Stochastic simulation methods

Transcription

1 tot ulto etod oue oepe: oltá éd U-lu lu Ro o: ote-lo etod ttp:// obetve o te oue: - letue ote - opute ode - ee poet - ppe to du - pott oueet - teet l o te web o ve PRAIA toduto to ote lo etod p. o tud ode poble te eld. tt ee. llbu About ote lo etod Rdo ube eeto leet o tttl P tot Poee d tl Peoe ow d ote lo etod e I odel etopol d Glube d o te I odel e K lot owt et ote lo etod lute lot wee d W lot e to ote lo etod ool ote lo Qutu ote lo etod Futted te p-le Applto o etod wett oto depoto o to o ue owt do etwo.. Duo d peetto o ee poet Wt e te ote lo etod? ple: oleul d ulto opute ulto etod: - oleul d detet ulto bed o te teto o te equto o oto - ote lo etod tot ulto teque wee te do ube eeto pl ul ole - ellul utot ppo to ve peoe deted o ltte wt detet o tot updte ule D opee upeo o ubtte ulto o popto tot level ulto o popto le-le - I eel we pe bout ote lo ulto etod weeve te ue o te do ube e ul te lot! - ote lo teque e wdel ued poble o: tttl p ot odeed tte p tel ee -bod poble ople te lud e bop eoo-p ole peoe ptle p ev-o p ue p euoee et. olld elt bll olld opete vbtol d o oleule Uold pote ple: ellul utot ulto ellul utot ebt eloto d pl wve oe well-ow poble wee etod e ueul.rdo wl o ltte 3D ellul utot oet e odel A odulo ellul utot Dte evoluto 3D dple ellul utot < > ~

2 oe well-ow poble wee etod e ueul oe well-ow poble wee etod e ueul. Peolto poble: 3.e I odel: J µ ± < > oe well-ow poble wee etod e ueul. e p-tte Pott odel p J δ < > Feel wt ote lo ulto e -. tud te do wl - -A b odel tul ee ow oto lututo duo et -We ode t te ple D e P/ P/ e due lo poble peetl elto: te Glto bod Do owt oe- te Pott odel low tepetue Altl tud Qutte o teet: W P W W ; ~? P? W :ube o poble pt wt tep tt ed up t oodte W : totl ube o poble pt wt tep W /! [ / ]![ / ]! l[ P ] l! l[ /!] l[ /!] l We tud te >> d << lt d ue te ollow ppoto: l! l l tl oul P ep l... lo epo I te otuu lt: l... P ep l popete P P d D D D D te D d 3D e: / / 3D 3D D D D / / / e l epoet depedet tu o te deo! Iteet poble: -Rdo wl wt etto o eo pel e: el-vod do wl wt?

3 olv te poble b -tpe ulto -te de: epodu te do wl b u do ube d el te epeet wt do tep te lult uell tu < > # lude <tdo.> # lude <tdlb.> #dee _edet 5 #dee 5 #dee 5 FI *p; lot edu[]; t ; o; <; *_edet; o; < pope"eult.dt"""; o; <; ptp"%d loep; Reult.dt ; ; o; <; tlotd/rad_ax.**-; edu[]lot edu[]*-*/lot; %\" *_edet edu[]; ulto eult Pe tto oolol te I oo wt e tee e t. t be two tte: ete l tte o evou tte. e te o t obe te ollow dl ule:. e t dol u tou te wole oo. Fo te to te te top d loo oud. t detet ol toe t tt e wt dte lle t.. I evou t ee o ote t oud t beoe l. Otewe e evou. 3. I l t ee evou t oud t beoe evou. Otewe e l.. Wt ve ll p pobblt l t beoe evou detll. Poble: pove tt te teod lt d te t te ebt pe tto uto o te t det. I.e. tee tl t det ρ te te o tt o ρ<ρ te tble d equlbu tt te evou t oetto d o ρ>ρ >. Altl oluto otto: ρ det o evou t det o l t P pobblt/evet te tt evou t beoe l P pobblt/evet te tt l t beoe evou te te equto o te d: d P P dt P ; P otto: d d equlbu: [ ] eq. dt dt te ltl oluto o equlbu we ode te lt l / e e e te ode-pete te te I te vt o te tl pot << e < te tble oluto > te tble oluto >? ρ? e [ ] e e Alw eptble eq. eptble ol ulto o te poble e po be oud o te oue oe-pe te t e pled ew do poto t e ulto tep t uoelted do oto e lot:. We te ulto pete ube o tet tep p pobblt o et evou detll ube o tep o w ve o doe. We ode ulto wt deet t ube outot le.. Fo e e we tle te tte l o evou o e t 3. U te dl ule.-. we ve ew do poto o te t d updte te tte. We do t te t te tet tep e d t tep eeded o te vee. We tud te vee vlue o uto o uel oluto o eq. otuou le eq. ded le uel oluto o eq. otuou le eq. ded le Pot: ulto Pot: ulto Rdo ube - I ode to et do ube we eed el tot do poe le: tow de o to o - I elt tee o tot poe ou lulto o ulto we ue peudo-do ube eeted detetll b ou opute. ee ube wll ppo deed do bevo te ttt t oe popete. - I pple oe de tee w wll be ble to eete el do ube u o eple tuel dode et... e peed o tee eeto e oweve ve low. Rdo ube...te e to ulto - Uol dtbuted o ve tevl el ube o tee - Dtbuted od to ve dtbuto Uol dtbuted peudo-do ube e oe o ot o te e te odulo eeto P t to e tee o to - do ode : t doe b: od pott te pope oe o d to et equee wt peodt o - t doe b te uto: d te ple ue o d ot dted! Geet do lot te [ tevl: lot d / RAD _ AX. Geet do lot te [R R tevl: R d: lot d / RAD _ AX.* R R Geet do tee te[r R -] tevl: R - e do ode tee betwee d RAD_AX. RAD_AX uull te l tee- - t peod o epetto RAD_AX - t tt o te e eed t e u. - to tt te ee o ote do eed ue te doe uto t lot d / RAD _ AX.* R R 3

