From bfed53d2c76cc0449d2d6ef1963863092fb98d03 Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Tue, 6 Feb 2018 19:57:42 +0000 Subject: As presented at Princeton University on 2018-02-06 --- Jauslin_Princeton_2018.tex | 271 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 271 insertions(+) create mode 100644 Jauslin_Princeton_2018.tex (limited to 'Jauslin_Princeton_2018.tex') diff --git a/Jauslin_Princeton_2018.tex b/Jauslin_Princeton_2018.tex new file mode 100644 index 0000000..af99bbc --- /dev/null +++ b/Jauslin_Princeton_2018.tex @@ -0,0 +1,271 @@ +\documentclass{ian-presentation} + +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{dsfont} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf\Large +\hfil Crystalline ordering\par +\smallskip +\hfil in hard-core lattice particle systems\par +\vfil +\large +\hfil Ian Jauslin +\normalsize +\vfil +\hfil\rm joint with {\bf Joel L. Lebowitz}\par +\vfil +arXiv: \vbox{ + \hbox{\tt \href{http://arxiv.org/abs/1705.02032}{1705.02032}} + \hbox{\tt \href{http://arxiv.org/abs/1708.01912}{1708.01912}} +} +\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}} +\eject + +\setcounter{page}1 +\pagestyle{plain} + +\title{Gas-liquid-crystal} +\vfill +\hfil\includegraphics[width=3.0cm]{gas.png} +\hfil\includegraphics[width=3.0cm]{liquid.png} +\hfil\includegraphics[width=3.0cm]{crystal.png} +\vfill\eject + +\title{Hard-core lattice particle (HCLP) systems} +\vfill +\hfil\includegraphics[width=1.2cm]{diamond.pdf} +\hfil\includegraphics[width=1.2cm]{cross.pdf} +\hfil\includegraphics[width=1.2cm]{hexagon.pdf} +\par +\vfill +\hfil\includegraphics[width=0.9cm]{V_triomino.pdf} +\hfil\includegraphics[width=0.9cm]{T_tetromino.pdf} +\hfil\includegraphics[width=0.9cm]{L_tetromino.pdf} +\hfil\includegraphics[width=0.9cm]{P_pentomino.pdf} +\vfill +\eject + +\title{Non-sliding HCLPs} +\begin{itemize} + \item There exist a {\bf finite} number $\tau$ of tilings $\{\mathcal L_1,\cdots,\mathcal L_\tau\}$ which are \penalty-1000{\bf periodic} and {\bf isometric} to each other. +\end{itemize} +\hfil\includegraphics[width=4cm]{cross_packing_l.pdf} +\hfil\includegraphics[width=4cm]{cross_packing_r.pdf} +\vfill +\eject + +\title{Non-sliding HCLPs} +\begin{itemize} + \item Defects are {\bf localized}: for every connected particle configuration $X$ that is {\it not} the subset of a close packing and every $Y\supset X$, there is empty space in $Y$ neighboring $X$. +\end{itemize} +\vfill +\hfil\includegraphics[width=2.1cm]{cross_sliding_2.pdf} +\hfil\includegraphics[width=2.1cm]{cross_sliding_3a.pdf} +\hfil\includegraphics[width=2.4cm]{cross_sliding_3b.pdf} +\vfill +\eject + +\title{Observables} +\begin{itemize} + \item Gibbs measure: + $$ + \left_{\nu} + := + \lim_{\Lambda\to\Lambda_\infty} + \frac1{\Xi_{\Lambda,\nu}(z)} + \sum_{X\subset\Lambda}A(X)z^{|X|}\mathfrak B_\nu(X)\prod_{x\neq x'\in X}\varphi(x,x') + $$ + \vskip-10pt + \begin{itemize} + \item $\Lambda$: finite subset of lattice $\Lambda_\infty$. + \item $z\geqslant 0$: fugacity. + \item $\varphi(x,x')$: hard-core interaction. + \item $\mathfrak B_\nu$: boundary condition: favors the $\nu$-th tiling. + \end{itemize} + \vskip-5pt + + \item Pressure: + \vskip-10pt + $$ + p(z):=\lim_{\Lambda\to\Lambda_\infty}\frac1{|\Lambda|}\log\Xi_{\Lambda,\nu}(z). + $$ +\end{itemize} +\vfill\eject + +\title{Theorem} +\begin{itemize} + \item $p(z)-\rho_m\log z$ and $\left<\mathds 1_{x_1}\cdots\mathds 1_{x_n}\right>_\nu$ are {\bf analytic} functions of $1/z$ for large values of $z$. + \vfill + + \item There are $\tau$ distinct Gibbs states: + $$ + \left<\mathds 1_x\right>_\nu= + \left\{\begin{array}{ll} + 1+O(y)&\mathrm{\ if\ }x\in\mathcal L_\nu\\[0.3cm] + O(y)&\mathrm{\ if\ not} + . + \end{array}\right. + $$ +\end{itemize} +\vfill +\eject + +\title{Low-fugacity expansion} +\begin{itemize} + \item Formally, + $$ + \frac1{|\Lambda|}\log\Xi_\Lambda(z) + = + \sum_{k=1}^\infty b_k(\Lambda)z^k + $$ + where, if $Z_\Lambda(k_i)$ denotes the number of configurations with $k_i$ particles, then + $$ + b_k(\Lambda):=\frac1{|\Lambda|} + \sum_{j=1}^k\frac{(-1)^{j+1}}j + \sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_j\geqslant 1}_{k_1+\cdots+k_j=k}}Z_\Lambda(k_1)\cdots Z_\Lambda(k_j) + $$ +\end{itemize} +\eject + +\title{Low-fugacity expansion} +\begin{itemize} + \item Second term: + $$ + b_2(\Lambda)=\frac1{|\Lambda|}\left(Z_\Lambda(2)-\frac12Z_\Lambda^2(1)\right) + $$ + \item $\frac12 Z_\Lambda^2(1)$: counts non-interacting particle configurations. + \item $Z_\Lambda(2)$: counts interacting particle configurations. + \item The terms of order $|\Lambda|^2$ cancel out! +\end{itemize} +\eject + +\title{Low-fugacity expansion} +\begin{itemize} + \item \href{http://dx.doi.org/10.1017/S0305004100011191}{[Ursell, 1927]}, \href{http://dx.doi.org/10.1063/1.1749933}{[Mayer, 1937]}: $b_k(\Lambda)\to b_k$. + \item \href{http://dx.doi.org/10.1016/0031-9163(62)90198-1}{[Groeneveld, 1962]}, \href{http://dx.doi.org/10.1016/0003-4916(63)90336-1}{[Ruelle, 1963]}, \href{http://dx.doi.org/10.1063/1.1703906}{[Penrose, 1963]}: + $$ + p(z)=\sum_{k=1}^\infty b_kz^k + $$ + which has a positive radius of convergence. +\end{itemize} +\eject + +\title{High-fugacity expansion} +\begin{itemize} + \item Inverse fugacity $y\equiv z^{-1}$: + $$ + \Xi_\Lambda(z)= + z^{N_{\mathrm{max}}} + \sum_{X\subset\Lambda} + y^{N_{\mathrm{max}}-|X|}\prod_{x\neq x'\in X}\phi(x,x') + $$ +\end{itemize} +\eject + +\title{High-fugacity expansion} +\begin{itemize} + \item Formally, + $$ + \frac1{|\Lambda|}\log\Xi_\Lambda + = + \rho_m\log z + + + \sum_{k=1}^\infty c_k(\Lambda)y^k + + + o(1) + $$ + where, if $Q_\Lambda(k_i)$ denotes the number of configurations with $N_{\mathrm{max}}-k_i$ particles, then + $$ + c_k(\Lambda):=\frac1{|\Lambda|} + \sum_{j=1}^k\frac{(-1)^{j+1}}{j\tau^j} + \sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_j\geqslant 