Thermodynamic time asymmetry and the Boltzmann equation

An important result of statistical mechanics is the Boltzmann equation, which describes the evolution of the velocity distribution of a gas towards the equilibrium Maxwell distribution. We introduce the Boltzmann equation by considering a dynamical model of a two-dimensional gas consisting of hard disks. We derive the Boltzmann equation for the model and compare the behavior predicted by this equation against the actual behavior of the system as observed in computer simulations. A puzzling feature of the Boltzmann equation is that although the dynamical laws governing the gas are time-reversal invariant, the behavior predicted by the Boltzmann equation is time asymmetric. We show that this time asymmetry arises from assumptions made in the derivation of the Boltzmann equation, and we use computer simulations of the model system to investigate the circumstances under which these assumptions hold.