Diffusion and annihilation reactions of Levy flights with bounded long-range hoppings

Abstract

A new kind of random walk named bounded Levy flights (BLFs), where the step length is a bounded random variable, is proposed and their properties are studied with the aid of mean field and Monte Carlo techniques. BLFs are characterized by the Levy exponent ( sigma ) and the length of the longest possible flight (RM). It is found that in one dimension (1D), the mean number of distinct sites visited by the walker (SN) and the average square displacement (RN2) behave like SN varies as RQMNds(Q=fsigma ,ds=1/2) and R2N varies as RMf( sigma )Nnu (v=1), where f( sigma ) is a continuously tunable function of sigma with f( sigma )0.9 ( sigma 0.1) and f( sigma )0 ( sigma 2). In addition, the long-time behaviour of annihilation reactions between BLFs, which react via exchange in 1D is found to be anomalous because the density of walkers ( rho A) behaves like d rho A/dt-Rf( sigma )M rho AX with X=1+(1/ds)=3(t) while, shortly after the beginning of the reaction, the classical behaviour X=2(t0) holds.