Martynov Vompe

The Martynov-Vompe (Refs. 1 and 2) closure

$\left.B\right.=-a(\ln y^{*})-b(\ln y^{*})^{2}$

where

$\left.y^{*}(r)\right.=\ln y(r)-\beta \Phi _{p}$

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_p} is the perturbative part of the pair potential (Note: in the WCA separation for the Lennard-Jones system, the `perturbative part' is the attractive part). Martynov and Vompe have used the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dp_v-dP_c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dU-dP} thermodynamic consistencies in constructing their closures (Ref. 3).

References

1. G. A. Martynov and A. G. Vompe "Differential condition of thermodynamic consistency as a closure for the Ornstein-Zernike equation", Physical Review E, 47 pp. 1012 - 1017 (1993)
2. A. G. Vompe and G. A. Martynov "The bridge function expansion and the self-consistency problem of the Ornstein–Zernike equation solution", Journal of Chemical Physics 100 pp. 5249-5258 (1994)
3. Lloyd L. Lee, Dhananjay Ghonasgi, and Enrique Lomba "The fluid structures for soft-sphere potentials via the zero-separation theorems on molecular distribution functions", Journal of Chemical Physics 104 pp. 8058-8067 (1996)