Equilibrium between radiation and matter for classical relativistic multiperiodic systems . Derivation of Maxwell-Boltzmann distribution from Rayleigh-Jeans spectrum

The motion of a charged pointlike relativistic particle under the action of a given force field plus a random electromagnetic radiation is studied. It is assumed that the given force field alone should produce a multiply periodic motion, which is perturbed by the action of both the random radiation and the reaction damping. The random radiation is represented by a stochastic process and an equation is obtained for the equilibrium probability density of the particle in phase space. In the particular case of a random radiation with Rayleigh-Jeans spectrum, it is shown that the stationary solution, corresponding to radiation-matter equilibrium, is given by the Maxwell-Boltzmann distribution.


I. INTRODUCTION
The derivation of the classical blackbody spec- trum (Rayleigh-Jeans law) was made between 1900 and 1905 with the works of Rayleigh, Einstein, and Jeans, In the years that followed, Einstein himself and others studied different models to ob- tain the classical spectral law, a subject extremely important because the disagreement of that law with empirical evidence was historically the origin of the quantum revolution.' The subject was con- sidered to be definitely settled in 1924 with the work of Van Vleck.However, strangely enough, all papers dealt with nonrelativistic theory.Recently, Boyer has claimed that a classical relativis- tic treatment leads to a contradiction between the Rayleigh-Jeans law for thermal radiation and the Maxwell-Boltzmann distribution.The purpose of this paper is to report a consistent relativistic derivation which proves that the Rayleigh-Jeans law does indeed lead to the Maxwell-Boltzmann distribution.It is interesting to note that a similar situation has arisen with the quantal law.In fact, Boyer also claimed that there was a contradiction between quantum theory, relativity, and statistical mechanics, which has been shown to be not true.
The interest in making a classical relativistic derivation of the blackbody spectrum is the follow- ing.In the first place, it is different to say that classical laws do not agree with experiments (e.g. , they lead to the Rayleigh-Jeans law instead of the correct Planck law), which is well known, than to say that the classical postulates are self- contradictory, which should deny the possibility of a fully relativistic classical statistical mechanics.
In the second place, some doubts have arisen with respect to whether the Planck law, instead of the Rayleigh-Jeans law, could not be derived from classical postulates provided that one includes a zero-point radiation with an ~co ~spectrum (i.e. , the same spectrum of the fluctuation field of the vacuum in quantum electrodynamics).This might lead to a stochastic alternative to quantum theory (usually called stochastic or random electrodynam- ics7).
In this paper we show that the stationary state of a point charge without structure in the presence of a random electromagnetic radiation with spectrum constant&&co, and other forces, has a phase- space distribution function given by the Maxwell- Boltzmann law, 8'0(q, p) = const Xexp( -W'/g'0) where 8' stands for the deterministic relativistic energy of the charge.%ith regard to the other forces present in the problem, we only consider multiply periodic systems.In a second part of this work, to be published later, we shall prove that, with these conditions, the Rayleigh-Jeans and Maxwell- Boltzmann laws, the charge is in perfect energetic equilibrium with the random electromagnetic radi- ation at each frequency.
The plan of this paper is as follows.In Sec.II, we give the mathematical formulation of the prob- lem.We present the model of electromagnetic background radiation and several properties of multiply periodic motions, which will be needed later.In Sec.III, we review methods for the solu- tion of stochastic differential equations which are neither linear nor Markovian.
These are used in Sec.IV in order to obtain a Fokker-Planck equation.The coefficients of that equation are obtained in Sec.V and, finally, in Sec.
VI we calculate the solution of the Fokker-Planck equation.Several details of the calculations are given in Appendices A, B, and C, while in Appen- dix D we analyze the special case of a central po- tential.

