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The fluctuation theorem for currents in open quantum systems

Identifieur interne : 001A89 ( Istex/Corpus ); précédent : 001A88; suivant : 001A90

The fluctuation theorem for currents in open quantum systems

Auteurs : D. Andrieux ; P. Gaspard ; T. Monnai ; S. Tasaki

Source :

RBID : ISTEX:C4C2D904D831B205DB5678ACBB39DBD250EB9995

Abstract

A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the OnsagerCasimir reciprocity relations to the nonlinear response coefficients.

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DOI: 10.1088/1367-2630/11/4/043014

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ISTEX:C4C2D904D831B205DB5678ACBB39DBD250EB9995

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<title>The fluctuation theorem for currents in open quantum systems</title>
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<author address="nj299837ad1" email="nj299837ea1">
<first-names>D</first-names>
<second-name>Andrieux</second-name>
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<author address="nj299837ad1" alt-address="nj299837aad3" email="nj299837ea2">
<first-names>P</first-names>
<second-name>Gaspard</second-name>
</author>
<author address="nj299837ad2" email="nj299837ea3">
<first-names>T</first-names>
<second-name>Monnai</second-name>
</author>
<author address="nj299837ad2" email="nj299837ea4">
<first-names>S</first-names>
<second-name>Tasaki</second-name>
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<short-author-list>D Andrieux
<italic>et al</italic>
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<orgname>Université Libre de Bruxelles</orgname>
, Code Postal 231, Campus Plaine, B-1050 Brussels,
<country>Belgium</country>
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<address id="nj299837ad2" showid="yes">Department of Applied Physics,
<orgname>Waseda University</orgname>
, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555,
<country>Japan</country>
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<heading>Abstract</heading>
<p indent="no">A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the Onsager–Casimir reciprocity relations to the nonlinear response coefficients.</p>
</abstract>
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<body refstyle="numeric">
<sec-level1 id="nj299837s1" label="1">
<heading>Introduction</heading>
<p indent="no">Quantum systems can be driven out of equilibrium by time-dependent perturbations, by interaction with reservoirs at different chemical potentials or temperatures, or by combining both. In the latter cases, the quantum system is open and currents of energy and particles flow across the system. Such processes take place in mesoscopic electronic conductors as well as in chemical reactions. These nonequilibrium processes can be characterized by the relations linking their currents to the differences of chemical potentials. These relations may be linear in the case of Ohm's law, but are typically nonlinear, which defines the nonlinear response coefficients.</p>
<p>Alternatively, the full counting statistics of the particles transferred between the reservoirs can be considered. This statistics aims to characterize the transfers of particles in terms of the functions generating all the statistical moments of the fluctuating numbers of particles. The knowledge of this generating function gives access not only to the conductance and the noise power but also to higher-order moments and thus to the properties of nonlinear response. The full counting statistics has attracted considerable theoretical interest and is also envisaged in experimental measurements [
<cite linkend="nj299837bib1">1</cite>
]–[
<cite linkend="nj299837bib3">3</cite>
]. After the pioneering work of Levitov and Lesovik [
<cite linkend="nj299837bib4">4</cite>
], several methods have been developed in order to obtain the full counting statistics in mesoscopic conductors. One of them is based on Keldysh Green's function formalism, in which an expression for the generating function has been obtained within a semiclassical approximation [
<cite linkend="nj299837bib5">5</cite>
]–[
<cite linkend="nj299837bib9">9</cite>
]. The full counting statistics can also be obtained on the basis of quantum Markovian master equations describing the fluctuations of the currents [
<cite linkend="nj299837bib10">10</cite>
], as well as in terms of stochastic path integrals [
<cite linkend="nj299837bib11">11</cite>
]. The generating function obtained in the approaches using the semiclassical approximation or the Markovian master equation has been shown to obey a symmetry relation as the consequence of time reversibility [
<cite linkend="nj299837bib12">12</cite>
]. In nonequilibrium statistical mechanics, this relation is known as the fluctuation theorem which has been established for several classes of systems. These latter are either time-independent deterministic [
<cite linkend="nj299837bib13">13</cite>
]–[
<cite linkend="nj299837bib15">15</cite>
] or stochastic systems sustaining nonequilibrium steady states [
<cite linkend="nj299837bib16">16</cite>
]–[
<cite linkend="nj299837bib23">23</cite>
], or time-dependent Hamiltonian or stochastic systems, in which case the fluctuation theorem is closely related to the Jarzynski nonequilibrium work theorem [
<cite linkend="nj299837bib24">24</cite>
]–[
<cite linkend="nj299837bib27">27</cite>
]. Similar symmetry relations have been considered for continuous-time random walks [
<cite linkend="nj299837bib28">28</cite>
]. Quantum versions of the fluctuation theorem and the Jarzynski nonequilibrium work theorem have also been obtained [
<cite linkend="nj299837bib29">29</cite>
]–[
<cite linkend="nj299837bib47">47</cite>
]. Moreover, a further relationship has recently been proved for time-dependent quantum Hamiltonian systems [
<cite linkend="nj299837bib48">48</cite>
], allowing the derivation of the Green–Kubo formulae and the Onsager–Casimir reciprocity relations for the linear response coefficients [
<cite linkend="nj299837bib49">49</cite>
]–[
<cite linkend="nj299837bib52">52</cite>
].</p>
<p>An open question is to bridge the gap separating the time-dependent situations from the nonequilibrium steady states, which are expected to be reached in the long-time limit. The problem is to deal with nonequilibrium steady states without relying on the semiclassical or Markovian approximations or on the neglect of the energy or particle content of the subsystem coupling the reservoirs.</p>
<p>In the present paper, our aim is to prove the fluctuation theorem for the currents in open systems obeying the Hamiltonian quantum dynamics and sustaining nonequilibrium steady states in the long-time limit. We start by considering a time-dependent quantum system in contact with energy and particle reservoirs at different temperatures and chemical potentials. The amounts of energy and particles that are exchanged between the initial and final times are determined by quantum measurements. This framework is similar to the one considered by Kurchan to obtain a fluctuation theorem for quantum systems [
<cite linkend="nj299837bib29">29</cite>
]. Here, this framework is extended by taking the initial states as grand-canonical instead of canonical equilibrium states, which allows us to deal with transfers of particles between the reservoirs. In this way, we obtain an exact relationship which is the consequence of microreversibility for the probability of a certain exchange of energy and particles between the reservoirs during the time-dependent external drive. An equivalent symmetry relation is obtained for the generating function of all the fluctuating variables. However, these symmetry relations are expressed in terms of the temperatures and chemical potentials of the reservoirs. The problem is that we need a symmetry relation in terms of the
<italic>differences</italic>
of temperatures and chemical potentials, which define the thermodynamic forces (also called the affinities) driving the currents across the system. The importance of this point has recently been discussed in the review [
<cite linkend="nj299837bib42">42</cite>
].</p>
<p>The central contribution of the present paper is the proof that, in the long-time limit, the aforementioned generating function only depends on the
<italic>differences</italic>
between the parameters of the reservoirs. This proof is carried out by obtaining lower and upper bounds on the generating function in terms of a new generating function which only depends on the
<italic>differences</italic>
of parameters and further functions which are bounded in the long-time limit. Combining this fundamental result with the previously established symmetry relation of the generating function, the fluctuation theorem is proved for nonequilibrium steady states in open quantum systems. Thanks to this quantum fluctuation theorem, the Onsager–Casimir reciprocity relations and their generalizations to the nonlinear response coefficients can be inferred [
<cite linkend="nj299837bib21">21</cite>
,
<cite linkend="nj299837bib53">53</cite>
].</p>
<p>The plan of the paper is as follows. The protocols for the forward and reversed drives of the open system are introduced in section 
<secref linkend="nj299837s2">2</secref>
. The symmetry relations for the probability and the generating function are proved in section 
<secref linkend="nj299837s3">3</secref>
. In section 
<secref linkend="nj299837s4">4</secref>
, we obtain the quantum fluctuation theorem for the currents in the steady state reached in the long-time limit. In section 
<secref linkend="nj299837s5">5</secref>
, the consequences of the fluctuation theorem on the linear and nonlinear response coefficients are deduced. The conclusions are drawn in section 
<secref linkend="nj299837s6">6</secref>
.</p>
</sec-level1>
<sec-level1 id="nj299837s2" label="2">
<heading>Open quantum system and time-dependent protocols</heading>
<sec-level2 id="nj299837s2.1" label="2.1">
<heading>The total Hamiltonian</heading>
<p indent="no">We consider a total quantum system composed of a subsystem in contact with several reservoirs of energy and particles. Initially, the reservoirs are decoupled from each other. During a time interval
<inline-eqn></inline-eqn>
