Moderately dense gas quantum kinetic theory: Transport coefficient expressions

Transcription

1 Modratly dns gas quantu kintic thory: Transport cofficint xprssions R. F. Snidr and G. W. Wi Dpartnt of Chistry, Univrsity of British Colubia, Vancouvr V6T 1Z1, Canada J. G. Muga Dpartanto d Física Fundantal y Exprintal, Facultad d Física, Univrsidad d La Laguna, Tnrif, Spain Rcivd 24 April 1996; accptd 15 May 1996 Exprssions for th transport cofficints of a odratly dns gas ar obtaind, basd on a rcntly drivd dnsity corrctd quantu Boltzann quation. Linarization of th quations dtrining th pair corrlation and th fr singlt dnsity oprators about local quilibriu is discussd first. Th rat of chang of th pair corrlations is tratd as dynaic ffcts for pairs of particls rlaxing to local quilibriu via a rlaxation ti odl arising fro intractions with third particls. In contrast, th singlt dnsity oprator satisfis a Boltzann quation with binary collisions. Spatially inhoognous corrctions to th collision suproprator ar includd. Contributions to th transport cofficints aris fro th prturbation fro local quilibriu through fluxs associatd with kintic, collisional and, for th thral conductivity, potntial nrgy chaniss. A coparison is ad btwn th classical liit of th transport cofficint xprssions obtaind hr and th classical xprssions prviously drivd fro th Boltzann quation with th nonlocal collision corrctions of Grn and Bogoliubov Arican Institut of Physics. S I. INTRODUCTION A dnsity corrctd quantu Boltzann quation has rcntly bn proposd. 1 Th novl fatur of this quation is th rol of pair corrlations, not only in dtrining th dtaild natur of binary collisions, but also contributing to a dnsity corrction to th singlt dnsity oprator. Th prcding papr 2 Equations lablld I. ) rfr to papr I. dscribs th local quilibriu proprtis of such a syst in th prsnc of wak inhoognitis. Th objct of th prsnt papr is to linariz th pair corrlation and Boltzann quations, solv th rsulting quations following th Chapan Enskog procdur 3 7 and to obtain xprssions for th transport cofficints. It should b phasizd that for a odratly dns gas on that includs dnsity ffcts up to th scond virial cofficint, this givs only part of th dnsity scond virial corrctions to th transport cofficints. Thr ar also bound stat 8,9 and tripl collision ffcts 9,10 that contribut to th transport cofficints at th sa ordr of dnsity. A tratnt of all ths ffcts basd on a unifid starting point is still to b accoplishd. Th inclusion of th ffcts of both bound stats and pair corrlations is th objct of th rcnt thsis by Wi. 11 As prviously phasizd, 2 th sparation of acroscopic and icroscopic otion is convnintly forulatd with a phas spac dscription of th translational otion, naly using th Wignr quivalnt rprsntation of th various dnsity oprators. Th corrlations ar dtrind by an quation govrnd, in part, by a dcay to thir local quilibriu for, odlld using a dcay ti. Linarization of this quation about local quilibriu introducs th prturbation and its solution by a Chapan Enskog 3 7 typ of approach is carrid out in Sction II. That is, th ti drivativs ar liinatd using an appropriat st of quations of chang for th variabls that paratriz th local quilibriu stat for th pair corrlations. Only th siplst approxiat solution for kping only th lowst Sonin polynoials is discussd in this work. Th corrsponding linarization and solution of th Boltzann quation including contributions fro pair corrlations is carrid out in Sction III. Hr it is th standard thod of liinating th ti drivativs by th us of th quations of chang for th hydrodynaic variabls of ass, ontu and nrgy dnsitis that is ployd. Ths rlations ar coplicatd by th prsnc of scond virial typ dnsity corrctions. Th rsulting intgral Chapan Enskog quation for th prturbation thus contains as part of its driving tr, contributions fro pair corrlation ffcts, both dirctly as nw trs fro th lft hand sid of th Boltzann quation and indirctly fro th liination of th ti drivativs. Exprssions for th transport cofficints ar obtaind in Sction IV. Ths ar affctd by th prsnc of pair corrlations. Th papr nds with a short gnral discussion on th approach that has bn takn in this work to obtain th dnsity corrctions to th transport cofficints, coparing with th prvious classical xprssions and pointing out thos addd contributions that aris by xplicitly including pair corrlations. II. LINEARIZED EQUATION FOR THE PAIR CORRELATIONS Equation I.17 govrning th pair corrlation dnsity oprator is first cast into Wignr function for for th cntr of ass otion, raining an oprator in rlativ otion, copar Eqs. I.29 and I.37. This is writtn as 3066 J. Ch. Phys. 105 (8), 22 August /96/105(8)/3066/13/$ Arican Institut of Physics

