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authorJaron Kent-Dobias <jaron@kent-dobias.com>2017-05-26 10:20:59 -0400
committerJaron Kent-Dobias <jaron@kent-dobias.com>2017-05-26 10:20:59 -0400
commit7b4f4ee4813f8afc3e9949a368fe708351d3b89c (patch)
tree33b6d66b24cf57c193d3dabc8b891ca382940c92
downloadpaper-7b4f4ee4813f8afc3e9949a368fe708351d3b89c.tar.gz
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paper-7b4f4ee4813f8afc3e9949a368fe708351d3b89c.zip
changes
-rw-r--r--.gitignore5
-rw-r--r--essential_ising.bib99
-rw-r--r--essential_ising.tex206
-rw-r--r--figs/scaling_func.gplot16
-rw-r--r--figs/scaling_func.tex260
-rw-r--r--makefile13
6 files changed, 599 insertions, 0 deletions
diff --git a/.gitignore b/.gitignore
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--- /dev/null
+++ b/.gitignore
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+*.pdf
+*.aux
+*.log
+*.bbl
+*.blg
diff --git a/essential_ising.bib b/essential_ising.bib
new file mode 100644
index 0000000..c84c1a7
--- /dev/null
+++ b/essential_ising.bib
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+
+@article{houghton.1980.metastable,
+ title={The metastable Ising magnet in a negative field},
+ author={Hougton, Anthony and Lubensky, Tom C},
+ journal={Physics Letters A},
+ volume={77},
+ number={6},
+ pages={479--480},
+ year={1980},
+ publisher={North-Holland}
+}
+
+@article{langer.1967.condensation,
+ title={Theory of the condensation point},
+ author={Langer, James S},
+ journal={Annals of Physics},
+ volume={41},
+ number={1},
+ pages={108--157},
+ year={1967},
+ publisher={Elsevier}
+}
+
+@article{langer.1969.metastable,
+ title={Statistical theory of the decay of metastable states},
+ author={Langer, JS},
+ journal={Annals of Physics},
+ volume={54},
+ number={2},
+ pages={258--275},
+ year={1969},
+ publisher={Elsevier}
+}
+
+@article{gaveau.1989.analytic,
+ title={Metastable decay rates and analytic continuation},
+ author={Gaveau, B and Schulman, LS},
+ journal={Letters in Mathematical Physics},
+ volume={18},
+ number={3},
+ pages={201--208},
+ year={1989},
+ publisher={Springer}
+}
+
+@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}
+}
+
+@article{gunther.1993.transfer-matrix,
+ title={Numerical transfer-matrix study of metastability in the d= 2 Ising model},
+ author={G{\"u}nther, Christoph CA and Rikvold, Per Arne and Novotny, MA},
+ journal={Physical review letters},
+ volume={71},
+ number={24},
+ pages={3898},
+ year={1993},
+ publisher={APS}
+}
+
+@inproceedings{widom.1981.interface,
+ title={Structure of the interface between fluid phases},
+ author={Widom, Benjamin},
+ booktitle={Faraday Symposia of the Chemical Society},
+ volume={16},
+ pages={7--21},
+ year={1981},
+ organization={Royal Society of Chemistry}
+}
+
+@article{gunther.1980.goldstone,
+ title={Goldstone modes in vacuum decay and first-order phase transitions},
+ author={G{\"u}nther, NJ and Wallace, DJ and Nicole, DA},
+ journal={Journal of Physics A: Mathematical and General},
+ volume={13},
+ number={5},
+ pages={1755},
+ year={1980},
+ publisher={IOP Publishing}
+}
+
+@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}
+}
+
diff --git a/essential_ising.tex b/essential_ising.tex
new file mode 100644
index 0000000..d6a92cc
--- /dev/null
+++ b/essential_ising.tex
@@ -0,0 +1,206 @@
+% Ising model abrupt transition.
+%
+% 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}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic}
+
+\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\im{\mathop{\mathrm{Im}}\nolimits}
+\def\dd{\mathrm d}
+\def\O{\mathcal O}
+\def\ei{\mathop{\mathrm{Ei}}\nolimits}
+\def\b{\mathrm b}
+
+\newcommand\pd[3][]{
+ \ifthenelse{\isempty{#1}}
+ {\def\tmp{}}
+ {\def\tmp{^#1}}
+ \frac{\partial\tmp#2}{\partial#3\tmp}
+}
+
+\maketitle
+
+\begin{abstract}
+\end{abstract}
+
+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
+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.
