Anomalous transport equations and their application to fractal walking

Generalized transport equations covering both normal transport processes (such as neutron transport) and anomalous ones including Lévy walk and trapping are derived. The behaviours of the spatial distribution and mean-square displacement are investigated for fractal space–time processes. Exact solutions for the spatial distribution have been found in the one-dimensional case and numerical results are obtained for the case when the free paths and waiting times obey the one-sided stable distributions with the same characteristic exponent α=β=1/2. The principal result of the performed analysis is that the finite speed of a moving particle has a direct bearing on the spatial distribution. In the case of anomalous diffusion this effect does not vanish with time and has an influence on the asymptotic behaviour of the distribution form in contrast with normal diffusion when the speed affects only diffusivity but does not affect the form.