ (a.u.) | $D_{e}$ ($cm^{-1}$) | $\omega_{e}$($cm^{-1}$) | $\langle d\rangle$(D)
---|---|---|---|---
DMC | 7.58(5) | $3.8(3)\times 10^{3}$ | 77(2) | 0.58(2)
Ref [78] | 7.63 | $4.199\times 10^{3}$ | 76 | 0.615
## 5 Conclusions.
We have carried out careful calculations of the dipole moments and potential
energy curves for the KRb and LiSr molecules using Hartree-Fock, Configuration
Interaction, and fixed-node diffusion quantum Monte Carlo methods. The
calculations show significant effects of the electronic correlations on the
magnitude of the dipole moments over the whole range of the investigated
interatomic distances.
Although single determinant QMC already captures most of the correlation
energy, the low-lying excitations need to be explicitly included into the
trial function for the LiSr dimer, particularly for accurate computation of
the dipole moment. The need is clear from the fact that the weights of some
configurations beyond the reference HF configuration are significant.
Comparisons of CI(SD) with CI(SDTQ) clearly illustrates the multi-reference
nature of the ground state. The accuracy of our results is supported by the
consistency between our most extensively correlated CI approach and our
methodologically quite distinct QMC results.
In addition, the high percentage of the correlation energy recovered by QMC
($\approx$ 90%) captures most of the dynamical correlation, and therefore
provides much more extensive insight into how correlations affect sensitive
molecular properties.
Acknowledgments. We gratefully acknowledge support by the U.S. Army Research
Office.
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|
arxiv-papers
| 2013-01-08T23:51:10 |
2024-09-04T02:49:40.016615
|
{
"license": "Public Domain",
"authors": "Shi Guo, Michal Bajdich, Lubos Mitas, and Peter J. Reynolds",
"submitter": "Peter Reynolds",
"url": "https://arxiv.org/abs/1301.1723"
}
|
1301.1746
|
# Generalized Secure Transmission Protocol for Flexible Load-Balance Control
with Cooperative Relays in Two-Hop Wireless Networks
Yulong Shen13, Xiaohong Jiang2 and Jianfeng Ma1
1School of Computer Science and Technology, Xidian University, China 2School
of Systems Information Science, Future University Hakodate, Japan
3Email:[email protected]
###### Abstract
This work considers secure transmission protocol for flexible load-balance
control in two-hop relay wireless networks without the information of both
eavesdropper channels and locations. The available secure transmission
protocols via relay cooperation in physical layer secrecy framework cannot
provide a flexible load-balance control, which may significantly limit their
application scopes. This paper extends the conventional works and proposes a
general transmission protocol with considering load-balance control, in which
the relay is randomly selected from the first $k$ preferable assistant relays
located in the circle area with the radius $r$ and the center at the middle
between source and destination (2HR-($r,k$) for short). This protocol covers
the available works as special cases, like ones with the optimal relay
selection ($r=\infty$, $k=1$) and with the random relay selection ($r=\infty$,
$k=n$ i.e. the number of system nodes) in the case of equal path-loss, ones
with relay selected from relay selection region ($r\in(0,\infty),k=1$) in the
case of distance-dependent path-loss. The theoretic analysis is further
provided to determine the maximum number of eavesdroppers one network can
tolerate to ensure a desired performance in terms of the secrecy outage
probability and transmission outage probability. The analysis results also
show the proposed protocol can balance load distributed among the relays by a
proper setting of $r$ and $k$ under the premise of specified secure and
reliable requirements.
###### Index Terms:
Two-Hop Wireless Networks, Relay Cooperation, Physical Layer Secrecy,
Transmission outage, Secrecy Outage.
## I Introduction
Wireless networks have the promising applications of in many important
scenarios (like battlefield networks, emergency networks, disaster recovery
networks). However, Due to the energy constrained and broadcast properties,
the consideration of secrecy and lifetime optimization in such networks is of
great importance for ensuring the high transmission efficiency and
confidentiality requirements of these applications. Two-hop wireless networks,
as a building block for large multi-hop network system, have been a class of
basic and important networking scenarios [1]. The analysis and design of
transmission protocol in basic two-hop relay networks serves as the foundation
for secure information exchange of general multi-hop network system.
For the lifetime optimization, an uneven use of the nodes may cause some nodes
die much earlier, thus creating holes in the network, or worse, leaving the
network disconnected, which is critical in military or emergency networks. For
this problem, a lot of protocols were proposed to balance the traffic across
the various relay nodes and avoids overloading any relay node in various
wireless networks, especially energy constrained wireless environments (like
wireless sensor networks) [7-16](see Section V for related works). We notice
there is tradeoff between the load-balance capacity and transmission
efficiency and still no approaches can flexibly control it. Regarding the
secrecy, the traditional cryptographic approach can provide a standard
information security. However, the everlasting secrecy can not be achieved by
such approach, because the adversary can record the transmitted messages and
try any way to break them [12]. Especially, recent advances in high-
performance computation (e.g. quantum computing) further complicate acquiring
long-lasting security via cryptographic approaches [13]. This motivates the
consideration of signaling scheme in physical layer secrecy framework to
provide a strong form of security, where a degraded signal at an eavesdropper
is always ensured such that the original data can be hardly recovered
regardless of how the signal is processed at the eavesdropper [14][15][16].
The secure and reliable transmission in physical layer secrecy framework for
two-hop relay wireless networks has been studied and a lot of secure
transmission protocols were proposed in [17-28](see Section V for related
works). These works mainly focus on the maximum secrecy capacity and minimum
energy consumption, in which the system node with the best link condition to
source and destination is selected as information relay. These protocols are
attractive in the sense that provides very effective resistance against
eavesdroppers. However, since the channel state is relatively constant during
a fixed time period, some relay nodes with good link conditions always prefer
to relay packages, which results in a severe load-balance problem and a quick
node energy depletion. Such, these protocol is not suitable for energy-limited
wireless networks (like wireless sensor networks). In order to realize load-
balance, Y. Shen et al. further proposed a random relay selection protocol
[29][30], in which the relay node is random selected from the system nodes.
However, this protocol has lower transmission efficiency. Such, it is only
suitable for large scale wireless network environment with stringent energy
consumption constraint.
In summary, the available secure transmission protocols cannot provide a
flexible load-balance control, which may significantly limit their application
scopes. This paper extends conventional secure cooperative transmission
protocols to a general case to enable the load-balance to be flexibly
controlled in the two-hop relay wireless networks without the knowledge of
eavesdropper channels and locations. The main contributions of this paper are
as follows:
* •
This paper proposes a new transmission protocol 2HR-($r,k$) for two-hop relay
wireless network without the knowledge eavesdropper channels and locations,
where the relay is randomly selected from the first $k$ preferable assistant
relays located in the circle area with the radius $r$ and the center at the
middle between source and destination. This protocol is general protocol, and
can flexibly control the tradeoff between the load-balance among relays and
the transmission efficiency by a proper setting of $k$ and $r$ under the
premise of specified secure and reliable requirements.
* •
In case that the path-loss is identical between all pairs of nodes, theoretic
analysis of 2HR-($r,k$) protocol is provided to determine the corresponding
exact results on the number of eavesdroppers one network can tolerate to
satisfy a specified requirement and shows that the 2HR-($r,k$) protocol covers
all the available secure transmission protocols as special cases, like ones
with the optimal relay selection ($r=\infty$, $k=1$) [19][20][27][29] and with
the random relay selection ($d=\infty$, $k=n$ i.e. the number of system
nodes)[29][30].
* •
In case that the path-loss between each pair of nodes also depends on the
distance between them, a coordinate system is presented and the theoretic
analysis of 2HR-($r,k$) protocol is provided to determine the corresponding
exact results on the number of eavesdroppers one network can tolerate to
satisfy a specified requirement and shows that the 2HR-($r,k$) protocol covers
all the available secure transmission protocols as special cases, like ones
with relay selected from relay selection region ($r\in(0,\infty),k=1$)[30].
The remainder of this paper is organized as follows. Section II presents
system models and the 2HR-($r,k$) protocol. Section III presents the theoretic
analysis in case of equal path-loss between all node pairs. Section IV
presents the theoretic analysis in case that path-loss between each node pair
also depends on their relative locations. Section V is related works and
Section VI concludes this paper.
## II System Models and 2HR-($r,k$) Protocol
### II-A Network Model
A Two-hop wireless network scenario is considered where a source node $S$
wishes to communicate securely with its destination node $D$ with the help of
multiple relay nodes $R_{1}$, $R_{2}$, $\cdots$, $R_{n}$. Also present in the
environment are $m$ eavesdroppers $E_{1}$, $E_{2}$, $\cdots$, $E_{m}$ without
knowledge of channels and locations. The relay nodes and eavesdroppers are
independent and also uniformly distributed in the network, as illustrated in
Fig.1. Our goal here is to design a general protocol to ensure the secure and
reliable information transmission from source $S$ to destination $D$ and
provide flexible load-balance control among the relays.
Figure 1: System scenario: Source $S$ wishes to communicate securely with
destination $D$ with the assistance of finite relays $R_{1}$, $R_{2}$,
$\cdots$, $R_{n}$ ($n$=5 in the figure) in the presence of passive
eavesdroppers $E_{1}$, $E_{2}$, $\cdots$, $E_{m}$ ($m$=5 in the figure).
Cooperative relay scheme is used in the two-hop transmission.
### II-B Transmission Model
Consider the transmission from a transmitter $A$ to a receiver $B$, and denote
the $i^{th}$ symbol transmitted by node $A$ by $x_{i}^{\left(A\right)}$. We
assume that all nodes transmit with the same power $E_{s}$ and path-loss
between all pairs of nodes is independent. We denote the frequency-
nonselective multi-path fading from $A$ to $B$ by $h_{A,B}$. Under the
condition that all nodes in a group of nodes, $\mathcal{R}$, are generating
noises, the $i^{th}$ signal received at node $B$ from node $A$, denoted by
$y_{i}^{\left(B\right)}$, is determined as:
$y_{i}^{\left(B\right)}=\frac{h_{A,B}}{d_{A,B}^{\alpha/2}}\sqrt{E_{s}}x_{i}^{\left(A\right)}+\sum_{A_{j}\in\mathcal{R}}\frac{h_{A_{j},B}}{d_{A_{j},B}^{\alpha/2}}\sqrt{E_{s}}x_{i}^{\left(A_{j}\right)}+n_{i}^{\left(B\right)}$
where $d_{A,B}$ is the distance between node $A$ and $B$, $\alpha\geq 2$ is
the path-loss exponent, $\left|h_{A,B}\right|^{2}$ is exponentially
distributed and without loss of generality, we assume that
$E{\left[\left|h_{A,B}\right|^{2}\right]}=1$. The noise
$n_{i}^{\left(B\right)}$ at receiver $B$ is assumed to be i.i.d complex
Gaussian random variables with mean $N_{0}$. The SINR $C_{A,B}$ from $A$ to
$B$ is then given by
$C_{A,B}=\frac{E_{s}\left|h_{A,B}\right|^{2}d_{A,B}^{-\alpha}}{\sum_{A_{j}\in\mathcal{R}}E_{s}{\left|h_{A_{j},B}\right|^{2}d_{A_{j},B}^{-\alpha}}+N_{0}/2}$
For a legitimate node and an eavesdropper, we use two separate SINR thresholds
$\gamma_{R}$ and $\gamma_{E}$ to define the minimum SINR required to recover
the transmitted messages for legitimate node and eavesdropper, respectively.
Therefore, a system node (the selected relay or destination) is able to decode
a packet if and only if its received SINR is greater than $\gamma_{R}$,
whereas each eavesdropper try to achieve target SINR $\gamma_{E}$ to recover
the transmitted message. However, from an information-theoretic perspective,
we can map to a secrecy rate formulation
$R\geq\frac{1}{2}\log(1+\gamma_{R})-\frac{1}{2}\log(1+\gamma_{E})$ [31].
Hence, we can also think the $\gamma_{R}$ and $\gamma_{E}$ can be set by the
desired secrecy rate of the system.
### II-C 2HR-($r,k$) Protocol
Notice the available transmission protocols have their own advantages and
disadvantages in terms of the transmission efficiency and energy consumption,
and thus are suitable for different network scenarios. With respect to these
protocols as special cases, a general transmission protocol 2HR-($r,k$) is
proposed to control the balance of load distributed among the relays and works
as follows.
1. 1.
_Relay selection region determination_ : The circle area, with radius $r$ and
the center at the middle point between source $S$ and destination $D$, is
determined as relay selection region.
2. 2.
_Channel measurement_ : The source $S$ and destination $D$ broadcast a pilot
signal to allow each relay to measure the channel from $S$ and $D$ to itself.
The relays, which receive the pilot signal, can accurately calculate
$h_{S,R_{j}},j=1,2,\cdots,n$ and $h_{D,R_{j}},j=1,2,\cdots,n$.
3. 3.
_Candidate relay selection_ : The relays with the first $k$ large
$min\left(|h_{S,R_{j}^{r}}|^{2},|h_{D,R_{j}^{r}}|^{2}\right)$ form the
candidate relay set $\mathfrak{R}$. Here, $R_{j}^{r}$ denotes the $j$-th relay
node in the relay selection region.
4. 4.
_Relay selection_ : The relay, indexed by $j^{\ast}$, is selected randomly
from candidate relay set $\mathfrak{R}$. Using the same method with Step 2,
each of the other relays $R_{j},j=1,2,\cdots,n,j\neq j^{\ast}$ in network
exactly knows $h_{R_{j},R_{j^{\ast}}}$.
5. 5.
_Two-Hop transmission_ : The source $S$ transmits the message to
$R_{j^{\ast}}$, and concurrently, the relay nodes with indexes in
$\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$ transmit noise to
generate interference at eavesdroppers. The relay $R_{j^{\ast}}$ then
transmits the message to destination $D$, and concurrently, the relay nodes
with indexes in $\mathcal{R}_{2}={\left\\{j\neq
j^{\ast}:|h_{R_{j},D}|^{2}<\tau\right\\}}$ transmit noise to generate
interference at eavesdroppers.
_Remark 1_ : The load is completely balanced among the relays in the candidate
relay set $\mathfrak{R}$ whose size is determined by parameter $r$ and $k$ in
the 2HR-($r,k$) protocol. Notice that a too larger $r$ and $k$ may lead to
larger size of the candidate relay set $\mathfrak{R}$. Thus, the load-balance
can be flexibly controlled by a proper setting of the parameter $r$ and $k$ in
terms of network performance requirements.
_Remark 2_ : The parameter $\tau$ involved in the 2HR-($r,k$) protocol serves
as the threshold on path-loss, based on which the set of noise generating
relay nodes can be identified. Notice that a too large $\tau$ may disable
legitimate transmission, while a too small $\tau$ may not be sufficient for
interrupting all eavesdroppers. Thus, the parameter $\tau$ should be set
properly to ensure both secrecy requirement and reliability requirement.
_Remark 3_ : In the case that there is equal path-loss between all pairs of
nodes, i.e., $d_{A,B}=1$ for all $A\neq B$, the channel state information is
independent of the parameter $r$ in 2HR-($r,k$) protocol. Since the parameter
$r$ is no effect on relay selection, the relay selection region is the whole
network area with $r=\infty$. Therefore, 2HR-($r,k$) protocol is castrated as
2HR-($\infty,k$) in case of equal path-loss between all node pairs.
### II-D Transmission Outage and Secrecy Outage
For a Two-hop relay transmission from the source $S$ to destination $D$, we
call transmission outage happens if $D$ can not receive the transmitted
packet. We define the transmission outage probability, denoted by
$P_{out}^{\left(T\right)}$, as the probability that transmission outage from
$S$ to $D$ happens. For a predefined upper bound $\varepsilon_{t}$ on
$P_{out}^{\left(T\right)}$, we call the communication between $S$ and $D$ is
reliable if $P_{out}^{\left(T\right)}\leq\varepsilon_{t}$. Similarly, we
define the transmission outage events $O_{S\rightarrow R_{j^{\ast}}}^{(T)}$
and $O_{R_{j^{\ast}}\rightarrow D}^{(T)}$ for the transmissions from $S$ to
the selected relay $R_{j^{\ast}}$ and from $R_{j^{\ast}}$ to $D$,
respectively. Due to the link independence assumption, we have
$\displaystyle P_{out}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\right)$
(1) $\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\right)$
Regarding the secrecy outage, we call secrecy outage happens for a
transmission from $S$ to $D$ if at least one eavesdropper can recover the
transmitted packets during the process of this two-hop transmission. We define
the secrecy outage probability, denoted by $P_{out}^{\left(S\right)}$, as the
probability that secrecy outage happens during the transmission from $S$ to
$D$. For a predefined upper bound $\varepsilon_{s}$ on
$P_{out}^{\left(S\right)}$, we call the communication between $S$ and $D$ is
secure if $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$. Similarly, we define
the secrecy outage events $O_{S\rightarrow R_{j^{\ast}}}^{(S)}$ and
$O_{R_{j^{\ast}}\rightarrow D}^{(S)}$ for the transmissions from $S$ to the
selected relay $R_{j^{\ast}}$ and from $R_{j^{\ast}}$ to $D$, respectively.
Due to the link independence assumption, we have
$\displaystyle P_{out}^{\left(S\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)+P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$
(2) $\displaystyle\ \ \ \ \ \ \ \ \ -P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\cdot P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(S)}\right)$
## III Equal Path-Loss Between All Node Pairs
In this section, we analyze 2HR-($r,k$) protocol in the case where the path-
loss is equal between all pairs of nodes in the system. The _Remark 3_ shows
2HR-($r,k$) protocol is castrated as 2HR-($\infty,k$) in case of equal path-
loss between all node pairs. We now analyze that under the 2HR-($\infty,k$)
protocol the number of eavesdroppers one network can tolerate subject to
specified requirements on transmission outage and secrecy outage. The
following two lemmas regarding some basic properties of
$P_{out}^{\left(T\right)}$, $P_{out}^{\left(S\right)}$ and $\tau$ are first
presented, which will help us to derive the main result in Theorem 1.
_Lemma 1_ : Consider the network scenario of Fig 1 with equal path-loss
between all pairs of nodes, under the 2HR-($r,k$) protocol the transmission
outage probability $P_{out}^{\left(T\right)}$ and secrecy outage probability
$P_{out}^{\left(S\right)}$ there satisfy the following conditions.
$\displaystyle P_{out}^{\left(T\right)}\leq
2\left(\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}\right)$
(3) $\displaystyle\ \ \ \ \ \ \ \ \ \
-\left(\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}\right)^{2}$
here $\Psi=e^{-2\gamma_{R}{\left(n-1\right)\left(1-e^{-\tau}\right)}\tau}$,
and
$\displaystyle P_{out}^{\left(S\right)}\leq
2m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}$
(4) $\displaystyle\ \ \ \ \ \ \ \ \
-\left[m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\right]^{2}$
The proof of Lemma 1 can be found in the Appendix A.
_Lemma 2_ : Consider the network scenario of Fig 1 with equal path-loss
between all pairs of nodes, to ensure
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$ and
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ under the 2HR-($r,k$) protocol,
the parameter $\tau$ must satisfy the following condition.
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1\right)}{2\gamma_{R}\left(n-1\right)}}$
and
$\displaystyle\tau\geq-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]}$
here, $\lfloor\cdot\rfloor$ is the floor function.
###### Proof.
The parameter $\tau$ should be set properly to satisfy both reliability and
secrecy requirements.
$\bullet$ Reliability Guarantee
To ensure the reliability requirement
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$, we know from formula (3) in the
Lemma 1, that we just need
$\displaystyle
2\left(\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}\right)$
$\displaystyle-\left(\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}\right)^{2}$
$\displaystyle\leq\varepsilon_{t}$
Thus,
$\displaystyle\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}\leq
1-\sqrt{1-\varepsilon_{t}}$ (5)
Notice that
$\displaystyle\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left(1-\Psi\right)^{i}\Psi^{n-i}\bigg{]}$
(6)
$\displaystyle=\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}1-\sum\limits_{i=0}^{n-j}\binom{n}{i}\left(1-\Psi\right)^{i}\Psi^{n-i}\bigg{]}$
$\displaystyle=\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}1-\sum\limits_{i=0}^{n-j}\frac{\binom{n}{i}}{\binom{n-j}{i}}\binom{n-j}{i}\left(1-\Psi\right)^{i}\Psi^{n-j-i}\Psi^{j}\bigg{]}$
We also notice the $i$ can take from $0$ to $n-j$, then we have
$\displaystyle
1\leq\frac{\binom{n}{i}}{\binom{n-j}{i}}\leq\frac{n!}{(n-j)!j!}$
Substituting into formula (6), we have
$\displaystyle\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}1-\sum\limits_{i=0}^{n-j}\frac{\binom{n}{i}}{\binom{n-j}{i}}\binom{n-j}{i}\left(1-\Psi\right)^{i}\Psi^{n-j-i}\Psi^{j}\bigg{]}$
(7)
$\displaystyle\leq\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}1-\Psi^{j}\cdot\sum\limits_{i=0}^{n-j}\binom{n-j}{i}\left(1-\Psi\right)^{i}\Psi^{n-j-i}\bigg{]}$
$\displaystyle=1-\frac{1}{k}\sum\limits_{j=1}^{k}\Psi^{j}$
$\displaystyle=1-\frac{1}{k}\bigg{[}\sum\limits_{j=0}^{k}\frac{1}{\binom{k}{j}}\binom{k}{j}\Psi^{j}-1\bigg{]}$
$\displaystyle\leq
1-\frac{1}{k}\bigg{[}\frac{1}{\binom{k}{\lfloor\frac{k}{2}\rfloor}}\sum\limits_{j=0}^{k}\binom{k}{j}\Psi^{j}-1\bigg{]}$
$\displaystyle=1-\frac{1}{k}\bigg{[}\frac{1}{\binom{k}{\lfloor\frac{k}{2}\rfloor}}(1+\Psi)^{k}-1\bigg{]}$
According to formula (5), (6) and (7), in order to ensure the reliability, we
need
$1-\frac{1}{k}\bigg{[}\frac{1}{\binom{k}{\lfloor\frac{k}{2}\rfloor}}(1+\Psi)^{k}-1\bigg{]}\leq
1-\sqrt{1-\varepsilon_{t}}$
or equally,
$\Psi\geq\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1$
that is,
$e^{-2\gamma_{R}\left(n-1\right)\cdot\left(1-e^{-\tau}\right)\tau}\geq\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1$
Therefore
$\left(1-e^{-\tau}\right)\tau\leq\frac{-\log\left(\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1\right)}{2\gamma_{R}\left(n-1\right)}$
By using Taylor formula, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1\right)}{2\gamma_{R}\left(n-1\right)}}$
$\bullet$ Secrecy Guarantee
To ensure the secrecy requirement
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$, we know from Lemma 1 that we
just need
$\displaystyle
2m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}$
$\displaystyle-\left[m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\right]^{2}$
$\displaystyle\leq\varepsilon_{s}$
Thus,
$\displaystyle
m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\leq
1-\sqrt{1-\varepsilon_{s}}$
That is,
$\displaystyle\tau\geq-\log{\left[1+\frac{\log{\left(\frac{1-\sqrt{1-\varepsilon_{s}}}{m}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\right)}}\right]}$
∎
Based on the results of Lemma 2, we now can establish the following theorem
regarding the performance of the proposed protocol in case of equal path-loss
between all node pairs.
Theorem 1. Consider the network scenario of Fig 1 with equal path-loss between
all pairs of nodes. To guarantee $P_{out}^{\left(T\right)}\leq\varepsilon_{t}$
and $P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ under 2HR-($r,k$) protocol,
the number of eavesdroppers $m$ the network can tolerate must satisfy the
following condition.
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left(\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1\right)}{2\gamma_{R}}}}}$
###### Proof.
From Lemma 2, we know that to ensure the reliability requirement, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log\left(\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1\right)}{2\gamma_{R}\left(n-1\right)}}$
(8)
and
$\displaystyle\left(n-1\right)\left(1-e^{-\tau}\right)\leq\frac{-\log\left(\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1\right)}{2\gamma_{R}\tau}$
(9)
To ensure the secrecy requirement, we need
$\displaystyle\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{m}$
(10)
From formula (9) and (10), we can get
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}}$
(11) $\displaystyle\ \ \ \
\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\frac{-\log\left(\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1\right)}{2\gamma_{R}\tau}}}$
By letting $\tau$ take its maximum value for maximum interference at
eavesdroppers, from formula (8) and (11), we get the following bound
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\left(\frac{1}{1+\gamma_{E}}\right)^{\sqrt{\frac{-\left(n-1\right)\log\left(\left[\binom{k}{\lfloor\frac{k}{2}\rfloor}\left(1+k\sqrt{1-\varepsilon_{t}}\right)\right]^{\frac{1}{k}}-1\right)}{2\gamma_{R}}}}}$
∎
Based on the above analysis, by simple derivation, we can get the follow
corollary to show our proposal is a general protocol.
Corollary 1. Consider the network scenario of Fig 1 with equal path-loss
between all pairs of nodes, the analysis results of the proposed protocol is
identical to that of protocols with the optimal relay selection presented in
[19][20] by setting of $k=1$ and $r=\infty$, and is identical to that of
protocols with the random relay selection presented in [29][30] by setting of
$k=n$ and $r=\infty$.
_Remark 4_ : In case of equal path-loss of all pairs of nodes and the
parameter $r=\infty$, we notice that the larger $k$ means the better load-
balance among the relays and the lower transmission efficiency, and vice
versa. In particular, when $k=1$, 2HR-($r,k$) protocol has the worse load-
balance among the relays and the highest transmission efficiency, and when
$k=n$, 2HR-($r,k$) protocol has the best load-balance among the relays and the
lower transmission efficiency.
## IV General Case To Addressing Path-Loss
In this section, we consider the more general scenario where the path-loss
between each pair of nodes also depends on the distance between them. The
related theoretic analysis is further provided to determine the number of
eavesdroppers one network can tolerant by adopting the 2HR-($r,k$) protocol.
To address the distance-dependent path-loss, we consider a coordination system
shown in Fig 2, in which the two-hop relay wireless networks employed in the
2-D plane of unit area, consisting of the square
$\left[-0.5,0.5\right]\times\left[-0.5,0.5\right]$. The source $S$ located at
coordinate $\left(-0.5,0\right)$ wishes to establish two-hop transmission with
destination $D$ located at coordinate $\left(0.5,0\right)$.
Figure 2: Coordinate system for the scenario where path-loss between pairs of
nodes is based on their relative locations.
To address the near eavesdropper problem and also to simply the analysis for
the 2HR-($r,k$) protocol, we assume that there exits a constant $d_{0}>0$ such
that any eavesdropper falling within a circle area with radius $d_{0}$ and
center $S$ or $R_{j^{\ast}}$ can eavesdrop the transmitted messages
successfully with probability 1, while any eavesdropper beyond such area can
only successfully eavesdropper the transmitted messages with a probability
less than 1. Based on such a simplification, we can establish the following
two lemmas regarding some basic properties of $P_{out}^{\left(T\right)}$,
$P_{out}^{\left(S\right)}$ and $\tau$ under this protocol.
_Lemma 3_ : Consider the network scenario of Fig 2, under the 2HR-($r,k$)
protocol the transmission outage probability $P_{out}^{\left(T\right)}$ and
secrecy outage probability $P_{out}^{\left(S\right)}$ there satisfy the
following condition.
$\displaystyle P_{out}^{\left(T\right)}\leq
1-\Upsilon^{\varphi_{1}+\varphi_{2}}\sum\limits_{l=1}^{k}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$ (12) $\displaystyle\ \ \ \ \ \
\ \ \
-\frac{\Upsilon^{2\left(\varphi_{1}+\varphi_{2}\right)}}{k^{2}}\sum\limits_{l=k+1}^{n}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$
$\displaystyle P_{out}^{\left(S\right)}\leq$ (13) $\displaystyle
2m\left[\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right]$
$\displaystyle-\left[m\left(\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right)\right]^{2}$
here,
$\displaystyle\Upsilon=e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}}{\left(0.5+r\right)^{-\alpha}}}$
$\displaystyle\varphi_{1}=\int_{-0.5}^{0.5}\int_{-0.5}^{0.5}\frac{1}{\left(x^{2}+y^{2}\right)^{\frac{\alpha}{2}}}dxdy$
$\displaystyle\varphi_{2}=\int_{-0.5}^{0.5}\int_{-0.5}^{0.5}\frac{1}{\left[\left(x-0.5\right)^{2}+y^{2}\right]^{\frac{\alpha}{2}}}dxdy$
$\displaystyle\psi=\int_{-0.5}^{0.5}\int_{-0.5}^{0.5}\frac{1}{\left[\left(x-0.5\right)^{2}+\left(y-0.5\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy$
The proof of Lemma 3 can be found in the Appendix B.
_Lemma 4_ : Consider the network scenario of Fig 2, to ensure
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$ and
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ by applying 2HR-($r,k$)
protocol, the parameter $\tau$ must satisfy the following condition.
$\displaystyle\tau\leq\sqrt{\frac{-\log{\left[\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}\right]}}{\gamma_{R}\left(n-1\right)\left(\varphi_{1}+\varphi_{2}\right)\left(0.5+r\right)^{\alpha}}}$
and
$\tau\geq-\log\left[1+\frac{\log{\left(\frac{\frac{1-\sqrt{1-\varepsilon_{s}}}{m}-\pi{d_{0}}^{2}}{1-\pi{d_{0}}^{2}}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\psi{d_{0}}^{\alpha}\right)}}\right]$
here, $\varphi_{1}$, $\varphi_{2}$, and $\psi$ are defined in the same way as
that in Lemma 3, and
$\displaystyle\nu_{1}=k^{2}\sum\limits_{l=1}^{k}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$
$\displaystyle\nu_{2}=k^{2}\sum\limits_{l=k+1}^{n}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$
###### Proof.
The parameter $\tau$ should be set properly to satisfy both reliability and
secrecy requirements.
$\bullet$ Reliability Guarantee
To ensure the reliability requirement
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$, we know from formula (12) in
Lemma 3 that we just need
$\displaystyle
1-\Upsilon^{\varphi_{1}+\varphi_{2}}\sum\limits_{l=1}^{k}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$ $\displaystyle\ \ \ \ \ \ \ \
\
-\frac{\Upsilon^{2\left(\varphi_{1}+\varphi_{2}\right)}}{k^{2}}\sum\limits_{l=k+1}^{n}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$
$\displaystyle\leq\varepsilon_{t}$
Thus,
$\displaystyle\Upsilon^{\varphi_{1}+\varphi_{2}}\geq\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}$
here
$\displaystyle\nu_{1}=k^{2}\sum\limits_{l=1}^{k}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$
$\displaystyle\nu_{2}=k^{2}\sum\limits_{l=k+1}^{n}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$
That is,
$\displaystyle
e^{-\frac{\gamma_{R}\tau\left(n-1\right)\left(1-e^{-\tau}\right)\left(\varphi_{1}+\varphi_{2}\right)}{\left(0.5+r\right)^{-\alpha}}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \
\geq\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}$
Thus,
$\displaystyle\tau\left(1-e^{-\tau}\right)\leq\frac{-\log{\left[\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}\right]}}{\gamma_{R}\left(n-1\right)\left(\varphi_{1}+\varphi_{2}\right)\left(0.5+r\right)^{\alpha}}$
By using Taylor formula, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log{\left[\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}\right]}}{\gamma_{R}\left(n-1\right)\left(\varphi_{1}+\varphi_{2}\right)\left(0.5+r\right)^{\alpha}}}$
$\bullet$ Secrecy Guarantee
To ensure the secrecy requirement
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$, we know from formula (13) in
Lemma 3 that we just need
$\displaystyle
2m\left[\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right]$
$\displaystyle-\left[m\left(\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right)\right]^{2}$
$\displaystyle\leq\varepsilon_{s}$
Thus,
$\displaystyle
m\cdot\left[\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\leq 1-\sqrt{1-\varepsilon_{s}}$
that is,
$\tau\geq-\log\left[1+\frac{\log{\left(\frac{\frac{1-\sqrt{1-\varepsilon_{s}}}{m}-\pi{d_{0}}^{2}}{1-\pi{d_{0}}^{2}}\right)}}{\left(n-1\right)\log{\left(1+\gamma_{E}\psi{d_{0}}^{\alpha}\right)}}\right]$
∎
Based on the results of Lemma 4, we now can establish the following theorem
regarding the performance of 2HR-($r,k$) protocol.
Theorem 2. Consider the network scenario of Fig 2. To guarantee
$P_{out}^{\left(T\right)}\leq\varepsilon_{t}$ and
$P_{out}^{\left(S\right)}\leq\varepsilon_{s}$ based on the proposed
2HR-($r,k$) protocol, the number of eavesdroppers $m$ the network can tolerate
must satisfy the following condition.
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\pi{d_{0}}^{2}+\left(1-\pi{d_{0}}^{2}\right)\omega}$
here
$\displaystyle\omega=\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\sqrt{\frac{-\left(n-1\right)\log{\left[\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}\right]}}{\gamma_{R}\left(\varphi_{1}+\varphi_{2}\right)\left(0.5+r\right)^{\alpha}}}}$
$\varphi_{1}$, $\varphi_{2}$, $\nu_{1}$,$\nu_{2}$ and $\psi$ are defined in
the same way as that in Lemma 3 and Lemma 4.
###### Proof.
From Lemma 4, we know that to ensure the reliability requirement, we have
$\displaystyle\tau\leq\sqrt{\frac{-\log{\left[\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}\right]}}{\gamma_{R}\left(n-1\right)\left(\varphi_{1}+\varphi_{2}\right)\left(0.5+r\right)^{\alpha}}}$
(14)
and
$\displaystyle\left(n-1\right)\left(1-e^{-\tau}\right)\leq\frac{-\log{\left[\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}\right]}}{\gamma_{R}\tau\left(\varphi_{1}+\varphi_{2}\right)\left(0.5+r\right)^{\alpha}}$
(15)
To ensure the secrecy requirement, we need
$\displaystyle
m\cdot\left[\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right]$
(16) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \leq 1-\sqrt{1-\varepsilon_{s}}$
From formula (15) and (16), we can get
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)}$
(17)
$\displaystyle\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\frac{-\log{\left[\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}\right]}}{\gamma_{R}\tau\left(\varphi_{1}+\varphi_{2}\right)\left(0.5+r\right)^{\alpha}}}\left(1-\pi{d_{0}}^{2}\right)}$
By letting $\tau$ take its maximum value for maximum interference at
eavesdroppers, from formula (14) and (17), we get the following bound
$\displaystyle
m\leq\frac{1-\sqrt{1-\varepsilon_{s}}}{\pi{d_{0}}^{2}+\left(1-\pi{d_{0}}^{2}\right)\omega}$
here
$\displaystyle\omega=\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\sqrt{\frac{-\left(n-1\right)\log{\left[\frac{k^{2}\sqrt{{\nu_{1}}^{2}+4\left(1-\varepsilon_{t}\right)\nu_{2}}-k^{2}\nu_{1}}{2\nu_{2}}\right]}}{\gamma_{R}\left(\varphi_{1}+\varphi_{2}\right)\left(0.5+r\right)^{\alpha}}}}$
∎
_Remark 5_ : The parameter $r$ determines the relay selection region. When
parameter $r$ tends to $0$, few system nodes locate in relay selection region,
and the relay selection process tends to optimal from the view of relay
selection region with less load-balance capacity. With increasing of parameter
$r$, the more relays are in relay selection region, which can ensure better
load-balance.
_Remark 6_ : The SINR at the receiver depends on channel state information and
the distance between the transmitter and receiver. The _Remark 4_ and _Remark
5_ show that the parameter $r$ and $k$ in 2HR-($r,k$) protocol can be flexibly
set to control the tradeoff the load-balance and the transmission efficiency
in terms of channel state information and the distance between the transmitter
and receiver respectively.
_Remark 7_ : In order to get the better load-balance, set a larger $r$ and $k$
which will result in a lower transmission efficiency. The Theorem 1 and
Theorem 3 show that the number of eavesdroppers one network can tolerant is
decreasing as the increasing $r$ and $k$.
_Remark 8_ : In the initial stage of the network operation, the parameter $r$
and $k$ can be set small values to ensure the high efficiency, since all
relays are energetic which load-balance among the relays is not first
considered. With the passage of time of the network operation, the parameter
$r$ and $k$ can be gradually set higher values for better load-balance among
the relays to extend the network lifetime.
Based on the above analysis, by simple derivation, we can get the follow
corollary to show our proposal is a general protocol.
Corollary 2. Consider the network scenario of Fig 2, the analysis results of
the proposed protocol with $r\rightarrow\infty$ and $k=n$ is identical to that
of Protocol 3 with $a=0$ and $b=0$ (the parameters $a$ and $b$ determine the
relay selection region) proposed in [30], and the analysis results of the
proposed protocol with $r\rightarrow 0$ and $k=n$ is identical to that of
Protocol 3 with $a\rightarrow 0.5$ and $b\rightarrow 0.5$ proposed in [30].
_Remark 9_ : The protocol proposed in [30] have the ability to control load-
balance among the relays by only control on the relay selection region.
Whereas, 2HR-($r,k$) protocol can realize load-balance by control on both
relay selection set and relay selection region.
## V Related Works
A lot of research works have been dedicated to load-balance transmission
scheme for balanced energy consumption among system nodes to prolong the
network lifetime in wireless networks. A few dynamic load balancing strategies
and schemes were proposed in [2][3] for distributed systems. For wireless mesh
network, a multi-hop transmission scheme is proposed in [4], in which
information relay is selected based on the current load of the relay nodes.
For wireless access networks, a distributed routing algorithm that performs
dynamic load-balance by constructs a load-balanced backbone tree [5]. J. Gao
et al. extended the shortest path routing to support load-balance [6]. In
particular, for energy constrained wireless sensor networks, load-balance is
significant important, and a lot of transmission schemes were proposed for
load-balance among relays and prolonging the network lifetime [7][8][9].
Lifetime optimization and security of multi-hop wireless networks was further
considered and the secure transmission scheme with load-balance is proposed in
[10][11].
Recently, attention is turning to achieve physical layer secrecy and secure
transmission scheme via cooperative relays is considered in large wireless
networks. Some transmission protocols are proposed to select the optimal relay
in terms of the maximum secrecy capacity or minimum transmit power. In case
that eavesdropper channels or locations is known, node cooperation is used to
improve the performance of secure wireless communications and a few
cooperative transmission protocols were proposed to jam eavesdroppers
[17][18]. In case that eavesdropper channels or locations is unknown, D.
Goeckel et al. proposed a transmission protocol based on optimal relay
selection [19][20]. For both one-dimensional and two-dimensional networks, a
secure transmission protocol is proposed in [21]. Z. Ding et al. considered
the opportunistic use of relays and proposed two secrecy transmission
protocols [22]. The ”two-way secrecy scheme” was studied in [23] [24] and M.
Dehghan et al. explored the energy efficiency of cooperative jamming scheme
[25]. A. Sheikholeslami et al. proposed a protocol, where the signal of a
given transmitter is protected by the aggregate interference produced by the
other transmitters [26]. A secure transmission protocol are presented in case
where the eavesdroppers collude [27]. J. Li et al. proposed two secure
transmission protocols to confound the eavesdroppers [28]. The above works
mainly focus on the maximum the secrecy capacity, in which the system nodes
with best link condition is always selected as information relay. Such, these
protocols have less load-balance capacity. In order to address this problem,
Y. Shen et al. further proposed a protocol with random relay selection in
[29][30]. This protocol can provide good load-balance capacity and balanced
energy consumption among the relays, whereas it has low transmission
efficiency.
## VI Conclusion
This paper proposed a general 2HR-($r,k$) protocol to ensure secure and
reliable information transmission through multiple cooperative system nodes
for two-hop relay wireless networks without the knowledge of eavesdropper
channels and locations. We proved that the 2HR-($r,k$) protocol has the
capability of flexible control over the tradeoff between the load-balance
capacity and the transmission efficiency by a proper setting of the radius $r$
of relay selection region and the size $k$ of candidate relay set. Such, in
general it is possible for us to set proper value of parameters according to
network scenario to support various applications. The results in this paper
indicate that the parameters $r$ and $k$ of the 2HR-($r,k$) protocol do also
affect the number of eavesdroppers one networks can tolerant under the premise
of specified secure and reliable requirements.
## Appendix A Proof of Lemma 1
###### Proof.
Based on the definition of transmission outage probability, we have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)$
$\displaystyle\ \ \ \ \ =P\left(C_{S,R_{j^{\ast}}}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \
=P\left(\frac{E_{s}\cdot|h_{S,R_{j^{\ast}}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}E_{s}\cdot|h_{R_{j},R_{j^{\ast}}}|^{2}+N_{0}/2}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \ \doteq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}|h_{R_{j},R_{j^{\ast}}}|^{2}}\leq\gamma_{R}\right)$
$\displaystyle\ \ \ \ \ \leq
P\left(\frac{H}{{|\mathcal{R}_{1}|}\tau}\leq\gamma_{R}\right)$ $\displaystyle\
\ \ \ \ =P\left(H\leq\gamma_{R}{|\mathcal{R}_{1}|}\tau\right)$
Here, $H=min\left(|h_{S,R_{j^{\ast}}}|^{2},|h_{D,R_{j^{\ast}}}|^{2}\right)$.
Compared to the noise generated by multiple system nodes, the environment
noise is negligible and thus is omitted here to simply the analysis. Notice
that $\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$.
Employing Appendix C, we should have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\right)\leq
F_{H}\left(\gamma_{R}{|\mathcal{R}_{1}|}\tau\right)$ $\displaystyle\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \
=\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\cdot$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left[1-e^{-2\gamma_{R}{|\mathcal{R}_{1}|}\tau}\right]^{i}\left[e^{-2\gamma_{R}{|\mathcal{R}_{1}|}\tau}\right]^{n-i}\bigg{]}$
Since there are $n-1$ other relays except $R_{j^{\ast}}$, the expected number
of noise-generation nodes is given by $|\mathcal{R}_{1}|=\left(n-1\right)\cdot
P\left(|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right)=\left(n-1\right)\left(1-e^{-\tau}\right)$.
Then we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\leq\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\cdot$
$\displaystyle\left[1-e^{-2\gamma_{R}{\left(n-1\right)\left(1-e^{-\tau}\right)}\tau}\right]^{i}\left[e^{-2\gamma_{R}{\left(n-1\right)\left(1-e^{-\tau}\right)}\tau}\right]^{n-i}\bigg{]}$
For convenience of the description, let
$\Psi=e^{-2\gamma_{R}{\left(n-1\right)\cdot\left(1-e^{-\tau}\right)}\tau}$,
and we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\right)\leq\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}$
(18)
Employing the same method, we can get
$\displaystyle P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\right)\leq\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}$
(19)
Substituting formula (18) and (19) into formula (1), we have
$\displaystyle P_{out}^{\left(T\right)}\leq
2\left(\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \
-\left(\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-\Psi\right]^{i}\Psi^{n-i}\bigg{]}\right)^{2}$
According to the definition of secrecy outage probability, we know that
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)=P\left(\bigcup_{i=1}^{m}\left\\{C_{S,E_{i}}\geq\gamma_{E}\right\\}\right)$
Thus, we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\leq\sum_{i=1}^{m}P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$
(20)
Based on the Markov inequality,
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$ $\displaystyle\ \ \ \
\ \leq
P\left(\frac{E_{s}\cdot|h_{S,E_{i}}|^{2}}{\sum_{R_{j}\in\mathcal{R}_{1}}E_{s}\cdot|h_{R_{j},E_{i}}|^{2}}\geq\gamma_{E}\right)$
$\displaystyle\ \ \ \ \ =E_{\left\\{h_{R_{j},E_{i}},j=0,1,\cdots,n+mp,j\neq
j^{\ast}\right\\},\mathcal{R}_{1}}$ $\displaystyle\ \ \ \ \ \ \ \ \
\left[P\left(|h_{S,E_{i}}|^{2}>\gamma_{E}\cdot\sum_{R_{j}\in\mathcal{R}_{1}}|h_{R_{j},E_{i}}|^{2}\right)\right]$
$\displaystyle\ \ \ \ \ \leq
E_{\mathcal{R}_{1}}\left[\prod_{R_{j}\in\mathcal{R}_{1}}E_{h_{R_{j},E_{i}}}\left[e^{-\gamma_{E}|h_{R_{j},E_{i}}|^{2}}\right]\right]$
$\displaystyle\ \ \ \ \
=E_{\mathcal{R}_{1}}\left[\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}\right]$
Substituting into formula (20), we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\leq\sum_{i=1}^{m}\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}=m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{1}|}$
(21)
employing the same method, we can get
$\displaystyle P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)\leq
m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{|\mathcal{R}_{2}|}$ (22)
Since the expected number of noise-generation nodes is given by
$|\mathcal{R}_{1}|=|\mathcal{R}_{2}|=\left(n-1\right)\left(1-e^{-\tau}\right)$,
thus, substituting formula (21) and (22) into formula (2), we can get
$\displaystyle P_{out}^{\left(S\right)}\leq
2m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}$
$\displaystyle\ \ \ \ \ \ \ \ \
-\left[m\cdot\left(\frac{1}{1+\gamma_{E}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\right]^{2}$
∎
## Appendix B Proof of Lemma 3
###### Proof.
Notice that two ways leading to transmission outage are: 1) there are no
candidate relays in the relay selection region; 2) the SINR at the selected
relay or the destination is less than $\gamma_{R}$. We also notice that if the
number of the eligible relays in candidate relay region less than or equal to
$k$, the relay will be random selected from candidate relay set
$\mathfrak{R}$.
Let $A_{l}$, $l=0,1,\cdots,n$, be the event that there are just $l$ system
nodes in the relay selection region. We have
$\displaystyle
P_{out}^{\left(T\right)}=\sum\limits_{l=0}^{n}P_{out|A_{l}}^{\left(T\right)}\cdot
P(A_{l})$ (23)
Since the relay is uniformly distributed, the number of relays in candidate
relay region is a binomial distribution $\left(n,\pi r^{2}\right)$. We have
$\displaystyle P(A_{l})=\binom{n}{l}\left(\pi r^{2}\right)^{l}\left(1-\pi
r^{2}\right)^{n-l}$ (24)
$P_{out|A_{l}}^{\left(T\right)}$ is discussed from the following three
aspects.
1) $l=0$
In this case, there are no relays in the relay selection region, then, we have
$\displaystyle P_{out|A_{l}}^{\left(T\right)}=1$ (25)
2) $1\leq l\leq k$
Since the number of candidate relay nodes is less than or equal to $k$. The
relay selection process is to select relay randomly in the candidate relay set
$\mathfrak{R}$ which consists of these $l$ relays located in the relay
selection region.
Notice $P_{out|A_{l}}^{\left(T\right)}$ is determined as
$\displaystyle P_{out|A_{l}}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)+P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\bigg{|}A_{l}\right)$ (26) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
-P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)\cdot
P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\bigg{|}A_{l}\right)$
Based on the definition of transmission outage probability, we have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)$
$\displaystyle\ \ \ \ \
=P\left(C_{S,R_{j^{\ast}}}\leq\gamma_{R}\bigg{|}A_{l}\right)$ $\displaystyle\
\ \ \ \
=P\left(\frac{E_{s}\cdot\frac{|h_{S,R_{j^{\ast}}}|^{2}}{d_{S,R_{j^{\ast}}}^{\alpha}}}{\sum_{R_{j}\in\mathcal{R}_{1}}E_{s}\cdot\frac{|h_{R_{j},R_{j^{\ast}}}|^{2}}{d_{R_{j},R_{j^{\ast}}}^{\alpha}}+\frac{N_{0}}{2}}\leq\gamma_{R}\bigg{|}A_{l}\right)$
$\displaystyle\ \ \ \ \ \doteq
P\left(\frac{\frac{|h_{S,R_{j^{\ast}}}|^{2}}{d_{S,R_{j^{\ast}}}^{\alpha}}}{\sum_{R_{j}\in\mathcal{R}_{1}}\frac{|h_{R_{j},R_{j^{\ast}}}|^{2}}{d_{R_{j},R_{j^{\ast}}}^{\alpha}}}\leq\gamma_{R}\bigg{|}A_{l}\right)$
Compared to the noise generated by multiple system nodes, the environment
noise is negligible and thus is omitted here to simply the analysis. Notice
that $\mathcal{R}_{1}={\left\\{j\neq
j^{\ast}:|h_{R_{j},R_{j^{\ast}}}|^{2}<\tau\right\\}}$, then
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)\leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}d_{S,R_{j^{\ast}}}^{-\alpha}}{\sum_{R_{j}\in\mathcal{R}_{1}}\tau
d_{R_{j},R_{j^{\ast}}}^{-\alpha}}\leq\gamma_{R}\bigg{|}A_{l}\right)$
Without loss of generality, Let $\left(x,y\right)$ be the coordinate of
$R_{j}$, shown in Fig 2. The number of noise generation nodes in square
$\left[x,x+dx\right]\times\left[y,y+dy\right]$ is
$\left(n-1\right)\left(1-e^{-\tau}\right)dxdy$. Then, we have
$\displaystyle\sum_{R_{j}\in\mathcal{R}_{1}}\frac{\tau}{d_{R_{j},R_{j^{\ast}}}^{\alpha}}$
$\displaystyle\ \ \ \ \ \ \ \
=\int_{0}^{1}\int_{0}^{1}\frac{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}{\left[\left(x-x_{R_{j^{\ast}}}\right)^{2}+\left(y-y_{R_{j^{\ast}}}\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy$
where $\left(x_{R_{j^{\ast}}},y_{R_{j^{\ast}}}\right)$ is the coordinate of
the selected relay $R_{j^{\ast}}$ which locates in the relay selection region.
Because the relays are uniformly distributed, it is the worst case that the
selected relay $R_{j^{\ast}}$ is located on the point $\left(0,0\right)$,
where the interference at $R_{j^{\ast}}$ from the noise generation nodes is
largest, and the best case with the selected relay $R_{j^{\ast}}$ located in
the edge of the circular relay selection region, where the interference at
$R_{j^{\ast}}$ from the noise generation nodes is lowest. Then, we consider
the worst case and have
$P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)\leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}d_{S,R_{j^{\ast}}}^{-\alpha}}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}\leq\gamma_{R}\bigg{|}A_{l}\right)$
here,
$\varphi_{1}=\int_{-0.5}^{0.5}\int_{-0.5}^{0.5}\frac{1}{\left(x^{2}+y^{2}\right)^{\frac{\alpha}{2}}}dxdy$
Due to $0.5-r\leq d_{S,R_{j^{\ast}}}\leq 0.5+r$, then,
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)$
$\displaystyle\ \ \ \leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}\left(0.5+r\right)^{-\alpha}}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}\leq\gamma_{R}\bigg{|}A_{l}\right)$
$\displaystyle\ \ \
=P\left(|h_{S,R_{j^{\ast}}}|^{2}\leq\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}}{\left(0.5+r\right)^{-\alpha}}\bigg{|}A_{l}\right)$
$\displaystyle\ \ \
=1-e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}}{\left(0.5+r\right)^{-\alpha}}}$
For convenience of description, let
$\Upsilon=e^{-\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)}}{\left(0.5+r\right)^{-\alpha}}}$,
we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)\leq 1-\Upsilon^{\varphi_{1}}$ (27)
Employing the same method, we can get
$\displaystyle P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\bigg{|}A_{l}\right)\leq 1-\Upsilon^{\varphi_{2}}$ (28)
here,
$\varphi_{2}=\int_{-0.5}^{0.5}\int_{-0.5}^{0.5}\frac{1}{\left[\left(x-0.5\right)^{2}+y^{2}\right]^{\frac{\alpha}{2}}}dxdy$
Substituting formula (27) and (28) into formula (26), we have
$\displaystyle
P_{out|A_{l}}^{\left(T\right)}\leq\left[1-\Upsilon^{\varphi_{1}}\right]+\left[1-\Upsilon^{\varphi_{2}}\right]-\left[1-\Upsilon^{\varphi_{1}}\right]\left[1-\Upsilon^{\varphi_{2}}\right]$
(29) $\displaystyle\ \ \ \ \ \ \ \ \ \ =1-\Upsilon^{\varphi_{1}+\varphi_{2}}$
3) $k<l\leq n$
In this case, the relay selection process is to select relay randomly in the
candidate relay set $\mathfrak{R}$ which consists of the relays with the first
$k$ large $min\left(|h_{S,R_{j}}|^{2},|h_{D,R_{j}}|^{2}\right)$ in the relay
selection region.
Notice $P_{out|A_{l}}^{\left(T\right)}$ is determined as
$\displaystyle P_{out|A_{l}}^{\left(T\right)}=P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)+P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\bigg{|}A_{l}\right)$ (30) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
-P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)\cdot
P\left(O_{R_{j^{\ast}}\rightarrow D}^{(T)}\bigg{|}A_{l}\right)$
Let the random variable
$H=min\left(|h_{S,R_{j^{\ast}}}|^{2},|h_{D,R_{j^{\ast}}}|^{2}\right)$ and from
Appendix C, the distribution function of $H$ is
$F_{H}\left(x\right)=\begin{cases}\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=l-j+1}^{l}\binom{l}{i}\\\
\ \ \ \ \ \ \ \left[1-e^{-2x}\right]^{i}\left[e^{-2x}\right]^{l-i}\bigg{]}\ \
\ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$ (31)
Based on the definition of transmission outage probability, employing the
similar method above, we have
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)$
$\displaystyle\ \ \ \leq
P\left(\frac{|h_{S,R_{j^{\ast}}}|^{2}\left(0.5+r\right)^{-\alpha}}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}\leq\gamma_{R}\bigg{|}A_{l}\right)$
$\displaystyle\ \ \ \leq
P\left(H\leq\frac{\gamma_{R}{\tau\left(n-1\right)\left(1-e^{-\tau}\right)\varphi_{1}}}{\left(0.5+r\right)^{-\alpha}}\bigg{|}A_{l}\right)$
From formula (31), we can get
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(T)}\bigg{|}A_{l}\right)$
(32) $\displaystyle\ \ \
\leq\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=l-j+1}^{l}\binom{l}{i}\left(1-\Upsilon^{2\varphi_{1}}\right)^{i}\cdot\left(\Upsilon^{2\varphi_{1}}\right)^{l-i}\bigg{]}$
$\displaystyle\ \ \
=\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}1-\sum\limits_{i=0}^{l-j}\frac{\binom{l}{i}}{\binom{l-j}{i}}\binom{l-j}{i}\left(1-\Upsilon^{2\varphi_{1}}\right)^{i}\cdot$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\
\left(\Upsilon^{2\varphi_{1}}\right)^{l-j-i}\left(\Upsilon^{2\varphi_{1}}\right)^{j}\bigg{]}$
$\displaystyle\ \ \
\leq\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}1-\left(\Upsilon^{2\varphi_{1}}\right)^{j}\bigg{]}$
$\displaystyle\ \ \
=1-\frac{1}{k}\sum\limits_{j=1}^{k}\left(\Upsilon^{2\varphi_{1}}\right)^{j}$
$\displaystyle\ \ \
=1-\frac{\Upsilon^{2\varphi_{1}}\left(1-\Upsilon^{2k\varphi_{1}}\right)}{k\left(1-\Upsilon^{2\varphi_{1}}\right)}$
Employing the same method, we can get
$\displaystyle P\left(O_{R_{j^{\ast}}\rightarrow
D}^{(T)}\bigg{|}A_{l}\right)\leq
1-\frac{\Upsilon^{2\varphi_{2}}\left(1-\Upsilon^{2k\varphi_{2}}\right)}{k\left(1-\Upsilon^{2\varphi_{2}}\right)}$
(33)
Substituting formula (32) and (33) into formula (30), we have
$\displaystyle P_{out|A_{l}}^{\left(T\right)}\leq
1-\frac{\Upsilon^{2\left(\varphi_{1}+\varphi_{2}\right)}\left(1-\Upsilon^{2k\varphi_{1}}\right)\left(1-\Upsilon^{2k\varphi_{2}}\right)}{k^{2}\left(1-\Upsilon^{2\varphi_{1}}\right)\left(1-\Upsilon^{2\varphi_{2}}\right)}$
(34)
Substituting formula (24), (25), (29) and (34) into formula (23), we have
$\displaystyle
P_{out}^{\left(T\right)}=\sum\limits_{l=0}^{n}P_{out|A_{l}}^{\left(T\right)}\cdot
P(A_{l})$ $\displaystyle\ \ \ \ \ \ =P_{out|A_{0}}^{\left(T\right)}\cdot
P(A_{0})+\sum\limits_{l=1}^{k}P_{out|A_{l}}^{\left(T\right)}\cdot P(A_{l})$
$\displaystyle\ \ \ \ \ \ \ \ \
+\sum\limits_{l=k+1}^{n}P_{out|A_{l}}^{\left(T\right)}\cdot P(A_{l})$
$\displaystyle\ \ \ \ \ \ \leq 1\cdot\left(1-\pi r^{2}\right)^{n}$
$\displaystyle\ \ \ \ \ \ \ \ \
+\left(1-\Upsilon^{\varphi_{1}+\varphi_{2}}\right)\sum\limits_{l=1}^{k}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$ $\displaystyle\ \ \ \ \ \ \ \
\
+\left[1-\frac{\Upsilon^{2\left(\varphi_{1}+\varphi_{2}\right)}\left(1-\Upsilon^{2k\varphi_{1}}\right)\left(1-\Upsilon^{2k\varphi_{2}}\right)}{k^{2}\left(1-\Upsilon^{2\varphi_{1}}\right)\left(1-\Upsilon^{2\varphi_{2}}\right)}\right]\cdot$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
\sum\limits_{l=k+1}^{n}\binom{n}{l}\left(\pi r^{2}\right)^{l}\left(1-\pi
r^{2}\right)^{n-l}$
$\displaystyle\ \ \ \ \ \ \leq
1-\Upsilon^{\varphi_{1}+\varphi_{2}}\sum\limits_{l=1}^{k}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$ $\displaystyle\ \ \ \ \ \ \ \
\
-\frac{\Upsilon^{2\left(\varphi_{1}+\varphi_{2}\right)}\left(1-\Upsilon^{2k\varphi_{1}}\right)\left(1-\Upsilon^{2k\varphi_{2}}\right)}{k^{2}\left(1-\Upsilon^{2\varphi_{1}}\right)\left(1-\Upsilon^{2\varphi_{2}}\right)}\cdot$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
\sum\limits_{l=k+1}^{n}\binom{n}{l}\left(\pi r^{2}\right)^{l}\left(1-\pi
r^{2}\right)^{n-l}$ $\displaystyle\ \ \ \ \ \ \leq
1-\Upsilon^{\varphi_{1}+\varphi_{2}}\sum\limits_{l=1}^{k}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$ $\displaystyle\ \ \ \ \ \ \ \
\
-\frac{\Upsilon^{2\left(\varphi_{1}+\varphi_{2}\right)}}{k^{2}}\sum\limits_{l=k+1}^{n}\binom{n}{l}\left(\pi
r^{2}\right)^{l}\left(1-\pi r^{2}\right)^{n-l}$
According to the definition of secrecy outage probability, we know that
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)=P\left(\bigcup_{i=1}^{m}\left\\{C_{S,E_{i}}\geq\gamma_{E}\right\\}\right)$
Thus, we have
$\displaystyle P\left(O_{S\rightarrow
R_{j^{\ast}}}^{(S)}\right)\leq\sum_{i=1}^{m}P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$
(35)
Based on the definition of $d_{0}$, we denote by $G_{1}^{(i)}$ the event that
the distance between $E_{i}$ and the source is less than $d_{0}$, and denote
by $G_{2}^{(i)}$ the event that distance between $E_{i}$ and the source is
lager than or equal to $d_{0}$. We have
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$ $\displaystyle\ \ \ \
=P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{1}^{(i)}\right)P\left(G_{1}^{(i)}\right)$
$\displaystyle\ \ \ \ \ \ \ \
+P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)P\left(G_{2}^{(i)}\right)$
$\displaystyle\ \ \ \ \leq 1\cdot\frac{1}{2}\pi{d_{0}}^{2}$ $\displaystyle\ \
\ \ \ \ \ \
+P\left(\frac{\frac{|h_{S,E_{i}}|^{2}}{d_{S,E_{i}}^{\alpha}}}{\sum\limits_{R_{j}\in\mathcal{R}_{1}}\frac{|h_{R_{j},E_{i}}|^{2}}{d_{R_{j},E_{i}}^{\alpha}}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)\left(1-\frac{1}{2}\pi{d_{0}}^{2}\right)$
of which
$\displaystyle
P\left(\frac{\frac{|h_{S,E_{i}}|^{2}}{d_{S,E_{i}}^{\alpha}}}{\sum_{R_{j}\in\mathcal{R}_{1}}\frac{|h_{R_{j},E_{i}}|^{2}}{d_{R_{j},E_{i}}^{\alpha}}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)$
$\displaystyle\ \ \leq
P\left(\frac{|h_{S,E_{i}}|^{2}{d_{0}}^{-\alpha}}{\Gamma\int_{0}^{1}\int_{0}^{1}\frac{1}{\left[\left(x-x_{E_{i}}\right)^{2}+\left(y-y_{E_{i}}\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)$
where $\left(x_{E_{i}},y_{E_{i}}\right)$ is the coordinate of the eavesdropper
$E_{i}$. $\Gamma$ is the sum of $\left(n-1\right)\left(1-e^{-\tau}\right)$
independent exponential random variables.
From Fig 2 we know that the strongest interference at eavesdropper $E_{i}$
happens when $E_{i}$ is located in the point $(0,0)$, while the smallest
interference at $E_{i}$ happens it is located at four corners of the network
region. By considering the smallest interference at eavesdroppers, we then
have
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)$
$\displaystyle\ \ \ \ \ \ \leq
P\left(\frac{|h_{S,E_{i}}|^{2}{d_{0}}^{-\alpha}}{\Gamma\psi}\geq\gamma_{E}\right)$
$\displaystyle\ \ \ \ \ \
=P\left(|h_{S,E_{i}}|^{2}\geq\Gamma\gamma_{E}\cdot\psi\cdot{d_{0}}^{\alpha}\right)$
here
$\psi=\int_{-0.5}^{0.5}\int_{-0.5}^{0.5}\frac{1}{\left[\left(x-0.5\right)^{2}+\left(y-0.5\right)^{2}\right]^{\frac{\alpha}{2}}}dxdy$
Based on the Markov inequality,
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\bigg{|}G_{2}^{(i)}\right)$
$\displaystyle\ \ \ \ \ \leq
E_{\Gamma}\left[e^{-\Gamma\gamma_{E}\psi{d_{0}}^{\alpha}}\right]$
$\displaystyle\ \ \ \ \
=\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}$
Then, we have
$\displaystyle P\left(C_{S,E_{i}}\geq\gamma_{E}\right)$ (36)
$\displaystyle\leq\frac{1}{2}\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\frac{1}{2}\pi{d_{0}}^{2}\right)$
Employee the same method, we have
$\displaystyle P\left(C_{R_{j^{\ast}},E_{i}}\geq\gamma_{E}\right)$ (37)
$\displaystyle\leq\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)$
Notice that
$\displaystyle\frac{1}{2}\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\frac{1}{2}\pi{d_{0}}^{2}\right)$
(38)
$\displaystyle=\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)$
$\displaystyle\ \ \ \
-\frac{1}{2}\pi{d_{0}}^{2}\left[1-\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\right]$
$\displaystyle\leq\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)$
From formula (36), (37) and (38), we can get
$\displaystyle P\left(O_{S\rightarrow R_{j^{\ast}}}^{(S)}\right)\leq
P\left(O_{R_{j^{\ast}}\rightarrow D}^{(S)}\right)$ (39) $\displaystyle\leq
m\left[\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right]$
Substituting formula (39) into formula (2), we have
$\displaystyle P_{out}^{\left(S\right)}\leq$ (40) $\displaystyle
2m\left[\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right]$
$\displaystyle-\left[m\left(\pi{d_{0}}^{2}+\left(\frac{1}{1+\gamma_{E}\psi{d_{0}}^{\alpha}}\right)^{\left(n-1\right)\left(1-e^{-\tau}\right)}\left(1-\pi{d_{0}}^{2}\right)\right)\right]^{2}$
∎
## Appendix C The Distribution Function and Probability Density of
$H=min\left(|h_{S,R_{j^{\ast}}}|^{2},|h_{D,R_{j^{\ast}}}|^{2}\right)$
Let the random variable
$H=min\left(|h_{S,R_{j^{\ast}}}|^{2},|h_{D,R_{j^{\ast}}}|^{2}\right)$. The
node $R_{j^{\ast}}$ is randomly selected from the relay selection set
consisting of system nodes with the first $k$ large
$min\left(|h_{S,R_{j}}|^{2},|h_{D,R_{j}}|^{2}\right)$, $j=1,2,\cdots,n$. The
distribution function and probability density of $H$ is given by
$F_{H}\left(x\right)=\begin{cases}\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\\\
\ \ \ \ \ \ \ \left[1-e^{-2x}\right]^{i}\left[e^{-2x}\right]^{n-i}\bigg{]}\ \
\ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
and
$f_{H}\left(x\right)=\begin{cases}\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\frac{n!}{(j-1)!(n-j)!}\cdot\\\
\ \
\left[1-e^{-2x}\right]^{n-j}\left[e^{-2x}\right]^{j-1}\left[2e^{-2x}\right]\bigg{]}\
\ \ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
###### Proof.
Because the random variable
$H=min\left(|h_{S,R_{j^{\ast}}}|^{2},|h_{D,R_{j^{\ast}}}|^{2}\right)$ is the
random selection relay from the first $k$ large random variable
$min\left(|h_{S,R_{j}}|^{2},|h_{D,R_{j}}|^{2}\right)$, $j=1,2,\cdots,n$. From
Appendix D,
$\displaystyle
F_{H}\left(x\right)=\frac{1}{k}\sum\limits_{j=1}^{k}F_{H_{j}^{l}}(x)$
$\displaystyle
f_{H}\left(x\right)=\frac{1}{k}\sum\limits_{j=1}^{k}f_{H_{j}^{l}}(x)$
According to Appendix E, we have
$F_{H}\left(x\right)=\begin{cases}\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\\\
\ \ \ \ \ \ \ \left[1-e^{-2x}\right]^{i}\left[e^{-2x}\right]^{n-i}\bigg{]}\ \
\ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
and
$f_{H}\left(x\right)=\begin{cases}\frac{1}{k}\sum\limits_{j=1}^{k}\bigg{[}\frac{n!}{(j-1)!(n-j)!}\cdot\\\
\ \ \
\left[1-e^{-2x}\right]^{n-j}\left[e^{-2x}\right]^{j-1}\left[2e^{-2x}\right]\bigg{]}\
\ \ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
∎
## Appendix D The Randomly Selected Variable from the Random Variable Set
Let $X_{1},\cdots,X_{n}$ be continuous random variables, with density
$f_{X_{1}}(x),\cdots,f_{X_{n}}(x)$ and distribution function
$F_{X_{1}}(x),\cdots,F_{X_{n}}(x)$. The random variable, indexed by $Y$, is
selected randomly from $X_{1},\cdots,X_{n}$. The distribution function and
probability density of $Y$ is given by
$\displaystyle F_{Y}\left(y\right)=\frac{1}{n}\sum_{i=1}^{n}F_{X_{i}}(y)$
$\displaystyle f_{Y}\left(y\right)=\frac{1}{n}\sum_{i=1}^{n}f_{X_{i}}(y)$
###### Proof.
We assume the $s$-th random variable is selected as $Y$,
$P(s=i)=\frac{1}{n},i=1,\cdots,n$. Then we have
$\displaystyle F_{Y}\left(y\right)=P\left(Y\leq y\right)$ $\displaystyle\ \ \
\ \ \ \ \ =\sum_{i=1}^{n}P\left(X_{s}\leq y|s=i\right)P\left(s=i\right)$
$\displaystyle\ \ \ \ \ \ \ \ =\sum_{i=1}^{n}\frac{1}{n}P\left(X_{s}\leq
y|s=i\right)$ $\displaystyle\ \ \ \ \ \ \ \
=\sum_{i=1}^{n}\frac{1}{n}P\left(X_{i}\leq y\right)$ $\displaystyle\ \ \ \ \ \
\ \ =\frac{1}{n}\sum_{i=1}^{n}F_{X_{i}}(y)$
$\displaystyle f_{Y}\left(y\right)=F_{Y}^{\prime}\left(y\right)$
$\displaystyle\ \ \ \ \ \ \ \ =\frac{1}{n}\sum_{i=1}^{n}F_{X_{i}}^{\prime}(y)$
$\displaystyle\ \ \ \ \ \ \ \ =\frac{1}{n}\sum_{i=1}^{n}f_{X_{i}}(y)$
∎
## Appendix E The Distribution Function and Probability Density of the $k$-th
Largest Random Variable
The $|h_{A,B}|^{2}$ is path-loss between any node $A$ and $B$ with the
Rayleigh fading, and is exponentially distributed with
$E\left[|h_{A,B}|^{2}\right]=1$. The
$min\left(|h_{S,R_{j}}|^{2},|h_{D,R_{j}}|^{2}\right)$, $j=1,2,\cdots,n$, are
$n$ random variables in which the $j$-th largest random variable is denoted by
$H_{j}^{l}$. The distribution function and probability density of the random
variable $H_{j}^{l}$, $j=1,2,\cdots,n$, are given by
$F_{H_{j}^{l}}\left(x\right)=\begin{cases}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-e^{-2x}\right]^{i}\left[e^{-2x}\right]^{n-i}\
\ \ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
$f_{H_{j}^{l}}\left(x\right)=\begin{cases}\frac{n!}{(j-1)!(n-j)!}\cdot\\\
\left[1-e^{-2x}\right]^{n-j}\left[e^{-2x}\right]^{j-1}\left[2e^{-2x}\right]\ \
\ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
###### Proof.
Because the $|h_{A,B}|^{2}$ is exponentially distributed with
$E\left[|h_{A,B}|^{2}\right]=1$ between any node $A$ and $B$, according to
order statistics in [32], we can get the distribution function of the
$min\left(|h_{S,R_{j}}|^{2},|h_{D,R_{j}}|^{2}\right)$ for each relay
$R_{j},j=1,2,\cdots,n$, indexed by $H_{j}$, as following,
$f_{H_{j}}(x)=\begin{cases}2e^{-2x}\ \ \ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq
0$}\\\ \end{cases}$
$F_{H_{j}}(x)=\begin{cases}1-e^{-2x}\ \ \ &\text{$x>0$}\\\ 0\ \ \
&\text{$x\leq 0$}\\\ \end{cases}$
According to order statistics in [32], The distribution function and
probability density of the $j$-th smallest in $H_{j}$, $j=1,2,\cdots,n$,
indexed by $H_{j}^{s}$, are given by
$F_{H_{j}^{s}}\left(x\right)=\begin{cases}\sum\limits_{i=j}^{n}\binom{n}{i}\left[1-e^{-2x}\right]^{i}\left[e^{-2x}\right]^{n-i}\
\ \ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
$f_{H_{j}^{s}}\left(x\right)=\begin{cases}\frac{n!}{(j-1)!(n-j)!}\cdot\\\
\left[1-e^{-2x}\right]^{j-1}\left[e^{-2x}\right]^{n-j}\left[2e^{-2x}\right]\ \
&\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
Since the $j$-th largest, indexed by $H_{j}^{l}$, is equal to the
$\left(n-j+1\right)$-th smallest in $H_{j}$, $j=1,2,\cdots,n$, we should have
$\displaystyle F_{H_{j}^{l}}\left(x\right)=F_{H_{n-j+1}^{s}}\left(x\right)$
$\displaystyle\ \ \ \
=\begin{cases}\sum\limits_{i=n-j+1}^{n}\binom{n}{i}\left[1-e^{-2x}\right]^{i}\left[e^{-2x}\right]^{n-i}\
\ &\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
$\displaystyle f_{H_{j}^{l}}\left(x\right)=f_{H_{n-j+1}^{s}}\left(x\right)$
$\displaystyle\ \ \ \ =\begin{cases}\frac{n!}{(j-1)!(n-j)!}\cdot\\\
\left[1-e^{-2x}\right]^{n-j}\left[e^{-2x}\right]^{j-1}\left[2e^{-2x}\right]\ \
&\text{$x>0$}\\\ 0\ \ \ &\text{$x\leq 0$}\\\ \end{cases}$
∎
## References
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* [11] S. Ozdemir, _”Secure Load Balancing via Hierarchical Data Aggregation in Heterogeneous Sensor Networks,”_ Journal of Information Science and Engineering vol.25, pp.1691-1705, 2009.
* [12] J. Talbot and D. Welsh, _”Complexity and Crytography : An Introduction,”_ , Cambridge University Press, 2006.
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|
arxiv-papers
| 2013-01-09T03:21:23 |
2024-09-04T02:49:40.028687
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yulong Shen, Xiaohong Jiang and Jianfeng Ma",
"submitter": "Yulong Shen",
"url": "https://arxiv.org/abs/1301.1746"
}
|
1301.1826
|
# Strongly aligned and oriented molecular samples at a kHz repetition rate
Sebastian Trippel Terry Mullins Nele L. M. Müller Jens S. Kienitz Karol
Długołęcki Center for Free-Electron Laser Science, DESY, Notkestrasse 85,
22607 Hamburg, Germany Jochen Küpper [email protected]
http://desy.cfel.de/cid/cmi Center for Free-Electron Laser Science, DESY,
Notkestrasse 85, 22607 Hamburg, Germany Department of Physics, University of
Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany The Hamburg Centre for
Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
###### Abstract
_Dedicated to Bretislav Friedrich on the occasion of his 60 ${}^{\text{th}}$
birthday_
We demonstrate strong adiabatic laser alignment and mixed-field orientation at
kHz repetition rates. We observe degrees of alignment as large as
$\langle\cos^{2}\theta\rangle_{\text{2D}}=0.94$ at 1 kHz operation for
iodobenzene. The experimental setup consist of a kHz laser system
simultaneously producing pulses of 30 fs (1.3 mJ) and 450 ps (9 mJ). A cold 1
K state-selected molecular beam is produced at the same rate by appropriate
operation of an Even-Lavie valve. Quantum state selection has been obtained
using an electrostatic deflector. A camera and data acquisition system records
and analyzes the images on a single-shot basis. The system is capable of
producing, controlling (translation and rotation) and analyzing cold molecular
beams at kHz repetition rates and is, therefore, ideally suited for the
recording of ultrafast dynamics in so-called “molecular movies”.
## I Introduction
Aligned and oriented molecules serve as ideal samples to study steric effects
in chemical reactions Brooks (1976); Stapelfeldt and Seideman (2003) and to
image the structure and dynamics of complex molecules directly in the
molecular frame, if that is strongly confined, i. e., linked to the laboratory
frame of the measurement. This would yield so-called “molecular movies” of the
ongoing dynamics, conceivably without prior knowledge on the investigated
system.
Bretislav Friedrich has been at the forefront of the development of methods to
control complex molecules, including brute force orientation Friedrich and
Herschbach (1991); Loesch and Remscheid (1990), laser alignment Friedrich and
Herschbach (1995) and mixed-field orientation Friedrich and Herschbach (1999a,
b). At that time, the degree of alignment and orientation was too weak to
image molecular dynamics directly in the molecular frame. However, over the
last two decades, the available degree of control has been constantly
increased. The combination with rotational-state selection Holmegaard _et
al._ (2009); Putzke _et al._ (2011) has improved the achievable control
dramatically, with the strongest demonstrated degree of alignment so far of
$\langle\cos^{2}\theta\rangle_{\text{2D}}>0.97$ Holmegaard _et al._ (2009).
In addition, recent results on the orientation of molecules in mixed fields
show that the adiabaticity and the resulting degree of orientation strongly
depend on the applied electric fields, i. e., the laser-pulse duration Nielsen
_et al._ (2012).
Strong laser alignment and mixed-field orientation has been exploited in the
recording of molecular frame photoelectron angular distributions of complex
molecules Holmegaard _et al._ (2010). Controlled samples increase the
contrast in all direct imaging experiments. They allow the simple averaging of
many individual experiments, they simplify the data analysis, and no
orientation relationship between patterns from randomly oriented molecules
need to be derived numerically Spence and Doak (2004); Filsinger _et al._
(2011). Moreover, they are crucial to various advanced “photography”
experiments: Tomographic reconstruction approaches for X-ray Pabst _et al._
(2010); Filsinger _et al._ (2011) or electron diffraction Reckenthaeler _et
al._ (2009); Hensley _et al._ (2012) and photoelectron holography experiments
of aligned molecules require typically
$\langle\cos^{2}\theta\rangle_{\text{2D}}\approx 0.9$ Filsinger _et al._
(2011); Krasniqi _et al._ (2010).
Such a strong degree of alignment has, so far, only been achieved in adiabatic
alignment experiments making use of very cold molecular beams and nanosecond
laser pulses at a repetition rate of a few 10 Hz Kumarappan _et al._ (2006);
Holmegaard _et al._ (2009). However, the low repetition rates renders time
resolved studies of molecular dynamics, with their generally low count rates,
tedious, at least, or even infeasible. Therefore, experimental setups which
provide strong alignment at high repetition rates are highly desirable. This
requires the production of cold molecular beams of complex molecules and
strong laser fields with pulse durations that are comparable or longer than
the rotational period of the molecules. The lack of nanosecond lasers with
sufficient pulse energies at kHz repetition rates suggests the use of laser
pulses generated by an amplified Ti:Sa laser system. However, the generation
of such laser pulses with pulse durations on the order of 1 ns and the
required peak intensities of $>10^{11}$ W/cm3 is challenging. Moreover, _a
priori_ it has been unclear whether “heavy” molecules with a rotation time in
the order of a few hundred picoseconds (like iodobenzene) can be strongly
aligned by a sub-nanosecond, strongly linearly chirped broadband 800 nm laser
pulse.
Recently, some relevant molecular beam setups with high repetition rates have
been developed and some individual ingredients of the necessary control and
detection details have been demonstrated Horio and Suzuki (2009); Ren _et
al._ (2012); Irimia _et al._ (2009). A benchmark experiment demonstrating
long-pulse alignment of molecules in a low-pressure continuous beam at 1 kHz
repetition rate, probed with short ps x-ray pulses from a synchrotron source,
has demonstrated weak alignment of $\langle\cos^{2}\theta\rangle\approx 0.4$
Peterson _et al._ (2008). Impulsive alignment experiments have been
performed, again exploiting continuous molecular beams, with lasers operating
at kHz repetition rates Peronne _et al._ (2003); Rouzee _et al._ (2012). The
achieved degree of alignment however is typically also only moderately strong
($\langle\cos^{2}\theta\rangle\leq 0.85$). This makes these approaches not
very well suited for molecular-frame imaging studies of complex molecules.
Moreover, in these experiments the alignment typically only persists for short
periods of time ($\sim\\!1$ ps), which severely limits the time-window for
time-resolved experiments, esp. for large amplitude dynamics, such as
conformer interconversion or folding motions.
Here, we present a new experimental setup that provides strongly aligned and
oriented samples of state-selected molecules at a repetition rate of 1 kHz.
This rate is a good compromise between current table-top laser systems, pulsed
molecular beam sources, and high speed camera systems. In addition, it
demonstrates a clear pathway for the sample preparation at upcoming light
sources with high-repetition rates, such as x-ray free-electron lasers
(XFELs), synchrotrons, and laser based high-harmonic generation (HHG) sources.
In order to efficiently use these light sources, the capability of high
repetition rate adiabatic alignment and orientation is highly desirable.
Whereas the availability of synchronized pulses from high-power table-top
laser systems is practically an intrinsic feature, XFEL facilities, such as
the European XFEL in Hamburg, are actively pursuing the setup of high-
repetition rate lasers that meet the requirements set by the current work.
## II Experimental setup
The schematic of the experimental setup is shown in Figure 1.
Figure 1: Schematic of the experimental setup. The pulsed molecular beam
passes two skimmers before it enters the electrostatic deflector where it is
dispersed, depending on its quantum states, along the Y-axis. The alignment or
orientation laser pulses as well as the probe pulse cross the molecular beam
inside of a velocity map imaging spectrometer. The ions are mapped on a
position sensitive detector consisting of an MCP and a phosphor screen. The
angle $\theta$ is defined as the angle between the Y-axis and the most-
polarizable axis of the molecule, i. e., the C–I bond axis.
Details will be published in a longer account and only a brief description
will be presented here. A pulsed molecular beam is provided by expanding 10
mbar of iodobenzene seeded in 120 bar of helium through an Even-Lavie valve
Even _et al._ (2000) cooled to -20 ∘C. After passing two skimmers the
molecular beam enters an electric deflector, where the molecules are dispersed
according to their quantum state Filsinger _et al._ (2009a). The state
selected molecular ensembles are aligned or oriented by laser or mixed dc-
electric and laser fields Friedrich and Herschbach (1999a, b); Holmegaard _et
al._ (2009); Ghafur _et al._ (2009), respectively, inside a velocity map
imaging spectrometer (VMI) Eppink and Parker (1997). The angular confinement
is probed through strong-field multiple ionization by a short laser pulse
followed by Coulomb explosion of the molecule. The resulting I+ ions are
velocity mapped onto a 40 mm diameter position sensitive detector (Photonis)
consisting of a multi-channel-plate (MCP), a fast phosphor screen (P-46), and
a high frame-rate camera. High speed oil-free pumping (4000 l/s for the source
and 2000 l/s for deflection and detection chambers, respectively) and
optimized operation conditions of the Even-Lavie pulsed valve allow for the
generation of dense ($>10^{9}$ molecules/cm3 in the interaction volume) and
cold (1 K) molecular beams, whose population distribution is further reduced
to some ten rotational states by state-selection Filsinger _et al._ (2009a).
Alignment and ionization laser pulses are provided by an amplified femtosecond
laser system (Coherent Legend Elite Duo HE USX NSI) at a repetition rate of 1
kHz. The total output power of the system is larger than 10 W with a bandwidth
of $\geq 72$ nm centered at 800 nm. Directly behind the amplification stages
the laser beam is split into two parts, an alignment (orientation) beam
($\approx 9$ mJ/pulse) and a probe beam ($\approx 1.3$ mJ/pulse). The duration
of the alignment pulse can be compressed or stretched (negatively chirped)
with an external compressor continuously from 40 fs up to 520 ps. The probe
beam is compressed to 30 fs using the standard grating based compression
setup. Since both beams are produced by the same amplifier system they are
inherently synchronized. The two beams are incident on a 60 cm focal length
lens parallel to each other with a transverse distance of 10 mm, resulting in
field strength of up to $5\times 10^{11}$ W/cm2 and $5\times 10^{14}$ W/cm2
for alignment and probe pulse, respectively. The foci are overlapped in space
and time in the molecular beam and in the center of the velocity map imaging
spectrometer. Vertically scanning the lens allows probing different parts of
the, typically, quantum state dispersed, molecular beam, i. e., to probe
ensembles with varying rotational excitation and correspondingly varying
effective dipole moments and effective polarizabilities. This motion is
automatized and thus one can completely automatically measure the ion-
distribution in the VMI as a function of vertical beam position. From this
data, one can determine the molecular beam density, the degree of alignment,
and the orientation all at once. A typical scan over the molecular beam at 1
kHz repetition rate takes about 1000 s (_vide infra_).
The imaging system CMOS camera (Optronis CL600x2) is operated at a camera-
link-readout-limited resolution of $480\times 480$ pixel at sustained 1 kHz
repetition rate. This corresponds to a spatial resolution of 80 µm on the
phosphor screen. The typical spatial illumination on the camera corresponding
to one ion is four pixels. The background corrected camera images are analyzed
for every single shot with a centroiding algorithm on a standard PC computer,
making use of eight available cores by sequentially distributing the images
onto different cores. The coordinates of the ion hits are passed to the main
data acquisition system. The single-shot analysis allows high signal rates
exploiting the high saturation limit of the detection system.
## III Results
### III.1 1D Alignment
Figure 2 a) shows the degree of alignment
$\langle\cos^{2}\theta\rangle_{\text{2D}}$ 111The degree of alignment
$\langle\cos^{2}\theta\rangle_{\text{2D}}$ is defined as
$\int_{0}^{\pi}\int_{v1}^{v2}\cos^{2}\theta f(\theta,v)d\theta dv$ with
$f(\theta,v)$ being the two dimensional ion distribution on the detector. The
angle $\theta$ is defined as the angle between the polarization of the
alignment laser and the projection of the three dimensional ion velocity
vector onto the two dimensional detector plane. The velocity is given by
$v=\sqrt{v_{x}^{2}+v_{y}^{2}}$. as a function of the peak intensity of the
alignment pulse; see the inset of Figure 1 for a definition of $\theta$. The
results are shown for a repetition rate of 1 kHz in solid black (deflector
off) and dashed black (deflected beam) as well as for a repetition rate of 100
Hz in solid blue/grey (deflector off) and dashed blue/grey (deflected).
Figure 2: (Color online). a) The degree of alignment as a function of the peak
intensity of the alignment pulse for a repetition rate of 1 kHz in solid black
(deflector off) and dashed black (deflected) as well as for a repetition rate
of 100 Hz in solid blue/grey (deflector off) and dashed blue/grey (deflected).
The insets show b) the 2D velocity distribution without alignment laser pulse
and c) with an alignment pulse with a peak intensity of $0.42\times 10^{12}$
W/cm2 at a repetition rate of 1 kHz in the deflected part of the beam.
The observed power dependence of the degree of alignment for a cold beam is as
expected Kumarappan _et al._ (2006). For randomly aligned molecules a degree
of alignment $\langle\cos^{2}\theta\rangle_{\text{2D}}=0.5$ is expected. The
degree of alignment increases with increasing laser intensity, but is limited
by $\langle\cos^{2}\theta\rangle_{\text{2D}}=1$. The 2D momentum image for I+
ions for the alignment-field-free case at 1 kHz repetition rate is shown in
Figure 2 b). The distribution is circularly symmetric as expected for the case
of an isotropic sample and the polarization of the probe laser pulse linear
and perpendicular to the detector plane. The degree of alignment is given by
$\langle\cos^{2}\theta\rangle_{\text{2D}}=0.5007(5)$. Figure 2 c) shows the
corresponding ion distribution when the molecules are aligned at 1 kHz
repetition rate along the alignment pulse polarization axis, i. e., linear,
vertical and parallel to the detector plane. The number of ions is about three
ions/pulse and the image has been recorded in less than 10 minutes. The peak
laser intensity is $4.2\times 10^{11}$ W/cm2. The degree of alignment
determined from the outer structure (between $v_{1}$=1.7 km/s and $v_{2}$=2.2
km/s) of the two dimensional ion distribution is
$\langle\cos^{2}\theta\rangle_{\text{2D}}=0.935(1)$. The radial structures in
the ion images indicate that there are two fragmentation channels present in
the Coulomb explosion of iodobenzene: the inner ring corresponds to
$\text{I}^{+}$ recoiling from a singly charged phenyl, whereas the outer ring
corresponds to $\text{I}^{+}$ recoiling from doubly charged phenyl Larsen _et
al._ (1999). Using the outer ring for the determination of
$\langle\cos^{2}\theta\rangle_{\text{2D}}$ is favorable as it corresponds to
the most sudden fragmentation channel and, therefore, to the best axial recoil
conditions. The derived value underestimates the degree of alignment since the
probe and alignment pulse have perpendicular polarization and, therefore, the
least-aligned molecules are ionized with the highest efficiency Kumarappan
_et al._ (2006).
A slightly higher degree of alignment is obtained when the valve is operated
at 100 Hz repetition rate. The degree of alignment with the deflector off at
100 Hz repetition rate is comparable to the one obtained at 1 kHz repetition
rate in the deflected beam. A slight increase of the degree of alignment to
$\langle\cos^{2}\theta\rangle_{\text{2D}}=0.942(1)$ is observed in the
deflected part of the beam at 100 Hz repetition rate. In order to investigate
the reason for this behavior we have operated the valve at 100 Hz at a higher
temperature and leaked helium gas into the chamber. Both, temperature and
source chamber pressure, have been adjusted to match the conditions found at 1
kHz repetition rate. Under these conditions we observed a degree of alignment
that matches the 1 kHz results. Therefore, the slightly smaller degree of
alignment is attributed to a higher valve temperature and a higher background
pressure in the source chamber when the valve is operated at 1 kHz. The lower
peak intensity of the employed alignment pulse is the reasons for the smaller
value of $\langle\cos^{2}\theta\rangle_{\text{2D}}$ compared to previous
reported best value obtained using a 20 Hz 10 ns injection seeded Nd:YAG laser
Holmegaard _et al._ (2009). Obviously, these experimental limits could be
solved by exploiting higher-power lasers and considerably larger vacuum pumps,
which have not been available for this study.
The pulse length of the alignment pulse is 450 ps. The rotational period of
iodobenzene (J-type revival) is 707.7 ps Poulsen _et al._ (2004); Dorosh _et
al._ (2007). Thus, the laser pulse length is shorter than the rotational
period of iodobenzene, placing this study in the intermediate regime between
adiabatic and impulsive alignment – closer to the adiabatic case. The observed
degree of alignment clearly indicates comparable control as in previously
reported adiabatic alignment experiments Larsen _et al._ (1999); Holmegaard
_et al._ (2009). The detailed influence of non-adiabatic effects for the
studied system is not clear and a more detailed investigation of the alignment
dynamics as a function of the laser pulse duration is in preparation.
### III.2 1D Orientation
Figure 3 shows the ratio of ions in the upper half of the detector images
divided by the total number of ions
$\text{N}_{\text{up}}/\text{N}_{\text{tot}}$ as a function of the angle
$\beta$.
Figure 3: (Color online) The ratio
$\text{N}_{\text{up}}/\text{N}_{\text{tot}}$ as a function of the angle
$\beta$. The color scheme is as in Figure 2 a). The inset shows the definition
of the angle $\beta$. The typical time scale of one full $\beta$-angle scan at
1 kHz is in the order of 1000 s.
The color scheme is as in Figure 2a). The angle $\beta$ is defined as the
angle between the polarization of the alignment laser and the static electric
field of the VMI spectrometer, $\text{E}_{\text{static}}=840$ V/cm, as shown
in the inset of Figure 3. The probe beam is circularly polarized. The observed
dependence of the orientation as a function of the angle beta is as expected
from previous low-repetition rate experiments Holmegaard _et al._ (2009);
Filsinger _et al._ (2009a). The molecules are better oriented in the
deflected part of the beam, where the population is confined to the
energetically lowest, i. e., the most polar, rotational states. In addition we
observe better orientation at 100 Hz than at 1 kHz repetition rates. The
reason for this is the slightly increased rotational temperature of the 1 kHz
molecular beam, as discussed above. In comparison with the orientation for the
same molecule obtained using 10 ns alignment-laser pulses Holmegaard _et al._
(2009) we obtain a considerably smaller degree of orientation in the current
experiment. This can be attributed to the nonadiabatic mixing of levels in the
near-degenerate doubles created by the strong laser field, which has been
shown to be more prominent for shorter laser pulses Nielsen _et al._ (2012).
This is also in agreement with earlier theoretical studies that have shown
that the degree of impulsive orientation, using short laser pulses, is limited
by the magnitude of the applied DC electric field Rouzee _et al._ (2009).
### III.3 Molecular-beam deflection dependence
In Figure 4 the vertical molecular beam profiles (Figure 4 c) and the
dependence of the degrees of alignment (Figure 4 a) and orientation (Figure 4
b) on the position in the molecular beam, measured in a single lens scan as
described above, are shown.
Figure 4: (Color online) The degree of alignment (a), the ratio
$\text{N}_{\text{up}}/\text{N}_{\text{tot}}$ (b) and the molecular beam
density (c) as a function of the position in the molecular beam. The color
scheme is as in Figure 2a). The positions for the best alignment and
orientation are marked by vertical lines. The time for the data acquisition at
1 kHz for a single curve is in the order of half an hour.
The color scheme of the plots is the same as in Figure 2a). The molecular beam
profiles and the observed deflection, for an applied deflector voltage of 12
kV, are in good agreement with previous measurements Filsinger _et al._
(2009a). For the beam with the deflector off, the observed degrees of
alignment and orientation are practically constant over the main part of
molecular beam profile. Towards the sides, however, the achievable control
decreases, which is especially pronounced for the alignment measurement. This
demonstrates that the molecular beam is considerably warmer at the sides than
in the center. This is attributed to collisions with the rest gas and to
interference with mechanical apertures, i. e., skimmers. When applying an
inhomogeneous electric field, the molecules are deflected upwards. Moreover,
the molecular beam is dispersed according to the molecules’ effective dipole
moments, or, correspondingly, according to their rotational states. The most
polar, lowest-energy rotational states are deflected the most, and this is
reflected in the increased degrees of alignment and orientation in the
deflected part of the molecular beam. This accounts for the larger
contributions of lower quantum states which show a large deflection. In
addition, a decrease of the degree of alignment and orientation in regions
where the high rotational states remain (right part of the molecular beam
profile) is observed. Helium is not present anymore in the deflected part of
the molecular beam since its trajectory is not influenced by the electric
deflector. As the degree of alignment and orientation is dependent on the
position in the dispersed molecular beam, the properties of the beam change as
a function of the vertical laser probe position. This opens up the possibility
for advanced multidimensional investigations, i. e., the observation of state-
selective dynamics, in the future. It can also be used to increase the
understanding of the deflection process of complex molecules, including the
investigation of nonadiabatic couplings of rotational states by the “slowly”
changing electric fields. Moreover, it enables the study of the nature of
ensembles of molecules in a single molecular quantum states in mixed fields,
as in a previous study on impulsive alignment and mixed-field orientation of a
nearly pure ground state ensemble of OCS Nielsen _et al._ (2011, 2012).
## IV Conclusions and Outlook
In conclusion, a high-repetition-rate experimental molecular physics setup has
been developed. It allows the preparation of very strongly aligned and
oriented samples of quantum-state-selected cold molecules at a 1 kHz
repetition rate. The dependence of the degree of alignment and orientation on
the position in the molecular beam has been analyzed. Both parameters are
enhanced when probing the state-selected molecules in the deflected part of
the beam. A maximum degree of alignment at 1 kHz of
$\langle\cos^{2}\theta\rangle_{\text{2D}}=0.935$ has been obtained for a 450
ps long pulse with a peak intensity of $0.42\times 10^{12}$ W/cm2. The
obtained degree of orientation is
$\text{N}_{\text{up}}/\text{N}_{\text{tot}}=0.56$. This value is lower than
what was demonstrated in previous studies Holmegaard _et al._ (2009), and it
is limited by the combination of the applied DC electric field in the VMI
spectrometer and the pulse duration of the alignment pulse, in accordance with
previous analyses Nielsen _et al._ (2012); Rouzee _et al._ (2009).
The pulse duration of our alignment pulse is continuously tunable over a wide
range from $<\\!50$ fs to $>\\!500$ ps; this will allow the investigation of
the influence of this duration on the (non)adiabaticity on the alignment and
orientation Nielsen _et al._ (2012). Preliminary measurements show that the
degree of alignment is increasing when the pulse duration is shortened, due to
the stronger peak intensity available in our setup. At the same time, revivals
structures start to appear, indicating clear nonadiabatic effects in the
alignment dynamics. It is likely, that for each molecular sample a trade-off
between adiabatic and non-adiabatic driving of the alignment can be found to
ensure an optimal degree of alignment and orientation.
The demonstrated strong control over molecules at high repetition rates
promises the feasibility of novel investigations of “weak” processes, such as
chemical dynamics, using molecular-frame photoelectron angular distributions,
photoelectron holography, or X-ray or electron diffraction imaging. For the
applied, relatively short, alignment laser pulses, only moderate pulse
energies, on the order of 10 mJ are necessary. It is envisioned, that such
pulses will be available at hundreds of kHz or even MHz repetition rates in
the near future. For instance, similar setups are envisioned for the European
XFEL, where burst mode lasers with similar pulse energies are under
consideration and electric state selectors can be implemented for arbitrary
repetition rate molecular beams. This opens up the possibility of performing
time resolved dynamics studies using molecular frame photoelectron angular
distributions Hansen _et al._ (2011) or photoelectron holography Landers _et
al._ (2001); Krasniqi _et al._ (2010). Moreover, the controlled samples serve
as ideal targets in x-ray or electron diffraction experiments Filsinger _et
al._ (2011); Reckenthaeler _et al._ (2009); Hensley _et al._ (2012), and
they hold great promise toward attosecond and high-harmonic-generation
experiments of complex molecules, where the molecular alignment and
orientation can be exploited to modulate and enhance the output Velotta _et
al._ (2001). Moreover, the state-selection process inherently separates
structural isomers Filsinger _et al._ (2008, 2009b), which is a necessary
ingredient to investigate ultrafast dynamics of structural isomers, such as
charge migration in various conformers of glycine Kuleff and Cederbaum (2007).
###### Acknowledgements.
We thank the DESY-FS infrastructure groups for support, HASYLAB/Petra III for
hosting our laboratory, the CFEL staff for administrative and technical
support and Vinod Kumarappan for the camera software. This work has been
supported by the excellence cluster “The Hamburg Centre for Ultrafast Imaging
- Structure, Dynamics and Control of Matter at the Atomic Scale” of the
Deutsche Forschungsgemeinschaft. N. L. M. M. acknowledges financial support by
the Joachim Herz Stiftung.
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|
arxiv-papers
| 2013-01-09T12:23:36 |
2024-09-04T02:49:40.040029
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sebastian Trippel, Terry Mullins, Nele L. M. M\\\"uller, Jens S.\n Kienitz, Karol D{\\l}ugo{\\l}ecki, and Jochen K\\\"upper",
"submitter": "Jochen K\\\"upper",
"url": "https://arxiv.org/abs/1301.1826"
}
|
1301.1839
|
# Jet-parton inelastic interaction beyond eikonal approximation
Raktim Abir Theory Division, Saha Institute of Nuclear Physics, 1/AF
Bidhannagar, Kolkata 700064, India.
###### Abstract
Most of the models of jet quenching generally assumes that a jet always
travels in a straight eikonal path, which is indeed true for sufficiently hard
jet but may not be a good one for moderate and low momentum jet. In this
article an attempt has been made to relax part of this approximation for
$2\rightarrow 3$ processes and found a (15-20)$\%$ suppression in the
differential cross-section for moderately hard jets because of the noneikonal
effects. In particular, for the process $qq^{\prime}\rightarrow qq^{\prime}g$
in the centre of momentum frame the scattering with an angle wider than $\pm
0.52\pi$ is literally forbidden unlike the process $gg\rightarrow ggg$ that
allows an angular range $\pm\pi$. This may have consequence on the suppression
of hadronic spectra at low transverse momenta.
## I Introduction
Constituent quark number scaling of elliptic flow and jet quenching are
supposed to be the most prominent signatures that favour the partonic degrees
of freedom in the deconfined QCD matter. Overwhelming evidences of both
signatures coming from dedicated heavy ion experiments $viz.,$ STAR and
PHENIX@ RHIC BNL Adams:2005dq ; Adcox:2004mh , ALICE @ LHC CERN Aamodt:2010jd
; Aamodt:2010pa established the fact that, the primordial hot-soup of nuclear
matter, produced in those experiments, indeed contain partonic degrees of
freedom, before freeze out to hadrons in later stage, instead of hadronic
degrees of freedom throughout. Observation of strong suppression of inclusive
yields of high momentum hadrons and semi-inclusive rate of azimuthal back-to-
back high momentum hadron pairs relative to p-p collisions, are expectations
from jet quenching. Both of them are extensively explored in collisions of Au-
Au nuclei at $\sqrt{s}=200$ A GeV in RHIC. ALICE, the dedicated heavy Ion
collider experiment at CERN, seems to appear as a factory of Jets. Evidence
for jet quenching has also been observed recently in Pb-Pb collision at ALICE
Aamodt:2010jd .
There are a few well known models in the literature Salgado:2003gb ;
Wicks:2005gt ; Wang:2001ifa ; Jeon:2003gi ; Mustafa:1997pm ; Mustafa:2003vh ;
Qin:2007rn ; Qin:2009gw ; Abir:2012pu ; Abir:2011jb ; Zapp:2012ak that aim to
quantify energy loss (mainly radiative) and jet quenching phenomena within
perturbative QCD. Most of these formalisms for energy loss of a high momentum
parton through gluon radiations, have some common technical approximations in
order to make the calculation simpler. These approximations are implemented
both at the level of single emission kernel calculations and at multiple gluon
emission estimation schemes. Main kinematic approximations at the level of
single emission kernel, are listed below (also see Refs. Armesto:2011ht ;
Mehtar-Tani:2013pia ; Renk:2011aa for a comprehensive discussion):
* •
Eikonal parton trajectory I : Leading parton is having energy $E$ ($i.e.$
$p_{z}=E$ and $p_{\bot}=0$, by definition to start with) is much larger than
the transverse momentum exchanged gluon $q_{\perp}$ with the medium, $E\gg
q_{\perp}$, so that it does not give sufficient transverse kick to deflect the
parton from straight trajectory along $z$ axis. To relax this approximation it
is therefore important to keep track of the terms of ${\cal O}(q_{\bot}/E)$ in
the formalism.
* •
Eikonal parton trajectory II : Energy of the leading parton is sufficiently
high, $E\gg k_{\perp}$ ($k_{\perp}$ being transverse component of the emitted
gluon) so that it does not get enough transverse kick also from emitted gluon.
This ensures that the leading parton is in eikonal trajectory. However it does
not fix any definite direction for gluon emission, which requires comparison
of $k_{\perp}$ with longitudinal component $k_{z}$ or with energy $\omega$ of
the emitted gluon. Therefore, it is important not to neglect terms of ${\cal
O}(k_{\bot}/E)$ in the formalism to relax this approximation in the jet
studies.
* •
Soft gluon emission : One often uses the additional approximation that the
gluon energy is much smaller than the leading parton energy $\omega\ll E$.
When $x$ is the fraction of energy carried out by the emitted gluon, $i.e.$,
$x=w/E$, this approximation ensures that $x\rightarrow 0$. When $x$ is typical
light cone variable, defined by the fraction of light cone $(+)ve$ momentum
carried out by the emitted gluon, $i.e.$,
$x=k^{+}/p^{+}=(k_{\bot}/\sqrt{s})e^{\eta}$, this approximation ensures
$x\rightarrow 0$, only in the mid rapidity and backward rapidity regions
$(-\infty\geq\eta\geq 0)$ but not in the forward regions where $\eta$ could be
a large positive number.
