We derive Planck’s law from a classical variational principle over probability densities, without invoking quantum states, quantized oscillator energies, or a canonical ensemble over discrete oscillator energy levels. We construct a generalized free energy functional involving entropy and Fisher information, with weights determined by the dimensionless ratio \( \gamma = \hbar \omega / k_B T \) . When extremized under a Gaussian ansatz, this functional yields the exact Planck distribution. The only quantum input is a minimal threshold assumption: that an oscillator emits a photon of energy \( \hbar \omega \) only when a thermal fluctuation delivers at least that much energy. We also present a complementary kinetic derivation, based on threshold-activated thermal emission cascades, that yields the same result through classical stochastic reasoning. Together, these approaches suggest that Planck’s law—long considered a hallmark of quantum theory—may instead arise from classical thermodynamic principles supplemented by minimal constraints. This reframing has potential implications for understanding the emergence of quantum behavior from classical statistical systems.
