Classic thermodynamic stability theory states that heat capacities have to be positive [1,2]. For a system in thermal contact with a larger system, the second order virtual variation in entropy around an equilibrium state is \[ \delta^2 S = - \frac{C_V (\delta T)^2}{T^2} \lt 0 , \] with heat capacity at constant volume \[ C_V = \left( \frac{\partial E}{\partial T} \right)_V , \] and the negative sign required by stability implying that \( C_V \gt 0 \). Recall that the quantities used in thermodynamics are averages over microscopic realizations. For instance, the thermodynamic internal energy is given by \( E = \langle E^\mathrm{micr} \rangle \).

Statistical mechanics introduces a different proof of the positivity of heat capacities [3]. It starts from the expression for the energy average over a canonical ensemble \[ E = \langle E^\mathrm{micr} \rangle = \frac{\sum_i E_i^\mathrm{micr} \exp(-E_i^\mathrm{micr}/k_\mathrm{B}T)}{\sum_i \exp(-E_i^\mathrm{micr}/k_\mathrm{B}T)} , \] and gets \[ \frac{\diff E}{\diff T} = \frac{1}{k_\mathrm{B}T^2} \bigg\langle \big(E_i^\mathrm{micr} - \langle E^\mathrm{micr} \rangle \big)^2 \bigg\rangle , \] which is clearly a positive quantity. Notice that the canonical ensemble does not include the volume as variable, and the above total derivative is equivalent to the partial derivative that defines \( C_V \).

Now, astronomers claim that the above proofs are wrong and that heat capacity can be negative in 'exotic' systems [3]. Their starting point is the virial theorem that relates kinetic energy \( K = \langle K^\mathrm{micr} \rangle \) and potential energy \( \Phi = \langle \Phi^\mathrm{micr} \rangle \) for an isolated gravitational system with energy \( E = K + \Phi \) \[ 2 K + \Phi = 0 . \] Using \( K = \frac{3}{2} N k_\mathrm{B} T \) yields \[ \frac{\diff E}{\diff T} = - \frac{\diff K}{\diff T} = - \frac{3}{2} N k_\mathrm{B} . \] Their 'proof' concludes by associating \( C_V \) to this negative quantity. Astronomers and nuclear physicists not only pretend that negative heat capacities are theoretically admissible, but they also claim those capacities are routinely measured in gravitational systems and nuclear clusters [3], by plotting \( E \) vs \( T \) and then finding regions where energy decreases when temperature increases; i.e., finding regions where \( (\diff E /\diff T) \lt 0 \). A more accurate description of their claim is

when heat is absorbed by a star, or star cluster,I bolded the relevant part that disproves their claims. The problem is that the quantity \( (\diff E/\diff T) \) that they are measuring is not \( C_V \) because volume is not held constant during differentiation. If we write a detailed model of the energy we find that the potential energy depends on volume through the interparticle distances \[ E(T,V) = K(T) + \Phi(V) = \frac{3}{2} N k_\mathrm{B} T + \Phi(V) . \] Now, using the definition (2), we find \[ C_V = \left( \frac{\partial E(T,V)}{\partial T} \right)_V = \frac{\diff K(T)}{\diff T} = \frac{3}{2} N k_\mathrm{B} \gt 0 . \] Therefore, what astronomers and others are really measuring is not the heat capacity \( C_V \) but the abstract quantity \[ \frac{\diff E}{\diff T} = C_V + \frac{\Phi(V(T))}{\diff T} . \] This quantity can be positive, negative, or zero depending of the nature of the interactions. For systems satisfying the virial theorem this quantity is justit will expandand cool down.

*minus*the heat capacity \( (\diff E/\diff T) = - C_V \).

#### References

**[1]**Modern Thermodynamics

**1998:**

*John Wiley & Sons Ltd.; Chichester.*Kondepudi, D. K.; Prigogine, I.

**[2]**Critical analysis of negative heat capacity in nanoclusters

**2007:**

*EPL, 79, 43001.*Michaelian K.; SantamarĂa-Holek I.

**[3]**Negative Specific Heat in Astronomy, Physics and Chemistry

**1998:**

*arXiv:cond-mat/9812172 v1.*Lynden-Bell, D.