Abstract
Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM 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 (MSD) with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.
Highlights
Diffusive transport is quite ubiquitous, ranging from quantum processes such as laser cooling over thermally activated transport in living biological cells, to the dispersal of tracer chemicals in geophysical aquifers
Mean squared displacement We show that the effects observed for the probability density function translate to the behaviour of the mean squared displacement (MSD)
We studied by simulations the stochastic process of reflected fractional Brownian motion (FBM), which is confined to a finite interval with reflecting boundary conditions, a situation that is typical for tracer particles in the viscoelastic confines of biological cells or their organelles, as well as in artificially crowded liquids in microfluidic devises and similar
Summary
Any further distribution of Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time this work must maintain correlated increments and is frequently used to model anomalous diffusion processes. Density function of this reflected FBM 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. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α.
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