2017/06/16 by T. V. Dudnikova, Dudnikova, T. V.
Mathematics · Physics and Astronomy · #35Lxx #37A25 #37K60 #60Fxx #60G60 #82Cxx #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #math-ph #math.MP #msc:35Lxx #msc:37A25 #msc:37K60 #msc:60Fxx #msc:60G60 #msc:82Cxx
paper · pdf · doi:10.48550/arxiv.1706.06429
29 pages; added examples in Secs 3,4; corrected typos; results unchanged
openalex publication_date 2017/06/16 · arxiv created 2018/04/16 · arxiv updated 2018/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a d-dimensional harmonic crystal, d≥ 1, and study the Cauchy problem with random initial data. We assume that the random initial function is close to different translation-invariant processes for large values of x1,…,xk with some k∈\1,…,d\. The distribution μt of the solution at time t∈ℝ is studied. We prove the convergence of correlation functions of the measures μt to a limit for large times. The explicit formulas for the limiting correlation functions and for the energy current density (in mean) are obtained in the terms of the initial covariance. We give the application to the case of the Gibbs initial measures with different temperatures. In particular, we find stationary states in which there is a constant non-zero energy current flowing through the harmonic crystal. Furthermore, the weak convergence of μt to a limit measure is proved. We also study the initial boundary value problem for the harmonic crystal with zero boundary condition and obtain the similar results.