4 et te uo do ube eeto. Dete te epetto peod te ow ll te ee wll epet. ut be b poble.. et te uot o te dtbuto --> te to tet. bot o tee d lot eeto de: otut to o te ube eeted oe ed ott tevl. Deote b te eque o eet ube te -t b : /... e vlue o ut ovee to / o uo dtbuto -->. o:: χ / χ 3. e etu p tet --> tet bot te uot d te bee o oelto te [ tevl. A vul tet b plott o - oodte te uto o. I te eeto pope oe te pot ut ove uol te [ [ que. e bee o ot-e oelto --> te oelto tet o te totl bee o ode oelto we ut ve --> < > < >< < > < > > Peudo-do ube dtbuted od to deed dtbuto We e loo o Ge do ube eeto tt ve do ube od to te dtbuto uto o te [R R tevl. et u uppoe tt Ge ve do ube dtbuted uol o te [ tevl. R G G R G d [ ]; R R Ge Ge R Ge Ge Ge Ge R Ge G [ Ge G R ] Popete o ood do ube eeto te b eeto ould ve lo peod o detetble oelto betwee te te dtbuto loe to te te deed oe led o eltvel ot ee ould be ve t! ould ot ot tetl uto le ep. ould ue ll out o eo ould be teted beoe te ue ould be epetble o optl debu pupoe ee. Wte ple de-tow po. U te d uto wte ple IGO-e po ve te ube o -9 do ode 3. U te d uto wte lot do ube eeto o te [ tevl d tet t! - e te etu-p tet - e te to tet - lulte te d 6 oelto vlue. Wte do ube eeto tt eete lot do ube o te [ tevl od to te ^ dtbuto uto. 5. Wte do ube eeto tt eete lot do ube o te [ tevl od to Gu dtbuto leet o tttl p -tttl p del wt te o le ube o ptle o tot poee - te 3D oodte d 3D pe-pe o oe ptle - te 6D tte-pe o oe ptle - te 6D tte-pe o ptle te tte o te te teed b tet pot t 6D pe - te llowed eo o te tte-pe eo o te tte-pe wee te tet pot ove; pot petted b te etell poed odto - te etell poed odto --> te eeble w te te - te eod pple: ve ot te u ote t te te eeded o pl eueet te tet pot o te te vt ll te llowed pot o te tte-pe. ot ll te epet te eod pple! - we we eue oe pl qutt we uull eue t te-vee o te tte tt e vted du te eueet te b te tet pot o te te - te tt epet eodt: te vee --> eeble vee eble vee vee ove te llowed pot o te tte-pe - o o eod te: te vee ot be epled b eeble vee! e ool eeble All llowed pot o te tte-pe e equll pobbl eled. Ipott oop qutt: W : te ube o llowed otte lw olt equto Geelto o te olt equto: Re etop vld o ll eeble: Ueul equto o dl ltll o uell W--> p l p U ed o et te: U ed elevt teod potetl: etop P U U µ U U U µ l! l l tl oul U

5 ol eeble Deet pot o te tte-pe e ot vted wt te e pobblte e pobblt p o vt o-tte wt ee : Ipott oop qutt; te o ptto uto p e Avee vlue o pl qutt: < A > F l A e ed o et te: ed elevt teod potetl: F ee-ee F F P e F µ χ F F F µ < > < > < > < > leet o tl peoeo tl peoeo: bevo e pe-tto pot -pe-tto e uull poble ol te wt teto o t ottuet. - el pe-tto e ol te te! Pe tto Ft ode tto te t ode devtve o te elevt teod potetl dotut eple: bol ee doto... eod ode pe tto te eod ode devtve o te elevt teod potetl ult o dotut eple: p-eoet tto peolto. ote tpe... e ode pete - tee te deee o te ode te te d te pe - It uull t devtve o te elevt teod potetl - It uull deol - It uull te [] tevl - Dotut t te tto pot o t-ode pe-tto - It devtve dotut o eod-ode pe-tto - ple o ode pete: <>: o te I odel eltve e o te let lute: o peolto ρ/ρ : o ee o bol tl epoet - deed o eod-ode pe-tto - ll elevt pl qutte ve powe-lw bevo te vt o te tto pot. e qutte ete ovee to o dvee. - pott te uvelt o te tl epoet te do ot deped o te oop detl o te odel ut o te et popete o te ode pete d te deolt o te te - oe pott tl epoet: : tl epoet o te ode pete : tl epoet o γ: tl epoet o te ueptblt υ: tl epoet o te oelto let tl epoet e ul qutte ode to udetd d odel petto! A ~ t t - te tl epoet o qutt A > A--> we --> < A-->/- we --> we dee > - < tl epoet e ot depedet o e ote! --> l lw: uvel equto betwee te tl epoet. o D te d γ pel equto vld ll deo dν leet o tot Poee ov poee/ ov ov poee teed b l o eo.e. te tttl popete o te edte utue e uquel deteed o te peet edle o te pt ple: do wl --> ov poe; el-vod wl O ov poe et be te tte o te tot te t tep tot vble e te- evoluto o te te debed b equee o tte:... e odtol pobblt tt eled pevoul we d:.. - : P... Deto: Fo ov poe we ve: P... P P... P. P... P. P P P oe-tep tto pobblte eleet o te tot t Deto: A pobblt dtbuto ove te poble tte w lled vt o tto o ve ov t: w -->P w > ; w w ; w P odt o ov - A ov eduble d ol eve tte be eed o eve tte! te tot t eduble - A ov peod ll tte e peod. A tte peod > P ule : tee d te llet tee wt t popet. A tte peod o u > et. Deto: A eduble d peod ov lled eod e b teoe o ov poee: A eod ov poe vt dtbuto w ove te poble tte depedetl o ow w tll oe 5

