Current fluctuations for anharmonic chains in thermal equilibrium

We study the total current correlations for anharmonic chains in thermal equilibrium, putting forward predictions based on the second moment sum rule and on nonlinear fluctuating hydrodynamics. We compare with molecular dynamics simulations for hard collision models. For the first time we investigate the full statistics of time-integrated currents. Generically such a quantity has Gaussian statistics on a scale $\sqrt{t}$. But if the time integration has its endpoint at a moving sound peak, then the fluctuations are suppressed and only of order $t^{1\mathrm{/}3}$. The statistics is governed by the Baik-Rains distribution, known already from the fluctuating Burgers equation.