Thermodynamics and time-average

TL;DR

This paper explores how to formulate a consistent statistical thermodynamics for far-from-equilibrium dynamical systems using time-averaged empirical probabilities, leading to generalized entropy expressions beyond Boltzmann-Gibbs.

Contribution

It introduces a natural extension of Gibbs and Khinchin's entropy definition using empirical probabilities, deriving a generalized entropy functional that includes Tsallis entropy as a special case.

Findings

Gibbs entropy is recovered for Poisson-type empirical probabilities.

Tsallis entropies emerge from deformations of the Poisson distribution.

The generalized entropy framework applies to non-equilibrium dynamical systems.

Abstract

For a dynamical system far from equilibrium, one has to deal with empirical probabilities defined through time-averages, and the main problem is then how to formulate an appropriate statistical thermodynamics. The common answer is that the standard functional expression of Boltzmann-Gibbs for the entropy should be used, the empirical probabilities being substituted for the Gibbs measure. Other functional expressions have been suggested, but apparently with no clear mechanical foundation. Here it is shown how a natural extension of the original procedure employed by Gibbs and Khinchin in defining entropy, with the only proviso of using the empirical probabilities, leads for the entropy to a functional expression which is in general different from that of Boltzmann--Gibbs. In particular, the Gibbs entropy is recovered for empirical probabilities of Poisson type, while the Tsallis entropies are recovered for a deformation of the Poisson distribution.