Relativistic ring equations

Physica

43 (1969)

321352

o North-Holland

RELATIVISTIC

Publishing

RING

Co., Amsterdam

EQUATIONS

I. 11 PERTURBATIVE

APPROACH

M. BAUS* Faculte’ des Sciences,

Universitd

Received

Libre

de Bruxelles,

31 October

Uelgique

1968

Synopsis Using geneous ring

a perturbative relativistic

with

The

kinetic

approximation.

and the radiation cases.

approach

plasma transfer

For a strongly

incomplete.

cuts and undamped We also derive

equation

stable

Specifically

the finite

system

relativistic

plasma

the coupled

frequency

electromagnetic

waves system

equation

for

collision

interactions the

particle

for the electromagnetic previous

results

contributions which

have

operator

momentum

hitherto

within

been neglected

for unstable

for various

but are shown equation

the

distribution

field are derived

are recovered

to the kinetic

of ring equations

for a homo-

is computed

to be

from branch are discussed.

relativistic

plasmas.

The culmination of the kinetic theory of 1. Introduction and summary. classical plasmas with only Coulomb interactions has been the derivation of a kinetic equation within the ring approximation, i.e., the BalescuLenardr) equation. The interest in radiation processes and relativistic effects in plasmas has directed much of the later investigations towards an extension of the original Coulomb plasma formalism to plasmas with electromagnetic interactionss-7). S ome of these investigations contributed to the derivation of kinetic equations within the ring approximation for such electromagnetic plasmas. Klimontovich derived a kinetic equation for a strongly stable, homogeneous and isotropic (in position and momentum space) plasma with electromagnetic interactions and relativistic particle dynamics using his formalism and the integral equation methods). An equation including anisotropic effects was obtained by Silins) but via a rather ad hoc procedure. Using the test particle method and non-relativistic particle dynamics Rostoker, Aamodt and Eldridge4) obtained a different expression in form but equivalent result. On the other hand the equation obtained by Dupree5) contains a friction coefficient which does not agree with the one of the previous results. In this paper, see also ref. 6, we consider a perturbative approach to the ring equations for a homogeneous plasma with electromagnetic interactions * Aspirant

du Fonds

National

Belge

de la Recherche 321

Scientifique.

322

RI. RAUS

and relativistic particle dynamics. The retarded interactions will be described with the aid of the oscillator formalism developed by iMangeney7) within the framework of the Prigogine theory8). In section 2 the relativistic ring diagrams generalizing Balescu’s ring diagramsI) will be summed with the aid of a method used by Rksibois”) in the Coulomb case and based on his so-called factorization theorem. This yields then the finite-frequency collision operator within the ring approximation and is the starting point for the kinetic equations to be derived. Next we consider the kinetic equations in three typical cases. In section 3 we consider the asymptotic collision operator and the subsequent kinetic equations for a strongly stable plasma without account of non-classical effects. By strongly stable L\:C mean a plasma for which the dispersion equation for electromagnetic waves has only roots corresponding to strongly damped modes i.e., damped on a collision-time scale so that we can neglect their influence on the relaxation. By nonclassical effects we mean those contributions which arise from the branch cuts in the complex planes in which the finite-frequency collision operator is defined and which appear only Lvhen the relativistic modifications to the non-relativistic theory are properly taken into account. These contributions come from outside the classical region /Re ~1 < ck of the complex z plane for any wave vector k and consequently vanish in the non-relativistic limit. Another non-classical effect arises from the undamped modes allowed for by the dispersion equation of a relativistic plasma. When these two types of non-classical effects arc neglected, which is not necessarily a consistent approximation for a relativistic treatment, together with the contributions from the strongly damped modes, the collision operator simplifies drastically and a kinetic equation for the one-particle momentum distribution is obtained which is the relativistic generalization of Balescu’s ring equation and which agrees with the results of refs. 2-5 when the limitations stated in the beginning of the introduction are taken into account. In this approximation the kinetic equation for the action variable distribution of the field oscillators has a vanishing r.h.s. The first moment of this equation could have been inferred easily from the fact that the particle equation conserves the particle energy while the different field modes remain independent within this approximation. To obtain these results it has also been necessary to neglect rapidly oscillating contributions to the time-dependent collision operator from which the non-vanishing leading approximation in the sense of refs. 1 and 10 should still be extracted. In section 4 we consider the modifications to the previous case introduced when the dispersion equation for electromagnetic waves has roots corresponding to modes either damped or growing on a relaxation time scale and give explicitly the expressions for the cast of a weakly unstable system. For a weakly stable plasma the situation would be quite similar. The kinetic equations describing this situation form a coupled system of

RELATIVISTIC

equations relativistic

which is rather case

RING

complicated

EQUATIONS.

I

even when compared

1). The particle equation,

323

with the non-

while still of the Fokker-Planck

type, contains now diffusion and friction coefficients oscillating at the frequencies of the normal modes of the plasma and coupled to the field through the average energy in the field modes. The field equation is no longer stationary but describes emission, absorption, scattering and mode conversion processes for the longitudinal and transverse field modes. The explicit form of the cross sections for these different radiation processes governed by the particle momentum distribution are given. Assuming the damping rates to be very small compared to the (real) frequency of the electromagnetic plasma modes, the system of kinetic equations will possibly transform into the so-called quasilinear form as shown for the non-relativistic limit in ref. 10. It should be noticed, however, that when this is done the stabilization mechanism usually discussed within this framework is completely modified because now not only a redistribution of the particle energy becomes possible, but also a redistribution of the field energy, for example as radiation. In section 5 we consider the specifically relativistic effects arising from the non-classical regions of the complex planes on which the finite frequency collision operators are defined. The origin of these contributions is the relativistic limit for the particle velocities which yield Cauchy integrals, such as the dielectric tensor, with a finite cut in the complex plane. The end-points of the cut are branch points and it is thus necessary to introduce branch cuts. However, the branch cuts can be introduced in such a way that the region between them (jRe z/ < ck) is completely analogous to the non-relativistic or classical complex plane. The contributions from thenon-classicalregions consists of cut-contributions, undamped pole contributions and eventually additional weakly stable and unstable poles. As, moreover, the collision operator involves two complex variables these contributions are not additive because the contributions from the classical region of one variable can be evaluated afterwards in a non-classical region of the other variable. The resulting collision term is extremely complicated and only the radiation transfer equation is qualitatively discussed. However, it is indicated that there are non-classical contributions even to the asymptotic collision operator while the time-dependent terms may possibly decay but will do so on a time scale which cannot be precised. Equilibrium evaluationsill is) of different but related expressions show a non-exponential decay with a crucial wave vector and temperature dependence. From this it follows that even for a strongly stable plasma these contributions can in general not be omitted. On the other hand it should be noticed that the thorough modification of the structure of the complex plane by relativistic effects invalidates the often made statement43 5) that relativistic effects can be incorporated into a non-relativistic treatment through a suitable transformation of variables from velocities to (relativistic) momenta. Clearly our main emphasis

324

M. BAIJS

is on the qualitative aspects of the relativistic kinetic equations because due to the extremely complicated collision terms a quantitative physical insight is only possible on the basis of further approximations. This, however, is outside the scope of this paper. The appendix a contains a simple complex plane proof of the factorization theorem used in the summation of the relativistic ring diagrams. Some explicit expressions for quantities abbrcviated in the text are given in appendix b. In view of the rather complicated algebra used in this perturbative approach, a second paperra) will be devoted to a non-perturbative approach with the aid of which the derivation of the relativistic ring equations is reconsidered. 2. Summation of the relativistic ring diagrams. The relativistic plasma will be described with the aid of Mangeney’s oscillator formalism 7). The particle phase space variables will be the positions and mechanical momenta {xi, pi}(i = 1, . . ., N) while the field variables will be the angle and action variables {[A, VA} (A = k~, ,UA)of the vacuum field oscillators of wave vector kn and polarization ,UA. The Liouville operator, its matrix elements, the diagram technique and all other information about the Prigogine theory can be found in ref. 14 and will not be repeated here. As is well knownr) the nil approximation (vz,~= number of particles in a Debye sphere) to the collision operator Y of the Prigogine theory can be obtained by summing all the ring diagrams. These diagrams of order e”(eU)n (n = 0, 1, . . ., co) describe also the collective processes exhibited by a homogeneous relativistic plasma. Here e is the electron charge and d the mean number density. The relativistic extension of the Coulomb ring diagrams is straightforward and yields, in diagrammatic language, the following expression for the collision operator in the Lorentz gauge:

(la)

where the full line describes

a particle

state and the dotted line an oscillator

RELATIVISTIC

state

as explained

RING

EQUATIONS.