* •
Small angle/collinear gluon emission : Energy of the emitted gluon $\omega$
is much larger than its transverse momentum $k_{\bot}$, $\omega\gg k_{\bot}$
and $\omega\simeq k_{z}$. For any $2\rightarrow 3$ process, without loss of
generality, one can take $k_{\bot}=\omega\sin\theta_{g}$ and
$k_{z}=\omega\cos\theta_{g}$, where $\theta_{g}$ being the angle between
direction of propagation of leading parton and direction of emitted gluon.
This particular approximation therefore implies $\theta_{g}\simeq 0$.
At this point it is worth mentioning that soft gluon emission approximation is
a broader approximation as it automatically encompasses the eikonal parton
trajectory II approximation, because energy $w$ should always be more than the
transverse momentum $k_{\bot}$ for a massless emitted gluons.
In this article we make an effort to relax the eikonal parton trajectory I
approximation in some extent for the radiative/inelastic process $2\rightarrow
3$. Investigation of all the matrix elements in ${\cal{O}}(\alpha_{s}^{3})$
for $2\rightarrow 3$ radiative processes have been done keeping terms up to
$\cal{O}$ $(t/s)$. The first order in eikonal expansion, i.e., $\cal{O}$
$(t/s)$, is termed here as noneikonal, since the calculation is performed in
Feynman gauge with Mandelstum variables instead of most extensively used light
cone gauge with light cone variables111After connecting $t$ to $q_{\perp}$ and
$s$ to $E$ in the centre of momentum frame, term of $\cal{O}$ $(t/s)$ ensures
the relaxation of the approximation $E\gg q_{\perp}$ (eikonal parton
trajectory I).. We have neglected terms of ${\cal{O}}(t^{2}/s^{2})$ and higher
orders. Throughout our study we also do not use any approximation related to
small angle/collinear gluon emission. Nevertheless, we have used soft gluon
emission approximation which automatically include Eikonal parton trajectory
II assumption.
Here we relaxed the kinematic constrains associated with eikonal propagation.
Nevertheless (non)eikonal propagation is directly related to space-time tracks
that the partons take when going through the medium. While an eikonal path is
a straight line, a non-eikonal path is generically any trajectory that have
deviation from straight line and obviously always somewhat longer. This longer
path, taken by the parton may matter, in the soft sector when the problem is
embedded into a hydrodynamically evolving density distribution Renk:2011gj .
We note that there is transport model, although uses the cross sections which
are kind of eikonal, takes all the trajectories into account, so particles do
not need to move on straight lines Fochler:2010wn .
## II Inelastic Quark-Quark Scattering ($qq^{\prime}\rightarrow
qq^{\prime}g$)
The process $qq^{\prime}\rightarrow qq^{\prime}g$ (prime denotes different
quark flavour) in ${\cal{O}}(\alpha_{s}^{3})$ appears in five $t$ channel
Feynman diagrams, which are shown in the Fig. 1 (see also Fig. 2 for other
details). Note that $k_{1}$ and $k_{2}$ are momenta of the different quark
flavours in the entrance channel whereas $k_{3}$ and $k_{4}$ are those for
exit channel and $k_{5}$ is that of the emitted gluon. Scattering angle
between ${\bf k_{1}}$ and ${\bf k_{3}}$ is $\theta_{q}$ whereas $\theta_{g}$
is the angle between direction of emission of soft gluon $\bf\hat{k}_{5}$ and
direction of incoming projectile quark $\bf\hat{k}_{1}$. We now define the
relevant Mandelstum variables for this $2\rightarrow 3$ process as
$\displaystyle s=(k_{1}+k_{2})^{2}\,,\hskip
8.5359pts^{\prime}=(k_{3}+k_{4})^{2},$ $\displaystyle
u=(k_{1}-k_{4})^{2}\,,\hskip 8.5359ptu^{\prime}=(k_{2}-k_{3})^{2},$
$\displaystyle t=(k_{1}-k_{3})^{2}\,,\hskip
8.5359ptt^{\prime}=(k_{2}-k_{4})^{2},$ (1)
with
$\displaystyle s+t+u+s^{\prime}+t^{\prime}+u^{\prime}=0\ .$ (2)
When the emitted gluon is soft ($k_{5}\rightarrow 0$) compare to other
external legs, one can assume : $t^{\prime}\rightarrow t,\ \
s^{\prime}\rightarrow s,\ \ u^{\prime}\rightarrow u$ and we can express the
transverse component of the momentum of the emitted gluon in the centre of
momentum frame of $k_{1}$ and $k_{2}$ as
$\displaystyle k_{\perp}^{2}$ $\displaystyle=$
$\displaystyle\frac{4(k_{1}\cdot k_{5})(k_{2}\cdot k_{5})}{s}$ (3)
$\displaystyle=$ $\displaystyle\frac{(s+t+u)(s+u^{\prime}+t^{\prime})}{s}$
$\displaystyle=$ $\displaystyle\frac{(s+t+u)^{2}}{s}\ .$
\begin{overpic}[width=260.17464pt]{qQ_qQg.eps} \put(-8.0,67.0){(1)}
\put(-8.0,38.0){(2)} \put(51.0,67.0){(3)} \put(51.0,38.0){(4)}
\put(25.0,10.0){(0)} \end{overpic} Figure 1: Five tree level Feynman diagrams
for the process $qq^{\prime}\rightarrow qq^{\prime}g$.
The hierarchy among various scale of momentum, employed in the present work is
stated as
$\sqrt{s},E>\sqrt{|t|},q_{\bot}\gg w\geq k_{\bot}\geq m_{d}\,,$ (4)
where $m_{d}$ is the Debye screening mass acts as an infrared cut-off. We note
that the above hierarchy relaxes the aproximations
$\sqrt{s},E\gg\sqrt{|t|},q_{\bot}$ (Eikonal parton trajectories I ) and $w\gg
k_{\bot}$ (small angle/collinear gluon emission ) in some extent.
Figure 2: Non-zero angular deviation from eikonal trajectory. Angle between
incoming and outgoing momentum of projectile is $\theta_{q}$ and direction of
emition of gluon with that of incoming momentum of projectile parton is
$\theta_{g}$.
### II.1 Matrix Elements, Amplitude and Cross-section
The gauge invariant amplitude summed over all the final states and averaged
over initial states for the process, $qq^{\prime}\rightarrow qq^{\prime}g$, is
$\left|{\cal M}_{qq^{\prime}\rightarrow qq^{\prime}g}\right|^{2}=\sum_{i\geq
j}{\cal M}_{ij}^{2}\ ,$ (5)
where $i$ and $j$ run from $0$ to $4$. We note that the index $0$ represents
the diagram where the soft gluon emits from the exchanged gluon line whereas
the indices $m=1,2,3,4$ represent the diagrams where emission of the soft
gluons are being from external fermion lines having momenta $k_{m}$ (see Fig.
2). Equation (5) contains total fifteen terms in which there are five self
interfering ‘genuine amplitudes’ for $i=j$ and ten cross interfering
‘interference amplitudes’ associated with gluon emissions involving two
diagrams for $i\neq j$. Here we have extensively used the FORM, REDUCE and
CalcHep Programs Belyaev:2012qa . Results are given below up to ${\cal
O}(1/k_{\bot}^{2})$ and ${\cal O}(t/s)$, for soft gluon emission.
#### II.1.1 Genuine amplitudes
By genuine amplitudes we are referring those are coming from each diagram that
interfers with itself. In the Feynman gauge 222In light cone gauge amplitudes
coming from diagrams that involve gluon emission from target partons can only
be neglected, others are not. all of them vanish within soft gluon emission
approximations and in ${\cal O}(1/k_{\bot}^{2})$:
$\displaystyle{\cal M_{\rm 11}^{\rm 2}}$ $\displaystyle=$ $\displaystyle{\cal
M_{\rm 33}^{\rm 2}}=0;\,\,{\cal M_{\rm 22}^{\rm 2}}={\cal M_{\rm 44}^{\rm
2}}=0;\,\,{\cal M_{\rm 00}^{\rm 2}}=0.$ (6)
However, they may contribute in ${\cal O}(1)$, ${\cal O}(k_{\bot}^{2})$,
${\cal O}(k_{\bot}^{4})$ etc, and in ${\cal{O}}(t^{2}/s^{2})$ and higher
orders. All of them can safely be neglected in the soft emission limit as our
aim is to go beyond the eikonal approximation-I, $E\gg q_{\perp}$.
#### II.1.2 Interference amplitudes
Within the approximations employed above the amplitudes corresponding to the
matrix elements of $({\rm 1}\bf\otimes{\rm 4})$ and $({\rm 2}\bf\otimes{\rm
3})$ are identical in the leading order (${\cal O}(1/k_{\bot}^{2})$) as well
in ${\cal O}(t/s)$ and given as
$\displaystyle{\cal M_{\rm 14}^{\rm 2}}$ $\displaystyle=$ $\displaystyle{\cal
M_{\rm 23}^{\rm
2}}=\frac{7}{8}\frac{128}{27}g^{6}\frac{s^{2}}{t^{2}}\frac{1}{k_{\bot}^{2}}\left[1+2\frac{t}{s}\right],$
where ${\cal O}(t/s)$ is purely noneikonal (i.e., the first order in eikonal
approximation) in nature as noted earlier. Also the amplitudes for $({\rm
1}\bf\otimes{\rm 2})$ and $({\rm 3}\bf\otimes{\rm 4})$ are identical and the
contribution is obtained in ${\cal O}(1/k_{\bot}^{2})$ and ${\cal O}(t/s)$ as
$\displaystyle{\cal M_{\rm 12}^{\rm 2}}$ $\displaystyle=$ $\displaystyle{\cal
M_{\rm 34}^{\rm
2}}=\frac{1}{4}\frac{128}{27}g^{6}\frac{s^{2}}{t^{2}}\frac{1}{k_{\bot}^{2}}\left[1+\frac{t}{s}\right].$
Both $({\rm 1}\bf\otimes{\rm 3})$ and $({\rm 2}\bf\otimes{\rm 4})$ are also
identical within the employed hierarchy. However, they do not contribute in
leading order but only in ${\cal O}(t/s)$. Hence, in Feynman gauge the
contribution from the interference between initial state and final state
radiations, is exclusively noneikonal in nature and given as
$\displaystyle{\cal M_{\rm 13}^{\rm 2}}$ $\displaystyle=$ $\displaystyle{\cal
M_{\rm 24}^{\rm
2}}=\frac{1}{4}\frac{128}{27}g^{6}\frac{s^{2}}{t^{2}}\frac{1}{k_{\bot}^{2}}\left[\frac{1}{2}\frac{t}{s}\right],$
Finally any diagram interfering with ${\rm 0}$, i.e., $({\rm 0}\bf\otimes{\it
l})$ with $l=1,2,3,4$, does not contribute in ${\cal O}(1/k_{\bot}^{2})$ but
contributes in $(1/k_{\bot}\sqrt{t})$ and higher orders. In the limit
$|\sqrt{t}|\sim q_{\bot}\gg w$, amplitudes of ${\cal O}(1/k_{\bot}\sqrt{t})$
are subleading, in comparison to ${\cal O}(1/k_{\bot}^{2})$. Therefore, all of
these (${\cal M_{\rm 10}^{\rm 2}}$, ${\cal M_{\rm 20}^{\rm 2}}$, ${\cal M_{\rm
30}^{\rm 2}}$ and ${\cal M_{\rm 40}^{\rm 2}}$) do not contribute within the
approximation used in this work.
Here we note that color coherence pattern between initial and final state
radiation in the presence of a QCD medium derived in light cone gauge shows
angular distribution of the induced gluon spectrum is broadly modified when
one includes interference terms Armesto:2012qa ; MehtarTani:2012cy . In
Feynman gauge within our hierarchy only interferance terms are contributing to
the amplitude.
The gauge invariant amplitude for the process, $qq^{\prime}\rightarrow
qq^{\prime}g$, can now be obtained by summing all the subamplitudes as
$\displaystyle\left|{\cal M}_{qq^{\prime}\rightarrow
qq^{\prime}g}\right|^{2}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!12g^{2}\left|{\cal M}_{qq^{\prime}\rightarrow
qq^{\prime}}\right|^{2}_{eknl}\frac{1}{k_{\perp}^{2}}\left(1+\frac{17}{9}\frac{t}{s}\right),$
(7)
where the two body amplitude is given as
$\displaystyle\left|{\cal M}_{qq^{\prime}\rightarrow
qq^{\prime}}\right|^{2}_{eknl}=\frac{8}{9}g^{4}\frac{s^{2}}{t^{2}}.$ (8)
The three-body amplitude in (7) for the inelastic process,
$qq^{\prime}\rightarrow qq^{\prime}g$, contains the two-body amplitude for the
elastic process, an infrared factor for the emission of a soft gluon and a
noneikonal correction factor. The expression in (7) will lead to $q_{\bot}\gg
k_{\bot}$ limit of the Gunion and Bertsch formula Gunion:1981qs . If the
emitted gluon is much softer than others it can then be regulated by the Debye
screening mass, $m_{d}$. Terms within the parenthesis in (7) would correspond
to noneikonal correction over the eikonal Gunion Bertsch formula. Eq. (7) is
complete upto ${\cal O}(1/k_{\bot}^{2})$ and ${\cal O}(t/s)$, for emission of
a soft gluon in the process $qq^{\prime}\rightarrow qq^{\prime}g$. Similar
investigation have been done earlier for the process $gg\rightarrow ggg$
Das:2010hs ; Abir:2010kc ; Bhattacharyya:2011vy .
Figure 3: Dependence of inverse cross-section ${\sigma^{-1}}$ on scattering
angle $\theta_{q}$, all prefactor that are independent of $\theta_{q}$ taken
to be unity for convenience.
#### II.1.3 Rutherford scattering beyond eikonal approximation
It is interesting to note how noneikonality gives way to probe beyond
Rutherford scattering limits. In the centre of momentum frame for a typical
$2\rightarrow 2$ (or even in case of $2\rightarrow 3$ when fifth particle is
ultra soft!) process
$\displaystyle\frac{t}{s}$ $\displaystyle=$
$\displaystyle-\sin^{2}\frac{\theta_{q}}{2}$ (9)
The eikonal cross-section $(\sigma_{eknl})$ is directly connected to the
Rutherford scattering scattering cross-section as
$\displaystyle\sigma_{eknl}\propto\frac{s^{2}}{t^{2}}=\frac{1}{\sin^{4}(\theta_{q}/2)}\,.$
(10)
When one relaxes the eikonal approximation then the cross-ection can be
written as
$\displaystyle\\!\\!\\!\sigma_{ne}\\!\\!$ $\displaystyle\propto$
$\displaystyle\\!\\!\frac{s^{2}}{t^{2}}\left(1+\frac{17}{9}\frac{t}{s}\right)\\!\\!=\\!\\!\frac{1}{\sin^{4}\frac{\theta_{q}}{2}}\left(1-\frac{17}{9}\sin^{2}\frac{\theta_{q}}{2}\right),$
(11)
which puts a restriction on the scattering angle, $\theta_{q}$. In Fig.3 we
have plotted inverse cross-section $(\sigma^{-1})$ for both eikonal and
noneikonal case. Even though both behave identically with a similar plateau in
the small angle scattering but noneikonal cross-section has a very sharp fall
in comparison with the eikonal one for large angle scattering. As seen the
noneikonal inverse matrix element is bounded by the scattering angle,
$\theta_{q}=\pm 2\sin^{-1}(3/\sqrt{17})\simeq\pm 0.52\pi$, in centre of
momentum frame, in contrary to that of eikonal one having a natural bound of
$\pm\pi$. This indicates that the back scattering is forbidden for the case of
$qq^{\prime}\rightarrow qq^{\prime}g$ when the emitted gluon is soft.
#### II.1.4 Cross-section in the first order in eikonal (viz.,noneikonal)
approximation
The cross-section for the process ${qq^{\prime}\rightarrow qq^{\prime}g}$ can
be obtained as
$\displaystyle\sigma_{qq^{\prime}\rightarrow qq^{\prime}g}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\int\prod_{i=3}^{5}\frac{d^{3}k_{i}}{(2\pi)^{3}2E_{i}}\left|{\cal
M}_{qq^{\prime}\rightarrow
qq^{\prime}g}\right|(2\pi)^{4}\delta^{4}(k_{1}+k_{2}-k_{3}+k_{4}+k_{5})\,.$
(12)
In the centre of momentum frame, ${\bf k_{1}}+{\bf k_{2}}={\bf k_{3}}+{\bf
k_{4}}+{\bf k_{5}}={\bf 0}$, and one obtains
$\displaystyle\sigma_{qq^{\prime}\rightarrow qq^{\prime}g}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\int\frac{d^{3}k_{3}}{(2\pi)^{3}2E_{3}}\frac{1}{(2\pi)^{3}2E_{4}}\frac{d^{3}k_{5}}{(2\pi)^{3}2E_{5}}\left|{\cal
M}_{qq^{\prime}\rightarrow
qq^{\prime}g}\right|(2\pi)^{4}\delta(E_{1}+E_{2}-E_{3}+E_{4}+E_{5})$ (13)
$\displaystyle=$
$\displaystyle\frac{1}{2s}\left[-\frac{1}{2}\frac{1}{(2\pi)^{2}}\int\frac{dq_{\bot}^{2}dq_{z}}{E_{3}}\right]\frac{1}{(2\pi)^{3}2E_{4}}\left[\frac{1}{4}\frac{1}{(2\pi)^{2}}\int\frac{dk_{\bot}^{2}d\theta_{g}}{\sin\theta_{g}}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
12g^{2}\frac{8}{9}g^{4}\frac{s^{2}}{t^{2}}\frac{1}{k_{\bot}^{2}}\left(1+\frac{17}{9}\frac{t}{s}\right)(2\pi)^{4}\delta(E_{1}+E_{2}-E_{3}+E_{4}+\omega)$
$\displaystyle=$
$\displaystyle\frac{1}{2s}\left[-\frac{1}{2}\frac{1}{(2\pi)^{2}}\int\frac{dq_{\bot}^{2}}{E_{1}}\right]\frac{1}{(2\pi)^{3}2E_{1}}\left[-\frac{1}{4}\frac{1}{(2\pi)^{2}}\int
dk_{\bot}^{2}d\eta\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
12g^{2}\frac{8}{9}g^{4}\frac{s^{2}}{(q_{\bot}^{2})^{2}}\left(1+\frac{q_{\bot}^{2}}{s}\right)^{-2}\frac{1}{k_{\bot}^{2}}\left(1-\frac{17}{9}\frac{q_{\bot}^{2}}{s}\right)(2\pi)^{4}\,,$
where we have used rapidity
$\eta=-\ln\left[\tan\left(\theta_{g}/2\right)\right]$, $\bf q=k_{1}-k_{3}$,
$\bf q_{\bot}=q~{}{\sin\theta_{q}}$. The cross-section contains factors,
having term like $q_{\bot}^{2}/s$, that are responsible for non-eikonal
effects.
In thermal medium taking debye mass as infrared regulator, the differential
cross-section can be expressed as
$\displaystyle\frac{d~{}\sigma^{qq^{\prime}\rightarrow
qq^{\prime}g}}{dq_{\bot}^{2}dk_{\bot}^{2}d\eta}=2C_{A}C_{qq^{\prime}}\alpha^{3}\frac{\Gamma_{ab}}{(q_{\bot}^{2}+m_{d}^{2})^{2}}\frac{1}{k_{\bot}^{2}+m_{d}^{2}}\,,$
(14)
where $C_{A}=3$ and $C_{qq^{\prime}}=8/9$ are Casimir factors, and
$\Gamma_{ab}=\zeta_{\it a}\zeta_{\it b}$, with various factors are,
explicitly, given as
$\displaystyle\zeta_{\it a}$ $\displaystyle=$
$\displaystyle\left(1+\frac{q_{\bot}^{2}}{s}\right)^{-2}\,,$
$\displaystyle\zeta_{\it b}$ $\displaystyle=$
$\displaystyle\left(1-\frac{17}{9}\frac{q_{\bot}^{2}}{s}\right)\ .$ (15)
The factor $\zeta_{\it a}$ comes from eikonal part of the matrix elements, and
$\zeta_{\it b}$ is the noneikonal factor originated from noneikonal part of
matrix elements. The differential cross-section for the process
$qq^{\prime}\rightarrow qq^{\prime}g$ as given in Eq.(14) correctly reproduces
the result of Biro:1993qt in the limit $q_{\bot}^{2}\gg k_{\bot}^{2}$ and in
the eikonal limit, $\sqrt{s},E\gg q_{\bot}^{2}$, as all the noneikonal
factors, $viz.,$ $\zeta_{\it a}$ and $\zeta_{\it b}$ become identically unity
and so as $\Gamma_{ab}$.
## III Inelastic gluon-gluon fusion ($gg\rightarrow ggg$)
The three gluon production via gluon-gluon fusion $gg\rightarrow ggg$ is
extremely important in the context of heavy-ion phenomenology. For a sequence
of events: hot glue scenario of glasma field, thermal equilibration, gluon
chemical equilibration in later time, parton matter viscosity, radiative
energy-loss of high energy partons jet propagating through thermalised QGP,
this process plays a crucial role. Matrix elements for the process
$gg\rightarrow ggg$ have been computed up to ${\cal O}(t^{3}/s^{3})$ in
Abir:2010kc . Considering ${\cal O}(t/s)$ result it is now straightforward to
evaluate the differential cross-section for this process in first order in
eikonal approximation as
$\displaystyle\frac{d~{}\sigma^{gg\rightarrow
ggg}}{dq_{\bot}^{2}dk_{\bot}^{2}d\eta}=2C_{A}C_{gg}\alpha^{3}\frac{\Gamma_{ab}}{(q_{\bot}^{2}+m_{d}^{2})^{2}}\frac{1}{k_{\bot}^{2}+m_{d}^{2}}\,$
(16)
where $C_{gg}=9/2$ and the factor, $\Gamma_{ab}=\zeta_{a}\zeta_{b}$, with its
various components
$\displaystyle\zeta_{\it a}$ $\displaystyle=$
$\displaystyle\left(1+\frac{q_{\bot}^{2}}{s}\right)^{-2}\,,$
$\displaystyle\zeta_{\it b}$ $\displaystyle=$
$\displaystyle\left(1-\frac{1}{2}\frac{q_{\bot}^{2}}{s}\right)\ .$ (17)
Figure 4: Typical estimation of noneikonal factors at $T=300MeV$ with
$\alpha=0.3$ for the process $qq^{\prime}\rightarrow qq^{\prime}g$. First
order noneikonal factors $\zeta_{a}=\left(1+{q_{\bot}^{2}}/{s}\right)^{-2}$,
$\zeta_{b}=\left(1-{17q_{\bot}^{2}}/{9s}\right)$, and the full contribution
${\Gamma}_{ab}=\zeta_{a}\zeta_{b}$. Figure 5: Typical estimation of noneikonal
factors at $T=300MeV$ with $\alpha=0.3$ for the process $gg\rightarrow ggg$.
First order noneikonal factors
$\zeta_{a}=\left(1+{q_{\bot}^{2}}/{s}\right)^{-2}$,
$\zeta_{b}=\left(1-{q_{\bot}^{2}}/{2s}\right)$ and the full contribution
${\Gamma}_{ab}=\zeta_{a}\zeta_{b}$.
The factor comming from eikonal part of matrix element $\zeta_{\it a}$ is same
for both processes $qq^{\prime}\rightarrow qq^{\prime}g$ and $gg\rightarrow
ggg$ whereas the noneikonal factor $\zeta_{\it b}$ is different. This
noneikonal factor, obviously, does not put any restriction on the scattering
angle ($\theta=\pm\pi$) for the process $gg\rightarrow ggg$, and allows it to
go in full natural range $\pm\pi$ as compared to the process
$qq^{\prime}\rightarrow qq^{\prime}g$.
Unlike $gg\rightarrow ggg$ where a Park-Taylor type formula Parke:1986gb is
available to compute the matrix element, the computation of matrix elements up
to ${\cal O}(t/s)$ is quite cumbersome in case of $qg\rightarrow qgg$. Also in
this article we have performed our study on inelastic quark-quark scattering
but with different flavours. In case of same flavour $qq\rightarrow qqg$
things would be a more involved one. We leave them for future study.
## IV Results and Discussion
For quantitative estimation of the noneikonal effects, we have taken the
average value of the momentum transfer squared which can be obtained
Abir:2012pu as
$\displaystyle\langle q_{\bot}^{2}\rangle$ $\displaystyle\simeq$
$\displaystyle\left({\int\limits_{m_{g}^{2}}^{E^{2}}dq_{\bot}^{2}\
q_{\bot}^{2}\ \frac{d\sigma_{2\rightarrow
3}}{dq_{\bot}^{2}}}\right)/\left({\int\limits_{m_{g}^{2}}^{E^{2}}dq_{\bot}^{2}\
\frac{d\sigma_{2\rightarrow 3}}{dq_{\bot}^{2}}}\right)$ (18)
$\displaystyle\simeq$ $\displaystyle 2g^{2}T^{2}\ln\left(E/gT\right).$
$\langle q_{\bot}^{2}\rangle$ has then been embedded in $\zeta_{a}$,
$\zeta_{b}$ to have a qualitative estimation of noneikonal effects over
eikonal crosssections.
In Fig.4 and Fig.5 the first order noneikonal factors: $\zeta_{a}$,
$\zeta_{b}$ and the full contribution $\Gamma_{ab}$ for both processes
$qq^{\prime}\rightarrow qq^{\prime}g$ and $gg\rightarrow ggg$, respectively,
displayed. It can be seen that the noneikonal effects are $\sim(15-20)\%$ over
eikonal one for moderately hard jets. However, the noneikonal effect gradually
becomes mild for very high energetic jets. Typically for charged hadrons in
the momentum range of $8$ to $50$ GeV, i.e. typical parton kinematics of $16$
to $150$ GeV, way off the scale of Fig.4 and Fig.5, indicate that non-eikonal
effects are largely absent above $(10-15)$ GeV parton kinematics. In the
literature attempts have already been made to address the noneikonal
propagation of partons for collision/elastic processes in a monte-carlo
approach Auvinen:2009qm by considering full ${\cal{O}}(\alpha_{s}^{2})$
matrix elements for relevant $2\rightarrow 2$ processes. Eikonal propagation
approximation was found to be good on the $10\%$ level. Present study also
revels that for radiative/inelastic process, Eikonal parton trajectory I
approximation seems to be within $(15-20)\%$ level. This approximation should
be crude only in soft and moderate momentum regimes. We also note that
splitting kernels for partons produced in large virtuality scattering
processes that subsequently traverse a region of strongly-interacting matter
have been investigated early in the literature within effective theory
formalism Ovanesyan:2011kn .
There has always been a quest for large angle radiations. In the present study
we do not assume small angle/collinear emission approximation either in the
course of calculating matrix elements or in the calculations of kinematics. In
the present calculation the kinematic relation $k_{\bot}=\omega\sin\theta_{g}$
ensures that one can go safely to the limit where $k_{\bot}\simeq\omega$.
Most of the pQCD based model calculations for describing the medium is not
able to account the almost ideal fluid behaviour, which seems to be a
manifestation of long range correlations. It also would lead to a large
elastic contribution to energy loss Auvinen:2010yt for reasonable values of
coupling which is not supported by the data. Sort of single hard scenario
Gyulassy:2000fs ; Wicks:2005gt has been discussed here at the level of single
emission karnel in which gluons are induced by a single hard scattering with
the medium, but many widely used quenching models, for instance Salgado:2003gb
, make the opposite assumption of multiple soft interactions with the medium
leading to induced gluon radiation. Such classes of models can probably be
expected to have somewhat milder non-eikonal corrections due to scattering
with the medium.
The kinematic constraints, $E\gg\omega\gg k_{\bot},q_{\bot}$ referred in the
literature as soft eikonal approximation that neglects any change in parton
trajectory due to multiple scatterings but assumes a straight line tragectory
throughout. The diffusion of partons in a hot and dense medium can have an
unavoidable link beyond the eikonal approximation and it is worth to relax
eikonal approximation. In this work an attempt has been made to relax part of
this approximation for some of the inelastic processes and their differential
cross-sections in first order noneikonal approximation have been obtained.
Primary estimation indicates $15-20\%$ reduction in the cross section due to
first order noneikonal effect for both the processes in the soft and
intermediate parton energies. These cross-sections naturally reproduce
eikonally approximated results in the eikonal limit for soft emission, i.e.,
$\sqrt{s},E\gg q_{\bot}$ and $q_{\bot}\gg k_{\bot}$. QGP produced at LHC,
where large virtuality scattering processes may be dominant one, is seems to
be $`$less opaque to jets than predicted’ by constrained extrapolations from
RHIC Horowitz:2011gd . There are however other views also, for instance
Renk:2011aa ; Renk:2011gj ; Renk:2012wr , where another set of constrained
extrapolations show considerable variation in the postdictions of RHIC-
constrained scenarios with LHC data. Here this have been taken as a constraint
and cause to disregard class of models which fail to predict/postdict
correctly the uprising behaviour of nuclear modification factor rather than
assigning it as generic surprising feature of LHC data.
Our results indicate some reductions in interaction strengths of jets due to
non-eikonal effects, in soft and intermediate sector. In the soft sector when
the problem is embedded into a hydrodynamically evolving density distribution
this could lead to non-trivial effects. We also show that wide back scattering
with scattering angle more than $\simeq\pm 0.52\pi$ is forbidden in case of
$qq^{\prime}\rightarrow qq^{\prime}g$ when the emitted gluon in soft. This,
however, is not the case for $gg\rightarrow ggg$. In future study we intend
quantitative estimation of this noneikonal effect in jet quenching and other
consequences in heavy-ion collisions phenomenology.
###### Acknowledgements.
Acknowledgments : I thank Munshi G. Mustafa for valuable discussions with
numerous help during the course of this work and critically reading the
manuscript. I also thank Jan Uphoff for his valuable suggestions and comments.
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|
arxiv-papers
| 2013-01-09T13:00:11 |
2024-09-04T02:49:40.047376
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Raktim Abir",
"submitter": "Raktim Abir",
"url": "https://arxiv.org/abs/1301.1839"
}
|
1301.1987
|
# A Polynomial Invariant
for Rank 3 Weakly-Colored Stranded Graphs111Preprint: pi-mathphys-313, ICMPA-
MPA/2012/36.
Remi C. Avohou International Chair in Mathematical Physics and Applications,
ICMPA-UNESCO Chair, 072BP50, Cotonou, Rep. of Benin [email protected]
, Joseph Ben Geloun Perimeter Institute for Theoretical Physics, 31 Caroline
St, Waterloo, ON N2L 2Y5, Canada and International Chair in Mathematical
Physics and Applications, ICMPA-UNESCO Chair, 072BP50, Cotonou, Rep. of Benin
[email protected] and Mahouton N. Hounkonnou International
Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, 072BP50,
Cotonou, Rep. of Benin [email protected]
###### Abstract.
The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal
polynomial invariant for ribbon graphs. We find an extension of this
polynomial for a particular family of graphs called rank 3 weakly-colored
stranded graphs. These graphs live in a 3D space and appear as the gluing
stranded vertices with stranded edges according to a definite rule (ordinary
graphs and ribbon graphs can be understood in terms of stranded graphs as
well). They also possess a color structure in a specific sense [Gurau, Commun.
Math. Phys. 304, 69 (2011)]. The polynomial constructed is a seven
indeterminate polynomial invariant of these graphs which responds to a similar
contraction/deletion recurrence relation as obeyed by the Tutte and Bollobas-
Riordan polynomials. It is however new due to the particular cellular
structure of the graphs on which it relies. The present polynomial encodes
therefore additional data that neither the Tutte nor the Bollobas-Riordan
polynomials can capture for the type of graphs described in the present work.
MSC(2010): 05C10, 57M15
###### Contents
1. 1 Background and main results
2. 2 Graphs and Tutte polynomial
1. 2.1 Graphs
2. 2.2 Graphs with flags
3. 3 Ribbon graphs and Bollobas-Riordan polynomial
1. 3.1 Ribbon graphs
2. 3.2 Ribbon graphs with flags
4. 4 Rank $D$ stranded graphs with flags and a generalized polynomial invariant
1. 4.1 Stranded, tensor, colored graphs with flags
2. 4.2 W-colored stranded graphs
3. 4.3 Polynomial invariant for $3D$ w-colored tensor graphs
## 1\. Background and main results
The Tutte polynomial is a universal polynomial invariant defined on a graph
which obeys a particular recurrence relation for the contraction and deletion
of the edges of this graph. This universal property turns out to have several
implications in particular in statistical mechanics [17]. The Bollobas-Riordan
(BR) polynomial [6] defines an authentic extension of the Tutte polynomial for
simple graphs [19] to ribbon graphs or neighborhoods of graphs embedded in
surfaces.
In this work, we investigate a family of graphs for which both the Tutte and
BR polynomial invariants find a meaningful extension. The family of such
graphs are issued from the so-called colored tensor graphs representing
simplicial complexes in any dimension $D$ [10]. The present study is mainly
addressed for $D=3$, although, at the beginning and for the sake of
generality, we carry out the analysis for any dimension $D$. We focus on $D=3$
because we aim at making a first step towards the definition of a universal
polynomial invariant precisely of the Tutte or BR form for particular graphs
having a (3D) volume which is still missing to the best of our knowledge.
It can be underlined that the graphs that we shall study are somehow recently
found. They naturally appear as Feynman graphs in a particular quantum field
theoretical framework called colored tensor field theory introduced by Gurau
[10][12] and further investigated afterwards [11, 3, 2, 13]. For a version of
these graphs without color, we refer the reader to more references therein.
According to quantum field procedures, these graphs can be dually associated
with gluing of basic $D$ simplexes (vertices) along their faces or $D-1$
boundary simplexes (propagators or edges). The graphs in question are called
stranded and can be depicted in a three dimensional space. Fields in this
framework are rank $D$ tensors. Hence the notion of (tensor) rank and the
dimension of the space generated by the gluing are related. We shall not
distinguish these in the following.
There are two main contributions addressing an extension of the BR polynomial
for a similar family of graphs that will be presented here: the contribution
by Gurau [12] and the one by Tanasa [18]. Let us recall that for tensor
graphs, the vertex is stranded and the simplest way of contracting an edge
leads immediately to another type of vertex which differs from the initial. As
a consequence, the simplest prescriptions of contracting and deleting edges
are not well defined without any further considerations. And indeed, the two
aforementioned works undertake the definition of the contraction and deletion
of edges in a very peculiar manner.
There are several layers of difficulties in order to obtain the correct
definition of the type of graphs that ought to be considered, the type of
topological polynomial defined for this family of graphs and the type of
recurrence relations that the polynomial needs to satisfy. Our method is
radically different from that of Tanasa since we will use the Gurau colored
prescription in order to have a control on the type of graphs that could be
generated. Moreover, and in contrast with the work by Gurau, we propose to
enlarge the family of graphs from the colored tensor graphs to what will be
called weakly-colored (w-colored) stranded graphs for which contraction and a
similar notion of deletion make a sense. Strictly speaking these w-colored
stranded graphs are equivalence classes of graphs. Within these classes, the
contraction operation becomes stable in some sense. From this stability, we
can define a polynomial containing all specific data pertaining to the
extended dimension 3 and the associated cellular decomposition of the graph.
We can establish as well a recurrence relation satisfied by the new polynomial
for w-colored stranded graphs.
Our main result appears in Theorem 4 of Section 4. This statement successfully
determines the contraction and cut rule (an operation intuitively clear which
replaces the deletion) with respect to an edge that should be fulfilled by a
new polynomial spelled out in Definition 36. The representatives of the
equivalence class of graphs on which the polynomial is defined have several
topological ingredients. Some of these ingredients will be introduced step by
step in the following program as they turn out to be well defined for simple
graphs and ribbon graphs.
A first ingredient is captured by the notion of flag [19] which can be
introduced at the simplest graph level. This analysis is the purpose of
Section 2. Theorem 1 yields another interesting result of this work since it
mainly establishes that the Tutte polynomial for graphs with flags satisfies
the universal equations of the Tutte polynomial. This point is original, to
our knowledge, since it proves that the universal equations for the Tutte
polynomial with particular weights, apart from their more general meaning and
usefulness in statistical mechanics [5], can be found as equations for simple
graphs decorated with flags.
A second key notion is that of open and closed faces of a graph which can be
already introduced for ribbon graphs. The consequences that open and closed
faces in a ribbon graph have on the related BR polynomial are addressed in
Section 3. Theorem 3 is another partial important contributing in establishing
our final result. We emphasize that in the work by Krajewski and co-workers
[15] ribbon graphs with flags and their topological polynomial have been
introduced in a different manner (the notion of open faces is definitely
absent therein, for instance). These authors used the framework of the partial
or Chmutov duality [9] for establishing several interesting results pertaining
to ribbon graphs with flags. Our approach is more “conventional” and much in
the spirit of the contribution [6]. Thus, as far as we are concerned with
ribbon graph flags and to the extent of this work, Theorem 3 is new and
explicitly describes the implications that might have flags on the BR
polynomial.
A closing appendix provides explicit examples of the polynomial obtained for
typical graphs.
## 2\. Graphs and Tutte polynomial
This section starts the review of the main theorem that we aim to generalize
in the next sections.
### 2.1. Graphs
We review basic definitions for graphs (called sometimes simple graphs in the
following) and their Tutte polynomial [19, 5, 16].
###### Definition 1 (Graph [19]).
A graph $\mathcal{G}$ is defined by a set $\mathcal{V}$ of vertices and a set
$\mathcal{E}$ of edges and a relation of incidence which associates with each
edge either one or two vertices called its ends. We denote it as
$\mathcal{G}(\mathcal{V},\mathcal{E})$ or simply $\mathcal{G}$.
###### Definition 2 (Edges and terminal forms [19][16]).
Let $e$ be an edge of a graph $\mathcal{G}$.
$\bullet$ $e$ is called a self-loop in $\mathcal{G}$ if the two ends of $e$
are adjacent to the same vertex $v$ of $\mathcal{G}.$
$\bullet$ $e$ is called a bridge in $\mathcal{G}$ if its removal disconnects a
component of $\mathcal{G}.$
$\bullet$ $e$ is called an ordinary or regular edge of $\mathcal{G}$ if it is
neither a bridge nor a self-loop.
$\bullet$ A graph which does not contain any regular edge is called a terminal
form.
###### Definition 3 (Deletion and contraction [19], Chap. I and II).
Let $\mathcal{G}$ be a graph and $e$ one of its edges.
$\bullet$ We call $\mathcal{G}-e$ the graph obtained from $\mathcal{G}$ by
removing $e$.
$\bullet$ If $e$ is not a self-loop, the graph $\mathcal{G}/e$ obtained by
contracting $e$ is defined from $\mathcal{G}$ by deleting $e$ and identifying
its end vertices into a new vertex.
$\bullet$ If $e$ is a self-loop, $\mathcal{G}/e$ is by definition the same as
$\mathcal{G}-e.$
One notices that after an arbitrary full contraction-deletion of regular edges
of a given graph the end result is necessarily given by a collection of graphs
composed by bridges and/or self-loops, hence a terminal form.
###### Definition 4 (Subgraphs, spanning subgraphs [19]).
$\bullet$ A subgraph $A$ of $\mathcal{G}(\mathcal{V},\mathcal{E})$ is defined
as a graph $A(\mathcal{V}_{A},\mathcal{E}_{A})$ the vertex set of which is a
subset of $\mathcal{V}$ and the edge set of which is a subset of $\mathcal{E}$
together with their end vertices. Hence, $\mathcal{E}_{A}\subseteq\mathcal{E}$
and $\mathcal{V}_{A}\subseteq\mathcal{V}$ and we denote
$A\subseteq\mathcal{G}$.
$\bullet$ A spanning subgraph $A$ of $\mathcal{G}(\mathcal{V},\mathcal{E})$ is
defined as a subgraph $A(\mathcal{V}_{A},\mathcal{E}_{A})$ of $\mathcal{G}$
with all vertices of $\mathcal{G}$. Hence
$\mathcal{E}_{A}\subseteq\mathcal{E}$ and $\mathcal{V}_{A}=\mathcal{V}$ and we
denote $A\Subset\mathcal{G}$.
###### Definition 5 (Graph operations [19][6]).
Consider two disjoint graphs
$\mathcal{G}_{1}(\mathcal{V}_{1},\mathcal{E}_{1})$ and
$\mathcal{G}_{2}(\mathcal{V}_{2},\mathcal{E}_{2})$. We define
$\bullet$ The disjoint union graph $\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ as
the graph defined by the disjoint union of the two graphs;
$\bullet$ The product graph
$\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ of the two graphs simply
merges at the vertices $v_{1}\in\mathcal{V}_{1}$ and $v_{2}\in\mathcal{V}_{2}$
the two graphs by removing $v_{1}$ and $v_{2}$ and insert a new vertex $v$
which has all edges incident to $v_{1}$ and to $v_{2}$.
Note that if a graph $\mathcal{G}$ can be written as
$\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$, this also means that
$\mathcal{G}$ is one-to-one with $\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ (we
will come back on this bijection later). It is simple to check that
$A\Subset\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ can be written as $A_{1}\sqcup
A_{2}$, where $A_{i}\Subset\mathcal{G}_{i}$. A similar relation holds for
subgraphs $A\subset\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$. Idem,
$A\Subset\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ implies that
$A=A_{1}\cdot_{v_{1},v_{2}}A_{2}$ where $A_{i}\Subset\mathcal{G}_{i}$. However
such a relation holds for subgraphs of
$\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ only if each
$A_{i}\subset\mathcal{G}_{i}$ contains the vertex $v_{i}$.
We are now in position to define the Tutte polynomial and specify its main
properties.
###### Definition 6 (Tutte polynomial 1 [6][16]).
Let $\mathcal{G}(\mathcal{V},\mathcal{E})$ be a graph, then the Tutte
polynomial $T_{\mathcal{G}}$ of $\mathcal{G}$ admits the following expansion:
$T_{\mathcal{G}}(x,y)=\sum_{A\Subset\mathcal{G}}(x-1)^{{\rm
r}\,(\mathcal{G})-{\rm r}\,(A)}(y-1)^{n(A)},$ (1)
where ${\rm r}\,(A)=|\mathcal{V}|-k(A)$ is the rank of the spanning subgraph
$A$, $k(A)$ its number of connected component and
$n(A)=|A|+k(A)-|\mathcal{V}|$ is its nullity or cyclomatic number.
There is an equivalent definition of this polynomial.
###### Definition 7 (Tutte polynomial 2 [5]).
Let $\mathcal{G}(\mathcal{V},\mathcal{E})$ be a graph. The Tutte polynomial of
$\mathcal{G}$ denoted $T_{\mathcal{G}}(x,y)$ is defined in the following way:
$\bullet$ If $\mathcal{G}$ has no edges, then $T_{\mathcal{G}}(x,y)=1.$
Otherwise, for any edge $e\in\mathcal{E}$
$\bullet$
$T_{\mathcal{G}}(x,y)=T_{\mathcal{G}-e}(x,y)+T_{\mathcal{G}/e}(x,y)$, if $e$
is a regular edge.
$\bullet$ $T_{\mathcal{G}}(x,y)=xT_{\mathcal{G}-e}(x,y)=xT_{\mathcal{G}/e}$,
if $e$ is a bridge.
$\bullet$
$T_{\mathcal{G}}(x,y)=yT_{\mathcal{G}-e}(x,y)=yT_{\mathcal{G}/e}(x,y)$, if $e$
is a self-loop.
Some comments are in order at this point.
(1) In several works, the variable $(y-1)$ appearing in (1) is replaced simply
by $y$222Using $y$ instead of $(y-1)$ leads to non negative coefficients of
the polynomial [6].. The above convention (for instance used in [16]) is
simpler for the following reason. In order to compute the Tutte polynomial of
$\mathcal{G}$, one performs a full contraction and deletion of regular edges
of this graph which yields terminal forms. The Tutte polynomial of
$\mathcal{G}$ is obtained by summing the contributions of its terminal forms.
According to the above convention, $(x-1)$ and $(y-1)$, for a terminal form
composed with $m$ bridges and $p$ self-loops, the Tutte polynomial is
$x^{m}y^{p}$.
(2) By definition, for a self-loop $e$, $\mathcal{G}-e=\mathcal{G}/e$ hence
$T_{\mathcal{G}-e}=T_{\mathcal{G}/e}$ holds. It is however not trivial to
prove that, for a bridge $e$, $T_{\mathcal{G}-e}=T_{\mathcal{G}/e}$ holds. A
way to agree with this is to remark that
$\mathcal{G}-e=\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ and
$\mathcal{G}/e=\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ where
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are the two maximal333These graphs are
maximal in the sense that they contain all vertices and all lines on a given
side of the bridge. subgraphs on each side of the bridge and $v_{1}$ and
$v_{2}$ are the end vertices of $e$. The claim
$T_{\mathcal{G}-e}=T_{\mathcal{G}/e}$ is a direct consequence of the next
proposition following from the additivity property of the rank and the nullity
with respect to the disjoint union and product operations for graphs which
implies a term by term monomial factorization.
###### Proposition 1 (Operations on Tutte polynomial 1 [19]).
Let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be two disjoint graphs, then
$\displaystyle
T_{\mathcal{G}_{1}\sqcup\mathcal{G}_{2}}=T_{\mathcal{G}_{1}}T_{\mathcal{G}_{2}}=T_{\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}}\,.$
(2)
for any vertices $v_{1,2}$ in $\mathcal{G}_{1,2}$, respectively.
### 2.2. Graphs with flags
There exists a notion of “decorated” graph encompassing the above graph
consideration. This new category includes graphs with flags or half-edges [16]
hooked to vertices.
###### Definition 8 (Flag).
A flag or half-edge (or half-line) is a line incident to a unique vertex and
without forming a loop. (See Figure 1.)
Figure 1. A flag incident to a unique vertex.
###### Definition 9 (Cut of an edge [16]).
Let $e\in\mathcal{E}$ be an edge of $\mathcal{G}(\mathcal{V},\mathcal{E})$.
The cut graph $\mathcal{G}\vee e$ is the graph obtained from $\mathcal{G}$ by
“cutting” $e$, namely by removing $e$ and let two flags attached to the end
vertices of $e$. If $e$ is a self-loop then the resulting two flags are
attached to the same vertex. (See an illustration in Figure 2)
Figure 2. Cutting an edge.
###### Definition 10 (Graph and subgraph with flags).
$\bullet$ A graph $\mathcal{G}$ with flags is a graph
$\mathcal{G}(\mathcal{V},\mathcal{E})$ with a set $\mathfrak{f}$ of flags
defined by the disjoint union of $\mathfrak{f}^{1}$ the set of flags obtained
only from the cut of all edges of $\mathcal{G}$ and a set $\mathfrak{f}^{0}$
of “additional” flags together with a relation which associates with each
additional flag a unique vertex. We denote a graph with set $\mathfrak{f}^{0}$
of additional flags as
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$. (See Figure 3.)
Figure 3. A graph with flags, its set $\mathfrak{f}^{0}$ of additional flags
in red and set $\mathfrak{f}^{1}$ in blue after cutting its edges and removing
$\mathfrak{f}^{0}$.
$\bullet$ A c-subgraph $A$ of
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ is defined as a graph
$A(\mathcal{V}_{A},\mathcal{E}_{A},\mathfrak{f}^{0}_{A})$ the vertex set of
which is a subset of $\mathcal{V}$, the edge set of which is a subset of
$\mathcal{E}$ together with their end vertices. Call
$\mathcal{E}_{A}^{\prime}$ the set of edges incident to the vertices of $A$
and not contained in $\mathcal{E}_{A}$. The flag set of $A$ contains a subset
of $\mathfrak{f}^{0}$ plus additional flags attached to the vertices of $A$
obtained by cutting all edges in $\mathcal{E}_{A}^{\prime}$. In symbols,
$\mathcal{E}_{A}\subseteq\mathcal{E}$ and
$\mathcal{V}_{A}\subseteq\mathcal{V}$,
$\mathfrak{f}^{0}_{A}=\mathfrak{f}^{0;0}_{A}\cup\mathfrak{f}^{0;1}_{A}(\mathcal{E}_{A})$
with $\mathfrak{f}^{0;0}_{A}\subseteq\mathfrak{f}^{0}$ and
$\mathfrak{f}^{0;1}_{A}(\mathcal{E}_{A})\subseteq\mathfrak{f}^{1}$, where
$\mathfrak{f}^{0;1}_{A}(\mathcal{E}_{A})$ is the set of flags obtained by
cutting all edges in $\mathcal{E}_{A}^{\prime}$ and incident to vertices of
$A$. We write $A\subseteq\mathcal{G}$. (See $A$ as illustrated in Figure 4.)
$\bullet$ A spanning c-subgraph $A$ of
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ is defined as a
c-subgraph $A(\mathcal{V}_{A},\mathcal{E}_{A},\mathfrak{f}^{0}_{A})$ of
$\mathcal{G}$ with all vertices and all additional flags of $\mathcal{G}$.
Hence $\mathcal{E}_{A}\subseteq\mathcal{E}$ and $\mathcal{V}_{A}=\mathcal{V}$,
$\mathfrak{f}^{0}_{A}=\mathfrak{f}^{0}\cup\mathfrak{f}^{0;1}_{A}(\mathcal{E}_{A})$.
We denote it $A\Subset\mathcal{G}$. (See $\tilde{A}$ as an illustration in
Figure 4.)
Figure 4. A graph $\mathcal{G}$, a c-subgraph $A$ defined by the edge $e$ and
the spanning c-subgraph $\tilde{A}$ associated with $e$.
$\mathcal{G}$$e$$A$$e$$\tilde{A}$$e$
Remark that a spanning c-subgraph has always a greater number of additional
flags than the initial graph which spans it. Moreover, one can define the rank
and nullity of any (spanning c-sub-) graph with flags as in the ordinary
situation.
###### Definition 11 (Edge contraction of graphs with flags).
Let $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a graph with
flags. We define the contraction of a non self-loop edge $e$ in $\mathcal{G}$,
i.e. $\mathcal{G}/e$, by the graph obtained from $\mathcal{G}$ by removing $e$
and identifying the two end vertices into a new vertex having all their
additional flags and remaining incident lines. For a self-loop $e$,
contraction $\mathcal{G}/e$ and deletion $\mathcal{G}-e$ coincide.
Noting that the above contraction does not change the number of additional
flags of a graph, the following proposition is straightforward.
###### Proposition 2.
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a graph with
flags and $e$ be an edge. Then
$\displaystyle\mathfrak{f}^{0}(\mathcal{G}\vee
e)=\mathfrak{f}^{0}\cup\\{e_{1},e_{2}\\}\,,\qquad\mathfrak{f}^{0}(\mathcal{G}/e)=\mathfrak{f}^{0}\,,$
(3)
where the two flags $e_{1,2}$ are obtained from cutting $e$.
Operations for graphs in the sense of Definition 5 can be reported for graphs
with flags. The disjoint union graph $\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$
and product graph $\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ keep the
same sense with set of additional flags the disjoint union of additional flag
sets. Spanning c-subgraphs of $\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ and
$\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ are of the form
$A_{1}\sqcup A_{2}$ and $A_{1}\cdot_{v_{1},v_{2}}A_{2}$, respectively, with
$A_{i}\Subset\mathcal{G}_{i}$. The only point to check is that, given
$A_{i}(\mathcal{V}(\mathcal{G}_{i}),\mathcal{E}_{A_{i}},\mathfrak{f}^{0}_{A_{i}}=\mathfrak{f}^{0}_{\mathcal{G}_{i}}\cup\mathfrak{f}^{0;1}(\mathcal{E}_{A_{i}}))$
$\displaystyle\mathfrak{f}^{0}_{A_{1}\sqcup
A_{2}}=\mathfrak{f}^{0}_{\mathcal{G}_{1}}\cup\mathfrak{f}^{0}_{\mathcal{G}_{2}}\,,\qquad\mathfrak{f}^{0;1}_{A_{1}\sqcup
A_{2}}(\mathcal{E}_{A_{1}}\cup\mathcal{E}_{A_{2}})=\mathfrak{f}^{0;1}_{A_{1}}(\mathcal{E}_{A_{1}})\cup\mathfrak{f}^{0;1}_{A_{2}}(\mathcal{E}_{A_{2}})\,$
(4)
$\displaystyle\mathfrak{f}^{0}_{A_{1}\cdot_{v_{1},v_{2}}A_{2}}=\mathfrak{f}^{0}_{\mathcal{G}_{1}}\cup\mathfrak{f}^{0}_{\mathcal{G}_{2}}\,,\qquad\mathfrak{f}^{0;1}_{A_{1}\cdot_{v_{1},v_{2}}A_{2}}(\mathcal{E}_{A_{1}}\cup\mathcal{E}_{A_{2}})=\mathfrak{f}^{0;1}_{A_{1}}(\mathcal{E}_{A_{1}})\cup\mathfrak{f}^{0;1}_{A_{2}}(\mathcal{E}_{A_{2}})\,.$
(5)
This holds because the flags obtained by cutting
$(\mathcal{E}_{A_{1}}\cup\mathcal{E}_{A_{2}})^{\prime}$ in
$\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ or by cutting
$(\mathcal{E}_{A_{1}}\cup\mathcal{E}_{A_{2}})^{\prime}$ in
$\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ turns out to be identical
to the flags obtained from cutting $\mathcal{E}_{A_{1}}^{\prime}$ in
$\mathcal{G}_{1}$ (letting $\mathcal{G}_{2}$ untouched) with those from
cutting $\mathcal{E}_{A_{2}}^{\prime}$ in $\mathcal{G}_{2}$ (letting
$\mathcal{G}_{1}$ untouched). One finally shows that spanning c-subgraphs of
$\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ and of
$\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ are one-to-one.
We can now identify a new polynomial for graphs with flags and list its main
properties.
###### Definition 12 (Tutte polynomial for graphs with flags).
Let $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a graph with
flags. The Tutte polynomial of
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ is given by
$\mathcal{T}_{\mathcal{G}}(x,y,t)=\sum_{A\Subset\mathcal{G}}(x-1)^{{\rm
r}\,(\mathcal{G})-{\rm r}\,(A)}(y-1)^{n(A)}\,t^{f(A)},$ (6)
with $f(A):=|\mathfrak{f}^{0}|+|\mathfrak{f}^{0;1}_{A}(\mathcal{E}_{A})|$.
The following proposition holds.
###### Proposition 3 (Operations on Tutte polynomial 2).
Let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be two disjoint graphs with flags,
then
$\displaystyle\mathcal{T}_{\mathcal{G}_{1}\sqcup\mathcal{G}_{2}}=\mathcal{T}_{\mathcal{G}_{1}}\mathcal{T}_{\mathcal{G}_{2}}=\mathcal{T}_{\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}}\,.$
(7)
for any vertices $v_{1,2}$ in $\mathcal{G}_{1,2}$, respectively.
###### Proof.
From Proposition 1, we must check only the behavior of the sets of flags and
prove that they factorize properly. Consider
$\mathcal{G}=\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ or
$\mathcal{G}=\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ their spanning
c-subgraphs $A\Subset\mathcal{G}$ can be simply expressed as subsets
$A=A_{1}\sqcup A_{2}\subset\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ or
$A_{1}\cdot_{v_{1},v_{2}}A_{2}\subset\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$
where $A_{1,2}\Subset\mathcal{G}_{1,2}$. The proof is then immediate from the
fact that $\mathfrak{f}^{0}_{A_{1}\sqcup
A_{2}}=\mathfrak{f}^{0}_{A_{1}\cdot_{v_{1},v_{2}}A_{2}}=\mathfrak{f}^{0}_{A_{1}}\cup\mathfrak{f}^{0}_{A_{2}}$
such that
$f(A_{1}\sqcup A_{2})=f(A_{1})+f(A_{2})=f(A_{1}\cdot_{v_{1},v_{2}}A_{2})\,.$
(8)
∎
We are now in position to prove our first new result.
###### Theorem 1 (Contraction and cut on Tutte polynomial).
Let $e$ be a regular edge of
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$. The Tutte polynomial
of $\mathcal{G}$ satisfies the relations
$\displaystyle\mathcal{T}_{\mathcal{G}}$ $\displaystyle=$
$\displaystyle\mathcal{T}_{\mathcal{G}\vee e}+\mathcal{T}_{\mathcal{G}/e}\,.$
(9) $\displaystyle\mathcal{T}_{\mathcal{G}\vee e}$ $\displaystyle=$
$\displaystyle t^{2}\,\mathcal{T}_{\mathcal{G}-e}\,.$ (10)
For a bridge $e$, one has
$\mathcal{T}_{\mathcal{G}/e}=\mathcal{T}_{\mathcal{G}-e}=t^{-2}\,\mathcal{T}_{\mathcal{G}\vee
e}$, and
$\mathcal{T}_{\mathcal{G}}=(1+(x-1)t^{2})\mathcal{T}_{\mathcal{G}/e}\,.$ (11)
For a self-loop $e$, one has
$\mathcal{T}_{\mathcal{G}/e}=\mathcal{T}_{\mathcal{G}-e}=t^{-2}\,\mathcal{T}_{\mathcal{G}\vee
e}$ and
$\mathcal{T}_{\mathcal{G}}=(y-1+t^{2})\mathcal{T}_{\mathcal{G}-e}\,.$ (12)
###### Proof.
The proofs of these statements are similar to the ordinary one for graph
without flags. However, a special care has to be paid on the flags.
Take a regular edge $e$. $\mathcal{G}\vee e$ has a set of vertex
$\tilde{\mathcal{V}}=\mathcal{V}$ and the number of connected components
$k(\mathcal{G}\vee e)=k(\mathcal{G})$. Then, it is direct to get ${\rm
r}\,(\mathcal{G}\vee e)={\rm r}\,\mathcal{G}={\rm r}\,(\mathcal{G}-e)$.
Meanwhile, contracting $e$ in $\mathcal{G}$ keeps the same meaning, hence
${\rm r}\,(\mathcal{G}/e)=(|\mathcal{V}|-1)-k(\mathcal{G})={\rm
r}\,(\mathcal{G})-1$. Furthermore, $f(\mathcal{G}\vee e)=f(\mathcal{G})+2$ and
$f(\mathcal{G}/e)=f(\mathcal{G})$.
We first focus on (9) and decompose the Tutte polynomial in the ordinary form
as a sum on subsets containing $e$ and a remainder that we must compare with
$\mathcal{T}_{\mathcal{G}/e}$ and $\mathcal{T}_{\mathcal{G}\vee e}$,
respectively.
Let us choose a bijection between the two sets $\\{A\Subset\mathcal{G};e\notin
A\\}$ and $\\{A\Subset\mathcal{G}\vee e\\}$. For a regular edge $e$,
$A\Subset\mathcal{G}$, with $e\notin A$, uniquely defines
$A^{\prime}\Subset\mathcal{G}\vee e$ such that $A^{\prime}=A$ namely they both
have the same vertices, edges and flags. It is also simple to check that
$\\{A\Subset\mathcal{G}\vee e\\}$ and $\\{A\Subset\mathcal{G}-e\\}$ are in
one-to-one correspondence by just removing from any $A\Subset\mathcal{G}\vee
e$ the two additional flags introduced by the cut of $e$ in $\mathcal{G}\vee
e$ and get a spanning subset of $\mathcal{G}-e$. The mapping is clearly
invertible hence bijective.
Given $e$, a regular edge, consider $A\Subset\mathcal{G}\vee e$, its unique
corresponding element $A^{\prime}$ in $\\{B\Subset\mathcal{G};e\notin B\\}$
and $A^{\prime\prime}$ its corresponding element in
$\\{B\Subset\mathcal{G}-e\\}$. The monomial in $\mathcal{T}_{\mathcal{G}\vee
e}$ related to $A$ can be recast in the form
$\displaystyle(x-1)^{{\rm r}\,(\mathcal{G}\vee e)-{\rm
r}\,(A)}(y-1)^{n(A)}t^{f(A)}$ $\displaystyle=$ $\displaystyle(x-1)^{{\rm
r}\,(\mathcal{G})-{\rm
r}\,(A^{\prime})}(y-1)^{n(A^{\prime})}t^{f(A^{\prime})}\,,$ (13)
$\displaystyle=$ $\displaystyle(x-1)^{{\rm r}\,(\mathcal{G}-e)-{\rm
r}\,(A^{\prime\prime})}(y-1)^{n(A^{\prime\prime})}t^{f(A^{\prime\prime})+2}$
(14)
This achieves the proof that (a) $\mathcal{T}_{\mathcal{G}\vee e}$ and
$\sum_{A\Subset\mathcal{G};e\notin A}(\cdot)$ has the same number of terms and
the same monomials and (b) $\mathcal{T}_{\mathcal{G}\vee
e}=t^{2}\,\mathcal{T}_{\mathcal{G}-e}$.
Let us deal with the contraction now. We prove that
$\\{A\Subset\mathcal{G};e\in A\\}$ is one-to-one with
$\\{A\Subset\mathcal{G}/e\\}$ for a regular edge $e$. Given a regular edge
$e$, $A\Subset\mathcal{G}$, with $e\in A$, uniquely defines
$A^{\prime}\Subset\mathcal{G}/e$ defined by the contraction of $e$ in $A$ (in
the sense of Definition 11).
Given $e$ a regular edge, $A\Subset\mathcal{G}/e$ and its unique corresponding
subset $A^{\prime}\Subset\mathcal{G}$, $e\in A^{\prime}$, we can write each
monomial of $\mathcal{T}_{\mathcal{G}/e}$ as
$\displaystyle(x-1)^{{\rm r}\,(\mathcal{G}/e)-{\rm
r}\,(A)}(y-1)^{n(A)}t^{f(A)}=(x-1)^{{\rm r}\,(\mathcal{G})-1-({\rm
r}\,(A^{\prime})-1)}(y-1)^{n(A^{\prime})}t^{f(A^{\prime})}$ (16)
which achieves the proof that $\mathcal{T}_{\mathcal{G}/e}$ and
$\sum_{A\Subset\mathcal{G};e\in A}(\cdot)$ has the same number of terms and
the same terms. The proof of (9) and (10) is then complete.
We now prove $\mathcal{T}_{\mathcal{G}-e}=\mathcal{T}_{\mathcal{G}/e}$ and
(11). Given a bridge $e$,
$\mathcal{G}/e=\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ and
$\mathcal{G}-e=\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ for the maximal444Maximal
is understood in the sense that the subgraph contains all vertices, all edges
and all additional flags of one bridge side. subgraphs $\mathcal{G}_{i}$ on
each side of the bridge $e$ with end vertices $v_{1,2}$. The fact that
$\mathcal{T}_{\mathcal{G}-e}=\mathcal{T}_{\mathcal{G}/e}$ follows from
Proposition 3.
Second, $\\{A\Subset\mathcal{G};e\notin A\\}$ is one-to-one with
$\\{A\Subset\mathcal{G}/e\\}$ since $\\{A\Subset\mathcal{G};e\notin A\\}$ is
one-to-one with $\\{A\Subset\mathcal{G}_{1}\sqcup\mathcal{G}_{2}\\}$ (each
$A\Subset\mathcal{G}$, $e\notin A$ is uniquely mapped on
$A^{\prime}\Subset\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ by removing simply the
flags issued from the cut of $e$ in $A$) and also
$\\{A\Subset\mathcal{G}/e\\}=\\{A\Subset\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}\\}$.
Consider a bridge $e$ in $\mathcal{G}$ and decompose the Tutte polynomial of
$\mathcal{G}$ as
$\displaystyle\mathcal{T}_{\mathcal{G}}(x,y,t)$ $\displaystyle=$
$\displaystyle\sum_{A\Subset\mathcal{G};e\notin A}(x-1)^{{\rm
r}\,(\mathcal{G})-{\rm
r}\,(A)}(y-1)^{n(A)}t^{f(A)}+\mathcal{T}_{\mathcal{G}/e}(x,y,t)$ (17)
$\displaystyle=$ $\displaystyle\sum_{A\Subset\mathcal{G}/e}(x-1)^{{\rm
r}\,(\mathcal{G}/e)+1-{\rm
r}\,(A)}(y-1)^{n(A)}t^{f(A)+2}+\mathcal{T}_{\mathcal{G}/e}(x,y,t)$ (18)
$\displaystyle=$
$\displaystyle[1+(x-1)t^{2}]\;\mathcal{T}_{\mathcal{G}/e}(x,y,t)\,.$ (19)
Furthermore, consider the bijection between $\\{A\Subset\mathcal{G}\vee e\\}$
and $\\{A\Subset\mathcal{G}-e\\}$ which maps any $A\Subset\mathcal{G}\vee e$
uniquely on $A^{\prime}\Subset\mathcal{G}-e$ by removing two flags from the
cut of $e$. Then, the monomial in $\mathcal{T}_{\mathcal{G}\vee e}$
$(x-1)^{{\rm r}\,(\mathcal{G}\vee e)-{\rm
r}\,(A)}(y-1)^{n(A)}t^{f(A)}=(x-1)^{{\rm r}\,(\mathcal{G}-e)-{\rm
r}\,(A^{\prime})}(y-1)^{n(A^{\prime})}t^{f(A^{\prime})+2}$ (20)
yielding $\mathcal{T}_{\mathcal{G}\vee
e}=t^{2}\,\mathcal{T}_{\mathcal{G}-e}=t^{2}\,\mathcal{T}_{\mathcal{G}/e}$
which achieves the proof of (11).
Last, for a self-loop $e$, we prove (12). For a self-loop $e$,
$\mathcal{T}_{\mathcal{G}-e}=\mathcal{T}_{\mathcal{G}/e}$ holds by definition.
Using the bijection between $\\{A\Subset\mathcal{G};e\notin A\\}$ and
$\\{A\Subset\mathcal{G}-e\\}$ mapping $A$ to $A^{\prime}$ the same subgraph
without the two additional flags of $A$ from the cut of $e$, and the bijection
between $\\{A\Subset\mathcal{G};e\in A\\}$ and $\\{A\Subset\mathcal{G}-e\\}$
given by just deleting $e$ in $A\Subset\mathcal{G}$, we have
$\displaystyle\mathcal{T}_{\mathcal{G}}(x,y,t)$ $\displaystyle=$
$\displaystyle\sum_{A\Subset\mathcal{G}-e}(x-1)^{{\rm r}\,(\mathcal{G}-e)-{\rm
r}\,(A^{\prime})}(y-1)^{n(A^{\prime})}t^{f(A^{\prime})+2}$ (21)
$\displaystyle+$ $\displaystyle\sum_{A\Subset\mathcal{G}-e}(x-1)^{{\rm
r}\,(\mathcal{G}-e)-{\rm
r}\,(A^{\prime})}(y-1)^{n(A^{\prime})+1}t^{f(A^{\prime})}$ (22)
$\displaystyle=$
$\displaystyle[y-1+t^{2}]\;\mathcal{T}_{\mathcal{G}-e}(x,y,t)\,.$ (23)
The last equality from (12) comes from the bijection between
$\\{A\Subset\mathcal{G}\vee e\\}$ and $\\{A\Subset\mathcal{G}-e\\}$ given by
removing in $A$ the two flags coming from $e$ and note that in
$\mathcal{T}_{\mathcal{G}\vee e}$ each monomial can be recast in the form
$(x-1)^{{\rm r}\,(\mathcal{G}\vee e)-{\rm
r}\,(A)}(y-1)^{n(A)}t^{f(A)}=(x-1)^{{\rm r}\,(\mathcal{G}-e)-{\rm
r}\,(A^{\prime})}(y-1)^{n(A^{\prime})}t^{f(A^{\prime})+2}$ (24)
so that, once again, $\mathcal{T}_{\mathcal{G}\vee
e}=t^{2}\,\mathcal{T}_{\mathcal{G}-e}$ which achieves the proof of (12).
∎
Few remarks concerning this definition of Tutte polynomial for graphs with
flags follow:
$\bullet$ For any edge $e$, we note that $\mathcal{T}_{\mathcal{G}\vee
e}=t^{2}\,\mathcal{T}_{\mathcal{G}-e}$.
$\bullet$ The contraction/cut rule seems to be the natural one in the
formalism including the notion of flags.
$\bullet$ The polynomial $\mathcal{T}_{\mathcal{G}}$ has always a factor of
$t^{|\mathfrak{f}^{0}|}$ which should be removed by a new normalization. Note
that the naive prescription to introduce the variable
$t^{f(A)-f(\mathcal{G})}$ does not work. In order to remove this factor, we
need another property of $\mathcal{T}$ which follows.
$\bullet$ Universality: We know that the universal form of the Tutte
polynomial (1) is given by the following statement [Theorem 2, Chap. X [5]]:
There is a unique map
$U:\mathcal{G}\rightarrow\mathbb{Z}[x,y,\alpha,\sigma,\tau]$ such that for the
graph $E_{n}$ made uniquely with $n$ vertices
$U(E_{n})=\alpha^{n}\,,$ (25)
for every $n\geq 1$ and for every $e\in E(G)$, we have
$\displaystyle U(G)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}xU(G-e)\mbox{ if $e$ is a bridge},\\\
yU(G-e)\mbox{ if $e$ is a loop},\\\ \sigma U(G-e)+\tau U(G/e)\mbox{ if $e$ is
neither a bridge nor a loop}.\end{array}\right.$ (29)
Furthermore,
$U(G)=\alpha^{k(G)}\sigma^{n(G)}\tau^{r(G)}T_{G}(\alpha x/\tau,y/\sigma)\,.$
(30)
The Tutte polynomial for
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ satisfies the
recurrence relation given by Theorem 1 which is remarkably of the form (29).
Indeed, after a change of variables as
$\displaystyle\left\\{\begin{array}[]{ll}X=(x-1)t^{2}+1\\\
Y=y-1+t^{2}.\end{array}\right.$ (33)
and given the fact that, for a given graph
$\mathcal{G}(\mathcal{V},\mathcal{E})$ and $A\Subset\mathcal{G}$,
$f(A)=|\mathfrak{f}^{0}|+2(|\mathcal{E}|-|\mathcal{E}_{A}|)\,,$ (34)
we get
$\mathcal{T}_{\mathcal{G}}(x,y,t)=t^{|\mathfrak{f}^{0}|}t^{2n(\mathcal{G})}\,T_{\mathcal{G}}(X,\frac{Y}{t^{2}})\,.$
(35)
If we set $\sigma=t^{2}$, $\alpha=\tau=1$ in (29), $\mathcal{T}_{\mathcal{G}}$
is merely one solution of (29) modified by a factor $t^{|\mathfrak{f}^{0}|}$
and thus defines a $t$-deformed version of $T_{\mathcal{G}}$.
$\bullet$ The exponent of $t$ in
$\mathcal{T}_{\mathcal{G}}(x,y,t)/t^{|\mathfrak{f}^{0}|}$ is always even this
is justified by the fact that each c-spanning subgraph is defined via
successive cut of edges yielding each two flags. This can be easily removed by
performing a change in the variable $t\to t^{1/2}$.
$\bullet$ The terminal form made with $m$ bridges, $n$ self-loops and $q$
additional flags admits the Tutte polynomial
$(1+(x-1)t^{2})^{m}\,(y-1+t^{2})^{n}\,t^{q}=X^{m}Y^{n}t^{q}\,,$ (36)
after some change of variables.
$\bullet$ Putting $t=1$ in $\mathcal{T}_{\mathcal{G}}(x,y,t)$ yields the
ordinary Tutte polynomial
$\mathcal{T}_{\mathcal{G}}(x,y,1)=T_{\mathcal{G}}(x,y)$ (37)
obeying the ordinary contraction/deletion rule.
## 3\. Ribbon graphs and Bollobas-Riordan polynomial
In this section, we first recall the generalization of the Tutte polynomial to
ribbon graphs known as the BR polynomial for ribbon graphs [6, 7]. Then, we
investigate ribbon with flags according to our previous developments of
Subsection 2.2.
### 3.1. Ribbon graphs
We start by some basic facts on the ribbon graphs and their BR polynomial.
###### Definition 13 (Ribbon graphs [6][15]).
A ribbon graph $\mathcal{G}$ is a (not necessarily orientable) surface with
boundary represented as the union of two sets of closed topological discs
called vertices $\mathcal{V}$ and edges $\mathcal{E}.$ These sets satisfy the
following:
$\bullet$ Vertices and edges intersect by disjoint line segment,
$\bullet$ each such line segment lies on the boundary of precisely one vertex
and one edge,
$\bullet$ every edge contains exactly two such line segments.
The previous notion of self-loops, bridges, regular edges, terminal forms (of
Definition 2) can be easily reported here when the ribbon graph is seen as a
simple graph namely when forgetting its ribbon character. Ribbon edges can be
twisted as well. Introducing twisted edges has some consequences on the
orientability of the ribbon graph. In addition, there is a new topological
notion that we now describe.
###### Definition 14 (Faces [6]).
A face is a component of a boundary of $\mathcal{G}$ considered as a geometric
ribbon graph, and hence as surface with boundary.
If $\mathcal{G}$ is regarded as the neighborhood of a graph embedded into a
surface, $\mathcal{F}$ is the set of faces of the embedding. A ribbon graph is
denoted by $\mathcal{G}(\mathcal{V},\mathcal{E})$.
###### Definition 15 (Deletion and contraction [6]).
Let $\mathcal{G}$ be a ribbon graph and $e$ one of its edges.
$\bullet$ We call $\mathcal{G}-e$ the ribbon graph obtained from $\mathcal{G}$
by deleting $e$ and keeping the end vertices as closed discs.
$\bullet$ If $e$ is not a self-loop, the graph $\mathcal{G}/e$ obtained by
contracting $e$ is defined from $\mathcal{G}$ by deleting $e$ and identifying
its end vertices $v_{1,2}$ into a new vertex which possesses all edges in the
same cyclic order as their appeared on $v_{1,2}$.
$\bullet$ If $e$ is a trivial twisted self-loop, contraction is deletion:
$\mathcal{G}-e=\mathcal{G}/e$. The contraction of a trivial untwisted self-
loop $e$ is the deletion of the self-loop and the addition of a new connected
component vertex $v_{0}$ to the graph $\mathcal{G}-e$. We write
$\mathcal{G}/e=(\mathcal{G}-e)\sqcup\\{v_{0}\\}$.
The notion of contraction of a twisted or untwisted self-loop $e$ in
$\mathcal{G}$ is subtle since it coincides with the edge deletion in the graph
$\mathcal{G}^{*}$ dual of $\mathcal{G}$, namely $(\mathcal{G}^{*}-e^{*})^{*}$
[6]. The above point on the contraction of these self-loops is compatible with
this dual notion of deletion.
Spanning subgraphs in the present context keep the sense of Definition 4:
$A\Subset\mathcal{G}$ if $A$ is defined by a subset of edges
$\mathcal{E}_{A}\subseteq\mathcal{E}$ and possesses all vertices $\mathcal{V}$
of $\mathcal{G}$.
###### Definition 16 (BR polynomial 1 [6]).
Let $\mathcal{G}$ be a ribbon graph. We define the ribbon graph polynomial of
$\mathcal{G}$ to be
$R_{\mathcal{G}}(x,y,z)=\sum_{A\Subset\mathcal{G}}(x-1)^{{\rm
r}\,(\mathcal{G})-{\rm r}\,(A)}(y-1)^{n(A)}z^{k(A)-F(A)+n(A)},$ (38)
where $F(A)$ is the number of faces of $A$.
Notice that we use the same convention, with $(y-1)$ parametrizing the nullity
of the subgraph, as in the Tutte polynomials defined so far. This is different
from the convention of BR in [6] which rather uses $y$. It is however simple
to change variable at any moment and recover the convention used therein.
The BR polynomial obeys as well a contraction and deletion rule.
###### Theorem 2 (Contraction and deletion [6]).
Let $\mathcal{G}$ be a ribbon graph. If $e$ is a regular edge, then
$R_{\mathcal{G}}=R_{\mathcal{G}/e}+R_{\mathcal{G}-e}\,,$ (39)
for every bridge $e$ of $\mathcal{G}$, one has
$R_{\mathcal{G}}=x\,R_{\mathcal{G}/e}\,,$ (40)
for a trivial untwisted self-loop
$R_{\mathcal{G}}=y\,R_{\mathcal{G}-e}\,,$ (41)
and for a trivial twisted self-loop, the following holds
$R_{\mathcal{G}}=(1+(y-1)z)\,R_{\mathcal{G}-e}\,.$ (42)
We emphasize that the relations (40)-(42) are useful for the determination of
the BR polynomial from terminal forms which play the role of boundary
conditions. The explicit formula (which has to be distinguished from the sum
over spanning subgraphs) of BR polynomials of one-vertex ribbon graphs are not
entirely known, to our knowledge [1]. We will only restrict to these trivial
self-loops in the remaining analysis.
The following fact turns out to be crucial. During contraction and deletion
moves of a regular edge, the exponent of $z$, namely ${k(A)-F(A)+n(A)}$, in
the BR polynomial is invariant. Indeed, for a regular edge $e$, $k(A)$ does
not change whether or not $e\in A$; $n(A)$, as a difference
$|\mathcal{E}_{A}|-{\rm r}\,(A)$, is also invariant whether or not $e\in A$.
Finally, $F(A)$ is never affected by the contraction if $e\in A$ whereas, if
$e\notin A$, $F(A)$ remains constant after deletion of $e$ in the graph
$\mathcal{G}$. One could have chosen different types of exponent functions of
these ingredients for defining a different polynomial invariant under
contraction and deletion. However, the above choice is motivated by the fact
that the combination
$k(A)-F(A)+n(A)=2k(A)-(|\mathcal{V}|-|\mathcal{E}_{A}|-F(A))$ (43)
is nothing but the genus or twice the genus (for oriented surfaces) of the
subgraph $A$. Furthermore, writing the exponent in this form also helps for
the determination of the terminal forms because $k(A)-F(A)$ and $n(A)$ turn
out to be additive quantities with respect to the product of disjoint graphs.
More motivations can be found in [6]. These remarks will be exploited in the
following in order to uncover other types of BR polynomials.
### 3.2. Ribbon graphs with flags
This section starts another main result of this work. We identify an extended
form of the BR polynomial for ribbon graphs with flags. If such a polynomial
must be related with the polynomial by Krajewski et al. [15] introduced for
the same type of graphs that mapping will be only clear after finding the
multivariate form of the polynomial found in the present work and then
recasting the intertwined sums of spanning subgraphs related by Chmutov
duality as used in [15] in terms of a unique spanning c-subgraph sum. These
subtleties will be not treated of the present work since they will not yield
much insights for our final goal.
###### Definition 17 (Ribbon flags and external points).
A ribbon flag or half-ribbon (or half-edge or simply flag, without ambiguity)
is a ribbon incident to a unique vertex by a unique segment and without
forming loops. A flag has two segments one touching a vertex and another free
or external segment. The end-points of any free segment are called external
points of the flag.
A ribbon flag is drawn in Figure 5.
Figure 5. A ribbon flag $f$ incident to one vertex disc. The two segments of
$f$, $s_{1}$ touching the vertex and $s_{2}$ a free segment with its end
points $a$ and $b$.
$s_{1}$$s_{2}$$a$$b$
It is clear that this definition is purely combinatorial. The notion of flag
can be introduced as a topological ingredient of the ribbon graph as well.
Such a flag is nothing but a topological closed disc. It intersects a vertex
at one segment $s_{1}$ (see Figure 5). Consider on the boundary of the flag
another segment $s_{2}$ disjoint from $s_{1}$. The segment $s_{2}$ is the
external segment and it is introduced in order to distinguish the two
connected segments of the boundary of the flag.
###### Definition 18 (Cut of a ribbon edge [15]).
Let $\mathcal{G}(\mathcal{V},\mathcal{E})$ be a ribbon graph and let $e$ be an
edge. The cut graph $\mathcal{G}\vee e$, is the graph obtained by removing $e$
and let two flags attached at the end vertices of $e$. If $e$ is a self-loop,
the two flags are on the same vertex.
The definitions of a ribbon graph with flags
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ and of its spanning
c-subgraphs follow naturally from Definition 10 and will be refrained at this
point. We denote again the spanning c-subgraph inclusion as
$A\Subset\mathcal{G}$.
Considered as a geometric surface, note that cutting an edge on a graph
modifies the boundary faces of this graph. One can say that the new boundary
faces follow the contour of the flags. But combinatorially, we introduce a
discrepancy between this type of faces and the initial ones (which come from
boundary of well formed edges). This will be useful in the following section
dealing with the tensor situation.
###### Definition 19 (Closed, open faces, strands and pinching [12]).
Consider $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ a ribbon
graph with flags.
$\bullet$ A closed or internal face is a boundary face component of a ribbon
graph (regarded as a geometric ribbon) which never passes through any free
segment of additional flags. The set of closed faces is denoted
$\mathcal{F}_{{\rm int}}$. (See the closed face $f_{1}$ in Figure 6)
$\bullet$ An open or external face is a boundary face component leaving an
external point of some flag rejoining another external point. The set of open
faces is denoted $\mathcal{F}_{{\rm ext}}$. (Examples of open faces are
provided in Figure 6.)
$\bullet$ The two boundaries lines of a ribbon edge or a flag are called
strands. Each strand may belong to a closed or open face.
$\bullet$ The set of faces $\mathcal{F}$ of a graph having a set of flags is
defined by $\mathcal{F}_{{\rm int}}\cup\mathcal{F}_{{\rm ext}}$.
$\bullet$ A graph is said to be open if $\mathcal{F}_{{\rm ext}}\neq\emptyset$
i.e. $\mathfrak{f}^{0}\neq\emptyset$. It is closed otherwise.
$\bullet$ The pinching of a flag is a combinatorial procedure performed at one
flag which identifies its external points and hence its strands.
$\bullet$ The pinched graph
$\widetilde{\mathcal{G}}(\widetilde{\mathcal{V}},\widetilde{\mathcal{E}},\widetilde{\mathfrak{f}}^{0})$
of a ribbon graph $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ is a
closed ribbon graph obtained after the pinching of all its flags. Thus
$\widetilde{\mathcal{V}}=\mathcal{V}$, $\widetilde{\mathcal{E}}=\mathcal{E}$,
and $|\widetilde{\mathfrak{f}}^{0}|=|\mathfrak{f}^{0}|$ where
$\widetilde{\mathfrak{f}}^{0}$ are now pinched flags;
$\widetilde{\mathcal{F}}=\mathcal{F}_{{\rm int}}\cup\mathcal{F}^{\prime}$
where $\mathcal{F}^{\prime}$ are additional faces obtained from
$\mathcal{F}_{{\rm ext}}$ after the pinching of $\mathcal{G}$.
Figure 6. A ribbon graph $\mathcal{G}$ with a closed face $f_{1}$ (in red)
and open faces $f_{2,3,4}$ (in black, green and blue, resp.). The pinched
graph $\widetilde{\mathcal{G}}$ and its two closed faces $f_{1}$ and
$\tilde{f}$.
$f_{1}$$f_{2}$$f_{3}$$f_{4}$$\mathcal{G}$$\tilde{f}$$f_{1}$$\widetilde{\mathcal{G}}$
Note that, for a graph without flags $\mathfrak{f}^{0}=\emptyset$,
$\mathcal{G}$ is closed and
$\widetilde{\mathcal{F}}=\mathcal{F}=\mathcal{F}_{{\rm int}}$ and
$\widetilde{\mathcal{G}}=\mathcal{G}$.
###### Definition 20 (Boundary graph [12]).
$\bullet$ The boundary $\partial{\mathcal{G}}$ of a ribbon graph
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ is a simple graph
$\partial{\mathcal{G}}({\mathcal{V}}_{\partial},{\mathcal{E}}_{\partial})$ in
the sense of Definition 1 such that ${\mathcal{V}}_{\partial}$ is one-to-one
with $\mathfrak{f}^{0}$ and ${\mathcal{E}}_{\partial}$ is one-to-one with
$\mathcal{F}_{{\rm ext}}$. (The boundary of the graph given in Figure 6 is
provided in Figure 7.)
$\bullet$ The boundary of a closed graph is empty.
Figure 7. The boundary $\partial{\mathcal{G}}$ of $\mathcal{G}$ of Figure 6
is represented in dashed lines.
As a result, $\partial\widetilde{\mathcal{G}}=\emptyset$ since the pinched
graph $\widetilde{\mathcal{G}}$ is closed. In practice, the boundary graph
$\partial{\mathcal{G}}$ can be read off the graph $\mathcal{G}$ by inserting a
vertex with degree two at each flag. External faces which are the edges of
$\partial{\mathcal{G}}$ are incident to these vertices. Hence, a boundary
graph realizes combinatorially what is expected from the total pinching of a
ribbon graph. Since $\partial{\mathcal{G}}$ has only vertices with always two
incident lines or one incident line if the two sides of the flag defined in
fact the same external face, then it is not difficult to prove the following
statement.
###### Proposition 4.
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a ribbon graph
with flags,
$\widetilde{\mathcal{G}}(\widetilde{\mathcal{V}},\widetilde{\mathcal{E}},\widetilde{\mathfrak{f}}^{0})$
its pinching and
$\partial{\mathcal{G}}({\mathcal{V}}_{\partial},{\mathcal{E}}_{\partial})$ its
boundary. One has
$\mathcal{F}^{\prime}={\mathcal{C}}_{\partial}\,,$ (44)
where $\mathcal{F}^{\prime}$ are the additional closed faces obtained by the
pinching of $\mathcal{G}$ and ${\mathcal{C}}_{\partial}$ are the connected
components of the boundary graph $\partial{\mathcal{G}}$.
The notion of edge contraction and deletion for graphs with flags can be
simply understood as in Definition 15 without mentioning flags.
###### Definition 21 (BR polynomial for ribbon graphs with flags).
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a ribbon graph
with flags. We define the ribbon graph polynomial of $\mathcal{G}$ to be
$\mathcal{R}_{\mathcal{G}}(x,y,z,s,t)=\sum_{A\Subset\mathcal{G}}(x-1)^{{\rm
r}\,(\mathcal{G})-{\rm r}\,(A)}(y-1)^{n(A)}z^{k(A)-F_{{\rm
int}}(A)+n(A)}\,s^{C_{\partial}(A)}\,t^{f(A)},$ (45)
where $C_{\partial}(A)=|{\mathcal{C}}_{\partial}(A)|$ is the number of
connected component of the boundary of $A$ and $F_{{\rm
int}}(A)=|\mathcal{F}_{{\rm int}}(A)|$.
The polynomial $\mathcal{R}$ (45) generalizes the BR polynomial $R$ (38). The
latter $R$ can be only recovered from $\mathcal{R}$ for closed ribbon graphs
and at the limit $t=1$. After performing the change of variable $s\to z^{-1}$,
we are led to another extension of the BR polynomial for ribbon graphs with
flags. We will refer the second polynomial to as $\mathcal{R}^{\prime}$. In
symbol, for a graph $\mathcal{G}$, we write
$\mathcal{R}_{\mathcal{G}}(x,y,z,z^{-1},t)=\mathcal{R}^{\prime}_{\mathcal{G}}(x,y,z,t)\,,\qquad\mathcal{R}^{\prime}_{\mathcal{G}}(x,y,z,t=1)=R_{\widetilde{\mathcal{G}}}(x,y,z)\,,$
(46)
where $R_{(\cdot)}$ is given by Definition 16.
Graph operations ($\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ and
$\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$) extend to ribbon graphs
[6] and ribbon graphs with flags. The product
$\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$ at the vertex resulting
from merging $v_{1}$ and $v_{2}$ (in the sense of Definition 15) respects the
cyclic order of all edges and flags on the previous vertices $v_{1}$ and
$v_{2}$. The fact that
$R_{\mathcal{G}_{1}\sqcup\mathcal{G}_{2}}=R_{\mathcal{G}_{1}}R_{\mathcal{G}_{2}}=R_{\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}}$
holds for ribbon graphs without flags [6] can be extended for ribbon graphs
with flags under particular conditions. The following proposition holds.
###### Proposition 5 (Operations on BR polynomials).
Let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be two disjoint ribbon graphs with
flags, then
$\displaystyle\mathcal{R}_{\mathcal{G}_{1}\sqcup\mathcal{G}_{2}}$
$\displaystyle=$
$\displaystyle\mathcal{R}_{\mathcal{G}_{1}}\mathcal{R}_{\mathcal{G}_{2}}\,,$
(47)
$\displaystyle\mathcal{R}^{\prime}_{\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}}$
$\displaystyle=$
$\displaystyle\mathcal{R}^{\prime}_{\mathcal{G}_{1}}\mathcal{R}^{\prime}_{\mathcal{G}_{2}}\,,$
(48)
for any disjoint vertices $v_{1,2}$ in $\mathcal{G}_{1,2}$, respectively.
###### Proof.
Using Propositions 1 and 3, one must only check the behavior of the exponents
$k(A)-F_{{\rm int}}(A)+n(A)$ and $C_{\partial}(A)$, for any
$A\Subset\mathcal{G}$.
Consider first $\mathcal{G}=\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$. A spanning
c-subgraph $A\Subset\mathcal{G}$ expresses as $A=A_{1}\sqcup
A_{2}\Subset\mathcal{G}_{1}\sqcup\mathcal{G}_{2}$ with
$A_{1,2}\Subset\mathcal{G}_{1,2}$. It is straightforward to see that $k(A)$,
$F_{{\rm int}}(A)$, $n(A)$ and $C_{\partial}(A)$ are all additive quantities
and therefore (47) is recovered.
Consider now $\mathcal{G}=\mathcal{G}_{1}\cdot_{v_{1},v_{2}}\mathcal{G}_{2}$
and $A=A_{1}\cdot_{v_{1},v_{2}}A_{2}\Subset\mathcal{G}$. We have the following
relations
$k(A)=k(A_{1})+k(A_{2})-1\,,\qquad n(A)=n(A_{1})+n(A_{2})\,,$ (49)
where the last equality follows from
$|\mathcal{V}(A)|=|\mathcal{V}(A_{1})|+|\mathcal{V}(A_{2})|-1$. More issues
arise for the faces. Taking product at two vertices $v_{1}$ of $A_{1}$ and
$v_{2}$ of $A_{2}$, one face $f_{1}$ of $A_{1}$ and one face $f_{2}$ of
$A_{2}$ enter in contact and merge. Three different situations may occur:
\- Both $f_{i}\in\mathcal{F}_{{\rm int}}(A_{i})$, $i=1,2$, then
$\displaystyle F_{{\rm int}}(A)=F_{{\rm int}}(A_{1})+F_{{\rm
int}}(A_{2})-1\,,\qquad
C_{\partial}(A)=C_{\partial}(A_{1})+C_{\partial}(A_{2})\,.$ (50)
\- $f_{1}\in\mathcal{F}_{{\rm int}}(A_{1})$ and $f_{2}\in\mathcal{F}_{{\rm
ext}}(A_{2})$ (or vice-versa), then (50) still holds because the internal face
$f_{1}$ is lost (becoming external by touching $f_{2}$) whereas the connected
component of the boundary graph $\partial A_{2}$ which contains $f_{2}$ is
conserved.
\- Both $f_{i}\in\mathcal{F}_{{\rm ext}}(A_{i})$, then
$\displaystyle F_{{\rm int}}(A)=F_{{\rm int}}(A_{1})+F_{{\rm
int}}(A_{2})\,,\qquad
C_{\partial}(A)=C_{\partial}(A_{1})+C_{\partial}(A_{2})-1\,.$ (51)
It is clear that, by changing $s\to z^{-1}$, the two quantities $F_{{\rm
int}}(A)$ and $C_{\partial}(A)$ add up as exponent of the $z$ variable and
always cancel the contribution from the $k(A)$, see (49).
∎
We come to the properties of $\mathcal{R}_{\mathcal{G}}$.
###### Theorem 3 (Contraction and cut on BR polynomial).
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a ribbon graph
with flags. Then, for a regular edge $e$,
$\mathcal{R}_{\mathcal{G}}=\mathcal{R}_{\mathcal{G}\vee
e}+\mathcal{R}_{\mathcal{G}/e}\,,$ (52)
for a bridge $e$, we have
$\mathcal{R}_{\mathcal{G}}=(x-1)\mathcal{R}_{\mathcal{G}\vee
e}+\mathcal{R}_{\mathcal{G}/e}\,;$ (53)
for a trivial twisted self-loop $e$, the following holds
$\mathcal{R}_{\mathcal{G}}=\mathcal{R}_{\mathcal{G}\vee
e}+(y-1)z\,\mathcal{R}_{\mathcal{G}/e}\,,$ (54)
whereas for a trivial untwisted self-loop $e$, we have
$\mathcal{R}_{\mathcal{G}}=\mathcal{R}_{\mathcal{G}\vee
e}+(y-1)\mathcal{R}_{\mathcal{G}/e}\,.$ (55)
###### Proof.
In the following proof, spanning c-subgraphs are simply called subgraphs. Let
us make two preliminary remarks. (A) The subgraphs $A$ of $\mathcal{G}$ which
do not contain $e$ are precisely the subgraphs of $\mathcal{G}\vee e$. (B)
Also, if $e$ is not a self-loop then the map $A\mapsto A/e$ provides a
bijection from the subgraphs of $\mathcal{G}$ which contain $e$ to the
subgraphs of $\mathcal{G}/e$. Note importantly that $A$ and $A/e$ does not
have the same vertices and edges but have the same flags and so same faces.
Let us prove (52). For a regular edge $e$, let $A\Subset\mathcal{G}\vee e$,
$A^{\prime}$ its partner in $\mathcal{G}$ such that $e\notin A^{\prime}$ by
remark (A). The fact that the monomial of $A$ in $\mathcal{R}_{\mathcal{G}\vee
e}$ and the monomial corresponding to $A^{\prime}$ in
$\mathcal{R}_{\mathcal{G}}$ are identical is simple to check. Then
$\sum_{A\Subset\mathcal{G};e\notin A}(\cdot)=\mathcal{R}_{\mathcal{G}\vee e}$.
We concentrate now on the remaining sum related to the contraction of $e$. In
particular, we focus on sets of faces during the contraction. Choose
$A\Subset\mathcal{G}$ with $e\in A$ and, by remark (B), let
$A^{\prime}\Subset\mathcal{G}/e$ be its corresponding subgraph. One has
$\displaystyle F_{{\rm int}}(A)=F_{{\rm int}}(A^{\prime})\,\,\mbox{ and
}\,\,C_{\partial}(A)=C_{\partial}(A^{\prime})\,.$ (56)
The monomial of $\mathcal{R}_{\mathcal{G}/e}$ related to $A$ is of the form
$\displaystyle(x-1)^{{\rm r}\,(\mathcal{G}/e)-{\rm
r}\,(A)}(y-1)^{n(A)}z^{k(A)-F_{{\rm
int}}(A)+n(A)}\,s^{C_{\partial}(A)}\,t^{f(A)}$ (57) $\displaystyle=(x-1)^{{\rm
r}\,(\mathcal{G})-1-({\rm
r}\,(A^{\prime})-1)}(y-1)^{n(A^{\prime})}z^{k(A^{\prime})-F_{{\rm
int}}(A^{\prime})+n(A^{\prime})}\,s^{C_{\partial}(A^{\prime})}\,t^{f(A^{\prime})}$
(58)
which achieves the proof that
$\mathcal{R}_{\mathcal{G}/e}=\sum_{A\Subset\mathcal{G};e\in A}(\cdot)$ and
then (52) is obtained.
We prove now (53). Let $e$ be a bridge in $\mathcal{G}$ and decompose
$\mathcal{R}_{\mathcal{G}}$ as $\sum_{A\Subset\mathcal{G};e\notin
A}(\cdot)+\mathcal{R}_{\mathcal{G}/e}$. It remains to prove that the first sum
can be mapped on $(x-1)\mathcal{R}_{\mathcal{G}\vee e}$ but this is
straightforward from ${\rm r}\,(\mathcal{G})={\rm r}\,(\mathcal{G}\vee e)+1$
and since all other terms remain unchanged.
The proofs of relations (54) and (55) are now provided. Consider a trivial
(twisted or not) self-loop $e$ in $\mathcal{G}$, then
$\sum_{A\Subset\mathcal{G};e\notin A}(\cdot)=\mathcal{R}_{\mathcal{G}\vee e}$
still holds in any case. Moreover, if $e$ is a self-loop, the subgraphs of
$\mathcal{G}-e$ are one-to-one with the subgraphs of $\mathcal{G}$ containing
$e.$ The passage from the subgraphs of $\mathcal{G}$ containing $e$ to those
of $\mathcal{G}-e$ or conversely is just by deleting $e$ or gluing $e$ to the
corresponding subgraph.
Consider a trivial twisted self-loop $e$ in $\mathcal{G}$. The contraction of
$e$ coincides with its deletion. Hence to each $A\Subset\mathcal{G}$ with
$e\in A$ and its corresponding $A^{\prime}\Subset\mathcal{G}/e=\mathcal{G}-e$
(obtained by just deleting $e$ in $A$), one finds that $F_{{\rm
int}}(A)=F_{{\rm int}}(A^{\prime})$ and
$C_{\partial}(A)=C_{\partial}(A^{\prime})$. Therefore, one has the relation
between the terms:
$\displaystyle(x-1)^{{\rm r}\,(\mathcal{G})-{\rm
r}\,(A)}(y-1)^{n(A)}z^{k(A)-F_{{\rm
int}}(A)+n(A)}\,s^{C_{\partial}(A)}\,t^{f(A)}$ (59) $\displaystyle=(x-1)^{{\rm
r}\,(\mathcal{G}-e)-{\rm
r}\,(A^{\prime})}(y-1)^{n(A^{\prime})+1}z^{k(A^{\prime})-F_{{\rm
int}}(A^{\prime})+n(A^{\prime})+1}\,s^{C_{\partial}(A^{\prime})}\,t^{f(A^{\prime})}\,.$
(60)
Thus (54) is satisfied.
Because $e$ is a trivial untwisted self-loop, the contraction of $e$ is its
deletion supplemented by an addition of a new vertex on $\mathcal{G}-e$ that
we denote $\mathcal{G}/e=(\mathcal{G}-e)\cup\\{v_{0}\\}$. Given
$A\Subset\mathcal{G}$ with $e\in A$, we associate a unique corresponding
element $A^{\prime}$ in $(\mathcal{G}-e)\cup\\{v_{0}\\}$ such that we delete
$e$ in $A$ and add to it $v_{0}$ as a new vertex. We can infer that $F_{{\rm
int}}(A)=F_{{\rm int}}(A^{\prime})$, and
$C_{\partial}(A)=C_{\partial}(A^{\prime})$. Thus, the following relation
between the terms corresponding to $A$ and $A^{\prime}$ holds:
$\displaystyle(x-1)^{{\rm r}\,(\mathcal{G})-{\rm
r}\,(A)}(y-1)^{n(A)}z^{k(A)-F_{{\rm
int}}(A)+n(A)}\,s^{C_{\partial}(A)}\,t^{f(A)}$ (61) $\displaystyle(x-1)^{{\rm
r}\,(\mathcal{G}/e)-{\rm
r}\,(A^{\prime})}(y-1)^{n(A^{\prime})+1}z^{(k(A^{\prime})-1)-F_{{\rm
int}}(A^{\prime})+(n(A^{\prime})+1)}\,s^{C_{\partial}(A^{\prime})}\,t^{f(A^{\prime})}\,,$
(62)
so that (55) is obtained.
∎
Importantly, we cannot map $\mathcal{R}_{\mathcal{G}\vee e}$ and
$\mathcal{R}_{\mathcal{G}/e}$ for a bridge or a trivial self-loop $e$. Such a
mapping can be recovered only after a reduction.
###### Corollary 1.
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a ribbon graph
with flags. Then, for a regular edge $e$,
$\mathcal{R}^{\prime}_{\mathcal{G}}=\mathcal{R}^{\prime}_{\mathcal{G}\vee
e}+\mathcal{R}^{\prime}_{\mathcal{G}/e}\,,\qquad\mathcal{R}^{\prime}_{\mathcal{G}\vee
e}=t^{2}\mathcal{R}^{\prime}_{\mathcal{G}-e}\,;$ (63)
for a bridge $e$, we have
$\mathcal{R}^{\prime}_{\mathcal{G}/e}=\mathcal{R}^{\prime}_{\mathcal{G}-e}=t^{-2}\,\mathcal{R}^{\prime}_{\mathcal{G}\vee
e}$
$\mathcal{R}^{\prime}_{\mathcal{G}}=[(x-1)t^{2}+1]\mathcal{R}^{\prime}_{\mathcal{G}/e}\,;$
(64)
for a trivial twisted self-loop,
$\mathcal{R}^{\prime}_{\mathcal{G}/e}=\mathcal{R}^{\prime}_{\mathcal{G}-e}=t^{-2}\mathcal{R}^{\prime}_{\mathcal{G}\vee
e}$ and
$\mathcal{R}_{\mathcal{G}}=[t^{2}+(y-1)z]\,\mathcal{R}_{\mathcal{G}/e}\,,$
(65)
whereas for a trivial untwisted self-loop, we have
$\mathcal{R}^{\prime}_{\mathcal{G}/e}=\mathcal{R}^{\prime}_{\mathcal{G}-e}=t^{-2}\mathcal{R}^{\prime}_{\mathcal{G}\vee
e}$ and
$\mathcal{R}_{\mathcal{G}}=[t^{2}+(y-1)]\mathcal{R}_{\mathcal{G}/e}\,.$ (66)
###### Proof.
This is a consequence of Theorem 3, Proposition 5 and Theorem 1 and the fact
that $\mathcal{R}^{\prime}_{\mathcal{G}\vee
e}=t^{2}\mathcal{R}^{\prime}_{\mathcal{G}-e}$ that we need to detail.
