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Diffstat (limited to 'ising_scaling.tex')
| -rw-r--r-- | ising_scaling.tex | 48 |
1 files changed, 43 insertions, 5 deletions
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 \\ @@ -621,6 +639,26 @@ A Levenburg--Marquardt algorithm is performed \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' dat11 = 'data/h_series_ours_11.dat' dat13 = 'data/h_series_ours_13.dat' |