THE UNIVERSITY OF MICHIGAN 274;, -I -T 2764-8-T = RL-2076 KINETIC EQUATIONS FOR PLASMAS by R. K. Osborn October 1961 Report No. 2764-8-T on Contract DA 36-039 SC-75041 The work described in this report was partially supported by the ADVANCED RESEARCH PROJECTS AGENCY, ARPA Order Nf. 120-61, Project Code Nr. 7400 Prepared For The Advanced Research Projects Agency and the U. S. Army Signal Research and Development Laboratory Ft. Monmouth, New Jersey

THE UNIVERSITY 2734 -b-Tb OF MICHIGAN I ABSTRACT This paper presents a unified development of kinetic equations describing particle and photon transport In plasmas subjected to non-constant and non-uniform external fields. It is found that the equations for the particle distributions are a generalization to the plasma of the equations postulated by Uehling and Uhlenbeck+ for neutral quantum gases. As the equations describing photon transport have been developed and discussed previously, only a brief discussion Is incorporated here for completeness It is then shown that the present description of the plasma is sufficiently complete and consistently developed that an H-theoremn is (iinmlnstrabkle + E. A. Uehling and G, E, Uhlenbeck, Phy Rev. 43, 552(1933), iv

THE UNIVERSITY OF MICHIGAN 12764-b-T INTRODUCTION The purpose of this report is to present - from one particular point of view - a summary of some preliminary investigations of particle and photon transport in fully ionized gases, 1,2, 3]. The emphasis on a particular point of v iw Is not intended to sugrgest that it is necessarily the best vantage point from which to inspect the subject, but rather that it is a seemingly simplifying and clarifying - and yet so far somewhat unexploited - vantage point. Furthermore, mention of this emphasis serves to warn that no attempt shall be made herein to review the many interesting and different approaches to this problem that have been developed In the past few years, [4], Because recourse to experiment to test seml-intuitive models of the plasma is not oftc-n feasible, It seems necessary at the present time to investigate the validity (or range of approximate validity) of such models from strictly theoretical considerations. The accomplishment of such an objective requires firstly a comprehensive axiomatic statement of the problem (the axioms being reasonably widely agreed upon, of course, followed secondly by a deduction of descriptions of the plasma to which the various models purportedlv correspond. Needless to say, no suc h ambitious progranm has v, t been achiev(ed,...., ---- 1 Iu

THE UNIVERSITY OF MICHIGAN' 12764-8-TI The present discussion is restricted to the delineation of an approach to the problem of determining the validity of Boltzmanni type equations for the description of particle and photon balance in the fully ionized plasma. It is admitted at the outset that this approach essentially fails with respect to both of the main points indicated above. In the first place the selection of axioms is hardly universally agreed upon and in the second place the deduction of consequences from the chosen axioms Is far less rigorous than is desirable. Nevertheless the results seem suggestive and represent somewhat of a generalization of those usually discussed in the context of the present problem. Furthermore, though the deductions herein proceed via many approximations (none of which have been investigated in detail), the steps required for their testing are usually discernible. The discussion will be divided into several sections. Section I will incorporate a statement of the axioms and some discuss ion thereof. Section II will be devoted to an approximate deduction )f a balance relation for the particles in the plasma and some consideration of Max ull's equations. Section III will present a similar development of a transport equation for photons. In Section IV some of the implications of these balance relations thr the therm,o(dvnam i stale of the plasma will be-examined. In particular, an H-theorem tor the particle photon system will be sketched. 2

THE UNIVERSITY OF MICHIGAN,2764-8-T - I THE AXIOMS The axioms required for the description of systems of the type presently under consideration are usually considered to be of two kinds. The first of these is for the purpose of specifying the dynamics of the interactions between theoparticles that comprise the plasma, whereas the second is for the purpose of introducing statistical concepts into a description of a system characterized by a huge number of degrees of freedom The dynamical axiom is conveniently expressed in terms of a Hamiltonian for the system; from which, according to the canonical equations whether classically or quantum mechanically intorprotd, il 11 infhmrnation may be deduced. Since we are here concerned with electrodynami (s, aw may expect that the dynamn ical axioml will b reasonallv firm and non ejnltrne Trs ial; at least within certain self evident limitations such as, for example, non-relativistic treatment of the particles. The statistical axiom is usually introduced via the conce pt of ensembles of systems In terms of which the probability of finding the given system in a given state at a given instant can be meaningfully formulated. Though usually considered necessary (whether the system be dealt with in classical or quantum terms), we shall avoid the explicit introduction of such concepts into the present discussion, It is for this reason that our axiomatlzation of the system may be considered controversial 3

THE UNIVERSITY OF MICHIGAN 12764-8-Ti to say the least. Instead we shall treat the s'stem quantum mechanically and apparently rely solely upon the stllsi>ai concepts inherent in such a treatment. The equivocation Is a recognition of the possibility that ussttficati )n of some of the approximations to be invoked subsequently may require the ensemble concept - but such a necessity is not evident at the moment. We will note that all of the results of the conventional statistical treatments of systems -lniiai to the one considered here are forthcoming from the present analysis. The dynamical axiom will be stated in the form of an energy density for fields of interacting charged particles and photons, and the Schroedinger equation for the wave function which characterizes the states of such a system. The field theoretic formalism is dictated by the desire to deal with photon transport on the same footing as one deals with particle transport, and so far there has been no indication that this i.:> feasible in the classical, or semi-classical context [5 ]. It has the slight, further, formal advantage that the singlet densities whose equations we seek can be defined In terms of expectation values. In non-degenerate plasmas it is not expected that quantum effects will play a significant role In the description of particle transport. In view of these remarks we have [6] H~ = Ift a ' (1) at where H = / (x) d3x, (2) X _. -... - - 4

THE UNIVERSITY OF MICHIGAN j2764-8-TT * - - -'-~ -**- - and e e +1 g. (x) = - I(hV — A- A) j i 2m c a' e X [(itV+ - A + + ( A- c + 1- (VX A)2 + + 8v - - + er Ae ). ] c jo + [27rc (3) o' 0TCt + + ee,' 3 Y (x) (x) ( x) (, (x_') + - d X' -- 1 + S J x - xi In equation (3), q- Is a wave operator for a field of particles of the oth kind, Ae and 0 are the vector and scalar potentials of the "external fields", and A and P are the "transverse" magnetic and electric operators for the photon field. The external fields are presumed known and hence are unquantized, whereas the "internal' fields are described by operators which satisfy the c(rmmutation relations, (x), Lf,(x') +.. + (-) ) r ), S(X) + j(Y') yt(x) = 4 S(x-x') 5

- THE UNIVERSITY OF MICHIGAN 2764-8-T and [A X), P (xI)] = A.(x) P, (x') (4) - P (x') Aj(x) = i 4 ( - x-) -Ih t ( ) all theit r vuatllties commuting. The parti.lI.',)ttistv anti commutation or commutation rules depending upon whether they are fermlons or bosons, This distinction Is of no Importance for the non-degenerate plasma, but will be maintained throughout the early stages of the analysis, for the sake of generality. The assertion of the transversality of the field operators is rendered formally precise by the statements (V' A) = (V- P ) = 0. (5) The energy density, equation (3), may be rewritten variously by regrouping terms or by transforming coordinates. Expression of the energy density In various ways is desirable since some calculations proceed most naturally from one form of equation (3) whereas others require other ways of writing it, For this reason we expend a little effort here upon the rewriting of (3) in two different ways - one most suitable for the discussion of particle transport while the other simplifies the treatment of photon transport. Both ways of exhibiting (3) are, of course, equivalent. For the purpose of dealing with particle transport It Is convenient first of all to 6

