%#! platex master_thesis-M2.tex
\chapter{$BM}O@E*9M;!(B}

$BK\>O$G$O(B, $BGH?t6u4V$K$*$1$kJ}Dx<0$r7W;;$9$k$3$H$K$h$C$F(B (\ref{eq:futousiki})
$B$,@.N)$9$k$+H]$+$r5DO@$9$k(B. (\ref{eq:futousiki}) $B$N:8JU$NId9f$O?t3XE*$K(B
$B3NDj$G$-$J$$$3$H$rL@$i$+$K$7(B, $B8e$N>O$N?tCM7W;;$NI,MW@-$r=R$Y$k(B. 

\section{$B%9%Z%/%H%k$NH/E8J}Dx<0(B}

$B%(%M%k%.!<%9%Z%/%H%k$NH/E8J}Dx<0$O(B
\begin{align}
 \DP{}{t}E(k) = -\DP{}{k}\Pi_{\mathrm{E}}(k) + F^{\mathcal{E}}(k) +
  D^{\mathcal{E}}_{\mathrm{Ek}}(k) + D^{\mathcal{E}}_{\mathrm{vis}}(k)
 \label{eq:evo_eksp} 
\end{align}
$B$G$"$k(B. $B$3$3$G(B, $E(k)$ $B$O%(%M%k%.!<%9%Z%/%H%k(B, $\Pi_{\mathrm{E}}(k)$ $B$O(B
$B%(%M%k%.!<%U%i%C%/%9(B, $F^{\mathcal{E}}(k)$ $B$O%(%M%k%.!<$N6/@)$rI=$99`(B,
$D^{\mathcal{E}}_{\mathrm{Ek}}(k)$ $B$O%(%/%^%s%@%s%T%s%0$K$h$k%(%M%k%.!<(B
$B$N;60o$rI=$99`(B, $D^{\mathcal{E}}_{\mathrm{vis}}(k)$ $B$OG4@-$K$h$k%(%M%k(B
$B%.!<$N;60o$rI=$99`$G$"$k(B. $B1&JU$N3F9`$NDj5A$O$=$l$>$l0J2<$N$H$*$j$G$"$k(B:
\begin{align*}
 \Pi_{\mathrm{E}}(k) &\equiv \int_{k}^{\infty} T(k') \,\dd k'
 \hspace{20pt} ($B$?$@$7(B, \, T_{\mathrm{E}}(k) \, $B$O%(%M%k%.!<EAC#H!?t(B ), \\
 F^{\mathcal{E}}(k) &\equiv -\sum_{k\leq\bm{k}<k+\Delta k} \biggl\{k_x
 Uk_d^2\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}]\biggr\},\\ 
 D^{\mathcal{E}}_{\mathrm{Ek}}(k) &\equiv
 -\sum_{k\leq\bm{k}<k+\Delta k}
 \biggl\{\frac{1}{2}\kappa|\bvec{k}|^2\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\{\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}}\}]  
 + \frac{\kappa}{2}|\bvec{k}|^2
  \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}-\hat{\psi}_{\bvec{k}}\}\hat{\tau}^{\ast}_{\bvec{k}}]
 \biggr\},\\ 
 D^{\mathcal{E}}_{\mathrm{vis}}(k) &\equiv
 -\sum_{k\leq\bm{k}<k+\Delta k} \biggl\{
 \nu|\bvec{k}|^{10}|\hat{\psi}_{\bvec{k}}|^2 + \nu|\bvec{k}|^8 
 (|\bvec{k}|^2+k_d^2)|\hat{\tau}_{\bvec{k}}|^2 \\
 &\hspace{60pt} \quad
 +\frac{1}{2}\Delta\nu|\bvec{k}|^8\{|\bvec{k}|^2
 |\hat{\psi}_{\bvec{k}}|^2-(|\bvec{k}|^2+k_d^2)\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}]\}
 \\
 &\hspace{60pt} \qquad
 +
 \frac{1}{2}\Delta\nu|\bvec{k}|^8\{(|\bvec{k}|^2+k_d^2)|\hat{\tau}_{\bvec{k}}|^2-|\bvec{k}|^2\mathrm{Re}[\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}_{\bvec{k}}]\}
 \biggr\}.  
\end{align*}

