arXiv math-ph/02110172002-11-11math-phmath.MPmath.PR

On two-temperature problem for harmonic crystals

T. V. Dudnikova, A. I. Komech, N. J. Mauser

发表信息：: J. Stat. Phys. 114 (2004), no.3/4, 1035-1083
DOI：: 10.1023/B:JOSS.0000012516.89488.20

英文摘要：

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components,$d,n \ge 1$. 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 converges to different space-homogeneous processes as $x_d\to\pm\infty$, with the distributions $μ_\pm$. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian translation-invariant measure as $t\to\infty$. 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 $μ_\pm=g_\pm$ with two different temperatures $T_{\pm}$ 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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