
\sect{Mechanical model for symmetry breaking}

%Baruch's C05 revised by DC

An airtight piston of mass ${M}$ is free to move inside a cylindrical tube 
of cross sectional area ${a}$. 
The tube is bent into a semicircular shape of radius ${R}$. 
On each side of the piston there is an ideal gas of ${N}$ atoms at a temperature ${T}$. 
The angular position of the piston is~$\varphi$ (see figure).
The gravitation field of Earth exerts a force $Mg$ on the piston, 
while its effect on the gas particles can be neglected.    

The partition function of the system can be written as $d\varphi$ 
integral over $\exp[-A(\varphi)]$. The variable $\varphi$ is regarded     
as the ``order parameter" of the system. A small difference $\Delta N$ 
in the occupation of the two sides is regarded as the conjugate field. 
The susceptibility is defined via the relation ${\langle\varphi\rangle \approx \chi \Delta N}$.   

\begin{itemize}

\item [(1)]
Write an explicit expression for $A(\varphi)$.
 
\item [(2)]
Find the coefficients in the expansion ${A(\varphi)=(a/2)\varphi^2+(u/4)\varphi^4-h\varphi}$.
 
\item [(3)]
Deduce what is the critical temperature $T_c$. 

\item [(4)]
Using Gaussian approximation find what is $\chi$ for ${T > T_c}$.

\item [(5)]
Using Gaussian approximation find what is $\chi$ for $T < T_c$.
 
\item [(6)]
Sketch a plot of $\chi$ versus $T$ indicating by dashed lines the Gaussian 
approximations and by solid line the expected exact result.   
Write what is the range $\Delta T$ around $T_c$ where the Gaussian approximation fails. 

\item [(7)] 
What is the way to take the ``thermodynamic limit" 
such as to have a phase transition at finite temperature? 

\item [(8)] 
In reality, as the temperature is lowered, droplets condense on the walls 
of the left (larger) chamber. What do you expect to find in the right chamber (gas? liquid? both?).   

\end{itemize}

{\bf Guidelines:} In items (4) and (5) simplify the result assuming ${T \sim T_c}$ 
and express it in terms of $T_c$ and ${T-T_c}$.  
The final answer should include one term only.
Care about numerical prefactors - their correctness indicates that the algebra is done properly. 
In item (7) you are requested to identify the parameter that should be taken 
to infinity in order to get a "phase transition". Please specify what are 
the other parameters that should be kept constant while taking this limit.  

\begin{center}
\includegraphics[scale=0.7]{C05}
\end{center}


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