Analytical and semi-analytical solutions to the kinetic equation with Coulomb collision term and a monoenergetic source function

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1 Aalytical ad mi-aalytical oltio to th kitic qatio with Colomb colliio trm ad a moorgtic orc fctio P R Gocharov Sait-Ptrbrg Polytchic Uivrity, 955, Ria ad RRC "Krchatov Ititt", Mocow, 8, Ria gocharov@phtf.t.va.r Abtract. Complt ad phyically adqat aalytical ad mi-aalytical oltio hav b obtaid ig a practical dimiol form of kitic qatio amig azimthal ymmtry ad Maxwllia ditribtio of targt plama pci. Formrly coidrd implifid qatio with trcatd Colomb colliio trm do ot corv th mbr of particl, ar iapplicabl to dcrib high rgy ditribtio tail, ad ar alo tially abl to dmotrat th Maxwllizatio proc atrally obrvd i th low rgy rgio of corrct ditribtio. Th rlt may b fl i mrical modlig ad i xprimtal data aalyi, pcially cocrig clar proc ad advacd localizd, agl-rolvd prathrmal particl diagotic.. Itrodctio Lt f ( v ) b th oght vlocity ditribtio fctio of tt particl of typ, ad f ( ) v b th kow vlocity ditribtio fctio of targt plama pci cotd by idx. Fctio f ( v ) ad f ( ) v ar ormalizd to ity. To obtai a practical form of th qatio to b olvd, w Colomb colliio trm [] xprd via partial pottial fctio [,] corrpodig to th itractio of tt particl with particlar pci of targt plama f Φ = 4π ( v ) d v, () Ψ = f ( ) d 8π v v, () whr dot th dity of particl, ad = v v dot th magitd of th rlativ vlocity of particl ad. A how i [4], amptio of azimthal ymmtry (i.. = ) ϕ

2 ad agl iotropy of ditribtio fctio f ( v ) of targt plama pci lad to th followig xprio for th partial colliio trm i phrical polar coordiat i vlocity pac: ( f ) L iϑ ( f ) L m Φ Ψ Ψ C = v ( f ), () v v m v v v viϑ ϑ v v ϑ whr m ad m ar th ma of particl of pci ad, rpctivly, dot th dity of particl, L ( 4π Z Z ) Λ =, (4) m Z ad Z ar th lctric charg mbr of particl of pci ad, rpctivly, i th lmtary charg, Λ i Colomb logarithm. Th fll colliio trm i C = C. (5). Dimiol otatio W follow th dimiol approach of moograph [4], ig a lightly diffrt otatio. amly, w do ot itrodc th ijctio vlocity ito th xprio for th colliio trm, ic thi i a xtral paramtr, ad it i mor atral to rtai it i th tt particl orc fctio oly. Ulik [4], i or otatio a mall factor ( m ) / m <., wh pci ar io, appar atrally i th vlocity diffio trm withot itrodcig th ratio of th lctro tmpratr to th tt particl ijctio rgy. Dfiig gralizd tmpratr for all targt plama pci m v m v T = = f v v dv, (6) + ( ) 4π thr partial (i.. corrpodig to th particlar pci ) dimiol fctio 4π m a ( v) = v T 4π b ( v) = v Ψ Φ v v, (7), (8) 4π T Ψ c ( v) =, (9) m v v

3 ad th dimioal cotat v c [cm/] ad τ [] whr v = m T c m m / m τ = Zω p Λm i th lctro plama frqcy, ad a dimiol paramtr, (), () 4π ω p = () m m ε = m w th writ thr dimiol fctio mmd ovr all pci /, () m Z T a( v) = ε a ( v), (4) m T m Z b( v) = b ( v), (5) m v c Z c( v) = c ( v), (6) T m ad, fially, th colliio trm qivalt to (5) i th form ( f ) ( f ) v c a( v) c( v) C = v ( ) c + b v ( f ) + iϑ. (7) τ v v v v iϑ ϑ ϑ For th particlar ca wh all targt plama pci ar Maxwllia th drivativ Φ, v Ψ v, ad xprd via Chadrakhar fctio / m v T m f ( v ) = (8) πt Ψ wr calclatd i [4], ad fctio (7)-(9) wr v a follow: z x z (9) z π z G( z) = x dx = rf ( z) π z ( ) ( ) υ ( υ ) a v = b v = G, () υ c ( v) = + υ G( υ ), () π υ

