Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries

Abstract

Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected fractional Brownian motion at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude $1/L$ in an interval of length $L$ found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement $⟨X^2(t)⟩\simeq t^{\alpha }$ with $1<\alpha <2$, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion ($0<\alpha <1$) this behaviour is reversed and the particle density is depleted close to the boundaries. The mean squared displacement in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent $\alpha$. Our a priori surprising results may have interesting consequences for the application of fractional Brownian motion for processes such as molecule or tracer diffusion in the confined of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

19 pages, 8 figures, IOP LaTeX

References in corpus (6)

• Anomalous transport in the crowded world of biological cells
• Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion
• Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins
• Spectral content of a single non-Brownian trajectory
• Collective dynamics effect transient subdiffusion of inert tracers in gel networks
• Superdiffusion dominates intracellular particle motion in the supercrowded space of pathogenic Acanthamoeba castellanii

Cited by in corpus (6)

• Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions
• First passage time moments of asymmetric Lévy flights
• Lévy walk dynamics in an external harmonic potential
• Anomalous diffusion and long-range memory in the scaled voter model
• Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls
• Tempered fractional Brownian motion on finite intervals