From b1090e768378b0f563e2bffd29a81ded180f6f1e Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Thu, 22 Sep 2016 00:15:35 +0000 Subject: As presented at Rutgers University on 2016-09-22 --- Jauslin_Rutgers_2016.tex | 197 ++++++++++++++++++++++++++++++++++++++++++++++ README | 32 ++++++++ figs/phase_largealpha.pdf | Bin 0 -> 55189 bytes figs/phase_result.pdf | Bin 0 -> 63057 bytes figs/phase_smallalpha.pdf | Bin 0 -> 48249 bytes figs/plate.pdf | Bin 0 -> 23294 bytes figs/plates.pdf | Bin 0 -> 18263 bytes header.sty | 13 +++ kiss.cls | 51 ++++++++++++ presentation.sty | 110 ++++++++++++++++++++++++++ toolbox.sty | 32 ++++++++ 11 files changed, 435 insertions(+) create mode 100644 Jauslin_Rutgers_2016.tex create mode 100644 README create mode 100644 figs/phase_largealpha.pdf create mode 100644 figs/phase_result.pdf create mode 100644 figs/phase_smallalpha.pdf create mode 100644 figs/plate.pdf create mode 100644 figs/plates.pdf create mode 100644 header.sty create mode 100644 kiss.cls create mode 100644 presentation.sty create mode 100644 toolbox.sty diff --git a/Jauslin_Rutgers_2016.tex b/Jauslin_Rutgers_2016.tex new file mode 100644 index 0000000..253f5ce --- /dev/null +++ b/Jauslin_Rutgers_2016.tex @@ -0,0 +1,197 @@ +\documentclass{kiss} +\usepackage{presentation} +\usepackage{header} +\usepackage{toolbox} +\usepackage{tikz} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf\Large +\hfil Emergence of a nematic phase\par +\smallskip +\hfil in a system of hard plates\par +\vfil +\large +\hfil Ian Jauslin +\normalsize +\vfil +\hfil\rm with {\bf Margherita Disertori} and {\bf Alessandro Giuliani}\par +\vfil +\hfil\href{http://ian.jauslin.org}{\tt http://ian.jauslin.org} +\eject + +\setcounter{page}1 +\pagestyle{plain} + +\title{Hard plates} + \begin{itemize} + \item Parallelepiped $k\times k^\alpha\times 1$, $\alpha\in[0,1]$, $k\gg1$ + \end{itemize} + \bigskip + \includegraphics[width=\textwidth]{figs/plate.pdf} + \begin{itemize} + \item Center in $\mathbb R^3$ + \end{itemize} +\eject + +\title{Hard plates} + \begin{itemize} + \item 6 orientations + \end{itemize} + \vfil + \hfil\includegraphics[width=220pt]{figs/plates.pdf} +\eject + +\title{Heuristics} + \begin{itemize} + \item For $\frac12\leqslant\alpha\leqslant1$ + \end{itemize} + \vfil + \includegraphics[width=\textwidth]{figs/phase_largealpha.pdf} + \begin{itemize} + \itemptchange{$\scriptstyle\blacktriangleright$} + \begin{itemize} + \item $N_b$: biaxial nematic + \item $N_-$: plate-like nematic + \item $I$: isotropic + \item $?$: ? + \end{itemize} + \itemptreset + \end{itemize} +\eject + +\title{Heuristics} + \begin{itemize} + \item For $0\leqslant\alpha\leqslant\frac12$ + \end{itemize} + \vfil + \includegraphics[width=\textwidth]{figs/phase_smallalpha.pdf} + \begin{itemize} + \itemptchange{$\scriptstyle\blacktriangleright$} + \begin{itemize} + \item $N_b$: biaxial nematic + \item $N_+$: rod-like nematic + \item $I$: isotropic + \item $?