
\sect{Quantum Bose Gas with an oscillating piston}

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 ${T}$. \\

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

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

\begin {itemize}
\item[(a)]
Find the equilibrium position of the piston.

\item[(b)]
What is the condition for (*) to be valid?

\item[(c)]
Below which temperature (**) holds?

\item[(d)]
What is the frequency of small oscillations of the piston.

\end {itemize}

Express your answers using ${L, A, N_{a}, N_{b}, \mathsf{m}_{a},\mathsf{m}_{b}, T, M}$.

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