II. FORMULATION OF THE PROBLEM
where v is the proper time, u" and a", are the four-velocity and the four-acceleration, respective- ly, and FI' is the force four-vector.The relations with the usual three-vectors are u" =(yc, y v), a = y,y a+y v FI'= yv F yF c The first term on the right-hand side of (2.1} represents the force due to the potential V(r ) which we shall call deterministic force.The second term is the force of the radiation field on the particle, which could be written as a Lorentz force A. Equations of motion maI'=F" +F" + We wish to study the motion of a charged point particle (i.e. , structureless} interacting with an elec- tromagnetic radiation field.We assume that there are other forces, besides the ones of the radiation field, acting upon the particle.We shall consider conservative forces satisfying the following condi- tions: (i) The potential increases at infinity quickly enough.
(ii) Under the action of this potential alone, the motion should be multiply periodic.
With respect to the first condition, it is true that the usual potentials do not fulfil it, but, when we consider equilibrium with radiation, we must take the system as being enclosed in some "box", which is equivalent to putting an infinite potential outside the box.On the other hand, the second condition will allow us to analyze the motion in terms of fre- quencies, which will be needed in the second part of this work.In particular, every central force field is included under the second condition.Finally, condition (i} guarantees that all states of the system are bounded, i.e. , for any energy ~r(t; I') ĩs bounded by a finite value.
The differential equation of the motion of the charged point particle is the Lorentz-Dirac equa- tion (here, the signature is + where an overdot means d/dt and a is the spatial part of a".
Although we consider here deterministic forces which derive from a velocity-independent potential, it is straightforward to generalize to potentials which depend linearly on the velocities, provided that we use the canonical momentum of the deter- The fields E( r, t) and 8( r, t) provide the thermal bath in which our system is immersed, and they will be considered random fields whose statistical properties will be stated below.
Finally, the last term is the damping due to the radiation reaction.The Lorentz-Dirac expression used in Equation (2.1) is the standard one for a point particle.
If we use the time t of a fixed frame instead of the proper time ~, Eq. ( 2.1) can be written in terms of the (mechanical) linear momentum p=myv=m(1 -U /c ) '/ v ministic motion and that the deterministic equa- tions fulfil the condition of being multiperiodic.Any radiation field in a space without charges can be written as a sum of plane waves [the field of the charge under consideration is taken into ac- count in the last term of Eq. (2.1), not in the second one].We assume that the amplitudes of the plane waves are statistically independent of each other.Then, there are two primary models for these amplitudes.Either we assume amplitudes with a fixed modulus and phases at random (i.e. , with the same probability for each y between 0 and Zn. ), or we assume that the moduli are Gaussian random variables with random phases.' A Gaussian random variable with zero mean is fully characterized by its standard derivation, therefore a single number gives all information about the probability distribution of the amplitudes of a plane wave.Actually, both models, although apparently different, lead to the same correlations for the random fields, so that both models are equivalent.We shall use the first model, following Boyer.
The random field will be written With all this, the following correlations are ob- tained: where 0 is a random variable satisfying By homogeneity and isotropy the quantity %(k) depends only on the frequency.This dependence can be found by comparison between the energy density of the field per unit frequency interval and the spectrum of the field defined as the Fourier transform of the correlation.
(a) The energy density of an electromagnetic field is given by U= (E2+82) . 8m We wish to calculate the energy density corresponding to all the plane waves which have frequencies be- tween to and ro+b, co..3) and (2.4), we obtain a constant of the motion.Choosing now J as the new momenta, we define 1 U, ", ", = (E"'+B"') which is the desired relation between the energy density and the spectrum.

C. Properties of multiperiodic motions
Here we summarize the most relevant properties in order to fix the notation we use.
When the Hamilton-Jacobi equation is separable and the paths in each plane (q;,p;} are closed or periodic, it is possible to define angle and action variables in the following way.