4 whr ot, that υ =, v = T m. () v vt v T a ( v), a ( v) ; () c ( v), c v ( v). (4) v π To compt (), () i th viciity of v = it i fl to apply th dcompoitio giv i [5] v k k + z z z 4 5 rf ( z) = = z + z + z (5) π k = (k + )!! π 5 Fctio b( v ) i rlatd to th dyamic frictio forc ad i rpoibl for th lowig-dow proc. Fctio a( v ) ad c( v ) ar both rlatd to th diffio tor i vlocity pac. Th trm with c( v ) i (7) cotai oly th agl drivativ ad i rpoibl for th pitch agl cattrig. Th trm with a( v ) dcrib th vlocity diffio proc. For a iothrmal Maxwllia plama a( v) = εb( v), ad ε i a mall paramtr wh th tt particl ar igificatly havir tha lctro, whil for lctro a( v) = b( v). To calclat th colliio trm (7) for vlociti mch gratr tha thrmal vlociti of targt plama io v = T m ad mch mallr tha th thrmal vlocity of targt plama lctro T Ti i i v = T m, i.. for v v v, th followig implifid formla may b d itad of Ti applyig (4)-(6) ad (), (): whr T ( a) m 4 a( v) = ε Z + v v m π ( ) T, (6) ( b) m 4 b( v) = Z + ( v v ) T, (7) m π ff c( v) = Z v + π v, (8) T Z m Z T =, (9) T m ( a) i i i Z i i m Z =, () m ( b) i i Z ff i i = Zi i, () i 4

5 ad th mmatio i (9)-() i ovr all io pci of th targt plama. Th firt trm i righthad id of xprio (6)-(8) rprt th cotribtio of targt plama io, ad th cod trm rprt th cotribtio of targt plama lctro. Io ad lctro cotribtio to th implifid lowig-dow trm govrd by (7) ar qal wh thrfor, () i oft calld a critical vlocity. ( ( b) π 4 ) / v = Z v c, (). Workig form of th qatio Coidr Boltzma kitic qatio for th oght ditribtio fctio f, glctig th patial ihomogity ad th lctric fild ( f ) t = C + S, () whr C i th colliio trm corrpodig to colliio of particl, origiatig from a moorgtic bam i a magtically cofid plama, with particl of all pci of th targt plama, ad S i th orc fctio of particl. Th colliio trm C calclatd by (4)-(7) ad (), () i a xact a [] with oly two amptio, viz., that th azimthal ymmtry tak plac ad that th targt plama i Maxwllia. Th azimthal ymmtry i a raoabl amptio ic Larmor gyratio td to avrag-ot th agl ϕ dpdc, ad th ditribtio fctio f ( v ) i a trog magtic fild i axially ymmtric, i.. it i a fctio of th vlocity magitd v ad th pitch agl ϑ. ff Simplifid qatio olvd i [6,7] corrpod to b( v ) giv by (7), ad c( v) = Z v v, i.. th firt trm i (8), whil vlocity diffio trm with a( v ) i icorrct i both [6] ad [7]. I ca of iothrmal Maxwllia targt plama, i.. T = T, th corrct Colomb colliio oprator applid to th Maxwllia ditribtio fctio with th qilibrim tmpratr T c = T rlt i llificatio of th colliio trm. A oppod to [6,7], thi fdamtal phyical proprty prrv if w th corrct xprio for dimiol fctio a( v ), b( v ) ad c( v ) giv i ctio. Th prpo of th bqt ctio i to obtai th xact ad phyically adqat oltio of () withot implificatio. Th, th workig form of qatio () i 5

6 φ a( ) φ b( ) a( ) a φ b c( ) φ = φ + ( ζ ) + τ S (, ζ, τ ), (4) τ ζ ζ whr φ(, ζ, τ ) f i th oght fctio, = v / i th dimiol vlocity, τ = t / τ i th dimiol tim, ad ζ = coϑ i th pitch agl coi. Th tatioary moorgtic iotropic orc fctio i S S ( ) = δ ( ), (5) 4π ad th tatioary moorgtic aiotropic orc fctio i S S (, ζ ) = δ ( ) ( ζ ), (6) π whr δ ( ) i dlta-fctio, / = v i th dimiol ijctio vlocity, ad ( ζ ) i th ity-ormalizd agl ditribtio of th orc. Th orc fctio giv by ithr (5) or (6) i ormalizd to th orc rat S [cm - - ], i.. th mbr of particl of typ ijctd i it volm i it tim, c S d v = πv d dζ S (, ζ ) = S. (7) 4. Iotropic problm 4.. Slowig-dow Aalytical tady tat oltio of qatio (4) takig ito accot oly th dyamic frictio forc, bt ot th diffio i vlocity pac, i.. with a( ) =, c( ) =, ad iotropic S ( ) giv by (5), ca b obtaid by variabl paratio mthod for th corrpodig tatioary homogo firt ordr ordiary diffrtial qatio ad th variatio of cotat. Th rltig iotropic ditribtio φ S τ =, (8) ( ) H ( ) 4 π b( ) whr H ( ) i Haviid tp fctio, i typically d plggig th implifid formla (7) for b( ) itad of (5) ad (). W rprodc thi impl tatioary lowig-dow ditribtio imilar to [8-] hr a a rfrc for compario with or oltio blow. 6

