Checking for isotropy

Comparing temperatures measured along orthogonal directions not only provides a useful check for the interactions' isotropy, but also can be used to appraise the imaging system. For an isotropic sample, we expect that

$$\frac{\Delta T}{\left< T \right>} \equiv 2 \, \frac{T_x - T_y}{T_x + T_y}$$ (34)

will vanish for arbitrary choices of $u(r)$. More usually, the combination of a commercial video camera and video frame grabber results in slightly different length scale calibrations. Even quite small differences measurably affect the temperatures calculated from the associated force components if the interaction is assumed to be isotropic. For example, we found that $\Delta T = -0.025~\left< T \right>$ for the confined sample at $H = 9$ and $\Delta T = 0.06~\left< T \right>$ for the sample at $H = 195~\ensuremath{\unit{\mu m}}\xspace$. Even such small differences are plainly visible in Fig. 7(c).

Apparent anisotropies of this magnitude appear consistently in our data sets regardless of the samples' composition, concentration, degree of confinement, and so are unlikely to reflect statistical errors. Nor are they likely to signal a real anisotropy in our samples' interactions. Instead, they result from the hyperconfigurational temperatures' sensitivity to subtle geometric distortion in our imaging system. Rescaling the measured $x$ and $y$ coordinates slightly can substantially reduce the apparent anisotropy in the entire hierarchy of hyperconfigurational temperatures, as the data in Table 1 show. For the system used in this study, a 0.7% correction of the $x:y$ scale ratio is enough to account for the 5.8% anisotropy of $T_h^{(1)}$ in the $H = 195~\ensuremath{\unit{\mu m}}\xspace$ data.

The same scaling factor also corrects the apparent anisotropy in the other samples we have studied, and thus appears to be correctly interpreted as a correction to the calibration of our imaging system. Furthermore, differences in the scaling factors as small as $\pm 0.1\%$ perform substantially less well, as shown by the data in Table 1. This level of sensitivity greatly exceeds the typical 1% calibration accuracy obtained by imaging test patterns, and thus provides a new tool for assessing and correcting geometric defects in the digital video microscopy system.

Table: Correcting apparent anisotropy in the hyperconfigurational temperatures of the $H = 195~\ensuremath{\unit{\mu m}}\xspace$ data set by rescaling coordinates. In each case, $g(r)$, $u(r)$ and $T_h^{(s)}$ were recalculated with revised particle locations.

		$x:y$ scaling factor
$\frac{\Delta T}{\left< T \right>}$		1:1	1.003:0.997	1.003:0.9963	1.003:0.996
	1	5.8%	0.61%	-0.07%	-0.82%
$s$	3	4.4%	0.87%	0.009%	-0.20%
	5	2.6%	1.3%	0.16%	0.90%
	7	1.4%	1.8%	0.65%	0.71%

Successfully correcting apparent anisotropy in the hyperconfigurational temperatures also provides insight into the nature of the system's interactions. If replacing $u(r)$ by an arbitrary function causes $T_x$ to differ from $T_y$ then the system indeed may be anisotropic, either because its interactions are anisotropic, or else because of its response to an external field. In the latter case, the external field contributes an additional configuration-dependent term to the Hamiltonian, ${\mathcal H}_{ext}({\mathbf \Gamma})$, which contributes, in turn, to the definitions of the configurational temperatures. If $\partial_x {\mathcal H}_{ext}({\mathbf \Gamma}) \neq \partial_y {\mathcal H}_{ext}({\mathbf \Gamma})$, then the configurational temperatures along the two directions generally will differ.