Multifractal spectrum of phase space related to generalized thermostatistics

TL;DR

This paper explores the multifractal spectrum of phase space and its connection to generalized thermostatistics, demonstrating how non-extensive Tsallis statistics describe complex systems with self-similar structures.

Contribution

It introduces a model linking multifractal phase space spectra with Tsallis and Kaniadakis formalisms, providing explicit descriptions of the statistical weight and system complexity.

Findings

The multifractal spectrum function $f(d)$ increases monotonically from -1 to 1.

The statistical weight exponent $ au(q)$ can be modeled by hyperbolic tangent deformed exponentials.

The number of monofractals varies with phase space volume and dimension.

Abstract

We consider a self-similar phase space with specific fractal dimension $d$ being distributed with spectrum function $f(d)$. Related thermostatistics is shown to be governed by the Tsallis formalism of the non-extensive statistics, where the non-additivity parameter is equal to ${\bar\tau}(q)\equiv 1/\tau(q)>1$, and the multifractal function $\tau(q)= qd_q-f(d_q)$ is the specific heat determined with multifractal parameter $q\in [1,\infty)$. In this way, the equipartition law is shown to take place. Optimization of the multifractal spectrum function $f(d)$ derives the relation between the statistical weight and the system complexity. It is shown the statistical weight exponent $\tau(q)$ can be modeled by hyperbolic tangent deformed in accordance with both Tsallis and Kaniadakis exponentials to describe arbitrary multifractal phase space explicitly. The spectrum function $f(d)$ is proved to increase monotonically from minimum value $f=-1$ at $d=0$ to maximum one $f=1$ at $d=1$. At the same time, the number of monofractals increases with growth of the phase space volume at small dimensions $d$ and falls down in the limit $d\to 1$.