Generalized thermostatistics based on multifractal phase space

TL;DR

This paper develops a generalized thermostatistics framework based on multifractal phase space, linking the spectrum of fractal dimensions to Tsallis' non-extensive statistics and deriving relations between system complexity and statistical weight.

Contribution

It introduces a novel approach connecting multifractal phase space properties with Tsallis' formalism, including a new interpretation of the non-additivity parameter.

Findings

Thermostatistics governed by Tsallis' formalism with an inverted non-additivity parameter.

Equipartition law holds within this multifractal framework.

Relation between statistical weight and system complexity derived from spectrum optimization.

Abstract

We consider the self-similar phase space with reduced fractal dimension $d$ being distributed within domain $0<d<1$ with spectrum $f(d)$. Related thermostatistics is shown to be governed by the Tsallis' formalism of the non-extensive statistics, where role of the non-additivity parameter plays inverted value ${\bar\tau}(q)\equiv 1/\tau(q)>1$ of the multifractal function $\tau(q)= qd(q)-f(d(q))$, being the specific heat, $q\in(1,\infty)$ is multifractal parameter. In this way, the equipartition law is shown to take place. Optimization of the multifractal spectrum $f(d)$ derives the relation between the statistical weight and the system complexity.