HARMONIC CRYSTALS: STATIONARY NON-EQUILIBRIUM STATES AND DERIVATION OF THE ENERGY TRANSPORT EQUATION

This paper will discuss open problems in nonequilibrium statistical physics.Using one model (an infinite crystal lattice in the harmonic approximation) as an example, the following three problems will be considered: 1. Stabilization of statistical solutions at long times. For the model under study, the Cauchy problem is considered. Assuming that the initial data are a random function, the asymptotic behavior of the distributions of random solutions at long times is studied, and their convergence to a Gaussian distribution is demonstrated. 2. Existence of nonequilibrium states. The states of the system under study are understood as Borel probability measures on a suitable phase space. Nonequilibrium states are those in which there is a nonzero heat flux passing through the system. A class of stationary nonequilibrium states will be constructed for the model under consideration. 3. Derivation of macroscopic evolution equations (such as the energy transport equation, the Euler and Navier-Stokes equations) from the Hamiltonian dynamics of interacting particles. Applying R.L. Dobrushin's approach and the methods of the problems listed above, a hydrodynamic description of this model will be constructed. It will be shown that the Wigner limit function satisfies the energy transport equation.