Dependence of chaotic diffusion on the size and position of holes

TL;DR

This paper investigates how the size and position of phase space 'holes' influence the diffusion coefficient in chaotic maps, revealing non-monotonic dependencies and deriving analytical formulas for small holes.

Contribution

It provides an analytical framework linking hole parameters to diffusion behavior, including formulas for small holes and insights into deviations from random walk models.

Findings

Diffusion coefficient varies non-monotonically with hole size and position.

Analytical formulas for small holes based on periodic orbits.

Asymptotic regimes deviate from standard stochastic approximations.

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 one-dimensional, periodically lifted maps. The parameter dependent diffusion coefficient can be obtained analytically via a Taylor-Green-Kubo formula in terms of a functional recursion relation. We find that the diffusion coefficient varies non-monotonically 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.