Dependence of chaotic diffusion on the size and position of holes
Abstract
A particle driven by deterministic chaos and moving in a spatially extended environment can exhibit normal diffusion, with its mean square displacement growing proportional to the time. Here, we consider the dependence of the diffusion coefficient on the size and the position of areas of phase space linking spatial regions (`holes') in a class of simple onedimensional, periodically lifted maps. The parameter dependent diffusion coefficient can be obtained analytically via a TaylorGreenKubo formula in terms of a functional recursion relation. We find that the diffusion coefficient varies nonmonotonically with the size of a hole and its position, which implies that a diffusion coefficient can increase by making the hole smaller. We derive analytic formulas for small holes in terms of periodic orbits covered by the holes. The asymptotic regimes that we observe show deviations from the standard stochastic random walk approximation. The escape rate of the corresponding open system is also calculated. The resulting parameter dependencies are compared with the ones for the diffusion coefficient and explained in terms of periodic orbits.
 Publication:

Chaos
 Pub Date:
 June 2012
 DOI:
 10.1063/1.4721887
 arXiv:
 arXiv:1112.3922
 Bibcode:
 2012Chaos..22b3132K
 Keywords:

 chaos;
 diffusion;
 random processes;
 recursion method;
 stochastic processes;
 05.45.a;
 05.60.k;
 05.40.a;
 Nonlinear dynamics and chaos;
 Transport processes;
 Fluctuation phenomena random processes noise and Brownian motion;
 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 12 pages, 5 figures