\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 ${aR\left(\frac{\pi}{ 2} +\phi \right)}$. \begin{itemize} \item [(a)] Find the free energy ${F}$. 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%