On work and heat in time-dependent strong coupling

TL;DR

This paper analyzes the effects of time-dependent strong coupling between a system and a thermal bath, revealing additional forces and correction terms in work and heat functionals relevant to stochastic thermodynamics and quantum entropy changes.

Contribution

It introduces the impact of time-dependent coupling on work, heat, and forces in system-bath interactions, extending stochastic thermodynamics to strong, dynamic couplings.

Findings

Additional harmonic force due to time-dependent coupling.

Large correction terms in work and heat during rapid coupling changes.

Finite contributions to work in Jarzynski's equality despite large corrections.

Abstract

This paper revisits the classical problem of representing a thermal bath interacting with a system as a large collection of harmonic oscillators initially in thermal equilibrium. As is well known the system then obeys an equation, which in the bulk and in the suitable limit tends to the Kramers-Langevin equation of physical kinetics. I consider time-dependent system-bath coupling and show that this leads to an additional harmonic force acting on the system. When the coupling is switched on and switched off rapidly the force has delta-function support at the initial and final time, an effect that has been observed previously. I further show that the work and heat functionals as recently defined in stochastic thermodynamics at strong coupling contain additional terms depending on the time derivative of the system-bath coupling. I discuss these terms and show that while they can be very large if the system-bath coupling changes quickly they only give a finite contribution to the work that enter in Jarzynski's equality. I also discuss that these corrections to standard work and heat functionals provide an explanation for non-standard terms in the change of the von Neumann entropy of a quantum bath interacting with a quantum system found in an earlier contribution (Aurell \& Eichhorn, 2015).

Equations

$$H_B=\sum _b\frac{p_b^2}{2m_b}+\frac{1}{2}{m}_b{\omega }_b^2{q}_b^2.$$

$$H_S=\frac{P^2}{2M}+V(Q,t)$$

$$H_I(t)=-\sum _b{C}_b(t)q_{b}Q$$

$$H_C(t)=\sum _b\frac{C_b^2(t)Q^2}{2m_b{\omega }_b^2}.$$

$$H_{TOT}=H_S+H_B+H_I+H_C.$$

$$\dot{p_b}$$

$$\dot{q_b}$$

$$\dot{P}$$

$$\dot{Q}$$

$$q_b(t)$$

$$\dot{Q}$$

$$\xi (t)$$

$$\gamma (t,s)$$

$$\chi (t,s)$$

$$\begin{array}[]{lcll}f(\omega)&=&\frac{2}{\pi}\omega_{c}^{-3}\omega^{2}&\omega<\Omega\\ f(\omega)&=&0&\omega>\Omega\end{array}$$

$$C_{\omega }(s)=\sqrt{{\omega }_c^3{m}_{\omega }\eta (s)}$$

$${\dot{C}}_b(t)=\frac{1}{2}\frac{\dot{\eta }(t)}{\eta (t)}{C}_b(t).$$

$$\begin{array}[]{lcll}\frac{C_{\omega}(t)C_{\omega}(s)f(\omega)}{m_{\omega}}&=&\frac{2\sqrt{\eta(t)\eta(s)}}{\pi}\omega^{2}&\omega<\Omega\end{array}$$

$$\gamma (t,s)$$

$$\chi (t,s)$$

$$\gamma (t,s)$$

$$\chi (t,s)$$

$$F_{fric}$$

$$F_{if}$$

$$E\left[\xi (t)\xi (s)\right]$$

$$F_S=\xi +F_{fric}+F_{if}$$

$${\dot{Q}}_B=-\frac{1}{2}\frac{\dot{\eta }(t)}{\eta (t)}Q$$

$$F_{fric}+F_{if}=-\eta \left(\dot{Q}-{\dot{Q}}_B\right).$$

$$\Delta H_{TOT}=\Delta H_S+\Delta H_B+\Delta H_I+\Delta H_C.$$

$${⟨e^{-\beta \Delta H_{TOT}}⟩}_{eq}=e^{-\beta \Delta F_{TOT}}$$

$$\delta W=\Delta H_{TOT}=\Delta H_S+\Delta H_B$$

$$\delta W=\int {\partial }_t{H}_S+{\partial }_t{H}_I+{\partial }_t{H}_C$$

$$\mathcal{H}=H_S+H_C-{\beta }^{-1}\log {⟨e^{-\beta H_I}⟩}_B$$

$$E=\mathcal{H}+\beta {\partial }_{\beta }\mathcal{H}$$

Full text

Stochastic thermodynamics, strong coupling, Zwanzig model, quantum-classical correspondence for heat

