From ee4dad82736f44449bbc32d11b3edf8a94ce308d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 27 Oct 2021 18:14:42 +0200 Subject: New data in plots with much better convergence. --- data/ghigh_series_ours_2.dat | 14 +++++++------- data/ghigh_series_ours_6.dat | 15 +++++++++++++++ data/glow_series_ours_2.dat | 28 ++++++++++++++-------------- data/glow_series_ours_6.dat | 15 +++++++++++++++ data/phi_comparison.dat | 11 +++++------ data/phi_series_ours_2.dat | 30 +++++++++++++++--------------- data/phi_series_ours_6.dat | 15 +++++++++++++++ ising_scaling.tex | 44 +++++++++++++++++++++++++++++--------------- 8 files changed, 115 insertions(+), 57 deletions(-) create mode 100644 data/ghigh_series_ours_6.dat create mode 100644 data/glow_series_ours_6.dat create mode 100644 data/phi_series_ours_6.dat 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{ -- cgit v1.2.3-70-g09d2