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\begin{document}
\title[]{The Quantum Flow of Electronic transport I: The finite volume case}%
\author{J. Bellissard} \address{Institut de Recherche sur les Syst\`emes Atomiques ou
Mol\'eculaires Complexes, Universit\'e Paul Sabatier, Toulouse,
France}
%
%\email{}%
\author{R. Rebolledo }\address{Facultad de Matem\'aticas, Universidad Cat\'olica de
Chile, Casilla 306, Santiago 22, Chile}
\author{D. Spehner }\address{Facultad de F\'{\i}sica, Universidad Cat\'olica de
Chile, Casilla 306, Santiago 22, Chile}
\author{W. von Waldenfels }\address{Institut f\"ur Angewandte Mathematik,
Universit\"at Heidelberg, Germany}
%\email{}%
% 
\begin{abstract}
This article provides a model for the dissipative dynamics and
electronic transport in Anderson insulators. This model is based
upon quantum stochastic flows which satisfy a given stochastic
differential equation. The Lindblad master equation follows from
a projection of the quantum flow. For fixed parameters of the
model, the Gibbs state is the unique stationary state of the
quantum dynamical semigroup associated to the Lindblad generator.
The current produced by an applied constant electric field is
computed within the linear response approach. This yields a novel
expression for the DC conductivity.
\end{abstract}
\maketitle
\tableofcontents
% 
\section{Introduction}
\label{secintro}
This work is aimed at describing a family of quantum kinetic
models for dissipative transport of electrons in a solid with
finite volume. An important consequence is the rigorous proof of
the validity of the linear response theory leading to a formula
for the electrical conductivity of a GreenKubo type. This is a
preliminary step towards a generalization of our study to the
case of an infinite volume aperiodic solid.
\vspace{.1cm}
The construction of our model follows similar lines than those
currently used in Quantum Optics
\cite{CohenDupondRoc,Or,Gardiner,fagrebosaa,squeezing,
fagrebo,Haake}. We start from an {\em ab initio} theory involving
all degrees of freedom from both, the atom and the quantized
electromagnetic field. Tracing out the degrees of freedom of the
field in the corresponding Liouvillevon Neumann equation, an
integrodifferential equation is obtained for the density matrix
of the atom (NakajimaZwanzig equation) \cite{Haake}. By using a
Markov approximation and second order perturbation theory, a
master equation follows \cite{CohenDupondRoc,Or,Gardiner,Haake}.
This differential equation is generally of the Lindblad type
\cite{Lin}. It describes the single dynamics of the atom. It has
been shown that this kind of master equation appears in some
specific models involving the atom alone, submitted to random
timedependent external forces (noises)
\cite{Da,Carmichael,Hegerfeldt,PeGi,GRW,Ghirardi,Gorini76}. These
models involve classical stochastic noises and give rise to
effective non linear stochastic Schr\"odinger equations. Indeed,
these equations are non linear due to a renormalization
procedure introduced to keep the norm of the atomic wave packet
conserved at all time. In a nonequilibrium statistical approach,
one may also use equivalent linear stochastic Schr\"odinger
equations, for which the square norm of the wave packet is
conserved only after averaging over the noise
\cite{SpBe,SpBe2,Ghirardi,vanKampen}. The use of stochastic
Schr\"odinger equations is quite efficient as far as numerical
computations are concerned. Indeed, let $N$ denotes the number of
relevant atomic eigenstates. Then, while the master equation is a
system of $N^2$ coupled linear differential equations for the
density matrix, within the stochastic approach instead, one is
reduced to solve $N$ equations for each realization of the
noises, saving dramatically
for large $N$ both memory and computing time.
Electrons in a solid are also submitted to dissipative
mechanisms, such as electronphonon or electronelectron
collisions. Similarly, a master equation may be derived too,
whenever the coupling between electrons and the other degrees of
freedom are small enough for perturbation theory to hold. There
is, however, a major difference compared to atoms: while atoms are
usually represented by a system with a finite number of levels,
electrons in a solid may eventually explore an infinite lattice,
requiring the use of a Hamiltonian in an infinite dimensional
Hilbert space. For perfect crystals, however, a translation
symmetry argument allows to reduce the problem to a finite number
of bands, leading to a situation analogous to that of Quantum
Optics. For aperiodic solids, no translation symmetry survives in
general, so that the infinite volume limit becomes more involved.
In a series of previous works \cite{BeSBvE,SBBe98,SpBe,SpBe2,these},
kinetic models were proposed, along the lines of the Drude
\cite{Dru} and Lorentz models \cite{Lor}, in which the
dissipation mechanism was represented through a classical
external timedependent noise. The Relaxation Time Approximation
(RTA) was first investigated in \cite{BeSBvE} giving rise to a
Kubo formula that was used to analyze the width of plateaux in
the Integer Quantum Hall Effect, and in \cite{SBBe98} to analyze
the effect of anomalous transport arising in quasicrystals
\cite{SBBe95}. In \cite{SpBe}, the kinetic model was fully
developed for the impurity band electrons in a semiconductor. It
was proved that the instantaneous random evolution operators
exist and are almost surely unbounded at the infinite volume
limit, even though the evolution is squarenorm preserving after
averaging over the noise.
\vspace{.1cm}
In the present work, the classical noises are replaced by their
quantum counterpart, following the ideas developed in
Noncommutative Stochastic Calculus \cite{partha92,meyer95}. For indeed, the master equation of Lindblad type
may be obtained from a Schr\"odinger equation with a quantum
noise as will be shown here. This fact has been put forward in Quantum Optics by
Gardiner and Collet \cite{Gardiner85} (see \cite{Gardiner}).
This leads to the characterization of the socalled {\em Quantum Flow},
from which the {\em Quantum Dynamical Semigroup} is obtained by taking the average.
The Quantum Dynamical Semigroup has a generator of the Lindblad type \cite{Lin},
which is the adjoint of the generator appearing in the master equation.
The main advantages of this approach
are two. First, the instantaneous evolution is unitary at all
time, making the mathematical analysis much easier. This is
because the degrees of freedom that are the source of dissipation
in the real problem, are represented here by a Bose gas of
virtual particles driven by the same master equation. Secondly,
since Poisson and Wiener processes can be all obtained from
quantum noises defined over a suitable Fock space \cite{partha92,meyer95}, all previous
stochastic Schr\"odinger equations involving classical noises
are contained in the quantum stochastic differential
equation (see also \cite{Barchielli}). As long as quantum computing is not available,
this quantum stochastic differential equation approach will however
not be efficient in numerical calculations!
\vspace{.2cm}
We focus ourselves on
a specific family of models which are physically relevant for transport in
disordered solids in the strong localization regime (Anderson insulators) \cite{Shkolvskii}.
It is known that the electronic spectrum in such solids
is pure point near the Fermi energy and the corresponding
eigenfunctions are exponentially localized,
even in the infinite volume limit \cite{Anderson,Frolich}.
The electrons are coupled to low energy acoustic phonons. If an electric field is applied,
this coupling induces electronic hopping
from one localized state into another (hopping transport).
The corresponding Lindblad equation is
similar to the Quantum Optical Master
Equation~\cite{these}, although it is more conveniently written in the
second quantization framework.
\vspace{.2cm}
This paper is organized as follows. We start by motivating our
model by some heuristic considerations on open quantum systems in
section 2. These are well known results, connected with the
derivation of master equations, which throw light on the
underlying physical approximations. Section 3 provides the
mathematical framework of our analysis and introduce our model.
The specific quantum flow and its average, the quantum dynamical
semigroup, are obtained in the foregoing section. The generator
of the semigroup yields the master equation in Lindblad form. The
analysis of the equilibrium and stationary states is the subject
of the fifth section. The final section is devoted to the
computation of the conductivity in our model.
\vspace{.3cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Heuristics}
\label{secheuristics}
This section is devoted to a brief review of the basic ideas and
different approximations involved from {\em ab initio} in the
derivation of master equations for open quantum systems coupled
to infinite reservoirs. Excellent books on the subject are
available where the reader can find more details (see e.g.
\cite{Haake,CohenDupondRoc, Gardiner,Daviesbook}). Even though,
for the sake of clarity we feel necessary to state here our basic
approximations and assumptions on times scales, to precise for
which physical systems the kinetic models of the next section
could be relevant.
\subsection{Starting from Quantum Statistical Mechanics}
Within the framework of Algebraic Field Theory in Quantum
Statistical Mechanics \cite{Bratteli96,Haa}, a quantum system $S$
is described by its \CS $\Aa$ of observables, which will be
unital. One of the basic facts of Statistical Mechanics is that
there are conserved quantities beside the total energy, such as
the number of particles of a given species, the volume, or the
total momentum. In the algebraic framework, these conserved
quantities are the infinitesimal generators of a locally compact
Lie group $G$, called the {\em gauge group} acting on $\Aa$ via
automorphisms, $g\in G \mapsto \theta_g \in \Aut (\Aa)$. The
conservation of these physical quantities is then expressed by
demanding that $G$ be a symmetry of the evolution, namely
$\alpha_t^{(S)} \circ\theta_g = \theta_g \circ \alpha_t^{(S)}$ for
all $t\in\RM,\, g\in G$, where $\alpha_t^{(S)}$ is the 1parameter
group of automorphisms of $\Aa$ associated to the time evolution.
It is convenient to consider the time itself as one component of
$G$, so that $\theta_{(t,0)}=\alpha_t^{(S)}$. The corresponding
conserved quantity is the total energy. In most applications, $G$
can be taken as an Abelian group of the form $G= {\RM}^l \times
\TM^m$, where the first $l$ component correspond to first
integrals which are continuous whereas the $m$ others correspond
to the discrete ones. In such case, it is meaningful to consider
holomorphic continuation of $\theta$ in the complexification of
$G$. Then $\Aa^{(0)}$ will denote a dense subalgebra made of
elements $A \in \Aa$ such that the map $g\in G \mapsto
\theta_g(A)\in \Aa^{(0)}$ can be continued as an entire
holomorphic function on $\CM_{l,m}= \CM^{l}\times (\TM + i
\RM)^m$. The equilibrium state of $S$ is described by a KMS
state. Namely, it is a $G$invariant state $\omega$ on $\Aa$ for
which there exists $\beta \in \RM_+^{l+m}$ and, for any $A, \, B
\in \Aa^{(0)}$, a function $F_{A,B}(z)$ holomorphic and bounded
in the strip $\{ z \in \CM_{l,m} ; 0< \Im (z_i) <\beta_i,\,
i=1,\ldots ,l+m \}$, such that
%
$$F_{A,B}(g) = \omega (A\theta_g(B)) , \hspace{1cm}
F_{A,B} (g+i \beta) = \omega (\theta_g (B) A),
\hspace{1cm} g\in G \,. $$
%
The parameters $\beta = (\beta_1,\cdots, \beta_{l+m})$ are the
Lagrange multipliers associated to the conserved quantities, such
as the inverse temperature $1/k_B T$ ($k_B$ is the Boltzmann
constant), the chemical potential $\mu$, the pressure, etc...)
In the particular situation investigated in this work, the system
$S$ is an open system that is, it interacts with thermal baths
generically denoted $B$. An evolution towards a non equilibrium
steady state is created by turning on, at initial time, some
external forces, which may depend on time and space, and/or by
creating a gradient in the family of Lagrange multipliers $\beta$
defining the equilibrium of the system. In practice, such a
gradient is usually created by coupling the system to two or more
thermal bath with different values of $\beta$. The \CS of
observables for the total system $S \cup B$ is the tensor product
$\Aa \otimes \bigotimes_r\vN{B}_r$ of the observable algebras
$\Aa$ of the system and $\vN{B}_r$ of each bath. The quantum
evolution of the system and the family of baths is described
through a $1$parameter group of automorphisms $t\in\RR \mapsto
\alpha_t \in\Aut (\Aa \otimes \Bb)$. The automorphisms could
include external forces, denoted $\Ee$, in some cases. If the
external forces depend on time, the new evolution is no longer a
group, but it is a $2$parameters family of automorphisms
$\alpha_{t,s}$ such that:
%
\begin{enumerate}
\item \label{a1}$\alpha_{t,s} \in \Aut (\Aa \otimes \Bb)$ for all $t,s\in\RM$.
\item \label{a2}$\alpha_{s,s} = id\,$ for all $s\in\RM$.
\item \label{a3}$\alpha_{t,s} =\alpha_{t,u}\circ \alpha_{u,s}$ for
all $t,u,s \in\RM$.
\end{enumerate}
The equilibrium state of the bath is the tensor product
$\omega^{(B)}
= \bigotimes_r \omega^{(B_r)}_{\beta^{(r)}}$ of
the KMS state of each bath with Lagrange multipliers
$\beta^{(r)}$. These Lagrange multipliers may also depend on time.
Starting at time $t_0$, the state $\omega_0^{(S \, \cup B)} =
\omega^{(S)}_{\beta} \otimes \omega_{\beta}^{(B)}$, is
transformed into $\omega_0^{(S \, \cup B)} \circ \alpha_{t,t_0}$
at time $t$.
\subsection{The Markov approximation}
The next step consists in restricting the evolution to the system
$S$ only. In the Heisenberg picture, this can be done by defining
the unitypreserving linear maps $\Phi_{t,t_0}$ from $\Aa$ into
itself such that, for any $A\in \Aa$,
%
\begin{equation} \label{eqdyn_red}
\omega^{(S)}_\beta \circ \Phi_{t,t_0} (A)
=
\omega_0^{(S\cup B)} \circ \alpha_{t,t_0} (A\otimes \1)\, .
\end{equation}
%
The evolution of the observables is given by $A(t) =
\Phi_{t,t_0}(A(t_0))$. As shown by Kraus \cite{Kraus},
$\Phi_{t,t_0}$ is a {\em completely positive map} for any $t \geq
t_0$. Namely, for any matrix
$\hat{A}= ( A_{i,j} )_{i,j=1}^n$, the matrix
$(\Phi_{t,t_0} ((\hat{A}^{\ast}\hat{A})_{i,j}) )_{i,j=1}^n$
is positive in $\Aa \otimes M_n(\CM)$. In general, $\Phi_{t,t_0}$
is not an automorphism of $\Aa$ and do not satisfy the property
\eqref{a3} above.
