From kanao@stupid03.kugi.kyoto-u.ac.jp Mon Jul 12 15:53:10 1999
To: yoden@stupid03.kugi.kyoto-u.ac.jp
Subject: tex
Date: Mon, 12 Jul 1999 15:53:08 +0900
From: Manabu Kanao <kanao@stupid03.kugi.kyoto-u.ac.jp>


֥饦ưTexǤ

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\begin{document}

\pagenumbering{roman}
\tableofcontents
\pagenumbering{arabic}
\Dchapter{֥饦ư---ȥƥåˤγҳȻ}
\Dchapterhead

饰󥸥ŪγұưŪ˷ޤΤǤʤ, ΨŪ
(stochasitic)˷ޤºݤ˷׻. ֥饦ưȤΤ
ƤǤ. ˽ä, 뤤¿γҷפ.


%------------------------------------------------------------------------------
\section{Ūа}

֥饦ưȯȸˤĤ, (1983; 2-8)˴ŤޤȤ.

åȥɤοʪؼԤǿʪκ˦ˤȯԤȤƤ褯ΤƤ 
R.Brown, ⤫ʴǴѻƤ
ݤ, ʴФγҤդդ𻨤ʱư򤷤Ƥ뤳Ȥȯ.
1827ǯΤȤǤ. ϤβʴγҤαưġ˦αư
ȹͤ. , ޤʤ, ̵ʪγҤϤϱǤúγ
αưƱ褦ʿ񤤤򼨤Ȥ狼, αưñ˵ʬ
Ǯư˲᤮ʤȤե󥹤L.G.GouyˤΤ
줿. º, αư®٤޼αΤǴۤ礭, 
ޤͷγҤ礭ۤ礭, ˲٤⤤ۤ礭
줿. 1905ǯˤA.Einsteinˤä, ޤ1906ǯ
Marian von Smoluchowskiˤä, αưФǮʬұ
ưŸ줿.


%------------------------------------------------------------------------------
\section{ǥ}

֥饦ư򵭽ҤŪʥǥˤ, 󥸥ǥȥ
Ѱ̥ǥ뤬. 󥸥ǥǤ, 饰󥸥®٤Υ
Ѳʬ뤳Ȥˤä, γҤεפ. ,
Ѱ̥ǥǤ, γҤεפѰ̥٥ȥ¤ˤä
.

Ǥ, 󥸥ǥͤ. Langevin , 1908ǯ˼Τ褦
Ƴ, ֥饦ưβϤԤä(Rodean, 1996; Section
2.1). ֥饦γҤϤκѤʤ˱ư, Ϥϴ˥
ͤ. ֥饦γҤ, ʿѤȤƤ®٤㤹Ǵ(졼
꡼໤)뤬, ƽִ֤Ǥߤ, Τۤ޼ʬҤȤξͤˤ
Ϥ. ʤ, ֥饦ưαưȤ, Τ
꼰Ǥ롧
\begin{equation}
   \DD{\Dvect{u}}{t} = - a_1 \Dvect{u} + b \Dvect{\xi}(t),    \label{eq:brown1}
\end{equation}
, $t$ ϻ, $\Dvect{u}$ ϥ֥饦γҤ®, $a_1$ γҤƯ
Ǵ˴ؤ뷸Ǥ. $b$ ȥؿ $\Dvect{\xi}$ѤʬҤ
ͤˤ֥饦γҤԵ§ʲ®ʬ򤢤魯. Langevin Ϥ
鼡Τ褦ʷ̤Ƴ
%\vspace{-1.5em}
\vspace{-1.0em}
\begin{Dnitemize}
  \item $t \ll 1/a_1$ ʤ, γҤѰ̤2ʿѤ $t^2$ 㤹.
  \item $t \gg 1/a_1$ ʤ, γҤѰ̤2ʿѤ $t$ 㤹.
\end{Dnitemize}
%\vspace{-1em}
Τ褦ʷ̤, ʪγȻݤ̩ܤʴطäƤ.

