Equilibrium fluctuation for an anharmonic chain with boundary conditions in the Euler scaling limit

TL;DR

This paper proves that fluctuations in an anharmonic oscillator chain with boundary conditions evolve according to linearized Euler equations in the hyperbolic scaling limit, even over extended time scales.

Contribution

It establishes the macroscopic evolution of equilibrium fluctuations for anharmonic chains with boundary conditions under hyperbolic scaling, including extended time scales.

Findings

Fluctuations follow linearized Euler equations with boundary conditions.

Boundary conditions are set by constant tension and fixed endpoints.

Results hold even for time scales larger than hyperbolic.

Abstract

We study the evolution in equilibrium of the fluctuations for the conserved quantities of a chain of anharmonic oscillators in the hyperbolic space-time scaling. Boundary conditions are determined by applying a constant tension at one side, while the position of the other side is kept fixed. The Hamiltonian dynamics is perturbed by random terms conservative of such quantities. We prove that these fluctuations evolve macroscopically following the linearized Euler equations with the corresponding boundary conditions, even in some time scales larger than the hyperbolic one.