Equipartition and the Virial Theorem

Equipartition assigns a fixed mean energy to each quadratic term in the Hamiltonian, independent of the coefficient in front of it. It is the reason a monatomic gas has $U = \frac{3}{2} N k_{B} T$ , a diatomic gas at room temperature has $\frac{5}{2} N k_{B} T$ , and a classical solid obeys the Dulong–Petit law. The theorem is a one-line consequence of a Gaussian integral, but its scope is broader than the translational kinetic energy: the same argument covers rotational and vibrational modes, the potential energy of an oscillator, and, in generalized form, the virial theorem that links kinetic energy to the forces that confine a system. The places where equipartition fails are as informative as where it holds, and they are all failures of the classical assumption that energy varies continuously.

Equipartition for one quadratic coordinate

Let one of the phase-space variables — call it $x$ , standing for any coordinate or momentum — enter the Hamiltonian through a single quadratic term,

H = a x^{2} + H^{'} (rest),

where $a > 0$ is a constant and $H^{'}$ does not depend on $x$ . The canonical average of the energy stored in that term is a ratio of Gaussian integrals,

⟨ a x^{2} ⟩ = \frac{- \infty \int \infty a x ^{2} e ^{- β a x^{2}} d x}{- \infty \int \infty e ^{- β a x^{2}} d x},

with the $H^{'}$ factors cancelling between numerator and denominator. A compact way to evaluate the ratio differentiates the denominator with respect to $β$ . With $I (β) = \int e^{- β a x^{2}} d x = π / β a$ ,

⟨ a x^{2} ⟩ = - \frac{1}{I} \frac{\partial I}{\partial β} = - \frac{\partial ln I}{\partial β} = - \frac{\partial}{\partial β} (- \frac{1}{2} ln β + const) = \frac{1}{2 β} = \frac{1}{2} k_{B} T .

The coefficient $a$ has dropped out entirely: a stiff mode and a soft mode store the same mean thermal energy. This is the essential content of the theorem.

The Boltzmann weight

e^{- β a x^{2}}

over a quadratic coordinate is a Gaussian; its variance sets

⟨ a x^{2} ⟩ = \frac{1}{2} k_{B} T

regardless of the stiffness

a

. A stiffer mode (narrow curve) and a softer mode (wide curve) store the same mean energy.

Heat capacities from counting quadratic terms

The power of the theorem is that it reduces the internal energy to a count. A monatomic gas has three quadratic kinetic terms per particle, $H_{i} = (p_{i x}^{2} + p_{i y}^{2} + p_{i z}^{2}) /2 m$ , so $f = 3 N$ and $U = \frac{3}{2} N k_{B} T$ , $C_{V} = \frac{3}{2} N k_{B}$ . A rigid diatomic molecule adds two rotational terms, $\frac{1}{2} I ω_{1}^{2} + \frac{1}{2} I ω_{2}^{2}$ , for the two axes perpendicular to the bond, giving $f = 5$ per molecule, $U = \frac{5}{2} N k_{B} T$ , $C_{V} = \frac{5}{2} N k_{B}$ .

A classical solid is the cleanest count. Each atom sits in a three-dimensional harmonic well, contributing three quadratic kinetic terms and three quadratic potential terms,

H_{i} = \frac{p _{i} ^{2}}{2 m} + \frac{1}{2} m ω^{2} r_{i}^{2}, f = 6 per atom .

Then $U = 3 N k_{B} T$ and $C_{V} = 3 N k_{B}$ , the Dulong–Petit law: the molar heat capacity of a monatomic solid is $3 R$ , independent of the material. Each atom carries $\frac{1}{2} k_{B} T$ six times over, three from motion and three from the springs that hold it in place.

Equipartition as a ledger of quadratic terms, each worth

\frac{1}{2} k_{B} T

. A monatomic gas has three (translation); a rigid diatomic adds two (rotation); a classical solid has six per atom (three kinetic, three potential), giving Dulong–Petit

C_{V} = 3 N k_{B}

The generalized equipartition theorem

The single-coordinate result generalizes to a statement about any phase-space variable and any derivative of the Hamiltonian. Let $x_{i}$ be one of the $6 N$ canonical coordinates ${q, p}$ , and consider the average $⟨ x_{i} \partial H / \partial x_{j} ⟩$ over the canonical distribution $ρ \propto e^{- β H}$ . Writing the average as a phase-space integral and integrating by parts in $x_{j}$ ,

⟨ x_{i} \frac{\partial H}{\partial x _{j}} ⟩ = \frac{\int x _{i} \frac{\partial H}{\partial x _{j}} e ^{- β H} d Γ}{\int e ^{- β H} d Γ} = - \frac{1}{β} \frac{\int x _{i} \frac{\partial}{\partial x _{j}} e ^{- β H} d Γ}{\int e ^{- β H} d Γ} .

Integrating the numerator by parts, the boundary term vanishes because $e^{- β H} \to 0$ as any momentum grows without bound and at the walls of the container, leaving $\int (\partial x_{i} / \partial x_{j}) e^{- β H} d Γ = δ_{ij} \int e^{- β H} d Γ$ . Hence the generalized equipartition theorem,

⟨ x_{i} \frac{\partial H}{\partial x _{j}} ⟩ = k_{B} T δ_{ij} .

