General linear thermodynamics for periodically driven systems with multiple reservoirs

TL;DR

This paper develops a linear thermodynamics framework for Markov systems with periodic driving and multiple reservoirs, providing explicit formulas for thermodynamic quantities and verifying Onsager relations, with applications to quantum dots.

Contribution

It introduces a comprehensive linear thermodynamics theory for periodically driven Markov systems, including explicit formulas and verification of Onsager relations, applicable to quantum systems with multiple reservoirs.

Findings

Derived explicit formulas for thermodynamic quantities.

Verified Onsager-Casimir reciprocal relations.

Applied theory to a quantum dot system.

Abstract

We derive a linear thermodynamics theory for general Markov dynamics with both steady-state and time-periodic drivings. Expressions for thermodynamic quantities, such as mechanical and chemical work, heat and entropy production are obtained in terms of equilibrium probability distribution and the drivings. The entropy production is derived as a bilinear function of thermodynamic forces and the associated fluxes. We derive explicit formulae for the Onsager coefficients and use them to verify the Onsager-Casimir reciprocal relations. Our results are illustrated on a periodically driven quantum dot in contact with two electron reservoirs and optimization protocols are discussed.

Equations

$${\dot{p}}_m(t)=\sum _{n,j}{W}_{mn}^j(t)p_n(t),$$

$$\sum _m{W}_{mn}^j(t)=0,$$

$$P_m^j=\frac{1}{Z_{\mathrm{eq}}^j}{e}^{\frac{-({ϵ}_m-{\mu }_j{n}_m)}{T_j}},$$

$$\frac{W_{mn}^{eq;j}}{W_{nm}^{eq;j}}=e^{-\{({ϵ}_m-{ϵ}_n)-{\mu }_j(n_m-n_n)\}/T_j}.$$

$$W_{mn}^{eq;j}=C_{mn}^j{\lambda }_n^{eq;j},$$

$$C_{mn}^j=C_{nm}^j,C_{nn}^j=-\sum _{m\ne n}{C}_{mn}^j.$$

$$\frac{1}{T_j(t)}$$

$${\mu }_j(t)$$

$${ϵ}_n(t)={ϵ}_{0,n}+T_{0,j}{F}_{ϵ}{\gamma }_{ϵ,n}{g}_{ϵ}(t),$$

$$C_{mn}^j(t)=C_{mn}^{eq,j}+\sum _{\alpha ,j}{F}_{{\alpha }_j}{g}_{{\alpha }_j}(t)C_{mn}^{{\alpha }_j}.$$

$$W_{mn}^j(t)=W_{mn}^{eq,j}+\sum _{\alpha }{F}_{{\alpha }_j}{W}_{mn}^{{\alpha }_j}(t),$$

$$W_{mn}^{\alpha ;j}(t)=g_{{\alpha }_j}(t)[W_{mn}^{eq,j}{\gamma }_n^{\alpha }+C_{mn}^{{\alpha }_j}{\lambda }_n^{eq;j}],$$

$${\gamma }_n^{\alpha }=\{\begin{array}{ll}({ϵ}_{0,n}-{\mu }_0{n}_n)& \alpha =T_j\\ {\gamma }_{ϵ,n}& \alpha =ϵ\\ -n_n& \alpha ={\mu }_j.\end{array}$$

$$p(t)=p^{eq}+\sum _{\alpha ,j}{F}_{{\alpha }_j}{p}^{{\alpha }_j}(t),$$

$${\dot{p}}^{{\alpha }_j}(t)=W^{eq}{p}^{{\alpha }_j}(t)+W^{{\alpha }_j}(t)p^{eq}.$$

$$p^{{\alpha }_j}(t)={\int }_0^{\infty }\mathrm{d}\tau e^{W^{eq}\tau }{W}^{{\alpha }_j}(t-\tau )p^{eq},$$

$$p^{{\alpha }_j}(t)={\int }_0^{\infty }\mathrm{d}\tau e^{W^{eq}\tau }{W}^{eq,j}{\gamma }^{\alpha }{p}^{eq}{g}_{{\alpha }_j}(t-\tau ),$$

