Historical research guide · 0458 · 2024
Entropy Considerations in Stochastic Electrodynamics
Daniel C. Cole
In one minute
Daniel Cole, at Boston University, reviews how entropy is actually calculated in stochastic electrodynamics — the classical theory that keeps Maxwell’s equations and Newton’s law of motion exactly as they are, and adds one thing: a real, fluctuating zero-point radiation field that is still there at absolute zero. He separates two ideas usually run together. Caloric entropy is built from heat flow, adiabatic surfaces and isothermal steps; probabilistic entropy counts microstates in phase space. Once zero-point fluctuations are in the picture, Boyer showed in 1969 that the two no longer agree, and Cole’s survey shows that essentially all the successful work has used the caloric version. The central result he traces is a derivation rather than a postulate: demand that no heat flows during slow, reversible rearrangements at zero temperature, and the spectrum is forced to rise as the cube of frequency — energy per mode proportional to frequency — with the coefficient fixed at half the reduced Planck constant by matching to measured van der Waals and Casimir forces.
Connection
Chapter 2 asks what the vacuum is, and this paper answers with thermodynamics rather than quantum postulates: the zero-point spectrum is what the second law forces on a classical field theory once you stop assuming radiation vanishes at absolute zero. That matters for chapter 6 too, because it says precisely what zero-point radiation is — the equilibrium at zero temperature — and therefore that a device drawing work from it is working with a driven, non-equilibrium arrangement rather than an isothermal one.
From the source
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, the zero-point field, which serves as the source-free boundary condition for Maxwell’s equations.Section 1, Introduction, paragraph 1
The zero-point spectrum is derived, not assumed. Requiring that the ensemble average of heat flow be zero at zero temperature during slow reversible displacement operations forces the spectral energy density to be proportional to the cube of the angular frequency — equivalently, the average energy per normal mode is a constant times the frequency — and matching the result to van der Waals and Casimir forces fixes that constant at one half of the reduced Planck constant.Section 1, equations (2), (3) and (4); Section 3.1.2, equations (62) and (63)
The same zero-temperature spectrum is forced by a second, independent route: radiation alone inside a cavity whose walls are deformed or moved, with Casimir forces between the walls included, gives energy per mode proportional to frequency with the same coefficient, and requires generalising Wien’s displacement law and the Stefan–Boltzmann law because the early blackbody work implicitly assumed cavity radiation vanishes at zero temperature.Section 3.2
With the zero-point field included, classical stochastic electrodynamics reproduces results normally thought to need quantum theory: the harmonic oscillator agrees with quantum electrodynamics at all temperatures once operator orderings are symmetrised, and so do fully retarded van der Waals forces between atoms, Casimir forces between continuum materials, oscillator specific heats, blackbody radiation, diamagnetism, and effects of acceleration through the vacuum.Section 1, Introduction, paragraphs 3 and 8
Counting phase-space microstates gives an unexpected result for the one-dimensional harmonic oscillator: the number of phase-space regions within an energy shell is independent of the energy, a counter-example to the usual assumption that the microstate count rises rapidly with energy. For N such oscillators the count scales as energy to the power N minus one, and for the non-relativistic classical hydrogen model it rises as the orbit widens.Sections 2.2.2, 2.2.3 and 2.2.4, equations (23), (42) and (56)
Two open questions are named. 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 — more accurate relativistic calculations and chaotic-orbit effects are the proposed routes to settling it.Section 1, paragraph 4; Section 4, Concluding Remarks, paragraph 4
The way in
https://doi.org/10.3390/physics6040075How to cite it
Daniel C. Cole (2024) Entropy Considerations in Stochastic Electrodynamics doi:10.3390/physics6040075