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\title{Cluster-flip colloidal \& atomistic algorithms with background potentials}
\author{Jaron Kent-Dobias \and James P Sethna}
\institute{\includegraphics[width=7em]{figs/bold_cornell_seal_cmyk_red.pdf}}
\date{}

\begin{document}

\begin{frame}
  \maketitle
\end{frame}

\begin{frame}
  \frametitle{Introduction}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\columnwidth]{../statphys_27/figs/ising_hl_00_1}
        \onslide<2>\includegraphics[width=\columnwidth]{../statphys_27/figs/ising_hl_00_2}
        \onslide<3>\includegraphics[width=\columnwidth]{../statphys_27/figs/ising_hl_00_3}
      \end{overprint}
    \end{column}
    \begin{column}{0.5\textwidth}
      Cluster Monte Carlo famously speeds up simulation of lattice models.

      \hfill{\scriptsize Wang \& Swendsen, Physica A \textbf{167} (1990) 565}\\
      \hfill{\scriptsize Wolff, PRL \textbf{62} 4 (1989) 361}

      \vspace{1em}

      Recent extension for lattice models extends this to arbitrary potentials.

      \hfill{\scriptsize Kent-Dobias \& Sethna, PRE \textbf{98} 063306}

      \vspace{1em}

      That method also happens to work for spatial degrees of freedom!
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Hard spheres without potential}
  
  \begin{columns}
    \begin{column}{0.5\textwidth}
      \begin{enumerate}
        \item \alert<2>{Pick a symmetry transformation.}
        \item \alert<3>{Pick a seed.}
        \item \alert<4>{Transform the seed.}
        \item \alert<5>{Identify particles with intersections.}
        \item \alert<6>{Transform each intersecting particle.}
        \item \alert<7-12>{Repeat 5--6 until exhausted.}
      \end{enumerate}

      \vspace{1em}

      \hfill{\scriptsize Dress \& Krauth \textit{J Phys A: Math Gen} \textbf{28} (1995) 597} 
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\textwidth]{figs/hard_free_1.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/hard_free_2.pdf}
        \onslide<3>\includegraphics[width=\textwidth]{figs/hard_free_3.pdf}
        \onslide<4>\includegraphics[width=\textwidth]{figs/hard_free_4.pdf}
        \onslide<5>\includegraphics[width=\textwidth]{figs/hard_free_5.pdf}
        \onslide<6>\includegraphics[width=\textwidth]{figs/hard_free_6.pdf}
        \onslide<7>\includegraphics[width=\textwidth]{figs/hard_free_7.pdf}
        \onslide<8>\includegraphics[width=\textwidth]{figs/hard_free_8.pdf}
        \onslide<9>\includegraphics[width=\textwidth]{figs/hard_free_9.pdf}
        \onslide<10>\includegraphics[width=\textwidth]{figs/hard_free_10.pdf}
        \onslide<11>\includegraphics[width=\textwidth]{figs/hard_free_11.pdf}
        \onslide<12>\includegraphics[width=\textwidth]{figs/hard_free_12.pdf}
        \onslide<13>\includegraphics[width=\textwidth]{figs/hard_free_13.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Soft spheres without potential}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\textwidth]{figs/soft_free_1.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/soft_free_2.pdf}
        \onslide<3>\includegraphics[width=\textwidth]{figs/soft_free_3.pdf}
        \onslide<4,6>\includegraphics[width=\textwidth]{figs/soft_free_4.pdf}
        \onslide<5>\includegraphics[width=\textwidth]{figs/soft_free_4-2.pdf}
        \onslide<7>\includegraphics[width=\textwidth]{figs/soft_free_5.pdf}
        \onslide<8>\includegraphics[width=\textwidth]{figs/soft_free_6.pdf}
        \onslide<9>\includegraphics[width=\textwidth]{figs/soft_free_7.pdf}
        \onslide<10>\includegraphics[width=\textwidth]{figs/soft_free_8.pdf}
        \onslide<11>\includegraphics[width=\textwidth]{figs/soft_free_9.pdf}
        \onslide<12,14>\includegraphics[width=\textwidth]{figs/soft_free_10.pdf}
        \onslide<13>\includegraphics[width=\textwidth]{figs/soft_free_10-2.pdf}
        \onslide<15>\includegraphics[width=\textwidth]{figs/soft_free_11.pdf}
      \end{overprint}
    \end{column}
    \begin{column}{0.5\textwidth}
      Pair potential $V$ gives `Ising' Hamiltonian:
      \[
        H=\sum_{ij}V_{ij}(\vec r_i, \vec r_j)
      \]

