Sampling cost of fluctuation-theorem free-energy estimation under reservoir local heating: measurable-indicator bounds, dimensional dependence, and non-Markovian extensions

For a driven test system coupled to an explicitly modeled spatially extended heat reservoir, Pradhan, Kafri and Levine (Phys. Rev. E 77, 041129 (2008)) showed that the Crooks theorem and the Jarzynski equality remain valid even though the reservoir region in contact with the driven system goes out of equilibrium during the irreversible process (local heating). The price of local heating appears instead in the convergence of free-energy estimates: the stronger the local heating, the larger the variance of the exponential work average and hence the number of experimental realizations needed for a target accuracy. Later literature has resolved several parts of that paper's open agenda — the identification of the free-energy difference when equilibrium additivity fails (strong-coupling and Hamiltonian-of-mean-force theory), fluctuation theorems for finite baths, for arbitrary quantum reservoirs, and for time-reversal-symmetry-breaking fields — but the quantitative sampling questions remain open. The open problem is: derive analytic, quantitative relations between physically measurable indicators of local heating (deviations of near-contact reservoir energy distributions from the equilibrium exponential form) and the variance and required number of realizations of free-energy estimates; determine the dimensional dependence of this sampling cost, of which the one-dimensional case treated in the source paper is expected to be the worst case; and extend the validity and variance analysis to combined-system dynamics with genuine memory.

Background

The Jarzynski equality states that for a test system (TS) driven from equilibrium state A to state B by an external parameter λ(t), the average over nonequilibrium trajectories of $e^{-\beta W}$, where $W$ is the work performed and $\beta$ the inverse reservoir temperature, equals $e^{-\beta\Delta F_{TS}}$, the exponential of the equilibrium free-energy difference. The Crooks theorem is the stronger statement that the forward and time-reversed work distributions obey $P_F(W)/P_R(-W)=e^{\beta(W-\Delta F_{TS})}$. Standard proofs assume an ideal heat reservoir: Markovian dynamics for which the TS satisfies detailed balance with respect to a canonical distribution, or Langevin noise obeying a fluctuation-dissipation relation, both of which presume the reservoir remains in thermal equilibrium throughout. In real experiments the reservoir is a physical object with its own degrees of freedom, and the work performed on the TS dissipates into the reservoir, transiently heating and dis-equilibrating the region in contact with the TS — an effect called local heating. Whether and under what conditions the fluctuation theorems survive local heating is a fundamental question about their domain of validity.

Pradhan, Kafri and Levine (2008) addressed this by retaining the reservoir degrees of freedom explicitly: the driven TS is a portion of an isolated combined system (CS) consisting of TS plus reservoir. They showed that the Crooks theorem and Jarzynski equality hold under three sufficient conditions: (1) the closed CS obeys time-reversal-symmetric dynamics when the driving is switched off; (2) extensive equilibrium properties are additive, i.e. $F_{CS}=F_{TS}+F_B$ with $F_B$ the reservoir free energy; (3) the control parameter couples directly only to the TS. Condition (2) requires either a thermodynamic-limit TS or weak TS–reservoir interaction energy; the paper notes that when it fails, the theorems still hold with $\Delta F_{TS}$ replaced by the free-energy difference $\Delta F_{CS}$ of the full combined system, whose value depends on the interaction energy. For a proof in the stochastic case the CS is assumed to evolve by Markovian dynamics with symmetric rates $w_{j,i}=w_{i,j}$, and a large-reservoir limit $E\gg W$ is used; the authors also flag that for quantum reservoirs an effective stochastic description may be inadequate, since noise of quantum origin is not captured by either the Hamiltonian or the Markovian-stochastic proofs.

