On the convergence to statistical equilibrium for harmonic crystals

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components, $d,n$ arbitrary, $d,n\ge 1$, and study the distribution $\mu_t$ of the solution at time $t\in\R$. The initial measure $\mu_0$ has a translation-invariant correlation matrix, zero mean, and finite mean energy density. It also satisfies a Rosenblatt- resp. Ibragimov-Linnik type mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green's function and on Bernstein's ``room-corridors'' method

Towards Time - Dynamics for Bosonic Systems in Quantum Statistical Mechanics

Consider a one-dimensional lattice boson system with the Hamiltonian in a finite box Λ, H_Λ = K_Λ + U_Λ. Here K_Λ is the kinetic energy and U_Λ is the potential energy corresponding to a finite-range pair interaction. For a class of states T of the infinite system, we prove the existence of the limit T_t(A) = lim_(Λ→Z) T(e^(itH_Λ)*Ae^(-itH_Λ)) for any t ϵ R^4 and any local observable A. Thereby a family {T_t, t ϵ R^4} of locally normal states is determined which describes the time-evolution of the initial state T

Estabilización de la solución estadística de la ecuación parabólica.

We study the convergence of the statistical solutions of the parabolic equation. Under some mixing condition (in the sense of Rosenblatt) for initial measure and natural assumptions on the coefficients of the equation we prove weak convergence to the Gaussian distribution. Similar results for the hyperbolic equations were obtained in [1–4]