7 4.. Slowig-dow ad vlocity diffio i iothrmal plama Simplifid oltio (8), glctig th diffio i vlocity pac, i ihrtly abl to dcrib th Maxwllizatio proc. It i alo cttig off th high rgy ditribtio tail ad thrfor i iapplicabl at >. To obtai a phyically adqat oltio, lt firt coidr a iotropic problm amig that c( ) =, S ( ) i giv by (5), ad all targt plama pci ar i thrmal qilibrim i.. T = T. Eqatio (4) with c( ) = rdc to whr φ φ p( ) + q( ) + r( ) φ( ) = f ( ), (9) a( ) p( ) =, (4) b( ) a( ) a q( ) = + 4, (4) r( ) = b, (4) S τ =. (4) ( ) f ( ) δ 4π A mtiod abov, i thi iothrmal ca a( ) = εb( ), ad it ca b aily chckd by btittio that Maxwllia fctio ε ( ) φ = (44) i a partial oltio of th homogo qatio corrpodig to (9). It ca alo b aily vrifid by btittio, that th cod idpdt oltio of th homogo qatio i ( ) ε φ = d. (45) a( ) l ow that φ ( ) ad φ ( ) ar dtrmid, w ca fid th oltio of th ihomogo qatio (9) i th form ε φ( ) = C ( ) φ ( ) + C ( ) φ ( ), (46) ig Lagrag mthod of variatio of cotat. To atify (9) w rqir that C ( ) φ( ) + C ( ) φ( ) =. (47) C ( ) φ ( ) + C ( ) φ ( ) = f ( ) p( ) Thi i a ytm of liar algbraic qatio with rpct to C ( ) ad C ( ). It oltio i 7

8 Itgratig (48) ad (49), w obtai whr K ad K ar arbitrary cotat. ε S τ C ( ) = δ ( ) d, (48) a( ) π l S τ C ( ) = δ. (49) ( ) π ε Sτ ( ) = ( ) + π a( ) l C H d K, (5) S τ C ( ) = H + K, (5) ( ) π Fially, th partial oltio of th ihomogo qatio (9) i ε τ ε Sτ ( ) ( ) c π l c S ε φp ( ) = H d + H d πv a( ) v a( ), (5) ad th gral oltio of th homogo qatio corrpodig to (9) i l ε ε ε a l ε φh( ) = K + K d. (5) ( ) Th gral oltio of (9) i φ( ) = φ ( ) + φ ( ). (54) h p Two othr idpdt qatio ar rqird to fid th cotat K ad K. A raoabl coditio to dtrmi K i that φ( ), thrfor, K =. Thr i o particlar bodary coditio at =. Th maig of th mltiplir i th Maxwllia trm ε K ca b xplaid ig th ormalizatio coditio. Sic or ditribtio fctio i ormalizd to th mbr of particl, th itgral ovr th tir vlocity pac hold b qal to th dity of particl of typ, which, i tr, qal th orc rat S tim th dratio actio, i.. κτ of orc 4 π φ( ) d = κτ S, (55) whr κ i a dimiol cotat, ad κτ i th tim rqird to attai th tady tat. Uig th fact that π / ε d = ε 4, (56) 8

9 w obtai th rlatiohip btw K ad κ κs τ 4 K = ( ) / φ p d πε 4π. (57) 4.. Slowig-dow ad vlocity diffio i oiothrmal plama It i mor difficlt to obtai th xact aalytical oltio, wh targt plama pci hav diffrt tmpratr. I thi bctio w dcrib a mrical oltio of th iotropic problm (9)-(4). For th mrical tratmt itad of δ ( ) w a dlta-lik fctio ( ) ( ) =, (58) π whr i a mall dimiol paramtr corrpodig th pak width, ad th orc fctio S ( ) giv by S S ( ) =. (59) 4π ( ) Th right-had id of (9) i th Sτ f ( ) = 4π v π c ( ). (6) ot that a( ) ad b( ) ar diffrt fctio giv by (4), (5), ad (), ad thr i o impl proportioality btw thm i cotrat to th iothrmal ca coidrd i bctio 4.. To olv th problm formlatd by (9)-(4), ad (6) ovr th itrval [, ] a iform grid whr k,, w itrodc = + ( k ) h, (6) k L R L h =, (6) ad i th grid dimio. Uig forward diffrc drivativ at = = L, i.. for k =, ctral diffrc drivativ at th ir grid poit, i.. for k,( ), ad backward diffrc drivativ at = = R, i.. k =, w approximat qatio (9) by a ytm of liar algbraic L R qatio Aφ = f, (6) 9

10 T whr φ = (,,..., ) i th oght tor of th oltio ovr th grid, (,,..., ) φ φ φ right had id tor, ad matrix f = f f f i th T b c µ... a b c... a b c... a4 b4 c4... A = a b c... a b c... η a b (64) appar to b almot tridiagoal xcpt for th two xtrao lmt A, = µ ad A, = η. Th mai diagoal lmt ar th lowr diagoal lmt ar th ppr diagoal lmt ar p q b = r h h +, p k p q bk = rk, k,( ), ad b h = r h + h +, (65) ad th rmaiig two lmt ar a k = pk qk h h, k,( ) p, ad q a = h h, (66) q p c = h h, ad pk qk c = k h + h, k,( ), (67) µ = p ad p h η =. (68) h To mak th ytm trly tridiagoal, w prmltiply both id of (6) by almot ity matrix ( µ c) Q = (69) ( η a ) Th rltig ytm radily olbl by dobl wp mthod i

