From edce6eb1bb92a2c9531ab9fb346a50da527ccc2a Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Mon, 17 Oct 2022 20:11:41 -0400 Subject: As presented at Rutgers university on 2022-08-26 --- Jauslin_Glimpse_2022.tex | 317 +++++++++++++++++++++++++++++++++++ Makefile | 48 ++++++ README | 34 ++++ figs/atoms.fig/Makefile | 15 ++ figs/atoms.fig/crystal-base.gp | 21 +++ figs/atoms.fig/crystal.py | 24 +++ figs/atoms.fig/gas-base.gp | 21 +++ figs/atoms.fig/gas-rods-base.gp | 21 +++ figs/atoms.fig/gas-rods.py | 88 ++++++++++ figs/atoms.fig/gas.py | 39 +++++ figs/atoms.fig/nematic-base.gp | 21 +++ figs/atoms.fig/nematic.py | 90 ++++++++++ figs/atoms.fig/smectic-base.gp | 21 +++ figs/atoms.fig/smectic.py | 92 ++++++++++ figs/diamonds.fig/Makefile | 20 +++ figs/diamonds.fig/diamonds.py | 65 +++++++ figs/diamonds.fig/shapes.sty | 1 + figs/libs/shapes.sty | 109 ++++++++++++ figs/rods.fig/Makefile | 28 ++++ figs/rods.fig/libs/shapes.sty | 1 + figs/rods.fig/rods.tikz.tex | 30 ++++ figs/shapes.fig/L_tetromino.tikz.tex | 11 ++ figs/shapes.fig/Makefile | 28 ++++ figs/shapes.fig/P_pentomino.tikz.tex | 11 ++ figs/shapes.fig/T_tetromino.tikz.tex | 11 ++ figs/shapes.fig/V_triomino.tikz.tex | 11 ++ figs/shapes.fig/cross.tikz.tex | 11 ++ figs/shapes.fig/diamond.tikz.tex | 11 ++ figs/shapes.fig/hexagon.tikz.tex | 37 ++++ figs/shapes.fig/libs/shapes.sty | 1 + libs/ian-presentation.cls | 187 +++++++++++++++++++++ 31 files changed, 1425 insertions(+) create mode 100644 Jauslin_Glimpse_2022.tex create mode 100644 Makefile create mode 100644 README create mode 100644 figs/atoms.fig/Makefile create mode 100644 figs/atoms.fig/crystal-base.gp create mode 100644 figs/atoms.fig/crystal.py create mode 100644 figs/atoms.fig/gas-base.gp create mode 100644 figs/atoms.fig/gas-rods-base.gp create mode 100644 figs/atoms.fig/gas-rods.py create mode 100644 figs/atoms.fig/gas.py create mode 100644 figs/atoms.fig/nematic-base.gp create mode 100644 figs/atoms.fig/nematic.py create mode 100644 figs/atoms.fig/smectic-base.gp create mode 100644 figs/atoms.fig/smectic.py create mode 100644 figs/diamonds.fig/Makefile create mode 100644 figs/diamonds.fig/diamonds.py create mode 120000 figs/diamonds.fig/shapes.sty create mode 100644 figs/libs/shapes.sty create mode 100644 figs/rods.fig/Makefile create mode 120000 figs/rods.fig/libs/shapes.sty create mode 100644 figs/rods.fig/rods.tikz.tex create mode 100644 figs/shapes.fig/L_tetromino.tikz.tex create mode 100644 figs/shapes.fig/Makefile create mode 100644 figs/shapes.fig/P_pentomino.tikz.tex create mode 100644 figs/shapes.fig/T_tetromino.tikz.tex create mode 100644 figs/shapes.fig/V_triomino.tikz.tex create mode 100644 figs/shapes.fig/cross.tikz.tex create mode 100644 figs/shapes.fig/diamond.tikz.tex create mode 100644 figs/shapes.fig/hexagon.tikz.tex create mode 120000 figs/shapes.fig/libs/shapes.sty create mode 100644 libs/ian-presentation.cls diff --git a/Jauslin_Glimpse_2022.tex b/Jauslin_Glimpse_2022.tex new file mode 100644 index 0000000..fb89d6a --- /dev/null +++ b/Jauslin_Glimpse_2022.tex @@ -0,0 +1,317 @@ +\documentclass{ian-presentation} + +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{xcolor} + +\definecolor{highlight}{HTML}{981414} +\def\high#1{{\color{highlight}#1}} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf\Large +\hfil Statistical Mechanics\par +\smallskip +\hfil from microscopic to macroscopic\par +\vfil +\large +\hfil Ian Jauslin\par +\rm\normalsize +\smallskip +\hfil{\tiny Hill 602, 534} +\vfil +{\tt \href{mailto:ian.jauslin@rutgers.edu}{ian.jauslin@rutgers.edu}} +\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}} +\eject + +\setcounter{page}1 +\pagestyle{plain} + +\title{Statistical mechanics at Rutgers (math)} +\begin{itemize} + \item Eric Carlen + \item Shelly Goldstein + \item Ian Jauslin + \item Michael Kiessling + \item Joel Lebowitz +\end{itemize} + +\vfill +\eject + +\title{What is Statistical Mechanics?