diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzicyg" "b/data_all_eng_slimpj/shuffled/split2/finalzzicyg" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzicyg" @@ -0,0 +1,5 @@ +{"text":"\\section{Supplemental Material}\n\n\\setcounter{equation}{0} \\setcounter{figure}{0} \\setcounter{table}{0} %\n\\renewcommand{\\theequation}{S\\arabic{equation}} \\renewcommand{\\thefigure}{S%\n\\arabic{figure}} \\renewcommand{\\bibnumfmt}[1]{[S#1]} \\renewcommand{%\n\\citenumfont}[1]{S#1}\n\nIn the supplementary material, we will show the presence of singularities in the energy of the highest excited state in a\nsubspace with zero magnetization and show the effects of $c_2$ on the relaxation process.\n\nTo illustrate the existence of a singularity in the energy of the highest excited state, we plot the level's energy in Fig.~\\ref{figs1}. Clearly, the second derivative of the energy\nwith respect to $q$ exhibits a discontinuous jump at $q=\\pm 2 c_2$, implying the existence of a second-order excited state\nquantum phase transition there. This is consistent with the existence of a discontinuous jump for the first derivative of the order parameter $\\langle \\rho_0 \\rangle$ with respect to $q$ [see Fig. 1($\\text{a}_\\text{2}$)].\n\t\nTo show the effects of $c_2$ on the relaxation process, in Fig.~\\ref{figs2}, we plot the measured $\\langle \\rho_0\\rangle $ as a function of time after $q$ is suddenly quenched to $q_f=2.1c_2$\nfor different $c_2$, which is controlled by tuning the atom number $N$, given $c_2\\propto N^{2\/5}$ under Thomas-Fermi approximation.\nThe figure demonstrates that while $\\langle \\rho_0\\rangle$\nremains smaller than $0.4\\%$ for small $c_2$ (there are no observable atoms for $\\rho_0$ except for the noise of a camera), it increases from zero for sufficiently large values of $c_2$, implying that the system decays\ntoward the ground state of the final Hamiltonian with $\\langle{\\rho}_0\\rangle = 1$.\nOur results are\nconsistent with previous observation that the relaxation is stronger for larger $q_f$ and $c_2$~\\cite{Liu2009PRL,Yang2019PRA}.\n\n\n\n\\begin{figure}[t]\n\\includegraphics[width=6.4in]{figS1.pdf}\n\\caption{(Color online) Theoretically calculated (a) energy per particle of the highest excited state in a subspace with zero\nmagnetization, (b) its first and (c) second derivative with respect to $q$. The units of $E$ and $q$ are $c_2$.}\n\\label{figs1}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=2.6in]{figs2.pdf}\n \\caption{(Color online) Experimentally measured $\\langle \\rho_0\\rangle$ as a function of time for distinct $c_2$. As\n the interaction strength $c_2$ is increased by raising the atom number, $\\langle \\rho_0\\rangle$ develops nonzero values instead of remaining zero as time progresses, reflecting that the atoms tend to decay into the ground state with $\\rho_0=1$ of the final Hamiltonian. Here, $q_f\\approx2.1c_2$.\n } \\label{figs2}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{The OS* algorithm}\n\\label{sec:algorithm}\n\nOS\\ensuremath{^{*}}\\xspace is a unified algorithm for optimization and sampling. For simplicity, we first present its sampling version, then move to its optimization version, and finally get to the unified view. We start with some background and notations about rejection sampling.\n\n\\subsection{Background}\n\\label{sec:background}\n\n\n\nLet $\\ensuremath{p}\\xspace:X\\rightarrow \\mathbb{R}_{+}$ be a measurable $L_1$ function with respect to a base measure $\\mu$ on a space $X$, i.e. \n$\\int_X p(x) d\\mu(x) < \\infty$. We define $\\bar{\\p}\\xspace(x) \\equiv \\frac{\\ensuremath{p}\\xspace(x)}{\\int_X p(x) d\\mu(x)}$. The function \\ensuremath{p}\\xspace can be seen as an unnormalized density over $X$, and $\\bar{\\p}\\xspace$ as a normalized density which defines a probability distribution over $X$, called the \\emph{target distribution}, from which we \nwant to sample from\\footnote{By abuse of language, we will also say that a sample from $\\bar{\\p}\\xspace$ is a sample from $\\ensuremath{p}\\xspace$.}. While we may not be able to sample directly from the target distribution $\\bar{\\p}\\xspace$, let us assume that we can easily compute $p(x)$ for any given $x$. \n\\emph{Rejection Sampling} (RS) \\citep{Robert2004} then works as follows. We define a certain unnormalized \\emph{proposal density} $\\ensuremath{q}\\xspace$ over $X$, which is such that (i) we know how to directly sample from it (more precisely, from $\\bar{\\q}\\xspace$), and (ii) $q$ dominates $p$, i.e. for all $x\\in X, p(x) \\leq q(x)$. We then repeat the following process: (1) we draw a sample $x$ from $q$, (2) we compute the ratio $r(x)\\equiv p(x)\/q(x) \\leq 1$, (3) with probability $r(x)$ we accept $x$, otherwise we reject it, (4) we repeat the process with a new $x$. It can then be shown that this procedure produces an exact sample from $\\ensuremath{p}\\xspace$. Furthermore, the average rate at which it produces these samples, the \\emph{acceptance rate}, is equal to $P(X)\/Q(X)$ \\citep{Robert2004}, where for a (measurable) subset $A$ of $X$, we define $P(A) \\equiv \\int_A p(x) d\\mu(x)$ and similarly with $Q$. In Fig.~\\ref{fig:panel1}, panel (S1), the acceptance rate is equal to the ratio of the area below the $p$ curve with that below the $q$ curve.\n\n\n\n\n\n\n\n\n\\subsection{Sampling with OS\\ensuremath{^{*}}\\xspace}\nThe way OS\\ensuremath{^{*}}\\xspace does sampling is illustrated on the top of Fig.~\\ref{fig:panel1}. In this illustration, we start sampling with an initial proposal density $q$ (see (S1)). Our first attempt produces $x_1$, for which the ratio $r_{q}(x_1) = p(x_1)\/q(x_1)$ is close to $1$; this leads, say, to the acceptance of $x_1$. Our second attempt produces $x_2$, for which the ratio $r_{q}(x_2) = p(x_2)\/q(x_2)$ is much lower than $1$, leading, say, to a rejection. Although we have a rejection, we have gained some useful information, namely that $p(x_2)$ is much lower than $q(x_2)$, and we are going to use that ``evidence'' to define a new proposal $q'$ (see (S2)), which has the following properties:\n\\begin{itemize}\n\\item One has $p(x) \\leq q'(x) \\leq q(x)$ everywhere on $X$.\n\\item One has $q'(x_2) < q(x_2)$.\n\\end{itemize}\nOne extreme way of obtaining such a $q'$ is to take:\n$$\nq'(x) \\equiv\n\\begin{cases} \np(x) & \\text{if } x=x_{2}\\\\\nq(x) & \\text{if } x\\neq x_{2}\n\\end{cases}\n$$ \nwhich, when the space $X$ is discrete,\nhas the effect of improving the acceptance rate, but only slightly so, by insuring that any time $q'$ happens to select $x_2$, it will accept it.\n\nA better generic way to find a $q'$ is the following. Suppose that we are provided with a small finite set of ``one-step refinement actions\" $a_j$, depending on $q$ and $x_2$, which are able to move from $q$ to a new $q_j' = a_j(q,x_2)$ such that for any such $a_j$ one has $p(x) \\leq q_j'(x) \\leq q(x)$ everywhere on $X$ and also $q_j'(x_2) < q(x_2)$. Then we will select among these $a_j$ moves the one that is such that the $L_1$ norm of $q_j'$ is minimal among the possible $j$'s, or in other words, such that $\\int_X q_j'(x) d\\mu(x)$ is minimal in $j$. The idea there is that, by doing so, we will improve the acceptance rate of $q'_j$ (which depends directly on $\\|q_j'\\|_1$) as much as possible, while (i) not having to explore a too large space of possible refinements, and (ii) moving from a representation for $q$ to an only slightly more complex representation for $q'_j$, rather than to a much more complex representation for a $q'$ that could result from exploring a larger space of possible refinements for $q$.\\footnote{In particular, even if we could find a refinement $q'$ that would exactly coincide with $p$, and therefore would have the smallest possible $L_1$ norm, we might not want to use such a refinement if this involved an overly complex representation for $q'$.}\nThe intuition behind such one-step refinement actions $a_j$ will become clearer when we consider concrete examples later in this paper.\\mdcomment{The intuition could be made more formal by explaining the stuff about exponential-family models with some active features and a lot of inactive features. However this would take some care and also some space.}\n\n\\begin{figure*}\n\\includegraphics[clip=false, draft=false, trim=0cm 3cm 4cm 8cm, scale=.5]{panelonebis}\n\\caption{Sampling with OS\\ensuremath{^{*}}\\xspace (S1, S2), and optimization with OS\\ensuremath{^{*}}\\xspace (O1, O2).}\n\t\\label{fig:panel1}\n\\end{figure*}\n\n\n\n\\subsection{Optimization with OS\\ensuremath{^{*}}\\xspace}\nThe optimization version of OS\\ensuremath{^{*}}\\xspace is illustrated on the bottom of Fig.~\\ref{fig:panel1}, where (O1) shows on the one hand the function $p$ that we are trying to maximize from, along with its (unknown) maximum $p^{*}$, indicated by a black circle on the $p$ curve, and corresponding to $x^{*}$ in $X$. It also shows a ``proposal'' function $q$ which is such --- analogously to the sampling case --- that (1) the function $q$ is above $p$ on the whole of the space $X$ and (2) it is easy to directly find the point $x_1$ in $X$ at which it reaches its maximum $q^{*}$, shown as a black circle on the $q$ curve.\n\n\nA simple, but central, observation is the following one. Suppose that the distance between $q(x_1)$ and $p(x_1)$ is smaller than $\\epsilon$, then the distance between $q(x_1)$ and $p^{*}$ is also smaller than $\\epsilon$. This can be checked immediately on the figure, and is due to the fact that on the one hand $p^{*}$ is higher than $p(x_1)$, and that on the other hand it is below $q(x^{*})$, and \\emph{a fortiori} below $q(x_1)$. In other words, if the maximum that we have found for $q$ is at a coordinate $x_1$ and we observe that $q(x_1)-p(x_1) < \\epsilon$, then we can conclude that we have found the maximum of $p$ up to $\\epsilon$.\n\nIn the case of $x_1$ in the figure, we are still far from the maximum, and so we ``reject'' $x_1$, and refine $q$ into $q'$ (see (O2)), using exactly the same approach as in the sampling case, but for one difference: the one-step refinement option $a_j$ that is selected is now chosen on the basis of how much it decreases, not the $L_1$ norm of $q$, but the max of $q$ --- where, as a reminder, this max can also be notated $\\|q\\|_\\infty$, using the $L_\\infty$ norm notation.\\footnote{A formal definition of that norm is that $\\|q\\|_\\infty$ is equal to the ``essential supremum'' of $q$ over $(X,\\mu)$ (see below),\nbut for all practical purposes here, it is enough to think of this essential supremum as being the max, when it exists.}\n\n\nOnce this $q'$ has been selected, one can then find its maximum at $x_2$ and then the process can be repeated with $q_{1}=q, q_{2}=q', ...$ until the difference between $q_{k}(x_k)$ and $p(x_k)$ is smaller than a certain predefined threshold.\\mdc{Show the equivalence between multiplicative and subtractive thresholds.}\n\n\n\n\\subsection{Sampling $L_1$ vs. Optimization $L_\\infty$}\nWhile sampling and optimization are usually seen as two completely distinct tasks, they can actually be viewed as two extremities of a continuous range, when considered in the context of $L_p$ spaces \\citep{Ash1999}. \n\nIf $(X, \\mu)$ is a measure space, and if $f$ is a real-valued function on this space, one defines the $L_p$ norm $\\norm{f}_p$, for $1 \\leq p < \\infty$ as:\n$$\n\\norm{f}_p \\equiv \\left(\\int_X |f|^p(x) d\\mu(x)\\right)^{1\/p}.\n$$\nOne also defines the the $L_\\infty$ norm $\\norm{f}_\\infty$ as:\n$$\n\\norm{f}_\\infty \\equiv \\inf \\{ C\\ge 0 : |f(x)| \\le C \\mbox{ for almost every } x\\},\n$$\nwhere the right term is called the \\emph{essential supremum} of $|f|$, and can be thought of roughly as the ``max'' of the function. So, with some abuse of language, we can simply write:\n$\n\\norm{f}_\\infty \\equiv \\max_{x\\in X}|f| .\n$\nThe space $L_p$, for $1 \\leq p \\leq \\infty$, is then defined as being the space of all functions $f$ for which $\\norm{f}_p < \\infty$.\n\nUnder the simple condition that $\\norm{f}_p < \\infty$ for some $p < \\infty$, we have:\n$\n\\lim_{p \\ensuremath{\\rightarrow}\\xspace \\infty} \\norm{f}_p = \\norm{f}_\\infty.\n$\n\nThe standard notion of sampling is relative to $L_1$. However we can introduce the following generalization --- where we use the notation $L_\\alpha$ instead of $L_p$ in order to avoid confusion with our use of $p$ for denoting the target distribution. We will say that we are performing \\emph{sampling of a non-negative function $f$ relative to $L_\\alpha(X,\\mu)$}, for $1\\leq \\alpha <\\infty$, if $f \\in L_\\alpha(X,\\mu)$ and if we sample --- in the standard sense --- according to the normalized density distribution $\\bar{f}(x) \\equiv \\frac{f(x)^\\alpha}{\\int_X f(x)^\\alpha d\\mu(x)}$. In the case $\\alpha = \\infty$, we will say that we are sampling relative to $L_\\infty(X,\\mu)$, if $f \\in L_\\infty(X,\\mu)$ and if we are performing optimization relative to $f$, more precisely, if for any $\\epsilon > 0$, we are able to find an $x$ such that $|\\norm{f}_\\infty-f(x)| < \\epsilon$.\\mdc{Check this definition.}\n\nInformally, sampling relative to $L_\\alpha$ ``tends'' to sampling with $L_\\infty$ (i.e. optimization), for $\\alpha$ tending to $\\infty$, in the sense that for a large $\\alpha$, an $x$ sampled relative to $L_\\alpha$ ``tends'' to be close to a maximum for $f$. We will not attempt to give a precise formal meaning to that observation here, but just note the connection with the idea of \\emph{simulated annealing} \\citep{Kirkpatrick1983}, which we can view as a mix between the MCMC Metropolis-Hastings sampling technique \\citep{Robert2004} and the idea of sampling in $L_\\alpha$ spaces with larger and larger $\\alpha$'s.\n\nIn summary, we thus can view optimization as an extreme form of sampling. In the sequel we will often use this generalized sense of sampling in our algorithms.\\footnote{Note: While our two experiments in section \\ref{sec:experiments} are based on discrete spaces, the OS\\ensuremath{^{*}}\\xspace algorithm is more general, and \\emph{can be applied to any measurable space (in particular continuous spaces)}; in such cases, $p$ and $q$ have to be measurable functions, and the relation $p \\leq q$ should be read as $p(x) \\leq q(x)$ a.e. (almost everywhere) relative to the base measure $\\mu$.}\n\n\n\\subsection{OS\\ensuremath{^{*}}\\xspace as a unified algorithm}\nThe general design of OS\\ensuremath{^{*}}\\xspace can be described as follows:\n\\begin{itemize}\n\\item Our goal is to OS-sample from $p$, where we take the expression ``OS-sample'' to refer to a generalized sense that covers both sampling (in the standard sense) and optimization.\n\\item We have at our disposal a family $\\mathcal{Q}$ of proposal densities over the space $(X,\\mu)$, such that, for every $q\\in \\mathcal{Q}$, we are able to OS-sample efficiently from $q$.\n\\item Given a reject $x_1$ relative to a proposal $q$, with $p(x_1) < q(x_1)$, we have at our disposal a (limited) number of possible ``one-step'' refinement options $q'$, with $p \\leq q' \\leq q$, and such that $q'(x_1) < q(x_1)$.\n\\item We then select one such $q'$. One possible selection criterion is to prefer the $q'$ which has the smallest $L_1$ norm (sampling case) or $L_\\infty$ norm (optimization). In one sense, this is the most natural criterion, as it means we are directly lowering the norm that controls the efficiency of the OS-sampling; for instance, for sampling, if $q_1'$ and $q_2'$ are two candidates refinements with $\\norm{q_1'}_1 < \\norm{q_2'}_1$, then the acceptance rate of $q_1'$ is larger than that of $q_2'$, simply because then $P(X)\/Q_1'(X) > P(X)\/Q_2'(X)$ , and similarly, in optimization, if $\\norm{q_1'}_\\infty < \\norm{q_2'}_\\infty$, then the gap between $\\max_x(q_1'(x))$ and $p^*$ is smaller than that between $\\max_x(q_2'(x))$ and $p^*$, simply because then $\\max_x(q_1'(x)) < \\max_x(q_2'(x))$. \nHowever, using this criterion may require the computation of the norm of each of the possible one-step refinements, which can be costly, and one can prefer simpler criteria, for instance simply selecting the $q'$ that minimizes $q'(x_1)$.\\mdcomment{Maybe this should be expanded and linked to examples in the experiments ? (not sure)}\n\n\n\\item We iterate until we settle on a ``good'' $q$: either (in sampling) one which is such that the cumulative acceptance rate until this point is above a certain threshold; or (in optimization) one for which the ratio $p(x_1)\/q(x_1)$ is closer to $1$ than a certain threshold, with $x_1$ being the maximum for $q$.\n\\end{itemize}\n\nThe following algorithm gives a unified view of OS\\ensuremath{^{*}}\\xspace, valid for both sampling and optimization. This is a high-level view, with some of the content delegated to the subroutines \\texttt{OS-Sample, Accept-or-Reject, Update, Refine, Stop}, which are described in the text.\n\\begin{algorithm}[H]\n\\caption{The OS\\ensuremath{^{*}}\\xspace algorithm}\n\\label{algo:OSstar}\n\\begin{algorithmic}[1] \n\\WHILE { \\NOT Stop($h$) }\n\\STATE OS-Sample $x \\sim q$ \n\\STATE $r \\ensuremath{\\leftarrow}\\xspace p(x)\/q(x)$\n\\STATE Accept-or-Reject$(x,r)$\n\\STATE Update($h,x$)\n\\IF {Rejected($x$)}\n\\STATE $q$ \\ensuremath{\\leftarrow}\\xspace Refine$(q,x)$\n\\ENDIF\n\\ENDWHILE\n\\RETURN \\!\\!$q$ along with accepted $x$'s in $h$\n\\end{algorithmic}\n\\end{algorithm}\n\nOn entry into the algorithm, we assume that we are either in sample mode or in optimization mode, and also that we are starting from a proposal $q$ which (1) dominates $p$ and (2) from which we can sample or optimize directly. We use the terminology \\emph{OS-Sample} to represent either of these cases, where \\texttt{OS-Sample $x \\sim q$} refers to sampling an $x$ according to the proposal $q$ or optimizing $x$ on $q$ (namely finding an $x$ which is an argmax of $q$), depending on the case. On line (1), $h$ refers to the history of the sampling so far, namely to the set of trials $x_1, x_2, ...$ that have been done so far, each being marked for acceptance or rejection (in the case of sampling, this is the usual notion, in the case of optimization, all but the last proposal will be marked as rejects). The stopping criterion \\texttt{Stop($h$)} will be to stop: \n(i) \\emph{in sampling mode}, if the number of acceptances so far relative to the number of trials is larger than a certain predefined threshold, and in this case will return on line (8), first, the list of accepted $x$'s so far, which is already a valid sample from $p$, and second, the last refinement $q$, which can then be used to produce any number of future samples as desired with an acceptance ratio similar to the one observed so far;\n(ii) \\emph{in optimization mode}, if the last element $x$ of the history is an accept, and in this case will return on line (8), first the value $x$, which in this mode is the only accepted trial in the history, and second, the last refinement $q$ (which can be used for such purposes as providing a ``certificate of optimality of $x$ relative to $p$'', but we do not detail this here).\n\n\nOn line (3), we compute the ratio $r$, and then on line (4) we decide to accept $x$ or not based on this ratio; in optimization mode, we accept $x$ if the ratio is close enough to $1$, as determined by a threshold\\footnote{When $X$ is a finite domain, it makes sense to stop on a ratio \\emph{equal} to 1, in which case we have found an exact maximum. This is what we do in some of our experiments in section \\ref{sec:experiments}.}; in sampling mode, we accept $x$ based on a Bernoulli trial of probability $r$.\n\nOn line (5), we update the history by recording the trial $x$ and whether it was accepted or not.\n\nIf $x$ was rejected (line (6)), then on line (7), we perform a refinement of $q$, based on the principles that we have explained.\n\n\n\\input Piecewise.tex\n\n\n\\subsection{Background}\n\\label{sec:background}\n\n\n\nLet $\\ensuremath{p}\\xspace:X\\rightarrow \\mathbb{R}_{+}$ be a measurable $L_1$ function with respect to a base measure $\\mu$ on a space $X$, i.e. \n$\\int_X p(x) d\\mu(x) < \\infty$. We define $\\bar{\\p}\\xspace(x) \\equiv \\frac{\\ensuremath{p}\\xspace(x)}{\\int_X p(x) d\\mu(x)}$. The function \\ensuremath{p}\\xspace can be seen as an unnormalized density over $X$, and $\\bar{\\p}\\xspace$ as a normalized density which defines a probability distribution over $X$, called the \\emph{target distribution}, from which we \nwant to sample from\\footnote{By abuse of language, we will also say that a sample from $\\bar{\\p}\\xspace$ is a sample from $\\ensuremath{p}\\xspace$.}. While we may not be able to sample directly from the target distribution $\\bar{\\p}\\xspace$, let us assume that we can easily compute $p(x)$ for any given $x$. \n\\emph{Rejection Sampling} (RS) \\cite{Robert2004} then works as follows. We define a certain unnormalized \\emph{proposal density} $\\ensuremath{q}\\xspace$ over $X$, which is such that (i) we know how to directly sample from it (more precisely, from $\\bar{\\q}\\xspace$), and (ii) $q$ dominates $p$, i.e. for all $x\\in X, p(x) \\leq q(x)$. We then repeat the following process: (1) we draw a sample $x$ from $q$, (2) we compute the ratio $r(x)\\equiv p(x)\/q(x) \\leq 1$, (3) with probability $r(x)$ we accept $x$, otherwise we reject it, (4) we repeat the process with a new $x$. It can then be shown that this procedure produces an exact sample from $\\ensuremath{p}\\xspace$. Furthermore, the average rate at which it produces these samples, the \\emph{acceptance rate}, is equal to $P(X)\/Q(X)$ \\cite{Robert2004}, where for a (measurable) subset $A$ of $X$, we define $P(A) \\equiv \\int_A p(x) d\\mu(x)$ and similarly with $Q$. In Fig.~\\ref{fig:RS1-ARS1} (left), the acceptance rate is equal to the ratio of the area below the $p$ curve with that below the $q$ curve.\n\n\n\n\\begin{figure}[H]\n\\includegraphics[clip=true, draft=false, trim=6cm 6cm 3cm 7cm, scale=.55]{RS1}\n\\caption{Rejection Sampling}\n\t\\label{fig:RS1}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Conclusions and Perspectives}\n\\label{sec:conclusion}\n\n\\changed{In this paper, we have proposed a unified viewpoint for rejection sampling and heuristic optimization, by using functional upper-bounds for both. While in sampling, the upper-bounds are refined by decreasing their integral ($L_1$ norm), in optimization they are refined by decreasing their maximum ($L_\\infty$ norm).}\nDepending on the problem, several classes of upper bounds can be used. We \nshowed that variable-order max-backoff bounds on $n$-gram probabilities gave state-of-the art performances\non the exact decoding of high-order HMMs. For many practical problems, simpler piecewise bounds can be derived, which we illustrated on the example of sampling and decoding on a large tree-width graphical model.\nOne interesting property of the proposed approach is the \nadaptive nature of the algorithm: the rejected sample is used to quickly choose an effective refinement, \nan approach which can be computational attractive compared to the computation of the $L_p$ norm of all the potential one-step refinements. In the case of graphical model sampling, we showed that this can lead to a speedup factor of an order of magnitude.\n\nThe results presented in this paper motivate\nfurther research in the development of domain-specific functional bounds. \nOne important extension will be to derive bounds for \\emph{agreement-based} models \n$p(.)=p_1(.)\\,p_2(.)$ corresponding to the product of two (or more) simple models $p_1$ and $p_2$.\nOne typical example corresponds to the agreement between an HMM $p_1$ and a probabilistic context-free grammar $p_2$\nin order to take into account both the syntactic and the low-level $n$-gram structures of the language \\cite{DD-nlp}.\n\nAnother research direction is to improve those models that are based on piecewise bounds: \ntheir quality can be limited since their refinement is always local,\n i.e. they can only improve the bound on a single element of the partition. In the example of graphical model sampling, if we condition on the value of a first variable, say $x_1$, and then further refine the partition $\\{x_1=k\\}$ by conditioning on the value of a second variable $x_2$, the complementary region of the space $\\{x_1\\neq k\\}$ will not be impacted. Intuitively, the quality of the bound could be improved if we could refine it \\emph{jointly} for $x_1$ and $x_2$, in a way analogous to what was done in the context of HMMs, where the incorporation of one higher-order n-gram into the proposal had a global impact on the whole event space.\n\n\n\\section{Experiments}\\label{sec:experiments}\n\\label{sec:experiments}\n\n\\input{ExperimentsHMM.tex}\n\n\\input{ExperimentsGraphical.tex}\n\n\n\n\n\n\\subsection{Discrete Probabilistic Graphical Models}\n\\label{sec:ExperimentsGraphical}\n\n\\def\\mathcal{E}{\\mathcal{E}}\n\\def\\mathcal{T}{\\mathcal{T}}\n\\def\\mathcal{N}{\\mathcal{N}}\n\\defC{C}\n\n\n\n\\paragraph{Approach}\n\nThe OS\\ensuremath{^{*}}\\xspace approach \ncan be applied to exact sampling and optimization on graphical models with loops, where the objective function takes the form of a product of local potentials:\n\\begin{eqnarray}\np(x) = \\prod_{n\\in\\mathcal{N}} \\psi_n(x) \\prod_{e\\in\\mathcal{E}} \\phi_e(x),\n\\label{eq:pgm}\n\\end{eqnarray}\nwhere $(\\mathcal{N},\\mathcal{E})$ defines an undirected graph with nodes $\\mathcal{N}$ and \\changed{edges} $\\mathcal{E}$. The unary potential functions are denoted by $\\psi_n, n\\in\\mathcal{N}$ and the binary potentials by $\\phi_e, e\\in\\mathcal{E}$. Since integrating and sampling from (\\ref{eq:pgm}) can be done efficiently for \ntrees, we \\changed{first determine a spanning tree $\\mathcal{T}$ of the graph $\\mathcal{E}$. Let us denote by $\\phi_e^{\\text{max}}$ and $\\phi_e^{\\text{min}}$ the maximal and minimal values of the potential for edge $e$. If we define:}\n\\begin{eqnarray}\nq(x) = \\prod_{n\\in\\mathcal{N}} \\psi_n(x) \\prod_{e\\in\\mathcal{T}} \\phi_e(x) \\prod_{e\\in\\mathcal{E} - \\mathcal{T}} \\phi_e^{\\text{max}},\n\\label{eq:pgm-ub}\n\\end{eqnarray}\nthen $q$ is an upper-bound for $p$ over $X$.\\footnote{\\changed{Any choice of tree produces an upper-bound, but it is advantageous to choose one for which $q$ is as ``close'' as possible to $p$, which we heuristically do by using Prim's algorithm~\\citep{Prim1957} for selecting a maximum spanning tree on the graph having weights $\\log({\\phi_e^{\\text{max}}}\/{\\phi_e^{\\text{min}}})$, $e\\in\\mathcal{E}$. The intuition is that edges with nearly constant potentials create a small gap between the exact value\n$\\phi_e(x)$ and the the bound $\\phi_e^{\\text{max}}$ and can be left outside the tree.}}\n\nTo improve this upper bound, we use the \\emph{conditioning} idea (see e.g. \\citep{Koller2009}, chap. 9.5) which corresponds to partitioning the configuration space $X$. Assume we observe all the possible values $1,2,\\cdots,K$ of a node, say $x_i$, having the set of incident edges $C_i$. \\changed{The restriction of $p$ to the subspace $X_{i,k} =\\{x\\in X;x_i=k\\}$ can be written as:} \n\\begin{eqnarray*}\n\\changed{p_{i,k}(x)} = \\psi_i(k) \\underbrace{ \\prod_{n\\in\\mathcal{N}-\\{i\\}} \\psi_n(x) \n\\prod_{e\\inC_i} \\phi_e(x)\n}_{\\textrm{unary potentials}}\\quad\n\\underbrace{ \\prod_{e\\in\\mathcal{E}-C_i} \\phi_e(x) }_{\\textrm{binary potentials}} ,\n\\label{eq:pgm-cond}\n\\end{eqnarray*}\nwhere the binary potentials of edges incident to $x_i$ have now been absorbed into unary potentials associated with neighboring nodes to $x_i$, and where we have used the informal notation $\\psi_i(k)$ in an obvious way.\nNote that, by doing so, we have eliminated from the graph the edges incident to $x_i$, which may have been participating in loops; in particular, if in equation (\\ref{eq:pgm-ub}) we had some edge $e\\in\\mathcal{E} - \\mathcal{T}$ incident to $x_i$, then this edge is now absorbed in one of the neighbors of $x_i$, which implies that the maximum $\\phi_e^{\\text{max}}$ has now been replaced by its exact value, which is necessarily lower, implicitly refining $q$. By conditioning with respect to all the possible values of $x_i$, we obtain $K$ different graphs with the node $x_i$ removed, in other words we have partitioned the event space into $K$ subspaces; \\changed{we then define on each such subspace a proposal $q_{i,k}(x)$ as in equation~(\\ref{eq:pgm-ub}), which is lower than the restriction of $q$ to $X_{i,k}$.\nWe then define the refinement $q'$ of $q$ on the whole of $X$ as $q'(x) = \\sum_{k} q_{i,k}(x)$, where $q_{i,k}(x)$ is taken to be null for $x\\not\\in X_{i,k}$. This scheme can be seen to be an instance of $OS\\ensuremath{^{*}}\\xspace$ with piecewise bounds.}\n\nIf we repeat this process iteratively based on observed rejections in one of the subspaces, we obtain a hierarchical partition of $X$ which is more fine-grained in some regions of $X$ than in others.\nThe refinements obtained have the form~(\\ref{eq:pgm-ub}), but on reduced graphs in which we have introduced the evidences given by the conditioned variables. While the cardinality of the hierarchical partition may grow exponentially, we can monitor the acceptance rate\/computation time ratio, as we will see below.\n\n\n\n\\paragraph{Ising model experiment} In what follows, we only consider exact sampling, the problem of MAP estimation \\changed{(i.e. optimization)} in graphical models having been thoroughly investigated over the past decade~(e.g. \\citep{SontagEtAl_uai08}). \n\\changed{One interesting question is to understand the trade-off between improving the acceptance rate and incurring the cost of performing a refinement}. \n\\changed{An \\emph{a priori} possible policy could be to first refine the proposals up to a certain point and only then sample until we get the required number of samples.\nAs the experiments will show, there are however two reasons to interleave sampling with refining: the first is that observing rejected samples from $q$ helps to choose the refinements, as argued previously; the second is to have a criterion for stopping the refinement process: the computation of this criterion requires an estimate of the current acceptance rate which can be estimated from the samples.}\n\nWe consider a Ising model of 100 binary variables on a 10x10 uniform grid\nwith unary potentials and binary coupling strengths drawn according to a centered normal distribution with standard deviation 0.1.\nNote that, \\changed{in certain cases}, exact sampling for Ising models can be done in polynomial time~\\citep{Ullrich2010}\nby using an elegant MCMC approach called \\emph{coupling from the past}~\\citep{Propp1996}.\nIt is based on the fact that two properly coupled Markov chains follow exactly the target distribution at the time they coaelesce. However, \\changed{applications of this approach rely on certain strong assumptions on the potentials, typically that they are either all attractive or all repulsive~\\citep{Craiu2011}}, while we sample models with random positive or negative coupling strengths, making the problem much harder. One interesting extension of our work would be to use these algorithms as proposal distributions for more general models. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.9\\linewidth,trim=0cm 4cm 0cm 6cm]{Fig5.pdf}\n\\caption{Comparison of different refinement policies for the piecewise bound on a 10x10 Ising model grid.}\n\\label{fig:strategies}\n\\end{figure*}\n\n\n\nThe $OS^*$ algorithm was run using 4 different policies\nfor the choice of the refinements, where a policy is a function that takes as input the current proposal and returns \na (subspace, conditioning-variable) pair. The first two policies in addition use a rejected sample $x$ as input, while the last two \\changed{do not and are deterministic}:\n(i)~\\emph{random split in the region of the rejected sample} means that once an observation has been rejected, we refine the subspace which contains the sample by conditioning with respect to one of the remaining variables \\changed{$x_i$} selected at random;\n(ii)~\\emph{highest bound improvement on rejected sample} is the same but the \\changed{index $i$ of the conditioning variable is selected such that refining on $x_i$ leads to the largest decrease in the value of $q(x)$ (for this specific rejected sample $x$); this is a similar refinement strategy to the one we used in the HMM experiments;} \n(iii)~\\emph{most probable region} refines a variable (selected uniformly at random) in the most probable subspace of the piecewise bound $q$; \n(iv)~\\emph{highest acceptance rate} is the \\changed{most ``ambitious''} greedy policy where the largest reduction of the total mass $q(X)$ of the proposal is identified among all the possible choices of (subspace, conditioning-variable). This has been implemented efficiently by maintaining a priority queue containing all the possible triples (subspace, conditioning-variable, maximal-bound-improvement).\nThe acceptance rates obtained by following these policies are compared on Figure~\\ref{fig:strategies} (left) by running 2000 refinements\nwith policies (i) and (iii), 800 refinements with policy \\changed{(ii)} and 400 refinement with policy (iv).\nResults confirm that the best refinement is obtained using the deterministic policy (iv).\nThe policy (ii) based on rejected samples reaches the same acceptance rates using twice as\nmany refinements. The other two policies (i) and (iii) are very naive and do not reach a significant \nacceptance rate, even after 400 iterations. Based on these results, one might conclude that the \npolicy (iv) is the best. However, we will now see that this is not the case when the computation time is taken into account.\n\nAfter $T$ trials, the partition function of \\changed{$p$, namely $p(X)$,} can be estimated (without bias) by $\\hat Z_T\\equiv \\frac{1}{T}\\sum_{t=1}^T r_t\\:q_t(X)$\nwhere \\changed{the observed accept ratios $\\{r_t\\}_{t=1}^T$ and refinement weights $\\{q(X)_t\\}_{t=1}^T$ are used}.\nHence, \\changed{$\\hat\\pi_T = \\frac{\\hat Z_T}{q_T(X)}$} is an unbiased estimate of the current acceptance rate $\\pi_T \\equiv p(X)\/q_T(X)$.\nNote that these estimators are based on \\emph{all} the samples $x_t$, accepted or not. Hence, if we decide to stop refining now,\nthe expected time to obtain $n$ additional exact samples from the target distribution $p$ is\napproximately $\\frac{n\\tau_T^{\\textrm{samp}}}{\\hat\\pi_T}$, where $\\tau_T^{\\textrm{samp}}$ is the average time to obtain one \\changed{trial} from the \ndistribution $q_T$. \nIf we add the total time $\\tau_T^{\\textrm{ref}}$ \\changed{spent in computing} the set of refinements up to the current refinement $q_T$, we obtain an estimate $\\hat\\tau^{\\textrm{tot}}_T$ of the expected total time to obtain $n$ samples:\n\\(\n\\hat\\tau^{\\textrm{tot}}_T = \\frac{n\\tau_T^{\\textrm{samp}}}{\\hat\\pi_T} + \\tau_T^{\\textrm{ref}}\\enspace.\n\\label{eq:total-time}\n\\)\nThis quantity computed for $n=1$ sample is plotted against the acceptance rate on Figure~\\ref{fig:strategies}~(right). \nFor each policy, the expected computation time starts to decrease as the acceptance rate increases; in this regime, \nthe refinement time is small compared to the time reduction due to a higher acceptance rate. For large values of the acceptance rate,\nthe refinement time is no more negligeable, leading to an increase of the total computation time.\nWe see (Figure~\\ref{fig:strategies}~(right)) that the \\emph{highest acceptance rate} policy \\changed{(iv)}, despite its very good acceptance rate for a fixed number of refinements, requires more time in total than alternative refinements policies. The difference is \nstriking: if we look at the minimum of each curve (\\changed{which corresponds to the optimal stopping time for the refinements}), it\nis at least 10 times faster to use the policy (ii) based on a refinement rule applied only to the rejected sample.\nThis experiment confirms the benefit of using rejected samples: by adaptively choosing the refinements, we spot the regions of the space that are important to refine much faster than by computing the best possible alternative refinement. \n\\subsection{HMMs}\n\\label{sec:hmms}\n\n\\def\\ensuremath{p}\\xspace_{\\textrm{lm}}{\\ensuremath{p}\\xspace_{\\textrm{lm}}}\n\\def\\ensuremath{p}\\xspace_{\\textrm{obs}}{\\ensuremath{p}\\xspace_{\\textrm{obs}}}\n\n\\noindent\\emph{Note: An extended and more detailed version of these experiments is provided in \\citep{Carter2012}}.\n\n\\bigskip\n\nThe objective in our HMM experiments is to sample a word sequence with density $\\bar{\\p}\\xspace(x)$ proportional to $\\ensuremath{p}\\xspace(x)=\\ensuremath{p}\\xspace_{\\textrm{lm}}(x)\\ \\ensuremath{p}\\xspace_{\\textrm{obs}}(o|x)$, where $\\ensuremath{p}\\xspace_{\\textrm{lm}}$ is\nthe probability of the sequence $x$ under an $n$-gram model and $\\ensuremath{p}\\xspace_{\\textrm{obs}}(o|x)$ is the probability of \nobserving the noisy sequence of observations $o$ given that the word sequence is $x$. \nAssuming that the observations depend only on the current state, this probability can be written:\n\\begin{eqnarray}\np(x) &=& \\prod_{i=1}^\\ell \\ensuremath{p}\\xspace_{\\textrm{lm}}(x_i|x^{i-1}_{i-n+1})\\ \\ensuremath{p}\\xspace_{\\textrm{obs}}(o_i|x_i) \\enspace.