I Introduction

Consider a collection of $N$ harmonic oscillators governed by the Hamiltonian

[TABLE]

The oscillators start in thermal equilibrium at some inverse temperature $\beta$. We will call this the bath. Consider also another externally driven system described by a Hamiltonian

[TABLE]

and consider the case when the system and the bath interact linearly during some finite time interval. The interaction Hamiltonian is

[TABLE]

where the $C_{b}(t)$ are functions of time, and to this we add the counter-term Hamiltonian

[TABLE]

The total Hamiltonian of the bath and the system is

[TABLE]

We will be concerned with the setting that $C_{b}(t)$ rise rapidly from zero to some steady values $C_{b}$ after an initial time $t_{i}$ and decrease rapidly back to zero before some final time $t_{f}$. The system and the bath are therefore disconnected before and at time $t_{i}$ as well as at and after time $t_{f}$. It is natural to postulate that the heat exchanged between the system and the bath during the process is $\Delta H_{B}$, the change of the energy of the bath in the time interval $[t_{i}:t_{f}]$.

The paper has two goals. The first is to compare $\Delta H_{B}$ to heat, work and internal energy in the recently introduced Stochastic Thermodynamics at strong coupling Seifert (2016); Jarzynski (2017). The new effect that will appear is a work which arises from the change of the system-bath coupling, and which is entirely dissipated into heat. When the change of the system-bath coupling is fast this work is large, and can pathwise and on average be (much) larger than the standard Jarzynski work. Due to a cancellation effect most of it however does not contribute to the average in Jarzynski’s equality. The second goal is to explain results on the classical limit of the corresponding quantum problem which were found in an earlier contribution Aurell and Eichhorn (2015). It will be shown that the main extra term found there was an artefact from assuming that the total density matrix of the system and the bath being initially factorized, while the system and the bath are nevertheless interacting from the beginning of the process. If instead the system-bath coupling goes to zero smoothly at the beginning and the end of the process the classical limit of heat is recovered fully, and quantum-classical correspondence holds on the level of expectation values.

The paper is organized as follows. In Sections II and III I describe how interaction with a bath as defined here leads to a Kramers-Langevin dynamics for the system, with the additional terms that appear when the system-bath coupling depends on time. In Section IV I consider work, internal energy and heat as recently defined in Stochastic Thermodynamics at strong coupling and relate them to $\Delta H_{B}$, and in Section V I show how these functionals of the system history are modified when the system-bath coupling depend on time. In Section VI I discuss the extra work referred to above, and in Section VII I compare the expectation value of $\Delta H_{B}$ to the earlier computed classical limit of the change of the von Neumann entropy of the bath. In Section VIII I sum up the results. In Appendix A I discuss for completeness an additional term in the classical limit of the change of the von Neumann entropy that was also found in Aurell and Eichhorn (2015), but which has a different origin than the main focus of this paper.

II Recall of the Zwanzig theory

All the calculations in this section are straightforward, and mostly re-statements of known results Zwanzig (1973). The equations of motion of the bath and the system that follow from (1), (2) (3) and (4) are

[TABLE]

The initial position and momentum of the system $(Q^{i},P^{i})$ are assumed known. All randomness in this problem is hence due to lack of knowledge of the initial positions and momenta of the bath. These are $2N$ real numbers while the time history of the system over the finite time interval, $\{Q,P\}_{t_{i}}^{t_{f}}$, needs infinitely many variables to be completely specified. One may therefore assume that a complete knowledge of the system history uniquely specifies the initial positions and momenta of the bath; to the equilibrium probability $P^{eq}$ in the bath then corresponds a probability distribution $P\left[\{Q,P\}_{t_{i}}^{t_{f}}\right]$ on system histories. This will be assumed throughout the following.