325

I

in ref. 14, and where the box in (1) is defined

by the

equation : -l.j-p_

= A._+,

n’ + _!+__

II’

+n

_-[,__q-JLt;__L

a (2.4 or else: q-yl

= ~s~,,.+-__r__.d_

+

ci

n

----I,

m

_____“‘+etc, [P (2.b)

where 6,, 12,in (2) is a Kronecker delta. Moreover, it is implicitly understood that (1 .b) contains all possible relative time orders of the two boxes. Indeed, the first and last vertices in (1) are determined by the homogeneity condition and in between one can use only vertices which introduce a new particle for each es-factor so as to obtain powers of (e%) and retain thus all powers of (cad)” e‘4?N q,t in the solution of the kinetic equation. Here (up is the plasma frequency defining the short time scale. Moreover, all intermediate states have to be correlated states according to the so called irreducibility condition 14). Clearly, the only vertices which satisfy these conditions

are -

----

diagrams of (1). It is also recognized

, and _____-

and from these one constructs

that (1) is the relativistic

generalization

ring diagramsi)

in which the Coulomb interaction

by

gauged)

a (Lorentz

retarded

interaction

knowledge of the previous investigations that (la) corresponds to the self-energy

o

the

of Balescu’s is now replaced

-______.

With

some

in this fieldi*) it is also recognized diagram while (1 b) corresponds to

the diagrams of the Landau equation but now with the single particle propagator ( ) replaced by a screened or Vlasov propagator (---c)-).

h/loreover, it should

be pointed

out that

(la)

has no non-

relativistic analogue. The processes described by these diagrams are thus the self-interaction of a screened particle (1 a) and a Landau type of collision through a between two screened particles, i.e., two particles interacting screened field. The summation of (1) will thus yield all the possible information about the screening or collective effects within the n;’ approximation. However, this information is obtained only, it is fair to say so, after elementary but tricky algebra. The form of the diagrams (1) suggests to perform the summation with the aid of the relativistic extension of a factorization theorem already used by Resiboiss) in the non-relativistic case. This theorem is reconsidered in appendix a where a simpler proof is given. Using the factorization theorem we can state that the sum over all possible relative time orders of twotin-

RI. NAUS

326

dependent (commuting) branches of the ring diagrams equals their product at the final time (or, when Laplace-transformed, equals the convolution of the two branches). In order now to make the subsequent notations easier we introduce the following conventions : a) The upper line labels will be (I, z’, j, A) with further specification such as (ii, j’, . .), whereas the lower line labels will be (-2, z - z’, i, y). See for example ( 1) . b) the contribution of a vertex (the A(j, A) of ref. 14) will be denoted here V,,(I/nj) when the oscillator-line enters the vertex from the right (left). c) the numerical coefficients will be grouped around the (complex plane) propagators as follows :

(3) where e,, , v, are the charge and velocity of particle n, ,Q the normalization volume, TV, E, are the action variable and the free motion frequency of oscillator 01== (kK, ,u~) with 14) :

1 = ej.kl

=Y

-q,ky;

v =

We now consider the Laplace Within the ring approximation factorization theorem (a.5) as :

where C denotes ,u and the signs butions from the of the diagrams

c III =

v/l

=

v,,.

(4)

transform Y(z) of the collision operator14). (1) we can rewrite this operator on using the

a sum over the wave vector 1, the particles, the polarizations F of the oscillators. Moreover Ra, Rb denote the contri(la)-rings and (1 b)-rings respectively. The very swzmation reads then:

Rb = VjyP., V,tPi;,P,;( where we introduced

Vi,y,P,V,,j,

the following

+ Vj,A,P; VAy,),

abbreviation

for the Vlasov

(6) propagator

KELATIVlSTIC

KING EQUATIONS.

I

327

P,ir + etc.

(7)

(the box (2)):

= Pifc?j,j’ + Pi’(Vj,,Pg&.,) + P,+(V’jL,Pg%,j,)

P?

+

P~(Vjln,P:yn,jf)

and a similar definition for Pii,. As next step we shall reduce the general expression (6). A simplification will occur. then if, in agreement with the discussion of ref. 14, we assume pa(t) to be completely factorized, i.e., we neglect momentum and action correlations. This assumption is clearly meaningful only in a large systemId). We now use the fact that the vertex operator I/,,, is a sum of two differential operators, one in the particle momentum (n) and one in the action variable of oscillator 01, together with the fact that the momentum and action distribution function po vanishes at the boundaries of phase space, in order to write:

vm-= v::,+ v;,;

(8)

s dPnvna~o(Pn) = J dp,V,“,po(pn) ! d%~n~~~~:,,,~o(~d

=

j

(9)

;

d~aV;,P,V&po(qLy)

=

= V;,p,A,,

(10)

with :

pa= (-) qp,;

v;,,,=

A,,

A my& = 27W,. E& ,~a = 274&L*% -

-A-; a%

qLm,“);

‘0 = (v, c); E/i = (Q, d,, 0). Using (9) for the last vertex

(11) (in time) at which a given particle

can drop the particle part of that pearing always in the combination (6) will thus read:

appears,

we

vertex and are left with oscillators apindicated in (10). The reduced forms of

(12) (13) (14 (15)

nl.

328

PUS

where Jj df indicates an integration over the whole phase space except j ancl \ve introduced the following abbreviations for the reduced boxes (see (7)) : J dl’l’,;, i

= S dr’fi;, ;

j dW$ :’

p,g = I’;h;,j.

+ P~(vj,,pgA,,,,)

7rjjT = Pj+hj,jt

+ Pj+ (Vj~.,P~lV&f)

:

j dr7ij;., ; ..

Pi! -1. etc. ; I’,? +

(16)

etc.

(17)

The next step consists in the extraction of geometric series from (16, 17). The repeating units are, however, connected by a four-dimensional scalar product :

This is a consequence of our using the Lorentz gauge [pn =z 0, 1, 2, 3). The geometric series vvill thus repeat a four by four tensorial unit. Such series are, however, irrelevant here because as is well known 2,15) the collective processes are functionals of the dielectric tensor which is a three by three tensor. Therefore we shall regroup the longitudinal (,B = 3) and scalar (,u = 0) polarization contributions before summing the series (12-15). As the arithmetics involved are quite cumbersome and tricky we have indicated the main steps in appendix b. The kinetic equation for degree of freedom CX(U= j or y) can be written:

r’tpo(ol;t)

1

with a collision

(194

C‘npo(ol;t) + IL operator:

where pa(t) is the momenta and action distribution of the whole system while ~“(cx; i) is the momentum distribution (a: == j) or action distribution (a = y) of degree of freedom ~1. The explicit expression of the destruction term D, figuring in (19a) Lvill not be used here. Using the factorization theorem the collision operator (19~) can be written: dz’C,,(z, z’; i -

c&; t - T) = ?ii-

r).

