Equilibrium between radiation and matter for classical relativistic multiperiodic systems . II . Study of radiative equilibrium with Rayleigh-Jeans radiation

We continue the study of the problem of equilibrium between radiation and classical relativistic systems begun previously [Phys. Rev. D 27, 1254 {1983)].We consider the emission and absorption of energy by a relativistic pointlike particle immersed in a Rayleigh-Jeans radiation field. The particle is acted upon by a force which, if alone, would produce a multiply periodic motion. It is shown that radiative balance at each frequency holds. A discussion is given of the results reported in both papers.


I. INTRODUCTION
This is the second paper of a series dealing with the study of the equilibrium between radiation and classical relativistic material systems.
In the first paper' we showed that, if a relativistic multiperiodic system is im- mersed in a random radiation having a Rayleigh-Jeans spectrum at a given temperature, then the equilibrium dis- tribution of the system is the Maxwell-Boltzmann distribution.In this paper we shall show that, under these con- ditions, the radiation field is also in equilibrium at each frequency.That is, the power emitted and the power ab- sorbed by the system exactly cancel at each frequency on the average.
As is well known, the problem of the equilibrium be- tween radiation and matter (i.e. , the derivation of the blackbody spectrum) was actively studied at the beginning of the century.In the last few years there has been renewed interest in the subject.All previous studies were made in the nonrelativistic approximation.
In this context, Boyer has shown that if we impose energy balance at each frequency and for each state of the mechanical system, we are led necessarily to the Rayleigh-Jeans (RJ) spectrum for the radiation and the Maxwell-Boltzmann (MB) distribution for the material system.The same con- clusions are obtained if we impose the less stringent and more physical condition that the energy balance is satis- fied at each frequency (" radiative equilibrium" ).Howev- er, for relativistic systems, Boyer has suggested recently that the MB and RJ distributions might be incompatible.
We have shown in Ref. 1, as mentioned above, that this is not the case.
In order to give a complete proof of the existence of an equilibrium state, it is necessary to show that not only does global equilibrium exist"but so does radiative equili- brium, that is, equilibrium at each frequency.This guarantees that the spectrum of the radiation does not change with time.The proof of the radiative equilibrium is the main purpose of this work.It is necessary to point out that we are considering the equilibrium between radia- J tion and matter at a classical level.Therefore, we consider an isolated cavity with perfectly reflecting walls and a ma- terial system inside.With these conditions, the radiative equilibrium is equivalent to Kirchhoff's law, which is a well-known consequence of the general principles of ther- modynamics.However, it has been conjectured ' that the Kirchhoff law might not be valid if one assumes the ex- istence of a nonthermal zero-point field such that any wall would be partially transparent to it.The plan of the paper is as follows.In Sec.II, we shall calculate the emitted power per unit frequency interval and in Sec.III, the absorbed power.Everywhere we fol- low the notations and the techniques of Ref. 1.The calcu- lation of the absorbed power rests upon a method which is explained in Appendix A. In Sec.IV, we discuss the re- sults obtained.Finally, in Appendix B we discuss the relation between the emitted and absorbed power at every frequency and the reduced Fokker-Planck equation.

II. EMITTED POWER
As in Ref. 1, we consider a particle which, if unper- turbed, would describe a multiply periodic motion.That motion is actually perturbed by the action of the random radiation and by the radiation reaction.However, we as- sume that these perturbations are small, in the sense that their effect in a deterministic period is negligible, so that, to lowest order, the power emitted can be calculated from the deterministic (i.e. , unperturbed) motion averaged over all orbits with a suitable distribution function Wo in phase space.
Now, according to Ref. 1, for a multiply periodic sys- tem immersed in a Rayleigh-Jeans radiation, 8'0 depends to lowest order only on the energy, and then on J. Also, for multiply periodic motions the ensemble average over an orbit equals the time average' which we must use in order to decouple the Fourier components of the motion.
