Quantum Statistics of the Distribution Functions

Abstract

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 <f> = Swad (ri-r2) 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.Recently a quantum-statistical method of evaluating the pair distribution function has been presented by Fujita, Isihara, and Montroll.

Low-density expansions for the homogeneous dipolar Bose gas at zero temperature

“…of Eq. ( 37) reproduces the well-known decay 1/r 4 for a radially symmetric short-range potential [30]. Note that the 1/r 4 decay of g(r)−1 arises due to the linear dependence of the static structure factor at small momentum.…”

“…From the physical point of view, Figure 1. Polar plot of the anisotropic prefactor f (ǫ dd , θ) in the asymptotics of the one-particle correlation functions (29) and (30) at different values of the dipolar ratio ǫ dd (that is, f is shown as a distance between the origin and a point on the plot as a function of the angle; see the upper panel). The prefactor is given by Eq.…”

“…Apparently, the idea of universality was first expressed by Lev Landau back in 1947 [see the footnote after equation (30) in the original paper by Bogoliubov [10]]. A discussion of universal and non-universal effects for Bose gases with shortrange potentials can be found in, e.g., Refs.…”

Correlation Functions in Ring - Diagram Approximation

“…Quantum gases with hard-core potentials have been treated by the binary-kernel method and the pseudopotential method. 5 In the next section we shall introduce conjugate diagrams which play important roles in developing our theory. These diagrams are classified and treated separately in accordance with the number of interaction lines and the topological structure.…”