When do generalized entropies apply? How phase space volume determines entropy

TL;DR

This paper establishes how the phase space volume of a classical system determines whether its entropy is of Boltzmann-Gibbs type or a generalized form, especially in systems with frozen degrees of freedom.

Contribution

It provides a criterion linking phase space volume growth to the applicability of generalized entropies in systems with vanishing relevant degrees of freedom.

Findings

Generalized entropies occur when relevant degrees of freedom vanish in the thermodynamic limit.

Phase space volume collapse to a lower-dimensional surface indicates non-Boltzmann-Gibbs entropy.

Applicable to systems like self-organized critical systems, vortices, and anomalous diffusion.

Abstract

We show how the dependence of phase space volume $\Omega(N)$ of a classical system on its size $N$ uniquely determines its extensive entropy. We give a concise criterion when this entropy is not of Boltzmann-Gibbs type but has to assume a {\em generalized} (non-additive) form. We show that generalized entropies can only exist when the dynamically (statistically) relevant fraction of degrees of freedom in the system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen and is practically statistically inactive. Systems governed by generalized entropies are therefore systems whose phase space volume effectively collapses to a lower-dimensional 'surface'. We explicitly illustrate the situation for binomial processes and argue that generalized entropies could be relevant for self organized critical systems such as sand piles, for spin systems which form meta-structures such as vortices, domains, instantons, etc., and for problems associated with anomalous diffusion.