Lévy flights in quenched random force fields

Abstract

Lévy flights, characterized by the microscopic step index $f,$ are for $f<\mathrm{}2\mathrm{}$ (the case of rare events) considered in short-range and long-range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension $2f-2$ in the short-range case and the critical fall-off exponent $2f-2$ in the long-range case. By means of a dynamic renormalization-group analysis based on the momentum shell integration method, we determine flows, fixed point, and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient. Unlike the case of ordinary Brownian motion in a quenched force field characterized by a single critical dimension or fall-off exponent $d=2,$ two critical dimensions appear in the Lévy case. A critical dimension (or fall-off exponent) $d=f$ below which the diffusion coefficient exhibits anomalous scaling behavior, i.e., algebraic spatial behavior and long time tails, and a critical dimension (or fall-off exponent) $d=2f-2$ below which the force correlations characterized by a nontrivial fixed point become relevant. As a general result we find in all cases that the dynamic exponent $z,$ characterizing the mean square displacement, locks onto the Lévy index $f,$ independent of dimension and independent of the presence of weak quenched disorder.