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.