Probability density of the fractional Langevin equation with reflecting walls

Abstract

We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while anti-persistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for non-thermal fractional Brownian motion with reflecting walls, and we discuss broader implications.

12 pages, 13 figures, final version as published

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Cited by in corpus (13)

• Reflected fractional Brownian motion in one and higher dimensions
• Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions
• First passage time moments of asymmetric Lévy flights
• First passage properties of asymmetric Lévy flights
• Time Glass: A Fractional Calculus Approach
• Dynamics and steady states of a tracer particle in a confined critical fluid
• Absence of confinement and non-Boltzmann stationary states of fractional Brownian motion in shallow external potentials
• Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls
• Distribution and pressure of active Lévy swimmers under confinement
• Langevin analogy between particle trajectories and polymer configurations
• Levy flights in steep potential wells: Langevin modeling versus direct response to energy landscapes
• Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement
• Tempered fractional Brownian motion on finite intervals