Entropy Considerations in Stochastic Electrodynamics

Abstract

The use of entropy concepts in the field of stochastic electrodynamics is briefly reviewed here. Entropy calculations that have been fully carried out to date are discussed in two main cases: first, where electric dipole oscillators interact with zero-point, or zero-point plus Planckian, or Rayleigh–Jeans radiation; and second, where only these radiation fields exist within a cavity. The emphasis here is on the first, more complicated, case, where both charged particles and radiation fields are present and interacting. Unlike the usual exposition on entropy in classical statistical mechanics, involving probabilistic notions of phase-space occupation, the calculations to date for both particles and fields, or for fields alone, follow the caloric entropy method, where the notions of heat flow, adiabatic surfaces, and isothermal conditions are utilized. Probability notions certainly still enter into the calculations, as the fields and charged particles interact stochastically together, following Maxwellian electrodynamics. Examples of phase-space calculations for harmonic oscillators and classical hydrogen atoms are carried out, emphasizing how much farther caloric entropy calculations have successfully gone.

Keywords: stochastic electrodynamics; classical physics; harmonic oscillator; entropy; thermodynamics; statistical mechanics; electromagnetic radiation

1. Introduction

This paper discusses aspects of entropy calculations in the theory of nature usually referred to as “stochastic electrodynamics” (SED). This theory involves only classical physics, where by this is meant the electrodynamics described by the microscopic classical Maxwell’s equations plus the relativistic version of Newton’s equation of motion for a charged point particle, or, in other words, the Lorentz–Dirac equation. However, it is recognized in SED that to properly describe nature requires that electromagnetic fields and particle motion must allow for a fluctuating behavior, even at temperature zero. For electromagnetic fields, this consists of nonzero fluctuating classical radiation at temperature zero, or “zero-point” (ZP) radiation, that satisfies Maxwell’s equations. These ZP fields serve as the homogeneous or source free boundary conditions for Maxwell’s equations, present even when radiation sources of charges and currents equal zero, just as occurs when classical thermal radiation fields exist in a cavity with no free charges present.

ZP radiation obeys a number of important physical properties such as the more obvious requirements of homogeneity in space and time and isotropy in direction in every inertial frame, but also much more complex properties such as being scale invariant and conformal invariant. This paper concentrates on two of these ZP properties, namely, that the spectrum must be Lorentz invariant, so that all inertial frames see the same spectrum; and that the fundamental definition of zero temperature must be obeyed by ZP radiation of no heat flow during reversible thermodynamic operations. In addition, ZP radiation obeys a number of other interesting and important properties that are discussed more in some of the reviews of SED.

The classical theory of SED is able to successfully describe a range of natural phenomena in agreement with quantum mechanical (QM) theory. The results are unexpected in that many quantum phenomena can be understood qualitatively and quantitatively with this inclusion of the ZP behavior of fields and particles. As an example of the comparison between SED and QM and quantum electrodynamics (QED), Boyer showed that for the simple harmonic oscillator (SHO) system, SED agreed with QED for all temperatures greater than or equal to zero, provided the quantum operator orders were symmetrized, and as well agreed with QM in the “resonance approximation” of small charge. The complicated fully retarded, valid at all distances, van der Waals forces between atoms, as modeled by electric dipole SHOs, also share this agreement, as do Casimir forces between continuum materials. These agreements hold for all temperature conditions. Even the “atomic collapse” problem of Rutherford’s classical “satellite model” seems to be resolved once ZP radiation is taken into account. The qualitative mechanism was first proposed by Boyer and has since been shown in numerical simulations. However, interestingly enough, ionization problems, rather than atomic collapse, are now the concern. Possibilities of more accurate relativistic calculations and consideration of numerical-based “chaotic effects” when the classical electron’s orbit becomes relatively large, may be points that could rectify this case.

