From fc8d1b9c4678dbc5370641f6f7a75e29e6a82b04 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 5 Aug 2017 16:04:49 -0400
Subject: changed paper and figures to incorporate new definition of scaling
functions
---
.gitignore | 1 +
essential-ising.tex | 11 +++++------
figs/fig-susmag.gplot | 34 +++++++++++++++++-----------------
3 files changed, 23 insertions(+), 23 deletions(-)
diff --git a/.gitignore b/.gitignore
index 137e59e..e413095 100644
--- a/.gitignore
+++ b/.gitignore
@@ -4,6 +4,7 @@
*.blg
*Notes.bib
*.dvi
+essential-ising.pdf
figs/*.tex
fig-*.tex
diff --git a/essential-ising.tex b/essential-ising.tex
index c98e938..b7f432c 100644
--- a/essential-ising.tex
+++ b/essential-ising.tex
@@ -267,10 +267,10 @@ moments can still be extracted, e.g., the susceptibility, by taking
\chi=\pd MH=-\frac1{T_\c}\pd[2]Fh
=-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'.
\]
-With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$,
+With a scaling form defined by $T_\c\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$,
this yields
\[
- \fX^\twodee(Y/B)=\frac{AB^2}{\pi T_\c Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big]
+ \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big]
\label{eq:sus_scaling}
\]
Scaling forms for the free energy can then be extracted by direct integration
@@ -320,13 +320,12 @@ single curve, is plotted in Fig.~\ref{fig:scaling_fits}.
For the \twodee Ising model on a square lattice, exact results at zero
temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$
-\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-/T_\c$ with
-$C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that
+\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that
$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we
assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of
the susceptibility scaling function, then
-$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi T_\c$ and the constant
-$A$ is fixed to $A=\pi T_\c\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The
+$T_\c\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant
+$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The
resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in
Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement
between our proposed functional forms and what is measured. However, there
diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot
index db33997..a639f4d 100644
--- a/figs/fig-susmag.gplot
+++ b/figs/fig-susmag.gplot
@@ -29,7 +29,7 @@ GC(i) = G(i) * (2 * asinh(1))**2 * (Ch * (2 * asinh(1))**(-Delta))**i
M0 = (2**2.5 * asinh(1))**0.125
B = Tc**2 * M0 / (16 * pi)
C0 = 0.0255369719
-A = pi / 2 * C0 / (B**2 * Tc)
+A = pi / 2 * C0 / (B**2)
#c0 = -0.012384
#lamb = 1.76962
@@ -41,10 +41,10 @@ A = pi / 2 * C0 / (B**2 * Tc)
#A2 = pi / 2 * C0 / (B2**2 * Tc)
n = 1
-c(i) = i == 1 ? 0.0037735 : 0
+c(i) = i == 1 ? 0.00856277 : 0
lamb = 10.487
B2 = B
-A2 = 0.749317
+A2 = 1.70034
#n2 = 2
#c2(i) = i == 1 ? -0.177238 : \
@@ -71,14 +71,14 @@ A2 = 0.749317
#A3 = 0.88157
n2 = 5
-c2(i) = i == 1 ? 0.00281829 : \
- i == 2 ? 0.000632215 : \
- i == 3 ? -0.0911689 : \
- i == 4 ? 1.93584 : \
- i == 5 ? -24.7397 : 0
+c2(i) = i == 1 ? 0.00639522 : \
+ i == 2 ? 0.00143461 : \
+ i == 3 ? -0.206879 : \
+ i == 4 ? 4.39278 : \
+ i == 5 ? -56.1389 : 0
lamb2 = 13.2489
B3 = B
-A3 = 0.845002
+A3 = 1.91747
susfunc = "figs/fig-sus_scaling-func.dat"
magfunc = "figs/fig-mag_scaling-func.dat"
@@ -91,15 +91,15 @@ set key off
set size 1,1 - 0.52
set origin 0,0.52
set xrange [-40:1200]
-set yrange [-0.4:14.4]
-set ylabel offset 1,0 '$\chi|t|^{\gamma}\times 10^3$'
+set yrange [-1:33]
+set ylabel offset 1,0 '$T_\c\chi|t|^{\gamma}\times 10^3$'
set lmargin 6
set xtics format ''
set mxtics 5
set mytics 5
set bmargin 0.2
-plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B2):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B2):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B2)**(i-1) / Tc)) with lines dt 5 lw 2 lc black
+plot num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B2):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B2):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B2)**(i-1))) with lines dt 5 lw 2 lc black
set bmargin -1
set tmargin 0.2
@@ -110,7 +110,7 @@ set ylabel offset 1,0 '$M|t|^{-\beta}$'
set xlabel '$h|t|^{-\beta\delta}$'
set xtics format '%g'
-plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black
+plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black
set logscale xy
set tmargin -1
@@ -118,8 +118,8 @@ set lmargin -1
set size 0.65,0.325
set origin 0.31,0.5 + 0.29 / 2
set xrange [0.0015:1900]
-set yrange [0.00002:0.02]
-set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$'
+set yrange [0.00002:0.08]
+set ylabel offset 2.5,0 '\footnotesize$T_\c\chi|t|^\gamma$'
set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$'
set mxtics 5
set xtics format '' -2,10,1000
@@ -127,7 +127,7 @@ set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$'
set mytics 5
set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01
-plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B):(-sum[i=2:8] GC(i) * i * (i-1) * (10**$1 / B)**(i-2) / Tc) with lines dt 5 lw 2 lc black
+plot num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B):(-sum[i=2:8] GC(i) * i * (i-1) * (10**$1 / B)**(i-2)) with lines dt 5 lw 2 lc black
unset logscale xy
set logscale x
@@ -138,5 +138,5 @@ set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$'
set ytics format '\footnotesize {%g}' 1.2,0.2,1.8
set mytics 5
-plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B2) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black
+plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B2) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black
--
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