Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers

TL;DR

This paper investigates the crossover from anomalous to normal diffusion in complex systems using tempered fractional Gaussian noise, deriving explicit expressions for MSD and correlations, and applying the model to lipid bilayer dynamics.

Contribution

It introduces a comprehensive model of tempered fractional Gaussian noise with explicit formulas for crossover behaviors, enhancing understanding of diffusive dynamics in biological systems.

Findings

Derived explicit MSD and correlation functions for tempered noise.

Demonstrated model's applicability to lipid bilayer subdiffusion and crossover.

Showed differences between tempered fractional Brownian motion and previous models.

Abstract

The emerging diffusive dynamics in many complex systems shows a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive-diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive-diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively described the subdiffusion-diffusion and subdiffusion-subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling.