New data in plots with much better convergence.

8 files changed, 115 insertions, 57 deletions

diff --git a/data/ghigh_series_ours_2.dat b/data/ghigh_series_ours_2.dat
index 4f3dd27..47e089f 100644
--- a/data/ghigh_series_ours_2.dat
+++ b/data/ghigh_series_ours_2.dat
@@ -1,15 +1,15 @@
 0	0
 1	0
-2	-1.845228078232858
+2	-1.8452280782328416
 3	0
-4	8.30011667399619
+4	8.377514122546064
 5	0
-6	-93.91176560012791
+6	-96.52148911058097
 7	0
-8	1429.3561030820913
+8	1493.9976737243942
 9	0
-10	-25294.653782288846
+10	-26818.45130191777
 11	0
-12	490304.7125789622
+12	526377.7118305506
 13	0
-14	-1.0099250773153434e7
\ No newline at end of file
+14	-1.0966384870468803e7
\ No newline at end of file
diff --git a/data/ghigh_series_ours_6.dat b/data/ghigh_series_ours_6.dat
new file mode 100644
index 0000000..efa49ac
--- /dev/null
+++ b/data/ghigh_series_ours_6.dat
@@ -0,0 +1,15 @@
+0	0
+1	0
+2	-1.8452280782328523
+3	0
+4	8.333711749926277
+5	0
+6	-95.16896371253469
+7	0
+8	1457.5503638974826
+9	0
+10	-25869.797578475864
+11	0
+12	501951.0982257904
+13	0
+14	-1.033917515714893e7
\ No newline at end of file
diff --git a/data/glow_series_ours_2.dat b/data/glow_series_ours_2.dat
index f9927e8..3385257 100644
--- a/data/glow_series_ours_2.dat
+++ b/data/glow_series_ours_2.dat
@@ -1,15 +1,15 @@
 0	0
-1	-1.3578383417064466
-2	-0.048953289719890325
-3	0.04145804952829153
-4	-0.07859966533925074
-5	0.2198470108765085
-6	-0.790971823869045
-7	3.4711729442114767
-8	-18.069245794661075
-9	109.29036827414234
-10	-754.7713051203647
-11	5864.4413713769845
-12	-50640.97000480466
-13	481133.8191124413
-14	-4.987584839224958e6
\ No newline at end of file
+1	-1.3578383417062967
+2	-0.04895328972041035
+3	0.040152590897806796
+4	-0.07634150977872403
+5	0.22553840411972031
+6	-0.8705719288881231
+7	4.0294426461487145
+8	-21.503792136560655
+9	130.1609902246785
+10	-886.8731778001119
+11	6762.939912929589
+12	-57330.54580849964
+13	536110.1451178609
+14	-5.486439916757801e6
\ No newline at end of file
diff --git a/data/glow_series_ours_6.dat b/data/glow_series_ours_6.dat
new file mode 100644
index 0000000..555f1c3
--- /dev/null
+++ b/data/glow_series_ours_6.dat
@@ -0,0 +1,15 @@
+0	0
+1	-1.357838341706558
+2	-0.04895328972070345
+3	0.038863930625903244
+4	-0.06836211971140452
+5	0.1838836984300239
+6	-0.6591714228830016
+7	2.938563888747659
+8	-15.659413828476081
+9	97.6039830158417
+10	-705.9286232118421
+11	5933.433358981995
+12	-58038.580008779725
+13	653987.4117822968
+14	-8.280938794522655e6
\ No newline at end of file
diff --git a/data/phi_comparison.dat b/data/phi_comparison.dat
index d111f53..7821568 100644
--- a/data/phi_comparison.dat
+++ b/data/phi_comparison.dat
@@ -1,6 +1,5 @@
-2	0.0015185988806263317	0.0016424512339846542	0.00010931409748554666	0.0014019918834638455	0.0009799159023033766
-3	0.0007353808953864949	0.0014547698806950948	0.0011618922642676333	0.0001926030528925822	0.0004716780917582721
-4	0.0006832169448554026	0.0015493096608610868	0.001202074914739018	0.00024563197897510894	0.0011902522778089069
-5	0.0001498262988313126	0.00034058740497194817	0.0003177652580080431	0.00008748091339953398	0.00012570023181056306
-6	9.052199169801156e-6	0.00003198782581631532	0.00005362053295303315	0.00005162642890723171	0.00002227019515451439
-7	5.551832441996751e-6	0.000013915434556921458	9.941126498061448e-6	0.000011350077847509937	0.00003260973372059311
\ No newline at end of file
+2	0.0002955829708120028	0.0002625212059513826	0.000028239602411814224	0.00008275586984924158	0.00005011431128301583
+3	0.00004906258666981245	0.00006482894953951934	0.000028531647414078987	6.68686237088878e-6	0.00001369543769757386
+4	2.148412799218846e-6	2.603799919176364e-6	6.527516478171957e-7	8.371111562578692e-7	7.601714303177897e-7
+5	7.595046134234451e-7	1.303033890176497e-6	8.262734071118327e-7	1.228725780497153e-8	3.3812614840688813e-7
+6	6.932139018367423e-8	1.2168572560700142e-7	7.317986536192755e-8	1.1355372860760715e-8	4.499344289949237e-8
\ No newline at end of file
diff --git a/data/phi_series_ours_2.dat b/data/phi_series_ours_2.dat
index f16c881..beedc59 100644
--- a/data/phi_series_ours_2.dat
+++ b/data/phi_series_ours_2.dat
@@ -1,15 +1,15 @@
-0	-1.1962147849173665
-1	-0.3204525761245943
-2	0.11077688258551482
-3	0.017828886533464094
-4	-0.0012439137023040104
-5	-0.0006436835676704981
-6	0.0008119273579228864
-7	-0.0003697032267769685
-8	-0.00006515301480336557
-9	0.0001753495754505909
-10	-0.00007628792820886294
-11	-0.000016198436209520737
-12	0.000032017927181286665
-13	-0.000012952206779860703
-14	-2.4795512728255315e-7
\ No newline at end of file
+0	-1.197437800827181
+1	-0.3190726460965614
+2	0.11091443628541181
+3	0.016509650519849282
+4	-0.0003141121112829187
+5	-0.0005146254923651025
+6	0.0002221366986115451
+7	-0.00005010818239482555
+8	-5.828936646513926e-7
+9	5.746698725714958e-6
+10	-2.317572453533117e-6
+11	2.8319049007218993e-7
+12	1.3282081081853632e-7
+13	-7.401803626641128e-8
+14	1.6095359608980834e-8
\ No newline at end of file
diff --git a/data/phi_series_ours_6.dat b/data/phi_series_ours_6.dat
new file mode 100644
index 0000000..4d46109
--- /dev/null
+++ b/data/phi_series_ours_6.dat
@@ -0,0 +1,15 @@
+0	-1.197733314476603
+1	-0.3188102465763356
+2	0.11088626986286532
+3	0.016426906005372888
+4	-0.00026404279344288684
+5	-0.0005140260159366009
+6	0.00020886534275493685
+7	-0.000044828373716923474
+8	3.2472263422728396e-7
+9	4.312795231984893e-6
+10	-1.989930723631579e-6
+11	4.3398967612728695e-7
+12	3.0343615844906865e-8
+13	-6.741502699179006e-8
+14	2.8695485833852726e-8
\ No newline at end of file
diff --git a/ising_scaling.tex b/ising_scaling.tex
index 909d99f..744f2d5 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -37,6 +37,17 @@ linkcolor=purple
 \date\today
 