\n\\label{eq:hmm}\n\\end{eqnarray}\n\n\\paragraph{Approach}\n\nTaking a tri-gram language model for simplicity, let us define $w_3(x_i|x_{i-2} x_{i-1}) = \\ensuremath{p}\\xspace_{\\textrm{lm}}(x_i|x_{i-2} x_{i-1})\\ \\ensuremath{p}\\xspace_{\\textrm{obs}}(o_i|x_i)$. \nThen consider the\n observation $o$ be fixed, and write $p(x) = \\prod_i w_3(x_i|x_{i-2} x_{i-1})$. In optimization\/decoding,\n we want to find the argmax of $p(x)$, and in sampling, to sample from $p(x)$. Note that the state space \nassociated with $p$ can be huge, as we need to represent explicitly all contexts $(x_{i-2}, x_{i-1})$ in\n the case of a trigram model, and even more contexts for higher-order models.\n\n\\sloppy\nWe define $w_2(x_i| x_{i-1}) = \\max_{x_{i-2}} w_3(x_i|x_{i-2} x_{i-1})$, along with \n$w_1(x_i) = \\max_{x_{i-1}} w_2(x_i|x_{i-1})$, where the maxima are taken over all possible \ncontext words in the vocabulary.\nThese quantities, which can be precomputed efficiently, \ncan be seen as optimistic ``max-backoffs'' \nof the trigram $x_{i-2}^i$, where we have forgotten some part of the context.\nOur initial proposal is then $q_{0}(x) = \\prod_i w_1(x_i)$. Clearly, for any sequence\n$x$ of words, we have $p(x) \\leq q_{0}(x)$. \nThe state space of $q_0$ is much less costly to represent than that of $p(x)$.\n\\fussy\n\nThe proposals $q_{t}$, which incorporate n-grams of variable orders, can be represented efficiently through {WFSAs} (weighted FSAs). In Fig.~\\ref{fig:HMMab}(a), we show a WFSA representing the initial\nproposal $q_0$ corresponding to an example with four observations, which we take to be the acoustic\nrealizations of the words `the, two, dogs, barked'. The weights on edges correspond \nonly to unigram max-backoffs, and thus each state corresponds to a NULL-context. Over\nthis WFSA, both optimization and sampling can be done efficiently by the standard dynamic\nprogramming techniques (Viterbi~\\citep{rabiner}and ``backward filtering-forward sampling''~\\citep{Scott2002}), where the forward weights \nassociated to states are computed similarly, either in the max-product or in the sum-product semiring.\n\n\n\\begin{figure}\n\\includegraphics[clip=true, draft=false, trim=0cm 3.2cm 2.8cm 4cm, scale=.5]{HMMab.pdf}\n\\caption{{\\it An example of an initial q-automaton (a), and its refinement (b).}}\n\\label{fig:HMMab}\n\\vspace{-10pt}\n\\end{figure}\n\n\nConsider first sampling, and suppose that the first sample from $q_0$ produces\n$x_1 =$ \\textit{the two dog barked}, marked with bold edges in the drawing. Now, \ncomputing the ratio $p(x_1)\/q_0(x_1)$ gives a result much smaller than $1$, in part because \nfrom the viewpoint of the full model $p$, the trigram \\textit{the two dog} is \nvery unlikely; i.e. the ratio $w_3(dog| the\\ two)\/w_1(dog)$ is very low. \nThus, with high probability, $x_1$ is rejected. \nWhen this is the case, we produce a refined proposal $q_1$, \nrepresented by the WFSA in Fig.~\\ref{fig:HMMab}(b), which takes into account the more\nrealistic bigram weight $w_2(dog| two)$ by adding a node (node 6) for the context \\textit{two}.\nWe then perform a sampling trial with $q_1$, which this time tends to avoid producing\n\\textit{dog} in the context of \\textit{two}; if we experience a reject later on some sequence $x_2$, we refine again, meaning that we identify an n-gram in $x_2$, which, if we extend its context by one (e.g from a unigram to a bigram or from a bigram to a trigram), accounts for some significant part of the gap between $q_1(x_2)$ and $p(x_2)$. We stop the refinement process when we start observing acceptance rates above a certain fixed threshold.\n\nThe case of optimization is similar. Suppose that with $q_0$ the maximum is $x_1 =$ \n\\textit{the two dog barked}, then we observe that $p(x_1)$ is lower than $q_0(x_1)$, \nreject $x_1$ and refine $q_0$ into $q_1$. We stop the process at the point where \nthe value of $q_t$, at its maximum $x_{q_{t}}$, is equal to the value of $p$ at $x_{q_{t}}$,\nwhich implies that we have found the maximum for $p$.\n\n\\paragraph{Setup}\n\n\\def\\textrm{num}{\\textrm{num}}\nWe evaluate our approach on an SMS-message retrieval task. \nLet $N$ be the number of possible words in the vocabulary. \nA latent variable $x\\in\\{1,\\cdots,N\\}^\\ell$ represents a sentence defined as a sequence of $\\ell$ words.\nEach word is converted into a sequence of numbers based on a mobile phone numeric keypad, assuming some level of random noise in the conversion. \nThe task is then to recover the original message. \n\nWe use the English side of the Europarl corpus~\\citep{europarl} for training and test data (1.3 million sentences).\nA 5-gram language model is trained using SRILM~\\citep{srilm} on 90\\% of\nthe sentences. On the remaining 10\\%, we randomly select 100 sequences\nfor lengths 1 to 10 to obtain 1000 sequences\nfrom which we remove the ones containing numbers, \nobtaining a test set of size 926.\n\n \n\\begin{figure}\n\\centering{\\includegraphics[width=1.2\\linewidth,trim=2cm 3cm 0cm 6cm]{fig3.pdf}}\n\\caption{\\label{fig:sms-n-iter-states} SMS-retrieval experiment. \n(a): optimization; (b) and (c): sampling.}\n\\label{fig:sms-decoding}\n\\end{figure}\n\n\\paragraph{Optimization}\n\nWe limit the average number of latent tokens in our decoding experiments to 1000.\n In the top plot (a) of Fig.~\\ref{fig:sms-decoding} \nwe show the number of iterations (running Viterbi then updating $q$) that the\ndifferent n-gram models of size 3, 4 and 5 take to do exact decoding of the test-set.\nFor a fixed sentence length, we can see that decoding with larger n-gram models leads to a sub-linear\nincrease w.r.t. $n$ in the number of iterations taken.\n\nTo demonstrate the reduced nature of our q-automaton, we \nshow in Table~\\ref{tab:sms-ngram-diff} the distribution of n-grams\nin our final model for a specific input sentence of length 10. \nThe total number of n-grams in the full model would be ${\\sim} 3.0 {\\times} 10^{15}$; exact decoding here is not tractable using existing techniques. By comparison, our HMM has only 118 five-grams and 9008 n-grams in total.\n\n\\begin{table}[h]\\footnotesize\n\\begin{tabular*}{1\\linewidth}{@{\\extracolsep{\\fill}} c | c | c | c | c | c }\nn: & 1 & 2 & 3 & 4 & 5 \\\\\n\\hline\nq: & 7868 & 615 & 231 & 176 & 118 \\\\\n\\end{tabular*}\n\\caption{\\label{tab:sms-ngram-diff} {\\it\\# of n-grams in our variable-order HMM.}}\n\\end{table}\n\\vspace{-10pt}\n\n\\paragraph{Sampling}\n\nFor the sampling experiments, we limit the number of latent tokens\nto 100. We refine our $q$ automaton until we reach \na certain fixed cumulative acceptance rate (AR). We also compute a rate based only\non the last 100 trials (AR-100), as this tends\nto better reflect the current acceptance rate.\nIn plot (b), at the bottom of Fig.~\\ref{fig:sms-decoding}, \nwe show a single sampling run using a 5-gram model \nfor an example input, and the cumulative \\# of accepts (middle curve). \nIt took 500 iterations before the first successful\nsample from $p$. \n\nWe noted that there is a trade-off between the time needed to compute\nthe forward probability weights needed for sampling,\nand the time it takes to adapt the variable-order HMM. To resolve this, we use batch-updates: making $B$ trials from the same\n$q$-automaton, and then updating our model in one step. By doing this,\nwe noted significant speed-ups in sampling times. Empirically,\nwe found $B=100$ to be a good value.\nIn plot (c),\nwe show the average \\# of iterations in our models\nonce refinements are finished (AR-100=20\\%) for different\norders $n$ over different lengths. We note a sub-linear increase in\nthe number of trials when moving to \nhigher $n$; for length${=}$10, and for $n=3,4,5$, average number of trials: 3-1105, \\, 4-1238, \\, 5-1274.\n\n\n\\section{Introduction}\n\nCommon algorithms for sampling high-dimensional distributions are based on MCMC techniques \\citep{Andrieu2003,Robert2004}, which are approximate in the sense that they produce valid samples only asymptotically. By contrast, the elementary technique of Rejection Sampling \\citep{Robert2004} directly produces exact samples, but, if applied naively to high-dimensional spaces, typically requires unacceptable time before producing a first sample.\n\nThe algorithm that we propose, OS\\ensuremath{^{*}}\\xspace, is a joint exact \\underline{O}ptimization and \\underline{S}ampling algorithm that is inspired both by rejection sampling and by classical A\\underline{\\mbox{\\(^{*}\\)}} optimization, and which can be applied to high-dimensional spaces. The main idea is to upper-bound the complex target distribution $p$ by a simpler proposal distribution $q$, such that a dynamic programming (or another low-complexity) method can be applied to $q$ in order to efficiently sample or maximize from it. In the case of sampling, rejection sampling is then applied to $q$, and on a reject at point $x$, $q$ is refined into a slightly more complex $q'$ in an adaptive way. This is done by using the evidence of the reject at $x$, implying a gap between $q(x)$ and $p(x)$, to identify a (soft) constraint implicit in $p$ which is not accounted for by $q$, and by integrating this constraint in $q$ to obtain $q'$.\n\nThe constraint which is integrated tends to be highly relevant and to increase the acceptance rate of the algorithm. By contrast, many constraints that are constitutive of $p$ are never ``activated'' by sampling from $q$, because $q$ never explores regions where they would become visible. For example, anticipating on our HMM experiments in section \\ref{sec:hmms}, there is little point in explicitly including in $q$ a 5-gram constraint on a certain latent sequence in the HMM if this sequence is already unlikely at the bigram level: the bigram constraints present in the proposal $q$ will ensure that this sequence will never (or very rarely) be explored by $q$.\n\nThe case of optimization is treated in exactly the same way as sampling. Formally, this consists in moving from assessing proposals in terms of the $L_1$ norm to assessing them in terms of the $L_\\infty$ norm. Typically, when a dynamic programming procedure is available for sampling ($L_1$ norm) with $q$, it is also available for maximizing from $q$ ($L_\\infty$ norm), and the main difference between the two cases is then in the criteria for selecting among possible refinements.\n\n\\paragraph{Related work}\nIn an heuristic optimization context the two interesting, but apparently little known, papers \\citep{Kam1996,did-lin} , discuss a technique for decoding images based on high-order language models for which upper-bounds are constructed in terms of simpler variable-order models. Our application of OS\\ensuremath{^{*}}\\xspace in section \\ref{sec:hmms} to the problem of maximizing a high-order HMM is similar to their technique; however \\citep{Kam1996,did-lin} do not attempt to generalize their approach to other optimization problems amenable to dynamic programming or discuss any connection to sampling.\n\nIn order to improve the acceptance rate of rejection sampling, one has to lower the proposal $q$ curve as much as possible while keeping it above the $p$ curve. In order to do that, some authors \\citep{Gilks1992,Gorur2008}, have proposed \\emph{Adaptive Rejection Sampling (ARS)} where, based on rejections, the $q$ curve is updated to a lower curve $q'$ with a better acceptance rate. These techniques have predominantly been applied to continuous distributions on the one-dimensional real line where convexity assumptions on the target distribution can be exploited to progressively approximate it tighter and tighter through upper bounds consisting of piecewise linear envelopes.\n\nAlso in the context of rejection sampling, \\citep{Mansinghka2009} considers the case of a probabilistic graphical model; it introduces an heuristically determined order of the variables in this model and uses this (fixed) order to define a sequence of exact samplers over an increasing set of variables, where the exact sampler over the first $k+1$ variables is recursively obtained by using the preceding exact sampler over the first $k$ variables and accepting or rejecting its samples based on the $(k+1)^{\\text{th}}$ variable. While our experiments on graphical models in section \\ref{sec:ExperimentsGraphical} have some similarities to this approach, they do not use a cascade of exact samplers, but rather they partition the space of configurations dynamically based on rejects experienced by the current proposal.\n\n\\begin{comment}\n\\medskip\nThe remainder of this paper is structured as follows. In section \\ref{sec:background}, we start by a brief reminder about rejection sampling and its adaptive version. \nIn section \\ref{sec:algorithm}, we introduce OS\\ensuremath{^{*}}\\xspace, starting by describing its sampling mode, then it optimization mode, and then moving to the unified view; we also explain the connections to the A\\ensuremath{^{*}}\\xspace algorithm.\nWe then move in section \\ref{sec:experiments} to our two experiments: the first concerned with high-order HMMs, the second with large pairwise graphical models.\nWe finally discuss perspectives and conclude in section \\ref{sec:conclusion}.\n\\end{comment}\n\n\\section{Experiments}\\label{sec:experiments}\n\n\\input{ExperimentsHMM.tex}\n\n\\input{ExperimentsGraphical.tex}\n\n\n\n\n\\input{Conclusion.tex}\n\n\n\n\n\n\n\n\\clearpage\n\n\\bibliographystyle{plainnat}\n\\input{OSstar_arXiv.bbl}\n\n\\end{document}\n\n\n\n\n\n\\subsection{A connection with A*}\n\\label{sec:piecewise}\n\n\\begin{changedEnv}\n\n\\begin{figure}\n\\centering{\\includegraphics[clip=true, draft=false, trim=6cm 5cm 6cm 5cm, scale=.5]{ARSstar1}}\n\\caption{A connection with A\\ensuremath{^{*}}\\xspace.}\n\t\\label{fig:ARSstar1}\n\\end{figure}\n\nA special case of the OS\\ensuremath{^{*}}\\xspace algorithm, which we call ``OS\\ensuremath{^{*}}\\xspace with piecewise bounds'', shows a deep connection with the classical A\\ensuremath{^{*}}\\xspace optimization algorithm \\citep{Hart1968} and is interesting in its own right. Let us first focus on sampling, and let us suppose that $q_0$ represents an initial proposal density, which upper-bounds the target density $p$ over $X$. We start by sampling with $q_0$, and on a first reject somewhere in $X$, we split the set $X$ into two disjoint subsets $X_1,X_2$, obtaining a partition of $X$. \nBy using the more precise context provided by this partition, we may be able to improve our upper bound $q_0$ over the whole of $X$ into tighter upper bounds on each of $X_1$ and $X_2$, resulting then in a new upper bound $q_1$ over the whole of $X$. We then sample using $q_1$, and experience at some later time another reject, say on a point in $X_1$; at this point we again partition $X_1$ into two subsets $X_{11}$ and $X_{12}$, tighten the bounds on these subsets, and obtain a refined proposal $q_2$ over $X$; we then iterate the process of building this ``hierarchical partition'' until we reach a certain acceptance-rate threshold.\n\n\nIf we now move our focus to optimization, we see that the refinements that we have just proposed present an analogy to the technique used by A\\ensuremath{^{*}}\\xspace. This is illustrated in Fig.~\\ref{fig:ARSstar1}.\nIn A\\ensuremath{^{*}}\\xspace, we start with a \\emph{constant} optimistic bound --- corresponding to our $q_0$ --- for the objective function which is computed at the root of the search tree, which we can assume to be binary. We then expand the two daughters of the root, re-evaluate the optimistic bounds there to new constants, obtaining the piecewise constant proposal $q_1$, and move to the daughter with the highest bound. We continue by expanding at each step the leaf of the partial search tree with the highest optimistic bound (e.g. moving from $q_1$ to $q_2$, etc.).\\footnote{\\changed{OS\\ensuremath{^{*}}\\xspace, when used in optimization mode, is in fact \\emph{strictly more general than A\\ensuremath{^{*}}\\xspace,} for two reasons: (i) it does not assume a piecewise refinement strategy, namely that the refinements follow a hierarchical partition of the space, where a given refinement is limited to a leaf of the current partition,\nand (ii) even if such a stategy is followed, it does not assume that the piecewise upper-bounds are constant. Both points will become clearer in the HMM experiments of section \\ref{sec:hmms}, where including an higher-order n-gram in $q$ has impact on several regions of $X$ simultaneously, possibly overlapping in complex ways with regions touched by previous refinements; in addition, the impact of a single n-gram is non-constant even in the regions it touches, because it depends of the multiplicity of the n-gram, not only on its presence or absence.}}\n\n\nWe will illustrate OS\\ensuremath{^{*}}\\xspace sampling with (non-constant) piece-wise bounds in the experiments of section \\ref{sec:ExperimentsGraphical}.\n\n\\end{changedEnv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAt the crossroad of quantum dynamics and thermodynamics, the theory of energy exchanges between quantum systems has attracted a lot of interest. Several articles focused on the fluctuations of energy and entropy fluxes, in equilibrium and non-equilibrium steady states, either in the two-time measurement protocol \\cite{Kurchan00},\\cite{EHM_nonequilibrium_2009},\\cite{BJPPP15},\\cite{BPP18},\\cite{BPR19} or for continuous-time measurements \\cite{DerezinskiRoeckMaes08_fluctuations},\\cite{JPW14}. We may also cite the lecture notes \\cite{JOPP_entropic_2012}, which contains a detailed treatment of Fermionic systems, and the article \\cite{JPPP_energy_2015}, which does not consider fluctuations but energy conservations in the two-time measurement protocol. The definition of the fluctuations in classical systems is well-established and linked with large deviations (note the article \\cite{BJP_energy_2017} with a comparison with experimental data), while there are several quantum analogues for the fluctuations of currents, as noted in \\cite{DerezinskiRoeckMaes08_fluctuations}; the abstract way is to study a function $e(\\alpha)$ linked with some Renyi entropy, as in \\cite{JLP_fluctuations_2013}. The most mundane way (and the one of this article) is to study the large deviations of a random variable obtained by measuring some energy observables on the systems. Besides the study of fluctuations, we may also note some works on the Landauer principle (\\cite{Landauer_reebWolf},\\cite{JP_landauer_2014},\\cite{BFJP16} \\cite{HJPR18}) or in link with resource theory (\\cite{fully_quantum_second_law_CSHO}, \\cite{thermal_operations_Mazurek18}).\n\nA convenient approach to this subject is the one of Markovian dynamics: we consider a small system $S$ in contact with an exterior system $B$, and assume that the effect of the exterior systems on $S$ at a time $t$ does not depends on the past interactions between them.\nThen, the density matrix on the system $S$ evolves according to a linear equation; the dynamic is described by a so-called quantum Markov semigroup, whose generator is a linear super-operator $\\mathcal{L}$ called Lindbladian \\cite{Lindblad1976} or GKSL operator.\nThis type of evolution are used as effective models for open systems, notably in quantum optics \\cite{optics_gardiner_zoller_qt_noise}, and they where considered since the beginnings of quantum mechanics (see Landau \\cite{landau27} or its English translation in \\cite{landau_collected}), though their general theory really started in the '70 (see \\cite{briefHistory} for a history).\nA problem is to rigorously derive a Markovian evolution from a standard Hamiltonian evolution on complete system $SB$; it may be obtained as some limit for a convenient scaling, which is sometimes called a stochastic limit (\\cite{accardi_stochastic_limit}, \\cite{accardiLu_first_40_years_GKSL}) and encompasses the low density limit and most importantly the weak coupling limit (\\cite{Davies74} \\cite{Derezinski2007}).\nThe fluctuation of currents in models obtained from the weak coupling limit has notably been studied in \\cite{DerezinskiRoeckMaes08_fluctuations}, \\cite{RoeckMaes06_fluctuations}.\n\n In this article, we use another derivation of Markovian dynamics, the continuous-time limit of repeated interactions (\\cite{AttalPautrat}, \\cite{AttalJoye}, \\cite{repeated_interactions_BAM14}). This allows to make a direct relation between discrete-time dynamics and the quantum Markov semigroup. In the repeated interaction framework, the exterior system is divided as the sum of identical subsystems which interact one after the other with the system $S$ during a time $\\tau$. As $\\tau \\rightarrow 0$ and under suitable normalization of the Hamiltonian, the obtained dynamics converges to a continuous-time Markovian semigroup $(\\Lambda^t)_{t\\in [0, +\\infty)}$ . Importantly, it is possible to measure some observable before and after each measurements, making the evolution a random process. The limit evolution is then a stochastic process $(\\tilde{\\rho_t})_{t\\in [0, +\\infty)}$ which is linked with the semigroup $(\\Lambda^t)_{t\\in [0, +\\infty)}$ through the relation\n \\[\n \\mathbb{E}(\\tilde{\\rho_t})=\\Lambda^t(\\rho_0)~.\n \\] \n and is called an unraveling of the semigroup(\\cite{srinivas_davies_81}\\cite{Belavkin_07_eventum}\\cite{Carmichael_lecture93}\\cite{Breuer2002}). The convergence in distribution of the process has notably been studied by Pellegrini \\cite{PellegriniDiffusive08}\\cite{Pellegrini_Nechita09}\\cite{pellegrini_jumps_10}.\n \nUnder some assumptions of detailed balance on the Lindbladian and its unraveling, it is possible to interpret the measured observables as energy exchanged between the systems and some subsystems $B_1, \\cdots, B_n$ of $B$, and to study their large deviations. This is the subject of the article \\cite{JPW14}, which is the main inspiration of this article. In this context, the Lindbladian of the system is expressed as \n\\[\n\\mathcal{L}=i[H_S, \\bullet]+\\sum_{i=1}^n \\mathcal{L}_i\n\\]\nwhere the $\\mathcal{L}_i$ represents the effect of the $i$-th bath, and there is some pseudo-potential energy $K_S$ which commutes with $H_S$ and such that the $\\mathcal{L}_i$'s satisfy a detailed balance condition with respect to the Gibbs state \n\\[\n\\sigma_{\\beta_i}=\\frac{e^{-\\beta_i K_S}}{\\tr{e^{-\\beta_i K_S}}}~.\n\\]\nIt is then possible to define the mean energy flux entering the $i$-th bath as \n\\[\nJ_i=-\\tr{\\mathcal{L}_i(K_S)\\rho_\\infty}\n\\]\nand some random processes $(N^i_t)_{t\\in [0, +\\infty)}$ where $N_t^i$ represents the energy increase measured in the $i$-th bath, with the relation\n\\[\n\\lim_{t\\rightarrow \\infty} \\frac{1}{t} \\mathbb{E}\\left(N_t^i\\right)=J_i~.\n\\]\n\nThe contributions of the present work are first, to explore the link between the continuous-time and the discrete-time approaches to the fluctuations of energy fluxes through the limit of repeated interaction; then, to study both the mean energy fluxes $J_i$ and the large deviations of the $N_t^i$ in the particular case of quasi-free fermionic systems.\n\nA free fermionic dynamics describes non-interacting fermions which may jump between the system and the baths, of which quasi-free fermionic semigroups is a generalization. Quasi-free fermionic dynamics on bosonic and fermionic spaces has long been studied (see for example \\cite{AlickiLendi} or \\cite{fagnolaRebolledo2002}). These models have the advantage to be explicitly solvable in many cases, and still show non-trivial behavior which make them good toy models, for example to test quantum functional inequalities as in \\cite{TPK14}. Moreover, they are formally similar to classical networks of harmonic oscillators driven by Langevin noise \\cite{MNV03_heat_network}\\cite{EckmannZabey04}\\cite{JPS16}. Some recent works of Prosen \\cite{Prosen2008} \\cite{Prosen2010} introduced methods to study the convergence as $t\\rightarrow +\\infty$ to a unique stationary state, and the author of the present article established a necessary criterion for the convergence and uniqueness \\cite{AndreysFermions1}. The repeated interaction model on fermionic spaces was introduced by Platini and Karevski \\cite{Platini2008} \\cite{KarevskiPlatini2009} in the case of the XY model.\n\\vspace{0.5cm}\n\nIn the second section of this article, we concentrate on the study of the mean energy fluxes $J_i$. We describe a general framework of repeated interaction models with a globally conserved quantity $K$ (which is for example used in \\cite{HJPR18}), and show (Proposition \\ref{prop:thermal=db}) how the conservation of $K$ corresponds in the continuous-time limit to a property of detailed balance on the Lindbladian $\\mathcal{L}$ (see Alicki \\cite{alicki_76} or the introduction of \\cite{carlen_maas_17}). Under the condition that the baths $B_i$ are described by Gibbs states at temperatures $\\beta_i$, we express the first and second principles of thermodynamics in terms of the mean energy fluxes: \n\\begin{align*}\n\\sum_{i=1}^n J_i&=0 & \\sum_{i=1}^n \\beta_i J_i & \\geq 0 ~.\n\\end{align*}\nWe show that for any list of fluxes $(J_1, \\cdots, J_n)$ satisfying the above conditions (with a strict inequality) there exists a thermal model yielding theses energy fluxes (Proposition \\ref{prop:thermal_machines}). The proof makes use of a design of quantum fridge borrowed from \\cite{thermal_machine_small_maximal_efficiency_jpa11} and \\cite{thermal_machine_small_prl10}. We apply this framework to the case of quasi-free fermionic systems, and prove that they satisfy a stronger inequality (Theorem \\ref{theo:no_fridge}): provided $\\beta_1\\leq \\beta_2\\leq \\cdots \\leq \\beta_n$ we have\n\\[\n\\sum_{i=1}^k J_i \\leq 0\n\\]\nfor any $k\\in \\{1,\\cdots, n\\}$. \nIn particular, there cannot be energy entering the bath of highest temperature. This theorem is inspired by the article of Eckmann and Zabey \\cite{EckmannZabey04} on energy fluxes in classical harmonic networks. \n\nIn the third section, we describe the large deviations of the random energy fluxes $N^i_t$ for quasi-free fermionic models. Using the results of \\cite{JPW14}, we express the cumulant generating functional \n\\[\ne(\\alpha)=\\lim_{t\\rightarrow +\\infty} \\frac{1}{t} \\log \\mathbb{E}\\left(e^{\\sum_{i=1}^n \\alpha_i N_i}\\right)\n\\]\nin terms of the largest eigenvalue of a deformed Lindblad operator $\\mathcal{L}_{\\alpha}$. Note that if $L$ is the dimension of the one-particle space, the fermionic space is of dimension $2^L$, so $\\mathcal{L}_\\alpha$ is of size $2^{2L}\\times Z^{2L}$. We are able to reduce the computation of $e(\\alpha)$ to the computation of the eigenvalues of an operator of dimension $4 L \\times 4 L$, through the resolution of a Riccati equation (Theorem \\ref{theo:e_alpha_fermionic}). This is formally similar to the study of the large deviations of entropy in classical harmonic networks \\cite{JPS16}. We apply this to the numerical computation of the rate functional of large deviations for the fermionic chain, and show that the longest the chain is, the larger the fluctuations are.\n\n\n\n\\section{Mean energy exchanges for quantum Markovian unravelings }\n\nWe will use the following general notations: \n\n\\begin{itemize}\n\\item Most Hilbert spaces considered will be finite dimensional. We write $\\mathcal{H}_S$ a Hilbert space associated with a system $S$, and $\\mathds{1}_S$ the identity on this space; the space of operators on $\\mathcal{H}_S$ is written $\\mathcal{B}(\\mathcal{H}_S)$, and the set of states on $\\mathcal{H}_S$ is written $\\mathfrak{S}(\\mathcal{H}_S)$. An operator on $\\mathcal{B}(\\mathcal{H}_S)$ is called a super-operator. \n\\item For any state $\\rho$ we write $S(\\rho)=-\\tr{\\rho\\log\\rho}$ the Von Neumann entropy and $S(\\rho|\\sigma)=\\tr{\\rho(\\log \\rho-\\log \\sigma}$ the relative entropy with respect to a state $\\sigma$.\n\\item For any state $\\sigma>0$ we write $\\Delta_\\sigma(A)=\\sigma A \\sigma^{-1}$ the corresponding modular operator.\n\\item For two operators $A, B$ we write $[A, B]=AB-BA$ and $\\set{A, B}=AB+BA$.\n\\end{itemize}\n\n\\subsection{The repeated interactions model in the continuous-time limit}\n\nWe consider a quantum system $S$ represented by a finite-dimensional Hilbert space $\\mathcal{H}_S$, in interaction with a bath $B$. In the Markovian approximation, the evolution on $\\mathcal{H}_S$ can be modeled by a quantum Markov semigroup.\n\n\\begin{defi}\nA \\emph{quantum Markov semigroup} (QMS) on a finite-dimensional Hilbert space $\\mathcal{H}_S$ is a family of linear maps on $\\mathcal{B}(\\mathcal{H}_S)$ which are completely positive and unity-preserving maps $(\\Lambda^t)_{t\\in I}$ where $I$ is $\\mathbb{N}$ or $[0, +\\infty)$, satisfying $\\Lambda^{s+t}=\\Lambda^s \\Lambda^t$ for $s, t\\in I$ such that $t\\rightarrow \\Lambda^t$ is continuous if $I=[0, +\\infty)$.\n\\end{defi} \n\nNote that $\\Lambda^t$ describes the evolution in the Heisenberg representation, the evolution in the Schr\\\"odinger representation being described by $(\\Lambda^t)^*$.\n\nIn the case of discrete time $I=\\mathbb{N}$ we can construct such a model by a repeated interaction process: the bath is decomposed as a series of identical and independent sub-baths, interacting one after another with the system. The sub-bath model is a finite-dimensional Hilbert space $\\mathcal{H}_B$, and we fix a unitary $U$ on $\\mathcal{H}_S\\otimes \\mathcal{H}_B$. Each bath subsystem is in the same state $\\rho_B\\in \\mathcal{B}(\\mathcal{H}_B)$, thus the evolution on the system is \n\\[\n(\\Lambda^1)^*(\\rho_S)=\\text{Tr} _{\\mathcal{H}_B}\\left(U(\\rho_S\\otimes \\rho_B)U^*\\right)~.\n\\]\nIn this article, we consider a QMS in continuous time $I=[0, T)$ obtained as the limit of discrete-time QMS when a parameter $\\tau>0$ goes to zero, under suitable renormalization. We will take advantage of the easy interpretation of the repeated interaction model in discrete time to define quantities such as the entropy production and the energy exchanges between the bath and the system. The continuous-time limit of repeated interactions has been introduced by Attal in \\cite{AttalToy}, and developed by Attal and Pautrat \\cite{AttalPautrat}.\n\n\\begin{prop}[Adaptation of Theorem 22 of \\cite{AttalPautrat}]\nFix some self-adjoint operators $H_S\\in \\mathcal{B}(\\mathcal{H}_S)$ and $H_{SB}\\in \\mathcal{B}(\\mathcal{H}_S\\otimes \\mathcal{H}_B)$ and a state $\\rho_B \\in \\mathfrak{S}(\\mathcal{H}_B)$ such that \n\\begin{align}\\label{eq:conditionV}\n\\text{Tr} _{B}(H_{SB}(\\mathds{1}_S\\otimes \\rho_B))=0~.\n\\end{align}\nFor any time scale $\\tau>0$ write $U_\\tau$ the self-adjoint operator\n\\[\nU_\\tau=\\exp(-i\\tau H_S\\otimes \\mathds{1}_B -i\\sqrt{\\tau} H_{SB})\n\\]\nand define the map $\\Lambda_\\tau$ by $(\\Lambda_\\tau)^*(\\rho)=\\text{Tr} _{B}(U_\\tau (\\rho\\otimes \\rho_B)U_\\tau^*)$. Then for any $t\\in [0, +\\infty)$ the map $\\Lambda^\\ent{t\/\\tau}_\\tau$ converges to a map $\\Lambda^t$ as $\\tau \\rightarrow 0$ on the trace-norm topology over trace-class operators, locally uniformly in $t$. The family $(\\Lambda^t)_{t\\geq 0}$ is a continuous QMS with generator\n\\begin{align}\\label{eq:lindblad}\n\\mathcal{L}(A)=i[H_S, A]+\\Phi(A)-\\frac{1}{2}\\set{\\Phi(\\mathds{1}_S), A}\n\\end{align}\nwhere $\\Phi$ is the completely positive map defined by $\\Phi^*(\\rho)=\\text{Tr} _{B}\\left(H_{SB}(\\rho\\otimes \\rho_B) H_{SB}\\right)$.\n\nMoreover, any continuous QMS can be obtained that way. We will call the triple $(H_S, H_{SB}, \\rho_B)$ a repeated interaction model for the QMS $\\Lambda$.\n\\end{prop}\nThis theorem was proved by Attal and Pautrat in \\cite{AttalPautrat} in the case where $\\rho_B$ is a pure state. We introduced this generalization with the condition of Equation \\eqref{eq:conditionV} in \\cite{AndreysFermions1}.\n\n\\begin{proof}\nThe idea to derive the continuous-time limit is to use the formula \n\\[\n\\lim_{n\\rightarrow +\\infty} \\left(I+\\frac{A}{n}+o\\left(\\frac{1}{n}\\right)\\right)^n=e^A\n\\]\napplied to $n=\\ent{t\/\\tau}$, and $A=t\\mathcal{L}$. \n\nDeveloping $U_\\tau$ we obtain\n\\[\nU_\\tau=\\mathds{1}_{SB}-i\\sqrt{\\tau} H_{SB}-i\\tau H_S\\otimes \\mathds{1}_B-\\frac{\\tau}{2} H_{SB}^2+O(\\tau^{3\/2})\n\\]\nwhere $O(\\tau^{3\/2})=\\tau^{3\/2} R(\\tau)$ for an operator $R(\\tau)$ which is uniformly bounded as $\\tau\\rightarrow 0$. Thus, we have\n\\[\n(\\Lambda_\\tau)^*(\\rho)=\\rho-i\\sqrt{\\tau} \\text{Tr} _B\\left(H_{SB}(\\rho\\otimes \\rho_B)-(\\rho\\otimes \\rho_B)H_{SB} \\right)+\\tau \\mathcal{L}(\\rho)+O(\\tau^{3\/2})\n\\]\nwhere $\\mathcal{L}$ is defined by \\ref{eq:lindblad}. The term in $\\sqrt{\\tau}$ is zero because of Equation \\eqref{eq:conditionV}. Thus, \n\\[\n(\\Lambda_\\tau)^{\\ent{t\/\\tau}}=(\\mathds{1}_{\\mathcal{B}(\\mathcal{H}_S)} +\\tau \\mathcal{L}+o(\\tau) )^{\\ent{t\/\\tau}}\n\\]\nand $\\tau=t\/\\ent{t\/\\tau}+o(\\tau)$, so we obtain the convergence part of the Theorem.\n\nLet us prove that any norm-continuous QMS $(\\Lambda^t)_{t\\geq 0}$ can be obtained by a repeated interaction model. By a theorem of Lindblad \\cite{Lindblad1976} $(\\Lambda^t)_{t\\geq 0}$ admits a generator $\\mathcal{L}$ which is of the form given by Equation \\eqref{eq:lindblad} for some completely positive map $\\Phi$. Since $\\Phi$ is completely positive and norm-continuous we can write $\\Phi(\\rho)=\\sum_{i=1}^{+\\infty} L_i^* \\rho L_i$ where the $L_i$ are bounded operators and $\\sum_{i=1}^{+\\infty} L_i^* L_i$ is bounded. To model $\\Lambda^t$ as the limit of a repeated interaction model, we choose $\\mathcal{H}_B=l^2(\\mathbb{N})$ and $\\rho=\\ket{0}\\bra{0}$ and \n\\[\nH_{SB}=\\sum_{i=1}^{+\\infty} L_i \\otimes \\ket{i}\\bra{0}+L_i^*\\otimes \\ket{0}\\bra{i}~.\n\\]\n\\end{proof}\n\\begin{rem}\nNote that the operator $H_{SB}$ constructed in the last part of the proof satisfies the stronger condition \n\\begin{align}\\label{eq:strongConditionV}\n\\tr{H_{SB} (\\mathds{1}_S\\otimes f(\\rho_B)}=0\n\\end{align}\nfor any function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ (or equivalently, $\\bra{i}H_{SB}\\ket{i}=0$ for a Hilbert basis $\\ket{i}$ in which $\\rho_B$ is diagonal). This condition will be useful later.\n\\end{rem}\n\n\n\n\n\n\\subsection{Detailed balance} \\label{subseq:energy_conservation}\n\nWe will consider only a special case of QMS, arising from a bath which is composed of several parts, each of them at thermal equilibrium with respect to a globally conserved pseudo-energy.\n\n\\begin{defi}\nWe call a \\emph{thermal repeated interaction model} a repeated interaction model $(H_S,H_{SB},\\rho_B)$ with a bath decomposed as $\\mathcal{H}_B=\\bigotimes_{i=1}^n \\mathcal{H}_{B_i}$ in the state $\\rho_B=\\bigotimes_{i=1}^n \\rho_{B_i}$, where the $\\rho_{B_i}$ are Gibbs states:\n\\[\n\\rho_{B_i}=\\frac{e^{-\\beta_i K_{B_i}}}{\\tr{e^{-\\beta_i K_{B_i}}}}\n\\]\nfor some inverse temperatures $\\beta_i \\in \\mathbb{R}$ and some self-adjoint operators $K_{B_i}\\in \\mathcal{B}_{sa}(\\mathcal{H}_{B_i})$, with the following assumptions: $H_{SB}$ can be decomposed as $\\sum_{i=1}^n H_{S B_i}$ with $H_{SB_i}$ acting only on $\\mathcal{H}_S\\otimes \\mathcal{H}_{B_i}$, and there exists a self adjoint operator $K_S$ on $\\mathcal{H}_S$ with\n\\begin{align}\\label{eq:invariantN}\n[H_S, K_S]&=0 & \\left[H_{SB_i},~ K_S\\otimes \\mathds{1}_B + \\mathds{1}_S\\otimes K_{B_i}\\right]&=0~.\n\\end{align}\nWe call the operator $K_S$ the \\emph{pseudo-energy} of the model.\n\\end{defi}\n\nWe choose the name pseudo-energy for $K_S$ because it is invariant and has the same dimension as the energy $K_{B_i}$, but it does not generate the dynamic.\n\nThe QMS arising from a thermal repeated interaction model are characterized by a detailed balance condition, defined as follow: \n\n\\begin{defi}\nA continuous-time QMS is said to satisfy the detailed balance condition with respect to a state $\\sigma$ if its generator $\\mathcal{L}$ can be written \n\\[\n\\mathcal{L}(A)=i[H_S, A]+\\Phi(A)-\\frac{1}{2}\\set{\\Phi(\\mathds{1}), A}\n\\]\nwhere $H_S$ is a self-adjoint operator commuting with $\\sigma$ and $\\Phi$ is a completely positive map satisfying\n\\begin{align}\\label{eq:db}\n\\Phi^*( A\\sigma)\\sigma^{-1}=\\Phi(A)\n\\end{align}\nfor any operator $A$. \n\\end{defi}\nThis condition is related to weaker conditions such as the time-reversal invariance (see \\cite{JPW14}). Note that Equation \\eqref{eq:db} means that $\\Phi$ is self-adjoint with respect to the scalar product $\\scal{A, B}_{\\sigma, 0}=\\tr{\\sigma A^* B}$. It implies that $\\Phi$ commutes with the modular operator $\\Delta_\\sigma$, and that for any $s$ it is self-adjoint with the scalar product $\\scal{A, B}_{\\sigma, s}=\\tr{\\sigma^{1-s}A^*\\sigma^s B}$. See for example Carlen and Maas \\cite{carlen_maas_17} or the original article of Alicki \\cite{alicki_76}.\n\\newline\n\n\nAlicki proved the following characterization of the strong detailed balance:\n\n\n\\begin{theo}\\label{theo:alickidb}\nLet $\\beta\\in \\mathbb{R}$ and consider the Gibbs state $\\sigma=\\exp{-\\beta K_S}\/Z$. Let $\\Lambda$ be a continuous-time QMS. Then $\\Lambda^t$ satisfies the detailed balance for all $t$ if and only if it can be written in the Lindblad form \\ref{eq:lindblad} with operator $H_S$ commuting with $K_S$, and completely positive map $\\Phi$ of the form\n\n\\begin{align}\n\\Phi(A)&=\\sum_{\\delta\\in sp([K_S, \\bullet]), \\delta \\geq 0}~\\sum_{i=1}^{n_{\\delta}} e^{-\\frac{\\beta}{2}\\delta~}L_{(\\delta, i)}^* A L_{(\\delta, i)}+e^{\\frac{\\beta}{2}\\delta~}L_{(\\delta, i)} A L_{(\\delta,i)}^*\n\\end{align}\nfor some integers $n_\\delta$, where the $L_{(\\delta,i)}$ are operators satisfying $[K_S, L_{(\\delta,i)}]=\\delta L_{(\\delta, i)}$ and $\\tr{L_{(\\delta,i)}}=0$.