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	2015�N�x���v�͊wII �h��9 (6��18���o��A6��25����o)
	��
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	�S�� �g�X ��
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 [���1.]
 $\varepsilon>0,D(\varepsilon)=D_{0}V\varepsilon^{\frac{1}{2}}$($V$�͑̐�)
$\varepsilon\le 0,D(\varepsilon)=0$�̎��A
  $x=\beta\epsilon$�ƕϐ��ϊ�����
\begin{enumerate}
\item[\maru{1}] $T_{B}$��$D_{0}^{-\frac{2}{3}}$�ɔ�Ⴗ�邱�Ƃ������B
\item[\maru{2}] $T<T_{B}$��$B_{1}(1)<1$�������B
\item[\maru{3}] $T<T_{B}$�̂Ƃ�$\varepsilon>0$��
$\varepsilon=0$�̗��q��$N_e$��$N_{0}$��S���q��$N$��$T$�A$T_{B}$�ŕ\���B
\item[\maru{4}] $T<T_{B}$�̈��͂�$T^{n}$�ɔ�Ⴗ�邱�Ƃ������A$n$�����߂�B
\end{enumerate}

\bigskip

\noindent[��] \maru{1} $T_B$��
\begin{equation}
	B_1(1)	= \int^\infty_{-\infty} b(\eps)\frac{D(\eps)}
			{N}d\eps = 1
\end{equation}
  �̏����𖞂������x�Ƃ��Ē�`����Ă���B
 �����ŁA$b(\eps)$�̓{�[�X���z��\���A�������A$z=1$�A$T=T_B$�A�܂�
\begin{equation}
	b(\eps) = \frac{1}
			{\exp[\eps/\kB T_B]-1}
\end{equation}
 ���ŗ^�����Ă���$D(\eps)$��������ƁA
\begin{equation}
	\int^\infty_0 \frac{d\eps}
			{\exp[\eps/\kB T_B]-1}
		\frac{D_{0}V\epsilon^{1/2}}							{N} = 1
\end{equation}
 $\eps/\kB T_B=x$�Ƃ��ĕϐ��ϊ�����B
 $dx=d\eps/\kB T_B$������A
\begin{equation}
	\int^\infty_0 \frac{\kB T_Bdx}
			{\exp[x]-1}\frac{D_{0}V(\kB T_Bx)^{1/2}}
												{N} = 1	\label{hensuuhenkan}
\end{equation}
 $\kB T_B$�������肾���ƁA
\begin{equation}
	(\kB T_B)^{3/2}D_{0}\frac{V}
							{N}\int^\infty_0 \frac{x^{1/2}dx}
			{\exp[x]-1} = 1
\end{equation}
 ���������āA$T_B$�ɂ��ĉ����ƁA
\begin{equation}
	T_B	= \kB^{-1}\left\{D_{0}\frac{V}
				{N}I\right\}^{-2/3}
					\label{nipai}
\end{equation}
 �����ŁA
\begin{equation}
	I	= \int^\infty_0 \frac{x^{1/2}dx}
			{\exp[x]-1}
\end{equation}
  �Ƃ����B
 $I$��$V$�ɂ�$N$�ɂ��˂�Ȃ��萔��\���B
 
\bigskip

\maru{2} $T\ne T_B$�̎��́A
\begin{equation}
	B_1(1)=(\kB T)^{3/2}D_0\frac{V}
				{N}I=\left(\frac{T}
					{T_B}\right)^{3/2}
\end{equation}
  ������A$T<T_B$�̎��́A$B_1(1)=(T/T_B)^{3/2}<1$�����藧�B
 ����䂦�A$T<T_B$��BEC���N����ƌ�����B

\bigskip


\maru{3} ���q��$N$�͉��x$T$�Ɖ��w�|�e���V����$\mu$�ŕ\���ƁA
\begin{equation}
	N = \int D(\eps)\frac{d\eps}
			{\exp[(\eps-\mu)/\kB T]-1} +N_0	
						\label{NB}
\end{equation}
 $T<T_B$�ł́A$\mu=0$������
\begin{equation}
	N = \int D(\eps)\frac{d\eps}
			{\exp[\eps/\kB T]-1} +N_0
\end{equation}
 $D(\eps)$�ɗ^����ꂽ�����������
\begin{equation}
	N	 = \int D_{0}V\epsilon^{1/2}\frac{d\eps}
			{\exp[\eps/\kB T]-1} +N_0
\end{equation}
 �E�ӂ�1���ڂ�$\eps>0$�̗��q���A�܂�$N_e$�A
   2���ڂ�$\eps=0$�̗��q��������A
\begin{align}
	N_e
	 &= \int D_{0}V\epsilon^{1/2}\frac{d\eps}
			{\exp[\eps/\kB T]-1} \label{Ne}\\
	N_0	&= N-\int D_{0}V\epsilon^{1/2}\frac{d\eps}
			{\exp[\eps/\kB T]-1}	\label{Nzero}
\end  {align}

