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Planck's law from a classical free energy extremum involving fisher information
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DOI：10.1007/s40509-026-00383-0.png)
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Abstract
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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=& hstrok;omega/kBT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gamma = \hbar \omega / k_B T $$\end{document}. 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 & hstrok;omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hbar \omega $$\end{document} 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.
Keywords:
Planck's law
Blackbody radiation
Fisher information
Variational principle
Free energy extremization
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