Planck Law from a Classical Free Energy Extremum Involving Fisher Information

TL;DR

The paper shows that Planck's blackbody spectrum can arise from a purely classical variational principle by introducing a generalized free energy $F_\gamma[p]$ that combines potential energy, Shannon entropy, and Fisher information with weights set by $\gamma = \frac{\hbar \omega}{k_B T}$. A Gaussian ansatz for a harmonic oscillator yields the exact Planck mean energy per mode $\langle E_{\text{rad}}\rangle = \frac{\hbar \omega}{e^{\gamma}-1}$, and a complementary kinetic derivation based on threshold-activated emission cascades reproduces the same result, providing two classical routes to blackbody radiation. The only explicit quantum input is a threshold $E \geq \hbar \omega$ for photon emission, with zero-point energy $\frac{\hbar \omega}{2}$ emerging in the zero-temperature limit. Together, these results offer a thermodynamic and information-geometric perspective on Planck’s law, connecting classical optimization with quantum-like behavior without invoking quantized energy levels or Bose statistics.

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.

Figures (1)

• Figure 1: Time-domain diagnostics for the bursty cluster realization compared to a matched-mean Poisson baseline. Dashed curves show analytic expectations (for the cluster model via Eqs. \ref{['eq:g2_tau_cluster']}--\ref{['eq:fano_cluster']}), and markers the simulation results. (a) Binned estimate of $g^{(2)}(\tau)$ versus delay $\tau$. (b) Binned estimate of $g^{(2)}_{\Delta t}(0)$ versus bin width $\Delta t$. (c) Corresponding Fano factor $F(\Delta t)$. For Poisson, $g^{(2)}(\tau)=1$ and $F(\Delta t)=1$ for all $\Delta t$.