Content-Type: multipart/mixed; boundary="-------------0505030608515" This is a multi-part message in MIME format. ---------------0505030608515 Content-Type: text/plain; name="05-159.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-159.comments" 26 pages, revised version ---------------0505030608515 Content-Type: text/plain; name="05-159.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-159.keywords" entropy production, fluctuation theorem, quantum stochastic calculus ---------------0505030608515 Content-Type: application/x-tex; name="fluctuation dissipated heat.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fluctuation dissipated heat.tex" \documentclass[a4paper,12pt]{article} \usepackage[centertags]{amsmath} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{newlfont} \usepackage{bbm} % THEOREM-LIKE ENVIRONMENTS ----------------------------------------- \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}{theorem}[section] \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} \newtheorem{notes}[theorem]{Notes} \newtheorem{example}[theorem]{Example} \numberwithin{equation}{section} % \MATHOPERATOR ----------------------------------------------------- \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Prob}{\boldsymbol{Prob} } \DeclareMathOperator{\ex}{Ex} \DeclareMathOperator{\ext}{Ext} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\supp}{Supp} \newcommand{\re}{\Re\,} \newcommand{\im}{\Im\,} \newcommand{\HH}{\mathsf{H}} \newcommand{\PP}{\mathsf{P}} \newcommand{\RR}{\mathsf{R}} % GREEK - 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\bs ------------------------------------------------- \newcommand{\bsa}{{\boldsymbol a}} \newcommand{\bsb}{{\boldsymbol b}} \newcommand{\bsc}{{\boldsymbol c}} \newcommand{\bsd}{{\boldsymbol d}} \newcommand{\bse}{{\boldsymbol e}} \newcommand{\bsf}{{\boldsymbol f}} \newcommand{\bsg}{{\boldsymbol g}} \newcommand{\bsh}{{\boldsymbol h}} \newcommand{\bsi}{{\boldsymbol i}} \newcommand{\bsj}{{\boldsymbol j}} \newcommand{\bsk}{{\boldsymbol k}} \newcommand{\bsl}{{\boldsymbol l}} \newcommand{\bsm}{{\boldsymbol m}} \newcommand{\bsn}{{\boldsymbol n}} \newcommand{\bso}{{\boldsymbol o}} \newcommand{\bsp}{{\boldsymbol p}} \newcommand{\bsq}{{\boldsymbol q}} \newcommand{\bsr}{{\boldsymbol r}} \newcommand{\bss}{{\boldsymbol s}} \newcommand{\bst}{{\boldsymbol t}} \newcommand{\bsu}{{\boldsymbol u}} \newcommand{\bsv}{{\boldsymbol v}} \newcommand{\bsw}{{\boldsymbol w}} \newcommand{\bsx}{{\boldsymbol x}} \newcommand{\bsy}{{\boldsymbol y}} \newcommand{\bsz}{{\boldsymbol z}} \newcommand{\bsA}{{\boldsymbol A}} \newcommand{\bsB}{{\boldsymbol B}} \newcommand{\bsC}{{\boldsymbol C}} \newcommand{\bsD}{{\boldsymbol D}} \newcommand{\bsE}{{\boldsymbol E}} \newcommand{\bsF}{{\boldsymbol F}} \newcommand{\bsG}{{\boldsymbol G}} \newcommand{\bsH}{{\boldsymbol H}} \newcommand{\bsI}{{\boldsymbol I}} \newcommand{\bsJ}{{\boldsymbol J}} \newcommand{\bsK}{{\boldsymbol K}} \newcommand{\bsL}{{\boldsymbol L}} \newcommand{\bsM}{{\boldsymbol M}} \newcommand{\bsN}{{\boldsymbol N}} \newcommand{\bsO}{{\boldsymbol O}} \newcommand{\bsP}{{\boldsymbol P}} \newcommand{\bsQ}{{\boldsymbol Q}} \newcommand{\bsR}{{\boldsymbol R}} \newcommand{\bsS}{{\boldsymbol S}} \newcommand{\bsT}{{\boldsymbol T}} \newcommand{\bsU}{{\boldsymbol U}} \newcommand{\bsV}{{\boldsymbol V}} \newcommand{\bsW}{{\boldsymbol W}} \newcommand{\bsX}{{\boldsymbol X}} \newcommand{\bsY}{{\boldsymbol Y}} \newcommand{\bsZ}{{\boldsymbol Z}} \newcommand{\bsalpha}{{\boldsymbol \alpha}} \newcommand{\bsbeta}{{\boldsymbol \beta}} \newcommand{\bsgamma}{{\boldsymbol \gamma}} \newcommand{\bsdelta}{{\boldsymbol \delta}} \newcommand{\bsepsilon}{{\boldsymbol \epsilon}} \newcommand{\bsmu}{{\boldsymbol \mu}} \newcommand{\bsomega}{{\boldsymbol \omega}} % ABBREVIATION ------------------------------------------------------ \newcommand{\fig}{Fig.\;} \newcommand{\cf}{cf.\;} \newcommand{\eg}{e.g.\;} \newcommand{\ie}{i.e.