Derivation of Maxwell's equations

,I(R, t) =

(37)

for the charge-current density, P(R, t) =

, M(R, t) = Cm>

(38)

for the polarization tensor and a0 =

alact,o = a/ax, alay, ala2

(39)

for the time and space derivatives. From (l)-(4), with the help of the lemma (33), (35), we obtain now the macroscopic equations V.B = 0, (40) aoB + V A E = 0, (41) V-E = Q - V.P,

(42)

--aoE + V A B = I/c + i&P + V A M,

(43) which are Maxwell’s equations. They have manifestly the same form and covariance properties as their atomic counterparts. We may notice that, although the charge-current density vector and the polarization tensor are found here as retarded averages of quantities containing the function (1 -

Rjj’. (R -

Rp)/c

IR -

RpI}-ld(Rp

-

I?),

they are simply equal to the averages of the expressions (5), (6) and (7), (8), containing 6(R,(O)- R), with “ordinary distribution functions”. As a matter of fact it can be shown (appendix I) that the following relation exists between retarded and ordinary distribution functions: fret(Ri’),

..-,

Rf’, E&O’, . .., lbm’IR, t) = f(

1

1-

Rf’.(R - Rho’) c ,R _ Rio’,

R’o’ k > ***, Rj‘?‘, tie’ , *a*,&m’., t -

\R - Rio’1 c

>

*

(44)

where f(Ri”‘, . .., Rp’, [Lo’, . . ., lim); tt) is the ordinary distribution function for the system, which is defined by writing the number of atoms in the volume element dRi”’ . . . dRp) d5i”’ . . . d.@’ around the point Rjf’, . . . . Rp’, go, k , . . ., tirn’ in fluxion space at the time tk as: f(Ri”), . . . . Rp’, lie’, . . . . @“‘;

tk) dRi”’ . . . dRII”’ d$”

. . . d@‘?

(45)

The factor between curly brackets in (44) also appears in the denominator of { 1 - RF). (R - Ri”))/cIR - Rr)I}-l d(Rf) - R). Therefore, if we average this quantity with a retarded distribution function, we get the same result as by averaging d(R,co)- R) with an ordinary distribution function.

DERIVATION

OF MAXWELL’S

263

EQUATIONS

An alternative form of equations (42) and (43) is obtained when the field tensor D=E+P,H=B-M (46) is introduced*). Then (42) and (43) can be written as V-D = Q,

(47)

-aaoD + V A H = I/c.

(48)

The quantities (37) and (38), and therefore also (46) contain (ordinary) averages over (5), (6), (7) and (8). We shall discuss a few aspects of the Maxwell equations, involving these averages, in the next section. The lemma also permits to derive macroscopic charge conservation from microscopic charge conservation (appendix II). 3 8. On special forms of the Maxwell equations. The Maxwell equations (40)-(43), as well as the atomic field equations (l)-(4), were written with a polarization tensor developed up to second order in the internal coordinates. Then one has as atomic multipoles: the electric dipole moment pk, the electric quadrupole moment qk and the magnetic dipole moment vk, which appear in the expressions (7) and (8). However, one can also choose to classify the terms of the polarization tensor according to their multipole order, and study, for instance, the case of the dipoles only 3). Then (38) with (7) and (8) has the form p =
(ak'pk

+

Pk * vk)d(Rk

-

R)>,

(49)


Pk A pk)d(Rk

-

R)>.

(50)

k M=

The polarization (49) contains a part arising from the motion of magnetic moments, which does not occur in non-relativistic theory (see $9). One notices that the forms of both P and M are affected by the atomic velocity . Rk = pkc.

If all atomic velocities tik are equal, say to /?c, we obtain for (49) and (50), p = fi.
pkd(Rk

~*<;vk~(& k

-

-

R))

+

p*


R)>

-

p A
-

R)>,

(51)

-

R)),

(52)

because s2 =

u+

(%Q - j!?s-

1) p/3/p”

(53)

is now a constant quantity. These formulae may be applied to the case of a conductor with a rigid ion lattice. (Note that the arbitrarily moving con*) The fields E, B, D and H are thus defined in a general theoretical

way as averages

scopic quantities. Since they satisfy the Maxwell equations, one can derive “operational” definition of these fields, as measured in certain cavities.

from

of the micro-

these the

usual

264

S.

R. DE GROOT

duction electrons do not contribute and vk = 0.)