More recent interest in SED has been generated by hydrogen-based simulations and analytical explorations of this nonlinear system. The nonlinearity of this atomic system makes the SED calculations particularly complicated. Other nonlinear systems are known to have different physical predictions between SED and QM. However, some researchers feel that agreement will be found to hold when accurately considering physical atomic cases that exist in nature, such as hydrogen, as opposed to arbitrary nonlinear binding systems not found in nature.

It should be noted that the electromagnetic radiation fields in SED are indeed the fields of classical electrodynamics, so that the field fluctuations are also those of classical physics. Surprisingly, the list of physical effects just discussed, which are normally thought to be explainable only by quantum theory, can be understood via SED. The reason for this, at least regarding linear systems, is shown in detail by Timothy Boyer, where expectation values of various physical quantities are compared between SED and QM or QED. In the SED case, the expectation values of the radiation fields are based on classical probability methods arising from the classical stochastic fluctuations of the radiation fields. In the QM and QED cases, Boyer showed that these expectation values, calculated using the very different physical notions of photon annihilation and creation operators and upon being symmetrized, agree exactly with the corresponding SED quantities.

The possibility has been raised by researchers that SED may be a more fundamental theory than QM, in that QM may be derivable from SED, but not vice-versa. However, there are many unsolved problems in SED, including a full understanding of hydrogen line spectra and a deeper understanding of excited states, diffraction and interference patterns of charged particles, and creation and annihilation operations of charged particles. For some of these, qualitative, and sometimes deeper, explanations exist, such as for the wavelike behavior of diffraction and interference of particles, photon-like behavior, and superfluid behavior. As emphasized by Boyer, SED provides a classical physics description with the recognition that ZP electromagnetic fields need to be included, resulting in a stochastic classical physics theory that greatly widens the physical phenomena that are addressable, including SHO behavior, Casimir forces, van der Waals forces, oscillator specific heats, blackbody radiation, diamagnetism, and effects of acceleration through the vacuum, all of which agree with QM results.

The present review examines how entropy effects have been included in the analysis of classical electrodynamic systems in SED. In an early SED paper in 1969, Boyer presented physical arguments that there was a need to distinguish between what he referred to as “caloric entropy” and “probabilistic entropy”. Boyer argued that when ZP energy was included the two approaches yield different results. The most detailed analyses in SED have dealt with the former, caloric entropy, which is what this review concentrates on. The main focus in this paper is to contrast the two entropies and to note the rather large number of physical systems that have been appropriately addressed in SED using the caloric one.

As for an outline, Section 2 discusses general concepts of these two entropies. The remainder of the paper deals with calculations involving caloric entropy. Section 3 turns to thermodynamic processes involving displacement operations and temperature changes for interacting electric dipole SHOs bathed in ZP plus thermal radiation. Section 3.1.1 covers the “all distance” case between SHO electric dipoles, while Section 3.1.2 turns to the shorter distance scenario, which results in more recognizable formulae. Section 3.2 discusses the thermodynamics of radiation within cavities that can change in size and shape. The paper ends with concluding remarks in Section 4.

Before proceeding with these discussions, a brief outline of the main results of the thermodynamic operations analyzed in the cited work follows. Let the classical electromagnetic radiation spectrum in thermal equilibrium be a function of angular frequency and temperature. Equation (1) states that one eighth of pi times the ensemble average of the squared electric and magnetic thermal fields, with ZP fields included, equals the integral of that spectrum over all angular frequencies. The change in that quantity is what was actually found, due to displacement and temperature changes. The angular brackets represent an ensemble average over the radiation fields.

Each of the cited references contains a demonstration that for no heat to flow at temperature zero during slow reversible displacement operations of the discussed systems, the spectrum at zero temperature must be proportional to the cube of the angular frequency. Expressing the spectrum in terms of the average energy per normal mode at a given temperature and frequency, equation (2) states that the spectral density is the average energy per mode times the angular frequency squared divided by pi squared times the cube of the speed of light — the prefactor being the number of normal modes per unit volume and per unit angular frequency interval. Hence, at zero temperature, equation (3): the average energy per mode is a constant times the angular frequency.

