On the Einstein–Smoluchowski relation in the framework of generalized statistical mechanics

Abstract

Anomalous statistical distributions that exhibit asymptotic behavior different from the exponential Boltzmann–Gibbs tail are typical of complex systems constrained by long-range interactions or time-persistent memory effects at the stationary non-equilibrium or meta-equilibrium. In this framework, a nonlinear Smoluchowski equation, which models the system’s time evolution towards its steady state, is obtained using the gradient flow method based on a free-energy potential related to a given generalized entropic form. Comparison of the stationary distribution resulting from the maximization of entropy for a canonical ensemble with the steady state distribution resulting from the Smoluchowski equation gives an Einstein-Smoluchowski-like relation. Despite this relationship between the mobility of particle μ and the diffusion coefficient D retains its original expression: μ=βD, appropriate considerations, physically motivated, force us an interpretation of the parameter β different from the traditional meaning of inverse temperature.