(2.2) + lim -f dt lim r f dQ5'E, . (2.9) Therefore, the average in phase space can be put as a time average over each orbit ensemble averaged over phase space.
The emitted power per unit frequency interval is given by the energy carried per unit time by the Fourier com- ponents of the field created by the particle with frequencies between co and co+d~.In order to calculate it, we take a sphere X centered on the origin of coordinate 0, that we take in such a way that the deterministic motion develops around it.The power emitted is given by P, = lim -f dt lim r f dQE, T o .-4~( 2.10) A procedure similar to that used in Ref. 1 with h~a nd g shows that E, can be expanded in the form The second integral is of order I/r because in a multiply periodic motion the acceleration and the velocity are bounded, and the RE~is also bounded.Therefore that in- tegral goes to zero in the limit r ~oo.Consequently P, = lim f der S= .lim r f dQr S, .
Now, in the limit r~oo, we have the corresponding average emitted power is given by P, (no)= lim -f dt lim r f dQ In view of that, the emitted power at a frequency ~, Ie(~) can be written, after averaging over phase space, n=r +5, 5=0 (2.8a) with 6-r"'=0 so that 1,(~)= f dgWo(g) X g'5(con ore)[ P( ego)+Pe(n.coo)] ñ (2.15) where g' is the sum extended to those values of n such that M ll co p = g n; cot )0 .
Let us calculate the term P, (n cop) given by (2.14).To do that, we must calculate the Fourier coefficients of the electric field E(t), As is usual (see Ref. 1), we make the change in the integral t~t'=t", (t, r ).With the reasoning of Ref. 1 we write and 1 r, e n X [(n P) Pn-. dt Now, in the limit r~no, R =r rg and n =r +6 with 5 given by (2.8a), whence we have, neglecting terms in llr, l,er X[(r p)X-p] in t-0 t' -in tn rit in ta (rn g/n) E = hmdt'- (1 Pr ) -dt 1 -P r whence we obtain, after an integration by parts, and, finally, it results in The product between heavy parentheses can be written P(ti) (I rr ) 13(t~)with the notation of Ref. 1.With the change t( t2 -u and using -(2.9) of that reference we get (2.22)P, (n cop)=-f dw lim -f du e f dQP. (I rr ) P "ex -in p7.0[8'](co,w, u;cop)'=c f dQexp i r(r r") -P (I r-r ) P "= g cr -[8'](to, cop)e'" e, (2.24)   C n, m and therefore e (n'cop) . (n cop f dw lim -f du e '"'" " g cr [N'](n.
where we have used and the fact that the term under the integral sign does not depend on w.
III. ABSORBED PO%'ER 2 f dQ e(k, A, ) A, =1 Xcos[ k. r -co't +8( k, A, ) ], (3.1) where 8 is a random phase with uniform distribution.If we consider a given sample of the background stochastic field, we assume that the state of the system is given at time t by the phase-space distribution p(g, t) so that the absorbed power from E"~" is given by P, (co,co+bco, t)= f dgV.eE c, (g, t)p(g, t) . (3.2) If we average to all samples of the stochastic field we have (P, (co,co+bco, t)) = f dgeV (E"o, "(g,t)p(g, t)) . (3.3)We must obtain the energy absorbed by the particle from the background field.As the magnetic field does not produce work, we should consider only the electric field.We must obtain the absorbed power by the particle from the component E"~"(r, t) of the field whose angular frequency is between co and co+A,co.This can be written The absorbed power at the stationary state will be ob- tained taking the limit P, (co,co+ hco) = lim (P, (co, co+bco, t) ) t~ce = lim f dgeV (E"~"(g, t)p(g, t)) . (3.4) t~00 If we write the probability density as a sum of the equilibrium density 8'ouncorrelated with the field- plus fluctuations 5p, we see that the absorption of energy is due to the correlation between the stochastic field and the density fluctuations.
