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26 pages, revised version
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entropy production, fluctuation theorem, quantum stochastic calculus
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\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.
%
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\section{The Model}
%
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%
\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}
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\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."\\
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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.
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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

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\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 $s<t$
\begin{equation}\label{factor}
\mathcal{F}
=\mathcal{F}_{s]}\otimes\mathcal{F}_{[s,t]}\otimes\mathcal{F}_{[t}
\qquad \mathcal{F}_{[s,t]} \equiv  \Gamma_s[(L^2[s,t])^{\ell}]
\end{equation}
The operators $\mathbf{A}_{k,\omega,t}$ and
$\mathbf{A}^*_{k,\omega,t}$ on $\mathcal{F}$ are the usual
annihilator and creator of the test function $\chi_{\omega,k,t}$.
We omit here the precise definitions which are well-known since we
will only need some common wisdom about the solution of
(\ref{dilation1}).\\
 Let $\mathcal{B}(\mathcal{H} \otimes \mathcal{F}
)$ be the algebra of bounded operators on $\mathcal{H} \otimes
\mathcal{F}$. That suffices to give meaning to the following
Quantum Stochastic Differential Equation (QSDE) on
$\mathcal{B}(\mathcal{H} \otimes \mathcal{F} )$, see \cite{HP},
\begin{eqnarray}\label{dilation1}
d\mathbf{U}_t &=& \sum_{\omega,k}  \big( V_{\omega,k}
d\mathbf{A}^{*}_{\omega,k,t}-V^{*}_{\omega,k}
d\mathbf{A}_{\omega,k,t}\big)\mathbf{U}_t\\
&-&(iH_{f}dt+\frac{1}{2}\sum_{\omega,k}V^{*}_{\omega,k}V_{\omega,k}dt)
\mathbf{U}_t, \quad \mathbf{U}_0=\mathbbm{1} \nonumber
\end{eqnarray}
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The solution to (\ref{dilation1}) is unitary. Let $Y_t$ be the
unitary on $\mathcal{G}$ which implements the left shift
\begin{equation}
(Y_t(f))(s)=f(s+t)
\end{equation}
then we obtain a strongly continuous unitary group
$\widehat{\mathbf{U}}_t$ by putting
\begin{equation}\label{group}
\widehat{\mathbf{U}}_t= \left\{
\begin{array}{ll}d\Gamma_s(Y_t)\mathbf{U}_t & \textrm{ if } t \geq
0
\\
\mathbf{U}_{-t}^{-1}d\Gamma_s(Y_t) & \textrm{ if } t < 0
\end{array} \right.
\end{equation}
with
$\mathbf{U}_t$ the solution of (\ref{dilation1}).\\
Equation (\ref{dilation1}) is again our nonequilibrium model
describing several heat baths at different temperatures connected
to the subsystem. In Section 4 we motivate the model physically.
The connection with Section \ref{weak} and in particular with the
path-space measure (\ref{maat}) is the subject of Lemma
\ref{connection1}. Note that by taking each $V_{\omega,k}=0$ or
$V_k=0$ in (\ref{newv}) the subsystem decouples from the reservoir
and (\ref{dilation1}) is solved by
\begin{equation}\label{decoupled}
\mathbf{U}_t= e^{-itH_f}\otimes \mathbbm{1}
\end{equation}
Remark that in (\ref{dilation1}), the reservoirs are now not only
labelled by $k$, as in the original physical picture, but also by
$\omega$; each transition has its own mathematical reservoir.\\
To formulate our results we also need to specify the state. It is
taken of the form
\begin{equation}
\overline{\rho} \equiv \rho \otimes \phi  \textrm{  on  }
\mathcal{H}\otimes \mathcal{F}
\end{equation}
where $\rho$ is the unique stationary state of the dynamics
restricted to the subsystem, see assumption {\bf A1}. The state
$\phi$ denotes the vacuum on all Fock spaces involved: with $\Psi
