\sect{Quantum Bose Gas with an oscillating piston (Exam2002 Q2)} 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 ${L, A, N_{a}, N_{b}, \mathsf{m}_{a},\mathsf{m}_{b}, T, M}$. %%begin{figure} \putgraph{Ex404} %%\caption{}\label{} %%end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%