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+{"text":"\\section{Introduction}\n\\label{introduction}\nThe physical state and structure of the interstellar medium (ISM) are important parameters for understanding the star formation in a galaxy. In a typical star-forming region, young massive stars are born and start to illuminate their parental cloud. UV photons ionize the surrounding medium, creating H\\,{\\sc ii}\\xspace regions, while the transition to the neutral atomic or molecular phase occurs at higher visual extinction, where the material is more effectively shielded. Far-UV (FUV) photons control the chemical activity in these regions, namely the photodissociation region (PDR; \\citealt{tielens-1985}). By studying the latter, we can investigate the conditions of the molecular clouds, which in turn will be potential sites for the next episode of star formation. \n\nHow does the propagation of radiation and the ISM composition affect\nISM observables in low-metallicity galaxies? Addressing this question is important to understand the evolution of low-metallicity galaxies, which undergo more bursty star formation than normal galaxies. Nearby star-forming dwarf galaxies present distinct\nobservational signatures compared to well-studied disk galaxies. Dwarfs are usually metal poor, H\\,{\\sc i}\\xspace\\ rich, and molecule poor as\na result of large-scale photodissociation \\citep[e.g.,][]{kunth-2000,hunter-2012,schruba-2012}. Mid-IR (MIR) and far-IR (FIR) observations have revealed bright atomic lines from H\\,{\\sc ii}\\xspace regions ([S\\,{\\sc iii}]\\xspace, [Ne\\,{\\sc iii}]\\xspace, [Ne\\,{\\sc ii}]\\xspace, [O\\,{\\sc iii}]\\xspace, etc.) and PDRs ([C\\,{\\sc ii}]\\xspace, [O\\,{\\sc i}]\\xspace) \\citep[e.g.,][]{hunter-2001,madden-2006,wu-2008,hunt-2010,cormier-2015}. Their spectral energy distributions (SEDs) are also different from spiral and elliptical galaxies and indicative of altered dust properties, with a relatively low abundance of polycyclic aromatic hydrocarbons (PAHs) and perhaps a different dust composition \\citep[e.g.,][]{madden-2006,galliano-2008,remy-2013}. \nIt is still unknown, however, whether these differences between dwarf and disk galaxies are the direct result of recent star formation activity shaping the ISM or instead a consequence of the low-metallicity ISM that is independent of star formation activity. To answer this, one needs to observe tracers of the interplay between the ISM and various stages of star formation activity. While there are now a number of important studies available on PDR properties modeling FIR lines on large scales in various extragalactic environments \\citep[e.g.,][]{kaufman-2006,vasta-2010,gracia-carpio-2011,cormier-2012,parkin-2013} or in our Galaxy under solar-metallicity conditions \\citep[e.g.,][]{cubick-2008,bernard-salas-2012,bernard-salas-2015}, only a few studies are published on individual extragalactic regions \\citep{mookerjea-2011,lebouteiller-2012}. Of particular interest are dwarf galaxies, where the effect due to radiative feedback is expected to be most significant. \nThe goal of this paper is to investigate how the low-metallicity ISM reacts under the effects of star formation in regions that have undergone different histories. The nearby low-metallicity galaxy NGC\\,4214 provides an excellent environment to perform this experiment because it has well-separated star-forming centers, one hosting a super star cluster, which allows us to study the effects of extreme star-forming conditions on the surrounding ISM. \n\nNGC\\,4214 is a nearby irregular galaxy located 3\\,Mpc away \\citep{dalcanton-2009} with a metallicity of $\\sim$0.3\\,Z$_{\\odot}$ \\citep{kobulnicky-1996} and a wealth of ancillary data. It shows various morphological characteristics such as H\\,{\\sc i}\\xspace holes and shells and a spiral pattern \\citep{mcintyre-1998}. \nNGC\\,4214 is known to host two main, well-defined star-forming regions with recent activity (Fig.\\,\\ref{3color}). The largest of the two regions is found in the center of the galaxy (also referred to as NW or region~I) and contains several clusters, including a super star cluster, while the second region is found to the southeast (also referred to as SE or region~II) and is  younger and more compact. \nUsing near-IR, optical, and UV data, several studies have constrained the ages of the clusters in the two main regions, which show evidence for recent star formation \\citep{ubeda-2007,sollima-2013,sollima-2014}. \\cite{schruba-2012} have measured the ongoing SFR of NGC\\,4214 to be 0.12\\,M$_{\\odot}$\\,yr$^{-1}$. \nThe galaxy seems to have maintained its star formation in the past 10\\,Gyr at an average rate of $\\sim$0.02\\,M$_{\\odot}$\\,yr$^{-1}$, with a prolonged star formation episode that occurred about 3\\,Gyr ago and several shorter bursty events within the past Gyr at a rate of 0.05-0.12\\,M$_{\\odot}$\\,yr$^{-1}$ \\citep{mcquinn-2010,williams-2011}. \\par\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.52]{fig1.eps}\n\\centering\n\\caption{Three-color image of NGC\\,4214 using the HST WFC3 filters F438W (B, blue), F502N ([O\\,{\\sc iii}]\\xspace, green), and F657N (H$\\alpha$+[N\\,{\\sc ii}]\\xspace, red), downloaded from the Hubble Legacy Archive (\\protect\\url{http:\/\/hla.stsci.edu\/}).}\n\\label{3color}\n\\end{figure}\n\nIn this paper, we present observations of MIR and FIR fine-structure cooling lines in NGC\\,4214, which provide key diagnostics of the physical conditions of the ISM. We focus our analysis on the two main star-forming complexes. The line emission is analyzed with radiative transfer models to characterize the ISM conditions. We take into account directly observed star formation histories and explore how they affect the IR line emission. Photometry is used for the energy budget of the models. The structure of this paper is the following: Sect.~\\ref{data} describes the data, Sect.~\\ref{method} describes the model, and the results are presented in Sect.~\\ref{results}. We summarize and discuss our results in Sect.~\\ref{discussion}. \n\n\n\\section{Data}\n\\label{data}\n\\subsection{{\\it Herschel} data}\nWe used observations of NGC\\,4214 obtained by the PACS instrument \\citep{poglitsch-2010} onboard the \\textit{Herschel}\\xspace Space Observatory \\citep{pilbratt-2010} as part of the Dwarf Galaxy Survey \\citep{madden-2013}. The list of observations can be found in Table~\\ref{AOR}. The photometry data at 70$\\mu$m\\xspace, 100$\\mu$m\\xspace, and 160$\\mu$m\\xspace, with respective beam sizes (FWHM) of 5.6$^{\\prime\\prime}$\\xspace, 6.7$^{\\prime\\prime}$\\xspace, and 11.3$^{\\prime\\prime}$\\xspace, were published by \\cite{remy-2013}. These bands cover the peak of the SED originating from the reprocessed stellar light by the dust. \nThe spectroscopy comprises observations of the \\oiii88$\\mu$m\\xspace and \\nii122$\\mu$m\\xspace lines, which trace the ionized gas, as well as the \\cii157$\\mu$m\\xspace, \\oi63$\\mu$m\\xspace, and \\oi145$\\mu$m\\xspace lines, which trace the PDR. The data consist of small mappings of $5\\times5$ rasters separated by $\\sim$16$^{\\prime\\prime}$\\xspace for \\oiii88$\\mu$m\\xspace and \\oi63$\\mu$m\\xspace and $3\\times3$ rasters separated by $\\sim$24$^{\\prime\\prime}$\\xspace for the other lines, ensuring a uniform coverage of $1.6^{\\prime}\\times1.6^{\\prime}$. Originally presented in \\cite{cormier-2010}, the PACS spectral data were re-processed with the reduction and analysis software HIPE user release v.11 \\citep{ott-2010} and PACSman v.3.5 \\citep{lebouteiller-2012}. With the improved calibration and definition of the regions, flux maps are globally consistent with those published in \\cite{cormier-2010} and line ratios agree within 30\\%. Flux maps of the \\cii157$\\mu$m\\xspace and \\oi63$\\mu$m\\xspace lines are shown in Fig.\\,\\ref{mask}. The associated error maps include data and line-fitting uncertainties, but not calibration uncertainties, which are on the order of 15\\%. \nThe FWHM is 9.5$^{\\prime\\prime}$\\xspace below 100$\\mu$m\\xspace and 10$^{\\prime\\prime}$\\xspace, 11$^{\\prime\\prime}$\\xspace, 12$^{\\prime\\prime}$\\xspace at 122$\\mu$m\\xspace, 145$\\mu$m\\xspace, 160$\\mu$m\\xspace, respectively. All maps were convolved to the \\cii157$\\mu$m\\xspace resolution of $\\sim$12$^{\\prime\\prime}$\\xspace , which at the distance of NGC\\,4214 corresponds to a physical scale of 175\\,pc. \nIn both the photometry and spectroscopy data sets, the convolutions were performed using kernels provided by \\cite{aniano-2011}\\footnote{\\url{http:\/\/www.astro.princeton.edu\/~ganiano\/Kernels\/}}.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5.585cm]{fig2a.eps}\\hspace{-1mm}\n\\includegraphics[width=4.25cm]{fig2b.eps}\\hspace{-1mm}\n\\includegraphics[width=4.25cm]{fig2c.eps}\\hspace{-1mm}\n\\includegraphics[width=4.25cm]{fig2d.eps}\n\\centering\n\\caption{\nMaps of the \\cii157$\\mu$m\\xspace, \\oi63$\\mu$m\\xspace, \nand TIR emission in NGC\\,4214. Units are W\\,m$^{-2}$\\,sr$^{-1}$. \nThe two star-forming regions, as defined in Sect.~\\ref{sect:definesf}, \nare outlined with red contours. \nThe right panel shows the \\textit{Spitzer}\\xspace IRS mapping strategy. \nOrange: Long-High module coverage; \ncyan: Short-High module coverage; \ngray background: \\cii157$\\mu$m\\xspace map. \n}\n\\label{mask}\n\\end{figure*}\n\n\n\n\\subsection{{\\it Spitzer} data}\nNGC\\,4214 was observed with the three instruments onboard the \\textit{Spitzer}\\xspace space telescope \\citep{werner-2004}. We used the MIPS 24$\\mu$m\\xspace observations obtained within the Local Volume Legacy Survey \\citep{dale-2009} that were processed by \\cite{bendo-2012}. The MIPS 24$\\mu$m\\xspace map, which has an original FWHM of 5.9$^{\\prime\\prime}$\\xspace, was convolved to a resolution of $\\sim$12$^{\\prime\\prime}$\\xspace to match that of the PACS data. \\par\n\nThe IRS observations (program ID 3177, PI. Skillman) consist of small mappings of the two main star-forming regions in high-resolution mode \\citep{houck-2004}. We extracted the data from the \\textit{Spitzer}\\xspace Heritage Archive (see Table~\\ref{AOR}) and processed them with the software CUBISM v1.8 \\citep{smith-2007}. We used the default mapping procedure and bad pixel removal to produce spectral cubes with pixel sizes 2.26$^{\\prime\\prime}$\\xspace for the Short-High module and 4.46$^{\\prime\\prime}$\\xspace for the Long-High module. \nWe then created surface brightness maps for all spectral lines of interest -- \\siv10.5$\\mu$m\\xspace, \\neii12.8$\\mu$m\\xspace, \\neiii15.6$\\mu$m\\xspace, \\siii18.7$\\mu$m\\xspace, \\siii33.5$\\mu$m\\xspace, which all trace H\\,{\\sc ii}\\xspace regions -- in the following way: for each pixel of the cube, we extracted the signal with a range of $\\pm$0.7$\\mu$m\\xspace around the line and fit a polynomial of order two for the baseline and a Gaussian for the line with the IDL routine \\texttt{mpfit}. For a more stable fit, the peak of the Gaussian is required to be positive, the position of the peak is expected within one instrumental FWHM of the rest wavelength, and the width is limited to the instrumental resolution ($R=600$). Finally, we added random noise and iterated the fit $300$ times to estimate the best-fit parameters as the median of the resulting parameters and the error on those parameters as the standard deviation. \nError maps again include data and line-fitting uncertainties, but not calibration uncertainties, which are on the order of 5\\%. \nThe coverage of the star-forming regions of the galaxy in the IRS maps is only partial, as shown in Fig.\\,\\ref{mask}. No integrated values for the flux of the whole regions could be retrieved. To obtain a representative value for the line flux in each region, we regridded the IRS maps to that of the \\oiii88$\\mu$m\\xspace map. Then we selected the pixels that appear in both maps and scaled the emission of these pixels to the \\oiii88$\\mu$m\\xspace line to infer the corresponding line fluxes for the star-forming regions as a whole. \n\n\\begin{table}\n\\caption{List of \\textit{Herschel}\\xspace and \\textit{Spitzer}\\xspace observations.}\n\\label{AOR}\n \\centering\n\\vspace{-5pt}\n  \\begin{tabular}{l l}\n\\hline \\hline\n    \\vspace{-10pt}\\\\\n\\multicolumn{2}{c}{\\textit{Herschel}\\xspace data}\\\\\nInstrument & Observation Identification number (OBSID) \\\\\n\\hline\n    \\vspace{-10pt}\\\\\nPACS phot. & $1342211803$, $1342211804$, $1342211805$, \\\\\n & $1342211806$ \\\\\nPACS spec. & $1342187843$, $1342187844$, $1342187845$, \\\\\n & $1342188034$, $1342188035$, $1342188036$ \\\\\n\\hline \\hline\n    \\vspace{-10pt}\\\\\n\\multicolumn{2}{c}{\\textit{Spitzer}\\xspace data}\\\\\nInstrument & Astronomical Observation Request (AOR) \\\\\n\\hline\n    \\vspace{-10pt}\\\\\nMIPS 24$\\mu$m\\xspace & $22652672$, $22652928$, $22710528$, $22710784$ \\\\\n\\multicolumn{2}{l}{IRS Short-High} \\\\\n~On-source: & $10426368$, $10426624$, $10426880$, $10427136$ \\\\\n~Background: & $13728256$, $13729792$, $13730304$, $13730816$, \\\\\n& $13733120$, $13762304$, $13767424$, $13767936$, \\\\\n& $13768448$, $13768704$, $13769728$, $13770496$, \\\\\n& $13773568$\\\\\n\\multicolumn{2}{l}{IRS Long-High} \\\\\n~On-source: &$10424832$, $10425088$, $10425344$, $10425600$, \\\\\n& $10425856$, $10426112$, $10427392$\\\\\n~Background: & $13728768$, $13729792$, $13730304$, $13730816$, \\\\\n& $13733120$, $13763328$, $13764352$, $13765376$, \\\\\n&$13767424$, $13767936$, $13768448$, $13768704$,\\\\\n&  $13769728$, $13770496$, $13773568$\\\\\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\nWe focus on these selected IRS lines because they are among the brightest MIR fine-structure cooling lines and can be used as reliable diagnostics of the physical conditions in H\\,{\\sc ii}\\xspace regions. In general, the intensity or luminosity ratio of two lines of the same element but different ionization level is indicative of the radiation field hardness. Such diagnostics are the \\neiii15.6$\\mu$m\\xspace\/\\neii12.8$\\mu$m\\xspace or the \\siv10.5$\\mu$m\\xspace\/\\siii18.7$\\mu$m\\xspace ratios \\citep[e.g.,][]{verma-2003}, which are insensitive to the density because of their high critical densities (see Table~\\ref{fluxes}). Accordingly, species of the same ionization level but different transition are indicative of the electron density as a result of the different critical densities for each transition \\citep{osterbrock}. Examples are the \\siii18.7$\\mu$m\\xspace\/\\siii33.5$\\mu$m\\xspace, \\neiii15.6$\\mu$m\\xspace\/\\neiii36.0$\\mu$m\\xspace, or \\nii122$\\mu$m\\xspace\/\\nii205$\\mu$m\\xspace ratios \\citep[e.g.,][]{rubin-1994}. These diagnostics are insensitive to the temperature inside the H\\,{\\sc ii}\\xspace region. Unfortunately, the \\neiii36.0$\\mu$m\\xspace and \\nii205$\\mu$m\\xspace lines fall at the edge of the IRS and PACS wavelength ranges, respectively, where the spectra are too noisy to detect or derive a reliable line ratio for the two star-forming regions. Therefore we relied on the [S\\,{\\sc iii}]\\xspace line to probe the electron density.\n\n\n\n\\subsection{Total infrared luminosity map}\nTo construct a total infrared (TIR) luminosity map of the galaxy, we combined the MIPS 24$\\mu$m\\xspace and PACS 70, 100, and 160$\\mu$m\\xspace data, following \\cite{galametz-2013}: \n\\begin{equation}\nL_{\\rm TIR}=\\int_{3\\mu m}^{1\\,100\\mu m}L_{\\nu}d\\nu=\\sum c_i L_i\n.\\end{equation}\nWe used the values of the coefficients, $c_i$, from Table~3 of their paper: $[c_{24},c_{70},c_{100},c_{160}] = [2.064,0.539,0.277,0.938]$. This method, although slightly less accurate than a direct integration of a well-sampled SED, does not require degrading the resolution of our data beyond the PACS 160$\\mu$m\\xspace beam and is sufficient for our modeling purposes to estimate the energy budget in the star-forming regions. The $L_{\\rm TIR}$ map is shown in Fig.\\,\\ref{mask}.\n\n\n\n\\subsection{Defining the star-forming regions}\n\\label{sect:definesf}\nTo define the apertures for the main star-forming regions, we set a threshold for the signal-to-noise ratio (S\/N) equal to 5 in each individual PACS 70$\\mu$m\\xspace, 100$\\mu$m\\xspace, and 160$\\mu$m\\xspace photometry and PACS spectral map. We masked all pixels below this S\/N and drew the contours, which include all the remaining unmasked pixels separately for the photometry and the spectroscopy maps. Because the emission in the photometry maps is more extended, we kept the contours from the photometry and used these apertures throughout the analysis to define the two star-forming regions, as shown in Fig.\\,\\ref{mask}. This means that pixels in the spectroscopy maps that are below the S\/N threshold but within the region contours are still counted. \nThe fluxes and uncertainties for the line and TIR emission were measured taking into account all pixels in each region. They\nare reported in Table~\\ref{fluxes}. \nThe ISM emission (gas and dust) peaks in these two regions, and most of the line fluxes are twice as high in the central region as$\\text{ in}$ the southern region, except for \\neiii15.6$\\mu$m\\xspace (factor 1.3) and \\siv10.5$\\mu$m\\xspace, which have lower fluxes toward the central region. This hints at different physical conditions in the two regions, which we investigate with radiative transfer models. \n\n\\begin{table*}\n\\caption{Observed MIR and FIR fluxes for the line and broadband emission.\nUncertainties on the fluxes include data and line-fitting uncertainties, but not \ncalibration uncertainties, which are on the order of 5\\% for the \\textit{Spitzer}\\xspace lines and 15\\% for the \\textit{Herschel}\\xspace lines.\nCritical density and ionization potential values are taken from \\citet{cormier-2012}. \nCritical densities are noted [e] for collisions with electrons and [H] for collisions with hydrogen atoms.}\n\\label{fluxes}\n \\centering\n\\vspace{-5pt}\n  \\begin{tabular}{l c c c c c}\n\\hline\\hline\n    \\vspace{-10pt}\\\\\n &   & \\multicolumn{2}{c}{Flux $\\pm$ uncertainty} & & Ionization \\\\ \n Line   &  Wavelength   &       \\multicolumn{2}{c}{($\\times10^{-16}$ W~m$^{-2}$)} & Critical density  &  potential  \\\\ \n                &       ($\\mu$m\\xspace)          &       Region I          &       Region II & (cm$^{-3}$) &      (eV)    \\\\\n\\hline\n    \\vspace{-10pt}\\\\\n\\lbrack \\textsc{S\\,iv}] &       10.51 & $5.68\\pm0.21$           & $8.40\\pm0.10$   & $5\\times10^4$ [e]     & 34.79\\\\       \n\\lbrack Ne\\textsc{\\,ii}] &      12.81 & $8.98\\pm0.22$           & $4.13\\pm0.11$   & $7\\times10^5$ [e]     & 21.56\\\\\n\\lbrack Ne\\textsc{\\,iii}] &     15.56 & $18.70\\pm0.14$  & $14.25\\pm0.08$        & $3\\times10^5$ [e]       &40.96\\\\\n\\lbrack \\textsc{S\\,iii}] &      18.71 & $11.78\\pm0.20$  & $6.85\\pm0.08$ & $2\\times10^4$ [e]       & 23.34 \\\\\n\\lbrack \\textsc{S\\,iii}] &      33.48 & $18.71\\pm0.27$  & $8.20\\pm0.12$ & $7\\times10^3$ [e]       & 23.34\\\\\n\\lbrack \\textsc{O\\,i}] &        63.18 & $10.11\\pm0.35$  & $4.06\\pm0.21$ & $5\\times10^5$ [H]       & -\\\\\n\\lbrack \\textsc{O\\,iii}] &      88.36 & $31.86\\pm0.62$  & $13.50\\pm0.40$        & $5\\times10^2$ [e]       & 35.12\\\\\n\\lbrack \\textsc{N\\,ii}] &       121.90 &        $0.44\\pm0.20$           & $0.14\\pm0.08$   & $3\\times10^2$ [e]     & 14.53\\\\ \n\\lbrack \\textsc{O\\,i}] &        145.52 &        $0.65\\pm0.09$           & $0.32\\pm0.07$   & $1\\times10^5$ [H]     & -\\\\\n\\lbrack \\textsc{C\\,ii}] &       157.74 &        $26.34\\pm0.33$  & $10.05\\pm0.21$  & 50 [e], $3\\times10^3$ [H]     & 11.26\\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{l c c c}\n\\hline\n    \\vspace{-10pt}\\\\\n   &  & \\multicolumn{2}{c}{Flux density $\\pm$ uncertainty} \\\\\n Broadband      &       Wavelength      & \\multicolumn{2}{c}{(Jy)}\\\\\n                &               ($\\mu$m\\xspace)  &       Region I          &       Region II              \\\\\n\\hline\n    \\vspace{-10pt}\\\\\nMIPS  & 24 & $0.67\\pm0.01$ & $0.48\\pm0.01$ \\\\\nPACS & 70 & $7.36\\pm0.21$ &     $3.72\\pm0.12$ \\\\\nPACS & 100 & $7.91\\pm0.19$ & $4.10\\pm0.11$ \\\\\nPACS & 160 & $6.07\\pm0.10$ & $3.23\\pm0.06$ \\\\\n\\hline\n    \\vspace{-10pt}\\\\\n{$L_{\\rm TIR}$ (erg\\,s$^{-1}$)} & 3 -- 1100 & $5.25\\times10^{41}$ &       $3.02\\times10^{41}$     \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{table*}\n\n\\section{Description of the model}\n\\label{method}\n\\subsection{Model geometry and strategy}\nOur objective is to characterize the physical conditions of the ISM phases from which the IR emission arises in NGC\\,4214. To that end, we used the spectral synthesis code \\textsc{Cloudy} v.13, last described by \\cite{ferland-2013}. We performed a multiphase detailed modeling of the ISM for which we combined line and continuum emission, following the method described in \\cite{cormier-2012}. We considered the two main star-forming regions of NGC\\,4214: the most evolved central region (NW-I) and the southern region (SE-II). Here we present the main aspects of the model and how it is applied to each region. The components\/ISM phases of the model are \n\\begin{enumerate}\n         \\item a central source of radiation,\n         \\item an ionized medium component (H\\,{\\sc ii}\\xspace region) that surrounds the central source, \n         \\item a neutral medium (PDR) surrounding the H\\,{\\sc ii}\\xspace region.\n\\end{enumerate}\n\\noindent \nThis method assumes a single radiation source responsible for the observed SED of the studied region. In other words, we took all of the different sources (star clusters) and the surrounding clouds from which they have formed and represented them with one central source and one surrounding cloud. We thus targeted the integrated properties of each region. In practice, we mixed components of the ISM that have different composition and properties\nand blend them in a single system. \nThe applied geometry is spherical. The source is in the center and the illuminated face of the cloud lies at a certain distance that we call inner radius. In our case, the effective geometry is one-dimensional plane-parallel because the cloud forms a thin shell and its distance from the radiation source is large. \\par\n\nThe radiation source, representative of the stars that populate the clusters of the star-forming region, illuminates a cloud of dust and gas. It controls the ionization parameter, $U$, which characterizes the field and is defined as the ratio of the incident ionizing photon density to the hydrogen density. Hard UV photons from the source ionize hydrogen and form the H\\,{\\sc ii}\\xspace region. As this radiation is transmitted through the cloud, it is attenuated\nand thus becomes softer, which decreases its influence on ionization. However, it still controls the processes further in the cloud (in the PDR).\\par\n\nThe adopted strategy is to treat the H\\,{\\sc ii}\\xspace region first and then use the H\\,{\\sc ii}\\xspace region parameters as input for the PDR modeling. This allows for a self-consistent approach \\citep{abel-2005}, which is usually not directly available in standard PDR codes (see \\citealt{roellig-2007} for a comparison of PDR codes), and is important to accurately derive the radiation field that impinges\non the PDR. We first ran the simulation until the end of the H\\,{\\sc ii}\\xspace region, choosing to stop the simulation when the (electron) temperature reached 500\\,K to ensure that the model had transitioned to the atomic phase. We iterated to optimize our parameters so\nthat they matched the observed emission of known H\\,{\\sc ii}\\xspace region-diagnostic lines ([\\textsc{S\\,iv}]\\,10.5$\\mu$m\\xspace, [Ne\\textsc{\\,ii}]\\,12.8$\\mu$m\\xspace, [Ne\\textsc{\\,iii}]\\,15.6$\\mu$m\\xspace, [\\textsc{S\\,iii}]\\,18.7$\\mu$m\\xspace, [\\textsc{S\\,iii}]\\,33.5$\\mu$m\\xspace, \\oiii88$\\mu$m\\xspace, and \\nii122$\\mu$m\\xspace). Then we fed the result of this model to the PDR and compared the predictions to the remaining three PDR lines observed: \\oi63$\\mu$m\\xspace, \\oi145$\\mu$m\\xspace, and \\cii157$\\mu$m\\xspace, choosing a visual extinction of 10\\,mag as the stopping criterion. At this point, the gas temperature had fallen to roughly 10\\,K. \n\n\n\n\\subsection{Model parameters}\nWe constrained the properties of the star-forming regions by varying some of the parameters that control the physics of the models while keeping others fixed. The main parameters that we consider are\n\\begin{enumerate}\n  \\item a radiation field source: shape, age, luminosity (varied),\n  \\item the hydrogen density of the ISM, $n_{\\rm H}$ (varied),\n  \\item the ISM gas elemental abundances (fixed),\n  \\item the inner radius, $r_{\\rm in}$ (varied),\n  \\item the magnetic field, B (fixed),\n  \\item the turbulent velocity, $v_{\\rm turb}$ (fixed).\n\\end{enumerate}\n\\noindent Parameters that are fixed were set to values from the literature. The other parameters were varied inside a range whose width reflects the dispersion of published measurements or of the data. The main parameter of interest for this study is the radiation field, which was varied within a range guided by studies of the star formation history.\n\n\n\\subsubsection{Hydrogen density ($n_{\\rm H}$)}\n\\label{density}\nWe performed our simulations assuming pressure equilibrium. As the model proceeds through consecutive zones of the cloud, it keeps the pressure constant. Thus, the density of the medium varies to satisfy this equilibrium. The initial density that we specified in the models is the density at the illuminated face of the cloud, where the H\\,{\\sc ii}\\xspace region starts. The initial values and the range we probed are motivated by the observed \\siii18.7$\\mu$m\\xspace\/\\siii33.5$\\mu$m\\xspace ratio in the H\\,{\\sc ii}\\xspace region, which is\nknown to be sensitive to the electron density in the range 10$^2$-10$^4$\\,cm$^{-3}$ \\citep{rubin-1994}. We therefore let the initial density vary in the range 100-300\\,cm$^{-3}$ for the central region and 300-600\\,cm$^{-3}$ for the southern region with a common step of 25\\,cm$^{-3}$. By iterating this procedure, we constrained the best values for the density at the beginning of the H\\,{\\sc ii}\\xspace region.    \n\n\n\\subsubsection{Inner radius ($r_{\\rm in}$)}\nIn our spherical geometry, the source is at the center and is surrounded by a cloud. The illuminated face of the cloud lies at a certain distance $r_{\\rm in}$. This is not a strictly physically constrained parameter because the setup we used does not realistically model each cluster, but instead tries to mimic a whole region and reproduce its emission. The variation of this radius changes the photons flux and thus is expected to affect our results. We let $r_{\\rm in}$ vary from 1 to 100\\,pc for both regions.\n\n\n\\subsubsection{Elemental abundances}\nElemental abundances in the models were set to the observed values for oxygen, sulfur, nitrogen, and neon, taken from \\cite{kobulnicky-1996}. Some measurements partially cover our defined regions, and we adopted them as representative. Exclusively for carbon, we scaled its abundance according to the study on the dependence of $\\log(C\/O)$ on metallicity by \\cite{izotov-1999}. For other elements, we used the default ISM composition of \\textsc{Cloudy} and scaled the abundances to our metallicity (1\/3). The values used are indicated in Table~\\ref{elements}. \n\n\\begin{table}\n  \\caption{Elemental abundances in NGC\\,4214. \n  Values (in logarithmic scale) for NGC\\,4214 are taken from \\citet{kobulnicky-1996} and solar values from \\citet{asplund-2009}.}\n\\label{elements}\n \\centering\n\\vspace{-5pt}\n   \\begin{tabular}{c c c c}\n   \\hline \n    \\vspace{-10pt}\\\\\n        Abundance               & Region I      & Region II  & Solar value \\\\\n        \\hline\n    \\vspace{-10pt}\\\\\n        \\lbrack O\/H]            & $-3.795\\pm0.05$       & $-3.64\\pm0.04$        & $-3.31\\pm0.05$ \\\\\n        \\lbrack S\/H]            & $-5.380\\pm0.06$       & $-5.21\\pm0.06$        & $-4.88\\pm0.03$ \\\\\n        \\lbrack N\/H]            & $-5.094\\pm1.00$       & $-5.02\\pm0.10$        & $-4.17\\pm0.05$ \\\\\n        \\lbrack Ne\/H]           & $-4.535\\pm0.11$       & $-4.51\\pm0.08$        & $-4.07\\pm0.10$ \\\\\n        \\lbrack C\/H]            & $-4.295\\pm0.30$       & $-4.14\\pm0.30$        & $-3.57\\pm0.05$ \\\\\n        \\hline \n   \\end{tabular}\n \\end{table}\n\n\\subsubsection{Radiation field - shape and energy}\nWe used the code \\textsc{Starburst99}  \\citep{leitherer-2010} to produce a stellar population spectrum that serves as input for our models. We chose a Kroupa initial mass function between 0.08 and 120\\,M$_{\\odot}$ \\citep{kroupa-2001}, as done in \\cite{andrews-2013}, and Padova asymptotic giant branch tracks with Z=0.004. \nAs discussed above, we did not model each cluster individually, but we used integrated emission from the entire star-forming regions instead. We tried to be as close to the shape and intensity of the radiation field of the H\\,{\\sc ii}\\xspace regions as possible. Motivated by the star formation history of the galaxy presented in \\cite{mcquinn-2010} and \\cite{williams-2011}, we tested the following cases:\n\\begin{itemize}\n\\item For the central region, we considered two limiting scenarios: \n(i)~a single-burst star formation event and (ii)~a continuous star formation model, with SFR=0.07\\,${\\rm M_{\\odot}\\,yr^{-1}}$. The ages of the clusters were varied within a range of (i)~1-20\\,Myr (in steps of 0.5\\,Myr) and (ii)~200-1000\\,Myr (in steps of 200\\,Myr).\n\\item For the southern region, we considered a single-burst event with an age that varied from 1 to 20\\,Myr (in steps\nof 0.5\\,Myr).\n\\end{itemize}\n\\noindent\nA fixed mass of 10$^5$\\,M$_{\\odot}$ (typical cluster mass in \\citealt{sollima-2013}) was considered for the single bursts, where the stars are created at once (delta burst). Our value for the SFR (0.07\\,${\\rm M_{\\odot}\\,yr^{-1}}$) is representative of the `average' rate at which this galaxy formed stars within the past 1\\,Gyr of its history. The ages of the clusters in both regions were guided by values from \\cite{ubeda-2007}, \\cite{sollima-2013}, and \\cite{sollima-2014}. \\cite{ubeda-2007} found 2-7\\,Myr for region~I, along with extended clusters in the same region with ages of 150 to 190\\,Myr. In region~II, they found ages spreading around 2\\,Myr. \\cite{sollima-2013,sollima-2014} reported a larger spread in ages. In region~I, they obtained ages around a median of 14\\,Myr, and for the more extended clusters the ages lie between 10 and 300\\,Myr. For region~II, they found a median age of $\\sim$20\\,Myr. \\par\n\nFor the luminosity emitted from each region, we chose to use the TIR luminosity as a first approximation of the luminosity of the starburst. This choice implies the assumption that all radiation from the clusters in the region is reprocessed by the dust and thus emitted at longer wavelengths. In doing so, we kept in mind that there can be processes that we did not model (UV escape fraction or a diffuse ionized medium, for example) and that can contribute to this radiated energy (see Sect.~\\ref{discussion}). \n\n\n\\subsubsection{Magnetic field strength (B) and turbulent velocity ($v_{\\rm turb}$)}\nMagnetic fields and turbulence play an important role in structuring the ISM \\citep[e.g.,][]{mckee-2007}. When \\textsc{Cloudy} solves the pressure equilibrium for each zone of the modeled cloud, a magnetic pressure term equal to $\\displaystyle P_{\\rm B}=\\frac{B^2}{8\\pi}$ is included in the equation of state along with a turbulent pressure term equal to $\\displaystyle P_{\\rm turb}=2.8 \\cdot 10^6 \\cdot 3 \\cdot (\\frac{n_{\\rm H}}{10^5\\,{\\rm cm^{-3}}})(\\frac{v_{\\rm turb}}{\\rm {1\\,km\\,s^{-1}}})^2$~[cm$^{-3}$\\,K], for isotropic turbulent motions, where $n_{\\rm H}$ is the total hydrogen density and $v_{\\rm turb}$ is the turbulent velocity (see the {\\sc Hazy} documentation of \\textsc{Cloudy} for more information). \\par\n\nThe magnetic field of NGC\\,4214 was measured by \\cite{kepley-2011} using multiwavelength radio emission. The reported field strength in the center of the galaxy is 30\\,$\\mu$G, and the pressure term due to this field has the same order of magnitude as the hot gas and the gravitational contributions. Since it is not well known how the observed magnetic field might affect our observed line intensities, we excluded it from our default models and tested one case with a magnetic field strength of 30\\,$\\mu$G. \\par \n\nAnother potential energy source to consider can arise from the dissipation of turbulence. Turbulent energy is converted into thermal energy as it cascades from large scales to small scales through dissipation. However, we did not resolve size scales for which we can measure this. \n\\textsc{Cloudy} does not attempt to model the dissipation mechanism, but assumes a simple thermal energy source based on line width. The turbulent velocity was set to a value of 1.5\\,km\\,s$^{-1}$ by default in our models, and we tested two other cases: one case with an intermediate turbulent velocity ($v_{\\rm turb}$=3\\,km\\,s$^{-1}$ or FWHM=5\\,km\\,s$^{-1}$) that corresponds to the approximate line width observed in the CO(1-0) data by \\cite{walter-2001}, and one case with a high turbulent velocity ($v_{\\rm turb}$=50\\,km\\,s$^{-1}$) as found in the diffuse ionized gas by \\cite{WilcotsThurow-2001} and used also in \\cite{kepley-2011}.\n\nNevertheless, we explore the effects of excluding or including magnetic fields and turbulence in Sect.~\\ref{magn}. \n\n\n\\subsection{Determination of best-fitting models}\n\\label{minchi}\nWe aim to converge on a unique parameter set that best describes the conditions of the regions. We first ran models for which\nwe varied the parameters in a coarse grid to narrow down the parameter space, using ranges of values found in the literature to start with. We then used the \\textit{optimization} option of \\textsc{Cloudy}, which automatically varies the specified parameters in a finer grid to find the optimal solution.\n\nWe computed the average $\\chi ^2$, denoted $\\bar\\chi ^2$, for each model by comparing the observed fluxes of \\siv10.5$\\mu$m\\xspace, \\neiii15.6$\\mu$m\\xspace, \\siii18.7$\\mu$m\\xspace, \\siii33.5$\\mu$m\\xspace, and \\oiii88$\\mu$m\\xspace to the fluxes predicted from the radiative transfer calculation. These lines are the most luminous and most strongly correlated with the H\\,{\\sc ii}\\xspace region (as opposed to [N\\,{\\sc ii}]\\xspace and [Ne\\,{\\sc ii}]\\xspace which, from experience, can arise from other phases). We refer to these as the optimized lines in Table~\\ref{chi2ionic}. The goodness of the line emission fit is given by low $\\bar\\chi ^2$ values.\nThe \\textit{optimization} method of \\textsc{Cloudy} searches the minimum of $\\bar\\chi ^2$ that is defined as\n\\begin{align}\n   \\bar\\chi ^2=\\frac{1}{n}\\sum{\\chi_i ^2}=\\frac{1}{n}\\sum{\\frac{( M_i - O_i)^2}{(min\\{O_i;M_i\\} \\times \\sigma_i)^2}},\n\\end{align}\n\\noindent where $n$ is the number of lines optimized and $\\chi_i^2$ are the $\\chi^2$ values of the individual optimized lines.\nM$_i$ and O$_i$ are the modeled and observed fluxes, and $\\sigma_i$ is the fractional error on the observed flux (uncertainty\/flux) with calibration uncertainties added in quadrature to the measured uncertainties described in Sect.~\\ref{data}. We have five observables (the ionic lines listed above) and varied four parameters (cluster age, source luminosity, hydrogen density, and inner radius). \n\n\n\n\\section{Results}\n\\label{results}\nIn this section we present the model results for the two regions according to the star formation histories considered. Parameters of the best-fitting models and their corresponding $\\chi^2$ values are reported in Tables~\\ref{model} and \\ref{chi2ionic}, respectively. \n\n\n\\begin{table*}[!t]\n\\caption{Input parameters for the best-fitting models. \nValues are given at the illuminated face of the cloud. \nValues in parenthesis for the magnetic field strength \nand turbulent velocities are tested in Sect.~\\ref{magn}.}\n\\label{model}\n  \\centering\n\\vspace{-5pt}\n  \\begin{tabular}{l c c c}\n  \\hline \n    \\vspace{-10pt}\\\\\n  Parameter                             & \\multicolumn{2}{c}{Central region} & Southern region\\\\\n                                                & Single burst & Continuous & Single burst \\\\\n  \\hline\n    \\vspace{-10pt}\\\\\n  Density $n_{\\rm H}$ [cm$^{-3}$]               & 155           & 180             & 440   \\\\\n  Inner radius $r_{\\rm in}$ [pc]                        & 85.4                 & 62.1          & 22.3  \\\\\n  Stellar age t [Myr]                                   & 4.1                 & 440           & 3.9            \\\\\n  Total luminosity L [erg\\,s$^{-1}$]            & $1.15\\times10^{42}$ & $1.91\\times10^{42}$   & $5.25\\times10^{41}$   \\\\\n  Magnetic field B [$\\mu$G]                     & -                     & - (30)          & -             \\\\\n  Turbulent velocity $v_{\\rm turb}$ [km\\,s$^{-1}$]      & 1.5         & 1.5 (3, 50)   & 1.5           \\\\\n  \\hline \n  \\end{tabular}\n\\end{table*}\n\\begin{figure*}\n\\centering\n\\includegraphics[clip,trim=0 20mm 0 0,width=17cm,height=5cm]{fig3a}\n\\includegraphics[clip,trim=0 20mm 0 0,width=17cm,height=5cm]{fig3b}\n\\caption{Results for the central region (top panel) \nand for the southern region (bottom panel): line emission \nfor the H\\,{\\sc ii}\\xspace region (left side) and the PDR (right side), assuming \npressure equilibrium.\nGreen bars represent the observations, blue bars our single-burst model predictions, \nand gray bars with a dashed outline the continuous star formation model predictions.}\n\\label{h2bars}\n\\end{figure*}\n\n\\subsection{Line emission}\n\\label{modelresults}\n\\subsubsection{Central region (I): The single-burst model}\n\\label{SB}\nFor the single-burst star formation event, the best-fitting model of the central region has the following parameters: burst age of 4.1\\,Myr with luminosity $1.1\\times10^{42}$\\,erg\\,s$^{-1}$, density of 155\\,cm$^{-3}$, and $r_{\\rm in}\\simeq85$\\,pc. The corresponding ionization parameter at the illuminating face of the cloud is $\\log(U)=-2.7$. For the H\\,{\\sc ii}\\xspace region, all optimized lines are matched within $\\pm$30\\%, and \\neii12.8$\\mu$m\\xspace and \\nii122$\\mu$m\\xspace are underpredicted by a factor of $\\sim$8 and 3, respectively (see Fig.\\,\\ref{h2bars}, blue bars in the top panel). \nIn the PDR, the \\oi145$\\mu$m\\xspace and 63$\\mu$m\\xspace lines are overpredicted by a factor of $\\sim$2.5, and the \\cii157$\\mu$m\\xspace line is matched within 20\\%. We note that [C\\,{\\sc ii}]\\xspace emission can arise from both the neutral and the ionized phases of the ISM, with a potentially non-negligible contribution from the warm ionized medium in the Milky Way \\citep{heiles-1994}. The H\\,{\\sc ii}\\xspace region of our best-fitting model contributes negligibly to the predicted \\cii157$\\mu$m\\xspace and [O\\,{\\sc i}]\\xspace emission ($<$3\\%). The ultraviolet radiation field strength $G_0$ inside the PDR is $455$ in units of the equivalent \\cite{habing-1968} flux ($1\\,G_0 = 1.6\\times10^{-3}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$). The input luminosity required by the model, which all comes out as $L_{\\rm TIR}$ in the PDR, is $\\text{about twice as}$  high\nas the observed $L_{\\rm TIR}$ in that region. \n\nThese results represent the simplifying case that the recent starburst dominates the star formation in this central region, so the line emission can be explained with a single burst. The age of the burst in the model nicely agrees with the range of ages from \\cite{ubeda-2007} and is at the younger end of ages from \\cite{sollima-2013,sollima-2014}.  \n\n\n\n\\subsubsection{Central region (I): The continuous star formation model}\nIn the continuous star formation scenario, the best-fitting model of the central region has the following parameters: stellar age of 440\\,Myr with luminosity $1.9\\times10^{42}$\\,erg\\,s$^{-1}$, density of 180\\,cm$^{-3}$, and $r_{\\rm in}\\simeq62$\\,pc. The ionization parameter is $\\log(U)=-2.5$. Line predictions for the H\\,{\\sc ii}\\xspace region and the PDR from this model solution are shown in Fig.\\,\\ref{h2bars} (gray bars). For all optimized lines (\\oiii88$\\mu$m\\xspace, \\siii18.7 and 33.5$\\mu$m\\xspace, \\siv10.5$\\mu$m\\xspace, and \\neiii15.6$\\mu$m\\xspace), the model matches the observations within $\\pm$20\\%. The two other ionic lines, \\neii12.8$\\mu$m\\xspace and \\nii122$\\mu$m\\xspace, are underpredicted by a factor of $\\sim$7 and 4. \nIn the PDR, [C\\,{\\sc ii}]\\xspace and the [O\\,{\\sc i}]\\xspace lines are matched within a factor of $\\sim$2. The contribution of the H\\,{\\sc ii}\\xspace region to the predicted PDR emission is only 1\\%. \nWe find $G_0\\simeq1.2\\times10^3$, which is higher than in the single-burst case. The luminosity of the model exceeds the observed $L_{\\rm TIR}$ by a factor of 3.5. \\par\n\nThe results represent the simplifying case of continuous star formation dominating this region, with the starbursts being embedded in it. The age of the model agrees well with the star formation event in the window 400-500\\,Myr ago reported by \\cite{mcquinn-2010}. \n\n\n \n\\subsubsection{Southern region (II)}\n\\label{southres}\nThe best-fitting model for the southern region is characterized by a burst age of 3.9\\,Myr with luminosity $5.3\\times10^{41}$\\,erg\\,s$^{-1}$, density of 440\\,cm$^{-3}$, and $r_{\\rm in}\\simeq22$\\,pc. The ionization parameter is $\\log(U)=-2.3$. Line predictions for the H\\,{\\sc ii}\\xspace region and the PDR are shown in the bottom panel of Fig.\\,\\ref{h2bars}. This burst found for the southern region is slightly younger than the burst in the central region, in agreement with \\cite{ubeda-2007}. In the H\\,{\\sc ii}\\xspace region, the \\oiii88$\\mu$m\\xspace, \\siii18.7 and 33.5$\\mu$m\\xspace, and \\siv10.5$\\mu$m\\xspace lines are reproduced within 30\\%, while \\neiii15.6$\\mu$m\\xspace, \\neii12.8$\\mu$m\\xspace, and  \\nii122$\\mu$m\\xspace are underpredicted by a factor of 1.7, 10, and 6, respectively. \nFeeding this model to the PDR, the \\cii157$\\mu$m\\xspace line is underpredicted by a factor of 3.4, while the [O\\,{\\sc i}]\\xspace lines are both overpredicted by a factor of $\\sim$2. The contribution of the H\\,{\\sc ii}\\xspace region to the PDR line emission is only 1\\%. $G_0$ is found to be about $3.2\\times10^3$, which is higher than in the central region. The luminosity of the model is 1.7 times higher than the observed $L_{\\rm TIR}$ in this region. \n\n\n\n\\subsubsection{Comparison to empirical line ratios}\nPhysical conditions in the H\\,{\\sc ii}\\xspace region are mainly determined by tracers of the radiation field strength ([Ne\\,{\\sc iii}]\\xspace\/[Ne\\,{\\sc ii}]\\xspace, [S\\,{\\sc iv}]\\xspace\/\\siii18.7$\\mu$m\\xspace) and of density (\\siii18.7$\\mu$m\\xspace\/\\siii33.5$\\mu$m\\xspace). We compare these well-known diagnostic ratios in the two star-forming regions in Table~\\ref{ratios}. In the southern region, ratios of [Ne\\,{\\sc iii}]\\xspace\/[Ne\\,{\\sc ii}]\\xspace and [S\\,{\\sc iv}]\\xspace\/\\siii18.7$\\mu$m\\xspace are observed to be about twice as high and \\siii18.7$\\mu$m\\xspace\/\\siii33.5$\\mu$m\\xspace marginally higher than in the central region, indicating that the radiation field is harder and the medium denser. This is indeed what we recover with our best-fitting models (Table~\\ref{model}), as they match the sulfur ratios well. \n\n\n\\begin{table}\n\\caption{$\\chi^2$ values for the two star-forming regions.}\n\\label{chi2ionic}\n \\centering\n\\vspace{-5pt}\n   \\begin{tabular}{l c c c}\n   \\hline \\hline\n    \\vspace{-10pt}\\\\\n        H\\,{\\sc ii}\\xspace region~~~  &\\multicolumn{2}{c}{Central region}    & \\hspace{-2mm}Southern region \\\\             \n                                & Burst         & \\hspace{-1mm}Continuous & Burst \\\\\n        \\hline\n    \\vspace{-10pt}\\\\\n        \\multicolumn{2}{l}{Individual $\\chi^2$ values:}         &                 &               \\\\\n        \\lbrack \\textsc{O\\,iii}] \\,88$\\mu$m\\xspace               & 0.01  & 0.03    & 0.01  \\\\\n        \\lbrack \\textsc{N\\,ii}]\\,122$\\mu$m\\xspace                & 12.58 & 44.63   & 82.56 \\\\\n        \\lbrack \\textsc{S\\,iii}]\\,18.7$\\mu$m\\xspace              & 2.24  & 2.38    & 37.69 \\\\\n        \\lbrack \\textsc{S\\,iii}]\\,33.5$\\mu$m\\xspace              & 8.10  & 6.99    & 31.89 \\\\\n        \\lbrack \\textsc{S\\,iv}]\\,10.5$\\mu$m\\xspace       & 0.76  & 0.70  & 4.94    \\\\\n        \\lbrack Ne\\textsc{\\,ii}]\\,12.8$\\mu$m\\xspace      & 14442.2       & 13842.6 & 58101.7       \\\\\n        \\lbrack Ne\\textsc{\\,iii}]\\,15.6$\\mu$m\\xspace     & 8.14  & 24.65         & 224.51        \\\\\n        \\hline\n    \\vspace{-10pt}\\\\\n        $\\bar\\chi^2$ (all ionic lines)          & 2067.72       & 1988.85 & 8354.76       \\\\\n        $\\bar\\chi^2$ (optimized lines) \\hspace{-4mm}          & 3.85  & 6.95  & 59.81   \\\\\n        \\hline\n        \\hline\n    \\vspace{-10pt}\\\\\n        PDR~~~  & \\multicolumn{2}{c}{Central region}    & \\hspace{-2mm}Southern region \\\\               \n                                & Burst         & \\hspace{-1mm}Continuous & Burst \\\\\n        \\hline\n    \\vspace{-10pt}\\\\\n        \\multicolumn{2}{l}{Individual $\\chi^2$ values:}         &                 &               \\\\\n        \\lbrack \\textsc{C\\,ii}]\\,157$\\mu$m\\xspace        & 1.57  & 31.83 & 258.63  \\\\\n        \\lbrack \\textsc{O\\,i}]\\,63$\\mu$m\\xspace  & 149.78        & 57.05 & 85.37   \\\\\n        \\lbrack \\textsc{O\\,i}]\\,145$\\mu$m\\xspace & 34.14 & 11.83 & 21.27 \\\\\n        \\hline\n    \\vspace{-10pt}\\\\\n        $\\bar\\chi^2$ (all PDR lines)    & 61.83 & 33.57 & 121.76        \\\\\n        \\hline \\hline\n   \\end{tabular}\n \\end{table}\n\\begin{table*}\n\\caption{Observed and predicted MIR line ratios for the two star-forming regions.\nThe [Ne\\,{\\sc iii}]\\xspace\/[Ne\\,{\\sc ii}]\\xspace and [S\\,{\\sc iv}]\\xspace\/[S\\,{\\sc iii}]\\xspace line ratios are indicative of the radiation field strength, \nand the ratio of the two [S\\,{\\sc iii}]\\xspace lines is a density diagnostic.