Influence of the environment on anomalous diffusion

We study the effect of weak environmental fluctuations on a deterministic map dynamics, which, in the unperturbed case, is characterized by anomalous diffusion. We show, with the help of numerical calculations, that there is a crossover time 1/ε, at which the waiting time distributions change from an inverse power-law distribution into an exponential behavior. We prove theoretically that the diffusion coefficient of the long-time process is proportional to 1/${\mathrm{\epsilon }}^{\mathrm{\alpha }}$, with α positive or negative, according to whether we consider the superdiffusive or the subdiffusive case. With very weak environmental fluctuations the diffusion coefficient of the former case becomes anomalously large and that of the latter case anomalously small. The theoretical predictions are confirmed by the numerical results.