EXPONENTIALLY LONG EQUILIBRIUM TIMES IN A ONE-DIMENSIONAL COLLISIONAL MODEL OF CLASSICAL GAS

Abstract

Around 1900, J. H. Jeans suggested that the “abnormal” specific heats observed in diatomic gases, specifically the lack of contribution to the heat capacity from the internal vibrational degrees of freedom, in apparent violation of the equipartition theorem, might be caused by the large separation between the time scale for the vibration and the time scale associated with a typical binary collision in the gas. We consider here a simple 1D model and show how, when these time scales are well separated, the collisional dynamics is constrained by a many- particle adiabatic invariant. The effect is that the collisional energy exchanges betgween the translational and the vibrational degrees of freedom are slowed down by an exponential factor (as Jeans conjectured). A metastable situation thus occurs, in which the fast vibrational degrees of freedom effectively do not contribute to the specific heat. Hence, the observed “freezing out” of the vibrational degrees of freedom could in principle be explained in terms of classical mechanics. We discuss the phenomenon analytically, on the basis of an approximation introduced by Landau and Teller (1936) for a related phenomenon, and estimate the time scale for the evolution to statistical equilibrium. The theoretical analysis is supported by numerical examples.