  We revisit the equilibrium statistical mechanics of a classical fluid of
point-like particles with repulsive power-law pair interactions, focusing on
density and energy fluctuations. Such long-range interactions, decaying with
inter-particle distance $r$ as $1/r^s$ in $d$ dimensions, are known to fall
into two qualitatively different categories. For $s<d$ ("strongly" long-range
interactions) there are screening of correlations and suppression of
large-wavelength density fluctuations (hyperuniformity). These effects
eliminate density modes with arbitrarily large energy. For $s>d$ ("weakly"
long-range interactions) screening and hyperuniformity do not occur. Using
scaling arguments, variational analysis, and Monte Carlo simulations, we find
another qualitative distinction. For $s \geq d/2$ the strong repulsion at short
distances suppresses small-wavelength density fluctuations. This "small-scale
hyperuniformity" prevents indefinitely negative entropy and large energy
fluctuations. The distinct behaviors for $s \geq d/2$ and $s<d/2$ give rise to
qualitatively different dependencies of the entropy, heat capacity, and energy
fluctuations on temperature and density. We investigate the effect of
introducing an upper cutoff distance in the pair-potential. The effect of the
cutoff on energy fluctuations is strong for $s<d/2$ and negligible for $s \geq
d/2$.
