Continuous Reformulation of Planck's Law

Continuous Reformulation of Planck’s Law. Jaafar Gaber1,∗ 1Université Marie et Louis Pasteur, UTBM, CNRS, institut FEMTO-ST, F-90000 Belfort, France Abstract. In this paper, we present a continuous reformulation of Planck’s quantization law within a continuous and geometric thermodynamic framework. The discrete spectrum En=nhνthat underlies the traditional Planck distribution is generalized into a functional form En=E0f(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 λencodes an intrinsic geometric scale, the Planck distribution is recovered in the flattening limit λ→0 with E0λheld fixed, so that En≈nhνup 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. 1 Introduction The birth of quantum theory was marked by Planck’s discrete quantization [1,2], together with the associated canonical averaging that led to the blackbody radiation law: En=nhν, (1) where each oscillator of frequency νin a thermal cavity can only hold integer multiples of the elementary quantum hν. Thermal equilibrium is then entirely characterized by the temperature T, and the material details of the cavity walls play no dynamical role; the spectrum is assumed universal and purely thermal. However, many radiative systems of physical interest are not purely thermal or universal. Plasmonic cavities, nanoresonators, driven emitters, near-field thermal devices, or structured photonic media display spectra that are neither perfectly Planckian nor strictly discrete-line atomic spectra. Their spectra combine continuous emission envelopes with scale-dependent resonant peaks. Such systems therefore fall outside the purely discrete assumption of Eq. (1). ∗e-mail: [email protected] E3S Web of Conferences 680, 00091 (2025) https://doi.org/10.1051/e3sconf/202568000091 ICEGC'2025 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (https://creativecommons.org/licenses/by/4.0/). In recent work, we introduced a generalized statistical framework in which the canonical Boltzmann–Gibbs weight is preserved, but the spectrum of accessible energies is no longer assumed to be additive or evenly spaced [5]. This framework was subsequently extended into a full thermodynamic theory of spectral energy, where the energy levels follow an exponential law and the natural analytic tool is the Mellin transform rather than the standard Laplace transform [6]. Within this approach, the resulting formalism describes a model-based selfsimilar spectral thermodynamics, in which self-similarity emerges from the chosen spectral geometry rather than from direct experimental observation. In the present paper, we reformulate Planck quantization and extract a geometric interpretation of the radiation law. Planck’s law can be understood as the limiting case of a broader, geometrically organized energy spectrum. Accordingly, instead of postulating discrete additive quanta, we introduce a continuous energy law that varies across scales. In this picture, quantization is not imposed by hand as a linear ladder but emerges, within the present model, as a manifestation of curvature within an abstract energy manifold. 2 From discrete quanta to continuous spectra In Planck’s original derivation, the allowed energy values of an electromagnetic mode of frequency νare discrete and uniformly spaced, following the linear law (1). The corresponding canonical partition function reads ZPlanck = ∞ X n=0 exp −nhν kBT!=1 1−exp(−hν/kBT).(2) From ln ZPlanck, one obtains the mean energy ⟨E⟩and recovers the standard Planck distribution for blackbody radiation. The central assumption behind (2) is additivity: each energy level differs from the previous one by the same increment hν. We now relax this constraint. Let the spectrum be defined more generally by En=E0f(n),(3) where f(n) need not be linear in n, but must ensure the convergence of the canonical partition function Zf(s). The corresponding partition function becomes Zf(s)= ∞ X n=0 exp"−sE0f(n) kBT#,(4) where sis a dimensionless scaling parameter (e.g., mode degeneracy, coupling weight, or geometric multiplicity). The canonical Boltzmann weighting is preserved, while the spectral law is generalized. Equation (4) reduces to (2) when f(n)=nand s=1. The expressions En=E0f(n) and Zf(s) go beyond a simple reparametrization of existing models; they redefine the structure of statistical mechanics by treating the energy spectrum as a geometric property of the thermodynamic manifold. Moving from fixed linear spectra to continuous laws that vary across scales extends Boltzmann and Planck formulations into a coherent geometric framework. A particularly relevant case, introduced in [5–7], is the exponential spectrum f(n)=eλn=⇒En=E0eλn,(5) E3S Web of Conferences 680, 00091 (2025) https://doi.org/10.1051/e3sconf/202568000091 ICEGC'2025 2 where λis a (dimensionless) scale parameter. Instead of a uniform spacing En+1−En=hν, the levels form a geometric progression with the ratio En+1 En =eλ.