Molecular theory of anomalous diffusion

TL;DR

This paper develops a molecular theory of anomalous diffusion using a Markovian random walk model, capturing sub-, classical, and super-diffusion through particle interactions, with analytical and simulation results aligning across regimes.

Contribution

It introduces a Master Equation framework that models different diffusion regimes via particle interactions, leading to a nonlinear Fokker-Planck equation with self-similar solutions.

Findings

The model reproduces sub-, classical, and super-diffusion behaviors.

Analytical solutions match simulation results across regimes.

Physical interpretation links interactions to diffusion exponents.

Abstract

We present a Master Equation formulation based on a Markovian random walk model that exhibits sub-diffusion, classical diffusion and super-diffusion as a function of a single parameter. The non-classical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean-squared displacement ($<r^2(t)>\sim t^{\gamma}$ with $0 <\gamma \leq 1.5$) but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and super-diffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent $\gamma$. Simulations based on the Master Equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.