Multiplicative noise: A mechanism leading to nonextensive statistical mechanics

Abstract

A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann-Gibbs statistical mechanics (based on $S_1\equiv -k\int dup(u)\ln p(u)$). Similarly, other classes of models point toward nonextensive statistical mechanics (based on $S_q\equiv k[1-\int du[p(u){]}^q]/[q-1]$, where the value of the entropic index $q\in \Re$ depends on the specific model). We show here a family of models, with multiplicative noise, which belongs to the nonextensive class. More specifically, we consider Langevin equations of the type $\dot{u}=f(u)+g(u)\xi (t)+\eta (t)$, where $\xi (t)$ and $\eta (t)$ are independent zero-mean Gaussian white noises with respective amplitudes $M$ and $A$. This leads to the Fokker-Planck equation ${\partial }_{t}P(u,t)=-{\partial }_u[f(u)P(u,t)]+M{\partial }_u\{g(u){\partial }_u[g(u)P(u,t)]\}+A{\partial }_{uu}P(u,t)$. Whenever the deterministic drift is proportional to the noise induced one, i.e., $f(u)=-\tau g(u)g^{\prime }(u)$, the stationary solution is shown to be $P(u,\infty )\propto \{1-(1-q)\beta [g(u){]}^2{\}}^{\frac{1}{1-q}}$ (with $q\equiv \frac{\tau +3M}{\tau +M}$ and $\beta =\frac{\tau +M}{2A}$). This distribution is precisely the one optimizing $S_q$ with the constraint $<[g(u){]}^2{>}_q\equiv \{\int du[g(u){]}^2[P(u){]}^q\}/\{\int du[P(u){]}^q\}=$constant. We also introduce and discuss various characterizations of the width of the distributions.

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