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 ${\Gamma }_q$ for phonons. For the cubic anharmonicity, one finds that for small $q$, ${\Gamma }_q\propto q^{3∕2}$. The self-consistent procedure is extended to potentials with higher odd powers and one finds that the leading order behavior is still ${\Gamma }_q\propto q^{3∕2}+O\left(q^2\right)$. 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 $\kappa \propto N^{1∕3}$ for this class of potentials. Thus, our calculations provide a microscopic basis for one class of universal behavior.

• Received 21 September 2007

DOI:https://doi.org/10.1103/PhysRevE.77.011113