THE UNIVERSITY OF MICHIGAN 2764-8-T, decompose the vector potential, A, Into two parts,,1 e,, A = A + A, (6) where AS refers to the "slowly varying" part of A, while Af denotes the "rapidly varying" part df A. The distinction between "slow' and 'rapi(d' variation w ill be determined only in the context of a givent situation and hence A ill not be discussed further at this point. We then introduce the rotatiun, e s R =A + A, (7) i.e., R represents the superposition of the known, externally applied field and the "slowly varying" part of the Internal field. Given this decomposition of the internally induced electromagnetic field, it Is convenient to rewrite equation (3) as, Zm c - c - )T] j.- e- c - R) r CC C oT + [rc2 P2 + (VX A) + e0 87r (continued) 7

THE UNIVERSITY OF MICHIGAN 2764-8-TI (-T' e er, 1 + I-a T~r / dx d x',(x) %, (x') %(x).,(x') x - xT (8) For convenient treatment of photon transport, it is useful to employ a tdfftr - ent regrouping of the terms in equation (3). Introducing the operator, II = - i h V - - A, c (9) we express the energy density as, (x)= [27rc P + - (V AA)j - 087T i 1 2m, a ' 2mn c - ' i r+ + 21 a ' (II-" ~ )' (II UJ) ) (1I T +) A Y + A.... (IJ).0 r(Tv 3d ~T (x) I,(X') ) (x) (,(x') x — - X 2 ar cr T or 2 A + e +0 0a- 'cT ra, o-' - va (10) 8

THE UNIVERSITY OF MICHIGAN 12764-8-Tj For some calculational purposes It Is convenient to transform to momentum space. Accordingly we Introduce the fourier analyses, 2ih \r c ik x + l A (-) P= I -- 1, (k) (11) 87rcV -- kX Tcr W- a (K) e - K In the relations (11), V represents the volume of quantization, It is the volume with respect to which the quantities A, P, and t obey periodic boundary conditions. + (k (k (k )+ a (-k)( (-k), (12) where the E (k) are the unit polarization vectors of the photon field. The Indices A, take on two values corresponding to the two states of polarization for the photons. Us;ums over k and K are the usual sums over the integers permitted by the reu *n that A, P, and 9 be periodic on the boundaries of the volume V. The -- 9

THE UNIVERSITY OF MICHIGAN 12764-8-T quantities (k) and a.(k) are the creation and destruction operators for photons of momentum h k and polarization X while a (K) and a (K) are the creation and destruction operators for particles of kind a- and momentum h K. The commutatlon rules governing the creation and destruction operators are, o (, aA _ - =(k-' )> and a (K i. a,(K (K') 13) a '- - Note that in equation (13) the functions c (k - k') and S(K - K'i represent Kronecker delta's since the arguments take on discrete values, whereas the function (x-x') appearing in equation (4) is a Pirac delta function. We shall continue to use the same notation for the two kinds of delta functions, letting the context reveal which interpretation of the symbol is appropriate in a given case, 10

THE UNIVERSITY OF MICHIGAN 2764-8-T II BALANCE RELATIONS FOR TIlE PARTICLES (ur iw irn' concern in this section shall be for the deduction of a Boltzmanntype equation forl a singlet particle density. Thus our initial task must be the identification of a quantity which, in some sense, may be interpreted as the expected number of particles to be found in a small element of volume in phase space. Since we have formulated the problem in quantum mechanical terms, it is evident that some difficulty will be encountered here, as it is impossible to localize particles with arbitrary precision in phase space. A way out (:.nd the t, ont chos<tn hr&et as it has been many times elsewhere) is simply to give up the notion ot ' ine,ir' int meaningfulness" of the' singlet density. Alternatively (and equivalently) we may solely require of the singlet density that it be a proper weight function for the calculation of observable averages. With these remarks in mind, we define a singlet density for particles of kind a by f (, K 8 d3ze-2iKz, z (x-z) y(Y(x+z) ) (14) 8 4 e-2ix' -, aQ+(K++)a (K -9)Q) Q The density defined by (14) is not everywhere positive but assumes negative values - - 11

.- THE UNIVERSITY OF MICHIGAN [2764-8-Ti as well because of the impossibility of simultaneously specifying the (x, K) coordinates of a particle within arbitrarily small ranges. Nevertheless, it is convenient (when meaningful) to interpret f as the expected number of part iles of kind a to be found at the point (x, K) per unit volume in configuration.pa(. Fo'rce is given to this interpretation by the observations that f (x,t) = f (x, K, t) ( +(x) (x) ), (15) K and t (Kt) =fd xi (x, K,t) —(,a (K) a (K) ), (16) a- J y — - 1 (y -~ oi. e., that f (x, t) and f (K, t) are indeed the expected particle densities or particle numbers in either configuration space or momentum space separately. Further reinforcement follows from the observation to be tnadc.uVth-sf(4uc!v that f (x. K, t) is persumably truly interpretable as a particle density in the classical limit. It should be noted in passing that the density defined by equation (14) is simply the field theoretic equivalent of the Wigner distribution function [7, 8J, and has been employed for purposes similar to those that concern us here many times previously. + Note that f is not a density in K-space (momentum space). This is a direct consequence of Raving defined the Fourier transforms (11) one of which connects the two expressions for fc given in equation (14) in a finite volume in configuration space. However, in spite of the fact that the variable, K(and k for photons), is discretely distributed, we shall assume that the operation of differentiation of functions of K with respect to K is meaningful.

THE UNIVERSITY OF MICHIGAN 1 2764-8-t Some more notation will prove useful. Define the operators i) (x, z) = + (x- z) ( (x +z) cr~~ '- -- -- and p (K,)= a (K+_)a (K- (17) Then equation (14) may also be written as jfd3 -2iK T z ) (x, e - - 8 3 e 9 z Tr P (x,z) D 8 e2ix Tr p (K,9) D, (18) 9 where Tr AB means take the trace of the product of the matrices A and B, and D is the density matrix for the svstcm in the Schroedinger representation. Because of the invariance of traces to uniLv\ transformations, the forms (18) for f (x,K,t) will prove most useful lotr' al'ulational purposes.Now recall the expression for the energy density given in equation (18) and write the total energy as x x (x) H + +, (19) x --------------- 13 f W

I. THE UNIVERSITY 4 12764-8-Tf OF MICHIGAN where H0= H =U fa3 e. + e +21 e f3x 0 qi~y a.9 Z cx, CTI ecrer x [d 3~~ 4(x)+ W J (x'1) qP() I/i_(x u (xx) 1 + I J u r - ac Ix, I (19) ~/3x[27r c2P2 + 1 (Vx A) 23 pi ea Z7 f{F + 2m 7+i a a (20a) I e cxi q(i C 1-i.c e - itv - C )4+iAF4} j! 4). (20b) xd x' + r~II -- IJI arty1+gaJl - cf'x T ' X SWx S(x') 1 (20c) 14

- THE UNIVERSITY OF MICHIGAN '2764-8-T where u(x, x )= S(x) s(x') +S(x') s(s) + s(x) s(x') -. (21) The step functions S and s require definition, and the breakup of the coulomb energy according to the parts appearing in equations (20a) and (20c) requires some explanation. We are envisaging the selection within the system of a sub-volume, V, whose size, shape, and location will remain largely unspecified for the moment. We may hope that under some practical circumstances V can be chosen small compared to regions over which macroscopic spatial variations are significant. We have then defined step functions according to S(x) 1, x A V, 0, otherwise and s(x) -, x 6 V,, otherwise, (22) and decomposed the coulomb energy of the system into two parts. The part incorporated into II, equation ('0a), corresponds to the interaction energy between all particles not both in V, whereas the other part. HIP. is the remainder, i. e., the energy of interaction between particles both in V. The utility and validity of this breakup of the coulomb energy will be manifested and investigated as we go along. We now transform to an interaction representation by -- 15

