
\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 squre 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 $-q_iq_j\ln|r_i-r_j|$, 
where $q_i,q_j=\pm q$. 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 $T<T_{c}$ 
the system is unstable. Determine what is $T_{c}$. 
Comment on the reason for this instability. 

\item [{{(c)}}] Determine what is $C$ if the interaction 
can be neglected. Handle the Gibbs factor correctly. 

\item [{{(d)}}] Find the chemical potential $\mu(T,N,L)$, 
and solve for $N(\mu,T,L)$. In particular write what is $N$ for a fixed $\mu$ 
in the limit $L\rightarrow\infty$ for both $T>T_{c}$ and $T<T_{c}$. 

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
