Phase transition in the Jarzynski estimator of free energy differences

Abstract

The transition between a regime in which thermodynamic relations apply only to ensembles of small systems coupled to a large environment and a regime in which they can be used to characterize individual macroscopic systems is analyzed in terms of the change in behavior of the Jarzynski estimator of equilibrium free energy differences from nonequilibrium work measurements. Given a fixed number of measurements, the Jarzynski estimator is unbiased for sufficiently small systems. In these systems the directionality of time is poorly defined and the configurations that dominate the empirical average, but which are in fact typical of the reverse process, are sufficiently well sampled. As the system size increases the arrow of time becomes better defined. The dominant atypical fluctuations become rare and eventually cannot be sampled with the limited resources that are available. Asymptotically, only typical work values are measured. The Jarzynski estimator becomes maximally biased and approaches the exponential of minus the average work, which is the result that is expected from standard macroscopic thermodynamics. In the proper scaling limit, this regime change has been recently described in terms of a phase transition in variants of the random energy model. In this paper this correspondence is further demonstrated in two examples of physical interest: the sudden compression of an ideal gas and adiabatic quasistatic volume changes in a dilute real gas.

Scaling Law for Irreversible Entropy Production in Critical Systems

“…As noted previously1819202122232425262728, the Jarzynski equality may break down for many practical and intrinsic reasons. Its breakdown in our case is peculiar as the deviation is determined by an universal combination of L and ε 0 , which is inherited from the equilibrium scaling behavior of second-order phase transitions.…”

Convergence of large-deviation estimators

“…We illustrate in this section the previous results about estimator convergence for four types of distributions: Gaussian, exponential, Bernoulli, and power-law. Gaussian distributions have been extensively studied in the context of the Jarzynski estimator [31][32][33][34][35][36] and are revisited here to illustrate the case of unbounded random variables. The exponential distribution is considered as a limiting case of the saddle-point analysis, whereas Bernoulli random variables illustrate our results for the bounded case and are relevant for data network applications [37][38][39][40].…”

“…Convergence problems have also been studied for the so-called Jarzynski estimator, which is an estimator similar to (2) used to obtain free energy differences from nonequilibrium experiments [24][25][26][27][28][29][30]. The focus of these studies, however, is mostly on the statistical bias ofĜ M (k) [31][32][33][34][35][36], which disappears in the limit M → ∞, rather than the convergence ofĜ M (k) as a function of k and M .…”

Fluctuation theorems for non-Markovian quantum processes

“…as, respectively, the entropy flux between system and environment and the total entropy production. Equations (7), (8) and (9) describe the time dependence of entropy due to the ensemble dynamics described by a TCL master equation. The irreversibility of the process is characterized by a nonzero rate of entropy productionṠ i (t) inside the system which, furthermore, in the case of a Markovian dynamics never becomes negative.…”