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 insertions(+) create mode 100644 essential-ising.tex (limited to 'essential-ising.tex') 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 $10$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. +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 v1.2.3-54-g00ecf