PHYSICAL REVIEW D VOLUME 27, NUMBER 6 15 MARCH 1983
Equilibrium between radiation and matter
for classical relativistic multiperiodic systems. Derivation
of Maxwell-Boltzmann distribution from Rayleigh-Jeans spectrum
R. Blanco, L. Pesquera, and E. Santos
Departamento de Fisica Teorica, UniUersidad de Santander, Santander, Spain
(Received 24 June 1982)
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 ac-
tion 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 proba-
bility 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. Re-
cently, 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 spec-
trum 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 en-
ergy 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,
1254
EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1255
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'= y?v F yFc
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"+ 2e~ da&det st 3 3
1
a a"u&2 vc
(2.1)
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. Fi-
nally, condition (i} guarantees that all states of the
system are bounded, i.e., for any energy
~
r(t; I')
~
is 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 + 2)
Fsi I( ) v XB(i' t)
c
as
r= (=v},
(p 2+ 2 2)1/2
et 2e 1p=F "+ a ? a?a v3c c
(2.2a}
v t&B(r, t)
c
(2.2b}
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
1256 R. BLANCO, L. PESQUERA, AND E. SANTOS
ministic motion and that the deterministic equa-
tions fulfil the condition of being multiperiodic.
2
E(r, t)= g f d ke(k, j()5'(k)
B. Stochastic properties
of the radiation field
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 Gauss-
ian 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
Xcos[k r c?ot +8(k, A )],
2
B(r,t)= g f d'k ? ' e(k)
A, =1
Xcos[k r cot+?8(k, A, )],
=
?,522 52(k ?k'),
(cos8(k, A, )sin8(k', A,')) =0 .
Polarization vectors are such that
k.e(k, A, ) =0,
e(k, iL) e(k, A, ') =522
2g c;(k,A, )ej(k, A, )=5J?
(2.3)
With all this, the following correlations are ob-
tained:
where 0 is a random variable satisfying
dPs(8) = d8,12'
(cos8(k, A )cos8( k', A, ') ) = (sin8(k, A )sin8(k', A, ') )
(E'(ri tl )Ej(r2 t2) ~ (~'(rl ti )~j(r2 t2) ~
k;kj +2(k)d k 5"? cos[k (ri ?r2) ?c0(ti ?t2)],k
kt 8' (k)(E;(r?t,)8 (r2, t2)) = d ketjt k cos[k (ri r2) to(ti ?t2)]?.
(2.5a)
(2.5b)
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. The fields corresponding to those waves are
~2 I l2E?=f den' & f dope(k, ~)cos[k r ?co't+8(k, ~)],
27 EQUII.IBRIUM BETWEEN RADIATION AND MATTER FOR. . . 12S7
where co'=c
~
k
~
. Using Eqs. (2.3) and (2.4), we
obtain
a constant of the motion. Choosing now J as the
new momenta, we define
1
U,?, ?,= (E?'+B?')8m
co+ha +i(co')co'2
dN
N C
Finally, taking hN ~0, it turns out that
It is possible to show that if one of the q; moves
through a period, the corresponding wj changes by
2m while the remaining w do not change. Also
U( ) I (cllylv+lkfll)
U
h,co-+0 kN
9 (co }co
C
w;= and S'=g'(Ji, . . . , J?}BS'BJ;
4n co 8' (co)S;co =
C 3
independently of the coordinate, which is a conse-
quence of the isotropy. Then, we have
4m'S(co)=S;(co)= U(co),
3
(2.7)
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,
(2.6}
(b) The spectrum of the radiation is defined by
+oo
S;(co )= (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.,
Mg co;n; =0 - ?n; =0, i (M
k &M ~Nk ??0.
Then, since we have
BS'wk=Nk=O=, k)M
BJI,
'
it follows that ? is only a function of the M first
J;.
We shall use the following notation: w and J
inw;4= +%,ne
n
inIO; in', .t
= gqi, ne
n
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,
1258 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
represent the first M coordinates and momenta
w =(wl, . . . , wM ), J =(sl, . . . , zM )
and w', J', the rest, i.e.,
=(+M+l~ ' ~ +n ~t J =(JM+l~ '
~ ~n)
BX;
Bwj
8JJ.
Bp;
Bw Bp
BXJ BJl
Bx; Bw~
ap,
'
Bp; BJ.
Bwj Bx
(2.11)
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;
l W n l w n
n, n
e/w 'n (2.8')
with
M f""f(' p)(2~ )M
= lim ?f dt f(r(t), p(t)),T +eo T 0
where
(2.9)
f d"=II f, di=1
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'
Bgl dPJ' BlI( BQJ Bpl BPJ.
ag, =ap, ' aP,
=
ap,
' ag,
=
a~,
'
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 densi-
ty, 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
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1259
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 possi-
ble, 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 equation for the stationary density
Wp. The technical details of its obtention may be
found in Ref. 19. In a "current" form this equa-
tion reads
[G"Wp]+ g GI'8 ?8Wp
p
=0,
(3.1)
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, . . . , 6, (3.2}
where P ; represents the deterministic force, a2P;~
the 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?, . 6 ah?WC, (g-",-uG"=?a gT P~ " +T a f du g "K?((,0)
~k "
Bh?Bh?G""=a T f du g " " (K?((,0)Kt(g ",?u))
(3.3a)
(3.3b)
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 time ?u, if they
1260 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
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,(p2+ 2 2)1/2 '
2 d -d 2e . 1a a;+3?(F );= a; ? a~ "v;3c c
(v XB);
uK;+3 ??Fg"?e Eg+
C
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~ ~+ (v &&B); a=X, VJ.Bk
B
c ~ k gp; (p +~2c2)1/2
=0.
d Bh?G1 yT F,
~ ) Bpj.
Bhp Bh,g&"= T I du g (F~ (r, p, 0)Fj (r ",p ",?u))
~Bk~ &;, pp,
V (p2+m2c2)1/2 (p2+m2c2)3/2
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 )/8'0form 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
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1261
8Gi(y)= ?g T F) J gp (4.1a)
3
G2(qr, g)= T f du g ?(F?"(r,p, 0)Fi"(r ",p ",?u))I (4.1b)
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, I , , u (4.2)
and the Fokker-Planck equation can be written in the form
M 3 B[Gi(Ji)WO+G2(Ji)Wo]+ g, [Gi(J' )Wo+G2(Ji )Wo]1BJ; '=M+&
3
+ g, [Gi(w&' )Wp+G2(ioi' )Wo]
+r Bw;
(4.3)
V. CALCULATION OF THE COEFFICIENTS
OF THE FOKKER-PLANCK EQUATION
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
6 i(y) = ?T [F g~)
28
3c' c'dw (a ??a aij v).g
(5.1)
where we have written y& for Vzy. On the other
hand, the following relations hold:
T
v'a v aa&=(a', a) = y' ' ',y'a+ y' " '- v
C .
