Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls

Fractional Brownian motion and the fractional Langevin equation are models of anomalous diffusion processes characterized by long-range power-law correlations in time. We employ large-scale computer simulations to study these models in two geometries, (i) the spreading of particles on a semi-infinite domain with an absorbing wall at one end and (ii) the stationary state on a finite interval with absorbing boundaries at both ends and a source in the center. We demonstrate that the probability density and other properties of the fractional Langevin equation can be mapped onto the corresponding quantities of fractional Brownian motion driven by the same noise if the anomalous diffusion exponent alpha is replaced by 2 - alpha. In contrast, the properties of fractional Brownian motion and the fractional Langevin equation with reflecting boundaries were recently shown to differ from each other qualitatively. Specifically, we find that the probability density close to an absorbing wall behaves as P(x) similar to x(kappa) with the distance x from the wall in the long-time limit. In the case of fractional Brownian motion, kappa varies with the anomalous diffusion exponent alpha as kappa = 2/alpha - 1, as was conjectured previously. We also compare our simulation results to a perturbative analytical approach to fractional Brownian motion.

Automated summary

Research shows that in the models of fractional Brownian motion and fractional Langevin equation, probability density and other properties can be mapped onto each other, but they have qualitatively different behaviors when reflecting boundaries are present.