Generalization of Chapman-Enskog Solution to the Quantum Boltzmann Transport Equation and Some Applications.
The aim of this work is to develop a new method for solving the quantum Boltzmann equation for the distribution functions of both Bose-Einstein and Fermi-Dirac particles, subject to collisions in which the effect of exchange is incorporated in the collision terms. A generalization of the Chapman-Enskog method is used to solve this equation which involves development of a set of orthogonal polynomials, designated by A(,n)('m)(x), which are appropriate for solution of a transport equation with exchange. The following simplifying assumptions are made in this work: The gas is simple, monatomic and dilute, therefore, only binary collisions are considered; velocity of a molecule is independent of its position; temperature, density and velocity of particles relative to the mass motion are considered to vary slowly in space as compared to the mean free path of the molecules. Subject to the above assumptions, the Boltzmann equation is treated to first-order in temperature gradients, velocity gradients relative to the mass motion and density gradients for both Bose-Einstein and Fermi-Dirac particles. As applications, the coefficients of viscosity and thermal conductivity for monatomic gases, and the electrical conductivity for a metal are derived. Numerical results for the coefficient of thermal conductivity for a rigid -elastic sphere model of Helium gas are in good agreement with the experiment.
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- Physics: General