Mode counting and the ultraviolet catastrophe

Put the field in a cubical cavity of side L with conducting walls. Tangential E must vanish on every wall, which forces Eₓ ∝ cos(kₓ x) sin(k_y y) sin(k_z z) and its two cyclic partners — a sine in each of the two directions along which that component runs tangential — and the allowed wavevectors are k = (π/L)(nₓ, n_y, n_z) with the nᵢ integers. Every mode is one point of a cubic lattice in n-space, one point per unit cell, with frequency ν = (c/2L)√(nₓ² + n_y² + n_z²). Two features of the count are worth defending, because both get challenged. Only the positive octant counts, since −n labels the same standing wave as +n. And each lattice point with all three indices non-zero carries two modes rather than three: ∇⋅E = 0 kills the longitudinal polarisation and leaves two transverse states. Set one index to zero and two of the three components collapse with their sines, leaving one surviving polarisation instead of two — those are the lowest resonances a real cavity has, and they are why the nᵢ ≥ 1 bookkeeping used below is an asymptotic count rather than an exact one. Swap the conducting walls for periodic boundary conditions and the lattice spacing doubles to 2π/L while all eight octants open up — the same answer. Weyl's theorem goes further: for any cavity large compared with the wavelength, the leading count depends on the volume alone and not on the shape.

Counting every mode below ν means counting lattice points in the positive octant of a sphere of radius R = 2Lν/c. Replace the point count by the octant's volume and double it for polarisation: N(ν) = 2 × (1/8)(4π/3)R³ = (8π/3)(ν³/c³)V, and differentiating gives the density everything else is built on, dN/(V dν) = 8πν²/c³. That smoothing step needs R ≫ 1, and it is worth watching it fail. Take a 1.0 cm cube below 30 GHz, where R = 2.00: the smooth formula claims 8.4 modes, while an exact tally finds five — three at (c/2L)√2 = 21.2 GHz, each with one index zero and so one polarisation, and two at (c/2L)√3 = 26.0 GHz. Go up to R = 8.0 and the tally gives 515 against the formula's 537, 4% high; the surface corrections cancel for a conducting cavity, so what survives is only a term of order R, and its share dies away as R grows. At optical frequencies there is nothing left to worry about, since R = 2L/λ reaches 4 × 10⁵ for a 10 cm cube at 500 nm. And the density doubles for every 41% rise in frequency, without ever turning over.

A mode of the field is dynamically a harmonic oscillator. Expand the field in the cavity as a sum over modes and the energy separates into terms ½q̇² + ½ω²q² in the mode amplitudes — equivalently, half electric and half magnetic. Two quadratic terms, so classical equipartition gives ⟨E⟩ = 2 × ½kT = kT. It is worth seeing where that comes from, because that is where the trouble lives: the mean energy is ∫E e^(−E/kT)dE ÷ ∫e^(−E/kT)dE taken over a continuous E from 0 to ∞, and the integral returns kT no matter what ω is. Nothing classical says a fast mode should be harder to excite than a slow one — the Boltzmann factor cares only about energy, and a classical oscillator may carry any energy at any frequency. So the 10¹⁵ Hz modes are stocked exactly as generously as the 10¹⁰ Hz ones: at 2000 K, 2.76 × 10⁻²⁰ J apiece.

Multiply the two factors: u(ν, T) = (8πν²/c³)kT, in joules per cubic metre per hertz. There is no adjustable constant — geometry and equipartition fix it completely, which makes it a prediction rather than a fit. And in the far infrared and radio it is right. Against the measured spectrum the classical value is too large by the factor (e^x − 1)/x with x = hν/kT, which expands as 1 + x/2 + …, so Rayleigh–Jeans is within 1% for x < 0.02. At 2000 K that means wavelengths beyond about 360 µm. Radio astronomy lives entirely inside that window and still quotes source strengths as brightness temperatures defined from this law. Convert to wavelength carefully: u(λ, T) = u(ν, T)|dν/dλ| = u(ν, T)c/λ², and that Jacobian is what turns ν² into λ⁻⁴. Substituting ν = c/λ without it returns λ⁻² — a different power law, not merely a different constant.

Now add the modes up. The energy density below a frequency ν is 8πkTν³/3c³, growing without limit because neither classical electromagnetism nor classical statistics supplies a highest frequency, and growing per unit volume, so it is not an artefact of a large box. Numbers show the scale of the failure. At 2000 K the measured total is aT⁴ = 1.21 × 10⁻² J m⁻³. The classical running total passes that value at 1.12 × 10¹⁴ Hz — a wavelength of 2.67 µm, still in the near infrared — and by the time the integral reaches the violet edge at 400 nm it stands at 3.61 J m⁻³, about three hundred times the entire measured budget. Carry it to X-rays at 3 × 10¹⁸ Hz and it is 2 × 10¹¹ J m⁻³. Ehrenfest named this the ultraviolet catastrophe in 1911, and the name is precise: the trouble is not at any one frequency but in the tail, where a ν² density of modes times a fixed kT can never converge.

A derivation with two ingredients and a fatal conclusion has exactly two suspects, and they can be tried separately. The mode count is checkable on its own: resolve the resonances of a microwave cavity and count them, one line at a time. They are there, in the predicted number. Equipartition gets no such direct test inside the cavity, and its second failure was already on the table elsewhere. Molar heat capacities of solids sit at 3R near room temperature, exactly as equipartition predicts, then collapse towards zero on cooling; diatomic gases shed their vibrational and then their rotational contributions the same way. Both failures share one signature: degrees of freedom stop accepting energy once kT is small compared with something. Naming that something, and the constant that sets its scale, is the next step.