$BF1MM$K(B, $B%(%s%9%H%m%U%#!<%9%Z%/%H%k$NH/E8J}Dx<0$O(B
\begin{align}
 \DP{}{t}Q(k) = -\DP{}{k}\Pi_{\mathrm{Q}}(k) + F^{\mathcal{Q}}(k) +
  D^{\mathcal{Q}}_{\mathrm{Ek}}(k) + D^{\mathcal{Q}}_{\mathrm{vis}}(k)
 \label{eq:evo_enssp} 
\end{align}
$B$G$"$k(B. $B$3$3$G(B, $Q(k)$ $B$O%(%s%9%H%m%U%#!<%9%Z%/%H%k(B, $\Pi_{\mathrm{Q}}(k)$ $B$O(B
$B%(%s%9%H%m%U%#!<%U%i%C%/%9(B, $F^{\mathcal{Q}}(k)$ $B$O%(%s%9%H%m%U%#!<$N6/@)$rI=$99`(B,
$D^{\mathcal{Q}}_{\mathrm{Ek}}(k)$ $B$O%(%/%^%s%@%s%T%s%0$K$h$k%(%s%9%H%m%U%#!<(B
$B$N;60o$rI=$99`(B, $D^{\mathcal{Q}}_{\mathrm{vis}}(k)$ $B$OG4@-$K$h$k%(%s%9%H(B
$B%m%U%#!<$N;60o$rI=$99`$G$"$k(B. $B1&JU$N9`$NDj5A$O$=$l$>$l0J2<$N$H$*$j$G$"$k(B:
\begin{align*}
 \Pi_{\mathrm{Q}}(k) &\equiv \int_{k}^{\infty} C(k') \,\dd k'
 \hspace{20pt} ($B$?$@$7(B, \, C_{\mathrm{E}}(k) \, $B$O%(%s%9%H%m%U%#!<EAC#H!?t(B), \\
 F^{\mathcal{Q}}(k) &\equiv -\sum_{k\leq\bm{k}<k+\Delta k} \biggl\{ - k_x
 U|\bvec{k}|^2 \, k_d^2 )\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}]
 + k_x
 Uk_d^2(|\bvec{k}|^2+k_d^2)\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}]\biggr\},\\ 
 D^{\mathcal{Q}}_{\mathrm{Ek}}(k) &\equiv
 -\sum_{k\leq\bm{k}<k+\Delta k} \biggl\{\frac{1}{2}\kappa|\bvec{k}|^4\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\{\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}}\}]
 + \frac{\kappa}{2}|\bvec{k}|^2(|\bm{k}|^2+k_d^2)
  \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}-\hat{\psi}_{\bvec{k}}\}\hat{\tau}^{\ast}_{\bvec{k}}]\biggr\},\\ 
 D^{\mathcal{Q}}_{\mathrm{vis}}(k) &\equiv -\sum_{k\leq\bm{k}<k+\Delta k} \biggl\{
 \nu|\bvec{k}|^{12}|\hat{\psi}_{\bvec{k}}|^2 + \nu|\bvec{k}|^8
 (|\bvec{k}|^2+k_d^2)^2|\hat{\tau}_{\bvec{k}}|^2 \\
 &\hspace{60pt}\quad -\frac{1}{2}\Delta\nu|\bvec{k}|^{10}\{|\bvec{k}|^2
 |\hat{\psi}_{\bvec{k}}|^2-(|\bvec{k}|^2+k_d^2)\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}]\}
 \\
 &\hspace{60pt} \qquad -\frac{1}{2}\Delta\nu|\bvec{k}|^8
 (|\bvec{k}|^2+k_d^2)\{(|\bvec{k}|^2+k_d^2)|\hat{\tau}_{\bvec{k}}|^2-|\bvec{k}|^2\mathrm{Re}[\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}_{\bvec{k}}]\}\biggr\}.
\end{align*}
  
