Universality of thermal conduction in vibrating chains for a class of potentials

We present calculations for the relaxation rates of phonons in one-dimensional chains in which atoms interact with a class of pairwise potentials which are anharmonic with odd powers. The calculations are based on a self-consistent procedure for second order processes and lead to integral equations for the wave-vector-dependent on-shell relaxation rate Gammaq for phonons. For the cubic anharmonicity, one finds that for small q, Gammaq proportional, q3/2. The self-consistent procedure is extended to potentials with higher odd powers and one finds that the leading order behavior is still Gammaq proportional, q3/2+O(q2). With the assumption that the transport relaxation rate has the same wave-vector dependence, this result implies that the thermal conductivity, kappa diverges with the chain size, N, as kappa proportional, N1/3 for this class of potentials. Thus, our calculations provide a microscopic basis for one class of universal behavior.