%ParentFile :A00compgfdE.tex%\documentclass[11pt]{book}%\setcounter{chapter}{3}%\begin{document}%\chapter{Brownian motion -- particle diffusion by stochastic force}In this chapter, the motions of Lagrangian particles not indeterministic but in {\it stochastic } process is studied.This phenomenon is well known as {\it Brownian motion}.We will calculate the tracks of particles forced by random numbersequence.%======================================================================\section{Theoretical Background}We summarize the discovery and following researches of Brownianmotion according to Kobayashi [1983; section 2-8].Robert Brown, who is a famous Scottish botanist for his discovery of thenucleus in the plant cell, first observed Brownian motion in 1827.He studied pollen grains suspended in a liquid with a microscope,and noticed a random oscillatory motion of the microparticles.He interpreted that the motion belonged to the biotic particlesthemselves.Similar phenomena were found later for inorganic microparticles, andcharcoal microparticles in a solution of India ink.These oscillations were clarified nothing but thermal fluctuations ofhuge molecules by L. G. Gouy in France.He showed that the motion was more rapid for smaller particles, forsmaller viscosity of suspending medium, and for higher temperature.The molecular kinetic theory of heat was introduced and developed totreat these fluctuations of Brownian particles by A. Einstein in 1905and by M. von Smoluchowski in 1906.%======================================================================\section{Models}There are two basic concepts to describe the Brownian motion;{\it Langevin model} and {\it random displacement model}.In the former, a particle trajectory is defined by time integral ofrandom variations of its Lagrangian velocity.In the latter, the trajectory is given by sum of random displacement vector.Hereafter, we treat the Langevin model which is introduced by P. Langevin and used to analyze Brownian motion in 1908 [refer section 2.1 of {\it Rodean}, 1996].Here we assume that the Brownian particles are controlled by absolutelyrandom force with no external force.The particles are moved by the random force due to the collision with molecules of suspending medium, and drifted by the viscosity which is linearly related to the velocity (Rayleigh friction) for average.Brownian motion is expressed by using the following equation,\begin{equation}   \frac{d\mathbf{u}}{dt} = - a_1 \mathbf{u} + b \mathbf{\xi}(t),    \label{eq:brown1}\end{equation}where $t$ is time, $\mathbf{u}$ is velocity of a Brownian particle, $a_1$is the coefficient of viscosity, and $\mathbf{\xi}$ denotes the randomfunction.The second term of the right hand side shows the random acceleration ofparticles by collisions.Langevin deduced from (\ref{eq:brown1}) that;%\vspace{-1.5em}\begin{itemize}  \item if $t \ll 1/a_1$, root mean square of particle displacement has	 linear relation to $t^2$,  \item if $t \gg 1/a_1$, root mean square of particle displacement has	 linear relation to $t$.\end{itemize}\vspace{-1em}These results are closely related with {\it diffusion process}.In subsequent studies, acceleration terms by gravitational field andelectric field were added to {\it Langevin equation} (\ref{eq:brown1}) as\begin{equation}   \frac{d\mathbf{u}}{dt} = \mathbf{a_0} - a_1 \mathbf{u} + b \mathbf{\xi}(t).	 \label{eq:brown2}\end{equation}A finite difference equation can be derived from (\ref{eq:brown2}),\begin{equation}   \mathbf{u}_{i+1} =    \mathbf{u}_i + ( \mathbf{a_0} - a_1 \mathbf{u}_i + b \mathbf{\xi}_i ) \Delta t,	 \label{eq:brown3}\end{equation}where $\mathbf{u}_i$ denotes the velocity of a Brownian particle at time$i$, and a random function $\mathbf{\xi}_i$ is the $i$-th value of Gaussiandeviates.The particle location, denoted $\mathbf{x}_i$, is defined by the following backward difference equation,\begin{equation}   \mathbf{x}_{i+1} = \mathbf{x}_i + \mathbf{u}_{i+1} \Delta t.  \label{eq:brown4}\end{equation}In the next section, we solve these finite difference equations numerically.%======================================================================\section{Exercises}\subsection*{Example 1  Tracks of Brownian particles}Run the program "C5-1" to trace trajectories of Brownian particles.Click the {\tt START} button in the application window, and display thetime development of locations of 6 particles with 6 colors.Both parameters {\tt External force} ($\mathbf{a_0}$) and {\tt Rayleighfriction} ($a_1$) are set 0 for default, and begin this experiment withthe basic condition that neither external field nor viscosityaccelerates the particles.\begin{figure}[t]  \begin{center}    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C04_01.epsf}    \caption{Result of Example 1.}    \label{fig:C04_01}  \end{center}\end{figure}Click the graph window or the {\tt NEXT} button, and new results fordifferent random sequences will be shown (for 9 times).Change the value of {\tt External force} or {\tt Rayleighfriction}, and plot graphs to observe the difference of particle tracksamong various cases.It can be seen that each particle oscillates randomly, and is drifted byexternal force.Put a nonzero integer into {\tt seed}, and you can assign another sequenceof deviates for calculation.Each of particles shows random track independently, andthese particle motions depend not on the past status but on whathappened last.This stochastic process is called {\it Markov process} [e.g. {\itKitahara}, 1997; section 4-6].\subsection*{Example 2  Particle diffusion process}Run "C5-2" to observe the behavior of 3000 particles.For the initial condition, particles are distributed uniformly in acircle in the center.Same as "C5-1", set both parameters 0 for default, and begin testwith the basic condition.\begin{figure}[t]  \begin{center}    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C04_02.epsf}    \caption{Result of Example 2.}    \label{fig:C04_02}  \end{center}\end{figure}Click the {\tt START}, and you can see the movie of particle motions.The particles, which are localized in the initial condition, becomespatially expanded gradually, and the particle distribution functionprojected on each axis becomes like a Gaussian distribution.Random oscillation of each particles give rise to {\it diffusionprocess}.Change the parameters, and compare the difference of particle diffusionprocesses in various conditions.Set the parameter {\tt step} smaller, if you want to change the timeresolution of plot output higher.%======================================================================\section*{References}\begin{description}\itemsep   -1.55mm\topsep    0mm\parsep    0mm\item Kitahara, K., 1997: {\it Nonequilibrium statistical mechanics},      Iwanami Basic Series in Physics 8,       Iwanami Shoten Publishers, 279pp (in Japanese).\item Kobayashi, K., 1983: {\it Thermal Statistical Physics I}, Asakura Shoten      Publishers, 239pp (in Japanese).\item Rodean, H.C., 1996:      {\it Stochastic Lagrangian models of turbulent diffusion},      Meteorological Monographs {\bf 26 (No.48)},      American Meteorological Society, 84pp.\end{description}%\end{document}