Two specializations recover the familiar results. Taking $x_{i} = x_{j} = p$ for a momentum that appears as $p^{2} /2 m$ gives $⟨ p \partial H / \partial p ⟩ = ⟨ p^{2} / m ⟩ = k_{B} T$ , so $⟨ p^{2} /2 m ⟩ = \frac{1}{2} k_{B} T$ , the kinetic form. Taking $x_{i} = x_{j} = q$ for a coordinate in a potential term $\frac{1}{2} κ q^{2}$ gives $⟨ q \partial H / \partial q ⟩ = ⟨ κ q^{2} ⟩ = k_{B} T$ , so the potential energy $⟨ \frac{1}{2} κ q^{2} ⟩ = \frac{1}{2} k_{B} T$ . The harmonic oscillator therefore holds $k_{B} T$ in total, split evenly between kinetic and potential energy — the double count that gives the solid its factor of six.

The virial theorem

Summing the generalized theorem over the coordinate part yields the classical virial theorem. For $N$ particles with positions $r_{i}$ and the total force $F_{i} = - \partial H / \partial r_{i}$ on each, set $x_{i} = q$ and $x_{j} = q$ for each Cartesian position component and add. The diagonal generalized-equipartition identity $⟨ r_{i} \cdot \partial H / \partial r_{i} ⟩ = 3 k_{B} T$ per particle becomes, after summing over the $N$ particles,

⟨ i \sum r_{i} \cdot F_{i} ⟩ = - 3 N k_{B} T .

The quantity $V = \frac{1}{2} \sum_{i} r_{i} \cdot F_{i}$ is the virial of Clausius, and the theorem relates its average to the temperature. For an ideal gas the only forces are the wall reactions confining the particles, and evaluating the wall virial reproduces $P V = N k_{B} T$ ; the same relation, extended to interparticle forces, generates the virial expansion of a non-ideal gas.

The virial theorem splits the confining forces on the particles into the wall reaction (ideal-gas pressure) and the interparticle forces. For an ideal gas only the wall term survives and

⟨ \sum r_{i} \cdot F_{i} ⟩ = - 3 N k_{B} T

reproduces

P V = N k_{B} T

Quantum freeze-out

Equipartition is a classical theorem, and its assumption is that the energy of a mode varies continuously so that the Gaussian integral is over an unrestricted real variable. Real modes are quantized, with a level spacing $Δ ε$ . When $k_{B} T ≫ Δ ε$ the levels are effectively continuous and equipartition holds; when $k_{B} T ≪ Δ ε$ the mode cannot be excited beyond its ground state, its mean energy saturates, and its contribution to the heat capacity vanishes. The mode is frozen out.

The vibrational mode of a diatomic molecule is the standard example. Its level spacing $ℏ ω$ corresponds to a temperature $θ_{vib} = ℏ ω / k_{B}$ of order $1 0^{3}$ K, far above room temperature, so vibration is frozen and does not contribute the $k_{B} T$ that equipartition would assign. Rotation, with a much smaller spacing $θ_{rot}$ of order $10$ K, is active at room temperature but freezes below it. The heat capacity therefore rises in steps as temperature crosses each characteristic scale, and only in the high-temperature plateau of every mode does the full equipartition count apply.

A quantized mode with level spacing

Δ ε

. For

k_{B} T ≫ Δ ε

many levels are populated, the spectrum is effectively continuous, and equipartition gives

\frac{1}{2} k_{B} T

per quadratic term. For

k_{B} T ≪ Δ ε

only the ground level is occupied and the mode is frozen, contributing nothing to

C_{V}

The relativistic gas as a contrast

Equipartition depends on the energy being quadratic in the phase-space variable, so a mode whose energy is not quadratic carries a different mean energy. The ultrarelativistic gas is the sharp case: a particle with $ε = c ∣ p ∣$ has energy linear in the momentum magnitude. The generalized theorem still applies with $⟨ p \partial H / \partial p ⟩ = k_{B} T$ per momentum, but now $p \partial ε / \partial p = p \cdot c = ε$ for each of the three Cartesian directions gives, on summing, $⟨ ε ⟩ = 3 k_{B} T$ per particle rather than $\frac{3}{2} k_{B} T$ . A homogeneous energy of degree $s$ in the momentum, $ε \propto p^{s}$ , gives $⟨ ε ⟩ = (3/ s) k_{B} T$ ; the nonrelativistic quadratic case is $s = 2$ and the ultrarelativistic linear case is $s = 1$ . The mean energy per particle, and with it the equation of state relating pressure to energy density, tracks the degree of the dispersion relation, not the mere count of momentum components.

Summary

Each quadratic term in the classical Hamiltonian contributes $\frac{1}{2} k_{B} T$ to the mean energy, independent of its coefficient: a Gaussian-integral fact.
Counting quadratic terms gives $U = \frac{1}{2} f k_{B} T$ and $C_{V} = \frac{1}{2} f k_{B}$ : $f = 3$ per monatomic particle, $5$ for a rigid diatomic, $6$ per atom in a classical solid (Dulong–Petit, $C_{V} = 3 N k_{B}$ ).
The generalized theorem $⟨ x_{i} \partial H / \partial x_{j} ⟩ = k_{B} T δ_{ij}$ contains both the kinetic and potential forms; summed over positions it gives the virial theorem $⟨ \sum r_{i} \cdot F_{i} ⟩ = - 3 N k_{B} T$ , which returns $P V = N k_{B} T$ for an ideal gas.
Equipartition fails by quantum freeze-out when a level gap exceeds $k_{B} T$ : the mode saturates and drops out of $C_{V}$ , producing the temperature staircase of the molecular heat capacity.
For a dispersion $ε \propto p^{s}$ the mean energy is $(3/ s) k_{B} T$ per particle; the nonrelativistic gas is $s = 2$ , the ultrarelativistic gas $s = 1$ with $⟨ ε ⟩ = 3 k_{B} T$ .