Diffusive hydrodynamic limit for general anharmonic oscillator chains

Establish the diffusive hydrodynamic limit, with macroscopic energy transport described by the heat equation, for typical anharmonic oscillator chains whose energy current is non-gradient, beyond the equilibrium-fluctuation setting currently treated by adaptations of Varadhan’s non-gradient method.

Background

The paper explains that entropy methods face serious difficulties for oscillator chains with conservative energy-preserving noise because the microscopic energy current can grow at the same order as the local energy, preventing the higher-moment estimates required by standard cutoff arguments. Although analogous difficulties can be bypassed at the hyperbolic scale using relative entropy methods, the diffusive scale is substantially harder.

For typical anharmonic chains, the energy current is non-gradient. The cited prior work addresses only equilibrium energy fluctuations through adaptations of Varadhan’s non-gradient method, leaving the general diffusive hydrodynamic limit unresolved. The present article instead proves a law of large numbers for the empirical energy distribution of a pinned harmonic chain with random velocity flips, so it does not resolve the broader anharmonic-chain problem.