}\n\\label{ratios}\n\\centering\n\\vspace{-5pt}\n   \\begin{tabular}{l c c c c c c c}\n   \\hline\n    \\vspace{-10pt}\\\\\nRatio           & & \\multicolumn{3}{c}{Central region}          & & \\multicolumn{2}{c}{Southern region} \\\\\n   & &   Observed &    Burst & Continuous                        & & Observed      & Burst \\\\\n   \\hline\n    \\vspace{-10pt}\\\\\n\\lbrack Ne\\textsc{\\,iii}]\/[Ne\\,{\\sc ii}]\\xspace& & 2.082        & 13.72 & 12.34          & & 3.448               &       28.955          \\\\  \n\\lbrack \\textsc{S\\,iv}]\/[\\textsc{S\\,iii}]18.7 & &       0.481 & 0.479 & 0.477                 & & 1.227  &  1.044\\\\\n\\lbrack \\textsc{S\\,iii}]18.7\/[\\textsc{S\\,iii}]33 & &  0.630  & 0.593         & 0.601         & & 0.835 & 0.846\\\\\n\\hline\n   \\end{tabular}\n  \\end{table*}\n\n\n\\subsection{Effects of magnetic fields and turbulence on the PDR temperature and density}\n\\label{magn}\nAfter determining the best parameters for the H\\,{\\sc ii}\\xspace regions, we explored the effect of cloud density on the PDR emission in more\ndetail. \nDensity is of critical importance in the emission output of the simulation because of the different critical densities of the observed lines. Density values quoted so far are representative of the H\\,{\\sc ii}\\xspace region and evolve inside the modeled cloud. Figure~\\ref{dens} shows hydrogen density profiles in the clouds for each case presented in Sect.~\\ref{modelresults}. The density starts at the initial value we set for each model in the H\\,{\\sc ii}\\xspace region, remaining practically at the same level throughout the H\\,{\\sc ii}\\xspace region. At the interface between the H\\,{\\sc ii}\\xspace region and the PDR, there is a jump in density required to keep the model in pressure equilibrium. When pressure is only determined by the gas pressure ($P_{\\rm gas}=n_{\\rm H}kT$; i.e., magnetic fields and turbulence are omitted), the temperature difference at the phase transition is balanced by a rise in density of 2-3 orders of magnitude. Within this frame, the effects of magnetic fields or turbulence, implemented as pressure terms in \\textsc{Cloudy}, can be understood as follows: when total pressure equilibrium is assumed, they give more support at the phase transition, thus preventing a large difference in density between the H\\,{\\sc ii}\\xspace region and the PDR and moderating the density increase at large optical depths. \nFor the model presented in Sect.~\\ref{modelresults}, where a low turbulence value is included but no magnetic fields, representative PDR densities are $2\\times10^4$\\,cm$^{-3}$ in the central region for both the single-burst and continuous star formation models, and $7\\times10^4$\\,cm$^{-3}$ in the southern region (see Fig.\\,\\ref{dens}). \n\n\\begin{figure}\n\\centering\n\\includegraphics[clip,width=8.2cm]{fig4.eps}\n\\vspace{-5pt}\n\\caption{Density profiles in the modeled clouds for the central and southern region, which include a turbulence pressure term ($v_{\\rm turb}$=1.5\\,km\\,s$^{-1}$). Note that the x-axis is logarithmic, so the H\\,{\\sc ii}\\xspace region (with a constant low density) occupies a thin layer of the cloud, stopping at low visual extinction ($\\sim$0.1\\,mag).}\n\\label{dens}\n\\end{figure}\n\n\nThe effects of magnetic field and turbulence on the cloud density, temperature, and line emission are shown in Fig.\\,\\ref{turb}. \nWe present a set of runs for our best-fitting model in the central region single-burst case (note that we recover similar behaviors for the central continuous and southern single-burst cases, as shown in Appendix~\\ref{appendixa}) with the magnetic field (B=30\\,$\\mu$G) and\/or the turbulence pressure ($v_{\\rm turb}$=1.5, 3, 50\\,km\\,s$^{-1}$) terms switched on. These terms have no impact on the ionic line emission because thermal pressure dominates the pressure balance in the H\\,{\\sc ii}\\xspace region, but they noticeably change the emission of the PDR lines. When only thermal pressure is considered, the gas density jumps to values $>3\\times10^4$\\,cm$^{-3}$ in the PDR and the [O\\,{\\sc i}]\\xspace lines are overpredicted by an order of magnitude (black bars and dotted line). \nWith only the magnetic field on, all three PDR lines are underpredicted by a factor of $\\sim$3 (orange bars), but their ratios are kept in the range observed thanks to the lower densities achieved ($2\\times10^3$\\,cm$^{-3}$). Comparing models with different turbulent velocities, we see that the [C\\,{\\sc ii}]\\xspace line is best matched for low\nor intermediate velocities because of their moderate densities ($\\sim10^4$\\,cm$^{-3}$) and slightly lower PDR temperatures (at $A_{\\rm V} \\simeq 1-3$\\,mag). Increasing the turbulent velocity reduces the predicted [O\\,{\\sc i}]\\xspace emission and PDR density. The high-turbulence model performs poorly because it has the most dramatic effect on the density and line emission. \n\nTo summarize, we find that the best case that simultaneously matches all three PDR lines in the central region is the model with intermediate turbulent velocity ($v_{\\rm turb}$=3\\,km\\,s$^{-1}$), which has a density of $8\\times10^3$\\,cm$^{-3}$, but we stress that the main effect of turbulence is to reduce the PDR density.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[clip,width=\\textwidth,height=6cm]{fig5a.eps}\n\\includegraphics[clip,width=8cm]{fig5b.eps} \\hspace{1cm}\n\\includegraphics[clip,width=8cm]{fig5c.eps}\n\\vspace{-5pt}\n\\caption{Effect of turbulence and magnetic fields on the predicted line intensities (top panel), density, and temperature (bottom panels) in the modeled cloud for the central region single-burst case. Green bars: observations. Gray bars and solid lines: only low turbulence switched on ($v_{\\rm turb}$=1.5\\,km\\,s$^{-1}$, default model). Black bars and dotted lines: no magnetic fields and no turbulence. Orange bars and dashed lines: only magnetic field switched on (B=30\\,$\\mu$G). Red bars and dash-dotted line: only moderate turbulence switched on ($v_{\\rm turb}$=3\\,km\\,s$^{-1}$). Cyan bars and triple-dotted-dashed line: only high turbulence switched on ($v_{\\rm turb}$=50\\,km\\,s$^{-1}$).}\n\\label{turb}\n\\end{figure*}\n\n\n\n\\subsection{Input spectra and SED}\n\\label{insed}\nFigure~\\ref{spectra} shows the input and output SEDs of the models for the two regions. The input SED is the stellar spectrum of the illuminating source modeled with \\textsc{Starburst99} and also includes the CMB at millimeter wavelengths. In the central region, the stellar spectrum has a wider distribution for the continuous star formation model (red curve) than the single-burst star formation model (black curve), and it is more luminous in the near-IR regime due to the presence of old stars.  \nCompared to observations, all input SEDs fall above the GALEX FUV data because they are unattenuated, and the single-burst input SEDs fall below the 2MASS data because they lack old stars. In the central region, the input SED of the continuous model, on the other hand, agrees well with the 2MASS data. For better agreement with the FUV data, we estimated the level of extinction required to attenuate the input SEDs. We considered average extinction values $E(B-V)$ of 0.1, 0.05, and 0.1\\,mag for the central continuous central single-burst and southern single-burst models, respectively (dotted lines in Fig.\\,\\ref{spectra}), which are in the range of values found by \\cite{ubeda-2007b}. \n\nFocusing on the output SEDs of the models for the central region, it is anticipated that the level of the FIR continuum is different. The higher the input luminosity of the source, the higher the peak of the output SED. Moreover, as the dust temperature rises, the peak of the output SED is expected to shift to shorter wavelengths. For the continuous model, the higher FUV luminosity therefore\nprovides more dust heating, explaining the slight shift of its peak to shorter wavelengths. \nWe can compare the output SEDs to the observed PACS 70$\\mu$m\\xspace, 100$\\mu$m\\xspace, and 160$\\mu$m\\xspace fluxes. The models agree relatively well with observations for the southern region, but overpredict the FIR continuum emission in the central region.\nThis is not surprising because we used as input luminosity a higher value than the observed TIR flux and the modeled PDR has a high $A_{\\rm V}$. Better agreement with continuum observations requires a model that predicts a TIR flux lower by a factor of 2-3, for example, by reducing the covering factor of the PDR. We return to this in Sect.~\\ref{modelum}.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[clip,trim=0 0 0.1cm 0,width=6.7cm]{fig6a.eps}\n\\includegraphics[clip,trim=1.95cm 0 0.1cm 0,width=5.75cm]{fig6b.eps}\n\\includegraphics[clip,trim=1.95cm 0 0.1cm 0,width=5.75cm]{fig6c.eps}\n\\vspace{-10pt}\n\\caption{Spectral energy distributions of the two star-forming regions: central region single-burst case (left panel), central region continuous case (middle panel), and southern region single-burst model (right panel). The black and blue curves correspond to the input and output SEDs, respectively. The dotted lines are the attenuated input SEDs. The data points are the photometry measurements from GALEX FUV, 2MASS J, H, K bands, MIPS 24\\,$\\mu$m\\xspace, and PACS at 70$\\mu$m\\xspace, 100$\\mu$m\\xspace, and 160\\,$\\mu$m\\xspace. In panels for the central region, the dashed curves are scaled versions of the output SEDs, considering a covering factor of $0.5$ for the PDR. }\n\\label{spectra}\n\\end{figure*}\n\n\n\n\\section{Discussion}\n\\label{discussion}\nWe have presented models for the two star-forming regions of NGC\\,4214 that work for most of the observed MIR and FIR lines. Some discrepancies remain between our models and observations ([Ne\\,{\\sc ii}]\\xspace, [N\\,{\\sc ii}]\\xspace, [C\\,{\\sc ii}]\\xspace\/[O\\,{\\sc i}]\\xspace, and $L_{\\rm TIR}$); these are not due to the choice of parameter space but rather to missing physics or components in our models. In this section, we discuss various aspects of our analysis: 1)~the discrepancies with observations, and we give clues to improve our models, 2)~which star formation scenario describes the data better, and 3)~how our ISM results relate to the known properties and evolution of the two regions.\n\n\n\\subsection{Discrepancies between models and observations}\n\\label{disclines}\n\\subsubsection{Ionic lines}\nIn our best-fitting models of the H\\,{\\sc ii}\\xspace region, the \\neii12.8$\\mu$m\\xspace and \\nii122$\\mu$m\\xspace lines are systematically underpredicted. These lines have lower excitation potentials (21.56 and 14.53\\,eV, respectively) than the other, better matched ionic lines, and can therefore be partially excited outside of the main H\\,{\\sc ii}\\xspace region. This effect can be significant in NGC\\,4214 because of the poor spatial resolution. \n\nDiscrepancies between the observed and modeled [Ne\\,{\\sc iii}]\\xspace\/[Ne\\,{\\sc ii}]\\xspace ratio, with the same amplitude as we found for NGC\\,4214, were reported by \\cite{martin-hernandez-2002} for H\\,{\\sc ii}\\xspace regions observed by ISO\/SWS (see their figure~2). These could again be related to a mixture of physical conditions within the ISO beam, which is also relatively large. For the starburst galaxy Haro\\,11, we have examined the effect of an additional low-ionization component (star of effective temperature 35\\,000\\,K and $n_{\\rm H}\\simeq 10^{1-3}$\\,cm$^{-3}$). This reproduced the observed \\neii12.8$\\mu$m\\xspace and \\nii122$\\mu$m\\xspace emission without significantly affecting the other ionic lines \\citep{cormier-2012}.\nAlternatively, given that the neon lines have the largest energy difference, they are more sensitive to the underlying stellar atmosphere models than the sulfur lines. Constraining those models is beyond the scope of this paper, and we relied on the sulfur lines as being more robust diagnostics of the H\\,{\\sc ii}\\xspace region conditions in NGC\\,4214 (i.e., models presented in Sect.~\\ref{modelresults}). \n\nWe also assessed the effect of including the \\neii12.8$\\mu$m\\xspace line in the best-fitting solution-tracking procedure for the southern region. This resulted in a solution where the \\neii12.8$\\mu$m\\xspace absolute flux was better reproduced (within a factor of 3). This model uses an older burst (5\\,Myr), higher density (500\\,cm$^{-3}$), and the same inner radius (20\\,pc). The corresponding ionization parameter is $\\log(U)=-2.4$, which is 0.1\\,dex lower than previously found. However, the model underpredicts the neon intensities by a factor of 2 and the [\\textsc{S\\,iv}]\/[\\textsc{S\\,iii}]\\,18.7$\\mu$m\\xspace ratio by a factor of $\\sim$4, while that ratio was matched within 20\\% without the [Ne\\,{\\sc ii}]\\xspace constraint. \n\n\n\\subsubsection{PDR lines}\nIn our default PDR solutions for both regions, [C\\,{\\sc ii}]\\xspace is systematically underpredicted compared to [O\\,{\\sc i}]\\xspace. The best PDR model, found in Sect.~\\ref{magn} for the central region single-burst case, includes a turbulent velocity of 3\\,km\\,s$^{-1}$. In the southern and central region continuous case, similar turbulent velocities also lead to better agreement with the [C\\,{\\sc ii}]\\xspace\/[O\\,{\\sc i}]\\xspace line ratios, but still underpredict the observed emission in absolute values (Appendix~\\ref{appendixa}).\n\nIn addition to the stars, X-rays can be a source of heating in the PDR and affect the FIR line emission. Point sources and diffuse X-ray emission have been reported in \\cite{hartwell-2004} and \\cite{ghosh-2006}. The identified point sources are not coincident with the peak of the FIR emission, and we therefore ignored them. The diffuse emission is mostly detected in the central region, with a luminosity of $3\\times 10^{38}$\\,erg\\,s$^{-1}$ \\citep{hartwell-2004}, which is lower than that of the starburst. We have tested the effect of this diffuse X-ray component on the PDR lines in the central region and found that it increases the predicted intensity of the [O\\,{\\sc i}]\\xspace lines by $\\sim$30\\% and the [C\\,{\\sc ii}]\\xspace intensity by less than 10\\%. As X-rays are not the main source of heating in the PDR, they do not help to produce significantly more [C\\,{\\sc ii}]\\xspace emission. \n\nWe further explored the possibility of [C\\,{\\sc ii}]\\xspace originating from a diffuse ionized component. We compared the [C\\,{\\sc ii}]\\xspace and \\nii122$\\mu$m\\xspace intensities and the PACS upper limit on the \\nii205$\\mu$m\\xspace line, which gives \\nii122$\\mu$m\\xspace\/\\nii205$\\mu$m\\xspace$>$1, to theoretical predictions assuming pure collisional regime. Following \\cite{bernard-salas-2012} and applying $C$ and $N$ elemental abundances observed in NGC\\,4214, we found that less than 16\\% of the total observed [C\\,{\\sc ii}]\\xspace emission arises in diffuse ionized gas. \\textit{Herschel}\\xspace SPIRE FTS observations of \\nii205$\\mu$m\\xspace toward the central region also indicate that \\nii122$\\mu$m\\xspace\/\\nii205$\\mu$m\\xspace$=$2.5 (priv. comm. R. Wu), which gives an ionized gas density of $\\sim$60\\,cm$^{-3}$ and a contribution of only 8\\% to the total [C\\,{\\sc ii}]\\xspace emission. Therefore, if a low-density ionized component is added to our current models to account for the missing [N\\,{\\sc ii}]\\xspace and [Ne\\,{\\sc ii}]\\xspace emission, this component will not contribute significantly to the [C\\,{\\sc ii}]\\xspace emission. Our best-fitting models for the central region also predict that the H\\,{\\sc ii}\\xspace region contributes less than a few percent to the [C\\,{\\sc ii}]\\xspace emission. \n\nThe [C\\,{\\sc ii}]\\xspace emission most likely arises from a neutral phase, but its conditions are not well described by our default PDR models. With the PDR tests performed in Sect.~\\ref{magn}, we have explored the effect of density on the PDR emission lines, but these lines are also sensitive to the radiation field strength. To reduce the radiation field intensity $G_0$ (not the hardness), we placed the PDR farther away by stopping the model at the H$^+$\/H\\,{\\sc i}\\xspace phase transition and resuming the calculation at a larger distance in the H\\,{\\sc i}\\xspace phase (note that this breaks the pressure equilibrium). The direct effect is to dilute the UV field before it reaches the PDR, as proposed by \\cite{israel-2011} and \\cite{cormier-2015}, which is equivalent to increasing the porosity of the medium. For the central single-burst central continuous and the southern single-burst cases, we increased the PDR distance by a factor of 2, 3, and 5 ($r_{\\rm in}$$\\simeq$170, 186, and 115\\,pc), respectively. This way, $G_0$ decreases to $\\sim$120 in all three cases, boosting the [C\\,{\\sc ii}]\\xspace emission, and the predicted [C\\,{\\sc ii}]\\xspace\/[O\\,{\\sc i}]\\xspace ratios match the observed ratios within 40\\% (Fig.~\\ref{pdrdist}). In absolute values, the [C\\,{\\sc ii}]\\xspace and [O\\,{\\sc i}]\\xspace intensities are 2 to 3 times too high,\nhowever. This can be compensated by a PDR covering factor lower than unity (see Sect.~\\ref{modelum} below). \n\nWe conclude that a moderate density, moderate $G_0$ neutral medium (compared to our dense, high $G_0$ default PDR model) with a\nlow turbulent velocity and a covering factor lower than unity is the most plausible origin for the observed [C\\,{\\sc ii}]\\xspace emission. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[clip,trim=0 10mm 0 0,width=\\textwidth,height=5.8cm]{fig7.eps}\n\\vspace{-10pt}\n\\caption{Effect of changing the distance to the modeled cloud for the PDR calculation. The PDR distance is increased by a factor of 2, 3, and 5 for the central single-burst (left panel), central continuous (middle panel), and southern single-burst cases (right panel), respectively.}\n\\label{pdrdist}\n\\end{figure*}\n\n\n\\subsubsection{Model luminosity}\n\\label{modelum}\n\\cite{hermelo-2013} have fit the dust SED of the whole galaxy. In particular, their models require less UV luminosity than that observed to match the IR emission. They discussed possible explanations for this disagreement, proposing an escape of unattenuated UV radiation along with a particular geometry and dust properties in the galaxy. Such an argument of a UV escape fraction could also apply in our case. As seen in Fig.\\,\\ref{spectra}, the observed photometry data points in the FIR, which correspond to emission originating from the PDR, are lower than the modeled SED for the central region. This disagreement can indicate a different covering factor for the PDR. In our model, the PDR fully covers the sphere around the source (i.e., covering factor of unity). For the best-fitting models to better match the photometry, it could be that the PDR component is more porous, allowing radiation to escape the cloud. To illustrate this, we plot the resulting SEDs for the two models of the central region considering a PDR covering factor of 0.5 (dashed curves in Fig.\\,\\ref{spectra}).\n\nPart of the discrepancies in our results that we have discussed in this section~\\ref{disclines} originate in modeling each complex as a single cloud, which is imposed by the lack of spatial resolution in the observations. Clearly, future improvements are expected from observations with better spatial resolution. \n\n\n\\subsection{Central region: bursty or continuous star formation?}\n\\label{borc}\nThe star formation history of NGC\\,4214 over the last Gyr is complex. It shows bursts lasting for shorter or longer periods and a continuous `background', which takes place throughout this whole time window \\citep{mcquinn-2010,williams-2011}. In that sense, we could say that it is a rather hybrid star formation pattern. \nIn the central region, we have investigated cases of both a single-burst and a continuous star formation mode. However, when we use a single model to reproduce the observed line emission, we simplify the problem, since we do not take into account both modes. Hence arises the question of which of the two approaches is the most adequate to model the MIR-FIR line emission. \nWe have shown that both modes can satisfyingly reproduce the observed mid- and far-infrared line emission. By comparing the $\\bar\\chi^2$ values of the best-fitting models (Table~\\ref{chi2ionic}), we found that the single-burst case seems to globally\nperform better when considering the lines used in the optimization method. When including [N\\,{\\sc ii}]\\xspace and [Ne\\,{\\sc ii}]\\xspace, the two modes perform similarly, although the high $\\bar\\chi^2$ values are driven by the poor fit to the \\neii12.8$\\mu$m\\xspace line. \nFor the PDR, the continuous scenario gives a lower $\\bar\\chi^2$ , but the default PDR solutions are not optimum and can be fine-tuned for both modes (by lowering the density and $G_0$, see Sects.~\\ref{magn} and \\ref{disclines}). We conclude that both are limiting, simplifying cases of modeling the ISM in NGC\\,4214, with the continuous star formation model being marginally more accurate inside the PDR and the single-burst in the H\\,{\\sc ii}\\xspace region (without further refinement).\n\n\n\\subsection{Comparison between the two star-forming regions}\nHow do the ISM conditions that we have characterized in the two star-forming regions relate to their star formation properties? \nWe have found that the modeled cluster (radiation source) in the southern region contains younger stellar populations with a harder radiation field than that in the central region, in agreement with the results of \\cite{ubeda-2007}, for instance. The hydrogen density is also higher in the southern star-forming region, but the metallicities of the regions are very similar. The southern region is observed to be at a younger, more compact stage than the central region. The central region is more evolved\nand had time to expand, as observed by the presence of shells that may have swept away the dense material \\citep{walter-2001}, and is thus consistent with a more diffuse ISM. \n\nWe calculated the star formation rate surface densities for the two regions combining the GALEX FUV map and the \\textit{Spitzer}\\xspace 24$\\mu$m\\xspace map, as done in \\cite{leroy-2008}, with%\n\\begin{equation}\n\\label{sfcomp}\n\\begin{split}\n\\Sigma_{\\rm SFR} {\\rm [M_{\\odot}\\,yr^{-1}\\,kpc^{-2}]} = 3.2 \\cdot 10^{-3} \\cdot {\\rm I_{24}~[MJy\\,sr^{-1}]} \\\\ ~+~ 8.1 \\cdot 10^{-2} \\cdot {\\rm I_{\\rm FUV}~[MJy\\,sr^{-1}]}\n\\end{split}\n,\\end{equation}\n\\noindent where $\\Sigma_{\\rm SFR}$ is the SFR surface density and I$_{24}$ and I$_{\\rm FUV}$ the 24$\\mu$m\\xspace and FUV intensities. We also measured the atomic and molecular hydrogen content of the two regions using the 21cm map from THINGS\\footnote{\\url{http:\/\/www.mpia-hd.mpg.de\/THINGS\/Data.html}} \\citep{walter-2008} and the CO(1-0), CO(2-1) transition maps from \\cite{walter-2001} and HERACLES\\footnote{\\url{http:\/\/www.cv.nrao.edu\/~aleroy\/heracles_data\/}} \\citep{leroy-2009}, respectively. We used a conversion factor of $\\alpha_{\\rm CO}$=4.38~[M$_\\odot$\\,pc$^{-1}$\\,(K\\,km\\,s$^{-1}$)$^{-1}$] from CO(1-0) luminosity to H$_2$ mass. If we were to use a different conversion factor due to the low metallicity of these regions \\citep[e.g.,][]{schruba-2012}, this would not affect the relative comparison of the regions (see Table~\\ref{sfprops}). \nThe central region (I) has a total hydrogen content of M$_{\\rm gas,I}$=M$_{\\rm HI}$+M$_{\\rm H_2}$=2.05$\\times$10$^6\\,{\\rm M_\\odot}$ and a molecular (mass) fraction of $f_{\\rm mol,I}$=M$_{\\rm H_2}$\/M$_{\\rm HI}$=0.35. Integrating Eq.~\\ref{sfcomp} in the region, we find SFR$_{\\rm I}$=2.2$\\times$10$^{-2}\\,{\\rm M_\\odot\\,yr^{-1}}$. The southern region (II) has a higher total hydrogen content of M$_{\\rm gas,II}$=2.68$\\times$10$^6\\,{\\rm M_\\odot}$, an H$_2$ fraction $f_{\\rm mol,II}$=0.32, and SFR$_{\\rm II}$=1.9$\\times$10$^{-2}\\,{\\rm M_\\odot\\,yr^{-1}}$. \nTherefore the southern region has relatively more gas compared to its SFR than the central region. In terms of efficiency, SFR\/M$_{\\rm gas}$, it is about 50\\% lower in the southern region. \nThis could reflect a slightly more efficient, cluster-like star formation episode in the central region or simply encode a different evolutionary state as the SFR and gas masses are a strong function of time on scales of individual star-forming regions \\citep[e.g.,][]{schruba-2010}. The southern region being younger, it may still be in the process of forming stars. \n\n\n\\begin{table}\n  \\caption{Comparison of star formation properties.}\n\\label{sfprops}\n \\centering\n\\vspace{-5pt}\n   \\begin{tabular}{l c c c}\n   \\hline \n    \\vspace{-10pt}\\\\\n        Quantity                        & & Region I    & Region II   \\\\\n        \\hline\n    \\vspace{-10pt}\\\\\n        {$L_{\\rm CII}\/L_{\\rm TIR}$}     & & $5.4\\times10^{-3}$  & $3.6\\times10^{-3}$ \\\\\n        {$L_{\\rm CII}\/L_{\\rm CO(1-0)}$}         & & $6.7\\times10^4$     & $2.5\\times10^4$ \\\\\n        {M$_{\\rm gas=H_2+HI}$~$[$M$_\\odot]$}    & & $2.05\\times10^6$    & $2.68\\times10^6$ \\\\\n        {M$_{\\rm H_2}$\/M$_{\\rm HI}$}    & & 0.35        & 0.32 \\\\\n        {SFR~$[$M$_\\odot$\\,yr$^{-1}]$}  & & $2.2\\times10^{-2}$  & $1.9\\times10^{-2}$ \\\\\n        {SFE~$[$Gyr$^{-1}]$}    & & 10.7        & 7.1 \\\\\n        \\hline \n   \\end{tabular}\n \\end{table}\n\n\nThe main differences in ISM conditions that we extracted from our modeling relate to the H\\,{\\sc ii}\\xspace region properties. The emission lines are a factor of two lower in luminosity in the southern region, except [Ne\\,{\\sc iii}]\\xspace and [S\\,{\\sc iv}]\\xspace, which are proportionally higher in the southern region. By contrast, the PDR conditions in the two regions are similar ($n_{\\rm H}\\simeq10^4$\\,cm$^{-3}$ and $G_0\\simeq150$). Our modeling reflects conditions resulting from the recent star-forming event and has little predictive power regarding a different, future star-forming event. In particular, at the linear scale that we probe ($\\sim$175\\,pc), the PDR conditions are averaged over multiple star-forming clouds and not representative of the underlying, possibly different, substructure in individual molecular clouds. \nHowever, there is more PDR emission relative to CO in the central region, that is, high [C\\,{\\sc ii}]\\xspace\/CO and [O\\,{\\sc i}]\\xspace\/CO ratios \\citep[see also][]{cormier-2010}. \nAs found by \\cite{walter-2001}, CO emission is centrally concentrated in the south and more diffuse in the center. The concentration of molecular gas may be nourishing the current star formation episode in the south or is being observed at a pre-disruption stage with the same fate as the central region. \nThe increased porosity, which evidently is an intrinsic property of the low-metallicity ISM, is seen in both the central and southern star-forming regions. \nThe main evolution within the dense medium is seen in the covering factor of the PDR, which is found to be lower in the central, more evolved region than in the southern region, and in CO, which probably suffers more from photodissociation with time and its emission is seen farther away from the cluster center, but this cannot be modeled with our static approach. Observing an intermediate PDR tracer at the C$^+$\/CO transition, such as C\\,{\\sc i}, would help to test this evolution scenario. \n\n\n\n\\section{Conclusion}\nWe have investigated the physical conditions characterizing \nthe ISM of the dwarf irregular galaxy NGC\\,4214 by modeling \\textit{Spitzer}\\xspace \nand \\textit{Herschel}\\xspace observations of MIR and FIR fine-structure cooling lines. \nWe used the spectral synthesis code \\textsc{Cloudy} to \nself-consistently model the H\\,{\\sc ii}\\xspace region and PDR properties of the two main \nstar-forming regions in NGC\\,4214. \nWe summarize our results as follows: \n\\begin{itemize}\n\\item The ionized gas in the southern region is found to be $2.5$ \ntimes denser than in the central region (440\\,cm$^{-3}$ \nversus 170\\,cm$^{-3}$) and typified by a harder radiation field. \nOur best-fitting models of the H\\,{\\sc ii}\\xspace region+PDR reproduce most \nionic and neutral atomic lines, namely the \\oiii88$\\mu$m\\xspace, \\siii18.7 \nand 33.5$\\mu$m\\xspace, \\siv10.5$\\mu$m\\xspace, \\neiii15.6$\\mu$m\\xspace, and \\oi63$\\mu$m\\xspace lines, \nwithin a factor of $\\sim$2. \n\\item The observed \\nii122$\\mu$m\\xspace and \\neii12.8$\\mu$m\\xspace lines are \nthe most discrepant with our model solutions for the H\\,{\\sc ii}\\xspace region. \nA single model component seems too simplistic to account for all \nobserved lines simultaneously. Given the complexity of these star-forming regions, \na multi-component modeling would be more appropriate. \nIn particular, a lower excitation ionized gas component may be \nrequired to match the [N\\,{\\sc ii}]\\xspace and [Ne\\,{\\sc ii}]\\xspace emission in both regions. \n\\item Our H\\,{\\sc ii}\\xspace region models and the established observational \n[C\\,{\\sc ii}]\\xspace\/[N\\,{\\sc ii}]\\xspace line ratio used as a proxy for the fraction of [C\\,{\\sc ii}]\\xspace arising in the ionized gas \nboth indicate that the [C\\,{\\sc ii}]\\xspace emission is mostly associated with the PDR gas, \nwith only a $\\sim$10\\% contribution from the ionized gas. \n\\item Constant pressure models where thermal pressure dominates \nthe pressure equilibrium perform rather poorly for the PDR lines \nmostly because of the high densities and high $G_0$ values \nreached in the PDR. Including additional pressure terms, such as \nweak turbulent or magnetic pressure, or placing the \nPDR cloud farther away and reducing its covering factor, \nleads to a much improved reproduction of the observed line intensities. \n\\item Star formation histories have an effect on the predicted \nMIR-FIR line emission. We have explored the two simplifying cases \nof a bursty and a continuous star formation scenario. \nIn the central region, we found that the bursty scenario works marginally \nbetter for the H\\,{\\sc ii}\\xspace region and the continuous scenario for the PDR, \nalthough both modes can reproduce \nthe observations after refining the PDR conditions. \n\\item The H\\,{\\sc ii}\\xspace region modeling from IR emission is consistent with \nthe evolutionary stages of the regions found in previous studies: \nthe southern region is younger and more compact, while the central region \nis more evolved and diffuse. \nOn the linear scale of our study ($\\sim$175\\,pc), the PDR conditions \nof individual star-forming clouds are averaged out and do not echo \nthe observed differences between the two regions (stellar ages, H\\,{\\sc ii}\\xspace conditions, etc.). \nThe increased porosity of the star-forming regions appears \nas an intrinsic characteristic of the low-metallicity ISM, \nwhile the covering factor of the PDR, which is reduced \nin the central region, stands out as the main evolution tracer.\n\\end{itemize}\n\\vspace{3mm}\n\n\\begin{acknowledgements}\nWe would like to thank S. Hony for his help with the IRS data and \nfor fruitful discussion, F. Walter for providing us the CO(1-0) data, \nand S. Glover for his advice on turbulence issues.\nWe thank the referee for his or her comments on the manuscript.\nDC and FB acknowledge support from DFG grant BI 1546\/1-1.\nThis work is based in part on archival data obtained with the \nSpitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, \nCalifornia Institute of Technology under a contract with NASA.\n\\textit{Herschel}\\xspace is an ESA space observatory with science instruments \nprovided by European-led Principal Investigator consortia and \nwith important participation from NASA.\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA precise determination of Parton Distribution Functions (PDFs) with \nreliable estimate of their uncertainties is crucial for the success of \nthe physics program at the LHC experiments. \nOn the one hand PDF uncertainties are often the dominant theoretical \nuncertainties for many relevant signal and background \nprocesses~\\cite{Campbell:2006wx}. \nOn the other hand overestimated PDF errors might hinder \nthe discovery of new physics effects, as shown for example \nin~\\cite{Ferrag:2004ca}.\nTop physics at the LHC is no exception and both top pair and single-top\nproduction present complementary and interesting properties as far as PDF\ndetermination\/effects are concerned. This contribution aims to review\nsome of the implications of PDFs for top quark physics at the LHC.\n\nIn the first part of the contribution we summarize the present \nstatus of the predictions for $t\\bar{t}$, $t-$ and $s-$channel single-top \ncross-sections, computed using different PDF sets. We show how differences \nin the predictions can directly be traced to both differences in the parton \nluminosities, and the values of physical parameters used in the PDF \nanalyses, such as the strong coupling constant $\\alpha_s$ or the  \nthe $b$-quark mass, $m_b$.\nIn particular, we highlight the importance to account for the uncertainty \non the $b$-quark mass for accurate predictions of single-top production at \nthe LHC.\n\nIn the second part we study PDF induced correlations between PDFs and \n$t\\bar{t}$\/single-top cross-sections and between top and $W^\\pm$\/$Z^0$ \ncross-sections. \nWe briefly discuss how these correlations could be used in order to \nimprove the accuracy of top cross-section measurements with early data \nat the LHC.\nThese correlation studies are performed within the framework of the \nNNPDF parton analysis~\\cite{DelDebbio:2007ee,Ball:2008by,Ball:2009mk,\nBall:2010de}  which, by relying on Monte Carlo techniques \nfor the estimation of uncertainties, provides an ideal tool for such \nstatistical studies.\n\nThe baseline PDF set for the studies presented in this contribution\nis the recently released NNPDF2.0~\\cite{Ball:2010de}, the first NLO global \nfit using the NNPDF methodology. \n\n\\section{Top-quark production at the LHC}\n\n\\subsection{$t\\bar{t}$ production}\n\nTop pair production is the main channel for top quark production at Tevatron \nand LHC.\nIn Table~\\ref{tab:ttbar} we collect the predictions for the top pair\ncross-section at LHC 7 TeV at NLO computed with the MCFM code~\\cite{ref:MCFM} \nusing different PDF sets.\n\n\\begin{table}[ht!]\n  \\caption{Top pair cross-section at NLO with different PDF sets at \n    LHC 7 TeV.}\n  \\label{tab:ttbar}\n  \\begin{narrowtabular}{2cm}{c|c}\n    \\hline\n      CTEQ6.6~\\cite{Nadolsky:2008zw}  & 147.7 $\\pm$ 6.4 pb \\\\\n      MSTW2008~\\cite{Martin:2009iq}   & 159.0 $\\pm$ 4.7 pb \\\\\n      NNPDF2.0~\\cite{Ball:2010de}     & 160.0 $\\pm$ 5.9 pb \\\\\n    \\hline\n      ABKM09~\\cite{Alekhin:2009ni}    & 131.9 $\\pm$ 4.8 pb \\\\\n      HERAPDF1.0~\\cite{:2009wt}       & 136.4 $\\pm$ 4.7 pb \\\\\n    \\hline\n  \\end{narrowtabular}\n\\end{table}\n\nWe notice that the predictions from the three global fits, NNPDF2.0, CTEQ6.6 \nand MSTW08 agree at the 1-sigma level.\nThe differences with PDF sets based on reduced datasets, ABKM09 and HERAPDF1.0, \nare larger. \nWe note that top pair production depends strongly on the large-$x$ gluon, \nand thus using sets which do not include Tevatron jet data might lead to rather \ndifferent predictions for this observable.\nOne should notice that differences between the predictions from different PDF \nsets also arise from the use of different values for the strong coupling \nconstant $\\alpha_s$.  \nIt has been shown that using a common value of $\\alpha_s$ brings predictions\nfrom different groups for various LHC observables, including top pair \nproduction, in better agreement~\\cite{Demartin:2010er,Ubiali:2010xc}.\n\nOnce we subtract the difference introduced by different choices for $\\alpha_s$, \nthe remaining differences can be directly traced to differences in the PDF \nluminosities at the typical scale of the process.\nThis is illustrated in Fig.~\\ref{fig:gglumi}, where the gluon-gluon luminosity\nfor LHC at 7 TeV is plotted for the CTEQ6.6, MSTW2008 and NNPDF2.0 NLO sets.\nFor example, the lower value for the cross-section obtained using the CTEQ6.6\nset reflects the smaller $gg$ luminosity as compared to the other sets at\n$Q^2=m_t^2$.\n\\begin{figure}[ht!]\n  \\centering\n  \\includegraphics[width=7cm]{gglumi-7-rel-global}\n  \\caption{Gluon-gluon parton luminosities for CTEQ6.6, MSTW2008 and NNPDF2.0\n    including the associated PDF uncertainties. Results are showed as ratios\n    to NNPDF2.0.}\n  \\label{fig:gglumi}\n\\end{figure}\n\n\\subsection{Single-top production}\n\nNext we present predictions for single-top production at the LHC at 7 TeV.\nThese predictions for various PDF sets are collected in \nTable~\\ref{tab:singletop}, where we present results for both the $t$- \nand $s$-channel single-top cross-sections computed at NLO in QCD with \nthe MCFM code. We have used the $N_f=5$ (massless) calculation \nfor the results presented here, recently the $N_f=4$ calculation, which\nproperly takes into account the effects due to finite $b$-quark mass, also \nbecame available~\\cite{Campbell:2009ss}.\n\nWhile at the Tevatron the contributions of $t$- and $s$-channel $W$ exchange \nto single-top production are comparable in size, at the LHC $t$-channel \nproduction is by far the dominant production mechanism.\n\nFrom the point of view of testing the predictions from different PDF sets \n$t$-channel single-top is also very interesting due to the fact that, in the \nso called 5-flavour scheme ({\\it i.e.} a scheme where the $b$ is assumed to\nbe a parton in the proton) the cross-section at LO probes directly the \n$b$-quark PDF, which in turn is closely related to the gluon distribution \nfrom which it is generated radiatively.\n\nFrom Table~\\ref{tab:singletop} one notices that the central predictions from \nthe various PDF sets can differ by several times the quoted 1-sigma PDF \nuncertainty\\footnote{Note that for the ABKM09 prediction the uncertainty \nincludes the associated $\\alpha_s$, $m_c$ and $m_b$ uncertainties, which \ncannot be disentangled from the PDF uncertainties, and is thus much larger \nthan the one obtained using other PDF sets.}.\nThere are different contributions to this discrepancy. The first stems from \nthe different values of the strong coupling constant $\\alpha_S$ which are \nused by different parton sets. Since single-top production is mediated\nby electroweak gauge bosons, $\\alpha_s$ enters only in radiative corrections,\nunlike the case of $t\\bar{t}$ production discussed above, this\neffect is rather small.\n\nIn order to separate the differences in the single-top production \ncross-section which arise from the differences in the PDFs themselves\nand those from other physical parameters (like $m_b$ or $\\alpha_s$) which \nalso enter in the PDF analyses and in the computation of the partonic matrix \nelement, we plot in Fig.~\\ref{fig:bglumi} the $b$-gluon parton luminosity, \nwhich determines the LO cross-section, for the CTEQ6.6, MSTW2008 and \nNNPDF2.0 NLO sets. \nIt is clear that parton luminosities in the kinematic region relevant for \nsingle-top production differ by an amount much smaller than the cross-sections \nthemselves, suggesting that the differences indeed come from variations of \nother physical parameters which enter the PDF analysis.\n\nIndeed, it can be seen that the bulk of this difference is related to the \ndifferent values of the $b$-quark mass used in the fits by the different\ncollaborations. The NNPDF collaboration sets $m_b=4.3$ GeV, CTEQ uses \n$m_b=4.5$ GeV while the MSTW08 fit is performed setting $m_b=4.75$ GeV. \nIn order to substantiate our claim that the different values used the $b$-quark \nmass explain the bulk of the difference for the $t$-channel single-top \ncross-section, we produced two NNPDF2.0 sets with $m_b = 3.7$ GeV \nand  $m_b = 5.0$ GeV respectively. \nThe results for the $b$-gluon parton luminosities for these modified \nsets are shown in the right plot in Fig.~\\ref{fig:bglumi} and the \ncorresponding cross-sections for the $t$-channel single-top cross-section \nare collected in Table~\\ref{tab:singletop-mb}.\nIt is clear that the value of $m_b$ is anti-correlated with the $bg$ luminosity \nand the $t$-channel single-top cross-section.\n\nFrom Table~\\ref{tab:singletop-mb} one sees that variations of the $b$-quark \nmass of the order or $\\delta m_b\\sim 0.7$ GeV induce an uncertainty in the \ncross-section of $\\delta\\sigma \\sim 3$ pb.\nIf we take the PDG average as the best available determination of the $b$-quark\nmass and convert it from the \\hbox{$\\overline{\\rm MS}$} scheme to the pole mass scheme we obtain an\nuncertainty of approximately $\\delta m_b({\\rm PDG})\\sim 0.2$ GeV. Rescaling \nerrors, we are still left with an uncertainty in the $t$-channel single-top \ncross-section of $\\delta\\sigma \\sim 0.8$ pb, still larger than the typical \nnominal PDF uncertainties quoted in Table~\\ref{tab:singletop}. \nIt is also clear from Table~\\ref{tab:singletop-mb} and Fig.~\\ref{fig:bglumi} \nthat using a similar value of $m_b$ would bring the predictions from different \nPDF sets into much better agreement. The uncertainty due to $m_b$ should \nthus always be accounted for in the theoretical predictions for LHC single-top \nproduction.\n\n\\begin{table}[t!]\n  \\caption{Single-top cross-section at NLO with different PDF sets at \n    LHC 7 TeV.}\n  \\label{tab:singletop}\n  \\begin{narrowtabular}{2cm}{c|c|c}\n    \\hline\n                                    & $t$-channel            & $s$-channel   \\\\ \n    \\hline\n    CTEQ6.6~\\cite{Nadolsky:2008zw}  &  40.85 $\\pm$  0.50 pb & 2.33 $\\pm$ 0.05 pb\\\\\n    MSTW2008~\\cite{Martin:2009iq}   &  41.96 $\\pm$  0.26 pb & 2.38 $\\pm$ 0.04 pb\\\\\n    NNPDF2.0~\\cite{Ball:2010de}     &  44.33 $\\pm$  0.32 pb & 2.38 $\\pm$ 0.06 pb\\\\\n    \\hline\n    ABKM09~\\cite{Alekhin:2009ni}    &  43.17 $\\pm$  1.98 pb & 2.40 $\\pm$ 0.03 pb\\\\\n    HERAPDF1.0~\\cite{:2009wt}       &  40.04 $\\pm$  0.33 pb & 2.38 $\\pm$ 0.05 pb\\\\\n    \\hline\n  \\end{narrowtabular}\n\\end{table} \n\n\n\\begin{figure}[ht!]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{bglumi-7-rel-global.eps}\\qquad\n  \\includegraphics[width=0.45\\textwidth]{bglumi-7-rel-nnpdf.eps}\n  \\caption{(left) $b$-gluon parton luminosities for CTEQ6.6, MSTW2008 and \n    NNPDF2.0, normalized to the NNPDF2.0 value. (right) $b$-gluon parton \n    luminosities for NNPDF2.0 fits with different values of the $b$-quark \n    mass normalized to the standard NNPDF2.0.}\n  \\label{fig:bglumi}\n\\end{figure}\n\n\\begin{table}[t!]\n  \\caption{$t$-channel single-top cross-section at NLO computed using NNPDF2.0 \n    sets with different values of the $b$-quark mass.}\n  \\label{tab:singletop-mb}\n  \\begin{narrowtabular}{2cm}{l|c}\n    \\hline\n    NNPDF2.0 ($m_b=3.7$ GeV) &  46.77 $\\pm$ 0.36 pb \\\\\n    NNPDF2.0 ($m_b=4.3$ GeV) &  44.33 $\\pm$ 0.32 pb \\\\\n    NNPDF2.0 ($m_b=5.0$ GeV) &  41.04 $\\pm$ 0.32 pb \\\\\n    \\hline\n  \\end{narrowtabular}\n\\end{table}\n\n\\section{PDF-induced correlations}\n\nIt is well known (see for example the discussion in \nRef.~\\cite{Nadolsky:2008zw,Ball:2008by}) that parton densities\ninduce correlations among different observables measured at hadron \ncolliders. \nThese can be the PDFs themselves, one PDF and a physical observable or \ntwo physical observables. The latter case is especially important from \nthe experimental point of view, since it allows to define measurement \nstrategies in which the PDF uncertainties between two observables cancel, \nfor example in the case in which these correlation between the two observables \nis maximal.\n\nIn the case of a PDF set based on the Monte Carlo method, like NNPDF, \nthe correlation coefficient $\\rho[A,B]$ for two observables $A$ and $B$ which \ndepend on PDFs is given by the standard expression for the correlation of two\nstochastic variables~\\cite{Ball:2008by,Demartin:2010er}\n\\begin{equation}\n  \\label{eq:correlation}\n  \\rho[A,B]=\\frac{\\langle A B\\rangle_{\\mathrm{rep}}\n    - \\langle A\\rangle_{\\mathrm{rep}}\\langle B\\rangle_{\\mathrm{rep}} }\n  {\\sigma_A\\sigma_B}\n\\end{equation}\nwhere the averages are taken over the ensemble of the $N_{\\mathrm{rep}}$ values \nof the observables computed with the different replicas of the PDF set, \nand $\\sigma_{A,B}$ are the standard deviations for the observables as computed \nfrom the MC ensemble.\nThe value of $\\rho$ characterizes whether two observables are correlated \n($\\rho \\approx 1$), anti-correlated ($\\rho \\approx -1$) or uncorrelated \n($\\rho\\approx 0$). In the following we present results for the NNPDF2.0 set, \nthe LHC cross-sections have been obtained as before using the MCFM code.\n\nAs a first example, we compute the correlation between the $t\\bar{t}$ and\n$t$-channel single-top cross-section at the LHC (7 TeV) and different PDFs\nat the factorization scale $\\mu_f=m_{\\mathrm{top}}$ as a function of $x$.\nThe results are plotted in Fig.~\\ref{fig:pdf_obs_corr}. \nThe most remarkable features are that the $t\\bar{t}$ cross-section at the \nLHC at 7 TeV is mostly correlated to the gluon distribution at $x\\sim 0.1$ \nand anti-correlated with it at small-$x$, with the same behaviour present \nfor sea quark PDFs,  generated radiatively from the gluon. \nWe note also that the $u$- and $d$-quark distributions are anti-correlated \nwith the cross-section at medium-\/large-$x$. \n\nAs for the case of the $t$-channel single-top cross-section, we point out \nthat the strong correlation with the gluon (and therefore with the $c$- \nand $b$-quark distributions) present for $t\\bar{t}$ is now milder \nand peaked at medium-$x$, $x\\sim 0.01$, and now we find a moderate \ncorrelation at medium-\/small-$x$ with the $u$ and $d$ PDFs, of the \nopposite sign as in the $t\\bar{t}$ cross-section. \nWe would like to stress as well the correlation of single-top\ncross-section and the $s$-,$\\bar{s}$-quark PDFs at medium-\/small-$x$, which \nis notably absent in the $t\\bar{t}$ case.\n\n\\begin{figure}[ht!]\n  \\centering\n  \\includegraphics[width=0.33\\textwidth,angle=270]{pdf_ttbar_corr}\\qquad\n  \\includegraphics[width=0.33\\textwidth,angle=270]{pdf_singletop_corr}\n  \\caption{Correlation between parton densities and $t\\bar{t}$ (left) \n    and $t$-channel single-top (right) cross-sections at the LHC 7 TeV. \n    The PDF set used is NNPDF2.0 and cross-sections have been\n    computed with MCFM.