\;} % MISCELLANEOUS ----------------------------------------------------- \newcommand{\un}[1]{\underline{#1}} \newcommand{\defin}{\stackrel{\text{def}}{=}} \newcommand{\bound}{\partial} \newcommand{\sbound}{\hat{\partial}} \newcommand{\rel}{\,|\,} \newcommand{\pnt}{\rightsquigarrow} \newcommand{\pa}{_\bullet} \newcommand{\nb}[1]{\marginpar{\tiny {#1}}} \newcommand{\pair}[1]{\langle{#1}\rangle} \newcommand{\0}{^{(0)}} \newcommand{\1}{^{(1)}} \newcommand{\con}{_{\text{con}}} \newcommand{\print}[1]{${#1} \quad {\mathcal {#1}} \quad {\mathfrak {#1}} \quad {\mathbb {#1}} \quad {\boldsymbol {#1}}$ \newline} \DeclareMathOperator{\prob}{Prob} \begin{document} \begin{center} \noindent{\large \bf Steady state fluctuations of the dissipated heat for a quantum stochastic model} \\ \vspace{15pt} {\bf Wojciech De Roeck}\footnote{Aspirant FWO, U.Antwerpen, email: {\tt wojciech.deroeck@ua.ac.be}} and {\bf Christian Maes}\footnote{email: {\tt christian.maes@fys.kuleuven.ac.be}}\\ Instituut voor Theoretische Fysica\\ K.U.Leuven, Belgium. \end{center} \vspace{20pt} \footnotesize \noindent {\bf Abstract: } We introduce a quantum stochastic dynamics for heat conduction. A multi-level subsystem is coupled to reservoirs at different temperatures. Energy quanta are detected in the reservoirs allowing the study of steady state fluctuations of the entropy production. Our main result states a symmetry in its large deviation rate function. \vspace{5pt} \footnotesize \noindent {\bf KEY WORDS: } entropy production, fluctuation theorem, quantum stochastic calculus \vspace{20pt} \normalsize \section{Introduction} Steady state statistical mechanics wants to construct and to characterize the stationary distribution of a subsystem in contact with several reservoirs. By nature the required scenario is an idealization as some essential specifications of the reservoirs must be kept constant. For example, intensive quantities such as temperature or (electro-)chemical potential of the different reservoirs are defined and unchanged for an extensive amount of time, ideally ad infinitum. Reservoirs do not interact directly with each other but only via the subsystem; they remain at their same spatial location and can be identified at all times. That does not mean that nothing happens to the reservoirs; flows of energy or matter reach them and they are like sinks and sources of currents that flow through the subsystem. Concrete realizations and models of steady states vary widely depending on the type of substances and on the nature of the driving mechanism.\\ An old and standard problem takes the subsystem as a solid in contact at its ends with two heat reservoirs and wants to investigate properties of the energy flow. Beloved by many is a classical model consisting of a chain or an array of coupled anharmonic oscillators connected to thermal noises at the boundaries. The reservoirs are there effectively modeled by Langevin forces while the bulk of the subsystem undergoes a Hamiltonian dynamics, see e.g. \cite{luc,eck,mnv}. Our model to be specified below is a quantum analogue of that scenario in the sense that we also consider a combination of Hamiltonian dynamics and Markovian thermal forces.\\ We imagine a chain of coupled two(or multi)-level systems. The dynamics of the isolated subsystem is unitary with Hamiltonian $H_S$. Quanta of energy $\omega$ are associated to the elementary transitions between energy levels. Two physical reservoirs at inverse temperatures $\beta_k, k=1,2$, are now attached to the subsystem. The total dynamics is described by a quantum stochastic differential equation through which we can observe the number $N_{\omega,k}$ of quanta with energy $\omega$ that are piled up in the $k-$th reservoir. The total energy $N \equiv H_S + \sum_{\omega,k} \omega N_{\omega,k}$ is conserved under the dynamics (Proposition \ref{conserv}). The change in the second term corresponds to the flow of energy quanta in and out of the reservoirs and specifies the dissipated heat. Our main result is obtaining a symmetry in the fluctuations of that dissipated heat that extends the so called steady state fluctuation theorem to a quantum regime (Proposition \ref{ft}).