From 0000fd8c04698cc1406e6a26dd20a7ce025d7450 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 19 Oct 2021 17:43:53 +0200
Subject: Writing.
---
data/phi_comparison.dat | 10 ++++++++++
ising_scaling.tex | 48 +++++++++++++++++++++++++++++++++++++++++++-----
2 files changed, 53 insertions(+), 5 deletions(-)
create mode 100644 data/phi_comparison.dat
diff --git a/data/phi_comparison.dat b/data/phi_comparison.dat
new file mode 100644
index 0000000..9193688
--- /dev/null
+++ b/data/phi_comparison.dat
@@ -0,0 +1,10 @@
+2 0.002860955318525926 0.004496459219585747 0.0025781014469987568 0.0004361990091461404
+3 0.0005720429508622171 0.0010847239134089692 0.000805365486839224 0.00006427186448818359
+4 0.00003961608489011503 0.0001278039316774393 0.00018174532718064074 0.00013408467605927413
+5 0.0000622987443403833 0.00016919055775577174 0.0002085264051783775 0.0001300161350704411
+6 0.00005016392362722222 0.00014150435296594877 0.00016732830408854038 0.00007562595035311148
+7 0.000015703311988302104 0.00005762691693961264 0.00009304388663239349 0.00007579125219901034
+8 8.173890766238756e-6 0.000021607891761421527 0.000022883111337773654 6.9772437447900015e-6
+9 2.4873158455118727e-6 7.768088409076945e-6 9.816343265717231e-6 3.88602466879287e-6
+10 3.7198724605058686e-6 0.000011866786064407275 0.00001594714070687897 8.677026311975505e-6
+11 0.00006257848162638524 0.00006629563696586294 0.0000134012475765527 0.0000386060655095978
\ No newline at end of file
diff --git a/ising_scaling.tex b/ising_scaling.tex
index 2301e73..be5364d 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -491,11 +491,29 @@ This leaves as unknown variables the positions $\theta_0$ and
$\theta_{\mathrm{YL}}$ of the abrupt transition and Yang--Lee edge singularity,
the amplitude $A_\mathrm{YL}$ of the latter, and the unknown functions $F$ and
$h$. We determine these approximately by iteration in the polynomial order at
-which the free energy and its derivative matches known results. Gradients can be computed with
-
-A Levenburg--Marquardt algorithm is performed
-
-\begin{table}
+which the free energy and its derivative matches known results. We write as a
+cost function the difference between the known series coefficients of the
+scaling functions $\mathcal F_\pm$ and the series coefficients of our
+parametric form evaluated at the same points, $\theta=0$ and $\theta=\theta_c$,
+weighted by the uncertainty in the value of the known coefficients or by a
+machine-precision cutoff, whichever is larger. A Levenburg--Marquardt algorithm
+is performed on the cost function to find a parameter combination which
+minimizes it. As larger polynomial order in the series are fit, the truncations
+of $F$ and $h$ are extended to higher order so that the codimension of the fit
+is constant. A term is added to $F$ whenever a new coefficient of the high
+temperature series is added, and one is added to $h$ whenever a new coefficient
+of the low temperature series is added.
+
+We performed this procedure starting with $n=2$, or matching the scaling
+function at the low and high temperature zero field points to quadratic order,
+through $n=9$. The resulting fit coefficients can be found in Table
+\ref{tab:fits} without any sort of uncertainty, which is difficult to quantify
+directly due to the truncation of series. However, precise results exist for
+the value of the scaling function at the critical isotherm, or equivalently for
+the series coefficients of the scaling function $\mathcal F_0$, and the
+accuracy of the fit results can be checked against the known values here.
+
+\begin{table}\label{tab:fits}
\begin{tabular}{c|ccc}
$n$ & $\mathcal F_-^{(n)}$ & $\mathcal F_0^{(n)}$ & $\mathcal F_+^{(n)}$ \\\hline
0 & 0 & $-1.197733383797993$ & 0 \\
@@ -619,6 +637,26 @@ A Levenburg--Marquardt algorithm is performed
\end{tabular}
\end{table}
+\begin{figure}
+ \begin{gnuplot}[terminal=epslatex, terminaloptions={size 8.65cm,5.35cm}]
+ dat = 'data/phi_comparison.dat'
+
+ set xlabel '$n$'
+ set ylabel '$|\mathcal F_0^{(n)}-|$'
+
+ set style data linespoints
+ set logscale y
+
+ plot \
+ dat using 1:2 title '0', \
+ dat using 1:3 title '1', \
+ dat using 1:4 title '2', \
+ dat using 1:5 title '3'
+ \end{gnuplot}
+ \caption{
+ }
+\end{figure}
+
\begin{figure}
\begin{gnuplot}[terminal=epslatex, terminaloptions={size 8.65cm,5.35cm}]
dat9 = 'data/h_series_ours_9.dat'
--
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