To go forward, it is necessary to discuss more precisely the
physical situation. For indeed, different time scales can be well
distinguished in practice. The shortest one is the {\em
correlation time $\tau_c$ of the bath}, also called the memory
time. This time is of the order of, or is greater than $\hbar/(k_B
T)$. It depends only on the bath and on its interaction
Hamiltonian with $S$ (see \cite{CohenDupondRoc}). For times
$tt_0$ much bigger than $\tau_c$, one can neglect memory
effects \cite{Haake} and approximate the evolution $\Phi_{t,t_0}$
by some linear maps $\Phi_{t,t_0}^{(M)} :\Aa \rightarrow \Aa$
satisfying the properties \eqref{a2} and \eqref{a3} above for any
$t \geq u \geq s$. This approximation is called the {\em Markov
approximation}. Clearly, $\Phi_{t,t_0}^{(M)}$ must be completely
positive for times $t t_0 \gg \tau_c$, but its positivity could
fail for times $tt_0 \simeq \tau_c$ for which the Markov
approximation is not valid~\cite{Gnutzmann}. The evolution of
observables for times $tt_0 \gg \tau_c$ is described via a
linear master equation~:
%
\begin{equation} \label{eqmaster_eq}
\frac{d A }{dt}(t) = \Ll_t(A(t)) \,,
\end{equation}
%
where $\Ll_t$ is the generator of the Markov evolution. In
concrete cases, $\Ll_t$ is often computed to lowest order in the
systembath coupling constant $\lambda$, starting from the
NakajimaZwanzig equations and using a ``MarkovBorn''
approximation (see \cite{Haake} and references therein). This
approximation is valid provided $\lambda \tau_c \ll \hbar$. If
timedependent forces are applied, one must assume that the $x_i\in\Lambda$ and $x_j\in\Lambda$
typical variation time $\Delta t_\Ee$ of these forces is much
bigger than $\tau_c$. Then the generator $\Ll_t$ depends on time
$t$. Let $\alpha^{(S)}_{t,t_0}$ be the twoparameter group of
automorphisms of $\Aa$ describing the evolution of the system $S$
uncoupled to the baths and $H(t)$ be the associated Hamiltonian
(the time argument indicates that timedependent external forces
must be included). One can always write $\Ll_t$ as the sum of the
generator $\Ll_{H(t)}$ of $\alpha^{(S)}_{t,t_0}$ and of a {\em
dissipative generator} $\Dd_t$ describing the effect of the baths
on the dynamics of $S$,
%
$$ \Ll_t(A) = \Ll_{H(t)} + \Dd_t (A) \mbox{ , }\;\;\;\;\;
\Ll_{H(t)} (A) = i [ H(t), A ] \,. $$
%
By the timedependent version of Lindblad's theorem
\cite{Lin,Chebotarev}, if $\Aa$ is the von Neumann algebra of all
bounded operators on a given Hilbert space, the unity preserving
maps $\Phi_{t,t_0}^{(M)}$ are completely positive and ultraweakly
continuous (normal) for any $t \geq t_0$ and $(t,t_0) \mapsto
\Phi_{t,t_0}^{(M)}(A)$ is ultraweakly continuous for any $A \in
\Aa$, and $\Dd_t$ can be written in the canonical Lindblad form,
namely there exist some operators $L_{\ell}(t)$ acting on the
Hilbert space such that:
%%%%%%%%
\begin{equation}
\label{eqLindblad} \Dd_t (A) =
\sum_{\ell} \Bigl(
L_{\ell}^{\ast}(t) A L_{\ell}(t) 
\frac{1}{2} L_{\ell}^{\ast}(t) L_{\ell}(t) A
 \frac{1}{2} A \,L_{\ell}^{\ast}(t) L_{\ell}(t)
\Bigr) \,.
\end{equation}
%%%%%%%%
In general, the bath is also responsible for some renormalization
of the energies of $S$ (e.g. the Lamb shift in Quantum Optics),
even though, this effect can always be included in $H(t)$.
\subsection{The adiabatic approximation} \label{secadiabatic}
Other relevant time scales are given by the {\em Heisenberg time}
$\tau_{H(t)} = \hbar/\Delta E(t)$, where $\Delta E(t)$ is the
typical energy interval between nearest eigenvalues of $H(t)$, and
by the typical {\em relaxation time} $\tau_{rel}$ of the system.
That time is the typical rate of variation of a state of the
system under the dissipative generator $(\Dd_t)_\ast$. In the
perturbative regime $\lambda \tau_c \ll \hbar$, $\tau_{rel}$ is
of order $\hbar^2 \lambda^{2} \tau_c^{1}$ and is much bigger
than $\tau_c$~\cite{CohenDupondRoc}. If the Hamiltonian $H(t)$
have a discrete spectrum, $\lambda$ is small enough and the force
variation time $\Delta t_\Ee$ big enough for the inequalities
%
$$ \tau_{H(t)} \ll \tau_{rel} \mbox{ , }\;\;\;\; \tau_{H(t)} \ll
\Delta t_\Ee $$
%
to hold for any $t \geq 0$, one can further simplify
(\ref{eqmaster_eq}) by means of an adiabatic approximation. This
approximation is performed by making a local averaging in time as
follows. Let $\delta t$ be a time interval satisfying:
%
\begin{equation} \label{eqtimes}
\sup_{t \geq 0} \tau_{H(t)} \ll \delta t \ll \tau_{rel} \mbox{ ,
}\;\;\;\; \delta t \ll \Delta t_\Ee \,.
\end{equation}
%
One writes equation (\ref{eqmaster_eq}) in the interaction
picture by defining~:
$$ \tilde{A}_t(s) = e^{(ts) \Ll_{H(s)}} A(s) \,. $$
%with $x_{ij}$, $x_i$ given by \eqref{eqx_ij}.
Then the variation $\delta \tilde{A}_t$ of $\tilde{A}(s)$
between $t$ and $t+\delta t$ is given by~:
%
\begin{equation*} \label{eqmaster_eq_inter}
\frac{\delta \tilde{A}_t}{\delta t}
=
\int_t^{t+\delta t} \frac{d s}{\delta t}
\Bigl( e^{(st) \Ll_{H(s)}} \Dd_{s} \, e^{(st) \Ll_{H(s)}} \tilde{A}_t (s)
 (st) \frac{d \Ll_{H(s)}}{d s} \tilde{A}_t (s) \Bigr)
\,.
\end{equation*}
%
Since $\tilde{A}_t(s)$ and $H(s)$ vary in time respectively on
scales $\tau_{rel}$ and $\Delta t_\Ee$ much bigger than $\delta
t$ one can replace, up to a small error, $\tilde{A}(s)$ and
$H(s)$ in the integrand by their values at $s=t$. The last term
inside the parenthesis is of order $\delta t / \Delta t_\Ee$ and
can be neglected with respect to the first. Thus:
%
\begin{equation} \label{eqmaster_eq_inter2}
\frac{\delta \tilde{A}_t}{\delta t}
\simeq
\int_0^{\delta t} \frac{d s}{\delta t} \,
e^{s \Ll_{H(t)}} \Dd_{t} \, e^{s \Ll_{H(t)}} A(t)
\,.
\end{equation}
%
If one is only interested in the evolution of the observable $A$
on the time scale $\delta t$, one can approximate the left hand
side by the exact derivation $d \tilde{A}/ds _{t}=\Ll_{H(t)}
A(t) + d A/dt$ (coarse graining in time). The last step consists
in letting $\delta t$ (or, more precisely, $\delta t/
\tau_{H(t)}$) tend to infinity in the right hand side of
(\ref{eqmaster_eq_inter2}). The error in doing so is small,
because the matrix elements of the integrand in the basis of
eigenvectors of $H(t)$ are constant or oscillating functions of
$s$ with period $\tau_{H(t)} \gg \delta t$ or more. Therefore, the
separation of time scales (\ref{eqtimes}) allows to approximate
the master equation(\ref{eqmaster_eq}) by the often simpler
expression:
%
\begin{equation} \label{eqad_master_eq}
\frac{d A}{d t}
=
\Ll^{(ad)}_t A(t)
\mbox{ , }\;\;\;\; \Ll^{(ad)}_t
=
\Ll_{H(t)} + \widetilde{\Dd_t}
\,,
\end{equation}
%
where
%
\begin{equation} \label{eqexpect}
\widetilde{\Dd_t}
=
\lim_{\delta t \rightarrow \infty}
\int_0^{\delta t} \frac{d s}{\delta t}\,
e^{s \Ll_{H(t)}} \Dd_s \, e^{s \Ll_{H(t)}}
\mbox{ , } \Ll \in \Bb(\Aa)\,.
\end{equation}
%
We do not enter here into the mathematical details about the
topology involved in this definition (a more precise analysis is
postponed for a separate publication).
The adiabatic approximation is thus equivalent to replace the
generator $\Ll_t=\Ll_{H(t)}+\Dd_t$ by $\widetilde{ \Ll_t} =
\Ll_{H(t)} +\widetilde{\Dd_t}$. The dissipative part
$\widetilde{\Dd_t}$ is of Lindblad form (\ref{eqLindblad}) even
if $\Dd_t$ is not (i.e., if $\Phi^{(M)}_{t,t_0}$ violate
positivity for $t  t_0 \simeq \tau_c$). It is clear from
(\ref{eqexpect}) that $\Ll_{H(t)}$ and $\widetilde{\Ll_t}$
commute.
%
\begin{defn}
A generator $\Ll$ of a quantum dynamical semigroup is said to be
adiabatic with respect to $\Ll_H= i [H,\cdot]$ if $\Ll \circ\Ll_H
= \Ll_H \circ\Ll$.
\end{defn}
\vspace{0.1cm}
Thus, the system $S$ coupled to the baths $B_r$ can be described
by an adiabatic generator if the time scales of the problem are
such that~:
%
\begin{equation} \label{eqtime_separations}
\tau_c \ll \delta t \ll \Delta t_\Ee \mbox{ , }\;\;\;\; \sup_{t
\geq 0} \tau_{H(t)} \ll \delta t \ll \tau_{rel} \,,
\end{equation}
%
and if the derivative $d A/dt$ in eq. (\ref{eqad_master_eq}) is
understood as a coarsegrained rate of variation $\delta A/\delta
t$ on the time scale $\delta t$.
The second inequality in (\ref{eqtime_separations}) can never be
satisfied in any macroscopic solid, since $\tau_{H(t)}$ tends to
zero in the thermodynamic limits. However, the adiabatic
approximation still holds if any pair of eigenfunctions of $H(t)$,
associated with eigenvalues whose difference is of the order of,
or less than $\hbar/\delta t$, are nearly uncoupled by the
dissipative generator $(\Dd_t)_\ast$. This situation is realized
at very low temperature in disordered solids in the Anderson
localization regime. Indeed, the spectrum of $H$ is pure point at
the infinite volume limit and its eigenfunctions are
exponentially localized in this regime \cite{Frolich}. Lack of
symmetry makes degeneracies highly improbable, so that one can
ignore the effect of weak external forces on the spectrum, at
least at finite volume. The probability of bathdriven transition
from an eigenfunction into another is exponentially small with the
distance between the two localization centers~\cite{Shkolvskii}.
As a consequence of the finite density of states at the Fermi
level, eigenfunctions associated to close eigenvalues around the
Fermi energy are localized far apart. Thus, matrix elements of
the right hand side of (\ref{eqmaster_eq_inter2})
between two eigenfunctions with
eigenvalues very close to each other can be neglected. A more
quantitative analysis in the variable range hopping regime shows
that the adiabatic approximation is indeed valid at low enough
temperatures~\cite{these}. On the other hand, for disordered or
perfect metals and perfect quasicrystals, the eigenfunctions are
not exponentially localized and the adiabatic approximation
leading to (\ref{eqad_master_eq}) is expected to fail.
\subsection{The current and the conductivity}
The aim of linear transport theory is the study of the average
value of observables $\omega^{(S)}_\beta (A(t))$ or of their
derivative (currents) in the limit of weak external forces and/or
gradients. To be specific, let $S$ be a finite system composed of
charge carriers, of charge $q$, coupled to two baths of chemical
potentials $\mu_\diamond$, $\mu_\star$, respectively. At $t=0$,
these two chemical potentials coincide and $S$ is at equilibrium.
For $t>0$ they evolve towards different values. Let us study the
response of $S$ to an external timedependent electric field
${\Ee}(t)$ applied at $t=0$. The timedependent Hamiltonian is
$\secq{H}(t)= \secq{H}(0)  q \,{\Ee}(t)\, \XV$, where $\secq{H}(0)$
and $\XV$ are respectively the Hamiltonian of $S$ in absence of
field and the position operator of the charge carriers in second
quantization. The evolution is given by an adiabatic quantum
dynamical semigroup whose generator is $\Ll$. Charge transport can
only occur in a finite sample if there are exchanges of carriers
with the baths. Therefore it makes sense to separate $\Ll$ into
two parts:
%
$$ \Ll_t = \Ll^{(int)}_t + \Ll^{(ext)}_t = \Ll_{\secq{H}(t)} +
\Dd^{(int)}_t + \Dd^{(ext)}_t\,, $$
%
where $\Dd^{(int)}_t$ corresponds to the dissipative evolution
which keeps the number of particles in $S$ unchanged, while
$\Dd^{(ext)}_t$ instead, involves a variation in the number of
particles in $S$. In other words, $\Dd^{(int)}_t$ is a generator
characterized by operators $L_\ell$ which commute with the number
operator $\secq{N}$, and $\Dd^{(ext)}_t$ is the remaining part;
$\Ll_t=\Ll_{\secq{H}(t)} + \Dd^{(int)}_t $.
The adiabatic current operator is:
%
\begin{equation}
\Jad(t) = q \,\frac{\delta \XV(t)}{\delta t} \Bigr_{int}
=
q \, \Phi_{t,0}\circ \Ll^{(int)}_t (\XV)
\,.
\end{equation}
%
If one uses the adiabatic master equation (\ref{eqad_master_eq})
for the evolution of observables, the coarse grained velocity
$\delta \XV /\delta t= \Ll^{(int)}_t (\XV)$ must be used to
compute the current, instead of the usual ``instantaneous''
velocity $d \XV/dt= i [ \secq{H}(t), \XV ] = \Ll_{\secq{H}}(\XV)$.
$\Jad(t)$ is only defined for systems with finite volume $V$. In
the thermodynamic limit, the limit of $\Jad(t)/V$ as $V \uparrow
\RM^d$ should be studied. The current density at time $t$ is
\cite{these}:
%
\begin{equation}
{j}(t)
=
\omega^{(S)}_\beta ( \Jad(t) )
=
\omega^{(S)}_\beta \circ
\Phi_{t,0} \bigl( \Ll^{(int)}_t (\XV) \bigr) \,.
\end{equation}
%
In section \ref{sechopping_conduct}, we will be interested in the
DC current, i.e, to the case of an applied electric
field $\Ee(t)$
growing rapidly after $t=0$ and attaining a constant value $\Ee$ for $t \geq
\Delta t_\Ee >0$, where $\Delta t_\Ee$ satisfies
(\ref{eqtime_separations}). Similarly, the difference
$\mu_\star\mu_\diamond$ between the chemical potentials of the
two baths is assumed to be constant after a fixed time which is
supposed to be $>> \delta t$. More precisely, one assumes that
$f(t)$ is a uniformly increasing function equal to $1$ for $t >
\Delta t_\Ee$. The steadystate current density is:
%
\begin{equation}
j = \lim_{T \rightarrow\infty}
\frac{1}{T}\, \int_0^T \omega^{(S)}_\beta \circ
\bigl( \Phi_{\Ee,\mu_\diamond,\mu_\star} \bigr)_t
\bigl( \Ll^{(int)}_{\Ee} (\XV)dt
\bigr)
\,.