The first equation of (63) needs no comment after Theorem 1. Let us prove the
second equation of (63). There is a one-to-one assignment between spanning
c-subgraphs $A\Subset\mathcal{G}\vee e$ and $A^{\prime}\Subset\mathcal{G}-e$
given by the following: $A^{\prime}$ is obtained from $A$ by removing the two
flags issued from the cut of $e$. Note immediately that $\mathcal{F}_{{\rm
int}}(A)\subset\mathcal{F}_{{\rm int}}(A^{\prime})$. We have
$\mathcal{F}_{{\rm
int}}(A)\cup{\mathcal{C}}_{\partial}(A)=\mathcal{F}(\tilde{A})$, where
$\tilde{A}$ is the pinched graph obtained from $A$. But the pinched graph
$\tilde{A}$ coincides with the pinched graph $\tilde{A}^{\prime}$ with,
however, two additional flags. Indeed, they have the same vertices, edges,
$\mathfrak{f}^{0}(\tilde{A})=\mathfrak{f}^{0}(\tilde{A}^{\prime})\cup\\{f_{e,1},f_{e,2}\\}$,
where $f_{e,i}$, $i=1,2$, are obtained from cutting $e$, and for any face in
$\tilde{A}$, one has a unique corresponding face in $\tilde{A}^{\prime}$. This
is because, given a face of $\tilde{A}$, either this face comes from
$\mathcal{F}_{{\rm int}}(A)$ and then should be in $\mathcal{F}_{{\rm
int}}(A^{\prime})$, or this face comes from ${\mathcal{C}}_{\partial}(A)$ and
therefore two cases may occur: (i) the removal of the flags $f_{e;i}$ in $A$
yields directly a closed face of $A^{\prime}$ and this before pinching
$A^{\prime}$, such that this faces belongs to $\mathcal{F}_{{\rm
int}}(A^{\prime})$; (ii) the removal of the flags $f_{e;i}$ in $A$ yields
still an open face in $A^{\prime}$ and, so, the face belongs to
${\mathcal{C}}_{\partial}(A^{\prime})$. The unicity of the face correspondence
follows immediately by reversing the reasoning. Hence,
$F(\tilde{A})=F(\tilde{A}^{\prime})$ and $f(A)=f(A^{\prime})+2$, such that
using Theorem 1 for the remaining part, one can map the terms of
$\mathcal{R}^{\prime}_{\mathcal{G}\vee e}$ to those of
$t^{2}\mathcal{R}^{\prime}_{\mathcal{G}-e}$.
One notices that $\mathcal{R}^{\prime}_{\mathcal{G}\vee
e}=t^{2}\mathcal{R}^{\prime}_{\mathcal{G}-e}$ is true for any edge and not
only for regular ones. For special edges, $\mathcal{R}^{\prime}_{\mathcal{G}}$
can be computed using Proposition 5 along the lines of Theorem 1 and Theorem
3. Equations in (65) and (66) simply follows from (54) and (55) of Theorem 3,
respectively, and simple operations on flags as performed in Theorem 1.
∎
We summarize the main features of the BR polynomial for ribbon graphs with
flags:
$\bullet$ In the ribbon formulation, once again, the cut and contraction are
natural.
$\bullet$ $\mathcal{R}$, $\mathcal{R}^{\prime}$ and $\mathcal{T}$ have similar
properties concerning the number of flags $|\mathfrak{f}^{0}|$ and the even
exponent of the $t$ variable.
$\bullet$ Universality: Only the polynomial $\mathcal{R}^{\prime}$ is
guaranteed to have a universal property. This follows from the ordinary
arguments for the universal property of the BR polynomials for closed graphs
with weighted in the recurrence relation, for ordinary edges
$\displaystyle\phi(\mathcal{G})=\sigma\phi(\mathcal{G}-e)+\tau\phi(\mathcal{G}/e)\,,$
(67)
with initial conditions, for any bridge $e$
$\phi(\mathcal{G})=X\phi(\mathcal{G}/e)$ and the data of all BR polynomials
for one-vertex ribbon graphs [6]. Given (63), once again by performing some
change of variables, we can recast the relations satisfied by
$\mathcal{R}^{\prime}$ in this form.
$\bullet$ The polynomial $\mathcal{R}^{\prime}$ evaluated on terminal forms
made with $p$ bridges, $m$ trivial untwisted self-loops, $n$ twisted self-
loops and $q$ flags is easily computable as
$[(x-1)t^{2}+1]^{p}\,[t^{2}+y-1]^{m}\,[t^{2}+(y-1)z]^{n}\,t^{q}=X^{p}Y^{m}Z^{n}t^{q}\,.$
(68)
$\bullet$ Note that, although there is a one-to-one map between
$\\{A\Subset\mathcal{G}\vee e\\}$ and $\\{A^{\prime}\Subset\mathcal{G}-e\\}$
(see in the proof above), $A^{\prime}$ has the same vertices and edges but
different flags and so different faces. There is a priori no relation between
$\mathcal{R}_{\mathcal{G}\vee e}$ and $\mathcal{R}_{\mathcal{G}-e}$ due to the
discrepancy between their number of internal faces and boundary components. A
relation between the polynomial of $\mathcal{G}\vee e$ and $\mathcal{G}-e$ can
be only recovered for the $\mathcal{R}^{\prime}$ polynomial. For any edge $e$,
$\mathcal{R}^{\prime}_{\mathcal{G}\vee
e}=t^{2}\mathcal{R}^{\prime}_{\mathcal{G}-e}$.
$\bullet$ We have
$\mathcal{R}_{\mathcal{G}}(x,y,z,z^{-1},t)=\mathcal{R}^{\prime}_{\mathcal{G}}(x,y,z,t)\,,\qquad\mathcal{R}^{\prime}_{\mathcal{G}}(x,y,z,t=1)=R_{\mathcal{G}}(x,y,z)\,,$
(69)
hence $\mathcal{R}_{\mathcal{G}}$ and $\mathcal{R}^{\prime}_{\mathcal{G}}$ are
generalized BR polynomials satisfying contraction and deletion operations.
Finally, to recover the Tutte polynomial we set
$\mathcal{R}_{\mathcal{G}}(x,y,z=1,s=1,t)=\mathcal{T}_{\mathcal{G}}(x,y,t)\,.$
(70)
from which the ordinary Tutte polynomial $T_{\mathcal{G}}$ is directly
inferred.
## 4\. Rank $D$ stranded graphs with flags and a generalized polynomial
invariant
This section undertakes the definition of a new polynomial for particular
graphs which aims at generalizing the BR polynomial $\mathcal{R}$ for graphs
on surfaces with boundaries in its most expanded form as treated in the
Section 3.2. The central notion of graphs discussed below is combinatorial and
can be always pictured in a 3D space. Our main result appears in Theorem 4
after a thorough definition of the type of graphs extending ribbon graphs with
flags for which a generalized polynomial invariant turns out to exist.
The primary notion of rank $D$ colored tensor graphs considered here has been
introduced by Gurau in [10]. As ribbon graphs can be dually mapped onto
triangulations of surfaces in 2D, colored tensor graphs can be interpreted as
simplicial complexes or dual of triangulations of topological spaces in any
dimension. They are of particular interest in certain quantum field
theoretical frameworks defined with tensor fields hence the name tensor
graphs555Ribbon graphs are, in this sense, rank 2 or matrix graphs.. The
importance of these graphs has been highlighted by Gurau which proved that the
cellular structure associated with these colored graphs generates dually only
simplicial pseudo-manifolds of $D$ dimensions [11]. It has been also proved
that a colored tensor graph which is bipartite induces naturally an
orientation of the dual complex [8].
Some anterior studies have addressed the extension of the BR polynomial for
higher dimensional object within the framework of such graphs. Mainly, two
authors Gurau [12] and Tanasa [18] have defined two separate notions of
extended BR polynomial. Let us do a blitz review of their results and compare
these to the one obtained in the present work.
Gurau defined a multivariate polynomial invariant for colored tensor graphs
dually associated with simplicial complexes with boundaries for any dimension
$D$. The polynomial that we obtain in the present work admits a multivariate
form which is related to the polynomial obtained by this author restricted to
3D but extends it to a larger type of graph than only the colored tensor
graphs. We emphasize that the difficulties encountered by Gurau (as well as
Tanasa, see below) for defining a contraction procedure for such type of
graphs without destroying the entire graph structure will be improved our
scheme.
In a different perspective, the work by Tanasa in [18] deals with tensor
graphs without colors which are equipped with another vertex. The polynomial
as worked out by this author is only valid for graphs triangulating
topological objects without boundary. The polynomial that we define is
radically different from that one in several features, since mainly, it relies
on the presence of colors in the graph.
### 4.1. Stranded, tensor, colored graphs with flags
After some operations, the initial type of rank $D$ colored tensor graphs
generates a final class of graphs which will be the one of interest in our
analysis. Like ribbon graphs, colored tensor graphs have both a topological
meaning (realized in the dual triangulation) and a combinatorial formulation
that will be our main concern here. Before defining colored tensor graphs, let
us introduce the combinatorial concept of stranded graph structure, the true
backbone of this theory.
Stranded and tensor graphs. We start by primary notions of stranded structures
and graphs.
###### Definition 22 (Stranded vertex and edge).
$\bullet$ A rank $D$ stranded vertex is a collection of segments called bows
or strands which can be drawn in a 3-ball hemisphere (or a spherical cup in a
three dimensional space) such that:
1. (a)
the bows are not intersecting;
2. (b)
all end points of the bows are drawn on the boundary (circle) of the
equatorial disc (or section) called the vertex frontier;
3. (c)
these end points can be partitioned in sets called pre-flags with
$0,1,2,\dots,D$ elements;
4. (d)
the pre-flags should form a connected collection that is each pre-flag should
be connected to any other pre-flag by a tree of bows.
The coordination (or valence or degree) of a rank $D$ stranded vertex is the
number of its pre-flags. By convention, (C1) we include a particular vertex
made with one disc and assume that it is a stranded vertex of any rank made
with a unique closed strand and (C2) a point is a rank 0 stranded vertex.
$\bullet$ A rank $D$ stranded edge is a collection of segments called again
strands such that:
1. (a’)
the strands are not intersecting (but can cross, i.e. can be non parallel);
2. (b’)
the end points of the strands can be partitioned in two disjoint parts called
sets of end segments of the edge such that a strand cannot have its end points
in the same set of end segments;
3. (c’)
the number of strands is $D$.
Rank $D>0$ stranded vertices just enforce that any entering strand in the
vertex should be exiting by another point at the frontier of the vertex.
According to our convention, the ordinary disc and ribbon edges of ribbon
graphs are valid stranded vertex and stranded edges.
The notion of connectivity should be clarified. Vertices (with connected pre-
flags) and edges are by convention connected objects even though graphically
they sometimes appear disconnected in the ordinary topological sense. This is
will be justified later on based on the dual that can represent such objects
after few more constraints. Examples of stranded vertices and edges with rank
$D=4$ and $D=5$, respectively, have been provided in Figure 8.
Figure 8. A rank 4 stranded vertex $v$ of coordination 6, with connected pre-
flags (highlighted with different colors) with crossing bows; a trivial disc
vertex $d$; a rank 5 edge $e$ with non parallel strands.
$v$$d$$e$
###### Definition 23 (Stranded and tensor graphs).
$\bullet$ A rank $D$ stranded graph $\mathcal{G}$ is a graph
$\mathcal{G}(\mathcal{V},\mathcal{E})$ which admits:
1. (i)
rank $D$ stranded vertices;
2. (ii)
rank at most $D$ stranded edges;
3. (iii)
One vertex and one edge intersect by one set of end segments of the edge which
should coincide with a pre-flag at the frontier vertex. All intersections of
vertices and edges are pairwise distinct.
$\bullet$ A rank $D$ tensor graph $\mathcal{G}$ is a rank $D$ stranded graph
such that:
1. (i’)
the vertices of $\mathcal{G}$ have a fixed coordination $D+1$ and their pre-
flags have a fixed cardinal $D$. From the point of view of the pre-flags, the
pattern followed by each stranded vertex is that of the complete graph
$K_{D+1}$;
2. (ii’)
the edges of $\mathcal{G}$ are of rank $D$ and their strands are parallel.
Some illustrations of a rank 3 stranded and tensor graphs are given in Figure
9 and 10, respectively.
Figure 9. A rank 3 stranded graph. Figure 10. A rank 3 tensor graph with
rank 3 vertices as with fixed coordination 4, pre-flags with 3 points linked
by bows according to the pattern of $K_{4}$; edges are rank 3 with parallel
strands.
It should be pointed out that, although the vertices and edges of stranded and
tensor graphs are drawn in a three dimensional space, we do not treat these as
embedded graphs. The reason for this follows. There is an equivalent way to
construct rank $D$ stranded vertices, edges and graphs using uniquely
combinatorial (permutation) maps [19] (see Chap. X). For this we need to label
each pre-flag point (or cross in the jargon of [15]) and define to which it is
associated. There should be $\sigma_{0}$ a vertex map, $\sigma_{1}$ an edge
map and $\theta$ a pre-flag map partitioning the set of crosses. For simple
and ribbon graphs such permutation triple exists [19]. For instance, they have
been extensively used in [15] in order to extend Chmutov duality to ribbon
graphs with flags. At this moment, there is no equivalent formulation for
higher rank stranded graphs in the way we introduce it here. However, one
realizes that these maps can be defined point by point for each graph
constituent. In other words, $\sigma_{0}$ should be defined for each vertex,
$\theta$ for each pre-flag, $\sigma_{1}$ for each edge. Remark that to capture
the general features that these maps should satisfy for stranded or even
tensor graphs can be a non trivial task.
At the end, we will always consider as identical two rank $D$ stranded graphs
which are defined by the same $(\sigma_{0},\theta,\sigma_{1})$. Examples of
identical vertices and edges have be given in Figures 11 and 12. Thus we can
modify a stranded graph and still obtain the same graph after moving the pre-
flags at the frontier vertex (the order of their points can be also changed),
deform and cross arbitrarily bows and strands. The sole point to be required
is to keep the incidence relations of strand segments between the labeled
points. For these reasons, henceforth, we will always use a “minimal”
graphical representation for stranded vertices and edges which is the one
defined by strands using the “shortest” path between the pre-flags points.
A side remark is that, in Definition 23, the precision in (ii’) that the
strands of edges in tensor graphs are parallel seems now superfluous since,
using the above combinatorial maps, we can always make parallel non parallel
strands. The price to pay for satisfying that is to get a more peculiar vertex
with more bow crossings following the fact that we should preserve the same
incidence of bows. Hence, the word parallel in (ii’) is seen as the simplest
choice possible for edges making as simple as possible the vertex in the case
of tensor graphs.
Figure 11. Two identical vertices, $v_{1}$ and $v_{2}$, defined up to a
rotation of their frontier; note that the pairing of pre-flag points (labeled
by colors here) is the same (deformation and crossing of their bows are
meaningless). $v_{2}$ will be not preferred though because one of its bows,
the one between the two green pre-flag points, do not use the shortest path
between pre-flags points.
$v_{1}$$v_{2}$
Figure 12. Two identical rank 3 edges between two graphs (shaded).
$\mathcal{G}_{1}$$\mathcal{G}_{2}$$\mathcal{G}_{1}$$\mathcal{G}_{2}$
Given a stranded graph and collapsing its stranded vertices to points and its
edges to simple lines, one obtains a simple graph in the sense of Definition
1. This justifies the fact that stranded graphs are graphs. Moreover, it is
with respect to this simple graph point of view that we will say that a
stranded graph is connected or not. In other words, a stranded graph is
connected if its corresponding collapsed graph is connected. For instance,
both graphs of Figure 9 and 10 are connected.
We denote, when no confusion is possible,
$\mathcal{G}(\mathcal{V},\mathcal{E})$ a rank $D$ stranded graph with vertex
set $\mathcal{V}$ and edge set $\mathcal{E}$. It can be seen that any rank $D$
stranded graph is a rank $D^{\prime}$ stranded graph, with $D^{\prime}\geq D$.
This is however not true for tensor graphs. Calling a rank $D$ stranded graph
in the remaining will often refer to the minimal rank for which this graph is
well defined.
Lower rank stranded graphs. Let us give low rank examples of the above graphs
which turns out to be either trivial or known.
1. (0)
Rank 0 stranded graphs are simple graphs made with only points as vertices and
no edges.
2. (1)
Rank 1 stranded graphs as for vertices segments (note that any pre-flag
reduces to a point and, in order to form a connected collection, one has to
consider a unique segment between two pre-flags) and for edges segments as
well. Such rank 1 stranded graphs cannot be directly identified with simple
graphs in the sense of Definition 1. They can be viewed as simple graphs if
one collapses all vertices to points. Hence, neither rank 0 nor rank 1
stranded graphs allows to define simple graphs in general. In fact, for any
rank $D$, the simple graph structure cannot be directly achieved without using
the collapsing procedure described above.
3. (2)
General ribbon graphs, in the sense of Definition 13, are one-to-one with
particular rank 2 stranded graphs. Given a ribbon graph
$\mathcal{G}(\mathcal{V},\mathcal{E})$ (Definition 13) and an edge $e$ of
$\mathcal{G}$. The ribbon edge $e$ touches its end vertex or vertices in two
segments $s$ and $s^{\prime}$. Consider the two distinct segments on the
boundary of $e$ which are not $s$ and $s^{\prime}$ (denoted by $s_{1}$ and
$s_{2}$ in Figure 13A). These two segments define the two (not intersecting
and potentially twisted) strands of a rank 2 stranded edge with $s$ and
$s^{\prime}$ as end segments. By convention, a vertex with no incident line is
a stranded vertex. So it is a stranded graph of any rank in particular $D=2$.
Consider now a vertex $v$ of $\mathcal{G}$ with incident edges $e_{1}$,
$e_{2}$, …, $e_{p}$, $p\in\mathbb{N}$. We can construct a stranded vertex
$v^{\prime}$ of rank $2$ in the following manner. The frontier vertex of
$v^{\prime}$ is simply the boundary of $v$. The end points of end segments of
$e_{k}$, $1\leq k\leq p$, define the pre-flags (with exactly 2 points) of
$v^{\prime}$. From these pre-flags, one defines the bows of $v^{\prime}$ as
the segments between the end segments of $e_{k}$ which lies in $v$. See Figure
13B.
Figure 13. Ribbon edge and vertex of a ribbon graph as a rank 2 stranded edge
and vertex of a stranded graph.
$s_{1}$$s_{2}$AB
Ribbon graphs equipped with vertices with fixed coordination equals to 3 are
rank 2 tensor graphs. The reason why the converse is not true is that, in any
rank 2 tensor graph, the bows might be not cyclically disposed on the vertex
frontier. Thus the definition of any ribbon graph vertex as a disc may be not
always satisfied.
While a stranded graph can be regarded as an abstract object, tensor graphs
actually possess a topological content.
###### Definition 24 (Dual of a tensor graph).
Let $\mathcal{G}$ be a rank $D$ tensor graph. A vertex of $\mathcal{G}$
represents a $D$ simplex and an edge of $\mathcal{G}$ represents a $(D-1)$
simplex. The graph $\mathcal{G}$ is dual to a simplicial complex obtained from
the gluing of $D$ simplexes along $(D-1)$ simplexes lying at their boundary.
As an illustration of the above definition, a rank 3 tensor graph represents a
simplicial complex in 3D composed by tetrahedra (3-simplexes) which are glued
along their boundary triangles (2-simplexes). For rank $D$ tensor graph, a
pre-flag is of cardinal $D$. Each couple of points induces a segment in the
pre-flag, we obtain $D(D-1)/2$ disjoint segments. These pre-flag segments are
graphically pictured by $D$ points or $D-1$ joined (adjacent) segments between
these $D$ points at the frontier vertex. One may interpret that these
$D(D-1)/2$ disjoint segments as the number of $(D-2)$-simplexes of the
boundary of a $D-1$ simplex. For $D=3$, there are $D(D-1)/2=3$ edges in a
triangle. Combinatorially, we draw these edges as $D=3$ points or $D-1=2$
joined segments.
We now come back to our previous convention that stranded vertices and edges
are connected. Because the dual of these two entities are connected
topological simplexes only under tensor graph axioms, we simply assume that,
in tensor graphs and by extension in stranded graphs, these objects are also
connected.
###### Definition 25 (Rank $D$ flag).
A rank $D$ stranded flag (or half-edge or simply flags, when no confusion may
occur) is a collection of $D$ parallel segments called strands satisfying the
same properties of strands of rank $D$ edges but the flag is incident to a
unique rank $D^{\prime}$ stranded vertex, with $D^{\prime}\geq D$, by one of
its set of end segments without forming a loop.
A rank $D$ flag has two sets of end segments: one touching a vertex and
another called free or external set of end segments, the elements of which
called themselves free or external segments. The $D$ end-points of all free
segments are called external points of the rank $D$ flag. (See Figure 14.)
Figure 14. A rank 3 stranded flag with its external points $a$, $b$ and $c$.
$a$$b$$c$
###### Definition 26 (Cut of an edge).
Let $\mathcal{G}(\mathcal{V},\mathcal{E})$ be a rank D stranded graph and $e$
a rank $d$ edge of $\mathcal{G}$, $1\leq d\leq D$. The cut graph
$\mathcal{G}\vee e$ or the graph obtained from $\mathcal{G}$ by cutting $e$ is
obtained by replacing the edge $e$ by two rank $d$ flags at the end vertices
of $e$ and respecting the strand structure of $e$. (See Figure 15.) If $e$ is
a self-loop, the two flags are on the same vertex.
Figure 15. Cutting a rank 3 stranded edge.
The cut graph $\mathcal{G}\vee e$ is called a rank $D$ stranded graph with
flags. The presence of flags is compatible with rank $D$ stranded vertices and
edges. Furthermore, in a stranded graph, we will always consider that all pre-
flags are used. In other words, all pre-flags are necessarily either in
contact with edges or in contact with flags. An example of such a graph is
provided in Figure 16.
Figure 16. A rank 3 stranded graph with all pre-flags used.
Hence, given a set of additional flags $\mathfrak{f}^{0}$ in some stranded
graph, assuming that $\mathfrak{f}^{0}=\emptyset$ simply means that all pre-
flags are in contact with edges. Any rank $D$ stranded graph with additional
set of flag $\mathfrak{f}^{0}$ is denoted
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$.
The notion of c-subgraph and spanning c-subgraph follows naturally from
Definition 10 with however the new meaning of the cut operation. The spanning
c-subgraph inclusion will be again denoted by $A\Subset\mathcal{G}$. To keep
track of the internal structure of the vertices, c-subgraphs will be preferred
to subgraphs which adds to the fact that we will only use stranded graphs with
all pre-flags fully connected to flags. This also justifies why in the
formulation of [12] the deletion an edge is in fact the cut of the edge in the
sense precisely defined above. Thus, the cut operation should be the natural
operation within this framework.
Particular edges as self-loops, bridges (defined through a cut of an edge
which brings an additional connected component in the graph), regular edges
and terminal forms are extended in the present context because the stranded
graph admits a graph structure.
Most of the concepts of Definition 19 are valid for stranded graphs. Faces of
a stranded graph $\mathcal{G}$ are the maximal connected components made with
strands of $\mathcal{G}$. Closed faces never pass through any free segment of
additional flags. $\mathcal{F}_{{\rm int}}$ denotes the set of these closed
faces. Open or external faces are maximally connected component leaving an
external point of some flag rejoining another external point. The set of open
faces is denoted $\mathcal{F}_{{\rm ext}}$. As usual, the set of faces
$\mathcal{F}$ of a graph is defined by $\mathcal{F}_{{\rm
int}}\cup\mathcal{F}_{{\rm ext}}$. Open and closed stranded graphs follows the
same ideas as in Definition 19 as well.
Several notions (such as open and closed face) which are totally combinatorial
in the stranded situation bear a true topological content in the tensor graph
case. For instance, a closed face in a rank 3 tensor graph is a 2D surface in
the bulk (interior) of the dual triangulation. An open face is a surface
intersecting the boundary of the simplicial complex dual to the graph.
Let us discuss in more details the structure of an edge $e$. Assuming that $e$
is of rank $d$, consider the two pre-flags $f_{1}$ and $f_{2}$ where $e$ is
branched to its end vertex $v$ (a self-loop situation) or vertices $v_{1,2}$
(a non self-loop case). It may happen that, after branching $e$, $p$ closed
faces are formed666If $e$ is a self-loop, $p\leq d$; if $e$ is not a self-
loop, $p\leq d/2$ if $d$ is even or $p\leq(d-1)/2$, if $d$ is odd. such that
these closed faces are completely contained in $e$ and $v$ or $v_{1,2}$. These
closed faces will be called inner faces of the edge. An edge with $p$ inner
faces is called $p$–inner edge. See Figure 17 A,B and C for an illustration in
the rank 3 situation. The remaining strands passing through $f_{1}$ and
$f_{2}$ which are not used in the inner faces are called outer strands. They
connect $f_{1}$ and $f_{2}$ to other pre-flag families $\\{f_{1;i}\\}$ and
$\\{f_{2;j}\\}$, called neighbors of $f_{1}$ and $f_{2}$ respectively (this
has been also illustrated in Figure 17). Dealing with a self-loop, we add the
condition that neighbor families should satisfy $f_{1}\notin\\{f_{2;j}\\}$ and
$f_{2}\notin\\{f_{1;i}\\}$. Note that these two families may have pre-flags in
common for a self-loop situation. If one of the neighbor families is empty, it
means that for a non self-loop case, $f_{1}$ or $f_{2}$ define a vertex. For
the self-loop case, the families can be empty but then there should exist some
bows between $f_{1}$ and $f_{2}$ and, if both families are empty, $f_{1}$ and
$f_{2}$ define together a vertex.
For any type of edge,
\- Assuming that $\\{f_{1;i}\\}$ and $\\{f_{2;j}\\}$ are not empty, this
implies that the outer strands of $f_{1}$ and $f_{2}$ are related by strands
via $e$. Consequently, $\\{f_{1;i}\\}$ and $\\{f_{2;j}\\}$ are connected via
strands in $e$.
\- Assuming that $\\{f_{1;i}\\}$ is empty but $\\{f_{2;j}\\}\neq\emptyset$ (or
vice-versa without loss of generality) the outer strands of $f_{2}$ are all
related via $e$ (and in the case of a self-loop, these outer strands may be
related with bows in common between $f_{1}$ and $f_{2}$). Therefore
$\\{f_{2;j}\\}$ is connected via strands in $e$.
\- Assuming that both $\\{f_{1;i}\\}$ and $\\{f_{2;j}\\}$ are empty, then $e$
generates only inner faces.
The notion of edge contraction can be defined at this point.
Figure 17. Some rank 3 $p$–inner edges: 0–inner $e_{1}$ (A), 1–inner $e_{2}$
(B) and 2–inner $e_{3}$ (C) edges (with inner faces highlighted in red). Outer
strands highlighted (in green and blue) for each $p$–inner edge case of A, B,
C and their attached neighbor pre-flags.
A$f_{1}$$f_{1;1}$$f_{1;2}$$f_{1;3}$$f_{2}$$f_{2;1}$$e_{1}$B$f_{1}$$f_{1;1}$$f_{2}$$f_{2;1}$$e_{2}$C$f_{1}$$f_{1;1}$$f_{2}$$e_{3}$$(v_{1})$$(v_{2})$$(v_{1})$$(v_{2})$$(v)$
###### Definition 27 (Stranded edge contractions).
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ a rank $D$
stranded graph with flags. Let $e$ be a rank $d$ edge with pre-flags $f_{1}$
and $f_{2}$ and consider their neighbor families $\\{f_{1;i}\\}$ and
$\\{f_{2;j}\\}$.
$\bullet$ If $e$ is a rank $d$ $p$–inner edge but not a self-loop, the graph
$\mathcal{G}/e$ obtained by contracting $e$ ‘softly’ is defined from
$\mathcal{G}$ by removing the $p$ inner faces generated by $e$, replacing $e$
and its end vertices $v_{1,2}$ by $p$ disjoint disc vertices and a new vertex
$v^{\prime}$ which possesses all edges but $e$ and all flags in the same
cyclic order as their appear on $v_{1}$ and $v_{2}$, and strands obtained by
connecting directly $\\{f_{1;i}\\}$ and/or $\\{f_{2;j}\\}$ via the outer
strands of $f_{1}$ and $f_{2}$. (See examples of rank 3 edge contraction in
Figure 18 A’ and B’.)
$\bullet$ If $e$ is a rank $d$ $p$–inner self-loop, the graph $\mathcal{G}/e$
obtained by contracting $e$ ‘softly’ is defined from $\mathcal{G}$ by removing
the $p$ inner faces of $e$, by replacing $e$ and its end vertex $v$ by $p$
disjoint disc vertices and one vertex $v^{\prime}$ having all edges as $v$ but
$e$ and all flags and strands built in the similar way as previously done by
connecting the neighbor families of $f_{1}$ and/or $f_{2}$. (See an example of
rank 3 self-loop contraction in Figure 18C’.)
If there is no outer strand left after removing the $p$ inner faces of $e$
then the vertex $v^{\prime}$ is empty.
$\bullet$ For any type of $p$–inner edges, the graph
$(\mathcal{G}/e)^{\natural}$ obtained by contracting $e$ ‘hardly’ is defined
in the same way as above for each type of edge but without adding the extra
$p$-discs.
Figure 18. Graphs A’,B’ and C’ obtained after edge contraction of A, B and C
of Figure 17, respectively.
A’B’C’
Soft and hard contractions agree obviously on $0$–inner edges. The result of a
hard contraction can be always inferred from that one of a soft contraction
performed on the same graph by simply removing the extra discs introduced at
the end of the soft procedure. Introducing this discrepancy in the edge
contraction will be justified in the following.
The terms “cyclic order” can be removed from the definition provided we keep
track of the pairing of the pre-flags points. The cyclic order preservation is
a simple and convenient choice for recovering the final vertex after the
contraction.
One may directly check that contracting a non self-loop in a ribbon graph can
be immediately seen as a rank 2 edge (soft or hard) contraction. For self-loop
in ribbon graphs, we conjecture that the soft contraction procedure agrees
with ordinary contraction of ribbon self-loop once again. In fact, one may
check that a trivial self-loop contraction for a ribbon graph coincides with
the notion contraction of a $p$–inner self-loop contraction in the soft sense
of Definition 27, with $p=0,1,2$, when the ribbon graph is viewed as a rank 2
stranded graph777 Indeed, for a ribbon graph, a trivial untwisted self loop is
equivalent to a 1– or 2–inner self-loop. It is a 1–inner self-loop if one of
the face of the self-loop does not immediately close in the vertex and pass
through another edge. A trivial untwisted self-loop with one vertex and one
edge is a 2–inner self-loop. A trivial twisted self-loop can be 1– or 0–inner
self-loop, depending on the number of edges of the vertex having the self-
loop. If there is more than one edge in the vertex, it is a 0–inner self-loop,
otherwise it is a 1–inner self-loop. Treating systematically these
configurations of trivial self-loops in a ribbon graph, one can prove the
above claim for these cases. . For a general self-loop $e$, we have strong
hints that the soft contraction still coincides with the simple contraction of
$e$ in a ribbon graph $\mathcal{G}$ defined through the dual graph
$(\mathcal{G}^{*}-e^{*})^{*}=\mathcal{G}/e$ [6] hence, our above conjecture.
###### Proposition 6.
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ a rank $D$
stranded graph with flags. Let $e$ be a rank $d$ edge. The graphs
$\mathcal{G}/e$ and $(\mathcal{G}/e)^{\natural}$ obtained by soft or hard
contraction of $e$, respectively, admit a rank $D$ stranded structure.
###### Proof.
The following does not depend on the type of contraction. After contraction of
an edge $e$, remark that the remaining edges in the graph $\mathcal{G}$ are
untouched. Hence they are at most rank $D$ stranded edges. The gluing of these
edges and vertices of the $\mathcal{G}/e$ is unchanged as well. Ignoring the
potentially present $p$ discs which are rank $D$ stranded graphs for any $D$,
the unique issue that we need to check is whether or not the vertex
$v^{\prime}$ obtained after edge contraction admits a rank $D$ stranded
structure.
To proceed with this, observe first that bows in $v^{\prime}$ are defined to
be previous non intersecting bows or merged bows from neighbors pre-flags.
Obviously all these bows are not intersecting before and after contraction.
The frontier vertex can be easily extended (by deformation) to contain all
pre-flags of the vertex $v_{1}$ and $v_{2}$ except $f_{1}$ and $f_{2}$ the
pre-flags of $e$. The key point is that the vertex $v^{\prime}$ may be
disconnected with respect to its pre-flags. One must check that every
connected component in $v^{\prime}$ are rank $D$ stranded vertices.
The remaining point is then quickly solved from the fact that before and after
contraction all points are connected via bows in the graph. This means that,
before contraction, all pre-flags have all of their points connected by bows
to other pre-flag points. After contraction, some of the bows are removed. But
whenever a bow is removed its end points are removed as well. Thus all
remaining bows (if exist) linking all pre-flags are either untouched or
carefully reconnected between themselves in the neighbor families of $f_{1}$
and/or $f_{2}$ (if exist). Hence there is no point in any remaining pre-flags
without bow attached. Now if the vertex $v^{\prime}$ gets disconnected and
some of the pre-flags form a separate set, all points of these pre-flags must
be fully connected between themselves by bows. This ensures that any connected
set of pre-flags forms a new stranded vertex with rank at most $D$.
∎
After the above discussion, the following proposition is straightforward.
###### Proposition 7 (Rank evolution under contraction).
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ a rank $D$
stranded graph with flags. Let $e$ be a rank $d$ edge. The ranks of the graphs
$\mathcal{G}/e$ and $(\mathcal{G}/e)^{\natural}$ are less than or equal to
$D$.
At this point, one may wonder if the additional discs obtained after soft edge
contraction are not inessential features of the graphs for the remaining
analysis. In fact, they will be very useful for the preservation of number of
faces during the edge contraction of the graph as it is the case for ribbon
graphs. Furthermore as discussed above, if one wants to match with the
definition of a deletion in the dual, they could become totally relevant. Hard
contractions are also useful but in another context, for instance, contracting
particular tensor subgraphs called dipoles (see for instance [4]). We will
however focus only soft contractions. What we shall need is just to partially
remove the disc which could be generated and work “up to trivial discs” hence
in the framework of equivalence class of graphs.
###### Definition 28 (Graph equivalence class).
Let $D_{\mathcal{G}}$ be the subgraph in a rank $D$ stranded graph
$\mathcal{G}$ defined by all of its trivial disc vertices and
$\mathcal{G}\setminus D_{\mathcal{G}}$ the rank $D$ stranded graph obtained
after removing $D_{\mathcal{G}}$ from $\mathcal{G}$.
Two rank $D$ stranded graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are
“equivalent up to trivial discs” if and only if $\mathcal{G}_{1}\setminus
D_{\mathcal{G}_{1}}=\mathcal{G}_{2}\setminus D_{\mathcal{G}_{2}}$. We note
$\mathcal{G}_{1}\sim\mathcal{G}_{2}$.
One can check that $\sim$ is indeed an equivalence relation. The next
important notion to identify is that of (spanning) c-subgraphs of an
equivalence class $\mathfrak{G}$ of a graph $\mathcal{G}$. This can be done as
follows. A (spanning) c-subgraph $\mathfrak{g}$ of $\mathfrak{G}$ is the
equivalence class of a (spanning) c-subgraph of any of its representative. To
see why this concept is independent of the representative holds by definition:
any (spanning) c-subgraphs of some representative coincides with some
(spanning) c-subgraphs of any another representative up to discs. The fact
that one c-subgraph is spanning in some $\mathcal{G}$ means that it includes
all vertices of $\mathcal{G}$. Removing all discs, it becomes spanning in
$\mathcal{G}\setminus D_{\mathcal{G}}$, for any $\mathcal{G}$.
Colored tensor graphs. Apart from the stranded structure, the second important
feature of the type of graphs which we will study is the coloring.
###### Definition 29 (Colored and bipartite graphs).
$\bullet$ A $(D+1)$ colored graph is a graph together with an assignment of a
color belonging to the set $\\{0,\cdots,D\\}$ to each of its edges such that
no two adjacent edges share same color.
$\bullet$ A bipartite graph is a graph whose set $\mathcal{V}$ of vertices is
split into two disjoint sets, i.e.
$\mathcal{V}=\mathcal{V}^{+}\cup\mathcal{V}^{-}$ with
$\mathcal{V}^{+}\cap\mathcal{V}^{-}=\emptyset$, such that each edge connects a
vertex $v^{+}\in\mathcal{V}^{+}$ and a vertex $v^{-}\in\mathcal{V}^{-}$.
We are in position to define a colored tensor graph.
###### Definition 30 (Colored tensor graph [10, 13]).
A rank $D$ colored tensor graph $\mathcal{G}$ is a graph such that:
$\bullet$ $\mathcal{G}$ is $(D+1)$ colored and bipartite;
$\bullet$ $\mathcal{G}$ is a rank $D$ tensor graph.
The type of restriction introduced in Definition 30 allows us to have a
control on the type of graphs which are generated. This restriction is
therefore useful to discuss a specific class of these graphs but still
containing a large (infinite) number of them.
As an illustration, a rank 3 colored tensor graph is pictured in Figure 19.
Each vertex (looking like the $K_{4}$ complete graph again) is the dual of a
tetrahedron and an edge represents a triangle endowed with a color
$i\in\\{0,1,2,3\\}$. The graph is also bi-partite.
We denote a rank $D$ colored tensor graph as
$\mathcal{G}(\mathcal{V},\mathcal{E})$ where, as usual, $\mathcal{V}$ is its
set of vertices and $\mathcal{E}$ is its set of edges. In a rank $D$ colored
tensor graph, to each pre-flag could be assigned a color so that an edge can
only connect pre-flags of the same color. In such a situation, edges will
inherit pre-flag colors. Equivalently, one could infer a color for each pre-
flag as the color of the edge touching this pre-flag. There are certainly more
data worthwhile to be discussed in such a graph.
Figure 19. A rank 3 colored tensor graph.
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###### Definition 31 ($p$-bubbles [10]).
A $p$-bubble with colors $i_{1}<i_{2}<\dots<i_{p}$, $p\leq D$, and
$i_{k}\in\\{0,\dots,D\\}$ of a rank $D$ colored tensor graph $\mathcal{G}$ is
a maximally connected component made of strands of edges of colors
$\\{i_{1},\dots,i_{p}\\}$.
For any $D$, vertices are 0-bubbles, edges are 1-bubbles. Restricting to
$D=3$, there are two other types of $p$-bubbles that we shall need in the
following:
$\bullet$ The connected components of $\mathcal{G}$ made with two colors are
cycles of lines along which the colors alternate. These are 2-bubbles called
the faces of the graph (See the face $f_{03}$ (in red) in Figure 20).
$\bullet$ A 3-bubble (or simply bubble in $D=3$) is a connected component of
the graph which has three colors (See Figure 20).
Figure 20. The face $f_{03}$ (in red) and bubbles of the graph of Figure 19.
$f_{03}$$\mathbf{b}_{012}$$\mathbf{b}_{013}$$\mathbf{b}_{023}$$\mathbf{b}_{123}$
Faces of stranded graphs are made with strands of edges and vertices. We say
that “a face passes through an edge (or vertex) $x$ time(s)” if it is built
with $x\geq 0$ strands of this edge (or vertex). Of course, $x=0$, means that
the face never passes through the element.
Remark that, importantly, having bicolored faces prevents faces to pass more
than once in an edge. As in the definition of a face, any strand of the rank
$D$ colored tensor graph is also bicolored. This crucial feature will be used
several times in the following.
Any 3-bubble can be seen as a $2D$ ribbon graph. The vertices of a bubble are
three valent vertices obtained by decomposing the vertex of the graph in the
way of Figure 20. The edges of a 3-bubble are colored ribbon edges generated
by the decomposition of the colored edges of the rank $D$ colored graph. As
ordinary ribbon graphs, 3-bubbles have faces as well. These faces are endowed
with a pair of colors like the faces of the initial graph. Thus a bubble is
simply a rank 2 colored tensor graph lying inside the rank 3 colored tensor
graph. The set of 3-bubbles will be denoted by $\mathcal{B}_{3}$ and
$|\mathcal{B}_{3}|=B_{3}$. The index 3 will be omitted when discussing $D=3$.
In the dual picture, a face is a (boundary) surface and a bubble is a
collection of surfaces which close and form a $3D$ region in the simplex.
Often, one associates directly the $3D$ region to the bubble.
A rank $D$ colored tensor graph admits another equivalent representation. By
collapsing the entire rank $D$ colored tensor graph in the way previously
mentioned, namely by forgetting its stranded structure, we obtain another
representation of the same graph. This representation, called _compact_ in the
following, is merely obtained by regarding the tensor graph as a simple
bipartite colored graph in the sense of Definition 29. It is worth underlining
that one representation of the graph determines unambiguously the other (this
is not the case for the collapsed graph which cannot fully characterize the
stranded structure of its generating graph). For instance, the graph of Figure
19 can be also drawn in the compact form of Figure 21.
Figure 21. Compact representation of the graph of Figure 19.
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Rank $D$ flags can be considered as well on rank $D$ colored tensor graph,
provided these flags possess a color and their gluing respects the graph
coloring at each vertex. Thus to axioms (i’) and (ii’) of Definition 23, we
add:
(iii’) to each edge and flag, one assigns a color $i\in\\{0,1,\dots,D\\}$.
The cut of an edge can be done in the same sense of Definition 26 for colored
tensor graphs. The crucial point is solely to respect the color structure of
the graph after the cut such that each of resulting flags possess the same
color structure of the former edge.
The definitions of a rank $D$ colored tensor graph with flags
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ (see Figure 22 for an
illustration for $D=3$) and its spanning c-subgraphs of $\mathcal{G}$ follow
from spanning c-subgraphs of the stranded graph $\mathcal{G}$ and from
Definition 10. The only point to be added is the color structure. The
following is straightforward:
###### Proposition 8.
(Spanning) c-subgraphs of a rank $D$ colored tensor graph are rank $D$ colored
tensor graphs.
Figure 22. An open rank 3 colored tensor graph and its bubbles; $f_{01}$
(highlighted in red) is an open face; $\mathbf{b}_{012}$, $\mathbf{b}_{013}$
and $\mathbf{b}_{123}$ are open bubbles and $\mathbf{b}_{023}$ is closed.
1230
Let us come back to the elements of the graphs. An edge $e$ is called bridge
if $\mathcal{G}\vee e$ becomes disconnected. There is no self-loop at this
moment.
As in the case of ribbon graphs with flags, introducing tensor graphs with
flags requires a special treatment for faces. Using Definition 19, faces can
be open or closed connected components if they pass through external points or
not. An open face with color pair (01) is highlighted in red in Figure 22. The
sets of closed and open faces will be denoted by $\mathcal{F}_{{\rm int}}$ and
$\mathcal{F}_{{\rm ext}}$, respectively. Hence, for a rank $D$ colored graph
with flags, the set $\mathcal{F}$ of faces is the disjoint union
$\mathcal{F}_{{\rm int}}\cup\mathcal{F}_{{\rm ext}}$. The notion of closed or
open rank 3 colored tensor graph can be reported accordingly if
$\mathcal{F}_{{\rm ext}}=\emptyset$ or not, respectively.
A bubble is open or external if it contains open faces otherwise it is closed
or internal. Open and closed bubbles for a rank 3 colored tensor graph with
flags given in Figure 22 have been illustrated therein. The sets of closed and
open bubbles will be denoted by $\mathcal{B}_{{\rm int}}$ and
$\mathcal{B}_{{\rm ext}}$, respectively.
The notions of pinched graph $\widetilde{\mathcal{G}}$ and of boundary graph
is of great significance and we can address the pinching procedure colored
tensor graph with flags. First, pinching a tensor flag can be defined by the
identification of the external points of the flag. The pinched graph
$\widetilde{\mathcal{G}}(\widetilde{\mathcal{V}},\widetilde{\mathcal{E}},\widetilde{\mathfrak{f}}^{0})$
is a closed graph obtained from $\mathcal{G}$ after pinching of all of its
flags and still the following relations are valid
$\widetilde{\mathcal{V}}=\mathcal{V}\,,\qquad\widetilde{\mathcal{E}}=\mathcal{E}\,,\qquad|\widetilde{\mathfrak{f}}^{0}|=|\mathfrak{f}^{0}|\,.$
(71)
###### Definition 32 (Boundary tensor graph).
$\bullet$ The boundary graph
$\partial{\mathcal{G}}({\mathcal{V}}_{\partial},{\mathcal{E}}_{\partial})$ of
a rank $D$ colored tensor graph
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ is a graph obtained by
inserting a vertex with degree $D$ at each additional flags of $\mathcal{G}$
and taking the external faces of $\mathcal{G}$ as its edges. Thus,
$|{\mathcal{V}}_{\partial}|=|\mathfrak{f}^{0}|$ and
${\mathcal{E}}_{\partial}=\mathcal{F}_{{\rm ext}}$.
$\bullet$ The boundary of a closed rank $D$ colored tensor graph is empty.
Figure 23. Boundary graph (in dashed lines) of the graph of Figure 22.
11230
The boundary of a tensor graph without colors is obtained in the same way as
defined above since the colored feature of the graph is not used in the
definition. To the graph of Figure 22, we associate its boundary depicted in
Figure 23.
The dual of rank 3 colored tensor graph with flags is nothing but a 3D
simplicial complex with tetrahedra (as vertices) made with colored triangles
having a colored 2D boundary [12]. This colored boundary surface is obtained
by gluing colored triangles dually associated with colored flags.
Let us stress that the boundary graph $\partial{\mathcal{G}}$ of a rank $D$
(colored or not) tensor graph $\mathcal{G}$ encodes, in the dual picture, the
triangulation of the boundary of the $D$ simplicial complex dual to
$\mathcal{G}$. In the specific instance of 3D, $\partial{\mathcal{G}}$
represents a 2D triangulation of the boundary of a 3-simplicial complex dual
to $\mathcal{G}$. In the same vein that colors improve the topology of the
triangulated object associated with $\mathcal{G}$, a colored boundary graph
$\partial{\mathcal{G}}$ improves the regular feature of the boundary manifold
for that triangulated object. The construction of boundary graph
$\partial{\mathcal{G}}$ by Definition 32 should be nothing but a compact
representation of a more elaborate graph which can encode the triangulation of
a surface. We first need another type of colored structure for graphs in order
to discuss this.
###### Definition 33 (Ve-colored graphs).
$\bullet$ A $(D+1)$ ve-colored graph is a graph such that to each vertex is
assigned a color in $\\{0,\dots,D\\}$ and to each edge is assigned a pair
$(ab)$ of colors, $a,b$ in $\\{0,\dots,D\\}$, $a\neq b$, such that
\- either the edge is incident to vertices with the same color $c$, in this
case, $a$ (or $b$) equals $c$ and $b\in\\{0,\dots,D\\}\setminus\\{a\\}$
(respectively, $a\in\\{0,\dots,D\\}\setminus\\{b\\}$);
\- the edge is incident to vertices with different colors $a$ and $b$
and such that two adjacent edges cannot share the same couple of colors.
Some 4 ve-colored graphs are drawn in Figure 24.
Figure 24. Examples of 4 ve-colored graphs.
One notices that a $(D+1)$ ve-colored graph is a $D(D+1)/2$ colored graph for
edges equipped color pairs $ab$, $a<b$, chosen in
$\\{01,02,\dots,0D,12,\dots,D(D-1)\\}$. Thus in a ve-colored graph there is no
self-loops.
###### Definition 34 (Ve-colored tensor graph).
A rank $D$ ve-colored tensor graph $\mathcal{G}$ is a graph such that:
$\bullet$ $\mathcal{G}$ is $(D+2)$ ve-colored;
$\bullet$ $\mathcal{G}$ is a rank $D$ tensor graph.
The following statement holds.
###### Proposition 9 (Color structure of $\partial{\mathcal{G}}$ [12]).
The boundary graph
$\partial{\mathcal{G}}({\mathcal{V}}_{\partial},{\mathcal{E}}_{\partial})$ of
a rank $D$ colored tensor graph
$\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ is a rank $(D-1)$ ve-
colored tensor graph.
###### Proof.
The colors of vertices in $\partial{\mathcal{G}}$ inherits the colors of the
flags of $\mathcal{G}$. The edge coloring should clearly coincide with the
external face coloring defined by a couple $(ab)$. At one flag, the color
pairs of any external face never coincide. This makes the graph
$\partial{\mathcal{G}}$ as $(D+1)$ ve-colored. To construct the extra tensor
structure on $\partial{\mathcal{G}}$, one expands its vertices and edges into
stranded vertices with fixed coordination $D$ and edges with $D-1$ parallel
strands. The connecting pattern of the pre-flags of its vertices follows the
one of the complete graph $K_{D}$. The proposition follows.
∎
For instance in 3D, each vertex of $\partial{\mathcal{G}}$ is colored and is
three valent. Each edge of $\partial{\mathcal{G}}$ is a bicolored ribbon. See
Figure 25.
Figure 25. Rank 2 ve-colored stranded structure of the boundary graph of
Figure 23.
1112131011121310
###### Remark 1.
Importantly, any face of the boundary graph coincide with a connected
component of the boundary $\partial\mathbf{b}_{aa_{1}a_{2}}$ of some open
bubble $\mathbf{b}_{aa_{1}a_{2}}$ viewed as an open ribbon graph. Indeed, any
(closed) face of a boundary graph is made with strands coming from open faces
belonging necessarily to open bubbles. The boundary graph being closed, in
order to get all of its faces, one has first to pinch all external bubbles
$\mathbf{b}$, yielding $\widetilde{\mathbf{b}}$, and collect closed faces
coming uniquely from the pinching as performed in Proposition 4.
As an illustration of the above remark, consider the graph $\mathcal{G}$
Figure 22 with boundary $\partial{\mathcal{G}}$ drawn in Figure 23. Let us
pick the face $f_{012}$ of $\partial{\mathcal{G}}$ and consider the open
bubble $\mathbf{b}_{012}$ and its pinched version
$\widetilde{\mathbf{b}}_{012}$. The closed face $\widetilde{f}_{012}$ of
$\widetilde{\mathbf{b}}_{012}$ coincide with $f_{012}$. Furthermore, any such
closed face cannot belong to two different pinched bubbles by color exclusion
and by the fact that there are only two bubbles passing through an edge which
could use a given edge strand.
### 4.2. W-colored stranded graphs
It has been discussed in length that the contraction of an edge in a tensor
colored graph yields another type of graph for which neither the color
(Definition 29) nor the tensor axioms (Definition 23) apply [12]. In order to
define the topological polynomial and circumvent such an odd feature of tensor
graphs, some proposals have been made to redefine the notion of contraction of
an edge or redefine subgraphs for which the contraction applies [12, 18].
Using Definition 27, we can now contract any rank $D$ edge provided the fact
that we are working in the extended framework of stranded graphs. This
definition therefore applies to a colored tensor graph. Our task is to
characterize the new class of stranded graph obtained after an edge
contraction of a colored tensor graph. From now on edge contraction means
always soft edge contraction in the sense Definition 27.
The contraction of an edge $e$ in a rank $D$ colored tensor graph
$\mathcal{G}$ (possibly with flags) yields a rank $D$ stranded graph
$\mathcal{G}/e$ obtained from $\mathcal{G}$ by the procedure described in
Definition 27 and respecting the coloring. By respecting the coloring, we
understand that edges which are not contracted keep their color and all
strands keep their color pair. For a rank $D$ colored tensor graph, it is
straightforward to see that any edge contraction in $\mathcal{G}$ brings in
$\mathcal{G}/e$ a vertex possessing at least two pre-flags of the same color
(see an example in $D=3$ in Figure 26). This breaks the color axiom on
vertices since it allows that edges with the same color to be branched on the
same vertex. But still in $\mathcal{G}/e$ there is a color structure defined
in a sense weaker than Definition 29. Such a colored structure is mainly based
on the feature that any edge contraction preserves strands and their color
pair.
Figure 26. Contraction of an edge in a rank 3 colored tensor graph with
flags.
112301120
In the following, we first spell out the object of interest as defined by the
result of the contraction of a rank $D$ colored tensor graph. Then we find a
characterization of it.
###### Definition 35 (Rank $D$ w-colored graph I).
A rank $D$ weakly colored or w-colored graph is the equivalence class (up to
trivial discs) of a rank $D$ stranded graph obtained by successive edge
contractions of some rank $D$ colored tensor graph.
The equivalence class of any rank $D$ colored tensor graph up to trivial discs
is a rank $D$ w-colored graph. This holds by convention when no contraction is
performed. Another property is that several rank $D$ colored tensor graphs may
have the same contracted graph. Hence to one rank $D$ w-colored graph may
correspond several rank $D$ colored tensor graphs which could generate it. On
the other hand, the existence of a representative $\mathcal{G}$ of some rank
$D$ w-colored graph $\mathfrak{G}$ as being the contraction of some rank $D$
colored tensor graph does not imply necessarily that any other representative
results from a graph contraction. Given $\mathcal{G}$ the representative of
$\mathfrak{G}$ and $\mathcal{G}_{\text{color}}$ the rank $D$ colored tensor
graph which gives $\mathcal{G}$ after contraction, let us assume that this
contraction yields $\ell_{0}$ extra discs with particular bi-colored strands.
Any representative $\mathcal{G}^{\prime}$ of $\mathfrak{G}$,
$\mathcal{G}^{\prime}$ could come from the contraction of a rank $D$ colored
tensor graph
$\mathcal{G}^{\prime}_{\text{color}}\sim\mathcal{G}_{\text{color}}$ if the
number $k$ of discs of $\mathcal{G}^{\prime}$ is larger than $\ell_{0}$, in
which case
$\mathcal{G}^{\prime}_{\text{color}}=(\mathcal{G}_{\text{color}}\setminus
D_{\mathcal{G}_{\text{color}}})\cup D_{k-\ell_{0}}$, where
$D_{k-\ell_{0}}=D_{\mathcal{G}^{\prime}}\setminus D_{\ell_{0}}$ and where
$D_{\ell_{0}}$ is the set of additional discs coming uniquely from the
contraction of $\mathcal{G}_{\text{color}}/D_{\mathcal{G}_{\text{color}}}$.
One should pay attention of the type of color pairs that should occur on discs
of $\mathcal{G}^{\prime}$ to match with the color pairs of discs obtained
after contractions.
It is relevant to discuss in greater detail the type of (non trivial) vertices
that will generate any representative of any rank $D$ w-colored graphs. To
address this, we observe the following result from Lemma 3 in [4] and slightly
adapted to the present context.
###### Lemma 1 (Full contraction of a tensor graph).
$\bullet$ Contracting an edge in a rank $D$ (colored) tensor graph does not
change its boundary. $\bullet$ The contraction of all edges in arbitrary order
of a rank $D$ (colored) tensor graph $\mathcal{G}$ yields a graph
$\mathcal{G}^{0}$ defined by the boundary $\partial{\mathcal{G}}$ up to
additional discs.
###### Proof.
Contracting one edge in $\mathcal{G}$, all flags and all external faces remain
untouched, and no other external face can be created by such a move (we remove
all present inner faces and all remaining faces get shorter but are never
cut). Hence for the resulting stranded graph $\mathcal{G}/e$, one has
$\partial(\mathcal{G}/e)=\partial{\mathcal{G}}$. By iteration, contracting all
edges of $\mathcal{G}$ in arbitrary order, the resulting graph
$\mathcal{G}^{0}$ should contain $\partial{\mathcal{G}}$ as its boundary. But
contracting all edges of $\mathcal{G}$, one obtains a single vertex graph per
connected component plus trivial discs and no other closed faced left
otherwise it would mean that there exist still edges where these closed faces
pass. Hence necessarily $\mathcal{G}^{0}$ is nothing but a graph made only
with vertices (without edges) defined therefore by
$\partial{\mathcal{G}}^{0}=\partial{\mathcal{G}}$ up to trivial discs.
∎
Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a connected
rank $D$ colored tensor graph with a nonempty set of flags,
$\mathfrak{f}^{0}\neq\emptyset$. After a full contraction of all edges of
$\mathcal{G}$, Lemma 1 tells us that the end result is totally encoded in the
boundary graph $\partial{\mathcal{G}}$ up to some discs. This graph
$\mathcal{G}^{0}$ appears as one or a collection of rank $D$ stranded vertices
$v_{\partial{\mathcal{G}};k}$, $k$ parametrizing the number of connected
components of $\partial{\mathcal{G}}$. Note that if
$\mathfrak{f}^{0}=\emptyset$ then there is no boundary in $\mathcal{G}$ and
$v_{\partial{\mathcal{G}}}$ is empty.
Consider a stranded graph made with such vertices
$v_{\partial{\mathcal{G}}_{i};k_{i}}$ for a finite family of boundaries
$\partial{\mathcal{G}}_{i}$ of some connected rank $D$ colored tensor graphs
$\mathcal{G}_{i}$ with a nonempty set of flags, $i\in I$, $I$ finite. In order
to glue these vertices, one needs edges of rank $D$.
###### Proposition 10 (Rank $D$ w-colored stranded graphs II).
A rank $D$ stranded graph $\mathcal{G}$ is a representative of a rank $D$
w-colored graph $\mathfrak{G}$ if and only if it satisfies
1. (i”)
vertices $v_{i}$ of $\mathcal{G}$ attached with flags are graphs $V_{i}$
coming from the boundaries $\partial{\mathcal{G}}_{i}$ of a family of
connected rank $D$ colored tensor graph
$\mathcal{G}_{i}(\mathcal{V}_{i},\mathcal{E}_{i},\mathfrak{f}_{i}^{0})$ with
$\mathfrak{f}_{i}^{0}\neq\emptyset$. All pre-flags of $\mathcal{G}$ have a
fixed cardinal $D$;
2. (ii’)
the edges of $\mathcal{G}$ are of rank $D$ and with parallel strands;
3. (iii’)
to edges and flags of $\mathcal{G}$ are assigned a color
$\in\\{0,1,\dots,D\\}$ inherited from the pre-flags of the $v_{i}$’s.
###### Proof.
Let us consider the representative $\mathcal{G}$ of a rank $D$ w-colored graph
$\mathfrak{G}$ and $\mathcal{G}_{\text{color}}$ a rank $D$ colored tensor
graph the contraction of which yields $\mathcal{G}$. Let us pick another
representative $\mathcal{G}^{\prime}$ of $\mathfrak{G}$. Properties (ii”) and
(iii’) for edges and flags of $\mathcal{G}^{\prime}$ are trivially satisfied.
Next, observe that to any rank $D$ colored tensor graph corresponds a rank $D$
w-colored tensor graph which is the class of this same graph. Hence if
$\mathcal{G}$ contains the vertices $v_{0}$ and $\bar{v}_{0}$ (from bi-partite
property) of ordinary colored tensor graphs, we can define these two vertices
as the boundaries of the graphs defined by themselves attached with flags; the
cardinal of their pre-flags is fixed to $D$ by definition. It is obvious that
if $v_{0}$ and $\bar{v}_{0}$ (which are not trivial discs) are in
$\mathcal{G}$ they are also in $\mathcal{G}^{\prime}$ and so the same
construction applies to $\mathcal{G}^{\prime}$.
Any vertex $v$ in $\mathcal{G}$ which is not a disc and not of the form of
$v_{0}$ and $\bar{v}_{0}$ comes necessarily from some contractions of some
subset $\\{e_{l}\\}_{l\in I_{v}}$ of edges in $\mathcal{G}_{\text{color}}$.
Note that for all $v$, $\\{e_{l}\\}_{l\in I_{v}}$ should define a partition of
edges of $\mathcal{G}_{\text{color}}$ which do not belong to $\mathcal{G}$.
Consider the c-subgraph $\mathcal{G}_{v}$ in $\mathcal{G}_{\text{color}}$
defined by $\\{e_{l}\\}_{l\in I_{v}}$ and their end vertices. By Proposition
8, $\mathcal{G}_{v}$ is a rank $D$ colored tensor graph. Then, after the full
contraction of all the $e_{l}$’s, by Lemma 1, we obtain
$\partial{\mathcal{G}}_{v}$ up to some discs. $\partial{\mathcal{G}}_{v}$ also
coincides with the new vertex $v$ in $\mathcal{G}$ attached with flags.
Proceeding by iteration on all vertices of the representative $\mathcal{G}$,
we get a family of rank $D$ colored of c-subgraphs
$\mathcal{G}_{v}\subset\mathcal{G}_{\text{color}}$, the overall contraction of
which yields the vertices $v$ (plus flags) of $\mathcal{G}$ up to some trivial
discs. The cardinal of all present pre-flags is obviously fixed to $D$. Any
vertex $v$ in $\mathcal{G}$ which is not a disc is also present in
$\mathcal{G}^{\prime}$ therefore all vertices of $\mathcal{G}^{\prime}$ come
from boundaries of some families of colored graphs.
Now consider a rank $D$ stranded graph $\mathcal{G}$ satisfying (i”), (ii’)
and (iii’) and its equivalence class $\mathfrak{G}$. Then by replacing in
$\mathcal{G}$ all the vertices $v_{i}$ by the colored graphs $\mathcal{G}_{i}$
and respecting the coloring, we obtain a rank $D$ colored tensor graph
$\mathcal{G}_{\text{color}}$. Contracting in $\mathcal{G}_{\text{color}}$ all
edges in all $\mathcal{G}_{i}$’s yields again the $v_{i}$’s determined by
their boundary (Lemma 1), up to some discs. Thus the end result
$\mathcal{G}^{\prime}$ of these edge contractions is $\mathcal{G}$ up to some
discs therefore belongs to $\mathfrak{G}$. Hence $\mathfrak{G}$ is a rank $D$
w-colored graph.
∎
From now on, since the rank of edges, flags and pre-flags is fixed to $D$, we
omit to mention it. Thus, when no confusion may occur, rank $D$ colored tensor
graph are named colored graphs and rank $D$ w-colored graphs are named
w-colored graphs.
The key notion of interest here pertains again to the spanning c-subgraphs for
a w-colored graph. This notion has been already mentioned for equivalent
classes, the sole new point here is to add colors.
###### Proposition 11 (c-subgraph).
Let $\mathfrak{G}$ be a rank $D$ w-colored graph. The c-subgraphs of
$\mathfrak{G}$ admit a rank $D$ w-colored graph structure.
###### Proof.
Consider $\mathfrak{G}$ a w-colored graph and $\mathcal{G}$ the representative
in $\mathfrak{G}$ obtained after contraction of some colored graph
$\mathcal{G}_{\text{color}}$. Let $\mathfrak{g}\subset\mathfrak{G}$ be a
c-subgraph of $\mathfrak{G}$. For any such $\mathfrak{g}$, we shall show that
there exists a subgraph $g$ of $\mathcal{G}$ such that $g$ is a representative
of $\mathfrak{g}$. Then the proof will be complete because $g$ is exactly the
colored graph that we seek since it comes from a contraction of some subgraph
of $\mathcal{G}_{\text{color}}$ admitting itself a colored structure by
Proposition 8. Now, the existence of $g$ is guaranteed by the following.
Choose any representative $g^{\prime}$ of $\mathfrak{g}$ in any
$\mathcal{G}^{\prime}$, then $g^{\prime}$ is a c-subgraph in
$\mathcal{G}^{\prime}$. From $g^{\prime}$, we built
$\tilde{g}=g^{\prime}\setminus D_{g^{\prime}}$ which is a c-subgraph of
$\mathcal{G}$. Expanding $\tilde{g}\subset\mathcal{G}$ back in
$\mathcal{G}_{\text{color}}$, we get a colored c-subgraph of
$\mathcal{G}_{\text{color}}$ the contraction of which yields finally a
c-subgraph $g$ of $\mathcal{G}$. To see that $g\sim g^{\prime}$ is direct
since $g\sim\tilde{g}$.
∎
We give now few more properties of the vertices of representatives of rank $D$
w-colored graphs. First, there are infinitely many vertices for these graphs.
However, some constraints should be imposed on vertices for being the
boundaries of some rank $D$ colored tensor graphs.
###### Lemma 2.
Consider a rank $D$ w-colored graph with representative made with a vertex
with $N$ pre-flags. If $D$ is odd then $N$ should be always even.
###### Proof.
This simply holds from the fact that the number $ND$ of pre-flag points in the
vertex considered of the representative should be even in order to properly
form bows.
∎
For rank $D$ tensor graphs, the above lemma is obvious since $N$ is always
fixed to $D-1$. The point of the above proposition is to establish that a
vertex of rank $D$ w-colored graph can be of arbitrary degree but with even
parity in odd dimension.
The next lemma follows arguments given in Lemma 2.1 in [3] (as a dictionary,
external legs in quantum field theory corresponds to flags). For sake of
completeness, let us restate this proposition in the present language.
###### Lemma 3 (Lemma 2.1 in [3]).
Given a rank $D$ colored tensor graph with $n$ vertices and $N$ flags. If one
color is missing on its flags, then
\- $n$ is an even number
\- the number $N$ of flags is even and the colors of flags should appear in
pair.
###### Proof.
If one color, say $a$, is missing on the flags, it means that $a$ appear only
on edges. Each edge occupies two pre-flags of the same color and that one has
$n$ pre-flags of each color. Now since $a$ should only appear on edges, it
follows that the $n$ pre-flags of color $a$ should be all paired to give
edges. This proves the first point.
We henceforth know that $n$ is even. Assuming that there is an odd number of
flags possessing a particular color, this implies that the number of pre-flags
with that color which should be paired for making edges would be odd as well,
hence would be impossible. The second point follows.
∎
Lemma 3 entails other properties of the vertices of representatives of a
w-colored graph. Indeed, we infer from this statement that the vertices in a
rank $D$ w-colored graph can be only of some types. Some vertices attached
with flags have been depicted in Figure 27 using the compact representation.
Note that, in contrast with colored graphs, the compact notation becomes
ambiguous for general w-colored graphs. We use it here for displaying the
color feature of the different flags attached to the vertices. For
convenience, we also introduce the following notation: each pre-flag $f$ may
have a color $a$ or $\bar{a}$ which is reminiscent of the fact that $a$ and
$\bar{a}$ come from different types of vertices in some bipartite graph. We
call $a$ and $\bar{a}$ conjugate colors and call conjugate flags, the flags
which possess $a$ and $\bar{a}$ as colors.
When a vertex has uniquely one pair of conjugate colors $a$ and $\bar{a}$ for
all pre-flags, it is related to a so-called tensor unitary invariant object
[14]. We call it a 1-color vertex. When the vertex has 2 conjugate color
pairs, $2<D$, then it can be seen as a gluing of two 1-color vertices $v_{1}$
and $v_{2}$ after cutting some bows in $v_{1}$ and $v_{2}$ and then gluing
these (still respecting the coloring). Then one may figure out that the trace
invariant object above gets a weaker sense because the invariance is broken in
some sector (induced by the cut bows). For more colors, one proceeds by
iteration gluing two 1-color vertices and then gluing the result with another
1-color vertex with a different color. Performing such a gluing is possible up
to a number of $D-1$ colors. Though beyond the scope of this work, an
interesting task would be to count the number of such $2,3,\dots,D-1$
w-colored vertices for a fixed number $N$ of flags. For the last type of
vertices with all $D$ colors, it does not necessarily follow from the gluing
of 1-color vertices. The same counting will become highly non trivial, if
exists.
Figure 27. Some vertices of w-colored graphs.
$\bar{1}$10132$\bar{0}$$\bar{3}$$\bar{1}$$\bar{2}$$a$$\bar{a}$$b$$\bar{b}$$a$$\bar{a}$$b$$\bar{b}$$\bar{c}$$c$$a$$2$$0$$3$$1$$\bar{a}$$a$$\bar{3}$$\bar{0}$$\bar{1}$$\bar{2}$$\bar{b}$$b$$\bar{a}$$\bar{0}$$3$$\bar{1}$$\bar{2}$$\bar{3}$$2$$1$$0$
###### Lemma 4.
Let $e$ be a regular edge or a bridge with end vertices $v_{1}$ and $v_{2}$ of
any representative of a rank $3$ w-colored graph. Contracting $e$ in the
representative yields a new vertex $v$ which is connected.
###### Proof.
Consider $e$ which is not a self-loop, $e$ is common to all $\mathcal{G}$
representative of a rank 3 w-colored graph. Its strands join in $v_{1}$ and in
$v_{2}$ a collection of connected families of pre-flags. The initial vertices
$v_{1}$ and $v_{2}$ being connected, two different cases may occur:
(i) All strands of $e$ are connected to a unique pre-flag either in $v_{1}$ or
$v_{2}$. This implies that there are just two pre-flags in $v_{1}$ or in
$v_{2}$. Contracting $e$, the vertex $v$ obtained is connected.
(ii) Assume now that attached to each strand of $e$, we have $d$ families of
connected pre-flags in $v_{1}$ or in $v_{2}$, $2\leq d\leq 3$ (see an
illustration in Figure 28). This is enforced by the coloring which prevents
faces to pass through $e$ more than once. If contracting $e$ gives a
disconnected vertex then, in each $v_{1}$ and $v_{2}$, the connected families
of pre-flags do not have any bow between them888Note that this is only a
necessary condition for obtaining a disconnected vertex.. Notice that there is
necessarily at least one connected family of pre-flags joining $e$ by a unique
bow $b$. For such a family $\xi$, all pre-flags are fully connected via bows
and there is a unique pre-flag point, the one related to $e$ via $b$, not
linked to any other pre-flag in $\xi$. By an argument of parity, we aim at
showing that it is impossible.
Figure 28. Families $F_{1},F_{2}$ and $F_{3}$ of connected pre-flags attached
to each strand of a non self-loop edge.
$F_{1}$$F_{2}$$F_{3}$$e$
For simplicity, let us fix that $e$ is of color $0$ and that the strand $s$ of
color pair $(01)$ in $e$ touches the connected and separate family $\xi$ a
pre-flag $f$. Two cases may occur: either $f$ is of color $1$ or of color
$\bar{0}$.
(A) If $f$ of color $1$: Denote $N_{i}$ and $\bar{N}_{i}$ the number of pre-
flags in $\xi$ of color $i=0,1,2,3$ and
$\bar{i}=\bar{0},\bar{1},\bar{2},\bar{3}$, respectively. There are exactly
$N_{i}+\bar{N}_{i}$ pre-flag points labeled by color pairs $(ij)$, $i=0,2,3$,
$j\in\\{0,1,2,3\\}$ but $j\neq i$. These points come from pre-flags of colors
$0,2,3$ and $\bar{0},\bar{2},\bar{3}$. Meanwhile, there are
$(N_{1}-1)+\bar{N}_{1}$ pre-flag points of color pairs $(01)$, and
$N_{1}+\bar{N}_{1}$ of color pairs $(1j)$, $j\in{2,3}$. These last points come
uniquely from pre-flags with color $1$ and $\bar{1}$. Assuming that all pre-
flag points but one in $\xi$ should be totally connected by bows implies the
following equations:
$\displaystyle N_{0}+\bar{N}_{0}+(N_{1}-1)+\bar{N}_{1}=2k^{01}\,,$ (72)
$\displaystyle N_{0}+\bar{N}_{0}+N_{2}+\bar{N}_{2}=2k^{02}\,,$ (73)
$\displaystyle N_{0}+\bar{N}_{0}+N_{3}+\bar{N}_{3}=2k^{03}\,,$ (74)
$\displaystyle N_{1}+\bar{N}_{1}+N_{2}+\bar{N}_{2}=2k^{12}\,,$ (75)
$\displaystyle N_{1}+\bar{N}_{1}+N_{3}+\bar{N}_{3}=2k^{13}\,,$ (76)
$\displaystyle N_{2}+\bar{N}_{2}+N_{3}+\bar{N}_{3}=2k^{23}\,,$ (77)
for some $k^{ij}$ positive integers counting the number of bow with colors
$(ij)$. It is simple to show that this system leads to the inconsistent
equation
$\displaystyle 2(N_{0}+\bar{N}_{0})+2(k^{12}-k^{02})-1=2k^{01}\,.$ (78)
(B) If $f$ is of color $\bar{0}$: $N_{0}+(\bar{N}_{0}-1)$ becomes the number
of pre-flag points of color pairs $(01)$ in $\xi$ induced by pre-flags of
color $0$ and $\bar{0}$ and $N_{1}+\bar{N}_{1}$ is the number of pre-flags
points of colors $01$ induced by pre-flags of color $1$ and $\bar{1}$. This is
the same as switching the role of $1$ and $\bar{0}$ in the above case. The
above system holds again as constraints on the number of pre-flag points and
yields the same conclusion.
∎
The above lemma should clearly admit an extension to any dimension $D$ by
partitioning $D$ and analyzing the connection of the strands of $e$ and the
connected pre-flag families. We do not need however such a stronger and much
technical result for the following.
In contrast, the contraction of a self-loop may disconnect the vertex. In any
case, since self-loop graphs are terminal forms, a special treatment for them
is required.
The whole construction above is mainly made in order to maintain the
consistency of the $p$-bubble definition. For instance, edges are still
1-colored objects, (closed and open) faces are bicolored connected objects,
(closed and open) 3-bubbles are connected objects with 3 colors. However,
3-bubble are no longer built uniquely with three valent vertices. They can be
made with vertices with lower or greater valence as illustrated in Figure 29.
In any case, Definition 31 is still valid and we shall focus on this.
Figure 29. Two bubbles graphs $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ of
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$, respectively: $\mathbf{b}_{1}$ has a
vertex with valence 2 and $\mathbf{b}_{2}$ a vertex with valence 4.
$\mathcal{G}_{1}$$\mathbf{b}_{1}$$0$$\bar{0}$$\mathcal{G}_{2}$$\mathbf{b}_{2}$$0$$\bar{0}$
###### Lemma 5.
Let $\mathcal{G}$ be a representative of $\mathfrak{G}$ a rank $D$ w-colored
graph. $\bullet$ Cutting an edge $e$ in $\mathcal{G}$ yields a representative
$\mathcal{G}\vee e$ of another rank $D$ w-colored graph denoted
$\mathfrak{G}\vee e$. $\bullet$ Contracting an edge $e$ in $\mathcal{G}$
yields a representative $\mathcal{G}/e$ of another rank $D$ w-colored graph
denoted $\mathfrak{G}/e$.
###### Proof.
Let $e$ be an edge in $\mathcal{G}^{\prime}$ representative of some w-colored
graph $\mathfrak{G}$ and $\mathcal{G}$ the representative of $\mathfrak{G}$
resulting from a contraction of some colored graph
$\mathcal{G}_{\text{color}}$.
The cut graph $\mathcal{G}^{\prime}\vee e$ is equivalent to $\mathcal{G}\vee
e$. We use again here the expansion and contraction move: starting from
$\mathcal{G}\vee e$, we expand this graph in $\mathcal{G}_{\text{color}}\vee
e$ and contract it back to nothing but itself. One finds that $\mathcal{G}\vee
e$ is therefore a graph resulting from the contraction of a colored graph. Its
equivalence class is the w-colored graph desired.
Contracting now $e$ in $\mathcal{G}^{\prime}$, one gets a graph
$\mathcal{G}^{\prime}/e$ equivalent to $\mathcal{G}/e$. There exists a colored
graph $\mathcal{G}^{0}_{\text{color}}$ contracting to $\mathcal{G}/e$. This is
nothing but $\mathcal{G}_{\text{color}}$ on which after all contraction
yielding $\mathcal{G}$, one perform another contraction on $e$. Note that we
could have defined
$\mathcal{G}^{0}_{\text{color}}=\mathcal{G}_{\text{color}}/e$ and perform the
same series of contraction bringing initially $\mathcal{G}$. The end result is
$\mathcal{G}/e$. Thus the equivalence class of $\mathcal{G}/e$ is the
w-colored graph that we seek.
∎
We henceforth restrict the rank of graphs to $D=3$ and aim at studying the
universal invariant polynomial extending BR polynomial for this new category
of graphs. The extension for any $D$ should require more work but could be
almost inferred from the subsequent analysis. From now rank 3 w-colored graphs
will be called more simply w-colored graphs.
Mostly, we have studied non self-loop edges so far and their main properties
under contraction and cut can be guessed in any dimension. Self-loops are more
subtle and their behavior under contraction is much more involved. Restricting
to dimension 3 will simplify the analysis.
Self-loops find also a meaning in the present situation. A self-loop can be at
most a 3–inner edge. A 3–inner edge determines by itself a graph, see Figure
30O. If it is a 2–inner self-loop then $e$ should be of the form illustrated
in Figure 30A. Otherwise, $e$ may be a 1–inner self-loop (Figure 30B) or a
0–inner (Figure 30C).
Figure 30. $p$–inner self-loops: a 3–inner (O) and a 2–inner (A) self-loop
with their unique possible configuration; a 1–inner self-loop with possible
two sectors $v_{1}$ and $v_{2}$ (B) and a 0–inner self-loop with three
possible sectors $v_{1},v_{2}$ and $v_{3}$ (C).
OABC$v_{1}$$v_{2}$$v_{1}$$v_{2}$$v_{3}$
For a 2–inner self-loop $e$, the outer strand of $e$ should be in contact with
other flags or edges, otherwise $e$ would be a 3–inner edge. We represent the
rest of the vertex with possible edges and flags attached by a black diagram
in Figure 30A.
A 1–inner self-loop $e$ has two outer strands which should be in contact with
two sectors of the vertex, $v_{1}$ and $v_{2}$ as illustrated in Figure 30B,
such that $v_{1}$ and $v_{2}$ have at least two flags or edges. Remark that
nothing prevents $v_{1}$ and $v_{2}$ to have pre-flags in common or to be
linked by bows. In this situation, we shall say that these sectors coincide
otherwise we call them separate. A 2–inner self-loop $e$ has three such
sectors, $v_{1},v_{2}$ and $v_{3}$, which can pairwise coincide or can be
pairwise separate. Separate sectors should contain each at least two flags or
edges.
The notion of trivial self-loop can be addressed at this point for this
category of graphs. We call a self-loop $e$ trivial if it is a 3–inner or
2–inner self-loop or if it is a 1–inner or 0–inner self-loop such that there
is no edge between separate sectors $v_{i}$.
For a 3–inner self-loop, the contraction gives three trivial discs, see Figure
31O. For a trivial 2–inner self-loop the contraction is still straightforward
and yields Figure 31A. For a 1–inner self loop contraction (see Figure 31B),
one gets one extra disc and has two possible configurations: either the vertex
remains connected or it gets disconnected with two (possibly non trivial)
vertices in both situations. Concerning the contraction of a trivial 1–inner
self-loop, it is immediate that the vertex gets disconnected in two non
trivial vertices. For a 0–inner self-loop contraction (see Figure 31C), we
have no additional disc but three types of configurations with up to three
disconnected (and possibly non trivial) vertices. The contraction of a trivial
0–inner self-loop yields directly three disconnected and non trivial vertices.
Figure 31. $p$–inner self-loop contractions corresponding to O, A, B and C of
Figure 30, respectively.
OABC$v_{1}$$v_{2}$$v_{1}$$v_{2}$$v_{3}$
###### Lemma 6 (Rank 3 trivial self-loop contraction).
Let $\mathfrak{G}$ a w-colored graph and $\mathcal{G}$ any of its
representative with boundary $\partial{\mathcal{G}}$, $e$ be a trivial self-
loop of $\mathcal{G}$, and $\mathcal{G}^{\prime}=\mathcal{G}/e$ the result of
the contraction of $\mathcal{G}$ by $e$ with boundary denoted by
$\partial{\mathcal{G}}^{\prime}$. Let $k$ denote the number of connected
components of $\mathcal{G}$, $V$ its number of vertices, $E$ its number of
edges, $F_{{\rm int}}$ its number of faces, $B_{\rm int}$ its number of closed
bubbles and $B_{\rm ext}$ its number of open bubbles, $C_{\partial}$ the
number of connected component of $\partial{\mathcal{G}}$, $f=V_{\partial}$ its
number of flags, $E_{\partial}=F_{{\rm ext}}$ the number of edges and
$F_{\partial}$ the number of faces of $\partial{\mathcal{G}}$, and let
$k^{\prime},V^{\prime},E^{\prime},F_{{\rm
int}},C^{\prime}_{\partial},B^{\prime}_{{\rm int}},B^{\prime}_{\rm
ext},C^{\prime}_{\partial},f^{\prime},E^{\prime}_{\partial}$, and
$F^{\prime}_{\partial}$ denote the similar numbers for $\mathcal{G}^{\prime}$
and its boundary $\partial{\mathcal{G}}^{\prime}$.
$\bullet$ If $e$ is a 3–inner self-loop, then
$\displaystyle k^{\prime}=k+2\,,\quad V^{\prime}=V+2\,,\quad
E^{\prime}=E-1\,,\quad F^{\prime}_{{\rm int}}=F_{{\rm int}}\,,$ (79)
$\displaystyle C^{\prime}_{\partial}=C_{\partial}\,,\quad f^{\prime}=f\,,\quad
E^{\prime}_{\partial}=E_{\partial}\,,\quad
F^{\prime}_{\partial}=F_{\partial}\,,$ (80) $\displaystyle B^{\prime}_{\rm
int}=B_{{\rm int}}-3\,,\qquad B^{\prime}_{\rm ext}=B_{\rm ext}\,.$ (81)
$\bullet$ If $e$ is a 2–inner self-loop, then
$\displaystyle k^{\prime}=k+2\,,\quad V^{\prime}=V+2\,,\quad
E^{\prime}=E-1\,,\quad F^{\prime}_{{\rm int}}=F_{{\rm int}}\,,$ (82)
$\displaystyle C^{\prime}_{\partial}=C_{\partial}\,,\quad f^{\prime}=f\,,\quad
E^{\prime}_{\partial}=E_{\partial}\,,\quad
F^{\prime}_{\partial}=F_{\partial}\,,$ (83) $\displaystyle B^{\prime}_{\rm
int}=B_{{\rm int}}-1\,,\qquad B^{\prime}_{\rm ext}=B_{\rm ext}\,.$ (84)
$\bullet$ If $e$ is a trivial $p$–inner self-loop such that $p=0,1$, then
$\displaystyle k^{\prime}=k+2\,,\quad V^{\prime}=V+2\,,\quad
E^{\prime}=E-1\,,\quad F^{\prime}_{{\rm int}}=F_{{\rm int}}\,,$ (85)
$\displaystyle C^{\prime}_{\partial}=C_{\partial}\,,\quad f^{\prime}=f\,,\quad
E^{\prime}_{\partial}=E_{\partial}\,,\quad
F^{\prime}_{\partial}=F_{\partial}\,,$ (86) $\displaystyle B^{\prime}_{\rm
int}+B^{\prime}_{\rm ext}=B_{{\rm int}}+B_{\rm ext}+\alpha_{p}\,,$ (87)
where $\alpha_{p}=3-2p$.
###### Proof.
$\bullet$ The first situation of a 3–inner self-loop defines a particular
closed graph made of a single vertex with a unique self-loop. The contraction
destroys the vertex and the three closed bubbles and generate three discs. The
result follows.
For $p$–inner self loops, $p\leq 2$, there should be other flags or edges on
the same end vertex. The first lines of (84)-(87) should not cause any
difficulty using the definition of the contraction which preserve (open and
closed) strands. Let us focus on the number of bubbles of the graph.
$\bullet$ Consider that $e$ is 2–inner. One first should notice that there is
a bubble generated by the two inner faces of $e$ and that contracting $e$
destroys this closed bubble. Besides, the other (open or closed) bubbles which
passed through $e$ and which were defined by one of the two inner faces are
simply deformed. They loose one face (the inner) but remain in the contracted
graph. The second line of (84) follows.
Let us assume now that $e$ is trivial and 1– or 0–inner with color $a$.
Consider in the initial vertex $v$, the two or three separate sectors of pre-
flags $v_{1}$, $v_{2}$ and $v_{3}$ determined by the strands of $e$. We remind
that $v_{1}$, $v_{2}$ and $v_{3}$ should contain each at least two flags or
edges. Consider as well the three bubbles labeled by colors $(aa_{1}a_{2})$,
$(aa_{1}a_{3})$ and $(aa_{2}a_{3})$ passing through $e$.
$\bullet$ If $e$ is 1–inner, let us assume that the face $(aa_{3})$ is the one
which is inner, without loss of generality. The edge $e$ decomposes in three
ribbon edges each determined by a couple of pairs $(aa_{i};aa_{j})$, $i<j$,
such that the strand $(aa_{1})$ has an end point in the sector $v_{1}$ and
$(aa_{2})$ an end point in $v_{2}$. If $e$ is trivial and 1–inner, its
contraction yields two connected vertices plus one disc. Note that only the
bubbles $\mathbf{b}_{aa_{1}a_{3}}$ and $\mathbf{b}_{aa_{2}a_{3}}$ (in the
whole graph) include the face $(aa_{3})$. Contracting $e$, the bubbles
$\mathbf{b}_{aa_{1}a_{3}}$ and $\mathbf{b}_{aa_{2}a_{3}}$ just loose that face
but are still present since in $v_{1}$ and $v_{2}$, there exist still faces
where $\mathbf{b}_{aa_{1}a_{3}}$ and $\mathbf{b}_{aa_{2}a_{3}}$ passed before.
Now the last bubble $\mathbf{b}_{aa_{1}a_{2}}$ gets split in two parts: these
parts are disjoint bubbles with the same triple colors since there is no edge
in the graph relating them. These bubbles can be open or closed depending on
the nature of $\mathbf{b}_{aa_{1}a_{3}}$ in $v_{1}$ and $v_{2}$. Hence,
$B^{\prime}_{{\rm int}}+B^{\prime}_{\rm ext}=B_{{\rm int}}+B_{\rm ext}+1$.
$\bullet$ Now let us discuss the 0–inner self-loop situation with its three
separate sectors $v_{1}$, $v_{2}$ and $v_{3}$. The discussion is somehow
similar to the 1–inner case. The contraction of $e$ yields three connected
vertices. Using the similar routine explained above, one finds that each of
the bubbles $\mathbf{b}_{aa_{i}a_{j}}$, $i<j$ gets split into two parts after
the contraction. Each of these pairs of parts are not related at all by any
edge or bow. Thus each bubble produces two bubbles. The resulting bubbles may
be open or closed depending on the nature of $\mathbf{b}_{aa_{i}a_{j}}$ in
each sector. We have $B^{\prime}_{{\rm int}}+B^{\prime}_{\rm ext}=B_{{\rm
int}}+B_{\rm ext}+3$.
∎
It is clear that the contraction of a 2–inner self-loop also obeys (87) as
well for $p=2$. In fact (84) is a stronger result because it mainly separates
the variation of the numbers of closed and open bubbles. For a general 1–inner
self-loop contraction, it turns out that the number $B_{\rm int}+B_{\rm ext}$
may vary of 1 or not. Similarly, for a non necessarily trivial 0–inner self-
loop contraction, the same number of bubbles may vary from 0,1,2 up to 3.
###### Lemma 7 (Faces of a bridge).
$\bullet$ The faces passing through a bridge of any representative of a rank 3
w-colored graph are necessarily open. $\bullet$ These faces belong to the same
connected component of the boundary graph of the representative.
###### Proof.
A bridge $e$ with color $a$ has three strands of color pairs $(aa_{1})$,
$(aa_{2})$ and $(aa_{3})$ defining three faces passing through $e$.
The first point is quite clear: following a strand inside the graph, for
instance, on the left side of the bridge, before closing that face one has to
go on the other side of the bridge without passing by $e$ (we recall that by
the presence of strands with different color pair, we cannot pass twice by a
bridge). This is impossible by definition of a bridge. Hence each strand
should terminate inside of each side of the bridge hence are open.
Assume that one the faces, say $(aa_{1})$, does not belong to the same
connected component of the boundary of the others. Consider in its own
connected component all boundary vertices. Each of these vertices are of
valence 3 equipped with edges with different color pairs $(a_{i}a_{j})$. Since
$(aa_{1})$ should connect one of these boundary vertices, there exists at
least one vertex with half edges of color $a$ or $a_{1}$. Consider all
possible vertices where the strand $(aa_{1})$ could connect. There are $N_{a}$
and $N_{a_{1}}$ such vertices whereas there are $N_{a_{2}}$ and $N_{a_{3}}$
vertices which cannot be joined by $(aa_{1})$. Each of these vertices have
$N_{a_{\alpha}}$, $\alpha=\emptyset,1,2,3$, half edges of each color pair
$(a_{\alpha}a_{j})$, $j\neq\alpha$. Since all these half-edges should be fully
paired in order to have a connected component therefore each of the following
sums should give an even positive integer
$\displaystyle N_{a}+N_{a_{1}}+1\,,\qquad N_{a}+N_{a_{j}}\,,\qquad
N_{a_{1}}+N_{a_{j}}\,,\quad j=2,3\,.$ (88)
This system is clearly inconsistent.
∎
###### Lemma 8 (Cut/contraction of special edges).
Let $\mathcal{G}$ be a representative of $\mathfrak{G}$ a rank 3 w-colored
graph and $e$ an edge in $\mathcal{G}$. Then,
$\bullet$ if $e$ is a bridge, in the above notations, we have
$\displaystyle k(\mathcal{G}\vee e)=k(\mathcal{G}/e)+1,\;V(\mathcal{G}\vee
e)=V(\mathcal{G}/e)+1,\;E(\mathcal{G}\vee
e)=E(\mathcal{G}/e),\;f(\mathcal{G}\vee e)=f(\mathcal{G}/e)+2,$ (89) (90)
$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G}/e)\,,\quad B_{{\rm int}}(\mathcal{G}\vee e)=B_{{\rm
int}}(\mathcal{G}/e)\,,$ (91) $\displaystyle C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G}/e)+1\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G}/e)+3\,,\quad F_{\partial}(\mathcal{G}\vee
e)=F_{\partial}(\mathcal{G}/e)+3\,,$ (92) $\displaystyle B_{{\rm
ext}}(\mathcal{G}\vee e)=B_{{\rm ext}}(\mathcal{G}/e)+3\,;$ (93)
$\bullet$ if $e$ is a trivial $p$–inner self-loop, $p=0,1,2$, we have
$\displaystyle k(\mathcal{G}\vee e)=k(\mathcal{G}/e)-2,\;V(\mathcal{G}\vee
e)=V(\mathcal{G}/e)-2,\;E(\mathcal{G}\vee
e)=E(\mathcal{G}/e),\;f(\mathcal{G}\vee e)=f(\mathcal{G}/e)+2,$ (94) (95)
$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)+C_{\partial}(\mathcal{G}\vee
e)=F_{{\rm int}}(\mathcal{G}/e)+C_{\partial}(\mathcal{G}/e)-2\,,\quad
E_{\partial}(\mathcal{G}\vee e)=E_{\partial}(\mathcal{G}/e)+3\,,$ (96)
$\displaystyle B_{{\rm int}}(\mathcal{G}\vee e)+B_{{\rm ext}}(\mathcal{G}\vee
e)=B_{{\rm int}}(\mathcal{G}/e)+B_{{\rm ext}}(\mathcal{G}/e)-(3-2p)\,.$ (97)
Moreover, given $\partial\mathcal{G}$ the boundary of $\mathcal{G}$ and a
bridge or trivial $p$–inner self-loop $e$, $p=0,1,2,3$,
$2C_{\partial}(\mathcal{G}\vee e)-\chi(\partial(\mathcal{G}\vee
e))=2C_{\partial}(\mathcal{G})-\chi(\mathcal{G})=2C_{\partial}(\mathcal{G}/e)-\chi(\partial(\mathcal{G}/e))\,,$
(98)
where $\chi(\partial{\mathcal{G}})$ denote the Euler characteristics of the
boundary of $\mathcal{G}$.
###### Proof.
We start by the bridge case. The equations in (90) are easily found. Let us
focus on (91). By Lemma 7, we know that necessarily the faces passing through
$e$ are open. All closed faces on each side of the bridge are conserved after
cutting $e$. The same are still conserved after edge contraction. Hence
$F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm int}}(\mathcal{G}/e)$ and $B_{{\rm
int}}(\mathcal{G}\vee e)=B_{{\rm int}}(\mathcal{G}/e)$. We now prove (92).
Still by the second point of Lemma 7, the three external faces belong to the
same boundary component. After cutting $e$, this unique component yields two
boundary components. It is direct to get $C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G}/e)+1$, $E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G}/e)+3$ (the cut of $e$ divides each external face
into two different external strands) and $F_{\partial}(\mathcal{G}\vee
e)=F_{\partial}(\mathcal{G}/e)+3$ because
$C_{\partial}(\mathcal{G}/e)=C_{\partial}(\mathcal{G})$,
$E_{\partial}(\mathcal{G}/e)=E_{\partial}(\mathcal{G})$ and
$F_{\partial}(\mathcal{G}/e)=F_{\partial}(\mathcal{G})$ which are immediate
from Lemma 1. Concerning the number of external bubbles, there are three
bubbles in $\mathcal{G}$ passing through the bridge. Each of these is
associated with two color pairs $(aa_{i};aa_{j})$, $i<j$. These bubbles are
clearly in $\mathcal{G}/e$ and cutting the bridge each of these bubble splits
in two. This yields (93).
We focus now on a trivial $p$–inner self-loop $e$. The relations (95) can be
determined without difficulty. We concentrate on the rest of the equations.
Consider the faces $f_{i}$ in $e$ made with outer strands. For $p=0,1,2$, we
have $f_{i}$, $1\leq i\leq 3-p$. These faces can be open or closed. We do a
case by case study according to the number of open or closed faces among the
$f_{i}$’s.
\- Assume that $3-p$ of $f_{i}$’s are closed. Cutting $e$ entails
$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G})-3\,,\quad C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})+1\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3\,,$ (99) $\displaystyle B_{{\rm
int}}(\mathcal{G}\vee e)+B_{{\rm ext}}(\mathcal{G}\vee e)=B_{{\rm
int}}(\mathcal{G})+B_{{\rm ext}}(\mathcal{G})\,,$ (100) $\displaystyle
F_{\partial}(\mathcal{G}\vee e)=F_{\partial}(\mathcal{G})+3\,.$ (101)
Note that in this situation only the variation of the total number of bubbles
can be known.
\- Assume that $3-p-1$ of the $f_{i}$’s are closed and one is open. Cutting
$e$ gives
$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G})-2\,,\quad C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3\,,$ (102) $\displaystyle B_{{\rm
int}}(\mathcal{G}\vee e)=B_{{\rm int}}(\mathcal{G})-1\,,\quad B_{{\rm
ext}}(\mathcal{G}\vee e)=B_{{\rm ext}}(\mathcal{G})+1\,,$ (103) $\displaystyle
F_{\partial}(\mathcal{G}\vee e)=F_{\partial}(\mathcal{G})+1\,.$ (104)
\- Assume that $3-p-2$ of the $f_{i}$’s are closed and two are open. Cutting
$e$ gives
$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G})-1\,,\quad C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})-1\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3\,,$ (105) $\displaystyle B_{{\rm
int}}(\mathcal{G}\vee e)=B_{{\rm int}}(\mathcal{G})\,,\quad B_{{\rm
ext}}(\mathcal{G}\vee e)=B_{{\rm ext}}(\mathcal{G})\,,$ (106) $\displaystyle
F_{\partial}(\mathcal{G}\vee e)=F_{\partial}(\mathcal{G})-1\,.$ (107)
Note that this case does not apply for $p=2$.
\- For $p=0$ an additional situation applies: assume that all three $f_{i}$’s
are open. Cutting $e$ gives
$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G})\,,\quad C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})-2\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3\,,$ (108) $\displaystyle B_{{\rm
int}}(\mathcal{G}\vee e)=B_{{\rm int}}(\mathcal{G})\,,\quad B_{{\rm
ext}}(\mathcal{G}\vee e)=B_{{\rm ext}}(\mathcal{G})\,,$ (109) $\displaystyle
F_{\partial}(\mathcal{G}\vee e)=F_{\partial}(\mathcal{G})-3\,.$ (110)
Lemma 6 relates the same numbers for $\mathcal{G}/e$ and $\mathcal{G}$ from
which one is able to prove (96) and (97).
Last, we prove the relation (98) on the Euler characteristics of the different
boundaries. We first note that, from (90), (92), (95) and (96), for any
special (bridge or trivial $p$–inner, $p=0,1,2$) edge,
$f(\mathcal{G}\vee e)=f(\mathcal{G})+2=f(\mathcal{G}/e)+2\,,\qquad
E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3=E_{\partial}(\mathcal{G}/e)+3\,.$ (111)
For the bridge case, (98) follows from the relations
$F_{\partial}(\mathcal{G}\vee
e)=F_{\partial}(\mathcal{G})+3=F_{\partial}(\mathcal{G}/e)+3$ and
$C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})+1=C_{\partial}(\mathcal{G}/e)+1$ in (92). The
result holds also for trivial $0,1,2$–inner self-loops, after the case by case
study giving (100)–(110). Last, for the 3–inner self-loop, in addition to
(111) which still holds, the following relations are valid
$\displaystyle C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})+1=C_{\partial}(\mathcal{G}/e)+1\,,\quad
F_{\partial}(\mathcal{G}\vee
e)=F_{\partial}(\mathcal{G})+3=F_{\partial}(\mathcal{G}/e)+3\,,$ (112)
and allow us to conclude.
∎
### 4.3. Polynomial invariant for $3D$ w-colored tensor graphs
We shall define first an invariant for rank 3 w-colored graphs, check its
consistency and then state our main result.
###### Definition 36 (Topological invariant for rank 3 w-colored graph).
Let $\mathfrak{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a rank 3
w-colored stranded graph with flags. The generalized topological invariant
associated with $\mathfrak{G}$ is given by the following function associated
with any of its representatives $\mathcal{G}$ (using the above notations)
$\displaystyle\mathfrak{T}_{\mathfrak{G}}(x,y,z,s,w,q,t)=\mathfrak{T}_{\mathcal{G};a}(x,y,z,s,w,q,t)=$
(113) $\displaystyle\sum_{A\Subset\mathcal{G}}(x-1)^{{\rm
r}\,(\mathcal{G})-{\rm r}\,(A)}(y-1)^{n(A)}z^{5k(A)-[3(V-E(A))+2(F_{{\rm
int}}(A)-B_{{\rm int}}(A)-B_{{\rm ext}}(A))]}$ (114)
$\displaystyle\qquad\times\quad
s^{C_{\partial}(A)}\,w^{F_{\partial}(A)}q^{E_{\partial}(A)}t^{f(A)}\,.$
Our initial task is to show that the above definition does not actually depend
on the representative $\mathcal{G}$ of $\mathfrak{G}$. Given any other
representative $\mathcal{G}^{\prime}$ of $\mathfrak{G}$, there is a one to one
map between spanning c-subgraphs of $\mathcal{G}$ and those of
$\mathcal{G}^{\prime}$. This can be achieved by mapping any
$A\Subset\mathcal{G}$ onto $A^{\prime}\Subset\mathcal{G}^{\prime}$ such that
$A^{\prime}=(A\setminus D_{\mathcal{G}})\cup D_{\mathcal{G}^{\prime}}$ which
can be easily inverted. Now, one checks that ${\rm r}\,(\mathcal{G})$, ${\rm
r}\,(A)$, $n(A)$ and $5k(A)-(3V+2F_{\rm int}(A))$ do not depend of the
representative. The remaining exponents $B_{\rm int}(A)$, $B_{\rm ext}(A)$,
$C_{\partial}(A)$, $f(A)$, $E_{\partial}(A)$ and $F_{\partial}(A)$ depend only
on $A\setminus D_{\mathcal{G}}=A^{\prime}\setminus D_{\mathcal{G}^{\prime}}$
hence do not depend on the representative.
One notices that the above consistency checking establishes that only the
class of the spanning c-subgraphs of some representative matters in the
definition of $\mathfrak{T}_{\mathfrak{G}}$. Thus, we could have introduced a
state sum on $\mathfrak{g}\Subset\mathfrak{G}$ in order to define this
invariant directly from the equivalent classes point of view using Proposition
11. We will keep however the more explicit formulation in terms of the
spanning c-subgraphs of representatives.
Another crucial point is to show that the exponent of $z$ in (113) is always a
non negative integer.
###### Proposition 12.
Let $\mathcal{G}$ be any representative of $\mathfrak{G}$ a rank 3 w-colored
graph. Then
$\zeta(\mathcal{G})=3(E(\mathcal{G})-V(\mathcal{G}))+2[B_{{\rm
int}}(\mathcal{G})+B_{{\rm ext}}(\mathcal{G})-F_{{\rm int}}(\mathcal{G})]\geq
0\,.$ (115)
###### Proof.
Let us consider the set of closed and open bubbles $\mathcal{B}_{{\rm int}}$
and $\mathcal{B}_{{\rm ext}}$ of $\mathcal{G}$, respectively (in this proof,
we drop the dependence in the graph $\mathcal{G}$ in all quantities). Let
$B_{{\rm int}}$ and $B_{{\rm ext}}$ be their respective cardinal. Each open or
closed bubble $\mathbf{b}$ is an open or a closed ribbon graph with
$V_{\mathbf{b}}$ number of vertices, $E_{\mathbf{b}}$ number of edges,
$F_{{\rm int};\mathbf{b}}$ number of closed faces and
$C_{\partial}(\mathbf{b})$ number of closed faces obtained uniquely from open
strands after pinching (coinciding with the number of connected component of
the boundary of the open bubble $\mathbf{b}$).