(20)

(” As a result

of the above

computations

we arrive

then

at the

following

KELATIVISTIC

collision

RING

EQUATIONS.

1

329

operator :

c&z, 2’ ; t -

T) = cy + b1 + cy

C,(z, z’; t -

T) = c; + c;,

+ cy ; (21)

where Cyl L “I, Cy2, Cfz, C;, Ct are given explicitly in (b. 21-25) of appendix b. Although we can obtain all the information about the plasma within the n;’ approximation when there are no ternary or higher order initial correlations present (the contributions to D, (19a) can then be obtained from (1) by discarding the first vertex), the expressions will become soon too complicated to be manageable. We shall therefore restrict the desired information as follows: 1) as the particle equations are coupled only to we need only the first moment of the field equations and we shall restrict the field as a consequence all contributions become equations to this moment, functionals of the dielectric tensor expressing the collective behaviour, 2) we shall neglect the modifications of po during a collision (po(t - T) = PO(~)), this is correct only when there are no strong instabilities in the system, 3) although the destruction fragments can become important in unstable plasmas in which case they behave somewhat as the collision term, in the following we neglect them. This is an assumption on the initial correlations. The next step in the computation of the collision operators will be the evaluation of the z and z’ integrations in (19) and (20). In the z’-integration (20) we shall have contributions* from the poles of the free propagators and from the roots of the dispersion equation for electromagnetic waves: det D-l(k,

w) = 0,

(234

where D-l is the inverse of D. This tensor characterizes perties of the plasma and is given by (see (b. 18)) : D(k, 0)) = [&(k,

co-

GT]-1,

the dispersive

pro-

(23b)

where : T = 1-

I%; (234

and where E is a dielectric

tensor defined as:

4nez,d, E(k, co; t) = 1 + C ~ co !I .

&I-

f

*) A more detailed is given in the various

s

1

dpnva

(21% x dn) x

k

co -

_a

k.v,

,oo(pn; t). >

discussion of the possible sections below.

contributions

to the L-integrations

=

CL%+i0) + C,(t),

(24)

where zj are the poles* of the collision operator C,(z). Notice that in (24) C,(+ i0) depends on time only through po whereas the second term, sal C,(t), depends moreover explicitely on time. The nature of the time dependence of C,(t) depends strongly on the position of the roots of the dispersion equation. For roots in the upper half plane (unstable plasma waves) C,(t) grows, for roots in the lower plane (weakly stable plasma waves) C,(t) decays and for roots far down in the lower half plane (damped plasma waves) C,(t) is negligeable for long times. As roughly speaking the damping for small jkl is proportional to /kl one can define two complementary region+) in the k-integral contained in the collision operator Cj. The radiation region corresponding to small Jkj, say from 0 to the inverse Debye length L,’ and a collision or opacity region from L,’ to some inverse minimum impact parameter b-1. In the radiation region the dominant contributions will come from the emission of (longitudinal and transverse) plasma waves, i.e., from C,(t) whereas this emission will be strongly damped in the collision region where C,(+ i0) will be dominant. In order to get some more precise information we will consider separately the case of stable and unstable plasmas. Moreover, when the relativistic velocity limits are taken into account it is seen that the imaginary parts of (23d) vanish for J(oj > c Jkl, CO3 Ke LC). As the imagninary parts of (23d) determine the damping of the plasma waves in this approximation it follows that undamped plasma waves can exist in a relativistic plasma. And indeed, roots of the dispersion equation can be found in the region /(!)I> c Ikj, 01 = Re (I) when for example (23d) is evaluated with the relativistic equilibrium momentum distribution. A discussion of these roots can be found for instance in ref. 16. Their importance was discussed recently in ref. 12. As such undamped waves, except for collisional damping, persist forever we incorporate them naturally into the radiation region and discuss them further in section 5 together with the other specifically relativistic contributions although the system is really stable in this case. 3. Ring equations section we consider * ‘l%c other possible

in the collision

region for strongly

the ring equations singularities

of C,(z)

arc discussed

In this region for a strongly

stable systems.

in the collision

below.

stable system without taking into account non-classical effects. This means from the plasma waves (23a) we neglect C,(t) (see (24)), the contributions and the contributions from the specifically relativistic effects discussed in section 5. The roots of (23a) correspond to the normal modes or plasma waves of our system. In special cases these plasma waves reduce to the usual plasmons (see (33) below). In the case of a strongly stable system we suppose the dispersion equation (23a) yields only roots corresponding to strongly damped modes, i.e., damped on a collision time scale (-~,l). In this case a fantastic simplification occurs, only one term of (b. 21-25) survives in (lo) for the approximation considered here, namely the contribution to Cj(+iO) of the residue at z’ = Z.uj of the first term of (b.21). In order to obtain this result one has to perform some manipulations based mainly upon the following remarks: 1) the convolutions in the complex plane are commutative, 2) we integrate over 2 in Ci and 3) D(Z, w), V,(Z, CO),u,(Z, CC)) (I, W) + are even while D,(Z, w), P,(Z, cu) are odd in the transformation --f (--1, -CC)) with cc)complex. Moreover, D can be rewritten as: D = (det D-i)-1

D’,

(25)

where D’ is the matrix of the cofactors of D-1. As D-i is a plus function, 1) D’ will be a plus function and thus regular in the upper half plane. The singularities of D will thus come mainly from the roots of the dispersion equation (23a) which in this case are all strongly damped by assumption. Finally we can also rewrite the last factors of the terms of (b.23, 25) as:

;)E,‘” n&m

+

c3

v1,l?!?l) =

(

%lv~:,,,v:::,,,

E /J&i

l IV - ,7’

1 &V-

2’ + z 1 ’ (26)

which after the operations on z’ are performed will vanish in the limit z’ --f + i0. Using the above hints one finally arrives at the following kinetic equations

for stable systems

&pa (pj; t) = C 2efej& . Q+Q-k.;nm(pn;

under the above approximations: j dk s dpn. k.c$8(k.vf,). t) PO(P~; t) ;

&po(qy; t) = 0,

(27)

where the sum over 1~ runs over the different particle species of charge c,, and number density d, and where the standard notations of refs. l-8 have been used, i.e., & = 8/apj, din = dj - dn, qn = vi - vn. Moreover, Q+ is defined as : Q’ = q.D(&k,

&k.vn).v,

= (QF)“,

(28)

where D was defined in (b. 18) and (23). The product Q+Q- = IQ+12 appearing

M. BAUS

332

in (27) clearly corresponds to K” times a scattering cross section for a (soft) collision between screened particlesa). Some special forms of this cross section are of particular importance: 1) when the medium is isotropic vector available

to decompose

in r-space

there is only one independent

~(k, u), namely k. Hence we can write:

E(k, (0) = . . . 1 + . . . L = FJJK, Cc))L + F+?,

Lu) T,

(29)

where CL is the usual or longitudinal dielectric constant and FT the transverse dielectric constant related to the magnetic permeability ,U by the relation 2l 159r7) :

(30) This relation (30) shows explicitly that supposing e to be a scalar, E = sl, which is consistent only when neglecting spatial dispersion is equivalent to the neglect of the magnetic properties of the plasma, i.e., p = 1. Lsing now the decomposition (29) in the cross section we get:

2)

in the extreme case e(k, w) = 1, eq. (31) re d uces to the relativistic cross section 7, :

&i$ _vj*v,,

IQ'I"

Landau

-

(32)

(kev,)”

3) when the system is isotropic in velocity space, i.e., po(p; t) = pa(/pl; t) for all t, we get (29) and for the cross section we obtain:

(33) where furthermore

the magnetic

term in 611 will vanish.

