On the convergence to statistical equilibrium for harmonic crystals

Abstract

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

Exact Relaxation in a Class of Nonequilibrium Quantum Lattice Systems

“…Thus, our approach is considerably simpler and more physically intuitive than a potential approach based on the C * -algebraic arguments developed in Ref. [14] to study the local relaxation of free fermions and bosons freely moving in R n and classical systems [15]. Interestingly, the convergence is not only true in the time average, but actually true for any large instant of time.…”

On Two-Temperature Problem for Harmonic Crystals

“…9 we have extended the results to the wave equation with the two-temperature initial measures. The present paper develops our previous results, (10) where the harmonic crystal has been considered for all d \ 1 in the case of translation invariant initial measures. Here we extend the results to the two-temperature initial measures.…”

“…generally is a discontinuous function by (4.10). Therefore, q 10 . (x) decays as a negative power of |x|.…”

“…follow by the same arguments as in ref. 10. We apply our results to a particular case when m ± =g ± are Gibbs measures with two distinct temperatures T ± \ 0 (we adjust the definition of the Gibbs measures g ± in Section 4).…”

“…Namely, we develop our ''cutoff '' strategy from ref. 10 which more carefully exploits the mixing condition in Fourier space. This approach allows us to cover all d \ 1.…”