THE UNIVERSITY OF MICHIGAN 2764-8-Ti = U U+= (23) where U satisfies the equation at so that > satisfies the equation, (pInP + 11 )Because U is a unitary transformation. tide singlet density, equation (18), as f -dl ze Tr p (x. z, t)D a V - - (24): i ft -a-. (25) (25) we may rewrite the expressions for the par8) -2ix 'r (K (2 )v T....-)G v where + + D= U D U and pJ - U p U It then follows that p and D satisfy the equations (27) a - at i [ ' 'h ) f) I (28a) and __D _ i I pX + pp,at i. I + 1 ] (28b) Thus by means of the transformation (23) we have invested p with an explicit time dependence resulting from the interaction of particles of the o'th kind with the externally applied electromagnetic fields, (A and 0), with the "slowly varying" part of 16 I

THE UNIVERSITY OF MICHIGAN 2764-8-TT -- i the transverse internal field, (A ), and with the coulomb fields of other particles outside of a volume V surrounding the point x at which f is to be evaluated. Conversely, the same transformation implies thai the time depen(lence of the density matrix D results I rom the interaictions. o( part icles with the "rapidly varying" part of the internal transverse field and between particles in the sub-volume V. We may anticipate that it is essentially the time dependence of p that describes the influence of external and self-consistent internal fields upon the temporal variation of f, whereas the time dependence of D will contribute the effects of coulomb a "collisions" and particle-photon collisions to the temporal variation of f To realize the content of these remichrks. consider 9 a f -2iK. z '.~Jx __ 3 ( -2- 3D a d ze Tr.. 1) + _ e Tr p (29) where the different contributions to the time derivative of f have beeQ expressed in different coordinate systems for calculational convenience. Noting that [p D i jo - _ i 0 at t one finds after tedious but wholly straightforward manipulations, including the transition to the continuum in K-space, that equation (29) may be written as 17

THE UNIVERSITY OF MICHIGAN 12764-8-Ti 3f -hK. 0f e + C _ Tr R cos(. 7 ) P D t V 3m dx. m e CK t m Jpx. m c J z o j CT j.c 0sin - / T sin 2c _ __ (2r (prtr r d b r31 )oe sic i w E furt comresin)f eota tion, wes av itroducd th zs o (X/') (x z) 4'p (X, -A J, x') fd 2 / +elatd to Tr R sillreco e an u, m, ns p m a j j 2 Kza m C" j CT V 8 C ~ -2x ^ (2e) where, for further compression of notation, we have introduced thc sYnbl )( for the Fourier operator, i.e., r2D- -2iK c z j 3 d zC (32) Transcendental operators are defined by their power series representations. Equation (31) is exact, but almost contentless as it stands, as it is no more than a relation between the time derivative of fand diverse funttionals not obviously related to f To reduce (31) to recognizable and usable. ftornm, numerous ac 18 I

THE UNIVERSITY OF MICHIGAN 2764-8-i approximations must be invoked. Our attitude at this point will be to introduce the approximations with operational precision, but to expend little effort in attempts to evaluate or justitf then U We first note that K is particle momentum (and h Kj/mj = vj, the j'th colnponent of the velocity of the o'th particle) aud hence that the trigonometric functions of the operator YK have a natural representation as power series in t. Since the left-hand-side of equation (.31) describes the influence on f of explicit time variation, transport. andi the interactions of the particles repa resented by f with external and "slowly v;,rving'' internal electromagnetic fields, it is not expected that specifically quantum eflectl. will be significant. Thus we retain only the explicit, lowest order dependence upon t in these terms. We are then led immediately to af p. af e a dl -- + - Ua Tr R p D - e - ( at m ax. m c j x. Kz a ox j 3 3 2 e e 3 3 1 -2iK. z d x' d e - ' ((x-z) (c,(x') (x+z) o,(x') ) U!1 It - - - X V ) U(x, x')+ K TrR.(V V) 7 p D -K -x - - n c j j -x -K z a e2 0 — 2ix Tr0 rR )7 V 8_ aD - -c- ^ Tr R (^7 e -Tr p(K~ ) 2 T n c2 x -1 Kz o (27r) Trp at (33) 19

---- THE UNIVERSITY OF MICHIGAN 2764-8-11 To simplify further, the left-hand-side of (:33) requires a more serious and subtle sort of approximation. First consider the terms of the form Tr (R) K p D, (34) where ((R) stands for ap)propriate operator functionals of the portion of the electromagnetic field represented by R. Note that if R stood solely for the external (given) field, (34) could be written as '(R)Tr rzpc D = 4(R) t, (35) i.e., all such terms as (34) could be written as explicit functionals of f - the singlet density for which we are attempting to deduce a balance relation. In actuality, of course. R is dependent in part upon a portion of the internal vector potential (AS), and consequently the replacement of (34) by(35) involves the approximation of replacing averages of products by the product of averages, i. e., instead of (35) we have Tr (R)'K p D -[Trr(R)D] [Tr Kz Pp D] = Tr(R)D f. (3 G) Introducing the notation LR)= Tr R D, and making the replacements (3(0) in (33) 20

THE UNIVERSITY OF MICHIGAN j2764-8-T as well as the replacement +((, cy(x-z),(x') o (x+z),(x') ) (37) h, (?so(,- / _ ((X) (+z) ) (, ot(/), ) which is facilitated by the nature of the plotential u(x, x'), i. e., the replacement of doublet densities by products of singlet densities is presumably the more justified the farther apart the space points upon which they depend, we obtain af + 3t m aa e + Ca m c j o cy -- <.. Ci., p. m c bjx.. \. jp. af -e o ap. J x e o xfu( xX)f (x t) J CT ' -- -xI 3f a 9 e2 2m c o',) <R.> axe Hf Of. P1 v 8 -2ix = (27r t) 3 V 9 Tr p (K (. t) (y - I ) it (,8) where we have also approximated < R>=<-R (39) and employed the notation of equation (15). Now define a total, longitudinal electric + For consistency with later analysis which explicitly incorlxporates exchange effects in the representation of collisions, the exchange contributions to the replacements (37) should be retained here. For a discussion of this matter, see Oldwig von Roos, Phys. Rev., 119, 1174 (1960). 21

-- THE UNIVERSITY OF MICHIGAN 12764-8-Tj field by EL - a e d3 x'u( x')f,(x', t) ax. ax., a' /e x' 3d x' s(x') (40) ')X ax. X XI where we make explicit use of the nature of the function u(x, x'), equation (21), and of the fact that the point x is in V. Evidently the second term in (40) is the coulomb field at the point x due to all charged particles in the system outside of the volume V about x. We then may write equation (38) as 2 f e f e a<r e2 aR? af + _- R _ +e EL+ __ _ __ at ax. j m c c ax. 2m c2 ax. & \~ j a j. j 8pj 2V 8 V~ -2ix 9: ()V e Trp (K t) aD (41) (2 - 'h)3 u at' t' The left-hand-side of equation\(41) has now assumed the conventional form of the rat equation for a singlet density in configuration and momentum (instead of velocity) space without collisions and with self-consistent fields. The fact that the fields, L,\ E andR>, satisfy appropriate Maxwell's equations with sources in the plasma is not necessarily obvious and will be discussed in some detail later. At the moment we turn our attention to the right-hand-side of (41) which should, in some sense, I- - - --- l 22

