Power Series of Kinetic Theory. I. Perturbation Expansion

In recent years intensive efforts have been made to develop, from first principles, systematic corrections to the established kinetic equations, and thereby obtain an understanding of the approach to thermal equilibrium for arbitrary macroscopic systems. These efforts, dominated by Bogoliubov's synchronization technique and "functional assumption," have met with only partial success. In fact, the method of synchronization has been shown to lead to serious difficulties when carried beyond the lowest order results, so that an $H$ theorem is lacking for the higher order terms. To discuss the problem in full generality, we construct in this paper the direct perturbation series (and in the follow paper, Bogoliubov's synchronized series) to all orders in a parameter $\epsilon$ that can be identified with the potential strength. An explicit expression is obtained for the $\nu \mathrm{th}$-order term of the $s$-body distribution function and a simple, systematic graphical representation of all the terms is derived. The result is obtained by the use of a matrix formalism that allows an effective decoupling of the Bogoliubov-Born-Green- Kirkwood- Yvon equations, and thereby, for a detailed analysis of the perturbation series. Bogoliubov's basic result concerning the secular behavior of perturbation theory ($F^{12}\sim t$) is deduced here as a special case of a general theorem: The $\nu \mathrm{th}$-order term for the $s$-body distribution grows for large times as a polynomial in time whose leading power is [$\frac{\nu }{2}$] independent of $s$.