# “Realizability of iso-\(g_2\) processes via effective pair interactions” is Published in The Journal of Chemical Physics

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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 \(\phi_c=1/2d\), 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 \(κ^{−1}(\phi)\) 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.