}\n  \\label{fig:pdf_obs_corr}\n\\end{figure}\n\nAs previously pointed out, the correlation coefficient \nEq.~(\\ref{eq:correlation}) can also be computed between two cross-sections, \nwhich is potentially relevant since in the case of a sizable correlation \nthe measurement of one of these observables would provide useful information \non the value of the other one. In this respect we have computed the \ncorrelation between the $t\\bar{t}$ and $t$-channel single-top cross-sections \non one side and $W^{\\pm}$ or $Z^0$ cross-sections at the LHC on the other.\nThe values for the correlation coefficients for the different pairs of \nobservables are collected in Table~\\ref{tab:obs_obs_corr} and the correlation \nellipses are plotted in Fig.~\\ref{tab:obs_obs_corr} for the $t\\bar{t}$ \ncross-section and in Fig.~\\ref{fig:singletop_VB_corr} for $t$-channel \nsingle-top.\n\n\\begin{table}[t!]\n  \\centering\n  \\begin{narrowtabular}{3cm}{c|c|c|c}\n    \\hline\n    $\\mathbf{\\rho}$     & $\\sigma_{W^+}$ & $\\sigma_{W^-}$ & $\\sigma_{Z^0}$\\\\\n    \\hline\n    $\\sigma_{t\\bar{t}}$ &   -0.716       &    -0.694      &   -0.773      \\\\\n    \\hline\n    $\\sigma_{t}$        &    0.330       &     0.140      &    0.240      \\\\\n    \\hline\n  \\end{narrowtabular}\n  \\caption{Correlation coefficients between $t\\bar{t}$ or $t$-channel \n    single-top and $W^{\\pm}$ or $Z^0$ cross-sections at the LHC 7 TeV in the\n    NNPDF2.0 analysis. }\n  \\label{tab:obs_obs_corr}\n\\end{table}\n\nBoth the values of $\\rho$ and the shape of the correlation ellipses show \na significant anti-correlation between the $t\\bar{t}$ cross-section and \nthe $W^\\pm$ and $Z^0$ ones.\nGiven the fact that the vector boson cross-sections at the LHC are \n${\\cal{O}}(10)$ times larger than the $t\\bar{t}$ cross-section and \nmore accurately known from the theoretical point of view, it is foreseeable \nto use those in order to better calibrate the top pair cross-section \nmeasurement in early data.\n\nOn the other hand the single-top cross-section shows a very mild \ncorrelation to the vector boson one thus rendering a similar approach based \non the precision measurement of EW bosons difficult. However, if one is able\nto identify other observables which should be measured with a similar\nprecision and that are correlated to the single-top cross-section, the\ndiscussion of the $t\\bar{t}$ case would also apply here.\n \n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=0.31\\textwidth]{Wminus-ttbar}\\quad\n  \\includegraphics[width=0.31\\textwidth]{Wplus-ttbar}\\quad\n  \\includegraphics[width=0.31\\textwidth]{Z-ttbar}\\quad\n  \\caption{Correlation between $t\\bar{t}$ and Electroweak Vector Boson cross \n    sections at the LHC 7 TeV. The cross-sections have been computed\nwith MCFM and the NNPDF2.0 parton set.}\n  \\label{fig:ttbar_VB_corr}\n\\end{figure}\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=0.31\\textwidth,angle=90]{Wminus-singletop}\\quad\n  \\includegraphics[width=0.31\\textwidth]{Wplus-singletop}\\quad\n  \\includegraphics[width=0.31\\textwidth]{Z-singletop}\\quad\n  \\caption{Correlation between $t$-channel single-top and Electroweak Vector \n    Boson cross-sections at the LHC 7 TeV.  The cross-sections have\n been computed\nwith MCFM and the NNPDF2.0 parton set.}\n  \\label{fig:singletop_VB_corr}\n\\end{figure}\n\n\\section{Conclusions}\n\nThe quality of top physics results from the LHC experiments will be affected, \namong other factors, by our knowledge of Parton Distribution Functions and \ntheir uncertainties. In this contribution we have reviewed the present status \nof predictions for $t\\bar{t}$ and single-top cross-sections evaluated at \nNLO in QCD with different PDF sets and pointed out differences among them, \ntrying to elucidate the reasons for these differences. For the case of \nsingle-top production, we have shown that one important source of difference\namong predictions obtained using different PDF sets is the value of the \n$b$-quark mass $m_b$ used by the different collaborations. \n\nIn the second part we briefly discussed PDF-induced correlations between \nparton densities and top cross-sections and between the latter and \nelectroweak vector boson production cross-sections at the LHC at 7 TeV. \nThese correlation could be useful to define experimental strategies to \nmeasure the top quark cross-section in a way in which PDF uncertainties are \nreduced.\n\n\\acknowledgments\nAG would like to thank the organizers, and in particular Fabio Maltoni,\nfor the kind invitation to participate in a very nice and stimulating \nWorkshop and the HEPTOOLS European Network for providing the financial \nsupport for his participation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{1. ISM computational complexity derivation}\n\n\n\nFor ISM, DG and SM, the bottleneck resides in the computation of the gradient.\n\\[ f ( W) = \\sum_{i, j} \\gamma_{i, j} e^{- \\frac{\\tmop{Tr} ( W^T A_{i, j}\n   W)}{2 \\sigma^2}} \\]\n\\[ \\frac{\\partial f}{\\partial W} = \\left[ \\sum_{i, j}  \\frac{\\gamma_{i,\n   j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( W^T A_{i, j} W)}{2 \\sigma^2}} A_{i, j}\n   \\right] W \\]\n\\[ \\frac{\\partial f}{\\partial W} = \\left[ \\sum_{i, j}  \\frac{\\gamma_{i,\n   j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( W^T \\Delta x_{i, j} \\Delta x_{i, j}^T\n   W)}{2 \\sigma^2}} A_{i, j} \\right] W \\]\nWhere $A_{i, j} = \\Delta x_{i, j} \\Delta x_{i, j}^T$. The variables have the\nfollowing dimensions.\n\\[ \\begin{array}{l}\n     x_{i, j} \\in \\mathbbm{R}^{d \\times 1}\\\\\n     W \\in \\mathbbm{R}^{d \\times q}\n   \\end{array} \\]\nTo compute a new $W$ with DG, we first mulitply $\\Delta x_{i, j}^T W$, which\nis $O ( d)$. Note that $W$ in DG is always 1 single column. Next, it\nmultiplies with its own transpose to yied $O ( d  + q^2)$. Then we\ncompute $A_{i, j}$ to get $O ( d + q^2 + d^2)$. Since this operation needs to\nbe added $n^2$ times, we get, $O ( n^2 ( d + q^2 + d^2))$. Since $d \\gg q$,\nthis notation reduces down to $O ( n^2 d^2)$. Let $T_1$ be the number of\niterations until convergence, then it becomes $O ( T_1 n^2 d^2)$. Lastly, in\nDG, this operation needs to be repeated $q$ times, hence, $O ( T_1 n^2 d^2\nq)$.\n\n\n\nTo compute a new $W$ with SM, we first mulitply $\\Delta x_{i, j}^T W$, which\nis $O ( d q)$. Next, it multiplies with its own transpose to yied $O ( d q\n + q^2)$. Then we compute $A_{i, j}$ to get $O ( d q + q^2 + d^2)$.\nSince this operation needs to be added $n^2$ times, we get, $O ( n^2 ( d q +\nq^2 + d^2))$. Since $d \\gg q$, this notation reduces down to $O ( n^2 d^2)$.\nThe SM method requires the computation of the inverse of $d \\times d$ matrix.\nSince inverses is cubic, it becomes $O ( n^2 d^2  + d^3)$. Lastly, let\n$T_2$ be the number of iterations until convergence, then it becomes $O ( T_2\n( n^2 d^2 + d^3))$.\n\n\n\nTo compute a new $W$ with ISM, we first mulitply $\\Delta x_{i, j}^T W$, which\nis $O ( d q)$. Next, it multiplies with its own transpose to yied $O ( d q\n + q^2)$. Then we compute $A_{i, j}$ to get $O ( d q + q^2 + d^2)$.\nSince this operation needs to be added $n^2$ times, we get, $O ( n^2 ( d q +\nq^2 + d^2))$. Since $d \\gg q$, this notation reduces down to $O ( n^2 d^2)$.\nThe ISM method requires the computation of the eigen decomposition of $d\n\\times d$ matrix. Since inverses is cubic, it becomes $O ( n^2 d^2  +\nd^3)$. Lastly, let $T_3$ be the number of iterations until convergence, then\nit becomes $O ( T_3 ( n^2 d^2 + d^3))$.\n\n\\end{document}\n\n\\section{GENERAL FORMATTING INSTRUCTIONS}\n\n\n\\end{document}\n\n\\section{GENERAL FORMATTING INSTRUCTIONS}\n\n\\end{document}\n\n\n\n\\section{Introduction}\n\nClustering, i.e., the process of grouping similar objects in a dataset together, is a classic problem. It is extensively used for exploratory data analysis.  Traditional clustering algorithms typically identify a single partitioning of a given dataset.  However, data is often multi-faceted and can be both interpreted and clustered through multiple viewpoints (or, {\\em views}).  For example, the same face data can be clustered based on either identity or based on pose.  In real applications, partitions generated by a clustering algorithm may not correspond to the view a user is interested in. \n\n\n\n\\iffalse\n\\fi\nIn this paper, we address the problem of finding an {\\em alternative clustering}, given a dataset and an existing, pre-computed clustering.  Ideally, one would like the alternative clustering to be {\\em novel} (i.e., non-redundant) w.r.t. the existing clustering to reveal a new viewpoint to the user.  Simultaneously, one would like the result to reveal partitions of high clustering {\\em quality}.\n Several recent papers propose algorithms for  alternative clustering~\\cite{gondek2007non,cui2010learning,dang2010generation,davidson2008finding,cui2007non,niu2014iterative}.  Among them, Kernel Dimension Alternative Clustering (KDAC) is a flexible approach, shown to have superior performance compared to several competitors~\\cite{niu2014iterative}. KDAC is as powerful as spectral clustering in discovering arbitrarily-shaped clusters (including ones that are not linearly separable) that are non-redundant w.r.t.~an existing clustering.  As an additional advantage, KDAC can simultaneously learn the subspace in which the alternative clustering resides. \n\nThe flexibility of KDAC comes at a price: the KDAC formulation involves optimizing a non-convex cost function constrained over the space of orthogonal matrices (i.e, the Stiefel manifold). Niu et al.~\\cite{niu2014iterative} proposed a Dimension Growth (DG) heuristic for solving this optimization problem, which is nevertheless highly computationally intensive. We elaborate on its complexity in Section~\\ref{gen_inst}; experimentally, DG is quite slow, with a convergence time of roughly $46$ hours on an Intel Xeon CPU, for a $624$ sample-sized face data (c.f.~Section~\\ref{sec:exp}).  This limits the applicability of KDAC in interactive exploratory data analysis settings, which often require results to be presented to a user within a few seconds.  It also limits the scalability of KDAC to large data.  Alternately, one can solve the KDAC optimization problem by gradient descent on a Stiefel manifold (SM)~\\cite{wen2013feasible}. However, given the lack of convexity, both DG or SM are prone to get trapped to local minima. Multiple iterations  with random initializations are required to ameliorate the effect of locality.  This increases computation time, and also decreases in effectiveness as the dimensionality of the data increases: the increase in dimension rapidly expands the search space and the abundance of local minima. As such, with both DG and SM, the clustering quality is negatively affected by an increase in dimension.\n\n\n\\iffalse\n\\fi\n\n{\\bf Our Contributions.}   \nMotivated by the above issues, we make the following contributions:\n\\begin{packeditemize}\n\\item We propose an Iterative Spectral Method (ISM), a \\emph{novel algorithm} for solving the non-convex optimization constrained on a Stiefel manifold problem inherent in KDAC. Our algorithm has  several highly desirable properties. First, it \\emph{significantly outperforms} traditional methods such as DG and SM in terms of both computation time and quality of the produced alternative clustering.  Second, the algorithm relies on an \\emph{intuitive use of iterative spectral decompositions}, making it  both easy  to  understand as well as easy to implement, using off-the-shelf libraries. \n\\item  ISM has a natural initialization, constructed through a Taylor approximation of the problem's Lagrangian. Therefore, high quality results can be obtained without random restarts in search of a better initialization. We show that this initialization is a contribution in its own right, as its use improves performance of competitor algorithms.\n\\item We provide \\emph{theoretical guarantees} on its fixed point.  In particular, we establish conditions under which  the fixed point of ISM satisfies both the 1st and 2nd order necessary conditions for local optimality. \n\\item We extensively evaluate the performance of ISM in solving KDAC with synthetic and real data under various clustering quality and cost measures.  Our results show an improvement in execution time by up to a factor of roughly $70$ and $10^5$, compared to SM and DG, respectively. At the same time, ISM  outperforms SM and DG in clustering quality measures along with significantly lower computational cost. \n\n\n\n\\end{packeditemize}\n\\section{Kernel Dimension Alternative Clustering (KDAC) }\n\\label{gen_inst}\n\nIn alternative clustering, a dataset is provided along with existing clustering labels. Given this\nas input, we seek a $\\emph{new}$ clustering that is (a) distinct from the existing clustering, and (b) has high quality with respect to a clustering quality measure.\nAn example illustrating this is shown in Figure   \\ref{fig:moon_illustrate}.  \nThis dataset comprises 400  points in $\\mathcal{R}^4$. Projected to the first two dimensions, the dataset contains two clusters of intertwining parabolas shown as Clustering A. Projected to the last two dimensions, the dataset contains two Gaussian clusters shown as Clustering B. Points clustered together in one view can be in different clusters in the alternative view. In alternative clustering, given (a) the dataset, and (b) one of the two possible clusterings (e.g., Clustering B), we wish to discover the alternative clustering illustrated by the different view.\n\n\\begin{figure}[ht] \n  \\centering\n  \\includegraphics[width=7cm,height=3cm]{{extras\/moon}}\n  \\caption{Four-dimensional moon dataset. Projection into the first two dimensions reveals different clusters than projection to the latter two dimensions.}\n  \\label{fig:moon_illustrate}\n\\end{figure}\n\n\n\nFormally, let $X \\in \\mathcal{R}^{n \\times d}$ be a dataset with $n$ samples and\n$d$ features, along with an existing clustering \n$Y \\in \\mathcal{R}^{n \\times k}$, where $k$ is\nthe number of clusters. If  $x_i$ belongs\nto cluster $j$, then $Y_{i, j}= 1$; otherwise, $Y_{i,j}=0$.\nWe wish to discover an alternative clustering $U \\in \\mathcal{R}^{n \\times k}$ \non some lower dimensional subspace of dimension $q \\ll d$.\nLet $W \\in \\mathcal{R}^{d \\times q}$ be a projection\nmatrix such that $ X W \\in \\mathcal{R}^{n \\times q}$.  \n\n\n\n\nWe seek the optimal projection $W$ and clustering  $U$ that \nmaximizes the statistical dependence between $X W$ with $U$, yielding a high clustering quality, while minimizing the\ndependence between $X W$ and $Y$, ensuring the novelty of the new clustering. \nDenoting DM as a Dependence Measure function, and using $\\lambda$ as a weighing constant, this optimization can be written as:  \n\\begin{subequations}\\label{eq:orig_objective}\n  \\begin{align}\n\\text{Maximize:}& \\quad\\tmop{DM} ( X W , U) - \\lambda \\tmop{DM} ( X W, Y),\\\\\n   \\text{s.t :}& \\quad W^T W = I, U^T U = I.\n\\end{align}\n\\end{subequations}  \nAs in spectral clustering, the labels of the alternative clustering are retrieved by performing $K$-means on matrix $U$, treating its rows as samples. \nThere are many potential choices for DM. The most well-known measures are correlation and mutual information (MI). While correlation performs well in many applications, it lacks the ability to measure non-linear relationships. Although there is clear relationship in Clustering A in Figure \\ref{fig:moon_illustrate}, correlation would mistakenly yield a value of nearly 0. \nAs a dependence measure, MI is superior in that it also measures non-linear relationships. However, due to the probabilistic nature of its formulation, a joint distribution is required. Depending on the distribution, the computation of MI can be prohibitive. \n\nFor these reasons, the Hilbert Schmidt Independence Criterion (HSIC) \\cite{gretton2005measuring} has been proposed for KDAC \\cite{niu2014iterative}. Like MI, it captures non-linear relationships. Unlike MI, HSIC does not require estimating a joint distribution, and it relaxes the need to discretize continuous variables. In addition, as shown by Niu et al.~\\cite{niu2014iterative}, \nHSIC is mathematically equivalent to spectral clustering, further implying that a high HSIC between the data and $U$ yields  high clustering quality.\nA visual comparison of HSIC and correlation  can be found in\nFigure \\ref{HSIC_capture_nonlinear} of Appendix \\ref{App:HSIC} in the supplement.\n\n\nUsing HSIC as a dependence measure, the objective of KDAC becomes \n\\begin{subequations}\\label{eq:main_objective}\n  \\begin{align}\n\\text{Maximize:}& \\quad\\tmop{HSIC} ( X W , U) - \\lambda \\tmop{HSIC} ( X W, Y),\\\\\n   \\text{subject to:}& \\quad W^T W = I, U^T U = I.\n\\end{align}\n\\end{subequations}  \nwhere\n$\\tmop{HSIC} ( X, Y) \\equiv \\frac{1}{(n-1)^2} \\tmop{Tr} ( K_{X} H K_Y H).$\nHere, the variables $K_{X}$ and $K_Y$ are Gram matrices, \nand the $H$ matrix is a\ncentering matrix where $H = I - \\frac{1}{n} \\bm{1}_n \\bm{1}^T_n$ with \n$\\bm{1}$ the $n$-sized  vector of all ones. \nThe elements of $K_{X}$ and $K_Y$\nare calculated by kernel functions $k_{X} ( x_i, x_j)$ and $k_Y ( y_i, y_j)$.  \n The kernel functions for $Y$ and $U$ used in KDAC are $K_Y = Y Y^T$ and $K_U = U U^T$, and the kernel function for $XW$ is  the Gaussian $k_{XW} ( x_i, x_j) = \\exp(- {\\tmop{Tr} [ ( x_i - x_j)^T\nW W^T ( x_i - x_j)]}\/{(2 \\sigma^2)})$. \nDue to the equivalence of HSIC and spectral clustering, the practice of normalizing the kernel $K_{XW}$ is adopted from spectral clustering by Niu et al.~\\cite{niu2014iterative}.   \nThat is, for $K_{XW}$ the unnormalized\nGram matrix, the normalized matrix is defined as $D^{- 1 \/ 2} K_{XW} D^{- 1 \/ 2}$ where $D=\\mathrm{diag}(\\bm{1}_n^TK_{XW})$ is a diagonal matrix whose elements are the  column-sums of $K_{XW}$.\n\n \n\\textbf{KDAC Algorithm.}\nThe optimization problem \\eqref{eq:main_objective} is non-convex. The KDAC algorithm solves (\\ref{eq:main_objective}) using alternate maximization between the variables $U$, $W$ and $D$, updating each while holding the other two fixed. After convergence, motivated by spectral clustering, $U$ is discretized via $K$-means to provide the alternative clustering. The algorithm proceeds in an iterative fashion,  summarized in Algorithm \\ref{KDAC_algorithm}. In each iteration, variables $D$, $U$, and $W$ are updated as follows:\n\n\\textbf{Updating D:} \nWhile holding $U$ and $W$ constant, $D$ is computed as  \n$D=\\mathrm{diag}(\\bm{1}_n^TK_{XW})$.\nMatrix $D$ is subsequently treated as a scaling constant throughout the rest of the iteration. \n\n\\textbf{Updating U:} \nHolding $W$ and $D$ constant and solving for $U$, (\\ref{eq:main_objective}) reduces to :\n\\begin{align} \\label{eq:spectral_clustering}\n     \\textstyle\\max_{U:U^TU=I}  \\tmop{Tr} ( U^T \\mathcal{Q} U), \n\\end{align}\nwhere  $\\mathcal{Q}= H D^{- 1 \/ 2} K_{XW} D^{- 1 \/ 2} H$. This is precisely spectral clustering~\\cite{von2007tutorial}: \n \\eqref{eq:spectral_clustering} can be solved by setting $U$'s columns to the $k$ most dominant eigenvectors of $\\mathcal{Q}$, which can be done in $O(n^3)$ time.\n\n\n   \n\\begin{algorithm}[t]\n    \\scriptsize\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n    \\Input{dataset $X$, original clustering $Y$}\n    \\Output{alternative clustering $U$ }\n    Initialize $W_0$ using $W_{\\mathrm{init}}$ from (\\ref{eq:winit})\\\\\n    Initialize $U_0$ from original clustering\\\\\n    Initialize $D_0$ from $W$ and original clustering\\\\\n    \\While{($U$ not converged) or ($W$ not converged)}{\n        Update $D$ \\\\\n        Update $W$ by solving Equation (\\ref{eq:main_cost_function})\\\\\n        Update $U$ by solving Equation (\\ref{eq:spectral_clustering})\\\\\n\t}\n\tClustering Result $\\gets$ Apply K-means to $U$\n\t\\caption{KDAC Algorithm} \n\\label{KDAC_algorithm}\n\\end{algorithm}\n\\begin{algorithm}[t]\n    \\scriptsize\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n    \\Input{$U$,$D$,$X$, $Y$}\n    \\Output{ $W^*$ }\n    Initialize $W_0$ to the previous value of $W$ in the master loop of KDAC.\\\\\n\t\\While{$W$ not converged}{\n\t\t$W \\gets \\mathop{\\mathrm{eig}}_{\\min} ( \\Phi ( W));$\\\\\n\t\n\t} \t\n\t\\caption{ISM Algorithm} \\label{masteralg}   \n\\end{algorithm}\n   \n   \n\n\n\\textbf{Updating W:} \nWhile holding $U$ and $D$ constant to solve for $W$, (\\ref{eq:main_objective}) reduces to:\n\\begin{subequations}\n\\label{eq:main_cost_function}\n\\begin{align} \n   \\text{Minimize:}\\quad&F(W) =- \\textstyle\\sum_{i, j} \\gamma_{i, j} e^{- \\frac{\\tmop{Tr} [ W^T A_{i, j}\n   W]}{2 \\sigma^2}}\\label{eq:mainobj}\\\\\n   \\text{subject to:}\\quad &W^TW=I\\label{eq:mainconstr}\n\\end{align}\n\\end{subequations}\nwhere $\\gamma_{i,j}$ are the elements of matrix $\\gamma = D^{- 1 \/ 2} H ( U U^T - \\lambda Y Y^T) H D^{- 1 \/ 2}$, and \n$A_{i, j} = ( x_i - x_j) ( x_i - x_j)^T$ (see Appendix \\ref{Cost_Derivation} in the supplement for the derivation). \nThis objective, along with a Stiefel Manifold constraint, $W^TW=I$, pose a challenging optimization problem as neither is convex. Niu et al.~\\cite{niu2014iterative} propose solving \\eqref{eq:main_cost_function} through an algorithm termed Dimensional Growth (DG). This algorithm solves for $W$ by computing individual columns of $W$ separately through gradient descent (GD). Given a set of computed columns, the next column is computed by GD projected to a subspace orthogonal to the span of computed set. Since DG is based on GD, the computational complexity is dominated by computing the gradient of \\eqref{eq:mainobj}. The latter is given by:\n\\begin{equation} \\label{eq:gradient_of_main_cost_function}\n\\nabla F(W) =\\textstyle\\sum^n_i \\sum_j^n  \\frac{\\gamma_{i, j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} [\n   W^T A_{i, j} W]}{2 \\sigma^2}} A_{i, j} W. \n\\end{equation}\nThe complexity of DG is $O(t_{DG} n^2d^2q)$, where $n$, $d$ are the dataset size and dimension, respectively, $q$ is the dimension of the subspace of the alternative clustering, and $t_{DG}$ is the number of iterations of gradient descent. The calculation of the gradient contributes the term $O(n^2d^2q)$. Although this computation is highly parallelizable, the algorithm still suffers from slow convergence rate. Therefore, $t_{DG}$ often dominates the computation cost.  \n\n\n\n\nAn alternative approach to optimize \\eqref{eq:main_cost_function} is through classic methods for performing optimization on the Stiefel Manifold (SM) \\cite{wen2013feasible}. The computational complexity of this algorithm is dominated by the computation of the gradient and a matrix inversion with $t_{SM}$ iterations.   This yields a complexity of $O(t_{SM} n^2 d^2 + t_{SM}d^3)$ for SM. Finally, as gradient methods applied to a non-convex objective, both SM and DG require multiple executions from random initialization points to find improved local minima. This approach becomes less effective as the dimension $d$ increases.\n\n\n\n\n\\section{An Iterative Spectral Method}\n\nThe computation of KDAC is dominated by the $W$ updates in Algorithm~\\ref{KDAC_algorithm}. Instead of using DG or SM to solve the optimization problem for $W$ in KDAC, we propose an Iterative Spectral Method (ISM).  Our algorithm is motivated from the following observations. The Lagrangian of \\eqref{eq:main_cost_function} is:\n \\begin{align*}\n      \\mathcal{L} ( W, \\Lambda) =& - \\textstyle\\sum_{i, j} \\gamma_{i, j} \\exp\\left(- \\frac{\\tmop{Tr}\n      (W^T A_{i, j} W)}{2 \\sigma^2}\\right)\\\\\n      &- \\frac{1}{2} \\tmop{Tr} ( \\Lambda ( W^T W -\n      I)) \\numberthis \\label{eq:lag}\n\\end{align*}\n    Setting $\\nabla_W \\mathcal{L}(W,\\Lambda)=0$ gives us the equation:\n\\begin{align}\\Phi(W)W = W \\Lambda,\\label{eq:balance} \\end{align}\nwhere \n\\begin{equation} \\label{eq:phi_equation}\n  \\Phi ( W) = \\textstyle\\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^2} \\exp(- \\frac{\\tmop{Tr} [ W^T A_{i, j}\n  W]}{2 \\sigma^2}) A_{i, j},\n\\end{equation}\nand $\\Lambda$ is a diagonal matrix.\n\nRecall that a feasible $W$, satisfying \\eqref{eq:mainconstr}, is orthonormal. \n\\eqref{eq:balance} is an eigenequation; thus, a stationary point $W$ of the Lagrangian \\eqref{eq:lag}  comprises of $q$ eigenvectors of $\\Phi(W)$ as columns.\nMotivated by this observation, ISM attempts to find such a $W$ in the following iterative fashion. Let $W_0$ be an initial  matrix. Given $W_k$ at iteration $k$, the matrix $W_{k+1}$  is computed as:\n$$W_{k+1} = \\textstyle\\mathop{\\mathrm{eig}}_{\\min} ( \\Phi ( W_k)), \\quad k=0,1,2,\\ldots,$$\nwhere the operator $\\mathop{\\mathrm{eig}}_{\\min} (A)$ returns a matrix whose columns are\nthe $q$ eigenvectors corresponding to the smallest eigenvalues of $A$.\n\n\nISM is summarized in Alg.~\\ref{masteralg}. Several important observations are in order. First, the algorithm ensures that $W_k$, for $k\\geq 1$, is feasible, by construction: selected eigenvectors are orthonormal and satisfy \\eqref{eq:mainconstr}. Second, it is also easy to see that a fixed point of the algorithm will also be a stationary point of the Lagrangian \\eqref{eq:lag} (see also~Lemma~\\ref{lemma:eig}). Though it is harder to prove,  selecting  eigenvectors corresponding to the \\emph{smallest} eigenvalues is key: we show that this is precisely the property that relates a fixed point of the algorithm to the local minimum  conditions (see Thm.~\\ref{thm:stationary}).\nFinally, ISM has several computational advantages. For $t_{ISM}$ iterations, the calculation of $\\Phi(W)$, and the ensuing eigendecomposition yields a complexity of $O(t_{ISM}( n^2 d^2 +  d^3))$. Since $q\\ll d$, various approximation methods \\cite{vladymyrov2016variational}\\cite{richtarikgeneralized}\\cite{lei2016coordinate} can be employed to find the few eigenvectors. For example, the Coordinate-wise Power Method\\cite{lei2016coordinate}, approximates the most dominant eigenvalue at $O(d)$ time, reducing ISM's complexity to $O(t_{ISM} n^2 d^2)$. This improvement is further confirmed experimentally (see Figure \\ref{fig:ND_vs_time}). Lastly, $t_{ISM}$ is magnitudes smaller than both $t_{DG}$ and $t_{SM}$. In general $t_{ISM} < 10$, while $t_{SM} > 50$ and $t_{DG} > 200$.\n\n \n\\subsection{Convergence Guarantees}\n\nAs mentioned above, the selection of the eigenvectors corresponding to the \\emph{smallest} eigenvalues of $\\Phi(W_k)$ is crucial for the establishment of a stationary point. Namely, we establish the following theorem:\n\\begin{theorem}\\label{thm:stationary}\n  For large enough $\\sigma$ (satisfying Inequality \\eqref{ineq:large_sigma_orig}), a fixed point $W^*$ of  Algorithm~\\ref{masteralg} satisfies the necessary conditions of a local minimum of \\eqref{eq:main_cost_function} if $\\Phi(W^*)$ is full rank.\n\\end{theorem}\n\\begin{proof}\nThe main body of the proof is organized into a series of lemmas proved in the supplement.\n\nOur first auxiliary lemma (from~\\cite{wright1999numerical}), establishes conditions necessary for a stationary point of the Lagrangian to  constitute local minimum. \n\\begin{lemma} \\label{lemma:2nd_order}\n  [Nocedal,Wright, Theorem 12.5~{\\cite{wright1999numerical}}] (2nd Order Necessary Conditions)\n    Consider the optimization problem:\n  $ \\min_{W : h (W) = 0} f (W), $\nwhere $f : \\mathbb{R}^{d \\times q} \\to \\mathbb{R}$ and $h :\n  \\mathcal{R}^{d \\times q} \\to \\mathbb{R}^{q \\times q}$ are twice continuously\n  differentiable. Let   $\\mathcal{L}$ be the Lagrangian of this optimization problem. Then, a local minimum must satisfy the following  conditions:\n  \\begin{subequations}\n \n    \\begin{align} \n    &\\nabla_W \\mathcal{L} (W^{\\ast}, \\Lambda^{\\ast}) = 0, \\label{eq:1st_W}\\\\\n    &\\nabla_{\\Lambda} \\mathcal{L} (W^{\\ast}, \\Lambda^{\\ast}) = 0, \\label{eq:1st_lambda}\\\\\n  \n \n \n \n   \n    \\begin{split}\n    \\tmop{Tr} ( Z^T &\\nabla_{W W}^2 \\mathcal{L}(W^{\\ast}, \\Lambda^{\\ast}) Z) \\geq 0 \\\\&\\tmop{for}   \\tmop{all} Z \\neq 0 , \\tmop{with} \n       \\nabla h (W^{\\ast})^T Z = 0.   \\label{eq:2nd_W}\n       \\end{split}\n    \\end{align}   \n \n  \\end{subequations}\n\\end{lemma}\nArmed with this result, we next characterize the properties of a fixed point of Algorithm \\ref{masteralg}:\n\\begin{lemma} \\label{basic_lemma}\n  Let $W^{\\ast}$ be a fixed point of Algorithm \\ref{masteralg}. Then it satisfies:\n \n   $ \\Phi ( W^{\\ast}) W^{\\ast} = W^{\\ast} \\Lambda^{\\ast},$\n \n  where $\\Lambda^{\\ast} \\in \\mathcal{R}^{q \\times q}$ is a diagonal matrix\n  containing the $q$ smallest eigenvalues of $\\Phi ( W^{\\ast})$ and\n\n  $W^{\\ast^T} W^{\\ast} = I. $\n \n\\end{lemma}\nThe proof can be found in Appendix \\ref{proof_of_lemma_2}.\nOur next result, whose proof is  in Appendix \\ref{proof_of_lemma_3}, states that \n a fixed point satisfies the 1st order conditions of Lemma \\ref{lemma:2nd_order}. \n\\begin{lemma} \\label{lemma:eig}\n  If $W^{\\ast}$ is a fixed point and $\\Lambda^{\\ast}$ is as defined in Lemma~\\ref{basic_lemma},\n  then $W^{\\ast}$, $\\Lambda^*$ satisfy the 1st order conditions (\\ref{eq:1st_W})(\\ref{eq:1st_lambda}) of Lemma \\ref{lemma:2nd_order}.\n\\end{lemma}\nOur last lemma, whose proof is in Appendix \\ref{proof_of_lemma_4}, establishes that a fixed point satisfies the 2nd order conditions of Lemma \\ref{lemma:2nd_order}, for large enough $\\sigma$.  \n\\begin{lemma} \\label{lemma:2nd_order_lemma}\n  If $W^{\\ast}$ is a fixed point, $\\Lambda^{\\ast}$ is as defined in Lemma~\\ref{basic_lemma}, and $\\Phi(W^*)$ is full rank,\n  then given a large enough $\\sigma$ (satisfying Inequality \\eqref{ineq:large_sigma_orig}), $W^{\\ast}$ and $\\Lambda^*$ satisfy the 2nd order condition (\\ref{eq:2nd_W}) of Lemma \\ref{lemma:2nd_order}.\n\\end{lemma}\n Thm.~\\ref{thm:stationary} therefore follows.\n\\end{proof}\n\n\nThm.~\\ref{thm:stationary} is stated in terms of a large enough $\\sigma$; we can characterize this constraint precisely. In the proof of Lemma~\\ref{lemma:2nd_order_lemma} we establish the following condition on $\\sigma$:\n\\begin{multline}\n   \\sigma^2 [\\min_i ( \\bar{\\Lambda^*}_i) -\\max_j(\\Lambda_j^*)] \\geq \\\\\n    \\sum_{i, j} \\frac{|\\gamma_{i, j}|}{\\sigma^2} e^{-\n   \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}}\\tmop{Tr} (A^T_{i,j}A_{ij}). \\label{ineq:large_sigma_orig}\n\\end{multline}\nHere, $\\Lambda^*$ is the set of $q$ smallest eigenvalues of $\\Phi(W)$, and $\\bar{\\Lambda^*}$ is the set of the remaining eigenvalues. The left-hand side (LHS) of the equation further motivates ISM's choice of  eigenvectors corresponding to the $q$ smallest eigenvalues. This selection guarantees that the LHS of the inequality is positive. Therefore, given a large enough $\\sigma$, Inequality \\eqref{ineq:large_sigma_orig} and the 2nd order condition is satisfied. \n\nFurthermore, this equation provides a reasonable suggestion for the value of $q$.  Since we wish to maximize the term $(\\min_i ( \\bar{\\Lambda^*}_i) -\\max_j(\\Lambda_j^*))$ to satisfy the inequality, the value $q$ should be set where this gap is maximized. More formally, we will defined\n\\begin{align}\n   \\delta_{gap}=\\min_i ( \\bar{\\Lambda^*}_i) -\\max_j(\\Lambda_j^*)\n   \\label{eq:eiggap}.\n\\end{align}\nas the eigengap.\n\n\n\n\n\n\n\\subsection{Spectral Initialization via Taylor Approximation}\n\\label{sec:initialization}\nISM admits a natural initialization point, constructed via a Taylor approximation of the objective. As we show experimentally in Section~\\ref{sec:exp}, this initialization is a contribution in its own right: it improves both clustering quality and convergence time for ISM \\emph{as well as} competitor algorithms.  To obtain a good initialization, observe that by using the 2nd order Taylor approximation of the objective function \\eqref{eq:mainobj} at $W=0$, the Lagrangian can be approximated by \n\\begin{align*}\n\\tilde{\\mathcal{L}}(W,\\Lambda) \\approx & -\\textstyle \\sum_{i, j} \\gamma_{i, j}  \\left( 1 - \\frac{\\tmop{Tr}\n   (W^T A_{i, j} W)}{2 \\sigma^2} \\right)\\\\\n   & + \\frac{1}{2} \\tmop{Tr} (\\Lambda (I -\n   W^T W)). \n\\end{align*}\n  Setting $\\nabla_W \\tilde{\\mathcal{L}}(W,\\Lambda)=0$ reduces the problem into a simple eigendecomposition, namely, the one defined by the system \n$\\left[ \\textstyle \\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^2} A_{i,\n   j} \\right] W = W \\Lambda .$\nHence, the 2nd order Taylor approximation of the original cost objective has a closed form global minimum that can be used as an initialization point, namely:\n\\begin{align}\n   \\textstyle W_{\\mathrm{init}}=\\mathrm{eig}_{\\min}( \\sum_{i, j} {\\gamma_{i, j}} A_{i,\n   j}\/{\\sigma^2})\\label{eq:winit}.\n\\end{align}\n\nWe use this spectral initialization (SI) in the first master iteration of KDAC. In subsequent master iterations, $W_0$ (the starting point of ISM) is set to be the last value to which ISM converged to previously.\n\n\n\n\n\\section{Scalability}\\label{sec:scalability}\n\\chieh{The scalability adventage of ISM is two folds. First, the most computational complex portion of the algorithm is the eigen decomposition of $\\Phi(W) \\in \\mathbb{R}^{d \\times d}$. Since $d<<N$, the scalability is decoupled from the size of the data. Even if high dimensional data is used, the eigendecomposition within ISM only requires q eigenvectors. Since $q << N$, there exist many techniques to approximate eigenvectors, e.g, the power method. There also exist off the shelf GPU implementations of eigendecomposition (MAGMA, cuBLAS). To further improve the speed, updates of U can also be improved via the Variational Nystrom method, Vladymyrov et al. ICML16. Our key contribution is to provide theoretical guarantees for the spectral method to leverage existing research and technology. Second,   the proposed initialization is intimately tied to the ISM method. It results from applying the 2nd order Taylor approximation to the objective to approximate a closed form solution of the global minimum. Having a starting point close to the global minimum simultaneously improves convergence rate and the quality of the solution.}\n\n\n\\section{Experimental Results}\\label{sec:exp}\nWe experimentally validate the performance of ISM in terms of both speed and clustering quality. The source code is publicly available at github\\footnote{https:\/\/github.com\/neu-spiral\/ISM}. Because \\cite{niu2014iterative} has already performed extensive comparisons of KDAC against other alternative clustering methods, this section will concentrate on comparing  ISM to competing models for optimizing KDAC: Dimensional Growth (DG)~\\cite{niu2014iterative} and gradient descent on the Stiefel Manifold (SM)~\\cite{wen2013feasible}. SM is a generic approach, while DG is the approach originally proposed to solve KDAC~\\cite{niu2014iterative}. \n\nIn addition to introducing a new algorithm for optimizing KDAC, we also proposed an intelligent initialization scheme based on Taylor approximation in Section \\ref{sec:initialization}, we call Spectral Initialization (SI).  We also investigate how SI effects the performance of the various algorithms compared to the standard random initialization (RI).\n\n\n\n\\begin{table}[h]\n\\tiny\n\\centering\n\\setlength\\tabcolsep{0.2pt}\n    \\begin{tabular}{c|c|c|c|c|c}\n    \\textbf{SG} \n        & NMI       $\\uparrow$\n        & CQ        $\\uparrow$\n        & Novelty   $\\downarrow$\n        & Cost      $\\downarrow$\n        & Time      $\\downarrow$\\\\\n    \\midrule\n        ISM SI\n        \t& \\textbf{1} \n        \t& \\textbf{2} \n        \t& \\textbf{0} \n        \t& -1.2\n        \t& \\textbf{0.02} \\\\\n        ISM RI\n        \t& 0.4$\\pm$0.49\n        \t& 1.6$\\pm$0.478\n        \t& \\textbf{0.0$\\pm$0.0} \n        \t& -1.48$\\pm$0.381\n        \t& 0.04$\\pm$0.01\\\\\n        SM SI\n        \t& \\textbf{1} \n        \t& 1.99\n        \t& \\textbf{0} \n        \t& \\textbf{-1.791} \n        \t& 0.404\\\\\n        SM RI\n        \t& \\textbf{1$\\pm$0.0} \n        \t& 1.79$\\pm$0.398\n        \t& \\textbf{0.0$\\pm$0.0} \n        \t& -1.59$\\pm$0.398\n        \t& 1.47$\\pm$1.521\\\\\n        DG SI\n        \t& 0\n        \t& 1.629\n        \t& 1\n        \t& -1.316\n        \t& 1.732\\\\\n        DG RI\n        \t& 0.93$\\pm$0.196\n        \t& 0.99$\\pm$0.0\n        \t& 0.03$\\pm$0.104\n        \t& -0.99$\\pm$0.0\n        \t& 1.51$\\pm$0.102\\\\\n    \\midrule\n    \\textbf{LG} \\\\\n        ISM SI\n        \t& \\textbf{1} \n        \t& \\textbf{1.9}\n        \t& \\textbf{0} \n        \t& 509\n        \t& 0.239\\\\\n        ISM RI\n        \t& 0.8$\\pm$0.4\n        \t& \\textbf{1.9$\\pm$0.011}\n        \t& \\textbf{0.0$\\pm$0.0} \n        \t& 605$\\pm$193\n        \t& \\textbf{0.18$\\pm$0.038} \\\\\n        SM SI\n        \t& \\textbf{1} \n        \t& 1.9\n        \t& \\textbf{0} \n        \t& 509\n        \t& 0.413\\\\\n        SM RI\n        \t& 0.4$\\pm$0.49\n        \t& 1.9$\\pm$0.031\n        \t& 0.6$\\pm$0.49\n        \t& 1095$\\pm$479\n        \t& 11.21$\\pm$11.01\\\\\n        DG SI\n        \t& \\textbf{1} \n        \t& \\textbf{1.9} \n        \t& \\textbf{0} \n        \t& 509.975\n        \t& 1595.174\\\\\n        DG RI\n        \t& 0.1$\\pm$0.32\n        \t& 0.61$\\pm$0.17\n        \t& 0.9$\\pm$0.32\n        \t& \\textbf{101.5$\\pm$30.8} \n        \t& 1688$\\pm$551\\\\\n     \\midrule\n    \\textbf{Moon} \\\\       \t\n        ISM SI\n        \t& \\textbf{1} \n        \t& \\textbf{2} \n        \t& \\textbf{0} \n        \t& \\textbf{149} \n        \t& \\textbf{0.575} \\\\\n        ISM RI\n        \t& 0.4$\\pm$0.4\n        \t& \\textbf{2.0$\\pm$0.0} \n        \t& 0.5$\\pm$0.5\n        \t& 269.98$\\pm$43\n        \t& 3.07$\\pm$1.4\\\\\n        SM SI\n        \t& \\textbf{1} \n        \t& 1.998\n        \t& \\textbf{0} \n        \t& \\textbf{149} \n        \t& 2.653\\\\\n        SM RI\n        \t& 0.27$\\pm$0.335\n        \t& \\textbf{2.0$\\pm$0.0} \n        \t& 0.5$\\pm$0.5\n        \t& 277$\\pm$18\n        \t& 258$\\pm$374\\\\\n        DG SI\n        \t& \\textbf{1} \n        \t& 1.998\n        \t& \\textbf{0} \n        \t& 149.697\n        \t& 359.188\\\\\n        DG RI\n        \t& 0.09$\\pm$0.3\n        \t& 1.27$\\pm$0.6\n        \t& 0.8$\\pm$0.4\n        \t& 161.65$\\pm$102\n        \t& 3212$\\pm$1368\\\\\n    \\midrule\n    \\textbf{MoonN} \\\\       \t       \t\n        ISM SI\n        \t& \\textbf{1} \n        \t& \\textbf{2} \n        \t& \\textbf{0} \n        \t& \\textbf{15.3} \n        \t& \\textbf{0.3} \\\\\n        ISM RI\n        \t& 0.91$\\pm$0.2\n        \t& \\textbf{2.0$\\pm$0} \n        \t& 0.04$\\pm$0.1\n        \t& 15.59$\\pm$0.8\n        \t& 0.55$\\pm$0.2\\\\\n        SM SI\n        \t& \\textbf{1} \n        \t& \\textbf{2} \n        \t& \\textbf{0} \n        \t& \\textbf{15.3} \n        \t& 0.452\\\\\n        SM RI\n        \t& 0.22$\\pm$0.4\n        \t& \\textbf{2.0$\\pm$0} \n        \t& 0.72$\\pm$0.4\n        \t& 17.7$\\pm$1\n        \t& 102$\\pm$48.5\\\\\n        DG SI\n        \t& \\textbf{1} \n        \t& \\textbf{2} \n        \t& \\textbf{0} \n        \t& 15.4\n        \t& 3836.3\\\\\n        DG RI\n        \t& 0$\\pm$0.0\n        \t& 1.6$\\pm$0.04\n        \t& 1.0$\\pm$0.0\n        \t& 16.2$\\pm$0.06\n        \t& 3588$\\pm$304\\\\\n    \\midrule\n    \\textbf{Flower} \\\\       \t\n        ISM SI\n        \t& \\textbf{-} \n        \t& \\textbf{0.41} \n        \t& \\textbf{0} \n        \t& 20\n        \t& \\textbf{0.04}\\\\\n        ISM RI\n        \t& \\textbf{-} \n        \t& \\textbf{0.41$\\pm$0.0} \n        \t& \\textbf{0.0$\\pm$0.0} \n        \t& 19.51$\\pm$0.0\n        \t& 0.1$\\pm$0.01 \\\\\n        SM SI\n        \t& \\textbf{-} \n        \t& \\textbf{0.41} \n        \t& \\textbf{0} \n        \t& 20\n        \t& 0.153\\\\\n        SM RI\n        \t& \\textbf{-} \n        \t& 0.3$\\pm$0.2\n        \t& 0.01$\\pm$0.012\n        \t& \\textbf{0.7$\\pm$0.5} \n        \t& 0.8$\\pm$0.6\\\\\n        DG SI\n        \t& \\textbf{-} \n        \t& \\textbf{0.41} \n        \t& \\textbf{0} \n        \t& 20\n        \t& 27.251\\\\\n        DG RI\n        \t& \\textbf{-} \n        \t& 0.35$\\pm$0.02\n        \t& 0.2$\\pm$0.4\n        \t& 20.6$\\pm$1\n        \t& 37.0$\\pm$5\\\\\n    \\midrule\n    \\textbf{Faces} \\\\       \t       \t       \t\n        ISM SI\n        \t& \\textbf{0.57} \n        \t& \\textbf{0.61} \n        \t& 0.004\n        \t& 59.6\n        \t& \\textbf{1.5} \\\\\n        ISM RI\n        \t& 0.56$\\pm$0.002\n        \t& \\textbf{0.61$\\pm$0.0} \n        \t& \\textbf{0.0$\\pm$0.0} \n        \t& 59.6$\\pm$0.118\n        \t& 1.51$\\pm$0.146\\\\\n        SM SI\n        \t& 0.562\n        \t& 0.608\n        \t& 0.004\n        \t& 59.559\n        \t& 7.96\\\\\n        SM RI\n        \t& \\textbf{0.57$\\pm$0.002} \n        \t& \\textbf{0.61$\\pm$0} \n        \t& 0.004$\\pm$0\n        \t& 59.7$\\pm$0.1\n        \t& 109$\\pm$35.2\\\\\n        DG SI\n        \t& 0.564\n        \t& 0.6\n        \t& 0.004\n        \t& 59.8\n        \t& 100591.071\\\\\n        DG RI\n        \t& 0.458$\\pm$0.045\n        \t& 0.565$\\pm$0.018\n        \t& 0.024$\\pm$0.018\n        \t& \\textbf{54.28$\\pm$3} \n        \t& 166070$\\pm$4342\\\\\n    \\midrule\n    \\textbf{WebKb} \\\\\n        ISM SI\n        \t& \\textbf{0.37} \n        \t& 0.286\n        \t& \\textbf{0} \n        \t& -0.273\n        \t& 231\\\\\n        ISM RI\n        \t& 0.29$\\pm$0.07\n        \t& 0.54$\\pm$0.2\n        \t& 0.01$\\pm$0.002\n        \t& -0.52$\\pm$0.2\n        \t& 137.45$\\pm$15.7\\\\\n        SM SI\n        \t& 0.048\n        \t& \\textbf{3.296} \n        \t& 0.034\n        \t& \\textbf{-3.159} \n        \t& \\textbf{9.945} \\\\\n        SM RI\n        \t& 0.33$\\pm$0.06\n        \t& 0.12$\\pm$0.005\n        \t& 0.008$\\pm$0.004\n        \t& -0.11$\\pm$0.004\n        \t& 13511$\\pm$13342\\\\\n        DG SI\n        \t& 0.048\n        \t& 1.066\n        \t& 0.034\n        \t& -1.019\n        \t& 199887.134\\\\\n        DG RI\n        \t& 0.23$\\pm$0\n        \t& 1.06$\\pm$0\n        \t& 0.1$\\pm$0.005\n        \t& 0.616$\\pm$0.04\n        \t& 727694$\\pm$41068\\\\    \n\\end{tabular}\n\\caption{\n The Normalized Mutual Information, Clustering Quality, and clustering novelty are abbreviated in the first 3 columns as NMI, CQ, and Novelty. The cost of the objective and run time is displayed in the last two columns. For each optimization technique, spectral initialization (SI) and random initialization (RI) are separately tested. With RI, 10 random initial points have been tested with their mean and std displayed. } \\label{experiment_table_and_figures}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\\begin{figure}[!t] \n  \\centering\n  \\includegraphics[width=7.7cm,height=8.7cm]{{extras\/synthetic_data_results}} \n  \\caption{ Figure includes all original and alternative displayable clusterings. The face data is originally clustered by the identity of the individual. Here the average image of the alternative clustering is displayed. It is observable that the original and alternative clusters are all visually obvious clusters and provide alternative views.} \n  \\label{fig:results_figure}\n\\end{figure}\n\n\n\n\\begin{figure}[!t] \n  \\centering\n  \\includegraphics[width=0.7\\columnwidth]{{extras\/DimVTime_PolyFIt_slope}}\n  \\caption{Growth in dimension vs time in log\/log scale. A slope of 1 is linear growth.}\n  \\label{fig:ND_vs_time}\n\\end{figure}\n\n\\noindent\\textbf{Datasets.} We perform experiments on four synthetic and three real datasets.  The synthetic data are displayed in Figure \\ref{fig:results_figure}. Small Gaussian (SG) contains four Gaussian clusters with $40$ samples and two features shown at the top left of the figure. When this data is projected down to feature 2, the original clustering is created. Rotating this projection onto feature 1, the alternative clustering emerges. Large Gaussian (LG) contains four Gaussian clusters with $1000$ samples and four dimensions is shown in the top center location. The Gaussian clusters are rotated by $45$ degrees to reside in 3D, and the fourth dimension is generated from a uniform noise distribution. This synthetic data is designed to test the response of KDAC within a noisy environment.  \nWe generate two additional synthetic datasets shown in top right: Moon and Moon+Noise (MoonN). Both datasets have the first two dimensions as two parabolas and the second two dimensions as Gaussian clusters. The MoonN dataset further includes three noisy dimensions generated from a uniform distribution with 1000 samples.\nSince the Gaussian clusters have a compact structure and the parabolas have a non-linear structure, these datasets demonstrate KDAC's ability to handle mixed clustering structures in a uniformly noisy environment.\nDue to the novelty of alternative clustering, there has been very few public repository data that have at least two alternative labels. \nThe two traditional benchmark datasets are CMU's WebKB dataset~\\cite{cmu4universities} and face images from the UCI KDD repository~\\cite{bay2000uci}.  CMU's WebKB dataset consists of 1041 html pages from 4 universities. One labelling is based on universities, and an alternative labelling based on topic (course, faculty, project, student). After preprocessing by removing rare and stop words, we are left with 500 words. \nThe face dataset consists of $640$ images from $20$ people in four different poses. Each image has 32x30 pixels. Images are vectorized and PCA is then used to further condense the dataset by keeping 85\\% of the variance, resulting in a dataset of 624 samples and 20 features. \nWe set the existing clustering based on identity, and\nseek an alternative clustering   based on pose. The last real dataset is the Flower image by Alain Nicolas \\cite{particleoft}, a 350x256 pixel image. The RGB values of each pixel is taken as a single sample, with repeated samples removed. This results in a dataset of 256 samples and 3 features.\nAlthough this dataset does not have labels, the quality of the alternative clustering can be visually observed. With the exception of the high dimensional WebKB data, the remaining datasets are visualized in Figure \\ref{experiment_table_and_figures} (b).\n\n\\noindent\\textbf{Hyperparameter Tuning.