' C2
(5.2)
v a
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
1262 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
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~ also
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
TI6$'I = lim lim ?f dt f dQ(R S n)?, ,r-+oo T~ao T 0
where
R(r, t) = r ?g(t), n =
R
g(t} is the solution of the equations of motion with
only the deterministic forces and the subscript
"ret" means that the quantities must be considered
at the retarded time. If we take into account that
only the "acceleration fields" give rise to a loss of
energy by the charge, it results in
T 2G~($')= ?(2m) Ib, $'I =?lim lim ?f dt f dQ I g'I?,(2m)Mr~w T~m T 4mc '
tp
(5.3)
where
n x[(n ?p) x p]
(1?P n)
(p npn?)~0
Ir ?4?} I
and therefore
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[(r ?p)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 )?
nx(nxp)
= lim
~1?Pn dt 1 Pn?
Let us define
h+( r, t) =
g(r, t) =
I
After that, we shall prove the following:
n(r, t) X I [n(r, t) ?p(t)] X gz(t) I
1 ?P'n 1 ?P(t) n(r, t)
a n(r, t}X[n(r,t}Xp(t)]
1 pn ~t ? 1 ?p(t).n(r, t)
(5.4a)
(5.4b)
then
and
hN ~ cgf~oo
G~(tp) = z (2n ) lim lim-e ~ . . 14~c2 r ooT ~ 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 h~
and 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.
(1}The first relation is
continuous (Ref. 22).
(3}Let f be a continuous and bounded function
in R XI (I =phase space); then
lim ?f dt f(r, g(t), p(t))?,
T
= lim ? t r, t, t 1 ? 'n
T ~ T
with
28 T[&l
3c
(5.6) (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). Then
lim f dw f dQ f(r, Q, p)P~ ao
= f dw f dQf(00, Q, p) . (5.g)
This is true because the integrals have finite lim-
its and the expression under the integral sign is
[remember that t?, verifies
c(t ?t?,)= I R(r, t?,) I
=Ir ?g(t,.)I]
so that
dt =dt'[1 p(t') ?n(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
I
=
I
??r t-t) I ?+RM ~
As a consequence,
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,
1264 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
which proves Eq. (5.9).
After that, we may go to the proof of Eq. (5.5).
If we apply (5.9) to the function
dQh~ g,
we obtain
f dQ(hg g ")(1?p r )= (5.13)
Taking into account (2.9), (2.10), and (5.6) we
obtain finally Eq. (5.5).
B. Frequency analysis of G~(q)
= lim I T f dQ(h& g)(1?p n)(2n }
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 )?)M
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~ being
similar. We take the Fourier transform of g I ?,
(5.11) S(co)= f g I?,e '"'dt . (5.14)
where, as n ?+ r and n ?+ 0,
r X[(r ?p)Xqp]
1 ?P r dt 1 Pr?
(5.12a)
Then, we perform the change t'=t?, so that,
I
r g(t)
I
being b?ounded, t =+ 00 ~t'=+ ao
and, therefore,
+ 00S(to)= ~g(t~)e imt(t )? '
27T
r X(r Xp)
g
1 ?Pr dt 1 ?P r
(5.12b)
Now, from the definition of t?, the result is
In (5.11) the solid angle integration is straight-
forward, although lengthy. Details are given in
Appendix A. The result is so that
1 +~, , r ? (t')9'(co)= f g(t')exp iso t'+-28 C (1?P n)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 ? [1?p(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' " ' "" '
n
I
Putting these expressions in (5.15)
2'
~ +
= g rt-?(r)e '5(n top ?to) .
n
(5.16)
~ ~tn w0 tn. cootq-jrje e
n
If we perform now the inverse Fourier transform
in (5.14), taking (5.16) into account,
EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . .
+ 00
g (?(? S(~)e'"'dho=grt-?(r)e'" 'e " ""
(~r)ei n ~ w (t)~
~I 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~ have 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 -?) . .4mc &~ ce -+
n
(5.18)
In fact, from Eqs. (5.17a) and (5.17b) and (5.5),
we obtain
G)(ip) = z (2m ) lim lim?M4mc 1~00 T~OO T
T
x f, dt f dQ+gp
n m
+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 ?(rj= lim ? dt'e en T y' 0
X g(t')[1 ?p(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
g?= limT~ 00
I'
it X(tt XP) ?i n w0 i n ni0i(i'?)
e
1 Pn?
1 r, n X(n X P) d ?i n w, ?in pu0t(t')0 0
T o 1 p & dt'
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?= limT~ 00
1 f d g it X()t XP) ( in w0 ?in?m0i(i)}(, )(1 g1 Pn?
= lim in coce '.?f dt'[n X(+ X p)]eT~ 00 0
In the limit r~ao
1266 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
i
r ?g(t') i, r ?n g(t')
C C
where, for large r,
~, l Il ' W p / II ' CO pf /C n g(t')
?lim in co pe e ? dt'[n X (n X p)]exp in?cop(t'T~ 00 0 C
With a similar reasoning for p - we havey, n
?i n w& ?i n ~ m&rlc 1 T, - . , n. g(t')p -??lim in cope e ? dt'In X[(n ?p)X+&]Iexp in ?copq, n T 0 C
(5.20a)
Finally, putting (5.20a) and (5.20b) into (5.18) and taking into account that n ~ r,I'~ 00
2
Gi(p)= (2m) g f dQ(n cop) lim lim4m.c T~oo T ~00 T
n
Tl
X f, ?i, f, ?~[r"pX[(? ?P)Xq, ]](t&).[r X(r Xp)](t, )
(5.20b)
[k(t? ?k(t2)]
Xexp in co?p (ti ?t2) ?r .
C
Now, we relabel t& ?t2 ?u and t& ~t, which, taking into account that1, . 1lim, f dt = lim, f dt' Vt,T' ?+ cN T 0 T' ?+ ao T
leads to
2
Gi(y)= ?2 (2') g(n cop)4mc
X fdQlim?1
T~OC T
TXf dt limT'~m T'
X f, duXI? , X[(r ?p)Xg, ]j(t) [r X(r X13)](t?u)
Xexp ?in cop u r?
C
Finally, expanding the scalar product, taking Eq. (2.9) into account, we have
2 0
G, (p)= g f dw(n cop) lim ?f du e " "'"o.[y],4mc T~oo T
n
where
(5.21)
c?[q&](co, w, u;cop)= f dQ exp i r(r r")? ?C
(5.22)
EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1267
+ ~ + +
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 ?~
n, 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'"'"e4mc T~oo T
n k, m
2
(2ir) g(n cop) crp ?[y](n cop, cop) .
n
On the other hand, taking into account the definition of cr, we have
~*[q](~,w, u;~p) =~[~]( ~,w, u; ap?)
so that
(5.24)
~'
-.
,
- [v](~;~p}=~-.
,
- [m]( ?~'~p?
and then
Op -?[y](n cop'cop}=0'p -, [y](?n cop'cop} .
