We consider a d-dimensional harmonic crystal, d ⩾ 1, and study the Cauchy problem with random initial data. The distribution μ_t of the solution at time t ∈ ℝ is studied. We prove the convergence of correlation functions of the measures μ_t to a limit for large times. The explicit formulas for the limiting correlation functions and for the energy current density (in the mean) are obtained in terms of the initial covariance. Furthermore, we prove the weak convergence of μ_t to a limit measure as t → ∞. We apply these results to the case when initially some infinite “parts” of the crystal have Gibbs distributions with different temperatures. In particular, we find stationary states in which there is a constant nonzero energy current flowing through the crystal. We also study the initial boundary value problem for the harmonic crystal in the half-space with zero boundary condition and obtain similar results.

中文翻译：

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无限次谐波晶体的平稳态和能量流的收敛

我们考虑一个d维谐波晶体，d ⩾1，与随机初始数据研究柯西问题。分布μ_吨在时刻溶液的吨∈ℝ进行了研究。我们证明的措施的相关函数的收敛μ_牛逼为大时代的限制。根据初始协方差获得了极限相关函数和能量电流密度（平均值）的明确公式。此外，我们证明了弱收敛μ_牛逼的限制措施，因为ŧ→∞。我们将这些结果应用于最初的晶体的无限“部分”具有不同温度的吉布斯分布的情况。特别是，我们发现其中有恒定的非零能量电流流过晶体的稳态。我们还研究了零边界条件下半空间中谐波晶体的初始边值问题，并获得了相似的结果。