  The strength of fluctuations, as measured by their variance, is paramount in
the quantitative description of a large class of physical systems, ranging from
simple and complex liquids to active fluids and solids. Fluctuations originate
from the irregular motion of thermal degrees of freedom and statistical
mechanics facilitates their description. Here we demonstrate that fluctuations
are constrained by the inherent symmetries of the given system. For
particle-based classical many-body systems, Noether invariance at second order
in the symmetry parameter leads to exact sum rules. These identities
interrelate the global force variance with the mean potential energy curvature.
Noether invariance is restored by an exact balance between these distinct
mechanisms. The sum rules provide a practical guide for assessing and
constructing theories, for ensuring self-consistency in simulation work, and
for providing a systematic pathway to the theoretical quantification of
fluctuations.