$${\dot{W}}_{mech}(t)$$

$${\dot{W}}_{chem}(t)$$

$$\dot{Q}(t)$$

$${\dot{p}}_m^j(t)=\sum _n{W}_{mn}^j(t)p_n(t).$$

$$\dot{U}(t)={\dot{W}}_{mech}(t)+{\dot{W}}_{chem}(t)+\dot{Q}(t),$$

$${\dot{\bar{W}}}_{mech}=\frac{T_0{F}_{ϵ}}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt{\dot{g}}_{ϵ}(t)\sum _m{\gamma }_{ϵ,m}{p}_m(t).$$

$$J_{ϵ}=-\frac{1}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt{g}_{ϵ}(t)\sum _m{\gamma }_{ϵ,m}{\dot{p}}_m(t),$$

$$J_{ϵ}=\sum _j{J}_{{ϵ}_j},$$

$$J_{{ϵ}_j}=-\frac{1}{\mathcal{T}}{\int }_0^{\mathcal{T}}dt{g}_{ϵ}(t)\sum _{m,n}{\gamma }_{{ϵ}_j,m}{W}_{mn}^j(t)p_n(t),$$

$${\dot{\bar{W}}}_{chem}=T_0\sum _j{F}_{{\mu }_j}{J}_{{\mu }_j},$$

$$J_{{\mu }_j}$$

$$\dot{\bar{Q}}=-\dot{\bar{W}}-{\dot{\bar{W}}}_{chem}=-T_0\sum _j\left(F_{{ϵ}_j}{J}_{{ϵ}_j}+F_{{\mu }_j}{J}_{{\mu }_j}\right).$$

$$\bar{\sigma }=\sum _j{\bar{\sigma }}^j,$$

$${\bar{\sigma }}^j=\frac{1}{\mathcal{T}}\sum _{m,n}{\int }_0^{\mathcal{T}}dt{W}_{mn}^j{p}_n\ln \frac{W_{mn}^j{p}_n}{W_{nm}^j{p}_m}.$$

$${\bar{\sigma }}^j=\frac{1}{\mathcal{T}}\sum _{m,n}{\int }_0^{\mathcal{T}}dt{W}_{mn}^j{p}_n\ln \frac{W_{mn}^j}{W_{nm}^j}.$$

Full text

I Introduction

Due to the seminal work of primarily Onsager and Prigogine, the theory of linear irreversible thermodynamics has become one of the cornerstones of modern statistical physics. Close to equilibrium, one can use this framework to determine the thermodynamic fluxes, such as heat and work, and show that they satisfy general properties, such as Onsager symmetry and the Green-Kubo relations de2013non .

Over the last two decades, a somewhat different approach has been undertaken to study the thermodynamics of small-scaled systems seifert2012stochastic ; van2015ensemble . This theory, known as stochastic thermodynamics, uses Markov dynamics to model systems at the mesoscale, where fluctuations in the thermodynamic fluxes become important. The assumption of local detailed balance then leads to a consistent definition of the thermodynamic properties of the system. The stochastic fluxes of the system satisfy general relations such as the Jarzynski equality jarzynski1997nonequilibrium ; saira2012test . Furthermore, this theory has lead to applications in several other branches of science, such as information theory parrondo2015thermodynamics , chemical reaction networks rao2016nonequilibrium , and active matter mandal2017entropy ; dabelow2019irreversibility .

A natural question to ask is how the classical ideas of linear irreversible thermodynamics can be incorporated in the theory of stochastic thermodynamics. This problem has been addressed for several case-studies izumida2009onsager ; izumida2010onsager ; rosas2016onsager2 ; rosas2016onsager ; proesmans2016brownian ; cerino2016linear ; brandner2016periodic ; yamamoto2016linear ; proesmans2017underdamped ; rosas2017stochastic . Furthermore, general theories have been derived for periodically driven systems in contact with a single reservoir brandner2015thermodynamics ; proesmans2015onsager ; proesmans2016linear , and for steady-state systems in contact with two reservoirs tome2012entropy ; tome2015stochastic , leading to bounds the power and efficiency of thermodynamic engines bauer2016optimal ; proesmans2016power ; iyyappan2018relations . A more general approach for systems with any number of reservoirs and time-dependent driving has not been studied thoroughly.

In this paper, we close this gap by deriving a general formalism for the linear thermodynamics of stochastic systems with both steady-state and time-periodic drivings. Our study is carried out by taking into account multiple heat and particle reservoirs. We obtain expressions for thermodynamic quantities, such as mechanical and chemical work, heat and entropy production in terms of equilibrium probability distribution and the drivings. In particular, we show that general results of linear irreversible thermodynamics, such as the structure of entropy production rate and Onsager symmetry are valid for this general class of systems.

