From 7b4f4ee4813f8afc3e9949a368fe708351d3b89c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 10:20:59 -0400 Subject: changes --- .gitignore | 5 +++++ 1 file changed, 5 insertions(+) create mode 100644 .gitignore (limited to '.gitignore') diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..6e3b920 --- /dev/null +++ b/.gitignore @@ -0,0 +1,5 @@ +*.pdf +*.aux +*.log +*.bbl +*.blg -- cgit v1.2.3-70-g09d2 From 6a4e2f31e40a563333812d1d17cb32efbbdfdbf7 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 29 May 2017 01:39:21 -0400 Subject: many changes, moved to revtex format and rewrote much. added free energies for all dimensions --- .gitignore | 1 + essential-ising.bib | 24 ++++- essential-ising.tex | 269 +++++++++++++++++++++++++--------------------------- 3 files changed, 153 insertions(+), 141 deletions(-) (limited to '.gitignore') diff --git a/.gitignore b/.gitignore index 6e3b920..330fdf6 100644 --- a/.gitignore +++ b/.gitignore @@ -3,3 +3,4 @@ *.log *.bbl *.blg +*Notes.bib diff --git a/essential-ising.bib b/essential-ising.bib index bc154ea..ceb1b85 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -71,6 +71,17 @@ publisher={IOP Publishing} } +@article{dimitrovic.1991.finite, + title={Finite-size effects, goldstone bosons and critical exponents in the d= 3 Heisenberg model}, + author={Dimitrovi{\'c}, I and Hasenfratz, P and Nager, J and Niedermayer, Ferenc}, + journal={Nuclear Physics B}, + volume={350}, + number={3}, + pages={893--905}, + year={1991}, + publisher={Elsevier} +} + @article{fisher.1967.condensation, title={The theory of condensation and the critical point}, author={Fisher, Michael E}, @@ -205,7 +216,7 @@ @article{langer.1969.metastable, title={Statistical theory of the decay of metastable states}, - author={Langer, JS}, + author={Langer, James S}, journal={Annals of Physics}, volume={54}, number={2}, @@ -289,3 +300,14 @@ organization={Royal Society of Chemistry} } +@article{wilson.1971.renormalization, + title={Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior}, + author={Wilson, Kenneth G}, + journal={Physical Review B}, + volume={4}, + number={9}, + pages={3184}, + year={1971}, + publisher={APS} +} + diff --git a/essential-ising.tex b/essential-ising.tex index 0b631e5..9e4d2c3 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -3,7 +3,7 @@ % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[fleqn]{article} +\documentclass[aps,prl,reprint,fleqn]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} @@ -11,21 +11,17 @@ \mathtoolsset{showonlyrefs=true} -\title{Essential Singularity in the Ising Abrupt Transition} -\author{Jaron Kent-Dobias} - -\date{April 20, 2017} - -\begin{document} - \def\[{\begin{equation}} \def\]{\end{equation}} +\def\re{\mathop{\mathrm{Re}}\nolimits} \def\im{\mathop{\mathrm{Im}}\nolimits} \def\dd{\mathrm d} \def\O{\mathcal O} +\def\o{\mathcal o} \def\ei{\mathop{\mathrm{Ei}}\nolimits} \def\b{\mathrm b} +\def\c{\mathrm c} \newcommand\pd[3][]{ \ifthenelse{\isempty{#1}} @@ -34,18 +30,58 @@ \frac{\partial\tmp#2}{\partial#3\tmp} } +\begin{document} + +\title{Essential Singularity in the Ising Abrupt Transition} +\author{Jaron Kent-Dobias} +\author{James P.~Sethna} +\affiliation{Cornell University} + +\date{April 20, 2017} + +\begin{abstract} + Test +\end{abstract} + \maketitle +The Ising model is the canonical example of a system with a continuous phase +transition, and the study of its singular properties marked the first success +of the renormalization group ({\sc rg}) method in statistical physics +\cite{wilson.1971.renormalization}. This status makes sense: it's a simple +model whose phase transition admits {\sc rg} methods in a straightforward way, +and has exact solutions in certain dimensions and for certain parameter +restrictions. However, in one respect the Ising critical point is not simply a +continuous transition: it ends the line of abrupt phase transitions at zero +field below the critical temperature. Though typically neglected in {\sc rg} +scaling analysis of the critical point, we demonstrate that there are +numerically measurable contributions to scaling due to the abrupt transition +line that cannot be accounted for by analytic changes of control or +thermodynamic variables. + +{\sc Rg} analysis predicts that the singular part of the free energy per site +$F$ as a function of reduced temperature $t=1-\frac{T_c}T$ and field $h=H/T$ in the vicinity of the critical point takes the scaling form +$F(t,h)=g_t^{2-\alpha}\mathcal F(g_hg_t^{-\Delta})$, where +$\Delta=\beta\delta$ and $g_t$, $g_h$ are analytic functions of $t$, $h$ that +transform exactly linearly under {\sc rg}. When studying the properties of the +Ising critical point, it is nearly always assumed that $\mathcal F(X)$, the +universal scaling function, is an analytic function of $X$. However, it has +long been known that there exists an essential singularity in $\mathcal F$ at +$X=0$, though its effects have long been believed to be unobservable +\cite{fisher.1967.condensation}. With careful analysis, we have found that +assuming the presence of the essential singularity is predictive of the +scaling form of e.g. the susceptibility. + +The providence of the essential singularity can be understood using the +methods of critical droplet theory for the decay of an Ising system in a +metastable state, i.e., an equilibrium Ising state for $T0$ +subjected to a small negative external field $H<0$. It's long been known that the decay rate $\Gamma$ of metastable states in statistical mechanics is often related to the metastable free energy $F$ by -\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic} -\[ - \Gamma\propto\im F -\] -What exactly is meant by `metastable free energy' is important to establish, -since formally the free energy relies on the existence of an equilibrium -state. Here one can imagine either analytic continuation of the free energy -through an abrupt phase transition, or restriction of the partition function +$\Gamma\propto\im F$ +\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic}. +`Metastable free energy' can be thought of as either an analytic continuation of the free energy +through the abrupt phase transition, or restriction of the partition function trace to states in the vicinity of the local free energy minimum that characterizes the metastable state. In any case, the free energy develops a nonzero imaginary part in the metastable region. Heuristically, this can be @@ -53,151 +89,104 @@ thought of as similar to what happens in quantum mechanics with a non-unitary Hamiltonian: the imaginary part describes loss of probability in the system that corresponds to decay. -One can estimate the scaling of the decay rate of the {\sc 2d} Ising model -using ideas from nucleation theory. In this framework, the metastable state -decays when a sufficiently large domain in the stable state forms to grow -stably to fill out the whole system. The free energy of a domain of $N$ spins -causes a free energy change -\[ - \Delta F=\Sigma N^\sigma-MHN -\] -where $\Sigma$ is the surface tension and $1-\frac1d\leq\sigma<1$. This is -maximized by -\[ - N_c=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)} -\] -which corresponds to a free energy change -\[ - \Delta F_c\sim\bigg(\frac\Sigma{(MH)^\sigma}\bigg)^{1/(1-\sigma)} -\] -The rate of formation is proportional to the Boltzmann factor, -\[ - \Gamma\sim e^{-\beta \Delta - F_c}=e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}} -\] -For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes -\[ - \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}} -\] -There are known scaling forms for the surface tension and magnetization, $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ -\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$. -Since both the surface tension and magnetization have nonzero finite values at -the first-order transition $h=0$, $\mathcal S(0)=\O(1)$ and $\mathcal -M(0)=\O(1)$. It follows that +In critical droplet theory, the metastable state decays when a domain of the +equilibrium state forms that is sufficiently large to grow and envelop the +whole system. Assuming the free energy cost of the surface of the droplet +scales with the number of spins $N$ like $\Sigma N^\sigma$ and that of its +bulk scales like $-MHN$, the critical droplet size scales like +$N_\c\sim(MH/\Sigma)^{-1/(1-\sigma)}$ and the free energy of the critical +droplet scales like $\Delta F_\c\sim\Sigma^{1/(1-\sigma)}(HM)^{-\sigma/(1-\sigma)}$. +Assuming domains have minimal surfaces, $\sigma=1-\frac1d$ and +$\Delta F_\c\sim\Sigma^d(HM)^{-(d-1)}$. Assuming the scaling forms +$\Sigma=g_t^\mu\mathcal S(g_hg_t^{-\Delta})$ and $M=g_t^\beta\mathcal +M(g_hg_t^{-\Delta})$ and using known hyperscaling relations +\cite{widom.1981.interface}, this implies a scaling form +\begin{align} + \Delta F_c& + \sim\mathcal S^d(g_hg_t^{-\Delta})(g_hg_t^{-\Delta}\mathcal + M(g_hg_t^{-\Delta}))^{-(d-1)}\\ + &\sim\mathcal G^{-(d-1)}(g_h g_t^{-\Delta}) +\end{align} +Since both surface tension and magnetization are finite and nonzero for $H=0$ +at $T1$ this function has an essential singularity in the invariant +combination $g_hg_t^{-\Delta}$. + +This form of $\im F$ for small $h$ is known. Henceforth we will assume $h$ and +$t$ are sufficiently small that $g_t\simeq t$, $g_h\simeq h$, and all +functions of both variables can be truncated at lowest order. We make the scaling ansatz that +the imaginary part of the metastable free energy has the same singular +behavior as the real part of the equilibrium free energy, and that for small +$t$, $h$, $\im F(t,h)=t^{2-\alpha}\mathcal H(ht^{-\Delta})$ for \[ - \im F\simeq - \begin{cases} - \big(\frac - h{t^\Delta}\big)^{-(d-3)d/2}(g^*)^{-d(d-1)/4}\exp\big[-B\big(\frac - h{|t|^\Delta}\big)^{-(d-1)}(g^*)^{-(d+1)/2}\big] & d=2,4\\ - \big(\frac - h{t^\Delta}\big)^{-7/3}(g^*)^{-8/3}\exp\big[-B\big(\frac - h{t^\Delta}\big)^{-2}(g^*)^{-2}\big] - & d=3 - \end{cases} + \mathcal H(X)=A\Theta(-X)X^\zeta e^{-1/(BX)^{d-1}} \] -with $\Delta=3-\frac\epsilon2$, $g^*=2\pi^2\frac\epsilon{n+8}$ -\cite{houghton.1980.metastable,gunther.1980.goldstone}. This is consistent -with our form above. We therefore predict that +where $\Theta$ is the Heaviside function and with $\zeta=-(d-3)d/2$ for $d=2,4$ and $\zeta=-7/3$ for $d=3$ +\cite{houghton.1980.metastable,gunther.1980.goldstone}. Assuming that $F$ is +analytic in the upper complex-$h$ plane, the real part of $F$ in the +equilibrium state can be extracted from this imaginary metastable free energy +using the Kramers--Kronig relation \[ - \im F=t^{2-\alpha}\mathcal F(ht^{-\beta\delta})^{-(d-3)d/2}e^{-1/\mathcal - G(ht^{-\beta\delta})^{d-1}} + \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h' \] -In {\sc 2d} we have +In {\sc 3d} and {\sc 4d} this can be computed explicitly given our scaling +ansatz, yielding \[ - \im F=t^2\mathcal F(ht^{-\Delta})e^{-1/\mathcal G(ht^{-\Delta})} + \begin{aligned} + \mathcal F(X)&= + \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} + \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2})\\ + &\hspace{10em}-4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] + \end{aligned} + \notag \] -with $\Delta=\beta\delta=\frac{15}8$. In terms of $X=ht^{-\Delta}$, this is \[ - \im F=t^2\mathcal F(X)e^{-1/\mathcal G(X)}\simeq At^2|X|e^{-1/B|X|} + \begin{aligned} + \mathcal F(X)&= + \frac{A}{9\pi X^2}e^{1/(BX)^3} + \Big[3\Gamma(0,(BX)^{-3})\\ + &\hspace{2em}-3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) + -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] + \end{aligned} + \notag \] - -\begin{align} - F(X) - &=\frac1\pi\int_{-\infty}^\infty\frac{\im F(X')}{X'-X}\,\dd X' - =\frac{At^2}\pi\int_{-\infty}^0\frac{|X'|e^{-1/B|X'|}}{X'-X}\,\dd X'\\ - &=-\frac{At^2}\pi\int_0^\infty\frac{X'e^{-1/BX'}}{X'+X}\,\dd X' -\end{align} -since $\im F=0$ for $X>0$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. -Unfortunately this integral doesn't converge, and it seems we cannot evaluate -this result at the level of truncation we've chosen. However, - +At the level of truncation we are working at, the Kramers--Kronig relation +does not converge in {\sc 2d}. However, the higher moments can still be +extract, e.g., the susceptibility, by taking \[ - F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n + \chi\propto\pd[2]Fh + =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h' \] +This yields \[ - f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X - =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X - =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1) + \chi=|t|^{-\gamma}\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big] \] -for $n>1$. - -\begin{align} - \chi - &=\pd[2]Fh - =t^{-2\Delta}\pd[2]FX - =-\frac{2}\pi At^{2-2\Delta}\int_0^\infty\frac{X'e^{-1/BX'}}{(X+X')^3}\,\dd - X'\\ - &=\frac2\pi - \frac{ABt^{-\gamma}}{(BX)^3}\big[BX(1-BX)+e^{1/BX}\ei(-1/BX)\big] -\end{align} - +Scaling forms for the free energy can then be extracted by integration and +comparison with known exact results at zero field, yielding \[ - \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma} + \mathcal M(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0) \] - +with $\mathcal +M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$, and \[ - \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1) + \mathcal F(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX)) \] -$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$ -\cite{barouch.1973.susceptibility} -CORRECTIONS TO SCALING, $u_t$ and $u_h$ instead of $t$ and $h$. - -\begin{align} - u_h - &=h[1+c_ht+dht^2+e_hh^2+f_ht^3+\O(t^4,th^2)]\\ - u_t - &=t+b_th^2+c_t^2+d_t^3+e_tth^2+f_tt^4+\O(t^5,t^2h^2,h^4) -\end{align} -\begin{align} - c_h=\frac{\beta_c}{\sqrt2} - && - d_h=\frac{23\beta_c^2}{16} - && - f_h=\frac{191\beta_c^3}{48\sqrt2}\\ - c_t=\frac{\beta_c}{\sqrt2} - && - d_t=\frac{7\beta_c^2}6 - && - f_t=\frac{17\beta_c^3}{6\sqrt2}\\ - e_t=b_t\beta_c\sqrt2 - && - b_t=-\frac{E_0\pi}{16\beta_c^2} -\end{align} -$E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$ -\[ - F(t,h)-F(t,0)=\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2n}(t)h^{2n} -\] -\[ - \chi(t,h)=\pd[2]Fh=\chi_2(t)+\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2(n+1)}h^{2n} -\] - -\begin{align} - \chi - &=\pd[2]Fh - =\pd[2]{F_\b}h - +\frac d{y_t}\bigg(\frac d{y_t}-1\bigg)|u_t|^{d/y_t-2}\bigg(\pd{u_t}h\bigg)^2 -\end{align} +How predictive are these scaling forms in the proximity of the critical point +and the abrupt transition line? We used a form of the Wolff algorithm modified +to remain efficient in the presence of an external field by incorporating the +field as another spin with coupling $|H|$ to all others +\cite{dimitrovic.1991.finite}. Data was then taken for susceptibility and +magnetization for $|t|,h\leq0.1$ -\input{figs/scaling_func.tex} -\bibliographystyle{plain} +\bibliographystyle{apsrev4-1} \bibliography{essential-ising} \end{document} -- cgit v1.2.3-70-g09d2 From 76499e55eb813b66284b16423693700e86d86ae8 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 30 May 2017 22:11:46 -0400 Subject: added draft figures --- .gitignore | 2 +- essential-ising.tex | 16 +++++++++++++--- figs/fig_mag-collapse.pdf | Bin 0 -> 156076 bytes figs/fig_sus-collapse.pdf | Bin 0 -> 159531 bytes 4 files changed, 14 insertions(+), 4 deletions(-) create mode 100644 figs/fig_mag-collapse.pdf create mode 100644 figs/fig_sus-collapse.pdf (limited to '.gitignore') diff --git a/.gitignore b/.gitignore index 330fdf6..ee70ba3 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,4 @@ -*.pdf +essential-ising.pdf *.aux *.log *.bbl diff --git a/essential-ising.tex b/essential-ising.tex index cbb9866..2d8af31 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -163,7 +163,7 @@ ansatz, yielding for {\sc 4d}. At the level of truncation we are working at, the Kramers--Kronig relation does not converge in {\sc 2d}. However, the higher moments can still be -extract, e.g., the susceptibility, by taking +extracted, e.g., the susceptibility, by taking \[ \chi\propto\pd[2]Fh =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h' @@ -202,12 +202,22 @@ in the insets of those figures. As can be seen, there is very good agreement between our proposed functional forms and what is measured. \begin{figure} - \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data.} + \includegraphics{figs/fig_sus-collapse} + \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data from a + $L=1024$ square-lattice Ising model. +Different colored points show different values of $t$, which vary from +$0.01$, $0.02$, \dots, $0.1$. Solid black line shows fitted form, with +$C=0.0107\pm0.0011$ and $B=0.172\pm0.036$.} \label{fig:sus} \end{figure} \begin{figure} - \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data.} + \includegraphics{figs/fig_mag-collapse} + \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data from a +$L=1024$ square-lattice Ising model. Different colored points show different +values of $t$, which vary from $0.01$, $0.02$, \dots, $0.1$. Solid black lines +shows fitted form, with $\mathcal M(0)=1.21055\pm0.00031$, +$D=0.09440\pm0.00033$, and $B=0.085249\pm0.00099$.