, the reservoirs are put in contact with each other through the subsystem by some time-dependent interaction which has the effect of changing the energy and the particle numbers in each reservoir. The total Hamiltonian of the system is thus given by
<display-eqn id="nj299837eqn1" textype="equation" notation="LaTeX" eqnnum="1"></display-eqn>
where
<inline-eqn></inline-eqn>
denotes the Hamiltonian of the isolated subsystem,
<inline-eqn></inline-eqn>
the Hamiltonian of the
<italic>j</italic>
th isolated reservoir before the interaction is switched on, and
<inline-eqn></inline-eqn>
the time-dependent interaction to which the total system is submitted during the time interval
<inline-eqn></inline-eqn>
. The interaction is supposed to vanish before the initial time
<inline-eqn>
<math-text>
<italic>t</italic>
=0</math-text>
</inline-eqn>
so that
<inline-eqn></inline-eqn>
for
<inline-eqn>
<math-text>
<italic>t</italic>
<0</math-text>
</inline-eqn>
. Beyond the final time
<inline-eqn></inline-eqn>
, the subsystems are decoupled into the Hamiltonians
<inline-eqn></inline-eqn>
and
<inline-eqn></inline-eqn>
, whereupon the total Hamiltonian becomes
<display-eqn id="nj299837eqn2" textype="equation" notation="LaTeX" eqnnum="2"></display-eqn>
Furthermore, we suppose that the whole system is placed in a magnetic field
<inline-eqn>
<math-text>
<italic>B</italic>
</math-text>
</inline-eqn>
.</p>
<p>The observables of the total system include not only the Hamiltonian operators but also the numbers of particles of species
<inline-eqn>
<math-text>α=1, 2, …,
<italic>c</italic>
</math-text>
</inline-eqn>
inside the subsystem
<inline-eqn></inline-eqn>
and the reservoirs
<inline-eqn></inline-eqn>
with
<inline-eqn>
<math-text>
<italic>j</italic>
=1, 2, …,
<italic>r</italic>
</math-text>
</inline-eqn>
. The total number of particles of species
<inline-eqn>
<math-text>α</math-text>
</inline-eqn>
is thus given by
<display-eqn id="nj299837eqn3" textype="equation" notation="LaTeX" eqnnum="3"></display-eqn>
</p>
<p>We suppose that the total Hamiltonian operator has the symmetry
<display-eqn id="nj299837eqn4" textype="equation" notation="LaTeX" eqnnum="4"></display-eqn>
under the time-reversal operator
<inline-eqn>
<math-text>Θ</math-text>
</inline-eqn>
. This latter is an antilinear operator such that
<inline-eqn>
<math-text>Θ
<sup>2</sup>
=
<italic>I</italic>
</math-text>
</inline-eqn>
and which has the effect of changing the sign of all odd parameters such as magnetic fields. Equation (
<eqnref linkend="nj299837eqn4">4</eqnref>
) expresses the microreversibility in an external magnetic field. The numbers of particles are symmetric under time reversal:
<display-eqn id="nj299837eqn5" textype="equation" notation="LaTeX" eqnnum="5"></display-eqn>
</p>
<p>Since the numbers of particles of species
<inline-eqn>
<math-text>α</math-text>
</inline-eqn>
are conserved within each isolated subsystem before and after their coupling, we have that
<display-eqn id="nj299837eqn6" textype="equation" notation="LaTeX" eqnnum="6"></display-eqn>
for all
<inline-eqn>
<math-text>
<italic>j</italic>
,
<italic>j</italic>
′=
<upright>s</upright>
, 1, 2, …,
<italic>r</italic>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>α=1, 2, …,
<italic>c</italic>
</math-text>
</inline-eqn>
.</p>
<p>For our purposes, we are going to introduce a temperature and a chemical potential associated with each reservoir. Such quantities can be introduced as characterizing the statistical distribution of initial conditions and, as such, can be applied not only to the reservoirs but also to the subsystem. However, in a nonequilibrium system with currents flowing between reservoirs, the thermodynamic forces or affinities are determined by the differences of temperature and chemical potentials between the reservoirs, in which case the initial temperature and chemical potentials of the subsystem are not relevant. In this regard, we can simplify the formulation of the problem by regrouping the subsystem with one of the reservoirs, for instance the first one, and redefine the Hamiltonian and particle-number operators as follows:
<display-eqn id="nj299837eqn7" textype="equation" notation="LaTeX" eqnnum="7"></display-eqn>
In this case, the total Hamiltonian given by equations (
<eqnref linkend="nj299837eqn1">1</eqnref>
) and (
<eqnref linkend="nj299837eqn2">2</eqnref>
) can be rewritten as
<display-eqn id="nj299837eqn8" textype="equation" notation="LaTeX" eqnnum="8"></display-eqn>
and the total number of particles of species
<inline-eqn>
<math-text>α=1, 2, …,
<italic>c</italic>
</math-text>
</inline-eqn>
as
<display-eqn id="nj299837eqn9" textype="equation" notation="LaTeX" eqnnum="9"></display-eqn>
</p>
<p>An alternative formulation would be to consider that the subsystem Hamiltonian is included in the interaction potential, in which case
<inline-eqn></inline-eqn>
and
<inline-eqn></inline-eqn>
in equation (
<eqnref linkend="nj299837eqn8">8</eqnref>
), while
<inline-eqn></inline-eqn>
in equation (
<eqnref linkend="nj299837eqn9">9</eqnref>
).</p>
<p>We notice that, in the further case where the subsystem is considered on equal footing with the reservoirs, the sums should be extended to
<inline-eqn>
<math-text>
<italic>j</italic>
=
<upright>s</upright>
, 1, 2, …,
<italic>r</italic>
</math-text>
</inline-eqn>
in equations (
<eqnref linkend="nj299837eqn8">8</eqnref>
) and (
<eqnref linkend="nj299837eqn9">9</eqnref>
).</p>
<p>In the following, we define protocols with two quantum measurements before and after a unitary time evolution (see [
<cite linkend="nj299837bib42">42</cite>
] for a review).</p>
</sec-level2>
<sec-level2 id="nj299837s2.2" label="2.2">
<heading>The forward protocol</heading>
<p indent="no">The forward time evolution is defined as
<display-eqn id="nj299837eqn10" textype="equation" notation="LaTeX" eqnnum="10"></display-eqn>
with the initial condition
<inline-eqn>
<math-text>
<italic>U</italic>
<sub>
<upright>F</upright>
</sub>
(0;
<italic>B</italic>
)=
<italic>I</italic>
</math-text>
</inline-eqn>
. In the Heisenberg representation, the observables evolve according to
<display-eqn id="nj299837eqn11" textype="equation" notation="LaTeX" eqnnum="11"></display-eqn>
which also concerns the time-dependent Hamiltonian
<display-eqn id="nj299837eqn12" textype="equation" notation="LaTeX" eqnnum="12"></display-eqn>
The average of an observable is thus obtained from
<display-eqn id="nj299837eqn13" textype="equation" notation="LaTeX" eqnnum="13"></display-eqn>
We note that the dependence on the magnetic field is implicit in these expressions.</p>
<p>The initial state of the system is taken as the following grand-canonical equilibrium state of the decoupled subsystems at the different inverse temperatures
<inline-eqn>
<math-text>β
<sub>
<italic>j</italic>
</sub>
=1/(
<italic>k</italic>
<sub>
<upright>B</upright>
</sub>
<italic>T</italic>
<sub>
<italic>j</italic>
</sub>
)</math-text>
</inline-eqn>
and chemical potentials
<inline-eqn>
<math-text>μ
<sub>
<italic>j</italic>
α</sub>
</math-text>
</inline-eqn>
:
<display-eqn id="nj299837eqn14" textype="equation" notation="LaTeX" eqnnum="14"></display-eqn>
where
<inline-eqn>
<math-text>Φ
<sub>
<italic>j</italic>
</sub>
(
<italic>B</italic>
)=−
<italic>k</italic>
<sub>
<upright>B</upright>
</sub>
<italic>T</italic>
<sub>
<italic>j</italic>
</sub>
<upright>ln</upright>
Ξ
<sub>
<italic>j</italic>
</sub>
(
<italic>B</italic>
)</math-text>
</inline-eqn>
denotes the thermodynamic grand-potential of the
<inline-eqn>
<math-text>
<italic>j</italic>
</math-text>
</inline-eqn>
th subsystem in the initial equilibrium state.</p>
<p>An
<italic>initial quantum measurement</italic>
is performed that prepares the system in the eigenstate
<inline-eqn>
<math-text>|Ψ
<sub>
<italic>k</italic>
</sub>
⟩</math-text>
</inline-eqn>
of the subsystem operators of energy and particle numbers:
<display-eqn id="nj299837eqn15" textype="equation" notation="LaTeX" eqnnum="15"></display-eqn>
<display-eqn id="nj299837eqn16" textype="equation" notation="LaTeX" eqnnum="16"></display-eqn>
</p>
<p>After the time interval
<inline-eqn></inline-eqn>
, a
<italic>final quantum measurement</italic>
is performed in which the system is observed in the eigenstate
<inline-eqn></inline-eqn>
of the subsystem operators of energy and particle numbers:
<display-eqn id="nj299837eqn17" textype="equation" notation="LaTeX" eqnnum="17"></display-eqn>
<display-eqn id="nj299837eqn18" textype="equation" notation="LaTeX" eqnnum="18"></display-eqn>
We notice that semi-infinite time intervals are available to perform the initial and final quantum measurements of well-defined eigenvalues. Accordingly, this scheme based on two quantum measurements provides a way to measure the energies and the numbers of particles exchanged between the reservoirs during the time interval
<inline-eqn></inline-eqn>
of their mutual interaction. Indeed, the energy in the
<inline-eqn>
<math-text>
<italic>j</italic>
</math-text>
</inline-eqn>
th subsystem is observed to change by the amount
<display-eqn id="nj299837eqn19" textype="equation" notation="LaTeX" eqnnum="19"></display-eqn>
while the number of particles of species
<inline-eqn>
<math-text>α</math-text>
</inline-eqn>
in the
<inline-eqn>
<math-text>
<italic>j</italic>
</math-text>
</inline-eqn>
th subsystem changes by
<display-eqn id="nj299837eqn20" textype="equation" notation="LaTeX" eqnnum="20"></display-eqn>
during the forward protocol.</p>
</sec-level2>
<sec-level2 id="nj299837s2.3" label="2.3">
<heading>The reversed protocol</heading>
<p indent="no">The evolution operator of the reversed process is defined as
<display-eqn id="nj299837eqn21" textype="equation" notation="LaTeX" eqnnum="21"></display-eqn>
with the initial condition
<inline-eqn>
<math-text>
<italic>U</italic>
<sub>
<upright>R</upright>
</sub>
(0;
<italic>B</italic>
)=
<italic>I</italic>
</math-text>
</inline-eqn>
, and is related to the one of the forward process by the following:</p>
<proclaim id="nj299837pro1" type="lemma" format="num" style="italic">
<heading>Lemma 1</heading>
<p indent="no">The forward and reversed time evolution operators at the final time
<inline-eqn></inline-eqn>
are related to each other by
<display-eqn id="nj299837eqn22" textype="equation" notation="LaTeX" eqnnum="22"></display-eqn>
</p>
</proclaim>
<p>This lemma is proved by noting that the forward time evolution in the magnetic field
<inline-eqn>
