Description of Spectral Particle-in-Cell Codes from the UPIC Framework

Transcription

1 Dspton of Sptal Patl-n-Cll Cods fom th UPIC Famwok Vkto K. Dyk Dpatmnt of Physs and Astonomy Unvsty of Calfona, Los Angls Los Angls, Calfona I. Intoduton hs doumnt psnts th mathmatal foundaton of th pod Patl-n-Cll ods n th UCLA Patl-n-Cll Famwok. h a man knds of ods dsbd h. h fst s an ltostat od whh uss only th Coulomb fo of ntaton btwn patls. hs s th most fundamntal of plasma modls and s usful whn ndutv lt and magnt flds a not mpotant. h sond s th ltomagnt od, whh nluds all th lt and magnt flds dsbd by Maxwll s quaton. hs s th most omplt of th plasma modls, and nluds both plasma wavs and ltomagnt wavs. h thd s th Dawn modl. It s an xampl of a adatonlss, o na-fld, ltomagnt modl. It nluds th ndud lt and magnt flds dsbd by Faaday s and Amp s laws, but xluds tadaton ffts and thfo lght wavs. It s pmaly usful whn th thmal vloty of patls s muh small than th spd of lght and lght wavs a not mpotant. It s mo omplx than th ltomagnt od, but th tm stp an b muh lag. 1

2 II. ltostat Plasma Modl h smplst modl s th ltostat modl, wh th fo of ntaton s dtmnd by solvng only th Posson quaton n Maxwll s quaton. h man ntaton loop s as follows: 1. Calulat hag dnsty on a msh fom th patls: t ( x) = qs( x - x). Solv Posson s quaton: d : = 4t. Advan patl o-odnats usng Nwton s Law: dv m dt = q # ( x) S( x - x) dx dx dt = v h funton S(x) s th patl shap funton. Fo pont patls, ths would b a dlta funton, but n omput modlng xtndd shaps a ommonly usd. h ods dsbd h a sptal and solv th lt fld usng Fou tansfoms. Fo th ltostat as and pod bounday ondtons, a podu fo a gdlss systm s as follows: 1. Fou ansfom th hag dnsty: # 1 - : -k: x ( t ) k x t ( k, t) = V qs( x - x( t)) dx = qs( k). Solv Posson s quaton n Fou spa: k ( k) = - 4 t( k) k Not that ths quaton mpls that (k=0) = 0. hs mans that sttly pod systms a hag nutal.. Fou ansfom th Smoothd lt Fld to al spa: k S( x) = V ( k) S( k) k: x Fo dlta funton patls shaps, S(k) = 1V. Whn solvng ths quatons on th omput, w gnally us dst spa and tm o-odnats. In dstzng tm, th xplt lap-fog ntgaton shm s ommonly usd, baus t s sond od auat. In ths shm, th patl o-odnats a known at staggd tms.

3 h dst quatons of moton a as follows: q v( t + t) = v( t - t) + m S( x( t)) t x( t + t) = x( t) + v( t + t) t Although t s possbl to us th gdlss sptal fld solv shown abov n a vy auat patl od, ths s qut slow. Mo ommonly, hag dnsty s aumulatd on a gd fom th patl o-odnats aodng to som ntpolaton shm. h flds a thn alulatd at th gd ponts, thn ntpolatd to obtan th fo at th patl s poston. If w hav (Nx,Ny,Nz) gd ponts fo a systm of sz (Lx,Ly,Lz), thn th gd spangs a: x = Lx Nx z = Lz Nz = L N z z z and th hag dpost s thn dfnd to b: t( ) = q W( - x ) d, sl sl wh, s a dfnd at ntg valus n, m, l, as follows: = ( n, m, l ) s l = ( nl, ml, ll) x y z and th vto dlta funton s th podut of th Konk dlta funtons: d = d d d, sl n, nl m, ml l, ll hs s analogous to th gdlss as w had bfo: t ( x) = qs( x - x) h mpotant fatu s that th ntpolaton funton W b smooth and hav lmtd suppot, that s, t s zo outsd a small ang. h ntpolaton funton s usually th podut of th ntpolaton funtons n ah o-odnat. Fo xampl, th most ommon ntpolaton funton s lna, gvn by: Wx ( x) = ( x + x) x, -x 1 x # 0 ( x - x) x, 0 # x 1 x and smlaly fo th oth o-odnats. Howv, quadat and ub B-spln funtons a somtms usd.