}\nISM has 3 hyperparameters that require tuning, $q$, $\\lambda$, and $\\sigma$. The hyperparameter $q$ with a potential range of (0,$d$) is initially set to $k$, the number of clusters. A grid search was then conducted for $\\lambda \\in (0,10]$ and $\\sigma \\in (0,10]$ to find the highest CQ ($\\lambda$, $\\sigma$) pair that satisfies inequality (\\ref{ineq:large_sigma_orig}) at an increment of 0.01. In the event that Inequality (\\ref{ineq:large_sigma_orig}) cannot be satisfied, the highest CQ closest to satisfying (\\ref{ineq:large_sigma_orig}) is used. At this point, since $\\Phi(W^*)$ is computed, the $q$ value that maximizes the eigengap as defined in Eq(\\ref{eq:eiggap}) can be used if (\\ref{ineq:large_sigma_orig}) is not yet satisfied. Note that competing models do not have a natural way of selecting hyperparameters. Since all competing models optimize the same objective, we report all results using the same hyperparameters. To ensure reproducible experiments, the hyperparameters utilized in each experiment is provided in Appendix \\ref{App:Hyperparameters}.\n\n\\noindent\\textbf{Evaluation Method.} \nTo exhaustively evaluate ISM on KDAC, both external and internal measures have been recorded in Table~\\ref{experiment_table_and_figures}. More specifically, column 1 and 3 (NMI and Novelty) are external measures because they compare alternative cluster assignments against an externally known ground truth. To make the comparison, Normalized Mutual Information (NMI) as suggested by \\cite{strehl33knowledge} was used.  The NMI is a measure between 0 to 1 with 0 denoting no relationship and 1 as maximum relationship. Column 1 (NMI) is calculated by computing the NMI between the ground truth and the alternative cluster assignments. Column 3 (Novelty) is calculated by computing the NMI between the alternative cluster assignments against the original label. Ideally, we wish for the NMI of the alternative cluster assignments against the ground truth to be 1, and 0 against the original clustering.  If we let $U$ and $L$ be two clustering assignments, NMI can be calculated with $NMI(L,U) = \\frac{I(L,U)}{\\sqrt{H(L)H(U)}}$, where $I(L,U)$ is the mutual information between $L$ and $U$, and $H(L)$ and $H(U)$ are the entropies of $L$ and $U$ respectively. Lastly, since no ground truth exists for the Flower dataset, the NMI field is not applicable and indicated with a dash (-).\n\nColumns 2, 4 and 5 (CQ, Cost, Time) are internal measures used to evaluate ISM. They are internal measures since they are computed solely from the data and the algorithm with no external knowledge applied during the comparison. The clustering quality (CQ) is computed with $HSIC(XW,U)$ with $U$ as the clustering solution. This is equivalent to the spectral clustering objective with high values denoting high clustering quality. The cost quality records the objective function of KDAC in Eq. (\\ref{eq:main_objective}). The last column, Time, measures the execution time of the entire KDAC algorithm in seconds on an Intel Xeon E7 processor.\n\n\nIn Table~\\ref{experiment_table_and_figures}, we report the performance of the various methods: our ISM, SM, and DG, paired with two initialization schemes: our spectral initialization (SI) and random initialization (RI).  For random initialization, we repeat these 10 times and report the mean and standard deviation for each measure.  The optimal direction of each measure is denoted by the $\\uparrow\\downarrow$, with $\\uparrow$ denoting a preference towards larger values and $\\downarrow$ otherwise. For each field, the optimal result is printed in bold font.\n\n\n\n\\noindent\\textbf{Performance Comparison Results.} \nTable \\ref{experiment_table_and_figures} illustrates that ISM with SI outperforms its competitors in both internal and external quality measure for the first 3 columns. Since it is possible for the objective cost to achieve a low cost with a trivial solution based on different $(\\sigma, \\lambda)$ pairs, the objective should be low, but it is not always indicative of better clustering quality. This is demonstrated in the case of WebKb dataset, where SM+SI achieved a lower cost with a faster convergence. However, upon inspecting the clustering allocation in this case, we observed that it was a trivial classification, with almost all points in the same cluster.  Since the demand for a faster KDAC was the original motivator, special attention should be paid to the Time column. The execution time of ISM significantly outperforms the original approach (DG) as well as the standard approach (SM). This speed improvement is especially true in the Faces dataset where ISM improves the speed by 5 folds. \n\n\n\n\n\n\\noindent\\textbf{The Effect of Spectral Initialization.}\n By comparing SI against RI, we can isolate the effect of our proposed spectral initialization (SI). Comparing the rows, the spectral initialization improved both time and clustering quality for all methods. From this observation, we conclude that SI contributes to clustering quality by starting each algorithm at a desirable initialization. The convergence improvement came from placing each algorithm closer to the local minimum. Comparing all methods when using SI, we observed that ISM optimization still outperformed other algorithms in terms of time. From this, we conclude that the proposed SI has a greater impact on the clustering quality while the ISM optimization technique contribute towards faster convergence.\n\n\n\\iffalse\nIt is also important to point out that since ISM is a spectral method requiring only a few eigenvalues, it is possible to use  techniques such as the power method \\cite{richtarikgeneralized}, has shown strong improvement as the data scale.  However, the table contains a version purely based on the merits of ISM without extra improvements. Therefore, the source code that produced Table \\ref{experiment_table_and_figures} used only an off the shelf Numpy function for the eigendecomposition. \n\\fi\n\n\\noindent\\textbf{Scalability.}\nNote that KDAC consists of optimizing $U$ and $W$ from Eq. (\\ref{eq:main_objective}). Since the computational bottleneck resided in the optimization of $W$, ISM was designed to speed up this portion. The scalability analysis, therefore, will concentrate on only the optimization of $W$. ISM has a computational complexity of $O(t_{ISM}(n^2d^2+d^3)$ and the complexity can be divided into the calculation of the derivative and the eigendecomposition of $\\Phi(W)$. Although the derivative contributes to $O(n^2d^2)$, it is a highly parallelizable operation that can be rendered trivial through the usage of GPUs. Therefore, ISM's true bottleneck is not the number of samples ($n$), but the number of dimensions due to the $O(d^3)$ operation of eigendecomposition. As the dimensionality of the data increase, the complexity grows at a cubic rate. Yet, while this growth of dimensionality may exacerbate algorithms such as SM, the negative influence on ISM is limited. This is because the $O(d^3)$ term from SM came from a matrix inversion while ISM uses a spectral method. Since only very few eigenvalues are required, there exists many approximation algorithms to speed up the eigendecomposition \\cite{vladymyrov2016variational} \\cite{richtarikgeneralized}\\cite{lei2016coordinate}.  To demonstrate this point, Coordinate-wise Power Method (CPM) \\cite{lei2016coordinate} was implemented and compared to the eigendecomposition operation from Numpy. In this experiment, noisy dimensions of uniform distributions are added to the synthetic dataset of Large Gauss (LG) such that the total dimension increase in multiples of 2. As the dimension of the data grows exponentially, the time of execution is recorded in Figure \\ref{fig:ND_vs_time} in log scale. Given a log\/log scale, a slope of 1 is linear while a slope of 2 is quadratic. From studying the slope of each operation, the eigendecomposition from Numpy grows at a rate between linear and quadratic. However, by utilizing CPM, the growth becomes roughly linear. Therefore, the refinement of utilizing a spectral optimization method is a key reason for speed improvement.\n\n\n\n\n\n\\section{Conclusions}\n\nWe have proposed an iterative spectral optimization technique for solving a non-convex optimization problem while being constrained on a Stiefel manifold. This new technique demonstrates speed improvement for any algorithm that could be formulated into Eq (\\ref{eq:main_cost_function}). The ISM algorithm is easy to implement with existing software libraries. Due to the usage of the spectral method, approximation techniques from existing research could be deployed to further speed up the eigendecomposition. Accompanied with the algorithm are the theoretical guarantees that satisfy the first and second order necessary conditions for a local minimum. Besides these guarantees, we also proposed a natural initialization point based on Taylor approximation of the original cost function. Experiments on synthetic and real data confirmed that our proposed spectral initialization improved the performance of all the optimization algorithms, ISM, SM and DG. Simultaneously, the experiments demonstrate that our proposed ISM algorithm had the best convergence rate compared to competing models with up to 5 folds of speed improvement. Although we focus this paper on an important application of ISM, alternative clustering, the optimization algorithm proposed and guarantees can be extended to other optimization problems involving Gaussian-kernel like objectives constrained over the Stiefel manifold, which is a common formulation for dimensionality reduction with a kernel-based objective. \nUnderstanding whether ISM can be applied to such objectives and be leveraged to solve broader classes of problems is a natural future direction for this work. \n\n\n\\subsubsection*{Acknowledgements}\nWe would like to acknowledge support for this project from the NSF grant IIS-1546428. We would also like to thank Yale Chang for his insightful discussions. \n\n\n\\section{Derivation for Equation \\ref{eq:main_cost_function} }\nGiven the objective function,\n\\[ \\begin{array}{ll}\n     \\max & \\tmop{HSIC} ( X W , U) - \\lambda \\tmop{HSIC} ( X W, Y)\\\\\n     U, W & \\\\\n     s.t & W^T W = I , U^T U = I.\n   \\end{array} \\]\nUsing the HSIC measure defined, the objective function can be\nrewritten as\n\\[ \\begin{array}{lll}\n     \\tmop{HSIC} ( X W , U) - \\lambda \\tmop{HSIC} ( X W, Y) & = &\n     \\tmop{Tr} ( H  U U^T H D^{\\frac{-1}{2}} K_{XW} D^{\\frac{-1}{2}}) - \\lambda \\tmop{Tr} ( H Y Y^T H\n     D^{\\frac{-1}{2}} K_{XW} D^{\\frac{-1}{2}})\\\\\n     & = & \\tmop{Tr} ( D^{\\frac{-1}{2}} H ( U U^T - \\lambda Y Y^T) H D^{\\frac{-1}{2}} K_{XW})\\\\\n     & = & \\tmop{Tr} ( \\gamma K_{XW})\\\\\n     & = & \\sum_{i, j} \\gamma_{i, j} K_{X_{i, j}}.\n   \\end{array}  \\]\nwhere $\\gamma$ is a symmetric matrix and $\\gamma = H ( U U^T - \\lambda Y Y^T)\nH$. By substituting the Gaussian kernel for $K_{X_{i, j}}$, the objective\nfunction becomes\n\\[ \\begin{array}{l}\n     \\min\\\\\n     W\n   \\end{array} - \\sum_{i, j} \\gamma_{i, j} e^{- \\frac{\\tmop{Tr} [ W^T A_{i, j}\n   W]}{2 \\sigma^2}} \\begin{array}{llll}\n     &  & s.t & W^T W = I.\n   \\end{array} \\]\n\\label{Cost_Derivation}\n\\end{appendices}\n\n\n\n\n\n\n\n\n\n\\begin{appendices} \n    \\section{Proof for Lemma \\ref{basic_lemma} }\n    \\begin{proof}\n        Algorithm \\ref{masteralg} sets the smallest $q$ eigenvectors of $\\Phi ( W_k)$ as\n        $W_{k + 1}$. Since a fixed point $W^{\\ast}$ is reached when $W_k = W_{k + 1}$,\n        therefore $W^{\\ast}$ consists of the smallest eigenvectors of $\\Phi (\n        W^{\\ast})$ and $\\Lambda^{\\ast}$ corresponds with a diagonal matrix of eigenvavlues.\n        Since the eigenvectors of $\\Phi ( W^{\\ast})$ are orthonormal\n        , $W^{\\ast^T} W^{\\ast} = I$ is also satisfied.\n    \\end{proof}\n    \\label{proof_of_lemma_2}\n\\end{appendices}\n\n\n\n\n\\begin{appendices}\n\\section{Proof for Lemma \\ref{lemma:eig} }\n\\begin{proof}\n\n    Using Equation (\\ref{eq:main_cost_function}) as the objective function, the corresponding Lagrangian and its gradient is written as \n    \n    \\begin{align}\n      \\mathcal{L} ( W, \\Lambda) &= - \\sum_{i, j} \\gamma_{i, j} e^{- \\frac{\\tmop{Tr}\n      (W^T A_{i, j} W)}{2 \\sigma^2}} - \\frac{1}{2} \\tmop{Tr} ( \\Lambda ( W^T W -\n      I)) \\label{eq:lagrangian},\n    \\end{align}\n   and \n   \\begin{equation} \\label{eq:gradient_of_lagrangian}\n    \\nabla_W \\mathcal{L} ( W, \\Lambda) = \\sum_{i, j} \\frac{\\gamma_{i,\n      j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} (W^T A_{i, j} W)}{2 \\sigma^2}} A_{i, j} W\n      - W \\Lambda. \n    \\end{equation}\n   \n   \n  By setting the gradient of the Lagrangian to zero, and using the definition of $\\Phi(W)$ from Equation (\\ref{eq:phi_equation}), Equation (\\ref{eq:gradient_of_lagrangian}) can be written as\n    \\begin{equation}\\label{eig_relation} \n     \\Phi(W) W = W \\Lambda. \n    \\end{equation} \n    The gradient with respect to $\\Lambda$ is\n    \n    \\begin{align}\n        \\nabla_\\Lambda \\mathcal{L} ( W, \\Lambda) =  W^T W - I.\n    \\end{align}\n    \n    Setting this gradient of the Lagrangian also to zero, condition (\\ref{eq:1st_lambda}) is equivalent to\n     \n    \\begin{equation} \n        W^T W = I. \\label{prof:orth}\n    \\end{equation}\n     \n     By Lemma \\ref{basic_lemma}, a fixed point $W^*$ and its corresponding $\\Lambda^*$ satisfy (\\ref{eig_relation}) and (\\ref{prof:orth}), and the lemma follows.\n\\end{proof}\n\\label{proof_of_lemma_3}\n\\end{appendices}\n\n\n\\begin{appendices}\n\\section{Proof for Lemma \\ref{lemma:2nd_order_lemma} }\n\nThe proof for Lemma \\ref{lemma:2nd_order_lemma} relies on the following three sublemmas. The first two sublemmas demonstrate how the 2nd order conditions can be rewritten into a simpler form.  \nWith the simpler form, the third lemma demonstrates how the 2nd order conditions of a local minimum are satisfied given a large enough $\\sigma$.\n\n\\begin{sublemma}\\label{lemma:directional_derivate}\n\nLet the directional derivative in the direction of $Z$ be defined as \n\\begin{equation}\n  \\mathcal{D} f (W) [Z] := \\begin{array}{l}\n    \\lim\\\\\n    t \\rightarrow 0\n  \\end{array} \\frac{f (W + tZ) - f (W)}{t}.\n\\end{equation}\n\nThen the 2nd order condition of Lemma \\ref{lemma:2nd_order_lemma} can be written as \n\n\\begin{equation} \\label{cond:large_inequality}\n\\tmop{Tr} ( Z^T \\mathcal{D} \\nabla \\mathcal{L} [ Z]) = \\left\\{ \\sum_{i, j}\n   \\frac{\\gamma_{i, j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^*\n   )}{2 \\sigma^2}}  \\left[ \\tmop{Tr} ( Z^T A_{i, j} Z) -\n   \\frac{1}{\\sigma^2} \\tmop{Tr} ( Z^T A_{i, j} W^*)^2 \\right] \\right\\} -\n   \\tmop{Tr} ( Z^T Z \\Lambda^*),\n\\end{equation}\n\nfor all $Z$ such that \n\\begin{equation} \\label{cond:Zcondition}\nZ^T W^* + W^{*^T} Z = 0.\n\\end{equation}\n\n\\end{sublemma}\n\n\n\\begin{proof}\nObserve first that \n\\begin{equation}\n    \\nabla^2_{W^* W^*} \\mathcal{L} ( W^*, \\Lambda^*) Z = \n    \\mathcal{D} \\nabla \\mathcal{L} [ Z],\n\\end{equation}\n\nwhere the directional derivative of the gradient $\\mathcal{D} \\nabla \\mathcal{L} [ Z]$\nis given by \n\n\\[ \\mathcal{D} \\nabla \\mathcal{L} [ Z] = \\begin{array}{l}\n     \\lim\\\\\n     t \\rightarrow 0\n   \\end{array} \\frac{\\partial}{\\partial t} \\sum_{i, j} \\frac{\\gamma_{i,\n   j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( ( W^* + t Z)^T A_{i, j}  ( W^* + t Z))}{2\n   \\sigma^2}} A_{i, j}  ( W^* + t Z) - ( W^* + t Z) \\Lambda. \\]\n\nThis can be written as\n\\[ \\mathcal{D} \\nabla \\mathcal{L} [ Z] = T_1 + T_2 - T_3, \\]\nwhere\n\\begin{align}\n    T_1 &= \\begin{array}{l}\n        \\lim\\\\ t \\rightarrow 0\n        \\end{array} \\frac{\\partial}{\\partial t} \\sum_{i, j} \\frac{\\gamma_{i,\n        j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( ( W^* + t Z)^T A_{i, j}  ( W^* + t Z))}{2\n        \\sigma^2}} A_{i, j} W^*\\\\\n    &= \\begin{array}{l}\n        \\lim\\\\\n        t \\rightarrow 0\n        \\end{array} \\frac{\\partial}{\\partial t} \\sum_{i, j} \\frac{\\gamma_{i,\n        j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* + t Z^T A_{i, j} W^* +\n        t W^{*^T} A_{i, j} Z + t^2 Z^T A_{i, j} Z )}{2 \\sigma^2}} A_{i, j} W^*  \\\\\n    &= - \\sum_{i, j} \\frac{\\gamma_{i, j}}{2 \\sigma^4} e^{- \\frac{\\tmop{Tr} (\n        ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}} \\tmop{Tr} ( Z^T A_{i, j} W^* + W^{*^T}\n        A_{i, j} Z) A_{i, j} W^* \\\\\n    &= - \\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^4} e^{- \\frac{\\tmop{Tr} (\n   ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}} \\tmop{Tr} ( Z^T A_{i, j} W^*)\n   A_{i, j} W^* &    \\text{as $A_{i, j}=A_{i,j}^T$},\\\\\n    T_2 &= \\begin{array}{l}\n     \\lim\\\\\n     t \\rightarrow 0\n   \\end{array} \\frac{\\partial}{\\partial t} \\sum_{i, j} \\frac{\\gamma_{i,\n   j}}{\\sigma^2} t e^{- \\frac{\\tmop{Tr} ( ( W^* + t Z)^T A_{i, j}  ( W^* + t\n   Z))}{2 \\sigma^2}} A_{i, j} Z\\\\\n   &= \\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} (W^{*^T}\n   A_{i, j} W^*)}{2 \\sigma^2}} A_{i, j} Z, \\\\\n    T_3 &= \\begin{array}{l} \\lim\\\\ t \\rightarrow 0 \\end{array} \\frac{\\partial}{\\partial t}  ( W^* + t Z) \\Lambda\\\\\n    &= Z \\Lambda.\n\\end{align}\n\nHence, putting all three terms together yields\n\n\n\\begin{equation}\\label{eq:directional_gradient}\n\\mathcal{D} \\nabla \\mathcal{L} [ Z] = \\left\\{ \\sum_{i, j} \\frac{\\gamma_{i,\n   j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2\n   \\sigma^2}}  \\left[ A_{i, j} Z - \\frac{1}{\\sigma^2} \\tmop{Tr} ( Z^T A_{i, j}\n   W^*) A_{i, j} W^* \\right] \\right\\} - Z \\Lambda.\n\\end{equation}\n   \n     \n\n\\iffalse\nTo produce a formulation of condition (\\ref{eq:2nd_W}), Equation (\\ref{eq:directional_gradient}) needs to multiply by $Z^T$ and place trace around the expression. This results in \n\\fi\n\nHence,\n\n\\begin{equation}\n    \\tmop{Tr} (Z^T \\nabla^2_{W^* W^*} \\mathcal{L} ( W^*, \\Lambda^*) Z) = \n    \\tmop{Tr} ( Z^T \\mathcal{D} \\nabla \\mathcal{L} [ Z]) ,\n\\end{equation}\n\n\\begin{equation} \\label{trace_2nd_condition}\n   = \\left\\{ \\sum_{i, j}\n   \\frac{\\gamma_{i, j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^*\n   )}{2 \\sigma^2}}  \\left[ \\tmop{Tr} ( Z^T A_{i, j} Z) -\n   \\frac{1}{\\sigma^2} \\tmop{Tr} ( Z^T A_{i, j} W^*)^2 \\right] \\right\\} -\n   \\tmop{Tr} ( Z^T Z \\Lambda_W).   \n\\end{equation}\n\n\nNext, let $Z$ be such that $Z\\neq 0$ and $\\nabla h (W^{\\ast})^T Z = 0$, where\n\\begin{equation}\nh ( W^*) = W^{*^T} W^* - I.\n\\end{equation}\n\nTherefore, the constraint condition can be written on $Z$ in (\\ref{eq:2nd_W}) can be written as\n\n\n\\begin{align}\n\\begin{split}\n\\nabla h ( W^*)^T Z &= \\begin{array}{l} \\lim\\\\\n     t \\rightarrow 0\n   \\end{array} \\frac{\\partial}{\\partial t}  \\frac{( W^* + t Z)^T ( W^* + t Z) -\n   W^{*^T} W^*}{t}\\\\\n    &= Z^T W^* + W^{*^T} Z = 0. \\label{Zconclusion}\n\\end{split}\n\\end{align}\n\nUsing Equations (\\ref{trace_2nd_condition}) and (\\ref{Zconclusion})  lemma \\ref{lemma:directional_derivate} follows.\n\\end{proof}\n\n\n\nRecall from Lemma \\ref{basic_lemma} that $W^*$ consists\nof the $q$ eigenvectors \\ of $\\Phi (W^*)$ with the smallest eigenvalues. We define $\\bar{W^*} \\in \\mathcal{R}^{d \\times d - q}$ as all other eigenvectors of $\\Phi(W^*)$. \\ Because $Z$ has the same dimension as $W^*$, each\ncolumn of $Z$ resides in the space of $\\mathcal{R}^d$. Since the eigenvectors\nof $\\Phi (W^*)$ span $\\mathcal{R}^d$, each column of $Z$ can be represented\nas a linear combination of the eigenvectors of $\\Phi (W^*)$. In other words, each\ncolumn $z_i$ can therefore be written as $z_i = W^*P^{(i)}_W + \\bar{W^*}\nP^{(i)}_{\\bar{W^*}}$, where $P^{(i)}_{W^*} \\in \\mathcal{R}^{q \\times 1}$ and \\\n$P^{(i)}_{\\bar{W^*}} \\in \\mathcal{R}^{d - q \\times 1}$ \\ represents the\ncoordinates for the two sets of eigenvectors. Using the same notation, we also define $\\Lambda^* \\in\n\\mathcal{R}^{q \\times q}$ as the eigenvalues corresponding to $W^*$ and\n$\\bar{\\Lambda^*} \\in \\mathcal{R}^{d - q \\times d - q}$ as the eigenvalues\ncorresponding to $\\bar{W^*}$. The entire matrix $Z$ can therefore be\nrepresented as \n\\begin{equation} \\label{def:Z}\n  Z = \\bar{W^*} P_{\\bar{W^*}} + W^*P_{W^*}.\n\\end{equation}\n\n\nFurthermore, it can be easily shown that $P_{W^*}$ is a skew symmetric matrix, or $- P_{W^*} = P_{W^*}^T$. By  setting $Z$ from Equation (\\ref{cond:Zcondition}) into  (\\ref{def:Z}), the constraint can be rewritten as\n\n\\begin{align}\n[ P_{\\bar{W^*}}^T \\bar{W^*}^T + P_W^{*^T} W^{*^T}] W^* + W^{*^T} [ \\bar{W^*} P_{\\bar{W^*}} + W^* P_{W^*}] &= 0.\n\\end{align}\n\nSimplifying the equation yields the relationship\n\n\\begin{equation}\nP_W^{*^T} + P_{W^*} = 0.\n\\end{equation}\n\n\n\n\n\nUsing these definitions, we define the following sublemma.\n\n\\begin{sublemma}\\label{lemma:To_final_form}\nGiven a fixed point $W^*$ and a $Z$ satisfying condition (\\ref{cond:Zcondition}), the condition \n$\\tmop{Tr} ( Z^T \\mathcal{D} \\nabla \\mathcal{L} [ Z]) \\geq 0$ is equivalent to \n\\begin{equation} \\label{eq:inequality}\n   \\tmop{Tr} ( P_{\\bar{W^*}}^T \\bar{\\Lambda^*} P_{\\bar{W^*}}) - \\tmop{Tr} (\n   P_{\\bar{W^*}} \\Lambda^* P_{\\bar{W^*}}^T) \\geq C_2,\n\\end{equation}\nwhere \n\\begin{equation} \\label{def:C2}\n   C_2 = \\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^4} e^{-\n   \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}} \\tmop{Tr} (\n   Z^T A_{i, j} W^*)^2,\n\\end{equation}\n\n$P_{W^*}, P_{\\bar{W^*}}$ are given by Equation (\\ref{def:Z}), and $\\Lambda^*,\\bar{\\Lambda^*}$ are the diagonal matrices containing the bottom and top eigenvalues of $\\Phi(W^*)$ respectively.\n\n\n\n\\end{sublemma}\n\n\n\n\\begin{proof}\nBy condition  (\\ref{cond:large_inequality}), \n\\begin{align}\n    \\tmop{Tr} ( Z^T \\mathcal{D} \\nabla \\mathcal{L} [ Z]) &= C_1 - C_2 + C_3, \\label{eq:41} \n\\end{align}\nwhere\n\\begin{align*}\n    C_1  &= \\tmop{Tr} \\left( Z^T \\sum_{i, j} \\frac{\\gamma_{i,\n   j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2\n   \\sigma^2}} A_{i, j} Z \\right), \\\\\n   C_2 &= \\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^4} e^{-\n   \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}} \\tmop{Tr} (\n   Z^T A_{i, j} W^*)^2  \\label{eq:43},\\\\\n   C_3 &= - \\tmop{Tr} ( Z^T Z \\Lambda^*).\n\\end{align*}\n\n\n\n$C_1$ can be written as \n\n\\begin{align*}\n C_1 &= \\tmop{Tr} \\left( Z^T \\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^2} e^{-\n   \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}} A_{i, j} Z\n   \\right) &\\\\\n &=\\tmop{Tr} ( Z^T \\Phi ( W^*) [ \\bar{W^*} P_{\\bar{W^*}} + W^* P_{W^*}]) &\\\\\n &=\\tmop{Tr} ( Z^T [ \\Phi ( W^*)  \\bar{W^*} P_{\\bar{W^*}} + \\Phi ( W^*) W^* P_{W^*}])  & \\\\\n &=\\tmop{Tr} ( Z^T [ \\bar{W^*}  \\bar{\\Lambda} P_{\\bar{W^*}} + W^* \\Lambda P_{W^*}]) \n    & \\text{By definition of eigenvalues.} \\\\\n &= \\tmop{Tr} ( [ P_{\\bar{W^*}}^T \\bar{W^*} ^T + P_W^{*^T} W^{*^T} ] [ \\bar{W^*} \n   \\bar{\\Lambda} P_{\\bar{W^*}} + W^* \\Lambda P_{W^*}]) & \\text{Substitute for $Z$} \\\\\n &=\\tmop{Tr} ( P_{\\bar{W^*}}^T  \\bar{\\Lambda} P_{\\bar{W^*}} ) + \\tmop{Tr} ( P_{W^*}^T\n   \\Lambda P_W ) & \\text{Given $W^{*^T}W^*=I$, $\\bar{W^*}^TW^* =0 $}.\n\\end{align*}\n\n\n\nSimilarly\n\\begin{align*}\n    C_3 &= - \\tmop{Tr} ( Z^T Z \\Lambda) \\\\\n    &= - \\tmop{Tr} ( [ P_{\\bar{W^*}}^T \\bar{W^*} ^T + P_{W^*}^T W^{*^T}] \n        [ \\bar{W^*} P_{\\bar{W^*}} + W^* P_{W^*}] \\Lambda) \\\\\n    &= - \\tmop{Tr} ( [ P_{\\bar{W^*}}^T P_{\\bar{W^*}} + P_{W^*}^T P_{W^*}] \\Lambda)\\\\\n    &= - \\tmop{Tr} ( P_{\\bar{W^*}}^T P_{\\bar{W^*}} \\Lambda) - \n        \\tmop{Tr} ( P_{W^*}^T P_{W^*} \\Lambda).\n\\end{align*}\n\n\nBecause $P_{W^*}$ is a square skew symmetric matrix, the diagonal elements of $P_{W^*}\nP_{W^*}^T$ is the same as the diagonal of $P_{W^*} P_{W^*}^T$. From this observation, we\nconclude that $\\tmop{Tr} ( P_{W^*} P_{W^*}^T \\Lambda) = \\tmop{Tr} ( P_{W^*}^T P_{W^*}\n\\Lambda)$. Hence, \n\n\n\\begin{align*}\nC_3 = - \\tmop{Tr} ( P_{\\bar{W^*}} \\Lambda P_{\\bar{W^*}}^T ) -\n   \\tmop{Tr} ( P_{W^*}^T \\Lambda P_{W^*} ).\n\\end{align*}\n   \n\nPutting all 3 parts together yields\n\n\\begin{align}\n\\begin{split}\n    \\tmop{Tr} ( Z^T \\mathcal{D} \\nabla \\mathcal{L} [ Z]) &= \\tmop{Tr} (\n   P_{\\bar{W^*}}^T  \\bar{\\Lambda} P_{\\bar{W^*}} ) + \\tmop{Tr} ( P_{W^*}^T \\Lambda P_{W^*}\n   ) - C_2 - \\tmop{Tr} ( P_{\\bar{W^*}} \\Lambda P_{\\bar{W^*}}^T ) -\n   \\tmop{Tr} ( P_{W^*}^T \\Lambda P_{W^*} ) \\\\\n   &=\\tmop{Tr} ( P_{\\bar{W^*}}^T \\bar{\\Lambda} P_{\\bar{W^*}}) - \\tmop{Tr} (\n   P_{\\bar{W^*}} \\Lambda P_{\\bar{W^*}}^T) - C_2 .\n\\end{split}\n\\end{align}\n\n\n\nThe 2nd order condition (\\ref{eq:2nd_W}) is, therefore, satisfied, when   \n\\begin{equation} \\label{eq:58}\n   \\tmop{Tr} ( P_{\\bar{W^*}}^T \\bar{\\Lambda} P_{\\bar{W^*}}) - \\tmop{Tr} (\n   P_{\\bar{W^*}} \\Lambda P_{\\bar{W^*}}^T) \\geq C_2.\n\\end{equation}\n\n\\end{proof}\n\n\n\n\n\n\n\\begin{sublemma} \\label{large_enough_sigma}\n    Given $W^*$,$\\bar{W^*}$,$\\bar{\\Lambda^*}$, and $\\Lambda^*$ \n    as defined in Equation (\\ref{def:Z}), \n   if the corresponding smallest eigenvalue of $\\bar{\\Lambda^*}$ \n   is larger than the largest eigenvalue of $\\Lambda^*$, then given a large enough  $\\sigma$ the \n   condition (\\ref{eq:2nd_W}) of Lemma \\ref{lemma:2nd_order} is satisfied.\n\\end{sublemma}\n\n\\begin{proof}\nTo proof sublemma (\\ref{large_enough_sigma}), we provide bounds on each of the terms in \\eqref{eq:58}. Starting with $C_2$ defined at (\\ref{def:C2}). It has a trace term, $(\\tmop{Tr}(Z^TA_{ij}W^*))^2$ that can be rewritten as \n\n\\begin{equation}\\label{eq:60}\n  (\\tmop{Tr}(A_{ij}W^*Z^T))^2 = (\\tmop{Tr}(A_{ij}W^*P^T_{W^*}W^{*^T}+ A_{ij}W^*P^T_{\\bar{W^*}}\\bar{W^*}^T))^2.\n\\end{equation}\n\nSince $A_{ij}$ is symmetric and $W^*P^T_{W^*}W^{*^T}$ is skew-symmetric, then\nTr$(A_{ij}W^*P^T_{W^*}W^{*^T})=0$. Hence \n\n\\begin{align}\n   (\\tmop{Tr}(Z^TA_{ij}W^*))^2 & =  (\\tmop{Tr}(A_{ij}W^*Z^T))^2 \n    = (\\tmop{Tr}(A_{ij}W^*P^T_{\\bar{W^*}}\\bar{W^*}^T))^2 \\\\\n    &\\leq \\tmop{Tr}(A^T_{i,j}A_{ij})\\tmop{Tr}(P^T_{\\bar{W^*}}P_{\\bar{W^*}}) \n\\end{align}\n\nwhere the last inequality follows from Cauchy-Schwartz inequality and that fact that $W^{*^T}W^*=I$ and $\\bar{W^*}^T\\bar{W^*}=I$.\nThus, $C_2$ in \\eqref{eq:43} is bounded by\n\\begin{equation}\\label{eq:c2bound}\n   C_2 \\leq  \\sum_{i, j} \\frac{|\\gamma_{i, j}|}{\\sigma^4} e^{-\n   \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}} \\tmop{Tr} (\n    A^T_{i,j}A_{ij})\\tmop{Tr}(P^T_{\\bar{W^*}}P_{\\bar{W^*}})\n\\end{equation}\nSimilarly, the remaining terms in \\eqref{eq:41} can be bounded by\n\\begin{equation}\nC_1= \\tmop{Tr} ( P_{\\bar{W^*}}^T \\bar{\\Lambda^*} P_{\\bar{W^*}}) \\geq \\min_i\n    ( \\bar{\\Lambda^*}_i) \\tmop{Tr} ( P_{\\bar{W^*}} P_{\\bar{W^*}}^T) \n\\end{equation}\n\\begin{equation}\n C_3 = -\\tmop{Tr} ( P_{\\bar{W^*}} \\Lambda^* P^T_{\\bar{W^*}}) \\geq -\n     \\max_i ( \\Lambda_i^*) \\tmop{Tr} ( P_{\\bar{W^*}}^T P_{\\bar{W^*}}).\n\\end{equation}\n     \nUsing the bounds for each term, the Equation (\\ref{eq:58}) can be rewritten as\n\\begin{align}\n    [\\min_i ( \\bar{\\Lambda^*}_i) -\\max_j(\\Lambda_j^*)] \\tmop{Tr}(P^T_{\\bar{W^*}}P_{\\bar{W^*}}) &\\geq \n    \\sum_{i, j} \\frac{|\\gamma_{i, j}|}{\\sigma^4} e^{-\n   \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}}\n   \\tmop{Tr} (A^T_{i,j}A_{ij})\n   \\tmop{Tr}(P^T_{\\bar{W^*}}P_{\\bar{W^*}})\\\\\n   [\\min_i ( \\bar{\\Lambda^*}_i) -\\max_j(\\Lambda_j^*)] &\\geq \n    \\sum_{i, j} \\frac{|\\gamma_{i, j}|}{\\sigma^4} e^{-\n   \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}} \\tmop{Tr} (A^T_{i,j}A_{ij})\\label{eq:conclusion}\n\\end{align}\n\nIt should be noted that $\\Lambda^*$ is a function of $\\frac{1}{\\sigma^2}$. This relationship could be removed by multiplying both sides of the inequality by $\\sigma^*$ to yield\n\\begin{equation}\n   \\sigma^2 [\\min_i ( \\bar{\\Lambda^*}_i) -\\max_j(\\Lambda_j^*)] \\geq \n    \\sum_{i, j} \\frac{|\\gamma_{i, j}|}{\\sigma^2} e^{-\n   \\frac{\\tmop{Tr} ( ( W^{*^T} A_{i, j} W^* )}{2 \\sigma^2}}\\tmop{Tr} (A^T_{i,j}A_{ij}). \\label{ineq:large_sigma}\n\\end{equation} \n\nSince $\\sigma^2$ is always a positive value, as long as all the eigenvalues from $\\bar{\\Lambda^*}$ is larger than all the eigenvalues from $\\Lambda^*$, the left hand side of the equation will always be greater than 0. As $\\sigma \\rightarrow \\infty$, the right hand side approaches 0, and the  condition (\\ref{eq:2nd_W}) of Lemma \\ref{lemma:2nd_order} is satisfied. \n\\end{proof} \\label{proof_of_lemma_4}\n\n\nAs a side note, the eigen gap between min($\\bar{\\Lambda^*}$) and max($\\Lambda^*$) controls the range of potential $\\sigma$ values \u2014i.e. the larger the eigen gap the easier for $\\sigma$ to satisfy \\eqref{ineq:large_sigma}. Therefore, the ideal cutoff point should have a large eigen gap.\n\n\n\\end{appendices}\n\n\n\n\n\n\n\n\\newpage\n\\begin{appendices} \\label{Appendix:Convergence_plot}\n\\section{Convergence Plot from Experiments}\n\nFigure \\ref{fig:convergence} summarizes the convergence activity of various experiments. For each experiment, the top figure \nprovides the magnitude of the objective function. It can be seen \nthat the values converges towards a fixed point. The middle plot provide updates of the gradient of the Lagrangian. It can be seen that the gradient converges towards 0. The bottom plot shows the changes in $W$ during each iteration. The change in $W$ converge towards 0.\n\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=14cm,height=14cm]{{extras\/Convergence_plot}}\n  \\caption{Convergence Results from the Experiments.}\n  \\label{fig:convergence}\n\\end{figure}\n\\end{appendices}\n\\newpage\n\n\n\n\n\n\\begin{appendices}\n\\section{Proof of Convergence}\n\nThe convergence property of ISM has been analyzed and yields the following theorem.\n\n\\begin{theorem}\n  A sequence $\\{W_k\\}_{k\\in \\mathbb{N}}$ generated by Algorithm~\\ref{masteralg} contains a convergent subsequence. \n \\end{theorem}\n \n\n\n\\begin{proof}\n\nAccording to Bolzano-Weierstrass theorem, if we can show that the sequences generated from\nthe 1st order relaxation is bounded, it has a convergent subsequence. If we\nstudy the Equation $\\Phi(W)$ more closely, the key driver of the sequence of $W_k$\nis the matrix $\\Phi$, therefore, if we can show that if this matrix is\nbounded, the sequence itself is also bounded. We look inside the construction\nof the matrix itself.\n\n\\[ \\Phi_{n + 1} = \\left[ \\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^2} e^{-\n   \\frac{\\tmop{Tr} (W_n^T A_{i, j} W_n)}{2 \\sigma^2}} A_{i, j} \\right] \\]\n\nFrom this equation, start with the matrix $A_{i, j} = ( x_i - x_j) ( x_i -\nx_j)^T$. Since $x_i, x_j$ are data points that are always centered and scaled\nto a variance of 1, the size of this matrix is always constrained. It also\nimplies that $A_{i, j}$ is a PSD matrix. From this, the exponential term is always limited between\nthe value of 0 and 1. The value of $\\sigma$ is a constant given from the\ninitialization stage. Lastly, we have the $\\gamma_{i, j}$ term. Since $\\gamma\n= D^{- 1 \/ 2} H ( U U^T - \\lambda Y Y^T) H D^{- 1 \/ 2}$. The degree matrix\ncame from the exponential kernel. Since the kernels are bounded, $D$ is also\nbounded. The centering matrix $H$ and the previous clustering result $Y$ can\nbe considered as bounded constants. Since the spectral embedding $U$ is a\northonormal matrix, it is always bounded. From this, given that the components\nof $\\Phi$ is bounded, the infinity norm of the $\\Phi$ is always bounded. The\neigenvalue matrix of $\\Lambda$ is therefore also bounded. Using the\nBolzano-Weierstrass Theorem, the sequence contains a convergent sub-sequence. Given that $\\Phi$ is a continuous function of $W$, by continuity, $W$ also has a convergent sub-sequence.\n\\end{proof}\n\\end{appendices}\n\n\n\n\\begin{appendices}\n\\section{Proof for the initialization}\nAlthough the proof was originally shown through the usage of the 2nd order Taylor Approximation. A simpler approach was later discovered to arrive to the same formulation faster. We first note that Taylor's Expansion around 0 of an exponential is \n\\[e^x=1+x+\\frac{x^2}{2!}+...  .\\]\n\nGiven the objective Lagrangian in eq \\eqref{eq:lag}, \\ we simplify the Lagrangian by using the Taylor approximation only on the problematic exponential term. The approximation is expanded up to the 1st order centering around 0 to yield \n\\[ \\mathcal{L} \\approx - \\sum_{i, j} \\gamma_{i, j}  \\left( 1 - \\frac{\\tmop{Tr}\n   (W^T A_{i, j} W)}{2 \\sigma^2} \\right) + \\frac{1}{2} \\tmop{Tr} (\\Lambda (I -\n   W^T W)) .\\]\nBy taking the derivative of the approximated Lagrangian and setting the derivative to zero, an eigenvalue\/eigenvector relationship emerges as\n\\[ \\Phi W =  \\left[ \\sum_{i, j} \\frac{\\gamma_{i, j}}{\\sigma^2} A_{i,\n   j} \\right] W_0 = W_0 \\Lambda .\\]\n   \nFrom this, we see that $\\Phi_0$ is no longer a function of $W$. Using this $\\Phi_0$ we can then calculate a closed form solution for $W_0$\n\\end{appendices}\n\n\n\n\\begin{appendices}\n\\section{Proof for the computational complexity}\n\n\nFor ISM, DG and SM, the bottleneck resides in the computation of the gradient.\n\\[ f ( W) = \\sum_{i, j} \\gamma_{i, j} e^{- \\frac{\\tmop{Tr} ( W^T A_{i, j}\n   W)}{2 \\sigma^2}} \\]\n\\[ \\frac{\\partial f}{\\partial W} = \\left[ \\sum_{i, j}  \\frac{\\gamma_{i,\n   j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( W^T A_{i, j} W)}{2 \\sigma^2}} A_{i, j}\n   \\right] W \\]\n\\[ \\frac{\\partial f}{\\partial W} = \\left[ \\sum_{i, j}  \\frac{\\gamma_{i,\n   j}}{\\sigma^2} e^{- \\frac{\\tmop{Tr} ( W^T \\Delta x_{i, j} \\Delta x_{i, j}^T\n   W)}{2 \\sigma^2}} A_{i, j} \\right] W \\]\nWhere $A_{i, j} = \\Delta x_{i, j} \\Delta x_{i, j}^T$. The variables have the\nfollowing dimensions.\n\\[ \\begin{array}{l}\n     x_{i, j} \\in \\mathcal{R}^{d \\times 1}\\\\\n     W \\in \\mathcal{R}^{d \\times q}\n   \\end{array} \\]\nTo compute a new $W$ with DG, we first mulitply $\\Delta x_{i, j}^T W$, which\nis $O ( d)$. Note that $W$ in DG is always 1 single column. Next, it\nmultiplies with its own transpose to yied $O ( d  + q^2)$. Then we\ncompute $A_{i, j}$ to get $O ( d + q^2 + d^2)$. Since this operation needs to\nbe added $n^2$ times, we get, $O ( n^2 ( d + q^2 + d^2))$. Since $d \\gg q$,\nthis notation reduces down to $O ( n^2 d^2)$. Let $T_1$ be the number of\niterations until convergence, then it becomes $O ( T_1 n^2 d^2)$. Lastly, in\nDG, this operation needs to be repeated $q$ times, hence, $O ( T_1 n^2 d^2\nq)$.\n\n\n\nTo compute a new $W$ with SM, we first mulitply $\\Delta x_{i, j}^T W$, which\nis $O ( d q)$. Next, it multiplies with its own transpose to yied $O ( d q\n + q^2)$. Then we compute $A_{i, j}$ to get $O ( d q + q^2 + d^2)$.\nSince this operation needs to be added $n^2$ times, we get, $O ( n^2 ( d q +\nq^2 + d^2))$. Since $d \\gg q$, this notation reduces down to $O ( n^2 d^2)$.\nThe SM method requires the computation of the inverse of $d \\times d$ matrix.\nSince inverses is cubic, it becomes $O ( n^2 d^2  + d^3)$. Lastly, let\n$T_2$ be the number of iterations until convergence, then it becomes $O ( T_2\n( n^2 d^2 + d^3))$.\n\n\n\nTo compute a new $W$ with ISM, we first mulitply $\\Delta x_{i, j}^T W$, which\nis $O ( d q)$. Next, it multiplies with its own transpose to yied $O ( d q\n + q^2)$. Then we compute $A_{i, j}$ to get $O ( d q + q^2 + d^2)$.\nSince this operation needs to be added $n^2$ times, we get, $O ( n^2 ( d q +\nq^2 + d^2))$. Since $d \\gg q$, this notation reduces down to $O ( n^2 d^2)$.\nThe ISM method requires the computation of the eigen decomposition of $d\n\\times d$ matrix. Since inverses is cubic, it becomes $O ( n^2 d^2  +\nd^3)$. Lastly, let $T_3$ be the number of iterations until convergence, then\nit becomes $O ( T_3 ( n^2 d^2 + d^3))$.\n\n\\end{appendices}\n\n\n\n\n\n\\begin{appendices} \n\\section{Measure of Non-linear Relationship by HSIC Versus Correlation}\n\nThe figure below demonstrates a visual comparison of HSIC and correlation. It can be seen that HSIC  measures non-linear relationships, while correlation does not. \n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=14cm,height=3cm]{{extras\/HSIC_example}}\n  \\caption{Showing that HSIC captures non-linear information.}\n  \\label{HSIC_capture_nonlinear}\n\\end{figure}\n\\label{App:HSIC}\n\\end{appendices}\n\n\n\n\n\n\n\n\n\n\\begin{appendices}\n\\section{Implementation Details of the Cost function}\n\nThe computation of the cost using the formulation below is slow if it is implemented using a loop. \n\\begin{equation}\n  \\begin{array}{ll}\n    \\min & - \\sum_{i, j} \\gamma_{i, j} e^{- \\frac{\\tmop{tr} ( W^T A_{i, j}\n    W)}{2 \\sigma^2}}\\\\\n    W & \\\\\n    s.t & W^T W = I\\\\\n    & W \\in \\mathbb{R}^{d \\times q}\\\\\n    & A \\in \\mathbb{R}^{d \\times d}\\\\\n    & \\gamma_{i, j} \\in \\mathbb{R}\n  \\end{array}\n\\end{equation}\n\nInstead, we use the original formulation to derive a faster way to compute the cost.\n\nStarting with the original cost function as\n\\[ \\tmop{cost} = \\tmop{HSIC} ( X W, U) - \\lambda \\tmop{HSIC} ( X W, Y) \\]\n\\[ \\tmop{cost} = \\tmop{Tr} ( D^{-1\/2}K_{XW} D^{-1\/2} H U U^T H) - \\lambda \\tmop{Tr} (D^{-1\/2}K_{XW} D^{-1\/2} H Y Y^T H) \\]\n\nWhen optimizing $U$, it is obvious that the 2nd portion does not effect the\noptimization. Therefore, $U$ can be solved using the following form.\n\\[ U = \\begin{array}{l}\n     \\tmop{argmin}\\\\\n     U\n   \\end{array} \\tmop{Tr} ( U^T H D^{- 1 \/ 2} K_{X W} D^{- 1 \/ 2} H U) \\]\n\n\nIf we are optimization for $W$, using the combination of the rotation property and the combination of the 2 traces, the cost can be written as \n\\[ \\tmop{cost} = \\tmop{Tr} ( [ D^{- 1 \/ 2} H ( U U^T - \\lambda Y Y^T) H D^{-\n   1 \/ 2}] K). \\]\n\n\nIn this form, it can be seen that the update of $W$ matrix will only affect\nthe kernel $K$ and the degree matrix $D$. Therefore, it makes sens to treat\nthe middle portion as a constant which we refer as $\\Psi$.\n\\[ \\tmop{cost} = \\tmop{Tr} ( [ D^{- 1 \/ 2} \\Psi D^{- 1 \/ 2}] K) \\]\n\n\nGiven that $[ D^{- 1 \/ 2} \\Psi D^{- 1 \/ 2}]$ is a symmetric matrix, from this\nform, we can convert the trace into an element wise product $\\odot$.\n\\[ \\tmop{cost} = \\sum_{i,j} ([ D^{- 1 \/ 2} \\Psi D^{- 1 \/ 2}] \\odot K )_{i,j} \\]\n\n\nTo further reduction the amount of operation, we let $d$ be a vector of the\ndiagonal elements of $D^{- 1 \/ 2}$, hence $d = \\tmop{diag} ( D^{- 1 \/ 2})$,\nthis equality hold.\n\\[ D^{- 1 \/ 2} \\Psi D^{- 1 \/ 2} = [ d d^T] \\odot \\Psi \\]\n\n\nTherefore, the final cost function can be written in its simplest form as :\n\\[ \\tmop{cost} = \\sum_{i,j} \\Gamma_{i,j} = \\sum_{i,j} (\\Psi \\odot [ d d^T] \\odot K )_{i,j} \\]\n\n\nDuring update, as $W$ update during each iteration, the matrix $\\Psi$ stays as\na constant while $d d^T$ and $K$ update. The benefit of this form minimize the\ncomplexity of the equation, while simplify cost into easily parallelizable\nmatrix multiplications. The equation also clearly separates the elements into\nportions that require an update and portions that does not.\n\\end{appendices}\n\n\n\n\\begin{appendices}\n\\section{Implementation Details of the Derivative}\nAs it was shown from previous sections, the gradient of our cost function using the Gaussian Kernel is \n\\[ \\nabla f ( W) = \\left[ \\frac{1}{\\sigma^2} \\sum \\gamma_{i, j} K_{i, j} A_{i, j} \\right] W  - 2W\\Lambda\\].\n\n\nIf we let $\\Psi = \\left[  \\frac{1}{2\\sigma^2}  \\gamma_{i, j} K_{i, j}  \\right]$ , it can be rewritten as \n\\[ \\nabla f ( W) = \n\\left[ \\sum \\Psi_{i, j} A_{i, j} \\right] W - W\\Lambda.\\]\n\nFrom this formulation, the optimal $W$ is equivalent to the eigenvectors of the $\\left[ \\sum \\Psi_{i, j} A_{i, j} \\right]$. According to ISM, the $q$ eigenvectors corresponding to the smallest eigenvalues is used for $W$. Since solving $\\sum \\Psi_{i, j} A_{i, j}$ using a loop is slow, we vectorize the formulation so that\n\n\\[ \\sum \\Psi_{i, j} A_{i, j} = X^T[D_{\\Psi} - \\Psi]X \\], \n\nwhere $D_\\Psi$ is the degree matrix of $Psi$ such that the diagonal elements are defined as\n\n\\[ d_{i,i} = \\sum_{j} \\Psi_{i,j} \\], \n\nand $X \\in \\mathbb{R}^{n \\times d}$ is the original data.\n\n\\end{appendices}\n\n\n\n\n\n\\begin{appendices} \n\\section{Hyperparameters Used in Each Experiment}\n    \\begin{table}[h]\n    \\centering\n        \\begin{tabular}{l|ccc}\n        & $\\sigma$ & $\\lambda$ & $q$ \\\\ \\cline{1-4}\n        Gauss A & 1 & 0.04 & 1 \\\\\n        Gauss B 200 & 5 & 2 & 3 \\\\\n        Moon 400 & 0.1 & 1 & 3 \\\\\n        Moon+N 200 & 0.2 & 0.1 & 6 \\\\\n        Flower & 2 & 10 & 2 \\\\\n        Face & 3.1 & 1 & 17 \\\\\n        Web KB & 18.7 & 0.057 & 4 \\\\\n        \\end{tabular}\n    \\end{table}\n    \\label{App:Hyperparameters}\n\\end{appendices}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThis paper is about the Newtonian model of gravitation,\nalso known as the classical $N$-body problem.\nWe start recalling the standard notation.\nLet $E$ be an Euclidean space,\nin which the punctual masses $m_1,\\dots,m_N>0$\nare moving under the action of the inverse-square law of universal gravitation.\nIf the components of $x=(r_1,\\dots,r_N)\\in E^N$\nare the positions of the bodies,\nthen we shall denote $r_{ij}=\\norm{r_i-r_j}_E$\nthe distance between bodies $i$ and $j$\nfor any pair $1\\leq i<j\\leq N$.\nThe Newton's equations can be written as $\\ddot x=\\nabla U(x)$,\nwhere $U:E^N\\to\\mathbb{R}\\cup\\set{+\\infty}$ is the \\emph{Newtonian potential}, \n\\[\nU(x)=\\;\\sum_{i<j}\\;m_i\\,m_j\\;r_{ij}^{-1}\\;,\n\\]\nand the gradient is taken with respect to the mass scalar product.\nA configuration $x\\in E^N$ is said to be without collisions if\n$U(x)<+\\infty$,\nthat is to say, whenever we have $r_{ij}\\neq 0$ for all $i\\neq j$.\nWe denote $\\Omega\\subset E^N$ the open and dense set of configurations without collisions.\nTherefore Newton's equations define an analytic local flow on\n$T\\Omega=\\Omega\\times E^N$,\nwith a first integral given by the energy constant\n\\[h=\\frac{1}{2}\\norm{\\dot x}^2-U(x)\\,.\\]\n\nOne of the main difficulties\nfor the analysis of the dynamics in this model is the uncertainty,\nfor a given motion,\nabout the presence of singularities after a finite amount of time.\nThat is to say,\nwe can not predict whether a certain evolution of the bodies\nwill be defined for all future time or not.\nWe recall that maximal solutions that end in finite time\nmust undergo collisions at the last moment,\nor to have an extremely complex behaviour called\npseudocollision (\\cite{Dia}, p.39).\nNotwithstanding,\nthe classification of all possible final evolutions was developed,\nfor motions assumed to be without singularities in the future,\nessentially in terms of\nthe asymptotic behaviour of the distance between the bodies.\nSome of the greatest contributions in this direction\nare undoubtedly those due to Chazy,\nand especially those that he obtained in the works\n\\cite{Cha1, Cha2} that we comment below.\nHowever,\nthis approach does not provide the existence of motions\nfor any type of final evolution.\n\nIn this paper we will be concerned with the class of\nhyperbolic motions,\nas defined by Chazy by analogy with the Keplerian case.\n\n\\begin{definition}\nHyperbolic motions are those such that\neach body has a different limit velocity vector, that is\n$\\dot{r}_i(t)\\to a_i\\in E$ as $t\\to +\\infty$,\nand $a_i\\neq a_j$ whenever $i\\neq j$. \n\\end{definition}\n\nIf $V$ is a normed vector space\nand $x(t)$ is a smooth curve in $V$\nwith asymptotic velocity $a\\in V$,\nthen we must have $x(t)=ta+o(t)$ as $t\\to +\\infty$,\nbut the converse is of course not true.\nHowever, for $V=E^N$ and $a=(a_1,\\dots,a_N)\\in\\Omega$,\nthe converse is satisfied\nby solutions of the Newtonian $N$-body problem\n(see Lemma \\ref{lema-cont.limitshape}).\nThus, hyperbolic motions are characterized as\n\\emph{motions without singularities in the future and such that\n $x(t)=ta+o(t)$ for some configuration $a\\in\\Omega$.}\n\nIt follows that for any hyperbolic motion we have\n$\\alpha t < r_{ij}(t) <  \\beta t$ for some positive constants,\nfor all $i<j$, and for all $t$ big enough.\nAs we will see,\nChazy proved that this weaker property\nalso characterizes hyperbolic motions. \n\nAs usual,\n$I(x)=\\inner{x}{x}=\\sum_i m_i\\inner{r_i}{r_i}_E$ will denote the\n\\emph{moment of inertia} of the configuration $x\\in E^N$\nwith respect to the origin of $E$.\nWhen the motion $x(t)$ is given,\nwe will use the notation $U(t)$ and $I(t)$\nfor the compositions $U(x(t))$ and $I(x(t))$ respectively.\nThus for an hyperbolic motion such that\n$x(t)=a t + o(t)$ as $t\\to +\\infty$ we have\n$U(t)\\to 0$, $I(t)\\sim I(a)\\,t^2$ and $2h=I(a)>0$.\n\nWe say that a motion $x(t)$ has \\emph{limit shape}\nwhen there is a time dependent similitude $S(t)$ of the space $E$\nsuch that $S(t)x(t)$ converges to some configuration $a\\neq 0$\n(here the action of $S(t)$ on $E^N$ is the diagonal one).\nThus the limit shape of an hyperbolic motion is the shape\nof his asymptotic velocity $a=\\lim_{t\\to +\\infty}t^{-1}x(t)$.\nNote that, in fact,\nthis represents a stronger way of having a limit shape,\nsince in this case the similarities are given by homotheties.\n\n\n\\subsection{Existence of hyperbolic motions}\n\\label{s-existence}\n\nThe only explicitly known hyperbolic motions\nare of the homographic type,\nmeaning that the configuration is\nall the time in the same similarity class.\nFor this kind of motion,\n$x(t)$ is all the time a central configuration,\nthat is, a critical point of $I^{1\/2}U$.\nThis is a strong limitation,\nfor instance the only central configurations for $N=3$\nare either equilateral or collinear.\nMoreover,\nthe Painlev\u00e9-Wintner conjecture states that up to similarity\nthere are always a finite number of central configurations.\nThe conjecture was confirmed by Hampton and Moeckel\n\\cite{HamMoe} in the case of four bodies,\nand by Albouy and Kaloshin \\cite{AlbKal}\nfor generic values of the masses in the planar five-body problem.