
\sect{Baruch's  C05.}

The following mechanical model illustrates the symmetry breaking
aspect of second order phase transitions. An airtight piston of
mass ${M}$ is inside a tube of cross sectional area ${a}$ (see
figure). 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 ${aR\left( \frac{\pi}{2}-\phi \right)}$ while to the
left is ${aR\left(\frac{\pi}{ 2} +\phi \right)}$. The free energy
of the system has the form
\[ \ F = MgRcos\phi   - Nk_{B}T \left[ \ln \frac{aR\left( \frac{\pi}{2}-\phi\right)}
{N\lambda ^{3}}  + \ln \frac{aR(
\frac{\pi}{2}+\phi)}{N\lambda^{3}}+ 2 \right]\]
\begin{itemize}
\item [(a)]
Explain the terms in ${F}$. Interpret the minimum condition for
${F\left( \phi\right)}$ in terms of the pressures in the two
chambers.
\item [(b)]
Expand ${F}$ to 4th order in ${\phi}$ , show that there is a
symmetry breaking transition and find the critical temperature
${T_{c}}$.
\item [(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. (It is
sufficient to consider ${|\delta |<<1}$ and include a term
${\sim\phi\delta}$   in the expansion (b)).

\item [(d)]
At a certain temperature the left chamber (containing
${N\left(1+\delta \right)}$ atoms) is found to contain a droplet
of liquid coexisting with its vapor. Which of the following
statements may be true at equilibrium:
\begin{itemize}
\item[(i)]  The right chamber contains a liquid coexisting with its vapor.
\item[(ii)] The right chamber contains only vapor.
\item[(iii)] The right chamber contains only liquid.\\
\end{itemize}
\end{itemize}


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


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