4 h Dst Fou ansfom an now b usd to obtan th Fou tansfom of th dnsty: 1 kl 1 t( kl) = N t( ) = N qw( - x) - : -kl : wh N = NxNyNz, and w dfn k as follows: ( L n l k, L m l, L l l l = ) x y z hs s analogous to th gdlss as w had bfo: # 1 -k: x t ( k) = V qs( x - x( t)) dx h majo omplaton of usng a gd s that non-physal gd fos an as that w absnt bfo. hs as fom alasng, whh ous whn th ontnuous patl o-odnats x hav spatal vaatons lss than th gd spang. Suh vaatons annot b solvd, but gt mappd onto long wavlngths. o s xpltly how ths ous, w an wt th ntpolaton funton as an nfnt Fou ss, as follows: W( x) = W( k) k : x W ( k ) V 1 W ( ) - x k : = # x dx ths lads to: k 1 - k: x ( k-kl ) : t ( kl) = N q W( k) k h last sum ov s a gomt ss whh an b summd xpltly to gv: ( k-kl) : = N k N d k, kl + k N wh k N psnts wavlngths whh annot b solvd: nm mm lm kn = (,, ) x y z As a sult, on an wt th Fou tansfom of th dnsty as follows: N t ( kl) = q 9W( k') + W( k' + kn) C k N! 0 Compa ths wth th gdlss as w had bfo: -k: x t ( k) = qs( k) - k : x -kl : x 4

5 On an s that W(k ) ats lk a patl shap fato, smla to th funton S(k) n th gdlss as. h tms nvolvng non-zo valus of k N a th non-physal alasd tms. h lt fld s solvd at th gdponts as n th gdlss as, xpt fo th us of th Dst Fou ansfom. k ( k ) = - l l 4 t( kl) kl kl : ( ) = ( kl) kl wh Obtanng th lt fld at th patl s loaton nvolvs anoth ntpolaton. S( x) = ( ) W( x - ) = xyz Podng as bfo, on an show that: N S( x) = V ( kl) 9W( k') + W( k' + kn) C kl hs s analogous to th gdlss as: k S( x) = V ( k) S( k) k k: x N! 0 k : x kl : x Not that wth a gd, th fo alatng a patl no long dpnds mly on th spaaton of patls, but also on th dstan of ah patl fom th gd. In oth wods, th patls a not only ntatng wth ah oth, but also wth a pod stutu fomd by th gd tslf. hs non-onsvatv fo usually lads to slf-hatng, somtms to nstablty. h alasng an b mnmzd n on of two ways. On s to us a hgh od ntpolaton funton whos Fou ss s a small as possbl fo k > k N. hs s mo xpnsv n omput tm. Altnatvly, on an us an addtonal shap funton S(x) n addton to th ntpolaton funton. Both ths mthods fftvly mak th patls fatt, and t s had to mantan dnsty vaatons that a small than th patl sz. On an also gad thm as flt funtons. h most ommon ntpolaton funtons n us a th B-splns. hy hav a Fou tansfoms fo ah omponnt gvn by: Wn( k) = 1 sn( k ) L ; k n+ 1 hs funtons hav maxma na k = (p+1)k N, wh p s an ntg > 1. h wost alasng ous fo p = 1, whh maps dnsty vaatons at k = k N to k = k N. 5