\end{equation}
%
Here $\Phi^{\Ee,\mu_\diamond,\mu_\star}$ is the quantum
dynamical semigroup of generator
$\Ll_{\Ee,\mu_\diamond,\mu_\star}= \Ll^{(int)}_{\Ee} +
\Ll^{(ext)}_{\Ee,\mu_\diamond,\mu_\star}$ describing the adiabatic
Markovian evolution of the system submitted to the
(timeindependent) electric field ${\Ee}$ and coupled to the two
bath of chemical potential $\mu_\diamond$, $\mu_\star$.
\section{Introducing the model}\label{secmat_framework}
\subsection{Preliminary notations}
Consider a fixed finite lattice $\Lambda\subset\Zd$ and let
denote $\h = \CC^\Lambda$ the complex finitedimensional Hilbert
space used to represent a single electron on this lattice, in the
tight binding representation. Throughout this paper, the customary notation $\bo{\h}$ will be used for the algebra of all bounded linear operators on the given Hilbert space $\h$.
We now go through the second quantization procedure on the
FermiFock space $\ffock{\h}$ \cite{Bratteli96}. We denote
$\ket{0}$ the vacuum vector in $\ffock{\h}$. Given $\psi\in\h$,
the associated {\em creation} (respectively {\em annihilation})
operator on $\ffock{\h}$ is denoted $\vcrea{c}{\psi}$ (resp.
$\vann{c}{\psi}$). To simplify notations we simply write
$\crea{c}{i}=\vcrea{c}{\psi_i}$, $\ann{c}{i}=\vann{c}{\psi_i}$,
for all $i\in I$. The second quantization of a given selfadjoint
operator $B\in\bo{\h}$ is selfadjoint and denoted $\secq{B}$: $$
\secq{B}=\sum_{i,j}\langle\psi_j,B\psi_i\rangle
\crea{c}{j}\ann{c}{i}\; \in\bo{\ffock{\h}}.$$
The space $\ffock{\h}$ will be referred from now on as the {\em
initial space} for the dynamics.
\subsection{Describing the dynamics}
The electrons on the lattice $\Lambda$ are coupled with two
baths denoted $\star$ and $\diamond$, which have the
same inverse temperature $\beta$ but may have different chemical
potentials $\mu_\star$ and $\mu_\diamond$.
To describe the dynamics of the electrons, we will consider three
main dynamical phenomena. Firstly, the Hamiltonian dynamics. Secondly, the
dissipative effects due to the coupling of electrons with phonons.
Thirdly, the exchanges of electrons between the system and the baths.
The non dissipative dynamics of the single
particle is described by a {\em bare} Hamiltonian $H$ acting on
$\h$, which has a set of eigenvectors $\seque{\psi}{i}{I}$ where
$I$ is a finite set of indexes, with corresponding eigenvalues
$\seque{E}{i}{I}$.
The second quantized $\secq{H}$ of $H$,
\begin{equation}\label{rep_hamiltonian}
\secq{H}=\sum_{i\in I}E_i\crea{c}{i}\ann{c}{i},
\end{equation}
describes the Hamiltonian dynamics of the whole system of electrons.
The selfadjoint bounded operator $\secq{H}$ has an orthonormal basis of
eigenvectors in $\ffock{\h}$, which we construct as follows. Call
$\CONFIG=\set{0,1}^I$ the set of {\em configurations} (also known
as the set of {\em occupation numbers}). Given any
$\eta=\seque{\eta}{i}{I}\in \CONFIG$, the vector
\begin{equation}\label{eta}
\ket{\eta}=\prod_{i\in I}{\crea{c}{i}}^{\eta_i}\ket{0},
\end{equation}
is an eigenvector of $\secq{H}$.
It is convenient to add
two indices $\star\not\in I$ and $\diamondsuit\not\in I$ referring to the two baths,
to the set $I$ labelling the eigenstates of $H$.
Call
$L=(I\cup\{\star,\diamondsuit\})\times (I\cup\{\star,
\diamondsuit\})$. Each couple $\ell=(i,j)\in L$ will be
interpreted as an oriented segment, labeling a transition from
$i$ to $j$.
The description of the dissipative effects and the exchanges of electrons with the baths
are obtained by means of
the collection of {\em jump operators}
$\joper{i}{j}$, and the corresponding {\em jump rates}
$\jrate{i}{j}\geq 0$. The jump operators are defined on the space
$\ffock{\h}$ as follows:
\begin{equation}\label{joperator}
\joper{i}{j}=
\begin{cases} \;\crea{c}{j}\ann{c}{i} &\;\text{if $i,j\in I,\;i\not=j$}\\
\;\crea{c}{j} &\;\text{if $i\in\set{\diamond,\star}$, $j\in I$}\\
\;\ann{c}{i} &\;\text{if $i\in I$, $j\in\set{\diamond,\star}$}\\
\;0 & \;\text{otherwise}.
\end{cases}
\end{equation}
The jump rates $\jrate{i}{j}$ will be supposed to satisfy the
following detailed balance hypothesis:
\begin{eqnarray}
\jrate{i}{j}&=&
e^{\beta( E_i  E_j )} \jrate{j}{i}
\label{eqnew_det_bal1}
\\
\jrate{\star}{j}&=&e^{\beta( E_j
\mu_\star)}\jrate{j}{\star}
\label{eqnew_det_bal2}
\\
\jrate{\diamond}{j}&=&e^{\beta( E_j
\mu_\diamond)}\jrate{j}{\diamond}
\label{eqnew_det_bal3}
\end{eqnarray}
for $i,j\in I,
\;i\not=j$.
The quantum evolution is given via a {\em quantum cocycle}
$\func{U}{t}{\RR_+}$, defined on $\bo{\htot}$, that we will
characterize in differential form according to the following
steps.
\begin{itemize}
\item Firstly, the jump operators $\joper{i}{j}$ will be used
to model the action of both the coupling with phonons and the loss or gain of electrons
in the baths. If $i,j\in I$, $\joper{i}{j}$ implement a quantum jump
of one electron from an eigenfunction $\psi_i$ into
another $\psi_j$. If $i$ or $j \in \{ \star, \diamond \}$,
$\joper{i}{j}$ implement a loss or a
gain of an electron in the baths.
The probability per unit of time that a
jump $\ell=(i,j)$ occurs is given by the jump rates $\jrate{i}{j}$.
\item Secondly, the evolution includes stochastic terms which
will be expressed through {\em quantum noises}, namely, by bosonic creation
and annihilation operator, $\crea{A}{\ell}(t)$ and $\ann{A}{\ell}(t)$ respectively, depending
on the time $t$ and on $\ell=(i,j)$. These stochastic terms represent a Bose
gas of virtual particles which mimics the effects of both the phonons and the baths
on the electrons.
\item Thirdly, $\secq{H}$, $\joper{i}{j}$ and $\jrate{i}{j}$ determine
a unique quantum cocycle $\func{U}{t}{0}$, which is the solution of the
{\em quantum stochastic differential equation} (QSDE):
%
\begin{equation}\label{eqQSDE}
dU (t)=
\left[ \left(  i \secq{H} + \secq{K} \right) dt + \sum_\ell
\left(
L_\ell^\ast \, d \ann{A}{\ell}(t)  L_\ell \, d \crea{A}{\ell}(t)
\right)
\right] U(t),
\end{equation}
%
with:
%
\begin{equation}\label{coefL}
L_\ell=\sqrt{\gamma_\ell} \,C_\ell, \;(\ell\in L)
\end{equation}
%
and
%
\begin{equation} \label{eqK}
\secq{K}
=
\frac{1}{2}\sum_\ell L^*_\ell L_\ell
=\frac{1}{2}\sum_\ell\gamma _\ell\, C_\ell^*C_\ell.
\end{equation}
%
The evolution operators $U(t)$ are bounded on the bigger space:
%
$$\htot=\ffock{\h}\otimes\bfock{L^2(\RR_+,\CC^L)},$$
%
where the
symbol $\bfock{\cdot}$ is used for the BosonFock space.
\item The cocycle determines a {\em quantum flow} $B\mapsto \flow{k}{t}{B}$
from $\bo{\ffock{\h}}$ into $\bo{\htot}$. The projection of this
flow on the algebra $\bo{\ffock{\h}}$ gives the {\em quantum
dynamical (or Markov) semigroup} used to describe the averaged
dynamics. This semigroup has an infinitesimal generator expressed
in Lindblad form (see \cite{Lin}).
\end{itemize}
\subsection{Physical discussion} \label{secphysdisc}
In this paper we are concerned with a disordered solid of finite volume in the
strong localization regime (Anderson insulator).
The oneelectron Hamiltonian $H$ is e.g. given by the Anderson Hamiltonian:
%
$$
H
=
\sum_{x \in \Lambda} \epsilon_x \ketbra{x}
+ \sum_{x,y \in \Lambda, x,y \;\text{n.n.}} t_{xy} \vketbra{x}{y} ,
$$
%
where $\ket{x}$, ($x \in \Lambda$) are the canonical basis vectors,
$t_{xy}$ are hopping
terms between nearest neighbors $x,y \in \Lambda$,
and $\epsilon_x$, ($x \in \Lambda$)
are independent identically distributed random energies.
The randomness of $\epsilon_x$ is due to disorder in
the solid and must be distinguished from the randomness
introduced above, which describes dissipative effects.
It is wellknown \cite{Anderson,Frolich} that, if the disorder is strong enough (i.e., if
$\langle \Delta \epsilon^2_x \rangle/t_{xy}$ is large enough),
the eigenfunctions $\psi_i$ of $H$ with energies close to the Fermi energy
$\mu$ are exponentially localized.
As a result, the jump rates $\jrate{i}{j}$
decrease exponentially with the distance $\vert x_ix_j\vert$ between their localization centers $x_i\in\Lambda$ and $x_j\in\Lambda$.
The two baths are considered to be connected to the solid through two electric wires. This induces the existence of two privileged wavefunctions $\psi_{i(\star)}$ and $\psi_{i(\diamond)}$, whose localization centers $x_{i(\star)}$ and $x_{i(\diamond)}$ are closer to the wires than those of the other wavefunctions. The
rates $\jrate{i}{\star}$ and $\jrate{i}{\diamond}$, $i \in I$,
decrease exponentially with the distance between
$x_i$ and the wires. Hence the rates $\jrate{i(\star)}{\star}$ and $\jrate{i(\diamond)}{\diamond}$
are much bigger than the other rates $\jrate{i}{\star}$ and $\jrate{i}{\diamond}$, respectively.
\subsection{The basic spaces}
Consider the BosonFock space $\bfock{L^2(\mathbb{R}_+,\CC^L)}$.
We denote coherent vectors there by the symbol $e(f)$, where $f\in
L^2(\mathbb{R}_+,\CC^L)$, that is $$e(f)=\sum_{n\geq
0}\frac{1}{\sqrt{n!}}f^{\otimes n}.$$ Moreover, we denote
$({\delta_\ell};\;\ell\in L)$ an orthonormal basis of $\CC^L$.
As we mentioned before, the total Hilbert space for the evolution
is $\htot=\ffock{\h}\otimes \bfock{L^2(\mathbb{R}_+,\CC^L)}$. The
space $\htot$ is generated by vectors of the form $\Psi\otimes
e(f)$, ($\Psi\in\ffock{\h},\;f\in L^2(\mathbb{R}_+,\CC^L)$).
Furthermore, given a set $A$, the notation $1_A$ is used for the
characteristic function of the set $A$. Given any
$\Psi\in\ffock{\h}$ and $f\in L^2(\mathbb{R}_+,\CC^L)$, the
elements of the form $\Psi\otimes e(f 1_{[0,t]})$ generate the
space:
%
$$
\htot_{t]}=\ffock{\h}\otimes \bfock{L^2([0,t],\CC^L)} ,\; ( t\geq 0 ).
$$
%
$\htot$ has the {\em continuous tensor
product property}, i.e.
%
$$\htot=\htot_{t]}\otimes\htot_{[t},$$
%
where $\htot_{[t}=\bfock {L^2([t,\infty[,\CC^L)}$.
For any $t\geq 0$,
the algebra of all linear bounded operators on $\htot_{t]}$
is denoted $\mathcal{B}_t=\bo{\htot_{t]}}$.
Any element $x_t \in \htot_{t]}$ will be identified with $x_t\otimes e(0_{[t}) \in \htot$,
where $e(0_{[t})$ is the vacuum in
$\htot_{[t}$.
Similarly, any
operator $B_t \in \mathcal{B}_t$ will
be identified with $B_t\otimes\1_{[t}$, where $\1_{[t}$ is the
identity operator acting on $\htot_{[t}$.
In particular, $\ffock{\h}$ is embedded in $\htot$
through the identification of $\Psi\in\ffock{\h}$ with
$\Psi\otimes e(0)\in\htot$ and, for any operator
$B\in\mathcal{B}_0=\mathcal{B}(\ffock{\h})$,
$B (\Psi\otimes e(f))$ means $(B\Psi)\otimes e(f)$.
With these
identifications, $(\mathcal{B}_t)_{\geq 0}$ is an increasing family of subalgebras
of $\mathcal{B}$.
Given a
family $V=\func{V}{t}{0}$ of operators, such that
$V(t)\in\mathcal{B}_t$ for any $t\geq 0$, then $B\mapsto
\flow{k}{t}{B}=V(t)BV(t)^*$ maps $\mathcal{B}_0$ into the algebra
$\mathcal{B}_t$.
For any $t\geq 0$, we define projections $E_t$ from $\htot$ onto
$\htot_{t]}$ by:
%
$$E_t\Psi\otimes e(f)=\Psi\otimes
e(f1_{[0,t]})\otimes e(0_{[t}) .$$
%
In addition, a projection
$\mathbb{E}_{t]}$ between the algebras of operators $\mathcal{B}$
and $\mathcal{B}_t$ is associated to each $E_t$ as follows:
%
$$
\mathbb {E}_{t]}(Z)=E_tZE_t, (Z\in\mathcal{B}).
$$
%
In
particular, $E_0$ and $\mathbb{E}_{0]}$ will be written simply as
$E$ and $\mathbb{E}$.
\subsection{Quantum Noises} \label{secQnoise}
With these notations, creation, annihilation and number (or
conservation) processes are respectively characterized as follows:
\begin{equation}\label{creation}
A^\dag_\ell (t)\Psi\otimes e(f)=\frac{d}{d\epsilon}\Psi\otimes e(f+\epsilon
1_{[0,t]}{\delta_\ell})\mid_{\epsilon =0},
\end{equation}
\begin{equation}\label{annihilation}
A_\ell(t) \Psi\otimes e(f)=\langle {\delta_\ell} 1_{[0,t]},f\rangle \Psi\otimes e(f),
\end{equation}
\begin{equation}\label{number}
N_{\ell, m}(t)=A^\dag_\ell (t)A_m(t),
\end{equation}
for all $t\geq 0$, $u\in\ffock{\h}$, $f\in
L^2(\mathbb{R}_+,\CC^L)$, $\ell, m\in\mathbb{L}$ where the scalar
product $\langle\cdot,\cdot\rangle$ is that of
$L^2(\mathbb{R}_+,\CC^L)$.
Notice that for any fixed $t\geq 0$, the ranges of $A^\dag_\ell
(t)$, $A_\ell (t)$, $N_{\ell m}(t)$ are included in $\htot_{t]}$.