Action variables are defined by 1 Pcd% 2m where the integral goes over a circle or period of the path.If the Hamilton characteristic function" is written as W= g W;(qi, ai a"), then, by the transformation equations, we have 8' =W (q;, ai a"), whereby J; depends only on a and it is, therefore, (E;( r, t)E; ( r, t r) ) e'-"'dr Using Eq. (2.5) and performing the integration, we obtain so that w; is a constant whence w;(t) =w; +co;( J )t .
Then, each q; is a periodic function of w; and therefore of t, i.e. , f(r p) gf elw(E) ll (2.8)   where n w means g n;w; and n; is an integer.Some care is needed if there is degeneration, i.e. , when a relation exists of the type g, .n;co; =0 with not all n; zero.In this case it can be shown" that a linear canonical transformation exists such that the new coordinates and momenta have the same properties as before, but for each of the above- mentioned relations, it is possible to replace one of the w; by one constant.Then, we can assume that only M of the frequencies are incommensurable, i.e. , M g co;n; =0 --n; =0, i (M k &M ~Nk --0.As p;=W'i(q;, J&, . . ., J"), p; is also a periodic function of w; and, finally, every function defined in phase space will be a periodic function of the w, represent the first M coordinates and momenta w =(wl, . . ., wM ), J =(sl, . . ., zM ) and w', J', the rest, i.e.Now, w n means g, ltp;n; N. ote that w', J, and J ' are constant while W=W +PYPt, COP=(Col, . . ., COM) .
Similarly, Eq. ( 2.8) will be written f(r, p)= g f""exp i g w;n; T +eo T 0 where (2.9) The proof is trivial starting with (2.8 ).This ex- pression will be written as 7 [f]= f d wf (r, p) .
(2.10) Finally, a general property which is valid for any canonical transformation is' which, in our case, gives the relations because w '(t)= wp does not depend on t.
An important property for the future calcula- tions is the following.For any phase-space func- tion f (or, equivalently, any function depending on positions and velocities), we have

III. METHODS OF RESOLUTION OF NON-MARKOVIAN NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS
As we have seen above, a relativistic charged system in a radiation field is described by a dif- ferential equation with a stochastic term, which corresponds to the electromagnetic field.Usually, we say that a solution of this stochastic differential equation is found when we obtain an equation for the probability density of the phase-space variables of the system.In most of the physical problems it is enough to know the stationary probability density, and this will be indeed the case in our problem because we are interested in the equilibrium state of a relativistic system with a radiation field.
When the spectral density of the stochastic field is a constant (white-noise), standard methods may be used to obtain an equation for the probability density (Fokker-Planck equation).' However, in our problem we must consider spec- tra which may not be constant, since we are searching for the one corresponding to equilibrium.
In the case of a stochastic field which is not a white noise, the solution of Eq. (2.2) is not a Mar- kov process.Accordingly, there is no Fokker-Planck equation of the usual type (second-order partial derivatives).However, it is possible to ob- tain several approximate equations' containing second-order partial derivatives for the probability density in phase space.The basic feature underly- ing these approximations is the fact that the damp- ing and stochastic forces are small, of order (=2e /3mc ) with respect to the "deterministic unperturbed" Hamiltonian [i.e., H =(m c +p c )' + V], which makes it possible to use per- turbation methods.
The essential idea of these methods is to switch to an "interaction representation", namely to intro- duce as new variables a set of "constants of motion" corresponding to the "unperturbed" prob- lem (i.e. , without the damping and the stochastic forces).The "initial conditions" provide a very general kind of constants of motion (noticeably, they may be used for any kind of dynamical sys- tem, Hamiltonian or not).' Once this change of variables has been performed, the remaining forces (damping and stochastic forces} are small and pro- duce only a small diffusion in the space of the new variables.This feature makes it possible to use perturbation methods.Several methods are possible, and the result is a Fokker-Planck-"type" equa- tion for the probability density in phase space.
