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diff --git a/Jauslin_RUMA_2022.tex b/Jauslin_RUMA_2022.tex new file mode 100644 index 0000000..0cab260 --- /dev/null +++ b/Jauslin_RUMA_2022.tex @@ -0,0 +1,349 @@ +\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 the microscopic to the macroscopic\par +\vfil +\large +\hfil Ian Jauslin\par +\rm\normalsize +\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{Macroscopic laws: phases of water} +\begin{itemize} + \item Phenomena that are \high{directly observable} are \high{Macroscopic}. + + \item For example, water at ambient pressure freezes \high{at} $0^\circ\mathrm{C}$ and boils \high{at} $100^\circ\mathrm{C}$. + + \item Liquid water, vapor and ice all have \high{very different properties}, and yet one can \high{easily transition} between these states, simply by changing the \high{temperature} + \begin{itemize} + \item A gas fills the entire volume available. + \item A liquid is incompressible, but flows. + \item A solid is rigid, and moves only as a whole. + \end{itemize} + + \item Melting ice is \high{exactly} at $0^\circ\mathrm{C}$, and boiling water is \high{exactly} at $100^\circ\mathrm{C}$. +\end{itemize} +\vfill +\eject + +\title{Macroscopic laws: gasses} +\begin{itemize} + \item The state of a (ideal) gas is entirely characterized by \high{three} quantities: + \begin{itemize} + \item $p$: pressure + \item $T$: temperature + \item $n$: density + \end{itemize} + + \item Ideal gas law: + $$p=\frac{k_B}\mu nT$$ + \vskip-5pt + + \item Energy density: + $$e=\frac32k_B T$$ +\end{itemize} +\vfill +\eject + +\title{Microscopic Theories: phases of water} +\begin{itemize} + \item Understand macroscopic laws from \high{first principles}: \high{Microscopic} theories. + \vskip-5pt + \item Freezing and boiling: \high{ordering transitions}. + + \hfil + \includegraphics[width=3cm]{gas.png} + \hfil + \includegraphics[width=3cm]{liquid.png} + \hfil + \includegraphics[width=3cm]{crystal.png} + + \begin{itemize} + \item Gases expand because the molecules are far apart. + \vskip-5pt + \item Liquids are jammed, but molecules can still move around each other. + \vskip-5pt + \item Solids are constrained by the rigid pattern of their molecules. + \end{itemize} +\end{itemize} +\vfill +\eject + +\title{Microscopic Theories: gasses} +\begin{itemize} + \item Ideal gas: non-interacting molecules. + + \hfil + \includegraphics[width=3cm]{gas.png} + + \item We will discuss later how this predicts the laws discussed earlier. +\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 Microscopic dynamics are \high{reversible}. + + \item Consider the motion of a point particle, which follows the laws of (conservative) Newtonian mechanics. If time is \high{reversed}, the motion still satisfies the \high{same} laws of Newtonian mechanics. + + \item In fact, Newtonian mechanics has a \high{recurrence time}: any (bounded, conservative) mechanical system will return \high{arbitrarily close} to its original state in \high{finite} time. +\end{itemize} +\vfill +\eject + +\title{The arrow of time} +\begin{itemize} + \item Yet, many macroscopic phenomena are \high{irreversible}. + \item Friction: the law of friction is not invariant under time reversal. + \item The expansion of a gas in a container. + \item How can \high{reversible} microscopic dynamics give rise to \high{irreversible} macroscopic phenomena? +\end{itemize} +\vfill +\eject + +\title{The thermodynamic limit} +\begin{itemize} + \item One mole $\approx\ 6.02\times10^{23}$. + + \item Rough estimate of the recurrence time for a mechanical system containing $10^{23}$ particles: $\approx 10^{10^{23}}\ \mathrm{s}$. (Time since the big bang: $\approx 10^{17}\ \mathrm s$.) + + \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 interacting particles, we use \high{probability theory}. + + \item Simple example: the ideal gas: + \begin{itemize} + \item Each particle is a point, and no two particles interact. + \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 ideal 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 + $$ + Z=\int d\mathbf x d\mathbf v\ e^{-\beta H(\mathbf x,\mathbf v)} + =\int d\mathbf x\int d\mathbf v\ e^{-\frac{\beta m}2\mathbf v^2}=V^N\left(\frac{2\pi}{\beta m}\right)^{\frac32N} + . + $$ + \vskip-5pt + + \item The average energy is + $$ + \mathbb E(H)=\frac1Z\int d\mathbf xd\mathbf v\ H(\mathbf x,\mathbf v)e^{-\beta H(\mathbf x,\mathbf v)} + = + -\frac\partial{\partial\beta}\log Z + = + \frac{3N}{2\beta} + =\frac32Nk_BT + . + $$ + + \item + The ideal gas law can also be proved for this model. +\end{itemize} +\vfill +\eject + +\title{Hard sphere model} +\begin{itemize} + \item The ideal gas does \high{not} form a liquid or a solid phase. + + \item In order to have such phase transitions, we need an \high{interaction} between particles. + + \item \high{Hard sphere model}: each particle is a sphere of radius $R$, and the interaction is such that no two spheres can overlap. + + \item Parameter: density. +\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 at low density and a \high{crystalline} one at high density. +\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 ideal gas. +\end{itemize} + +\vfill +\eject + +\title{Hard sphere model} +\begin{itemize} + \item The \high{gaseous phase} is very well understood. + + \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. + + \item Even at zero temperature, it was only proved that they crystallize in 2005, and that proof is computer-assisted. + + \item This is very difficult: even tiny fluctuations in the positions of the spheres could destroy the crystalline structure. +\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, 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} +\begin{itemize} + \item Here again, the gas phase is well understood, but neither the nematic nor the smectic have yet been proved to exist. +\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} |