\n\\end{theo}\n\n\nThis theorem allows to make the link between the existence of a thermal model and the strong detailed balance condition, as follows.\n\n\\begin{prop}\\label{prop:thermal=db}\nA continuous-time QMS $\\Lambda$ satisfies the strong detailed balance condition with respect to $\\sigma=e^{-\\beta K_S}\/Z$ if and only if it admits a thermal repeated interaction model $(H_S,H_{SB}, \\rho_B)$ with only one bath at inverse temperature $\\beta$ and energy operator $K_S$. The model may be assumed to satisfy Condition \\ref{eq:conditionV}.\n\\end{prop}\n \n \\begin{proof}\nTo prove the sufficiency of our condition, assume that $\\Lambda$ admits a thermal repeated interaction model with one bath $\\mathcal{H}_B$ at inverse temperature $\\beta$, with energy operator $K_B$. Then $H_S$ commutes with $K_S$ so it commutes with $\\sigma$, and using the formula \n\\[\n\\Phi(A)=\\text{Tr} _B\\left(H_{SB} (A\\otimes \\mathds{1}_B) H_{SB} (\\mathds{1}_S\\otimes \\rho_B)\\right)~\n\\]\nwe obtain that for any operators $A, B\\in \\mathcal{B}(\\mathcal{H}_S)$ we have\n\\begin{align*}\n\\scal{A, \\Phi(B)}_{\\sigma, 0}&=\\tr{(\\sigma\\otimes \\mathds{1}_B) (A^*\\otimes \\mathds{1}_B) H_{SB} (B\\otimes \\mathds{1}_B) H_{SB} (\\mathds{1}_S\\otimes \\rho_B)} \\\\\n\\end{align*}\nand since $[H_{SB}, \\sigma\\otimes \\rho_B]=\\left[H_{SB}, \\frac{e^{-\\beta(K_S+K_B)}}{Z_B}\\right]=0$ this gives\n\\begin{align*}\n\\scal{A, \\Phi(B)}_{\\sigma, 0}&=\\tr{(\\sigma\\otimes \\rho_B)H_{SB}(A^*\\otimes \\mathds{1}_B)H_{SB} (B\\otimes \\mathds{1}_B)}\\\\\n&=\\scal{\\Phi(A), B}_{\\sigma, 0}~.\n\\end{align*}\nThus $\\Phi$ is self-adjoint for the scalar product $\\scal{A, B}_{\\sigma, 0}$ so the detailed balance condition is satisfied.\n\\newline\n\nTo prove the necessary condition, we apply Theorem \\ref{theo:alickidb}, obtaining an operator $H_S$ and operators $L_{(\\delta,i)}$~. To construct $\\mathcal{H}_B$, we consider the set \n\\[\n\\mathcal{D}_+(K_S)=\\bigcup_{\\delta\\in sp([K_S, \\bullet])} \\bigcup_{k=1}^{n\\delta} \\set{(\\delta, k)}\n\\]\n and we take $\\mathcal{H}_B=\\mathbb{C}^{\\mathcal{D}_+(K_S)}\\otimes \\mathbb{C}^2$, with Hilbert basis $\\ket{\\delta,k}\\otimes\\ket{i}$ for $(\\delta, k)\\in \\mathcal{D}_+(K_S)$ and $i\\in \\{{\\pmb{-}},{\\pmb{+}}\\}$. We consider the pseudo-energy operator\n\n\\[\nK_B=\\frac{1}{2} \\sum_{(\\delta, k)\\in \\mathcal{D}_+(K_S)} \\delta \\ket{\\delta, k}\\otimes \\Big(\\ket{{\\pmb{+}}}\\bra{{\\pmb{+}}}-\\ket{{\\pmb{-}}}\\bra{{\\pmb{-}}}\\Big)~\n\\]\nand take $\\rho_B=e^{-\\beta K_B}\/Z$ with $Z=\\tr{e^{-\\beta K_B}}$. We define the interaction operator $H_{SB}$ by\n\\[\nH_{SB}=Z\\sum_{(\\delta, k)\\in \\mathcal{D}_+(K_S)}L_{(\\delta,k)}\\otimes \\ket{\\delta, k}\\bra{\\delta, k}\\otimes \\ket{{\\pmb{-}}}\\bra{{\\pmb{+}}}+L_{(\\delta,k)}^*\\otimes \\ket{\\delta, k}\\bra{\\delta, k}\\otimes \\ket{{\\pmb{+}}}\\bra{{\\pmb{-}}}\n\\]\nThe relation $[K_S, L_{(\\delta, k)}]=\\delta L_{(\\delta, k)}$ implies the conservation of pseudo-energy\n\\[\n\\left[H_{SB}, K_S\\otimes \\mathds{1}_B+\\mathds{1}_S\\otimes K_B\\right]=0~.\n\\]\nIt is easily checked that\n\\[\n\\text{Tr} _B\\left(H_{SB}(A\\otimes \\mathds{1}_B)H_{SB} (\\mathds{1}_S\\otimes \\rho_B)\\right)=\\sum_{(\\delta,k)\\in \\mathcal{D}_+(K_S)} e^{-\\frac{\\beta}{2}\\delta}L_{(\\delta, k)}^*\\rho L_{(\\delta, k)}+e^{\\frac{\\beta}{2}\\delta}L_{(\\delta, k)} \\rho L_{(\\delta, k)}^*~.\n\\]\n \\end{proof}\n\n\n\\begin{rem}\nThe detailed balance condition implies the time-reversal invariance: there exists an anti-linear involution of algebras $\\Theta$ on $\\mathcal{B}(\\mathcal{H}_S)$ (called a time-reversal) with $\\Theta(\\sigma)=\\sigma$ such that $\\Theta(\\mathcal{L}^*(\\Theta(A)\\sigma)\\sigma^{-1})=\\mathcal{L}$. This condition is strictly weaker than the detailed balance, indeed it is satisfied if and only if $\\Phi$ can be written as\n\\[\n\\sum_{\\delta\\in sp([K_S, \\bullet]), \\delta\\geq 0} \\sum_{i=1}^{n_\\delta} e^{-\\frac{\\beta}{2}\\delta~}L_{(\\delta, i)}^* A L_{(\\delta, i)}+e^{\\frac{\\beta}{2}\\delta~}M_{(\\delta, i)} A M_{(\\delta,i)}^*\n\\] \nwhere $L_{(\\delta, i)}$ and $M_{(\\delta, i)}$ both satisfy the same conditions as the $L_{(\\delta,i)}$'s of Theorem \\ref{theo:alickidb} and are related by $M_{(\\delta, i)}=\\Theta\\left(L_{(\\delta,i)}\\right)$, which includes cases where $L_{(\\delta,i)}\\neq M_{(\\delta,i)}$ and the detailed balance condition is not satisfied. \n\\end{rem}\n\n\\begin{rem}\nIt is not clear how to define the detailed balance for a discrete-time Quantum Markov Semigroup $\\Lambda^n$, since it is not possible to separate the generator in a unitary and a dissipative part. The best way to define it is probably the existence of a thermal repeated interaction model (with only one bath).\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Energy and entropy fluxes}\\label{subseq:energy_fluxes}\n\nWe now consider a thermal repeated interaction model $(H_S, H_{SB})$ with pseudo-energy operator $K_S$ and $n$ baths with inverse temperatures $\\beta_1\\leq \\beta_2 \\leq \\cdots \\leq \\beta_n$ and energy operators $K_{B_i}$. We write $K_{tot}=K_S\\otimes \\mathds{1}_B+ \\sum_{i=1}^n \\mathds{1}_S\\otimes K_{B_i}$ the global preserved pseudo-energy operator, define \n\\[\n\\Phi_i(A)=\\tr{H_{SB_i} (A\\otimes \\mathds{1}_B) H_{SB_i} \\mathds{1}_S\\otimes \\rho_{B_i}}\n\\]\nand write\n\\[\n\\mathcal{L}_i(A)=\\Phi_i(A)-\\frac{1}{2}\\set{\\Phi_i(\\mathds{1}_S), A}\n\\]\nso that each $\\mathcal{L}_i$ generates a QMS with the strong detailed balance with respect to $\\sigma_i=e^{-\\beta_i K_S}\/Z_i$ and $\\mathcal{L}=-i[H_S, \\bullet]+\\sum_i \\mathcal{L}_i$.\n\\newline\n\n In discrete times, we can define the energy increase of the $i$-th bath during one interaction as\n\\[\nD_{\\tau, i}(\\rho)=\\tr{(\\mathds{1}_S\\otimes K_{B_i}) \\left(U_\\tau \\rho\\otimes \\rho_B U_\\tau^*-\\rho\\otimes \\rho_B\\right)}\n\\]\nand the total energy increase at time $\\tau k$ as \n\\[\nD_{\\tau, i, 0\\rightarrow k}(\\rho)=\\sum_{l=0}^{k-1} D_{\\tau, i}(\\rho(l))\n\\]\nwhere $\\rho(l)=(\\Lambda_\\tau^*)^l(\\rho)$. Let us study the limit as $\\tau\\rightarrow 0$ of $D_{\\tau, i, 0\\rightarrow \\ent{t\/\\tau}}(\\rho)$ for fixed $t\\geq 0$. We have\n\\begin{align}\\label{eq:estimate_Dtaui}\nD_{\\tau, i}(\\rho)&=i\\sqrt{\\tau}\\tr{[H_{SB_i}, K_{B_i}]\\rho\\otimes \\rho_{B_i}}-\\tau \\tr{\\mathcal{L}_i(K_S)\\rho} +O(\\tau^{\\frac{3}{2}}) ~.\n\\end{align}\nThe term in $\\tau$ is obtained by using the fact that $K_S\\otimes \\mathds{1}_B+\\mathds{1}_S\\otimes K_{B_i}$ commutes with $H_{SB_i}$. Thus, for the quantity $D_{\\tau, i, 0\\rightarrow \\ent{t\/\\tau}}(\\rho)$ to have a limit, we will impose the following assumption: \n\n\\begin{assumption}\\label{ass:strongConditionV}\nFor any bath index $i\\in \\{1, \\cdots, n\\}$ there is a basis $(\\ket{j})_{j\\in \\{1, \\cdots, \\dim (\\mathcal{H}_{B_i})\\}}$ of $\\mathcal{H}_{B_i}$ in which $K_{B_i}$ is diagonal and $\\bra{j} H_{SB_i} \\ket{j}=0$ for all $j$. Equivalently for any $\\alpha\\in \\mathbb{R}$ we have\n\\[\n\\text{Tr} _{B_i}\\left( H_{SB_i} \\mathds{1}_{\\mathcal{H}_S}\\otimes \\rho_{B_i}^\\alpha\\right)=0~.\n\\]\n\\end{assumption}\nThis assumption is satisfied by the interaction $H_{SB}$ constructed in Proposition \\ref{prop:thermal=db}. It is used to ensure the convergence of the energy fluxes, as in the following lemma.\n\n\\begin{lem}\nUnder Assumption \\ref{ass:strongConditionV} the quantity $D_{\\tau, i, \\ent{t\/\\tau}}(\\rho)$ converges for all $t$ and $\\rho$ as $\\tau\\rightarrow 0$ to a limit\n\\[\nD_{i, t}(\\rho)=\\int_0^t J_{i}(\\rho(s))ds\n\\]\nwhere $\\rho(s)=\\Lambda^s(\\rho)$ and $J_i$ is the flux of energy entering the $i$-th bath, equal to \n\\begin{align}\\label{eq:def_J}\nJ_i(\\rho)=-\\tr{\\rho \\mathcal{L}_i(K_S)}~.\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\n Assumption \\ref{ass:strongConditionV} implies that for any state $\\rho$ the quantity $\\tr{[H_{SB_i}, K_{B_i}]\\rho\\otimes \\rho_{B_i}}$ is equal to zero, so Equation \\eqref{eq:estimate_Dtaui} has no term in $\\sqrt{\\tau}$, so\n \\[\n D_{\\tau, i, 0\\rightarrow \\ent{t\/\\tau}}(\\rho)=\\tau \\sum_{l=0}^{\\ent{t\/\\tau}-1} J_i(\\Lambda^{l\\tau}(\\rho)) +O\\left(\\tau^{3\/2}\\right)\n \\] \n converges, and has for differential $J_{i}(\\rho(t))$.\n \\end{proof}\n\nNote that $\\sum_{i=1}^n J_i(\\rho)=-\\tr{\\rho\\mathcal{L}(K_S)}$ so at equilibrium ($\\mathcal{L}^*(\\rho)=0$) the fluxes satisfies the first law of thermodynamics: $\\sum_{i=1}^n J_i(\\rho)=0$. As we shall see, it also satisfies the second law. \n\n\\begin{prop}\\label{prop:second_principle}\nUnder assumption \\ref{ass:strongConditionV}, the fluxes satisfies\n\\begin{align*}\nS(\\rho(t))-S(\\rho)+\\sum_{i=0}^n \\beta_i D_{i, t}(\\rho)\\geq 0~.\n\\end{align*}\n\\end{prop}\n\n\n\n\\begin{proof}\nLet us consider the discrete-time model. The total entropy before the interaction is\n\\begin{align*}\nS(\\rho\\otimes \\rho_B)=S(\\rho)+\\sum_{i=1}^n \\beta_i \\tr{\\rho_{B_i} K_{B_i}}-\\beta_i \\log Z_i~.\n\\end{align*}\n\nNow, let us consider the complete state after the interaction $\\rho_{tot}'=U_\\tau (\\rho\\otimes \\rho_B) U_\\tau^*$ and the partial state $\\rho'=\\text{Tr} _B(\\rho_{\\tau}')=\\Lambda_\\tau(\\rho)$. Since the entropy is preserved by unitary evolution, $S(\\rho_{tot}')=S(\\rho\\otimes \\rho_B)$. Now we use a trick which goes back to \\cite{pusz_passive_1978} (and has been used many times, see Section III of \\cite{JP_landauer_2014}) : the relative entropy $S(\\rho_{tot}'|\\rho'\\otimes \\rho_B)$ is always positive, and it is equal to \n\\begin{align*}\nS(\\rho_{tot}'|\\rho'\\otimes \\rho_B)&=\\tr{\\rho_{tot}'\\left(\\log(\\rho_{tot}')-\\log(\\rho')\\otimes \\mathds{1}_B-\\mathds{1}_S\\otimes \\log(\\rho_B)\\right)}\\\\\n&=-S(\\rho_{tot}')+S(\\rho')+\\beta_i\\tr{\\rho_{tot}'\\mathds{1}_S\\otimes K_{B_i}}~.\n\\end{align*}\nThus we have\n\\[\nS(\\rho')-S(\\rho)+\\sum_{i=1}^n \\beta_i D_{\\tau,i}(\\rho)=S(\\rho_{tot}'|\\rho'\\otimes \\rho_B) \\geq 0~.\n\\]\nSumming over all $k\\in \\{0, \\ent{t\/\\tau}\\}$ and taking the limit as $\\tau\\rightarrow 0$ allows to conclude.\n\\end{proof}\n\n{\\bf Remark: } The quantities $\\mathcal{L}_i$, $J_i$ and $S(\\rho)$ can all be defined without the help of the repeated interaction model, and also originate from other models such as the weak coupling limit. However, the repeated interaction model has the advantage of being easy to interpret; moreover some results on the repeated interaction models pass to the limit, as in the proof of Proposition \\ref{prop:second_principle} above.\n\n\n\\subsection{Energy fluxes for stationary states and thermal machines}\n\nThe system being of finite dimension, there exists a trace-preserving projection (not necessarily an orthogonal one) $E$ on $\\ker(\\mathcal{L})$ such that\n\\[\n\\lim_{T\\rightarrow \\infty} \\frac{1}{T} \\int_{0}^T e^{t\\mathcal{L}^*}(\\rho) dt=E(\\rho)~.\n\\]\nHere are some properties that the QMS may enjoy:\n\\begin{enumerate}\n\\item The system is ergodic if there is a unique state $\\rho_\\infty$ with $\\mathcal{L}(\\rho_\\infty)=0$. Then $E(\\rho)=\\rho_\\infty$ for any initial state $\\rho$. \n\\item The system is primitive if $0$ is a simple eigenvalue and is the only eigenvalue with real part zero. Then $E(\\rho)=\\rho_\\infty=\\lim_{t\\rightarrow \\infty} e^{t\\mathcal{L}^*}(\\rho)$ for any initial state $\\rho$.\n\\item We will be interested in systems which are positivity improving, that is, primitive with a stationary state $\\rho_\\infty>0$. Equivalently (in finite dimension) for any nonzero operator $A$ with $A\\geq 0$, we have $e^{t\\mathcal{L}}(A)>0$ for all $t>0$.\n\\end{enumerate}\n\nLet us consider the asymptotic of the mean energy flux \n\\[\nJ_i=\\lim_{t\\rightarrow \\infty}\\frac{1}{T}D_{i, T}(\\rho)=J_i(E(\\rho))~.\n\\]\nBy Proposition \\ref{prop:second_principle} they satisfy the first and second law of thermodynamics\n\\begin{align}\n\\sum_{i=1}^n J_i=0 \\label{eq:firstlaw}\\\\\n\\sum_{i=1}^n \\beta_i J_i\\geq 0~.\\label{eq:secondlaw}\n\\end{align}\nWe call \\enquote{entropy production} the quantity $\\sum_{i=1}^n \\beta_i J_i$.\n\nAs we shall see, any list $(J_i)_{1\\leq i \\leq n}$ satisfying these conditions can be attained for a specific model, if the inequality in the second law is strict:\n\n\\begin{prop}\\label{prop:thermal_machines}\nLet $J_1, \\cdots, J_n\\in \\mathbb{R}$ such that $\\sum_{i=1}^n J_i=0$ and $\\sum_{i=1}^n \\beta_i J_i>0$. Then there exists a thermal repeated interaction model $(H_S, H_{SB})$ with $n$ baths, for which the complete Lindbladian $\\mathcal{L}$ is positivity improving and such that $J_i(\\rho_\\infty)=J_i$ for all $i$ (where $\\rho_\\infty$ is the unique stationary state of the QMS).\n\\end{prop}\n\nFor two baths, the conditions implies is $J_2=-J_1\\geq 0$ if $\\beta_1\\leq \\beta_2$, that is the energy flows from the hottest bath to the coldest one. The first nontrivial case arise with three baths, where we can have $J_3<0$ : the energy flowing from the hottest bath to the mild bath allows to pump energy from the coldest bath, as in a camping fridge (in which the hot bath is a gas stove, the mild bath is the ambient air and the cold bath is the inner of the fridge).\n\nLater in the article we describe the class of quasi-free fermionic semigroups, for which this is not true, and only some very specific fluxes can be obtained. \n\\vspace{0.5cm}\n\nIn order to prove this proposition, we introduce some special cases of thermal repeated interactions models.\n\n{\\bf The generalized depolarizing channel on a qubit: } This is the simplest non-trivial example of a thermal QMS. For any state $\\sigma$ on $\\mathcal{H}_S$ the corresponding depolarizing channel has for generator\n\\[\n\\mathcal{L}_{\\sigma, \\lambda}^*(\\rho)=\\lambda(\\sigma\\tr{\\rho}-\\rho)\n\\]\nfor some positive real number $\\lambda$ called the rate of the depolarizing channel. When $\\sigma$ is faithful then $\\mathcal{L}_{\\sigma, \\lambda}$ satisfies the detailed balance with respect to $\\sigma$; indeed, it admits the following thermal repeated interaction model: take $\\mathcal{H}_B\\simeq \\mathcal{H}_S$ and let $\\rho_B=\\sigma$ and let $H_{SB}$ be proportional to the swap operator: \n\\[\nH_{SB}=\\sqrt{\\lambda}\\sum_{i,j=1}^{d_S} \\ket{i}\\bra{j}\\otimes\\ket{j}\\bra{i}\n\\]\nwhere $(\\ket{i})_{i=1}^{d_S}$ is a Hilbert basis of $\\mathcal{H}_S$. \n\nMoreover, the semigroup is positivity improving when $\\sigma$ is faithful and it is ergodic in general. The semigroup can be described explicitly: \n\\[\n\\Lambda^t(\\rho)=\\left(\\rho-\\sigma\\right)e^{-\\lambda t}+\\sigma~.\n\\]\n\\vspace{0.5cm}\n\n{\\bf Two depolarizing channels at different temperature:}\n\nLet us consider two states $\\sigma_1=e^{-\\beta_1 K_S}\/Z_1$ and $\\sigma_2=e^{-\\beta_2 K_S}\/Z_2$, with $\\beta_1\\leq \\beta_2$. Then we can combine two depolarizing channels corresponding to these states: \n\\[\n\\mathcal{L}^*(\\rho)=\\lambda_1(\\sigma_1\\tr{\\rho}-\\rho)+\\lambda_2(\\sigma_2\\tr{\\rho}-\\rho)\n\\]\nfor some positive real numbers $\\lambda_1$ and $\\lambda_2$. It is actually the depolarizing channel of rate $\\lambda_1+\\lambda_2$ with respect to the state\n\\[\n\\rho_\\infty=\\frac{\\lambda_1 \\sigma_1+\\lambda_2\\sigma_2}{\\lambda_1+\\lambda_2}~.\n\\]\nNote that $\\rho_\\infty$ commutes with $K_S$ but it is not a Gibbs state with respect to the $K_S$ except in trivial cases. Associating one bath to each channel, we have\n\\[\nJ_1(\\rho_\\infty)=\\frac{\\lambda_1\\lambda_2}{\\lambda_1+\\lambda_2} \\tr{K_S (\\sigma_1-\\sigma_2)}~.\n\\]\nWhenever $\\beta_1<\\beta_2$ it is possible to choose any negative value for $J_1(\\rho_\\infty)$ by tuning the rates $\\lambda_1$ and $\\lambda_2$, hence proving the proposition in the case of two baths. \n\\vspace{0.5cm}\n\n{\\bf The quantum fridge:} This example was introduced by Linden, Popescu and Skrzypczyp in \\cite{thermal_machine_small_prl10} (see also \\cite{thermal_machine_small_maximal_efficiency_jpa11} where the solution is more detailed, and \\cite{cooling_entanglement_bhlp} for more developments). It is a simple model of quantum fridge, where the energy of the coolest bath is pumped out by the use of two other baths. The solution can be explicitly computed but the description is more involved, and we refer to \\cite{thermal_machine_small_maximal_efficiency_jpa11} for a complete discussion. Let us just describe the setup: the system is composed of three qubits: $\\mathcal{H}_S=(\\mathbb{C}^2)^{\\otimes 3}$. Write $P_1=\\ket{1}\\bra{1}\\otimes \\mathds{1}_{\\mathbb{C}^2\\otimes \\mathbb{C}^2}$ the projector on the state $\\ket{1}$ on the first qubit, and define $P_2$, $P_3$ similarly. We chose a pseudo-energy $K_S$ which acts independently on the qubits; \n\\[\nK_S=E_1 P_1+E_2 P_2+E_3~ P_3~.\n\\]\nThe energies $E_1, E_2, E_3$ are supposed nonzero.\nWe consider three baths with inverse temperatures $\\beta_1>\\beta_2>\\beta_3$; the Hamiltonian $H_S$ on the system is defined by\n\\[\nH_S=h K_S+g(\\ket{010}\\bra{101}+\\ket{101}\\bra{010})~.\n\\]\nWe assume $E_1+E_3=E_2$ so that $[H_S, K_S]=0$. This way, energy can flow from the hot bath to the middle bath only if some energy is pumped out of the cold bath. We take each bath acting on one qubit with the depolarizing channel corresponding to the Gibbs state $\\sigma_i$ on this qubit at inverse temperature $\\beta_i$, that is\n\\[\n\\sigma_i=\\frac{1}{1+e^{-\\beta_i E_i}}\\left(\\ket{0}\\bra{0}+e^{-\\beta_i E_i}\\ket{1}\\bra{1}\\right)\n\\]\nand \n\\[\n\\mathcal{L}=i[H_S, \\bullet]+\\lambda_1 \\mathcal{L}_1\\otimes \\mathds{1}_{\\mathcal{B}(\\mathbb{C}^2\\otimes \\mathbb{C}^2)}+\\lambda_2 \\mathds{1}_{\\mathcal{B}(\\mathbb{C}^2)}\\otimes \\mathcal{L}_2\\otimes \\mathds{1}_{\\mathcal{B}(\\mathbb{C}^2)}+\\lambda_3\\mathds{1}_{\\mathcal{B}(\\mathbb{C}^2\\otimes \\mathbb{C}^2)}\\otimes \\mathcal{L}_3~\n\\]\nwhere $\\mathcal{L}_i^*(\\rho)=\\sigma_i-\\rho$ for any state $\\rho$ on $\\mathbb{C}^2$. Note that $\\mathcal{L}$ is positivity improving since each $\\mathcal{L}_i$ is positivity improving on its respective qubit. In \\cite{thermal_machine_small_maximal_efficiency_jpa11} the stationary state $\\rho_\\infty$ is explicitly described, and the following facts are observed: \n\\begin{lem}\nThere exists a parameter $\\alpha\\in \\mathbb{R}$ depending on the $E_i$, $\\beta_i$, $\\lambda_i$ and $g,h$ such that\n\\begin{align}\\label{eq:prop_flux}\nJ_1(\\rho_\\infty)=&\\alpha E_1 & J_2(\\rho_\\infty)&=-\\alpha E_2 & J_3(\\rho_\\infty)&=\\alpha E_3\n\\end{align}\n Moreover, if all the parameters $E_i, \\lambda_i, g, f$ are nonzero and if \n \\[\n \\sum_{i=1}^n \\beta_i E_i\\neq 0\n \\]\n then $\\alpha \\neq 0$ and $\\alpha$ has the same sign as $\\sum_{i=1}^n \\beta_i E_i$. \n\\end{lem}\n\n\\begin{proof}[Elements of proof]\nThe proportionality relation \\ref{eq:prop_flux} can be proved directly: consider the observable $Q=P_1+2P_2+P_3$. Then $[H, Q]=0$ which implies\n\\[\n\\tr{\\rho_\\infty\\sum_{i=1}^3\\mathcal{L}_i^*(Q)}=\\tr{\\rho_\\infty\\mathcal{L}^*(Q)}=\\tr{\\mathcal{L}(\\rho_\\infty)Q}=0\n\\]\nbut the left-hand side is equal to\n\\[\n\\frac{J_1(\\rho_\\infty)}{E_1}+2\\frac{J_2(\\rho_\\infty)}{E_2}+\\frac{J_3(\\rho_\\infty)}{E_3}=0~.\n\\]\nSince the sum of the fluxes equals $0$ and $E_1+E_3=E_2$ this implies the proportionality relation. The sign of $\\alpha$ is constrained by Proposition \\ref{prop:second_principle}. To prove that $\\alpha$ is nonzero whenever $(\\beta_1-\\beta_2)E_1\\neq(\\beta_2-\\beta_3)E_3$ we need the explicit solution described in \\cite{thermal_machine_small_maximal_efficiency_jpa11}. We may just observe that $\\rho_\\infty$ is equal to the thermal equilibrium state $\\sigma_1\\otimes\\sigma_2\\otimes\\sigma_3$ if and only if $(\\beta_1-\\beta_2)E_1=(\\beta_2-\\beta_3)E_3$, since $[H, \\sigma_1\\otimes\\sigma_2\\otimes\\sigma_3]$ is equal to a nonzero coefficient times $e^{-\\beta_1 E_1-\\beta_3 E_3}-e^{-\\beta_2E_2}$.\n\\end{proof}\n\n Note that by multiplying the rates $\\lambda_i, f, g$ by some positive number $\\mu$ the stationary state does not change, so the energy fluxes $J_i(\\rho_\\infty)$ are all multiplied by $\\mu$. Thus, any fluxes $J_1, J_2, J_3$ satisfying $J_1+J_2+J_3=0$ and $\\beta_1J_1+\\beta_2 J_2+\\beta_3J_3>0$ and such that $J_1<0$ or $J_3>0$ can be attained by tuning the parameters of the model.\n\\vspace{0.5cm}\n\nWe can now conclude the proof of Proposition \\ref{prop:thermal_machines}.\n\\begin{proof}[Proof of Proposition \\ref{prop:thermal_machines}]\nThe idea is to combine systems: consider $\\mathcal{H}_{S_1}$ and $\\mathcal{H}_{S_2}$ two thermal models, each with $n$ baths at the same respective temperatures $\\beta_1, \\cdots, \\beta_n$ and Lindbladians $\\mathcal{L}_1, \\mathcal{L}_2$ and fluxes $J_k^1$ and $J_k^2$.\nThen the system $\\mathcal{H}_{S_1}\\otimes \\mathcal{H}_{S_2}$ with Lindbladian $\\mathcal{L}_1\\otimes \\mathds{1}_{S_2}+\\mathds{1}_{S_1}\\otimes \\mathcal{L}_2$ can also be considered as a thermal model with $n$ baths, and the energy fluxes are additive: $J_k=J^1_k+J^2_k$. The key of the proof is that quantum fridges allow to construct systems with arbitrary small entropy production.\n\nLet us show the Proposition by induction on $n$. Consider a list of fluxes $J_i$ satisfying the conditions of Proposition \\ref{prop:thermal_machines}. \n\nIf $n=2$ we can always obtain the fluxes with two depolarizing channels acting on the same qubit as in the example above.\n\nIf $n\\geq 3$, let $\\epsilon=\\sum_{i=1}^n \\beta_i J_i>0$. Define some fluxes $\\tilde{J}_i$ for $i=1, 2, 3$ by\n\\[\n\\left\\{\\begin{array}{ll}\n\\tilde{J}_1&=J_1 \\\\\n\\tilde{J}_2&=\\frac{\\beta_1-\\beta_3}{\\beta_3-\\beta_2} J_1-\\frac{\\epsilon}{2(\\beta_3-\\beta_2)}\\\\\n\\tilde{J}_3&=\\frac{\\beta_2-\\beta_1}{\\beta_3-\\beta_2} J_1+\\frac{\\epsilon}{2(\\beta_3-\\beta_2)}\n\\end{array}\\right.\n\\]\nThen $\\sum_{i=1}^3\\beta_i J_i=\\epsilon\/2>0$ and $\\sum_{i=1}^3 J_i=0$. Thus, the fluxes $\\tilde{J}_i$ can be obtained from a model of quantum fridge as above. Moreover, consider the list of fluxes $\\hat{J}_i$ defined by\n\\[\n\\hat{J}_i=\\left\\{\\begin{array}{cc}\nJ_i-\\tilde{J}_i & \\text{ if $1\\leq i \\leq 3$} \\\\\nJ_i & \\text{ if $i>3$}.\n\\end{array}\\right.\n\\]\nNote that $J_1=0$. Then $\\sum_{i=1}^n \\beta_i \\hat{J}_i=\\epsilon-\\epsilon\/2=\\epsilon\/2>0$ so the list $(\\hat{J}_2, \\cdots, \\hat{J}_n)$ also satisfies the conditions of Proposition \\ref{prop:thermal_machines} so by induction it admits a thermal model. By combining it to the quantum fridge with fluxes $(\\tilde{J}_i)_{1\\leq i\\leq 3}$ we obtain the desired model.\n\\end{proof}\n\nAs shown by this proof, complex and rich examples can be obtained by making generalized depolarizing channels interact with the help of a Hamiltonian. The rest of this article is focused on another family of thermal models, arising from non-interacting fermions.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Quasi-free fermionic semigroups arising from a repeated interaction model}\n\nIn this subsection we briefly introduce quasi-free fermionic systems and we describe a class of quantum semigroups on such systems. We then show that the asymptotic energy fluxes in thermal quasi-free fermionic systems are always trivial, in the sense that they can be decomposed as the sum of fluxes which stream between two of the baths, always from the hottest bath to the coldest one. Thus, it is never possible to pump energy from the coldest bath.\n\n\\subsubsection{Fermionic systems, quadratic Hamiltonians and quasi-free states}\n\nIn this part we recall the basic definitions and fix notations on fermionic systems. This is essentially a shorter version of the introduction of \\cite{AndreysFermions1}. For a more general introduction to fermionic and bosonic spaces, see Derezi\\'nski and G\\'erard \\cite{DerezinskiGerard}.\n\\vspace{0.5cm}\n\n Let us consider a (finite-dimensional) Hilbert space $\\mathcal{H}_0$, called the \\enquote{one-particle space}, and let us fix a Hilbert basis $\\ket{1},\\cdots, \\ket{L}$ of this space. The fermionic space constructed from $\\mathcal{H}_0$ is written $\\mathcal{H}=\\Gamma(\\mathcal{H}_0)$. It is of dimension $2^L$, and has for orthonormal basis $\\set{\\ket{u_1, \\cdots,u_L}~|~u_1, \\cdots, u_L\\in \\{0, 1\\}}$. We write $c_i$ the annihilation operator and $c_i^*$ the creation operator corresponding to the one-particle state $\\ket{i}$, so that\n\\[\nc_i\\ket{u_1, \\cdots, u_L}=(-1)^i \\delta_{u_i=1} \\ket{u_1, \\cdots, u_i-1, \\cdots, u_L}\n\\]\nand the anticommutation relations are satisfied: \n\\begin{align}\n\\set{c_i, c_j}&=\\set{c_i^*, c_j^*}=0 \\\\\n\\set{c_i^*, c_j}&=\\delta_{i,j} \\mathds{1}~\n\\end{align}\nwhere $\\set{A, B}=AB+BA$. More generally for any vector $v\\in \\mathcal{H}_0$ we have two operators $c_{v}^*=\\sum_{i=1}^L v_i c_i^*$ and $c_{v}=\\sum_{i=1}^L \\overline{v_i} c_i$. \n\nWe write $\\gamma_1, \\cdots, \\gamma_{2L}$ the Majorana operators, defined by\n\\begin{align*}\n\\gamma_i&=c_i+c_i^* & \\gamma_{i+L}&=-i (c_i -c_i^*)\\\\\\\n\\end{align*}\nfor $i\\leq L$. They are self-adjoint and satisfy the anticommutation relation \n\\begin{align}\n\\set{\\gamma_i, \\gamma_j}=2\\delta_{i,j}~.\n\\end{align}\n\nWe consider the Hilbert space $\\mathcal{Y}=\\mathcal{H}_0\\oplus \\overline{\\mathcal{H}_0}$, where $\\overline{\\mathcal{H}_0}$ is a Hilbert space endowed with an anti-unitary map $s$ to $\\mathcal{H}_0$. It is called the \\emph{phase space} ; it has a Hilbert basis $e_1, \\cdots, e_{2L}$ defined by $e_i=\\ket{i}\\oplus 0$ and $e_{i+L}=0\\oplus s(\\ket{i})$ for $i\\leq L$. We write $\\varphi$ the field operator, defined as a linear application from $\\mathcal{Y}$ to $\\mathcal{B}(\\mathcal{H})$ by\n\\begin{align*}\n\\varphi(e_i)&=c_i^* & \\varphi(e_{i+L})&=c_i~.\n\\end{align*}\nThe space $\\mathcal{Y}$ is endowed with the anti-linear involution $\\xi(x\\oplus s(y))=y\\oplus s(x)$, with the property $\\varphi(\\xi(z))=\\varphi(z)^*$. The anticommutation relations writes:\n\\[\n\\{\\varphi(x), \\varphi(y)\\}=\\scal{\\xi(x), y}~.\n\\]\n\nAnother interesting basis of $\\mathcal{Y}$ is the orthogonal basis $f_1, \\cdots, f_{2L}$ defined by $f_i=e_i+e_{i+L}$ and $f_{i+L}=-i(e_i-e_{i+_L})$, so that $\\varphi(f_i)=\\gamma_i$. In this basis, $\\xi$ is just the componentwise complex conjugation. The basis $e_1, \\cdots, e_{2L}$ will be called the creation\/annihilation basis while the basis $f_1, \\cdots, f_{2L}$ will be called the Majorana basis. \n\n For any operator $M: \\mathcal{Y}\\rightarrow \\mathcal{Y}$ we will write $M^T=\\xi M^* \\xi$. In the Majorana basis, it corresponds to the transposition, while in the creation annihilation basis, we have\n \\begin{align*}\n \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix}^T=\\begin{pmatrix} D^T & B^T \\\\ C^T & A^T \\end{pmatrix}~.\n \\end{align*}\n\n\n{\\bf Row and column operators:} Another useful way of seeing $\\varphi$ is as a row of operators: \n\\begin{defi}\n\nThe row operator is the operator $F^*: \\mathcal{H}\\otimes \\mathcal{Y} \\rightarrow \\mathcal{H}$ defined by \n\\[\nF^*(\\ket{u_1, \\cdots, u_L}\\otimes x)=\\varphi(x)\\ket{u_1, \\cdots, u_L}~.\n\\]\nIts adjoint is the operator $F: \\mathcal{H}\\rightarrow \\mathcal{H}\\otimes \\mathcal{Y}$ with\n\\[\nF(\\ket{u_1, \\cdots, u_L})=\\sum_{i=1}^{2L} \\Big(\\varphi(e_i) \\ket{u_1, \\cdots, u_n}\\Big)\\otimes e_i~.\n\\]\n\n\\end{defi}\n\nExpressed in the creation\/annihilation basis of $\\mathcal{Y}$, the operator $F$ forms a column of operators: \n\\[\nF_{c}^*=\\begin{pmatrix} c_1 \\\\ \\vdots \\\\ c_L \\\\ c_1^* \\\\ \\vdots \\\\ c_L^* \\end{pmatrix}~.\n\\]\nIn the Majorana basis its form is \n\\[\nF_{f}^*=\\begin{pmatrix} \\gamma_1 \\\\ \\vdots \\\\ \\gamma_{2L} \\end{pmatrix}~.\n\\]\n\n\nIn what follows, we recall the definition of quadratic operators, Bogoliubov transform and quasi-free states.\n\n\\begin{defi}\nA quadratic operator on $\\mathcal{H}=\\Gamma(\\mathcal{H}_0)$ is an operator of the form\n\\[\nA=\\frac{1}{2}F^*\\left(\\mathds{1}_{\\mathcal{H}}\\otimes T\\right) F~\n\\]\nfor some operator $T$ on $\\mathcal{Y}$. If $T^f$ is its matrix in the Majorana basis, we have\n\\[\nA=\\sum_{1\\leq i, j \\leq 2L} \\frac{1}{2}[T^f]_{i,j} \\gamma_i \\gamma_j\n\\] \nUp to replacing $A$ by $A+\\alpha \\mathds{1}$ for some $\\alpha\\in \\mathbb{R}$ we may always assume that $T^T=-T$. Under this condition, $A$ is self-adjoint if and only if $T^*=T$, equivalently $\\xi T \\xi=-T$, or equivalently there exists a real antisymmetric matrix $R$ such that $iR$ is the matrix of $T$ in the Majorana basis. \n\\end{defi}\n\nWe will implicitly write $T$ for $\\mathds{1}_{\\mathcal{H}}\\otimes T$ when there is no possible confusion, and we will write $d\\Gamma(T)=A=\\frac{1}{2} F^* T F$. \n\n A quadratic observable can also be expressed in the creation\/annihilation basis:\n\\[\nA=\\sum_{1\\leq i, j\\leq 2L} T^{c}_{i, j} (c^\\sharp_i)^* c^\\sharp_j\n\\]\nwhere for $i\\leq L$ we define $c_i^\\sharp=\\varphi(e_i)=c_i$ and $c_{i+L}^\\sharp=\\varphi(e_{i+L})=c_i^*$, and $T^{c}$ is the matrix of $T$ in the creation\/annihilation basis. We have $\\xi T \\xi=-T$ if and only if $T^c$ is of the form\n\\[\nT^{c}=\\begin{pmatrix}\nA & B\\\\\n-\\overline{B} & -\\overline{A}\n\\end{pmatrix}=\\frac{1}{2} \\begin{pmatrix}\n1 & i \\\\\n1 & -i\n\\end{pmatrix} T^{f}\\begin{pmatrix}\n1 & 1 \\\\\n-i & i \n\\end{pmatrix}\n\\]\nand under this condition $T$ is self-adjoint if and only if $A$ is self-adjoint and $B$ is antisymmetric (in the sense that $B^T=-B$).\n\nThe exponential of quadratic operators are characterized the following way: \n\n\\begin{prop}\\label{prop:commutation_formula_exp_F}\nFor any operator $M=\\exp{T}$ on $\\mathcal{Y}$ with $\\xi T=-T\\xi$ the operator $\\Gamma(M)=\\exp{\\frac{1}{2} F^*T F}$ on $\\mathcal{H}$ satisfies\n\\[\n\\varphi(Mx)=\\Gamma(M)\\varphi(x)\\Gamma(M)^{-1}~.\n\\]\nIn terms of the column operator, \n\\begin{align}\\label{eq:commutation_F_M}\n(\\mathds{1}_{\\mathcal{H}}\\otimes M) F=(\\Gamma(M)^{-1}\\otimes \\mathds{1}_{\\mathcal{Y}}) F~ \\Gamma(M)~.\n\\end{align}\n\\end{prop}\n{\\bf Remark: } There is a redundancy in the expression $F^*TF$ since the terms $\\gamma_i\\gamma_j$ and $\\gamma_j\\gamma_i$ can be regrouped, which is at the origin of the factor 2 in $\\Gamma(M)=\\exp{\\frac{1}{2} F^*TF}$.\n\nThe Bogoliubov transforms are a very important class of unitary operators on the phase space.\n\\begin{defi}\nA \\emph{Bogoliubov transform} is a unitary operator $U: \\mathcal{Y}\\rightarrow \\mathcal{Y}$ satisfying one of the equivalent conditions: \n\\begin{enumerate}\n\\item We have $\\xi U=U\\xi$.\n\\item The matrix of $U$ in the Majorana basis is real.\n\\item The matrix of $U$ in the creation\/annihilation basis is of the form\n\\[\nU_c=\\begin{pmatrix} \\nu & \\gamma \\\\ \\overline{\\gamma} & \\overline{\\nu}\n\\end{pmatrix}~.\n\\]\n\\item There exists a self-adjoint operator $T$ on $\\mathcal{Y}$ such that $\\xi T=-T\\xi$ and $U=\\exp{iT}$.\n\\item There exists a unitary operator $V$ on $\\mathcal{H}$ such that $\\varphi(Ux)=V \\varphi(x) V^*$ for any $x\\in \\mathcal{Y}$. \n\\end{enumerate}\n\\end{defi}\n\n\nThe interest of Bogoliubov transforms is that the operators $\\tilde{c_i}=\\varphi(Ue_i)$ also satisfy the anticommutation relations. Moreover, Bogoliubov transforms are a generalization of unitary transformations on the one-particle space: if $V$ is any unitary on $\\mathcal{H}_0$, we can define the Bogoliubov transform $U=V\\oplus \\overline{V}$ on $\\mathcal{Y}$, and the creation operators $\\tilde{c}_i$ corresponding to the new basis $V\\ket{1}, \\cdots, V\\ket{L}$ are simply the $\\Gamma(U) c_i \\Gamma(U)^*$. \n\\vspace{0.5cm}\n\nLet us now turn to the study of states on $\\mathcal{H}$. Much information on a state can be obtained by studying its covariance matrix: \n\\begin{defi}\nThe \\emph{covariance matrix} of a state $\\rho \\in \\mathfrak{S}(\\mathcal{H})$ is the matrix $\\cov(\\rho)$ on $\\mathcal{Y}$ defined by\n\\[\n\\cov(\\rho)=\\text{Tr} _{\\mathcal{H}}\\left(\\rho F F^*\\right)~.\n\\]\nIn the Majorana basis, \n\\[\n[\\cov(\\rho)_f]_{i, j}=\\tr{\\rho \\gamma_i\\gamma_j}~.\n\\]\n\\end{defi}\n\nThe covariance matrix is covariant under the evolution by a Bogoliubov transform, in the following sense: if $U$ is a Bogoliubov transform then \n\\[\n\\cov(\\Gamma(U) \\rho \\Gamma(U)^*)=U \\cov(\\rho) U^*~.\n\\]\nAny covariance matrix is of the form\n\\[\n\\cov(\\rho)=\\frac{1}{2} \\mathds{1} +M\n\\]\nwhere $M$ is a self-adjoint operator with $\\xi M \\xi=-M$.\n\nIn the creation annihilation basis, for $i\\leq L$ we have $\\cov(\\rho)_{i,i}=\\tr{\\rho c_i c_i^*}=1-\\tr{\\rho c_i^*c_i}$. The number $\\tr{\\rho c_i^* c_i}$ can be interpreted as the mean number of particles in the mode $i$. For this reason, some author prefer to define the covariance matrix as $\\mathds{1}_\\mathcal{Y}-\\text{Tr} _{\\mathcal{H}}\\left(\\rho F F^*\\right)$ (which is also the transpose of our definition). \n\\vspace{0.5cm}\n\nThe quasi-free states form a class of states which are fully determined by their covariance matrix. \n\\begin{defi}\nA state $\\rho$ is called a \\emph{quasi-free state} if it satisfies the Wick formula: for any $i_1, \\cdots, i_n \\in \\{1, \\cdots, 2L\\}$ we have\n\\begin{align}\\label{eq:wick}\n\\tr{\\rho~ \\gamma_{i_1}\\cdots \\gamma_{i_n}}&=\\left\\{\\begin{array}{cc}\n\\sum_{\\sigma\\in \\mathcal{P}_n} (-1)^{\\epsilon(\\sigma)} \\prod_{l=1}^{n\/2} ~\\tr{\\rho ~\\gamma_{\\sigma(2l)}\\gamma_{\\sigma(2l-1)}} & \\text{if $n$ is even} \\\\\n0& \\text{if $n$ is odd}~\n\\end{array}\\right.\n\\end{align}\nwhere $\\mathcal{P}_n$ is the set of pairings of the set $\\{1, \\cdots, n\\}$, that is the set of permutations $\\sigma$ of $\\set{1, \\cdots, n}$ with $\\sigma(2i) < \\sigma(2i+1)$ for all $i\\in \\{1, \\cdots, n\/2\\}$, and $\\epsilon(\\sigma)$ is the signature of the permutation $\\sigma$.\n\\end{defi}\n\nAny quasi-free state which is faithful is a Gibbs state for a quadratic Hamiltonian.\n\n\\begin{prop}\nAny faithful state $\\rho$ is quasi-free if and only if there exists a quadratic Hamiltonian $H=\\frac{1}{2} F^* T F$ such that $\\rho=e^{-\\beta H}\/\\tr{e^{-\\beta H}}$ for some $\\beta\\in \\mathbb{R}$. The covariance matrix of $\\rho$ is related to $T$ the following way: \n\\begin{align}\\label{eq:cov_T}\n\\cov\\left(\\frac{e^{-\\frac{\\beta}{2} F^* T F}}{\\tr{e^{-\\frac{\\beta}{2} F^* T F}}}\\right)=(\\mathds{1}+e^{-\\beta T})^{-1}~.\n\\end{align}\n\\end{prop}\n\n\n\nFinally, we define the number operator: \n\\begin{defi}\nThe \\emph{number operator} is the operator on $\\mathcal{H}$ defined by\n\\begin{align}\nN=\\sum_{i=1}^L c_i^*c_i=\\frac{1}{2} F^*\\left( \\mathds{1}_{\\mathcal{H}_0} \\oplus (-\\mathds{1}_{\\overline{\\mathcal{H}_0}})\\right) F+\\frac{L}{2} \\mathds{1}_{\\mathcal{H}_S}~.\n\\end{align}\n Any operator commuting with $N$ is called a gauge-invariant operator, and any operator commuting with $(-1)^N$ is called an even operator, the operators anti-commuting with $(-1)^N$ being called odd. We write $Even(\\mathcal{H})$ the space of even operators and $Odd(\\mathcal{H})$ the space of odd operators. \n \\end{defi}\n \n \n\n\n\\subsubsection{Quasi-free fermionic semigroups}\n\nLet us consider a fermionic system $\\mathcal{H}_S=d\\Gamma(\\mathcal{H}_{S, 0})$ with $\\mathcal{H}_{S,0}$ of finite dimension $L_S$, and some reference basis $\\ket{1^S}, \\cdots, \\ket{L_S^S}$, for which the creation operators are written $c_{S, i}$ and Majorana operators $\\gamma_{S, i}$. We also write $N_S$ the number operator on $S$, and $F_S$ the column operator and $\\mathcal{Y}_S$ its phase space. A quasi-free fermionic semigroups on $\\mathcal{H}_S$ is a quantum semigroup whose Lindbladian is of the form\n\\[\n\\mathcal{L}(\\rho)=-i [d\\Gamma(T_S), \\rho]+\\sum_{1\\leq i, j\\leq L} A_{i,j} \\left(\\gamma_{S, i} \\rho \\gamma_{S,j}-\\frac{1}{2}\\set{\\gamma_{S, j}\\gamma_{S, i}, \\rho}\\right)~\n\\]\nfor some self-adjoint matrix $A_{i,j}$. We are specifically interested in quasi-free semigroups arising from a quasi-free repeated interaction model: we consider a \\enquote{bath} system $\\mathcal{H}_B=d\\Gamma(\\mathcal{H}_B)$, with $\\mathcal{H}_{B, 0}$ of finite dimension $L_B$ and some reference basis $\\ket{1^B}, \\cdots, \\ket{L_B^B}$ and creation and Majorana operators $c_{B, i}$ and $\\gamma_{B, i}$. We wish to make it interact with $\\mathcal{H}_S$ by the use of a quadratic Hamiltonian. For this, we need to see $\\mathcal{H}_S\\otimes \\mathcal{H}_B$ as a fermionic system. There are many possible isomorphisms between $\\mathcal{H}_S\\otimes \\mathcal{H}_B$ and $\\mathcal{H}_{SB}=\\Gamma(\\mathcal{H}_{S, 0}\\oplus \\mathcal{H}_{B, 0})$, the two standard ones being $\\ket{u_S}\\otimes \\ket{u_B}\\mapsto \\ket{u_S}\\wedge \\ket{u_B}$ and $\\ket{u_S}\\otimes \\ket{u_B}\\mapsto \\ket{u_B}\\wedge \\ket{u_S}$. The second isomorphism is the most convenient in our case. A basis of $\\mathcal{H}_{S, 0}\\oplus \\mathcal{H}_{B, 0}$ is the basis\n\\[\n0_S\\oplus \\ket{1^B}, \\cdots, 0_S\\oplus\\ket{L_B^B},~ \\ket{1^S}\\oplus 0_B, \\cdots, \\ket{L_S^S}\\oplus 0_B~.