diff options
| -rw-r--r-- | ref_response.tex | 53 |
1 files changed, 11 insertions, 42 deletions
diff --git a/ref_response.tex b/ref_response.tex index 69764dd..78524bc 100644 --- a/ref_response.tex +++ b/ref_response.tex @@ -86,49 +86,18 @@ As suggested, an inset with a zoom on the critical region has been added to the }\\ In order to check the agreement in our fit, we performed the fit with a moving -temperature window that cuts off at $T_\text{max}$. Our fits' parameters are -$x_i$ for $i = 1, \ldots, 5$ for $C_0 = x_1 - x_2 (T / \mathrm K)$, $x_3 = -b^2/a$, $x_4 = b^2/Dq_*^4$, and $x_5 = b \sqrt{-g/u}$. The variation of these -parameters as a function of $T_\text{max}$ are shown on the top of -Figure \ref{fig:parameter_cutoff}. The parameter $x_1$ is fairly stable at all -temperature cutoffs, while the rest vary by 1.5--$2.5\times$ their $275\,\mathrm K$ value -down to cutoffs of $\sim90\,\mathrm K$. The fit functions that result from varying the cutoff are shown in Figure \ref{fig:parameter_curve}. +temperature window that cuts off at $T_\text{max}$. The fit functions that +result from varying the cutoff are shown in Figure \ref{fig:parameter_curve}. +The result is fairly robust until $T_\text{max}$ is reduced to around +$90\,\mathrm K$. As with any nonlinear least-squares fit, some linear +combinations of parameters are stiff and well-constrained while others are +sloppy and very poorly constrained. The parameters that vary most with the +temperature window are those that belong to the sloppiest linear combinations. +Notably, the parameters that make up the bare modulus $C_0$---and therefore +those responsible for transforming the experimental data in Figure 2(d)---are +robust as a function of the temperature cutoff and therefore so is the modified +Curie--Weiss behavior observed in that figure.\\ -\begin{figure} - \centering - \includegraphics[width=0.7\textwidth]{referee_response_cutoff_parameters.pdf} - \includegraphics[width=0.7\textwidth]{referee_response_cutoff_eigenvectors.pdf} - \caption{ - Fit parameters as a function of the cutoff temperature $T_\text{max}$. - (Top) Bare fit parameters corresponding to ratios of Landau coefficients. - (Bottom) Linear combinations of bare fit parameters corresponding to - eigendirections of the covariance matrix at $T_\text{max}=275\,\mathrm K$. - } - \label{fig:parameter_cutoff} -\end{figure} - -More insight into the consistency of the fit comes from examining the linear -combinations of parameters that form eigenvectors of the fit covariance matrix, -since---unlike the natural parameters of the mean field theory---these have -uncorrelated uncertainties. For the fit including all temperatures (up to -$275\,\mathrm K$), these are (in order of fit uncertainty): -\begin{align*} - y_1 &= -0.0020 x_1 + 2.2 \times 10^{-6} x_2 - 1.0 x_3 - 0.0023 x_4 - 0.0056 x_5 \\ - y_2 &= -0.015 x_1 + 0.000042 x_2 - 0.0055 x_3 - 0.021 x_4 + 1.0 x_5 \\ - y_3 &= -0.64 x_1 + 0.0020 x_2 + 0.0032 x_3 - 0.77 x_4 - 0.025 x_5 \\ - y_4 &= -0.77 x_1 + 0.0066 x_2 + 0.000075 x_3 + 0.64 x_4 + 0.0014 x_5 \\ - y_5 &= 0.0064 x_1 + 1.0 x_2 - 4.3 \times 10^{-6} x_3 - 0.0027 x_4 - 4.9 \times 10^{-7} x_5 -\end{align*} -The variation of these parameter combinations as a function of $T_\text{max}$ -are shown on the bottom of Figure \ref{fig:parameter_cutoff}. The parameter -$y_1$, which is principally $x_3 = a/b^2$, varies the most with the cutoff, at -most around $2\times$ its value until $\sim90\,\mathrm K$. The parameter $y_2$, -which is principally $x_5 = b \sqrt{-g/u}$, varies at most around $1.25\times$ -its value until $\sim90\,\mathrm K$. The other three parameters are stable at -any cutoff, and are mixed combinations of $x_1$, $x_2$, and $x_4$. -Notably, $x_1$ and $x_2$ are the only parameters involved in transforming the -experimental data in Figure 2(d), and their stability as a function of the data -window means that transformation is likewise stable.\\ \begin{figure} \centering |