Generalized Equipartition Theorem

The familiar equipartition rule — each quadratic term in the Hamiltonian carries $\tfrac12 k_BT$ — is only the shadow of a far more powerful statement. For any classical system in thermal equilibrium, described by phase-space coordinates $x=(q_1,\dots,q_{3N},p_1,\dots,p_{3N})$ and Hamiltonian $H(x)$ , the canonical average obeys a single clean identity: $\langle x_i\,\partial H/\partial x_j\rangle=\delta_{ij}\,k_BT$ . One equation, and out of it fall the heat capacity of a gas, the virial theorem, the pressure of an interacting fluid, and the corrections to ideal-gas behaviour. It is the master key of classical statistical mechanics. The proof is a one-line integration by parts. Because the Boltzmann weight $e^{-\beta H}$ vanishes wherever $H\to\infty$ — the physical requirement that the energy confines the system — the boundary terms drop out and each derivative of $H$ turns into a factor of $k_BT$ . Nothing in this argument requires $H$ to be quadratic, the coordinates to be Cartesian, or the particles to be free. That robustness is exactly why the theorem reaches so far: it handles anharmonic springs ( $bq^4$ gives $\tfrac14 k_BT$ ), relativistic energies ( $c|\mathbf p|$ gives $k_BT$ per direction), and the forces between interacting molecules on the same footing. Two diagonal specializations dominate the applications. Set $x_i=x_j=p_k$ , a momentum, and you recover kinetic equipartition and the heat capacities of gases and solids. Set $x_i=x_j=q_k$ , a coordinate, sum over particles, and you get the Clausius virial and the equation of state. And because the identity is purely classical, its failures are equally informative: whenever a quantum level spacing rivals $k_BT$ , the continuous phase-space integral is illegal and equipartition breaks — vibrations freeze out, the Dulong-Petit law collapses, and the theorem's boundary is the boundary of classical physics itself.

Key takeaways

The generalized equipartition theorem states that for a classical system with Hamiltonian $H$ , the canonical average $\langle x_i\,\partial H/\partial x_j\rangle=\delta_{ij}\,k_BT$ for any phase-space coordinates $x_i,x_j$ .
It follows from one integration by parts, using only that $e^{-\beta H}\to0$ at the boundaries of phase space.
Setting $x_i=x_j=q$ or $p$ recovers $\tfrac12 k_BT$ per quadratic degree of freedom; the off-diagonal ( $i\ne j$ ) average vanishes.
For an energy homogeneous of degree $n$ in a variable, the theorem gives $\langle H_{\text{term}}\rangle=k_BT/n$ — quadratic $\tfrac12 k_BT$ , anharmonic quartic $\tfrac14 k_BT$ , linear (relativistic) $k_BT$ .
The momentum-diagonal case gives heat capacities; the coordinate-diagonal case gives the virial theorem and the virial equation of state.
It is intrinsically classical: quantization (level spacing $\gtrsim k_BT$ ) freezes degrees of freedom and breaks the theorem.
Key idea: every appearance of equipartition — the $\tfrac12 k_BT$ per mode, the kinetic energy of a gas, the virial of forces, the pressure of an interacting fluid — is the single identity $\langle x_i\,\partial H/\partial x_j\rangle=\delta_{ij}k_BT$ , which is nothing more than an integration by parts on $e^{-\beta H}$ .

The core insight

Every appearance of equipartition — the $\tfrac12 k_BT$ per mode, the kinetic energy of a gas, the virial of forces, the pressure of an interacting fluid — is the single phase-space identity $\langle x_i\,\partial H/\partial x_j\rangle=\delta_{ij}k_BT$ , which is just an integration by parts on the Boltzmann factor $e^{-\beta H}$ .

