diff options
| -rw-r--r-- | data/phi_comparison.dat | 10 | ||||
| -rw-r--r-- | ising_scaling.tex | 48 |
2 files changed, 53 insertions, 5 deletions
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 \\ @@ -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' |