} \label{fig:mag} \end{figure} diff --git a/figs/fig_mag-collapse.pdf b/figs/fig_mag-collapse.pdf new file mode 100644 index 0000000..dd82cf5 Binary files /dev/null and b/figs/fig_mag-collapse.pdf differ diff --git a/figs/fig_sus-collapse.pdf b/figs/fig_sus-collapse.pdf new file mode 100644 index 0000000..03b561e Binary files /dev/null and b/figs/fig_sus-collapse.pdf differ -- cgit v1.2.3-70-g09d2 From 8b2c04f7245783ab2f1788f698fabd0bc4f8067a Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 5 Jun 2017 12:15:45 -0400 Subject: changed figures to gnuplot figures --- .gitignore | 5 +- essential-ising.tex | 12 +- figs/fig-mag.gplot | 33 ++ figs/fig-mag_data-func.dat | 1001 ++++++++++++++++++++++++++++++++++++++++++++ figs/fig-mag_data-num.dat | 500 ++++++++++++++++++++++ figs/fig-sus.gplot | 32 ++ figs/fig-sus_data-func.dat | 1001 ++++++++++++++++++++++++++++++++++++++++++++ figs/fig-sus_data-num.dat | 500 ++++++++++++++++++++++ figs/fig_mag-collapse.pdf | Bin 158224 -> 0 bytes figs/fig_sus-collapse.pdf | Bin 163684 -> 0 bytes figs/scaling_func.gplot | 16 - figs/scaling_func.tex | 260 ------------ makefile | 7 +- 13 files changed, 3082 insertions(+), 285 deletions(-) create mode 100644 figs/fig-mag.gplot create mode 100644 figs/fig-mag_data-func.dat create mode 100644 figs/fig-mag_data-num.dat create mode 100644 figs/fig-sus.gplot create mode 100644 figs/fig-sus_data-func.dat create mode 100644 figs/fig-sus_data-num.dat delete mode 100644 figs/fig_mag-collapse.pdf delete mode 100644 figs/fig_sus-collapse.pdf delete mode 100644 figs/scaling_func.gplot delete mode 100644 figs/scaling_func.tex (limited to '.gitignore') diff --git a/.gitignore b/.gitignore index ee70ba3..10c9386 100644 --- a/.gitignore +++ b/.gitignore @@ -1,6 +1,9 @@ -essential-ising.pdf +*.pdf *.aux *.log *.bbl *.blg *Notes.bib +*.dvi + +figs/*.tex diff --git a/essential-ising.tex b/essential-ising.tex index 2d8af31..b9ee8a2 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -202,22 +202,24 @@ in the insets of those figures. As can be seen, there is very good agreement between our proposed functional forms and what is measured. \begin{figure} - \includegraphics{figs/fig_sus-collapse} + \input{figs/fig-sus} + %\includegraphics{figs/fig_sus-collapse} \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data from a $L=1024$ square-lattice Ising model. Different colored points show different values of $t$, which vary from $0.01$, $0.02$, \dots, $0.1$. Solid black line shows fitted form, with -$C=0.0107\pm0.0011$ and $B=0.172\pm0.036$.} +$C=0.0111\pm0.0023$ and $B=0.173\pm0.072$.} \label{fig:sus} \end{figure} \begin{figure} - \includegraphics{figs/fig_mag-collapse} + \input{figs/fig-mag} + %\includegraphics{figs/fig_mag-collapse} \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data from a $L=1024$ square-lattice Ising model. Different colored points show different values of $t$, which vary from $0.01$, $0.02$, \dots, $0.1$. Solid black lines -shows fitted form, with $\mathcal M(0)=1.21055\pm0.00031$, -$D=0.09440\pm0.00033$, and $B=0.085249\pm0.00099$.} +shows fitted form, with $\mathcal M(0)=1.21039\pm0.00031$, +$D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$.} \label{fig:mag} \end{figure} diff --git a/figs/fig-mag.gplot b/figs/fig-mag.gplot new file mode 100644 index 0000000..ada3d95 --- /dev/null +++ b/figs/fig-mag.gplot @@ -0,0 +1,33 @@ + +set terminal pslatex rotate size 3.417,2.111 +set multiplot + +B = 0.0861 +A = 0.094 +M0 = 1.21039 +func = "figs/fig-mag_data-func.dat" +num = "figs/fig-mag_data-num.dat" + +set samples 10000 +set key off + +set xrange [-20:1200] +set yrange [1.18:1.75] +set ylabel offset 1,0 '$Mt^{-\beta}$' +set xlabel "$ht^{-\\beta\\delta}$" + +plot num with yerrorbars pt 0, func using (10**$1 / B):(M0 + 10**($2) * A) with linespoints pt 0 lw 3 + +set size 0.6,0.6 +set origin 0.38,0.2 +set xrange [-3:3] +set yrange [1.2:1.75] +set ylabel offset 4,0 '\footnotesize$Mt^{-\beta}$' +set xlabel offset 0,0.5 '\footnotesize$ht^{-\beta\delta}$' +set xtics format '\footnotesize$10^{%g}$' -2,2,3 +set ytics format '\footnotesize %g' + +plot num using (log10($1)):2:3 with yerrorbars pt 0, func using ($1-log10(B)):(M0 + A * 10**$2) with linespoints pt 0 lw 3 + + + diff --git a/figs/fig-mag_data-func.dat b/figs/fig-mag_data-func.dat new file mode 100644 index 0000000..29653f0 --- /dev/null +++ b/figs/fig-mag_data-func.dat @@ -0,0 +1,1001 @@ +-5. -4.698976620880227 +-4.99 -4.688976675606296 +-4.98 -4.678976738233927 +-4.97 -4.668977103542201 +-4.96 -4.658977175055251 +-4.95 -4.64897725444246 +-4.94 -4.6389776168638575 +-4.93 -4.628977705544413 +-4.92 -4.618977802145736 +-4.91 -4.608977906594962 +-4.9 -4.598978269856612 +-4.89 -4.588978384162751 +-4.88 -4.578978506414977 +-4.87 -4.568978870802035 +-4.86 -4.558979003497131 +-4.85 -4.548979144167584 +-4.84 -4.538979511480103 +-4.83 -4.528979663220596 +-4.82 -4.518979823098883 +-4.81 -4.508980093110478 +-4.8 -4.498980366682533 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1.3396628534382948 0.000020923681639429064 \ No newline at end of file diff --git a/figs/fig-sus.gplot b/figs/fig-sus.gplot new file mode 100644 index 0000000..78c73f1 --- /dev/null +++ b/figs/fig-sus.gplot @@ -0,0 +1,32 @@ + +set terminal pslatex rotate size 3.417,2.111 +set multiplot + +B = 0.173 +A = 0.0111 +func = "figs/fig-sus_data-func.dat" +num = "figs/fig-sus_data-num.dat" + +set samples 10000 +set key off + +set xrange [-20:1200] +set yrange [-0.2:13] +set ylabel offset 1,0 "$\\chi t^{\\gamma}\\times 10^3$" +set xlabel "$ht^{-\\beta\\delta}$" + +plot num using 1:(10**3 * $2):(10**3 * $3) with yerrorbars pt 0, func using (10**$1 / B):(10**(3+$2) * A) with linespoints pt 0 lw 3 + +set size 0.65,0.65 +set origin 0.3,0.3 +set xrange [-3:3] +set yrange [-4.5:-1.5] +set ylabel offset 2.5,0 '\footnotesize$\chi t^\gamma$' +set xlabel offset 0,0.5 '\footnotesize$ht^{-\beta\delta}$' +set xtics format '\footnotesize$10^{%g}$' -2,2,3 +set ytics format '\footnotesize$10^{%g}$' -4,1,-2 + +plot num using (log10($1)):(log10($2)):($3/$2) with yerrorbars pt 0, func using ($1-log10(B)):($2+log10(A)) with linespoints pt 0 lw 3 + + + diff --git a/figs/fig-sus_data-func.dat b/figs/fig-sus_data-func.dat new file mode 100644 index 0000000..972c77b --- /dev/null +++ b/figs/fig-sus_data-func.dat @@ -0,0 +1,1001 @@ +-5. 0.010094128168228536 +-4.99 0.0004986394100467353 +-4.98 -0.008441735254514486 +-4.97 0.010954806238037133 +-4.96 0.002527852467923256 +-4.95 -0.005308016370920201 +-4.94 0.0114293828525401 +-4.93 0.004018232985220573 +-4.92 -0.002858226194217986 +-4.91 -0.009219894501557 +-4.9 0.005081385417224557 +-4.89 -0.0009618799060500723 +-4.88 -0.006542816311444805 +-4.87 0.005798926687913552 +-4.86 0.0004888958038767844 +-4.85 -0.004414469974308102 +-4.84 0.006246050655940169 +-4.83 0.0015752929121863565 +-4.82 -0.0027318837493295303 +-4.81 -0.00005661662281440823 +-4.8 0.002370862728926167 +-4.79 -0.0014169159468057889 +-4.78 0.0008653476550368721 +-4.77 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-\put(151.0,859.0){\rule[-0.200pt]{310.279pt}{0.400pt}} -\end{picture} diff --git a/makefile b/makefile index f9421b5..0ec0215 100644 --- a/makefile +++ b/makefile @@ -1,9 +1,10 @@ -DOC=essential-ising.tex -FIGS=scaling_func +DOC=essential-ising +FIGS=fig-sus fig-mag all: ${FIGS:%=figs/%.tex} - rubber --pdf $(DOC) + rubber $(DOC).tex + dvipdf $(DOC).dvi figs/%.tex: figs/%.gplot gnuplot $< > $@ -- cgit v1.2.3-70-g09d2 From 91cf18f2f55706c7b764fdc5f048b0c6106c698e Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 12 Jul 2017 12:26:38 -0400 Subject: many final changes --- .gitignore | 1 + essential-ising.bib | 7 ++ essential-ising.tex | 288 +++++++++++++++++++++++++------------------------- figs/fig-susmag.gplot | 15 ++- makefile | 6 +- 5 files changed, 162 insertions(+), 155 deletions(-) (limited to '.gitignore') diff --git a/.gitignore b/.gitignore index 10c9386..be37e5a 100644 --- a/.gitignore +++ b/.gitignore @@ -7,3 +7,4 @@ *.dvi figs/*.tex +fig-*.tex diff --git a/essential-ising.bib b/essential-ising.bib index 556c4e7..df4a96b 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -134,6 +134,13 @@ publisher={APS} } +@book{flanigan.1972.complex, + title={Complex variables: harmonic and analytic functions}, + author={Flanigan, Francis J}, + year={1972}, + publisher={Courier Corporation} +} + @article{gaveau.1989.analytic, title={Metastable decay rates and analytic continuation}, author={Gaveau, B and Schulman, LS}, diff --git a/essential-ising.tex b/essential-ising.tex index 1aa5ea1..4673847 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -76,11 +76,10 @@ used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine - scaling functions for the free energy in the vicinity of the - critical point and the abrupt transition line. These functions have - essential singularities at zero field. Analogous forms for the magnetization - and susceptibility in two-dimensions are fit to numeric data and show good - agreement. + scaling functions for the free energy in the vicinity of the critical point + and the abrupt transition line. These functions have essential singularities + at zero field. Analogous forms for the magnetization and susceptibility in + two dimensions are fit to numeric data and show good agreement. \end{abstract} \maketitle @@ -88,30 +87,32 @@ The Ising model is the canonical example of a system with a continuous phase transition, and the study of its singular properties marked the first success of the renormalization group (\textsc{rg}) method in statistical physics -\cite{wilson.1971.renormalization}. This status makes sense: it's a simple -model whose phase transition admits \textsc{rg} methods in a straightforward way, -and has exact solutions in certain dimensions and for certain parameter -restrictions. However, in one respect the Ising critical point is not simply a -continuous transition: it ends the line of abrupt phase transitions at zero -field below the critical temperature. Though typically neglected in \textsc{rg} -scaling analyses of the critical point, we demonstrate that there are -numerically measurable contributions to scaling due to the abrupt transition -line that cannot be accounted for by analytic changes of control or +\cite{wilson.1971.renormalization}. Its status makes sense: it's a simple +model whose continuous phase transition contains all the essential features of +more complex ones, but admits \textsc{rg} methods in a straightforward way and +has exact solutions in certain dimensions and for certain parameter +restrictions. However, the Ising critical point is not simply a continuous +transition: it also ends a line of abrupt phase transitions extending from it +at zero field below the critical temperature. Though typically neglected in +\textsc{rg} scaling analyses of the critical point, we demonstrate that there +are numerically measurable contributions to scaling due to the abrupt +transition line that cannot be accounted for by analytic changes of control or thermodynamic variables. \textsc{Rg} analysis predicts that the singular part of the free energy per -site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ in -the vicinity of the critical point takes the scaling form -$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase $t<0$ \cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of -the Ising critical point, it is nearly always assumed that $\fF(X)$, -the universal scaling function, is an analytic function of $X$. However, it -has long been known that there exists an essential singularity in $\fF$ -at $X=0$, though its effects have long been believed to be unobservable +site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ +in the vicinity of the critical point takes the scaling form +$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase +$t<0$ \cite{cardy.1996.scaling}. When studying the properties of the Ising +critical point, it is nearly always assumed that the universal scaling +function $\fF$ is analytic, i.e., has a convergent Taylor series. However, it +has long been known that there exists an essential singularity in $\fF$ at +zero argument, though its effects have long been believed to be unobservable \cite{fisher.1967.condensation}, or simply just neglected \cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. With careful analysis, we have found that assuming the presence of the essential singularity is predictive of the scaling form of e.g. the -susceptibility. +susceptibility and magnetization. The provenance of the essential singularity can be understood using the methods of critical droplet theory for the decay of an Ising system in a @@ -136,19 +137,21 @@ In critical droplet theory, the metastable state decays when a domain of the equilibrium state forms whose surface-energy cost for growth is outweighed by bulk-energy gains. There is numerical evidence that, near the critical point, droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy -cost of the surface of a droplet of radius $R$ -is $\Sigma S_\dim R^{\dim-1}$ and that of its -bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are the surface area and -volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ is the surface tension of the equilibrium--metastable interface. The critical droplet size then is -$R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the critical -droplet is $\Delta -F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. +cost of the surface of a droplet of radius $R$ is $\Sigma S_\dim R^{\dim-1}$ +and that of its bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are +the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ +is the surface tension of the equilibrium--metastable interface. The critical +droplet size then is $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the +critical droplet is $\Delta +F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. Assuming the typical singular scaling forms $\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal M(h|t|^{-\beta\delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form \def\eqcritformone{ - T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)}\frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta}\fM(h|t|^{-\beta\delta}))^{\dim-1}} + T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)} + \frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta} + \fM(h|t|^{-\beta\delta}))^{\dim-1}} } \def\eqcritformtwo{ T\fG^{-(\dim-1)}(h|t|^{-\beta\delta}) @@ -169,70 +172,55 @@ M(h|t|^{-\beta\delta})$ and using known hyperscaling relations Since both surface tension and magnetization are finite and nonzero for $H=0$ at $T1$ this function has an essential singularity in the invariant combination $h|t|^{-\beta\delta}$. -% $\Gamma/\Gamma_\sq=(D_\sq/D)(\fM(0)^{1/\beta}/\fM_\sq(0)^{1/\beta})^{-7/4}$ - This form of $\im F$ for small $h$ is well known -\cite{langer.1967.condensation,harris.1984.metastability}. We make the scaling -ansatz that the imaginary part of the metastable free energy has the same -singular behavior as the real part of the equilibrium free energy, and that for -small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$ for +\cite{langer.1967.condensation,harris.1984.metastability}. We make the +scaling ansatz that the imaginary part of the metastable free energy has the +same singular behavior as the real part of the equilibrium free energy, and +that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$, +where \[ - \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}}, + \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} \label{eq:im.scaling} \] -where $\Theta$ is the Heaviside function. Results from combining an analysis -of fluctuations on the surface of critical droplets with \textsc{rg} recursion -relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for -$d=3$ +and $\Theta$ is the Heaviside function. Results from combining an analysis of +fluctuations on the surface of critical droplets with \textsc{rg} recursion +relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for $d=3$ \cite{houghton.1980.metastable,rudnick.1976.equations,gunther.1980.goldstone}. Assuming that $F$ is analytic in the upper complex-$h$ plane, the real part of $F$ in the equilibrium state can be extracted from this imaginary metastable free energy using the Kramers--Kronig relation \[ \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'. + \label{eq:kram-kron} \] This relationship has been used to compute high-order moments of the free energy in $H$ in good agreement with transfer matrix expansions \cite{lowe.1980.instantons}. Here, we compute the integral to come to explicit -functional forms. In \threedee and \fourdee this can be computed -explicitly given our scaling ansatz, yielding +functional forms. In \threedee and \fourdee this can be computed explicitly +given our scaling ansatz, yielding \def\eqthreedeeone{ \fF^\threedee(Y/B)&= \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} @@ -254,14 +242,14 @@ explicitly given our scaling ansatz, yielding \begin{align} &\begin{aligned} \eqthreedeeone\\ - &\hspace{6em} + &\hspace{8em} \eqthreedeetwo \end{aligned} \\ &\begin{aligned} \eqfourdeeone \\ - &\hspace{2em} + &\hspace{-0.5em} \eqfourdeetwo. \end{aligned} \end{align} @@ -272,119 +260,131 @@ explicitly given our scaling ansatz, yielding \eqfourdeeone\eqfourdeetwo. \end{align} \fi -At the level of truncation we are working at, the Kramers--Kronig relation -does not converge in \twodee. However, the higher moments can still be -extracted, e.g., the susceptibility, by taking +At the level of truncation of \eqref{eq:im.scaling} at which we are working +the Kramers--Kronig relation does