\n\\]\nThe corresponding creation operators $c_{0_S\\oplus \\ket{i^b}}$ and $c_{\\ket{i^S}\\oplus 0_B}$ on $\\mathcal{H}_SB$ are identified respectively with the operators $c_{S, i}\\otimes (-1)^N_B$ and $\\mathds{1}_{\\mathcal{H}}\\otimes c_{B, i}$ on $\\mathcal{H}_S\\otimes \\mathcal{H}_B$. We likewise consider $F_S$ and $F_B$ as column operators acting on $\\mathcal{H}_S\\otimes \\mathcal{H}_B$, and identify $\\mathcal{H}_S\\otimes \\mathcal{H}_B$ with $\\mathcal{H}_{SB}$ in what follows.\n\nThe repeated interaction model we consider is the following: \n\\begin{itemize}\n\\item The Hamiltonian on $\\mathcal{H}_S$ is \n\\[\nH_S=\\frac{1}{2} F_S^* T_S F_S\n\\]\n for some self-adjoint $T_S$ on $\\mathcal{Y}_S$ with $\\xi T_S\\xi=-T_S$. We write $iR_S$ its matrix in the Majorana basis, where $R_S$ is an antisymmetric $2L_S\\times 2L_S$ matrix.\n\\item The interaction Hamiltonian is \n\\[\nH_{SB}= F_S^* \\Theta F_B=\\frac{1}{2} \\left( F_S^* \\Theta F_B+F_B^* \\Theta^* F_S \\right)\n\\]\nwhere $\\Theta: \\mathcal{Y}_B\\rightarrow \\mathcal{Y}_S$ is a linear operator with $\\xi \\Theta \\xi=-\\Theta$. We write $iW$ its matrix in the Majorana basis, where $W$ is a $2L_S\\times 2L_B$ matrix.\n\\item The state $\\rho_B$ on $\\mathcal{H}_B$ is a quasi-free state. We write its covariance matrix $\\cov(\\rho_B)=M_B$, in the Majorana basis it is of the form $\\frac{1}{2}\\mathds{1}+i R_B$ where $R_B$ is a real antisymmetric matrix.\n\\end{itemize}\n\nFirst we check that the condition are satisfied for the continuous time-limit to exist: \n\\begin{lem}\nAssumption \\ref{ass:strongConditionV} is satisfied for $H_{SB}$ and $\\rho$, that is: for any $\\alpha \\in \\mathbb{R}$ we have\n\\[\n\\text{Tr} _{B}\\left(\\rho_B^\\alpha H_{SB} \\right)=0~.\n\\]\n\\end{lem}\n\\begin{proof}\nThe operator $\\rho_B^\\alpha$ is even and $H_{SB}\\in Odd(\\mathcal{H}_S)\\otimes Odd(\\mathcal{H}_S)$ so the operator $\\rho_B^\\alpha H_{SB}$ is in $Odd(\\mathcal{H}_S)\\otimes Odd(\\mathcal{H}_B)$, and so its partial trace with respect to $\\mathcal{H}_B$ is zero.\n\\end{proof}\n\nThe continuous-time QMS $(\\Lambda^t)_{0\\leq t \\leq \\infty}$ constructed from this model has for generator $\\mathcal{L}$ defined by\n\\[\n\\mathcal{L}(A)=i[H_S, \\rho]+\\frac{1}{2}\\set{\\Phi(\\mathds{1}_S), A}+\\Phi(A)\n\\]\nfor any $A\\in \\mathcal{B}(\\mathcal{H}_S)$, where $\\Phi(A)=\\text{Tr} _{B}\\left(\\mathds{1}_S\\otimes \\rho_B)H_{SB} (A\\otimes \\mathds{1}_B) H_{SB}\\right)$. The form of $\\Phi$ can be made more explicit: \n\\begin{align}\\label{eq:phi}\n\\Phi(A)=F_S^* \\left(A\\otimes \\Theta M_B \\Theta^*\\right) F_S=\\sum_{1\\leq i,j\\leq L_S} [\\Theta M_B \\Theta^*]^f_{i,j} \\gamma_{S, i} A \\gamma_{S, j}~\n\\end{align}\nwhere $[\\Theta M_B \\Theta^*]^f$ is the matrix of $\\Theta M_B \\Theta^*$ in the Majorana basis. \n\nLet us write $M_S(t)$ the covariance matrix of $\\rho_S(t)$. Using Equation \\eqref{eq:commutation_F_M} with $\\Gamma(U(\\tau))=\\exp\\left(-i\\tau H_S-i\\sqrt{\\tau} H_{SB}\\right)$ and passing to the limit as $\\tau\\rightarrow 0$ we obtain\n\\begin{align}\\label{eq:cov_evolution}\n\\frac{d}{dt} M_S(t)=\\left(-iT_S-\\frac{1}{2} \\Theta \\Theta^*\\right) M_S(t)+M_S(t)\\left(iT_S-\\frac{1}{2}\\Theta \\Theta^*\\right)+\\Theta M_B \\Theta^*~.\n\\end{align}\nThus, knowing $M_t$ is sufficient to compute $M_s$ for $s\\geq t$, even when we do not know anything else on $\\rho_S(t)$. Moreover, if $\\rho_S(0)$ is a quasi-free state, then $\\rho_S(t)$ is a quasi-free state for all $t$, and we have the following: \n\n\\begin{theo}\\label{theo:kalman}\nConsider the subspace $K(T_S, \\Theta)$ of $\\mathcal{Y}$ generated by the ranges of $T_S^k \\Theta$ for $k\\in\\mathbb{N}$. Then \nthe QMS is ergodic if and only if there is a unique solution $M_\\infty$ to the Lyapunov equation\n\\[\n\\left(-iT_S-\\frac{1}{2} \\Theta \\Theta^*\\right) M_\\infty+M_\\infty\\left(iT_S-\\frac{1}{2}\\Theta \\Theta^*\\right)+\\Theta M_B \\Theta^*=0~.\n\\]\n This is equivalent to the condition $K(T_S, \\Theta)=\\mathcal{Y}$. The equilibrium state is then the quasi-free state of covariance matrix $M_\\infty$. If the state $\\rho_B$ in the repeated interactions model of the QMS is faithful, then the stationary state $\\rho_\\infty$ of the QMS is also faithful, and the QMS is positivity improving.\n\\end{theo}\n\n This criterion is called the Kalman criterion (from the theory of control of linear systems) and $K(T_S, \\Theta)$ is called the Kalman space. See \\cite{AndreysFermions1} for a proof; a similar theorem was also proved by Prosen in \\cite{Prosen2008} in the case where the QMS is restricted to even operators. This theorem is just algebraic when restricted on quasi-free states, the hard part being to treat the case where the initial state is not quasi-free, particularly when the state $\\rho_\\infty$ is not faithful.\n \n Note that the condition is independent of $M_B$. If it is satisfied, then the operator $G=-iT_S-\\frac{1}{2} \\Theta \\Theta^*$ has all its eigenvalues with strictly negative real part, and the solution $M_\\infty$ is given by\n \\[\n M_\\infty=\\int_0^{+\\infty} e^{t G} \\Theta^* M_B \\Theta e^{t G^*} dt~.\n \\]\n\n\n\n\n\\subsubsection{Thermal quasi-free fermionic semigroups} \\label{subsub:qf_semi}\n\nWe consider quasi-free fermionic semigroups as described above for which there is a conserved quadratic pseudo-energy: we take\n \\[\n \\mathcal{H}_{B,0}=\\bigoplus_{i=1}^n \\mathcal{H}_{B_i, 0}\n\\]\nwith dimensions $L_{B_i}$, phase space $\\mathcal{Y}_{B_i}$ and $M_B=\\bigoplus_{i=1}^n M_{B_i}$, such that $\\Theta=\\sum_{i=1}^n \\Theta_i$ for some operators $\\Theta_i: \\mathcal{Y}_{B_i}\\rightarrow \\mathcal{Y}_S$ with $\\xi_i \\Theta_i\\xi_i=-\\Theta_i$. We fix some self-adjoint operators $\\kappa_S$ on $\\mathcal{Y}_S$ and $\\kappa_{i}$ on $\\mathcal{Y}_{B_i}$, all anti-commuting with the conjugation $\\xi_i$, and we define\n\\begin{align}\nK_S&=\\frac{1}{2}F_S^*\\kappa_S F_S & K_{B_i}&=\\frac{1}{2} F_{B_i}^* \\kappa_{i} F_{B_i}~.\n\\end{align}\nWe assume that they are conserved by the dynamic generated by $H_S$ and $H_{SB}$, which is equivalent to\n\\begin{align}\\label{eq:db_quadratic_operators}\n[T_S, \\kappa_S]&=0 & \\Theta_i \\kappa_{i}=\\kappa_S \\Theta_i~\\text{for all i}.\n\\end{align}\nMoreover, we assume that the $\\rho_{B_i}$ are thermal with respect to the $K_{B_i}$, that is $\\rho_{B_i}=e^{-\\beta_i K_{B_i}}\/Z_i$, or in terms of the covariance matrix, \n\\begin{align}\nM_{B_i}&=\\left(\\mathds{1}+e^{-\\beta_i \\kappa_{B_i}}\\right)^{-1}~.\n\\end{align}\n\nLet us write $M_\\beta$ the covariance matrix of the Gibbs' state $e^{-\\beta K_S}\/Z$, that is\n\\begin{align}\\label{eq:defi_mbeta}\nM_{\\beta}=\\left(1+e^{-\\beta \\kappa_S}\\right)^{-1}\n\\end{align}\nNote that by the detailed balance condition \\ref{eq:db_quadratic_operators} we have\n\\[\n\\Theta_i M_{B_i} \\Theta_i^*=\\Theta_i \\Theta_i^* M_{\\beta_i}~.\n\\]\nWe define\n\\begin{align}\nD_i(A)=\\frac{1}{2}\\set{\\Theta_i \\Theta_i^*, A}\n\\end{align}\nso that for any state $\\rho$ of covariance matrix $M_S$ we have\n\\begin{align}\n\\tr{\\mathcal{L}_i^*(\\rho) F F^*}=D_i(M_{\\beta_i}-M_S)~.\n\\end{align}\n\n The flux $J_i(\\rho)$ entering the i-th reservoir is then\n\\begin{align}\\label{eq:flux_fermions}\nJ_i(\\rho)=\\frac{1}{2}\\text{Tr} _{\\mathcal{Y}_S}\\left(\\kappa_S D_i(M_{\\beta_i}-M_S\\right))~.\n\\end{align}\nIndeed, we have\n\\begin{align*}\nJ_i(\\rho)&=-\\frac{1}{2}\\tr{\\mathcal{L}_i^*(\\rho) F^* \\kappa_S F}\\\\\n&=-\\frac{1}{2}\\sum_{1\\leq k, l\\leq L_S}[\\kappa_S]^f_{k,l} \\tr{\\mathcal{L}_i^*(\\rho)\\gamma_k \\gamma_l} \\\\\n&=-\\frac{1}{2}\\sum_{1\\leq k, l\\leq L_S} [\\kappa_S]^f_{k,l} [D_i(M_{\\beta_i}-M_S)]^f_{k,l} \\\\\n&=+ \\frac{1}{2} \\text{Tr} _{\\mathbb{C}^{2L_S}}\\left(([\\kappa_S]^f)D_i(M_{\\beta_i}-M_S)]^f\\right)\\\\\n\\end{align*}\nsince the matrix $[\\kappa_S]^f$ of $\\kappa_S$ in the Majorana basis is of the form $i R$ with $R$ real antisymmetric.\n\nBy Proposition \\ref{prop:second_principle}, if $\\rho_\\infty$ is a stationary state then $\\sum_{i=1}^n \\beta_i J_i(\\sigma)\\geq 0$. In the following theorem we show that there is a stronger constraint on the $J_i(\\sigma)$, which prevent non-trivial thermal machines such as the quantum heat pump to be designed.\n\n\\begin{theo}\\label{theo:no_fridge}\nConsider a thermal quasi-free QMS as above. Let $\\rho_\\infty$ be a stationary state and $J_i=J_i(\\rho_\\infty)$. Then there exists a family of fluxes $(J_{i,j})_{1\\leq i,j\\leq n}$ with $J_{j,i}=-J_{i,j}$ and $J_{i,j}\\geq 0$ if $\\beta_i \\geq \\beta_j$ such that for any $i\\leq n$,\n\\[\nJ_i=\\sum_{j=1}^n J_{i,j}~.\n\\]\nIn other words, the fluxes $J_i$ can be obtained from a combination of systems, each of them involving only two of the bath.\n\\end{theo}\n\nThis theorem is inspired by Lemma 1 of Eckmann and Zabey \\cite{EckmannZabey04}, in which they consider a system of oscillators coupled by springs and driven by Gaussian heat bath. They show something weaker that this theorem, namely that the bath of lower temperature cannot be pumped out of energy. Our proof is an elaboration of theirs, and also applies to the system they consider.\n\n\\begin{proof}\nFirst, we reformulate the theorem with a majorization condition: \n\\begin{lem}\nLet us assume that $\\beta_1\\leq \\beta_2\\leq \\cdots \\leq \\beta_n$. Then the fluxes $J_i$ can be decomposed as a sum of $J_{i,j}$ as above if and only if they satisfy that for all $k\\leq n$, \n\\[\n\\sum_{i=1}^k J_i\\leq 0~.\n\\]\n\\end{lem}\n\nThis is easily proved by induction. We assume $\\beta_1\\leq \\cdots \\leq \\beta_n$ in the rest of the proof.\n\n\\vspace{0.5cm}\n\nWe shall start by assuming that $\\Lambda^t$ is positivity improving. Then the eigenvalues of the operator $G=-iT_S-\\frac{1}{2} \\Theta \\Theta^*=-iT_S-\\frac{1}{2} \\sum_{i=1}^n \\Theta_i \\Theta_i^*$ have strictly negative real part. We define the function $F$ on $\\mathcal{B}(\\mathcal{Y})$ which is the inverse of $M\\rightarrow GM+MG^*$, that is\n\\[\nF(M)=\\int_0^\\infty e^{tG}M e^{tG^*} dt~.\n\\]\nThus, $M_S=F\\left(\\sum_{i=1}^n D_i(M_{\\beta_i})\\right)$ is the solution of \n\\[\nG M_S+M_SG^*+\\sum_{i=1}^n D_i(M_{\\beta_i})=0\n\\]\nso it is the covariance matrix of $\\rho_\\infty$. Note that $F$ is linear as a map on $\\mathcal{B}(\\mathcal{Y}_S)$, and for any $\\beta\\in \\mathbb{R}$ it satisfies \n\\begin{align}\\label{eq:zprope_F}\nF(\\sum_{i=1}^n D_i(M_\\beta))=M_\\beta~.\n\\end{align}\n\nLet us fix a $k\\leq n$ and show that\n\\[\n\\sum_{i=1}^k \\tr{\\kappa_S D_i(M_{\\beta_i}-M_S)}\\leq 0~.\n\\]\nWe define the following order relation between self-adjoint operators on $\\mathcal{Y}$. \n\\begin{defi}\nWrite $P_+$ the projection on the positive eigenspace of $\\kappa_S$, and $P_-$ the projection on the negative eigenspace of $\\kappa_S$. We say a self-adjoint matrix $M$ on $\\mathcal{Y}$ is $\\kappa_S$-positive (and we write $0 \\leq_{\\kappa_S}~M$ if $P_+ M P_+$ is a positive operator and $P_- M P_-$ is a negative operator.\n\\end{defi}\n\nWe have the following properties\n\\begin{enumerate}\n\\item If $0\\leq_{\\kappa_S}~M$ then $\\tr{M \\kappa_S}\\geq 0$. \n\\item The maps $D_i$ and $F$ are nondecreasing with respect to $\\geq_{\\kappa_S}$. \n\\item If $\\beta_1\\leq \\beta_2$ then $M_{\\beta_1}\\leq_{\\kappa_S}~M_{\\beta_2}$. \n\\end{enumerate}\nThe first property is trivial, the second property is a consequence of the fact that $T_S$ and $\\Theta_i\\Theta_i^*$ commutes with $\\kappa_S$ and that $\\Theta_i\\Theta_i^*$ is a positive operator. The third property is a consequence of the definition of $M_{\\beta}$.\n\nLet $\\Delta$ be the operator defined by\n\\[\n\\Delta=F\\left(\\sum_{i=k+1}^n D_i( M_{\\beta_k}-M_{\\beta_{i}}) \\right)~.\n\\]\n Then by the linearity of $F$ we have\n\\[\nM_S+\\Delta=F\\left(\\sum_{i\\leq k} D_i(M_{\\beta_i})+\\sum_{i>k}D_i(M_{\\beta_k})\\right)~.\n\\]\n By properties 2 and 3 above we have\n\\[\n\\Delta \\leq_{\\kappa_S} 0~\n\\]\nand by Equation \\eqref{eq:zprope_F} we have\n\\begin{align}\\label{eq:zmajoration_M_delta}\nM_S+\\Delta\\leq_{\\kappa_S} F\\left(\\sum_{i=1}^n D_i(M_{\\beta_k})\\right)=M_{\\beta_k}~.\n\\end{align}\nBy definition of $F$, we have \n\\[\n\\sum_{i=1}^n D_i(M_S+\\Delta)-i[T_S, M_S+\\Delta]=\\sum_{i\\leq k} D_i(M_{\\beta_i})+\\sum_{i>k} D_i(M_{\\beta_k})~\n\\]\nwhere $-i[T_S, M_S+\\Delta]$ satisfies $\\tr{-i[T_S, M_S+\\Delta]\\kappa_S}=0$ since $T_S$ commutes with $\\kappa_S$. Thus\n\\[\n\\sum_{i=1}^k J_i=\\tr{\\kappa_S \\sum_{i>k} D_i(M_S+\\Delta-M_{\\beta_k})+\\sum_{i\\geq k} D_i(\\Delta)}~.\n\\]\nIn the trace, the first sum is $\\leq_{\\kappa_S} 0$ because of Equation \\eqref{eq:zmajoration_M_delta} and the second sum is $\\leq_{\\kappa_S} 0$ because $\\Delta \\leq_{\\kappa_S}0$. This concludes the proof in the case where $(\\Lambda^t)$ is positivity improving. \n\\vspace{0.5cm}\n\nIn the general case, we can take the limit for some perturbation of the $\\Theta_i$ which makes the map positivity improving. Alternatively, let us decompose $\\mathcal{Y}=V_1\\oplus V_2$ where $V_1=K(T_S, \\Theta)$ and $V_2=V_1^\\perp$. Note that $range (\\Theta_i)\\subset V_1$ for any $i$, and $V_1$ is stable by $T_S$; let us decompose $M_S$ according to the decomposition of $\\mathcal{Y}$, of the form\n\\[\nM_S=\\begin{pmatrix} M_{11} & M_{12} \\\\ M_{12}^* & M_{22} \\end{pmatrix}\n\\]\nwhere $M_{11}$ acts only on $V_1$ and so on, and decompose $K_S$ into blocks written $K_{i,j}$ the same way. Then the fact that $K_S$ commutes with $\\Theta_i \\Theta_i^*$ implies that $\\Theta_i\\Theta_i^* K_{12}=0$ and so \n\\[\nJ_i=\\tr{D_i([M_{\\beta_i}]_{11}-M_{11})}.\n\\]\nThus it is sufficient to consider the restriction on the space of matrices on $V_1$, which is preserved by the map $M\\rightarrow GM+MG^*$ and on which this map is invertible; the proof in the positivity improving case applies.\n\\end{proof}\n\n\n\\subsubsection{Gauge-invariant quasi-free fermionic semigroups}\\label{subsub:gauge_invariant}\n\nIn the context of fermionic system, Gauge invariance means commuting with the number operator $N=\\sum_{i=1}^L c_i^*c_i$. This property really depends on the subspace $\\mathcal{H}_0$ of $\\mathcal{Y}$, and not just on the couple $(\\mathcal{Y}, \\xi)$. The properties of Gauge-invariant operators are best described in the creation\/annihilation basis. In what follows we use the \\enquote{small} row operator $C^*: \\mathcal{H}_0 \\otimes \\mathcal{H} \\rightarrow \\mathcal{H}$ defined by \n\\[\nC^*(x\\otimes \\ket{u})=c^*_{x}\\ket{u}~.\n\\]\nAny quadratic gauge-invariant operator can be written $\\lambda \\mathds{1}+ C^* T^0 C$ for some operator $T: \\mathcal{H}_0\\rightarrow \\mathcal{H}_0$ and some constant $\\lambda \\in \\mathbb{C}$. This kind of operator can be interpreted as acting independently on each fermionic particle, with no interactions between them. Similarly, any gauge-invariant state has a covariance matrix which is block-diagonal in the creation\/annihilation basis; since $\\cov(\\rho)+\\xi \\cov(\\rho)\\xi=\\mathds{1}$ the covariance matrix is of the form\n\\begin{align*}\n\\cov(\\rho)=\\begin{pmatrix} \\cov_0 (\\rho) & 0 \\\\ 0 & \\mathds{1}-\\cov_0(\\rho) \\end{pmatrix}\n\\end{align*}\nwhere $\\cov_0(\\rho)=\\text{Tr} _{\\mathcal{H}}\\left(\\rho C C^*\\right)$ will be called the \\enquote{small covariance matrix} in this article. Any gauge-invariant quasi-free state is fully described by its small covariance matrix. Gauge-invariant quasi-free fermionic semigroups can be fully described only in terms of operators on $\\mathcal{Y}$: \n\n\\begin{prop}\nLet $(\\Lambda^t)_{t\\geq 0}$ be a quasi-free fermionic semigroup as above, and assume that the operators $H_S$ and $H_{SB}$ and the state $\\rho_B$ are gauge-invariant. Let $T_S^0, \\Theta^0$ be such that\n\\begin{align*}\nH_S&=C_S^* T_S^0 C_S +\\lambda_S \\mathds{1} & H_{SB}&=C_S^* \\Theta^0 C_B+C_B^* \\left(\\Theta^0\\right)^* C_S +\\lambda_{SB} \\mathds{1}\n\\end{align*}\nand let $M_B^0=\\cov_0(\\rho_B)$. Then the small covariance matrix $M_S^0(t)=\\cov_0(\\Lambda^t(\\rho))$ satisfies the equation\n\\begin{align}\n\\frac{d}{dt} M_S^0(t)=\\left(-iT^0_S-\\frac{1}{2} \\Theta^0 \\left(\\Theta^0\\right)^*\\right) M_S^0(t)+M_S^0(t)\\left(iT^0_S-\\frac{1}{2}\\Theta^0 \\left(\\Theta^0\\right)^*\\right)+\\Theta^0 M_B^0 \\left(\\Theta^0\\right)^*~.\n\\end{align}\nThe map $\\Phi$ writes\n\\[\n\\Phi(A)= \\sum_{i, j} \\left[\\Theta^0 M_B^0 \\left(\\Theta^0\\right)^*\\right]_{i,j} c_i^* A c_j +\\left[\\Theta^0 (\\mathds{1}-M_B^0) \\left(\\Theta^0\\right)^*\\right]_{i,j} c_i A c_j^*~.\n\\]\nIf the semigroup is thermal with gauge-invariant conserved quantity $K_S=C_S \\kappa_S^0 C_S$ then the flux of \nenergy entering the i-th baths is \n\\[\nJ_i(\\rho)=\\tr{\\Theta^0 \\left(\\Theta^0\\right)^*\\left(M_{\\beta_i}^0-M_S^0\\right)}~\n\\]\nwhere $M_{\\beta_i}^0=\\left(1+e^{-\\beta_i \\kappa_S^0}\\right)^{-1}$.\n\\end{prop}\nNote that the number of fermions $N$ is always a globally conserved quantity in a gauge-invariant quasi-free fermionic system. However, the states $\\rho_{Bi}$ of the sub-baths are not always thermal with respect to the number operator $N_{Bi}$, so we cannot always take $K_S=N_S$. \n\\vspace{0.5cm}\n\n{\\bf The example of the fermionic chain :} Let us treat an example where $K_S=N_S$: the fermionic chain. We take two baths, indexed by $0$ and $L+1$, and put the $L$ sites of the system \\enquote{between them}. \nWe choose $\\mathcal{H}_{B0}$ and $\\mathcal{H}_{B(L+1)}$ with one site each, with energies $K_{0}=N_{B0}=c_{B0}^* c_{B 0}$ and $K_{L}=N_{BL}=c_{B (L+1)}^* c_{B(L+1)}$ at temperature $\\beta_1, \\beta_L$, and consider\n\\begin{align*}\n\\Theta^0&=\n\\begin{pmatrix}\n\\theta_0 & 0 \\\\\n0 & 0 \\\\\n\\vdots & \\vdots \\\\\n0 & \\theta_{L+1}\n\\end{pmatrix}\\\\\nT_{S}^0&= D+D^T\n\\end{align*}\nwhere $D$ is the upper-diagonal matrix\n\\[\nD=\\begin{pmatrix}\n0 & 1 & & \\\\\n0 & 0 & 1 & & 0\\\\\n& 0 & 0 & 1 & \\\\\n & &\\ddots & \\ddots & \\ddots \n\\end{pmatrix}~.\n\\]\nThus every site of the bath is in contact only with the nearest sites, with intensity $1$ inside of the system and intensity $\\theta_0, \\theta_{L+1}$ at the interface between the system and the baths. \n\nThis system is positivity improving; and the stationary state can be described explicitly (see \\cite{AndreysFermions1} for a more detailed treatment). Let us write $n_0=(1+e^{-\\beta_0})^{-1}$ and $n_{L+1}=(1+e^{-\\beta_1})^{-1}$ the unique elements of the (small) covariance matrices of the baths. Then the small covariance matrix $M_\\infty^0$ of the stationary state $\\rho_\\infty$ is of the form\n\\begin{align*}\nM_\\infty^0&=\\begin{pmatrix}\np_1 & ij & 0 & ... \\\\\n-ij & p_m & ij ... \\\\\n0& -ij & p_m & ... \\\\\n&&& \\ddots& \\\\\n0 & ...& &-ij &p_m & ij\\\\\n0 & ... &&& -ij & p_L\n\\end{pmatrix}\n\\end{align*}\nwhere $p_1, p_m, p_L $ and $j$ are real numbers that are independent of $L$. They are defined as follows.\nLet $s=4(\\theta_0^2+\\theta_{L+1}^2)+\\theta_1^2\\theta_{L+1}^2(\\theta_1^2+\\theta_{L+1}^2)$. Then \n\\begin{align*}\np_0&=\\frac{1}{s}\\Big(\\theta_0^2\\left(\\theta_{L+1}^4+\\theta_0^2\\theta_{L+1}^2+4\\right)n_0 +4\\theta_0^2n_{L+1}\\Big) \\\\\np_m&=\\frac{1}{s}\\Big(\\theta_0^2\\left( \\theta_{L+1}^4+4\\right)n_0+\\theta_{L+1}^2\\left(\\theta_0^4 +4\\right)n_{L+1}\\Big) \\\\\np_{L+1}&= \\frac{1}{s}\\Big( 4\\theta_{L+1}^2n_0+\\theta_0^2\\left(\\theta_{L+1}^4+\\theta_{L+1}^2\\theta_0^2+4\\right)n_{L+1}\\Big)\\\\\nj&= \\frac{2}{s} \\theta_0^2\\theta_{L+1}^2 (n_0-n_{L+1}) \\,.\n\\end{align*}\n\nThe energy fluxes are also independent of $L$, and\n\\begin{align}\\label{eq:flux_chain}\nJ_0=-J_1=\\theta_0^2 \\left(n_0-p_1\\right)=2 j ~.\n\\end{align}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Large deviations of energy exchanges for fermionic semigroups}\n\nIn the repeated interaction model, it is possible to measure the energy of each sub-bath before and after the interaction, thus measuring the energy exchanged during this interaction. In the continuous-time limit, this allows to treat the energy fluxes between the bath and the system as random variables, whose average over time converges to the stationary fluxes $J_i$. The purpose of this section is to study the large deviations of these random variables in the case of quasi-free fermionic systems; we first present the large deviation results in the case of thermal models, based on \\cite{pellegrini_jumps_10} and \\cite{JPW14}, and then describe more precisely the quasi-free fermionic case. The explicit computation of the large deviation rate involves the resolution of an algebraic Riccati equation, which is reminiscent of \\cite{JPS16}.\n\n\n\\subsection{Repeated measurement process for thermal models}\n\n\nA Markovian evolution under some indirect continual measurement can be described by the notion of \\emph{unraveling}:\n\n\\begin{defi}\nLet $I=[0, +\\infty)$ or $I=\\mathbb{N}$ be a set of times, and let $(\\Omega_t, \\nu_t)_{t\\in I}$ be a family of standard measured spaces (where $\\nu_t$ are Radon measures). We assume that there is a measure-preserving bijection \n\\[\n\\begin{array}{lll}\n(\\Omega_t\\times \\Omega_s,~ \\nu_t\\otimes \\nu_s)&\\rightarrow &(\\Omega_{t+s},~ \\nu_{t+s}) \\\\\n(\\omega_t, ~ \\omega_s) & \\mapsto &\\omega_t.\\omega_s\n\\end{array}\n\\]\n and for $\\omega\\in \\Omega_{t+s}$ we write $\\omega_{t]}$ the projection of $\\omega$ on $\\Omega_t$, i.e; the event such that $\\omega=\\omega_{t]}\\cdot\\omega_{(t,t+s]}$ for some event $\\omega_{(t,s]}\\in \\Omega_s$. Let $\\mathcal{H}_S$ be a separable Hilbert space, and $\\mathcal{L}$ the generator of a QMS $(\\Lambda^t)_{t\\in I}$. A \\emph{Markovian unraveling} on $\\mathcal{H}_S, \\Omega_t$ is a measurable map \n\\[\n(t\\in I, \\omega\\in \\Omega_t) \\mapsto \\Psi_t[\\omega]\n\\]\nwhere $\\Psi_t[\\omega]$ is a completely positive map for a.e. $\\omega\\in \\Omega_t$, with the following properties: \n\\begin{enumerate}\n\\item For all $t\\in I$ we have\n\\[\n\\int_{\\Omega_t} d\\nu_t(\\omega) \\Psi_t[\\omega]=\\Lambda^t~.\n\\]\n\\item For all $s, t\\in I$ and $\\omega_t\\in \\Omega_t, \\omega_s\\in \\Omega_s$, we have\n\\[\n\\Psi_{t+s}[\\omega_t.\\omega_s]=\\Psi_t[\\omega_t]\\circ\\Psi_s[\\omega_s]~.\n\\]\n\\end{enumerate}\n\nTo any Markovian unraveling, any initial state $\\rho$ and any $T$ corresponds a probability measure $\\P_T$ on $\\Omega_T$ and a process $(\\tilde{\\rho}_t)_{0\\leq t\\leq T}$ defined by\n\\begin{align}\nd\\P_T(\\omega)&=\\tr{\\rho\\Psi_T[\\omega](\\mathds{1})}d\\nu_T(\\omega) \\\\\n\\tilde{\\rho}_t&=\\frac{\\Psi_t[\\omega_{t]}]^*(\\rho)}{\\tr{\\Psi_t[\\omega_{t]}]^*(\\rho)}}~.\n\\end{align}\n\n\\end{defi}\n\nThe first property ensures that $\\mathbb{E}_{\\P_T}(\\tilde{\\rho}_t)=\\left(\\Lambda^t\\right)^*(\\rho)$, while the second property ensures that $\\P_t$ is the pushforward measure of $\\P_T$ under the projection $\\omega\\mapsto \\omega_{t]}$. In discrete time, such an unraveling can be obtained by performing measures on the bath, as follows.\n\\vspace{0.5cm}\n\n\nLet us consider a thermal model as above (subsection \\ref{subseq:energy_conservation}). Let us write $P_{i, E_i}$ the projector on the eigenspace of $K_{B_i}$ for eigenvalue $E_i$. For any list of eigenvalues $E=(E_1, \\cdots, E_n)$ consider the projector\n\\[\nP_E=\\bigotimes_{i=1}^n P_{i, E_i}~.\n\\]\n\n\nIf we perform one interaction and we measure $K_{B_i}$ before and after the interaction, with initial state $\\rho$, the state after the interaction is \n\\[\n\\tilde{\\rho}_S(\\tau)=\\frac{{\\Psi}_{\\tau}^*[E, F](\\rho)}{\\tr{{\\Psi}_{\\tau}^*[E, F](\\rho)}}\n\\]\nwhere $E=(E_1, \\cdots, E_n)$ is the result of the first measurement of $(K_{B_1}, \\cdots, K_{B_n})$, where $F$ is the result of the second measurement of the $K_{B_i}$, and\n\\[\n{\\Psi}_{\\tau}^*[E, F](\\rho)=\\text{Tr} _{B}\\left(P_F U_\\tau ~\\rho\\otimes \\left(P_E\\rho_B P_E \\right)~ U_\\tau^* P_F \\right)~.\n\\]\nThe outcome $(E, F)$ of the measurement appears with probability $\\tr{{\\Psi}_{\\tau}^*[E, F](\\rho)}$. Since we are interested only in the fluxes, we can forget about the precise outcome $(E, F)$ and just retain the difference $\\delta^i=F_i-E_i$. The state is then \n\\[\n\\frac{\\Psi_\\tau^*[\\delta](\\rho)}{\\tr{\\Psi_\\tau^*[\\delta](\\rho)}}\n\\]\nwhere \n\\[\n\\Psi_\\tau^*[\\delta](\\rho)=\\sum_{E, F \\text{ with } F-E=\\delta} \\Psi_\\tau^*[E, F](\\rho)~.\n\\]\n\nApplying this measurement and interaction repeatedly, we obtain a random process $(\\delta_\\tau(k))_{k\\in \\mathbb{N}}$ coupled with a random process of states $(\\tilde{\\rho}_{\\tau}(k))_{k\\in \\mathbb{N}}$. It is a Markovian unraveling of the QMS $(\\Lambda_\\tau^k)_{k\\in \\mathbb{N}}$ of repeated interaction. Indeed, write \n\\[\nN_\\tau^i(k)=\\sum_{i=1}^k \\delta^i_\\tau(i)\n\\]\nthe total energy exchange up to time $k\\tau$, and $N_\\tau(k)=(N_\\tau^1(k), \\cdots, N_\\tau^n(k))$. For $t\\in \\mathbb{N}$ the trajectory $(N_\\tau(k))_{k\\leq t}$ is in the space $\\Omega_{\\tau, t}=\\left(\\mathbb{R}^n\\right)^t$ which we endow with the counting measure $\\nu_t$~. Then the process $\\tilde{\\rho}_{\\tau}(k)$ is a Markovian unraveling of $\\Lambda_\\tau^k$ with maps\n\\[\n \\Psi_{\\tau,t}[(N_\\tau(k))_{k\\leq t}]=\\Psi_\\tau[N_\\tau(1)-N_\\tau(0)]\\circ \\cdots \\circ \\Psi_\\tau[N_\\tau(t)-N_\\tau(t-1)]~.\n\\]\n\n\n\nThe following theorem describes the limit in distribution of this process as $\\tau\\rightarrow 0$. It is a generalization of a theorem of Nechita and Pellegrini \\cite{Pellegrini_Nechita09}.\n\n\\begin{theo}[Nechita and Pellegrini]\nSuppose that Assumption \\ref{ass:strongConditionV} is satisfied. Then for any $T>0$ the process \n\\[\n(\\tilde{\\rho}_\\tau(\\ent{t\/\\tau}), N_{\\tau}(\\ent{t\/\\tau}))_{t\\in [0, T]}\n\\]\nconverges in distribution in the space of c\\`adl\\`ag functions to a process $(\\tilde{\\rho}_t, N_t)_{t\\in [0, T]}$ where $t\\mapsto N_t^i$ is piecewise constant, with a finite number of jumps, all in the set $ sp(N_{B_i})-sp(N_{B_i})$. The distribution of this process is described as a Markovian unraveling, the following way: we describe any trajectory by the list of jumps, so our universe $\\Omega_T$ is\n\n\\[\n\\Omega_T=\\left\\{ \\Big((t_1, i_1, \\delta_1), \\cdots, (t_k, i_k, \\delta_k)\\Big)~|~ 0K} \\frac{C_3^k T^k}{k!}\n\\]\nconverges to zero as $K\\rightarrow +\\infty$, uniformly in $\\tau$, and the same for the continuous-time version.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Large deviation of the energy fluxes}\n\nIn this section, we consider the large deviations of the random variables $\\frac{N_T^i}{T}$ as $T\\rightarrow \\infty$. We assume that $\\mathcal{L}$ is positivity improving, thus it has a unique stationary state $\\rho_\\infty$, and\n\\[\n\\lim \\mathbb{E}_{\\P_T} \\left(\\frac{N_T^i}{T}\\right)=J_i(\\rho_\\infty)~.\n\\]\nThis section is entirely based on Jak\\v{s}i\\'c, Pillet and Westrich \\cite{JPW14}. In this article, the authors construct the random variables $N_T^i$ directly from the semigroup, and study its large deviations. Our notations differ from their by one notable point: the authors express the large deviation of the entropy exchange $\\beta_i N_T^i$, while we consider the large deviations of the energy exchange $N_T^i$. We just recall their main results, giving only elements of proofs.\n\\vspace{0.5cm}\n\nIn the case of finite-dimensional systems, the large deviations of $N_T\/T$ fall into the simplest case of the G\\\"artner-Ellis theorem: it is sufficient to study the limit \n\\begin{align}\\label{eq:def_ealpha}\ne(\\alpha)=\\lim_{T\\rightarrow \\infty} \\frac{1}{T} \\log\\mathbb{E}\\left(e^{\\scal{ \\alpha, N_T}}\\right)~\n\\end{align}\nfor $\\alpha=(\\alpha_1, \\cdots, \\alpha_n)\\in \\mathbb{R}^n$ (and where $\\scal{ \\alpha, N_T}=\\sum_{i=1}^n \\alpha_i N_T^i$. The authors of \\cite{JPW14} prove the following large deviation principle: \n\n\\begin{theo}[Jak\\v{s}i\\'c, Pillet, Westrich] \\label{theo:large_deviations}\nThere exists a convex and continuous rate function $I$ on $\\mathbb{R}^n$ such that for any open set $E\\subset \\mathbb{R}^n$ we have\n\\[\n\\lim_{T\\rightarrow \\infty} \\frac{1}{T}\\log\\P\\left(\\frac{1}{T} N_T \\in E\\right)=-\\inf_{\\zeta \\in \\mathbb{E}} I(\\zeta)\n\\]\nThe function $I$ vanishes only at $\\zeta=(J_1, \\cdots, J_n)$, it satisfies\n\\begin{align}\n&I(\\zeta)=+\\infty~~ \\text{ if $\\sum_{i=1}^n \\zeta_i \\neq 0$} \\label{eq:I_sum}\\\\\n&I(\\zeta)-I(-\\zeta)=\\sum_{i=1}^n \\beta_i \\zeta_i \\label{eq:I_sym}\n\\end{align}\n\n and it is the Legendre transform of $\\alpha\\mapsto e(\\alpha)$: \n\\begin{align}\\label{eq:legendre}\nI(\\zeta)=\\sup_{\\alpha\\in \\mathbb{R}^n} \\big(\\scal{\\alpha, \\zeta}-e(\\alpha)\\big)~.\n\\end{align}\n\\end{theo} \n\nRelation \\ref{eq:I_sum} is a manifestation of the first principle (the sum of the fluxes must be $0$), and has been remarked the first time in \\cite{AndrieuGMT_09_fluctuations} (Proposition 1) while relation \\ref{eq:I_sym} is linked to the second principle, and is called the Gallavotti-Cohen symmetry. The quantity $\\sum_{i=1}^n \\beta_i \\zeta_i$ is interpreted as the mean entropy production, and heuristically the large deviations principle says that\n\\[\n\\frac{d\\P(N_T=\\zeta)}{d\\P(N_T=-\\zeta)}\\simeq e^{-T\\sum_{i=1}^n \\beta_i \\zeta_i}\n\\]\nas $T\\rightarrow \\infty$.\n\n\n\nThis theorem is the consequence of the G\\\"artner-Ellis theorem and the following properties of $e(\\alpha)$: \n\n\\begin{prop}[Jak\\v{s}i\\'c, Pillet, Westrich]\nFor any $\\alpha\\in \\mathbb{R}^n$ the limit \\ref{eq:def_ealpha} exists, and it satisfies the following properties: \n\\begin{enumerate}\n\\item The function $\\alpha \\mapsto e(\\alpha)$ is convex and real analytic on $\\mathbb{R}^n$. \n\\item For any $\\alpha \\in \\mathbb{R}^n$ and $\\lambda\\in \\mathbb{R}$, writing $1_n=(1, \\cdots, 1)$ we have\n\\[\ne(\\alpha+\\lambda 1_n)=e(\\alpha)~.\n\\]\n\\item \\label{prope:Gallavoti_cohen} For any $\\alpha\\in \\mathbb{R}^n$, writing $\\beta=(\\beta_1, \\cdots, \\beta_n)$ we have\n\\[\ne(\\alpha-\\beta)=e(-\\alpha)~.\n\\]\n\\item For all $i$ we have\n\\[\n\\left.\\frac{\\partial e(\\alpha)}{\\partial \\alpha_i}\\right|_{\\alpha=0}=-J_i~.\n\\]\n\\end{enumerate}\n\\end{prop}\n\n\n\nThe proof of these properties goes through the study of the \\enquote{deformed semigroup}, of generator $\\mathcal{L}_\\alpha^*$, as follows: \n\n\\begin{prop}[Jak\\v{s}i\\'c, Pillet, Westrich]\nWe define the super-operator $\\mathcal{L}_\\alpha$ by\n\\begin{align}\\label{eq:def_deformed_semi}\n\\mathcal{L}_\\alpha(A)&=\\mathcal{L}_G(A)+\\sum_{i=1}^n \\Phi_i\\left(A e^{-\\alpha_i K_S}\\right)e^{\\alpha_i N_S}\n\\end{align}\nThen\n\\begin{enumerate}\n\\item For any $T\\in \\mathbb{R}$ and $\\alpha\\in \\mathbb{R}^n$ we have\n\\[\n\\mathbb{E}\\left(e^{\\scal{\\alpha, N_T}}\\right)=\\tr{e^{T\\mathcal{L}_\\alpha^*}(\\rho_0)}~.\n\\]\n\\item For all $\\alpha\\in \\mathbb{R}^n$ the super-operator $e^{t\\mathcal{L}_\\alpha}$ is positivity improving. In particular the dominant eigenvalue of $\\mathcal{L}_\\alpha^*$ is real and of multiplicity $1$, and the corresponding eigenvector is a positive operator.\n\\item $e(\\alpha)$ is the dominant eigenvalue of $\\mathcal{L}_\\alpha$. \n\\item The super-operator $\\mathcal{L}_\\alpha$ is equal to\n\\begin{align}\n\\mathcal{L}_\\alpha(A)&=\\mathcal{L}_G(A)+\\sum_{i=1}^n \\sum_{\\delta^i} e^{-\\alpha\\delta^i} \\Phi_{i, \\delta^i}(A) \\\\\n&=\\mathcal{L}_G(A)+\\sum_{i=1}^n e^{\\frac{\\alpha_i}{2} N_S} \\Phi_i\\left( e^{-\\frac{\\alpha_i}{2} K_S}A e^{-\\frac{\\alpha_i}{2} K_S}\\right)e^{\\frac{\\alpha_i}{2} N_S}\n\\end{align}\n\\end{enumerate} \n\\end{prop}\n\nThe third assertion implies the analyticity of $\\alpha\\rightarrow e(\\alpha)$, since $\\alpha\\rightarrow \\mathcal{L}_\\alpha^*$ is analytic and the dominant eigenvalue of $\\mathcal{L}_\\alpha^*$ is simple for all $\\alpha$.\n\n\\begin{proof}[Elements of proof:]\nFor the first part, the definition of $N_T$ and of $\\P_T$ as unraveling of $\\Lambda^t$ and Formula \\ref{eq:defi_unraveling} allows to express $\\mathbb{E}\\left(e^{\\scal{\\alpha, N_T}}\\right)$ as a sum of multiple integrals, which happens to be the Dyson expansion of $\\mathcal{L}_\\alpha$. Another derivation of this formula goes through the discrete-time limit: in the repeated interaction procedure with interaction time $\\tau$, we have\n\\begin{align*}\n\\mathbb{E}\\left(e^{\\scal{\\alpha, N_\\tau(k)}}\\right)&=\\sum_{(N_\\tau(l))_{l\\leq k}\\in \\Omega_{\\tau, k}} e^{\\scal{\\alpha, N_\\tau(k)}} \\tr{\\rho\\Psi_{\\tau, k}\\left[(N_\\tau(l))_{l\\leq k}\\right](\\mathds{1})} \\\\\n&= \\tr{\\rho\\left(\\prod_{l=1}^k \\sum_{\\delta_\\tau} e^{\\scal{\\alpha, \\delta_\\tau}} \\Psi_\\tau[\\delta_\\tau]\\right)(\\mathds{1}) }~.\\\\\n\\end{align*}\n Write $\\alpha\\cdot K_B=\\sum_{i=1}^n \\alpha_i K_{B_i}$ and define $\\Lambda_{\\tau, \\alpha}$ by\n\\begin{align}\\label{eq:discrete_l_alpha}\n\\Lambda_{\\tau, \\alpha}^*(\\rho)=\\sum_{\\delta_\\tau} e^{\\scal{\\alpha, \\delta_\\tau}} \\Psi_\\tau^*[\\delta_\\tau](\\rho)=\\text{Tr} _{\\mathcal{H}_B} \\left( e^{-\\frac{\\alpha}{2}\\cdot K_B}U_\\tau \\left(\\rho\\otimes\\rho_B e^{\\alpha \\cdot K_B} \\right)U_\\tau^* e^{-\\frac{\\alpha}{2}\\cdot K_B} \\right)~.\n\\end{align}\nThen the following holds\n\\[\n\\Lambda_{\\tau, \\alpha}^*(\\rho)=\\rho+\\tau \\mathcal{L}_\\alpha^*(\\rho)+o(\\tau)\n\\]\nso we have\n\\[\n\\lim_{\\tau\\rightarrow 0}\\left(\\Lambda_\\tau\\right)^{\\ent{t\/\\tau}}=e^{t\\mathcal{L}_\\alpha}\n\\]\nand so \n\\[\n\\lim_{\\tau\\rightarrow 0}\\mathbb{E}\\left(e^{\\scal{\\alpha, N_\\tau\\left(\\ent{t\/\\tau}\\right)}}\\right)=\\tr{\\rho e^{t\\mathcal{L}_\\alpha(\\mathds{1})}}~.\n\\]\n\nFor the second part, since $e^{t\\mathcal{L}^*}$ is positivity improving, for any nonzero positive operators $A, B$ on $\\mathcal{H}_S$ we have\n\\begin{align*}\n\\tr{A e^{t\\mathcal{L}}(B)}>0~.\n\\end{align*}\nApplying the Dyson formula we get\n\\begin{align*}\n\\tr{A e^{t\\mathcal{L}}(B)}=\\sum_{k=0}^{+\\infty} \\sum_{i_1, \\delta_1, \\cdots, i_n, \\delta_n} \\int_{0\\leq t_1<\\cdots < t_n < t} \\tr{A e^{(t-t_n)\\mathcal{L}_G}\\Phi_{i_n, \\delta_n}\\cdots e^{t_1\\mathcal{L}_G}(B)}~.\n\\end{align*}\nThis implies that one of the terms in this sum is strictly positive. Moreover, \n\\begin{align*}\n\\tr{A e^{t\\mathcal{L}_\\alpha}(B)}=\\sum_{k=0}^{+\\infty} \\sum_{i_1, \\delta_1, \\cdots, i_n, \\delta_n} \\exp\\left(-{\\sum_{i=1}^n \\alpha_i \\delta^i} \\right)\\int_{0\\leq t_1<\\cdots < t_n < t} \\tr{A e^{(t-t_n)\\mathcal{L}_G}\\Phi_{i_n, \\delta_n}\\cdots e^{t_1\\mathcal{L}_G}(B)}~.\n\\end{align*}\nAll the terms in this sum are nonnegative, and one of them is strictly positive, so it is strictly positive. This implies the complete positivity of $e^{t\\mathcal{L}_\\alpha}$. \n\nThe third assertion is a consequence of two first assertions: if $\\lambda_\\alpha$ is the dominant eigenvalue of $e^{t\\mathcal{L}_\\alpha^*}$, then the corresponding eigenvector $\\rho_\\alpha$ is positive definite (since $e^{t\\mathcal{L}_\\alpha^*}$ is positivity improving), and we may assume that it is of trace $1$. There there exists $\\epsilon>0$ such that $e^{\\mathcal{L}_\\alpha^*}(\\rho_0)\\geq \\epsilon \\rho_\\alpha$ hence as $t\\rightarrow \\infty$,\n\\[\n\\tr{e^{t\\mathcal{L}_\\alpha^*}(\\rho_0)}\\geq \\epsilon e^{t\\lambda_\\alpha}+o(e^{t\\lambda_\\alpha})\n\\]\nwhich implies the third assertion. \n\nFor the last assertion, we just remark that\n\\begin{align*}\n\\Phi^*_{i, \\delta^i}(A e^{-\\alpha K_S}))&=\\sum_{E_i, F_i,~F_i-E_i=\\delta^i} \\text{Tr} _{B_i}\\left( H_{SB_i} A e^{-\\alpha K_S} P_{F_i} H_{SB_i} P_{E_i} \\rho_{B_i}\\right)\\\\\n&=\\sum_{E_i, F_i,~F_i-E_i=\\delta^i}\\text{Tr} _{B_i}\\left( H_{SB_i} Ae^{-\\alpha (K_S+K_{B_i})}e^{\\alpha K_{B_i}} P_{F_i} H_{SB_i} P_{E_i} \\rho_{B_i}\\right) \\\\\n&=\\sum_{E_i, F_i,~F_i-E_i=\\delta^i}\\text{Tr} _{B_i}\\left( H_{SB_i} A e^{\\alpha K_{B_i}} P_{F_i} H_{SB_i} P_{E_i} e^{-\\alpha (K_S+K_{B_i})}\\rho_{B_i}\\right) \\\\\n&=\\sum_{E_i, F_i,~F_i-E_i=\\delta^i}\\text{Tr} _{B_i}\\left( H_{SB_i} A e^{\\alpha F_i } P_{F_i} H_{SB_i} P_{E_i} e^{\\alpha E_i}\\rho_{B_i}\\right)e^{\\alpha K_S}\n\\end{align*} \nthe last line being obtained by $P_{E_i} e^{-\\alpha K_{B_i}}=e^{-\\alpha E_i}P_{E_i}$. This gives the first reformulation, the second comes from the fact that the $\\Phi_i$ satisfy the detailed balance, so it commutes with the modular operator $\\Delta_{e^{-\\alpha K_S\/2}}$. \n\\end{proof}\n\n\n\n\n\n\n\\subsection{The quasi-free fermionic case}\n\nIn this subsection we apply the formalism described above to the case of a quasi-free fermionic system. Hence, we consider a thermal quasi-free fermionic semigroup which is positivity improving; the idea is then to study the maximal eigenvalue of the deformed generator $\\mathcal{L}_\\alpha$. As shown above, the study of $\\mathcal{L}$ is greatly simplified by the existence of a closed equation for its covariance matrix, and the fact that it preserves the set of quasi-free states. In the case of $\\mathcal{L}_\\alpha$, the covariance matrix does not satisfies a closed equation in general, but the set of multiples of quasi-free states is still preserved, and restricted on this set the covariance matrix evolves according to a closed equation, admittedly more complex than the affine equation of the non-deformed semigroup. This allows to reduce the computation of $e(\\alpha)$ to the resolution of an algebraic Riccati equation; the outcome of this study is the following theorem: \n\n\n\\begin{theo}\\label{theo:e_alpha_fermionic}\nLet us consider a thermal quasi-free fermionic semigroup $\\mathcal{L}$ defined as in Paragraph \\ref{subsub:qf_semi}, and assume that it is positivity improving. For any $\\alpha, \\beta\\in \\mathbb{R}^n$ consider the operators on $\\mathcal{Y}$\n\\begin{align*}\nA_\\beta&=-iT_S+\\sum_{i=1}^n \\left(M_{\\beta_i}-\\frac{1}{2}\\mathds{1}\\right)\\Theta_i \\Theta_i^*\\\\\nB_{\\alpha, \\beta}&=\\sum_{i=1}^n e^{\\alpha_i \\kappa_S}M_{\\beta_i} \\Theta_i \\Theta_i^*\n\\end{align*}\nwhere we wrote $M_{\\beta_i}=\\left(\\mathds{1}+e^{-\\beta_i \\kappa_S}\\right)^{-1}$ the covariance matrix of the Gibbs state at temperature $\\beta_i$.