These quantum noises satisfy two important families of relations:
\begin{itemize}
\item {\em The canonical commutation relations (CCR):}
\begin{equation}\label{CCR1}
[A_\ell(s),A^\dag_m (t)]=\langle
1_{[0,s]}{\delta_\ell},1_{[0,t]}e_m\rangle \1=\delta_{\ell m}\min
(s,t)\1,\;(s,t\geq 0).\end{equation}
\begin{equation}\label{CCR2}
[A_\ell(s),A_m(t)]=[A^\dag_\ell (s),A^\dag_m (t)]=0.
\end{equation}
%
From this two fundamental relations, it follows that
%
\begin{eqnarray*}
[A_k(s),N_{\ell m}(t)]
& = &
\delta_{\ell k} \min (s,t) A_m(t) \\
{[A^\dag_k(s),N_{\ell m}(t)]}
& = &
\delta_{\ell k}\min
(s,t) A^\dag_\ell (t).
\end{eqnarray*}
%
\item {\em The It\^o table for differentials:} An integration
theory for these quantum noises has been developed by Hudson and
Parthasarathy (see e.g. \cite{partha92}, which allows to give an
interpretation to $dA_\ell(s)$, $dA^\dag_\ell(t)$ in terms of
stochastic integrals). This leads to the following table which
synthesizes the fundamental formulae of this non commutative
extension of It\^o calculus.
$$\begin{array}{ccccc} \hline
\nearrow & dA_\ell (t) & dN_{\ell m} (t) & dA^\dag_\ell (t) & dt\1 \\
\hline
dA_\ell (t) & 0 & dA_\ell (t) & dt\1 & 0 \\
\hline
dN_{\ell m} (t) & 0 & dN_{\ell m} (t) & dA^\dag_\ell (t) & 0 \\
\hline
dA^\dag_\ell (t) & 0 & 0 & 0 & 0 \\
\hline
dt\1 & 0 & 0 & 0 & \\
\hline
\end{array}$$
\end{itemize}
\subsection{Cocycles}
The evolution of our system will be characterized by a quantum
cocycle which will lead to a quantum dynamical semigroup and a
flow. The reader who is less aware of this kind of description may
first think to the simplest case of a hamiltonian dynamics, described
by a unitary group $\func{U}{t}{0}$ defined by $U(t)=e^{it \secq{H}}$.
The evolution operators $U(t)$
satisfy the group property $U(t+s)=U(t)U(s)$,
which is a particular case of cocycle.
They are the solutions of the differential equation
$dU(t)=i \secq{H}\, dt \,U(t)$, with $U(0)=\1$ (see e.g. \cite{reedsimon}).
In our case, due to dissipative effects,
the process $U(t)$ will not correspond to a group,
but to a less simpler cocycle, and the above differential
equation will be replaced by:
%
\begin{equation} \label{schgen}
dU(t)=dY(t)\,U(t)\;, \;\; U(0) = \1,
\end{equation}
%
where $Y(t)$ is a process involving bosonic quantum noises. Since bosonic quantum
noises are unbounded operators,
it is not clear whether this equation is wellposed, as $U(t)$
may not preserve their domains.
Thus,
the rigorous treatment of
\eqref{schgen} is easier to achieve through solving the adjoint equation:
%
$$dV(t)=V(t)\,dZ(t) \;, \;\; V(0)=\1$$
%
with $Z(t)=Y(t)^*$. Indeed, it will be shown below that
its integral form:
%
\begin{equation}\label{integral_form}
V(t)=\1+\int_0^t V(s) \,d Z(s),
\end{equation}
%
is wellposed in the strong topology and
has a unique solution $V(t) \in \bo{\htot}$,
where all the stochastic integrals are computed
following the theory of Hudson and Parthasarathy (see
\cite{partha92,meyer95}).
Although this implies that \eqref{schgen} is welldefined in the weak sense (its solution
being $U(t) = V(t)^*$), it
is not necessarily wellposed in the strong sense as \eqref{integral_form} is. This is why,
whereas \eqref{schgen} is more according to the physical tradition,
mathematicians do prefer to work with \eqref{integral_form} instead
(see e.g. \cite{partha92,meyer95}).
To express the cocycle property rigorously, one needs to introduce
first the usual time shift in $L^2(\mathbb{R}_+,\CC^L)$:
%
$$\sigma_tf(s)=
\begin{cases} f(st) & \;\text{if $s>t$}\\
0 & \;\text{otherwise}\end{cases}. $$
%
Take then the second
quantization $\Gamma (\sigma_t)$ of this shift which yields to a
map $Z\mapsto\Gamma (\sigma_t)Z\Gamma (\sigma_t)^*$. This map can
be canonically extended to a map $\theta_t$, called {\em covariant quantum
shift} over the algebra $\mathcal{B}$.
A straightforward computation shows that
the projections $\mathbb{E}_{t]}$ satisfy the following {\em covariant property}:
\begin{equation}\label{cov_property}
\mathbb{E}=\mathbb{E}_{t]}\circ\theta_t, \;\;(t\geq 0).
\end{equation}
\begin{defn}
A family $(V(t))_{t\geq 0}$ (respectively $(U(t))_{t\geq 0}$) of
linear bounded operators in $\htot$ is a {\bf left cocycle}
(resp. {\bf rightcocycle}) if for every $s,t\geq 0$ it holds:
$$V(t+s)=V(s)\,\theta_s (V (t)),\;(\text{resp. } U(t+s)=\theta_s \,(U
(t))U(s)).$$
\end{defn}
Notice that if $V$ is a leftcocycle, its adjoint $U(t)=V(t)^*$ is a rightcocycle.
Cocycles are related to solutions of some classes of stochastic
differential equations, as it has been shown in the seminal
articles of Hudson and Parthasarathy (see e.g. \cite{partha92,meyer95}),
further extended by Fagnola to the case of
equations with unbounded coefficients (see \cite{fagnola90,fftesi}).
In our model, according to \eqref{eqQSDE},
the stochastic differential equation is given by
\eqref{integral_form} with:
%
\begin{equation}\label{def_x}
dZ(t)=\sem{\secq{H}}{\secq{K}}{\ell}{L}{A}
\end{equation}
%
and $L_\ell$ and $\secq{K}$ are defined in \eqref{coefL} and \eqref{eqK}, respectively.
It is worth noticing here that, like $\secq{H}$, all the
$L_\ell$'s and $\secq{K}$ are bounded operators in $\ffock{\h}$.
We now precise the conditions under which the equation \eqref{integral_form}
makes sense and has a solution which is a unitary cocycle.
\begin{thm}
Given $Z$ defined by \eqref{def_x}, the equation \eqref{integral_form} is
well posed and has a unique solution $V(t)$,
which is a unitary cocycle.
\end{thm}
\begin{pf}
This theorem is a corollary of Propositions 26.1, 26.2
and Theorem 26.3 of \cite{partha92}, from
which it also follows that
%
$$\sup_{0\leq t\leq
T}\;\;\sup_{\Psi \in\ffock{\h}:\;\norm{\Psi}\leq 1}\norm{V(t)\Psi\otimes
e(f)}<\infty,$$ for every $f\in L^2(\mathbb{R}_+,\CC^L)$, $T>0$.
Although the unitary property is a consequence of Theorem 26.3 of \cite{partha92},
it is instructive to check this for $V(t)$ in our
case. Indeed, since $V(0)=\1$, it suffices to prove that
$d(\adj{V}(t)V(t))=0=d(V(t)\adj{V}(t))$. We first apply the
integration by parts formula:
%
\begin{eqnarray*}
d(V(t)\adj{V}(t))
& = &
(dV(t))\adj{V}(t)+V(t)d\adj{V}(t)+dV(t)d\adj{V}(t) \\
& = &
V(t)\left[dZ(t)+dZ^*(t)+dZ(t)dZ^*(t)\right ]\adj{V}(t) \\
& = &
V(t)\left [2\secq{K}+\sum_\ell L^*_\ell L_\ell\right ]\adj{V}(t) \\
& = & 0.
\end{eqnarray*}
%
>From this it follows that $$V(t)\adj{V}(t)=\1,\;\text{for all
$t\geq 0$}.$$ Thus, each $V(t)$ is a coisometry.
To verify that $V(t)$ is unitary, it remains to show that it is
an isometry. For this we need to introduce the so called {\em
dual cocycle}. We start by taking the customary {\em time
reversal} operator in $L^2(\mathbb{R}_+,\ka)$:
\begin{equation}\label{rev}
(\mathbf{r}_tf)=\begin{cases}
f(ts) & \;\text{if $t\geq s$}\\
f(s) & \;\text{otherwise},
\end{cases}
\end{equation}
for any $f\in L^2(\mathbb{R}_+,\ka)$. Now, take the second
quantization $\Gamma (\mathbf{r}_t)$ of the above operator:
\begin{equation}\label{rev2q}
\Gamma (\mathbf{r}_t)e(f)=e(\mathbf{r}_tf),
\end{equation}
and finally define the {\em time reversal map} in $\mathcal{B(H)}$
as
\begin{equation}\label{revmap}
\mathcal{R}_t(X)=\Gamma (\mathbf{r}_t) X\Gamma (\mathbf{r}_t)^*,
\end{equation}
for any $X\in \mathcal{B(H)}$.
The {\em dual cocycle} of $V$ is defined as
\begin{equation}\label{dualcocycle}
\tilde{V}(t)=\mathcal{R}_t(V(t)^*),\;(t\geq 0).
\end{equation}
%
It is shown in e.g. \cite{fftesi} that $\tilde{V}$ is
indeed a cocycle that satisfies the stochastic differential
equation:
\begin{equation}\label{dualeq}
d\tilde{V}(t)=\tilde{V}\,dZ(t)^*.
\end{equation}
%
Moreover, \eqref{dualcocycle} yields
$\tilde{V}(t)\tilde{V}(t)^*=\mathcal{R}_t(V(t)^*V(t))$.This allows
to prove that $V(t)$ is an isometry if and only if $\tilde{V}(t)$
is a coisometry. For that we use \eqref{dualeq} and the formula of
integration by parts again:
%
\begin{equation}
d(\tilde{V}(t)\tilde{V}(t)^*)
=
\tilde{V}(t)\left
[dZ(t)^*+dZ(t)+dZ(t)^*dZ(t)\right ]\tilde{V}(t)^*
= 0.
\end{equation}
%
Therefore $d(\tilde{V}(t)\tilde{V}(t)^*)=0$ and $V$ is an
isometry. Since $V$ is also a coisometry, it is unitary.
\end{pf}
\section{The quantum dynamical semigroup}
\subsection{The quantum Markov flow and the associated semigroup}
\label{secMarkov_flow}
The cocycle property of $V(t)$ and $U(t)=V(t)^*$ and the
structure of projections on the given Fock space are related to
an extension of the concept of Markov process to a non
commutative framework. Indeed, let us introduce the {\em Quantum
Flow} $(k_t )_{t \geq 0}$ associated to the cocycle $U$
as follows: for any $t\geq 0$, $\flow{k}{t}{\cdot}$ is the map
$\mathcal{B}_0\to\mathcal{B}_t$ defined by:
%
\begin{equation}\label{flow}
\flow{k}{t}{B}=U(t)^*BU(t)
\end{equation}
for all $B\in\mathcal{B}_0$. It is clear that
$\flow{k}{t}{\cdot}\circ
\mathbb{E}_{t]}=\mathbb{E}_{t]}\circ\flow{k}{t}{\cdot}$.
The cocycle property implies that:
%
\begin{equation}\label{qds}
\qds{t}{B}=\mean{\flow{k}{t}{B}},\;\;(B\in\vN{B}_0,\;t\geq 0).
\end{equation}
%
defines a semigroup $\Phi$. It can be shown \cite{partha92,meyer95}
(see also \cite{fftesi})
that $\Phi$ is a {\em Quantum Dynamical Semigroup}, i.e., it is a
$w^\ast$continuous semigroup of completely positive
normal maps (the complete positivity is a consequence of Stinespring's theorem
\cite{Stinespring55}).
The stochastic differential equation satisfied by
$\flow{k}{t}{B}$ can be readily derived from the stochastic differential
equation \eqref{schgen} for $U(t)$. Notice first that $\flow{k}{0}{B}=B$ and
$\flow{k}{t}{\1}=\1$, since $U$ is unitary. Moreover, each
$\flow{k}{t}{\cdot}$ is a *homomorphism of algebras.
By means of the
integration by parts formula, one obtains formally:
%
\begin{eqnarray*}
d\flow{k}{t}{B}
& = &
dU(t)^*BU(t)+U(t)^* B \,dU(t)+dU(t)^* B\,d U(t)
\\
& = &
U(t)^* \bigl( dY(t)^*B + B\,dY(t) + dY(t)^*B \,dY(t) \bigr) U(t).
\end{eqnarray*}
%
In our model,
%
\begin{equation}\label{yeq}
d Y(t)= \secq{G}\,dt+\sum_{\ell\in L}\left(
\adj{L_\ell}\, d \ann{A}{\ell}(t)L_\ell\, d\crea{A}{\ell}(t)\right ),
\end{equation}
with:
\begin{equation}\label{G}
\secq{G}=i\secq{H}+\secq{K} = i\secq{H}  \frac{1}{2} \sum_{\ell \in L}
L_\ell^\ast L_\ell \,.
\end{equation}
%
Hence, since $U(t)$ commutes with $d \ann{A}{\ell}(t)$ and $d\crea{A}{\ell}(t)$,
%
\begin{eqnarray*}
U(t)^*dY(t)^*BU(t)
&=&\flow{k}{t}{G^* B} dt + \sum_\ell \bigl(
\flow{k}{t}{L_\ell B}dA^\dag_\ell(t)  \flow{k}{t}{L_\ell^*B}dA_\ell(t)
\bigr).
\\
U(t)^* B\,dY(t)U(t)
&=&
\flow{k}{t}{B G} dt + \sum_\ell
\bigl( \flow{k}{t}{BL_\ell^*}dA_\ell(t) \flow{k}{t}{BL_\ell} dA^\dag_\ell(t)
\bigr) .
\end{eqnarray*}
%
Moreover, by using the It\^{o} rules (section~\ref{secQnoise}),
%
$$
U(t)^*dY(t)^*BdY(t)U(t)
= \flow{k}{t}{\sum_\ell L_\ell^*BL_\ell}dt \,.
$$
%
We summarize the discussion before as follows.
%
\begin{thm}\label{Theflow}
The quantum flow
$k=(k_t)_{t \geq 0}$ defined by \eqref{flow} solves the
structure equation:
%
\begin{equation}\label{eqflow}
d\flow{k}{t}{B}=\flow{k}{t}{\lind{B}}dt+\sum_{\ell\in
L}\flow{k}{t}{\alpha_\ell(B)}dA^\dag_\ell(t)+
\sum_{\ell\in L}\flow{k}{t}{\beta_\ell(B)}dA_\ell(t),
\end{equation}
where:
\begin{equation} \label{L}
\lind{B} =
\liou{\secq{H}}{B}\diss{\ell}{L}{B},
\end{equation}
%
the operators $L_\ell$ being defined by \eqref{coefL} above,
and
%
\begin{eqnarray}\label{alpha}
\alpha_\ell (B)
& = &
[L_\ell,B]
\\
\label{beta}
\beta_\ell(B)
& = & [B,L_\ell^*].
