Biased diffusion in a piecewise linear random potential

Abstract

We study the biased diffusion of particles moving in one direction under the action of a constant force in the presence of a piecewise linear random potential. Using the overdamped equation of motion, we represent the first and second moments of the particle position as inverse Laplace transforms. By applying to these transforms the ordinary and the modified Tauberian theorem, we determine the short-and longtime behavior of the mean-square displacement of particles. Our results show that while at short times the biased diffusion is always ballistic, at long times it can be either normal or anomalous. We formulate the conditions for normal and anomalous behavior and derive the laws of biased diffusion in both these cases. PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion -05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) -02.50.-r Probability theory, stochastic processes, and statistics S.I. Denisov, E.S. Denisova, H. Kantz: Biased diffusion in a piecewise linear random potential O O O O O [ J J J[

Weak Disorder in Periodic Potentials: Anomalous Transport and Diffusion

“…Such an analysis has recently been presented for the case of a piecewise linear random potential [22], but seems not yet to be available for the more realistic potentials considered here.…”

Weak subordination breaking for the quenched trap model

“…The particle returning to a lattice point visited already, "remembers" its waiting time there. Mathematicians have rigorously shown that in dimensions higher than one [28] or in the presence of a bias [55] (see also [56][57][58]) the CTRW approach describes well the quenched dynamics since the particle does not tend to revisit the same lattice points many times, thus confirming physical insight in [16,17,26] (for dimension d = 2 logarithmic corrections are also important). While the three dimensional QTM belongs to the domain of attraction of the CTRW the calculation of the anomalous diffusion constant is not trivial (see discussion in the summary).…”

Normal-to-anomalous diffusion transition in disordered correlated potentials: From the central limit theorem to stable laws

“…Clearly this situation can occur in the case in which the external forcing F 0 is exactly the critical tilt, as it was pointed out in Ref. [8,9].…”

“…A more recent approach to the problem of deterministic diffusion of overdamped particles in disordered potentials was carried by S. I. Denisov et al in Ref. [9]. They consider a class of piece-wise linear random potentials and this simplification let them find analytical expression for the diffusion coefficient (when normal diffusion occurs) as well as for the long-time and the short time behavior of the mean square displacement.…”