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Energy Growth of Infinite Harmonic Chain under Microscopic Random Influence

Infinite harmonic chains of point particles with finite range translation invariant interaction have considered. It is assumed that the only one particle influenced by the white noise. We studied microscopic and macroscopic behavior of the system's energies (potential, kinetic, total) when time goes to infinity. We proved that under quite general condition on interaction potential the energies grow linearly with time on macroscopic scale, and grow as $\ln(t)$ on microscopic scale. Moreover it is turned out that the system exhibit some equipartition properties in this non equilibrium settings.