Nonextensive It\^o-Langevin Dynamics

TL;DR

This paper explores generalized Itô-Langevin equations within nonextensive thermostatistics, linking microscopic correlated random processes to macroscopic anomalous diffusion and explaining the frequent appearance of q-Gaussian distributions in nature.

Contribution

It introduces a framework connecting nonextensive stochastic dynamics with nonlinear Fokker-Planck equations and a generalized central limit theorem for correlated variables.

Findings

Derivation of nonlinear Fokker-Planck equations from generalized Langevin dynamics.

Demonstration of q-Gaussian distributions emerging as natural solutions.

Establishment of a microscopic-macroscopic connection in anomalous diffusion.

Abstract

We study generalizations of It\^{o}-Langevin dynamics consistent within nonextensive thermostatistics. The corresponding stochastic differential equations are shown to be connected with a wide class of nonlinear Fokker-Planck equations describing correlated anomalous diffusion in fractals. A generalized central limit theorem is proposed in order to demonstrate how such equations emerge as a limit of correlated random variables. In doing so, we connect microscopic and macroscopic descriptions of correlated anomalous diffusion in a mathematically sound way and shed some light in explaining why $q$-Gaussian distributions appear quite often in nature.