(6) Equivalently, the spacing grows multiplicatively, En+1−En=(eλ−1) En,(7) so the spectrum is self-similar across scales rather than evenly spaced. This multiplicative hierarchy signals an intrinsic curvature within the space of accessible energies. In the flattening limit λ→0 with E0λheld fixed, one obtains En=E0(1 +λn)−→ En≈nhν+const.,(8) so that the exponential spectrum continuously reduces to the linear Planck quantization up to a constant offset. Thus, Planck’s law appears as the zero-curvature limit of a broader, scale-dependent thermodynamic geometry. In the flat or weak-curvature limit, any admissible spectral function f(n) that is differentiable and approximately linear in nrecovers the classical Planck form. Indeed, expanding f(n) to first order gives f(n)≃f(0) +f′(0) n,(9) so that En=E0f(n)≃E0f(0) +E0f′(0) n.(10) By identifying E0f′(0) =hνand neglecting the constant offset E0f(0), one obtains the linear quantization rule En≃nhν, (11) which defines the Planck regime. Therefore, the discrete linear law arises as the zerocurvature limit of any continuous spectral geometry admitting a first-order linear expansion. The exponential case f(n)=eλnillustrates this principle explicitly [7], with λacting as a curvature parameter whose vanishing restores the linear spectrum. 3 Continuous partition and Mellin structure In the previous section, we introduced the general spectral form (3) and the corresponding canonical sum (4). Different choices of f(n) define distinct thermodynamic geometries. Among them, the exponential law f(n)=eλn=⇒En=E0eλn, plays a special role. Among the possible spectral laws, the exponential form f(n)=eλnplays a distinguished role: it is the simplest functional form for which the canonical partition sum admits a Mellintype continuum approximation via the change of variables x=eλn, where the weighting becomes multiplicative rather than additive. Other monotonic or power-law forms of f(n) may also lead to Mellin-like structures under suitable transformations, but the exponential case remains the simplest and most analytically tractable realization. This property enables a direct analytic continuation of the discrete sum into a continuous integral with a logarithmic measure, making explicit the geometric nature of the energy scaling. E3S Web of Conferences 680, 00091 (2025) https://doi.org/10.1051/e3sconf/202568000091 ICEGC'2025 3 Inserting f(n)=eλninto Zf(s) yields Zλ(s)= ∞ X n=0 exp"−sE0eλn kBT#,(12) which defines a “double-exponential” partition function characteristic of self-similar spectra. This is no longer a geometric series: the Boltzmann factor is applied to an exponentially growing spectrum. As shown in [6] and following the Mellin–zeta framework introduced by Riemann [4], Eq. (12) can be treated through a Mellin-type analysis. Let x=eλnso that n=1 λln xand dn=1 λ dx x. Applying the Euler–Maclaurin formula to replace the discrete sum by its continuum approximation gives Zλ(s)≈1 λZ∞ 1 x−1exp"−sE0x kBT#dx.(13) This integral has the structure of a Mellin transform, i.e. an integral of the form Rxu−1F(x) dx. In this sense, Eq. (12) admits a Mellin-type representation rather than the usual Laplace structure of standard canonical ensembles. The thermodynamics thus ceases to be purely additive in energy and becomes multiplicative across scales. From Zλ(s), one can still define the mean energy ⟨E⟩λ=1 Zλ(s) ∞ X n=0 E0eλnexp"−sE0eλn kBT#,(14) and the associated entropy in canonical form, Sλ=kB ln Zλ(s)+⟨E⟩λ kBT!.(15) Equations (12)–(15) generalize the familiar Gibbs–Boltzmann relations to a non-additive, scale-structured spectrum. In the limit λ→0, the multiplicative increment eλ→1+λand the exponential ladder (5) flattens into a linear one. Then Zλ(s) continuously reduces to ZPlanck in (2), and the standard Planck law is recovered. Planck’s law therefore appears as the zero-curvature limit of a broader, scale-dependent thermodynamic structure. 4 Geometric interpretation of the spectrum The exponential law (5) admits a natural geometric interpretation. If consecutive energy levels differ by a constant ratio rather than a constant difference, the spectrum is organized into a hierarchy of scales. Such hierarchies are naturally associated, by analogy, with spaces of negative curvature [3], where distances grow multiplicatively rather than additively. A convenient analogy is provided by the Poincaré disk, which offers a simple model of multiplicative scaling in a negatively curved space. In this geometry, the hyperbolic distance between two radial points z1and z2is given by dH(z1,z2)=arcosh 1+2|z1−z2|2 (1 − |z1|2)(1 − |z2|2)!.