\n\nDefine the operator $Z_\\alpha$ on $\\mathcal{Y}\\oplus \\mathcal{Y}$ by\n\\[\nZ_\\alpha=\\begin{pmatrix} A_\\beta & B_{\\alpha, \\beta} \\\\ B_{-\\alpha, -\\beta} & -A_\\beta^* \\end{pmatrix}~.\n\\]\nThe set of eigenvalues of $Z_\\alpha$ is symmetric with respect to the imaginary axis, and the pure imaginary eigenvalues are of even multiplicity; let $\\lambda_1(\\alpha), \\cdots, \\lambda_{k}(\\alpha)$ be its eigenvalues of positive real part. Then \n\\[\ne(\\alpha)=\\frac{1}{2} \\sum_{i=1}^{k} \\lambda_i(\\alpha)-\\frac{1}{4}\\sum_{i=1}^n \\tr{\\Theta_i \\Theta_i^*}~.\n\\]\n\\end{theo}\n\nWe first prove the following proposition: \n\n\\begin{prop}\\label{prop:riccati_e_alpha}\n For any $\\alpha \\in \\mathbb{R}^n$, the deformed semigroup \n\\[\nt\\mapsto e^{t\\mathcal{L}_\\alpha^*}\n\\]\npreserves the vector space generated by the quasi-free state, and the eigenvector for the maximal eigenvalue $e(\\alpha)$ is proportional to a quasi-free state.\n\nMoreover, any quasi-free state $\\rho$ of covariance matrix $M$ is an eigenvector of $\\mathcal{L}_\\alpha^*$ if and only if $M$ is a solution to the Riccati equation\n\\begin{align}\\label{eq:riccati}\nG_{\\alpha, \\beta} M + M G_{\\alpha, \\beta}^* + M C_{\\alpha, \\beta} M +B_{\\alpha, \\beta}=0\n\\end{align}\n\nwhere $G_{\\alpha, \\beta}$ and $C_{\\alpha, \\beta}$ are defined the following way: let\n\\begin{align}\nQ_{\\alpha, \\beta} &=\\sum_{i=1}^n \\Theta_i \\Theta_i^* M_{\\beta_i}\\left(e^{\\alpha_i \\kappa_S}-1\\right)\n\\end{align}\nand \n\\begin{align}\nG_{\\alpha, \\beta} &=G-Q_{\\alpha, \\beta}=-iT_S-\\sum_{i=1}^n \\left(\\frac{1}{2}+\\left(e^{\\alpha_i \\kappa_S}-1\\right) M_{\\beta_i} \\right) \\Theta_i \\Theta_i^*\\\\\nC_{\\alpha, \\beta}&=Q_{\\alpha, \\beta}-Q_{\\alpha, \\beta}^T=\\sum_{i=1}^n \\Big(\\left(e^{\\alpha_i \\kappa_S}+e^{-\\alpha_i \\kappa_S}-2\\right)M_{\\beta_i}+\\mathds{1}-e^{\\alpha_i \\kappa_S}\\Big)\\Theta_i\\Theta_i^*~.\n\\end{align}\n\n\n\n\nThe corresponding eigenvalue is \n\\begin{align}\n\\lambda=\\frac{1}{2}\\tr{Q_{\\alpha, \\beta}^T M}=-\\frac{1}{2} \\tr{C_{\\alpha, \\beta} M}+\\frac{1}{2}\\tr{Q_{\\alpha, \\beta}}~.\n\\end{align}\n\\end{prop}\n\n\\begin{proof}\n\n{\\bf The vector space generated by quasi-free states is preserved: }\n\nWe use the discrete approximation: let us show that for any quasi-free state $\\rho$ the operator $\\Lambda_{\\tau, \\alpha}^*(\\rho)$ defined at Equation \\eqref{eq:discrete_l_alpha} is proportional to a quasi-free state. \n\n First, $\\rho_B e^{\\alpha.K_B}=\\frac{1}{Z}\\exp\\left(\\sum_{i=1}^n (-\\beta_i+\\alpha_i)K_{B_i}\\right)$ is proportional to a quasi-free state; thus its tensor product with $\\rho$ is also proportional to a quasi-free state. The unitary $U_\\tau$ is a Bogoliubov transform, so the following operator is proportional to a quasi-free state.\n\\[\nU_\\tau \\left(\\rho\\otimes\\rho_B e^{-\\alpha.K_B} \\right)U_\\tau^*~.\n\\]\n Moreover, for any quasi-free states $\\sigma, \\nu$ the operator $\\sigma^{\\alpha} \\nu \\sigma^\\alpha$ is proportional to a quasi-free state, and the partial trace of a quasi-free state is quasi-free, so\n\\[\n\\Lambda_{\\tau, \\alpha}^*(\\rho)\n\\] \nis proportional to a quasi-free state. Thus\n\\[\n\\left(\\Lambda_{\\tau,\\alpha}^*\\right)^{\\ent{t\/\\tau}}(\\rho)\n\\]\nis quasi-free for any $t$ and $\\tau$, and the set of quasi-free states is closed, hence we can pass to the limit as $\\tau\\rightarrow 0$ so $^{t\\mathcal{L}_\\alpha^*}(\\rho)$ is proportional to a quasi-free state.\n\\vspace{0.5cm}\n\n{\\bf The eigenvector for the maximal eigenvalue is quasi-free: }\n\nThis derives from the following lemma: \n\n\\begin{lem}\nLet $\\alpha\\in \\mathbb{R}^n \\mapsto L_\\alpha\\in M_{k,k}(\\mathbb{C})$ be a continuous map, write $\\lambda_\\alpha$ the dominant eigenvalue of $L_\\alpha$, and assume that it is simple for all $\\alpha$, of eigenvector $x_\\alpha$. Assume that there is a closed cone $\\mathcal{Q}$ which is stable by $L_\\alpha$ for any $\\alpha$, and that $x_0\\in \\mathcal{Q}$. Then for any $\\alpha \\in \\mathbb{R}^n$ we have $x_\\alpha\\in \\mathcal{Q}$.\n\\end{lem}\nApplying this lemma to $\\alpha\\mapsto \\mathcal{L}_\\alpha^*$ and $\\mathcal{Q}$ the set of operators proportional to quasi-free states gives that $\\rho_\\alpha$ is quasi-free for all $\\alpha\\in \\mathbb{R}^n$.\n\\begin{proof}\nWe assume $\\norm{x_\\alpha}=1$. \nUp to choosing the right phase for $x_\\alpha$ we can also assume that $\\alpha\\mapsto (\\lambda_\\alpha, x_\\alpha)$ is continuous since $\\alpha\\mapsto L_\\alpha$ is continuous. \nThus, the set $E=\\{\\alpha~|~x_\\alpha\\in \\mathcal{Q}\\}$ is closed. Let us show that it is open. \nConsider some $\\alpha_0\\in E$. \nWrite $V_\\alpha$ the vector space which is stable by $L_\\alpha$ and such that $V_\\alpha\\oplus (\\mathbb{C} x_\\alpha)=\\mathbb{C}^n$. \nThen $x_{\\alpha_0}=\\mu_\\alpha x_{\\alpha}+z_\\alpha$ where $z_\\alpha\\in V_{\\alpha}$ and $\\alpha\\mapsto \\mu_\\alpha$ is continuous, and nonzero for any $\\alpha$ close enough to $\\alpha_0$. Since $\\lambda_\\alpha$ is the maximal eigenvalue, we have\n\\[\n\\lim_{n\\rightarrow +\\infty} \\frac{L_\\alpha^n x_{\\alpha_0}}{\\lambda_\\alpha^n}=\\mu_\\alpha x_{\\alpha}\n\\]\nand $L_\\alpha^n x_{\\alpha_0}\\in \\mathcal{Q}$ for all $n$ so $x_\\alpha\\in \\mathcal{Q}$ when $\\mu_\\alpha \\neq 0$. \n\\end{proof}\n\n{\\bf Derivation of the equation for $M$ and $\\lambda$:}\n\nLet us first describe the action of $\\mathcal{L}_\\alpha$ more precisely: for all $i\\in \\{1, \\cdots, n\\}$, for any observable $A$, we have \n\\begin{align*}\n \\Phi_i\\left(A ~e^{-\\alpha_i K_S}\\right)e^{\\alpha_i K_S}\n&= F^* \\left(A \\otimes \\Theta_i M_{B_i} \\Theta_i^* \\right)\\left(e^{-\\alpha_i K_S}\\otimes \\mathds{1}_{\\mathcal{Y}}\\right)F e^{\\alpha_i K_S}~~\\text{by formula \\ref{eq:phi}} \\\\\n&= F^* \\left(A\\otimes \\Theta_i M_{B_i} \\Theta_i^*\\right) \\left(\\mathds{1}_{\\mathcal{H}_S}\\otimes e^{\\alpha_i \\kappa_S}\\right) F~~~\\text{by Proposition \\ref{prop:commutation_formula_exp_F}} \\\\\n&=F^*\\left(A\\otimes \\Theta_i M_{B_i}\\Theta_i e^{\\alpha_i\\kappa_S}\\right) F~.\n\\end{align*}\nBy the preservation of energy \\ref{eq:db_quadratic_operators} we have \n\\[\n\\Theta_i M_{B_i}\\Theta_i e^{\\alpha_i\\kappa_S}=\\Theta_i \\Theta_i^* M_{\\beta_i} e^{\\alpha_i \\kappa_S}~.\n\\]\n\nso\n\\[\n(\\mathcal{L}_\\alpha-\\mathcal{L})(A)=F^*(A\\otimes G) F=\\sum_{1\\leq k,l\\leq 2L_S} [Q_{\\alpha, \\beta}]^f_{k,l} \\gamma_k A \\gamma_l~.\n\\]\n\n\n\nLet us now consider a quasi-free state $\\rho$ of density matrix $M$, and let us assume that it is an eigenvector of $\\mathcal{L}_\\alpha^*$\n\\[\n\\mathcal{L}_\\alpha^*(\\rho)=\\lambda \\rho~.\n\\]\nThen we can express $\\lambda$ in terms of $M$, indeed\n\\begin{align*}\n\\lambda&=\\tr{\\mathcal{L}_\\alpha^*(\\rho)}\\\\\n&=\\tr{(\\mathcal{L}_\\alpha^*-\\mathcal{L}^*)(\\rho)}\\\\\n&=\\tr{\\rho F^* (\\mathds{1}_{\\mathcal{H}_S}\\otimes Q_{\\alpha, \\beta}) F} \\\\\n&=\\tr{M^T Q_{\\alpha, \\beta}}\\\\\n&=\\tr{M Q_{\\alpha, \\beta}^T}~.\n\\end{align*}\nSince $M^T=\\mathds{1}-M$ we have $\\lambda=\\frac{1}{2}\\tr{M Q_{\\alpha, \\beta}^T+M^T Q_{\\alpha, \\beta}}=\\frac{1}{2}\\tr{M(Q_{\\alpha, \\beta}^T-Q_{\\alpha, \\beta})}+\\frac{1}{2}\\tr{Q_{\\alpha, \\beta}}$, which is the formula of the theorem.\n\nLet us derive an equation for $M$, $\\tr{\\mathcal{L}_\\alpha^*(\\rho) F F^*}=\\lambda M$.\nBy Formula \\ref{eq:cov_evolution} we know that\n\\[\n\\text{Tr} _{\\mathcal{Y}}\\left(\\mathcal{L}^*(\\rho)FF^*\\right) =\\left(-iT_S-\\frac{1}{2} \\Theta \\Theta^*\\right) M+M\\left(iT_S-\\frac{1}{2}\\Theta \\Theta^*\\right)+\\Theta M_B \\Theta^*~.\n\\]\nThus we only need to compute $\\tr{(\\mathcal{L}_\\alpha^*-\\mathcal{L}^*)(\\rho)FF^*}$. For any $i,j\\in \\set{1, \\cdots, 2L_S}$, we have\n\\begin{align*}\n\\tr{(\\mathcal{L}_\\alpha^*-\\mathcal{L}^*)(\\rho)\\gamma_i\\gamma_j}&=\\sum_{i=1}^n \\tr{\\rho \\left(\\mathcal{L}-\\mathcal{L}_\\alpha\\right))(\\gamma_i\\gamma_j)}\\\\\n&= \\sum_{1\\leq k,l\\leq 2L_S} [Q_{\\alpha, \\beta}]^f_{k,l} \\tr{\\rho \\gamma_k \\gamma_i \\gamma_j \\gamma_l} \\\\\n&= \\sum_{1\\leq k,l\\leq 2L_S} [Q_{\\alpha, \\beta}]^f_{k,l} \\left( [M]^f_{k,i} [M]^f_{j,l}-[M]^f_{k,j} [M]^f_{i,l}+[M_S]^f_{k,l}[M_S]^f_{i,j} \\right)~~~\\text{by the Wick formula} \\\\\n&= [M^T Q_{\\alpha, \\beta} M^T-M Q_{\\alpha, \\beta}^T M+\\tr{Q_{\\alpha, \\beta}^T M}M]^f_{i,j}~.\n\\end{align*}\nSince $M^T=\\mathds{1}-M$ we obtain\n\\begin{align*}\n\\tr{(\\mathcal{L}_\\alpha^*-\\mathcal{L}^*)(\\rho)F F^*}&=Q_{\\alpha, \\beta}-M Q_{\\alpha, \\beta}-Q_{\\alpha, \\beta} M+M\\left(Q_{\\alpha, \\beta}-Q_{\\alpha, \\beta}^T\\right) M+\\tr{Q_{\\alpha, \\beta}^T M} M~.\n\\end{align*}\nFinally, \n\\begin{align}\n\\tr{\\mathcal{L}_\\alpha^*(\\rho)F F^*}=G_\\alpha M+M G_\\alpha^* +M C_{\\alpha, \\beta} M+B_{\\alpha, \\beta}+\\tr{Q_{\\alpha, \\beta}^T M} M ~\n\\end{align}\nwhere $G_\\alpha, B_{\\alpha, \\beta}, C_{\\alpha, \\beta}$ are defined in the theorem. Since $\\lambda =\\tr{Q_{\\alpha, \\beta}^T M}$, we have\n\\begin{align}\nG_\\alpha M+M G_\\alpha^* +M C_{\\alpha, \\beta} M+B_{\\alpha, \\beta}=0~.\n\\end{align}\n\\end{proof}\n\nWe now turn to the study of the Riccati equation and the proof of Theorem \\ref{theo:e_alpha_fermionic}. The problem is to find the solution of \\ref{eq:riccati} for which the eigenvalue is maximal. We will need the following properties of Riccati equations:\n\n\\begin{prop}\\label{prop:riccati_solution}\nLet us consider some matrices $A, B, Q$ on $\\mathbb{C}^d$ and consider the equation\n\\begin{align}\\label{eq:riccati_gen}\nXA+A^* X + XBX +C=0~.\n\\end{align}\nAssume that $B$ and $C$ are self-adjoint, that $B\\geq 0$, and that the Kalman space $K(A, B)$ is equal to $\\mathbb{C}^d$ (the Kalman space is defined in Theorem \\ref{theo:kalman}; we say that the pair $(A,B)$ is controllable).\n\nWrite $Z$ the matrix\n\\[\nZ=\\begin{pmatrix} A & B \\\\ -C & -A^*\\end{pmatrix}~.\n\\] \n Then for any solution $X$ of the Riccati equation the matrix $A+BX$ has for eigenvalues a subset of cardinal $d$ of the set of eigenvalues of $Z$ (counted with algebraic multiplicities). \n\n\nMoreover, if Equation \\eqref{eq:riccati_gen} admits a self-adjoint solution, then there is a self-adjoint solution $X_{max}$ such that $X\\leq X_{max}$ for any self-adjoint solution $X$. The maximal solution $X_{max}$ is the unique self-adjoint solution whose eigenvalues are the eigenvalues of $Z$ with positive real part (counted with algebraic multiplicities). The maximal solution is isolated in the set of self-adjoint solutions.\n\\end{prop}\n\t\nThis is extracted from results scattered in \\cite{Lancaster_Rodman95}. The fact that the maximal eigenvalue is isolated comes from Theorem 7.7.2. \n\\vspace{0.5cm}\n\nTo convert Equation \\eqref{eq:riccati} to an equation satisfying the hypothesis of this proposition, we note that $\\rho_\\alpha>0$ (since $\\mathcal{L}_\\alpha^*$ is positivity improving) so $M$ is of the form $(1+\\exp(-T))^{-1}$ for some operator $T$. We define\n\\[\nX=M^{-1}-\\mathds{1}~.\n\\]\nWe have $X>0$. Moreover,\n\\begin{align}\\label{eq:riccatiX}\nX A_\\beta+A_\\beta^* X +X B_{\\alpha, \\beta} X -B_{-\\alpha, -\\beta}=0~.\n\\end{align}\nThis formula is obtained by making the product of Equation \\ref{eq:riccati} with $M^{-1}$ on the left and the right, and using the relations\n\\begin{align*}\nA_\\beta&=G_{\\alpha, \\beta}+B_{\\alpha, \\beta}\\\\ \n-B_{-\\alpha, -\\beta}&=C_{\\alpha, \\beta}+B_{\\alpha, \\beta}+G_{\\alpha, \\beta}+G_{\\alpha, \\beta}^*~.\n\\end{align*}\n\nThe equation on $X$ satisfies the hypothesis of Proposition \\ref{prop:riccati_solution}. Indeed, the operator $B_{\\alpha, \\beta} =\\sum_{i=1}^n e^{\\alpha_i K_S}M_{\\beta_i} \\Theta_i \\Theta_i^*$ is positive; since the semigroup is positivity improving the Kalman space $K(T_S, \\Theta)$ is equal to $\\mathcal{Y}$. But $A_\\beta=-iT_S+R$ where $\\ran(R)=\\ran(\\Theta)$, and $\\ran(B_{\\alpha, \\beta})=\\ran(\\Theta)$, so $K(A_\\beta, B_{\\alpha, \\beta})=\\mathcal{Y}$.\n\nLet us express $e(\\alpha)$ in terms of $A_\\beta +XB_{\\alpha, \\beta}$. We have\n\\[\n-C_{\\alpha, \\beta}M=-M^{-1} M C_{\\alpha, \\beta} M=M^{-1}\\left(G_{\\alpha, \\beta} M + M G_{\\alpha, \\beta}^* +B_{\\alpha, \\beta}\\right)\n\\]\nthus\n\\begin{align*}\ne(\\alpha)&=\\frac{1}{2}\\left(-\\tr{C_{\\alpha, \\beta} M}+\\tr{Q_{\\alpha, \\beta}} \\right) \\\\\n&=\\frac{1}{2}\\tr{M ^{-1}G_{\\alpha, \\beta}M+G_{\\alpha, \\beta}^*+Q_{\\alpha, \\beta} +M^{-1} B_{\\alpha, \\beta}}\\\\\n&=\\frac{1}{2}\\tr{G_{\\alpha, \\beta}+G^*_{\\alpha, \\beta}+Q_{\\alpha, \\beta}+B_{\\alpha, \\beta}+X B_{\\alpha, \\beta}} \\\\\n&=\\frac{1}{2} \\tr{A_\\beta+X B_{\\alpha, \\beta}}-\\frac{1}{4}\\sum_{i=1}^n \\tr{\\Theta_i \\Theta_i^*}~.\n\\end{align*}\nThe last equality is due to the fact that $G_{\\alpha, \\beta}+G^*_{\\alpha, \\beta}+Q_{\\alpha, \\beta}+B_{\\alpha, \\beta}=A_\\beta+i T_S-\\frac{1}{2}\\sum_{i=1}^n \\Theta_i \\Theta_i^*$ and $\\tr{T_S}=0$.\n\nLet us show that $X$ is the maximal solution $X_{max}$ of the Riccati equation \\eqref{eq:riccatiX} (then the eigenvalues of $A_\\beta+X_{max} B_{\\alpha, \\beta}$ are the eigenvalues of $Z$ with positive real parts, and the theorem is proved). First, we have $\\tr{X B_{\\alpha, \\beta}}\\leq \\tr{X_{max} B_{\\alpha, \\beta}}$ since $X\\leq X_{max}$. Thus, it is sufficient to show that $M_{max}=(\\mathds{1}+X_{max})^{-1}$ is the covariance matrix of a state. This is equivalent to\n\\begin{align*}\nX_{max}>0~\\\\\n X_{max}^T=X_{max}^{-1}~.\n\\end{align*}\n\nWe have $X_{max}\\geq X >0$ so the inequality is satisfied. Moreover, we have $B_{\\alpha, \\beta}^T=B_{-\\alpha, -\\beta}$ and $A_\\beta^T=-A_\\beta^*$ so the map $X\\mapsto -(X^T)^{-1}$ preserves the set of solutions of Equation \\eqref{eq:riccatiX}. Since this map is increasing for the matrix order it sends a maximal solution on a maximal solution and $-\\left(X_{max}^T\\right)^{-1}=X_{max}$. Thus $(\\mathds{1}+X_{max})^{-1}$ is a covariance matrix, and it corresponds to the dominant eigenvector of $\\mathcal{L}_\\alpha^*$. This proves Theorem \\ref{theo:e_alpha_fermionic}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{The example of the fermionic chain}\n\nIn this subsection we describe the rate function of the large deviations on the fermionic chain of Paragraph \\ref{subsub:gauge_invariant}, which we compute numerically for different values of the length $L$. The rate function $I$ is a function of two variables, but as a consequence of the following lemma we can consider only one parameter. \n\n\\begin{lem}\nIf there are two baths, for any $\\alpha_1, \\alpha_2$ we have\n\\[\ne(\\alpha_1, \\alpha_2)=e(\\alpha_1-\\alpha_2, 0)~.\n\\]\nWriting $\\tilde{e}(\\alpha)=e(\\alpha, 0)$ for any $\\alpha\\in \\mathbb{R}$, we have\n\\[\nI(\\zeta_1, \\zeta_2)=\\left\\{\\begin{array}{cc} \n-\\underset{\\alpha \\in \\mathbb{R}}{\\inf} \\left(\\scal{\\alpha, \\zeta_1}-\\tilde{e}(\\alpha)\\right)& \\text{ if $\\zeta_1+\\zeta_2=0$} \\\\\n+\\infty & \\text{ if $\\zeta_1+\\zeta_2\\neq 0$}\n\\end{array} \\right.\n\\]\n\\end{lem}\nThis lemma is a straightforward consequence of \\eqref{eq:I_sum}.\n\nWe computed $I(\\zeta, -\\zeta)$ for the fermionic chain with $\\theta_0=\\theta_{L+1}=0$, and temperatures $\\beta_0=1, \\beta_{L+1}=0$, for chains of lengths $L$ from $2$ to $5$ (see Figure \\ref{fig:rate_1_0})\n\n\\begin{figure}[!h]\n\\includegraphics[scale=0.3]{rate_functional_b1=1_b2=0_l=2__10.pdf}\n\\caption{Rate functional $I(\\zeta, -\\zeta)$ for $\\beta_0=1, \\beta_{L+1}=0$ and for $L=2$ to $L=10$. The largest function corresponds to $L=2$ and the smallest corresponds to $L=10$.}\\label{fig:rate_1_0}\n\\end{figure}\n\nAs we can see, the rate functions have the same zero, which corresponds to the flux given in formula \\eqref{eq:flux_chain}, that is \n\\[\nJ_1=\\frac{4}{10}(n_0-n_1)=\\frac{4}{10}\\left(\\frac{1}{1+e^{-1}}-\\frac{1}{1+e^0}\\right)\\simeq 0.092~.\n\\]\n\nThe rate functions are very similar around this zero, and progressively separate for large values of $\\abs{\\zeta-J_1}$. The rate function is smaller for large values of $L$, which means that the fluctuations of the energy fluxes around their mean values are larger when the length of the chain is larger. This result is interesting: the mean energy flux is completely independent of the length of the chain, but the large deviations are sensible to this length. \n\\vspace{0.5cm}\n\n\nIn figure 2 we show the rate function in the case $\\beta_0=10$ and $\\beta_{L+1}=0$. Taking a high value of $\\beta_0-\\beta_{L+1}$ makes the asymmetry of $I$ under the change $\\zeta\\mapsto -\\zeta$ very visible. \n\n\\begin{figure}[!h]\n\\includegraphics[scale=0.3]{rate_functional_b1=10_b2=0_l=2__5.pdf}\n\\caption{Rate functional $I(\\zeta, -\\zeta)$ for $\\beta_0=10, \\beta_{L+1}=0$ and for $L=2$ to $L=5$. The largest function corresponds to $L=2$ and the smallest corresponds to $L=5$.}\\label{fig:rate_10_0}\n\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Notation} \\label{sec:intro}\n\nIn this paper we prove a series of results related to the geometric theory of the Jang equation and applications to classical problems in the theory of minimal and constant mean curvature graphs. \\\\\n\nIn Section \\ref{sec:Plateau} we show that there exist stable solutions of the Plateau problem for marginally outer trapped surfaces, answering a question of G. Galloway and N. O'Murchadha raised in \\cite{Galloway-OMurchadha:2008}. This result is subtle in view of the non-variational nature of these surfaces. The proof is based on the fact that using the existence theory in \\cite{Eichmair:2009-Plateau} we can construct \\emph{ordered} families of solutions of the Plateau problem. This result is used in the recent proof of the spacetime positive mass theorem by L.-H. Huang, D. Lee, R. Schoen and the first author in \\cite{spacetimePMT}. \\\\\n\nIn Sections \\ref{sec:scherk}, \\ref{sec:canonical}, and \\ref{sec:uniqueness} we develop the geometric theory of the Jang equation pioneered by R. Schoen and S.-T. Yau in \\cite{Schoen-Yau:1981-pmt2} to prove the existence of non-trivial and, in some cases, canonical Scherk-type solutions of the Jang equation in the complements of the total weakly future outer trapped region and the total weakly past outer trapped region. \\\\\n\nIn Section \\ref{sec:jss} we employ techniques from the geometric theory of the Jang equation, in particular the capillarity regularization and the geometric blow up analysis from \\cite{Schoen-Yau:1981-pmt2} and ideas from the solution of the non-variational Plateau problem for marginally outer trapped surfaces in \\cite{Eichmair:2009-Plateau}, to the classical Jenkins--Serrin problem \\cite{Jenkins-Serrin:1966, Jenkins-Serrin:1968} of finding necessary and sufficient conditions for a domain in a Riemannian surface to support a Scherk-type constant mean curvature graph. In the case of positive mean curvature, we are able to dispense with the a priori assumption that the domain admit a sub solution which is required in the foundational paper by J. Spruck \\cite{Spruck:1972} (in $\\R^2$) and its recent extension to domains in $\\mathbb{S}^2$ and $\\mathbb{H}^2$ by L. Hauswirth, H. Rosenberg, and J. Spruck \\cite{Hauswirth-Rosenberg-Spruck:2009}. Moreover, our results are valid in arbitrary complete Riemannian surfaces. In particular, the existence of a Scherk-type graph is curiously reduced to a (generically) finite set of inequalities relating area and circumference of certain polygons that can be inscribed into the domain: the Jenkins-Serrin-Spruck flux conditions. Our approach here does not distinguish between minimal and (positive) constant mean curvature graphs. In the case of minimal graphs, we recover the results by B. Nelli and H. Rosenberg \\cite{Nelli-Rosenberg:2002} (in $\\mathbb{H}^2$) and by A. Pinheiro \\cite{Pinheiro:2009} (in arbitrary Riemannian surfaces). Our methods carry over to higher dimensions. A detailed overview of the literature and a precise statement of our result are given in Section \\ref{sec:jss}. \\\\\n\nIn the appendices we collect several results that are needed in the paper. The simple indirect proof of the (known) interior gradient estimate for solutions of the prescribed mean curvature equation based on the regularity theory for almost minimal boundaries in low dimensions in Appendix \\ref{sec:ige} does not seem to appear in the literature, surprisingly. In Appendix \\ref{sec:boundary} we characterize the boundary of domains that support infinite boundary value solutions of the prescribed mean curvature equation \\emph{without} making an a priori assumption regarding the regularity of the boundary. In Appendix \\ref{sec:equicontinuous} we state another simple and useful consequence of the classical compactness and regularity theory for almost minimal boundaries: the horizontal parts of the unit normal vector fields of non-parametric solutions to the prescribed mean curvature equations are equicontinuous in low dimensions. This is an important ingredient for the geometric analysis in Section \\ref{sec:jss}. \\\\ \n\nWe proceed by introducing the notation employed throughout this paper, and by reminding the reader of some background material. \\\\\n\nLet $(M, g)$ be a connected Riemannian manifold of dimension $n$, with $3 \\leq n \\leq 7$, and let $k$ be a symmetric $(0, 2)$-tensor on $M$. In the context of the Cauchy problem for the Einstein equations in general relativity, $k$ is referred to as the (spacetime) second fundamental form tensor and the triple $(M, g, k)$ is called an initial data set. We often require that $(M, g, k)$ is asymptotically flat, i.e. that the complement of some compact subset of $M$ consists of finitely many connected components $N_1, \\ldots, N_m$, called the ends, each one diffeomorphic to $\\R^n \\setminus \\bar B_1(0)$ and such that in the corresponding coordinate systems the metric tensor $g_{ij}$ converges to the Euclidean metric $\\delta_{ij}$ and the second fundamental form tensor $k_{ij}$ to zero. More precisely, we require that\n\\begin{eqnarray*} \\label{eqn:AF}\n|g_{ij} - \\delta_{ij}| + |x||\\partial_k g_{ij}| = O(|x|^{-q}) \\text{ and } k_{ij} = O(|x|^{-q-1}) \\text{ as } |x| \\to \\infty, \\text{ for some } q > \\frac{n-2}{2}.\n\\end{eqnarray*}\nWhen $n=3$, we ask in addition that for some $\\beta >2$, \n\\begin{eqnarray*} \\label{eqn:AF3}\n\\tr_g(k) = g^{ij} k_{ij} = O(|x|^{-\\beta}) \\text{ as } x \\to \\infty. \n\\end{eqnarray*}\nThis last condition is imposed so that certain barriers for the Jang equation can be constructed far out in the asymptotically flat ends, cf. Appendix \\ref{sec:barriers}. \\\\\n\nLet $(M, g, k)$ be an initial data set, and let $\\Sigma \\subset M$ be a two-sided hypersurface with unit-normal vector field $\\nu$. The future outer and past outer expansion scalars of $\\Sigma$ are defined, respectively, as $\\theta^+_\\Sigma = H_\\Sigma + \\tr_\\Sigma (k)$ and $\\theta^-_\\Sigma = H_\\Sigma - \\tr_\\Sigma (k)$. Here, $H_\\Sigma$ is the mean curvature scalar of $\\Sigma$ computed as the tangential divergence of $\\nu$ and $\\tr_\\Sigma (k)$ is the trace of $k$ restricted to the tangent space of $\\Sigma$. The hypersurface $\\Sigma$ is called future outer trapped (respectively past outer trapped) if $\\theta^+_\\Sigma <0$ ($\\theta^-_\\Sigma <0$) everywhere on $\\Sigma$, weakly future outer trapped (respectively weakly past outer trapped) if $\\theta^+_\\Sigma \\leq 0$ ($\\theta^-_\\Sigma \\leq 0$) everywhere on $\\Sigma$, and is called a marginally future outer trapped surface or MOTS for short (respectively marginally past outer trapped surface or MITS for short) if $\\theta^+_\\Sigma = 0$ ($\\theta^-_\\Sigma =0$) everywhere on $\\Sigma$. Except for Section \\ref{sec:Plateau}, the MOTSs and MITSs appearing in this paper will be closed and in fact boundaries of sets $\\Omega$ that contain part of an end $\\{x \\in N_i : |x| \\geq r_0\\}$ for some $r_0 \\geq 1$ and $i \\in \\{1, \\ldots, m\\}$. If we write a hypersurface $\\Sigma$ as the (relative) boundary of a set $\\Omega$, say $\\Sigma = \\partial \\Omega$ in $U$ where $U$ is an open subset of $M$, we will use the unit normal field $\\nu$ of $\\Sigma$ pointing into $\\Omega$ to compute the scalar mean curvature $H_\\Sigma$ and the expansion scalars $\\theta^\\pm_\\Sigma$ unless otherwise noted. \\\\\n \nLet $(M, g, k)$ be a complete asymptotically flat manifold of dimension $3 \\leq n \\leq 7$. Fix one of the ends, say $N_1$. It is easy to see that $H_{S_r} > |\\tr_{S_r} k|$ for all $r \\geq r_0$, provided that $r_0 \\geq 1$ is sufficiently large. Here, $S_r := \\{x \\in N_1 : |x| = r\\}$ is the coordinate sphere of radius $r$ in $N_1$ and the mean curvature scalar $H_{S_r}$ is computed as the tangential divergence of the unit normal pointing into the end. Let $M_- \\subset M$ (respectively $M_+ \\subset M$) be the interior of the intersection of all open subsets $\\Omega \\subset M$ that contain $\\{x \\in N_1 : |x| \\geq r_0\\}$ and which have smooth properly embedded boundary satisfying $H_{\\partial \\Omega} + \\tr_{\\partial \\Omega} (k) \\leq 0$ ($H_{\\partial \\Omega} - \\tr_{\\partial \\Omega} (k) \\leq 0$). It follows from \\cite{Andersson-Metzger:2009} in dimension $n=3$ and from \\cite{Eichmair:2010} in dimensions $3 \\leq n \\leq 7$ that $M_-$ (respectively $M_+$) has compact embedded boundary and that $H_{\\partial M_-} + \\tr_{\\partial M_-} (k) = 0$ ($H_{\\partial M_+} -\\tr_{\\partial M_+} (k) = 0$). Thus $\\partial M_-$ is a MOTS and $\\partial M_+$ is a MITS. The complements of the regions $M_-$ and $M_+$ are the total weakly future outer trapped region and the total weakly past outer trapped region of $(M, g, k)$ with respect to the chosen end, respectively. If $(M, g, k)$ has more than one end, then both $\\partial M_-$ and $\\partial M_+$ are non-empty and each of them separates the portion $\\{x \\in N_1 : |x| \\geq r_0\\}$ of $N_1$ from $\\bigcup_{i=2}^m \\{x \\in N_i : |x| \\geq r_0\\}$. \\\\\n\nGiven an open subset $\\Omega \\subset M$ and a function $u \\in \\C^2(\\Omega)$ we define, in local coordinates near a point $x \\in \\Omega$, \n\\begin{eqnarray*} \nH(u) := \\left( g^{ij} - \\frac{u^i u^j}{1 + |D u|^2}\\right) \\frac{D^2_{ij} u }{\\sqrt{1 + |D u|^2}} \\text{ and } \\tr (k) (u)= \\left( g^{ij} - \\frac{u^i u^j}{1 + |D u|^2}\\right) k_{ij}. \n\\end{eqnarray*} \nHere, all geometric operations (raising indices, gradient, length of gradient, Hessian) are with respect to $g$. These definitions are independent of the particular coordinate system used. The function $H(u)$ is the scalar mean curvature of the graph $G = \\{(x, u(x)) : x \\in \\Omega\\}$ of $u$ in the Riemannian product $(\\Omega \\times \\R, g + (dx^{n+1})^2)$ at the point $(x, u(x))$ with respect to the downward pointing unit normal, and $\\tr(k)$ is the trace of $k$ (extended to $\\Omega \\times \\R$ by zero in the vertical direction) over the tangent space of this graph at $(x, u(x))$. If\n\\begin{eqnarray} \\label{Jang} H(u) + \\tr(k)(u) = 0,\\end{eqnarray}\nthen $G$, with its downward orientation, is a MOTS in the new initial data set $(M \\times \\R, g + (d x^{n+1})^2, k)$. Equation (\\ref{Jang}) is known as the Jang equation. \\\\\n\n\nFor background material on MOTSs, MITSs, and the Jang equation we refer the reader to the survey article \\cite{Andersson-Eichmair-Metzger:2010}.\\\\ \n\n\\section*{Acknowledgments} We would like to thank L. Andersson, G. Galloway, G. Huisken, and M. Mars for their interest in this work and for valuable discussions. The first author would like to thank R. Beig, P. Chru\\'sciel, and M. Heinzle of the group for Gravitational Physics at the University of Vienna, as well as the wonderful Erwin Schr\\\"odinger Institute, for kindly providing pleasant and very inspiring working conditions for him in the Summer of 2011. He also gratefully acknowledges the support of a Clay Liftoff Fellowship in the Summer of 2008, when some of the results in this paper were first conceived, and the support of NSF grant DMS-0906038 and of SNF grant 2-77348-12. \n\n\n\\section{Stability of solutions of the Plateau problem} \\label{sec:Plateau}\n\nThe following definition of stability for MOTSs spanning a non-empty boundary is\nthe natural analogue of the notion of stability for closed MOTSs that has been\nintroduced and studied systematically in \\cite{Andersson-Mars-Simon:2008}.\n\n\\begin{definition} [Cf. \\protect{\\cite[Section 2]{Galloway-OMurchadha:2008}}]\n \\label{def:stability_of_MOTS}\nLet $(M, g, k)$ be an initial data set and consider a two-sided hypersurface\n$\\Sigma \\subset M$ with boundary $\\partial \\Sigma = \\Gamma$ and with a designated\n``outward\" unit normal vector field $\\nu$. Assume that $\\Sigma$ is a MOTS. Then $\\Sigma$ is said\nto be {\\it stable in the sense of MOTSs} if there exists a smooth function $f$\nthat is positive in the interior of $\\Sigma$, vanishes on the boundary\n$\\Gamma$, and is such that $L_\\Sigma f \\geq 0$. Here,\n \\begin{equation*}\n L_\\Sigma \\phi := - \\Delta_\\Sigma \\phi + 2 \\langle X,\n D_\\Sigma \\phi \\rangle + \\left( \\tfrac{1}{2} \\Scal_\\Sigma - \\tfrac{1}{2} |h +\n k|_\\Sigma^2 - J (\\nu) - \\mu + \\div_\\Sigma X - |X|^2 \\right) \\phi,\n \\end{equation*}\n where $X$ is the tangential part of the vector field dual to $k(\\nu, \\cdot)$ on\n$\\Sigma$, $h$ is the second fundamental form of $\\Sigma$ (with trace\n$\\mc_\\Sigma$), $D_\\Sigma \\phi$ is the tangential gradient of $\\phi$ along\n$\\Sigma$, $\\Scal_\\Sigma$ is the scalar curvature of $\\Sigma$, $\\mu = \\frac{1}{2}\n\\left (\\Scal_M + (\\tr_M (k))^2 - |k|_M^2\\right)$ is the local mass density, and\nwhere $J := \\div(k - \\tr_M(k)g)$ is the local current density. Equivalently, $\\Sigma$ is stable in the sense of MOTSs if, and only if, the principal eigenvalue of $L_\\Sigma$ is non-negative. \n\\end{definition}\n\nFor a careful discussion of principal eigenvalues of (not necessarily self-adjoint) elliptic operators and their eigenfunctions, we refer the reader to \\cite[Sections 3.6 and 3.7]{Pinsky:1995}. \\\\\n\nWe recall the following theorem from \\cite{Eichmair:2009-Plateau}:\n\n\\begin{theorem} [\\cite{Eichmair:2009-Plateau}] \\label{thm:existencePlateau}\nLet $(M, g, k)$ be a complete initial data set of dimension $n$ with $2 \\leq n \\leq 7$. Let $\\Omega \\subsetneq M$ be a bounded open set with smooth boundary $ \\partial \\Omega$. Let $\\Gamma \\subset \\partial \\Omega$ be a non-empty smooth closed embedded submanifold of $\\partial \\Omega$ such that $\\partial \\Omega \\setminus \\Gamma = \\partial_- \\Omega \\dot \\cup \\partial_+ \\Omega$ for disjoint non-empty relatively open subsets $\\partial_- \\Omega, \\partial_+ \\Omega$ of $\\partial \\Omega$. Assume that $\\mc_{\\partial \\Omega} + \\tr_{\\partial \\Omega} k < 0$ near $\\partial_- \\Omega$ with the mean curvature computed as the tangential divergence pointing into $\\Omega$ and that $\\mc_{\\partial \\Omega} + \\tr_{\\partial \\Omega} k > 0$ near $\\partial_+ \\Omega$ with the mean curvature scalar computed as the tangential divergence of the unit normal pointing out of $\\Omega$. Then there exists a smooth hypersurface $\\Sigma \\subset \\Omega$ with boundary $\\Gamma$ that is an almost minimizing relative boundary in $\\Omega$ and such that $\\Sigma$ is a MOTS with respect to the unit normal pointing towards $\\partial_+ \\Omega$.\n\\end{theorem}\n\nThe following theorem answers a question posed in \\cite[Section 3]{Galloway-OMurchadha:2008}. It is an ingredient in the proof of the spacetime positive mass theorem given in \\cite{spacetimePMT}. \n\n\\begin{theorem}\nAssumptions as in Theorem \\ref{thm:existencePlateau}. Then there exists a solution $\\Sigma$ of the Plateau problem for MOTSs in $\\Omega$ with boundary $\\Gamma$ that is stable in the sense of MOTSs.\n\\begin{proof} Given $\\epsilon >0$ small, let $\\Gamma^\\epsilon := \\{ \\theta \\in \\partial_+ \\Omega: \\dist_{\\partial \\Omega} (\\theta, \\Gamma) = \\epsilon\\}$. From the explicit construction in the proof of Theorem \\ref{thm:existencePlateau} in \\cite[Chapter 4]{Eichmair:2009-Plateau} we see that the MOTSs $\\Sigma^\\epsilon \\subset \\Omega$ spanning $\\Gamma^\\epsilon$ are (strictly) ordered. To see this, recall that the open subset of $\\Omega$ whose relative boundary is $\\Sigma^\\epsilon$ is the geometric limit as $t \\searrow 0$ of downward translations of the regions lying above the graphs $u_{t}^\\epsilon \\in \\C^\\infty_{loc} (\\Omega)$ constructed in \\cite[Lemma 4.2]{Eichmair:2009-Plateau}. Given $t >0$ and $0 < \\epsilon < \\epsilon'$ small, we have that $\\mathcal{S}_{\\overline u^{\\epsilon'}_{t}} \\subset \\mathcal{S}_{\\overline u^{\\epsilon}_{t}}$ (in the notation of \\cite{Eichmair:2009-Plateau}) and hence $u^{\\epsilon'}_{t} \\leq u^\\epsilon_{t}$. It follows that the regions above the graphs are ordered so that $\\Sigma^{\\epsilon'}$ lies to one side (towards $\\partial_+ \\Omega$) of $\\Sigma^{\\epsilon}$. The geometric maximum principle shows that components of $\\Sigma^{\\epsilon'}$ and $\\Sigma^{\\epsilon}$ that span components of $\\Gamma^\\epsilon$ and $\\Gamma^{\\epsilon'}$ cannot touch unless they coincide. We will discard all extraneous closed components of $\\Sigma^\\epsilon$. This does not change that each $\\Sigma^\\epsilon$ is a relative boundary, nor that the $\\Sigma^\\epsilon$'s are ordered. \n\nThe geometric compactness properties of the almost minimizing relative boundaries $\\Sigma^\\epsilon$ show that as $\\epsilon \\searrow 0$, the $\\Sigma^\\epsilon$ converge smoothly and with multiplicity one to an embedded MOTS $\\Sigma$ that spans $\\Gamma$. We claim that this MOTS $\\Sigma$ is stable in the sense of MOTSs. To see this, let $U, V, W \\subset \\Sigma$ be non-empty open subsets with smooth boundaries such that $U \\Subset V \\Subset W \\Subset \\text{int } \\Sigma$. Let $\\nu$ be the unit normal vector field of $\\Sigma$ that points towards $\\partial_+ \\Omega$. (This makes sense because $\\Sigma$ is a relative boundary in $\\Omega$ spanning $\\Gamma$.) By assumption, there exist positive functions $f^\\epsilon \\in \\C^\\infty (W)$ for $\\epsilon >0$ small with $\\{\\exp_\\theta \\left( f^\\epsilon \\nu \\right) : \\theta \\in V \\} \\subset \\Sigma^\\epsilon$ and such that $f^\\epsilon \\to 0$ with all derivatives on compact subsets of $W$. Because $\\Sigma$ and $\\Sigma^\\epsilon$ both satisfy the MOTS equation, $f^\\epsilon$ is solution of a homogenous linear elliptic equation in divergence form on $V$. The operator describing the linearization of the equation at the function that vanishes identically is $L_\\Sigma$. Arguing exactly as in \\cite[p. 333]{Simon:1987}, using Harnack theory, it follows that the functions $f^\\epsilon$ can be rescaled (so their infimum is one on $V$, say) so as to converge smoothly to a positive function $f_V \\in \\mathcal{C}^\\infty(V)$ with $L_\\Sigma (f_V) = 0$. This implies that $U$ is stable in the sense of MOTSs. To see this, let $\\lambda$ be the first principal eigenvalue of $L_\\Sigma|_U$ and let $h \\in \\C^\\infty (\\bar U)$ be the corresponding first (Dirichlet) eigenfunction so that $L_\\Sigma h = \\lambda h$. We recall that the first principal eigenvalue is simple and that the corresponding eigenfunctions do not change signs. By scaling, using that $g$ vanishes on the boundary of $U$ and that $f_V$ is positive on $V$ and that $U \\Subset V$, we may assume that $0 < h \\leq f_V$ on $U$ with equality at some point. The maximum principle then implies that $\\lambda \\geq 0$. We conclude that every open subset $U \\Subset \\text{int } \\Sigma$ is stable in the sense of MOTSs. Using that the principal eigenvalue of an elliptic operator depends continuously on the operator and the domain, it follows that $\\Sigma$ is stable in the sense of MOTSs. \n\\end{proof}\n\\end{theorem}\n\n\n\\section{Scherk-type solutions of the Jang equation} \\label{sec:scherk}\n\nThe content of the following proposition is similar to that of Theorem 3.1 in \\cite{Metzger:2010-blowup}. We include an alternative proof here as preparation for the more general and difficult Theorem \\ref{thm:JangScherk} below. The modification of the data near the boundary in our proof is much less delicate than that in \\cite{Metzger:2010-blowup}. The regions $M_-$ and $M_+$ in the statement of Proposition \\ref{prop:existenceblowup} and Theorem \\ref{thm:JangScherk} below are defined in the introduction. \n\n\\begin{proposition} [Cf. Theorem 3.1 in \\cite{Metzger:2010-blowup} when $n=3$] \\label{prop:existenceblowup} Let $(M, g, k)$ be a complete asymptotically flat initial data set of dimension $n$, $3 \\leq n \\leq 7$, and assume that $\\partial M_-$ and $\\partial M_+$ are disjoint. There exists a smooth solution $u : M_- \\cap M_+ \\to \\mathbb {R}$ of the Jang equation $\\mc(u) + \\tr(k)(u) = 0$ such that $u(x) \\to 0$ as $x \\to \\infty$ in the end of $(M, g, k)$ contained in $M_- \\cap M_+$ and such that $u (x) \\to \\pm \\infty$ as $\\dist(x, \\partial M_\\pm) \\to 0$. \n\n\\begin{proof} We abbreviate $\\Omega := M_- \\cap M_+$. Let $\\chi \\in \\C^\\infty_c (M)$ be such that $\\chi \\equiv \\pm 1$ near $\\partial M_\\pm$. Given $\\epsilon \\in (0, 1)$ we define $k^\\epsilon := k + \\epsilon \\chi$. Note that $H_{\\partial M_+} + \\tr_{\\partial M_+}(k^\\epsilon) > 0$ and $H_{\\partial M_-} + \\tr_{\\partial M_-} (k^\\epsilon) < 0$. \n\nArguing exactly as in \\cite[Chapter 3]{Eichmair:2009-Plateau} and \\cite[Chapters 3 and 4]{Eichmair:2010}, using also the argument in Appendix \\ref{sec:barriers} and Footnote \\ref{footnote:noncompactdomains}, we see that for every $\\epsilon \\in (0, 1)$ there exists a connected open subset $\\Omega_0^\\epsilon \\subset \\Omega$ containing the chosen end of $(M, g, k)$ and a solution $u^\\epsilon \\in \\C^\\infty_{loc} (\\Omega_0^\\epsilon)$ of the Jang equation $H(u^\\epsilon) + \\tr(k^\\epsilon) (u^\\epsilon) = 0$ on $\\Omega_0^\\epsilon$ with the following properties: \n\\begin{enumerate} [(i)]\n\\item \\label{prop:topstab} We have that $\\{x \\in \\Omega : |x| > \\Lambda\\} \\subset \\Omega_0^\\epsilon$ for some $\\Lambda \\geq 1$ that is independent of $\\epsilon \\in (0, 1)$. The topological boundary $\\partial \\Omega_0^\\epsilon$ of $\\Omega_0^\\epsilon$ is a smooth properly embedded hypersurface in $\\Omega$ whose components are either marginally inner trapped or marginally outer trapped with respect to the unit normal pointing into $\\Omega_0^\\epsilon$. The components are $\\lambda$--minimizing with $\\lambda := 1 + 2 n \\sup_{x \\in \\Omega, \\epsilon \\in (0, 1) } |k^\\epsilon (x)|$ in $\\Omega$ (in the language of \\cite{Duzaar-Steffen:1993a}) and stable (in the sense of (\\ref{eqn:almoststability})).