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\begin{document}
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\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}}
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\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}