Any $\mathbf{b}_{i}\in\mathcal{B}_{{\rm int}}$ being a closed ribbon graph,
its Euler characteristics writes
$2-\kappa_{\mathbf{b}_{i}}=V_{\mathbf{b}_{i}}-E_{\mathbf{b}_{i}}+F_{{\rm
int};\mathbf{b}_{i}}\,,$ (116)
where $\kappa_{\mathbf{b}_{i}}$ refers to the genus of $\mathbf{b}_{i}$ or
twice its genus if $\mathbf{b}_{i}$ is oriented. Summing over all internal
bubbles, we get
$2B_{{\rm int}}-\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\kappa_{\mathbf{b}_{i}}=\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\left[V_{\mathbf{b}_{i}}-E_{\mathbf{b}_{i}}+F_{{\rm
int};\mathbf{b}_{i}}\right].$ (117)
Using the colors, one observes that each edge of $\mathcal{G}$ splits into
three ribbon edges belonging either to an open or a closed bubble, and each
internal face of $\mathcal{G}$ belongs to two bubbles which might be open or
closed. Thus we have
$\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}E_{\mathbf{b}_{i}}+\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}E_{\mathbf{b}_{x}}=3E\,,\qquad\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}F_{{\rm int};\mathbf{b}_{i}}+\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}F_{{\rm int};\mathbf{b}_{x}}=2F_{{\rm int}}\,.$ (118)
In addition, each vertex of the graph can be decomposed, at least, in three
vertices (3 vertices is the minimum given by the simplest vertex of the form
$\mathcal{G}_{1}$ in Figure 29) which could belong to an open or closed
bubble, we have
$\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}V_{\mathbf{b}_{i}}+\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}V_{\mathbf{b}_{x}}\geq 3V\,.$ (119)
Combining (118) and (119), we re-write (117) as
$3V-3E+2F_{{\rm int}}-2B_{{\rm int}}-\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}\left[V_{\mathbf{b}_{x}}-E_{\mathbf{b}_{x}}+F_{{\rm
int};\mathbf{b}_{x}}\right]\leq-\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\kappa_{\mathbf{b}_{i}}\,.$ (120)
We complete the last sum involving $\mathcal{B}_{{\rm ext}}$ by adding
$C_{\partial}(\mathbf{b}_{x})$ in order to get
$\displaystyle\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}\left[V_{\mathbf{b}_{x}}-E_{\mathbf{b}_{x}}+F_{{\rm
int};\mathbf{b}_{x}}+C_{\partial}(\mathbf{b}_{x})\right]=\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}(2-\kappa_{\widetilde{\mathbf{b}}_{x}})\,,$ (121)
which, substituted in (120), leads us to
$\displaystyle 3V-3E+2F_{{\rm int}}-2B_{{\rm int}}-2B_{{\rm
ext}}\leq-\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\kappa_{\mathbf{b}_{i}}-\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}\big{(}C_{\partial}(\mathbf{b}_{x})+\kappa_{\widetilde{\mathbf{b}}_{x}}\big{)}$
(122)
from which the lemma results. ∎
In fact, $\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}C_{\partial}(\mathbf{b}_{x})=F_{\partial}(\mathcal{G})$ is the number of
faces of the boundary graph $\mathcal{G}$ (see Remark 1). The above bound can
be refined since $F_{\partial}(\mathcal{G})\geq B_{{\rm ext}}(\mathcal{G})$
which merely follows from the fact that each
$\mathbf{b}_{x}\in\mathcal{B}_{{\rm ext}}$ has at least a boundary
contributing to $F_{\partial}(\mathcal{G})$ such that
$\displaystyle B_{{\rm
ext}}(\mathcal{G})\leq\sum_{\mathbf{b}_{x}}C_{\partial}(\mathbf{b}_{x})=F_{\partial}(\mathcal{G})\,.$
(123)
Thus we also have
$3V-3E+2F_{{\rm int}}-2B_{{\rm int}}-B_{{\rm
ext}}\leq-\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\kappa_{\mathbf{b}_{i}}-\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}\kappa_{\widetilde{\mathbf{b}}_{x}}\,.$ (124)
This may lead as well to yet another well defined invariant. However, we will
not use this relation in the following due to some rich relations that
$-\zeta(A)\geq 0$ entails. In particular, we have other positive combinations.
###### Proposition 13.
Let $\mathcal{G}$ be any representative of $\mathfrak{G}$ a rank 3 w-colored
graph. Then
$\displaystyle\zeta^{\prime}(\mathcal{G})=3[E(\mathcal{G})-V(\mathcal{G})]+2[B_{{\rm
int}}(\mathcal{G})+B_{{\rm ext}}(\mathcal{G})-F_{{\rm
int}}(\mathcal{G})-C_{\partial}(\mathcal{G})]\geq 0\,,$ (125)
$\displaystyle\zeta^{\prime\prime}(\mathcal{G})=3[E(\mathcal{G})-V(\mathcal{G})-C_{\partial}(\mathcal{G})]+2[B_{{\rm
int}}(\mathcal{G})+B_{{\rm ext}}(\mathcal{G})-F_{{\rm int}}(\mathcal{G})]\geq
0\,.$ (126)
###### Proof.
By the color prescription, each connected component of the boundary of
$\mathcal{G}$ has at least three faces. Therefore
$\displaystyle 3C_{\partial}\leq F_{\partial}\,.$ (127)
Then we also have $-F_{\partial}+2C_{\partial}\leq 0$. The proposition follows
from (122) and the fact that
$F_{\partial}=\sum_{\mathbf{b}_{x}}C_{\partial}(\mathbf{b}_{x})$ in the proof
of Proposition 12.
∎
Proposition 12 ensures $\zeta(A)\geq 0$, for any $A\Subset\mathcal{G}$, hence
the following statement.
###### Proposition 14 (Polynomial invariant).
$\mathfrak{T}_{\mathfrak{G}}=\mathfrak{T}_{\mathcal{G}}$ is a polynomial.
The quantity $V-E(A)+F_{{\rm int}}(A)-B_{{\rm int}}(A)$ is nothing but the
Euler characteristics for a closed colored graph $A$ understood as a cellular
complex [10]. This quantity always proves to be bounded by
$-\sum_{\mathbf{b}_{i}}\kappa_{\mathbf{b}_{i}}$. If the graph is not closed,
the same cellular complex has a boundary bearing itself a cellular
decomposition. The quantity $-\zeta(A)$ represents, in the present specific
instance, a weighted notion for Euler characteristics of the cellular complex
corresponding to $A$ which also take into account the boundary due to $B_{{\rm
ext}}$. The quantity $5k(A)-\zeta(A)$ will be however the one of interest in
the following.
Let us call $\mathcal{G}^{0}$ a representative graph made only with a finite
family of vertices with flags and without any edges possibly with discs. Then
$E(\mathcal{G}^{0})=B_{{\rm int}}(\mathcal{G}^{0})=0$, $F_{{\rm
int}}(\mathcal{G}^{0})=D$, $D$ being the number of trivial discs in
$\mathcal{G}^{0}$, $k(\mathcal{G}^{0})=V(\mathcal{G}^{0})$,
$k(\mathcal{G}^{0})-D=C_{\partial}(\mathcal{G}^{0})\geq 0$, we can check the
consistency of (113) as the following makes still a sense:
$\mathfrak{T}_{\mathcal{G}^{0}}(x,y,z,s,w,q,t)=z^{2[k(\mathcal{G}^{0})-D+B_{{\rm
ext}}(\mathcal{G}^{0})]}s^{k(\mathcal{G}^{0})-D}w^{F_{\partial}(\mathcal{G}^{0})}q^{E_{\partial}(\mathcal{G}^{0})}\,t^{|\mathfrak{f}^{0}(\mathcal{G}^{0})|}.$
(128)
It may exist several possible reductions of the above polynomial. We will
however focus on the most prominent ones given by
$\displaystyle\mathfrak{T}_{\mathcal{G}}(x,y,z,z^{-2},w,q,t)=\mathfrak{T}^{\prime}_{\mathcal{G}}(x,y,z,w,q,t)\,,\quad\mathfrak{T}_{\mathcal{G}}(x,y,z,z^{-2}s^{2},s^{-1},s,s^{-1})=\mathfrak{T}^{\prime\prime}_{\mathcal{G}}(x,y,z,s)\,,$
(129)
$\displaystyle\mathfrak{T}_{\mathcal{G}}(x,y,z,z^{2}z^{-2},z^{-1},z,z^{-1})=\mathfrak{T}^{\prime\prime\prime}_{\mathcal{G}}(x,y,z)\,.$
(130)
Proposition 13 ensures that $\mathfrak{T}^{\prime}_{\mathcal{G}}$ is a
polynomial. Meanwhile, $\mathfrak{T}^{\prime\prime}_{\mathcal{G}}$ is also a
polynomial with exponent of $s$ the Euler characteristics of the boundary
$\partial{\mathcal{G}}$.
$\mathfrak{T}^{\prime\prime\prime}_{\mathcal{G}}(x,y,z)$ combine both
invariants in a single exponent. These polynomials will be relevant in the
next analysis. Note that we could have introduced another polynomial between
$\mathfrak{T}^{\prime}$ and $\mathfrak{T}^{\prime\prime}$ which expresses as
$\mathfrak{T}_{\mathcal{G}}(x,y,z,s^{2},s^{-1},s,s^{-1})=\mathfrak{T}^{0}_{\mathcal{G}}(x,y,z,s)$.
But this latter turns out to satisfy the same properties as $\mathfrak{T}$ and
thus does not provide anything new.
We are now in position to prove our main theorem.
###### Theorem 4 (Contraction/cut rule for w-colored graphs).
Let $\mathfrak{G}$ be a rank 3 w-colored graph with flags. Then, for a regular
edge $e$ of any of the representative $\mathcal{G}$ of $\mathfrak{G}$, we have
$\mathfrak{T}_{\mathcal{G}}=\mathfrak{T}_{\mathcal{G}\vee
e}+\mathfrak{T}_{\mathcal{G}/e}\,,$ (132)
for a bridge $e$, we have $\mathfrak{T}_{\mathcal{G}\vee
e}=z^{8}s(wq)^{3}t^{2}\mathfrak{T}_{\mathcal{G}/e}$ and
$\mathfrak{T}_{\mathcal{G}}=[(x-1)z^{8}s(wq)^{3}t^{2}+1]\mathfrak{T}_{\mathcal{G}/e}\,;$
(133)
for a trivial p-inner self-loop $e$, $p=0,1,2$, we have
$\mathfrak{T}_{\mathcal{G}}=\mathfrak{T}_{\mathcal{G}\vee
e}+(y-1)z^{4p-7}\,\mathfrak{T}_{\mathcal{G}/e}\,.$ (134)
###### Proof.
Let $\mathcal{G}$ be any representative of the class $\mathfrak{G}$. Same
initial remarks as stated in the proof of Theorem 3 hold here for
$\mathcal{G}$. Our main concern is the change in the number of internal and
external bubbles and the independence of the final result on the
representative $\mathcal{G}$.
We concentrate first on (132). Consider an ordinary edge $e$ of $\mathcal{G}$,
the set of spanning c-subgraphs which do not contain $e$ being the same as the
set of spanning c-subgraphs of $\mathcal{G}\vee e$, the number of open and
closed bubbles on each subgraph is the same, it is direct to get
$\sum_{A\Subset\mathcal{G};e\notin A}(\cdot)=\mathfrak{T}_{\mathcal{G}\vee
e}$. Observe that the class of $\mathcal{G}\vee e$ defines $\mathfrak{G}\vee
e$ from Lemma 5 and we know that all exponents in
$\mathfrak{T}_{\tilde{\mathcal{G}}}$ do not depend on the representative
$\tilde{\mathcal{G}}$ of the class of $\tilde{\mathfrak{G}}$. Hence
$\mathfrak{T}_{\mathcal{G}\vee e}=\mathfrak{T}_{\mathfrak{G}\vee e}$ does not
depend on the representative.
Let us focus on the second term of (132). Consider $e$, its end vertices
$v_{1}$ and $v_{2}$ and its 3 strands with color pairs $(aa_{1})$, $(aa_{2})$
and $(aa_{3})$ and consider the set bubbles in $\mathfrak{G}$. Some bubbles do
not use any strand of $e$ and three bubbles can be formed using these strands
(these are of colors $(aa_{i}a_{j})$, $i<j$). Contracting now $e$, the vertex
obtained is connected by Lemma 4. The former three strands $e$ are clearly
preserved after the contraction. The bubbles not passing through $e$ are not
affected at all by the procedure. The three bubbles passing through $e$ are
also preserved since the contraction do not delete faces or strands. The faces
passing through $e$ getting only shortened, the net result from the point of
view of the bubbles passing through $e$ is simply an ordinary ribbon edge
contraction in the sense of ribbon graphs. From this results
$\sum_{A\Subset\mathcal{G};e\in A}(\cdot)=\mathfrak{T}_{\mathcal{G}/e}$. By
Lemma 5, we define $\mathfrak{G}/e$ and conclude that the polynomial
$\mathfrak{T}_{\mathfrak{G}/e}$ does not depend on the representative
$\mathcal{G}/e$ of $\mathfrak{G}/e$.
Let us focus now on the bridge case and (133). Cutting a bridge yields, as in
the ordinary case, from the sum $\sum_{A\Subset\mathcal{G};e\notin A}$ the
product $(x-1)\mathfrak{T}_{\mathcal{G}\vee e}$. The second sum remains as it
is using the mapping between $\\{A\Subset\mathcal{G};e\in A\\}$ and
$\\{A\Subset\mathcal{G}/e\\}$ and provided the fact that there is no change in
the vertex contracted which is ensured by Lemma 4. The last stage relates
$\mathfrak{T}_{\mathcal{G}\vee e}$ and $\mathfrak{T}_{\mathcal{G}/e}$. This
can be achieved by using the bijection between $A\Subset\mathcal{G}\vee e$ and
$A^{\prime}\Subset\mathcal{G}/e$ where each $A$ and $A^{\prime}$ are both
uniquely related to some $A_{0}\Subset\mathcal{G}$ as $A=A_{0}\vee e$ and
$A^{\prime}=A_{0}/e$. Using Lemma 8, the relation (133) follows. Then, Lemma 5
allows us to define ($\mathfrak{G}\vee e$ and) $\mathfrak{G}/e$ and to
conclude that the resulting polynomials ($\mathfrak{T}_{\mathcal{G}\vee e}$
and) $\mathfrak{T}_{\mathcal{G}/e}$ are independent on the representative
($\mathcal{G}\vee e$ and) $\mathcal{G}/e$.
Next, we discuss the trivial $p$–inner self-loop case and prove (134). The
property that the sum $\sum_{A\Subset\mathcal{G};e\notin
A}(\cdot)=\mathfrak{T}_{\mathcal{G}\vee e}$ should be direct. The second sum
is now studied.
The question is whether or not $e$ being a trivial $p$–inner self-loop in
$\mathcal{G}$ remains as such in $A$. The answer for that question is yes
because $A$ contains the end vertex of $e$. We may cut some edges in each
sectors $v_{i}$ (see Figure 30) for defining $A$ but the resulting sectors are
still totally separated in $A$.
Contracting a trivial $p$–inner self-loop generates $p$ discs and $3-p$ non
trivial vertices. Now, from Lemma 6, we know how evolve all numbers of
components in the graph: the nullity is again $n(A)=n(A^{\prime})+1$, and the
exponent of $z$ becomes
$\displaystyle 5k(A)-(3(V(\mathcal{G})-E(A))+2(F_{\rm int}(A)-B_{\rm
int}(A)-B_{{\rm ext}}(A))=$ (135) $\displaystyle
5(k(A^{\prime})-2)-\Big{[}3[V(\mathcal{G}/e)-2)-(E(A^{\prime})+1)]+2(F_{\rm
int}(A^{\prime})-B_{\rm int}(A^{\prime})-B_{{\rm
ext}}(A^{\prime})+\alpha_{p})\Big{]}$ (136)
$\displaystyle=5k(A^{\prime})-(3(V(\mathcal{G}/e)-E(A^{\prime}))+2(F_{\rm
int}(A^{\prime})-B_{\rm int}(A^{\prime})-B_{{\rm ext}}(A^{\prime}))-7+4p\,,$
(137)
where $\alpha_{p}=3-2p$, $A$ and $A^{\prime}$ refer to the subgraphs related
by bijection between spanning c-subgraphs of $\\{A\Subset\mathcal{G};e\in
A\\}$ and $\\{A^{\prime}\Subset\mathcal{G}/e\\}$. At the end, one gets
$(y-1)z^{4p-7}\mathfrak{T}_{\mathcal{G}/e}$. In the similar way as done
before, we are able to conclude that this result is independent of the
representative.
∎
Some comments are in order at this point.
\- The existence of a function defined on $\mathfrak{G}$ satisfying the
relations of Theorem 4 is guaranteed by the explicit formula of
$\mathfrak{T}_{\mathfrak{G}}$ given in Definition 36. The unicity of
$\mathfrak{T}_{\mathfrak{G}}$ follows from its existence and can be only
guaranteed by suitable “boundary condition” data which are given by the bridge
contraction (133) and the data of all one-vertex (rosette) w-colored graph
polynomials. These polynomials write again using the state sum (sum over
subgraphs) of such stranded rosette graphs (113). These conditions extend
boundary conditions making unique the BR polynomial [6].
\- Note that in the equality (137) the number of bubbles $B_{{\rm
int}}+B_{{\rm ext}}$ plays a crucial role in the determination of the exponent
of $z$. In addition, the contraction of a bridge for this category of graph
with flags is close to the Tutte/BR relation for bridge contraction since we
do not need any constrain in the polynomial (for instance putting $s=z^{-1}$
in the case of the BR polynomial for graphs with flags) in order to obtain a
one term relation for such a terminal form. For the last contraction/cut for a
trivial $p$–inner self-loop (134), we still cannot map
$\mathfrak{T}_{\mathcal{G}\vee e}$ on $\mathfrak{T}_{\mathcal{G}/e}$. These
features show that the polynomial $\mathfrak{T}$ is a genuine extension of the
BR polynomial.
###### Corollary 2 (Cut/contraction relations for $\mathfrak{T}^{\prime}$).
Let $\mathfrak{G}$ be a rank 3 w-colored stranded graph with flags and any of
its representative $\mathcal{G}$. Then, for a bridge $e$, we have
$\mathfrak{T}^{\prime}_{\mathcal{G}\vee
e}=z^{6}(wq)^{3}t^{2}\mathfrak{T}^{\prime}_{\mathcal{G}/e}$ and
$\mathfrak{T}^{\prime}_{\mathcal{G}}(x,y,z,w,q,t)=[(x-1)z^{6}(qw)^{3}t^{2}+1]\mathfrak{T}^{\prime}_{\mathcal{G}/e}(x,y,z,w,q,t)\,;$
(138)
for a trivial $p$–inner self-loop $e$ in $\mathcal{G}$, $0\leq p\leq 2$, we
have
$\mathfrak{T}^{\prime}_{\mathcal{G}}(x,y,z,1,q,t)=z^{4p-6}[q^{3}t^{2}+(y-1)z^{-1}]\,\mathfrak{T}^{\prime}_{\mathcal{G}/e}(x,y,z,1,q,t)\,;$
(139)
###### Proof.
Theorem 4 implies naturally (138) after the setting $s=z^{-2}$ in
$\mathfrak{T}_{\mathcal{G}}$. We now work out the contraction/cut of the
trivial self-loops.
Same arguments as in the proof of Theorem 4 should be invoked here. The
difference now is that by changing $s\to z^{-2}$, using (1) the one-to-one
mapping between $\\{A\Subset\mathcal{G}\vee e\\}$ and
$\\{A^{\prime}\Subset\mathcal{G}/e\\}$ such that to each
$A\Subset\mathcal{G}\vee e$ one associates $A^{\prime}\Subset\mathcal{G}/e$
defined by $A^{\prime}=\tilde{A}/e$, $\tilde{A}\Subset\mathcal{G}$, and
$\tilde{A}\vee e=A$ and (2) the relations (95)-(97) of Lemma 8, we can map,
for a trivial $0,1,2$–inner self-loop $e$, $\mathfrak{T}_{\mathcal{G}\vee e}$
on $\mathfrak{T}_{\mathcal{G}/e}$. We compute the variation of the exponent of
$z$ between $A$ and $A^{\prime}$ as
$\displaystyle 5k(A)-(3(V(\mathcal{G})-E(A))+2[F_{\rm
int}(A)+C_{\partial}(A)-B_{\rm int}(A)-B_{{\rm ext}}(A)])=$ (140)
$\displaystyle
5(k(A^{\prime})-2)-(3(V(\mathcal{G}^{\prime})-2-E(A^{\prime}))+2[F_{\rm
int}(A^{\prime})+C_{\partial}(A^{\prime})-2-B_{\rm int}(A^{\prime})-B_{\rm
ext}(A^{\prime})+\alpha_{p}])$ (141)
$\displaystyle=5k(A^{\prime})-(3(V(\mathcal{G}/e)-E(A^{\prime}))+2[F_{\rm
int}(A^{\prime})-B_{\rm int}(A^{\prime})-B_{\rm
ext}(A^{\prime})])-2\alpha_{p}\,.$ (142)
The rest of the variations can be easily identified using the same lemma. It
can be proved that the result does not depend on the representative.
∎
As a comment, for $w\neq 1$, $\mathfrak{T}^{\prime}_{\mathcal{G}\vee e}$
cannot be fully mapped onto $\mathfrak{T}^{\prime}_{\mathcal{G}/e}$ due to the
fact that the number $F_{\partial}$ of face of the boundary of each subgraph
cannot be determined in that general situation. Nevertheless, we have two
other noteworthy reductions which solve this issue without putting $w=1$.
###### Corollary 3 (Cut/contraction rules for $\mathfrak{T}^{\prime\prime}$
(and $\mathfrak{T}^{\prime\prime\prime}$)).
Let $\mathfrak{G}$ be a rank 3 w-colored stranded graph with flags and any of
its representative $\mathcal{G}$.
Then, for a bridge $e$, we have $\mathfrak{T}^{\prime\prime}_{\mathcal{G}\vee
e}=z^{6}\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}$ and
$\mathfrak{T}^{\prime\prime}_{\mathcal{G}}=[(x-1)z^{6}+1]\,\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}\,;$
(143)
for a trivial $p$–inner self-loop $e$ in $\mathcal{G}$, $0\leq p\leq 2$, we
have $\mathfrak{T}^{\prime\prime}_{\mathcal{G}\vee
e}=z^{4p-6}\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}$ and
$\mathfrak{T}^{\prime\prime}_{\mathcal{G}}=z^{4p-6}[1+(y-1)z^{-1}]\,\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}\,.$
(144)
The same contraction and cut rules applies for
$\mathfrak{T}^{\prime\prime\prime}_{\mathcal{G}}$.
###### Proof.
The new ingredient to achieve the proof of this statement is (98) of Lemma 8.
∎
More remarks can be made at this stage.
\- $\mathfrak{T}^{\prime\prime}$ and $\mathfrak{T}^{\prime\prime\prime}$
satisfy the same rules. Hence $\mathfrak{T}^{\prime\prime}$ which is more
general becomes more relevant.
\- We did not use the notion of product of graphs for mapping the polynomials
for $\mathcal{G}\vee e$ onto those of $\mathcal{G}/e$. In fact, even for
ribbon graphs our approach applies. The dot product of graphs is however an
elegant way to encode such a map. Certainly, this notion deserves to be
defined in the framework of stranded graphs in order to achieve more
calculations. For instance, how the calculation of more involved graphs with
many trivial self-loops can be inferred from the product of contributions of
much simpler trivial self-loops.
\- The exponents of $z^{4p-7}$ or of $z^{4p-6}$ can be negative in some
expressions. This simply implies that in the polynomial
$\mathfrak{T}^{\prime}_{\mathcal{G}/e}$ or
$\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}$ all monomials should contain an
enough large power of $z$ to render the overall expression a well defined
polynomial.
If the notion of point graph multiplication of two disjoint graphs is not
defined at this stage, in contrast, the disjoint union operation on graph
extends naturally in the present formulation.
###### Lemma 9 (Disjoint union).
Let $\mathfrak{G}_{1}$ and $\mathfrak{G}_{2}$ two disjoint rank 3 $w$-colored
graphs, then
$\mathfrak{T}_{\mathfrak{G}_{1}\sqcup\mathfrak{G}_{2}}=\mathfrak{T}_{\mathfrak{G}_{1}}\,\mathfrak{T}_{\mathfrak{G}_{2}}\,.$
(145)
###### Proof.
This is totally standard as in the ordinary procedure using additive
properties of exponents in $\mathfrak{T}_{\mathcal{G}}$.
∎
###### Corollary 4 (3–inner self-loop contraction).
Given a rank 3 w-colored graph $\mathfrak{G}$ and $\mathcal{G}$ one of its
representative containing a 3–inner self-loop $e$ then
$\mathfrak{T}_{\mathcal{G}}=z^{5}(z^{3}s(wq)^{3}t^{2}+y-1)\mathfrak{T}_{\mathcal{G}/e}\,.$
(146)
###### Proof.
We use the fact that $e$ is a 3–inner then it forms a separate subgraph $g$.
In order to compute the polynomial of $\mathcal{G}$, Lemma 9 can be applied
and a direct evaluation of the graph $g$ yields the desired factor.
∎
###### Definition 37 (Multivariate form).
The multivariate form associated with (113) is defined by:
$\displaystyle\widetilde{\mathfrak{T}}_{\mathfrak{G}}(x,\\{\beta_{e}\\},\\{z_{i}\\}_{i=1,2,3},s,w,q,t)=\widetilde{\mathfrak{T}}_{\mathcal{G}}(x,\\{\beta_{e}\\},\\{z_{i}\\}_{i=1,2,3},s,w,q,t)$
(147) $\displaystyle=\sum_{A\Subset\mathcal{G}}x^{{\rm r}\,(A)}(\prod_{e\in
A}\beta_{e})z_{1}^{F_{{\rm int}}(A)}z_{2}^{B_{{\rm int}}(A)}z_{3}^{B_{{\rm
ext}}(A)}\,s^{C_{\partial}(A)}\,w^{F_{\partial}(A)}q^{E_{\partial}(A)}t^{f(A)}\,,$
for $\\{\beta_{e}\\}_{e\in\mathcal{E}}$ labeling the edges of the graph
$\mathcal{G}$.
It is direct to prove the following statement by ordinary techniques.
###### Proposition 15.
For all regular edge $e$,
$\displaystyle\widetilde{\mathfrak{T}}_{\mathfrak{G}}=\widetilde{\mathfrak{T}}_{\mathfrak{G}\vee
e}+x\beta_{e}\,\widetilde{\mathfrak{T}}_{\mathfrak{G}/e}\,.$ (148)
###### Remark 2.
We can now compare $\widetilde{\mathfrak{T}}$ with Gurau’s polynomial denoted
in the following by $G$ [12]. We shall use however not the full form of
$G_{\mathcal{G}}$ denoted $P_{\mathcal{G}}$ in [12] but we will use instead
two improvements: (1) the normalized form
$P_{\mathcal{G}}(\\{\beta_{e}x_{1}\\},\dots)$ of $P_{\mathcal{G}}$, where
$x_{1}$ is the variable associated with the number of edges which brings an
inessential overall factor of $x_{1}^{E(A)}$ consistently absorbed by the
$\beta_{e}$ and (2) a rank formulation of
$G_{\mathcal{G}}(\\{\beta_{e}\\},\dots)=P_{\mathcal{G}}(\\{\beta_{e}x_{1}\\},\dots)$,
i.e. rather then using two variables $x_{0}$ for the vertices and $x_{4}$ for
the number of connected components of the rank 3 colored tensor graph, we will
simply use $x^{{\rm r}\,(A)}$, $A\Subset\mathcal{G}$. This notation is more
compact and ${\rm r}\,(A)$ is the quantity depending only on the class of $A$.
For a rank 3 colored tensor graph $\mathcal{G}$, the polynomials
$\widetilde{\mathfrak{T}}$ and $G$ are related by
$\widetilde{\mathfrak{T}}_{\mathcal{G}}(x,\\{\beta_{e}\\},z_{1},z_{2},z_{3}=1,s,w,q,t)=G_{\mathcal{G}}(x,\\{\beta_{e}\\},z_{1},z_{2},s,q,w,t)\,,$
(149)
with, according to the convention in [12], we have
$\displaystyle C_{\partial}=|\mathfrak{B}_{\partial}^{3}|,\quad
F_{\partial}=|\mathfrak{B}_{\partial}^{2}|,\quad
E_{\partial}=|\mathfrak{B}_{\partial}^{1}|,\quad
f=|\mathfrak{B}_{\partial}^{0}|,\quad B_{{\rm int}}=|\mathfrak{B}^{3}|,\quad
F_{{\rm int}}=|\mathfrak{B}^{2}|.$ (150)
As expected, for this category of graphs the polynomial
$\widetilde{\mathfrak{T}}$ is more general because it contains one additional
variable (denoted above by $z_{3}$) which is associated with the number of
external bubbles of the graph. This variable can be introduced by hand in $G$
making it a little more general. This additional variable does not lead to any
new features for the multivariate form however, as seen in our developments,
it plays an important role in the non-multivariate form.
We finally wrap up this section with additional insights:
$\bullet$ We have successfully defined the classes of rank 3 w-colored
stranded graphs for which a natural contraction/cut rule makes a sense and the
associated polynomial invariants $\mathfrak{T},\mathfrak{T}^{\prime}$ and
$\mathfrak{T}^{\prime\prime}$. Our results extend both BR polynomial to
arbitrary contractions of colored tensor graphs of rank 3. The achievement of
this requires to relax and to modify several initial concepts introduced by
Gurau (to mention a few, the meaning of 3-bubbles with now no fixed valence
and the very notion of contraction itself, for instance). The form of our
polynomial is not primarily given in the multivariate form (as the one by
Gurau). Thus, it has required a full-fledge proof in order to establish that
this polynomial is a genuine invariant for the present graphs. We have also
successfully identified terminal forms of the contraction/cut sequence for
this type of graphs. Such a study of terminal forms has been never addressed
in former works on stranded graphs [12][18].
$\bullet$ We stress that after the contraction of some $p$–inner self-loops,
other remaining trivial $p^{\prime}$–inner self-loops may change their number
of $p^{\prime}$ inner faces. This is the reason why given a certain number of
$m_{p}$ trivial $p$–inner self-loops, we cannot easily infer the number of
trivial $0,1,2$– or $3$–inner self-loops which shall really contribute to the
final evaluation of the terminal form. In fact, we can show that contracting a
trivial $p$–inner self-loop at some point of the sequence may only be neutral
for or increase by 1 the number of inner faces of other $p$–inner self-loops.
Hence, roughly speaking, after contraction there may or may not have a
“migration” of the elements of the set of $p$–inner self-loops to set of the
$(p+1)$–inner self-loops, $p+1\leq 3$.
In any case, the polynomial $\mathfrak{T}^{\prime\prime}$ of some terminal
form $\mathcal{G}$ representative of a w-colored graph made only with bridges
and trivial $p$–inner self-loops can be computed as
$x^{m}\,\left[\prod_{p=0}^{3}\,\\{z^{4p-6}[1+(y-1)z^{-1}]\\}^{n_{p}}\right]\times\mathfrak{T}^{\prime\prime}(\mathcal{G}^{0})\,,$
(151)
for some integer $m,n_{0,1,2,3}$, and where
$\mathfrak{T}^{\prime\prime}(\mathcal{G}^{0})$ is the polynomial of
$\mathcal{G}^{0}$ graph having only vertices and hence is given by (128).
After changing $s\to z^{-2}s^{2}$, one gets
$\mathfrak{T}^{\prime\prime}(\mathcal{G}^{0})=z^{2B_{{\rm
ext}}(\mathcal{G}^{0})}s^{2C_{\partial}(\mathcal{G}^{0})-(f(\mathcal{G}^{0})-E_{\partial}(\mathcal{G}^{0})+F_{\partial}(\mathcal{G}^{0}))}\,.$
(152)
Note that the contribution (151) only makes a sense as a polynomial if
exponents $n_{p}$ are not arbitrary. One must have indeed
$5n_{3}+n_{2}-3n_{1}-7n_{0}+2B_{\rm ext}(\mathcal{G}^{0})\geq 0\,,$ (153)
which should follow from Proposition 12. However it is interesting to show
that this is indeed true in a different (and quick) manner. The issue here is
to show that the total number of trivial 1–inner and 0–inner contributions
cannot exceed the number of 3–inner, 2–inner contributions and external
bubbles after the overall contraction. In fact, whenever we contract a trivial
1–inner or 0–inner self-loop, we split the one-vertex graph in $m_{1}=2$ or
$m_{0}=3$ one-vertex (non discs) graphs and get either a power of $z^{-3}$ or
$z^{-7}$, respectively. On each of these resulting graphs, we can pursue the
contraction. But, at the end, either we are led to a connected component of
the form of a 3–inner self-loop and we gain a power of $z^{5}$ per connected
component obtained, or to a final vertex with only flags (see Lemma 1). The
simplest non trivial vertex-graph with the minimal number of external bubbles
is given $\mathcal{G}_{1}$ in Figure 29 and yields $B_{{\rm
ext}}(\mathcal{G}_{1})=3$. Call $\beta_{3}$ the number of components yielding
a 3–inner self-loop and $\beta_{x}$ the number of components yielding an empty
vertex graph. Then
$\beta_{3}+\beta_{x}=2n_{1}+3n_{0}\,,\qquad\beta_{3}\leq n_{3}\,,\qquad
3\beta_{x}\leq B_{{\rm ext}}(\mathcal{G}^{0})\,,$ (154)
and hence
$\displaystyle-3n_{1}-7n_{0}+5n_{3}+2B_{{\rm ext}}(\mathcal{G}^{0})$
$\displaystyle\geq$ $\displaystyle-3n_{1}-7n_{0}+5\beta_{3}+2B_{{\rm
ext}}(\mathcal{G}^{0})$ (155) $\displaystyle\geq$ $\displaystyle
7n_{1}+8n_{0}+2(B_{{\rm ext}}(\mathcal{G}^{0})-5\beta_{x}/2)\geq 0\,.$ (156)
In all situation, we can always beat the negative power introduced by the
trivial $0-$ or $1$–inner edge.
$\bullet$ Studying the limit cases, we have the following reduced polynomials:
for a w-colored graph $\mathfrak{G}$ and any representative $\mathcal{G}$ of
$\mathfrak{G}$,
$\mathfrak{T}_{\mathcal{G}}(x,y,z=1,s=1,w=1,q=1,t)=\mathcal{T}_{\mathcal{G}}(x,y,t)\,,$
(157)
where, computing $\mathcal{T}$ for $\mathcal{G}$, we consider in its simple
graph (collapsed) form.
There is a priori no direct mapping between $\mathfrak{T}$ and BR polynomials
(with and without flags) because these two polynomials have a different
“dimensionality” or rank with respect to stranded structures. One might think
about a combinatorial manipulation by collapsing the tensor structure to a
ribbon graph by removing a strand. This deserves to be fully understood.
However, in order to find such a mapping with BR polynomials, a natural way
would be to define the classes of “rank 2” w-colored graphs and their
associated polynomials. These latter should coincide with $\mathcal{R}$ and
$\mathcal{R}^{\prime}$ after identifications and reduction of some variables.
There will be nothing more general than $\mathcal{R}$ and
$\mathcal{R}^{\prime}$ in such context. The point raised in rank $D\geq 3$ is
the presence of $p\geq 3$-bubbles.
Few important properties have to be addressed:
$\bullet$ The universality property of $\mathfrak{T}$, $\mathfrak{T}^{\prime}$
or $\mathfrak{T}^{\prime\prime}$ which is the fact that any other function
satisfying (132) and (133) can be expressed in terms of one of these
polynomial needs to be understood. This task is not performed in this work
which has mainly focused on the definition of the relevant graph objects for
which a polynomial invariant can be identified. The universality will be
addressed in a forthcoming work and will require a complete understanding of
the one-vertex w-colored graphs as they appear in the formalism. We think that
the introduction of colors will help in reducing the type of graph structures
that we need to classify. Hence, we aim at finding suitable classes of graphs
for which one of the polynomials $\mathfrak{T}$, $\mathfrak{T}^{\prime}$ or
$\mathfrak{T}^{\prime\prime}$ will prove to be universal.
$\bullet$ In fact, before addressing the universality issue on stranded
structures, one first needs to understand to what extent the universality
property can be recovered for the extension $\mathcal{R}$ of the BR polynomial
for ribbon graphs with flags (without pinching). This will amount to
generalize the notion of chord diagrams $\mathcal{D}$, their equivalence
relation under rotations about chords and finally their associated basic
canonical diagrams $\mathcal{D}_{ijk}$ (counting the nullity $i$, genus $j$ of
the diagram, $k$ is associated with the orientation of the surface associated
with diagram) [6, 5] to chord diagrams with flags. Such a preliminary task
will certainly imply the existence of a new canonical diagram
$\mathcal{D}_{ijkl}$, where $i$, $j$, $k$ keep their meaning and $l$ defines
the number of connected component associated with the boundary of the open
ribbon graph.
$\bullet$ There exists another formulation of the Tutte and BR polynomials in
terms of spanning tree expansions. Such a notion deserves to be investigated
as well in the context of w-colored stranded graphs.
Acknowledgements
Discussions with Vincent Rivasseau at early stage of this work are gratefully
acknowledged. RCA thanks the Perimeter Institute for its hospitality. Research
at Perimeter Institute is supported by the Government of Canada through
Industry Canada and by the Province of Ontario through the Ministry of
Research and Innovation. This work is partially supported by the Abdus Salam
International Centre for Theoretical Physics (ICTP, Trieste, Italy) through
the Office of External Activities (OEA)-Prj-15. The ICMPA is also in
partnership with the Daniel Iagolnitzer Foundation (DIF), France.
## Appendix: Examples
We carry out explicitly two examples in order illustrate our results in the
present appendix. We will not focus on the equivalence class $\mathfrak{G}$
but on a particular representative always denoted by $\mathcal{G}$.
Example 1: A simple colored graph. Consider the graph $\mathcal{G}$ given by
Figure 32. Computing the multivariate form of the polynomial, we obtain by the
spanning c-subgraph summation in (147) with $\beta_{i}$ is associated with the
edge of color $i$,
$\displaystyle\mathfrak{T}_{\mathcal{G}}(x,\\{\beta_{e}\\},z,s,q,t)$
$\displaystyle=$
$\displaystyle\beta_{0}\beta_{1}\beta_{2}\beta_{3}\,xz_{1}^{6}z_{2}^{4}$ (A.1)
$\displaystyle+$
$\displaystyle(\beta_{0}\beta_{1}\beta_{2}+\beta_{0}\beta_{2}\beta_{3}+\beta_{1}\beta_{2}\beta_{3}+\beta_{0}\beta_{1}\beta_{3})\,xz_{1}^{3}z_{2}z_{3}^{3}sw^{3}q^{3}t^{2}$
(A.2) $\displaystyle+$
$\displaystyle(\beta_{0}\beta_{1}+\beta_{0}\beta_{2}+\beta_{0}\beta_{3}+\beta_{1}\beta_{2}+\beta_{1}\beta_{3}+\beta_{2}\beta_{3})\,xz_{1}z_{3}^{4}sw^{4}q^{6}t^{4}$
(A.3) $\displaystyle+$
$\displaystyle(\beta_{0}+\beta_{1}+\beta_{2}+\beta_{3})\,xz_{3}^{5}sw^{5}q^{9}t^{6}+z_{3}^{8}s^{2}w^{8}q^{12}t^{8}\,.$
(A.4)
Then this polynomial coincides exactly with the normalized Gurau’s polynomial
$G_{\mathcal{G}}$ after setting the additional variable $z_{3}$ associated
with external bubbles to 1. Note that reversely, introducing a new variable
associated with the same component in $C_{\mathcal{G}}$ both polynomials
match, for this example. Note also that, in this specific example the exponent
of $z_{3}$ and of $w$ always coincide. However it is not always true that each
open bubble would have a unique boundary component.
Cutting one edge, say the one of color 0, yields $\mathcal{G}\vee e$, so that
we evaluate
$\displaystyle\mathfrak{T}_{\mathcal{G}\vee e}(x,\\{\beta_{e}\\},z,s,q,t)$
$\displaystyle=$
$\displaystyle\beta_{1}\beta_{2}\beta_{3}\,xz_{1}^{3}z_{2}z_{3}^{3}sw^{3}q^{3}t^{2}$
(A.5) $\displaystyle+$
$\displaystyle(\beta_{1}\beta_{2}+\beta_{1}\beta_{3}+\beta_{2}\beta_{3})xz_{1}z_{3}^{4}sw^{4}q^{6}t^{4}$
(A.6) $\displaystyle+$
$\displaystyle(\beta_{1}+\beta_{2}+\beta_{3})xz_{3}^{5}sw^{5}q^{9}t^{6}+z_{3}^{8}s^{2}w^{8}q^{12}t^{8}$
(A.7)
and contracting the same edge gives
$\displaystyle\mathfrak{T}_{\mathcal{G}/e}(x,\\{\beta_{e}\\},z,s,q,t)$
$\displaystyle=$
$\displaystyle\beta_{1}\beta_{2}\beta_{3}\,z_{1}^{6}z_{2}^{4}$ (A.8)
$\displaystyle+$
$\displaystyle(\beta_{1}\beta_{2}+\beta_{2}\beta_{3}+\beta_{1}\beta_{3})z_{1}^{3}z_{2}z_{3}^{3}sw^{3}q^{3}t^{2}$
(A.9) $\displaystyle+$
$\displaystyle(\beta_{1}+\beta_{2}+\beta_{3})z_{1}z_{3}^{4}sw^{4}q^{6}t^{4}+z_{3}^{5}sw^{5}q^{9}t^{6}\,.$
(A.10)
Thus, we get
$\displaystyle\mathfrak{T}_{\mathcal{G}}=\mathfrak{T}_{\mathcal{G}\vee
e}+x\beta_{0}\mathfrak{T}_{\mathcal{G}/e}\,.$ (A.11)
It should be also emphasized that the contraction of edge is performed in the
sense of Definition 27. This definition allows us to improve the
active/passive contraction scheme as used in [12].
Figure 32. A rank 3 colored tensor graph $\mathcal{G}$ and its cut
$\mathcal{G}\vee e$ and contraction $\mathcal{G}/e$ for a regular edge $e$ of
color 2.
$\mathcal{G}\vee e$$\mathcal{G}/e$$e$$3$$0$$1$$\mathcal{G}$
Example 2: A planar w-colored graph. We can compute $\mathfrak{T}$ for other
types of graphs which are not colored tensor graphs. In a specific instance,
consider the graph $\mathcal{G}$ of Figure 33. It combines both one colored
vertex and another type vertex. For simplicity, we change the variables
$(x-1)\to x$ and $(y-1)\to y$.
Figure 33. A representative graph $\mathcal{G}$ of a w-colored graph
(displaying conjugate colors at all flags), the cut graph $\mathcal{G}\vee
e_{2}$ and the contracted graph $\mathcal{G}/e_{2}$ with respect to $e_{2}$.
$e_{2}$$e_{1}$$e_{0}$$e_{0}$$e_{0}$$\mathcal{G}\vee
e_{2}$$\mathcal{G}/e_{2}$$e_{1}$$e_{1}$$\mathcal{G}$
By the spanning c-subgraph summation, we get
$\displaystyle\mathfrak{T}_{\mathcal{G}}(x,y,z,s,q,t)=[xz^{10}sw^{5}q^{9}t^{6}+xyz^{9}sw^{4}q^{6}t^{4}+2z^{2}w^{2}q^{6}t^{4}+3yzwq^{3}t^{2}+y^{2}]z^{14}sw^{5}q^{9}t^{6}\,.$
(A.12)
We want to compare (A.12) with the result obtained by contraction and cut
procedure using a much as possible results on terminal forms. Using the
notations of Figure 33, we must check that
$\mathfrak{T}_{\mathcal{G}}=\mathfrak{T}_{\mathcal{G}\vee
e_{2}}+\mathfrak{T}_{\mathcal{G}/e_{2}}\,.$ (A.13)
Evaluating $\mathfrak{T}_{\mathcal{G}\vee e_{2}}$ and
$\mathfrak{T}_{\mathcal{G}/e_{2}}$, one finds
$\displaystyle\mathfrak{T}_{\mathcal{G}\vee
e_{2}}=[xz^{8}s(wq)^{3}t^{2}+1]\mathfrak{T}_{(\mathcal{G}\vee
e_{2})/e_{0}}\,,\quad\mathfrak{T}_{(\mathcal{G}\vee
e_{2})/e_{0}}=\mathfrak{T}_{((\mathcal{G}\vee e_{2})/e_{0})\vee
e_{1}}+yz\,\mathfrak{T}_{((\mathcal{G}\vee e_{2})/e_{0})/e_{1}}\,,$ (A.14)
(A.15)
$\displaystyle\mathfrak{T}_{\mathcal{G}/e_{2}}=\mathfrak{T}_{(\mathcal{G}/e_{2})\vee
e_{1}}+yz\,\mathfrak{T}_{(\mathcal{G}/e_{2})/e_{1}}\,.$ (A.16)
We used in (A.15) the fact that $e_{0}$ is a bridge in $\mathcal{G}\vee e_{2}$
and that $e_{1}$ is a 2–inner self-loop in $(\mathcal{G}\vee e_{2})/e_{0}$.
Meanwhile, in (A.16), we used the fact that $e_{1}$ is a 2–inner self-loop in
$\mathcal{G}/e_{2}$.
A straightforward calculation yields
$\displaystyle\mathfrak{T}_{((\mathcal{G}\vee e_{2})/e_{0})\vee
e_{1}}=z^{16}sw^{7}q^{15}t^{10}\,,\qquad\mathfrak{T}_{((\mathcal{G}\vee
e_{2})/e_{0})/e_{1}}=z^{14}sw^{6}q^{12}t^{8}\,,$ (A.17)
$\displaystyle\mathfrak{T}_{(\mathcal{G}/e_{2})\vee
e_{1}}=z^{16}sw^{7}q^{15}t^{10}+yz^{15}sw^{6}q^{12}t^{8}\,,\qquad\mathfrak{T}_{(\mathcal{G}/e_{2})/e_{1}}=z^{14}sw^{6}q^{12}t^{8}+yz^{13}sw^{5}q^{9}t^{6}\,.$
(A.18)
Plugging these results on (A.15) and (A.16) and summing these contributions in
(A.13), we recover (A.12).
## References
* [1] R. C. Avohou, J. Ben Geloun and E. R. Livine, “On terminal forms for topological polynomials for ribbon graphs: The $N$-petal flower,” arXiv: 1212.5961[math.CO].
* [2] J. Ben Geloun, T. Krajewski, J. Magnen and V. Rivasseau, “Linearized Group Field Theory and Power Counting Theorems,” Class. Quant. Grav. 27, 155012 (2010) [arXiv:1002.3592 [hep-th]].
* [3] J. Ben Geloun, J. Magnen and V. Rivasseau, “Bosonic Colored Group Field Theory,” Eur. Phys. J. C 70, 1119 (2010) [arXiv:0911.1719 [hep-th]].
* [4] J. Ben Geloun and V. Rivasseau, “A Renormalizable 4-Dimensional Tensor Field Theory,” arXiv:1111.4997 [hep-th].
* [5] B. Bollobas, “Modern Graph Theory” (Springer, NY, 1998).
* [6] B. Bollobas and O. Riordan, “Polynomial of graphs on surfaces,” Math. Ann. 323, 81–96 (2002).
* [7] B. Bollobas and O. Riordan, “A polynomial invariant of graphs on orientable surfaces,” Proc. London Math. Soc. 83, 513–531 (2001).
* [8] F. Caravelli, “A Simple Proof of Orientability in Colored Group Field Theory,” SpringerPlus 1, 6 (2012) [arXiv:1012.4087 [math-ph]].
* [9] S. Chmutov, “Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial,” to appear in J. Combin. Theory Ser. B (2009), arXiv:0711.3490v3 [math.CO].
* [10] R. Gurau, “Colored Group Field Theory,” Commun. Math. Phys. 304, 69 (2011) [arXiv:0907.2582 [hep-th]].
* [11] R. Gurau, “Lost in Translation: Topological Singularities in Group Field Theory,” Class. Quant. Grav. 27, 235023 (2010) [arXiv:1006.0714 [hep-th]].
* [12] R. Gurau, “Topological Graph Polynomials in Colored Group Field Theory,” Annales Henri Poincare 11, 565 (2010) [arXiv:0911.1945 [hep-th]].
* [13] R. Gurau and J. P. Ryan, “Colored Tensor Models - a review,” SIGMA 8, 020 (2012) [arXiv:1109.4812 [hep-th]].
* [14] R. Gurau, “The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders,” Nucl. Phys. B 865, 133 (2012) [ar(x-1)iv:1203.4965 [hep-th]].
* [15] T. Krajewski, V. Rivasseau and F. Vignes-Tourneret, “Topological graph polynomials and quantum field theory. Part II. Mehler kernel theories,” Annales Henri Poincare 12, 483 (2011) [arXiv:0912.5438 [math-ph]].
* [16] T. Krajewski, V. Rivasseau, A. Tanasa and Zhituo Wang, “Topological Graph Polynomials and Quantum Field Theory. Part I. Heat Kernel Theories,” annales Henri Poincare ${\bf 12}$, 483 (2011) [arXiv:0811.0186v1 [math-ph]].
* [17] A. Sokal, “The multivariate Tutte polynomial (alias Potts model) for graphs and matroids,” Surveys in combinatorics 2005, 173-226, London Math. Soc. Lecture Note Ser., 327, (Cambridge Univ. Pr., Cambridge ,2005) arXiv:math/0503607; J. Ellis-Monaghan and C. Merino, “Graph polynomials and their applications I: The Tutte polynomial,” arXiv:0803.3079 [math.CO].
* [18] A. Tanasa, “Generalization of the Bollobas-Riordan polynomial for tensor graphs,” J. Math. Phys. 52, 073514 (2011) [arXiv:1012.1798 [math.CO]].
* [19] W. T. Tutte, “Graph theory”, vol. 21 of Encyclopedia of Mathematics and its Applications (Addison-Wesley, Massachusetts, 1984).
|
arxiv-papers
| 2013-01-09T21:09:43 |
2024-09-04T02:49:40.065661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Remi C. Avohou, Joseph Ben Geloun and Mahouton N. Hounkonnou",
"submitter": "Joseph Ben Geloun",
"url": "https://arxiv.org/abs/1301.1987"
}
|
1301.2016
|
# On the Weyl Tensor Classification in All Dimensions and its Relation with
Integrability Properties
Carlos Batista [email protected] Departamento de Física, Universidade
Federal de Pernambuco, 50670-901 Recife - PE, Brazil
###### Abstract
In this paper the Weyl tensor is used to define operators that act on the
space of forms. These operators are shown to have interesting properties and
are used to classify the Weyl tensor, the well known Petrov classification
emerging as a special case. Particularly, in the Euclidean signature this
classification turns out be really simple. Then it is shown that the
integrability condition of maximally isotropic distributions can be described
in terms of the invariance of certain subbundles under the action of these
operators. Here it is also proved a new generalization of the Goldberg-Sachs
theorem, valid in all even dimensions, stating that the existence of an
integrable maximally isotropic distribution imposes restrictions on the
optical matrix. Also the higher-dimensional versions of the self-dual
manifolds are investigated. These topics can shed light on the integrability
of Einstein’s equation in higher dimensions.
Weyl tensor, Goldberg-Sachs theorem, Isotropic structures, Integrability,
Petrov classification, Mariot-Robinson theorem, Self-dual manifolds
## I Introduction
The Petrov classification Petrov ; art1 is a scheme to classify the Weyl
tensor in four dimensions that has been responsible for much progress on
general relativity. In particular it was of fundamental importance for the
discovery of one of the most important solutions to Einstein’s equation, the
Kerr metric Kerr . The main reason behind the usefulness of such
classification is encoded on the Goldberg-Sachs(GS) theorem, which states that
in a vacuum(Ricci-flat) four-dimensional Lorentzian manifold the Weyl tensor
is algebraically special if, and only if, the spacetime admits a congruence of
null geodesics that is shear-free Goldberg-Sachs . Later it has been proved
that this theorem could be extended to all signatures if the concept of shear-
free geodesics is replaced by the existence of an integrable maximally
isotropic distribution Plebanski2 ; art2 , revealing the geometrical content
of the GS theorem.
The intent of the present article is to generalize the Petrov classification
to all dimensions and to find extensions of Goldberg-Sachs theorem (specially
in even dimensions). Hopefully this will help the search of new exact
solutions to Einstein’s equation in higher dimensions, with relevance to
string theory compactifications. Since Einstein’s equation is non-linear there
is usually no hope to find all its solutions, however with the help of GS
theorem Kinnersley was able to find all Petrov type D vacuum solutions in four
dimensions typeD - Kinnersley , a really impressive achievement. The topics
discussed here can, analogously, help to fully integrate Einstein’s equation
under certain circumstances in higher dimensions. Also the results obtained
here should promote more geometrical insight about the Weyl tensor with
possible applications in general relativity. Moreover this work has
applicability in differential geometry, specially in areas related to the
integrability of distributions and in spinor geometry, since maximally
isotropic distributions are ubiquitous in this article. Finally it is
pertinent to mention that recently it was made an explicit connection between
Navier-Stokes’ and Einstein’s equations Navier-Stokes , in this connection the
classification of the Weyl tensor makes a prominent role.
Different classification schemes for the Weyl tensor in dimensions greater
than four have been already defined during the last decade. In CMPP it was
put forward a form to classify any tensor in Lorentzian spaces based on the so
called boost weight, the well known CMPP classification. Extensions of the GS
theorem using such classification were looked for and some progress has been
made GS-CMPP ; GS-Ortaggio12 ; M. Ortaggio-Robinson-Trautman . A recent review
of this approach is available in reference CMPP-Review . In HigherGSisotropic1
; HigherGSisotropic2 it was defined a classification scheme for the Weyl
tensor valid in any dimension and signature based on the maximally isotropic
structures and it was found a generalization of the GS theorem that will be
used in the present article. In 5D classification the Weyl tensor was
classified in five dimensions using spinor techniques and applications were
made. In Spin6D it was worked out a classification scheme for the Weyl tensor
in six dimensions using spinors and the generalized GS theorem of
HigherGSisotropic2 was elegantly translated to the spinor language. Finally,
a very recent paper used spinorial language to define a classification for the
Weyl tensor in even dimensions Taghavi-Spinors . The classification presented
here is more refined than the previous ones and has the classification of
reference Spin6D and the Petrov classification as special cases. A short and
nice review of the previous literature can be found in Reall - Review , where
some possible applications of these subjects are also discussed.
In section II the Weyl tensor is used to define operators $\textbf{C}_{p}$
that act on the space of $p$-forms, it is also shown that such operators have
nice properties. For example, in the Euclidean signature they are Hermitean.
These operators are then used to classify the Weyl tensor in any dimension.
Section III shows that in even dimensions, $d=2n$, the operator
$\textbf{C}_{n}$ preserves the space of (anti-)self-dual $n$-forms and this is
used to split this operator into the direct sum of a self-dual part and an
anti-self-dual part. In section IV it is proved that in even dimensions the
integrability condition for maximally isotropic distributions found in
HigherGSisotropic2 can be nicely expressed in terms of the operator
$\textbf{C}_{n}$. Section V introduces a notation in which the map
$\textbf{C}_{p}$ takes a really simple and elegant form. Although not deeply
exploited here this notation seems to be promising. Then section VI gives a
brief review of some other important methods of classification for the
curvature and argues how the classification introduced here can be helpful.
Section VII provides a new and simple proof of the generalized Mariot-Robinson
theorem that will be important for future sections. Finally, in sections VIII
and IX are shown links between the integrability of null structures and the
optical matrix. In particular it is proved a new generalization of the GS
theorem, stating that in even dimensions the existence of integrable maximally
isotropic distributions imposes restrictions on the optical matrix. Appendix A
defines a refined version of the Segre classification that is used to classify
the operators $\textbf{C}_{p}$, while appendix B presents important results
about simple forms that are used throughout the article.
In the present work the vector bundles are assumed to be complexified so that
the results can be applied to any signature, on the same lines of references
art1 ; art2 ; Spin6D . All results obtained here are local and it is always
assumed that the manifold is endowed with a non-degenerate metric and its
Levi-Civita connection. Throughout the article some examples are worked out
with the intent of facilitating the comprehension and showing possible
applications of the concepts and tools introduced here.
## II The Map $\textbf{C}_{p}$
Let $(M,g)$ be a manifold of dimension $d$ endowed with a non-degenerate real
metric tensor $g_{\mu\nu}$ of arbitrary signature $s\,$111In this article the
metric tensor is assumed to be real just because sometimes it is convenient to
deal with a real Weyl tensor. But almost all results presented here are also
valid for the case of a complex metric.. When convenient the tangent bundle of
this manifold will be implicitly assumed to be complexified. Since all results
throughout this article will be local we can always assume the existence of a
volume form denoted by $\varepsilon_{\nu_{1}\nu_{2}\ldots\nu_{d}}$. This
$d$-form obeys to the following well known identity222As usual the indices
enclosed by square brackets are anti-symmetrized, while the indices inside
round brackets are symmetrized. For example,
$T_{[ab]}=\frac{1}{2}(T_{ab}-T_{ba})$ and
$T_{(ab)}=\frac{1}{2}(T_{ab}+T_{ba})$. Also, repeated indices are assumed to
be summed.:
$\varepsilon^{\mu_{1}\ldots\mu_{p}\,\nu_{p+1}\ldots\nu_{d}}\,\varepsilon_{\mu_{1}\ldots\mu_{p}\,\alpha_{p+1}\ldots\alpha_{d}}\;=\;p!(d-p)!\,(-1)^{\frac{d-s}{2}}\delta_{\alpha_{p+1}}^{\;[\nu_{p+1}}\ldots\delta_{\alpha_{d}}^{\;\nu_{d}]}\,.$
(1)
Given a $p$-form $K_{\nu_{1}\ldots\nu_{p}}=K_{[\nu_{1}\ldots\nu_{p}]}$ then
its Hodge dual is the $(d-p)$-form defined by:
$\widetilde{K}_{\mu_{1}\ldots\mu_{d-p}}\;=\;\frac{1}{p!}\,\varepsilon^{\nu_{1}\ldots\nu_{p}}_{\phantom{\nu_{1}\ldots\nu_{p}}\mu_{1}\ldots\mu_{d-p}}\,K_{\nu_{1}\ldots\nu_{p}}\,.$
(2)
Taking the Hodge dual twice and using equation (1) it is straightforward to
prove the following important relation:
$\widetilde{\widetilde{K}}_{\nu_{1}\ldots\nu_{p}}\;=\;(-1)^{[p(d-p)+\frac{d-s}{2}]}\,K_{\nu_{1}\ldots\nu_{p}}\,.$
(3)
The tangent bundle of $(M,g)$ is assumed to be endowed with the Levi-Civita
connection, whose trace-less part of the curvature, called the Weyl tensor,
will be denoted by $C_{\mu\nu\alpha\beta}$. This tensor has the following
symmetries:
$C_{\mu\nu\alpha\beta}=C_{[\mu\nu][\alpha\beta]}=C_{\alpha\beta\mu\nu}\;\;;\;\;C_{\mu[\nu\alpha\beta]}=0\;\;;\;\;C^{\mu}_{\phantom{\mu}\nu\mu\beta}=0\,.$
(4)
By means of the Weyl tensor we can define the following linear operator that
act on the space of $p$-forms with $p\geq 2$:
$\textbf{C}_{p}:\quad\;K_{\nu_{1}\ldots\nu_{p}}\;\mapsto\;K^{\prime}_{\nu_{1}\ldots\nu_{p}}=C^{\alpha\beta}_{\phantom{\alpha\beta}[\nu_{1}\nu_{2}}K_{\nu_{3}\ldots\nu_{p}]\alpha\beta}\,.$
(5)
The particular case $p=2$ of the above operator has been of fundamental
importance to the development of general relativity in four dimensions, since
it gives rise to the well known Petrov classification art1 ; Petrov . Such
classification has been used to find very important solutions to Einstein’s
equation, the most important examples being the Kerr metric Kerr and all type
$D$ vacuum solutions typeD - Kinnersley . The operator $\textbf{C}_{2}$ was
also investigated in higher dimensions on reference BivectHighDim , with the
intent of refining CMPP classification. It is also interesting to note that in
six dimensions the operator $\textbf{C}_{3}$ reduces to the Weyl operator
defined on Spin6D using spinors. But, as far as the present author knows, the
operators $\textbf{C}_{p}$, for arbitrary $p$, have not been defined before.
The goal of the present article is to study some properties of the maps
$\textbf{C}_{p}$ for $p>2$, define a classification for the Weyl tensor based
on them and, in some cases, relate these maps to a generalization of the
Goldberg-Sachs theorem.
Now let us work out a useful relation between the Weyl operator
$\textbf{C}_{p}$ and the Hodge dual map. If $K^{\prime}$ is the image of the
$p$-form $K$ under the operator $\textbf{C}_{p}$, as in equation (5), then we
have:
$\displaystyle\widetilde{K^{\prime}}^{\nu_{1}\ldots\nu_{(d-p)}}\;=\;\frac{1}{p!}\,\varepsilon^{\mu_{1}\ldots\mu_{p}\,\nu_{1}\ldots\nu_{d-p}}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\,\widetilde{\widetilde{K}}_{\mu_{3}\ldots\mu_{p}\alpha\beta}\,(-1)^{[(d-p)p+\frac{d-s}{2}]}\;=\;$
$\displaystyle\stackrel{{\scriptstyle(\ref{Hdge
Dual})}}{{=}}\;\frac{(-1)^{[(d-p)p+\frac{d-s}{2}]}}{p!\,(d-p)!}\,\varepsilon^{\mu_{1}\ldots\mu_{p}\,\nu_{1}\ldots\nu_{d-p}}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\,\,\varepsilon_{\sigma_{1}\ldots\sigma_{d-p}\mu_{3}\ldots\mu_{p}\alpha\beta}\,\widetilde{K}^{\sigma_{1}\ldots\sigma_{d-p}}\;=$
$\displaystyle\stackrel{{\scriptstyle(\ref{EpsilonEps})}}{{=}}\;\frac{(p-2)!\,(d-p+2)!}{p!\,(d-p)!}\,\delta_{\alpha}^{\;[\mu_{1}}\delta_{\beta}^{\;\mu_{2}}\delta_{\sigma_{1}}^{\;\nu_{1}}\ldots\delta_{\sigma_{d-p}}^{\;\nu_{d-p}]}\,C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\widetilde{K}^{\sigma_{1}\ldots\sigma_{d-p}}\;\stackrel{{\scriptstyle(\ref{Weyl
Symmetries})}}{{=}}\;C^{\phantom{\mu_{1}\mu_{2}}[\nu_{1}\nu_{2}}_{\mu_{1}\mu_{2}}\,\widetilde{K}^{\nu_{3}\ldots\nu_{d-p}]\mu_{1}\mu_{2}}\,.$
Where above it was used equations (2), (5) and (3) in the first equality and
the numbers above the other equal signs refer to the used equations. This
identity has been already known for the specific case $d=4$ and $p=2$, in
which case it has made a prominent role on the development of Petrov
classification Bel . The above equation is then the generalization of this
known fact to all dimensions and to all values of $p$. Such result can be put
in a more elegant form if an abstract notation is used. Denoting the bundle of
$p$-forms on $M$ by $\wedge^{p}M$ then
$\textbf{C}_{p}:\wedge^{p}M\rightarrow\wedge^{p}M$ is the abstract operator
that implements the map defined in equation (5), while
$\textbf{E}_{p}:\wedge^{p}M\rightarrow\wedge^{d-p}M$ will denote the operator
that when acts on a $p$-form gives its Hodge dual. In this notation the above
result is written in the following form:
$\textbf{E}_{p}\,\textbf{C}_{p}\;=\;\textbf{C}_{d-p}\,\textbf{E}_{p}\,.$ (6)
Now we are able to define an algebraic classification scheme for the Weyl
tensor valid in all dimensions, we just have to find matrix representations
for the operators $\textbf{C}_{p}$ and use the refined Segre classification
(see appendix A). Thus the classification of the Weyl tensor proposed here
amounts to gathering the refined Segre types of the operators $\textbf{C}_{p}$
for all possible values of $p$. But some of these calculations will be
redundant, since by equation (3) we have that the operator $\textbf{E}_{p}$ is
invertible, with inverse proportional to $\textbf{E}_{d-p}$. So using this
information in equation (6) we have that
$\textbf{C}_{p}=\textbf{E}_{p}^{\;-1}\textbf{C}_{d-p}\textbf{E}_{p}$, i.e.,
the operator $\textbf{C}_{p}$ can be obtained from $\textbf{C}_{d-p}$ by a
similarity transformation, therefore the algebraic type of these operators is
the same. Special attention shall be deserved for the cases $p=d$ and
$p=(d-1)$ since the “dual operators”, $\textbf{C}_{0}$ and $\textbf{C}_{1}$,
are not defined. But it is not difficult to prove that the operators
$\textbf{C}_{d}$ and $\textbf{C}_{d-1}$ are both zero, because of the
traceless property of the Weyl tensor. _So it is just necessary to classify
the Weyl operators $\textbf{C}_{p}$ for integer values of $p$ pertaining to
the interval $2\leq p\leq\frac{d}{2}$._
If $\textbf{K}_{p}$ and $\textbf{L}_{p}$ are $p$-forms then the following
symmetric inner product can be defined on the space $\wedge^{p}M$:
$<\textbf{K}_{p},\textbf{L}_{p}>\;=\;K^{\mu_{1}\ldots\mu_{p}}\,L_{\mu_{1}\ldots\mu_{p}}$
(7)
Such inner product is non-degenerate and its signature depends on the
signature of the metric $g$. If $\\{\textbf{B}^{i}\\}$ is a basis for the
space $\wedge^{p}M$ such that $<\textbf{B}^{i},\textbf{B}^{j}>\,=h^{ij}$ then
denoting by $h_{ij}$ the inverse matrix of $h^{ij}$ it follows that the
$p$-forms $\textbf{B}_{i}=h_{ij}\textbf{B}^{j}$ obey to the identity
$<\textbf{B}_{i},\textbf{B}^{j}>\,=\delta_{i}^{\phantom{i}j}$. Thus if
$\textbf{K}_{p}$ is a $p$-form then
$\textbf{K}_{p}=\,<\textbf{B}_{i},\textbf{K}_{p}>\textbf{B}^{i}$, from which
it follows that
$\textbf{B}_{i}^{\;\mu_{1}\ldots\mu_{p}}\textbf{B}^{i}_{\;\nu_{1}\ldots\nu_{p}}=\delta_{\nu_{1}}^{\;[\mu_{1}}\ldots\delta_{\nu_{p}}^{\;\mu_{p}]}$.
The matrix representation of the operator $\textbf{C}_{p}$ on the basis
$\\{\textbf{B}^{i}\\}$ is given by
$C_{p\,ij}=\,<\textbf{B}_{i},\textbf{C}_{p}(\textbf{B}^{j})>$, so that using
the previous results we see that the trace of this operator is zero:
$\textrm{tr}(\textbf{C}_{p})=C_{p\,ii}=\textbf{B}_{i}^{\;\mu_{1}\ldots\mu_{p}}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\textbf{B}^{i}_{\;\mu_{3}\ldots\mu_{p}\alpha\beta}=C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\delta_{\alpha}^{\;[\mu_{1}}\delta_{\beta}^{\;\mu_{2}}\delta_{\mu_{3}}^{\;\mu_{3}}\ldots\delta_{\mu_{p}}^{\;\mu_{p}]}\propto
C^{\alpha\beta}_{\phantom{\alpha\beta}\alpha\beta}=0\,.$ (8)
Also, using the symmetry $C_{\mu\nu\alpha\beta}=C_{\alpha\beta\mu\nu}$ of the
Weyl tensor it is simple matter to show that the Weyl operator
$\textbf{C}_{p}$ is self-adjoint with respect to this inner product, i.e, the
following equation is valid:
$<\textbf{K}_{p},\textbf{C}_{p}(\textbf{L}_{p})>\;=\;<\textbf{C}_{p}(\textbf{K}_{p}),\textbf{L}_{p}>\,.$
(9)
In the special case of $(M,g)$ being a real manifold of Euclidean signature it
follows that the inner product of $p$-forms defined on equation (7) is
positive definite. In this case the self-adjointness of operators
$\textbf{C}_{p}$ guarantees that they can be diagonalized and that the
eigenvalues are real. This property imposes huge restrictions on the possible
algebraic types that the Weyl tensor can have according the classification
defined above, since the refined Segre types of operators $\textbf{C}_{p}$
will depend just on the degeneracy of each eigenvalue and on the dimension of
the kernel of these operators. So we have the following lemma:
###### Lemma 1
If the metric $g$ has Euclidean signature then the operators $\textbf{C}_{p}$
admit a traceless diagonal matrix representation with real eigenvalues. In
particular this implies that on the refined Segre classification of these
operators all numbers inside the square bracket will be 1 (see appendix A).
## III Even Dimensions and the Self-duality
In this section the dimension of the manifold $M$ is assumed to be even,
$d=2n$. In such case plugging $p=n$ on equation (3) yields
$\textbf{E}_{n}\textbf{E}_{n}=(-1)^{n^{2}+n+\frac{s}{2}}\textbf{1}_{n}=(-1)^{\frac{s}{2}}\textbf{1}_{n}$,
where $\textbf{1}_{n}$ is the identity operator of space $\wedge^{n}M$. So the
space of $n$-forms can be split into a direct sum of the eigen-spaces of
$\textbf{E}_{n}$, $\wedge^{n}M=S^{+}\oplus S^{-}$, where $S^{+}$ is the space
of self-dual $n$-forms and $S^{-}$ is the space of anti-self-dual $n$-forms,
defined by:
$S^{\pm}=\\{\textbf{K}_{n}\;\in\;\wedge^{n}M\;|\;\textbf{E}_{n}(\textbf{K}_{n})=\pm\epsilon\textbf{K}_{n}\\}\,.$
Where $\epsilon$ is equal to $1$ or $i$ depending on whether $\frac{s}{2}$ is
respectively even or odd. The spaces $S^{+}$ and $S^{-}$ both have the same
dimension, $\frac{1}{2}\left({}^{2n}_{\,\,n}\right)$. Using equations (2) and
(7) a simple index rearrangement shows that the following property holds:
$<\textbf{K}_{n}\,,\,\textbf{E}_{n}(\textbf{L}_{n})>\;=\;(-1)^{n^{2}}\,<\textbf{E}_{n}(\textbf{K}_{n})\,,\,\textbf{L}_{n}>\,.$
Where $\textbf{K}_{n}$ and $\textbf{L}_{n}$ are arbitrary $n$-forms. Using
this equation we arrive at the following result:
###### Lemma 2
In a manifold of dimension $d=2n$ if $n$ is even then the spaces $S^{+}$ and
$S^{-}$ are orthogonal to each other with respect to the inner product $<,>$.
On the other hand if $n$ is odd then every element of $S^{+}$ is orthogonal to
every element of $S^{+}$ and every element of $S^{-}$ is orthogonal to every
element of $S^{-}$.
Now let us see that the operator $\textbf{C}_{n}$ has the important property
of preserving the spaces $S^{\pm}$. If $\textbf{K}_{n}^{\pm}$ pertain to
$S^{\pm}$ and $\textbf{L}_{n}^{\pm}=\textbf{C}_{n}(\textbf{K}_{n}^{\pm})$ then
equation (6) implies that:
$\textbf{E}_{n}(\textbf{L}_{n}^{\pm})=\textbf{E}_{n}\textbf{C}_{n}(\textbf{K}_{n}^{\pm})=\textbf{C}_{n}\textbf{E}_{n}(\textbf{K}_{n}^{\pm})=\textbf{C}_{n}(\pm\epsilon\textbf{K}_{n}^{\pm})=\pm\epsilon\textbf{L}_{n}^{\pm}\,.$
Therefore it is useful to define the following operators:
$\textbf{C}^{\pm}:\,\wedge^{n}M\rightarrow\wedge^{n}M\;\;,\;\;\;\textbf{C}^{\pm}\;\equiv\;\frac{1}{2}(\textbf{C}_{n}\,\pm\,\epsilon^{-1}\,\textbf{C}_{n}\textbf{E}_{n})\,.$
(10)
It is immediate to see that $\textbf{C}^{\pm}$ is zero when restricted to
$S^{\mp}$, i.e., $\textbf{C}_{n}=\textbf{C}^{+}\oplus\textbf{C}^{-}$. Thus the
matrix representation of $\textbf{C}_{n}$ can be put in block-diagonal form
with two blocks of the same dimension. This property restrict enormously the
possible algebraic types that the operator $\textbf{C}_{n}$ can have. This has
been already known in four dimensions and was of fundamental importance to the
development of the Petrov classification Petrov ; art1 . Also this has been
noted in six dimensions using spinor calculus Spin6D . This property of
$\textbf{C}_{n}$ guarantees that when convenient the operators
$\textbf{C}^{\pm}$ can be assumed to act on $S^{\pm}$ instead of
$\wedge^{n}M$.
Let us note that the trace of both operators $\textbf{C}^{+}$ and
$\textbf{C}^{-}$ is zero. From equation (8) it follows that the trace of
$\textbf{C}_{n}$ is zero, so that using the definition of operators
$\textbf{C}^{\pm}$, equation (10), we see that
$\operatorname{tr}(\textbf{C}^{\pm})=\pm\epsilon^{-1}\operatorname{tr}(\textbf{C}_{n}\textbf{E}_{n})$.
Using steps similar to the ones used to evaluate
$\operatorname{tr}(\textbf{C}_{p})$ we find for $n\geq 2$:
$\operatorname{tr}(\textbf{C}_{n}\textbf{E}_{n})=\frac{1}{n!}\textbf{B}_{i}^{\;\mu_{1}\ldots\mu_{n}}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\varepsilon^{\nu_{1}\ldots\nu_{n}}_{\phantom{\nu_{1}\ldots\nu_{n}}\mu_{3}\ldots\mu_{n}\alpha\beta}\textbf{B}^{i}_{\;\nu_{1}\ldots\nu_{n}}=\frac{1}{n!}C^{\alpha\beta}_{\phantom{\alpha\beta}\mu_{1}\mu_{2}}\varepsilon^{\nu_{1}\ldots\nu_{n}}_{\phantom{\nu_{1}\ldots\nu_{n}}\mu_{3}\ldots\mu_{n}\alpha\beta}\delta_{[\nu_{1}}^{\;\mu_{1}}\ldots\delta_{\nu_{n}]}^{\;\mu_{n}}=0\,.$
Where the last equality follows from the Bianchi identity,
$C_{[\alpha\beta\mu]\nu}=0$. So the operators $\textbf{C}^{\pm}$ have
vanishing trace.
From lemma 2 and from the fact that the inner product $<,>$ in $\wedge^{n}M$
is non-degenerate we have that when $n$ is odd it is possible to introduce a
basis $\\{\textbf{B}^{+i}\\}$ for $S^{+}$ and a basis $\\{\textbf{B}^{-i}\\}$
for $S^{-}$ such that $<\textbf{B}^{+i},\textbf{B}^{-j}>=\delta^{ij}$. To see
this just start with a basis for $S^{+}$ and a basis for $S^{-}$ and then use
the Gram-Schmidt process to conveniently redefine the basis of $S^{-}$. The
matrix representation of $\textbf{C}^{+}$ on the basis $\\{\textbf{B}^{+i}\\}$
is then $C^{+}_{ij}=<\textbf{B}^{-i},\textbf{C}_{n}(\textbf{B}^{+j})>$, while
the matrix representation of $\textbf{C}^{-}$ on the basis
$\\{\textbf{B}^{-i}\\}$ is
$C^{-}_{ij}=<\textbf{B}^{+i},\textbf{C}_{n}(\textbf{B}^{-j})>$. Then from
equation (9) it follows that $C^{+}_{ij}=C^{-}_{ji}$. So when $n$ is odd it
follows that operator $\textbf{C}^{+}$ can be obtained from $\textbf{C}^{-}$
and vice versa. On the other hand when $n$ is even the operators
$\textbf{C}^{+}$ and $\textbf{C}^{-}$ are in principle independent of each
other. The results obtained so far can be summarized by the following theorem.
###### Theorem 1
When the dimension of the manifold is $d=2n$ the operator $\textbf{C}_{n}$ is
the direct sum of an operator that acts on the space of self-dual $n$-forms,
$\textbf{C}^{+}:S^{+}\rightarrow S^{+}$, and an operator that acts on the
space of anti-self-dual $n$-forms, $\textbf{C}^{-}:S^{-}\rightarrow S^{-}$.
Both operators, $\textbf{C}^{+}$ and $\textbf{C}^{-}$, have vanishing trace.
In the particular case of odd $n$ it follows that the operator
$\textbf{C}^{-}$ is the adjoint of the operator $\textbf{C}^{+}$, i.e., in a
suitable basis the matrix representation of $\textbf{C}^{-}$ is the transpose
of the matrix that represents $\textbf{C}^{+}$.
This theorem implies that when $n$ is odd then to classify $\textbf{C}_{n}$ is
equivalent to classify $\textbf{C}^{+}$, since automatically the operator
$\textbf{C}^{-}$ has the same algebraic type of $\textbf{C}^{+}$. In the case
$n=3$ this was already proved using spinorial techniques on reference Spin6D .
In turn, when $n$ is even classify $\textbf{C}_{n}$ is equivalent to compute
the algebraic types of both operators $\textbf{C}^{+}$ and $\textbf{C}^{-}$.
In four dimensions a manifold is called self-dual when $\textbf{C}^{-}=0$ and
$\textbf{C}^{+}\neq 0$. These manifolds have been widely studied in the past
Plebanski3 , in particular it has been shown that Einstein’s vacuum equation
on self-dual manifolds reduces to a single second-order differential equation
Plebanski3 . Now it is natural to ask wether the notion of self-dual manifolds
can be extended to higher dimensions. Theorem 1 says that if the dimension is
$d=2n$ with odd $n$ then $\textbf{C}^{-}=0$ implies $\textbf{C}^{+}=0$, so
that the Weyl tensor is identically zero. But in principle if the dimension is
multiple of four, even $n$, the operators $\textbf{C}^{+}$ and
$\textbf{C}^{-}$ are independent of each other so that the self-dual manifolds
can be defined. But it turns out that a laborious investigation of the self-
dual constraint in 8 dimensions reveals that if $\textbf{C}^{+}=0$ then
$\textbf{C}^{-}=0$. Thus arriving at the following lemma:
###### Lemma 3
If the dimension of the manifold is $d=2n$ with odd $n$ or $n=4$ then the
constraint $\textbf{C}^{+}=0$ implies that the whole Weyl tensor vanishes. So
in these dimensions the notion of self-dual manifold cannot be introduced.
Although in the case of even $n$ this result has been worked out by the author
only for the case $n=4$ the calculations seems to indicate that lemma 3 is
valid for all $n\geq 4$. Thus, concerning the Weyl tensor, the dimension four
appears to be very special, since in this dimension the operators
$\textbf{C}^{+}$ and $\textbf{C}^{-}$ can be independent of each other art1 .
Now in order to get acquainted with the tools introduced so far let us see how
the Petrov classification emerges in the present formalism.
> Example: In four dimensions, $d=4$, the operators $\textbf{C}_{3}$ and
> $\textbf{C}_{4}$ are trivially zero, as explained in section II. So we can
> only use $\textbf{C}_{2}$ to algebraically classify the Weyl tensor.
> According to theorem 1 it follows that
> $\textbf{C}_{2}=\textbf{C}^{+}\oplus\textbf{C}^{-}$, where $\textbf{C}^{+}$
> is independent from $\textbf{C}^{-}$, and both have vanishing trace. Since
> the space of 2-forms has six dimensions it follows that the operators
> $\textbf{C}^{\pm}$ act on 3-dimensional spaces, $S^{\pm}$. Then using the
> refined Segre classification (appendix A) it follows that the possible types
> for $\textbf{C}^{\pm}$ are $[1,1,1|]$, $[2,1|]$, $[(1,1),1|]$, $[|3]$,
> $[|2,1]$ and $[|1,1,1]$, these types are respectively called $I$, $II$, $D$,
> $III$, $N$ and $O$ (see reference art1 ). Note that the refined Segre type
> $[1,1|1]$ is also possible and in the Petrov classification it also
> corresponds to the type $I$, so that the refined Segre classification
> provides one more algebraic type than the Petrov classification.
>
> Thus the operator $\textbf{C}^{+}$ can have six Petrov types, just as
> $\textbf{C}^{-}$. Then the Weyl tensor can have $6\times 6=36$ Petrov types.
> But from an intrinsic point of view just $21$ types are different, because
> the spaces $S^{+}$ and $S^{-}$ are interchanged by a simple multiplication
> of the volume form, $\boldsymbol{\varepsilon}$, by $-1$. When the Weyl
> tensor is real and the signature Euclidean the operators $\textbf{C}^{\pm}$
> admit diagonal representations, thus $\textbf{C}^{\pm}$ can just have types
> $I$, $D$ or $O$. While if the Weyl tensor is real and the signature
> Lorentzian then the operators $\textbf{C}^{\pm}$ can have any type, but the
> type of $\textbf{C}^{+}$ must be the same of $\textbf{C}^{-}$ art1 .
Before proceed some comments about reality conditions are in order. For a
manifold $(M,g)$ of dimension $d=2n$ and signature $s$ if $(s/2)$ is even then
the eigenvalues of $\textbf{E}_{n}$ are real, $\pm 1$. In this case it is
possible to find real bases for both spaces $S^{+}$ and $S^{-}$. On the other
hand if $(s/2)$ is odd the eigenvalues of $\textbf{E}_{n}$ are $\pm i$ so that
the elements of $S^{+}$ and $S^{-}$ must be complex. In this last case if an
$n$-form is self-dual then its complex-conjugate will be anti-self-dual, and
vice versa. It is also important to note that if the Weyl tensor is real, as
assumed in the present article, then this reality condition generally
constrains the possible algebraic types of $\textbf{C}_{n}$. These constraints
depend on the signature of the metric, just as happens to $\textbf{C}_{2}$ in
four dimensions art1 .
Particularly in the Euclidean signature we have $s=d=2n$, so that if $n$ is
even then it is possible to define real bases $\\{\textbf{B}^{+i}\\}$ and
$\\{\textbf{B}^{-i}\\}$ for $S^{+}$ and $S^{-}$ respectively such that
$<\textbf{B}^{+i},\textbf{B}^{+j}>=\delta^{ij}=<\textbf{B}^{-i},\textbf{B}^{-j}>$
(see lemma 2). Using equation (9) we see that in these bases the operators
$\textbf{C}^{\pm}$ have matrix representations that are real and symmetric. In
turn, if the signature is Euclidean and $n$ is odd it is always possible to
find a basis $\\{\textbf{B}^{+i}\\}$ for $S^{+}$ such that
$\\{\overline{\textbf{B}^{+i}}\\}$ is a basis for $S^{-}$ and
$<\textbf{B}^{+i},\overline{\textbf{B}^{+j}}>=\delta^{ij}$. The basis
$\\{\textbf{B}^{+i}\\}$ can also always be arranged to be such that the matrix
representation of $\textbf{C}^{+}$ is diagonal with real eigenvalues. To prove
this note that if $\\{\textbf{Q}^{+i}\\}$ is a basis for $S^{+}$ then, since
$(s/2)$ is assumed to be odd, the complex conjugate of this basis will be a
basis for $S^{-}$. Now the positive definiteness of $g$ implies that
$<\textbf{Q}^{+i},\overline{\textbf{Q}^{+i}}>$ is greater than zero so that by
the Gram-Schmidt process we can find a basis $\\{\textbf{B}^{\prime+i}\\}$
such that
$<\textbf{B}^{\prime+i},\overline{\textbf{B}^{\prime+j}}>=\delta^{ij}$. In
this basis, because of equation (9), the matrix representation of
$\textbf{C}^{+}$ is easily seen to be Hermitean, thus by an unitary
transformation we can change the basis $\\{\textbf{B}^{\prime+i}\\}$ to
another basis $\\{\textbf{B}^{+i}\\}$ such that
$<\textbf{B}^{+i},\overline{\textbf{B}^{+j}}>=\delta^{ij}$ and in which the
representation of $\textbf{C}^{+}$ takes a diagonal form.
## IV Integrability of Maximally Isotropic Distributions and the Operator
$\textbf{C}_{n}$
The celebrated Goldberg-Sachs theorem in its generalized version states that
in a Ricci-flat four-dimensional manifold a maximally isotropic distribution
$\\{V_{1},V_{2}\\}$ is integrable if, and only if, the 2-form
$B_{\mu\nu}=V_{1[\mu}V_{2\nu]}$ is an eigen-2-form of the Weyl operator
$\textbf{C}_{2}$ art2 . The intent of this section is to generalize in some
extent this theorem to manifolds of even dimension greater than four, more
precisely it will be made a connection between the integrability of maximally
isotropic distributions and some algebraic restrictions on the map
$\textbf{C}_{n}$. Throughout this section the manifold $(M,g)$ will be assumed
to have even dimension $d=2n$.
A distribution of vector fields $\\{V_{1},V_{2},\ldots,V_{p}\\}$ on the
manifold $(M,g)$ is called isotropic or totally null when every vector field
tangent to this distribution has zero norm or, equivalently,
$g(V_{i},V_{j})=0$ for all $i,j\in\\{1,2,...,p\\}$. In a manifold of dimension
$d=2n$ the maximum dimension that an isotropic distribution can have is $n$.
If the signature is different from zero, $s\neq 0$, then a maximally isotropic
distribution is necessarily complex Trautman . From now on it will always be
assumed that the tangent bundle is complexified, so that the maximally
isotropic distributions can exist. Now let us make the following important
definitions:
Definitions: A simple $p$-form
$K^{\mu_{1}\ldots\mu_{p}}=V_{1}^{[\mu_{1}}V_{2}^{\mu_{2}}\ldots
V_{p}^{\mu_{p}]}$ is said to generate the distribution
$\\{V_{1},\ldots,V_{p}\\}$. This form will be denoted abstractly by
$\textbf{K}_{p}=\frac{1}{p!}(V_{1}\wedge V_{2}\wedge\ldots\wedge V_{p})$. When
the distribution generated by a non-zero simple $p$-form $\textbf{N}_{p}$ is
isotropic this form is called a null $p$-form.
In order to deal with totally null structures it is useful to introduce a null
frame $\\{e_{1},\ldots,e_{n},e_{n+1}=\theta^{1},\ldots,e_{2n}=\theta^{n}\\}$
on the tangent bundle, defined to be such that the only non-zero inner
products are
$g(e_{a^{\prime}},\theta^{b^{\prime}})=\frac{1}{2}\delta_{a^{\prime}}^{\phantom{a}b^{\prime}}$,
where $a^{\prime}\in\\{1,2,\ldots,n\\}$. In particular all the frame vectors,
$e_{a}$, are null and the distribution $\\{e_{1},e_{2},\ldots,e_{n}\\}$ is
maximally isotropic. In this article it will always be assumed that the
indices $a,b,\ldots$ pertain to $\\{1,2,\ldots,2n\\}$, while the indices
$a^{\prime},b^{\prime},\ldots$ pertain to $\\{1,2,\ldots,n\\}$. Given a tensor
$T_{\mu\nu}$, for example, then $T_{ab}$ will denote the components of this
tensor on the null frame, $T_{ab}\equiv e_{a}^{\;\mu}e_{b}^{\;\nu}T_{\mu\nu}$.
In reference HigherGSisotropic2 part of the Goldberg-Sachs(GS) theorem was
generalized to all dimensions greater than four and to all signatures. The
content of this generalized GS theorem can be put in the following form: In a
Ricci-flat manifold if the Weyl tensor is such that
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ and generic otherwise then the
maximally isotropic distribution generated by $(e_{1}\wedge
e_{2}\wedge\ldots\wedge e_{n})$ is locally integrable. The main purpose of
this section is to express the integrability condition
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ in terms of algebraic constraints on
operator $\textbf{C}_{n}$. To this end it is important to define the following
subbundles of $\wedge^{n}M$:
$\mathcal{A}_{p}\,=\,\\{\,\textbf{K}_{n}\in\wedge^{n}M\;|\;e_{a^{\prime}_{p}}\lrcorner\ldots
e_{a^{\prime}_{2}}\lrcorner
e_{a^{\prime}_{1}}\lrcorner\textbf{K}_{n}\,=\,0\;\;\forall\;a^{\prime}_{1},\ldots,a^{\prime}_{p}\in(1,\ldots,n)\,\\}\,.$
Where $(V\lrcorner\textbf{K})$ is the interior product of the vector field $V$
into the differential form K. In index notation the subbundle
$\mathcal{A}_{p}$ is the one formed by the $n$-forms $\textbf{K}_{n}$ such
that $K_{a^{\prime}_{1}a^{\prime}_{2}\ldots a^{\prime}_{p}b_{1}b_{2}\ldots
b_{n-p}}=0$. For example, $\mathcal{A}_{1}$ has dimension one and is generated
by the $n$-form $(e_{1}\wedge e_{2}\wedge\ldots\wedge e_{n})$. Note that the
definition of the bundles $\mathcal{A}_{p}$ depends on the choice of null
frame. Now expanding the equation $\textbf{C}_{n}(\textbf{K})=\textbf{L}$
using the index notation in the null frame yields:
$L_{d_{1}d_{2}\ldots d_{n}}\,=\,C^{ab}_{\phantom{ab}[d_{1}d_{2}}K_{d_{3}\ldots
d_{n}]ab}\,=\,C^{a^{\prime}b^{\prime}}_{\phantom{a^{\prime}b^{\prime}}[d_{1}d_{2}}K_{d_{3}\ldots
d_{n}]a^{\prime}b^{\prime}}\,+\,2C^{a^{\prime}}_{\phantom{a^{\prime}}b^{\prime}[d_{1}d_{2}}K_{d_{3}\ldots
d_{n}]a^{\prime}}^{\phantom{d_{3}\ldots
d_{n}]a^{\prime}}b^{\prime}}\,+\,C_{a^{\prime}b^{\prime}[d_{1}d_{2}}K_{d_{3}\ldots
d_{n}]}^{\phantom{d_{3}\ldots d_{n}]}a^{\prime}b^{\prime}}$
Taking care of the different types of indices it is possible to see that if
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ then all subspaces $\mathcal{A}_{p}$
are invariant under the action of the operator $\textbf{C}_{n}$. Conversely,
careful calculations show that if both subbundles $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ are invariant under $\textbf{C}_{n}$ then
$C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ (this result is most easily seen using
the notation introduced in section V). So we can state the following theorem
and its immediate corollary:
###### Theorem 2
The following three statements are equivalent: (1) The Weyl tensor obeys to
the integrability condition $C_{a^{\prime}b^{\prime}c^{\prime}d}=0$; (2) The
subbundles $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are invariant under the
action of $\textbf{C}_{n}$; (3) All subbundles $\mathcal{A}_{p}$,
$p\in\\{1,2,\ldots,n\\}$, are invariant by the action of $\textbf{C}_{n}$.
Corollary _In a Ricci-flat manifold of dimension $d=2n$ if the spaces
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are invariant under the operator
$\textbf{C}_{n}$ and the Weyl tensor is otherwise generic then the maximally
isotropic distribution generated by $(e_{1}\wedge e_{2}\wedge\ldots\wedge
e_{n})$ is locally integrable._
Where the main theorem of reference HigherGSisotropic2 was used to arrive at
the above corollary. Since $\mathcal{A}_{1}$ is the one-dimensional subbundle
generated by $e_{1}\wedge e_{2}\wedge\ldots\wedge e_{n}$, the invariance of
$\mathcal{A}_{1}$ is equivalent to say that this null $n$-form is an eigen-
form of the operator $\textbf{C}_{n}$. So, in particular, theorem 2 implies
that if the integrability condition of the distribution generated by
$e_{1}\wedge e_{2}\wedge\ldots\wedge e_{n}$ is satisfied then we have,
$\textbf{C}_{n}(e_{1}\wedge e_{2}\wedge\ldots\wedge e_{n})\propto e_{1}\wedge
e_{2}\wedge\ldots\wedge e_{n}$. The converse of this implication is true only
in four dimensions. Note also that since the bundles of self-dual and anti-
self-dual $n$-forms, $S^{\pm}$, are invariant under $\textbf{C}_{n}$ then if
$\mathcal{A}_{p}$ is invariant by this operator then the subbundles
$\mathcal{A}^{\pm}_{p}\equiv\mathcal{A}_{p}\cap S^{\pm}$ are also invariant.
These results strengthen the relevance and the utility of the operators
$\textbf{C}_{p}$ introduced on section II. Now let us see explicitly the
application of this theorem in four and six dimensions:
> Example($d=4$): In four-dimensional manifolds, $n=2$, we have
> $\mathcal{A}_{1}=\mathcal{A}^{+}_{1}=\textrm{Span}\\{e_{1}\wedge e_{2}\\}$,
> $\mathcal{A}^{+}_{2}=\textrm{Span}\\{e_{1}\wedge e_{2},(e_{1}\wedge
> e_{3}+e_{2}\wedge e_{4})\\}$ and
> $\mathcal{A}^{-}_{2}=\textrm{Span}\\{e_{1}\wedge e_{4},(e_{1}\wedge
> e_{3}-e_{2}\wedge e_{4}),e_{2}\wedge e_{3}\\}=S^{-}$. Looking at the
> representation of the operators $\textbf{C}^{\pm}$ given in reference art1
> we conclude that the invariance of these spaces under the Weyl operator
> $\textbf{C}_{n}$ is equivalent to the vanishing of the Weyl scalars
> $\Psi_{0}^{+}=C_{1212}$ and $\Psi_{1}^{+}=C_{1213}$, this is the content of
> the well known Goldberg-Sachs theorem Plebanski2 ; art2 .
> Example($d=6$): When the dimension of the manifold is six we have
> $\mathcal{A}_{1}=\mathcal{A}^{+}_{1}=\textrm{Span}\\{e_{1}\wedge e_{2}\wedge
> e_{3}\\}$, $\mathcal{A}^{+}_{2}=\textrm{Span}\\{e_{1}\wedge e_{2}\wedge
> e_{3},\,e_{1}\wedge(e_{2}\wedge e_{5}+e_{3}\wedge
> e_{6}),\,e_{2}\wedge(e_{1}\wedge e_{4}+e_{3}\wedge
> e_{6}),\,e_{3}\wedge(e_{1}\wedge e_{4}+e_{2}\wedge e_{5})\\}$,
> $\mathcal{A}^{-}_{2}=(\mathcal{A}^{+}_{2})^{\bot}\cap S^{-}$,
> $\mathcal{A}^{+}_{3}=S^{+}$ and
> $\mathcal{A}^{-}_{3}=(\mathcal{A}^{+}_{1})^{\bot}\cap S^{-}$. Where
> $\mathcal{A}^{\bot}$ is the orthogonal complement of $\mathcal{A}$ with
> respect to the inner product defined on (7). Because of equation (9) it
> follows from these orthogonality relations that impose the invariance of
> $\mathcal{A}^{+}_{1}$ is equivalent to impose the invariance of
> $\mathcal{A}^{-}_{3}$ as well as the invariance of $\mathcal{A}^{+}_{2}$ is
> equivalent to the invariance of $\mathcal{A}^{-}_{2}$. The fact that the
> invariance of $\mathcal{A}^{+}_{p}$ is equivalent to the invariance of
> $\mathcal{A}^{-}_{n+1-p}$ and the invariance of $\mathcal{A}^{-}_{p}$ is
> equivalent to the invariance of $\mathcal{A}^{+}_{n+1-p}$, for
> $p\in\\{1,2,\ldots,\frac{n+1}{2}\\}$, seems to be a general feature of the
> cases in which $n$ is odd. Now using theorem 2 we find that the
> integrability condition $C_{a^{\prime}b^{\prime}c^{\prime}d}=0$ is
> equivalent to the invariance of the spaces $\mathcal{A}^{+}_{1}$ and
> $\mathcal{A}^{+}_{2}$. This agrees perfectly with the results found on
> reference Spin6D by means of spinorial calculus.
Note from the first example, $d=2n=4$, that the invariance of the spaces
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ impose no restrictions on the operator
$\textbf{C}^{-}$. This happens because in this case $\mathcal{A}^{-}_{1}=0$
and $\mathcal{A}^{-}_{2}=S^{-}$, so that the invariance of these spaces is
always guaranteed. This phenomenon is exclusive of the four-dimensional case,
in higher dimensions the invariance of $\mathcal{A}_{2}$ imposes restrictions
on both $\textbf{C}^{+}$ and $\mathbf{C}^{-}$. This is already clear when $n$
is odd, since in these cases $\textbf{C}^{+}$ and $\textbf{C}^{-}$ carry the
same degrees of freedom, one is the transpose of the other, as proved on
theorem 1. But even if $n$ is even, besides the constraints on
$\textbf{C}^{+}$, we have that $\textbf{C}^{-}$ is constrained by the
integrability condition $C_{a^{\prime}b^{\prime}c^{\prime}d}=0$, which stems
from the fact that
$\dim(\mathcal{A}^{-}_{2})=\frac{1}{2}(n+n^{2})\leq\dim(S^{-})=\frac{1}{2}\left({}^{2n}_{\,\,n}\right)$.
Thus if $n>2$ then $\mathcal{A}^{-}_{2}$ is a proper subspace of $S^{-}$,
which implies that $\textbf{C}^{-}$ must admit a non-trivial invariant
subspace if such integrability condition is satisfied. This remarkable
difference between the well known four-dimensional case and the higher-
dimensional cases will be enunciated as a lemma:
###### Lemma 4
In four dimensions the integrability condition of the distribution generated
by the self-dual null $2$-form $(e_{1}\wedge e_{2})$ imposes restrictions only
on $\textbf{C}^{+}$, so that $\textbf{C}^{-}$ can be arbitrary. But in higher
dimensions the integrability condition of the maximally isotropic distribution
generated by the self-dual null $n$-form $(e_{1}\wedge e_{2}\wedge\ldots\wedge
e_{n})$ constrains both operators, $\textbf{C}^{+}$ and $\textbf{C}^{-}$.
## V An Elegant Notation
In this section it will be introduced a notation to deal with the operators
$\textbf{C}_{p}$ that is elegant and promising. Such notation will make clear
that these operators are intimately related to the integrability of
distributions.
Let us denote by $\\{e^{a}\\}$ the frame of 1-forms dual to the null frame
$\\{e_{a}\\}$, $e^{a}(e_{b})=\delta^{a}_{\phantom{a}b}$. Then for a vanishing
Ricci tensor it follows that the Cartan structure equations are:
$\textbf{d}e^{a}+\omega^{a}_{\phantom{a}b}\wedge
e^{b}=0\quad\quad\textrm{and}\quad\quad\mathbb{C}^{a}_{\phantom{a}b}=\textbf{d}\omega^{a}_{\phantom{a}b}+\omega^{a}_{\phantom{a}c}\wedge\omega^{c}_{\phantom{c}b}\,.$
Where
$\mathbb{C}^{a}_{\phantom{a}b}\,\equiv\,\frac{1}{2}\,C^{a}_{\phantom{a}bcd}\,e^{c}\wedge
e^{d}$ are the curvature 2-forms when the Ricci tensor vanishes and
$\omega^{a}_{\phantom{a}b}$ are the connection 1-forms of the Levi-Civita
connection, i.e., 1-forms defined by the relation
$\nabla_{V}e_{b}=\omega^{a}_{\phantom{a}b}(V)e_{a}$ for any vector field $V$.
Taking the exterior derivative of the second Cartan equation it is simple
matter to see that:
$\textbf{d}\mathbb{C}^{a}_{\phantom{a}b}\,=\,\mathbb{C}^{a}_{\phantom{a}c}\wedge\omega^{c}_{\phantom{c}b}-\omega^{a}_{\phantom{a}c}\wedge\mathbb{C}^{c}_{\phantom{c}b}\,.$
(11)
If $\textbf{K}_{p}$ is a $p$-form then we can associate to it a set of
$(p-2)$-forms $\mathbb{K}^{\phantom{a}b}_{a}$, with
$a,b\in\\{1,2,\ldots,d\\}$, defined by:
$\mathbb{K}^{\phantom{a}b}_{a}\,\equiv\,\frac{2}{p!}\,K^{\phantom{a}b}_{a\phantom{ab}c_{1}c_{2}\ldots
c_{p-2}}\,e^{c_{1}}\wedge e^{c_{2}}\wedge\ldots\wedge e^{c_{p-2}}\,.$
Now using the just defined objects note that
$\mathbb{C}^{a}_{\phantom{a}b}\wedge\mathbb{K}^{\phantom{a}b}_{a}\,=\,\frac{1}{p!}C^{a}_{\phantom{a}bc_{1}c_{2}}K^{\phantom{a}b}_{a\phantom{b}c_{3}c_{4}\ldots
c_{p}}\,e^{c_{1}}\wedge e^{c_{2}}\wedge\ldots\wedge
e^{c_{p}}=\frac{1}{p!}C^{ab}_{\phantom{ab}[c_{1}c_{2}}K_{c_{3}c_{4}\ldots
c_{p}]ab}\,e^{c_{1}}\wedge e^{c_{2}}\wedge\ldots\wedge e^{c_{p}}\,.$
By definition of $\textbf{C}_{p}$, equation (5), the last relation implies the
following elegant form to express the action of this operator:
$\textbf{C}_{p}(\textbf{K}_{p})\,=\,\mathbb{C}^{a}_{\phantom{a}b}\wedge\mathbb{K}^{\phantom{a}b}_{a}\,.$
(12)
Now we are ready to see one more hint that the operator $\textbf{C}_{p}$ is
connected to the integrability of distributions. First let us define the
$(p-1)$-form $\textbf{D}\mathbb{K}^{\phantom{a}b}_{a}$ by
$\textbf{D}\mathbb{K}^{\phantom{a}b}_{a}\equiv\textbf{d}\mathbb{K}^{\phantom{a}b}_{a}+\omega^{b}_{\phantom{b}c}\wedge\mathbb{K}_{a}^{\phantom{a}c}-\omega^{c}_{\phantom{c}a}\wedge\mathbb{K}_{c}^{\phantom{c}b}$.
Then using the definition of $\mathbb{K}^{\phantom{a}b}_{a}$ and using the
metric to lower (or raise) the indices it easily follows that
$\textbf{K}_{p}=\frac{1}{2}\mathbb{K}_{ab}\wedge e^{a}\wedge e^{b}$. Taking
the exterior derivative of this relation and using the first structure
equation we find $\textbf{d}\textbf{K}_{p}=\frac{1}{2}e^{a}\wedge
e^{b}\wedge\textbf{D}\mathbb{K}_{ab}$, where $\textbf{D}\mathbb{K}_{ab}\equiv
g_{bc}\textbf{D}\mathbb{K}^{\phantom{a}c}_{a}$. Also taking the exterior
derivative of equation (12) and using equation (11) it follows that
$\textbf{d}\left[\textbf{C}_{p}(\textbf{K}_{p})\right]=\mathbb{C}^{ab}\wedge\textbf{D}\mathbb{K}_{ab}$.
Let us summarize these important relations:
$\textbf{d}\textbf{K}_{p}\,=\,\frac{1}{2}\,e^{a}\wedge
e^{b}\wedge\textbf{D}\mathbb{K}_{ab}\quad\quad;\quad\quad\textbf{d}\left[\textbf{C}_{p}(\textbf{K}_{p})\right]\,=\,\mathbb{C}^{ab}\wedge\textbf{D}\mathbb{K}_{ab}\,.$
(13)
Suppose that a simple $p$-form $\textbf{L}_{p}$ is in the kernel of
$\textbf{C}_{p}$, $\textbf{C}_{p}(\textbf{L}_{p})=0$, and that
$C_{ab}^{\phantom{ab}cd}\textbf{D}\mathbb{L}_{cd}=\lambda\textbf{D}\mathbb{L}_{ab}$,
with $\lambda\neq 0$. Then equation (13) implies that
$0=\mathbb{C}^{ab}\wedge\textbf{D}\mathbb{L}_{ab}=\frac{1}{2}C^{ab}_{\phantom{ab}cd}e^{c}\wedge
e^{d}\wedge\textbf{D}\mathbb{L}_{ab}=\frac{\lambda}{2}e^{c}\wedge
e^{d}\wedge\textbf{D}\mathbb{L}_{cd}$. Introducing this in the first relation
of (13) we see that the simple form $\textbf{L}_{p}$ is closed,
$\textbf{d}\textbf{L}_{p}=0$. This, in turn, implies that the distribution
annihilated by $\textbf{L}_{p}$ is integrable (more about this on section
VII).