Their illustrative model is a one-dimensional lattice of $L+1$ sites whose energies are stochastically redistributed between neighbors with uniformly random splitting fractions — the Kipnis–Marchioro–Presutti (KMP) stochastic energy-redistribution dynamics, an exactly solvable model of heat conduction. The TS is site $0$ with energy $\lambda x$; the reservoir is sites $1\ldots L$. At equilibrium each site's energy marginally follows the exponential (Boltzmann) distribution $P(\epsilon)=\beta e^{-\beta\epsilon}$. During driving, the marginal distributions of sites near the contact deviate strongly from this form (local heating), yet the Crooks theorem holds. The practical cost is quantified by the standard deviation $\sigma_X$ of $X\equiv e^{-\beta(W-\Delta F)}$: the error of the free-energy estimator $\Delta F_M$ obtained from $N_R$ realizations scales as $\sigma_X/\sqrt{N_R}$ by the central limit theorem, and the numerics show that stronger local heating (shorter experimental times, more violent driving) increases $\sigma_X$ and hence the required $N_R$. The authors note that a worst case can be bracketed by maximal local heating (sudden changes of the TS, effectively disconnecting the reservoir), but they provide no quantitative theory linking local heating to $\sigma_X$ or $N_R$, they treat only the one-dimensional case — which footnote-level physical reasoning suggests is the worst case since heat diffuses away less effectively than in higher dimensions — and the generalization to genuinely non-Markovian CS dynamics (evolution with memory at the level of the combined system, beyond the restricted non-Markovian noise classes known before 2008) is left open.

Subsequent literature has settled the free-energy identification question that motivated the additivity discussion (strong-coupling thermodynamics via the Hamiltonian of mean force, i.e. the effective TS Hamiltonian obtained by integrating out an equilibrated bath with the interaction included), established exact fluctuation theorems for finite and for arbitrary quantum reservoirs, and removed the microscopic time-reversal-symmetry requirement in the presence of symmetry-breaking fields. What no later work has supplied is the quantitative sampling theory in the presence of an explicitly modeled, locally heated reservoir: which measurable reservoir observables predict the slowdown, how the answer depends on spatial dimension, and what survives when the combined dynamics has memory.

Problem Statement

For a driven test system coupled to an explicitly modeled, spatially extended heat reservoir satisfying the source paper's sufficient conditions, the Crooks theorem and Jarzynski equality hold despite local heating of the reservoir near the contact region; the measurable consequence of local heating is a slowdown of free-energy estimation, quantified by the standard deviation $\sigma_X$ of $X\equiv e^{-\beta(W-\Delta F)}$, since the error of the estimator from $N_R$ realizations scales as $\sigma_X/\sqrt{N_R}$. The open problem, in the parts not resolved by later literature, is to build a quantitative theory of this slowdown:

1. Derive analytic relations between physically measurable indicators of local heating — for instance the deviation of the marginal energy distribution of reservoir sites near the contact from the equilibrium exponential form $P(\epsilon)=\beta e^{-\beta\epsilon}$, or other reservoir observables accessible during or before the driving — and the standard deviation $\sigma_X$, and thereby the number of realizations $N_R$ needed to reach a target accuracy in $\Delta F$. Such relations should be exhibited in a definite, solvable class of explicit-reservoir models (e.g. the stochastic energy-redistribution dynamics used in the source paper), reproduce the known limit that an equilibrated (Markovian) reservoir gives the standard dissipated-work-controlled variance, and clarify to what extent they are universal rather than model-specific.

2. Determine the dimensional dependence: extend the explicit-reservoir analysis to reservoirs of spatial dimension $d\ge 2$ and characterize how the local-heating-induced contribution to $\sigma_X$ and $N_R$ scales with $d$, testing the expectation that the one-dimensional case studied in the source paper is the worst case because heat diffuses away less effectively.

3. Extend the validity and variance analysis to combined-system dynamics with genuine memory, i.e. dynamics that is neither Hamiltonian nor Markovian-stochastic with symmetric rates — beyond the restricted non-Markovian noise classes already treated in the literature (e.g. the stochastic non-Markovian work relations of Speck and Seifert [5] and the non-Markovian work fluctuation theorem of Jiménez-Aquino [17]) — and establish under what conditions on the combined-system evolution the fluctuation relations and the sampling-cost analysis survive.

Deliverables 1–3 serve one objective: a quantitative account of how bath-side nonequilibrium structure induced by the driving controls the cost of fluctuation-theorem free-energy estimation. Explicitly out of scope, as resolved by later work: the formal modification of the theorems when the additivity condition $F_{CS}=F_{TS}+F_B$ fails (the relevant free energy is that of the combined system relative to the bare bath, the Hamiltonian-of-mean-force free energy, with no restriction on coupling strength); fluctuation theorems for finite baths with $E$ not much larger than $W$; exact fluctuation theorems for quantum reservoirs; and the presence of time-reversal-symmetry-breaking fields (field-reversed variants and weakened conditions are established).

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