Microscopic and macroscopic analysis of non-linear master equations: Vibrational relaxation of diatomic molecules

The thermodynamic and kinetic characteristics of vibrational relaxation of diatomic molecules are studied using HF gas as a model system. The non-linear master equation governing the relaxation is solved numerically using a comprehensive set of exponential gap rate constants. The results indicate a two-stage relaxation mechanism. A very fast V-V dominated stage leading to an intermediate quasi-equilibrium distribution which depends only on the initial mean number of vibrational quanta. During this stage the vibrational distribution can be described as a superposition of the initial and intermediate distributions. A second, very slow, V-T dominated stage ultimately brings the system to complete equilibrium with the heat bath. The relaxation is characterized microscopically by the time evolution of the vibrational distribution and macroscopically by the evolution of the moments. The bridge between the two levels of analysis is provided by the maximal entropy procedure. It is shown that the entropy deficiency is the only convex function which decays monotonically to equilibrium irrespective of the order of the relaxation mechanism. Using the maximal entropy form of the distribution it is shown that two moments, i.e. two macroscopic ob-servables, suffice to describe the distribution during the first stage while only a single moment is required to describe the final approach to equilibrium. During the intermediate stage more than two moments may be required.