This paper is organized as follows. We start in section II by introducing the model and by discussing its linearized dynamics. In section III, we define the work and heat fluxes and show how they are related to the entropy production rate. The evaluation of Onsager coefficients and the existence of an Onsager-Casimir symmetry relation are discussed in IV. In section V, we apply our formalism to a periodically driven two-level system. Conclusions and outlook are discussed in section VI.

II Model

Throughout this paper, we focus on systems with a discrete set of states in contact with multiple temperature and particle reservoirs that can induce transitions between distinct configurations. The system can be in a given state $m$, specified by its energy $\epsilon_{m}(t)$ and particle number $n_{m}$, with probability $p_{m}(t)$. The time-evolution of $p_{m}(t)$ is described by a master equation

[TABLE]

where $W^{j}_{mn}(t)$ is the probability per unit of time of a transition from a state $n$ to a state $m$ induced by reservoir $j$. Conservation of probability implies that

[TABLE]

valid for all $m$, and therefore $W^{j}_{mm}(t)=-\sum_{n\neq m}W^{j}_{nm}(t)$.

Each reservoir $j$ is characterized by a temperature $T_{j}(t)$ and chemical potential $\mu_{j}(t)$. If the system is in contact with a single reservoir with time-independent temperature $T_{j}$ and chemical potential $\mu_{j}$, it will converge to an equilibrium state given by the Boltzmann-Gibbs distribution:

[TABLE]

where $Z_{\rm eq}^{j}=\sum_{m}e^{-(\epsilon_{m}-\mu_{j}n_{m})/T_{j}}$ is the (grand-canonical) partition function. By definition, the above equilibrium distribution should satisfy the detailed balance condition, $W^{j,eq}_{mn}P^{j}_{n}-W^{j,eq}_{nm}P^{j}_{m}=0$, implying the following ratio between the transition rates $W^{j,eq}_{mn}$ and $W^{j,eq}_{nm}$

[TABLE]

This expression allows us to write the transition rate $W^{eq;j}_{mn}$ as follows:

[TABLE]

where $\lambda^{eq;j}_{n}=\exp\left({(\epsilon_{n}-\mu_{j}n_{n})/T_{j}}\right)$ and $C^{j}_{mn}$ is a matrix that quantifies the coupling strength between states $m$ and $n$. Due to the assumption of local detailed balance and the properties of the transition matrix, $C^{j}$ satisfies the following symmetry relations:

[TABLE]

II.1 Linear description

As stated before, the system will reach an equilibrium Boltzmann state when it is in contact with a single reservoir at constant temperature and chemical potential. This is generally not the case when the system is in contact with multiple reservoirs or when the temperature and chemical potential are modulated time-periodically. In those cases, detailed balance is broken, and the system starts dissipating heat and producing entropy. As each reservoir operates independently, the transition rate of each reservoir has the same form as in Eq. (5), but with time-dependent parameters $\epsilon_{m}(t),T_{j}(t),\mu_{j}(t)$ and $C^{j}_{mn}(t)$. The total transition matrix is obtained by summing over all reservoirs, $W_{mn}(t)=\sum_{j}W^{j}_{mn}(t)$, where $W^{j}_{mn}(t)$ is given by Eq. (5) for every reservoir $j$.

The temperatures and chemical potentials are modulated time-periodically. We introduce the driving functions $g_{T_{j}}(t)$ and $g_{\mu_{j}}(t)$ as

[TABLE]

where $F_{\alpha_{j}}$’s correspond to the strength of the thermodynamic drivings $\alpha_{j}\in\{T_{j},\mu_{j}\}$. The energy of each microscopic state is also driven periodically by an external work source,

[TABLE]

where $\gamma_{\epsilon,n}$ is the amplitude with which the level $n$ is modulated. As we are focusing on the regime close to equilibrium, both temperature and chemical potential modulations are assumed to be around the same equilibrium state for all reservoirs, $T_{0,j}=T_{0}$ and $\mu_{0,j}=\mu_{0}$ for all $j$.

