Langevin equation in complex media and anomalous diffusion

TL;DR

This paper introduces a stochastic Langevin equation model with variable parameters to describe complex anomalous diffusion in biological media, capturing various diffusion behaviors including Gaussian and fractional types.

Contribution

It develops a novel Langevin-based framework with parameter randomness to unify different models of anomalous diffusion in complex biological environments.

Findings

Model reproduces Gaussian anomalous diffusion

Captures fractional and generalized diffusion behaviors

Parameter distributions determine diffusion type

Abstract

The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modeling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can be also somewhat alternative to each other, e.g., Continuous Time Random Walk (CTRW) and Fractional Brownian Motion (FBM). To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modeling of velocity dynamics. The complexity of the medium is parameterized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particle's dynamics. We prove that, for proper distributions of these parameters, we can get both Gaussian anomalous diffusion, fractional diffusion and its generalizations.