\documentclass{article}
\usepackage{graphicx}
\title{Path Integrals and Monte Carlo}
\author{Keola Wierschem}
\date{September 2002}
\begin{document}
\raggedright
{\Large Path Integrals}
\begin{itemize}
\item {\large First formulated by R. P. Feynman}
\item {\large Introduces concept of "imaginary time", $\tau=\frac{1}{k_BT}$}
\end{itemize}
{\Large The Density Matrix, $\rho$}
\begin{itemize}
\item {\large "All static properties and, in principle, dynamic properties of a many-body system in thermal equilibrium are obtainable from the density matrix."}\footnote{D. M. Ceperley and E. L. Pollock, in Monte Carlo Methods in Theoretical Physics, 35 (1992).}
\item {\large $H=\sum_{i=1}^N-\frac{\hbar^2}{2m}\nabla^2_i+\sum_{i<j}v(\mid r_i-r_j\mid )$}
\item {\large $\rho(R,R';\beta)=\langle R\mid e^{-\beta H}\mid R'\rangle $}
\item {\large $e^{-\beta H}=(e^{-\frac{\beta}{m}H})^m$}
\item {\large $\rho(R,R';\beta)=\int...\int\rho(R,R_1;\frac{\beta}{m})\rho(R_1,R_2;\frac{\beta}{m})...\rho(R_{m-1},R_m;\frac{\beta}{m})\rho(R_m,R';\frac{\beta}{m})dR_1dR_2...dR_m$}
\item {\large $\rho(R,R';\beta)=\langle R_0\mid e^{-\frac{\beta}{m}H}\mid R_1\rangle...\langle R_{m-1}\mid e^{-\frac{\beta}{m}H}\mid R_m\rangle$}
\end{itemize}
\pagebreak
\begin{figure}
\includegraphics[height=60mm]{figures/ringpoly.eps}
\caption{Sketch of a "Ring Polymer"}
\end{figure}
\pagebreak
\begin{figure}
\includegraphics[height=115mm]{figures/timeslices.eps}
\end{figure}
\pagebreak
\begin{figure}
\includegraphics[height=150mm]{figures/fig2.ps}
\caption{PIMC simulation}
\end{figure}
\pagebreak
{\Large Monte Carlo}
\begin{itemize}
\item {\large Based on random numbers}
\item {\large New particle positions are proposed}
\item {\large New positions can be chosen randomly or with some specific distribution, as long as each particle can move to any point in configuration space with a finite number of moves}
\item {\large Probability of new positions are calculated in a systematic manner}
\item {\large Random number drawn to determine if move occurs}
\item {\large Process is repeated until relaxes}
\end{itemize}
\end{document}
