  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 $t^{1/3}$. The statistics is governed by the Baik-Rains distribution,
known already from the fluctuating Burgers equation.