\equiv 1 \oplus 0 \oplus 0 \otimes 0 \oplus \ldots$,
\begin{equation}\label{vac}
\phi(\cdot)=\langle \Psi,\cdot \Psi \rangle
\end{equation}
and, as in (\ref{factor})
\begin{equation}\label{state}
\Psi=\Psi_{s]} \otimes \Psi_{[s,t]} \otimes \Psi_{[t}
\end{equation}
For $ \mathbf{O} \in \mathcal{B}(\mathcal{H} \otimes
\mathcal{F})$, write $j_t(\mathbf{O}) \equiv
\widehat{\mathbf{U}}_{t} \mathbf{O} \widehat{\mathbf{U}}_{-t}$
with
$\widehat{U}_t$ as in (\ref{group}).\\
Remark that the state $\overline{\rho}$ is not invariant under the
dynamics $\widehat{\mathbf{U}}_{-t} \cdot
\widehat{\mathbf{U}}_{t}$, but it is invariant under the 'free
dynamics':\\
Define $\mathbf{V}_t= e^{itH_S} \otimes d\Gamma_s(Y_{t})$, for
$\mathbf{O} \in \mathcal{B}(\mathcal{H} \otimes \mathcal{F})$, we
have
\begin{equation}\label{time}
\overline{\rho}( \mathbf{O})=\overline{\rho}( \mathbf{V}_{t}
\mathbf{O}\mathbf{V}_{-t})
\end{equation}
and this remains true if we replace $H_S$ by $0$. For the
reservoir part this is trivial, for the system part it follows from (\ref{rhodiag}).\\
Let $\mathbf{N}_{\omega,k,t}$ be the number operator up to time
$t$ on the $(\omega,k)$'th Fock space
\[
\mathbf{N}_{\omega,k,t}=d \Gamma_s(\mathbbm{1}_{\omega,k,t})
\]
defined on a suitable dense domain. The connection of the QSDE
with the quantum counting process is provided by the following
Lemma, which we will not prove. It can be found for example in
\cite{Maassen2,M3}.\\
\begin{lemma}\label{projections}
Let $I_{\omega,k}$ be Borel sets in $\mathbb{R}^+$ and $S_{\omega,k}
\subset \mathbb{N}$. Let $E_{\omega,k}$ be the set of ordered
subsets  $\caC \subset I_{\omega,k}$ with $|\caC | \in S_{\omega,k}$ and put
$E \equiv \prod_{\omega,k}E_{\omega,k} \in \caA$. Define the
projector $\mathbf{1}_{\omega,k,I_{\omega,k}}$ as in
(\ref{intervalprojector}) and define the spectral projection
$\mathbf{P}_{I_{\omega,k}}(S_{\omega,k})$ on $\caF$, corresponding
to the operator $\Gamma_s[\mathbf{1}_{\omega,k,I_{\omega,k}}]$ and
the spectral subset $S_{\omega,k}$. Finally, Define $
\mathbf{P}_E\equiv
\otimes_{\omega,k}\mathbf{P}_{I_{\omega,k}}(S_{\omega,k})$ and let
$A \in \caB(\caH)$. Then
\begin{equation}\label{connection1}
\overline{\rho}( j_t[A \otimes \mathbf{P}_E])=\int_{\sigma \in E}
d\sigma \rho(\caW_t(\sigma)(A))
\end{equation}
\end{lemma}\noindent
In particular, one can take $A=\mathbbm{1}$ and
$I_{\omega,k}=[0,t]$, then
$\Gamma_s[\mathbf{1}_{\omega,k,I_{\omega,k}}] \equiv
\mathbf{N}_{\omega,k,t}$ and (\ref{connection1}) leads to
\begin{equation}
\overline{\rho}(
j_t[\mathbf{N}_{\omega,k,t}])=\sum_{n=0}^{+\infty}
n\mathbb{P}_{\rho}(E_{n,t})
\end{equation}
with $E_{n,t}=\{\sigma \in \Omega \big| n_{\omega,k,t}(\sigma)=n \}$ and
where the left-hand side is to be understood as a limit involving
spectral projections (see also the footnote in Proposition
\ref{ft}). One can also choose $I_{\omega,k}=[0,+\infty[$
and $N_{\omega,k}=\mathbb{N}$ for all $(\omega,k)$ (which leads to $E=\Omega_t$) in
(\ref{connection1}), hence
expressing that the QSDE dilates the semigroup
\begin{equation}
\overline{\rho}( j_t[A \otimes \mathbbm{1}])=\rho(e^{t\caL}(A))
\end{equation}












\subsection{Results}
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 We define the total energy\footnote{We actually mean a renormalized
energy: 'The infinite energy of the reservoirs at time 0 has
already been subtracted.'} of the $k$'th reservoir up to time $t$
\begin{equation}
\mathbf{N}_{k,t}=\sum_{\omega} \omega \mathbf{N}_{\omega,k,t}
\end{equation}
and a quantity which we interpret as the total energy of
subsystem+reservoirs
\begin{equation}\label{energy}
\mathbf{N}_t \equiv H_{S}+\sum_{\omega,k}\omega
\mathbf{N}_{\omega,k,t}
\end{equation}
We motivate this interpretation by noticing that $\mathbf{N}_t$ is
conserved.