After that (5.24) can be written
2
G2(y}= 2 (2m)4' (n cop)2[0 p ?[y](n cop~cop)+o p -?[q&](?n cop cop)]
n
( n .Pg) 0))0
2
, (2m )~g'(n cop) Re[0 p -?[y](n cop;cop)],27TC (5.25)
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-
tromagnetic field
1268 R. BLANCO, L. PESQUERA, AND E. SANTOS
Fsi E(~ ) v X (rpr)
C
(5.26)
A simplication is possible if we take into account the relation (2.11) so that
and, besides,
?+ ?+?gr ?+r
~?QP~P
Bx
BWj
?u
~+eg COj
whence it follows
BJ;
Wj =Ve
i ~Pe
Bx
Bwj Wj
=Xe =Ve
G ( )=T f "d g ~.-"(F"(-.-O)F"( " "?)&Q V
r,e=I
(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 ")?couC 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+2G2(y)= T f du f de f dQ cos r(r ?r")??eiu0 0 2c C
X[qp (I rr ) P "+(?Pp r )(P P ")?(g Pp")(P r )]).
(5.28)
Now, taking Eq. (5.22) into account, we have
2 oo 00G2(ip)= f dw f du f de cu 9' (co)[e '""o[p](co,w, u;coo)+e'""o[iip]( ?co, w, u;coo)] . (5.29)C
Inserting here the Fourier expansion of o[y) we have terms of the form
~
~?i(co+ m w 0)udu e =n5(cogm cop) lV.p?0 io+m co0
It is clear that the second term will not appear if the integral were from ?00 to + ao. It can be seen that
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1269
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 00Gz(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 G~
Before studying the possibility of replacing G2 instead of Gz we show that G2 has a rather simple relation
with G~.
Putting (5.23) into (5.30) and evaluating the integral we have
2
G2 ? 2 fdw f du f dcoco 3' (co) g [e '""0?[y](co;cop)e'"'"e ?"0"
+e'""cr?-[q&]( co;cop)e'?"'"e ?"ou]
00
2
(2n. ) f dco co 9' (co) g (2m )[5(co+m cop)o p -. [qr](co;cop)
m
+5(co?111' pc)opcr[gr]( ?co' cop) ]
2
(2n. )4c2 (m cop) P (m co p)o p - [y](?m cop,'cop)
m. Pea&0
(m cop)$' (rn. cop)o p - [y]( mcop , cop?)'
m ' Pyo&0
2
z
(2m) g'(n cop) 5' (n cop)Re[o p ?[y](n cop, cop)] .2c
n
(5.31)
Gq(p) =n O' G
~
(y) (5.32)
which is the desired relation.
Now we must consider the relation between G2
and G2.
E. Symmetrization of G2
It is unfortunate that in the analysis of the pos-
sibility 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.
1270 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
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.
Introducing (5.33) in Eq. (5.28) we perform the
solid-angle integral, whose details are given in Ap-
pendix B. We obtain
and
~?Qr ?r
ir ?r
(5.37)
The integrals are only needed to lowest order in
e which gives, after a tedious calculation summa-
rized in Appendix C,
f duIi(e)= Ti[qr]?+ ?,' f duIi(0)+r,',
(5.38a)
f du I,(e)=?T,[y]+?, f du I,(0)+r,",
(5.38b)
f du I3(e)= T3[y]?+ ?, f du I3(0)+r,"',
~
2+2 00G2(p)= 2 4m fdw f dco[Ii(e)+I2(e) where r,', r,", and r3" go to zero with e,
(5.38c)
where
+I3(e)],
(5.34)
and
Tl +T2+ T3 dl(?& i [v ]+d2(?&2[v l
(5.39)
Ii(e)=P[y](u) f dcoco e '"coscou0 Na
(S.3Sa)
&i[el=?, P,
(g~ p)(p. p)02'=
(5.40a)
(5.40b)
I2(e)=Q[q](u) f den co e
)(cosN u cosNa sinNaNa Na
(5.35b)
di(P) =? 1+ 12 p2 11
1+P
2P 1 ?P 2P 2
(5.41a)
I3(e)=R[y](u) f dc@ ei2e
slnNa cosNay slnNu
N a
d2(p)= ??, 1+1 3 1 1+P 3
2P 1 P 2p'
(5.41b)
(5.35c) Finally, we have
Q[q))=P~ P " 3(gr n?)(P "n?)?,
R[y]=(n?g)q)(p p ")?(ng p)( j)p p "),
(5.36a)
(5.36b)
(5.36c)
G2(q)= G2(p)+
c
dwT y +r, ,
which is the relation searched.
(5.42)
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . .
VI. SOLUTION OF THE FOKKER-PLANCK EQUATION
1271
From the expressions (5.32) and (5.42) for the coefficients, we have
M 2+2
G~(J~)Wp+7r 9 G](J;)Wp+ 2 2' f?dw T[J;]Wpi=1 c
Gi(J/)Wp+n 9' Gi(J/ )Wp+ ?fdw T[J/ ]Wp
3 8 e2@22 1+ g G](m )Wp+m. O' G&(uj)Wp+ ?fdw T[w/ ]Wp +r,c (6.1)
In the first place we shall prove that terms of order I/e do not contribute, i.e., that
M 3
Wp fdw T[J] + g Wp fdw T[J ] +, Wp fdw T[w ] =0.
M+~
As the energy depends on J&, . . . , JM and Wp is a function of the energy, it is clear that
ag 0 a8'0
?=0,BJ Bw
so that the previous relation can be written
M 3
Wp fdw T[J ] + g Wp, fdw T[J ]+? fdw T[w ] =0.i=1 I' =M+1 ~~i BM( (6.2)
fdw T[q]=?fdw(gz P)di(P) .
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) .
Now we prove that the last two terms together
give T[y]. To do that we define
(6.4)
d;(y)=d;(Vy ), i =1,2
which is possible because d; are even functions of
their arguments [see Eqs. (5.41a) and (5.41b)].
These functions fulfill the equality
We shall prove that the first and the second terms
are zero independently. We start by deriving the
following relation:
?d)(P)= ?d), (P )ddt dt
- -d2(p')=2P P
2 2
p pd (p)
Now, using again (2.9}
fdw [di(p)(q p)]?
(6.5)
= lim ?f dt [d&(p)(pp ?p)]1 T dT~m T o dt
taking (5.40a) and (5.40b) and (5.39}into account,
we see that
?[d~(P)(g~ P)]=(g~ P)d~(P)+T[q]
d&(y)= d2(y) .d-
dy
= lim d, (P)(gp. P)
~
p .
?
T~co T
(6.6)
Therefore, as It must be pointed out that d1 and d2 are well-
1272 R. BLANCO, L. PESQUERA, AND E. SANTOS
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.
(6.6) goes to zero, so that from (6.5), it follows
(6.3).
Let us now prove that the first term of (6.2) is
zero. For this term y=J; and yp ??VpJ;. Using
the relations (2.11),
Br
%p= g
If we put
r=gg?e'"'" ~qp ?gg?in;e'"'"
n
and
also zero. If q =J,
Br a~- ~ -+
eon w
BW BW
then
for the same reasons as before. Then, we have
a -
=
a
BJ BJfdw T[J ]=, ?f dw(jp P)di(P)
c Bd3(p)dw
2 Bwi
fdwd, (p),BJi BWi (6.8)
where we have used (6.3). For y=w,
gp = VpW)
g~ = g g ?in cooi n; e' " ' "=c ()Wi
and
dp
13(y)=d, (y)
d3(p) =di(p'),
because g ?depends only on J;,J, and w
whereas c00 depends only on J.