The equations of motion for the bath oscillators can be solved as

[TABLE]

By the assumption that the system and the bath are initially disconnected $C_{b}(t_{i})Q(t_{i})$ vanishes. Inserting $q_{b}(t)$ in the equations of motion for the system give

[TABLE]

using the ancillary quantities

[TABLE]

Equations (11) and (12-14) are the solutions to the general problem of a system interacting linearly with a bath of harmonic oscillators.

III The time-dependent Ohmic bath and its forces

In this section we introduce additional assumptions so that (11) is classic Kramers-Langevin equation when the system-bath coupling is constant. The starting point is to follow Caldeira and Leggett Caldeira and Leggett (1983) and assume an Ohmic bath i.e. that the spectrum of the bath oscillators is continuous up to an upper cut-off $\Omega$ and increases quadratically with frequency. The number of oscillators with frequencies in the interval $[\omega,\omega+d\omega]$ is $f(\omega)d\omega$ and

[TABLE]

where $\omega_{c}$ is some characteristic frequency less than $\Omega$. The total number of oscillators is hence $\frac{2}{3\pi}\left(\frac{\Omega}{\omega_{c}}\right)^{3}$. We also assume a function $\eta(s)$ with dimension (mass/time) such that

[TABLE]

which implies, for every bath oscillator,

[TABLE]

The two kernels in (13) and (14) then follow from

[TABLE]

and are

[TABLE]

We now further assume that $C_{b}(t)$ change between zero and their full value on a time scale much longer than $\Omega^{-1}$ so that the integrals in (19) and (20) can be approximated by delta functions. This leads to

[TABLE]

which means that the two terms inside the integral in (11) evaluate to

[TABLE]

The first of the above is a standard friction force while the second is a time-dependent harmonic force, proportional to the time derivative of the friction coefficient.

Turning now to force term $\xi$ it varies rapidly in time and depends directly on the initial positions and momenta och the bath oscillators. Its statistical properties follow from averaging over these initial positions and momenta, and it is immediate that $\hbox{E}\left[\xi(t)\right]=0$. The second moment is

[TABLE]

The random force $\xi$ therefore satisfies an Einstein relation with only the friction force $F_{fric}$.

For the following discussion it is convenient to introduce, in analogy with the “Sekimoto force” Sekimoto (1998, 2010),

[TABLE]

It is also convenient to introduce an auxiliary quantity with dimension velocity

[TABLE]

such that

[TABLE]

The interpretation of $\dot{Q}_{B}$ is that it is the time- and coordinate-dependent velocity of the system such that the force $F_{S}$ from the bath on the system vanishes in expectation. It is zero when the system-bath coupling is constant in time.

IV Work and heat of stochastic thermodynamics at strong coupling

We start by reviewing the derivation of Jarzynski’s equality at strong coupling following Jarzynski (2004), with adjustments arising from time-dependent system-bath coupling. The system and the bath together consititute a closed system and the work done on the combined system is the change of the total energy

[TABLE]

It is immediate that the combined system satisfies a Jarzynski equality

[TABLE]

where the average is over the equilibrium state of the combined system and $\Delta F_{TOT}$ is the change of total equilibrium free energy. As we assume that the system and the bath are uncoupled at the initial and the finite time $\Delta F_{TOT}$ is in fact here equal to $\Delta F$, the free energy change of the system only. For the same reason we can write

[TABLE]

the other two terms vanishing on the boundaries. On the other hand we can also use Liouville’s theory and write

[TABLE]

which has two extra terms compared to Jarzynski work when the system-bath coupling depends on time. The bath Hamiltonian $H_{B}$ does not contribute to (32) as it is independent of time.