(The reader interested in more technical details may consult Refs.14 -16.} We may give a qualitative idea of these methods by describing briefly one of them, the Markovian approximation.' We choose an intermediate time scale ht, large with respect to the correlation time t, of the stochastic force and small enough to have a small variation in the new variables (constants of motion} that we have introduced before.Then, us- ing this time scale, the process solution of Eqs. (2.2a) and (2.2b) may be approximated by a Marko- vian process and we may get an approximate Fokker-Planck equation in phase-space variables.
From these Fokker-Planck-type equations, we may obtain a "reduced" Fokker-Planck equation for the stationary density in terms of the constants of motion of the unperturbed system (i.e. , without the damping and the stochastic forces).Indeed, since the deterministic orbits are slowly perturbed by the dainping and stochastic forces, it is physi- cally intuitive that it will be possible to describe the stochastic motion as a "diffusion of the orbits" rather than a diffusion of the phase points them- selves.This reduced Fokker-Planck equation may be obtained by using a "reduction procedure" dev- ised by Haken.' It consists essentially in taking the average over the orbits which are characterized by prescribed values I hi I of the constants of motion of the unperturbed system.
The reduced equation for the stationary density obtained from the different Fokker-Planck-type equations in phase space is unique, which proves the consistency between the different approxima- tions: the same stationary density is obtained in the limit ~-+0.
We give now the final result for the reduced Fokker-Planck  where I h& J are the constants of motion of the deterministic system without the damping.
If we write the stochastic differential equations in the form g;=M;+a P;+aKt, i =1, . . ., where P ; represents the deterministic force, a2P;t he damping force due to radiation emission, and aK~the force of the random field, verifying (K; ) =0, the coefficients G" and G"' have the fol- lowing expressions: ah", . 6ah"WC, (g-", -u G"=-a gT P~" +T a f du g "K" ((, 0) ~k " Bh"Bh" G""=a T f du g " " (K"((, 0)Kt(g ",-u)) T represents the following operation on a function P( g } of the phase-space variables: T[y]= f y(g)dg g S(h, -h, (g)) The physical meaning of T is an average over the deterministic orbit, keeping the value of the constants of motion h;.
In the case of a multiperiodic system T consists of an integration over the variables w, since This last expression for T has been used in Eq. (2.10).
In (3.3a) and (3.3b) the notation g " means the value of the phase-space coordinates at timeu, if they are g at time 0, considering only the deterministic motion.
If we compare (3.2) with (2.2), it is easily seen that i =1,2,3 ~(1 --x;, P1 -- , P;=E;=0, It should be noted that we have introduced nonsymmetrical diffusion coefficients G"&Qg&".The reason for this is that in the case of a stochastic force with divergence zero g, BK,/'B(, =0, the contributions of the stochastic and the damping forces are separated.The damping contributes only to the drift term and the stochastic force to the diffusion.This is indeed the case for our stochastic force siilce Kr X~( =X~~+ Bhp Bh, g&"= T I du g (F~(r, p, 0)Fj (r ", p ",-u)) ~Bk~& ;, pp, Therefore we get finally the equation (3.1) with the following drift and diffusion coeff1c1ents: (3.4a) (3.4b)

IV. FOKKER-PLANCK EQUATION FOR MULTIPERIODIC SYSTEMS AND SOLUTIONS WHICH DEPEND ONLY ON THE ENERGY
We are going to study the problem of relativistic multiperiodic system interacting with a random ra- diation.The equilibrium state for such a system is given by the stationary density, which is a solution of a Fokker-Planck-type equation, as stated in Sec.
III.If there is a stationary solution for that equa- tion which is normalizable in the phase space, such a solution is unique.
Then, it corresponds to an ergodic process, more technically called recurrent.