THE UNIVERSITY OF MICHIGAN - 0764-8-T describe how close encounters between particles and the interaction of particles with the "high frequency" part of the electromagnetic field influences the temporal variation of f a We assume that we may so choose the volume V about x that the left-handside of (41) does not vary appreciably within it. We then integrate (41) over the volume V obtaining (employing the notation L f for the left-hand-side), 3 V / dxL f -VL f =Tr p (K, l t) (42) a a 3 a — t (2 7r h) xEV This integrating, or averaging, is often referred to as spatial course-graining, and is a procedure resorted to elsewhere in similar contexts [9]. Apparently the bulk of the task remaining to us is the estimation of the trace in (42) as some functional of f. This we shall accomplish by a cavalier recourse to approximations ao which, though reasonably well-defined, we shall make no attempt to justify herein. Recalling equations (17) and (27), we write (42) as 1D -3 aD -3 L f =(2r ) Tr Up (K) U (27r t) Tr p (K) U U. (43) a a at c- at We now approximate -D+ -D(t+ - 1(t)] +( UaD U D(t)s)- ( t)- (tUU(t)U (t+s)D(t+s)U(t+s)U+(t) -l(t)] (44) t s e s where s is some small but finite time interval which is long compared to collision time but short compared to intervals over which macroscopic quantities -vay --- 23

THE UNIVERSITY OF MICHIGAN!2764-8-TJ appreciably, e. g., the external fields. By virtue of the latter of these restrictions on s, we have -io o~s/ U(t+s) = e U(t): (45) so that (44) may be further approximated as ab + 1 f ills/th -iH0s/tl U -U - De D(t+s)c -D(t). (46) dt s L The "coarsening" of the time derivative explicit in (44) is also a procedure that has been employed before [10] and has essentially the same effect as the usual temporal coarse-graining [11]. Inserting (4(f) into (43) leads us to ill~ /! o iH~ t i{lS/ti L f (27rh) s-1 Tr p (K)c eIXs -Trp (K)It). (47) a a o- o -- Recall that H has been so defined as to include the kinetic energies of the particles and photons inside thal box of quantization (T), the kinetic energies of particles and photons outside the )(,\ plus nicir interaction energies with other particles outside the box and with the external fields (11 ), and the energy of interaction between the particles in the box and the external and "self-consistent" internal fields (H ) -- the latter generated by particles outside the box. Throughout the remainder-of the iscussion in this section, we shall neglect H compared to T and II. This approximation in the present contex c corresponds to the assumption that when two articles are sufficiently closely associated that their interaction may be described in collision terms they may be regarded as effectively decoupled from their 24

THE UNIVERSITY OF MICHIGAN 12764-8-I environment. Then noting that both T and H commute with p (K) (the number a - operator for particles in the box), we find that equation (47) becomes L'f (2 h)3 ) [Tr p (K)D(t+s)-Trp (K)D(t) iH s/l -ill s/t -(2-t) s- Tre p (K)e )(t!-Trp (K)D(t). (48) againfor appropriately small s. According to the above remarks, the Hamiltonian may be written as tp pOppp p), H=T+H +Hei+11tp =T+! +11^i+11PP+, (49) where we have redefined H and HP so that the portion of HP describing the intere tc actions of particles and photons outside of V has been added to He yielding 11, the remainder of Hp being designated Hp. As previously agreed, we shall ignore the coupling between particles inside and outside of V implied by Hi and will assume the effective commutivity of (T+HPP+HP ) with II. Then since Hc also commutes with p (K) (the interactions between particles outside of V cannot clange the number of\particelts iti a,lv.n state within V), we find for equation (48). L~f r(2rh)-3 s [-Trp (K)D(t) -i(T+HPHP )s/h i(T-tHPP +Hp ' + Trp (K)e D(t)e (50) a - -- 25

THE UNIVERSITY OF MICHIGAN 2764-8-T To continue from here it is useful to employ a specific representation for the explicit calculation of matrix elements. It will be convenient to choose a representation for the system which is in part the number representation for both particles and photons in the volume V and which has the following properties among others: a (K)a (K) | n.) o \,1 nn a o( - - I - oK \ (51) a (k)a (k) / n ' > f yl) nl1 a/> and <f y' | nn a> nns G,- ',. (52) I tiu Li nlat1)ls (n l)are the usual sets of numbers required for the specification of occupancy ot momentum states by particles and photons in V. The labels a are a sufficient se.t to complete the specification of the states of the whole system. The density matri\ has. of course, been defined for the whole system and must therefore depend upon the labels a. Nevertheless we shall suppress the dependence of all quantities upon these labels, as ihth p (K) and e [-i(T+ll PI) /h] are diagonal with respect to this part of our representation. In view of these remarks we find that -i(T+HPP+HP )s/ D(t) i(T+HPP+~I p )s/h) nf. n n n 1nn' ne' +(terms proportional to off-diagonal elements of D). (53) 26

THE UNIVERSITY OF MICHIGAN 12764-8-l, - rWe will ignore the contribution of the off-diagonal elements of D to the relation (53)., It is convenient to introduce the notation,P -3 i(T+HP +1 )t i (54) W =(2 7r h) s e n n' '1 /nn' nl so that equation (50) becomes Laf,L f n W D,(t)-(2 7r ) n D (t) a n'p ' n't,'n ) n nI nn' II qnrnY1 nK nj nn' ' K AKJ nnrn'' n nn nnn' T - - where the prime on the summation means that the term in the sum for which n=n' and = 1' is not to be included. Some manipulation is required in the development of equation (55), hinging primarily on the symmetry of W and the fact that (9 -3 n' n ' I i The explicit evaluation of the transition probabilities, W, is facilitated (at least perturbation-wise) by re-expressing the exponential operators in equation (54) as -i(T+HPP+lIP')s/h -i T s/h e = e (I+ Q) (57) where I is the identity operator. Recalling equation (54), we see that now --

THE UNIVERSITY OF MICHIGAN 2764-8-1 (27rh)3 -1 (e!>) nnf 2 =(2 -3 2 W s =(27rh) S-1 e I+Q,, (2r S1 7Q s n~n'~' nq n''1 nnn'r'' (58) because the kinetic energy operator T is diagonal in the number representation. The operator Q may be computed to various orders of approximation from the formula, Q(s) =..I j-1 Q(- ds.. dsj( PP+. (IPP p ),+ S1S S1 =0 Sj-0 (59) where iTs/h -iTs/h (IPP+H) = e (1PP+ HP )e (60) s A straightforward but tedious calculation reveals that through ternms of fourth degree in the interactions, QQ may be exprt.ss.d -is +,. 4 sin Q -- 4 i n L 1 (HI PP+ HP (HPP ) 1 + (| -HP ) J nn ma ma m vhere we have introduced tw = E. The complete disregard of the interactions n'q nq in H0 while calculating the effects of interactions between particles (or between particles and photons) in the volume V is equivalent to the assertion that the particles 28 41

THE UNIVERSITY OF MICHIGAN - 12764-8-Ti in V are decoupled from the environment created by other particles outside of V. A little reflection reveals that (H+HP n - Hp [2 n,2 (62) nqnrn' n nI nnn'n since the matrix elements of Hpp are non-zero only if the photon number does not change while those of H are non-vanishing only if the photon number changes. In fact, as we shall see, the mal i \ eleme, nts of IIpP describe particle-particle scattering, whereas the elements of Il — containing terms both linear and bilinear in the creation and destruction operators for photons -- describe either radiative particle transitions or particle-photon scattering. Since we are not in a position here to discuss particle transport in systems in which particle transitions to, from, or between bound states are of much importance: and since free particles cannot emit or abl)sorb a single photon, it follows that only photon scattering will be considered in the evaluation of IHp n I As it is our intention to consider thie influence of bremsstrahlung and inversebremsstrahlung on photon balance, we should here consider first order (one photon) radiative particle-particle scattering. Such processes are accounted for by the terms HPP HP +H P Hpp n nma, man'' ntrma man'I, Y,,:.1 h (a -u ) ma n a if we employ plane wave states for the description of the radiating particles, or by

