Markovian embedding of non-Markovian superdiffusion

TL;DR

This paper explores Markovian embeddings of non-Markovian superdiffusive processes described by generalized Langevin equations, demonstrating how potential barriers and bias forces influence long-term diffusion behavior and transient dynamics.

Contribution

It introduces multiple Markovian embedding schemes for superdiffusive GLE models and analyzes how external potentials and forces affect asymptotic and transient diffusion behaviors.

Findings

Periodic potentials convert superdiffusion to normal diffusion at large times.

Bias forces restore superdiffusive behavior asymptotically.

Transient superballistic currents can grow faster than ballistic, with giant superdiffusive enhancements.

Abstract

We consider different Markovian embedding schemes of non-Markovian stochastic processes that are described by generalized Langevin equations (GLE) and obey thermal detailed balance under equilibrium conditions. At thermal equilibrium superdiffusive behavior can emerge if the total integral of the memory kernel vanishes. Such a situation of vanishing static friction is caused by a super-Ohmic thermal bath. One of the simplest models of ballistic superdiffusion is determined by a bi-exponential memory kernel that was proposed by Bao [J.-D. Bao, J. Stat. Phys. 114, 503 (2004)]. We show that this non-Markovian model has infinitely many different 4-dimensional Markovian embeddings. Implementing numerically the simplest one, we demonstrate that (i) the presence of a periodic potential with arbitrarily low barriers changes the asymptotic large time behavior from free ballistic superdiffusion into normal diffusion; (ii) an additional biasing force renders the asymptotic dynamics superdiffusive again. The development of transients that display a qualitatively different behavior compared to the true large-time asymptotics presents a general feature of this non-Markovian dynamics. These transients though may be extremely long. As a consequence, they can be even mistaken as the true asymptotics. We find that such intermediate asymptotics exhibit a giant enhancement of superdiffusion in tilted washboard potentials and it is accompanied by a giant transient superballistic current growing proportional to $t^{\alpha_{{\rm eff}}}$ with an exponent $\alpha_{\rm eff}$ that can exceed the ballistic value of two.