= Solution
\b[<Fluctuation-induced forces> arise because separation changes the fluctuations available to a system and therefore changes its <free energy>.] The <force> is $F=-\partial\mathcal F/\partial d$ at fixed relevant control variables, not necessarily a <force> from a permanent charge or dipole.
For two neutral polarizable particles, correlated electromagnetic fluctuations give a short-distance dispersion attraction of the form $-C_6/r^6$. Retardation changes the asymptotic distance dependence. Boundaries produce a related <Casimir force>; for parallel ideal conducting plates at zero <temperature>,
$$
\frac{\mathcal E}{A}=-\frac{\pi^2\hbar c}{720d^3},\qquad\boxed{P=-\frac{\pi^2\hbar c}{240d^4}.}
$$
The negative <pressure> denotes attraction. Material response, nonplanar geometry and finite <temperature> alter the ideal result. The effect can be described through the separation-dependent electromagnetic mode spectrum rather than by treating a divergent absolute vacuum energy as observable.<Quantum Field Theory by David Tong>[Section2.3.2: Casimir effect and boundary-dependent fluctuation energy]{ref}
In a classical <depletion force>, smaller particles or <polymer> coils are excluded from a region near larger surfaces. Overlap of excluded regions increases their available volume. For ideal dilute depletants of osmotic <pressure> $\Pi$, $U(d)=-\Pi V_{\rm overlap}(d)$; the resulting attraction is entropic and scales with <temperature>. Confined interface fluctuations can instead generate repulsion when close approach removes <entropy>. Near a continuous phase transition, a large <correlation length> makes boundary-conditioned order-parameter fluctuations produce a <critical Casimir force>. Its sign depends on the boundary preferences, so fluctuation forces are not universally attractive.
The useful common idea is a free-energy difference with explicit boundary conditions; quantum vacuum, thermal field and excluded-volume mechanisms have different physical fluctuations and different ranges.
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