      \begin{enumerate}
        \item \alert<2>{Pick a symmetry transformation.}
        \item \alert<3>{Pick a seed.}
        \item \alert<4>{Transform the seed.}
        \item \alert<5>{Identify particles with $\Delta V_{ij}>0$.}
        \item \alert<6>{Transform each particle with probability $1-e^{-\Delta V_{ij}/T}$.}
        \item \alert<7-10>{Repeat 5--6 until exhausted.}
      \end{enumerate}
    \end{column}
  \end{columns}

\end{frame}

\begin{frame}
  \frametitle{Spheres in hard potential}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      Hard potential? Treat it like a particle!

      \begin{enumerate}
        \item \alert<2>{Pick a symmetry transformation.}
        \item \alert<3>{Pick a seed.}
        \item \alert<4>{Transform the seed.}
        \item \alert<5>{Identify `particles' with intersections.}
        \item \alert<6>{Transform each intersecting particle.}
        \item \alert<7-15>{Repeat 5--6 until exhausted.}
      \end{enumerate}
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\textwidth]{figs/hard_box_1.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/hard_box_2.pdf}
        \onslide<3>\includegraphics[width=\textwidth]{figs/hard_box_3.pdf}
        \onslide<4>\includegraphics[width=\textwidth]{figs/hard_box_4.pdf}
        \onslide<5>\includegraphics[width=\textwidth]{figs/hard_box_5.pdf}
        \onslide<6>\includegraphics[width=\textwidth]{figs/hard_box_6.pdf}
        \onslide<7>\includegraphics[width=\textwidth]{figs/hard_box_7.pdf}
        \onslide<8>\includegraphics[width=\textwidth]{figs/hard_box_8.pdf}
        \onslide<9>\includegraphics[width=\textwidth]{figs/hard_box_9.pdf}
        \onslide<10>\includegraphics[width=\textwidth]{figs/hard_box_10.pdf}
        \onslide<11>\includegraphics[width=\textwidth]{figs/hard_box_11.pdf}
        \onslide<12>\includegraphics[width=\textwidth]{figs/hard_box_12.pdf}
        \onslide<13>\includegraphics[width=\textwidth]{figs/hard_box_13.pdf}
        \onslide<14>\includegraphics[width=\textwidth]{figs/hard_box_14.pdf}
        \onslide<15>\includegraphics[width=\textwidth]{figs/hard_box_50.pdf}
        \onslide<16>\includegraphics[width=\textwidth]{figs/hard_box_51.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Spheres in soft potential}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      Soft potential? Treat it like a particle with effective pair potential $\tilde V$!

      \vspace{0.5em}

      \raisebox{-0.65em}{\includegraphics[width=0.1\textwidth]{figs/ex.pdf}} --- center of attractive potential

      \vspace{0.5em}

      \begin{enumerate}
        \item \alert<2>{Pick a symmetry transformation.}
        \item \alert<3>{Pick a seed.}
        \item \alert<4>{Transform the seed.}
        \item \alert<5>{Identify `particles' with $\Delta\tilde V_{ij}>0$.}
        \item \alert<6>{Transform each `particle' with probability $1-e^{-\Delta\tilde V_{ij}/T}$.}
        \item \alert<7-15>{Repeat 5--6 until exhausted.}
      \end{enumerate}
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\textwidth]{figs/hard_field_1.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/hard_field_2.pdf}
        \onslide<3>\includegraphics[width=\textwidth]{figs/hard_field_3.pdf}
        \onslide<4>\includegraphics[width=\textwidth]{figs/hard_field_4.pdf}
        \onslide<5>\includegraphics[width=\textwidth]{figs/hard_field_5.pdf}
        \onslide<6>\includegraphics[width=\textwidth]{figs/hard_field_6.pdf}
        \onslide<7>\includegraphics[width=\textwidth]{figs/hard_field_7.pdf}
        \onslide<8>\includegraphics[width=\textwidth]{figs/hard_field_8.pdf}
        \onslide<9>\includegraphics[width=\textwidth]{figs/hard_field_9.pdf}
        \onslide<10>\includegraphics[width=\textwidth]{figs/hard_field_10.pdf}
        \onslide<11>\includegraphics[width=\textwidth]{figs/hard_field_11.pdf}
        \onslide<12,14>\includegraphics[width=\textwidth]{figs/hard_field_12.pdf}
        \onslide<13>\includegraphics[width=\textwidth]{figs/hard_field_13.pdf}
        \onslide<15>\includegraphics[width=\textwidth]{figs/hard_field_14.pdf}
        \onslide<16>\includegraphics[width=\textwidth]{figs/hard_field_15.pdf}
        \onslide<17>\includegraphics[width=\textwidth]{figs/hard_field_16.pdf}
        \onslide<18,20>\includegraphics[width=\textwidth]{figs/hard_field_17.pdf}
        \onslide<19>\includegraphics[width=\textwidth]{figs/hard_field_18.pdf}
        \onslide<21>\includegraphics[width=\textwidth]{figs/hard_field_86.pdf}
        \onslide<22>\includegraphics[width=\textwidth]{figs/hard_field_87.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Caveats \& improvements}