THE UNIVERSITY OF MICHIGAN!2764-8-T| - - - the terms HP I (64) if we employ positive energy coulomb states. It is, in fact, the latter that we employ in the discussion of thermodynamics later. Utilizing equation (58) we may rewrite (55) as Laf = (2Tt)3sL / fy.S1NJ - n | Q n D, (65) oK h- 'h n nn, n nn Then the calculation of the quantities Q proceeds straightforwardly from the formula (61) and our knowledge of the IIamiltonian for the system. One finds after considerable manipulation that equation (65) becomes 'K K2K3 K, ' K2 t t-ht m n ion Sectio [ T1 t ransin r oa b i)(Ln ) crKn a'K3 (1 K G 112 njn Kj A-n 'A 2 A 'k' +Ani~ ) rK k) aK Ak 0 a 1 +A'k?) J nyi n -noKqLk(- +ncTK )(I +Ak ) ]D (Dn(i) where the quantities C and S are essentially the transition probabilities per unit time for elastic charged particle scattering and for the scattering of particles by photons respectively. Here we are ignoring the contributions to the particle balance relation due to radiative coulomb scattering. We retu'n to this qucestion when we, consider the thermodynamics of the system ir Section IN. The transltion probabilities 30

I I THE UNIVERSITY OF MICHIGAN I2764-8-Ti.-. -.......I - ^ represented by C and S art symmetric, ct g., oK, o'K3 (cK, o'K2 C =C.(67) caK, ' ItK2 aK, a'K3 The choice of sign in the factors (1 + n) depends upon whether the particles whose number is represented by n are bosons or fermions. These factors appear in our balance relation because of the dependence of reaction rates upon the densities of particles or photons in the final states. In particular, if the particles are fermions (e. g., electrons) so that the factors are of the form (1-n) and the only allowed values of n are 0 and 1, we see that transitions to occupied states are forbidden as must be the case because of the exclusion principle. However, as indicated earlier, we should not expect this dependence of reaction rates upon the density of particles in final states to be significant in the non-degenerate plasma (a system in which the number of available states greatly exceeds the number of particles). The direct, formal evaluation of the occupation number sums appearing in (66) leads to the introduction of higher order densities (higher order than the singlet density, e. g., doublet, triplet, and quartet densities) into the balance relation. To circumvent this complication at this point, we resort again to the approximation of replacing averages of products by products of averages -- bearing in mind that the average of an nK (or an AXk) with respect to the density matrix D is just the singlet density for particles of kind o having rylmt ntum T K I 31

i f I[ I r ------ THE UNIVERSITY OF MICHIGAN |2764-8-TJ (or for photons of polarization X and momentum ft k) multiplied by the volume of quantization, V. Thus we find that (66) may be expressed as Laf=T J d3pd3P2d3 p C(Up 'pU2;ap, a'p ) [f(p1)f^,(p2){(27rh) 3,Pa.P Ip'2,p_ +f (p a - +E XX' (2 rtl)- +fo,(p))] -f(P)f(P3) ((2 7 ft)- +tpl) (2 7r h)-3 +f,(P 2)] CFc i( 2 J Jd3pl dkd fdk'd l 'S (p X 'k';pX k) [f (pi)FX (k') (2 )-3 Pi k, k' I tf (P)] k2(2 -3+F (k) -f(p)F (k)[(27r t)-3 +f (P)} k'2(2.t)3+F (k) (68) The F 's represent the photon singlet densities -- to be iscussed in Part III. The transition probabilities are exhibitable as 0 [] 6 C(ap,, p2;ap, ap)= UE(2)) o(E )S(- E _ )_p+p p ) -a - - _ P P I - E p -P _1 2 / OCT( (69a) 32

THE UNIVERSITY OF MICHIGAN 12764-8-TI and. 2 (pl X'k;pX k)=(2xr) h4 (k)- (k') (E-Ek) X Sp ki' -p - k), (69b) where a, is the reduced mass of the scattering pair, and Ecand c are the energy of the scattering pair and the scattering angle in the center of mass coordinate system. We have introduced the symbol o(E C ) to represent the coulomb cross section, i.e., 2 2 f4 1 -cos (70) EC A)= ( ') sin-4 1 + (70) C 44 2 | G' 1+cos The factor modifying the usual formula for coulomb scattering which differs from unity only when the scattering particles are identical arises because o4f cxhange, i. e., because of the indistinguishability of targets from projectiles Analogously equation (69b) contains the Thomson scattering cross section (as would be expected since we are dealing solely with non-relativistic particles) as is readily demonstrated by averaging over photon polarization states. Recalling the definition of the operator L, equation (41), we re-express our densities in position and velocity space rather than position and momentum space, obtaining 33

i I THE UNIVERSITY OF MICHIGAN i2764-8-T| &f af e: f e af 3' +at v ~ a + Ej + (vxll) &j v d v2d v3 ((cpl, 'p2 P, c'P3) b j ( xj J av ) - - - - m m m )fC1 ^ t / P2 T / h \2 37r ) tlr ]+dkdf l d' dS' (1X' k '; pXk) [f0(vl)Fx (i.){ a [( \) +f T(v)j +FX(k) -f (V)F (k) -( )+fr(Vl)} +||+) 7 (3 ' 71, The{ vtlocitx variab le' was htrc l(' i i(x ( t t1 ac hiiti t(, tht i n tIllt at iri. vX (- (1-) <R>)/m; and we have employed the notation L 1 -E -L _ - (72a) and H 7 x<R. (72b) Recalling equations (7) and (40), we st e that E and H are interpretable as the superposition of the externally applied and a portion of the internally generated electric and magnetic fiejds. Thus if we ignore the scattering of particles by photons and take the classical limit of the terms describing the scattering of particles by I 34

THE UNIVERSITY OF MICHIGAN 12764-8-T particles, we find that equation (71) is just the conventional Boltzmann equation with self-consistent fields: except for tie implication of some restriction upon the interpretation of the sell-consisttent fields as well as upon the strength and variation of the permissible applied fields. The nature of these restrictions deserves a little:ttnltlion;it this point. In the present context they stein at least partially from tinh scem nlx necessity for the operations of spatial and teminx)ral coarse-graining. I het spatial coarse-graining required that the distribution functions for the particles and the fields E and H be essentially constant over an appropriately chosen volume V. Thus given V, this puts an obvious limitation on the space rates of change of the fields, whereas given the inhomogeneities of the fields a lower bound on the dimensions of V is immediately indicated. Furthermore the distribution functions are expected to represent the average number of particles in the volume V -- hence the particles must be presumed to be localized within V -- thus the least linear dimension of this volume must be large compared to the DeBroglic wavelength of the majority of particles of interest. It should be noted that the collision description of t!c inltterict ion of closely ssociated particles also requires that theitr )eBrogli( 1, l. -tiglhs.(.t:iually their elative DeBroglie wavelengths) be small compared to;::- i,.t ll, e.t-u (' of V. us rapid field variations require a fine-grained average, whereas systems of lowean-energy-particles require a coarse-grained average and in somne systems these 35

