Stochastic oscillators out of equilibrium: scaling limits and correlation estimates

Abstract We consider a chain of oscillators, which is driven by a purely harmonic potential, and is perturbed by a stochastic noise that exchanges two neighboring states, on the one-dimensional lattice with periodic boundary conditions. Under this perturbation, the system exhibits two conserved quantities: volume and energy. Our main result is the derivation of precise bounds on the two-point volume correlations and a uniform fourth-moment estimate for the volume. As a consequence, we are able to rigorously derive the hydrodynamic limit for both volume and energy. More precisely, under diffusive scaling, when the width of the lattice goes to zero, we show that, depending on the strength of the Hamiltonian dynamics, energy and volume evolve according to either a system of autonomous heat equations or a nonlinear system of coupled parabolic equations. Moreover, for general initial measures, again under diffusive scaling, we fully derive the non-equilibrium fluctuations for the volume.