θǤ, 󥸥(\ref{eq:brown1}), Ͼž
ˤ®ιबĤä줿
\begin{equation}
   \DD{\Dvect{u}}{t} = \Dvect{a_0} - a_1 \Dvect{u} + b \Dvect{\xi}(t),
	\label{eq:brown2}
\end{equation}

Ǥ, 󥸥(\ref{eq:brown2})ʬ, Ūȯ
Ѥ, ֥饦ưμ¸Ԥʤ. $i$ ܤλˤ
, ֥饦γҤ®٤$\Dvect{u}_i$Ȥ,
(\ref{eq:brown2})򼡤κʬɽ롧
\begin{equation}
   \Dvect{u}_{i+1} = 
   \Dvect{u}_i + ( \Dvect{a_0} - a_1 \Dvect{u}_i + b \Dvect{\xi}_i ) \Delta t,
	\label{eq:brown3}
\end{equation}
, ؿ $\Dvect{\xi}_i$, $i$ ܤǤ. 
ޤ, ֥饦γҤΰ$\Dvect{x}_i$, ʬͿ
Ȥ롧
\begin{equation}
   \Dvect{x}_{i+1} = \Dvect{x}_i + \Dvect{u}_{i+1} \Delta t.  \label{eq:brown4}
\end{equation}


%------------------------------------------------------------------------------
\section{ɸ¸}


ޤ, {\tt brown1}ǸġγҤˤĤƻȯŸԤʤ, 줾ε
ɤäƤߤ. ¸ե{\tt expar1.data}Խ{\tt
a\_0}{\tt a\_1}ͤ0Ȥ, Ͼžˤ®बʤǴƯ
ʤϤ褦.
\begin{description}
   \item[1] ({\tt brown1}  {\tt expar1.data})
	\begin{verbatim}	
	0.0     0.0     1.0     1       '֥饦ưܷ'
	a_0     a_1     b       iseed
	\end{verbatim}
\end{description}
6ĤγҤεפ٤Ƥߤ. դɽ줿ɥǥޥ
򥯥å, ̤Ȥä̤ɽ(10). 
Ϥ, 졼꡼໤ʤ, {\tt a\_0}{\tt a\_1}
򤤤Ѥ, פΰ㤤ѻ褦. γҤϳϤ˰ư
ʱư򤷤Ƥ뤳Ȥ狼. {\tt iseed} ȤƥǤʤ
Ϳ̤󤬤Ǥ.

γҤϤ줾Ω, ʵפȤ狼. Τ褦
γұư, ҤȤļλˤ֤ˤ¸, ˲
֤ˤޤǰ¸뤳ȤϤʤ. Τ褦ʳΨޥ륳ղȤ
(㤨, ̸(1997); 4-6).

, {\tt brown2}3000ĤγҷɤΤ褦ưƤߤ褦. 
Ȥ, 礭αѤưͤγҤʬۤƤ. 
Ϥ, {\tt a\_0}{\tt a\_1}ͤ0Ȥ, Ūʾ鸫Ƥߤ褦.
\begin{description}
   \item[2] ({\tt brown2}  {\tt expar2.data})
	\begin{verbatim}	
	0.0     0.0     1.0     1       10      '֥饦ưܷ'
	a_0     a_1     b       iseed   istep
	\end{verbatim}
\end{description}
ġγҤʱư򤹤, ˶ɺߤƤγҷϼ
Ū˹äƤ. ȻݤǤ. $x$  $y$ ˼
Ƥγҷʬ۴ؿ, 줾켡ʬۤ˶ŤƤ. Ĥ,
{\tt a\_0}{\tt a\_1}ͤ򤤤Ѥ, γҷȻƤͻҤ
㤤ѻ褦. {\tt istep}ϥսϤ륹ƥå״ֳ֤Ǥ.


%------------------------------------------------------------------------------
\section{ȯŸ}

ԥ塼;Ϥ, ץ{\tt brown2.f}parameterʸ
{\tt nmax}ͤ䤷Ƽ¸Ƥߤ褦(ѹ5Ľ). ʬ۴ؿ
餫ˤʤǤ. Ƽ˼ͱƤγʬۤ׻ݤΥܥå
{\tt jbox}, ǥե{\tt 30}䤹Ȥ褤.