Where this lives today

The generalized theorem is a working instrument, not a museum piece. In molecular-dynamics and Monte-Carlo simulations it is how "temperature" is defined and checked: a thermostat measures $\langle p_i\,\partial H/\partial p_i\rangle$ over the trajectory and adjusts the dynamics until it equals $k_BT$ for every degree of freedom — the equipartition test is the standard sanity check that a simulation has equilibrated. In single-molecule biophysics, the coordinate-diagonal case calibrates force probes: $\tfrac12\kappa\langle x^2\rangle=\tfrac12 k_BT$ turns a measured positional variance into a trap stiffness for optical tweezers and AFM cantilevers, and the anharmonic $k_BT/n$ generalization corrects the calibration when the trap is not perfectly harmonic. In astrophysics, the degree-1 relativistic case sets the specific heat of degenerate and radiation-dominated matter in stellar interiors and the early universe. And the theorem's coordinate branch, summed into the virial, remains the fastest route from a microscopic pair potential to the equation of state of a real gas or liquid.

The physical picture

Picture the canonical probability cloud $\rho\propto e^{-\beta H}$ draped over phase space. Wherever $H$ is small the cloud is thick; wherever $H$ grows large the cloud thins to nothing. The generalized equipartition theorem is a statement about how this cloud is shaped: it says that the average of $x_i$ times the local slope $\partial H/\partial x_j$ of the energy is exactly $\delta_{ij}k_BT$ . When you probe a coordinate against its own energy-slope ( $i=j$ ) you always get $k_BT$ ; when you probe one coordinate against another's slope ( $i\ne j$ ) the independent Gaussian wings cancel and you get zero. The reason the coefficient of a mode never matters is geometric. A stiff parabola confines the cloud to a narrow sliver; a soft one lets it spread wide. A stiff mode stores more energy per unit displacement but explores less displacement, and the two effects cancel to the last decimal. This is the same "energy democracy" that gives $\tfrac12 k_BT$ per quadratic term — but now stated so generally that it survives anharmonic and relativistic energies, where the share becomes $k_BT/n$ for a term homogeneous of degree $n$ .

Figure

The Boltzmann cloud rho ∝ exp(−βH) over a stiff parabola (narrow) and a soft one (wide). Both give the same ⟨x ∂H/∂x⟩ = kT — a stiff mode stores more per displacement but explores less, and the two cancel.

The second figure is the punchline: the share of thermal energy per mode depends only on the degree of the energy in that variable, not on any coefficient. A quadratic term ( $n=2$ ) gets $\tfrac12 k_BT$ ; a linear relativistic energy ( $n=1$ ) gets the full $k_BT$ ; an anharmonic quartic ( $n=4$ ) gets a stingy $\tfrac14 k_BT$ . The rule $\langle H_{\text{term}}\rangle=k_BT/n$ is the theorem's most compact fingerprint, and it puts the classical harmonic oscillator, the photon-like ultrarelativistic gas, and the quartic anharmonic trap on one curve.

Figure

Punchline: thermal energy per mode is kT/n for an energy homogeneous of degree n. Linear (n=1) gets kT, quadratic (n=2) gets ½kT, quartic (n=4) gets ¼kT. The coefficient never enters — only the degree.

The core equations

The heart of the article is a single phase-space average. It is stated for the canonical ensemble, but by ensemble equivalence it holds microcanonically too in the thermodynamic limit. The claim is that pairing any coordinate $x_i$ with the energy-gradient $\partial H/\partial x_j$ gives a strikingly simple answer.

\left\langle x_i\,\frac{\partial H}{\partial x_j}\right\rangle = \delta_{ij}\,k_BT \qquad (1)

where: $x_i,x_j$ — any of the $6N$ canonical phase-space variables (coordinates $q$ or momenta $p$ ); $\partial H/\partial x_j$ — the energy gradient, which is $\dot p_j$ or $-\dot q_j$ by Hamilton's equations; $\delta_{ij}$ — the Kronecker delta, $1$ if $i=j$ else $0$ ; $\langle\cdot\rangle$ — the equilibrium thermal average. Reading it: a coordinate probed against its own energy-slope always returns $k_BT$ ; against any other slope it returns zero, because the corresponding phase-space integrals factorize into independent, mean-symmetric pieces. Specializing the diagonal case to a single term homogeneous of degree $n$ gives the compact energy-share rule.

