Long memory constitutes a unified mesoscopic mechanism consistent 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.