\\ The quantum stochastic evolution that defines the model is a particular dilation of a semigroup dynamics that describes the weak coupling regime of our subsystem coupled to quasi-free boson fields. The dilation, a sort of quantum Langevin equation, is much richer and enables the introduction of a natural path space measure. One should remember here that a major conceptual difficulty in coming to terms with the notion of a variable entropy production for quantum steady states is to understand its path-dependence. One option is to interrupt the unitary dynamics with collapses, see e.g. \cite{Maes2}. Others have proposed an entropy production operator, avoiding the problem of path-dependence. Our set-up follows a procedure that is well-known in quantum optics with thermal noises formally replacing photon detectors, see \cite{Bouten,Maassen2}. In the resulting picture we record each energy quantum that is transferred between subsystem and reservoirs. It induces a stochastic process on quanta transferrals and there remains no problem to interpret the fluctuations of the entropy production. From the mathematical point of view, the model can be analyzed via standard probabilistic techniques. \subsection{Related Results} In the past decade, a lot of interest has been going to the Gallavotti-Cohen fluctuation relation \cite{Gal,evans}, see \cite{Poincare} for more recent references. In its simplest form that relation states that the steady state probability of observing a total entropy decrease $w_T=-wT$ in a time $T$, is exponentially damped with respect to the probability of observing an increase of $w_T$ as \begin{equation}\label{entproform2} \frac{\Prob(w_T=wT)}{\Prob(w_T=-wT)} \approx e^{+wT} \end{equation} at least for very large time spans $T$. The relation (\ref{entproform2}) is known as the steady state fluctuation theorem (SSFT) and states a symmetry in the fluctuations of the entropy dissipation in a stationary nonequilibrium steady state. The symmetry was first discovered in the context of dynamical systems and was applied to the phase space contraction rate in strongly chaotic dynamical systems, see \cite{ru,evans,Gal}. There is also a transient version of that symmetry, sometimes exactly verified for finite $T$ and known as the transient fluctuation theorem (TFT), see \cite{australie}. The basic underlying mechanism and general unifying principles connecting SSFT and TFT with statistical mechanical entropy have been explained in \cite{MN,Poincare}.\\ In the present paper we derive a quantum steady state fluctuation theorem. Monnai and Tasaki \cite{Monnai} have investigated an exactly solvable harmonic system and found quantum corrections to both SSFT and TFT. Matsui and Tasaki \cite{Matsui} prove a quantum TFT in a general $C^*$-algebraic setting. It is however unclear what is the meaning of their entropy production operator.\\ A related quantum Jarzynski relation was studied in \cite{DM}.\\ Besides the fluctuation theorem, we also describe a new approach to the study of heat conduction in the quantum weak coupling limit. In \cite{LS2} Lebowitz and Spohn studied the thermodynamics of the weak-coupling generator. They identified the mean currents, and they proved a Green-Kubo relation. At that time it was however impossible to conclude that these expressions are the first non-zero contributions to their counterparts at finite coupling $\lambda$. That has recently been shown in a series of papers by Jaksic and Pillet \cite{JP1,JP2,JP3,JP4}, who used spectral techniques to study the system at finite coupling $\lambda$. It was also shown that the stationary state of the weak coupling generator is the zeroth order contribution to the system part of the so called NESS, the natural nonequilibrium steady state. The current fluctuations we define in our model, agree with the expressions of \cite{LS2} as far as the mean currents and the Green-Kubo formula is concerned. Our entropy production operator is however new, it differs for example from the proposal of \cite{Matsui}.\\ The approach taken here also differs from the more standard route that has been followed and that was outlined by Ruelle in \cite{ru1}. Recently and within that approach and context of heat conduction, new results have been obtained in \cite{ashspohn,JP3,JP4,Frohlich}. To us it remains however very much unclear how to define and study in that scenario a fluctuating entropy; in contrast, that is exactly one of the things we can easily achieve via our approach but we remain in the weak coupling limit. % % % % % % \subsection{Basic strategies} \subsubsection{Microscopic approach} \label{micro} In general one would like to start from a microscopic quantum dynamics. The system is then represented by a finite dimensional Hilbert space $\mathcal{H}$ and system Hamiltonian $H_S$. The environment is made from $m$ thermal reservoirs, infinitely extended quantum mechanical