= Energy-temperature curve of a two-dimensional self-gravitating rod gas
{title2=$E=NkT_{\min}[\log(V/V_0)+2\Theta+\Theta^2\log(1-\Theta^{-1})]$}
With $\Theta=T/T_{\min}>1$, the <two-dimensional self-gravitating rod equilibrium> has the displayed energy per unit length for the potential reference $\psi(R_e)=-GM\log(V/V_0)$. Integration of $-\rho\psi/2$ gives its gravitational energy; its two-dimensional <Maxwell-Boltzmann velocity distribution> has kinetic energy $NkT$. At fixed area, differentiation gives $C=Nk[2+2\Theta\log(1-\Theta^{-1})+\Theta/(\Theta-1)]$. Expanding the logarithm shows $C/(Nk)=1+\sum_{j\geq3}(j-2)/(j\Theta^{j-1})>1$. The energy increases continuously from $-\infty$ to $+\infty$, so there is no finite-energy endpoint or turning point of the regular sequence. This excludes the classical <gravothermal catastrophe> in this model, while leaving a distinct canonical collapse at or below its minimum temperature.
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