Note that it is only in the two cases (31) and (33) of isotropic systems that one can distinguish between longitudinal and transverse plasma waves, i.e., plasmons and radiation, whereas the general case (27) still contains a coupling between the transverse and longitudinal fields. The particle equation of (27) was first obtained in the special form (33) for isotropic systems by Klimontovitcha) using his formalism and the integral equation method. Silina) derived the particle equation of (27) using an heuristic method (see also ref. 13). The non-relativistic equation obtained by Dupree5) contains a friction coefficient (see (34)) which does not agree with the one of our equation. The equation obtained by Rostoker, Aamodt and Eldridged) with the test particle method for nonrelativistic

RELATIVISTIC

particles

RING

EQUATIONS.

is of the same form but contains

which reduces to (23) after an integration

a different

I

333

dielectric

by parts in nonrelativistic

tensor

D

velocity

space. Finally it should be stressed that (27) are the relativistic ring equations for strongly stable systems in the collision region only when the specifically relativistic contributions discussed in section 5 are neglected. This point was not mentioned in the previous relativistic treatments23 3). On the other hand our equation for the field variables (27) states that the energy distribution of each normal mode remains constant within this approximation. The first moment of this equation can easily be inferred from the fact that in this approximation the particle energy is conserved while the different normal modes remain independent. With our assumptions most of the effects of the complicated collision operators (21) have disappeared. In this approximation the whole behaviour of the system is thus described by the particle equation of (27) whose properties will be briefly reviewed27 6) : 1. Eq. (27) strongly reminds the relativistic Landau equation7914) in the sense that the velocity dependent potential becomes now screened by the dispersion tensor containing the complete dielectric tensor ~(k, w; t) but the electromagnetic field still plays the role of a catalyst in the formation of the polarisation cloud leading to a screened potential. There is no radiation field in the system, only local induction fields and the particles interact only through virtual plasma waves. 2. Eq. (27) is of the Fokker-Planck type, i.e., a diffusion equation in momentum space : %JdPf)

3. 4. 5.

6. 7.

8.

= ~~~d*4Pom)

+ &‘(%O(P1))

(34)

containing diffusion d and friction a “constants”. It is a “long time” or kinetic equation valid for t > co;’ when nil Q 1. The relativistic Maxwell momentum distributions for electrons and ions are stationary solutions of (27) when taken at the same temperature. In analogy with the Boltzmann equation one can show that this equation conserves the number, the energy and the momentum of the particles. Moreover, one can derive an H-theorem for (27)) i.e., it drives the system to a canonical equilibrium state. Eq. (27) in fact forms a very complicated system of integro-differential equations for the different particle species. It is highly nonlinear because the unknown function PO(t) appears in the denominator in E. This fact expresses the constant adaptation of the polarization cloud to the state of the system yielding finally, in equilibrium, a Debye fourpotential. When the polarization of the medium is neglected the relativistic Landau equation is recovered. This is meaningful only for states

334

sufficiently close to equilibrium so that there are only few particles having velocities large compared to the thermal velocity and that the emission of plasma waves (at kmu) which are high-frequency waves is improbable. For weakly stable states, for instance, this is not possible

9. 10.

and one has to take into account consistently the polarization properties of the plasma. The relaxation time is of the order ~tntu,;’ and thus vvidely separated from the collision time LL);;’ for large values of no. This equation can be extended to weakly inhomogeneous systems and can be written in a four-vector forms). However, this does not imply its Lorentz invariance because neither the factorization of pa(t) nor the concept of weak inhomogeneity are invariant concepts (see also ref. 18).

\1Te shall now consider a less crude approximation weakly stable or weakly unstable modes.

in which vve allow for

4. Rirtg equations for weakly stable OY unstable systems. In the approximation complementary to the one considered in the previous section the dominant contribution to the collision operators Mill come from the radiation region, i.e., C(t) (see (24)). In this case one can first study the stabilization (see below) of the unstable modes in the radiation region (C e C(t)) and once stabilized the further evolution of the system can be studied in the collision region (C N C(+iO)). H owever, when weakly stable modes, i.e., modes decaying on a relaxation time scale, or different types of modes are present a separation into different regions is no longer possible and we are left with the original very complicated behaviour described by (b. 18-26) and (19-24). Moreover, in some of these cases it is no longer possible to neglect the destruction fragment. \Vhen only binary initial correlations are present one can include formally the contributions of the destruction fragment into the collision operator as done, for instance, in the non-relativistic case by Nishikawa and Osakala) (this paper contains misconceptions some of which were corrected in a paper by iMatsudaira20). Here, in view of the already very complicated collision operators, we shall neglect all initial correlations. For simplicity we assume that the dispersion relation (23) has only one simple root corresponding to a weakly unstable mode. The computations for a weakly stable root are quite similar. Using (25) we write by definition :

where lz is the unstable root and 5; its complex conjugate. Ey definition cc)* and Q lie in the upper half plane. The integration contours in this case are given in (36) (see figures).

RELATIVISTIC

RING

335

I

(36 b). Im z small.

(36 a). Im z large.

From

EQUATIONS.

(20) and (36b) we have for the L-integration: dz’Ca(z, z’) = (I =

&

s

dz’C&(z, z’) + {Res C&Z, z’) #=Cf

Imd-0

The contribution of the unstable results of the stable case:

Res Ca(z, z’)}. 2’=Z+l-

pole is thus additive

with respect

(37)

to the

C,(z) = CC(z) + C;nst(z) ; Cz”St(z) = ( FJS+ C,(z, z’) .

Res C,(z, 2’)). Z’=Z+t-

(38)

The s-integration can still be performed as indicated in (24). We recall that the specifically relativistic contributions will be considered separately in section 5. Taking (38) into account we have:

dT&

C, = s 0

C, = C,(+iO) C,(+iO)

dz eeiZ’C,(7) ; s c + C,(t);

= CS,t(+iO) + Cyfit(+iO); (39)

Remember that except for the time dependence explicitly indicated in (39) all the collision terms of (39) still depend implicitly on time through pa(t). We shall now consider the different terms of (39) separately. We first consider the particle equation, i.e., Cj. We have: &,0(j)

= iCy”(+iO) + iCyt(+iO)

e-iz’t + C Res {C;“(z) + Cp”“(z)}. Z=zj ZJ -izj

(40)

M. 13ACS

336

The first term in the r.h.s. of (40) is still gi\,ren by (27) or: dpt ~ Z.&Q-rfe$cd(l.vji) In order to write down the second term \vc introduce

Q+Z.djipo(i, j).

(41)

the following notation :

where K-l--, R+, R- denote the contributions of (b.21l23), featuring D+ and D-, D+ and D- respectively, but with D* replaced by S’ in accordance with (35). In writing (42) we discarded implicitly the part of (b.22) which is independent of D+ and which anyhow gives a vanishing contribution to (40). We can now write down the second term of (40) as follo\vs with the aid of (38) :

For the third term of (40) WC have:

s

dz’ C&z, z’),

(44)

In1 a’- 0

where zj are the poles of C;‘(z). As can be seen from the explicit

expressions

(b.21-23) these poles have real parts containing I-V or ~121 and will thus oscillate rapidly. These contributions will be neglected here. A more rigorous treatment of such terms is given in refs. 1 and 10. Finally the last term of (40) can be written as:

-1 R;-(z, 5-k) -

RJZ,

2

--t-

E-)

.