The more recent development of heat in stochastic thermodynamics at strong coupling starts from a Hamiltonian of the system at mean force Seifert (2016); Jarzynski (2017)

[TABLE]

where the average is over the Boltzmann distribution of the bath only. From this is derived an internal energy function at mean force defined as

[TABLE]

For a bath of Hamiltonian oscillators linearly coupled to the system the average in ${\cal H}$ cancels with the counter-term $H_{C}$ so that we have simply

[TABLE]

The change in this internal energy can be written as a time integral

[TABLE]

A definition of a path-wise heat functional which satisfies a First Law with (31) and (35) is then

[TABLE]

In a round-about way we have thus arrived at the same notion of heat as in Introduction. Using (32) and (36) we can write $\delta Q$ as a time integral as

[TABLE]

V Work and heat with a time-dependent Ohmic bath

We start by writing out (36), (32) and (38) explicitly for the model at hand:

[TABLE]

For the change of internal energy it is seen that the sum over the bath oscillators is the same as in (8), and using (26) we can write

[TABLE]

Equation (42) is the Jarzynski work ($\int\partial_{t}H_{S}dt$) plus a term expressing the work done on the system from the total force $F_{S}$ that arises from system-bath coupling. It can therefore read as the sum of the flow of energy into the system from the external system and from the bath, which is the same standard form in stochastic thermodynamics Sekimoto (1998). Equation (42) only involves the time-dependence of the system-bath coupling through the extra force $F_{if}$ entering $F_{S}$.

For the work functional, by the assumptions made in Section III, we have

[TABLE]

where $dQ_{B}=\dot{Q}_{B}dt$ and $\dot{Q}_{B}$ was introduced in (27). The work is therefore the Jarzynski work plus the work done by the Sekimoto force through the virtual increment of the system position $dQ_{B}$. The heat can similarly be written

[TABLE]

and is the work done by the Sekimoto reaction force ($-F_{S}$) through the difference between the actual increment $dQ$ and the virtual increment $dQ_{B}$.

An alternative way to divide up the different contributions to the heat which will be useful in Section VII below is

[TABLE]

When $\eta$ is constant in time this reduces to Sekimoto’s heat functional $\int-\xi\circ dQ+\eta\dot{Q}^{2}dt$.

VI The work done by the time-dependent system-bath coupling

It has been seen above that both the heat and the work contain a term $\int F_{S}\circ dQ_{B}$ which is not reflected in the internal energy change at all. Changing the system-bath coupling is a kind of external control acting on the system, which implies a kind of work. An original feature is that this work is entirely dissipated into heat (change of energy of the bath); nothing remains as change of internal energy. Since $\dot{Q}_{B}$ has been assumed non-zero only near the initial and final time this work is a kind of change of state function, and not a proper functional depending on the whole path.

The first part of this work, corresponding to the first of the three terms in $F_{S}$, is

[TABLE]

Assuming that $\eta$ changes from zero to a finite value $\overline{\eta}$ over a short time period $\Delta t$ the force $F_{if}$ is going to dominate all the other forces in the Kramers-Langevin equation (11). This means that we have

[TABLE]

while at the same time

[TABLE]

From this the contribution to (46), from the beginning of the process, is $\frac{Q_{i}P_{i}}{2M}\overline{\eta}-\frac{Q_{i}^{2}}{8M}\overline{\eta}^{2}$, and analogously just before the final time. The contribution of this part is therefore the change of an auxiliary friction-dependent energy:

[TABLE]

The second part of $\int F_{S}\circ dQ_{B}$ is

[TABLE]

Similarly to above this can be written as a functional of $Q_{i}$, $P_{i}$ and the function $\eta$ in the interval $[t_{i},t_{i}+\Delta t]$ and the same at the final time. We can therefore write

[TABLE]

with two functionals $A_{i}$ and $A_{f}$. The largest (most divergent) contributions to $A_{i}$ and $A_{f}$ are $\frac{1}{4}Q^{2}\int\frac{\dot{\eta}^{2}}{\eta}dt$ which diverge as $(\Delta t)^{-1}$. The third part of $\int F_{S}\circ dQ_{B}$ is

[TABLE]

This is a random variable which depends on the realization of $\xi$ just after the initial time and just before the final time of mean zero and variance

[TABLE]

The second and the third part of $\int F_{S}\circ dQ_{B}$ are hence potentially both large, and when $\Delta t$ tends to zero they can both be arbitrarily larger than standard Jarzynski work.