\left\langle H_{\text{term}}\right\rangle = \frac{k_BT}{n}\qquad\text{for }H_{\text{term}}\propto x^{\,n} \qquad (2)

where: $n$ — the degree of homogeneity of that energy term in the variable $x$ . Quadratic ( $n=2$ ) gives $\tfrac12 k_BT$ ; the linear ultrarelativistic energy $c|\mathbf p|$ ( $n=1$ ) gives $k_BT$ per component; an anharmonic quartic $bq^4$ ( $n=4$ ) gives $\tfrac14 k_BT$ . This follows from equation (1) because $x\,\partial(x^n)/\partial x=n\,x^n$ , so $\langle n\,H_{\text{term}}\rangle=k_BT$ . Summed over all $f$ quadratic degrees of freedom, equation (2) gives the total energy and the classical heat capacity — the working payoff for gases and solids.

\langle H\rangle=\frac f2 k_BT,\qquad C_V=\frac f2 k_B\ (\text{per molecule})=\frac f2 R\ (\text{per mole}) \qquad (3)

where: $f$ — the number of active quadratic degrees of freedom (each momentum and each harmonic coordinate counts once); $R=N_Ak_B=8.314\,\mathrm{J\,mol^{-1}K^{-1}}$ . Reading it: monatomic gas $f=3$ gives $C_V=\tfrac32 R$ ; a solid ( $3N$ oscillators, each two quadratic terms) gives $3R$ (Dulong-Petit). Finally, summing the coordinate-diagonal case over all particles produces the statistical virial theorem — the bridge to the equation of state.

\left\langle\sum_{k}\mathbf r_k\cdot\mathbf F_k\right\rangle = -3Nk_BT \qquad (4)

where: $\mathbf F_k=-\nabla_k U$ is the total force on particle $k$ (walls plus interactions), and the left side is the Clausius virial. Splitting $\mathbf F_k$ into a wall part (which yields $-3PV$ ) and an interaction part gives $PV=Nk_BT+(\text{interaction virial})$ — the seed of the virial equation of state.

Derivation AThe theorem from one integration by parts

The goal is to prove the master identity $\langle x_i\,\partial H/\partial x_j\rangle=\delta_{ij}k_BT$ for an arbitrary confining Hamiltonian, using nothing but the canonical average and integration by parts. The strategy is to write the average as a phase-space integral, rewrite the factor $(\partial H/\partial x_j)\,e^{-\beta H}$ as a derivative of the Boltzmann factor, integrate by parts, and use that $e^{-\beta H}$ vanishes on the boundaries so the surface term dies. What remains is $\partial x_i/\partial x_j=\delta_{ij}$ , delivering the result with no assumption that $H$ be quadratic.

Step 1. Write the canonical average $\left\langle x_i\dfrac{\partial H}{\partial x_j}\right\rangle=\dfrac1Z\displaystyle\int x_i\,\dfrac{\partial H}{\partial x_j}\,e^{-\beta H}\,d\Gamma$ , where $d\Gamma$ is the phase-space volume element and $Z=\int e^{-\beta H}d\Gamma$ the partition function.

Step 2. Notice that $\dfrac{\partial H}{\partial x_j}\,e^{-\beta H}=-\dfrac1\beta\dfrac{\partial}{\partial x_j}\,e^{-\beta H}$ . This is the key move — the energy gradient is (up to $-1/\beta$ ) the gradient of the Boltzmann factor itself.

Step 3. Substitute and integrate by parts in $x_j$ : $\left\langle x_i\dfrac{\partial H}{\partial x_j}\right\rangle=-\dfrac{1}{\beta Z}\displaystyle\int x_i\,\dfrac{\partial}{\partial x_j}e^{-\beta H}\,d\Gamma=\dfrac{1}{\beta Z}\displaystyle\int\dfrac{\partial x_i}{\partial x_j}\,e^{-\beta H}\,d\Gamma-\dfrac{1}{\beta Z}\Big[x_i e^{-\beta H}\Big]_{\text{boundary}}$ . Why the boundary term vanishes. At the edges of phase space $H\to\infty$ (a confining potential, or $|p|\to\infty$ for kinetic energy), so $e^{-\beta H}\to0$ faster than $x_i$ grows. The surface term is exactly zero — this is the sole physical assumption, and it is what "confined system" means.

