On work and heat in time-dependent strong coupling

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).

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).
The oscillators start in thermal equilibrium at some inverse temperature β. We will call this the bath. Consider also another externally driven system described by a Hamiltonian and consider the case when the system and the bath interact linearly during some finite time interval. The interaction Hamiltonian is where the C b (t) are functions of time, and to this we add the counter-term Hamiltonian The total Hamiltonian of the bath and the system is We will be concerned with the setting that C b (t) rise rapidly from zero to some steady values C b after an initial * eaurell@kth.se 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 ∆H B , the change of the energy of the bath in the time The paper has two goals. The first is to compare ∆H B to heat, work and internal energy in the recently introduced Stochastic Thermodynamics at strong coupling [1,2]. 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 [3]. 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 systembath 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 ∆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 ∆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 [3], 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 [4]. The equations of motion of the bath and the system that follow from (1), (2) (3) and (4) arė 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 f ti , 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 {Q, P } t f ti on system histories. This will be assumed throughout the following.
The equations of motion for the bath oscillators can be solved as 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 givė using the ancillary quantities Equations (11) and (12)(13)(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 [5] and assume an Ohmic bath i.e. that the spectrum of the bath oscillators is continuous up to an upper cut-off Ω and increases quadratically with frequency. The number of oscillators with frequencies in the interval [ω, ω + dω] is f (ω)dω and where ω c is some characteristic frequency less than Ω.
The total number of oscillators is hence 2 3π Ω ωc

3
. We also assume a function η(s) with dimension (mass/time) such that which implies, for every bath oscillator, The two kernels in (13) and (14) then follow from and are We now further assume that C b (t) change between zero and their full value on a time scale much longer than Ω −1 so that the integrals in (19) and (20) can be approximated by delta functions. This leads to which means that the two terms inside the integral in (11) evaluate to 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 ξ 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 E [ξ(t)] = 0. The second moment is The random force ξ therefore satisfies an Einstein relation with only the friction force F f ric . For the following discussion it is convenient to introduce, in analogy with the "Sekimoto force" [6,7], It is also convenient to introduce an auxiliary quantity with dimension velocitẏ such that The interpretation ofQ 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 [8], with adjust-ments 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 It is immediate that the combined system satisfies a Jarzynski equality where the average is over the equilibrium state of the combined system and ∆F T OT 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 ∆F T OT is in fact here equal to ∆F , the free energy change of the system only. For the same reason we can write the other two terms vanishing on the boundaries. On the other hand we can also use Liouville's theory and write 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 [1,2] 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 For a bath of Hamiltonian oscillators linearly coupled to the system the average in H cancels with the counterterm H C so that we have simply The change in this internal energy can be written as a time integral A definition of a path-wise heat functional which satisfies a First Law with (31) and (35) is then 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 δQ as a time integral as

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: 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 Equation (42) is the Jarzynski work ( ∂ 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 [6]. Equation (42) only involves the timedependence 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 where dQ B =Q B dt andQ 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 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 When η is constant in time this reduces to Sekimoto's heat functional −ξ • dQ + ηQ 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 F S • dQ B which is not reflected in the internal energy change at all. Changing the systembath 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. SinceQ 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 Assuming that η changes from zero to a finite value η over a short time period ∆t the force F if is going to dominate all the other forces in the Kramers-Langevin equation (11). This means that we have while at the same time From this the contribution to (46), from the beginning of the process, is QiPi 2M η − Q 2 i 8M η 2 , and analogously just before the final time. The contribution of this part is therefore the change of an auxiliary friction-dependent energy: Similarly to above this can be written as a functional of Q i , P i and the function η in the interval [t i , t i + ∆t] and the same at the final time. We can therefore write This is a random variable which depends on the realization of ξ just after the initial time and just before the final time of mean zero and variance The second and the third part of F S • dQ B are hence potentially both large, and when ∆t tends to zero they can both be arbitrarily larger than standard Jarzynski work.
Let us now consider the contribution of F S • dQ B to Jarzynski equality, which we write 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 −β(∆V f ric +Ai+A f ) . The third part (56) on the other hand contributes 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 ∆t), and must therefore cancel in Jarzynski equality overall.
Combining this result and (49) we have an alternative form of Jarzynski equality This may be compared to the strong-coupling form of Jarzynki equality which holds when η is strictly constant in time: 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 (∂ t H S ) are the same up to terms O(∆t). The differences between the two averages therefore stem from the additional term ∆V f rict , and the different distributions over the initial conditions.

