On two-temperature problem for harmonic crystals

Abstract

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components,$d,n\ge 1$. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_d\to \pm \infty$, with the distributions ${\mu }_{\pm }$. 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 translation-invariant measure as $t\to \infty$. The proof is based on the long time asymptotics of the Green function and on Bernstein's `room-corridor' argument. The application to the case of the Gibbs measures ${\mu }_{\pm }=g_{\pm }$ with two different temperatures $T_{\pm }$ is given. Limiting mean energy current density is $-(0,...,0,C(T_{+}-T_{-}))$ with some positive constant $C>0$ what corresponds to Second Law.

References in corpus (4)

• Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics
• On the convergence to statistical equilibrium 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 (11)

• Fast and slow thermal processes in harmonic scalar lattices
• Local stationarity for lattice dynamics in the harmonic approximation
• Harmonic Crystals in the Half-Space, I. Convergence to Equilibrium
• On the Convergence to a Statistical Equilibrium in the Crystal Coupled to a Scalar Field
• On convergence to equilibrium for one-dimensional chain of harmonic oscillators in the half-line
• Lattice Dynamics in the Half-Space, II. Energy Transport Equation
• Lectures on Quantum Mechanics (nonlinear PDE point of view)
• On the Convergence to a Statistical Equilibrium for the Dirac Equation
• On the convergence to a statistical equilibrium for the wave equations coupled to a particle
• Space-time statistical solutions for an inhomogeneous chain of harmonic oscillators
• On the energy current for harmonic crystals