Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions

TL;DR

This paper investigates how fractional Brownian motion in superharmonic potentials results in non-Boltzmann stationary distributions, revealing the effects of subdiffusive and superdiffusive dynamics on particle localization.

Contribution

It demonstrates the existence of non-Boltzmann stationary PDFs for fractional Brownian motion in superharmonic potentials and compares these with Markovian Lévy flights.

Findings

Subdiffusive motion leads to probability accumulation near the origin.

Superdiffusive motion results in bimodal stationary PDFs.

Fractional Langevin dynamics relax to Boltzmann distribution due to fluctuation-dissipation.

Abstract

We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise in a superharmonic external potential of the form $U(x)\propto x^{2n}$ ($n\in\mathbb{N}$). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential $U(x)$. We analyse in detail the temporal approach to stationarity as well as the shape of the non-Boltzmann stationary PDF. A typical characteristic is that subdiffusive, antipersistent (with negative autocorrelation) motion tends to effect an accumulation of probability close to the origin as compared to the corresponding Boltzmann distribution while the opposite trend occurs for superdiffusive (persistent) motion. For this latter case this leads to distinct bimodal shapes of the PDF. This property is compared to a similar phenomenon observed for Markovian L{\'e}vy flights in superharmonic potentials. We also demonstrate that the motion encoded in the fractional Langevin equation driven by fractional Gaussian noise always relaxes to the Boltzmann distribution, as in this case the fluctuation-dissipation theorem is fulfilled.