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

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.