11 whr Aɶ φ = fɶ, (7) bɶ cɶ... a b c... a b c... a4 b4 c4... Aɶ = QA = a b c... a b c... aɶ bɶ, (7) ad b ɶ µ = b a, µ η cɶ = c b, aɶ c = a b, bɶ η = b c, (7) a a c µ η f ɶ = Qf = f f, f, f,..., f, f f c a. (7) It i poibl to itrodc th bodary coditio aalogo to φ( ) o that th ytm rmai tridiagoal. If R, a raoabl approximatio of thi bodary coditio i T φ = φ( ) =. Thi corrpod to η =, a =, b =, ad f =. Thr i o pcific R bodary coditio at =. Thi ma that th mrical oltio of (7) will rprt a L particlar oltio of (9), aalogo to th aalytical rlt (54) with K = ad a idfiit val of K or κ, which, i pricipl, ca b dtrmid ig th ormalizatio coditio. Howvr, at high vlociti, roghly [, ], whr oltio (44) i mall, th particlar oltio aalogo to (5) will domiat, which i dtrmid by th orc fctio paramtr. Th, if w ar itrtd oly i th high rgy tail of th ditribtio, bt ot i th low rgy part, thr i o d to look for a dfiit val of K or κ. Th mrical oltio hold coicid with th aalytical rlt obtaid for th ca of iothrmal targt plama i th prvio bctio. Fig. how calclatio rlt for th paramtr giv i Tabl I ad Tabl II a a xampl. W how th rgy ditribtio fctio 4π E φ ( ) vr m m m E = itad of th oltio φ ( ) itlf, ic it i mor apprhibl from th practical viwpoit. Th xact aalytical oltio how by a olid gray crv corrpod to formla (5)-(54) with K = ad K giv by (57), whr th dimiol paramtr

12 κ =.4. Th ormalizatio coditio i xprd by (55). ot that, a it ca b from (5), at = fctio φ( ) go to mi ifiity bca of th cod trm i (5). Thi iglarity formally tak plac d to th of phrical polar coordiat i vlocity pac. Th probability dity for vlocity magitd φ ( ) ar both fiit, ic thy icld th appropriat Jacobia. φ( ) ad th probability dity for kitic rgy Th mrical oltio of th tridiagoal ytm of liar algbraic qatio (7) how by th dahd crv i Fig. wa obtaid for L = 8, R 8 =, ad = 496. Th olid black crv how th implifid oltio (8) with b( ) giv by (7). Th, th xact aalytical oltio for th iothrmal targt plama ad th corrpodig mrical oltio coicid at high rgi. Thir Maxwllia part at low rgi alo coicid for th cho val of paramtr κ. Th lowig-dow oltio (8) fail to dcrib th high rgy tail of th ditribtio corrctly ad i itriically iapplicabl to dmotrat th Maxwllizatio proc, ic importat phyical proprti ar miig i th implifid qatio glctig th diffio tor. Tabl I. Tt particl orc paramtr. Tabl II. Targt plama pci. Spci Dtro Elctro Z =, m 8 = 9.9 g Charg mbr Z = Ma 4 =.44 g m Ijctio rgy E = 5 kv Sorc rat 4 S =. cm - - Width paramtr = 4 =.8 cm -, 5 T = kv Hlio Z =, m = g H H 4 =.6 cm -, 5 H H T = kv Alpha Z = 4, m = g H 4 H 4 4 =. cm -, 4 5 H H T = kv Proto Z p =, m p 4 =.67 g 4 p =. cm -, p 5 T = kv

13 Fig.. Exact aalytical oltio (olid gray crv), mrical oltio (dahd black crv) ad implifid aalytical oltio (olid black crv) of th iotropic problm for 5 kv dtro ijctd ito iothrmal H : H : H (.6 :. :.) plama at T = 5 kv Aiotropic problm 5.. Slowig-dow ad pitch agl cattrig I thi bctio w obtai a aalytical tady tat oltio to qatio (4) with a( ) = ad aiotropic orc fctio (6). Diffrtial oprator = ( ζ ) ζ ζ (74) i th o that occr i Lgdr qatio. It idpdt oltio ar Lgdr fctio of th firt kid P ( ζ ) ad Lgdr fctio of th cod kid Q ( ζ ). Sic th lattr ar iglar at ζ = ± (i.. ϑ = ad ϑ = π ), it i maigfl to arch for a oltio i th form of a xpaio

