a^{2} \nabla^{2} S^{(3)}_i + \mu E_i,$$ with $J_0$ defined by equation . This can now be substituted into equation .

Keeping only terms that are directly comparable to those in equation , we obtain $$\begin{aligned}
A & = \frac{\Omega}{Sf_{S}(\Omega)} - \frac{J_{0}}{S}, \label{coeff_A} \\
B & = -\frac{1}{2 S f_{S}(\Omega)} \left( \frac{J_{0}}{S} \right)^{3} \frac{d}{d\Omega} \left( \frac{S f_S (\Omega)}{\Omega} \right), \label{coeff_B} \\
C & = \frac{J_{1} a^{2}}{S}, \label{coeff_C} \\
D & = \mu. \label{coeff_D}\end{aligned}$$ These equations show that $A$ and $B$ are determined by combinations of $J_0$ and $\Omega$, while $C$ and $D$ are determined by $J_1$ and $\mu$, respectively. The key point is that $J_0$ reduces to $ZJ_1$ for the conventional TIM, in which case $A$ and $C$ are not independent. Physically, this means that the correlation length, which sets the length scale over which the material responds to inhomogeneities, cannot be determined independently of the transition temperature and low-$T$ polarization. In other words, the four coefficients $A$, $B$, $C$ and $D$ are only described by three parameters, $\Omega$, $J_{1}$ and $\mu$.

In this case, the model predicts a significantly smaller correlation length at low temperatures than does the modified TIM. From equations (\[coeff\_A\]) and (\[coeff\_C\]), $$\xi = \sqrt{\frac{C}{A}} = \sqrt{\frac{J_1 