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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} + $$ + \left<A\right>_{\Lambda,q}:=\frac1{Z(\Lambda|q)}\int_{\Omega_{\Lambda,q}}\kern-10pt dP\ z^{|P|}\varphi(P)A(P) + $$ + \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} |