Now if $\textbf{K}_{p}$ is a $p$-form such that
$\textbf{D}\mathbb{K}_{a}^{\phantom{a}b}=\kappa\wedge\mathbb{K}_{a}^{\phantom{a}b}$
for some 1-form $\kappa$, then equation (13) implies that
$\textbf{d}\textbf{K}_{p}=\kappa\wedge\textbf{K}_{p}$ and
$\textbf{d}\left[\textbf{C}_{p}(\textbf{K}_{p})\right]=\kappa\wedge\textbf{C}_{p}(\textbf{K}_{p})$.
So if $\textbf{K}_{p}$ is a simple $p$-form then the first equation implies
that the distribution annihilated by $\textbf{K}_{p}$ is integrable and,
analogously, if $\textbf{C}_{p}(\textbf{K}_{p})$ is a simple $p$-form then
second equality implies that the distribution annihilated by
$\textbf{C}_{p}(\textbf{K}_{p})$ is integrable. Thus connecting the operator
$\textbf{C}_{p}$ with one more integrability property.
These are just simple results that follow from equation (13), but the elegance
of the expressions obtained in this section together with the results of
appendix B seems to denounce that the notation introduced here can be
exploited to find extensions of the Goldberg-Sachs theorem. Hopefully this
will be done elsewhere.
## VI Other Classification Methods
The intent of the present section is to briefly digress about other forms of
classifying the curvature of the tangent bundle and, when possible, relate in
some way these classification schemes to the classification for the Weyl
tensor defined in the preceding sections.
The CMPP classification is the most widespread and successful scheme of
classification for the Weyl tensor in higher-dimensional general relativity
CMPP ; Coley . Such scheme makes use of a one-dimensional subgroup of the
orthogonal transformations on the tangent bundle to split the Weyl tensor as a
sum of terms with different boost weights. This subgroup is determined once a
real null vector field, $l$, is chosen. If it is possible to choose $l$ in
order to annihilate the components of the Weyl tensor with boost weight 2 then
$l$ is said to be a Weyl aligned null direction(WAND). Further, if this null
direction is such that all Weyl tensor components with boost weights 2 and 1
vanish then $l$ is called a multiple WAND.
Let $e_{1}$ and $\theta^{1}$ be real vector fields such that
$\\{e_{1},e_{2},\theta^{1},\theta^{2}\\}$ is a null frame in a 4-dimensional
Lorentzian manifold. It is a established result that the null 2-form
$e_{1}\wedge e_{2}$ is an eigen-2-form of operator $\textbf{C}_{2}$ if, and
only if, the vector field $e_{1}$ is a multiple WAND art2 . Simple
calculations also reveal that in 5-dimensional spacetimes if $\textbf{C}_{2}$
admits a null eigen-2-form then the real null direction tangent to the
isotropic plane generated by this 2-form will be a multiple WAND. But the
converse of this result is not true anymore, i.e., in 5 dimensions the
existence of a multiple WAND does not imply the existence of a null
eigen-2-form for $\textbf{C}_{2}$. Now it is natural wondering whether similar
relations are verified in higher-dimensional Lorentzian manifolds. Using the
notation introduced in the last section it can be proved that if
$\boldsymbol{N}=e_{1}\wedge e_{2}\wedge\ldots\wedge e_{n}$ is a null $n$-form
in a Lorentzian manifold of dimension $d=2n+\epsilon$, with $\epsilon$ equal
to 0 or 1 and $n\geq 3$, such that $e_{1}$ is a real vector field then the
equation $\textbf{C}_{n}\left(\boldsymbol{N}\right)\propto\boldsymbol{N}$ does
not imply that $e_{1}$ is a WAND, let alone a multiple WAND. Conversely, if
$e_{1}$ is a multiple WAND then generally it is not true that
$\textbf{C}_{n}\left(\boldsymbol{N}\right)\propto\boldsymbol{N}$.
Particularly, these results can be easily verified in 6 dimensions by means of
the spinorial formalism of reference Spin6D .
Although it was argued in the last paragraph that there is no immediate
relation between a WAND and an eigen-$p$-form of the operator $\textbf{C}_{p}$
in higher-dimensional spacetimes, these Weyl operators are still valuable for
the CMPP classification. Just as reference BivectHighDim has used the well
known bivector operator $\textbf{C}_{2}$ to refine this classification, the
other operators $\textbf{C}_{p}$ can analogously be used to further refine the
CMPP classification. For this task the general aspects of the operator method
for classifying tensors in higher-dimensional manifolds described in
BivectHighDim ; Hervik-Coley could be helpful.
Another useful classification for the Weyl tensor is the Taghavi-Chabert
classification HigherGSisotropic2 . As well as CMPP classification is
associated to a null direction, the Taghavi-Chabert classification needs a
maximally isotropic distribution to be chosen. According to this
classification the Weyl tensor of an even-dimensional manifold ($d=2n\geq 6$)
can be of the following types, from the most general to the most special:
$\mathcal{C}^{-2}$, $\mathcal{C}^{-1}$, $\mathcal{C}^{0}$, $\mathcal{C}^{1}$,
$\mathcal{C}^{2}$ and $\mathcal{C}^{3}$. Using the formalism of section IV it
can be proved that if the subbundle $\mathcal{A}_{1}\subset\wedge^{n}M$ is
invariant under the operator $\textbf{C}_{n}$ then the Weyl tensor is more
special than type $\mathcal{C}^{-2}$, but the converse is not true.
Analogously, theorem 2 implies that the Weyl tensor is type $\mathcal{C}^{0}$,
or more special, if, and only if, both subbundles $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ are invariant under $\textbf{C}_{n}$. It would be
interesting, and hopefully valuable, to use the algebraic classification
introduced in section II to refine the Taghavi-Chabert classification, i.e.,
investigate how each of the types of the latter classification constrain the
refined Segre types of the operators $\textbf{C}_{p}$.
Finally, a geometric well known form of classifying the curvature of the
tangent bundle is using the holonomy group associated to the Levi-Civita
connection. In a connected manifold the holonomy is the same at all points
Nakahara , thus providing a global classification, whereas previously seen
classifications are local333For example, in Pravda05 it was shown that the
CMPP type on the horizon of a 5-dimensional black ring spacetime is different
from the CMPP type outside the horizon.. Particularly, in four-dimensional
Lorentzian manifolds the holonomy classification has been widely studied and
compared with Petrov classification Hall-Schell , in which case equation (6),
with $d=4$ and $p=2$, is behind some simplifications of this analysis. In
higher-dimensional Lorentzian manifolds important developments have recently
been achieved on this subject, see for example Galaev ; Leistner and
references therein.
Purely gravitational solutions of supergravity can be put in one-to-one
correspondence with Ricci-flat Lorentzian manifolds admitting a covariantly
constant spinor M-waves . By means of the Ambrose-Singer theorem it follows
that the integrability condition for a constant spinor imposes restrictions on
the holonomy444Generally the existence of a covariantly constant tensor on the
manifold imposes restrictions on the holonomy. For instance, Kähler manifolds
are characterized by the existence of a covariantly constant complex structure
and because of this their holonomy are subgroups of $U(n)\subset SO(2n)$,
where $n$ is the complex dimension of the manifold. M-waves . Therefore the
holonomy group is of fundamental importance to supergravity and, consequently,
to string theory. Holonomy classification also provides powerful tools to find
solutions to Einstein’s equation Galaev . Moreover, holonomy is of major
relevance to loop quantum gravity LoopQG , although in a broader sense, since
the holonomy may be associated to the curvature of a gauge field.
## VII Generalized Mariot-Robinson Theorem
The Mariot-Robinson theorem states that in four dimensions a null 2-form,
$\textbf{F}^{\prime}$, generates a locally integrable distribution of planes
if, and only if, there exists some function $h\neq 0$ such that
$\textbf{F}=h\textbf{F}^{\prime}$ satisfies Maxwell’s equations,
$\textbf{d}\textbf{F}=0$ and $\textbf{d}(\widetilde{\textbf{F}})=0$ Robinson ;
McIntoshII . Where $\widetilde{\textbf{F}}$ is the Hodge dual of F and d is
the exterior derivative operator. In the present section this theorem will be
generalized in a natural way to all dimensions.
Following similar steps to the ones taken in McIntoshII let
$\\{\omega^{1},\ldots,\omega^{p}\\}$ be linearly independent 1-forms of a
$d$-dimensional manifold such that
$\textbf{d}(h\omega^{1}\wedge\ldots\wedge\omega^{p})=0$ for some function
$h\neq 0$. Expanding this equation and taking the wedge product with
$\omega^{i}$ it is immediate to see that
$(\textbf{d}\omega^{i})\wedge\omega^{1}\wedge\ldots\wedge\omega^{p}=0$. It is
a standard result of differential geometry that this implies the local
integrability of the distribution of vector fields annihilated by
$\\{\omega^{1},\ldots,\omega^{p}\\}$ Frankel . Conversely, if the distribution
annihilated by $\\{\omega^{i}\\}$ is integrable then it follows that there
exist $p^{2}$ functions $f^{i}_{\phantom{i}j}$ and $p$ coordinates $x^{j}$
such that $\omega^{i}=f^{i}_{\phantom{i}j}\textbf{d}x^{j}$, with
$\det(f^{i}_{\phantom{i}j})=f\neq 0$. Then it is easy to see that
$\textbf{d}(\frac{1}{f}\omega^{1}\wedge\ldots\wedge\omega^{p})=\textbf{d}(\textbf{d}x^{1}\wedge\ldots\wedge\textbf{d}x^{p})=0$.
Thus we arrived at the following result: _The distribution of vector fields
annihilated by $\\{\omega^{1},\ldots,\omega^{p}\\}$ is integrable if, and only
if, there exists some function $h\neq 0$ such that_
$\textbf{d}(h\omega^{1}\wedge\ldots\wedge\omega^{p})=0$.
Now let us suppose that the dimension of the manifold is even, $d=2n$. As
defined in section IV, an $n$-form $\textbf{N}_{n}$ is called null if it can
be written as
$N^{\mu_{1}\mu_{2}\ldots\mu_{n}}=V_{1}^{[\mu_{2}}V_{1}^{\mu_{2}}\ldots
V_{n}^{\mu_{n}]}$ where the vector fields $\\{V_{1},\ldots,V_{n}\\}$ form an
isotropic distribution of dimension $n$. Such distribution is said to be
generated by $\textbf{N}_{n}=V_{1}\wedge\ldots\wedge V_{n}$ and in this case,
since the isotropic distribution is maximal, it is also the distribution
annihilated by $\textbf{N}_{n}$. As proved in appendix B null $n$-forms must
be self-dual or anti-self-dual, meaning that
$\textbf{E}_{n}(\textbf{N}_{n})\equiv\widetilde{\textbf{N}}_{n}=\pm\epsilon\,\textbf{N}_{n}$,
where $\epsilon$ is equal to $1$ or $i$ depending on the signature of the
manifold. So for a null $n$-form the equation $\textbf{d}\textbf{N}_{n}=0$ is
equivalent to the equation $\textbf{d}\widetilde{\textbf{N}}_{n}=0$. Thus
using the results of this and of the last paragraph we arrive at the following
theorem:
###### Theorem 3
In a space of dimension $d=2n$, the null n-form $\textbf{N}_{n}^{\prime}$
generates an integrable maximally isotropic distribution if, and only if,
there exists some function $h\neq 0$ such that
$\textbf{N}_{n}=h\textbf{N}_{n}^{\prime}$ obeys to the equations
$\textbf{d}\textbf{N}_{n}=0$ and $\textbf{d}\widetilde{\textbf{N}}_{n}=0$.
It is important to note that, contrary to the Goldberg-Sachs theorem, no
condition on the Ricci tensor was assumed. This theorem was proved before on
reference Kerr-Robinson , but there the proof uses spinor and twistor calculus
and is less concise.
## VIII Optical Matrix and the Forms that are Closed and Co-closed
Let $\\{l,n,m_{2},m_{3},\ldots,m_{(d-1)}\\}$ be a semi-null frame of the
tangent bundle such that the only non-zero inner products are $g(l,n)=1$ and
$g(m_{i},m_{j})=\delta_{ij}$, in particular the vectors $l$ and $n$ are null.
Then denoting by $\nabla$ the Levi-Civita connection let us define the
following quantities:
$M_{0}\,=\,l^{\nu}n^{\mu}\,\nabla_{\nu}l_{\mu}\
\;\;\;;\;\;M_{i}\,=\,l^{\nu}m_{i}^{\mu}\,\nabla_{\nu}l_{\mu}\
\;\;\;;\;\;M_{ij}\,=\,m_{j}^{\nu}m_{i}^{\mu}\,\nabla_{\nu}l_{\mu}\ \,.$ (14)
Note that in Lorentzian signature the vectors of the frame $\\{l,n,m_{i}\\}$
can be chosen to be real, so that the quantities $M_{0}$, $M_{i}$ and $M_{ij}$
are real in this case. The $(d-2)\times(d-2)$ matrix $M_{ij}$ is called the
optical matrix and is usually decomposed as the sum of a symmetric trace-less
matrix $\sigma_{ij}$, the shear, a skew-symmetric matrix $A_{ij}$, the twist,
and a term proportional to the identity matrix $\theta\delta_{ij}$, the
expansion.
$M_{ij}\,=\,\sigma_{ij}\,+\,A_{ij}\,+\,\theta\delta_{ij}\;;\quad\theta=\frac{1}{d-2}\delta^{ij}M_{ij}\;;\quad\sigma_{ij}=M_{(ij)}-\theta\delta_{ij}\;;\quad
A_{ij}=M_{[ij]}$ (15)
It is easy matter to establish that the null congruence generated by $l$ is
geodesic if, and only if, $M_{i}=0$ and the parametrization is affine when
$M_{0}=0$. Also, straightforward calculations show that the congruence
generated by $l$ is hyper-surface-orthogonal,
$l_{[\alpha}\nabla_{\mu}l_{\nu]}=0$, if, and only if, $M_{i}$ and $A_{ij}$
both vanish.
Now let $\textbf{K}_{p}$ be a non-zero $p$-form obeying to the equations
$\textbf{dK}_{p}=0$ and $\textbf{d}\widetilde{\textbf{K}}_{p}=0$ and such that
$l$ is a multiple aligned null direction CMPP ; Coley of $\textbf{K}_{p}$.
Using index notation these constraints are respectively given by:
$\nabla_{[\alpha}\,K_{\mu_{1}\mu_{2}\ldots\mu_{p}]}=0\;\;;\quad\nabla^{\alpha}\,K_{\alpha\mu_{2}\ldots\mu_{p}}=0\;\;;\quad
K^{\mu_{1}\mu_{2}\ldots\mu_{p}}=p!f_{j_{2}j_{3}\ldots
j_{p}}\,l^{[\mu_{1}}m_{j_{2}}^{\mu_{2}}m_{j_{3}}^{\mu_{3}}\ldots
m_{j_{p}}^{\mu_{p}]}\,.$ (16)
Where $f_{j_{2}j_{3}\ldots j_{p}}$ is completely antisymmetric and in the last
relation it is being assumed a sum over the indices $j_{2}j_{3}\ldots j_{p}$.
Let us prove that the existence of a non-zero $p$-form that satisfies equation
(16) imposes restrictions on the optical matrix as well as on $M_{i}$.
Developing the equation
$0=(\nabla^{\alpha}K_{\alpha\mu_{2}\ldots\mu_{p}})l^{\mu_{2}}=-K_{\alpha\mu_{2}\ldots\mu_{p}}\nabla^{\alpha}l^{\mu_{2}}$
the following results can be obtained:
$M_{i}\,f_{ij_{3}\ldots j_{p}}\,=\,0\;\;;\quad A_{ij}f_{ijk_{4}\ldots
k_{p}}\,=\,0\,.$ (17)
Analogously, expanding the equation
$0=\nabla_{[\alpha}\,K_{\mu_{1}\mu_{2}\ldots\mu_{p}]}l^{\alpha}m_{j_{1}}^{\phantom{j_{1}}\mu_{1}}\ldots
m_{j_{p}}^{\phantom{j_{p}}\mu_{p}}$ we get:
$M_{[j_{1}}\,f_{j_{2}\ldots j_{p}]}\,=\,0\,.$
Note that contracting the above equation with $M_{j_{1}}$ and using equation
(17) we get that $M_{j}M_{j}=0$. In the particular case of Lorentzian
signature the frame $\\{l,n,m_{i}\\}$ can be real, then $M_{i}$ is real and
the relation $M_{j}M_{j}=0$ implies that $M_{j}$ vanish, i.e., the null
congruence generated by $l$ is geodesic. On the same vein, after workout the
equality
$0=(\nabla^{\alpha}\,K_{\alpha\mu_{2}\ldots\mu_{p}})m_{j_{2}}^{\phantom{j_{2}}\mu_{2}}\ldots
m_{j_{p}}^{\phantom{j_{p}}\mu_{p}}$ it follows that:
$K_{\alpha\mu_{2}\ldots\mu_{p}}\nabla^{\alpha}(m_{j_{2}}^{\phantom{j_{2}}\mu_{2}}\ldots
m_{j_{p}}^{\phantom{j_{p}}\mu_{p}})\,=\,(p-1)!\,\,l^{\alpha}\nabla_{\alpha}f_{j_{2}\ldots
j_{p}}\,+\,(p-1)!\,f_{j_{2}\ldots j_{p}}\nabla^{\alpha}l_{\alpha}\,.$ (18)
Now expanding the relation
$0=\left(\nabla_{[\alpha}\,K_{\mu_{1}\mu_{2}\ldots\mu_{p}]}\right)l^{\alpha}n^{\mu_{1}}m_{j_{2}}^{\phantom{j_{2}}\mu_{2}}\ldots
m_{j_{p}}^{\phantom{j_{p}}\mu_{p}}$, using the identity
$\nabla^{\alpha}l_{\alpha}=M_{0}+(d-2)\theta$ and the equation (18) it
follows, after some algebra, that:
$2(p-1)\,f_{i[j_{3}\ldots
j_{p}}\,\sigma_{j_{2}]i}\;=\;(d-2p)\,\theta\,f_{j_{2}\ldots j_{p}}\,.$
These results are summarized by the following theorem:
###### Theorem 4
If $K^{\mu_{1}\mu_{2}\ldots\mu_{p}}=p!f_{j_{2}\ldots
j_{p}}\,l^{[\mu_{1}}m_{j_{2}}^{\mu_{2}}\ldots m_{j_{p}}^{\mu_{p}]}$ is a non-
zero $p$-form such that $\textbf{dK}_{p}=0$ and
$\textbf{d}\widetilde{\textbf{K}}_{p}=0$ then it follows that:
* •
$M_{i}\,f_{ij_{3}\ldots j_{p}}\,=\,0$
* •
$M_{[j_{1}}\,f_{j_{2}\ldots j_{p}]}\,=\,0$
* •
$2(p-1)\,f_{i[j_{3}\ldots
j_{p}}\,\sigma_{j_{2}]i}\;=\;(d-2p)\,\theta\,f_{j_{2}\ldots j_{p}}$
* •
$M_{i}M_{i}\,=\,0$
* •
$A_{ij}f_{ijk_{4}\ldots k_{p}}\,=\,0$
In the particular case of a real frame $\\{l,n,m_{i}\\}$ (Lorentzian
signature) it follows from the fourth point above that $M_{i}=0$, i.e., the
vector field $l$ is geodesic.
Particularly, if the signature is Lorentzian and the dimension is four, $d=4$,
then the case $p=2$ of the above theorem implies that if a real null bivector
F obeys to the source-free Maxwell’s equations then the real null direction
$l$ such that $F_{\mu\nu}l^{\nu}=0$ generates a congruence that is geodesic
and shear-free (shear-free meaning that $\sigma_{ij}=0$), a classical result
first obtained a long time ago by Ivor Robinson Robinson . In the Lorentzian
signature the case $p=2$ of the above theorem was obtained before on reference
M. Ortaggio-Robinson-Trautman . Also in the Lorentzian case these results can
be easily obtained using the generalized GHP formalism of reference GHP
555Thanks to Harvey S. Reall for gently pointing out this reference.. More
precisely the first three points of the above theorem can be found in the
proof of Lemma 3 of reference GHP , whereas the last two points of the above
theorem are not explicit in GHP , although can be easily derived using the GHP
formalism. But it is worth remarking that there is an important difference
between the Lorentzian and the non-Lorentzian signatures: while in the former
case the equation $M_{i}M_{i}=0$ implies that the null vector field $l$ is
geodesic in the latter this is not true in general. Note also that in the
Euclidean case we have that $n$ is the complex conjugate of $l$, therefore in
this signature any constraint on the optical matrix of $l$ imposes an
analogous constraint on the optical matrix of $n$.
## IX A Generalized Version of the Goldberg-Sachs Theorem
Using the previous results it will be proved in this section a generalized
version of the Goldberg-Sachs theorem that is valid in any even dimension and
that makes no assumption about the Ricci tensor, contrary to the usual
generalizations of such theorem. The theorem presented here states that if
there exists an integrable maximally isotropic distribution on a manifold of
even dimension then the shear matrix of the null directions tangent to such
distribution must be constrained.
Before proceeding let us introduce some notation that will be used in what
follows. In a manifold of dimension $d=2n$ let
$\\{e_{1},e_{2},\ldots,e_{n}\\}$ be a maximally isotropic distribution of
vector fields. We can complete this distribution to form a null frame
$\\{e_{1},\ldots,e_{n},e_{n+1}=\theta^{1},\ldots,e_{2n}=\theta^{n}\\}$ such
that the only non-zero inner products are
$g(e_{a^{\prime}},\theta^{b^{\prime}})=\frac{1}{2}\delta_{a^{\prime}}^{\phantom{a}b^{\prime}}$.
From this frame we can construct a semi-null frame, like the one used in the
latter section, $\\{l,n,m_{2},m_{3},\ldots,m_{(d-1)}\\}$ defined by:
$l=e_{1}\;\;;\;\;n=2\theta^{1}\;\;;\;\;m_{j}=(e_{j}+\theta^{j})\;\;;\;\;m_{j+n-1}=-i(e_{j}-\theta^{j})\;,\;\;\;\textrm{where}\;\;j\,\in\,\\{2,3,\ldots,n\\}\,.$
Note that in general the choice of the vector $e_{1}$ to be $l$ is arbitrary,
we could have chosen any vector on the distribution $\\{e_{1},\ldots,e_{n}\\}$
to take this place. In the special case of Lorentzian signature there is a
privileged choice, since in this case a maximally isotropic subspace admits
just one real direction Trautman , in the present section whenever the
signature is Lorentzian the real direction will be assumed to be tangent to
the vector field $e_{1}=l$.
Once defined such semi-null frame we can define the optical matrix, the shear,
the twist and the expansion associated to the distribution
$\\{e_{1},\ldots,e_{n}\\}$ to be just as defined in equations (14) and (15).
Also from this maximally isotropic distribution we can construct the null
$n$-form $\textbf{N}_{n}=e_{1}\wedge e_{2}\wedge\ldots\wedge e_{n}$. In the
semi-null frame just defined this $n$-form has the following expansion:
$N^{\mu_{1}\mu_{2}\ldots\mu_{n}}\,\equiv\,n!\,e_{1}^{\,[\mu_{1}}\ldots
e_{n}^{\,\mu_{n}]}\,=\,n!\,\widehat{f}_{j_{2}j_{3}\ldots
j_{n}}l^{[\mu_{1}}m_{j_{2}}^{\,\mu_{2}}\ldots m_{j_{n}}^{\,\mu_{n}]}.$ (19)
With $\widehat{f}_{j_{2}j_{3}\ldots j_{n}}=\widehat{f}_{[j_{2}j_{3}\ldots
j_{n}]}$ different from zero only when each index $j_{r}$ is equal to $r$ or
to $r+n-1$ (and permutations of this), in which case
$\widehat{f}_{j_{2}j_{3}\ldots j_{n}}=2^{1-n}\,i^{q}$ where $q$ is the number
of indices $j_{r}$ that are equal to $r+n-1$. For example, in six dimensions,
$n=3$, the non-zero components of $\widehat{f}_{j_{2}j_{3}}$ are:
$\widehat{f}_{23}=-\widehat{f}_{32}=\frac{1}{4}\;\;;\;\;\widehat{f}_{25}=-\widehat{f}_{52}=\frac{i}{4}\;\;;\;\;\widehat{f}_{43}=-\widehat{f}_{34}=\frac{i}{4}\;\;;\;\;\widehat{f}_{45}=-\widehat{f}_{54}=-\frac{1}{4}\,.$
(20)
Now we are able to prove the main result of this section. Suppose that the
maximally isotropic distribution $\\{e_{1},\ldots,e_{n}\\}$ is integrable. So
from what was seen in section VII there exists some function $h\neq 0$ such
that $\textbf{d}(h\textbf{N}_{n})=0$ and $\textbf{
d}(h\widetilde{\textbf{N}}_{n})=0$, where $\textbf{N}_{n}=e_{1}\wedge
e_{2}\wedge\ldots\wedge e_{n}$. Thus using theorem 4 and equation (19) it
follows that $h\widehat{f}_{i[j_{3}\ldots j_{n}}\,\sigma_{j_{2}]i}=0$, which
leads to the following theorem:
###### Theorem 5 (Generalized GS)
In a manifold of even dimension $d=2n$ if the maximally isotropic distribution
$\\{e_{1},\ldots,e_{n}\\}$ is integrable then the shear matrix associated to
this distribution, $\sigma_{ij}$, is constrained by the following equation:
$\widehat{f}_{i[j_{3}j_{4}\ldots j_{n}}\,\sigma_{j_{2}]i}\;=\;0\,.$
Also, the twist matrix, $A_{ij}$, and the scalars $M_{i}$ are constrained by
the following relations:
$A_{ij}\widehat{f}_{ijk_{4}\ldots k_{n}}\,=\,0\quad;\quad
M_{i}\,\widehat{f}_{ij_{3}\ldots j_{n}}=0\quad;\quad
M_{[j_{1}}\,\widehat{f}_{j_{2}\ldots j_{n}]}=0\quad;\quad M_{i}M_{i}=0.$
Particularly, from this last relation, $M_{i}M_{i}=0$, it follows that
$e_{1}=l$ must be geodesic in the Lorentzian signature.
It is worth noting that in this theorem no condition was imposed on the Ricci
tensor. It is also relevant to mention that in the appendix C of reference GS-
Ortaggio12 the integrability of a maximally isotropic distribution is
expressed in terms of the Ricci rotation coefficients of a null frame. Now let
us see an example of the use of theorem 5.
> Example: In six dimensions, $d=6$, if a maximally isotropic distribution
> $\\{e_{1},e_{2},e_{3}\\}$ is integrable then it follows from theorem 5 that
> $\widehat{f}_{i[j}\sigma_{k]i}=0$, i.e.,
> $\widehat{f}_{ij}\sigma_{ki}=\widehat{f}_{ik}\sigma_{ji}$. Denoting by
> $\boldsymbol{f}$ the matrix $\widehat{f}_{ij}$ and by $\boldsymbol{\sigma}$
> the matrix $\sigma_{ij}$ then since $\boldsymbol{f}$ is skew-symmetric and
> $\boldsymbol{\sigma}$ is symmetric it follows that this equation is
> equivalent to
> $\boldsymbol{f}\boldsymbol{\sigma}=-\boldsymbol{\sigma}\boldsymbol{f}$. From
> equation (20) it follows that we can write:
>
> $\boldsymbol{f}\,=\,\frac{1}{4}\left[\begin{array}[]{cccc}0&1&0&i\\\
> -1&0&-i&0\\\ 0&i&0&-1\\\ -i&0&1&0\\\ \end{array}\right]\,.$
>
> Then using the fact that the matrix $\boldsymbol{\sigma}$ is symmetric and
> imposing that it anti-commutes with $\boldsymbol{f}$ we find that the shear
> matrix must be of the following form:
>
> $\boldsymbol{\sigma}\,=\,\left[\begin{array}[]{cccc}\Delta&\Theta&0&\Phi\\\
> \Theta&-\Delta&-\Phi&0\\\ 0&-\Phi&\Delta&\Theta\\\ \Phi&0&\Theta&-\Delta\\\
> \end{array}\right]\,.$
>
> Where $\Delta$, $\Theta$ and $\Phi$ are arbitrary functions. Calculating the
> eigenvalues of $\boldsymbol{\sigma}$ we find
> $\pm\sqrt{\Delta^{2}+\Theta^{2}+\Phi^{2}}$, so the shear must have two
> degenerate eigenvalues. It must be stressed that by orthonormal
> transformations of the semi-null frame it is possible to simplify further
> the shear matrix. For example, in the Lorentzian signature in addition of
> being traceless and symmetric $\boldsymbol{\sigma}$ is also real, so that
> this matrix admits, in a suitable basis, the form
> $\operatorname{diag}(\lambda,\lambda,-\lambda,-\lambda)$ with
> $\lambda=\sqrt{\Delta^{2}+\Theta^{2}+\Phi^{2}}$. This result is perfectly
> compatible with the six-dimensional calculations, in Lorentzian signature,
> found in appendix C of GS-Ortaggio12 .
Despite being called here a generalized version of the Goldberg-Sachs theorem
the theorem just presented carries little resemblance with the usual version
of the GS theorem. The main differences being that in theorem 5 no mention is
made to the Weyl tensor and the Ricci tensor is not assumed to be constrained.
In order to understand why it is adequate to call this theorem a generalized
version of the GS theorem it is necessary to review some previous results on
the literature, this will be done in the next paragraph.
The first apparition of the Goldberg-Sachs theorem was in reference Goldberg-
Sachs , where it was proved that a Ricci-flat four-dimensional Lorentzian
manifold has an algebraically special Weyl tensor if, and only if, it admits a
null congruence that is geodesic and shear-free($\sigma_{ij}=0$). Later this
result has been put in a conformally invariant form by relaxing the constraint
on the Ricci tensor GS-CottonYork , particularly the GS theorem has been
proved to be valid for Einstein manifolds. A decade after this the GS theorem
was generalized to be valid in four-dimensional manifolds of all signatures in
Plebanski2 , where it was proved that the concept of geodesic and shear-free
should be substituted in the other signatures by the requirement of local
integrability of totally null 2-surfaces (see also Robinson Manifolds ).
Indeed, the leafs of integrable maximally isotropic distributions can be
proved to be totally geodesic Plebanski2 ; Mason-Chabert-KillingYano , so that
in the Lorentzian case the intersection of an integrable maximally isotropic
distribution with its complex conjugate yields a null congruence that is
geodesic and in four dimensions is shear-free.
Now we are in position to conclude that the theorem presented in this section
is deeply connected to the original version of the GS theorem. From the
reference Plebanski2 it follows, in particular, that in a Ricci-flat
4-dimensional manifold if the Weyl tensor is algebraically special then the
manifold admits an integrable maximally isotropic distribution of vector
fields. So because of theorem 5 this implies that in four dimensions if the
Weyl tensor is algebraically special then $\widehat{f}_{i}\sigma_{ji}=0$.
Since in the Lorentzian signature the $2\times 2$ matrix $\sigma_{ij}$ is
real, symmetric and traceless this last equation implies that $\sigma_{ji}=0$,
thus arriving at the usual form of the GS theorem. The converse of theorem 5
although valid in four dimensions probably is not valid in higher dimensions,
so that this theorem captures only one direction of the original GS theorem.
A connection between algebraically special Weyl tensors and restrictions on
the shear matrix can also be easily made in higher dimensions using previous
results. Combining theorem 5 with the corollary of theorem 2 we immediately
arrive at the following result:
Corollary _In a Ricci-flat manifold of even dimension $d=2n$ if the
subbundles $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are invariant under the
operator $\textbf{C}_{n}$ and this operator is generic otherwise then the
shear matrix of the maximally isotropic distribution
$\\{e_{1},\ldots,e_{n}\\}$ is constrained by the following equation:_
$\widehat{f}_{i[j_{3}j_{4}\ldots j_{n}}\,\sigma_{j_{2}]i}\;=\;0\,.$
_Where $\widehat{f}_{j_{2}j_{3}\ldots j_{n}}$ was defined below equation (19)
and the subbundles $\mathcal{A}_{p}$ were defined in section IV. In the
particular case of Lorentzian signature the vector field $e_{1}$ is geodesic._
Also, although not explicitly stated, the twist matrix and the scalars $M_{i}$
are also constrained if the hypotheses of this corollary are satisfied (see
theorem 5). This corollary is the reason of why the theorem 5 was called here
a generalized Goldberg-Sachs theorem.
## X Conclusions and Perspectives
In this article it was investigated several aspects of the Weyl tensor
classification and its relation with the integrability of vector
distributions, specially the maximally isotropic ones. At first the Weyl
operators $\textbf{C}_{p}$ were introduced and its properties investigated.
These operators are non-trivial generalizations of the already known bivector
map provided by the Weyl operator, here represented by $\textbf{C}_{2}$, that
can be used to classify the Weyl tensor. It was shown that such operators are
self-adjoint with respect to the natural inner product on the space of forms
and that in the Euclidean signature they can diagonalized, providing a simple
form to classify the Weyl tensor.
In even dimensions, $d=2n$, the operator $\textbf{C}_{n}$ has the special
property of preserving the space of (anti-)self-dual $n$-forms, so that we can
write $\textbf{C}_{n}=\textbf{C}^{+}\oplus\textbf{C}^{-}$. This allows us to
define the self-dual manifolds as the ones with $\textbf{C}^{-}=0$. But lemma
3 guarantees that if $n$ is odd or $n=4$ then the condition $\textbf{C}^{-}=0$
implies that the whole Weyl tensor vanishes. It was also proved that the
integrability condition of a maximally isotropic distribution is given by the
invariance of certain subbundles of $\wedge^{n}M$ under the operator
$\textbf{C}_{n}$.
In theorem 4 it was shown that if a manifold, of arbitrary dimension and
signature, admits a null $p$-form that is closed and co-closed then the
optical matrices of the null directions “tangent” to this form obey to several
constraints. This was then used to arrive at a generalized version of the
Goldberg-Sachs theorem stating that if an even-dimensional manifold admits an
integrable maximally isotropic distribution then the shear matrix is
constrained.
The objects and tools introduced here deserve further investigation and
probably the elegant notation introduced in section V will help on this
enterprise. On theoretical grounds it is important to search for new relations
between the operators $\textbf{C}_{p}$ and integrability properties. This can
shed light on the integration of higher-dimensional Einstein’s equation, as
happened in four dimensions typeD - Kinnersley . Also the practical
implications of theorem 4 and of the generalized Goldberg-Sachs theorem should
be analyzed more deeply. These theorems should be particularly useful in
situations where there are physical fields represented by $p$-forms whose
equations of motion implies that they are harmonic (closed and co-closed).
## Acknowledgments
Thanks to the Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq) for the financial support. I also want to thank the anonymous referee
of the Journal of Mathematical Physics for some valuable suggestions.
The published version is available on `http://dx.doi.org/10.1063/1.4802240`
## Appendix A Refined Segre Classification
In this appendix it will be presented the refined Segre classification, a
classification scheme for square matrices over the complex field that was
defined in Spin6D .
Given a square matrix $M$ over the complex field then by means of a similarity
transformation it can always be put in the so called Jordan canonical form.
This canonical form is a block-diagonal matrix such that each block is equal
to a single number or to a matrix of the following form:
$J=\left[\begin{array}[]{cccc}\lambda&1&0&\ldots\\\ 0&\lambda&1&\\\
\vdots&&\ddots&1\\\ 0&\ldots&0&\lambda\\\ \end{array}\right].$
The Jordan canonical form of a matrix is unique up to the ordering of the
Jordan blocks. Each block of the above form has only one eigenvector and its
eigenvalue is $\lambda$.
The Segre classification of the matrix $M$ is the list of the dimensions of
the Jordan blocks. This list of numbers is put inside a square bracket and the
dimensions of the Jordan blocks with the same eigenvalue are to be put
together inside a round bracket. This classification can be refined if the
numbers corresponding to the blocks with eigenvalue zero are made explicit by
putting them on the right of the others and separating by a vertical bar. For
example, suppose that the Jordan canonical form of a matrix $M$ is
$\left[\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&0&0&0\\\ 0&0&\alpha&0&0\\\
0&0&0&\beta&1\\\ 0&0&0&0&\beta\\\ \end{array}\right]\,.$ (21)
Then the next table summarizes the possible types that the matrix $M$ can have
on the Segre classification and on the refined version of this classification.
| $0\neq\alpha\neq\beta\neq 0$ | $\alpha=0\neq\beta$ | $\alpha=\beta\neq 0$ | $\beta=0\neq\alpha$ | $\alpha=\beta=0$
---|---|---|---|---|---
Segre Classification | $[2,2,1]$ | $[(2,1),2]$ | $[(2,1),2]$ | $[(2,2),1]$ | $[(2,2,1)]$
Refined Segre Class. | $[2,1|2]$ | $[2|2,1]$ | $[(2,1)|2]$ | $[1|2,2]$ | $[|2,2,1]$
Table 1: Possible algebraic types for the matrix of equation (21).
## Appendix B Simple Forms and the Integrability of Distributions
Let $\textbf{K}_{p}$ be a $p$-form that is simple, i.e., there exists
$1$-forms $\omega^{j}$ such that
$\textbf{K}_{p}=\omega^{1}\wedge\omega^{2}\wedge\ldots\wedge\omega^{p}$. Using
equation (2) it is possible to see that the Hodge dual of a simple form is a
simple form, so that we can write:
$\widetilde{\textbf{K}}_{p}\;=\;\widehat{\omega}^{1}\wedge\widehat{\omega}^{2}\wedge\ldots\wedge\widehat{\omega}^{d-p}\,.$
(22)
By means of the metric $g$ it is possible to make a one-to-one association
between vectors and 1-forms. For example, the vector field $V$ is associated
to the 1-form $\omega$ when $\omega(X)=g(V,X)$ for all vector fields $X$.
Using this correspondence let us denote by $V_{j}$ the vectors associated to
the 1-forms $\omega^{j}$ and by $\widehat{V}_{r}$ the vectors associated to
$\widehat{\omega}^{r}$. Now let us define the following vector field
distributions:
$\mathcal{N}=\\{V_{1},V_{2},\ldots,V_{p}\\}\quad\quad;\quad\quad\widetilde{\mathcal{N}}=\\{\widehat{V}_{1},\widehat{V}_{2},\ldots,\widehat{V}_{d-p}\\}\,.$
Using equations (2), (22) and the definition $V_{j}^{\;\mu}=\omega^{j\,\mu}$
it follows that:
$[V_{j}\lrcorner(\widehat{\omega}^{1}\wedge\ldots\wedge\widehat{\omega}^{d-p})]_{\mu_{2}\ldots\mu_{d-p}}\,=\,\frac{n!}{p!}\,V_{j}^{\;\mu_{1}}\varepsilon^{\nu_{1}\ldots\nu_{p}}_{\phantom{\nu_{1}\ldots\nu_{p}}\mu_{1}\ldots\mu_{d-p}}V_{1\,\nu_{1}}\ldots
V_{p\,\nu_{p}}\,=\,0\,,$
where $(V\lrcorner\mathbf{F})$ is the interior product of the vector field $V$
into the differential form $\mathbf{F}$. This implies that
$\widehat{\omega}^{r}(V_{j})=0$, which is equivalent to
$g(\widehat{V}_{r},V_{j})=0$. So we arrived at the following important result:
$\widetilde{\mathcal{N}}\;=\;\mathcal{N}^{\bot}\,.$ (23)
Where $\mathcal{N}^{\bot}$ is the orthogonal complement of $\mathcal{N}$.
Given a simple $p$-form $\textbf{K}_{p}$, the vector distribution annihilated
by it is defined by
$\mathcal{A}_{\mathbf{K}_{p}}=\\{U\,\in\,TM\,|\,U\lrcorner\textbf{K}_{p}=0\\}$.
Since $\widehat{\omega}^{r}(V_{j})=0$ it follows that $\mathcal{N}$ is the
vector distribution annihilated by the form $\widetilde{\textbf{K}}_{p}$,
while $\widetilde{\mathcal{N}}$ is the distribution annihilated by
$\textbf{K}_{p}$. It was proved in section VII that the distribution
$\mathcal{A}_{\mathbf{K}_{p}}$ is integrable if, and only if, there exists
some function $h\neq 0$ such that $\textbf{d}(h\textbf{K}_{p})=0$, therefore
we can state:
$\left\\{\begin{array}[]{ll}\mathcal{N}\quad\textrm{is
integrable}\quad\Leftrightarrow\quad\exists\;f\neq
0\;|\;\textbf{d}(f\widetilde{\textbf{K}}_{p})=0\\\
\widetilde{\mathcal{N}}\quad\textrm{is
integrable}\quad\Leftrightarrow\quad\exists\;g\neq
0\;\,|\;\textbf{d}(g\textbf{K}_{p})=0\,.\end{array}\right.$
In the special case in which $d=2n$, $p=n$ and $\textbf{K}_{n}$ is a null
$n$-form, i.e, the distribution $\mathcal{N}$ is maximally isotropic, it
follows that $\mathcal{N}^{\bot}=\mathcal{N}$. So by equation (23) we see that
$\widetilde{\textbf{K}}_{n}\propto\textbf{K}_{n}$, which implies that
$\textbf{K}_{n}$ must be self-dual or anti-self-dual. In this case we have
that $\mathcal{N}$ is integrable if, and only if, there exists some function
$h\neq 0$ such that $\textbf{K}^{\prime}_{n}=h\textbf{K}_{n}$ obeys to the
equations $\textbf{d}(\textbf{K}^{\prime}_{n})=0$ and
$\textbf{d}(\widetilde{\textbf{K}^{\prime}}_{n})=0$.
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|
arxiv-papers
| 2013-01-10T02:07:59 |
2024-09-04T02:49:40.090925
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Batista",
"submitter": "Carlos A. Batista da S. Filho",
"url": "https://arxiv.org/abs/1301.2016"
}
|
1301.2096
|
1]EarthByte Group, School of Geosciences, Madsen Buildg. F09, The University
of Sydney, NSW 2006, Australia 2]Global New Ventures, Statoil ASA, Forus,
Stavanger, Norway 3]formerly with GET GME TSG, Statoil ASA, Oslo, Norway
Christian Heine
([email protected])
# Kinematics of the South Atlantic rift
Christian Heine Jasper Zoethout R. Dietmar Müller [ [ [
###### Abstract
The South Atlantic rift basin evolved as branch of a large Jurassic-Cretaceous
intraplate rift zone between the African and South American plates during the
final breakup of western Gondwana. While the relative motions between South
America and Africa for post-breakup times are well resolved, many issues
pertaining to the fit reconstruction and particular the relation between
kinematics and lithosphere dynamics during pre-breakup remain unclear in
currently published plate models. We have compiled and assimilated data from
these intraplated rifts and constructed a revised plate kinematic model for
the pre-breakup evolution of the South Atlantic. Based on structural
restoration of the conjugate South Atlantic margins and intracontinental rift
basins in Africa and South America, we achieve a tight fit reconstruction
which eliminates the need for previously inferred large intracontinental shear
zones, in particular in Patagonian South America. By quantitatively accounting
for crustal deformation in the Central and West African rift zone, we have
been able to indirectly construct the kinematic history of the pre-breakup
evolution of the conjugate West African-Brazilian margins. Our model suggests
a causal link between changes in extension direction and velocity during
continental extension and the generation of marginal structures such as the
enigmatic pre-salt sag basin and the São Paulo High. We model an initial E–W
directed extension between South America and Africa (fixed in present-day
position) at very low extensional velocities from 140 Ma until late
Hauterivian times ($\approx$126 Ma) when rift activity along in the equatorial
Atlantic domain started to increase significantly. During this initial
$\approx$14 Myr-long stretching episode the pre-salt basin width on the
conjugate Brazilian and West African margins is generated. An intermediate
stage between $\approx$126 Ma and Base Aptian is characterised by strain
localisation, rapid lithospheric weakening in the equatorial Atlantic domain,
resulting in both progressively increasing extensional velocities as well as a
significant rotation of the extension direction to NE–SW. From base Aptian
onwards diachronous lithospheric breakup occurred along the central South
Atlantic rift, first in the Sergipe-Alagoas/Rio Muni margin segment in the
northernmost South Atlantic. Final breakup between South America and Africa
occurred in the conjugate Santos–Benguela margin segment at around 113 Ma and
in the Equatorial Atlantic domain between the Ghanaian Ridge and the Piauí-
Ceará margin at 103 Ma. We conclude that such a multi-velocity, multi-
directional rift history exerts primary control on the evolution of this
conjugate passive margins systems and can explain the first order tectonic
structures along the South Atlantic and possibly other passive margins.
The formation and evolution of rift basins and continental passive margins is
strongly depedendent on lithosphere rheology and strain rates (e.g. Buck et
al., 1999; Bassi, 1995). Strain rates are directly related to the relative
motions between larger, rigid lithospheric plates and thus the rules of plate
tectonics. A consistent, independent kinematic framework for the pre-breakup
deformation history of the South Atlantic rift allows to link changes in
relative plate velocities and direction between the main lithospheric plates
to events recorded at basin scale and might help to shed light on some of the
enigmatic aspects of the conjugate margin formation in the South Atlantic,
such as the pre-salt sag basins along the West African margin (e.g. Huismans
and Beaumont, 2011; Reston, 2010), the extinct Abimael ridge in the southern
Santos basin, and the formation of the São Paulo high. Over the past decades,
our knowledge of passive margin evolution and the sophistication of
lithospheric deformation modelling codes has substantially increased (e.g.
Huismans and Beaumont, 2011; Péron-Pinvidic and Manatschal, 2009; Rüpke et
al., 2008; Crosby et al., 2008; Lavier and Manatschal, 2006), as have the
accuracy and understanding of global and regional relative plate motion models
(e.g. Seton et al., 2012; Müller et al., 2008) for oceanic areas. However, the
connections between these two scales and and the construction of quantified
plate kinematic frameworks for pre-breakup lithospheric extension remains
limited due to the fact that no equivalent of oceanic isochrons and fracture
zones are generated during continental lithospheric extension to provide
spatio-temporal constraints on the progression of extension. Provision of such
kinematic frameworks would vastly help to improve our understanding of the
spatio-temporal dynamics of continental margin formation. The South Atlantic
basin with its conjugate South American and West African margins and
associated Late Jurassic/Early Cretaceous rift structures (Fig. 1) offers an
ideal testing ground to attempt to construct such a framework and link a
kinematic model to observations from marginal and failed rift basins. There
are relatively few major lithospheric plates involved, the motions between
these plates during the early extension phase can modeled using well-
documented intraplate rifts and the conjugate passive margins still exist in
situ. In addition, an extensive body of published literature exists which
documents detailed aspects of the conjugate passive margin architecture.
### 0.1 Aims and rationale
Following Reston (2010), previously published plate kinematic reconstructions
of the South Atlantic rift have only been able to address basic questions
related to the formation of the conjugate South Atlantic margins in pre-
seafloor spreading times and fall short of explaining the impact of plate
kinematic effects on rift margin evolution. In this paper, we present a
completely revised plate kinematic model for the pre-breakup and early
seafloor spreading history of the South Atlantic rift that integrates
observations from all major South American and African rifts. We alleviate
unnecessary complexity from existing plate tectonic models for the region –
such as a number of large, inferred intracontinental shear zones, for which
observational evidence is lacking – while providing a kinematic framework
which can consistently explain relative plate motions between the larger
lithospheric plates and smaller tectonic blocks. The model improves the full
fit reconstruction between South American plates and Africa, especially in the
southernmost South Atlantic and is able to explain the formation of the Salado
and Colorado Basins in Argentina, two Early Cretaceous-aged basins which
strike nearly orthogonal to the main South Atlantic rift, in the context of
larger scale plate motions. This allows us to link changes in plate motions
with well-documented regional tectonic events recorded in basins along the
major rift systems and to explain the formation of the enigmatic pre-salt sag
basins of the central South Atlantic in the context of a multi-direction,
multi-velocity plate kinematic history.
## 1 Plate reconstructions: data, regional elements, and methodology
We build a spatio-temporal kinematic framework of relative motions between
rigid lithospheric plates based on the inventory of continental rifts,
published and unplublished data from the conjugate passive margins and
information from oceanic spreading in the South Atlantic. We utilise the
interactive, open-source plate kinematic modelling software GPlates
(http://www.gplates.org) as an intergration platform and the Generic Mapping
Tools (http://gmt.soest.hawaii.edu; Wessel and Smith, 1998) to generate our
paleo-tectonic reconstructions (See Figs. 13–21 and electronic supplements).
High resolution vectorgraphics of our reconstructions along with the plate
model and geospatial data for use in GPlates are available in the
supplementary data as well as on the internet at
http://datahub.io/en/dataset/southatlanticrift.
### 1.1 Absolute plate motion model and timescale
The plate kinematic model is built on a hybrid absolute plate motion model
with a moving hotspot reference frame back to 100 Ma and an adjusted
paleomagnetic absolute reference frame for times before 100 Ma (Torsvik et
al., 2008). Global plate motion models are based on magnetic polarity
timescales to temporarily constrain plate motions using seafloor spreading
magnetic anomalies (e.g. Seton et al., 2012; Müller et al., 2008). Here, we
use the geomagnetic polarity timescale by Gee and Kent (2007, hereafter called
_GeeK07_) which places the young end of magnetic polarity chron M0 at 120.6 Ma
and the base of M0r at 121 Ma. Agreement across various incarnations of
geological time scales exists that the base of magnetic polarity chron M0r
represents the Barrêmian-Aptian boundary (e.g. Ogg and Hinnov, 2012; He et
al., 2008; Channell et al., 2000, and references therein). However,
considerable debate is ongoing about the absolute age of of the base of M0r
with proposed ages falling in two camps, one around 121 Ma (e.g. He et al.,
2008; Gee and Kent, 2007; Berggren et al., 1995) and a second one around
125-126 Ma (Ogg and Hinnov, 2012; Gradstein et al., 2004). Both choices affect
the absolute ages of the different pre-Aptian stratigraphic stages as
magnetostratigraphy is used to provide a framework for biostratigraphic
events. Our preference of using Gee and Kent (2007)’s 121.0 Ma as base for M0r
is based the arguments put forward by He et al. (2008) and on a review of
global seafloor spreading velocities, in which an older age assigned to the
base of M0r of 125.0 Ma (Gradstein et al., 2004) would result in significantly
higher spreading velocities or larger oceanic crust flux for the pre-M0
oceanic crust (Seton et al., 2009; Gee and Kent, 2007; Cogné and Humler,
2006). Additionally, Stone et al. (2008) report a reversed magnetic polarity
for a Cretaceous-aged dyke from Pony’s Pass quarry on the Falkland Islands
dated to 121.3$\pm$1.2 Ma (Ar-Ar dating). This further supports the notion of
tying the absolute age of M0r old and Base Aptian to closer to 121 Ma than to
125 or 126 Ma as postulated by Ogg (2012); Gradstein et al. (2004).
In pre-seafloor spreading, continental environments, well constrained direct
age control for relative plate motions does not exist. Here, stratigraphic
information from syn-rift sequences and subsidence data needs to be converted
to absolute ages in a consistent framework. As the correlation between
absolute ages and stratigraphic intervals has undergone several major
iterations over the past decades, published ages from South American and
African rift basins require readjustment. To tie stratigraphic ages to the
magnetic polarity timescale predominantly used for global plate kinematic
models, we have converted the estimates given by Gradstein et al. (1994, here
named _Geek07/G94_) and Gradstein et al. (2004, _Geek07/GTS04_) to the GeeK07
polarity chron ages (Fig. 2). We use Geek07/GTS04, which places Base Aptian
(Base M0r old) at 121 Ma (Fig. 2).
For published data we have converted both, stratigraphic and numerical ages to
the new hybrid timescale. A particular example for large differences in
absolute ages are the publications by Genik (1993, 1992) which are based on
the EXXON 1988 timescale (Fig. 2; Haq et al., 1987). Here, the base Cretaceous
is given as $\approx 133$ Ma, whereas the recent GTS timescales (Gradstein et
al., 2004) as well as our hybrid timescale place the base Cretaceous between
144.2 $\pm$ 2.5 and 142.42 Ma, respectively. Issues pertaining to correlate
biostratigraphic zonations across different basins further complicate the
conversion between stratigraphic and absolute ages regionally (e.g. Chaboureau
et al., in press; Poropat and Colin, 2012).
### 1.2 Continental Intraplate Deformation
Deformation between rigid continental plates has to adhere to the basic rules
of plate tectonics and can be expressed as relative rotations between a
conjugate plate pair for any given time (Dunbar and Sawyer, 1987). Depending
on data coverage and quality, a hierarchical plate model can be assembled from
this information (e.g. Ross and Scotese, 1988). In pre-seafloor spreading
environments, the quantification of horizontal deformation on regional scale
is complicated as discrete time markers such as oceanic magnetic anomalies or
clear structural features, such as oceanic fracture zones, are lacking.
However, rift infill, faulting and post rift subsidence allow – within
reasonable bounds – to quantify the amount of horizontal deformation (White,
1989; Gibbs, 1984; Le Pichon and Sibuet, 1981, e.g.). We assume that a
significant amount of horizontal displacement ($>$15 km) is preserved in the
geological record either as foldbelts (positive topographic features) or
fault-bounded sedimentary basins and recognised subsequently. Key elements of
our plate kinematic model are a set of lithospheric blocks which are non-
deforming during the rifting and breakup of western Gondwana (160–85 Ma; Fig.
1). These blocks are delineated using first order structures, such as main
basin-bounding faults, thrust belts or large gradients in reported sediment
thickness, indicative of subsurface faulting, in conjunction with potential
field and published data. The delineation of these blocks is described in
detail in later parts of this manuscript.
We then construct a relative plate motion model based on published data to
constrain the timing, direction, and accumulated strain in intraplate
deformation zones. This information is augmented with interpreted tectonic
lineaments and kinematic indicators (faults, strike slip zones) from various
publications (e.g. Matos, 2000; Genik, 1993; Matos, 1992; Genik, 1992; Exxon
Production Research Company, 1985) as well as potential field data from
publicly available sources (Andersen et al., 2010; Sandwell and Smith, 2009;
Maus et al., 2007) to further refine rigid block and deforming zone outlines.
Stage rotations were derived by identifying the predominant structural grain
and choosing an appropriate rotation pole which allows plate motions to let
relative displacement occur so that the inferred kinematics from strain
markers along the whole deforming zone (e.g. WARS/CARS) are satisfied. The
amount of relative horizontal displacement was then implemented by visual
fitting, using published data and computed extension estimates using total
sediment thickness.
Sediment thickness can serve as proxy for horizontal extension in rift basins
which experienced a single phase of crustal extension. Here, we calculate
total tectonic subsidence (TTS; Sawyer, 1985) by applying an isostatic
correction for varying sediment densities (Sykes, 1996) to the gridded total
sediment thickness (Exxon Production Research Company, 1985). Extension
factors based on total tectonic subsidence, assuming Airy isostasy throughout
the rifting process and not accounting for flexural rigidity, can be computed
along a set of parallel profiles across a rift basin, oriented parallel to the
main extension direction using the empirical relationship of Le Pichon and
Sibuet (1981):
$TTS=7.83(1-\frac{1}{\beta})$ (1) $\gamma=\frac{TTS}{7.83}$ (2)
where $\gamma=1-\frac{1}{\beta}$ and $\beta$ is the stretch factor. It has
been shown that this method allows to compute an upper bound for extension
factors in rift basins, in accordance with results from other methods (Barr,
1985; Heine et al., 2008) Values obtained using this methodology are likely to
overestimate actual extension estimates and main uncertainties in this
approach are input sediment thickness and the estimated sediment compaction
curves along with the assumption that a single rift phase created the basin.
We have applied this methodology for some of the major rift basins
(Muglad/Melut, Doba/Doseo/Bongor, Termit/Ténéré/Grein-Kafra, Gao, Salado,
Colorado; see Tab. Kinematics of the South Atlantic rift and supplementary
material) to constrain the amount of rift-related displacement. We assume the
bulk of the sediment thickness was accumulated during a single rift phase.
However, it is known that both CARS and WARS experienced at least one younger
phase of mild rifting and subsequent reactivation which has affected the total
sediment thickness (e.g. Genik, 1992; Guiraud et al., 2005) and hence will
affect the total amount of extension computed using this method. The extension
estimates extracted from individual rift segments are implemented by
considering the general kinematics of the whole rift system, as individual
rift segments can open obliquely and their margins are not necessarily
oriented orthogonally to the stage pole small circles.
Figure 5 shows estimates for a set of parallel profiles across the Termit
Basin, orthogonal to the main basin axis, ranging between c. 50–100 km. Where
possible, these estimates for extension were verified using published data
(Tab. Kinematics of the South Atlantic rift). Due to the inherent lack of
precise kinematic and temporal markers during continental deformation and
insufficient data we only describe continental deformation by a single stage
rotation (Table 1, Figs. 6 and 7).
### 1.3 Passive margins and oceanic domain
Over the past decade, a wealth of crustal-scale seismic data covering the
conjugate South Atlantic margins, both from industry and academic projects,
have been published (e.g. Blaich et al., 2011; Unternehr et al., 2010;
Greenroyd et al., 2007; Franke et al., 2007, 2006; Contrucci et al., 2004;
Mohriak and Rosendahl, 2003; Cainelli and Mohriak, 1999; Rosendahl and
Groschel-Becker, 1999). We made use of these data to redefine the location of
the continent-ocean boundary in conjunction with proprietary industry long
offset reflection seismic data (such as the ION GXT CongoSPAN lines,
http://www.iongeo.com/Data_Library/Africa/CongoSPAN/) as well as proprietary
and public potential field data and models (e.g. Sandwell and Smith, 2009;
Maus et al., 2007).
As some segments of this conjugate passive margin system show evidence for
hyperextended margins as well as for extensive volcanism and associated
seaward dipping reflectors sequences (SDRs), we introduce the “landward limit
of the oceanic crust” (LaLOC) as boundary which delimits relatively
homogeneous oceanic crust oceanward from either extended continental crust or
exhumed continental lithospheric mantle landward or SDRs where an
interpretation of the Moho and/or the extent of continental crust is not
possible. This definition has proven to be useful in areas where a classic
continent-ocean boundary (COB) cannot easily be defined such as in the distal
parts of the Kwanza basin offshore Angola (Unternehr et al., 2010), the
oceanward boundary of the Santos basin (Zalán et al., 2011) or along the
conjugate magmatic margins of the southern South Atlantic.
Area balancing of the Top Basement and Moho horizons has been used on
published crustal scale passive margin cross sections by Blaich et al. (2011)
to restore the initial pre-deformation stage of the margin, assuming no out-
of-plane motions and constant area. Initial crustal thickness estimates are
based on the CRUST2 crustal thickness model (Laske, 2004). While these
assumptions simplify the actual margin architecture and do not account for
alteration of crustal thickness during extension, Fig. 3 shows that the
differences between a choice of three different limits of the extent of
continental crust (minimum, COB based on Blaich et al., 2011, and LaLOC) only
has relatively limited effects on the width of the restored margin.
Considering the plate-scale approach of this study and inherent uncertainties
in the interpretation of subsalt structures on seismic data, these estimates
provide valid tie points for a fit reconstruction. The resulting fit matches
well with Chang et al. (1992)’s estimates for pre-extensional margin geometry
for the Brazilian margin (compare Figs. 13–21). Area balancing of the
continental basement (Top Basement to Moho) results in stretching estimates
ranging from 2.6-3.3 (Fig. 3). Some of the margin cross sections do not cover
the full margin width from unstretched continental to oceanic crust (e.g.
North Gabon and Orange sections). Here, we allow for a slight overlap in the
fit reconstruction. We have also carried out an extensive regional
interpretation of Moho, Top Basement and Base Salt reflectors on ION GXT
CongoSPAN seismic data to verify results from areal balancing of the published
data.
The conjugate South Atlantic passive margins are predominantely non-volcanic
in the northern and central part and volcanic south of the conjugate Santos/
Benguela segment (Blaich et al., 2011; Moulin et al., 2009, e.g). In the
volcanic margin segments, the delineation of the extent of stretched
continental crust is hampered by thick seaward-dipping reflector sequences
(SDRs) and volcanic build-ups. Previous workers have associated the landward
termination of the SDRs along the South American and southwestern African
continental margins with the “G” magnetic anomaly and a prominent positive
“large magnetic anomaly (LMA)” delineating the boundary between a transitional
crust domain of mixed extended and heavily intruded continental crust (Blaich
et al., 2011; Moulin et al., 2009; Gladczenko et al., 1997; Rabinowitz and
LaBrecque, 1979, e.g.). We follow previous workers in magmatically dominated
margin segments by using the seaward edge of SDRs and transition to normal
oceanic spreading for the location of our LaLOC.
Published stratigraphic data from the margins is integrated to constrain the
onset and dynamics of rifting along with possible extensional phases (e.g.
Karner and Gambôa, 2007). Little publicly available data from distal and
deeper parts of the margins exist which could further constrain the spatio-
temporal patterns of the late synrift subsidence.
To quantify oceanic spreading and relative plate motions between the South
American and Southern African plates before the Cretaceous Normal Polarity
Superchron (CNPS, 83.5–120.6 Ma) we use a pick database compiled by the
EarthByte Group at the University of Sydney, forming the base for the digital
ocean floor age grid (Seton et al., 2012).
We combine these data with the interpretations of Max et al. (1999) and Moulin
et al. (2009) and the WDMAM gridded magnetic data (Maus et al., 2007) to
create a set of isochrons for anomaly chrons M7n young (127.23 Ma), M4 old
(126.57 Ma), M2 old (124.05 Ma), and M0r young (120.6 Ma) using the magnetic
polarity timescale of Gee and Kent (2007). M sequence anomalies from M11 to M8
are only reported for the African side (Rabinowitz and LaBrecque, 1979)
whereas M7 has been identified on both conjugate abyssal plains closed to the
LaLOC (Moulin et al., 2009; Rabinowitz and LaBrecque, 1979). Oceanic spreading
and relative plate velocities during the CNPS are linearily interpolated with
plate motion paths only adjusted to follow prominent fracture zones in the
Equatorial and South Atlantic.
### 1.4 Reconstruction methodology
Deforming tectonic elements and their tectono-stratigraphic evolution from the
South Atlantic, Equatorial Atlantic, and intraplate rift systems in Africa and
South America such as intraplate basins, fault zones or passive margin
segments are synthesised using available published and non-published data to
constructed a hierarchical tectonic model (Fig. 9). Starting with the African
intraplate rifts, we iteratively refine the individual stage poles and fit
reconstructions based on published estimates and our own computations,
ensuring that the implied kinematic histories for a plate pair do not violate
geological and kinematic constraints in adajcent deforming domains. For
example, the choice of a rotation pole and rigid plate geometries which
describe the deformation between our Southern African and NE African plates to
explain the opening of the Muglad Basin is also required to match deformation
along the western end of the Central African Shear Zone in the Doba and Bongor
basins (e.g. Fig. 6).
After restoring the intra-African plate deformation, we iteratively refine the
tight fit reconstruction of South American plate against the NW African and
South African plates by reconstructing our restored profiles (Fig. 3), Chang
et al. (1992)’s restored COB, and key structural elements.
Subsequently, remaining basin elements and rigid blocks of the Patagonian
extensional domain (Salado, North Patagonian Massif, Deseado, Rawson, San
Julian and Falkland blocks) are restored to pre-rift/deformation stage and
integrated to achieve a full fit reconstruction for the latest Jurassic/Early
Cretaceous time in the southern South Atlantic (cf. Fig. 13).
## 2 Tectonic elements: Rigid blocks and deforming domains
Burke and Dewey (1974) pointed out that Africa did not behave as a single
rigid plate during Cretaceous rifting of the South Atlantic. They divided
Africa into two plates separated along the Benue Trough-Termit Graben (WARS in
Fig. 1). Subsequent work identified another rift system trending to the east
from the Benoue area (Genik, 1992; Fairhead, 1988, CARS in Fig. 1).
There is less evidence for Late Jurassic/Early Cretaceous deformation in South
America, although several continent-scale strike slip zones have been
postulated (see Moulin et al., 2009, and references therein). we define four
major plate boundary zones and extensional domains: the Central African
(CARS), West African (WARS), South Atlantic (SARS), and Equatorial Atlantic
Rift Systems (EqRS). We include a “Patagonian extensional domain”, composed of
Late Jurassic/Early Cretaceous aged basins and rigid blocks in southern South
America in our definition of the SARS (Fig. 1). From these four extensional
domains, only the SARS and EqRS transitioned from rifting to breakup, creating
the Equatorial and South Atlantic Ocean basins. In the next sections we review
timing, kinematics, type and amount of deformation for each of these domains.
### 2.1 Africa
The West African and Central African rift systems (Figs. 1 and 4; WARS & CARS)
and associated depocenters document extensional deformation between the
following continental lithospheric sub-plates in Africa, starting in the
Latest Jurassic/Early Cretaceous:
1. 1.
Northwest Africa (NWA), bound to the East by the WARS/East Niger Rift and
delimited by the Central and Equatorial Atlantic continental margins to the
West and South, respectively (e.g. Guiraud et al., 2005; Burke et al., 2003;
Genik, 1992; Fairhead, 1988).
2. 2.
Nubian/Northeast Africa (NEA), bound by the WARS to the West, and the
CARS/Central African Shear Zone to the South (Bosworth, 1992; Genik, 1992;
Popoff, 1988; Schull, 1988; Browne et al., 1985; Browne and Fairhead, 1983,
e.g.). To the East, Northeast and North this block is delimited by the East
African rift, the Red Sea and the Mediterranean margin, respectively.
3. 3.
Southern Africa (SAf) is separated from NEA through the CARS and bound by the
East African Rift system to the East and Southeast. Its southern and western
Margins are defined by the South Atlantic continental passive margins (e.g.
Nürnberg and Müller, 1991; Unternehr et al., 1988).
4. 4.
The Jos subplate, named after the Jos Plateau in Nigeria, is situated between
NWA, NEA and the Benoue Trough region. We define this plate along its western
margin by a graben system of Early Cretaceous age in the Gao Trough/Graben
area in Mali and the Bida/Nupe basin in NW Nigeria (Guiraud et al., 2005;
Guiraud and Maurin, 1992; Genik, 1992; Adeniyi, 1984; Cratchley et al., 1984;
Wright, 1968). The Benoue Trough and WARS delimit the the Jos subplate to the
South and East. As northern boundary we chose a diffuse zone through the
Iullemmeden/Sokoto Basin and Aïr massive, linking the WARS with the Gao Trough
area (Guiraud et al., 2005; Genik, 1992; Cratchley et al., 1984, e.g.).
5. 5.
The Adamaoua (Benoue), Bongor and Oban Highlands microplates (Fig. 1) are
situated south of the Benoue Trough and north of the sinistral Borogop fault
zone. This fault zone defines the western end of the CARS as it enters the
Adamaoua region of Cameroon (Genik, 1992; Benkhelil, 1982; Burke and Dewey,
1974). Together with the Benoue Trough in the north, the Atlantic margin in
the west and the Doba, Bongor, Bormu-Massenya basins it encompasses a
relatively small cratonic region in Nigeria/Cameroon which has been termed
“Benoue Subplate” by previous workers (e.g. Moulin et al., 2009; Torsvik et
al., 2009). The Yola rift branch (YB in Fig. 1) of the Benoue Trough as well
as the Mamfe Basin (Mf) indicate significant crustal thinning (Fairhead et
al., 1991; Stuart et al., 1985) justifying a subdivision of this region into
the two blocks.
The CARS and WARS are distinct from earlier Karoo-aged rift systems which
mainly affected the eastern and southern parts Africa (Bumby and Guiraud,
2005; Catuneanu et al., 2005) but have presumably formed along pre-existing
older tectonic lineaments of Panafrican age (Daly et al., 1989).
#### 2.1.1 Central African Rift System (CARS)
The eastern part of the CARS, consisting of the Sudanese Melut, Muglad and
Bagarra basins, forms a zone a few hundred kilometers wide with localised NW-
SE striking sedimentary basins which are sharply delimited to the north by the
so-called Central African Shear Zone (Bosworth, 1992; McHargue et al., 1992;
Schull, 1988; Exxon Production Research Company, 1985; Browne and Fairhead,
1983). Subsurface structures indicate NNW/NW-trending main basin-bounding
lineaments and crustal thinning with up to 13 km of Late Jurassic/Early
Cretaceous-Tertiary sediments (Fig. 4; Mohamed et al., 2001; McHargue et al.,
1992; Schull, 1988; Browne et al., 1985; Browne and Fairhead, 1983). Published
values for crustal extension in the Muglad and Melut basins in Sudan range
between 22-48 km (Browne and Fairhead, 1983), 15–27 km (McHargue et al., 1992)
in SW-NE direction and $\beta=1.61$ for an initial crustal thickness of 35 km,
resulting in 56 km of extension (Mohamed et al., 2001) with a first postrift
phase commencing in the Albian ($\approx$110 Ma; McHargue et al., 1992).
Dextral transtensional motions along the western part of the CARS during the
Early Cretaceous created the Salamat, Doseo and Doba basins (Fig. 4; Bosworth,
1992; McHargue et al., 1992; Schull, 1988). These depocenters locally contain
more than 8 km of Early Cretaceous to Tertiary sediments and are bound by
steeply dipping faults, indicative of pull-apart/transtensional kinematics for
the basin opening (Fig. 4; Genik, 1993; Maurin and Guiraud, 1993; Binks and
Fairhead, 1992; Genik, 1992; Guiraud et al., 1992; Exxon Production Research
Company, 1985; Browne and Fairhead, 1983). The structural inventory of these
basins largely follows old Panafrican-aged lineaments and has been reactived
during the Santonian compressive event (Guiraud et al., 2005; Janssen et al.,
1995; Maurin and Guiraud, 1993; Genik, 1992; Daly et al., 1989).
The Borogop Fault (Fig. 1 & 6) defines the western part of the Central African
Shear Zone and enters the cratonic area of the Adamaoua uplift in Cameroon,
where smaller, Early Cretaceous-aged basins such as the Ngaoundere Rift are
located (Fig. 4; Maurin and Guiraud, 1993; Plomerová et al., 1993; Guiraud et
al., 1992; Fairhead and Binks, 1991). The reported total dextral displacement
is estimated to be 40–50 km based on basement outcrops and around 35 km in the
Doseo Basin (Genik, 1992; Daly et al., 1989).
We have implemented a compounded total extension between 20–45 km between 140
and 110 Ma (early Albian) in the Muglad and Melut Basins between, generated
through rotation of the SAf block counterclockwise relative to NEA (Fig. 6,
Tab. Kinematics of the South Atlantic rift). The stage pole is located located
in the Somali Basin, south of the Anza Rift/Lamu Embayment, resulting in
moderate extension ($\approx$10 km) in the northern Anza Rift in Kenya which
is supported by observations faulting of Early Cretaceous age (Morley et al.,
1999; Reeves et al., 1987). The chosen stage rotation results in estimates of
distributed oblique extension/ sinistral transtension along the western end of
the CARS in the Bongor, Doba and Doseo Basins. Based on our choice of rigid
plates, this stage pole accounts for the opening of the Sudanese, Central
African Rifts as well as the Bongor basin.
Transtensional dextral displacement along the Borogop fault zone in our model
amounts to 40-45 km in the Doseo Basin which is in agreement with 25–56 km
extension reported from the Sudanese basins (Mohamed et al., 2001; McHargue et
al., 1992).
Stage poles and associated small circles for the NEA–SAf rotation are oriented
orthogonally to mapped Early Cretaceous extensional fault trends for the Doba
and Doseo Basin (Genik, 1992), and graben-bounding normal faults in the
Sudanese basins (Fig. 6). Other authors have used 70 km of strike
slip/extension for the CASZ and Sudan Basins, respectively (Moulin et al.,
2009), which is about double the amount reported (Genik, 1992; McHargue et
al., 1992). Torsvik et al. (2009) model the CARS but do not specify an exact
amount of displacement between their Southern African and the NE African sub-
plates.
#### 2.1.2 West African Rift System (WARS)
The West African/East Niger rift (WARS) extends northward from the eastern
Benoue Trough region through Chad and Niger towards southern Algeria and Lybia
(Fig. 1). The recent Chad basin is underlain by a series of N–S trending rift
basins, encompassing the Termit Trough, N’Dgel Edgi, Tefidet, Ténéré, and
Grein-Kafra Basins containing up to 12 km of Early Cretaceous to Tertiary
sediments (Figs. 4 & 5; Guiraud et al., 2005; Guiraud and Maurin, 1992; Genik,
1992; Exxon Production Research Company, 1985). These basins are extensional,
asymmetric rifts, initiated through block faulting in the Early Cretaceous,
with a dextral strike-slip component reported from the Tefidet region (“Tef”
in Fig. 1; Guiraud and Maurin, 1992; Genik, 1992). The infill is minor
Paleozoic to Jurassic pre-rift, non-marine sediments and a succession of non-
marine to marine Cretacous clastics of up to 6 km thickness with reactivation
of the structures during the Santonian (Bumby and Guiraud, 2005; Guiraud et
al., 2005; Genik, 1992). Towards the north of the WARS, N–S striking fault
zones of the El Biod-Gassi Touil High in the Algerian Sahara and associated
sediments indicate sinistral transpression during the Early Cretaceous
(Guiraud and Maurin, 1992). The main rift development occurred during Genik
(1992)’s Phase 3 from the Early Cretaceous to Top Albian (130–98 Ma) with full
rift development by 108 Ma (Genik, 1992, using the EXXON timescale).
Early Cretacous sedimentation and normal faulting in the Iullemmeden/Sokoto
and Bida Basins in NW Nigeria and the Gao Trough in mali indicates that
lithospheric extension also affected an area NW of the Jos subplate and
further west of the WARS sensu strictu (Guiraud et al., 2005; Obaje et al.,
2004; Genik, 1993, 1992; Guiraud and Maurin, 1992; Adeniyi, 1984; Cratchley et
al., 1984; Petters, 1981; Wright, 1968). Reported sediment thicknesses here
range between 3–3.5 km for the Bida Basin (Obaje et al., 2004, Fig. 4). Our
definition of the WARS hence encompasses this area of diffuse lithospheric
extension.
Palinspastic restoration of 2-D seismic profiles across the Termit basin part
of the WARS yields extension estimates between 40–80 km based (using Moho
depths of 26 km; Genik, 1992). Our computed maximum extension estimates for
the WARS rift basins using total tectonic subsidences results in a maximum
cumulative extension of 90–100 km (Fig. 5, Tab. Kinematics of the South
Atlantic rift). We use 70 km of extension in the Termit Basin region and 60 km
in the Grein-Kafra Basin to accommodate relative motions between the Jos
Subplate and NEA between Base Cretaceous and 110 Ma. Fault and sediment
isopach trends indicate an E–W to slightly oblique rifting, trending NNW-SSE
for the main branch of the WARS. Our stage rotation between NWA and NEA
results in oblique, NNE-SSW directed opening of the WARS (Fig. 7). Other
workers have used 130 km of E–W directed extension in the South (Termit Basin)
and 75 km for northern parts (Grein/Kafra Basins; Torsvik et al., 2009)
between 132–84 Ma or 80 km of SW-NE directed oblique extension (Moulin et al.,
2009). For the Bida (Nupe) Basin/Gao Trough, we estimate that approximately 15
km of extension occurred between the Jos Subplate and NWA, resulting in a
cumulative extension between NWA and NEA of 85–75 km between 143 Ma and 110
Ma.
#### 2.1.3 Benoue Trough
The Benoue Trough and associated basins like the Gongola Trough, Bornu and
Yola Basins are located in the convergence of the WARS and CARS in the
junction between the Northwest, Northeast and Southern African plates. The
tectonic position makes the Benoue Trough susceptible to changes in the
regional stress field, reflected by a complex structural inventory (Benkhelil,
1989; Popoff, 1988). Sediment thicknesses reach locally more than 10 km along,
with the oldest outcropping sediments reported as Albian age from anticlines
in the Upper Benoue Trough (Fairhead and Okereke, 1990; Benkhelil, 1989).
Subsidence in the Benoue Trough commences during Late Jurassic to Barrêmian as
documented by the Bima-1 formation in the Upper Benoue Trough (Guiraud et al.,
2005; Guiraud and Maurin, 1992).
The observed sinistral transtension in the Benoue Trough is linked to the
opening of the South Atlantic basin and extension in the WARS and CARS
(Guiraud et al., 2005; Genik, 1993, 1992; Fairhead and Binks, 1991; Fairhead
and Okereke, 1990; Benkhelil, 1989; Popoff, 1988; Benkhelil, 1982; Burke,
1976; Burke and Dewey, 1974). It follows that the onset of rifting and amount
of extension in the Benoue Trough is largely controlled by the relative
motions along the WARS and CARS. maximum crustal extension estimates based in
gravity inversion are 95 km, 65 km, and 55 km in the Benue and Gongola
Troughs, and Yola Rift with about 60 km of sinistral strike slip (Fairhead and
Okereke, 1990; Benkhelil, 1989). Rift activity is reported from the “Aptian
(or earlier)” to the Santonian (Fairhead and Okereke, 1990), synchronous with
the evolution of the CARS and WARS (Guiraud et al., 2005; Genik, 1992).
We here regard the Benoue Trough as product of differential motions between
NWA, NEA and the Adamaoua Microplate which is separated from South Africa by
the Borogop Fault zone. Deformation implemented in our model for the WARS and
CARS result in $\approx$20 km of N–S directed relative extension and about 50
km of sinistral strike-slip in the Benoue Trough between the Early Cretaceous
and early Albian.
The Mamfe Basin at the SW end of the Benoue Trough is separating the Oban
Highlands Block from the Adamaoua Microplate in the east (Fig. 1). It is
conjugate to NE Brazil. 20 km of extension are reported for this region which
we have restored in our plate model.
### 2.2 South America
The present-day South American continent is composed of a set of Archean and
Proterozoic cores which were assembled until the early Paleozoic, with its
southernmost extent defined by the Rio de la Plata craton (Fig. 8; Pángaro and
Ramos, 2012; Almeida et al., 2000). Large parts of South America, in contrast
to Africa, show little evidence for significant and well preserved, large
offset (10’s of km) intraplate crustal deformation during the Late Jurassic to
Mid-Cretaceous. In the region extending from the Guyana shield region in the
north, through Amazonia and São Francisco down to the Rio de la Plata Craton
there are no clearly identifiable sedimentary basins or compressional
structures with significant deformation reported in the literature which
initiated or became reactivated during this time interval.
The Amazon basin and the Transbrasiliano Lineament have been used as the two
major structural elements by various authors to accommodate intraplate
deformation of the main South American plate. Eagles (2007) suggests the
Solimões-Amazon-Marajó basins as location of a temporary, transpressional
plate boundary during South and Equatorial Atlantic rifting, where a southern
South America block is dextrally displaced by $\approx$200 km against a
northern block. The basin is underlain by old lithosphere of the Amazonia
Craton (Li et al., 2008; Almeida et al., 2000) which experienced one main
rifting phase in early Paleozoic times and subsequent, predominantely
Paleozoic sedimentary infill (Fig. 8; da Cruz Cunha et al., 2007; Gonzaga et
al., 2000; Matos and Brown, 1992; Nunn and Aires, 1988). It is covered by a
thin blanket of Mesozoic and Cenozoic sediments which show mild reactivation
with NE-trending reverse faults and minor dextral wrenching along its eastern
margin/Foz do Amazon/Marajó Basin during the Late Jurassic to Early Cretaceous
(da Cruz Cunha et al., 2007; Costa et al., 2001; Gonzaga et al., 2000).
Reactivation affecting the whole Amazon basin is reported only from the
Cenozoic (Azevedo, 1991; Costa et al., 2001). We do not regard this tectonic
element as a temporary plate boundary during formation of the South Atlantic.
The continental-scale Transbrasiliano lineament (TBL; Almeida et al., 2000)
formed during the Pan-African/Brasiliano orogenic cycle. It is a potential
candidate for a major accommodation zone for intraplate deformation in South
America (Feng et al., 2007; Pérez-Gussinyé et al., 2007), however, the amount
of accommodated deformation and the exact timing remain elusive (Almeida et
al., 2000). Some authors suggest strike slip motion along during the opening
of the South Atlantic between 60–100 km along this 3000 km long, continent-
wide shear zone, reaching from NE Brazil down into northern Argentina
(Aslanian and Moulin, 2010; Moulin et al., 2009; Fairhead et al., 2007). Along
undulating lineaments such as the TBL, any strike-slip motion would have
resulted in a succession of restraining and releasing bends (Mann, 2007),
creating either compressional or extensional structures in the geological
record. For comparison, the reported offset along the Borogop Shear zone in
the Central African Rift System ranges around 40 km during the Late
Jurassic–Early Cretaceous and created a series of deep ($>$6 km)
intracontinental basins (Doba, Doseo, Salamat – Fig. 4; Genik, 1992; McHargue
et al., 1992). While we do not refute evidence of reactivation of the TBL
during the opening of the South Atlantic, published geological and geophysical
data do not provide convincing support for the existence of a plate boundary
separating the South American platform along the Transbrasiliano lineament
during the opening of the South Atlantic.
In southern Brazil, previous authors have argued for the Carboniferous Paraná
Basin being the location for a large NW–SE striking intracontinental shear
zone to close the “underfit” problems in the southern part of the South
Atlantic (Moulin et al., 2012, 2009; Torsvik et al., 2009; Eagles, 2007;
Nürnberg and Müller, 1991; Unternehr et al., 1988; Sibuet et al., 1984). This
zone, obscured by one of the largest continental flood basalt provinces in the
world (Peate, 1997; White and McKenzie, 1989), has been characterised as R-R-R
triple junction with 100 km N-S extension (Sibuet et al., 1984), as Paraná-
Coehabamba shear zone with 150 km dextral offset (Unternehr et al., 1988), as
Parana-Chacos Deformation Zone with 60–70 km extension and 20-30 km of strike
slip (Nürnberg and Müller, 1991), as Paraná-Etendeka Fracture Zone – a
transtensional boundary with 175 km lateral offset (Torsvik et al., 2009), or
dextral strike-slip zone with 150 km strike slip and 70 km extension (Moulin
et al., 2009). Peate (1997) ruled out the possibility for a R-R-R triple
junction due to timing of magma emplacement and orientation of the associated
dyke swarm. The minimum amount of deformation proposed by previous authors is
20–30 km strike slip and 60–70 km extension (Nürnberg and Müller, 1991). In
analogy to the well documented CARS and the discussion of the TBL above, such
significant displacement should have resulted in a set of prominent basins
extending well beyond the cover of the Paraná flood basalt province and
manifested itself as major break along the South American continental margin,
similar to the Colorado and Salado basins further south. Although sub-basalt
basin structures have been reported (Eyles and Eyles, 1993; Exxon Production
Research Company, 1985), there is no evidence for a large scale continental
shear zone obscured by the Paraná large igneous province (LIP).
In our model we have subdivided the present-day South American continent in
the following tectonic blocks, partly following previous authors (Fig. 1; e.g.
Moulin et al., 2009; Torsvik et al., 2009; Macdonald et al., 2003; Nürnberg
and Müller, 1991; Unternehr et al., 1988):
1. 1.
The main South American Platform (SAm), extending from the Guyana Craton in
the North to the Rio de la Plata Craton in the South, with the exception of
the NE Brazilian Borborema Province.
2. 2.
The NE Brazilian Borborema Province block (BPB).
3. 3.
The Salado Subplate, located between the Salado and Colorado Basins.
4. 4.
The _Patagonian extensional domain_ south of the Colorado Basin, composed of
the rigid Pampean Terrane, North Patagonian Massif, Rawson block, San Julian
block, the Deseado block, and Malvinas/Falkland block, and the Maurice Ewing
Bank extended continental crust.
#### 2.2.1 Northeast Brazil
The Borborema Province block (BPB; Figs. 1 and 8) is located in NE Brazil and
separated from the South American plate along a N-S trending zone extending
from the Potiguar Basin on the eastern Brazilian Equatorial Atlantic margin
southwards to the Recôncavo-Tucano-Jatobá rift (RTJ, Fig. 8). It is an
exception in the otherwise tectonically stable South American plate and a set
of isolated Early Cretaceous rift basins (e.g. Araripe and Rio do Peixe
basins) as well as abundant evidence of reactivated Proterozoic aged basement
structures and shear zones indicates distributed, but highly localised
lithospheric deformation during the opening of the South Atlantic rift (de
Oliveira and Mohriak, 2003; Matos, 2000; Mohriak et al., 2000; Chang et al.,
1992; Matos, 1992; Milani and Davison, 1988; Castro, 1987). Rifting commenced
in the Latest Jurassic/Berriasian and lasted until the Mid-Barrêmian, well
documented through extensive hydrocarbon exploration (Matos, 2000, 1999;
Szatmari and Milani, 1999; Magnavita et al., 1994; Chang et al., 1992; Milani
and Davison, 1988).
Similar to the smaller tectonic blocks in the Benoue Trough region, we regard
the lithospheric deformation affecting this block caused by the motions of the
larger surrounding tectonic plates. The Borborema province is “crushed” during
the early translation of South America relative to Africa with extension in
the West Congo cratonic lithosphere localised along existing and reactivated
basement structures leading to small, spatially confined basins such as the
Araripe, Rio do Peixe, Iguatu and Lima Campos.
We model the rifting in the Recôncavo-Tucano-Jatobá and Potiguar basins by
allowing for $\approx$40 and 30 km extension, respectively, through relative
motions between South America and the Borborema Province block between 143 Ma
and 124 Ma (Mid-Barrêmian).
#### 2.2.2 Southern South America
The WNW-ESE striking Punta del Este Basin, the genetically related Salado
Basin adjacent to the South and the ENE-WSW trending Santa Lucia
Basin/Canelones Graben system, delimit our South American Block towards the
South (Fig. 8; Soto et al., 2011; Jacques, 2003; Kirstein et al., 2000;
Stoakes et al., 1991; Zambrano and Urien, 1970). Well data supports the onset
of syn-rift subsidence around the Latest Jurassic/Early Cretaceous and post-
rift commencing at Base Aptian (Stoakes et al., 1991). The rift-related
structural trend is predominantly parallel to the basin axis, indicating a
NNE-SSW directed extension. Sediment thicknesses reach 6 km with crustal
thicknesses around 20–23 km (Croveto et al., 2007) yielding stretching factors
of around 1.4. We have implemented 40 km of NE-SW transtension between 145 Ma
to Base Aptian for the eastern part of the basin, which is assumed to have
been linked towards the west by a zone of diffuse deformation with the General
Levalle basin. We have split the southern Rio de la Plata craton along the
syntaxis of the Salado/Punta del Este Basin between the South American block
and the Salado Sub-plate.
The rigid Salado block contains the Precambrian core of the Tandilia region
and the Paleozoic Ventania foldbelt (Pángaro and Ramos, 2012; Ramos, 2008) and
is delimited by the Late Jurassic/Early Cretaceous-aged Colorado, Macachín,
Laboulaye/General Levalle and San Luis basins in the South, Southwest and
West, respectively (Fig 8; Pángaro and Ramos, 2012; Franke et al., 2006;
Webster et al., 2004; Urien et al., 1995; Zambrano and Urien, 1970). Basement
trends deduced from seismic and potential field data indicate, similar to the
Salado Basin, E-W trending rift structures, orthogonal to the SARS and point
to to N-S directed extension/transtension (Pángaro and Ramos, 2012; Franke et
al., 2006, J. Autin personal communication, 2012). Pángaro and Ramos (2012)
estimate around 45 km (20%) N-S directed extension for the Colorado basin. Our
model assumes $\approx$50 km of NE-SW directed transtension for the basin from
150 Ma to Base Aptian when relative motions between Patagonian Plates and
South America cease (Somoza and Zaffarana, 2008).
The Colorado Basin marks the transition between the blocks related to the
South American Platform and the Patagonian part of South America (Pángaro and
Ramos, 2012), which we here summarise as Patagonian extensional domain. The
Patagonian lithosphere south of the Colorado Basin is composed of a series of
amalgamated magmatic arcs and terranes with interspersed Mesozoic sedimentary
basins (Ramos, 2008; Macdonald et al., 2003; Ramos, 1988; Forsythe, 1982). For
the purpose of this paper, the North Patagonian Massif, Rawson Block, Deseado
Block and Malvinas/Falkland Island Block are not separately discussed as
deformation in the Colorado and Salado basins largely accounts for clockwise
rotation of the Patagonian South America during the Late Jurassic to Aptian.
#### 2.2.3 The Gastre Shear Zone
The Gastre shear zone is used in previous plate tectonic models as major
intracontinental shear zone, separating Patagonian blocks from the main South
American plate (Torsvik et al., 2009; Macdonald et al., 2003). However, no
substantial transtensional or transpressional features along this proposed
faults zone are recognised in this part of Patagonia, nor is the geodynamic
framework of southern South America favouring the proposed kinematics. A
detailed geological study of the Gastre Fault zone area lead von Gosen and
Loske (2004) to conclude that there is no evidence for a late Jurassic–Early
Cretaceous shear zone in the Gastre area. Our model does not utilise a Gastre
Shear Zone to accommodate motions between the South American and Patagonian
blocks.
## 3 Plate reconstructions
The plate kinematic evolution of the South Atlantic rift and associated
intracontinental rifts preserved in the African and South American plates, is
presented as self-consistent kinematic model with a set of finite rotation
poles (Table 1). In the subsequent description of key timeslices we refer to
Southern Africa (SAf) fixed in present day position. We will focus on the
evolution of the conjugate South Atlantic margins. For paleo-tectonic maps in
1 Myr time intervals please refer to the electronic supplements.
### 3.1 Kinematic scenarios
Plate motions are expressed in the form of plate circuits or rotation trees in
which relative rotations compound in a time-dependent, non-commutative way.
The core of our plate tectonic model is the quantified intraplate deformation
which allows us to indirectly model the time-dependent velocities and
extension direction in the evolving South Atlantic rift. Between initiation
and onset of seafloor spreading, the plate circuit for the South America plate
is expressed by relative motions between the African sub-plates (Fig. 9). The
kinematics of rifting are well constrained through structural elements and
sedimentation patterns, however, the timing of extension carries significant
uncertainties due to predominantely continental and lacustrine sediment
infill. Limited direct information from drilling into the deepest parts of
these rifts is publicly available. The regional evolution allows for a
relatively robust dating of the onset of deformation at the Base Cretaceous in
all major rift basins (e.g. Janssen et al., 1995), whereas the onset of post-
rift subsidence in the CARS and WARS, is not as well constrained and further
complicated through which have experienced subsequent phases of significant
reactivation (e.g. Guiraud et al., 2005).
The design of our plate circuit (Fig. 9) implies that the timing of rift and
post-rift phases significantly affect the resulting relative plate motions
between South America and Southern Africa. We have tested five alternative
kinematic scenarios by varying the duration of the syn-rift phase in the
African intracontinental rifts, and along the Equatorial Atlantic margins to
evaluate the temporal sensitivity of our plate model (Tab.2). The onset of
syn-rift was changed between 140 Ma and 135 Ma in the CARS and WARS, along
with the end of the syn rift phase ranging between 110 Ma and 100 Ma. For the
Equatorial Atlantic rift, we evaluated syn-rift phase onset times between 140
Ma and 132 Ma (Top Valanginian). Additionally, we included the plate model of
Nürnberg and Müller (1991) with forced breakup at 112 Ma (“NT91”; Torsvik et
al., 2009) in our comparison. Flowlines for each alternative scenario were
plotted and evaluated against observed lineations and fracture zone patterns
of filtered free-air gravity. The implied extension history for the conjugate
South Atlantic passive margins was used as a primary criterion to eliminate
possible alternative scenarios. In the preferred model PM1, rifting along
SARS, CARS, WARS, and EqRS starts simultaneously around the early Berriasian
(here: 140 Ma). It satisfies geological (timing and sequence of events) and
geophysical observations (alignment of flowlines with lineaments identified in
the gravity data) for all marginal basins. Alternatively tested models
introduced kinematics such as transpression in certain parts of the margins
for which are not supported by the geological record. All tested scenarios
(PM1-PM5), however, show a good general agreement, confirming the robustness
of our methodology of constructing indirect plate motion paths for the
evolution of the SARS through accounting for intraplate deformation.
The largest difference between our model preferred model PM1 and model NT91
are the start of rifting and the implied kinematic history for the initial
phase of extension. Relative motions between SAm and the African plates
commences at 131 Ma in NT91. This results in higher extensional velocites and
strain rates in NT91 for all extensional domains due to a shorter duration
($\Delta t$ = 12 Myrs) between the onset of plate motions and key tiepoint at
Chron M0 (the same across all models).
In the northern Gabon region, the flowlines for model NT91 indicate a rapid
initial NNE-SSW translation of South America relative to Africa by about 100
km for the time from 131–126 Ma (Fig. 10). The resulting transpression along
the northern Gabon/Rio Muni margin during this time interval, is not evidenced
from the geological record (e.g. Brownfield and Charpentier, 2006; Turner et
al., 2003). This initial extension phase is followed by a sudden E-W kink in
plate motions from 126 Ma to 118 Ma before the flowlines turn SW-NE, parallel
to our model(s) for the time of the CNPS. Our modeled extension history for
the Gabon margin implies an initial 40-60 km E–W directed rifting between
South America and Africa until 127 Ma, followed by a 40∘ rotation of the
extension direction and subsequent increase in plate velocities.
Along the Nambian margin, predicted initial extension directions for the
relative motions between South America–Africa and Patagonian terranes–Africa
diverge in the central Orange Basin, conjugate to the Colorado and Salado
Basins (Fig. 11). Northward from here, relative motions for South
America–Africa are directed WNW–ESE, while the Patagonian Terranes–Africa
motions south of the central Orange Basin imply opening of the southernmost
SARS in ENE-WSW direction. This is a result of the relative clockwise rotation
of the Patagonian Terranes away from South America ue to the rifting in the
Salado/Punta del Este and Colorado basins and accordance with structural
observations from the southern Orange Basin (H. Koopmann, personal
communication, 2012). These model predictions are supported by the trend of
the main gravity lineaments in contrast to the plate motions paths of model
NT91 which are not reconcilable with gravity signatures (Fig. 11).
In the Santos basin our preferred model PM1 predicts an initial NW-SE directed
extension in the proximal part, oriented orthogonally to the main gravity
gradients, Moho topography and proximal structural elements (Fig. 12; e.g.
Stanton et al., 2010; Chang, 2004; Meisling et al., 2001). In the western
Santos basin we model the initial extension phase to be focussed between the
São Paulo High/Africa and South America, until the onset of the third
extensional phase around the Barrêmian/Aptian boundary, a time when the
inferred oceanic Abimael spreading becomes extinct (Figs. 9 and 12; Scotchman
et al., 2010) and the São Paulo High is translated from the African to the
South American plate.
### 3.2 Fit reconstruction and the influence of Antarctic plate motions
The fit reconstruction (Fig. 13) is generated by restoring the pre-rift stage
along the intraplate WARS and CARS for the African sub-plates, and in the
Recôncavo-Tucano-Jatobá, Colorado and Salado Basins for the South American
sub-plates using estimates of continental extension (see Sect. 2). We then use
area balanced crustal-scale cross sections along the South Atlantic
continental margins, (Sect. 1.3, Fig. 3) in combination with the restored
margin geometry published by Chang et al. (1992) to construct pre-rift
continental outlines. Margins along the Equatorial Atlantic are generally
narrow (Azevedo, 1991) and associated with complex transform fault tectonics
and few available published crustal scale seismic data, the location of the
LaLOC here is only based on potential field data, with the expection of the
Demerara Rise-French Guiana-Foz do Amazon segment where we have utilised
Greenroyd et al. (2008, 2007). We allow on average 100 km of overlap between
the present-day continental margins in the EqRS (Fig. 14), compared to a few
hundred km in the central part of the SARS. Total stretching estimates for
profiles across the margins for the central and southern South Atlantic
segment range between 2.3–3.8 for the 10 profiles shown in Fig. 3. South
America is subsequently visually fitted against the West African and African
Equatorial Atlantic margins using key tectonic lineaments such as fracture
zone end points and margin offsets (Fig. 13). Transpressional deformation has
affected the conjugate Demerara Rise/Guinea Plateau submerged promontories,
resulting in shortening of both conjugate continental margins during the
opening of the SARS (Basile et al., 2013, 2005; Benkhelil et al., 1995). Our
reconstructions hence show a gap of $\approx$50 km between the Demerara Rise
and the Guinea Plateau at the western EqRS as we have not restored this
shortening (Fig. 13).
The dispersal of Gondwana into a western and eastern part was initated with
continental rifting and breakup along the incipient Somali-Mozambique-Wedell
Sea Rift (e.g. König and Jokat, 2006; Norton and Sclater, 1979). NE-SW
directed displacement of Antarctica as part of eastern Gondwana southwards
relative to South America and Africa creates an extensional stress field which
affects southernmost Africa and the present-day South American continental
promontory comprised of the Ewing Bank and Malvinas/Falkland Island and the
Proto-Weddell sea from the Mid-Jurassic onwards (König and Jokat, 2006;
Macdonald et al., 2003). Our pre-rift reconstruction for the Patagonian
extensional domain takes into account possible earlier phases of extension
which have affected, for example, the San Jorge, Deseado or San Julían Basins
(Jones et al., 2004; Homovc and Constantini, 2001) and hence should represent
a snapshot at 140 Ma, rather than a complete fit reconstruction which restores
all cumulative extension affecting the area in Mesozoic times. Initial rifting
of this region preceeding relative motions between South America and Africa,
is recorded by Oxfordian-aged syn-rift sediments in the Outeniqua Basin in
South Africa as well as subsidence and crustal stretching in the North
Falkland Basin and the Maurice Ewing Bank region (Fig. 8; Jones et al., 2004;
Paton and Underhill, 2004; Macdonald et al., 2003; Bransden et al., 1999). The
overall NE-SW extension causes a clockwise rotation of the Patagonian blocks
away from SAf and SAm commencing at $\approx$150 Ma in our model, with an fit
reconstruction position of the Falkland Islands just south Aghulas
Arch/Outeniqua basin. This early motion results in the onset of crustal
stretching in the North Falkland (150 Ma), Colorado (150 Ma) and Salado (145
Ma) basins which preceeds relative motions between the main African and South
American plates (Fig. 13). The resulting relative motion in the southern part
of the SARS is NE-SW directed transtension (Fig. 11), and E-W extension
between the Malvinas/Ewing Bank promontory and Patagonia. Our fit
reconstruction for the southernmost South Atlantic places the Malvinas/
Falkland Islands immediately south of the Aghulas Arch/Outeniqua Basin. Based
on published extension estimates for relevant basins in southern
Patagonia/offshore Argentina (Jones et al., 2004, e.g.), alternative pre-rift
reconstructions placing the Falkland Islands further east and excessive
microplate rotation (Mitchell et al., 1986) can not be reconciled with the
currently available data.
### 3.3 Phase I: Initial opening – Berriasian to late Hauterivian
(140–126.57 Ma)
Extensional deformation along the WARS, CARS and SARS is documented to start
in the Early Cretaceous (Berriasian) by the formation of intracontinental rift
basins and deposition of lacustrine sediments along the conjugate South
Atlantic margins (Chaboureau et al., in press; Poropat and Colin, 2012; Dupré
et al., 2007; Brownfield and Charpentier, 2006; Bate, 1999; Cainelli and
Mohriak, 1999; Coward et al., 1999; Karner et al., 1997; Chang et al., 1992;
Guiraud et al., 1992). Here, we use 140 Ma as absolute starting age of the
onset of deformation in the South Atlantic, Central and West African rift
systems (lower Berriasian in our Geek07/GTS04 hybrid timescale, Fig. 2). In
the southern part of the SARS, SW-directed extension has already commenced in
the latest Jurassic with syn-rift subsidence in the Colorado, Salado, Orange
and North Falkland Basins (Séranne and Anka, 2005; Jones et al., 2004; Clemson
et al., 1999; Maslanyj et al., 1992; Stoakes et al., 1991). Evidence for
deformation and rifting along the EqRS during the Early Cretaceous is sparse,
however, indications for magmatism and transpressional deformation are found
in basins along the margin (e.g. Marajó Basin, São Luis Rift; Soares Júnior et
al., 2011; Costa et al., 2002; Azevedo, 1991). The Ferrer-Urbanos-Santos Arch
is an E–W trending basement high between the proximal parts of the Barrerinhas
basin and the onshore Parnaiba basin which was generated by transpression
during the Neocomian (Azevedo, 1991).
Extension in all major rift basins occurs at slow rates during the initial
phase, with separation velocities between SAm and SAf around 2 mm a-1 in the
Potiguar/Rio Muni segment and up to c. 20 mm a-1 in the southernmost SARS
segment, closer to the stage pole equator (Fig. 15). Extensional velocities in
the magmatically dominated margin segments of the southern South Atlantic are
well above 10 mm a-1. The predominant extension direction changes from NW-SE
in the northern SARS segment to WNW-ESE in the southern part, and SSW-NNE for
the conjugate Patagonia-SAf segment (Figs. 15 and 13–16. Along the central
SARS segment, this phase corresponds to Karner et al. (1997)’s “Rift Phase 1”
with broadly distributed rifting.
Flowlines derived from the plate model for the early extension phase correlate
well with patterns observed in the free air gravity field along the proximal
parts of the margins, such as in the northern Orange, and inner Santos Basins
(Figs. 11, 12). Along NE–SW trending margin segments, such as Rio Muni/Sergipe
Alagoas and Santos/Benguela, the extension is orthogonal to the margin,
whereas oblique rifting occurs in other segments. In this context we note that
the strike of the Taubaté Basin in SE Brazil (“Tau” in Fig. 8) as well as
observed onshore and offshore extensional structures and proximal Moho uplift
in the Santos Basin (Stanton et al., 2010; Fetter, 2009; Meisling et al.,
2001; Chang et al., 1992) are oriented orthogonal to our modeled initial
opening direction of the South Atlantic rift. The Recôncavo-Tucano-Jatobá
(RTJ) rift opens as we model the BPB attached to the West African Congo Craton
and Adamaoua subplate, with SAm being relatively displaced northwestward.
