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\sect{Mechanical model for symmetry breaking}

The following mechanical
model illustrates the symmetry breaking aspect of second order phase
transitions. An air tight piston of mass ${M}$ is inside a 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 volume to the right
of the piston is ${aR\left(\frac{\pi}{2}-\varphi\right)}$ while to
the left is ${aR\left(\frac{\pi}{2}+\varphi\right)}$.
The free
energy of the system has the form ${F\left(T,\varphi\right) = MgR
\cos\varphi -
NT\left[2+\ln\left(aR\left(\frac{\pi}{2}-\varphi\right)/N\lambda^{3}\right)+\ln\left(aR\left(\frac{\pi}{2}+\varphi\right)/N\lambda^{3}\right)\right]}$.

Explain the terms in ${F}$.
Interpret the minimum condition for ${F\left(\varphi\right)}$
in terms of the pressures in the two chambers.
Expand ${F}$ up to 4th
order in ${\varphi}$. Show that there is a symmetry breaking
transition and find the critical temperature ${T_{c}}$.
Describe
what happens to the phase transition if the number of atoms on the
left and right of the piston is ${N\left(1+\delta\right)}$ and
${N\left(1-\delta\right)}$ respectively. Note that it is sufficient
to consider ${|\delta|<<1}$ and include a term ${~\varphi \delta}$
in the expansion. At a certain temperature the left chamber
(containing ${N\left(1+\delta\right)}$ atoms) is found to contain a
droplet of liquid coing with its vapor.
Which of the following
statements may be true at equilibrium:



\begin {itemize}
\item[(a)] The right chamber contains a liquid coming with its
vapor;

\item[(b)] The right chamber contains only vapor;

\item[(c)] The right chamber contains only liquid.
\end {itemize}



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