\documentclass{beamer}
\usetheme{Singapore}
\title{Physics Simulation using the MonteCarlo Method}
\author{Safi Dewshi\\ 625818}
\date{\today}
\begin{document}
\section{Introduction}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}{What is Monte-Carlo}
\begin{itemize}
\item Coined by Stanislaw Ulam
\pause
%Mathematician, reference to uncle playing odds at Monte Carlo casino
\item Statistical technique
\pause
% Model probablistic (stochastic) systems
\item Model complicated or chaotic systems
%n the 1990s, for instance, the Environmental Protection Agency started using Monte Carlo simulations in its risk assessments
\pause
\item Requires a "good" random number generator
%Introducing small variations

\end{itemize}
\end{frame}
\section{Random Number Generators}
\begin{frame}

\begin{center}
\includegraphics[width=0.7\textwidth]{random_number.png}
\end{center}

\end{frame}

\begin{frame}{Randomisation}
What makes a bad Random Number Generator?


\pause
%predictabile, periodic. However comparatively efficient 
\includegraphics[width=0.8\textwidth]{badrng.png}
\end{frame}
\begin{frame}{"Good" Random number generator}
\pause
%Weather= nondeterministic/periodic
%Chaotic: butterfly effect: small changes in initial conditions leading to big changes in outcome
%Other possibilities: quantum systems eg interference patterns
%However, computationally expensive. Pseudorandom generators better for simulation/modelling because sufficiently unpredictable
\includegraphics[width=\textwidth]{FoggDam-NT.jpg}
\end{frame}
\begin{frame}{A better RNG for simulations}
\begin{center}
\includegraphics[width=0.9\textwidth]{rnd1.png}
\end{center}
\end{frame}
\section{Eliminating Bias}
\begin{frame}{Bias}

\[
Bias \propto \mathcal{O}\left(\frac{1}{N}\right)
\]
%As number of measurements -> \inf, becomes irrelevant. Obv inf N unobtainable
% Order of bias important when averaging
\end{frame}
\begin{frame}{Thermalisation}
How can we remove this bias?
%initial bias because arbitrary starting point. equilibrium attained after time t_eq
%Discard initial sweaps until program has settled. more earlier results discarded=better, but discarding results. if t_eqilibrium comparable to N, need careful estimate to remove correlations

\begin{itemize}
\item Initial sweeps: $$\frac{n}{N}$$
\item Overall Bias: $$\frac{1}{\sqrt{N}}$$
\end{itemize}

\end{frame}
\begin{frame}{Binning}
Method of pre-processing to reduce the effect of observation errors.
\begin{center}
\includegraphics[width=0.5\textwidth]{binning.png}
\end{center}
%method of eliminating bias. eg image processing- combining cluster of pixels into single pixel, reducing number of data points, however reducing noise as well
\end{frame}
\begin{frame}{Jackknife method}
We define the Jackknife averages as
\[
x_i^J=\frac{1}{N-1}\sum\limits_{j\neq i}(x_j)
\]
% Then define f_i^J=f(x_i^J)
\pause
\[
\bar{f}^J=\frac{1}{N}\sum\limits_{i=1}^{N} f^J
\]
%average of all the x-values except x_i

% effect of jackknife+binning= eliminate outlying elements/biases from calculation. 
\end{frame}
\begin{frame}
Any Questions?
\end{frame}
\end{document}