\n\\item We have that $u(x) \\to 0$ as $x \\to \\infty$ in $\\Omega$. We have that $u^\\epsilon(x)$ diverges to plus infinity if $x \\in \\Omega_0^\\epsilon$ approaches a marginally inner trapped component of the boundary of $\\Omega_0^\\epsilon$, and to minus infinity if $x \\in \\Omega$ converges to a marginally outer trapped boundary component.\n\\item \\label{prop:stablegraph} The graphs $\\{(x, u^\\epsilon (x)) : x \\in \\Omega_0^\\epsilon \\}$ of $u^\\epsilon$ are complete hypersurfaces of $M \\times \\R$ that are stable and $\\lambda$--minimizing in $\\Omega$. \n\\end{enumerate}\nThe $\\lambda$--minimizing property and stability of the graphs asserted in (\\ref{prop:stablegraph}) implies that of the components of $\\partial \\Omega_0^\\epsilon$ in (\\ref{prop:topstab}), cf. \\cite[p. 254]{Schoen-Yau:1981-pmt2}, \\cite[Lemma A.1] {Eichmair:2009-Plateau}, and the discussion in Appendix \\ref{sec:boundary}. \n\nAs in \\cite{Eichmair:2010} or the discussion in Appendix \\ref{sec:boundary}, we see that the stability and the almost minimizing property (via uniform local mass bounds) lead to curvature estimates for these graphs that are independent of $\\epsilon \\in (0, 1)$. We now let $\\epsilon \\searrow 0$ and pass the graphs of $u^\\epsilon$ to a smooth, properly embedded\\footnote{The almost minimizing property does not imply uniform curvature estimates near $\\partial \\Omega$. As in Appendix \\ref{sec:boundary}, we use the stability and the completeness for that. Once we know that a smooth limit exists, we can use the almost minimizing property to rule out sheeting.} subsequential limit that contains a connected complete graphical component whose domain contains the asymptotically flat end and which satisfies all the above properties with $\\epsilon = 0$. Using the mean value theorem and that there are no MOTSs or MITSs (with respect to $k$) in $\\bar \\Omega$ besides $\\partial M_-$ and $\\partial M_+$ we see that this graphical component has all the properties asserted in the conclusion of the theorem. \n\\end{proof}\n\\end{proposition} \n\n\\begin{remark} Both in Proposition \\ref{prop:existenceblowup} above and in Theorem \\ref{thm:JangScherk} below, the outermost property of $M_-$ and $M_+$ prevents the domain of the sought-after graphical solution of the Jang equation from ``popping outward\". If we think of the results in this section in analogy with the classical Jenkins--Serrin theory \\cite{Jenkins-Serrin:1966, Jenkins-Serrin:1968}, then this outermost property takes the place of the Jenkins--Serrin flux conditions (\\ref{eqn:totalflux}), (\\ref{eqn:partialfluxA}), and (\\ref{eqn:partialfluxB}) in the statement of Theorem \\ref{thm:mainjss} below (with $H_0=0$). \n\\end{remark}\n\n\\begin{theorem} \\label{thm:JangScherk} Let $(M, g, k)$ be a complete asymptotically flat initial data set of dimension $n$ where $3 \\leq n \\leq 7$. Assume that $\\partial M_+$ and $\\partial M_-$ are both non-empty and that they intersect transversely. There exists a smooth solution $u : M_- \\cap M_+ \\to \\R$ of the Jang equation \n$H(u) + \\tr(k) (u) = 0$ such that $u(x) \\to 0$ as $x \\to \\infty$ in the end of $(M, g, k)$ contained in $M_- \\cap M_+$ and such that $u(x) \\to \\pm \\infty$ as $x \\in M_- \\cap M_+$ approaches an interior point $y$ of a component of $M_{\\mp} \\cap \\partial M_{\\pm} \\subset \\partial (M_- \\cap M_+)$. The topological closure of the graph $\\{ (x, u(x)) : x \\in M_- \\cap M_+ \\}$ of $u$ in $M \\times \\R$ is a smooth properly embedded hypersurface with manifold boundary $(\\partial M_+ \\cap \\partial M_-) \\times \\R$. \n\\begin{proof} Let $\\Omega := M_- \\cap M_+$.\nWe denote by $\\nu$ the unit normal vector field of the hypersurface $M_{\\mp} \\cap \\partial M_\\pm$ pointing into $\\Omega$. Let $ \\partial_- \\Omega := M_+ \\cap \\partial M_-$ and $\\partial_+ \\Omega := M_- \\cap \\partial M_+$. Note that $\\partial_- \\Omega$ and $\\partial_+\\Omega$ are manifolds with boundary and that their manifold boundaries coincide. The topological boundary of $\\Omega$ is the disjoint union of $\\partial_- \\Omega$, $\\partial_+ \\Omega$, and $\\partial M_- \\cap \\partial M_+$. Given a component $\\gamma$ of $\\partial_\\pm \\Omega$, let $\\Theta_\\gamma \\in \\C^\\infty(\\bar \\gamma)$ be positive in the interior of $\\gamma$ and zero on its manifold boundary. \n\nLet $\\epsilon \\in (0, 1)$ be so small that the sets $\\{ \\exp_{y} t \\Theta_\\gamma (y) \\nu(y) : y \\in \\text{int } \\gamma \\text { and } t \\in (0, 2\\epsilon) \\} \\subset \\Omega$ are disjoint as $\\gamma$ ranges over the components of $\\partial_\\pm \\Omega$. Let $\\chi^\\epsilon \\in \\C^\\infty_{loc} (\\Omega)$ with values in $[-1, 1]$ be supported in the union of all these sets and such that $\\chi^\\epsilon (y) \\equiv \\pm 1$ on $\\text{Cr}_{\\gamma}^\\epsilon := \\{ \\exp_{y} t \\Theta_\\gamma (y) \\nu(y) : y \\in \\text{int } \\gamma \\text { and } t \\in (0, \\epsilon)\\}$ when $\\gamma$ is a component of $\\partial_\\pm \\Omega$. \n\nLet $\\gamma$ be a component of $\\partial_- \\Omega$. Consider the hypersurface $\\{ (\\exp_y \\epsilon ( 1 - e^{-h}) \\Theta(y) \\nu (y), h) : y \\in \\text{int } \\gamma \\text{ and } h \\in (0, \\infty)\\} \\subset \\Omega \\times \\R$. It has piecewise smooth manifold boundary consisting of $\\gamma \\times \\{0\\}$ and $(\\partial \\gamma) \\times [0, \\infty)$. It is the graph of a positive function $\\overline u_{\\gamma}^\\epsilon \\in \\C^\\infty_{loc} (\\text{Cr}_{\\gamma}^\\epsilon)$. We have that $H(\\overline u_{\\gamma}^\\epsilon) + \\tr(k) (\\overline u_{\\gamma}^\\epsilon) = O(\\epsilon)$. \n\nSimilarly, let $\\gamma$ be a component of $\\partial_+ \\Omega$. Consider the hypersurface $\\{ (\\exp_y \\epsilon ( 1 - e^{h}) \\Theta(y) \\nu (y), h) : y \\in \\text{int } \\gamma \\text{ and } h \\in (- \\infty, 0)\\} \\subset \\Omega \\times \\R$. Its topological boundary is the union of $\\gamma \\times \\{0\\}$ and $(\\partial \\gamma) \\times (- \\infty, 0]$. This hypersurface is the graph of a negative function $\\underline u_{\\gamma}^\\epsilon \\in \\C^\\infty_{loc} (\\text{Cr}_{\\gamma}^\\epsilon)$. We have that $H(\\underline u_{\\gamma}^\\epsilon) + \\tr(k) (\\underline u_{\\gamma}^\\epsilon) = O(\\epsilon)$.\n\nChoose $c >0$ so that for all $\\epsilon >0$ sufficiently small $H(\\overline u_{\\gamma}^\\epsilon) + \\tr(k^\\epsilon) (\\overline u_{\\gamma}^\\epsilon) < - 2\\epsilon$ on $\\text{Cr}_{\\gamma}^\\epsilon$ when $\\gamma$ is a component of $\\partial_-\\Omega$ and such that $H(\\underline u_{\\gamma}^\\epsilon) + \\tr(k^\\epsilon) (\\underline u_{\\gamma}^\\epsilon) > 2\\epsilon$ on $\\text{Cr}_{\\gamma}^\\epsilon$ when $\\gamma$ is a component of $\\partial_+ \\Omega$. Here, $k^\\epsilon := k + c \\epsilon \\chi^\\epsilon$.\n\nGiven $t>0$, the functions $- \\frac{\\epsilon}{t} + \\overline u_{\\gamma}^\\epsilon$ and $\\frac{\\epsilon}{t} + \\underline u_{\\gamma}^\\epsilon$ are, respectively, super and sub solutions of the regularized Jang equation $H(u) + \\tr(k^\\epsilon)(u) = t \\cdot u$ on $\\text{Cr}_{\\gamma}^\\epsilon$. Let $C >0$ be a constant greater than $n \\sup_{\\epsilon \\in (0, 1), x \\in \\Omega} |k^\\epsilon(x)|$. Then $\\frac{C}{t}$ and $- \\frac{C}{t}$ are constant super and sub solutions of this equation. \n\nAs in \\cite[Chapters 3 or 4]{Eichmair:2009-Plateau} one sees that for every $t>0$ there exists a (Perron) solution $u_{t}^\\epsilon \\in \\C^\\infty _{loc} (\\Omega)$ of $H(u_{t}^\\epsilon) + \\tr(k^{\\epsilon}) (u_{t}^\\epsilon) = t \\cdot u_{t}^\\epsilon$ such that $- \\frac{C}{t} \\leq u_{t}^\\epsilon \\leq \\frac{C}{t}$ on $\\Omega$, such that $u_{t}^\\epsilon \\leq - \\frac{\\epsilon}{t} + \\overline u_{\\gamma}^\\epsilon $ on $\\text{Cr}_{\\gamma}^\\epsilon$ for components $\\gamma$ of $\\partial_-\\Omega$, such that $\\frac{\\epsilon}{t} + \\underline u_{\\gamma}^\\epsilon \\leq u_{t}^\\epsilon$ on $\\text{Cr}_{\\gamma}^\\epsilon$ for components $\\gamma$ of $\\partial_+ \\Omega$, and such that $|u_t^\\epsilon (x)| \\leq b_\\Lambda(|x|)$ on $\\{x \\in \\Omega : |x| > \\Lambda\\}$ for some $\\Lambda \\geq 1$ sufficiently large. Here, $b_\\Lambda$ is as in Appendix \\ref{sec:barriers}.\\footnote{\\label{footnote:noncompactdomains} The results in \\cite{Eichmair:2009-Plateau} are for compact domains. One way to obtain $u_t^\\epsilon$ that decays to zero in the asymptotically flat end is as a limit of solutions $u_t^{\\epsilon, r}$ on $\\{x \\in \\Omega : |x| \\leq r\\}$ with zero boundary values on $\\{x \\in \\Omega : |x| = r\\}$ as $1 \\leq r \\to \\infty$. Note that $|u_{t}^{\\epsilon, r}(x)| \\leq b_\\Lambda (|x|)$ on $\\{x \\in \\Omega : \\Lambda < |x| \\leq r\\}$ for a fixed $\\Lambda \\geq 1$ sufficiently large.} \n\nIt follows that if $y$ is an interior point of $\\partial_-\\Omega$, then $\\limsup_{x \\to y, x \\in \\Omega} u_{t}^\\epsilon (x) \\leq - \\frac{\\epsilon}{t}$ and that if $y$ is an interior point of $\\partial_+ \\Omega$, then $\\liminf_{x \\to y, x \\in \\Omega} u_{t}^\\epsilon (x) \\geq \\frac{\\epsilon}{t}$.\n\nThe mean curvature of the graphs of $u_{t}^\\epsilon$ is bounded by $2C$ so that they are $2C$--minimizing (in the language of \\cite{Duzaar-Steffen:1993a}) in $\\Omega \\times \\R$, cf. \\cite[Appendix A]{Eichmair:2009-Plateau}. A standard application of Allard's boundary regularity theorem exactly as in \\cite[Chapter 4]{Eichmair:2009-Plateau}, using that the intersection $\\partial M_- \\cap \\partial M_+$ is transverse, shows that the closure of the graph of $u_{t}^\\epsilon$ in $M \\times (- \\frac{\\epsilon}{t}, \\frac{\\epsilon}{t})$ is a $\\C^{1, \\alpha}$ manifold with boundary $(\\partial M_- \\cap \\partial M_+) \\times (-\\frac{\\epsilon}{t}, \\frac{\\epsilon}{t})$. The $\\C^{1, \\alpha}$ estimates near the boundary depend only on $C$ and the geometry of $\\partial M_- \\cap \\partial M_+$; they are independent of $\\epsilon, t >0$. \n\nWe now pass the graphs of $u_{t}^\\epsilon$ to a geometric subsequential limit as $\\epsilon, t \\searrow 0$. The existence and analysis of such limits is exactly as in \\cite[Chapters 3, 4]{Eichmair:2009-Plateau} (which in turn are largely based on \\cite{Schoen-Yau:1981-pmt2}). If in particular the limit along the subsequence $(t_n, \\epsilon_n) \\to (0, 0)$ were not a graph with the properties asserted in the theorem, there would be some $x \\in \\Omega$ such that the sequence $\\{u_{t_n}^{\\epsilon_n} (x)\\}_{n=1}^\\infty$ is unbounded. For definiteness, let us assume that $u_{t_n}^{\\epsilon_n} (x) \\to - \\infty$, possibly after passing to a further subsequence. There exists a sequence of upward translations of the graphs of $u_{t_n}^{\\epsilon_n}$ that converge, possibly after passing to a further subsequence, locally smoothly as hypersurfaces to a vertical cylinder $\\Sigma \\times \\R$, where $\\Sigma \\subset \\bar \\Omega$ is a smooth properly embedded submanifold with boundary $\\partial_- M \\cap \\partial_+ M$ that encloses a bounded region $\\tilde \\Omega$ with $\\partial \\Omega$ with $x \\in \\tilde \\Omega$, and such that $H_\\Sigma + \\tr_\\Sigma(k) = 0$. Here, the mean curvature is computed with respect to the unit normal pointing out of $\\tilde \\Omega$. \nWe can argue exactly as in \\cite[Proposition 4.1 and Remark 4.1]{Eichmair:2010} that there exists a MOTS in $M_-$ that is homologous to $\\partial M_-$ and which encloses $\\{x\\} \\cup M_-$. (The point is that we can force a blow down of the Jang equation in the complement of the closure of $\\tilde \\Omega$ in $M_-$.) This contradicts the assumption that $\\partial M_-$ is the outermost MOTS. \n\\end{proof}\n\\end{theorem}\n\n\n\\section{Canonical blow up of the Jang equation} \\label{sec:canonical}\n\nIn view of analogous results for Scherk-type minimal and constant mean curvature graphs on bounded domains, it is tempting to conjecture that the solutions of the Jang equation constructed in Proposition \\ref{prop:existenceblowup} and Theorem \\ref{thm:JangScherk} are unique with their properties. In Section \\ref{sec:uniqueness} we prove such a uniqueness result in the special case where $k\\equiv0$. In the case of general second fundamental form $k$, we will show in Theorem \\ref{thm:maximalsolution} below that there exist \\emph{canonical, pointwise maximal} solutions of the Jang equation for example when $M_- \\subset M_+$. The proof of this result proceeds via a curious geometric variant of the Perron method that uses the outermost condition built into the definition of $M_+$ in lieu of a super solution for the problem. The basic ingredients are variations of classical PDE techniques, cf. in particular \\cite{Jenkins-Serrin:1966, Jenkins-Serrin:1968, Serrin:1969, Serrin:1970} and also \\cite{Eichmair:2009-Plateau} and the references therein, and the analysis of geometric limits of the Jang equation developed in \\cite{Schoen-Yau:1981-pmt2}.\n\n\\begin{theorem} \\label{thm:maximalsolution} Let $(M, g, k)$ be a complete asymptotically flat initial data set of dimension $n$, $3\\leq n \\leq 7$, fix an end, and let $M_+ \\subset M$ be the complement of the total inner trapped region of $(M, g, k)$ with respect to that end. Assume that there exists a solution $u : M_+ \\to \\mathbb{R}$ of the Jang equation $\\mc(u) + \\tr(k)(u)=0$ such that $u(x) \\to \\infty$ as $\\dist(x, \\partial M) \\to 0$ and such that $u(x) \\to 0$ as $x \\to \\infty$ in the asymptotically flat end. The pointwise supremum of all such solutions is again a solution with the same properties. \n\\begin{proof}\nThe results in Appendix \\ref{sec:barriers} show that there exists $\\Lambda \\geq 1$ such that $|u(x)| \\leq b_\\Lambda (|x|)$ on $N:= \\{x \\in M^+ : |x| > \\Lambda\\}$ for every solution $u$ of the Jang equation as in the statement of the theorem. \n\nFix a smooth function $u : M_+ \\to \\R$ as in the statement of the theorem. Using translates of $u$ to obtain a priori oscillation bounds\n and standard methods as in \\cite[Lemma 2.2]{Eichmair:2009-Plateau}, one sees that for every $x \\in M_+$ there exists $\\rho^D(x) \\in (0, \\dist(x, \\partial M_+))$ small such that for every $\\rho \\in (0, \\rho^D(x))$ the equation $\\mc(v) + \\tr(k)(v)=0$ on $B_\\rho(x)$ with continuous boundary data on $S_\\rho(x)$ admits a solution $v \\in \\C^{\\infty} (B_\\rho(x)) \\cap \\C^0(\\bar B_\\rho(x))$. The notion of (Perron) sub solutions $\\underline u \\in \\C(M_+)$ and super solutions $\\overline u \\in \\C(M_+)$ of the Jang equation $\\mc(v) + \\tr(k)(v) = 0$ on $M_+$ can thus be defined in the usual way. Consider the class of functions $\\mathcal {S}_u = \\{ \\underline u \\in \\C (M_+) : \\underline u \\text{ is a Perron sub solution of the Jang equation, } \\underline u \\geq u \\text{ on } M_+, \\text{ and } |u(x)| \\leq \\beta_\\Lambda (|x|) \\text{ for all } x \\in M_+ \\text { with } |x| > \\Lambda\\}$. This class is closed under taking pointwise maximum and under lifting $\\underline u \\in \\mathcal{S}_{u}$ to the function $\\hat {\\underline u} \\in \\C(M_+)$ that equals $\\underline u$ on the complement of $B_\\rho(x)$ and equals the solution $v$ of $H(v) + \\tr(k)(v) = 0$ on $B_\\rho(x)$ such that $v = u$ on $S_\\rho(x)$, for every $r \\in (0, \\rho^D(x))$ and every $x \\in M_+$. Note that $u \\in \\mathcal S_u$. The function $u^P : M_+ \\to \\R \\cup \\{\\infty\\}$ is defined pointwise by $u^P (x) := \\sup_{\\underline u \\in \\mathcal{S}_u} \\underline u (x)$. Let $\\Omega := \\{x \\in M_+ : u^P(x) < \\infty\\}$. Note that $N \\subset \\Omega$.\n\nWe claim that $\\Omega$ is open, that $u^P|_\\Omega$ is a smooth solution of the Jang equation, and that $\\lim_{x \\to y, x \\in \\Omega} u^P(x) = \\infty$ for every $y \\in \\partial \\Omega$. To see this, fix $x \\in \\Omega$. Following through the standard proof of the regularity of the Perron solution (cf. e.g. \\cite[p. 25]{Gilbarg-Trudinger:1998}) we see that given $\\rho \\in (0, \\rho^D(x))$, there exist $\\{\\underline u_i\\}_{i=1}^\\infty \\subset \\mathcal S_u$ such that $u \\leq \\underline u_1 \\leq \\underline u_2 \\leq \\ldots \\leq u^P$ on $B_\\rho (x)$, such that $H(\\underline u_i) + \\tr(k)(\\underline u_i) = 0$ on $B_\\rho (x)$ for all $i = 1, 2, \\ldots$, and such that $\\lim_{i \\to \\infty} \\underline u_i (x) = u^P (x) < \\infty$. The analysis of geometric limits of solutions of the Jang equation shows that the geometric limit of the graphs of $\\underline u_i$ in $B_\\frac{\\rho}{2}(x) \\times \\R$ contains the graph of a smooth solution $\\tilde u^{x, r} \\in \\C^\\infty_{loc}(\\Omega_0^{x, \\rho})$ of the Jang equation above some open subset $\\Omega_0^{x, \\rho} \\subset B_\\rho(x)$. Moreover, $\\lim_{i \\to \\infty} \\underline u_i(y) = \\infty$ for all $y \\in B_{\\frac{\\rho}{2}} (x) \\setminus \\Omega_0^{x, \\rho}$, $\\lim_{z \\to y, z \\in \\Omega_0^{x, \\rho}} \\tilde u^{x, \\rho}(z) = \\infty$ for all $y \\in B_{\\frac{\\rho}{2}}(x) \\cap \\partial \\Omega_0^{x, \\rho}$, and $B_{\\frac{\\rho}{2}}(x) \\cap \\partial \\Omega_0^{x, \\rho}$ is a smooth properly embedded MITS in $B_{\\frac{\\rho}{2}}(x)$. (The point is that the functions $\\underline u_i$ are bounded below by $u$ so that there can be no cylindrical components in their geometric limit, cf. the argument in Appendix \\ref{sec:ige} and the properties listed in Step 4 in Subsection \\ref{sec:OmegaNullEmpty}. Note that $\\Omega_0^{x, \\rho}$ might have several components.) Clearly, $\\tilde u^{x, \\rho} \\leq u^P$ on $\\Omega_0^{x, \\rho}$. That $\\tilde u^{x, \\rho} = u^P$ on the connected component of $\\Omega_0^{x, \\rho}$ containing $x$ follows from the strong maximum principle for differences of solutions of the Jang equation, as in the standard proof of the regularity of Perron solutions. Since also $u^P \\geq u$ everywhere on $M_+$, we can deduce all the properties of $\\Omega$ and $u^P$ asserted at the beginning of this paragraph. \n\nThe argument above shows that away from $\\partial M_+$ the boundary of $\\Omega$ is a smooth properly embedded MITS. That the boundary of $\\Omega$ is smooth and embedded up to $\\partial_+ M$ follows from the characterization in Appendix \\ref{sec:boundary} of boundaries of domains that support solutions of prescribed mean curvature equations with infinite boundary data. The definition of $M_+$ implies that $\\Omega = M_+$. Clearly, $u^P$ has all the properties asserted in the conclusion of the theorem. \n\\end{proof}\n\\end{theorem}\n\n\\begin{remark} By taking the least super solution instead of the largest sub solution in the proof of Theorem \\ref{thm:maximalsolution} we obtain an analogous result where $M_+$ is replaced by $M_-$ and blow up to plus infinity at the boundary is replaced by blow down to minus infinity. \n\\end{remark}\n\n\n\\section{Uniqueness of a blow up function $u$ at the outermost minimal surface} \\label{sec:uniqueness}\n\nLet $(M, g)$ be a complete asymptotically flat initial data set of dimension $n$, $2 \\leq n \\leq 7$, with $k \\equiv 0$. Fix one of the ends and let $\\Omega:= M_- = M_+$ be as in the introduction. In this case, $\\partial \\Omega$ is called the horizon of $(M, g)$. Note that $\\partial \\Omega$ is a minimal surface. Let $\\partial_- \\Omega$ and $\\partial_+ \\Omega$ be unions of different components of $\\partial \\Omega$ such that $\\partial \\Omega = \\partial_- \\Omega \\dot \\cup \\partial_+ \\Omega$. \\\\\n\nThe arguments proving Proposition \\ref{prop:existenceblowup} show that there exists a smooth solution $u : \\Omega \\to \\R$ of the minimal surface equation $$\\div \\left( \\frac{D u}{ \\sqrt {1 + |D u|^2}} \\right) = 0$$ such that $\\lim_{x \\to y, x \\in \\Omega} u(x) = \\pm \\infty$ for $y \\in \\partial_{\\pm} \\Omega$, and such that $u(x) \\to 0$ as $|x| \\to \\infty$ in $\\Omega$. In this section we show that there is a unique function $u$ with these properties. The proof is a straightforward adaption to our situation of a general argument due to J. Nitsche \\cite{Nitsche:1965} as applied in e.g. \\cite{Jenkins-Serrin:1966, Jenkins-Serrin:1968, Spruck:1972, Hauswirth-Rosenberg-Spruck:2009} to establish uniqueness of the Scherk-type graphs constructed there. We give the complete argument since the asymptotically flat ends require some care. \\\\\n\nTo see that $u$ is unique under the present assumptions, note first that the results in Appendix \\ref{sec:boundary} show that the divergence of $u$ near $\\partial \\Omega$ is uniform in the distance to the respective components of the boundary, and that the upward and downward solutions of the graph converge geometrically to the vertical cylinders $\\partial_+ \\Omega \\times \\R$ and $\\partial_- \\Omega \\times \\R$ respectively. In particular, \\begin{eqnarray} \\label{eqn:convergencenormal} \\lim_{x \\to y, x \\in \\Omega} \\frac{D u}{\\sqrt {1 + |D u|^2}} (x) = \\mp \\nu(y) \\text{ for } y \\in \\partial_\\pm \\Omega,\\end{eqnarray} where $\\nu$ is the unit normal of $\\partial \\Omega$ pointing into $\\Omega$. Using the argument in Appendix \\ref{sec:barriers} we see that \\begin{eqnarray} \\label{eqn:decay} |u (x)| + |x| |D u(x)| = O(|x|^{2-\\beta}) \\text{ as } |x| \\to \\infty \\text{ in } \\Omega \\end{eqnarray} for every $\\beta \\in (2, n)$. \\\\\n\nSuppose that $v : \\Omega \\to \\R$ is a second solution with these properties. Fix $T \\in (0, \\infty)$ that is a regular value of both $(u-v)$ and $(v-u)$. Using (\\ref{eqn:convergencenormal}) we see that\n\\begin{eqnarray*}\n\\lim _ {s \\searrow 0} \\int_{ \\{x \\in \\Omega : \\dist (x, \\partial \\Omega) = s, |(u - v)(x)| < T\n\\}} (u - v) g \\left( \\frac{D u}{\\sqrt{1 + |D u|^2}} - \\frac{D\nv}{ \\sqrt {1 + |D v|^2}}, \\eta \\right) d \\mathcal{H}^{n-1}= 0. \n\\end{eqnarray*}\nHere, $\\eta$ denotes the unit normal of $\\{x \\in \\Omega : \\dist (x, \\partial \\Omega) =\ns\\}$ pointing towards $\\partial \\Omega$. Using the decay estimates (\\ref{eqn:decay}) for $u$ and $v$ we obtain that \n\\begin{eqnarray*}\n\\lim_{r \\to \\infty} \\int_{\\{x \\in \\Omega : |x| = r, |u - v| < T\\}} (u - v) g \\left(\n\\frac{D u}{\\sqrt{1 + |D u|^2}} - \\frac{D v}{\\sqrt{1 + |D\nv|^2}}, \\eta\\right) d \\mathcal{H}^{n-1}= 0\n\\end{eqnarray*}\nwhere $\\eta$ is the unit normal of $\\{x \\in \\Omega: |x| = r\\}$ pointing towards the\nend. Using the divergence theorem and that $u, v$ satisfy the minimal surface equation, this implies that\n\\begin{equation*}\n0 = \\int_{\\{x \\in \\Omega : |u - v| < T\\}} g\\left(D u - D v, \\frac{D u\n}{\\sqrt{1 + |D u|^2}} - \\frac{D v}{\\sqrt{1 + |D v|^2}}\\right) d\n\\mathcal{L}^n.\n\\end{equation*}\nUsing the strict convexity of the functions $\\xi \\to \\sqrt{1 + g(x)(\\xi, \\xi)}$ on $T_x M$ for all $x \\in M$ we conclude that the integrand is pointwise non-negative with equality at $x \\in \\Omega$ if only if $D u = D v$ at $x$. \nIt follows that $u$ and $v$ can only differ by a constant. Since we assume that they both tend to zero on the asymptotically\nflat end, we obtain that $u = v$, as desired. \\\\\n\nWe see no way to extend this argument to non-zero $k$ at this point and have to contend with the existence of the canonical solution guaranteed under the hypotheses of Theorem \\ref{thm:maximalsolution}. \\\\\n\n\\begin{remark} Let $u : \\Omega \\to \\R$ be as above. It follows that $$\\H^{n-1} (\\partial_+ \\Omega) - \\H^{n-1} (\\partial_-\\Omega) = \\lim_{r \\to \\infty} \\int_{\\{x \\in \\Omega : |x| = r\\}} g (D u, D |x|) d \\H^{n-1}.$$ Thus the unique blow up solution witnesses the area of the horizon at infinity. \\end{remark}\n\n\n\\section{Extensions of the classical Jenkins--Serrin theory} \\label{sec:jss}\n\n\\subsection{Introduction}\n\nThe classical Jenkins--Serrin theory \\cite{Jenkins-Serrin:1966, Jenkins-Serrin:1968} characterizes those bounded domains $\\Omega \\subset \\R^2$ with piecewise smooth boundary for which there exists a solution $u : \\Omega \\to \\R$ of the minimal surface equation such that \\begin{eqnarray*} \\label{eqn:ScherkA} \\lim_{(x, y) \\to (x_0, y_0) \\in \\partial_\\pm\\Omega} u(x, y) = \\pm \\infty \\text{ and } \\\\ \\label{eqn:ScherkB} \\lim_{(x, y) \\to (x_0, y_0) \\in \\partial_0 \\Omega} u(x, y) = \\phi(x_0, y_0). \\end{eqnarray*} Here, the sets $\\partial_0 \\Omega$, $\\partial_- \\Omega$, and $\\partial_+ \\Omega$ are unions of the smooth components of the boundary such that $\\partial \\Omega = \\partial_+ \\Omega \\dot {\\cup} \\partial_- \\Omega \\dot {\\cup} \\partial_0 \\Omega \\dot \\cup \\{\\text{corners}\\}$, and $\\phi \\in \\mathcal{C}(\\partial_0 \\Omega)$ is a given function. A basic example of such a configuration is when $\\Omega = (- \\frac{\\pi}{2}, \\frac{\\pi}{2}) \\times (- \\frac{\\pi}{2}, \\frac{\\pi}{2})$ and $u(x, y) = \\log \\frac{\\cos(x)}{\\cos(y)}$. The graph of $u$ in $\\R^3$ is of course the classical Scherk surface. In general, these domains are precisely those for which the connected components of $\\partial_\\pm \\Omega$ are straight line segments such that no two segments in $\\partial_- \\Omega$ and no two segments in $\\partial_+ \\Omega$ have an endpoint in common, for which the geodesic curvature of $\\partial_0 \\Omega$ is non-negative, and which satisfy the Jenkins--Serrin condition \\cite{Jenkins-Serrin:1966, Jenkins-Serrin:1968}. When $\\partial_0 \\Omega \\neq \\emptyset$ these conditions demand that the circumference of every polygon inscribed in $\\Omega$ whose endpoints are chosen from the finitely many corner points is strictly greater than twice the total length of its sides that coincide with segments in $\\partial_+ \\Omega$ and also greater than twice the total length of its sides that coincide with segments in $\\partial_- \\Omega$. When $\\partial_0 \\Omega = \\emptyset$ the Jenkins--Serrin condition is the same except for the inscribed polygon that is the whole domain; one demands that the length of $\\partial_+ \\Omega$ equals the length of $\\partial_- \\Omega$. \\\\\n\nAn important and influential development of the field was accomplished by J. Spruck \\cite{Spruck:1972}, who has extended the classical Jenkins--Serrin theory to graphs $u : \\Omega \\subset \\R^2 \\to \\R$ of constant mean curvature two. Provided that the piecewise smooth boundary $\\partial \\Omega$ consists of a union $\\partial_+ \\Omega$ of circular arcs of unit radius that are convex towards $\\Omega$, a union of circular arcs $\\partial_- \\Omega$ of unit radius that are concave towards $\\Omega$, and a union of boundary arcs $\\partial_0 \\Omega$ whose geodesic curvature is greater than or equal to $1$, he finds necessary and sufficient conditions for the existence of solutions $u$ assuming (arbitrary) continuous boundary values on $\\partial_0 \\Omega$ and tending to $\\infty$ on approach towards $\\partial_+ \\Omega$ and $- \\infty$ on approach towards $\\partial_- \\Omega$ \\emph{provided} a sub solution $\\underline u : \\Omega^* \\to \\R$ \\begin{eqnarray} \\label{eqn:subsolution} \\div \\left( \\frac{D \\underline u}{\\sqrt{1 + |D \\underline {u}|^2}}\\right) \\geq 2\\end{eqnarray} exists on the domain $\\Omega^*$ obtained from flapping the negatively curved components of the boundary outward. (That this process gives a domain is a further additional assumption in \\cite{Spruck:1972}.) The advantage of the piecewise smooth domain $\\Omega^*$ over $\\Omega$ is that its boundary arcs are all convex, so that solutions of the constant mean curvature equation with prescribed boundary values away from the corners can be constructed on it. \\\\\n\nA further important contribution to the theory is due to U. Massari \\cite{Massari:1977}, who has extended the Jenkins--Serrin theory to arbitrary dimensions and variable (Lipschitz) mean curvature in the case when $\\partial_0 \\Omega \\neq \\emptyset$. His techniques are different from J. Spruck's. In particular, no flapping of the boundary is required. In the special case where the mean curvature is constant, the necessary and sufficient conditions he provides for the existence of a solution are not obviously the same as those given in \\cite{Spruck:1972}. We note that condition $(3.3)$ in \\cite{Massari:1977} stands in, morally and technically, for the requirement (\\ref{eqn:subsolution}) above that a sub solution of the prescribed mean curvature equation exists on the domain. In the book of E. Giusti \\cite[Chapter 16]{Giusti:1984}, an extension of the technique of U. Massari to the ``slightly more complex\" case where $\\partial_0\\Omega = \\emptyset$ is presented in the minimal surface case. \\\\\n\nMore recently, the Jenkins--Serrin theory for minimal graphs has been extended to $\\mathbb{H}^2 \\times \\R$ in \\cite{Nelli-Rosenberg:2002} and to $M \\times \\R$ in \\cite{Pinheiro:2009, Mazet-Rodriguez-Rosenberg:2011} (where $(M, g)$ is a general complete Riemannian surface). The Jenkins--Serrin--Spruck theory for constant mean curvature graphs has been developed for $\\mathbb{H}^2 \\times \\mathbb{R}$ and $\\mathbb{S}^2 \\times \\mathbb{R}$ in \\cite{Hauswirth-Rosenberg-Spruck:2009}. A further important recent development are the results of P. Collin and H. Rosenberg \\cite{Collin-Rosenberg:2010} and A. Folha and S. Melo \\cite{Folha-Melo:2011} who give necessary and sufficient conditions for the existence of Scherk-type minimal and constant mean curvature graphs on ideal polygons (with infinite area) in hyperbolic space. \\\\\n\nAs an example of a particularly interesting application of the construction of Scherk-type graphs on Riemannian surfaces, we mention the surprising construction of harmonic diffeomorphisms between the complex plane and the hyperbolic space by P. Collin and H. Rosenberg \\cite{Collin-Rosenberg:2010}. \\\\\n\nIn this section, we prove an extension of the Jenkins--Serrin--Spruck theory for domains $\\Omega \\subset M$ with $\\partial_0 \\Omega = \\emptyset$ in Riemannian surfaces $(M, g)$ and for general $H_0 \\in [0, \\infty)$. When $H_0 = 0$, our result is marginally different from the corresponding result in \\cite{Pinheiro:2009} because we consider the possibility of closed geodesics in the blow up\/blow down analysis (in Step $4$ below). In the case where $H_0 >0$, neither do we hypothesize the existence of a sub solution on $\\Omega$, nor do we make any additional assumptions regarding the existence of an auxiliary domain as in \\cite{Spruck:1972, Hauswirth-Rosenberg-Spruck:2009}. \\\\\n\nIt will be apparent from our proof that our approach extends (for the most part) verbatim to appropriate domains in Riemannian manifolds of dimension $2 \\leq n \\leq 7$. The description of admissible domains in higher dimensions is cumbersome. Also, the Jenkins--Serrin--Spruck conditions become impossible to verify in examples that are not very symmetric, limiting the application of these results in higher dimension. For this reason, we omit a detailed discussion here. \\\\\n\nIn our proof we work with solutions of the (finite boundary value) Dirichlet problem that will in general not assume any particular boundary data but only have the right asymptotic behavior. This is an important difference with the construction in \\cite{Spruck:1972} where solutions of the Dirichlet problem are constructed on an auxiliary domains $\\Omega^*$ whose boundary is sufficiently convex away from finitely many points to construct solutions that take on prescribed continuous data. The construction of $\\Omega^*$ in \\cite{Spruck:1972} proceeds by flapping the negatively curved boundary components $\\partial_- \\Omega$ outward so they become convex arcs. This construction requires symmetry of the ambient manifold that is not in general available. For this reason, the construction in \\cite{Spruck:1972, Hauswirth-Rosenberg-Spruck:2009} is carried out only in the simply connected space forms of dimension two. \\\\\n\n\n\\subsection {The case where $\\partial_0 \\Omega = \\emptyset$} \\label{sec:OmegaNullEmpty}\n\nLet $(M, g)$ be a complete boundaryless Riemannian surface, let $H_0 \\in [0, \\infty)$, and let $\\Omega \\subsetneq M$ be a connected bounded open set such that $\\partial \\Omega = \\partial \\bar \\Omega$. Here and below, we will use ``$\\partial$\" to denote the topological boundary of a set. We assume that $\\partial \\Omega$ is piecewise smooth and in fact the union of finitely many properly embedded arcs $\\{A_i, B_j\\}$ and properly embedded closed curves $\\{E_k, F_l\\}$ such that the outward geodesic curvature of each arc $A_i$ and closed curve $E_k$ is constant and equal to $H_0$ and such that the outward geodesic curvature of each arc $B_j$ and closed curve $F_l$ is constant and equal to $- H_0$. We assume that all the curves and the interior of all the arcs are pairwise disjoint, and that no two arcs $A_i$ and $A_{i'}$ and no two arcs $B_j$ and $B_{j'}$ have an endpoint in common. \\\\\n\nThe union of the closed curves $\\{E_k\\}$ and the interiors of the positively curved arcs $\\{A_i\\}$ is denoted by $\\partial_+ \\Omega$. The union of the closed curves $\\{F_l\\}$ and the interiors of the negatively curved arcs $\\{B_j\\}$ is denoted by $\\partial_-\\Omega$. The endpoints of the arcs $\\{A_i, B_j\\}$ are called the \\emph{corners} of $\\partial \\Omega$. The assumptions imply (via the strong maximum principle when $H_0 >0$) that any two arcs that share an endpoint meet at a non-zero angle. \\\\\n\nA \\emph{generalized polygon} is a non-empty open subset $P\\subset \\Omega$ with $\\partial P = \\partial \\bar P$ and such that $\\partial P$ is piecewise smooth and consists of finitely many of the following building blocks: \\begin{enumerate}\n\\item Finitely many arcs of constant geodesic curvature $\nH_0$ whose endpoints are amongst the corners of $\\partial\\Omega$ and whose interiors are embedded and pairwise disjoint. We also require that each arc whose interior intersects $\\partial \\Omega$ is one of $\\{A_i, B_j\\}$. \n\\item Pairwise disjoint embedded closed curves of constant geodesic curvature $\\pm H_0$ that either lie entirely in $\\Omega$ or coincide with one of $\\{E_k, F_l\\}$ and which are disjoint from the boundary arcs.