<math-text>
<italic>B</italic>
</math-text>
</inline-eqn>
, followed by the operation of time reversal, by the reversed time evolution in the magnetic field
<inline-eqn>
<math-text>−
<italic>B</italic>
</math-text>
</inline-eqn>
, and finally by time reversal again is equal to the identical operator:
<display-eqn id="nj299837eqn23" textype="equation" notation="LaTeX" eqnnum="23"></display-eqn>
from which we deduce equation (
<eqnref linkend="nj299837eqn22">22</eqnref>
).</p>
<p>The reversed protocol is supposed to start with the following grand-canonical equilibrium state of the final decoupled subsystems:
<display-eqn id="nj299837eqn24" textype="equation" notation="LaTeX" eqnnum="24"></display-eqn>
at the same inverse temperatures
<inline-eqn>
<math-text>β
<sub>
<italic>j</italic>
</sub>
=1/(
<italic>k</italic>
<sub>
<upright>B</upright>
</sub>
<italic>T</italic>
<sub>
<italic>j</italic>
</sub>
)</math-text>
</inline-eqn>
and chemical potentials
<inline-eqn>
<math-text>μ
<sub>
<italic>j</italic>
α</sub>
</math-text>
</inline-eqn>
as in the forward protocol and where
<inline-eqn></inline-eqn>
denotes the grand-canonical thermodynamic potential of the
<inline-eqn>
<math-text>
<italic>j</italic>
</math-text>
</inline-eqn>
th subsystem in the final equilibrium state and the reversed magnetic field.</p>
<p>Similarly to the forward protocol, initial and final quantum measurements are performed to determine the changes of energies and particle numbers in the subsystems.</p>
</sec-level2>
</sec-level1>
<sec-level1 id="nj299837s3" label="3">
<heading>Consequences of microreversibility</heading>
<sec-level2 id="nj299837s3.1" label="3.1">
<heading>The symmetry relation for the probability of the fluctuations</heading>
<p indent="no">The probability distribution function to observe the energy (
<eqnref linkend="nj299837eqn19">19</eqnref>
) and particle transfers (
<eqnref linkend="nj299837eqn20">20</eqnref>
) during the forward protocol is defined as
<display-eqn id="nj299837eqn25" textype="equation" notation="LaTeX" eqnnum="25" lines="multiline"></display-eqn>
We notice that this function is a probability density because the quantities
<inline-eqn>
<math-text>δ(·)</math-text>
</inline-eqn>
are Dirac distributions for both the energy and the particle numbers.</p>
<p>Inserting the expression of the initial density matrix (
<eqnref linkend="nj299837eqn14">14</eqnref>
), using the Dirac delta distributions to replace the initial energies and numbers into the final ones, we find that
<display-eqn id="nj299837eqn26" textype="equation" notation="LaTeX" eqnnum="26" lines="multiline"></display-eqn>
where we have introduced the difference of the thermodynamic grand-potential of the
<inline-eqn>
<math-text>
<italic>j</italic>
</math-text>
</inline-eqn>
th subsystem as
<display-eqn id="nj299837eqn27" textype="equation" notation="LaTeX" eqnnum="27"></display-eqn>
According to the lemma (
<eqnref linkend="nj299837eqn22">22</eqnref>
), the probability of the transition
<inline-eqn>
<math-text>
<italic>k</italic>
→
<italic>l</italic>
</math-text>
</inline-eqn>
during the forward process is equal to the probability of the transition
<inline-eqn>
<math-text>
<italic>l</italic>
→
<italic>k</italic>
</math-text>
</inline-eqn>
in the reversed process and magnetic field:
<display-eqn id="nj299837eqn28" textype="equation" notation="LaTeX" eqnnum="28" lines="multiline"></display-eqn>
Substituting this identity into equation (
<eqnref linkend="nj299837eqn26">26</eqnref>
) and introducing the probability of negative changes in the energies and particle numbers during the reversed process as
<display-eqn id="nj299837eqn29" textype="equation" notation="LaTeX" eqnnum="29" lines="multiline"></display-eqn>
with the final density matrix (
<eqnref linkend="nj299837eqn24">24</eqnref>
), we obtain the following symmetry relation:
<display-eqn id="nj299837eqn30" textype="equation" notation="LaTeX" eqnnum="30"></display-eqn>
If this relation is restricted to the energy change in a single system, this fluctuating quantity is the work
<inline-eqn>
<math-text>
<italic>W</italic>
</math-text>
</inline-eqn>
performed on the system and we recover the quantum version of Crooks' fluctuation theorem
<display-eqn id="nj299837eqn31" textype="equation" notation="LaTeX" eqnnum="31"></display-eqn>
with the corresponding difference of free energy
<inline-eqn></inline-eqn>
 [
<cite linkend="nj299837bib25">25</cite>
]. The relation (
<eqnref linkend="nj299837eqn30">30</eqnref>
) extends this result to the transfer of particles under the effect of the differences of chemical potentials driving the system out of equilibrium.</p>
</sec-level2>
<sec-level2 id="nj299837s3.2" label="3.2">
<heading>The symmetry relation for the generating function</heading>
<p indent="no">The generating functions of the statistical moments of the exchanges of energy and particles are defined by
<display-eqn id="nj299837eqn32" textype="equation" notation="LaTeX" eqnnum="32"></display-eqn>
for the forward and reversed processes. The knowledge of these generating functions provides the full counting statistics of the process. We notice that the generating function of the forward protocol is alternatively defined as
<display-eqn id="nj299837eqn33" textype="equation" notation="LaTeX" eqnnum="33"></display-eqn>
with
<display-eqn id="nj299837eqn34" textype="equation" notation="LaTeX" eqnnum="34"></display-eqn>
<display-eqn id="nj299837eqn35" textype="equation" notation="LaTeX" eqnnum="35"></display-eqn>
and the generating function of the reversed protocol as
<display-eqn id="nj299837eqn36" textype="equation" notation="LaTeX" eqnnum="36"></display-eqn>
with
<display-eqn id="nj299837eqn37" textype="equation" notation="LaTeX" eqnnum="37"></display-eqn>
<display-eqn id="nj299837eqn38" textype="equation" notation="LaTeX" eqnnum="38"></display-eqn>
</p>
<p>Taking the Laplace transforms of the symmetry relation (
<eqnref linkend="nj299837eqn30">30</eqnref>
), we obtain an equivalent symmetry relation in terms of the generating functions:
<display-eqn id="nj299837eqn39" textype="equation" notation="LaTeX" eqnnum="39"></display-eqn>
in terms of the temperatures and chemical potentials of the subsystems. As mentioned in the introduction, this symmetry relation has not yet the appropriate form because the thermodynamic forces or affinities do not appear.</p>
</sec-level2>
</sec-level1>
<sec-level1 id="nj299837s4" label="4">
<heading>Quantum fluctuation theorem for the currents</heading>
<p indent="no">In this section, we prove that, in the long-time limit, the generating functions entering into the symmetry relation (
<eqnref linkend="nj299837eqn39">39</eqnref>
) only depend on the differences of the parameters
<inline-eqn>
<math-text>ξ
<sub>
<italic>j</italic>
</sub>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>η
<sub>
<italic>j</italic>
α</sub>
</math-text>
</inline-eqn>
, leading to the announced symmetry. In the long-time limit, a nonequilibrium steady state can be reached between the reservoirs if the coupling remains constant over the whole time interval except finite transients. In this respect, it is important to suppose that the temperatures and chemical potentials that have been introduced here above concern the reservoirs themselves. Accordingly, from now on, the subsystem can no longer be considered as a subsystem on equal footing with the reservoirs, and the subsystem Hamiltonian is supposed to be included in either one of the reservoir Hamiltonians or in the interaction potential as formulated in equations (
<eqnref linkend="nj299837eqn8">8</eqnref>
) and (
<eqnref linkend="nj299837eqn9">9</eqnref>
) with the indices
<inline-eqn>
<math-text>
<italic>j</italic>
=1, 2, …,
<italic>r</italic>
</math-text>
</inline-eqn>
referring to the reservoirs themselves.</p>
<sec-level2 id="nj299837s4.1" label="4.1">
<heading>The theorem</heading>
<p indent="no">We consider a situation where two large quantum systems interact through a bounded time-dependent perturbation described by
<inline-eqn>
<math-text>
<italic>V</italic>
(
<italic>t</italic>
)</math-text>
</inline-eqn>
. Then, the generator of time evolution of the whole system is given by
<display-eqn id="nj299837eqn40" textype="equation" notation="LaTeX" eqnnum="40"></display-eqn>
where the subsystem Hamiltonian
<inline-eqn></inline-eqn>
is included in either
<inline-eqn>
<math-text>
<italic>H</italic>
<sub>1</sub>
</math-text>
</inline-eqn>
,
<inline-eqn>
<math-text>
<italic>H</italic>
<sub>2</sub>
</math-text>
</inline-eqn>
, or
<inline-eqn>
<math-text>
<italic>V</italic>
(
<italic>t</italic>
)</math-text>
</inline-eqn>
(imagine a quantum dot located between two electrodes). Hereafter, we assume that
<inline-eqn>
<math-text>
<italic>V</italic>
(0)=0</math-text>
</inline-eqn>
,
<inline-eqn></inline-eqn>
and
<inline-eqn>
<math-text>
<italic>V</italic>
(
<italic>t</italic>
)=
<italic>V</italic>
<sub>0</sub>
</math-text>
</inline-eqn>
for
<inline-eqn></inline-eqn>
, meaning that the interaction is switched on over a short time interval
<inline-eqn>
<math-text>
<italic>t</italic>
<sub>0</sub>
</math-text>
</inline-eqn>
, remains constant during the long lapse of time
<inline-eqn></inline-eqn>
, and is finally switched off again over a short time interval
<inline-eqn>
<math-text>
<italic>t</italic>
<sub>0</sub>
</math-text>
</inline-eqn>
.</p>
<p>Let
<inline-eqn>
<math-text>
<italic>N</italic>
<sub>
<italic>j</italic>
α</sub>
</math-text>
</inline-eqn>
(
<inline-eqn>
<math-text>
<italic>j</italic>
=1, 2</math-text>
</inline-eqn>
) be the number of
<inline-eqn>
<math-text>α</math-text>
</inline-eqn>
-particles in the
<inline-eqn>
<math-text>
<italic>j</italic>
</math-text>
</inline-eqn>
th large system and assume that
<inline-eqn>
<math-text>[
<italic>N</italic>
<sub>1α</sub>
+
<italic>N</italic>
<sub>2α</sub>
,
<italic>H</italic>
<sub>1</sub>
+
<italic>H</italic>
<sub>2</sub>
+
<italic>V</italic>
<sub>0</sub>
]=0</math-text>
</inline-eqn>
.</p>
<p>Since the interaction is symmetric under time reversal
<inline-eqn></inline-eqn>
, the evolution operator of the forward and reversed protocols are identical
<display-eqn id="nj299837eqn41" textype="equation" notation="LaTeX" eqnnum="41"></display-eqn>
and they are therefore solutions of one and the same equation:
<display-eqn id="nj299837eqn42" textype="equation" notation="LaTeX" eqnnum="42"></display-eqn>
