Random flights with quenched noise amplitudes
We study the random hopping motion of a particle, for which the jump directions vary randomly in time but the jump lengths l (“noise amplitudes”) are fixed in space (quenched disorder). Two cases are considered: I the distribution p(l) of jump lengths has a finite second moment, and II p(l) decays slowly according to a Lévy distribution, p(l)∼l−1−f with 0<f<2. For simplicity we will restrict our study to one-dimension and consider the jump lengths to be correlated over a short distance a around regularly spaced lattice sites. In case I we find that the diffusion coefficient strongly differs from the mean-field result due to the spatial fluctuations of the jump lengths. The diffusion coefficient can nevertheless be calculated from a modified mean field treatment, when the effective probability peff(l) for the particle to be at a site with jump length l is taken into account. In case II we find that for f≳0.7 the superdiffusion in the quenched case is slowed down in comparison with the annealed case (where the jump lengths are drawn anew at each time step), leading to a novel length-time scaling relation in the interval 0.7≲f≲1.3. This slowing down can again be explained by considering the effective jump-length distribution peff(l)∼l−1−g in the stationary state, which decays more rapidly than p(l), i.e. g≥f. For f≳1.3, g becomes larger than 2 and the diffusion becomes normal although p(l) has no finite second moment. A scaling theory is developed that describes the dynamical crossover from the annealed to the quenched situation.
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