The Rayleigh-Jeans Law
The Rayleigh-Jeans law was the first serious attempt to derive the black-body spectrum from classical physics. It works brilliantly at low frequencies and catastrophically at high frequencies. It is a law that tells you exactly where classical physics ends.
Rayleigh's original 1900 derivation contained an error — he missed a factor of two. Jeans corrected it in 1905, and the formula that emerged was:
u(ν,T) = (8πν²/c³) · kT
where u is the spectral energy density, ν is frequency, c is the speed of light, and k is Boltzmann's constant.
The derivation is elegant in its simplicity. You place electromagnetic standing waves in a cubic cavity of side L. The allowed wavevectors are quantized: k_x = n_xπ/L, and so on. The number of modes in a frequency interval dν is proportional to ν² dV dν — that is the famous ν² density of states. Then you invoke equipartition: each mode, being a harmonic oscillator, carries an average energy kT. Multiply and you get the law.
Everything is classically sound. The mode counting is correct. The equipartition theorem was the foundation of statistical mechanics. The result, however, diverges. The integral over all frequencies is infinite. The total energy in the cavity is infinite. The prediction is absurd.
At low frequencies, where hν ≪ kT, the Planck distribution reduces to the Rayleigh-Jeans law by Taylor expansion. The classical result is an approximation that works when the quantum nature of light is invisible. The cutoff is not a parameter — it is fundamental. When the photon energy hν becomes comparable to or larger than kT, classical physics has nothing to say.
The Rayleigh-Jeans law was important not only for its failure but for what that failure revealed. It showed that the classical equipartition theorem — the principle that every degree of freedom gets kT — cannot be universally valid. If it were, every mode of every electromagnetic field would carry infinite energy. The universe is selective; it does not distribute energy equally among all possible modes.
Trolla considers the Rayleigh-Jeans law to be one of the most instructively wrong formulas in physics. It is correct in its regime and spectacularly wrong in its domain of breakdown. It is a warning label painted on the boundary of classical theory, and it was one of the clearest signals that something fundamentally new was needed.
There is a peculiar quality to the Rayleigh-Jeans law that Trolla finds interesting. It is so close to being right. At low frequencies — the frequencies you can measure, the frequencies that matter for most practical purposes — it is correct. The divergence only appears when you extend the mathematics into regions where no experiment has looked. It is as if classical physics worked perfectly in the world it was designed to describe and only collapsed when asked to speak about territory it had never entered.