with the initial condition
<inline-eqn>
<math-text>
<italic>U</italic>
(0;
<italic>B</italic>
)=
<italic>I</italic>
</math-text>
</inline-eqn>
and
<inline-eqn></inline-eqn>
. For the same reason, the initial and final reservoir Hamiltonians are the same,
<inline-eqn></inline-eqn>
for all
<inline-eqn>
<math-text>
<italic>j</italic>
=1, 2, …,
<italic>r</italic>
</math-text>
</inline-eqn>
, so that the initial and final density matrices have the same definition
<display-eqn id="nj299837eqn43" textype="equation" notation="LaTeX" eqnnum="43"></display-eqn>
where
<inline-eqn>
<math-text>Φ
<sub>
<italic>j</italic>
</sub>
(
<italic>B</italic>
)</math-text>
</inline-eqn>
is the thermodynamic grand-potential of the
<inline-eqn>
<math-text>
<italic>j</italic>
</math-text>
</inline-eqn>
th reservoir in magnetic field
<inline-eqn>
<math-text>
<italic>B</italic>
</math-text>
</inline-eqn>
. Accordingly, the forward and reversed generating functions also have the same definition
<display-eqn id="nj299837eqn44" textype="equation" notation="LaTeX" eqnnum="44"></display-eqn>
with equations (
<eqnref linkend="nj299837eqn34">34</eqnref>
) and (
<eqnref linkend="nj299837eqn35">35</eqnref>
) and where the average
<inline-eqn>
<math-text>⟨·⟩</math-text>
</inline-eqn>
is carried out with respect to the density matrix (
<eqnref linkend="nj299837eqn43">43</eqnref>
).</p>
<p>According to equation (
<eqnref linkend="nj299837eqn39">39</eqnref>
), this generating function has the symmetry
<display-eqn id="nj299837eqn45" textype="equation" notation="LaTeX" eqnnum="45"></display-eqn>
in terms of the temperatures and chemical potentials of the reservoirs.</p>
<p>Our purpose in this section is to prove</p>
<proclaim id="nj299837pro2" type="proposition" format="num" style="italic">
<heading>Proposition 1</heading>
<p indent="no">Assume that the limit
<display-eqn id="nj299837eqn46" textype="equation" notation="LaTeX" eqnnum="46"></display-eqn>
exists, it is a function only of
<inline-eqn>
<math-text>ξ
<sub>1</sub>
−ξ
<sub>2</sub>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>η
<sub>1α</sub>
−η
<sub>2α</sub>
</math-text>
</inline-eqn>
:
<display-eqn id="nj299837eqn47" textype="equation" notation="LaTeX" eqnnum="47"></display-eqn>
</p>
</proclaim>
<p>We would like to remark that, if the system had very long but finite recurrent times and much shorter relaxation times, the quantities
<inline-eqn>
<math-text>
<italic>Q</italic>
</math-text>
</inline-eqn>
and
<inline-eqn></inline-eqn>
would exist provided
<inline-eqn></inline-eqn>
is sufficiently longer than the relaxation times but shorter than the recurrent times.</p>
<p>The interpretation of this proposition is that, because of the finiteness of the subsystem and the interaction
<inline-eqn>
<math-text>
<italic>V</italic>
<sub>0</sub>
</math-text>
</inline-eqn>
, the energy and particles lost by the left (respectively right) reservoir are transferred to the right (respectively left) reservoir within the overwhelming duration
<inline-eqn></inline-eqn>
and, as a result,
<inline-eqn>
<math-text>
<italic>Q</italic>
</math-text>
</inline-eqn>
becomes a function
<inline-eqn></inline-eqn>
depending only on the differences
<inline-eqn>
<math-text>ξ
<sub>1</sub>
−ξ
<sub>2</sub>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>η
<sub>1α</sub>
−η
<sub>2α</sub>
</math-text>
</inline-eqn>
. We remark that the explicit form of the generating function
<inline-eqn></inline-eqn>
is given by equation (
<eqnref linkend="nj299837eqn92">92</eqnref>
).</p>
<p>The above proposition implies that
<display-eqn id="nj299837eqn48" textype="equation" notation="LaTeX" eqnnum="48"></display-eqn>
<display-eqn id="nj299837eqn49" textype="equation" notation="LaTeX" eqnnum="49"></display-eqn>
<display-eqn id="nj299837eqn50" textype="equation" notation="LaTeX" eqnnum="50"></display-eqn>
<display-eqn id="nj299837eqn51" textype="equation" notation="LaTeX" eqnnum="51"></display-eqn>
where we have introduced the affinities:
<display-eqn id="nj299837eqn52" textype="equation" notation="LaTeX" eqnnum="52"></display-eqn>
<display-eqn id="nj299837eqn53" textype="equation" notation="LaTeX" eqnnum="53"></display-eqn>
driving respectively the heat current and the
<inline-eqn>
<math-text>α</math-text>
</inline-eqn>
-particle currents from reservoir 2 to reservoir 1. The result (
<eqnref linkend="nj299837eqn51">51</eqnref>
) is obtained by using the definition (
<eqnref linkend="nj299837eqn47">47</eqnref>
) at the line (
<eqnref linkend="nj299837eqn48">48</eqnref>
), the symmetry (
<eqnref linkend="nj299837eqn45">45</eqnref>
) and the independency of the quantities
<inline-eqn>
<math-text>ΔΦ
<sub>
<italic>j</italic>
</sub>
</math-text>
</inline-eqn>
on the time interval
<inline-eqn></inline-eqn>
at the line (
<eqnref linkend="nj299837eqn49">49</eqnref>
), again the definition (
<eqnref linkend="nj299837eqn47">47</eqnref>
) at the line (
<eqnref linkend="nj299837eqn50">50</eqnref>
), and finally the definitions of the affinities (
<eqnref linkend="nj299837eqn52">52</eqnref>
) and (
<eqnref linkend="nj299837eqn53">53</eqnref>
). Hence, we have</p>
<proclaim id="nj299837pro3" type="theorem" format="num" style="italic">
<heading>Fluctuation theorem</heading>
<p indent="no">The generating function of the independent currents satisfies the symmetry
<display-eqn id="nj299837eqn54" textype="equation" notation="LaTeX" eqnnum="54"></display-eqn>
</p>
</proclaim>
<p>In the particular case where the two systems have the same temperature,
<inline-eqn>
<math-text>β
<sub>1</sub>
=β
<sub>2</sub>
</math-text>
</inline-eqn>
, the generating function has the symmetry:
<display-eqn id="nj299837eqn55" textype="equation" notation="LaTeX" eqnnum="55"></display-eqn>
and we recover the symmetry
<display-eqn id="nj299837eqn56" textype="equation" notation="LaTeX" eqnnum="56"></display-eqn>
of the generating function of the independent particle currents, which has already been proved elsewhere for stochastic processes [
<cite linkend="nj299837bib23">23</cite>
].</p>
<p>We notice that the fluctuation theorem (
<eqnref linkend="nj299837eqn54">54</eqnref>
) which is here proved thanks to the proposition (
<eqnref linkend="nj299837eqn46">46</eqnref>
) and (
<eqnref linkend="nj299837eqn47">47</eqnref>
) reduces to the steady-state fluctuation theorem presented as equation (104) in the review [
<cite linkend="nj299837bib42">42</cite>
] for vanishing magnetic field,
<inline-eqn>
<math-text>
<italic>B</italic>
=0</math-text>
</inline-eqn>
. Accordingly, the proposition (
<eqnref linkend="nj299837eqn46">46</eqnref>
) and (
<eqnref linkend="nj299837eqn47">47</eqnref>
) also provides a rigorous proof of such steady-state fluctuation theorems.</p>
</sec-level2>
<sec-level2 id="nj299837s4.2" label="4.2">
<heading>Setting</heading>
<p indent="no">In order to demonstrate the above proposition, the time evolution is decomposed into different pieces corresponding to the short initial transient over
<inline-eqn>
<math-text>0<
<italic>t</italic>
<
<italic>t</italic>
<sub>0</sub>
</math-text>
</inline-eqn>
, the long steady interaction over
<inline-eqn></inline-eqn>
, and the final short transient over
<inline-eqn></inline-eqn>
. We introduce the lapse of time of the steady interaction
<display-eqn id="nj299837eqn57" textype="equation" notation="LaTeX" eqnnum="57"></display-eqn>
</p>
<p>In addition to
<inline-eqn>
<math-text>
<italic>U</italic>
(
<italic>t</italic>
;
<italic>B</italic>
)</math-text>
</inline-eqn>
defined by equation (
<eqnref linkend="nj299837eqn10">10</eqnref>
), we introduce
<inline-eqn>
<math-text>
<italic>U</italic>
<sub>1</sub>
(
<italic>t</italic>
;
<italic>B</italic>
)</math-text>
</inline-eqn>
as the solution of
<display-eqn id="nj299837eqn58" textype="equation" notation="LaTeX" eqnnum="58"></display-eqn>
It is then easy to show
<display-eqn id="nj299837eqn59" textype="equation" notation="LaTeX" eqnnum="59"></display-eqn>
where
<inline-eqn></inline-eqn>
,
<inline-eqn>
<math-text>
<italic>U</italic>
<sub>
<italic>f</italic>
</sub>
=
<italic>U</italic>
<sub>1</sub>
(
<italic>t</italic>
<sub>0</sub>
;
<italic>B</italic>
)</math-text>
</inline-eqn>
,
<inline-eqn></inline-eqn>
and
<inline-eqn>
<math-text>
<italic>U</italic>
<sub>
<italic>i</italic>
</sub>
=
<italic>U</italic>
(
<italic>t</italic>
<sub>0</sub>
;
<italic>B</italic>
)</math-text>
</inline-eqn>
. We further note that
<display-eqn id="nj299837eqn60" textype="equation" notation="LaTeX" eqnnum="60"></display-eqn>
where
<inline-eqn>
<math-text>
<italic>H</italic>
<sub>0</sub>
=
<italic>H</italic>
<sub>1</sub>
+
<italic>H</italic>
<sub>2</sub>
</math-text>
</inline-eqn>
,
<inline-eqn>
<math-text>Γ
<sub>
<italic>f</italic>
</sub>
=
<upright>e</upright>
<sup>
<upright>i</upright>
<italic>H</italic>
<sub>0</sub>
<italic>t</italic>
<sub>0</sub>
</sup>
<italic>U</italic>
<sub>
<italic>f</italic>
</sub>
</math-text>
</inline-eqn>
,
<inline-eqn>
<math-text>Γ
<sub>τ</sub>
=
<upright>e</upright>
<sup>
<upright>i</upright>
<italic>H</italic>
<sub>0</sub>
τ</sup>
<italic>U</italic>
<sub>τ</sub>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>Γ
<sub>
<italic>i</italic>
</sub>
=
<upright>e</upright>
<sup>
<upright>i</upright>
<italic>H</italic>
<sub>0</sub>
<italic>t</italic>
<sub>0</sub>
</sup>
<italic>U</italic>
<sub>
<italic>i</italic>
</sub>
</math-text>
</inline-eqn>
.</p>
<p>Therefore, with the aid of
<inline-eqn>
<math-text>[
<italic>H</italic>
<sub>0</sub>
,
<italic>H</italic>
<sub>
<italic>j</italic>
</sub>
]=[
<italic>H</italic>
<sub>0</sub>
,
<italic>N</italic>
<sub>
<italic>j</italic>
α</sub>
]=[
<italic>H</italic>
<sub>0</sub>
, ρ]=0</math-text>
</inline-eqn>
, we have
<display-eqn id="nj299837eqn61" textype="equation" notation="LaTeX" eqnnum="61" lines="multiline"></display-eqn>
where
<inline-eqn>
<math-text>Γ
<sub>λ</sub>
(τ)=
<upright>e</upright>
<sup>
<upright>i</upright>
<italic>H</italic>
<sub>0</sub>
τ</sup>
Γ
<sub>λ</sub>
<upright>e</upright>
<sup>−
<upright>i</upright>
<italic>H</italic>
<sub>0</sub>
τ</sup>
</math-text>
</inline-eqn>
(
<inline-eqn>
<math-text>λ=
<italic>i</italic>
</math-text>
</inline-eqn>
or
<inline-eqn>
<math-text>
<italic>f</italic>
</math-text>
</inline-eqn>
).</p>
<p>For later purposes, we introduce
<display-eqn id="nj299837eqn62" textype="equation" notation="LaTeX" eqnnum="62"></display-eqn>