6 ow D - It bd etod volv bot detet d tot d - oleul d etod ll deee o eedo wee epltl te to out --> ll equto o oto o ptle - ow d oe deee o eedo e epeeted ol tou te tot luee - we tud ow d ol eeble. de: e eet o te ott tepetue et-bt --> b tot oe-eld t o te ptle. - ow d ulto etod te te debed b tot deetl equto. Fo eple te equto o oto o ptle ow oto: dv R t µ v ev equto o oto tot equto o oto. e oupl to te dt et bt eled tou te Rt tot oe. Queto: Wt popete ould Rt ve ode to be equvlet wt etbt t tepetue? - we e loo o Rt tt wll led o v te ll vt well-olt dtbuto epeted ol eeble. - let u wo D! - b u te teo o tot poee d ov t be ow tt t eved we: P R < R < R > µ / < R t > > / < R t R > µ δ t ep R / < R > Alot o ult te ow d:. A tl poto d velot o te ptle : te te-tep o uell tet te equto o oto. oe o Rt led to: P v e v. Dw do ube o Gu dtbuto wt e eo d ve debed bove. wll ve u Rt. 3. Itete te equto o oto wt te obted vlue o R d et te ew poto d velote.. Poeed wt tep. Aote w o do ow d: - b t to out te oupl o te te to te et-bt b tttl ollo wt vtul ptle. I t ppo o to ee. - e tot ollo ued to be tteou evet - te olld vtul ptle ve well-olt oetu dtbuto - e te tevl t w ptle ue ollo dtbuted od to t P t λe λ λ te e ollo te Alot II. o ow D:. Get tl poto d velot o te ptle. ooe te tevl od to te bove dtbuto 3. Itete te equto o oto utl te te o tot ollo.. ooe oetu t do o te well-olt dtbuto t tepetue. 5. Poeed wt tep 3 ee. Pove b opute ulto tt te ve epe o Rt led to well- olt dtbuto o te ptle velote D. tud te oto o ptle o potetl d ubet to et-bt t tepetue. D 3. tud te oto o ptle W potetl vlle ott wt et-bt t tepetue. ot pt o te W potetl vlle e o. D. tud poble. 3 we te two u o te W potetl vlle odulted t-pe b te-le o opoet. lulte te oelto uto betwee te ptle poto d te etel odult eld. D e te peoeo o tot eoe e ote lo etod Deto: ote lo etod ue do equee o ube to lulte tttl ette o ple populto o deed pete ow eple: lult PI lult peolto teold.. ote eple: lult vee etto d ee o te I odel eel: pplto e eoou d t. e outle o etod:. Depto o te te te o lto. elet ppopte eeble o te poble tt pot: oe deol ote lo teto 3. Obevble e oputed u oted dtbuto uto. Ulttel te ol to opute qutte ppe eult o -deol teto o te tte-pe. e de to ple te otbuto to et ette o te obevble. Oe deol ote lo teto Poble: ve uto opute te tel: b I d e tel I be oputed b oo pot dol o te [b] tevl d wt uo dtbuto: b I ttowd pl e to lw o le ube utee u tt o uetl le ple oe oe bt loe to te deed tel! et be do ube eleted od I µ d; to oled pobblt det µ te :! te bove to lo tue te P l do ube e oelted o te tevl te ow pdl te etod ovee? --> o µ ot. ve bdl!!! etl lt teoe: : te: P µ d I ω I ω ep d I O ω 6