\end{eqnarray}
%
$\alpha_\ell$ and $\beta_\ell$
are known as the structure maps of the flow.
As a result, $\lind{\cdot}$ is the infinitesimal generator of
the quantum dynamical semigroup $\Phi$ defined by \eqref{qds} associated to the flow.
\end{thm}
Note that all of the above maps $\mathcal{L}$, $\alpha_\ell$, $\beta_\ell$
apply the algebra $\vN{B}_0$ into itself in the finite volume
case.
In the infinite volume case their interpretation is
rendered delicate by difficulties that arise with the involved
domains.
The canonical form of the generator $\mathcal{L}$ of a general uniformly norm
continuous quantum dynamical semigroup has been discovered by Lindblad \cite{Lin}.
The generator \eqref{L} is particular case of the Lindblad form.
Its most noticeable property is that it satisfies the adiabatic condition
discussed in subsection~\ref{secadiabatic}.
\begin{prop}\label{commadiab}
${\mathcal{L}}$ is adiabatic with respect to $\mathcal{L}_{\secq{H}} (.) = i [ \secq{H}, . ]$,
i.e., for all $B\in\vN{B}_0$ it holds
\begin{equation}\label{adiab1}
\liou{\secq{H}}{\lind{B}}=\lind{
\liou{\secq{H}}{B}}.
\end{equation}
\end{prop}
\begin{pf}
Recall that $\lind{B}=\liou{\secq{H}}{B}+\mathcal{D}(B)$, where
$$\mathcal{D}(B)=\diss{\ell}{L}{B}.$$
It is enough to show \eqref{adiab1} for $\mathcal{D}$ instead of
$\mathcal{L}$. A straightforward calculation gives:
\begin{eqnarray} \label{adiab2}
&&[\secq{H},{\mathcal{D}(B)}]\mathcal{D}\left
([\secq{H},{B}]\right )\nonumber
\\
& &
=\frac{1}{2}\sum_\ell
\left\{[\secq{H},L^*_\ell L_\ell]B+2[\secq{H},L^*_\ell]BL_\ell+2L^*_\ell
B[\secq{H},L_\ell]B[\secq{H},L^*_\ell
L_\ell]\right\}.
\end{eqnarray}
To compute the righthand side of this equation, we have to determine
$[\secq{H},L_{ij}]$ and
$[\secq{H},L^*_{ij}L_{ij}]$, using \eqref{coefL} and \eqref{joperator}.
The canonical anticommutation relations yields
\begin{equation}
[n_i,\crea{c}{j}]=\delta_{ij}\crea{c}{j},\qquad[n_i,\ann{c}{j}]=
\delta_{ij}\ann{c}{j}.
\end{equation}
%
Thus, it follows easily that
$[\secq{H},\crea{c}{j}]=E_j\crea{c}{j}$ and
$[\secq{H},\ann{c}{j}]=E_j\ann{c}{j}$, for all $j\in I$ .
As a result,
\begin{equation}\label{comm_hl}
[\secq{H},L_{i j}]=\bigl( E_j\charac{I}(j) +E_i \charac{I}(i) \bigr) L_{i j},
\end{equation}
%
where $\charac{A}$ is the customary notation for the
characteristic function of the set $A$ (i.e. $\charac{A}(i)=1$ if
$i\in A$, and is zero otherwise).
We also obtain:
\begin{equation} \label{comm_hll}
[\secq{H},\adj{L}_{i j} L_{i j}]=0\;,\;\;(i,j\in L).
\end{equation}
%
Thus, using \eqref{comm_hl} and \eqref{comm_hll},
the right hand side of \eqref{adiab2} is zero.
\end{pf}
Call $\vN{A}_0=\vN{A}_{\secq{H}}$ the Abelian subalgebra of $\mathcal{B}_0$
generated by the bounded selfadjoint operator $\secq{H}$, which is included in its commutant
$\vN{A}^\prime_0$. The adiabatic property clearly implies that
$\qds{t}{\vN{A}_0^\prime} \subseteq \vN{A}_0^\prime$ for all $t\geq 0$.
\subsection{The master equation}
The predual algebra $\mathcal{B}_{0*}$ of $\vN{B}_0$ is isomorphic
to the space of traceclass operators $\trclass{\ffock{\h}}$ \cite{Bratteli96}.
Thus, any normal state $\omega$ will be identified with the unit
trace positive operator $\rho\in\trclass{\ffock{\h}}$ such that
$\omega (B)=\tr{\rho B}$, which we call a {\em
density matrix} on $\ffock{\h}$ in the sequel.
The predual semigroup $\Phi_*=(\Phi_{*t})_{t\geq 0}$ defined
through
\begin{equation}
\qds{*t}{\omega}(B)=\omega (\qds{t}{B}),\
(\omega\in\vN{B}_{0*},\; B\in\vN{B}_0),
\end{equation}
induces a weakly continuous semigroup of tracepreserving completely positive
maps
on $\trclass{\ffock{\h}}$. This
semigroup is generated by the adjoint map $\prelind{.}$,
\begin{equation}\label{pred}
\prelind{\rho}=\liou{\secq{H}}{\rho}\prediss{\ell}{L}{\rho} \;,
\;\;(\rho\in\trclass{\ffock{\h}}).
\end{equation}
%
Thus, the {\em Master Equation} of the evolution of states is
\begin{equation}\label{master}
\frac{d\rho_t}{dt}=\prelind{\rho_t},
\end{equation}
where $\rho_t=\qds{*t}{\rho}$, $t\geq 0$, with $\rho$ a density matrix.
\subsection{Unravelings of the quantum flow}
The quantum flow obtained in Theorem \ref{Theflow} contains
several stochastic models for the electronic transport
constructed before, using classical stochastic noises. Indeed, it
is well known that both, Wiener and Poisson processes can be
obtained from quantum noises as follows (see e.g. \cite{meyer95}.
Consider first the canonical multidimensional Wiener process (or
Brownian motion) $W=\seque{W}{\ell}{L}$. Denote $x\cdot y$ the
scalar product of two vectors in $\CC^L$. The Wiener space
$L^2(W)$ is generated by the family of random variables
\begin{equation}\label{Wexp}
\expsem{f}{W}{}=\exp \left (\int_0^\infty f(s)\cdot
dW(s)\frac{1}{2}\int_0^\infty\norm{f(s)}^2ds\right ),\;f\in
L^2(\RR_+,\CC^L).
\end{equation}
Moreover, the stochastic process
$$\expsem{f}{W}{t}=\expsem{f\charac{[0,t]}}{W}{}, \;(t\in \RR_+)$$
satisfies the stochastic differential equation
\begin{equation}\label{expsde}
d\expsem{f}{W}{t}=\expsem{f}{W}{t}f(t)\cdot dW(t),\;(t\geq 0, f\in
L^2(\RR_+,\CC^L))
\end{equation}
We thus define $T_W:\htot\to L^2(W)\otimes \htot_{0]}$, through
the equation
\begin{equation}\label{Udef}
T_W\Psi\otimes e(f)=\expsem{f}{W}{}\Psi,
\end{equation}
for all $f\in L^2(\RR_+,\CC^L)$.
Now, compute the action of $T_W$ on $\crea{A}{\ell}(t)\Psi\otimes
e(f)$: $$T_W\crea{A}{\ell}(t)\Psi\otimes
e(f)=T_W\Psi\otimes\frac{d}{d\epsilon}e(f+\epsilon
1_{[0,t]}{\delta_\ell})\mid_{\epsilon =0}$$
$$=\left (W_\ell (t)\int_0^tf_\ell (s)ds\right )\expsem{f}{W}{}\Psi.$$
And, $$T_W\ann{A}{\ell}(t)\Psi\otimes e(f)=\left (\int_0^tf_\ell
(s)ds\right )\expsem{f}{W}{}\Psi.$$
So that, $$T_W (\crea{A}{\ell}(t)+\ann{A}{\ell}(t))\Psi\otimes
e(f)=W_\ell (t)T_W\Psi\otimes e(f).$$
Moreover, consider the process $$Y_W(t)=Gt\sum_\ell L_\ell W_\ell
(t),$$ which is defined on the same probability space than $W$
and takes values in $\bo{\ffock{\h}}$. This process coincides
with the restriction to $\ffock{\h}=\htot_{0]}$ of the process
$T_WY(t)$, with $Y(t)$ as introduced in \eqref{yeq}.
A cocycle $U_W$ can be associated to the process $Y_W$ as the
unique solution to the classical stochastic differential equation
\begin{equation}\label{classic}
dU_W(t)=dY_W(t)U_W(t),\;\;U_W(0)=\1.
\end{equation}
This is a classical stochastic equation, although each $U_W(t)\in
L^2(W)\otimes\htot_{0]}$, because we are concerned with classical
noises. Call $\mean{\cdot}$ the expectation in the underlying
probability space (the Wiener space). The flow associated to the
cocycle $U_W$, termed as {\em classical flow}, is
$$k^W_t(B)=U_W^*(t)BU_W(t),\;(t\geq 0).$$ And the semigroup
associated to this flow is the family of completely positive maps
$$B\mapsto \mean{k^W_t(B)}.$$
Although the classical and quantum flow are different, they
generate the {\em same} quantum dynamical semigroup.
\begin{prop}
The classical flow satisfies the stochastic differential equation
\begin{equation}\label{clflow1}
dk^W_t(B)=k^W_t\left (\lind{B}\right )dt+\sum_\ell k^W_t\left
(\theta_\ell (B)\right )dW_\ell (t),
\end{equation}
where $\lind{B}$ is given by \eqref{L} and
\begin{equation}\label{theta_l}
\theta_\ell (B)=(\adj{L_\ell}B+BL_\ell),
\end{equation}
for all $\ell\in L$.
As a result,
\begin{equation}\label{qds_w}
\qds{B}{t}=\mean{k^W_t (B)},
\end{equation}
for all $t\geq 0$ and $B\in\vN{B}_0$.
\end{prop}
\begin{pf}
Indeed, using the classical It\^o's formula, we obtain
$$dk^W_t(B)=dU_W(t)^*BU_W(t)+U_W(t)^*BdU_W(t)+dU_W^*(t)BdU_W(t)$$
$$=U_W^*(t)dY_W^*(t)BU_W(t)+U_W^*(t)BdY_W(t)U_W(t)+U_W^*(t)dY_W(t)^*BdY_W(t)U_W(t).$$
\begin{eqnarray}
U_W(t)^*dY_W^*(t)BU_W(t)&=&U_W^*(t)G^*BU_W(t)dt\label{clf1}\\
&&\sum_\ell U_W^*(t)L_\ell^*
BU_W(t)dW_\ell (t)\nonumber\\&=&k^W_t(G^*B)dt\\
&&\sum_\ell k^W_t(L_\ell^* B)dW_\ell (t).\nonumber
\end{eqnarray}
\begin{eqnarray}
U_W^*(t)BdY_W(t)^*U_W(t)&=&U_W^*(t)BGU_W(t)dt\label{clf2}\\
&&\sum_\ell U_W^*(t)BL_\ell U_W(t)dW_\ell(t)\nonumber\\
&=&k^W_t(BG)dt\nonumber\\
&&\sum_\ell k^W_t(BL_\ell)dW_\ell(t).\nonumber
\end{eqnarray}
\begin{eqnarray}
U_W^*(t)dY_W^*(t)BdY_W(t)U_W(t)&=&\sum_\ell U_W^*(t)L_\ell^* BL_\ell U_W(t)dt\label{clf3}\\
&=&\flow{k}{t}{\sum_\ell L_\ell^*BL_\ell}dt.\nonumber
\end{eqnarray}
The addition of \eqref{clf1}, \eqref{clf2} and \eqref{clf3}
yields \eqref{clflow1}.
From \eqref{clflow1} and taking the expectation of the flow, it
follows that its associated semigroup is generated by
$\lind{\cdot}$. Thus, it coincides with $\Phi$.
\end{pf}
\vspace{0.3in} Similar computations are involved in the Poisson
unraveling of the quantum flow. Indeed, following \cite{meyer95},
the BosonFock space is related with the Poisson space as
follows. We denote by $N(t)=(N_\ell (t);\;\ell\in L)$ a canonical
multidimensional Poisson process. That is, the $N_\ell$'s are
independent Poisson processes with intensity rate 1, thus each
process $M_\ell (t)=N_\ell (t)t$ is a martingale satisfying the
classical It\^o table $$dM_\ell (t)dM_k(t)=\delta_{\ell k}dt.$$
Like in the previous Wiener case, the Poisson space $L^2(N)$ has
a total family $(e_N(f);\;f\in L^2(\RR_+,\CC^L))$ defined by
\begin{equation}\label{poisson_exp}
e_N(f)=e^{\sum_\ell \int_0^\infty f_\ell (s)ds}\prod_{s\geq
0}\left (1+f(s)\cdot \Delta N(s)\right ),
\end{equation}
where $f(s)\cdot \Delta N(s)=\sum_\ell f_\ell (s)\Delta N_\ell
(s)$, and $\Delta N_\ell(s)=N_\ell (s)N_\ell(s)$ denotes the
jump size (1 or 0) of $N_\ell$ at time $s\geq 0$.
Define $T_N:\htot\to L^2(N)\otimes \htot_{0]}$ by
\begin{equation}\label{deftn}
T_N\psi\otimes e(f)=e_N(f)\otimes \psi,
\end{equation}
for all $\psi\in\ffock{\h}$, $f\in L^2(\RR_+,\CC^L)$.
Define
\begin{equation}
Y_N(t)=Gt\sum_{\ell\in L}L_\ell M_\ell (t),\;(t\geq 0).
\end{equation}
A straightforward computation, following Meyer \cite{meyer95}, p.
73 to 76, shows that the process $Y_N$ coincides with the
restriction to $\htot_{0]}$ of the process $T_NY(t)$, with $Y(t)$
as introduced in \eqref{yeq}.
We can again consider a classical stochastic differential
equation, with Poissonian noise, to obtain a classical cocycle
$U_N$ and a flow:
\begin{equation}\label{p_cocycle}
dU_N=dY_N(t)U_N(t),\;U_N(0)=\1,
\end{equation}
\begin{equation}\label{p_flow}
k^N_t(B)=U_N(t)^*BU_N(t),\;(t\geq 0,\;B\in{\vN{B}}_0).
\end{equation}
Paraphrasing the computations performed in Proposition
\ref{clflow2} we obtain
\begin{prop}
The classical Poissonian flow satisfies the stochastic
differential equation
\begin{equation}\label{clflow2}
dk^N_t(B)=k^N_t\left (\lind{B}\right )dt+\sum_\ell k^N_t\left
(\vartheta_\ell (B)\right )dM_\ell (t),
\end{equation}
where $\lind{B}$ is given by \eqref{L} and
\begin{equation}\label{vartheta_l}
\vartheta_\ell (B)=(\adj{L_\ell}B+BL_\ell),
\end{equation}
for all $\ell\in L$.
