О стационарных неравновесных состояниях линейных гамильтоновых систем

In the talk, we discuss the long-time behavior of distributions of solutions for infinite-dimensional Hamiltonian systems and the existence of a nonzero heat flux in them. As a model, we consider a linear Hamiltonian system consisting of a real scalar Klein-Gordon field coupled to an infinite harmonic crystal. This system can be considered as the description of the motion of electrons (so-called Bloch electrons) in the periodic medium that is generated by the ionic cores. For the coupled system, we study the Cauchy problem with random initial data. We prove that the distributions of the solutions weakly converge to a limiting measure for large times. Under the condition that the initial random function in the “left” and “right” parts of the space has the Gibbs distribution with different temperatures, we find the stationary states (i.e., the probability limiting measures) of the system in which the limiting energy current density does not vanish. Thus, for this system we construct a class of stationary non-equilibrium states.