\n\\end{enumerate}\n\n\\begin{theorem} \\label{thm:mainjss} Let $(M, g)$ and $\\Omega \\subset M$ be as above. A necessary and sufficient condition for the existence of a smooth function $u: \\Omega \\to \\R$ such that \n\\begin{eqnarray} \\label{eqn:jssgraph}\n\\div \\left (\\frac{D u}{ \\sqrt{1 + |D u|^2}} \\right) = H_0 \\text { with } \\\\ \\nonumber \\lim_{x \\to x_0, x \\in \\Omega} u(x) = \\left \\{ \\begin{array}{ll} \\infty & \\text{if } x_0 \\in \\partial_+ \\Omega \\\\ - \\infty & \\text{if } x_0 \\in \\partial_- \\Omega \\end{array} \\right. \n\\end{eqnarray}\nis that \\begin{eqnarray} \\label{eqn:totalflux} \\H^1_g(\\partial_+ \\Omega) = H_0 \\L^2_g(\\Omega) + \\H^1_g(\\partial_- \\Omega) \\end{eqnarray} and that \\begin{eqnarray} \\label{eqn:partialfluxA} 2 \\H^1_g(\\partial_+ \\Omega \\cap \\partial P) < \\H^1_g(\\partial P) + H_0 \\L^2_g(P) \\text{ and } \\\\ \\label{eqn:partialfluxB} 2 \\H^1_g(\\partial_- \\Omega \\cap \\partial P) < \\H^1_g (\\partial P) - H_0 \\L^2_g(P)\\end{eqnarray} for every generalized polygon $P \\subsetneq \\Omega$. \n\\end{theorem}\n\nThe conditions (\\ref{eqn:totalflux}), (\\ref{eqn:partialfluxA}), (\\ref{eqn:partialfluxB}) appear in the classical work of Jenkins--Serrin (when $H_0 = 0$ and $(M, g)$ is $\\R^2$ with the Euclidean metric) and its generalization due to J. Spruck (when $H_0 >0$ and $(M, g)$ is Euclidean space), and also in the work of L. Hauswirth, H. Rosenberg, and J. Spruck \\cite{Hauswirth-Rosenberg-Spruck:2009} (where $H_0 > 0$ and $(M, g)$ is one of $\\R^2, \\mathbb{S}^2, \\mathbb{H}^2$ with their constant curvature metrics). The necessity of these conditions follows from a standard argument, that we summarize briefly: \\\\\n\nLet $u : \\Omega \\to \\R$ be as in the statement of Theorem \\ref{thm:mainjss}. Let $U \\subset M$ be a non-empty and open subset such that $U \\cap \\partial_\\pm \\Omega = U \\cap \\partial \\Omega$. The discussion in Appendix \\ref{sec:boundary} shows that the graphs of the functions $u \\mp t$ converge as hypersurfaces smoothly on compact subsets of $U \\times \\R$ to $\\partial_\\pm \\Omega \\times \\R$ as $t \\to \\infty$. In particular, as $x \\in \\Omega$ approaches a point $x_0 \\in \\partial_\\pm \\Omega$, the horizontal part of the downward unit normal of these graphs, $X (x):= (1 + |D u|^2)^{-1\/2} Du$, converges to $\\pm$ the outward pointing unit normal of $\\partial_\\pm \\Omega$ at $x_0$. The necessity of condition (\\ref{eqn:totalflux}) follows from applying the divergence theorem to the vector field $X$ on smooth interior approximations of the domain $\\Omega$. The necessity of conditions (\\ref{eqn:partialfluxA}), (\\ref{eqn:partialfluxB}) follows from the same argument applied to generalized polygons $P \\subsetneq \\Omega$, using also that $|X (x)| < 1$ for $x \\in \\partial P \\cap \\Omega$. \\\\\n\n\\begin{remark} A consequence of the the existence of a graph as in (\\ref{eqn:jssgraph}) is that every $\\gamma \\in \\{A_i, B_j, E_k, F_l\\}$ is stable in that $\\int_\\gamma |\\bar D \\psi|^2 \\geq \\int_\\gamma (H_0^2 + \\kappa_g) \\phi^2$ for every $\\psi \\in \\mathcal{C}^1(\\gamma)$ that vanishes on the boundary of $\\gamma$ if $\\gamma \\in \\{A_i, B_j\\}$. Here, $\\bar D$ is the (tangential) gradient and $\\kappa_g$ is half the scalar curvature of $(M, g)$. This implies, for example, that in Euclidean space and when $H_0 = 1$, a domain that satisfies the Jenkins--Serrin--Spruck conditions (\\ref{eqn:totalflux}), (\\ref{eqn:partialfluxA}), (\\ref{eqn:partialfluxB}) must also satisfy $\\H^1_\\delta (A_i), \\H^1_\\delta(B_j) \\leq \\pi$. The condition that $\\H^1_\\delta (B_j) < \\pi$ was part of the assumptions in \\cite[p. 16]{Spruck:1972}. In fact, our proof (see property (\\ref{item10}) in Step 4 below) shows that the conditions (\\ref{eqn:totalflux}), (\\ref{eqn:partialfluxA}), (\\ref{eqn:partialfluxB}) need only be verified for generalized polygons whose boundary arcs and closed curves are stable. This sharpening of the classical Jenkins--Serrin--Spruck condition is useful when constructing examples.\n\\end{remark}\n\n\\begin{remark} The complement of a generalized polygon $P$ in $\\Omega$ is again a generalized polygon. Condition (\\ref{eqn:partialfluxB}) for a generalized polygon $P \\subsetneq \\Omega$ follows from condition (\\ref{eqn:partialfluxA}) applied to $\\Omega \\setminus \\bar P$ in view of (\\ref{eqn:totalflux}).\n\\end{remark}\n\n\n\\noindent {\\bf Step 1: Construction of the auxiliary domain $\\hat \\Omega$}\n\nFix a component $\\gamma$ of $\\partial_\\pm \\Omega$ and a smooth function $\\Theta_\\gamma \\in \\C^\\infty(\\bar \\gamma)$ that is positive on $\\gamma$ and which vanishes on its (manifold) boundary. Let $\\nu$ be the unit normal of $\\gamma$ pointing out of $\\Omega$. The piecewise smooth domain $\\hat \\Omega$ is obtained from $\\bar \\Omega \\setminus \\{\\text{vertices}\\}$ by adding the \\emph{crescents} $\\text{Cr}_\\gamma := \\{ \\exp_\\theta (t \\Theta_\\gamma(\\theta) \\nu (\\theta) ) : t \\in (0, \\epsilon) \\text{ and } \\theta \\in \\text{int}(\\gamma) \\}$ as $\\gamma$ ranges over all components of $\\partial_\\pm \\Omega$. Here, $\\epsilon > 0$ small is chosen so that there are no issues with the regularity of the exponential map and such that the crescents $\\text{Cr}_\\gamma$ are pairwise disjoint. \\\\\n\n\n\\noindent {\\bf Step 2: Construction of barriers on $\\text{Cr}_\\gamma$ and the functions $H_k(x)$} \n\nLet $\\gamma$ be a component of $\\partial_+ \\Omega$. We would like to find solutions that tend to $\\infty$ on approach to $\\gamma$ and hence require a sub solution. The hypersurface $\\{ (\\exp_\\theta (\\epsilon e^h \\Theta_\\gamma (\\theta) \\nu (\\theta)), h) : h \\in (- \\infty, 0) \\text{ and } \\theta \\in \\text{int} (\\gamma)\\}$ of $M \\times \\R$ is the graph of a (locally) smooth function $\\underline u_\\gamma : \\text{Cr}_\\gamma \\to \\R$ whose downward unit normal corresponds to the outward unit normal of the cylinder. \\\\\n\nLet $\\gamma$ be a component of $\\partial_- \\Omega$. We would like to find solutions that tend to $- \\infty$ (and hence require a super solution). As above, the hypersurface $\\{ (\\exp_\\theta (\\epsilon e^{-h} \\Theta_\\gamma (\\theta) \\nu (\\theta)), h) : h \\in (0, \\infty) \\text{ and } \\theta \\in \\text{int} (\\gamma) \\}$ is the vertical graph of a locally smooth function $\\overline u_\\gamma : \\text{Cr}_\\gamma \\to \\R$.\\\\\n\nThe function $H : \\hat \\Omega \\to \\R$ defined as \n\\begin{eqnarray*}\nH(x) = \\left \\{ \\begin{array}{lll} H_0 & \\text{if } x \\in \\bar \\Omega \\setminus \\{ \\text{corners}\\}, \\\\ \\div \\left( \\frac{D \\overline u_\\gamma }{\\sqrt{1 + |D \\overline u_\\gamma|^2}}\\right) (x) & \\text{if } x \\in \\text{Cr}_\\gamma \\text{ if } \\gamma \\text{ is a component of } \\partial_- \\Omega, \\text{ and}\\\\ \\div \\left( \\frac{D \\underline u_\\gamma }{\\sqrt{1 + |D \\underline u_\\gamma|^2}}\\right) (x) & \\text{if } x \\in \\text{Cr}_\\gamma \\text{ if } \\gamma \\text{ is a component of } \\partial_+ \\Omega \\end{array} \\right. \n\\end{eqnarray*}\nis locally Lipschitz. Moreover, $H(x) = H_0 + O(\\epsilon)$ uniformly on $\\hat \\Omega$. \\\\\n\nLet $\\chi : \\hat \\Omega \\to [-1, 1]$ be a locally smooth function such that $\\chi \\equiv \\pm 1$ near $\\text{Cr}_\\gamma$ for $\\gamma \\in \\partial_\\pm \\Omega$. Let $k \\geq 1$. We define a locally Lipschitz function $H_k$ on $\\hat \\Omega$ by $H_k (x) := H (x) - k^{-1\/2} \\chi (x)$. Note that $\\sqrt{k} + \\underline u_\\gamma$ is a sub solution of the equation \n\\begin{eqnarray} \\label{eqn:regularized} \n\\div \\left( \\frac{D u}{\\sqrt{1 + |D u|^2}} \\right)= H_k + \\frac{1}{k} \\cdot u\n\\end{eqnarray} \non $\\text{Cr}_\\gamma$ when $\\gamma$ is a component of $\\partial_+ \\Omega$, and that $- \\sqrt{k} + \\overline u_\\gamma$ is a super solution for this equation on $\\text{Cr}_\\gamma$ when $\\gamma$ is a component of $\\partial_- \\Omega$. \\\\\n\nFix a constant $C > \\sup_{k \\geq 1, x \\in \\hat \\Omega} |H_k(x)|$. Then $- C k$ and $Ck$ are, respectively, sub and super solutions for (\\ref{eqn:regularized}) on $\\hat \\Omega$. The introduction of the capillarity regularization (\\ref{eqn:regularized}) of the prescribed mean curvature equation so that large constants become barriers is exactly as in \\cite{Schoen-Yau:1981-pmt2}. \\\\\n\n\n\\noindent {\\bf Step 3: The construction of $u_k$}\n\nLet $u_k \\in \\C^{2, \\alpha}_{loc} (\\hat \\Omega)$ be the largest (Perron) sub solution of equation (\\ref{eqn:regularized})\nthat lies below $C k$ on all of $\\hat \\Omega$ and below $- \\sqrt{k} + \\overline u_\\gamma$ on all crescents $\\text{Cr}_\\gamma$ corresponding to components $\\gamma$ of $\\partial_- \\Omega$. To justify the existence of such a solution, we refer to \\cite[p. 375]{Serrin:1970}, the interior gradient estimate stated in Appendix \\ref{sec:ige}, and also \\cite[Theorems 1.1 and 1.4]{Spruck:2007}, \\cite[Chapter 16]{Gilbarg-Trudinger:1998}, and \\cite[Chapter 3]{Eichmair:2009-Plateau}. The maximum principle implies that $u_k$ lies above $- C k$ on all of $\\hat \\Omega$ and above $\\sqrt{k} + \\underline{u}_\\gamma$ on all crescents $\\text{Cr}_\\gamma$ corresponding to components $\\gamma$ of $\\partial_+ \\Omega$. From (\\ref{eqn:regularized}) we see that the mean curvature of the graph of $u_k$ is bounded uniformly by $2C$ on $\\hat \\Omega$. \\\\\n\n\n\\noindent {\\bf Step 4: Geometric limits of $\\graph(u_k)$}\n\nWe claim that there is a subsequence $\\{u_{k_i}\\}$ of $\\{u_k\\}$ and there exist disjoint open subsets $\\Omega_0, \\Omega_+, \\Omega_-$ of $\\hat \\Omega$ and $u \\in \\mathcal{C}^{2, \\alpha}_{loc} (\\Omega_0)$ with the following properties: \n\n\\begin{enumerate} [(a)]\n\\item \\label{item0} $\\hat \\Omega = (\\overline \\Omega_0 \\cup \\overline {\\Omega}_- \\cup \\overline{\\Omega}_+) \\cap \\hat \\Omega$. In particular, the topological boundaries $\\partial \\Omega_0$, $\\partial \\Omega_-$, $\\partial \\Omega_+$ of $\\Omega_0$, $\\Omega_-$, $\\Omega_+$ locally separate $\\Omega_0$, $\\Omega_-$, $\\Omega_+$ from their respective complements $\\hat \\Omega \\setminus \\Omega_0$, $\\hat \\Omega \\setminus \\Omega_-$, $\\hat \\Omega \\setminus \\Omega_+$ in $\\hat \\Omega$. Moreover, $\\partial \\Omega_0 \\cap \\hat \\Omega$, $\\partial \\Omega_- \\cap \\hat \\Omega$, and $\\partial \\Omega_+ \\cap \\hat \\Omega$ are properly embedded $\\C^{2, \\alpha}$ hypersurfaces in $\\hat \\Omega$. \n\\item \\label{item1} For every $x \\in \\Omega_+$ there exists an open neighborhood of $x$ in $\\Omega_+$ so that $u_{k_i} (y)$ exceeds a given constant for all $y$ in this neighborhood, provided $i$ is sufficiently large. Put differently, $u_{k_i}$ diverges to plus infinity locally uniformly on $\\Omega_+$. \n\\item For every $x \\in \\Omega_-$ there exists an open neighborhood of $x$ in $\\Omega_-$ so that $u_{k_i} (y)$ lies below a given constant for all points $y$ in this neighborhood, provided $i$ is sufficiently large. Put differently, $u_{k_i}$ diverges to minus infinity locally uniformly on $\\Omega_-$. \n\\item We have that $u_{k_i} \\to u$ in $\\mathcal{C}^{2, \\alpha}_{loc}(\\Omega_0)$. In particular, $$\\div \\left(\\frac{D u}{ \\sqrt{1 + |D u|^2}}\\right) = H \\text{ on } \\Omega_0.$$ \n\\item \\label{item2} $\\text{Cr}_\\gamma \\subset \\Omega_\\pm$ when $\\gamma$ is a component of $\\partial_\\pm \\Omega$. \n\\item The sets $\\partial \\Omega_0 \\cap (\\hat \\Omega \\cup \\{\\text{corners}\\})$, $\\partial \\Omega_- \\cap (\\hat \\Omega \\cup \\{\\text{corners}\\})$, and $\\partial \\Omega_- \\cap (\\hat \\Omega \\cup \\{\\text{corners}\\})$ consist of finitely many arcs and closed curves of constant geodesic curvature in $\\overline {\\Omega}$. These arcs and closed curves are pairwise disjoint in $\\Omega$. The arcs are properly immersed and embedded in $\\Omega$. Their endpoints are corners of $\\Omega$, and the endpoints of any one arc may coincide. The closed curves are contained in $\\Omega$ and they are properly embedded. \n\\item \\label{item11} If $\\gamma$ is a component of $\\partial \\Omega_\\pm \\cap \\partial \\Omega_0$, then $\\lim_{x \\in \\Omega_0, x \\to x_0} u (x) = \\pm \\infty$ uniformly near $x_0 \\in \\text{int} (\\gamma)$. \n\\item \\label{item5} The geodesic curvature of a component $\\gamma$ of $\\partial \\Omega_0 \\cap \\partial \\Omega_+$ is constant and equal to $H_0$ when we orient $\\gamma$ by the unit normal $\\nu$ pointing into $\\Omega_+$. Every divergent series of downward translations of the hypersurface $\\text{graph} (u) = \\{(x, u(x)) : x \\in \\Omega_0\\}$ converges to $(\\partial \\Omega_0 \\cap \\partial \\Omega_+)\\times \\R$ in $\\C^{2, \\alpha}$ on compact subsets of $\\hat \\Omega \\times \\R$. \n\\item \\label{item6} The geodesic curvature of a component $\\gamma$ of $\\partial \\Omega_0 \\cap \\partial \\Omega_-$ is constant and equal to $H_0$ when we orient $\\gamma$ by the unit normal $\\nu$ pointing into $\\Omega_0$. Every divergent series of upward translations of the hypersurface $\\text{graph} (u) = \\{(x, u(x)) : x \\in \\Omega_0\\}$ converges to $(\\partial \\Omega_0 \\cap \\partial \\Omega_-)\\times \\R$ in $\\C^{2, \\alpha}$ on compact subsets of $\\hat \\Omega \\times \\R$. \n\\item \\label{item7} The geodesic curvature of a component $\\gamma$ of $\\partial \\Omega_- \\cap \\partial \\Omega_+$ is constant and equal to $H_0$ when we orient $\\gamma$ by the unit normal $\\nu$ pointing into $\\Omega_+$. There exists a cylindrical neighborhood of $\\gamma \\times \\R$ in $\\hat \\Omega \\times \\R$ in which the graphs $\\{(x, u_{k_i} (x)) : x \\in \\hat \\Omega\\}$ converge in $\\C^{2, \\alpha}$ to $\\gamma \\times \\R$ on compact subsets.\n\\item \\label{item16} The graphs $\\{(x, u_{k_i} (x)) : x \\in \\hat \\Omega\\}$ converge as embedded $\\C^{2, \\alpha}$ hypersurfaces on compact subsets of $\\hat \\Omega \\times \\R$ to the union of the cylinders $(\\Omega \\cap \\partial_0 \\Omega)\\times \\R$, $(\\partial_- \\Omega \\cap \\partial_+ \\Omega) \\times \\R$ and the graph $\\{(x, u(x)) : x \\in \\Omega_0\\}$, as $i \\to \\infty$. \n\\item \\label{item9} The vector fields $\\frac{D u_k}{\\sqrt {1 + |D u_k|^2}}$ are locally equicontinuous in $\\hat \\Omega$. \n\\item \\label{item8} With $\\gamma$ and $\\nu$ as in (\\ref{item5}) - (\\ref{item7}) we have that $$\\lim_{i \\to \\infty} \\int_\\gamma \\frac {g(\\nu, D u_{k_i})} {\\sqrt{1 + |D u_{k_i}|^2}} d \\H^1_g = \\H^1_g (\\gamma).$$ In fact, the integrand on the left converges to $1$ locally uniformly on $\\text{int}(\\gamma)$.\n\\item \\label{item10} The arcs and closed curves $\\gamma$ in (\\ref{item5}), (\\ref{item6}), and those in (\\ref{item7}) interior to $\\Omega$ are stable in the sense that \n\\begin{eqnarray*} \n \\int_\\gamma (H^2_0 + \\kappa _g) \\psi^2 d \\H^1_g \\leq \\int_\\gamma |\\bar D \\psi|^2 d \\H^1_g \\text{ for all } \\psi \\in \\C^1(\\gamma) \\text{ with } \\text{supp} (\\phi) \\subset \\text{int}(\\gamma).\n\\end{eqnarray*}\nHere, $\\kappa_g$ is half the scalar curvature of $(M, g)$ and $\\bar D$ is the (tangential) gradient of $\\psi$. \n\\end{enumerate}\nThe properties listed here extend classical results about limits of monotone sequences in the Jenkins--Serrin--Spruck theory, cf. \\cite{Jenkins-Serrin:1966, Jenkins-Serrin:1968, Spruck:1972, Hauswirth-Rosenberg-Spruck:2009, Pinheiro:2009}. For limits of general (i.e. not necessarily monotone) sequences, some of these properties can be inferred directly from the results of L. Mazet \\cite{Mazet:2007}. The ideas in \\cite{Mazet:2007} have been employed to prove Jenkins-Serrin-Spruck type results for minimal graphs supported on domains in Riemannian surfaces in \\cite{Mazet-Rodriguez-Rosenberg:2011} and for constant mean curvature graphs in \\cite{Folha-Melo:2011}. \n\nThe above properties are also variations of classical results on generalized solutions of the minimal surface equation \\cite{Massari-Miranda:1984} or the geometric theory of Jang equation \\cite{Schoen-Yau:1981-pmt2}. Below, we supply a few comments on the proofs of these properties to assist the reader. The proofs extend to domains of dimension $n \\leq 7$. \\\\\n\nThe particular barriers used in the construction of $u_k$ in Step 4 imply (\\ref{item2}). \\\\\n\nProperties (\\ref{item0}) - (\\ref{item8}) can be deduced from the compactness and regularity properties of almost minimizing boundaries, of which graphs of bounded mean curvature are a basic example, along with the geometric Harnack principle (as in Appendix \\ref{sec:ige}), which ensures that geometric limits of our graphs are made up of graphical and cylindrical components. For limits of minimal graphs, this is the approach of \\cite{Massari-Miranda:1984}. For the case of geometric limits of solutions of the regularized Jang equation (including the capillarity regularization), this approach has been worked out in detail in \\cite{Eichmair:2009-Plateau}, to which we refer the reader for more general statements statements and further references. \\\\\n\nProperty (\\ref{item9}) follows from Lemma \\ref{lem:equicontinuity} in Appendix \\ref{sec:equicontinuous}. \\\\\n\nThe argument leading to (\\ref{item10}) is very similar to that in Appendix \\ref{sec:boundary}. The Jacobi identity (\\ref{Jacobiidentity}) is replaced by the differential inequality \n\\begin{eqnarray*} \\Delta_{G_k} \\nu^{3}_k + (|h_k|^2 + \\Ric_{g + (dx^3)^2}(\\nu_k, \\nu_k)) \\nu^{3}_k = - g \\left( \\frac{ D u_k } {{\\sqrt{1 + |D u_k|^2}}}, D (H_k + \\frac{1}{k} \\cdot u_k)\\right) \\nu^3_k \\leq k^{-1\/2} |D \\chi| \\nu^3_k \\text{ on } \\Omega\n\\end{eqnarray*}\nwhere all geometric quantities on the left are computed for the graph $G_k := \\{(x, u_k(x)) : x \\in \\Omega\\}$. The additional contribution to the stability inequality (\\ref{eqn:stabilityG}) disappears when we take geometric subsequential limits of $G_k$ and its vertical translates as $n \\to \\infty$. Since the arcs $\\gamma$ for which (\\ref{item10}) is asserted appear as cross-sections of vertical cylinders that appear in such limits, and because $|h|^2 + \\Ric_{g+(dx^3)^2} (\\nu, \\nu)$ reduces to $H_0^2 + \\kappa_g$ on such cross-sections, we are done. \\\\\n\n\n\\noindent {\\bf Step 4: Analysis of the limit using the Jenkins--Serrin--Spruck conditions}\n\nThe analysis of the geometric limit of the graphs of the solutions $u_k : \\Omega \\to \\R$ using the Jenkins--Serrin--Spruck condition here shares structural features with the analysis in \\cite{Spruck:1972} (see in particular Sections 5 and 6 therein) or \\cite[Section 7]{Hauswirth-Rosenberg-Spruck:2009}. Because of the absence of a sub solution in particular, our main technical step, Case b below, is quite different. \\\\\n\nThe properties listed in Step 4 show that the components $P$ of $\\Omega_0 \\cap \\Omega$, $\\Omega_+\\cap \\Omega$, and $\\Omega_-\\cap \\Omega$ are generalized polygons in $\\Omega$. If $P$ is a component of $\\Omega_- \\cap \\Omega$, then \n\\begin{eqnarray} \\label{eqn:flux-}\n H_0 \\L^2_g(P) + \\limsup_{i \\to \\infty} \\int_P \\frac{u_{k_i}}{k_i} d\\L^2_g \\geq \\\\ \\H^1_g (\\partial P \\cap \\partial_+ \\Omega) + \\H^1_g (\\partial P \\cap \\Omega) + \\liminf_{i \\to \\infty} \\int_{\\partial P \\cap \\partial_- \\Omega} \\frac {g(\\nu, D u_{k_i})} {\\sqrt{1 + |D u_{k_i}|^2}}d \\H^1_g. \\nonumber\n\\end{eqnarray}\nHere, $\\nu$ is the unit normal pointing out of $\\Omega$. Note that the second term on the left is always non-positive. Similarly, if $P$ is a component of $\\Omega_+ \\cap \\Omega$, then \n\\begin{eqnarray} \\label{eqn:flux+}\n H_0 \\L^2_g(P) + \\liminf_{i \\to \\infty} \\int_P \\frac{u_{k_i}}{k_i} d \\L^2_g \\leq \\\\ - \\H^1_g (\\partial P \\cap \\partial_- \\Omega) - \\H^1_g (\\partial P \\cap \\Omega) + \\limsup_{i \\to \\infty} \\int_{\\partial P \\cap \\partial_+ \\Omega} \\frac {g(\\nu, D u_{k_i})} {\\sqrt{1 + |D u_{k_i}|^2}}d \\H^1_g, \\nonumber\n\\end{eqnarray}\nwhere again the unit normal $\\nu$ points out of $\\Omega$. The second term on the left is always non-negative. \\\\\n\nWe analyze the following cases: \\\\\n\n\\noindent {\\bf Case a: $\\emptyset \\neq \\Omega_- \\cap \\Omega \\subsetneq \\Omega$}\n\nLet $P$ be a component of $\\Omega_- \\cap \\Omega$. The Jenkins--Serrin--Spruck condition for $P$ implies that \n\\begin{eqnarray} \\nonumber\n H_0 \\L^2_g(P) < \\H^1_g (\\partial P \\cap \\partial_+ \\Omega) + \\H^1_g (\\partial P \\cap \\Omega) - \\H^1_g (\\partial P \\cap \\partial_- \\Omega)\n\\end{eqnarray} \nwhich leads to an immediate contradiction with (\\ref{eqn:flux-}), since clearly $$\\left| \\int_{\\partial P \\cap \\partial_- \\Omega} \\frac {g(\\nu, D u_{k_i})} {\\sqrt{1 + |D u_{k_i}|^2}} d \\H^1_g \\right| \\leq \\H^1_g(\\partial P \\cap \\partial_- \\Omega).$$ Thus Case a cannot occur. \\\\\n\n\\noindent {\\bf Case a': $\\emptyset \\neq \\Omega_+ \\cap \\Omega \\subsetneq \\Omega$} \n\nLet $P$ be a component of $\\Omega_+ \\cap \\Omega$. The Jenkins--Serrin--Spruck condition for $P$ implies that \n\\begin{eqnarray} \\nonumber\nH_0 \\L^2_g(P) > \\H^1_g (\\partial P \\cap \\partial_+ \\Omega) - \\H^1_g (\\partial P \\cap \\Omega) - \\H^1_g (\\partial P \\cap \\partial_- \\Omega)\n\\end{eqnarray} \nwhich leads to an immediate contradiction with (\\ref{eqn:flux+}), since clearly $$\\left| \\int_{\\partial P \\cap \\partial_+ \\Omega} \\frac {g(\\nu, D u_{k_i})} {\\sqrt{1 + |D u_{k_i}|^2}} d \\H^1_g \\right| \\leq \\H^1_g(\\partial P \\cap \\partial_+ \\Omega).$$ Thus Case a' cannot occur. \\\\\n\n\\noindent {\\bf Case b: $\\Omega \\subset \\Omega_-$}\n\nThe Jenkins--Serrin--Spruck condition for $P = \\Omega_- \\cap \\Omega = \\Omega$ implies that \n\\begin{eqnarray} \\nonumber\n H_0 \\L^2_g(\\Omega) = \\H^1_g (\\partial_+ \\Omega) - \\H^1_g (\\partial_- \\Omega). \n\\end{eqnarray}\nIn conjunction with (\\ref{eqn:flux-}) we conclude that $$\\limsup_{i \\to \\infty} \\int_P \\frac{u_{k_i}}{ {k_i}}d \\L^2_g = 0 \\text{ and that } \\liminf_{i \\to \\infty} \\int_{\\partial_- \\Omega} \\frac {g(\\nu, D u_{k_i})} {\\sqrt{1 + |D u_{k_i}|^2}} d \\H^1_g = - \\H^1_g (\\partial_- \\Omega).$$ Passing to a further subsequence, if necessary, we see that \n\\begin{eqnarray} \\label{eqn:conclusion1Caseb}\n\\lim_{i \\to \\infty} \\L^2_g (\\{ x \\in \\Omega : \\frac{u_{k_i}}{{k_i}} < - \\epsilon\\}) = 0 \\text{ for every } \\epsilon > 0\n\\end{eqnarray}\nand, using (\\ref{item9}), that \n\\begin{eqnarray} \\label {eqn:conclusion2Caseb}\n\\lim_{i \\to \\infty} \\frac {g(\\nu, D u_{k_i})} {\\sqrt{1 + |D u_{k_i}|^2}} = -1 \\text{ locally uniformly on } \\partial_- \\Omega\n\\end{eqnarray} \nwhere $\\nu$ is the unit normal pointing out of $\\Omega$. \\\\\n\nFix a component $\\gamma$ of $\\partial_-\\Omega$ and let $z \\in \\gamma$. It follows from the assumptions that $z \\in \\Omega_-$. Consider the functions $\\tilde u_{k_i} (x) := u_{k_i} (x) - u_{k_i} (z)$. (This is an upward translation for $k_i$ large.) Then\n$$\\div \\left( \\frac{D \\tilde u_{k_i}}{\\sqrt{1 + |D \\tilde u_{k_i}|^2}} \\right)= H_{k_i} + \\frac{\\tilde u_{k_i}}{k_i} + \\frac{u_{k_i}(z)}{k_i}.$$ Using that $ |u_{k_i} (x) | \\leq C k_i$ for all $x \\in \\hat \\Omega$ we see that the mean curvature of these graphs is uniformly bounded. We pass to a further subsequence so that $k_i^{-1} u_{k_i} (z)$ converges to a constant $c \\in [- C, 0]$. We pass to a further subsequence so that the graphs of the $\\tilde u_{k_i}$ converge geometrically in $\\C^{2, \\alpha}$ to a union of properly embedded graphs and cylinders on compact subsets in $\\hat \\Omega \\times \\R$. The mean curvature of these graphs and cylinders at a point $(x, x^{3}) \\in \\hat \\Omega \\times \\R$ in the geometric limit is $H(x) + c$. The point $(z, 0) \\in \\gamma \\times \\R$ is contained in the geometric limit. Using (\\ref{eqn:conclusion2Caseb}) we see that $(z, 0)$ is contained in a cylindrical component $\\tilde \\gamma \\times \\R$ of the limit, where $\\tilde \\gamma \\subset \\hat \\Omega$ is a properly embedded curve whose mean curvature at $x \\in \\gamma$ is given by $H(x) + c$. Moreover, the tangent spaces of $\\gamma$ and $\\tilde \\gamma$ agree together with their orientation at any point of $\\gamma \\cap \\tilde \\gamma$. \\\\\n\nWe claim that $\\gamma = \\tilde \\gamma$. In particular, $c = 0$. To see this, we distinguish two cases. \\\\\n\nFirst, assume that $\\tilde \\gamma \\subset \\overline {\\text{Cr}}_\\gamma$. Then the assertion is a consequence of the maximum principle. (We use that $c \\leq 0$ here.) \\\\\n\nSecond, assume that $\\tilde \\gamma \\cap \\Omega \\neq \\emptyset$. Recall that every $y \\in \\gamma$ has an open neighborhood in $\\hat \\Omega$ that is separated by $\\gamma$ into two components such that $\\tilde u_{k_i}$ tends to plus infinity locally uniformly in one component as $k \\to \\infty$, and such that $\\tilde u_{k_i}$ tends to minus infinity locally uniformly in the other component. Since $\\tilde \\gamma \\cap \\Omega \\neq \\emptyset$, we conclude that $\\{x \\in \\Omega : \\limsup_{i \\to \\infty} k_i^{-1 } \\tilde u_{k_i}(x) \\leq 0\\} = \\{x \\in \\Omega : \\limsup_{i \\to \\infty} k_i^{-1} u_{k_i}(x) \\leq c\\}$ contains a non-empty open subset. In conjunction with (\\ref{eqn:conclusion1Caseb}) we conclude that $c=0$. It follows that in this case, $\\gamma$ and $\\tilde \\gamma$ satisfy the same geometric equation. Further, we know that they intersect non-trivially, and that at any point of intersection they intersect tangentially with the same orientation. The Hopf boundary point lemma shows that $\\gamma = \\tilde \\gamma$, which contradicts the assumption that $\\tilde \\gamma \\cap \\Omega \\neq \\emptyset$. Hence $\\gamma = \\tilde \\gamma$ is the only possibility.\\\\\n\nThe argument in the preceding paragraph shows that there exists a relatively open neighborhood $U_\\gamma$ of $\\gamma$ in $\\hat \\Omega$ that is disjoint from the crescents corresponding to the positively curved boundary components such that $\\tilde u_{k_i}$ converges to $- \\infty$ locally uniformly in $U_\\gamma \\cap \\text{Cr}_\\gamma$, and to $\\infty$ in $U_\\gamma \\cap \\Omega$. \\\\\n\nWe can repeat the above reasoning for any of the components $\\gamma_1, \\ldots, \\gamma_m$ of $\\partial_- \\Omega$, choosing a point $z_i \\in \\gamma_i$ for each component. Passing to a further subsequence and relabeling if necessary, we may assume that $u_{k_i} (z) \\geq u_{k_i} (z_i)$ for all $i \\in \\{1, \\ldots, m\\}$ where $z = z_1$. Also, we may pick $y \\in \\Omega$ near $\\gamma_1$ so that $0 \\geq \\frac{u_{k_i} (y)}{k_i} \\geq \\frac{u_{k_i} (z)}{k_i} \\to 0$. Finally, let $\\hat u_{k_i} := u_{k_i} - u_{k_i} (y)$. For $k$ large, this is an upward translation. Then \n$$\\div \\left( \\frac{D \\hat u_{k_i}}{\\sqrt{1 + |D \\hat u_{k_i}|^2}} \\right)= H_{k_i} + \\frac{\\hat u_{k_i}}{k_i} + \\frac{u_{k_i}(y)}{k_i}.$$\nFor the remainder of the argument, we replace $\\text{Cr}_\\gamma$ by $U_\\gamma \\cap \\text{Cr}_\\gamma$ (keeping the same notation) for all components $\\gamma$ of $\\partial_- \\Omega$. This may shrink the auxiliary domain $\\hat \\Omega$ slightly. However, we gain that $\\tilde u_{k_i} (x) \\to - \\infty$ locally uniformly in these new crescents $\\text{Cr}_\\gamma$. Note that $\\tilde u_{k_i} \\to \\infty$ locally uniformly in $\\text{Cr}_\\gamma$ when $\\gamma$ is a component of $\\partial_+\\Omega$. We can take a subsequential geometric limit of the graphs of $\\tilde u_{k_i}$ just as we did for the original sequence $u_{k_i}$ so that (\\ref{item0}) - (\\ref{item9}) continue to hold. We use $\\hat \\Omega_0, \\hat \\Omega_-, \\hat \\Omega_+$ instead of $\\Omega_0, \\Omega_-, \\Omega_+$ to avoid confusion. As before, the components of $\\Omega \\cap \\hat \\Omega_0$ and $\\Omega \\cap \\hat \\Omega_\\pm$ are generalized polygons. We claim that $\\Omega \\subset \\hat \\Omega_0$. To see this, note that $(y, 0)$ is contained in the geometric limit. This implies that $y \\in \\partial \\hat \\Omega_- \\cup \\partial \\hat \\Omega_+ \\cup \\hat \\Omega_0$ so that $\\Omega \\subset \\hat \\Omega_\\pm$ is impossible. The cases $\\emptyset \\neq \\Omega \\cap \\hat \\Omega_\\pm \\subsetneq \\Omega$ can be ruled out exactly as in Cases a and a' above. It follows that $\\Omega = \\hat \\Omega_0$, and we can conclude as in Case c below. \\\\\n\n\\noindent {\\bf Case b': $\\Omega \\subset \\Omega_+$}\n\nExactly as in Case b, we can conclude that a sequence of downward translations of $u_{k_i}$ will converge to a solution of the original problem. We point out that in fact, the analysis can be shortened considerably in this case because large constants are super solutions of the equation. \\\\\n\n\\noindent {\\bf Case c: $\\Omega_0 = \\Omega$} \n \nIn this case, the solutions $u_{k_i}$ converge to the sought-after solution $u : \\Omega \\to \\R$ by (\\ref{item2}) and (\\ref{item11}). \n\n\\begin{remark} Let $(M, g)$ be a complete Riemannian manifold of dimension $n$ with $2 \\leq n \\leq 7$. Let $H \\in \\C^\\infty(M)$ and let $\\Omega \\subset M$ be a bounded domain whose boundary can be written as the disjoint union of hypersurfaces $\\partial_- \\Omega$ and $\\partial_+ \\Omega$ such that $H_{\\partial_- \\Omega} (x)> H(x)$ with respect to the unit normal pointing into $\\Omega$ and such that $H_{\\partial_+ \\Omega}(x) < H (x)$ with respect to the unit normal pointing out of $\\Omega$. There exists an open subset $U \\subset \\Omega$ containing a neighborhood of $\\partial_- \\Omega$ whose boundary in $\\Omega$ is a smooth hypersurface $\\Sigma$ whose mean curvature at $x \\in \\Sigma$ with respect to the unit normal pointing out of $U$ equals $H(x)$. Such a set $U$ can be found by minimizing the brane functional \n\\begin{eqnarray} \\label{eqn:brane} U \\to \\H^{n-1}_g (\\Omega \\cap \\partial^*U) - \\int_U H (x) d \\L^n_g (x).\\end{eqnarray}\nThis was proven by M. Fuchs in \\cite[Theorems 2.1 and 4.1]{Fuchs:1991}. The existence of a hypersurface with prescribed mean curvature $H$ also follows from the non-variational approach in \\cite{Andersson-Metzger:2009, Eichmair:2009-Plateau}; simply note that such surfaces are MOTSs in the initial data set $(M, g, k:= - \\frac{H}{n-1} g)$. The proofs in \\cite{Andersson-Metzger:2009, Eichmair:2009-Plateau} proceed by constructing a limit of solutions of regularized Jang equations whose boundary values diverge to plus and minus infinity near $\\partial_+ \\Omega$ and $\\partial_- \\Omega$ respectively. The observation we have exploited in the proof of Theorem \\ref{thm:mainjss} that the capillarity term in the regularized Jang equation contributes ``with a good sign\" in the flux integrals (\\ref{eqn:flux-}) and (\\ref{eqn:flux+}) can be used to show that the boundaries of prescribed mean curvature arising in this way also minimize (\\ref{eqn:brane}). Cf. \\cite[Remark 3.2]{Eichmair:2009-Plateau}.\n\\end{remark}\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nJosephson junctions are physical devices of prominent, wide spread scientific and practical use. Moreover, these can be used in testing the fundamentals of quantum mechanics and in studies for the many faces of chaotic complexity in classical physics. Scientists exploit them for a multitude of diverse theoretical and experimental studies. Applications in physics, electronics and other branches of engineering are well established: magnetometers, SQUIDs, superconducting qubits, RSFQ (rapid single flux quantum) circuitry - all use a Josephson junction as a primary building block. Here, we engineer a SQUID-device which is composed of three Josephson junctions and behaves as a physical ratchet system, i.e. a periodic structure which exhibits reflection-symmetry breaking \\cite{rmphm,astumianPT,annalenhmn,reimann}.\n\nA similar system was analyzed in Ref. \\cite{zapata1996prl} for the over-damped case of the resistively shunted Josephson junctions. Here, we extend the study to include inertial effects by accounting for a finite capacitance (mass). This therefore leads to a modeling of the capacitively and resistively shunted case. In terms of classical mechanics, the former corresponds to the over-damped Brownian motion dynamics while the latter includes both finite dissipation and observable inertial effects. This extension is non-trivial because in the latter case the system allows for classical chaos. When the SQUID is driven by both a time-periodic and a constant current, it exhibits anomalous transport behavior including an absolute negative conductance in the linear response regime and negative static resistance in the nonlinear response regime.\n\nThis paper is organized as follows. In Sec. II we describe the circuit with three Josephson junctions and derive an equation which governs the dynamics of the studied system. Sec. III contains a detailed analysis of the deterministic transport processes occurring in our working model. In Sec. IV we study the role of thermal noise on the dynamics of the system. In Sec. V we seek the regime for which the ratchet effect arising in the device is most pronounced. In Sec. VI we propose the method of controlling the voltage direction by the external magnetic flux. Last but not least, Sec. VII provides a summary and conclusions. In the Appendix we derive an expression for the voltage across the SQUID.\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.89\\linewidth]{fig1}\n \\caption{The asymmetric SQUID composed of three Josephson junctions and the equivalent circuit composed of two junctions, where the Josephson phase difference is $\\varphi_1 = \\varphi_u + \\varphi_d$. The physical quantity of interest is the long-time average voltage $V$ across the SQUID which is expressed by the relation: $V = \\hbar \\langle \\dot{\\varphi}_1 \\rangle\/2e = \\hbar \\langle \\dot{\\varphi}_2 \\rangle\/2e$, see Eq. (A1) in the Appendix.}\n \\label{fig1}\n\\end{figure}\n\\section{Model}\nWe study transport properties of an experimental realization of the rocking ratchet mechanism in an asymmetric superconducting quantum interference device (SQUID) \\cite{zapata1996prl, weiss2000, sterck2002, berger2004, sterck2005,sergey,sterck2009}. We analyze the current-voltage characteristics in the framework of the Stewart-McCumber theory \\cite{stewart,mccumber}. The Stewart-McCumber model describes the semi-classical regime of a small Josephson junction for which a spatial dependence of characteristics can be neglected. Let us remind that in this theory the current $I(t)$ flowing through the junction is split into three components: the displacement current associated with its capacitance $C$, the normal Ohmic current due the finite resistance $R$ of the junction and the super-current of Cooper pairs characterized by the critical current $J$. Its explicit form reads\n\\begin{equation}\n \\label{eq1}\n I(t) = C \\dot{V}(t) + \\frac{V(t)}{R} + J \\sin{\\varphi(t)},\n\\end{equation}\nwhere $\\varphi(t)$ is the phase difference between the macroscopic wave functions of the Cooper electrons in both sides of the junction, a dot denotes differentiation with respect to time $t$ and $V(t)$ is the voltage across the device which obeys the Josephson relation \\cite{josephson}\n\\begin{equation}\n \\label{eq2}\n V(t) = \\frac{\\hbar}{2e}\\dot{\\varphi}(t).\n\\end{equation}\nIf we insert (\\ref{eq2}) into (\\ref{eq1}) and include according to the fluctuation-dissipation relation \\cite{kubo} the effect of a non-zero temperature $T>0$ by adding Johnson-Nyquist noise, the above Stewart-McCumber equation takes the form\n\\begin{equation}\n \\label{eq2a}\n I(t) = \\frac{\\hbar}{2e} C \\ddot{\\varphi} + \\frac{\\hbar}{2e} \\frac{1}{R} \\dot{\\varphi} + J \\sin \\varphi + \\sqrt{\\frac{2k_B T}{R}}\\,\\xi(t),\n\\end{equation}\nwhere $\\varphi \\equiv \\varphi(t)$, $k_B$ is the Boltzmann constant and thermal fluctuations are modeled by $\\delta$-correlated Gaussian white noise $\\xi(t)$ of zero mean and unit intensity\n\\begin{equation}\n \\label{eq13}\n \\langle \\xi(t) \\rangle = 0, \\quad \\langle \\xi(t)\\xi(s) \\rangle = \\delta(t-s).\n\\end{equation}\nFollowing the proposal in Refs. \\cite{zapata1996prl,sterck2005} we consider a SQUID ratchet which is composed of three Josephson junctions as sketched in Fig. \\ref{fig1}. The loop contains two Josephson junctions in series in the left arm and one junction in the other arm. All elements are shunted with resistances ($R_u,R_d,R_2$) and corresponding capacitances ($C_u,C_d,C_2$). We safely can ignore the individual sub-gap resistances of the unshunted junctions, those being much larger than the shunt resistances. Moreover, the loop is pierced by an external magnetic flux $\\Phi_e$. For each element in the left arm, exposed to the Kirchhoff left-arm current $I_1(t)$, we can write the Stewart-McCumber relation\n\\begin{subequations}\n\t\\label{eq3}\n\t\\begin{align}\n\t I_1(t) &= \\frac{\\hbar}{2e} C_u \\ddot{\\varphi}_u + \\frac{\\hbar}{2e} \\frac{1}{R_u} \\dot{\\varphi}_u + J_u \\sin{\\varphi_u} +\\sqrt{\\frac{2k_B T}{R_u}}\\,\\xi_u(t), \\\\\n I_1(t) &= \\frac{\\hbar}{2e} C_d \\ddot{\\varphi}_d + \\frac{\\hbar}{2e} \\frac{1}{R_d} \\dot{\\varphi}_d + J_d \\sin{\\varphi_d} + \\sqrt{\\frac{2k_B T}{R_d}}\\,\\xi_d(t), \t\t\n\t\\end{align}\n\\end{subequations}\nwhere $\\xi_u(t)$ and $\\xi_d(t)$ are independent Gaussian white noises of the same statistics as in (\\ref{eq13}). The processes $\\xi_u(t)$ and $\\xi_d(t)$ have to be independent to ensure the physically correct equilibrium Gibbs state.\n\nNext, we consider the case when the two junctions in the left arm are identical, i.e. $J_u = J_d \\equiv J_1, R_u = R_d \\equiv R_1\/2, C_u = C_d \\equiv 2C_1$. Using these equal parameters we make us of the fact that ideally the super-current in the \\emph{left arm} is conserved.\nTherefore, we find that the realization of the two phase solutions are synchronous in absence of the two noise terms for same initial conditions and temperature $T=0$. This singles out the unique and equal phases $\\varphi_u=\\varphi_d$. It implies that a solution of (5a) also obeys (5b) with same imposed left arm current $I_1(t)$\\cite{zapata1996prl,sterck2005}. The Kirchhoff law remains valid also in presence of current noise noise with the identical (now random) left arm current $I_1(t)$. Because of the additional inhomogeneous Nyquist current noise term in each junction, however, the two solutions generally stay no longer perfectly synchronized. Assuming small noise intensities for the two thermal independent Gaussian noise sources of equal strength we approximate the phases as being synchronized nevertheless, i.e. $\\varphi_u=\\varphi_d=\\varphi_1\/2$, with $\\varphi_1 \\equiv \\varphi_u + \\varphi_d$\nTaking half of each relation in (\\ref{eq3}a) and (\\ref{eq3}b) and adding gives for the stochastic current $I_1(t)$ the result\n\\begin{align}\n\t\\label{eq4}\n\tI_1(t) &= \\frac{\\hbar}{2e} C_1 \\frac{d^2}{dt^2} (\\varphi_u + \\varphi_d) + \\frac{\\hbar}{2e} \\frac{1}{R_1} \\frac{d}{dt}(\\varphi_u + \\varphi_d) \\nonumber \\\\\n\t&+ J_1\\sin{\\left( \\frac{\\varphi_u + \\varphi_d}{2}\\right )} \\cos\\left( \\frac{\\varphi_u - \\varphi_d}{2} \\right) \\nonumber \\\\\n\t&+ \\sqrt{\\frac{k_B T}{R_1}}\\,\\xi_u(t) + \\sqrt{\\frac{k_B T}{ R_1}}\\,\\xi_d(t).