} +\begin{itemize} + \item Phenomena that are \high{directly observable} are \high{Macroscopic}: for example + \begin{itemize} + \item Ideal gas law: + $$pV=Nk_BT$$ + \item Freezing and other phase transitions. + \item Ohm's law: + $$V=RI$$ + \end{itemize} + \vskip-5pt + + \item How to understand these? \high{Microscopic} theories! +\end{itemize} +\vfill +\eject + +\addtocounter{page}{-1} +\title{What is Statistical Mechanics?} +\begin{itemize} + \item Phenomena that are \high{directly observable} are \high{Macroscopic}: for example + \begin{itemize} + \item Ideal gas law: \high{free molecules} + $$pV=Nk_BT$$ + \item Freezing and other phase transitions: \high{ordering of particles}. + \item Ohm's law: \high{electrons moving through a metal} + $$V=RI$$ + \end{itemize} + \vskip-5pt + + \item How to understand these? \high{Microscopic} theories! +\end{itemize} +\vfill +\eject + +\title{What is Statistical Mechanics?} +\vfill +\begin{itemize} + \item Statistical mechanics: understanding how the \high{macroscopic} properties follow from the \high{microscopic} laws of nature (``first principles''). +\end{itemize} +\vfill +\eject + +\title{The arrow of time} +\begin{itemize} + \item The microscopic dynamics are \high{reversible}. + \begin{itemize} + \item Consider the motion of a point particle, which follows the laws of Newtonian mechanics. + \item If time is \high{reversed}, the motion still satisfies the \high{same} laws of Newtonian mechanics. + \end{itemize} + + \item Many macroscopic phenomena are \high{irreversible}. + \begin{itemize} + \item For example: friction: the law of friction is not invariant under time reversal. + \item Or, consider the expansion of a gas in a container. + \end{itemize} +\end{itemize} +\vfill +\eject + +\title{The thermodynamic limit} +\begin{itemize} + \item One mole $\approx\ 6.02\times10^{23}$. + + \item Whereas a \high{finite} number of microscopic particles behaves reversibly, an \high{infinite} number of microscopic particles does not. + + \item Fundamental tool of statistical mechanics: the \high{thermodynamic limit}, in which the number of particles $\to\infty$. + +\end{itemize} +\vfill +\eject + +\title{Putting the Statistics in Statistical Mechanics} +\begin{itemize} + \item To understand these infinite particles interacting with each other, we use \high{probability theory}. + + \item Simple example: the free gas: + \begin{itemize} + \item Each particle is a point, and no two particles interact with each other. + \item Probability distribution: \high{Gibbs distribution} + $$ + p(\mathbf x,\mathbf v)=\frac1Z e^{-\beta H(\mathbf x,\mathbf v)} + ,\quad + \beta:=\frac1{k_BT} + $$ + where $H(\mathbf x,\mathbf v)$ is the energy of the configuration where the particles are located at $\mathbf x\equiv(x_1,\cdots,x_N)$ with velocities $\mathbf v\equiv(v_1,\cdots,v_N)$. + \end{itemize} +\end{itemize} +\vfill +\eject + +\title{The free gas} +\begin{itemize} + \item The energy is the kinetic energy: + $$ + H(\mathbf