Nonextensive diffusion as nonlinear response

TL;DR

This paper derives a generalized diffusion equation for classical fluids based on nonlinear response assumptions, linking it to nonextensive statistical mechanics and distinguishing it from the porous media equation.

Contribution

It provides a systematic derivation of a generalized diffusion equation incorporating nonlinear response and nonextensive statistics, expanding classical diffusion models.

Findings

The derived equation exhibits $r/\sqrt{Dt}$ scaling.

It incorporates a generalized Einstein relation.

It predicts power-law probability distributions.

Abstract

The porous media equation has been proposed as a phenomenological ``non-extensive'' generalization of classical diffusion. Here, we show that a very similar equation can be derived, in a systematic manner, for a classical fluid by assuming nonlinear response, i.e. that the diffusive flux depends on gradients of a power of the concentration. The present equation distinguishes from the porous media equation in that it describes \emph{% generalized classical} diffusion, i.e. with $r/\sqrt Dt$ scaling, but with a generalized Einstein relation, and with power-law probability distributions typical of nonextensive statistical mechanics.