6 1.0 Lna Intpolaton Funton W1 vs k W1=(sn(0.5*k)0.5*k)** Intpolaton Funton W W1*S W1 S=xp(-(k*a)**), a = k Fgu 1. Fou tansfom of fst od ntpolaton funton W1 wth and wthout a gaussan smoothng funton S. Mods wth k > mappd to mods wth k <. 1.0 Intpolaton Funtons W vs k W1=(sn(0.5*k)0.5*k)** Intpolaton Funtons W W W1 W=(sn(0.5*k)0.5*k)** S=xp(-(k*a)**), a = W1*S k Fgu. Fou tansfoms of fst and sond od ntpolaton funtons W1 and W, and W1 wth gaussan smoothng S. 6

7 A patl shap (o flt) funton S(k) whh s small n th vnty of k=k N wll suppss th alasng. Of ous, on s also suppssng som physal mods, so ths shm s lmtng th soluton of th modl. h lsson h s that whn usng gds, on must suppss, on way o anoth, nfomaton whh annot b solvd. Hgh od ntpolatons hav btt soluton, but a mo ostly. In Fou spa, a ommon flt funton s: S( k ) ( k a ) = - L wh a s th patl sz, whh osponds to a gaussan patl n spa. If fltng s usd to suppss alasng, thn th fftv patl shap s gvn by: Sff ( k) = V $ % W( k) S( k) In al spa ths osponds to a onvoluton of th ntpolaton funton wth th flt funton. Fo non-sptal ods, suh fltng s typally don n al spa. Whn fltng s usd, sptal ltostat ods an onsv ngy to pats p mllon ov thousands of tm stps, and onsv momntum to ound-off o. vn whn alasng s suppssd, gd ffts a stll psnt, pmaly du to th us of fat patls ath than pont patls. h asst way to undstand ths, s to not that n Fou spa on plas q => q S ff (k), both n th hag dpost and n th fo alulaton. In plasma thoy, hag nts only n th alulaton of th plasma fquny, so that f alasng s nglgbl: ~ p & ~ p( V $ Sff ( k)) Wh th plasma fquny now dpnds on k, and may not b sotop. Fo lna ntpolaton, gaussan smoothng wth a > 0.5, and an sotop gd, on has: - k ( a + 6) ~ p & ~ p. ~ p 61 - k ( a + hus to s how th gd affts th plasma, on an pla th plasma fquny whh appas n plasma thoy wth th abov xpsson. Whth ths k dpndn of th plasma fquny s mpotant o not dpnds on th plasma paamts and wavs und study. Fo xampl, th dspson laton fo plasma wavs: ~ = ~ p + k v boms: th a 6 ~ = ~ p + ; - + k v md th Whth ths s mpotant o not dpnds on th sz of th gd latv to th Dby lngth. 7

8 III. ltomagnt Plasma Modl Mo omplx s th ltomagnt modl, wh th fo of ntaton s dtmnd by Maxwll s quaton. h man ntaton loop s as follows: 1. Calulat hag and unt dnsty on a msh fom th patls: t ( x) = qs( x - x) j( x) = qvs( x - x) Not that wth ths dfnton of dnsts, th quaton of ontnuty s automatally satsfd: t d : j = qv : ds( x - x( t)) t. Solv Maxwll s quaton: 4 1 d # B = j + t 1 B d # t d : B = 0 d : = 4t. Advan patl o-odnats usng th Lontz Fo: dv m dt dx = q # 6 ( x) + v # B( S( x - x) dx dt = v h ods dsbd h a sptal and solv th lt and magnt flds usng Fou tansfoms. Fo th ltomagnt as, th podu fo a gdlss systm s as follows: 1. Fou ansfom th hag and unt dnsts: # 1 k x t ( k) = V qs( x - x( t)) dx = qs( k) - : - : # 1 k x j( k) = V qvs( x - x( t)) dx = qvs( k) h quaton of ontnuty n Fou spa satsfd: -k: x t( k) k : j = qk : vs( k) t k x - : - : k x 8