x,\mathbf v)= + \frac12m\sum_{i=1}^Nv_i^2 + . + $$ + \vskip-5pt + + \item Denoting the number of particles by $N$ and the volume by $V$, we have + $$ + Z=V^N\left(\frac{2\pi}{\beta m}\right)^{\frac32N} + . + $$ + \vskip-5pt + + \item The pressure can be computed to be + $$ + p + =\frac N{\beta V} + \equiv\frac{Nk_BT}V + $$ + that is, the ideal gas law. +\end{itemize} +\vfill +\eject + +\title{Hard sphere model} +\begin{itemize} + \item Let us now consider a system where the microscopic particles \high{interact}: the \high{hard sphere model}, in which each particle is a sphere of radius $R$, and the interaction is such that no two spheres can overlap. + + \item Probability distribution: + $$ + p(\mathbf x)=\frac1Ze^{\beta\mu N} + $$ + where $\mu$ is the \high{chemical potential} and $\beta=\frac1{k_BT}$. +\end{itemize} +\vfill +\eject + +\title{Hard sphere model} +\vskip-10pt +\begin{itemize} + \item We expect, from numerical simulations, to see two phases: a \high{gaseous} phase and a \high{crystalline} one. +\end{itemize} + +\hfil +\includegraphics[width=3cm]{gas.png} +\hfil +\includegraphics[width=3cm]{crystal.png} +\vskip-10pt + +\begin{itemize} + \item In the \high{gaseous phase}, the particles are almost decorrelated: they behave as if they did not interact. + + \item In the \high{crystalline phase}, they form large scale periodic structures: they behave very differently from the non-interacting gas. +\end{itemize} + +\vfill +\eject + +\title{Hard sphere model} +\begin{itemize} + \item The \high{gaseous phase} is very well understood. Much about it can be computed using analytic expansions (called ``cluster expansions'' or ``Mayer expansions''). + + \item The \high{crystalline phase} is much more of a mystery: we still lack a proof that it exists at positive temperatures! + + \item \high{Open Problem}: prove that hard spheres crystallize at sufficiently low temperatures. +\end{itemize} +\vfill +\eject + +\title{Liquid crystals} +\begin{itemize} + \item Phase of matter that shares properties of \high{liquids} (disorder) and \high{crystals} (order). + + \item Nematic liquid crystals: order in orientation, disorder in position. +\end{itemize} +\hfil\includegraphics[width=4cm]{nematic.png} +\vfill +\eject + +\title{Liquid crystals} +\begin{itemize} + \item Model: hard cylinders. + + \item Expected phases: \high{gas}, \high{nematic}, \high{smectic} +\end{itemize} +\hfil\includegraphics[height=4cm]{gas-rods.png} +\hfil\includegraphics[height=4cm]{nematic.png} +\hfil\includegraphics[height=4cm]{smectic.png} +\vfill +\eject + +\title{Liquid crystals} +\begin{itemize} + \item Here again, the gas phase is well understood, but neither the nematic nor the smectic have yet been proved to exist. + + \item \high{Open Problem}: Prove the existence of a nematic or smectic phase. +\end{itemize} +\vfill +\eject + +\title{Continuous symmetry breaking} +\begin{itemize} + \item Difficulty for both the hard spheres and liquid crystals: \high{breaking a continuous symmetry} (translation for the hard spheres, rotation for the liquid crystals). + + \item Continuous symmetries cannot\textsuperscript{$\ast$} be broken in one or two dimensions. + + \item Continuous symmetry breaking can, so far, only be proved in very special models. +\end{itemize} +\vfill +\eject + +\title{Lattice models} +\begin{itemize} + \item Many examples: +\end{itemize} +\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{Hard diamond