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Donald Lynden-Bell investigated the unusual thermodynamics of self-gravitating systems. A bound system can become hotter while losing total energy: contraction makes its gravitational energy more negative while increasing typical particle speeds. This negative heat capacity is compatible with a suitable isolated-system description, even though equilibrium heat capacity in a canonical ensemble is nonnegative. His account connects this ensemble distinction with gravitational thermal runaway and the stability of star clusters.^[ref]

For an equilibrium entropy $S(E,V,N)$ that is concave in energy at fixed volume and particle number, the thermodynamic relation $(\partial S/\partial E{)}_{V,N}=1/T$ gives, wherever the derivatives are finite and invertible,Strict concavity therefore implies positive heat capacity at constant volume. A straight coexistence segment can instead have constant temperature over an energy interval; finite strictly positive heat-capacity conclusions do not apply there.

There is also a direct canonical ensemble statement for a temperature-independent Hamiltonian operator:The numerator is the canonical energy variance. Positivity follows from a variance, without assuming an ideal gas or noninteracting particles.

The concavity hypothesis is essential. Isolated systems with long-range gravitational interactions can have negative heat capacity in a microcanonical ensemble; the familiar virialized-gas example does not belong to a stable canonical branch in contact with an infinite heat bath. A claim that every stable equilibrium has strictly positive heat capacity would erase this ensemble distinction.

A decrease in equilibrium temperature when the total energy increases along a specified family of equilibria. A virialized nonrelativistic self-gravitating ideal gas has $E=-U$ and $U$ proportional to its particle-weighted mean temperature, producing this behavior. It does not mean the local gas parcel has negative heat capacity at fixed volume. It occurs naturally for isolated gravitational systems rather than stable ordinary canonical equilibrium with an infinite heat bath.

For a fully ionized, nonrelativistic monatomic ideal gas, $\gamma =5/3$ and the virial theorem becomesSince $U$ increases with its particle-weighted mean temperature, the equilibrated self-gravitating object has negative heat capacity: adding total energy allows expansion, makes $\Omega$ less negative, and reduces $U$. Conversely, radiating energy permits contraction and raises the internal temperature. This refers to a sequence of gravitational equilibria, not to the positive local heat capacity of a parcel at fixed volume.

In an ordinary nondegenerate nuclear-burning star, a rise in nuclear heating tends to expand and cool the burning region, lowering its temperature-sensitive reaction rate. Reduced heating leads to contraction and reheating. This expansion feedback is the nuclear thermostat; it can fail in degenerate matter where pressure responds weakly to temperature. During contraction of a pre-main-sequence star, gravitational binding supplies the radiated luminosity and heats the interior. For a fixed-structure scaling, half the released gravitational energy increases $U$ and half escapes as radiation, until hydrogen burning can support a long-lived main sequence equilibrium.