2 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints 3067 f c12 P t 2 f c12il rl f c12 iv 12 f f 12 f c12 f c12, 1 whr f f 12 is th proprly transford product of fr dnsity oprators. Sinc it is th Wignr function for of th fr dnsity oprator, Eqs. I.24 and I.25, that has bn usd to xprss th local quilibriu and sall gradint bhaviour of th fr particls, th two particls can b localizd at diffrnt positions and a gradint xpansion of th product is appropriat. On carrying out this coputation f f 12 is givn by f f 12 R,P,t f lh f 12 f lnh f12 f lh f K rl /kt f K rl /kt f p 2kT op,r f op P2v f K rl /kt f n 2 f K CM /kt f 4kT f 3/2 r 3 4kT f K rl /kt f 1 2 : v f p op,r op lnt f whil th quilibriu for for th pair corrlation function is f c12 R,P,t n 2 f K CM /kt f 4kT f 3/2 r 3 UT f, on th basis that this is th for that would occur if th pair corrlations wr at coplt quilibriu with th fr particl otion through intractions with a third fr particl. Th particl onta p 1 and p 2 in th prturbation trs 1 and 2 ar to b intrprtd in trs of cntr of ass and rlativ onta according to 1 2 Pp op. Th corrlation quation 1 is now solvd within th philosophy of th Chapan Enskog procdur. Th ti drivativ in Eq. 1 is to scal as a gradint. Thus th lft hand sid of 1 is to b valuatd with th local quilibriu pair corrlation function I.29 in ordr to b linar in gradints iplying in particular that th ti drivativ is copltly xprssd in trs of th ti dpndnc of th paratrs spcifying th dnsity of corrlatd particls n c, thir an vlocity v c and thir local tpratur T c. Ths ar in turn obtaind fro th quations of chang drivd by taking th appropriat onts of Eq. 1. In writing down th Wignr function for th corrlations, Eqs. I.29 and I.37, th corrlation dnsity, tpratur and an vlocity hav all bn takn as indpndnt variabls. Sinc ths can diffr fro thir valus whn th corrlations ar in quilibriu with th frly oving particls, ths diffrncs dpnd on th nonunifority of th gas and thus scal with gradints as do th prturbations and. Rathr 2 3 than introducing sparat paratrs, it would b quivalnt to includ th diffrncs as part of th prturbation. But having introducd xplicit paratrs, it is ncssary to rcogniz that to avoid rdundancy, th prturbation ust not contribut to ths paratrs. As a consqunc th thr auxiliary conditions Tr rl f c l dp0, Tr rl Pf c l dp0, Tr rl P2v c 2 H 4 rl f l c dp0 on ar iposd. Ths ar alost th sa as th corrlation contributions to th gnral auxiliary conditions discussd in Sc. IV of Rf. 2. With ths conditions, intgrating Eq. 1 ovr P and tracing ovr th rlativ otion, an quation of chang for n c is obtaind 4 n c t n cv c n c n c. 5 Hr n c n f 2 r 3 (T f )Tr rl U(T f ) is th quilibriu dnsity of pair corrlations as dtrind by f. Th corrsponding quation for th vlocity v c is, fro th P/2 ont of Eq. 1 and aftr liinating a contribution fro n c /t, v c t v c v c 1 2n c P c n c v f v c, 6 n c with th corrlation prssur tnsor P c 1 2 Tr rl P2v c P2v c f c12 dp. Finally th quation of chang for th tpratur is obtaind fro th quation of chang for th corrlation nrgy corr CM rl 1 Tr n c rl P2v c 2 H 4 rl f c12 dp. 8 Th rat of th chang for this nrgy pr particl is corr v t c corr 1 P n c : v c 1 q c n c c c n c n c corr n c with corrlation nrgy hat flux corr 7, 9 q c Tr rl P2v c 2 P2v c 2 H 4 rl f c12 dp and corrlation nrgy production 10

3 3068 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints c 1 Tr rl p op V V p op f f 12 dp n 2 f 3 r T f Tr 2kT rl p op V V p op f K rl /kt fp op,r op : v f n f 2 Vdr rl v f. 11 Th approxiation is th rsult of using Eq. 2 and th subsqunt valuation of th trac. Equation 9 also involvs th quilibriu corrlation nrgy, which is obtaind fro Eqs. 3 and 8 as corr 3 2 kt f Tr rlh rl UT f. 12 Tr rl UT f At local quilibriu, th corrlation nrgy is dtrind solly by th tpratur T c so that with th corrlation hat capacity C corr corr /T c, th nrgy quation of chang is quivalnt to an quation of chang for T c. Eliinating th ti dpndnc in th corrlation quation by ans of th abov rsults and rtaining at ost linar in gradint trs, this quation can b writtn l f c12 P t 2 f n c12 l l c n c f c12 n c f l c12 L Tc lnt c L 2 vc : v c 2 L 0 vc v c n c l f n c12 c P2v c v fv c G 2 3 kt c 2 U 1 T c U Tc EU kt c corr corr C corr T c il rl f l c iv 12 f lh f ip2v f V 4kT 12 f lh f 12 p op,r op lnt f f i V 2kT 12 f lh f 12 p op,r op : v f f 1 f c12 n c f n c12 c l f c12 l Hr v c (2) is th sytric traclss part 12 of th scond rank tnsor v c, E U is th nrgy associatd with th Ursll oprator, s Eq. I.A11, and th dinsionlss cntr of ass ontu G (P2v c )/4kT c has bn introducd to siplify th notation. It is notd that thr is no tr involving th gradint of n c in this quation, consistnt with th notion that th prturbations ar du solly to tpratur and vlocity gradints. Th thr xpansion cofficints apparing in this quation ar L Tc G U1 T c U Tc EU kt c kt c G, 14 and L 2 vc 2G 2, L 0 vc k C corrg k C corr U 1 T c U Tc EU kt c G U 1 T c U Tc EU kt c n 2 fvdr rl. 16 n c C corr T c Consistnt with th gradint xpansions of th prturbations, Eqs. I.26 and I.38, all trs in Eq. 13 should b proportional to a gradint. Thos that ar not xplicitly of this for involv th diffrncs btwn th corrsponding corrlatd and fr paratrs, for xapl T c T f. Sinc ths diffr only bcaus of th spatial nonunifority of th gas, ths diffrncs ust b xprssibl in trs of gradints. As thr ar only th tpratur and vlocity gradints availabl, rotational sytry rquirs that ach diffrnc can only dpnd on on gradint, spcifically n c n c n c c v 0, T c T f T c v 0, v c v f D c lnt. 17 Th first cobination of quantitis that nds to b xplicitly xprssd in trs of th gradints is G f is th dinsionlss cntr of ass ontu using th fr particl tpratur T f il rl f l lh c12 iv 12 f f 12 iv 12 n c G 2 K rl /kt c 4kT c 3/2 Tr rl UT c n 2 f h 3 G 2 f K rl /kt f 2kT f 3 lh iv 12 f f 12 G K rle U kt c c v 0 P2v 0 D c lnt. kt Th othr cobination is f c12 n c n c f c12 l f c12 4 kt cg lnt D G U 1 T c U Tc EU kt c v c Subsqunt to aking this xpansion of th diffrnc btwn fr and corrlatd paratrs, thr is no nd for furthr distinction btwn ths paratrs so that, for siplification, all tpraturs T f T c T ar subsquntly xprssd by th sa sybol, as ar th an vlocitis v f v c v 0. With ths valuations, th linar in gradints for for th corrlation quation splits into thr sparat quations. This sparation rquirs th prturbations and to b