\begin{proposition}\label{conserv}
Let $\mathbf{N}_t$  be as in (\ref{energy}).For all bounded Borel
functions $f$:
\begin{equation}
\overline{\rho}(f(j_t[\mathbf{N}_t]))=\overline{\rho}(f(\mathbf{N}_t))
\end{equation}
\end{proposition} \noindent
Now we define the change of entropy in the environment up to time
$t$.
\begin{equation}\label{entpro1}
\mathbf{W}_t \equiv \sum_{k} \beta_k j_t(\mathbf{N}_{k,t})
\end{equation}
Its `steady state expectation' is the entropy production. Our main
result is a fluctuation theorem for $\mathbf{W}_t$.

\begin{proposition}\label{ft}
Assume ({\bf A1}). Let $\mathbf{W}_t$ be defined in (\ref{entpro1}).
There is a neighbourhood $\caU$ of $0$ in $\mathbb{R}$, such that
for all $\kappa \in \caU$, the limit
\begin{equation}\label{rate function}
\hat{e}(\kappa) \equiv \lim_{t \rightarrow +\infty}
\frac{1}{t}\log \overline{\rho} [e^{-\kappa \mathbf{W}_t}]
\end{equation}
exists\footnote{ The notation $\overline{\rho} [e^{-\kappa
\mathbf{W}_t}]$ is meant as a weak limit of sums of spectral
projections $\mathbf{P}_{\eta}$:
\begin{equation}
\overline{\rho} [e^{-\kappa \mathbf{W}_t}]=\sum_{\eta \in
\mathbf{Sp}(\mathbf{W}_0)} e^{-\kappa \eta}\overline{\rho}
[j_t(\mathbf{P}_{\eta})]
\end{equation}
One can in fact also prove that the vacuum $\Psi$ from (\ref{vac}) is in the domain of $j_t(e^{-\kappa \mathbf{W}_t})$ but we do not go into that.} and is real-analytic.\\
We write $\hat{e}(\kappa,H_f)$, indicating explicitly the
dependence on the effective Hamiltonian $H_f$. Similarly,
$e(\kappa,H_f)$ is defined by (\ref{rate function clas}). Then,
\begin{equation} \label{symmetryhat}
\hat{e}(\kappa,H_f)=e(\kappa,H_f)
\end{equation}
and thus all statements in Proposition \ref{ft1} carry over to
$\hat{e}(\kappa,H_f)$.
\end{proposition} \noindent
From  $\overline{\rho}$ we deduce probability measures
$\mathbb{T}_t$ on $\mathbb{R}$. Let $A \subset \mathbb{R}$ be
measurable, then
\begin{equation}
\mathbb{T}_t(A)=\overline{\rho} (\mathbf{P}_{A})
\end{equation}
where $\mathbf{P}_{A}$ is the spectral projection on $A$
associated to $\mathbf{W}_t$. Via Legendre-transformation
(\ref{symmetry}) implies
\begin{equation}\label{entproform}
-\lim_{t \rightarrow +\infty}\frac{1}{t}
\log{\frac{d\mathbb{T}_t(-a)}{d\mathbb{T}_t(a)}}=a
\end{equation}
which is (\ref{entproform2}).\\
Next in Propositions \ref{meancurrent}-\ref{green-kubo} we find
that our model satisfies the thermodynamics in the weak-coupling
limit.