 \siki{Ne}��\siki{Nzero}�ɑ΂���\siki{hensuuhenkan}�ƁA
   ���l�̕ϐ��ϊ�������ƁA
\begin{align}
	N_e
	 &= D_{0}V(\kB T)^{3/2}
	\int\frac{x^{1/2}dx}
			{\exp[x]-1} 	\\
	N_0	&= N-D_{0}V(\kB T)^{3/2}
	\int\frac{x^{1/2}dx}
			{\exp[x]-1}
\end  {align}
 \siki{nipai}���g���ƁA
\begin{align}
	N_e
	 &= N(\kB T_B)^{-3/2}(\kB T)^{3/2}
	=   N\left(\frac{T}
			{T_B}
		\right)^{3/2}
	\\
	N_0	&= N-N(\kB T_B)^{-3/2}(\kB T)^{3/2}
	=   N-N\left(\frac{T}
			{T_B}
		\right)^{3/2}
\end  {align}

\bigskip

\maru{4} ���͂ɂ́A$\varepsilon=0$�̗��q�͊�^���Ȃ��̂ŁA
\begin{align}
	P &= -\frac{\kB T}
			   {V}\int^\infty_{-\infty}D(\varepsilon)\ln(1-ze^{-\varepsilon/\kB T})d\varepsilon	\\
\intertext{�^�����Ă���$D(\varepsilon)$���������}
	&= -\frac{\kB T}
		{V}\int^\infty_0D_0 V \varepsilon ^{1/2}\ln(1-ze^{-\varepsilon/\kB T})d\varepsilon \\
\intertext{$\varepsilon/\kB T=x$�Ƃ����}
	&= -\frac{\kB T}
		{V}\int^\infty_0D_0 V (\kB Tx)^{1/2}\ln(1-ze^{-x})\kB Tdx
											\\
\intertext{�]�ډ��x�ȉ��Ȃ̂�$z=1$}
	&= -\frac{\kB T}
		{V}\int^\infty_0D_0 V (\kB Tx)^{1/2}\ln(1-e^{-x})\kB Tdx \\
\intertext{���������}
	&= -(\kB T)^{5/2}D_0
		\int^\infty_0 x^{1/2}\ln(1-e^{-x})dx
	\qquad	\propto T^{5/2}
\end  {align}
 �܂�A$n=5/2$

\bigskip

\noindent [���2.] $T<T_{B}$�ŁA���͂�$\varepsilon=0$�̗��q����^���Ȃ����Ƃ𕶌��𒲂ׂĎ����Ȃ����B


\bigskip

\noindent[��] $\varepsilon=0$�̗��q����^���鈳�͂̒l�́A
\begin{equation}
	P_0 = -\frac{\kB T}
			   {V}\ln(1-z)
\end   {equation}
  �ŗ^������B
 �����ŁA$z$�͂����̗ʂŁA$z=e^{\beta\mu}$�Œ�`�����B
 $\mu$�͉��w�|�e���V�����B
 $N/V$���ŁA$N\to\infty$�A$V\to\infty$�̋Ɍ���$P_0$�̑傫���𒲂ׂ�B
 ���̂܂ܑ������ƁA�s��B

 BEC�ł́A$N_0\approx N$�����炱�̋Ɍ��ɑ΂���$N/V$�����Ȃ�$N_0/V$�����B
 $n_0=N_0/V$�Ƃ��āA
\begin{equation}
	N_0=\frac{1}
	{1/z-1}
\end{equation}
 ��ό`
\begin{equation}
	1-z=\frac{1}
	{n_0V+1}
\end{equation}
 �����$P_0$�̎��ɑ��
\begin{align}
	\lim_{V\to\infty}P_0 &= \lim_{V\to\infty}\{-\frac{\kB T}
			   {V}\ln(1-z)\}		\\
				&= \lim_{V\to\infty}\{-\frac{\kB T}
			   {V}\ln\frac{1}
	{n_0V+1}\}	= 0
\end   {align}
\end  {document}