THE UNIVERSITY OF MICHIGAN 2764-8-T! opposing demands may not be met. This is unlikely however in the fully ionized plasma. A serious complication is introduced by the presence of strong magnetic fields, even if homogeneous. Evidently, in such a circumstance, it is required that the dimensions of the quantization volume must be small compared to the-radius of gyration of the lightest particle of interest in the system. Otherwise, the employment of the plane wave representation for the particles in V would be unsuitable. Finally, the assumption of the localizability of photons in the volume of quantization implies that its least linear dimension be large compared to the wavelengths of such photons. The equations governing the behavior of E and H remain to be developed in the present context. As they represent superpositions of internally generated and externally applied field, and as the external fields are presumed known, it suffices to consider only the portions of E and HI which arise from charges and currents within the plasma. If we designate these portions by & and W respectively, then by equation (40) we have cL (- t) (x, t( axt Ze d X' S(x') (73) |Xi a IX for the longitudinal part of the self-consistent electric field. For the transverse part of this field we have 36

THE UNIVERSITY OF MICHIGAN 12764-8-T! -a ---*_____ T x, t)= - c a Tr A( x) D(t) (74) ~j - at whereas for the self-consistent magnetic field we have,j(x, t)= Tr Vx A(x) D(t). (75) 3 J It is immediately evident that _ is divergenceless, for V *Tr V7xA8 l)= rr [V Vx A D= 0 (7G) It is perhaps also equally obvious that _L is the solution to Maxwell's equation V = 47r e. f (x,t) (77) -a a (o giving the longitudinal part of the electric field at x due to charges outside of the volume V about x. Recalling equation (74). it is seen that Vx T -t Tr xAI D, (78) which, according to equation (75), leads to Vx T- 1. (79) Thus three of Maxwell's equations are established almost trivially as descriptive of the "slowly varying" part of the internally generated electric and magnetic fields. But not so, apparently, for the fourth equation relating the fields to the currents in the system. The difficulty here seems to stem from the necessity of calculating 37

I THE UNIVERSITY OF MICHIGAN 12764-8-TI explicitly operators representing the time derivatives of A,i. e., of calculating commutators of A with the Hamiltonian. But such calculations have here been complicated by the fact that A represents only the "low frequency parl of the vector ptential generated by charges and currents in the plasma. A senmi intuitive circumvention of this difficulty is accomplished by considering an equation satisfied by the exact fields in the plasma. i. e., the fields which have not been decomposed into parts of the "fast" and "slow" variation. Labeling these "complete" fields by L' and _' one finds (as shown elsewhere) that they satisfy the equation: Vx -' C at - ' - TrJ D, (80) here:__ 2 J =Z[ i: ( )mc V q/) PaJ. (81) op L — a -1 c This is the anticipated relation between the exact fields and the exact currents in the plasma. However, it is desirable to translate the above description of the current to the conventional macroscopic description of plasma currents, i. e. J =Ze J m /c e v f d v. (82) - macro ao - a ais is readily accomplished, for This is readily accomplished, for I 38 38

THE UNIVERSITY OF MICHIGAN:2764-8-T e fv f V d p -- ( I) - M- - A e h 2 K f f d3KA> >K- (83) m - m c ' Y (5 where here of course, A is interpreted as the total vector potential in the plasma. A little calculation reveals that the first term on the right-hand(l-sid(le of equation (83) is the same as the average of the first term on the right-hand-side of equation (81): so that Tr J D -<J >= J -op -op macro +_ me p \ < <- (84) 2 (r -A a Entering this relation into equation (80), and decomposing r ' and __' into parts of "fast" and "slow" variation we obtain 47r + x F -1 4j7- <AF> < y>-<A pp>1]. (85) e m c LL ca'J CT o Note that A only enters into the "correlation term" in (85). since we have already 39

THE UNIVERSITY OF MICHIGAN 12764-8-T employed assumptions equivalent to the statement that qA> 4 y(>- - (C - 0. (86) It is at leoast intuitively reasonable to argue at this point that the macroscopic current J m contributes only to the slowly varying fields E and, whereas -macro the "correlation current" contributes only to the rapidly varying component of the internal fields. The latter however, have already presumably been adequately accounted for in terms of photon distributions, hence the only relevant part of equation (85) is Vx _-1 4. (87) c -t c -macro We now assert that equations (76), (77), (79), and (87) provide al appropriate description of the self-consistent fields appearing in the particle transport equations (71). 40

THE UNIVERSITY OF MICHIGAN i2764-8-t1 THE BALANCE RELATION FOR THE PHOTON DISTRIBUTIONS In order to complete our description of the plasma, we require the equations governing the photon distributions, Fk. The deduction of a set of such equations has been described in considerabl, detail elsewhere [2 ], hence only the highlights of that deduction will be sketched here, We defne a photon let density in close analogy to the definition (14) of the particle singltet density, i. e. F (x, k, 1) = k -21 x ) e - ( I (k+ q) a(k - q) ) \_ `'%- q% 7 Tr p (x. q ) D, A. - (88) where we have introduced the notation '-r _ 8 ei x - 2 xq V q (89) p for the fourler sum operator, and pX(k, q ) = A (k + q ) %(k - S ) (90) 41

THE UNIVERSITY OF MICHIGAN '2764-8-T for the photon singlet density operator. To illuminate the sense In which (88) defines an appropriate photon slnglet density, we first integrate Fk over the volume, V; obtain ing, 3 F (k, t)= /d x F (x, k, t) V - l' (k) a(k ) (kD. (91) Thus F (k, t) Is just the expected number of photons of polarizati:,n X having momX - entum h k to be found in the volume V at time t - suggesting the interpretation of 3 3 F (x, k, t d x as the corresponding expected number in d x. Next we observe that /3 x / d x (h c k) F (x, k, t) Xk k V (92) I (E ) +(H) = d.x (, - _ )- -h c k, k V T where we have made use of the relation E = - 4r c P. The second term on the right-hand-side of (92) Is just the zero-point energy impllicit in the first term on *1 --- -- 42

THE UN yERSITY OF MICHIGAN 12764-8-T! the right-han dde; hence the difference on the right represents the energy of the actual photons in the volume V. This suggests that, In some sense, we might Identify h c k F (x, k, t) kX T 2 2 (.r. (93) kX and hence - since ti c k Is the energy per photon of mom-ntum h k - further Lnterpret / F(x, k, t) as the expected nunmber of photons per unit volume Xk at the point x. Of course, the identification (93) is not quantitative, since thu functions Identified actually differ by quantitiets whose integral over V vanish Evidently, the concept of the spatial localization of photons is somewhat more obscure than Is the case for particles. In this instance also It is convenient to select an appropriate interaction representation in which to calculate the time rate of change of FX. To Identify the transformation leading to a useful interaction representation, it Is desirable first to regroup terms in the Hamiltonian (1(0) after Introducing the fourier analysis (11) of the vector potential A. One finds that 43

THE UNIVERSITY OF.2764-8-t MICHIGAN * H = (h c k)/Z kX k X 1+ o k) 22 /2m c k a 7 e c C Zrt tc m 2m cr\ kV ( t c k) c (k ) a (k) A f dx~~ +Izfdx(r~)+ r 1 d3x -ik T(_kI or + + (k) +(k) (II) d x e -(7l (k) j+ k ) - (q c.1 L -- Y ) ] (rcrl e0T eT,.1 1 + I d x d3 x V +W, xI ) 1- ( j YjIX, +Ze d3x ) a' Yu + -m ck- rx(k) ((-k) a+(k) ( -k) - m cVk - - L X Acrk)X 4> +a (-k) a(k)J I 6i d x L - YUt ai~kk'Ix I 7VB e ma.cV (- k) Jkk / 3 -ix(k-k') + dxe - a.. (94) 44

