Possibility of direct solution of the classical Liouville equation for inelastic molecular collisions; the reduced Liouville equation

Starting with the ordinary classical Liouville equation for the time evolution of phase space distribution functions, total energy conservation is used to obtain a reduced Liouville equation which determines the steady-state (i.e., time independent) reduced distribution function on the ''energy shell'' in phase space. Boundary conditions which correspond to time-independent scattering theory are easily imposed, and one sees clearly how to extract the time-independent transition probability matrix for a given total energy; this is the classical analog of the time-independent on-shell S matrix of quantum scattering theory. The reduced Liouville equation is of a form that is amenable to direct numerical solution, and ways for approaching this are described. A particular approximate version of the reduced Liouville equation is seen to be equivalent to a recently proposed stochastic model