Kinetic Theory of a Weakly Coupled Gas

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Abstract
Using the multiple‐time‐scale method on the BBGKY hierarchy, the weak coupling expansion is carried out to higher orders. It is found that there are two local breakdowns of the expansion. One occurs at small relative velocities between particles. The correct asymptotic representation for the small relative velocity region is given. The second breakdown occurs for particles having a large separation at t with their relative velocity oriented in such a way that they were in collision at t = 0. Such a breakdown indicates that in contrast to the Bogoliubov functional assumption, the higher‐order correlation functions should vary on the kinetic time scale in their own right. A sufficient condition on the smoothness of the initial correlation functions is given such that one obtains the Fokker‐Planck equation at the lowest‐order approximation in the expansion. The connection between irreversibility and the requirement of nonsecularity in the multiple‐time‐scale formulation is also indicated.

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