Integral Equations in the Kinetic Theory of Gases and Related Topics

Abstract

Integral equations occur in many areas of chemistry, physics and engineering. We consider in this chapter the integral equations that arise in radiative transfer theory and in the study of transport processes in dilute gases modeled with the Boltzmann equation. The first use of a collocation was the Gauss-Legendre quadrature for the solution of the integro-differential isotropic radiative transfer equation. The integral equations that are used to calculate the heat conductivity and viscosity of a dilute monatomic gas are derived with the Chapman-Enskog method of solution of the Boltzmann equation. The integral equations are solved with spectral and pseudospectral methods. These numerical methods are also used to calculate the eigenfunctions and eigenvalues for the linearized collision operator for a one component gas as well as for the linear collision operator for a binary mixture. The solution of the Boltzmann equation for many applications can be expressed in terms of the eigenfunctions and eigenvalues of the collision operators that in general possess an infinite number of discrete eigenvalues and a continuum. The eigenvalue spectra of these operators are calculated and discussed. A pseudospectral method of solution of the Boltzmann integral equation is used for the calculation of the nonequilibrium reaction rate for a model reactive system. A pseudospectral method is also used to solve the Chapman-Enskog integral equation that gives the viscosity of a dilute gas. The relaxation to equilibrium of an initial anisotropic nonequilibrium distribution for a binary gas mixture versus the mass ratio of the two components is studied. Also presented are the spectral solutions of Boltzmann equation for the Milne problem of rarefied gas dynamics, the escape of light atoms from a planetary atmosphere and the calculation of ion mobilities. Pseudospectral methods with nonclassical weight functions are used in some of these applications. The chapter concludes with the study of the relaxation to equilibrium of a one component gas as described by the nonlinear isotropic Boltzmann equation.