Small perturbations in hard balls dynamics.

摘要

This dissertation consists of three articles written in cooperation with F. Bonetto, N. Chernov, J. Lebowitz and N. Simanyi. The common theme is small perturbations of hard balls dynamics on a flat table with obstacles. While the original dynamics preserves the Lebesgue measure and is hyperbolic and ergodic for all models that we have considered, even smallest perturbations may significantly change the nature of the dynamics. Physically observable invariant measure may become singular with respect to Lebesgue measure, or even fail to exist. In some cases the dynamics collapses to a continuum of degenerate stable regimes.;In Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas we consider a single particle in a two-dimensional periodic array of convex obstacles. The particle is subject to a constant electric field as well as a Gaussian thermostat which keeps the energy constant. This model provides a local approximation to the one without thermostat. We study properties of projections of the invariant physically observable (SRB) measure—despite its singular nature, projections on the space coordinates have continuous densities.;In Stable regimes for hard disks in a channel with twisting walls we consider a system of N hard disks in a two-dimensional channel with slightly modified rules of collisions with walls. While with zero perturbation the system is ergodic (this is proven for N = 2 and expected to be true for N > 2), for arbitrarily small perturbations it tends to collapse to the attractive submanifolds of the phase space.;In Speed Distribution of N Particles in the Thermostated Periodic Lorentz Gas with a Field we study a system of N point particles moving on a two-dimensional torus with convex obstacles or random scatterers and a constant electric field. Particles are connected by a common Gaussian thermostat that keeps the kinetic energy constant. We show that for small fields the velocity distribution of the particles is independent of the nature of scattering.