RUSSIAN ROULETTE AND PARTICLE SPLITTING

Transcription

1 RUSSAN ROULETTE AND PARTCLE SPLTTNG M. Ragheb 3/7/203 NTRODUCTON To stuatos are ecoutered partcle trasport smulatos:. a multplyg medum, a partcle such as a eutro a cosmc ray partcle or a photo may geerate a large umber of secodary partcles. 2. The folloed partcle remas th the cosdered rego thout escape, leadg to a excessvely log scatterg cha. ether case a method for termatg the partcle chas s eeded, hle matag the process ubased s eeded. Russa Roulette ad partcle splttg are to complemetary mportace-samplg methods hch ca acheve that goal. RUSSAN ROULETTE the sprt of Tolsto s War ad Peace, the Russa Roulette game s played amog people ho have cosumed a substatal amout of sprts, hstorcally vodka, ad are the ufortuate possesso of a sx-shooter. To dsplay ther courage the royal salos stead of o the battle-feld, or maybe to sho ther dsda for lfe, a sgle bullet s loaded to the gu, the barrel flpped, more drks are cosumed to completely loose touch th realty, the the players take turs amg the gu at ther temples ad pressg the trgger. At each tral, the probablty of the perso pullg the trgger provg hs stupdty beyod a reasoable doubt ad thus deservg to shorte hs lfe ad promptly meetg hs creator s /6. He eeds to repeat hs brave feat for oly sx tmes to face a probablty of oe of blog off hs bra ad rug the expesve carpet uder hs feet. Luckly Mote Carlo ths game s appled to fcttous partcles partcle trasport smulatos. Cosder a partcle hose statstcal eght s:. The Russa Roulette radom varable ca be descrbed as: p ( p) kll partcle keep partcle e = 0 e ( p) ()

2 That s, th probablty p, the partcle s klled ad th probablty (-p), the partcle survves, but ts statstcal eght s adjusted to the e value: e ( p) (2) The mportat cosderato s that the expected eght of the partcle s preserved at the same that t had before the collso. fact the expected value of the statstcal eght s: E ( e) = p.0 + ( p).. ( p ) (3) As a example, let us cosder a choce of the kll probablty as: p=0.5, hch klls the partcles 50 percet of the tme, ad lets them survve 50 percet of the tme. The partcles that are klled are elmated from the smulato ad e source partcles are sampled. Those that survve ll o be assged a statstcal eght gve by Eq, 2 as: e = 2 ( 0.5) (4) That s the survvg partcles ll have ther statstcal eght doubled. aother case f e choose: p=0.9, partcles ll be klled 90 percet of the tme hle those survvg the process ll be assged a e eght gve by: e = 0 ( 0.9) (5) The process s repeated th dfferet values of the kll probablty p utl the umber of partcles s reduced to a maageable sze. t may eve be ecessary to use t some smulatos to ed up the geerated otherse fte chas. Thus t provdes a

3 coveet ay to termate partcle hstores, hle preservg ther orgal statstcal eght. PARTCLE SPLTTNG Coversely to the Russa Roulette techque, the sample sze ca be creased, hle avodg a bas by usg the Splttg method hch. Cosder a partcle of statstcal eght:. t ca be splt to ay umber of e detcal partcles each th statstcal eght such that: =, =,2,..., (6) The expected value of the e partcles eghts s ther sum: E( ) = E ( ) = =. 2 (7) hch preserves the tally splt partcle statstcal eght. By such a eghtg method, oe cotrols the total umber of tracks, ad the relatve umbers of tracks varous regos of phase space, hch s equvalet to a form of mportace Samplg. deally, the umber of paths ould be proportoal to ther cotrbuto to the fal result, th avodace of those paths that do ot cotrbute to the sought aser the sprt of ay ell thought mportace Samplg method. COUPLED RUSSAN ROULETTE AND PARTCLE SPLTTNG Whe Russa Roulette ad Partcle Splttg are combed, partcles ll ted to have early equal eghts, hch s advatageous reducg the varace the computed quatty of terest. Cosder a medum subdvded to boudares: r, r,..., r,... r 2 each of them assged a mportace:

4 ,,...,,..., 2 as sho Fg.. Fgure. Regos assgmet of the mportace values. Whe a partcle eters a e rego (+) from rego, e defe the relatve mportace of the regos as: = (8) + ν s calculated. f the e rego that s etered has greater mportace tha the prevous oe: + ν = > the partcle s splt to ν detcal partcles, each carryg a statstcal eght: e (9) ν f o the other had:

5 = <, + ν ' Russa Roulette s played, th the partcle klled th a probablty: pkll = ν ' (0) The partcle survves th a probablty: psurvval = ν ' () The survvg partcle s kept the smulato th a e statstcal eght: e = p survval. ν ' (2) f the regos have equal mportace: = =, + ν " ether Russa Roulette or Splttg are played. Notce that lke the case of Russa Roulette, the expected value of the statstcal eght for the partcle s preserved: E ( ).. p + 0. p ν '.. ν ' + 0.( ν ') ν ' e survval kll (3) As a example of the applcato of the methodology, f the relatve mportace of a rego relatve to the prevous rego s: = = 2>, ν + the comg partcle s splt to: ν = 2 partcles,

6 each carryg a statstcal eght: e 2 e f the relatve mportace of a rego s less tha the prevous rego: = = <, + ν ' 0.5 Russa Roulette s appled ad the partcle s klled th a probablty: the survval probablty beg: The survvg partcle eght s: EXERCSE p = 0.5 = 0.5, kll p survval = 0.5. = e. For the example o the applcato of the combed Russa Roulette ad Partcle Splttg techques, prove that the expected value of the partcle statstcal eght s preserved.