Exact solution for the spatially homogeneous nonlinear Kac model of the Boltzmann equation with an external force term

Abstract

The author studies the spatially homogeneous Kac model of the nonlinear Boltzmann equation in 1+1 dimensions (velocity v and time t) when an external force term is present. The author finds a closed solution. The force term decreases exponentially with time, like constant1 exp-constant2*t, where both constants depend explicitly on the moments of the cross section. The author finds, for the relaxation to equilibrium, that the Tjon overpopulation effect depends on both the initial condition and the microscopic model of cross section. For the existence of this effect, the author establishes a criterion which is a well defined linear combination of the moments of the author cross section.