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

Abstract

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 Γ_q for phonons. For the cubic anharmonicity, one finds that for small q, Γ_q∝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 Γ_q∝q^3/2+O(q²). With the assumption that the transport relaxation rate has the same wave-vector dependence, this result implies that the thermal conductivity, κ diverges with the chain size, N, as κ∝N^1/3 for this class of potentials. Thus, our calculations provide a microscopic basis for one class of universal behavior.