diff options
| -rw-r--r-- | ising_scaling.tex | 83 |
1 files changed, 51 insertions, 32 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index 5c1612d..668ba63 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -797,7 +797,13 @@ error in the function and its first several derivatives appears to trend towards zero exponentially in the polynomial order $n$. Even at $n=2$, where only six unknown parameters have been fit, the results are -accurate to within $2\times10^{-3}$. This approximation for the scaling functions also captures the singularities at the high- and low-temperature zero-field points well. +accurate to within $2\times10^{-3}$. This approximation for the scaling +functions also captures the singularities at the high- and low-temperature +zero-field points well. A direct comparison between the magnitudes of the +series coefficients known numerically and those given by the approximate +functions is shown for $\mathcal F_-$ in Fig.~\ref{fig:glow.series}, for +$\mathcal F_+$ in Fig.~\ref{fig:ghigh.series}, and for $\mathcal F_0$ in +Fig.~\ref{fig:phi.series}. \begin{figure} \begin{gnuplot}[terminal=epslatex] @@ -830,35 +836,6 @@ accurate to within $2\times10^{-3}$. This approximation for the scaling function \begin{figure} \begin{gnuplot}[terminal=epslatex] - dat1 = 'data/glow_numeric.dat' - dat2 = 'data/glow_series_ours_0.dat' - dat3 = 'data/glow_series_ours_7.dat' - dat4 = 'data/glow_series_caselle.dat' - ratLast(x) = (back2 = back1, back1 = x, back1 / back2) - back1 = 0 - back2 = 0 - - set xlabel '$1/m$' - set xrange [0:0.55] - set ylabel '$\mathcal F_-^{(m)}/\mathcal F_-^{(m-1)}$' - set yrange [0:15] - - 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$)', \ - dat4 using (1/$1):(abs(ratLast($2))) title 'Caselle \textit{et al.}' - \end{gnuplot} - \caption{ - Sequential ratios of the series coefficients of the scaling function - $\mathcal F_-$ as a function of inverse polynomial order $m$. The - extrapolated $y$-intercept of this plot gives the radius of convergence of - the series. - } -\end{figure} - -\begin{figure} - \begin{gnuplot}[terminal=epslatex] dat1 = 'data/ghigh_numeric.dat' dat2 = 'data/ghigh_series_ours_2.dat' dat3 = 'data/ghigh_series_ours_7.dat' @@ -882,7 +859,7 @@ accurate to within $2\times10^{-3}$. This approximation for the scaling function function of polynomial order $m$. The numeric values are from Table \ref{tab:data}, and those of Caselle \textit{et al.} are from the most accurate scaling function listed in \cite{Caselle_2001_The}. - } + } \label{fig:ghigh.series} \end{figure} \begin{figure} @@ -903,7 +880,49 @@ accurate to within $2\times10^{-3}$. This approximation for the scaling function dat3 using 1:(abs($2)) title 'This work ($n=7$)' \end{gnuplot} \caption{ - } + The series coefficients for the scaling function $\mathcal F_0$ as a + function of polynomial order $m$. The numeric values are from Table + \ref{tab:data}. + } \label{fig:phi.series} +\end{figure} + +Besides reproducing the high derivatives in the series well, the approximate +functions defined here feature the appropriate singularity at the abrupt +transition. Fig.~\ref{fig:glow.radius} shows the ratio of subsequent series +coefficients for $\mathcal F_-$ as a function of the inverse order, which +should converge in the limit of $m\to0$ to the inverse radius of convergence +for the series. Approximations for the function without the explicit +singularity have a nonzero radius of convergence, where both the numeric data +and the approximate functions defined here show the appropriate divergence in +the ratio. + +\begin{figure} + \begin{gnuplot}[terminal=epslatex] + dat1 = 'data/glow_numeric.dat' + dat2 = 'data/glow_series_ours_0.dat' + dat3 = 'data/glow_series_ours_7.dat' + dat4 = 'data/glow_series_caselle.dat' + ratLast(x) = (back2 = back1, back1 = x, back1 / back2) + back1 = 0 + back2 = 0 + + set xlabel '$1/m$' + set xrange [0:0.55] + set ylabel '$\mathcal F_-^{(m)}/\mathcal F_-^{(m-1)}$' + set yrange [0:15] + + 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$)', \ + dat4 using (1/$1):(abs(ratLast($2))) title 'Caselle \textit{et al.}' + \end{gnuplot} + \caption{ + Sequential ratios of the series coefficients of the scaling function + $\mathcal F_-$ as a function of inverse polynomial order $m$. The + extrapolated $y$-intercept of this plot gives the radius of convergence of + the series. + } \label{fig:glow.radius} \end{figure} \section{Outlook} |