From d16d1e6b3394f7cd604cb9c64630042c8c77d6dc Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 10:43:44 -0400 Subject: moved files to new name --- essential_ising.tex | 206 ---------------------------------------------------- 1 file changed, 206 deletions(-) delete mode 100644 essential_ising.tex (limited to 'essential_ising.tex') diff --git a/essential_ising.tex b/essential_ising.tex deleted file mode 100644 index d6a92cc..0000000 --- a/essential_ising.tex +++ /dev/null @@ -1,206 +0,0 @@ -% 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 $10$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. -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