Ordering principle for cluster expansions in the theory of quantum fluids, dense gases, and simple classical liquids

A study is made of a series-expansion procedure which gives the leading terms of the $n$-particle distribution function $p^{\mathrm{}\left(n\right)\mathrm{}}(1,2,\dots ,n)$ as explicit functionals in the radial distribution function $g\left(r\right)\mathrm{}$. The development of the series is based on the cluster-expansion formalism applied to the Abe form for $p^{\mathrm{}\left(n\right)\mathrm{}}$ expressed as a product of the generalized Kirkwood superposition approximation $p_K^{\mathrm{}\left(n\right)\mathrm{}}$ and a correction factor $\exp \left[A^{\mathrm{}\left(n\right)\mathrm{}}\right(1,2,\dots ,n\left)\right]$. An ordering parameter $\mu$ is introduced to determine $A^{\mathrm{}\left(n\right)\mathrm{}}$ and $p^{\mathrm{}\left(n\right)\mathrm{}}$ in the form of infinite power series in $\mu$, and the postulate of minimal complexity is employed to eliminate an infinite number of possible classes of solutions in a sequential relation connecting $A^{\mathrm{}(n-1\mathrm{})\mathrm{}}$ and $A^{\mathrm{}\left(n\right)\mathrm{}}$. Derivation of the series for $p^{\mathrm{}\left(n\right)\mathrm{}}$ and many other algebraic manipulations involving a large number of cluster integrals are greatly simplified by the use of a scheme which groups together all cluster terms having, in a certain way, the same source term. In particular, the scheme is useful in demonstrating that the nature of the series structure of $p^{\mathrm{}\left(3\right)\mathrm{}}$ is such that its three-point Fourier transform $S^{\mathrm{}\left(3\right)\mathrm{}}({\overrightarrow{\mathrm{k}}}_1,{\overrightarrow{\mathrm{k}}}_2,{\overrightarrow{\mathrm{k}}}_3)$ has as a factor the product of the three liquid-structure functions $S\left(k_1\right)S\left(k_2\right)S\left(k_3\right)$. The results obtained to order ${\mu }^4$ for $A^{\mathrm{}\left(3\right)\mathrm{}},p^{\mathrm{}\left(3\right)\mathrm{}}$, and $S^{\mathrm{}\left(3\right)\mathrm{}}$ agree with those derived earlier in a more straightforward but tedious approach. The result for $p^{\mathrm{}\left(4\right)\mathrm{}}$ shows that the convolution approximation $p_c^{\mathrm{}\left(4\right)\mathrm{}}$, which contains ${\mu }^3$ terms, must be supplemented by a correction of $O\left({\mu }^3\right)$ in order to be accurate through third order. The $\mu$-expansion approach is also examined for the cluster expansion of the correlation function in the Bijl-Dingle-Jastrow description of a manyboson system, and then compared with the number-density expansion formula by using the Gaussian model for $g\left(r\right)-1\mathrm{}$ to evaluate cluster integrals. A testing procedure based on the requirement $p^{\mathrm{}\left(3\right)\mathrm{}}\mathrm{}(1,2,2)=0\mathrm{}$ is developed to study accuracy of the $\mu$-ordered approximations for $p^{\mathrm{}\left(3\right)\mathrm{}}$. Numerical results obtained to orders ${\mu }^2,{\mu }^3$, and ${\mu }^4$ with the Gaussian model indicate substantial improvements with each increase in the order of truncation in the power series of $p^{\mathrm{}\left(3\right)\mathrm{}}$. A brief discussion is presented concerning the asymptotic behavior of $g\left(r\right)\mathrm{}$ in the context of equilibrium statistical mechanics.