What is the name of the theorem that states that energy is equally distributed among degrees of freedom?

The theorem stating that energy is equally distributed among degrees of freedom is the equipartition theorem.

In classical statistical mechanics, the equipartition theorem says that, at thermal equilibrium, each independent quadratic degree of freedom contributes an average energy of one-half kBT, where kB is the Boltzmann constant and T is absolute temperature. For a monatomic ideal gas, the three translational degrees of freedom therefore contribute an average total kinetic energy of three-halves kBT per particle.

The theorem connects microscopic motion with macroscopic quantities such as heat capacity. It also helps explain the classical ideal-gas law and the Dulong–Petit prediction for solid heat capacities. The phrase “equally distributed” refers to average energy contributions, not identical energy in every particle at every instant.

Equipartition is a classical result and fails when quantum energy-level spacings are large compared with thermal energy. Degrees of freedom can then become “frozen out,” especially at low temperatures. This limitation helped reveal problems in classical physics, including the ultraviolet catastrophe, and contributed to the development of quantum theory.