A statistical-mechanical theory of evaluating the distribution functions of a many-body system is presented. The theory is a natural extension of the quantum statistical theory of Lee and Yang for the grand partition function and gives a new formalism which is different from that developed recently by Fujita, Isihara, and Montroll. The density matrices are first developed in the Uhlenbeck-de Boer $U$ functions. A diagrammatical consideration separates out nonconnected products from connected products of the $U$ functions, yielding an expansion formula which is simpler than that reported by de Boer some time ago. Application of the resulting expression to free bosons and fermions is made. Then the distribution functions are developed in the binary kernel introduced by Yang and Lee. This expansion is used for the evaluation of the pair distribution function of a hard-sphere Bose gas at the lowest temperature. The results improve upon those reported previously by Lee, Huang, and Yang and others. The normalization and divergence difficulties encountered by Fujita and Hirota are removed. Actually, their interpretation of Lee, Huang, and Yang's results in terms of the chain diagrams is not satisfactory. Instead the chain diagram results may be compared with the more recent results by Wu. Use of the approximate pseudopotential $\phi =8\mathrm{}\pi a\delta (r_1-r_2)$ is reflected in insufficiency of the chain diagram approximation. Satisfactory and consistent results are obtained when a new set of diagrams is taken into consideration.

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