Fluctuation-induced force

Boundaries or inclusions change the available thermal or quantum fluctuations and thereby the free energy. Their force follows from $F=-\partial \mathcal{F}/\partial d$ under specified control conditions. Electromagnetic dispersion, depletion and critical fluctuations are distinct realizations, with signs and ranges set by their physical modes and boundary conditions.

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.

Confining a near-critical fluctuating order parameter changes its free energy. Its range is governed by the growing correlation length, and its sign depends on the boundary preferences. This is a thermal collective effect, distinct from the zero-temperature ideal-conductor electromagnetic formula.

Two parallel ideal conducting plates at zero temperature have separation-dependent electromagnetic energy per area $-{\pi }^2\hslash c/(720d^3)$ and pressure $-{\pi }^2\hslash c/(240d^4)$. The negative sign denotes attraction in this geometry. Finite temperature, material response and boundary geometry change the ideal result; an absolute divergent vacuum energy is not the measured quantity.

For a massless real scalar in one spatial dimension, applying $e^{-\omega /(\pi \Lambda )}$ to both the Dirichlet mode sum and the unbounded-line integral gives the same leading bulk energy $\pi L{\Lambda }^2/2$. Their finite difference is $-\pi /(24L)$ in units $\hslash =c=1$, producing the attractive force $-\pi /(24L^2)$. This scalar boundary force is distinct from the retarded atom-surface Casimir-Polder interaction.