\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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