Ito-Langevin equations within generalized thermostatistics

An entire family of microscopic Langevin equations is shown here to give rise to Tsallis-type macroscopic distributions (proportional to [1 - beta(1 - q) U(x)](1/(1-q)), U(x) being the potential) as exact stationary solution to the standard linear Fokker-Planck equation. This is the result of a specific interplay between the underlying deterministic and stochastic forces for which the stationary solution becomes identical to that of a nonlinear diffusion process. The fact that a process stemming from the standard theory of Fokker-Planck equations, usually associated with Boltzmann-Gibbs statistics, can equivalently be viewed within the non-extensive scenario of generalized thermostatistics is emphasized. This greatly enhances the understanding of the types of physical processes which can naturally be described within that framework. Furthermore, it is shown specifically that a slight change in the microscopic dynamics of a system following Boltzmann-Gibbs statistics may lead to Tsallis statistics. The Tsallis index q is directly determined by the coefficients of the underlying forces. (C) 1998 Elsevier Science B.V.