(45)

Once more we shall neglect the rapidly oscillating contributions from Ii;-, R/-and R,:.Retaining thus only the pole in z = E+ - [- we have for (45) e2(Im~~)t -x6-

Finally

“i’

- (p

-

the contributions

5-,

6’~).

to the kinetic

(46)

equation

(40) will consists

of the

:

RELATIVISTIC

stable contribution the unstable

RING

EQUATIONS.

I

337

(41) and of the two types (43, 46) of contributions

mode. Note that (46) depends explicitly

and implicitly

from

on time.

Moreover, in contrast to the stable case, (43) and (46) contain now contributions which are coupled to the field equation. However, as can be seen from the P_;, PT (3) dependence of expressions (b. 21-23) this coupling proceeds only via the first moment (q,> of the field equation, i.e. : (47)

(“/lv) = s ~w?YPOhV~ 4 J

where (47) represents (up to a factor (27t)-l+) the energy in the y-mode of the electromagnetic field. As this particle equation is still of the FokkerPlanck type (34) we shall thus obtain diffusion and friction coefficients proportional to the energy in the field. However, these contributions are so complicated that simplifying assumption are needed to study them. This, however, is a different subject which we shall not consider here. It should nevertheless be noticed that the general structure is of the quasilinear type and the equations will reduce to quasilinear ones when the approximations discussed in ref. 10 are introduced. We now turn to the field equation. In this case the complex integrations are difficult to perform. Indeed, in (b.25) appears a tensor Z whose poles are not directly connected with the roots of the dispersion equation (23a). Fortunately the particle equation is only coupled to the first moment of the field equation, i.e., the radiation transfer equation, which can be expressed completely as a function of the dielectric tensor D. Let us consider thus the radiation transfer equation. Therefore we multiply (47) by (2x)-i v,, to form the energy of mode y and moreover by the density of states (2xc)-3 vz to yield the energy density z.+ of normal mode y = {k,,, ,+I. We have then: &<~,> = j dy, G~,C,po(qv; We first take the corresponding following simplifications : S;(z, 2’) = j drlv&&X~,

moment

of (b.24,

.It-.D-.z.Q)

+ ya$n(&

-

25). This introduces

the

Pkb)&l

-

JQI,))

PtPj{V~,,~fiyQly~, -

p,.

V~,,
V;i)} PO@,i).

Where we used the notations

PO(i); (49)

2’) PO(Y) =

= d, C C J dpc J dp, c (2x&,+ .n+.D’-.29) 0, Y’) EVi,i * (-2X&S

(48)

4 PO(Y) =

= d, C C 1 d&(2x&,+. II+. D+ -4 i EY S;(G z’) = j dq,4+C;(z,

(2x)-4c-3v;.

d, =

t) = $44 ;

introduced

(50) in section 2 together

with those of

appendix

b. In analogy

with (42) we write:

where K.jmP and R,l- correspond tively but with D* replaced

to the contributions

of (50) and (49) respcc-

by S* (see (35)). I n analogy with (40) w:e write: -(+iO,

P)

+ iKJ_(+iO,

6’) -

In agreement with the stable case (27) WC have only contributions corresponding to (43), (46) and (44). The latter again contain rapidly oscillating terms which we neglect (see however, ref. 10). These contributions come from poles at for example z = 1. ZQ- 6~. However, other rapidly oscillating terms coming, e.g., from z = y1- E,- will have to be retained because here Y( = c (I( contratily to Z~ZJ~ is not an integration variable. After some cumbersome but elementary algebra we can write the contribution of (49) to (48), say s;, as:

.{E;-(E:) *II’qz;-) *S(Z, 51’) * .P(& El’W;!

(El’) -

e,!(q)

*l-I’(g)

*S(I, E;‘.) .EI,(I,

t: ) $<“llv)}

(53)

with :

(54)

where l, EM denote the electric E of (b. 18). Moreover, we have

and magnetic

parts of the dielectric

tensor

The summation over E in (53) has to be performed with explicit account of the Kronecker delta on 1. This summation can be simplified with the aid of the following remark. The symmetry property D(Z, w) = D( - I, - 0)) (U complex)

RELATIVISTIC

RING

EQUATIONS.

I

339

implies that if Et is a root of the dispersion equation; -Ef, is also a root. According to our hypothesis there is only one unstable root. The second root will thus be identical to the first or lie in the lower half plane. In the first case we have the symmetry relation: EI’ = --YE;,;

1 = -&Fyky,

(57)

while in the second case only one of the roots [c,, -tzV will be unstable. WC call EzYthe unstable root and incorporate this case in the first one by excluding E = + 1 in the summation in (53). In the following, the symbol Cl will mean a summation over F =.= 5 1 if E+ is an odd function of the wave vector (see (57)) and E = - 1 otherwise. Before rewriting (53) explicitly, we consider SE. The contribution form St(z = +iO) is odd in the interchange of particle labels i and j and will thus vanish. In the remaining contributions we neglect rapidly oscillating terms and retain only the contributions from z = E+ - E-, 2 = Ey + [-, z = t+ - BY, z = EV - t-. Adding the different contributions we arrive finally at the following kinetic equation describing the radiation transfer: Mu,)

= ey -

ay
with an emission coefficient

e-i4vgu+E,-)t

+ 5

-

I= i$,#+,,J il#V(Y

-

C 2f~&+-~~,J I’

(58)

given by:

1

+ (-v)

k(vg@+ t,T) (E,‘ - vgY)(Sy+ + VgJ %’

BY,IL(&J +

hl. HAUS

340

The cross section

for polarization

scattering

by:

1

e-idw,+ty-)t

+

iEji&~T;j~

The absorption

-(g=qgq

coefficient

appearing

~-~

appearing P,fTP

+ vg,)

$&-y)

in (58) is given

c&(&;)

.

(60)

in (58) reads:

uY = a; + al: ; 1 _ eidvvy-~y+)l qRC’m

i

~~

P

“(y&l --- q,

2

c&,,,(t;~ 1;

a:: = p;Tlly.

(61)

Finally we have the polarization scattering

cross :

section

,-iE(Ey---Yyll)f

+iff(,t--Where (59-62)

:’ -

%d

contain

(5;

for backscattering

1 _ vgu)([,

_ vg,) G,,@)

with

~G,K)

and without

.

(62)

the following abbreviations:

&! = t-2;; c$I((fl) = d ,‘(w) -E; (b, (0) $,,, P;,,(c’)) = n;(w)

* v,>

where we used (54-56)

;

Cfj)* ~~~,,,?

(63)

and (see (b.26) and (35)):

A; (u) = &: (co)* II*(w).

tqk,, c-0).

(64)

The various coefficients in (58) correspond to a unit volume and have the usual dimensions of the corresponding coefficients appearing in the phenomenological transfer equation 6) divided by the dimensions of a velocity. Eq. (58) describes all the radiation processes possible within the n,’ approximation, i.e., the emission and absorption (e,, uY) of longitudinal and transverse modes, and the scattering (p:,) between the different modes:

1IEIATIVISTIC

KING

EQUATIONS.