14 φ(, ζ ) = φ ( ) P ( ζ ) (75) = ggtd i [] ad applid i [6,7]. I cotrat to [6,7], w do ot hat to implify th qatio, ad obtai th oltio i gral form itabl for b( ) ad c( ) giv ithr by xact formla (5), (6), (), ad (), or by implifid formla (7), (8). Sbtittig (75) ito (4) with a( ) = ad ig th idtity lad to qatio P ( ζ ) = ( + ) P ( ζ ) (76) φ b S τ b φ c φ P ζ δ ζ =. (77) ( ) + ( ) ( + ) ( ) ( ) ( ) = ( ) ( ) π Mltiplyig both id of (77) by ( ) m whr δ m i Krockr ymbol, ad dotig P ζ, itgratig ovr [,], ig th orthogoality coditio P ( ζ ) P ( ζ ) dζ = δ, (78) m + m ( ζ ) P ( ζ ) dζ (79) = w arriv at firt ordr ordiary diffrtial qatio φ b ( + ) c( ) + S τ b b v b + φ ( ) = δ ( ) ( ) ( ) π c ( ). (8) Th gral oltio of th corrpodig homogo qatio obtaid by variabl paratio mthod i A ( + ) c( ɶ ) φ ( ) = xp d b( ) ɶ, (8) b( ɶ ) whr A i a arbitrary cotat. Th oltio of ihomogo qatio (8) ca b obtaid by variatio of cotat A. Rgardig it a a kow fctio A( ), ad btittig (8) ito (8) yild th drivativ + Sτ ( + ) c( ɶ ) A ( ) = ( δ ) xp d πv ɶ. (8) c b( ɶ ) Th, A ( ) = vrywhr xcpt =, whr th xpot i (8) qal ity. Thrfor, itgratig (8), w hav 4

15 + S τ A = H + A ɶ, (8) ( ) ( ) π whr A ɶ i a arbitrary cotat. Amig φ ( ), w fid A ɶ =. Fially, th oltio of (8) i Sτ H ( ) c( ɶ ) φ ( ) = ( + ) xp ( ) + d 4 π b( ) ɶ. (84) b( ) ɶ ot that φ ( ) coicid with (8) bca ( ζ ) i ormalizd to ity, ad P ( ζ ) =. If th orc i moodirctioal, ad th ijctio agl coi i ζ = coϑ, i.. th (79) giv ( ζ ) = δ ( ζ ζ ), (85) = P ( ) ζ. (86) 5.. Complt qatio with lowig-dow, vlocity diffio, ad pitch agl cattrig A mi-aalytical tady tat oltio to qatio (4), icldig a( ), with aiotropic orc fctio (6) ca b obtaid i th form (75). Applyig th procdr imilar to (76)-(79), w arriv at cod ordr ordiary diffrtial qatio a( ) φ b( ) a( ) a φ b c( ) + ( ) ( ) φ + S τ To olv it mrically, w rplac ( ) = δ π δ with ( ) ( ). (87) giv by (58), ad rwrit (87) a whr φ φ p( ) + q( ) + r( ) φ ( ) ( ) = f, (88) a( ) p( ) =, (89) b( ) a( ) a q( ) = + 4 (9) b c( ) r( ) = ( + ), (9) 5

16 Sτ f ( ) = 4π v ( + ) ( ) c π. (9) Eqatio (88) i formally aalogo to (9), th, w ca apply th mrical mthod dcribd i bctio 4. to obtai a oltio ( ) φ ovr a iform grid o a fiit itrval [, ]. Aftr that th fial rlt i calclatd ig (75). Each trm i th ri rqir qatio (88) to b olvd mrically. Th mmatio of covrgig ri i prformd til th rqird rlativ prciio i achivd. A ccfl vrificatio of th algorithm wa prformd a dcribd blow. ot that for = th problm xprd by (88)-(9) rdc to (9)-(4), ad th xact aalytical oltio obtaid i bctio 4. i valid. It ca b d to vrify th mrical algorithm. Aothr poibl way of vrificatio i to artificially rdc a( ) mltiplyig it by a mall cotat,.g., ad obtai a complt mi-aalytical oltio i thi pcial ca. Th rlt hold agr with th implifid aalytical oltio of bctio 5. obtaid for a( ) =. Fig. how calclatio rlt for th paramtr giv i Tabl I ad Tabl II a a xampl. Th orc i moodirctioal, o that (85) ad (86) hold. Ijctio agl i ϑ = i thi xampl. W how th rgy ditribtio fctio E φ(, ζ ) vr m m L m E = itad of th R oltio φ(, ζ ) itlf. mrical oltio of (88) wr obtaid for L = 4, R = 8, ad grid dimio = 496. Fctio φ(, ζ ) wa calclatd by (75). Solid gray crv how th complt mi-aalytical oltio corrpodig to (75) ad (88), takig ito accot lowig-dow, vlocity diffio, ad pitch agl cattrig. Dahd black crv how th implifid aalytical oltio corrpodig to (75) ad (84), takig ito accot lowig-dow ad pitch agl cattrig, ad ig (7), (8) to calclat b( ) ad c( ). Th implifid oltio at all agl fail to dcrib high rgy tail of th ditribtio. Bid, it i tially abl to dmotrat th Maxwllizatio proc obrvd i th low rgy part of th corrct ditribtio. 6

