Condensed Matter Physics Seminar: Sophie Hermann

Abstract:
Noether's theorem is familiar to most physicists due to its fundamental role
in linking the existence of conservation laws to the underlying symmetries
of a physical system. I will present how Noether's reasoning also applies
within statistical mechanics to thermal systems, where fluctuations are
paramount. Exact identities ("sum rules") follow thereby from functional
symmetries. The obtained sum rules contain both well-known relations,
such as the first order term of the Yvon-Born-Green (YBG) hierarchy (i.e.
the spatially resolved force balance), as well as previously unknown
identities, relating different correlations in many-body systems. The
identification of the underlying Noether concept enables their systematic
derivation.
Since Noether's theorem is quite general, it is possible to generalize to
arbitrary thermodynamic observables. This generalization yields sum rules
for hyperforces, i.e. the mean product between the considered observable
and the relevant forces that act in the system. Simulations of a range of
simple and complex liquids demonstrate the fundamental role of these
correlation functions in the characterization of spatial structure, such as
quantifying spatially inhomogeneous self-organization. Finally, we show
that the considered phase-space-shifting is a gauge transformation in
equilibrium statistical mechanics.