Noise-induced transition from anomalous to ordinary diffusion: The crossover time as a function of noise intensity

We study the interplay between a deterministic process of weak chaos, responsible for the anomalous diffusion of a variable x, and a white noise of intensity Ξ. The deterministic process of anomalous diffusion results from the correlated fluctuations of a statistical variable ξ between two distinct values +1 and -1, each of them characterized by the same waiting time distribution ψ(t), given by ψ(t)≃${\mathit{t}}^{\mathrm{-}\mathrm{\mu }}$ with 2<μ<3, in the long-time limit. We prove that under the influence of a weak white noise of intensity Ξ, the process of anomalous diffusion becomes normal at a time ${\mathit{t}}_{\mathit{c}}$ given by ${\mathit{t}}_{\mathit{c}}$∼1/${\mathrm{\Xi }}^{\mathrm{\beta }\left(\mathrm{\mu }\right)}$. Here β(μ) is a function of μ which depends on the dynamical generator of the waiting time distribution ψ(t). We derive an explicit expression for β(μ) in the case of two dynamical systems, a one-dimensional superdiffusive map and the standard map in the accelerating state. The theoretical prediction is supported by numerical calculations. (c) 1995 The American Physical Society