\section{(\ref{eq:futousiki}) $B$N@.N)2DG=@-(B}

$B0J2<(B, Tung and Orland (2003) $B$G$b<h$j>e$2$i$l$?(B, $B%(%s%9%H%m%U%#!<47@-NN(B
$B0h$K$*$1$k(B
\begin{equation}
 k^2 \Pi_{\mathrm{E}}(k) - \Pi_{\mathrm{Q}}(k) \label{eq:discuss}
\end{equation}
$B$NId9f$K$D$$$F5DO@$9$k(B. $BDj>o@-MpN.$r9M$($k(B. $B$3$N$H$-(B, 
$\partial\Pi_{\mathrm{E}}(k)/\partial t = 0,\,
\partial\Pi_{\mathrm{E}}(k)/\partial t=0$ $B$G$"$k(B. $B$h$C$F(B, (\ref{eq:evo_eksp}), (\ref{eq:evo_enssp}) $B$h$j(B
\begin{align}
 \Pi_{\mathrm{E}}(k)=-\int_{k}^{\infty}\sum_{q\leq\bvec{q}<q+\Delta q}
 \left\{F^{\mathcal{E}}(q) + D^{\mathcal{E}}_{\mathrm{Ek}}(q) +
 D^{\mathcal{E}}_{\mathrm{vis}}(q)\right\} \dd q, \label{eq:ekflux} \\
 \Pi_{\mathrm{Q}}(k)=-\int_{k}^{\infty}\sum_{q\leq\bvec{q}<q+\Delta q}
 \left\{F^{\mathcal{Q}}(q) + D^{\mathcal{Q}}_{\mathrm{Ek}}(q) +
 D^{\mathcal{Q}}_{\mathrm{vis}}(q)\right\} \dd q \label{eq:ensflux}
\end{align}
$B$H$J$k(B. $B$3$l$r(B (\ref{eq:discuss}) $B$KBeF~$9$k$H(B, 
\begin{align}
 \begin{split} \label{eq:discuss-2}
 & k^2\Pi_{\mathrm{E}}(k)-\Pi_{\mathrm{Q}}(k) \\ =&
 -\int_{k}^{\infty}\sum_{q\leq\bvec{q}\le q+\Delta q} \biggl\{ k_d^2 q_x U
 \underbrace{\mathrm{Im}\{\hat{\psi}_{\bvec{q}}\hat{\tau}_{\bvec{q}}^{\ast}\}}_{(\ast)}
 (k_d^2-k^2) 
 \\ &\hspace{70pt}-\frac{1}{2}\kappa
  q^2(k^2-q^2)|\hat{\psi}_{\bvec{q}}-\hat{\tau}_{\bvec{q}}|^2 \\
  &\hspace{70pt}+ \frac{1}{2}\kappa q^2 k_d^2(|\hat{\tau}_{\bvec{q}}|^2
 -\underbrace{\mathrm{Re}\{\hat{\psi}_{\bvec{q}}\hat{\tau}^{\ast}_{\bvec{q}}\}}_{(\ast\ast)}) \\
 &\hspace{70pt}- (k^2-q^2)\biggl(\nu q^{10}|\hat{\psi}_{\bvec{q}}|^2
  + \frac{1}{2}\Delta\nu
  q^8\{q^2|\hat{\psi}_{\bvec{q}}|^2-(q^2+k_d^2)\underbrace{\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{q}}\hat{\tau}_{\bvec{q}}]}_{(\ast\ast)}\}\biggr)
  \\ 
 &\hspace{70pt}- (k^2-q^2-k_d^2)\biggl(\nu
  q^8(q^2+k_d^2)|\hat{\tau}_{\bvec{q}}|^2 \\
 &\hspace{70pt}\hspace{75pt}+ \frac{1}{2}\Delta\nu q^8\{(q^2-k_d^2)|\hat{\tau}_{\bvec{q}}|^2-q^2\underbrace{\mathrm{Re}[\hat{\psi}_{\bvec{q}}^{\ast}\hat{\tau}_{\bvec{q}}]}_{(\ast\ast)}\}\biggr)\biggr\} \,\dd q
 \end{split} 
\end{align}
$B$H$J$k(B. $B$3$3$G$O(B, $k_d$ $B$h$j$b9bGH?tB&$K$*$1$k(B (\ref{eq:discuss}) $B$NId9f(B
$B$K$D$$$F5DO@$7$F$$$k$N$G(B, $k>k_d$ $B$G$"$k(B. $B$D$^$j(B, $-k^2+k_d^2<0$ $B$G$"$k(B.
$B$^$?(B, $B@QJ,HO0O$N@_Dj$+$i(B
$k^2-q^2\leq 0$, \, $k^2-k_d^2-q^2 < 0, \, q^2-k_d^2 > 0$ $B$G$"$k(B. $B$h$C$F(B, (\ref{eq:discuss-2})
$B$N9`$N$&$A(B, $BId9fIT3NDj$J$N$O(B $(\ast)$ $B$r4^$`9`$H(B $(\ast\ast)$ $B$r4^$`9`(B
($B$$$:$l$b(B, $B=g05%b!<%I$H7905%b!<%I$NAj4X$N9`(B)$B$N$_$G$"$k(B. $B$7$+$7(B, $B$3$l$i$N(B
$BId9f$O?t3XE*$K7hDj$9$k$3$H(B 
$B$O$G$-$J$$(B. $B$D$^$j(B, $B=`CO9U(B 2 $BAX%b%G%k$K$*$$$F$O(B, $B$3$l$i$N9`$N:nMQ$,$"$k(B
$B$?$a$K(B, (\ref{eq:discuss}) $B$NId9f$r?t3XE*$K7hDj$9$k$3$H$,$G$-$J$$(B. $B$h$C(B
$B$F(B, $B?tCM7W;;$K$h$C$F(B (\ref{eq:discuss}) $B$NId9f$,$I$N$h$&$K$J$k$+$rD4$Y$k(B
$BI,MW$,$"$k(B.

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