Time and Space Eigenvalues of the Boltzmann Equation with a Synthetic Thermalization Kernel

Abstract

The time and space eigenvalues of the Boltzmann equation have been obtained, particular attention being given to those eigenvalues which lie close to the limit point. This has been possible by the use of a synthetic kernel, which converts the usual integral equation to a differential one: the solution of this equation is obtained by the W.K.B. method.Results have been obtained for the infinite and finite medium time eigenvalues in the gas model approximation. The eigenvalues of the scattering operator have been shown to be infinite in number—also for the gas model. For the space eigenvalues it has been shown that, for a proton gas, only the fundamental exists, all higher eigenvalues are absent. It is found that as the mass of the gas increases, more space eigenvalues appear, but for any gas of finite mass these are finite in number.