Momentum conservation implies anomalous energy transport in 1D classical lattices

Under quite general conditions, we prove that for classical many-body latti ce Hamiltonians in one dimension (1D) total momentum conservation implies a nomalous conductivity in the sense of the divergence of the Kubo expression for the coefficient of thermal conductivity, kappa. Our results provide ri gorous confirmation and explanation of many of the existing "surprising" nu merical studies of anomalous conductivity in 1D classical lattices, includi ng the celebrated Fermi-Pasta-Ulam problem.