On the other hand, the function d i(y) is con-
tinuous and, therefore integrable. Let us define
d3(y) such that
+in ~ w
?e
BJ
a-
n. oec n ~ w-?aJ,'
a
=?c
Ji
Then, in this case
fdwT[w ]
BWi
which gives
(g~ P)di(P)=c Pd, (P)
a
d3(p) .C 8
2 Bw.
Then, Eq. (6.3) gives
BW.
?fd w( jr~ p)di(p)
82
, fdwd3(p) . (6.9)
T
dw ? ,d3(p)c 8BJ
fdw T[J;]=?fdw ? d3(p)2 ~wi
2~ af gd ?f d; d(p)Bw;
(6.7)
because d3(P) is a periodic function of w;.
Finally, we prove that the second term of (6.2) is
Hence, adding (6.8) and (6.9), we obtain the rela-
tion
8
BJ fdw T[J ]+,fdw T[io ]=0.BWi
From this equation, and (6.7} it follows Eq. (6.2}.
Once we have shown that all terms of order 1/e
in the Fokker-Planck equation (6.7) cancel out, we
may take the limit a~0, which gives
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1273
M 3
[Gi(Ji)(Wo+m' e Wo)]+ g, [Gi(J/)(Wo+n 9' Wo)1
,
BJ~ i=M+~
,
[Gi(w )(Wo+m'9"Wo)] (6.10)
It is easy to realize that the function
8'o ?const)& exp 2+2
is a solution of that equation. As the potential V goes to Do quickly eriough in the limit r~ oo, it is clear
that this solution is normalizable and, therefore, unique, which is the result we desired.
VII. CONCLUSIONS
We have shown that the stationary state of a re-
lativistic multiply periodic system in the presence
of an electromagnetic radiation with Rayleigh-
Jeans spectrum is given by the Maxwell-Boltzmann
distribution. This definitely contradicts the Boyer
claim that Maxwell-Boltzmann distribution and
Rayleigh-Jeans spectrum are incompatible in rela-
tivistic theory.
As an interesting byproduct, we have proven
that in the Fokker-Planck equation, written in the
form of the divergence of a current, the damping
and the stochastic force appear separately. In fact,
the damping gives rise to the drift and the stochas-
tic force to the diffusion. Also, the diffusion ma-
trix is symmetric, at least for the Rayleigh-Jeans
spectrum and for any spectrum in the case of a
central potential. The separation between drift and
diffusion is due to the fact that the drift term is
related to the dissipation of energy by the particle
(this can be seen in V A) and the diffusion term is
related to the increase in energy of the particle due
to the background field, which will be seen more
clearly in the second part of the paper.
Finally, we must point out that if the system is
on the equilibrium state corresponding to the solu-
tion of the stationary Fokker-Planck equation (in
our case, the relativistic Maxwell-Boltzmann distri-
bution), this fact only guarantees that there is glo-
bal energetic equilibrium with the radiation.
Nevertheless, the equilibrium should exist at each
frequency (radiative equilibrium, related to
Kirchhoff's law). If this is not fulfilled, the radia-
APPENDIX A
We shall prove the relation (5.13). Let
p=1 p r. A?n ea.sy calculation gives
0g"=
,
(r P?)(r"o . p) ??, p-,p' p'
1 d (r g~)(r p)?
pdt p
(A1)
= h'"+ h'" (A2)
where
h =?-j + (r"o'j1 ( o P) P (A3a)
1 d ro ?Ph' '= (ro j~)?.
p dt p
(A3b)
(i) co
Writing I =I~+I2 with I; = dQ h g p, we
calculate separately each integral. From (A 1) and
(A3a) we have
tion is not in equilibrium, but its spectrum is con-
tinuously changing. The study of this condition,
which is essential for the foundations of classical
relativistic statistical mechanics and the study of
the blackbody spectrum, will be made in the
second part of the paper. We shall show that there
exists radiative equilibrium between a radiation
with Rayleigh-Jeans spectrum and a relativistic
multiply periodic material system.
Ii ?fdQ p ?,(r P)(r ?P) ??,P
', P p'
pz+ [(r p)/p?](r ?q& ~ )
which, after a straightforward calculation, can be written as
1274 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
I~ ?fdQ ?I(r .p&)(r P)(P ?1)+p[(I3.P)(r .g&z)+(P Pz)(r P)]+p (P gz)J,p' (A5)
choosing the coordinate axes such that OZ is in the direction of P, it turns out that p= 1 ?P cos8 does not
depend on y and also P?=Pz ?0. Moreover, for any vector A,
A.
(A6)
Concerning the integration over the azimuthal angle qr, three kinds of terms appear in (A5):
(a) fdQ g(8) =2m f sin8g (8)d8, (A7a)
(b) fdQg(8)(r a)=2ma, f sin8cos8g(8)d8, (A7b)
(c) fdQ g(8)(rp. a)(r~. b) =2~ f d8 g(8)sin8[ ,' (a,b??+azbz)sin 8+a,b,cos 8] . (A7c)
After a few simple calculations and taking (A6) and (A7) into account, (A5) becomes
cos28)+2(1 f3 cos8)?
p 4
2~(q' p'~)(~ ~) ~ . (p ?1)(?1+3cos 8)+4p cos8(1 ?p cos8)d8 sin8
p2 p 2p
Finally, making the change y=cos8, and defining
1 K
c,= ay (1?Py)'
Ii ??rr(I3. q z)[(1+P )(cp+c2) 4Pci]?
Cp ?3C2 +4pC I+m'(P'gz)(P'P)
z
(cp+c2)?
~(2) oo
Now, we calculate I2 fdQ h g p?. After elementary calculations, it can be written as
(A10)
Considering the expressions (A7a) ?(A7c) and also
fdQg(8)(r .a)(r .b)(r .c )=2m f d8sin8g(8)[ ?,sin 8cos8[a,(b. c )+b,(a. c)+c,(a.b)]
+a,b, c,(cos 8 ?,sin 8cos8)] ?. (A11)
We perform the integration over the angle y and obtain
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1275
qP. P 2 - -. i ?- (qP P)(P P}I2 2m?f d8sin8' ?p cos8 + (?p p) ?, (pp p)?p 3 P 4 2 P P2
cos28 ( 0'p 'P }(P'P )
+ 3
r
P2) 1 yP. P (P P}(HAPP)
?sin 8 cos8 P +2
p 2
(gP P)(P.P)'+ P 3 (cos38- 2 sin28 cos8) (A12)
Making again the change cos8 =y and defining now
ykbk= f dy (1?Py)' (A13)
yk"k= f dy (1?Py)'
Iq can be written
(A14)
T
I2 2' (P?qrp)PP P
(b) ?b2}(1?P } 2 cp 3c2 (1?P ) S 3
(1?P')+ (g P)(P P) cp ?c2 ? (bt ?b3)P P (A15)
It remains to calculate the integrals bk, ck, and dk. For that, it is easy to prove the relations
bk ?? [(1+P) ?( ?1) (1?P) ]? Ck 1,1 4 k 44P(1 P2)4 4P
ck= [(1+P) ?( ?1) (1?P) ]? dk-i1 k3P( 1 P2)3 3P
which allow us to obtain the following values:
do=2y' di=2Py' co= Sy'(3+P'} ci= ,y'P?
c,= ?,y'(1+3P'), b, =2y'(1+P'), b, = ?,Py'(5+P'),
b, = ?,y'(1+5P'), b3 2Py'(1+P'} ?.