The references concern electrodynamic systems interacting via either van der Waals force or Casimir forces. To obtain the correct results for these cases requires that the constant scaling factor must be given by equation (4): one half of the reduced Planck constant.

The name “zero-point radiation” has to do with the thermodynamic radiation fields at the absolute temperature zero. It should also be noted that the cited references usually dealt with a related function defined by equation (5), rather than the average energy per mode of equation (2). Either is acceptable, but the average energy per mode has a more relatable physical meaning.

2. Two Forms of Entropy

2.1. Caloric Form of Entropy in Classical Physics

The concept of caloric entropy is a well-defined quantity due to the first and second laws of thermodynamics, with the second law being particularly important here. The second law is what ensures that the differential of caloric entropy is an exact differential and that the absolute temperature Kelvin scale exists and is well defined. Energy conservation holds that the change in internal energy of a system is equal to the heat energy that flows into the system plus the work done on the system — equation (6).

In what follows, ensembles of similarly prepared systems are considered. Taking the ensemble average, represented by angular brackets, and considering for now relatively small changes, reduces the first law of thermodynamics to equation (7): the differential of the ensemble-averaged internal energy is the sum of the inexact differentials of the ensemble-averaged heat and work. Here the heat and work differentials are inexact, dependent on the path of the thermodynamic process and not on the endpoints. In this paper, and in the papers cited here, the heat term is the small change in the ensemble average of the heat radiation that flows into the system of interest, while the work term is the same for work done on the system. As known in mathematics, the sum of two inexact differentials can sum to an exact differential, as is the case here for the internal energy.

From the second law of thermodynamics, an integrating factor can be proven to exist for the heat flow into the system during any reversible process, so that equation (8) — the reversible heat flow divided by that integrating factor — is an exact differential of the caloric entropy function. Here the differential is the difference between two entropy surfaces, and the subscript indicates that reversible thermodynamic processes are being considered. The differential relationship for the function of state exists because of the second law of thermodynamics. For the systems discussed in Section 3.1 in this paper, where there are N electric dipole SHOs in three-dimensional space, the caloric entropy is a function of 3N independent thermodynamic coordinates for the position coordinates of the oscillators, plus one more coordinate, namely, the temperature of the system, so 3N plus one independent thermodynamic coordinates.

Moreover, not only does an integrating factor exist for the heat flow of any system, but this integrating factor can be expressed as equation (9), an as yet undetermined function of temperature — the same function for any system — times a function of the caloric entropy. Because of equations (8) and (9), the ratio of the reversible isothermal heat flow at one temperature to the reversible isothermal heat flow at another, where both isothermal curves traverse between the same two entropy surfaces, is given by equation (10): the ratio of the temperature functions alone, the entropy integrals cancelling. Using the choice in which that function is the temperature itself, equation (11), results in the absolute temperature Kelvin scale and also results in the concept of being at a “zero-point” temperature with no heat flow at zero temperature. Note that systems can fluctuate in energy at zero temperature, but the ensemble average of heat flow at zero temperature is zero during this reversible thermodynamic process, as equation (12) states.

The calculations in Section 3 below proceed from this overview. In the case of electric dipole SHOs in Section 3.1, these calculations follow from the electromagnetic ZP fluctuations and how they propagate to the fluctuations of the SHOs. In Section 3.2, just field fluctuations are considered within cavities of electromagnetic radiation. The ensemble averages of the heat flow, the internal energy, and the work done are all carried out with respect to the impact of the radiation fluctuations on the net systems.

