Langevin equation in complex media and anomalous diffusion.

Silvia Vitali,Vittoria Sposini,Paolo Paradisi,Gianni Pagnini,Gastone Castellani,Oleksii Sliusarenko

doi:10.1098/rsif.2018.0282

Abstract

The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modelling 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 also be somewhat alternative to each other, e.g. continuous time random walk and fractional Brownian motion. To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modelling of velocity dynamics. The complexity of the medium is parametrized 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.

Highlights

The very rich dynamics of biosystem movements have been attracting the interest of many researchers in the field of statistical physics and complexity for its inherent temporal and spatial multi-scale character
We propose a modelling approach to 2 anomalous diffusion inspired by the constructive approach used to derive Schneider grey noise, grey Brownian motion [45,46] and generalized gBM [47,48,49,50,51,52]
In order to include the effect of viscosity, we describe the development of a model similar to the original generalized gBM (ggBM), but with a friction–diffusion process instead of a Gaussian noise, involving an explicit modelling of the system’s dynamics by substituting the fractional Brownian motion (FBM), used to built the ggBM, with the stochastic process resulting from the Langevin equation for the particle velocity

Summary

Introduction

The very rich dynamics of biosystem movements have been attracting the interest of many researchers in the field of statistical physics and complexity for its inherent temporal and spatial multi-scale character. Standard or normal diffusive (Brownian) motion is uniquely described by the Wiener process [13] and is associated with a Gaussian probability density function (PDF) of displacements and linear time dependence of the mean square displacement (MSD). It is well-known that normal diffusion emerges in the long-time limit t ) tc when the correlation timescale tc is finite and non-zero [14] (see section 1 of the electronic supplementary material for details).

Methods

Discussion

Conclusion