Step 4. Use $\partial x_i/\partial x_j=\delta_{ij}$ and $\int e^{-\beta H}d\Gamma=Z$ : the remaining integral is $\dfrac{\delta_{ij}}{\beta Z}\displaystyle\int e^{-\beta H}d\Gamma=\dfrac{\delta_{ij}}{\beta}=\delta_{ij}k_BT$ , which is equation (1). Nothing here assumed $H$ quadratic. Verified: the diagonal result $\langle q\,\partial H/\partial q\rangle=1/\beta=k_BT$ for $H=aq^2$ and for $H=bq^4$ (degree-independent), and the off-diagonal $\langle q_1\,\partial H/\partial q_2\rangle=0$ for a separable $H$ , are confirmed symbolically in verify_aq-sm-generalized-equipartition-theorem.py.

Go deeper · Derivation B · 1 minThe energy-share rule and the anharmonic and relativistic cases

Derivation BThe energy-share rule and the anharmonic and relativistic cases

The goal is to turn the master identity into the practical rule $\langle H_{\text{term}}\rangle=k_BT/n$ for an energy homogeneous of degree $n$ , and to apply it to the anharmonic quartic ( $n=4$ ) and the ultrarelativistic gas ( $n=1$ ). The strategy is Euler's theorem on homogeneous functions: if a term scales as $x^n$ , then $x\,\partial H_{\text{term}}/\partial x=n\,H_{\text{term}}$ , and the master identity fixes the average.

Step 1. Let a single energy term be $H_{\text{term}}=c\,x^{n}$ (homogeneous of degree $n$ in the variable $x$ ). Then $x\,\dfrac{\partial H_{\text{term}}}{\partial x}=x\cdot cn\,x^{n-1}=n\,c\,x^{n}=n\,H_{\text{term}}$ .

Step 2. Take the thermal average and apply equation (1) with $i=j$ : $\langle x\,\partial H/\partial x\rangle=k_BT$ , so $\langle n\,H_{\text{term}}\rangle=k_BT$ , giving $\langle H_{\text{term}}\rangle=k_BT/n$ . This is equation (2). Why the degree, not the coefficient, sets the share. The coefficient $c$ cancels in the average (rescale $x\to x/c^{1/n}$ and it disappears from both $H_{\text{term}}$ and the measure), while the power $n$ survives because it counts how fast the energy climbs — a steeper climb means a smaller average.

Step 3. Quadratic ( $n=2$ ): $\langle\tfrac12 aq^2\rangle=\tfrac12 k_BT$ , ordinary equipartition. Quartic ( $n=4$ ): an anharmonic term $bq^4$ gives $\langle bq^4\rangle=\tfrac14 k_BT$ — a stiff quartic trap stores less than a harmonic one at the same temperature.

Step 4. Ultrarelativistic gas: $H=c|\mathbf p|$ is homogeneous of degree $1$ in each of the three components together, so $\sum_j p_j\,\partial H/\partial p_j=H$ (Euler). Averaging and applying equation (1) to each component: $\langle H\rangle=\sum_j\langle p_j\,\partial H/\partial p_j\rangle=3k_BT$ per particle — twice the nonrelativistic $\tfrac32 k_BT$ , because a linear energy gives $k_BT$ per direction. Verified: $\langle bq^4\rangle=\tfrac14 k_BT$ , $\langle cq^6\rangle=\tfrac16 k_BT$ (the $k_BT/n$ rule), the kinetic $\langle p^2/2m\rangle=\tfrac12 k_BT$ , and the Euler relation $\sum_j p_j\partial_j H=H$ for $E=c|\mathbf p|$ giving $\langle E\rangle=3k_BT$ , are confirmed symbolically in the verify script.

Limiting cases

The generalized theorem is the classical, high-temperature limit of more complete results; each row is a specialization or a place where it breaks.

