Casimir versus Helmholtz forces in the Gaussian model: exact results for Dirichlet--Dirichlet, Neumann--Dirichlet, Neumann--Neumann, and periodic boundary conditions

TL;DR

This paper derives exact results for fluctuation-induced Casimir and Helmholtz forces near the critical point in the Gaussian model, analyzing their behavior under various boundary conditions and ensemble constraints.

Contribution

It provides the first exact comparison of Casimir and Helmholtz forces in the Gaussian model for multiple boundary conditions at criticality.

Findings

Casimir and Helmholtz forces follow finite-size scaling near criticality.

For Dirichlet-Dirichlet and Neumann-Dirichlet conditions, forces differ and can change sign.

Under periodic and Neumann-Neumann conditions, forces coincide and are always attractive.

Abstract

We present results and compare the behavior of two fluctuation-induced forces pertinent for their corresponding ensembles: the critical Casimir force in the grand canonical (fixed external field $h$) one and the critical Helmholtz force in the canonical (fixed average value of the order parameter $m$) one. We do so by deriving exact results for their behavior near the bulk critical point at $T=T_c$ in the three-dimensional Gaussian model. We consider Dirichlet-Dirichlet, Neumann-Dirichlet, Neumann-Neumann, and periodic boundary conditions. For every boundary condition examined, we confirm that both forces follow a finite-size scaling. We find that for Dirichlet-Dirichlet and Neumann-Dirichlet boundary conditions the Casimir and the Helmholtz force differ from each other. For Dirichlet-Dirichlet boundary conditions the Casimir force is always attractive, while the Helmholtz force can be both attractive and repulsive as a function of $T$ and $m$. For Neumann-Dirichlet boundary conditions the Casimir force changes sign from repulsive to attractive with increase of $h$, while the Helmholtz force stays always repulsive. Under periodic and Neumann-Neumann boundary conditions the Casimir force and the Helmholtz force coincide - the first does not depend on $h$, while the latter does not depend on $m$; they are always attractive.