Finally, substituting (A16) into (A9) and (A15) we obtain
fdn h" g "(1 p r"')=~(p?q,. )-, y'+~(.P j,)(P P) , y. -
+2m. (p g )P' , y +2m(P Pp?P)(P P)', y ='?
which is the relation we were looking for.
(A16)
1276 R. BLANCO, L. PESQUERA, AND E. SANTOS
APPENDIX 8
We must calculate the integral
L =fdQcos ?r (r ?r ") cpu ?[q&? (I r?r ) p +(g? r )(p p ")?(g? p ")(p ~ )] .C
J
(B1)
Let us call
V~=q?(p p ") (g? p?")p
Then (B1) can be written
L =fdQ cos ?r (r ?r "}?eau [pp? (I rr ) ?p "+r V&] (B1')
Choosing the OZ axis in the direction of r ?r and using the definitions (5.37) and the relations (A7), we
perform the integration over the angle p resulting in
L = sin8 d8 cos[coa cos8 cou?]2m ( gr P ") 1 +cos 00
T
+ ( Pp? n?)(p " n?) + V~,cos81
?3 cos 8
2
Now, taking into account that
sin8 cos[coa cos8]cos8 =00
and
2sin8 sin[tea cos8]f(cos 8 ) =0,0
the result for L
r r
L =2n sin8 d8 cos(cou )cos(cuba cos8 ) ( p? p ") + ( p n?)(p "n?)1 +cos 0 =?1?3 cos 00
+ sin(cou )sin(boa cos8 )(V~ n?)cos8 (B2)
We make again the change cos8= y and consider the following integrals:
1 sinNa
Cp ? dy cos(coay) =2
?1 NA'
1f d, ( ) g 2 slncoa 2 coscoa 2 slncoa
(B3a)
(B3b)
1 cosNa sinNaSI ? dy y sin(coay) =?2
?1 NCX
(B3c)
Note that these expressions are all well defined in the case that a ~0.
Now, (B2) can be written
L =277[cos(cpu)[(g?'P ) & (Cp+Cp)+(g)?'11+ )(P a+ ) (Cp ?3Cp)]+sin(cou )(V~ n?)SI I .
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1277
Substituting the values of C0, C2, and S~, and re-
grouping terms, we obtain finally
L = 4n. cos(cou ) P [y](u)sin(boa )
ma
+4m. cos(cou ) ? 1 Q[y](u)boa cuba
+4m sin(cou ) ? R [q ](u),
co a
where P, Q, and R are given in (5.36a) ?(5.36c).
This proves finally (5.34).
APPENDIX C
and
v=sgn(u) .
Let us calculate each of the integrals in (5.34)
separately.
(a} K& ? I&(u;e)du0
P[yj(u)
a(u)
X de toe '"cos(cou)sin(coa ) .0
The integration over co gives
P [y](u) u +a
a(u) [e +(u+a) ]
a =vpuII(u), (C1a)
We want to calculate (5.34) in the limit e~0. It
will be useful in the following to know the first
terms of an expansion in the parameter u, of the
several functions which appear in the integrals.
An easy calculation gives the following expres-
sions:
u ?a
[e +(u ?a) ]
Let us introduce a parameter 5 & 0 and write
5 00
IC) = I)( ;u)edu+ f I&(u;e)du
(C2)
(C3)
II(u) = 1?
a=vp 1?
u+O(u )
u+O(u )
(Clb)
(Clc)
We shall prove that the second integral vanishes
e~0 independently of the value of 5. Note firstly
that, for any value of e, I~ goes to zero as u ~ op,
at least faster than 11'u and then, the integral ex-
ists. On the other hand, choosing e g&5, then
f I&(u;e)du
I 2n?=v ?+ ?,ul, +O(u )
PP- PP ??3
(Cld)
(C le)
f ao P 1-e ?(u) (u+a)3 (u ?a)
P[yj= ??,up[a]+O(u ),
gI, P PP
(C1f)
= ?2E' P q u 2 235 (u ?a )
(C4)
g [qr]+ O (")a 2P
R [y]=vu [?,Pg[y]+O(u)],
R [q)] =
?,([%]+O(u),a
(Cih)
(Cli)
(Clj)
The integrand of this expression is continuous
Vu, and behaves like I/u as u ~ 00. Therefore
the integral of (C4) is finite and then
f I&(u;e)du =r, (e) ~ 0.5 @~0 (C5)
Consider now the first integral of (C3). Let us
define
0[el =Qo+ uQo+ o(u '?
Qo= ?2e~p P
(Cik)
(Cll}
5P&(e;5)=e f Ii(u;e)du,
5P~(0;5)= lime f I~(u;e)du .p~0 0
(C6a)
(C6b)
(Clm} Obviously, P&(e;5}EC'(0,ao ) that is, it is con-
tinuous and has a continuous derivative for e@0.
1278 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
%e shall prove that this is also true for @=0.
Let us calculate (C6b). For this, we make the
change u =ex, so v=sgnx. Taking (Cla) into ac-
count,
5/e
Pi(e;5) = dx x I i(ex)Fi (x;II(ex)},0
(C7)
where
F ( 11 ) '1 +vpII[1+x (1+vPII) ]
1 ?vPII
[1+x(1?vPII)]2
I i(u) F(x;II(u)) EC' as a function of u, so we
can apply the theorem of the mean value,
I (u)F, (x,11(u))?I,(0+.)F,(x;II(0+ ))
r
a
=u I i(u)F, (x;II(u))
(C9)
P [qr](u)I'iu = (CS)
where 0 & u' & u. Then (C7) can be written
5/e 5/e
y, (e;5)= f dxx I,(0+)F,(x;rr(0+)) + f [r,(u)F, (x;11(u))] . 0Q
(Clo)
The second integral of (C10) will vanish as @~0.
To see that note that I i(5)F, ?;II(5)
5 6 5
E'
ar,
r, (ex)F,(x;II(ex))+r,(ex) (x;II(ex))II(ex)
art
S/e
+ f dx x I,(u)F, (x;II(u))0 BQ
(C13)
is continuous in [0, oo ) and behaves at least as
1/x in the limit x ?+ao. This is true even for
a=0, so the integral
This expression is well defined if @+0. We take
now the limit e?+0. As
f dx x [I,(u)F, (x;II(u))]
,
a=o
Fi ?,'II(5)5 1+v@ii(5)
$2 21+?(1+vPII(5) )
g2
is finite. As it is multiplied by e, it goes to zero as
e?+0. Therefore
Pi(0;5)= lim Pi(e;5)
e~o
5/e
= lim f dxx I i(0+)Fi(x;1)e~O
1 ?vPII(5)
2$21+?(1?vPII(5) )
Q2
[1+vPII(5) ]
[E +5 (I +vf131( 5)) ]
Let
=I i(0+) f xF, (x;1)dx .