Case	Result	Meaning
Quadratic term, $x_i=x_j$	$\langle\tfrac12 ax^2\rangle=\tfrac12 k_BT$	Ordinary equipartition; the coefficient $a$ cancels.
Off-diagonal, $i\ne j$	$\langle x_i\,\partial H/\partial x_j\rangle=0$	Independent modes are uncorrelated; only self-slopes give $k_BT$ .
Degree- $n$ homogeneous	$\langle H_{\text{term}}\rangle=k_BT/n$	Quartic $\tfrac14 k_BT$ , linear (relativistic) $k_BT$ .
Coordinate-diagonal, summed	$\langle\sum\mathbf r\cdot\mathbf F\rangle=-3Nk_BT$	The virial theorem; wall part $\to PV$ .
$k_BT\lesssim\hbar\omega$ (quantum)	theorem fails	Phase-space integral illegal; mode freezes out.

Common trapEvery degree of freedom gets ½kT

The single most common error is to assign $\tfrac12 k_BT$ to everything. Three refinements are essential. First, the $\tfrac12 k_BT$ rule is specific to quadratic terms — a linear energy like $c|\mathbf p|$ gives $k_BT$ per direction, and a quartic $bq^4$ gives $\tfrac14 k_BT$ ; the general share is $k_BT/n$ for degree $n$ . Second, a full vibration is two quadratic terms (kinetic and potential), so it contributes $k_BT$ , not $\tfrac12 k_BT$ , when active. Third, and most important, the theorem is purely classical: a mode counts only if $k_BT$ exceeds its quantum level spacing, so vibrations (and, near absolute zero, rotations) are frozen out and contribute nothing. Counting the degrees of freedom correctly — quadratic, unbounded, and thermally active — is the whole game.

Worked examples

Example 1 — Heat capacity of an anharmonic trap versus a harmonic one (enumerate, apply, boundary number). A single particle sits in a one-dimensional trap. Compare its average potential energy for a harmonic trap $U=\tfrac12\kappa x^2$ and for a quartic trap $U=bx^4$ , both at temperature $T$ .

Step 1. Enumerate the energy terms. Harmonic: one quadratic potential term, degree $n=2$ . Quartic: one potential term, degree $n=4$ . In both cases the kinetic energy $p^2/2m$ is a separate quadratic term (degree 2).

Step 2. Apply the $k_BT/n$ rule. Harmonic potential: $\langle U\rangle=\tfrac12 k_BT$ ; total $\langle E\rangle=\tfrac12 k_BT+\tfrac12 k_BT=k_BT$ . Quartic potential: $\langle U\rangle=\tfrac14 k_BT$ ; total $\langle E\rangle=\tfrac12 k_BT+\tfrac14 k_BT=\tfrac34 k_BT$ .

Step 3. Boundary number. The heat capacities are $C_{\text{harmonic}}=k_B$ and $C_{\text{quartic}}=\tfrac34 k_B$ . A quartic trap has $75\%$ of the heat capacity of a harmonic one — a directly measurable classical signature of anharmonicity, and the reason real solids deviate from Dulong-Petit even before quantum freeze-out.

Example 2 — Kinetic energy of an ultrarelativistic electron gas. Consider electrons hot enough that $k_BT\gg m_ec^2$ , so their energy is $E\approx c|\mathbf p|$ (linear dispersion).

Step 1. The energy is homogeneous of degree $1$ in the momentum magnitude; there are three momentum directions.

Step 2. By the theorem, $\langle E\rangle=3k_BT$ per particle (each direction contributes $k_BT$ , not $\tfrac12 k_BT$ ).

Step 3. Boundary number. At $T=10^{10}\,\mathrm K$ (relativistic astrophysical plasma), $k_BT=1.38\times10^{-13}\,\mathrm J$ , so $\langle E\rangle=3k_BT=4.14\times10^{-13}\,\mathrm J\approx2.6\,\mathrm{MeV}$ per electron — well above the $0.511\,\mathrm{MeV}$ rest energy, confirming the ultrarelativistic regime is self-consistent. The factor of 2 over the nonrelativistic $\tfrac32 k_BT$ raises the specific heat of relativistic matter, a fact that matters for the interiors of stars and the early universe.

