Momentum conservation implies anomalous energy transport in 1D classical lattices

Under quite general conditions, we prove that for classical many-body lattice Hamiltonians in one dimension (1D) total momentum conservation implies anomalous conductivity in the sense of the divergence of the Kubo expression for the coefficient of thermal conductivity, kappa. Our results provide rigorous confirmation and explanation of many of the existing "surprising" numerical studies of anomalous conductivity in 1D classical lattices, including the celebrated Fermi-Pasta-Ulam problem.