Random Walks

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<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle #1\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\eikrwt{e^{i(\vec k\vec r-wt)}} \def\qscat{Q_{scat}} \def\qabs{Q_{abs}}

\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \def\lya{Ly\alpha}

\section{Random Walks}

If $\vec\phi=\phi_1,\phi_2,\dots$ is a random variable consisting of a series of random values, then a random walk $\vec w$ is a sum over this random variable: \begin{equation} w_i=\sum_{n=1}^{i}\phi_n \end{equation}

Random walks have the property that \begin{equation} |w_n|\propto\sqrt{n}, \end{equation} or more generally: \begin{equation} |w_n-w_m|\propto\sqrt{n-m} \end{equation}

\end{document} <\latex>