4 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints 3069 writtn as a cobination of th gradints, s Eqs. I.26 and I.38. First is th quation for th scalar ultiplying v 0, f c12 L 0 lh vc il rl f c12 C c12 iv 12 f f 12 G K rle U kt c i c 6kT V 12 f lh f 12 p op, r op iv 12 f lh f 12 C 1 C 2 1 f c12c c12. Scond is th quation for th vctor ultiplying lnt, f c12 i L Tc il rl f c12 A c12 G V 12 f lh f 12 p op,r op 4kT f iv 12 f lh f 12 A 1 A 2 id c 4 kt GV lh 12f f 12 1 f c12a c Lastly is th quation for th scond rank sytric traclss tnsor ultiplying v 0 (2), f c12 L 2 vc il rl f c12 B c12 i 2kT V 12 f lh 2 f 12 p op,r op iv 12 f lh f 12 B 1 B 2 1 f c12 B c Ths thr quations ar to dtrin th thr functions C c,a c and B c arising in th gradint xpansion of A c ln T c B c : v c 2 C c v c ; 23 s also Eq. I.38. This will b don by picking appropriat functional fors for ths functions and taking onts. It is to b notd that th ti drivativs of Eq. 1 wr liinatd by th us of onts associatd with th nubr, vlocity and nrgy avrags and ths rquird th auxiliary conditions 4. In trs of th thr functions C c,a c, and B c, ths auxiliary conditions bco and Tr rl f c l C c dp0, Tr rl G c l A c dp0, Tr rl K CM H rl f c l C c dp0. 24 It is asy to vrify that on taking appropriat onts of Eqs , ths auxiliary conditions ar consistnt with ths quations. A on ont approxiation to ach of th thr prturbation functions is now proposd. For this calculation it is assud that th appropriat onts ar proportional to th nrgy production fr parts of th corrsponding lft hand sid of Eqs Thus th for for C c is takn as C c 2 3 k C corrg k C corr U 1 TU T EU c ktc 0 25 with xpansion cofficint c 0 c. This satisfis th two rlvant auxiliary conditions of Eq. 24. Th ont of Eq. 20 is takn with rspct to 2 3 G H rl rl C rl T 26 to giv c c C rl T 3 Vdr r Tr rl U rl, 27 with th potntial factor arising fro th f lh f 12 p op, r op tr in Eq. 20. Th analogous choic for th A c ont satisfying Eq. 24 is A c G G U1 TU T EU kta c Taking th atrix lnt of Eq. 21 with G 2 5 H rl rl 2 kt G 29 givs a c 1 kt Vdr r Tr rl U2C rl 5kT rl. 30 Th obvious choic for B c is G (2) b c 0. Th xpansion cofficint is obtaind fro th G (2) atrix lnt of Eq. 22 to b b c 0 2. III. LINEARIZED BOLTZMANN EQUATION Th Boltzann quation I.2 is now cast into Wignr function for and linarizd about local quilibriu. Again ti drivativs ar to b liinatd by us of th quations of continuity, otion and nrgy, so thy ar to b tratd as linar in position gradint. Th collision tr in th Boltzann quation I.2 involvs th initial positions of th colliding pair, so that gradint corrctions can aris fro th collision tr bcaus th particls start at diffrnt positions. Thoas t al. 13 hav carrid out th gradint xpansion of th collision tr as wll as xprssing th rsult in phas spac rprsntation, s also th work of Barwinkl and Grossann. 14 It is convnint to writ this xpansion as th cobination J h J c J r, whr J h is th collision tr as if th syst was hoognous with all particl positions th sa and th othr two trs aris fro gradints in th local quilibriu Wignr functions. Th xpansion of th Boltzann quation thn can b writtn in phas spac rprsntation as f l f 1 f lh c1 p t f l f 1f lh c1 J h J c J r J n 31

5 3070 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints with all trs dtrind solly by th local quilibriu stat collctd togthr bfor th quality sign and th trs dpndnt on th prturbation aftr th quality sign. Th collision trs J h,j c and J r ar calculatd 13 using th local quilibriu fr dnsity oprator. Th purly local quilibriu contribution J h vanishs according to J h 8i2 3 2 dq dkqv,k f l f1 r,pqk,tf l f2 r,pqk,tk q in 2 6 h 3 Tr r K CM /kt V, H rl /kt in 2 6 h 3 Tr r K CM /kt K rl,u Th scond quality aks us of th assud lack of bound stats in that 1 whil th third quality dpnds on th idntity V, H rl /kt K rl,u. 33 Th corrction to th collision tr associatd with th displacnt of th cntr of ass fro th acroscopic position r is givn by J c 2h 3 dqdk q qqvk f l f1 r,pqk,tf l f2 r,pqk,tk qq qqkf l f1 r,pqk,tf l f 2 r,pqk,t k Vqq q 0 in 2 6 4h 3 Tr r K CM /kt r op,v, H rl /kt, 34 whr th drivativ with rspct to q has bn valuatd by considring a position rprsntation of th oprators and carrying out th drivativ to obtain th anticoutator r op,. Furthr siplification of this xprssion is accoplishd by using Eq. 33 and th oprator idntity A,B,C A,B,C B,A,C, in th for r op,v, H rl /kt r op,k rl,u 35 2 i p op,u K rl,r op,u. 36 On taking th trac, th collision tr J c can b writtn as J c n 2 6 h 3 Tr r K CM /kt p op U with J cn lnn f J cv : v 0 J ct lnt 37 J cn 2n 2 6 h 3 Tr r K CM /kt p op U, 38 J cv 2n 2 6 f r kth 3 Tr r K CM /kt p op pv 0 p op U, and J ct 3 2 J cn n 2 6 kth 3 Tr r K CM /kt p op K CM U kt 2 U T. Th xplicit xprssion for J r is J r 2h 3 dqdk k qv,kkf l f2 r,pq k,t f l f1 r,pqk,tf l f1 r,pq k,t f l f2 r,pqk,tkk q k in 2 6 2kTh 3 Tr r K CM /kt V, K rl /kt p op,r op : v 0 ln T pp opv 0 J rv : v 0 J rt ln T 41 with gradint corrctions having vlocity gradint cofficint J rv in 2 6 2kTh 3 Tr r K CM /kt V, K rl /kt p op,r op and tpratur gradint cofficint J rt in 2 6 2kTh 3 Tr r K CM /kt pp op v 0 V, K rl /kt p op,r op Finally th collision tr J n is writtn in trs of a linar rlaxation oprator J n r,p,tf f l R 8i2 3 2 dq dkqv,k f l f1 r,pqk,tf l f2 r,pqk,t 1 pqk 2 pqkk ] q 2in 2 6 h 4 Tr r K CM /kt V, K rl /kt This coplts th dscription of all th trs in th linarizd Boltzann quation 31. Th ti drivativ in Eq. 31 involvs both th fr and corrlatd local quilibriu Wignr functions which ar rspctivly paratrizd by n f,v f,t f and n c,v c,t c. Sinc