  \begin{columns}
    \begin{column}{0.5\textwidth}
      \alert<1-3>{Symmetries of period boundaries more restrictive than free space.}

      \vspace{1em}

      \alert<4-5>{Flips ergodic for non-spherical particles confined to finite volume with potential.}

      \vspace{1em}

      \alert<6-12>{Careful choice of flips likely important at high density.}

      \vspace{1em}

      Addition of confining field allows these at cost of boundaries.
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\textwidth]{figs/hard_torus_1.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/hard_torus_2.pdf}
        \onslide<3>\includegraphics[width=\textwidth]{figs/hard_torus_3.pdf}
        \onslide<4>\includegraphics[width=\textwidth]{figs/dimers_torus.pdf}
        \onslide<5>\includegraphics[width=\textwidth]{figs/dimers_field.pdf}
        \onslide<6>\includegraphics[width=\textwidth]{figs/same_swap_1.pdf}
        \onslide<7>\includegraphics[width=\textwidth]{figs/same_swap_2.pdf}
        \onslide<8>\includegraphics[width=\textwidth]{figs/same_swap_3.pdf}
        \onslide<9>\includegraphics[width=\textwidth]{figs/same_swap_4.pdf}
        \onslide<10>\includegraphics[width=\textwidth]{figs/same_swap_5.pdf}
        \onslide<11>\includegraphics[width=\textwidth]{figs/same_swap_6.pdf}
        \onslide<12>\includegraphics[width=\textwidth]{figs/same_swap_7.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{A better swap?}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      Swap famously speeds up formation of ultrastable computer glasses.

      \vspace{1em}

      \alert<2-4>{Chooses pair and attempts to exchange them with Metropolis rules.}

      \vspace{1em}

      \alert<5-10>{Cluster flip seeds can be constructed similarly, but fail differently.}

      \vspace{1em}

      Whether these can be tuned to improve efficiency is not known.
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1,4>\includegraphics[width=\textwidth]{figs/diff_swap_1.pdf}
        \onslide<2,5>\includegraphics[width=\textwidth]{figs/diff_swap_2.pdf}
        \onslide<3>\includegraphics[width=\textwidth]{figs/diff_swap_22.pdf}
        \onslide<6>\includegraphics[width=\textwidth]{figs/diff_swap_3.pdf}
        \onslide<7>\includegraphics[width=\textwidth]{figs/diff_swap_4.pdf}
        \onslide<8>\includegraphics[width=\textwidth]{figs/diff_swap_5.pdf}
        \onslide<9>\includegraphics[width=\textwidth]{figs/diff_swap_6.pdf}
        \onslide<10>\includegraphics[width=\textwidth]{figs/diff_swap_7.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Demo time}
\end{frame}

\begin{frame}
  \frametitle{The dirty deets}

  System with symmetry group $G$ and objects with state $s_i$ (including position, radius, orientation, spin, \dots)
  \[
    H=\sum_{ij}V(s_i, s_j)+\sum_iU(s_i)
  \]
  For $s_0\in G$ and new potential $\tilde V$ defined by $\tilde V_{0i}=U(s_0^{-1}\cdot s_i)$,
  \begin{align*}
    \tilde H&=\sum_{ij}V_{ij}(s_i, s_j)+\sum_iU_i(s_0^{-1}\cdot s_i) \\
     &=\sum_{ij}\tilde V_{ij}(s_i, s_j)
  \end{align*}
  has the form of a system without a potential.
\end{frame}

\end{document}