As a result,
\begin{equation}\label{qds_N}
\qds{B}{t}=\mean{k^N_t (B)},
\end{equation}
for all $t\geq 0$ and $B\in\vN{B}_0$, where $\mean{\cdot}$
denotes the mean taken in the Poisson space.
\end{prop}
It is worth noticing that in all cases we recover the same
quantum Markov semigroup, moreover, both classical flows have the
same structure maps, even though they are defined in different
spaces.
%Here we need to quote the work of Dalibard, Spehner, Bellissard, Belavkin.
\section{Analysis of the equilibrium and stationary states}
Under the finite volume assumption, there is an infinite number
of stationary states for the class of quantum dynamical semigroups
$\Phi$ associated to the quantum flows considered in the previous section.
On the other hand, it is well known (see
for instance \cite{Bratteli96}, section 5.3.2) that
for finite systems, the
KMS condition completely characterizes the Gibbs states
$\omega_\beta=\tr{\rho_\beta\;\cdot}$, which is given by a density matrix
$\rho_\beta\in\trclass{\ffock{\h}}$ of the form:
%
\begin{equation}\label{sts}
\rho_\beta=\frac{1}{Z(\beta,\mu)}\exp\left (\beta (\secq{H}\mu \secq{N})\right).
\end{equation}
%
$\secq{N}=\sum_{i\in I} n_i$
is the total number operator and
$Z(\beta,\mu)=\tr{\exp (\beta (\secq{H}\mu \ \secq{N}))}$ is the grand
canonical partition function.
It is our purpose to prove that, if the two baths have identical chemical
potentials $\mu_\star=\mu_\diamond =\mu$,
the detailed balance
conditions \eqref{eqnew_det_bal1}, \eqref{eqnew_det_bal2} and \eqref{eqnew_det_bal3}
imply that
the Gibbs state $\omega_\beta$ is the {\em unique} stationary state of the semigroup $\Phi$.
We will actually prove a slightly more general result, showing that the
reverse statement is also true provided \eqref{eqnew_det_bal1},
\eqref{eqnew_det_bal2} and
\eqref{eqnew_det_bal3} are replaced by more general detailed balance conditions
in the case where $H$ has degenerate eigenvalues.
Call $\secq{K}_\mu=\secq{H}\mu \secq{N}$ and $\Phi_{K_\mu}$ the
$^*$automorphism group on $\mathcal{B}(\ffock{\h})$
defined by the generalized Hamiltonian
$\secq{K}_\mu$. In our case, $\exp (\beta \secq{H})$ is a traceclass
operator since $\ffock{\h}$ is finitedimensional. Then, from
\cite{Bratteli96}, Proposition 5.2.23, the Gibbs state $\omega_\beta$
is the unique
$\Phi_{K_\mu}$KMS state at the value $\beta$.
It is clear from \eqref{sts} that $\omega_\beta$ is faithful.
%a result of Takesaki, which is proved as Theorem 5.3.10 in
%\cite{Bratteli96}, shows that $\omega_\beta$ is faithful.
Call $\Phi^{(int)}$ the semigroup generated by
$\mathcal{L}^{(int)}(\cdot)=\lind{\cdot}\mathcal{D}^{(ext)}(\cdot)$,
where
\begin{equation}
\mathcal{D}^{(ext)}(B)
=
\frac{1}{2} \sum_{\ell \in (\{ \star, \diamond \} \times I) \,\cup\,
(I \times \{ \star, \diamond \})}
\bigl( L_\ell^\ast L_\ell B  2 L_\ell^\ast B L_\ell + B L_\ell^\ast L_\ell
\bigr).
\end{equation}
%
Let say that two indices $i,k\in
I\cup\set{\star,\diamond}$ are {\em connected} if there exists a
collection of indices $i_1,\ldots,i_{n}$ in $I$ such that
$\jrate{i_m}{i_{m+1}}>0$ for all $m=0,\ldots,n$, with $i_0=i$ and $i_{n+1}=k$.
The theorem below is the main result of this section.
\begin{thm}\label{stat_state_phi_phint}
Let $\mu_\diamond=\mu_\star=\mu$. Then the Gibbs state $\omega_\beta$ is a
stationary state of the quantum dynamical semigroup $\Phi$
defined by \eqref{qds} if and
only if the following detailed balance equations are satisfied:
\begin{eqnarray} \label{cnsdb1}
& &
\jrate{k}{l}= \jrate{l}{k} \,e^{\beta (E_kE_l)}\;,\;\;(l,k\in I, E_k \not= E_l)
\\
\label{cnsdb2}
& &
(\jrate{l}{\star}+\jrate{l}{\diamond})\, e^{\beta (E_l\mu)}

(\jrate{\star}{l}+\jrate{\diamond}{l})
=
\frac{1}{e^{\beta (E_l  \mu )} +1} \sum_{k \in I, E_k = E_i}
( \jrate{k}{l}  \jrate{l}{k} ) \;,\;\;(l\in I)
\\
\label{cnsdb3}
& &
\sum_{k,l \in I, E_k = E_l} \frac{\jrate{k}{l}  \jrate{l}{k}}
{e^{\beta (E_k  \mu )} + 1}
= 0 .
\end{eqnarray}
%
If these equations hold and every index $i\in I$ is connected to at least
one bath $\star,\diamond$,
then $\omega_\beta$ is the unique stationary state of $\Phi$.
Moreover, if the detailed balance equations hold,
then $\omega_\beta$ is also a stationary state
of the quantum dynamical semigroup $\Phi^{(int)}$.
Furthermore, given any other normal state with density matrix $\rho$, both
$\qds{*t}{\rho}$ and $\Phi^{(int)}_{*t}(\rho)$ converge in
the $w^*$topology towards $\rho_\beta$ as $t\to\infty$, i.e.
%
\begin{equation}
\tr{ \rho\, \Phi_t (B)} = \tr{ \rho\, \mean{k_t(B)} }
\rightarrow \tr{\rho_\beta\, B}
\end{equation}
%
as $t \to \infty$ for any $B \in {\mathcal{B}}_0$ (similarly for $\Phi^{(int)}$).
\end{thm}
It is worth noticing that if all the eigenvalues of $H$ are nondegenerate then the above detailed balance equations \eqref{cnsdb1}, \eqref{cnsdb2}, \eqref{cnsdb3}, reduce to the former equations \eqref{eqnew_det_bal1}, \eqref{eqnew_det_bal2}, \eqref{eqnew_det_bal3}.
The proof of the theorem before will be performed in several steps,
starting by the following proposition.
\begin{prop}\label{detailed_balance}
Let $\mu_\diamond=\mu_\star=\mu$. Then a stationary state of the form
\eqref{sts} exists for the semigroup $\Phi$ if and
only if \eqref{cnsdb1} to \eqref{cnsdb3} hold.
\end{prop}
\begin{pf}
We have to prove that \eqref{cnsdb1}, \eqref{cnsdb2} and
\eqref{cnsdb3} are equivalent to
$\mathcal{L}_\ast ( \omega_\beta ) = 0$, where $\prelind{.}= \liou{\secq{H}}{.}
+ {\mathcal{D}}_\ast(.)$ is given by \eqref{pred}.
A density $\rho_\beta$ given by \eqref{sts} can be written in the
form
\begin{equation}\label{rhopeta}
\rho_\beta=\sum_{\eta\in\CONFIG}p(\eta)\ketbra{\eta},
\end{equation}
where the coefficients $p(\eta)$ are positive and satisfy
$\sum_{\eta\in\CONFIG}p(\eta)=1$. Clearly, $\liou{\secq{H}}{\rho_\beta} = 0$.
By \eqref{joperator} and \eqref{coefL},
%
\begin{equation*}
L_\ell=
\begin{cases} \;\jrate{i}{j} \,\crea{c}{j}\ann{c}{i} &\;\text{if $\ell=(i,j) \in I^2$, $i\not=j$}\\
\;\jrate{i}{j}\,\crea{c}{j} &\;\text{if $\ell=(i,j) \in\set{\diamond,\star} \times I$}\\
\;\jrate{i}{j}\,\ann{c}{i} &\;\text{if $\ell =(i,j) \in I \times \set{\diamond,\star}$}\\
\end{cases}
\end{equation*}
%
It then follows from \eqref{pred} that $\tr{ \rho_\beta\,{\mathcal{D}} ( \vketbra{\eta}{\xi} )}=0$
for any $\eta, \xi \in \CONFIG$, $\eta \not= \xi$.
Thus
$\mathcal{L}_\ast ( \omega_\beta ) = 0$ is equivalent to:
%
$$\tr{e^{\beta
\secq{K}_\mu}\mathcal{D}(\ketbra{\eta})}=0\;,\;\;(\eta\in \CONFIG).
$$
%
We first notice that
\begin{equation}\label{sts1}
\kappa(\eta)
=
\tr{e^{\beta \secq{K}_\mu} \ketbra{\eta} }=
\prod_{i\in I}e^{\beta\eta_i
(E_i\mu)} =p(\eta)Z(\beta,\mu).
\end{equation}
%
We easily obtain the following identities:
%
\begin{eqnarray*}
\tr{e^{\beta
\secq{K}_\mu}\crea{c}{i}\ketbra{\eta}\ann{c}{i}}
& = &
(1\eta_i)\kappa(\eta) e^{\beta(E_i\mu)}
\\
\tr{e^{\beta
\secq{K}_\mu}\ann{c}{j}\ketbra{\eta}\crea{c}{j}}
& = & \eta_j\kappa(\eta)
e^{\beta(E_j\mu)}
\\
\tr{e^{\beta
\secq{K}_\mu}\crea{c}{i}\ann{c}{j}\ketbra{\eta}\crea{c}{j}\ann{c}{i}}
& = &
(1\eta_i)\eta_j\kappa(\eta)e^{\beta(E_iE_j)}
\\
\tr{e^{\beta
\secq{K}_\mu}n_i(\1n_j)\ketbra{\eta}}
& = &
(1\eta_j)\eta_i\kappa(\eta)=\tr{e^{\beta
\secq{K}_\mu}\ketbra{\eta}n_i(\1n_j)}.
\end{eqnarray*}
%
Therefore, by using \eqref{pred} and the above expression for $L_\ell$,
%
\begin{eqnarray} \label{eqtr_eta}
\nonumber
\tr{e^{\beta
\secq{K}_\mu}\mathcal{D}(\ketbra{\eta})}
&=&
\kappa(\eta)
\sum_{i,j \in I} \left(
(1\eta_i)\eta_j\jrate{i}{j}\,e^{\beta(E_iE_j)} \eta_i(1\eta_j)\jrate{i}{j}
\right)
\\
\nonumber
& &
+ \kappa(\eta) \sum_{j\in I} \left(
\eta_j(\jrate{\star}{j} + \jrate{\diamond}{j}) e^{\beta(E_j\mu)}
(1\eta_j)(\jrate{\star}{j} + \jrate{\diamond}{j} )
\right)
\\
& &
+ \kappa(\eta) \sum_{i\in I} \left(
(1\eta_i)(\jrate{i}{\star}+\jrate{i}{\diamond}) e^{\beta (E_i\mu)}
 \eta_i(\jrate{i}{\star}+\jrate{i}{\diamond})
\right)
\end{eqnarray}
%
To abbreviate, call
%
\begin{eqnarray} \label{eqtheta1}
\theta_{ji}
& = &
\jrate{i}{j} \,e^{\beta (E_iE_j)}\jrate{i}{j}
\\ \label{eqtheta2}
\theta_{j}
& = &
(\jrate{\star}{j} +\jrate{\diamond}{j})
e^{\beta (E_j  \mu)}(\jrate{j}{\star}+\jrate{j}{\diamond}),
\end{eqnarray}
%
for all $i,j\in I$. Now
we exchange the indices $j$ and $i$ in the second term inside the first sum
and in the last sum of formula \eqref{eqtr_eta}.
This shows that $\mathcal{L}_\ast ( \omega_\beta ) = 0$ if and only if
%
\begin{equation}
\sum_{i,j\in I}(1\eta_i)\eta_j\theta_{ji}+\sum_{j\in
I}\eta_j\theta_j\sum_{j\in I}(1\eta_j)\theta_je^{\beta
(E_j\mu)}=0 \;,\;\;(\eta\in \CONFIG).
\end{equation}
%
Since any $\eta\in\CONFIG$ can be written as a characteristic
function $\eta=\charac{T}$ for a given subset $T$ of $I$, the
above equation is equivalent to have
%
\begin{equation}\label{cnsfund}
\sum_{j\in T} \sum_{i\in I \setminus T} \theta_{ji}+\sum_{j\in
T}\theta_j\sum_{j \in I \setminus T} \theta_je^{\beta(E_j\mu)}=0.
\end{equation}
%
From \eqref{eqtheta1}, \eqref{eqtheta2} and \eqref{cnsfund},
it is clear that \eqref{cnsdb1} and
\eqref{cnsdb2} are sufficient conditions to have $\omega_\beta$
as a normal stationary state. We prove their necessity. Assume
$\omega_\beta$ to be stationary, then \eqref{cnsfund} holds. Take
$T=I$ in \eqref{cnsfund}, then
%
\begin{equation}\label{1}
\sum_{j\in I}\theta_j=0,
\end{equation}
and the choice $T=\emptyset$ gives
\begin{equation}\label{2}
\sum_{j\in I}\theta_je^{\beta(E_j\mu)}=0.
\end{equation}
%
Using \eqref{2}, \eqref{cnsfund} may be written equivalently as
%
\begin{equation} \label{eqsum_theta}
\sum_{j\in T}\sum_{i\in I \setminus T} \theta_{ji}+\sum_{j\in T}\theta_j
(1+ e^{\beta(E_j\mu)})=0.
\end{equation}
%
On the other hand, as $\theta_{jj}=0$, replacing $T=\set{l}$ in \eqref{eqsum_theta}, for
any fixed $l\in I$, yields
%
\begin{equation}\label{3}
\sum_{i\in I}\theta_{li}+\theta_l(1+e^{\beta (E_l\mu)})=0.
\end{equation}
%
Fix, $k,l \in I$ and use \eqref{cnsfund} with $T=\set{k,l}$. Then
\begin{equation}\label{4}
\sum_{i\not\in\set{l,k}}(\theta_{li}+\theta_{li})+\theta_l (1+e^{\beta
(E_l\mu)})+\theta_k(1+e^{\beta (E_k\mu)})=0.
\end{equation}
%
From equation \eqref{3} we derive:
%
$$
\sum_{i\not\in\set{l,k}}(\theta_{li}+\theta_{ki}) + \theta_{lk}\theta_{kl}
+ \left
[\theta_l(1+e^{\beta (E_l\mu)})+\theta_k(1+e^{\beta
(E_k\mu)})\right ]
= 0,
$$
%
so that \eqref{4} yields
%
$$\theta_{lk}=\theta_{kl}.$$
%
But
$\theta_{lk}=\theta_{kl}e^{\beta (E_lE_k)}$, therefore
%
$$
\theta_{lk}( e^{\beta (E_lE_k)}+ 1)=0.
$$
%
It follows that $\theta_{lk}=0$ if $E_k \not= E_l$, which proves \eqref{cnsdb1}.