<display-eqn id="nj299837eqn63" textype="equation" notation="LaTeX" eqnnum="63"></display-eqn>
<display-eqn id="nj299837eqn64" textype="equation" notation="LaTeX" eqnnum="64"></display-eqn>
<display-eqn id="nj299837eqn65" textype="equation" notation="LaTeX" eqnnum="65"></display-eqn>
where
<inline-eqn>
<math-text>
<italic>H</italic>
<sub>0</sub>
=
<italic>H</italic>
<sub>1</sub>
+
<italic>H</italic>
<sub>2</sub>
</math-text>
</inline-eqn>
,
<inline-eqn>
<math-text>
<italic>N</italic>
<sub>0α</sub>
=
<italic>N</italic>
<sub>1α</sub>
+
<italic>N</italic>
<sub>2α</sub>
</math-text>
</inline-eqn>
,
<inline-eqn>
<math-text>Δ
<italic>H</italic>
<sub>0</sub>
=(
<italic>H</italic>
<sub>1</sub>
−
<italic>H</italic>
<sub>2</sub>
)/2</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>Δ
<italic>N</italic>
<sub>0α</sub>
=(
<italic>N</italic>
<sub>1α</sub>
−
<italic>N</italic>
<sub>2α</sub>
)/2</math-text>
</inline-eqn>
. Since
<inline-eqn>
<math-text>[
<italic>C</italic>
, ρ]=[
<italic>D</italic>
, ρ]=0</math-text>
</inline-eqn>
, we have
<display-eqn id="nj299837eqn66" textype="equation" notation="LaTeX" eqnnum="66"></display-eqn>
This is our starting point. Note that
<inline-eqn>
<math-text>
<italic>C</italic>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>
<italic>D</italic>
</math-text>
</inline-eqn>
are Hermitian for real
<inline-eqn>
<math-text>ξ
<sub>
<italic>j</italic>
</sub>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>η
<sub>
<italic>j</italic>
α</sub>
</math-text>
</inline-eqn>
and that
<inline-eqn>
<math-text>
<italic>D</italic>
</math-text>
</inline-eqn>
is the function only of
<inline-eqn>
<math-text>ξ
<sub>1</sub>
−ξ
<sub>2</sub>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>η
<sub>1α</sub>
−η
<sub>2α</sub>
</math-text>
</inline-eqn>
.</p>
</sec-level2>
<sec-level2 id="nj299837s4.3" label="4.3">
<heading>Some inequalities</heading>
<p indent="no">Here, for the sake of self-containedness, well-known equalities and inequalities [
<cite linkend="nj299837bib54">54</cite>
] necessary for the following proof are summarized. For an operator
<inline-eqn>
<math-text>
<italic>X</italic>
</math-text>
</inline-eqn>
, the operator norm
<inline-eqn>
<math-text>∥
<italic>X</italic>
∥</math-text>
</inline-eqn>
is defined by
<display-eqn id="nj299837eqn67" textype="equation" notation="LaTeX" eqnnum="67"></display-eqn>
and it satisfies:</p>
<proclaim id="nj299837pro4" type="Equality" format="num" style="upright">
<heading>Equality 1</heading>
<p indent="no">For any unitary
<inline-eqn>
<math-text>
<italic>U</italic>
</math-text>
</inline-eqn>
,
<inline-eqn>
<math-text>∥
<italic>U</italic>
<sup>†</sup>
<italic>XU</italic>
∥=∥
<italic>X</italic>
∥</math-text>
</inline-eqn>
.  □</p>
<p>Indeed, we find
<display-eqn id="nj299837eqn68" textype="equation" notation="LaTeX" eqnnum="68"></display-eqn>
where we have set
<inline-eqn>
<math-text>|ψ⟩=
<italic>U</italic>
|ϕ⟩</math-text>
</inline-eqn>
.  □</p>
</proclaim>
<proclaim id="nj299837pro5" type="Inequality" format="num" style="upright">
<heading>Inequality 1</heading>
<p indent="no">
<inline-eqn>
<math-text>⟨
<italic>X</italic>
<sup>†</sup>
<italic>Y</italic>
<sup>†</sup>
<italic>YX</italic>
⟩⩽∥
<italic>Y</italic>
∥
<sup>2</sup>
⟨
<italic>X</italic>
<sup>†</sup>
<italic>X</italic>
⟩</math-text>
</inline-eqn>
.</p>
<p>Let
<inline-eqn>
<math-text>{ϕ
<sub>σ</sub>
}</math-text>
</inline-eqn>
be a complete orthonormal basis of eigenvectors of
<inline-eqn>
<math-text>ρ</math-text>
</inline-eqn>
:
<inline-eqn>
<math-text>ρ|ϕ
<sub>σ</sub>
⟩=ρ
<sub>σ</sub>
|ϕ
<sub>σ</sub>
⟩</math-text>
</inline-eqn>
. Then, because of
<inline-eqn>
<math-text>⟨ϕ|
<italic>X</italic>
<sup>†</sup>
<italic>X</italic>
|ϕ⟩⩽∥
<italic>X</italic>
∥
<sup>2</sup>
⟨ϕ|ϕ⟩</math-text>
</inline-eqn>
,
<display-eqn id="nj299837eqn69" textype="equation" notation="LaTeX" eqnnum="69" lines="multiline"></display-eqn>
  □</p>
</proclaim>
<proclaim id="nj299837pro6" type="Inequality" format="num" style="upright">
<heading>Inequality 2</heading>
<p indent="no">
<inline-eqn>
<math-text>⟨
<italic>X</italic>
<sup>†</sup>
<italic>Y</italic>
<sup>†</sup>
<italic>YX</italic>
⟩⩽∥
<upright>e</upright>
<sup>−
<italic>A</italic>
</sup>
<italic>X</italic>
<sup>†</sup>
<upright>e</upright>
<sup>
<italic>A</italic>
</sup>
∥
<sup>2</sup>
⟨
<italic>Y</italic>
<sup>†</sup>
<italic>Y</italic>
⟩</math-text>
</inline-eqn>
where
<inline-eqn></inline-eqn>
.</p>
<p>Thanks to the cyclicity of the trace, we have the Kubo–Martin–Schwinger (KMS) condition
<inline-eqn>
<math-text>⟨
<italic>XY</italic>
⟩=⟨
<upright>e</upright>
<sup>
<italic>A</italic>
</sup>
<italic>Y</italic>
<upright>e</upright>
<sup>−
<italic>A</italic>
</sup>
<upright>e</upright>
<sup>−
<italic>A</italic>
</sup>
<italic>X</italic>
<upright>e</upright>
<sup>
<italic>A</italic>
</sup>
⟩</math-text>
</inline-eqn>
for canonical averages
<inline-eqn></inline-eqn>
with
<inline-eqn>
<math-text>Ξ=
<upright>Tr</upright>
 
<upright>e</upright>
<sup>−2
<italic>A</italic>
</sup>
</math-text>
</inline-eqn>
. The KMS condition and inequality 1 imply
<display-eqn id="nj299837eqn70" textype="equation" notation="LaTeX" eqnnum="70" lines="multiline"></display-eqn>
  □</p>
</proclaim>
</sec-level2>
<sec-level2 id="nj299837s4.4" label="4.4">
<heading>Proof</heading>
<proclaim id="nj299837pro7" type="step" format="num" style="upright">
<heading>Step 1</heading>
<p indent="no">Let
<inline-eqn>
<math-text>
<italic>X</italic>
<sub>1</sub>
=
<upright>e</upright>
<sup>−
<italic>C</italic>
−
<italic>D</italic>
</sup>
Γ
<sub>τ</sub>
Γ
<sub>
<italic>i</italic>
</sub>
(−
<italic>t</italic>
<sub>0</sub>
) 
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
</math-text>
</inline-eqn>
. Then inequality 1 leads to
<display-eqn id="nj299837eqn71" textype="equation" notation="LaTeX" eqnnum="71" lines="multiline"></display-eqn>
</p>
<p>Since
<inline-eqn>
<math-text>Γ
<sub>
<italic>f</italic>
</sub>
(τ)
<sup>†</sup>
Γ
<sub>
<italic>f</italic>
</sub>
(τ)=1</math-text>
</inline-eqn>
, we have
<display-eqn id="nj299837eqn72" textype="equation" notation="LaTeX" eqnnum="72" lines="multiline"></display-eqn>
where
<inline-eqn>
<math-text>
<italic>Y</italic>
<sub>1</sub>
=
<upright>e</upright>
<sup>−
<italic>C</italic>
−
<italic>D</italic>
</sup>
Γ
<sub>
<italic>f</italic>
</sub>
(τ)
<sup>†</sup>
 
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
</math-text>
</inline-eqn>
and inequality 1 has been used.</p>
<p>Since
<inline-eqn>
<math-text>Γ
<sub>
<italic>f</italic>
</sub>
(τ)=
<upright>e</upright>
<sup>
<upright>i</upright>
<italic>H</italic>
<sub>0</sub>
τ</sup>
Γ
<sub>
<italic>f</italic>
</sub>
<upright>e</upright>
<sup>−
<upright>i</upright>
<italic>H</italic>
<sub>0</sub>
τ</sup>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>[
<italic>H</italic>
<sub>0</sub>
,
<italic>C</italic>
]=[
<italic>H</italic>
<sub>0</sub>
,
<italic>D</italic>
]=0</math-text>
</inline-eqn>
,
<display-eqn id="nj299837eqn73" textype="equation" notation="LaTeX" eqnnum="73" lines="multiline"></display-eqn>
where we have used equality 1 for the norm. Similarly,
<inline-eqn>
<math-text>∥
<upright>e</upright>
<sup>−
<italic>C</italic>
−
<italic>D</italic>
</sup>
Γ
<sub>
<italic>f</italic>
</sub>
(τ)
<sup>†</sup>
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
∥=∥
<upright>e</upright>
<sup>−
<italic>C</italic>
−
<italic>D</italic>
</sup>
Γ
<sub>
<italic>f</italic>
</sub>
<sup>†</sup>
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
∥</math-text>
</inline-eqn>
.</p>
<p>In short, in terms of
<display-eqn id="nj299837eqn74" textype="equation" notation="LaTeX" eqnnum="74"></display-eqn>
we have
<display-eqn id="nj299837eqn75" textype="equation" notation="LaTeX" eqnnum="75"></display-eqn>
</p>
</proclaim>
<proclaim id="nj299837pro8" type="step" format="num" style="upright">
<heading>Step 2</heading>
<p indent="no">In terms of
<inline-eqn>
<math-text>
<italic>X</italic>
<sub>2</sub>
=
<upright>e</upright>
<sup>−
<italic>C</italic>
−
<italic>D</italic>
</sup>
Γ
<sub>
<italic>i</italic>
</sub>
(−
<italic>t</italic>
<sub>0</sub>
)
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
</math-text>
</inline-eqn>
, one has from inequality 2
<display-eqn id="nj299837eqn76" textype="equation" notation="LaTeX" eqnnum="76" lines="multiline"></display-eqn>
Conversely, in terms of
<inline-eqn>
<math-text>
<italic>Y</italic>
<sub>2</sub>
=
<upright>e</upright>
<sup>−
<italic>C</italic>
−
<italic>D</italic>
</sup>
Γ
<sub>
<italic>i</italic>
</sub>
(−
<italic>t</italic>
<sub>0</sub>
)
<sup>†</sup>
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
</math-text>
</inline-eqn>
, inequality 2 leads to
<display-eqn id="nj299837eqn77" textype="equation" notation="LaTeX" eqnnum="77" lines="multiline"></display-eqn>
In short, let
<inline-eqn>
<math-text>
<italic>G</italic>
<sub>2</sub>
(ξ
<sub>
<italic>j</italic>
</sub>
, η
<sub>
<italic>j</italic>
α</sub>
;
<italic>B</italic>
)≡⟨
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
Γ
<sub>τ</sub>
<sup>†</sup>
<upright>e</upright>
<sup>−2
<italic>C</italic>
−2
<italic>D</italic>
</sup>
Γ
<sub>τ</sub>
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
⟩</math-text>
</inline-eqn>
, then
<display-eqn id="nj299837eqn78" textype="equation" notation="LaTeX" eqnnum="78"></display-eqn>
</p>
</proclaim>
<proclaim id="nj299837pro9" type="step" format="num" style="upright">
<heading>Step 3</heading>
<p indent="no">We set
<display-eqn id="nj299837eqn79" textype="equation" notation="LaTeX" eqnnum="79"></display-eqn>
where
<inline-eqn></inline-eqn>
and
<inline-eqn></inline-eqn>
. Then, in terms of
<inline-eqn></inline-eqn>
, we have
<display-eqn id="nj299837eqn80" textype="equation" notation="LaTeX" eqnnum="80" lines="multiline"></display-eqn>
where
<inline-eqn></inline-eqn>
and
<inline-eqn>
<math-text>[
<italic>H</italic>
<sub>0</sub>
,
<italic>C</italic>
]=[
<italic>H</italic>
<sub>0</sub>
,
<italic>D</italic>
]=0</math-text>
</inline-eqn>
, inequality 1 and equality 1 have been used. Because of
<inline-eqn></inline-eqn>
, one has
<inline-eqn></inline-eqn>
and, thus,
<display-eqn id="nj299837eqn81" textype="equation" notation="LaTeX" eqnnum="81"></display-eqn>
</p>
<p>Furthermore, inequality 2 gives
<display-eqn id="nj299837eqn82" textype="equation" notation="LaTeX" eqnnum="82" lines="multiline"></display-eqn>
Thus,
<inline-eqn>