I

341

(*ky, p). There is no nonlinear mode coupling in (58). This process will The redistribution of energy described appear only in the no2 approximation. by (58) is completely governed by the particle distribution po(pj; t), i.e., the coefficients eY, uy, $;, are functionals of ~a(&; t). The coefficients of (58) have no definite sign and will mix thus the spontaneous and induced processes. In the non-relativistic case the stabilization of the plasma is usually explained as a redistribution of the relative drift energy (for the two stream instability) into thermal energy 21). However, it is worthwhile to notice that in the relativistic case there will exist a supplementary stabilization mechanism namely apart form the redistribution of the particle energy, the field energy can now be redistributed or radiated as described by (58). Finally the eqs. (40)) (58) are of the so-called quasilinear form, i.e., the nonlinear particle eq. (40) contains diffusion and friction coefficients proportional to the field energy and is coupled to a linear equation (58) (at least when the dependence of the coefficients is neglected) for the energy of the field modes. This should, however, not be confused with the quasilinear equations for an inhomogeneous collisionless plasma as derived from the nonlinear Vlasov equations’s). Nevertheless, we can take over the interpretation of the stabilization mechanism, namely when there is initially an unstable mode it will grow according to (58) (uY < 0) but when increases the diffusion and friction coefficient of the particle equation (40) will grow yielding a more effective redistribution of the energy between the particles by diffusion and a more effective slowing down of the rapid particles by increased friction. This process, in turn, will re-shape the momentum distribution from an unstable to a stable one. It should be noticed, however, that our quasilinear equations are long-time equations whereas the quasilinear equations for collisionless plasmas are short-time equations, the approximations such as neglecting the time dependence of the coefficients of (58) should therefore be handled more carefully in our case. In this section we have considered the relativistic ring equation for a weakly unstable system (a similar treatment holds for weakly stable systems) and shown how the quasilinear forms of the non-relativistic theory can be extended so as to include the complete electromagnetic field and an increasing number of physical processes such as proper radiation are now incorporated. In the derivations of this section we did, however, neglect the specifically relativistic contributions which will be discussed in the final section. 5. S$ecifically relativistic effects irt the ring equatiom. In the preceding sections we derived the form of the collision term for the pure collisional contribution (“strongly stable plasma”) and for the plasma wave radiation contributions (“weakly stable or unstable plasma”). The finite frequency collision term (19-21) contains, however, still different types of contributions

which we shall briefly discuss in this section. However, as no general statement about their time dependence can be made without the consideration of an explicit model, which is beJ*ond the scope of this paper, wre did not include them in the preceding sections although they can really belong there in some concrete situations. When the relativistic particle dynamics is properly taken into account two novel features show up in the dielectric tensor (23d) and the dispersion relation (23a). First, the dielectric tensor (23d) contains a Cauchy integral* of a different type than those appearing in the non-relativistic limit. Indeed, the relativistic velocity distribution vanishes exponentially 1”) as IV! -+ c so that the velocity integrations in (23d) extend after the change of variables p --f 2, is performed; over the finite domain jzll < c defining a Cauchy integral with a finite cut on the real axis instead of the infinite cut encountered in the non-relativistic treatment 1). The end points of the cut are branch points of this Cauchy integral and a new type of contribution coming from the branch cut will be added to those usually considered for non-relativistic particles. Finally, it follows from the structure of this Cauchy integral that the dispersion equation (23a) is real in the domain IRc (I)/> c lkl, Im cc)= 0 for all k and consequently’) this equation can have roots in this domain corresponding to undamped plasma ivaves. These two types of contributions can be distinguished from others by their cdependence because they are specifically relativistic effects vanishing as c -+ 00. Holrever, except for specific cases 11,12) no general argument about their time dependence can be given lvhich would ljermit to distinguish them from other contributions. When a type of RiemarnPLebesque lemma is invoked they will eventually damp in a non-exponential way but on a time scale which is not necessarily different from those of the preceding sections and which can only be studied in specific situations. However, because of their crucial c-dependence these contributions can expected to be important only for ultra-relativistic plasmas or plasmas containing sufficiently ultrarelativistic particles. Let us consider no\v these contributions in more detail. As stated above the collision term is a functional of a Cauchy integral with a finite cut, the ends of which are branch points. Therefore we first choose a branch cut. For example, e(k, O; t) is defined in a complex II) plane with one of the branch cuts of (65) (see figure). Where we have also indicated the possible Lal)lacc transform in\wsion Ccontours /‘. =\lthough these different branch cuts are mathematically quivalent, they do not all have the same interest from the physical point of * It is assumed here that the integrand entering

the definition

assumed valid.

of a Cauchy

integrall)

in (23d) satisfies

the regularity

conditions

for all times at bvhich the equations

are

KELXTlVISTIC

(65a).

Branch

cut:

(--oh,

RING

EQUATIONS.

(65b).

ck).

I

Branch and

(65~).

Branch

cut:

(-c/z

-

i0, --ck (ck -

iw,

-

im),

ck -

(-ck

-

ioo,

343

cut:

(-

00, --c/z)

(ck, + co).

ck -

icm) and

i0).

view. Indeed, consider the z’-integration of (20) along the contour defined in (a.5). When the different cuts and contours of (65) are used to compute (20), we shall not only get different cut contributions but also different pole contributions. With (65a) there is no need for an analytical continuation of ~(k, z’) because this function is continuous inside the contour ra. We shall thus pick up contributions from the undamped roots of the dispersion equation and, as a(k, z’) is real on the real z’ axis in Pa, pairs of complex conjugate contributions form the unstable roots of (23a). With (65b) we have to cross the discontinuity of ~(k, z’) by analytic continuation and we pick up contributions from the unstable and undamped roots but also from the weakly stable “Landau” roots gained by analytic continuation. The Landau pole contributions are thus hidden in the cut contribution of (65a). Moreover, it will be a nontrivial task to take the non-relativistic limit when (65a) is used. On the other hand with (65b) we have a clean separation between the contributions from the undamped poles and the Landau poles whereas the cut contribution clearly vanishes in the non-relativistic limit. However, it can be seen that with (65b) the analytic continuation of the velocity distribution in the unphysical region 1~1> c will be needed in order to evaluate the cut contribution. Finally, the branch cut which yields a clean separation between the unstable poles, the Landau poles, the specifically relativistic effects (and with which the non-relativistic limit is readily

taken) is given by (65~). Clearly, similar remarks hold for the z-integration (24). From the above it follows that, contrary to an often made statemen@, 5) the relativistic effects modify also qualitatively the collision integral. Through a suitable change of variablesd, 5) (U + p) one recovers eventually (apart for integration by parts, normalization constants and other subtle changes) the quantitative expressions of the kinetic equations in the classical region

(Re z’/ < c/r, (Re(z -

-‘)I < cii of (65c),

however,

the

qualitative

changes in the structure of the complex plane (65) are not readily recovered. On the other hand, as discussed above, some care has to be exercised when the contour of (65a) is used because contrary to [\-hat is claimed in ref. 12 there are no Landau poles in I’a because they all lie on a different Riemann sheet (see 65b, c). Finally we mention that the dispersion equations for an equilibrium plasma taking the relativistic modifications (65) of the complex plane into account have been discussed in ref. 16 whereas relativistic effects in the field auto-correlation function have been discussed in ref. 12. \l:e now turn to the evaluation of the specifically relativistic effects on the kinetic equations in the ring approximation. For (20) we have from (65~) and the above discussion :

where the stable and unstable contributions (CLt,, Cyt) have been discussed in the preceding sections. Indeed, as the discussion of sections 3 and 4 are restricted to the ,,classical region” of (65c), namely !Re z’l < ck and /Re(z - 2’); < ck no modification of these contributions is necessary. The third term of (66) corresponds to the cut contribution of the Y-integration (20) with (65c), i.c.: 00

I

dy{C’,(z, ck + 0 -iy)

-

C,,(z, ck -

0 -

iv) $

0

-+ (Y,(z, Finally

we

have

--c/z + the

0 -

iy) -

contribution

C,X(.z,-ck

-

0 -iy)).

from the undamped

(67) poles w:! := Re a$,

I(,); 1 > ck :

(68) and from the unstable poles & lying in the “relativistic region” /Re z’/ > ck, where they appear in complex conjugate pairs (l,: = (t,?)*) : C&ust,rel(~) = C {$Z~;~+c,(z, z’) :,+

Res C,(z, z’) + Res C,(z,z’)j. tr _zj-_‘=Z+ti-

(69)

Finally the z-integration (24) will not only yield “classical” pole contributions (see (24)) but also relativistic cut contributions and mixed contributions

RELATIVISIYC

such as the evaluation

RING

of the classical

EQUATIONS.

