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| -rw-r--r-- | ising_scaling.tex | 53 |
1 files changed, 29 insertions, 24 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index b758534..8488f9d 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -641,9 +641,10 @@ abrupt transition and the Yang--Lee point, the coefficients in the analytic part $G$ of the scaling function, and the coefficients in the undetermined coordinate function $g$. -The other parameters $B$, $C_0$, $\theta_{YL}$, and $C_{YL}$ are determined by known properties. {\color{blue} XXX Is this right?} -For $\theta>\theta_0$, the form \eqref{eq:essential.singularity} can be -expanded around $\theta=\theta_0$ to yield +The other parameters $B$, $C_0$, $\theta_{YL}$, and $C_{YL}$ are determined or +further constrained by known properties. For $\theta>\theta_0$, the form +\eqref{eq:essential.singularity} can be expanded around $\theta=\theta_0$ to +yield \begin{equation} \begin{aligned} \operatorname{Im}u_f @@ -703,8 +704,11 @@ the known coefficients or by a 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 weak residuals -$j!g_j$ and $j!G_j$ encouraging the coefficients of the analytic functions $g$ and $G$ in -\eqref{eq:schofield.funcs} and \eqref{eq:analytic.free.enery} to stay small. {\color{blue} Are these multiplied by a small constant? Should they have an $1/R^n$ for the expected radius of convergence?} This can be interpreted as a prior which expects these functions to be analytic, and therefore have series coefficients which decay with a factorial. +$j!g_j$ and $j!G_j$ encouraging the coefficients of the analytic functions $g$ +and $G$ in \eqref{eq:schofield.funcs} and \eqref{eq:analytic.free.enery} to +stay small. This can be interpreted as a prior which expects these functions +to be analytic, and therefore have series coefficients which decay with a +factorial. A Levenberg--Marquardt algorithm is performed on the cost function to find a parameter combination which minimizes it. As larger polynomial order in the series are fit, the truncations of $G$ and $g$ are extended to higher order so @@ -714,21 +718,19 @@ function at the low and high temperature zero field points to quadratic order), through $n=6$. At higher order we began to have difficulty minimizing the cost. The resulting fit coefficients can be found in Table \ref{tab:fits}. -Precise results exist for the value of the -scaling function and its derivatives at the critical isotherm, or equivalently -for the series coefficients of the scaling function $\mathcal F_0$. Since we do -not use these coefficients in our fits, the error in the approximate scaling -functions and their derivatives can be evaluated by comparison to their known -values at the critical isotherm, or $\theta=1$. -The difference between the numeric values of the coefficients $\mathcal -F_0^{(m)}$ and those predicted by the iteratively fit scaling functions are -shown in Fig.~\ref{fig:error}. For the values for which we were able to make a -fit, the error in the function and its first several derivatives appear to -trend exponentially towards zero in the polynomial order $n$. The predictions -of our fits at the critical isotherm can be compared with the numeric values to -higher order in Fig.~\ref{fig:phi.series}, where the absolute values of both -are plotted. {\color{blue} XXX Is the critical isotherm still $\theta=1$ with our general coordinate change?} - +Precise results exist for the value of the scaling function and its derivatives +at the critical isotherm, or equivalently for the series coefficients of the +scaling function $\mathcal F_0$. Since we do not use these coefficients in our +fits, the error in the approximate scaling functions and their derivatives can +be evaluated by comparison to their known values at the critical isotherm, or +$\theta=1$. The difference between the numeric values of the coefficients +$\mathcal F_0^{(m)}$ and those predicted by the iteratively fit scaling +functions are shown in Fig.~\ref{fig:error}. For the values for which we were +able to make a fit, the error in the function and its first several derivatives +appear to trend exponentially towards zero in the polynomial order $n$. The +predictions of our fits at the critical isotherm can be compared with the +numeric values to higher order in Fig.~\ref{fig:phi.series}, where the absolute +values of both are plotted. \begin{table} \begin{tabular}{r|lll} \multicolumn1{c|}{$m$} & @@ -932,8 +934,9 @@ are plotted. {\color{blue} XXX Is the critical isotherm still $\theta=1$ with ou with respect to $\eta$ evaluated at $\eta=0$, as a function of the 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 two limits used in the fit, at $H=0$ and $T$ above and below $T_c$. Convergence here should reflect overall convergence of our scaling function at all $\theta$. {\color{blue} Thicker lines? Larger symbols?} - } \label{fig:error} + midway between the two limits used in the fit, at $H=0$ and $T$ above and + below $T_c$. Convergence here should reflect overall convergence of our + scaling function at all $\theta$.} \label{fig:error} \end{figure} @@ -965,7 +968,7 @@ are plotted. {\color{blue} XXX Is the critical isotherm still $\theta=1$ with ou \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}. {\color{blue} XXX Numerics is invisible? Why is the decay indicating a radius of convergence larger than $\theta_0$? Mention relation of decay to $\theta_0$? Or even plot $F_0^{(m)} \theta_0^m$ to make points easier to see?} + \ref{tab:data} and are partially obscured by the other data. } \label{fig:phi.series} \end{figure} @@ -1045,7 +1048,9 @@ Fig.~\ref{fig:phi.series}. The series coefficients for the scaling function $\mathcal F_+$ as a 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}. Note all agree well for $H$ near zero, $T > T_c$. {\color{blue} Caselle invisible?} + accurate scaling function listed in \cite{Caselle_2001_The}. Note all agree + well for $H$ near zero, $T > T_c$, as does the function of Caselle + \textit{et al}. } \label{fig:ghigh.series} \end{figure} |