Theory of Thermal Conductivity of Anharmonic Crystals

The thermal conductivity $\kappa$ of an anharmonic crystal is calculated based on the Green-Kubo linear-response theory, using the Zwanzig-Mori projection-operator method. The appropriate heat-current autocorrelation function is determined in the Van Hove long-time weak-coupling limit and is shown to be real and of exponential form. The final result for $\kappa$ has the well-known Debye-Peierls form, as expected. The relaxation time is shown to be the half-interval Fourier transform of the time correlation of a function which describes linear coupling of the phonon modes averaged over the harmonic lattice.