Absence of confinement and non-Boltzmann stationary states of fractional Brownian motion in shallow external potentials

TL;DR

This paper investigates the behavior of fractional Brownian motion in shallow potentials, revealing conditions under which stationary states exist and highlighting the absence of confinement in certain regimes.

Contribution

It provides a detailed numerical analysis of fractional Brownian motion in subharmonic potentials, identifying the specific conditions for the existence of stationary states.

Findings

Stationary states exist only if c > 2(1 - 1/α).

In shallow potentials, confinement is absent for certain parameters.

The study compares fractional Brownian motion with Levy flights in similar potentials.

Abstract

We study the diffusive motion of a particle in a subharmonic potential of the form $U(x)=|x|^c$ ($0<c<2$) driven by long-range correlated, stationary fractional Gaussian noise $\xi_{\alpha}(t)$ with $0<\alpha\le2$. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent $\alpha$. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation $c>2(1-1/\alpha)$ holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function (PDF). Moreover we discuss analogies of non-stationarity of L{\'e}vy flights in shallow external potentials.