THE UNIVERSITY OF MICHIGAN - i2764-8-T The prime on the last sum Implies that the terms for which X = ' and k = k' are not to be included in the sum. They appear explicitly In the terms immediately preceding and In the second set of terms which are proportional to the photon number operator, a,(k) ac(k) The first sum on the right-hand-side of (94) Is simply the zero-point energy of the radiation field. Now define i r t'. pho,ton Hamiltonian by H - H = (h c k) b (k) ca(k) kX X + - y1 d (95) x 1 + H = H~ + HI1 (96) The transformation to the desired interaction representation is now accomplished according to J = U, (97) where U = e(-i H~t/ti ). (98) In the new representation the expression for the photon density becomes 45

THE UNIVERSITY OF MICHIGAN 12764-8-Tj F(x, k, t) = q Tr p (k,, qt) D, (99) where = —, p (luOa) at t and D, H (1OOb) at h From here on the calculation of the time rate of change of FA proceeds In a manner formally iielttici to the corresponding calculations for the particle densities described in Section II. (It should be noted in passing that a more elaborate and more subtle treatment of photon transport in dispersive media is readily accessible here. However, as such Investigations are only in prigress and not complete, discussion of them will be deferred), Thus we shall dispcilsc with further discussion of deductive detail and go Immediately to the final results which are embodied in the equation F + v ftl- VF at g -X TX, (continued) 46 --

i THE UNIVERSITY OF MICHIGAN 12764-8-T - (v) f f (V) i F(k)} 2 F FM,(k') fMF (k) r +f (v + (kl LrX- cr - ___ -[f (k) F (k) (101) In the medium, and have neglected higher order space derivatives of F than the X first. To the extent that we may replace averages of products by products of averages, we find for v in this case (because of our chhuicte j H, equation (95), which Is the prime consideration in the determination of the manner in which photons propagate between Interactions with the particles of the plasma), __ r N _ _ _e 1 v - ck + (102) g k 2ck m J where N Is the average density of particles of kind a in the volume V. Since only the electrons will contribute appreciably to the sum In (102), we note that c - /ck2, (13) Vg _. c - WC/2ck (103) 47

- THE UNIVERSITY OF MICHIGAN 12764-8-Ti where O e = 4 N e /m. Thus photons of momentum hk such that we/2c k > 1 e ee do not propagate In the medium as is expected. Evidently the energy of the "free" photons in the medium is given by 2 2k2 E = t w(k) = c k(l + we/2 k ). (104) The vector -1Q is just the unit vector in the direction of propagation of the photons, I.e.,.. = k/k. The scattering frequency, S,, is the same as the one encountered earlier (69b) in the equations governing particle transport. The quantities EX(k) and a(k) are the transition probabilities per unit time for the emission or absorption of photons of polarization X and propagation vector k. Of course, these transition probabillties are space and time dependent as are the distributions F,. They may be explicitly evaluated In the sense of first order perturbation theory from the formulas, k ) = ) (K) 1 + V f (K' (105a) KK' r ) = 7 TK ( ) f (K) 1 + V f (K ), (105b) KK' r where 2r -Ik- x C 2 TK(X k |<K'| - e — 6K(k) (E -E - K- Ck -1K'k (106) - " 48

I -- - THE Ul IVERSITY OF MICHIGAN 12764-8-T Because the particle transition accompanying photon emission or absorption may be between, to, or from discrete states, It Is not generally feasible to convert the sums In (105a, b) into Integrals, Thus the K's In these formulas are simply to be Interpreted as a sufficient set of labels for the complete specification of particle elgenstates. The velocity operator appearing In equation (106) is to be represented as v.= (p - - Ae)/m The details of some calculations of E and 2 for specific c mechanisms of emission and absorption have been presented elsewhere [Z], and therefore will not be entered Into here. * -- -- 49 -- --

THE UNIVERSITY OF MICHIGAN 12764-8-TI IV THERMODYNAMICS OF THE FULLY IONIZED PLASMA The present description of the plasma is complete and irreversible (In the sense of the many and varied approximations that have been Introduced Into Its development). It is complete in the sense that we have as many equations as we have unknown functions described by them, and these equations Involve functional parameters (scattering frequencies, emission and absorption transition probabilItles etc.) whose analytical representations have been specified. It is Irreversible In the sense that the whole set Is not Invariant under the transformation (v, k, Hp, 9 — ( -v, -k, -H, -t). Thus this description of the plasma should contain the important implication that, under certain circumstances at least, the system progresses Irreversibly to a unique state referred to as the thermodynamic state; and that In that state the system variables assume specified forms. Thus It Is of s ane interest to explore these Impllcatlofis here, as this has not hitherto been /complished from the present point of view for the plasma. However, in order that such an Implication be realizable It Is also necessary that our description be consistent. That is, the equations must explicitly describe the full effect of any given Interaction upon all distributions concerned to the same 50

-- THE UNIVERSITY OF MICHIGAN 12764-8-TI level of approximation, But this has not yet been done here, for although due account has been taken of the effect of emission and absorption processes on the photon distributions (as is obviously essential), the effect of these same phenomena upon the time rate of change of the particle distributions has so far been neglected. It was pointed out earlier that for the sake of consistency Inelastic corrections to scattering should Indeed be retained In the equations for the particle distributions, (71), but that their retention implied only a small correction to the scattering cross section and hence would be Ignored. Thus their quantitative importance to the description of the fully Ionized plasma Is most probably negligible. But to achieve a proper qualitative appreciation of the approach to equilibrium, the influence of these mechanisms of Interaction between particles and photons on the particle distributions' cannot be Ignored, Since the description of the effects that concern us here (to the present level of approximation, i., first order perturbation theory) would simply enter additively Into equation (71), we merely sketch at this point the calculation of these requisite extra terms. According to equation (65), we should compute (n- n) W D (107) t ea TK Knvnt njnl where Wea Is appropriately chosen to represent particle interactions involvlnj single photon emission or absorption. Since the transition probabilities for these 51

I THE UNIVERSITY OF MICHIGAN 12764-8-T processes are inversely proportional to the masses of the emitting and absorbing particles, It Is evident that we need to evaluate (107) for the electrons In the plasma only. Furthermore, for the discussion of the particle distributions at least, we are restricted to fully ionized systems. Also, as we are solely concerned with the approach to the thterm — ax namic state in this section, we shall presume that our system is not exposed tui external fields uv any kindl Thus the only events that concern us here are the electroniic "free-free" transitions executed In ionic coulomb fields. In order to keep the manipulations to a minimum and the bring the results smoothly Into line with the formalism of equations (101), (105a, 1) and (106) for the photons,we shall calculate the desired transition probabilities by first older perturbation theory assuming positive energy coulomb wave-functions for the representation of the electronic states. Accordingly, the quantity to be calculated is ea 4 m 3 w= _ ) — _ - W nq n'q' ( nin'rSl n r 2-W sr h (nn ) 3 2 X n ed x (A- p/m)Y n'y' (108) Again, after tedious but straight forward manipulations, we find for (107), 1

THE UNIVERSITY OF MICHIGAN I ~ 764-8-T S f e 6 t )ea =2111(d3vI dk dflV x TKIt (X k) m k2 x L 2 ff 'h f (v') Fx (k) 3 - fc (v,, - fI C2 + Fx (k)} T 3 + VI v x r 3 Me x - 7r 'h Me - f (v) e -?- 7r 'h uf (i)} f K( TKv( k) CI I(v I) 2 k ( 2 ) 3 + F X(k)} - f (v?)} FXGk)j (109) I KY In this expressi[on, T K (X k ) Is- the same as Is given In equation (106),, 53