γҷѰ̤2ʿѤ׻, λѲץåȤ, Langevin 
ΤƤߤ褦.


%------------------------------------------------------------------------------
\section{Ρ}
ˡ

$[0,1]$֤ΰ $x$ ˤơ$x$$x+dx$ δ֤ο
Ψ $p(x)dx$ ϼμͿ롣
 \[
 \begin{array}{l}
    \Ddsty p(x)dx=\left \{ 
       \begin{array}{ll}
           \Ddsty dx & (0<x<1)\\
           \Ddsty 0  & (ʳ)\\
      \end{array}
 \right .
 \end{array}
 \]
 $x$ 餽򤢤ؿ$y(x)$ѴȤ롣
$y$γΨ̩$f(y)$ϡΨδѴ§
 \begin{equation}
    \Ddsty |f(y)dy|=|p(x)dx|
 \end{equation}
ʤ
 \begin{equation}
    \Ddsty f(y)=p(x) \left |\frac{dx}{dy} \right |
 \end{equation}
ǷǤ롣$y$ǤդʬۤΤ뤿ˤϡʬ
򤫤ʤФʤʤ$f(y)$ʬ$F(y)$Ȥȡ
βϡ$x=F(y)$Ǥ롣äơ
̩$f(y)$ؤѴ
\begin{equation}
    \Ddsty y(x)=F^{-1}(x)
\end{equation}
Ȥʤ롣ѴץȤƼ¸Ǥ뤫Ǥʤϡ
$F(y)$εմؿ׻Ǥ뤫ɤˤäƤ롣 
Ѵˡϡ󼡸ʾˤ̲Ǥ롣⤷$x_1,x_2,\cdots $
ƱΨ̩ $ p(x_1,x_2,\cdots)dx_1dx_2 \cdots $
⤷$y_1,y_2,\cdots$ΰİĤ٤Ƥ $x$ δؿǤꡢ
$x$$y$ȤƱС$y$ ƱΨ̩٤
 \begin{equation}
    \Ddsty f(y_1,y_2,\cdots)dy_1dy_2=
           p(x_1,x_2,\cdots)|J|dx_1,dx_2,\cdots
 \end{equation}
Ϳ롣$|J|$$x$$y$ˤĤƤΥ䥳ӹ󼰤Ǥ롣
$y$γΨ̩٤ʬۤƤȤˤĤƹͤ롣
 \begin{equation}
    \Ddsty f(y)dy=\frac{1}{\sqrt{2 \pi}} e^{-y^2/2}dy
 \end{equation}
$[0,1]$֤ΣĤΰ$x_1,x_2$ȣĤ $y_1,y_2$Ȥ
֤Ѵ
\begin{equation}
    \Ddsty y_1=\sqrt{-2 \ln x_1}\cos 2 \pi x_2 \\
\end{equation}
\begin{equation}
    \Ddsty y_2=\sqrt{-2 \ln x_1}\sin 2 \pi x_2 
\end{equation}
ͤ롣ϳΨδѴ§Ȥ狼롣ä
ؤѴ(1.10)(1.11)Ϳ롣

%------------------------------------------------------------------------------
\section*{ʸ}

\begin{description}
\labelsep  0 mm
\itemsep  -1.55 mm
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\parsep    0 mm

\item  , 1983: Ǯϳ I, īҽŹ, 239pp.

\item ̸ , 1997: ʿշϤϳء, ȴʪ꡼ 8, 
      ȽŹ, 279 pp.

\item Rodean, H.C., 1996:
      {\it Stochastic Lagrangian models of turbulent diffusion},
      METEOROLOGICAL MONOGRAPHS {\bf 26-48},American Meteorological
      Society, 84pp.

\item W.H.Press,B.P.Flannery,S.A.Teukolsky,W.T.Vetterling,1993:
      Numerical Recipes in C,ɾ,685pp.
\end{description}

\end{document}


