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

TL;DR

This paper investigates the realizability of specific pair correlation functions in many-body systems, demonstrating that certain functions like the unit-step $g_2$ are achievable up to high densities in various dimensions, supporting the Zhang-Torquato conjecture.

Contribution

The study provides numerical evidence that the unit-step $g_2$ function is realizable in 1, 2, and 3 dimensions up to specific packing fractions, confirming the sufficiency of known conditions for realizability.

Findings

Unit-step $g_2$ is realizable up to $ ho=0.49$ in 1D.

In 2D and 3D, realizability extends up to the maximum packing fraction $rac{1}{2^d}$.

Large-$r$ effective potentials follow specific exponential and Yukawa forms near the terminal packing fraction.

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 $\rho$ 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 $\phi=0.49$ 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 [\textit{Phys. Rev. E}, 68, 041113 1-25 (2003)].