THE UNIVERSITY OF MICHIGAN 72764-8-T' We now undertake the task of proving an H-theorem for systems that are not exposed to external fields, that have already become spatially homogeneous, that exhibit no internal electric currents, but that are still ten porall dry(ting,. Bv virtue of these assumptions we may presume the absence of -elf consistent fields and may take for our system equations: if =3 3 3 / v d v2d v C3 (o- p cr' P; P, 0' P3) it X [PT + f(v) [TI +fT(v3 )] [ Po' + fT(V [ 1 PT +fT (V2 f(vi ) f,(vz ) PT, + f,(v f (v ) P +f (v) -- (: f,(v3) Po+ f-,(l ) Pa, + f,(v3) + 3d v dk dfidk' df' S (pl A' k '; p k ) XX ' x [Pa +f( >] [P + F(k)l] [p t f (v + Fh (k )J f (v1 ) FX'(k') P + ft(V ) Pk, + FX,(k') f (v ) FX() P+ f(v ) -Pk + FX(k ) (fe0) (110) 54

THE UNIVERSITY OF MICHIGAN i2764-8-T and 3 Fk 3 3 - " = I d v d v dk' dQ' S ( p kX' k'; e X k ) t 1 f- ' - TX' [ P f,(Vl)] [pk,+ Fk' [ P f f( )] [Pk + F(k) f(1) F,(k') pfG(V) F (k) P + f (Vl) Pk' + FX-(1') P + f (v) Pk + F(k) - 0' -._ - 0 - * 3 3 K' + d v d v' VT K k XkI K r [: pl. + F ( p — rf (,v " -f (v't) I x(k) Pe c [ e j L '+. fe(Y ) Pe - t (v) f (Vt) Fx (k) Pe -i(V ) pk + FX(k) (111) These equations follow directly from (71) and (101), with due regard for (105 a, b) and (106), by an obvious regrouping of terms, the Introduction of the symbols 3 Z 3 p = (ma /27T 'h) and p = k /(27r),and by a conversion of the sums In (105a, b) 01C 55

THE UNIVERSITY OF MICHIGAN 2764-8-Ti to Integrals on the assumption that the particle states between which emission and absorption transitions occur are sufficiently densely distributed. We now define a function S according to S= -K d v f (v),f (v )+ p + p *- Po- L or ~ 0 — ~ ~" (T T P U + ^(v)}; {Pc-, f }' ' - k k -d S Fl). k [FktFX(k) t F(k) + pkt pk - {Pk + F(k) J kX + Fx(k)}]. (11z) It is then readily shown that f (v) dt /+ f (v) i {, } (F> ( k) - )d.d-. (113).J ^^ - x 56

THE UNIVERSITY OF MICHIGAN 2764-8-T Substituting (110) and (111) for the derivatives of the distributions In (113) and performing the usual nlanipulations on the variables of integration while noting the symmetries explicit in the scattering frequencies C and S, one finds that dS = - / d v d v d vdv3 C(cr p, T' p,; o-p, cr' P3) dt... [ f ] [ t(3] [P+ (V ] [ ' ff (9 ] fiv) f (v3 ) P + f(V 1 + f f- r-(vI ) { Po + f (K) {^' + f,(vz ) + (v, a +' fC- (v ) f (v3 ) p +f(v ) p +f ( ) oa - a - o r - +" (3 p +f (v1 ) I -r _ - 3 3 - K d v dd vd dk dnldk' dl' S (P X' k'; p X k ) aIXX ' (continued) * -,7

THE UNIVERSITY OF MICHIGANI Z2764-8-T, ( [PC. ii fa' (9][v + Fx (k.) PC'~ fT~(v1 )j[k? + FX, ( kI 2vF f(V) FX (k) +P0 tf (vi) {k + FX1?(kt )~ La f(vl )F f(k') &a'.+f (v) Ik+FXQ(i J f (v1 ) Fx? (k?) fT(v p + f (v ) O. - - FxA.(k ) Pk+ F x(k) I.KI -flIdd~v d'Idk df VT (Xk ) K x 10 X Pe - fe(%, [p -f (vI) [Pk + Fx>1(k )] fc( )FAL(k f (v Q C-v) - + A f ( V ) e fe~ (I) F x(k ) x (114) Pe f(v ) e - p -f (v' I) e eC Pk+ FX(k-) 58

THE UNIVERSITY OF MICHIGAN 2764-8-T Since f / -- 1 for all fermlons (because the number of fermions in V must never exceed the number of available states) it Is seen that all of thj Integrands in equation (114) are everywhere negative or zero. Thus it follows that dS 0, (115) dt and vanishes only when f (vl) ) ^ ) f (v3) P' +f(vl) s;; t? 'r " - f a' " + fo-'( 3 f (v) FX (k') f (v F ) P+f( Pk+ FX(k) p+ F(') v P+F (k) f (v) f(v') F (k) e- e- - (116) P -f (v ) - Pk + F(k) e- e e- - By virtue of the conservation of energy implied In C, S, and T, It follows that all of the relations (116) Imply that 3 2 3 (m /2'h ) k 2r) f (v ):= - F (k) = (117) +- E(v)/ - Be -1 e -1 cr + 59

THE UNIVERSITIY OF MICHIGAN where E(v) = mv2/2, (k) = c k [1 + w /2c k ] and B is a nolnalizatloa constant for the particle distributions to be determined by requirements of conservation of particles. The particle distributions are recognized to be those appropriate for Bosons or Fermions as the case may be, while that for the photons is a modified Planck distribution - the modification disappearing in the vacuum since there we vanish(s. The fact that S Is an alwavs increasing function of the time as the syste. anc!es staLtc w ith time suggests Its identification as the system entropy; and, hence, the further identification of the distributions (117) as those appropriate to the thermo dynamic state. Of course, these identifications are not complete until the so far arbitrary constants K (equation (112) ) and 0 (equation (117) ) have been specified, The fact that 0 Is the only parameter common to all of the distributions, and the fact that temperature Is defined to be the property of systems in equilibrium which is the same for all systems, suggests that 0 is a function of temperature. Appeal to experiment is required for the explicit Identification, 0 = C T. ACKNOWLEDGEMENT The author gratefully acknowlt,tdges the many clarifying discussions with and the considerable assistance obtained froml Messrs E. H, Klevans and E. Ozizmir throughout the course of these Investigations. 60

THE UNIVERSITY OF MICHIGAN 2764-8-T' REFERENCES 1. Osborn, R. K., "Theory of Plasmas, Part 1", University of Michigan Radiation Laboratory Report 02756-1-T, March 1960, AF 33(616)-5585. 2. Osborn, R. K. and E. H. Klevans, "Photon Transport Theory", Annals of Physics, 15, 105(1961). 3. Osborn, R. K, "Particle and Photon Transport in Plasmas", to be published In the January 1962 issue of the PGAP Transactions on Plasma Physics, Much of the content of the present article appears In this reference In abbreviated form. 4< For a considerable bibliography of investigations of the theory of plasmas consult for example J. Drummond. Plasma Physics, McGraw-Hill, New York, 1961.. For classical, microscopic treatmnents of radiation in plasmas see A. Simon and E. Harris, Pys_. of Fluids, 3, 245;255(1960). A quantum analogue to this has been given by P. Burt, Doctoral Thesis, University of Tennessee, June 1961. 6. Schiff, L., Quantum Mechanics, Znd Edition, McGraw-Hill, New York, 1955. 7. Wigner, EP,, Phys. Rev., 40, 749(1932), 8. Ono, So, Proceedings of the International Symposium on Transport Processes in Statistical Mechanics, p. 229, Interscience, New York (1959). 9. Ono, S., Prog. Theor. Phys, (Japan), 12,113(1954). 10, Mort, H. and J. Ross, Phys. Rev,, 109, 1877(1958). 11. Kirkwood, J. G,, J, Chem. Phys, 14, 120(1946). 61