$: ? + \end{itemize} + \itemptreset + \end{itemize} +\eject + + +\title{Result} + \begin{itemize} + \item $\frac56<\alpha<1$ (we can also do $\alpha=1$). \end{itemize} + \vfil + \includegraphics[width=\textwidth]{figs/phase_result.pdf} +\eject + +\title{Previous results} + \begin{itemize} + \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: 2-dimensional hard rods (on a lattice). + \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: 2-dimensional hard needles. + \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: elongated molecules in 3 dimensions: 1st order phase transition! (non-rigorous) + \end{itemize} +\eject + +\title{Setup} + \begin{itemize} + \item {\it Type} of a plate: $q\in\{1,2,3\}$. + \item (Grand-canonical) Gibbs average, in a box $\Lambda$, with $q$-boundary conditions: + \end{itemize} + $$ + \left_{\Lambda,q}:=\frac1{Z(\Lambda|q)}\int_{\Omega_{\Lambda,q}}\kern-10pt dP\ z^{|P|}\varphi(P)A(P) + $$ + \begin{itemize} + \itemptchange{$\scriptstyle\blacktriangleright$} + \begin{itemize} + \item $\Omega_{\Lambda,q}$: plate configurations with $q$-boundary conditions, $|P|$: number of plates in $P$, + \item $z$: {\it activity}: $z=e^{\beta\mu}$, + \item $\varphi(P)$: hard-core potential, + \item $Z(\Lambda|q)$: partition function (normalization), + \item $A$: {\it local} observable. + \end{itemize} + \itemptreset + \end{itemize} +\eject + +\title{Setup} + \begin{itemize} + \item Boundary condition: any plate, centered at $x\in\Lambda$, that satisfies + $$ + d_\infty(x,\ \mathbb R^3\setminus\Lambda)\leqslant\left(\frac4{1-\alpha}+3\right)\frac k2 + $$ + is of type $q$. + + \item Local observable: + $$ + A(P)=\sum_{p\in P}a(p) + $$ + and $a$ is {\it compactly supported}. + \end{itemize} +\eject + +\title{Theorem} + \begin{itemize} + \item If $zk^{3-\alpha}\ll1\ll zk^{5\alpha-2}$ + \itemptchange{$\scriptstyle\blacktriangleright$} + \begin{itemize} + \item $\mathds1_x(P)$: indicator that $\exists p\in P$: $d_\infty(x,p)\leqslant\frac12$: + $$ + \left<\mathds1_x(P)\right>_{\Lambda,q}\equiv\rho=z(1+o(1)) + $$ + \vskip5pt + \item $\mathcal N_{x,q}(P)$: number of plates $p\in P$ of type $q$ with $d_\infty(p,x)\leqslant\frac k4$: for $m\neq q$, + $$ + \left<\mathcal N_{x,q}(P)\right>_{\Lambda,q}\geqslant C zk^3\gg1,\quad + \left<\mathcal N_{x,m}(P)\right>_{\Lambda,q}=o(1) + $$ + \item There exists $\eta_k\to0$ as $k\to\infty$ such that + $$ + \left<\mathds1_x(P);\mathds1_y(P)\right>_{\Lambda,q}^T\leqslant C\rho^2 \eta_k^{\frac{|x-y|}k} + $$ + \end{itemize} + \itemptreset + \end{itemize} +\eject + +\title{Cluster expansion} + \begin{itemize} + \item For $\mathbf A\equiv(A_1,\cdots,A_n)$, $\mathbf s\equiv(s_1,\cdots,s_n)\in\mathbb R^n$ + $$ + F_{\Lambda,q}(\mathbf s\cdot\mathbf A):=\log\int_{\Omega_{\Lambda,q}}\kern-10pt dP\ z^{|P|}\varphi(P)e^{\sum_{i=1}^ns_iA_i(P)} + $$ + \item Generating function: + $$ + \left_{\Lambda,q}^T=\left.\partial_{s_1}\cdots\partial_{s_n}F_{\Lambda,q}(\mathbf s\cdot\mathbf A)\right|_{\mathbf s=0} + $$ + \end{itemize} +\eject + +\title{Cluster expansion} + $$ + F_{\Lambda,q}(\mathbf s\cdot\mathbf A)=F^{(0)}_{\Lambda,q}(\mathbf s\cdot\mathbf A)+\sum_{\mathcal X\in\Xi(\Lambda)}\phi^T(\mathcal X)\prod_{X\in\mathcal X} K_{\Lambda,q}^{(\mathbf s\cdot\mathbf A)}(X) + $$ + \begin{itemize} + \itemptchange{$\scriptstyle\blacktriangleright$} + \begin{itemize} + \item $F_{\Lambda,q}^{(0)}(\mathbf s\cdot\mathbf A)$: {\it all} plates are of type $q$, + \item $\Xi(\Lambda)$: collections of {\it polymers}: connected unions of $\frac k2\times\frac k2\times\frac k2$ cubes, + \item $\phi^T$: {\it Mayer coefficient}, + \item $K_{\Lambda,q}^{(\mathbf s\cdot\mathbf A)}(X)$: {\it activity} of $X$. + \end{itemize} + \itemptreset + \end{itemize} +\eject + +\title{Cluster