Hyperbolic scaling limit of non-equilibrium fluctuations for a weakly anharmonic chain

TL;DR

This paper studies the non-equilibrium fluctuations of a weakly anharmonic oscillator chain, showing convergence to a linear hyperbolic system under certain conditions and providing bounds on the convergence speed.

Contribution

It establishes the hyperbolic scaling limit of fluctuations for a weakly anharmonic chain with vanishing anharmonicity, using relative entropy techniques.

Findings

Fluctuation fields converge to a linear p-system in the hyperbolic limit.

Convergence speed to hydrodynamic limit is quantitatively bounded.

Transition speed remains spatially homogeneous due to weak anharmonicity.

Abstract

We consider a chain of $n$ coupled oscillators placed on a one-dimensional lattice with periodic boundary conditions. The interaction between particles is determined by a weakly anharmonic potential $V_n = r^2/2 + \sigma_nU(r)$, where $U$ has bounded second derivative and $\sigma_n$ vanishes as $n \to \infty$. The dynamics is perturbed by noises acting only on the positions, such that the total momentum and length are the only conserved quantities. With relative entropy technique, we prove for dynamics out of equilibrium that, if $\sigma_n$ decays sufficiently fast, the fluctuation field of the conserved quantities converges in law to a linear $p$-system in the hyperbolic space-time scaling limit. The transition speed is spatially homogeneous due to the vanishing anharmonicity. We also present a quantitative bound for the speed of convergence to the corresponding hydrodynamic limit.