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

Abstract

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 p for the infinite chain is reduced by integrating over the "outside" variables to a function pN of the variables of the N-particle segment that is the thermodynamic system. The reduced Liouville function pn, 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 pN measures the evolution of knowledge of the system (more accurately, the growth of missing pertinent information) in the sense of information theory. As p t [ -+ 0% energy is equipartitioned, the entropy evolves to the value expected from equilibrium statistical mechanics, and p~evolves to an equilibrium distribution function. The simple chain exhibits diffusion in coordinate space, i.e., Brownian motion, and the diffusivity is shown to depend only on the initial distribution of momenta (not of coordinates) in the heat bath. The harmonically bound chain, in the limit of weak coupling, serves as an excellent model for the approach to equilibrium of a canonical ensemble of weakly interacting particles.