\sect{2D Coulomb gas}
$N$ ions of positive charge $q$ and $N$ ions of negative charge $-q$ are constrained
to move in a two dimensional square of side $L$ and area ${\mathsf{A}=L^{2}}$.
The interaction energy of charge $q_{i}$ at position $r_i$
with another charge $q_j$ at position $r_j$ is $-q_iq_j\ln[|r_i-r_j|/a]$,
where $q_i,q_j=\pm q$ and $a$ is a microscopic length scale.
The mass of the ions is $\mass$.
\begin{itemize}
\item [{{(a)}}]
By rescaling space variables to $r_{i}:=r_{i}/L$,
the partition function can be written as $Z(L)=CL^{\alpha}$,
where $C$ does not depend on $L$. Find $\alpha$.
Hint: $\sum_{ij} q_i q_j$ has a very simple dependence on $N$.
\item [{{(b)}}] Calculate the pressure, and show that for $TT_{c}$ and $T