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

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 $ ho$ can be achieved by $d$-dimensional many-body systems. 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 $g_2(r)$ at all $r$ and $S(k)$ at all $k$, we show that the unit-step function $g_2$, which is the zero-density limit of the hard-sphere potential, is 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. For $phi