VII. COMPARISON TO QUANTUM ENTROPY PRODUCTION
In [3] 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 β. The central quantity computed in that paper (Eq. 3 in [3]) was where i and f denote the initial and final states of the system, P if is the transition probability of and E if is the quantum expectation value of bath Hamiltonian at the final time projected on these same states of the system, see (A6) for the full formula in standard notation. The bath is originally at equilibrium at inverse temperature β and U (β) 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 [9]. In slightly abbreviated form where if stands for integration over the initial and final positions of the system and projections on initial and final states, this quantity is and S i and S r are real and imaginary parts of the Feynman-Vernon action. The first main result of [3] (Eq. 15) was that where I (1) , I (2) and I (3) are three quadratic functionals. In this section will be discussed the classical limit of I (2) + I (3) 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 where the last term is given in standard notation in (A5). 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 [10,11]. In appendix A the term I (1) if , also found in [3], is shown to be equal to the this discrepancy between E if Ĥ B (t i ) and U (β)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 [3] by β so that they express heat and not entropy production. This leads to where the corresponding two kernels are given in [3] Eq. 16 (with one factor β less): In the Caldeira-Leggett limit these kernels tend to Assuming that both η andη vanishes at the initial and final time we can integrate by parts freely to find 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 (68) is obviously the same as the expectation value of the corresponding term in (45). The other term can be re-expressed as The separation in (70) is analogous to the split into 2i S mid i and ∆S b in [3], Section 5, with the difference that ∆S b (the two last terms) now vanishes since η is zero at the initial and final time.
To discuss the first two terms in (70) we use the procedure of [3], Sections 6 & 7. The first step is to express the quantum averages through the Wigner trans-form which for the first term in (70) leads to the result in [3], Eq. 34. For the second term in (70) we have that 1 2 (Y 2 − X 2 ) = Qα where Q = 1 2 (X + Y ) is the average (eventually classical) path and α = (Y − X) is the quantum deviation; a term iα translates to a partial derivative with respect to momentum variable P in the Wigner transform. Collecting both terms we have hence P (q ′ , p ′ , 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) 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 where Prob(path) is a weight over paths and dΞ is the increment of the aggregated random force, a Gaussian random variable with mean zero and variance 2k B T ηdt.
The short-time propagator of the Kramers-Langevin pro-cess is and has the property that ∆ΞP ∆s (Q ′ , P ′ , Q, P ) = −2k B T η∆s∂ P ′ P ∆s (Q ′ , P ′ , Q, P ) + O ∆s 2 Integration by parts moves over the derivative with respect to P ′ to respectively P (Q f , P f , Q ′ , P ′ ) and P ′ +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 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 only [12]. It could therefore be argued that the Caldeira-Leggett theory of quantum Brownian motion [5], 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 coworkers in [13] investigated the possibility that the bath is initially in equilibrium conditional on the position of the system i.e. with respect to Hamiltonian . 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 F S •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 [3]. 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 cor-respondence [14], on the level of the expected heat. The extra term ∆S b found in [3] 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.
with the kernel (a factor β less compared to [3]) As noted in [3] (A1) equals ∂ β 1 S r where S r is the real part of the Feynman-Vernon action. Writing the quantum expectation value of I (1) in the Feynman-Vernon theory therefore means where the notation is as in Section VII above. The second equality follows because in the exponent only S r depends on β.
We want to relate the term in (A3) to the difference be-tween E if Ĥ B (t i ) and P if U (β), where U (β) is the unconditioned expected energy of the bath at inverse temperature β. To do so we write more formally where V is the total unitary operation on the system and the bath, and ρ eq B (β) is the initial (equilibrium) density matrix of the bath. Using that ρ eq B (β) = e −βĤB /Z(β) and −∂ β log Z = U we have E if Ĥ B (t i ) = −∂ β P if + U P if . Therefore which was to be shown.