\documentclass[11pt,fleqn]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% template that does not use Revtex4
%%% but allows special fonts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Please use this template.
%%% Edit it using e.g. Notepad
%%% Ignore the header (do not change it)
%%% Process the file in the Latex site
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page setup
\topmargin -1.5cm
\oddsidemargin -0.04cm
\evensidemargin -0.04cm
\textwidth 16.59cm
\textheight 24cm
\setlength{\parindent}{0cm}
\setlength{\parskip}{0cm}
% Fonts
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}
% Math symbols I
\newcommand{\sinc}{\mbox{sinc}}
\newcommand{\const}{\mbox{const}}
\newcommand{\trc}{\mbox{trace}}
\newcommand{\intt}{\int\!\!\!\!\int }
\newcommand{\ointt}{\int\!\!\!\!\int\!\!\!\!\!\circ\ }
\newcommand{\ar}{\mathsf r}
\newcommand{\im}{\mbox{Im}}
\newcommand{\re}{\mbox{Re}}
% Math symbols II
\newcommand{\eexp}{\mbox{e}^}
\newcommand{\bra}{\left\langle}
\newcommand{\ket}{\right\rangle}
% Mass symbol
\newcommand{\mass}{\mathsf{m}}
\newcommand{\Mass}{\mathsf{M}}
% More math commands
\newcommand{\tbox}[1]{\mbox{\tiny #1}}
\newcommand{\bmsf}[1]{\bm{\mathsf{#1}}}
\newcommand{\amatrix}[1]{\begin{matrix} #1 \end{matrix}}
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
% Other commands
\newcommand{\hide}[1]{}
\newcommand{\drawline}{\begin{picture}(500,1)\line(1,0){500}\end{picture}}
\newcommand{\bitem}{$\bullet$ \ \ \ }
\newcommand{\Cn}[1]{\begin{center} #1 \end{center}}
\newcommand{\mpg}[2][1.0\hsize]{\begin{minipage}[b]{#1}{#2}\end{minipage}}
\newcommand{\Dn}{\vspace*{3mm}}
% Figures
\newcommand{\putgraph}[2][0.30\hsize]{\includegraphics[width=#1]{#2}}
% heading
\newcommand{\heading}[1]{\begin{center} \Large {#1} \end{center}}
\newcommand{\auname}[1]{\begin{center} \bf Submitted by: #1 \end{center}}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\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
$x_0=\frac{N_AT^\frac{-3}2}{A\zeta\left(\frac{5}2\right)\left (\frac{m_b}{2\pi}\right)^\frac{3}2}$
%%%%%%%%%%%%%%%
\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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}