To make further progress, we assume that the thermodynamic forces are sufficiently small so that we can perform a linear approximation. This is the crucial assumption for the theory of linear irreversible thermodynamics de2013non . By expanding the coupling matrix $C$ up to first-order with respect to modulations of temperature, chemical potential and energy, we have that

[TABLE]

The perturbed coupling matrices $C^{\alpha_{j}}$ should satisfy the same symmetry relations as the unperturbed coupling matrices, Eq. (6). They are used for obtaining the following linear expression for $W^{j}_{mn}(t)$ in terms of thermodynamic forces:

[TABLE]

where

[TABLE]

and the vector $\gamma^{\alpha}$ has elements given by

[TABLE]

Since the driving functions $g_{\alpha_{j}}(t)$ are assumed to be time periodic, $g_{\alpha_{j}}(t+\mathscr{T})=g_{\alpha_{j}}(t)$, with $\mathscr{T}$ being the period of the driving, the system will relax to a time-periodic steady-state distribution. This distribution can be expanded up to linear order in terms of the thermodynamic forces proesmans2016linear ,

[TABLE]

where $p^{eq}$ is the Boltzmann-Gibbs distribution associated to the reference energy, temperature and chemical potential, $\epsilon_{0,n},T_{0}$ and $\mu_{0}$, respectively. Substituting Eq. (14) into the master equation (1) leads to

[TABLE]

Thus, the first-order contribution in the probability ${\dot{p}}^{\alpha_{j}}(t)$ depends only on the total equilibrium matrix $W^{eq}_{mn}=\sum_{j}W^{eq,j}_{mn}$ and on the linear perturbation matrix $W^{\alpha_{j}}(t)$ evaluated over the equilibrium probability. The above expression can be integrated, leading to

[TABLE]

which is time-periodic. Plugging in the explicit formula for $W^{\alpha_{j}}(t)$ and once again taking into account the properties of the coupling matrix given by Eq. (6), we arrive at the final expression for the component $p^{\alpha_{j}}(t)$

[TABLE]

i.e., the first order correction in the coupling matrix does not lead to corrections in the probability distribution. We conclude that, up to first-order, the time-periodic steady-state distribution can be obtained exactly in terms of the equilibrium properties, the driving function $g_{\alpha}(t)$’s and the $\gamma_{\alpha}$’s.

Finally, it is worth mentioning that Eq. (17) reduces to the one obtained in Ref. proesmans2016linear for the one reservoir case.

III Thermodynamic fluxes

Having developed a general formalism to derive the linear dynamics of the system under study, we are now ready to evaluate the thermodynamic properties, using the framework of stochastic thermodynamics van2015ensemble ; seifert2012stochastic . In particular, the mechanical work flux ${\dot{W}}_{mech}(t)$, chemical work flux ${\dot{W}_{chem}(t)}$ and the heat flux ${\dot{Q}(t)}$ are given by

[TABLE]

with

[TABLE]

The time evolution of the mean internal energy of the system $U(t)=\sum_{m}\epsilon_{m}(t)p_{m}(t)$ is given by

[TABLE]

in agreement with the first law of thermodynamics. By inserting Eq. (9) into Eq. (18), the average mechanical work per unit of time can be written as

[TABLE]

Eq. (23) is conveniently rewritten as a product of forces and flux, $\dot{\bar{W}}_{mech}=T_{0}F_{\epsilon}J_{\epsilon}$, with work flux $J_{\epsilon}$ given by

[TABLE]

where a partial integration was performed taking into account the periodicity of $p_{m}(t)$. One can decompose it further as

[TABLE]

with $J_{\epsilon_{j}}$ given by

[TABLE]

and $T_{0}F_{\epsilon}J_{\epsilon_{j}}$ can be interpreted as the mechanical work delivered to reservoir $j$.

Proceeding analogously, the total mean chemical work per cycle can be obtained by integrating Eq. (19) over one period and subtracting $\mu_{0}\sum_{j}\int_{0}^{\mathscr{T}}{\dot{p}}^{j}_{m}(t)dt=0$, which gives

[TABLE]

where $J_{\mu_{j}}$ is defined as

[TABLE]

Since the driving is periodic, the average internal energy of the system does not change over a period of the driving, ${\dot{\bar{U}}}=0$. The first law of thermodynamics then leads to an expression for the average heat in terms of fluxes,

[TABLE]

This expression can also be evaluated by a direct integration of Eq. (20) over one period and summing over the contribution of all reservoirs.