9 . Solv Maxwll s quaton n Fou spa: In a sptal od, on gnally spaats th lt fld nto longtudnal and tansvs pats L and, whh hav th popty that k x L = 0 and k = 0, and solvs thm spaatly. W mak us of th quaton of ontnuty to lmnat th longtudnal lt fld: ( k) k k 4 1 t L t ( ) k t k : j k hs sults n th followng sts of quatons: k L( k) = - k 4 t( k) ( k : j j j ) = - k k ( k) B( k) t k # B( k) 4 j = - ( k) t k # ( k) Not that ths quaton mpls that j(k=0) = 0. hs mans that sttly pod systms hav no nt unt.. Fou ansfom th lt and Magnt Flds to al spa: k: xj S( xj) = V 6 ( k) + L( S( k) BS( xj) = V B( k) S( k) k k k: x j In dstzng tm fo th fld quatons, on uss th followng shm: fst advan th magnt fld half a stp usng th old lt fld. hn lap-fog th lt fld a whol stp usng th nw magnt fld. Fnally advan th magnt fld th manng half stp usng th nw lt fld: t B( k, t - ) = B( k, t - ) - k # ( k, t - ) ( k, t) = ( k, t - ) + 9 k # B( k, t - ) - 4 j ( k, t - ) C t B( k, t) = B( k, t - ) - k # ( k, t) h tm stp must b shot nough to solv lght wavs. hs s known as th Couant ondton: K 9

10 h dst quatons of moton fo th patls a as follows: t t q v( t ) v ( t ) v ( t ) v ( t ) m ( x ( t)) ( + = - + > S + ) # BS ( x ( t )) H x( t + t) = x( t) + v( t + ) t h fst quaton s an mplt quaton wh th nw vloty appas on both sd of th quaton. h soluton s known as th Bos Mov. It onssts of an alaton a half tm stp usng only th lt fld: q v( t) = v( t - ) + m S( x( t)) Followd by a otaton about th magnt fld: R X ( ) X v ( t) = ' v( t) ; 1 - ( ) + v( t) # X + 6 v( t) : X@ X1 ; 1 + ( ) wh th yloton fquny s dfnd to b: X = qbs( x( t)) m Fnally, th s anoth alaton a half tm stp usng only th lt fld: q v( t + ) = v R ( t) + m S( x( t)) h us of th gd n th sptal ltomagnt od s analogous to ts us n th ltostat od. h hag dnsty and longtudnal lt fld a th sam whl th unt s gvn by: j( ) = qv W( - x ) d, sl sl h ntpolatd lt and magnt flds a gvn by: S( x) = 6 ( ) + L( W( x - ) BS( x) = B( ) W( x - ) 10

11 IV. Dawn Plasma Modl Most omplx s th Dawn (adatonlss ltomagnt) modl, wh th fo of ntaton s dtmnd by th Dawn subst of Maxwll s quaton. h dffn btwn th two s n th xpsson fo Amp s law. Maxwll s quaton has: 4 1 d # B = j + t whas th Dawn subst has: 4 1 L d # B = j + t hs small dffn s sgnfant baus t tuns th quatons fom hypbol fom to llpt fom and lmnats lght wavs. h man ntaton loop s as follows: 1. Calulat hag, unt and dvatv of unt dnsty on a msh fom th patls: t ( x) = qs( x - x) j( x, t) = qv( t) S( x - x( t)) j( x) t dv = q 9 dt S ( x - x) - vd $ v S ( x - x) C In th od, w atually dpost two quantts spaatly, an alaton dnsty and a vloty flux: dv a ( x ) = q dt S ( x - x) M( x) = qvvs( x - x) and thn dffntat: j( x) t = a - d : M 11

