On stochastic reductions in molecular collision theory: Projection operator formalism; application to classical and quantum forced oscillator model

A method is developed for treating complex molecular collision processes through the application of stochastic reduction formalisms. We begin by describing a projection operator method for decomposing a complicated collision system into two (or more) subsystems, each of which is assumed to be weakly correlated (not necessarily weakly interacting) with the others. Approximations to this correlation are then introduced, and this results in a set of coupled equations for the reduced density operators (or classical phase space distributions) associated with each subsystem. We then examine the classical mechanical application of this theory to the forced oscillator model of V-T energy transfer. Arguments of multiple time scales are used to uncouple the stochastically reduced equations of motion, and thus we may evaluate the memory kernel analytically. This leads to a single diffusion equation for the time evolution of the action in the oscillator during the collision. Comparison with the corresponding exact results indicates excellent agreement of low order moments of the classical distributions of action in the limit of small energy transfer (i.e., ΔE/ℏω<1). Of particular note is the fact that our stochastic theory predicts an average energy transfer (first moment) in exact agreement with the exact result independent of magnitude of the energy transfer. In a related application of our general stochastic formalism, we consider the quantum mechanical forced oscillator model. This problem is treated in two different ways: (a) through the use of reduced density matrices (which leads to master equations), and (b) through the Wigner equivalent formalism (which is formally analogous to the classical treatment). The resulting transition probabilities obtained from these two equivalent applications are shown to be identical. Comparison of stochastic and exact quantum results indicates quantitative agreement of the probabilities for ΔE/ ℏω≤0.1, and average agreement of the probabilities for larger values of the energy transfer.