Fractal Diffusion Coefficients in Simple Dynamical Systems.

Abstract

Deterministic diffusion is studied in simple, parameter-dependent dynamical systems. The diffusion coefficient is often a fractal function of the control parameter, exhibiting regions of scaling and self-similarity. Firstly, the concepts of chaos and deterministic diffusion are introduced in the context of dynamical systems. The link between deterministic diffusion and physical diffusion is made via random walk theory. Secondly, parameter-dependent diffusion coefficients are analytically derived by solving the Taylor-Green-Kubo formula. This is done via a recursion relation solution of fractal ‘generalised Takagi functions’. This method is applied to simple one-dimensional maps and for the first time worked out fully analytically. The fractal parameter dependence of the diffusion coefficient is explained via Markov partitions. Linear parameter dependence is observed which in some cases is due to ergodicity breaking. However, other cases are due to a previously unobserved phenomenon called the ‘dominating-branch’ effect. A numerical investigation of the two-dimensional ‘sawtooth map’ yields evidence for a possible fractal structure. Thirdly, a study of different techniques for approximating the diffusion coefficient of a parameter-dependent dynamical system is then performed. The practicability of these methods, as well as their capability in exposing a fractal structure is compared. Fourthly, an analytical investigation into the dependence of the diffusion coefficient on the size and position of the escape holes is then undertaken. It is shown that varying the position has a strong effect on diffusion, whilst the asymptotic regime of small-hole size is dependent on the limiting behaviour of the escape holes. Finally, an exploration of a method which involves evaluating the zeros of a system’s dynamical zeta function via the 5 weighted Milnor-Thurston kneading determinant is performed. It is shown how to relate the diffusion coefficient to a zero of the dynamical zeta function before analytically deriving the diffusion coefficient via the kneading determinant.