Our aim is to show that a spectrum of Rayleigh-Jeans type for the random radiation im- plies a Maxwell-Boltzmann distribution at equili- brium for the system.For a free particle, the Maxwell-Boltzmann distribution does not depend on position, and it is a constant over space.That distribution is not normalizable and it is not a true probability distribution in phase space.We are then forced to assume that the potential V(r ) goes to infinity quickly enough for a function of the -V( r )/8'0 form e ' to be integrable.This is fulfilled if it is infinity everywhere except in a finite region, but this is not necessary.A weaker condition is, for example, the existence of a real number a & 0 such that V(r) &constXr for r~ao.Assuming, then, that V behaves adequately, we shall prove that the Maxwell-Boltzmann distribution is a solu- tion of the stationary Fokker-Planck equation, which, therefore, is the searched solution.
We begin defining the funtionals 8 Gi(y)=g T F ) J gp (4.1a) 3 G2(qr, g)= T f du g "(F""(r, p, 0)Fi"(r ", p ",-u)) for any two functions p and f.In our problem of a multiperiodic motion, these functions will be the action constants J&, J2, J3 and angle constants w~+i, . . ., io3 (i.e. , I, J ', and w ' according to our notation of Sec.II C), that is, 6 -M constants.Now we shall search for solutions of the Fokker-Planck equation of the orm Wp = Wp [ S' ( r p ) ] i.e. , depending on r and p only through the energy.As 8' depends only on J and not on I', nor w' (see Sec. II C), then the second term of the Fokker-Planck equation (3.1), includes only a sum for v = 1, . . ., M. Also Therefore in (3.4a) and (3.4b) appear only the functions G"" corresponding to h, = J"v = 1, . . ., M. Be- sides, G""=G2(hq, h") .
So we can define G2(q ) y G2('pi J/ )ro/ v=1 M 3 g ()J = T I du g g "or"(F, "(r, p, 0)Fi"(r ", p ",-u)) u "co, , r, p, I r,p, -u (4.2) and the Fokker-Planck equation can be written in the form We begin with the calculation of the general form of the functionals G& and G2.

A. Calculation of G~{(p)
Taking into account Eqs. (4.1a) and (2.2a) and (2.2b), we have where we have written y& for Vzy.On the other hand, the following relations hold: a a~-ya +y c In order to show that the Maxwell-Boltzmann distribution is a solution of the Fokker-Planck equation, it is convenient to write G~and G2 in similar form.In Eq. (4.2) it is seen that the coeffi- cient 62 is written in terms of the correlation function of the random field.If we perform the Fourier transform of that, it results in an expres- sion related to the spectral density of the random field.Therefore, it is convenient to write G~a lso in terms of frequencies.It is easy to see, however, that this cannot be made directly from the expres- sion F yz which is averaged over an orbit in G~.
Actually, we need new expressions for (4.1).
In order to see more easily the way to do that, let us consider the particular case in which y is the energy (which is not true in general).Then I = UJ. Bpj.
G&($')= -f dwF .v, that is, G&($'} is the average energy lost per unit time as a consequence of radiation by the charged particle.On the other hand, we know that this en- ergy is given by the flux of the Poynting vector S through a spherical surface at infinity.With the notation of Ref. 21 we have T I6$'I = lim lim -f dt f dQ(R S n)", , r-+oo T~ao T 0 where R(r, t) = r -g(t), n = R g(t} is the solution of the equations of motion with only the deterministic forces and the subscript "ret" means that the quantities must be considered at the retarded time.If we take into account that only the "acceleration fields" give rise to a loss of energy by the charge, it results in The spectral analysis of the emitted energy im- plies making a Fourier expansion of the Poynting vector, and to calculate the energy corresponding to each component.Here we are not interested in this calculation for the particular case of the ener- gy, which will be the aim of a forthcoming paper.We are rather interested in the generalization of (5.3) for any q&.
In order to derive an expression similar to (5.3}, we begin noting that, in the limit r ~ao, n~r =rlr, r X[(rp)Xp] r-+go (1 -P r~)3 Then, it can be shown by direct calculation that 1 d r X(r XP) (1 Pr ) dt (1 P-. r )-

~T
X f dt f dQ[hq(r, t).g(r, t)]",, (5.5) EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . .which is the desired generalization of (5.3).From (5.5) we shall obtain the expression of Gi(y) in terms of frequencies by expanding hã nd g in Fourier series.But before that we first prove Eq. (5.5).To do this, we need three rela- tions, which are proven below.
