Abstract
The connection between a number of derivations of the Boltzmann equation from the Liouville equation is investigated. In particular it is shown that the time-smoothed second distribution function, used by Kirkwood in his derivation of the Boltzmann equation, is identical with that obtained by Bogolubov by a systematic expansion in the two parameters 1/υ (density) and μ (uniformity) and keeping terms of O(1/υ) and of O(μ). This leads not only in both cases to the Boltzmann equation but also to identical correction terms of O(μ) on this equation. A similar analysis is carried out as regards to the derivations of Yvon, Born and Green, Hollinger and Curtiss, Schönberg and Frisch. Contrary to the methods of Bogolubov and Kirkwood, these derivations do not, however, introduce explicitly or correctly coarse-grained distribution functions.
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