Oi(0'?) =~i [q ]
(Cl 1)
(C12)
[1?vPII(5) ]
[e +5 (1?vPII(0)) ]
The first term of the right-hand side of (C13) van-
ishes as e ?+0. For the second term we make an
analysis similar to the one made above. Let
The integral in (Cl 1) is finite because xFi is
continuous in [0, ao ) and goes to zero like 1/x as
x goes to 00. The calculation of it will be made
below.
Now we are going to prove that the first deriva-
tive of P2 has a limit as e?+0. From (C7} we have
g(x, u) = 1,(u}F,(x;11(u)) .a
BQ
Using again the mean value theorem
EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1279
5/e 5/ef dx x g(x, ex) ?f dx x g(x,0+)
5/e Bg
= f dxx2ex du
5/e Bg
=e f dxx . (C14)
where
0
T, =r, (O ?) f dxxZ, (x;I),
dxx g(x;0?) .
Now, from (CS) and (Clh) we find that
Now Bg/Bu behaves like 1/x, as x ?+ oo and the
integration gives rise to a term of type inc, if
e ?5, then (C14) results in e inc which goes to
zero as a~0. Therefore
BP
~
s/e
Be
xx'g x,o+ +r, e
and
and
I' (o+ }=? k[%112
0[q)12
ay,
lim = dxx g(x, O+}
0 Be 0
B
We have then shown that
a{(l, ay) +r2(e)=D&+r2(e), (C15)
Be Be 0
where
r2(e) ?+ 0.
@~0
Once this is made, we can apply the mean value
theorem to the function P (,
Also from (C9) we see that I', is an odd func-
tion and then it is immediately clear that
Tl Tl
On the other hand, from (Clh) again,
I',(0+ ) = ?I,(0?) and, as F is odd,
g(x, O+)=g( ?x,O ?). Then D& D&. Wit?h all
this, adding (C17) and (C18) we find that
E&+E&?f I~(u;e)du =2D~+r&+r2+r&+r2
+ 00 Ii(u;0)du+R, (e)
and finally
Ei ? Ti[p]+?,f? Ii(u.;0)du+Re(e)
(C19)
P)(e;5) =P)(0;5)+e i}p )
Be
and taking (C12) and (C15) into account,
{ I?'5 }=Ti [m) +eD i +?2?)
and finally
E, =?{t,(e;5)+r i(e )=?T, [q ]+D i1 . 1
+r2(~)+r~(e) .
(C16) with R~(e) ~0.e-+0
(b) Eq ??f I3(u;e)du
=f duR [{p](u)
00 sin(coa )g f dco coe '"sin(cou }0 COAX
?cos(coa )
(C17)
If we consider now the expression of E& but in-
tegrated over values of u g0,
0
E& ??f It(u;e)du,
and making the same calculations, we obtain
E,= T,[P]+D)+ 2+?),
e
A first integration over co gives
r
R [q))(u)3=
CX
1 1 1
e
e +(u+a) e +(u ?a)
R[y](u) u +a u ?a
[e2+( + )2]2 [ 2+( )2]2
Introduring again the parameter 5, and choosing
e &&5 we have
1280 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
f "I3( u;e)du-e f"du5 5 1 1(u+a) (u ?a) 1 1(u+a) (u ?a)3+ 3
=e Ry u ?8
5
K3 ?f I3(u;e)du+r3(e) .
We define now
(C20)
which vanishes as a~0 because the integrand is
continuous in [5,oo) and behaves like 1/us as
u ~ ao. Therefore,
5
lfl3(0;5) = lim e f I3(u;e)dua~0 (C21b)
Again, $3C C'(0, Oo ) and we shall prove that also
p3 EC'[0, oo ). We show first that (C2 lb) exists.
Let us make again the change u =ex; it results in
5/e
Ijk 3 (e;5 ) =f x I 3(ex )F3(x;II (ex ) )dx
5$3(e;5 )=e f I3(u;e)du, (C21a) where
(C22)
R [q&](u}I3u = (C23)
1 1 1F3(x;II)=?
2vpx II 1+x (1+vpII) 1+x (1?vpII)
(1+vPII ) (1?vPII )22+[1+x (1+vPII) ] [1+x (1?vPII) ]
(C24)
We make now the same calculations as we made for E~. Note that, again I 3(0+} is finite, F3(x,II) behaves
like 1/x as x~oo and I 3(u). F3(x;II(u)) is continuous for u E[0,ca ). We only give here the result
with
E3 T3[p]+?, f ?I?3(u;0)du+83(G)
T [y]=I' (0+)f xF (x;1)dx .
(C25)
(C26)
Now, we are going to calculate the sum T~+ T3. Substituting (Clh) and (C9) in (Cl 1), and (Clj) and (C24)
in (C26), adding (Cl 1) and (C26) and regrouping terms, we obtain
Cfm] (1+P)'T1+~3 dx x
2p 0 [1+x (1+p) ]
00 1+ dx 1+x (1+p)
Using the integrals
(1?P)'
[1+x (1?P) ]
1
1+x (1?P) (C27)
S&(a)= fdx z z 2 ?? ? 2 2, Sz(a)= dx 2 2 ?lnx ??,ln(1+x a ),xa 1 1(1+x a ) 2(1+x a ) x (1+x a )
the integration in (C27) is immediate and the result obtained is
EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1281
(C28)Ti[t]+T3[t]= ln I+A
(c) K2 ??f Iq(u;e)du= f du Q[tp](u) f dw w e '"sin(wu)[cos(wa)/w a ?sin(wa)/w a ]. The in-
tegration over m gives
Q2[y](u)I2(u;e) = 2'
e 1 u+o,2+ 2 2 ??arctane +(u+a) e +(u ?a)~ a e
Q ?CX
?arctan
E'
We introduce again the parameter 5 and choose e ?5. Considering that arctan 1/e-m/2 ?e+ , E3?+
f Q 1 1 2I2(u;e)du -e 2+ 2 ? 2 2 +0 42a' (u +a) (u ?a)' (u ' ?a') u
f Q 2 22+0 4 du.(u ?a) u
K2 ? I2(u;e)du+r4(e) .0
Now, we define
$2(e;5)=2p e f I2(u;e)de,
Pq(0;5)=2P lim e f Iq(u;e)de .e~O
(C29)
(C30a)
(C30b)
The integral is finite, and then f I2(u;e)du ?+ 0,e~O
therefore
In this case we are interested in considering up
to the second derivative. We shall prove that
$26 C [0, ao } in the variable e. For that let us be-
gin by showing that (C30b) exist. Making again
the change u =ex we have
5/e
yg(e;5) =f Q[q)](ex)F2(x;II(ex))dx,
(C31)
where
1 1 1F2(x,II}= 2+x~II2 1+x (1+vPII) 1+x2(1?vPII)
1 [arctanx (1+vf311) ?arctanx ( 1 ?vPII )]vyllx
From that expression we see that Fq is finite in the limit e ?+0 and behaves like 1/x as x ~ ao.
