Long memory constitutes a unified mesoscopic mechanism consistent with nonextensive statistical mechanics

TL;DR

This paper unifies two mesoscopic mechanisms for nonextensive statistics, showing that long memory effects can produce q-exponential stationary distributions through a generalized Fokker-Planck framework.

Contribution

It introduces a unified approach linking multiplicative noise and non-Markovian processes via a generalized Fokker-Planck equation, demonstrating the emergence of q-exponentials as stationary solutions.

Findings

Existence of a function ta(x,p) unifying linear and nonlinear Fokker-Planck equations.

Derivation of q-exponential stationary distributions for a broad class of systems.

Unification of long memory effects with nonextensive statistical mechanics.

Abstract

We unify two paradigmatic mesoscopic mechanisms for the emergence of nonextensive statistics, namely the multiplicative noise mechanism leading to a {\it linear} Fokker-Planck (FP) equation with {\it inhomogenous} diffusion coefficient, and the non-Markovian process leading to the {\it nonlinear} FP equation with {\it homogeneous} diffusion coefficient. More precisely, we consider the equation $\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}[F(x) p(x,t)] + 1/2D \frac{\partial^2}{\partial x^2} [\phi(x,p)p(x,t)]$, where $D \in {\cal R}$ and $F(x)=-\partial V(x) /\partial x$, $V(x)$ being the potential under which diffusion occurs. Our aim is to find whether $\phi(x,p)$ exists such that the inhomogeneous linear and the homogeneous nonlinear FP equations become unified in such a way that the (ubiquitously observed) $q$-exponentials remain as stationary solutions. It turns out that such solutions indeed exist for a wide class of systems, namely when $\phi(x,p)=[A+BV(x)]^\theta [p(x,t)]^{\eta}$, where $A$, $B$, $\theta$ and $\eta$ are (real) constants. Our main result can be sumarized as follows: For $\theta \neq 1$ and arbitrary confining potential $V(x)$, $p(x,\infty) \propto \lbrace 1-\beta(1-q)V(x)\rbrace ^{1/(1-q)} \equiv e_q^{-\beta V(x)}$, where $q= 1+ \eta/(\theta-1)$. The present approach unifies into a single mechanism, essentially {\it long memory}, results currently discussed and applied in the literature.