= 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> $n\pi/L$, $n\geq1$. Replacing one end by a <Neumann boundary condition> gives $(n+1/2)\pi/L$, $n\geq0$. With the same bulk subtraction, their finite zero-point <energies> are respectively
$$
E_{DD}=-\frac{\pi\hbar c}{24L},\qquad E_{DN}=+\frac{\pi\hbar c}{48L}.
$$
Differentiation 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.
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