  Noether's Theorem is familiar to most physicists due its fundamental role in
linking the existence of conservation laws to the underlying symmetries of a
physical system. Typically the systems are described in the particle-based
context of classical mechanics or on the basis of field theory. We have
recently shown [Commun. Phys. $\textbf{4}$, 176 (2021)] that Noether's
reasoning also applies to thermal systems, where fluctuations are paramount and
one aims for a statistical mechanical description. Here we give a pedagogical
introduction based on the canonical ensemble and apply it explicitly to ideal
sedimentation. The relevant mathematical objects, such as the free energy, are
viewed as functionals. This vantage point allows for systematic functional
differentiation and the resulting identities express properties of both
macroscopic average forces and molecularly resolved correlations in many-body
systems, both in and out-of-equilibrium, and for active Brownian particles. To
provide further background, we briefly describe the variational principles of
classical density functional theory, of power functional theory, and of
classical mechanics.