7 Fo ttowd pl µ ot. te eo ~ / /!! e eo depedet o te oe o d µ! --> lue A bette etod o lult We eete do pot od to te p dtbuto b b I d I p d p I p b p d ote lo o tttl p poble We wt to opute tel le: --> ppopte eeble dtbuto -->eleet o te tte-pe Ω--> te ete tte-pe --> te lto o te te < A > Ω Ω A [ ] d [ ] d e deol tel w etl oputble ol o lted ube o poble!!! e b de: I we ooe p loe poble to: we et --> d te etod ovee pdl o ll vlue o!!! Poble: e etod eed dve owlede o I! p b d Oe w to oveoe te poble b ue oe p uto tt well te bevo o!. e eo lo odebl edued! pote pl pl te eboood wee le! de: to ue te pote pl o lult tee tel I te teto we ooe te tte wt pobblt P----> oo < A > [ ] P --> d tu te eo--> < A > A A P [ ] P [ ] Poble: we tll do t ow! e etopol et l. de... A lot to be deved tt eete tte od to te deed P! de: u ov u tt tt o tl tte ute tte tte e eeted w e ulttel dtbuted od to P Fo t ov eed to pe te W tto pobblte o tte to tte. I ode tt te lt dtbuto be P we eed:. Fo ll opleet p o et o pe pot tee et d u tt W eodt. Fo ll : W 3. Fo ll : W '. Fo ll : ' ' W ' P ' P etee o te lt dtbuto Ited o. A toe but ple odto be ued te o lled detled ble: W ' P W ' P ' Reult: We otut ov led to te deed P dtbuto wtout te po owlede o!!! ple: te ol eeble: etopol d: W ep[- / ] >; W < Glube d: [ ] ep[ ] - W ep[- / ] / ep[- / ] - Alot o ote lo ulto:. pe tl pot te pe pe. Geete ew tte 3. opute te W tto pobblt. Geete uo do ube betwee []. 5. I <W --> up to te ew tte d etu to. I >W --> out te old tte ew d etu. 6. Avee te deed A qutt o ll tte te te tl >> tet tte Detled ble ted e I odel J µ ± < > Iteto wt eet ebo ol! Ode pete: < > - poteou etto poble o - t odel o udetd eod t-eoet o loled p - o J> --> eoet ode - o J< --> t-eoet ode - o pe tto D - eo-pet pe tto o D> - eod ode pe tto odedode - I D d D etl olvble! - Due to te lol teto lult dult. - et oluto ve dult D - o et oluto 3D - Appoto etod: e-eld teo eolto d low tepetue epo Ipott qutte - uve - - te tl epoet o ueptbltγ ode pete pe et d oelto letυ t eult: D ; D.698J/ que ltte; /8; lot dveee!; γ7/; υ 3D o et eult beleved tt:. J/ que ltte; /8; 5/6; γ5/; υ5/8 7

8 e te t oluto to te D I t eult D: F < > e J J 3 J... ep J ep J ± ± < P > < P > < P > < P > P Fo : ξ < > t J o pe tto t >!!! Fo we ve tt ξ e ξ te oelto let [ l t ] ξ J > > ep[ J ] P ep J ep J ep[ J ] F l l λ λ l λ F l P λ λ λ te eevlue o P! we d λ >λ J J J e o J e J e e e-eld ppoto o te I odel pl oluto: <> ep[ Jq < > µ ] e Jq < > µ e < > P P e e t < > t[ Jq < > µ ] o t Jq e teto betwee te p de-oupled! qj < > µ o < > t t < > A plt equto o <> o t< te ol poble oluto: <> -->pet bevo o t>two oluto;<> utble oluto d <> > tble oluto --> eoet bevo o te que ltte q; Jq t te tl pot --> o te ub ltte q6;.. te eboood o < : 3 < > 3 eult o e-eld ppo J/ que ltte et:.69 J/ 6 J/ ub ltte beleved. J/ J/ I et:! /; et D: /8; beleved 3D:.3 ; et D: ; beleved 3D:. γ; et D: 7/; beleved 3D:.5 υ/; et D: ; beleved 3D:.6 - te deolt o te poble ee e-eld ppoe beoe bette d bette! - e-eld totll wo D!! Ipleet te etopol d Glube ote lo o te D I odel Poble: tud <> <> < v > <χ> d o D I odel b u te etopol o Glube lot. We ode d J. e ut e odeed tt. -et u ue ltte wt ee boud odto < > -We ode ol eeble d tu d ± We would le to lulte: < > < >< > < > < > [ < > < ] < v > > [ < > < ] < χ > > wll be deteed o te o < v > d <χ> 8

9 I ode to et te deed qutte we ve to lulte te ollow deol u tel: < >< > ep < >< > ep < > < > < > < > ep < >< > ep < > < > We wll ue te etopol etod to lulte tee u tel! e lot etopol d Glube o te I odel:. F ve tepetue. F tl p outo 3. lulte te tl vlue o d. ode ew p outo b vtull lpp oe dol eleted p 5. lulte te ee o te ew outo d te ee e due to t p-lp 6. lulte te etopol Glube W--> pobblte o t ove 7. Geete do ube betwee d 8. I <W--> ept te lp d updte te vlue o te ee to d etto to 9. Repet te tep - 8 te dve te te to te deed ol dtbuto o te tte. Repet te tep -8 b ollet te vlue o d lulte te vee. opute t vee o le ube o tep. lulte te vlue o <> <> < v > d <χ> b te ve oul 3. e te tepetue d epet te lot o te ew tepetue well.. otut te deed <> <> < v > <χ> uve wt te ode--> ee te opute ode A ple D Glube d ode to o <> uto o o deet te e to o < v > uto o o vou te e to o <χ> uto o o vou te e tte o o vou te e tep. 9

