Thermodynamics of a Set of Closed Curved Surfaces and Noether's Theorem

Abstract

For a set of closed curved surfaces that resemble a Langmuir monolayer an energy is defined that depends only on the curvature of the surfaces interacting with the dimensions normal to the surfaces, that is, the thickness of these surfaces. The thickness, or depth, of the surfaces originates because of surface-particles with chains, that move freely on the inside of the surfaces at a certain depth. This is a purely geometrical description that does not depend on the introduction of ad hoc constants like the elastic constant. With a statistical mechanical model the equations of state are calculated for a gas of non-connected sphere-like surfaces in a volume. There are two states of which the lower temperature state is depending mostly on the interaction energy and surface properties, and the higher temperature state is depending mostly on sphere kinetic and volume properties. This results in aggregation of the sphere-like surfaces from many small ones to few large ones when lowering temperature and vice versa. The model allows for the calculation of the partition function and, when the emergence of the curvature - depth interaction is described as a phase-transformation, for the application of Noether's theorem. Because of these properties, the model is interesting in its own right apart from being an addition to existing elastic descriptions of surfaces.