12 . Solv Maxwll s quaton: As n th ltomagnt od, w spaat th lt fld nto longtudnal and tansvs pats, = L + and solv thm spaatly: d # L = 0 d : = L 1 B 4 j d # B = j = j + d = t d # t = t d : B = 0 d : L = 4t. Advan patl o-odnats usng th Lontz Fo: dv m dt dx = q # 6 ( x) + v # B( S( x - x) dx dt = v Fo th Dawn as, th podu fo solvng ths quatons fo a gdlss systm s as follows: 1. Fou ansfom th hag, unt, and dvatv of unt dnsts # 1 k x t ( k) = V qs( x - x( t)) dx = qs( k) - : - : # 1 k x j( k) = V qvs( x - x( t)) dx = qvs( k) j( k) t dv = q 9 dt - ( k : v) vcs( k) k x - : - :. Solv th Dawn subst of Maxwll s quaton n Fou spa: k L( k) = - 4 t( k) 4 k # j( k) B( k) k k j ( k) j( k) k j( k) t = t - ( k : ) k t ( k) k - k: x k x j ( k) t 4. Fou ansfom th lt and Magnt Flds to al spa: k: xj S( xj) = V 6 ( k) + L( S( k) BS( xj) = V B( k) S( k) k k k: x j 1

13 Dstzng tm fo ths fld quatons s muh mo omplx than fo th ltomagnt modl, sn on annot us th lap-fog algothm fo. In fat, dpnds on th alaton dv j dt of all th patls, but th alaton of a patl dpnds on, so w hav a vy lag systm of oupld quatons! A smpl tatv shm wh on uss old valus of dv j dt on th ght hand sd to fnd nw valus of : ( xj) s unstabl whn k < ~ p k o 4 j ( t) ; k ; t k: x j o stablz th taton, on an modfy th quaton by subtatng a onstant fom both sds: d ~ 0 n 4 j ~ - = t - n p p0 o wh th shft onstant s th avag plasma fquny: ~ p0 4 q = V m and th supspts n and o f to nw and old valus of th taton. h soluton to ths nw quaton s: n ( xj) k 4 j ( t) ; k + ~ ; p0 t - ~ 0 4 p o k: x Not whn th soluton has onvgd, ths quaton dus to th ognal on. Solvng ths quaton qus knowldg of th vlots and alatons of th patls at tm t. hs s obtand fom th lap-fog shm follows: v j( t + ) + v j( t - ) v j( t) = ; h taton stats by fst alulatng L (t) fom x(t), and sttng v j( t + ) = v j( t - ) dv j( t) v j( t + ) - v j( t - ) dt = ; hs s quvalnt to assumng th fos a small and that hangs n th unts a domnatd by onvton. Nxt solv fo ntal (t) and B(t). h taton loop thn has two pats. Fst, advan patls, alulat dv j (t)dt and v j (t), and dpost dj(t)dt and j(t). Do not updat patls. Sond, solv fo mpovd (t) and B(t). Rpat. Whn onvgd, us th ltomagnt Bos Mov to updat th patls. j 1

14 hs taton shm woks wll and onvgs n about tatons so long as th plasma dnsty dos not vay too muh, spfally f max( ~ p( x )) < 1.5~ p0 Byond that, th numb of tatons ndd nass, and vntually th algothm boms unstabl agan. It an b stablzd by modfyng th shft onstant as follows: 1 ~ po = 6 max( ~ p( x)) + mn( ~ p( As th dnsty vaaton boms mo xtm, th numb of tatons nass, but t sms to man stabl. 14

15 V. Radatv and Dawn ltomagnt Flds h tansvs pats of Maxwll s quatons a: wh 4 1 d # B = j + t = L + 1 B d # t h tansvs pats of th Dawn subst an b wttn: d # B D 4 1 = j + t L d # D 1 B t h subspt D has bn addd to ndat that th Dawn flds a dffnt than th Maxwll flds. Lt us spaat th tansvs Maxwll flds nto two pats, th Dawn pat dfnd by th abov quatons and a Radatv pat, gvn th subspt R. B = BD + BR = + D R Subtatng th Dawn quatons fom th Maxwll quatons, gvs us an quaton fo th Radatv pats: d # B R 1 = t d # R 1 B t hs spaaton allows us to s mo laly that th Dawn pat of th ltomagnt fld s dvn by th plasma unt, whas th Radatv pat s dvn by th dsplamnt unt. h fld D n th ltomagnt and Dawn modls a not xatly th sam, howv. In th ltomagnt as, th unt j nluds a spons to th fld R, whh s mssng n th Dawn unt. Nvthlss, spaatng th total lt fld nto th pats, = L + D + R s a vy usful dagnost n llumnatng physal posss n plasmas. D R 15

