Realizability of Iso-$g_2$ Processes via Effective Pair Interactions

Abstract

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 $ρ$ can be achieved by $d$-dimensional many-body systems. 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. In light of 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 methodology 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 numerically 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 [\textit{Phys. Rev. E}, 68, 041113 1-25 (2003)].