systems, with environment Hamiltonian, formally, \[ H_R \equiv H^R_1 +\ldots+ H^R_m \] The coupling between system and reservoirs is local and via some bounded interaction term $\lambda H^{SR}$ so that the total Hamiltonian takes the form \[ H_\lambda \equiv H_S \otimes 1 + 1 \otimes H_R + \lambda \sum_{k=1}^m V_k \otimes R_k \] where we have already inserted a specific form for the coupling $H^{SR}$ using self-adjoint reservoir operators $R_k$ and $V_k$ acting on $\mathcal{H}$. On the same formal level, which can however easily be made precise, the total quantum dynamics is then just \[ U_t^\lambda \equiv e^{-iH_\lambda t}\, \cdot \, e^{+iH_\lambda t} \] We will not follow the beautiful spectral or scattering approach that has recently been exploited for that nonequilibrium problem. We refer the reader to the specialized references such as \cite{JP3,ru1,ashspohn} and we only outline the main steps totally ignoring essential assumptions and technicalities:\\ One starts the dynamics from an initial state \[ \omega \equiv \mbox{tr} \otimes \omega^1 \otimes \ldots \otimes \omega^m \] where $\mbox{tr}$ stands for the normalized trace-state in the system and the $\omega^k$ are equilibrium KMS states at inverse temperature $\beta_k$ for the $k$-th reservoir, $k=1,\ldots,m$. The quantum dynamics takes that initial state to the new (now coupled) state $\omega_t$ at time $t>0$. The NESS is obtained via an ergodic average \[ \bar \rho \equiv \lim_{T\uparrow+\infty} \frac 1{T} \int_0^Tdt\, \omega_t \] One of the first questions (and partially solved elsewhere, see e.g. \cite{JP3,ru1,ashspohn}) is then to derive the natural conditions under which the mean entropy production rate \[ \dot{S} \equiv \sum_{k=1}^m\beta_k \,\bar{\rho}( i[H_\lambda,H_k^R]) \] is strictly positive. While that mean entropy production certainly coincides with conventional wisdom, we do not however believe that the operator \[ i[H_\lambda,H_k^R] \] or equivalent expressions, is the physically correct candidate for the study of current fluctuations which would obey the SSFT. That is not even the case for the simplest (classical) stochastic dynamics; one needs to go to path-space and study current fluctuations in terms of (fluctuating) trajectories.\\ \subsubsection{Weak coupling approach}\label{weakapproach} Starting from the microscopic dynamics above, we can of course always look at the reduced dynamics $\Lambda^\lambda_t$ on the system \[ \Lambda_t^\lambda\rho = \mbox{Tr}_R[U_t^\lambda(\rho\otimes\omega^1 \otimes \ldots \otimes \omega^m)] \] for a density matrix $\rho$ on the system. Obviously, the microscopic evolution couples the system with the environment and the product form of the state will in general not be preserved. One can however attempt a Boltzmann-type Ansatz or projection technique to enforce a repeated randomization. That can be made rigorous in the so called weak coupling limit. For that one needs the interaction picture and one keeps $\lambda^2 t =\tau$ fixed. That is the Van Hove-Davies-limit \cite{vanhove,Dav2}: \[ \lim_{\lambda\rightarrow 0} \Lambda_t^0\,\Lambda_t^\lambda\rho \equiv e^{\tau L}\rho \] where $L$ is a linear operator acting on density matrices for the system. The generator will be written out more explicitly in Section \ref{weak} but its dual $\mathcal{L}$ acting on $\mathcal{B}(\mathcal{H})$ is of the form, see \eqref{decomposition gen}, \[ \mathcal{L}(\cdot)=-i [H_{f},\cdot]+\sum_{k} \mathcal{L}_k(\cdot) \] where the $\mathcal{L}_k$ can be identified with the contribution to the dissipation from the $k$th reservoir. $H_f$ is an effective, renormalized Hamiltonian depending on details of the reservoirs and the coupling.\\ From now on we write $\rho$ for the (assumed) unique invariant state (see \cite{LS2} for sufficient conditions to have a unique invariant state): \[ e^{\tau L}\rho =\rho,\quad \tau\geq 0 \] Again one can study here the mean entropy production, as for example done in \cite{LS2} and argue that \[ \mbox{Tr}[\rho\,\mathcal{L}_k(H^S)] \] represents the stationary heat flow into the $k$'th reservoir, at least in the weak coupling regime. Nothing tells us here however about the physical fluctuations in the heat current for which higher moments should be considered. In fact, the reservoirs are no longer visible as the weak coupling dynamics is really a jump process