From d16d1e6b3394f7cd604cb9c64630042c8c77d6dc Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 26 May 2017 10:43:44 -0400
Subject: moved files to new name

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 essential_ising.tex | 206 ----------------------------------------------------
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-%  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}
-
-- 
cgit v1.2.3-70-g09d2