We apply to the integrand of (C31), the mean value theorem, and obtain
(C32)
5/e 5/ef Q(ex)F (x, II(ex))dx =f Q(0+)F (x;1)dx +ef dx x [Q(u)F, (x, ll(u))] (C33)
The last integral is finite when we put E=0 because the integrand is of order 1/x as x~ao. Then,
equally as in the cases above,
5/e 5/e 00
lim f Q(e )Fxz( IxI(e ))xdx=lim f Q(0+)Fz(x;1)dx= f Q(0+)Fz(x;l)dx .e~O g~o 0
That expression can be integrated immediately, substituting (C32) and using the integrals
(C34)
1 1S3(a)=f dx = ???aarctan(ax),x (1+x a ) (C35}
1282 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
I
?
1 aS4(a)= fdx arctan(xa)= ? arctan(xa)+ ?S3(a) .X 2x 2
The result is
tI}2(0;5)= lim $2(E;5)=0 .
p ?+0
We calculate now the first derivative in E?+0 Fr.om (C31) we have
5 5/e
= ??Q[q](5}F, ?;11(5)+ f dxx Q[y](Ex)F (x;II(Ex))+Q[p)(Ex) (x;II(Ex))~ II(Ex)BE E' 0 BII
(C36)
(C37)
From (C32) and the expansion of arctan I/E we have
e4 1 1
F2 ?,m (5) 2+ 25 II (5) 5 (1+vPII(5)} 5 (1?vPII(5))
1 1 1
vPII (5)5 5(1?vt311 (5)) 5(1+vPII (5))
where the first term of the right-hand side of (C38) vanishes, as E ?+0. Therefore
BP2 s/e aF,(0;5)=lim f dxx Q(Ex)F2(x;II(Ex))+Q(Ex) (x;II(Ex))II(Ex)BE e~o
+O(E },
(C38)
(C39)
We apply to that integrand the mean value theorem, and after the same arguments as in cases (a) and (b),
taking into account that the derivative with respect to E is of order of 1/x, as x ?+ ao, we obtain
Defining
BF2(0;5)=f x Q(0+)F,(x;1)+Q(D+) (x;1)II(0) dx .0 8'7l (C40)
C, =f xF2(x;1)dx,
/F2C2= f x (x;1)dx,
we have
(C41a)
(C41b)
(0;5)=Q(0+ )C)+Q(0+.)II(0)C2=2p2T2[g)] .
E'
(C42)
The exp?ssions Ci and C2 will be calculated later. Now, consider the second derivative of p2. From (C38),
we have
25 Q[t](5)F, ?;11(5)? Q[q ](5) ?,11(5)5 BF2
BE E ~2 Bx 2
r
5 5 5. BF,
,
?Q[v)(5)F2 ?,11(5) +Q[y)(5) ?;II(5)II(5)BII
5/e BFg
+ f, x'dx Q[ml?x)Fz(x;11(Ex))+2Q[q](Ex) (x;II(Ex))II(Ex)
8 F2 ()F2 ~ ~+Q[q](Ex), (x;11(Ex))II'(Ex)+Q[q ](Ex) ?(x;11(Ex))11(Ex}
BH
(C43)
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1283
Now, F2(x,II)-1/x, BF2/Bx ?1/x, and BFz/BII- I/x as x~ao, then
5 4 aE2 5 S aF2 5 4F2 ?', H ?;II-e, and ?;rI -e4.
Bx 6' C1II E
Therefore, the two first terms of (C43) vanish as e~O. With respect to the integral we apply the mean
value theorem and use the same arguments as in previous cases. The result we obtain is
-, ~. BF2
lim =f x Q(0+)F2(x;1)+2Q(0+) (x;1)II(0)
?+0 ()g 0
Q F BF)+Q(0+)
~
(x;1)II (0)+Q(0+) (x;1)II(0) dx =2p2D2 .
err'
Now, we apply the Taylor formula up to the second derivative, to the function $2,
B$2 e~ 8 p2$2(e;6)=$2(0;5)+e (0;5)+, e'&ede 2
and using (C29), (C30a) ?(C30c), (C37), (C42), (C44), and (C45)
1 T,[q)+a?,+r, (e) .E'
Using the same notation as in the case of IC& it turns out that
(C44)
(C45)
(C46)
+2 T2[q ]+D2+&s?) . (C47)
Easily it can be seen that T2 ??T2 and D2 ?D2, whereby
1 1 +~
K2 ??T2+ ? duI2 u;0 +82 e
E' 00
It remains to calculate (C41a) and (C41b). From (C32),
1 1 1 1
C~ ? ?? z z + 2 z ? [arctanx(1+p) ?arctanx(1 ?p)] dx1+x'(1+P)' 1+x'(1?P)' Px
2 2 2 + 2 2 2 + arctaox 1 + ?arctanx 1?
"1 2x (1+ ) 2x (1? ) 3
[1+x'(1+p)']' [1+x'(1?p)']' Px
(C48)
Px 1+x (1+p) 1+x (1?p)
Using the integrals S],Sq, S3, S4, and also
1 arctan (xa)g~(a)= f dx arctan (xa)= ?? +a&q(a),
we obtain the following expressions:
C& ? ?2+ ?ln1 1+P1? (C49a)
Cq ? 2p y +6??lnz z 3 1+P1? (C49b)
Substituting (Clg), (Cl 1), and (Clm) and considering (C28) and (C42) we obtain
1284 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
T[p)=T, +T2+T3=(P~ P) ln + z C& +(p~ P) 21 1+P 1 -PP4p 1 ?p 4p' ' p2
1+Pln
4p 1 ?p 4p2 2p2
=di(p)Qi[9)+d2(p)Q2[V)
where we have used (C49a) and (C49b), and d i, dq, Q i, and Q2 are given by (5.40a) and (5.40b) and (5.41a)
and (5.4lb). This ends the proof of (5.38) and (5.39) and, consequently, (5.42).