6 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints 3071 a ti drivativ is to scal as a position gradint in th Chapan Enskog thory for th calculation of transport cofficints, thn any dviation btwn th fr and pair corrlation sts of paratrs is proportional to a gradint and can b ignord whn stiating a ti drivativ. Thus both sts of paratrs can b rplacd by th fluid paratr st n, v 0 and T. Morovr, it is notd that th ti drivativ always occurs in th cobination /t p/ in th Boltzann quation. Th appropriat st of quations to b usd in valuating th ti drivativs is xaind first, thn th rsulting for of th Chapan Enskog quation is discussd. Fro th quation of continuity I.5, th rat of chang of th full dnsity is n t p n pv 0 nn v Th quation of otion I.6 givs, to trs linar in th gradints, v 0 t p v 0 pv 0 v 0 1 n P, 46 whr PnkT(1nB) is th local quilibriu prssur with n 2 f n 2 appropriat to kping only scond ordr in dnsity trs. For th local nrgy pr particl K V, nrgy consrvation, s Eqs. I.7 and I.8, iplis that to trs linar in th gradints t p pv 0 P n v Now it is T and n f that ntr xplicitly into th linarizd Boltzann quation. Th rat of chang of tpratur is obtaind fro th nrgy quation whil th quation for n f is drivd fro th rlation btwn n f and n, Eqs. I.39 and I.A15. Th tpratur quation is considrd first. According to Eq. I.43 th kintic nrgy dpnds on both tpratur T and dnsity n f, likwis th potntial nrgy ignoring gradint corrctions n V r,t 1 2Tr 12 V 12 r 1 r f 1 f n f 2 r 3 Tr rl V H rl /kt, 48 copar Eq. I.A8. By throdynaics, and consistnt with th rlations btwn th local quilibriu quantitis, th quilibriu nrgy pr particl dpnds on volu (1/n) and tpratur T according to dc v dt d1/n 1/n T C v dt P T P 1 n 2 dnc v dtkt db 2 dt dn, Tn 49 with constant volu hat capacity C v (/T) n, s Eq. I.A5. On cobining th quations of chang for nrgy and nubr dnsity, th quation T t p T pv 0 T T nc v P Tn v 0 50 for th tpratur is obtaind. In a siilar annr, th fr dnsity is also dpndnt on n and T through th quilibriu constraint I.39, so that n f t p n f 14nB n t p n 2n 2 db dt T t p T 14nB pv 0 n14nb 2n2 kt C v db n2n2 dt db dt v 0. pv 0 T 51 Sinc both th local quilibriu fr Wignr function as wll as f lh c1 ar paratrizd by n f,v 0, and T, th lattr through n c n 2 f 3 r Tr rl U, th abov st of quations is sufficint for valuating th ti drivativs apparing in Eq. 31. On carrying out th indicatd ti and spac drivativs in Eq. 31 and liinating th ti drivativs with th rlations in th last paragraph, th lft hand sid of this quation is forally linar in th gradints n, v 0 and T. Furthror it is a ncssary condition for consistncy with th quation of continuity that thr b no dpndnc on n. As an aid to writing down th dtaild xprssions involving th gradints and to prov th lack of dpndnc on n, th individual contributions ar organizd as follows: d f l f 1 f lh c1 F n d ln n f F v dv 0 F T d ln T 52 with drivativs with rspct to ln n f,v 0 and ln T givn by and F n f l f 1 2f lh c1, F v pv 0 kt 53 f l f1 2n 2 6 kth 3 Tr r p op pv 0 K CM /kt U 54 F T W 2 2 /kt 3 f l f 1 n 2 6 h 3 Tr r K K CM CM kt 3 U TU T, 55 whrin th dinsionlss ontu 3 W(p v 0 )/2kT has bn introducd. With this brakdown of th various contributions, th dpndnc of th lft hand sid of Eq. 31 on th gradints can b xplicitly obtaind. But as statd arlir, thr should b no dpndnc on n. That this is th cas is now donstratd. Dpndnc on n ariss fro th ti and spac drivativs of n f and also through th prssur gradint associatd with th ti drivativ of v 0. Collcting ths contributions togthr givs as th cofficint of n,

7 3072 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints pv 0 n f 14nBF n n 1 P nt F v 1 n f 14nBJ cn 12nB n pv 0F n ktf v J cn 0, 56 l to trs linar in n. Th trs in F n and F v involving f f 1 clarly cancl and on xaining th various contributions fro pair corrlations ths ar also asily shown to cancl. On collcting th various contributions, th linarizd Boltzann quation 31 can b writtn in th for L T ln T L v 2 : v 0 2 L v 0 v 0 R, 57 whr 12 v 0 (2) is th sytric traclss part of th scond rank tnsor v 0. Th gnral wight factor f l f 1 has bn rovd so that th linarizd Boltzann quation can b tratd as an oprator quation in a Hilbrt spac with innr product AB Af l f1 r,p,tbdp. 58 In th gnral cas th cofficints of th gnralizd forcs ar givn xplicitly by rtaining only trs up to scond ordr in th dnsity f l f 1 L T F T 2n 2 T db F n dt n f pv 0 W 2 5 db nbnt 2 dt pv 0 l pv L 2 0 v F v f f 1 F kt v 1nBnT db f l f1 J rt n2 6 r dtj cn 2n 2 n f T db dt J ctj rt h 3 Tr r p oppv K CM /kt 0 J cv J rv 2 2W 1 2 f l f1 2n2 6 r kth 3 Tr r K CM /kt p op pv 0 2 U K CM kt 5 UTU T, in 2 6 2kTh 3 Tr r K CM /kt V, K rl /kt p op,r op 2, whr irrducibl Cartsian tnsors such as W (2) ar dfind in Rf. 12, and l L 0 v 1 3 v F pv 0 f f 1 kf l f1 C v F n n n14nb 2n2 kt db f C v dt k F db C T1nBnT v dt 1 3 U: J cv J rv 2C v db 1nBnT 3k dtw 2 23nB2nT 3 db in2 r 6 6kTh 3 Tr r K CM /kt V, K rl /kt p op, r op. 6 dt n2 k r C v h 3 Tr r K CM /kt TU T This coplts th idntification of th trs in th Chapan Enskog quation 57. Th linar quation 57 is to b solvd for th prturbation. According to th Frdhol altrnativ, 15,16 a solution xists only if th givn function lft hand sid of th quation is orthogonal to th lft invariants