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\sect{2D coulomb gas}

${N}$ ions of positive charge ${q}$, and ${N}$ ions with negative
charge ${-q}$ are constrained to move in a two dimensional square of
side ${L}$. The interaction energy of charge ${qi}$ at position
${ri}$ with another charge ${qj}$ at ${rj}$ is� ${-qiqj ln|ri-rj|}$
where ${qi,qj=�q}$. Prove that� ${Z\left(\beta ,L\right) =
L^{\left(N(4-\beta q^{2}\right)} f(\beta)}$.

Estimate ${f\left(\beta\right)}$ for the case ${N=1}$, and explain
what happens if� ${\left(\frac{1}{\beta}\right) < \frac{q^{2}}{2}}$.
Discuss now the case ${N>>1}$.� Explain what happens if
${\left(\frac{1}{\beta}\right)< \frac{q^{2}}{4}}$.

Hint: The partition function is in general a monotonic increasing
function of the volume. It follows, for this particular model
system, that� ${f\left(\beta \right)=\infty}$.at low temperatures.
The ${N=1}$ case can be used in order to illuminate the reason for
this divergence. Explain what is the additional ingredient that is
required in order to stabilize the physics of this model.



