Abstract
A Kubo formula for the lattice thermal conductivity of the anharmonic crystal is derived. In the limit of weak anharmonicity it can be simplified by using van Hove's formal techniques for treating the time evolution of dissipative systems. It then reduces to the phonon-Boltzmann equation of Peierls. Since the system is inhomogeneous, a local representation in terms of wave packets formed from phonon states is explicitly constructed and used. The question of smoothness of the matrix elements of the anharmonic phonon interaction is discussed. It is pointed out that some of the approximations made in van Hove's theory for many-particle systems are not in general valid because strong cancellations occur among the terms assumed there to be dominant. This is related to a certain lack of invariance of the “diagonal singularity” property. The results however are seen to be correct, at least for weak coupling, except in cases where phonon branches or, in the case of electrical conductivity, electronic bands are degenerate over an appreciable volume of k-space.
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