on the energy levels of the system Hamiltonian, see further in Section \ref{clasweak}. The heat flow and the energy changes in the individual reservoirs cannot be reconstructed from the changes in the system. The present paper will use a new idea for the study of the fluctuations of the heat dissipation in a reservoir. \subsubsection{Dilation} While the weak coupling dynamics is very useful for problems of thermal relaxation (one reservoir) and for identifying the conditions of microscopic reversibility (detailed balance) characterizing an equilibrium dynamics, not sufficient information is left in the weak coupling limit to identify the variable heat dissipated in the various reservoirs. Heat is path-dependent and we need at least a notion of energy-trajectories.\footnote{At least, if one has a stochastic or effective description of the system dynamics, as is the case in the weak-coupling limit. We do {\bf not} claim at all that the trajectory-picture is microscopically fundamental.} The good news is that we can obtain such a representation at the same time as we obtain a particular dilation of the weak coupling dynamics. The representation is basically achieved via an unravelling of the weak coupling generator $\mathcal{L}$ and the corresponding Dyson-expansion of the semigroup dynamics. That will be explained under \ref{unravel}.\\ There are many possible dilations of a quantum dissipation. It turns out that there is a dilation whose restriction to the system coincides with the Dyson-representation in terms of energy-trajectories of the weak coupling dynamics. That dilation is well studied and goes under the name of a quantum stochastic dynamics. The associated quantum stochastic calculus was invented by Hudson and Parathasaraty, \cite{HP,Par}. It has been extensively employed for the purpose of quantum counting processes, see e.g. \cite{Maassen2,M3}. Various representations and simplifications have been added, such as in \cite{MarkAlicki} where a (classical) Brownian motion extends the weak coupling dynamics. \subsubsection{Results} We prove a symmetry in the large deviation generating function (Proposition \ref{ft}). In the case of non-degenerated free subsystem Hamiltonians $H_S$, this rate function is analytic and this implies the large deviation principle. The symmetry is recongnized as the fluctuation theorem for the entropy production. For degenerate Hamiltonians, we can only establish analyticity in a neighbourhood of $0$, this implies the central limit theorem for the currents (but we do not stress this point). Here the symmetry does not translate in a standard fluctuation theorem. However, in both cases, the symmetry implies a Green-Kubo relation and (modified) Onsager reciprocity (Proposition \ref{green-kubo}). \subsection{Outline of the paper} In section 2, we introduce the quantum stochastic model and state the result. In section 3 follows a discussion where the main points and novelties are emphasized. Proofs are postponed to section 4. % % % % \section{The Model} % % % % % \subsection{Weak Coupling}\label{weak} We briefly introduce here the weak-coupling dynamics without speaking about its derivation, which is not relevant for the discussion here. Some of that was briefly adressed in Sections \ref{micro}-\ref{weakapproach}.\\ Let $\mathcal{H}$ be a $d$-dimensional Hilbert space assigned to a small subsystem, called system in what follows. Let $H_S$ be a Hamiltonian on $\mathcal{H}$ with eigenvalues $\ep_x,\ldots, \ep_y$ labelled via $x,y \in \Lambda$, some finite set and spectral projections $P_x,P_y, \ldots$. \begin{equation}\label{setomega} F \equiv \{\omega \in \mathbb{R} | \quad \exists x, y \in \Lambda : \omega= \ep_x -\ep_y \} \end{equation} is the set of eigenvalues of the derivation $-i[H_S,\cdot]$. We label by $k=1,\ldots,m$ different heat reservoirs at inverse temperatures $\beta_k$. To each reservoir $k$ is assigned a self-adjoint operator $V_k$\footnote{One can take more couplings per reservoir, and they need not be self-adjoint, but for notational clarity, we restrict to that case.} on $\mathcal{H}$ and for each $\omega \in F$, we put \begin{equation} \label{newv} V_{\omega,k}= e^{\frac{\beta_k \omega}{4}} \psi_{\omega,k} P_{y}V_k P_{x}, \qquad \ep_x -\ep_y=\omega \end{equation} with $\psi_{\omega,k} \in \mathbb{C}$ arbitrary, but obeying the constraint $\bar{\psi}_{\omega,k}=\psi_{-\omega,k}$.