\n\nOn the other hand,\nChazy proved in \\cite{Cha2} that the set of initial conditions\ngiving rise to hyperbolic motions is an open subset of $T\\Omega$,\nand moreover,\nthat the limit shape depends continuously on the initial condition\n(see Lemma \\ref{lema-cont.limitshape}).\nIn particular,\na motion close enough to some hyperbolic homographic motion\nis still hyperbolic.\nHowever, this does not allow us to draw conclusions\nabout the set of configurations that are realised as limit shapes.\nIn this paper we prove that \\emph{any} configuration\nwithout collisions is the limit shape of some hyperbolic motion.\nAt our knowledge, there are no results in this direction\nin the literature of the subject.\n\n\n\nAn important novelty in this work is the use of global viscosity solutions, in the sense introduced by Crandall, Evans and Lions \\cite{CraLio,CraEvaLio},\nfor the supercritical Hamilton-Jacobi equation\n\\begin{equation}\\tag{HJ}\\label{HJh}\nH(x,d_xu)=h \\qquad x\\in E^N,\n\\end{equation}\nwhere $H$\nis the Hamiltonian of the Newtonian $N$-body problem,\nand $h>0$.\n\nWe will found global viscosity solutions through a limit process\ninspired by the Gromov's construction of the ideal boundary\nof a complete locally compact metric space.\nTo do this,\nwe will have to generalize to the case $h>0$ the H\u00f6lder estimate\nfor the action potential discovered by the first author in\n\\cite{Mad1} in the case $h=0$.\nWith this new estimate we will remedy the loss of\nthe Lipschitz character of the viscosity subsolutions,\nwhich is due to the singularities of the Newtonian potential.\n\nIn a second step, we will show that the functions thus obtained\nare in fact fixed points of the Lax-Oleinik semigroup. \nMoreover,\nwe will prove that given any configuration without collisions\n$a\\in\\Omega$,\nthere are solutions of Equation (\\ref{HJh}) such that\nall its calibrating curves are hyperbolic motions\nhaving the shape of $a$ as limit shape.\nFollowing this method (developed in Sect. \\ref{s-HJ})\nwe get to our main result.\n\n\\begin{theorem}\n\\label{thm-princ}\nFor the Newtonian $N$-body problem in a space $E$\nof dimension at least two,\nthere are hyperbolic motions $x:[0,+\\infty)\\to E^N$ such that\n\\[x(t)=\\sqrt{2h}\\,t\\;a+o(t)\\quad\\text{ as }\\quad t\\to +\\infty,\\]\nfor any choice of $x_0=x(0)\\in E^N$,\nfor any configuration without collisions $a\\in\\Omega$ normalized by $\\norm{a}=1$,\nand for any choice of the energy constant $h>0$.  \n\\end{theorem}\n\nWe emphasize the fact that the initial configuration can be chosen \\emph{with} collisions.\nThis means that in such a case, the motion $x$ given by the theorem is continuous at $t=0$,\nand defines a maximal solution $x(t)\\in\\Omega$ for $t>0$. \nFor instance,\nchoosing $x_0=0\\in E^N$,\nthe theorem gives the existence of ejections from the total collision configuration,\nwith prescribed positive energy and arbitrarily chosen limit shape.\n\nMoreover, the well known Sundman's inequality (see Wintner \\cite{Win}) implies that motions with total collisions have zero angular momentum. \nTherefore, we deduce the following non trivial corollary.\n\n\\begin{corollary}\nFor any configuration without collisions $a\\in\\Omega$\nthere is a hyperbolic motion with zero angular momentum\nand asymptotic velocity $a$.  \n\\end{corollary}\n\n\nIt should be said that the hypothesis that excludes\nthe collinear case $\\dim E=1$ is only required to ensure\nthat action minimizing curves do not suffer collisions.\nThe avoidance of collisions is thus assured by the celebrated\nMarchal's Theorem that we state below in Sect. \\ref{s-var.setting}.\nThe collinear case could eventually be analyzed\nin the light of the results obtained by Yu and Zhang \\cite{YuZha}.\n\nTheorem \\ref{thm-princ} should be compared with that obtained\nby the authors in \\cite{MadVen} which concerns completely parabolic motions.\nWe recall that completely parabolic motions (as well as total collisions)\nhave a very special asymptotic behaviour.\nIn his work of 1918 \\cite{Cha1},\nChazy proves that the normalized configuration\nmust approximate the set of normal central configurations.\nUnder a hypothesis of non-degeneracy,\nhe also deduces the convergence to a particular central configuration. \nThis hypothesis is always satisfied in the three body problem.\nHowever,\na first counterexample with four bodies in the plane\nwas founded by Palmore \\cite{Pal},\nallowing thus the possibility of motions\nwith infinite spin (see Chenciner \\cite{Che1} p.281).\n\nIn all the cases, Chazy's Theorem prevents arbitrary limit shapes\nfor completely parabolic motions as well as for total collisions.\nIn this sense,\nlet us mention for instance the general result by Shub \\cite{Shu}\non the localisation of central configurations,\nshowing that they are isolated from the diagonals.\n\nWe should also mention that the confinement of the\nasymptotic configuration to the set of central configurations,\nboth for completely parabolic motions and for total collisions,\nextends to homogeneous potentials of degree $\\alpha\\in (-2,0)$.\nFor these potentials the mutual distances must grow like $t^{2\/(2-\\alpha)}$ in\nthe parabolic case, and must decay like $\\abs{t-t_0}^{2\/(2-\\alpha)}$\nin the case of a total collision at time $t=t_0$.\nOn the other hand,\nit is known that potentials giving rise to strong forces near collisions\ncan present motions of total collision with non-central asymptotic configurations.\nWe refer the reader to the comments on the subject by Chenciner in \\cite{Che3}\nabout the Jacobi-Banachiewitz potential, and to Arredondo et al. \\cite{ArPeChSt}\nfor related results on the dynamics of total collisions in the case of\nSchwarzschild and Manev potentials. \n\n\nLet us say that there is another natural way to prove the existence of hyperbolic motions, \nusing the fact that the Newtonian force vanishes when all mutual distances diverge.\nWe could call these motions \\emph{almost linear}.\nThe way to do that is as follows.\nSuppose first that $(x_0,a)\\in \\Omega\\times\\Omega$\nis such that the half-straight line given by $\\bar{x}(t)=x_0+ta$, $t>0$\nhas no collisions ($\\bar{x}(t)\\in\\Omega$ for all $t>0$).\nConsider now the motion $x(t)$ with initial condition\n$x(0)=x_0$ and $\\dot x(0)=\\alpha a$ for some positive constant $\\alpha$.\nIt is not difficult to prove that, for $\\alpha>0$ chosen big enough,\nthe trajectory $x(t)$ is defined for all $t>0$, and moreover,\nit is a hyperbolic motion with limit velocity $b\\in\\Omega$ close to $\\alpha a$.\nIn particular,\nthe limit shape of such a motion can be obtained\nas close as we want from the shape of $a$.\n\nThe previous construction is unsatisfactory for several reasons.\nFirst, we do not get exactly the desired limit shape but a close one.\nThis approximation can be made as good as we want,\nbut we lose the control of the energy constant $h$ of the motion,\nwhose order of magnitude is that of $\\alpha^2$.\nSecondly,\nit is not possible to apply this method when the half-straight line\n$\\bar{x}$ presents collisions.\nFor instance this is the case if we take $a=z_0-x_0$\nfor any choice of $z_0\\in E^N\\setminus\\Omega$.\nFinally,\neven if the homogeneity of the potential can be exploited\nto find a new hyperbolic motion\nwith a prescribed positive energy constant,\nand the same limit shape,\nwe lose the control on the initial configuration.\nIndeed,\nif $x$ is a hyperbolic motion defined for all $t\\geq 0$\nwith energy constant $h$,\nthen the motion $x_\\lambda$ defined by\n$x_\\lambda(t)=\\lambda\\,x(\\lambda^{-3\/2}t)$\nis still hyperbolic with energy constant $\\lambda^{-1}h$.\nMoreover, the limit shapes of $x$ and $x_\\lambda$ are the same,\nbut $x_\\lambda(0)=\\lambda x(0)$ meaning that\nthe initial configuration is dilated by the factor $\\lambda$.\n\n\n\n\n\n\n\\subsection{Other expansive motions}\n\\label{s-other}\n\nHyperbolic motions are part of the family of\n\\emph{expansive motions} which we define now.\nIn order to classify them,\nas well as for further later uses,\nwe summarize below a set of well-known facts\nabout the possible evolutions of the motions\nin the Newtonian $N$-body problem.\n\n\\begin{definition}\n[Expansive motion]\nA motion $x:[0,+\\infty)\\to \\Omega$ is said to be expansive\nwhen all the mutual distances diverge.\nThat is, when $r_{ij}(t)\\to+\\infty$ for all $i<j$.\nEquivalently, the motion is expansive if $U(t)\\to 0$.\n\\end{definition}\n\nWe will see that there are three well defined classes of expansive motions. \nFirst of all we must observe that,\nsince we $U(t)\\to 0$ implies $\\norm{\\dot x(t)}\\to\\sqrt{2h}$,\nexpansive motions can only occur with $h\\geq 0$.\n\nIn his pioneering work,\nJean Chazy proposed a classification of motions in terms of their final evolution.\nIn the Keplerian case there is only one distance function to consider,\nand the three classes of motions are elliptic, parabolic and hyperbolic.\nExtending the analysis for $N\\geq 3$,\nhe introduced several hybrid classes of motions,\nsuch as hyperbolic-elliptical in which some distances diverge and others remain bounded.\nIn his attempt to achieve a full classification,\nhe obtains the theoretical possibility of complex behaviours\nsuch as the so-called oscillatory motions or the superhyperbolic motions,\nsee Saari and Xia \\cite{SaaXia}.\nAfter the works of Chazy, and for quite some time,\nspecialists have doubted the existence of such motions\nbecause of his complex and paradoxical appearance. \nThe same can be said about the existence of pseudo-collision\nsingularities, which, as is well known, are impossible if $N=3$.\n\nLet us say that the existence results of oscillatory motions goes\nback to the work of Sitnikov \\cite{Sit} for the spatial restricted\nthree-body problem.\nThen, the main idea in this paper was extended to\nthe unrestricted problem by Alekseev\n(see Moser \\cite{Mos} for a more detailed explanation of this\nand other related developments).\nSitnikov's ideas were undoubtedly very important\nfor the construction of the first example of a motion\nwith a pseudo-collision singularity with five bodies by Xia \\cite{Xia}.\nWith respect to superhyperbolic motions we must say that,\nalthough there are no known examples of them,\nthey exist at least in a weak sense\nfor the collinear four-body problem\n(with regularisation of binary collisions) \\cite{SaaXia}.\n\nAs we will see, to achieve the proof of the announced results,\nit will be crucial to show that certain motions that will be obtained\nare not superhyperbolic,\nand that they do not suffer collisions nor pseudo-collisions.\n\nWe need to introduce two functions which play an important role\nin the classical description of the dynamics.\nFor a given motion, these two functions are\n\\[\nr(t)=\\min_{i<j}r_{ij}(t)\\quad \\text{ and }\\quad\nR(t)=\\max_{i<j}r_{ij}(t)\\,\n\\]\nthe minimum and the maximum separation between the bodies\nat time $t$.\nWe now recall some  facts concerning the possible behaviours\nof the trajectories as $t\\to +\\infty$ in terms of the behaviours\nof these functions.\n\nWe start by fixing some notation and making some remarks.\n\n\\begin{notation}\nGiven positive functions $f$ and $g$,\nwe will write $f\\approx g$ when the quotient of them\nis bounded between two positive constants.\n\\end{notation}\n\n\\begin{remark}\nIt is easy to see that $r\\approx U^{-1}$.\nMoreover, \n$R^2\\approx I_G$ where $I_G$  denotes the moment of inertia\nwith respect to the center of mass $G$ of the configuration.\nTo see this it suffices to write $I_G$\nin terms of the mutual distances.\n\\end{remark}\n\n\\begin{remark}\n\\label{rmk-configur.measure}\nThe function $\\mu=U\\,I_G\\,^{1\/2}$ is homogeneous\nof degree zero.\nSome authors call this function the\n\\emph{configurational measure}.\nAccording to the previous remark we have $\\mu\\approx R\\,r^{-1}$.\n\\end{remark}\n\n\\begin{remark}\nBy K\u00f6nig's decomposition we have that $I=I_G+M\\norm{G}_E^2$\nwhere $M$ is the total mass of the system.\nTherefore, using the Largange-Jacobi identity $\\ddot I=4h+2U$ we deduce that,\nif $h>0$ and the center of mass is at rest,\nthen $R(t)>At$ for some constant $A>0$.\n\\end{remark}\n\n\n\\begin{theorem*}[1922, Chazy \\cite{Cha2} pp. 39 -- 49]\nLet $x(t)$ be a motion with energy constant $h>0$ and defined for all $t>t_0$.\n\n\\begin{enumerate}\n\\item[(i)] The limit\n\\[\\lim_{t\\to +\\infty}R(t)\\,r(t)^{-1}=L\\in [1,+\\infty]\\]\nalways exists.\n\n\\item[(ii)] If $L<+\\infty$ then there is a configuration $a\\in\\Omega$,\nand some function $P$, which is analytic in a neighbourhood of $(0,0)$,\nsuch that for every $t$ large enough we have\n\\[x(t)=ta-\\log(t)\\,\\nabla U(a)+P(u,v)\\]\nwhere $u=1\/t$ and $v=\\log(t)\/t$.\n\\end{enumerate}\n\\end{theorem*}\n\nAs Chazy pointed out, surprisingly Poincar\u00e9 made the mistake of \nomitting the $\\log(t)$ order term in his\n\\emph{``M\u00e9thodes Nouvelles de la M\u00e9canique C\u00e9leste\"}.\n\nSubsequent advances in this subject were recorded much later,\nwhen Chazy's results on final evolutions were included\nin a more general description of motions.\nFrom this development we must recall the following theorems.\nNotice that none of them make assumptions\non the sign of the energy constant $h$.\n\n\\begin{theorem*}[1967, Pollard \\cite{Pol}]\nLet $x$ be a motion defined for all $t>t_0$.\nIf $r$ is bounded away from zero then we have that\n$R=O(t)$ as $t\\to +\\infty$.\nIn addition $R(t)\/t\\to +\\infty$ if and only if $r(t)\\to 0$.\n\\end{theorem*}\n\nThis leads to the following definition.\n\n\\begin{definition} A motion is said to be superhyperbolic when\n\\[\\limsup_{t\\to +\\infty}\\;R(t)\/t=+\\infty.\\]\n\\end{definition}\n\nA short time later it was proven that,\neither the quotient $R(t)\/t\\to +\\infty$,\nor $R=O(t)$ and the system expansion can be described more accurately.\n\n\\begin{theorem*}[1976, Marchal-Saari \\cite{MarSaa}]\nLet $x$ be a motion defined for all $t>t_0$.\nThen either $R(t)\/t \\to +\\infty$ and $r(t)\\to 0$,\nor there is a configuration $a\\in E^N$ such that $x(t)=ta+O(t^{2\/3})$.\nIn particular, for superhyperbolic motions the quotient $R(t)\/t$ diverges.\n\\end{theorem*}\n\nOf course this theorem does not provide much information\nin some cases, for instance if the motion is bounded then\nwe must have $a=0$.\nOn the other hand,\nit admits an interesting refinement concerning the\nthe behaviour of the subsystems.\nMore precisely,\nwhen $R(t)=O(t)$ and the configuration $a$ given by the theorem\nhas collisions the system decomposes naturally into subsystems,\nwithin which the distances between the bodies\ngrow at most like $t^{2\/3}$.\nConsidering the internal energy of each subsystem,\nMarchal and Saari (Ibid, Theorem 2 and corollary 4 pp.165-166)\ngave a decription of the possible dynamics\nthat can occur within the subsystems.\nFrom these results we can easily deduce the following.\n\n\\begin{theorem*}[1976, Marchal-Saari \\cite{MarSaa}]\nSuppose that $x(t)=ta+O(t^{2\/3})$ for some $a\\in E^N$,\nand that the motion is expansive.\nThen, for each pair $i<j$ such that $a_i=a_j$ we have $r_{ij}\\approx t^{2\/3}$.\n\\end{theorem*}\n\nNotice that we can always consider the \\emph{internal motion}\nof the system, that is, looking at the relative positions\nof the bodies with respect to their center of mass.\nThis gives a new motion with the same distance functions.\nMoreover,\nthe internal motion of an expansive motion is also expansive.\n\nAll the previous considerations allow us to classify expansive motions\naccording to the asymptotic order of growth of the distances between the bodies.\nSince an expansive motion is not superhyperbolic,\nwe can assume that it is of the form $x(t)=ta+O(t^{2\/3})$ for some $a\\in E^N$.\nMoreover, we can assume that the center of mass is at rest,\nmeaning that $G(a)=0$.\nWe get then the following three types.\n\n\\begin{enumerate}\n\\item[(H) ] \\emph{Hyperbolic} :\n$a\\in\\Omega$, and $r_{ij}\\approx t$ for all $i<j$\n\\item[(PH)] \\emph{Partially hyperbolic} :\n$a\\in E^N\\setminus\\Omega$ but $a\\neq 0$.\n\\item[(P) ] \\emph{Completely parabolic} :\n$a=0$, and $r_{ij}\\approx t^{2\/3}$ for all $i<j$.\n\\end{enumerate}\n\nLet $h_0$ be the energy constant of the above defined\ninternal motion.\nIt is clear that the first two types can only occur if $h_0>0$,\nwhile the third requires $h_0=0$.\n\nFinally,\nwe observe that Chazy's Theorem applies in the first two cases .\nIn these cases,\nthe limit shape of $x(t)$ is the shape of the configuration $a$\nand moreover,\nwe have $L<+\\infty$ if and only if $x$ is hyperbolic.\nOf course if $h_0>0$ and $L=+\\infty$ then\neither the motion is partially hyperbolic or it is not expansive.\n\n\n\n\n\n\n\n\n\\subsection{The geometric viewpoint}\n\\label{s-geom.view}\n\nWe explain now the geometric formulation\nand the geometrical meaning of this work\nwith respect to the Jacobi-Maupertuis metrics associated\nto the positive energy levels.\nSeveral technical details concerning these metrics are given\nin Sect. \\ref{s-jm.dist}.\nThe boundary notions are also discussed in Sect. \\ref{s-busemann}.\nIt may be useful for the reader to keep in mind\nthat reading this section can be postponed to the end.\n\nWe recall that for each $h\\geq 0$, the Jacobi-Maupertuis metric\nof level $h$ is a Riemannian metric defined\non the open set of configurations without collisions $\\Omega$.\nMore precisely,\nit is the metric defined by $j_h=2(h+U)\\,g_m$,\nwhere $g_m$ is the Euclidean metric in $E^N$\ngiven by the mass inner product.\nOur main theorem has a stronger version\nin geometric terms.\nActually Theorem \\ref{thm-princ} can be reformulated\nin the following way.\n\n\\begin{theorem}\\label{thm-princ.rays}\nFor any $h>0$, $p\\in E^N$ and $a\\in\\Omega$,\nthere is  geodesic ray of the Jacobi-Maupertuis metric of level $h$\nwith asymptotic direction $a$ and starting at $p$.\n\\end{theorem}\nThis formulation requires some explanations.\nThe Riemannian distance $d_h$ in $\\Omega$ is defined as usual\nas the infimum of the length functional among all the\npiecewise $C^1$ curves in $\\Omega$ joining two given points.\nWe will prove that $d_h$ can be extended\nto a distance $\\phi_h$ in $E^N$,\nwhich is a metric completion of $(\\Omega,d_h)$,\nand which also we call the Jacobi-Maupertuis distance.\nMoreover, we will prove that $\\phi_h$ is precisely the action\npotential defined in Sect. \\ref{s-var.setting}.\n\nThe minimizing geodesic curves can then be defined\nas the isometric immersions of compact intervals\n$[a,b]\\subset\\mathbb{R}$ within $E^N$.\nMoreover, we will say that a curve $\\gamma:[0,+\\infty)\\to E^N$\nis a geodesic ray\nfrom $p\\in E^N$, if $\\gamma(0)=p$ and each restriction\nto a compact interval is a minimizing geodesic.\nTo deduce this geometric version of our main theorem\nit will be enough to observe that the obtained\nhyperbolic motions can be reparametrized\ntaking the action as parameter to obtain geodesic rays.\n\nWe observe now the following interesting implication of\nChazy's Theorem.\n\\begin{remark}\n\\label{rmk-Chazy.implic}\nIf two given hyperbolic motions have the same asymptotic\ndirection, then they have a bounded difference.\nIndeed,\nif $x$ and $y$ are hyperbolic motions with the same asymptotic\ndirection, then the two unbounded terms\nof the Chazy's asymptotic development of $x$ and $y$ also agree.\n\\end{remark}\n\n\nWe recall that the Gromov boundary of a geodesic space\nis defined as the quotient set of the set of geodesic rays\nby the equivalence that identifies rays that are kept\nat bounded distance.\nFrom the previous remark,\nwe can deduce that two geodesic rays with asymptotic direction\ngiven by the same configuration $a\\in\\Omega$\ndefine the same point at the Gromov boundary.\n\n\\begin{notation}\nLet $\\phi_h:E^N\\times E^N\\to\\mathbb{R}^+$ be the Jacobi-Maupertuis\ndistance for the energy level $h\\geq 0$\nin the full space of configurations.\nWe will write $\\mathcal{G}_h$ for the corresponding Gromov boundary.\n\\end{notation}\n\nThe proof of the following corollary is given in Sect. \\ref{s-jm.dist}.\n\n\\begin{corollary}\\label{cor-Gr.boundary}\nIf $h>0$, then each class in $\\Omega_1=\\Omega\/\\mathbb{R}^+$\ndetermines a point in $\\mathcal{G}_h$ which is composed\nby all geodesic rays with asymptotic direction in this class.\n\\end{corollary}\n\n\nOn the other hand,\nif instead of the arc length we parametrize the geodesics\nby the dynamical parameter,\nthen it is natural to question the existence\nof non-hyperbolic geodesic rays.\nWe do not know if there are partially hyperbolic geodesic rays. \nNor do we know if a geodesic ray should be an expansive motion. \n\nIn what follows we denote $\\norm{v}_h$\nfor the norm  of $v\\in T\\Omega$ with respect to the metric $j_h$,\nand $\\norm{p}_h$ the dual norm of an element $p\\in T^*\\Omega$.\nIf $\\gamma:(a,b)\\to\\Omega$ is a geodesic\nparametrized by the arc length, then\n\\[\\norm{\\dot\\gamma(s)}_h^2=\n2(h+U(\\gamma(s))\\,\\norm{\\dot\\gamma(s)}^2=1\\]\nfor all $s\\in (a,b)$.\nTaking into account that $U\\approx r^{-1}$ we see that\nthe parametrization of motions as geodesics\nleads to slowed evolutions over passages near collisions.\nWe also note that for expansive geodesics we have\n$\\norm{\\dot\\gamma(s)}\\to 1\/\\sqrt{2h}$.\n\nFinally we make the following observations about the\nHamilton-Jacobi equation that we will solve in the weak sense.\nFirst, the equation (\\ref{HJh}) , that explicitly reads\n\\[\\frac{1}{2}\\norm{d_xu}^2-U(x)=h\\]\ncan be written in geometric terms, precisely as the eikonal equation\n\\[\n\\norm{d_xu}_h=\\frac{1}{\\sqrt{2(h+U(x))}}\\,\\norm{d_xu}=1\n\\]\nfor the Jacobi-Maupertuis metric.\nOn the other hand,\nthe solutions will be obtained as limits of weak subsolutions,\nwhich  can be viewed as $1$-Lipschitz functions for the\nJacobi-Maupertuis distance.\nWe will see that the set of viscosity subsolutions\nis the set of functions\n$u:E^N\\to\\mathbb{R}$ such that $u(x)-u(y)\\leq \\phi_h(x,y)$\nfor all $x,y\\in E^N$.\n\n\n\n\n\n\n\n\\section{Viscosity solutions of the Hamilton-Jacobi equation}\n\\label{s-HJ}\n\n\n\nThe \\emph{Hamiltonian} $H$ is defined over\n$T^*E^N\\simeq E^N\\times (E^*)^N$ as usual by\n\\[\nH(x,p)=\\frac{1}{2}\\norm{p}^2-U(x)\n\\]\nand taking the value $H(x,p)=-\\infty$\nwhenever the configuration $x$ has collisions.\nHere the norm is the dual norm with respect to the mass product,\n that is,\nfor $p=(p_1,\\dots,p_N)\\in (E^*)^N$\n\\[\n\\norm{p}^2=\nm_1^{-1}\\,\\norm{p_1}^2+\\dots+m_N^{-1}\\,\\norm{p_N}^2\\,,\n\\]\nthus in terms of the positions of the bodies\nequation (\\ref{HJh}) becomes\n\\[H(x,d_xu)=\n\\sum_{i=1}^N\\frac{1}{2\\,m_i}\\norm{\\frac{\\partial u}{\\partial r_i}}^2-\n\\;\\sum_{i<j}\\frac{m_im_j}{r_{ij}}\\; = h\\,.\\]\nAs is known,\nthe method of characteristics for this type of equations\nconsists in reducing the problem to the resolution\nof an ordinary differential equation,\nwhose solutions are precisely the characteristic curves.\nOnce these curves are determined,\nwe can obtain solutions by integration along these curves,\nfrom a cross section in which the solution value is given.\nOf course,\nhere the characteristics are precisely the solutions\nof the $N$-body problem and can not be computed.\nOur method will be the other way around:\nfirst we build a solution as a limit of subsolutions,\nand then we find characteristic curves\nassociated with that solution.\n\nWe will start by recalling\nthe notion of viscosity solution in our context.\nThere is an extremely wide literature on viscosity solutions due to\n the great diversity of situations in which they can be applied.\nFor a general and introductory presentation the books of Evans\n \\cite{Eva1} and Barles \\cite{Bar} are recommended.\nFor a broad view on the Lax-Oleinik semigroups\nwe suggest references \\cite{Ber,Eva2, Fat}.  \n\n\\begin{definition}[Viscosity solutions]\nWith respect to the Hamilton-Jacobi equation (\\ref{HJh}),\nwe say that a continuous function $u:E^N\\to\\mathbb{R}$ is a\n\\begin{enumerate}\n\\item \\emph{viscosity \\bf{subsolution}},\nif for any $\\psi\\in C^1(E^N)$ and for any configuration $x_0$\nat which $u-\\psi$ has a local maximum we have\n$H(x_0,d_{x_0}\\psi)\\leq h$.\n\n\\item \\emph{viscosity \\bf{supersolution}},\nif for any $\\psi\\in C^1(E^N)$ and for any configuration $x_0$\nat which $u-\\psi$ has a local minimum we have\n$H(x_0,d_{x_0}\\psi)\\geq h$.\n\n\\item \\emph{viscosity \\bf{solution}},\nas long as is both a subsolution and a supersolution.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\nIt is clear that we get the same notions by taking test functions\n$\\psi$ defined on open subsets of $E^N$.\n\\end{remark}\n\n\\begin{remark}\nThe notion of viscosity solution is a generalization\nof the notion of classical solution.\nIndeed, if $u\\in C^1(E^N)$\nsatisfies the Hamilton-Jacobi equation everywhere,\nthen $u$ is a viscosity solution\nsince we can take $\\psi=u$ as a test function.\n\\end{remark}\n\nIf $u\\in C^0(E^N)$ is a viscosity solution,\nthen we have $H(x,d_xu)=h$\nat any point where $u$ is differentiable.\nThis follows from the fact that for any $C^0$ function $u$,\nthe differentiability at $x_0\\in E^N$\nimplies the existence of $C^1$ functions\n$\\psi^-$ and $\\psi^+$ such that $\\psi^-\\leq u \\leq \\psi^+$ and\n$\\psi^-(x_0)=u(x_0)=\\psi^+(x_0)$.\nAs we will see (Lemma \\ref{lema-visc.subsol.locLip}),\nin our case viscosity subsolutions are locally Lipschitz\nover the open and dense set $\\Omega\\subset E^N$\nof configurations without collisions.\nTherefore by Rademacher's Theorem\nthey are differentiable almost everywhere.\nBut, as is well known to the reader familiar with the subject,\nbeing a viscosity solution is a much more demanding property\nthan satisfying the equation almost everywhere.\n\n\n\\begin{remark}\nWe note that the participation of the unknown $u$\nin equation (\\ref{HJh}) is only through the derivatives $d_xu$.\nTherefore the set of classical solutions\nis preserved by addition of constants.\nAlso note that the same applies for the set of\nviscosity subsolutions and the set of viscosity supersolutions.\n\\end{remark}\n\nFrom now on,\nwe will use of the powerful interaction between\nthe Hamiltonian view of dynamics and the Lagrangian view.\nThe Hamilton-Jacobi equation provides a great bridge\nbetween the symplectic aspects of dynamics\nand the variational properties of trajectories.\n\nOnce the Lagrangian action is defined,\nwe will characterize the set of viscosity subsolutions\nas the set of functions satisfying a property of domination\nwith respect to the action.\nThen, the next step will be to prove the equicontinuity\nof the family of viscosity subsolutions\nby finding an estimate for an action potential.\n\n\n\n\n\\subsection{Action potentials and viscosity solutions} \n\\label{s-var.setting}\n\n\n\nThe \\emph{Lagrangian} is defined on\n$TE^N\\simeq E^N\\times E^N$ fiberwise\nas the convex dual of the Hamiltonian, that is\n\\[\nL(x,v)=\\sup\\set{p(v)-H(x,p)\\mid p\\in (E^*)^N}\n\\]\nor equivalently,\n\\[\nL(x,v)=\\frac{1}{2}\\norm{v}^2 + U(x)\\,,\n\\]\nso in particular it takes the value $L(x,v)=+\\infty$\nif $x$ has collisions.\nThe Lagrangian action will be considered\non absolutely continuous curves,\nand its value could be infinite.\nWe will use the following notation.\nFor $x,y\\in E^N$ and $\\tau>0$, let\n\\[\n\\mathcal{C}(x,y,\\tau)=\n\\set{\\gamma:[a,b]\\to E^N \\mid\\gamma(a)=\nx,\\,\\gamma(b)=y,\\,b-a=\\tau}\n\\]\nbe the set of absolutely continuous curves\ngoing from $x$ to $y$ in time $\\tau$,\nand\n\\[\n\\mathcal{C}(x,y)=\\bigcup_{\\tau>0}\\mathcal{C}(x,y,\\tau)\\,.\n\\]\nThe Lagrangian action of a curve $\\gamma\\in\\mathcal{C}(x,y,\\tau)$\nwill be denoted\n\\[\n\\mathcal{A}_L(\\gamma)=\\int_a^b L(\\gamma,\\dot\\gamma)\\,dt=\n\\int_a^b \\tfrac{1}{2}\\norm{\\dot\\gamma}^2+U(\\gamma)\\,dt\n\\]\n\nIt is well known that Tonelli's Theorem\non the lower semicontinuity of the action for convex Lagrangians\nextends to this setting.\nA proof can for instance be found in \\cite{daLMad} (Theorem 2.3).\nIn particular we have, for any pair of configurations $x,y\\in E^N$,\nthe existence of curves achieving the minimum value\n\\[\n\\phi(x,y,\\tau)=\n\\min\\set{\\mathcal{A}_L(\\gamma)\\mid \\gamma\\in\\mathcal{C}(x,y,\\tau)}\n\\]\nfor any $\\tau>0$.\nWhen $x\\neq y$ there also are curves reaching the minimum\n\\[\n\\phi(x,y)=\\min\\set{\\mathcal{A}_L(\\gamma)\\mid \\gamma\\in\\mathcal{C}(x,y)}\n=\\min\\set{\\phi(x,y,\\tau)\\mid \\tau>0}\n\\]\nIn the case $x=y$ we have\n$\\phi(x,x)=\\inf\\set{\\mathcal{A}_L(\\gamma)\\mid \\gamma\\in\\mathcal{C}(x,y)}=0$.\nWe call these functions on $E^N\\times E^N$ respectively\nthe \\emph{fixed time action potential} and the\n\\emph{free time} (or \\emph{critical}) \\emph{action potential}.\n\nAccording to the Hamilton's principle of least action,\nif a curve $\\gamma:[a,b]\\to E^N$ is a minimizer of the Lagrangian\naction in $\\mathcal{C}(x,y,\\tau)$ then $\\gamma$ satisfy Newton's\nequations at every time $t\\in [a,b]$ in which $\\gamma(t)$\nhas no collisions, i.e. whenever $\\gamma(t)\\in\\Omega$.\n\nOn the other hand,\nit is easy to see that there are curves\nboth with isolated collisions and finite action.  \nThis phenomenon,\nalready noticed by Poincar\u00e9 in \\cite{Poi}, prevented the use\nof the direct method of the calculus of variations\nin the $N$-body problem for a long time. \n\nA big breakthrough came from Marchal,\nwho gave the main idea needed to prove the following theorem.\nComplete proofs of this and more general versions were\nestablished by Chenciner \\cite{Che1}\nand  by Ferrario and Terracini \\cite{FerTer}.\n\n\\begin{theorem*}[2002, Marchal \\cite{Mar}]\nIf $\\gamma\\in\\mathcal{C}(x,y)$ is defined on some interval $[a,b]$,\nand satisfies $\\mathcal{A}_L(\\gamma)=\\phi(x,y,b-a)$,\nthen $\\gamma(t)\\in\\Omega$ for all $t\\in (a,b)$.\n\\end{theorem*}\nThanks to this advance,\nthe existence of countless periodic orbits has been established\nusing variational methods.\nAmong them, the celebrated three-body figure eight\ndue to Chenciner and Montgomery \\cite{CheMon}\nis undoubtedly the most representative example,\nalthough it was discovered somewhat before.\nMarchal's Theorem was also used to prove the nonexistence\nof entire free time minimizers \\cite{daLMad},\nor in geometric terms,\nthat the zero energy level has no straight lines.\nThe proof we provide below for our main result\ndepends crucially on Marchal's Theorem.\nOur results can thus be considered as a new application\nof Marchal's Theorem, this time in positive energy levels.\n\nWe must also define for $h>0$ the\n\\emph{supercritical action potential} as the function \n\\[\n\\phi_h(x,y)=\n\\inf\\set{\\mathcal{A}_{L+h}(\\gamma)\\mid \\gamma\\in\\mathcal{C}(x,y)}=\n\\inf\\set{\\phi(x,y,\\tau)+h\\tau\\mid \\tau>0}.\n\\]\n\nFor the reader familiar with the Aubry-Mather theory,\nthis definition should be reminiscent of the supercritical\naction potentials used by Ma\u00f1\u00e9 to define the critical value\nof a Tonelli Lagrangian on a compact manifold. \n\nAs before we prove\n(see Lemma \\ref{lema-JM.geod.complet} below),\nnow for $h>0$,\nthat given any pair of different configurations $x,y\\in E^N$,\nthe infimum in the definition of $\\phi_h(x,y)$\nis achieved by some curve $\\gamma\\in\\mathcal{C}(x,y)$,\nthat is, we have $\\phi_h(x,y)=\\mathcal{A}_{L+h}(\\gamma)$.\nIt follows that if $\\gamma$ is defined in $[0,\\tau]$,\nthen $\\gamma$ also minimizes $\\mathcal{A}_L$ in $\\mathcal{C}(x,y,\\tau)$\nand by Marchal's Theorem we conclude that $\\gamma$\navoid collisions,\ni.e. $\\gamma(t)\\in\\Omega$ for every $t\\in (0,\\tau)$.\n\n\n\\subsubsection{Dominated functions and viscosity subsolutions}\n\nLet us fix $h>0$\nand take a $C^1$ subsolution $u$ of $H(x,d_xu)=h$,\nthat is, such that $H(x,d_xu)\\leq h$ for all $x\\in E^N$.\nNotice now that,\nsince for any absolutely continuous curve\n$\\gamma:[a,b]\\to E^N$ we have\n\\[\nu(\\gamma(b))-u(\\gamma(a))=\n\\int_a^b d_{\\gamma}u(\\dot\\gamma)\\,dt\\,,\n\\]\nby Fenchel's inequality we also have\n\\[\nu(\\gamma(b))-u(\\gamma(a))\\leq\n\\int_a^b L(\\gamma,\\dot\\gamma)+\nH(\\gamma,d_{\\gamma}u)\\;dt\\;\\leq\n\\;\\mathcal{A}_{L+h}(\\gamma)\\,.\n\\]\nTherefore we can say that if $u$ is a $C^1$ subsolution,\nthen\n\\[\nu(x)-u(y)\\leq\n\\mathcal{A}_{L+h}(\\gamma)\n\\]\nfor any curve\n$\\gamma\\in\\mathcal{C}(x,y)$.\nThis motivates the following definition.\n\n\\begin{definition}[Dominated functions]\nWe said that $u\\in C^0(E^N)$ is dominated by $L+h$,\nand we will denote it by $u\\prec L+h$,\nif we have\n\\[\nu(y)-u(x)\\leq\n\\phi_h(x,y)\\quad\\text{ for all }\\quad x,y\\in E^N.\n\\]\n\\end{definition}\n\nThus we know that $C^1$ subsolutions are dominated functions.\nWe prove now the well-known fact that dominated functions\nare indeed viscosity subsolutions.\n\n\\begin{proposition}\n\\label{prop-dom.are.visc.ssol}\nIf $u\\prec L+h$ then $u$ is a viscosity subsolution of (\\ref{HJh}).\n\\end{proposition}\n\n\\begin{proof} \nLet $u\\prec L+h$ and consider a test function $\\psi\\in C^1(E^N)$.\nAssume that $u-\\psi$ has a local maximum\nat some configuration $x_0\\in E^N$.\nTherefore,\nfor all $x\\in E^N$ we have $\\psi(x_0)-\\psi(x) \\leq u(x_0)-u(x)$.\n\nOn the other hand,\nthe convexity and superlinearity of the Lagrangian\nimplies that there is a unique $v\\in E^N$ such that\n$H(x_0,d_{x_0}\\psi)=d_{x_0}\\psi (v)-L(x_0,v)$.\nTaking any smooth curve $x:(-\\delta,0]\\to E^N$\nsuch that $x(0)=x_0$ and\n$\\dot x(0)=v$ we can write, for $t\\in (-\\delta,0)$\n\\[\\frac{\\psi(x_0)-\\psi(x(t))}{-t}\\leq \\frac{u(x_0)-u(x(t))}{-t}\\leq\n\\;-\\frac{1}{t}\\;\\mathcal{A}_{L+h}\\left(x\\mid_{[t,0\\,]}\\right)\\]\nthus for $t\\to 0^-$ we get\n$d_{x_0}\\psi (v)\\leq L(x_0,v)+h$,\nthat is to say, $H(x_0,d_{x_0}\\psi)\\leq h$ as we had to prove.\n\\end{proof}\nActually, the converse can be proved.\nFor all that follows,\nwe will only need to consider dominated functions,\nand for this reason,\nit will no longer be necessary to manipulate test functions\nto verify the subsolution condition in the viscosity sense.\nHowever,\nfor the sake of completeness we give a proof of this converse.\n\nA first step is to prove that viscosity subsolutions\nare locally Lipschitz on the open, dense,\nand full measure set of configurations without collisions\n(for this we follow the book of\nBardi and Capuzzo-Dolcetta \\cite{BarCap}, Proposition 4.1, p. 62).\n\n\\begin{lemma}\n\\label{lema-visc.subsol.locLip}\nThe viscosity subsolutions of (\\ref{HJh})\nare locally Lipschitz on $\\Omega$.\n\\end{lemma}\n\n\\begin{proof}\nLet $u\\in C^0(E^N)$ be a viscosity subsolution and\nlet $z\\in\\Omega$.\nWe take a compact neighbourhood $W$ of $z$\nin which the Newtonian potential is bounded,\ni.e. such that $W\\subset\\Omega$.\nThus our Hamiltonian is coercive on $T^*W$,\nmeaning that given $h>0$ we can choose $\\rho>0$ for which,\nif $\\norm{p}>\\rho$ and $w\\in W$ then $H(w,p)>h$.\n\nWe choose now $r>0$ such that\nthe open ball $B(z,3r)$ is contained in $W$.\nLet $M=\\max\\set{u(x)-u(y)\\mid\\;x,y\\in W}$\nand take $k>0$ such that $2kr>M$.\n\nWe take now any configuration $y\\in B(z,r)$ and we define,\nin the closed ball $\\overline{B}_y=\\overline{B}(y,2r)$,\nthe function $\\psi_y(x)=u(y)+k\\norm{x-y}$.\nWe will use the function $\\psi_y$ as a test function in the open set\n$B^*_y=B(y,2r)\\setminus\\set{y}$.\nWe observe first that $u(y)-\\psi_y(y)=0$ and that\n$u-\\psi_y$ is negative in the boundary of $\\overline{B}_y$.\nTherefore the maximum of $u-\\psi_y$\nis achieved at some interior point\n$x_0\\in B(y,2r)$.\n\nSuppose that $x_0\\neq y$.\nSince $\\psi_y$ is smooth on $B^*_y$,\nand $u$ is a viscosity subsolution,\nwe must have $H(x_0,d_{x_0}\\psi_y)\\leq h$.\nTherefore we must also have $k=\\norm{d_{x_0}\\psi_y}\\leq\\rho$.\n\nWe conclude that, if we choose $k>\\rho$ such that $2rk>M$,\nthen for any $y\\in B(z,r)$\nthe maximum of $u-\\psi$ in $\\overline{B}_y$\nis achieved at $y$, meaning that $u(x)-u(y)\\leq k\\norm{x-y}$\nfor all $x\\in\\overline{B}_y$.\nThis proves that $u$ is $k$-Lipschitz on $B(z,r)$.\n\\end{proof}\n\n\\begin{remark}\n\\label{rmk-vssol.are.subsol.ae}\nBy Rademacher's Theorem, we know that any viscosity subsolution\nis differentiable almost everywhere in the open set $\\Omega$.\nIn addition, at every point of differentiability $x\\in\\Omega$\nwe have $H(x,d_xu)\\leq h$.\nTherefore, since $\\Omega$ has full measure in $E^N$,\nwe can say that viscosity subsolutions satisfies $H(x,d_xu)\\leq h$\nalmost everywhere in $E^N$.\n\\end{remark}\n\n\\begin{remark}\nWe observe that the local Lipschitz constant $k$ we have obtained\nin the proof depends, a priori,\non the chosen viscosity subsolution $u$.\nWe will see that this is not really the case.\nThis fact will result immediatly from the following proposition and\nTheorem \\ref{thm-phih.estim}.\n\\end{remark}\n\nWe can prove now that\nthe set of viscosity subsolutions of $H(x,d_xu)=h$\nand the set of dominated functions $u\\prec L+h$ coincide.\n\n\\begin{proposition}\n\\label{prop-visc.ssol.are.dom}\nIf $u$ is a viscosity subsolution of (\\ref{HJh}) then $u\\prec L+h$.\n\\end{proposition}\n\n\\begin{proof}\nLet $u:E^N\\to\\mathbb{R}$ be a viscosity subsolution.\nWe have to prove that\n\\[u(y)-u(x)\\leq \\mathcal{A}_{L+h}(\\gamma)\\quad\n\\text{ for all }x,y\\in E^N,\\;\\gamma\\in\\mathcal{C}(x,y)\\,.\\]\nWe start by showing the inequality for any segment\n$s(t)=x+t(y-x)$, $t\\in [0,1]$.\nNote first that in the case $y=x$ there is nothing to prove,\nsince the action is always positive.\nThus we can assume that $r=\\norm{y-x}>0$.\n\nWe know $H(x,d_xu)\\leq h$ is satisfied\non a full measure set $\\mathcal{D}\\subset E^N$\nin which $u$ is differentiable,\nsee Lemma \\ref{lema-visc.subsol.locLip}\nand Remark \\ref{rmk-vssol.are.subsol.ae}.\nAssuming that $s(t)\\in\\mathcal{D}$ for almost every $t\\in[0,1]$\nwe can write\n\\[\n\\frac{d}{dt}\\,u(s(t))=d_{s(t)}u\\,(y-x)\\quad\\text{ a.e. in }[0,1]\n\\]\nfrom which we deduce,\napplying Fenchel's inequality and integrating,\n\\[\nu(y)-u(x)\\leq\n\\int_0^1L(s(t),y-x)+H(s(t),d_{s(t)}u)\\;dt\\leq \\mathcal{A}_{L+h}(s)\\,.\n\\]\nOur assumption may not be satisfied.\nMoreover,\nit could even happen that all the segment\nis outside the set $\\mathcal{D}$ in which the derivatives of $u$ exist. \nThis happens for instance if $x$ and $y$ are configurations\nwith collisions and with the same colliding bodies.\nHowever Fubini's Theorem say us that our assumption is verified\nfor almost every $y\\in S_r=\\set{y\\in E^N\\mid \\norm{y-x}=r}$.\nThen \n\\[\nu(y)-u(x)\\leq \\mathcal{A}_{L+h}(s)\\quad\\text{ for almost }y\\in S_r\n\\]\nTaking into account that both $u$ and $\\mathcal{A}_{L+h}(s)$\nare continuous as functions of $y$,\nwe conclude that the previous inequality holds in fact,\nfor all $y\\in S_r$.\n\nWe remark that the same argument applies\nto any segment with constant speed,\nthat is to say, to any curve $s(t)=x+tv$, $t\\in [a,b]$.\nConcatenating these segments we deduce that the inequality\nalso holds for any piecewise affine curve\n$p\\in\\mathcal{C}(x,y)$. The proof is then achieved as follows.\n\nLet $\\gamma\\in\\mathcal{C}(x,y)$ be a curve such that\n$\\mathcal{A}_{L+h}(\\gamma)=\\phi_h(x,y)$.\nThe existence of such a curve is guaranteed by Lemma\n\\ref{lema-JM.geod.complet}.\nSince this curve is a minimizer of the Lagrangian action,\nMarchal's Theorem assures that,\nif $\\gamma$ is defined on $[a,b]$,\nthen $\\gamma(t)\\in\\Omega$ for all $t\\in(a,b)$.\nIn consequence, the restriction of $\\gamma$\nto $(a,b)$ must be a true motion.\n\nSuppose that there are no collisions at configurations\n$x$ and $y$.\nSince in this case $\\gamma$ is thus $C^1$ on $[a,b]$,\nwe can approximate it by sequence of piecewise affine curves\n$p_n\\in\\mathcal{C}(x,y)$,\nin such a way that $\\dot p_n(t)\\to\\dot\\gamma(t)$\nuniformly for $t$ over some full measure subset $D\\subset [a,b]$.\nIn order to be explicit, let us define for each $n>0$\nthe polygonal $p_n$ with vertices at configurations\n$\\gamma(a+k(b-a)n^{-1})$ for $k=0,\\dots,n$.\nThen $D$ can be taken as the complement in $[a,b]$\nof the countable set $a+\\mathbb{Q}(b-a)$. \nTherefore, we have\n$u(y)-u(x)\\leq \\mathcal{A}_{L+h}(p_n)$ for all $n\\geq 0$,\nas well as\n\\[\n\\lim_{n\\to \\infty}\\mathcal{A}_{L+h}(p_n)=\n\\mathcal{A}_{L+h}(\\gamma)=\\phi_h(x,y)\\,,\n\\]\nThis implies that $u(y)-u(x)\\leq \\phi_h(x,y)$.\nIf there are collisions at $x$ or $y$,\nthen we apply what we have proved\nto the configurations without collisions\n$x_\\epsilon=\\gamma(a+\\epsilon)$ and\n$y_\\epsilon=\\gamma(b-\\epsilon)$,\nand we get the same conclusion\ntaking the limit as $\\epsilon\\to 0$. \nThis proves that $u\\prec L+h$.\n\\end{proof}\n\n\\begin{remark}\nThe use of Marchal's Theorem in the last proof\nseems to be required by the argument.\nIn fact, the argument works well for non singular Hamiltonians\nfor which it is known \\emph{a priori} that the corresponding\nminimizers are of class $C^1$. \n\\end{remark}\n\n\\begin{notation}\nWe will denote $\\mathcal{S}_h$\nthe set of viscosity subsolutions of (\\ref{HJh}).\n\\end{notation}\n\nObserve that,\nnot only we have proved that $\\mathcal{S}_h$ is precisely the set\nof dominated functions $u\\prec L+h$,\nbut also that $\\mathcal{S}_h$ agrees with the set of functions\nsatisfying $H(x,d_xu)\\leq h$ almost everywhere in $E^N$. \n\n\\subsubsection{Estimates for the action potentials}\n\nWe give now an estimate for $\\phi_h$ which implies that\nthe viscosity subsolutions form an equicontinuous family\nof functions.\nTherefore, if we normalize subsolutions by imposing $u(0)=0$,\nthen according to the Ascoli's Theorem\nwe get to the compactness of the set of normalized subsolutions.\n\nThe estimate will be deduced from the basic estimates for\n$\\phi(x,y,\\tau)$ and $\\phi(x,y)$\nfound by the first author for homogeneous potentials\nand that we recall now.\nThey correspond in the reference\nto Theorems 1 \\& 2 and Proposition 9,\nconsidering that in the original formulation\nthe value $\\kappa=1\/2$ is for the Newtonian potential.\n\nWe will say that a given configuration $x=(r_1,\\dots,r_N)$\nis contained in a ball of radius $R>0$ if we have\n$\\norm{r_i-r_0}_E<R$ for all $1\\leq i \\leq N$\nand for some $r_0\\in E$.\n\n\\begin{theorem*}[\\cite{Mad1}]\nThere are positive constants $\\alpha_0$ and $\\beta_0$ such that,\nif $x$ and $y$ are two configurations contained\nin the same ball of radius $R>0$, then for any $\\tau>0$\n\\[\\phi(x,y,\\tau)\\leq \\;\n\\alpha_0\\; \\frac{\\;R^2}{\\tau}\\;+\\;\\beta_0\\;\\frac{\\;\\tau}{R}\\;.\\]\n\\end{theorem*}\n\nIf a configurations $y$ is close enough\nto a given configuration $x$,\nthe minimal radius of a ball containing both configurations\nis greater than $\\norm{x-y}$. \nHowever,\nthis result was successfully combined with an argument\nproviding suitable cluster partitions,\nin order to obtain the following theorem.\n\n\\begin{theorem*}[\\cite{Mad1}]\nThere are positive constants $\\alpha_1$ and $\\beta_1$ such that,\nif $x$ and $y$ are any two configurations, and $r>\\norm{x-y}$,\nthen for all $\\tau>0$\n\\begin{equation}\n\\tag{*}\n\\label{ineq-basic}\\phi(x,y,\\tau)\\leq \\;\n\\alpha_1\\; \\frac{\\;r^2}{\\tau}\\;+\\;\\beta_1\\;\\frac{\\tau}{r}\\;.\n\\end{equation}\n\\end{theorem*}\n\nNote that the right side of the inequality\nis continuous for $\\tau,\\rho>0$.\nTherefore,\nwe can replace $r$ by $\\norm{x-y}$ whenever $x\\neq y$.\n\n\\begin{remark}\n\\label{rmk-bound.phixxt}\nIf $x=y$ then the upper bound (\\ref{ineq-basic})\nholds for every $r>0$.\nChoosing $r=\\tau^{2\/3}$, we get to the upper bound\n$\\phi(x,x,\\tau)\\leq\\mu\\,\\tau^{1\/3}$ which holds for any $\\tau>0$,\nany $x\\in E^N$,\nand for the positive constant $\\mu=\\alpha_1+\\beta_1$.\n\\end{remark}\n \nTherefore we can now bound the critical potential.\nThe previous remark leads to $\\phi(x,x)=0$ for all $x\\in E^N$.\nOn the other hand,\nfor the case $x\\neq y$ we can bound $\\phi(x,y)$\nwith the bound for $\\phi(x,y,\\tau)$,\ntaking $r=\\norm{x-y}$ and $\\tau=\\norm{x-y}^{3\/2}$.\n\n\\begin{theorem*}\n[H\u00f6lder estimate for the critical action potential, \\cite{Mad1}]\nThere exist a positive constant $\\eta>0$\nsuch that for any pair of configurations\n$x,y \\in E^N$\n\\[\\phi(x,y)\\leq \\;\\eta\\;\\norm{x-y}^\\frac{1}{2}\\;.\\]\n\\end{theorem*}\n\nThese estimates for the action potentials have been used\nfirstly to prove the existence of parabolic motions\n\\cite{Mad1,MadVen} and were the starting point\nfor the study of free time minimizers \\cite{daLMad,Mad2},\nas well as their associated Busemann functions\nby Percino and S\u00e1nchez \\cite{Per, PerSan},\nand later by Moeckel, Montgomery and S\u00e1nchez \\cite{MoMoSa}\nin the planar three-body problem.\n \nFor our current purposes,\nwe need to generalize the H\u00f6lder estimate\nof the critical action potential\nin order to also include supercritical potentials.\nAs expected, the upper bound we found\nis of the form $\\xi(\\norm{x-y})$,\nwhere $\\xi:[0,+\\infty)\\to\\mathbb{R}^+$\nis such that $\\xi(r)\\approx r^\\frac{1}{2}$\nfor $r\\to 0$ and $\\xi(r)\\approx r$ for $r\\to +\\infty$.\n\n\\begin{theorem}\n\\label{thm-phih.estim}\nThere are positive constants $\\alpha$ and $\\beta$ such that,\nif $x$ and $y$ are any two configurations and $h\\geq 0$, then\n\\[\n\\phi_h(x,y)\\leq \\;\n\\left(\\alpha\\norm{x-y}+h\\;\\beta\\norm{x-y}^2\\right)^{1\/2}\\;.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nWe have to bound\n$\\phi_h(x,y)=\n\\inf\\set{\\phi(x,y,t)+h\\tau\\mid\\tau>0}$.\nFix any two configurations $x$ and $y$ and let $r>\\norm{x-y}$.\nWe already know by (\\ref{ineq-basic})\nthat for any $\\tau>0$ we have\n\\begin{equation}\n\\tag{**}\n\\label{ineq-basic.h}\n\\phi(x,y,\\tau) + h\\tau\\;\\leq \\;\nA\\; \\frac{1}{\\tau}\\;+B\\;\\tau\n\\end{equation}\n\\[A=\\alpha_1\\,r^2\\quad\\text{ and }\\quad B=\\beta_1\\,r^{-1} + h\\,,\\]\n$\\alpha_1$ and $\\beta_1>0$ being two positive constants.\nSince the minimal value of the right side of inequality\n(\\ref{ineq-basic.h})\nas a function of $\\tau$ is $2(AB)^{1\/2}$ we conclude that\n\\begin{eqnarray*}\n\\phi_h(x,y)&=&\\inf\\set{\\phi(x,y,t)+h\\tau\\mid\\tau>0}\\\\\n&\\leq&\\left(\\alpha\\,r+h\\;\\beta\\,r^2\\right)^{1\/2}\n\\end{eqnarray*}\nfor\n$\\alpha=4\\,\\alpha_1\\beta_1$ and $\\beta=4\\,\\alpha_1$.\nBy continuity,\nwe have that the last inequality also holds for $r=\\norm{x-y}$\nas we wanted to prove.