TRANSPORT IN POLYGONAL BILLIARD SYSTEMS by Matthew

Title of dissertation: TRANSPORT IN POLYGONAL BILLIARD SYSTEMS Matthew L. Reames, Doctor of Philosophy, 2009 Dissertation directed by: Professor J. Robert Dorfman Department of Physics The aim of this work is to explore the connections between chaos and diffusion by examining the properties of particle motion in non-chaotic systems. To this end, particle transport and diffusion are studied for point particles moving in systems with fixed polygonal scatterers of four types: (i) a periodic lattice containing manysided polygonal scatterers; (ii) a periodic lattice containing few-sided polygonal scatterers; (iii) a periodic lattice containing randomly oriented polygonal scatterers; and (iv) a periodic lattice containing polygonal scatterers with irrational angles. The motion of a point particle in each of these system is non-chaotic, with Lyapunov exponents strictly equal to zero. For many-sided polygons, greater than 100 sides, we present the results of our study that shows that the systems appear to be diffusive with a transport coefficient nearly equal to that of a periodic Lorentz gas with circular scatterers at the same density. The partial van Hove function for the polygonal system has, numerically, a fractal dimension equal to that of the partial van Hove function for the periodic Lorentz gas with circular scatterers. Further, we show that a non-zero average Lyapunov exponent for the system can be defined, numerically, in spite of the fact that the actual Lyapunov exponent is zero. It is also possible to verify a relationship, valid for chaotic systems, between the diffusion coefficient, the average Lyapunov exponent, and the fractal dimension of the partial van Hove function. We also report results of a study of the transport properties and dynamical properties of a system with few-sided polygons, of less than 100 sides. These systems always appear to be super-diffusive, and non-chaotic, with a value of zero for the Lyapunov exponent. The partial van Hove function has the same fractal dimension as that for a periodic Lorentz gas with circular scatterers. For randomly oriented scatterers and scatterers with irrational angles, we construct a simple channel model that allows us to isolate individual features of the polygonal Lorentz gases and study their effects on transport properties. The systems have a value of zero for their Lyapunov exponents, and, depending on the orientation of the scatterers, the systems can appear to be either diffusive or superdiffusive. Although there does not seem to be a direct link between mathematical chaos and ordinary diffusion in these models, the non-chaotic systems show that if any such connection exists, it must be very subtle. Even a weak form of random walk motion may result in ordinary diffusion. TRANSPORT IN POLYGONAL BILLIARD SYSTEMS