The equation of state of two dimensional Yang-Mills theory

We study the pressure, $P$, of SU($N$) gauge theory on a two-dimensional torus as a function of area, $A=l\mathrm{/}t$. We find a cross-over scale that separates the system on a large circle from a system on a small circle at any finite temperature. The cross-over scale approaches zero with increasing $N$ and the cross-over becomes a first order transition as $N\to \mathrm{\infty }$ and $l\to 0$ with the limiting value of $2Pl\mathrm{/}(N-1)t$ depending on the fixed value of $Nl$.