Nonergodicity, accelerator modes, and asymptotic quadratic-in-time diffusion in a class of volume-preserving maps

Abstract

Using an elementary application of Birkhoff’s ergodic theorem, we give necessary and sufficient conditions for the existence of asymptotically n^2 diffusion (where n is an integer representing discrete time in the angle variables in a class of volume-preserving twist maps. We show that nonergodicity is the dynamical mechanism giving rise to this behavior. The influence of accelerator modes on diffusion is described. We discuss how additive noise changes the diffusive behavior and we investigate the effective-diffusivity dependence on bare diffusivity and accelerator modes. In particular, we find that the dependence of the effective-diffusivity coefficient on bare diffusivity is universal for small values of bare-diffusivity coefficient σ if asymptotic n^2 diffusion is present in the σ=0 case.