Applying the thermodynamic sum rule

Even when the configurational and hyperconfigurational temperatures converge to the thermodynamic value for a reasonable choice of pair potential, the system's degree of equilibration still has to be assessed. Figure 11 shows the integrand of the sum rule in Eq. (32) for two slightly different effective potentials, both of which are consistent with the interactions measured in the confined colloidal monolayer at $H = 9~\ensuremath{\unit{\mu m}}\xspace$. Even differences in the input potential too small to affect the configurational temperatures can change the sum rule's integrand substantially. These changes affect whether or not the sum rule as a whole is satisfied.

Normalizing the integral in Eq. (32) by the integral of the integrand's absolute value provides a useful measure of convergence. The best-fit pair potential obtained from the liquid structure inversion (plotted as circles in the inset to Fig. 11) yields an unacceptably large relative error of 0.5. At such low areal densities ( $n \sigma ^2 = 0.0684$), $u(r)$ is very similar to the potential of mean force, $w(r) = - k_B T \, \ln g(r)$, and very small changes in $u(r)$ can influence the sum rule's integrand substantially. In fact, adjusting the potential just slightly to the function plotted as squares in Fig. 11 improves the sum rule's convergence to 0.001. This modified potential still successfully collapses the configurational and hyperconfigurational temperatures, and thus may be considered an improved estimate for the pair potential. This successful application of the thermodynamic sum rule suggests that the system is indeed in equilibrium, and that the candidate pair potential, including its long-ranged attraction, accurately describes the confined particles' interactions.

Figure 11: Integrand of the sum rule in Eq. (32) for the confined silica monolayer at $H = 9~\ensuremath{\unit{\mu m}}\xspace$. Inset: Best fit pair potential from Fig. 1 (circles) and another estimate (squares) consistent with both the confidence interval of $u(r)$ and with convergence of the configurational temperatures to the thermodynamic value. The former gives a relative error of 0.5 in the sum rule, and the latter 0.001.