model} +\hfil\includegraphics[height=6cm]{diamonds.pdf} +\vfill +\eject + +\addtocounter{page}{-1} +\title{Hard diamond model} +\hfil\includegraphics[height=6cm]{diamonds_color.pdf} +\vfill +\eject + +\title{Hard diamond model} +\vfill +\begin{itemize} + \item Idea: treat the vacancies as a gas of ``virtual particles''. + + \item Can prove crystallization for a large class of lattice models. +\end{itemize} +\vfill +\eject + +\title{Hard rods on a lattice} +\begin{itemize} + \item Model: rods of length $k$ on $\mathbb Z^2$. +\end{itemize} +\hfil\includegraphics[height=5cm]{rods.pdf} +\vfill +\eject + +\title{Hard rods on a lattice} +\begin{itemize} + \item Can prove that, when $k^{-2}\ll\rho\ll k^{-1}$, the system forms a nematic phase. + + \item For larger densities, one expects yet another phase, in which there are tiles of horizontal and vertical rods. + + \item \high{Open Problem}: generalization to 3 dimensions. +\end{itemize} +\vfill +\eject + +\title{Conclusion} +\begin{itemize} + \item Statistical Mechanics establishes a \high{link} between \high{Microscopic} theories and \high{Macroscopic} behavior. + + \item (In equilibrium) it consists in studying the properties of special probability distributions called \high{Gibbs Measures}. + + \item Even simple models pose significant mathematical challenges. + + \item Still, much can be said about \high{lattice models}, even though there are many problems that are \high{still open}! +\end{itemize} + +\end{document} diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..8571f86 --- /dev/null +++ b/Makefile @@ -0,0 +1,48 @@ +PROJECTNAME=$(basename $(wildcard *.tex)) +LIBS=$(notdir $(wildcard libs/*)) +FIGS=$(notdir $(wildcard figs/*.fig)) + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +SYNCTEXS=$(addsuffix .synctex.gz, $(PROJECTNAME)) + +all: $(PROJECTNAME) + +$(PROJECTNAME): $(LIBS) $(FIGS) + pdflatex -file-line-error $@.tex + pdflatex -synctex=1 $@.tex + +$(SYNCTEXS): $(LIBS) $(FIGS) + pdflatex -synctex=1 $(patsubst %.synctex.gz, %.tex, $@) + + +libs: $(LIBS) + +$(LIBS): + ln -fs libs/$@ ./ + +figs: $(FIGS) + +$(FIGS): + make -C figs/$@ + for pdf in $$(find figs/$@/ -name '*.pdf'); do ln -fs "$$pdf" ./ ; done + for png in $$(find figs/$@/ -name '*.png'); do ln -fs "$$png" ./ ; done + +clean-aux: clean-figs-aux + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + rm -f $(addsuffix .out, $(PROJECTNAME)) + +clean-libs: + rm -f $(LIBS) + +clean-figs: + $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean; ) + rm -f $(notdir $(wildcard figs/*.fig/*.pdf)) + +clean-figs-aux: + $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean-aux; ) + +clean-tex: + rm -f $(PDFS) $(SYNCTEXS) + +clean: clean-aux clean-tex clean-libs clean-figs diff --git a/README b/README new file mode 100644 index 0000000..198b6c6 --- /dev/null +++ b/README @@ -0,0 +1,34 @@ +This directory contains the source files to typeset the presentation, and +generate the figures. This can be accomplished by running + make + +This document uses a custom class file, located in the 'libs' directory, which +defines a number of commands. + + +* Dependencies: + + pdflatex + TeXlive packages: + amsfonts + graphics + hyperref + latex + pgf + standalone + xcolor + GNU make + python + gnuplot + +* Files: + + Jauslin_Glimpse_2022.tex: + main LaTeX file + + libs: + custom LaTeX class file + + figs: + source code for the figures + diff --git a/figs/atoms.fig/Makefile b/figs/atoms.fig/Makefile new file