^aDepartment of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
^bDepartment of Chemistry, Princeton University, Princeton, NJ 08544, USA
^cPrinceton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08544, USA

*Corresponding author. Princeton University, Department of Mechanical Engineering, Princeton, NJ 08544, USA.
E-mail address: torquato@princeton.edu (S. Torquato).
¹Supported by the U.S. Department of Energy Computational Science Graduate Fellowship.
²Supported by the Office of Basic Energy Sciences, DOE, under Grant no. DE-FG02-04ER46108.

Available online 15 June 2005.

Physica A: Statistical Mechanics and its Applications, Volume 360, Issue 1 , 15 January 2006, Pages 21-36.

Abstract

The pair correlation function g₂(r) provides a basic geometric descriptor for many-particle systems. It must obey two necessary conditions: (i) non-negativity for all distances r, and (ii) non-negativity of its associated structure factor S(k) for all k. Here we utilize an improved stochastic construction algorithm for particle configurations to establish conditions in which (i) and (ii) are also sufficient, i.e., g₂(r) is in fact realizable.
Two types of target pair correlation functions have been investigated in one, two, and three dimensions for hard-core particles, specifically a unit step function, and a contact d plus step pair correlation function. Results indicate that the former target function is realizable up to a terminal density set by necessary condition (ii), at which the particle core packing fraction equals 2^-d in d dimensions. Furthermore, results are consistent with the proposition that for d>1 the contact d plus step function is realizable up to a terminal density due to condition (ii) at which the packing fraction of cores is (d+2)/2^d+1 [Torquato and Stillinger, J. Phys. Chem. B 106 (2000) 8354, 11406].