Exact Equilibration of Harmonically Bound Oscillator Chains

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Abstract
The approach to equilibrium of a finite segment of an infinite chain of harmonically coupled, harmonically bound oscillators is treated exactly, both when the initial description of the rest of the chain is canonical and when it is Gaussian. The necessary mathematical properties of the bound oscillator functions are developed and used to demonstrate exact equipartition of energy. The entropy of the finite segment, or system, is shown to evolve to a time-independent equilibrium state that is, in the limit of weak coupling, the correct one for a system of noninteracting harmonic oscillators.