An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g(2)(r) [or equivalently, structure factor S(k)] at some number density rho can be achieved by many-body systems in d-dimensional Euclidean space. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. To further test this conjecture, we study the realizability problem of the nonequilibrium iso-g(2) process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g(2) remains invariant for a positive range of densities. Using a precise inverse algorithm that determines effective potentials that match hypothesized functional forms of g(2)(r) for all r and S(k) for all k, we show that the unit-step function g(2), which is the zero-density limit of the hard-sphere potential, is remarkably realizable up to the packing fraction phi = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal " packing fraction phi(c) = 1/2(d), at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through phi(c). For phi near but below phi(c), the large-r behaviors of the effective potentials are given exactly by the functional forms exp[ - kappa(phi)r] for d = 1, r(-1/2) exp[ - kappa(phi)r] for d = 2, and r(-1) exp[ - kappa(phi)r] (Yukawa form) for d = 3, where kappa(-1)(phi) is a screening length, and for phi = phi(c), the potentials at large r are given by the pure Coulomb forms in the respective dimensions as predicted by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost " random sequential addition at the maximum packing fraction phi(c) = 1/8 is much shorter ranged than that for the 3D unit-step function g(2) at phi(c); thus, it does not constrain the realizability of the unit-step function g(2). Our inverse methodology yields effective potentials for realizable targets, and, as expected, it does not reach convergence for a target that is known to be non-realizable, despite the fact that it satisfies all known explicit necessary conditions. Our findings demonstrate that exploring the iso-g(2) process via our inverse methodology is an effective and robust means to tackle the realizability problem and is expected to facilitate the design of novel nanoparticle systems with density-dependent effective potentials, including exotic hyperuniform states of matter.
展开▼