I

stable term C:(z)

345

around a relativistic

contour. Clearly, as we are interested here only in the qualitative aspects of the relativistic ring equations there is no need for a detailed computation (with the aid of (21) and (b.21-25)) of the various collision terms of the different kinetic equations. This would indeed lead to some rather fantastic expressions the physical content of which can become clear only when further approximations are introduced. However, in order to discuss in more detail the qualitative modifications introduced by the relativistic effects we consider the radiation transfer equation with full account of these effects. In general, the source term S, of (48) contains contributions from weakly damped Landau-poles, unstable pole contributions already discussed in section 4, rapidly oscillating terms not explicitly considered but analogous to those considered in refs. 1 and 10 and finally the specifically relativistic effects discussed above (see (66)). We now consider these latter for the case of a strongly stable system. For simplicity we consider only the transverse modes ,u? = 1, 2 in which case (49, 50) read:

$2,

z’) = d, 2 C J dpt dp:~~,(2rcsy.D+.vl)(o:

- 9) f~,(cu‘! - 9).

I,

i.i 6, *(-27-t&,* D--vi)

PiPi{V$,.f&l,~j

+ (i y’ --f i, 4)

p0(6

-

V;&,,>

+

i).

(70)

By assumption we shall neglect here all unstable poles and stable Landau poles and retain only the undamped poles (68) in which case we can write: e,.D+.q When

= C Wnzf -

(20 is evaluated

S;(z) =

1 on

s,.D,t *vi.

(71)

with the aid of (65~) we obtain in this case for (70) :

c Res S;(z, z’) + Uln z’=w,

Res SF(z, z’) Z’=l.Uj

S&“,,(z);

(7%

St(z) = 2 Res St(z, z’) + Res Si(z, z’) + Res St(z, 2’) Wn z’=wn z’=l.uj z’=E* where the cut contribution

Sf,&),

(72b)

reads in both cases (see (67)) :

m

s

dy{S;‘“(z,

ck

+

0 -

iy) -

S;jD(z, ck -

0 -

iy) 4

0

+ S;!b(z, -ck From

(72)

the

+ 0 -

asymptotic

iy) -

S$b(~, -ck

contributions

-

0 -

S;(z = +iO),

iy)}. S”,(.z = $_iO)

(73) are

readily computed. It is seen that when the symmetry properties of (70) are taken into account the contributions from the residues at Z’ := l.vj to S~““‘(Z= +iO) vanish, whereas

The other contributions

of (72) to .S~:‘“(Z= +iO) remain, however,

and from

this we conclude that, even for a strongly stable system, the relativistic corrections yield a nonzero asymptotic collision term SY(5 = +iO). Let us now consider the contributions of (72) to the (explicitly) time dependent collision term, say S,(t) (see (24)). For (72a) we obtain a residue at z = u + Z-vi which is a rapidly oscillating term containing the integration variable l.vi and when treated as indicated in refs. 1 and 10 its leading contribution vanishes (such terms have been neglected in the previous sections). Finally we have the contribution from the residue at z == &Y + ~~~~~ which describes an interference effect between the vacuum mode and the undamped plasma wave, and the contribution from the singularities of SF,,,(x) which when the residue can be commuted with the y-integration of (73) yields a term oscillating at (V li v - iy) and will eventually damp on a time scale governed by the particle distribution. For s;(t) similar contributions arise but due to its bilinear character in D a new type of contribution appears. Indeed, it is seen that S!,(Z) contains terms with branch points at can then be performed with 2 = 01, iv and z = I.vj + Y. The’z-integration a contour similar to the one of (6.5~) but displaced an amount ~1)~~ or 1.vj. The collision term SE(t) will thus contain contributions from the poles in the Z’ plane evaluated around the cut of the z plane and even the cut contribution of the 2’ plane evaluated around the cut of the z plane. Useless to say that at least qualitatively the relativistic kinetic equations are completely different from the non-relativistic ones or c\‘en from the evaluations in the classical region IRe ~‘1 < ck, 1Re (Z - z’) 1 < clz for relativistic systems such as those of refs. 2 and 3. To illustrate this fact a non-ccluilibrium velocity distribution which yields unstable cut-contributions but no unstable Landau poles will be considered

in a separate

paper”“J).

Acknowledgements. It is a pleasure for mc to thank Professors I. Prigogine and R. Balescu for fruitful discussions and encouragements. JVe are also grateful to our colleagues of the Brussels group.

factorization theorem for non-relativistic systems was derived by Resibois”) within the time-dependent formalism. However, as his proof is difficult to follow we give the following simpler one. The

factorization

theorem.

A

RELATIVISTIC

Consider the Laplace

RING

transform

W) = (POLPC) p

c

EQUATlONS.

of the collision

L; _z

347

1

operator:

(a4

(PCLPO).

c

Here we have used the compact expression of the collision operator as obtained in ref. 23, with the aid of the so called projection operator method. Let us split now L as follows: L =

L-1 +

L2

+

Ll2,

(a.4

in which L1, L2 denote the Liouville operators of two subsets of the degrees of freedom of L, and Lr2 denotes the interactions between the two groups. Define now an approximation to Y(Z) in which the contribution of Liz to the propagator of (a.2) is neglected, i.e.:

ql,,&)

= (POLPC) -

P,(Ll

1 +

L2)

P, -

-

z

(PCLPO).

As L1 and Ls trivially commute we can use Cauchy’s write for the collision operator in this approximation: Y/n,,r,,.(z)= -&

i’dzf(PoLP,)