 \begin{abstract}
+  We describe a method for approximating the universal scaling functions for
+  the Ising model in a field. By making use of parametric coordinates, the free
+  energy scaling function has a polynomial series everywhere. Its form is
+  taken to be a sum of the simplest functions that contain the singularities
+  which must be present: the Langer essential singularity and the Yang--Lee
+  edge singularity. Requiring that the function match series expansions in
+  the low- and high-temperature zero-field limits fixes the parametric
+  coordinate transformation. For the two-dimensional Ising model, we show that
+  this procedure converges exponentially with the order to which the series are
+  matched. We speculate that with appropriately modified parametric
+  coordinates, the method might also converge deep in the metastable phase.
 \end{abstract}
 
 \maketitle
@@ -494,7 +505,7 @@ and
 \end{equation}
 We have also included the analytic part $G$, which we assume has a simple
 series expansion
-\begin{equation}
+\begin{equation} \label{eq:analytic.free.enery}
   G(\theta)=\sum_{i=1}^\infty G_i\theta^{2i}
 \end{equation}
 From the form of the real part, we can infer the form of $\mathcal F$ that is
@@ -517,8 +528,8 @@ where
 The scaling function has a number of free parameters: the position $\theta_0$
 of the abrupt transition, prefactors in front of singular functions from the
 abrupt transition and the Yang--Lee point, the coefficients in the analytic
-part $G$ of $\mathcal F$, and the coefficients in the undetermined function
-$g$.  Other parameters are determined by known properties.
+part $G$ of $\mathcal F$, and the coefficients in the undetermined coordinate
+function $g$.  Other parameters are determined by known properties.
 
 For $\theta>\theta_0$, the form \eqref{eq:essential.singularity} can be
 expanded around $\theta=\theta_0$ to yield
@@ -581,7 +592,8 @@ machine-precision cutoff, whichever is larger. We also add the difference
 between the predictions for $A_\mathrm{YL}$ and $\xi_\mathrm{YL}$ and their
 known numeric values, again weighted by their uncertainty. In order to
 encourage convergence, we also add to the cost the weighted coefficients
-$j!g_j$ and $j!G_j$.
+$j!g_j$ and $j!G_j$ defining the function $g$ and $G$ in
+\eqref{eq:schofield.funcs} and \eqref{eq:analytic.free.enery}.
 