10 Ipott obevto: te odeed W--> tto led to eod ov poe oe tep deed p lp tl! ppl te bove lot o < oe lo ollow ow te ode e te te. d t ot eel be te el oe. e etopol etod teded to eld equlbu popete d ot dl ulto o te te! It beleved tt te Glube tto pobblte ve elt ptue o te d well! Iluee o te ltte e to --> te e eet te ltte e ut te oelto let! o el pe tto d o el dveee! --> pott poble o l: et te deed qutte o te --> lt. Oe w o te te qu-te to poe peod boud odto. ee te eee te opute ode! e K ote lo o: et o edet te etod Ued o: - oput qul equlbu popete t low tepetue te etopol o Glube lot - ult up-le tot poee wt ep- tpe tvto pobblt. eple: - duo o to o tl ubtte - owt poe - d o deet tl - d o p t low tepetue Dulte tt e olved b te K etod: Reeee: ot Klo ebowt; J. oputtol P vol. 7 pp e etopol d Glube d ve eetve t low tepetue wee ep- <<!! too eeted tep wee ot ed te te t low tepetue uull d e bot b!. Fo up-le tot poee te Glube d etopol d lo eetve due to te lel deet te-le tt e peet el poble oe oete eple t ee lte. e K etod o low tepetue I te - We we wt to opute te pe et o te ueptblt t low tepetue we eed to ollow te lututo o te ee o etto o tep - At low tepetue te etopol o Glube lot ve eetve e te te ot o te te wt d ot ppe. - A qu oluto: e tto t e lp-ttept d updte te edet te o e tte eeded te vee od to te pobblt o e tto. - We ve ow te lot d e te teoetl b lte.. et te te t.. Fo lt o te etopol o Glube te pobblte o ll te poble W--> tto p lp te te. 3. lulte te uultve uto R o I wee te totl ube o poble tto te te let RR. Get uo do ube u [. 5.Fd te evet to out b d te vlue o o w 6. out evet. 7. Relulte ll vlue tt t ve bee ed due to te tto 8. Get ew uo do ube u [. 9. Updte te te ulto tep o wet te vee b ttt wee t-lou/r. etu to tep. R < ur R otvto o te b lot - et u peue we ve te wt 3 poble tto: w --> wt te pobblt.; w --> wt te pobblt.5 w3 ---> wt te pobblt I te b etopol o Glube lot we e te tto b ttept wt te ve pobblte. e pobblt to ooe ve tto o te poble tee tto popotol to te vlue o. - Wt d 3 we et te uultve uto R R d R3: R.; R.6; R33. - We plot ow te vlue o eo d R pot o le ollow 3 R. R.6 R3. - I we eete do ube u [ d ultpl t b R3. t wll oepod to oe pot o te le. e pobblt to et tto w wll be popotol wt te dte betwee R d R w /R--> t wt we wted to do! - At e ulto tep we e t tto d te te t w deet tto ou te deed oe! Poble: ow to updte te te ode to et te ood vlue o te vee otvto o te equto o te -It e to detee te t pobblt det tt te tto w wt te dd t oued up to te t: d t t dt t ep t e ott eult o te odto t ep t - te dvdul tto o Poo poe.e. te pobblt to ve tto te dt lel popotol wt te let o te tevl dt. - ueul etue o Poo poee tt le ube o Poo poee wt te wll beve ote le Poo poe wt popete l to te le poee. --> deot b Ft te pobblt tt o w tto ou up to te t we et: F t ep Rt wt R I ode to ve te tep t t pobblt dtbutowe eed to eete tu q do ube od to te tr ep-rt dtbuto! be doe wt te ow etod te eet uol dtbuted u do ube ee do ube eeto: e pobblt to ve te t tto betwee te t t [ d tdt df t P t dt t dt dt R ep Rt dt l u l u' q R R dt ult d poee. ult -owt b u tepetue Pott- odel tvl pplto G-owt etl: owt o te e o te oo-tlle do. ll do e eted up b le oe te -boud ove.peetll obeved tt te <d> e- e ee : / < d t > ~ < d > t e Pott odel: qδ < >... q - we ue que o tul ltte - we tt wt le q ube o tl tte - e ltte te wll ve dol oe Pott vble - e etopol o Glube d t ept ol tep tt do O ee te ee o te te

