Analytical results for long time behavior in anomalous diffusion

TL;DR

This paper analytically investigates the long-term behavior of anomalous diffusion using a generalized Langevin approach, introducing a universal scaling factor and deriving exact expressions for diffusion parameters.

Contribution

It proposes a novel analytical method to determine the diffusion coefficient and scaling factor in anomalous diffusion, extending the concept of the diffusion exponent.

Findings

Derived an exact expression for the scaling factor $rac{rac{$ for all diffusion types

Showed that the scaling factor is a universal parameter linked to the diffusion exponent

Validated the analytical results with numerical simulations showing excellent agreement

Abstract

We investigate through a Generalized Langevin formalism the phenomenon of anomalous diffusion for asymptotic times, and we generalized the concept of the diffusion exponent. A method is proposed to obtain the diffusion coefficient analytically through the introduction of a time scaling factor $\lambda$. We obtain as well an exact expression for $\lambda$ for all kinds of diffusion. Moreover, we show that $\lambda$ is a universal parameter determined by the diffusion exponent. The results are then compared with numerical calculations and very good agreement is observed. The method is general and may be applied to many types of stochastic problem.