Let us now consider the contribution of $\int F_{S}\circ dQ_{B}$ to Jarzynski equality, which we write

[TABLE]

to emphasize that the initial equilibrium distribution is factorized between the system and the bath. For given initial and final coordinates and momenta the first and second parts contribute simply $e^{-\beta\left(\Delta V_{fric}+A_{i}+A_{f}\right)}$. The third part (56) on the other hand contributes

[TABLE]

The (potentially divergent) contributions from the second and the third part hence cancel for each given initial and final coordinates and momenta (up to terms small as $\Delta t$), and must therefore cancel in Jarzynski equality overall.

Combining this result and (49) we have an alternative form of Jarzynski equality

[TABLE]

This may be compared to the strong-coupling form of Jarzynki equality which holds when $\eta$ is strictly constant in time:

[TABLE]

In above the average is over the system and the bath initially in joint equilibrium with the terms $H_{I}$ and $H_{C}$ included, and the free energy change is of the combined system and bath. Pathwise the contributions to (57) and (58) from the Jarzynski work $\int(\partial_{t}H_{S})$ are the same up to terms ${\cal O}(\Delta t)$. The differences between the two averages therefore stem from the additional term $\Delta V_{frict}$, and the different distributions over the initial conditions.

VII Comparison to quantum entropy production

In Aurell and Eichhorn (2015) was computed the first-order change of the von Neumann entropy in a (quantum) bath of harmonic oscillators, between two measurements on the system when the bath is initially in equilibrium at inverse temperature $\beta$. The central quantity computed in that paper (Eq. 3 in Aurell and Eichhorn (2015)) was

[TABLE]

where $i$ and $f$ denote the initial and final states of the system, $P_{if}$ is the transition probability of and $\hbox{E}_{if}$ is the quantum expectation value of bath Hamiltonian at the final time projected on these same states of the system, see (80) for the full formula in standard notation. The bath is originally at equilibrium at inverse temperature $\beta$ and $U(\beta)$ is its equilibrium internal energy. The calculation of (59) is a straight-forward generalization of the transition probability $P_{if}$ as computed by Feynman and Vernon by integrating out the bath variables Feynman and Vernon (1963). In slightly abbreviated form where $\int_{if}$ stands for integration over the initial and final positions of the system and projections on initial and final states, this quantity is

[TABLE]

and $S_{i}$ and $S_{r}$ are real and imaginary parts of the Feynman-Vernon action. The first main result of Aurell and Eichhorn (2015) (Eq. 15) was that

[TABLE]

where ${\cal I}^{(1)}$, ${\cal I}^{(2)}$ and ${\cal I}^{(3)}$ are three quadratic functionals. In this section will be discussed the classical limit of $\left<{\cal I}^{(2)}+{\cal I}^{(3)}\right>_{if}$ and shown to be identical to the expected value of the classical heat discussed above in Section V.

The change of expected bath energy can be written

[TABLE]

where the last term is given in standard notation in (79). The expression in (61) differs from (59) by the absence of $P_{if}$ in the denominator and by the quantity subtracted at the initial time. In fact, the expected energy of the bath energy measured at the initial time, conditioned on the future observation of the final state $f$ of the system, is not the same as the unconditioned equilibrium internal energy of the bath times the transition probability, a (simple) example of quantum retrodiction Gammelmark et al. (2013); Rybarczyk et al. (2015). In appendix A the term $\left<{\cal I}^{(1)}\right>_{if}$, also found in Aurell and Eichhorn (2015), is shown to be equal to the this discrepancy between $\hbox{E}_{if}\left[\hat{H}_{B}(t_{i})\right]$ and $U(\beta)P_{if}$.