AND

J. VLIEGER

to P and M because they have pk = 0

$ 9. Discussion of previous work. Relativistic theories in which it was tried to derive the Maxwell equations from the microscopic field equations were given by Dgllenbachy), Kaufmans), Bacryg) and Nyborglo). Dgllenbach stressed the desirability of expressing the polarization tensor in terms of the atomic multipole moments in a covariant way. (In this paper (38) with (7) and (8).) He gave, however, only approximate formulae, involving the dipole moments. His equations, expressed in terms of microscopic quantities, are then no longer covariant. Kaufman and Bacry give a multipole expansion, which is based on an extra hypothesis concerning the internal times of the atomic system. One would then arrive at formulae different from (7) and (8). (N o such extra assumption has to be made in our work.) Neither Dgllenbach, nor Bacry or Nyborg express the polarization tensor in terms of the proper atomic momentss) pk, qk, vk, etc. (see $2). Kaufman, however, finds in the dipole approximation expressions (49) and(50) correctly. The other papers on the subject consider the Maxwell equations with non-relativistic expressions for the polarization and the magnetization. In these papers averaging procedures are carried out along with a multipole expansion of some kind. Lorentz’ original derivationl) (which was refined by Rosenfeldl) and extended to arbitrary multipole order in Voisin’sll) paper) and Mazur and Nij boer’s treatmentz) were already mentioned in 0 1. Mazur and Nijboer considered the fields as dynamical quantities which depend on the particle phases (positions and velocities), and also explicitly on R and t. As Mazurl2) showed, this requires particular initial conditions to be fulfilled by the fields. Brittinls) and Schraml4) therefore used distribution functions which depend not only on the phases of the particles, but also on those of the field oscillators. DSllenbach7) performed a covariant averaging over small space-time volumes. In the present paper we introduced a covariant averaging over the motion of the particles in a physically small cell. We described this averaging by means of a distribution function which may depend on the atomic positions and multipole moments and on the first and higher time derivatives of these quantities, also taking retardation of the fields into account. The lemma (33), (35), according to which differentiations commute with averaging, was stated already by Rosenfeldl). It was proved in nonrelativistic theory for phase space averages by Irving and Kirkwood6) and, for the same case, derived from conservation of probability alone by Mazur and Nijboerz)*). *) One should note. that these authors write the conservation of probability equation which has the form of a partial differential equation for the probability

as a continuity density in phase

DERIVATION

The treatment

of Mazur

p =
k

M

=

OF MAXWELL’S

and Nijboers)

(pk


-

-

‘Jk’v)

Pk

*

which is indeed the non-relativistic

leads to Maxwell equations with

d(Rk

(pk

265

EQUATIONS

-

-

R>,

qk.V)}

limiting

(54) d(Rk

-

R)>,

(55)

case of (38) with (7) and (8),

as can be seen by neglecting squares of (atomic and intra-atomic) velocities. Results obtained prior to Mazur and Nijboer were not quite so general since they involved only systems at rest or systems moving with uniform velocity, whereas (54) and (55) allow for arbitrary, non-relativistic, motion, just as (38) with (7) and (8) are valid for arbitrary, relativistic, motion.

APPENDIX

Connexion

I

between retarded and ordinary distribution fmctiom

in fluxion

Let us consider a shrinking light wave front in fluxion space arriving in Rk”) = R at the time t, and a volume element dV (around the point Rio’, . . . . Rp’ , t(O) k , ***>lp)) of which two surface elements coincide with the wave front at two different times and the others are perpendicular to the wave front. Now it follows from (25) that the number of atoms, encountered by this wave front in dV is equal to space.

fret(R~“), . . . . RF’,

$“,

. . . . @:“‘jR, t) dV,

(A 1)

This number must be equal to the number of particles present in dV at the time t - IR - Rio’1 / c, when the light wave passes through it:

f(

Rio), . . . . Rp), &O’, . . ., Ep’; t -

IR -

Rio)1 C

>

dV

(A 2)

(where f is the ordinary distribution function defined by (45)), minus the number of particles leaving dV without being struck by the wave. For the latter number we have to consider only the particles escaping through the surface element dS at which the wave front arrives last. (We can disregard the contributions of atoms disappearing through other surface elements, since they are of higher order in the differentials.) The number of atoms disappearing is therefore equal to the number of particles in the volume element (Rp& dt dS, where (Rf’)n is the projection of Ri” in the direction normal to the surface element dS and dt the time which it takes the light to space.

For the proof

conservation

of the lemma,

law, expressing

however,

one could

the fact that, if one follows

alternatively

use a different

form of this

a region of phase space points in its natural

motion, the number of systems with their phases in this region is conserved (cf. eq. (29) in our case). Conversely we could also have used for the proof of our lemma the conservation of particles in the form of a continuity equation. This equation follows for instance from equation (29) and has the form of an integro-differential straightforward.

equations).

The

method

of 5 6 is, however,

somewhat

more

266

S. R. DE GROOT

AND

1. VLIEGER

travel through dV. Since

and cdtdS

= dV,

we find that

so that the number of atoms disappearing

f(

Rp, .

..)

RF), go’, .

is given by

JR ..)

pp;

t

-

Rio’\ c

RLl’.(R c IR -

-

Rio)) dV

(A3)

Rk”‘I

for Rio) # R. For R,(O)= R this number is zero, since the wave front now approaches the volume element in such a way that no particles can escape it. From (A I)-(A 3) we obtain te desired relation (44).

APPENDIX

II

Conservation of charge. From the definitions of the atomic current density (5) and (6), follows the microscopic conservation electric charge dpldt = -Q-i.

chargelaw of (A 4)

With the definition (37) of the macroscopic charge-current density and the fact that differentiation commutes with averaging, we then obtain directly the macroscopic law of conservation of charge +/at = -Q

.I.