2.2. Probabilistic-Based Entropy in Classical Physics

2.2.1. Main Points of Probabilistic Entropy and SED

Here, the suggestion by Boyer is used to refer to the “probabilistic entropy” as being related to the number of energy microstates of a system that provides the same macrostate, such that equation (13) holds: the entropy is Boltzmann’s constant times the natural logarithm of that number. In the original developments of statistical mechanics by Ludwig Boltzmann, James Maxwell, Josiah Gibbs, and others, the notions of probabilistic entropy and the microstate count occurred before the start of QM. As originally conceived, the measure or size of the phase space of a mechanical system at a constant value of energy was key to determining the probability of the system being in this net “energy state”. All phase-space points with the same energy value were presumed to be equally probable to occur. A key example of such a mechanical system was a point particle with a given mass linearly bound by position in three-dimensional space to an equilibrium point. If this quite simple classical mechanical system was in equilibrium with a heat reservoir at a constant temperature, then all phase-space points for particles — a six-dimensional phase space with three dimensions in position and three in momentum — with the same energy were assumed to be equally probable when computing the probabilistic entropy in statistical mechanics for classical physics.

Once one alters the SHOs to being N electric dipole oscillators, each interacting electromagnetically with each other as well as with ZP radiation or ZP plus Planckian radiation, the case certainly becomes significantly more complicated regarding the probability and “counting” of phase-space points with equal energy in this 6N-dimensional phase space. This leads us to recognize one important point in SED: that to date there is no direct connection between the probabilistic entropy of equation (13) and the ensemble average of heat flow and work done, as was discussed for the caloric entropy in Section 2.1. The probabilistic entropy may not satisfy equation (8) with it substituted for the caloric one, especially with the zero-point fluctuations needing to be taken into account. In contrast, when combining QM with statistical mechanics notions, the situation is quite different. Counting of equal energy microstates is well defined in this case and derivations are available and taught on how to calculate internal energy and work done; the use of the partition function for particles’ energies is a common tool to simplify this process.

The point to make here is that in SED, except in the simplest of cases such as a single SHO, the phase-space idea has not been used to calculate ensemble averages of internal energy, heat flow, and work done. This does not mean that probability values for equilibrium cases, ensemble averages, and so on, have not been calculated in SED. They certainly have. The literature provides a method and examples of such calculations, starting from the probability notions of ZP plus Planckian radiation and propagating to the interaction behavior of charged point particles. As early as in 1963 the probability distribution for a single particle in an SHO binding potential was calculated with ZP radiation present. More complicated systems, such as described in Section 3, have not been tackled in any sort of detail using phase-space notions, with part of the problem being how to include phase-space notions with ZP fields included in the analysis.

As an aside here, the original ideas of Boltzmann and others for “counting” or including phase-space states in classical physics are illustrated here, before turning to caloric entropy notions in Section 3. The expressions follow the early classical ideas of these quantities, although I am not aware of these calculations being published or examined in the past in any sort of detail. Let us consider three cases for illustration, namely, the one-dimensional SHO phase space, then the phase space for N independent one-dimensional SHOs, which includes the case of a single three-dimensional SHO, and finally, the phase space for the nonrelativistic classical hydrogen model. In each case, the microstate count is computed as a function of the energy of the system, as if the single system was in thermodynamic equilibrium with a heat reservoir.

2.2.2. Microstate count for a one-dimensional harmonic oscillator

Let the infinitesimal phase-space “area” for a one-dimensional SHO be denoted by the product of an infinitesimal length and momentum in phase space, with the nonrelativistic momentum being mass times velocity. This example is discussed by Frederick Reif in a helpful qualitative sense, but the analysis is not carried out fully, which leads to a somewhat unexpected finding that is pointed out at the end of the present calculation.

The “number” of phase-space regions of that size within an energy interval is given by equation (14), an integral over position and momentum of a Dirac delta function selecting the states of the chosen energy, divided by the phase-space cell size. (The evaluation — equations (15) to (22), selecting the two real roots of the momentum, integrating first over momentum and then over position out to the turning points of the ellipse, and reducing the position integral to a standard arcsine form — is omitted for length; the complete text is at the source.)

Hence, equation (23): the number of phase-space regions of the given size within an energy interval is two pi times the square root of the mass over the spring constant, times the width of the energy interval, divided by the phase-space cell size. Thus, for the one-dimensional SHO, the somewhat unexpected result is that this number is independent of the energy, so for any value of the energy, the number of microstates does not change. This serves as a counter example to the usual notion that the microstate count increases rapidly with energy.

