summaryrefslogtreecommitdiff
path: root/ising_scaling.tex
diff options
context:
space:
mode:
authorJaron Kent-Dobias <jaron@kent-dobias.com>2021-10-25 13:52:09 +0200
committerJaron Kent-Dobias <jaron@kent-dobias.com>2021-10-25 13:52:09 +0200
commite595268052c136ead38b223022a5c19de0ee1d1c (patch)
treeace9ceaa8e5ccc30ea929098320080b64cd46bb0 /ising_scaling.tex
parent3d09f2ae676ad17423038e554e34079ea51d89a7 (diff)
downloadpaper-e595268052c136ead38b223022a5c19de0ee1d1c.tar.gz
paper-e595268052c136ead38b223022a5c19de0ee1d1c.tar.bz2
paper-e595268052c136ead38b223022a5c19de0ee1d1c.zip
Removed last traces of h and t functions.
Diffstat (limited to 'ising_scaling.tex')
-rw-r--r--ising_scaling.tex44
1 files changed, 23 insertions, 21 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex
index 5630fe4..ca1bd15 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -328,13 +328,15 @@ argument for all real $\theta$ by
\end{equation}
For small $\theta$, $\mathcal F(\theta)$ will
resemble $\mathcal F_+$, for $\theta$ near one it will resemble $\mathcal F_0$,
-and for $\theta$ near $\theta_0$ it will resemble $\mathcal F_-$. This can be seen explicitly using the definitions \eqref{eq:schofield} to relate the above form to the original scaling functions, giving
+and for $\theta$ near $\theta_0$ it will resemble $\mathcal F_-$. This can be
+seen explicitly using the definitions \eqref{eq:schofield} to relate the above
+form to the original scaling functions, giving
\begin{equation} \label{eq:scaling.function.equivalences.2d}
\begin{aligned}
\mathcal F(\theta)
- &=|t(\theta)|^{D\nu}\mathcal F_\pm\left[g(\theta)|1-\theta^2|^{-\Delta}\right]
- +\frac{(1-\theta^2)^2}{8\pi}\log t(\theta)^2\\
- &=|h(\theta)|^{D\nu/\Delta}\mathcal F_0\left[(1-\theta^2)|g(\theta)|^{-1/\Delta}\right]
+ &=|1-\theta^2|^{D\nu}\mathcal F_\pm\left[g(\theta)|1-\theta^2|^{-\Delta}\right]
+ +\frac{(1-\theta^2)^2}{8\pi}\log(1-\theta^2)^2\\
+ &=|g(\theta)|^{D\nu/\Delta}\mathcal F_0\left[(1-\theta^2)|g(\theta)|^{-1/\Delta}\right]
+\frac{(1-\theta^2)^2}{8\pi}\log g(\theta)^{2/\Delta}
\end{aligned}
\end{equation}
@@ -578,22 +580,22 @@ 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}
- $m$ & $\mathcal F_-^{(m)}$ & $\mathcal F_0^{(m)}$ & $\mathcal F_+^{(m)}$ \\\hline
- 0 & 0 & $-1.197733383797993$ & 0 \\
- 1 & $-1.35783834$ & $-0.318810124891$ & 0 \\
- 2 & $-0.048953289720$ & $0.110886196683$ & $-1.84522807823$ \\
- 3 & 0.0388639290 & $0.01642689465$ & 0 \\
- 4 & $-0.068362121$ & $-2.639978\times10^{-4}$ & 8.3337117508 \\
- 5 & 0.18388371 & $-5.140526\times10^{-4}$ & 0 \\
- 6 & $-0.659170$ & $2.08856\times 10^{-4}$ & $-95.16897$ \\
- 7 & 2.937665 & $-4.4819\times10^{-5}$ & 0 \\
- 8 & $-15.61$ & $3.16\times10^{-7}$ & 1457.62 \\
- 9 & 96.76 & $4.31\times10^{-6}$ & 0 \\
- 10 & $-679$ & $-1.99\times10^{-6}$ & -25891 \\
- 11 & $5.34\times10^3$ & & 0 \\
- 12 & $-4.66\times10^4$ & & $5.02\times10^5$ \\
- 13 & $4.46\times10^5$ & & 0 \\
+ \begin{tabular}{c|lll}
+ $m$ & \multicolumn{1}{c}{$\mathcal F_-^{(m)}$} & $\mathcal F_0^{(m)}$ & $\mathcal F_+^{(m)}$ \\\hline
+ 0 & \hphantom{$-$}0 & $-1.197\,733\,383\,797\ldots$ & \hphantom{$-$}0 \\
+ 1 & $-1.357\,838\,341\,707\ldots$ & \hphantom{$-$}$0.318\,810\,124\,891\ldots$ & \hphantom{$-$}0 \\
+ 2 & $-0.048\,953\,289\,720\ldots$ & \hphantom{$-$}$0.110\,886\,196\,683(2)$ & $-1.845\,228\,078\,233\ldots$ \\
+ 3 & \hphantom{$-$}$0.038\,863\,932(3)$ & $-0.016\,426\,894\,65(2)$ & \hphantom{$-$}0 \\
+ 4 & $-0.068\,362\,119(2)$ & $-2.639\,978(1)\times10^{-4}$ & \hphantom{$-$}$8.333\,711\,750(5)$ \\
+ 5 & \hphantom{$-$}$0.183\,883\,70(1)$ & \hphantom{$-$}$5.140\,526(1)\times10^{-4}$ & \hphantom{$-$}0 \\
+ 6 & $-0.659\,171\,4(1)$ & \hphantom{$-$}$2.088\,65(1)\times 10^{-4}$ & $-95.168\,96(1)$ \\
+ 7 & \hphantom{$-$}$2.937\,665(3)$ & \hphantom{$-$}$4.481\,9(1)\times10^{-5}$ & \hphantom{$-$}0 \\
+ 8 & $-15.61(1)$ & \hphantom{$-$}$3.16\times10^{-7}$ & \hphantom{$-$}1457.62(3) \\
+ 9 & \hphantom{$-$}96.76 & $-4.31\times10^{-6}$ & \hphantom{$-$}0 \\
+ 10 & $-679$ & $-1.99\times10^{-6}$ & $-25\,891(2)$ \\
+ 11 & \hphantom{$-$}$5.34\times10^3$ & & \hphantom{$-$}0 \\
+ 12 & $-4.66\times10^4$ & & \hphantom{$-$}$5.02\times10^5$ \\
+ 13 & \hphantom{$-$}$4.46\times10^5$ & & \hphantom{$-$}0 \\
14 & $-4.66\times10^6$ & & $-1.04\times10^7$
\end{tabular}
\end{table}
@@ -823,7 +825,7 @@ accuracy of the fit results can be checked against the known values here.
\section{Outlook}
-The successful smooth description of the Ising free energy produced in part by analytically continuing the singular imaginary part of the metastable free energy inspires an extension of this work: a smooth function that captures the universal scaling \emph{through the coexistence line and into the metastable phase}. Indeed, the tools exist to produce this: by writing $t(\theta)=(1-\theta^2)(1-(\theta/\theta_m)^2)$ for some $\theta_m>\theta_0$, the invariant scaling combination
+The successful smooth description of the Ising free energy produced in part by analytically continuing the singular imaginary part of the metastable free energy inspires an extension of this work: a smooth function that captures the universal scaling \emph{through the coexistence line and into the metastable phase}. Indeed, the tools exist to produce this: by writing $t=R(1-\theta^2)(1-(\theta/\theta_m)^2)$ for some $\theta_m>\theta_0$, the invariant scaling combination
\begin{acknowledgments}
The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien