Exact time evolution of a classical harmonic-oscillator chain

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Abstract
We investigate the dynamical behavior of a classical harmonic-oscillator chain with periodic and fixed-end boundary conditions. The displacement and velocity autocorrelation functions are obtained by a recurrence relations method. We show that the finite diffusion constant and the divergence in the mean-square displacement of a tagged oscillator arise from the zero-frequency mode present in the chain with periodic boundary conditions. For the chain with fixed-end boundary conditions, the diffusion constant vanishes and there is no divergence in the mean-square displacement. These results should hold for the harmonic-oscillator model in higher dimensions.