\n\\end{proof}\n\n\\begin{corollary} \\label{comp-visc-subsol}\n\\label{coro-visc.ssol.comp}\nThe set of viscosity subsolutions\n$\\mathcal{S}^0_h=\\set{u\\in\\mathcal{S}_h\\mid u(0)=0}$\nis compact for the topology of the uniform convergence\non compact sets.\n\\end{corollary}\n\n\\begin{proof}\nBy Propositions \\ref{prop-dom.are.visc.ssol} and\n\\ref{prop-visc.ssol.are.dom}\nwe know that $u\\in\\mathcal{S}_h$ if and only if $u\\prec L+h$.\nThus by Theorem \\ref{thm-phih.estim} we have that,\nfor any $u\\in\\mathcal{S}_h$, and for all $x,y\\in E^N$, \n\\[u(x)-u(y)\\leq \\phi_h(x,y)\\leq \\xi(\\norm{x-y})\\]\nwhere $\\xi:[0,+\\infty)\\to\\mathbb{R}^+$ is given by\n$\\xi(\\rho)=\\left(\\alpha\\,\\rho+h\\,\\beta\\,\\rho^2\\right)^{1\/2}$.\n\nSince $\\xi$ is uniformly continuous,\nwe conclude that the family of functions $\\mathcal{S}_h$\nis indeed equicontinuous.\nTherefore,\nthe compactness of $\\mathcal{S}^0_h$ is actually\na consequence of Ascoli's Theorem.\n\\end{proof}\n\n\n\n\\subsection{The Lax-Oleinik semigroup}\n\nWe recall that a solution of $H(x,d_xu)=h$\ncorresponds to a stationary solution\n$U(t,x)=u(x)-ht$ of the evolution equation\n\\[\\partial_tU+H(x,\\partial_xU)=0\\,,\\]\nfor which the Hopf-Lax formula reads\n\\[U(t,x)=\n\\inf\\set{u_0(y)+\\mathcal{A}_L(\\gamma)\\mid\ny\\in E^N,\\,\\gamma\\in\\mathcal{C}(y,x,t)}\\,.\\]\nIn a wide range of situations,\nthis formula provides the \\emph{unique}\nviscosity solution satisfying the initial condition $U(0,x)=u_0(x)$.\nUsing the action potential we can also write the formula as\n\\[U(t,x)=\\inf\\set{u_0(y)+\\phi(y,x,t)\\mid y\\in E^N}.\\]\nIf the initial data $u_0$ is bounded,\nthen $U(t,x)$ is clearly well defined and bounded.\nIn our case, we know that solutions will not be bounded,\nthus we need a condition ensuring that the function\n$y\\mapsto u_0(y)+\\phi(y,x,t)$ is bounded by below.\nAssuming $u_0\\prec L+h$ we have the lower bound\n\\[u_0(x)-ht \\leq u_0(y)+\\phi(y,x,t)\\]\nfor all $t>0$  and all $x\\in E^N$,\nbut this is in fact an equivalent formulation\nfor the domination condition\n$u_0\\prec L+h$, that is to say $u\\in\\mathcal{S}_h$.\n\n\\begin{definition}\n[Lax-Oleinik semigroup] The backward\\footnote{\nThe \\emph{forward} semigroup is defined in a similar way,\nsee \\cite{Eva2}.\nThis other semigroup gives the opposite solutions\nof the reversed Hamiltonian $\\tilde{H}(x,p)=H(x,-p)$.\nIn our case the Hamiltonian is reversible,\nmeaning that $\\tilde{H}=H$.}\nLax-Oleinik semigroup is the map\n$T:[0,+\\infty)\\times \\mathcal{S}_h\\to\\mathcal{S}_h$,\ngiven by $T(t,u)=T_tu$, where\n\\[\nT_tu(x)=\\inf\\set{u(y)+\\phi(y,x,t)\\mid y\\in E^N}\n\\]\nfor $t>0$, and $T_0u=u$.\n\\end{definition}\n\nObserve that $u\\prec L+h$ if and only if\n$u\\leq T_tu+ht$ for all $t>0$.\nAlso,\nwe note that $T_tu-u\\to 0$ as $t\\to 0$, uniformly in $E^N$.\nThis is clear since for all $x\\in E^N$ and $t>0$ we have\n$-ht\\leq T_tu(x)-u(x)\\leq \\phi(x,x,t)\\leq \\mu\\,t^{1\/3}$,\nwhere the last inequality is justified by Remark\n\\ref{rmk-bound.phixxt}.\n\nIt is not difficult to see that $T$ defines an action on $\\mathcal{S}_h$,\nthat is to say, that the semigroup property $T_t\\circ T_s=T_{t+s}$\nis always satisfied.\nThus the continuity at $t=0$ spreads throughout all the domain.\n\nOther important properties of this semigroup are the\n\\emph{monotonicity},\nthat is to say, that $u\\leq v$ implies $T_tu\\leq T_tv$,\nand the \\emph{commutation with constants},\nsaying that for every constant $k\\in\\mathbb{R}$,\nwe have $T_t(u+k)=T_tu+k$.\n\nThus,\nfor $u\\in\\mathcal{S}_h$ and $s,t>0$ we can write\n$T_su\\leq T_s(T_tu+ht)=T_t(T_su)+ht$,\nwhich implies that we have $T_su\\in\\mathcal{S}_h$ for all $s>0$.\n\n\\begin{definition}\n[Lax-Oleinik quotient semigroup]\nThe semigroup $(T_t)_{t\\geq 0}$\ndefines a semigroup $(\\hat{T}_t)_{t\\geq 0}$\non the quotient space $\\hat{\\mathcal{S}}_h=\\mathcal{S}_h\/\\mathbb{R}$,\ngiven by $\\hat{T}_t[u]=[T_tu]$.\n\\end{definition}\n\n\\begin{proposition}\n\\label{prop-LO.fixed.points}\nGiven $h\\geq 0$ and $u\\in\\mathcal{S}_h$ we have that,\n$[u]\\in\\hat{\\mathcal{S}}_h$ is a fixed point of $(\\hat{T}_t)_{t\\geq 0}$\nif and only if\nthere is $h'\\in [0,h]$ such that $T_tu=u-h't$ for all $t\\geq 0$.\n\\end{proposition}\n\n\\begin{proof}\nThe sufficiency of the condition is trivial.\nIt is enough then to prove that it is necessary.\nThat $[u]$ is a fixed point of $\\hat{T}$ means that we have\n$\\hat{T}_t[u]=[u]$ for all $t>0$.\nThat is to say, there is a function $c:\\mathbb{R}^+\\to\\mathbb{R}^+$ such that\n$T_tu=u+c(t)$ for each $t\\in\\mathbb{R}^+$.\nFrom the semigroup property, we can easily deduce that\nthe function $c(t)$ must be additive,\nmeaning that $c(t+s)=c(t)+c(s)$\nfor all $t,s\\geq 0$.\nMoreover,\nthe continuity of the semigroup implies the continuity of $c(t)$.\nAs it is well known,\na continuous and additive function from $\\mathbb{R}^+$ into itself is linear,\ntherefore we must have $c(t)=c(1)t$.\nNow, since $u\\leq T_tu+ht$ for all $t\\in \\mathbb{R}^+$,\nwe get $0\\leq c(1)+h$.\nOn the other hand, since $u\\prec L-c(1)$\nand $\\mathcal{S}_h=\\emptyset$ for $h<0$, hence $-c(1)\\geq 0$.\nWe conclude that $c(t)=-h't$ for some $h'\\in [0,h]$.\n\\end{proof}\n\n\n\\subsubsection{Calibrating curves and supersolutions}\n\nWe finish this section by relating the fixed points\nof the quotient Lax-Oleinik semigroup and\nthe viscosity supersolutions of (\\ref{HJh}).\nThis relationship is closely linked\nto the existence of certain minimizers,\nwhich will ultimately allow us to obtain\nthe hyperbolic motions we seek.\n\n\\begin{definition}\n[calibrating curves]\nLet $u\\in\\mathcal{S}_h$ be a given subsolution.\nWe say that a curve $\\gamma:[a,b]\\to E^N$\nis an \\emph{$h$-calibrating}\ncurve of $u$,\nif $u(\\gamma(b))-u(\\gamma(a))=\\mathcal{A}_{L+h}(\\gamma)$.\n\\end{definition}\n\n\\begin{definition}\n[h-minimizers]\nA curve $\\gamma:[a,b]\\to E^N$\nis said to be an \\emph{$h$-minimizer} if\nit verifies $A_{L+h}(\\gamma)=\\phi_h(\\gamma(a),\\gamma(b))$.\n\\end{definition}\n\n\\begin{remark}\n\\label{rmk-hcalib.hmin}\nAs we have see,\nthe fact that $u\\in\\mathcal{S}_h$ is characterized by $u\\prec L+h$.\nTherefore for all $x,y\\in E^N$ we have\n\\[\nu(x)-u(y)\\leq \\phi_h(x,y)\\leq \\mathcal{A}_{L+h}(\\gamma)\n\\]\nfor any $\\gamma\\in\\mathcal{C}(x,y)$.\nIt follows that every $h$-calibrating curve of $u$\nis an $h$-minimizer.\n\\end{remark}\n\nIt is easy to prove that restrictions of $h$-calibrating curves\nof a given subsolution $u\\in\\mathcal{S}_h$\nare themselves $h$-calibrating curves of $u$.\nThis is also true, and even more easy to see, for $h$-minimizers.\nBut nevertheless,\nthere is a property valid for the calibrating curves\nof a given subsolution but which is not satisfied in general\nby the minimizing curves.\nThe concatenation of two calibrating curves is again calibrating.\n\n\\begin{lemma}\n\\label{lema-concat.calib}\nLet $u\\in\\mathcal{S}_h$.\nIf $\\gamma_1\\in\\mathcal{C}(x,y)$ and $\\gamma_2\\in\\mathcal{C}(y,z)$\nare both $h$-calibrating curves of $u$,\nand $\\gamma\\in\\mathcal{C}(x,z)$ is a concatenation of\n$\\gamma_1$ and $\\gamma_2$,\nthen $\\gamma$ is also an $h$-calibrating curve of $u$.\n\\end{lemma}\n\n\\begin{proof}\nWe have $u(y)-u(x)=\\mathcal{A}_{L+h}(\\gamma_1)$,\nand $u(z)-u(y)=\\mathcal{A}_{L+h}(\\gamma_2)$.\nAdding both equations we get $u(z)-u(x)=\\mathcal{A}_{L+h}(\\gamma)$.\n\\end{proof}\n\nWe give now a criterion for a subsolution to be a viscosity solution.\nFrom here on, a curve defined on a noncompact interval will be said\n$h$-calibrating if all its restrictions to compact intervals are too.\nIn the same way we define $h$-minimizers\nover noncompact intervals.\n\nWe start by proving a lemma on calibrating curves of subsolutions.\n\\begin{lemma}\n\\label{lema-no.test.col}\nLet $u\\in\\mathcal{S}_h$, and let $\\gamma:[a,b]\\to E^N$ be an\n$h$-calibrating curve of $u$.\nIf $x_0=\\gamma(b)$ is a configuration with collisions,\nthen there is no Lipschitz function\n$\\psi$ defined on a neighbourhood of $x_0$\nsuch that $\\psi\\le u$ and $\\psi(x_0)=u(x_0)$.\n\\end{lemma}\n\n\\begin{proof}\nSince our system is autonomous,\nwe can assume without loss of generality that $b=0$.\nThus the $h$-calibrating property of $\\gamma$ says that\nfor every $t\\in [a,0]$\n\\[\n\\int_t^0 \\tfrac{1}{2}\\norm{\\dot\\gamma}^2\\,dt\\,+\n\\int_t^0 U(\\gamma)\\,dt\\,+ \\,h\\abs{t}=\n\\mathcal{A}_{L+h}(\\gamma\\mid_{[t,0]})=u(x_0)-u(\\gamma(t))\\,.\n\\]\nOn the other hand, if $\\psi\\leq u$ is a $k$-Lipschitz function\non a neighbourhood of $x_0$ such that $\\psi(x_0)=u(x_0)$\nthen we also have, for $t$ close enough to $0$,\n\\[\nu(x_0)-u(\\gamma(t))\\leq\n\\psi(x_0)-\\psi(\\gamma(t))\\leq\nk\\norm{\\gamma(t)-x_0}.\n\\]\nTherefore we also have\n\\[\n\\int_{t}^0\\norm{\\dot\\gamma}^2\\,dt\\leq\\,2k\\norm{\\gamma(t)-x_0}.\n\\]\nNow, applying Cauchy-Schwarz we can write\n\\[\n\\int_{t}^0\\norm{\\dot\\gamma}\\,dt\\leq\n\\abs{t}^{1\/2}\\left(\\int_{t}^0\\norm{\\dot\\gamma}^2\\,dt\\right)^{1\/2}\\]\nand thus we deduce that\n\\[\n\\norm{\\gamma(t)-x_0}^2\\leq\n\\left(\\int_{t}^0\\norm{\\dot\\gamma}\\,dt\\right)^2\\leq \n\\,2k\\norm{\\gamma(t)-x_0}\\,\\abs{t}\n\\]\nhence that\n\\[\n\\norm{\\gamma(t)-x_0}\\leq\n\\,2k\\abs{t}.\n\\]\nFinally, since\n\\[\n\\int_{t}^0U(\\gamma)\\,dt\\leq u(x_0)-u(\\gamma(t))\\leq\n\\,k\\norm{\\gamma(t)-x_0}\n\\]\nwe conclude that\n\\[\n\\int_{t}^0U(\\gamma)\\,dt\\leq\n\\,2k^2\\abs{t}.\n\\]\nTherefore, dividing by $\\abs{t}$ and taking the limit for $t\\to 0$\nwe get $U(x_0)\\leq 2k^2$.\nThis proves that $x_0$ has no collisions.\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop-criterion.visc.sol}\nIf $u\\in\\mathcal{S}_h$ is a viscosity subsolution of (\\ref{HJh}), and\nfor each $x\\in E^N$ there is at least one $h$-calibrating curve\n$\\gamma:(-\\delta,0]\\to E^N$  with $\\gamma(0)=x$,\nthen $u$ is in fact a viscosity solution.\n\\end{proposition}\n\n\\begin{proof}\nWe only have to prove that $u$ is a viscosity supersolution.\nThus we assume that $\\psi\\in C^1(E^N)$ and $x_0\\in E^N$\nare such that $u-\\psi$ has a local minimum in $x_0$.\nWe must prove that $H(x_0,d_{x_0}\\psi)\\geq h$.\nFirst of all, we exclude the possibility that $x_0$\nis a configuration with collisions.\nTo do this, it suffices to apply Lemma \\ref{lema-no.test.col},\ntaking the locally Lipschitz function $\\psi+u(x_0)-\\psi(x_0)$.\n\nLet now $\\gamma:(-\\delta,0]\\to E^N$ with $\\gamma(0)=x_0$\nand $h$-calibrating.\nThus for $t\\in(-\\delta,0]$\n\\[\n\\int_t^0L(\\gamma,\\dot\\gamma)\\,dt\\,-\\,ht\n=u(x_0)-u(\\gamma(t))\n\\]\nand also, given that $x_0$ is a local minimum of $u-\\psi$,\nfor $t$ close enough to $0$\n\\[\nu(x_0)-u(\\gamma(t))\\leq\n\\psi(x_0)-\\psi(\\gamma(t))\\,.\n\\]\nSince $x_0\\in\\Omega$ and $\\gamma$ is a minimizer,\nwe know that $\\gamma$ can be extended beyond $t=0$\nas solution of Newton's equation.\nIn particular $v=\\dot\\gamma(0)$ is well defined,\nand moreover, using the previous inequality we find\n\\[\nd_{x_0}\\psi(v)=\n\\;\\lim_{t\\to 0^-}\\,\\frac{\\psi(x_0)-\\psi(\\gamma(t))}{-t}\\geq\nL(x_0,v)+h\n\\]\nwhich implies, by Fenchel's inequality,\nthat $H(x_0,d_{x_0}\\psi)\\geq h$.\n\\end{proof}\n\nThe following proposition complements the previous one.\nIt states that under a stronger condition,\nthe viscosity solution is in addition a fixed point\nof the quotient Lax-Oleinik semigroup.\n\n\\begin{proposition}\n\\label{prop-fixedLO.if.calib}\nLet $u\\in\\mathcal{S}_h$ be a viscosity subsolution of (\\ref{HJh}).\nIf for each $x\\in E^N$ there is\nan $h$-calibrating curve of $u$, say \n$\\gamma_x:(-\\infty,0]\\to E^N$,\nsuch that $\\gamma_x(0)=x$,\nthen $T_tu=u-ht$ for all $t\\geq 0$.\n\\end{proposition}\n\n\\begin{proof}\nFor each $x\\in E^N$, for $t\\geq 0$ we have\n\\[\nT_tu(x)-u(x)=\\inf\\set{u(y)-u(x)+\\phi(y,x,t)\\mid y\\in E^N} \n\\]\nthus it is clear that $T_tu(x)-u(x)\\geq -ht$\nsince we know that $u\\prec L+h$.\nOn the other hand,\ngiven that $\\gamma_x$ is  an $h$-calibrating curve of $u$,\n\\[\nu(x)-u(\\gamma_x(-t))=\\phi(\\gamma_x(-t),x,t)+ht.\n\\]\nWriting $y_t=\\gamma_x(-t)$ we have that\n$u(y_t)-u(x)+\\phi(y_t,x,t)=-ht$\nand we conclude that $T_tu(x)-u(x)\\leq -ht$.\nWe have proved that $T_tu=u-ht$ for all $t\\geq 0$.\n\\end{proof}\n\n\\begin{remark}\n\\label{rmk-inverse.sens.lamin}\nThe formulation of the previous condition can confuse a little,\nsince the calibrating curves are parametrized on negative intervals.\nHere the Lagrangian is symmetric,\nthus reversing the time of a curve always preserves the action.\nMore precisely, \ngiven an absolutely continuous curve $\\gamma:[a,b]\\to E^N$,\nif we define $\\tilde{\\gamma}$ on $[-b,-a]$\nby $\\tilde{\\gamma}(t)=\\gamma(-t)$,\nthen $\\mathcal{A}_L(\\tilde{\\gamma})=\\mathcal{A}_L(\\gamma)$.\n\nWe can reformulate the calibrating condition of the previous\nproposition in this equivalent way:\n\\emph{For each $x\\in E^N$,\nthere is a curve $\\gamma_x:[0,+\\infty)\\to E^N$ such that\n$\\gamma_x(0)=x$, and such that\n$u(x)-u(\\gamma_x(t))=\\mathcal{A}_{L+h}(\\gamma_x\\mid_{[0,t]})$\nfor all $t>0$}.\n\\end{remark}\n\\begin{remark}\nThe hypothesis of\nProposition \\ref{prop-criterion.visc.sol}\nimplies the hypothesis of\nProposition \\ref{prop-fixedLO.if.calib}.\nThis is exactly what we do in the proof of\nTheorem \\ref{thm-horofuns.have.lamin} below.\n\\end{remark}\n\n\n\n\\section{Ideal boundary of a positive energy level}\n\n\nThis section is devoted to the construction of global viscosity\nsolutions for the  Hamilton-Jacobi equations (\\ref{HJh}).\nThe method is quite similar to that developed by Gromov\nin \\cite{Gro} to compactify locally compact metric spaces\n(see also \\cite{BaGrSch}, chpt. 3).\n\n\\subsection{Horofunctions as viscosity solutions}\n\nThe underlying idea giving rise to the construction of horofunctions\nis that each point in a metric space $(X,d)$ can be identified \nwith the distance function to that point.\nMore precisely,\nthe map $X\\to C(X)$ which associates to each point $x\\in X$\nthe function $d_x(y)=d(y,x)$\nis an embedding such that for all $x_0,x_1\\in X$\nwe have $\\max\\abs{d_{x_0}(y)-d_{x_1}(y)}=d(x_0,x_1)$.\n\nIt is clear that\nany sequence of functions $d_{x_n}$ diverges if $x_n\\to\\infty$,\nthat is to say,\nif the sequence $x_n$ escapes from any compact subset of $X$. \nHowever, for a noncompact space $X$,\nthe induced embedding of $X$ into the quotient space\n$C(X)\/\\mathbb{R}$ has in general an image with a non trivial boundary.\nThis boundary can thus be considered\nas an ideal boundary of $X$.\n\nHere the metric space will be $(E^N,\\phi_h)$ with $h>0$,\nand the set of continuous functions $C^0(E^N)$\nwill be endowed with\nthe topology of the uniform convergence on compact sets.\nInstead of looking at equivalence classes of functions,\nwe will take as the representative of each class\nthe only one vanishing at $0\\in E^N$. \n\n\\begin{definition}\n[Ideal boundary]\nWe say that a function $u\\in C^0(E^N)$ is in the ideal boundary\nof level $h$ if there is a sequence of configurations $p_n$,\nwith $\\norm{p_n}\\to +\\infty$ and such that for all $x\\in E^N$\n\\[\nu(x)=\\lim_{n\\to\\infty}\\phi_h(x,p_n)-\\phi_h(0,p_n).\n\\]\nWe will denote $\\mathcal{B}_h$ the set of all these functions,\nthat we will also call horofunctions.\n\\end{definition}\n\nThe first observation is that $\\mathcal{B}_h\\neq\\emptyset$\nfor any value of $h\\geq 0$.\nThis can be seen as a consequence of the estimate for the\npotential $\\phi_h$ we proved,\nsee Theorem \\ref{thm-phih.estim}.\n\nActually for any $p\\in E^N$,\nthe function $x\\to \\phi_h(x,p)-\\phi_h(0,p)$ is in $\\mathcal{S}^0_h$,\nthe set of viscosity subsolutions vanishing at $x=0$.\nSince by Corollary \\ref{coro-visc.ssol.comp}\nwe know that $\\mathcal{S}^0_h$ is compact,\nfor any sequence of configurations $p_n$\nsuch that $\\norm{p_n}\\to +\\infty$\nthere is a subsequence\nwhich defines a function in $\\mathcal{B}_h$ as above.\n\nIt is also clear that $\\mathcal{B}_h\\subset\\mathcal{S}_h$.\nFunctions in $\\mathcal{B}_h$ are limits of functions in $\\mathcal{S}_h$,\nand this set is closed in $E^N$\neven for the topology of pointwise convergence.\nBut, since we already know that\nthe family $\\mathcal{S}_h$ is equicontinuous,\nthe convergence is indeed uniform on compact sets.\n\n\\begin{notation}\nWhen the value of $h$ is understood,\nwe will denote $u_p$ the function\ndefined by $u_p(x)=\\phi_h(x,p)$\nwhere $p$ is a given configuration.\n\\end{notation}\n\nOne fact that should be clarifying is that for any $p\\in E^N$,\nthe subsolution given by $u_p$ fails to be a viscosity solution\nprecisely at $x=p$.\nIf $x\\neq p$, then there is a minimizing curve of $\\mathcal{A}_{L+h}$\nin $\\mathcal{C}(p,x)$\n(see Lemma \\ref{lema-JM.geod.complet} below),\nand clearly this curve is $h$-calibrating of $u_p$.\nOn the other hand, there are no $h$-calibrating curves of $u_p$\ndefined over an interval $(-\\delta,0]$\nand ending at $x=p$.\nThis is because $u_p\\geq 0$, $u_p(p)=0$,\nand $h$-calibrating curves, as $h$-minimizers,\nhave strictly increasing action.\nActually, this property of the $u_p$ functions occurs\nfor all energy levels greater than or equal to the critical one,\nin a wide class of Lagrangian systems.\nThe simplest case to visualize is surely\nthe case of absence of potential energy in an Euclidean space,\nin which we have $u_p(x)=h\\,\\norm{x-p}$ and\nhis $h$-calibrating curves are segments of the half-lines\nemanating from $p$ with a constant speed (gradient curves).\n\nThis suggest that the horofunctions must be viscosity solutions,\nwhich is what we will prove now.\n\n\\begin{theorem}\n\\label{thm-horofuns.are.viscsol}\nGiven $u\\in\\mathcal{B}_h$ and $r>0$ there is,\nfor each $x\\in E^N$, some $y\\in E^N$ with $\\norm{y-x}=r$,\nand a curve $\\gamma_x\\in\\mathcal{C}(y,x)$ such that\n$u(x)-u(y)=\\mathcal{A}_{L+h}(\\gamma_x)$.\nIn particular, every function $u\\in\\mathcal{B}_h$\nis a global viscosity solution of (\\ref{HJh}).\n\\end{theorem}\n\n\\begin{proof}\nLet $u\\in\\mathcal{B}_h$, that is to say\n$u=\\lim_n(u_{p_n}-u_{p_n}(0))$\nfor some sequence of configurations $p_n$ such that\n$\\norm{p_n}\\to +\\infty$, and $u_{p_n}(x)=\\phi_h(x,p_n)$.\n\nLet $x\\in E^N$ be any configuration, and fix $r>0$.\nUsing Lemma \\ref{lema-JM.geod.complet} we get,\nfor each $n>0$, a curve $\\gamma_n\\in\\mathcal{C}(p_n,x)$\nsuch that $\\mathcal{A}_{L+h}(\\gamma_n)=\\phi_h(p_n,x)$.\nEach curve $\\gamma_n$ is thus\nan $h$-calibrating curve of $u_{p_n}$.\n\nIf $\\norm{p_n-x}>r$,\nthen the curve $\\gamma_n$ must pass through a configuration\n$y_n$ with $\\norm{y_n-x}=r$.\nExtracting a subsequence if necessary,\nwe may assume that this is the case for all $n>0$,\nand that $y_n\\to y$, with $\\norm{y-x}=r$.\nSince the arc of $\\gamma_n$ joining\n$y_n$ to $x$ also $h$-calibrates $u_{p_n}$ we can write\n\\[\nu_{p_n}(x)-u_{p_n}(y_n)=\\phi_h(y_n,x)\n\\]\nfor all $n$ big enough. We conclude that\n\\[\nu(x)-u(y)=\\lim_{n\\to\\infty}u_{p_n}(x)-u_{p_n}(y)=\\phi_h(y,x)\n\\]\nwhich proves the first statement.\nThe second one follows now from the criterion for\nviscosity solutions given in Proposition\n\\ref{prop-criterion.visc.sol}.\t\n\\end{proof}\nOur next goal is to prove that horofunctions are actually\nfixed points of the quotient Lax-Oleinik semigroup.\nWe will achieve this goal by showing the existence\nof calibrating curves allowing the use of\nProposition \\ref{prop-fixedLO.if.calib}.\nThese calibrating curves will be the key to the proof of\nthe existence of hyperbolic motions.\n\nThanks to the previous theorem\nwe can build maximal calibrating curves.\nThen, Marchal's Theorem will allow us to assert that\nthese curves are in fact true motions of the $N$-body problem.\nNext we have to prove\nthat these motions\nare defined over unbounded above time intervals,\nthat is to say,\nwe must exclude the possibility of\ncollisions or pseudocollisions.\nIt is for this reason that we will also invoke\nthe famous von Zeipel's theorem\\footnote{\nThis theorem had no major impact on the theory\nuntil it was rediscovered after at least half a century later,\nand proved to be essential for the understanding\nof pseudocollision singularities, see for instance Chenciner's\nBourbaki seminar \\cite{Che2}.\nAmong other proofs, there is a modern version due to McGehee \n \\cite{McG} of the proof originally outlined by von Zeipel.}\n that we recall now.\n\n\\begin{theorem*}\n[1908, von Zeipel \\cite{Zei}]\nLet $x:(a,t^*)\\to E^N$ be a maximal solution of the\nNewton's equations of the $N$-body problem with $t^*<+\\infty$.\nIf $\\norm{x(t)}$ is bounded in some neighbourhood of $t^*$, then\nthe limit $\\,x_c=\\lim_{t\\to t^*}x(t)$\nexists and the singularity is therefore due to collisions. \n\\end{theorem*}\n\n\n\\begin{theorem}\n\\label{thm-horofuns.have.lamin}\nIf $u\\in\\mathcal{B}_h$ then for each $x\\in E^N$ there is a curve\n$\\gamma_x:[0,+\\infty)\\to E^N$ with $\\gamma_x(0)=x$,\nand such that for all $t>0$\n\\[\nu(x)-u(\\gamma_x(t))=\\mathcal{A}_{L+h}(\\gamma_x\\mid_{[0,t]}).\n\\]\nIn particular,\nevery function $u\\in\\mathcal{B}_h$ satisfies $T_tu=u-ht$ for all $t>0$.\n\\end{theorem}\n\n\\begin{proof}\nLet us fix a configuration $x\\in E^N$.\nBy Theorem \\ref{thm-horofuns.are.viscsol} we know that\n$u$ has at least one $h$-calibrating curve\n$\\gamma:(-\\delta,0]\\to E^N$ such that $\\gamma(0)=x$.\nBy application of Zorn's Lemma\nwe get a maximal $h$-calibrating curve\nof the form $\\gamma:(t^*,0]\\to E^N$ with $\\gamma(0)=x$.\nWe will prove that $t^*=-\\infty$,\nand thus the required curve can be defined on $[0,+\\infty)$ by\n$\\gamma_x(t)=\\gamma(-t)$.\n\nSuppose by contradiction that $t^*>-\\infty$.\nSince $\\gamma$ is an $h$-minimizing curve, we know that its\nrestriction to $(t^*,0)$ is a true motion with energy constant $h$.\nEither the curve can be extended as a motion for values\nless than $t^*$, or it presents a singularity at $t=t^*$.\nIn the case of singularity, we have at $t=t^*$ either a collision,\nor a pseudocollision.\nAccording to von Zeipel's Theorem,\nin the pseudocollision case we must have\n$\\sup\\set{\\norm{\\gamma(t)}\\mid t\\in (t^*,0]}=+\\infty$.\n\nSuppose that the limit $y=\\lim_{t\\to t^*}\\gamma(t)$ exists.\nThen by Theorem \\ref{thm-horofuns.are.viscsol} we can choose\na calibrating curve $\\tilde{\\gamma}$ defined on $(-\\delta,0]$\nand such that $\\tilde{\\gamma}(0)=y$.\nThus the concatenation of $\\tilde{\\gamma}$ with $\\gamma$\ndefines a calibrating curve $\\gamma^+$ defined on\n$(t^*-\\delta,0]$ and such that $\\gamma^+(0)=x$.\nBut this contradicts the maximality of $\\gamma$.\n\nOn the other hand, if we suppose that $\\norm{\\gamma(t)}$\nis unbounded, we can choose a sequence $y_n=\\gamma(t_n)$\nsuch that $\\norm{y_n-x}\\to +\\infty$.\nLet us define $A_n=\\mathcal{A}_L(\\gamma\\mid_{[t_n,0]})$.\n\nA standard way to obtain a lower bound for $A_n$\nis by neglecting the potential term which is positive.\nThen by using  the Cauchy-Schwarz inequality we obtain that\nfor all $n>0$ we have $2\\abs{t_n}A_n\\geq \\norm{y_n-n}^2$.\nSince $\\gamma$ is $h$-minimizing we deduce that\n\\[\n\\phi_h(y_n,x)\\geq\n\\frac{\\norm{y_n-x}^2}{2\\abs{t_n}}+h\\abs{t_n}\n\\]\nfor all $n>0$.\nSince $\\norm{y_n-x}\\to +\\infty$ and $t_n\\to t^*>-\\infty$\nwe get a contradiction with the upper estimate given by Theorem\n\\ref{thm-phih.estim}.\nIndeed that theorem implies that\n$\\phi_h (y_n,x)$ is bounded above by a function which is\nof order $O(\\norm{y_n - x})$ as $\\norm{y_n-x} \\to +\\infty$,  \nwhich contradicts the displayed inequality.\n\nThe last assertion is a consequence of\nProposition \\ref{prop-fixedLO.if.calib} and\nRemark \\ref{rmk-inverse.sens.lamin}.\n\\end{proof}\n\n\n\n\\subsection{Busemann functions}\n\\label{s-busemann}\n\nWe recall that a length space $(X,d)$ is say to be a\n\\emph{geodesic space}\nif the distance between any two points is realized as the length\nof a curve joining them.\nA \\emph{ray} in $X$ is an isometric embedding\n$\\gamma:[0,+\\infty)\\to X$.\nAs we already say in Sect. \\ref{s-geom.view},\nthe Gromov boundary of a geodesic space is defined as\nthe quotient space of the set of rays of $X$ under the\nequivalence relation: $\\gamma\\sim\\gamma'$ if and only if the\nfunction given by $d(\\gamma(t),\\gamma'(t))$ on $[0,+\\infty)$\nis bounded.\n\nThere is a natural way to associate a horofunction to each ray.\nLet us write $d_p$ for the function measuring the distance to\nthe point $p\\in X$ that is, $d_p(x)=d(x,p)$.\nOnce $\\gamma$ is fixed, we define\n\\[\nu_t(x)=d_{\\gamma(t)}(x)-d_{\\gamma(t)}(\\gamma(0))\\,,\n\\quad\\text{ and }\\quad\nu_\\gamma=\\lim_{t\\to\\infty}u_t\\,.\n\\]\n\nThese horofunctions $u_\\gamma$ are called\n\\emph{Busemann functions}\nand are well defined because of\nthe geodesic characteristic property of rays.\nMore precisely, for any ray $\\gamma$ and for all $0\\leq s\\leq t$,\nwe have $d(\\gamma(t),\\gamma(s))=t-s$,\nhence that $u_t\\leq u_s$.\nMoreover, \nit is also clear that $u_t\\geq -d_{\\gamma(0)}$,\nwhich implies that $u_\\gamma\\ge -d_{\\gamma(0)}$.\nWe also note that $u_\\gamma=\\lim_n u_{t_n}$ whenever\n$(t_n)_{n>0}$ is a sequence such that $t_n\\to\\infty$.\n\nIt is well known that under some hypothesis on $X$ we have that,\nfor any two equivalent rays $\\gamma\\sim\\gamma'$,\nthe corresponding Busemann functions are the same\nup to a constant, that is $[u_\\gamma]=[u_{\\gamma'}]$.\nTherefore in these cases a map is defined\nfrom the Gromov boundary into the ideal boundary,\nand it is thus natural to ask about the injectivity and the surjectivity\nof this map.\nHowever, the following simple and enlightening example\nshows a geodesic space in which there are equivalent rays\n$\\gamma\\sim\\gamma'$ for which\n$[u_\\gamma]\\neq [u_{\\gamma'}]$.\n\n\\begin{example}\n[The infinite ladder]\nWe define $X\\subset\\mathbb{R}^2$ as the union of the two straight lines\n$\\mathbb{R}\\times\\set{-1,1}$ with the segments $\\mathbb{Z}\\times [-1,1]$,\nsee figure \\ref{ladder}. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=1]{ladder.pdf}\n\\caption{The infinite ladder.} \n\\label{ladder}\n\\end{figure}\n\nWe endow $X$ with the length distance induced by the standard\nmetric in $\\mathbb{R}^2$.\nIt is not difficult to see that every ray in $X$\nis eventually of the form\n$x(t)=(\\pm t+c, \\pm 1)$.\nEach possibility for the two signs determines one of the four\ndifferent Busemann functions which indeed compose the\nideal boundary.\nTherefore,\n there are four points in the ideal boundary of $X$,\nwhile there is only two classes of rays\ncomposing the Gromov boundary of $X$.\n\\end{example}\n\nLet us return to the context of the $N$-body problem,\nthat is to say,\nlet us take as metric space the set of configurations $E^N$,\nwith the action potential $\\phi_h$ as the distance function.\nActually $(E^N,\\phi_h)$ becomes a length space, and $\\phi_h$\ncoincides with the length distance of the Jacobi-Maupertuis\nmetric when restricted to $\\Omega$.\nProofs of all these facts are given in Sect. \\ref{s-jm.dist}.\nWe are interested in the study of the ideal and Gromov\nboundaries of this space,\nin particular we need to understand the rays in\nthis space having prescribed asymptotic direction.\nAs we will see, they will be found as calibrating curves\nof horofunctions in a special class.\n\n\\begin{definition}\n[Directed horofunctions]\nGiven a configuration $a\\neq 0$ we define the set\nof horofunctions directed by $a$ as the set\n\\[\n\\mathcal{B}_h(a)=\n\\set{u\\in\\mathcal{B}_h\\mid\nu=\\lim_n(u_{p_n}-u_{p_n}(0))\\,,\\;\np_n=\\lambda_na+o(\\lambda_n),\\;\n\\lambda_n\\to +\\infty}.\n\\]\n\\end{definition}\n\\begin{remark}\nTheorem \\ref{thm-phih.estim} implies,\nin a manner identical to the proof of\nCorollary \\ref{comp-visc-subsol},\nthat $\\mathcal{B}_h(a)\\neq\\emptyset$.\n\\end{remark}\n  \n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=1.1]{HyperKepler2.pdf}\n\\caption{Calibrating curves of a hyperbolic Busemann function\n$u(x)=\\lim_n (\\phi_h(x,na)-\\phi_h(0,na))$ in the Kepler problem.} \n\\label{HyperKepler2}\n\\end{figure}\n\nThe following theorem is the key\nfor the proof of Theorem \\ref{thm-princ}\nand its proof is given in Sect. \\ref{s-proofs.main}.\n\n\\begin{theorem}\n\\label{thm-calib.direct.horo}\nLet $a\\in\\Omega$ and $u\\in\\mathcal{B}_h(a)$.\nIf $\\gamma:[0,+\\infty)\\to E^N$ satisfies\n\\[\nu(\\gamma(0))-u(\\gamma(t))=\\mathcal{A}_{L+h}(\\gamma\\mid_{[0,t]})\n\\]\nfor all $t>0$,\nthen $\\gamma$ is a hyperbolic motion of energy $h$\nwith asymptotic direction $a$.\n\\end{theorem}\n\nWe can thus deduce the following corollary,\nwhose proof is a very easy application of the Chazy's Theorem on\nhyperbolic motions, see Remark \\ref{rmk-Chazy.implic}.\n\\begin{corollary}\n\\label{coro-calib.bus.equiv}\nIf $a\\in\\Omega$ and $u\\in \\mathcal{B}_h(a)$ then\nthe distance between any two $h$-calibrating curves for $u$\nis bounded on their common domain.\n\\end{corollary}\n\nWe can also apply Theorem \\ref{thm-calib.direct.horo} to deduce\nthat calibrating curves of a hyperbolic Busemann function are\nmutually asymptotic hyperbolic motions.\n\\begin{corollary}\n\\label{coro-calib.of.busemann}\nIf $\\gamma$ is an hyperbolic $h$-minimizer,\nand $u_\\gamma$ its associated Busemann function,\nthen all the calibrating curves of $u_\\gamma$ are hyperbolic\nmotions with the same limit shape and direction as $\\gamma$.\n\\end{corollary}\n\n\\begin{proof}\nSince $\\gamma$ is hyperbolic, we know that\nthere is a configuration without collisions $a\\in\\Omega$\nsuch that $\\gamma(t)=ta+o(t)$ as $t\\to +\\infty$.\nTaking the sequence \n$p_n=\\gamma(n)$ we have that $p_n=na+o(n)$,\nand also that\n\\[\nu_\\gamma-u_\\gamma(0)=\n\\lim_{n\\to +\\infty}[u_{p_n}-u_{p_n}(0)].\n\\]\nThis implies that $u_\\gamma-u_\\gamma(0)\\in\\mathcal{B}_h(a)$,\nhence that $u_\\gamma$ is a viscosity solution and\nmoreover, Theorem \\ref{thm-calib.direct.horo} says that\nthe calibrating curves of $u_\\gamma$ all of the form $ta+o(t)$.\nOn the other hand, clearly $\\gamma$ calibrates $u_\\gamma$\nsince for any $0\\leq s \\leq t$ we have that\n\\[\nu_{\\gamma(t)}(\\gamma(s))-u_{\\gamma(t)}(\\gamma(0))=\n-\\phi_h(\\gamma(0),\\gamma(s)),\n\\]\nwhich in turn implies, taking the limit for $t\\to +\\infty$, that\n\\[\nu_\\gamma(\\gamma(0))-u_\\gamma(\\gamma(s))=\n-u_\\gamma(\\gamma(s))=\n\\phi_h(\\gamma(0),\\gamma(s)).\\]\n\\end{proof}\n\n\n\n\\section{Proof of the main results on hyperbolic motions}\n\n\n\nThis part of the paper contains the proofs that so far it has been\npostponed for different reasons.\nIn the first part we deal with several lemmas and technical results,\nafter which we complete the proof of the main results in Sect.\n\\ref{s-proofs.main}.\n\n\n\\subsection{Chazy's Lemma}\n\\label{s-Chazy.lema}\n\nThe first lemma that we will prove states that the set\n$\\mathcal{H}^+\\subset T\\Omega$ of initial conditions in the\nphase space given rise to hyperbolic motions is an open set.\nMoreover,\nit also says that the map defined in this set\nwhich gives the asymptotic velocity in the future\nis continuous. \nThis is precisely what in Chazy's work appears as\n\\emph{continuit\u00e9 de l'instabilit\u00e9}.\nWe give a slightly more general version for homogeneous\n potentials of degree $-1$,\nbut the proof works the same for potentials\nof negative degree in any Banach space.\n\nIntuitively what happens is that,\nif an orbit is sufficiently close to some given hyperbolic motion,\nthen after some time the bodies will be so far away each other,\nthat the action of the gravitational forces\nwill not be able to perturb their velocities too much.\n\n\\begin{lemma}\n\\label{lema-cont.limitshape}\nLet $U:E^N\\to\\mathbb{R}\\cup\\set{+\\infty}$ be a homogeneous potential\nof degree $-1$ of class $C^2$ on the open set\n$\\Omega=\\set{x\\in E^N\\mid U(x)<+\\infty}$.\nLet $x:[0,+\\infty)\\to\\Omega$ be a given solution of\n$\\ddot x=\\nabla U(x)$ satisfying $x(t)=ta+o(t)$\nas $t\\to +\\infty$ with $a\\in\\Omega$.\n\nThen we have:\n\\begin{enumerate}\n\\item[(1)] The solution $x$ has asymptotic velocity $a$,\nmeaning that\n\\[\n\\lim_{t\\to+\\infty}\\dot x(t)=a\\,.\n\\]\n\\item[(2)] (Chazy's continuity of the limit shape)\nGiven $\\epsilon>0$,\nthere are constants $t_1>0$ and $\\delta>0$ such that,\nfor any maximal solution $y:[0,T)\\to\\Omega$ satisfying\n$\\norm{y(0)-x(0)}<\\delta$ and\n$\\norm{\\dot y(0)-\\dot x(0)}<\\delta$,\nwe have:\n\\begin{enumerate}\n\\item[(i)] $T=+\\infty$, $\\norm{y(t)-ta}<t\\epsilon$ for all $t>t_1$,\nand moreover\n\\item[(ii)] there is $b\\in\\Omega$ with $\\norm{b-a}<\\epsilon$ \nfor which $y(t)=tb+o(t)$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $0<\\rho<\\epsilon$ such that the closed ball\n$B=\\overline B(a,\\rho)$ is contained in $\\Omega$.\nLet $k=\\max \\set{\\norm{\\nabla U(z)}\\mid z\\in B}$, and choose\n$t_0>0$ in such a way that for any $t\\geq t_0$ we have\n$\\norm{x(t)-ta}<t\\rho$. Therefore, since $\\nabla U$ is\nhomogeneous of degree $-2$, for each $t\\geq t_0$ we have\n$t^{-1}x(t)\\in B$ and\n\\[\n\\norm{\\nabla U(x(t))} \\leq\nt^{-2}\\norm{\\nabla U(t^{-1}x(t))} \\leq\nkt^{-2}\\,.\n\\]\nThus, for $t_0<t_1<t_2$ we can write\n\\[\n\\norm{\\dot x(t_2)-\\dot x(t_1)} \\leq\n\\int_{t_1}^{t_2}\\norm{\\nabla U(x(s))}ds \\leq\n\\int_{t_1}^{+\\infty}ks^{-2}\\,ds =\n\\frac{k}{t_1}\n\\]\nfrom which we deduce that $\\dot x(t)$ has a limit for $t\\to +\\infty$.\nThis limit can not be other than $\\lim \\dot x= a$,\nsince otherwise we would have that the\nderivative of $x(t)-ta$ has a non null limit contradicting\nthe hypothesis $x(t)-ta=o(t)$.\n\nWriting $x_1=x(t_1)$ and $\\dot x_1=\\dot x(t_1)$, we see that\nwe can fix $t_1>t_0$ large enough such that\n\\[\n\\norm{x_1-t_1a}<t_1\\,\\frac{\\rho}{3}\\,.\n\\]\nIf in addition $t_1>3k\/\\rho$ we also have\n\\[\n\\norm{\\dot x_1-a}\\leq \\frac{k}{t_1}<\\frac{\\rho}{3}\\,.\n\\]\n\nOn the other hand, since the vector field $X(x,v)=(v,\\nabla U(x))$\nis of class $C^1$, it defines a local flow on $T\\Omega$.\nLet us denote by $(x_0,\\dot x_0)$ the initial condition\n$(x(0),\\dot x(0))$ of $x(t)$.\nWe can choose $\\delta>0$ such that, for any choice of\n$(y_0,\\dot y_0)\\in T\\Omega$ verifying\n$\\norm{y_0-x_0}<\\delta$ and $\\norm{\\dot y_0-\\dot x_0}<\\delta$,\nthe maximal solution $y:[0,T)\\to\\Omega$\nwith $y(0)=y_0$  and $\\dot y(0)=\\dot y_0$\nsatisfies the following two conditions: $T>t_1$, and\n\\[\n\\norm{y_1-t_1a}<t_1\\,\\frac{\\rho}{3},\n\\;\\textrm{ and }\\;\n\\norm{\\dot y_1-a}<\\frac{\\rho}{3}\n\\]\nwhere $y_1=y(t_1)$ and $\\dot y_1=\\dot y(t_1)$.\n\nNow, assume that $t\\in[t_1,T)$ is such that $y(s)\\in sB$\nfor all $s\\in[t_1,t]$.\nAs before we have $\\norm{\\dot y(s)-\\dot y_1}\\leq k\/t_1<\\rho\/3$,\nand thus $\\norm{\\dot y(s)-a}<2\\rho\/3$.\nTherefore we can deduce that\n\\begin{eqnarray*}\n\\norm{y(t)-ta}&\\leq& \\norm{y_1-t_1a}\n                              +\\int_{t_1}^t\\norm{\\dot y(s)-a}\\,ds\\\\\n &<& t_1\\,\\frac{\\rho}{3} +(t-t_1)\\frac{2\\rho}{3}<t\\rho\n\\end{eqnarray*}\nSince the last inequality is strict, in fact we have proved that\n$y(s)\\in s\\interior{B}$ for all $s\\in [t_1,t]$,\nwhere $\\interior{B}$ denotes the open ball $B(a,\\rho)$.\nThus, the set of $t\\in[t_1,T)$ such that $y(s)\\in sB$\nfor all $s\\in[t_1,t]$ is an open subset, and easily we conclude\nthat we must have $y(t)\\in t\\interior{B}$ for all $t\\in[t_1,T)$.\n\n$T=+\\infty$.\nOtherwise $K=\\cup_{t\\in[0,T]}\\,tB$ would be compact and\n$(y(t),\\dot y(t))\\in K\\times B$ for all $t\\in[t_1,T)$,\nwhich is impossible for a maximal solution.\nBy the same argument used for the motion $x$,\nwe have that $\\dot y(t)$ has a limit $b\\in B$.\nIn particular $\\norm{b-a}<\\epsilon$ and $y(t)=tb+o(t)$.\n\\end{proof}\n\n\n\n\\subsection{Existence and properties of h-minimizers}\n\nThe following lemma ensures that for $h>0$,\nthe length space $(E^N,\\phi_h)$ is indeed geodesically convex.\n\nActually the lemma give us minimizing curves\nfor any pair of configurations, even with collisions,\nand it follows from Marchal's Theorem that\nsuch curves avoid collisions at intermediary times.\nThe proof is a well-known argument based on the\nTonelli's Theorem for convex Lagrangians,\ncombined with Fatou's Lemma\nfor dealing with the singularities of the potential.\n\n\\begin{lemma}[Existence of minimizers for $\\phi_h$]\n\\label{lema-JM.geod.complet}\nGiven $h>0$ and $x\\neq y\\in E^N$ there is a curve\n$\\gamma\\in\\mathcal{C}(x,y)$  such that\n$\\mathcal{A}_{L+h}(\\gamma)=\\phi_h(x,y)$.\n\\end{lemma}\n\nWe need to introduce before some notation\nand make a simple remark that we will use several times.\nIt is worth noting that the remark applies whenever we consider\na system defined by a potential $U>0$.\n\n\\begin{notation}\nGiven $h\\geq 0$, for $x,y\\in E^N$ and $\\tau>0$ we will write\n\\[\n\\Phi_{x,y}(\\tau)=\\tfrac{1}{2}\\norm{x-y}^2\\tau^{-1}+h\\,\\tau\\,.\n\\]\n\\end{notation}\n\n\\begin{remark}\n\\label{rmk-usual.bound}\nGiven $h\\geq 0$ we have, for any pair of configurations\n$x,y\\in E^N$ and any $\\tau>0$\n\\[\n\\phi(x,y,\\tau)+ h \\tau \\;\\geq\\; \\Phi_{x,y}(\\tau).\n\\]\nIndeed, given any pair of configurations\n$x,y\\in E^N$ and for any $\\sigma\\in\\mathcal{C}(x,y,\\tau)$,\nthe Cauchy-Schwarz inequality implies\n\\[\n\\norm{x-y}^2\\leq\n(\\;\\int_a^b\\norm{\\dot \\sigma}\\,dt\\;)^2\\leq\n\\;\\tau\\,\\int_a^b\\norm{\\dot \\sigma}^2\\,dt\\,,\n\\]\nthus, since $U>0$,\n\\[\n\\mathcal{A}_L(\\sigma)>\n\\tfrac{1}{2}\\int_a^b\\norm{\\dot \\sigma}^2\\,dt \\geq\n\\tfrac{1}{2}\\,\\norm{x-y}^2\\tau^{-1}.\n\\]\nThis justifies the assertion,\nsince this lower bound does not depend on the curve $\\sigma$.\n\\end{remark}\n\n\n\\begin{proof}\n[Proof of Lemma \\ref{lema-JM.geod.complet}]\nLet $x,y\\in E^N$ be two given configurations, with $x\\neq y$.\nWe start by taking a minimizing sequence of\n$\\mathcal{A}_{L+h}$ in $\\mathcal{C}(x,y)$, that is to say,\na sequence of curves $(\\sigma_n)_{n> 0}$ such that\n\\[\n\\lim_{n\\to\\infty}\\mathcal{A}_{L+h}(\\sigma_n)=\\phi_h(x,y)\\,.\n\\]\nThen from this minimizing sequence we build a new one,\nbut this time composed by curves with the same domain.\nTo do this, we first observe that,\nif each $\\sigma_n$ is defined on an interval $[0,\\tau_n]$,\nthen by the previous remark we know that\n\\[\n\\mathcal{A}_{L+h}(\\sigma_n)\\geq\n\\phi(x,y,\\tau_n)+h\\tau_n \\geq\n\\Phi_{x,y}(\\tau_n)\n\\]\nwhere $\\Phi_{x,y}$ is the above defined function.\nSince clearly $\\Phi_{x,y}$ is a proper function on $\\mathbb{R}^+$,\nwe deduce that\n$0<\\liminf \\tau_n\\leq \\limsup \\tau_n<+\\infty$,\nand therefore we can suppose without loss of generality\nthat $\\tau_n\\to\\tau_0$ as $n\\to\\infty$.\nIt is not difficult to see that reparametrizing linearly\neach curve $\\sigma_n$ over the\ninterval $[0,\\tau_0]$ we get a new minimizing sequence.\nMore precisely, for each $n>0$ the reparametrization is\nthe curve\n$\\gamma_n:[0,\\tau_0]\\to E^N$ defined by\n$\\gamma_n(t)=\\sigma_n(\\tau_n\\tau_0^{-1}\\, t)$.\nComputing the action of the curves $\\gamma_n$ we get\n\\[\n\\int_0^{\\tau_0}\\tfrac{1}{2}\\norm{\\dot\\gamma_n}^2\\,dt=\n\\tau_n\\tau_0^{-1}\n\\int_0^{\\tau_n}\\tfrac{1}{2}\\norm{\\dot\\sigma_n}^2\\,dt\n\\]\nand\n\\[\n\\int_0^{\\tau_0} U(\\gamma)\\,dt=\n\\tau_0\\tau_n^{-1}\n\\int_0^{\\tau_n} U(\\sigma)\\,dt\n\\]\nthus we have that\n\\[\n\\lim_{n\\to\\infty}\\mathcal{A}_{L+h}(\\gamma)=\n\\lim_{n\\to\\infty}\\mathcal{A}_{L+h}(\\sigma)=\n\\phi_h(x,y).\n\\]\n\nOn the other hand, It is easy to see that a uniform bound\nfor the action of the family of curves $\\gamma_n$\nimplies the equicontinuity of the family.\nMore precisely,\nif the bound $\\mathcal{A}_L(\\gamma_n)\\leq \\tfrac{1}{2}\\,M^2$\nholds for all $n>0$,\nthen using Cauchy-Schwarz inequality as in Remark\n\\ref{rmk-usual.bound} we have\n\\[\n\\norm{\\gamma_n(t)-\\gamma_n(s)}\\leq\nM\\abs{t-s}^\\frac{1}{2}\n\\]\nfor all $t,s\\in [0,t_0]$ and for all $n>0$.\nThus by Ascoli's Theorem we can assume that\nthe sequence $(\\gamma_n)$ converges uniformly to\na curve $\\gamma\\in\\mathcal{C}(x,y,\\tau_0)$.\nFinally, we apply Tonelli's Theorem for convex Lagrangians to get\n\\[\n\\int_0^{\\tau_0}\\tfrac{1}{2}\\norm{\\dot\\gamma}^2\\,dt\\;\\leq\\;\n\\liminf_{n\\to\\infty}\n\\int_0^{\\tau_0}\\tfrac{1}{2}\\norm{\\dot\\gamma_n}^2\\,dt\n\\]\nand Fatou's Lemma to obtain that\n\\[\n\\int_0^{\\tau_0} U(\\gamma)\\,dt\\;\\leq\\;\n\\liminf_{n\\to\\infty}\n\\int_0^{\\tau_0} U(\\gamma_n)\\,dt.\n\\]\nTherefore $\\mathcal{A}_L(\\gamma)\\leq \\phi(x,y,\\tau_0)$,\nwhich is only possible if the equality holds,\nand thus we deduce that $\\mathcal{A}_{L+h}(\\gamma)=\\phi_h(x,y)$.\n\\end{proof}\n\nThe next lemma is quite elementary and provides a\nrough lower bound for $\\phi_h$.\nHowever it has an interesting consequence, namely that\nreparametrizations of the $h$-minimizers by arc length\nof the metric $\\phi_h$ are Lipschitz\nwith the same Lipschitz constant.\nWe point out that this lower bound only depends\non the positivity of the Newtonian potential.\n\n\\begin{lemma}\\label{lema-geod.are.lipschitz}\nLet $h>0$.\nFor any pair of configurations $x,y\\in E^N$ we have\n\\[\n\\phi_h(x,y)\\geq \\sqrt{2h}\\norm{x-y}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nWe note that\n\\[\n\\phi_h(x,y)=\n\\min\\set{\\phi(x,y,\\tau)+\\tau h\\mid \\tau>0}\\geq\n\\min\\set{\\Phi_{x,y}(\\tau)\\mid\\tau>0},\n\\]\nand that\n\\[\n\\min\\set{\\Phi_{x,y}(\\tau)\\mid \\tau>0}=\n\\sqrt{2h}\\norm{x-y}.\n\\]\n\\end{proof}\n\n\\begin{remark}\n\\label{rmk-reparam.are.Lip}\nIf $\\gamma(s)$ is a reparametrization of an $h$-minimizer\nand the parameter is the arc length for the metric $\\phi_h$, \nthen we have\n\\[\n\\sqrt{2h}\\,\\norm{\\gamma(s_2)-\\gamma(s_1)}\\leq\n\\phi_h(\\gamma(s_1),\\gamma(s_2))=\\abs{s_2-s_1}.\n\\]\nTherefore all these reparametrizations are Lipschitz\nwith Lipschitz constant $1\/\\sqrt{2h}$. \n\\end{remark}\n\nFinally, the following and last lemma will be used to estimate\nthe time needed by an $h$-minimizer to join two given\nconfigurations.\n\n\\begin{lemma}\n\\label{lema-time.estim}\nLet $h>0$, $x,y\\in E^N$ two given configurations,\nand let $\\sigma\\in\\mathcal{C}(x,y,\\tau)$ be an $h$-minimizer.\nThen we have\n\\[\n\\tau_-(x,y)\\leq \\tau \\leq \\tau_+(x,y)\n\\]\nwhere $\\tau_-(x,y)$ and $\\tau_+(x,y)$\nare the roots of the polynomial\n\\[\nP(\\tau)=2h\\,\\tau^2-2\\phi_h(x,y)\\,\\tau+\\norm{x-y}^2.\n\\] \n\\end{lemma}\n\n\\begin{proof}\nSince $\\sigma$ minimizes $A_{L+h}$,\nin view of Remark \\ref{rmk-usual.bound} we have\n\\[\n\\phi_h(x,y)=\n\\phi(x,y,\\tau)+\\tau h\\geq\n\\Phi_{x,y}(\\tau)\n\\]\nthat is,\n\\[\\phi_h(x,y)\\geq \\frac{\\norm{x-y}^2}{2\\tau}+\\tau h\\,,\\]\nwhich is equivalent to say that $P(\\tau)<0$.\n\\end{proof}\n\n\n\\subsection{Proof of Theorems \\ref{thm-princ} and \\ref{thm-calib.direct.horo} }\n\\label{s-proofs.main}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm-princ}]\nGiven $h>0$, $a\\in\\Omega$ and $x_0\\in E^N$ we proceed as\nfollows.\nFirst, we define the sequence of functions\n\\[\nu_n(x)=\n\\phi_h(x,na)-\\phi_h(0,na)\\,,\\qquad x\\in E^N.\n\\]\nEach one of this functions is a viscosity subsolution\nof the Hamilton-Jacobi equation $H(x,d_xu)=h$,\nthat is to say, we have $u_n\\prec L+h$ for all $n>0$.\nSince the estimate for the action potential $\\phi_h$\ngiven by Theorem \\ref{thm-phih.estim} implies that\nthe set of such subsolutions is an equicontinuous family,\nand since we have $u_n(0)=0$ for all $n>0$,\nwe can extract a subsequence converging to a function\n\\[\n\\mathfrak{u} (x)=\\lim_{k\\to +\\infty}u_{n_k}(x),\n\\]\nand the convergence is uniform on compact subsets of $E^N$.\nActually the limit is a directed horofunction\n$\\mathfrak{u}\\in\\mathcal{B}_h(a)$.\n\nBy Theorem \\ref{thm-horofuns.have.lamin}\nwe know that there is at least one curve $x:[0,+\\infty)\\to E^N$,\nsuch that\n\\[\n\\phi_h(x_0,x(t))=\n\\mathcal{A}_L(x\\mid_{[0,t]})+ht=\n\\mathfrak{u} (x_0)-\\mathfrak{u} (x(t)).\n\\]\nfor any $t>0$, and such that $x(0)=x_0$.\nProposition \\ref{prop-criterion.visc.sol} now implies\nthat $\\mathfrak{u}$ is a viscosity solution of\nthe Hamilton-Jacobi equation $H(x,d_xu)=h$, and moreover,\nthat $\\mathfrak{u}$ is a fixed point of the quotient Lax-Oleinik semigroup.\n\nFinally, by Theorem \\ref{thm-calib.direct.horo} we have that\nthe curve $x(t)$ is a hyperbolic motion, with energy constant $h$,\nand whose asymptotic direction is given by the configuration $a$.\nMore precisely, we have that\n\\[\nx(t)=t\\;\\frac{\\sqrt{2h}}{\\norm{a}}\\;a \\,+\\,o(t)\n\\]\nas $t\\to +\\infty$, as we wanted to prove.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm-calib.direct.horo}]\nFor $h>0$ and $a\\in\\Omega$,\nlet $u\\in\\mathcal{B}_h(a)$ be a given horofunction directed by $a$.\nThis means that there is a sequence of configurations\n$(p_n)_{n>0}$, such that $p_n=\\lambda_na+o(\\lambda_n)$\nwith $\\lambda_n\\to+\\infty$ as $n\\to\\infty$, and such that\n\\[\nu(x)=\\lim_{n\\to\\infty}(u_{p_n}(x)-u_{p_n}(0))\n\\]\nwhere $u_p$ denotes the function $u_p(x)=\\phi_h(x,p)$.\nLet also $\\gamma:[0,+\\infty)\\to E^N$ be the curve\ngiven by the hypothesis and satisfying\n\\[\nu(\\gamma(0))-u(\\gamma(t))=\\mathcal{A}_{L+h}(\\gamma\\mid_{[0,t]})\n\\]\nfor all $t>0$.\nIn particular $\\gamma$ is an $h$-minimizer.\nWe recall that this means that the restrictions of\n$\\gamma$ to compact intervals are global\nminimizers of $\\mathcal{A}_{L+h}$.\nThus the restriction of $\\gamma$ to $(0,+\\infty)$ is\na genuine motion of the $N$-body problem,\nwith energy constant $h$,\nand it is a maximal solution if and only if\n$\\gamma(0)$ has collisions,\notherwise the motion defined by $\\gamma$\ncan be extended as a motion \nto some interval $(-\\epsilon, +\\infty)$.\n\nThe proof is divided into three steps.\nThe first one will be to prove that the curve $\\gamma$\nis not a superhyperbolic motion. This will be deduced from\nthe minimization property of $\\gamma$.\nThen we will apply the Marchal-Saari theorem to conclude\nthat there is a configuration $b\\neq 0$ such that\n$\\gamma(t)=tb+O(t^{2\/3})$.\nThe second and most sophisticated step will be to\nexclude the possibility of having collisions in $b$,\nthat is to say, in the limit shape of the motion $\\gamma$.\nFinally, once it is known that $\\gamma$ is a hyperbolic motion,\nan easy application of the Chazy's Lemma\n\\ref{lema-cont.limitshape} will allow us to conclude\nthat we must have $b=\\lambda a$ for some $\\lambda>0$.\nThen the proof will be achieved by observing that, since\n$\\norm{b}=\\sqrt{2h}$,\nwe must also have $\\lambda=\\sqrt{2h}\\norm{a}^{-1}$.