Here, the Pernambuco shear zone acts as continuation of the Borogop Fault Zone
into NE Brazil (Matos, 1999). Potiguar Basin and Benoue Trough make up one
depositional axis while the RTJ and inner/northern Gabon belong to the central
South Atlantic segment.
Around 138 Ma (Mid-Berriasian), break up and seafloor spreading starts in
isolated compartments between the Rawson Block and the continental margin
south of the Orange Basin. The Tristan da Cuñha hotspot has been located
beneath the Pelotas-Walvis segment of the SARS since 145 Ma (Figs. 13–16, and
paleo-tectonic maps in electronic supplement) and likely caused significant
alteration of the lithosphere during the early phase rifting, with the
eruption of the Paraná-Etendeka Continental Flood basalt province occurring
between 138–129 Ma (Peate, 1997; Stewart et al., 1996; Turner et al., 1994).
While we have not included a temporary plate boundary in the Paraná Basin in
our model, evidence from the Punta Grossa and Paraguay dyke swarms (Peate,
1997; Oliveira, 1989) points to NE-SW directed extension, similar to the
“Colorado Basin-style” lithospheric extension orthogonal to the main SARS
(Turner et al., 1994). Emplacement of massive seaward dipping reflector
sequences dominate the early evolution of the southern SARS segment and are
well documented by seismic and potential field data (Blaich et al., 2011;
Moulin et al., 2009; Bauer et al., 2000; Gladczenko et al., 1997, e.g.). While
the delineation of the extent of continental crust along volcanic margins
remains contentious, we model a near-synchronous transition to seafloor
spreading south of the Walvis Ridge/Florianopolis High around 127 Ma (Figs.
16–17 and supplementary material).
Between 136–132 Ma, the plume center is located beneath the present-day South
American coastline in the northern Pelotas Basin. Our reconstructions indicate
that the westernmost position of a Tristan plume (assumed to be fixed) with a
diameter of 400 km is more than 500 km east of the oldest Paraná flood
basalts, at the northern and western extremities of the outcrop (Turner et
al., 1994). Oceanic magnetic anomalies off the Pelotas/northern Namibe margins
indicate asymmetric spreading with a predomiant accretion of oceanic crust
along the African side (Moulin et al., 2009; Rabinowitz and LaBrecque, 1979),
which in our reconstructions can be explained through plume-ridge interactions
of high magma volume fluxes and relatively low spreading rates (Fig. 15;
Mittelstaedt et al., 2008). The São Paulo – Río de Janeiro coastal dyke
swarms, emplaced between 133–129 Ma, have any extrusive equivalents in the
Paraná LIP (Peate, 1997). The dykes are oriented orthogonal to our modeled
initial extension direction, both along the Brazilian as well as along the
African margin in southern Angola. Along with a predominant NE–SW striking
metamorphic basement grain presenting inherited weaknesses (Almeida et al.,
2000), these dykes are probably related to lithospheric extension along the
conjugate southern Campos/Santos–Benguela margin segment of the SARS.
In the southern segment, the Falkland-Aghulas Fracture zone is established
when the Maurice Ewing block starts moving with the Patagonian plate around
134 Ma. Strain rate estimates from the Orange basin support the onset of
rifting around 150-145 Ma (Jones et al., 2004).
Towards the late Hauterivian (127 Ma), the width of the enigmatic Pre-salt Sag
basin width has been created by slow, relative extension between SAm and SAf,
with extension velocities ranging between 7–9 mm a-1 (Campos/Jatobá, Fig. 15).
### 3.4 Phase II: Equatorial Atlantic rupture (126.57–120.6 Ma)
Following the initial rifting phase which is characterised by low strain
rates, deformation along the EqRS between NWA and northern SAm intensifies due
to strain localisation and lithospheric weakening (Heine and Brune, 2011).
Marginal basins of the EqRS record increasing rates of
subsidence/transpression (Azevedo, 1991) and in the southern South Atlantic
seafloor spreading anomalies M4 and M0 indicate a 3-fold increase in relative
plate velocities between SAm and African Plates (Figs. 17 and 15). Velocity
increases of similar magnitude, albeit with a different timing, are reported
by Torsvik et al. (2009); Nürnberg and Müller (1991). Along the EqRs, Azevedo
(1991) describes a set of transpressional structures in the Barreirinhas Basin
which caused folding of Albian and older strata and reports 50–120 km of
dextral strike slip along the Sobradinho Fault in the proximal Barrerinhas
basin, pointing towards an early dextral displacement. With the changed
kinematics, SAm now rotates clockwise around NWA, causing transtensional
opening of basins in the eastern part of the EqRS and about 20 km of
transpression along the Demerara Rise and Guinea Plateau at the western end of
the EqRS. This is in accordance with observed compressional structures along
the southern margin of the Guinea Plateau and the northeastern margin of the
Demerara Rise (Basile et al., 2013; Benkhelil et al., 1995).
The increase in extensional velocities jointly occurs with a sudden change in
extension direction from NW–SE to more E–W (Fig. 15). Depending on the
distance from the stage pole for this time interval, the directional change is
between 75∘ in the northern SARS (Rio Muni-Gabon/Potiguar-Sergipe Alagoas
segment) and 30∘ in the southern Pelotas/Walvis segment. The directional
change visible in the flowlines agrees very well with a pronounced outer
gravity high along the Namibian margin (Fig. 11) and lineaments in the Santos
Basin (Fig. 12). This change in the plate motions had severe effects on the
patterns and distribution of extension in the SARS. Karner et al. (1997)
report a 100 km westward step of the main axis of lithospheric extension in
the Gabon/Cabinda margin segment during their second rift phase (Hauterivian
to late Barrêmian). Along the Rio Muni/ North Gabon margin a mild inversion in
the pre-breakup sediment is observed (Lawrence et al., 2002) which we relate
to the change in plate kinematics, manifested in this margin segment as change
from orthogonal/slightly oblique extension to transform/strike slip. In the
Santos Basin, the change in extensional direction results in transtensional
motions, most likely responsible for the creation of the Cabo Frio fault zone
and localised thinning in proximal parts of the SW Santos basin, now
manifested as an “Axis of Basement Low” (Modica and Brush, 2004) north of the
São Paulo High. By the time of change in plate motions ($\approx$127 Ma), the
pre-salt sag basin width had been fully generated (Fig. 17). Increasing
extensional velocities resulted in fast, highly asymmetric localisation of
lithosphere deformation in the northern and central SARS segments, most likely
additionally influenced by the inherited basement grain/structures. This has
left large parts of the Phase I rift basin preserved along the Gabon-Kwanza
margin, whereas south of the Benguela/Cabo Frio transform the Phase I rift is
largely preserved on the Brazilian side.
Davison (2007) reports that from about 124 Ma (Early Aptian using Gradstein et
al., 2004), extensive evaporite sequences are deposited in areas north of the
Walvis Ridge, starting with the Paripuiera salt in the northern
Gabon/Jequitinhonha segment and the slightly younger Loeme evaporites in the
Kwanza basin. We note that in our reconstructions, the massive volcanic build
ups of the Walvis Ridge/ Florianopolis Platform form a large barrier toward
the Santos basin in the north. North of the ridge, exension localised and
propagated northwards into the southwestern part of the Santos Basin (the now
aborted Abimael ridge; Scotchman et al., 2010; Mohriak et al., 2010; Gomes et
al., 2009) towards the end of this phase.
The São Paulo High remains part of the African lithosphere, which explains the
“Cabo Frio counterregional fault trend” in the northern Santos Basin (Modica
and Brush, 2004). Prior NW-SE extension, paired with additional thinning and
transform motions along the Cabo Frio fault system (Stanton et al., 2010;
Meisling et al., 2001) could have formed a shallow gateway through the inner
parts of the Santos Basin, allowing the supply of seawater into the isolated
central SARS. Dynamic, plume-induced topography (Rudge et al., 2008) from the
extensive magmatic activity in the Paraná-Etendeka Igneous province might have
caused additional lithospheric buoycancy of the extending lithosphere in the
Santos basin.
Seafloor spreading is fully established in a Proto-South Atlantic ocean basin
(Pelotas-Walvis segment to Falkland/Aghulas fracture zone), while oceanic
circulation in this segment was probably quite restricted as the
Falkland/Malvinas–Maurice Ewing Bank continental promontory had not cleared
the Southern African margin (Fig. 18).
### 3.5 Phase III: Break up – Aptian to late Albian (120.6–100 Ma)
Magnetic anomaly chron M0 (120.6 Ma) is clearly identified in large parts of
the southern South Atlantic and provides, along with established fracture
zones, one major tiepoint for our reconstruction. Further lithospheric weaking
and strain localisation along the EqRS resulted in widespread transtensional
motions and related complex basin formation in the marginal basins (Basile et
al., 2005; Matos, 2000, 1999; Azevedo, 1991) causing a second increase in
extensional velocities between the African and SAm plates and a minor change
in separation direction from 120.6 Ma onwards (Fig. 15). We here linearly
interpolate the plate velocities during the CNPS (120.6–83.5 Ma) only
adjusting plate motion paths along well established fracture zones.
By 120 Ma (early Aptian, Fig. 18) continental breakup has occurred in the
northernmost SARS between the Potiguar/Benin and Pernambuco–Paraiba/Rio Muni
margin segments and between the conjugate northern Gabon/Jequitinhonha–Camamu
margins with incipient breakup in the remaining part of the central SARS
(Espirito Santo/Cabinda-Campos/Kwanza). South of the Cabo Frio/Benguela
transform, changes in plate motions resulted in deformation to jump from the
Avedis and Abimael ridges in the southwestern Santos basin (Scotchman et al.,
2010; Mohriak et al., 2010; Gomes et al., 2009) towards the African side,
rifting the São Paulo High away from the Namibian/Benguela margin. Extensive
evaporite deposition continues in most parts of the deforming central SARS,
peaking in Mid-Aptian times (Karner et al., 2007; Davison, 2007).
By 115 Ma (Middle Aptian, Fig. 19), large parts of the SARS and EqRS have
broken up diachronously and entered post-rift thermal subsidence. Our model
predicts breakup in the Campos/Kwanza segment by 119 Ma, in the westernmost
part of the EqRS at 118 Ma, in line with deep water basin conditions reported
between the Guinea and St. Paul Fracture zones (Jones, 1987). In the EqRS our
model predicts accretion of oceanic lithosphere in the Deep Ghanaian Basin
from 117 Ma onwards, and break up along the southern Campos/Benguela margin
segment in the SARS around 115 Ma. For the Gabon margins, Dupré et al. (2007)
and Karner et al. (1997) assume the onset of rifting at around 118 Ma, and
Late Barrêmian–Early Aptian, respectively, which is in agreement with our
model. Subsidence data from nearly all margin segments in the SARS indicate
cessation of fault activity and a change to post-rift thermal subsidence by
this time, with the exception of the outer Santos Basin, where the final
breakup between SAf and SAm occurs between 113–112 Ma. This timing is in
agreement with the deposition of the youngest evaporites in the outer Santos
Basin around 113 Ma (Davison, 2007). We hypothesise that the early salt
movement in the Gabon, Kwanza, Espirito Santo, Campos and Santos basins and
the observed chaotic salt in the distal part of these basins is related to the
fast localisation of lithospheric deformation, break-up and subsequent rapid
subsidence during the early Aptian, introducing topographic gradients
favouring gravitational sliding and downslope compression in the earliest
postrift (Fort et al., 2004).
Relative plate motions in the CARS and WARS cease around 110 Ma in the early
Albian, with most intracontinental rift basins entering a phase of thermal
subsidence before subsequent minor reactivation occurred in Post-Early
Cretaceous times (Janssen et al., 1995; Genik, 1993; Maurin and Guiraud, 1993;
Genik, 1992; Guiraud and Maurin, 1992; McHargue et al., 1992). The cessation
of rifting in WARS and CARS results in the onset of transpression along the
Côte d’Ivoire-Ghanaian Ridge in the EqRS as the trailing edges of SAm continue
to move westward, while NWA now remains stationary with respect to the other
African plates. The NWA transform margins now provide a backstop to the
westward-directed motions of SAm, causing compression associated with up to 2
km of uplift to occur along the Ghanaian transform margin between the middle
to late Albian (Antobreh et al., 2009; Basile et al., 2005; Pletsch et al.,
2001; Clift et al., 1997). Complete separation between African and South
American continental lithospheres is achieved at 104 Ma (Fig. 21), while the
oceanic spreading ridge clears the Côte d’Ivoire/Ghana Ridge by 100–99 Ma.
We present a new plate kinematic model for the evolution of the South Atlantic
rifts. Our model integrates intraplate deformation from temporary plate
boundary zones along the West African and Central African Rift Systems as well
as from Late Jurassic/Early Cretaceous rift basins in South America to achieve
a tight fit reconstruction between the major plates. Our plate motion
hierarchy describes the motions of South America to Southern Africa through
relative plate motions between African sub-plates, allowing to to model the
time-dependent pre-breakup extension history of the South Atlantic rift
system. Three main phases with distinct velocity and kinematics result through
the extension along the Central African, West African and Equatorial Atlantic
rift systems exerting significant control on the dynamics of continental
lithospheric extension in the evolving South Atlantic rift from the Early
Cretaceous to final separation of Africa and South America in the late Albian
(104 Ma). An intial phase of slow E–W extension in the South Atlantic rift
basin from 140 Ma (Base Cretaceous) to 127 Ma (late Hauterivian) causes
distributed extension in W–E direction. The second phase from 126–121 Ma (late
Hauterivian to base Aptian) is characterised by rapid lithospheric weakening
along the Equatorial Atlantic rift resulting in increased extensional
velocities and a change in extension direction to SW–NE. In the last phase
commences at 120 Ma and culminates in diachronous breakup along the South
Atlantic rift and formation of the South Atlantic ocean basin. It is
characterised by a further increase in plate velocities with a minor change in
extension direction.
We argue that our proposed three-stage kinematic history can account for most
basin forming events in the Early Cretaceous South Atlantic, Equatorial
Atlantic, Central African and West African Rift Systems and provide a robust
quantitative tectonic framework for the formation of the conjugate South
Atlantic margins. In particular, our model addresses the following issues
related to the South Atlantic basin formation:
* •
We achieve a tight fit reconstruction based on structural restoration of the
conjugate South Atlantic margins and intraplate rifts, without the need for
complex intracontinental shear zones.
* •
The orientation of the Colorado and Salado Basins, oriented perpendicular to
the main South Atlantic rift are explained through clockwise rotation of the
Patagonian sub-plates. This also implies that the southernmost South Atlantic
rift opened obliquely in a NE–SW direction.
* •
A rigid South American plate, spanning the Camamu to Pelotas basin margin
segments, requires that rifting started contemporaneously in this segment of
the South Atlantic Rift (Gabon, Kwanza, Benguela and Camamu, Espirito Santo,
Campos and Santos basins), albeit so far stratigraphic data (e.g. Chaboureau
et al., in press) indicates differential onset of the syn rift phase in the
various marginal basins.
* •
The formation of the enigmatic pre-salt sag basin along the West African
margin is explained through a multi-phase and multi-directional extension
history in which the initial sag basin width was created through slow (7–9 mm
a-1) continental lithospheric extension until 126 Ma (Late Hauterivian).
* •
Normal seafloor spreading in the southern segment of the South Atlantic rift
commenced at around 127 Ma, after a prolonged phase of volcanism affecting the
southern South Atlantic conjugate margins.
* •
Strain weakening along the Equatorial Atlantic Rift caused an at least 3-fold
increase of plate velocities between South America and Africa between 126–121
Ma, resulting in rapid localisation of extension along the central South
Atlantic rift.
* •
Linear interpolation of plate motions between 120.6 Ma and 83.5 Ma yield
break-up ages for the conjugate Brazilian and West African margins which are
corroborated by the onset of post-rift subsidence in those basins.
We conclude that our multi-direction, multi-velocity extension history can
consistently explain the key events related to continental breakup in the
South Atlantic rift realm, including the formation of the enigmatic pre-salt
sag basin.
The plate kinematic framework presented in this paper is robust and
comprehensive, shedding new light on the spatio-temporal evolution of the
evolving South Atlantic rift system. However, problems remain with regard to
the absolute timing of events in the evolution of the South Atlantic rift due
to a lacking coherent stratigraphy for both conjugate margin systems, and
limited access to crustal scale seismic data. Relatively old and limited data
for the African intraplate rifts results in large uncertainties in both
absolute timing as well as structural constraints on the kinematics of rifting
which we have attempted to minimise in this comprehensive approach.
Strengthening the kinematic framework for the South Atlantic rift also offers
an ideal laboratory to investigate the interaction between large scale plate
tectonics and lithosphere dynamics during rifting, and the relationship
between plume activity, rift kinematics and the formation of volcanic/non-
volcanic margins.
Data and high resolution images related to the publication are provided with
the manuscript and can be freely accessed under a Creative Commons license at
the Datahub under the following URL:
http://datahub.io/en/dataset/southatlanticrift.
###### Acknowledgements.
We would like to thank Statoil for permission to publish this work. CH and JZ
are indebted to many colleagues in Statoil’s South Atlantic new venture teams,
specialist groups, and R&D who have contributed in various aspects to an
inital version of the model. In particular we would like to thank Hans
Christian Briseid, Leo Duerto, Rob Hunsdale, Rosie Fletcher, Eric Blanc, Tommy
Mogensen Egebjerg, Erling Vågnes, Jakob Skogseid, and Ana Serrano-Oñate for
discussions and valuable feedback. Feedback from Patrick Unternehr, Ritske
Huismans, Julia Autin and Hannes Koopmann on various aspects of our work are
kindly acknowledged. CH thanks John Cannon and Paul Wessel for numerous fast
software bug fixes. Editor Douwe J. J. v. Hinsbergen, an anonymous reviewer
and Dieter Franke are thanked for constructive reviews and comments which
improved the discussion version of the paper. Dieter Franke also kindly
supplied digitised SDR outlines and structural data for the Argentine margin.
Plate reconstructions were made using the open-source software GPlates
(http://www.gplates.org), all figures were generated using GMT
(http://gmt.soest.hawaii.edu; Wessel and Smith, 1998). Structural restorations
were partly done using Midland Valley’s Move software with an academic
software license to The University of Sydney. CH is funded by ARC Linkage
Project L1759 supported by Shell International E&P and TOTAL. JZ publishes
with permission from Statoil ASA. RDM is supported by ARC Laureate Fellowship
FL0992245.
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Figure 1: Rigid crustal blocks, Jurassic-Cretaceous rift structures, and
sediment isopachs, reconstructed to 83.5 Ma position (magnetic anomaly chron
C34y) with Africa held fixed in present coordinates using the rotation poles
of Nürnberg and Müller (1991). Proterozoic rocks are based on United States
Geological Survey (2012), metamorphic basement grain as thin white lines
(Exxon Production Research Company, 1985). Oceanic fractures zones are shown
as gray lines. Tristan da Cuñha hotspot (TdC) indicated by volcano symbol.
Rigid crustal blocks denoted as black boxes, rift basins outlined by red fault
symbols and denoted in green font and white boxes. Abbreviations: A - Adamaoua
Highlands/Benoue Microplate, B - Bongor Block, Ben - Benoue Trough, BPB -
Borborema Province Block, Bida - Bida Basin, Bon - Bongor Trough, CaG/CaGN -
Canelones Graben (Sta. Lucia)/-North, Col - Colorado Rift, Dob - Doba Basin,
Des - Deseado Massif in Patagonia, Dos - Doseo Basin, EWB - Maurice Ewing
Bank, Gao - Gao Trough, GrK - Grein-Kafra Rift, Jat - Jatobá Rift, Lev -
General Levalle (Leboulaye) Rift, Mac - Maccachin Rift, MAL - Malvinas Block,
Mal - Malvinas Basin, Mar - Maranau Rift, Mel - Melut Rift, Mf - Mamfe Basin,
MOR - South Atlantic mid-oceanic ridge, Mug - Muglad Rift, NEA - Northeast
Africa, NBR - Northeast Brazilian rifts, NF - Malvinas Norte/North Falklands,
NFB - North Falklands Block, NPM - North Patagonian massif, NWA - Northwest
Africa, O - Oban Highlands Block, PAM - Pampean Terrane, PdE - Punta del Este
Basin, RAW - Rawson Block, Raw - Rawson Rift, SAf - Southern Africa, Sal -
Salado Rift, SAM - South America, SJED - San Jorge Extensional Domain, SJuB -
San Julian Block, SLC - São Luis Craton, SLR - Sao Luis Rift SPH - Sao Paulo
High, Tef - Tefidet Rift, Ten - Ténéré Rift, Ter - Termit Rift, Tuc - Tucano
Rift, UBe- Upper Benoue Rift, Yola - Yola rift branch, Vald - Valdes Rift.
Sediment isopachs from Exxon Production Research Company (1985) are shown as
thin gray lines. Younger deformation and plates related rifting of East Africa
Rift system are not shown. Lambert Azimuthal Equal Area projection centered at
20∘W/12∘N.
Figure 2: Comparison of the timescales used in publications related to the
South Atlantic marginal basins and the African intraplate basins. GeeK07 is
Gee and Kent (2007), Exxon88 is Haq et al. (1987), G94 is Gradstein et al.
(1994) and GTS2004 is Gradstein et al. (2004). GeeK07+GTS04 and GeeK07+G94
shows magnetic polarity time scale with stratigraphic intervals from GTS04 and
G94, respectively, adjusted to tiepoints annotated on the right hand side of
the stratigraphic columns (gray font), where *.N indicates the relative
position from the base of the chron (e.g. Base Barrêmian at 125.85 ma - Base
M4n young).
Figure 3: Margin cross sections and area balancing based on Blaich et al.
(2009, 2011) and our synthesised data based on depth-migrated and gridded
CongoSPAN data along various segments of the conjugate South Atlantic margins.
Tr: restored thickness. Extend of continental crust used for area balancing: L
- LaLOC (maximum estimate, landward limit oceanic crust), M - Minimum estimate
(conservative), B - COB based on Blaich et al. (2009, 2011). Note that our
interpretation of the CongoSPAN data (CS Grid North and South sections)
includes a zone where no Moho could be identified on seismic data, which is
here tentatively interpreted as exhumed mantle. Further note the difference in
length of the Colorado Basin section. The profile data are available as plain
text files at http://datahub.io/en/dataset/southatlanticrift
Figure 4: Oblique Mercator map of present day Africa, with onshore topography
(ETOPO1; Amante and Eakins, 2009), offshore free air gravity (Sandwell and
Smith, 2009), both grayscale, and gridded sediment thickness (Exxon Production
Research Company, 1985) superimposed. map legend: LaLOC - landward limit of
oceanic crust, Proterozoic rocks and Mesozoic extrusive rocks based on United
States Geological Survey (2012). Seaward-dipping reflectors (SDRs) based on
UTIG PLATES open data
(https://www.ig.utexas.edu/research/projects/plates/data.htm). White polygons
indicate sedimentary basins (N to S): SiL - Sierra Leone marginal, C’dI - Côte
d’Ivorie marginal, Sp - Saltpond, Bnn - Benin, Bid - Bida, Iull -
Iullememmden, Chad - Chad Basin, B’ue - Benoue Trough, Bgr - Bongor Trough,
Dob - Doba, Dos - Doseo, BgG - Banggara Graben, Mel - Melut, Mug - Muglad, Smt
- Salamat, Ng - Ngaoundere, Dou - Douala, RM - Rio Muni, Ga - Gabon Coastal,
Cuv - Cuvette Central, LC - Lower Congo, Kw - Kwanza/Cuanza, IKw - Inner
Kwanza, Na/Bga - Namibe/Benguela, Wal - Walvis, SWA - South West African
marginal, Orange - Orange, Oud - Oudtshoorn, Otq - Outeniqua. Fz: Oceanic
fracture zone. Colored lines are oceanic isochrons from (Müller et al., 2008),
M0 - M0 magnetic anomaly (120.6 ma). Plate and sub-plates: NWA - Northwest
Africa, NUB–Nubia/Northeast Africa, JOS- Jos Plateau Subplate and SAf –
Austral Africa. Origin of map projection at 10${}^{\circ}S$ 5∘W, azimuth of
oblique equator 80∘.
Figure 5: Computed extension for the Termit Basin in Chad based on total
sediment thickness. a) sediment thickness based on Exxon Production Research
Company (1985); b) stretching factor grid and computed amount of extension
using the method of Le Pichon and Sibuet (1981), see text for details.
Estimates of horizontal extension for the Termit basin. Cyan colored lines are
main rift bounding faults, light blue lines are profile lines with original
length, red lines are restored length based on extension estimates from
sediment thickness. Number indicate individual amount of computed extension
along each line in kilometers.
Figure 6: Stage pole rotation and associated small circles (dashed grey lines)
at 2∘ spacing for 145–110 ma interval for the Central African Rift System
(CARS) and associated relative plate motions at 110 Ma. main rigid plates: SAf
- Southern Africa, NEA - Northeastern Africa. main basins and rifts: Bgr -
Bongor Basin, Dob - Doba, Dos - Doseo, Smt - Salamat, BgG - Bangara Graben,
Mug- Muglad, Mel - Melut, Anza - Anza Rift. Red boxes and text indicate
published profiles for Melut and Muglad Basins and estimated extension values
based on McHargue et al. (1992): M-A – Profile A-A’, M-E – Profile E-E’, M-X –
Profile X-X’. Vectors shown on map only show relative motions, white core of
published profiles and of green flowlines indicates actual amount of extension
in km. These are $\approx$24 km for profile M-A, $\approx$27 km for profile
M-E and $\approx$15 km for Profile M-X during the intial basin formin phase
F1(McHargue et al., 1992, red;), with modeled extension of $\approx$30 km,
$\approx$17 km, and $\approx$17 km (green), respectively. Red vectors indicate
angular displacement of rigid block around stage pole back to fit
reconstruction position.
Figure 7: Stage pole rotation and associated small circles (dashed and
stippled grey lines) at 2∘ spacing for 145–110 Ma interval for the West
African Rift System (WARS) and associated relative plate motions at 110 ma.
main plates: NEA - Northeast Africa, NWA - Nortwest Africa, Jos - Jos
subplate. main rifts: Gao - Gao Trough, UBe- Upper Benoue, Ter - Termit, Bgr -
Bongor, Tef - Tefidet, Ten - Ténéré, GrK - Grein-Kafra. Basins: Chad - Chad
Basin, Iullemmeden - Iullemmeden Basin, Bida - Bida Basin. Red boxes and text
indicate published profiles for Grein-Kafra and Termit Rifts and estimated
extension values based on Genik (1992): G-A – Profile A-A’, G-B – Profile
B-B’. Vectors shown on map only show relative motions, white core of published
profiles and of green flowlines indicates actual amount of extension in km.
These are $\approx$45 km for profile G-A and 40–80 km for Profile G-B, (Genik,
1992, red;), with modeled extension of $\approx$28 km and $\approx$43 km
(green), respectively. Red vectors indicate angular displacement of rigid
block around stage poles back to fit reconstruction position. Vectors
northeast of the Grein-Kafra rift indicate possible early Cretaceous basin-
and-swell topography related to strike-slip reactivation of basement
structures in southern Lybia based on Guiraud et al. (2005).
Figure 8: Oblique Mercator map of present day South America (SAM), with
onshore topography (ETOPO1; Amante and Eakins, 2009), offshore free air
gravity (Sandwell and Smith, 2009), both grayscale, and gridded sediment
thickness (Exxon Production Research Company, 1985) superimposed. map legend
as in Fig. 4. White polygons denote sedimentary basins (N to S): FdA - Foz do
Amazon, PM - Para-Maranhao, Ba - Barreirinhas, PC - Piaui-Ceara, SLu - Sao
Luis Graben, Px - Rio do Peixe, Pot - Potiguar, PP - Pernambuco–Paraiba, Ar -
Araripe, SeA - Sergipe-Alagoas, Tu - Tucano, Re - Recôncavo, Jac - Jacuipe, Am
- Amazonas, C’mu - Camamu, Cx - Cumuruxatiba, ES - Espirito Santo, Cam -
Campos, San - Santos, Tau - Taubaté, Parana - Parana Basin, Pel - Pelotas, StL
- Santa Lucia, Sal - Salado, Lev - General Levalle, Col - Colorado, mac -
maccachin, Nq - Neuqén, Raw - Rawson, NF - North Falklands, SJ - San Jorge
Colored lines are oceanic crust ages from (Müller et al., 2008). Plate
boundaries (thin back decorated lines) after Bird (2003). Origin of map
projection at 50${}^{\circ}S$ 20∘W, azimuth of oblique equator -65∘.
Figure 9: Plate circuits used for the plate model for two main stages: a)
140–120.6 Ma, and b) 120.6–110 Ma for the main lithospheric plates. Integers
indicate the Plate-ID number. Integers on connection lines refer deforming
zones (Tab. Kinematics of the South Atlantic rift). Dashed lines indicate
areas where extension is indirectly constrained by plate circuit, connections
with dots indicate no relative motion. Plate abbreviations
Figure 10: Age-coded flowlines plotted on filtered free air gravity basemap
(offshore) for the North Gabon/Rio Muni segment of the African margin. Initial
phase flowline direction of our preferred model (PM1, thin black line with
circles) is perpendicular to the trend of the present-day coastline, striking
NW-SE and only turns parallel to oceanic transform zones (trending SW-NE)
after $\approx$ 126 ma. Note that models PM4 & PM 5 induce a significant
amount of compression for the intial phase of relative plate motions and do
not agree with geological observations from the area. Relative motions between
SAf and SAm in Model NT91 commence at 131 ma, Circles are plotted in 2 Myr
time intervals starting at 144 ma and from 132 for NT91 (Nürnberg and Müller,
1991; Torsvik et al., 2009). Legend abbreviations: LaLOC - Landward limit of
oceanic crust, C. - Contour. Free air gravity anomalies (Sandwell and Smith,
2009) filtered with 5th order Butterworth lowpass filter with 35 km
wavelength.
Figure 11: Age-coded flowlines plotted on filtered free air gravity basemap
(offshore) for the Orange Basin segment along the West African margin. Early
phase opening is oblique to present day margin, with oblique initial extension
of SAm relative to SAf (4 northern flowlines) in WNW-ESE direction and intial
extension between Patagonian South American blocks and SAf in SW-NE direction
(southern 2 flowlines). Note that the Orange Basin is located between the two
divergent flowline populations and that a positive gravity anomaly is
contemporaneous with the inflection point at 126.57 ma (M4) and associated
velocity increase and extension direction change (cf. Fig. 15) along the
margin. Legend/symbology and filter as in Fig. 10.
Figure 12: Age-coded flowlines plotted on filtered free air gravity basemap
(offshore) for the Santos Basin segment on South American margin. SPH- São
Paulo High in the outer Santos basin. Note that initial NW-SE relative
extensional direction as predicted by the plate model is perpendicular to the
Santos margin hingeline (cf. Meisling et al., 2001, for details on the crustal
structure in along the inner Santos basin margin) and does conform to observed
structural patterns in the extended continental crust of the SPH (cf. Chang,
2004). Legend/symbology and filter as in Fig. 10
Figure 13: Plate tectonic reconstruction at 140 Ma, Africa fixed in present-
day coordinates. For map legend see Fig. 22, bottom color scales indicate
geological stages and magnetic reversals with red diamond indicating time of
reconstruction, abbreviations here: T - Tithonian, Val. - Valaginian, Haut’n -
Hauterivian, Barrem. - Barrêmian. Restored continental margin is indicated by
dashed magenta colored line, restored profiles (Fig. 3) by solid magenta
lines. Rigid lithospheric blocks denoted by black labels: SAf - Austral
African Plate, BoP - NE Brazilian Borborema Province plate, Jos - Jos Plateau
Subplate, NEA - NE African Plate, NPM - North Patagonian massif, PT - Pampean
Terrane, RaB - Rawson Block, Sal - Salado Subplate, SAm - main South American
Plate, SJuB - San Julian Block, SLC - São Luis Craton Block. Actively
extending basins are indicated by red background in label, postrift basins are
indicated by light gray background in label, abbreviations: BeT - Benoue
Trough, CaR - Canelones Rift, D/D B. - Doba and Doseo Basins, ColB - Colorado
Basin, IuB - Iullemmeden Basin, marB - marajó Basin, MugB - Muglad Basin, NFB
- North Falkland Basin, PotB - Potiguar Basin, TRJ - Reconavo, Jatoba, Tucano
Basins, SaB - Salado Basin. Other abbreviations: RdJ - Rio de Janeiro, B. -
Basin, GPl. - Guyana Plateau, DeR - Demerara Rise. Present-day hotspots are
shown as volcano symbol and plotted with 400 km diameter (dashed magenta
colored circle with hachured fill), assuming that they are stationary over
time: A - Ascension, B - Bouvet, Ch - Chad/Tibesti, Ca - Mount Cameroon, Cd -
Cardno Seamount, D - Discovery, T - Tristan da Cuñha. Figure 14: Overlap
between the conjugate present-day landward limits of the oceanic crust (LaLOC)
for the South America (SAm), Southern Africa (SAf) and Northwest Africa (NWA).
Shown is the fit reconstruction at 143 with SAf fixed in present-day position.
Spacing between contour lines is 50 km, present-day coastlines as thick black
lines. Note that the position of the LaLOC along the volcanic margins in
southern part of the South Atlantic rift is not well constrained. Largest
overlap is created between the Campos/Santos basin margin and the southern
Kwanza/Benguela margin implying up to 500 km extension for the SE’most Santos
Basin.
Figure 15: Relative separation velocities and directions between 6 South
American and Patagonian points relative to a fixed Southern African plate
using preferred plate kinematic model. Upper portion of figure shows
extensional velocities over time, lower portion shows extension direction
relative to a fixed Southern African plate.
Figure 16: Plate tectonic reconstruction at 132 Ma, with Africa fixed in
present-day coordinates. For map legend see Fig. 22, abbreviations as in Fig.
13. Note initial extension directions along the margin rotate from NW-SE in
Gabon/Sergipe-Alagoas segment to W-E in Pelotas/Walvis Basin segment with
increasing distance from stage pole location. Flowlines between Patagonian
blocks in southern South America and southern Austral Africa indicate and
initial SW-NE directed motions between these plates (cf. Fig 11). In West
Africa, the Iullemmeden and Bida Basin as well as the Gao Trough are
undergoing active extension. Tectonism along the Equatorial Atlantic rift
increases. Additional abbreviations: DGB - Deep Ghanian Basin, CdIGR - Côte
d’Ivoire/Ghana Ridge and associated marginal basins. Figure 17: Plate
tectonic reconstruction at 125 Ma, with Africa fixed in present-day
coordinates. For map legend see Fig. 22, abbreviations as in Fig. 13. Note
that at this time, the West African pre-salt basin width is generated and
seafloor spreading is abutting against the Walvis ridge/Florianopolis ridge at
the southern margin of the Santos Basin, with extension and possible rifting
along the Abimael ridge in the inner SW part of the Santos basin (cf.
Scotchman et al., 2010). Relative rotation of SAm to NWA results in
$\approx$20 km of transpression between the Guinea Plateau and the Demerara
Rise. Rift basins along the Equatorial Atlantic rift are all actively
subsiding. Additional abbreviations: SPH - São Paulo High, EB - Maurice Ewing
Bank. Figure 18: Plate tectonic reconstruction at 120 Ma, with Africa fixed
in present-day coordinates.For map legend see Fig. 22, abbreviations as in
Fig. 13. Significant increase in extensional velocities causes break up and
subsequent seafloor spreading in the northernmost part of the South Atlantic
rift extending down to the conjugate Cabinda/Espirito Santo segment. Due to
changed kinematics, rifting and extension in the Santos/Benguela segment
focusses on the African side, causing the transfer of the Sao Paulo High block
onto the South American Plate. Breakup also occurs between Guinea Plateau and
Demerara rise in the westernmost part of the Equatorial Atlantic. Figure 19:
Plate tectonic reconstruction at 115 Ma, with Africa fixed in present-day
coordinates. For map legend see Fig. 22, abbreviations as in Fig. 13.
Continental break-up has occurred along most parts of the conjugate South
American/Southern African and NW African margins apart from the “Amazon” and
Côte d’Ivoire/Ghana transform segment of the Equatorial Atlantic. Oceanic
accrection has already commenced in the Deep Ghanaian Basin. In the SARS, only
the Benguela-Walvis/Santos segment have not yet broken up. In this segment,
extension is focussed asymmetrically close to the African margin, east of the
São Paulo High. Basins in southern South America have entered the post rift
phase and no significant relative motions between the rigid lithospheric
blocks occur in post-Barrêmian times. Post-rift thermal subsidence and
possible gravitationally-induced flow of evaporite deposits towards the basin
axis occurs in conjugate margin segments of the central SARS. Figure 20:
Plate tectonic reconstruction at 110 Ma, with Africa fixed in present-day
coordinates. For map legend see Fig. 22, abbreviations as in Fig. 13. Full
continental separation is achieved at this time, with narrow oceanic gateways
now opening between the Côte d’Ivoire/Ghana Ridge and the Piaui-Céara margin
in the proto-Equatorial Atlantic and between the Ewing Bank and Aghulas Arch
in the southernmost South Atlantic. Deformation related to the break up
between Africa and South America in the African intracontinental rifts ceases
in post-Aptian times. Towards the Top Aptian, break up between South America
and Africa has largely been finalised. The only remaining connections are
between major offset transfer faults in the Equatorial Atlantic rift and
between the outermost Santos Basin and the Benguela margin where a
successively deeping oceanic gateway between the northern and southern Proto-
South Atlantic is proposed. Seafloor spreading is predicted for the conjugate
passive margin segments such as the Deep Ghanaian Basin (DGB). The stage pole
rotation between SAm and NWA predicts compression alon gthe CdIGR in
accordance with observed uplift during this time (e.g. Pletsch et al., 2001;
Clift et al., 1997). Abbreviations: CdIGR - Côte d’Ivoire-Ghana Ridge, EB -
Maurice Ewing Bank. Figure 21: Plate tectonic reconstruction at 104 Ma, with
Africa fixed in present-day coordinates. For map legend see Fig. 22,
abbreviations as in Fig. 13. Full continental separation is achieved at this
time, with narrow oceanic gateways now opening between the Côte d’Ivoire/Ghana
Ridge and the Piaui-Céara margin in the proto-Equatorial Atlantic and between
the Ewing Bank and Aghulas Arch in the southernmost South Atlantic.
Deformation related to the break up between Africa and South America in the
African intracontinental rifts ceases in post-Aptian times. Figure 22: Map
legend for Figs. 13-21.
Table 1: Finite rotations parameters for main rigid plates for preferred South
Atlantic rift model M1. Abbreviations are Abs - absolute reference frame, SAf
- Southern Africa, NEA - Northeast Africa, NWA - Northwest Africa, Sal -
Salado Subplate, NPM - North Patagonian massif block, CM - Mesozoic magnetic
anomaly chron, y - young, o - old. magnetic anomaly timescale after Gee and
Kent (2007). For microplate rotation parameters please see electronic
supplements.
Moving | | Rotation pole | Fixed | Comment
---|---|---|---|---
plate | Age | Lon | Lat | Angle | plate |
SAf | 100.00 | 14.40 | -29.63 | -20.08 | Abs | O’Neill et al. (2005)
SAf | 110.00 | 6.61 | -29.50 | -26.77 | Abs | Steinberger and Torsvik (2008)
SAf | 120.00 | 6.11 | -25.08 | -30.45 | Abs | Steinberger and Torsvik (2008)
SAf | 130.00 | 5.89 | -25.36 | -33.75 | Abs | Steinberger and Torsvik (2008)
SAf | 140.00 | 7.58 | -25.91 | -38.53 | Abs | Steinberger and Torsvik (2008)
SAf | 150.00 | 10.31 | -27.71 | -37.25 | Abs | Steinberger and Torsvik (2008)
NEA | 110.0 | 0.0 | 0.0 | 0.0 | SAf |
NEA | 140.0 | -0.13 | 39.91 | 1.35 | SAf | Fit
NWA | 110.0 | 0.0 | 0.0 | 0.0 | NEA |
NWA | 140.0 | 25.21 | 5.47 | 2.87 | NEA | Fit
SAm | 83.5 | 61.88 | -34.26 | 33.51 | SAf | Nürnberg and Müller (1991), CAn34
SAm | 96.0 | 57.46 | -34.02 | 39.79 | SAf | Linear interpolation
SAm | 120.6 | 51.28 | -33.67 | 52.35 | SAf | Anomaly CM0ry
SAm | 120.6 | 52.26 | -34.83 | 51.48 | NWA | Crossover CM0ry
SAm | 126.57 | 50.91 | -34.59 | 52.92 | NWA | Anomaly CM4no
SAm | 127.23 | 50.78 | -34.54 | 53.04 | NWA | Anomaly CM7ny
SAm | 140.0 | 50.44 | -34.38 | 53.40 | NWA | Fit
NPM | 124.05 | 0.0 | 0.0 | 0.0 | Sal |
NPM | 150.00 | -35.94 | -63.06 | 4.10 | Sal | Fit - Extension based on Pángaro and Ramos (2012)
Sal | 124.05 | 0.0 | 0.0 | 0.0 | SAm |
Sal | 145.00 | -33.02 | -60.52 | 4.40 | SAm | Fit
| | | | | |
Table 2: Alternative kinematic scenarios tested for onset and cessation of
syn-rift phases in main rift zones.
Model | Syn-rift onset | Syn-rift cessation in WARS and CARS
---|---|---
PM1 | SARS, CARS, WARS, and EqRS simultaneously: around Base Cretaceous (140 Ma). | 110 Ma (early Albian)
PM2 | SARS, CARS, WARS, and EqRS simultaneously: around Base Cretaceous (140 Ma). | 100 Ma (late Albian)
PM3 | SARS, CARS and WARS simultaneously: around Base Cretaceous (140 Ma); EqRS: Top Valangian (132 Ma) | 110 Ma (early Albian)
PM4 | SARS, EqRS, and CARS simultaneously: around Base Cretaceous (140 Ma); WARS: Base Valangian (135 Ma) | 110 Ma (early Albian)
PM5 | SARS and EqRS simultaneously around Base Cretaceous (140 Ma); WARS and CARS: 135 Ma | 110 Ma (early Albian)
NT91 | Model parameters of Nürnberg and Müller (1991) with forced breakup at 112 Ma after Torsvik et al. (2009)
| |
Table 3: Deformation estimates used for major intraplate deformation zones.
“Computed” extension is based on the our method as described in Sec. 1 and
represents a maximum estimate as it is based on total sediment thickness,
always measured orthogonal to major basin/rift axis. “Implemented” deformation
is actual displacement as implemented in plate kinematic model. Integers in
column “Deforming zone” refer to plate circuit (Fig. 9), abbreviations refer
to plate pairs. $\perp$: extension measured orthogonal to long rift/basin
axis, sc: along small circle around corresponding stage pole.
Deforming | Basin | Extension [km] | Syn-rift duration
---|---|---|---
zone | names | Published | Computed | Implemented | [Myrs]
CARS - Central African Rift System
1 - NEA-SAf | Melut Basin | 15 kma | 76 km (max.) | see Muglad | 140–110 Ma
1 - NEA-SAf | Muglad Basin | 24–27 kma,22–48 kmb, 56$\pm 6$c km | 83 km (max.) | 50 km (N), 27 km (S) | 140–110 Ma
1 - NEA-SAf | Doseo Basin | 35-40 kmd (dextral) | 41 km (Doseo) $\perp$ | 52 km (oblique), 30 km (E-W) | 140–110 Ma
2 - BON-NEA | Bongor/Doba Basins | | 31 km (Bongor, $\perp$), 20 km (Doba, $\perp$) | 57 km for Doba & Bongor basins comb. (sc) | 140–100 Ma
WARS - West African Rift System
3 NWA-NEA | Termit Basin | 40 kmd | 27–98 km | 60-65 km $\perp$ , 80-85 km (sc) | 140–110 Ma
3 NWA-NEA | Grein-Kafra Basin | (?) | 61 km (max.) | 22 km $\perp$, 60-65 km (sc) | 140–110 Ma
4 JOS-NWA | Bida/Gao/Iullemmeden | (?) | 28 km (Bida/Iullemmeden, max.) | 15-20 km (Gao Trough, sc) | 135–110 Ma
JOS $|$ BEN | (Central) Benoue Trough | | 61 km (max.) | 36 km $\perp$, 40 km (sc) | 140–110 Ma
South America
5 - SAL-SAm | Salado/Punta Del Este Basins | (?) | 65 km (max.) | 45 km (sc, $\perp$) | 140–120.6 Ma
6 - NPM-SAL | Colorado Basin | 45 kmb | 106 km (max.) | 40 km (sc, $\perp$) | 140–120.6 Ma
aMcHargue et al. (1992), bBrowne and Fairhead (1983), cMohamed et al.
(2001),dGenik (1992),ePángaro and Ramos (2012)
|
arxiv-papers
| 2013-01-10T11:53:17 |
2024-09-04T02:49:40.109287
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christian Heine and Jasper Zoethout and R. Dietmar M\\\"uller",
"submitter": "Christian Heine",
"url": "https://arxiv.org/abs/1301.2096"
}
|
1301.2130
|
# Distributed soft thresholding
for sparse signal recovery
C. Ravazzi, S.M. Fosson, and E. Magli
Department of Electronics and Telecommunications,
Politecnico di Torino, Italy
###### Abstract
In this paper, we address the problem of distributed sparse recovery of
signals acquired via compressed measurements in a sensor network. We propose a
new class of distributed algorithms to solve Lasso regression problems, when
the communication to a fusion center is not possible, e.g., due to
communication cost or privacy reasons. More precisely, we introduce a
distributed iterative soft thresholding algorithm (DISTA) that consists of
three steps: an averaging step, a gradient step, and a soft thresholding
operation. We prove the convergence of DISTA in networks represented by
regular graphs, and we compare it with existing methods in terms of
performance, memory, and complexity.
###### Index Terms:
Distributed compressed sensing, distributed optimization, consensus
algorithms, gradient-thresholding algorithms.
## I Introduction
Compressed sensing [1] is a new technique for nonadaptive compressed
acquisition, which takes advantage of the signal’s sparsity in some domain and
allows the signal recovery starting from few linear measurements. In this
framework, distributed compressed sensing [2] has emerged in the last few
years with the aim of decentralizing data acquisition and processing. Most
attention has been devoted to study how to perform decentralized acquisition,
e.g., in sensor networks, assuming that recovery can be performed by a
powerful fusion center that gathers and processes all the data. This model,
however, has some significant drawbacks. First, data collection at a fusion
center may be prohibitive in terms of energy utilization, particularly in
large-scale networks, and may also introduce delays, which reduce the sensor
network performance. Second, robustness is critical: a failure of the fusion
center would stop the whole process, while some sensors’ faults are generally
tolerated in networks of considerable dimensions. Third, in several
applications sensors may not be willing to convey their information to a
fusion center for privacy reasons [3].
In this work, we consider the problem of in-network processing and recovery in
distributed compressed sensing as formulated in [4]. Specifically, we assume
that no fusion center is available and we consider networks formed by sensors
that can store a limited amount of information, perform a low number of
operations, and communicate under some constraints. Our aim is to study how to
achieve compressed acquisition and recovery leveraging on these seemingly
scarce resources, with no computational support from an external processor.
The key point is to suitably exploit local communication among sensors, which
allows to spread selected information through the network. Based on this, we
develop an iterative algorithm that achieves distributed recovery thanks to
sensors’ collaboration.
In a single sensor setting, the problem of reconstruction can be addressed in
different ways. The $\ell_{1}$-norm minimization and Lasso are known to
achieve optimal reconstruction under a sparsity model, but they are
computationally expensive. E.g., the complexity of $\ell_{1}$-norm
minimization is cubic in the length of the vector to be reconstructed.
Therefore, several iterative methods have been developed, which yield
suboptimal results at a fraction of the computation cost of the optimal
methods [5]. On the other hand, the reconstruction problem in a distributed
setting has received much less attention so far [4]. The complexity of optimal
methods is rather large also in the distributed case. This is even more
important in a sensor network scenario, where the limited amount of energy and
computation resources available at each node calls for algorithms with low
complexity and low memory usage. This paper fills this gap, presenting a
distributed iterative algorithm for compressed sensing reconstruction. In
particular, we propose a decentralized version of iterative thresholding
methods [5]. The proposed technique consists of a gradient step that seeks to
minimize the Lasso functional, a thresholding step that promotes sparsity and,
as a key ingredient, a consensus step to share information among neighboring
sensors.
Our aim in the design of this reconstruction algorithm is to achieve a
favorable trade-off between the reconstruction performance, which should be as
close as possible to that of the optimal distributed reconstruction [4], and
the complexity and memory requirements. Memory usage is particularly critical,
as microcontrollers for sensor networks applications typically have a few kB
of RAM. As will be seen, the proposed algorithm is indeed only slightly
suboptimal with respect to [4], in terms of the overall number of measurements
required for a successful recovery. On the other hand, it features a much
lower memory usage, making it suitable also for low-energy environments such
as wireless sensor networks.
In this paper, we theoretically prove the convergence of the algorithm for
networks that can be represented by regular graphs. Moreover, we numerically
verify its good performance, comparing it with that of existing methods, such
as distributed subgradient method (DSM, [6]) and alternating direction method
of multipliers (ADMM, [4] and [7]).
## II Problem formulation
### II-A Notation
Throughout this paper, we use the following notation. We denote column vectors
with small letters, and matrices with capital letters. Given a matrix ${X}$,
${X}^{\mathsf{T}}$ denotes its transpose and $(X)_{v}$ (or $x_{v}$) denotes
the $v$-th column of $X$. We consider $\mathbb{R}^{n}$ as a Banach space
endowed with the following norms:
$\|x\|_{p}=\left(\sum_{i=1}^{n}|x_{i}|^{p}\right)^{1/p}$ with $p=1,2.$ For a
rectangular matrix $M\in\mathbb{R}^{m\times n}$, we consider the Frobenius
norm
$\left\|M\right\|_{F}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}M_{ij}^{2}}=\sqrt{\sum_{j=1}^{n}\left\|(M)_{j}\right\|_{2}^{2}},$
and the operator norm $\left\|M\right\|_{2}=\sup_{z\neq
0}{\|Mz\|_{2}}/{\|z\|_{2}}.$ We define the sign function as
$\text{sgn}(x)=1\text{ if }x>0$, $\mathrm{sgn}(0)=0$ and $\text{sgn}(x)=-1\
\text{otherwise}$. If $x$ is a vector in $\mathbb{R}^{n}$, $\text{sgn}(x)$ is
intended as a function to be applied elementwise. A symmetric graph is a pair
$\mathcal{G}=(\mathcal{V,E})$ where $\mathcal{V}$ is the set of nodes, and
$\mathcal{E}\subseteq\mathcal{V\times V}$ is the set of edges with the
property that $(i,i)\in\mathcal{E}$ for all $i\in\mathcal{V}$ and
$(i,j)\in\mathcal{E}$ implies $(j,i)\in\mathcal{E}$. A $d$-regular graph is a
graph where each node has $d$ neighbors. A matrix with non-negative elements
$P$ is said to be stochastic if $\sum_{j\in\mathcal{V}}P_{ij}=1$ for every
$i\in\mathcal{V}$. Equivalently, $P$ is stochastic if
$P{\mathds{1}}={\mathds{1}}$. The matrix $P$ is said to be adapted to a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ if $P_{v,w}=0$ for all
$(w,v)\notin{\mathcal{E}}.$
### II-B Model and assumptions
We consider a sensor network whose topology is represented by a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$. We assume that each node
$v\in\mathcal{V}$ acquires linear measurements of the form
$y_{v}=A_{v}x_{0}+\xi_{v}$ (1)
where $x_{0}\in\mathbb{R}^{n}$ is a $k$-sparse signal (i.e., the number of its
nonzero components is not larger than $k$), $\xi_{v}\in\mathbb{R}^{m}$ is an
additive noise process independent from $x_{0}$, and
$A_{v}\in\mathbb{R}^{m\times n}$ (with $n>\\!\\!>m$) is a random projection
operator. If the data $(y_{v},A_{v})$ taken by all sensors were available at
once in a single fusion center that performs joint decoding, a solution would
be to solve the Lasso problem [8]. The Lasso refers to the minimization of the
convex function $\mathcal{J}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ defined by
$\mathcal{J}(x,\lambda):=\sum_{v\in\mathcal{V}}\left\|y_{v}-{A}_{v}x\right\|_{2}^{2}+\frac{2\lambda}{\tau}\left\|x\right\|_{1}$
(2)
where $\lambda>0$ is a scalar regularization parameter that is usually chosen
by cross validation [8] and $\tau>0$. Let us denote the solution of (2) as
$\widehat{x}=\widehat{x}(\lambda)=\underset{x\in\mathbb{R}^{N}}{\mathrm{argmin\,}}\mathcal{J}(x,\lambda).$
(3)
This optimization problem is shown to provide an approximation with a bounded
error, which is controlled by $\lambda$ [1]. A large amount of literature has
been devoted to developing fast algorithms for solving (2) and characterizing
the performance and optimality conditions. We refer to [5] for an overview of
these methods.
### II-C Iterative soft thresholding
A popular approach to solve (2) is the iterative soft thresholding algorithm
(ISTA). ISTA is based on moving at each iteration in the direction of the
steepest descent followed by thresholding to promote sparsity [9].
Let us collect the measurements in the vector
$y=(y_{1}^{\mathsf{T}},\ldots,y_{|\mathcal{V}|}^{\mathsf{T}})^{\mathsf{T}}$
and let $A$ be the complete sensing matrix
$A=(A_{1}^{\mathsf{T}},\ldots,A_{|\mathcal{V}|}^{\mathsf{T}})^{\mathsf{T}}$.
Given $x{(0)}$, iterate for $t\in\mathbb{N}$
$\displaystyle x{(t+1)}$ $\displaystyle=\eta_{\lambda}(x(t)+\tau
A^{\mathsf{T}}(y-Ax))$
where $\tau$ is the stepsize in the direction of the steepest descent. The
operator $\eta$ is a thresholding function to be applied elementwise, i.e.
$\eta_{\lambda}(x)=\text{sgn}(x)(|x|-\lambda)$ if $|x|<\lambda$ and
$\eta_{\lambda}(x)=0$ otherwise. The convergence of this algorithm was proved
in [9], under the assumption that $\|A\|_{2}^{2}<{1}/{\tau}.$
## III Proposed distributed iterative soft thresholding algorithm
In this section, we introduce our distributed iterative soft thresholding
algorithm (DISTA), which has been developed following the idea of minimizing a
suitable distributed version of the Lasso functional. In the next, we first
discuss such a functional and then we show the algorithm.
### III-A DISTA description
We recast the optimization problem in (2) into a separable form which
facilitates distributed implementation. The goal is to split this problem into
simpler subtasks executed locally at each node. Let us replace the global
variable $x$ in (2) with local variables $\\{x_{v}\\}_{v\in\mathcal{V}}$,
representing estimates of $x_{0}$ provided by each node. While the
conventional centralized Lasso problem attempts to minimize
$\mathcal{J}(x,\lambda)$, we recast the distributed problem as an iterative
minimization of the functional
$\mathcal{F}:\mathbb{R}^{n\times|\mathcal{V}|}\longmapsto\mathbb{R}^{+}$
defined as follows
$\displaystyle\begin{split}\mathcal{F}(x_{1},\ldots,x_{|\mathcal{V}|}):=\sum_{v\in\mathcal{V}}&\left[q\|y_{v}-A_{v}x_{v}\|_{2}^{2}+\frac{2\alpha}{\tau_{v}|\mathcal{V}|}\|x_{v}\|_{1}\right.\\\
&\left.+\frac{1-q}{\tau_{v}}\sum_{w\in\mathcal{V}}P_{v,w}\|\overline{x}_{w}-{x}_{v}\|_{2}^{2}\right]\end{split}$
(4)
where $P=[P_{v,w}]_{v,w\in\mathcal{V}}$ is a stochastic matrix adapted to the
graph $\mathcal{G}$, and for some $q\in(0,1)$.
By minimizing $\mathcal{F}$, each node seeks to recover the sparse vector
$x_{0}$ from its own linear measurements, and to enforce agreement with the
estimates calculated by other sensors in the network. It should also be noted
that $\mathcal{F}(\bar{x},\ldots,\bar{x})=q\mathcal{J}(\bar{x},\lambda)$ if
and only if $x_{v}=\overline{x}$, $\alpha=q\lambda$ and $\tau_{v}=\tau$ for
all $v\in\mathcal{V}$.
Note that $q$ can be viewed as a temperature parameter; as $q$ decreases,
estimates $x_{v}$ associated with adjacent nodes become increasingly
correlated. Let us denote as $\\{\widehat{x}_{v}^{q}\\}_{v\in\mathcal{V}}$ a
minimizer of (4). If $\mathcal{G}$ is connected, then we expect that
$\lim_{q\rightarrow 0}\widehat{x}_{v}^{q}=\widehat{x}$, $\forall
v\in\mathcal{V}.$ This fact suggests that if $q$ is sufficiently small, then
each vector $\widehat{x}_{v}^{q}$ can be used as an estimate of $x_{0}$.
DISTA seeks to minimize (4) in an iterative, distributed way. The key idea is
as follows. Starting from $x_{v}(0)=0$ for any $v\in\mathcal{V}$, each node
$v$ stores two messages at each time $t\in\mathbb{N}$, ${x}_{v}(t)$ and
$\overline{x}_{v}(t)$. The update is performed in an alternating fashion: at
even $t$, $\overline{x}_{v}(t+1)$ is obtained by a weighted average of the
estimates $x_{w}(t)$ for each $w$ communicating with $v$; at odd $t$,
$x_{v}(t+1)$ is computed as a thresholded convex combination of a consensus
and a gradient term. More precisely, the pattern is summarized in Algorithm 1.
Algorithm 1 DISTA
Given symmetric, row-stochastic matrix $P$ adapted to the graph, $\alpha$,
$\tau_{v}>0$, $x_{v}(0)=0$, $y_{v}=A_{v}x_{0}$ for any $v\in\mathcal{V}$,
iterate
* •
$t\in 2\mathbb{N}$, $v\in\mathcal{V}$,
$\displaystyle\overline{x}_{v}(t+1)$
$\displaystyle=\sum_{w\in\mathcal{V}}P_{v,w}{x}_{w}(t)$ $\displaystyle
x_{v}(t+1)$ $\displaystyle=x_{v}(t)$
* •
$t\in 2\mathbb{N}+1$, $v\in\mathcal{V}$,
$\displaystyle\overline{x}_{v}(t+1)$ $\displaystyle=\overline{x}_{v}(t)$
$\displaystyle{x}_{v}(t+1)$
$\displaystyle=\eta_{\alpha}\left[(1-q)\sum_{w\in\mathcal{V}}P_{v,w}\overline{x}_{w}(t)\right.$
$\displaystyle+q\Big{(}{x}_{v}(t)+\tau_{v}A_{v}^{\mathsf{T}}(y_{v}-A_{v}{x}_{v}(t))\Big{)}\Bigg{]}$
It should be noted that DISTA provides a distributed protocol: each node only
needs to be aware of its neighbors and no further information about the
network topology is required. Moreover, if $|\mathcal{V}|=1$, DISTA coincides
with ISTA. In section IV-A, we will prove its convergence and show that it
minimizes $\mathcal{F}$.
### III-B Discussion and comparison with related work
Algorithms for distributed sparse recovery (with no central processing unit)
in sensor networks have been proposed in the literature in the last few years.
We distinguish two classes:
1. 1.
algorithms based on the decentralization of subgradient methods for convex
optimization (DSM, [6]);
2. 2.
distributed implementation of the alternate method of multipliers (ADMM, [4,
10, 7]);
#### III-B1 DSM
Our proposed approach leverages distributed algorithms for multi-agent
optimization that have been proposed in the literature in the last few years
[6]. The main goal of these algorithms is to minimize over a convex set the
sum of cost functions that are convex and differentiable almost everywhere. It
should be noted that these methods can be applied to the Lasso functional. The
memory storage requirements and the computational complexity are similar to
DISTA, as it does not require to solve linear systems, to invert matrices, or
to operate on the matrices $A_{v}$. However, we emphasize some substantial
differences.
DSM is not guaranteed to converge while DISTA will be proved to converge to a
minimum of (4). The convergence can be achieved by considering the related
“stopped” model (see page 56 in [6]), whereby the nodes stop computing the
subgradient at some time, but they keep exchanging their information and
averaging their estimates only with neighboring messages for subsequent time.
However, the tricky point of such techniques is the optimal choice of the
number of iterations to stop the computation of the subgradient. Moreover, the
limit point cannot be variationally characterized and depends on the time we
stop the model. In [11], the stepsize for the subgradient computation
decreases to zero along the iterations. This choice, however, requires to fix
an initial time and is not be feasible in case of time-variant input:
introducing a new input would require some resynchronization. For this reason
the parameters $q,\tau$ in distributed iterative thresholding algorithms are
kept fixed and will be compared with DSM with constant stepsize.
#### III-B2 Consensus ADMM
The consensus ADMM [4, 7] is a method for solving problems in which the
objective and the constraints are distributed across multiple processors. The
problem in (2) is solved by introducing dual variables $\omega_{v}$ and
minimizing the augmented Lagrangian in a iterative way with respect to the
primal and dual variables. The algorithm entails the following steps for each
$t\in\mathbb{N}$: node $v$ receives the local estimates from its neighbors,
uses them to evaluate the dual price vector and the new estimate via
coordinate descent and thresholding. The bottleneck of the consensus ADMM is
the inversion of the $n\times n$ matrices $(A_{v}^{\mathsf{T}}A_{v}+\rho I)$
for some $\rho>0$.
Although the inversion of the matrices in consensus ADMM is performed off-
line, the storage of an $n\times n$ matrix may be prohibitive for a low power
node with a small amount of available memory. More precisely, for the
consensus ADMM each node has to store $2+m+mn+n^{2}+3n$ real values. DISTA,
instead, requires only $3+m+mn+2n$ real values, that correspond to $q$,
$\alpha$, $\tau_{v}$, $y_{v}$, $A_{v}$, $x_{v}(t)$, and $\overline{x}_{v}(t)$.
Suppose that the node can store $S$ real values: for the consensus ADMM, the
maximum allowed $n$ is of the order of $\sqrt{S}$, while for DISTA is of the
order of $S$. For example, let us consider the implementation of DISTA on
STM32F microcontrollers with Contiki operating system. Each node has 16 kB of
RAM; as the static memory occupied by consensus ADMM and DISTA is almost the
same, let us neglect it along with the memory used by the operating system
(the total is of the order of hundreds of byte). Using a single-precision
floating-point format, $2^{12}$ real values can be stored in 16 kB. Therefore,
even assuming that the nodes take just one measurement ($m=1$), consensus ADMM
can handle signals with length up to 61 samples, while DISTA up to 1364. This
clearly shows that DISTA is much more efficient in low memory devices.
## IV Main results
### IV-A Theoretical results
In this section, we summarize our convergence analysis of DISTA.
Let $X(t)=(x_{1}(t),\dots,x_{|\mathcal{V}|}(t))$,
$\overline{X}(t)=XP^{\mathsf{T}}$, and define the operator
$\Gamma:\mathbb{R}^{n\times|\mathcal{V}|}\longmapsto\mathbb{R}^{n\times|\mathcal{V}|}$
where
$(\Gamma
X)_{v}=\eta_{\alpha}\left[(1-q)(\overline{X}P^{\mathsf{T}})_{v}+q(x_{v}+\tau_{v}A_{v}^{\mathsf{T}}(y_{v}-A_{v}x_{v}))\right]$
with $v\in\mathcal{V}$. DISTA can be equivalently rewritten as $X(t+1)=\Gamma
X(t)$ with any initial condition $X(0)$.
Our goal is to find sufficient conditions that guarantee the convergence of
the estimates $X(t)$ to a finite limit point, and to characterize this limit
point in terms of the properties of the function $\mathcal{F}$ in (4).
In our analysis, we adopt the following assumptions.
###### Assumption 1.
$\mathcal{G}$ is a $d$-regular graph, $d$ being the degree of the nodes.
###### Assumption 2.
The nodes in $\mathcal{V}$ use uniform weights, i.e., $P_{u,v}=1/d$ if
$(u,v)\in\mathcal{E}$ and zero otherwise.
The following theorem ensures that, under certain conditions on the stepsize
$\\{\tau_{v}\\}_{v\in\mathcal{V}}$, the problem of minimizing the function in
(4) is well-posed.
###### Theorem 1 (Characterization of minima).
If $\tau_{v}<\|A_{v}\|_{2}^{-2}$ for all $v\in\mathcal{V}$, the set of
minimizers of $\mathcal{F}(X)$ is not empty and coincides with the set
$\mathrm{Fix}(\Gamma)=\\{Z\in\mathbb{R}^{n\times\mathcal{V}}:\Gamma Z=Z\\}$.
The proof is rather technical and is omitted for brevity. The interested
reader can refer to [12].
Moreover,
###### Theorem 2 (Convergence).
If $\tau_{v}<\|A_{v}\|_{2}^{-2}$ for all $v\in\mathcal{V}$, DISTA produces a
sequence $\\{X(t)\\}_{t\in\mathbb{N}}$ such that
$\lim_{t\rightarrow\infty}\left\|{X}(t)-X^{\star}\right\|_{F}=0$
where the limit point $X^{\star}$ is a fixed point of $\Gamma$.
These theorems guarantee that DISTA produces a sequence of estimates
converging to a minimum of the function $\mathcal{F}$ in (4). The sketch of
the proof is deferred to the Appendix.
It is worth mentioning that Theorem 2 does not imply that DISTA achieves
consensus: local estimates are very close to each other, but do not
necessarily coincide at convergence. Consensus can be obtained by letting $q$
go to zero or considering the related “stopped” model whereby the nodes stop
computing the subgradient at some time, but they keep exchanging their
information and averaging their estimates only with neighbors messages for the
rest of the time. Stopped models were introduced in [6].
### IV-B Numerical results
To demonstrate the good performance of DISTA, we conduct a series of
experiments for the complete graph architecture and for a variety of total
number of measurements. We consider the complete topology where
$P_{ij}=\frac{1}{N}$ for every $i,j=1,\dots,N$. For a fixed $n$, we construct
random recovery scenarios for sparse vector $x_{0}$. For each $n$, we vary the
number of measurements $m$ per node and the number of nodes in the network.
For each $(N,m,|\mathcal{V}|)$, we repeat the following procedure 50 times. A
signal is generated by choosing $k$ nonzero components uniformly among the $n$
elements and sampling the entries from a Gaussian distribution
$\mathsf{N(0,1)}$. Matrices $(A_{v})_{v\in\mathcal{V}}$ are sampled from the
Gaussian ensemble with $m$ rows, $n$ columns, zero mean and variance
$\frac{1}{m}$. We fix $n=150$, $k=15$, $\alpha=10^{-4}$, and $\tau=0.02$.
In the noise-free case, we show the performance of DISTA in terms of
reconstruction probability as a function of the number of measurements (see
Figure 1). In particular, we declare $x_{0}$ to be recovered if
$\sum_{v\in\mathcal{V}}\|x_{0}-x^{\star}_{v}\|^{2}_{2}\big{/}(n|\mathcal{V}|)<10^{-4}$.
Figure 1: Noise-free case: recovery probability of DISTA for a complete graph,
$n=150$, $k=15$. The red curve represents $m|\mathcal{V}|=70.$
The color of the cell reflects the empirical recovery rate of the 50 runs
(scaled between 0 and 1). White and black respectively denote perfect recovery
and failure for all experiments. It should be noted that the number of total
measurements $m|\mathcal{V}|$, which are sufficient for successfully recovery,
is constant: the red curve collects the points $(m,|\mathcal{V}|)$ such that
$m|\mathcal{V}|=70$, which turns out to be a sufficient value to obtain good
reconstruction.
In Figure 3 the probability of success of DISTA, consensus ADMM, and DSM are
compared as a function of the number of measurements per node. The curves are
depicted for different numbers of sensors. We notice that the number of
measurements needed for success by DISTA is smaller with respect to DSM. On
the other hand, DISTA has performance close to the optimal ADMM: we obtain an
almost perfect match in the curves obtained in the same scenario. In [12], we
have compared also the time of convergence: as known, DSM has problems of
slowness [4] and thousands of steps are not sufficient to converge. DISTA,
instead, is significantly faster, hence feasible. It does not reach the
quickness of consensus ADMM, which however has the price of the inversion of a
$n\times n$ matrix to start the algorithm. The interested reader can refer to
[12] for a test of the algorithm with a real dataset.
Finally, let us consider the noisy case. In Figure 3, the mean square error
$\textsf{MSE}=\frac{\sum_{v\in\mathcal{V}}\|{x_{0}}-x^{\star}_{v}\|^{2}_{2}}{n|\mathcal{V}|},$
averaged over 50 runs is plotted as a function of the signal-to-noise ratio
$\textsf{SNR}=\frac{\mathbb{E}\left[\sum_{v\in\mathcal{V}}\|y_{v}\|_{2}^{2}\right]}{\mathbb{E}\left[\sum_{v\in\mathcal{V}}\|\xi_{v}\|_{2}^{2}\right]}.$
The number of sensors is $|\mathcal{V}|=10$. The graph shows that DISTA
performs better then DSM, even at larger compression level: taking $m=8$ is
sufficient for DISTA to obtain a MSE lower than that obtained by DSM with
$m=12$ measurements. Notice that this is the best performance that can be
obtained by DSM in this setting, that is, even without compression we do not
see any improvement. On the other hand, DISTA with $m=12$ is worse than
consensus ADMM with same $m$ or with $m=8$, but it is better than consensus
ADMM with $m=6$ for sufficiently large SNR. In conclusion, DISTA can achieve
the optimal performance of consensus ADMM at the price of a smaller
compression level, which is not achievable by DSM.
Figure 2: Noise-free case: DISTA vs DSM and consensus ADMM, complete graph,
$n=150$, $k=15$.
Figure 3: Noise case: DISTA vs DSM and consensus ADMM, complete graph,
$n=150$, $k=15$, $|\mathcal{V}|=10$.
## V Concluding remarks
The problem of distributed estimation of sparse signals from compressed
measurements in sensor networks with limited communication capability has been
studied. We have proposed the first iterative algorithm for distributed
reconstruction, based on the iterative soft thresholding principle. This
algorithm has low complexity and memory requirements, making it suitable for
low energy scenarios such as wireless sensor networks. Numerical results show
that the proposed algorithm outperforms existing distributed schemes in terms
of memory and complexity, and is almost as good as the consensus ADMM method.
We have also provided proof of convergence in the case of regular graphs.
## VI Acknowledgment
This work is supported by the European Research Council under the European
Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement
n.279848.
## References
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* [2] D. Baron, M. F. Duarte, M. B. Wakin, S. Sarvotham, and R. G. Baraniuk, “Distributed compressive sensing,” http://arxiv.org/abs/0901.3403, 2005 (revised 2009).
* [3] P. A. Forero, A. Cano, and G. B. Giannakis, “Consensus-based distributed support vector machines,” J. Mach. Learn. Res., vol. 99, pp. 1663 – 1707, 2010.
* [4] G. Mateos, J. A. Bazerque, and G. B. Giannakis, “Distributed sparse linear regression,” IEEE Trans. Signal Proc., vol. 58, no. 10, pp. 5262 – 5276, 2010\.
* [5] M. Fornasier, Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Series on Computational and Applied Mathematics, 2010.
* [6] A. Nedic, A. Ozdaglar, and P. A. Parrillo, “Constrained consensus and optimization in multi-agent networks,” IEEE Trans. Automat. Contr., vol. 55, pp. 922 – 938, 2010.
* [7] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn., vol. 3, no. 1, pp. 1 – 122, 2011.
* [8] R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the Royal Statistical Society, Series B, vol. 58, pp. 267 – 288, 1994.
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## Appendix A Proof of Theorem 2
We provide a sketch of the proof that the sequence of the
$\\{X(t)\\}_{t\in\mathbb{N}}$ converges to a fixed point of $\Gamma$, applying
the Opial’s Theorem to the operator $\Gamma$ (see Theorem 2).
###### Theorem 3 (Opial’s theorem [13]).
Let $T$ be an operator from a finite-dimensional space $\mathbb{S}$ to itself
that satisfies the following conditions:
1. 1.
$T$ is asymptotically regular (i.e., for any $x\in\mathbb{S}$, and for
$t\in\mathbb{N}$, $\left\|T^{t+1}x-T^{t}x\right\|_{2}\to 0$ as $t\to\infty$);
2. 2.
$T$ is nonexpansive (i.e., $\left\|Tx-
Tz\right\|_{2}\leq\left\|x-z\right\|_{2}$ for any $x,z\in\mathbb{S}$);
3. 3.
$\mathrm{Fix}(T)\neq\emptyset$, $\mathrm{Fix}(T)$ being the set of fixed point
of $T$.
Then, for any $x\in\mathbb{S}$, the sequence $\\{T^{t}(x)\\}_{t\in\mathbb{N}}$
converges weakly to a fixed point of $T$.
It should be noticed that in $\mathbb{R}^{n}$ the weak convergence coincides
with the strong convergence.
Let us now prove that $\Gamma$ satisfies the Opial’s conditions. Instead of
optimizing (4), let us introduce a surrogate objective function:
$\begin{split}&\mathcal{F}^{\mathcal{S}}(X,C,B):=\sum_{v\in\mathcal{V}}\bigg{(}q\left\|A_{v}x_{v}-y_{v}\right\|_{2}^{2}+\frac{2\alpha}{\tau_{v}}\left\|x_{v}\right\|_{1}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{1-q}{d\tau_{v}}\sum_{w\in\mathcal{N}_{v}}\left\|x_{v}-c_{w}\right\|_{2}^{2}+\frac{q}{\tau_{v}}\left\|x_{v}-b_{v}\right\|_{2}^{2}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-q\left\|A_{v}(x_{v}-b_{v})\right\|_{2}^{2}\bigg{)}\end{split}$ (5)
where $C=(c_{1},\dots,c_{|\mathcal{V}|})\in\mathbb{R}^{n\times|\mathcal{V}|}$,
$B=(b_{1},\dots,b_{|\mathcal{V}|})\in\mathbb{R}^{n\times|\mathcal{V}|}$. It
should be noted that, defining $\overline{X}=XP^{\mathsf{T}}$,
$\mathcal{F}^{\mathcal{S}}(X,\overline{X},X)=\mathcal{F}(X).$
If we suppose $\alpha>0$ and $\tau_{v}<\|A_{v}\|_{2}^{-2}$ for all
$v\in\mathcal{V}$, then $\mathcal{F}^{\mathcal{S}}$ is a majorization of
$\mathcal{F}$, and minimizing $\mathcal{F}^{\mathcal{S}}$ leads to a
majorization-minimization (MM) algorithm. The optimization of (5) can be
computed by minimizing with respect to each $x_{v}$ separately.
###### Proposition 4.
The following fact holds:
$\displaystyle\underset{x_{v}\in\mathbb{R}^{n}}{\mathrm{argmin\,}}\mathcal{F}^{\mathcal{S}}(X,C,B)=\eta_{\alpha}\left[(1-q)\overline{c}_{v}+qb_{v}+q\tau
A_{v}^{\mathsf{T}}(y_{v}-A_{v}b_{v})\right]$
where $\overline{c}_{v}=\frac{1}{d}\sum_{w\in\mathcal{N}_{v}}c_{w}$.
###### Proposition 5.
If $\tau_{v}<\left\|A_{v}\right\|_{2}^{-2}$ for all $v\in\mathcal{V}$ the
following facts hold:
$\displaystyle\underset{c_{v}\in\mathbb{R}^{n}}{\mathrm{argmin\,}}\mathcal{F}^{\mathcal{S}}(X,C,B)$
$\displaystyle=\frac{1}{d}\sum_{w\in\mathcal{N}_{v}}{x}_{w},$ (6)
$\displaystyle\underset{b_{v}\in\mathbb{R}^{n}}{\mathrm{argmin\,}}\mathcal{F}^{\mathcal{S}}(X,C,B)$
$\displaystyle=x_{v}.$ (7)
In few words, in DISTA the vectors $x_{v}$ and $\overline{x}_{v}$ are updated
in an alternating fashion, separating the minimization of the consensus part
from the regularized least square function in (4).
We now prove that $\Gamma$ is asymptotically regular, i.e., that
$X(t+1)-X(t)\rightarrow 0$ for $t\rightarrow\infty.$ In particular, this
property guarantees the numerical convergence of the algorithm.
###### Lemma 6.
If $\tau_{v}<\left\|A_{v}\right\|_{2}^{-2}$ for all $v\in\mathcal{V}$, then
the sequence $\\{\mathcal{F}(X(t))\\}_{t\in\mathbb{N}}$ is non increasing and
admits the limit.
###### Proof.
The function is lower bounded ($\mathcal{F}(X)\geq 0$) and the sequence
$\\{\mathcal{F}_{p}(X(t)\\}_{t\in\mathbb{N}}$ is decreasing and therefore
admits the limit.
From Lemma 4 and Lemma 5 we obtain the following inequalities:
$\displaystyle\mathcal{F}(X(t+1))$
$\displaystyle\leq\mathcal{F}^{\mathcal{S}}(X(t+1),\overline{X}(t+1),X(t))$
$\displaystyle\leq\mathcal{F}^{\mathcal{S}}(X(t+1),\overline{X}(t),X(t))$
$\displaystyle\leq\mathcal{F}^{\mathcal{S}}(X(t),\overline{X}(t),X(t))=\mathcal{F}(X(t)).$
∎
###### Proposition 7.
For any $\tau_{v}\leq\min_{v\in\mathcal{V}}\left\|A_{v}\right\|_{2}^{-2}$ the
sequence $\\{X(t)\\}_{t\in\mathbb{N}}$ is bounded and
$\lim_{t\rightarrow+\infty}\left\|X(t+1)-X(t)\right\|_{F}^{2}=0.$
###### Proof.
By Lemma 4 we obtain
$\displaystyle\mathcal{F}(X(t))-\mathcal{F}(X(t+1))$
$\displaystyle\quad\geq\mathcal{F}^{\mathcal{S}}(X(t+1),\overline{X}(t+1),X(t))$
$\displaystyle\quad-\mathcal{F}^{\mathcal{S}}(X(t+1),\overline{X}(t+1),X(t+1))$
$\displaystyle\quad\geq\frac{q}{\tau_{v}}\sum_{v\in\mathcal{V}}(x_{v}(t+1)-x_{v}(t))^{\mathsf{T}}M_{v}(x_{v}(t+1)-x_{v}(t))\geq
0.$
Notice that the last expression is nonnegative as
$M_{v}=I-\tau_{v}A_{v}^{\mathsf{T}}A_{v}$ are positive definite for all
$v\in\mathcal{V}$.
As $\mathcal{F}^{\mathcal{S}}(X(t))-\mathcal{F}^{\mathcal{S}}(X(t+1)\to 0$ we
thus conclude that $\left\|x_{v}(t+1)-x_{v}(t)\right\|_{2}^{2}\to 0$ for any
$v\in\mathcal{V}$ and
$\lim_{t\rightarrow+\infty}\left\|X(t+1)-X(t)\right\|_{F}^{2}=0.$
∎
We now prove that $\Gamma$ is nonexpansive.
###### Lemma 8.
For any $\tau\leq\min_{v\in\mathcal{V}}\left\|A_{v}\right\|_{2}^{-2}$,
$\Gamma$ is nonexpansive.
###### Proof.
Since $\eta_{\alpha}$ is nonexpansive, for any
$X,Z\in\mathbb{R}^{n\times|\mathcal{V}|}$,
$\displaystyle\left\|(\Gamma X)_{v}-(\Gamma Z)_{v}\right\|^{2}_{2}$
$\displaystyle\leq\left\|(1-q)(\overline{\overline{x}}_{v}-\overline{\overline{z}}_{v})+q(I-\tau
A_{v}^{\mathsf{T}}A_{v})(x_{v}-z_{v})\right\|^{2}_{2}$
$\displaystyle\leq\left[(1-q)\left\|\overline{\overline{x}}_{v}-\overline{\overline{z}}_{v}\right\|_{2}+q\left\|I-\tau
A_{v}^{\mathsf{T}}A_{v}\right\|_{2}\left\|x_{v}-z_{v}\right\|_{2}\right]^{2}.$
Notice that $I-\tau A_{v}^{\mathsf{T}}A_{v}$ always has the eigenvalue 1 with
algebraic multiplicity $n-m$, as the rank of $A_{v}$ is $m$. Moreover, if
$\tau<\left\|A_{v}\right\|_{2}^{-2}$, $I-\tau A_{v}^{\mathsf{T}}A_{v}$ is
positive definite and its spectral radius is 1. Since $I-\tau
A_{v}^{\mathsf{T}}A_{v}$ is a symmetric matrix, we then have $\left\|I-\tau
A_{v}^{\mathsf{T}}A_{v}\right\|_{2}=1$. Thus, applying the triangular
inequality,
$\displaystyle\left\|(\Gamma X)_{v}-(\Gamma
Z)_{v}\right\|^{2}_{2}\leq\left[(1-q)\left\|(\overline{\overline{x}}_{v}-\overline{\overline{z}}_{v})\right\|_{2}+q\left\|x_{v}-z_{v}\right\|_{2}\right]^{2}$
$\displaystyle\leq\frac{(1-q)^{2}}{d^{4}}\left(\sum_{w\in\mathcal{N}_{v}}\sum_{w^{\prime}\in\mathcal{N}_{w}}\left\|x_{w^{\prime}}-z_{w^{\prime}}\right\|_{2}\right)^{2}+q^{2}\left\|x_{v}-z_{v}\right\|_{2}^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{2(1-q)q}{d^{2}}\sum_{w\in\mathcal{N}_{v}}\sum_{w^{\prime}\in\mathcal{N}_{w}}\left\|x_{w^{\prime}}-z_{w^{\prime}}\right\|_{2}\left\|x_{v}-z_{v}\right\|_{2}.$
Applying the Cauchy-Schwarz inequality, we obtain
$\displaystyle\left\|(\Gamma X)_{v}-(\Gamma Z)_{v}\right\|^{2}_{2}$
$\displaystyle\leq\frac{(1-q)^{2}}{d^{2}}\sum_{w\in\mathcal{N}_{v}}\sum_{w^{\prime}\in\mathcal{N}_{w}}\left\|x_{w^{\prime}}-z_{w^{\prime}}\right\|_{2}^{2}+q^{2}\left\|x_{v}-z_{v}\right\|_{2}^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{2(1-q)q}{d^{2}}\sum_{w\in\mathcal{N}_{v}}\sum_{w^{\prime}\in\mathcal{N}_{w}}\left\|x_{w^{\prime}}-z_{w^{\prime}}\right\|_{2}\left\|x_{v}-z_{v}\right\|_{2}.$
Finally, summing over all $v\in\mathcal{V}$ and considering that
$2\left\|x_{w^{\prime}}-z_{w^{\prime}}\right\|_{2}\left\|x_{v}-z_{v}\right\|_{2}\leq\left\|x_{w^{\prime}}-z_{w^{\prime}}\right\|^{2}_{2}+\left\|x_{v}-z_{v}\right\|^{2}_{2}$,
$\displaystyle\left\|(\Gamma X)-(\Gamma
Z)\right\|^{2}_{F}=\sum_{v\in\mathcal{V}}\left\|(\Gamma X)_{v}-(\Gamma
Z)_{v}\right\|^{2}_{2}$
$\displaystyle\leq(1-q)^{2}\left\|X-Z\right\|_{F}^{2}+q^{2}\left\|X-Z\right\|_{F}^{2}+2(1-q)q\left\|X-Z\right\|_{F}^{2}$
$\displaystyle\leq\left\|X-Z\right\|_{F}^{2}.$
∎
Proof of Theorem 2 Given the numerical convergence (proved in Proposition 7),
the existence of fixed points (guaranteed by Theorem 1), and the
nonexpansivity (Lemma 8) of the operator $\Gamma$, the assertion follows from
a direct application of the Opial’s theorem.∎
|
arxiv-papers
| 2013-01-10T14:16:33 |
2024-09-04T02:49:40.130726
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chiara Ravazzi, Sophie M. Fosson, and Enrico Magli",
"submitter": "Sophie Fosson",
"url": "https://arxiv.org/abs/1301.2130"
}
|
1301.2198
|
# On the double-ridge effect at the LHC
S.M. Troshin, N.E. Tyurin
Institute for High Energy Physics,
Protvino, Moscow Region, 142281, Russia
We discuss a possible explanation of the double-ridge observation in
pPb–collisions at the LHC emphasizing that this double structure in the two-
particle correlation function can result from the rotation of the transient
state of matter.
The ridge-like structure in correlation function of two particles has been
observed for the first time at RHIC in the peripheral collisions of nuclei in
the near-side jet production (cf. recent paper [1] and references therein for
the earlier ones). It was observed that the secondary particles have a narrow
$\Delta\phi$ two-particle correlation distribution (where $\phi$ is an
azimuthal angle) and wide distribution over $\Delta\eta$ ($\eta$ is a
pseudorapidity). This phenomenon was called ridge effect and associated
usually with the collective properties of a medium produced under interaction.
For the first time, a similar effect was revealed by the CMS Collaboration [2]
in pp–collisions and it becomes clear that a deconfined phase formation has
the qualitative similarity in pp- and AA-collisions. This feature has been
confirmed by the observation of the ridge effect in PbPb-collisions by ALICE,
ATLAS and CMS [3, 4, 5]. We emphasize further that significant role belongs to
peripheral collision dynamics controlled in AA–collisions by the experimental
conditions and in pp–collisions by the transition into the reflective
scattering mode [6] at the LHC energies. Of course, there should be a
quantitative difference between pp- and AA-collisions and the signal of ridge
should be more prominent in AA-interactions.
The peripheral interactions and quark-pion liquid formation in the transient
state of interaction [7] would lead to the coherent rotation which can provide
explanation for the double-ridge effect observed recently by the ALICE and
ATLAS collaboration in pPb-interactions at $\sqrt{s_{NN}}=5.02$ TeV [8, 9].
Those most recent experimental data probe net effects related to the matter
formed under interaction since they were obtained by subtraction of the low
multiplicity events (ALICE) or contributions from recoiling dijets (ATLAS).
Thus, they provide quantitative estimates of the collective effects
contribution.
It should be noted that peripheral mechanism of pPb-interactions is controlled
dynamically by the reflective scattering mode which is prominent at the LHC
energies [6]. The details will be described further. Now we briefly repeat the
main points of the production mechanism proposed in [10]. It is based on the
geometry of the overlap region and dynamical properties of the transient state
in hadron interaction111Of course, this is not unique mechanism leading to the
appearance of the ridge, description of the other ones can be found in the
recent review paper [11]. The above picture assumes that the deconfinement
takes place at the initial stage of interaction. The transient state of matter
appears then as a rotating medium of massive quarks and pions which hadronize
and form multiparticle state at the final stage of interaction. The essential
point which is needed for the presence of this rotation is a non-zero impact
parameter in the collision. We believe that this qualitative picture seems
applicable to pp, pA and AA-interactions. However, the peripheral mechanism of
pp-interaction (and of pA-interactions also), as it was already mentioned, is
controlled dynamically by the transition to the reflective scattering mode at
the LHC energies. Due to the reflective scattering [6] the inelastic overlap
function $h_{inel}(s,b)$,
$h_{inel}(s,b)\equiv\frac{1}{4\pi}\frac{d\sigma_{inel}}{db^{2}},$
has a peripheral impact parameter dependence in the region of $\sqrt{s}>2$ TeV
[6]. Unitarity equation for the elastic amplitude $f(s,b)$ in the impact
parameter representation rewritten at high energies has the simple form
$\mbox{Im}f(s,b)=h_{el}(s,b)+h_{inel}(s,b)$ (1)
and $h_{inel}(s,b)$ is the sum of all inelastic channel contributions to the
unitarity equation. Consider for ilustration the case of pure imaginary
elastic scattering amplitude, i.e. $f(s,b)\to if(s,b)$. Then the inelastic
overlap function is related to the elastic scattering amplitude by the simple
relation
$h_{inel}(s,b)=f(s,b)(1-f(s,b)).$
Reflective scattering mode implies the saturation of the unitarity limit at
small impact parameters, e.g. $f(s,b=0)\to 1$ at $s\to\infty$. Thus, the
peripheral form of the inelastic overlap function is just an effect of the
reflective scattering222The elastic scattering amplitude $f(s,b)$ is supposed
to be a monotonically decreasing function of the impact parameter. According
to peripheral impact parameter dependence of $h_{inel}(s,b)$, the mean
multiplicity
$\langle n\rangle(s)=\frac{\int_{0}^{\infty}bdb\langle
n\rangle(s,b)h_{inel}(s,b)}{\int_{0}^{\infty}bdbh_{inel}(s,b)}$
obtains the maximal input from the collisions with non-zero impact parameter
values. Thus, the events with signicant values of multiplicity at the LHC
energies would correspond to the peripheral hadron collisions [6]. At such
high energies there is a dynamical selection mechanism of peripheral region in
impact parameter space. In the nuclear collisions such selection is provided
by the relevant experimental conditions.
The geometrical picture of hadron collision at non-zero impact parameters
implies [7] that the generated massive virtual quarks in the overlap region
could obtain very large initial orbital angular momentum at high energies
(Fig. 1).
Figure 1: Overlap region in the initial stage of the hadron interaction,
valence constituent quarks are located in the central part of hadrons.
Due to strong interaction between the quarks this orbital angular momentum
leads to the coherent rotation of the quark system located in the overlap
region in the $xz$-plane (Fig. 2). This rotation is similar to the rotation of
the liquid where strong correlations between particles momenta exist. Thus,
the non-zero orbital angular momentum should be realized as a coherent
rotation of the quark-pion liquid as a whole. The assumed particle production
mechanism at moderate transverse momenta is the simultaneuos excitations of
the parts of the rotating transient state (of massive constituent quarks
interacting by pion exchanges) by the valence constituent quarks. We refer
here to proton-proton interactions, in pA-interactions the effects should be
enhansed due to many collisions but would remain qualitatively the same.
Since the transient matter is strongly interacting, the excited parts should
be located closely to the periphery of the rotating transient state otherwise
absorption would not allow quarks and pions leave the interaction region
(quenching).
The mechanism is sensitive to the particular direction of rotation and to the
rotation plane orientatation leads to the narrow distribution of the two-
particle correlations in $\Delta\phi$. However, these correlation could have a
broad distribution versus polar angle ($\Delta\eta$) (Fig. 2). Quarks in the
exited part of the cloud could have different values of the two components of
the momentum (with its third component lying in the rotation $xz$-plane) since
the exited region is spatially extended. The rotation with orbital angular
momentum $L(s,b)$ contributes to the $x$-component of the transverse momentum
and does not contribute to the $y$-component of the transverse momentum, i.e.
$\Delta p_{x}=\kappa L(s,b)$ (2)
while
$\Delta p_{y}=0.$ (3)
Figure 2: Interaction of the valence constituent quarks with rotating quark-
pion liquid, dashed lines indicate the clockwise rotation of the transient
matter.
It should be noted that the ALICE and ATLAS are observing the double-ridge
structure with two maximums in the two-particle correlation function at
$\Delta\phi=0$ and $\Delta\phi=\pi$ after subtraction of the low-multiplicity
events or expected dijet events, respectively. In terms of Fig.2 this
observation means the presence of the correlations between particles in the
upper cluster ($\Delta\phi=0$) and between particles from the upper and lower
clusters ($\Delta\phi=\pi$). Lower cluster of particles is the result of the
excitation of the transient matter by the constituent valence quark from the
second colliding hadron. Since both particle clusters originate from the
coherent rotating quark-pion liquid, it is evident that the particles from
lower and upper clusters should be correlated and this will produce the second
observed ridge.
Thus, the double-ridge structure observed in the high multiplicity events by
the ALICE and ATLAS can serve as an experimental manifestation of the coherent
rotation of the transient state of matter generated during the hadron
collisions. The narrowness of the two-particle correlation distribution in the
asimuthal angle is the distinctive feature of this mechanism which reflects
the presesence of the rotation plane in every event of particle production at
the LHC.
Other experimentally observed effects of this collective effect are described
in [7].
It should be noted that we believe that the nature of the state of matter
revealed at the LHC in proton collisions is the same as the nature of the
state revealed at RHIC under collisions of nuclei. But, since the LHC energies
are significantly higher that the RHIC energies, the differences should be
expected due to the energy-dependent dynamics related to the peripheral form
of inelastic collisons generated by the reflective scattering mode. There
should be no ridge effect in pp- and pA-interations at RHIC, while the double-
ridge effect should be observed in the events of particle production in the
peripheral AA-collisions at RHIC and in pp-collisions at the LHC (after
subtraction procedure of the low-multiplicity or dijet events being
performed). Evidently, the double-ridge effect should be observed by the CMS
experiment also in pPb-collisions with the use of the similar subtractions.
## References
* [1] STAR Collaboration, B. I. Abelev et al., Phys. Rev. Lett. 105, 022301 (2010).
* [2] CMS Collaboration, V. Khachatryan et al., JHEP 1009, 091 (2010).
* [3] ALICE Collaboration, K. Aamodt et al., Phys. Lett. B708, 249 (2012).
* [4] ATLAS Collaboration, G. Aad et al., Phys. Rev. C86, 014907 (2012).
* [5] CMS Collaboration, S. Chatrchyan et al., Eur. Phys. J. C72, 2012 (2012).
* [6] S.M. Troshin, N.E. Tyurin, Int. J. Mod. Phys. A 22, 4437 (2007).
* [7] S.M. Troshin, N.E. Tyurin, Int. J. Mod. Phys. A 26, 4703 (2011).
* [8] ALICE Collaboration, B. Abelev et al., arXiv: 1212.2001v1 [nucl-ex].
* [9] ATLAS Collaboration, G. Aad et al., arXiv: 1212.5198v2 [hep-ex].
* [10] S.M. Troshin, N.E. Tyurin, Int. J. Mod. Phys. E 17, 1619 (2008).
* [11] Wei Li, Mod. Phys. Lett. A 27, 1230018 (2012).
|
arxiv-papers
| 2013-01-09T09:05:21 |
2024-09-04T02:49:40.139873
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. M. Troshin, N. E. Tyurin",
"submitter": "Sergey Troshin",
"url": "https://arxiv.org/abs/1301.2198"
}
|
1301.2379
|
# Extremal property of a simple cycle
Alexander N. Gorban Department of Mathematics, University of Leicester,
United Kingdom [email protected]
###### Abstract
We study systems with finite number of states $A_{i}$ ($i=1,\ldots,n$), which
obey the first order kinetics (master equation) without detailed balance. For
any nonzero complex eigenvalue $\lambda$ we prove the inequality
$\frac{|\Im\lambda|}{|\Re\lambda|}\leq\cot\frac{\pi}{n}$. This bound is sharp
and it becomes an equality for an eigenvalue of a simple irreversible cycle
$A_{1}\to A_{2}\to\ldots\to A_{n}\to A_{1}$ with equal rate constants of all
transitions. Therefore, the simple cycle with the equal rate constants has the
slowest decay of the oscillations among all first order kinetic systems with
the same number of states.
###### pacs:
82.20.Rp, 82.40.Bj, 02.50.Ga
## Damped oscillations in kinetic systems.
We study systems with finite number of states $A_{i}$ ($i=1,\ldots,n$). A non-
negative variable $p_{i}$ corresponds to every state $A_{i}$. This $p_{i}$ may
be probability, or population, or concentration of $A_{i}$. For the vector
with coordinates $(p_{i})$ we use the notation $P$.
We assume that the time evolution of $P$ obey first order kinetics (the master
equation)
$\frac{{\mathrm{d}}p_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}(q_{ij}p_{j}-q_{ji}p_{i})\;\;(i=1,\ldots,n),$ (1)
where $q_{ij}$ ($i,j=1,\ldots,n$, $i\neq j$) are non-negative. In this
notation, $q_{ij}$ is the rate constant for the transition $A_{j}\to A_{i}$.
Any set of non-negative coefficients $q_{ij}$ ($i\neq j$) corresponds to a
master equation.
Let us rewrite (1) in the vector notations, $\dot{P}=KP$, where the matrix
$K=(k_{ij})$ has the elements
$k_{ij}=\left\\{\begin{array}[]{cl}q_{ij}&\mbox{ if }i\neq j;\\\
-\sum_{m,\,m\neq i}q_{mi}&\mbox{ if }i=j.\end{array}\right.$ (2)
The standard simplex $\Delta_{n}=\\{P\ |\ p_{i}\geq 0,\,\sum_{i}p_{i}=1\\}$ is
forward-invariant with respect to the master equation (1) because it preserves
positivity and has the conservation law $\sum_{i}p_{i}=const$. This means that
any solution of (1) $P(t)$ with the initial conditions $P(t_{0})\in\Delta_{n}$
remains in $\Delta_{n}$ after $t_{0}$: $P(t)\in\Delta_{n}$ for $t\geq t_{0}$.
One can use this forward invariance to prove some known important properties
of $K$. For example, there exists a non-negative vector $P^{*}\in\Delta_{n}$
($p^{*}_{i}\geq 0$) such that $KP^{*}=0$ (equilibrium). Indeed, any continuous
map of $\Delta_{n}\to\Delta_{n}$ has a fixed point, therefore $\exp(Kt)$ has a
fixed point in $\Delta_{n}$ for any $t>0$. If $\exp(Kt)P^{*}=P^{*}$ for some
$P^{*}\in\Delta_{n}$ and sufficiently small $t>0$, then $KP^{*}=0$ because
$\exp(Kt)P=P+tKP+o(t)$.
The proof that $K$ has no nonzero imaginary eigenvalues for the systems with
positive equilibria gives another simple example. We exclude one zero
eigenvector and consider $K$ on the invariant hyperplane where
$\sum_{i}p_{i}=0$. If $K$ has a nonzero imaginary eigenvalue $\lambda$, then
there exists a 2$D$ $K$-invariant subspace $L$, where $K$ has two conjugated
imaginary eigenvalues, $\lambda$ and $\overline{\lambda}=-\lambda$.
Restriction of $\exp(Kt)$ on $L$ is one-parametric group of rotations. For the
positive equilibrium $P^{*}$ the intersection $(L+P^{*})\cap\Delta_{n}$ is a
convex polygon. It is forward invariant with respect to the master equation
(1) because $L$ is invariant, $P^{*}$ is equilibrium and $\Delta_{n}$ is
forward invariant. A polygon on a plane cannot be invariant with respect to
one-parametric semigroup of rotations $\exp(Kt)$ ($t\geq 0$). This
contradiction proves the absence of imaginary eigenvalues. We use the
reasoning based on forward invariance of $\Delta_{n}$ below in the proof of
the main result.
The master equation obeys the principle of detailed balance if there exists a
positive equilibrium $P^{*}$ ($p^{*}_{i}>0$) such that for each pair $i,j$
($i\neq j$)
$q_{ij}p^{*}_{j}=q_{ji}p^{*}_{i}.$ (3)
After Onsager Ons , it is well known that for the systems with detailed
balance the eigenvalues of $K$ are real because $K$ is under conditions (3) a
self-adjoined matrix with respect to the entropic inner product
$\langle x,y\rangle=\sum_{i}\frac{x_{i}y_{i}}{p^{*}_{i}}$
(see, for example, VanKampen1973 ; YBGE1991 ).
Detailed balance is a well known consequence of microreversibility. In 1872 it
was introduced by Boltzmann for collisions Boltzmann1964 . In 1901 Wegscheider
proposed this principle for chemical kinetics Wegscheider1901 . Einstein had
used this principle for the quantum theory of light emission and absorbtion
(1916, 1917). The backgrounds of detailed balance had been analyzed by Tolman
Tolman1938 . This principle was studied further and generalized by many
authors YangHlavacek2006 ; GorbYabCES2012 .
Systems without detailed balance appear in applications rather often. Usually,
they represent a subsystem of a larger systems, where concentrations of some
of the components are considered as constant. For example, the simple cycle
$A_{1}\to A_{2}\to\ldots\to A_{n}\to A_{1}$ (4)
is a typical subsystem of a catalytic reaction (a catalytic cycle). The
complete reaction may have the form
$S+A_{1}\to A_{2}\to\ldots\to A_{n}\to A_{1}+P,$ (5)
where $S$ is a substrate and $P$ is a product of reaction.
The irreversible cycle (4) cannot appear as a limit of systems with detailed
balance when some of the constants tend to zero, whereas the whole catalytic
reaction (5) can GorbYabCES2012 . The simple cycle (4) can be produced from
the whole reaction (5) if we assume that concentrations of $S$ and $P$ are
constant. This is possible in an open system, where we continually add the
substrate and remove the product. Another situation when such an approximation
makes sense is a significant excess of substrate in the system,
$[S]\gg[A_{i}]$ (here we use the square brackets for the amount of the
component in the system). Such an excess implies separation of time and the
system of intermediates $\\{A_{i}\\}$ relaxes much faster than the
concentration of substrate changes.
In systems without detailed balance the damped oscillations are possible. For
example, let all the reaction rate constants in the simple cycle be equal,
$q_{j+1\,j}=q_{1n}=q>0$. Then the characteristic equation for $K$ is
$\det(K-\lambda I)=(-q-\lambda)^{n}+q^{n}(-1)^{n+1}=0$
and $\lambda=-q+q\exp\left(\frac{2\pi ik}{n}\right)\;(k=0,\ldots,n-1)$. For
nonzero $\lambda$, the ratio of imaginary and real parts of $\lambda$ is
$\frac{|\Im\lambda|}{|\Re\lambda|}=\frac{\left|\sin\frac{2\pi
k}{n}\right|}{\left|1-\cos\frac{2\pi k}{n}\right|}=\left|\cot\frac{\pi
k}{n}\right|\leq\cot\frac{\pi}{n}\,.$
The maximal value, $\cot\frac{\pi}{n}$, corresponds to $k=1$. For large $n$,
$\cot\frac{\pi}{n}\approx\frac{n}{\pi}$ and oscillations in the simple cycle
decay rather slowly.