<math-text>
<italic>G</italic>
<sub>3</sub>
(ξ
<sub>
<italic>j</italic>
</sub>
, η
<sub>
<italic>j</italic>
α</sub>
;
<italic>B</italic>
)≡⟨
<upright>e</upright>
<sup>
<italic>D</italic>
</sup>
Γ
<sub>τ</sub>
<sup>†</sup>
<upright>e</upright>
<sup>−2
<italic>D</italic>
</sup>
Γ
<sub>τ</sub>
<upright>e</upright>
<sup>
<italic>D</italic>
</sup>
⟩</math-text>
</inline-eqn>
satisfies
<display-eqn id="nj299837eqn83" textype="equation" notation="LaTeX" eqnnum="83"></display-eqn>
</p>
<p>Conversely, we have
<display-eqn id="nj299837eqn84" textype="equation" notation="LaTeX" eqnnum="84" lines="multiline"></display-eqn>
where we have used inequality 2. Let
<inline-eqn></inline-eqn>
, then inequality 1 and equality 1 lead to
<display-eqn id="nj299837eqn85" textype="equation" notation="LaTeX" eqnnum="85" lines="multiline"></display-eqn>
Thus,
<display-eqn id="nj299837eqn86" textype="equation" notation="LaTeX" eqnnum="86"></display-eqn>
</p>
</proclaim>
<proclaim id="nj299837pro10" type="step" format="num" style="upright">
<heading>Step 4</heading>
<p indent="no">From steps 1–3, in terms of
<display-eqn id="nj299837eqn87" textype="equation" notation="LaTeX" eqnnum="87"></display-eqn>
<display-eqn id="nj299837eqn88" textype="equation" notation="LaTeX" eqnnum="88"></display-eqn>
we have
<display-eqn id="nj299837eqn89" textype="equation" notation="LaTeX" eqnnum="89"></display-eqn>
Note that the constants
<inline-eqn>
<math-text>
<italic>L</italic>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>
<italic>K</italic>
</math-text>
</inline-eqn>
are independent of
<inline-eqn>
<math-text>τ</math-text>
</inline-eqn>
and that
<inline-eqn>
<math-text>
<italic>G</italic>
<sub>3</sub>
(ξ
<sub>
<italic>j</italic>
</sub>
, η
<sub>
<italic>j</italic>
α</sub>
;
<italic>B</italic>
)</math-text>
</inline-eqn>
is a function only of
<inline-eqn>
<math-text>ξ
<sub>1</sub>
−ξ
<sub>2</sub>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>η
<sub>1α</sub>
−η
<sub>2α</sub>
</math-text>
</inline-eqn>
since the operator
<inline-eqn>
<math-text>
<italic>D</italic>
</math-text>
</inline-eqn>
depends only on them.</p>
</proclaim>
<proclaim id="nj299837pro11" type="step" format="num" style="upright">
<heading>Step 5</heading>
<p indent="no">We have, here, to notice that all the norms appearing in equations (
<eqnref linkend="nj299837eqn87">87</eqnref>
) and (
<eqnref linkend="nj299837eqn88">88</eqnref>
) are bounded by constants independent of the reservoir volumes
<inline-eqn>
<math-text>Ω</math-text>
</inline-eqn>
if the interaction operator
<inline-eqn>
<math-text>
<italic>V</italic>
(
<italic>t</italic>
)</math-text>
</inline-eqn>
has a finite norm. For instance, the operator
<inline-eqn></inline-eqn>
defined with the time-ordered exponential ‘T exp’ is unitary and its norm is thus equal to unity [
<cite linkend="nj299837bib54">54</cite>
]. Moreover,
<inline-eqn>
<math-text>
<upright>e</upright>
<sup>−
<italic>C</italic>
−
<italic>D</italic>
</sup>
</math-text>
</inline-eqn>
is
<inline-eqn>
<math-text>
<upright>O</upright>
(
<upright>e</upright>
<sup>∓Ω</sup>
)</math-text>
</inline-eqn>
if
<inline-eqn>
<math-text>
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
</math-text>
</inline-eqn>
is
<inline-eqn>
<math-text>
<upright>O</upright>
(
<upright>e</upright>
<sup>±Ω</sup>
)</math-text>
</inline-eqn>
so that the product
<inline-eqn>
<math-text>
<upright>e</upright>
<sup>
<italic>C</italic>
+
<italic>D</italic>
</sup>
Γ
<sub>
<italic>i</italic>
</sub>
(−
<italic>t</italic>
<sub>0</sub>
)
<upright>e</upright>
<sup>−
<italic>C</italic>
−
<italic>D</italic>
</sup>
</math-text>
</inline-eqn>
is independent of
<inline-eqn>
<math-text>Ω</math-text>
</inline-eqn>
and has a finite norm in the large-system limit. By similar arguments, all the norms in equations (
<eqnref linkend="nj299837eqn87">87</eqnref>
) and (
<eqnref linkend="nj299837eqn88">88</eqnref>
) are found to be independent of
<inline-eqn>
<math-text>Ω</math-text>
</inline-eqn>
. Reservoirs of arbitrarily large size can thus be considered in parallel with arbitrarily long interaction time in order to achieve a steady state.</p>
<p>Accordingly, if the following limit exists
<display-eqn id="nj299837eqn90" textype="equation" notation="LaTeX" eqnnum="90"></display-eqn>
one has
<display-eqn id="nj299837eqn91" textype="equation" notation="LaTeX" eqnnum="91" lines="multiline"></display-eqn>
In short, we have shown
<display-eqn id="nj299837eqn92" textype="equation" notation="LaTeX" eqnnum="92"></display-eqn>
The left-most term only contains
<inline-eqn>
<math-text>
<italic>D</italic>
</math-text>
</inline-eqn>
, which is a function only of
<inline-eqn>
<math-text>ξ
<sub>1</sub>
−ξ
<sub>2</sub>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>η
<sub>1α</sub>
−η
<sub>2α</sub>
</math-text>
</inline-eqn>
, or
<display-eqn id="nj299837eqn93" textype="equation" notation="LaTeX" eqnnum="93"></display-eqn>
QED.</p>
</proclaim>
</sec-level2>
<sec-level2 id="nj299837s4.5" label="4.5">
<heading>Generalization</heading>
<p indent="no">The previous results can be generalized to the case of
<inline-eqn>
<math-text>
<italic>r</italic>
>2</math-text>
</inline-eqn>
reservoirs. In this case, the proposition (
<eqnref linkend="nj299837eqn47">47</eqnref>
) is that the generating function is a function
<display-eqn id="nj299837eqn94" textype="equation" notation="LaTeX" eqnnum="94"></display-eqn>
depending only on the independent parameters:
<display-eqn id="nj299837eqn95" textype="equation" notation="LaTeX" eqnnum="95"></display-eqn>
<display-eqn id="nj299837eqn96" textype="equation" notation="LaTeX" eqnnum="96"></display-eqn>
with
<inline-eqn>
<math-text>
<italic>j</italic>
=1, 2, …,
<italic>r</italic>
−1</math-text>
</inline-eqn>
. The proof is similar to that in the case
<inline-eqn>
<math-text>
<italic>r</italic>
=2</math-text>
</inline-eqn>
with the operators:
<display-eqn id="nj299837eqn97" textype="equation" notation="LaTeX" eqnnum="97"></display-eqn>
<display-eqn id="nj299837eqn98" textype="equation" notation="LaTeX" eqnnum="98"></display-eqn>
replacing equations (
<eqnref linkend="nj299837eqn63">63</eqnref>
) and (
<eqnref linkend="nj299837eqn64">64</eqnref>
), where
<inline-eqn></inline-eqn>
and
<inline-eqn></inline-eqn>
.</p>
<p>In the general case, the fluctuation theorem should read
<display-eqn id="nj299837eqn99" textype="equation" notation="LaTeX" eqnnum="99"></display-eqn>
in terms of the independent affinities
<display-eqn id="nj299837eqn100" textype="equation" notation="LaTeX" eqnnum="100"></display-eqn>
<display-eqn id="nj299837eqn101" textype="equation" notation="LaTeX" eqnnum="101"></display-eqn>
with
<inline-eqn>
<math-text>
<italic>j</italic>
=1, 2, …,
<italic>r</italic>
−1</math-text>
</inline-eqn>
.</p>
</sec-level2>
</sec-level1>
<sec-level1 id="nj299837s5" label="5">
<heading>Symmetry relations for the response coefficients</heading>
<sec-level2 id="nj299837s5.1" label="5.1">
<heading>The fluctuation theorem and response coefficients</heading>
<p indent="no">If we gather the independent parameters and affinities in the case of
<inline-eqn>
<math-text>
<italic>r</italic>
=2</math-text>
</inline-eqn>
reservoirs as
<display-eqn id="nj299837eqn102" textype="equation" notation="LaTeX" eqnnum="102"></display-eqn>
<display-eqn id="nj299837eqn103" textype="equation" notation="LaTeX" eqnnum="103"></display-eqn>
or in the general case of
<inline-eqn>
<math-text>
<italic>r</italic>
>2</math-text>
</inline-eqn>
reservoirs as
<display-eqn id="nj299837eqn104" textype="equation" notation="LaTeX" eqnnum="104"></display-eqn>
<display-eqn id="nj299837eqn105" textype="equation" notation="LaTeX" eqnnum="105"></display-eqn>
with
<inline-eqn>
<math-text>α=1, 2, …,
<italic>c</italic>
</math-text>
</inline-eqn>
and
<inline-eqn>
<math-text>
<italic>j</italic>
=1, 2, …,
<italic>r</italic>
−1</math-text>
</inline-eqn>
, the fluctuation theorem (
<eqnref linkend="nj299837eqn54">54</eqnref>
) reads
<display-eqn id="nj299837eqn106" textype="equation" notation="LaTeX" eqnnum="106"></display-eqn>
where we have explicitly written the dependence of the generating function on the affinities defining the nonequilibrium steady state.</p>
<p>The idea is to differentiate successively the fluctuation theorem with respect to both
<inline-eqn></inline-eqn>
and
<inline-eqn>
<math-text>
<bold-italic>
<italic>A</italic>
</bold-italic>
</math-text>
</inline-eqn>
to obtain symmetry relations for the linear and nonlinear response coefficients as well as further coefficients characterizing the statistics of the current fluctuations [
<cite linkend="nj299837bib21">21</cite>
].</p>
<p>On the one hand, the mean currents can be obtained from the generating function and, on the other hand, expanded in powers of the affinities:
<display-eqn id="nj299837eqn107" textype="equation" notation="LaTeX" eqnnum="107"></display-eqn>
which defines the response coefficients:
<display-eqn id="nj299837eqn108" textype="equation" notation="LaTeX" eqnnum="108"></display-eqn>
<display-eqn id="nj299837eqn109" textype="equation" notation="LaTeX" eqnnum="109"></display-eqn>
<display-eqn id="nj299837eqn110" textype="equation" notation="LaTeX" eqnnum="110"></display-eqn>
around the state of thermodynamic equilibrium.</p>
<p>We notice that, if we set
<inline-eqn></inline-eqn>
in the fluctuation theorem (
<eqnref linkend="nj299837eqn54">54</eqnref>
), we obtain the identities
<display-eqn id="nj299837eqn111" textype="equation" notation="LaTeX" eqnnum="111"></display-eqn>
<display-eqn id="nj299837eqn112" textype="equation" notation="LaTeX" eqnnum="112"></display-eqn>
The former is a condition of normalization and the latter a condition of global detailed balancing, which is a consequence of the fluctuation theorem (
<eqnref linkend="nj299837eqn54">54</eqnref>
) but may be assumed for itself as a weaker property than the fluctuation theorem [
<cite linkend="nj299837bib55">55</cite>
]. On the other hand, the generating function of the cumulants of the fluctuating currents at equilibrium satisfies
<display-eqn id="nj299837eqn113" textype="equation" notation="LaTeX" eqnnum="113"></display-eqn>
obtained from the fluctuation theorem (
<eqnref linkend="nj299837eqn54">54</eqnref>
) at the equilibrium
<inline-eqn>
<math-text>
<bold-italic>A</bold-italic>
=
<bold>0</bold>
</math-text>
</inline-eqn>
.</p>