\n\\end{align}\nWith equal solutions $\\varphi_u=\\varphi_d$ this expression yields the Langevin equation\n\\begin{equation}\n \\label{eq5}\n I_1(t) = \\frac{\\hbar}{2e} C_1 \\ddot{\\varphi}_1 + \\frac{\\hbar}{2e} \\frac{1}{R_1} \\dot{\\varphi}_1 + J_1\\sin{\\left( \\frac{\\varphi_1}{2} \\right)} + \\sqrt{\\frac{2k_B T}{R_1}}\\,\\xi_1(t).\n\\end{equation}\nHere, we used the fact that the linear combination of two independent Gaussian white noises of intensities $D_u = k_BT\/2R_u$ and $D_d = k_BT\/2R_d$ gives again Gaussian white noise with the total intensity described by $D_1=D_u+D_d=2k_BT\/R_1$. Note that the stochastic process in (\\ref{eq5}) amounts to a Johnson-Nyquist thermal noise for an overall shunt resistance $R_1 \\equiv 2R_u = 2R_d$.\n\nFrom the above analysis it follows that two identical junctions in series can be considered as one for which the supercurrent-phase relation assumes the form: $J_1\\sin{\\left(\\varphi_1\/2\\right)}$ \\cite{zapata1996prb,zapata1996prl,reimannpr}. This result was obtained in Ref.\\cite{zapata1996prb} in the framework of the Ginzburg-Landau theory, cf. Eq. (23) therein. \n\nLet us discuss the above assumed synchronized phase approximation in presence of small current noise in more detail. In the over-damped limit ($C_1=0$) this result agrees for identical junctions in the left arm as used for the three-junction SQUID rocking ratchet experiment investigated by Sterck \\emph{et al.} \\cite{sterck2005}, see Eqs. (4)-(7) therein. In reality, however, slight different junction parameters will physically lead to asynchronous phase variations in the two junctions in the left arm. Likewise, finite temperatures will, as indicated above, destroy as well the perfect synchronous motion of the noisy solutions $\\varphi_u= \\varphi_d$, as assumed above at all times. However, the actual temperatures are experimentally very {\\it small} \\cite{sterck2002,sterck2005}. As it turns out, the physical ratchet effect for the average voltage emerging from this approximation remains itself \\emph{robust}. The latter has been verified before with simulations in the overdamped limit and also has been tested from experimental evidence in the corresponding low temperature limit. It was validated explicitly (i) numerically in \\cite{zapata1996prl,sterck2005} and also (ii) experimentally for the three junction SQUID ratchet setup realized in the works \\cite{sterck2005,sterck2009}.\nPut differently, because we focus here on the Josephson voltage across the device, i.e. the \\emph{average behavior} of the rate of change of the phase $\\varphi_1$ but not on explicit stochastic values, the substitution of the $\\cos[\\left(\\varphi_u - \\varphi_d\\right)\/2]$-term by unity is justified in practice, as the corrections due to higher moments of the asynchronous phase difference can be safely neglected. In addition it must be kept in mind that the use of the Stewart-McCumber model is itself an approximation. Therefore, our theoretical predictions following from (\\ref{eq5}) must be used as a guide towards \"physical reality\" for the experimenter rather than taken as granted without \"error\" \\cite{zapata1996prl,reimannpr,sterck2005}.\n\nFor the junction in the right arm, the Stewart-McCumber equation reads\n\\begin{equation}\n \\label{eq6}\nI_2 (t) = \\frac{\\hbar}{2e} C_2 \\ddot{\\varphi}_2 + \\frac{\\hbar}{2e} \\frac{1}{R_2} \\dot{\\varphi}_2 + J_2\\sin \\varphi_2 + \\sqrt{\\frac{2k_B T}{R_2}}\\,\\xi_2(t).\n \\end{equation}\nWe next add the constraint for the phases in the loop threaded by the magnetic flux \\cite{barone}\n\\begin{equation}\n \\label{eq7}\n \\varphi_2 - \\varphi_1 = 2 \\pi \\frac{\\Phi}{\\Phi_0},\n\\end{equation}\nwhere $\\Phi_0 = h\/2e$ is the flux quantum and the actual flux $\\Phi$ is a sum of the external flux $\\Phi_e$ and the flux due to the flow of currents\n\\begin{equation}\n \\label{eq8}\n \\Phi = \\Phi_e + L i(t),\n\\end{equation}\nwhere $L$ is the loop inductance and $i(t)$ is the \\emph{circulating} current which tends to screen the magnetic flux. If the current is fed to the loop symmetrically then $i(t)=I_1(t) -I_2(t)$. An asymmetric case is presented in the Appendix. We consider the scenario when the second contribution is small, namely\n\\begin{equation}\n \\label{eq9}\n\t|Li(t)| << \\Phi_{0}.\n\\end{equation}\nIn this regime the internal flux increases monotonically with the external one and this operating mode is often called \"dispersive\" \\cite{barone}. Then, from Eqs. (\\ref{eq7})-(\\ref{eq9}) we find\n\\begin{equation}\n \\label{eq10}\n \\varphi_2 = \\varphi_1 + \\tilde{\\Phi}_e, \\quad \\tilde{\\Phi}_e= 2\\pi \\frac{\\Phi_{e}}{\\Phi_0}.\n\\end{equation}\nThe total current $I(t)$ flowing through the SQUID is\n\\begin{equation}\n I(t) = I_1(t) + I_2(t).\n\t\\end{equation}\nWe insert $I_1(t)$ and $I_2(t)$ from (\\ref{eq5}) and (\\ref{eq6}) and use (\\ref{eq10}) to eliminate $\\varphi_2$. The result is\n\\begin{align}\n \\label{eq12}\n\t\\frac{\\hbar}{2e} C \\ddot{\\varphi_1} + \\frac{\\hbar}{2e} \\frac{1}{R} \\dot{\\varphi}_1 &= -J_1\\sin{\\left( \\frac{\\varphi_1}{2} \\right)} - J_2\\sin{(\\varphi_1 + \\tilde{\\Phi}_e)} \\nonumber\\\\ &+ I(t) -\n\t\\sqrt{\\frac{2k_B T}{R}}\\,\\xi(t),\n\\end{align}\nwhere $C = C_1 + C_2$ and $R^{-1} = R_1^{-1} + R_2^{-1}$.\nThe Gaussian white noise $\\xi(t)$ is a linear combinations of $\\xi_1(t)$ and $\\xi_2(t)$ and has the same statistics as in (\\ref{eq13}), cf. the similar transformation from (\\ref{eq4}) to (\\ref{eq5}).\n\nLet the device be driven by an additional external current $I(t)$ which is composed of the static DC bias $I_0$ and the AC driving of amplitude $A$ and angular frequency $\\Omega$, i.e.\n\\begin{equation}\n \\label{eq14}\n I(t) = I_0 + A\\cos(\\Omega t).\n\\end{equation}\nThe mean value over the period $2\\pi\/\\Omega$ is constant, $\\langle I(t)\\rangle = I_0$. As a consequence we obtain that\n\\begin{equation}\n \\label{eq15}\n \\frac{d \\langle I(t)\\rangle}{dt} = \\frac{d \\langle I_1(t)\\rangle}{dt} + \\frac{d \\langle I_2(t)\\rangle}{dt} = 0.\n\\end{equation}\nIn the Appendix, we show that in this case the voltage $V$ across the SQUID, averaged over the period of the AC current, is given by the relation\n\\begin{equation}\n \\label{eq16}\n\tV = \\frac{\\hbar}{2e} \\langle \\dot{\\varphi}_1 \\rangle,\n\\end{equation}\nwhere $\\varphi_1$ is a solution of (\\ref{eq12}) and $\\langle \\cdot \\rangle$ denotes a temporal average over one period of the AC current.\n\n\\subsection{Going to a dimensionless formulation}\nWe next transform (\\ref{eq12}) into its dimensionless form. This can be achieved in several ways. It is known \\cite{kautz} that for such a system there are four characteristic frequencies: plasma frequency $\\omega_p^2 = 2eJ_1\/\\hbar C$, the characteristic frequency of the junction $\\omega_c = 2eR J_1\/\\hbar$, the frequency $\\omega_r =1\/RC$ related to the relaxation time and the frequency $\\Omega$ of the AC current. There are three independent characteristic time scales related to these frequencies (note that $\\omega_p^2=\\omega_c \\omega_r$). Here, we follow \\cite{zapata1996prl} and define the new phase $x$ and the dimensionless time $\\hat{t}$ as\n\\begin{equation}\n \\label{eq17}\n x = \\frac{\\varphi + \\pi}{2}, \\quad \\hat{t} = \\frac{t}{\\tau_c}, \\quad \\tau_c = \\frac{\\hbar}{eRJ_1}.\n\\end{equation}\nThe corresponding dimensionless form of (\\ref{eq12}) reads\n\\begin{equation}\n \\label{eq18}\n \\tilde C \\ddot{x}(\\hat{t}) + \\dot{x}(\\hat{t}) = -U'(x(\\hat{t})) + F + a\\cos(\\omega \\hat{t}) + \\sqrt{2D}\\,\\hat{\\xi}(\\hat{t}),\n\\end{equation}\nwhere the dot and prime denotes a differentiation over the dimensionless time $\\hat t$ and the phase $x$, respectively. We introduced a spatially periodic potential $U(x)$ of period $2\\pi$ of the following form \\cite{zapata1996prl}\n\\begin{equation}\n \\label{eq19}\n U(x) = - \\sin(x) - \\frac{j}{2} \\sin(2x + \\tilde\\Phi_e - \\pi\/2).\n\\end{equation}\nThis potential is reflection-symmetric if there exists $x_0$ such that $U(x_0+x)=U(x_0-x)$ for any $x$. If $j \\neq 0$, it is generally asymmetric and its reflection symmetry is broken. We classify this characteristics as a ratchet potential. However, even for $j \\neq 0$ there are certain values of the external flux $\\tilde \\Phi_e$ for which it is is still symmetric. The dimensionless capacitance $\\tilde C$ is the ratio between two characteristic time scales $\\tilde C = \\tau_r\/\\tau_c$, where the relaxation time is $\\tau_r = RC$. Other re-scaled parameters are $j = J_2\/J_1$, $F = I_0\/J_1$, $a = A\/J_1$ and $\\omega = \\Omega\\tau_c$. It is worth to note that the noise intensity $D = e k_B T\/\\hbar J_1$ is the quotient of the thermal energy and the Josephson coupling energy. The re-scaled Gaussian white noise is of vanishing mean and the auto-correlation function \\mbox{$\\langle \\hat{\\xi}(\\hat{t})\\hat{\\xi}(\\hat{s}) \\rangle = \\delta(\\hat{t} - \\hat{s})$}. Hereafter, we will use only dimensionless variables and shall omit the 'hat' notation in all quantities appearing in (\\ref{eq18}). In Fig. \\ref{fig2}, the ratchet potential $U(x)$ is shown for $j=1\/2$ and two values of the external magnetic flux $\\tilde \\Phi_e = \\pi\/2$ (positive \"polarity\") and $\\tilde \\Phi_e = -\\pi\/2$ (negative \"polarity\"). The symmetric potential for $j=0$ is also depicted. We would like to add that (\\ref{eq18}) has a mechanical interpretation: it is identical to the Langevin equation of a classical Brownian particle of mass $m=\\tilde C$ (i) moving in spatially periodic ratchet potential $U(x)$, (ii) being rocked by an unbiased harmonic force $a\\cos(\\omega t)$ and (iii) exposed to a static force $F$. In this mechanical framework the phase $x$ and the voltage $V$ translates to the space coordinate and the velocity of the Brownian particle, respectively.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.89\\linewidth]{fig2}\n \\caption{The symmetric potential $U(x) = -\\sin(x)$ for $j=0$ (solid red line) is depicted in comparison with the ratchet potential given by (\\ref{eq19}) for $j=1\/2$ and two values of the external magnetic flux $\\tilde\\Phi_e = \\pi\/2$ (dashed green line) and $\\tilde\\Phi_e = -\\pi\/2$ (dotted blue line).}\n \\label{fig2}\n\\end{figure}\n\nThe most important characteristic of transport behavior of the SQUID is the current-voltage curve in the stationary regime. The voltage (\\ref{eq16}) or its dimensionless counterpart $\\langle v \\rangle = \\langle \\dot{x} \\rangle$ is determined by (\\ref{eq18}). In the long time limit, it takes the form of a Fourier series over all harmonics \\cite{jung1993}, namely,\n\\begin{equation}\n\t\\label{eq20}\n\t\\lim_{t\\to\\infty} \\langle {\\dot x(t)} \\rangle = \\langle v \\rangle + v_{\\omega}(t) + v_{2\\omega}(t) + \\dots,\n\\end{equation}\nwhere $\\langle v \\rangle $ is a DC (time-independent) component and $v_{n \\omega}(t)$ are time-periodic functions of zero average over a basic period $2\\pi\/\\omega$. In this case the DC component $\\langle v \\rangle$ is obtained after averaging over both the temporal period of the driving and the corresponding ensemble \\cite{jung1993}\n\\begin{equation}\n\t\\label{eq21}\n\t\\langle v \\rangle = \\lim_{t\\to\\infty} \\frac{\\omega}{2\\pi} \\int_{t}^{t+2\\pi\/\\omega} \\mathbb{E}[\\dot{x}(s)] \\, ds,\n\\end{equation}\nwhere $\\mathbb{E}[\\dot{x}(s)]$ denotes an average over initial conditions and all realizations of the thermal noise. The actual stationary voltage is then given as\n\\begin{equation}\n \\label{eq22}\n V = R J_1 \\langle v \\rangle.\n\\end{equation}\nBecause the SQUID is driven by the external current (\\ref{eq14}), the system is far away from thermal equilibrium and a time-dependent non-equilibrium state is reached in the long time limit. The key ingredient for the occurrence of directed transport $\\langle v \\rangle \\neq 0$ is the symmetry breaking. It is the case when the DC current $F \\neq 0$ or the reflection symmetry of the potential $U(x)$ is broken.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.85\\linewidth]{fig3a} \\\\\n \\includegraphics[width=0.85\\linewidth]{fig3b} \\\\\n \\includegraphics[width=0.85\\linewidth]{fig3c}\n \\caption{The deterministic transport behavior as a function of AC driving strength $a$ and its angular frequency $\\omega$ of the dynamics in (\\ref{eq18}) within three distinct regimes: (a) over-damped regime ($\\tilde C = 0.2$), (b) moderate damping regime ($\\tilde C = 2$) and (c) under-damped regime ($\\tilde C = 7$). The average voltage $\\langle v \\rangle$ is presented for vanishing bias $F = 0$ and thermal noise $D = 0$. The periodic potential $U(x)$ has a positive \"polarity\": $j=1\/2$ and $\\tilde \\Phi_e=\\pi\/2$.}\n \\label{fig3}\n\\end{figure}\n\\section{Deterministic dynamics}\nFirst, let us consider the corresponding deterministic version of the Langevin equation (\\ref{eq18}), i.e. we set formally $D=0$. This is not the manifest realistic physical situation as thermal noise is present as well. However, it can help to understand general properties of the system. When $D=0$, (\\ref{eq18}) is equivalent to a system of three autonomous differential equations of the first order and the phase space is three dimensional. It is a minimal dimension for chaotic behavior to occur. Indeed, periodic, quasi periodic and chaotic trajectories can be detected. A rough classification can be made into locked states in which the motion of $x$ is bounded to a few spatial periods and running states in which it is unlimited in space of $x$. The latter are crucial for the occurrence of the deterministic transport. For some regimes, ergodicity is broken and the systematic non-zero voltage emerges with its sign depending on the choice of selected initial conditions.\nHowever, in the presence of small noise the system typically becomes ergodic and transitions between possibly coexisting deterministic disjoint attractors are probable. In particular, this give rise to diffusive directed transport.\n\nIn order to obtain the relevant transport characteristics we have to resort to comprehensive numerical simulations of driven Langevin dynamics. We integrated (\\ref{eq18}) by employing a weak version of the stochastic second order predictor corrector algorithm \\cite{platen} with a time step typically set to about $10^{-3} \\cdot 2\\pi\/\\omega$. Since (\\ref{eq18}) is a second-order differential equation, we have to specify two initial conditions $x(0)$ and $\\dot{x}(0)$. Moreover, because for some regimes the system may be non ergodic in order to avoid the dependence of the presented results on the specific selection of initial conditions we have chosen phases $x(0)$ and dimensionless voltages $\\dot{x}(0)$ equally distributed over interval $[0, 2\\pi]$ and $[-2,2]$, respectively. All quantities of interest were ensemble-averaged over $10^3 - 10^4$ different trajectories which evolved over $10^3 - 10^4$ periods of the external AC driving. Numerical calculations were done by use of a CUDA environment implemented on a modern desktop GPU. This scheme allowed for a speed-up of a factor of the order $10^3$ times as compared to a common present-day CPU method \\cite{januszewski2009}.\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.37\\linewidth]{fig4a}\n \\includegraphics[width=0.37\\linewidth]{fig4b} \\\\\n \\includegraphics[width=0.37\\linewidth]{fig4c}\n \\includegraphics[width=0.37\\linewidth]{fig4d}\n \\caption{Representative transport characteristics of the rocked SQUID in the deterministic regime ($D = 0$) and for the potential $U(x)$ with a positive \"polarity\": $j=1\/2$ and $\\tilde \\Phi_e=\\pi\/2$. Panel (a): current-voltage curve for driving strength $a = 2.4$, driving angular frequency $\\omega = 0.31$ and capacitance $\\tilde C = 6.31$. Panel (b): the dependence of the average voltage $\\langle v \\rangle$ on the AC-driving frequency $\\omega$. Panel (c): Influence of the AC-driving amplitude $a$ on the DC voltage. Panel (d): the dependence of the average voltage $\\langle v \\rangle$ on the SQUID capacitance $\\tilde C$. Remaining parameters in panels (b)-(d) are the same as in (a).}\n \\label{fig4}\n\\end{figure*}\n\\subsection{General behavior}\nThe system described by (\\ref{eq18}) possesses a 5-dimensional parameter space $\\{\\tilde C, a, \\omega, F, D\\}$. In this section we consider the deterministic case $D = 0$. Let us study the non-trivial ratchet effect by putting $F = 0$. Then, all forces on the right hand side of (\\ref{eq18}) are zero on average: the mean potential force $-\\langle U'(x)\\rangle =0 $ on the interval $[x, x+2\\pi]$ and the average AC driving\n$\\langle a \\cos(\\omega t) \\rangle = 0$ on the time interval $[t, t+2\\pi\/\\omega]$. If $\\langle v \\rangle \\neq 0$, we detect the ratchet effect. Now, the parameter space $\\{\\tilde C, a, \\omega\\}$ is 3-dimensional and its exploration is tractable numerically with the currently available personal GPU computers. Depending on the value of the dimensionless capacitance $\\tilde C$ the device can operate in three distinct regimes: over-damped ($\\tilde C << 1$), moderate ($\\tilde C \\sim 1$) and under-damped ($\\tilde C >> 1$). The first regime has been extensively studied in Refs. \\cite{zapata1996prl,weiss2000,sterck2005,sergey,sterck2009}. In particular, it is known that in the deterministic case the average voltage $\\langle v \\rangle$ is almost quantized, displaying Shapiro-like steps in the current-voltage characteristic for the adiabatic and non-adiabatic AC driving frequencies $\\omega$. As long as the potential $U(x)$ is asymmetric, generally $\\langle v \\rangle \\neq 0$ with $F = 0$ \\cite{bartussek1994,borromeo2002}. Since very fast positive and negative changes of the driving current cannot induce a non-zero average voltage, it is sufficient to limit our considerations to low and moderate AC driving frequencies $\\omega$. We have performed scans of the parameter space: $\\tilde C \\times a \\times \\omega \\in [0.1;10] \\times [0;10] \\times [0.1;1]$ at a resolution of 200 points per interval to determine the general behavior of the system. The results are depicted in Fig. \\ref{fig3} for the positive \"polarity\" of the potential $U(x)$, i.e. for the external magnetic flux $\\tilde \\Phi_e = \\pi\/2$, cf. Fig. \\ref{fig2}.\n\nOn all $(a, \\omega)$ cuts, there occurs no ratchet effect for $a < 1$ and high driving frequencies $\\omega$. The domains of non zero average voltage $\\langle v \\rangle$ have a striped structure. Although there is no obvious direct connection to chaotic properties of the system, we have found that for regimes where the ratchet effect is present a chaotic behavior is typically observed. For a fixed amplitude $a$, the ratchet behavior generally tends to disappear as the frequency $\\omega$ grows. On the other hand, for a fixed frequency $\\omega$, there is the optimal amplitude $a$ that maximizes the ratchet effect. The increase of the capacitance $\\tilde C$ causes the appearance of regions for which the average voltage $\\langle v \\rangle$ reverses its sign. This should be contrasted with the over-damped regime in which the average voltage drop across the device is never negative for the potential with the positive polarity. Consequently, the capacitance $\\tilde C$ of the device together with the amplitude $a$ and frequency $\\omega$ of the AC-driving can serve as convenient parameters to manipulate the direction of transport processes occurring in the system (\\ref{eq18}).\n\\subsection{Voltage {\\it vs} DC current: Negative conductance}\nBecause the dynamics determined by (\\ref{eq18}) is non-linear and the system is multidimensional, it should not come as surprise that the current-voltage curve is also non-linear and often depicts a non-monotonic function of the system parameters. Typically, the average voltage $\\langle v \\rangle$ is an increasing function of the DC-current $F$. This is true especially for large $F$. Such regimes correspond in the parameter space to normal, Ohmic like transport behavior. However, there are also regimes of anomalous transport exhibiting negative conductance \\cite{kostur2006}: If the average voltage $\\langle v \\rangle$ is a decreasing function of the static bias $F$, the differential conductance\n\\begin{equation}\n\\label{eq23}\n\t\\mu(F) = \\left[\\frac{dv(F)}{dF}\\right]^{-1}\n\\end{equation}\ncan take negative values within some interval of $F$. Such a situation is depicted in panel (a) of Fig. \\ref{fig4}. Clearly, there are several windows of the static current $F$ for which this effect is observed. It is worth to notice that this phenomenon is missing in the over-damped regime ($\\tilde C \\to 0$) or in the absence of the AC driving \\cite{machura2007, speer2007epl}. Moreover, in panel (a) we show the ratchet effect: for $F=0$ the voltage is non-zero and for small negative DC current, $F<0$, the voltage is positive, $\\langle v \\rangle >0$. The latter phenomenon is named Absolute Negative Conductance (ANC) \\cite{kostur2008}.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.73\\linewidth]{fig5a} \\\\\n \\includegraphics[width=0.73\\linewidth]{fig5b}\n \\caption{Destructive influence of thermal noise on the ratchet effect. Panel (a): The current-voltage characteristics is presented in the deterministic limit ($D = 0$, solid red line) and for the thermal noise driven case ($D = 9.7 \\cdot 10^{-5}$, dashed green line) case. Panel (b): The dependence of the average voltage $\\langle v \\rangle$ on the thermal noise intensity $D$ for vanishing bias $F=0$. The remaining parameters are: $a = 1.9$, $\\omega = 0.6$ and $\\tilde C = 0.645$. The potential $U(x)$ has the positive \"polarity\": $j=1\/2$ and $\\tilde \\Phi_e=\\pi\/2$.}\n \\label{fig5}\n\\end{figure}\n\\subsection{Multiple voltage reversals}\nAccording to the previous statement on the basis of general scans in the parameter space typical transport characteristics depicted in Fig. \\ref{fig4} exhibit multiple reversals of the voltage $\\langle v \\rangle$ for the zero DC current, $F=0$. However, it should be stressed that this effect is not present in the over-damped regime ($\\tilde C \\to 0$) when for a fixed potential polarity the voltage has a fixed sign. The phenomenon of multiple voltage reversal \\cite{jung1996,mateos2000,mateos2003,kostur2000} is most pronounced for moderate values of the amplitude $a$ and the frequency $\\omega$ of the time-oscillating harmonic driving. In panel (b)-(d) we observe several local extrema and one global maximum of the voltage. For the increasing capacitance $\\tilde C$ there are more regions in the parameter space for which this effect occurs. One can conveniently manipulate the direction of transport processes occurring in the system just by variation of its capacitance $\\tilde C$, amplitude $a$ or frequency $\\omega$.\n\\section{Role of thermal noise}\nWe can expect that thermal noise perturbs deterministic dynamics and can thus reduce or even destroy some deterministic effects. However, more interesting are the regimes for which it can enhance or induce new features for the system dynamics. We analyze the role of thermal fluctuations and discuss the influence of temperature on the stationary voltage. The regimes presented below are optimal in the sense that the effects are most pronounced for the illustrated parameter domains.\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.73\\linewidth]{fig6a} \\\\\n \\includegraphics[width=0.73\\linewidth]{fig6b}\n \\caption{Constructive influence of thermal noise on the ratchet effect. Panel (a): The current-voltage characteristics is presented in the deterministic limit ($D = 0$, solid red line) and noise driven case ($D = 0.0008$, dashed green line) case. Panel (b): The dependence of the average voltage $\\langle v \\rangle$ on the thermal noise intensity $D$. The remaining parameters are: $a = 2.3$, $\\omega = 0.63$, $\\tilde C = 1.98$, $j=1\/2$ and $\\tilde \\Phi_e=\\pi\/2$.}\n \\label{fig6}\n\\end{figure}\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.37\\linewidth]{fig7a}\n \\includegraphics[width=0.37\\linewidth]{fig7b} \\\\\n \\includegraphics[width=0.37\\linewidth]{fig7c}\n \\includegraphics[width=0.37\\linewidth]{fig7d}\n \\caption{Destructive influence of thermal noise on the ratchet effect. In the panels (a) and (b) thermal noise may reverse the sign of the DC-voltage from negative to positive in comparison to the deterministic case. The remaining two panels (c) and (d) depict an opposite situation when the sign is shifted from positive to negative. Parameters for (a) and (b) read: $a = 1.7$, $\\omega = 0.2$ and $\\tilde C = 7.97$. For (c) and (d) they are as follows $a = 3.2$, $\\omega = 0.417$, $\\tilde C = 7$. The parameters of the potential $U(x)$ are: $j=1\/2$ and $\\tilde \\Phi_e=\\pi\/2$.}\n \\label{fig7}\n\\end{figure*}\n\\subsection{Destructive role of thermal fluctuations}\nAn example of a regime where thermal fluctuations play a destructive role is illustrated with Fig. \\ref{fig5}. Panel (b) shows the dependence of the stationary average voltage $\\langle v \\rangle$ on the thermal noise intensity or temperature, $D \\propto T$. A careful inspection of that figure reveals that indeed there is a window of temperature for which the DC-voltage is practically zero. One should also note that small increase of temperature causes a sharp reduction of the voltage $\\langle v \\rangle$ and therefore this phenomenon can be useful to trap the phase $x$ in one of the potential wells \\cite{goldobin2007, sickinger2012, goldobin2013}. In panel (a) of the same figure we depict the current-voltage curves for the deterministic $D = 0$ and noisy $D = 9.7 \\cdot 10^{-5}$ cases. Essentially, temperature plays a destructive role: there is no a ratchet effect for the noise intensity $D$ corresponding to the minimum of the curve in panel (b).\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.37\\linewidth]{fig8a}\n \\includegraphics[width=0.37\\linewidth]{fig8b}\n \\caption{Noise induced ratchet effect. Panel (a) shows the current-voltage characteristic in this regime. Panel (b) depicts the dependence of the average voltage $\\langle v \\rangle$ on the thermal noise intensity $D$ in absence of a bias $F=0$. Other parameters are: $a = 1.8$, $\\omega = 0.56$, $\\tilde C = 1$, $j=1\/2$ and $\\tilde \\Phi_e=\\pi\/2$.}\n \\label{fig8}\n\\end{figure*}\n\\subsection{Constructive role of thermal fluctuations}\nThe opposite scenario occurs when thermal noise has a positive effect on relevant transport characteristics. It means that the voltage exhibits a maximum as a function of the thermal noise intensity $D$. In the mechanical framework it is equivalent to the situation when the mean first passage time for the particle to escape over the potential barrier is shortened by the increase of the thermal noise intensity $D$. This effect is exemplified in Fig. \\ref{fig6}. Panel (b) shows the dependence of the average voltage $\\langle v \\rangle$ on temperature. Evidently, its increase causes an increase in the voltage. There is an optimal temperature, corresponding to $D \\approx 0.0008$, for which the voltage $\\langle v \\rangle$ assumes a maximal value. This finding is confirmed in the current-voltage curve presented in the panel (a) of the same figure. Temperature plays a constructive role, the ratchet effect is strengthened by noise.\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.3\\linewidth]{fig9a}\n \\includegraphics[width=0.3\\linewidth]{fig9b}\n \\includegraphics[width=0.3\\linewidth]{fig9c} \\\\\n \\includegraphics[width=0.3\\linewidth]{fig9d}\n \\includegraphics[width=0.3\\linewidth]{fig9e}\n \\caption{Optimal regime for the occurrence of the ratchet transport. The dependence of the average voltage $\\langle v \\rangle$ on the static DC-bias $F$, the angular frequency $\\omega$, amplitude $a$, capacitance $\\tilde C$ and thermal noise intensity $D$ is presented in panels (a)-(e). The chosen parameters are: $D=0$, $F = 0$, $a = 2.25$, $\\omega = 0.638$, $\\tilde C = 1.65$, $j=1\/2$ and $\\tilde \\Phi_e=\\pi\/2$.}\n \\label{fig9}\n\\end{figure*}\n\\subsection{Noise induced voltage reversals}\nWe have found a regime where thermal fluctuations are able to reverse the DC-voltage from positive to negative values and vice versa. Such regimes are illustrated in Fig. \\ref{fig7}. In panel (b) of this figure the first situation is presented: the average voltage $\\langle v \\rangle$ is negative in the deterministic limit and increases with increasing temperature. There is a critical value of the thermal noise intensity $D$ for which the DC-voltage changes its sign and becomes positive for higher temperature. Panel (a) of the same figure shows the current-voltage curves corresponding to this regime. At this point it is worth to note that for this set of parameters the phenomenon of negative differential conductance is also detected. In particular, we can observe that this effect is robust with respect to small changes of the thermal noise intensity $D$. Panels (c) and (d) depict the opposite scenario: starting from low temperature the increase of $D$ changes the voltage from positive to negative values.\n\\subsection{Noise induced ratchet effect}\nThe next interesting phenomenon, which is activated by thermal fluctuations, is the noise induced ratchet effect. It corresponds to the situation when there is no directed transport in the deterministic regime $D = 0$ for vanishing static DC-bias $F = 0$ but it is observed when the thermal noise intensity is non-zero $D \\neq 0$. Such a scenario is depicted in Fig. \\ref{fig8}: The average voltage $\\langle v \\rangle$ vanishes for low thermal noise intensity $D$ and starts to increase with increasing temperature. There emerges also an optimal value of the thermal noise intensity $D \\approx 0.0056$ for which the ratchet effect becomes most pronounced. This regime can be considered as a special case of a constructive influence of thermal noise on the ratchet phenomenon. Our finding is confirmed in the current-voltage curve which is presented in panel (a) of the same figure.\n\\section{Tailoring the ratchet current}\nModern personal GPU computers have given us opportunity to scan the parameter space of the system with high resolution in a reasonable time and therefore we were able to find a regime for which the ratchet effect is {\\it globally} maximal, see in Fig. \\ref{fig9}. All transport characteristics corresponding to this set of parameters are presented below. It turns out that the ratchet effect is optimal in the moderate capacitance regime $\\tilde C \\approx 1.65$, for the moderate amplitude $a \\approx 2.25$ and the frequency $\\omega \\approx 0.638$ of the time-oscillating current. Moreover, there are several clearly indicated peaks in the dependence of the DC-voltage on the system parameters. The effect of thermal noise on the ratchet effect is destructive for this set of parameters. However, it is worth to note that this regime is temperature robust because the average voltage $\\langle v \\rangle$ starts to decrease significantly only for the thermal noise intensities higher than $D \\approx 5 \\cdot 10^{-4}$, cf. Fig. \\ref{fig9}(e).\n\\section{Control of transport by external magnetic flux}\nTransport measured as the stationary DC-voltage can be controlled in a several ways. It seems that from the experimental point of view the simplest way is to vary a DC current $F$ or an external constant magnetic flux $\\tilde \\Phi_e$. We first consider the unbiased domain with $F=0$. In Fig. \\ref{fig10} we depict how the DC-voltage behaves in the parameter plane $\\{\\tilde \\Phi_e, \\tilde C\\}$ for two cases: $D=0$ (panel (a)) and $D= 10^{-3}$ (panel (b)). The most important feature of these plots is the symmetry with respect to the magnetic flux $\\tilde\\Phi_e$. For an arbitrary integer number $n$, the transformation $\\tilde\\Phi_e \\to 2\\pi n - \\tilde\\Phi_e$ reverses the polarity of the potential (\\ref{eq19}) and as a consequence reverses also the voltage sign. The geometric structure of the domains in the depicted regime of the $\\{\\tilde \\Phi_e, \\tilde C\\}$-variation is complex. There are islands of positive and negative voltage.\n\nFor the deterministic case ($D=0$) we reveal the refined structure. Some of these regions survive when the temperature is increased while others disappear. We detect a few robust regimes for which \"islands\" of non-zero voltage persist. It is seen that if the capacitance is fixed at the proper value the direction of transport can be changed by the magnetic field. In some regions, several voltage reversals can be obtained by use of this method. If the DC-current is applied, the above symmetry is destroyed. This case is shown in Fig. \\ref{fig11}. However, there are still regimes where the magnetic field is a relevant control parameter for the direction of transport.\n\\begin{figure}[t]\n\t\\centering\n \\includegraphics[width=0.85\\linewidth]{fig10a}\n \\includegraphics[width=0.85\\linewidth]{fig10b}\n \\caption{Voltage across the the rocked SQUID in the parameter plane $\\{\\tilde\\Phi_e, \\tilde C\\}$. Upper panel: The deterministic case $D=0$. Bottom panel: The role of temperature $D=10^{-3}$. The DC current is absent, i.e. $F=0$. The remaining parameters are: $a = 2.4$, $\\omega = 0.31, j=1\/2$.}\n \\label{fig10}\n\\end{figure}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.85\\linewidth]{fig11a}\n \\includegraphics[width=0.85\\linewidth]{fig11b}\n \\caption{Voltage across the the rocked SQUID in the parameter plane $\\{\\Phi_e, \\tilde C\\}$. Upper panel: the deterministic case $D=0$. Bottom panel: influence of temperature $D=10^{-3}$. The DC current $F=0.1$. The remaining parameters are: $a = 2.4$, $\\omega = 0.31, j = 1\/2$.}\n \\label{fig11}\n\\end{figure}\n\\section{Summary}\nWe analyzed the characteristics of the voltage across an asymmetric SQUID device composed of three capacitively and resistively shunted Josephson junctions which are threaded by a magnetic flux. We derived the evolution equation which governs the dynamics of the phase across the SQUID. The effective potential experienced by the phase displays a symmetry breaking in the form of the ratchet potential. Under the influence of an oscillating current source, the current-voltage characteristics yields the possibility to obtain a finite DC-voltage in presence of a vanishing DC-current, i.e. a ratchet effect is obtained. Within a tailored ranges of parameters, the same sign of the DC-voltage can be obtained regardless of the sign of the external DC-current.\n\nWith this comprehensive study we have taken into consideration the role of a finite capacitance of the SQUID. As a consequence, the resulting ratchet dynamics becomes rather rich, giving rise to features which are absent in the over-damped limit. With the help of the computational power of modern GPU computers we have identified a whole range of novel phenomena inherent for the ratchet current. These are a negative (differential) conductance, repeated DC-voltage reversals, noise induced DC-voltage reversals and particular forms of solely noise-induced ratchet features. For given tailored sets of parameters the ratchet voltage assumes optimal values.\nLast but not least, we have been able to detect the set of parameters for which the ratchet effect is globally maximal and demonstrated how the direction of transport can be manipulated by tailoring the threading external magnetic flux.\n\nThe main goal of this work was the exploration and identification of parameter regimes for directed ratchet transport in realistic SQUID devices possessing {\\it finite} capacitances. Such a study is of relevance for applications which make use of a generation and its control of the induced ratchet-voltages, their direction (sign), magnitude and their intrinsic sensitive dependence on system parameters.\n\nOther transport quantifiers concerning the overall quality of the inertia-induced transport, such as the nature of the ratchet-voltage fluctuations (yielding in turn a diffusion dynamics of the phase across the SQUID), or the efficiency of the device \\cite{machura2004, machura2005, machura2006,machura2010} have not been addressed here. Given the underlying complexity of the inertial ratchet dynamics these numerical studies are even more cumbersome than the presented ones.\n\nFinally, an interesting question concerns the robustness of our results with respect to slightly different junction parameters in series; an assumed exact mathematically equality of parameters for two junctions is practically difficult to achieve. This issue has been addressed in the positive for the case of the over-damped regime \\cite{zapata1996prl}, where it was found that the corresponding results remain robust. For the under-damped regime, the complexity of the problem becomes even more higher multi-dimensional and therefore this task is presently beyond the scope of this work. Nevertheless, those additional aspects are on our agenda when the corresponding cumbersome numerical investigations become technically more feasible.\n\\section*{Acknowledgments}\nThis work was supported in part by the MNiSW program \"Diamond Grant\" (J. S.), NCN grant DEC-2013\/09\/B\/ST3\/01659 (J. {\\L}.), and by a grant HA1517\/-2 from the Deutsche Forschungsgemeinschaft (DFG) (P. H.). The authors also like to thank Peter Talkner for constructive discussions.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}