## Estimate of eigenvalues.
Let us consider the general master equation (1) without any assumption of
detailed balance.
###### Theorem 1.
For every nonzero eigenvalue $\lambda$ of matrix $K$
$\frac{|\Im\lambda|}{|\Re\lambda|}\leq\cot\frac{\pi}{n}$ (6)
###### Proof.
Let us assume that the master equation (1) has a positive equilibrium
$P^{*}\in\Delta_{n}$: for all $i=1,\ldots,n$ $p_{i}^{*}>0$ and
$\sum_{j}q_{ij}p^{*}_{j}=\sum_{j}q_{ji}p^{*}_{i}.$
The systems with non-negative equilibria may be considered as limits of the
systems with positive equilibria.
Let $\lambda$ be a complex eigenvalue of $K$ and let $L$ be a 2D real subspace
of the hyperplane $\sum_{i}p_{i}=0$ that corresponds to the pair of complex
conjugated eigenvalues, $(\lambda,\overline{\lambda})$. Let us select a
coordinate system in the plane $L+P^{*}$ with the origin at $P^{*}$ such that
restriction of $K$ on this plane has the following matrix
$\mathcal{K}=\left[\begin{array}[]{cc}\Re\lambda&-\Im\lambda\\\
\Im\lambda&\Re\lambda\end{array}\right]\,.$
In this coordinate system
$\exp(t\mathcal{K})=\left[\begin{array}[]{cc}\exp(t\Re\lambda)\cos(t\Im\lambda)&-\exp(t\Re\lambda)\sin(t\Im\lambda)\\\
\exp(t\Re\lambda)\sin(t\Im\lambda)&\exp(t\Re\lambda)\cos(t\Im\lambda)\end{array}\right]\,.$
The intersection $\mathcal{A}=(L+P^{*})\cap\Delta_{n}$ is a polygon. It has
not more than $n$ sides because $\Delta_{n}$ has $n$ $(n-2)$-dimensional faces
(each of them is given in $\Delta_{n}$ by an equation $p_{i}=0$). For the
transversal intersections (the generic case) this is obvious. Non-generic
situations can be obtained as limits of generic cases when the subspace $L$
tends to a non-generic position. This limit of a sequence of polygons cannot
have more than $n$ sides if the number of sides for every polygon in the
sequence does nor exceed $n$.
Let the polygon $\mathcal{A}$ have $m$ vertices $\mathbf{v}_{j}$ ($m\leq n$).
We move the origin to $P^{*}$ and enumerate these vectors
$\mathbf{x}_{i}=\mathbf{v}_{i}-P^{*}$ anticlockwise (Fig 1). Each pair of
vectors $\mathbf{x}_{i},\mathbf{x}_{i+1}$ (and
$\mathbf{x}_{m},\mathbf{x}_{1}$) form a triangle with the angles $\alpha_{i}$,
$\beta_{i}$ and $\gamma_{i}$, where $\beta_{i}$ is the angle between
$\mathbf{x}_{i}$ and $\mathbf{x}_{i+1}$, and $\beta_{m}$ is the angle between
$\mathbf{x}_{m}$ and $\mathbf{x}_{1}$. The Sine theorem gives
$\frac{|\mathbf{x}_{i}|}{\sin\alpha_{i}}=\frac{|\mathbf{x}_{i+1}|}{\sin\gamma_{i}}$,
$\frac{|\mathbf{x}_{m}|}{\sin\alpha_{m}}=\frac{|\mathbf{x}_{1}|}{\sin\gamma_{1}}$.
Several elementary identities and inequalities hold:
$\begin{split}&0<\alpha_{i},\beta_{i},\gamma_{i}<\pi;\;\;\sum_{i}\beta_{i}=2\pi;\;\;\alpha_{i}+\beta_{i}+\gamma_{i}=\pi;\\\
&\prod_{i}\sin\alpha_{i}=\prod_{i}\sin\gamma_{i}\mbox{ (the closeness
condition).}\end{split}$ (7)
These conditions (7) are necessary and sufficient for the existence of a
polygon $\mathcal{A}$ with these angles which is star-shaped with respect to
the origin.
Let us consider the anticlockwise rotation ($\Im\lambda<0$, Fig. 1). The case
of clockwise rotations differs only in notations. For the angle $\delta$
between $K\mathbf{x}_{i}$ and $\mathbf{x}_{i}$, $\sin\delta=-\Im\lambda$,
$\cos\delta=-\Re\lambda$ and $\tan\delta=\frac{\Im\lambda}{\Re\lambda}$.
Figure 1: The polygon $\mathcal{A}$ is presented as a sequence of vectors
$\mathbf{x}_{i}$. The angle $\beta_{i}$ between vectors $\mathbf{x}_{i}$ and
$\mathbf{x}_{i+1}$ and the angles $\alpha_{i}$ and $\gamma_{i}$ of the
triangle with sides $\mathbf{x}_{i}$ and $\mathbf{x}_{i+1}$ are shown. In the
Fig., rotation goes anticlockwise, i.e. $\Im\lambda<0$. In this case, the
polygon $\mathcal{A}$ is invariant with respect to the semigroup
$\exp(t\mathcal{K})$ ($t\geq 0$) if and only if $\delta\leq\alpha_{i}$ for all
$i=1,\ldots,m$, where $\delta$ is the angle between the vector field
$\mathcal{K}\mathbf{x}$ and the radius-vector $\mathbf{x}$.
For each point $\mathbf{x}\in L+P^{*}$ ($\mathbf{x}\neq P^{*}$), the straight
line
$\\{\mathbf{x}+\epsilon\mathcal{K}\mathbf{x}\,|\,\epsilon\in\mathbb{R}\\}$
divides the plane $L+P^{*}$ in two half-plane (Fig. 1, dotted line). Direct
calculation shows that the semi-trajectory
$\\{\exp(t\mathcal{K})\mathbf{x}\,|\,t\geq 0\\}$ belongs to the same half-
plane as the origin $P^{*}$ does. Therefore, if $\delta\leq\alpha_{i}$ for all
$i=1,\ldots,m$ then the polygon $\mathcal{A}$ is forward-invariant with
respect to the semigroup $\exp(t\mathcal{K})$ ($t\geq 0$). If
$\delta>\alpha_{i}$ for some $i$ then for sufficiently small $t>0$
$\exp(t\mathcal{K})\mathbf{x}_{i}\notin\mathcal{A}$ because
$\mathcal{K}\mathbf{x}_{i}$ is the tangent vector to the semi-trajectory at
$t=0$.
Thus, for the anticlockwise rotation ($\Im\lambda<0$), the polygon
$\mathcal{A}$ is forward-invariant with respect to the semigroup
$\exp(t\mathcal{K})$ ($t\geq 0$) if and only if $\delta\leq\alpha_{i}$ for all
$i=1,\ldots,m$. The maximal $\delta$ for which $\mathcal{A}$ is still forward-
invariant is $\delta_{\max}=\min_{i}\\{\alpha_{i}\\}$. We have to find the
polygon with $m\leq n$ and the maximal value of $\min_{i}\\{\alpha_{i}\\}$.
Let us prove that this is a regular polygon with $n$ sides. Let us find the
maximizers $\alpha_{i},\beta_{i},\gamma_{i}$ ($i=1,\ldots,m$) for the
optimization problem:
$\min_{i}\\{\alpha_{i}\\}\to\max\;\mbox{subject to conditions (\ref{Cond}).}$
(8)
For solution of this problem, all $\alpha_{i}$ are equal. To prove this
equality, let us mention that $\min_{i}\\{\alpha_{i}\\}<\frac{\pi}{2}$ under
conditions (7) (if all $\alpha_{i}\geq\frac{\pi}{2}$ then the polygonal chain
$\mathcal{A}$ cannot be closed). Let $\min_{i}\alpha_{i}=\alpha$. Let us
substitute in (7) the variables $\alpha_{i}$ which take this minimal value by
$\alpha$. The derivative of the left hand part of the last condition in (7)
with respect to $\alpha$ is not zero because $\alpha<\frac{\pi}{2}$. Assume
that there are some $\alpha_{j}>\alpha$. Let us fix the values of $\beta_{i}$
($i=1,\ldots,m$). Then $\gamma_{i}$ is a function of $\alpha_{i}$,
$\gamma_{i}=\pi-\beta_{i}-\alpha_{i}$. We can use the implicit function
theorem to increase $\alpha$ by a sufficiently small number $\varepsilon>0$
and to change the non-minimal $\alpha_{j}$ by a small number too,
$\alpha_{j}\mapsto\alpha_{j}-\theta$; $\theta=\theta(\varepsilon)$. Therefore,
at the solution of (8) all $\alpha_{j}=\alpha$ ($j=1,\ldots,m$).
Now, let us prove that for solution of the problem (8) all $\beta_{i}$ are
equal. We exclude $\gamma_{i}$ from conditions (7) and write
$\beta_{i}+\alpha<\pi$; $0<\beta_{i},\alpha$;
$m\log\sin\alpha=\sum_{i}\log\sin(\beta_{i}+\alpha).$ (9)
Let us consider this equality as equation with respect to unknown $\alpha$.
The function $\log\sin x$ is strictly concave on $(0,\pi)$. Therefore, for
$x_{i}\in(0,\pi)$
$\log\sin\left(\frac{1}{m}\sum_{i=1}^{m}x_{i}\right)\geq\frac{1}{m}\sum_{i=1}^{m}\log\sin
x_{i}$
and the equality here is possible only if all $x_{i}$ are equal. Let
$\alpha^{*}\in(0,\pi/2)$ be a solution of (9). If not all the values of
$\beta_{i}$ are equal and we replace $\beta_{i}$ in (9) by the average value,
$\beta=\frac{2\pi}{m}$, then the value of the right hand part of (9) increases
and $\sin\alpha^{*}<\sin(\beta+\alpha^{*})$. If we take all the $\beta_{i}$
equal then (9) transforms into elementary trigonometric equation
$\sin\alpha=\sin(\beta+\alpha)$. The solution $\alpha$ of equation (9)
increases when we replace $\beta_{i}$ by the average value:
$\alpha>\alpha^{*}$ because $\sin\alpha^{*}<\sin(\beta+\alpha^{*})$,
$\alpha\in(0,\pi/2)$ and $\sin\alpha$ monotonically increases on this
interval. So, for the maximizers of the conditional optimization problem (8)
all $\beta_{i}=\frac{2\pi}{m}$ and
$\alpha_{i}=\gamma_{i}=\frac{\pi}{2}-\frac{\pi}{m}$. The maximum of $\alpha$
corresponds to the maximum of $m$. Therefore, $m=n$. Finally,
$\max\\{\delta\\}=\frac{\pi}{2}-\frac{\pi}{n}$ and
$\max\left\\{\frac{|\Im\lambda|}{|\Re\lambda|}\right\\}=\cot\frac{\pi}{n}.$
This is exactly the same value as for an eigenvalue of the simple cycle of the
lengths $n$ with equal rate constants, $\lambda=-q(1-\exp(\frac{2\pi i}{n}))$.
∎
## Discussion.
The simple cycle with the equal rate constants gives the slowest decay of
oscillations or, in some sense, the slowest relaxation among all first order
kinetic systems with the same number of components. The extremal properties of
the simple cycle with equal constants were noticed in numerical experiments 25
years ago BochByk1987 . V.I. Bykov formulated the hypothesis that this system
has extremal spectral properties. This paper gives the answer: yes, it has.
## References
* (1) L. Onsager, Reciprocal relations in irreversible processes. I. Phys. Rev. 37 (1931), 405–426.
* (2) N.G. van Kampen, Nonlinear irreversible processes, Physica 67 (1) (1973) 1–22
* (3) G.S. Yablonskii, V.I. Bykov, A.N. Gorban, V.I. Elokhin, Kinetic Models of Catalytic Reactions (Series “Comprehensive Chemical Kinetics,” Volume 32); Elsevier: Amsterdam, The Netherlands, 1991.
* (4) L. Boltzmann, Lectures on gas theory, Univ. of California Press, Berkeley, CA, USA, 1964.
* (5) R. Wegscheider (1901), Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme, Monatshefte für Chemie / Chemical Monthly 32 (8) (1901) 849–906.
* (6) R.C. Tolman, The Principles of Statistical Mechanics. Oxford University Press, London, 1938.
* (7) J. Yang, W.J. Bruno, W.S. Hlavacek, J. Pearson, On imposing detailed balance in complex reaction mechanisms, Biophys J. 91 (2006) 1136–1141.
* (8) A.N. Gorban, G.S. Yablonskii, Extended detailed balance for systems with irreversible reactions, Chem. Eng. Sci. 66 (2011) 5388–5399. arXiv:1101.5280 [cond-mat.stat-mech].
* (9) A.N. Bocharov, V.I. Bykov, Parametric analysis of eigenvalues of matrices corresponding to linear one-route catalytic reaction mechanism, React. Kinet. Catal. Lett. 34 (1) (1987) 75–80.
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arxiv-papers
| 2013-01-11T02:00:18 |
2024-09-04T02:49:40.151889
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. N. Gorban",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1301.2379"
}
|
1301.2443
|
11institutetext: Computer Science III
University of Bonn, Germany11institutetext: University of Bonn, Computer
Science III,
Römerstraße 164, 53117 Bonn, Germany
{tantius, dsp, behrend}@cs.uni-bonn.de
# Towards an Application of
Update Propagation on Logic Programs
Representing Java Source Code
Richard Tantius Daniel Speicher Andreas Behrend
(17 May 2012)
###### Abstract
Logic programs are now used as a representation of object-oriented source code
in academic prototypes for about a decade. This representation allows a clear
and concise implementation of analyses of the object-oriented source code. The
full potential of this approach is far from being explored. In this paper, we
report about an application of the well-established theory of update
propagation within logic programs. Given the representation of the object-
oriented code as facts in a logic program, a change to the code corresponds to
an update of these facts. We demonstrate how update propagation provides a
generic way to generate incremental versions of such analyses.
###### Keywords:
logic program, update propagation, application, logic meta programming,
refactoring impact
## 1 Introduction
In this paper, we show how update propagation can be employed for efficiently
computing-derived information within the domain of logic fact-bases
representing object-oriented source code.
### 1.1 Logic Meta Programming
Logic Meta Programming approaches build a detailed representation of a program
(typically) written in another programming language in a logic fact-base [14],
[8], [10]. This representation allows analysing the orginal program by means
of a logic program built on those facts. This approach has been used to detect
code locations that need design improvement [13]. It provides a suitable basis
for different static analyses from the implementation of code quality metrics
to the implementation of type constraints. Recently, we have been arguing that
logic meta programming should be used to integrate the knowledge about good
design structures and suspicious design structures, creating a database of
code quality knowledge, which can be evolved over time [13].
### 1.2 Update Propagation
Update propagation (UP) is an established database research topic, which has
been studied over the last 40 years mainly in the context of integrity
checking and materialized views maintenance, e.g., [4], [2], [9], [7], [11],
[12]. Therefore, UP is known from the SQL and Datalog world. UP makes a
contribution to efficiently compute implicit changes of derived relations
resulting from explicitly performed updates of extensional facts of a logic
fact-base. SQL view specifications and Datalog rules, as well as Prolog rules
are related forms of deductive rules. This supports the idea to adapt UP to
the Prolog world, and also use sets of deltas, together with specialized
update statements to incrementally maintain derived predicates (which can be a
software analysis, implemented in Prolog). The original predicates are
required only once for materializing their initial answers, the specialized
delta versions are used in update statements afterwards for continuously
updating the materialized results. Assuming that a great portion of the
materialized content of the original logic rules remains unchanged, the
application of such update statements may considerably enhance the efficiency
of computing the state of such relations after an update of the fact-base.
### 1.3 Refactoring Impact Prediction
UP enables computing the results of an analysis after an update of the fact-
base without actually executing the change. Such an update of the fact-base
can be induced for example by applying a structural improvement like a
refactoring [6] to the source code. This allows to efficiently execute a
”what-if”-style analysis. The approach enables us to evaluate a potential
refactoring of Java programs, by computing quality attributes like software
metrics before the refactoring and simulate via UP how the metric result
changes due to the refactoring.
### 1.4 Automation
Figure 1 introduces our implementation which automates several aspects of the
refactoring impact prediction via UP. The picture shows the different
components of the system. Those components may consist of several Prolog
modules. The application covers the derivation of a suitable abstract model,
on which we build the software analysis we intend to employ. We use the Logic
Meta Programming approach JTransformer presented in [8], to derive such a
model. The implementation also handles the refactoring simulation on the level
of that abstract model, metric computation and UP rule generation. We have
used SWI-Prolog (in the version 6.0.2111The Project homepage of SWI-Prolog:
http://www.swi-prolog.org/ (accessed 17.08.12).) for the implementation.
Beside the metric definitions that had to be implemented in a strictly
declarative syntax, so that we can apply UP, we used the full feature set of
the SWI-Prolog environment for the other parts of the implementation.
Figure 1: Overview Update Propagation
## 2 Model and Analysis
Software analyses like software metrics are a frequently-studied approach to
detect lack of quality and are also capable of making improvements of quality
measurable. Logic Meta Programming Approaches provide the capability to
represent software systems as logic programs. A refactoring in this context,
therefore, can be understood as a transformation imposing changes on an
extensional fact-base. We discuss the structural cohesion metric Lack of
Cohesion in Methods $LCOM1$ [5], as an example of such a software analysis.
Cohesion can be defined as the degree of how closely module components are
related to each other. A unified framework for structural metrics was
presented by Briand in [3], who created a common model for existing metrics.
The model unified the syntactical representation and operational semantic of
those metrics. In order to provide the information about the source code the
metric relies on, we also present a simplified abstract model as basis for
$LCOM1$. This meta model will be directly derived from the Logic Meta
Programming fact-base.
### 2.1 Abstract Cohesion Model
The Logic Meta Programming approach [8], which we use to derive our abstract
cohesion model, is based on Prolog. For this reason, we represent the relevant
information for the $LCOM1$ metric as Prolog facts. We will also present the
$LCOM1$ metric itself as a logic program in the following. To sufficiently
describe the information relevant for cohesion metric, we need to take the
following information into account: Which class contains a certain method or
field? Which methods are called and which fields are accessed by a method? The
presented model is based on the cohesion model as presented in Briand [3] and
was adapted to Prolog. We consider the following Prolog predicates:
`c(M).` | ``class
---|---
`cm(C, M).` | ``class contains method
`cf(C, F).` | ``class contains field
`mf(M, F).` | ``method accesses field
`mm(M, N).` | ``method invokes method
The related Prolog facts are ground versions of the predicates from above and
the variables `C,M,F,...` are bound to unique identifiers for the
corresponding elements. In the next subsection, we demonstrate how we extract
this model from the JTransformer fact-base. In the following section, we build
the $LCOM1$ metric on top of those facts as a logic program.
### 2.2 Model Fact Derivation
We derive the abstract cohesion model directly from the fact-base created by
the JTransformer Logic Meta Programming approach. JTransformer builds a so
called Abstract Syntax Tree (AST) that already is a full model representation
in Prolog of the Java language. The JTransformer facts are called Program
Element Facts PEF, they are the starting point for creating the cohesion
model.
Figure 1 gives an overview of the various components of our Implementation.
The architecture consists of three layers. In the first layer, we have the
metric rules and the facts created by JTransformer. The layer in the middle is
the meta programming layer, in which we process the rules and facts. Here we
compile the UP rules. In the layer at the bottom, we have the subsystems which
actually contain the executable code. Each subsystem may consist of several
Prolog modules.
Based on generator predicates, the Model Facts Converter component creates
cohesion model facts (from the JTransformer facts) and asserts them to a
prolog module (`cohesion_model`). Additionally, we perform checks, if an
element should be included at all. In the case of classes, we do not consider
the three following class types. Interface classes do not provide method calls
and attribute references. Cohesion cannot be examined here. Classes from
external dependencies are supposed to be examined elsewhere. Anonymous classes
are not supposed to be analysed standalone.
The implementation of the class generator predicates is as follows:
generate(FactsModule, c, [ClassId]) :-
% Here we only use the class id
classT(ClassId, _, _, _),
source_class(ClassId).
For the free variable `FactsModule`, we use the module `cohesion_model`
mentioned above as a default value. The implementation of the class analysis
considers different JTransfomer facts to determine the class type:
source_class(ClassId) :-
% JTransformer facts
not(externT(ClassId)),
not(interfaceT(ClassId)),
% See below
not(anonymous_class__(ClassId)).
anonymous_class__(Class):-
classT(Class, _, ClassName, _),
string_concat(’ANONYMOUS$’, _, ClassName).
On top of the provided model facts, we create our software analyses, for
example the cohesion metric presented before. The deductive Metric rule
definition is the starting point (also shown in Figure 1). We define the
deductive part of the metric in a Prolog module:
`:- module(` $metricName$`_deductive_ruleset, []).`
` `$rule_{1}$`(A, B, `$...$`) :- `$...$
` `
For example:
` :- module(lcom1_deductive_ruleset, []).`
` ...`
### 2.3 Declarative Metric Implementation
The $LCOM1$ metric definition we present in the following is based on the
definition given by Briand in [3].
#### 2.3.1 Query and Mapping
The computation of structural metrics can be divided into two steps. First, a
query step collects the elements or relations that are relevant for the
metric. Second, a mapping maps the result of the query to a number. Separating
these steps has the benefit that we can discuss both steps on their own. A
query result may be evaluated with different mappings. A mapping may be
applied to the result of different queries. The separation of query and
mapping has the advantage, that we can apply the update propagation approach
to the deductive rules defining the query.
##### Query
$LCOM1$ counts the number of method pairs within a class that do not access
even one common field. We split this definition into two predicates. Each
predicate will be first described in natural language, then as a logic
program.
$M$ and $N$ in $C$ are connected, if $M$ accesses a field $F$, that belongs to
the class $C$, and $N$ accesses that field $F$ as well.
cp(C, M, N) :-
mf(M,F), cf(C, F), mf(N,F).
$M$ and $N$ in $C$ are a pair of methods lacking cohesion, if $M$ is a method
in $C$, $N$ is as well a method in $C$ and $M$ and $N$ are not a connected
pair in $C$:
lp(C, M, N) :-
cm(C, M), cm(C, N), not(cp(C, M, N)).
##### Mapping
To complete the $LCOM1$ computation rule, we need to perform some additional
steps after the deductive part.
lcom1(C, R):-
findall([M,N], (cp(C, M, N), not(M=N)), E),
length(E, T),
R is T/2.
## 3 Refactoring as Cohesion Model Update
We model a refactoring as an update on our cohesion model. Performing the
refactoring only on the abstract level of the cohesion model, helps to
concentrate the computation only on aspects relevant for the metric
computation. Because we use update propagation in the following, we do not
directly perform model updates, rather every update will generate a so called
delta fact, which depicts the actual change and will be discussed in detail in
Section 4. In the following, we briefly discuss the refactorings we use and
describe their effects on the level of the model. Both refactorings assume in
our setting, that the moved element will be extracted into a new class,
creating the following delta fact: `add_c(#newClassId)`
Since we exclude constructor methods from our model and do not consider
modificators as public, private and protected, the following refactorings
require no special preconditions to be applied.
### 3.1 Move Method, Move Field
The move method refactoring moves a method from one class to another. Though
in a real world refactoring, we would need to adjust the code in several ways,
so that it remains functioning and compileable we add two simple delta facts
to our model:
add_cm(C, M)
del_cm(C, M)
Similar to the move method refactoring the move field refactoring moves a
field from one class to another, the resulting delta facts are as follows:
add_cf(C, F)
del_cf(C, F)
## 4 Update Propagation
In this section, we show how to apply update propagation in Prolog. Because of
the different evaluation mechanisms for SQL views (set-oriented, bottom-up)
and Prolog rules (instance-oriented, top-down), however, the transformation
techniques from UP could not be applied directly. Instead, the specific
properties of Prolog rules have to be taken into account in order to achieve a
complete and sound update propagation based on delta predicates.
### 4.1 Rule Transformation in Prolog
% Derived from: lp(C, M, N) :- cm(C, M), cm(C, N), not(cp(C, M, N)).
add_lp(C, M, N) :- add_cm(C, M), nwd_cm(C, N), not(nwi_cp(C, M, N)), not( lp(C, M, N)).
add_lp(C, M, N) :- add_cm(C, N), nwd_cm(C, M), not(nwi_cp(C, M, N)), not( lp(C, M, N)).
add_lp(C, M, N) :- del_cp(C, M, N), nwd_cm(C, M), nwd_cm(C, N), not( lp(C, M, N)).
del_lp(C, M, N) :- del_cm(C, M), cm(C, N), not( cp(C, M, N)), not(nwi_lp(C, M, N)).
del_lp(C, M, N) :- del_cm(C, N), cm(C, M), not( cp(C, M, N)), not(nwi_lp(C, M, N)).
del_lp(C, M, N) :- add_cp(C, M, N), cm(C, M), cm(C, N), not(nwi_lp(C, M, N)).
nwi_lp(C, M, N) :- nwd_cm(C, M), nwd_cm(C, N), not(nwi_cp(C, M, N)).
nwd_lp(C, M, N) :- lp(C, M, N), not(del_lp(C, M, N)).
nwd_lp(C, M, N) :- add_lp(C, M, N).
% Derived from: cp(C, M, N) :- mf(M, F), cf(C, F), mf(N, F).
add_cp(C, M, N) :- add_mf(M, F), nwd_cf(C, F), nwd_mf(N, F), not( cp(C, M, N)).
add_cp(C, M, N) :- add_cf(C, F), nwd_mf(M, F), nwd_mf(N, F), not( cp(C, M, N)).
add_cp(C, M, N) :- add_mf(N, F), nwd_mf(M, F), nwd_cf(C, F), not( cp(C, M, N)).
del_cp(C, M, N) :- del_mf(M, F), cf(C, F), mf(N, F), not(nwi_cp(C, M, N)).
del_cp(C, M, N) :- del_cf(C, F), mf(M, F), mf(N, F), not(nwi_cp(C, M, N)).
del_cp(C, M, N) :- del_mf(N, F), mf(M, F), cf(C, F), not(nwi_cp(C, M, N)).
nwi_cp(C, M, N) :- nwd_mf(M, F), nwd_cf(C, F), nwd_mf(N, F).
nwd_cp(C, M, N) :- cp(C, M, N), not(del_cp(C, M, N)).
nwd_cp(C, M, N) :- add_cp(C, M, N).
% Direct transition rules for the model facts mf/2 and cf/2
nwd_mf(C, M) :- mf(C, M), not(del_mf(C, M)).
nwd_mf(C, M) :- add_mf(C, M).
nwd_cf(C, F) :- cf(C, F), not(del_cf(C, F)).
nwd_cf(C, F) :- add_cf(C, F).
Figure 2: The derived update propagation and indirect and direct transition
rules for the $LCOM1$ definition.
The task of UP is to systematically compute the set of all induced changes,
starting from the physical changes of base data. Technically, this is a set of
delta facts for any affected predicate which may be stored in corresponding
delta relations. For each predicate symbol p, we will use a pair of delta
predicates <add_p, del_p> representing the insertions and deletions induced on
p by an update. The initial set of delta facts represents the so-called _UP
seeds_.
In the following, we briefly review a transformation-based approach to UP
where the Prolog rules and the UP seeds are employed to derive _propagation
rules_ for computing delta relations. A propagation rule refers to at least
one delta predicate in its body in order to provide a focus on the underlying
changes when computing induced updates. For showing the effectiveness of an
induced update, however, references to the state of a predicate before and
after the base update has been performed are necessary.
For each predicate ${\tt p}$ we use ${\tt old\\_p}$ to refer to its old state
before the changes given in the delta sets have been applied (technically the
rule behind ${\tt old\\_p}$ is the unmodified version of ${\tt p}$). We use
${\tt new\\_p}$ to refer to the new state of ${\tt p}$. These state relations
are never completely computed, but are queried with bindings from the delta
sets in the propagation rule body, and thus act as a test of effectiveness. An
induced insertion or induced deletion can be simply represented by the
difference between the two consecutive database states. We consider the
following Prolog rule:
p(X) :- q(Y),r(Z),not s(C).
The difference rules may look as follows:
add_p(X) :- add_q(Y), new_r(Z), not(new_m(C)), not(old_p(X)).
add_p(X) :- new_q(Y), add_r(Z), not(new_m(C)), not(old_p(X)).
add_p(X) :- new_q(Y), new_r(Z), del_m(C), not(old_p(X)).
The propagation rules basically perform a comparison of the `old` and `new`
versions of the predicates. While providing a focus on insertions into q and
r, all necessary combinations of delta and state predicates are considered.
Because of the negative referenced predicate s, an additional rule has to be
considered, which covers new derivations for p due to a deletion from s.
All propagation rules also contain the additional effectiveness test `not
old_p(X)`, to check for the effectiveness of the induced insertions in case of
alternative derivations of the same fact `p` in the old state. As an
optimization we can drop the test, in case there are no alternative
derivations, or if the set of, insertions can be overestimated. To avoid the
full determination of state predicates, we should move the delta predicates as
far left as possible in the rule body. This way the bindings provided by the
delta facts can be used for restricting the evaluation of state predicates.
This leads to the following rules:
add_p(X) :- add_q(Y), new_r(Z), not(new_m(C)).
add_p(X) :- add_r(Z), new_q(Y), not(new_m(C)).
add_p(X) :- del_m(C), new_q(Y), new_r(Z).
For simulating the new predicate state from a given update and the old state,
so called _transition rules_ [12] can be used. The transition rules of a
derived predicate infer its new state from the new states of the underlying
predicates. Thus, for a rule $A:-L_{1},\ldots,L_{n}$ a transition rule of the
form $new\\_A:-new\\_L_{1},\ldots,new\\_L_{n}$ is considered. In contrast, for
every extensional predicate $A$ so-called incremental transition rules are
used:
new_A :- old_A, not(del_A).
new_A :- add_A.
which explicitly refer to the computed changes to A.
For a rule $A:-L_{1},\ldots,L_{n}$ we may also use the direct transition rules
if there are no mutual dependencies between the predicate and the predicates
in the body. In any case, the indirect transition rules of the form
`new`$\\_A$` :- new`$\\_L_{1}$`,`$\ldots,$`new`$\\_L_{n}$
are used in the effectiveness test of negative propagation rules.
As an example, consider the following Prolog program
p(X) :- q(X,Y),r(Y),not(s(Y)).
q(1,2). r(3). s(4).
q(2,3). r(4). s(5).
q(3,4). r(5). s(6).
and the insertion `r(2)` into relation r. The following propagation and
transition rules
p(X) :- add_r(Y),new_q(X,Y),not(new_s(Y)).
new_q(X,Y) :- q(X,Y), not(del_q(X,Y)).
new_q(X,Y) :- add_q(X,Y).
new_s(X,Y) :- s(X,Y), not(del_s(X,Y)).
new_s(X,Y) :- add_s(X,Y).
were derived using the scheme described above. These rules allow for
efficiently computing the induced insertion p(1) (represented by the fact
$add\\_p(1)$) by avoiding any redundant recomputations.
## 5 Propagation of Cohesion Model Updates
In this section, we apply UP as presented in Section 4 to our cohesion model
and the metric rules from Section 2.
Before we employ UP to simulate the model state and metric results after the
refactoring, we first need to generate delta facts as described in Section 3.
At definition time of the metric rules, we may also derive the propagation and
transition rules for UP. Figure 2 shows the result of the rule derivation
process. We can see that for each body literal of a rule that appears in the
metric definition, we create a positive and a negative propagation rule, so
that for the cohesive pair rule `cp` we obtain six propagation rules in total.
We replace negated delta literals simply by their opposite versions, for
example: ` not(del_cp) ` $\Leftrightarrow$ ` add_cp`. For both rules `lp` and
`cp`, we also see two derived transition rules ` nwi_lp ` and ` nwi_cp`,
simulating the versions of those predicates in the new state, which operate on
the propagation rules.
### 5.1 Rule Generation
The UP rule generator component consists of two subcomponents, which can be
seen in the middle of Figure 1. The Rule Analyser component collects meta
information about the metric rules from the deductive rule modules and the UP
Rules Generator, which creates the UP rules, based on the collected meta
information. SWI-Prolog provides various meta predicates to examine loaded
programs. It is important that UP ensures that the augmented rule set (which
includes UP rules as shown in Figure 2 and the original rules like those for
$LCOM1$) generated for a set of logic rules which is guaranteed to terminate,
still keeps this property. This was shown for the language set of Datalog [1],
[7], which only allows straight forward declarative rules, in comparison to
Prolog. We also need to assure this in Prolog, it would be unfavourable if the
UP rules got stuck in infinite loops. Prolog allows a broad variety of syntax
constructs. For the metric definition, therefore, we only allow the model
predicates and predicates defined in the template module itself and those from
the cohesion model. We also allow a narrow list of built-in predicates, namely
`=/2`, `member/2` and the negation predicate `not/1`. We also do not allow any
complex terms in the head of the rules, like: ` cp(a(A),B,[H`$|$`T]) :- ...`.
Though this is a sharp restriction of Prolog, we were able to describe several
structural cohesion metrics, as long as they contained a deductive part.
##### Rule Analyser
First, the Rule Analyser determines all predicates defined in the template
module which contains the $LCOM1$ metric rules we presented before. For
$LCOM1$ those are `cp` and `lp`. The analyser collects various meta
information, which we need to build the UP rules. An overview of the collected
meta information:
` head_of_rule(`$headId$`, `$groundHead$`, [`$name$`, `$arity$`])`
` body_predicate_of_rule(`$headId$`, `$bodyId$`,`
` `$positonInBody$`, `$groundPredicate$`)`
` rule_variables(`$headId$`, `$bodyId$`, `$headBodyPrefix$`, `
` `$positonInBody$`, `$groundPredicate$`)`
` `
` predicate_dependencies_transitive_closure(`
` [`$name$`, `$arity$`], `$dependencies$`)`
` `
We need to determine the free variables in the head and the body of the rules.
We also need to check, if the rules contain self references. This is relevant
to determine the positioning of the delta terms (`del_` and `add_` in the body
of propagation rules). Second, in order to create the indirect transition
rules `nwi_P` of a predicate `P`, we need to analyse, if there are transitive
mutual dependencies between `P` and its body predicates `L`i (where 0 $\leq$ i
¡ #number of body predicates of `P` ). The result will determine if we rather
use `nwi_L`i or `nwd_L`i in `nwi_P`, this is important to ensure that the
rules still terminate. We therefore create a predicate dependency graph.
##### UP Rules Generator
Based on the information collected by the rule analyser, the UP Rules
Generator creates the UP rules. As mentioned before, we create negative and
positive Propagation Rules (for the metric), the direct Transition Rules
(metric and model facts) and indirect Transition Rules (metric only). The
rules created are asserted to a module. After completing the generation
process, we also compile all rules to static predicates by using the Prolog
`compile_predicates` predicate. The rule creation is based on string
concatenation, therefore we convert all predicates to atoms, and prepend the
necessary augmentations to each predicate.
## 6 Conclusions and Future Work
Earlier research had shown that the representation of object-oriented source
code as a fact-base in a logic program allows for clear and concise
implementation of static analyses of the object-oriented code. We explored the
potential of this approach further by applying the well-established theory of
update propagation to it. Update propagation gives us a generic way to
transform any analysis represented as a (sufficiently well-formed) logic
program into an incremental version. Given an actual or hypothetical small
change to the fact-base, this incremental version of the analysis provides an
efficient way to caculate the new result of the analysis after the
(hypothetical) change.
We implemented a transformation, generating the propagation and transition
rules based on the original rules. Besides the applicability of update
propagation to our setting, we already conducted several experiments to
explore the performance benefits. In our experiments, update propagation was
significantly faster than actually transforming our model and re-evaluating
the original definition. As a future work, we intend a detailed study of the
performance benefits.
We focused on the precalculation of the refactoring impact on metrics as this
is in line with our research interest. Nevertheless, there is no reason to
limit update propagation for logic meta programming to metrics or not even to
refactoring. In the context of refactoring, update propagation could for
example be used to verify that certain constraints will still hold true after
the refactoring. This would be in line with the original motivation of update
propagation.
A precise definition of the pairs of abstract refactorings on the model and
the corresponding refactorings on the source code, could ease and clarify the
nature of the induced updates model updates. More sophisticated refactorings
also need complex preconditions, before they may be applied legally. We should
be able to check those conditions on the level of sour model.
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* [11] R. Manthey: Reflections on some fundamental issues of rule-based incremental update propagation, DAISD 1994: pp. 255-276, September 19-21.
* [12] A. Olivé: Integrity Constraints Checking In Deductive Databases, VLDB 1991: 513-523.
* [13] D. Speicher: Evolving Software Quality Knowledge, IC3K 2nd International Workshop on Software Knowledge - SKY 2011, Paris, France, 2011.
* [14] Roel Wuyts. A logic meta-programming approach to support the co-evolution of object-oriented design and implementation. PhD thesis, Vrije Universiteit Brussel, 2001\.
|
arxiv-papers
| 2013-01-11T10:26:48 |
2024-09-04T02:49:40.159748
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Richard Tantius, Daniel Speicher, Andreas Behrend",
"submitter": "Andreas Behrend",
"url": "https://arxiv.org/abs/1301.2443"
}
|
1301.2462
|
Measurement of time-dependent $\mathcal{CP}$ violation in charmless $B$ decays
at LHCb
Stefano Perazzini on behalf of the LHCb Collaboration
Istituto Nazionale di Fisica Nucleare (INFN)
Sezione di Bologna
Via Irnerio 46, 40126 Bologna, Italy
## 1 Abstract
In the following we present the measurements of time-dependent $\mathcal{CP}$
violation in charmless $B$ meson decays performed by LHCb analyzing the $p-p$
collision data collected at a center-of-mass energy of 7 TeV during the 2010
and 2011 LHC runs. In particular we will focus on the analysis of charmless
two-body $B$ decays where the direct and mixing-induced $CP$ asymmetry terms
of the $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ decays have been
measured using 0.69 fb-1 of data collected during 2011. The measurement of the
branching ratio of the $B_{s}^{0}\to K^{*0}\overline{K}^{*0}$ decay, using 35
pb-1 collected during 2010, is also reported. In the end we show the relative
branching ratios of all the decay modes of $B_{\left(s\right)}^{0}\to
K_{S}h^{+}h^{\prime-}$ decays (where $h^{\left(\prime\right)}=\pi,K$),
measured analyzing 1 fb-1 of data collected during 2011.
## 2 Introduction
In this paper we report measurements of time-dependent $\mathcal{CP}$
violation in charmless $B$ meson decays performed at LHCb [1]. We will also
briefly discuss the LHCb potential with other measurements in this sector. All
the results here shown are obtained analyzing the $p-p$ collision data
collected at a center-of-mass energy of 7 TeV during the 2010 and 2011 LHC
runs. The time-dependent $\mathcal{CP}$ asymmetry of either a $B^{0}$ or a
$B_{s}^{0}$ meson decaying into a $\mathcal{CP}$-eigenstate $f$ can be written
as:
$A_{\mathcal{CP}}\left(t\right)=\frac{\Gamma_{\overline{B}\to
f}\left(t\right)-\Gamma_{B\to f}\left(t\right)}{\Gamma_{\overline{B}\to
f}\left(t\right)+\Gamma_{B\to f}\left(t\right)}=\frac{A_{dir}\cos{\left(\Delta
mt\right)}+A_{mix}\sin{\left(\Delta
mt\right)}}{\cosh{\left(\frac{\Delta\Gamma}{2}t\right)}+A_{\Delta}\sinh{\left(\frac{\Delta\Gamma}{2}t\right)}},$
(1)
where $\Gamma\left(t\right)$ represents the time dependent decay rate of the
initial $B$ or $\overline{B}$ meson to the final state $f$, $\Delta m$ and
$\Delta\Gamma$ are the $B$ meson oscillation frequency and decay width
difference respectively, and where the relation
$A_{dir}^{2}+A_{mix}^{2}+A_{\Delta}^{2}=1$ holds. Within this
parameterization, $A_{dir}$ and $A_{mix}$ account for $\mathcal{CP}$ violation
in the decay and in the interference between mixing and decay, respectively.
Since the decay amplitudes of charmless $B$ decays receive important
contribution from penguin diagrams, such a class of decays represents a
powerful tool in the search for physics beyond the Standard Model. In fact new
particles may appear as virtual contributions inside the loop diagrams,
altering the Standard Model expectation for $A_{dir}$ and $A_{mix}$.
The LHCb experiment has a great potential in this sector thanks to its
capabilities of efficiently triggering and reconstructing charmless $B$
decays, to its excellent decay-time resolution allowing to follow the fast
$B_{s}^{0}$ oscillations and to the large cross section for $B$ hadron
production at LHC.
## 3 $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$
The $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ decay amplitudes
receive contributions from tree, electroweak penguin, strong penguin and
annihilation topologies. The presence of penguin diagrams makes these decays
sensitive to New Physics, but with the drawback of theoretical uncertainties
in the determination of relevant hadronic parameters. In Refs. [3] strategies
are presented in order to use the $U$-spin symmetry (i.e. the invariance of
strong interaction dynamics under the quark exchange $d\leftrightarrow s$)
relating the $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ decays, in
order to constraint hadronic parameters and determine the CKM phase $\gamma$
and the $B_{s}^{0}$ mixing phase $\phi_{s}$.
Crucial aspects for measuring time dependent $\mathcal{CP}$ asymmetries in
$B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ decays are the event
selection, the calibration of the particle identification (PID) and the
flavour tagging. These decays are mainly selected by the hadronic trigger of
LHCb, that exploits high transverse momentum and large impact parameter with
respect to primary vertices, typical of $B$ hadron decay products. Then the
sample is further refined applying a set of kinematic and topological
criteria. Finally the 8 final states ($\pi^{+}\pi^{-}$, $K^{+}K^{-}$,
$K^{+}\pi^{-}$, $\pi^{+}K^{-}$, $p\pi^{-}$, $\overline{p}\pi^{+}$, $pK^{-}$
and $\overline{p}K^{+}$) are separated into exclusive subsamples by means of
the PID information provided by the two ring-imaging Cherenkov detectors of
LHCb. The calibration of the PID observables is performed using large samples
of pions, kaons and protons selected from $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$ and $\Lambda\rightarrow p\pi^{-}$ decays (and
their charge conjugates). As a demonstration of its PID capabilities, using
$0.37$ fb-1 of integrated luminosity collected during 2011, LHCb measured the
relative branching ratios of various two-body charmless $B$ decays [5]. The
measurements of the branching ratios of the $B_{s}^{0}\to K^{+}K^{-}$ and
$B_{s}^{0}\to\pi^{+}K^{-}$ decays are the most precise to date. In addition,
the $B_{s}^{0}\to\pi^{+}\pi^{-}$ decay is observed for the first time ever
with a significance of more than 5$\sigma$.
The determination of the initial flavour of the signal $B$ meson (the so-
called ”flavour tagging”) is obtained using a multivariate algorithm that
analyzes the decay products of the other $B$ hadron in the event [6]. The
response of the algorithm is calibrated by measuring the oscillation of the
flavour specific decay $B^{0}\rightarrow K^{+}\pi^{-}$, in which the amplitude
is related to the effective mistag rate. Using the measured mistag rate as an
external input to a two dimensional (invariant mass and decay time) maximum
likelihood fit of the $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ spectra, the direct
and mixing-induced $CP$ asymmetry terms $A_{\pi\pi}^{dir}$,
$A_{\pi\pi}^{mix}$, $A_{KK}^{dir}$ and $A_{KK}^{mix}$ have been measured. The
results, obtained using 0.69 fb-1 of data collected during 2011, are [7]:
$\displaystyle A_{\pi\pi}^{dir}=0.11\pm 0.21\mathrm{(stat.)}\pm
0.03\mathrm{(syst.)}$ (2) $\displaystyle A_{\pi\pi}^{mix}=-0.56\pm
0.17\mathrm{(stat.)}\pm 0.03\mathrm{(syst.)}$ (3) $\displaystyle
A_{KK}^{dir}=0.02\pm 0.18\mathrm{(stat.)}\pm 0.04\mathrm{(syst.)}$ (4)
$\displaystyle A_{KK}^{mix}=0.17\pm 0.18\mathrm{(stat.)}\pm
0.05\mathrm{(syst.)},$ (5)
with correlations $\rho\left(A_{\pi\pi}^{dir},A_{\pi\pi}^{mix}\right)=-0.34$
and $\rho\left(A_{KK}^{dir},A_{KK}^{mix}\right)=-0.10$. $A_{\pi\pi}^{dir}$ and
$A_{\pi\pi}^{mix}$ are measured for the first time at a hadronic machine and
are compatible with the previous results from $B$-Factories [8]. The time-
dependent $\mathcal{CP}$ asymmetry terms $A_{KK}^{dir}$ and $A_{KK}^{mix}$ are
measured for the first time ever and are compatible with theoretical
expectations [4].
## 4 Other charmless decays
Using other charmless $B$ decays where time-dependent $\mathcal{CP}$
asymmetries can be measured, LHCb already produced results. Using just 35 pb-1
collected during 2010, LHCb observed for the first time the $B_{s}^{0}\to
K^{*0}\overline{K}^{*0}$ decay [9]. This decay is governed by pure penguin
topologies offering an interesting potential in the quest for New Physics. It
has been pointed out [10] that the time dependent $\mathcal{CP}$ asymmetries
in this decay can be used to measure the CKM phase $\gamma$ and the
$B_{s}^{0}$ mixing phase $\phi_{s}$ owing to the information provided by the
$U$-spin related decay $B^{0}\to K^{*0}\overline{K}^{*0}$. The signal yield
obtained from the fit to the invariant mass distribution is equal to $49.8\pm
7.5$ candidates and the corresponding branching ratio is
$\mathcal{BR}\left(B_{s}^{0}\to K^{*0}\overline{K}^{*0}\right)=\left(2.81\pm
0.46\mathrm{(stat.)}\pm 0.45\mathrm{(syst.)}\pm
0.34\left(f_{s}/f_{d}\right)\right)\times 10^{-5}$, where the last error comes
from the knowledge of the $b$-quark hadronization probabilities.
Decay | $\mathcal{BR}\times 10^{-6}$
---|---
$\mathcal{BR}\left(B^{0}\to K_{S}K^{\pm}\pi^{\mp}\right)$ | $5.8\pm 0.9\pm 0.9\pm 0.2$
$\mathcal{BR}\left(B^{0}\to K_{S}K^{+}K^{-}\right)$ | $26.3\pm 2.0\pm 2.0\pm 1.1$
$\mathcal{BR}\left(B_{s}^{0}\to K_{S}\pi^{+}\pi^{-}\right)$ | $11.9\pm 3.0\pm 2.0\pm 0.5$
$\mathcal{BR}\left(B_{s}^{0}\to K_{S}K^{\pm}\pi^{\mp}\right)$ | $97\pm 7\pm 10\pm 4$
$\mathcal{BR}\left(B_{s}^{0}\to K_{S}K^{+}K^{-}\right)$ | $4.2\pm 1.5\pm 0.9\pm 0.2$
Table 1: Branching ratios of the $B_{\left(s\right)}^{0}\to
K_{S}h^{+}h^{\prime-}$ decays measured by LHCb. The first error is
statistical, the second is systematic and the third is due to the uncertainty
on $\mathcal{BR}\left(B^{0}\to K_{s}\pi^{+}\pi^{-}\right)$ that is used as a
normalization.
Another important result is the measurement of the relative branching ratios
of all the decay modes of $B_{\left(s\right)}^{0}\to K_{S}h^{+}h^{\prime-}$
decays (where $h^{\left(\prime\right)}=\pi,K$) [11]. The dominant topologies
governing these decays are $b\rightarrow q\overline{q}s$ loop transitions
(with $q=u,d,s$), where new particles in several extensions of the Standard
Model may appear as virtual contributions. Several strategies have been
proposed in order to extract information on the CKM phase $\gamma$ and also on
the $B^{0}$ and $B_{s}^{0}$ mixing phases $\phi_{d}$ and $\phi_{s}$ [12] from
the time-dependent analysis of these decays in the Dalitz plane. As a first
step, using the full 1 fb-1 collected during the 2011, LHCb measured the
relative branching ratios of all the $B_{\left(s\right)}^{0}\to
K_{S}h^{+}h^{\prime-}$ modes with respect to the well established
$\mathcal{BR}\left(B^{0}\to K_{s}\pi^{+}\pi^{-}\right)$. The results are
reported in Tab. 1 where the first error is statistical, the second is
systematic and the third is due to the uncertainty on
$\mathcal{BR}\left(B^{0}\to K_{s}\pi^{+}\pi^{-}\right)$. First evidence of the
two decays $B_{s}^{0}\to K_{S}\pi^{+}\pi^{-}$ and $B_{s}^{0}\to
K_{S}K^{+}K^{-}$ is obtained with a significance of 4.3$\sigma$ and
3.3$\sigma$ respectively. Furthermore, LHCb observed for the first time the
$B_{s}^{0}\to K_{S}K^{\pm}\pi^{\mp}$ decay.
## References
* [1] A. A. Alves et al., JINST 3, S08005 (2008).
* [2] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963); M. Kobayashi and T. Maskawa, Prog. Theor. Phys 49, 652 (1973).
* [3] R. Fleischer, Phys. Lett. B 459, 306 (1999); R. Fleischer, Eur. Phys. J. C 52, 267 (2007); R. Fleischer and R. Knegjens, Eur. Phys. J. C 71, 1532 (2011); M. Ciuchini et al., JHEP 1210, 029 (2012).
* [4] B. Adeva et al. [LHCb Collaboration], arXiv:0912.4179v3 [hep-ex]
* [5] R. Aaij et al. [LHCb Collaboration], arXiv:1206.2794v1 [hep-ex].
* [6] R. Aaij et al. [LHCb Collaboration], Eur. Phys. J. C72, (2012); R. Aaij et al. [LHCb Collaboration], LHCb-CONF-2012-026.
* [7] R. Aaij et al. [LHCb Collaboration], LHCb-CONF-2012-007.
* [8] H. Ishino et al. [Belle Collaboration], Phys. Rev. Lett. 98, 211801 (2007); B. Aubert et al. [BaBar Collaboration], arXiv:0807.4226v2 [hep-ex].
* [9] R. Aaij et al. [LHCb Collaboration], Phys.Lett. B709, 50 (2011).
* [10] M. Ciuchini et al., Phys.Rev.Lett. 100, 031802 (2008); B. Bhattacharya et al., Phys.Lett. B717, 403 (2012); S. Descotes-Genon et al., Phys.Rev. D85, 034010 (2012).
* [11] R. Aaij et al. [LHCb Collaboration], LHCb-CONF-2012-023.
* [12] M. Ciuchini et al., Phys.Rev. D74, 051301 (2006); M. Gronau et. al, Phys.Rev D75, 014002 (2007); M. Ciuchini et al., Phys.Lett. B645, 201 (2007).
|
arxiv-papers
| 2013-01-11T11:38:46 |
2024-09-04T02:49:40.167317
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stefano Perazzini (for the LHCb Collaboration)",
"submitter": "Stefano Perazzini Stefano Perazzini",
"url": "https://arxiv.org/abs/1301.2462"
}
|
1301.2465
|
# H2CO and N2H+ in Protoplanetary Disks: Evidence for a CO-ice Regulated
Chemistry
Chunhua Qi Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA 02138, USA Karin I. Öberg University of Virginia, Departments
of Chemistry and Astronomy, Charlottesville, VA 22904, USA David J. Wilner
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
###### Abstract
We present Submillimeter Array observations of H2CO and N2H+ emission in the
disks around the T Tauri star TW Hya and the Herbig Ae star HD 163296 at
2′′–6′′ resolution and discuss the distribution of these species with respect
to CO freeze-out. The H2CO and N2H+ emission toward HD 163296 does not peak at
the continuum emission center that marks the stellar position but is instead
significantly offset. Using a previously developed model for the physical
structure of this disk, we show that the H2CO observations are reproduced if
H2CO is present predominantly in the cold outer disk regions. A model where
H2CO is present only beyond the CO snow line (estimated at a radius of 160 AU)
matches the observations well. We also show that the average H2CO excitation
temperature, calculated from two transitions of H2CO observed in these two
disks and a larger sample of disks around T Tauri stars in the DISCS (the Disk
Imaging Survey of Chemistry with SMA) program, is consistent with the CO
freeze-out temperature of $\sim$20 K. In addition, we show that N2H+ and H2CO
line fluxes in disks are strongly correlated, indicative of co-formation of
these species across the sample. Taken together, these results imply that H2CO
and N2H+ are generally present in disks only at low temperatures where CO
depletes onto grains, consistent with fast destruction of N2H+ by gas-phase
CO, and in situ formation of H2CO through hydrogenation of CO ice. In this
scenario H2CO, CH3OH and N2H+ emission in disks should appear as rings with
the inner edge at the CO midplane snow line. This prediction can be tested
directly using observations from ALMA with higher resolution and better
sensitivity.
protoplanetary disks; astrochemistry; stars: formation; ISM: molecules;
techniques: high angular resolution; radio lines: ISM
## 1 Introduction
Planetary systems are assembled from dust and gas in the disks surrounding
pre-main sequence stars. The nature of the formed planets are intimately
linked to the structure, composition and evolution of the parent circumstellar
disk. Molecular emission lines serve as probes of disk characteristics, such
as density, temperature and ionization fraction, that are not accessible by
other observations. For example, N2H+ along with the deuterated ions like H2D+
and DCO+ are believed to trace the ionization fraction near the midplane of
the disks (Öberg et al., 2011c). Molecular distributions in disks are also
important to characterize because of their connection with the composition of
forming planetesimals. This especially true of organic molecules, of which
H2CO is an important representative. While most molecules are expected to be
reprocessed in larger planetary bodies, the disk-chemical composition may
survive quite intact in icy planetesimals, including comets (Mumma & Charnley,
2011). Such planetesimals may have seeded the Earth with water and organics,
connecting disk chemistry with the origins of life. Predicting the organic
composition of these planetesimals depend on our understanding of the
distribution of the organic composition of grains in the disks.
The overall disk chemical structure is set by a combination of photochemistry
at the disk surface and sequential freeze-out of molecules in the disk
interior (e.g. Aikawa et al., 2002). The distributions of N2H+ and H2CO and
their relationship to CO present important test cases of the chemical models.
The CO molecule is one of the last to freeze out and predicted to deplete
quickly from the gas-phase at T$<20-25$ K for typical disk mid-plane
densities. Abundant N2H+ is expected only where CO is depleted because N2H+
forms from protonation of N2, which remains in the gas-phase at temperatures a
few degrees lower than CO (Öberg et al., 2005), and is destroyed mainly by
reactions with CO (Bergin et al., 2002). In dense cores in star forming
regions, the abundance of CO is observed to show a strong anti-correlation
with N2H+ (e.g. Caselli et al., 1999; Bergin et al., 2002; Jørgensen, 2004).
In disk models, this effect is manifested as a jump in the N2H+ column density
at the CO “snow line”, one order of magnitude in a recent calculation (Walsh
et al., 2012). The snow line is here defined as the disk radius where the
midplane dust temperature is cold enough for volatiles to condense into ice
grains.
The H2CO chemistry is more complicated than N2H+ because H2CO can form through
several different pathways, both in the gas-phase and on grain-surfaces.
Grain-surface formation of H2CO should depend directly on CO freeze-out;
constrained by theory (Tielens & Hagen, 1982; Cuppen et al., 2009) and
experiments (e.g. Watanabe et al., 2003; Fuchs et al., 2009), H2CO (and CH3OH)
form readily from CO ice hydrogenation. If this is the dominant formation
pathway of H2CO in disks, and if ices are partially desorbed non-thermally
(e.g. Garrod et al., 2007; Öberg et al., 2009b, a), then H2CO gas should
coincide with N2H+ exterior to the CO snow line.
To date, emission from millimeter wavelength H2CO and N2H+ lines has been
detected toward 8 and 6 protoplanetary disks, respectively (Dutrey et al.,
1997; Aikawa et al., 2003; Qi et al., 2003; Thi et al., 2004; Dutrey et al.,
2007; Henning & Semenov, 2008; Öberg et al., 2010, 2011b). Most detections are
toward T Tauri stars with massive disks, and the detection fraction toward
more luminous Herbig Ae stars is low (Öberg et al., 2011b). Based on these
observations, H2CO and N2H+ are mainly abundant in disks with large reservoirs
of cold dust and gas, where CO freeze-out is expected to occur. A direct
connection between CO freeze-out and N2H+ and H2CO in disks has yet to be
observationally established, however.
In this paper we present Submillimeter Array (SMA) observations of H2CO and
N2H+ toward the disks around HD 163296 and TW Hya, and we use these new
observations along with H2CO and N2H+ observations from DISCS (Disk Imaging
Survey of Chemistry with SMA) to constrain the H2CO and N2H+ distributions.
The new data and their calibration are described in §2. In §3, we present the
H2CO and N2H+ images and spectra toward HD 163296 and TW Hya, models of the
H2CO distribution toward HD 163296, H2CO excitation temperature calculations,
and examine the relationship between H2CO and N2H+ emission across the sample
of disks. In §4, we discuss the implications of these results, summarize the
mounting evidence for CO-ice regulated chemistry and make predictions for
future observations of H2CO and N2H+ emission from disks with better
sensitivity and resolution.
## 2 Observations
The observations of HD 163296 (${\rm R.A.=17^{h}56^{m}21.279^{s}}$, ${\rm
decl.=-21^{\circ}57\arcmin 22\farcs 38}$; J2000.0) were made between 2008 and
2012, and of TW Hydrae (${\rm R.A.=11^{h}01^{m}51.875^{s}}$, ${\rm
decl.=-34^{\circ}42\arcmin 17\farcs 155}$; J2000.0) between 2008 and 2012,
using the eight-antenna Submillimeter Array (SMA) located atop Mauna Kea,
Hawaii. Table 1 provides a summary of the observational parameters and
results. For the 2007 and 2008 observations, the SMA receivers operated in a
double-sideband mode with an intermediate frequency (IF) band of 4–6 GHz from
the local oscillator frequency, sent over fiber optic transmission lines to 24
overlapping “chunks” of the digital correlator. The 2012 observations were
made after an upgrade that enabled a second IF band of 6–8 GHz, effectively
doubling the bandwidth.
The SMA observations of HD 163296 were carried out in the compact-north
(COM-N), compact (COM) and subcompact (SUB) array configurations. The 2007
observations included the DCO+ 3–2 line at 216.1126 GHz and the H2CO
$3_{1,2}-2_{1,1}$ line at 225.698 GHz. The 2012 observations included the N2H+
3–2 at 279.512 GHz and the H2CO $4_{1,4}-3_{1,3}$ line at 281.527 GHz. The
observing loops used J1733–130 as the main gain calibrator and observed
J1744–312 every other cycle to check the phase calibration. Flux calibration
was done using observations of Titan and Uranus. The derived fluxes of
J1733-130 were 1.19 Jy (2007 Mar 20), 1.25 Jy (2012 Jun 10), 1.40 Jy (2012 Aug
12 and 14). The bandpass response was calibrated using observations of 3C279,
Uranus and J1924–292.
The SMA observations of TW Hya were carried out in the compact (COM) and
subcompact (SUB) array configurations. The 2008 observations included the H2CO
$5_{1,5}-4_{1,4}$ line at 351.769 GHz. The 2012 Jan 13 SUB observation
included the H2CO $4_{1,4}-3_{1,3}$ and N2H+ 3–2 lines, like HD 163296, but
unfortunately the chunk containing N2H+ was corrupted and unusable. The 2012
Jun 04 COM observation using a similar setting successfully included the N2H+
line. The observing loops used J1037–295 as the gain calibrator. Flux
calibration was done using observations of Titan and Callisto. The derived
fluxes of J1037–295 were 0.73 Jy (2008 Feb 23), 0.73 Jy (2012 Jan 13) and 0.82
Jy (2012 Jun 4). The bandpass response was calibrated using observations of
3C279 and 3C273.
Routine calibration tasks were performed using the MIR software package
111http://www.cfa.harvard.edu/$\sim$cqi/mircook.html, and imaging and
deconvolution were accomplished in the MIRIAD software package.
## 3 Results
In this section we present detections of H2CO and N2H+ emission lines toward
HD 163296 and TW Hya, display their respective distributions (§3.1), and
compare the higher quality observations of H2CO in HD 163296 with models
(§3.2). We then combine the new data with previously reported H2CO and N2H+
detections in disks to examine trends with respect to H2CO excitation
temperature (§3.3) and with each other (§3.4).
### 3.1 H2CO and N2H+ towards HD 163296 and TW Hya
Figure 1 shows images of the spectrally integrated emission toward TW Hya and
HD 163296 at the rest frequencies of two H2CO lines and the N2H+ $J=3-2$ line.
H2CO $4_{1,4}-3_{1,3}$ and $5_{1,5}-4_{1,4}$ and N2H+ are detected toward TW
Hya, and H2CO $3_{1,2}-2_{1,1}$ and H2CO $4_{1,4}-3_{1,3}$ and N2H+ are
detected toward HD 163296. The emission toward TW Hya appears to be centrally
peaked at the size scale of the beams (FWHM $>2\arcsec$).
Toward HD 163296, however, neither the H2CO lines nor the N2H+ line emission
peaks at the location of the continuum peak that marks the stellar position,
but instead show significant offsets. This is most readily apparent for the
H2CO $3_{1,2}-2_{1,1}$ line that was observed with a slightly smaller and more
advantageously rotated beam, where the emission appears ring-like. This is the
second reported observation of a ring-like H2CO distribution after DM Tau
(Henning & Semenov, 2008). Interpreting the H2CO emission toward DM Tau is
however complicated by a large central cavity in dust emission (Andrews et
al., 2011).
Figure 2 shows the spatially integrated spectra. The line shapes and central
velocities agree with what has been previously observed for other molecular
lines toward these disks. The line fluxes are listed in Table 1, and the
values are comparable to detections of these lines toward other large
protoplanetary disks (Öberg et al., 2010, 2011b).
### 3.2 HD 163296 H2CO Model Results
Figure 1 demonstrates that H2CO emission toward HD 163296 is spatially
resolved and thus contains information on the radial distribution of H2CO in
the disk. The relative excitation of the two H2CO transitions should probe
primarily the vertical distribution and thus provide complementary
constraints. We explore the H2CO distribution based on a previously developed
accretion disk model with a well-defined temperature and density structure,
constrained by the HD 163296 broadband spectral energy distribution, spatially
resolved millimeter dust continuum, and multiple CO and CO isotopologue line
observations (Qi et al., 2011). We adopt the same methods as Qi et al. (2008)
for constraining the H2CO distribution, here fitting models that assume a
radial power-law (§3.2.1) and a simple ring with inner boundary at the CO
“snow line” (§3.2.2).
#### 3.2.1 Power-law Model
For a first-order analysis of the distribution of H2CO, we model the radial
variation in the column density as a power law N${}_{100}\times(r/100)^{p}$
between an inner radius Rin and outer radius Rout, where N100 is the column
density at 100 AU in cm-2, $r$ is the distance from the star in AU, and $p$ is
the power-law index.
For the vertical distribution, we assume that H2CO is present with a constant
abundance in a layer with boundaries toward the midplane and toward the
surface of the disk (similar to Qi et al. (2008)). This assumption is
motivated by chemical models (e.g. Aikawa & Nomura, 2006) that predict a
three-layered structure where most molecules are photodissociated in the
surface layer, frozen out in the midplane, and have an abundance that peaks at
intermediate disk heights. The surface ($\sigma_{s}$) and midplane
($\sigma_{m}$) boundaries are presented in terms of
$\Sigma_{21}=\Sigma_{H}/(1.59\times 10^{21}cm^{-2})$, where $\Sigma_{H}$ is
the hydrogen column density measured from the disk surface. This simple model
approximates the vertical location where H2CO is most abundant. The excitation
of multiple transitions can constrain both $\sigma_{s}$ and $\sigma_{m}$, but
in this case of very modest signal-to-noise, we fix $\sigma_{s}$ to 0.79, the
surface boundary found for CO by Qi et al. (2011), and we fit $\sigma_{m}$ for
the midplane boundary and the power-law parameters (N100, $p$, Rin and Rout).
Using the structure model, we compute a grid of synthetic H2CO visibility
datasets over a range of Rout, Rin, $p$, $\sigma_{m}$ and N100 values and
compare with the observations. The best-fit model is obtained by minimizing
$\chi^{2}$, the weighted difference between the real and imaginary part of the
complex visibility measured in the ($u,v$)-plane sampled by the SMA
observations of both H2CO transitions. We use the two-dimensional Monte Carlo
model RATRAN (Hogerheijde & van der Tak, 2000) to calculate the radiative
transfer and molecular excitation. The collisional rates are taken from the
Leiden Atomic and Molecular Database (Schöier et al., 2005).
Table 2 lists the best-fit parameters of the model. The power-law index of 2
implies an H2CO column density that strongly increases with radius. Figure 3
shows the best-fit radial distribution of the H2CO column density. Figures 4
and 5 present comparisons between the observed channel maps and the best-fit
model. The model reproduces the main features of the observations remarkably
well, in particular the flux ratio between the inner and outer channels, and
the lack of emission at the location of the continuum peak.
Figure 6 shows the $\chi^{2}$ surfaces for the Rin and Rout versus the power
law index $p$, which enables us to quantify the uncertainties associated with
the inner and outer region sizes and the power-law index. We find that $p$ is
constrained between 0.5–3.0 (within 1$\sigma$) while Rin is constrained to be
$<$200 AU. The $\chi^{2}$ value does not change significantly for inner radii
$<$90 AU, as expected for the 2$\arcsec$ beam size of the observations. The
outer radius is better constrained, since the emission is very sensitive to
the value of Rout with a positive power-law index. Figure 7 shows the H2CO 3–2
line spectrum compared with the spectra derived from models with different
radial column densities power-law indices. The spectra suggest a lack of high
velocity line wings associated with emission originating in the inner regions
of the disk, consistent with the results of the $\chi^{2}$ analysis.
#### 3.2.2 Ring Model
The positive power-law index found for the H2CO radial distribution implies
that H2CO is present mainly in the outer disk. This is expected if H2CO forms
in situ from CO ice hydrogenation and is therefore present mainly beyond the
CO snow-line, previously determined to be at 160 AU by Qi et al. (2011).
Guided by this astrochemical ansatz, we have tried a second “ring” model where
the H2CO gas is only present where CO has frozen out. The vertical surface
boundary is then defined by the CO freeze-out temperature of 19 K (Qi et al.,
2011), while the midplane boundary can be constrained by the excitation of
multiple H2CO transitions, as in the power-law model. Within this layer, the
abundance of H2CO is assumed to follow that of H nuclei with a constant
fractional abundance, which is also a parameter fit to the data.
The best-fit abundance is $5.5\times 10^{-11}$ and the midplane boundary
$\sigma_{m}$ is consistent with what we find in the power-law model (Table 2).
Figure 3 shows that the vertical surface boundary at 19 K effectively results
in a ring-like radial structure, where the inner edge of the ring is at CO
snow line. Figure 3 also shows that the best-fit power-law and ring models
result in similar H2CO column densities beyond the CO snow line. The profile
of the ring model is considerably flatter than the power-law model and even
drops outside of 300 AU exponentially to the edge of CO emission. Figures 4–5
show that the power-law model and the ring model channel maps display some
subtle differences. But both provide good fits to the data within the noise of
the SMA observations. The same model also provides a good match to the N2H+
flux, but the combination of low signal-to-noise and observations of just one
transition preclude any independent modeling of the N2H+ distribution.
### 3.3 H2CO Excitation Temperatures
HD 163296 is very favorable for H2CO and N2H+ imaging since its relatively
high luminosity and massive disk puts the CO snow line at a large angular
distance compared to other disks, enabling us to resolve the H2CO and N2H+
emission. Without such spatial information, however, we can still obtain a
constraint on where H2CO emission originates in disks based on the average
H2CO excitation temperatures. As a gross approximation, H2CO that coexists
with CO ice should be cold, i.e. present at an excitation temperature
comparable to the CO freeze-out temperature. To test the viability of using
excitation temperatures to constrain the H2CO distribution we extracted
spectra from three of the HD 163296 simulations presented in §3.2, selecting
models with the best outer radius, no inner hole and power-law indices of
$-2$, $0$, and $2$. These distributions approximately correspond to a H2CO
abundance that follows the H2 column, that keeps constant with radius and that
increases steeply, forming a ring. Assuming LTE, that both H2CO transitions
trace the same underlying populations, and a single rotational excitation
temperature, $T_{\rm rot}$, we calculate
$T_{\rm rot}=\frac{E_{\rm 1}-E_{\rm 0}}{ln\left((\nu_{1}S\mu_{1}^{2}\int
T_{\rm 0}dv)/(\nu_{0}S\mu_{0}^{2}\int T_{\rm 1}dv)\right)},$ (1)
where $E_{\rm 0}$ and $E_{\rm 1}$ are the upper energy levels for the low and
high H2CO transitions used in the calculation (H2CO $3_{0,3}-2_{0,2}$ and
$4_{1,4}-3_{1,3}$ for most disks), $\nu$ and $S\mu^{2}$ the corresponding line
frequencies and temperature independent transition strengths and dipole
moments, $\int Tdv$ the integrated line intensity, which is calculated from
the integrated fluxes based on $F/T=13.6\lambda^{2}/(a\times b)$, where $F$ is
the flux in Jy, $T$ the intensity in K, $\lambda$ the line wavelength in
millimeters, and $a$ and $b$ the emission diameters in $\arcsec$. Because of
both vertical and radial temperature gradients in the disk, the size
dependence of the emission regions from two transitions is complicated but the
emission area is not expected to be very different. For simplicity we assume
the extent of the emission is the same for both transitions on account of the
model dependent effects of the temperature gradients. All of the line
parameters were gathered from Splatalogue (a transition-resolved compilation
of several spectroscopic databases), with the data originating from CDMS
(Müller et al., 2005). LTE is a reasonable approximation if H2CO is mainly
present at high densities. The critical densities for the observed H2CO
transitions vary between 105 and 5$\times$106 cm-3 (Troscomptet al., 2009),
dependent on assumed kinetic temperatures. At radii $<$300 AU, all gas colder
than 25 K is at densities higher than 107 cm-3 (Qi et al., 2011), justifying
this assumption.
Using the simulated spectra we derive an excitation temperature of 27 K for
the model with $p=-2$ and excitation temperatures close to or below 20 K for
the other two models. Excitation temperatures thus provide some constraints on
the H2CO distribution, but a temperature close to the expected CO freeze-out
temperature only implies an increasing abundance with radius; it cannot be
used to assess how steeply the abundance increases.
By combining the new H2CO detections toward TW Hya and HD 163296 with H2CO
data from DISCS (Öberg et al., 2010, 2011b), we have a sample of 10 disks with
two H2CO line detections or one H2CO line detection and one upper limit (Table
3). These data are sufficient to calculate excitation temperatures, albeit
with substantial uncertainties. Figure 8 shows the calculated excitation
temperatures for 9 of the 10 disks; the chemically peculiar Herbig Ae star HD
142527 is not included because its excitation temperature of 250 K is probably
not due to thermal excitation.
The H2CO excitation temperature, listed in Table 3, is consistent with, or
lower than, the CO freeze-out temperature of $\sim$20 K for all of these
disks. The average H2CO temperature in the sample (excluding HD 142527) is
$18\pm 6$ K. We note that the relatively high excitation temperature toward HD
163296 is most likely due to the fact that some H2CO 3–2 emission is resolved
out by the SMA observations, as the compact-north antenna configuration has
few short baselines.
### 3.4 H2CO/N2H+ Correlations
We examine the disk sources to test if N2H+ and H2CO emission are correlated
across the sample, as would be expected if 1) the two molecules form under the
same physio-chemical conditions, i.e. only in the regions where CO has frozen
out, 2) the line emission trace the total N2H+ and H2CO column well, and 3)
midplane ionization levels and CO hydrogenation efficiencies do not differ
‘too much’ across the sample (i.e. the size of the CO freeze-out region is the
most important regulator of N2H+ and H2CO column across the sample). It should
be noted that there are other scenarios that could produce a correlation as
well, and that the correlation analysis below should only be considered as a
constraint on the H2CO distribution in combination with the results in the
previous sections. In particular a constant H2CO/N2H+ across a disk sample
would be expected if the relative fractions of chemically characteristic disk
regions is always similar in disks. To conclusively test this requires a
larger sample than currently available, but the fact that we did not find that
H2CO emission correlates with any other molecular emission than N2H+ already
challenges this scenario.
Where possible, we base the comparison on the H2CO $3_{0,3}-2_{0,2}$ line that
has $E_{\rm up}=21$ K, similar to N2H+ $J=3-2$ ($E_{\rm up}=27$ K), to
minimize variations in fluxes caused by the different detailed temperature
structures in different disks. Excluding HD 142527, this line has been
observed toward 6/9 of the sample. For the remaining 3/9 disks, we calculate
the expected H2CO $3_{0,3}-2_{0,2}$ line flux based on the H2CO excitation
temperatures and fluxes of other H2CO lines toward each source. We then
normalize the flux of each H2CO $3_{0,3}-2_{0,2}$ line to the (Taurus)
distance of 140 pc, and we further normalize to a disk mass of 0.01 M⊙ to
account for the fact that more nearby and more massive disks tend to have
overall stronger line emission. Figure 9 shows that there is a strong
correlation between the normalized H2CO and N2H+ fluxes in the disk sample;
the rank correlation is statistically significant at the 95% level. As
expected, Figure 10 shows that this implies a nearly constant N2H+ 3-2 / H2CO
3${}_{0,3}-2_{0,2}$ flux ratio across the sample.
## 4 Discussion
Our modeling strategy in this study and in Qi et al. (2008) has been to first
constrain the overall structure of molecular emission in disks using a
parametric model with a minimum of free parameters, i.e. to determine whether
the radial column density profile of a species decreases, increases or is flat
as a function of disk radius. As demonstrated in Öberg et al. (2012), the
slope of the radial column density profile already can place significant
constraints on the formation pathway of a molecule. Here we find that H2CO
toward HD 163296 belongs to the family of molecules that display an increasing
column density with radius. This first-order constraint on the H2CO
distribution motivated us to consider H2CO formation pathways that would
result in an increase of H2CO with disk radius. H2CO formation through CO-ice
hydrogenation results in a simple prediction that H2CO should be present in a
ring, with the inner edge at the CO snow line. We therefore set up a second
model, based on this prediction, to test if the observations are consistent
with this hypothesis for H2CO formation. We propose that this combination of
backward and forward modeling both provides a fair view of the constraints
obtained by fitting the data, and challenges our basic understanding of disk
chemistry.
### 4.1 H2CO Formation
H2CO can form through multiple chemical pathways. We have shown that the H2CO
distribution towards HD 163296 and the sample statistics are consistent with
formation through in situ CO ice hydrogenation. Here we consider the effects
of additional pathways for H2CO formation, in particular (1) in situ gas phase
formation, and (2) formation in the pre- and proto-stellar phases, followed by
incorporation into the disk.
H2CO can form in the gas-phase through ion-neutral reactions involving, e.g.
CH${}_{3}^{+}$ or through neutral-neutral reactions between CH3 and O (e.g.
Aikawa & Herbst, 1999). The neutral-neutral formation pathway is expected to
result in a radially flat column density structure for a typical T Tauri disk
(Aikawa & Herbst, 1999), with most emission originating at temperatures of
20-40 K (Aikawa et al., 2003). It is not clear from existing disk models
whether the structure will look substantially different if the neutral-ion
reactions dominate. In a protostellar chemistry model (Bergin & Langer, 1997),
CH${}_{3}^{+}$ disappears when CO depletes (E. Bergin, private communication).
This suggests that H2CO forming through this pathway should be anti-correlated
with CO-freeze-out. Neither of these gas-phase pathways thus predicts excess
H2CO column densities in the outer disk, or at low (T$<$20 K) temperatures.
The observed low excitation temperature could on its own be explained by
efficient turbulent mixing of H2CO formed in the gas-phase and then cooled
down in the midplane regions, similarly to what has been proposed to explain
cold CO gas in disks (Aikawa, 2007). Turbulent mixing would not, however,
explain the observed deficiency of H2CO towards the inner disk in HD 163296.
Gas-phase formation of H2CO then seems an unlikely dominant source of H2CO in
disks in light of the new observations, but the case is unlikely to be
conclusively settled until the exclusive grain-surface product CH3OH is
observed to display a similar distribution.
H2CO in disks could be a product of protostellar or molecular cloud chemistry,
as H2CO is commonly observed in pre- and protostellar sources, and this
molecular content may be preserved, at least in part, through the process of
disk formation. Willacy (2007) has modeled this scenario, starting with a H2CO
ice abundance of 10${}^{-6}n_{\rm H}$ inherited from the cold cloud. The model
also includes H2CO formation through gas-phase processes in the disk and
results are presented with and without photodesorption. In the model without
photodesorption, H2CO follows CO in the inner disk and has an additional outer
disk component with an abundance that decreases with radius beyond the CO snow
line. When photodesorption is included, the H2CO column is flat across the
disk, corresponding to a power-law index of 0. Neither predicted abundance
pattern is consistent with the new observations. In addition, H2CO is common
towards protostellar sources of a range of luminosities (e.g. Schöier et al.,
2004; Bisschop et al., 2007), while it is pre-dominantly detected towards
disks around the low luminosity T Tauri stars.
In short, H2CO formation through in situ CO-ice hydrogenation is not only
consistent with the observations, but it is the only pathway proposed (so far)
that naturally explains the observations. To conclusively demonstrate a CO-ice
hydrogenation origin would, however, require the detection of co-spatial
emission of CH3OH; CH3OH has no known efficient gas-phase formation pathway
and is predicted to form together with H2CO whenever CO ice is hydrogenated
(Cuppen et al., 2009).
### 4.2 Locating the CO “Snow Line”
The “snow line” is typically used to denote the midplane disk radius at which
the temperature is low enough for water to condense out on dust grains.
Outside of the snow line, grain accretion will be faster because of larger and
stickier grains, which may substantially speed up the formation of
planetesimals and eventually planets (e.g. Hayashi, 1981; Ida & Lin, 2004,
2008; Ciesla & Cuzzi, 2006; Kretke & Lin, 2007). In the Solar System, the
dividing line between rocky planets and gas giants coincide with the H2O snow
line (Lewis, 1974). CO is another abundant volatile in disks and its snow line
may boost planet formation in the outer disk by providing extra solid masses
(Dodson-Robinson et al., 2009) and inducing planet traps (Masset et al., 2006;
Hasegawa & Pudritz, 2012), and could affect the elemental make-up of the
forming gas-giants (Öberg et al., 2011a). Because of its high volatility
($T_{\rm freeze-out}\sim 20$ K), the CO snow line is expected at disk radii of
10s–100s of AU. This should make it a far more accessible target than the H2O
snow line for millimeter interferometry studies aimed at examining the general
effects of snow lines on disk structures.
Localizing the CO snow line directly from millimeter CO data is challenging,
however. Disks have both a radial temperature gradient away from the central
star, and a vertical one set by radiative heating at the disk surface (e.g.
Aikawa & Herbst, 1999). This results in a CO condensation front that is
located at different radii at different disk heights (Fig. 11), and also that
some CO is present at all radii in the upper disk layers. This fact, together
with a complex radiative transfer (most CO lines are expected to be optically
thick in the disk center and optically thin in the outer parts of the disk),
means that the location of a CO snow line in a disk cannot be inferred from
simply inspecting a CO disk image. The best constraint that exists to date on
a CO snow line radius is toward the Herbig Ae star HD 163296 based on the
analysis of multi-transition, multi-isotope, spatially resolved CO line data
on a self-consistent physical disk model (Qi et al., 2011). The temperature
structure of the model has been constrained by optically thick multiple CO
lines and detailed analysis of the optically thinner 13CO emission reveals a
significant column density reduction at around 165 AU that cannot be explained
by the overall disk column distribution as traced by the dust. This is
interpreted as the result of CO freeze-out and the location as the CO snow
line. Uncertainties in the temperature structure will mainly affect the
determination of the CO freeze-out temperature at the location of the CO snow
line, rather than the location itself , which is constrained between 135 and
175 AU in the disk of HD 163296. While fruitful, this is a time consuming
approach that will be difficult to apply to larger samples of disks and will
always include some degree of model dependency.
Another approach to constrain the CO snow line location is to identify trace
species that are only present where CO has begun to freeze out. Such molecules
should display a ring-like structure with the inner edge corresponding to the
midplane CO snow line. As we have described, N2H+ and H2CO are good candidate
probes of CO snow lines. In principle, these species can be used as powerful
chemical imaging tools to constrain CO snow line locations in large samples of
disks, rather than relying on complex analysis of the CO isotopologue
observations.
#### 4.2.1 Simulated ALMA Observations
It seems clear that both N2H+ and H2CO are outer-disk species from the
chemical perspective. To connect their formation to the onset of CO freeze-out
conclusively requires a combination of higher sensitivity, and higher
resolution imaging. This kind of imaging can be done with the newly available
capabilities of the ALMA telescope in Chile, which is nearing completion of
construction.
To demonstrate the astrochemical predictions generated by our analysis of the
SMA data, and the ease at which they can be tested with ALMA, we present a set
of simulations of HD 163296 ALMA observations using the antenna configuration
5 (corresponding to 0.3″ resolution) in Figure 12. The predicted H2CO and N2H+
rings are readily detected at this resolution, and the power-law and ring
models are clearly distinguished. For both models, the diameter of the
emission maximum can be estimated directly from the images within a fraction
of the beam size, enabling us to determine if the H2CO and N2H+ emission truly
trace the CO snow line. In addition, ALMA should have sufficient sensitivity
to detect CH3OH if it is present with a similar abundance and distribution as
H2CO, as expected if both of these species are formed through CO hydrogenation
and then non-thermally desorbed.
## 5 Conclusions
We have presented three observational results that support the idea that CO
freeze-out regulates the H2CO and N2H+ chemistry in disks:
1. 1.
H2CO and N2H+ emission towards HD 163296 appears offset from the continuum
peak at a size scale consistent with the CO snow line at 160 AU. These
observations are matched well by a ring model where H2CO is present only in
the disk regions with CO freeze-out.
2. 2.
The H2CO excitation temperature in a sample of 9 disks is typically below
$\sim 20$ K, consistent with the bulk of H2CO emission originating in disk
regions where CO is expected to freeze-out.
3. 3.
The H2CO and N2H+ emission are correlated across the disk sample, consistent
with the hypothesis of coexistence beyond the CO snow line.
These results suggest that both N2H+ and H2CO should be present in rings, with
the inner edge at the CO snow line. This may be used as a probe of CO snow
line locations across samples of disk and is also important for predicting the
organic content of comets forming at different disk radii. In general, the
radial and vertical distributions of molecules constitute strong probes of the
basic chemistry used into astrochemical models, while molecular abundances and
column densities are probably best used to test our understanding of the
structure and history of individual objects.
Facilities: SMA
The SMA is a joint project between the Smithsonian Astrophysical Observatory
and the Academia Sinica Institute of Astronomy and Astrophysics and is funded
by the Smithsonian Institution and the Academia Sinica. We thank Edwin Bergin
and Paola D’Alessio for their helpful suggestions, and a referee for
constructive comments on the paper. Support for K. I. O. is provided by NASA
through a Hubble Fellowship grant awarded by the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., for NASA, under contract NAS 5-26555. We also acknowledge
NASA Origins of Solar Systems grant No. NNX11AK63.
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Figure 1: H2CO and N2H+ observations toward TW Hya and HD 163296 with the SMA.
The emission toward TW Hya is centrally peaked, while there is a clear offset
in the H2CO emission toward HD 163296. The first two contour levels are 3 and
5$\sigma$, with 1$\sigma$ measured to be 0.10 (H2CO 4–3), 0.13 (H2CO 5–4), and
0.36 (N2H+) Jy km s-1 per beam toward TW Hya, and 0.06 (H2CO 3–2), 0.17 (H2CO
4–3), and 0.13 (N2H+) Jy km s-1 per beam toward HD 163296. Figure 2:
Spatially integrated spectra of H2CO and N2H+ toward TW Hya and HD 163296. The
red dashed lines mark VLSR toward each source, based on CO observations (Qi et
al. 2006, 2011). The double-peak structure typical for rotating disks is not
resolved toward TW Hya with the applied spectral resolution because of its
face-on orientation. Figure 3: The hydrogen nuclei column density toward HD
163296 (Qi et al. 2011) is plotted together with the H2CO column density from
the best-fit power law and ‘ring’ models. In the ring model, the H2CO
abundance is defined to be zero when the temperature is less than 19 K, the CO
freeze-out temperature, which leads to the sharp drop in the H2CO column
density interior to the CO snow line at 160 AU. Figure 4: Observed and
simulated channel maps for H2CO $3_{1,2}-2_{1,1}$ toward HD 163296, using the
best-fit power-law (p=2) and ring model. Both models fit the data within the
uncertainties. The first contour is 2$\sigma$ and each following contour step
is 1$\sigma$. The channel velocity in km s-1 is in the upper right corner of
each panel and the synthesized beam is displayed in the lower left panel.
Figure 5: As Fig. 4 but for the H2CO $4_{1,4}-3_{1,3}$ line. Figure 6:
Iso-$\chi^{2}$ surfaces of $R_{\rm out}$ and $R_{\rm in}$ versus $p$. Contours
correspond to the 1–5 $\sigma$ errors. Figure 7: Simulated spectra for H2CO
$3_{1,2}-2_{1,1}$ for models with radial column densities power-law indices
$p=2$(best-fit, red line), $0.5$(within 1$\sigma$ noise level, green line) and
$-2$(blue line), overlaid with the HD 163296 spectra in gray shade. Figure 8:
The calculated H2CO excitation temperatures for all disks observed with the
SMA that have at least one H2CO detection and one upper limit, except for the
disk around HD 142527, which has a very high excitation temperature of $>$100
K. All remaining disks have excitation temperatures H2CO consistent with or
lower than the expected CO freeze-out temperature of $\sim$20 K (red dashed
horizontal line). Figure 9: Correlation of N2H+ 3–2 (E${}_{\rm u}=27$ K) and
H2CO 3${}_{0,3}-2_{0,2}$ (E${}_{\rm u}=21$ K) emission normalized to disk mass
(based on dust modeling) and source distance. The red symbols mark disks where
the H2CO 3${}_{0,3}-2_{0,2}$ flux has been calculated based on the flux from
other H2CO transitions. Figure 10: The N2H+/H2CO ratio, demonstrating that
it is almost constant, as would be expected from the strong correlation
between the normalized fluxes. Figure 11: Illustration of the expected
distribution of CO in the gas-phase and on grain-surfaces. Co freeze-out
beyond the CO-snow line ($\rm R_{20K}$) and interior to the dashed contour is
predicted to result in a large increase of N2H+ and the onset of H2CO
production from CO ice. The ice may then non-thermally desorb to produce gas-
phase H2CO. Figure 12: The predicted morphology of H2CO 3–2 emission toward
HD 163296 for the power-law and ring model, when using ALMA antenna
configuration 5 (corresponding to 0.3″ resolution) and 1h integration. The
H2CO ring structure should be accompanied by similar CH3OH and N2H+ rings if
the observations reported in this paper are due the exclusive presence of H2CO
in the outer disk, exterior to the CO snow line. For the CH3OH simulation we
assumed the same column density and excitation temperature for CH3OH as has
been observed for H2CO, and imaged the 241.767 GHz 5${}_{0,5}-4_{0,4}$ line.
White contours are [0.025,0.05,0.1] Jy km s-1 per beam.
Table 1: Observational Parametersa
Parameter | HD 163296
---|---
| 2007 Mar 20 (COM-N) | 2012 Jun 10 (COM) | 2012 Jun 10 (COM)
| | | 2012 Aug 12 (SUB)
| | | 2012 Aug 14 (SUB)
Baseline (m) | 16.4–139.2 | 16.4–77.0 | 9.5–77.0
Lines | H2CO 31,2–21,1 | H2CO 41,4–31,3 | N2H+ 3–2
Rest frequency (GHz) | 225.69778 | 281.52693 | 279.51170
Beam Size (FWHM) | 3$\farcs$4$\times$2$\farcs$2 | 3$\farcs$9$\times$2$\farcs$5 | 4$\farcs$9$\times$3$\farcs$4
P.A. | 54∘ | 15∘ | 14∘
Channel spacing (km s-1) | 0.54 | 0.87 | 0.60
RMS Noise (Jy beam-1) | 0.063 | 0.096 | 0.072
Integrated Flux (Jy km s-1) | 0.89 | 1.55 | 0.53
Continuum (GHz) | 221 | 273 | 273
Beam Size (FWHM) | 3$\farcs$5$\times$2$\farcs$2 | 4$\farcs$1$\times$2$\farcs$6 | 5$\farcs$2$\times$3$\farcs$5
P.A. | 56∘ | 15∘ | 14∘
RMS Noise (mJy beam-1) | 3.4 | 9.4 | 3.8
Flux Density (Jy beam-1) | 0.49 | 0.96 | 0.95
Integrated Flux (Jy) | 0.61 | 1.06 | 1.07
Parameter | TW Hya
| 2012 Jan 13(SUB) | 2008 Feb 23 (COM) | 2012 Jun 4 (COM)
Baseline (m) | 9.5–45.2 | 16.4–77.0 | 16.4–77.0
Lines | H2CO 41,4–31,3 | H2CO 51,5–41,4 | N2H+ 3–2
Rest frequency (GHz) | 281.52693 | 351.76866 | 279.51170
Beam Size (FWHM) | 9$\farcs$1$\times$5$\farcs$6 | 3$\farcs$3$\times$2$\farcs$0 | 3$\farcs$5$\times$2$\farcs$0
P.A. | 24∘ | 1∘ | 10∘
Channel spacing (km s-1) | 0.43 | 0.17 | 0.11
RMS Noise (Jy beam-1) | 0.16 | 0.23 | 0.55
Integrated Flux (Jy km s-1) | 1.22 | 0.54 | 2.2
Continuum (GHz) | 273 | 346 | 273
Beam Size (FWHM) | 9$\farcs$2$\times$5$\farcs$7 | 3$\farcs$5$\times$2$\farcs$0 | 4$\farcs$8$\times$2$\farcs$4
P.A. | 25∘ | -2∘ | 10∘
RMS Noise (mJy beam-1) | 6.2 | 12 | 14
Flux Density (Jy beam-1) | 0.90 | 1.20 | 0.70
Integrated Flux (Jy) | 0.93 | 1.52 | 0.92
aafootnotetext: All quoted values assume natural weighting.
Table 2: Best-Fit Model Parametersa
Parameter | HD 163296
---|---
Power-law Model
$\rm N_{100}$ (cm-2) | 1.7$\times$1011
$\rm p$ | 2
$\rm R_{out}$ (AU) | 410
$\rm R_{in}$ (AU) | $<$200
Surface boundary $\sigma_{s}$ | 10-0.1=0.79
Midplane boundary $\sigma_{m}$ | 101.5=31.6
$\chi^{2}$ | 720101
Ring Model
Fractional abundance | 5.5$\times 10^{-11}$
$R_{out}$ (AU) | 500
Surface boundary | T$<$19 K
Midplane boundary $\sigma_{m}$ | 101.5=31.6
$\chi^{2}$ | 720101
aParameters in italics were fixed from CO modeling ($\sigma_{s}$ for the
power-law model and $R_{out}$ for the ring model).
Table 3: Central star and disk, and H2CO data.
Source | RA | DEC | Spec. | d | mdisk | Detected H2COi | Calc. $3_{0,3}-2_{0,2}$ | Trot
---|---|---|---|---|---|---|---|---
| | | type | [pc] | [M⊙] | | [Jy km s-1] | [K]
TW Hyaa | 11 01 51.91 | $-$34 42 17.0 | K7 | 56 | 0.005 | $4_{1,4}-3_{1,3}$, $5_{1,5}-4_{1,4}$ | $\sim$1.3j | 11(6)
HD 163296b | 17 56 21.29 | $-$21 57 21.9 | A1 | 122 | 0.089 | $3_{1,2}-2_{1,1}$, $4_{1,4}-3_{1,3}$ | $\sim$0.46j | 29(10)
DM Tauc | 04 33 48.73 | $+$18 10 10.0 | M1 | 140 | 0.02 | $3_{0,3}-2_{0,2}$, $4_{1,4}-3_{1,3}$ | – | 11(4)
AA Tauc | 04:34:55.42 | $+$24:28:53.2 | K7 | 140 | 0.01 | $3_{0,3}-2_{0,2}$, ($4_{1,4}-3_{1,3}$)k | – | $<$18
LkCa 15c | 04 39 17.78 | $+$22 21 03.5 | K5 | 140 | 0.05 | $3_{0,3}-2_{0,2}$, $4_{1,4}-3_{1,3}$ | – | 16(3)
GM Aurc | 04 55 10.98 | $+$30 21 59.4 | K3 | 140 | 0.03 | $3_{0,3}-2_{0,2}$, $4_{1,4}-3_{1,3}$ | – | 12(5)
IM Lup d,e | 15 56 09.23 | $-$37 56 05.9 | M0 | 150 | 0.08 | $3_{0,3}-2_{0,2}$, $4_{1,4}-3_{1,3}$ | – | 22(6)
AS 209f | 16 49 15.29 | $-$14 22 08.6 | K5 | 125 | 0.028 | $3_{0,3}-2_{0,2}$, ($4_{1,4}-3_{1,3}$)k | – | $<$30
V4046 Sgrg | 18 14 10.47 | $-$32 47 34.5 | K5 | 73 | 0.01 | $3_{1,2}-2_{1,1}$, $4_{1,4}-3_{1,3}$ | $\sim$0.92j | 12(3)
HD 142527h | 15 56 41.89 | $-$42 19 23.3 | F6 | 145 | 0.1 | $3_{0,3}-2_{0,2}$, $4_{1,4}-3_{1,3}$ | – | 250(120)
Star and disk data from: aQi et al. (2004), bQi et al. (2011), cAndrews &
Williams (2005), dLombardi et al. (2008), ePinte et al. (2008), fAndrews et
al. (2009), gRodriguez et al. (2010), hVerhoeff et al. (2011).
iFrom this work and Öberg et al. (2010, 2011b). Upper limits in parentheses.
jCalculated from other H2CO transition strengths and the determined excitation
temperature. kThe $3-2$ lines were reported as upper limits in Öberg et al.
(2010, 2011b), but after re-examination of the emission maps, we conclude that
these lines are better described as tentative detections (2-3$\sigma$).
|
arxiv-papers
| 2013-01-11T12:02:13 |
2024-09-04T02:49:40.173515
|
{
"license": "Public Domain",
"authors": "Chunhua Qi, Karin Oberg and David Wilner",
"submitter": "Chunhua Qi",
"url": "https://arxiv.org/abs/1301.2465"
}
|
1301.2533
|
# A Novel Analytical Method for Evolutionary Graph Theory Problems
Paulo Shakarian Network Science Center and Dept. of Electrical Engineering
and Computer Science, United States Military Academy, West Point, NY 10996
Patrick Roos Dept. of Computer Science, University of Maryland, College Park,
MD 20740 Geoffrey Moores Network Science Center and Dept. of Electrical
Engineering and Computer Science, United States Military Academy, West Point,
NY 10996
###### Abstract
Evolutionary graph theory studies the evolutionary dynamics of populations
structured on graphs. A central problem is determining the probability that a
small number of mutants overtake a population. Currently, Monte Carlo
simulations are used for estimating such fixation probabilities on general
directed graphs, since no good analytical methods exist. In this paper, we
introduce a novel deterministic framework for computing fixation probabilities
for strongly connected, directed, weighted evolutionary graphs under neutral
drift. We show how this framework can also be used to calculate the expected
number of mutants at a given time step (even if we relax the assumption that
the graph is strongly connected), how it can extend to other related models
(e.g. voter model), how our framework can provide non-trivial bounds for
fixation probability in the case of an advantageous mutant, and how it can be
used to find a non-trivial lower bound on the mean time to fixation. We
provide various experimental results determining fixation probabilities and
expected number of mutants on different graphs. Among these, we show that our
method consistently outperforms Monte Carlo simulations in speed by several
orders of magnitude. Finally we show how our approach can provide insight into
synaptic competition in neurology.
###### keywords:
evolutionary dynamics, Moran process, complex networks
††journal: BioSystems
## 1 Introduction
Evolutionary graph theory (EGT), introduced by [14], studies the problems
related to population dynamics when the underlying structure of the population
is represented as a directed, weighted graph. This model has been applied to
problems in evolutionary biology [28], physics [25], game theory [20],
neurology [26], and distributes systems [13]. A central problem in this
research area is computing the fixation probability \- the probability that a
certain subset of mutants overtakes the population. Although good analytical
approximations are available for the undirected/unweighted case [1, 6], these
break down for directed, weighted graphs as shown by [16]. As a result, most
work dealing with evolutionary graphs rely on Monte Carlo simulations to
approximate the fixation probability [22, 7, 4]. In this paper we develop a
novel deterministic framework to compute fixation probability in the case of
neutral drift (when mutants and residents have equal fitness) in directed,
weighted evolutionary graphs based on the convergence of “vertex
probabilities” to the fixation probability as time approaches infinity. We
then show how this framework can be used to calculate the expected number of
mutants at a given time, how the framework can be modified to do the same for
related models, how it can provide non-trivial bounds for fixation probability
in the case of an advantageous mutant, and how it can provide a non-trivial
lower bound on the mean time to fixation. We also provide various experiments
that show how our method can outperform Monte Carlo simulations by several
orders of magnitude. Additionally, we show that the results of this paper can
provide direct insight into the problem of synaptic competition in neurology.
Our method also fills a few holes in the literature. First, it allows for
deterministic computation of fixation probability when there is an initial set
of mutants – not just a singleton (the majority of current research on
evolutionary graph theory only considers singletons). Second, it allows us to
study how the mutant population changes as a function of time. Third, we show
(by way of rigorous proof) that fixation probability, under the case of
neutral drift is a lower bound for the case of the advantageous mutant -
confirming simulation observations by [15]. Fourth, we show (also by way of
rigorous proof) that fixation probability under neutral drift is additive
(even for weighted, directed graphs), which extends the work of [6] that
proved this for undirected/unweighted graphs. Fifth, we provide a non-trivial
lower bound for the computation of mean time to fixation in the general case -
which has only previously been explored for well-mixed populations [2] and
special cases of graphs [5].
This paper is organized as follows. In Section 2 we review the original model
of Lieberman et al., introduce the idea of “vertex probabilities” and show how
they can be used to find the fixation probability. We then show how this can
be used to determine the expected number of mutants at a given time in Section
3. This is followed by a discussion of how the framework can be extended to
other update rules in Section 4 and then for bounding fixation probability in
the case of an advantageous mutant in Section 5. We then discuss how our
approach can be adopted to bound mean time to fixation in Section 6. We use
the results of the previous sections to create an algorithm for computing
fixation probability and introduce a heuristic technique to significantly
decrease the run-time. The algorithm and several experimental evaluations are
described in Section 7. In Section 8, we show how our framework can be applied
to neurology to gain insights into synaptic competition. Finally, we discuss
related work in Section 9 and conclude.
## 2 Directly Calculating Fixation Probability
The classic evolutionary model known as the Moran Process is a stochastic
process used to describe evolution in a well-mixed population [18]. All the
individuals in the population are either mutants or residents. The aim of such
work was to determine if a set of mutants could take over a population of
residents (achieving “fixation”). In [14], evolutionary graph theory (EGT) is
introduced, which generalizes the model of the Moran Process by specifying
relationships between the $N$ individuals of the population in the form of a
directed, weighted graph. Here, the graph will be specified in the usual way
as $G=(V,E)$ where $V$ is a set of nodes (individuals) and $E\subseteq V\times
V$. In most literature on evolutionary graph theory, the evolutionary graph is
assumed to be strongly connected. We make the same assumption and state when
it can be relaxed.
For any node $i$, the numbers
${k_{\textit{in}}^{(i)}},{k_{\textit{out}}^{(i)}}$ are the in- and out-
degrees respectively. We will use the symbol $N$ to denote the sized of $V$.
Additionally, we will specify weights on the edges in set $E$ using a square
matrix denoted $W=[w_{ij}]$ whose side is of size $N$. Intuitively, $w_{ij}$
is the probability that member of the population $j$ is replaced by $i$ given
that member $i$ is selected. We require $\sum_{j}w_{ij}=1$ and that $(i,j)\in
E$ iff $w_{ij}>0$. If for all $i,j$, we have
$w_{ij}=1/{k_{\textit{out}}^{(i)}}$, then the graph is said to be
“unweighted.” If for all $(i,j)\in E$, we have $(j,i)\in E$ the graph is said
to be “undirected.” Though our results primarily focus on the general case, we
will often refer to the special case of undirected/unweighted graphs as this
special case is quite common in the literature [1, 6].
In this paper we will often consider the outcome of the evolutionary process
when there is a set of initial mutants as opposed to a singleton. Hence, we
say some set (often denoted $C$) is a configuration if that set specifies the
set of mutants in the population (all other members in the population then are
residents). We assume all members in the population are either mutants or
residents and have a fitness specified by a parameter $r>0$. Mutants have a
fitness $r$ and residents have a fitness of $1$. At each time step, some
individual $i\in V$ is selected for “birth” with a probability proportional to
its fitness. Then, an outgoing neighbor $j$ of $i$ is selected with
probability $w_{ij}$ and replaced by a clone of $i$. Note if $r=1$, we say we
are in the special case of neutral drift.
We will use the notation $P_{V^{\prime},t}$ to refer to the probability of
being in configuration $V^{\prime}$ after $t$ timesteps and
$P_{V^{\prime},t|C}$ to be the probability of being in configuration
$V^{\prime}$ at time $t$ conditioned upon initial configuration $C$. Perhaps
the most widely studied problem in evolutionary graph theory is to determine
the fixation probability. Given set of mutants $C$ at time $0$, the fixation
probabilty is defined as follows.
