Derivations of the Planck Blackbody Spectrum from Thermodynamic Ideas in Classical Physics with Classical Zero-Point Radiation

TL;DR

This paper presents two classical physics-based derivations of the Planck blackbody spectrum incorporating zero-point radiation, linking thermodynamics, zero-point energy, and spacetime structure.

Contribution

It introduces novel classical derivations of the Planck spectrum that include zero-point radiation, emphasizing thermodynamic principles and energy minimization.

Findings

Derivations based on monotonic properties of the canonical potential.

Helmholtz free energy minimization at thermal equilibrium.

Zero-point energy embedded in the traditional Planck spectrum.

Abstract

Based upon thermodynamic ideas, two new derivations of the Planck blackbody spectrum are given within classical physics which includes classical zero-point radiation. The first and second laws of thermodynamics, applied to a harmonic oscillator or a radiation normal mode, require that the canonical potential $\phi(\omega/T)$ is a function of a single variable corresponding to the ratio of the oscillation frequency to the temperature. The second law of thermodynamics involves extremum ideas which may be applied to thermal radiation. Our first derivation of the Planck spectrum is based upon the idea that the canonical potential $\phi(\omega/T)$ is a monotonic function and all its derivatives are monotonic when interpolating between zero-point energy at low temperature and energy equipartition at high temperature; the monotonic behavior precludes the canonical potential from giving a preferred value for the ratio $\omega/T.$ Our second derivation of the Planck spectrum is based upon the requirement that the change in the Helmholtz free energy of the radiation in a partitioned box held at constant temperature should be a minimum at thermal equilibrium. Finally, the change in Casimir energy with change in partition position for the radiation in a partitioned box is shown to correspond at high temperature to the absence of zero-point energy when the spectral energy per normal mode is chosen as the traditional Planck spectrum which omits zero-point energy at low temperature; thus the idea of zero-point energy is embedded in the traditional Planck spectrum. It is emphasized that thermal radiation is intimately connected with zero-point radiation and the structure of spacetime in classical physics.