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\heading{E5969: 2D Coulomb gas} \auname{Michael Epshtein}
\textbf{The problem:}\\
\begin{flushleft}
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 $L^{2}.$ the
interaction energy of charge $q_{i}$ at position r$_{i}$ with another
charge $q_{j}$ at position r$_{j}$ is $-q_{_{i}}q_{_{j}}\; ln\left|r_{i}-r_{j}\right|$
where $q_{_{i}},q_{_{j}}=\pm q$ . mass of the ions is $m$.
\par\end{flushleft}
\Dn
\begin{description}
\item [{{(a)}}] By rescaling space variables to $r_{i}=\frac{1}{L}r_{i}$,
The partition function can be written as $Z(L)=CL^{\alpha}$ find
$\alpha$ where C dont depend on L. Note that: $\underset{iT_{c}$ and $T0$
\par\end{flushleft}
\[
\left(\cfrac{\partial P}{\partial L^{2}}\right)_{T,N}=\frac{N}{L^{2}}(\frac{1}{2}q^{2}-2\tau)>0\]
\begin{flushleft}
we define low $T$ by $k_{B}TT_{c}\;,\;\; N\rightarrow\infty$ the system acting as ideal gas
$PV=NTk_{B}$
\par\end{flushleft}
\begin{flushleft}
$T