not converge in \twodee. However, higher +moments can still be extracted, e.g., the susceptibility, by taking \[ \chi=\pd[2]Fh =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields +With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, +this yields \[ \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} \] -The constant $A$ can be fixed by zero-field results, with $\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=\frac{2AB^2}\pi$. Scaling forms for the free energy can then be extracted by direct integration and their constants of integration fixed by known zero field values, yielding \begin{align} \label{eq:mag_scaling} \fM^\twodee(Y/B) - &=\fM(0)+\frac{ABT_c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ + &=\fM(0)+\frac{ABT_\c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ \fF^\twodee(Y/B) - &=\fF(0)+T_cY\bigg(\frac{\fM(0)}B+\frac{AT_c}\pi e^{1/Y}\ei(-1/Y)\bigg) + &=\fF(0)+T_\c Y\bigg(\frac{\fM(0)}B+\frac{AT_\c}\pi e^{1/Y}\ei(-1/Y)\bigg) \end{align} -with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. +with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in +two dimensions. -How are these functional forms to be interpreted? They are not asymptotic -forms in any sense, as there is no limit of $t$ or $h$ in which they become -aribitrarily correct. It is well established that this method of using +How are these functional forms to be interpreted? Though the scaling function +\eqref{eq:im.scaling} for the imaginary free energy of the metastable state is +asymptotically correct sufficiently close to the critical point, the results +of the integral relation \eqref{eq:kram-kron} are not, since there is no limit +of $t$ or $h$ in which it becomes arbitrarily correct for a given truncation +of \eqref{eq:im.scaling}. It is well established that this method of using unphysical or metastable elements of a theory to extract properties of the -stable or equilibrium theory only works for high moments of those predictions -\cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. -These functions should be understood as possessing exactly the correct +stable or equilibrium theory is only accurate for high moments of those +predictions \cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. The +functions above should be understood as possessing exactly the correct singularity at the coexistence line, but requiring polynomial corrections, especially for smaller integer powers. Using these forms in conjunction with existing methods of describing the critical equation of state or critical properties with analytic functions in $h$ will incorporate these low-order -corrections while preserving the correct singular structure. +corrections while preserving the correct singular structure. In other words, +the scaling functions can be \emph{exactly} described by +$\tilde\fF(X)=\fF(X)+f(X)$ for some analytic function $f$. Higher order terms +in the expansion of $\tilde\fF$ become asymptotically equal to those of $\fF$ +because, as an analytic function, progressively higher order terms of $f$ must +eventually become arbitrarily small \cite{flanigan.1972.complex}. How predictive are these scaling forms in the proximity of the critical point -and the abrupt transition line? We simulated the \twodee Ising model on square lattice using a form of the Wolff algorithm modified -to remain efficient in the presence of an external field. Briefly, the external field $H$ is applied by adding an extra spin $s_0$ with coupling $|H|$ to all others -\cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking $M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. 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 $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. -Data was then taken for susceptibility and -magnetization for $T_\c-T,H\leq0.1$. The resulting scaling functions $\fX$ and -$\fM$ are plotted using this data in +and the abrupt transition line? We simulated the \twodee Ising model on square +lattice using a form of the Wolff algorithm modified to remain efficient in +the presence of an external field. Briefly, the external field $H$ is applied +by adding an extra spin $s_0$ with coupling $|H|$ to all others +\cite{dimitrovic.1991.finite}. A quickly converging estimate for the +magnetization in the finite-size system was then made by taking +$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. +Data was then taken for susceptibility and magnetization for +$T_\c-T,H\leq0.1$. This data, rescaled as appropriate to collapse onto a +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 +$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we +assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of +the susceptibility scaling function, then +$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant +$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_0^-$. 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 are systematic differences that can be seen most clearly in the -magnetization. Since our method is known to only be accurate for high moments -of the free energy, we should expect that low moments require corrections. -Therefore, we also fit those corrections using +between our proposed functional forms and what is measured. However, there +are systematic differences that can be seen most clearly in the magnetization. +Since our method is known to only be accurate for high moments of the free +energy, we should expect that low moments require corrections. Therefore, we +also fit those corrections using \begin{align} - \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\label{eq:sus_scaling_poly}\\ - \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX)\label{eq:mag_scaling_poly} + \tilde\fX^\twodee(X)&=\fX^\twodee(X)+\sum_{n=0}^Nf_n(BX)\label{eq:sus_scaling_poly}\\ + \tilde\fM^\twodee(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=0}^NF_n(BX)\label{eq:mag_scaling_poly} \end{align} -where $F_n'(x)=f_n(x)$ and +where $F_n'(Y)=f_n(Y)$ and \[ \begin{aligned} - f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\ - F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}). + f_n(Y)&=\frac{c_nx^n}{1+(\lambda Y)^{n+1}}\\ + F_n(Y)&=\frac{c_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda Y)^{n+1}). \end{aligned} \label{eq:poly} \] The functions $f_n$ have been chosen to be pure integer power laws for small -argument, but vanish appropriately at large argument. -We fit these functions to our numeric data for $N=3$. The resulting curves are -also plotted in Fig.~\ref{fig:scaling_fits} as dashed lines. - -%\begin{table} -% \centering -% \begin{tabular}{c|llc} -% Lattice & $T_\c$ & $\fM(0)^{1/\beta}$ & $D/D_\sq$ \\ -% \hline % ------------------------------------------------------------------- -% Square & $2/\log(1+\sqrt2)$ & $2^{5/2}\arcsinh1$ & 1 \\ -% Triangular & $4/\log3$ & $4\log3$ & $3^{3/2}/4$ \\ -% Hexagonal & $2/\log(2+\sqrt3)$ & $\frac8{\sqrt3}\arccosh2$ & $3^{3/2}/8$ -% \end{tabular} -% \caption{ -% The critical temperatures and amplitudes for the magetization along both the coexistence line and the critical isotherm, for three different lattices. -% } -% \label{tab:consts} -%\end{table} +argument, but vanish appropriately at large argument. This is necessary +because the susceptibility vanishes with $h|t|^{-\beta\delta}$ while bare +polynomial corrections would not. We fit these functions to our numeric data +for $N=0$ while requiring that $C_0^-/T_\c=\fX'(0)=c_0+2AB^2/\pi$. The +resulting curves are also plotted as dashed lines in +Fig.~\ref{fig:scaling_fits}. Our singular scaling function with one low-order +correction appears to match data quite well. \begin{figure} - \input{figs/fig-susmag} + \input{fig-susmag} \caption{ Scaling functions for (top) the susceptibility and (bottom) the - magnetization plotted in terms of the invarient combination - $h|t|^{-\beta\delta}$. Points with error bars show data with - sampling error taken from simulations of a $4096\times4096$ - square-lattice Ising model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ and - $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of - $T$. The solid lines show our - analytic results \eqref{eq:sus_scaling} and \eqref{eq:mag_scaling}, while - the dashed lines show fits of \eqref{eq:sus_scaling_poly} and - \eqref{eq:mag_scaling_poly} to the data for $N=3$, with $C_1=-0.00368$, - $C_2=-0.0191$, $C_3=0.0350$, and $\lambda=2.42$. + magnetization plotted in terms of the invariant combination + $h|t|^{-\beta\delta}$. Points with error bars show data with sampling + error taken from simulations of a $4096\times4096$ square-lattice Ising + model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ + and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of + $T$, with red corresponding to $0.01$ and violet to $0.1$. The solid lines + show our analytic results \eqref{eq:sus_scaling} and + \eqref{eq:mag_scaling}, while the dashed lines show fits of + \eqref{eq:sus_scaling_poly} and \eqref{eq:mag_scaling_poly} to the data + for $N=0$, with $c_0=-0.0124$ and $\lambda=1.77$. } \label{fig:scaling_fits} \end{figure} We have used results from the properties of the metastable Ising ferromagnet -and the analytic nature of the free energy to derive the universal scaling +and the analytic nature of the free energy to derive universal scaling functions for the free energy, and in \twodee the magnetization and -susceptibility, in the limit of small $t$ and $h$. Because of an essential +susceptibility, in the limit of small $t<0$ and $h$. Because of an essential singularity in these functions at $h=0$---the abrupt transition line---their -form cannot be brought into that of regular functions by analytic redefinition of control or thermodynamic -variables. These predictions match the results of simulations well. Having -demonstrated that the essential singularity in thermodynamic functions at the -abrupt singularity leads to observable effects. we hope that these functional -forms will be used in conjunction with traditional perturbation methods to -better express the equation of state of the Ising model in the whole of its -parameter space. +form cannot be brought into that of any regular function by analytic +redefinition of control or thermodynamic variables. These predictions match +the results of simulations well. Having demonstrated that the essential +singularity in thermodynamic functions at the abrupt transition leads to +observable scaling effects, we hope that these functional forms will be used in +conjunction with traditional perturbation methods to better express the +equation of state of the Ising model in the whole of its parameter space. \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index dfdd2a1..9682953 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -19,10 +19,9 @@ B = Tc**2 * M0 / (16 * pi) C0 = 0.0255369719 A = pi / 2 * C0 / (B**2 * Tc) -C1 = -0.00368218 -C2 = -0.0190753 -C3 = 0.0349601 -lamb = 2.41718 +c0 = -0.012384 +lamb = 1.76962 +A2 = pi / 2 * (C0 / Tc - c0) / B**2 susfunc = "figs/fig-sus_scaling-func.dat" magfunc = "figs/fig-mag_scaling-func.dat" @@ -43,7 +42,7 @@ 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 / B):(10**(3+$2) * A * B**2 + 10**3 * (poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1))) with lines dt 2 lw 2 lc black +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 / B):(10**(3+$2) * A2 * B**2 + 10**3 * poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black set bmargin -1 set tmargin 0.2 @@ -54,7 +53,7 @@ set ylabel offset 1,0 '$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 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) smooth csplines with lines dt 2 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) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) smooth csplines with lines dt 2 lw 2 lc black set logscale xy set tmargin -1 @@ -71,7 +70,7 @@ set xtics add ('$\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 / B):(10**$2 * A * B**2 + poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1)) with lines dt 2 lw 2 lc black +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 / B):(10**$2 * A2 * B**2 + poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black unset logscale xy set logscale x @@ -82,5 +81,5 @@ set ylabel offset 4,0 '\footnotesize$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 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) with lines dt 2 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 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) with lines dt 2 lw 2 lc black diff --git a/makefile b/makefile index b726870..32f4b74 100644 --- a/makefile +++ b/makefile @@ -6,10 +6,10 @@ FUNCS=fig-mag_scaling-func fig-sus_scaling-func all: ${DOC}.pdf -figs/%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} +%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} gnuplot $< > $@ -${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=figs/%.tex} +${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=%.tex} rubber $(DOC).tex dvipdf $(DOC).dvi @@ -17,5 +17,5 @@ clean: rubber --clean $(DOC) rm -f $(DOC).pdf rm -f $(DOC)Notes.bib - rm -f ${FIGS:%=figs/%.tex} + rm -f ${FIGS:%=%.tex} -- cgit v1.2.3-70-g09d2 From 0c6de182fd0b2b933741bba6a79253a99c350488 Mon Sep 17 00:00:00 2001 From: "James P. Sethna" Date: Thu, 3 Aug 2017 12:30:20 -0400 Subject: Added ref and changed .gitignore so I can see the current version --- .gitignore | 1 - refs/PhysRevB.40.9006.pdf | Bin 0 -> 670094 bytes 2 files changed, 1 deletion(-) create mode 100644 refs/PhysRevB.40.9006.pdf (limited to '.gitignore') diff --git a/.gitignore b/.gitignore index be37e5a..137e59e 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,3 @@ -*.pdf *.aux *.log *.bbl diff --git a/refs/PhysRevB.40.9006.pdf b/refs/PhysRevB.40.9006.pdf new file mode 100644 index 0000000..d455145 Binary files /dev/null and b/refs/PhysRevB.40.9006.pdf differ -- cgit v1.2.3-70-g09d2 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(-) (limited to '.gitignore') 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 v1.2.3-70-g09d2 From e11ae4bdaacb3c3e386039ab013ce1ed514b9792 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 22 Dec 2020 19:41:25 +0100 Subject: Deleted a lot. --- .gitignore | 5 +- essential-ising.bib | 570 -- essential-ising_prl.txt | 18 - essential-ising_prl2.txt | 1 - figs/fig-mag_scaling-func-mod.dat | 11514 ----------------------------------- figs/fig-mag_scaling-func.dat | 10001 ------------------------------- figs/fig-series-data.dat | 15 - figs/fig-series.gplot | 29 - figs/fig-sus_scaling-func-mod.dat | 11515 ------------------------------------ figs/fig-sus_scaling-func.dat | 10001 ------------------------------- figs/fig-susmag.gplot | 133 - makefile | 21 - refs/PhysRevB.40.9006.pdf | Bin 670094 -> 0 bytes 13 files changed, 1 insertion(+), 43822 deletions(-) delete mode 100644 essential-ising.bib delete mode 100644 essential-ising_prl.txt delete mode 100644 essential-ising_prl2.txt delete mode 100644 figs/fig-mag_scaling-func-mod.dat delete mode 100644 figs/fig-mag_scaling-func.dat delete mode 100644 figs/fig-series-data.dat delete mode 100644 figs/fig-series.gplot delete mode 100644 figs/fig-sus_scaling-func-mod.dat delete mode 100644 figs/fig-sus_scaling-func.dat delete mode 100644 figs/fig-susmag.gplot delete mode 100644 makefile delete mode 100644 refs/PhysRevB.40.9006.pdf (limited to '.gitignore') diff --git a/.gitignore b/.gitignore index e413095..034b355 100644 --- a/.gitignore +++ b/.gitignore @@ -4,7 +4,4 @@ *.blg *Notes.bib *.dvi -essential-ising.pdf - -figs/*.tex -fig-*.tex +/*.pdf diff --git a/essential-ising.bib b/essential-ising.bib deleted file mode 100644 index 64dacb0..0000000 --- a/essential-ising.bib +++ /dev/null @@ -1,570 +0,0 @@ - -@article{ambegaokar.1978.dissipation, - author = "V. 