Adapting to the setting of the present paper we from now on take the system-bath coupling coefficients $C_{b}$ to depend on time as in Section II and we will divide the formula in Aurell and Eichhorn (2015) by $\beta$ so that they express heat and not entropy production. This leads to

[TABLE]

where the corresponding two kernels are given in Aurell and Eichhorn (2015) Eq. 16 (with one factor $\beta$ less):

[TABLE]

In the Caldeira-Leggett limit these kernels tend to

[TABLE]

Assuming that both $\eta$ and $\dot{\eta}$ vanishes at the initial and final time we can integrate by parts freely to find

[TABLE]

In above the (quantum) expectation is the same as in (60) and $X$ and $Y$ are the forward and backward quantum paths. If we substitute for $X$ and $Y$ a classical path $Q$ and average over its probability distribution the term $\left<{\cal I}^{(3)}\right>_{if}$ in (68) is obviously the same as the expectation value of the corresponding term in (45). The other term can be re-expressed as

[TABLE]

The separation in (70) is analogous to the split into $\frac{2i}{\hbar}S_{i}^{mid}$ and $\Delta S_{b}$ in Aurell and Eichhorn (2015), Section 5, with the difference that $\Delta S_{b}$ (the two last terms) now vanishes since $\eta$ is zero at the initial and final time.

To discuss the first two terms in (70) we use the procedure of Aurell and Eichhorn (2015), Sections 6 & 7. The first step is to express the quantum averages through the Wigner transform which for the first term in (70) leads to the result in Aurell and Eichhorn (2015), Eq. 34. For the second term in (70) we have that $\frac{1}{2}(Y^{2}-X^{2})=Q\alpha$ where $Q=\frac{1}{2}(X+Y)$ is the average (eventually classical) path and $\alpha=(Y-X)$ is the quantum deviation; a term $\frac{i\alpha}{\hbar}$ translates to a partial derivative with respect to momentum variable $P$ in the Wigner transform. Collecting both terms we have hence

[TABLE]

$P(q^{\prime},p^{\prime},q,p)$ in above is the Wigner function which in the Caldeira-Leggett limit tends to the transition kernel of the classical stochastic process (11).

The second step is to compare the right-hand side of (70) to the expectation value of the first term in (45) $\left<\int-\xi\circ(dQ-dQ_{B})\right>_{if}$. The expectation is conditioned on the initial and final states which are assumed to be given by definite classical coordinates and momenta. It is convenient to re-write this as

[TABLE]

where $\hbox{Prob}(\hbox{path})$ is a weight over paths and $d\Xi$ is the increment of the aggregated random force, a Gaussian random variable with mean zero and variance $2k_{B}T\eta dt$. One can therefore discretize the inner integral in (71) and consider the contribution from the short time interval $[s,s+\Delta s]$

[TABLE]

The short-time propagator of the Kramers-Langevin process is

[TABLE]

and has the property that

[TABLE]

Integration by parts moves over the derivative with respect to $P^{\prime}$ to respectively $P(Q_{f},P_{f},Q^{\prime},P^{\prime})$ and $\frac{P^{\prime}+P}{2M}$ which gives the expression in (70). The same analysis through the Wigner function as outlined here can of course also be applied to the last (simpler) term in (68), with the same result as given above up to terms of order $\hbar^{2}$.

VIII Discussion

The study of the interaction of a classical or quantum system with a bath of harmonic oscillators has a long history, and it behoves the author to here motivate the need for another paper on the subject.

Consider the assumption of the Feynman-Vernon theory that the system and the bath are originally decoupled and the bath is at equilibrium. If the coupling is not weak this is questionable because if the system and the bath were in contact precisely at the initial time they should have been so also slightly before, and then they could not have been initially fully decoupled. Nevertheless, this assumption is needed to integrate out the bath variables and arrive at the Feynman-Vernon open system development operator of the system only111Although the bath oscillators can be integrated out also if the system and the bath are originally in equilibrium together, the analysis is then considerably more complicated Grabert et al. (1988); that possibility has not been considered here.. It could therefore be argued that the Caldeira-Leggett theory of quantum Brownian motion Caldeira and Leggett (1983), which is based on Feynman-Vernon, is only valid at weak coupling. Indeed, it has been noted several times that the classical limit (Kramers-Langevin equation) of an Ohmic bath has a delta-function force proportional to the friction acting at the initial time, and that the random and deterministic forces from the bath therefore do not obey an Einstein relation. It has also been noted that it is a difference whether one assumes that the decoupled initial conditions pertain exactly at the initial time or only slightly after.

