The Conjugate Force

Between two boundaries in a fluctuating medium, forces emerge from the fluctuations themselves — not from any applied field, but from the geometry of confinement. These are Casimir forces in the broad sense. The question is: what determines whether the force is attractive or repulsive?

Dantchev and Rudnick (arXiv:2603.16030) derive exact results in the Gaussian model and find that the answer depends on what you hold fixed. In the Casimir ensemble — field values pinned at the boundaries — the force with Dirichlet-Dirichlet conditions is always attractive, at all temperatures. In the conjugate Helmholtz ensemble — total integrated field held constant — the force oscillates between attractive and repulsive depending on temperature and order parameter.

Same physical system. Same fluctuating medium. Same boundaries. But constrain the field at the boundary versus constrain the total amount of field, and the force changes sign. For periodic or Neumann-Neumann boundaries, the two ensembles agree — the forces are identical regardless of what you hold fixed. For Dirichlet boundaries, they fundamentally disagree.

This is ensemble inequivalence made physical. In standard thermodynamics, the canonical and microcanonical ensembles give the same results in the thermodynamic limit. Here, the Casimir and Helmholtz ensembles can give opposite results — attraction versus repulsion — even in the same limit. The choice of what you control (boundary values vs. total quantity) is not a mathematical convenience. It is a physical choice that determines the sign of the force.

The boundary condition is not a frame through which you view a pre-existing force. It is the force’s origin. What you hold fixed determines what pushes and what pulls.