Equipartition Theorem
The equipartition theorem states that, for a classical system in thermal equilibrium at temperature T, each independent quadratic term in the energy, such as (1/2)mvx² or (1/2)kx², has an average value of (1/2)kBT. It predicts heat capacities of gases and solids and the size of thermal fluctuations, but fails when quantum energy-level spacing is comparable to or larger than kBT.
Curious
A gas molecule can move left-right, up-down and forward-back, and a molecule like nitrogen can also tumble about two axes. The equipartition theorem says each of these motions holds, on average, the same amount of energy, (1/2)kBT. Count the motions and you can predict how much energy a gas stores and how much heat it needs to warm up (see Internal Energy and Specific Heat Capacity).
The theorem also explains why small particles jitter: a pollen grain in water receives (1/2)kBT of kinetic energy per direction just like a water molecule, so it moves randomly, the Brownian motion that Einstein used in 1905 to prove atoms exist.
Exploring
| System | Quadratic terms | Average energy (classical) |
|---|---|---|
| Monatomic gas atom | 3 (translation) | (3/2)kBT |
| Diatomic molecule (room T) | 3 translation + 2 rotation | (5/2)kBT |
| 1D harmonic oscillator | 1 kinetic + 1 potential | kBT |
| Atom in a solid (3D) | 3 kinetic + 3 potential | 3kBT (Dulong–Petit) |
Deep Dive
Derivation. For an energy term E = Aq², the Boltzmann-weighted average ⟨Aq²⟩ = ∫Aq² e−Aq²/kBTdq / ∫e−Aq²/kBTdq = kBT/2, independent of A. A generalised form states ⟨qi∂H/∂qj⟩ = δijkBT, which also gives the virial theorem used in astrophysics (see Galaxy Clusters).
Failure and quantum theory. Equipartition predicted heat capacities of 7/2 R for diatomic gases (including vibration), but measurements gave 5/2 R at room temperature; for hydrogen, rotation also freezes out below about 100 K (rotational temperature 85 K) and vibration only becomes active above a few thousand kelvin (6,330 K). Applied to the infinite number of electromagnetic modes in a cavity, it predicted infinite energy, the ultraviolet catastrophe (see Thermal Radiation). These failures, which Kelvin listed among his “clouds” of 1900, were resolved by quantisation of energy (see Quantum Harmonic Oscillator): a mode of frequency ν is excited only when kBT is comparable to hν.
Thermal noise. Equipartition sets the size of thermal fluctuations in any mode: a capacitor holds ⟨q²⟩/2C = kBT/2, giving Johnson–Nyquist noise in resistors, and suspended mirrors and test masses jitter by √(kBT/k), one of the noise sources that gravitational-wave detectors must suppress (see LIGO and Virgo).
Concept Relationship Map
Requires: kinetic theory, Maxwell-Boltzmann distribution
Enables: heat capacities, Brownian motion, thermal noise
Commonly confused with: counting coordinates instead of quadratic terms
Sources
- Boltzmann, L. (1876). Wiener Berichte, 74, 553–560.
- Einstein, A. (1905). “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen.” Annalen der Physik, 17, 549–560.
- Li, T., Kheifets, S., Medellin, D., Raizen, M.G. (2010). “Measurement of the Instantaneous Velocity of a Brownian Particle.” Science, 328, 1673–1675.