Exponential or power law? How to select a stable distribution of probability in a physical system

TL;DR

This paper presents a unified framework for understanding the stability of relaxed states in nonextensive and Gibbs' statistical mechanics, linking power law and exponential distributions through a generalized fluctuation formula, and applies it to nonlinear Fokker-Planck systems.

Contribution

It introduces a unified treatment of stability for relaxed states in both nonextensive and Gibbs' statistical mechanics, connecting power law and exponential distributions without numerical orbit computations.

Findings

Results align with previous simulations of Pareto-like distributions.

Provides a method to determine the distribution type from map dynamics and noise level.

No assumptions needed on noise type or numerical orbit calculations.

Abstract

A mapping of nonextensive statistical mechanics into Gibbs' statistical mechanics exists, which leads to a generalization of Einstein's formula for fluctuations. A unified treatment of stability of relaxed states in nonextensive statistical mechanics and Gibbs' statistical mechanics follows. The former and the latter are endowed with probability distribution of microstates ruled by power laws and Boltzmann exponentials respectively. We apply our treatment to the relaxed states described by a 1D nonlinear FokkerPlanck equation. If the latter is associated to the stochastic differential equation obtained in the continuous limit from a 1D, autonomous, discrete map affected by noise, then we may ascertain whether if a relaxed state follow a power law distribution (and with which exponent) by looking at both map dynamics and noise level, with no assumptions concerning the additive or multiplicative nature of the noise and with no numerical computation of the orbits. Results agree with the simulations of Sanchez et al. EPJ 143.1 (2007) 141-143 concerning relaxation to a Pareto-like distribution.