Boundary-condition dependence of a fluctuation force

A fluctuation-induced force depends on how boundaries restrict fluctuating fields. Changing the boundary conditions changes the permitted normal modes, their frequencies and the separation-dependent free energy. The resulting force is $-\partial \mathcal{F}/\partial L$ at the stated control conditions; confinement alone does not determine its sign.

For a concrete ideal example, take a massless scalar field of wave speed $c$ on a one-dimensional interval of length $L$ at zero temperature. Two Dirichlet boundary conditions give mode wavenumbers $n\pi /L$, $n\ge 1$. Replacing one end by a Neumann boundary condition gives $(n+1/2)\pi /L$, $n\ge 0$. With the same bulk subtraction, their finite zero-point energies are respectivelyDifferentiation gives attraction in the first case and repulsion in the second. These scalar examples are distinct from the three-dimensional electromagnetic Casimir force. The comparison illustrates why the boundary prescription must accompany any reported fluctuation-force law.