\n\nWe start now by proving that $\\gamma$ is not superhyperbolic.\nWe will give a proof by contradiction.\nSupposing that $\\gamma$ is superhyperbolic\nwe can choose $t_n\\to +\\infty$ such\nthat $R(t_n)\/t_n\\to +\\infty$.\nWe recall that $R(t)=\\max\\set{ r_{ij}(t)\\mid i<j}$\ndenotes the maximal distance between the bodies at time $t$,\nand that $R(t)=O(\\norm{\\gamma(t)})$.\nThus we can assume that\n$\\norm{\\gamma(t_n)-\\gamma(0)}\/t_n\\to +\\infty$.\nGiven that the calibrating property implies that the curve\n$\\gamma$ is an $h$-minimizer,\nfor each $n>0$ we have\n\\[\n\\mathcal{A}_L(\\gamma\\mid_{[0,t_n]})+ht_n=\n\\phi_h(\\gamma(0),\\gamma(t_n)).\n\\]\nLet us write for short $r_n=\\norm{\\gamma(0)-\\gamma(t_n)}$.\nIn view of the observation we made in\nRemark \\ref{rmk-usual.bound},\nand using Theorem \\ref{thm-phih.estim},\n we have the lower and upper bounds\n\\[\n\\tfrac{1}{2}\\,r_n^2\\;t_n^{-1} +ht_n\\;\\leq\\;\n\\phi_h(\\gamma(0),\\gamma(t_n))\\;\\leq\\;\n\\left(\\alpha\\;r_n+h\\beta\\; r_n^2\\right)^{1\/2}\n\\]\nfor some constants $\\alpha,\\beta >0$ and for any $n>0$.\nIt is not difficult to see that this is impossible for $n$ large\nenough using the fact that $r_n\\,t_n^{-1}\\to +\\infty$.\nThus by the Marchal-Saari theorem there is a configuration\n$b\\in E^N$ such that $\\gamma(t)=tb+O(t^{2\/3})$.\nSince by the Lagrange-Jacobi identity $b=0$ forces $h=0$,\nwe know that $b\\neq 0$.\n\nWe prove now that $b$ has no collisions, that is to say,\nthat $b\\in\\Omega$. This is our second step in the proof.\nLet us write $p=\\gamma(0)$, $q_0=\\gamma(1)$\nand let us also define $\\sigma_0\\in\\mathcal{C}(q_0,p,1)$\nby reversing the parametrization of\n$\\gamma_0=\\gamma\\mid_{[0,1]}$.\nThus $\\sigma_0$ calibrates the function $u$,\nthat is to say, we have\n$u(p)-u(q_0)=\\mathcal{A}_{L+h}(\\sigma_0)$.\n\nNow, using Lemma \\ref{lema-JM.geod.complet} we can define\na sequence of curves $\\sigma'_n\\in\\mathcal{C}(p_n,q_0)$,\nsuch that $\\mathcal{A}_{L+h}(\\sigma'_n)=\\phi_h(p_n,q_0)$ for all $n>0$.\nThus each curve $\\sigma'_n$\nis an $h$-calibrating curve of the function\n$u_{p_n}(x)=\\phi_h(x,p_n)$.\nIt will be convenient to also consider the curves $\\gamma'_n$\nobtained by reversing the parametrizations of\nthe curves $\\sigma'_n$.\nIf for each $n>0$ the curve $\\sigma'_n$ is defined over\nan interval $[-s_n,0]$, then we get a sequence of curves\n$\\gamma'_n\\in\\mathcal{C}(q_0,p_n,s_n)$,\nrespectively defined over the intervals $[0,s_n]$. \n\nSince $q_0$ is an interior point of $\\gamma$,\nMarchal's Theorem implies that $q_0\\in\\Omega$.\nThus for each curve $\\gamma'_n$ the velocity\n$w_n=\\dot\\gamma'_n(0)$ is well defined.\nSince $h$-minimizers have energy constant $h$,\nwe also have $\\norm{w_n}^2=2(h+U(q_0))$ for all $n>0$.\nThis allow us to choose a subsequence $n_k$ such that\n$w_{n_k}\\to v_0$ as $k\\to\\infty$.\nAt this point we need to prove\nthat $\\lim s_n=+\\infty$.\nThis can be done by application of Lemma \\ref{lema-time.estim}\nto the $h$-minimizers $\\gamma'_n$ as follows.\nGiven two configurations $x,y\\in E^N$,\nthe polynomial given by the lemma satisfies\n$P(\\tau)\\geq \\norm{x-y}^2-2\\phi_h(x,y)\\tau$ for all $\\tau>0$.\nTherefore,\nwhen $x\\neq y$ its roots can be bounded below by\n$\\norm{x-y}^2\/2\\phi_h(x,y)$.\nUsing this fact, we have that for all $n>0$,\n\\[\ns_n>\\frac{\\norm{q_0-p_n}^2}{2\\,\\phi(q_0,p_n)}.\n\\]\nThen the upper bound for $\\phi_h$ given by\nTheorem \\ref{thm-phih.estim} implies that $\\lim s_n=+\\infty$. \n\nLet us summarize what we have built so far.\nFrom now on, let us write for short $q_k=p_{n_k}$, $t_k=s_{n_k}$,\n$v_k=w_{n_k}$, and also $\\gamma_k=\\gamma'_{n_k}$ and\n$\\sigma_k=\\sigma'_{n_k}$.\nFirst, there is a sequence of configurations $(q_k)_{k>0}$,\nsuch that, for some increasing sequence $n_k$\nof positive integers, we have\n$q_k=\\lambda_{n_k}a+o(\\lambda_{n_k})$ as $k\\to\\infty$.\nAssociated to each $q_k$ there is an $h$-minimizer\n$\\gamma_k:[0,t_k]\\to E^N$, with $t_k\\to +\\infty$,\nsuch that $\\gamma_k\\in\\mathcal{C}(q_0,q_k)$.\nMoreover, $v_k=\\dot\\gamma_k(0)$ and we have\n$v_k\\to v_0$ as $k\\to\\infty$.\nIn addition, each reversed curve $\\sigma_k\\in\\mathcal{C}(q_k,q_0)$\nis an $h$-calibrating curve of the function\n$u_{q_k}(x)=\\phi_h(x,q_k)$.\n\nWe will prove that $v_0=\\dot\\gamma(1)$. \nTo do this, we start by considering the maximal solution\nof Newton's equations with initial conditions $(q_0,v_0)$\nand by calling $\\zeta$ its restriction to positive times,\nlet us say for $t\\in [0,t^*)$.\nNext, we choose $\\tau\\in (0,t^*)$ and we observe that\nwe have $t_k>\\tau$ for any $k$ big enough.\nThus, for these values of $k$, we have that\n$\\gamma_k(t)$ and $\\dot\\gamma_k(t)$ converge respectively\nto $\\zeta(t)$ and $\\dot\\zeta(t)$, and the convergence is uniform\nfor $t\\in[0,\\tau]$.\nTherefore,\n\\[\n\\lim_{k\\to\\infty}\\mathcal{A}_{L+h}(\\gamma_k\\mid_{[0,\\tau]})=\n\\mathcal{A}_{L+h}(\\zeta\\mid_{[0,\\tau]}).\n\\]\nOn the other hand, since on each compact set our function\n$u(x)$ is the uniform limit of the functions\n$u_k(x)=u_{q_k}(x)-u_{q_k}(0)$, we can also write\n\\[\nu(q_0)-u(\\zeta(\\tau))=\n\\lim_{k\\to\\infty}\\,(\\,u_k(q_0)-u_k(\\gamma_k(\\tau))\\,).\n\\]\nWe use now the fact that for each one of these values of $k$\nwe have, by the calibration property, that\n\\[\nu_k(q_0)-u_k(\\gamma_k(\\tau))=\n\\mathcal{A}_{L+h}(\\gamma_k\\mid_{[0,\\tau]}),\n\\]\nto conclude then that\n\\[\nu(q_0)-u(\\zeta(\\tau))=\\mathcal{A}_{L+h}(\\zeta\\mid_{[0,\\tau]}).\n\\]\nNotice that what we have proved is that the reversed curve\n$\\zeta(-t)$ defined on $[-\\tau,0]$\nis indeed an $h$-calibrating curve of $u$.\nThe concatenation of this calibrating curve with the\ncalibrating curve $\\sigma_0$\nresults, according to Lemma \\ref{lema-concat.calib},\nin a new calibrating curve,\ndefined on $[-\\tau,1]$ and passing by $q_0$ at $t=0$.\nTherefore this concatenation of curves is an $h$-minimizer,\nwhich implies that it is smooth at $t=0$.\nWe have proved that $\\dot\\zeta(0)=v_0=\\dot\\gamma(1)$.\nThis also implies that $t^*=+\\infty$ and that\n$\\zeta(t)=\\gamma(t+1)$ for all $t\\geq 0$.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=1.1]{proof33a.pdf}\n\\caption{The $C^1$ approximation of the curve $\\gamma$ by\n$h$-minimizers from $q_0$ to $q_k=p_{n_k}$.\nHere $\\lambda=\\lambda_{n_k}$ and\n$\\norm{q_k-\\lambda a}<r=o(\\lambda)$.} \n\\label{Proof33a}\n\\end{figure}\n\nFor simplicity,\nin the rest of the proof we will call $\\gamma$ the curve $\\zeta$,\nassuming then that the original curve $\\gamma$ was\nreparametrized to be defined on the interval $[-1,+\\infty)$.\nMaking this abuse of notation we can then write\n$\\gamma_k(t)\\to\\gamma(t)$,\nand $\\dot\\gamma_k(t)\\to\\dot\\gamma(t)$,\nuniformly on any compact interval $[0,T]$.\n\nWe continue now with the proof that the limit shape $b$\nof $\\gamma $ has no collisions.\nWe will make use of the function $\\mu$ that we have mentioned\nin Remark \\ref{rmk-configur.measure} which is\ncalled the configurational measure.\nIt is defined as the homogeneous function of degree zero\n$\\mu:E^N\\setminus\\set{0}\\to\\mathbb{R}^+$ given by\n$\\mu(x)=\\norm{x}U(x)=U(\\norm{x}^{-1}x)$,\nthat is to say, $\\mu=UI^{1\/2}$. \nNotice that $\\mu(x)<+\\infty$ if and only if $x\\in\\Omega$.\n\nUnder the assumption that $b$ has collisions,\nwe will construct a new sequence of curves\n$\\eta_k\\in\\mathcal{C}(q_0,q_k)$ in such a way that\n$\\mathcal{A}_{L+h}(\\eta_k)<\\mathcal{A}_{L+h}(\\gamma_k)$\nfor all $k$ big enough.\nSince this contradicts the minimality of the curves $\\gamma_k$\nwe will conclude that $b\\in\\Omega$.\nThe construction of the curves $\\eta_k$ will be done in terms\nof the polar components of the curves $\\gamma_k$.\nMore precisely, for each $k>1$ we define the functions\n\\[\n\\rho_k:[0,t_k]\\to\\mathbb{R}^+,\n\\quad\n\\rho_k(t)=\\norm{\\gamma_k(t)}\n\\]\n\\[\n\\theta_k:[0,t_k]\\to\\mathbb{S},\n\\quad\n\\theta_k(t)=\\norm{\\gamma_k(t)}^{-1}\\gamma_k(t)\n\\]\nwhere $\\mathbb{S}=\\set{x\\in E^N\\mid \\inner{x}{x}=1}$ is the unit sphere\nfor the mass inner product. \nThus, for each $k>0$ we can write $\\gamma_k=\\rho_k\\theta_k$,\nand the Lagrangian action in polar coordinates writes\n\\[\n\\mathcal{A}_{L+h}(\\gamma_k)=\n\\int_0^{t_k}\\tfrac{1}{2}\\;\\dot\\rho_k^{\\,2}\\,dt\\,+\n\\int_0^{t_k}\\tfrac{1}{2}\\;\\rho_k\\,\\dot\\theta_k^{\\,2}\\,dt\\,+\n\\int_0^{t_k}\\rho_k^{-1}\\,\\mu(\\gamma_k)\\,dt\\;+\nht_k.\n\\]\nAssuming that $\\mu(b)=+\\infty$, we can find $\\epsilon>0$\nsuch that, if $\\norm{x-b}<\\epsilon$,\nthen $\\mu(x)>3\\mu(a)$.\nOn the other hand, since we have that $\\gamma(t)=tb+o(t)$,\nthere is $T_0>0$ such that\n$\\norm{\\gamma(t)t^{-1}-b}<\\epsilon\/2$ for all $t\\geq T_0$.\n\nWe use now the approximation of $\\gamma$ by the curves\n$\\gamma_k$. For each $T\\geq T_0$ there is a positive integer\n$k_T$ such that, if $k>k_T$,\nthen $t_k>T$ and\n$\\norm{\\gamma_k(t)-\\gamma(t)}<T_0\\epsilon\/2$\nfor all $t\\in[T_0,T]$.\nIt follows that, for $k>k_T$ and for any $t\\in[T_0,T]$\nwe have\n\\[\n\\norm{\\frac{\\gamma_k(t)}{t}-\\frac{\\gamma(t)}{t}}<\n\\frac{\\epsilon}{2},\n\\]\nand then $\\norm{\\gamma_k(t)t^{-1}-b}<\\epsilon$.\nIn turn, since $\\mu$ is homogeneous, this implies that\n\\[\n\\mu(\\gamma_k(t))=\n\\mu(\\gamma_k(t)t^{-1})>3\\mu(a).\n\\]\n\nNow we are almost able to define the sequence of curves\n$\\eta_k\\in\\mathcal{C}(q_0,q_n)$.\nLet us write $k_0$ for $k_{T_0}$.\nFor $k\\geq k_0$ we know that\n$\\mu(\\gamma_k(T_0))>3\\mu(a)$.\nMoreover, since the extreme $p_k$ of the\ncurve $\\gamma_k$ lies in a ball $B_r(\\lambda a)$ with\n$r=o(\\lambda)$, we can assume that $k_0$ is big enough\nin order to have $\\mu(p_k)<2\\mu(a)$ for all $k\\geq k_0$.\nThen we define\n\\[\nT_k=\n\\max\\set{T\\geq T_0\\mid\n\\mu(\\gamma_k(t))\\geq 2\\mu(a)\n\\text{ for all } t\\in [T_0,T]},\n\\]\nand $c_k=\\theta_k(T_k)$.\nGiven $T>T_0$,\nby the previous considerations we have that\n$k>k_T$ implies $T_k>T$.\nThus,\nwe can take $T_k$ as large as we want\nby choosing $k$ large enough.\nThe last ingredient for building the curve $\\eta_k$\nis a minimizer $\\delta_k$ of $\\mathcal{A}_{L+h}$ in\n$\\mathcal{C}(\\gamma_k(T_0),\\rho_k(T_0)c_k)$\nwhose existence is guaranteed by Theorem\n\\ref{lema-JM.geod.complet}.\nThen we define $\\eta_k$ as follows.\nFor $k<k_0$ we set $\\eta_k=\\gamma_k$.\nFor $k\\geq k_0$ the curve $\\eta_k$ is the concatenation\nof the following four curves: (i) the restriction of $\\gamma_k$\nto $[0,T_0]$, (ii) the minimizer $\\delta_k$ above defined,\n(iii) the homothetic curve $\\rho_k(t)c_k$ for $t\\in [T_0,T_k]$,\nand (iv) the restriction of $\\gamma_k$ to $[T_k,t_k]$\n(see Figure \\ref{Proof33b}).\n\nWe will show that\n$\\Delta_k=\\mathcal{A}_{L+h}(\\gamma_k)-\\mathcal{A}_{L+h}(\\eta_k)>0$\nfor $k$ large enough.\n\nWe start by observing that the first and the last components\nof $\\eta_k$ are also segments of $\\gamma_k$ so that\ntheir contributions to $\\Delta_k$ cancel each other out.\n\nAlso we have\n\\[\n\\mathcal{A}_{L+h}(\\gamma_k\\mid_{[T_0,T_k]}) = \n\\int_{T_0}^{T_k}\\,\\tfrac{1}{2}\\,\\dot\\rho_k^{\\,2}\\,dt +\n\\int_{T_0}^{T_k}\\,\\tfrac{1}{2}\\,\\rho_k\\dot\\theta^{\\,2}\\,dt +\n\\int_{T_0}^{T_k}\\,\\rho_k^{-1}\\mu(\\gamma_k)\\,dt +\nh(T_k-T_0),\n\\]\nand\n\\[\n\\mathcal{A}_{L+h}(\\,\\rho_kc_k\\mid_{[T_0,T_k]}) =\n\\int_{T_0}^{T_k}\\,\\tfrac{1}{2}\\,\\dot\\rho_k^{\\,2}\\,dt +\n\\int_{T_0}^{T_k}\\,\\rho_k^{-1}\\,2\\mu(a)\\,dt +\nh(T_k-T_0).\n\\]\nWe recall that\n$\\mu(\\gamma_k(t))\\geq 2\\mu(a)$ for all\n$t\\in [T_0,T_k]$.\nTherefore, so far we can say that\n\\[\n\\Delta_k>\n\\int_{T_0}^{T_k}\\rho_k^{-1}\\,\n\\left(\\mu(\\gamma_k(t))-2\\mu(a)\\right)\\,dt\\;-\\;\n\\mathcal{A}_{L+h}(\\delta_k).\n\\]\nThis part of the proof is essentially done.\nTo conclude we only need to establish estimates\nfor the two terms on the right side of the previous inequality.\nMore precisely,\nwe will prove that the the integral diverges as $k\\to\\infty$,\nand that the second term is bounded as a function of $k$.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=1.1]{proof33b.pdf}\n\\caption{For $k$ large enough, the $\\mathcal{A}_{L+h}$ action\nof the green curve $\\eta_k$\nis less than that of the curve $\\gamma_k$.\nThe intermediate points are \n$b_k=\\gamma_k(T_0)$,\n$d_k=\\rho_k(T_0)c_k$,\nand $e_k=\\rho_k(T_k)c_k=\\gamma_k(T_k)$.} \n\\label{Proof33b}\n\\end{figure}\n\n\\begin{claim}\nThe sequence $\\mathcal{A}_{L+h}(\\delta_k)$ is bounded.\n\\end{claim}\n\\begin{proof}\nIndeed, the curve $\\delta_k$ is a minimizer of $\\mathcal{A}_{L+h}$\nbetween curves binding two configurations of size\n$\\rho_k(T_0)$, and\n\\[\n\\rho_k(T_0)\\to\\rho(T_0)=\\norm{\\gamma(T_0)}\n\\]\nas $k\\to\\infty$. Therefore there is $R>0$ such that\nthe endpoints of the curves $\\delta_k$ are all contained\nin the compact ball $B_R(0)\\subset E^N$.\nOn the other hand, \nsince by Theorem \\ref{thm-phih.estim} we know that\nthe action potential $\\phi_h$ is continuous, we can conclude that\n$\\sup \\mathcal{A}_{L+h}(\\delta_k)<+\\infty$.\n\\end{proof}\n\n\\begin{claim}\nThe sequence\n$\\int_{T_0}^{T_k}\\rho_k^{-1}\\,\n\\left(\\mu(\\gamma_k(t))-2\\mu(a)\\right)\\,dt$\ndiverges as $k\\to\\infty$.\n\\end{claim}\n\\begin{proof}\nIn order to get a lower bound for the integral of $\\rho_k^{-1}$,\nwe make the following considerations.\nWe note first that $\\rho(t)=\\norm{\\gamma(t)}< \\alpha t+\\beta$\nfor some constants $\\alpha, \\beta>0$.\nThis is because we know that\n$\\gamma(t)=tb+o(t)$ as $t\\to +\\infty$.\nThus we have that for any $T>T_0$\n\\[\n\\int_{T_0}^T\\rho^{-1}dt \\;\\geq\\;\n\\log(\\alpha T+\\beta)-\\log(\\alpha T_0 +\\beta).\n\\]\nTherefore, for any choice of $K>0$ there is $T>0$ such that\nthe integral at the left side is bigger than\n$\\mu(a)^{-1}K$.\n\nOn the other hand,\nsince for $k>k_T$ we have that $T_k>T$, and since\n$\\gamma_k(t)$ uniformly converges to $\\gamma(t)$\non $[T_0,T]$, we can assume that we have\n$\\mu(\\gamma_k(t))>3\\mu(a)$ for all $t\\in [T_0,T]$\nand then,\nneglecting the part of the integral between $T$ and $T_k$\nwhich is positive, to conclude that\n\\[\n\\int_{T_0}^{T_k}\\rho_k^{-1}dt\\,\n\\left(\\mu(\\gamma_k(t))-2\\mu(a)\\right)\\,dt \\;>\n\\;\\mu(a)\\int_{T_0}^{T}\\rho_k^{-1}dt \\;>\\;K\n\\]\nfor every $k$ sufficiently large.\n\\end{proof}\nIt follows that for large values of $k$ the difference $\\Delta_k$ is\npositive, meaning that the corresponding curves $\\gamma_k$\nare not $h$-minimizers because the curves $\\eta_k$ have\nsmaller action.\nTherefore we have proved by contradiction that $b\\in\\Omega$.\n\nThe last step to finish the proof is to show that $b=\\lambda a$\nfor some $\\lambda >0$.\nIf not, we can choose two disjoint cones $C_a$ and $C_b$\nin $E^N$, centered at the origin and with axes directed by\nthe configurations $a$ and $b$ respectively.\nSince we know that $b\\in\\Omega$,\nwe can apply Chazy's Lemma to get that for $k$ large enough\nthe curves $\\gamma_k$ are defined for all $t>0$, and that there\nis $T^*>0$ for which we must have $\\gamma_k(t)\\in C_b$\nfor all $t>T^*$ and any $k$ large enough.\nBut this produces a contradiction, because we know that\n$q_k=\\gamma_k(t_k)=\\lambda_{n_k}a+o(\\lambda_{n_k})$\nas $k\\to\\infty$, which\nforces to have $q_k\\in C_a$ for $k$ large enough.\n\\end{proof}\n\n\n\n\\section{The Jacobi-Maupertuis distance for nonnegative energy}\n\\label{s-jm.dist}\n\nIn this section we develop the geometric viewpoint\nand we show, for $h\\geq 0$, that when restricted to $\\Omega$\nthe action potential $\\phi_h$ is exactly the Riemannian distance\nassociated to the Jacobi-Maupertuis metric $j_h=2(h+U)g_m$,\nwhere $g_m$ is the mass scalar product.\nMoreover,\nwe will see that the metric space $(E^N,\\phi_h)$\nis the completion of $(\\Omega,j_h)$.\nThe fact that $\\phi_h$ is a distance over $E^N$ is a\nstraightforward consequence of the definition and of\nLemma \\ref{lema-geod.are.lipschitz} or\nLemma \\ref{lema-compar.pot.action.kepler}\ndepending on whether $h>0$ or $h=0$.  \nIt is also immediate to see that $(E^N,\\phi_h)$ is a length space,\nthat is to say $\\phi_h$ coincides with the induced length distance.\nFrom now on,\nwe denote by $\\mathcal{L}_h(\\gamma)$ the Riemannian length\nof a $C^1$ curve $\\gamma$,\nand we denote by $d_h$ the Riemannian distance on $\\Omega$.\n \n\\begin{proposition}\nFor all $h\\geq 0$,\nthe space $(E^N,\\phi_h)$ is the completion of $(\\Omega,d_h)$.    \n\\end{proposition}\n\n\\begin{proof}\nIn the case $h>0$,\nthe fact that $(E^N,\\phi_h)$ is a complete length space\ncomes directly from the definition of $\\phi_h$\nand from Lemma \\ref{lema-geod.are.lipschitz}\nand Theorem \\ref{thm-phih.estim}.\nMoreover, we have that $\\phi_h$ generates the topology of $E^N$\nand that $\\Omega$ is thus a dense subset.\n\nFor the case $h=0$ the argument is exactly the same,\nbut instead of Lemma \\ref{lema-geod.are.lipschitz},\nwhich becomes meaningless,\nwe have to use Lemma \\ref{lema-compar.pot.action.kepler} below.\n\nThe proof will be achieved now by showing that\nthe inclusion of $\\Omega$ into  $E^N$ is an isometry,\nthat is to say,\nthat $\\phi_h$ coincides with $d_h$ when restricted to $\\Omega$.\nGiven $(x,v)\\in \\Omega\\times E^N$,\nwe have\n\\[\n\\norm{v}_h=j_h(x)(v,v)^{1\/2}\\leq L(x,v)+h\n\\]\nwith equality if and only if $\\mathcal{E}(x,v)=h$,\nwhere $\\mathcal{E}(x,v)=\\frac{1}{2}\\norm{v}^2-U(x)$\nis the energy function in $T\\Omega$.\nIt follows that if $\\gamma$ is an absolutely continuous curve in\n$\\Omega$ it holds $\\mathcal{L}_h(\\gamma)\\leq A_{L+h}(\\gamma)$,\nwith equality if and only if\n$\\mathcal{E}(\\gamma(t),\\dot{\\gamma}(t))=h$  for almost all $t$.\nGiven now $x,y\\in\\Omega$,\nby Marchal's Theorem any $h$-minimizer joining $x$ to $y$\nis a genuine motion,\nin particular it is a $C^1$ curve.\nSince $d_h$ is defined as the infimum of $\\mathcal{L}_h(\\gamma)$\nover all $\\mathcal{C}^1$ curves in $\\Omega$ joining $x$ to $y$,\nwe have that $d_h(x,y)\\leq\\phi_h(x,y)$.\n\nIn order to prove the converse inequality,\nlet $\\epsilon>0$ and $\\gamma:[0,1]\\to \\Omega$\nbe a $\\mathcal{C}^1$ curve joining $x$ to $y$ such that\n$\\mathcal{L}_h(\\gamma)\\leq d_h(x,y)+\\epsilon$.\nWe can now find a finite sequence $0=t_0<...<t_N=1$\nsuch that for any $i=1,...,N$ the points\n$\\gamma(t_{i-1})$ and $\\gamma(t_i)$ can be joined by a\nminimizing geodesic in $\\Omega$, here denoted $\\sigma_i$.\nWe will assume that each $\\sigma_i$ is parametrized by arclength,\nthus $\\dot{\\sigma}_i(t)\\neq 0$ for all $t$.\nLet us reparametrize now each $\\sigma_i$ so that,\ndenoting $\\delta_i$ the reparametrization,\nwe have $\\mathcal{E}(\\delta_i(t),\\dot\\delta_i(t))=h$ for all $t$,\nand let $\\delta$ be the concatenation of all $\\delta_i$. \nBy construction\n\\[\n\\phi_h(x,y)\\leq A_{L+h}(\\delta)=\n\\mathcal{L}_h(\\delta)\\leq\n\\mathcal{L}_h(\\gamma)\\leq\nd_h(x,y)+\\epsilon,\n\\]  \nand by arbitrariness of $\\epsilon$ we conclude that\n$\\phi_h(x,y)\\leq d_h(x,y)$.\n\\end{proof}  \n\n\\begin{lemma}\n\\label{lema-compar.pot.action.kepler}\nThere exist a constant $\\mu_0>0$ such that\nfor all $x,y\\in E^N$ satisfying $x\\neq y$, we have\n\\[\n\\phi_0(x,y)\\geq \\frac{\\mu_0}{\\rho}\\norm{x-y}\n\\]\nwhere\n$\\rho=\\max\\set{\\norm{x},\\norm{y}}^\\frac{1}{2}$.\n\\end{lemma}\n\n\\begin{proof}\nThe main idea of the proof is to estimate $\\phi_0$\nby comparing it with the action of some Kepler problem in $E^N$.\nSince $U$ is a continuous function with values in $(0,+\\infty]$,\nthe minimum of $U$ on the unit sphere of $E^N$,\nhere denoted $U_0$, is strictly positive.\nThus, by homogeneity of the potential,\nif $x$ is any nonzero configuration we have\n\\[\nU(x)=\\frac{1}{\\norm{x}}\\,U\\left(\\frac{x}{\\norm{x}}\\right)\n\\geq \\frac{U_0}{\\norm{x}}.\n\\]\nLet us consider now the Lagrangian function\nassociated to the Kepler problem in $E^N$\nwith potential $U_0\/\\norm{x}$, that is to say \n\\[\nL_{\\kappa}(x,v)=\n\\frac{1}{2}\\norm{v}^2+\\frac{U_0}{\\norm{x}}\\;.\n\\]\nBy the previous inequality we know that $L_\\kappa(x,v) \\le L(x,v)$.\nThe critical action potential associated to $L_\\kappa$\nis defined on $E^N\\times E^N$ by\n\\[\n\\Phi_0(x,y)=\n\\min\\set{\\mathcal{A}_{L_\\kappa}(\\gamma)\\mid \\gamma\\in\\mathcal{C}(x,y)},\n\\]\nand it follows immediately from the definition that\n$\\Phi_0(x,y)\\le \\phi_0(x,y)$.\nAssume now $x\\neq y$, and let\n$\\gamma : [0,\\tau]\\rightarrow E^N$ be a free-time minimizer for\n$\\mathcal{A}_{L_\\kappa}$ in $\\mathcal{C}(x,y)$.\nThus $\\gamma$ is an absolutely continuous curve satisfying\n$\\mathcal{A}_{L_\\kappa}(\\gamma)=\\Phi_0(x,y)$.\nAs a zero energy motion of the Kepler problem,\nwe know that $\\gamma$ is an arc of Keplerian parabola,\nand in particular we know that\n\\[\n\\max_{t\\in[0,\\tau]}\\norm{\\gamma(t)}=\n\\max\\set{\\norm{x},\\norm{y}}\n\\]\nwhich in turn implies that\n\\[\n\\frac{U_0}{\\norm{\\gamma(t)}}\\,\\geq\\,\n\\frac{U_0}{\\rho^2}\n\\]\nfor all $t\\in[0,\\tau]$. \nThus, using this lower bound and Cauchy-Schwarz inequality\nfor the kinetic part of the action of $\\gamma$ we deduce that \n$\\Phi_0(x,y)\\geq g(\\tau)$,\nwhere $g:\\mathbb{R}^+\\to\\mathbb{R}$ is the function defined by \n\\[\ng(s)=\n\\frac{\\norm{x-y}^2}{2s}+\\frac{U_0}{\\rho^2}\\,s.\n\\]\nObserving now that $g$ is convex and proper,\nand replacing $g(\\tau)$ in the previous inequality\nby the minimum of $g(s)$ for $s>0$, we obtain\n\\[\n\\phi_0(x,y)\\geq\n\\Phi_0(x,y)\\geq\n\\frac{\\mu_0}{\\rho}\\,\\norm{x-y}.\n\\]\nfor $\\mu_0=\\sqrt{2U_0}$.\n\\end{proof} \n\nNow we have all the necessary elements to give\nthe proof of the corollary stated in Sect. \\ref{s-geom.view}.\nWe have to prove that if two geodesic rays have the same\nasymptotic limit, then they are equivalent in the sense of\nhaving bounded difference.\n\n\\begin{proof}\n[Proof of Corollary \\ref{cor-Gr.boundary}]\nLet $\\gamma:[0,+\\infty)\\to E^N$ be a geodesic ray \nof the distance $\\phi_h$, with $h>0$.\nWe assume that $\\gamma(s)=sa+o(s)$\nas $s\\to +\\infty$ for some $a\\in\\Omega$.\nThus, we know that $\\gamma(s)$ is without collisions\nfor all $s$ sufficiently big. \nBy performing a time translation we can assume that\n$\\gamma(s)\\in \\Omega$ for all $s\\geq 0$, hence that\n$\\gamma$ is a geodesic ray of the\nJacobi-Maupertuis metric $j_h$ in $\\Omega$.\nNow we know that $\\gamma$ admits a factorization\n$\\gamma(s)=x(t_\\gamma(s))$ where\n$x(t)$ is a motion of energy $h$.\nMore precisely,\nthe inverse of the new parameter $t_\\gamma$\nis a function $s_x$ satisfying $x(t)=\\gamma(s_x(t))$.\nSince $\\gamma$ is arclength parametrized,\nwe have $\\norm{\\dot\\gamma(s)}_h=1$ for all $s\\geq 0$,\nand we deduce that $s_x$ is the solution\nof the differential equation \n\\begin{equation}\\tag{$\\star$}\\label{eq-edo.reparam}\n\\dot s_x(t)=2h+2U(\\gamma(s_x(t)))\n\\end{equation}\nwith intial condition $s_x(0)=0$.\nThis implies that $s_x(t)\\to +\\infty$ and $\\dot s_x(t)\\to 2h$\nas $t\\to +\\infty$,\nhence we also have $s_x(t)=2ht+o(t)$\nand $x(t)=2ht\\,a+o(t)$ as $t\\to +\\infty$.\nIn particular $x(t)$ is a hyperbolic motion.\nWe claim now that\n\\[\ns_x(t)=2ht+\\frac{U(a)}{h}\\log t+O(1).\n\\]\nand the proof is as follows.\nFrom (\\ref{eq-edo.reparam}) we have, for $t>1$,\n\\begin{equation}\\tag{$\\star\\star$}\\label{eq-sx.asymptotic}\ns_x(t)=2ht+\\int_0^1 2U(x(\\nu))\\,d\\nu+\\int_1^t 2U(x(\\nu))\\,d\\nu.\n\\end{equation}\nOn the other hand, by Chazy's Theorem we have that\n\\[\nx(t)=\n2ht\\,a-\\frac{\\log t}{4h^2}\\,\\nabla U(a)+O(1).\n\\]\nWe observe then that\n\\begin{eqnarray*}\nU(x(\\nu))&=&\n\\frac{1}{2h\\,\\nu}\\,U\\left(a +O\\left(\\frac{\\log \\nu}{\\nu}\\right)\\right)\\\\\n& &\\\\\n&=&\\frac{U(a)}{2h}\\,\\frac{1}{\\nu}+\nO\\left(\\frac{\\log \\nu}{\\nu^2}\\right)\n\\end{eqnarray*}\nNow the claim can be verified by replacing this last expression of\n$U(x(\\nu))$ in the last term of (\\ref{eq-sx.asymptotic}).\n\nGiven now another gedodesic ray\n$\\sigma : [0,+\\infty)\\to E^N$,\ndenoting $\\sigma(s)=y(t_\\sigma(s))$\nthe reparametrization such that $y(t)$ is a motion\nof energy constant $h$,\nand denoting $s_y$ the inverse of $t_\\sigma$,\nit is clear from the previous asymptotic estimates that the\ndifference $s_x(t)-s_y(t)$ is bounded.\nSince the derivative of $s_x$ and $s_y$\nare both bounded below by the same positive constant,\nwe easily conclude that\n$t_\\gamma(s)-t_\\sigma(s)$ is also bounded.\nBy replacing in the asymptotic expansion of $x(t)$ and $y(t)$\nwe find that $\\gamma(s)-\\sigma(s)$ is bounded. \n\\end{proof}\n\n\n\n\n\n\\section{Open questions on bi-hyperbolic motions}\n\\label{s-bi.hyp}\n\nWe finish with some general open questions.\nThey are closely related to the recent advances made by\nDuignan {\\it et al.} \\cite{DuMoMoYu}\nin which the authors show in particular that\nthe limit shape map $(x,v)\\mapsto (a^-,a^+)$\ndefined below is actually real analytic.\n\nWe define bi-hyperbolic motions\nas those which are defined for all $t\\in\\mathbb{R}$,\nand are hyperbolic both in the past and in the future.\nThe orbits of these entire solutions\ndefine a non-empty open set in the phase space,\nnamely the intersection of the two open set\n\\[\\mathcal{H}=\\mathcal{H}^ +\\cap\\mathcal{H}^-\\]\nwhere\n$\\mathcal{H}^+\\subset T\\Omega=\\Omega\\times E^N$\nis the set of the initial conditions giving rise\nto hyperbolic motions in the future, and\n$\\mathcal{H}^-=\\set{(x,v)\\in T\\Omega\\mid (x,-v)\\in\\mathcal{H}^+}$\nis the set of the initial conditions\ngiving rise to hyperbolic motions in the past.\nNewton's equations define a complete vector field\nin the open set $\\mathcal{H}\\subset\\Omega\\times E^N$.\nWe will denote by $\\varphi^t$ the corresponding flow\nand $\\pi:\\Omega\\times E^N\\to\\Omega$\nthe projection onto the first factor.\n \nWe also note that this open and completely invariant set\nhas a natural global section,\ngiven by the section of \\emph{perihelia}:\n\\[\\mathcal{P}=\\mathcal{H}\\cap\\set{(x,v)\\in T\\Omega\\mid \\inner{x}{v}=0}\\,.\\]\n\n\\begin{proposition}\nThe flow $\\varphi^t$ in $\\mathcal{H}$\nis conjugated to the shift in $\\mathcal{P}\\times\\mathbb{R}$.\n\\end{proposition}\n\n\\begin{proof}\nGiven $(x_0,v_0)\\in\\mathcal{H}$,\nlet $x(t)=\\pi(\\varphi^t(x_0,v_0))$ be\nthe generated bi-hyperbolic motion.\nSince $I=\\inner{x}{x}$,\nit follows from the Lagrange-Jacobi identity $\\ddot I=4h+2U$,\nthat $I$ is a proper and strictly convex function.\nThus, there is a unique $t_p\\in\\mathbb{R}$ such that \n$\\varphi^{t_p}(x_0,v_0)\\in\\mathcal{P}$.\nMoreover,\nthe sign of $\\dot I=\\inner{x}{\\dot x}$ is the sign of $t-t_p$\nand $\\norm{x(t)}$ reaches its minimal value at $t=t_p$.\nThe conjugacy is thus given by the map\n$(x_0,v_0)\\mapsto (p(x_0,v_0),-t_p)$,\nwhere $p:\\mathcal{H}\\to\\mathcal{P}$ gives the phase point at perihelion\n$p(x_0,v_0)=(x(t_p),\\dot x(t_p))$.\n\\end{proof}\n\nNaturally associated with each bi-hyperbolic motion,\nthere is the pair of limit shapes that it produces\nboth in the past and in the future.\nMore precisely, we can define the \\emph{limit shape map}\n$S:\\mathcal{H}\\to\\Omega\\times\\Omega$ by\n\\[S(x,v)=(a^-(x,v),a^+(x,v))\\]\n\\[a^\\pm(x,v)=\\lim_{t\\to\\pm\\infty} \\;\\abs{t}^{-1}\\pi(\\varphi^t(x,v))\\,.\\]\nAs a consequence of Chazy's \\emph{continuity of the instability}\n(Lemma \\ref{lema-cont.limitshape}) we have that\nthe limit shape map is actually a continuous map.\nIt is also clear that\n\\[\\norm{a^-(x,v)}=\\norm{a^+(x,v)}\\] for all $(x,v)\\in\\mathcal{H}$.\nIn fact, we have\n\\[\\norm{a^\\pm(x,v)}^2=2h=\\norm{v}^2-2U(x)\\]\nwhere $h>0$ is the energy constant\nof the generated bi-hyperbolic motion.\nHence the image of $S$ is contained in the manifold\n\\[\n\\mathcal{S}=\n\\set{(a,b)\\in\\Omega\\times\\Omega\\mid \\norm{a}=\n\\norm{b}}\\,.\n\\]\nClearly, we have $S\\circ\\varphi^t=S$ for all $t\\in\\mathbb{R}$.\nTherefore the study of the limit shape map can be restricted\nto the section of perihelia $\\mathcal{P}$.\nCounting dimensions we get\n\\[\\dim\\mathcal{P}=2dN-1=\\dim\\mathcal{S}\\]\nwhere $d=\\dim E$.\n\nWe will see now that the center of mass\ncan be reduced to the origin.\nLet us call $G:E^N\\to E$ the linear map that associates\nto each configuration its center of mass.\nMore precisely,\nif $M=m_1+\\dots+m_N$ is the total mass of the system,\nthen the center of mass $G(x)$ of $x=(r_1,\\dots,r_N)\\in E^ N$\nis well defined by the condition\n$MG(x)=m_1r_1+\\dots+m_Nr_N$.\nJust as we did for the quantities U and I,\nwe will write $G(t)$ instead of $G(x(t))$ when\nthe motion $x(t)$ is understood.\nWe observe now that if $x(t)=ta^++o(t)$ as $t\\to+\\infty$,\nthen $G(t)=tG(a^+)+o(t)$.\nMoreover, since $\\ddot G(t)=0$ for all $t\\in\\mathbb{R}$ we know that\nthe velocity of the center of mass $\\dot G(t)=v_G$ is constant,\nhence $G(t)=tv_G+G(0)$.\nTherefore we must have $G(a^+)=v_G$.\nIf in addition $x(t)=-ta^-+o(t)$ as $t\\to-\\infty$,\nthen we also have $G(a^-)=-v_G$.\nWe conclude that\n\\[G(a^-(x,v))=-\\;G(a^+(x,v))\\]\nfor all $(x,v)\\in\\mathcal{H}$.\nThis allows to reduce in $d$ dimensions\nthe codomain of the limit shape map.\nOn the other hand,\na constant translation of a bi-hyperbolic motion\ngives a new bi-hyperbolic motion with the same limit shapes.\nThus the domain can also be reduced of $d$ dimensions\nby imposing the condition $G(x(0))=0$.\n\nFinally,\nwe note that bi-hyperbolic motions are preserved\nby addition of uniform translations.\nLet $\\Delta\\subset E^N$ be the diagonal subspace,\nthat is the set of configurations of total collision. \nFor any bi-hyperbolic motion $x(t)$\nwith limit shapes $a^-$ and $a^+$, \nand any $v\\in\\Delta$,\nwe get a new bi-hyperbolic motion $x_v(t)=x(t)+tv$,\nwhose limit shapes are precisely $a^--v$ and $a^++v$.\nIn particular these configurations without collisions\nhave opposite center of mass and the same norm.\nThe equality of the norms can also be deduced\nfrom the orthogonal decomposition\n$E^N=\\Delta\\oplus \\ker G$ and using the fact that\n$G(a^+-a^-)=0$.\n\nIn sum,\nwe can perform the total reduction of the center of mass\nby setting $G(x(0))=G(\\dot x(0))$,\nwhich leads to $G(a^-)=G(a^+)=0$.\nWe define\n\\[\\mathcal{P}_0=\\set{(x,v)\\in\\mathcal{H}\\mid\\;\nG(x)=G(v)=0\\text{ and }\\inner{x}{v}=0}\\]\n\\[\\mathcal{S}_0=\\set{(a,b)\\in\\Omega\\times\\Omega\\mid\\;\nG(a)=G(b)=0\\text{ and }\\norm{a}=\\norm{b}}\\]\nand we maintain the balance of dimensions.\n\n\\begin{question}\nIs the limit shape map $S:\\mathcal{P}_0\\to\\mathcal{S}_0$\na local diffeomorphism?\n\\end{question}\n\nThe answer is yes in the Kepler case\n(see Figure \\ref{HyperKepler1}).\nBut in the general case,\nthis property must depend on the potential $U$.\nFor instance,\nin the extremal case of $U=0$,\nin which motions are thus straight lines,\nwe get the restriction $a^-=-a^+$ for all hyperbolic motion.\nIn this case the shape map loses half of the dimensions.\n\nIt is therefore natural to ask, for the general $N$-body problem,\nwhether or not\nthere is some relationship between these two functions.\n\n\\begin{question}\nHow big is the image of the limit shape map?\n\\end{question}\n\nIn the Kepler case, only the pairs $(a,b)$ such that\n$\\norm{a}=\\norm{b}$ and $a\\neq\\pm\\, b$\nare realized as asymptotic velocities of some hyperbolic trajectory.\nThis can be generalized for $N\\geq 3$.\nIf $a\\in\\Omega$ is a planar central configuration\nand $R\\in SO(E)$ keeps invariant the plane containing $a$,\nthe pair $(a,Ra)$ is realized as the limit shapes\nof a unique homographic hyperbolic motion,\nexcept in the cases $R=\\pm\\,Id$.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=1.1]{HyperKepler1.pdf}\n\\caption{Hyperbolic motions of the Kepler problem\nwith fixed value of the energy constant $h>0$\nand asymptotic velocity $a$ in the future.\nAll but one of these motions are bi-hyperbolic.\nThe blue curve $\\mathcal{P}$ is composed of the\ncorresponding perihelia.} \n\\label{HyperKepler1}\n\\end{figure}\n\nWe now devote attention to the effect of homogeneity.\nRecall that\nif $x(t)$ is a bi-hyperbolic motion of energy constant $h$,\nthen  for every $\\lambda>0$ the solution given by\n$x_\\lambda(t)=\\lambda\\,x(\\lambda^{-3\/2}t)$\nis still bi-hyperbolic with energy constant $\\lambda^{-1}h$.\nMoreover,\nif we note $x_0=x(0)$ and $v_0=\\dot x(0)$ then we have\n\\[\n(x_\\lambda(t),\\dot x_\\lambda(t))=\n\\varphi^t(\\lambda\\, x_0,\\lambda^{-1\/2}\\,v_0)\n\\]\nfor all $t\\in\\mathbb{R}$.\nThese considerations prove the following remark.\n\n\\begin{remark}\nFor any $(x,v)\\in\\mathcal{H}$ and for any $\\lambda>0$, we have\n\\[S(\\lambda\\, x,\\lambda^{-1\/2}\\,v)=\\lambda^{-1\/2}\\,S(x,v)\\,.\\]\n\\end{remark}\n\nLet us introduce the following question with an example.\nConsider the planar three-body problem with equal masses.\nThat is, $E=\\mathbb{R}^2\\simeq\\mathbb{C}$, $N=3$ and $m_i=1$ for $i=1,2,3$.\nFor $h>0$, define the equilateral and collinear configurations\n\\[a_h=\\sqrt\\frac{2h}{3}\\,(1,z,z^2)\\qquad b_h=\\sqrt h\\,(-1,0,1)\\]\nwhere $z$ is a primitive root of $z^3-1$.\nThus we have $\\norm{a_h}=\\norm{b_h}=\\sqrt{2h}$ and also\n$G(a_h)=G(b_h)=0$ for all $h>0$.\n\n\\begin{question}\nIs the pair $(a_h,b_h)$ in the image of the limit shape map?\n\\end{question}\n\nIn other words,\nis there a bi-hyperbolic motion whose dynamics originates\nin the past with a contraction from a big equilateral triangle,\nand then, after a period of strong interaction between the particles,\nthe evolution ends with an almost collinear expansion?\n\nIn our view, the method of viscosity solutions\ncould be useful to answer this question. \nIn particular, we consider it necessary to push forward\nthe understanding of the regions of differentiability\nof these weak solutions.\nIt seems reasonable that an orbit like this\ncan be found by looking for critical points\nof a sum of two Busemann functions (see Sect. \\ref{s-busemann}).\n\n\\begin{question}\nIf the answer to Question 3 is yes,\nwhat is the infimum of the norm of the perihelia\nof the bi-hyperbolic motions having these limit shapes? \n\\end{question}\n\nObserve that once we have a bi-hyperbolic motion which is\nequilateral in the past and collinear in the future,\nwe can play with the homogeneity in order to obtain a new one,\nbut having a perihelion contained in an arbitrarily small ball.\nThat is to say, it would be possible to make, at some point,\nall bodies pass as close as we want from a total collision.\nOf course, to do this\nwe must increase the value of the energy constant indefinitely.\nThus we preserve the limit shapes in the weak sense,\nbut not the size of the asymptotic velocities.\n In the family of motions $(x_\\lambda)$ described above,\nthe product of the energy constant $h$\nand the norm of the perihelion is constant.\nIn the Kepler case,\nonce we fix the value of $h>0$\nthere is only one bi-hyperbolic motion\nconnecting a given pair $(a,b)$ (see \\cite{Alb}).\nTherefore we can see the norm of the perihelion\nas a function of the limit shapes.\nWe can see that the norm of the perihelion\ntends to $0$ for $a\\to b$,\nand tends to $+\\infty$ for $a\\to -b$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Acknowledgements}\nThe authors would like to express\na very special thank to Albert Fathi,\nwho suggested the use of the method\nof viscosity solutions, as well as the way\nto construct the maximal calibrating curves\nof the horofunctions. \nThe first author is also grateful to\nAlain Albouy, Alain Chenciner and Andreas Knauf\nfor several helpful conversations at the IMCCE.\nFinally, we would like to thank the referee\nfor his accurate and helpful comments,\nand Juan Manuel Burgos\nfor pointing out a subtle mistake in one of the proofs.\n\n\n\n\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUnderstanding open system dynamics and decoherence is important in several areas of quantum physics~\\cite{BreuerBook_2002,RivasBook_2012}. During the last ten years, there have been significant developments in both understanding the role of non-Markovian memory-effects~\\cite{Rivas_2014, Breuer_2016, Hall_2018, Li_2019,Li_2019_EPL2} and in developing improved tools and techniques to treat open system \ndynamics~\\cite{Vega_2017}.\nHere, one of the common themes is the role that various types of correlations play in open system dynamics. In particular, understanding\n initial correlations between composite environments~\\cite{Laine_2012,Liu_2013,Laine_2014,Xiang_2014,Liu_2016,HamedaniRaja_2017} and the role of initial system-environment correlations~\\cite{pechukas94,Alicki_1995,pechukas95,jordan04,shaji05a,Linta-Shaji,Wiseman_2019,Lyyra_2018,Alipour_2019}  have led to fundamental insights as well as practical knowledge regarding open systems.\n\nPhotons  provide a common and  highly controllable system where the influence of correlations can be studied both conceptually and practically~\\cite{Laine_2012,Liu_2013,Laine_2014,Xiang_2014,Liu_2016,HamedaniRaja_2017,Lyyra_2018}. Here, the polarization state of the photon is the open system and its frequency is the environment. Polarization and  frequency are coupled via birefringence leading to dephasing of a polarization state of the photon(s). The control of initial frequency distribution allows for the engineering of the decoherence and it is also possible to exploit various correlations for single photon or composite two-photon systems~\\cite{ElsiNature2011,ElsiPRL2012,Lyyra_2018}.\n\nOn the one hand, dephasing dynamics of photons has often been described using the concept of decoherence functions and subsequent family of completely positive (CP) dynamical maps, in the past. On the other hand, master equations are one of the most common tools to treat open system dynamics~\\cite{BreuerBook_2002}. However, master equations have not been used extensively when considering multipartite photonic systems and dephasing.\nWe consider first a bipartite two-photon system where the initial system-enviroment state is factorized whilst there exist initial correlations between the environmental states. \nIt has been shown earlier that this induces non-local memory effects in open system dynamics~\\cite{ElsiPRL2012,Liu_2013}. However,  the role of these types of initial correlations and non-local memory effects have not been considered on the level of master equations before, to the best of our knowledge. We derive generic master equations which display explicitly the role of initial correlations both on the dephasing rates and on the operator form of the master equation.  This allows also to reveal how even quite straightforward changes in the initial environmental state change drastically the description of photonic dephasing and increases the number of jump operators in the master equation. \n\nContinuing within the framework of correlations and open systems, we also study another long-standing problem in this context.\nThis is the role that initial system-environment correlations play in open system dynamics. Here, our interest is to see, what kind of insight the recently developed\n\\textit{bath positive decomposition} method~\\cite{Wiseman_2019} allows when studying the open dynamics of the polarization states. This very general method is based on decomposing initial arbitrary system-environment \nstate to a number of terms where each term can be treated with its individual CP-map. We show that for single-photon dephasing, this decomposition allows to describe, in a insightful way, how initial correlations influence the dynamics beyond the contribution arising from the factorized part.\n\nThe structure of the paper is the following. In the next section we describe briefly the basics of photonic dephasing.\nIn Section \\ref{sec3} we focus on the correlations within the composite environment and derive various master equations in this context and discuss the insight they provide.\nSection \\ref{sec4}, in turn, describes the initially correlated system-environment case for single photon and Sec.~\\ref{sec5} concludes the paper.\n\n\\section{Preliminaries with single photon dephasing dynamics}\n\nWe start with a brief recall of the single-photon dephasing model~\\cite{ElsiNature2011}.\nPolarization degree of freedom and frequency degree of freedom of a photon correspond to the open system and its environment, respectively. To begin with, we consider initially factorized joint polarization-frequency state \n\\begin{equation}\n\\hat{\\rho}_{SE}(0)=\\hat{\\rho}_S(0)\\otimes \\ket{\\Omega}\\bra{\\Omega}.\n\\end{equation}\nHere,  $\\hat{\\rho}_S(0)$ is the density operator of the initial polarization state and \n\\begin{equation}\n\\ket{\\Omega}= \\int d \\omega \\; g(\\omega)\\ket{\\omega},\n\\end{equation}\nis the initial  frequency state where $g(\\omega)$ is the probability amplitude that the photon has frequency $\\omega$. \nThe polarization Hilbert space is discrete and spanned by the horizontal-vertical polarization basis $\\{\\ket{h},\\ket{v}\\}$, while the Hilbert space of the frequency degree of freedom is spanned by the continuous frequency basis $\\{\\ket{\\omega}\\}$. \n\nThe system-environment -- or polarization-frequency -- interaction is provided by the Hamiltonian ($\\hbar=1$)\n\\begin{equation}\\label{eq:Hamiltonian}\n\\hat{H}=(n_h\\ket{h}\\bra{h}+n_v\\ket{v}\\bra{v})\\otimes\\int d \\omega \\; \\omega \\; \\ket{\\omega}\\bra{\\omega} ,\n\\end{equation}\nwhere $n_h$ ($n_v$) is the refraction index for polarization component $h$ ($v$).\nFor interaction time $t$, and tracing over the frequency, the reduced polarization state is\n\\begin{equation}\\label{eq:MapSinglePhoton}\n\\hat{\\rho}_S(t)=\\begin{pmatrix}\n\\bra{ h} \\rho_S(0)\\ket{h} & \\kappa(t)\\bra{ h} \\rho_S(0)\\ket{v} \\\\\n\\kappa(t)^{*}\\bra{ v} \\rho_S(0)\\ket{h} & \\bra{ v} \\rho_S(0)\\ket{v}.\n\\end{pmatrix},\n\\end{equation}\nHere, the dephasing dynamics is given by the decoherence function\n\\begin{equation}\n\\label{eq:kappa}\n \\kappa(t)=\\int d\\omega \\; \\vert g(\\omega)\\vert^2 \\mathrm{e}^{-i \\Delta n \\omega t},\n \\end{equation}\nwhere $\\Delta n \\equiv n_v-n_h$. Note that $0\\leq \\vert \\kappa(t) \\vert \\leq 1 $ for all times $t\\geq 0$ and $\\vert \\kappa(0) \\vert=1$. \n\nEquation \\eqref{eq:MapSinglePhoton} describes a $t$-parametrized completely positive (CP) map $\\hat{\\Phi}_t$, such that $\\hat{\\rho}_S(t)=\\hat{\\Phi}_t[\\hat{\\rho}_S(0)]$, and its corresponding master equation takes the form\n \\begin{equation}\\label{eq:MasterEq}\n\\frac{d}{dt}\\hat{\\rho}_S(t)=-i\\frac{\\nu(t)}{2}[\\hat{\\sigma}_z,\\hat{\\rho}_S(t)]+\\frac{\\gamma(t)}{2}[\\hat{\\sigma}_z \\hat{\\rho}_S(t) \\hat{\\sigma}_z-\\hat{\\rho}_S(t)].\n\\end{equation}\nHere,  $\\hat{\\sigma}_z$ is the Pauli $z$ operator and the rates $\\nu(t)$ and $\\gamma(t)$ can be expressed in terms of the decoherence function $\\kappa(t)$ as\n\\begin{equation}\\label{eq:Rates}\n\\gamma(t)=-\\Re \\bigg[\\frac{1}{\\kappa(t)} \\frac{d\\kappa(t)}{dt}\\bigg], \\quad \\nu(t)=-\\Im \\bigg[\\frac{1}{\\kappa(t)} \\frac{d\\kappa(t)}{dt}\\bigg],\n\\end{equation}\nwhere, $\\Re[\\cdot]$ and $\\Im[\\cdot]$ indicate the real and imaginary parts, respectively. \n\nEquation~\\eqref{eq:Rates} shows that once the decoherence function $\\kappa(t)$ is obtained from Eq.~\\eqref{eq:kappa}, then we can derive the corresponding rates in master equation ~\\eqref{eq:MasterEq}. Indeed,  the decoherence function $\\kappa(t)$ in Eq.~\\eqref{eq:kappa} is the Fourier transformation of the initial frequency \nprobability distribution $P(\\omega) =|g(\\omega)|^2$, and therefore the control of this distribution allows to study various types of dephasing maps and to engineer  the form and time dependence of the dephasing rate $\\gamma(t)$ in master equation ~\\eqref{eq:MasterEq}.\n\nFor example, a Gaussian frequency distribution with variance $\\sigma^2$ and mean value $\\bar{\\omega}$, i.e., \n\\[P(\\omega)=\\frac{\\mathrm{exp}[-(\\omega-\\bar{\\omega})^2\/2\\sigma^2]}{\\sqrt{2\\pi}\\sigma},\\]\nleads to a positive and time dependent dephasing rate $\\gamma(t)= \\Delta n ^2 \\sigma^2 t$ which presents a time-dependent Markovian dynamics. On the other hand, a Lorentzian distribution \n\\[P(\\omega)=\\frac{\\lambda}{\\pi[(\\omega-\\omega_0)^2+\\lambda^2]},\\]\nresults in a constant decay rate $\\gamma=\\lambda \\Delta n $, corresponding to dynamical semi-group and Lindbad-Gorini-Kossakowski-Sudarshan (LGKS) dynamics~\\cite{GoriniJmathPhys1976,LindbladCommun1976}. We note that the latter case has been also reported in \\cite{Smirne2014, Smirne2018}. \nThe transition from Markovian to non-Markovian regime, in turn,  is observed with further modifications of the frequency distribution~\\cite{ElsiNature2011}.\n\nIn the following, we generalize the master equation~in Eq.\\eqref{eq:MasterEq} to two-photon case. In particular, we are interested in how the initial correlations between the frequencies of the two photons influence the various dephasing rates and the operator form of the corresponding master equation for a bipartite open system. \n\n\\section{Master equation for two-photon dephasing dynamics: role of initially correlated joint frequency distribution} \\label{sec3}\n\nConsider a pair of photons, labeled $a$ and $b$, whose total polarization-frequency initial state is again in a factorized form\n\\begin{equation}\n\\hat{\\rho}_{SE}(0)=\\hat{\\rho}_S(0)\\otimes \\ket{\\Omega}\\bra{\\Omega},\n\\end{equation}\nwhere now\n\\begin{equation}\n\\ket{\\Omega}=\\int d \\omega_a \\int  d \\omega_b \\; g(\\omega_a,\\omega_b)\\ket{\\omega_a,\\omega_b},\n\\end{equation}\nis the initial state of the two-photon frequency degree of freedom and the corresponding joint probability distribution is $P(\\omega_a,\\omega_b)=\\vert g(\\omega_a,\\omega_b)\\vert ^2$. Initial polarization state is $\\hat{\\rho}_S(0)$, whose Hilbert space is spanned by the bipartite basis $\\{\\ket{hh},\\ket{hv},\\ket{vh},\\ket{vv}\\}$. \n\nThe polarization of each photon interacts locally with its own frequency and therefore system-environment interaction Hamiltonian for the two photons is a sum of the two local contributions~\\cite{ElsiPRL2012}\n\\begin{equation}\\label{eq:HamiltonianTwoPhoton}\n\\hat{H}=\\hat{H}_a \\otimes \\hat{I}_b +  \\hat{I}_a \\otimes \\hat{H}_b.\n\\end{equation}\nHere, each local term is given by Eq.~\\eqref{eq:Hamiltonian} and $\\hat{I}_a$ ($\\hat{I}_b$) is the identity operator for photon a (b).