Non-exponential relaxation for anomalous diffusion

TL;DR

This paper investigates the relaxation dynamics in normal and anomalous diffusion using a generalized Langevin equation, revealing non-exponential correlation functions and the transition from non-Markovian to Markovian behavior.

Contribution

It introduces a general correlation function for relaxation phenomena that is non-exponential and demonstrates its applicability to both normal and anomalous diffusion regimes.

Findings

Correlation function is even and non-exponential.

Transition from non-Markovian to Markovian behavior in normal diffusion.

Power-law decay of correlation function in anomalous diffusion for long times.

Abstract

We study the relaxation process in normal and anomalous diffusion regimes for systems described by a generalized Langevin equation (GLE). We demonstrate the existence of a very general correlation function which describes the relaxation phenomena. Such function is even; therefore, it cannot be an exponential or a stretched exponential. However, for a proper choice of the parameters, those functions can be reproduced within certain intervals with good precision. We also show the passage from the non-Markovian to the Markovian behaviour in the normal diffusion regime. For times longer than the relaxation time, the correlation function for anomalous diffusion becomes a power law for broad-band noise.