\begin{proposition}\label{meancurrent}
Consider $\caL_k$ as in (\ref{decomposition gen}).  For all $k$,
\begin{equation}\label{rhs}
 \frac{d}{du }
 \overline{\rho}(j_{u}[\mathbf{N}_{k,u}])\big|_{u=0}=\rho(\mathcal{L}_k(H_S))
\end{equation}
\end{proposition} \noindent
The righthand side of \eqref{rhs} are the mean currents as defined
in \cite{LS2}.\\
We finally get a Green-Kubo relation between the linear response
coefficients $L_{k,l}$ and the current-current correlations
\begin{proposition}\label{green-kubo}
Assume ({\bf A1}) with $(\beta,\ldots,\beta)\in \caV$. Define the linear
response coefficients for $k,l \in K$
\begin{equation}
L_{k,l} \equiv
\frac{d}{d\beta_{l}}\frac{d}{dt}\overline{\rho}(j_t[\mathbf{N}_{k,t}])
\big|_{t=0, \overline{\beta}=\beta}
\end{equation}
Then,
\begin{equation}
L_{k,l}(H_f)+ L_{k,l}(-H_f)\equiv  \beta  \lim_{t \rightarrow
+\infty}\frac{1}{t}\overline{\rho}_{\beta}(j_t[\mathbf{N}_{k,t}\mathbf{N}_{l,t}])(H_f)
\end{equation}
and also
\begin{equation}
L_{k,l}(H_f)+ L_{k,l}(-H_f) \equiv \beta
\int_{-\infty}^{+\infty}dt \frac{d^2}{dv_1
dv_2}\overline{\rho}_{\beta}(j_{v_1}[\mathbf{N}_{k,v_1}]j_{t+v_2}[\mathbf{N}_{l,t+v_2}])\big|_{v_1,v_2=0}(H_f)
\end{equation}
Whenever $H_S$ is non-degenerate, we have $L_{k,l}(H_f)=
L_{k,l}(-H_f)$ and thus
\begin{equation}
2L_{k,l} \equiv \beta \int_{-\infty}^{+\infty}dt \frac{d^2}{dv_1
dv_2}\overline{\rho}_{\beta}(j_{v_1}[\mathbf{N}_{k,v_1}]j_{t+v_2}[\mathbf{N}_{l,t+v_2}])\big|_{v_1,v_2=0}
\end{equation}
\end{proposition} \noindent









\section{Discussion}
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\subsection{Entropy production for Markov
processes}\label{clasweak}
 It is well known that the weak coupling generator
is 'classical' in the sense that the commutant algebra $\caA_{cl}
\equiv \big\{[A,H_S]=0 \big| A \in \caB(\caH) \big\}$ is
invariant. In case the Hamiltonian $H_S$ is non-degenerate,
$\caA_{cl}$ is a commutative algebra. Then we can construct a
Markov process with state space $\Lambda$ which is the restriction
of (the dual of) the semigroup $e^{t\mathcal{L}}$ to $\caA_{cl}
\cong \caC(\Lambda)$. Loosely speaking, let $\mu$ be the
stationary state, $\Omega^T \equiv \Lambda^{[0,T]} $ the pathspace
up to time $T$, and $\caP^T_{\mu}$ the pathspace measure (starting
from $\mu$)of this Markov process. The time reversal operation
$\Theta$ acts on
$\Omega^T$ as  $(\Omega \xi)(t)=\xi(T-t)$.\\
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For such Markov processes describing a nonequilibrium dynamics, we
dispose of a general strategy for identifying the entropy
production. It turns out in a lot of interesting cases
\cite{Poincare,Maes1,MRV} that
\begin{equation}\label{clasft}
\ln{\frac{d\caP^T_{\mu}}{d\Theta
\caP^T_{\mu}}}(\xi)=S_t(\xi)+\caO(1)
\end{equation}
where $S_t(\xi)$ is the random variable that one physically
identifies as the entropy production. The second term in the
righthand side is non-extensive in time. The algorithm allows to
derive (\ref{entproform2}) from (\ref{clasft}).\\
Since we also have a Markov generator, we can apply the same
scheme to our setup. To evaluate the result, we however need a
physical notion of entropy production in our model. As mentioned
earlier, such a notion is rather unambiguous here, see also
\cite{LS2,JP3}:
\begin{equation}\label{spohns mean}
\langle\textrm{current into k'th reservoir} \rangle =
 \rho(\caL_k (H_S))
\end{equation}
But the mean entropy production based on these currents is {\bf
not} equal to the expectation value of (\ref{clasft}):
\begin{equation}\label{inequality}
\sum_k \beta_k \rho(\caL_k (H_S))  \neq \mathbb{E}_{\caP_{\mu}}
\big( \ln{\frac{d\caP_{\mu}}{d\Theta \caP_{\mu}}}(\xi) \big)
\end{equation}
For example, take two reservoirs $(k=L,R)$, the system Hamiltonian
$H_S$ left-right symmetric, and $\psi(\omega,L)=\psi(\omega,R)$
for all $\omega$. One easily checks that the righthand side of
(\ref{inequality}) depends on the temperatures $\beta_L,\beta_R$
only through the combination $\frac{\beta_L+\beta_R}{2}$.\\
This trivial remark shows that it is not enough to look at the
semigroup to identify the entropy production.

\subsection{Integrated currents within the semigroup approach }
Starting from (\ref{spohns mean}) one could define the integrated
currents  operators $\hat{J}_{k,t}$  as
\begin{equation}\label{Spohns current}
\hat{J}_{k,t}= \int_0^{t}du e^{u \caL}(\caL_k(H_S))
\end{equation}
and study their fluctuations.\\
One can ask whether these fluctuations coincide with these in our
model? The answer is partially positive because
\begin{proposition}\label{current-current}
Take $\overline{\beta}=(\beta,\ldots,\beta)$. For all $k,l \in K$
and all $u \in \mathbb{R}^+_0$,
\begin{equation}\label{currenteq}
\frac{d^2}{dv_1 dv_2} \overline{\rho}_{\beta}
[ j_{v_1}(\mathbf{N}_{k,v_1})j_{u+v_2}(\mathbf{N}_{l,u+v_2})
]\big|_{v_1=0,v_2=0}=-\rho_{\beta} \big(\caL_{l}(H_S) e^{u
\caL}(\caL_{k}(H_S)) \big)
\end{equation}
\end{proposition}\noindent
which gives a relation between Proposition \ref{green-kubo} and
the Green-Kubo relation in \cite{LS2}. Also the averages coincide,
leading to (\ref{spohns mean}). However it is {\bf not} true that
for a reasonable class of functions $f$
\begin{equation}
 \overline{\rho} \big(
j_{t}[f(\mathbf{N}_{k,t}) ]\big)=\rho \big(f(\hat{J}_{k,t})\big)
\end{equation}
So the mean entropy production and the Green-Kubo-formula can
correctly be expressed in terms of the operators $\hat{J}_{k,t}$,
but higher order fluctuations of the dissipated heat cannot.
%
%
%
\subsection{Connection to microscopic dynamics}
We know of two derivations in the literature of
this dilation from a microscopic setup:\\
{\bf 1 Stochastic Limit}\\
Accardi et al. prove in \cite{Acc} that the weak-coupling limit
can be extended to the total evolution of subsystem observables.
Let $\mathcal{U}^{\lambda}_{t}$ be the evolution (in the
interaction picture) on the total system with $\lambda$ the
coupling between subsystem and reservoirs. Then, in a certain
sense,
\begin{equation}
\mathcal{U}^{\lambda}_{-t/\lambda^2} (\mathbbm{1}\otimes
O)\mathcal{U}^{\lambda}_{t/\lambda^2} \rightarrow_{\lambda
\rightarrow 0} \mathbf{U}^*_{t} (\mathbbm{1}\otimes O)\mathbf{U}_{t}
\end{equation}
whereas the traditional weak coupling limit only speaks about
convergence in expectation of the left-hand side. The unitary
$U_t$ is the solution of (\ref{dilation1}).\\
{\bf 2 Repeated Interactions}\\
In \cite{Attal}, Attal and Pautrat describe a subsystem with
Hilbertspace $\mathcal{H}$ interacting repeatedly for a time $h$
with a small reservoir with Hilbertspace $\mathcal{R}$. After each
time $h$, $\mathcal{R}$ is replaced by an identical copy. This
procedure ensures that at any time, the subsystem sees a `fresh'
reservoir. In the limit $h \rightarrow 0$ the dynamics (in the
interaction picture) converges in a certain sense to the solution
of a QSDE. One can choose a particular QSDE by tuning the
parameters of the interaction. Assume that
\begin{equation}
\mathcal{R}= \bigotimes_{\omega,k} \mathcal{R}_{\omega,k}
\end{equation}
Each $\mathcal{R}_{\omega,k}$ is 2-dimensional with basevectors
$(\theta,\omega)$. Define $a_{\omega,k}$ on
$\mathcal{R}_{\omega,k}$ by
\begin{equation}
a_{\omega,k}(\omega)= \theta  \qquad a_{\omega,k}\theta= 0
\end{equation}
Choose the dynamics on $\mathcal{H} \otimes \mathcal{R}$ as
$e^{-itH(h)}$ for $0\leq t \leq h$ with
\begin{equation}
H(h)=H_{f}+ \frac{1}{\sqrt{h}}\sum_{\omega,k}
\big(V_{\omega,k}a^*_{\omega,k}+ V^*_{\omega,k}a_{\omega,k} \big)
\end{equation}
Then, through the limiting procedure of \cite{Attal}, equation
(\ref{dilation1}) obtains.
