Ian Jauslin
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-rw-r--r--Jauslin_Rutgers_2016.tex197
-rw-r--r--README32
-rw-r--r--figs/phase_largealpha.pdfbin0 -> 55189 bytes
-rw-r--r--figs/phase_result.pdfbin0 -> 63057 bytes
-rw-r--r--figs/phase_smallalpha.pdfbin0 -> 48249 bytes
-rw-r--r--figs/plate.pdfbin0 -> 23294 bytes
-rw-r--r--figs/plates.pdfbin0 -> 18263 bytes
-rw-r--r--header.sty13
-rw-r--r--kiss.cls51
-rw-r--r--presentation.sty110
-rw-r--r--toolbox.sty32
11 files changed, 435 insertions, 0 deletions
diff --git a/Jauslin_Rutgers_2016.tex b/Jauslin_Rutgers_2016.tex
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+\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}
+ $$
+ \left<A\right>_{\Lambda,q}:=\frac1{Z(\Lambda|q)}\int_{\Omega_{\Lambda,q}}\kern-10pt dP\ z^{|P|}\varphi(P)A(P)
+ $$
+ \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:
+ $$
+ \left<A_1,\cdots,A_n\right>_{\Lambda,q}^T=\left.\partial_{s_1}\cdots\partial_{s_n}F_{\Lambda,q}(\mathbf s\cdot\mathbf A)\right|_{\mathbf s=0}
+ $$
+ \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
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+++ b/README
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+* 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
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diff --git a/figs/phase_result.pdf b/figs/phase_result.pdf
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diff --git a/figs/phase_smallalpha.pdf b/figs/phase_smallalpha.pdf
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+++ b/figs/phase_smallalpha.pdf
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diff --git a/figs/plate.pdf b/figs/plate.pdf
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+++ b/figs/plate.pdf
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diff --git a/figs/plates.pdf b/figs/plates.pdf
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diff --git a/header.sty b/header.sty
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+%%
+%% 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
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--- /dev/null
+++ b/kiss.cls
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+%%
+%% 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
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--- /dev/null
+++ b/presentation.sty
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+%%
+%% 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
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+%%
+%% 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$\square$}