mode 100644 index 0000000..49923c0 --- /dev/null +++ b/figs/atoms.fig/Makefile @@ -0,0 +1,15 @@ +PROJECTNAME=crystal gas nematic smectic gas-rods +PNGS=$(addsuffix .png, $(PROJECTNAME)) + +all: $(PNGS) + +$(PNGS): + cp $(patsubst %.png, %, $@)-base.gp $(patsubst %.png, %, $@).gp + python $(patsubst %.png, %, $@).py >> $(patsubst %.png, %, $@).gp + gnuplot $(patsubst %.png, %, $@).gp > $@ + +clean-aux: + rm -f $(addsuffix .gp, $(PROJECTNAME)) + +clean: clean-aux + rm -f $(PNGS) diff --git a/figs/atoms.fig/crystal-base.gp b/figs/atoms.fig/crystal-base.gp new file mode 100644 index 0000000..4de0ee2 --- /dev/null +++ b/figs/atoms.fig/crystal-base.gp @@ -0,0 +1,21 @@ +set terminal pngcairo size 2048,2048 + +set key off +unset colorbox +unset border +unset xtics +unset ytics +unset ztics + +set parametric + +set view equal xyz + +set isosample 100 + +set pm3d depthorder +set pm3d lighting primary 0.50 specular 0.6 + +set palette defined (0 "#339999", 1 "#339999") + +splot \ diff --git a/figs/atoms.fig/crystal.py b/figs/atoms.fig/crystal.py new file mode 100644 index 0000000..4b1ba3c --- /dev/null +++ b/figs/atoms.fig/crystal.py @@ -0,0 +1,24 @@ +#!/usr/bin/env python3 + +from math import * +import random +import sys + +# size of lattice +N=5 + +# configuration +config=[] +for i in range(N): + for j in range(N): + for k in range(N): + config.append([2*i+((j+k)%2),sqrt(3)*(j+(k%2)/3),2*sqrt(6)/3*k]) + + +for i in range(len(config)): + print(str(config[i][0])+"+cos(u)*sin(v)", end=",") + print(str(config[i][1])+"+sin(u)*sin(v)", end=",") + print(str(config[i][2])+"+cos(v)", end=" ") + print("with pm3d", end="") + if i1 and abs(c)<1.0001): + if(c>0): + return([w[0],0]) + else: + return([w[0],pi]) + if(s>=0): + return([w[0],acos(c)]) + return([w[0],2*pi-acos(c)]) + +# configuration +config=[] +# add rods +while len(config)1 and abs(c)<1.0001): + if(c>0): + return([w[0],0]) + else: + return([w[0],pi]) + if(s>=0): + return([w[0],acos(c)]) + return([w[0],2*pi-acos(c)]) + +# configuration +config=[] +# add rods +while len(config)1 and abs(c)<1.0001): + if(c>0): + return([w[0],0]) + else: + return([w[0],pi]) + if(s>=0): + return([w[0],acos(c)]) + return([w[0],2*pi-acos(c)]) + +# configuration +config=[] +# add rods +while len(config) diamonds.tikz.tex + cat diamonds.tikz.tex | sed 's/%1%/red/g;s/%2%/blue/g' > diamonds_color.tikz.tex + sed -i 's/%1%/teal/g;s/%2%/teal/g' diamonds.tikz.tex + pdflatex -jobname diamonds diamonds.tikz.tex + pdflatex -jobname diamonds_color diamonds_color.tikz.tex + +clean-aux: + rm -f diamonds.tikz.tex + rm -f diamonds.log + rm -f diamonds.aux + rm -f diamonds_color.tikz.tex + rm -f diamonds_color.log + rm -f diamonds_color.aux + +clean: clean-aux + rm -f diamonds.pdf + rm -f diamonds_color.pdf diff --git a/figs/diamonds.fig/diamonds.py b/figs/diamonds.fig/diamonds.py new file mode 100644 index 0000000..2bc4d9c --- /dev/null +++ b/figs/diamonds.fig/diamonds.py @@ -0,0 +1,65 @@ +#!/usr/bin/env python3 + +from math import * +import random + +# size of space (must be int) +L=30 +# number of particles +N=int(L*L/2*0.97) + +# check whether two diamonds overlap +def check_overlap(x1,x2): + if(sqrt((x1[0]-x2[0])**2+(x1[1]-x2[1])**2)<=1): + return(True) + return(False) + +# configuration +config=[] + +# put particles on odd lattice manually +for i in range(4): + for j in range(4): + if (i!=2 or j!=1): + config.append([2*int(L/2/2)+1+i+j,2*int(L/2/2)+i-j]) + +# add particles +while len(config)