From fc8d1b9c4678dbc5370641f6f7a75e29e6a82b04 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias 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 -- cgit v1.2.3-54-g00ecf