\\ We further consider an effective self-adjoint Hamiltonian $H_f$ satisfying \begin{equation} [H_{f},H_S]=0 \end{equation} We work with the following weak coupling generator $\mathcal{L}$ on $\mathcal{B}(\mathcal{H})$ \begin{equation} \label{gen2} \mathcal{L}(\cdot)=-i [H_{f},\cdot]+\sum_{\omega,k}\big( V_{\omega,k}^{*}\cdot V_{\omega,k}-\frac{1}{2}\{V_{\omega,k}^{*}V_{\omega,k},\cdot\} \big) \end{equation} By grouping all terms with the same $k$, we can write \begin{equation} \label{decomposition gen} \mathcal{L}(\cdot)=-i [H_{f},\cdot]+\sum_{k} \mathcal{L}_k(\cdot) \end{equation} For $\caL$ to be really the generator derived in \cite{Dav1}, we would need to relate the factors $\phi_{\omega,k}$ to correlation functions in the reservoir and we should fix the renormalized hamiltonian $H_f$. However, since this is not necessary for our discussion, we omit these specifications.\\ We will need the following assumption {\bf A1}\\ \textit{There is an open set $ \caV \subset \mathbb{R}^{m}$ such that for all $\overline{\beta}=(\beta_1,\ldots,\beta_{m}) \in \caV$, the generator $\caL\equiv \caL_{\overline{\beta}}$ has a unique invariant steady state $\rho =\rho_{\overline{\beta}}$. This implies that $0$ is a non-degenerate eigenvalue (corresponding to $\mathbbm{1}$) of $\caL_{\overline{\beta}}$. Moreover, none of the $V_{\omega,k}$ should be zero} \footnote{This ensures also that every part of the system is effectively coupled to all reservoirs.}.\\ Remark that if $\overline{\beta}=(\beta,\ldots,\beta)$, then $\rho_{\overline{\beta}}=\exp[-\beta H_S]/\mbox{tr}(\exp[-\beta H_S])\equiv \rho_{\beta}$\\ Furthermore, one can easily see that \begin{equation}\label{rhodiag} \sum_{x \in \Lambda} P_x \rho P_x=\rho \end{equation} % % % % % % % \subsection{Unravelling of the generator}\label{unravel} We associate to that semigroup dynamics a path space measure by a procedure which is known as "unravelling of the generator."\\ % % % Write the weak coupling generator as \[ \caL=\caL_0+\sum_{\omega,k} \caJ_{\omega,k} \] with \begin{equation}\label{j} \caJ_{\omega,k}(\cdot)= V^*_{\omega,k} \cdot V_{\omega,k} \end{equation} and \[ \caL_0(\cdot)=-i [H_{f},\cdot] -\frac{1}{2}\sum_{\omega,k} \{V_{\omega,k}^{*}V_{\omega,k},\cdot\} \big) \] Let $\Omega^1_{t,n}$ be the set of ordered $n$-tupels $\{ (t_1,\ldots,t_n ) \big| t_1 \leq t_2 \leq \ldots \leq t_n \leq t \}$. Put $\Omega^1_{t}=\bigcup_{n \in \mathbb{N}} \Omega^1_{t,n}$. One can make $\Omega^1_t$ into a measure space $(\Omega^1_t,d\sigma)$ by embedding $\Omega^1_{t,n}$ in $([0,t]^n,\mathcal{A}^1_{t},dx)$ with $dx$ the Lesbegue measure and $\mathcal{A}^1_t$ the $\sigma$-algebra. In the same way, one defines $\Omega^1 \equiv \Omega^1_{t=+\infty}$ and $\mathcal{A}^1 \equiv \mathcal{A}^1_{t= +\infty}$. Finally let $\Omega \equiv (\Omega ^1)^{\ell}= \prod_{k,\omega} \Omega^1_{k,\omega} $ with $\Omega^1_{k,\omega} $ denoting identical copies of $\Omega^1$, similarly we introduce $\mathcal{A} \equiv (\mathcal{A}^1)^{\ell}$ and $\Omega_t,\mathcal{A}_t $ as products of $\Omega^1_t,\mathcal{A}^1_t$. Often, $\Omega$ is called the Guichardet space, see \cite{Guich}.\\ An element $\sigma \in \Omega$ looks as \begin{equation}\label{ordered} \sigma=(\omega_1,k_1,t_1;\ldots;\omega_n,k_n,t_n) \textrm{ with } n \in \mathbb{N} \textrm{ and } (t_1 \leq t_2 \leq \ldots \leq t_n) \end{equation} For future use, we also define 'number functions' $\overline{n}_t \in \mathbb{N}^{\ell}$ with components \begin{equation}\label{numbers} n_{\omega,k,t}(\sigma)\equiv |\sigma \cap \Omega_{\omega,k,t}| \end{equation} The notation $\Omega_{\omega,k,t}$ should be understood as $\Omega^1_t$ where $\Omega_{\omega,k}$ is an indexed copy of $\Omega^1$.