Realizability of iso-g₂ processes via effective pair interactions

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g₂(r) [or equivalently, structure factor S(k)] at some number density ρ 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-g₂ process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g₂ remains invariant for a positive range of densities. Using a precise inverse algorithm that determines effective potentials that match hypothesized functional forms of g₂(r) for all r and S(k) for all k, we show that the unit-step function g₂, which is the zero-density limit of the hard-sphere potential, is remarkably realizable up to the packing fraction ϕ = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum “terminal” packing fraction ϕ_c = 1/2^d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through ϕ_c. For ϕ near but below ϕ_c, the large-r behaviors of the effective potentials are given exactly by the functional forms exp[ − κ(ϕ)r] for d = 1, r^−1/2 exp[ − κ(ϕ)r] for d = 2, and r⁻¹ exp[ − κ(ϕ)r] (Yukawa form) for d = 3, where κ⁻¹(ϕ) is a screening length, and for ϕ = ϕ_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 ϕ_c = 1/8 is much shorter ranged than that for the 3D unit-step function g₂ at ϕ_c; thus, it does not constrain the realizability of the unit-step function g₂. 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-g₂ 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.