# On the energy current for harmonic crystals

## TL;DR

This paper analyzes the long-term behavior of energy currents in a $d$-dimensional harmonic crystal with random initial data, deriving explicit formulas for correlation functions and energy flow, including stationary states with non-zero energy currents.

## Contribution

It provides explicit formulas for the limiting correlation functions and energy current density in harmonic crystals, including cases with non-zero stationary energy flow.

## Key findings

- Correlation functions converge to explicit limits over time
- Explicit formulas for energy current density are derived
- Stationary states with constant non-zero energy flow are identified

## Abstract

We consider a $d$-dimensional harmonic crystal, $d\ge 1$, 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 $x_1,\dots,x_k$ with some $k\in\{1,\dots,d\}$. The distribution $\mu_t$ of the solution at time $t\in\mathbb{R}$ is studied. We prove the convergence of correlation functions of the measures $\mu_t$ 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 $\mu_t$ 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.