(3} Let f be a continuous and bounded function in R XI (I =phase space); then limf dt f(r, g(t), p(t))", T = limt r, t, t 1 -'n (5.9) In order to prove this, let us make in the first integral the change (5.7) In order to prove that, it is enough to combine (5.1) with (5.2) and to apply the property (2.9) to the first term in the integrand, T &[a y~]=(2m ) lim -f dt(a gz).T~~T T (2m) lim -a.gz I o -f a. pzdt because, by the nature of the motion, a and pp are bounded in time.Finally, a=cp and replacing (5.2) we see that (5.6) follows immediately.
(2) Let f (r, p) be a continuous function in phase space such that lim, "f(r,p) =f(oo,Q, p) exists (Q is the solid angle)..g)This is true because the integrals have finite lim- its and the expression under the integral sign is [remember that t", verifies c(tt", )= I R(r, t",) Then f dt f(r, g(t), P(t))", 'ret~' T] r, t', t' 1 -n dt'. (5.10) Now, as g(t ) is constrained within a finite region due to the condition on the potential (region whose limits will depend on the constants of the motion), the results is that there exists some constant RM such that I g(t) I &RM for any t and, therefore, c I t t"t f, , f(r, g(t'), p(t'))( I pn)dt' & f-, , I f(r, g(t'), p(t'))(I pn) I dt-' r, t, t 1 -n dt'.
So that, after dividing by T and taking the limit T +ao the correspond-ing term is zero.The same is true for the upper limit of the integral on the right-hand side of (5.10), t",(r, T), so that we have finally lim -f, , f(r, ((t'), p(t'))(1 -p n}dt'= lim -f f(r, g(t},p(t))(1 -p n)dt, which proves Eq. (5.9).
After that, we may go to the proof of Eq. (5.5).
B. Frequency analysis of G~(q) = lim where we have used (2.9) and (2.10).Now, taking Eqs.(5.8) and (2.9) we have lim lim -f dt f dQ[h+.g]", M f dw f dQ(h+ g ")(1 P.r )- We must perform a frequency analysis of the vectors h+ and g.We shall start proving that the only frequencies involved are of the form M N =No.n= with n integer.
We shall consider only g, the proof for h~b eing similar.We take the Fourier transform of g I ", (5.11) S(co)= f g I", e '"'dt .
(5.15)Note that all quantities under the integral are calculated at time t' (except r, which is fixed).Let rt = g(t'}exp ico -[1p(t') n(r, t')], r -(t') C rt depends on t' through g and p, which are periodic functions of the ur; and, therefore, rt will be also a periodic function of the angle variables.Then, it is possible to construct a Fourier series ex- pansion in the form rt = f ( r, w ( t') ) = g r7-"(r ) e' " ' ""' g ("( --S(~)e'"'dho=grt-"(r)e'" 'e " "" (~r)ei n ~w (t) ~~Ĩ n Similarly, for h~~""we obtain h~~", = g(M -"(r)e'"'"'" (5.17a) (5.17b)Note that it is the time t, and not the retarded time, which appears in (5.17a) and (5.17b), in spite of the fact that g and h~h ave been calculated at the retarded time.