(A 5)

The macroscopic charge-current density (Q, Z/c) is a four-vector because its atomic counterpart (5) and (6) is a four-vector. Therefore if Q = 0 in a certain reference frame, the transformed charge density Q’ in a different reference frame will in general not vanish, although the charges ek: are invariant quantities 15). APPENDIX

III

The plasma. In a fully ionized plasma, there are no stable groups of point particles. In the terminology of this paper we have to consider all single electrons and nuclei as sub-units (called “atoms” in $2) numbered by-k. These sub-units have no multipole moments in this theory. Hence the atomic polarization tensor (p, m), given by (7) and (8), vanishes. The atomic field

DERIVATION

OF MAXWELL’S

267

EQUATIONS

equations (3) and (4) become simply V.e=p,

(-4 6)

--doe + V A b = i/c,

(A 7)

with the atomic charge-current density vector (5), (6), where k now numbers all electrons and nuclei. The Maxwell equations (42) and (43) become then V-E =Q,

(A 8)

-&-,E + V A~B;= I/c,

(A 9)

with the macroscopic charge-current vector given by (37). Since P and M vanish, the fields D and H are equal to E and B in this case. Even for plasmas which are not fully ionized, equations (A 8) and (A 9) usually form the starting point. This means that polarizations of the ions are neglected*). APPENDIX

IV

The Lienard-Wiechert retarded Averaged Lihard-Wieckert potentials. potential**) of a set of point particles k has the form

’ = f vk =

2

IR -

Rk I{1 -

&*;.-

&)/Cl

R

- Rk))

(A 10)

for the scalar potential and a similar expression for the vector potential. These are the solutions of (A 6) and (A 7) with (l), (2), (5) and (6). We write the macroscopic potential @(R, t) as the averaged LienardWiechert potential: @(R, t) = ff

q’kfret(& &ciR, t) d& dhk,

(A 11)

where fretis the retarded distribution function in the fluxion space spanned by Rk and tik. We can write (A 11) with (A 10) as

@tR,t, = jj

JR

yRk,f(&,

lilk; t - IR - &l/C) d& diik,

(A 12)

because of the relation (44) between retarded and ordinary distribution functions fret and f in fluxion space. The integration over the velocity gives

@lR,t, =

l’,RTRk,f(Rk; t -

*) We note that in the theory of plasmas, a refractive index, although the polarization **)

Advanced

potentials

could be formally

IR - &l/c) d&,

with (A 8) and (A 9) as Maxwell equations, vanishes. treated

in a completely

similar theory.

(A 13) one derives

268

DERIVATION

which, upon introduction 1

@(I?, t) =

IR - R’j

Now we recognize

OF MAXWELL’S

EQUATIONS

of a delta-function

e&R

can be written as

- R/J f(Rk; t - jR - WI/c) dRk dR’.

here the average charge density _/-Q&R -

Rk) f(&;

t) d&

(A 14)

(37) with (5):

= e(R, t),

(A 15)

so that (A 14) becomes

se(R’,t -

@(R,t) =

IR - R’llc)

dR,

(R - R’1

We have thus derived the expression for the retarded tinuous charge distribution from the Lienard-Wiechert particles. Similarly one finds for the vector potential

A(R, t) =

s

(A 16)

’

potential of a conpotential of point

l(R’, t - IR - R’llc) dR, IR - R’I ’

(A 17)

where I is the averaged electric current density. The authors are indebted stimulating remarks. Received

to Professor

G. E. Uhlenbeck

for some

9-10-64

REFERENCES

1)

Lorentz,

H. A., Proc. roy. Acad.,

Rosenfeld,

L., Theory

2)

Mazur,

3)

de Groo t, S. R. and Vlieger, (1964)71, 294; Nuovo Cimento

4)

de Groot,

5)

Vlieger,

6)

Irving,

P. and Nijboer,

(1951).

19 (1953) 971.

J., Physica 31 (1965) 33 (1964) 1225. P., Non-equilibrium

P. and de Groot,

J. H. and Kirkwood,

(1902) 254;

Amsterdam

B. R. A., Physica

S. R. and Mazur, J., Mazur,

Amsterdam

of electrons,

thermodynamics,

S. R., Physica

J. G., J. them.

125; for results see also Phys.

12) Mazur, 13) Brittin,

10

(1962).

27 (1961) 353, 957, 974.

Phys. 18 (1950) 817.

Dsllenbach, W., Ann. Physik 58 (1919) 523; cf. Pauli, A. N., Ann. Physics 18 (1962) 264. 8) Kaufman, H., Ann. Physique 8 (1963) 197. 9) Bacry, P., Nuovo Cimento 31 (1964) 1209; 32 (1964) ‘0) Nyborg, J., Physica 25 (1959) 195. 11) Voisin, 7)

Amsterdam

Letters

W., reference

15.

1131.

P., Adv. them. Phys. l(l958) 310. W. E., Phys. Rev. 108 (1957) 843.

K., Physica 26 (1960) 1080. 14) Schram, W., Enc. math. Wiss. V 2, fast. 4; RelativitBtstheorie, 151 Pauli, vity, London (1958).

Leipzig

(1921);

Theory

of relati-