17 Fig.. Complt mi-aalytical oltio (olid gray crv), ad implifid aalytical oltio (dahd black crv) of th aiotropic problm for 5 kv dtro ijctd ito iothrmal H : H : H (.6 :. :.) plama at T = 5 kv Tim-dpdt problm 6.. Slowig-dow ad pitch agl cattrig I thi bctio th otatioary problm i olvd. Coidr qatio (4) with timdpdt orc fctio S S (, ζ, τ ) = δ ( ) ( ζ )( H ( τ τ ) H ( τ τ) ), (9) π whr τ ad τ digat th orc actio tart ad top tim rpctivly, ad < τ < τ. A implifid qatio with a( ) = ca b radily olvd ig aalytical tchiq. A oppod to [7], w obtai th oltio i gral form itabl for b( ) ad c( ) giv ithr by 7

18 xact formla (5), (6), (), ad (), or by implifid formla (7), (8). Amig th iitial coditio xpadig φ(, ζ, τ ) =, (94) = τ φ(, ζ, τ ) = φ (, τ ) P ( ζ ), (95) = rcallig (76) ad (78), ad applyig Laplac traform pτ φ (, p) = φ (, τ ) dτ, (96) w rdc (4) with a( ) = to firt ordr ordiary diffrtial qatio pτ pτ ( ) φ b p ( + ) c( ) Sτ ( + ) δ ( ) + φ(, p) = b( ) b( ) b( ) 4 π b( ) p. (97) xt, w olv th corrpodig homogo qatio by variabl paratio mthod ad th by variatio of cotat w fid ( ) ( ) d d + c ɶ d p ɶ ɶ τ p ɶ ɶ τ + + S ( ) ( ) ( ) τ ( ) ( ) b b b H ɶ + ɶ ɶ ɶ φ (, p) = 4 π b( ) p. (98) Sic < rgio i coidrd i thi implifid problm with o vlocity diffio, ad b( ) >, th itgral d ɶ ɶ b( ɶ) >, ad th w ca th kow Laplac traform p p H ( τ ) for >. (99) Fially, th tim-dpdt oltio i c( ɶ ) ( + ) dɶ b( ɶ ) Sτ ( + ) ( ) ɶ ɶ = H H 4 π b( ) b ɶ H d ɶ dɶ φ(, τ ) τ τ τ τ. () ( ) b( ) ɶ For =, τ =, ad τ = fctio Sτ H ( ) d φ(, τ ) = H τ 4 π b( ) ɶ ɶ b( ) ɶ () 8

19 i th otatioary lowig-dow oltio of (4) with a( ) = ad c( ) =, which i imilar to otatioary oltio obtaid i [8,9]. Sic th tim-dpdt Haviid tp fctio qal ity for τ, thi paag to th limit mak () coicid with tady tat oltio (84), ad alo mak () coicid with th implt tady tat lowig-dow ditribtio (8). 6.. Complt qatio with lowig-dow, vlocity diffio, ad pitch agl cattrig h τ Th problm (4), (9) i olvd mi-aalytically ovr a iform grid τ j = ( j ) h τ, j, M, = Τ ( M ), o a fiit tim itrval τ [, Τ ] ad a iform grid = + ( i ) h, i,, =, o a fiit vlocity itrval [, ] h ( ) ( ) R L applyig Crak icolo mthod, propod i [], to th qatio for φ (, τ ) L R i L, mployig xpaio (95) ad whr φ φ φ = φ,,, τ, () τ φ φ φ φ φ,,, τ p( ) q( ) r( ) ( ) (, ) = + + φ + f τ, () a( ) p( ) =, (4) b( ) a( ) a q( ) = + 4 (5) b c( ) r( ) = ( + ), (6) Th dicrt cotrpart of () ( + ) ( ) S τ f H H ( ) (, τ ) = ( τ τ ) ( τ τ) 4π π. (7) i, j+ i, j φ φ i, j+ φ φ i, j φ φ = φ,,, τ φ,,, τ (8) hτ ivolvig forward diffrc drivativ at = = L, i.. for i =, ctral diffrc drivativ at th ir vlocity grid poit, i.. for i,( ), ad backward diffrc drivativ at 9

20 = =, i.. i =, aftr applyig a impl tridiagoalizatio rmblig (69), lad to a R tridiagoal ytm of liar algbraic qatio ( Φ ) j+ j j j+ Φ = Φ + +, (9) A Φ B D D which d to b olvd at vry tp i tim to gt th oltio tor at th bqt tim grid poit ( j + ) ig th oltio tor zro iitial coditio. ( φ, φ,..., φ ) j+, j+, j+, j+ T Φ =, () j Φ prvioly obtaid at th prcdig tim grid poit j ad amig Tim-idpdt tridiagoal matric i (9) ar ad bɶ cɶ... a bɶ c... a bɶ c... a4 bɶ 4 c4... A = a bɶ c... a bɶ c... aɶ bɶ () bˆ cˆ... a bˆ c... a bˆ c... a ˆ 4 b4 c4... B = a ˆ b c... a ˆ b c... aˆ ˆ b. () Mai diagoal lmt of th matric ar calclatd a r( i ) p( i ) bɶ i = hτ, ˆ r( i ) p( ) i bi = hτ + h h Uppr ad lowr diagoal lmt ar for i,( ). ()

