Decay of Correlations. II. Relaxation of Momentum Correlations and Momentum Distributions in Harmonic Oscillator Chains

Abstract

We have studied the relaxation of the n-particle momentum distribution function fn(pn; t) and the corresponding n-particle Ursell function Un(pn; t) for both an isolated infinite chain of coupled harmonic oscillators and an infinite chain of coupled harmonic oscillators in contact with a heat bath. The time development of the Ursell function gives us the desired information on the decay of initial momentum correlations. For both of these systems we have found the asymptotic relaxation laws fn(pn; t) − fn(0)(pn) ∼ f1(p; t) − f1(0)(p) and Un(pn; t)→0 as [f1(p; t) − f1(0)(p)]n, where the superscript zero denotes the equilibrium distribution. The momentum correlation function Un(pn; t) thus goes to zero (no correlations) much more rapidly than the n-particle distribution function approaches its equilibrium value. These results are identical with those obtained by us previously for initially correlated, noninteracting many-particle systems in contact with a heat bath and by Glauber for the relaxation of spins in an Ising lattice. It is interesting to speculate on the reason for this identity of results in such apparently different relaxation models, but no answer can be provided as yet.