Solutions of the nonlinear Boltzmann equation describing relaxation to equilibrium

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Abstract
A moment method is used to obtain exact solutions of the nonlinear Boltzmann equation for the relaxation to equilibrium of a spatially uniform gas of Maxwell-model molecules. The results contradict a recent conjecture of Krook and Wu concerning similarity solutions. Unless the initial conditions are very special, long-time relaxation follows linear modes corresponding to solutions of the linearized Boltzmann equation, rather than similarity solutions. Application of the moment method to other collision models gives approximate rather than exact results, but shows that there is coupling between the linear relaxation modes, which are independent for the Maxwell model. At long times all moments, therefore, decay together along the lowest linear mode. Numerical calculations for rigid spheres illustrate this behavior.