⟨2K⟩ + ⟨V⟩ = 0 at virial equilibrium — statistical mechanics of self-gravitating systems
Clausius (1870): for a stable bound gravitational system in equilibrium,
2⟨K⟩ + ⟨V⟩ = 0
so ⟨E⟩ = ⟨K⟩ + ⟨V⟩ = −⟨K⟩ < 0. Adding energy (heating) makes the system expand and cool — negative heat capacity! Used to estimate cluster masses from observed velocities (dark matter detection), stellar stability (white dwarfs, neutron stars).