Basis of the Functional Assumption in the Theory of the Boltzmann Equation

The long-time behavior of the $n$-particle probability densities for a large, dilute system of point particles interacting with short-range repulsive forces is studied. The main result is an exact series for the $n$-particle density which consists of two parts. The first part is a time-independent functional of the singlet density which is expressed as a functional power series and which is a direct analog of the equilibrium density series. The second part is also a functional power series in the singlet density but the coefficients depend on time and on the initial correlations. The coefficients of both series are given explicitly in terms of operators which are determined by the dynamics of isolated groups of particles. It is demonstrated that these operators vanish for phase points corresponding to motions during which there are two or more groups of particles which either are statistically and dynamically independent or are such that each of them is dynamically connected to the rest by no more than one particle. It is argued that all the terms of the exact series are finite and that the terms of second part (the error) decrease with increasing time so that the first part is the asymptotic form proposed by Bogoliubov. The relevance of the results for the Boltzmann equation is indicated. A form of the Boltzmann collision integral which is valid in the steady state and to all orders of the density is described.