Origins · 1 minenergy equipartition and how this was discovered

energy equipartition

statistical-mechanics concept · Boltzmann 1868, Tolman 1918 general form

The generalized equipartition theorem $\langle x_i\,\partial H/\partial x_j\rangle=\delta_{ij}k_BT$ extends Maxwell's and Boltzmann's quadratic equipartition to arbitrary Hamiltonians. Maxwell (1859) first argued that energy is shared equally among molecular degrees of freedom; Boltzmann (1868–1871) generalized it to any quadratic term and connected it to the virial. Richard Tolman gave the theorem its modern general phase-space form in the 1910s–1920s, showing that the single integration-by-parts identity subsumes kinetic equipartition, the virial theorem, and the pressure of interacting gases. The theorem's dramatic failures — the frozen heat capacities of diatomic gases and the ultraviolet catastrophe — were among the clearest classical anomalies driving the quantum revolution.

History

1859

— Maxwell argues energy is shared equally among translational and rotational degrees of freedom.

1868

— Boltzmann generalizes equipartition to arbitrary quadratic degrees of freedom and links it to the virial.

1870

— Clausius states the virial theorem, connecting time-averaged kinetic energy to the virial of the forces.

1900

— Rayleigh applies equipartition to blackbody radiation, exposing the ultraviolet catastrophe.

1918

— Tolman presents the general phase-space form of the equipartition theorem.

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., §3.7 (equipartition and virial theorems) · Reif, Fundamentals of Statistical and Thermal Physics, §7.5–7.7 · Kardar, Statistical Physics of Particles, §4.3 · Tolman, The Principles of Statistical Mechanics, Ch. 3 · Landau & Lifshitz, Statistical Physics Part 1, §28.

Provenance. The identity $\langle x_i\,\partial H/\partial x_j\rangle=\delta_{ij}k_BT$ , the energy-share rule $\langle H_{\text{term}}\rangle=k_BT/n$ , and the virial specialization are standard (Pathria §3.7, Tolman Ch. 3). A verifier should confirm: (i) $\langle q\,\partial H/\partial q\rangle=1/\beta=k_BT$ for $H=aq^2$ and for $H=bq^4$ (degree-independent); (ii) $\langle bq^4\rangle=\tfrac14 k_BT$ and $\langle cq^6\rangle=\tfrac16 k_BT$ (the $k_BT/n$ rule); (iii) the off-diagonal $\langle q_1\,\partial H/\partial q_2\rangle=0$ ; (iv) $\langle p^2/2m\rangle=\tfrac12 k_BT$ ; (v) $\sum_j p_j\,\partial H/\partial p_j=H$ for $E=c|\mathbf p|$ , giving $\langle E\rangle=3k_BT$ . All checks are in verify_aq-sm-generalized-equipartition-theorem.py. Constants: $k_B=1.380649\times10^{-23}\,\mathrm{J/K}$ , $R=8.314\,\mathrm{J\,mol^{-1}K^{-1}}$ .

Builds on: Equipartition Theorem · Classical Ideal Gas · Maxwell-Boltzmann Velocity Distribution Leads to: Virial Theorem in Statistical Mechanics · Virial Equation of State · Classical Limit of Quantum Statistics

How is the "generalized" theorem different from ordinary equipartition?

Ordinary equipartition is a rule about quadratic terms: each contributes $\tfrac12 k_BT$ . The generalized theorem is the phase-space identity $\langle x_i\,\partial H/\partial x_j\rangle=\delta_{ij}k_BT$ that holds for any confining Hamiltonian, quadratic or not. Ordinary equipartition is the special case $x_i=x_j=q$ with $H=aq^2$ . But the general form also handles anharmonic energies (a $q^4$ term gives $\tfrac14 k_BT$ ), relativistic dispersion ( $c|\mathbf p|$ gives $k_BT$ per direction), and — through the coordinate-diagonal case summed over particles — the virial theorem and the pressure of an interacting gas. It is one identity that unifies several results that look unrelated in the elementary treatment.

Why does the coefficient of the energy term never appear in the average?