2.2.3. Microstate count for N one-dimensional harmonic oscillators

Let the infinitesimal phase space for N one-dimensional SHOs be denoted by the product of all the individual position and momentum cells. The “number” of phase-space regions of that size within an energy interval is then given by equation (24). It should be noted that when N equals three, the result will be the same as for three one-dimensional oscillators, which is the same as for a single three-dimensional isotropic SHO.

(The evaluation — equations (25) to (41), integrating over the momenta one at a time, each integral reducing to the same standard form as in the single-oscillator case, with the successive integration limits set by the requirement that the remaining energy stay positive — is omitted for length; the complete text is at the source.)

The result is equation (42): the number of microstates within the energy interval scales as the energy raised to the power N minus one, times two to the power N, times pi to the power N, times the mass over spring constant ratio raised to the power N over two, divided by the factorial of N minus one, and by the phase-space cell size. As an aside, let us note that the electromagnetic radiation in a cavity can be expressed as a sum of oscillators of similar mathematical form to the material SHOs discussed above. The relationship of phase space and energy of these “radiation oscillators” should follow the similar development here. The present calculation may prove helpful in pursuing the thermodynamics of this point further.

When N equals one, one recovers the earlier result of equation (23). When N equals three, one obtains the result for a single isotropic three-dimensional SHO, equation (43), which grows as the square of the energy.

2.2.4. Microstate count for the nonrelativistic classical hydrogen model

Of interest now is equation (44), the same phase-space count for a particle in the Coulomb potential of a proton. Let us focus only on the elliptical orbits, or the “bound orbits”. (The evaluation — equations (45) to (54), carrying out the momentum integral at fixed position and then the position integral over the region accessible to a bound orbit — is omitted for length; the complete text is at the source.)

Substituting the semimajor axis in terms of the binding energy gives equation (55), and hence equation (56): the microstate count goes as the mass to the three-halves power, times the electron charge to the sixth power, divided by the magnitude of the energy raised to the five-halves power. Since the energy of an elliptical orbit is minus the squared charge divided by twice the semimajor axis, as the semimajor axis increases the energy decreases and the microstate count increases, as might be expected.

As mentioned in Section 1, hydrogen has become a key interest in further developments of SED. The above result and development may be helpful here.

2.2.5. Some Summary Points on Probabilistic Entropy

In conventional statistical mechanics with QM systems, the microstate count for a system in thermodynamic equilibrium with a heat reservoir at constant temperature is arrived at by “counting” the quantized energy microstates of a system. In classical physics, the microstates are the “size” of the net phase space divided by a relatively small phase-space size, such as the position-momentum cell for the one-dimensional SHO, such that all phase-space points in this phase space have an energy within the chosen interval. Three such examples were analyzed here to illustrate the dependence of the size of the microstate count as a function of energy. Although illustrative of the basic ideas, these examples do not bring us closer to handling the types of systems in SED that are discussed in Section 3 below using caloric entropy. The methods in the cited literature could be used to properly propagate the stochastic nature of the ZP or ZP plus Planckian fields to the particle motion. For a single SHO, or multiple SHOs that are not interacting, the calculations would not be so complicated and have actually been performed for a single SED oscillator some time ago. However, when N electric dipole SHOs are interacting, as is the case in Section 3, this method seems quite prohibitive.

