Abstract
Noether’s calculus of invariant variations yields exact identities from functional symmetries. The standard application to an action integral allows to identify conservation laws. Here we rather consider generating functionals, such as the free energy and the power functional, for equilibrium and driven many-body systems. Translational and rotational symmetry operations yield mechanical laws. These global identities express vanishing of total internal and total external forces and torques. We show that functional differentiation then leads to hierarchies of local sum rules that interrelate density correlators as well as static and time direct correlation functions, including memory. For anisotropic particles, orbital and spin motion become systematically coupled. The theory allows us to shed new light on the spatio-temporal coupling of correlations in complex systems. As applications we consider active Brownian particles, where the theory clarifies the role of interfacial forces in motility-induced phase separation. For active sedimentation, the center-of-mass motion is constrained by an internal Noether sum rule.
Highlights
Noether’s calculus of invariant variations yields exact identities from functional symmetries
Considering the symmetries of the partition sum does not require to engage with density functional concepts; the elementary definition suffices. We demonstrate that this approach is consistent with the earlier work in equilibrium[9,10,11,12,13,14,15,16], and that it enables one to go, with relative ease, beyond the sum rules that these authors formulated
We consider spatial translations of the position coordinate r at fixed chemical distribution, and the average is over the equilibrium distribution at fixed μ and T; the sum runs over all particles i = 1...N
Summary
Noether’s calculus of invariant variations yields exact identities from functional symmetries. Exploiting energy conservation in the effective system yields a first integral, which constitutes a nontrivial identity in the statistical problem This reasoning has been generalized to the delicate problem of the three-phase contact line that occurs at a triple point of a fluid mixture[6,7]. This shifting enables him, as well as Lovett et al.[10] and Wertheim[11] in earlier work, to identify systematically the effects that result from the displacement and formulate these as highly nontrivial interrelations (“sum rules”) between correlation functions This approach was subsequently generalized to higher than two-body direct[12] and density[13] correlation functions and the relationship to integral equation theory was addressed[14,15]. The system is displaced instantaneously only at the latest time t, such that the differential displacement is ε_dt (purple dotted)
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