 A Levenberg--Marquardt algorithm is performed on the cost function to find a
 parameter combination which minimizes it. As larger polynomial order in the
@@ -781,12 +793,12 @@ to their known values at the critical isotherm, or $\theta=1$.
     dat = 'data/phi_comparison.dat'
 
     set xlabel '$n$'
-    set xrange [1.5:7.5]
+    set xrange [1.5:6.5]
 
     set logscale y
     set format y '$10^{%T}$'
     set ylabel '$|\Delta\mathcal F_0^{(m)}|$'
-    set yrange [0.000002:0.003]
+    set yrange [0.000000005:0.0005]
 
     set style data linespoints
     set key title '\raisebox{0.5em}{$m$}' bottom left
@@ -801,7 +813,9 @@ to their known values at the critical isotherm, or $\theta=1$.
   \caption{
     The error in the $m$th derivative of the scaling function $\mathcal F_0$
     with respect to $\eta$ evaluated at $\eta=0$, as a function of the
-    polynomial order $n$ at which the scaling function was fit.
+    polynomial order $n$ at which the scaling function was fit. The point
+    $\eta=0$ corresponds to the critical isotherm at $T=T_c$ and $H>0$, roughly
+    midway between the limits used in the fit at $H=0$ and $T\neq T_c$.
   } \label{fig:error}
 \end{figure}
 
@@ -824,7 +838,7 @@ Fig.~\ref{fig:phi.series}.
   \begin{gnuplot}[terminal=epslatex]
     dat1 = 'data/glow_numeric.dat'
     dat2 = 'data/glow_series_ours_0.dat'
-    dat3 = 'data/glow_series_ours_7.dat'
+    dat3 = 'data/glow_series_ours_6.dat'
     dat4 = 'data/glow_series_caselle.dat'
 
     set xlabel '$m$'
@@ -838,7 +852,7 @@ Fig.~\ref{fig:phi.series}.
     plot \
       dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \
       dat2 using 1:(abs($2)) title 'This work ($n=2$)', \
-      dat3 using 1:(abs($2)) title 'This work ($n=7$)', \
+      dat3 using 1:(abs($2)) title 'This work ($n=6$)', \
       dat4 using 1:(abs($2)) title 'Caselle \textit{et al.}'
   \end{gnuplot}
   \caption{
@@ -853,7 +867,7 @@ Fig.~\ref{fig:phi.series}.
   \begin{gnuplot}[terminal=epslatex]
     dat1 = 'data/ghigh_numeric.dat'
     dat2 = 'data/ghigh_series_ours_2.dat'
-    dat3 = 'data/ghigh_series_ours_7.dat'
+    dat3 = 'data/ghigh_series_ours_6.dat'
     dat4 = 'data/ghigh_caselle.dat'
 
     set key top left Left reverse
@@ -866,7 +880,7 @@ Fig.~\ref{fig:phi.series}.
     plot \
       dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \
       dat2 using 1:(abs($2)) title 'This work ($n=2$)', \
-      dat3 using 1:(abs($2)) title 'This work ($n=7$)', \
+      dat3 using 1:(abs($2)) title 'This work ($n=6$)', \
       dat4 using 1:(abs($2)) title 'Caselle \textit{et al.}'
   \end{gnuplot}
   \caption{
@@ -881,7 +895,7 @@ Fig.~\ref{fig:phi.series}.
   \begin{gnuplot}[terminal=epslatex]
     dat1 = 'data/phi_numeric.dat'
     dat2 = 'data/phi_series_ours_2.dat'
-    dat3 = 'data/phi_series_ours_9.dat'
+    dat3 = 'data/phi_series_ours_6.dat'
     set key top right
     set logscale y
     set xlabel '$m$'
@@ -892,7 +906,7 @@ Fig.~\ref{fig:phi.series}.
     plot \
       dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \
       dat2 using 1:(abs($2)) title 'This work ($n=2$)', \
-      dat3 using 1:(abs($2)) title 'This work ($n=7$)'
+      dat3 using 1:(abs($2)) title 'This work ($n=6$)'
   \end{gnuplot}
   \caption{
     The series coefficients for the scaling function $\mathcal F_0$ as a
@@ -915,7 +929,7 @@ the ratio.
   \begin{gnuplot}[terminal=epslatex]
     dat1 = 'data/glow_numeric.dat'
     dat2 = 'data/glow_series_ours_0.dat'
-    dat3 = 'data/glow_series_ours_7.dat'
+    dat3 = 'data/glow_series_ours_6.dat'
     dat4 = 'data/glow_series_caselle.dat'
     ratLast(x) = (back2 = back1, back1 = x, back1 / back2)
     back1 = 0
@@ -929,7 +943,7 @@ the ratio.
     plot \
       dat1 using (1/$1):(abs(ratLast($2))) title 'Numeric', \
       dat2 using (1/$1):(abs(ratLast($2))) title 'This work ($n=2$)', \
-      dat3 using (1/$1):(abs(ratLast($2))) title 'This work ($n=7$)', \
+      dat3 using (1/$1):(abs(ratLast($2))) title 'This work ($n=6$)', \
       dat4 using (1/$1):(abs(ratLast($2))) title 'Caselle \textit{et al.}'
   \end{gnuplot}
   \caption{