Entropy and macroscopic disequilibrium. I. Isothermal time evolution with applications to vibrational relaxation

A practical method for using bulk averaged values of observables for the characterization and prediction of the molecular population time evolution during isothermal relaxation is presented. In practical applications to realistic examples of vibrational relaxation very few bulk averages were required to accurately predict the population distribution even when the initial population was very strongly inverted. The time dependence of the macroscopic observables which are employed as input is conveniently formulated in terms of sum rules. The bulk average values are used as constraints in a maximal entropy procedure for the determination of the population distribution. It is shown that the procedure is of a variational type. Monotonic convergence of the information theoretic predicted distribution to the exact one is guaranteed upon inclusion of additional macroscopic input. The concept of "independent moments" is introduced for this purpose. Only independent observables are informative, i.e., provide independent data which are required for convergence. The number of informative observables decreases with time and is typically very much smaller than the number of significantly populated molecular energy states. The method is illustrated by comparing its predictions to the results of a numerical solution of the master equation, with a realistic set of rate constants and for different initial conditions. The application of the surprisal analysis to the interpretation, characterization, and compaction of the population distribution is demonstrated. Turning to predictions ("surprisal synthesis"), only strongly inverted initial populations required three independent moments, during the initial stages. Over much of the relaxation a single moment ("vibrational temperature") sufficed for an accurate prediction. The limits where the characteristic vibrational temperature is high or low compared to the temperature of the buffer gas are discussed. Special reference is made to rotational relaxation. The rate of internal entropy production due to the irreversible relaxation and the rate of increase of the global entropy are discussed and shown to be positive.