From 1c392812f20bd2b49cb98cfff5e9445953f7b8fb Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 30 Sep 2017 20:37:23 -0400 Subject: many changes --- essential-ising.tex | 111 +- figs/fig-mag_scaling-func-mod.dat | 12608 ++++++++++++++++++++++++++++++++---- figs/fig-sus_scaling-func-mod.dat | 11130 +++++++++++++++++++++++++++++-- 3 files changed, 22266 insertions(+), 1583 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index 7dd6686..dd2c083 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -32,6 +32,7 @@ % 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 @@ -44,9 +45,9 @@ % dimensions \def\dim{d} -\def\twodee{\textsc{2d} } -\def\threedee{\textsc{3d} } -\def\fourdee{\textsc{4d} } +\def\twodee{\textsc{2d}} +\def\threedee{\textsc{3d}} +\def\fourdee{\textsc{4d}} % fancy partial derivative \newcommand\pd[3][]{ @@ -78,8 +79,9 @@ 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. + at the abrupt transition. Analogous forms for the magnetization and susceptibility in + two dimensions are fit to numeric data and show good agreement, especial + when their nonsingular behavior is modified to match existing numeric results. \end{abstract} \maketitle @@ -111,7 +113,7 @@ 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 +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 @@ -132,20 +134,19 @@ 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 of probability in -the system that corresponds to decay. +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, -droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy +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 then is $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the -critical droplet is $\Delta -F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. +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 @@ -163,14 +164,15 @@ M(h|t|^{-\beta\delta})$ and using known hyperscaling relations \begin{aligned} \Delta F_\c &=\eqcritformone\\ - &\sim\eqcritformtwo. + &\sim\eqcritformtwo \end{aligned} \] \else \[ - \Delta F_\c=\eqcritformone\sim\eqcritformtwo. + \Delta F_\c=\eqcritformone\sim\eqcritformtwo \] \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 $T1$ 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 $f$ and an +arbitrary analytic additive function $Y$, +\begin{align} + f(\theta)=\bigg(1-\frac{\theta^2}{\theta_\c^2}\bigg)\sum_{n=0}^\infty + f_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 thees +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. + \begin{figure} \input{fig-susmag} diff --git a/figs/fig-mag_scaling-func-mod.dat b/figs/fig-mag_scaling-func-mod.dat index 5702933..657dda5 100644 --- a/figs/fig-mag_scaling-func-mod.dat +++ b/figs/fig-mag_scaling-func-mod.dat @@ -1,1144 +1,11464 @@ -0.009568286124210458 2^(5/16)*ArcSinh[1]^(1/8) -0.014105474133770269 1.222524728271578 -0.018669057029485486 1.2226398161196887 -0.0232591226867679 1.2227552165109419 -0.027875759620704425 1.2228709287034474 -0.032519056989508764 1.2229869519647474 -0.03718910459804977 1.2231032855717547 -0.04188599290129413 1.2232199288105705 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