Historical research guide · 0390 · 2026
Applying the thermodynamics of a stochastic classical point charge in a spherical charge to derive classical electromagnetic zero-point radiation
Daniel C. Cole
In one minute
Daniel Cole, at Boston University, works in stochastic electrodynamics — the theory that keeps Maxwell’s equations and Newton’s laws exactly as they are and adds one thing: a real, fluctuating electromagnetic field that is still present at absolute zero. The hard question that theory has always faced is where that field’s spectrum comes from. Cole’s answer, developed over years and set out in his 2024 review, is thermodynamics. Absolute zero means no heat flows during slow reversible operations, and demanding exactly that of a classical charged system bathed in radiation forces the spectrum to rise as the cube of frequency, with its constant fixed by measured van der Waals and Casimir forces. This 2026 paper applies that machinery to a new and more physical system: a point charge moving stochastically inside a spherical charge, bound by real electrostatics rather than by an idealised spring. The paper is behind the publisher’s paywall, so the account here is read from its title and from Cole’s own published statement of the method.
Connection
Chapter 2 asks what the vacuum is, and this line of work answers with thermodynamics instead of quantum postulates: the zero-point spectrum is what the second law forces on a classical field theory once you stop assuming that radiation vanishes at absolute zero. Chapter 3 needs that field to be real enough to hold an atom in its ground state, and chapter 6 needs to know exactly what it is — the equilibrium at zero temperature, which is why a device drawing work from it must be a driven, non-equilibrium arrangement rather than an isothermal one.
From the source
Classical electromagnetic zero-point radiation can be derived, rather than assumed, from the thermodynamics of a stochastic classical point charge moving inside a spherical charge — a charged particle bound by a real, extended charge distribution.Title; The European Physical Journal Plus, volume 141, article 174, 17 February 2026
The method is the caloric one. The fundamental thermodynamic definition of zero temperature is that the ensemble average of heat flow is zero during reversible thermodynamic operations, and it is the second law that guarantees an integrating factor exists, that the caloric entropy differential is exact, and that the absolute Kelvin scale is well defined.The author’s own account of the method, Physics 6 (2024), Section 2.1
Applied to interacting electric dipole oscillators, and again to radiation in cavities whose walls are deformed or moved, the demand of no heat flow at zero temperature yields the same answer both times: the average energy per mode must be a constant times frequency, and the constant is fixed at one half of the reduced Planck constant by matching the measured form of van der Waals and Casimir forces.The author’s own account of the method, Physics 6 (2024), Sections 3.1.2 and 3.2, and Conclusions
The system treated here is a different and more physical starting point from the electric dipole simple harmonic oscillators of that earlier work: a point charge inside a spherical charge is bound by electrostatics itself rather than by an idealised spring, so the derivation runs on a system made only of charges and fields.Title, read against the systems treated in Physics 6 (2024), Section 3
Stochastic electrodynamics is entirely classical — the microscopic Maxwell equations plus the Lorentz–Dirac equation of motion for a charged point particle — with one addition: a nonzero fluctuating classical radiation field at zero temperature, serving as the source-free boundary condition for Maxwell’s equations. Deriving that field’s spectrum from thermodynamics is what makes the theory self-contained.The author’s own account of the method, Physics 6 (2024), Section 1
Two questions stay open in the programme and are the ones to watch: no precise relationship between caloric and probabilistic entropy has yet been established once zero-point fluctuations are included, and the classical hydrogen simulations that resolved the old atomic-collapse problem now face an ionisation problem instead, with relativistic calculations and chaotic-orbit effects named as the routes to settling it.The author’s own account of the method, Physics 6 (2024), Conclusions
The way in
https://doi.org/10.1140/epjp/s13360-026-07392-3How to cite it
Daniel C. Cole (2026) Applying the thermodynamics of a stochastic classical point charge in a spherical charge to derive classical electromagnetic zero-point radiation doi:10.1140/epjp/s13360-026-07392-3