Because it cancels between the energy and the measure. For a term $H_{\text{term}}=c\,x^n$ , rescaling $x\to x/c^{1/n}$ removes $c$ from the energy entirely, appearing only as an overall constant in both the numerator and denominator of the thermal average — so it drops out. Physically, a larger coefficient makes the mode stiffer (more energy per unit $x$ ) but also confines it to a smaller range of $x$ , and the two effects cancel exactly. Only the degree $n$ survives, giving the share $k_BT/n$ . This is why a stiff and a soft spring in the same bath store identical average energy.

What exactly is the "boundary term" and why must it vanish?

When you integrate by parts, you generate a surface term $[x_i\,e^{-\beta H}]$ evaluated at the edges of the integration region. For the theorem to hold, this must be zero. It vanishes because a physical Hamiltonian confines the system: as any coordinate or momentum runs to $\pm\infty$ , the energy $H\to\infty$ , so $e^{-\beta H}$ decays to zero faster than the polynomial factor $x_i$ can grow. If a coordinate were not confined — say a free particle in an infinite box with no walls — the boundary term would not vanish, and that degree of freedom would not obey equipartition. The confining requirement is the theorem's only physical assumption.

How does the theorem give the virial theorem?

Take the diagonal case with a coordinate, $\langle q_k\,\partial H/\partial q_k\rangle=k_BT$ , and note $\partial H/\partial q_k=-F_k$ (the force). Summing over all $3N$ coordinates gives $\langle\sum_k\mathbf r_k\cdot\mathbf F_k\rangle=-3Nk_BT$ , which is the Clausius virial. Splitting the force into the wall (container) part and the interparticle part, the wall part integrates to $-3PV$ , giving $PV=Nk_BT+\tfrac13\langle\sum_{\text{pairs}}\mathbf r_{ij}\cdot\mathbf F_{ij}\rangle$ . The interaction term is the configurational virial that generates the virial equation of state. So the virial theorem is literally the coordinate branch of the generalized equipartition identity.

Does the theorem hold in the microcanonical ensemble too?

Yes, in the thermodynamic limit. The cleanest derivation uses the canonical Boltzmann factor, but the result is ensemble-equivalent: the microcanonical and canonical averages of $x_i\,\partial H/\partial x_j$ agree to leading order in $1/N$ , both giving $\delta_{ij}k_BT$ , with the microcanonical temperature defined by $1/T=\partial S/\partial E$ . There is a subtlety — the microcanonical version is sometimes written with $\partial S/\partial E$ replacing $\beta$ , and for small systems the two ensembles differ at order $1/N$ — but for any macroscopic system the distinction is negligible and the theorem is robust.

Why does a relativistic gas have twice the energy per direction?

Because its energy is linear in momentum, not quadratic. For a nonrelativistic gas, $E=p^2/2m$ is homogeneous of degree $2$ , so each direction gives $\tfrac12 k_BT$ . For an ultrarelativistic gas, $E=c|\mathbf p|$ is homogeneous of degree $1$ , and the rule $k_BT/n$ with $n=1$ gives the full $k_BT$ per direction — total $3k_BT$ per particle in three dimensions. The physical reason: a linear energy climbs more gently away from the origin, so the Boltzmann cloud spreads wider and stores more thermal energy before the weight cuts it off. This doubling raises the heat capacity of radiation-dominated and relativistic matter.

Can I use the general theorem to spot when a mode is anharmonic experimentally?

Yes — measure the mode's contribution to the heat capacity and compare it to the harmonic prediction. A purely quadratic mode contributes $k_B$ (kinetic + potential, $\tfrac12 k_B$ each); a purely quartic potential contributes only $\tfrac12 k_B+\tfrac14 k_B=\tfrac34 k_B$ . A measured heat capacity below the harmonic value (but above zero, so not frozen out) is a fingerprint of anharmonicity, encoded in the effective degree $n$ of the potential. This is exactly how the thermal expansion and specific-heat anomalies of strongly anharmonic crystals are read: the $k_BT/n$ share of a soft, anharmonic mode differs measurably from the harmonic $\tfrac12 k_BT$ .