  The isothermal compressibility (i.e., the asymptotic number variance) of
equilibrium liquid water as a function of temperature is minimal near ambient
conditions. This anomalous non-monotonic temperature dependence is due to a
balance between thermal fluctuations and the formation of tetrahedral
hydrogen-bond networks. Since tetrahedrality is a many-body property, it will
also influence the higher-order moments of density fluctuations, including the
skewness and kurtosis. To gain a more complete picture, we examine these
higher-order moments that encapsulate many-body correlations using a recently
developed, advanced platform for local density fluctuations. We study an
extensive set of simulated phases of water across a range of temperatures (80 K
to 1600 K) with various degrees of tetrahedrality, including ice phases,
equilibrium liquid water, supercritical water, and disordered nonequilibrium
quenches. We find clear signatures of tetrahedrality in the higher-order
moments, including the skewness and excess kurtosis, that scale for all cases
with the degree of tetrahedrality. More importantly, this scaling behavior
leads to non-monotonic temperature dependencies in the higher-order moments for
both equilibrium and non-equilibrium phases. Specifically, at near-ambient
conditions, the higher-order moments vanish most rapidly for large length
scales, and the distribution quickly converges to a Gaussian in our metric.
However, at non-ambient conditions, higher-order moments vanish more slowly and
hence become more relevant especially for improving information-theoretic
approximations of hydrophobic solubility. The temperature non-monotonicity that
we observe in the full distribution across length-scales could shed light on
water's nested anomalies, i.e., reveal new links between structural, dynamic,
and thermodynamic anomalies.
