Solution of the Boltzmann Equationfor Phonon Transport

Abstract

We discuss two popular methods for solving the radiative transfer equation in the field of thermal radiation, which can be used to calculate conduction on nanoscales under certain hypotheses. After a brief summary of the theory leading to a radiative transfer equation for phonons, we present the P ₁ method and the discrete ordinate method. The first is based on a global approach to transfer in which the only unknown is the local internal energy. In particular, it uses an approximate treatment of the boundary conditions and for this reason becomes somewhat inaccurate when transfer is dominated by ballistic phonons from the bounding surfaces. The second method takes into account the directional aspect of the transfer and yields better results than the P ₁ method, except near the diffusive regime. It solves a transport equation in a discrete set of directions. Integrated quantities such as the internal energy and flux are evaluated using quadrature formulas. Whereas the partial differential equation derived in the P ₁ approach can be solved by standard methods, the numerical system associated with the discrete ordinate method is more specific, particularly in cylindrical geometries.