%
%
%
%
%
%



\section{Proofs}
%
%
%
\subsection{Proof of Lemma \ref{prob} }
The proof can be found in \cite{Maassen2}. From
\begin{equation}
 \int_{\Omega}d\sigma
\mathcal{W}_t(\sigma)1  =e^{t\caL}1=1
\end{equation}
and $\mathcal{W}_t(\sigma)=\mathcal{W}_t(\sigma \cap \Omega_t)$
for all $t$ follows that $\mathbb{P}_{\rho,t}$ is indeed a family
of probability measures (for all $\rho$). Further one checks that
for $s < t$ and $\sigma \in \Omega_s$
\begin{equation}
\mathcal{W}_t(\sigma)=\mathcal{W}_s(\sigma)e^{(t-s)\caL}
\end{equation}
which yields consistence and hence existence of
$\mathbb{P}_{\rho}$.
\subsection{Proof of Proposition \ref{conserv} }
Fix $t \geq 0$. Remark that $\mathbf{N}_t$ has purely discrete
spectrum. Call $\mathbf{P}^t(e)$ a spectral projection of
$\mathbf{N}_t$ associated to the eigenvalue $e$ and
$\mathbf{P}_{\omega,k,t}(N_{\omega,k})$ a spectral projection of
$\mathbf{N}_{\omega,k,t}$ associated to the eigenvalue
$N_{\omega,k} \in \mathbb{N}$. Using the notation
$\overline{N}\equiv (N_{\omega,k})_{\omega,k} \in
\mathbb{N}^{\ell}$, one has
\begin{equation}\label{terrible}
\mathbf{P}^t(e)=\sum_{\begin{array}{c}x \in \Lambda ,\overline{N} \in \mathbb{N}^{\ell}\\
\epsilon_x+\sum_{\omega,k} \omega N_{\omega,k}=e
\end{array}} P_x \otimes ( \otimes_{\omega,k}
\mathbf{P}_{\omega,k,t}(N_{\omega,k}) )
\end{equation}
It suffices to prove
\begin{equation}\label{sufficeproof}
\overline{\rho}(j_t[\mathbf{P}^t(e)])=\overline{\rho}(\mathbf{P}^t(e)))
\end{equation}
Remark, using $[V_{\omega,k}^{*}V_{\omega,k},H_S]=0$ and the
notation (\ref{numbers})
\begin{equation}\label{remark}
P_y\caW(\sigma)(P_x)P_y=\left\{
\begin{array}{ll}\caW_t(\sigma)(P_x) & \textrm{ if } \sum_{\omega,k} \omega n_{\omega,k,t} =\epsilon_y-\epsilon_x
\\
0 & \textrm{ otherwise }\\
\end{array} \right.