16 A smla spaaton an b don n tms of th vto potntal A. In th Coulomb Gaug, wh A = AD + AR d : A D = 0 d : A R = 0 h full st of Maxwll s quaton fo th vto potntal an b wttn: d 4 1 A j - t h Dawn subst an b wttn: d A D 4 j j = j t Subtatng th two gvs th quaton fo adatv pat of th vto potntal: d A R 1 4 t j = - d # B h Radatv and Dawn flds an thn b dvd: and B D D = d # AD BR = d # A 1 A t D R 1 A t R R hs dompostons a usd pmaly as flts fo dagnosts. Fo xampl, on an s th lght wav mo laly whn A R s analyzd, than whn A tslf s analyzd. L 16

17 VI. ngy and Momntum Flux Fo th ltomagnt modl, th ngy flux s wll known to b gvn by th Poyntng vto S: wh : B : B d : S + t C j : S = 4 # B hs quaton dsbs th onsvaton of ngy: th tm at of hang of ltomagnt fld ngy plus th outflow of th ngy s qual to th ngatv of th wok don on th patls. hs quaton s not unqu and oth ngy flux quatons an also b dvd: only dffns n ngy and flux a sgnfant. It s lss wll known that analogous ngy flux quatons an b dvd fo th ltostat and Dawn modls. wh Fo th ltostat modl, an ngy flux quaton s gvn by: L : L d : S + t 9 8 C j : S j 4 1 z = ; - d t z and L dz L hs quaton an b asly shown by makng us of th quaton of ontnuty and th dntty: d : ( fv) = V : df + f( d : V) An altnat fom of ths quaton an b dvd by usng th sult, zdz L : L d : ; = - tz to obtan: d : Sl + t 1 j : L 9 tzc wh th altnatv ngy flux vto s Sl j 8 1 z z = z + ; t dz - zd t h ltostat ngy n th fom s usful fo solatd systms. 17

18 Fo th Dawn modl, an ngy flux quaton s gvn by: wh L : L B : B d : S + t 8 8 j : ( L 9 + C + ) 1 z S = ( L 4 ) # B ; + - t hs an b vfd by takng th dvgn of S, and makng us of th Dawn quatons. In th Dawn modl, th man pont to not s that th tansvs lt fld dos not nt nto th dfnton of th fld ngy. An altnat fom fo th Dawn as an b dvd by usng th sult, wh B A # B B : B d : j C = - : A = d # A along wth th pvous altnat fom fo ltostats to obtan: d : S l + t 1 1 j A j ( : : L 9 tz + C + ) wh th altnatv ngy flux vto s: S l = S t 6 zdz + A # 18

19 In addton to th ngy flux, th momntum flux quaton s also usful. Fo th ltomagnt as, th quaton s wll known: wh 1 S d : - t = t + j # B = BB - 1 ( : + B : B) I C s th Maxwll Stss nso. h quantty S s th momntum n th ltomagnt fld. In th ltostat as, th s no momntum n th longtudnal fld and th magnt fld vanshs, so th momntum flux quaton dus to: wh d : = t = ( : ) I C In th Dawn as, th momntum flux quaton s fomally th sam as n th ltomagnt as: 1 S d : - ( ) t L = t + + j # B But th fld momntum vto s: S = 4 L # B and th stss tnso s: = 4 1 BB 1 L L L L ( L L L : + : + B : B) I C Not that th Dawn modl dos hav momntum n th ltomagnt fld, vn though th s no adaton. h tansvs lt fld dos not ontbut to ths momntum, just as t dos not ontbut to th Dawn fld ngy. Not also that, unlk th ltomagnt as, th Poyntng vto fo ngy s not th sam as th Poyntng vto fo momntum. hs ngy and momntum flux quatons a not unqu, and altnatv foms a possbl and usful. 19

