Phase transition in the Jarzynski estimator of free energy differences

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.

Figure 1

Approach of the normalized bias to the curve (24) as $M\to \infty$. The random energy model corresponds to the dash-dotted line.

Figure 2

Approach of the normalized bias to the curve (25) as $M\to \infty$. The random energy model corresponds to the dash-dotted curve.

Figure 3

Dependence of the normalized bias on $N$ for $\epsilon =0.5$ (top left), $\epsilon =1$ (top right), $\epsilon =2$ (bottom left), and $\epsilon =5$ (bottom right). The asymptotic value for the discrete random energy model (DREM) corresponds to the dash-dotted curve in the plots.

Figure 4

Dependence of the normalized bias on $N$ for $M=10$ (top left), $M=100$ (top right), $M=1000$ (bottom left), and $M=100000$ (bottom right). The dash-dotted curve corresponds to the discrete random energy model and gives the limiting behavior as $M\to \infty$.

Figure 5

Transition between the low- and high-temperature regimes in the random energy model with energy levels sampled from a gamma distribution (see the text for details).