Hydrodynamic Limit for an Hamiltonian System with Boundary Conditions and Conservative Noise

TL;DR

This paper investigates the macroscopic behavior of a chain of coupled anharmonic oscillators with boundary conditions, showing that under hyperbolic scaling, the system's distributions converge to solutions of the Euler equations, aided by stochastic perturbations.

Contribution

It establishes the hydrodynamic limit for a Hamiltonian system with boundary conditions and conservative noise, connecting microscopic dynamics to macroscopic Euler equations.

Findings

Convergence of microscopic distributions to Euler system solutions

Effective ergodic properties due to velocity exchanges

Validation of hydrodynamic limit in smooth regime

Abstract

We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is attached to a point on the left and there is a force (tension) $\tau$ acting on the right. In order to provide good ergodic properties to the system, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distributions of the elongation, momentum and energy, converge to the solution of the Euler system of equations, in the smooth regime.