Family of additive entropy functions out of thermodynamic limit

TL;DR

This paper introduces a new family of entropy functions derived from the additivity condition of Lyapunov functions, which can better describe finite systems with long-tailed distributions, differing from Tsallis entropy.

Contribution

It derives a one-parametric family of entropy functions combining Boltzmann-Gibbs-Shannon and Burg entropies, extending the modeling of finite systems with long tails.

Findings

The new entropy family effectively models long-tailed distributions.

It provides explicit methods to generalize master equations.

The approach differs from and extends Tsallis entropy.

Abstract

Starting with the additivity condition for Lyapunov functions of master equation, we derive a one-parametric family of entropy functions which may be appropriate for a description of certain effects of finiteness of statistical systems, in particular, distribution functions with long tails. This one-parametric family is different from Tsallis entropies, and is essentially a convex combination of the Boltzmann-Gibbs-Shannon entropy and the entropy function introduced by Burg. An example of how longer tails are described within the present approach is worked out for the canonical ensemble. In addition, we discuss a possible origin of a hidden statistical dependence, and give explicit recipes how to construct corresponding generalizations of master equation.