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Siggia", - year = "1980", - journal = "Physical Review B", - volume = "21", - pages = "1806--26", - title = "Dynamics of superfluid films", -} - -@article{aharony.1983.fields, - title={Nonlinear scaling fields and corrections to scaling near criticality}, - author={Aharony, Amnon and Fisher, Michael E}, - journal={Physical Review B}, - volume={27}, - number={7}, - pages={4394}, - year={1983}, - publisher={APS} -} - -@article{assis.2017.analyticity, - title={Analyticity of the Ising susceptibility: An interpretation}, - author={Assis, M and Jacobsen, JL and Jensen, I and Maillard, JM and McCoy, BM}, - journal={arXiv preprint arXiv:1705.02541}, - year={2017} -} - -@article{barouch.1973.susceptibility, - title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$}, - author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun}, - journal={Physical Review Letters}, - volume={31}, - number={23}, - pages={1409}, - year={1973}, - publisher={APS} -} - -@article{baxter.1979.series, - title={Series expansions from corner transfer matrices: the square lattice ising model}, - author={Baxter, RJ and Enting, IG}, - journal={Journal of Statistical Physics}, - volume={21}, - number={2}, - pages={103--123}, - year={1979}, - publisher={Springer} -} - -@article{baker.1980.ising, - title={Ising model phase boundary}, - author={Baker Jr, George A and Kim, Doochul}, - journal={Journal of Physics A: Mathematical and General}, - volume={13}, - number={4}, - pages={L103}, - year={1980}, - publisher={IOP Publishing} -} - -@article{betts.1971.lattice, - title={Lattice-lattice scaling and the generalized law of corresponding states}, - author={Betts, DD and Guttmann, AJ and Joyce, GS}, - journal={Journal of Physics C: Solid State Physics}, - volume={4}, - number={14}, - pages={1994}, - year={1971}, - publisher={IOP Publishing} -} - -@article{bogomolny.1977.dispersion, - title={Calculation of the green functions by the coupling constant dispersion relations}, - author={Bogomolny, Evgeny B}, - journal={Physics Letters B}, - volume={67}, - number={2}, - pages={193--194}, - year={1977}, - publisher={Elsevier} -} - -@techreport{brezin.1976.perturbation, - title={Perturbation theory at large order; 2, role of the vacuum instability}, - author={Br{\'e}zin, E and Le Guillou, Jean Claude and Zinn-Justin, Jean}, - year={1976} -} - -@book{cardy.1996.scaling, - title={Scaling and renormalization in statistical physics}, - author={Cardy, John}, - volume={5}, - year={1996}, - publisher={Cambridge university press} -} - -@article{caselle.2001.critical, - title={The critical equation of state of the two-dimensional Ising model}, - author={Caselle, Michele and Hasenbusch, Martin and Pelissetto, Andrea and Vicari, Ettore}, - journal={Journal of Physics A: Mathematical and General}, - volume={34}, - number={14}, - pages={2923}, - year={2001}, - publisher={IOP Publishing} -} - -@article{chan.2011.ising, - title={The Ising susceptibility scaling function}, - author={Chan, Y and Guttmann, Anthony J and Nickel, BG and Perk, JHH}, - 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Despite this, they -have long been viewed as irrelevant to the quantitative study of abrupt phase -transitions, and likewise the presence of an abrupt phase transition has -typically been considered irrelevant to scaling analysis. Our work -demonstrates, for one common model with both a continuous and a -line of abrupt transitions, that well-known singularities in thermodynamic -functions at the abrupt transition long assumed to be unobservable are in fact -part of the scaling theory of the critical point, and make explicit -predictions confirmed by numerics. This result indicates that many published -statistical models whose abrupt transitions were not incorporated into scaling -theories should be revisited, and introduces a novel new method for -determining new universal scaling properties in previously poorly-understood -regimes. - - diff --git a/essential-ising_prl2.txt b/essential-ising_prl2.txt deleted file mode 100644 index 1cc1f44..0000000 --- a/essential-ising_prl2.txt +++ /dev/null @@ -1 +0,0 @@ -The Ising model critical point governs the termination of a line of abrupt transitions (e.g., the liquid-gas transition endpoint at high pressure). The universal scaling functions describing the critical behavior in non-zero field have long been a challenge. We show that these scaling functions inherit an essential singularity from the abrupt transition line, associated with the nucleation of droplets in the metastable phase (e.g. the supercooled gas), giving an analytic formula for the singularity and fitting to simulations. This same basic method may be useful in studying other critical points with metastability, such as in glasses and disordered systems. diff --git a/figs/fig-mag_scaling-func-mod.dat b/figs/fig-mag_scaling-func-mod.dat deleted file mode 100644 index fc2947a..0000000 --- a/figs/fig-mag_scaling-func-mod.dat +++ /dev/null @@ -1,11514 +0,0 @@ -0.0007052661693227589 1.222409953716735 -0.001122417188767094 1.222420598483419 -0.0015398054716768543 1.222431246058725 -0.001957431099433065 1.2224418966267505 -0.0023752941534654344 1.2224525501764913 -0.0027933947152594856 1.2224632067121646 -0.00321173286636588 1.2224738662400279 -0.0036303086883853297 1.2224845287591373 -0.00404912226296338 1.2224951942680957 -0.004468173671817067 1.222505862767504 -0.004887462996716529 1.2225165342554214 -0.005306990319480478 1.2225272087320074 -0.00572675572197753 1.2225378861967173 -0.006146759286147286 1.2225485666489038 -0.006567001093987319 1.2225592500879203 -0.006987481227529069 1.222569936513495 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-5.08212835343 --4.885 -5.08112843823 --4.884 -5.0801282809 --4.883 -5.07912836575 --4.882 -5.07812845035 --4.881 -5.07712853472 --4.88 -5.07612837911 --4.879 -5.0751284635 --4.878 -5.07412854767 --4.877 -5.07312863156 --4.876 -5.07212847773 --4.875 -5.07112856167 --4.874 -5.07012864537 --4.873 -5.06912849297 --4.872 -5.06812857672 --4.871 -5.06712866025 --4.87 -5.0661287435 --4.869 -5.0651285928 --4.868 -5.06412867613 --4.867 -5.0631287592 --4.866 -5.06212884202 --4.865 -5.06112869306 --4.864 -5.06012877593 --4.8629999999999995 -5.05912885857 --4.862 -5.058128941 --4.861 -5.05712879371 --4.86 -5.05612887619 --4.859 -5.05512895841 --4.858 -5.05412881252 --4.857 -5.05312889482 --4.856 -5.05212897685 --4.855 -5.05112905869 --4.854 -5.05012891449 --4.853 -5.04912899634 --4.852 -5.048129078 --4.851 -5.0471291594 --4.85 -5.04612901686 --4.849 -5.04512909835 --4.848 -5.04412917957 --4.847 -5.04312903844 --4.846 -5.04212911972 --4.845 -5.04112920078 --4.844 -5.04012928165 --4.843 -5.03912914213 --4.842 -5.03812922302 --4.841 -5.03712930373 --4.84 -5.03612938417 --4.839 -5.03512924633 --4.838 -5.03412932684 --4.837 -5.03312940713 --4.836 -5.03212948723 --4.835 -5.03112935097 --4.834 -5.03012943111 --4.833 -5.02912951106 --4.832 -5.02812937614 --4.831 -5.02712945615 --4.83 -5.02612953591 --4.829 -5.02512961547 --4.828 -5.02412948218 --4.827 -5.0231295618 --4.826 -5.02212964121 --4.825 -5.02112972043 --4.824 -5.02012958871 --4.823 -5.019129668 --4.822 -5.01812974705 --4.821 -5.01712961664 --4.82 -5.01612969579 --4.819 -5.0151297747 --4.818 -5.01412985341 --4.817 -5.01312972461 --4.816 -5.01212980338 --4.815 -5.01112988198 --4.814 -5.01012985739 --4.813 -5.0091299358 --4.812 -5.00812991154 --4.811 -5.00712998978 --4.81 -5.0061299658 --4.809 -5.00512994213 --4.808 -5.00413002024 --4.807 -5.00312999686 --4.806 -5.00213007481 --4.805 -5.00113005172 --4.804 -5.00013012953 --4.803 -4.99913010674 --4.802 -4.99813018437 --4.801 -4.99713016191 --4.8 -4.99613023938 --4.799 -4.99513021722 --4.798 -4.99413029454 --4.797 -4.99313027266 --4.796 -4.99213025108 --4.795 -4.99113032827 --4.794 -4.99013030696 --4.793 -4.98913038404 --4.792 -4.98813036303 --4.791 -4.98713043996 --4.79 -4.98613041925 --4.789 -4.985130496 --4.788 -4.98413047562 --4.787 -4.98313055223 --4.786 -4.98213053213 --4.785 -4.98113060861 --4.784 -4.98013058881 --4.783 -4.9791305693 --4.782 -4.97813064566 --4.781 -4.97713062642 --4.78 -4.97613070266 --4.779 -4.97513068372 --4.778 -4.9741307598 --4.777 -4.97313074119 --4.776 -4.97213081713 --4.775 -4.97113079882 --4.774 -4.97013087463 --4.773 -4.96913085659 --4.772 -4.96813093229 --4.771 -4.96713091455 --4.77 -4.96613089706 --4.769 -4.96513097267 --4.768 -4.96413095549 --4.767 -4.96313103098 --4.766 -4.96213101408 --4.765 -4.96113108942 --4.764 -4.96013107285 --4.763 -4.95913114806 --4.762 -4.95813113176 --4.761 -4.95713120687 --4.76 -4.95613119087 --4.759 -4.95513126586 --4.758 -4.95413125016 --4.757 -4.95313123471 --4.756 -4.95213130962 --4.755 -4.95113129447 --4.754 -4.95013136924 --4.753 -4.94913135441 --4.752 -4.94813142906 --4.751 -4.94713141453 --4.75 -4.94613148906 --4.749 -4.9451314748 --4.748 -4.94413154925 --4.747 -4.94313153528 --4.746 -4.94213160962 --4.745 -4.94113159594 --4.744 -4.94013158252 --4.743 -4.93913165679 --4.742 -4.93813164367 --4.741 -4.93713171781 --4.74 -4.936131705 --4.739 -4.93513177904 --4.7379999999999995 -4.9341317665 --4.737 -4.93313184046 --4.736 -4.93213182821 --4.735 -4.93113190207 --4.734 -4.93013189012 --4.733 -4.92913196385 --4.732 -4.92813195221 --4.731 -4.9271319408 --4.73 -4.92613201451 --4.729 -4.92513200339 --4.728 -4.92413207697 --4.727 -4.92313206618 --4.726 -4.92213213966 --4.725 -4.92113212914 --4.724 -4.92013220255 --4.723 -4.91913219232 --4.7219999999999995 -4.91813226562 --4.721 -4.9171322557 --4.72 -4.91613232892 --4.719 -4.91513231929 --4.718 -4.91413230988 --4.717 -4.91313238306 --4.716 -4.91213237397 --4.715 -4.91113244706 --4.714 -4.91013243826 --4.713 -4.90913251127 --4.712 -4.90813250274 --4.711 -4.90713257568 --4.71 -4.90613256744 --4.709 -4.90513264029 --4.708 -4.90413263237 --4.707 -4.90313270513 --4.706 -4.9021326975 --4.705 -4.90113269009 --4.704 -4.90013276283 --4.703 -4.89913275573 --4.702 -4.89813282839 --4.701 -4.89713282156 --4.7 -4.89613289418 --4.699 -4.89513288764 --4.698 -4.89413296018 --4.697 -4.89313295393 --4.696 -4.89213302638 --4.695 -4.89113302045 --4.694 -4.89013309283 --4.693 -4.88913308716 --4.692 -4.88813308176 --4.691 -4.88713315413 --4.69 -4.88613314901 --4.689 -4.88513322132 --4.688 -4.88413321647 --4.687 -4.88313328873 --4.686 -4.88213328418 --4.685 -4.88113335636 --4.684 -4.88013335212 --4.683 -4.87913342424 --4.682 -4.87813342029 --4.681 -4.87713349235 --4.68 -4.87613348867 --4.679 -4.87513348525 --4.678 -4.87413355731 --4.677 -4.87313355416 --4.676 -4.87213362618 --4.675 -4.87113362332 --4.674 -4.87013369529 --4.673 -4.86913369273 --4.672 -4.86813376463 --4.671 -4.86713376237 --4.67 -4.86613383422 --4.669 -4.86513383223 --4.668 -4.86413390406 --4.667 -4.86313390236 --4.666 -4.86213390092 --4.665 -4.86113397274 --4.664 -4.86013397156 --4.663 -4.85913404336 --4.662 -4.85813404248 --4.661 -4.85713411421 --4.66 -4.85613411364 --4.659 -4.85513418533 --4.658 -4.85413418505 --4.657 -4.8531342567 --4.656 -4.85213425669 --4.655 -4.85113432832 --4.654 -4.85013432861 --4.653 -4.84913432912 --4.652 -4.84813440078 --4.651 -4.84713440159 --4.65 -4.84613447321 --4.649 -4.84513447431 --4.648 -4.84413454588 --4.647 -4.84313454729 --4.646 -4.84213461883 --4.645 -4.84113462051 --4.644 -4.84013469205 --4.643 -4.83913469402 --4.642 -4.83813476552 --4.641 -4.83713476779 --4.64 -4.83613477028 --4.639 -4.83513484183 --4.638 -4.83413484461 --4.6370000000000005 -4.83313491613 --4.636 -4.83213491921 --4.635 -4.83113499069 --4.634 -4.83013499408 --4.633 -4.82913506554 --4.632 -4.8281350692 --4.631 -4.82713514066 --4.63 -4.82613514461 --4.629 -4.82513521604 --4.628 -4.8241352203 --4.627 -4.82313522478 --4.626 -4.82213529627 --4.625 -4.82113530104 --4.624 -4.8201353725 --4.623 -4.81913537758 --4.622 -4.81813544903 --4.621 -4.8171354544 --4.62 -4.81613552584 --4.619 -4.81513553149 --4.618 -4.81413560293 --4.617 -4.81313560887 --4.616 -4.8121356803 --4.615 -4.81113568655 --4.614 -4.81013569301 --4.6129999999999995 -4.8091357645 --4.612 -4.80813577127 --4.611 -4.80713584275 --4.61 -4.80613584982 --4.609 -4.80513592131 --4.608 -4.80413592865 --4.607 -4.80313600015 --4.606 -4.80213600779 --4.605 -4.80113607929 --4.604 -4.80013608722 --4.603 -4.79913615871 --4.602 -4.79813616695 --4.601 -4.7971361754 --4.6 -4.79613624697 --4.599 -4.79513625573 --4.598 -4.79413632731 --4.5969999999999995 -4.79313633635 --4.596 -4.79213640795 --4.595 -4.79113641729 --4.594 -4.79013648889 --4.593 -4.78913649853 --4.592 -4.78813657013 --4.591 -4.78713658008 --4.59 -4.78613659023 --4.589 -4.78513666194 --4.588 -4.78413667239 --4.587 -4.7831367441 --4.586 -4.78213675486 --4.585 -4.78113682659 --4.584 -4.78013683763 --4.583 -4.77913690939 --4.582 -4.77813692073 --4.581 -4.77713699251 --4.58 -4.77613700415 --4.579 -4.77513707594 --4.578 -4.77413708789 --4.577 -4.77313710005 --4.576 -4.77213717194 --4.575 -4.77113718441 --4.574 -4.77013725633 --4.573 -4.7691372691 --4.572 -4.76813734104 --4.571 -4.7671373541 --4.57 -4.76613742609 --4.569 -4.76513743944 --4.568 -4.76413751144 --4.567 -4.76313752511 --4.566 -4.76213759715 --4.565 -4.76113761112 --4.564 -4.76013762529 --4.563 -4.75913769744 --4.562 -4.75813771193 --4.561 -4.75713778412 --4.56 -4.7561377989 --4.559 -4.75513787114 --4.558 -4.75413788621 --4.557 -4.75313795849 --4.556 -4.75213797387 --4.555 -4.75113804618 --4.554 -4.75013806187 --4.553 -4.74913813422 --4.552 -4.74813815021 --4.551 -4.74713816642 --4.55 -4.7461382389 --4.549 -4.74513825542 --4.548 -4.74413832794 --4.547 -4.74313834476 --4.546 -4.74213841734 --4.545 -4.74113843446 --4.544 -4.74013850708 --4.543 -4.73913852451 --4.542 -4.73813859718 --4.541 -4.73713861492 --4.54 -4.73613868764 --4.539 -4.73513870567 --4.538 -4.73413872393 --4.537 -4.7331387968 --4.536 -4.73213881535 --4.535 -4.73113888829 --4.534 -4.73013890715 --4.533 -4.72913898015 --4.532 -4.72813899931 --4.531 -4.72713907236 --4.53 -4.72613909185 --4.529 -4.72513916495 --4.528 -4.72413918473 --4.527 -4.72313925791 --4.526 -4.722139278 --4.525 -4.72113929832 --4.524 -4.72013937164 --4.523 -4.71913939226 --4.522 -4.71813946566 --4.521 -4.71713948659 --4.52 -4.71613956005 --4.519 -4.7151395813 --4.518 -4.71413965483 --4.517 -4.71313967639 --4.516 -4.71213974999 --4.515 -4.71113977184 --4.514 -4.71013981973 --4.513 -4.7091398677 --4.5120000000000005 -4.70813991578 --4.511 -4.70713996394 --4.51 -4.7061400122 --4.509 -4.70514006058 --4.508 -4.70414010903 --4.507 -4.70314015759 --4.506 -4.70214018093 --4.505 -4.70114022974 --4.504 -4.70014027865 --4.503 -4.69914032767 --4.502 -4.69814037677 --4.501 -4.69714042599 --4.5 -4.6961404753 --4.499 -4.69514052471 --4.498 -4.69414057423 --4.497 -4.69314062384 --4.496 -4.69214067356 --4.495 -4.69114072338 --4.494 -4.69014077329 --4.493 -4.68914079874 --4.492 -4.68814084891 --4.491 -4.68714089919 --4.49 -4.68614094958 --4.489 -4.68514100006 --4.4879999999999995 -4.68414105065 --4.487 -4.68314110135 --4.486 -4.68214115214 --4.485 -4.68114120305 --4.484 -4.68014125406 --4.483 -4.67914130516 --4.482 -4.67814135639 --4.481 -4.6771414077 --4.48 -4.67614143528 --4.479 -4.67514148686 --4.478 -4.67414153856 --4.477 -4.67314159036 --4.476 -4.67214164227 --4.475 -4.67114169428 --4.474 -4.67014174641 --4.473 -4.66914179863 --4.4719999999999995 -4.66814185096 --4.471 -4.66714190341 --4.47 -4.66614195596 --4.469 -4.66514200863 --4.468 -4.6641420614 --4.467 -4.66314209112 --4.466 -4.66214214417 --4.465 -4.66114219732 --4.464 -4.66014225059 --4.463 -4.65914230396 --4.462 -4.65814235744 --4.461 -4.65714241105 --4.46 -4.65614246475 --4.459 -4.65514251856 --4.458 -4.6541425725 --4.457 -4.65314262654 --4.4559999999999995 -4.65214268069 --4.455 -4.65114273496 --4.454 -4.65014276687 --4.453 -4.64914282142 --4.452 -4.64814287608 --4.451 -4.64714293085 --4.45 -4.64614298574 --4.449 -4.64514304074 --4.448 -4.64414309587 --4.447 -4.6431431511 --4.446 -4.64214320644 --4.445 -4.64114326191 --4.444 -4.64014331749 --4.443 -4.63914337318 --4.442 -4.638143429 --4.441 -4.63714346312 --4.4399999999999995 -4.63614351921 --4.439 -4.63514357543 --4.438 -4.63414363176 --4.437 -4.63314368822 --4.436 -4.63214374479 --4.435 -4.63114380147 --4.434 -4.63014385829 --4.433 -4.62914391521 --4.432 -4.62814397227 --4.431 -4.62714402943 --4.43 -4.62614408671 --4.429 -4.62514414413 --4.428 -4.62414418049 --4.427 -4.62314423819 --4.426 -4.62214429601 --4.425 -4.62114435395 --4.4239999999999995 -4.62014441201 --4.423 -4.61914447021 --4.422 -4.61814452852 --4.421 -4.61714458696 --4.42 -4.61614464551 --4.4190000000000005 -4.6151447042 --4.418 -4.614144763 --4.417 -4.61314482193 --4.416 -4.61214488099 --4.415 -4.61114491963 --4.414 -4.61014497897 --4.413 -4.60914503846 --4.412 -4.60814509806 --4.411 -4.60714515779 --4.41 -4.60614521766 --4.409 -4.60514527764 --4.408 -4.60414533776 --4.407 -4.60314539799 --4.406 -4.60214545836 --4.405 -4.60114551886 --4.404 -4.60014557949 --4.4030000000000005 -4.59914564025 --4.402 -4.59814568119 --4.401 -4.59714574225 --4.4 -4.59614580345 --4.399 -4.59514586476 --4.398 -4.59414592621 --4.397 -4.5931459878 --4.396 -4.59214604951 --4.395 -4.59114611135 --4.394 -4.59014617334 --4.393 -4.58914623544 --4.392 -4.58814629769 --4.391 -4.58714636006 --4.39 -4.58614642257 --4.389 -4.58514646587 --4.388 -4.58414652869 --4.3870000000000005 -4.58314659165 --4.386 -4.58214665474 --4.385 -4.58114671796 --4.384 -4.58014678133 --4.383 -4.57914684482 --4.382 -4.57814690845 --4.381 -4.57714697223 --4.38 -4.57614703613 --4.379 -4.57514710017 --4.378 -4.57414716436 --4.377 -4.57314722867 --4.376 -4.57214727436 --4.375 -4.571147339 --4.374 -4.57014740377 --4.373 -4.5691474687 --4.372 -4.56814753375 --4.371 -4.56714759895 --4.37 -4.56614766429 --4.369 -4.56514772976 --4.368 -4.56414779539 --4.367 -4.56314786114 --4.366 -4.56214792704 --4.365 -4.56114799309 --4.364 -4.56014805928 --4.3629999999999995 -4.55914810738 --4.362 -4.5581481739 --4.361 -4.55714824056 --4.36 -4.55614830737 --4.359 -4.55514837431 --4.358 -4.5541484414 --4.357 -4.55314850864 --4.356 -4.55214857602 --4.355 -4.55114864355 --4.354 -4.55014871122 --4.353 -4.54914877904 --4.352 -4.54814884701 --4.351 -4.54714891512 --4.35 -4.54614896569 --4.349 -4.54514903415 --4.348 -4.54414910275 --4.3469999999999995 -4.54314917149 --4.346 -4.54214924039 --4.345 -4.54114930944 --4.344 -4.54014937864 --4.343 -4.53914944798 --4.342 -4.53814951747 --4.341 -4.53714958712 --4.34 -4.53614965692 --4.339 -4.53514972688 --4.338 -4.53414979698 --4.337 -4.53314985006 --4.336 -4.53214992051 --4.335 -4.53114999111 --4.334 -4.53015006186 --4.333 -4.52915013278 --4.332 -4.52815020383 --4.3309999999999995 -4.52715027505 --4.33 -4.52615034643 --4.329 -4.52515041795 --4.328 -4.52415048964 --4.327 -4.52315056147 --4.326 -4.52215063347 --4.325 -4.52115070563 --4.324 -4.52015076128 --4.323 -4.51915083379 --4.322 -4.51815090645 --4.321 -4.51715097927 --4.32 -4.51615105226 --4.319 -4.5151511254 --4.318 -4.51415119869 --4.317 -4.51315127216 --4.316 -4.512151345781 --4.3149999999999995 -4.511151419562 --4.314 -4.510151493512 --4.313 -4.509151567616 --4.312 -4.508151625686 --4.311 -4.507151700151 --4.31 -4.506151774779 --4.309 -4.50515184958 --4.308 -4.504151924535 --4.307 -4.503151999663 --4.306 -4.502152074947 --4.305 -4.501152150396 --4.304 -4.500152226019 --4.303 -4.499152301799 --4.302 -4.498152377746 --4.301 -4.497152453867 --4.3 -4.496152530147 --4.2989999999999995 -4.495152590869 --4.298 -4.49415266753 --4.297 -4.493152744349 --4.296 -4.492152821347 --4.295 -4.491152898503 --4.2940000000000005 -4.490152975839 --4.293 -4.489153053336 --4.292 -4.488153131004 --4.291 -4.48715320885 --4.29 -4.48615328686 --4.289 -4.48515336504 --4.288 -4.484153443401 --4.287 -4.483153521927 --4.286 -4.482153585363 --4.285 -4.481153664279 --4.284 -4.480153743359 --4.283 -4.479153822621 --4.282 -4.47815390205 --4.281 -4.477153981653 --4.28 -4.476154061441 --4.279 -4.475154141395 --4.2780000000000005 -4.474154221537 --4.277 -4.473154301844 --4.276 -4.472154382329 --4.275 -4.471154463 --4.274 -4.470154543841 --4.273 -4.469154610048 --4.272 -4.468154691288 --4.271 -4.467154772699 --4.27 -4.466154854291 --4.269 -4.46515493607 --4.268 -4.464155018021 --4.267 -4.463155100163 --4.266 -4.462155182477 --4.265 -4.461155264974 --4.264 -4.46015534766 --4.263 -4.459155430522 --4.2620000000000005 -4.458155513576 --4.261 -4.457155596803 --4.26 -4.456155665842 --4.259 -4.45515574948 --4.258 -4.454155833296 --4.257 -4.453155917297 --4.256 -4.452156001492 --4.255 -4.451156085865 --4.254 -4.450156170426 --4.253 -4.449156255181 --4.252 -4.448156340116 --4.251 -4.447156425249 --4.25 -4.446156510561 --4.249 -4.445156596061 --4.248 -4.444156681761 --4.247 -4.443156753691 --4.246 -4.442156839802 --4.245 -4.441156926097 --4.244 -4.440157012583 --4.243 -4.43915709927 --4.242 -4.43815718614 --4.241 -4.437157273203 --4.24 -4.436157360467 --4.239 -4.435157447918 --4.2379999999999995 -4.434157535562 --4.237 -4.43315762341 --4.236 -4.432157711445 --4.235 -4.431157799682 --4.234 -4.43015787457 --4.233 -4.429157963224 --4.232 -4.428158052082 --4.231 -4.427158141132 --4.23 -4.426158230386 --4.229 -4.425158319832 --4.228 -4.424158409476 --4.227 -4.423158499328 --4.226 -4.422158589371 --4.225 -4.421158679615 --4.224 -4.420158770068 --4.223 -4.419158860715 --4.2219999999999995 -4.418158951563 --4.221 -4.417159029481 --4.22 -4.416159120766 --4.219 -4.415159212261 --4.218 -4.414159303952 --4.217 -4.413159395847 --4.216 -4.412159487955 --4.215 -4.411159580261 --4.214 -4.410159672779 --4.213 -4.409159765498 --4.212 -4.408159858422 --4.211 -4.40715994514 --4.21 -4.406160038495 --4.209 -4.405160132059 --4.208 -4.404160225836 --4.207 -4.403160319819 --4.2059999999999995 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--4.165 -4.361164422597 --4.164 -4.360164526167 --4.163 -4.359164629977 --4.162 -4.358164734012 --4.161 -4.357164838282 --4.16 -4.356164937083 --4.159 -4.355165041834 --4.158 -4.354165146828 --4.157 -4.35316525205 --4.156 -4.352165357509 --4.155 -4.351165463212 --4.154 -4.350165569146 --4.1530000000000005 -4.349165669706 --4.152 -4.34816577613 --4.151 -4.347165882793 --4.15 -4.346165989702 --4.149 -4.345166096845 --4.148 -4.344166204229 --4.147 -4.34316630632 --4.146 -4.342166414201 --4.145 -4.341166522324 --4.144 -4.340166630697 --4.143 -4.339166739307 --4.142 -4.338166848168 --4.141 -4.337166957268 --4.14 -4.336167061161 --4.139 -4.335167170772 --4.138 -4.334167280623 --4.1370000000000005 -4.333167390728 --4.136 -4.332167501074 --4.135 -4.331167611669 --4.134 -4.330167717142 --4.133 -4.329167828248 --4.132 -4.328167939604 --4.131 -4.327168051217 --4.13 -4.326168163077 --4.129 -4.325168275188 --4.128 -4.324168387557 --4.127 -4.323168494883 --4.126 -4.322168607772 --4.125 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--4.003 -4.199184559676 --4.002 -4.198184709372 --4.001 -4.197184859413 --4. -4.196185009791 --3.9989999999999997 -4.195185160512 --3.998 -4.194185307648 --3.997 -4.193185459065 --3.996 -4.192185610829 --3.995 -4.191185762939 --3.9939999999999998 -4.190185915391 --3.993 -4.189186068194 --3.992 -4.188186217467 --3.9909999999999997 -4.187186370973 --3.99 -4.186186524834 --3.989 -4.185186679039 --3.988 -4.184186833599 --3.987 -4.183186988511 --3.9859999999999998 -4.182187143773 --3.985 -4.181187295576 --3.984 -4.180187451557 --3.9829999999999997 -4.179187607892 --3.982 -4.178187764587 --3.981 -4.177187921635 --3.98 -4.176188079042 --3.979 -4.175188233046 --3.9779999999999998 -4.174188391178 --3.977 -4.173188549674 --3.976 -4.17218870853 --3.975 -4.171188867748 --3.974 -4.170189027329 --3.973 -4.169189187272 --3.972 -4.168189343879 --3.971 -4.167189504563 --3.9699999999999998 -4.16618966561 --3.9690000000000003 -4.16518982703 --3.968 -4.164189988813 --3.967 -4.163190150967 --3.966 -4.162190309839 --3.965 -4.161190472739 --3.964 -4.160190636013 --3.963 -4.159190799659 --3.9619999999999997 -4.158190963677 --3.9610000000000003 -4.157191128073 --3.96 -4.156191292839 --3.959 -4.155191454391 --3.958 -4.15419161992 --3.957 -4.153191785825 --3.956 -4.152191952111 --3.955 -4.151192118776 --3.9539999999999997 -4.150192285819 --3.9530000000000003 -4.149192449703 --3.952 -4.148192617517 --3.951 -4.147192785717 --3.95 -4.1461929543 --3.949 -4.145193123266 --3.948 -4.144193292619 --3.947 -4.143193462359 --3.9459999999999997 -4.142193628997 --3.9450000000000003 -4.141193799522 --3.944 -4.140193970433 --3.943 -4.139194141736 --3.942 -4.13819431343 --3.941 -4.137194485514 --3.94 -4.136194654553 --3.939 -4.135194827433 --3.9379999999999997 -4.134195000706 --3.9370000000000003 -4.133195174379 --3.936 -4.132195348445 --3.935 -4.131195522909 --3.934 -4.130195697772 --3.933 -4.129195869648 --3.932 -4.128196045318 --3.931 -4.12719622139 --3.9299999999999997 -4.126196397861 --3.9290000000000003 -4.12519657474 --3.928 -4.124196752019 --3.927 -4.123196926365 --3.926 -4.122197104465 --3.925 -4.121197282969 --3.924 -4.120197461883 --3.923 -4.119197641205 --3.9219999999999997 -4.118197820935 --3.9210000000000003 -4.11719800108 --3.92 -4.116198178347 --3.919 -4.115198359321 --3.918 -4.114198540709 --3.917 -4.113198722509 --3.916 -4.112198904727 --3.915 -4.111199087361 --3.9139999999999997 -4.110199267171 --3.9130000000000003 -4.10919945065 --3.912 -4.108199634545 --3.911 -4.107199818861 --3.91 -4.106200003599 --3.909 -4.105200187156 --3.908 -4.104200372743 --3.907 -4.103200558755 --3.9059999999999997 -4.102200743599 --3.9050000000000002 -4.101200930467 --3.904 -4.100201117758 --3.903 -4.099201303901 --3.902 -4.098201492056 --3.901 -4.097201680639 --3.9 -4.096201869654 --3.899 -4.095202057537 --3.8979999999999997 -4.09420224742 --3.8970000000000002 -4.093202437741 --3.896 -4.092202626939 --3.895 -4.091202818133 --3.894 -4.090203009764 --3.893 -4.089203200288 --3.892 -4.0882033928 --3.891 -4.087203585753 --3.8899999999999997 -4.086203777612 --3.8890000000000002 -4.085203971453 --3.888 -4.084204165734 --3.887 -4.083204358939 --3.886 -4.082204554115 --3.885 -4.081204749736 --3.884 -4.080204945806 --3.883 -4.079205140817 --3.8819999999999997 -4.078205337787 --3.8810000000000002 -4.077205535211 --3.88 -4.076205731585 --3.879 -4.075205929915 --3.878 -4.074206128699 --3.877 -4.073206326449 --3.876 -4.072206526147 --3.875 -4.071206726302 --3.8739999999999997 -4.070206925436 --3.873 -4.069207126513 --3.872 -4.068207328047 --3.871 -4.067207530044 --3.87 -4.066207731039 --3.8689999999999998 -4.065207933963 --3.868 -4.064208137353 --3.867 -4.063208339756 --3.8659999999999997 -4.062208544079 --3.865 -4.061208748874 --3.864 -4.060208952691 --3.863 -4.059209158426 --3.862 -4.058209364632 --3.8609999999999998 -4.057209569875 --3.86 -4.05620977703 --3.859 -4.055209984659 --3.8579999999999997 -4.054210192762 --3.857 -4.053210399923 --3.856 -4.052210608983 --3.855 -4.051210818523 --3.854 -4.050211027132 --3.8529999999999998 -4.049211237634 --3.852 -4.04821144862 --3.851 -4.047211658688 --3.8499999999999996 -4.046211870643 --3.849 -4.045212083086 --3.848 -4.044212294622 --3.847 -4.043212508041 --3.846 -4.042212721949 --3.8449999999999998 -4.041212936345 --3.8440000000000003 -4.040213149857 --3.843 -4.03921336524 --3.842 -4.038213581117 --3.841 -4.037213796119 --3.84 -4.036214012988 --3.839 -4.035214230356 --3.838 -4.034214446861 --3.8369999999999997 -4.033214665228 --3.8360000000000003 -4.032214884097 --3.835 -4.031215102114 --3.834 -4.030215321988 --3.833 -4.029215542367 --3.832 -4.028215761907 --3.831 -4.0272159833 --3.83 -4.026216205199 --3.8289999999999997 -4.025216427605 --3.8280000000000003 -4.024216649195 --3.827 -4.023216872625 --3.826 -4.022217096568 --3.825 -4.021217319705 --3.824 -4.020217544677 --3.823 -4.019217770166 --3.822 -4.018217994861 --3.8209999999999997 -4.017218221387 --3.8200000000000003 -4.016218448434 --3.819 -4.015218674697 --3.818 -4.014218902788 --3.817 -4.013219131402 --3.816 -4.012219360537 --3.815 -4.01121958891 --3.814 -4.010219819102 --3.8129999999999997 -4.00922004982 --3.8120000000000003 -4.008220279789 --3.811 -4.007220511569 --3.81 -4.006220743881 --3.809 -4.005220975454 --3.808 -4.004221208834 --3.807 -4.003221442751 --3.806 -4.00222167594 --3.8049999999999997 -4.001221910932 --3.8040000000000003 -4.000222146465 --3.803 -3.999222382536 --3.802 -3.998222617897 --3.801 -3.997222855055 --3.8 -3.996223092756 --3.799 -3.99522332976 --3.798 -3.994223568556 --3.7969999999999997 -3.993223807898 --3.7960000000000003 -3.992224046557 --3.795 -3.991224287001 --3.794 -3.990224527997 --3.793 -3.98922476832 --3.792 -3.988225010425 --3.791 -3.987225253086 --3.79 -3.986225495086 --3.7889999999999997 -3.985225738862 --3.7880000000000003 -3.9842259832 --3.787 -3.983226228096 --3.786 -3.982226472349 --3.785 -3.981226718373 --3.784 -3.98022696496 --3.783 -3.979227210917 --3.782 -3.97822745864 --3.7809999999999997 -3.977227706931 --3.7800000000000002 -3.976227954604 --3.779 -3.975228204038 --3.778 -3.974228454044 --3.777 -3.973228703443 --3.776 -3.9722289546 --3.775 -3.971229206334 --3.774 -3.970229458645 --3.7729999999999997 -3.969229710363 --3.7720000000000002 -3.968229963838 --3.771 -3.967230217891 --3.77 -3.966230471365 --3.769 -3.965230726589 --3.768 -3.964230982397 --3.767 -3.963231237639 --3.766 -3.962231494626 --3.7649999999999997 -3.961231752202 --3.7640000000000002 -3.960232009224 --3.763 -3.959232267985 --3.762 -3.958232527342 --3.761 -3.957232787293 --3.76 -3.956233046703 --3.759 -3.955233307851 --3.758 -3.954233569599 --3.7569999999999997 -3.953233830817 --3.7560000000000002 -3.952234093771 --3.755 -3.951234357325 --3.754 -3.950234620365 --3.753 -3.949234885134 --3.752 -3.94823515051 --3.751 -3.947235415382 --3.75 -3.946235681981 --3.7489999999999997 -3.945235949191 --3.748 -3.94423621591 --3.747 -3.94323648435 --3.746 -3.942236753407 --3.745 -3.941237023081 --3.7439999999999998 -3.940237292278 --3.743 -3.939237563194 --3.742 -3.938237834732 --3.7409999999999997 -3.937238105804 --3.74 -3.936238378593 --3.739 -3.935238652006 --3.738 -3.934238924968 --3.737 -3.933239199641 --3.7359999999999998 -3.932239474944 --3.735 -3.931239749807 --3.734 -3.930240026378 --3.7329999999999997 -3.929240303583 --3.732 -3.928240581427 --3.731 -3.927240858843 --3.73 -3.926241137965 --3.729 -3.925241417728 --3.7279999999999998 -3.924241697076 --3.727 -3.923241978127 --3.726 -3.922242259823 --3.7249999999999996 -3.921242541116 --3.724 -3.920242824111 --3.723 -3.919243107753 --3.722 -3.918243391006 --3.721 -3.917243675955 --3.7199999999999998 -3.916243961557 --3.7190000000000003 -3.915244247818 --3.718 -3.914244533701 --3.717 -3.913244821279 --3.716 -3.912245109517 --3.715 -3.91124539739 --3.714 -3.910245686955 --3.713 -3.909245977186 --3.7119999999999997 -3.908246267063 --3.7110000000000003 -3.907246558631 --3.71 -3.906246850866 --3.709 -3.905247142762 --3.708 -3.904247436344 --3.707 -3.903247730599 --3.706 -3.902248025531 --3.705 -3.901248320139 --3.7039999999999997 -3.900248616428 --3.7030000000000003 -3.899248913399 --3.702 -3.898249210056 --3.701 -3.897249508394 --3.7 -3.896249807418 --3.699 -3.895250106139 --3.698 -3.89425040654 --3.697 -3.89325070763 --3.6959999999999997 -3.89225100843 --3.6950000000000003 -3.891251310908 --3.694 -3.890251614079 --3.693 -3.889251916973 --3.692 -3.888252221541 --3.691 -3.887252526807 --3.69 -3.886252832776 --3.689 -3.885253138482 --3.6879999999999997 -3.88425344586 --3.6870000000000003 -3.883253753944 --3.686 -3.882254061777 --3.685 -3.881254371279 --3.684 -3.880254681493 --3.683 -3.879254991467 --3.682 -3.878255303109 --3.681 -3.877255615468 --3.6799999999999997 -3.876255927597 --3.6790000000000003 -3.875256241396 --3.678 -3.874256555913 --3.677 -3.873256871153 --3.676 -3.872257186181 --3.675 -3.871257502872 --3.674 -3.870257820292 --3.673 -3.869258137512 --3.6719999999999997 -3.868258456392 --3.6710000000000003 -3.867258776008 --3.67 -3.866259095432 --3.669 -3.865259416519 --3.668 -3.864259738343 --3.667 -3.863260059989 --3.666 -3.862260383296 --3.665 -3.861260707345 --3.6639999999999997 -3.860261032138 --3.6630000000000003 -3.85926135677 --3.662 -3.85826168306 --3.661 -3.857262010099 --3.66 -3.856262336987 --3.659 -3.855262665533 --3.658 -3.854262994834 --3.657 -3.853263323995 --3.6559999999999997 -3.852263654812 --3.6550000000000002 -3.851263986391 --3.654 -3.85026431784 --3.653 -3.849264650945 --3.652 -3.848264984816 --3.651 -3.847265318569 --3.65 -3.846265653977 --3.649 -3.845265990157 --3.6479999999999997 -3.844266327108 --3.6470000000000002 -3.843266663959 --3.646 -3.842267002462 --3.645 -3.841267341744 --3.644 -3.840267680935 --3.643 -3.839268021779 --3.642 -3.838268363407 --3.641 -3.837268704956 --3.6399999999999997 -3.836269048157 --3.6390000000000002 -3.835269392149 --3.638 -3.83426973607 --3.637 -3.833270081645 --3.636 -3.832270428015 --3.635 -3.831270775179 --3.634 -3.830271122292 --3.633 -3.829271471056 --3.6319999999999997 -3.828271820621 --3.6310000000000002 -3.827272170147 --3.63 -3.826272521323 --3.629 -3.825272873306 --3.628 -3.82427322526 --3.627 -3.823273578864 --3.626 -3.822273933281 --3.625 -3.82127428768 --3.6239999999999997 -3.82027464373 --3.623 -3.8192750006 --3.622 -3.818275358287 --3.621 -3.817275715973 --3.62 -3.816276075309 --3.6189999999999998 -3.815276435471 --3.618 -3.814276795642 --3.617 -3.813277157463 --3.6159999999999997 -3.812277520116 --3.615 -3.81127788279 --3.614 -3.810278247113 --3.613 -3.809278612274 --3.612 -3.808278977467 --3.6109999999999998 -3.807279344309 --3.61 -3.806279711996 --3.609 -3.805280080127 --3.6079999999999997 -3.804280449105 --3.607 -3.803280819336 --3.606 -3.802281190017 --3.605 -3.801281561952 --3.604 -3.800281934345 --3.6029999999999998 -3.799282307595 --3.602 -3.798282682102 --3.601 -3.797283057076 --3.5999999999999996 -3.796283432914 --3.599 -3.795283810012 --3.598 -3.794284187583 --3.597 -3.793284566417 --3.596 -3.792284945732 --3.5949999999999998 -3.791285325921 --3.594 -3.790285707374 --3.593 -3.789286089319 --3.592 -3.788286472143 --3.591 -3.787286856235 --3.59 -3.786287240825 --3.589 -3.785287626686 --3.588 -3.784288013051 --3.5869999999999997 -3.783288400307 --3.5860000000000003 -3.782288788838 --3.585 -3.78128917788 --3.584 -3.780289568199 --3.583 -3.779289959037 --3.582 -3.778290350775 --3.581 -3.777290743794 --3.58 -3.776291137341 --3.5789999999999997 -3.775291531794 --3.5780000000000003 -3.774291927532 --3.577 -3.773292323805 --3.576 -3.772292721365 --3.575 -3.771293119467 --3.574 -3.770293518487 --3.573 -3.769293918796 --3.572 -3.768294319657 --3.5709999999999997 -3.767294721442 --3.5700000000000003 -3.766295124521 --3.569 -3.765295528159 --3.568 -3.764295933094 --3.567 -3.763296338595 --3.566 -3.76229674503 --3.565 -3.761297152766 --3.564 -3.760297561077 --3.5629999999999997 -3.759297970689 --3.5620000000000003 -3.758298380885 --3.561 -3.757298792025 --3.56 -3.756299204472 --3.559 -3.755299617509 --3.558 -3.754300031498 --3.557 -3.753300446797 --3.556 -3.752300862697 --3.5549999999999997 -3.751301279908 --3.5540000000000003 -3.750301697727 --3.553 -3.749302116507 --3.552 -3.748302536606 --3.551 -3.747302957319 --3.55 -3.746303379001 --3.549 -3.745303802005 --3.548 -3.744304225633 --3.5469999999999997 -3.743304650585 --3.5460000000000003 -3.742305076168 --3.545 -3.741305502731 --3.544 -3.740305930623 --3.543 -3.739306359154 --3.542 -3.738306789016 --3.541 -3.737307219524 --3.54 -3.736307651024 --3.5389999999999997 -3.73530808386 --3.5380000000000003 -3.734308517351 --3.537 -3.73330895184 --3.536 -3.732309387671 --3.535 -3.731309824165 --3.534 -3.730310262002 --3.533 -3.729310700509 --3.532 -3.728311140027 --3.5309999999999997 -3.727311580893 --3.5300000000000002 -3.72631202244 --3.529 -3.725312465002 --3.528 -3.724312908918 --3.527 -3.723313353523 --3.526 -3.722313799483 --3.525 -3.721314246139 --3.524 -3.720314693823 --3.5229999999999997 -3.719315142869 --3.5220000000000002 -3.718315592619 --3.521 -3.717316043733 --3.52 -3.716316495557 --3.519 -3.715316948423 --3.518 -3.714317402657 --3.517 -3.713317857611 --3.516 -3.712318313614 --3.5149999999999997 -3.711318770991 --3.5140000000000002 -3.710319229097 --3.513 -3.70931968858 --3.512 -3.7083201488 --3.511 -3.707320610079 --3.51 -3.706321072741 --3.509 -3.705321536148 --3.508 -3.704322000623 --3.5069999999999997 -3.703322466485 --3.5060000000000002 -3.702322933104 --3.505 -3.701323401112 --3.504 -3.700323869881 --3.503 -3.699324339731 --3.502 -3.698324810976 --3.501 -3.697325282994 --3.5 -3.69632575641 --3.4989999999999997 -3.695326230603 --3.498 -3.694326705891 --3.497 -3.693327182582 --3.496 -3.69232766006 --3.495 -3.691328138639 --3.4939999999999998 -3.690328618628 --3.493 -3.689329099413 --3.492 -3.688329581614 --3.4909999999999997 -3.687330064616 --3.49 -3.686330548733 --3.489 -3.685331034269 --3.488 -3.684331520618 --3.487 -3.68333200839 --3.4859999999999998 -3.682332496981 --3.485 -3.681332986699 --3.484 -3.680333477846 --3.4829999999999997 -3.679333969821 --3.482 -3.678334462932 --3.481 -3.677334957475 --3.48 -3.676335452859 --3.479 -3.675335949682 --3.4779999999999998 -3.674336447349 --3.477 -3.673336946164 --3.476 -3.672337446424 --3.4749999999999996 -3.671337947538 --3.474 -3.670338449808 --3.473 -3.669338953528 --3.472 -3.668339458114 --3.471 -3.667339964154 --3.4699999999999998 -3.666340471066 --3.469 -3.665340979147 --3.468 -3.664341488688 --3.4669999999999996 -3.663341999111 --3.466 -3.662342511 --3.465 -3.661343023777 --3.464 -3.660343537736 --3.463 -3.659344053165 --3.4619999999999997 -3.658344569493 --3.4610000000000003 -3.657345087012 --3.46 -3.656345606006 --3.459 -3.655346125911 --3.458 -3.654346647297 --3.457 -3.653347169598 --3.456 -3.652347693103 --3.455 -3.651348218096 --3.4539999999999997 -3.650348744014 --3.4530000000000003 -3.649349271146 --3.452 -3.64834979977 --3.451 -3.647350329331 --3.45 -3.64635086039 --3.449 -3.645351392393 --3.448 -3.644351925621 --3.447 -3.643352460355 --3.4459999999999997 -3.642352996041 --3.4450000000000003 -3.641353533238 --3.444 -3.640354071395 --3.443 -3.639354610792 --3.442 -3.638355151704 --3.441 -3.637355693588 --3.44 -3.63635623672 --3.439 -3.635356781375 --3.4379999999999997 -3.634357327011 --3.4370000000000003 -3.633357874175 --3.436 -3.632358422327 --3.435 -3.631358971742 --3.434 -3.63035952269 --3.433 -3.629360074638 --3.432 -3.628360627857 --3.431 -3.627361182617 --3.4299999999999997 -3.626361738386 --3.4290000000000003 -3.625362295702 --3.428 -3.624362854034 --3.427 -3.623363413653 --3.426 -3.622363974823 --3.425 -3.621364537022 --3.424 -3.620365100777 --3.423 -3.619365665568 --3.4219999999999997 -3.618366231659 --3.4210000000000003 -3.617366799315 --3.42 -3.616367368016 --3.419 -3.615367938027 --3.418 -3.614368509609 --3.417 -3.613369082248 --3.416 -3.612369656463 --3.415 -3.611370231742 --3.4139999999999997 -3.610370808346 --3.4130000000000003 -3.609371386534 --3.412 -3.608371965796 --3.411 -3.607372546392 --3.41 -3.60637312858 --3.409 -3.605373711852 --3.408 -3.604374296721 --3.407 -3.603374882683 --3.4059999999999997 -3.602375469994 --3.4050000000000002 -3.60137605891 --3.404 -3.600376648928 --3.403 -3.599377240556 --3.402 -3.598377833295 --3.401 -3.597378427398 --3.4 -3.596379023118 --3.399 -3.595379619961 --3.3979999999999997 -3.594380218178 --3.3970000000000002 -3.59338081802 --3.396 -3.592381418994 --3.395 -3.591382021599 --3.394 -3.590382625344 --3.393 -3.589383230479 --3.392 -3.588383837252 --3.391 -3.587384445177 --3.3899999999999997 -3.586385054502 --3.3890000000000002 -3.585385665473 --3.388 -3.584386277606 --3.387 -3.58338689139 --3.386 -3.582387506346 --3.385 -3.581388122717 --3.384 -3.580388740748 --3.383 -3.579389359961 --3.3819999999999997 -3.578389980837 --3.3810000000000002 -3.577390602905 --3.38 -3.576391226404 --3.379 -3.575391851576 --3.378 -3.57439247795 --3.377 -3.573393105765 --3.376 -3.572393735261 --3.375 -3.571394365971 --3.3739999999999997 -3.570394998366 --3.373 -3.569395631984 --3.372 -3.56839626706 --3.371 -3.56739690383 --3.37 -3.566397541833 --3.3689999999999998 -3.565398181305 --3.368 -3.56439882248 --3.367 -3.5633994649 --3.3659999999999997 -3.562400109028 --3.365 -3.561400754409 --3.364 -3.560401401275 --3.363 -3.559402049859 --3.362 -3.558402699707 --3.3609999999999998 -3.557403351278 --3.36 -3.556404004122 --3.359 -3.555404658468 --3.3579999999999997 -3.554405314546 --3.357 -3.553405971909 --3.356 -3.552406630784 --3.355 -3.551407291401 --3.354 -3.550407953313 --3.3529999999999998 -3.549408616972 --3.352 -3.548409281935 --3.351 -3.54740994843 --3.3499999999999996 -3.546410616679 --3.349 -3.545411286245 --3.348 -3.544411957572 --3.347 -3.543412630224 --3.346 -3.542413304425 --3.3449999999999998 -3.541413980396 --3.344 -3.540414657703 --3.343 -3.53941533657 --3.3419999999999996 -3.538416017217 --3.341 -3.537416699213 --3.34 -3.536417382994 --3.339 -3.535418068133 --3.338 -3.534418754849 --3.3369999999999997 -3.53341944336 --3.336 -3.53242013324 --3.335 -3.531420824709 --3.334 -3.530421517983 --3.333 -3.529422212638 --3.332 -3.528422909105 --3.331 -3.527423606961 --3.33 -3.526424306425 --3.3289999999999997 -3.525425007708 --3.3280000000000003 -3.524425710395 --3.327 -3.523426414909 --3.326 -3.522427120833 --3.325 -3.521427828383 --3.324 -3.520428537769 --3.323 -3.51942924858 --3.322 -3.518429961026 --3.3209999999999997 -3.517430675319 --3.3200000000000003 -3.516431391049 --3.319 -3.515432108632 --3.318 -3.51443282766 --3.317 -3.513433548343 --3.316 -3.5124342708882 --3.315 -3.5114349948921 --3.314 -3.5104357205623 --3.3129999999999997 -3.5094364481054 --3.3120000000000003 -3.5084371771201 --3.311 -3.507437907913 --3.31 -3.5064386402867 --3.309 -3.5054393745472 --3.308 -3.5044401103956 --3.307 -3.5034408478372 --3.306 -3.5024415870761 --3.3049999999999997 -3.5014423280152 --3.3040000000000003 -3.5004430706598 --3.303 -3.499443814913 --3.302 -3.4984445609785 --3.301 -3.4974453087604 --3.3 -3.4964460582622 --3.299 -3.4954468093903 --3.298 -3.4944475623454 --3.2969999999999997 -3.4934483170323 --3.2960000000000003 -3.4924490734562 --3.295 -3.4914498315221 --3.294 -3.4904505914303 --3.293 -3.4894513530866 --3.292 -3.4884521164948 --3.291 -3.487452881563 --3.29 -3.4864536484883 --3.2889999999999997 -3.4854544171774 --3.2880000000000003 -3.4844551876359 --3.287 -3.4834559598659 --3.286 -3.4824567337775 --3.285 -3.481457509565 --3.284 -3.4804582871371 --3.283 -3.4794590664981 --3.282 -3.4784598475577 --3.2809999999999997 -3.4774606305086 --3.2800000000000002 -3.4764614152613 --3.279 -3.4754622018185 --3.278 -3.4744629900919 --3.277 -3.4734637802726 --3.276 -3.4724645722708 --3.275 -3.4714653660911 --3.274 -3.4704661616449 --3.2729999999999997 -3.4694669591214 --3.2720000000000002 -3.4684677584333 --3.271 -3.467468559583 --3.27 -3.4664693624841 --3.269 -3.4654701673244 --3.268 -3.4644709740158 --3.267 -3.4634717825632 --3.266 -3.4624725929704 --3.2649999999999997 -3.4614734051506 --3.2640000000000002 -3.460474219291 --3.263 -3.4594750353036 --3.262 -3.4584758531933 --3.261 -3.4574766728744 --3.26 -3.4564774945312 --3.259 -3.4554783180782 --3.258 -3.4544791435195 --3.2569999999999997 -3.4534799707699 --3.2560000000000002 -3.4524808000135 --3.255 -3.4514816311639 --3.254 -3.4504824642263 --3.253 -3.4494832991165 --3.252 -3.4484841360155 --3.251 -3.4474849748399 --3.25 -3.4464858155937 --3.2489999999999997 -3.4454866582809 --3.248 -3.4444875028198 --3.247 -3.4434883493882 --3.246 -3.4424891979042 --3.245 -3.4414900483717 --3.2439999999999998 -3.4404909007085 --3.243 -3.4394917550931 --3.242 -3.4384926114427 --3.2409999999999997 -3.4374934697616 --3.24 -3.4364943299696 --3.239 -3.4354951922415 --3.238 -3.4344960564971 --3.237 -3.4334969227403 --3.2359999999999998 -3.4324977908906 --3.235 -3.4314986611233 --3.234 -3.4304995333576 --3.2329999999999997 -3.4295004075974 --3.232 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--2.319 -2.518216799628877 --2.318 -2.517223874547581 --2.3169999999999997 -2.5162309654853043 --2.316 -2.515238072475185 --2.315 -2.514245195554135 --2.314 -2.5132523347579547 --2.3129999999999997 -2.5122594901223745 --2.312 -2.51126666168349 --2.311 -2.5102738494771724 --2.31 -2.509281053539533 --2.309 -2.508288273906702 --2.308 -2.5072955106148633 --2.307 -2.5063027637016924 --2.306 -2.5053100332009994 --2.305 -2.5043173191504757 --2.304 -2.5033246215866334 --2.303 -2.502331940546181 --2.302 -2.501339276065757 --2.301 -2.5003466281821423 --2.3 -2.499353996932131 --2.299 -2.49836138235274 --2.298 -2.4973687844809556 --2.297 -2.496376203353844 --2.296 -2.4953836390097415 --2.295 -2.4943910914835485 --2.294 -2.493398560813768 --2.293 -2.4924060470378135 --2.292 -2.4914135501931107 --2.291 -2.490421070317305 --2.29 -2.4894286074480116 --2.289 -2.4884361616229533 --2.288 -2.4874437328798873 --2.287 -2.4864513212579777 --2.286 -2.4854589267927625 --2.285 -2.48446654952347 --2.284 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-2.447761034521498 --2.247 -2.446769339706407 --2.246 -2.445777663634764 --2.245 -2.4447860063490556 --2.2439999999999998 -2.443794367890765 --2.243 -2.4428027483015384 --2.242 -2.441811147623011 --2.241 -2.4408195658969882 --2.2399999999999998 -2.439828003165232 --2.239 -2.438836459469773 --2.238 -2.437844934852629 --2.237 -2.436853429356962 --2.2359999999999998 -2.4358619430228026 --2.235 -2.4348704758936015 --2.234 -2.4338790280116984 --2.233 -2.432887599419588 --2.2319999999999998 -2.431896190159766 --2.231 -2.4309048002749845 --2.23 -2.4299134298079332 --2.229 -2.4289220788014534 --2.2279999999999998 -2.427930747298435 --2.227 -2.426939435343001 --2.226 -2.425948142976144 --2.225 -2.4249568702421014 --2.2239999999999998 -2.4239656171841073 --2.223 -2.4229743838456232 --2.222 -2.4219831702701025 --2.221 -2.4209919765010817 --2.2199999999999998 -2.420000802582161 --2.219 -2.419009648557172 --2.218 -2.4180185144699338 --2.217 -2.417027400365372 --2.2159999999999997 -2.416036306285443 --2.215 -2.41504523227534 --2.214 -2.4140541783792426 --2.213 -2.4130631446414497 --2.2119999999999997 -2.412072131106298 --2.211 -2.4110811378183645 --2.21 -2.4100901648221837 --2.209 -2.409099212162422 --2.2079999999999997 -2.408108279883818 --2.207 -2.407117368032274 --2.206 -2.406126476650743 --2.205 -2.405135605785279 --2.2039999999999997 -2.4041447554809405 --2.203 -2.403153925783067 --2.202 -2.402163116736953 --2.201 -2.401172328388017 --2.1999999999999997 -2.400181560781737 --2.199 -2.3991908139638345 --2.198 -2.398200087979939 --2.197 -2.39720938287686 --2.1959999999999997 -2.3962186986985126 --2.195 -2.3952280354919426 --2.194 -2.3942373933032 --2.193 -2.3932467721784763 --2.1919999999999997 -2.392256172164044 --2.191 -2.391265593306347 --2.19 -2.3902750356518294 --2.189 -2.389284499247063 --2.1879999999999997 -2.388293984139646 --2.187 -2.3873034903745305 --2.186 -2.38631301799949 --2.185 -2.3853225670615017 --2.184 -2.3843321376076374 --2.183 -2.383341729685005 --2.182 -2.382351343340981 --2.181 -2.3813609786228374 --2.18 -2.3803706355780823 --2.179 -2.3793803142542096 --2.178 -2.378390014698957 --2.177 -2.377399736960986 --2.176 -2.3764094810862892 --2.175 -2.3754192471237214 --2.174 -2.3744290351214024 --2.173 -2.37343884512746 --2.172 -2.372448677190154 --2.171 -2.3714585313578 --2.17 -2.370468407678935 --2.169 -2.3694783062020837 --2.168 -2.3684882269758774 --2.167 -2.367498170050005 --2.166 -2.366508135471538 --2.165 -2.365518123290311 --2.164 -2.3645281335554342 --2.163 -2.363538166315992 --2.162 -2.3625482216213527 --2.161 -2.3615582995208437 --2.16 -2.360568400063922 --2.159 -2.3595785233001854 --2.158 -2.3585886692793068 --2.157 -2.3575988380519863 --2.156 -2.3566090296662923 --2.155 -2.3556192441730937 --2.154 -2.3546294816225313 --2.153 -2.3536397420647854 --2.152 -2.352650025550145 --2.151 -2.351660332128986 --2.15 -2.350670661851883 --2.149 -2.3496810147694127 --2.148 -2.348691390932301 --2.147 -2.347701790392212 --2.146 -2.346712213198366 --2.145 -2.345722659402672 --2.144 -2.3447331290562556 --2.143 -2.343743622210336 --2.142 -2.342754138916306 --2.141 -2.3417646792256126 --2.14 -2.3407752431898063 --2.139 -2.339785830860535 --2.138 -2.3387964422904743 --2.137 -2.337807077529779 --2.136 -2.3368177366313123 --2.135 -2.3358284196471035 --2.134 -2.3348391266294226 --2.133 -2.333849857630577 --2.132 -2.332860612702962 --2.131 -2.3318713918990834 --2.13 -2.3308821952716317 --2.129 -2.3298930228733377 --2.128 -2.3289038747570348 --2.127 -2.3279147509764937 --2.126 -2.326925651583177 --2.125 -2.3259365766310682 --2.124 -2.324947526173453 --2.1229999999999998 -2.323958500263713 --2.122 -2.3229694989554 --2.121 -2.3219805223020975 --2.12 -2.3209915703575548 --2.1189999999999998 -2.320002643175566 --2.118 -2.319013740810152 --2.117 -2.3180248633161127 --2.116 -2.317036010746048 --2.1149999999999998 -2.316047183154991 --2.114 -2.315058380597399 --2.113 -2.314069603127761 --2.112 -2.3130808508006786 --2.1109999999999998 -2.3120921236708414 --2.11 -2.3111034217931694 --2.109 -2.310114745222577 --2.108 -2.3091260940141156 --2.1069999999999998 -2.3081374682233577 --2.106 -2.3071488679052172 --2.105 -2.306160293114293 --2.104 -2.305171743907314 --2.1029999999999998 -2.3041832203398402 --2.102 -2.3031947224670053 --2.101 -2.302206250344884 --2.1 -2.3012178040295477 --2.0989999999999998 -2.3002293835770016 --2.098 -2.299240989043463 --2.097 -2.2982526204852944 --2.096 -2.2972642779588512 --2.0949999999999998 -2.296275961520691 --2.094 -2.295287671227568 --2.093 -2.2942994071361715 --2.092 -2.2933111693034003 --2.0909999999999997 -2.2923229577862783 --2.09 -2.2913347726419087 --2.089 -2.290346613927555 --2.088 -2.2893584817005737 --2.0869999999999997 -2.2883703760184133 --2.086 -2.2873822969387207 --2.085 -2.2863942445191237 --2.084 -2.28540621881751 --2.0829999999999997 -2.2844182198917755 --2.082 -2.2834302478000303 --2.081 -2.2824423026003955 --2.08 -2.28145438435125 --2.0789999999999997 -2.280466493110919 --2.078 -2.279478628937973 --2.077 -2.2784907918910537 --2.076 -2.2775029820289463 --2.0749999999999997 -2.276515199410534 --2.074 -2.2755274440948297 --2.073 -2.274539716140978 --2.072 -2.27355201560819 --2.0709999999999997 -2.2725643425558584 --2.07 -2.2715766970434914 --2.069 -2.270589079130629 --2.068 -2.2696014888770875 --2.0669999999999997 -2.2686139263426446 --2.066 -2.2676263915873314 --2.065 -2.2666388846712087 --2.064 -2.2656514056544834 --2.0629999999999997 -2.2646639545975247 --2.062 -2.263676531560747 --2.061 -2.262689136604793 --2.06 -2.2617017697902915 --2.0589999999999997 -2.260714431178137 --2.058 -2.2597271208292145 --2.057 -2.2587398388046425 --2.056 -2.2577525851656 --2.055 -2.256765359973406 --2.054 -2.2557781632894494 --2.053 -2.2547909951753953 --2.052 -2.2538038556928703 --2.051 -2.2528167449036576 --2.05 -2.2518296628697723 --2.049 -2.2508426096532403 --2.048 -2.2498555853162356 --2.047 -2.248868589921046 --2.046 -2.2478816235301933 --2.045 -2.2468946862061587 --2.044 -2.2459077780116883 --2.043 -2.2449208990095126 --2.042 -2.2439340492626694 --2.041 -2.2429472288341787 --2.04 -2.2419604377872258 --2.039 -2.2409736761851486 --2.038 -2.239986944091394 --2.037 -2.2390002415695043 --2.036 -2.238013568683227 --2.035 -2.237026925496358 --2.034 -2.23604031207287 --2.033 -2.2350537284768284 --2.032 -2.2340671747724956 --2.031 -2.233080651024132 --2.03 -2.2320941572962854 --2.029 -2.2311076936535033 --2.028 -2.2301212601605345 --2.027 -2.2291348568822578 --2.026 -2.2281484838836176 --2.025 -2.227162141229753 --2.024 -2.2261758289858933 --2.023 -2.2251895472174557 --2.022 -2.224203295989892 --2.021 -2.223217075368897 --2.02 -2.2222308854202093 --2.019 -2.2212447262097035 --2.018 -2.2202585978034364 --2.017 -2.219272500267578 --2.016 -2.2182864336684376 --2.015 -2.2173003980723847 --2.014 -2.2163143935460337 --2.013 -2.2153284201560526 --2.012 -2.2143424779692658 --2.011 -2.2133565670526045 --2.01 -2.2123706874732005 --2.009 -2.211384839298257 --2.008 -2.210399022595116 --2.007 -2.2094132374312907 --2.006 -2.2084274838743942 --2.005 -2.207441761992161 --2.004 -2.20645607185252 --2.003 -2.2054704135234755 --2.002 -2.2044847870731963 --2.001 -2.2034991925699563 --2. -2.2025136300822377 --1.999 -2.201528099678546 --1.9979999999999998 -2.2005426014276424 --1.9969999999999999 -2.199557135398316 --1.996 -2.1985717016595636 --1.995 -2.1975863002804634 --1.9939999999999998 -2.1966009313303037 --1.9929999999999999 -2.195615594878462 --1.992 -2.194630290994431 --1.991 -2.193645019747894 --1.9899999999999998 -2.192659781208641 --1.9889999999999999 -2.1916745754465934 --1.988 -2.1906894025318353 --1.987 -2.189704262534574 --1.9859999999999998 -2.188719155525159 --1.9849999999999999 -2.187734081574056 --1.984 -2.1867490407519106 --1.983 -2.1857640331294625 --1.9819999999999998 -2.184779058777636 --1.9809999999999999 -2.1837941177674507 --1.98 -2.1828092101701317 --1.979 -2.181824336056947 --1.9779999999999998 -2.1808394954993995 --1.9769999999999999 -2.1798546885690624 --1.976 -2.1788699153376765 --1.975 -2.1778851758771354 --1.9739999999999998 -2.176900470259476 --1.9729999999999999 -2.175915798556842 --1.972 -2.174931160841542 --1.971 -2.1739465571860372 --1.9699999999999998 -2.1729619876629016 --1.9689999999999999 -2.1719774523448745 --1.968 -2.1709929513048523 --1.967 -2.1700084846158028 --1.9659999999999997 -2.1690240523508963 --1.9649999999999999 -2.1680396545834864 --1.964 -2.1670552913869643 --1.963 -2.166070962834937 --1.9619999999999997 -2.165086669001135 --1.9609999999999999 -2.1641024099594555 --1.96 -2.163118185783866 --1.959 -2.1621339965485955 --1.9579999999999997 -2.1611498423279207 --1.9569999999999999 -2.160165723196297 --1.956 -2.1591816392283163 --1.955 -2.158197590498745 --1.9539999999999997 -2.157213577082446 --1.9529999999999998 -2.156229599054493 --1.952 -2.1552456564900027 --1.951 -2.154261749464383 --1.9499999999999997 -2.1532778780530646 --1.9489999999999998 -2.1522940423316843 --1.948 -2.1513102423759753 --1.947 -2.1503264782618627 --1.9459999999999997 -2.1493427500654514 --1.9449999999999998 -2.148359057862903 --1.944 -2.1473754017305837 --1.943 -2.1463917817450193 --1.9419999999999997 -2.1454081979828628 --1.9409999999999998 -2.144424650520893 --1.94 -2.1434411394360597 --1.939 -2.1424576648054887 --1.9379999999999997 -2.1414742267063813 --1.9369999999999998 -2.1404908252161814 --1.936 -2.1395074604124424 --1.935 -2.138524132372827 --1.9339999999999997 -2.1375408411751753 --1.9329999999999998 -2.1365575868975233 --1.932 -2.1355743696179865 --1.931 -2.134591189414893 --1.9299999999999997 -2.133608046366639 --1.9289999999999998 -2.1326249405518958 --1.928 -2.131641872049356 --1.927 -2.130658840937957 --1.9260000000000002 -2.129675847296738 --1.9249999999999998 -2.128692891204903 --1.924 -2.1277099727418305 --1.923 -2.1267270919869996 --1.9220000000000002 -2.1257442490201095 --1.9209999999999998 -2.124761443920974 --1.92 -2.1237786767695392 --1.919 -2.1227959476459493 --1.9180000000000001 -2.1218132566304666 --1.9169999999999998 -2.1208306038035527 --1.916 -2.119847989245758 --1.915 -2.1188654130378373 --1.9140000000000001 -2.1178828752606993 --1.9129999999999998 -2.1169003759953706 --1.912 -2.1159179153230743 --1.911 -2.1149354933251576 --1.9100000000000001 -2.1139531100831364 --1.9089999999999998 -2.1129707656786905 --1.908 -2.111988460193632 --1.907 -2.1110061937099647 --1.9060000000000001 -2.1100239663098193 --1.9049999999999998 -2.1090417780754693 --1.904 -2.1080596290894085 --1.903 -2.1070775194342235 --1.9020000000000001 -2.1060954491926895 --1.9009999999999998 -2.1051134184477274 --1.9 -2.1041314272824256 --1.899 -2.1031494757800067 --1.8980000000000001 -2.1021675640238864 --1.8969999999999998 -2.1011856920976264 --1.896 -2.100203860084932 --1.895 -2.0992220680696696 --1.8940000000000001 -2.0982403161359207 --1.8929999999999998 -2.097258604367833 --1.892 -2.0962769328497672 --1.891 -2.0952953016662352 --1.8900000000000001 -2.094313710901913 --1.8889999999999998 -2.0933321606416486 --1.888 -2.0923506509704297 --1.887 -2.091369181973396 --1.8860000000000001 -2.0903877537358815 --1.8849999999999998 -2.0894063663433657 --1.884 -2.088425019881479 --1.883 -2.087443714436008 --1.8820000000000001 -2.086462450092942 --1.8809999999999998 -2.0854812269383842 --1.88 -2.08450004505863 --1.879 -2.0835189045401385 --1.8780000000000001 -2.0825378054695203 --1.8769999999999998 -2.081556747933533 --1.876 -2.0805757320191285 --1.875 -2.079594757813409 --1.874 -2.078613825403638 --1.8729999999999998 -2.0776329348772538 --1.8719999999999999 -2.0766520863218294 --1.871 -2.0756712798251677 --1.87 -2.0746905154751456 --1.8689999999999998 -2.0737097933598645 --1.8679999999999999 -2.072729113567589 --1.867 -2.0717484761867353 --1.866 -2.0707678813059025 --1.8649999999999998 -2.069787329013789 --1.8639999999999999 -2.0688068193993745 --1.863 -2.067826352551699 --1.862 -2.06684592856003 --1.8609999999999998 -2.0658655475137953 --1.8599999999999999 -2.0648852095025494 --1.859 -2.0639049146160793 --1.858 -2.0629246629442717 --1.8569999999999998 -2.061944454577221 --1.8559999999999999 -2.0609642896051787 --1.855 -2.059984168118574 --1.854 -2.059004090207995 --1.8529999999999998 -2.058024055964203 --1.8519999999999999 -2.0570440654781144 --1.851 -2.0560641188408315 --1.85 -2.0550842161436353 --1.8489999999999998 -2.0541043574779523 --1.8479999999999999 -2.053124542935375 --1.847 -2.0521447726076913 --1.846 -2.051165046586841 --1.8449999999999998 -2.050185364964948 --1.8439999999999999 -2.049205727834289 --1.843 -2.0482261352873254 --1.842 -2.0472465874167076 --1.8409999999999997 -2.0462670843152013 --1.8399999999999999 -2.045287626075789 --1.839 -2.0443082127916274 --1.838 -2.043328844556019 --1.8369999999999997 -2.0423495214624663 --1.8359999999999999 -2.041370243604611 --1.835 -2.0403910110762777 --1.834 -2.0394118239715184 --1.8329999999999997 -2.0384326823844745 --1.8319999999999999 -2.0374535864095087 --1.831 -2.0364745361411485 --1.83 -2.0354955316740897 --1.8289999999999997 -2.034516573103205 --1.8279999999999998 -2.0335376605235678 --1.827 -2.032558794030379 --1.826 -2.0315799737190408 --1.8249999999999997 -2.0306011996851105 --1.8239999999999998 -2.029622472024365 --1.823 -2.0286437908327155 --1.822 -2.0276651562062638 --1.8209999999999997 -2.026686568241287 --1.8199999999999998 -2.0257080270342342 --1.819 -2.024729532681726 --1.818 -2.023751085280584 --1.8169999999999997 -2.0227726849277743 --1.8159999999999998 -2.0217943317204705 --1.815 -2.0208160257559884 --1.814 -2.019837767131869 --1.8129999999999997 -2.018859555945788 --1.8119999999999998 -2.0178813922956094 --1.811 -2.0169032762794252 --1.81 -2.0159252079953767 --1.8089999999999997 -2.0149471875419556 --1.8079999999999998 -2.013969215017682 --1.807 -2.0129912905213465 --1.806 -2.012013414151892 --1.8049999999999997 -2.011035586008439 --1.8039999999999998 -2.0100578061902863 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"#8fb032" -cc4 = "#eb6235" - -set logscale y - -data = "figs/fig-series-data.dat" - -set xrange [0.5:15.5] -set yrange [0.000005:5] -set key off -set xlabel '$n$' -set ylabel offset 2 '$|f_n|$' - -set lmargin 7 - -set ytics format '' - -set ytics add ('$\footnotesize10^{-5}$' 10**(-5),'$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3),'$\footnotesize10^{-2}$' 10**(-2), '$\footnotesize10^{-1}$' 10**(-1),'$\footnotesize10^{0}$' 10**(0)) - -plot \ - data using 1:3 with points lc rgb cc1, \ - data using 1:2 with points lc rgb cc2, \ - data using 1:5 with points lc rgb cc3 - diff --git a/figs/fig-sus_scaling-func-mod.dat b/figs/fig-sus_scaling-func-mod.dat deleted file mode 100644 index 5d49f9b..0000000 --- a/figs/fig-sus_scaling-func-mod.dat +++ /dev/null @@ -1,11515 +0,0 @@ -0.0004967499652960352 0.02552916136980551 -0.0009137823733494827 0.02552090801819801 -0.0013310520041847056 0.025513600697881153 -0.0017485589391690029 0.025506236977519796 -0.0021663032596971267 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terminal pslatex rotate size 3.417,4.222 -set multiplot - -cc1 = "#5e81b5" -cc2 = "#e19c24" -cc3 = "#8fb032" -cc4 = "#eb6235" - -Tc = 2 / log(1 + sqrt(2)) -Delta = 15. / 8 -gamma = 7. / 4 -beta = 1. / 8 - -Ch = 0.838677624411 - -t(T) = abs((T - Tc) / T) -h(T, H) = H / T -X(T, H) = h(T, H) * t(T)**(-Delta) - -poly(A, l, m, X) = A * X**m / (1 + (l * X)**(m + 1)) -polyint(A, l, m, X) = A * l**(-(m + 1)) * log(1 + (l * X)**(m + 1)) / (m + 1) - -G(i) = i == 1 ? -1.3578383417066 : \ - i == 2 ? -0.048953289720 : \ - i == 3 ? 0.038863932 : \ - i == 4 ? -0.068362119 : \ - i == 5 ? 0.18388370 : \ - i == 6 ? -0.6591714 : \ - i == 7 ? 2.937665 : \ - i == 8 ? -15.61 : 0 -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) - -n = 1 -c(i) = i == 1 ? 0.00856277 : 0 -lamb = 10.487 -A2 = 1.70034 -B2 = B - -n2 = 5 -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 -A3 = 1.91747 -B3 = B - -susfunc = "figs/fig-sus_scaling-func.dat" -susfuncm = "figs/fig-sus_scaling-func-mod.dat" -magfunc = "figs/fig-mag_scaling-func.dat" -magfuncm = "figs/fig-mag_scaling-func-mod.dat" - -num = "data/data_square-4096.dat" - -set samples 20000 -set key off - -set size 1,1 - 0.52 -set origin 0,0.52 -set xrange [-40:1200] -set yrange [-1:33] -set ylabel offset 1,0 '$T\chi|t|^{\gamma}\times 10^3$' -set lmargin 6 -set xtics format '' -set mxtics 5 -set mytics 5 -set bmargin 0.2 - -plot \ - num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ - susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ - susfuncm using 1:(10**3 * $2) with lines lw 3 lc rgb cc2 dt 2, \ - -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 3 lc rgb cc3 dt 3 - -set bmargin -1 -set tmargin 0.2 -set size 1,0.52 -set origin 0,0 -set yrange [1.18:1.74] -set ylabel offset 1,0 '$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)) with yerrorbars pt 0 lc black, \ - magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ - magfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 3 lc rgb cc3 dt 3 - -set logscale xy -set tmargin -1 -set lmargin -1 -set size 0.7,0.325 -set origin 0.26,0.5 + 0.29 / 2 -set xrange [0.0015:1900] -set yrange [0.00002:0.08] -set ylabel offset 4,0 '\footnotesize$T\chi|t|^\gamma$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' -set mxtics 5 -#set xtics format '' -2,10,1000 -set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' 1, "" 10, '$\footnotesize10^2$' 100, "" 1000) -set mytics 5 -#set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -set ytics add ('$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3), '$\footnotesize10^{-2}$' 10**(-2)) - -plot \ - num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ - susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ - susfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ - -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 3 lw 3 lc rgb cc3 - -unset logscale xy -set logscale x -set size 0.65,0.325 -set origin 0.31,0.23 / 2 -set yrange [1.18:1.74] -set ylabel offset 4,0 '\footnotesize$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)) with yerrorbars pt 0 lc black, \ - magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ - magfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 3 lc rgb cc3 dt 3 - diff --git a/makefile b/makefile deleted file mode 100644 index a04aaa6..0000000 --- a/makefile +++ /dev/null @@ -1,21 +0,0 @@ - -DOC=essential-ising -FIGS=fig-susmag fig-series -DATA=data_square-4096 -FUNCS=fig-mag_scaling-func fig-sus_scaling-func fig-series-data - -all: ${DOC}.pdf - -%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} - gnuplot $< > $@ - -${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=%.tex} - rubber $(DOC).tex - dvipdf $(DOC).dvi - -clean: - rubber --clean $(DOC) - rm -f $(DOC).pdf - rm -f $(DOC)Notes.bib - rm -f ${FIGS:%=%.tex} - diff --git a/refs/PhysRevB.40.9006.pdf b/refs/PhysRevB.40.9006.pdf deleted file mode 100644 index d455145..0000000 Binary files a/refs/PhysRevB.40.9006.pdf and /dev/null differ -- cgit v1.2.3-70-g09d2 From 8c7ba65464178104b3bb68e1a15b1e6fbf3e3231 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 22 Dec 2020 19:45:59 +0100 Subject: Started new paper. --- .gitignore | 4 + essential-ising.tex | 478 ---------------------------------------------------- ising_scaling.bib | 57 +++++++ ising_scaling.tex | 42 +++++ 4 files changed, 103 insertions(+), 478 deletions(-) delete mode 100644 essential-ising.tex create mode 100644 ising_scaling.bib create mode 100644 ising_scaling.tex (limited to '.gitignore') diff --git a/.gitignore b/.gitignore index 034b355..84b12c2 100644 --- a/.gitignore +++ b/.gitignore @@ -4,4 +4,8 @@ *.blg *Notes.bib *.dvi +*.fdb_latexmk +*.synctex.gz +*.fls +*.out /*.pdf diff --git a/essential-ising.tex b/essential-ising.tex deleted file mode 100644 index e125229..0000000 --- a/essential-ising.tex +++ /dev/null @@ -1,478 +0,0 @@ - -% -% Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. -% Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. -% -\documentclass[aps,prl,reprint]{revtex4-1} - -\usepackage[utf8]{inputenc} -\usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} - -% uncomment to label only equations that are referenced in the text -%\mathtoolsset{showonlyrefs=true} - -% I want labels but don't want to type out ``equation'' -\def\[{\begin{equation}} -\def\]{\end{equation}} - -% math not built-in -\def\arcsinh{\mathop{\mathrm{arcsinh}}\nolimits} -\def\arccosh{\mathop{\mathrm{arccosh}}\nolimits} -\def\ei{\mathop{\mathrm{Ei}}\nolimits} % exponential integral Ei -\def\re{\mathop{\mathrm{Re}}\nolimits} -\def\im{\mathop{\mathrm{Im}}\nolimits} -\def\sgn{\mathop{\mathrm{sgn}}\nolimits} -\def\dd{d} % differential -\def\O{O} % big O -\def\o{o} % little O - -% subscript for ``critical'' values, e.g., T_\c -\def\c{\mathrm c} - -% scaling functions -\def\fM{\mathcal M} % magnetization -\def\fX{\mathcal Y} % susceptibility -\def\fXt{\check{\mathcal Y}} % susceptibility -\def\fF{\mathcal F} % free energy -\def\fiF{\mathcal H} % imaginary free energy -\def\fS{\mathcal S} % surface tension -\def\fG{\mathcal G} % exponential factor - -% lattice types -\def\sq{\mathrm{sq}} -\def\tri{\mathrm{tri}} -\def\hex{\mathrm{hex}} - -% dimensions -\def\dim{d} -\def\twodee{\textsc{2d}} -\def\threedee{\textsc{3d}} -\def\fourdee{\textsc{4d}} - -% fancy partial derivative -\newcommand\pd[3][]{ - \ifthenelse{\isempty{#1}} - {\def\tmp{}} - {\def\tmp{^#1}} - \frac{\partial\tmp#2}{\partial#3\tmp} -} - -% used to reformat display math to fit in two-column ``reprint'' mode -\makeatletter -\newif\ifreprint -\@ifclasswith{revtex4-1}{reprint}{\reprinttrue}{\reprintfalse} -\makeatother - -\begin{document} - -\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} -\author{Jaron Kent-Dobias} -\author{James P.~Sethna} -\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} - -\date\today - -\begin{abstract} - Renormalization group ideas and results from critical droplet theory are - used to construct a scaling ansatz for the imaginary component of the free - energy of an Ising model in its metastable state close to the critical - point. The analytic properties of the free energy are used to determine - scaling functions for the free energy in the vicinity of the critical point - and the abrupt transition line. These functions have essential singularities - at the abrupt transition. Analogous forms for the magnetization and susceptibility in - two dimensions are fit to numeric data and show good agreement, especially - when their nonsingular behavior is modified to match existing numeric results. -\end{abstract} - -\maketitle - -The Ising model is the canonical example of a system with a continuous phase -transition, and the study of its singular properties marked the first success -of the renormalization group (\textsc{rg}) method in statistical physics -\cite{wilson.1971.renormalization}. Its status makes sense: it's a simple -model whose continuous phase transition contains all the essential features of -more complex ones, but admits \textsc{rg} methods in a straightforward way and -has exact solutions in certain dimensions and for certain parameter -restrictions. However, the Ising critical point is not simply a continuous -transition: it also ends a line of abrupt phase transitions extending from it -at zero field below the critical temperature. Though typically neglected in -\textsc{rg} scaling analyses of the critical point, we demonstrate that there -are numerically measurable contributions to scaling due to the abrupt -transition line that cannot be accounted for by analytic changes of control or -thermodynamic variables. - -\textsc{Rg} analysis predicts that the singular part of the free energy per -site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ -in the vicinity of the critical point takes the scaling form -$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase -$t<0$ \cite{cardy.1996.scaling}. When studying the properties of the Ising -critical point, it is nearly always assumed that the universal scaling -function $\fF$ is analytic, i.e., has a convergent Taylor series. However, it -has long been known that there exists an essential singularity in $\fF$ at -zero argument, though its effects have long been believed to be unobservable -\cite{fisher.1967.condensation}, or simply just neglected -\cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. -With careful analysis, we have found that assuming the presence of the -essential singularity is predictive of the scaling form of, for instance, the -susceptibility and magnetization. - -The provenance of the essential singularity can be understood using the -methods of critical droplet theory for the decay of an Ising system in a -metastable state, i.e., an equilibrium Ising state for $T0$ -subjected to a small negative external field $H<0$. The existence of an -essential singularity has also been suggested by transfer matrix -\cite{mccraw.1978.metastability,enting.1980.investigation,mangazeev.2008.variational,mangazeev.2010.scaling} and \textsc{rg} -methods \cite{klein.1976.essential}, and a different kind of essential -singularity is known to exist in the zero-temperature susceptibility -\cite{orrick.2001.susceptibility,chan.2011.ising,guttmann.1996.solvability,nickel.1999.singularity,nickel.2000.addendum,assis.2017.analyticity}. It has long been known that the decay -rate $\Gamma$ of metastable states in statistical mechanics is often related -to the metastable free energy $F$ by $\Gamma\propto\im F$ -\cite{langer.1969.metastable,penrose.1987.rigorous,gaveau.1989.analytic,privman.1982.analytic}. -`Metastable free energy' can be thought of as either an analytic continuation -of the free energy through the abrupt phase transition, or restriction of the -partition function trace to states in the vicinity of the local free energy -minimum that characterizes the metastable state. In any case, the free energy -develops a nonzero imaginary part in the metastable region. Heuristically, -this can be thought of as similar to what happens in quantum mechanics with a -non-unitary Hamiltonian: the imaginary part describes loss rate of probability -that the system occupies any `accessible' state, which corresponds to decay. - -In critical droplet theory, the metastable state decays when a domain of the -equilibrium state forms whose surface-energy cost for growth is outweighed by -bulk-energy gains. There is numerical evidence that, near the critical point, -these droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy -cost of the surface of a droplet of radius $R$ is $\Sigma S_\dim R^{\dim-1}$ -and that of its bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are -the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ -is the surface tension of the equilibrium--metastable interface. The critical -droplet size is then $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the -critical droplet is $\Delta F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. -Assuming the typical singular scaling forms -$\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal -M(h|t|^{-\beta\delta})$ and using known hyperscaling relations -\cite{widom.1981.interface}, this implies a scaling form -\def\eqcritformone{ - T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)} - \frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta} - \fM(h|t|^{-\beta\delta}))^{\dim-1}} -} -\def\eqcritformtwo{ - T\fG^{-(\dim-1)}(h|t|^{-\beta\delta}) -} -\ifreprint -\[ - \begin{aligned} - \Delta F_\c - &=\eqcritformone\\ - &\sim\eqcritformtwo - \end{aligned} -\] -\else -\[ - \Delta F_\c=\eqcritformone\sim\eqcritformtwo -\] -\fi -for the free energy change due to the critical droplet. -Since both surface tension and magnetization are finite and nonzero for $H=0$ -at $T1$ this function has an essential singularity in the invariant -combination $h|t|^{-\beta\delta}$. - -This form of $\im F$ for small $h$ is well known -\cite{langer.1967.condensation,harris.1984.metastability}. We make the -scaling ansatz that the imaginary part of the metastable free energy has the -same singular behavior as the real part of the equilibrium free energy, and -that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$, -where -\[ - \fiF(X)=A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} - \label{eq:im.scaling} -\] -and $\Theta$ is the Heaviside function. Results from combining an analysis of -fluctuations on the surface of critical droplets with \textsc{rg} recursion -relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for $d=3$ -\cite{houghton.1980.metastable,rudnick.1976.equations,gunther.1980.goldstone}. -Assuming that $F$ is analytic in the upper complex-$h$ plane, the real part of -$F$ in the equilibrium state can be extracted from this imaginary metastable -free energy using the Kramers--Kronig relation -\[ - \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'. - \label{eq:kram-kron} -\] -This relationship has been used to compute high-order moments of the free -energy with $H$ in good agreement with transfer matrix expansions -\cite{lowe.1980.instantons}. Here, we evaluate the integral explicitly to come -to -functional forms. In \threedee\ and \fourdee\ this can be done directly -given our scaling ansatz, yielding -\def\eqthreedeeone{ - \fF^\threedee(Y/B)&= - \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} - \bigg[\Gamma(\tfrac16)E_{7/6}(-Y^{-2}) -} -\def\eqthreedeetwo{ - -4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})\bigg] -} -\def\eqfourdeeone{ - \fF^\fourdee(Y/B)&= - -\frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} - \Big[3\ei(-Y^{-3}) -} -\def\eqfourdeetwo{ - +3\Gamma(\tfrac23)\Gamma(\tfrac13,Y^{-3}) - +\Gamma(\tfrac13)\Gamma(-\tfrac13,Y^{-3})\Big] -} -\ifreprint -\begin{align} - &\begin{aligned} - \eqthreedeeone\\ - &\hspace{7em} - \eqthreedeetwo - \end{aligned} - \\ - &\begin{aligned} - \eqfourdeeone - \\ - &\hspace{-0.5em} - \eqfourdeetwo, - \end{aligned} -\end{align} -\else -\begin{align} - \eqthreedeeone\eqthreedeetwo - \\ - \eqfourdeeone\eqfourdeetwo, -\end{align} -\fi -where $E_n$ is the generalized exponential integral and $\Gamma(x,y)$ is the -incomplete gamma function. -At the level of truncation of \eqref{eq:im.scaling} at which we are working -the Kramers--Kronig relation does not converge in \twodee. However, higher -moments can still be extracted, e.g., the susceptibility, by taking -\[ - \chi=\pd MH=-\frac1{T}\pd[2]Fh - =-\frac2{\pi T}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. -\] -With a scaling form defined by $T\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, -this yields -\[ - \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] - \label{eq:sus_scaling} -\] -Scaling forms for the free energy can then be extracted by direct integration -and their constants of integration fixed by known zero-field values, yielding -\begin{align} - \label{eq:mag_scaling} - \fM^\twodee(Y/B) - &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ - \fF^\twodee(Y/B) - &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg) - \label{eq:2d_free_scaling} -\end{align} -with $F(t,h)=t^2\fF(h|t|^{-15/8})+t^2\log t^2$ in -two dimensions, as $\alpha=0$ and $\beta\delta=\frac{15}8$. - -How are these functional forms to be interpreted? Though the scaling function -\eqref{eq:im.scaling} for the imaginary free energy of the metastable state is -asymptotically correct sufficiently close to the critical point, the results -of the integral relation \eqref{eq:kram-kron} are not, since there is no limit -of $t$ or $h$ in which it becomes arbitrarily correct for a given truncation -of \eqref{eq:im.scaling}. It is well established that this method of using -unphysical or metastable elements of a theory to extract properties of the -stable or equilibrium theory is only accurate for high moments of those -predictions \cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. The -functions above should be understood as possessing exactly the correct -singularity at the coexistence line, but requiring polynomial corrections, -especially for smaller integer powers. Using these forms in conjunction with -existing methods of describing the critical equation of state or critical -properties with analytic functions in $h$ will incorporate these low-order -corrections while preserving the correct singular structure. In other words, -the scaling functions can be \emph{exactly} described by -$\tilde\fF(X)=\fF(X)+f(X)$ for some analytic function $f$. Higher order terms -in the expansion of $\tilde\fF$ become asymptotically equal to those of $\fF$ -because, as an analytic function, progressively higher order terms of $f$ must -eventually become arbitrarily small \cite{flanigan.1972.complex}. - -How predictive are these scaling forms in the proximity of the critical point -and the abrupt transition line? We simulated the \twodee\ Ising model on square -lattice using a form of the Wolff algorithm modified to remain efficient in -the presence of an external field. Briefly, the external field $H$ is applied -by adding an extra spin $s_0$ with coupling $|H|$ to all others -\cite{dimitrovic.1991.finite}. A quickly converging estimate for the -magnetization in the finite-size system was then made by taking -$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin -\cite{kent-dobias.2018.wolff}. -Data was then taken for susceptibility and magnetization for -$T_\c-T,H\leq0.1$. This data, rescaled as appropriate to collapse onto a -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 $\fX(0)=C_0^-=0.025\,536\,971\,9$ \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 -$T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ 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 blue lines in -Fig.~\ref{fig:scaling_fits}. Though there is good agreement -between our functional forms and what is measured, there -are systematic differences that can be seen most clearly in the -magnetization. This is to be expected based on our earlier discussion: these -scaling forms should only be expected to well-describe the singularity at the -abrupt transition. Our forms both exhibit incorrect low-order -coefficients at the transition (Fig.~\ref{fig:series}) and incorrect -asymptotics as $h|t|^{-\beta\delta}$ becomes very large. - -In forthcoming work, -we develop a method to incorporate the essential singularity in the scaling -functions into a form that also incorporates known properties of the scaling -functions in the rest of the configuration space using a Schofield-like -parameterization \cite{schofield.1969.parametric,caselle.2001.critical,kent-dobias.2018.parametric}. Briefly, we define -parameters $R$ and $\theta$ by -\begin{align} - t=R(1-\theta^2) - && - h=h_0R^{\beta\delta}g(\theta) -\end{align} -where $h_0$ is a constant and $f$ is an arbitrary odd function whose first -finite zero $\theta_\c>1$ corresponds to the abrupt transition. In these -coordinates the invariant combination $h|t|^{-\beta\delta}$ is given by -\[ - h|t|^{-\beta\delta}=\frac{h_0g(\theta)}{|1-\theta^2|^{\beta\delta}}=\frac{h_0(-g'(\theta_\c))}{(\theta_c^2-1)^{\beta\delta}}(\theta_\c-\theta) - +\O\big( (\theta_\c-\theta)^2\big), -\] -an analytic function of $\theta$ about $\theta_\c$. -The simplest -function of the coordinate $\theta$ that exhibits the correct singularity at -the abrupt transitions at $h=\pm0$, $t<0$ is -\[ - \fX(\theta)=\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c-\theta)\bigg)+\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c+\theta)\bigg) -\] -This function is analytic in the range $-\theta_\c<\theta<\theta_\c$. -In order to correct its low-order behavior to match that expected, we -make use of both the freedom of the coordinate transformation $g$ and an -arbitrary analytic additive function $Y$, -\begin{align} - g(\theta)=\bigg(1-\frac{\theta^2}{\theta_\c^2}\bigg)\sum_{n=0}^\infty - g_n\theta^{2n+1} - && - Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n} -\end{align} -so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating these -coefficients, we can attempt to give the resulting scaling form a series -expansion consistent with known values. One such prediction---made by fixing -the first four terms in the low-temperature, critical isotherm, and -high-temperature expansions -of $\tilde\fX$---is shown as a dashed yellow line in -Fig.~\ref{fig:scaling_fits}. -As shown in Fig.~\ref{fig:series}, the low-order free energy coefficients of this prediction match known values -exactly up to $n=5$, and improve the agreement with higher-order coefficients. -Unlike scaling forms which treat $\fX$ as analytic at the coexistence line, -the series coefficients of the scaling form developed here increase without -bound at high order. - - -\begin{figure} - \input{fig-susmag} - \caption{ - Scaling functions for (top) the susceptibility and (bottom) the - magnetization plotted in terms of the invariant combination - $h|t|^{-\beta\delta}$. Points with error bars show data with sampling - error taken from simulations of a $4096\times4096$ square-lattice Ising - model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ - and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines - show our analytic results \eqref{eq:sus_scaling} and - \eqref{eq:mag_scaling}, the dashed yellow lines show - a scaling function modified to match known series expansions of the - susceptibility - to third order, and the - dotted green lines show the - polynomial resulting from truncating the known series expansion after the eight terms - reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. - } - \label{fig:scaling_fits} -\end{figure} - -\begin{figure} - \input{fig-series} - \caption{ - The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue - pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the - yellow saltires correspond to a scaling function modified to match known - series expansions of the susceptibility to third order---and therefore - the free energy to fifth order---and the green - stars - correspond to the first eight coefficients from - \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. The modified - scaling function and the known coefficients match exactly up to $n=5$. - } - \label{fig:series} -\end{figure} - -Abrupt phase transitions, such as the jump in magnetization in the Ising -model below $T_\c$, are known to imply essential singularities in the free -energy that are usually thought to be unobservable in practice. We have -argued that this essential singularity controls the universal scaling -behavior near continuous phase transitions, and have derived an explicit -analytical form for the singularity in the free energy, magnetization, -and susceptibility for the Ising model. We have developed a Wolff algorithm -for the Ising model in a field, and showed that incorporating our singularity -into the scaling function gives good convergence to the simulations in \twodee. - -Our results should allow improved high-precision functional forms for the free -energy~\cite{caselle.2001.critical}, and should have implications for the scaling -of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized -to predict similar singularities in systems where nucleation and metastability -are proximate to continuous phase transitions, such as 2D superfluid -transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{dahm.1989.dynamics}, and -freezing transitions in glasses, spin glasses, and other disordered systems. - - -%We have used results from the properties of the metastable Ising ferromagnet -%and the analytic nature of the free energy to derive universal scaling -%functions for the free energy, and in \twodee the magnetization and -%susceptibility, in the limit of small $t<0$ and $h$. Because of an essential -%singularity in these functions at $h=0$---the abrupt transition line---their -%form cannot be brought into that of any regular function by analytic -%redefinition of control or thermodynamic variables. These predictions match -%the results of simulations well. Having demonstrated that the essential -%singularity in thermodynamic functions at the abrupt transition leads to -%observable scaling effects, we hope that these functional forms will be used in -%conjunction with traditional perturbation methods to better express the -%equation of state of the Ising model in the whole of its parameter space. - -\begin{acknowledgments} - The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien - for helpful conversations. The authors would also like to think Jacques Perk - for pointing us to several insightful studies. JPS thanks Jim Langer for past inspiration, - guidance, and encouragement. This work was supported by NSF grants - DMR-1312160 and DMR-1719490. -\end{acknowledgments} - -\bibliography{essential-ising} - -\end{document} - diff --git a/ising_scaling.bib b/ising_scaling.bib new file mode 100644 index 0000000..36fdbe9 --- /dev/null +++ b/ising_scaling.bib @@ -0,0 +1,57 @@ +@article{Butera_2011_Free, + author = {Butera, P. and Pernici, M.}, + title = {Free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model}, + journal = {Phys. Rev. B}, + publisher = {American Physical Society}, + year = {2011}, + month = {2}, + volume = {83}, + pages = {054433}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.83.054433}, + doi = {10.1103/PhysRevB.83.054433}, + issue = {5}, + numpages = {15} +} + +@article{Campostrini_2000_Critical, + author = {Campostrini, Massimo and Pelissetto, Andrea and Rossi, Paolo and Vicari, Ettore}, + title = {Critical equation of state of three-dimensional {$XY$} systems}, + journal = {Physical Review B}, + publisher = {American Physical Society ({APS})}, + year = {2000}, + month = {9}, + number = {9}, + volume = {62}, + pages = {5843--5854}, + url = {https://doi.org/10.1103%2Fphysrevb.62.5843}, + doi = {10.1103/physrevb.62.5843} +} + +@article{Mangazeev_2008_Variational, + author = {Mangazeev, Vladimir V and Batchelor, Murray T and Bazhanov, Vladimir V and Dudalev, Michael Yu}, + title = {Variational approach to the scaling function of the 2D Ising model in a magnetic field}, + journal = {Journal of Physics A: Mathematical and Theoretical}, + publisher = {IOP Publishing}, + year = {2008}, + month = {12}, + number = {4}, + volume = {42}, + pages = {042005}, + url = {https://doi.org/10.1088%2F1751-8113%2F42%2F4%2F042005}, + doi = {10.1088/1751-8113/42/4/042005} +} + +@article{Mangazeev_2010_Scaling, + author = {Mangazeev, Vladimir V. and Dudalev, Michael Yu. and Bazhanov, Vladimir V. and Batchelor, Murray T.}, + title = {Scaling and universality in the two-dimensional Ising model with a magnetic field}, + journal = {Physical Review E}, + publisher = {American Physical Society (APS)}, + year = {2010}, + month = {6}, + number = {6}, + volume = {81}, + url = {https://doi.org/10.1103%2Fphysreve.81.060103}, + doi = {10.1103/physreve.81.060103} +} + + diff --git a/ising_scaling.tex b/ising_scaling.tex new file mode 100644 index 0000000..b7b76b2 --- /dev/null +++ b/ising_scaling.tex @@ -0,0 +1,42 @@ +\documentclass[aps,prb,reprint,longbibliography,floatfix]{revtex4-2} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{newtxtext, newtxmath} +\usepackage[ + colorlinks=true, + urlcolor=purple, + citecolor=purple, + filecolor=purple, + linkcolor=purple +]{hyperref} % ref and cite links with pretty colors + +\usepackage{amsmath, graphicx, xcolor} + +\begin{document} + +\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} +\author{Jaron Kent-Dobias} +\author{James P.~Sethna} +\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} + +\date\today + +\begin{abstract} +\end{abstract} + +\maketitle + +\cite{Campostrini_2000_Critical} + +\begin{acknowledgments} + The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien + for helpful conversations. The authors would also like to think Jacques Perk + for pointing us to several insightful studies. JPS thanks Jim Langer for past + inspiration, guidance, and encouragement. This work was supported by NSF + grants DMR-1312160 and DMR-1719490. +\end{acknowledgments} + +\bibliography{ising_scaling} + +\end{document} -- cgit v1.2.3-70-g09d2