In our problem, the stochastic force is not a white noise and the correlations must be calculated with some approx- imation procedure.Problems of this type are usually solved using Kubo' linear response theory which, in fact, has been applied to the nonrelativistic theory of radiation-matter equilibrium.
Kubo theory allows the calculation of the power absorbed by a system in statisti- cal equilibrium from an external electromagnetic field up to lowest order.It is assumed that statistical equilibrium exists before the external field is applied.However, in our case this condition is not fulfilled because the equilibrium is produced by the same external field (the background radiation) from which the absorption must be calculated.
Therefore it should be necessary to perform a dynamical calculation of the correlation between E ~"and p.In Appendix A, we develop this calculation by a method which allows to obtain the correlation from a series of cu- mulants which converges asymptotically.
If the spectrum of the radiation is of the RJ type -i.e. , 8' =constant -Wo is to lowest order a function of the deter- ministic energy alone.Then "Wo --VJ "Wo[8'] .Bpj Therefore 3 ~u g T, , (V ") "Wo(g)=(I rr ).V- ap, " whence it follows (3.6) (3.7) and the absorbed power per unit frequency is f dg f du f dQV. (I rr ) V -"cos r(r r-") -eau -Wo . (3.9) It is possible to extend the integral over u to negative values.To do that we consider Eq. (3.9) for u &0, 2 0 R = e3' -f dg f du f dQV (I rr ) V "c-os r(r -r")--cou Wo C -00 c (3.10) Then, we can change u'= -u and g'=P, s'o that dg=dg' and g=g " with the result that Eq. (3.10) agrees with (3.9).
Therefore, we can write f -dg f "du f dQV. (I rr ) V -"cos -r (r -r ")-cou Wo.We shall study in Appendix B the relation between the emitted and the absorbed power per unit frequency, and the coefficients of the reduced Fokker-Planck equation of Ref. 1.Moreover, we shall show that in the case of the Rayleigh-Jeans spectrum a stronger type of balance which implies not only radiative balance (3.14), but also detailed balance, is satisfied.

IV. DISCUSSION
The conclusion of our work (including Ref. 1) is twofold: First, that a relativistic material system immersed in a background radiation with a Rayleigh-Jeans spectrum approaches a stationary state given by the Maxwell- Boltzrnann distribution secondly, in this paper we have proven that in this state the system is also in radiative equilibrium, which means that the energy of the system does not change on the average and that the spectrum of the radiation does not change either.In this paper we have not proven that it is not possible to have equilibrium with a radiation having a different spectrum.The proof of this statement can be done with a particular system showing that equilibrium does not exist if the radiation st &c «&r d &c «&r ~(4.1a) (4.1b)where r, is the correlation time of the stochastic force, r" is the relaxation time, that is the time needed for the ef- fect of the force to be non-negligible.Here st stands for stochastic and d for damping.Also, in order to use the Haken' reduction procedure, it is necessary for the deter- ministic time to be small in comparison with both relaxa- tion times: The deterministic time T is essentially the period of the deterministic motion.This condition is necessary in order that the orbit is not changed too much by the action of the stochastic and damping forces.
The first problem we have is that the Rayleigh-Jeans spectrum does not have a correlation time because it diverges at high frequencies (ultraviolet catastrophe).The problem of the divergences is a fundamental one of classi- cal electrodynamics and we will not study it here in detail.
A possible solution is to use extendedinstead of pointlikeparticles.The extended particle produces an effective cutoff in the frequencies of the stochastic force whose inverse can be taken as a correlation time.For some reasonable models of extended particles' we have c -&O&O (4.3) spectrum is different from the RJ one.Actually, this has been made with the nonrelativistic anharmonic oscillator.
