Entropy, information theory, and the approach to equilibrium of coupled harmonic oscillator systems

Finite segments of infinite chains of classical coupled harmonic oscillators are treated as models of thermodynamic systems in contact with a heat bath, i.e., canonical ensembles. The Liouville function ρ for the infinite chain is reduced by integrating over the "outside" variables to a function ρ_N of the variables of the N-particle segment that is the thermodynamic system. The reduced Liouville function ρ_N which is calculated from the dynamics of the infinite chain and the statistical knowledge of the coordinates and momenta at t = 0, is a time-dependent probability density in the 2N-dimensional phase space of the system. A Gibbs entropy defined in terms of ρ_N measures the evolution of knowledge of the system (more accurately, the growth of missing pertinent information) in the sense of information theory. As |t