3. Calculations in SED Using Caloric Entropy

3.1. N Three-Dimensional Electric Dipole Oscillators Bathed in Zero-Point Plus Planckian Radiation

3.1.1. All Distances and Temperatures van der Waals Conditions

An arbitrary, but fixed, number N of electric dipole oscillators were studied in earlier work, in interaction with stochastic “incident” electromagnetic radiation fields. Specific incident and thermal radiation examples were examined for the possibility of satisfying various demands such as “no heat flow” during reversible displacement operations, the third law of thermodynamics, and restrictions on specific heats. ZP and ZP plus Planckian radiation satisfied these constraints. The positions of the oscillators were arbitrary. The resonant frequencies could readily have been made arbitrary but for convenient purposes were all held to the same value. All distances were allowed and interactions between electric dipole oscillators were executed by the van der Waals force expressions, valid for all distances. The ensemble average of the full retarded force between oscillators was obtained previously, which was used for determining the “work done” when slowly displacing the oscillators. Calculations for ensemble averages of changes in internal energy were carried out for quite a large volume surrounding the oscillators, where the surface of this volume was far from any of the oscillators. By finding the change in ensemble averages of internal energy and work done during slow reversible displacements of the oscillators from each other, the ensemble average of the heat flow due to radiation across the surface bounding the volume was deduced.

The expectation value of the change in internal energy within the volume due to displacement and temperature changes was calculated in that work. This change in internal energy was due to changes in the kinetic and potential energy changes of the oscillators as well as changes in the far more complicated electromagnetic field energy — in centimetre-gram-second units, one eighth of pi times the volume integral of the ensemble average of the squared total electric and magnetic fields — where the total fields were due to the “incident” radiation plus the radiated fields from each of the N oscillators. Later sections of that work considered the specific “incident radiation” cases of thermodynamic candidates of ZP, ZP plus Planckian, and Rayleigh–Jeans radiation. Because of the square of the fields in the above expression, cross-terms existed for the oscillator fields and the oscillator-incident fields. These represented the longest of the calculations, published as thirty-nine pages of an appendix. For isothermal reversible thermodynamic displacements, the only expectation value of a nonzero spectrum of radiation that resulted in no heat radiated in or out of the volume, was found to be the ZP radiation spectrum of equation (3).

A rather unexpected result was that the radiated energy from the oscillators, as if they were single oscillators radiating, increased in accordance with the size of the volume, but was canceled out by a similar term, a cross-term between the incident fields and the dipole fields. A similar cancellation occurred between the cross-term field energy of the oscillators and the cross-term field energy of the oscillator fields and incident fields. No approximation was made that the interaction energy was small or large enough between dipoles, which is often made in statistical mechanics assumptions of particle interaction versus interaction with a heat reservoir.

In this manner, the expectation value of the work done and change in internal energy was found for quite a large volume enclosing the oscillators and their displacement positions. The proof of no heat flow for nonzero “incident” radiation during reversible displacement operations for N oscillators was found valid only when a ZP radiation spectrum existed, coinciding with the definition of zero temperature. To show this, the “steady-state” solution to the radiation fields acting on the dipole oscillators was calculated, including the interaction in this solution of all dipoles acting on each other. The stochastic nature of the fields then propagated to the stochastic behavior of the SHOs.

Later work turned to taking the heat flow calculations to obtain the caloric entropy differential from equation (8) during reversible displacement operations for isothermal conditions of constant temperature. This process was then combined with taking into account heat flows when the dipole positions were held fixed, but the temperature within the volume of both fields and dipoles was changed reversibly by considering accumulated infinitesimal interactions with heat reservoirs of slowly changing temperature. The net result was an expression for the caloric entropy that was a function of the 3N plus one thermodynamic coordinates. The result enables one to analyze arbitrary reversible processes for this system, including a cycle such as the Carnot cycle of two processes performed isothermally, bordered by two adiabatic processes with no heat flow. The latter is achieved by changing the position of the oscillators while also changing the temperature, to ensure that the caloric entropy differential remains zero. Adiabatic surfaces in the 3N plus one thermodynamic coordinate space can then be constructed.

