Non-extensive random walks

TL;DR

This paper investigates the properties of non-extensive random walks leading to q-Gaussian distributions, revealing their additive-multiplicative structure and linking them to nonlinear diffusion, with implications for non-extensive statistical mechanics.

Contribution

It demonstrates the simple additive-multiplicative structure of relevant random walks and connects these sequences to nonlinear diffusion equations, advancing understanding of non-extensive statistical mechanics.

Findings

Random walks exhibit additive-multiplicative structure.

Connections established with nonlinear diffusion equations.

Results support the applicability of non-extensive statistical mechanics.

Abstract

The stochastic properties of variables whose addition leads to $q$-Gaussian distributions $G_q(x)=[1+(q-1)x^2]_+^{1/(1-q)}$ (with $q\in\mathbb{R}$ and where $[f(x)]_+=max\{f(x),0\}$) as limit law for a large number of terms are investigated. These distributions have special relevance within the framework of non-extensive statistical mechanics, a generalization of the standard Boltzmann-Gibbs formalism, introduced by Tsallis over one decade ago. Therefore, the present findings may have important consequences for a deeper understanding of the domain of applicability of such generalization. Basically, it is shown that the random walk sequences, that are relevant to this problem, possess a simple additive-multiplicative structure commonly found in many contexts, thus justifying the ubiquity of those distributions. Furthermore, a connection is established between such sequences and the nonlinear diffusion equation $\partial_t \rho=\partial^2_{xx}\rho^\nu$ ($\nu\neq1$).