lft ignvctors having zro ignvalu of R, and to ak th solution uniqu, so condition on th prturbation is rquird for ach right invariant. Just as for th standard Boltzann quation, it is first vrifid that th lft and right invariants ar th ass, ontu and kintic nrgy. It follows asily that th lft hand sid of Eq. 57 is orthogonal to ths invariants and th standard Chapan Enskog thod rquirs that th prturbation dos not contribut to th ass, ontu or nrgy dnsitis, s Eqs. I.40, I.42 and I.45 togthr with Eqs. 4, thus aking th solution uniqu. In ordr to vrify th invariants of R it is appropriat to bgin by studying atrix lnts of th linar suproprator R. Fro th dfinition of innr product 58, th contribution to th collisional rat of chang of obsrvabl associatd with prturbation is R in2 r 6 2h 3 dp Tr r 1 2 K CM /kt V, K rl /kt Hr a sytrization of btwn particls has bn carrid out. Sinc V couts with th cntr of ass otion, th coutator can b changd into th corrsponding coutator with th cobination of s, naly 1 2,V. For ass and linar ontu this coutator idiatly vanishs, thus ths four quantitis ar lft invariants of R. For qual to th kintic nrgy H (1), th cobination of s is qual to th su of cntr of ass and rlativ kintic nrgis. Th forr couts with V so can not contribut. Th rlativ kintic nrgy contribution can b writtn in th following for:

8 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints 3073 dp Tr r K rl K CM /kt V,X dp Tr r K CM /kt K rl,v X dp Tr r K CM /kt H rl,v X dp Tr r K CM /kt VH rl,x dp Tr r K CM /kt VK rl,x. 63 Thus, if X couts with K rl, quivalntly th prturbation bing a function solly of th particls ontu, thn th kintic nrgy is a lft invariant of R. It should b phasizd that any position dpndnc of is to b tratd as a paratr for th linarizd collision oprator, thus a constant as far as th collision is concrnd. Collision nonlocality ffcts, which ar in particular rsponsibl for convrsion btwn kintic and potntial nrgy, hav alrady bn accountd for by th prsnc of J c and J r in Eq. 31. Thus th kintic nrgy provids a fifth lft invariant for th linarizd Boltzann quation. Th idntification of right invariants rquirs looking at which s giv a zro contribution to th atrix lnt 62. For ach of ass, ontu and kintic nrgy, th su of s is ithr constant, th cntr of ass ontu or th su of cntr of ass and rlativ kintic nrgis. In all cass, th Mo llr oprators act to chang th cobination K rl /kt ( 1 2 ) into a function of cntr of ass ontu and total rlativ (H rl ) nrgy oprator. Such a quantity couts with H rl, so that for ach of ass, ontu and kintic nrgy, th R atrix lnt can b rplacd by R in2 r 6 2h 3 dp Tr r 1 2 K CM /kt K rl, K rl /kt This vanishs providd is a function of ontu, sinc th cobination of s thn couts with K rl. Thus, sinc th collision oprator R is localizd at on position and acts only to transfor th ontu dpndnc of th Wignr function, its lft and right invariants ar th standard ass, ontu and kintic nrgy. Th possibility of a collision inhoognity coplicats this idntification. In this work this probl is ovrco by xplicitly xpanding th inhoognity ffcts and trating th in a diffrnt annr. As statd arlir, th solvability of th linarizd Boltzann quation 57 rquirs that th lft hand sid b orthogonal to th lft invariants of R. Ths rquirnts ar now discussd. Sinc th quations of chang for n and v 0 hav bn basd on th onts of Eq. 31, ths quations ar xactly th rquirnts that th lft hand sid of Eq. 57 b orthogonal to th lft invariants. Dtaild valuation of th orthogonality conditions using th cofficints L T, L (2) v and L (0) v confir ths proprtis. Th quation of chang for th kintic nrgy was not usd in th liination of th ti drivativs to arriv at Eq. 57 so th orthogonality of th lft hand sid of Eq. 57 to th kintic nrgy nds to b xplicitly xaind. For th kintic nrgy only th L (0) v orthogonality is not autoatically satisfid by rotational invarianc. Th idiat intgral of this orthogonality condition lads to W 2 1 f l f 1 L 0 v dp 1 3n fk 2C v 2C v db 14nB3nT 3k dt n2 3 kt r Tr 2C v rl 3 2 K rl kt U T n2 r 3 12kT 2 Tr rl p op, V K rl /kt p op, r op. 65 Aftr aking us of th xplicit for for B(T), Eq. I.A2, and rtaining at ost trs of scond ordr in th dnsity, this is siplifid to W 2 1 f l f 1 L 0 v dp 1 n2 3 r 3kT 2 Tr rl VH rl H rl /kt n2 r 3 12kT 2 Tr rl p op, V K rl /kt p op, r op. 66 Th idntity of Appndix A shows that this cobination vanishs, thus vrifying that all of th orthogonality conditions ar satisfid. Thus a solution to th linarizd Boltzann quation xists. Th prsnc of right invariants for th collision oprator R iplis that th solution to Eq. 57 is not uniqu. Uniqunss is obtaind fro th cobination of auxiliary conditions I.40, I.42, I.45 and 4. Th auxiliary conditions 4 for th pair corrlation Wignr function iply that th auxiliary conditions for rduc to th rquirnts that b orthogonal to ass, ontu and kintic nrgy, naly th right invariants of R. For th ass, this is an idiat consqunc of Eqs. 4 and I.40. For th ontu, th scond quation in th st of Eq. 4 dos not liinat th total contribution of to Eq. I.42. But th spcific for chosn for only th vctor part, A c, could contribut to a ontu xpctation valu, s Eq. 28, givs no contribution to Eq. I.42. Finally, to obtain th condition for th kintic nrgy fro Eq. I.45, it is rcognizd that th indpndnt particl contribution to th potntial nrgy V 12 f l f 1 f l f2 1 2 dr 12 dp 1 dp 2 67 vanishs bcaus of th sparat auxiliary conditions I.40 for 1 and 2 aftr subtracting th appropriat corrlation condition of Eq. 4.