21 hτ p( i ) q( i ) ai = h h Th rmaiig lmt ar giv by h p( ) q( i ) = + h i, c τ i h for i,( ). (4) h p( ) q( ) h p( ) a bɶ τ τ = + r( ), h h h c (5) h p( ) q( ) h p( ) c bɶ τ τ = + + r( ), h h h a (6) ˆ h p( ) q( ) h p( ) a b = + r( ) + τ τ h h h c, (7) ˆ h p( ) q( ) h p( ) c b = + + r( ) + τ τ h h h a, (8) p( ) q( ) h p( ) bɶ τ p( ˆ ) q( ) hτ p( ) b aɶ = hτ +, aˆ = hτ +, (9) h h h a h h h a q( ) p( ) h p( ) bɶ τ q( ) p( ) h ˆ τ p( ) b cɶ = hτ, cˆ = hτ. () h h h c h h h c j j j j Tim-dpdt tor = ( D, D,..., D ) D i calclatd a T j hτ Di = f ( i, τ j ) for i,( ), () j h h p( ) τ τ j h h p( ) τ τ D = f (, τ j ) f (, τ j ), D = f (, τ j ) f (, τ j ). () c h a h Th rltig ditribtio fctio i calclatd ig (95). Each trm i th ri rqir qatio (9) to b olvd mrically withi th tim loop. Th mmatio of covrgig ri i prformd til th rqird rlativ prciio i achivd. A ccfl qatitativ vrificatio of th algorithm wa prformd i two way. Sic th ditribtio fctio i ormalizd to tt particl dity, th itgral of th ditribtio fctio ovr th tir vlocity pac hold b qal to th orc rat S mltiplid by th orc opratio dratio τ ( τ τ ), whil th orc (9) i actig (i.. for τ < τ < τ ). Aftr th orc trmiatio (i.. for τ > τ) th dity of tt particl hold rmai cotat ad qal to th orc rat S mltiplid by th total opratio dratio τ ( τ τ ). Aothr way to vrify th mi-aalytical oltio i to artificially rdc th fctio a( ) rpoibl for vlocity diffio, mltiplyig it by a mall cotat,.g.. I thi pcial ca th complt mi-aalytical oltio hold agr with th xact aalytical oltio giv by (95) ad () obtaid with a( ) =.

22 (a) (b)

23 (c) (d) Fig.. Complt mi-aalytical tim-dpdt oltio of th aiotropic problm for 5 kv dtro ijctd ito iothrmal H : H : H (.6 :. :.) plama at T = 5 kv. Evoltio 4 drig th orc actio (a) for τ =.5, (b) for τ =.5, (c) for τ =., ad (d) for τ =

24 (a) (b) 4

25 (c) (d) Fig. 4. Complt mi-aalytical tim-dpdt oltio of th aiotropic problm for 5 kv dtro ijctd ito iothrmal H : H : H (.6 :. :.) plama at T = 5 kv. Rlaxatio to 4 tatitical qilibrim aftr th orc trmiatio (a) for τ = 5.95, (b) for τ = 5., (c) for τ = 5.4, ad (d) for τ = 6.. 5

26 Fig. ad Fig. 4 how calclatio rlt for th paramtr giv i Tabl I ad Tabl II a a xampl, amig th orc fctio (9) with τ =. ad τ = 5.. Th orc i moodirctioal i thi xampl, ad th ijctio agl i ϑ =. Agai, w how th rgy ditribtio fctio E φ(, ζ, τ ) vr m m m E = itad of th oltio φ(, ζ, τ ) itlf. mrical oltio of (8) wr obtaid for L =, R = 7, ad vlocity grid dimio = 89. Tim grid dimio wa M = Fctio φ(, ζ, τ ) wa calclatd by (95). Fig. illtrat th tim voltio of th ditribtio fctio drig th orc actio. At th bgiig, Fig (a), th ditribtio i pakd i th viciity of th ijctio vlocity ad ijctio agl. Aftr that tt particl cattr i agl ad low dow to lowr rgi, Fig (b), ad thrmaliz, Fig (c), (d), approachig at low rgi th tatitical qilibrim with targt plama pci. Whil th orc with cotat rat i actig, th poplatio of thrmalizd particl i gradally icraig, ad th high rgy tail of th ditribtio, oc dvlopd, i ot chagig, a i Fig (c) ad (d), ad corrpod to th high rgy part of th tady tat oltio obtaid i bctio 5.. I or xampl at τ = tt particl dity i =.55 cm -, which i abot o ordr of magitd lowr tha diti of targt plama pci. W thrfor coidr matric A ad B a tim-idpdt ad glct th colliio of tt particl with thmlv. If th orc opratio dratio i logr, thrmalizd tt particl ca b tak ito accot a o of th compot of targt plama. Fig. 4 illtrat th rlaxatio of th tt particl ditribtio to tatitical qilibrim aftr th orc trmiatio. A gradal rdctio i th high rgy tail, Fig. 4 (a), (b), ca b, followd by iotropiatio, Fig. 4 (c), ad Maxwllizatio, Fig. 4 (d). ot that prvioly kow tim-dpdt aalytical oltio, ch a [7], do ot tak ito accot vlocity diffio, ad thrfor ar iapplicabl to dcrib high-rgy tail of th ditribtio corrctly, ad alo caot dmotrat thrmalizatio. I additio, implifid oltio do ot corv th mbr of particl owig to th of trcatd Colomb colliio oprator. 6

