Random walk in a random multiplicative environment

We investigate the dynamics of a random walk in a random multiplicative medium. This results in a random, but correlated, multiplicative process for the spatial distribution of random walkers. We show how the details of these correlations determine the asymptotic properties of the walk, i.e., the central limit theorem does not apply to these multiplicative processes. We also study a periodic source-trap medium in which a unit cell contains one source, followed byL–1 traps. We calculate the asymptotic behavior of the number of particles, and determine the conditions for which there is growth or decay in this average number. Finally, we discuss the asymptotic behavior of a random walk in the presence of randomly distributed, partially-absoprbing traps. For this case, a temporal regime of purely exponential decay of the density can occur, before the asymptotic stretched exponential decay, exp(–at ^1/3), sets in.     

Williams-watts dielectric relaxation: A fractal time stochastic process

Dielectric relaxation in amorphous materials is treated in a defect-diffusion model where relaxation occurs when a mobile defect, such as a vacancy, reaches a frozen-in dipole. The random motion of the defect is assumed to be governed by a fractal time stochastic process where the mean duration between defect movements is infinite. When there are many more defects than dipoles, the Williams-Watts decaying fractional exponential relaxation law is derived. The argument of the exponential is related to the number of distinct sites visited by the random walk of the defect. For the same reaction dynamics but with more traps than walkers, an algebraically decaying relaxation is found.