Boltzman distribution. For example, $\langle h \rangle_T$ is given by: $$\langle h \rangle_T = \dfrac{\int_0^\infty dz\, z
    \exp\left(U(z)/kT\right)}{\int_0^\infty dz\,
    \exp\left(U(z)/kT\right)},$$ where $U(z)$ is the total energy (gravity included) of the sphere–plate system at a surface–surface height $z$. (A similar expression yields $\langle d \rangle_T$). In the case of teflon, the short-range attraction means that the suspension is only metastable under fluctuations; here, we only average over separations prior to stiction by restriction $z$ to be $\geq$ the unstable equilibrium, and consider the stiction timescale separately below. In addition to the average equilibrium separations, we are also interested in quantifying the extent of the fluctuations of the spheres, which we do here by computing the 95% confidence interval $\{\sigma_{-},\sigma_{+}\}$, defined as the spatial region over which the sphere is found with 95% probability around the equilibria, where $\sigma_{\pm}$ denotes the lower/upper bound of that interval. These results are shown in [Fig. \[fig:fig7\]]{} for $h$, with $d$ shown in the inset, in which shaded regions indicate the confidence intervals, as a function of $R$ where $\alpha$ is chosen to yield approximately equal $h_e$ ($\alpha =
0.142$ for teflon and $\alpha=0.13$ for Si). (Note that the horizontal separation $\langle d \rangle$ is a 