The next step is to show that, as in G) (i(0), it is made a time average, each frequency contributing separately, i.e. , G)(y)= QG) -", where 6, "means the same expression as G~but putting, instead of hz ~", and g ~"" the corre- sponding Fourier components.This is equivalent to saying that each plane wave contributes in- dependently of the others to the value of the coef- ficient G((ip) in the same way that the radiated en- ergy, when averaged over the time, is equal to the sum of energies carried by the different plane waves.We shall prove then the following: 2 G)(y)= 2 (2n } lim g f dQ()LY -"rt -") . .+ei n. w(t)ei m w (t) 7 which after performing the time integral, leads to Eq. (5.18}.Now, we must calculate explicitly the Fourier components of h+ and g in order to introduce them into (5.18).Inverting the relations (5.17a) and (5.17b) we have r7-"(r}= lim -f dtg ~", e ~T (5.19a) p -"(r}= lim -f dt h~", e (5.19b)Consider (5.19a).We make the same change as be- fore, namely t'=t«, (t), which gives i n wo -i n Foot(t ) g "(r j= limdt'e e n T y' 0 X g(t')[1p(t') tt(r, t )] after several steps similar to the ones leading to Eq. (5.9).Now (1 p ) «X(it X p) dt 1 pn-.
which, after an integration by parts, leads to Again, all quantities are calculated in t .The first term in the sum is zero for T~00 for the quantity within brackets is bounded.Then g"= lim T~00 1 f d g it X()t XP) ( in w0 -in -m0i(i)}(, In the limit r~ao i r -g(t') i, r -n g(t') (5.20a) Finally, putting (5.20a) and (5.20b) into (5.18) and taking into account that n ~r, C Now, we relabel t& -t2 --u and t& ~t, which, taking into account that 1, . 1 lim, f dt = lim, f dt' Vt, Xexpin cop u r-C Finally, expanding the scalar product, taking Eq. ( 2.9) into account, we have Here, I represents the identity operator, r r a dyadic product, r means r (w} (i.e. , r = g "g "e' " ' " ), and, as stated at the end of Sec.II, i( n. w -n.agog) ~r -Q &n and similarly for P. Equation (5.21) is essentially the expression we are searching for, Gi.However a more convenient form is obtained if we note that o can be expressed as a Fourier series in the form cr[qr](co, w, u;cop)= g 0 "[q)]( co; cop)e'" "e' (5.23)This can be seen by performing the changes W -NOQ = Wu in the expressions of r "and P ".Hence if r=r(w), then r "=r(w") and similarly for P. Therefore o [p] depends on w and w" through r and P, and the consequence is a periodic function: ~fr]= g ~-", -[m]e'" "e "= X ~-", -[ale'" "e ™ñ , m where we have redefined the coefficients and expressed w"explicitly, which proves (5.23).Putting now that expression in (5.21) and performing the integral 2 0 2 g J dw(n cop} lim -I du g o-"[y](n cop, 'cop)e'"'"e Op -"[y](n cop'cop}=0'p -, [y](n cop'cop} .
After that (5.24) can be written 2 G2(y}= which is the final expression for the drift coefficient 6 i(y).Here g' "means a sum ove«hose values «n that verify (n cop) &O.

C. Calculation of G~(y)
We start with Eq. (4.2), where the random force F" is given by the Lorentz force of the random elec- A simplication is possible if we take into account the relation (2.11) so that and, besides, (5.27) The correlations must be calculated from the expression (5.26) of the random force and the field correla- tions (2.5a) and (2.5b).A straightforward calculation gives 2 oo 2@2( (F""(r, p, O)F, "(r " " -u) ) =f de ' f dQ cos -r (r -r ")-cou C C (5re "r re )+ P (eeijerjspi r s+enj eejspir s ) /JS + geje, p p "(5j r~r )ijao.
where we have changed the notation of (2.5a) and (2.5b) to obtain one similar to the one of G&(q&).Hence, and from (5.27), we obtain 2 2+2 G2(y)= T f du f de f dQ cos r(r -r")--eiu 0 0 2c C X[qp (I rr ) P "+(-Pp r )(P P ")-(g P p")(P r )]). (5.28) Now, taking Eq. (5.22) into account, we have f dw f du f de cu 9' (co)[e '""o[p](co, w, u;coo)+e'""o[iip]( -co, w, u; It is clear that the second term will not appear if the integral were from -00 to + ao.It can be seen that this is only possible if G&"=G"", which is not true in general.However, it is convenient to avoid that term, which will be possible because, as we shall prove, it does not contribute to the Fokker-Planck equation. In the following we shall call Gz (symmetrized) an expression like G2 but with the integral in u going from ao to + ao (with a factor -, inserted), i.e. , 2 + 00 00 Gz(y)= f dw f du f dcoco 9' (tJ)[e '""cr[y](co, w, u;cop)+e'""o[cp]( co-, w, u;cop)] .8c 00 (5.30) D. Relation between G2 and GB efore studying the possibility of replacing G2 instead of Gz we show that G2 has a rather simple relation with G~.

