Coherent States and Irreversible Processes in Anharmonic Crystals

Abstract

Irreversible phenomena involving phonons in a slightly anharmonic crystal are investigated using coherent states as a basis. This basis retains the advantages of the classical theory, in which phase relations are clearly exhibited, for a quantum-mechanical system. In the coherent-state basis the equation of motion for the density matrix has an obvious correspondence with the classical Liouville equation. In particular, the connection with the action-angle variables of Brout and Prigogine is elucidated. A new, rapid proof of the Brout-Prigogine equation is given using the method of semi-invariants. This technique exhibits higher-order corrections in a usable form. The quantum corrections to the classical equation of motion for the density function are shown to be due to extra terms involving second derivatives in the action-angle variables. The Peierls master equation is then derived from the Brout-Prigogine equation. The important problem of elastic scattering of phonons is treated by our method in an appendix.