\sect{Baruch's  C26.}

${N}$ ions of positive charge ${q}$ and ${N}$ with negative
charge ${-q}$ are constrained to move in a two dimensional square
of side ${L}$. The interaction energy of charge ${q_{i}}$ at
position ${{\bf r}_{i}}$ with another charge ${q_{j}}$ at ${{\bf
r}_{j}}$ is $-q_{i}q_{j} \ln|{\bf r}_{i}-{\bf r}_{j}|$ where
${q_{i},q_{j}=\pm q}$. The Hamiltonian is then ($m$ is the mass of each ion and ${\bf p}_i$ are momenta)
\[ {\cal H}=\sum_{i=1}^{2N}{\bf p}_i^2/2m - \sum_{i<j}^{2N}
q_{i}q_{j} \ln|{\bf r}_{i}-{\bf r}_{j}| \]
\begin{itemize}
\item [(a)]
By rescaling space variables to ${{\bf r}_{i}'=C{\bf r}_{i}}$,
where ${C}$ is an arbitrary constant, show that the partition
function ${Z\left(L\right)}$ satisfies:
${Z\left(L\right)=C^{N(\beta q^2- 4)}Z(CL)}$ . Deduce that
$Z(L)= A^{N(2-\beta q^2/2)}Z(1)$ where $A=L^2$ is the area.
[Hint: $\sum_{i<j}^{2N}q_iq_j=-q^2N$].
\item[(b)] Calculate the pressure
and show that at low T the system is unstable. Comment on the reason for
this instability and on how the model should be modified.
\item[(c)] Assume that $Z(1)$ has $N$ dependent
factors only from the momentum integrals
and from the Gibbs factors (this neglects a short range part of the interaction).
Find the chemical potential $\mu(T,N, A)$ and solve for $N(\mu,T,A)$. Find the limit of $N$ for a fixed $\mu$ when $A\rightarrow \infty$ for both $T>T_c=q^2/4$ and $T<T_c$. Interpret these results.
\end{itemize}

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