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author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2017-05-26 10:20:59 -0400 |
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committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2017-05-26 10:20:59 -0400 |
commit | 7b4f4ee4813f8afc3e9949a368fe708351d3b89c (patch) | |
tree | 33b6d66b24cf57c193d3dabc8b891ca382940c92 | |
download | paper-7b4f4ee4813f8afc3e9949a368fe708351d3b89c.tar.gz paper-7b4f4ee4813f8afc3e9949a368fe708351d3b89c.tar.bz2 paper-7b4f4ee4813f8afc3e9949a368fe708351d3b89c.zip |
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-rw-r--r-- | .gitignore | 5 | ||||
-rw-r--r-- | essential_ising.bib | 99 | ||||
-rw-r--r-- | essential_ising.tex | 206 | ||||
-rw-r--r-- | figs/scaling_func.gplot | 16 | ||||
-rw-r--r-- | figs/scaling_func.tex | 260 | ||||
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diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..6e3b920 --- /dev/null +++ b/.gitignore @@ -0,0 +1,5 @@ +*.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 @@ -0,0 +1,99 @@ + +@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 +\setlength{\unitlength}{0.240900pt} +\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi +\sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}% +\begin{picture}(1500,900)(0,0) +\sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}% +\put(151.0,130.0){\rule[-0.200pt]{4.818pt}{0.400pt}} +\put(131,130){\makebox(0,0)[r]{$0$}} +\put(1419.0,130.0){\rule[-0.200pt]{4.818pt}{0.400pt}} +\put(151.0,251.0){\rule[-0.200pt]{4.818pt}{0.400pt}} +\put(131,251){\makebox(0,0)[r]{$0.1$}} 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