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 \( C_V \).

Part of the nanocluster community claims that small systems are not bound by the above macroscopic proofs, and that negative heat capacities exist at small scales. Michaelian and SantamarĂa-Holek show that these incorrect results (both experimental and theoretical) derive from two basic problems: (

**i**) the system is non-ergodic or (

**ii**) the model used to represent the system does not obey quantum laws [2].

On the other side of the size spectrum, very large systems, we find to astronomers claiming that the above macroscopic proofs are wrong and that heat capacity can be negative in gravitational 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 the astronomers' 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 \( \Phi \) 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 \[ \frac{\diff E}{\diff T} = C_V + \frac{\diff\Phi(V(T))}{\diff T} . \] The sign of this abstract quantity depends of the nature of the interactions. For systems satisfying the virial theorem this quantity is justit will expandand cool down.

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

#### Acknowledgement

I thank Prof. Karo Michaelian for further remarks.

#### 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.