9 3074 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints Th auxiliary conditions iply crtain constraints on th possibl fors for A, B and C, in th gradint xpansion I.26 of. In particular th vctor A is affctd only by th ontu constraint, whil th scalar C is affctd by both th nubr dnsity and nrgy constraint. Ths constraints ar idntical to thos apparing in dilut gas kintic thory, 3 or or coparably to thos for a prvious classical tratnt 17 of odratly dns gas kintic thory. As in that tratnt, only th lowst ordr Sonin polynoials that ar ncssary to approxiat th prturbation will b rtaind in this work. Thus th xpansion functions ar approxiatd by Aa 1 W 5 2 W 2, Bb 0 W 2, Cc W W 4, 68 having unknown cofficints a 1, b 0 and c 2. Th Chapan Enskog quation 57 togthr with th for of th prturbation function I.26 can b split into thr quations according to th thr diffrnt gradints. This givs sparat quations for ach of A, B and C. Th agnitud a 1 of AAW is found by taking th coponnt of th A quation along th A dirction, thus X dpw 2 5 2W f f l L T a 1 WW RWW n f 2 a 1 v S, 69 whr v 16kT/ is th an vlocity of rlativ otion and S is th appropriat kintic cross sction for thral conductivity. Th driving tr X can b partially valuatd. Aftr so siplification, this can b writtn as, nglcting trs of ordr n 3, X kT n f 5n 2 3 kt 4 2 rl Tr TU T U in kT Tr rl 2 U b 0, 2GG,V : K rl /kt p op,r op. 70 Th quation for B rducs to an quation for th scalar X dpw 2 : L v 2 f f l b 0 W 2 :RW n f 2 b 0 v S Sinc th polarization is th sa hr as in th dilut gas cas, th cross sction for viscosity is writtn in standard 7 notation as S(20). For th dtrination of b 0, thr rains only th rduction of X to siplr for. This is accoplishd in th sa annr as prvious rductions, to giv X 5n f 1n f B n 2 3 f r 4kT 2 Tr rl p, V 2 : K rl /kt p op,r op Finally, for th calculation of c 2, thr is th dtrining quation X dp W W 4 f f l L v 0 c W W 4 R W W c 2 n f 2 v S. 73 Th driving tr to ordr n 2 is calculatd to b X n rl Tr TU T in kT Tr rl 4 5 2,V K rl /kt p op, r op. 74 Hr p op /kt is th dinsionlss rlativ ontu oprator. This coplts a listing and rduction of all th quantitis for a first ordr approxiation of th prturbation xpansion cofficints a 1,b 0 and c 2. IV. TRANSPORT COEFFICIENTS Dnsity oprator xprssions for th hat flux and prssur tnsor ar givn in Sc. IV of Rf. 2. Spcifically th prssur tnsor ariss by ans of kintic and collisional transfr chaniss whil th hat flux has th additional contribution of a potntial nrgy flux. As wll, th kintic contributions involv both fr and corrlatd pair particl ffcts. Fro th standard dilut gas kintic thory point of viw, fluxs ar xpandd in powrs of th spatial inhoognity. For a locally hoognous syst thr is no hat flux but thr is a local quilibriu prssur. It is th objct of th prsnt sction to obtain xprssions for ths quantitis. Th prsnt stiats ar ad assuing thr is only on tpratur T f T c T and stra vlocity v f v c v 0, with th corrlation dnsity n c n f 2 r 3 Tr rl U paratrizing th local quilibriu with th fr particl dnsity n f. This appars appropriat sinc it is noticd fro Sc. II that all conditions can b satisfid with ths qualitis, quivalntly that thr ar no constraints rquiring diffrnt tpraturs or an vlocitis for th fr and pair corrlations, or out of quilibriu pair corrlation dnsity. Th kintic prssur tnsor is calculatd according to

10 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints 3075 P 2kT K WWf dp 1 nkt 1 3 n 2 fkt 1 U 2n f kt WW W 2 3/2 dw 2n 2 f kt r 3 G 2 dgtr rl 3/2 W 1 W 1 U c P K U2 K v 0 2 K v 0 U, 75 to giv contributions K and K to th shar and bulk viscosity cofficints. On idntifying th cofficints of v 0 (2) and v 0 and siplifying th rsulting xprssions, th kintic contributions to th shar and bulk viscosity cofficints bco and K 1 2 ktn f b n c b 0 c 76 K 2n c cktc 0 C V 3C U. 77 corr Hr th hat capacity C V U d V U /dt associatd with th potntial nrgy V U 2Tr 1 rl VU/Tr rl U pr particl for th pair corrlations has bn introducd. Th collisional transfr part of th prssur tnsor rducs to P coll P virial U2 coll v 0 2 coll v 0 U 78 with virial prssur P virial n 2 f kt(b 1 /3), shar viscosity coll n 2 3 f r 40kT Tr rlr op V 2 : K rl /kt p op,r op n 2 3 f r b 0 20 Tr rlr op V 2 : K rl /kt 2, 79 and bulk viscosity coll n 2 3 f r 36kT Tr rlr op V K rl /kt p op, r op n 2 3 f r 24 /kt c 2 Tr rl r op V H 2 H rl rl kt 2 5 H rl kt Th kintic hat flux vctor is q 2kT K kt dpww 2 f l f 2kT kt dp Tr rl W 1 W 2 1 f l c12 c12 K T with kintic contribution to th thral conductivity K 5 4 n fk 2kT a n ck kt a 1 c1 4 3 whil th collisional hat flux q coll coll T Tr rl K rl U T ktr rl U, has thral conductivity contribution coll n 2 3 f r 24T Tr rlr op V: K rl /kt p op,r op ka 1 22kTT dp virial dt P virial n 2 3 f r 3 Tr rlr op V: K rl /kt GG. 84 Finally, th potntial nrgy hat flux q V 1 2 n 2 f r 3 kt rl Tr dg G 2 3/2 G V K rl /kt 1 2 U c12 V T has thral conductivity contribution V 5 12 n 2 f 3 r a 1 /kt k 2T Tr rlv K K rl rl kt c n kt C U V a c Svral diffrnt contributions to th transport cofficints hav bn found. Collctd togthr, th shar viscosity is K coll, 87 with dtaild forula givn by Eqs. 76 and 79. For th bulk viscosity, K coll, 88 with forula Eqs. 77 and 80. Finally, for th thral conductivity, K coll V, 89 with forula fro Eqs. 82, 84 and 86. V. DISCUSSION A rcntly proposd 1 dnsity corrctd quantu Boltzann quation phasizs th distinction btwn fr particl and pair corrlation ffcts. Th singlt dnsity oprator is influncd by both ffcts. Th fr dnsity oprator is dtrind by solving th quantu Boltzann quation whil a sparat quation I.17 basd on th scond BBGKY quation has bn introducd 2 for th dtrination of th oprator dscribing th pair corrlations. In unpublishd work, a foral solution of that quation involving a long ti liit, spcifically a gnralization of Eq. I.3, was usd for th pair corrlations but had th consqunc that th drivd xprssions for th transport cofficints did not rduc at low dnsity to thos associatd with th solution of th dilut gas Boltzann quation. Thus it was considrd that an altrnat thod for solving quation I.17 for th pair corrlations was ndd. Sction II has cast this quation into Wignr function for, linarizd it about local quilib-

