  We explore quantitative descriptors that herald when a many-particle system
in $d$-dimensional Euclidean space $\mathbb{R}^d$ approaches a hyperuniform
state as a function of the relevant control parameter. We establish
quantitative criteria to ascertain the extent of hyperuniform and
nonhyperuniform distance-scaling regimes n terms of the ratio $B/A$, where $A$
is "volume" coefficient and $B$ is "surface-area" coefficient associated with
the local number variance $\sigma^2(R)$ for a spherical window of radius $R$.
To complement the known direct-space representation of the coefficient $B$ in
terms of the total correlation function $h({\bf r})$, we derive its
corresponding Fourier representation in terms of the structure factor $S({\bf
k})$, which is especially useful when scattering information is available
experimentally or theoretically. We show that the free-volume theory of the
pressure of equilibrium packings of identical hard spheres that approach a
strictly jammed state either along the stable crystal or metastable disordered
branch dictates that such end states be exactly hyperuniform. Using the ratio
$B/A$, the hyperuniformity index $H$ and the direct-correlation function length
scale $\xi_c$, we study three different exactly solvable models as a function
of the relevant control parameter, either density or temperature, with end
states that are perfectly hyperuniform. We analyze equilibrium hard rods and
"sticky" hard-sphere systems in arbitrary space dimension $d$ as a function of
density. We also examine low-temperature excited states of many-particle
systems interacting with "stealthy" long-ranged pair interactions as the
temperature tends to zero. The capacity to identify hyperuniform scaling
regimes should be particularly useful in analyzing experimentally- or
computationally-generated samples that are necessarily of finite size.