To resolve these (and other) issues Caldeira and co-workers in Rosenau da Costa et al. (2000) investigated the possibility that the bath is initially in equilibrium conditional on the position of the system i.e. with respect to Hamiltonian $\sum_{b}\frac{p_{b}^{2}}{2m_{b}}+\frac{1}{2}m_{b}\omega_{b}^{2}\left(q_{b}-\frac{C_{b}(t)Q}{m_{b}\omega_{b}^{2}}\right)^{2}$. These initial conditions are mathematically possible and physically allowed, but raise the question of how the system and the bath would find themselves in such a state. If some control would have been exercised on the bath prior to the process this control must have been aware of the position of the system, and only a very exquisite procedure would have resulted in exactly the assumed initial state.

In this paper I have aimed to address these issees anew by allowing the system-bath coupling depend on time. It is then not a problem to have the system and the bath initially decoupled, but the price to pay are new effects that arise from the time-dependent friction. On the level of classical equations these effects are just the extra force $F_{if}$ of (24) which when the switching on and off of the bath-system coupling is fast reduces to the previously known delta-function force. On the level of classical heat and work functionals the situation is more involved and described by (43) and (44). There appears a part of the work denoted $\int F_{S}\circ dQ_{B}$ which is entirely dissipated into heat, which has support only at the beginning and the end of the process, and which is potentially quite large. Nevertheless, it only makes a finite contribution to the work functional that enters Jarzynski’s equality, see (57).

On the quantum level of expected change of the bath energy the analysis can be carried out using the Feynman-Vernon formalism, as done previously for constant system-bath coupling in Aurell and Eichhorn (2015). The classical limit of these expressions can then be seen to be the same as the classical heat functional with time-dependent friction. This is hence an example of quantum-classical correspondence Jarzynski et al. (2015), on the level of the expected heat. The extra term $\Delta S_{b}$ found in Aurell and Eichhorn (2015) arises from using factorized initial conditions with a finite system-bath interaction present from the very beginning of the process, and has therefore here been shown to be a kind of artefact.

Acknowledgments

This paper was motivated by earlier joint work with Prof. Ralf Eichhorn whom I thank for many and stimulating discussions on the subject matter. I thank Prof. H.T. Quan and Dr. Ken Funo for discussions and useful remarks, and the Institute of Theoretical Physics at the Chinese Academy of Sciences (Beijing) for hospitality. This research was supported by the Academy of Finland through its Center of Excellence COIN and by the Chinese Academy of Sciences CAS President’s International Fellowship Initiative (PIFI) grant No. 2016VMA002.

Appendix A An extra term found in Aurell and Eichhorn (2015)

We here discuss for completeness the first term in Aurell and Eichhorn (2015) Eq. 15

[TABLE]

with the kernel (a factor $\beta$ less compared to Aurell and Eichhorn (2015))

[TABLE]

As noted in Aurell and Eichhorn (2015) (75) equals $\partial_{\beta}\frac{1}{\hbar}S_{r}$ where $S_{r}$ is the real part of the Feynman-Vernon action. Writing the quantum expectation value of ${\cal I}^{(1)}$ in the Feynman-Vernon theory therefore means

[TABLE]

where the notation is as in Section VII above. The second equality follows because in the exponent only $S_{r}$ depends on $\beta$.

We want to relate the term in (77) to the difference between $\hbox{E}_{if}\left[\hat{H}_{B}(t_{i})\right]$ and $P_{if}U(\beta)$, where $U(\beta)$ is the unconditioned expected energy of the bath at inverse temperature $\beta$. To do so we write more formally

[TABLE]

where $V$ is the total unitary operation on the system and the bath, and $\rho_{B}^{eq}(\beta)$ is the initial (equilibrium) density matrix of the bath. Using that $\rho_{B}^{eq}(\beta)=e^{-\beta\hat{H}_{B}}/Z(\beta)$ and $-\partial_{\beta}\log Z=U$ we have $\hbox{E}_{if}\left[\hat{H}_{B}(t_{i})\right]=-\partial_{\beta}P_{if}+UP_{if}$. Therefore

[TABLE]

which was to be shown.