height (in units of nm) of a hollowed teflon/Si (blue/red lines) sphere suspended above a $H = 15$ nm ITO layered gold plate, for $\alpha = 0.142/0.13$, as a function of sphere radius $R$ (in units of $\mu$m). Solid/dashed lines correspond to stable/unstable equilibria. The red/blue shaded regions indicate positions where the teflon/Si spheres are found with $95\%$ probability. The inset shows $\langle d \rangle $ (thick line) and $d_e$ (thin line) separations as a function of their radius for two equal radii teflon Si spheres. The gray shaded region indicate the separations which the teflon and Si spheres are found with $95\%$ probability.[]{data-label="fig:fig7"}](teflon_flucs_prime){width="1.0\columnwidth"}

Statistics of Brownian motion
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As mentioned above, Brownian motion will disturb the spheres by causing them to move randomly about their stable equilibrium positions, and this can cause the Si sphere to move past its unstable equilibrium point, inducing it to stick top the plate. To quantify the range of motion of both spheres about their equilibria, we consider the statistical properties of their fluctuations. In particular, we consider the average plate–sphere separations $\langle h \rangle_T$ and average sphere–sphere separations $\langle d \rangle_T$ near room temperature ($T=300$ K), determined by an ensemble average over a 