Our results show that there is no inconsistency between MB and RJ distributions, contrary to Boyer's conjecture, thus putting on a firm basis classical relativistic statistical mechanics.However, there is a difficulty with the fact that the RJ distribution is not a true spectrum, because it gives rise to a divergent energy density.We shall com- ment on this problem below, but before that we shall study the validity of the approximations involved in our study.
Obviously, the first limitation comes from the difficul- ties of the classical electrodynamics of point particles.In our work, we have considered all deterministic orbits, in- cluding the ones with an energy near the minimum of the potential (assuming that it exists).When the size of the orbit is of the order of the classical radius of the electron, the model of a point particle lacks its validity, i e , t.he.
Lorentz-Dirac equation is no longer correct.In order that this difficulty may be overcome it is necessary that such orbits have a small probability.As the average energy of the MB distribution is k8=m 9' (8 being the absolute temperature), we see that our results will be not valid for too low temperatures.
Of course, this problem is not specific of a relativistic theory.
The main approximation involved in the solution of the stochastic differential equations of motion is the Markovi- an one."' This approximation rests upon the assump- tion that the forces due to the damping (radiation reac- tion) and the background field are small.More precisely, the following inequalities should hold: y «mc (y -1) .
3c T Writing kO q -6, mc As we shall see later, an estimate of condition (4.2b) leads to the inequality T »~, for moderately relativistic temperatures, which guarantees that, under those condi- tions, inequality (4.1b) also holds.Moreover, at the equilibrium state, both relaxation times r'"' and ~"should be of the same order.Therefore, the estimation of the condition (4.2b) alone is enough to guarantee the validity of the approximation.
It is to be noted that the spectrum of the background fie1d appears multiplied by the absolute temperature 8.As long as the spectrum gives the order of magnitude of the random force intensity, it seems reasonable to think that, if the temperature is too high, the effect of the ran- dom force will no longer be perturbative.An important question is whether or not the bound for 8 allows for rela- tivistic velocities, because, if it does, the results obtained will be valid, at least, for moderately relativistic systems.
What we have to verify is that inequality (4.2b) holds for those orbits for which energies are of the order of the most probable value, that is, kO.Clearly, this condition depends on the deterministic force both because the radiation of energy depends on the acceleration of the charge, and because the average energy depends on the external field.Consequently, our estimation is not necessarily valid for all the systems.
First, we suppose that the averages of the kinetic ener- gy, E, =mc (y -1), and the total one are of the same or- der; and, secondly, we calculate the accelerations as if the orbits were circular.It is clear that only in each particular case can more precise estimates be made.Condition (4.2b) means that the energy lost by the parti- cle in a period of the motion is small in comparison with the total energy, i.e. , we obtain y= 1+5, U /c =5(5+2)(5+1) So that Eq. (4.8) can be written (2+5)(1+5) « T/rp .
(4.8) (4.9) Therefore we see that it is compatible to take T»7"p and 5 of order unity, which means relativistic velocities.
In fact, 5-1 corresponds to U -0.75c .We must see whether or not this value for T is reasonable.With the considerations made above, we can relate the period with the radius of the orbit.In fact, for a circular orbit and, if we take 5-1 =-U /c --, , As we have said above, the radius of the orbit must be much bigger than the classical radius of the electron.
Consequently, we have tion at high frequencies is slower due to the fact that a Fourier analysis of a deterministic orbit gives a small con- tribution to high harmonics.Therefore, we arrive at a nonstationary state such that the spectrum of the radia- tion is of the form cp Q(co), where A(cp) is a function which decreases at high co more quickly than m .The function changes slightly with time so that it decreases at low frequencies and increases at high frequencies.
To see this clearly, we point out that, considering the spectrum with the cutoff, as has in fact been worked out in this paper, for e small enough, the equilibrium solution is still  and also the radiative equili- brium still holds when one neglects terms of order e.
Then, the exchange of' energy between the system and the radiation will be slower and slower and the equilibrium is never attained.Obviously, this type of problem is not specific of a relativistic theory and it was extensively dis- cussed at the beginning of the century giving rise to the birth of quantum theory.