\n\nWe write initial bipartite polarization state $\\hat{\\rho}_S(0)$ as\n\\[\\hat{\\rho}_S(0)=\\sum_{\\alpha,\\beta}\\sum_{\\alpha',\\beta'}p_{\\alpha\\beta,\\alpha'\\beta'}\\ket{\\alpha \\beta}\\bra{\\alpha' \\beta'},\\]\nwith sums over $h$ and $v$. After interaction time $t$, the polarization state is \\citep{ElsiPRL2012}\n\\begin{flalign}\\label{eq:MapTwoPhotoon}\n&\\hat{\\rho}_S(t)=\n\\\\ \n&\\begin{pmatrix} \\nonumber\np_{hh,hh} & \\kappa_b(t)p_{hh,hv} & \\kappa_a(t)p_{hh,vh} & \\kappa_{ab}(t)p_{hh,vv} \\\\\n\\kappa_b^{*}(t) p_{hv,hh} & p_{hv,hv} & \\Lambda_{ab}(t)p_{hv,vh} & \\kappa_a(t)p_{hv,vv} \\\\\n\\kappa_a^{*}(t) p_{vh,hh} & \\Lambda_{ab}^{*}(t)p_{vh,hv} & p_{vh,vh} & \\kappa_b(t)p_{vh,vv} \\\\\n\\kappa_{ab}^{*}(t) p_{vv,hh} & \\kappa_a^{*}(t)p_{vv,hv} & \\kappa_b(t)^{*} p_{vv,vh} & p_{vv,vv}\n\\end{pmatrix}.\n\\end{flalign}\nHere, the local decoherence functions for photon $j=a,b$ are given by\n\\begin{equation}\n\\label{k-loc}\n\\kappa_j(t)=\\int d\\omega_a  \\int \\;d\\omega_b \\; \\vert g(\\omega_a, \\omega_b)\\vert^2 \\mathrm{e}^{-i \\Delta n \\omega_j t},\n\\end{equation}\nand the non-local ones by\n\\begin{equation}\n\\label{k1-nloc}\n\\kappa_{ab}(t)=\\int d\\omega_a \\int \\; d\\omega_b \\; \\vert g(\\omega_a, \\omega_b)\\vert^2 \\mathrm{e}^{-i \\Delta n(\\omega_a+\\omega_b)t},\n\\end{equation}\nand \n\\begin{equation}\n\\label{k2-nloc}\n\\Lambda_{ab}(t)=\\int d\\omega_a \\int \\; d\\omega_b \\;\\vert g(\\omega_a, \\omega_b)\\vert^2 \\mathrm{e}^{-i \\Delta n(\\omega_a-\\omega_b)t}.\n\\end{equation}\nThe density matrix evolution given by Eqs.~(\\ref{eq:MapTwoPhotoon}-\\ref{k2-nloc})\ncan also be described by a $t$-parametrized completely positive dynamical map $\\hat{\\Phi}_t$, such that \n\\begin{equation}\\label{eq:DynMap}\n\\hat{\\rho}_S(t)=\\hat{\\Phi}_t[\\hat{\\rho}_S(0)].\n\\end{equation} It is important to note that when the initial joint frequency distribution factorizes,  $P(\\omega_a,\\omega_b)=P_a(\\omega_a)\\times P_b(\\omega_b)$, then the global decoherence functions are products of the local ones, i.e.,\n$\\kappa_{ab} (t) = \\kappa_a (t)  \\kappa_b (t)$ and $\\Lambda_{ab} (t)=  \\kappa_a (t)  \\kappa^{\\ast}_b (t)$. Subsequently, the map for the bipartite photon system is tensor product of the local CP maps $\\hat{\\Phi}_t=\\hat{\\Phi}_t^{(a)} \\otimes \\hat{\\Phi}_t^{(b)}$. However, when the initial frequency distribution does not factorize, $P(\\omega_a,\\omega_b)\\neq P_a(\\omega_a)\\times P_b(\\omega_b)$, and contains correlations, then the map for the bipartite system is not anymore product of the local maps, $\\hat{\\Phi}_t\\neq\n\\hat{\\Phi}_t^{(a)} \\otimes \\hat{\\Phi}_t^{(b)}$~\\cite{ElsiPRL2012}. Now, we are interested in how to derive the generator of the corresponding non-local bipartite dynamical map and  what are the modifications in the corresponding  dephasing master equations when the amount of initial frequency correlations change.\n\nWe begin our derivation by writing the dynamical map formally as\n\\begin{equation}\\label{eq:dynMap}\n\\hat{\\Phi}_t = \\exp \\Big[\\int_0^t d\\tau \\hat{\\mathcal{L}}_{\\tau}\\Big],\n\\end{equation}\nwhere $\\hat{\\mathcal{L}}_t$ is the generator of the dynamics. Finding an expression for the generator then provides us the master equation we want to construct as\n\\begin{equation}\\label{eq:formalME}\n\\frac{d}{dt}\\hat{\\rho}_S(t)=\\hat{\\mathcal{L}}_t [\\hat{\\rho_S}(t)].\n\\end{equation}\nProvided that the map in Eq.~\\eqref{eq:dynMap} is invertible and its derivative is well-defined, one can obtain the generator as\n\\begin{equation}\\label{eq:genMapInvMap}\n\\mathcal{\\hat{L}}_t=\\frac{d}{dt}\\hat{\\Phi}_t \\circ \\hat{\\Phi}^{-1}_t.\n\\end{equation}\nTo find the generator in Eq.~\\eqref{eq:genMapInvMap} we need a suitable representation for the dynamical map $\\hat{\\Phi}_t$. With this in mind, we expand the two-photon density matrix $\\hat{\\rho}_S(t)$ in terms of a complete and orthonormal operator basis $\\{\\hat{F_{\\alpha}}\\}$. Specifically, we choose here fifteen generators of $\\mathrm{SU}(4)$, whose exact  expressions can be found in~\\cite{Alicki1987}, plus $\\hat{F}_1=\\hat{I}\/\\sqrt{4}$, such that $\\mathrm{Tr}[\\hat{F}_i^{\\dagger} \\hat{F}_j]=\\delta_{i j}$. It is worth mentioning that one can alternatively use the basis constructed by the tensor product of Pauli matrices plus the identity.  Fixing the basis for the representation, then the two-photon polarization state at time $t$ is\n\\begin{equation}\\label{eq:BlochVec1}\n\\hat{\\rho}_S(t)=\\sum_{\\alpha=1}^{16} r_{\\alpha} (t)\\hat{F}_{\\alpha}, \\qquad r_{\\alpha}(t)= \\mathrm{Tr}[\\hat{F}_{\\alpha}\\hat{\\rho}_S(t)],\n\\end{equation}\nwhere coefficients $\\{r_{\\alpha}\\}$ form the generalized Bloch vector corresponding to the state $\\hat{\\rho}_S(t)$ as\n\\begin{equation}\\label{eq:BlochVec2}\n\\vec{r}(t)=(1\/2,r_2(t),...,r_{16}(t))^{\\mathrm{T}}.\n\\end{equation} \nBy using Eq.~\\eqref{eq:BlochVec1} for both $\\hat{\\rho}_S(t)$ and $\\hat{\\rho}_S(0)$, we can write Eq.~\\eqref{eq:DynMap} as\n\\begin{equation}\\label{eq:MapBlochRep}\nr_{\\alpha}(t)=\\sum_{\\beta}[\\hat{\\Phi}_t]_{\\alpha \\beta} r_{\\beta}(0),\n\\end{equation}\nwhere $[\\hat{\\Phi}_t]$ is the transformation matrix corresponding to the map $\\hat{\\Phi}_t$ represented in the basis $\\{\\hat{F_{\\alpha}}\\}$. Elements of this matrix depend on the decoherence functions given in \nEqs.~(\\ref{k-loc}-\\ref{k2-nloc}) and each column can be systematically calculated by using a proper pair of initial and evolved states (c.f.~Eq.~\\eqref{eq:MapTwoPhotoon}). One can proceed to find the matrix representation of the generator by calculating the derivative and inverse of $[\\hat{\\Phi}_t]$ and using them in Eq.\\eqref{eq:genMapInvMap}, such that\n\\begin{equation}\\label{eq:genMat}\n[\\hat{\\mathcal{L}}_t]=\\frac{d}{dt}[\\hat{\\Phi}_t].[\\hat{\\Phi}_t]^{-1},\n\\end{equation}\nwhere we have replaced operator multiplication by matrix multiplication.\n\nLet us now consider the generator in a Lindblad operator form\n\\begin{eqnarray}\\label{eq:LindbladGen}\n\\hat{\\mathcal{L}}_t[\\hat{\\rho}_S(t)]&&=-i[\\hat{H}(t),\\hat{\\rho}_S(t)]+\n\\\\\n&&\\sum_{\\alpha=2}^{16}\\sum_{\\beta=2}^{16} R_{\\alpha \\beta}(t)\\Big(\\hat{F}_{\\alpha}\\hat{\\rho}_S(t) \\hat{F}_{\\beta}^{\\dagger}-\\frac{1}{2}\\{\\hat{F}_{\\beta}^{\\dagger}\\hat{F}_{\\alpha},\\hat{\\rho}_S(t)\\}\\Big), \\nonumber\n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\hat{H}(t)=\\frac{-1}{2i}\\sum_{\\alpha=2}^{16}\\Big[R_{\\alpha1}(t)\\hat{F}_{\\alpha}-R_{1\\alpha}(t)^{*}\\hat{F}^{\\dagger}_{\\alpha}\\Big],\n\\end{equation}\ncaptures the environment induced coherent dynamics and $R_{\\alpha \\beta}(t)$ with $\\alpha, \\beta =2,3,...,16$ are elements of a $15\\times 15$ matrix providing the decay rates. Each element in the matrix representation of the generator then reads\n\\begin{equation}\\label{eq:equality}\n[\\hat{\\mathcal{L}}_t]_{\\alpha\\beta}=\\mathrm{Tr}[\\hat{F}_{\\alpha}^{\\dagger} \\hat{\\mathcal{L}}_t[\\hat{F}_{\\beta}]].\n\\end{equation}\nHere we use Eq.~\\eqref{eq:LindbladGen} in the right hand side. Finally, by elementwise comparison of Eq.~\\eqref{eq:equality} with Eq.~\\eqref{eq:genMat} we find the decay rates of the Lindblad master equation in Eq.~\\eqref{eq:LindbladGen} in terms of the decoherence functions in Eqs.~(\\ref{k-loc}-\\ref{k2-nloc}).\nBefore proceeding further, let us note that generator of a CP-divisible map always has a Linblad form \\cite{Alicki1987,GoriniJmathPhys1976,LindbladCommun1976,RivasBook_2012}. A map $\\hat{\\Phi}_t$ is CP-divisible  if it can be decomposed as $\\hat{\\Phi}_t=\\hat{\\Phi}_{t,s}\\hat{\\Phi}_s$ where the intermediate map $\\hat{\\Phi}_{t,s}$ is also a legitimate CP-map for all $t \\geqslant s \\geqslant0 $ \\cite{RivasPRL2010}. In this paper, however, we do not restrict ourselves to the CP-divisible maps and as we show later we also take non-Markovian dynamics into account.\n\nAfter finding the general expression for the decay rate matrix, it turns out that it is quite sparse and can be reduced to a $3\\times3$ matrix, which we denote by $R(t)$. The corresponding subspace is spanned by only three generators of $\\mathrm{SU}(4)$, which are linearly dependent on the operators $\\hat{I}_2 \\otimes \\hat{\\sigma}_z$, $\\hat{\\sigma}_z \\otimes \\hat{I}_2$, and $\\hat{\\sigma}_z \\otimes \\hat{\\sigma}_z$. This is indeed intuitive because population elements of the density matrix are invariant upon a dephasing channel, so those terms that couple the levels must be absent. \nThe explicit expression for the matrix $R(t)$, corresponding to a general frequency distribution, is provided in the Appendix. Considering this general result, we diagonalize it to rewrite the second term on r.h.s of Eq.~\\eqref{eq:LindbladGen} in the form\n\\begin{equation}\\label{eq:NormalizedME}\n\\hat{\\mathcal{D}}[\\hat{\\rho}_S(t)]=\\sum_{\\alpha=1}^{3} \\gamma_{\\alpha}(t)\\Big[\\hat{J}_{\\alpha}\\hat{\\rho}_S(t)\\hat{J}_{\\alpha}^{\\dagger}-\\frac{1}{2}\\{\\hat{J}_{\\alpha}^{\\dagger}\\hat{J}_{\\alpha},\\hat{\\rho}_S(t)\\}\\Big],\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{pmatrix}\n\\gamma_1(t) & 0 & 0 \\\\\n0 & \\gamma_2(t) & 0 \\\\\n0 & 0 & \\gamma_3(t)\n\\end{pmatrix}=U R(t) U^{\\dagger}, \\quad \\quad \\hat{J}_{\\alpha}=\\sum_j U_{\\alpha j}\\hat{F}_j,\n\\end{equation}\nand $U$ is the orthogonal transformation which diagonalizes the matrix $R(t)$. It is worth stressing that if the dynamical map in hand is CP-divisible, then all decay rates will be non-negative, i.e. $\\gamma_i(t)\\geq 0$ for all interaction times $t\\geq 0$. \n\nAbove general results hold for arbitrary initial frequency distributions.\nIn the following, we discuss explicitly initially correlated joint frequency distributions for bivariate single- and double-peak Gaussian cases.\nThese choices are motivated by their use in recent theoretical and experimental works, see e.g.~\\cite{ElsiNature2011,ElsiPRL2012,HamedaniRaja_2017}, and due to their ability to account for the explicit influence of frequency correlations in the dephasing dynamics.\n\n\\subsection{Single-peak bivariate Gaussian distribution}\n\nConsider the joint bivariate  Gaussian frequency distribution $P_{ab}(\\omega_a, \\omega_b)$\nand its covariance matrix $C$, such that $C_{ij}=\\langle \\omega_i \\omega_j\\rangle-\\langle\\omega_i\\rangle\\langle \\omega_j \\rangle$ for $i,j=a,b$ \\citep{ElsiPRL2012}.\nThe correlation coefficient is now given by $ K=C_{ab}\/\\sqrt{C_{aa}C_{bb}}$, such that $-1\\leq K \\leq 1$. A fully anti-correlated initial frequency distribution has $K=-1$, which dictates that for any pair of $\\omega_a$ and $\\omega_b$ we have $\\omega_a+\\omega_b\\equiv \\omega_0$, with some constant frequency $\\omega_0$.\nThe means of the local single photon frequency distributions are given by $(\\bar{\\omega}_a,\\bar{\\omega}_b)^{\\mathrm{T}}$ and we denote the difference between the local means as $\\bar{\\omega}_a-\\bar{\\omega}_b =\\Delta\\omega$ and their sum as $\\bar{\\omega}_a+\\bar{\\omega}_b =\\omega_0$. \nUsing Eqs.~(\\ref{k-loc}-\\ref{k2-nloc}) and denoting the variance of the distribution by $\\sigma^2$,  the decoherence functions become\n\\begin{flalign}\n&\\kappa_a(t)=\\mathrm{exp}\\Big[\\frac{-\\sigma^2 \\Delta n^2 t^2-i \\Delta n t (\\omega_0+\\Delta\\omega)}{2}\\Big],\n\\\\\n&\\kappa_b(t)=\\mathrm{exp}\\Big[\\frac{-\\sigma^2 \\Delta n^2 t^2-i \\Delta n t (\\omega_0-\\Delta\\omega)}{2}\\Big],\n\\\\\n&\\kappa_{ab}(t)=\\mathrm{exp}\\big[-\\sigma^2 \\Delta n^2 t^2(1+K)-i \\Delta n t \\omega_0\\big],\n\\\\\n&\\Lambda_{ab}(t)=\\mathrm{exp}\\big[-\\sigma^2 \\Delta n^2 t^2(1-K)-i \\Delta n t \\Delta\\omega\\big].\n\\end{flalign}\nIt is straightforward to check that the corresponding transformation matrix $[\\hat{\\Phi}_t]$ for the generalized Bloch vector is always invertible when time $t$ is finite. \nAfter inserting the above expressions for the decay rate matrix $R(t)$, see the Appendix, and followed by diagonalization, we obtain the rates  appearing in the master equation~\\eqref{eq:NormalizedME} as follows\n\\begin{flalign}\n&\\gamma_1(t)=2(1-K)\\sigma^2 \\Delta n^2 t,\\label{eq:gamma1}\n\\\\\n&\\gamma_2(t)=2(1+K)\\sigma^2 \\Delta n^2 t,\\label{eq:gamma2}\n\\\\\n&\\gamma_3(t)=0,\\label{eq:gamma3}\n\\end{flalign}\nand the corresponding jump operators \n\\begin{flalign}\n&\\hat{J}_1=\\frac{1}{2\\sqrt{2}}(\\hat{I}_2\\otimes \\hat{\\sigma}_z + \\hat{\\sigma}_z \\otimes \\hat{I}_2),\\label{eq:J1}\n\\\\\n&\\hat{J}_2=\\frac{1}{2\\sqrt{2}}(\\hat{I}_2\\otimes \\hat{\\sigma}_z - \\hat{\\sigma}_z \\otimes \\hat{I}_2),\\label{eq:J2}\n\\\\\n&\\hat{J}_3=\\frac{1}{2}\\hat{\\sigma}_z \\otimes \\hat{\\sigma}_z.\\label{eq:J3}\n\\end{flalign}\n\nDephasing rates $\\gamma_1$ and $\\gamma_2$ are linear functions of time and their slopes depend on the correlation coefficient $K$. Figure~\\ref{fig1} displays the rates for  $K=-1,0,1$. Since all the rates are non-negative and the first two are time dependent, this leads to CP-divisible dynamics which, however, does not fulfil the LGKS semigroup property. It is also interesting to note here the absence of the jump operator $\\hat{\\sigma}_z \\otimes \\hat{\\sigma}_z$ since the corresponding rate $\\gamma_3$ is always equal to zero. \nMoreover, the role of the environmental correlation coefficient $K$ of the initial joint frequency distribution is now explicit in expressions~(\\ref{eq:gamma1}-\\ref{eq:gamma3}). When $K=1$ ($K=-1$) the rate $\\gamma_1=0$ ($\\gamma_2=0$) and we are left with only one dephasing channel given by $\\hat{J}_2$ ($\\hat{J}_1$).  When there are no initial correlations between the two environments, $K=0$, then \n$\\gamma_1(t)=\\gamma_2(t)$. Subsequently, the corresponding generator and master equation contain equally weighted contributions of \n the two local jump operators $\\hat{J}_1$ and $\\hat{J}_2$. Changing the value of  the initial correlations $K$ allows then to tune the dynamics between the above mentioned extreme cases.\n\nIt is also worth discussing similarities and differences between our photonic model and the two-qubit model interacting with a common environment\n \\cite{PalmaPROC1996, ReinaPRA2002, CironeNEWJPhys2009, AddisPRA2013}. In the latter model two qubits are spatially separated by  a distance $D$, while they both interact with same physical and common bosonic environment. It is interesting that the master equation describing this model has the exact same operator form and jump operators \\cite{AddisPRA2013} obtained in equations \\eqref{eq:J1} to \\eqref{eq:J3}. In addition, decay rates derived in \\cite{AddisPRA2013} exhibit similar dependence on the distance $D$, as our decay rates here depend on the correlation coefficient $K$. Moreover, when $D \\rightarrow \\infty$, the dynamical map will be factorized to $\\hat{\\Phi}_t=\\hat{\\Phi}_t^{(a)} \\otimes  \\hat{\\Phi}_t^{(b)}$, with the superscripts corresponding to each qubit. The same behavior is also captured here when $K \\rightarrow 0$.\n However, it is worth keeping in mind that in our case the two environments are distinct physical entities and the tuning of the generator -- or form of the master equation -- is obtained by changing the initial bipartite environmental state. Furthermore, we can tune the generator continuously between the fully correlated and anti-correlated cases.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.95 \\columnwidth, height = 1.5 cm]{gamma1SP.pdf}\n\\includegraphics[width=0.95 \\columnwidth, height = 1.5 cm]{gamma2.pdf}\n\\includegraphics[width=0.95 \\columnwidth ,height = 2.4 cm]{gamma3SP.pdf}\n\\caption{(Color online) Decay rates as a function of normalized interaction time in the case of single-peak Gaussian frequency distribution. Large-dashed green when $K=-1$, solid black when $K=0$, and small-dashed red when $K=-1$. Here we set $\\Delta \\omega \/\\sigma=2$.}\n\\label{fig1}\n\\end{figure} \n\n\n\\subsection{Double-peak bivariate Gaussian distribution}\n\nWe consider a double-peak frequency distribution as sum of two single-peak bivariate Gaussian distributions, already used in \\cite{HamedaniRaja_2017}, such that \n\\begin{equation}\nP(\\omega_a,\\omega_b)=[P_1(\\omega_a, \\omega_b)+ P_2(\\omega_a, \\omega_b)]\/2.\n\\end{equation}\nWe assume that both single-peak terms have the same correlation coefficient $K$ and standard deviation $\\sigma$, but their means are located at $(\\omega_0\/2-\\Delta \\omega\/2,\\omega_0\/2+\\Delta \\omega\/2)^{\\mathrm{T}}$ and $(\\omega_0\/2+\\Delta \\omega\/2,\\omega_0\/2-\\Delta \\omega\/2)^{\\mathrm{T}}$, respectively. Please note that the correlation coefficient $K$ of each single-peak distribution $P_1(P_2)$ does not equal to the actual correlation coefficient of the bivariate distribution $P$, obtained by its covariance matrix. In more detail, whenever we have nonzero $K$ for each single-peak, we have non-zero correlation in $P$. But note that if $K=0$, then we still have correlation in $P$ as long as we have non-zero peak separation, $\\Delta \\omega \\neq 0$.\n\nThe decoherence functions calculated from Eqs.~(\\ref{k-loc}-\\ref{k2-nloc}) become\n\\begin{flalign}\n&\\kappa_a(t)=\\mathrm{exp}\\bigg[\\frac{-\\sigma^2 \\Delta n^2 t^2-i t \\Delta n \\omega_0}{2}\\bigg]\\cos\\bigg(\\frac{t \\Delta n \\Delta \\omega}{2}\\bigg),\n\\\\\n&\\kappa_b(t)=\\kappa_a(t),\n\\\\\n&\\kappa_{ab}(t)=\\mathrm{exp}\\big[-\\sigma^2 \\Delta n^2 t^2 (1+K)-i t \\Delta n \\omega_0\\big],\n\\\\\n&\\Lambda_{ab}(t)=\\mathrm{exp}\\big[-\\sigma^2 \\Delta n^2 t^2 (1-K)\\big]\\cos(t \\Delta n \\Delta \\omega).\n\\end{flalign}\nBy using the earlier obtained general results, in a similar manner compared to single-peak case, we obtain the dephasing\nrates\n\\begin{flalign}\n&\\gamma_1(t)=2(1-K)\\sigma^2 \\Delta n^2 t+ \\tan(t \\Delta n  \\Delta \\omega )\\Delta n \\Delta \\omega ,\\label{eq:gamma1Two}\n\\\\\n&\\gamma_2(t)=2(1+K)\\sigma^2 \\Delta n^2 t,\\label{eq:gamma2Two}\n\\\\\n&\\gamma_3(t)= \\frac{1}{2} \\tan \\bigg(\\frac{t \\Delta n \\Delta \\omega}{2} \\bigg)\\big[1-\\sec (t \\Delta n \\Delta \\omega)\\big]\\Delta n \\Delta \\omega.\\label{eq:gamma3Two}\n\\end{flalign}\nThe corresponding jump operators $\\{\\hat{J}_1, \\hat{J}_2, \\hat{J}_3\\}$ are the same as in the single-peak case, see Eqs.~(\\ref{eq:J1}-\\ref{eq:J3}).\nIn the limit $\\Delta \\omega \\rightarrow 0$ corresponding to single peak case, the rates~(\\ref{eq:gamma1Two}-\\ref{eq:gamma3Two}) reduce to those given by\n Eqs.~(\\ref{eq:gamma1}-\\ref{eq:gamma3}).\n\nFigure~\\ref{fig2} displays the rates for $K=-1,0,1$.\nDephasing rate $\\gamma_2$ remains the same as in the single peak case. However, rate $\\gamma_1$ -- corresponding to $\\hat{J}_1$ including the sum of the local jump operators -- changes. The rate includes now an extra term, coming from the peak separation $\\Delta \\omega$,  \nand an oscillatory part displaying negative values of the rate as a function of time. This also leads to non-Markovian dephasing dynamics which is not CP-divisible. It is even more striking that introducing the double-peak frequency structure, opens now an additional dephasing channel since the rate $\\gamma_3$ is non-zero. Here, the corresponding jump operator $\\hat{J}_3=\\frac{1}{2}\\hat{\\sigma}_z \\otimes \\hat{\\sigma}_z$ displays a joint bipartite structure, in contrast to local features of $\\hat{J}_1$ and $\\hat{J}_2$. This is an interesting observation since the system-environment interaction Hamiltonian is the same as before having only local interactions, see Eq.~(\\ref{eq:HamiltonianTwoPhoton}), whilst the only change introduced was going from single- to double-peak structure of the initial bipartite environmental state. It is also worth noting that even though $\\gamma_3$ is independent \nof $K$, its functional form is non-trivial since it contains the peak separation $\\Delta \\omega$ and trigonometric functions. \n\nThere is a somewhat subtle mathematical point related to the behavior of rates $\\gamma_1$ [Eq.~\\eqref{eq:gamma1Two}] and $\\gamma_3$ [Eq.~\\eqref{eq:gamma3Two}] which needs an attention. Indeed, $\\gamma_1(t)$ and $\\gamma_3(t)$ diverge at isolated points of times. Subsequently, the corresponding dynamical maps are non-invertible at these points. According to the Eq.~\\eqref{eq:MapBlochRep}, the generalized Bloch vector of the two-photon polarization state at time $t$ reads\n\n\\begin{equation}\n\\vec{r}(t)=\\left(\n\\begin{array}{c}\n \\frac{1}{2} \\\\\n  \\Gamma_0\\cos(t\\Delta\\Omega\/2)[\\cos(t\\Omega_0\/2)r_2-\\sin(t\\Omega_0\/2)r_3] \\\\\n   \\Gamma_0\\cos(t\\Delta\\Omega\/2)[\\cos(t\\Omega_0\/2)r_3+\\sin(t\\Omega_0\/2)r_2] \\\\\n r_4 \\\\\n   \\Gamma_0\\cos(t\\Delta\\Omega\/2)[\\cos(t\\Omega_0\/2)r_5-\\sin(t\\Omega_0\/2)r_6] \\\\\n   \\Gamma_0\\cos(t\\Delta\\Omega\/2)[\\cos(t\\Omega_0\/2)r_6+\\sin(t\\Omega_0\/2)r_5] \\\\\n  \\Gamma_-\\cos(t\\Delta \\Omega)r_7 \\\\\n  \\Gamma_-\\cos(t\\Delta \\Omega)r_8 \\\\\n r_9 \\\\\n  \\Gamma_+[\\cos(t\\Omega_0)r_{10}-\\sin(t\\Omega_0)r_{11}] \\\\\n  \\Gamma_+[\\cos(t\\Omega_0)r_{11}+\\sin(t\\Omega_0)r_{10}] \\\\\n  \\Gamma_0\\cos(t\\Delta\\Omega\/2)[\\cos(t\\Omega_0\/2)r_{12}-\\sin(t\\Omega_0\/2)r_{13}] \\\\\n   \\Gamma_0\\cos(t\\Delta\\Omega\/2)[\\cos(t\\Omega_0\/2)r_{13}+\\sin(t\\Omega_0\/2)r_{12}] \\\\\n   \\Gamma_0\\cos(t\\Delta\\Omega\/2)[\\cos(t\\Omega_0\/2)r_{14}-\\sin(t\\Omega_0\/2)r_{15}] \\\\\n   \\Gamma_0\\cos(t\\Delta\\Omega\/2)[\\cos(t\\Omega_0\/2)r_{15}+\\sin(t\\Omega_0\/2)r_{14}] \\\\\n r_{16} \\\\\n\\end{array}\n\\right),\n\\end{equation}\nwhere we have defined $\\Gamma_0=\\exp[-\\sigma^2 \\Delta n^2 t^2\/2]$, $\\Gamma_{\\pm}=\\exp[-\\sigma^2 \\Delta n^2 t^2(1\\pm K)]$, $ \\Delta\\Omega= \\Delta n \\Delta \\omega$, $\\Omega_0=\\Delta n \\omega_0$, and $\\vec{r}(0)=(1\/2,r_2,r_3,...,r_{16})^T$ is the initial Bloch vector. One can check that all of the different initial vectors (states) that share the same values of $r_4,r_7,r_8,r_9,r_{10},r_{11}$, and $r_{16}$ are mapped to the same vector (state) at $t=\\pi \/\\Delta \\Omega$. This many to one nature of the map --  at these isolated times -- makes it non-invertible. Although all the trajectories corresponding to the aforementioned initial vectors end up together at the isolated points, it is evident that they continue their different paths immediately after this. This can be seen in the following way. Consider the generator of the master equation in matrix form and its action on the generalized Bloch vector.\nWe see that while some rates diverge at certain points in time, it is precisely at these points that the generalized Bloch vector components -- with which the rates get multiplied -- all go to zero. In more detail, we have\n\\begin{equation}\n\\frac{d}{dt} r_{\\alpha}(t)=\\sum_{\\beta}[\\hat{\\mathcal{L}}_t]_{\\alpha \\beta} \\, r_{\\beta}(t),\n\\end{equation}\ntherefore, the product of the divergent rate with the zero value component leads to a finite rate of change of the Bloch vector which allows us to continue propagation of each state forward in time. Accordingly, following the trajectories immediately before they unite at a single point, lets us identify each one of them immediately after that, when they separate again. We see therefore that in spite of the divergences in the rates, the master equation we have obtained describes the dephasing evolution of the two-photon polarization state in meaningful way.\nIt is also worth noting that the divergent decoherence rates in master equations have appeared in earlier literature many times, e.g., in the prominent resonant Jaynes-Cummings model~\\cite{BreuerBook_2002}.\n \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.95 \\columnwidth, height = 1.5 cm]{gamma1DP.pdf}\n\\includegraphics[width=0.95 \\columnwidth, height = 1.5 cm]{gamma2.pdf}\n\\includegraphics[width=0.95 \\columnwidth ,height = 2.4 cm]{gamma3DP.pdf}\n\\caption{(Color online) Decay rates as a function of normalized interaction time in the case of double-peak Gaussian frequency distribution. Dashed green when $K=-1$, solid black when $K=0$, and dot-dashed when $K=-1$. Here we set $\\Delta \\omega \/\\sigma=2$.}\n\\label{fig2}\n\\end{figure} \n\\section{Single-photon dephasing with initial polarization-frequency correlations} \\label{sec4}\n\nWe described above how initial correlations between the composite environmental states influence the generator of the dynamical map and the corresponding master equation for photonic dephasing. In this section we continue with initial correlations but \ntake a different perspective by considering non-factorized initial system-evironmental state for single qubit.\nThis is motivated by the recent observation that initial system-environment correlations can be exploited for arbitrary control of single qubit dephasing~\\cite{Lyyra_2018}. We revisit this problem and obtain new insight by exploiting the very recently developed general method of \\textit{bath positive decomposition} ($\\mathrm{B+}$ decomposition) ~\\cite{Wiseman_2019}. \nIn general,  presence of initial system-environment correlations implies that the open system evolution is not described by a CP dynamical map   \\cite{pechukas94,Alicki_1995,pechukas95,jordan04,shaji05a,Linta-Shaji}. However, $\\mathrm{B+}$ decomposition method allows to treat this case with a set of CP maps, where each term of the decomposition is evolved over time with its individual CP map~\\cite{Wiseman_2019}. \n\n\\subsection{Preliminaries on $\\mathrm{B+}$ decomposition for initially correlated system-environment state}\nFollowing~\\cite{Wiseman_2019} we begin by considering arbitrary system-environment state -- in the corresponding Hilbert space $\\mathcal{H}=\\mathcal{H}_S\\otimes \\mathcal{H}_E$ -- and write it as \n\\begin{equation}\\label{eq:BplusDecom}\n\\hat{\\rho}_{SE}(0)=\\sum_{\\alpha}w_{\\alpha}\\hat{Q}_{\\alpha}\\otimes \\hat{\\rho}_{\\alpha}.\n\\end{equation} \nHere, $\\{\\hat{Q}_{\\alpha}\\}$ forms a basis (possibly overcomplete) for operators on $\\mathcal{H}_S$ and $\\{\\hat{\\rho}_{\\alpha}\\}$ are valid environmental density operators on $\\mathcal{H}_E$. Note that $\\hat{Q}_{\\alpha}$ need not be positive or trace orthogonal, so they may not constitute proper density matrices on the system Hilbert space. However when the initial state is factorized, this summation reduces to a single term $\\hat{\\rho}_{SE}(0)=\\hat{\\rho}_S(0)\\otimes \\hat{\\rho}_E(0)$ corresponding to reduced states of the open system and environment, respectively. In general, number of terms in this summation is restricted by $1\\leq N\\leq d^2$ where $d$ is the dimension of the system Hilbert space \\cite{Wiseman_2019}. All the information about initial state of the open system is incorporated in the weights $w_{\\alpha}$, such that $\\hat{\\rho}_S(0)=\\mathrm{Tr}_E[\\hat{\\rho}_{SE}(0)]=\\sum w_{\\alpha}\\hat{Q}_{\\alpha}$. Although $\\hat{Q}_{\\alpha}$ may not be legitimate density operators for the open system, those expressed by $\\hat{\\rho}_{\\alpha}$ are valid density operators for the environment. This means that the factorized form of the terms in \\eqref{eq:BplusDecom} allows to write the dynamics of the open system state as the weighted sum of legitimate CP-maps acting on $\\hat{Q}_{\\alpha}$. In more detail, if the total system-environment evolves due to a unitary operator $\\hat{U}(t)$, one has\n\\begin{eqnarray}\n\\hat{\\rho}_S(t)&&=\\sum_{\\alpha}w_{\\alpha}\\mathrm{Tr}_E[\\hat{U}(t)(\\hat{Q}_{\\alpha} \\otimes \\hat{\\rho}_{\\alpha})\\hat{U}(t)^{\\dagger}] \\nonumber\n\\\\\n&&=\\sum_{\\alpha}w_{\\alpha}\\hat{\\Phi}^{(\\alpha)}_t[\\hat{Q}_{\\alpha}],\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{eq:BPlusDecMap}\n\\hat{\\Phi}_t^{(\\alpha)}[\\cdot]:=\\mathrm{Tr}_E[\\hat{U}(t)(\\cdot\\otimes\\hat{\\rho}_{\\alpha})\\hat{U}(t)^{\\dagger}].\n\\end{equation}\nSince all maps of the form given in Eq.\\eqref{eq:BPlusDecMap} are CP, all previous tools for studying CP-maps are applicable here. In particular, one can investigate properties of each CP-map $\\Phi^{(\\alpha)}_t$ and see how they are connected to the presence of initial correlations.\n\nFor example, consider single qubit dynamics in the presence of initial system-environment correlations~\\cite{Wiseman_2019} . Using completeness of Pauli sigma basis $\\{I_2,\\hat{\\sigma}_x,\\hat{\\sigma}_y,\\hat{\\sigma}_z\\}$, we have \n\\begin{equation}\\label{eq:BplusDecQubit}\n\\hat{\\rho}_{SE}(0)=\\sum_{\\alpha=0,x,y,z} w_{\\alpha} \\hat{Q}_{\\alpha}\\otimes \\hat{\\rho}_{\\alpha},\n\\end{equation}\nin which\n\\begin{eqnarray}\n&&\\hat{Q}_0=\\frac{1}{2}(\\hat{I}_2 - \\hat{\\sigma}_x - \\hat{\\sigma}_y - \\hat{\\sigma}_z),\\label{eq:Q0}\\\\\n&&\\hat{Q}_{\\alpha}=\\frac{1}{2}\\hat{\\sigma}_{\\alpha} \\quad for \\; \\alpha = x,y,z,\\label{eq:Qalpha}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n&&\\hat{\\rho}_0=\\mathrm{Tr}_S[\\hat{\\rho}_{SE}(0)]=\\hat{\\rho}_E(0),\\label{rho1}\n\\\\\n&&\\hat{\\rho}_{\\alpha}=\\frac{\\mathrm{Tr}_S[((\\hat{I}_2+\\hat{\\sigma}_{\\alpha})\\otimes \\hat{I}_E)\\hat{\\rho}_{SE}(0)]}{w_{\\alpha}},\\label{rho2}\n\\end{eqnarray}\nwith $w_0=1$, and $w_{\\alpha}=\\mathrm{Tr}[((\\hat{I}_2+\\hat{\\sigma}_{\\alpha})\\otimes \\hat{I}_E)\\hat{\\rho}_{SE}(0)]$ for ${\\alpha}=x,y,z$.\nWe exploit these generic expressions below.\n\n\\subsection{Initial polarization-frequency correlation and $\\mathrm{B+}$ decomposition for single photon}\n\nWe consider initial polarization-frequency correlations by following the recent results and experimental work on generating, in principle, arbitrary single-photon dephasing dynamics~\\cite{Lyyra_2018}. Generic initial polarization-frequency state can be written as\n\\begin{eqnarray}\\label{eq:InStateCorrelated}\n\\ket{\\psi(0)}_{SE}=&& C_v\\ket{v}\\otimes\\int d \\omega g(\\omega)\\ket{\\omega}\\nonumber\n\\\\\n&&\\quad+C_h\\ket{h}\\otimes\\int d \\omega g(\\omega)\\mathrm{e}^{i \\theta(\\omega)}\\ket{\\omega},\n\\end{eqnarray}\nwhere $\\vert C_h \\vert ^2+\\vert C_v \\vert ^2=1$ and $\\int d \\omega \\vert g(\\omega ) \\vert ^2=1$.\nAbove, the crucial ingredient is the frequency dependent initial phase $\\theta(\\omega)$ for the component including the polarization $h$.\nIf $\\theta(\\omega)$ is a constant function, then there are no initial system-environment correlations. However, controlling the non-constant functional form of $\\theta(\\omega)$\nallows to control the initial correlations and their amount.\n  \nWhen the initial state evolves according to the interaction Hamiltonian in Eg.~\\eqref{eq:Hamiltonian}, the reduced polarization state at time $t$ is\n\\begin{equation}\\label{eq:MapInCorrel}\n\\hat{\\rho}(t)=\\begin{pmatrix}\n\\vert C_h \\vert ^2 & \\kappa(t)C_h C_v^{*} \\\\\n\\kappa(t)^{*}C_h^{*} C_v & \\vert C_v \\vert ^2\n\\end{pmatrix},\n\\end{equation}\nwhere the decoherence function is \n\\begin{equation}\\label{eq:decFuncCorrel}\n\\kappa(t)=\\int d \\omega \\vert g(\\omega) \\vert ^2  \\mathrm{e}^{i \\theta(\\omega)}\\mathrm{e}^{-i \\Delta n \\omega t}.\n\\end{equation}\nNote that in addition to the frequency probability distribution $|g(\\omega)|^2$, one can now use also $\\theta(\\omega)$, and subsequent initial correlations, to control the dephasing dynamics.\n\nThe dynamics given by Eqs.~(\\ref{eq:MapInCorrel}-\\ref{eq:decFuncCorrel}) can be equivalently formulated by using the $\\mathrm{B}+$ decomposition. \nConsidering the initial total state in Eq.~\\eqref{eq:InStateCorrelated}, and applying the $\\mathrm{B}+$ decomposition along Eq.~\\eqref{eq:BplusDecQubit} and  Eq.~\\eqref{rho1}-\\eqref{rho2}, we obtain environmental terms\n\\begin{widetext}\n\\begin{eqnarray}\n&&\\hat{\\rho}_0=\\int d\\omega \\int d\\omega'\\; g(\\omega)g(\\omega')^{*}\\;\\big(\\vert C_h\\vert ^2 \\mathrm{e}^{i[\\theta(\\omega)-\\theta(\\omega')]}+\\vert C_v\\vert ^2\\big)\\ket{\\omega}\\bra{\\omega'},\n\\\\\n&&\\hat{\\rho}_x=\\frac{1}{w_x}\\int d\\omega \\int d\\omega'\\; g(\\omega)g(\\omega')^{*}\\;\\big(\\vert C_h\\vert ^2 \\mathrm{e}^{i[\\theta(\\omega)-\\theta(\\omega')]}+\\vert C_v\\vert ^2+C_h C_v^{*}\\mathrm{e}^{i\\theta(\\omega')}+C_v C_h^{*}\\mathrm{e}^{-i\\theta(\\omega)}\\big)\\ket{\\omega}\\bra{\\omega'},\n\\\\\n&&\\hat{\\rho}_y=\\frac{1}{w_y}\\int d\\omega \\int d\\omega' \\;g(\\omega)g(\\omega')^{*}\\;\\big(\\vert C_h\\vert ^2 \\mathrm{e}^{i[\\theta(\\omega)-\\theta(\\omega')]}+\\vert C_v\\vert ^2+i C_h C_v^{*}\\mathrm{e}^{i\\theta(\\omega')}- i C_v C_h^{*}\\mathrm{e}^{-i\\theta(\\omega)}\\big)\\ket{\\omega}\\bra{\\omega'},\n\\\\\n&&\\hat{\\rho}_z=\\int d\\omega \\int d\\omega'\\; g(\\omega)g(\\omega')^{*}\\;\\ket{\\omega}\\bra{\\omega'},\n\\end{eqnarray}\n\\end{widetext}\nwith weights\n\\begin{eqnarray}\n&&w_x=1+2\\int d\\omega \\vert g(\\omega)\\vert ^2 \\Re{[C_v C_h^{*}\\mathrm{e}^{-i\\theta(\\omega)}]},\n\\\\\n&&w_y=1+2\\int d\\omega \\vert g(\\omega)\\vert ^2 \\Im{[C_v C_h^{*}\\mathrm{e}^{-i\\theta(\\omega)}]},\n\\\\\n&&w_z=2\\vert C_h\\vert ^2.\n\\end{eqnarray}\nEach specific term of the $\\mathrm{B}+$ decomposition is related to a frequency state ($\\hat{\\rho}_{\\alpha}$) above, and acts on its own input system operator $\\hat{Q}_{\\alpha}$, see Eqs~(\\ref{eq:Q0}-\\ref{eq:Qalpha}). In the current case, we can combine the contributions of $\\hat{\\rho}_0$ and $\\hat{\\rho}_z$ to simplify the decomposition into only three terms. Subsequently, polarization density matrix at time $t$ is given by  \n\\begin{eqnarray}\\label{eq:MapInCorrThreeTerms}\n\\hat{\\rho}(t)&=&\\frac{1}{2}\\begin{pmatrix} \\nonumber\nw_z & \\kappa_0(t)(i-1) \\\\\n\\kappa_0(t)^{*}(-i-1) & 2-w_z\n\\end{pmatrix}\n\\\\\n&&+\\frac{1}{2} w_x\\begin{pmatrix} \\nonumber\n0 & \\kappa_x(t) \\\\\n\\kappa_x(t)^{*} & 0\n\\end{pmatrix}\n\\\\\n&&+\\frac{1}{2} w_y\\begin{pmatrix}\n0 & -i \\kappa_y(t) \\\\\ni \\kappa_y(t)^{*} & 0\n\\end{pmatrix},\n\\end{eqnarray}\nwhere the three different decoherence functions are given by\n\\begin{flalign}\n&\\kappa_0(t)=\\int d \\omega \\vert g(\\omega)\\vert ^2 \\mathrm{e}^{-i \\Delta n \\omega t}, \\label{d1}\n\\\\\n&\\kappa_x(t)=\\frac{\\int d \\omega \\vert g(\\omega)\\vert ^2 (1+2 \\Re{[C_v C_h^{*}\\mathrm{e}^{-i\\theta(\\omega)}]})\\mathrm{e}^{-i \\Delta n \\omega t}}{w_x}, \\label{d2}\n\\\\\n&\\kappa_y(t)=\\frac{\\int d \\omega \\vert g(\\omega)\\vert ^2 (1+2 \\Im{[C_v C_h^{*}\\mathrm{e}^{-i\\theta(\\omega)}]})\\mathrm{e}^{-i \\Delta n \\omega t}}{w_y}.\\label{d3}\n\\end{flalign}\nIt is interesting to note here that the decoherence function $\\kappa_0$ is independent of $\\theta(\\omega)$ and actually corresponds directly to the case when there are no initial polarization-frequency correlations.\nThe other two functions, $\\kappa_x$ and $\\kappa_y$, depend also on $\\theta(\\omega)$ and describe in detail how the initial correlations change the dephasing dynamics.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{kSimDAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{k0DAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{kxDAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth ,height = 2 cm]{kyDAbs.pdf}\n\\caption{(Color online) Non-positive map decoherence functions. Magnitudes of the original decoherence function ($\\kappa$) and $\\mathrm{B}+$ decomposition decoherence functions ($\\kappa_0, \\kappa_x, \\kappa_y$) as a function of time. We set $C_h=C_v=1\/\\sqrt{2}$.}\\label{fig3}\n\\end{figure}\n\n\nIt is also interesting to compare Eq.~\\eqref{eq:MapInCorrThreeTerms} with the $\\mathrm{B+}$ decomposition for generic dephasing dynamics of a qubit coupled to a Bosonic bath, when qubit and bath are initially correlated \\citep{Wiseman_2019}. The total Hamiltonian of the qubit and the Bosonic bath reads\n\\begin{equation}\n\\hat{H}=\\omega_q \\; \\hat{\\sigma}_z+\\sum_i \\; \\omega_i \\hat{b}^{\\dagger}_i \\hat{b}_i+\\hat{\\sigma}_z\\otimes \\sum_i g_i (\\hat{b}^{\\dagger}_i+\\hat{b}_i),\n\\end{equation}\nwhere $\\omega_q$ is the qubit's energy level separation (in $\\ket{0}$, $\\ket{1}$ basis), $\\hat{b}^{\\dagger}_i$ and $\\hat{b}_i$ are bath mode creation and annihilation operators respectively, and $g_i$ is the coupling strength. Employing the $\\mathrm{B+}$ decomposition, dynamics of the off-diagonal element of the qubit's density matrix in the interaction picture reads \\citep{Wiseman_2019}\n\\begin{equation}\\label{eq:dephasScale}\n\\bra{0}\\rho_S(t)\\ket{1}=\\sum_{\\alpha}\\;w_{\\alpha} \\bra{0}\\hat{Q}_{\\alpha}\\ket{1}\\chi_{\\hat{\\rho}_{\\alpha}}(\\vec{\\xi}_t),\n\\end{equation}\nwhere $\\chi_{\\hat{\\rho}_{\\alpha}}(\\vec{\\xi}_t)=\\mathrm{Tr}_B[\\hat{\\rho}_{\\alpha}\\hat{D}(\\vec{\\xi}_t)]$ is the Wigner characteristic function of the bath state $\\hat{\\rho}_{\\alpha}$. Above $\\vec{\\xi}_t=(\\xi_1(t),\\xi_2(t),...)$ with\n\\[\\xi_j(t)=2g_j \\bigg(\\frac{1-\\mathrm{e}^{i\\omega_j t}}{\\omega_j}\\bigg) ,\\]\nand $\\hat{D}(\\vec{\\xi}_t)=\\exp(\\sum_i \\xi_i \\hat{b}^{\\dagger}_i+\\xi^{*}\\hat{b}_i)$ is the Glauber displacement operator. The comparison between \nEqs.~\\eqref{eq:MapInCorrThreeTerms} and \\eqref{eq:dephasScale} reveals that the decoherence functions in our photonic model -- corresponding to integral transformations of the frequency probability distribution and frequency dependent phase $\\theta(\\omega)$ --   play the exact same role as the characteristic functions in the dephasing dynamics of a qubit coupled to Bosonic bath.\n\nLet us go back to the photonic model and see in detail, for some examples, what is the relation between the orginal decoherence function \\eqref{eq:decFuncCorrel} and those appearing in the $\\mathrm{B}+$ decomposition in Eqs.~(\\ref{d1}-\\ref{d3}). In particular, we consider the similar cases as used in~\\citep{Lyyra_2018} to demonstrate arbitrary control of dephasing dynamics. These include a non-positive map, Markovian, non-Markovian, and coherence trapping dynamics.\nIn all of the cases below, the frequency distributions used and the values for $\\theta(\\omega)$ are similar to those considered in Ref.~\\citep{Lyyra_2018}, respectively.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.83 \\columnwidth, height = 1.5 cm]{k0DReIm.pdf}\n\\includegraphics[width=0.83 \\columnwidth, height = 1.5 cm]{kxDReIm.pdf}\n\\includegraphics[width=0.83 \\columnwidth, height = 2 cm]{kyDReIm.pdf}\n\\caption{(Color online) Non-positive map decoherence functions. Real and imaginary parts of the original decoherence function ($\\kappa$) and $\\mathrm{B}+$ decomposition decoherence functions ($\\kappa_0, \\kappa_x, \\kappa_y$) as a function of time. We set $C_h=C_v=1\/\\sqrt{2}$.}\n\\label{fig4}\n\\end{figure} \n\n\nFigure \\ref{fig3} shows the magnitude of various decoherence functions for the case of a non-positive (NP) map, i.e., $\\kappa (t) > \\kappa(0)$. \n It is easy to check that the off-diagonal term of the density matrix is obtained from $\\rho_{hv}(t)=(\\kappa_0(t)(i-1)+w_x \\kappa_x(t)-iw_y \\kappa_y(t))\/2$, and equivalently from $\\rho_{hv}(t)=\\kappa(t)C_h C_v^{*}$. Thereby, it is evident  that if $\\kappa_0(t)=\\kappa_x(t)=\\kappa_y(t)=0$, for some $t>0$, then $\\kappa(t)=0$. However, the reverse statement does not always hold. Instead, one can show that whenever $w_x=w_y=1$, then having identical decoherence functions, $\\kappa_0(t)=\\kappa_x(t)=\\kappa_y(t)$, is sufficient to have zero coherence, i.e., $\\kappa(t)=0$. This is an interesting result making a link between properties of the CP-maps obtained in $\\mathrm{B+}$ decomposition and  the original non-positive map. In fact, the case discussed in Fig. \\ref{fig3} demonstrates this situation.\nThis is even more evident when considering the real and imaginary parts of the decoherence functions explicitly, see Fig.~\\ref{fig4}. \nOne can see that the three decoherence functions $\\kappa_0$, $\\kappa_x$, and $\\kappa_y$ are identical when the interaction time is short. Therefore, since we also have $w_x = w_y = 1$, the decoherence function $\\kappa(t)$ has zero value in this regime.\n\nThe non-Markovian, Markovian, and coherence trapping cases are plotted respectively in Figs.~\\ref{fig5}, \\ref{fig6}, and \\ref{fig7}. Looking at the Fig.~\\ref{fig5}, one finds that $\\vert \\kappa(t) \\vert$ first decays to zero and then it revives again. This situation displays non-Markovian features, where coherence can revive after a period of disappearance. The Markovian case however illustrates a monotonically decaying  $\\vert \\kappa (t) \\vert$, see Fig.~\\ref{fig6}. Finally in the coherence trapping case we observe that $\\vert \\kappa(t) \\vert$ decays at first but mostly maintains its value later. \nMagnitudes of the other three decoherence \nfunctions, used in the $\\mathrm{B+}$ decomposition, are also plotted in the corresponding figures. We observe that these decoherence functions behave similarly, in contrast to the case of non-positive map. Again, whenever $\\kappa_0$, $\\kappa_x$, and $\\kappa_y$ are all zero, one has $\\kappa(t)=0$. However since $w_x$ and $w_y$ are not equal, we get a non-zero $\\kappa$, even though, $\\kappa_0$, $\\kappa_x$, and $\\kappa_y$ seem to be identical in some regions.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{kSimAAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{k0AAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{kxAAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth ,height = 2 cm]{kyAAbs.pdf}\n\\caption{(Color online) Non-Markovian dynamics. Magnitudes of the original decoherence function ($\\kappa$) and $\\mathrm{B}+$ decomposition decoherence functions ($\\kappa_0, \\kappa_x, \\kappa_y$) as a function of time. We set $C_h=C_v=1\/\\sqrt{2}$.}\\label{fig5}\n\\end{figure} \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{kSimBAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{k0BAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{kxBAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth ,height = 2 cm]{kyBAbs.pdf}\n\\caption{(Color online) Markovian dynamics. Magnitudes of the original decoherence function ($\\kappa$) and $\\mathrm{B}+$ decomposition decoherence functions ($\\kappa_0, \\kappa_x, \\kappa_y$) as a function of time. We set $C_h=C_v=1\/\\sqrt{2}$.}\\label{fig6}\n\\end{figure} \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{kSimCAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{k0CAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth, height = 1.5 cm]{kxCAbs.pdf}\n\\includegraphics[width=0.8 \\columnwidth ,height = 2 cm]{kyCAbs.pdf}\n\\caption{(Color online) Coherence trapping dynamics. Magnitudes of the original decoherence function ($\\kappa$) and $\\mathrm{B}+$ decomposition decoherence functions ($\\kappa_0, \\kappa_x, \\kappa_y$) as a function of time. We set $C_h=C_v=1\/\\sqrt{2}$.}\n\\label{fig7}\n\\end{figure}\n\n\n\\section{Discussion} \\label{sec5}\nWe have studied the influence of initial correlations on open system dynamics from two perspectives corresponding to master equation descriptions and recently introduced $\\mathrm{B+}$ decomposition method. By using a common two-photon dephasing scenario with local polarization-frequency interaction, our results show explicitly how initial correlations -- between the composite environments (frequencies) -- influence the decoherence rates and operator form of the master equation for the polarization state.  When the environment has a single-peak gaussian structure, the master equation contains two sets of jump operators, corresponding to sum and difference between the local interactions, and  whose weights can be controlled by changing the amount of initial environmental correlations. Here, the dephasing rates are non-negative and depend linearly on time for the considered case. Having a double-peak bivariate structure, the situation changes drastically. This opens an additional dephasing path with a non-local form for the corresponding operator, and the associated rate also has divergences. Moreover, the rates for the other two dephasing operators have distinctive functional forms.\n\n $\\mathrm{B+}$ decomposition method, in turn, allows to study such cases where the system and environment are initially correlated, preventing the use of conventional CP-maps.  We have used this decomposition to study dephasing, when polarization and frequency of a single photon are initially correlated.\nThe results display in detail how the initial correlations change the dephasing contribution arising solely on initial factorized state.\nIndeed, instead of having one decoherence function associated to dephasing, we have now three different decoherence functions corresponding to the elements of the \n $\\mathrm{B+}$  decomposition. Here, one of the functions arises due to initially factorized part and additional two decoherence functions include also contributions from initial polarization-frequency correlations. In general, our results shed light and help in understanding how different types of correlations influence the dephasing dynamics within the commonly used photonic framework. \n \n\n\n\\section*{Acknowledgments}\nSina Hamedani Raja acknowledges finical support from Finnish Cultural Foundation. Anil Shaji acknowledges the support of the EU through the Erasmus+ program and support of the Science and Engineering Research Board, Government of India through EMR grant No. EMR\/2016\/007221.\n\\section*{Appendix}\nGeneral expressions for the elements of the nonzero subspace of the decay rate matrix, denoted by $R(t)$, are\n\\begin{widetext}\n\\begin{eqnarray}\n&&R_{11}(t)=-\\Re{[\\bold{k}_b(t)]},\n\\\\\n&&R_{12}(t)=\\frac{1}{\\sqrt{3}}\\Big(i\\Im{[\\bold{k}_b(t)]}-i\\Im{[\\bold{k}_a(t)]}+i\\Im{[\\bold{\\Gamma}_{ab}(t)]}+\\Re{[\\bold{\\Gamma}_{ab}(t)]}-\\Re{[\\bold{k}_{a}(t)]}\\Big),\n\\\\\n&&R_{13}(t)=\\frac{1}{2\\sqrt{6}}\\Big(4i\\Im{[\\bold{k}_a(t)]}+2i\\Im{[\\bold{k}_b(t)]}-3i\\Im{[\\bold{k}_{ab}(t)]}-i\\Im{[\\bold{\\Gamma}_{ab}(t)]}+4\\Re{[\\bold{k}_{a}(t)]}-3\\Re{[\\bold{k}_{ab}(t)]}-\\Re{[\\bold{\\Gamma}_{ab}(t)]}\\Big),\n\\\\\n&&R_{21}(t)=R_{12}(t)^{*},\n\\\\\n&&R_{22}(t)=\\frac{1}{3}\\Big(-2\\Re{[\\bold{k}_a(t)]}+\\Re{[\\bold{k}_b(t)]}-2\\Re{[\\bold{\\Gamma}_{ab}(t)]}\\Big),\n\\\\\n&&R_{23}(t)=\\frac{1}{6\\sqrt{2}}\\Big(-3i\\Im{[\\bold{k}_{ab}(t)]}+6i\\Im{[\\bold{k}_b(t)]}+3i\\Im{[\\bold{\\Gamma}_{ab}(t)]}-4\\Re{[\\bold{k}_{a}(t)]}-3\\Re{[\\bold{k}_{ab}(t)]}+8\\Re{[\\bold{k}_{b}(t)]}-\\Re{[\\bold{\\Gamma}_{ab}(t)]}\\Big),\n\\\\\n&&R_{31}(t)=R_{13}(t)^{*},\n\\\\\n&&R_{32}(t)=R_{23}(t)^{*},\n\\\\\n&&R_{33}(t)=\\frac{1}{6}\\Big(-2\\Re{[\\bold{k}_a(t)]}-3\\Re{[\\bold{k}_{ab}(t)]}-2\\Re{[\\bold{k}_b(t)]}+\\Re{[\\bold{\\Gamma}_{ab}(t)]}\\Big),\n\\end{eqnarray}\nwhere we have defined $\\bold{k}_i(t)=\\frac{1}{\\kappa_i(t)}\\frac{d}{dt}\\kappa_i(t)$, $\\bold{\\Gamma}_{ab}(t)=\\frac{1}{\\Lambda_{ab}(t)}\\frac{d}{dt}\\Lambda_{ab}(t)$ and $\\bold{k}_{ab}(t)=\\frac{1}{\\kappa_{ab}(t)}\\frac{d}{dt}\\kappa_{ab}(t)$.\n\\end{widetext}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}