\\ Consider $\mathcal{W}_t(\sigma): \mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{H})$ as the completely positive map depending on $\sigma \in \Omega$ \begin{equation} \label{explicitmap} \mathcal{W}_t(\sigma)=I_{t}(\sigma) e ^{(t_1)\mathcal{L}_0} \mathcal{J}_{\omega_1,k_1}e^{(t_2-t_1)\mathcal{L}_0}\ldots e^{(t_n-t_{n-1})\mathcal{L}_0}\mathcal{J}_{\omega_n,k_n} e^{(t-t_n)\mathcal{L}_0} \end{equation} with $I_{t}(\cdot)$ the indicator function of $\Omega_{t} \subset \Omega$ and with the indices $(\omega_i,k_i), i=1,\ldots,n$ referring to the representation (\ref{ordered}) of $\sigma$. Complete positivity follows from (\ref{j}) and from \[ e^{t\mathcal{L}_0}=e^{-itH_f-\frac{t}{2}V_{\omega,k}V_{\omega,k}}\cdot e^{+itH_f-\frac{t}{2}V_{\omega,k}V_{\omega,k}} \] We can now write \begin{equation} e^{t \caL}=\int_{\Omega} d\sigma \mathcal{W}_t(\sigma) \end{equation} which is a Dyson expansion. % % % That expression induces a 'path space measure', or a notion of 'quantum trajectories'. \begin{lemma} \label{prob} Define \begin{equation}\label{maat} \mathbb{P}_{\rho,t}(E) \equiv \int_{E}d\sigma \quad \rho (\mathcal{W}_t(\sigma)1), \quad E \in \caA \end{equation} Then $\mathbb{P}_{\rho,t}$ are a consistent family of probability measures on $\Omega_t$ and thus we obtain a measure $\mathbb{P}_{\rho}$ on $\Omega$. \end{lemma}\noindent These probability measures are often called 'quantum counting processes' (see \cite{M3}). \subsubsection{Result} We define the entropy current $\mathbf{w}_t$ as a function on $\Omega$: \begin{equation}\label{entproclas} \mathbf{w}_t(\sigma) =\sum_{\omega,k} \beta_{k} \omega n_{\omega,k,t}(\sigma) \end{equation} with $n_{\omega,k,t}$ as in (\ref{numbers}). Now we can already formulate the main result of the paper: \begin{proposition}\label{ft1} Assume ({\bf A1}). Let $\mathbf{w}_t$ be defined in (\ref{entproclas}). There is a neighborhood $\caU$ of $0$ in $\mathbb{R}$, such that for all $\kappa \in \caU$, the limit \begin{equation}\label{rate function clas} e(\kappa) \equiv \lim_{t \rightarrow +\infty} \frac{1}{t}\log \int d\mathbb{P}_{\rho} e^{-\kappa \mathbf{w}_t} \end{equation} exists and is real-analytic.\\ Write $e(\kappa)\equiv e(\kappa,H_f)$, indicating explicitly the dependence on the effective Hamiltonian $H_f$. Then \begin{equation} \label{symmetry2} e(\kappa,H_f)=e(1-\kappa,-H_f) \end{equation} whenever the left-hand side exists.\\ The mean entropy production is non-negative \begin{equation} \label{nonneg} \int d\mathbb{P}_{\rho} \mathbf{w}_t \geq 0 \end{equation} Let now $H_S$ be non-degenerate, then (\ref{rate function clas}) exists and is real-analytic for all $\kappa$ and $e(\kappa,H_f)=e(\kappa,-H_f)$ and (\ref{rate function clas}) verifies \begin{equation} \label{symmetry} e(\kappa)=e(1-\kappa) \end{equation} for all $\kappa \in \mathbb{R}$\\ For non-degenerate $H_S$, the mean entropy production is strictly positive \begin{equation} \label{positive} \int d\mathbb{P}_{\rho} \mathbf{w}_t > 0 \end{equation} whenever there are $k,l$ such that $\beta_k \neq \beta_l$ \end{proposition} \noindent % % % % \subsection{QSDE} We will not prove nor even mention all technicalities in this subsection; recommended references are \cite{Par} for Quantum Stochastic Calculus and \cite{Fannes} for Canonical Commutation Relations.\\ Let $\ell \equiv |F| \times m$ (recall that $m$ is the number of reservoirs) and $\mathcal{G}\equiv (L^2(\mathbb{R}))^{\ell}$. An element $f=(f_1,\ldots,f_{\ell})$ of $\mathcal{G}$ consists of $\ell$ components which are labelled by the double index $(\omega,k)$. Let $\chi_t \in L^2(\mathbb{R})$ be the indicator function of the interval $[0,t]$ and \[ \chi_{\omega,k,t}=(0,\ldots,0,\underbrace{\chi_t}_{\omega,k },0,\ldots,0) \] Let $\mathbbm{1}_{[0,t]} \in \caB(L^2(\mathbb{R}))$ be the projector on $L^2([0,t])$ and \begin{equation}\label{intervalprojector} \mathbbm{1}_{\omega,k,t}=(0,\ldots,0,\underbrace{\mathbbm{1}_{[0,t]}}_{\omega,k },0,\ldots,0) \end{equation} We consider the bosonic Fock space ($\Gamma_s$ denotes symmetrized second quantization) \begin{equation}\label{fockspace} \mathcal{F} = \Gamma_s (\mathcal{G})= \otimes^{\ell}\Gamma_s (L^2(\mathbb{R})) \end{equation} with the factorization property for $s0$ yields. \begin{equation}\label{strict2} \int d\mathbb{P}^t_{\rho} (S^t_{\rho})=\int d\mathbb{P}^t_{\rho} (e^{-S^t_{\rho}}+S^t_{\rho}-1) \geq (e^{-\delta}+\delta -1) \mathbb{P}^t_{\rho}(|S^t_{\rho}| \geq \delta) \end{equation} Whenever not all $\beta_k$'s are equal, take $k,k'$ such that $\beta_k \neq \beta_{k'}$. Choose $\omega \in F$ and put $\sigma(s,u)=(\omega,k,s;-\omega,k',u)$ and $E=\{\sigma(s,u), s \leq u \leq t\} \in \mathcal{A}_t$. Then, \begin{equation} S_t(E)=\mathbf{w}_t(E)=(\beta_k-\beta_{k'})\omega \end{equation} and thus $\mathbb{P}^t_{\rho}\big(|S^t_{\rho}| \geq |(\beta_k-\beta_{k'})\omega|\big) \geq \mathbb{P}^t_{\rho}(E)>0$. Hence, (\ref{strict2}) implies (\ref{positive}). \subsection{Proof