3.1.2. Unretarded van der Waals Condition and Resonant Oscillator Approximation

To introduce the calculations that are presented in this section, a deeper physical understanding of the earlier work is first provided. The behavior of each of the N electric dipole three-dimensional SHOs was first written as the mass times acceleration being equal to all the forces on the oscillating electric dipole mass, which included the SHO binding force of the oscillator, the Lorentz force due to the electromagnetic radiation that the oscillators were bathed in, the radiation reaction term acting on the oscillating charge, and finally, the Lorentz force from all the other N minus one electric dipole oscillators. Thus, N coupled stochastic ordinary differential equations were written that needed to be solved simultaneously. These equations have previously been solved. In this way, the stochastic variation in the radiation fields in a large volume encompassing the N electric dipole oscillators created the corresponding stochastic behavior of the oscillating electric dipoles. The stochastic nature of the entire system of fields and charged particles resulted from thinking of each set of “volume of fields” and charged particles as but a member of an ensemble of similar systems. As prescribed in SED, each member of the ensemble evolves deterministically, meaning that once the initial conditions of the fields and charged particles in a single system are set, then the evolution of the system evolves according to Maxwell’s equations and the Lorentz–Dirac equation. For this study, the nonrelativistic version of the Lorentz–Dirac equation was used, but for SED in general, the relativistic version would be used.

As mentioned in Section 3.1.1, the earlier work and its appendix took these results and found the ensemble averages of the changes in the energies of all the parts of the problem, meaning the kinetic and SHO potential energies of the electric dipoles, plus the electromagnetic field energies of the oscillators and their cross-term energies with each other and with the incident radiation fields. To simplify the lengthy calculations and still see if no heat, or no change in caloric entropy, occurred in reversible position displacement operations at zero temperature, a later paper took those calculations and made two approximations: the small-charge limit was made so that a “resonant oscillator approximation” could be invoked; and the unretarded van der Waals approximation was made, in which the resonant frequency times a typical distance between oscillators, divided by the speed of light, is much smaller than one.

This resulted in the following sums, looking much more like QM results for sums of quantum SHO energy terms. The ensemble average of the potential energy of the oscillators was found to be equation (57), a double sum over the N oscillators and their three degrees of freedom, of the spectral function evaluated at the system eigenfrequencies times the ratio of the squared natural frequency to the squared eigenfrequency; while the corresponding kinetic energy was the sum of equation (58), the same double sum without that ratio. Here the first index sums over the N oscillators, the second over the three degrees of freedom of each oscillator, the natural frequency is the resonant frequency of each oscillator, and the eigenfrequencies are those of the system once electromagnetic interactions are taken into account. Finally, the unretarded electromagnetic dipole–dipole interaction energy was found to be equation (59), the same double sum weighted by the difference of the squared eigenfrequency and the squared natural frequency, over the squared eigenfrequency.

The result for the net of these became equation (60): the three energies sum to the plain double sum of pi squared times the spectral function at each eigenfrequency. Making the same approximations for the “work done” when slowly displacing the oscillators from each other resulted in equation (61), an integral over the eigenfrequencies of the spectral function divided by the eigenfrequency.

From equations (7), (60) and (61) one obtains equation (62): the change in the total oscillator-plus-field energy minus the work done equals a double sum of integrals of the derivative of the spectral function with respect to eigenfrequency, minus that function divided by the eigenfrequency. This equals zero when the spectral function is proportional to the eigenfrequency, no matter the positions of the oscillators. Since the average energy per mode is pi squared times that spectral function, equation (63), one again arrives at the statement that the average energy per mode at zero temperature is a constant times the frequency. The value of that constant as one half of the reduced Planck constant is then found by the required connection to the van der Waals force used throughout the calculations.

Although still not exactly simple to carry out, this analysis is far more succinct and less involved than the original. Moreover, it is good to see that even when the approximations of unretarded van der Waals expressions and the resonant approximation of the oscillators are made, the final result for zero temperature still holds.

