Linear response and transport equations in interacting phonon systems

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A derivation of a transport equation for phonons, in terms of the microscopic description of an anharmonic crystal, is given. The starting point is a complete set of equations for the phonon Green function, the self-energy, and the vertex part, as given by a functional method using real times. The role of the external source field for introducing nonequilibrium into the system is discussed in detail. The essential equivalence of some recent theories is shown, which have been proposed in connection with phonon transport and second sound. The integral equation for the vertex part is related to a Boltzmann equation for a position and time dependent phonon density. For this quantity an operator, namely the Wigner operator, can be extracted from the energy density. It is shown, however, that the energy density can only approximately be expressed in terms of this phonon density. With the aid of the Wigner operator the variable phonon density is represented by a Kubo formula as the linear response of the phonon quasiparticle gas to the external source field. Cubic and quartic anharmonicities are taken into account and their influence on the collision operator and on the drift term of the transport equation is discussed.