which set a Matsubara length-scale $2\pi
kT/\hbar$ [@Bordag] that can further interact with dispersion in order to yield strong temperature corrections [@RodriguezW10:PRL]. Experimentally, stable suspensions are potentially appealing in that one would be measuring static displacements rather than force between micro-scale objects. The stable configurations may be further modified, however, by three-body effects in sphere–sphere–plate geometries and by Brownian motion of the particles within the potential well created by the Casimir interaction, and these effects are studied in detail by the present paper.

Until the last few years, theoretical predictions of Casimir forces were limited to a small set of simple geometries (mainly planar geometries) amenable to analytical solution, but a number of computational schemes have recently been demonstrated that are capable of handling complicated (and, in principle, arbitrary) geometries and materials [@Rahi09:PRD; @Rodriguez07:PRA; @Homer09]. Here, since the geometries considered in this paper consist entirely of spheres and planes, we are able to adapt an existing technique [@Rahi09:PRD] based on Fourier-like (“spectral”) expansions that semi-analytically exploits the symmetries of this problem. This technique, formulated in terms of the scattering matrices of the objects in a basis of spherical or plane waves, was developed in various forms by multiple authors [@Emig07; 