of Proposition \ref{green-kubo} } Remark first that a slight generalization of Proposition \ref{ft} can be proven (in exactly the same way). Write $e(\kappa_1,\dots,\kappa_{m})= e^{-\beta \sum_k \kappa_k \mathbf{N}_k}$. Now (\ref{symmetry2}) reads \begin{equation}\label{ft2} e(\kappa_1,\ldots,\kappa_{m},H_f)=e(\frac{\beta_1}{\beta}-\kappa_1,\ldots,\frac{\beta_k}{\beta}-\kappa_{m},-H_f) \end{equation} By differentiating with respect to $\kappa_k$ and $\frac{\beta_l}{\beta}$ we obtain \begin{equation}\label{clt} L_{k,l}(H_f) +L_{k,l}(-H_f) \equiv \beta \lim_{t \rightarrow +\infty}\frac{2}{t}\overline{\rho}_{\beta}(j_t[\mathbf{N}_{k,t}\mathbf{N}_{l,t}])(-H_f) \end{equation} That computation is standard (see again \cite{LS1}, the interchange of the derivatives and the limit $t \rightarrow +\infty$ is justified by a theorem in \cite{Bryc}, stating that analyticity of the large deviation generating function implies the central limit theorem.). One sees that the right-hand side is the same for $H_f$ or $-H_f$. The Green-Kubo relation follows from (\ref{clt}) by time-translation invariance of the correlation function, which follows from (\ref{time}), and from the existence of the derivatives, which is proven in Proposition \ref{current-current}.\\ \subsection{Proof of Proposition \ref{current-current}} Let $\overline{N}=(N_{\omega})_{\omega},\overline{M}=(M_{\omega})_{\omega} \in \mathbb{N}^{|F|}$. We write the derivative on the right-hand side of (\ref{currenteq}) as a limit ($(v_1,v_2) \rightarrow (0,0)$)\footnote{For notational clarity, we assume $v_1,v_2$ positive, the full statement follows analogously with the help time-translation invariance (\ref{time}).} of the infinite sum \begin{equation}\label{doubleseries} \frac{1}{v_1 v_2} \overline{\rho} \big( j_{v_1}[\mathbf{N}_{k_1,v_1}](j_{u+v_2}[\mathbf{N}_{k_2,u+v_2}]-j_{u}[\mathbf{N}_{k_2,u}]) \big)=\sum_{\overline{N},\overline{M} } \frac{U^{(0,v_1)}_{k_1} U^{(u,v_2)}_{k_2}}{ v_1 v_2}\int_{E^{v_1,v_2}_{\overline{N};\overline{M}}} d\sigma \rho (\caW(\sigma)(\mathbbm{1})) \end{equation} with \[ U^{(0,v_1)}_{k_1}(\sigma)=\sum_{\omega} \omega n_{\omega,k_{1},v_1}(\sigma) \] \[ U^{(u,v_2)}_{k_2}(\sigma)=\sum_{\omega} \omega\big( n_{\omega,k_{2},u+v_2}(\sigma)-n_{\omega,k_{2},u}(\sigma)\big) \] and \[ E^{v_1,v_2}_{\overline{M},\overline{M}}=\{\sigma \in \Omega \big| n_{\omega,k_1,v_1}(\sigma)=N_{\omega}, \big(n_{\omega,k_2,u+v_2}(\sigma)-n_{\omega,k_2,u}(\sigma)\big)=M_{\omega}\} \] Put $C \equiv \max{ \| V_{\omega,k} \|^2}$, $D \equiv \max{|\omega| } $ and $|\overline{N}|=\sum_{\omega} N_{\omega},|\overline{M}|=\sum_{\omega} M_{\omega}$. By the Dyson series, the terms on the right-hand side of (\ref{doubleseries}) satisfy for all $v_1,v_2 \in [0,1]$ \begin{equation} \big|\frac{U^{(0,v_1)}_{k_1} U^{(u,v_2)}_{k_2}}{ v_1 v_2}\int_{\sigma \in E_{\overline{N},v_1;\overline{M};v_2}} \rho (\caW(\sigma)(\mathbbm{1}))\big| \leq D^2 |\overline{N}||\overline{M}| \frac{C^{|\overline{N}|+|\overline{M}|}}{|\overline{N}|! |\overline{M}|!} \end{equation} and hence (\ref{doubleseries}) can be bounded by the absolutely convergent series \begin{equation} \sum_{p=1}^{+\infty} C^p (p D)^2 \frac{1}{(\lfloor \frac{p}{2}\rfloor!)^2 } \end{equation} such that the $p$'th term bounds the sum of all terms in (\ref{doubleseries}) with $|\overline{N}|+|\overline{M}|=p$. Hence one can perform the limit $(v_1,v_2) \rightarrow (0,0)$ term by term to obtain \begin{equation} \sum_{\omega_1,\omega_2} \omega_1 \omega_2 \rho \big(\caJ_{\omega_2,k_2} e^{u \mathcal{L}}\caJ_{\omega_1,k_1} (\mathbbm{1})\big) \end{equation} Putting $\rho \equiv \rho_{\beta}$, (\ref{currenteq}) follows after some reshuffling, using \begin{equation} V_{\omega,k}^*=e^{-\frac{\beta_k \omega}{2}}V_{-\omega,k} \quad \sum_{\omega} \omega V_{\omega,k}^*V_{\omega,k}=\caL_k(H_S) \quad \rho_{\beta}V_{\omega,k}=V_{\omega,k}\rho_{\beta}e^{\beta_k \omega} \end{equation} % % % % % % {\bf Acknowledgments} \\ We thank Luc Bouten, Hans Maassen, Andr\'e Verbeure, Frank Redig and Karel Neto\v cn\'y for stimulating discussions. % \begin{thebibliography}{10} \raggedright \bibitem[AbF]{Frohlich} W.K. Abou-Salem and J. Fr\"ohlich: Adiabatic Theorems and Reversible Isothermal Processes, {\tt mp\_arc 05-19}. \bibitem[AF]{MarkAlicki} R. Alicki and M. 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