Equipartition theorem
Every quadratic degree of freedom carries an average energy of ½k_BT — the theorem that fixes heat capacities, and whose failures opened the door to quantum mechanics.
Definition
The equipartition theorem states that, in classical thermal equilibrium, each independent quadratic term in a system's energy — each squared velocity or squared displacement coordinate — carries an average energy of exactly (1/2)k_BT. A system with f such quadratic degrees of freedom therefore has mean energy ⟨E⟩ = (f/2)Nk_BT, and a molar heat capacity C_v = (f/2)R.
The theorem accounts for a sweep of heat capacities at a stroke: (3/2)R for a monatomic gas (three translational terms), (5/2)R for a diatomic gas at room temperature (adding two rotational terms), and 3R for a simple solid (three kinetic plus three potential terms — the Dulong–Petit value). It also yields the adiabatic exponent γ = (f+2)/f.
Its failures were historically decisive. Diatomic gases lose their rotational and vibrational contributions at low temperature; solids fall below the Dulong–Petit value beneath the Debye temperature; and applied to the electromagnetic field equipartition predicts the ultraviolet catastrophe. Each failure pointed to energy quantisation, and together they helped force the birth of quantum mechanics.
History
Stated in general form by James Clerk Maxwell (1860) and Ludwig Boltzmann; its breakdowns drove Einstein's 1907 quantum theory of solids, Debye's 1912 refinement, and ultimately Planck's quantum hypothesis of 1900.