1}_{k_1+\cdots+k_j=k}}Q_\Lambda(k_1)\cdots Q_\Lambda(k_j) + $$ +\end{itemize} +\eject + +\title{High-fugacity expansion} +\vfill +\hfil\includegraphics[width=2cm]{gf_diamond1.pdf} +\hfil\includegraphics[width=2.33cm]{gf_diamond2.pdf} +\par\vfill +\hfil\includegraphics[width=2.33cm]{gf_diamond3.pdf} +\vfill\eject + +\title{High-fugacity expansion} +\begin{itemize} + \item \href{http://dx.doi.org/10.1063/1.1697217}{[Gaunt, Fisher, 1965]}: diamonds: $c_k(\Lambda)\to c_k$ for $k\leqslant 9$. + \item \href{http://dx.doi.org/10.1098/rsta.1988.0077}{[Joyce, 1988]}: hexagons (integrable, \href{http://dx.doi.org/10.1088/0305-4470/13/3/007}{[Baxter, 1980]}). + \item \href{http://dx.doi.org/10.1209/epl/i2005-10166-3}{[Eisenberg, Baram, 2005]}: crosses: $c_k(\Lambda)\to c_k$ for $k\leqslant 6$. + \item Cannot be done {\it systematically}: there exist counter-examples: e.g. hard $2\times2$ squares on $\mathbb Z^2$: + $$ + c_1(\Lambda)\propto\sqrt{|\Lambda|} + $$ +\end{itemize} +\eject + +\title{Holes interact} +\begin{itemize} + \item Total volume of holes: $\in\rho_m^{-1}\mathbb N$. +\end{itemize} +\vfill +\hfil\includegraphics[height=4.5cm]{hole_example_cross.pdf} +\hfil\includegraphics[height=4.5cm]{hole_example_square.pdf} +\vfill +\eject + +\title{Non-sliding condition} +\begin{itemize} + \item Distinct defects are decorrelated. +\end{itemize} +\vfill +\hfil\includegraphics[height=5cm]{hole_example_cross_decorrelated.pdf} +\vfill +\eject + +\title{Gaunt-Fisher configurations} +\begin{itemize} + \item Group together empty space and neighboring particles. +\end{itemize} +\vfill +\hfil\includegraphics[width=2.5cm]{gaunt_fisher2.pdf} +\hfil\includegraphics[width=4cm]{gaunt_fisher3.pdf} +\vfill +\eject + +\title{Defect model} +\vskip-5pt +\begin{itemize} + \item Map particle system to a model of defects: + $$ + \Xi_{\Lambda,\nu}(z)=z^{\rho_m|\Lambda|}\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)} + \left(\prod_{\gamma\neq\gamma'\in\underline\gamma}\Phi(\gamma,\gamma')\right) + \prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma) + $$ + \begin{itemize} + \item $\Phi$: hard-core repulsion of defects. + \item $\zeta_\nu^{(z)}(\gamma)$: activity of defect. + \end{itemize} + \item The activity of a defect is exponentially small: $\exists\epsilon\ll 1$ + $$ + \zeta_\nu^{(z)}(\gamma)<\epsilon^{|\gamma|} + $$ + \vskip-5pt + \item Low-fugacity expansion for defects. +\end{itemize} +\eject + +\title{Crystallization} +\vfill +\begin{itemize} + \item Peierls argument: in order to have a particle at $x$ that is not compatible with the $\nu$-th perfect packing, it must be part of or surrounded by a defect. + \vfill + \item Note: a naive Peierls argument requires the partition function to be independent from the boundary condition. This is not necessarily the case here, and we need elements from Pirogov-Sinai theory. +\end{itemize} + + +\end{document} -- cgit v1.2.3-70-g09d2