Continuous Reformulation of Planck’s Law

Université Marie et Louis Pasteur, UTBM, CNRS, institut FEMTO-ST, F-90000 Belfort, France

^* e-mail: gaber@utbm.fr

Abstract

In this paper, we present a continuous reformulation of Planck’s quantization law within a continuous and geometric thermodynamic framework. The discrete spectrum E_n = nhν that underlies the raditional Planck distribution is generalized into a functional form E_n = E₀ f (n), where f (n) defines the intrinsic geometry of the accessible energy states. This continuous formalism embeds Boltzmann, Gibbs, and Planck statistics across arbitrary spectral geometries, where quantization emerges as a geometric reinterpretation of energy itself. For the exponential case f (n) = eλn, where λ encodes an intrinsic geometric scale, the Planck distribution is recovered in the flattening limit λ 0 with E_0λ held fixed, so that E_n nhν up to a constant offset. Non-zero curvature then describes self-similar and non-thermal spectra. More generally, any smooth spectral law that becomes locally linear recovers the standard Planck regime as its zero-curvature limit. Within this representation, the canonical partition function admits a Mellin-type continuum approximation for monotonic and differentiable spectral laws, and the blackbody law emerges as a limiting case of a more general, self-similar spectral thermodynamics. The formulation provides a coherent geometric interpretation of Boltzmann statistics, Gibbs’ partition function, and Planck’s quantization, extending them to systems with non-uniform confinement or curvature. Quantization thus appears not as a discrete postulate, but as a geometric property of the energy manifold.