+
+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
+\[
+ \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}}
+\]
+Since $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$
+\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$
+with $\mathcal S(0)=\O(1)$ and $\mathcal M(0)=\O(1)$,
+\[
+ \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}}
+\]
+with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$
+besides the prefactor. Results from field theory predict that, for small $H$
+and $1<d<5$, $d\neq 3$,
+\[
+ \im F\simeq\bigg(\frac h{t^\Delta}\bigg)^{-(d-3)d/2}(g^*)^{-d(d-1)/4}
+ \exp\bigg[-B\bigg(\frac h{|t|^\Delta}\bigg)^{-(d-1)}(g^*)^{-(d+1)/2}\bigg]
+\]
+\[
+ \im F\simeq\bigg(\frac
+ h{t^\Delta}\bigg)^{-7/3}(g^*)^{-8/3}\exp\bigg[-B\bigg(\frac
+ h{t^\Delta}\bigg)^{-2}(g^*)^{-2}\bigg]
+\]
+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
+\[
+ \im F=t^{2-\alpha}\mathcal F(ht^{-\beta\delta})^{-(d-3)d/2}e^{-1/\mathcal
+ G(ht^{-\beta\delta})^{d-1}}
+\]
+In {\sc 2d} we have
+\[
+ \im F=t^2\mathcal F(ht^{-\Delta})e^{-1/\mathcal G(ht^{-\Delta})}
+\]
+with $\Delta=\beta\delta=\frac{15}8$. In terms of $X=ht^{-\Delta}$, this is
+\[
+ \im F=t^2\mathcal F(X)e^{-1/\mathcal G(X)}\simeq At^2|X|e^{-1/B|X|}
+\]
+
+\cite{langer.1967.condensation}
+
+\[
+ F(X)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(X')}{X'-X}\,\dd X'
+ =\frac{At^2}\pi\int_{-\infty}^0\frac{|X'|e^{-1/B|X'|}}{X'-X}\,\dd
+ X'
+ =-\frac{At^2}\pi\int_0^\infty\frac{X'e^{-1/BX'}}{X'+X}\,\dd
+ X'
+\]
+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,
+
+\[
+ F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n
+\]
+\[
+ f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X
+ =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X
+ =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1)
+\]
+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}
+
+\[
+ \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma}
+\]
+
+\[
+ \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1)
+\]
+$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$
+\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}
+
+\input{figs/scaling_func.tex}
+
+\bibliographystyle{plain}
+\bibliography{essential_ising}
+
+\end{document}
+
diff --git a/figs/scaling_func.gplot b/figs/scaling_func.gplot
new file mode 100644
index 0000000..e25b340
--- /dev/null
+++ b/figs/scaling_func.gplot
@@ -0,0 +1,16 @@
+
+set terminal latex rotate
+set ylabel "$\\chi B/At^{-\\gamma}$"
+set xlabel "$h/Bt^{\\beta\\delta}$"
+set key off
+
+gamma = 0.57721566490153286060651209008240243104215933593992
+Ei(m, x) = gamma + log(abs(x)) + sum[k = 1 : m] x**k / (k * k!)
+Xi(m, A, B, x) = -A/(x * pi) * (1-B/x-(B/x)**2 * exp(B/x)* Ei(100, -B/x))
+
+set yrange [0.001:0.6]
+#set logscale xy
+
+plot [0.001:5] Xi(200, -1, 1, x)
+
+
diff --git a/figs/scaling_func.tex b/figs/scaling_func.tex
new file mode 100644
index 0000000..09448b7
--- /dev/null
+++ b/figs/scaling_func.tex
@@ -0,0 +1,260 @@
+% GNUPLOT: LaTeX picture
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diff --git a/makefile b/makefile
new file mode 100644
index 0000000..a2a0925
--- /dev/null
+++ b/makefile
@@ -0,0 +1,13 @@
+
+DOC=essential_ising.tex
+FIGS=scaling_func
+
+all: ${FIGS:%=figs/%.tex}
+ rubber --pdf $(DOC)
+
+figs/%.tex: figs/%.gplot
+ gnuplot $< > $@
+
+clean:
+ rubber --clean $(DOC)
+