’ ~ P,LlP, -

formula

~~~ 2’ P,L2P,

C’

(a.4) for (a.4) and

1 + z( -

z

(PCLPO)? (a.5)

where C’ is a contour enclosing all eigenvalues of (P,LlP,) and excluding a line anti-parallel to the real axes such that those of (z - P,LzP,) i.e., Im z > Im z’ > 0 and closed in the lower half plane. Using (a. I), eq. (a.5) can be rewritten as: ul,,,,,,(t)

= (PoLP,)

eCi(PCL1pc)te-i(“CLz”C)t (P,LPo),

(a4

i.e., the factorization theorem. We shall apply this theorem to the summation of the b-rings (1 b) with L1, LB corresponding to the upper and lower branch (line) contributions whereas the extreme vertices PoLP,, P,LPo are excluded. This theorem then states that for two commuting branches the ralative time order of the vertices is irrelevant and that the sum over all relative time orders equals the product of the two branches evaluation at the final time (a.6).

APPENDIX

b

Summation of the ring diagrams. In this appendix we sketch the summation of (12-15) in three-dimensional form. Consider first the repeating unit

Rl. BhlJS

348

of series (12, 13): C j dPn"nP,,V::,Pa2xEapo(pll).

(b.1)

(a)

The sum over the polarization c 2xV&&n w

and f

= .F(%, k, 0,) = (9,

sign ((a) = ,LL~,eLy)yields: Fa) ;

where (n, LY,k, co) denotes an element of (i, A, I, z’) or (i, y, -2, z where we have introduced the unit tensor in transverse l-space: T=

c El,,,&[,,, = 1 /1=1,2

2’) and

L;

1 = ii.

(b.3)

Note that the magnetic vn x (T x Z).d,

term can be written

= v,.ZT.d,

-

= vn*ld,, -

VJ.dll =

v,.Tl.d,,

in different

forms:

=

= (vn x d,,) x 2. Consider next the comrection (J dp,,aZRtipo(fi)).

between

tuo repeating

units:

Cj dPnLL17nl-‘,~,,po(m)).

Using :

(b.6) we rewrite

(b.5) as:

Straightforward application of (b.2) and (b.7) yields lations the following expressions for (12, 13) :

after

some manipu-

RELATIVISTIC

+

c

I

+ c V;,,P,P,eF;i+ . A’~vj~Pj~Vi,;.~}; .I

1 (7)

c C {(9T:.A-*7Vt:)

PiP#‘jnt

L (v) (A) i + + (9-y. A-. V,J PiP,(Fi+

l

349

P.9)

l

CT = c c {V~,P,PjV;y cF2 = r, c

EQUATIONS.

. V,,P,,P,(S---i+ A+. VT));

.(A--P-)

.q

RING

(b.lO)

+ Viyj) +

A-+. V;) PJ,( Vj’ni + Vi,j,)}.

(b.11)

3 Here we used the following three-dimensional tensors : 1)

9

= [I -

notations

to go from four- to

T]-r

T(k m) = C j dp,M’,S,po(rt) ?I

2)

and arithmetic

cF.t’t7.a = 9-A. A = P-S;

(b.12)

;

V,

A = P + P~T~A,

k-v - w V = v + c2k o2 _ 2 ;

(b.13)

V

3)

s.w

= 9.P.V

= s.v*

C2

> co+

=

k+ = 1,

2’;

u*=v-_--_k+;

w- = 2 - 2’;

%!7 4)

vmwn =

k- = -1,

(b. 14) (b.15)

v;,p,v;,.

Here clearly 9, S, 7, . . . denote the three-dimensional parts of 9, 9, -r, . . . and the labels f on a quantity indicate in which set the arguments are to be taken, for example (see b. 14) : A* = A(k*, LO,-)= A(+&

(b.16)

w*).

The splitting of C; into C;C and CT corresponds to a splitting in actionindependent (al) and action-dependent (as) contributions and similarly for CF’, CF”. We now notice the following forms for the A tensor: $.A

= D-D,

A = (os-9)

D;

.F =

1 09 -

vs

D,

(b.17)

*d-v,, Turning

x (TX k)*d?L po(p,;t).

(b. 18)

now to (14, 1.5) \vc write:

C, c

x s dfR,po

= C;, -1 CF.

(lx 19)

Y

operator

now we did not integrate series acting on Q. The

tensorial

operator:

As

up

to

c = [I -

q-1;

,~a== 1,2,3,

&r , Ei

over rjy, the Ct contribution remains an summation then leads to the following

-

Iy -

I Using moreover summation

(b.20)

,un = 0.

(b. 13) in (b.8-1 1) we can

write

the

final

result

of the

as:

+ c~(z,

k,,

I &a&

Di.D-.(I

z’) = J dr

-

v%-)J:P~}~~;

f; ,: {V;.,,P,P,V;:, + C V;,P,P,D;

j’

i Cf2(z, .z’) = J df x C c C {(D;.D-WV;) i

(b.21)

I (1’)(1) i

•Di-~~;PJ,J’~j~} PO; (b.22)

PJ’j(Vj~z

+ VW) i

RELATIVISTIC

RING

EQUATIONS.

351

I

(b.24)

C;(z, z’) = j dr C VT. l-I+. D+.v~P~P,I”,~~o; Y i C!(.Z, z’) = j dI’ x 2

C C (VT *C+*V~) Py(VI~*Z-*~i)

PiPjo

i,i (u’) (1) Ey . (w:,

where i is the transposed II = (09 -

(b.25)

VjAPAJf Ai)po ;

v’y,j +

matrix

of A and where: (b.26)

Y”) 1 + VZL.

Moreover, all distribution functions appearing in (b.21-25) are evaluated at time t - T. The summed expressions (b.21-25) are studied further in the main text.

REFERENCES

1) 2)

Balescu,

R., Statistical

York, 1963). Klimontovich, plasma,

Yu.

Pergamon

3)

Silin, V. P., J.E.T.P.

4)

Rostoker,

5)

Dupree,

6)

L., The Statistical

Theory 1969).

Fluids

6 (1963)

qualitative

review

of a Relativistic

Mathematics

Nonlinear

Physics”,

7)

Mangeney,

8)

See for example Resibois,

of Non-Equilibrium

O., Ann.

Publ.

(New

processes

Phys.

in the Proc.

in a

243.

of the Internat.

Mtinchen 1966 (appeared New York, 1968).

30 (1964)

461; Ann. Physique in Nonlinear in “Topics

: I. Prigogine

31 (1965)

of some of our results see: M. Baus:

Plasma”

as

“StatisSchool

of

“Topics

10 (1965) 191. Physics”, Springer

in

Verlag

1968). P., Phys.

Fluids

6 (1963)

817.

10)

Grecos,

A. P., Bull. Classe des Sci., Acad.

11)

Aaron,

R. and Curie,

12)

Lerche,

13)

Baus, M., Physica

I., Phys.

14)

Prigogine,

D., Phys.

Fluids

North

R.,

Holland

in Cargitse

Fluids

11 (1968)

43 (1969)

I., in “Statistical

J. Meixner, Balescu,

Interscience

1714.

and Physics, Springer Verlag

A., Physica

Particles,

1244.

R. and Eldridge,

Nonlinear

New York,

(New York,

13 (1961)

T. H., Phys.

For a preliminaiy

of Charged

Press,

N., Aamodt,

tical Mechanics

9)

Mechanics

Roy.

Belg. 53 (1967) 642;

9 (1966)

1007.

1423.

413.

517. Mechanics Publ.

of Equilibrium

Comp.,

Lectures

and Non-equilibrium”

ed.

1965;

1964,

Statistical

Mechanics,

Gordon-Breach

1966; Mangeney, Baus,

Balescu, 15) ‘6) 17)

A., see ref. 7;

M., Ph. D. Thesis, R., Baus,

of Brussels

1968 (see ref. 6);

A., Bull. Classe Sci., Acad.

Roy.

Belg. 53 (1967)

1043. Baus, M., Memoire de licence, Universite Libre de Bruxelles 1964. Prentice, A. J. R., Phys. Fluids 11 (1968) 1036. Properties of Plasmas Silin, V. P. and Rukhadze, A. A., “Electromagnetic Plasma-like

‘8)

Iiniversity

M. and Pytte,

Baus,

Media”

M., Bull.

Balescu,

(in russian)

atomizdat

Classe Sci., Acad.

R., Physica

38 (1969)

Roy.

119.

(Moscow,

Belg.

196 1).

53 (1967)

1352;

and

RELATIVISTIC

352 19)

Nishikawa,

20)

Matsudaira,

I(. and Osaka, N., Phys.

21)

Abraham,

22)

see for example Center

B. W.,

for Theor.

Phys.

23)

Baus,

M., Bull.

24)

Baus,

RI., J. math.

Classe

Y., Progr.

Fluids

J. math. chap.

RING

I.A.E.A. Sci.,

Acad.

Phys.

I

33 (1965)

402.

539.

6 (1965)

4 of “Plasma

Phys.

theor.

9 (1966)

Phys.

EgUATIONS.

630.

Physics”

(Vienna, Roy.

Trieste

seminar

1965). Bclg.

(to 1,~: p~~hlislrctl).

53 (1967)

129 1.

of the

Internat