Brownian motion in a non-isothermal solid

A general method for evaluating the friction constants of the non-isothermal Fokker-Planck equation is derived in the weak anharmonic-coupling limit for a solid, subject to an assumption that suitable angle-action variables can be found. One contribution is reduced to the solution of an integral equation of the Peierls-Boltzmann type and can be used to calculate the phonon scattering contribution q_p * to the heat of transport of a heavy impurity. The structure of the results for a one-dimensional nearest-neighbour chain, for which suitable angle-action variables can be found, is compared with that for a different theory of q_p * based on the special assuption that crystal momentum can be equated to particle momentum, and is found to be different. It is concluded that this special assumption can be dispensed with if suitable angle-action variables can be found, but to accomplish this for a stable three-dimensional crystal appears very difficult. The results for the one-dimensional model, where such variables are easily found, probably have little relevance to real crystals.