Ornstein-Zernike-like Equations in Statistical Geometry: Stable and Metastable Systems H. Reiss,* H. M. Ellerby, and J. A. Manzanares ² Department of Chemistry and Biochemistry, UniVersity of California, Los Angeles, California 90095-1569 ReceiVed: October 2, 1995; In Final Form: January 16, 1996 X Statistical geometric methods based on nearest-neighbor distributions are used, in connection with hard- particle systems, to develop Ornstein-Zernlike-like equations that have already been of considerable value in the statistical thermodynamic analysis of such systems and that promise to have even greater value. In this paper, we use these equations to (1) develop a relation that is valid for a hard particle system in unconstrained equilibrium and that shows that the insertion probability cannot vanish (short of closepacking) in such a system, (2) study the still incompletely settled issue concerning the equality of the hard-particle densities on the peripheries of cavities which are and are not occupied by hard particles and, in so doing, arrive at a relation that holds in a system in stable equilibrium but fails in a metastable system, (3) provide insight into the geometric mechanism of hard-particle phase transitions and allow simple estimates of the freezing densities, and (4) suggest a new physical interpretation for the direct correlation function. 1. Introduction There has been an increased interest in nearest-neighbor distributions (and neighbor distributions in general) in the statistical mechanics of fluids and related systems. 1-7 Interest has also been focused on the functions that comprise the neighbor distributions. In this paper, dealing with hard particle systems, we make use of these functions to develop Ornstein- Zernike-like equations and to derive relations that are valid on the stable branches of hard particle pressure-density isotherms and that are not necessarily valid on metastable branches. We also investigate some unresolved issues concerning the condi- tional particle density at the surface of a cavity and find a rather unique relation involving these densities that undergoes an abrupt change as the system passes from stable to metastable equilibrium. Byproducts of this study are some simple ideas concerning the “mechanism” of hard particle phase transitions and an alternative physical interpretation of the direct correlation function. (See Appendix C.) The term “statistical geometric characterization” is adopted to imply that the analysis is conducted with the aid of neighbor distributions and their component functions. 2. Nearest Neighbor Distribution Consider an arbitrary point in a uniform system of N particles (not necessarily hard particles) contained in a volume V. Selecting this point as the origin, we ask for the probability  0 (r)dr that the center of the particle nearest to the origin is located at the point r in the volume element dr. This probability can be expressed as where the notation ∫ r indicates that the integration is performed throughout the volume 4πr 3 /3 where r ) |r|. F is the uniform bulk number density N/V, and G 0 (r) is defined so that FG 0 (r) is the conditional density in dr (conditional on the interior volume 4πr 3 /3 being devoid of particle centers). The structure of eq 2.1 is obvious; the factor in square brackets is the probability that the interior volume is empty, while FG 0 (r)dr is the chance that at least one particle center will be found in dr, given that the interior volume is empty. In general, we shall be interested in nearest neighbor distributions that are spherically symmetric so that we can ask for the probability that the center of the particle nearest to the arbitrary point at the origin lies in the spherical shell of volume 4πr 2 dr. By denoting this probability by R 0 (r)dr, we have where From these equations, it is clear that R 0 (r) and  0 (r) are fully determined by G 0 (r) and vice versa. We can also place the center of a molecule at the origin and ask for the probability that the center of the nearest other molecule lies in dr at r. In this case, denote probability distributions by (r) and R(r), dropping the zero subscript, and eqs 2.1 and 2.3 are once again applicable to the unsubscripted quantities. If we consider a two-dimensional system, eqs 2.2 and 2.3 become while, for a one-dimensional system, we define the nearest- neighbor distribution to the right or left of the origin by R 0 (x), where x may be positive or negative, and the equations become For both the one- and two-dimensional systems, we denote (similar to the three-dimensional case) the distributions about ² Permanent address: Departament de Termodinamica, Facultat de Fisica, Universitat de Valencia, E-46100 Burjassot, Spain. X Abstract published in AdVance ACS Abstracts, March 1, 1996.  0 (r)dr ) [1 - ∫ r  0 (r′)dr′]FG 0 (r)dr (2.1) R 0 (r)dr ) [1 - ∫ 0 r R 0 (r′ )dr′]4πr 2 FG 0 (r)dr (2.2) R 0 (r) ) 4πr 2  0 (r) (2.3) R 0 (r)dr ) [1 - ∫ 0 r R 0 (r′ )dr′]2πrFG 0 (r)dr (2.4) R 0 (r) ) 2πr 0 (r) (2.5) R 0 (x)dx ) [1 - ∫ 0 x R 0 (x′ )dx′]FG 0 (x)dx (2.6) R 0 (x) )  0 (x) (2.7) 5970 J. Phys. Chem. 1996, 100, 5970-5981 0022-3654/96/20100-5970$12.00/0 © 1996 American Chemical Society