Paper detail

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

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 $ξ_α(t)$ with $0<α\le2$. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent $α$. 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 $c>2(1-1/α)$ 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{é}vy flights in shallow external potentials.