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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 $C/(Nk)=1+{\sum }_{j\ge 3}(j-2)/(j{\Theta }^{j-1})>1$. 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.

For an isolated, virialized three-dimensional Newtonian system with negligible boundary-pressure terms, the virial theorem gives $2K+W=0$. Hence its total energy is $E=K+W=-K$. Defining the kinetic temperature by $K=3NkT/2$ givesThus losing energy makes the system hotter, while adding energy makes it cooler. This negative heat capacity of a virialized gravitational system uses the inverse-distance potential and the stated boundary assumptions; it is not a dimension-independent consequence of self-gravity.

A gravothermal catastrophe occurs when a stellar core loses heat to its halo through two-body relaxation, contracts and becomes hotter because of its negative heat capacity. The larger temperature gradient then drives more heat outward and accelerates contraction. In the confined three-dimensional isothermal sequence this runaway is associated with loss of a regular thermodynamic equilibrium at a finite total energy. It is different from a fast collisionless instability.

For the rods, use energies and masses per unit axial length, consistently with their logarithmic interaction. DefineIntegrating the two-dimensional Maxwell-Boltzmann velocity distribution gives $\rho =A{e}^{\beta \psi }$, with a positive normalization $A$. The proposed potential has $\psi ={\psi }_e-2\log F/\beta$, so this requires $\rho \propto F^{-2}$. Direct differentiation givesConsequently the Poisson equation for Newtonian gravity is satisfied precisely byThe free normalization of the velocity distribution can be chosen to give this density. Its enclosed line mass verifies total-mass normalization:At the wall $F(1)=1$, so the required potential value is also satisfied. This establishes the two-dimensional self-gravitating rod equilibrium, including its field equation, distribution law, mass and boundary condition.

A regular positive solution needs $F(0)=1-q>0$. Since $\beta =m/(kT)$,The minimum is an infimum of regular equilibrium temperatures. At equality the central logarithm is singular; for $T<T_{\min }$, $F$ vanishes inside the box. Thus equality is not a regular isothermal rod configuration.

Each of the two velocity components has second moment $1/\beta$, so the kinetic energy is $K=M/\beta =NkT$. With the given reference potential, the gravitational potential energy is $W=-\frac{1}{2}\int \rho \psi d^{2}R$. Using $dM=M(1-q)dx/F^2$ givesThe remaining integral can be evaluated explicitly:Adding $K$ therefore yieldsNow $Nk{T}_{\min }=G{M}^2/2$, $\Theta =T/T_{\min }=1/q$ and $M/\beta =Nk{T}_{\min }\Theta$. The energy-temperature curve of a two-dimensional self-gravitating rod gas is consequentlyAt fixed box area, differentiation givesTo establish its sign rather than infer it from a plot, expand the logarithm for $\Theta >1$:The series and its derivative converge locally uniformly in $\Theta >1$. Moreover $E\to -\infty$ as $\Theta \downarrow 1$ and $E\to +\infty$ as $\Theta \to \infty$. Thus every finite energy on this mean-field isothermal sequence has a unique regular equilibrium temperature, and there is no finite-energy termination or turning point. The central density $M/[V(1-q)]$ diverges only in the infinite-negative-energy limit.

There is no finite-energy gravothermal catastrophe in this two-dimensional rod model. Its logarithmic interaction gives a different caloric curve from the three-dimensional inverse-distance problem. This microcanonical conclusion does not prohibit a distinct fixed-temperature collapse: in the canonical ensemble, a bath at or below $T_{\min }$ admits no regular equilibrium of this family.