Fractal Structures in the Phase Space of Simple Chaotic Systems with Transport

Abstract

Recent work by several authors has shown that non-equilibrium processes in simple, classical, chaotic systems can be described in terms of fractal structures that develop in the system’s phase space. These structures form exponentially rapidly in phase space as an initial non-equilibrium distribution evolves in time. Since the motion of a region in phase space, for a Hamiltonian system, is measure preserving, the phase space distribution is advected as a passive scalar in the motion of the phase points. Due to the chaotic nature of the motion, the stretching and folding motion in phase space produces very complicated fractal distributions which may vary greatly over regions of small measure. This mechanism is responsible for the formation of the fractals under discussion. Here we illustrate this phenomenon for a few simple models with deterministic diffusion. The origin of the fractals is explained and connected to the microscopic properties of the hydrodynamic modes of the system. These hydrodynamic modes are, in turn, closely related, on averaging, to the van Hove intermediate scattering function. Further we describe the connections of the properties of the fractals with important quantities for transport - transport coefficients and irreversible entropy production. One interesting result is a connection between the coefficient of diffusion for the moving particle in a chaotic Lorentz gas and the Hausdorff dimension of the hydrodynamic modes of diffusion at small wave numbers.