Abstract
We consider the dynamics of a harmonic crystal in d dimensions with n components, d,n≥1. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. The random function is translation-invariant in x 1,...,x d−1 and converges to different translation-invariant processes as x d →±∞, with the distributions μ ±. We study the distribution μ t of the solution at time \(t \in \mathbb{B}\). The main result is the convergence of μ t to a Gaussian translation-invariant measure as t→∞. The proof is based on the long time asymptotics of the Green function and on Bernstein's “room-corridor” argument. The application to the case of the Gibbs measures μ ±=g ± with two different temperatures T ± is given. Limiting mean energy current density is −(0,...,0,C(T +−T −)) with some positive constant C>0 what corresponds to Second Law.
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Dudnikova, T.V., Komech, A.I. & Mauser, N.J. On Two-Temperature Problem for Harmonic Crystals. Journal of Statistical Physics 114, 1035–1083 (2004). https://doi.org/10.1023/B:JOSS.0000012516.89488.20
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DOI: https://doi.org/10.1023/B:JOSS.0000012516.89488.20