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
r
d 2e ~ 1F = C7? QyCX V
3C C
whose vector character is straightforward from
(5.2). On the other hand, the correlation of the
stochastic force is given by
2 2 2(F?"(r,p, t)Fi"(r ', p', t')) =?J dao I dQcos r(r r')?co(t ?t')? ?C c
0 0 0 0(~rl rr rl )+g lij &rjspi rs +g rij jIspi~,
EJS 1JS
+ g flj Ic(Tpl pcs( jv rjor(T )
ij aa
(D2)
Here, the tensor character of the expression 5;j r; rj and the qua?ntity 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
be written
'2 2
V(r)+ m c +c BW l Bm l BW
Br I 2 BO r 2sin28 By
2 ' 1/2
= g', (D3)
which can be separated. The action variables J~ are the same as in the nonrelativistic case except that J~ is
given now by
1 1 "M (5' ?V(r)) z 2 l.J~ = p?dr= ? ?m2c2 ? dr .2' 7T Nl c2 f 2 (D4)
Through a linear canonical transformation, in the same way as in the nonrelativistic case, we define the new
variables as
27 EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1285
J) ?J),
J2 ??J2+J3?L,
J3 ?J3?Lz ~
so that the energy depends only on the first two, i.e.,
O'= 8'(Ji,J2)
(D5a)
(D5b)
(D5c)
(D6)
and, therefore, (03 ?0
Only the first two constants of action are invariant by rotations. Therefore, the reduction process must be
made by defining the operator
T[&p]= f dg pa(h; ?h;(g))g(g)= fdg&(J ?J (g))a(J ?J (g))&p(g)
+J2 2n' 2' 2'
dJs f dwi f dw, f dw3(p(Ji&J, &Ji&wi&w, &w3) (D7)
in such a way that we add the contributions of the motions described by each orbit in the reverse sense.
This is related to the way in which the coefficients are transformed by time reversal. We see that Ji and J2,
aside from being by rotations, are also time-reversal invariant. We shall use this property in order to prove
the symmetry of the diffusion matrix. To do that, we start by generalizing a result already proved in the
case that the stochastic force does not depend on the phase variables. ' Let
(Dg)
then the diffusion coefficients fulfill
B&J'(g, u) = (K(((,0)K~(g ",?u ) )
be the correlation of the stochastic force which appears in the expression of 6"'. Let g represent the same
spatial point as g, but with the velocity in the reverse sense. In our case g =(r, p) ~g =(r, ?p). We shall
write g; =e;g; with e; =+1 according to the value of the index i. We shall show the following general prop-
osition.
If B,J fulfills the relation
Bi (g, )u= s?eiBi?(g ",u ) (D9)
OTG" =eIJ,evG " (D10)
Let us assume also that h& e&h& with ?e& ?+1. If this is not true, as OT I and it has e?igenvalues +1, it
is possible to find linear combinations that fulfill it. In order to show (D10) we use (3.3b) and (D7) to ob-
tain
ah?aI,0 6""=6""(e;h;)=a f ding 5(e;h; h;(g)) f dug? B?i(g,u) .ag?ag-" "
We perform now the change of variables g;~g; =e;g; The follow. ing relations hold:
(Dl 1)
(D12a)
hq(e?g?)=eqhq(g, )=h~(g?),
(D12b)
(D12c)
ah?(g?) ag?ah?(g?)
a
(D12d)
Bh?
ag;"
ah?(g; ") ah?(e;g;") ah?(g;")
a(e,g, ") '"" ag, "
(D12e)
1286 R. BLANCO, L. PESQUERA, AND E. SANTOS 27
Then (Dl 1) becomes
OTG""=a f dg ff 5(h; ?h;(g)) f du g " ?e?e,ereiBri(e;g;, u) .
The new change g =g;" ?g, =g', "gives
Bh?t)h?OTG"'=a f dg'g5(h; ?h;(g')) f du g, ?e e e?eiB?i(E;g'; ",u) .
?, ag,
' ag;" " ' "
Now e;g', "=g and, by condition (D9),
E?eiBri(g,u) =Bi,(g,u),
whence
t)h, t)h?OrG&"=eqe, a f dg' ff 5(h; ?hi(g')) f du Q, , "?Bir(g/', u)=aqua, G'?ar' aC'?-"
which completes the proof.
Now, we apply this result to our particular case.
We start by considering the implications of Eq.
(D9) for this case. As g?=(r, p) and g=?(r, ?p),
then e~ ?e2 ?e3 ?1 and e4 ?e5 ?e6 ??1. More-
over, B,i is not zero only for r, l & 3. Putting this
into (D8), Eq. (D9) is written
{Fst(~~ 0)Fst(~ ?a ~?u
=&F (-r-",-p-",0)F,"(-r, ?p, ?u))
(D13)
because P=g ".
In order to understand the physical meaning of
(D13) let us consider a path and assume that the
particle follows it in a given sense which we label
with a plus. Then, the point r, p will be written
r i+ and the point r ",p "as r2+. On theleft-
hand side of (D13), we calculate the component r
of the force in the point r i+ at time r =0, and the
component l at the point r 2+ at time t= ?u. This
is the same as putting the component r in the point
r &+ and the component l in the point r 2+, the
former at a time u after. [Note in (D2) that the
explicit dependence with respect to the times in the
correlations is through their difference]. In the
motion with the stated sense, r 2+ is reached before
r i+, so that Fi is calculated before at the point
where the particle is earlier.
On the right-hand side of (D13), F? is calculated
at r~, but the sense of the motion is the contrary,
r
~, and at time ?u. Also I'I is calculated at r2,
also with the reverse sense of the motion, r 2, at
time 0.
We see, therefore, that F, and Fi are calculated
both at the same point of the path, in such a way
that the first one calculated is the first one to "see"
the particle. In summary, it gives the same result
the calculation of the correlation between two
points of the same path with independence of the
sense of the motion provided that we put the value
of the field at the time in which the particle is at
that point. Also, it is indifferent to calculate be-
fore either point provided that the sense of the
motion is such that we calculate before the field of
the point where the particle is first.
Let us see that relation (D13) is fulfilled in our
case. To do that we use (D2) which leads to
(Fst(~ ~ 0)Fst(~ ?g ~?tt ) )
e ~ co 3r (io)2 txt 2 2dN dQ cos r(r ?r ) ?~u (5.i rrri )+g(euj?erjsPt rs+eajeijsPirs )~?u
C c EJS
+ g erij elaaPi~a (5j a rj a )
EJECT(T
(D14)
EQUILIBRIUM BETWEEN RADIATION AND MATTER FOR. . . 1287
and
(Fst(~ ?Q ~ s 0)Fst(~ ~ ) )
dco dQ cos ?r (r "?r) ?toue' " co~8'2(co ) ~O ?u3 o 2 C
r
X (5tr rt r??)+g [e zetjs( pt?)rs+etzerjs( p?; )rs]
EJS
+ get;, E?(?p, ")( p, )(5?, r,'r ?)
ij acr
Performing the change r ' = r?, we have
iFt"( r ",?p ",0)F,"(r, ?p, ?u ) )
e ~ co 9' (co)2 oo 2 2 + ~o. ?u8N dQ cos ? r(r ?r ) ?cou
c 2 C
r
X (5?t rr rt )+g?(ertJEttsptrs +ettJ'Ejsp; "r, ) + g cia(pertjpa pt(5jo rj r(7)
1JS ij aa
where in the last terra we have changed the indices i by a and j by 0.
We see that (D15) and (D14) are equal and, therefore, relation (D13) is fulfilled. As a consequence we
may write
G""(Ephp) =e?e?G""(hp).
Now, as the only constants present are time-reversal invariant, it results ep =1,9=1,2 whence
G""(hp )=G'"(h p ),
which completes the proof.
(D15)
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