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\begin{document}
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\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

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\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}