Effect of interaction range on phonon relaxation in Fermi-Pasta-Ulam β chain

We study the effect of increasing the range of interactions on phonon relaxation in a chain of atoms with quartic anharmonicity. The study is motivated by recent numerical studies, showing that the value of the exponent α characterizing the divergence of conductivity with system size apparently depends on the presence of second neighbor couplings. We perform a quantum calculation of the wave-vector (q) dependent relaxation rate Γ(q) in the second order perturbation theory. The nonanalytic dependence of Γ(q) arises due to small-q singularity of the collision integral. We find that Γ(q)∝Aq^5∕3+Bq². This gives rise to an asymptotic value α=0.4, but the q² terms lead to a higher apparent value of α at small sizes of the chain.