E. Symmetrization of G2
It is unfortunate that in the analysis of the possibility of symmetrizing G2 it is unavoidable that Now, the Rayleigh-Jeans spectrum is u (co) =const Xco so that, taking (2.6) into account, 9' is a con- stant.Then, a comparison between (5.31) and (5.25) gives the explicit calculation of the integral (5.28) is rather lengthy.Also-, our result will be only valid for a Rayleigh-Jeans spectrum.Although this is all we need in the present paper, it would be desir- able to have a result independent of the spectrum.In the case of a central potential independent of the velocity, it is proved in Appendix D that, indeed, G2 --G2 for any spectrum.Also, in the case that the random force does not depend on the physical variables, if the system is multiperiodic, and nondegenerate, the diffusion matrix is symme- trical.' However, this is not proved in the general case, and we must restrict ourselves to the Ray- leigh-Jeans spectrum.
This spectrum present the difficulty that it gives rise to divergent integrals and it is necessary to in- troduce a cutoff, which may be removed at the end.We shall use a cutoff Rayleigh-Jeans spec- trum in the form S(co)=constXr0 e ' (e&0) .
(5.33)This fact can be related with the structure of the charge.Note that in the I.orentz-Dirac model we are using, it is supposed that particles are points.
We think that if an extended particle is considered, the effective spectrum will change considerably at high frequencies.This will be discussed in the second part of this work.
To do that, let us write (6.3) t [dl(P)(q p P)]=(q p P)dl(P)+(9 p P)di(P) +(qP P) d d((P) .p pd (p) Now, using again (2.9} f dw [di(p)(q p)]- It must be pointed out that d1 and d2 are well- defined functions at the origin and take values for p K[0, 1).On the other hand, the values of the constants J;, J, and w put bounds to the possible values of P so that d i and d2 are bounded func- tions.Finally, gz is a periodic function of the variables w.Its value at time T will be one of the values they take in the closed interval [0,2m ] This being compact and gz a continuous function, it is bounded.Therefore, in the limit T~ao, Eq.
On the other hand, the function d i(y) is con-  Then, Eq. ( 6.3) gives BW.

APPENDIX D
The case of central potentials has great interest because it can be shown that, if the Lorentz-Dirac damping term behaves like a vector with respect to rotations and the correlation of the stochastic force like a dyadic product, the solution of the Fokker- Planck equation depends only on constants which are invariant by rotations.Then, the process of reduction of the coefficients of that equation can be made to these constants alone.This is a great advantage because it allows the simplification of the problem of the symmetric character of the dif- fusion matrix G"", so that, for any spectral densi- ty, it can be shown that G&"=G"" if the potential is central.Therefore, in the particular case of a Rayleigh-Jeans spectrum, the proof that the equili- brium density is the Maxwell-Boltzmann one sim- plifies notably.(In the following we do not use the notation of Sec.II C.) The damping term in (2.2b) is (F""(r, p, t)Fi"(r ', p', t')) = -J dao I dQcos r(r r') -co(t -t')-- Here, the tensor character of the expression 5;j r; rj and the quantity eijk guarantee the tensor character of the correlation.
Consequently, the problem can be reduced to the constants which are invariant by rotations alone.Let us see which are these constants.The relativistic Hamilton-Jacobi equation for a particle in a central field can