27 7. Smmary Smi-aalytical tatioary ad otatioary oltio of th kitic qatio with Colomb colliio trm ad a moorgtic orc fctio hav b obtaid, glctig th patial ihomogity ad th lctric fild. Exact formlatio of Colomb colliio trm i d, amig that azimthal ymmtry tak plac ad that th targt plama pci ar Maxwllia. Th oltio rflct lowig-dow, vlocity diffio, ad pitch agl cattrig ffct. Exact aalytical iotropic tady tat oltio takig ito accot lowig-dow ad vlocity diffio, ad xact aalytical aiotropic tady tat ad tim-dpdt oltio takig ito accot lowig-dow ad pitch agl cattrig hav b obtaid ad d to vrify th miaalytical rlt. Prvio implifid oltio, ch a [6,7], ar iappropriat to dcrib high rgy tail of th ditribtio ad ar phyically iadqat at lowr rgi, whr th Maxwllizatio proc hold b obrvabl. Th rlt may b fl i mrical modlig, pcially cocrig clar proc i magtically cofid plama, ad alo advacd localizd, agl-rolvd prathrmal particl diagotic data aalyi ad itrprtatio. Ackowldgmt Thi work wa partially pportd by RFBR grat o офи_ц, Roatom Cotract o. Н.4б.45..., Cotract o of Miitry of Edcatio ad Scic of Ria ad alo Grat-i-Aid for JSPS Fllow o. 867 at atioal Ititt for Fio Scic, Japa. Th athor apprciat dicio of applid apct rlatd to clar fio tro orc dvlopmt with Prof. B.V. Ktv ad Prof. V.Y. Srgv, ad apct rlatd to or cotio collaboratio o advacd fat particl diagotic with Prof. S. Sdo, Dr. T. Ozaki, ad Dr.. Tamra. Th athor wold alo lik to xpr hi gratitd to Prof. Y.. Dtrovkii for a favorabl dicio of th macript. 7

28 Rfrc [] L. Lada, Di kitich Glichg für d Fall Colombchr Wchlwirkg, Phyikalich Zitchrift dr Sowjtio,, 54 (96). [] B.A. Trbikov, Th diffrtial form of th kitic qatio of a plama for th ca of Colomb colliio, Sov. Phy. - J. Expr. ad Thor. Phy., 7, 96 (958). [] B.A. Trbikov, Particl Itractio i a Flly Ioizd Plama. I M.A. Lotovich, d., Rviw of Plama Phyic, vol. (Coltat Bra, w York, 965). [4] Y.. Dtrovkii, D.P. Kotomarov, mrical Simlatio of Plama (Sprigr-Vrlag, Brli, 986). [5] I.S. Gradhty, I.M. Ryzhik, Tabl of Itgral, Sri, ad Prodct (Acadmic Pr, 7). [6] J.G. Cordy, W.G.F. Cor, Ergtic particl ditribtio i a toroidal plama with tral ijctio hatig, Phy. Flid, 7, 66 (974). [7] J.D. Gaffy, Ergtic io ditribtio rltig from tral bam ijctio i tokamak, J. Plama Phyic, 6, 49 (976). [8].V. Grihaov, Y.. Dtrovkii,.V. Kartkia, D.P. Kotomarov, Ergy trafr from fat io to a plama, Sov. J. Plama Phy.,, 4 (976). [9] J.G. Cordy, M.J. Hoghto, Problm aociatd with th ijctio of a high-rgy tral bam ito a plama, cl. Fio,, 5 (97). [] T.H. Stix, Stability of a toroidal plama bjct to tral ijctio, Phy. Flid, 6, 9 (97). [] M.. Roblth, W.M. MacDoald, D.L. Jdd, Fokkr-Plack Eqatio for a Ivr- Sqar Forc, Phy. Rv., 7, (957). [] J. Crak, P. icolo, A practical mthod for mrical valatio of oltio of partial diffrtial qatio of th hat-codctio typ, Proc. Camb. Phil. Soc., 4, 5 (947). 8