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 $\frac{\partial p(x,t)}{\partial t}=-\frac{\partial }{\partial x}[F(x)p(x,t)]+1/2D\frac{{\partial }^2}{\partial x^2}[\varphi (x,p)p(x,t)]$, where $D\in \mathcal{R}$ and $F(x)=-\partial V(x)/\partial x$, $V(x)$ being the potential under which diffusion occurs. Our aim is to find whether $\varphi (x,p)$ 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 $\varphi (x,p)=[A+BV(x){]}^{\theta }[p(x,t){]}^{\eta }$, where $A$, $B$, $\theta$ and $\eta$ are (real) constants. Our main result can be sumarized as follows: For $\theta \ne 1$ and arbitrary confining potential $V(x)$, $p(x,\infty )\propto \{1-\beta (1-q)V(x){\}}^{1/(1-q)}\equiv e_q^{-\beta V(x)}$, 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.

5 pages including 1 figure

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• Inter-occurrence times and universal laws in finance, earthquakes and genomes
• Vanishing largest Lyapunov exponent and Tsallis entropy
• A new structure entropy of complex networks based on Tsallis nonextensive statistical mechanics