Lévy decoupled random walks

Abstract

A diffusion model based on a continuous time random walk scheme with a separable transition probability density is introduced. The probability density for long jumps is proportional to x −1− γ (a Lévy-like probability density). Even when the probability density for the walker position at time t, P(x;t) , has not a finite second moment when 0< γ<2, it is possible to consider alternative estimators for the width of the distribution. It is then found that any reasonable width estimator will exhibit the same long-time behaviour, since in this limit P( x; t) goes to the distribution L γ ( x/ t α ), a Lévy distribution. The scaling property is verified numerically by means of Monte Carlo simulations. We find that if the waiting time density has a finite first moment then α=1/ γ, while for densities with asymptotic behaviour t −1− β with 0< β<1 (“long tail” densities) it is verified that α= β/ γ. This scaling property ensures that any reasonable estimator of the distribution width will grow as t α in the long-time limit. Based on this long-time behaviour we propose a generalized criterion for the classification in superdiffusive and subdiffusive processes, according to the value of α.