Solution of the Linearized Phonon Boltzmann Equation

Abstract

The linearized Boltzmann equation for the pure phonon field may be solved formally in terms of the eigenvectors of the normal-process collision operator. This representation is particularly convenient as a basis for solutions, since in the isotropic dispersionless case the temperature deviation $\ensuremath{\delta}T$ and the heat current Q are related to zero-eigenvalue eigenfunctions of this operator. The formal solution is summarized by two macroscopic equations relating $\ensuremath{\delta}T$ and Q. The first of these is the usual thermal-energy conservation condition; the second is a generalized phonon-thermal-conductivity relation involving a k- and $\ensuremath{\Omega}$-dependent thermal conductivity $\ensuremath{\kappa}(\mathbf{k},\ensuremath{\Omega})$. Examination of $\ensuremath{\kappa}(0,0)$ clarifies the role of normal processes and momentum-relaxing $R$ processes in determining the steady-state heat current. An alternative to the Callaway equation for the thermal conductivity is obtained. Examination of $\ensuremath{\kappa}(\mathbf{k},\ensuremath{\Omega})$ leads to a discussion of space-time-dependent phenomena in a phonon gas. A set of macroscopic equations which describe second sound with damping and Poiseuille flow are obtained. Second sound from the linear-response point of view discussed by Griffin is considered briefly. In the companion paper the problem of Poiseuille flow in a phonon gas is dealt with in considerable detail using these equations. The pure phonon field in a harmonic crystal is characterized by zero expectation value of the density variation of the crystal. However, in addition to the pure phonon field one may also have an elastic dilatation field in the harmonic approximation, which does lead to periodic density variation. Anharmonic effects will couple the phonon field and the dilatation field, leading to a coupling between elastic (sound waves) and thermal waves. The coupled-field dispersion relations are discussed.