The total entropy production $\bar{\sigma}$ is given by the sum of the contribution of all reservoirs

[TABLE]

where each term $\bar{\sigma}^{j}$ can be calculated through the microscopic formula schnakenberg1976network ; jiu1984stability

[TABLE]

Due to periodicity of the steady-state, the integral $\int_{0}^{\mathscr{T}}dt\,\sum_{m,n}W^{j}_{mn}p_{n}\ln(p_{n}/p_{m})$ is strictly zero and Eq. (31) then reduces to

[TABLE]

Since the ratio between $W^{j}_{mn}$ and $W^{j}_{nm}$ is given by the local detailed balance condition, we can derive an expression for $\bar{\sigma}^{j}$ in terms of thermodynamic variables;

[TABLE]

in agreement with the classical thermodynamic definition of entropy production callen1998thermodynamics .

By inserting the expressions for heat and temperature from Eqs. (7) and (20), we have that

[TABLE]

This suggests the introduction of a new thermodynamic flux,

[TABLE]

which allows us to rewrite the stochastic thermodynamics formula for entropy production to a bilinear function of thermodynamic forces and fluxes given by

[TABLE]

with $F_{\epsilon_{j}}=F_{\epsilon}$. This is in agreement with classical non-equilibrium thermodynamics de2013non .

It is worth noting that the structure of Eq. (35) clearly mimics that of the work and chemical fluxes, Eqs. (28) and (26). In fact one can easily verify that all three types of thermodynamic fluxes are of the form

[TABLE]

with $\alpha=T,\epsilon,\mu$.

IV Onsager coefficients

As the thermodynamic fluxes vanish in the absence of thermodynamic forces, one expects that they depend linearly on the thermodynamic forces $F_{\beta}$ near equilibrium, which implies the following form for a flux $J_{\alpha}$:

[TABLE]

where ${\it L}_{\alpha,\beta}$ are the so-called Onsager coefficients. From Eq. (36), the entropy production rate $\bar{\sigma}$ is depicted as a quadratic function of the thermodynamic forces,

[TABLE]

In the absence of odd parity variables (such as magnetic fields), the Onsager coefficients of steady-state systems generally satisfy the Onsager reciprocal relations, $L_{\alpha,\beta}=L_{\beta,\alpha}$. This is no longer the case for systems with time-dependent driving, since the driving breaks the time-reversal symmetry. In this instance, Onsager symmetry is replaced by the weaker Onsager-Casimir symmetry, which relates the Onsager coefficients under time-forward driving to the cross-coefficient of time-inverted driving,

[TABLE]

where the tilde stands for time inverted driving, $\tilde{g}_{\alpha}(t)=g_{\alpha}(-t)$.

Our aim here is to evaluate the Onsager coefficients and to prove the Onsager-Casimir reciprocal relations for a periodically driven system in contact with multiple reservoirs. By expanding Eq. (37) up to first order in the thermodynamic forces, we verify that $J_{\alpha}$ has two terms, one associated with the first order expansion of the transition matrix $W^{j}_{mn}$ (from Eqs. (11) and (12)) and the other with the expansion of the probability distribution $p_{n}(t)$ (from Eq. (17)). The total flux then reads $J_{\alpha_{j}}=J^{(1)}_{\alpha_{j}}+J^{(2)}_{\alpha_{j}}$, with

[TABLE]

and

[TABLE]

respectively. One can easily see that the first flux depends solely on the thermodynamic forces associated with the same reservoir, while the second flux is dependent on all thermodynamic forces. Using the linearized expressions, one can now associate an Onsager matrix with each of those fluxes, $J_{\alpha_{j}}^{(1)}=\sum_{\beta}L^{(1)}_{\alpha_{j},\beta_{j}}F_{\beta_{j}}$ and $J_{\alpha_{j}}^{(2)}=\sum_{\beta,j^{{}^{\prime}}}L^{(2)}_{\alpha_{j},\beta_{j^{{}^{\prime}}}}F_{\beta_{j^{{}^{\prime}}}}$, where the Onsager coefficients are given by

[TABLE]

and

[TABLE]

with $L^{(1)}_{\alpha_{j},\beta_{j^{\prime}}}=0$ for $j\neq j^{\prime}$. The total Onsager matrix is the sum of these two matrices, $L_{\alpha_{j},\beta_{j^{\prime}}}=L^{(1)}_{\alpha_{j},\beta_{j^{\prime}}}+L^{(2)}_{\alpha_{j},\beta_{j^{\prime}}}$. This structure for the Onsager coefficients resembles the one found for a class of quantum mechanical systems studied in brandner2016periodic .