11 - I te ll etopol o Glube lot oe would dol elet betwee ll te ltte te d ept toe ove tt do ot ee te ee --> u te ued etopol o Glube d el te elped te ee would be popotol wt te ube o ttept oe tep tl equl wt te ube o p would be odeed τ ut te. - I te K lot we vt OY te GRAI OUDARI d lp oe o t te tte equl to t ebo! e pobblt to lp te o te G te e. e te oweve to be updted od to te K ee! o te boud o lp to o te ebo tte te bul - I te ube o poble G tto q we et tt Rq. - e updte te te e G tto : l u - e ee o <dt> ollow te epeted o! dt τ R. ult d o to o tl ue o/d te depoto o to o ubtte - to depoted o ue up o oe te to eb te te ep o potetl be boud ee to eb to e epe pobblt b P ~ ep e tto te b wep w: te ttept eque pe ut te o te eque o te vbto o ode te boud ee popotol wt te ube o t eet ebo. - uull tee lo depoto poe oet wt te duo oe wt te dep ube o to depoted pe ut te pe ut te - te tee o-depoto o o-duo o oe tpe o to let A d. - poble: to ulte te d o te depoted to d te oed tutue. dult: te lel deet te-le tto te o te poble tot poee --> e te ulto ve let te lute Alot wede d W etod d te Wol lot t ot le p lp but lute lp lot - oe ulto tep we lp lute o p - tee lute beoe ve le deee d ppoe A ll lot: - ode ltte o te ubtte - te teto ott te boud eee to eb to te tepetue d depoto te - lulte te tto te o deet tuto deet pee d deet ube o eet ebo - updte te te ut τ lle t te llet peod o te tto o depoto - t e te-tep dol ooe te to e o te ubtte d ttept te tto to eb te wt te pobblt: P τ/ - t e te-tep depot to o dol oe te wt pobblt P dep τ/ dep - otue te ulto utl te deed te Poble d oluto: due to te lel deet tto te eet o te epoetl te tto te oul te ulto ve low d eetve. ot o te te ot ppe --> be u poved b te K etod! - t ve eetve te eboood o elte te tl low dow te eboood o ve ppopte to tud te tl epoet. R.. wede J-. W ouvel tl d ote lo ulto P. Rev. ett Abtt: A ew ppo to ote lo ulto peeted v l eet etod o ulto o le te e tlt. e lot volte d uvelt t eod-ode pe tto podu uuull ll vlue o te dl tl epoet.. U. Wol olletve ote lo Updt o p te P. Rev. ett Abtt: A ote lo lot peeted tt updte le lute o p ulteoul te t d e tlt. We deotte t ee te two-deol O odel o I d - t te tl tepetue d o 3 eebe wt oelto let oud d. O ltte up to 8 o o tl low dow vble wt utooelto te o - tep pe p o etto o lo-e qutte. p-p oelto d te oelto let e oelto betwee p d p eued b: < > te vee eeble vee. I two p e uoelted < >. - At te p e led up oe deto --> < > t tvl oelto beue lpp wll dl et ule t ot ebo o - e te tuto ve deet te p e ottl but ot depedetl -- > tee e le do o pllel p w pet o lo peod o te. u p pt o oe ote e tol oelted - At tepetue te p lutute pdl but lot depedetl o e ote. - o debe te el p oelto d elte te tvl oelto we dee te p oelto uto < > < >< > - I eel deped ol o te dte betwee te two p: --> o le te be odeed otuou vble. Fo >> ltte p we ve: ep / ξ ξ te oelto let ~ d η ξ dvee t ----> ξ ~ υ At we obt tu ~ d η η ote tl epoet η/ o te D I odel e tl low dow t ptue - e I odel doe ot ve el d bult to t tee o et ee te oted wt te p o tee o el te evoluto - I el dl te te dl vble e uto o te. e elto te tee te te le ove w te te ppoe equlbu. I At qutt tt ele towd t equlbu t A t A dt vlue A te elto te be teed : τ - e ote lo etod eete ueve p outo A t A dt - Altou tee o el te evoluto te ulto poe t be vewed d o d - We ollow ow outo o p ele to ote outo te ulto oe b --- > t dee elto te - e elto te led to te oelto let b te dl tl epoet : --> we ve ξ --> d et tt τ--> τ ~ ξ - e b poble: o te etopol o te Glube d!!! ---> te elto te dvee oe qul t te oelto let ---> tl low dow - ee ee tep to eete depedet uoelted outo --> te pl etted ol to ll poto o te tte-pe - e te lo eo.

12 e elto te etopol ulto --> > te utooelto te - e elto te te etopol tee ow tep to p ode to eete tttll depedet outo. - I te elto te o te ode o le tep eve outo be ued eu vee - I te elto te loe t ppotel τ tep ould be dded betwee eve pot. - to dee elto te we t dee te utooelto uto o qutt A: AA < A A > < A > A t vlue t te tep: - t utooelto uto de epoetll AA t ~ ep t / τ ep w dee te epoetl oelto elto te τ ep - tt o te equlte o below oe uet ote oelto te: t dt AA e AA t / τ τ t :teted oelto te; τ AA τ t AA AA wt we detee uull opute ulto tl low dow eod loo - o te ltte we ---> we et ξ--> e o te ltte ---> o el dveee - we et tu: τ ~ - o te etopol d Glube d d we et etopol Glube d K e lol lot. τ ~ d te ulto ve eet o le ltte o ll ltte o te ote d tee e pott te-e eet! - ood ew: te vlue o te dl tl epoet O uvel t deped o te lot ppe o wede d W! - te poble: elbote etod o w te vlue o te dl tl epoet lle! --> tee ut be o-lol lot! lute lot: - we lp toete ll oelted p. - e lp we eete tttll depedet outo. - te vlue o te d epoet beoe low.5! e wede d W lute lot o J> I odel - te b de: to det te lute o le d oelted p d tet te lute t p lpp t od to do teo. - t ee tt te lot ould led to eod ov poe d te detled ble odto ted! - te lot be eeled o bt J> o J< Pott odel otuto o te lute o oelted p: - te ple lute o le eet ebo p e O te lute o oelted p tee e too le At tee e tll p wt le oetto ltou te oelto betwee te t e ould v - te w o otut te lute o oelted p to put l betwee eet ebo d le p wt pobblt p-ep-j/. Flpp te lute: - ll te lute e lpped wt pobblt /! we ew oo vlue /- to ll p te lute - te p te wole ltte e t e updted! - te lot te detled ble --> ppopte o pott pl e wede d W lot o te D I odel:. ode ltte o p wt e. te pete J 3. ode tl outo o te p. put vtul bod wt pobblt -ep-j/ betwee eet ebo d le p 5. otut te lute o oelted p 6. lp te lute wt pobblt / t oe tep 7. et te ew outo o te p te d out t te lulto o te deed vee 8. Idet le eet ebo p d epet te lot tt o. dult: --> te otuto o te lute o oelted p eed o leve d t lute detto lot A eple ode --> po.???? detto o lute wt euo e Wol le lute lot - eve oe eet t te -W lot - deee: otut d lpp ol oe lute t te! - te w o otut te oelted p lute te e te -W lot e b o te Wol lot: - ooe p dol te ltte - otut te lute o oelted p tt o t p eed b oet eet ebo d le p wt pobblt p-ep-j/. Do t poe euvel utl te lute ot ow oe. - lp t lute o oelted p t wll be oe tep - updte te te popotoll wt te ube o lpped p - out te ew outo te vee o te deed qutte - e Wol lot oe eetve beue we otut ol oe lute lw lp t te pobblt to ooe lute popotol wt te e o te lute --> we wll uull lp be lute --> we eete tttll depedet outo! - A eple po ve ode.????? - A ple vul po to ope te eetvee o te etopol K d Wol lot:.ee e to ote lo teque Ide: Du ol etopol o W ote lo poedue b qutt o oto lot!!! we lulte ol te vee <> < > <> d < > - te dtbuto uto d be oweve lo be obted d te lot o pott tttl oto d P d d P d - wt te owlede o d t tepetue we te lulte vee t ote tepetue WIOU AY ddtol ulto t te deed tepetue. o te etod et P -/ / be te pobblt to et te equlbu outo t tepetue eee betwee -/ d / P D ep oepod to te ued ol eeble A.. Feebe R.. wede; P. Rev. ett. vol D te det o tte te eboood o depedet o tepetue Du te ulto t tepetue we otut to te P - / /. Wee te ube o p outo wt P ee te -/ d / tevl

