The solution to the 4-phonon Boltzmann equation for a 1D chain in a thermal gradient

Abstract

The phonon distribution function in a weakly anharmonic one-dimensional chain in a thermal gradient is determined in the high temperature or, equivalently, classical regime. Thermal relaxation is dominated by 4-phonon scattering processes in this limit and these are treated using Peierls' Boltzmann equation approach. An analytical analysis of the Boltzmann equation for small wave vector k shows that the distribution is singular and that this singularity at k = 0 is of the form of an infinite sum of power law branch points with powers that are rational and dense on the real axis above some critical value. The value of the smallest discrete power implies that the thermal conductivity is infinite in the thermodynamic limit. For finite systems, whenever 4-phonon scattering is the dominant mechanism inhibiting energy transport, the thermal conductivity will scale as (system length)2/5/(temperature)6/5.