Ultraviolet catastrophe

The ultraviolet catastrophe is the divergent total thermal radiation energy predicted by classical equipartition theorem applied to the electromagnetic mode density in a box. Each mode has mean energy $k_{B}T$, while the number of modes per volume per frequency is $8\pi {\nu }^2/c^3$. The Rayleigh-Jeans law therefore gives $u_{\nu }=8\pi k_{B}T{\nu }^2/c^3$, whose integral over all positive frequencies diverges at every $T>0$.

Planck's law replaces the classical mode energy by $h\nu /(e^{h\nu /(k_{B}T)}-1)$. It agrees with equipartition when $h\nu \ll k_{B}T$ but suppresses high-frequency excitations exponentially. The resulting spectral energy density is integrable and yields the finite Stefan–Boltzmann law. The failure concerns the classical equilibrium distribution of energy among modes, not the existence of electromagnetic wave propagation itself.