Markovian embedding of fractional superdiffusion

TL;DR

This paper introduces a Markovian embedding method for the Fractional Langevin Equation, enabling efficient simulation of superdiffusive processes and revealing transient hyperdiffusion phenomena in tilted potentials.

Contribution

It proposes a novel Markovian embedding technique for the FLE that accurately approximates superdiffusion over many decades, providing a flexible tool for modeling anomalous diffusion.

Findings

Efficient numerical realization of the Markovian embedding for FLE.

Accurate approximation of superdiffusion over many decades.

Observation of transient hyperdiffusion due to kinetic heating.

Abstract

The Fractional Langevin Equation (FLE) describes a non-Markovian Generalized Brownian Motion with long time persistence (superdiffusion), or anti-persistence (subdiffusion) of both velocity-velocity correlations, and position increments. It presents a case of the Generalized Langevin Equation (GLE) with a singular power law memory kernel. We propose and numerically realize a numerically efficient and reliable Markovian embedding of this superdiffusive GLE, which accurately approximates the FLE over many, about r=N lg b-2, time decades, where N denotes the number of exponentials used to approximate the power law kernel, and b>1 is a scaling parameter for the hierarchy of relaxation constants leading to this power law. Besides its relation to the FLE, our approach presents an independent and very flexible route to model anomalous diffusion. Studying such a superdiffusion in tilted washboard potentials, we demonstrate the phenomenon of transient hyperdiffusion which emerges due to transient kinetic heating effects.