Furthermore, \eqref{3} yields:
%
$$
\theta_l ( 1 + e^{\beta (E_l  \mu )} ) = \sum_{k, E_k = E_l}
\theta_{lk},
$$
%
from which \eqref{cnsdb2} follows. Finally, \eqref{cnsdb3} is a consequence of
\eqref{1}.
\end{pf}
We now show the uniqueness of the stationary state given by
\eqref{sts}. For this, we strongly use the fact that under the
hypotheses of Theorem \ref{detailed_balance} we have the
existence of a faithful stationary state, to apply the results of
\cite{frigerio78}. Indeed, in that case,
the set $\vN{A}(\Phi)$ of all fixed points of the
semigroup $\Phi$ is a von Neumann algebra. Let $\vN{N}(\Phi)$ be the algebra of all
elements $B\in\mathcal{B}_0$ such that
$\qds{t}{B^*B}=\qds{t}{B^*}\qds{t}{B}$, for all $t\geq 0$. Then, it is
clear that $\vN{A}(\Phi)\subseteq\vN{N}(\Phi)$. Frigerio (see
\cite{frigerio78}) was the first to notice that under the above hypothesis, the equality
between the two algebras implies the convergence towards the
equilibrium for the semigroup.
\begin{lem} [Frigerio] \label{lemFrigerio}
Let $\Phi$ be a quantum dynamical semigroup which admits a faithful normal stationary
state. Then
$\vN{A}(\Phi)$ is a von Neumann algebra
and there exists a unique $\Phi$invariant conditional expectation
$\condexp{\vN{A}(\Phi)}{\cdot}$ of $\mathcal{B}_0$ onto
$\vN{A}_0$ given by
\begin{equation}
\condexp{\vN{A}(\Phi)}{B}=w^*\lim_{T\to\infty}\frac{1}{T}\int_0^T\qds{t}{B}dt,
\end{equation}
for all $B\in\vN{B}_0$, where the integral is understood as a
weak* limit of Riemann sums.
Moreover, a normal state $\varphi$ on
$\mathcal{B}_0$ is $\Phi_*$invariant if and only if it is of the
form $$\varphi (\cdot)
=\varphi\vert_{\vN{A}(\Phi)}\circ\condexp{\vN{A}(\Phi)}{\cdot}.$$
In particular, if $\vN{A} = \CC\;\1$, the semigroup $\Phi$ has a unique stationary
state.
\end{lem}
For the proof of this lemma, see \cite{frigerio78}.
This reduces the study of normal $\Phi$invariant (stationary) states on
$\vN{B}_0$ to that of normal states on $\vN{A}(\Phi)$. It follows
that $\omega_\beta$ is the unique stationary state for $\Phi$ as soon as we prove that
$\vN{A}(\Phi)=\CC\1$.
In our case, since $\secq{H}$ is bounded and selfadjoint the
following result, borrowed to \cite{FagnolaRebolledo98} holds.
We use the customary notation $\{ B_1, \ldots, B_r \}^\prime$ for the algebra of
all operators $B \in {\mathcal{B}}_0$ commuting with all $B_i$, $i=1,\ldots ,r$
(generalized commutant).
\begin{lem} [FagnolaRebolledo] \label{atnt}
Under the hypotheses of Proposition \ref{detailed_balance}, the
semigroup $\Phi$ satisfies
\begin{eqnarray*}
\vN{A}(\Phi)
& = &
\set{\secq{H},L_\ell,L^*_\ell;\;\ell\in
L}^\prime,
\\
\vN{N}(\Phi)
& = &
\set{L_\ell,L^*_\ell;\;\ell\in
L}^\prime.
\end{eqnarray*}
\end{lem}
We refer the reader to \cite{FagnolaRebolledo98} for the proof
of this fact.
\begin{lem}\label{uniqueness}
Assume that all indices $i \in I$ are connected to at least one of the baths $\star,\diamond$.
Then
\begin{equation}\label{trl}
\set{L_\ell,L^*_\ell;\;\ell\in L}^\prime=\CC\1.
\end{equation}
%
As a result, $\vN{A}(\Phi)=\vN{N}(\Phi)=\CC\1$.
\end{lem}
\begin{pf} Let first assume that $\jrate{i}{\star} + \jrate{i}{\diamond} > 0$ for any $i \in I$.
Then $\set{L_\ell,L^*_\ell;\;\ell\in
L}^\prime\subseteq\set{\ann{c}{i},\crea{c}{i};\;i\in I}^\prime$.
But the set of bounded operators
$\set{\ann{c}{i},\crea{c}{i};\;i\in I}$ is irreducible on
$\ffock{\h}$ (see \cite{Bratteli96}, Proposition 5.2.2), so that
$\set{\ann{c}{i},\crea{c}{i};\;i\in I}^\prime=\CC\1$.
Let $i \in I$ be such that there is $k \in I$, $k \not= i$, for which $\jrate{i}{k} > 0$ and
$\jrate{k}{\star} + \jrate{k}{\diamond} > 0$.
Then $\crea{c}{k} \ann{c}{i}$, $\ann{c}{k}$ and thus
$\ann{c}{i} = \ann{c}{k} \crea{c}{k} \ann{c}{i} + \crea{c}{k} \ann{c}{i} \ann{c}{k}$
are in the algebra generated by the operators $L_\ell$ and $L_\ell^*$.
Therefore,
$\set{L_\ell,L^*_\ell;\;\ell\in L}^\prime \subset \set{\ann{c}{i},\crea{c}{i}}^\prime$.
Thus, if every $i$ is connected to $\star$ or $\diamond$, an iteration of the above
argument allows to complete the proof.
\end{pf}
\begin{pf} [of Theorem \ref{stat_state_phi_phint}]
This is a consequence of Lemmas \ref{lemFrigerio}, \ref{atnt} and \ref{uniqueness} and Proposition
\ref{detailed_balance}. The final statement is a straightforward
application of the main result of \cite{frigerio78} (see also
\cite{FagnolaRebolledo98}).
\end{pf}
\section{The hopping conductivity} \label{sechopping_conduct}
We derive in this section a formula for the DC conductivity. A
field $\Ee(t)$ is applied to the system at the initial time $t=0$
and becomes constant in time and equal to $\Ee$ for $t \geq
\Delta t_\Ee >0$. That is, we assume $\Ee (t)$ to be a continuous function which grows rapidly to the constant value $\Ee$. Since we are interested in the stationary regime, we will assume $t\geq \Delta t_\Ee$ or $t=0$
in any expression depending on the time $t$ below.
We denote by $\secq{H}$ the zerofield Hamiltonian and by $\secq{X}$
the position operator. $\secq{X}$ is the second quantized of the
usual position operator $X$ on $\h$, defined by $X \psi (x) = x
\,\psi(x)$, ($\psi \in \h$, $x \in \Lambda$). It is a
selfadjoint and bounded operator on $\ffock{\h}$.
The matrix elements of $X$ are denoted by:
%
\begin{equation} \label{eqx_ij}
x_{ij} = \langle \psi_i , X \psi_j \rangle,
\;\; x_{i} = x_{ii}, \;(i,j \in I).
\end{equation}
The chemical potentials of the two baths are initially equal to
$\mu$, so that the system is in the Gibbs equilibrium state
$\rho_\beta$ at $t=0$. They are then switched to different values
$\mu_\star$ and $\mu_\diamond$, allowing the system to evolve towards
the proper steady state.
To simplify the discussion, it is assumed that the only
eigenfunction $\psi_i$ from which electrons can be lost or gained
from the bath $\star$ (resp. from the bath $\diamond$) is
$\psi_{i(\star)}$ (resp. $\psi_{i(\diamond)}$). More precisely, the only
nonzero rates $\jrate{i}{j}$, ($i$ or $j \in \{ \star,
\diamond\}$) of jumps from or into the baths are
$\jrate{i(\star)}{\star}$, $\jrate{\star}{i(\star)}$,
$\jrate{i(\diamond)}{\diamond}$ and
$\jrate{\diamond}{i(\diamond)}$, with $i(\star), i(\diamond) \in I$,
$i(\star) \not= i(\diamond)$. This is a simplification of the real situation, in which
the rates $\jrate{i}{\star}$ (resp. $\jrate{i}{\diamond}$) are very small compared with
$\jrate{i(\star)}{\star}$ (resp. $\jrate{i(\diamond)}{\diamond}$) due to their exponential
decreasing (see subsection \ref{secphysdisc}).
The variations of chemical potential $\mu_\star\mu$ and
$\mu_\diamond\mu$ are chosen equal to the electrical potentials
$q \Ee x_\star$ and $q \Ee x_\diamond$ created by the field
$\Ee$ at the localization centers $x_{i(\star)}$ and
$x_{i(\diamond)}$,
respectively, where $q$ is the charge of the
electron.
The generalization to the case where several
eigenfunctions $\psi_i$ are connected to each bath is
straightforward, provided that the two sets $\{ x_i ; i \in
I,\jrate{i}{\star},\jrate{\star}{i} >0\}$ and $\{ x_i ; i \in
I,\jrate{i}{\diamond},\jrate{\diamond}{i} >0\}$ in $\RR^d$ are
perpendicular to the field $\Ee$.
In what follows, all quantities without an explicitly
mentioned $\Ee$dependence correspond to their values for
$\Ee=0$ and $\mu_\star=\mu_\diamond = \mu$.
The main result of this section is the following formula
for the hopping conductivity.
%
\begin{thm} \label{thKubo}
Assume that $\secq{H}$ and $H$ have nondegenerated eigenvalues. Let
moreover assume that
$\jrate{i}{j}(\Ee)=\jrate{i}{j} + \Oo(\Ee^2)$ if $i$ or $j \in \{
\star, \diamond \}$. Then $ j = \sigma \Ee + \Oo(\Ee^2) $ and
the conductivity $\sigma$ is given by the following formula:
%
\begin{equation}
\sigma
= \frac{q^2\beta}{V} \tr{ \rho_\beta\,\secq{P}
\bigl( \mathcal{\Ll}^{1} \lindint{\Xdiag}  \Xdiag \bigr)}
=  \frac{q^2\beta}{V} \tr{ \rho_\beta\,\secq{P}
\mathcal{L}^{1} \lindext{\Xdiag} },
\end{equation}
%
where $\secq{P}$ is the operator:
%
\begin{equation}
\secq{P} = \sum_{i,j \in I, i \not= j}
\jrate{j}{i} ( x_i  x_j ) ( 1  n_i ) n_j
\end{equation}
%
and $\Xdiag$ is the part of the position operator $\secq{X}$
which commutes with $\secq{H}$:
%
\begin{equation} \label{eqXdiag}
\Xdiag = \sum_{i \in I} x_i n_i \in \Aa_0^\prime.
\end{equation}
%
\end{thm}
This theorem is proved through the remain of this section.
Since only the part $\Xdiag \in \Aa_{0}^\prime$ contributes to the current,
one deals here with a ``classical current''. As noted in~\cite{these},
the contribution in $j$ of the
offdiagonal part of $\secq{X}$ (``coherent current'') vanishes because
of the
decoherence effects in the adiabatic regime,
which suppress for large times
the offdiagonal part of
observables in the preferential basis formed by the eigenvectors
of $\secq{H}$.
The physical interpretation of the operator $\secq{P}$
is the following. The
electrical power dissipated during
the evolution is given by the electronic observable
$ q \Ee \secq{P}$. Indeed, its matrix elements
$
q \Ee\, \bra{ \eta} \secq{P} \ket{\eta}
$ are the sum, over all $i,j \in I$, of the
rates $\jrate{i}{j}$ of an electronic jump from an {\em occupied} ($\eta_i=1$)
wavefunction $\psi_i \in \h$ into an {\em unoccupied} ($\eta_j=0$)
wavefunction
$\psi_j$, multiplied by the potential energy
$ q \Ee ( x_j  x_i )$ lost during this jump.
\subsection{Discussion}
Let us first discuss how the electric field $\Ee$
changes the parameters of the model studied in the previous sections.
%
\begin{itemize}
\item The Hamiltonian $\secq{H}$ becomes $\secq{H}_\Ee=\secq{H}  q \Ee \secq{X}$
in presence of the field. According to standard perturbation
theory, the eigenvectors $\psi_i(\Ee)$ and the (non
degenerated) eigenvalues $E_i(\Ee)$ of the oneelectron
Hamiltonian $H_\Ee=H  q \Ee X$ on $\h$ are given for any $i \in I$
by~\cite{Cohen}:
%
\begin{eqnarray}
\label{eqnewWF}
\psi_i(\Ee)
& = &
\psi_i  q \Ee \sum_{j \in I, j \not= i}
\frac{x_{ji} }{E_i  E_j} \psi_j + \Oo(\Ee^2)
\\
\label{eqnewE}
E_i(\Ee)
& = &
E_i q \Ee \, x_{i} + \Oo(\Ee^2).
\end{eqnarray}
%
\item The jump rates $\jrate{i}{j}$ are modified by the
electric field, in order to satisfy the conditions
\eqref{eqnew_det_bal1}, \eqref{eqnew_det_bal2} and
\eqref{eqnew_det_bal3} with the energies $E_i(\Ee)$. For the
choice $\mu_\star= \mu q \Ee x_\star$ and $\mu_\diamond= \mu q
\Ee x_\diamond$ of the two chemical potentials, the rates
$\jrate{i}{j}(\Ee)$, ($i$ or $j \in \{ \star , \diamond \}$) of
jumps from or into the baths differ from their zerofield values
only by terms of order $\Ee^2$. Actually, the change in the
chemical potential in \eqref{eqnew_det_bal2} for $j=i(\star)$
exactly compensates the change in the energy $E_j$ as given by
\eqref{eqnewE} (i.e., to first order in $\Ee$), and the same
holds true in \eqref{eqnew_det_bal3} for $j=i(\diamond)$. Thus
the conditions \eqref{eqnew_det_bal2} and
\eqref{eqnew_det_bal3} still hold for $\Ee \not= 0$ if
$\jrate{i}{j}(\Ee)=\jrate{i}{j} + \Oo(\Ee^2)$, ($i$ or $j \in \{
\star , \diamond \}$).
\item The jump operators $\joperl$ are built from the creation and annihilation
operators associated to the eigenvectors $\psi_i(\Ee)$ as explained in
section \ref{secmat_framework}.
It is easily seen from \eqref{joperator} and \eqref{eqnewWF} that
they are modified by the electric field as follows:
%
\begin{equation} \label{eqdeltajoper}
\joper{i}{j}(\Ee)
=
\joper{i}{j}  q \Ee
\sum_{k \in I, k \not= j} \frac{x_{kj}\,\charac{I}(j)}{E_jE_k}
\joper{i}{k}  q \Ee
\sum_{k \in I, k \not= i} \frac{x_{ik}\,\charac{I}( i)}{E_iE_k}
\joper{k}{j} + \Oo(\Ee^2).
\end{equation}
%
\end{itemize}
%
We denote by ${\Lle}$ the Lindbladian \eqref{L} corresponding to the values of
$\joper{i}{j}(\Ee)$ and $\jrate{i}{j}(\Ee)$ above.
