Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion

TL;DR

This paper introduces a superstatistical generalized Langevin equation model that explains non-Gaussian, viscoelastic anomalous diffusion observed in soft and biological matter, linking distribution shapes to memory kernel parameters.

Contribution

It develops a stochastic model connecting non-Gaussian distributions and anomalous diffusion through a random parametrization of the stochastic force, with detailed analytical and simulation validation.

Findings

Memory kernel distributions lead to exponential, power law, or power-log tails.

Velocity distribution remains Gaussian at a single point but joint distributions are non-Gaussian.

Position distribution transitions from Gaussian to non-Gaussian during relaxation.

Abstract

Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analytical analysis demonstrating how various types of parameter distributions for the memory kernel result in the exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in relaxation from Gaussian to non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with Monte Carlo simulations.