11 3076 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints riu and solvd it in a Chapan Enskog annr. Th quantu Boltzann quation I.2 was tratd in th sa annr in Sction III. In gnral ths quations would hav to b solvd siultanously but using th siplst for for th prturbation functions, th quations uncoupl and can b solvd indpndntly. Exprssions for th transport cofficints consistnt with ths solutions wr givn in Sction IV. A coparison of ths xprssions with th classical chanical rsults 17,18 obtaind fro solving th classical Boltzann quation with nonlocal collision corrctions first obtaind by Grn 19 and Bogoliubov 20 is givn in th following paragraphs. Rainwatr 23 has also considrd th rduction of th quantu foralis to th classical. His starting point is th quantu tratnt by Thoas and Snidr 13 so it dos not hav th pair corrlation ffcts as prsntd in this work. Th classical tratnt 17,18 had no sparat quations for th pair corrlations so all transport cofficints in that work ar xprssd in trs of th solution of th Boltzann quation. In both tratnts th solution of th Boltzann quation is paratrizd by th thr xpansion cofficints a 1, b 0 and c 2. Rduction of th quantu xprssions to thir classical liit givs xact agrnt with th classical rsult. It should b statd that any of th classical rsults ar partly xprssd in trs of th scond virial cofficint B(T) and its tpratur drivativs. To arriv at such xprssions, it was assud that th introlcular potntial supportd no bound stats and is orovr onotonically rpulsiv, so this also is an input into th coparison. Thus th prsnt solution of th quantu Boltzann quation is th quantu analog of th classical solution. This is rasonabl sinc th two quations xprss th sa ssntial faturs. Whr thy ight hav disagrd would b in th prsnc of th pair corrlations. But ths only appar in th quantu Chapan Enskog quation 57 through thir local quilibriu contribution to th singlt Wignr function, which ssntially rducs classically to part of th local Maxwllian distribution. As a consqunc, no diffrnc ariss whn th quantu solution is rducd to th classical. Coparison of th xprssions for th transport cofficints is not so straightforward. Exprssions 77 and 80 for th bulk viscosity ar asist to copar. Th classical tratnt has no c 0 c so thr is no kintic contribution to th bulk viscosity for sphrical particls. Morovr, th quantu collisional contribution to th bulk viscosity rducs in th classical liit to th rsult in Rf. 18. Sinc th bulk viscosity asurs th transfr of nrgy fro on for of nrgy to anothr, th prsnc of pair corrlations provids a possibility for such nrgy xchang whil th classical tratnt had no such possibility. Thus th diffrnc in K is a rasonabl rsult. Th prsnt tratnt has kintic contributions to th shar viscosity, Eq. 76, fro both th fr dnsity oprator Boltzann quation and th pair corrlations whil th classical tratnt has no xtra pair corrlation structur. Siply stting b 0 c 0 dos not lad to agrnt sinc in th prsnt tratnt b 0 is ultiplid by th fr dnsity whil in th classical cas th full gas dnsity appars. Agrnt is obtaind if it is assud that b 0 c 2b 0, with a possibl rational that th pair corrlations involv th ffct of two particls, thus th prturbation should b twic as big as th fr. Th quantu xprssion for th collisional contribution to th shar viscosity rducs to th classical xprssion givn in Rf. 18 xcpt that th xprssion for H R of that work should hav a factor of 2 ultiplying th Y contribution to this quantity. This issing 2 is an rror in that papr. Sinc its contribution to th shar viscosity and thral conductivity is scond ordr in th gas dnsity, this factor appars to not yt hav bn nurically usd in th calculation of th transport cofficints, for xapl, in Rfs. 21 and 23. Finally is th coparison of xprssions for th thral conductivity. Ths ar quit diffrnt in structur. First is th kintic contribution K. This can b ad to agr with th classical xprssion if a 1 c 2a 1, copar th discussion of th shar viscosity, togthr with ignoring th corrction factor in Eq. 82 involving U T. Th classical liit of Eq. 84 givs, in th notation of Rf. 18, coll class 1 3 n 2 f k 3 kth R 1 k 2kT 4 n2 a 1T db dt 43 N Not th rror in H R ntiond abov. This contains part of th dnsity corrctions in Eq. 26 of Rf. 18 for th thral conductivity. A dirct classical rduction of Eq. 86 givs zro for th first tr and th scond tr dos not corrspond to trs found in th classical tratnt. Th ain nw lnt hr is that th potntial nrgy cannot b drivd fro th Boltzann quation but involvs a sparat calculation using th pair dnsity oprator classically th pair distribution function. In th classical tratnt th pair distribution function was calculatd using th Grn- Bogoliubov procdur of following a pair of fr particls into th collision rgi, just as was don for th binary collision tr. Th quantu analog is to st th pair dnsity oprator qual to 2 12 f 1 f 2, 91 copar th collision tr in th Boltzann quation I.2. Within a linar-in-gradints valuation of th potntial hat flux, th localization of th pair of intracting particls can b placd at th cntr of ass of th pair, s th first two lins of Eq. I.50. Insrting th abov xprssion for th pair dnsity oprator into th potntial hat flux quation givs contributions both fro th prturbation functions 1 2 and fro th diffrnt localizations of th fr singlt dnsity oprators. Aftr taking th classical liit, th lattr givs th I tr in Eq. 26 of Rf. 18 whil th forr can b copltly xprssd in trs of th scond virial cofficint for onotonic rpulsiv potntials, thus V class 5n 2 f a 1 k kt 6 22T db d2 B 2 dt T2 dt

