An investigation of the survival probability for chaotic diffusion in a family of discrete Hamiltonian mappings

Abstract

Diffusive processes usually model the transport of particles in nonlinear systems. Complete chaos leads to normal diffusion, while mixed phase space gives rise to a phenomenon called stickiness, leading to anomalous diffusion. We investigate the survival probability that a particle moving along a chaotic region in a mixed-phase space has to survive a specific domain. We show along the chaotic part far from islands that an exponential decay describes the survival probability. Nonetheless, when the islands are incorporated into the domain, the survival probability exhibits an exponential decay for a short time, changing to a slower decay for a considerable enough time. This changeover is a signature of stickiness. We solve the diffusion equation by obtaining the probability density to observe a given particle along a specific region within a certain time interval. Integrating the probability density for a defined phase space area provides analytical survival probability. Numerical simulations fit well the analytical findings for the survival probability when the region is fully chaotic. However, the agreement could be better when mixed structure with islands and periodic areas are included in the domain.