13 3 u te pevou equto we et: ow we lulte t deed tepetue D ep D ] ep[ ep e / ott be obted o te olto odto - we et tu o vlue: ] ep[ ] ep[ A vee o te ee be t edtel lulted: > < > < d d A b lot o te to teque. ode ltte wt ppopte boud odto. te elevt pete tepetue ltte e ee ott et 3. ulto tepetue. e etopol Glube W Wol..et ulto t t tepetue. 5. te te tl tet tep ollet te vlue o d 6. te-tep ott etopol Glube W lot t pl ollet te d vlue veto W. 7. I te tep vble t ollet te *t vlue veto W. 8. otut te to o te dt W. 9. lulte te vee < > d < >.. u te poved oul o opute te vee <> d < > o vlue o te eboood o. ode ote tepetue o d epet te lot o. lult ote vee lult ote vee - lult <> o < > oe oplted - eed to otut te to t two deet tepetue - tee e two uto to detee: G d - G: te det o te etto tte - we et o ee two equto: G P ep P ] ep[ - we obt o tee d G - we otut tu te o tepetue d lulte te deed vee l lult : ] ep[ Ate ubttut d e vlue o be lulted b te olto odto o d ll we et: <> d < > be lulted t : > < > < d d - I t ple to teque te etted d vlue e ute ol o loe to te eeee tepetue!!! - A te vlue o - ee te eo te etted vee ee! - oue o te eo tt te -to ve lo eo. eo wll tol luee te lulto o te vee t tepetue o! - eet to tt ovelp oe wde te e o vlue! etod lled te UI IOGRA QU. [A.. Feebe d R.. wede P. Rev. ett. ol ] - UIAOIA pl teque tt e te ult to de ve eet e [.A. e d. euu; P. Rev. ett. ol ] e Qutu ote lo etod e Qutu ote lo etod Qutu-tttl te --> qutu el te ott wt etbt Qutu p odel Qutu p odel t odeto te qutu tue o te p J J ; ; ; J J --> eebe odel; J I odel ll ; J XY odel Itodu te d lowe opeto J J ; ; D J J

14 e ubbd odel e ubbd odel - bod lto eod quted o - debe tet eleto te peod potetl o ltte - eod qutto We tte qu-loled eleto tte o ve obtl d wt ve p oetto t o ve o t poto R - lto wtte wt te eto d detuto opeto t o tte debed wt ouptol ube - teto tol eeed b te eleto tu t etted t te e te ] ep[ ' ' ' ' ' ε R R t U t I D d ode up ol t eet ebo ltte te te lto beoe U t e te ptle ube opeto tte A ltte odel o tet eleto D A ltte odel o tet eleto D 8 o eleto: Det t o ltte wt te --> oupled le equto.l; l te let o te odeed pe; I te edu o bo te edu o bo todu t d W we et: 8 W t W t ; 8 - te bove equto be wtte eo quted o tte-veto --> oupto ube o te ell W t... ode oe t oe tet eleto o te e ltte poble b ode et te. o-te epulo d teto betwee ptle ebo ell. to out te p o te ptle --> b ddtol dete pete te te poeto o te p de o te Q etod de o te Q etod e te d deol qutu-tttl poble d deol ll tttl p poble we ode te te ol eeble... ep... ] [ep Ft poblt to tle te poble: dole d ode te tt te etopol o ote ee o te eulted tte poble: tee e ol poble ptle e o te poble ell --> we ve to dole X t w ptll poble o >>. eod poblt: e ll lot wtout dolto - ode tl outo - ode poble ew outo - ept te e --> wt pobblt: - otue te lot utl te teod equllbu eed - ollet peodll te elevt dt poble: lulto o te t eleet e RY I OUIG d t ot le o te tt w e wll eld eoble tto pobblte! ep ep ep P d poblt we oluto: to ewte o w te lulto o te P pobblte e e we ol ew ube e ed. - t poble b te ue o te otte-uu ppoto e otte e otte-uu ppoto uu ppoto We ode te eel o o te lto: d deote: W t ] [ W we dvde two pt: b : ; b t t wee t t t t t t e te de d b oute wt e ote but: ] [ b we wte tu: ep... ep ep ep 3 Idel would be to do t o te wole b! ot poble oweve beue d b do ot oute d ve te t o te e ube. - OUIO te otte-uu ppoto: A d e uetl A opeto: A A A A e e e e ] [ Appl te ppoto: >> tee o tt / d b / ll eou we wte ] ep [ep ] ep[ b b o wte podut o ple te we et - oplete et o tte veto betwee te epoetl te......