<p>We start by differentiating the fluctuation theorem with respect to the generating parameters
<inline-eqn>
<math-text>{λ
<sub>α</sub>
}</math-text>
</inline-eqn>
and also the affinities
<inline-eqn>
<math-text>{
<italic>A</italic>
<sub>α</sub>
}</math-text>
</inline-eqn>
to get
<display-eqn id="nj299837eqn114" textype="equation" notation="LaTeX" eqnnum="114"></display-eqn>
<display-eqn id="nj299837eqn115" textype="equation" notation="LaTeX" eqnnum="115"></display-eqn>
We set
<inline-eqn></inline-eqn>
in equation (
<eqnref linkend="nj299837eqn115">115</eqnref>
), use the conditions (
<eqnref linkend="nj299837eqn111">111</eqnref>
) and (
<eqnref linkend="nj299837eqn112">112</eqnref>
) and set
<inline-eqn>
<math-text>
<bold-italic>
<italic>A</italic>
</bold-italic>
=
<bold>0</bold>
</math-text>
</inline-eqn>
to get
<display-eqn id="nj299837eqn116" textype="equation" notation="LaTeX" eqnnum="116"></display-eqn>
<display-eqn id="nj299837eqn117" textype="equation" notation="LaTeX" eqnnum="117"></display-eqn>
from equations (
<eqnref linkend="nj299837eqn114">114</eqnref>
) and (
<eqnref linkend="nj299837eqn115">115</eqnref>
), which shows in particular that the mean currents vanish at equilibrium.</p>
</sec-level2>
<sec-level2 id="nj299837s5.2" label="5.2">
<heading>The Onsager–Casimir reciprocity relations and the Green–Kubo formulae</heading>
<p indent="no">Now, we differentiate equation (
<eqnref linkend="nj299837eqn114">114</eqnref>
) with respect to
<inline-eqn>
<math-text>λ
<sub>β</sub>
</math-text>
</inline-eqn>
to obtain
<display-eqn id="nj299837eqn118" textype="equation" notation="LaTeX" eqnnum="118"></display-eqn>
Setting
<inline-eqn></inline-eqn>
, we find the identity
<display-eqn id="nj299837eqn119" textype="equation" notation="LaTeX" eqnnum="119"></display-eqn>
for the second-order cumulant of the current fluctuations at equilibrium.</p>
<p>If we differentiate equation (
<eqnref linkend="nj299837eqn114">114</eqnref>
) with respect to
<inline-eqn>
<math-text>
<italic>A</italic>
<sub>β</sub>
</math-text>
</inline-eqn>
, we get
<display-eqn id="nj299837eqn120" textype="equation" notation="LaTeX" eqnnum="120"></display-eqn>
which reduces to
<display-eqn id="nj299837eqn121" textype="equation" notation="LaTeX" eqnnum="121"></display-eqn>
for
<inline-eqn></inline-eqn>
. We recover the formulae of the Green–Kubo type in the case
<inline-eqn>
<math-text>α=β</math-text>
</inline-eqn>
:
<display-eqn id="nj299837eqn122" textype="equation" notation="LaTeX" eqnnum="122"></display-eqn>
</p>
<p>The differentiation of equation (
<eqnref linkend="nj299837eqn115">115</eqnref>
) with respect to
<inline-eqn>
<math-text>
<italic>A</italic>
<sub>β</sub>
</math-text>
</inline-eqn>
leads to
<display-eqn id="nj299837eqn123" textype="equation" notation="LaTeX" eqnnum="123" lines="multiline"></display-eqn>
Using equations (
<eqnref linkend="nj299837eqn111">111</eqnref>
) and (
<eqnref linkend="nj299837eqn112">112</eqnref>
) in the limit
<inline-eqn></inline-eqn>
, we have
<display-eqn id="nj299837eqn124" textype="equation" notation="LaTeX" eqnnum="124"></display-eqn>
Combining with equation (
<eqnref linkend="nj299837eqn121">121</eqnref>
), we finally find the Onsager–Casimir reciprocity relations:
<display-eqn id="nj299837eqn125" textype="equation" notation="LaTeX" eqnnum="125"></display-eqn>
We notice that equation (
<eqnref linkend="nj299837eqn121">121</eqnref>
) leading to the Onsager–Casimir relation requires the link established by the fluctuation theorem (
<eqnref linkend="nj299837eqn54">54</eqnref>
) between the variables
<inline-eqn></inline-eqn>
and
<inline-eqn>
<math-text>
<bold-italic>
<italic>A</italic>
</bold-italic>
</math-text>
</inline-eqn>
and does not result from equations (
<eqnref linkend="nj299837eqn111">111</eqnref>
), (
<eqnref linkend="nj299837eqn112">112</eqnref>
) and (
<eqnref linkend="nj299837eqn113">113</eqnref>
) alone.</p>
</sec-level2>
<sec-level2 id="nj299837s5.3" label="5.3">
<heading>Symmetry relations at second order</heading>
<p indent="no">We proceed in a similar way to obtain relations for second-order response coefficients. The differentiation of equation (
<eqnref linkend="nj299837eqn118">118</eqnref>
) with respect to
<inline-eqn>
<math-text>λ
<sub>γ</sub>
</math-text>
</inline-eqn>
gives
<display-eqn id="nj299837eqn126" textype="equation" notation="LaTeX" eqnnum="126"></display-eqn>
which reduces to
<display-eqn id="nj299837eqn127" textype="equation" notation="LaTeX" eqnnum="127"></display-eqn>
for
<inline-eqn></inline-eqn>
.</p>
<p>On the other hand, the differentiation of equation (
<eqnref linkend="nj299837eqn118">118</eqnref>
) with respect to
<inline-eqn>
<math-text>
<italic>A</italic>
<sub>γ</sub>
</math-text>
</inline-eqn>
gives
<display-eqn id="nj299837eqn128" textype="equation" notation="LaTeX" eqnnum="128"></display-eqn>
Here, we introduce the coefficients
<display-eqn id="nj299837eqn129" textype="equation" notation="LaTeX" eqnnum="129"></display-eqn>
which characterizes the nonequilibrium response of the second cumulants of the current fluctuations. Setting
<inline-eqn></inline-eqn>
in equation (
<eqnref linkend="nj299837eqn128">128</eqnref>
) leads to the relation
<display-eqn id="nj299837eqn130" textype="equation" notation="LaTeX" eqnnum="130"></display-eqn>
</p>
<p>Now, if we differentiate equation (
<eqnref linkend="nj299837eqn120">120</eqnref>
) with respect to
<inline-eqn>
<math-text>
<italic>A</italic>
<sub>γ</sub>
</math-text>
</inline-eqn>
, we get
<display-eqn id="nj299837eqn131" textype="equation" notation="LaTeX" eqnnum="131" lines="multiline"></display-eqn>
whereupon we find for
<inline-eqn></inline-eqn>
that
<display-eqn id="nj299837eqn132" textype="equation" notation="LaTeX" eqnnum="132"></display-eqn>
involving the second-order response coefficient (
<eqnref linkend="nj299837eqn109">109</eqnref>
).</p>
<p>We end with the differentiation of equation (
<eqnref linkend="nj299837eqn123">123</eqnref>
) with respect to
<inline-eqn>
<math-text>
<italic>A</italic>
<sub>γ</sub>
</math-text>
</inline-eqn>
to obtain
<display-eqn id="nj299837eqn133" textype="equation" notation="LaTeX" eqnnum="133" lines="multiline"></display-eqn>
Taking
<inline-eqn></inline-eqn>
therein, we deduce
<display-eqn id="nj299837eqn134" textype="equation" notation="LaTeX" eqnnum="134"></display-eqn>
We notice that this relation is the consequence of the weaker condition of global detailed balancing (
<eqnref linkend="nj299837eqn112">112</eqnref>
) alone and could hold even though the fluctuation theorem (
<eqnref linkend="nj299837eqn54">54</eqnref>
) does not as recently shown in [
<cite linkend="nj299837bib55">55</cite>
], where the versions of equation (
<eqnref linkend="nj299837eqn134">134</eqnref>
), which are (anti)symmetrized with respect to the magnetic field, appear with the notations
<inline-eqn>
<math-text>
<italic>M</italic>
<sub>α, βγ</sub>
=(
<italic>k</italic>
<sub>
<upright>B</upright>
</sub>
<italic>T</italic>
)
<sup>2</sup>
<italic>G</italic>
<sub>α, βγ</sub>
<sup>(2)</sup>
</math-text>
</inline-eqn>
,
<inline-eqn>
<math-text>
<italic>R</italic>
<sub>αβ, γ</sub>
=
<italic>k</italic>
<sub>
<upright>B</upright>
</sub>
<italic>TS</italic>
<sub>αβ, γ</sub>
<sup>(1)</sup>
</math-text>
</inline-eqn>
and
<inline-eqn></inline-eqn>
. We point out that, on the other hand, equations (
<eqnref linkend="nj299837eqn130">130</eqnref>
) and (
<eqnref linkend="nj299837eqn132">132</eqnref>
) are consequences of microreversibility and the stronger fluctuation theorem, as it is the case for the Onsager–Casimir reciprocity relations.</p>
<p>Adding equation (
<eqnref linkend="nj299837eqn134">134</eqnref>
) to the same equation with
<inline-eqn>
<math-text>−
<italic>B</italic>
</math-text>
</inline-eqn>
instead of
<inline-eqn>
<math-text>
<italic>B</italic>
</math-text>
</inline-eqn>
and using equation (
<eqnref linkend="nj299837eqn132">132</eqnref>
), we moreover infer that
<display-eqn id="nj299837eqn135" textype="equation" notation="LaTeX" eqnnum="135"></display-eqn>
whereupon we finally obtain the symmetry relations:
<display-eqn id="nj299837eqn136" textype="equation" notation="LaTeX" eqnnum="136"></display-eqn>
<display-eqn id="nj299837eqn137" textype="equation" notation="LaTeX" eqnnum="137"></display-eqn>
<display-eqn id="nj299837eqn138" textype="equation" notation="LaTeX" eqnnum="138"></display-eqn>
If the magnetic field vanishes
<inline-eqn>
<math-text>
<italic>B</italic>
=0</math-text>
</inline-eqn>
, equations (
<eqnref linkend="nj299837eqn137">137</eqnref>
) reduces to the symmetry relations
<display-eqn id="nj299837eqn139" textype="equation" notation="LaTeX" eqnnum="139"></display-eqn>
which were previously deduced as the consequences of the fluctuation theorem [
<cite linkend="nj299837bib22">22</cite>
,
<cite linkend="nj299837bib53">53</cite>
].</p>
</sec-level2>
<sec-level2 id="nj299837s5.4" label="5.4">
<heading>Symmetry relations at third order</heading>
<p indent="no">Besides the third-order response coefficient (
<eqnref linkend="nj299837eqn110">110</eqnref>
), we introduce the coefficients
<display-eqn id="nj299837eqn140" textype="equation" notation="LaTeX" eqnnum="140"></display-eqn>
and
<display-eqn id="nj299837eqn141" textype="equation" notation="LaTeX" eqnnum="141"></display-eqn>
characterizing the nonequilibrium responses of, respectively, the second and third cumulants of the current fluctuations.</p>
<p>We continue the deduction by further differentiating the relations of the previous subsection with respect to
<inline-eqn>
<math-text>λ
<sub>δ</sub>
</math-text>
</inline-eqn>
or
<inline-eqn>
<math-text>
<italic>A</italic>
<sub>δ</sub>
</math-text>
</inline-eqn>
at
<inline-eqn></inline-eqn>
. We find successively from equation (
<eqnref linkend="nj299837eqn126">126</eqnref>
):
<display-eqn id="nj299837eqn142" textype="equation" notation="LaTeX" eqnnum="142"></display-eqn>
and
<display-eqn id="nj299837eqn143" textype="equation" notation="LaTeX" eqnnum="143"></display-eqn>
from equation (
<eqnref linkend="nj299837eqn128">128</eqnref>
):
<display-eqn id="nj299837eqn144" textype="equation" notation="LaTeX" eqnnum="144"></display-eqn>
from equation (
<eqnref linkend="nj299837eqn131">131</eqnref>
):
<display-eqn id="nj299837eqn145" textype="equation" notation="LaTeX" eqnnum="145" lines="multiline"></display-eqn>
and from equation (
<eqnref linkend="nj299837eqn133">133</eqnref>
