What is remarkable about the quantum Casimir force between perfect conductors is its very simple formula, which depends only on the geometry and on universal constants. Would this force be the same in the presence of different interactions?
For the electromagnetic case it is hard to imagine alternative interactions, but we can do it for elastic media. Remarkably, the large separation limit of the Casimir interaction has the same scaling for rods adsorbed onto an elastic membrane1, whether the dominant force is due to the surface tension \(\sigma\) or to the membrane rigidity \(\kappa\):
\begin{equation*}For the electromagnetic case it is hard to imagine alternative interactions, but we can do it for elastic media. Remarkably, the large separation limit of the Casimir interaction has the same scaling for rods adsorbed onto an elastic membrane1, whether the dominant force is due to the surface tension \(\sigma\) or to the membrane rigidity \(\kappa\):
V_C = - \frac{k_BT}{128} \frac{L_1^2 L_2^2}{R^4} f(\theta _1, \theta _2)
\end{equation*}
with \(L_1\) and \(L_2\) the length of the two rods and \(R\) the distance between theire centers. The orientation dependence \(f\) does differ between the \(\sigma\) and the \(\kappa\) cases, but is the same for rods perpendicular to their center-to-center vector (\(\theta _1 = \theta _2 = \pi /2\).)
The explanation is quite simple: the allowed fluctuation modes (thermally excited phonons here, instead of zero-temperature photons in Casimir's calculation) are uniquely defined by the boundary conditions, which are substantially the same for the tension- or rigidity-controlled systems. Their amplitude is of course different: of order \( k_B T/(\sigma q^2)\) and \( k_B T/(\kappa q^4)\), respectively, with \(q\) the wave vector, but (and this is where the miracle happens) we only need to count the energy contribution of the mode, which is always \(k_B T\), per the equipartition theorem!2
1. R. Golestanian, M. Goulian, and M. Kardar, Fluctuation-induced interactions between rods on a membrane, Phys. Rev. E 54, 6725-6734 (1996).↩
2. Just as it is \(\hbar \omega /2\) for photons, but in this case \(\omega\) does depend on \(q\). Can we conclude that thermal Casimir forces are "more universal" than the quantum variety?!↩
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