$F_{C}=\mathit{lim}_{t\rightarrow\infty}P_{V,t|C}$ (1)
This is the probability that an initial set $C$ of mutants takes over the
entire population as time approaches infinity. Similarly, we will use the term
the extinction probability, $\overline{F_{C}}$, to be
$\mathit{lim}_{t\rightarrow\infty}P_{\emptyset,t|C}$. If the graph if strongly
connected, then we have $F_{C}+\overline{F_{C}}=1$. Hence, for a strongly
connected graph, a mutant either fixates or becomes extinct. Typically, this
problem is studied using Monte Carlo simulation. This work uses the idea of a
vertex probabilities to create an alternative to such an approach. The vertex
probability is the probability that a certain vertex is a mutant at a certain
time given an initial configuration. For vertex $i$ at time $t$, we denote
this as $P_{i,t|C}$. Often, for ease of notation, we shall assume that the
probabilities are conditioned on some initial configuration and drop the
condition, writing $P_{i,t}$ instead of $P_{i,t|C}$. We note that $P_{i,t}$
can be expressed in terms of probabilities of configurations as follows.
$P_{i,t}=\mathop{\sum_{V^{\prime}\in 2^{V}}}_{\textit{s.t. }i\in
V^{\prime}}P_{V^{\prime},t}$ (2)
Viewing the probability that a specific vertex is a mutant at a given time has
not, to our knowledge, been studied before with respect to evolutionary graph
theory (or in related processes such as the voter model). The key insight of
this paper is that studying these probabilities sheds new light on the problem
of calculating fixation probabilities in addition to providing other insights
into EGT. For example, it is easy to show the following relationship.
###### Proposition 1
Let $V^{\prime}$ be a subset of $V$ and $t$ be an arbitrary time point. Iff
for all $i\in V^{\prime}$, $P_{i,t}=1$ and for all $i\notin V^{\prime}$,
$P_{i,t}=0$, then $P_{V^{\prime},t}=1$ and for all $V^{\prime\prime}\in 2^{V}$
s.t. $V^{\prime\prime}\not\equiv V^{\prime}$, $P_{V^{\prime\prime},t}=0$.
It is easy to verify that $F_{C}>0$ iff $\forall i\in V$,
$\mathit{lim}_{t\rightarrow\infty}P_{i,t}>0$. Hence, in this paper, we shall
generally assume that $\mathit{lim}_{t\rightarrow\infty}P_{i,t}>0$ holds for
all vertices $i$ and specifically state when it does not. As an aside, for a
given graph, this assumption can be easily checked: simply ensure for $j\in
V-C$ that exists some $i\in C$ s.t. there is a directed path from $i$ to $j$.
Now that we have introduced the model and the idea of vertex probabilities we
will show how to leverage this information to compute fixation probability. It
is easy to show that as time approaches infinity, the vertex probabilities for
all vertices converge to the fixation probability when the graph if strongly
connected.
###### Theorem 1
$\forall i,\mathop{\mathit{lim}}_{t\rightarrow\infty}P_{i,t|C}=F_{C}$
Now let us consider how to calculate $P_{i,t}$ for some $i$ and $t$. For
$t=0$, where we know that we are in the state where only vertices in a given
set are mutants, we need only appeal to Proposition 1 \- which tells us that
we assign a probability of $1$ to all elements in that set and $0$ otherwise.
For subsequent timesteps, we have developed Theorem 2 shown next (the proof of
which is included in the supplement).
###### Theorem 2
$\displaystyle P_{i,t}$ $\displaystyle=$ $\displaystyle
P_{i,t-1}+\mathop{\sum}_{(j,i)\in E}w_{ji}\left(P_{j,t-1}\cdot
S_{(j,t)|(j,t-1)}-P_{i,t-1}\cdot S_{(j,t)|(i,t-1)}\right)$
($S_{(j,t)|(i,t-1)}$ is the probability that $j$ is picked for reporduction at
time $t$ given that $i$ was a mutant at time $t-1$.)
We believe that a concise, tractable analytical solution for
$S_{(j,t)|(i,t-1)}$ is unlikely. However, for neutral drift ($r=1$), these
conditional probabilities are trivial - specifically, we have for all $i,j,t$,
$S_{(j,t)|(i,t-1)}=1/N$ as this probability of selection is independent of the
current set of mutants or residents in the graph. Hence, in the case of
neutral drift, we have the following:
$\displaystyle P_{i,t}$ $\displaystyle=$ $\displaystyle
P_{i,t-1}+\sum_{(j,i)\in
E}\frac{w_{ji}}{N}\cdot\left(P_{j,t-1}-P_{i,t-1}\right)$ (3)
Studying evolutionary graph theory under neutral drift was a central theme in
several papers on EGT in the past few years [6, 15] as it provides an
intuition on the effects of network topology on mutant spread. In Section 5 we
examine the case of the advantageous mutant ($r>1$). Neutral drift allows us
to strengthen the statement of Equation 1 to a necessary and sufficient
condition - showing that when the probabilities of all nodes are equal, then
we can determine the fixation probability.
###### Theorem 3
Assuming neutral drift ($r=1$), given initial configuration $C$ with fixation
probability $F_{C}$, if at time $t$ the quantities $P_{i,t|C}$ are equal (for
all $i\in V$), then they also equal $F_{C}$.
Therefore, under neutral drift, we can determine fixation probability when
Equation 3 causes all $P_{i,t}$’s to be equal. We can also use Equation 3 to
find bounds on the fixation probability for some time $t$ by the following
result that holds for any time $t$ under neutral drift.
$\min_{i}P_{i,t}\leq F_{C}\leq\max_{i}P_{i,t}$ (4)
Under neutral drift, we can show that fixation probability is additive for
disjoint sets. Broom et al. proved a similar result the a special case of
undirected/unweighted evolutionary graphs [6]. However, our proof (contained
in the supplement) differs from theirs in that we leverage Equation 3.
Further, unlike the result of Broom et al., our result applies to the more
general case of weighted, directed graphs.
###### Theorem 4
When $r=1$ for disjoint sets $C,D\subseteq V$, $F_{C}+F_{D}=F_{C\cup D}$.
## 3 Calculating the Expected Number of Mutants
In addition to allowing for the calculation of fixation probability, our
framework can also be used to observe how the expected number of mutants
changes over time. We will use the notation $\textbf{Ex}^{(t)}_{C}$ to denote
the expected number of mutants at time $t$ given initial set $C$. Formally,
this is defined below.
$\textbf{Ex}^{(t)}_{C}=\sum_{i\in V}P_{i,t}$ (5)
Unlike fixation probability, which only considers the probability that mutants
overtake a population, $\textbf{Ex}^{(t)}_{C}$ provides a probabilistic
average of the number of mutants in the population under a finite time
horizon. For example, is has been noted that graph structures which amplify
fixation normally also increase time to absorption [8, 21]. Hence, finding the
expected number of mutants may be a more viable topic in some areas of
research where time is known to be limited. Following from Equation 3 where we
showed how to compute $P_{i,t}$ for each node at a given time, we have the
following relationship concerning the expected number of mutants at a given
time under neutral drift.
$\displaystyle\textbf{Ex}^{(t)}_{C}$ $\displaystyle=$
$\displaystyle\textbf{Ex}^{(t-1)}_{C}+\frac{\textbf{Ex}^{(t-1)}_{C}}{N}-\frac{1}{N}\sum_{i\in
V}\sum_{(j,i)\in E}w_{ji}\cdot P_{i,t-1}$ (6)
Based on Equation 6, we notice that for $r=1$, at each time-step, the number
of expected mutants increases by at most the average fixation probability and
decreases by a quantity related to the average “temperature.” The temperature
of vertex $i$ (denoted $T_{i}$) is defined for a given node is the sum of the
incoming edge weights [14]: $T_{i}=\sum_{j}w_{ji}$. Intuitively, nodes with a
higher temperature change more often between being a mutant and being a
resident than those with lower temperature. Re-writing Equation 6 in terms of
temperature we have the following:
$\displaystyle\textbf{Ex}^{(t)}_{C}$ $\displaystyle=$
$\displaystyle\textbf{Ex}^{(t-1)}_{C}+\frac{\textbf{Ex}^{(t-1)}_{C}}{N}-\frac{1}{N}\sum_{i\in
V}T_{i}\cdot P_{i,t-1}$ (7)
Hence, if the preponderance of high temperature nodes are likely to be
mutants, then most likely the average number of mutants will decrease at the
next time step. We also note that Theorem 2, Equation 3, and Equation 6 do not
depend on the assumption that the underlying graph is strongly connected.
Therefore, as such is the case, we can study the relationship of time vs.
expected number of mutants for any evolutionary graph (under neutral drift).
This could be of particular interest to non-strongly connected evolutionary
graphs that may have trivial fixation probabilities (i.e. $1$ or $0$) but may
have varying levels of mutants before achieving an absorbing state.
## 4 Applying the Framework to Other Update Rules
The results of the last two sections not only apply to the original model of
[14], but several other related models in the literature. Viewing an
evolutionary graph problem as a stochastic process, where the states represent
different mutant-resident configurations, it is apparent that the original
model specifies the transition probabilities. However, there are other ways to
specify the transition probabilities known as update rules. Several works
address different update rules [1, 25, 15]. Overall, we have identified three
major families of update rules - birth-death (a.k.a. the invasion process)
where the node to reproduce is chosen first, death-birth (a.k.a. the voter
model) where the node to die is chosen first, and link dynamics, where an edge
is chosen. We summarize these in Table 1.
Table 1: Different families of update rules. Update Rule | Intuition
---|---
Birth-Death (BD) | (1) Node $i$ selected
(a.k.a. Invasion Process (IP)) | (2) neighbor of $i$, node $j$ selected
| (3) Offspring of $i$ replaces $j$
Death-Birth (DB) | (1) Node $i$ selected
(a.k.a. Voter Model (VM)) | (2) neighbor of $i$, node $j$ selected
| (3) Offspring of $j$ replaces $i$
Link Dynamics (LD) | (1) Edge $(i,j)$ selected
| (2) The offspring of one node in the
| edge replaces the other node
We have already shown how our methods can deal with the original model of
Lieberman et al., often referred to as the Birth-Death (BD) process. In this
section, we apply our methods to the neutral-drift (non-biased) cases of
death-birth and link-dynamics. In these models, the weights of the edges is
typically not considered. Hence, in order to align this work with the majority
of literature on those models, we will express vertex probabilities in terms
of node in-degree (${k_{\textit{in}}^{(i)}}$) and the set of directed edges
($E$). We note that these results can be easily extended to a more general
case with an edge-weight matrix as we used for the original model of EGT.
### 4.1 Death-Birth Updating
Under the death birth model (DB), at each time step, a vertex $i$ is selected
for death. With a death-bias (DB-D), it is selected proportional to the
inverse of its fitness, with a birth-bias (DB-B) it is selected with a
probability $1/N$, which is also the probability under neutral drift. Then, an
incoming neighbor ($j$) is selected either proportional to the fitness of all
incoming neighbors (birth-bias), or with a uniform probability (in the case
death-bias or neutral drift). The selected neighbor then replaces $i$. Here,
we compute $P_{i,t}$ under this dynamic with $r=1$.
$\displaystyle
P_{i,t}=(1-N^{-1})P_{i,t-1}+(N{k_{\textit{in}}^{(i)}})^{-1}\mathop{\sum}_{(j,i)\in
E}P_{j,t-1}$ (8)
We note that the proof of convergence still holds for death-birth - that is
for some time $t$, $\forall i$, the value $P_{i,t}$ is the same, then
$P_{i,t}=F_{C}$. Further, Theorem 4 holds for DB under neutral drift as well,
specifically, for disjoint sets $C,D\subseteq V$, $F_{C}+P_{D}=P_{C\cup D}$.
### 4.2 Link-Dynamics
With link dynamics (LD), at each time step an edge $(i,j)$ is selected either
proportional to the fitness of $i$ or the inverse of the fitness of $j$. It
has previously been shown that LD under birth bias is an equivalent process to
LD with a death bias [15]. Under neutral drift, the probability of edge
selection is $1/|E|$ (where $|E|$ is the cardinality of set $E$). Then, $i$
replaces $j$. Now, we compute $P_{i,t}$ under this dynamic with $r=1$.
$\displaystyle
P_{i,t}=(1-{k_{\textit{in}}^{(i)}}|E|^{-1})P_{i,t-1}+\frac{1}{|E|}\mathop{\sum}_{(j,i)\in
E}P_{j,t-1}$ (9)
Again, convergence and additivity of the fixation probability still hold under
link dynamics just as with BD and DB.
## 5 Bounding Fixation Probability for $r>1$
So far we have shown how our method can be used to find fixation probabilities
under the case of neutral drift. Here, we show how our framework can be useful
in the case of an advantageous mutant (when the value for $r$, the relative
fitness, is greater than $1$). First, we show that our method provides a lower
bound. We then provide an upper bound on the fixation probability that can be
used in conjunction with our framework when studying the case of the
advantageous mutant. We note that certain parts of these proofs are specific
for diffent update rules, and we identify them using the abbreviations from
the last section (DB-D, DB-B, and LD). The update of the original model of
[14] is known as the “birth-death” model and abbreviated BD. If the fitness
bias is on a birth event, we denote it as BD-B and if the bias is on a death
event we denote it as BD-D.
Naoki Masuda observes experimentally (through simulation) that the fixation
probability computed with neutral drift appears to be a lower bound on the
fixation probability for an advantageous mutant [15]. We were able to prove
this result analytically – the proof is included in the supplementary
materials.
###### Theorem 5
For a given set $C$, let $F^{(1)}_{C}$ be the fixation probability under
neutral drift and $F^{(r)}_{C}$ be the fixation probability calculated using a
mutant fitness $r>1$. Then, under BD-B, BD-D, DB-B, DB-D, or LD dynamics,
$F^{(1)}_{C}\leq F^{(r)}_{C}$.
This proof leads to the conjecture that $r^{\prime}>r$ implies
$F^{(r^{\prime})}_{C}\geq F^{(r)}_{C}$. However, we suspect that proving this
monotonicity property will require a different technique than used in Theorem
5. Next, to find an upper bound that corresponds with the lower bound above,
we use the proof technique introduced in [9], to obtain the following non-
trivial upper bounds of fixation probability for individual nodes in various
update rules.
$\displaystyle\mathbf{BD-B:}$ $\displaystyle F_{\\{i\\}}\leq$ $\displaystyle
r(r+\sum_{j}w_{ji})^{-1}$ (10) $\displaystyle\mathbf{BD-D:}$ $\displaystyle
F_{\\{i\\}}\leq$
$\displaystyle\left({\sum_{j}\frac{w_{ji}}{r-rw_{ji}+w_{ji}}}\right)^{-1}$
(11) $\displaystyle\mathbf{DB-B:}$ $\displaystyle F_{\\{i\\}}\leq$
$\displaystyle\sum_{j}rw_{ij}(1-w_{ij}+rw_{ij})^{-1}$ (12)
$\displaystyle\mathbf{DB-D:}$ $\displaystyle F_{\\{i\\}}\leq$ $\displaystyle
r\sum_{j}w_{ij}$ (13)
## 6 A Lower Bound for Mean Time to Fixation
Another important, although less-studied problem with respect to evolutionary
graph theory is the mean time to fixation - the average time it takes for a
mutant to take over the population. Closely related to this problem are mean
time to extinction (average time for the resident to take over) and mean time
to absorption (average time for either mutant or resident to take over). This
has been previously studied under the original Moran process for well mixed
populations [2] as well as some special cases of graphs [5]. However, to our
knowledge, a general method to compute these quantities (without resorting to
the use of simulation) have not been previously studied. Here we take a “first
step” toward developing such a method by showing how the techniques introduced
in this paper can be used to compute a non-trivial lower bound for mean time
to fixation (and easily modified to bound mean time to extinction and
absorption).
Let $F_{t|C}$ be the probability of fixation at time $t$. Therefore,
$F_{t|C}-F_{t-1|C}$ is the probability of entering fixation at time $t$. The
symbol $t_{C}$ is the mean time. By the results of [2], we have the following:
###### Theorem 6
$\displaystyle
t_{C}=\frac{1}{F_{C}}\sum_{t=1}^{\infty}t\cdot(F_{t|C}-F_{t-1|C})$
Our key intuition is noticing that at each time step $t$,
$F_{t|C}\leq\min_{i}P_{i,t}$. From this, we use the accounting method to
provide a rigorous proof for the following theorem that provides a non-trivial
lower-bound for the mean time to fixation. This result can be easily modified
for mean time to extinction and absorption as well.
###### Theorem 7
$\frac{1}{F_{C}}\sum_{t=1}^{\infty}t\cdot(P_{\min,t}-P_{\min,t-1})\leq\frac{1}{F_{C}}\sum_{t=1}^{\infty}t\cdot(F_{t|C}-F_{t-1|C})$
Where $P_{\min,t}=\min_{i}P_{i,t}$.
## 7 Algorithm and Experimental Evaluation
We leverage the finding of the previous sections in Algorithm 1. As described
earlier, our method has found the exact fixation probability when all the
probabilities in $\bigcup_{i}\\{P_{i,t}\\}$ (represented in the pseudo-code as
the vector $\mathbf{p}$) are equal. We use Equation 4 to provide a convergence
criteria based on value $\epsilon$, which we can prove to be the tolerance for
the fixation probability.
###### Proposition 2
Algorithm 1 returns the fixation probability $F_{C}$ within $\pm\epsilon$.
Algorithm 1 \- Our Novel Solution Method to Compute Fixation Probabilities
Input: Evolutionary Graph $\langle N,V,W\rangle$, configuration $C\subseteq
V$, natural number $R>0$, and real number $\epsilon\geq 0$.
Output: Estimate of fixation probability of mutant.
1: $p_{i}$ is the $i$th position in vector $p$ corresponding with vertex $i\in
V$.
2: Set $p_{i}=1$ if $i\in C$ and $p_{i}=0$ otherwise. {As per Proposition 1}
3: ${\mathbf{q}}\leftarrow{\mathbf{p}}$ {${\mathbf{q}}$ will be ${\mathbf{p}}$
from the previous time step.}
4: $\tau\leftarrow 1$
5: while $\tau>\epsilon$ do
6: for $i\in V$ {This loop carries out the calculation as per Equation 3} do
7: $sum\leftarrow 0$
8: ${\mathbf{m}}\leftarrow\\{j\in V|w_{ji}>0\\}$
9: for $j\in m$ do
10: $sum=sum+w_{ji}\cdot({\mathbf{q}}_{j}-{\mathbf{q}}_{i})$
11: end for
12: ${\mathbf{p}}_{i}\leftarrow{\mathbf{q}}_{i}+1/N\cdot sum$
13: end for
14: ${\mathbf{q}}\leftarrow{\mathbf{p}}$
15: $\tau\leftarrow(1/2)\cdot(\max p-\min p)$ {Ensures error bound based on
Equation 4}
16: end while
17: return $(\min p)+\tau$
Our novel method for computing fixation probabilities on strongly connected
directed graphs allows us to compute near-exact fixation probabilities within
a desired tolerance. The running time of the algorithm is highly dependent on
how fast the vertex probabilities converge. In this section we experimentally
evaluate how the vertex probabilities in our algorithms converge. We also
provide results from comparison experiments to support the claim that
Algorithm 1-ACC finds adequate fixation probabilities order of magnitudes
faster than Monte Carlo simulations. We also show how the algorithm can be
used to study the expected number of mutants as well as bound mean time to
fixation.
Figure 1: Left: Convergence of the minimum (MinP), maximum (MaxP), and average
(AvgP) of vertex probabilities towards the final fixation probability as a
function of our algorithm’s iterations t for a graph of 100 nodes. Right:
Average speedup (on a log scale) for finding fixation probabilities achieved
by our algorithm vs Monte Carlo simulation for graphs of different sizes.
Figure 2: Standard deviation of vertex probabilities as a function of our
algorithm’s iterations for the same 100 node graph of Figure 1 (left).
### 7.1 Convergence of Vertex Probabilities
We ran our algorithm to compute fixation probabilities on randomly weighted
and strongly connected directed graphs in order to experimentally evaluate the
convergence of the vertex probabilities. We generated the graphs to be scale-
free using the standard preferential attachment growth model [3] and randomly
assigned an initial mutant node. We replaced all edges in the graph given by
the growth model with two directed edges and then randomly assigned weights to
all the edges.
To compare Algorithm 1 with the Monte Carlo approach, we should set the
parameter $R$ in that algorithm to be comparable with $\epsilon$ in Algorithm
1. As $\epsilon$ is the provable error of a solution to Algorithm 1. Based on
the commonly-accepted definition of estimated standard error from statistics,
we can obtain the estimated standard error for the solution returned by Monte
Carlo approach with the following expression (where $R$ is the number of
simulation runs).
$\sqrt{\frac{F_{C}(1-F_{C})}{R-1}}$ (14)
We can use Equation 14 to estimate the parameter $R$ for the Monte Carlo
approach as follows. We set $\epsilon$ equal to the estimated standard error
as per Expression 14 and manipulate it algebraically. This gives us
$R\approx\frac{S(S-1)}{\epsilon^{2}}+1$ where $S$ is the solution to Algorithm
1, $\epsilon$ is the input parameter for Algorithm 1 and $R$ is the number of
simulation runs in the Monte Carlo approach that we estimate to provide a
comparable error bound. We also note, that as the vertex probabilities
converge, the standard deviation of the $p$ vector in Algorithm 1 could be a
potentially faster convergence criteria. Note that using standard deviation of
$p$ and returning the average vertex probability would no longer provide us of
the guarantee in Proposition 2, however it may provide good results in
practice. The modifications to the algorithm would be as follows: line 15
would be $\tau\leftarrow\textsf{st.dev}(p)$ and line 17 would be return
$\textsf{avg}(p)$. We will refer to this as Algorithm 1 with alternate
convergence criteria or Algorithm 1-ACC for short.
Figure 1 (left) shows the convergence of the minimum, maximum, and the average
of vertex probabilities towards the final fixation probability value for a
small graph of 100 nodes. We can observe that the average converges to the
final value at a logarithmic rate and much faster than the minimum and maximum
vertex probability values. This suggests that while Algorithm 1-ACC does not
give the same theoretical guarantees as Algorithm 1, it is much preferable for
speed since the minimum and maximum vertex probabilities take much longer to
converge to the final solution than the average. The fact that the average of
the vertex probabilities is much preferable as a fast estimation of fixation
probabilities is supported by the logarithmic decrease of the standard
deviation of vertex probabilities (see Figure 2). Convergences for other and
larger graphs are not shown here but are qualitatively similar to the relative
convergences shown in the provided graphs.
### 7.2 Speed Comparison to Monte Carlo Simulation
In order to compare our method’s speed compared to the standard Monte Carlo
simulation method, we must determine how many iterations our algorithm must be
run to find a fixation probability estimate comparable to that of the Monte
Carlo approach. Thankfully, as we have seen, we can get a standard error on
the fixation probability returned by the Monte Carlo approach as per Equation
14. While we did not theoretically prove anything about how smoothly fixation
probabilities from our methods approach the final solution, the convergences
of the average and standard deviation as shown above strongly suggest that
estimates from our method approach the final solution quite gracefully. In
fact, in the following experiments, once our method has arrived at a fixation
probability estimate within the standard error of simulations, the estimate
never again fell outside the window of standard error (although the estimate
did not always approach the final estimate monotonically). This is in stark
contrast to Monte Carlo simulations, from which estimations can vary greatly
before the method has completed enough single runs to achieve a good
probability estimate.
We generated a number of randomly weighted and strongly connected directed
graphs of various sizes on which we compare our solution method to Monte Carlo
approximation of fixation probabilities. The graphs were generated as in our
convergence experiments. For each graph of a different size, we generated a
number of different initial mutant configurations. We found fixation
probabilities both using Monte Carlo estimation with 2000 simulation runs and
our direct solution method, terminating when we have reached within the
standard error of the Monte Carlo estimation. Since the average vertex
probability proved to be such a good fast estimate of the true fixation
probability, we used Algorithm 1-ACC.
Figure 1 (right) shows the speedup our solution provides over Monte Carlo
simulation. Here speedup is defined as the ratio of the time it takes for
simulations to complete over the time it takes our algorithm to find a
fixation probability within the standard deviation. The often extremely low
number of iterations needed by our algorithm to find fixation probabilities
within the standard error of simulations may prompt the concern that the
probabilities fall within this window so soon by mere chance. However, our
experiments have shown that the fixation probability estimation found by our
algorithm at each iteration approaches the final fixation probability after
termination smooothly at a logarithmic rate, asymptotically approaching the
true fixation probability. While in this case the fixation probability
estimate slightly crosses over the true fixation probability and then slowly
approaches it again, none of the fixation probability estimates from our
algorithm exited the window of standard error (from simulations) once they
entered it.
We can observe from our experiments that computing fixation probabilities
using Monte Carlo simulations showed to be a very time-expensive process,
highlighting the need for faster solution methods as the one we have
presented. Especially for larger graph sizes, the time complexity of our
solution to achieve similar results to Monte Carlo simulation has shown to be
orders of magnitude smaller than the standard method.
### 7.3 Monitoring the Expected Number of Mutants
As observed in Section 3, our method not only allows for the calculation of
the fixation probability of a mutant, but also allows us to study how the
expected number of mutants change over time. In this section, we present
experimental results exploring the trajectory of the expected number of
mutants over time on various undirected/unweighted graphs and under different
initial mutant placement conditions.
First, we note that the expected number of mutants (as time approaches
infinity) in an unweighted/undirected graph with respect to a single initially
infected vertex $i$ can be computed by modifying the result of [6] (for BD
updating) to obtain the following.
$\mathop{\mathit{lim}}_{t\rightarrow\infty}\textbf{Ex}^{(t)}_{\\{i\\}}=\frac{1}{k_{i}\langle
k^{-1}\rangle}$ (15)
Where $\langle k^{-1}\rangle$ is the average inverse of the degree for the
graph. Hence, we can determine whether a node amplifies or suppresses
selection by observing if
$\mathop{\mathit{lim}}_{t\rightarrow\infty}\textbf{Ex}^{(t)}_{\\{i\\}}$ is
greater or less than $1$ respectively: if $k_{i}<\frac{1}{\langle
k^{-1}\rangle}$ selection is amplified and if $k_{i}>\frac{1}{\langle
k^{-1}\rangle}$ it is suppressed. We have used our algorithm to compare the
trajectory of the expected number of mutants over time when the initial mutant
is placed on amplifiers vs. suppressors under different graph topologies and
BD updating. We note that similar comparisons can be obtained with our
algorithm for the other update rules. We also note that by Theorem 5, an
amplifier for BD (with no bias) will also be an amplifier for the (biased)
BD-B and BD-D where $r>1$.
Figure 3: Expected number of mutants over time starting with a single mutant
placed on a graph for Barabási-Albert preferential attachment (BAR), Erdős-
Rényi (ERD), and Newmann-Watts-Strogatz small world (NWS) graphs. Lines are
averages over 50 random graphs of each type. In the left graph, mutants are
placed at the highest degree nodes, which are suppressors. In the right graph,
mutants are placed at lowest degree nodes, which are amplifiers. Figure 4:
Expected number of mutants over time for an Erdős-Rényi graph of 100 nodes,
with an extra muntant node ($mn$) and resident node($rn$) with directed edges
to $con\\_mn$ and $con\\_rn$ respectively. The value that the expected number
of mutants converges to depends on the relative degrees of $con\\_mn$ and
$con\\_rn$, as shown in the legend.
Figure 3 shows the trajectories of the expected number of mutants over time on
random [3] preferential attachment (BAR), [10] (ER), and [19] small world
graphs (NWS), each for when the initial mutant is placed on a suppressor
(highest degree node of graph) and amplifier (lowest degree node of graph).
Graphs are all of equal size at 100 nodes. We note that the highest degree
nodes are especially strong suppressors on BAR graphs, less so for NWS graphs,
and even less so for ER graphs. This makes sense when one considers the degree
distribution of the different graph topologies, which are scale-free or power-
law ($P(k)\sim{k}^{-3}$) for BR, roughly Poisson-shaped for NWS, and
relatively uniform for ER graphs. For lowest degree amplifiers, the expected
number of mutants grows faster early on in Barabási-Albert graphs, but it
plateaus earlier than and is eventually surpassed by the slower growing
expected number of mutants in the Erdős-Rényi, and Newmann-Watts-Strogatz
graphs. Such insights into the evolutionary process may be crucial in
applications, e.g. when one may be more interested in achieving highest number
of mutants in a short amount of time rather than highest number of mutants as
$t\to\infty$ or vice versa.
Finally, thus far we have only considered strongly connected graphs in which
the vertex probabilities converge as $t\to\infty$, but this is not the case
for some non-strongly connected graphs. We have thus also investigated the
expected number of mutants over time for some simple cases of such graphs.
Consider a random graph that is strongly connected, and then have a resident
node ($rn$) and mutant node ($mn$) connected with only directed edges into the
strongly connected graph. Clearly, the vertex probabilities cannot converge,
since $\forall\,t,P_{mn,t}=1$ and $P_{rn,t}=0$. Our experimental results in
Figure 4 show however that while the vertex probabilities do not converge, the
value for the expected number of mutants given by our algorithm seems to
converge. What value the expected number of mutants converges to depends on
the relative degrees of the nodes that the mutant node $mn$ and resident node
$rn$ connect to. We shall call these nodes $con\\_mn$ and $con\\_rn$,
respectively. If $k_{con\\_mn}\approx k_{con\\_rn}$, the expected value of
mutants converges at around 50% of the graph’s nodes. If
$k_{con\\_mn}>k_{con\\_rn}$, the expected value of mutants is less than 50% of
the graph’s nodes, and conversely, if $k_{con\\_mn}<k_{con\\_rn}$, it is
greater. These results are intuitive because lower degree nodes are better
spreaders under BD updating. These results are also interesting because the
expected value converges - even though the graphs are not strongly connected.
By an examination of Equation 6, this convergence is possible. However, we
have not proven that convergence always occurs. An interesting direction for
future work is to identify under what conditions will the expected number of
mutants converges in a non-strongly connected graph.
### 7.4 Experimentally Computing the Lower Bound of the Mean Time to Fixation
We also performed experiments to examine the lower bound on mean time to
fixation (discussed in Section 6) as compared to the average fixation time
determined from simulation run. In doing so, we were able to confirm the
lower-bound experimentally. We were able to use Algorithm 1-ACC to compute the
lower bound with a few changes (noted in the supplement).
We generated random (ER) graphs of size $10,20,50$ and $100$ nodes, creating
five different graphs for each number of nodes. The graphs were generated as
in our convergence experiments, and our comparison to Monte Carlo testing are
shown in Figure 5 where we demonstrate experimentally that our algorithm
produces a lower bound. Our algorithm was run until the standard deviation of
fixation probabilities for all vertices was $2.5\times 10^{-6}$. The Monte
Carlo simulations were each set at $10,000$ runs.
Figure 5: Mean-time-to-fixation comparison between algorithm and simulation.
Note that the y-axis is a logarithmic scale.
## 8 Application: Competition Among Neural Axons
In recent work, [26] created a model for synaptic competition based on death-
birth updating under neutral drift. They noted that the model aligns well with
their empirical observations. In the model, the graph represents a synaptic
junction and the nodes represent sites in the junction. For every two adjacent
sites in the synaptic junction, there is an undirected edge between the
corresponding two nodes in the graph. Hence, in- and out- degrees of each node
are the same. Initially, there are $K$ different axon types located in the
junction configured in a manner where all sites are initially occupied by one
axon type. At each time step, an axon occupying one of the sites is eliminated
- making the site open. The selection of the axon for elimination (death) is
with a uniform probability. Hence, there is no bias in this model. Following
the elimination of an axon, an adjacent axon grows into the site. The adjacent
axon is selected with a uniform probability of the eliminated axon’s
neighbors. Hence, based on the results of this paper, we can provide the
following insights into synaptic competition.
1. 1.
After $t$ axons are eliminated,111Note that the number of axons eliminated
corresponds directly to the number of timesteps in the model. the probability
of any site being occupied by an axon of a certain type can be calculated
directly by Theorem 8. Even though there are $K$ axon types, this theorem
still applies as it only considers the probability of a node being a mutant
(resp. a site being a one of the $K$ axon types).
2. 2.
Using point 1 above, we can determine the expected number of axons of a given
type after $t$ axons being eliminated.
3. 3.
After $t$ axons are eliminated, the probability of any set of sites being
occupied by a certain axon type is simply the sum of the probabilities of the
individual sites being occupied by that axon. As a result, the fixation
probability is additive.
4. 4.
Leveraging point 3 above combined with an easy modification of the result of
Broom et al. [6] for the BD model, the fixation probability of an axon
originating at site $i$ is $\frac{k_{i}}{2\cdot\Theta}$ where $k_{i}$ is the
number of sites adjacent to site $i$ (hence the degree of node $i$ in the
corresponding graph) and $\Theta$ is the total number of adjacencies in the
synapse (hence, half the number of directed edges in the corresponding graph).
5. 5.
Based on item 4 above and the results from Section 3, we can conclude that for
a given axon type (let’s call it “axon type A”) occupying a set of sites, that
if the average adjacencies of those sites is greater than (resp. less than)
the overall average adjacencies for the sites in the entire synaptic junction,
then as the number of eliminated axons approaches infinity, we can expect the
number of axon type A in the synaptic junction will increase (resp. decrease)
in expectation.
6. 6.
We can directly apply Theorem 7 to find a lower-bound on the number of
eliminated axons before fixation occurs.
We note that the results stated above are either precise mathematical
arguments or calculations that can be found exactly with a deterministic
algorithm. They are not theoretical approximations and do not rely on
simulation. As such is the case, we can make more precise statements about
synaptic competition (given the model) and can avoid the variance that
accompanies simulation results. Insights such as these may lead to future
biological experiments.
## 9 Related Work
Evolutionary graph theory was originally introduced in [14]. Previously, we
have compiled a comprehensive review [24] for a general overview of the work
in this exciting new area.
While most work dealing with evolutionary graphs rely on Monte Carlo
simulation, there are some good analytical approximations for the
undirected/unweighted cased based on the degree of the vertices in question.
Antal et al. [1] use the mean-field approach to create these approximations
for the undirected/unweighted case. Broom et al. [6] derive an exact
analytical result for the undirected/unweighted case in neutral drift, which
agrees with the results of Antal et al. They also show that fixation
probability is additive in that case (a result which we extend in this paper
using a different proof technique). However, the results of [16] demonstrate
that mean-field approximations break down in the case of weighted, directed
graphs. [15] also studied weighted, directed graphs, but does so by using
Monte Carlo simulation. [22] derive exact computation of fixation probability
through means of linear programming. However, that approach requires an
exponential number of both constraints and variables and is intractable. The
recent work of [27] introduces a parameter called graph determinacy which
measures the degree to which fixation or extinction is determined while
starting from a randomly choses initial configuration. This property is then
used to analyze some special cases of evolutionary graphs under birth-death
updating. There has been some work on algorithms for fixation probability
calculation that rely on a randomized approach [4, 9]. [4] present a heuristic
technique for speeding up Monte Carlo simulations by early termination while
[9] present utilize simulation runs in a fully-polynomial randomized
approximation scheme. However, our framework differs in that it does not rely
on simulation at all and provides a deterministic result. Further, our non-
randomized approach also allows for additional insights into the evolutionary
process - such as monitoring the expected number of mutants as a function of
time. Recently, [12] study the related problem of determining the probability
of fixation given a single, randomly placed mutant in the graph where the
vertices are “islands” and there are many individuals residing on each island
in a well-mixed population. They use quasi-fixed points of ODE’s to obtain an
approximation of the fixation probability and performed experiments with a
maximum of $5$ islands (vertices) containing $50$ individuals each. This
continuous approximation provides the best results when the number of
individuals in each island is much larger than the number of islands. As the
problem of this paper can be thought of as a special case where each island
has just one individual, it seems unlikely that the approximation of
Houchmandzadeh and Vallade’s approach will hold here.
Some of the results in this paper were previously presented in conferences by
the authors [23, 17]. The analysis and experiments concerning the expected
number of mutants at a given time, the extension of the framework for other
update rules (beyond birth-death), the use of the framework for the case of
$r>1$, and the neurology applications are all new results appearing for the
first time in this paper.
## 10 Conclusion
In this paper, we introduced a new approach to deal with problems relating to
evolutionary graphs that rely on “vertex probabilities.” Our presented
analytical method is the first deterministic method to compute fixation
probability and provides a number of novel uses and results for EGT problems:
* 1.
Our method can be used to solve for the fixation probability under neutral
drift orders of magnitude faster than Monte Carlo simulations, which is
currently the presiding employed method in EGT studies. We have extended the
method to all of the commonly used update processes in EGT. The special case
of neutral drift is not only of interest in the literature [6, 15] but also it
has been applied to problems in neurology [26].
* 2.
While the presented method is currently constrained to the case of neutral
drift, we have demonstrated how it can inform cases of non-neutral drift by
using it to provide both a lower and upper bound for this case. Combined with
our analytical method’s speed, this means that it can be used to acquire
useful knowledge to guide general EGT studies interested in the case of
advantageous mutants.
* 3.
We have shown how our analytical method can be used to calculate a non-trivial
lower bound to the mean time to fixation, providing a first step for a general
method to computing this and related quantities that is lacking in the current
literature.
* 4.
We have shown how our method can be used to calculate deterministically the
expected number of mutants, which is useful for applications that require
predictions on the number of mutants in the population under a specific finite
time horizon. We have also provided results on the expected number of mutants
on different common graph topologies, showing differences in the growth
trajectories of amplifiers and suppressors on these different topologies.
These results may prove highly significant in the recent application of EGT to
distributed systems [13] where the problem of information diffusion is
considered among computer systems. In such a domain, it may be insufficient to
guarantee fixation in the limit of time - which may be impractical - but
rather to make guarantees on the outcome of the process after a finite amount
of time.
* 5.
Finally, we have shown how our method can provide insight when applied to the
problem of synaptic competition in neuroscience.
Though evolutionary graph theory is still a relatively new research area, it
is actively being studied in a variety of disciplines [14, 24, 28, 25, 20, 26,
13]. We believe that more real-world applications will appear as this area
gains more popularity. As illustrated by recent work [26, 13], experimental
scientists with knowledge of EGT may be more likely to recognize situations
where the model may be appropriate. As these cases arise, deterministic
methods for addressing issues related to EGT may prove to be highly useful.
However, this paper is only a starting point - there are still many important
directions for future work. Foremost among such topics are scenarios where the
topology of the graph also changes over time or where additional attributes of
the nodes/edges in the graph affect the dynamics.
## Acknowledgments
P.S. is supported by ARO projects 611102B74F and 2GDATXR042 as well as OSD
project F1AF262025G001. P.R. is supported by ONR grant W911NF0810144. P.S.
would like to thank Stephen Turney (Harvard University) for several
discussions concerning his work. The opinions in this paper are those of the
authors and do not necessarily reflect the opinions of the funders, the U.S.
Military Academy, the U.S. Army, or the U.S. Navy.
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## Supplementary Material
## 11 Notes
Throughout this supplement, we will use an extended notation. Fixation
probability given initial configuration $C$ is denoted $F_{C}$. For vertex $i$
at time $t$, we denote this as $\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$. We will
use $\textbf{{S}}^{(t)}_{i}$ to denote the event that vertex $i$ was selected
for reproduction and $\textbf{{R}}^{(t)}_{ij}$ to denote the event of $i$
replacing $j$. We will often use conditional probabilities. For example,
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|C^{(0)})$ is the probability that $v_{i}$
is a mutant given the initial set $C$ of mutants. Throughout this supplement,
unless noted otherwise, all of our probabilities will be conditioned on
$C^{(0)}$. We will drop it for ease of notation with the understanding that
some set $C$ of $V$ were mutants at $t=0$. Hence,
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|C^{(0)})$.
## 12 Proof of Theorem 1
$\displaystyle\forall i,$
$\displaystyle\mathop{\mathit{lim}}_{t\rightarrow\infty}\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|C^{(0)})=F_{C}$
Proof. Consider the following definition property of
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|C^{(0)})$
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|C^{(0)})=\mathop{\sum_{V^{\prime}\in
2^{V}}}_{\textit{s.t. }v_{i}\in V^{\prime}}\textbf{Pr}(V^{\prime(t)}|C^{(0)})$
(16)
We note that as time approaches infinity, for all $V^{\prime}\in
2^{V}-\emptyset-V$ we have $\textbf{Pr}(V^{\prime(t)}|C^{(0)})=0$. As
$v_{i}\notin\emptyset$, the statement follows. Q.E.D.
## 13 Proof of Theroem 2
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})+\sum_{(v_{j},v_{i})\in
E}w_{ji}\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{j})-w_{ji}\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{i})$
Where $\textbf{{S}}^{(t)}_{i}$ is true iff $v_{i}$ is selected for
reproduction at time $t$.
Proof. Note we use the variable $\textbf{{R}}^{(t)}_{j}i$ is true iff $v_{j}$
replaces $v_{i}$ at time $t$.
CLAIM 1:
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$ $\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}\wedge\mathop{\bigwedge}_{(v_{j},v_{i})\in
E}\neg\textbf{{S}}^{(t)}_{j})+\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{S}}^{(t)}_{j}\wedge\textbf{{R}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{j})+$
$\displaystyle\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{S}}^{(t)}_{j}\wedge\neg\textbf{{R}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{i})$
This is shown by a simple examination of exhaustive and mutually exclusive
events based on the original model of [14].
CLAIM 2:
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}\wedge\mathop{\bigwedge}_{(v_{j},v_{i})\in
E}\neg\textbf{{S}}^{(t)}_{j})$ $\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\cdot\left(1-\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{i})\right)$
(Proof of claim 2) By exhaustive and mutual exclusive events, we have the
following.
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}\wedge\mathop{\bigwedge}_{(v_{j},v_{i})\in
E}\neg\textbf{{S}}^{(t)}_{j})$ $\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})-\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{S}}^{(t)}_{j}\wedge\textbf{{M}}^{(t-1)}_{i})$
By the definition of conditional probability, we have the following
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}\wedge\mathop{\bigwedge}_{(v_{j},v_{i})\in
E}\neg\textbf{{S}}^{(t)}_{j})$ $\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})-\sum_{(v_{j},v_{i})\in
E}\left(\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{i})\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\right)$
$\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})-\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\cdot\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{i})$
$\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\left(1-\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{i})\right)$
The claim immediately follows.
CLAIM 3: For all edges $(v_{j},v_{i})$, we have the following.
$\displaystyle\textbf{Pr}(\textbf{{S}}^{(t)}_{j}\wedge\textbf{{R}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{j})$
$\displaystyle=$ $\displaystyle
w_{ji}\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{j})$
(Proof of claim 3) The following is a direct application of the definition of
conditional probability.
$\displaystyle\textbf{Pr}(\textbf{{S}}^{(t)}_{j}\wedge\textbf{{R}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{j})$
$\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{R}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{j}|\textbf{{S}}^{(t)}_{j})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j})$
From our model, we note that given even $\textbf{{S}}^{(t)}_{j}$, the fitness
of the nodes is not considered in determining if the event associated with
$\textbf{{R}}^{(t)}_{ji}$ is to occur. Hence, it follows that
$\textbf{{M}}^{(t-1)}_{j}$ is independent of $\textbf{{R}}^{(t)}_{ji}$ given
$\textbf{{S}}^{(t)}_{j}$. As
$\textbf{Pr}(\textbf{{R}}^{(t)}_{ji}|\textbf{{S}}^{(t)}_{j})=w_{ji}$, we have
the following.
$\displaystyle\textbf{Pr}(\textbf{{S}}^{(t)}_{j}\wedge\textbf{{R}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{j})$
$\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{R}}^{(t)}_{ji}|\textbf{{S}}^{(t)}_{j})\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j}|\textbf{{S}}^{(t)}_{j})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j})$
$\displaystyle=$ $\displaystyle
w_{ji}\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j}|\textbf{{S}}^{(t)}_{j})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j})$
By Bayes Theorem, and that the model causes $\forall
i\textbf{Pr}(\textbf{{S}}^{(t)}_{i})>0$, we have the following.
$\displaystyle\textbf{Pr}(\textbf{{S}}^{(t)}_{j}\wedge\textbf{{R}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{j})$
$\displaystyle=$ $\displaystyle
w_{ji}\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{j})\cdot\frac{\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})}{\textbf{Pr}(\textbf{{S}}^{(t)}_{j})}\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j})$
$\displaystyle=$ $\displaystyle
w_{ji}\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{j})$
The claim follows immediately.
CLAIM 4: For all edges $(v_{j},v_{i})$, we have the following.
$\displaystyle\textbf{Pr}(\textbf{{S}}^{(t)}_{j}\wedge\neg\textbf{{R}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{i})$
$\displaystyle=$
$\displaystyle(1-w_{ji})\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{i})$
(Proof of claim 4) This mirrors claim 3.
(Proof of theorem) From claims 1-4, we have the following.
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$ $\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\cdot\left(1-\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{i})\right)+$ (19)
$\displaystyle\sum_{(v_{j},v_{i})\in
E}\left(w_{ji}\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{j})\right)+$
$\displaystyle\sum_{(v_{j},v_{i})\in
E}\left((1-w_{ji})\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\cdot\textbf{Pr}(\textbf{{S}}^{(t)}_{j}|\textbf{{M}}^{(t-1)}_{i})\right)$
Which, after re-arranging some terms, gives us the statement of the theorem.
Q.E.D.
## 14 Proof of Theorem 3
When $r=1$, if for some time $t$, $\forall i$, the value
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$ is the same, then
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=F_{C}$.
Proof Sketch. Consider
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})+\frac{1}{N}\sum_{(v_{j},v_{i})\in
E}w_{ji}\cdot(\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})-\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}))$
when for $t-1$, $\forall i,j$ we have
$\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})=\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$.
Clearly, in this case, the value for
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$.
As the probabilities of all vertices was the same at $t-1$, they remain so at
$t$. Therefore, in this case,
$\mathop{\mathit{lim}}_{t\rightarrow\infty}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$.
QED
## 15 Proof of Inequality 4
For any time $t$, under neutral drift ($r=1$),
$\min_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})\leq
F_{C}\leq\max_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$
Proof. PART 1: For any time $t$, under neutral drift ($r=1$),
$F_{C}\leq\max_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$.
We show that for each time step $t$,
$\max_{i}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\geq\max_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$.
Hence, by showing that, for any time $t^{\prime}$, we have
$\max_{i}\textbf{Pr}(\textbf{{M}}^{(t^{\prime})}_{i})\geq\mathit{lim}_{t\rightarrow\infty}\max_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$
which by allows us to apply Theorem 1 and obtain the statement of this
theorem. Suppose BWOC that at time $t$ we have
$\max_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})<\max_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$.
Then we have:
$\displaystyle\max_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})$
$\displaystyle<$ $\displaystyle\frac{1}{N}\sum_{(v_{j},v_{i})\in
E}w_{ji}\cdot\left(\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})-\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\right)$
$\displaystyle+\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$ $\displaystyle\leq$
$\displaystyle\frac{1}{N}\sum_{(v_{j},v_{i})\in
E}w_{ji}\cdot\left(\max_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})-\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\right)$
$\displaystyle+\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$ $\displaystyle=$
$\displaystyle\frac{\sum_{(v_{j},v_{i})\in
E}w_{ji}}{N}\left(\max_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})-\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\right)$
$\displaystyle+\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$
$\displaystyle\max_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})(1-\frac{\sum_{(v_{j},v_{i})\in
E}w_{ji}}{N})$ $\displaystyle<$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})(1-\frac{\sum_{(v_{j},v_{i})\in
E}w_{ji}}{N})$
$\displaystyle\max_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})$
$\displaystyle<$ $\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$
Which is clearly a contradiction and completes this part of the proof.
PART 2: For any time $t$, under neutral drift ($r=1$),
$F_{C}\geq\min_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$.
We show that for each time step $t$,
$\min_{i}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\leq\min_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$.
Hence, by showing that, for any time $t^{\prime}$, we have
$\min_{i}\textbf{Pr}(\textbf{{M}}^{(t^{\prime})}_{i})\leq\mathit{lim}_{t\rightarrow\infty}\max_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$
which by allows us to apply Theorem 1 and obtain the statement of this
theorerm. Suppose BWOC that at time $t$ we have
$\min_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})>\min_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$.
Then we have:
$\displaystyle\min_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})$
$\displaystyle>$ $\displaystyle\frac{1}{N}\sum_{(v_{j},v_{i})\in
E}w_{ji}\cdot\left(\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})-\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\right)$
$\displaystyle+\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$ $\displaystyle\geq$
$\displaystyle\frac{1}{N}\sum_{(v_{j},v_{i})\in
E}w_{ji}\cdot\left(\min_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})-\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\right)$
$\displaystyle+\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$ $\displaystyle=$
$\displaystyle\frac{\sum_{(v_{j},v_{i})\in
E}w_{ji}}{N}\left(\min_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})-\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})\right)$
$\displaystyle+\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$
$\displaystyle\min_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})(1-\frac{\sum_{(v_{j},v_{i})\in
E}w_{ji}}{N})$ $\displaystyle>$
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})(1-\frac{\sum_{(v_{j},v_{i})\in
E}w_{ji}}{N})$
$\displaystyle\min_{\ell}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{\ell})$
$\displaystyle>$ $\displaystyle\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$
Which is clearly a contradiction and completes this part of the proof. Q.E.D.
## 16 Proof of Theorem Theorem 4
When $r=1$ for disjoint sets $C,D\subseteq V$, $F_{C}+F_{D}=F_{C\cup D}$.
Proof. Consider some time $t$ and vertex $v_{i}$. Clearly, by Corollary 1,
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$ can be expressed as a linear combination
of the form $\sum_{v_{j}\in V}(C_{j}\cdot\textbf{Pr}(\textbf{{M}}^{(0)}_{j}))$
where $C_{j}$ is a coefficient. We note that these coefficients are the same
regardless of the initial configuration of mutants that
$\textbf{{M}}^{(t)}_{i}$ is conditioned on. Hence,
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|C^{(0)})$ is this positive function with
$\textbf{Pr}(\textbf{{M}}^{(0)}_{j})=1$ if $v_{j}\in C$ and $0$ otherwise (see
Proposition 3). Hence, for disjoint $C,D$, for any $v_{i}\in V$, we have
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|C^{(0)})+\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|D^{(0)})=\textbf{Pr}(\textbf{{M}}^{(t)}_{i}|(C\cup
D)^{(0)})$. The statement follows. Q.E.D.
## 17 Proof of Equation 9
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=\left(1-\frac{1}{N}\right)\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})+\frac{1}{N\cdot{k_{\textit{in}}^{(i)}}}\mathop{\sum}_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})$
Under death-birth dynamics with neutral drift ($r=1$).
Proof. $\textbf{{D}}^{(t)}_{i}$ and $\textbf{{B}}^{(t)}_{i}$ are random
variables associated with birth and death events for vertex $v_{i}$.
CLAIM 1:
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}\wedge\neg\textbf{{D}}^{(t)}_{i})+\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{D}}^{(t)}_{i}\wedge\textbf{{B}}^{(t)}_{j}\wedge\textbf{{M}}^{(t-1)}_{j})$
Follows directly form exhaustive and mutually exclusive events.
CLAIM 2:
$\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}\wedge\neg\textbf{{D}}^{(t)}_{i})=\left(1-\frac{1}{N}\right)\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$
By the definition of conditional probabilities, we have
$\textbf{Pr}(\neg\textbf{{D}}^{(t)}_{i}|\textbf{{M}}^{(t-1)}_{i})\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$.
Also, we know the probability of a given node dying is always $1/N$. Hence,
$\textbf{Pr}(\neg\textbf{{D}}^{(t)}_{i}|\textbf{{M}}^{(t-1)}_{i})=\textbf{Pr}(\neg\textbf{{D}}^{(t)}_{i})=1-\frac{1}{N}$
and the claim follows.
CLAIM 3: For any $(v_{j},v_{i})\in E$, we have
$\textbf{Pr}(\textbf{{D}}^{(t)}_{i}\wedge\textbf{{B}}^{(t)}_{j}\wedge\textbf{{M}}^{(t-1)}_{j})=\frac{1}{N\cdot{k_{\textit{in}}^{(i)}}}\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})$
As both birth and death events occur independent of any node being a mutant at
the previous time step, the definition of conditional probabilities gives us
the following:
$\displaystyle\textbf{Pr}(\textbf{{D}}^{(t)}_{i}\wedge\textbf{{B}}^{(t)}_{j}\wedge\textbf{{M}}^{(t-1)}_{j})$
$\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{D}}^{(t)}_{i}\wedge\textbf{{B}}^{(t)}_{j})\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})$
(20) $\displaystyle=$
$\displaystyle\textbf{Pr}(\textbf{{B}}^{(t)}_{j}|\textbf{{D}}^{(t)}_{i})\cdot\textbf{Pr}(\textbf{{D}}^{(t)}_{i})\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})$
(21)
From the model, we have the following:
$\displaystyle\textbf{Pr}(\textbf{{B}}^{(t)}_{j}|\textbf{{D}}^{(t)}_{i})$
$\displaystyle=$ $\displaystyle 1/{k_{\textit{in}}^{(i)}}$ (22)
$\displaystyle\textbf{Pr}(\textbf{{D}}^{(t)}_{i})$ $\displaystyle=$
$\displaystyle 1/N$ (23)
Hence, the claim follows. QED Q.E.D.
## 18 Proof of Equation 10
$\displaystyle\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=\left(1-\frac{{k_{\textit{in}}^{(i)}}}{|E|}\right)\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})+\frac{1}{|E|}\mathop{\sum}_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})$
Under link dynamics with neutral drift ($r=1$).
Proof. Here $\textbf{{S}}^{(t)}_{ij}$ is the random variable associated with
the selection of edge $(v_{i},v_{j})$.
CLAIM 1:
$\textbf{Pr}(\textbf{{M}}^{(t)}_{i})=\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}\wedge\bigwedge_{(v_{j},v_{i})\in
E}\neg\textbf{{S}}^{(t)}_{ji})+\sum_{(v_{j},v_{i})\in
E}\textbf{Pr}(\textbf{{S}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{j})$
Follows directly form exhaustive and mutually exclusive events.
CLAIM 2: $\textbf{Pr}(\bigwedge_{(v_{j},v_{i})\in
E}\neg\textbf{{S}}^{(t)}_{ji})=1-\frac{{k_{\textit{in}}^{(i)}}}{|E|}$
Clearly, we have $\textbf{Pr}(\bigwedge_{(v_{j},v_{i})\in
E}\neg\textbf{{S}}^{(t)}_{ji})=\textbf{Pr}(\bigvee_{\\{(v_{\beta},v_{\alpha})\in
E|\beta\neq i\\}}\textbf{{S}}^{(t)}_{\beta\alpha})$. As there are
${k_{\textit{in}}^{(i)}}$ incoming edges to $v_{i}$, we know that
$\textbf{Pr}(\bigvee_{\\{(v_{\beta},v_{\alpha})\in E|\beta\neq
i\\}}\textbf{{S}}^{(t)}_{\beta\alpha})=1-\frac{{k_{\textit{in}}^{(i)}}}{|E|}$,
giving us the claim.
CLAIM 3:
$\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i}\wedge\bigwedge_{(v_{j},v_{i})\in
E}\neg\textbf{{S}}^{(t)}_{ji})=\left(1-\frac{{k_{\textit{in}}^{(i)}}}{|E|}\right)\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{i})$
For any $\alpha,\beta$, the random variable $\textbf{{S}}^{(t)}_{\alpha\beta}$
is independent from $\textbf{{M}}^{(t-1)}_{i}$. Hence, the claim immediately
follows from this fact and claim 2.
CLAIM 4:
$\textbf{Pr}(\textbf{{S}}^{(t)}_{ji}\wedge\textbf{{M}}^{(t-1)}_{j})=\frac{1}{|E|}\cdot\textbf{Pr}(\textbf{{M}}^{(t-1)}_{j})$
As, by the definition of the model,
$\textbf{Pr}(\textbf{{S}}^{(t)}_{ji}|\textbf{{M}}^{(t-1)}_{j})=\textbf{Pr}(\textbf{{S}}^{(t)}_{ji})=\frac{1}{|E|}$,
the claim follows directly form the definition of conditional probabilities.
QED Q.E.D.
## 19 Proof of Theorem 5
For a given set $C$, let $F^{(1)}(C)$ be the fixation probability under
neutral drift and $F^{(r)}(C)$ be the fixation probability calculated using a
mutant fitness $r>1$. Then, under BD-B, BD-D, DB-B, DB-D, or LD dynamics,
$F^{(1)}(C)\leq F^{(r)}(C)$.
Proof. First, some notation.
* 1.
We define an interpretation, $\textbf{I}:2^{V}\rightarrow[0,1]$ as probability
distribution over mutant configurations. Hence, for some I we have
$\sum_{V^{\prime}\in 2^{V}}\textbf{I}(V^{\prime})=1$.
* 2.
Next, we define a transition function that maps configurations of mutants to
probabilities, $\chi:2^{V}\rightarrow[0,1]$ where for any $C\in 2^{V}$,
$\sum_{C^{\prime}\in 2^{V}}\chi(C,C^{\prime})=1$. We will use $\chi_{{+}}$ and
$\chi_{{-}}$ to indicate if the transition is made with a mutant being
selected for birth ($\chi_{{+}}$) or resident ($\chi_{{-}}$). Hence, for some
$C\in V$ and $v\notin C$, $\chi_{{-}}(C,C\cup\\{v\\})=0$ and
$\chi_{{+}}(C\cup\\{v\\},C)=0$. Hence, for all $C\in 2^{V}$,
$\sum_{C^{\prime}\in
2^{V}}(\chi_{{+}}(C,C^{\prime})+\chi_{{-}}(C,C^{\prime}))=1$.
* 3.
If the transitioon function is based on birth-death and computed with some
$r>1$, then we will write it as $\chi^{(r)}_{{+}},\chi^{(r)}_{{-}}$
respectively. If computed with $r=1$, then we write
$\chi^{(\mathit{nd})}_{{+}},\chi^{(\mathit{nd})}_{{-}}$ respectively.
* 4.
For some $C\in 2^{C}$, let $\textit{inc}(C)$ be the set of all elements $D\in
2^{V}$ s.t. $|D|\geq|C|$ and $\chi_{{+}}(C,D)>0$.
* 5.
For some $C\in 2^{C}$, let $\textit{dec}(C)$ be the set of all elements $D\in
2^{V}$ s.t. $|D|\leq|C|$ and $\chi_{{-}}(C,D)>0$.
* 6.
Given set $C\subseteq V$, we will use $F^{(r)}_{C}$ to denot the probability
of fixation given initial set of mutants $C$ where the value $r$ is used to
calculate all transition probabilities.
CLAIM 1: If a some time period, the probability distribution over mutant
configurations is I, the fixation probability is $\sum_{C\in
2^{V}}\textbf{I}(C)\cdot F^{(r)}_{C}$.
Clearly, for any time $t$,
$F^{(r)}_{C}=\mathit{lim}_{i\rightarrow\infty}\textbf{Pr}(V^{(i)}|C^{(t)})$.
Under the assumption that there exists some tim $\omega$ s.t. fixation is
reached, we have:
$\displaystyle F^{(r)}_{C}$ $\displaystyle=$
$\displaystyle\textbf{Pr}(V^{(\omega)}|C^{(t)})$ $\displaystyle=$
$\displaystyle\frac{\textbf{Pr}(V^{(\omega)}\wedge
C^{(t)})}{\textbf{Pr}(C^{(t)})}$
Hence, $F^{(r)}_{C}\cdot\textbf{Pr}(C^{(t)})=\textbf{Pr}(V^{(\omega)}\wedge
C^{(t)})$. The statement then follows by the summation of exhaustive and
mutually exclusive events.
CLAIM 2: If a some time period $t$, the probability distribution over mutant
configurations is I, and the transition functions used to reach the next time
step are $\chi_{{+}},\chi_{{-}}$, then the probability of being in some mutant
configuration $C$ at time $t+1$ is given by $\sum_{D\in
2^{V}}\textbf{I}(D)\cdot(\chi_{{+}}(D,C)+\chi_{{-}}(D,C))$.
Follows directly from the rules of dynamics.
CLAIM 3: If a some time period $t$, the probability distribution over mutant
configurations is I, mutant fitness $r$, and the transition functions used to
reach the next time step are $\chi^{(r)}_{{+}},\chi^{(r)}_{{-}}$, and all
subsequent transitions are computed using the same dynamics with neutral
drift, then the fixation probability is:
$\displaystyle\mathcal{P}(I,r)$ $\displaystyle=$ $\displaystyle\sum_{C\in
2^{V}}\textbf{I}(C)\cdot\left(\sum_{D\in\textit{inc}(C)}(\chi^{(r)}_{{+}}(C,D)\cdot
F^{(1)}_{D})+\sum_{D\in\textit{dec}(C)}(\chi^{(r)}_{{-}}(C,D)\cdot
F^{(1)}_{D}))\right)$
Follows directly from claims 1-2.
CLAIM 4: Under BD-B, BD-D, DB-B, DB-D, or LD dynamics, for some $r\leq
r^{\prime}$, for all $C,D\in 2^{V}$, we have
$\chi^{(r)}_{{+}}(C,D)\leq\chi^{(r^{\prime})}_{{+}}(C,D)$ and
$\chi^{(r)}_{{-}}(C,D)\geq\chi^{(r^{\prime})}_{{-}}(C,D)$.
CLAIM 4a: For some $r\leq r^{\prime}$, for all $C,D\in 2^{V}$, we have
$\chi^{(r)}_{{+}}(C,D)\leq\chi^{(r^{\prime})}_{{+}}(C,D)$.
Let $\\{v_{j}\\}=D-C$. For each vertex $v_{i}$, $f_{i}=1$ if $v_{i}\notin C$
(a resident) and $f_{i}=r$ if $v_{i}\in C$ (a mutant). When $D\equiv C$, the
following are all summed over the set $\\{v_{j}\in C|\exists v_{i}\in
C\wedge(v_{i},v_{j})\in E\\}$.
* 1.
Under BD-B,
$\displaystyle\chi^{(r)}_{{+}}(C,D)$ $\displaystyle=$
$\displaystyle\mathop{\sum_{v_{i}\in C|}}_{(v_{i},v_{j})\in E}\frac{r\cdot
w_{ij}}{r\cdot|C|+N-|C|}$
* 2.
Under BD-D,
$\displaystyle\mathop{\sum_{v_{i}\in C|}}_{(v_{i},v_{j})\in
E}\frac{w_{ij}}{N\cdot\sum_{v_{q}|(v_{i},v_{q})\in E}w_{iq}\cdot f_{q}^{-1}}$
* 3.
Under DB-B,
$\displaystyle\chi^{(r)}_{{+}}(C,D)$ $\displaystyle=$
$\displaystyle\mathop{\sum_{v_{i}\in C|}}_{(v_{i},v_{j})\in
E}\frac{w_{ij}\cdot r}{N\cdot\sum_{v_{q}|(v_{q},v_{j})\in E}w_{qj}\cdot
f_{q}}$
* 4.
Under DB-D,
$\displaystyle\chi^{(r)}_{{+}}(C,D)$ $\displaystyle=$
$\displaystyle\mathop{\sum_{v_{i}\in C|}}_{(v_{i},v_{j})\in
E}\frac{w_{ij}}{\sum_{v_{q}\in V}f_{q}^{-1}}$
* 5.
Under LD,
$\displaystyle\chi^{(r)}_{{+}}(C,D)$ $\displaystyle=$
$\displaystyle\sum_{v_{i}\in C|(v_{i},v_{j})\in E}\frac{w_{ij}\cdot
r}{\sum_{v_{q},v_{\ell}|(v_{q},v_{\ell})\in E}w_{q\ell}\cdot f_{q}}$
By simple algebraic manipulation, for each of these, when all values other
than $r$ are fixed, they increase as $r$ increases.
CLAIM 4b: For some $r\leq r^{\prime}$, for all $C,D\in 2^{V}$, we have
$\chi^{(r)}_{{-}}(C,D)\geq\chi^{(r^{\prime})}_{{-}}(C,D)$. Let
$\\{v_{j}\\}=C-D$. For each vertex $v_{i}$, $f_{i}=1$ if $v_{i}\notin C$ (a
resident) and $f_{i}=r$ if $v_{i}\in C$ (a mutant). When $D\equiv C$, the
following are all summed over the set $\\{v_{j}\in V-C|\exists v_{i}\in
V-C\wedge(v_{i},v_{j})\in E\\}$.
* 1.
Under BD-B,
$\displaystyle\chi^{(r)}_{{+}}(C,D)$ $\displaystyle=$
$\displaystyle\mathop{\sum_{v_{i}\in V-C|}}_{(v_{i},v_{j})\in
E}\frac{w_{ij}}{r\cdot|C|+N-|C|}$
* 2.
Under BD-D,
$\displaystyle\mathop{\sum_{v_{i}\in V-C|}}_{(v_{i},v_{j})\in
E}\frac{w_{ij}\cdot r^{-1}}{N\cdot\sum_{v_{q}|(v_{i},v_{q})\in E}w_{iq}\cdot
f_{q}^{-1}}$
* 3.
Under DB-B,
$\displaystyle\chi^{(r)}_{{+}}(C,D)$ $\displaystyle=$
$\displaystyle\mathop{\sum_{v_{i}\in V-C|}}_{(v_{i},v_{j})\in
E}\frac{w_{ij}}{N\cdot\sum_{v_{q}|(v_{q},v_{j})\in E}w_{qj}\cdot f_{q}}$
* 4.
Under DB-D,
$\displaystyle\chi^{(r)}_{{+}}(C,D)$ $\displaystyle=$
$\displaystyle\mathop{\sum_{v_{i}\in V-C|}}_{(v_{i},v_{j})\in
E}\frac{w_{ij}\cdot r^{-1}}{\sum_{v_{q}\in V}f_{q}^{-1}}$
* 5.
Under LD,
$\displaystyle\chi^{(r)}_{{+}}(C,D)$ $\displaystyle=$
$\displaystyle\sum_{v_{i}\in V-C|(v_{i},v_{j})\in
E}\frac{w_{ij}}{\sum_{v_{q},v_{\ell}|(v_{q},v_{\ell})\in E}w_{q\ell}\cdot
f_{q}}$
By simple algebraic manipulation, for each of these, when all values other
than $r$ are fixed, they decrease as $r$ increases.
CLAIM 5: Given some $C\in 2^{V}$, for all pairs $D,D^{\prime}$ where
$D\in\textit{inc}(C)$ and $D^{\prime}\in\textit{dec}(C)$, we have
$F^{(1)}_{D}\geq F^{(1)}_{D^{\prime}}$.
Follows directly from Theorem 5.
CLAIM 6: Given interpretation I, under BD-B, BD-D, DB-B, DB-D, or LD dynamics,
for some $r>1$, $\mathcal{P}(\textbf{I},r)\geq\mathcal{P}(\textbf{I},0)$.
Let us consider some set $C$ from the outermsot summation in the computation
of $\mathcal{P}(\textbf{I},r)$. Suppose, BWOC, there exists some $C\in 2^{V}$
s.t.:
$\displaystyle\sum_{D\in\textit{inc}(C)}(\chi^{(r)}_{{+}}(C,D)\cdot
F^{(1)}_{D})+\sum_{D\in\textit{dec}(C)}(\chi^{(r)}_{{-}}(C,D)\cdot
F^{(1)}_{D})$ $\displaystyle<$
$\displaystyle\sum_{D\in\textit{inc}(C)}(\chi^{(1)}_{{+}}(C,D)\cdot
F^{(1)}_{D})+\sum_{D\in\textit{dec}(C)}(\chi^{(1)}_{{-}}(C,D)\cdot
F^{(1)}_{D})$
This give us:
$\displaystyle\sum_{D\in\textit{inc}(C)}(\chi^{(r)}_{{+}}(C,D)\cdot
F^{(1)}_{D})-\sum_{D\in\textit{inc}(C)}(\chi^{(1)}_{{+}}(C,D)\cdot
F^{(1)}_{D})$ $\displaystyle<$
$\displaystyle\sum_{D\in\textit{dec}(C)}(\chi^{(1)}_{{-}}(C,D)\cdot
F^{(1)}_{D})-\sum_{D\in\textit{dec}(C)}(\chi^{(r)}_{{-}}(C,D)\cdot
F^{(1)}_{D})$
Let $F_{sm}=\mathit{inf}\\{F^{(1)}_{D}|D\in\textit{inc}(C)\\}$ and
$F_{lg}=\mathit{sup}\\{F^{(1)}_{D}|D\in\textit{dec}(C)\\}$, this give us:
$\displaystyle
F_{sm}\sum_{D\in\textit{inc}(C)}(\chi^{(r)}_{{+}}(C,D)-\chi^{(1)}_{{+}}(C,D))$
$\displaystyle<$ $\displaystyle
F_{lg}\sum_{D\in\textit{dec}(C)}(\chi^{(1)}_{{-}}(C,D)-\chi^{(r)}_{{-}}(C,D))$
Consider ther following:
$\displaystyle\sum_{D\in\textit{inc}(C)}\chi^{(r)}_{{+}}(C,D)+\sum_{D\in\textit{dec}(C)}\chi^{(r)}_{{-}}(C,D)$
$\displaystyle=$
$\displaystyle\sum_{D\in\textit{inc}(C)}\chi^{(1)}_{{+}}(C,D)+\sum_{D\in\textit{dec}(C)}\chi^{(1)}_{{-}}(C,D)$
$\displaystyle\sum_{D\in\textit{inc}(C)}\chi^{(r)}_{{+}}(C,D)-\sum_{D\in\textit{inc}(C)}\chi^{(1)}_{{+}}(C,D)$
$\displaystyle=$
$\displaystyle\sum_{D\in\textit{dec}(C)}\chi^{(1)}_{{-}}(C,D)-\sum_{D\in\textit{dec}(C)}\chi^{(r)}_{{-}}(C,D)$
Note that by claim 4, both sides of the above equation are positive numbers.
Hence, we have $F_{sm}<F_{lg}$, which contradicts claim 5.
PROOF OF THEOREM: Let
$\mathcal{P}^{(1)}(\textbf{I},r)=\mathcal{P}(\textbf{I},r)$ and
$\mathcal{P}^{(i+1)}(\textbf{I},r)=\mathcal{P}(\mathcal{P}^{(i)}(\textbf{I},r),r)$.
By claim 6, for any $i$,
$\mathcal{P}^{(i+1)}(\textbf{I},r)\geq\mathcal{P}(\mathcal{P}^{(i)}(\textbf{I},r),r)$.
Consider an interpretation I that describes the initial probability
distribution over mutant configurations. The fixation probability under
neutral drift is $\mathcal{P}(\textbf{I},1)$. For some value $r>1$, the
fixation probaiblity is
$\mathit{lim}_{i\rightarrow\infty}\mathcal{P}^{(i)}(\textbf{I},r)$. Clearly,
$\mathit{lim}_{i\rightarrow\infty}\mathcal{P}^{(i)}(\textbf{I},r)\geq\mathcal{P}(\textbf{I},1)$.
Q.E.D.
## 20 Proof of Theorem 6
$\displaystyle
t_{C}=\frac{1}{F_{C}}\sum_{t=1}^{\infty}t\cdot(F_{t|C}-F_{t-1|C})$
Proof. This proof was first presented in [2]. The mean time to fixation is
described as the expected time to fixation given that the process fixates. Let
$F_{t|C}$ be the probability that fixation is reached in exactly $t$ time-
steps or less. Hence, the probability of reaching fixation in exactly $t$ time
steps conditioned on the process reaching fixation is
$(F_{t|C}-F_{t-1|C})/F_{C}$. The remainder of the theorem follows from the
definition of an expected value. Q.E.D.
## 21 Proof of Theorem 7
We introduce two pieces of notation $t_{fix},t_{convg}$. We define $t_{fix}$
as a time s.t.
$t_{C}=\frac{1}{F_{C}}\sum_{t=1}^{t_{fix}}t\cdot(F_{t|C}-F_{t-1|C})$. and
$t_{convg}$ s.t. $\forall i$,
$\textbf{Pr}(\textbf{{M}}^{(t_{convg})}_{i})=F_{C}$. While in reality, both of
these values could be infinite, we note that the relationship $\infty\geq
t_{fix}\geq t_{convg}$ holds and that using a lower value for $t_{fix}$ and/or
$t_{convg}$ will still ensure we have a lower bound.
$\displaystyle\frac{1}{F_{C}}\sum_{t=1}^{\infty}t\cdot(P_{\min,t}-P_{\min,t-1})\leq\frac{1}{F_{C}}\sum_{t=1}^{\infty}t\cdot(F_{t|C}-F_{t-1|C})$
Where $P_{\min,t}=\min_{i}\textbf{Pr}(\textbf{{M}}^{(t)}_{i})$.
Proof. First, we have the following.
$\sum_{t=1}^{t_{convg}}t\cdot(P_{\min,t}-P_{\min,t-1})\leq\sum_{t=1}^{t_{fix}}t\cdot(F_{t|C}-F_{t-1|C})$
(24)
For any time $t$, let $\mathcal{P}_{C}^{(t)}=F_{t|C}-F_{t-1|C}$ and
$\textbf{Pr}(\Delta\textbf{{M}}^{(t)}_{\min})=P_{\min,t}-P_{\min,t-1}$. As for
each time $t$, we have $P_{\min,t}\geq F_{t|C}$, we can define
$\theta^{(t)}_{t^{\prime}}$ as the portion of $\mathcal{P}_{C}^{(t)}$
accounted for in $\textbf{Pr}(\Delta\textbf{{M}}^{(t^{\prime})}_{\min})$. This
results in having $\theta^{(t)}_{t^{\prime}}=0$ whenever $t^{\prime}>t$ or
$t>t_{convg}$ as well as the following:
$\displaystyle\mathcal{P}_{C}^{(t)}$ $\displaystyle=$
$\displaystyle\sum_{t^{\prime}=1}^{t}\theta^{(t)}_{t^{\prime}}$ (25)
$\displaystyle\textbf{Pr}(\Delta\textbf{{M}}^{(t)}_{\min})$ $\displaystyle=$
$\displaystyle\sum_{t^{\prime}=t}^{t_{fix}}\theta^{(t^{\prime})}_{t}$ (26)
$\displaystyle\sum_{t=1}^{t_{convg}}t\cdot(P_{\min,t}-P_{\min,t-1})$
$\displaystyle=$
$\displaystyle\sum_{t=1}^{t_{convg}}\sum_{t^{\prime}=t}^{t_{fix}}t\theta^{(t^{\prime})}_{t}$
(27) $\displaystyle=$
$\displaystyle\sum_{t=1}^{t_{convg}}\sum_{t^{\prime}=1}^{t_{fix}}t\theta^{(t^{\prime})}_{t}$
(28) $\displaystyle=$
$\displaystyle\sum_{t^{\prime}=1}^{t_{fix}}\sum_{t=1}^{t_{convg}}t\theta^{(t^{\prime})}_{t}$
(29) $\displaystyle=$
$\displaystyle\sum_{t^{\prime}=1}^{t_{fix}}\sum_{t=1}^{t^{\prime}}t\theta^{(t^{\prime})}_{t}$
(30)
We also note that the following is true:
$\displaystyle\sum_{t=1}^{t_{fix}}t\cdot(F_{t|C}-F_{t-1|C})$ $\displaystyle=$
$\displaystyle\sum_{t^{\prime}=1}^{t_{fix}}t^{\prime}\mathcal{P}_{C}^{(t^{\prime})}$
(31) $\displaystyle=$
$\displaystyle\sum_{t^{\prime}=1}^{t_{fix}}t^{\prime}\sum_{t=1}^{t^{\prime}}\theta^{(t^{\prime})}_{t}$
(32) $\displaystyle\geq$
$\displaystyle\sum_{t^{\prime}=1}^{t_{fix}}\sum_{t=1}^{t^{\prime}}t\theta^{(t^{\prime})}_{t}$
(33) $\displaystyle=$
$\displaystyle\sum_{t=1}^{t_{convg}}t\cdot(P_{\min,t}-P_{\min,t-1})$ (34)
Which concludes the proof. Q.E.D.
## 22 Materials and Methods
Except for the experiments dealing with time to fixation/extinction, all
algorithms were implemented in Python and run on a 2.33GHz Intel Xeon CPU. The
time-to-fixation experiments were run on a machine equipped with an Intel Core
i7 M620 processor running at $2.67$ GHz with $4$ GB RAM.
All graphs in the experiments were generated using the Python NetworkX package
[11]. Parameters used for the experiments concerning the expected number of
mutants were m=1 for BA, p = 0.5 for ER, and $k=2$ and $p=0.5$ for NWS graph
generator functions.
We modified Algorithm 1-ACC based on the results on mean time to fixation as
follows:
* 1.
Before line 14, insert: t += 1; Sum += t*(min(p)-min(q))
* 2.
Replace line 17 with: return = Sum/average(p), where average(q) is the
algorithm’s best estimate for Pc at termination.
|
arxiv-papers
| 2013-01-11T16:23:19 |
2024-09-04T02:49:40.185619
|
{
"license": "Public Domain",
"authors": "Paulo Shakarian, Patrick Roos, Geoffrey Moores",
"submitter": "Paulo Shakarian",
"url": "https://arxiv.org/abs/1301.2533"
}
|
1301.2604
|
# Structure and function in flow networks
Nicolás Rubido Institute for Complex Systems and Mathematical Biology,
University of Aberdeen, King’s College, AB24 3UE Aberdeen, UK Instituto de
Física, Facultad de Ciencias, Universidad de la República, Iguá 4225,
Montevideo, 11200, Uruguay Celso Grebogi Institute for Complex Systems and
Mathematical Biology, University of Aberdeen, King’s College, AB24 3UE
Aberdeen, UK Murilo S. Baptista Institute for Complex Systems and
Mathematical Biology, University of Aberdeen, King’s College, AB24 3UE
Aberdeen, UK
###### Abstract
This Letter presents a unified approach for the fundamental relationship
between structure and function in flow networks by solving analytically the
voltages in a resistor network, transforming the network structure to an
effective all-to-all topology, and then measuring the resultant flows.
Moreover, it defines a way to study the structural resilience of the graph and
to detect possible communities.
Resistor Networks, Maximum Flows, Generalized Inverse Laplacian Matrix,
Communities.
###### pacs:
41.20-q, 95.75.Pq, 89.40.-a, 89.75.Hc, 89.75.Fb
The relation between structure and function is one of the most studied topics
in the theory of complex networks Girvan ; Newman1 ; Newman2 ; Motter ; Lai1 ;
Lai2 ; Havlin . The structure is a topological representation of the
interacting elements forming a network and it is rigorously described by the
theory of graphs. Behaviour is a functional observable of the network and can
be measured by a variety of different approaches and methodologies, e.g., it
can be quantified as a statistical description of the weights of connections
of the structure of the network Girvan ; Newman1 ; Newman2 ; Motter or depend
on the dynamics of interactions among the elements.
This work deals with flow networks that satisfy conservation laws (Kirchhoff’s
laws). The model for the flow network is stated in terms of resistor networks
(weighted symmetric graphs), which have a source node $s$ and a sink node $t$
feeding the system, and a linear relationship between voltages and currents
VF_problem . The problem models a wide variety of physical phenomena, such as
river flow channels, transport networks, ion channels, and electrical
circuits. Its solutions establish a clear relationship between the topological
structure of the networks (namely, adjacency matrix and edge weights, assumed
known) and the functional flows passing through nodes and edges (that are a
consequence of solving the flow model). The foundation of network flow theory
roots back to Kirchhoff Kirchhoff , answering what are the current flows in an
electrical circuit when a set of voltages is applied. The solution is then
achieved by solving Kirchhoff’s equations Kirchhoff ; Ahuja ; Bollobas .
In this Letter, we present a unified approach for the fundamental relationship
between structure and function in flow networks by calculating voltages
(loads) when input currents (flows) are given. The main results of this
approach are the following.
First, a novel formula for the voltages in the flow network model, i.e., the
_Voltage-Flow (VF) problem_ [Eq. (2)], is analytically derived. The solution
[Eq. (6)] is expressed in terms of the known graph’s weighted Laplacian matrix
eigenspace base. In particular, we show that the behaviour of the flows is not
only dependent on the matrix spectra, but intrinsically connected to the
eigenvectors of this matrix.
Second, we show that the _equivalent resistance_ (also known as two-point
distance resistance) formula Wu-Arpita is retrieved [Eq. (7)] without further
assumptions from the analytical expression for the voltages induced by the
single $s$-$t$ pair. The equivalent resistance between any two nodes is a
topological measure that gives further information about the structure of any
graph. It is shown here how it constitutes a highly useful tool to reduce the
complexity of any network structure: an arbitrary topology is transformed to
an effective all-to-all complete graph.
Third, using both the voltage differences and the equivalent resistance, an
_effective functional flow adjacency_ (EFFA) between any two points in the
circuit is constructed [Eq. (8)]. These effective flows are the real
measurable observables that are obtained when a probe is placed between any
two nodes in the network. The EFFA matrix expresses the level of connectivity
in terms of flow transmission; the functional relationship among the network
components in the effective all-to-all topology. In particular, if two nodes
are not directly connected, an effective current between them may still exist.
Fourth, as an application of our approach, we show how to identify the nodes
that are mainly responsible for the transmission of flow, how to detect
qualitatively communities of the network, and how to infer maximal node
outflow for different graph topologies and resistor distributions. This
reveals topological issues that are related to node maximum capacities in a
natural manner.
The starting point for our unified approach is the calculation of voltages in
a network by solving the VF problem. The VF problem is set by interpreting the
graph structure as a linear resistor network circuit, which then receives an
input flow $I$ at node $s$ and leaves at node $t$ (a current $I$ enters the
circuit at a source and leaves at a sink). The network structure is given by a
connected strict graph $\mathcal{G}=\\{\mathcal{V},\,\mathcal{E}\\}$ (where
$\mathcal{V}$ and $\mathcal{E}$ are the sets of nodes and edges, respectively)
with symmetric edge weights $W_{ij}\equiv A_{ij}/R_{ij}$ for
$i,\,j=1,\ldots,\,N$ ($N$ being the number of nodes in $\mathcal{V}$, $A_{ij}$
the $ij$-th element of the adjacency matrix, and $R_{ij}$ the edge’s
resistance).
Imposing conservation of charge in every node of the network, i.e., a graph
where at each node the current arriving at it equals the amount leaving it
(first Kirchhoff’s law), the flow vector is
$(\vec{F}^{(st)})_{i}=I\,\left(\delta_{is}-\delta_{it}\right)$ for
$i=1,\,\ldots,\,N$, with $\delta_{ij}$ being the Kronecker delta. Thus, using
the laws of Kirchhoff and Ohm, the net current in any node is given by
$F_{i}^{(st)}=I\,\left(\delta_{is}-\delta_{it}\right)=\sum_{j=1}^{N}\frac{A_{ij}}{R_{ij}}\left(V_{i}^{(st)}-V_{j}^{(st)}\right)\,,$
(1)
where $V_{i}^{(st)}$ is the voltage potential at node $i$ given the particular
$s$-$t$ pair. Equation (1) is rearranged such that the VF problem is expressed
in matrix form
$\mathbf{G}\,\vec{V}^{(st)}=\vec{F}^{(st)}\,,$ (2)
where the upper-indexes indicate that the problem depends on the location of
the $s$-$t$ nodes on $\mathcal{V}$ and $\mathbf{G}$ is the weighted Laplacian
matrix. The entries of $\mathbf{G}$ are
$G_{ij}=\left\\{\begin{array}[]{lcl}\sum_{k=1}^{N}W_{ik}&\text{if}&i=j\,,\\\
-W_{ij}&\text{if}&i\neq j\,.\end{array}\right.$ (3)
Then, the VF solution is achieved once the voltages in each node are found
from inverting $\mathbf{G}$. However, the rank of $\mathbf{G}$ is $N-1$
($\det\left(\mathbf{G}\right)=\prod_{n=0}^{N-1}\lambda_{n}=0$) and direct
inversion is not possible Bollobas ; FanChung .
The Laplacian inversion is addressed here by a different interpretation of the
derivation of the generalized inverse matrix, also known as Moore-Penrose
matrix Bollobas ; Wu-Arpita .
The eigenvalue problem for the Laplacian is given by
$\mathbf{G}\,\mathbf{P}=\mathbf{P}\,\mathbf{\Lambda}$, where
$\Lambda_{ij}=\delta_{ij}\,\lambda_{n}$ ($n=i-1=0,\ldots,N-1$) is a diagonal
matrix containing all eigenvalues and
$\mathbf{P}=\\{\vec{v}_{0},\,\vec{v}_{1},\,\ldots,\,\vec{v}_{N-1}\\}$ is the
matrix whose columns are the eigenvectors of $\mathbf{G}$. We prove here that
to express any element of $\mathbf{G}$, the only elements needed from the
spectral decomposition are the non-null eigenvalues and corresponding
eigenvectors. This results in the following decomposition of $\mathbf{G}$
$\mathbf{G}=\mathbf{P}_{r}\,\mathbf{\Lambda}_{r}\,\mathbf{P}_{r}^{T}\,,$ (4)
where $T$ indicates transpose, $\mathbf{P}_{r}\in\mathbb{R}^{N\times(N-r)}$,
$\mathbf{\Lambda}_{r}\in\mathbb{R}^{(N-r)\times(N-r)}$, and $r$ being the
number of connected components of $\mathcal{G}$, which in this Letter is fixed
at $r=1$.
Equation (4) is derived by analyzing each Laplacian matrix entry using the
full spectral decomposition. In particular,
$G_{ij}=\sum_{k=1}^{N}P_{ik}\lambda_{k-1}P^{-1}_{kj}$, where for $k=1$,
$\lambda_{0}=0$, so no contribution is added to the sum. Thus, observing that
the inverse of an orthogonal matrix is its transpose
($P^{-1}_{kj}=P_{jk}=\left(\vec{v}_{k-1}\right)_{j}$), the $ij$’s entry of
$\mathbf{G}$ is given by:
$G_{ij}=\sum_{k=2}^{N}\left(\vec{v}_{k-1}\right)_{i}\,\lambda_{k-1}\,\left(\vec{v}_{k-1}\right)_{j}$,
which is easily extendible to various connected components making the lower
bound for the summation index $k=r+1$.
However, removing the null eigenspace of the spanning set of eigenvectors, the
rank of the new $\mathbf{G}$ in Eq. (4) is $N$ and inversion can be directly
performed. Consequently, each element of the generalized inverse Laplacian
matrix $Z_{ij}\equiv(\mathbf{G}^{-1})_{ij}$ is found from
$Z_{ij}=\sum_{n=1}^{N-1}\left(\vec{v}_{n}\right)_{i}\frac{1}{\lambda_{n}}\left(\vec{v}_{n}\right)_{j}\,,$
(5)
with the new index $n=k-1$ and $i,\,j=1,\ldots,N$. It is easy to show that
$\sum_{k=1}^{N}G_{ik}\,Z_{kj}=\delta_{ij}-1/N$, where the extra constant
factor comes from the incompleteness of the restricted eigenspace and it
cancels out when calculating voltage differences.
Returning to Eq. (2), the _solution for the VF problem_ at a given node
$i\in\mathcal{V}$ is
$\left(\vec{V}^{(st)}\right)_{i}=\sum_{j=1}^{N}Z_{ij}\,\left(\vec{F}^{(st)}\right)_{j}=I\,\left(Z_{is}-Z_{it}\right)$,
$\left(\vec{V}^{(st)}\right)_{i}=I\,\sum_{n=1}^{N-1}\frac{\left(\vec{v}_{n}\right)_{i}}{\lambda_{n}}\left[\left(\vec{v}_{n}\right)_{s}-\left(\vec{v}_{n}\right)_{t}\right]\,.$
(6)
This formula is the backbone of our unified approach. It is applicable to any
connected weighted graph and is extendible to many sources and sinks with
different inputs and outputs as long as flow conservation is fulfilled
($\sum_{i}F_{i}^{(st)}=0$ and $F_{i}=0\;\forall\,i\neq s$ or $t$).
Furthermore, it allows to derive the equation for the equivalent resistance
between any two nodes, to construct the effective flows, and later apply them
to, e.g., community detection.
The derivation of the _equivalent resistance_ $\rho_{ij}$ between any two
nodes Wu-Arpita is done as follows. Set the source at a starting node ($i=s$)
and the sink at an ending point ($j=t$), then Eq. (6) provides an exact closed
formula for $\rho_{ij}$ in terms of eigenvalues and eigenvectors of the
weighted Laplacian matrix. The reason for doing this is that the voltage
difference between these two nodes gives the incoming current times the
resistance between them, hence,
$(\vec{V}^{(st)})_{s}-(\vec{V}^{(st)})_{t}=I\,\rho_{st}$. Therefore, using the
same procedure for every pair of nodes in the graph all $\rho_{ij}$’s are
determined,
$\rho_{ij}=\sum_{n=1}^{N-1}\frac{1}{\lambda_{n}}\left[\left(\vec{v}_{n}\right)_{i}-\left(\vec{v}_{n}\right)_{j}\right]^{2}\,,$
(7)
providing a solution that is independent of where the $s$-$t$ pair is placed.
The matrix element $\rho_{ij}$ is a topological measure that weighs all paths
between any pair of nodes $i$ and $j$. All paths between these nodes that are
allowed by the adjacency structure of the graph are collapsed to one
equivalent link with weight $\rho_{ij}$. The result is that a single effective
link is created between $i$ and $j$. The important contribution is that _this
quantity presents the possibility to diminish the complexity of any graph to a
simpler all-to-all equivalent topology_ with a betweenness score given by
$\rho_{ij}$.
As a working example, results of applying the solution given by Eqs. (6) and
(7) to a ring graph, namely $C_{N}$, of $N=16$ nodes with resistors equal to
one ($R_{ij}=1\;\forall\,i,j$) and input current $I=1$ are shown in Fig. 1.
For this network, the set of unordered eigenvalues is given by
$\lambda_{n}=2-2\cos(2\pi\,n/N)$, with $n=0,\ldots,N-1$ FanChung .
The top left panel in Fig. 1 shows that for the $N\,(N-1)=240$ possibilities
of $s$-$t$ pairs in $C_{N}$ the maximum voltages achieved in the ring do not
surpass $2$ (arbitrary units). This is the maximum difference that the
eigenvector coordinates can achieve in the non-normalized frame (top right
panel in Fig. 1), hence giving a maximum voltage difference of $4$. This
result shows that not only the eigenvalues of the Laplacian are important to
infer the network’s functional behaviour, but eigenvectors are also needed.
Also, the maximum (minimum) is periodic, in the sense that the largest
difference happens every time the source and sink are separated by $8$ edges.
Figure 1: The top left panel has the maximum (black) and minimum (red)
voltages that are achieved in an unweighed ($R_{ij}=1\;\forall\,i,j$) ring
graph ($C_{N}$) with $N=16$ nodes for every $s$-$t$ pair in units of $I=1$ for
the input current at node $s$. The top right panel shows the eigenvectors
($\sqrt{N}\vec{v}_{n}$) coordinate values (color scale) of the corresponding
weighed Laplacian matrix. The bottom left panel exhibits in color coded, for
all these possible input-output configurations, the voltages $V_{i}^{(st)}$ of
every node. The bottom right panel shows the equivalent resistor matrix
($\rho_{ij}$) for every pair of nodes in $C_{N}$.
The bottom panels in Fig. 1 show the voltage solutions for all the $s$-$t$
pairs (left) and all equivalent resistances (right). The resulting values show
the functional and structural symmetry that this simple network has. As all
edge weights are unity ($R_{ij}=1$), the equivalent resistance $\rho_{ij}$
maximum can only be $4$. This is easily seen when calculating it directly by
means of series and parallel resistor calculations. For instance, for nodes
$i=1$ and $j=9$ there are two parallel paths with $8$ edges in series, hence:
$\rho_{1,9}^{-1}=\frac{1}{8}+\frac{1}{8}\;\Rightarrow\;\rho_{1,9}=4$, which is
the same value as element $\rho_{1,9}$ in the bottom right panel of Fig. 1
(found from Eq. (7)).
Having the voltage difference and equivalent resistance given by Eqs. (6) and
(7), the third result of our unified approach is addressed: the construction
of the _effective edge flows_ , namely, $\phi_{ij}^{(st)}$. We introduce these
flows by finding the voltage differences among any two nodes in the effective
topology given by $\rho_{ij}$. Thus, the EFFA matrix elements for each $s$-$t$
configuration are given by
$\phi_{ij}^{(st)}=\frac{1}{\rho_{ij}}\left[\left(\vec{V}^{(st)}\right)_{i}-\left(\vec{V}^{(st)}\right)_{j}\right]\,,$
(8)
where $(\vec{V}^{(st)})_{i}>(\vec{V}^{(st)})_{j}$ such that an effective
current is flowing from node $i$ to $j$, otherwise $\phi_{ij}^{(st)}=0$.
Hence, the $s$-$t$ dependent EFFA matrix defines an effective directed
network, which breaks the symmetry of the effective all-to-all topology
structure given by $\rho_{ij}$. In particular, the physical current on each
edge $ij\in\mathcal{E}$ of the graph is given by
$I_{ij}^{(st)}=\frac{A_{ij}}{R_{ij}}\,[(\vec{V}^{(st)})_{i}-(\vec{V}^{(st)})_{j}]$.
The voltage differences define a functional relationship among nodes and the
equivalent resistance an effective topology structure. The introduction of
_the EFFA relates both structure and functional characteristics of the graph_
, thus, it enables to find functional hubs that might not be there in the
topological structure, and detect highly connected areas (communities) in a
functional way. In this sense, the $s$-$t$ analysis acts similarly as how a
dye does when introduced to a circulatory system of a person: it highlights a
certain part of the network structure to detect what is being sought, e.g.,
clog blocking, arteries stiffness, etc.
Figure 2: The left panel shows the effective flow functional adjacency (EFFA)
matrix for the unweighed $C_{N}$ graph of Fig. 1, where all source-sink pairs
have been averaged, i.e., $\phi_{ij}$. The right panel exhibits the same EFFA
matrix for a particular source-sink pair, namely, $s=2$ and $t=10$
($\phi_{ij}^{(2,10)}$). Each matrix element’s magnitude is associated with a
color scale.
The mean EFFA matrix is defined to detect statistically relevant functional
observables. It is calculated by all the effective currents (Eq. (8)) among
the different pairs averaged over all possible realizations of source-sink
pairs. Its matrix elements are
$\phi_{ij}\equiv\left\langle\phi_{ij}^{(st)}\right\rangle_{(st)}=\sum_{s=1}^{N}\sum_{t=1\,(t\neq
s)}^{N}\frac{\phi_{ij}^{(st)}}{N\,(N-1)}\,,$ (9)
which define a matrix that constitutes an effective structural flow quantity
and is related to the equivalent resistance matrix. The EFFA matrix and its
averaged counterpart for the example of the ring graph are shown in Fig. 2,
where the left panel shows the mean EFFA matrix and the right panel exhibits
the symmetry breaking for a particular $s$-$t$ pair.
With the three quantities analytically expressed (voltages in Eq. (6),
equivalent resistances in Eq. (7), and the effective currents in Eq. (8)), we
investigate graphs random graphs (RN) Erdos , small-world (SW) networks
Strogatz , scale-free (SF) topologies Barabasi , and numerically generated
clustered networks (CN) of $N=2^{9}$ nodes. The following conclusions
constitute the application of our unified approach to these concrete
topologies. In particular, the differences between the respective adjacency
and effective topologies (equivalent resistance) are shown in the left and
centre columns of Figs. 3 and 4.
To study other features of the EFFA matrix that are not represented in its
averaged version and to answer how the flow is effectively distributed in any
given topology, the probability density function (PDF) of the effective edge
flows $\phi_{ij}^{(st)}$ for every particular $s$-$t$ is calculated. This does
not only tells how diffusive or localized the transport of currents is as a
function of the effective topology, but also gives the maximum flow values a
particular graph can have, i.e., the maximum edge capacity.
Results show that the EFFA PDFs for RN, SW, and SF networks are roughly
invariant under changes of the $s$-$t$ pair. However, SW and SF networks
produce more high flow magnitudes ($\phi_{ij}^{(st)}\simeq 1$) than RN for a
broad range of parameters graphs , hence, RNs distribute flows in a more
uniform way than SF or SW networks. These observations are seen in the example
shown on the right column of Fig. 3.
Figure 3: The figure shows the following unweighted graphs: a random (top
row), a scale-free (centre row), and a small-world (bottom row) network of
$N=2^{9}$ nodes and connecting probability $p=0.05$ graphs . The adjacency
matrix (left column), equivalent resistance matrix (centre column in color
coded), and effective functional flow adjacency PDFs (right column) for all
$s$-$t$ configurations (gray scale) are shown for each graph.
The detection of communities is done by analysis of the mean EFFA matrix and
the equivalent resistance. In general, _the application of the EFFA analysis
on numerically generated CNs qualitatively exhibits the communities that the
networks have_. This is seen in the centre and right panels of the example
exhibited in Fig. 4, where the communities are spotted as dark areas, both in
$\rho_{ij}$ and $\phi_{ij}$, though the contrast in the mean EFFA matrix is
higher than in the equivalent resistance, implying that communities are better
identified by $\phi_{ij}$. _The $\phi_{ij}$ matrix_ not only detects the
communities but also _shows that the effective flows between communities are
larger than the ones within each community_. Hence, the intra-edges connecting
communities must have a high flow capacity as they represent the vulnerable
links in the network. Removal of any of these edges generate drastic changes
in the flows, leading to the isolation of communities.
Figure 4: The figure shows the adjacency matrix (left column), equivalent
resistance (middle column), and the average effective functional adjacency
matrix (right column) for an unweighed clustered network formed by $4$ random
graphs of $N_{c}=2^{7}$ graphs . The network has a total of $N=2^{9}$ nodes
and each matrix element magnitude is associated with a color scale.
The observations on flow transmission and community detection have been
previously shown by doing structural analysis of networks, e.g., in Refs.
Girvan ; Newman1 ; Newman2 ; Motter ; Lai1 ; Lai2 ; Havlin . In particular,
the possibility of using voltages as a betweenness measure score is pointed
out in Refs. Girvan ; Newman1 ; Newman2 and is related to shortest-paths and
random walks. With our unified approach, and the four results that are
addressed in this Letter, we show the practicality of it. Moreover, the linear
relationship between loads and flows, and conservation of charge, are
restrictions needed to define the flow network on top of the structure but not
for the structure itself. Thus, the flow analysis is applicable to any network
structure outside the domain of flow networks as long as the graph to be
analysed is connected.
An estimate of the computational time that our approach requires is as
follows. To calculate the voltages one needs the eigenvectors and eigenvalues
of the Laplacian matrix. This is calculated in time $\mathcal{O}(N^{\alpha})$,
with $\alpha$ being an exponent that nowadays rounds $2$ and $N$ the number of
nodes in the network. Then, if all possible source-sink pairs are analysed,
the flows require $\mathcal{O}(N^{3})$ to be found. This means that the
algorithm is not as effective as other methods Andrea but its mathematical
formulas still provide great information for obtaining upper and lower bounds
for currents without the need for simulations. Furthermore, community
detection can be quantitatively described if other techniques are used, such
as PDF analysis of the EFFA matrix elements.
Authors acknowledge the Scottish University Physics Alliance (SUPA).
## References
* (1) M. Girvan and M. E. J. Newman, PNAS 99(12), 7821-7826 (2002).
* (2) M. E. J. Newman and M. Girvan, Phys. Rev. E 69, 026113 (2004).
* (3) M. E. J. Newman, Eur. Phys. J. B 38, 321-330 (2004).
* (4) A. E. Motter, Phys. Rev. Lett. 93(9), 098701 (2004).
* (5) Rui Yang, Wen-Xu Wang, Ying-Cheng Lai, and Guanrong Chen, Phys. Rev. E 79, 026112 (2009).
* (6) Wen-Xu Wang and Ying-Cheng Lai, Phys. Rev. E 80, 036109 (2009).
* (7) Roni Parshani, Sergey V. Buldyrev, and Shlomo Havlin, Phys. Rev. Lett. 105, 048701 (2010).
* (8) In particular, the approach that we take is derived from the maximum flow problem, which corresponds to finding a feasible flow through a single-source $s$ and a single-sink $t$ that is maximum when costs and capacities at the edges are given. The solution for this problem is equal to the minimum capacity of an $s$-$t$ cut in the network as stated in the max-flow min-cut theorem Ahuja ; Bollobas . On the other hand, if the $s$-$t$ flow is fixed, then the problem turns into finding how is the transmission given and what is its implications to the structure and function of the network.
* (9) G. Kirchhoff, Vorlesungen über Mechanik (Wien, Wilhelm, Ed. 1864-1928).
* (10) Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice-Hall, 1993).
* (11) B. Bollobás, Modern Graph Theory (Springer-Verlag, New York, 1998, chap. 9).
* (12) Fan R. K. Chung, Spectral Graph Theory (Am. Math. Soc. and CBMS 92, 1997).
* (13) F. Y. Wu, J. Phys. A: Math. Gen. 37, 6653-6673 (2004); Arpita Ghosh, Stephen Boyd, and Amin Saberi, SIAM Rev. 50(1), 37-66 (2008); and references therein.
* (14) Starting with a ring graph of $N=2^{9}$ nodes, a tuning parameter $p$ controls the addition of links. For the RN, SW, and SF graphs the process is straightforward (taken from Refs. Erdos ; Strogatz ; Barabasi ) and $p$ is related to the probability of adding an edge. The RN process follows the strategy of Erdos . The SF graph is constructed as in the rewiring procedure of Ref. Barabasi . The SW graph is constructed as in Ref. Strogatz . For the CN, four RN of $N_{c}=2^{7}$ are created with equal statistics (probability $p=0.01$), then approximately $10$ links are added between them randomly. In addition, weights are included from the uniform random distribution such that $R_{ij}\in(0,1]\;\forall\,i,j$.
* (15) P. Erdos and A. Renyi, Publ. Math. 6, 290 (1959).
* (16) D. J. Watts and S. H. Strogatz, Nature 393, 440-442 (1998).
* (17) A.-L. Barabási and R. Albert, Science 286, 509 (1999).
* (18) Andrea Lancichinetti, Santo Fortunato, and Filippo Radicchi, Phys. Rev. E 78, 046110 (2008).
|
arxiv-papers
| 2013-01-10T16:45:02 |
2024-09-04T02:49:40.199333
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nicol\\'as Rubido, Celso Grebogi, and Murilo S. Baptista",
"submitter": "Nicol\\'as Rubido",
"url": "https://arxiv.org/abs/1301.2604"
}
|
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