The quantum dynamical semigroup
generated by ${\Lle}$ is denoted by ${\Phe}$.
\subsection{The linear response}
Our starting point is the expression \eqref{eqcurrentj} below for
the current density:
%
\begin{equation} \label{eqcurrentj}
j
=
\frac{q}{V} \timeav{t}
\tr{ \rho_\beta \, \qdse{t}{\lindeint{\secq{X}}} }
.
\end{equation}
%
\begin{prop}
The limit in \eqref{eqcurrentj} is welldefined.
If the oneelectron Hamiltonian $H$ has nondegenerated eigenvalues,
then it is equal, to lowest order in $\Ee$, to:
%
\begin{equation}
\label{eqcurrentj2}
j
=
\frac{q}{V} \timeav{t}
\tr{ \rho_\beta \, \qdse{t}{\lindeint{\Xdiag}} } + \Oo(\Ee^2).
\end{equation}
%
where $\Xdiag$ is given by \eqref{eqXdiag}.
Moreover, for any value of the field $\Ee$, the current $j$
vanishes if $\mu_\star=\mu_\diamond = \mu$.
\end{prop}
\begin{pf}
It follows from the existence of a faithful
normal stationary state for $\Phe$
(Theorem \ref{stat_state_phi_phint})
and from Frigerio's ergodic theorem (Lemma~\ref{lemFrigerio})
that the $\star$weak limits:
%
$$ \mathbb{E}^{\Aa(\Phe)} (\secq{B}) =
\text{$w^\ast$}\timeav{t} \qdse{t}{\secq{B}} $$
%
define a conditional expectation
$\mathbb{E}^{\Aa(\Phe)} :\Bb_0 \rightarrow
\Aa(\Phe) = \ker \Lle$.
In particular, the limit \eqref{eqcurrentj} is welldefined.
Assume that $H$ has nondegenerated eigenvalues. By standard
perturbation theory, the same holds true for $H_\Ee$ if $ \Ee $
is suffisciently small.
Since $\prelind{\rho_\beta}=\prelindint{\rho_\beta}=0$,
one has for any $i, j \in I$:
%
\begin{equation} \label{eqtrick}
\tr{ \rho_\beta \,\qdse{t}{\lindeint{ \crea{c}{i}(\Ee) \ann{c}{j}(\Ee) }} }
=
\tr{ \rho_\beta \, \qdse{t}{\lindeint{ \crea{c}{i} \ann{c}{j} }} }
+ \Oo(\Ee^2).
\end{equation}
%
Let us
show that the timeaverage of the lefthand side
vanishes if $i \not= j$. Indeed, by lemma \ref{atnt},
$\Aa(\Phe)$ is included in the algebra
$\Aa_{\secq{H}_\Ee}^\prime$
of all bounded operators commuting with $\secq{H}_\Ee$.
The conditional expectation
$\mathbb{E}^{\secq{H}_\Ee} : \Bb_0 \rightarrow \Aa_{\secq{H}_\Ee}^\prime$
thus satisfies $\mathbb{E}^{\Aa(\Phe)} \mathbb{E}^{\secq{H}_\Ee}
= \mathbb{E}^{\Aa(\Phe)}$. Moreover, by proposition \ref{commadiab},
$\Lleint$ commutes with $\mathbb{E}^{\secq{H}_\Ee}$. Hence:
%
$$
\tr{ \rho_\beta\, \mathbb{E}^{\Aa(\Phe)}
\lindeint{ \crea{c}{i}(\Ee) \ann{c}{j}(\Ee) }}
=
\tr{ \rho_\beta\, \mathbb{E}^{\Aa(\Phe)}
\Lleint \mathbb{E}^{\secq{H}_\Ee}
\bigl( \crea{c}{i}(\Ee) \ann{c}{j}(\Ee) \bigr) }
=
0
$$
%
for any $i,j \in I$, $i \not=j$.
Together with \eqref{eqtrick}, this proves \eqref{eqcurrentj2}.
Let $\mu_\star=\mu_\diamond = \mu$. It follows from the detailed balance
and Theorem \ref{stat_state_phi_phint} that:
%
$$
j=\frac{q}{V} \tr{ \rho_\beta^{(\Ee)} \lindeint{\secq{X}} },
$$
%
where
%
$$
\rho_\beta^{(\Ee)} = \frac{1}{Z(\beta,\mu,\Ee)}
\exp \bigl( \beta(\secq{H}_\Ee  \mu \secq{N} ) \bigr)
$$
%
is the unique stationary state of the semigroup
$\Phe$.
But this state satisfies $\prelindeint{\rho_\beta^{(\Ee)}} = 0$
by theorem \ref{stat_state_phi_phint} again.
This implies $j=0$.
\end{pf}
By subtracting to the righthand side of \eqref{eqcurrentj} the
vanishing current obtained for $\Ee=0$ and
$\mu_\star=\mu_\diamond=\mu$, one gets to lowest order in $\Ee$:
%
\begin{equation} \label{eqjj}
j
=
\frac{q}{V} \timeav{t} \tr{ \rho_\beta \,
\delta \Phi_t\circ \lindint{\Xdiag}
+ \rho_\beta \,\Phi_t\circ \delta \lindint{\Xdiag}
} + \Oo(\Ee^2),
\end{equation}
%
with $\delta \Llint \equiv \Lleint  \Llint$ and, by Abel's formula,
%
$$ \delta \qds{t}{\cdot}
\equiv
\qdse{t}{\cdot}  \qds{t}{\cdot}
=
\int_0^t \qds{ts}{ \,\delta \Ll (\qds{s}{\cdot})}ds
+ \Oo(\Ee^2).
$$
%
A straightforward computation using $\prelind{\rho_\beta}=0$ yields:
%
\begin{eqnarray}
\nonumber
j
& = &
\frac{q}{V} \biggl\{ \lim_{T \rightarrow \infty} \frac{1}{T}
\tr{ \rho_\beta\, \delta \Ll \circ (\Phi_T  \1 ) \circ\Ll^{2}
\circ\lindint{\secq{X}} }
 \tr{ \rho_\beta\, \delta \Ll \circ\Ll^{1} \circ\lindint{\Xdiag} }
\\
\nonumber
& &
+ \tr{ \rho_\beta\,\delta \lindint{\Xdiag} }
\biggr\}.
\end{eqnarray}
%
The limit vanishes by the convergence to equilibrium (see the
last statement of theorem \ref{stat_state_phi_phint}), leading to:
%
\begin{equation} \label{eqj_linear}
j
=
 \frac{q}{V} \tr{ \rho_\beta\,
\delta \Ll \circ{\Ll^{1} }\circ\lindint{\Xdiag}
 \rho_\beta\,\delta \lindint{\Xdiag} }.
\end{equation}
\subsection{The computation of the conductivity}
To go further, one needs to compute $\delta \Ll= \Lle\Ll$ to the
first order in $\Ee$. Let $\eta \in \CONFIG$. Since $\rho_\beta$
commutes with $\secq{H}$, $\tr{ \rho_\beta [ \delta \secq{H},
\ketbra{\eta} \,] }
=0$.
By \eqref{eqdeltajoper}, the trace:
%
$$
\tr{ \rho_\beta
\bigl(
2 \,\delta \joperl^\star \ketbra{\eta} \joperl
+ 2 \joperl^\star \ketbra{\eta} \,\delta \joperl
 \delta (\joperl^\star \joperl)\ketbra{\eta}
 \ketbra{\eta} \,\delta ( \joperl^\star \joperl )
\bigr)}
$$
%
vanishes as well for any $\ell \in L$. Taking into account that
$\delta \jrate{i}{j} = \Oo(\Ee^2)$ for $i$ or $j$ in $\{ \star,
\diamond \}$, this proves that, disregarding terms of order
$\Ee^2$,
%
\begin{eqnarray}
\nonumber
\tr{ \rho_\beta \,\delta \lind{\ketbra{\eta}} }
& = &
\tr{ \rho_\beta \,\delta \lindint{\ketbra{\eta}} }
\\
\nonumber
& = &
\sum_{i,j, i \not= j}
\delta \jrate{i}{j}
\tr{ \rho_\beta \,
\joper{i}{j}^\star \ketbra{\eta} \joper{i}{j}
 \rho_\beta \,n_i (1n_j ) \ketbra{\eta} }.
\end{eqnarray}
%
The representation of $\rho_\beta$ in the form \eqref{rhopeta}
yields:
%
\begin{eqnarray}
\nonumber
\tr{ \rho_\beta \,\delta \lind{\ketbra{\eta}} }
& = &
\sum_{\dir{i}{j}{\eta}} \delta \jrate{i}{j} p(\eta+ \charac{i}  \charac{j} )
 \sum_{\dir{j}{i}{\eta}} \delta \jrate{i}{j} p(\eta) + \Oo(\Ee^2)
\\
& = &
p(\eta) \sum_{\dir{i}{j}{\eta}}
\bigl( \delta \jrate{i}{j} e^{\beta(E_i  E_j )}
 \delta \jrate{j}{i}
\bigr) + \Oo(\Ee^2).
\end{eqnarray}
%
Now, as a result of \eqref{eqnewE} and of the detailed balance condition
\eqref{eqnew_det_bal1} for $\Ee=0$ and $\Ee \not= 0$
with the energies $E_i(\Ee)$,
%
$$
\delta \jrate{i}{j} e^{\beta(E_i  E_j )}  \delta \jrate{j}{i}
= \jrate{j}{i}
\delta \ln \frac{\jrate{i}{j}}{\jrate{j}{i}}
=  q \Ee \beta\,\jrate{j}{i} ( x_i  x_j ) + \Oo(\Ee^2).
$$
%
Then:
%
\begin{equation} \label{eqdeltaLl}
\tr{ \rho_\beta \,\delta \lind{\secq{B}} }
=
\tr{ \rho_\beta \,\delta \lindint{\secq{B}} }
=
q \Ee \beta\,\tr{ \rho_\beta \,\secq{P} \secq{B} } + \Oo(\Ee^2),
\end{equation}
%
with $\secq{B}= \ketbra{\eta}$. Since all eigenvalues of $\secq{H}$
are assumed to be nondegenerated, the projectors $\ketbra{\eta}$, ($\eta \in
\CONFIG)$ span $\Aa_0^\prime$. Thus \eqref{eqdeltaLl} is actually true
for any $\secq{B} \in \Aa_0^\prime$.
But the algebra
$\Aa_0^\prime$ is invariant under the adiabatic generators
$\Ll$ and $\Llint$ (section~\ref{secMarkov_flow}).
Moreover, $\Xdiag$ belongs to $\Aa_0^\prime$
by definition.
The result \eqref{thKubo}
is obtained from \eqref{eqj_linear} and \eqref{eqdeltaLl}.
\subsection{Link with the resistor network approach}
According to the work of Miller and Abrahams~\cite{Miller}, the
hopping current should be equal to the current of a network of
electrical resistors connecting all points $x_i$, ($i \in I$). It
is interesting to check explicitly this affirmation in our model.
Consider the normal state $\rho_\infty^{(\Ee)}$ on
$\Bb_0=\Bb(\ffock{\h})$ defined as:
%
$$
\tr{\rho_\infty^{(\Ee)} \secq{B} }
=
\timeav{t} \tr{ \rho_\beta \, \qdse{t}{\secq{B}} },
( \secq{B} \in \Bb_0 ).
$$
%
It is convenient to {\em define} some local chemical potentials $\delta \mu_i$
as follows:
%
$$
p_\infty^{(\Ee)}(\eta)
\equiv
\bra{\eta} \rho_\infty^{(\Ee)} \ket{\eta}
=
\frac{1}{Z(\beta,\mu)} \prod_{i \in I} e^{\beta(E_i  \mu  \delta \mu_i)\eta_i},
(\eta \in \CONFIG). $$
%
One easily shows that, to lowest order in $\delta \mu_i$,
$
p_\infty^{(\Ee)} (\eta) = p(\eta) ( 1 + \sum_{i \in I} \delta \mu_i
\eta_i)$.
A straightforward computation then gives:
%
$$
\tr{ \bigl( \rho_\infty^{(\Ee)}  \rho_\beta \bigr) \lindint{\secq{B}} }
=
\beta \sum_{i,j \in I, i \not=j} \jrate{j}{i} ( \delta \mu_i  \delta \mu_{j} )
\, \tr{ \rho_\beta (1  n_i) n_j \secq{B}}
$$
%
for $\secq{B} = \ketbra{\eta}$, $\eta \in \CONFIG$ and thus for
any $\secq{B} \in \Aa_0^\prime$. It follows from this formula
and from \eqref{eqjj} and \eqref{eqdeltaLl} that:
%
\begin{eqnarray}
\nonumber
j
& = &
\frac{q}{V}
\tr{ \bigl( \rho_\infty^{(\Ee)}  \rho_\beta \bigr) \lindint{\Xdiag} }
+ \frac{q}{V} \tr{ \rho_\beta\,\delta \lindint{\Xdiag} }
\\
\nonumber
& = &
\frac{q \beta}{2V} \sum_{i,j,k \in I, i \not=j} x_k
\bigl( \delta\mu_i q \Ee x_i  \delta \mu_j + q \Ee x_j \bigr)
\bigl\{ \jrate{j}{i} \tr{ \rho_\beta\,(1n_i ) n_j n_k }
\\
\nonumber
& &
 \jrate{i}{j} \tr{ \rho_\beta\,(1n_j ) n_i n_k } \bigr\}.
\end{eqnarray}
%
The expression inside the brackets is equal to $( \charac{k}(j) 
\charac{k}(i) ) R_{ij}^{1}/(q^2 \beta)$, where:
%
$$
R_{ij}^{1} = q^2 \beta \,\jrate{i}{j} f_\mu(E_i) (1f_\mu(E_j) )
$$
%
and $f_\mu(E)$ is the FermiDirac function:
%
$$
f_\mu(E) = \frac{1}{1 + e^{\beta (E\mu)}}.
$$
%
As a result,
%
\begin{equation}
j
=
\frac{1}{2qV} \sum_{i,j \in J, i \not= j}
R_{ij}^{1} (x_i  x_j )
\bigl( \delta\mu_j q \Ee x_j  \delta \mu_i + q \Ee x_i \bigr)
\end{equation}
%
It is easy to show~\cite{these} that the expression one the righthand side
is the average, over all surfaces $S$
perpendicular to $\Ee$, of the current density
through $S$ obtained by Miller and Abrahams (\cite{Miller}, formula III1).
The quantity inside the second parenthesis is the total difference of potential
(electrical and chemical) between the localization centers $x_i$ and $x_j$.
Thus, the current is indeed the same as that of a resistor network,
each site $x_i$ being connected to all sites $x_j$,
($j \in I$) by some electric wires of resistances $R_{ij}$.
\vspace{.2cm}
\noindent {\bf Acknowledgements:} This research has been
partially supported by the ``C\'atedra Presidencial en Ciencias''
of R. Rebolledo, ``C\'atedra Presidencial en Ciencias'' of F.
Claro, FONDECYT grant 1990439, ECOS program, Institut
Universitaire de France. Moreover, D.S. acknowledge the financial
support by a Fondecyt postdoctoral grant no. 3000035.
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\end{document}