Convergence to Stationary States and Energy Current for Infinite Harmonic Crystals

Abstract

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.