The hard-sphere fluid: New exact results with applications

Abstract

A theorem for convolution integrals is proved and then applied to extend the “second zero-separation theorem” to the bridge functionb(r) and direct-correlation tail functionsd(r). This theorem allows us to exactly relate∂b(r)/∂r and∂d(r)/∂ratr=0 for the hard-sphere fluid to the “contact value” of the radial distribution functiong(r) atr=σ ⁺. From this we obtain immediately the exact values of ∂b(r)/∂r and ∂d(r)/∂r atr=0 through second order in number density ρ. Using our results to compare the exact and Percus-Yevick (PY) bridge function, we find that they differ significantly. After obtaining the bridge function and tail function and their derivatives atr=0 andr=σ through, we suggest new approximations forb(0) andd(0) as well as an analytical integral-equation theory to improve the PY approximation in the pure hard-sphere fluid. The major deficiency of that approximation has been its poor assessment of the cavity function inside the hard-core region. Our theory remedies this defect in a way that yields ay(r) that is self-consistent with respct to the virial and compressibility relations and also the two zero-separation relations involvingy(r) and its spatial derivative atr=0.