Power Series of Kinetic Theory. II. Expansion with the Functional Ansatz

Abstract

We discuss the synchronized expansion of the Liouville equation (i.e., the expansion with the "functional ansatz"). We utilize the multiple-time-scale method and the equivalence theorem proven in preceding paper, which specializes this method to Bogoliubov's functional method. The technique is shown to successfully eliminate the secular behavior discussed in the preceding paper, to all orders in the series expansion for the distribution function of an arbitrary number of particles. However, new divergent terms (supersecularities) are shown to appear in the higher order terms. The distribution functions and kinetic conditions are given explicitly for $\ensuremath{\nu}\mathrm{th}$ order, and are shown to be decomposed unambiguously as the sum of two contributions: the "normal" (convergent) part and the "supersecular" (divergent) part. The normal part is proved to be given exactly by the perturbation-theory result of the previous paper with all secular terms removed, while the supersecular part is constituted by divergent terms which are explicitly constructed. The $\ensuremath{\nu}\mathrm{th}$-order term can be written down without any knowledge of the lower order terms. Exploiting this result, we resum the kinetic condition to all orders. The result, if convergent, constitutes the exact kinetic equation applicable in principle to any system whose evolution can be characterized by the single-particle distribution function.