Nonlinear fluctuations for a chain of weakly anharmonic oscillators with stochastic perturbation

We study the fluctuations of the phonon modes in a one-dimensional chain of anharmonic oscillators where the deterministic Hamiltonian dynamics is perturbed by random exchanges of momentum between nearest neighbor particles. There are three locally conserved quantities: volume, momentum and energy. We study the evolution in equilibrium of the fluctuation fields of the two phonon modes (linear combination of the volume stretch and momentum), on a diffusive space-time scale after recentering on their sound velocities. We show that, weakening the anharmonicity with the scale parameter, the recentered phonon fluctuations fields converge to the stationary solutions of two uncoupled stochastic Burgers equations. The nonlinearity in the Burgers equation depends on the presence of a cubic term in the anharmonic potential (corresponding to the $\alpha$-FPUT dynamics). Main ingredients of the proof, based on a compactness argument for the Dynkin's martingale decomposition, are the second-order Boltzmann-Gibbs principle, as well as equipartition of energy, to characterize the nonlinear term and Riemann-Lebesgue estimates showing that fields with diverging velocity to different directions have no interaction in the limit.