15 to out tt t ol o te oupto ube d we wte - d lbel oupto ube - o D ltte --> D ltte - ll ube e depedet! - due to te ol te: We to ow te qutu-tttl poble to ll oe! P τ P e ; ;! wee : τ ep e e l P -e eetve eee be lulted o te oupto ube t te: - deo deed b --> ptl deo - deo deed b --> otte o te deo - te etod equvlet wt te PA tel oulto o te Qutu e Peo te ulto: - we ve ou-te teto betwee ebo te wt te vlue o d bot eve o odd. - te d pe t teto epeeted b e-bod ptte wee te teto oud te d plquet - te odto o P to t: eptble e: led to o-te l eee - te oupto ube o te let de o wte plquet e eed b ut d te vlue o te t de e deeed b ut - te oupto ube o te let de o wte plquet e deee b ut d te vlue o te t de e eed b ut. - te oupto ube ot beoe etve! e Q lot - we o tt te vlue o / d b / u lle t - we ode tl eptble outo o te ube - we e te vlue o te oupto ube oud wte plquet debed ele - we lulte te e te totl ee o te te te u be doe o te ebo d plquet - we ept t e wt pobblt Glube d etopol o ote d lo OK! - we otue te lot utl teod equlbu eed - te equlbu ppoed we ollet peodll te elevt dt - tep deed * tl ep P ep e lulto o te e to - eel tee eee e ol uell lulble te lo epo o te oul o P! - oe ple e oweve t ltll poble - eple: ple eo wt ol eet ebo teto. τ e e p e e l[o p] p e e l[ p] p tτ t 8 Deteto o te elevt pl qutte l ; [ e ]ep e e e D d 3D e - te ppoto pplble - te Q etod te deolt o te poble eed b oe e oul o d χ te deto te e te ll e! - o te pplblt o te toto to be b --> te poble beoe tell dult to opute-te ded e ool ote lo etod. eut ool ote lo ulto P. Rev. ett. ol tude te teod te ot ol but ool eeble e ed! lw; W: te ube o otte oepod to o-tte wt ed All otte ve te e pobblte eed o et do ube eeto to opute vee o te ve eeble < A>A / W wee te uto o ll poble otte A te vlue o te pl qutt A otte e poble ow to ple tte wt ed vlue! A edte poblt would be to dol t to e te tte o te te u eod ov d ept ol te e tt eep te totl ee o te te ott. te b poble tt tl e v d e deded loo lot o oputto te! e de o pov t te deo lot 5

16 e deo lot We dd et deee o eedo to te ol oop te o teet t te deo e totl ee o te deo teod te oeved! ee ee te betwee te deo d te teod te te deo eeve ee o te d ve ee to te. e ol ott tt te deo ee ot beoe etve! We ele te ollow eod d to p te o-ol eeble o te pe e o te I odel: e Deo lot -ooe tl p outo wt te deed ee -ooe ote p outo b lpp oe p -opute te e te ee o te te due to te odeed e - te te ve te out to te deo d te tl outo epted. -I > d te deo uet ee o t e te te deo ve te ee ee o te te d ew outo epted. I te deo \ ot eou ee te ew tl outo eeted. -te bove tep e otued utl epeettve ple obted d vee o t e lulted e te deo ol oe deee o eedo opo to te deee o eedo o te te te ee lututo o te te wll be o ode / w ve ll o >> o te lulted vee wll ppote well te odto o te o-ol eeble t te ed ee e deo ee dtbuto te deo teoete e deo epet t del teoete e t ol oe deee o eedo opo to te deee o eedo te wt w t e ee Fo ll tttl p we ow tt te odeed etup te pobblt tt te ee o te deo t ve tep d : ep wee d P d ep te ptto uto o te deo Oe w to tee tu te tepetue tt te deo teoete epeee b plott l[p d ] uto o d te lope wll ve dto o / A ple w to detee te tepetue epeeed b te deo b lult < d >. Fo te e o te I odel t be ow tt: / J l J / < d > o J/< d > << < d > Poble to olve b ool ote lo. opute o d d -8 o te vee lope o P d d < d >. ope You eult to te et eult o te te oe-deol ltte /tj/. ode 5 tep to et up te te d t ote lo tep to tud te ttt o d.. ulte te two-deol I odel o que ltte b u te deo lot. opute < d > < > d < > uto o. Fo ee tud te ueptblt uto o te tepetue o te te. ode ltte wt e 5 to et up te te d ddtol tep to tud te ee vee. 6