Temperature relaxation in a binary gas. II. Time dependent solution

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Abstract
The time dependent Boltzmann equations describing the temperature relaxation in a hard‐sphere gas are solved and the explicit time dependence of the distribution functions is determined. The time evolution of the perturbations of the distribution functions is studied in detail with a moment method. The range of validity of the earlier results obtained with a steady state assumption is determined. Provided that the intial temperature ratio is neither too large nor too small (102–10−2), it is found that the steady state assumption is valid only in the extreme disparate mass limit, i.e., m1/m2 of the order of 10−3–10−5. Qualitatively it appears that the ratio of the self‐relaxation times to the temperature equilibration time must be of the order of 10−3–10−4 or smaller for a steady state to occur. Since the temperature relaxation rate is slow for this range of mass and initial temperatures, there is an extremely small perturbation of the distribution functions from the Maxwellian form. Perturbations of the distribution function which result in departures of a few percent from the equilibrium estimates of the temperature ratio or the temperature difference are found to occur for only the fastest relaxation rates. Estimates of these small departures from equilibrium based on the steady state assumption are very much in error.