20 VII. Unts hs ods us dmnsonlss gd unts, whh mans that dstan s nomalzd to som dstan. Gnally, ths dstan s th smallst dstan whh nds to b solvd n th od, suh as a Dby lngth. m s nomalzd to som fquny 0. Gnally ths fquny s th hghst fquny that nds to b solvd n th od, suh as th plasma fquny. Chag s nomalzd to th absolut valu of th hag of an lton. Mass s nomalzd to th mass of an lton m. Oth vaabls a nomalzd fom som ombnaton of ths. In summay, dmnsonlss poston, tm, vloty, hag, and mass a gvn by: x K = x d I t = ~ t 0 v K = v d~ 0 q K = q mo m = m Dmnsonlss hag and unt dnsts a gvn by: tl = td J j = jd d~ 0 Dmnsonlss lt fld, potntal, magnt fld, and vto potntal a gvn by: M = m ~ 0 d zm = z m ~ 0 d B M = B m ~ 0 AM = A m d~ 0 Dmnsonlss ngy s gvn by: WO = Wm ~ 0 d Dmnsonlss ngy dnsty flux (Poyntng vto) s gvn by: SK = Sm~ 0 h dmnsonlss patl quatons of moton a: mo dv L dt I = ql 6 M + vl # h dmnsonlss Maxwll s quatons a: K d M # B M = A J f j + d M : M = A f t L M It h dmnsonlss ngy flux quaton s: dxm dt I = vl BM d M # M It 1 : B : B d M M M M M : S + ; A + K Jj : M f 0

21 wh A f 4 = m~ 0d dfns th laton btwn th sous and th flds. Whatv tm and spa sals a hosn, ths quatons hav th sam fom. Only th onstant A f hangs. In ths ods, th nomalzaton lngth s hosn to b th gd spang, d = L N = L N = L N x x y y z z and th nomalzaton fquny to b th plasma fquny p. In that as, on an show that: A f 1 = = nod N N N Np x y z wh Np s th numb of patls. h gd spang s thn latd to som oth dmnsonlss physal paamt, typally th Dby lngth. hus: m D vth d = = v d~ J p th wh th dmnsonlss thmal vloty s an nput to th od. Not that f th gd spa s qual to Dby lngth, thn A f s dntal to th plasma paamt g whh appas as an small xpanson paamt n plasma thoy. 1

22 Bblogaphy [1] V. K. Dyk, UPIC: A famwok fo massvly paalll patl-n-ll ods, Comput Phys. Comm. 177, 95 (007). [] R. W. Hokny and J. W. astwood, Comput Smulaton Usng Patls [MGaw- Hll, Nw Yok, 1981]. [] C. K. Bdsall and A. Bu Langdon, Plasma Physs va Comput Smulaton [MGaw-Hll, Nw Yok, 1985]. [4] V. K. Dyk, Patl Smulaton of RF Hatng and Cunt Dv, Comput Physs Rpots 4, 45 (1986). [5] A. N. Kaufman and P. S. Rostl, h Dawn Modl as a ool fo ltomagnt Plasma Smulaton, Phys. Fluds 14, 446 (1971). [6] C. W. Nlson and H. Ralph Lws, Patl-Cod Modls n th Nonadatv Lmt, Mthods n Computatonal Physs, vol. 6, Ald, Fnbah, and Rotnbg, d. [Aadm Pss, Nw Yok, 1976], p. 67. [7] J. Busnado-Nto, P. L. Pthtt, A.. Ln, and J. M. Dawson, A Slf-Consstnt Magntostat Patl Cod fo Numal Smulaton of Plasmas, J. Computatonal Phys., 00 (1977). [8] V. K. Dyk, ngy and Momntum Consvaton homs fo ltostat Smulatons, J. Computatonal Phys. 56, 461 (1984). [9] Dyk, V.K. and Slottow, J.., "Supomputs n th Classoom," Computs n Physs, vol., No., 1989, p. 50.