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

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 ? and the diffusion coefficient D retains its original expression: ? = ? D, appropriate considerations, physically motivated, force us an interpretation of the parameter ? different from the traditional meaning of inverse temperature.