Statistical-Mechanical Theory of the Boltzmann Equation and Fluctuations in μ Space

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The mean evolution and fluctuations of the particle density in μ space of low-density gases are rigorously formulated by using three new viewpoints recently developed. The first is to introduce a spatially coarse-grained particle density A(p, r; t) in μ space, the shortest distance b of whose spatial variation satisfies l_f≫b≫r_0, l_f being the mean free path and r_0 the diameter of molecule. The second is to use a time-convolutionless nonlinear reduced equation of motion for A(p, r; t). The third is to single out the kinetic processes with length scale of order l_f with the aid of the scaling r(>__=b)→Lr, l_f→Ll_f, r_0→r_0, c→c/L with L≫1, c being the mean particle density. Thus it is shown that the Boltzmann equation for the causal part of A(p, r; t), the linear Boltzmann-Langevin equation with a Gaussian white fluctuating force and the Gaussian distribution law for the fluctuating part hold in the following limit : c→o, b→∞, t→∞, cb^d→∞, (d>1) with cb and ct being kept constant, where d is the dimensionality. The present theory does not use any assumption like Bogoliubov's functional postulate, Kirkwood's time smoothing and the product assumption.
理論物理学刊行会の論文
1976-10-25