Some Properties of the Long-Time Values of the Probability Densities for Moderately Dense Gases

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Abstract
It has been argued that, for sufficiently large times, the n-particle probability densities of a moderately dense, simple gas become time-independent functionals of the one-particle probability densities. Proofs are given for several properties of the power series representation of these functionals. In particular, it is shown that the equilibrium value of the n-particle functional is identical to the usual equilibrium probability density term by term, and that the corresponding generalized Boltzmann collision integral vanishes as it should. The Green-Cohen form of the functional is shown to be a formal power series solution of Bogoliubov's functional differential hierarchy. Moreover, a proof is given that the Bogoliubov and Green-Cohen forms of the functional are formally identical term by term. It is argued that the higher terms of these two series probably diverge together. In the course of the discussion, several new properties of the coefficient operators of the power series for the functional are derived. Moreover, an integral equation for the n-particle functional is derived which may have solutions not representable as functional power series in the one-particle probability density.