\end{equation}
By Lemma \ref{connection1} one has
\begin{equation}
\overline{\rho}\big(j_t[P_x \otimes ( \otimes_{\omega,k}
\mathbf{P}_{\omega,k,t}(N_{\omega,k}) )]\big)=\rho
\big(\int_{E(\overline{N},t)}d\sigma \caW_t(\sigma) (P_x)\big)
\end{equation}
with $ E(\overline{N},t) \equiv \{\sigma \in \Omega \big|
n_{\omega,k,t}=N_{\omega,k} \} $. Writing out the left-hand side
of (\ref{sufficeproof}), using (\ref{remark}) and (\ref{rhodiag}) and adopting a shorthand for the summation
of (\ref{terrible})
\begin{eqnarray}
\overline{\rho}\big(j_t(\mathbf{P}^t(e))\big)&=&\sum_{\epsilon_x+\sum_{\omega,k}
\omega N_{\omega,k}=e} \rho \big(\int_{E(\overline{N},t)}
\caW(\sigma)(P_x)
\big)\\
&=&\sum_{\epsilon_x+\sum_{\omega,k} \omega N_{\omega,k,t}=e} \rho
\big(P_{e}\big[\int_{E(\overline{N},t)} \caW(\sigma)(P_x) \big] P_e\big)\\
&=&\sum_{x} \rho (P_{e}[e^{t\caL}(P_x)] P_e)=\rho
(P_{e}[e^{t\caL}(\mathbbm{1})]P_e)=\rho
(P_{e})\\
\end{eqnarray}
where we slightly extended the notation by putting $P_e \equiv 0$
whenever $e \notin \mathbf{Sp}(H_s)$. With the same convention,
the right-hand side of (\ref{sufficeproof}) can be expanded as
\begin{equation}
\sum_{\epsilon_x+\sum_{\omega,k} \omega N_{\omega,k,t}=e}(\rho
\otimes \phi)(P_x \otimes (\otimes_{\omega,k}
\mathbf{P}_{\omega,k,t}(N_{\omega,k}))=\rho(P_e)
\end{equation}
since only for $\overline{N}=0$, the expectation in the vacuum
state is non-zero. This proves (\ref{sufficeproof}) and thus also
(\ref{conserv}).
\subsection{Proof of Proposition \ref{ft1} and \ref{ft}}
For each $\kappa$, define on $\caB(\caH)$
\begin{equation} \label{deformedgen}
\mathcal{L}_{\kappa}(\cdot)=-i [H_{f},\cdot]+\sum_{\omega,k}\big(
e^{\kappa \beta_k \omega} V_{\omega,k}^{*}\cdot
V_{\omega,k}-\frac{1}{2}\{V_{\omega,k}^{*}V_{\omega,k},\cdot\}
\big)
\end{equation}

We rely on (\ref{maat}) to rewrite (\ref{rate function clas}) on
the space $\caB(\caH)$
\[
e(\kappa)=\lim_{t \rightarrow +\infty}
\log{\rho(e^{t\mathcal{L}_{\kappa}}(\mathbbm{1}))}
\]
Now the proof is a minor modification of the reasoning in
\cite{LS1}. Let $\caV \subset \caB(\caH)$ be the (real) subspace
of self-adjoint operators and remark that $\caL$ leaves $\caV$
invariant. Endow $\caV$ with the scalar product given by the
trace.
\[
\langle A,B \rangle= \mbox{tr}(A^*B) \in \mathbb{R}
\]
Remark that $e^{t\mathcal{L}_{\kappa}}$, viewed as a matrix on
$\caV$ has real entries. For $\kappa=0$, this matrix has a unique
maximal eigenvector (this eigenvector is $\mathbbm{1} \in \caV $
and the eigenvalue is $1$). It is clear that the entries of
$e^{\mathcal{L}_{\kappa}}$ depend on $\kappa$ through functions
like $e^{\kappa \beta_k \omega}$ for some $\beta_k,\omega \in
\mathbb{R}$. Hence the entries are real-analytic in $\kappa$ and
we can view $e^{\mathcal{L}_{\kappa}}$ as a analytic perturbation
of $e^{\mathcal{L}_{0}}$. Since both the multiplicity and the
dimension of the Jordan block associated to the eigenvalue $1$ is
$1$, one concludes from basic perturbation theory and from the
reality of the matrix entries, that there is a neighbourhood
$\caU$ of $0$ such that for $\kappa \in \caU$,
$e^{t\mathcal{L}_{\kappa}}$ has a unique maximal positive
eigenvalue $E_{\kappa}$ corresponding to a real eigenvector
$v_{\kappa}$. Then
