Generalized diffusion equation

TL;DR

This paper introduces a systematic method using a $q$-generating function to derive a generalized advection-diffusion equation, extending classical diffusion models to account for non-classical behaviors.

Contribution

The paper presents a novel $q$-generating function approach for deriving generalized diffusion equations from classical statistical mechanics.

Findings

Derivation of a generalized advection-diffusion equation in Fourier space.

Solutions indicate the $q$-generating function effectively generalizes classical diffusion.

Method provides a systematic framework for non-classical diffusion modeling.

Abstract

Modern analyses of diffusion processes have proposed nonlinear versions of the Fokker-Planck equation to account for non-classical diffusion. These nonlinear equations are usually constructed on a phenomenological basis. Here we introduce a nonlinear transformation by defining the $q$-generating function which, when applied to the intermediate scattering function of classical statistical mechanics, yields, in a mathematically systematic derivation, a generalized form of the advection-diffusion equation in Fourier space. Its solutions are discussed and suggest that the $q$-generating function approach should be a useful tool to generalize classical diffusive transport formulations.