(16) Moving radially outward corresponds to exploring successively higher “scales” rather than linear coordinates. This is directly analogous to the transition En→En+1in (5), where each step rescales the energy by a fixed factor eλ. E3S Web of Conferences 680, 00091 (2025) https://doi.org/10.1051/e3sconf/202568000091 ICEGC'2025 4 In this analogy, the parameter λacts as a curvature-like quantity controlling the deviation from the flat spectrum. Larger values of λcorrespond to stronger geometric deformation of the energy spacing, whereas the limit λ→0 represents a flattening of the spectrum toward linear quantization. By analogy with hyperbolic geometry, this behavior can be summarized as an effective relation κeff∼ −λ2,(17) where κeffdenotes an effective curvature parameter with no claim of direct physical measurement. When λ,0, the energy manifold thus behaves, by analogy with a hyperbolic space, as if it possessed a finite negative curvature: the accessible states populate a multiplicative, scalestructured ladder. In the limit λ→0, this curvature tends to zero, the ladder becomes linear, and one recovers the evenly spaced spectrum of Planck quantization. This geometric reading reinterprets quantization, within the present model, as a manifestation of curvature in an abstract energy space: discreteness is not a primitive postulate, but a special geometric regime corresponding to the flat limit of the thermodynamic manifold. 5 Discussion and physical implications Our continuous spectral formulation (5)–(17) provides a coherent framework for describing radiative systems beyond the thermal equilibrium assumption, including non-thermal and geometrically confined emitters [7]. First, it naturally accommodates non-thermal or quasi-thermal emitters. Classical blackbody radiation assumes a cavity in global equilibrium and a spectrum that depends only on the temperature T. In contrast, real systems such as nanoresonators, plasmonic gaps, and near-field thermal emitters exhibit spatial confinement and mode-dependent enhancement, leading to noticeably non-Planckian or programmable emissive properties [8–10]. In such contexts, energy exchange is not governed by a single global quantum hν, but by multiple local modes or geometric scales that shape the spectral response. Within the present framework, the curvature parameter λencodes this confinement or spectral hierarchy of electromagnetic modes. Second, the Mellin-type partition function (12)–(13) suggests that statistical mechanics in structured radiative media is not purely additive but organized by multiplicative energy scaling. This defines a thermodynamics of scales, where the relevant degrees of freedom are distributed across hierarchical energy manifolds rather than evenly spaced levels. The same reasoning could extend to fractal optical media, hierarchical resonator arrays, or multiscale plasmas. Finally, in the λ→0 limit, all these deformations continuously reduce to the standard Planck law. Planck’s spectrum is therefore not contradicted but embedded as the flat, additive limit of a broader, curved, and multiplicative thermodynamic geometry. 6 Conclusion We have shown that Planck’s discrete quantization can be reformulated as the limiting case of a continuous, geometrically organized energy spectrum. Replacing the linear law En= nhνby a general functional form En=E0f(n) allows the energy manifold itself to acquire a geometry. In this framework, the canonical partition function becomes a spectral object whose analytic structure depends on the intrinsic form of f(n). For exponential spectra, the partition assumes a Mellin-type form, introducing a scale-dependent entropy Sλand a curvature κeff∼ −λ2that characterizes the underlying hyperbolic geometry. More generally, E3S Web of Conferences 680, 00091 (2025) https://doi.org/10.1051/e3sconf/202568000091 ICEGC'2025 5 any smooth f(n) that admits a local linear expansion recovers the standard Planck regime in the zero-curvature limit. Quantization thus appears not as a fundamental discreteness, but as a geometric manifestation of curvature in the energy manifold. The classical blackbody spectrum emerges as the flat limit of a broader, continuous thermodynamic geometry. References [1] M. Planck, On the Theory of the Law of Energy Distribution in the Normal Spectrum, Verhandlungen der Deutschen Physikalischen Gesellschaft 2, 237 (1900). [2] M. Planck, On the Law of the Energy Distribution in the Normal Spectrum, Annalen der Physik 4, 553–563 (1901). [3] H. Poincare, Les méthodes nouvelles de la mécanique céleste, Vol. T.1 (Gauthier-Villars, Paris 1892). [4] B. Riemann, On the Number of Prime Numbers less than a Given Quantity, Monatsberichte der Königlichen Preussischen Akademie der Wissenschaften zu Berlin (1859); English translation by D. R. Wilkins (1998). [5] J. Gaber, Continuous Reformulation of Boltzmann Statistics and Geometric Partition Functions, Preprint (2025). [6] J. Gaber, Spectral Energy Thermodynamics, Preprint (2025). [7] J. Gaber, Continuous Reformulation of Planck Quantization, HAL preprint, hal05014506 (2025). [8] J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen,Coherent emission of light by thermal sources, Nature 416, 61–64 (2002). [9] C. R. Otey and S. Fan, Numerically exact calculation of electromagnetic heat transfer between a dielectric sphere and plate, Phys. Rev. B 84, 245431 (2011). [10] I. Latella, S.-A. Biehs, and P. Ben-Abdallah,Smart thermal management with near-field thermal radiation, Optics Express, vol. 29, no. 16, pp. 24816–24846, 2021. E3S Web of Conferences 680, 00091 (2025) https://doi.org/10.1051/e3sconf/202568000091 ICEGC'2025 6