Random Matrix Theory and the Master Equation for Finite Systems

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Abstract
In this paper we consider the justification for the use of a stationary Markov assumption to describe vibrational relaxation and dissociation in isolated, highly excited, polyatomic molecules. We discuss first a model in which only a single oscillator is allowed to break, the remaining vibrations serving as a heat bath which interacts weakly with the reactive degree of freedom. A random matrix hypothesis is used to derive a time‐independent form for the rates of internal energy exchange. In the limit of very ``hot'' molecules these transition probabilities are shown to have the expected linear and quadratic dependences on the individual vibrational quantum numbers. The constants of proportionality are simple functions of the size of the molecule and the interaction strengths coupling the nuclear motions. Finally, we return to the full dynamical theory of unimolecular reactions which was introduced in an earlier paper [J. Chem. Phys. 52, 5718 (1970)] as an alternative to the conventional statistical (e.g., transition‐state) approaches. We use the Zwanzig projection operator formalism to derive the sufficient conditions for which Liouville's equation for internal energy exchange assumes the Markov form appropriate to an irreversible relaxation process. The familiar Green's function (resolvent operator) techniques are generalized to the case of tetradic propagators and a stochastic description is shown to be justified for time scales which correspond to experimental observation of the vibrational dissociation.