3.2. Dynamics Involving Cavities and Fields Using Caloric Entropy

The analysis outlined in Section 3.1 for deducing the needed ZP radiation spectral form at zero temperature is important in that both radiation fields and charged particle interactions are included, yet still the result that the average energy per mode is a constant times frequency is deduced. However, this result also holds up if one only considers radiation fields between two parallel plates or within a cavity. Earlier references examined the case where the walls of the cavity are deformed in shape and size; in addition, one of them considered the case where the temperature of the radiation can be changed. Related material considered only temperature changes for a relatively large volume of radiation and where electric dipole oscillators were held fixed in space. Two forms of a generalization of Wien’s displacement law, a generalized derivation of the Stefan–Boltzmann law, and the third law of thermodynamics were investigated there. The cavity references also carried this work out but took it farther, uncovering more about the dynamics of thermodynamic radiation in cavities. Much of this needed to be studied because the early work on blackbody radiation, such as that by Max Planck and Wilhelm Wien, made the implicit assumption that thermodynamic radiation within a cavity would vanish at temperature zero. These generalizations, including being needed in the Stefan–Boltzmann law, were carried out in that work.

This research was performed making use of the concept of caloric entropy as opposed to probabilistic entropy. Changes in internal energy for a cavity were calculated. Infinities due to ZP radiation vanished because only changes in this energy contribution were taken into account. “Work done”, as when a piston moves within a cylinder, or when the walls of a cavity are deformed, where the cavity contains ZP plus Planckian radiation, were also carried out within the concept of caloric entropy. To be as general as possible, Casimir forces between walls were taken into account here.

Certainly, Casimir forces between walls of a cavity are normally too small to be taken into account experimentally, except perhaps in a thought experiment. Nevertheless, theoretically they can be included in the analysis, which they were. Possibly, as further experimental means are made possible for micron-sized cavities, such as considered in “quantum cavity electrodynamics”, then this consideration becomes more feasible experimentally.

A key difference between the two cavity references is that one focuses more on operations like a piston in a cylinder, as in Wien’s displacement theorem analysis, while the other considers more general operations like elementary movements or deformations to the walls. In both cases, though, the result still arises that in order to have no heat flow in or out of the cavity, the average energy per mode at zero temperature must be a constant times the frequency, with one half of the reduced Planck constant being the needed value to fit the form of Casimir forces. In addition, one of them calculates the caloric entropy as a function of temperature and distance between the walls. Many of the results were deduced from the demands of the second law of thermodynamics, the thermodynamic definition of temperature, and the recognition that laws such as Wien’s displacement law need to be generalized to apply to radiation in a cavity without the assumption that thermal radiation must vanish at zero temperature.

4. Concluding Remarks

This paper reviewed the cases for probabilistic and caloric entropy, recognizing, along with Boyer, that there is a difference in these quantities since ZP fluctuations need to be taken into account. Use of caloric entropy however can be carried out for highly complicated systems, with results that make reasonable sense. This quantity was discussed in Section 3.1.1 by changes in ensemble averages over oscillator energies plus field energies. Making unretarded van der Waals approximations and the resonant oscillator approximation, as in Section 3.1.2, greatly simplifies the expressions, but still results in the same condition at zero temperature of the ZP radiation spectrum, where the spectral density is proportional to the cube of the angular frequency. The same result occurs when considering the thermodynamics of deformable cavities, as in Section 3.2.

The zero-temperature condition of Lorentz invariance that yields this same result is in some ways more fundamental; each inertial reference frame should “see” the same ZP spectrum. But in another sense, the “no heat flow” at zero temperature for reversible displacement operations also holds in just cavities of radiation, thereby also serving as a fundamental stipulation. Moreover, this result needs to hold for an arbitrary number of dipole oscillators, so that fields plus oscillators must obey this result. If the analysis was possible for hydrogen or other atoms, one would also expect the same result to hold.

Without doubt, far more work has been performed in SED using the concept of caloric entropy as opposed to probabilistic entropy. A precise relationship between the two has yet to be determined, except for the recognition that the results of using either are quite different. Possibly, a deeper connection would provide more detailed information of aid to further SED exploration. Likely the use of concepts as discussed in Section 2 will be of help here.

Funding. This research received no external funding.

Data Availability Statement. The original contributions presented in the study are included in the article.

Conflicts of Interest. The author declares no conflicts of interest.

(The fifty-item reference list is omitted; the complete text is at the source.)