APPENDIX A
The equations of motion for the system that we consid- er can be written 2&7'~-23 T » -6/10 sec .
Let us see in more detail why the nonexistence of a correlation time poses no problem.The condition 7; «~, " is necessary in order that the cumulant expansion appear- ing in the theory of the Markov approximation is conver- gent, at least asymptotically.
If there is not a correlation time, we need another parameter in order to guarantee the convergence.
In order to see what this parameter is, let us note that our calculations are made by introducing a cutoff in the spectrum, which is written co exp(ecp) instead of co .
After obtaining the Fokker-Planck equation with that spectrum it is seen that everything in this equation is finite in the limit e -+0.Inspection of the expression for the diffusion coefficient shows that the convergence depends on the fact that the contribution of the high frequencies to the deterministic motion decreases quickly enough.In other words, the deterministic motion produces an effective cutoff and we have an effective correlation time of the order of the deterministic period.This allows us to work with a spectrum of co type which has not a definite correlation time.
Before concluding, a few words are convenient to clari- fy the real meaning of the results obtained.Let us consid- er a material system in a cavity with walls which reflect perfectly at all frequencies.We assume that at the initial time there is no radiation in the cavity and the material system has a finite energy.Then the system radiates and, after some time, the spectrum of the radiation should ap- proach the Rayleigh-Jeans spectrum at low frequencies.
The exchange of energy between the system and the radia-  and W~the perturbation and where F; is the deterministic force, a F; the radiation reaction, and aE; the background electromagnetic field.The constant a, which is of order e, gives a measure of the intensity of the radiation reaction and the background field.We shall assume that the effect of these forces is small in a deterministic period and besides, that the correlation of the stochastic field decreases quickly with a correlation time ~, such that a~, &&1, which corresponds to conditions (4.2a) and (4.2b).These are the conditions for the validity of the reduced Fokker-Planck equation used in Ref. 1 for the calculation of the probability distri- bution in the equilibrium state.
Gur aim is the calculation of the correlation (E,a"(g,t)p(g, t) ), E"a"being the component of the electric field with fre- quencies between m and ~+Am, and p the probability dis- tribution in phase space.To do that we write the con- tinuity equation for p, T t 'r (t) -Re f dco' -f ' dr, e'"" f du e ' '" f ' dr2 X f dr f dr3 .f .dr, ((eE(g, r) e ' W, ($,0)e 'Wi(g, -r2) &&e ' 'W~(g, -[r2+ +r i])))8p(g), (A33) where we have used and, due to the properties of the action-angle variables, so that = X'p"( J«J «w «r2« ., r~i)e n ~+ W()Q i n. w ~n'~OQ e p= p ( J, J ', w', rz, . . . ,r~i)e '"' e and we have taken into account that 8' is a constant with respect to the deterministic motion.
Now we can analyze the divergences that appear in (A33) when we take the limit t~00.We see that u ap- pears in (A33) due to the action of the classical Liouvillian on the phase-space coordinates.Therefore, we will have, WOQ on the right of e ', some function Therefore, the integral on u gives rise to a term of the form - As a consequence, the divergences appear at those fre- quencies co which agree with some frequency of the multi- ply periodic motion.The important thing is that in each term of the cumulant expansion all frequencies of the multiply periodic motion appear.Therefore, every one behaves as t if co coincides with some frequency n. cop and are bound in the opposite case.As the remaining contri- butions to C'" are of order (av;) ' we obtain, finally, that for frequencies co&n copen, C~r, (ar-, )™ I and if co=n cop for some n, C -t(ar, ) ' for all t.
. has values between y a and y a .Although A for a circular orbit a a~takes the smallest of these values, we consider the largest one which improves the estima- tion. i=1 iu(co -n .PoO) is bounded if co~n.cop and increases as t for co = n.coo. which