We are now ready to study the reciprocal relations for $L^{(1)}_{\alpha_{j},\beta_{j}}$ and $L^{(2)}_{\alpha_{j},\beta_{j^{{}^{\prime}}}}$. Remarkably, both Onsager matrices will satisfy an Onsager-Casimir relation separately, which implies that the total Onsager matrix will satisfy the same Onsager-Casimir symmetry. We first look at $L^{(1)}_{\alpha_{j},\beta_{j}}$. One can easily verify that these coefficients are invariant under time-reversal by replacing $g_{\alpha}(t)$ and $g_{\beta}(t)$ by $g_{\alpha}(-t)$ and $g_{\beta}(-t)$ and doing a change of integration variable to $t^{\prime}=-t$. Subsequently, taking into account the detailed balance condition, one can show that $\sum_{m,n}\gamma_{n}^{\beta}W_{nm}^{eq,j}\gamma_{m}^{\alpha}p_{m}^{eq}=\sum_{m,n}\gamma_{m}^{\alpha}W_{mn}^{eq,j}\gamma_{n}^{\beta}p_{n}^{eq}$ and hence the right side of Eq. (43) becomes

[TABLE]

which proves the Onsager-Casimir symmetry for $L^{(1)}_{\alpha_{j},\beta_{j}}$.

The proof of the second term can be done in a similar way. We start by taking the time transformation $-t=t^{\prime}-\tau$ in Eq. (44),

[TABLE]

Due to the periodicity of the drivings, the first integral can be shifted by $\mathscr{T}-\tau$, allowing us to rewrite it as

[TABLE]

By once again appealing to the detailed balance condition, one can show that

[TABLE]

and hence $L^{(2)}_{\alpha_{j},\beta_{j^{{}^{\prime}}}}$ becomes

[TABLE]

The right hand side of Eq. (49) is just the Onsager coefficient ${\tilde{L}}_{\beta_{j^{{}^{\prime}}},\alpha_{j}}$. This completes the proof of the Onsager-Casimir symmetry.

These results also implies an Onsager-Casimir relation for any combination of $J_{\alpha,i}$’s. For example, if we define,

[TABLE]

for some matrix $A$, then the associated thermodynamic forces are given by

[TABLE]

as the entropy production rate $\sigma=\sum_{i}F^{\prime}_{i}J^{\prime}_{i}$ is independent of the choice of $J$’s. The new Onsager matrix $L^{\prime}$ is of the form,

[TABLE]

and one can straightforwardly verify that this matrix should satisfy the same Onsager-Casimir relations as the original matrix $L$. In particular, we can conclude that Onsager-Casimir symmetry is also valid when one only looks at the total fluxes $J_{\epsilon}$, $J_{\mu}$ and $J_{T}$.

V Two level systems

As an example, we consider a quantum dot, with one active energy level, in contact with two electron reservoirs at temperatures $T_{1}(t)$ and $T_{2}(t)$ and chemical potentials $\mu_{1}(t)$ and $\mu_{2}(t)$, respectively. The quantum dot can be empty or occupied by a single electron with probabilities $p_{0}(t)$ and $p_{1}(t)=1-p_{0}(t)$, respectively. The temperature and chemical of the electron reservoirs as well as the energy of the quantum dot are modulated according to Eqs. (7)-(9), respectively. The total transition matrix ${W(t)}$ is the sum of both reservoir contributions ${W(t)}={W^{1}(t)+W^{(2)}(t)}$, with $W^{j}(t)$ given by

[TABLE]

where $\Gamma_{j}$ describes the interaction between the quantum dot and the $j$-th reservoir and $y(t)$ is the Fermi-Dirac distribution $y(t)=[1+\exp((\epsilon(t)-\mu(t))/T(t))]^{-1}$.

