Scarce defects induce anomalous diffusion

Abstract

We introduce a simple model of deterministic particles in weakly disordered media which exhibits a transition from normal to anomalous diffusion. The model consists of a set of non-interacting overdamped particles moving on a disordered potential. The disordered potential can be thought as a substrate having some "defects" scattered along a one-dimensional line. The distance between two contiguous defects is assumed to have a heavy-tailed distribution with a given exponent α, which means that the defects along the substrate are scarce if α is small. We prove that this system exhibits a transition from normal to anomalous diffusion when the distribution exponent α decreases, i.e., when the defects become scarcer. Thus we identify three distinct scenarios: a normal diffusive phase for α > 2, a superdiffusive phase for 1/2 < α ≤ 2, and a subdiffusive phase for α ≤ 1/2. We also prove that the particle current is finite for all the values of α, which means that the transport is normal independently of the diffusion regime (normal, subdiffusive, or superdiffusive). We give analytical expressions for the effective diffusion coefficient for the normal diffusive phase and analytical expressions for the diffusion exponent in the case of anomalous diffusion. We test all these predictions by means of numerical simulations.

Active particles in reactive disordered media: how does adsorption affects diffusion?

“…Now we shall present some calculations for obtaining an analytical expression for the effective diffusion coefficient. To this end we will made use of the central limit theorem in order to obtain the asymptotic statistical properties of the net displacement |r(t) − r(0)| for large t. This procedure is similar to the one used for obtaining the transport properties (such as the diffusion coefficient and the effective particle current) in one-dimensional disordered systems [24][25][26][27].…”

“…( 9) and ( 13). This technique has been proved to be useful for determining the exact diffusion coefficients in one-dimensional disordered media [24][25][26][27].…”

“…Nevertheless, such a sum of independent random variables still converges (appropriately normalized) to a stable law [31,32]. When the central limit theorem is broken in this class of systems, it has been show the occurrence of a transition from normal to anomalous diffusion [24][25][26][27] and the limit theorems to stable laws allow the computation of the anomalous diffusion exponent β in Eq. (34).…”

“…This fact is interesting because for these values of α the random adsorption times have no finite variance, i.e., even having a heavy-tailed distribution for the random adsorption times with infinite variance, the system yet exhibit normal diffusive behavior. There is a plethora of examples in the literature which infinite variance (of the random variables involved in the underlying dynamics of the system) is enough for the system to exhibit anomalous behavior [26,27,36,37].…”

Noise-induced rectification in out-of-equilibrium structures

“…Then we use the particlepolymer model that has been studied in Refs. [8][9][10][11][12] to study the dynamics of Brownian particles on random potentials. Particularly we show that, under the influence of an unbiased time-dependent periodic forcing, the spatial irreversibility of the medium induces a rectification phenomenon similar to the one occurring in the so-called rocked thermal ratchets.…”