On the Convergence to a Statistical Equilibrium in the Crystal Coupled to a Scalar Field

Abstract

We consider the dynamics of a field coupled to a harmonic crystal with $n$ components in dimension $d$, $d,n\ge 1$. The crystal and the dynamics are translation-invariant with respect to the subgroup ${\mathbb{Z}}^d$ of ${\mathbb{R}}^d$. The initial data is a random function with a finite mean density of energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. Moreover, initial correlation functions are translation-invariant with respect to the discrete subgroup ${\mathbb{Z}}^d$. We study the distribution ${\mu }_t$ of the solution at time $t\in \mathbb{R}$. The main result is the convergence of ${\mu }_t$ to a Gaussian measure as $t\to \infty$, where ${\mu }_{\infty }$ is translation-invariant with respect to the subgroup ${\mathbb{Z}}^d$.

33 pages

References in corpus (6)

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• Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics
• On the convergence to statistical equilibrium for harmonic crystals
• On two-temperature problem for harmonic crystals
• On Convergence to Equilibrium Distribution, I. The Klein - Gordon Equation with Mixing
• On a Two-Temperature Problem for Wave Equation

Cited by in corpus (6)

• Approach to thermal equilibrium in harmonic crystals with polyatomic lattice
• Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle
• On convergence to equilibrium for one-dimensional chain of harmonic oscillators in the half-line
• On linear stability of crystals in the Schroedinger-Poisson model
• On the convergence to a statistical equilibrium for the wave equations coupled to a particle
• On the dispersion decay for crystals in the linearized Schrödinger-Poisson model