For simplicity, we set $F_{\mu_{j}}=0$, implying that the chemical potentials of both reservoirs are the same and thereby $\dot{\bar{W}}_{chem}=0$. The thermodynamic variables are modulated via the driving functions $g_{\epsilon}(t)=\sqrt{2}\sin(\omega t+\phi)$ and $g_{T_{j}}(t)=\sqrt{2}\sin(\omega t)$, respectively, where we have assumed that both reservoirs and the electron level are all driven with the same frequency $\omega=2\pi/\mathscr{T}$, but with a phase-difference $\phi$ between the driving of the energy and the and temperature drivings.

The lowest order expressions for the energy and temperature fluxes $J_{\epsilon}$ and $J_{T_{j}}$ and for the entropy production rate ${\bar{\sigma}}$ can now be calculated using the expressions from the previous section and are given by

[TABLE]

[TABLE]

[TABLE]

respectively, where $\chi_{j}=\Gamma_{j}\mathscr{T}$ and ${\tilde{\chi}}=(\Gamma_{1}+\Gamma_{2})\mathscr{T}$. For the special case, $F_{T_{1}}=F_{T_{2}}$ and $\Gamma_{1}=\Gamma_{2}$, the above expressions reduce to the single reservoir case up to a factor $2$ proesmans2016linear . $J_{T_{2}}$ has a similar expression as $J_{T_{1}}$ with $T_{1}$ and $T_{2}$, $F_{1}$ and $F_{2}$ interchanged. One can easily verify that these expressions satisfy Onsager-Casimir symmetry.

These results can be used to optimize the amount of mechanical work, $\dot{\bar{W}}=T_{0}F_{\epsilon}J_{\epsilon}$ that can be done by the system. In particular, an optimization with respect to $F_{\epsilon}$ yields the following relation for the maximum mechanical power:

[TABLE]

Fig. 1 depicts the behavior of $\sigma$ and $\dot{\bar{W}}$ versus $\phi$ for distinct couplings $\chi_{1}$ and $\chi_{2}$ with $F_{\epsilon}$ given by its optimal value. In the limit of low (large) “effective” couplings, ${\tilde{\chi}}\ll 1$ (${\tilde{\chi}}\gg 1$), the work output is maximal (minimum) when the driving of the work source is in phase with that of the heat sources, $\phi=0$ and minimum (maximum) mwhen the driving is out of phase, $\phi=\pi/2$, in accordance with Eq. (56). Conversely, for the above choice of $F_{\epsilon}$, the positions of maxima and minima of the entropy production fulfill the above relation

[TABLE]

where $+(-)$ denote to the maximum (minimum). They approach $\pi/2$ (maximum) and $\phi=0$ (minimum) for ${\tilde{\chi}}\ll 1$ and ${\tilde{\chi}}\gg 1$, respectively and deviate from these limits for intermediate coupling sets.

Similar analytic optimizations for other thermodynamic fluxes can also be performed. For instance, by optimizing $\dot{\bar{W}}$ with respect to both $F_{\epsilon}$ and the phase-difference $\phi$, the expression for $\dot{\bar{W}}_{\mathrm{max}}$ becomes

[TABLE]

where the optimal phase-difference and amplitude are given by

[TABLE]

VI Conclusions

In this paper, we have derived a general linear description for the thermodynamics of Markov systems in contact to multiple reservoirs, using the framework of stochastic thermodynamics. We have shown that the thermodynamic fluxes such as mechanical and chemical work and heat, can be written in a general form, as functions of the driving and the equilibrium properties of the system. The entropy production is obtained as a bilinear function of thermodynamic forces and associated fluxes. Furthermore, we calculated all Onsager coefficients and showed that they satisfy a generalized Onsager-Casimir relation.

Finally, we mention some interesting directions for further research. Firstly, it would be interesting to extend our analysis to higher order response coefficients and to study the resulting constraints on heat engines cleuren2015universality ; vroylandt2018degree . Secondly, it would be very interesting to see if our results can be extended to quantum mechanical systems and systems with strong coupling vinjanampathy2016quantum ; jarzynski2017stochastic . Finally, it should be no problem to test our predictions, such as the generalized Onsager-Casimir relation, with state-of-the-art experimental setups pekola2015towards ; ciliberto2017experiments .

Acknowledgements.

We thank Bart Cleuren for a careful reading of the manuscript. KP is a postdoctoral fellow of the Research Foundation-Flanders (FWO). C.E.F. acknowledges the financial support from FAPESP under grant No 2018/02405-1