Generalized diffusion equation with nonlocality of space-time. Memory function modelling

TL;DR

This paper develops a generalized framework for nonlocal space-time diffusion equations using fractional derivatives, revealing wave-like behaviors and discontinuities in ion transport within complex environments.

Contribution

It introduces a novel approach to derive non-Markovian diffusion equations with fractional derivatives using the Liouville equation and NSO method, incorporating space-time nonlocality.

Findings

Derived new non-Markovian diffusion equations with fractional derivatives.

Identified wave behavior and discontinuities in the dispersion relations.

Calculated phase and group velocities showing nonlocal effects.

Abstract

We presented a general approach for obtaining the generalized transport equations with fractional derivatives by using the Liouville equation with fractional derivatives for a system of classical particles and Zubarev's nonequilibrium statistical operator (NSO) method within Gibbs statistics. The new non-Markovian diffusion equations of ions in spatially heterogeneous environment with fractal structure and generalized Cattaneo-Maxwell diffusion equation with taking into account the space-time nonlocality are obtained. Dispersion relations are found for the Cattaneo-Maxwell diffusion equation with taking into account the space-time nonlocality in fractional derivatives. The frequency spectrum, phase and group velocities are calculated. It is shown that it has a wave behaviour with discontinuities, which are also manifested in the behaviour of the phase velocity.