Multiplicative noise: A mechanism leading to nonextensive statistical mechanics

A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann-Gibbs statistical mechanics [based on S(1)equivalent to-kintegraldu p(u)ln p(u)]. Similarly, other classes of models point toward nonextensive statistical mechanics [based on S(q)equivalent tok[1-integraldu[p(u)](q)]/[q-1], where the value of the entropic index qis an element ofR 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 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 derivative(t)P(u,t)=-partial derivative(u)[f(u)P(u,t)]+Mpartial derivative(u){g(u)partial derivative(u)[g(u)P(u,t)]}+Apartial derivative(uu)P(u,t). Whenever the deterministic drift is proportional to the noise induced one, i.e., f(u)=-taug(u)g(')(u), the stationary solution is shown to be P(u,infinity)proportional to{1-(1-q)beta[g(u)](2)}(1/(1-q)) [with qequivalent to(tau+3M)/(tau+M) and beta=(tau+M/2A)]. This distribution is precisely the one optimizing S-q with the constraint <[g(u)](2)>(q)equivalent to{integraldu [g(u)](2)[P(u)](q)}/{integraldu [P(u)](q)}=const. We also introduce and discuss various characterizations of the width of the distributions. (C) 2003 American Institute of Physics.