title: Forces and symmetries in the statistical mechanics of active and thermal many-body systems
Creators: name: Hermann, Sophie
Publisher: name: University of Bayreuth
Year: 2022
Subjects: subject: Statistical Mechanics
subject: Noether's Theorem
subject: Sum Rules
subject: Density Funtional Theory
subject: Force Balance
subject: Active Brownian Particles
subject: 530

description: Noether's theorem of invariant variations is an important mathematical theorem in functional analysis that Emmy Noether derived in 1918 as part of her habilitation thesis. In the physics community her theorem is mainly known for linking the symmetries of a given system to corresponding exact conservation laws. In this thesis we use the invariance of statistical mechanics functionals with respect to several continuous symmetries to determine on the basis of Noether's theorem exact identities for many-body systems. These statistical mechanical identities are then exploited within various applications, in particular for active Brownian particles, which form a simple nonequilibrium model system of self-propelled entities. The self-propulsion leads to several interesting phenomena including a gas-liquid-like phase transition which even occurs for purely repulsive interparticle interactions and is hence motility-induced. Further application addresses the behaviour of active Brownian particles in a gravitational field confined by a lower bounding wall. Such active sedimentation is experimentally accessible. We lay out Noether's theorem to statistical mechanics both in the grand canonical and in the canonical ensemble. Therefore, we consider the invariances of various important and fundamental statistical functionals such as the grand potential and the free energy under symmetry operations including spatial shifts and rotations. The argument rests on two facts. On the one hand the invariant functional does not change under the symmetry transformation. On the other hand one can still expand the transformed functional around the original (i.e. non-transformed) functional with respect to the variation parameter. Comparing the results of either perspective, it becomes apparent that the linear as well as all higher order contributions in the expansion have to vanish individually. In linear order the procedure yields sum rules that describe the vanishing of mean values such as both global and spatially resolved ("local") forces and torques. In quadratic order the cancellation relates variances and curvatures, e.g. the variance of the external force with a mean curvature of the external potential. Functional differentiation of global first order sum rules gives a full hierarchy of local sum rules which relate several correlation functions that are essential in liquid state theory. While some of the hierarchies are already known in the literature, the identification of the underlying Noether concept enables their systematic derivation and it provides a constructive way to obtain new sum rules. Those sum rules include exact memory identities of nonequilibrium time-correlation functions. The Noether concept generalizes from classical to quantum statistical mechanics as we demonstrate. The sum rules that Noether's theorem generates hold quite generally in statistical mechanics as long as the system is closed by an impenetrable external potential and no boundary contributions arise. Considering boundary terms is necessary when we apply Noether's theorem to the thermal sedimentation-diffusion equilibrium, active sedimentation, and the phase separation of both active and thermal Brownian particles. For the later system we recover the well-known mechanical pressure balance at phase coexistence and show that this relationship also holds in nonequilibrium for active Brownian particles. Boundary contributions are also crucial for the presented proof of the viral hard wall contact theorem, which states that the density at a hard wall is determined by the virial bulk pressure. The proof itself is based on the global total force balance, which is a direct consequence of the Noether invariance under a global displacement. We take the continuity equation as the direct origin of an exact sum rule which relates the global polarization at the interface with the current at the system boundaries far away from the interface. This has since been verified experimentally and numerically. In systems that are bounded by bulk states such as sedimentation and motility-induces phase separation of active Brownian particles, the global polarization is solely determined by bulk values and hence constitutes a state function. In both examples we give explicit expressions for this state function. In combination with Noether's theorem the polarization sum rule is then applied to global force balance equations. This combined use of sum rules yields deeper insights into the dynamics of the center of mass motion as we demonstrated for the example of active Brownian particles. We demonstrate that all applicable sum rules are satisfied within a previously developed theoretical power functional description of the bulk and the interface at motility-induced phase separation of active Brownian particles. The variational theory works on the basis of forces and rests on the force density balance and the continuity equation. We consider the validity of the sum rules as a strong support of the theory and determine on the basis of this description the free interfacial tension. Using a square gradient approximation for the interfacial force contributions we obtain positive results for the nonequilibrium tension, which is in accordance with the observed mechanical stability of the interface in both simulations and experiments.