expansion} + \begin{itemize} + \item Absolutely convergent expansion: $\exists\epsilon_k\to0$ such that, for $m\geqslant0$, + $$ + \sum_{\mAthop{\mathcal X\in\Xi(\Lambda)}_{|\mathcal X|\ge m}}\left|\phi^T(\mathcal X)\prod_{X\in\mathcal X}K_{\Lambda,q}^{(\mathbf s\cdot\mathbf A)}(X)\right|\le + \epsilon_k^m + $$ + \end{itemize} + +\end{document} diff --git a/README b/README new file mode 100644 index 0000000..0adbefb --- /dev/null +++ b/README @@ -0,0 +1,32 @@ +* Typeset + +In order to typeset the LaTeX presentation, run + pdflatex Jauslin_Rutgers_2016.tex + pdflatex Jauslin_Rutgers_2016.tex + + +* Files + + Jauslin_Rutgers_2016.tex + body of the presentation. + + figs : + figures. + + header.sty : + packages and local definitions + + presentation.sty : + presentation style file. + + kiss.cls : + barebones class file + + +* Comment on the presentation style + +This presentation is typeset using the 'presentation.sty' style file, in which +the margins and sizes are set to make the output look like a presentation, and +several useful commands are defined. + +The LaTeX 'beamer' package is not used. diff --git a/figs/phase_largealpha.pdf b/figs/phase_largealpha.pdf new file mode 100644 index 0000000..3f7b47e Binary files /dev/null and b/figs/phase_largealpha.pdf differ diff --git a/figs/phase_result.pdf b/figs/phase_result.pdf new file mode 100644 index 0000000..db54007 Binary files /dev/null and b/figs/phase_result.pdf differ diff --git a/figs/phase_smallalpha.pdf b/figs/phase_smallalpha.pdf new file mode 100644 index 0000000..eaa4d31 Binary files /dev/null and b/figs/phase_smallalpha.pdf differ diff --git a/figs/plate.pdf b/figs/plate.pdf new file mode 100644 index 0000000..b500022 Binary files /dev/null and b/figs/plate.pdf differ diff --git a/figs/plates.pdf b/figs/plates.pdf new file mode 100644 index 0000000..6b3d0ed Binary files /dev/null and b/figs/plates.pdf differ diff --git a/header.sty b/header.sty new file mode 100644 index 0000000..4ebdea7 --- /dev/null +++ b/header.sty @@ -0,0 +1,13 @@ +%% +%% Load packages +%% + +\usepackage{color} +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{array} +\usepackage{etoolbox} +\usepackage{dsfont} + diff --git a/kiss.cls b/kiss.cls new file mode 100644 index 0000000..7f0029f --- /dev/null +++ b/kiss.cls @@ -0,0 +1,51 @@ +%% +%% Barebones class declaration +%% + +\NeedsTeXFormat{LaTeX2e}[1995/12/01] +\ProvidesClass{kiss} + +\setlength\paperheight {297mm} +\setlength\paperwidth {210mm} + +%% fonts +\input{size11.clo} +\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm} +\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} +\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} +\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} +\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} +\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl} +\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc} + +%% something is wrong with \thepage, redefine it +\gdef\thepage{\the\c@page} + +%% array lines +\setlength\arraycolsep{5\p@} +\setlength\arrayrulewidth{.4\p@} + + +%% default offsets: 1in, correct with \hoffset and \voffset +%\hoffset=0pt +\hoffset=-50pt +%\voffset=0pt +\voffset=-72pt +%% horizontal margins +%\oddsidemargin=31pt +%\evensidemargin=31pt +%% vertical margin +%\topmargin=20pt +%% body size +%\textwidth=390pt +\textwidth=460pt +%\textheight=592pt +\textheight=704pt +%% header size and margin +%\headheight=12pt +%\headsep=25pt +%% footer size +%\footskip=30pt +%% margin size and margin +%\marginparwidth=35pt +%\marginparsep=10pt diff --git a/presentation.sty b/presentation.sty new file mode 100644 index 0000000..4e07694 --- /dev/null +++ b/presentation.sty @@ -0,0 +1,110 @@ +%% +%% Presentation style +%% + +%% can call commands even when they are not defined +\def\safe#1{% + \ifdefined#1% + #1% + \else% + {\color{red}\bf?}% + \fi% +} + + +%% paper size +\setlength\paperheight{240pt} +\setlength\paperwidth{320pt} + +%% body size +% height=paperheight-2xtopmargin-footskip +\textheight=208pt +% width=paperwidth-2xoddsidemargin +\textwidth=272pt + +%% margins +\voffset=-1in +\hoffset=-1in +\oddsidemargin=24pt +\evensidemargin=24pt +\topmargin=8pt +\headheight=0pt +\headsep=0pt +\marginparsep=0pt +\marginparwidth=0pt +\footskip=16pt +\skip\footins=0pt + +%% reset skips +\parskip=0pt +\parindent=0pt +\baselineskip=0pt +\AtBeginDocument{ +} + +%% footer +\def\ps@plain{ + \def\@oddhead{} + \def\@evenhead{\@oddhead} + \def\@oddfoot{\tiny\hfill\thepage/\safe\slidecount\hfill} + \def\@evenfoot{\@oddfoot} +} +\def\ps@empty{ + \def\@oddhead{} + \def\@evenhead{\@oddhead} + \def\@oddfoot{} + \def\@evenfoot{\@oddfoot} +} + +%% save total slide count +\AtEndDocument{ + \immediate\write\@auxout{\noexpand\gdef\noexpand\slidecount{\thepage}} +} + +\pagestyle{plain} + +%% title of slide +\def\title#1{ + \hfil{\bf\large #1}\par + \hfil\vrule width0.75\textwidth height0.3pt\par + \vskip5pt +} + +%% itemize +\newlength\itemizeskip +% left margin for items +\setlength\itemizeskip{20pt} +% item symbol +\def\itemizept{\textbullet} +\newlength\itemizeseparator +% space between the item symbol and the text +\setlength\itemizeseparator{5pt} + +\newlength\current@itemizeskip +\setlength\current@itemizeskip{0pt} +\def\itemize{ + \vfil + \addtolength\current@itemizeskip{\itemizeskip} + \leftskip\current@itemizeskip +} +\def\enditemize{ + \addtolength\current@itemizeskip{-\itemizeskip} + \par\leftskip\current@itemizeskip + \vfil +} +\newlength\itempt@total +\def\item{ + \settowidth\itempt@total{\itemizept} + \addtolength\itempt@total{\itemizeseparator} + \par + \vfil + \hskip-\itempt@total\itemizept\hskip\itemizeseparator +} + +\def\itemptchange#1{ + \let\itempt@prev\itemizept + \def\itemizept{#1} +} +\def\itemptreset{ + \def\itemizept{\itempt@prev} +} diff --git a/toolbox.sty b/toolbox.sty new file mode 100644 index 0000000..b214a3e --- /dev/null +++ b/toolbox.sty @@ -0,0 +1,32 @@ +%% +%% A collection of useful commands +%% + +%% can call commands even when they are not defined +\def\safe#1{ + \ifdefined#1 + #1 + \else + {\color{red}\bf?} + \fi +} + + +%% larger skip +\newskip\hugeskipamount + \hugeskipamount=24pt plus8pt minus8pt +\def\hugeskip{\vskip\hugeskipamount} + + +%% stack relations in subscript or superscript +\def\mAthop#1{\displaystyle\mathop{\scriptstyle #1}} + +%% array spanning the entire line +\newlength\largearray@width +\setlength\largearray@width\textwidth +\addtolength\largearray@width{-10pt} +\def\largearray{\begin{array}{@{}>{\displaystyle}l@{}}\hphantom{\hspace{\largearray@width}}\\[-.5cm]} +\def\endlargearray{\end{array}} + +%% qedsquare +\def\qed{\penalty10000\hfill\penalty10000$\square$} -- cgit v1.2.3-54-g00ecf