\[
e(\kappa)=\lim_{t \rightarrow +\infty}\frac{1}{t}
\log{\rho(e^{t\mathcal{L}_{\kappa}}(\mathbbm{1}))}= \lim_{t \rightarrow +\infty}
\frac{1}{t}\log{\rho(v_{\kappa})e^{tE_{\kappa}}}=E_{\kappa}
\]
since we can choose $\caU$ small enough such that $\rho(v_{\kappa})$
is bounded away from zero. Because of
$V_{\omega,k}^*=e^{\frac{-\beta_k \omega}{2}}V_{-\omega,k}$, one
sees that
\begin{equation}\label{symmetry3}
L_{\kappa}(H_f)=L^*_{1-\kappa}(-H_f)
\end{equation}
Since the highest eigenvalues of $L_{\kappa}$ and $L^*_{\kappa}$
coincide, and since one can repeat the above argument for
$L^*_{\kappa}$, (\ref{symmetry3}) yields
\[
e_{\kappa}(H_f)=e_{1-\kappa}(-H_f)
\]
If the Hamiltonian $H_S$ is non-degenerate, one can strengthen
this result as follows (now completely following \cite{LS1}):
Since $\caL$ leaves the space of diagonal matrices in the
$H_S$-basis invariant, one can reduce $\caV$ to the space of these
diagonal matrices. Since we assume that the $V_{\omega,k}$ are
non-zero, we can apply the Perron-Frobenius theorem to establish
the existence of a unique maximal eigenvalue $E_{\kappa}$ and a
unique eigenvector $v_{\kappa}$ with positive entries.
(\ref{symmetry}) follows because of (\ref{symmetry3}), since on
diagonal matrices there is no difference between $i[H_f,\cdot]$
and $-i[H_f,\cdot]$, both being $0$. Finally, (\ref{symmetryhat})
is
a simple consequence of Lemma \ref{connection1}.\\
To prove (\ref{nonneg}) and (\ref{positive}), define the time
reversal $\theta$ on $\Omega_t$ as
\begin{equation}
\theta(t_1,\omega_1,k_1;\ldots, )=(t-t_n,-\omega_n,k_n;\ldots, )
\end{equation}
and the time-reversed path space measure $\Theta
\mathbb{P}^t(\cdot)=\mathbb{P}^t(\theta \cdot)$. Consider the
function $S_t:\Omega^{t} \rightarrow \mathbb{R}$
\begin{equation}
S^t_{\mu}(\sigma)=\log \frac{d\mathbb{P}^t_{\mu}(H_f)}{d(\Theta\mathbb{P}^t_{\mu})(-H_f) }(\sigma)
\end{equation}
with $\mu$ a density matrix on $\caB(\caH)$. Let $\mbox{Id} \equiv
\frac{\mathbbm{1}}{d}$ where $d$ is the dimension of the system space $\caH$. One checks explicitly
\begin{equation}
S^t_{\mbox{Id}}(\cdot)=\mathbf{w}_t(\cdot)
\end{equation}
Furthermore, because of Lemma \ref{prob}, one has for all $\mu$ on
$\caB(\caH)$
\begin{equation}
\int d\mathbb{P}^t_{\mu} e^{-S^t_{\mu}}=\int
d\Theta\mathbb{P}^t_{\mu}=1
\end{equation}
From Jensen's inequality one gets then for all $t \in
\mathbb{R}^+$
\begin{equation}
\int d\mathbb{P}^t_{\mbox{Id}}\mathbf{w}_t \geq 0
\end{equation}
Since by {\bf A1} the semigroup relaxes to $\rho$ exponentially
fast (see e.g. \cite{LS2}), we also have
\begin{equation}
\lim_{t \rightarrow +\infty}\frac{1}{t}\int
d\mathbb{P}^t_{\rho}\mathbf{w}_t= \lim_{t \rightarrow
+\infty}\frac{1}{t}\int d\mathbb{P}^t_{\mbox{Id}}\mathbf{w}_t \geq
0
\end{equation}
In the case of non-degenerate $H_S$, stationarity of $\rho$ yields
\begin{equation}
\int d\mathbb{P}^t_{\rho} S^t_{\rho}=\int d\mathbb{P}^t_{\rho}
\mathbf{w}_t
\end{equation}
Remark that the function $f: \mathbb{R} \rightarrow \mathbb{R}: x
\rightarrow e^{-x}+x-1 $ is positive for all $x$ and monotone for
$x \geq 0$. A Chebyshev inequality with $\delta >0$ 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.
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