Planck Law from a Classical Free Energy Extremum Involving Fisher Information

Abstract

We derive the Planck law from a classical variational principle over probability densities, without invoking quantum states, quantized oscillator energies, or ensemble averages. We construct a generalized free energy functional involving entropy and Fisher information, with weights determined by the dimensionless ratio of quantum to thermal energy. 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 only when a thermal fluctuation delivers at least as much energy as the photon has. We also present a complementary kinetic derivation, based on threshold-activated thermal cascades, that yields the same result through classical stochastic reasoning. Together, these approaches provide a thermodynamic and information-theoretic route to black-body radiation, grounded in classical principles and variational stability.