Beyond Boltzmann-Gibbs statistics: Maximum entropy hyperensembles out-of-equilibrium

TL;DR

This paper proposes a maximum entropy hyperensemble framework for describing out-of-equilibrium thermodynamic systems using non-equilibrium entropy and two parameters, providing a smooth transition from equilibrium certainty to non-equilibrium uncertainty.

Contribution

It introduces a novel hyperensemble approach with two parameters, including a non-equilibrium temperature, to better model systems away from equilibrium.

Findings

Large, rare fluctuations become more common out of equilibrium.

The hyperensemble smoothly interpolates between equilibrium and non-equilibrium states.

Non-equilibrium entropy serves as a key additional parameter.

Abstract

What is the best description that we can construct of a thermodynamic system that is not in equilibrium, given only one, or a few, extra parameters over and above those needed for a description of the same system at equilibrium? Here, we argue the most appropriate additional parameter is the non-equilibrium entropy of the system, and that we should not attempt to estimate the probability distribution of the system, but rather the metaprobability (or hyperensemble) that the system is described by a particular probability distribution. The result is an entropic distribution with two parameters, one a non-equilibrium temperature, and the other a measure of distance from equilibrium. This dispersion parameter smoothly interpolates between certainty of a canonical distribution at equilibrium and great uncertainty as to the probability distribution as we move away from equilibrium. We deduce that, in general, large, rare fluctuations become far more common as we move away from equilibrium.