)
<display-eqn id="nj299837eqn146" textype="equation" notation="LaTeX" eqnnum="146" lines="multiline"></display-eqn>
This last relation is the consequence of the weaker condition of global detailed balancing in the same way as equation (
<eqnref linkend="nj299837eqn134">134</eqnref>
).</p>
<p>In the case of a vanishing magnetic field
<inline-eqn>
<math-text>
<italic>B</italic>
=0</math-text>
</inline-eqn>
, equations (
<eqnref linkend="nj299837eqn143">143</eqnref>
) and (
<eqnref linkend="nj299837eqn145">145</eqnref>
) reduce, respectively, to
<display-eqn id="nj299837eqn147" textype="equation" notation="LaTeX" eqnnum="147"></display-eqn>
and
<display-eqn id="nj299837eqn148" textype="equation" notation="LaTeX" eqnnum="148"></display-eqn>
which has been obtained elsewhere as a consequence of the fluctuation theorem [
<cite linkend="nj299837bib53">53</cite>
].</p>
</sec-level2>
</sec-level1>
<sec-level1 id="nj299837s6" label="6">
<heading>Conclusions</heading>
<p indent="no">In this paper, a fluctuation theorem for the currents has been proved for open quantum systems reaching a nonequilibrium steady state in the long-time limit
<fnref linkend="nj299837fn1">
<sup>4</sup>
</fnref>
. In the considered protocol, the heat and particle currents are defined in terms of the exchanges of energy and particles between reservoirs, as measured at the initial and final times when the reservoirs are decoupled.</p>
<p>We start from a general symmetry relation for the generating function of the exchanges of energy and particles, which is the consequence of the microreversibility guaranteed by the measurement protocol. This symmetry relation is expressed in terms of the temperatures and chemical potentials of the reservoirs. However, the fluctuation theorem for the currents requires a symmetry with respect to the thermodynamic forces or affinities which are given in terms of the
<italic>differences</italic>
of temperatures and chemical potentials.</p>
<p>We show that this symmetry indeed holds by proving that, in the long-time limit, the generating function only depends on the
<italic>differences</italic>
between the parameters corresponding to the different reservoirs. A steady state is reached if the interaction established by the subsystem coupling the reservoirs together remains constant over a lapse of time which is sufficiently long with respect to the transient time intervals associated with the initial switch on of the interaction and its final switch off. Although the generating function introduced by time-dependent protocols depends on the absolute values of the temperatures and chemical potentials, suitable inequalities allow us to relate it to an equivalent function which depends on the
<italic>differences</italic>
of temperatures and chemical potentials. The ratio of the generating function to the equivalent function is bounded by constants that are independent of the lapse of time during which the interaction is constant and, moreover, of the volumes of the reservoirs. Therefore, the lapse of time of constant interaction can be taken to be arbitrarily long and the reservoirs arbitrarily large to reach a steady state. Accordingly, the generating function of the currents has the symmetry of the fluctuation theorem with respect to the affinities characterizing the steady state. A rigorous proof is thus established for such steady-state fluctuation theorems as considered in the review [
<cite linkend="nj299837bib42">42</cite>
].</p>
<p>As a consequence, the Onsager–Casimir reciprocity relations can be obtained for the linear response coefficients from the fluctuation theorem. Furthermore, generalizations of the reciprocity relations to the nonlinear response coefficients can also be deduced.</p>
<p>Finally, we notice that, besides the scheme based on two quantum measurements that we have here considered, another scheme can be envisaged where the currents of energy or particles are continuously monitored by ideal probes which are weakly coupled to the system. We hope to report on this further problem in a future publication.</p>
</sec-level1>
<acknowledgment>
<heading>Acknowledgment</heading>
<p indent="no">DA thanks the FRS-FNRS Belgium for financial support. This research is financially supported by the Belgian Federal Government (IAP project ‘NOSY’) and the ‘Communauté française de Belgique’ (contract ‘Actions de Recherche Concertées’ No. 04/09-312). This work was partially supported by a Grant-in-Aid for Scientific Research (C) (No. 17540365) from the Japan Society for the Promotion of Science, and by the ‘Academic Frontier’ Project at Waseda University from the Ministry of Education, Culture, Sports, Science and Technology of Japan, as well as by a Waseda University Grant for Special Research Projects (No. 2008A-850) from Waseda University.</p>
</acknowledgment>
</body>
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<cr_issn type="print">00346861</cr_issn>
<cr_issn type="electronic">15390756</cr_issn>
</crossref>
</journal-ref>
<journal-ref id="nj299837bib53" num="53">
<authors>
<au>
<second-name>Andrieux</second-name>
<first-names>D</first-names>
</au>
<au>
<second-name>Gaspard</second-name>
<first-names>P</first-names>
</au>
</authors>
<year>2007</year>
<jnl-title>J. Stat. Mech.</jnl-title>
<volume></volume>
<pages>P02006</pages>
<crossref>
<cr_doi>http://dx.doi.org/10.1088/1742-5468/2007/02/P02006</cr_doi>
<cr_issn type="electronic">17425468</cr_issn>
</crossref>
</journal-ref>
<book-ref id="nj299837bib54" num="54">
<authors>
<au>
<second-name>Bratteli</second-name>
<first-names>O</first-names>
</au>
<au>
<second-name>Robinson</second-name>
<first-names>D W</first-names>
</au>
</authors>
<year>2002</year>
<book-title>Operator Algebras and Quantum Statistical Mechanics</book-title>
<volume>vol 1 and 2</volume>
<publication>
<place>Berlin</place>
<publisher>Springer</publisher>
</publication>
<misc-text>corollary 2.3.4, proposition 5.4.1</misc-text>
</book-ref>
<journal-ref id="nj299837bib55" num="55">
<authors>
<au>
<second-name>Förster</second-name>
<first-names>H</first-names>
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<au>
<second-name>Büttiker</second-name>
<first-names>M</first-names>
</au>
</authors>
<year>2008</year>
<jnl-title>Phys. Rev. Lett.</jnl-title>
<volume>101</volume>
<pages>136805</pages>
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<cr_doi>http://dx.doi.org/10.1103/PhysRevLett.101.136805</cr_doi>
<cr_issn type="print">00319007</cr_issn>
<cr_issn type="electronic">10797114</cr_issn>
</crossref>
</journal-ref>
</reference-list>
</references>
<footnotes>
<footnote id="nj299837fn1" source="text" type="number" marker="4">
<p>We obtained the symmetry relation (
<eqnref linkend="nj299837eqn39">39</eqnref>
) in March 2006 and the relations of section 
<secref linkend="nj299837s5">5</secref>
in April 2006, both for systems in a magnetic field. We finally established the connection between the two thanks to the proposition (
<eqnref linkend="nj299837eqn47">47</eqnref>
) in June 2008.</p>
</footnote>
</footnotes>
</back>
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<titleInfo lang="eng">
<title>The fluctuation theorem for currents in open quantum systems</title>
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<titleInfo type="abbreviated">
<title>The fluctuation theorem for currents in open quantum systems</title>
</titleInfo>
<titleInfo type="alternative" lang="eng">
<title>The fluctuation theorem for currents in open quantum systems</title>
</titleInfo>
<name type="personal">
<namePart type="given">D</namePart>
<namePart type="family">Andrieux</namePart>
<affiliation>Center for Nonlinear Phenomena and Complex Systems, Universit Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium</affiliation>
<affiliation>E-mail: dandrieu@ulb.ac.be</affiliation>
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</name>
<name type="personal">
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<namePart type="family">Gaspard</namePart>
<affiliation>Center for Nonlinear Phenomena and Complex Systems, Universit Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium</affiliation>
<affiliation>Author to whom any correspondence should be addressed.</affiliation>
<affiliation>E-mail: gaspard@ulb.ac.be</affiliation>
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<roleTerm type="text">author</roleTerm>
</role>
</name>
<name type="personal">
<namePart type="given">T</namePart>
<namePart type="family">Monnai</namePart>
<affiliation>Department of Applied Physics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan</affiliation>
<affiliation>E-mail: monnai@aoni.waseda.jp</affiliation>
<role>
<roleTerm type="text">author</roleTerm>
</role>
</name>
<name type="personal">
<namePart type="given">S</namePart>
<namePart type="family">Tasaki</namePart>
<affiliation>Department of Applied Physics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan</affiliation>
<affiliation>E-mail: stasaki@waseda.jp</affiliation>
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<abstract>A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the OnsagerCasimir reciprocity relations to the nonlinear response coefficients.</abstract>
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<title>New Journal of Physics</title>
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<title>New J. Phys.</title>
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<genre type="journal">journal</genre>
<identifier type="ISSN">1367-2630</identifier>
<identifier type="eISSN">1367-2630</identifier>
<identifier type="PublisherID">nj</identifier>
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<identifier type="URL">stacks.iop.org/NJP</identifier>
<part>
<date>2009</date>
<detail type="volume">
<caption>vol.</caption>
<number>11</number>
</detail>
<detail type="issue">
<caption>no.</caption>
<number>4</number>
</detail>
<extent unit="pages">
<start>1</start>
<end>25</end>
<total>25</total>
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<identifier type="DOI">10.1088/1367-2630/11/4/043014</identifier>
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<accessCondition type="use and reproduction" contentType="copyright">IOP Publishing and Deutsche Physikalische Gesellschaft</accessCondition>
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