12 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints 3077 n2 k 4 3 kt I. 92 ACKNOWLEDGMENTS This work was supportd in part by th Natural Scincs and Enginring Rsarch Council of Canada. J.G.M. acknowldgs support by Gobirno Autónoo d Canarias Spain Grant No. 92/077 and Ministrio d Educación y Cincia Spain PB With this chang in intrprtation of how to stiat th potntial hat flux, th cobination K coll V rducs to th classical xprssion for th thral conductivity. Not that T is givn with an ovrall wrong sign, in Eq. 30 of Rf. 18, which was first notd by Bnntt and Curtiss. 21 Fro th abov discussion it is sn that th prsnt tratnt of th kintic thory of a odratly dns quantu gas has siilaritis to th prvious classical tratnt. Th xplicit nd in quantu chanics to account for pair corrlation corrctions to th singlt dnsity oprator vanishs in th classical liit and all othr quantitis in th quantu Boltzann quation hav classical analogs. It is in th valuation of xpctation valus whr th distinction btwn th tratnts ariss. In th prsnt quantu tratnt, th Boltzann quation dtrins only th non local-quilibriu part of th fr dnsity oprator, so that it is ncssary, vn for on particl obsrvabls, to calculat th nonquilibriu part of th pair corrlations so that thir contribution to all xpctation valus ay b obtaind. Th prsnt tratnt has considrd ths pair corrlation ffcts to b sowhat diffusiv, long rangd ffcts rathr than as hard, short rangd collision typ ffcts. Thus a sparat quation for th pair corrlations has bn proposd, and solvd within th Chapan Enskog approach. In contrast, th arlir classical tratnt usd th sa collisional ansatz, th classical liit of Eq. 91, for th full pair distribution function as for th collision tr in th Boltzann quation, which rflcts hard, prcussiv collisions. But in nding this discussion, it is phasizd that th two approachs ar not all that diffrnt. Whil it is th linarization, Eq. 13, of th pair corrlation quation I.17 within th Chapan Enskog procdur that is usd in this work, if th thr particl dcay ffcts ar ignord, thn it is asy to intgrat th pair corrlation quation to giv Eq. I.3, s Appndix B, a rsult consistnt with Eq. 91. This approach was not followd in th prsnt tratnt sinc it was found that th rsulting pair corrlation dnsity oprator was unboundd if th fr singlt dnsity oprator is paratrizd with a position dpndnt tpratur and stra vlocity. It is rarkd that th classical kintic thory of odratly dns gass has bn xtndd in a nubr of ways so it is not as liitd as th abov discussion ay iply. Th gnralization to includ non-onotonic introlcular potntials has bn carrid out by Rainwatr. 9 In coopration with co-workrs 22 h has also approxiatly incorporatd th rol of thr particl collisions and th prsnc of bound stats. Ths paprs should b rfrrd to for a dtaild discussion of th thods and litratur of th approachs that thy us. APPENDIX A: A TRACE IDENTITY It is to b shown that in th absnc of bound stats Z 1 4 Tr rlp op, V K rl /kt p op, r op Tr rl VH H rl /kt. A1 Th first anticoutator is idntifid as th rsult of th coutator K rl,v i p op V V p op, A2 which can b furthr writtn as H rl,v sinc th coutator of V with itslf vanishs. Insrting this rlation into Eq. A1 and using th cyclic invarianc of th trac to flip th coutator givs Z i 4 Tr rlvh rl, K rl /kt p op, r op i 4 Tr rlvk rl, K rl /kt p op, r op, A3 with th scond quality arising fro th intrtwining rlation whn thr ar no bound stats. Now th K rl coutator affcts only r op according to K rl,r op 2 i p op. Z now rducs to Z 1 2 Tr rl V K rl /kt 2K rl. A4 A5 Finally th intrtwining rlation changs any function of K rl into th corrsponding function of H rl. Th unitarity of in th absnc of bound stats iplis that 1 rsulting in th quality that was to b provn. APPENDIX B: COLLISIONAL INTEGRATION OF THE PAIR CORRELATION EQUATION Without th rlaxation tr, Eq. I.17 taks th for i c12 L 2 t 12 c12 V 12 f 1 f 2. This can idiatly b forally intgratd to c12 t il 2 12 tt0 c12 t 0 t 2 i il 12 tt V 12 f 1 t f 2 tdt. t0 B1 B2 If th ti t 0 is takn so that th pair of particls ar far apart, bfor intraction, thn it is appropriat to st

13 3078 Snidr, Wi, and Muga: Gas quantu kintic thory: Transport cofficints c12 (t 0 )0. Morovr, if th prsnc of othr particls is ignord, thn th fr dnsity oprators volv frly ovr th ti intrval t 0 t, that is, f 1 t il 1 1 tt f 1 t. Thn th intgral can b carrid out c12 t f1 t f2 t il 2 12 tt0 B3 il L2 t0 t f 1 t f 2 t. B4 Finally, if th ti intrval tt 0 can b takn larg copard to th ti of duration of a collision, th product of volution suproprators can b approxiatd by th Mo llr suproprator and Eq. I.3 is obtaind. 1 R. F. Snidr, J. Stat. Phys. 61, R. F. Snidr, G. W. Wi, and J. G. Muga, J. Ch. Phys. 105, , prcding papr. Equations in this papr ar lablld as I.. 3 J. O. Hirschfldr, C. F. Curtiss, and R. B. Bird, Molcular Thory of Gass and Liquids Wily, Nw York, S. Chapan and T. G. Cowling, Th Mathatical Thory of Non- Unifor Gass Cabridg Univrsity Prss, Cabridg, J. H. Frzigr and H. G. Kapr, Mathatical Thory of Transport Procsss in Gass North-Holland, Astrda, J. A. McLnnan, Introduction to Nonquilibriu Statistical Mchanics Prntic-Hall, Englwood Cliffs, F. R. W. McCourt, J. J. M. Bnakkr, W. E. Köhlr, and I. Kuščr, Nonquilibriu Phnona in Polyatoic Gass Oxford Univrsity Prss, Oxford, D. E. Stogryn and J. O. Hirschfldr, J. Ch. Phys. 31, 1531, ; 33, J. C. Rainwatr, J. Ch. Phys. 81, D. K. Hoffan and C. F. Curtiss, Phys. Fluids 7, ; 8, 667, G. W. Wi, Ph.D. dissrtation, Univrsity of British Colubia, J. A. R. Coop and R. F. Snidr, J. Math. Phys. 11, S also F. M. Chn, H. Moraal, and R. F. Snidr, J. Ch. Phys. 57, , Appndix A. 13 M. W. Thoas and R. F. Snidr, J. Stat. Phys. 2, Th gradint corrctions to th collision oprator that ar lablld in this rfrnc as J 11 and J 12 ar lablld in th prsnt papr as J c and J r. 14 K. Barwinkl and S. Grossann, Z. Physik 198, ; K. Barwinkl, Z. Naturforsch. Til A 24, 22, F. Risz and B. Nagy, Functional Analysis Ungar, Nw York, T. Kato, Prturbation Thory for Linar Oprators Springr, Nw York, R. F. Snidr and C. F. Curtiss, Phys. Fluids 1, R. F. Snidr and F. R. McCourt, Phys. Fluids 6, H. S. Grn, Molcular Thory of Fluids North Holland, Astrda, N. Bogoliubov, J. Phys. U.S.S.R. 10, D. E. Bnntt and C. F. Curtiss, J. Ch. Phys. 51, D. G. Frind and J. C. Rainwatr, Ch. Phys. Ltt. 107, ; J.C. Rainwatr and D. G. Frind, Phys. Rv. A 36, J. C. Rainwatr, in Rarfid Gas Dynaics, ditd by J. Harvy and G. Lord Oxford Univrsity Prss, Oxford, 1995, p. 114.