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\exnumber{3040}
\heading{Quantum Bose Gas with an oscillating piston} 
\auname{Nimrod Sherf}


{\bf The problem:}

\Dn

A cylinder of length ${L}$ and cross section ${A}$ is divided into
two compartments by a piston. The piston has mass ${M}$ and it is
free to move without friction. Its distance from the left basis of
the cylinder is denoted by ${x}$. In the left side of the piston
there is an ideal Bose gas of ${N_{a}}$ particles with mass
${\mathsf{m}_{a}}$. In the right side of the piston there is an
ideal Bose gas of ${N_{b}}$ particles with mass ${\mathsf{m}_{b}}$.
The temperature of the system is ${T}$. \\

\Dn
(*) Assume that the left gas can be treated within the framework of the Boltzmann approximation. 

(**) Assume that the right gas is in condensation.
\Dn

\Dn

(1)Find the equilibrium position of the piston.


(2)What is the condition for (*) to be valid?

(3)Below which temperature (**) holds?

(4)What is the frequency of small oscillations of the piston.

\Dn




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{\bf The solution:}

\Dn

(1) 
\Dn
In equilibrium $P_A=P_B$ .

\Dn

For an Ideal gas $P_AAx=N_AT$. 
\\

For a Bose gas in condensation $P_B=\zeta\left(\frac{5}2\right) \left(\frac{m_b}{2\pi}\right)^\frac{3}2T^\frac{5}2.$

\Dn
Let us define $x_0$ as the position of the piston at equilibrium.

\Dn

 Using the fact that in equilibrium $P_A=P_B$ we get
  $x_0=\frac{N_AT^\frac{-3}2}{A\zeta\left(\frac{5}2\right)\left (\frac{m_b}{2\pi}\right)^\frac{3}2}$



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\Dn
\Dn
(2)
In order to use the  Boltzmann approximation we demand low density in section A:
\\

\Dn
$\Rightarrow N_A\lambda_T^3{\ll}V=Ax_0$
\\


$\frac{1}{\sqrt{m_aT}}{\ll}\left(\frac{Ax_0}{N_A} \right)^\frac{1}{3}$
\\



\Dn
$\Rightarrow m_b{\ll}m_a$
\\
%%%%%%%%%%%%%%%
\Dn
\Dn

\Dn
(3)The total number of particles in side B is $N_b=N_b(\text{ground state}) +N_b(\text{excited states} )$.
\Dn
We want $ N_b>N_b(\text{excited states} )$
\\

$N_b>\int_0^\infty  \!g( \epsilon ) f( \epsilon ) \, \mathrm{d} \epsilon .$

\Dn

Where $f( \epsilon,\mu\to 0 )=\frac{1}{e^{\beta( \epsilon -\mu)}-1}$ 
\Dn

And $g(\epsilon)=\frac{1}{4\pi^2}(2m_b)^\frac{3}{2}A(L-x_0)\sqrt{\epsilon}$


\Dn

$\Rightarrow   N_b>A(L-x_0)\zeta\left(\frac{3}2\right) \left(\frac{m_b}{2\pi}\right)^\frac{3}2T^\frac{3}2$

\Dn
$\Rightarrow T^\frac{3}2<\left(\frac{m_b}{2\pi}\right)^\frac{3}2\left(\frac{N_b}{\zeta\left(\frac{3}2\right)}+\frac{N_a}{\zeta\left(\frac{5}2\right)}\right)\frac{1}{AL}$

\Dn


%%%%%%%%%%%%%%%
\Dn
\Dn  
(4)For small oscillations around the equilibrium position we get:  \Dn
\\
\Dn
$\frac{N_aT}{x_0+dx}-\frac{N_aT}{x_0}\approx \frac{-N_aTdx}{x_0^2}=-Mw^2dx$
\\


$\Rightarrow   w^2=\frac{N_aT}{Mx_0^2}  $


\Dn
$w=\zeta\left(\frac{5}2\right)\frac{A}{\sqrt{N_aM}}\left(\frac{m_b}{2\pi}\right)^\frac{3}2T^2$
\Dn


\Dn

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