At equilibrium, energy does not play favourites. Every square-law way a system can store it — each translation, each spring — is handed the same ration: half a kT. Count the quadratic degrees of freedom and you have the energy. Rendered, not quoted.
source J. C. Maxwell, “On Boltzmann’s Theorem on the average distribution of energy in a system of material points,” Transactions of the Cambridge Philosophical Society, vol. 12 (1879), pp. 547–570 — developed with L. Boltzmann. Scientific Papers, vol. 2 (archive.org) · no single stable article DOI — AMBER.
Write the energy as a sum of independent square-law terms, H = Σ cₓ qₓ². Weight each configuration by Boltzmann, e⁻ᴸᵀᵀ. The generalized equipartition theorem says ⟨q ∂H/∂q⟩ = kT.
For a purely quadratic term c·q² that forces ⟨c q²⟩ = ½kT — the constant c and the mass drop out. The share depends only on the power, not the stiffness.
General law: a term ∝|q|ⁿ carries kT/n. Quadratic (n=2) → ½kT. That is the whole engine.
Energy shared equally among the modes — this is why the-ideal-gas carries &frac32;kT per atom: three translational (quadratic) modes, each ½kT.
Add rotations and vibrations and the same accounting sets every classical heat capacity: Cᵛ = (f/2) N k for f quadratic freedoms. Monatomic → 3/2 R; rigid diatomic → 5/2 R. The counting IS the physics.
A live re-check, independent of the boot selfcheck. It recomputes the mode shares now and confirms the quartic mode reads T/4, the quadratic T/2. Trip the tamper (window 6) and this flips red.
WITNESS · initialising…A temperature T and a list of degrees of freedom, each labelled by the power of its coordinate in the energy. Natural units: k = 1 (Boltzmann constant set to one), so energies read directly in units of T.
Pick a system and a temperature. The panel counts each mode’s power and awards it kT/n live — verified against a direct numerical average over the Boltzmann distribution.
| mode | power n | share kT/n | energy |
|---|
A total mean energy, and the proven law behind it: ½kT per quadratic mode — kT per oscillator, &frac32;kT per monatomic atom, and kT/4 for a quartic (anharmonic) mode.
booting…Equipartition is classical, and it lies at low T. It assumes every mode is continuously excitable — that kT dwarfs the quantum level spacing.
When kT ≪ ħω a mode freezes out: it stops taking its share. Vibrations are frozen at room temperature; solid heat capacities fall to zero as T→0 (Debye), not to 3R. Push equipartition to a field with infinite modes and it predicts the ultraviolet catastrophe. WALL: the classical high-T limit is assumed, not universal.
Disclosed planted void. Press below to mis-assign ½kT to the quartic mode — treating a term ∝x⁴ as if it were ∝x². Its true share is kT/4. The Witness (window 7) recomputes and catches the lie.