}, the unrestricted process will see a transition to

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1 A u an Mod wt Rvs Inoaton Excang: ot Andx t : Dnng t Rstctd Pocss W gn wt ocusng on a stctd vson o t cot M dscd aov stctd vson osvs t ocss on ov stats o wc N W not tat wnv t stctd ocss s n stats { } t nxt tanston du to an xtna o ta dand ava can on sut n t t stctd ocss vtng to t sa stat { } o to t stat { } Wn a ta ava occus n stat { } t unstctd ocss w s a tanston to { } wc w nsu tat t st stat ntd n t stctd ocss w { } Wn an xtna ava occus n stat { } t unstctd ocss w s a tanston to { } Isctv o t scc sa at wc t unstctd ocss oows t s ctan tat t st stat t nts n t stctd stat sac w t { }o { } dndng on wt an ta ava occus twn two conscutv vsts to t stats wt N Not tat t stctd ocss s aso a M Its stat-tanston-at daga s sown n t Fgu 4 Not tat a t tanston ats xct tos out o stats { }an t sa as n t unstctd ocss In t oowng w dn t tanston ats o t oowng tanstons: { < } { } and { < } { } Lt us st ocus on t o tanston Dn: : tatng n a stat { < } and condtond on t nxt tanston ng du to xtna ava t oat tat t unstctd ocss w tun to stat { } o t st t o t acs stat { } us s t oat tat statng n { < } t s no ta ava unt t ocss s ac n t sa stat an ta ava w sut n IP u and to t ocss w st ac t stat { } Usng standad condtonng agunts w can wt t cuson: q w q snts t oat tat onc n stat { } a svc tanston w ng t ocss ac to { } and snts t oat tat onc n stat

2 { } an xtna ava w ta t ocss owad cuson aov stats tat onc n stat { } t on wa to go ac to stat { } s to t go toug a svc coton o an xtna ava occus tn t ocss ust co ac to { } and tn go toug a svc coton Nxt o t stuctu o unstctd Maov can t s ca tat Anot wa to s ts s to s not tat t aov cusv quaton stands o a and w ovd on vad souton to ts sst o cusv quatons Now w can s t quaton as: q and to 4 q As ts oat s t sa o a < w w s ca t Fna to cot t caactaton o t stctd Maov can t tanston at attacd to { } { } o < w and t at o { } { } w at o { } { } w W w noa ts ats dvdng t us a t acwad tanstons wt at w consdd as avng at ; tanston ats a cangd to ; to and to t : tad-tat Poats o t Rstctd Pocss o cs w soud dn t stad-stat oats sa ' o t stctd ocss as saat o t stad-stat oats o t unstctd ocss Howv w w on dvong quatons xssng ' as a ut o ' wc woud aso od o oats o avod cutt w w wo wt notaton vn w addssng t stctd ocss Fst ocus on t stats w ω aanc quatons a: o W not tat t scond-od oognous dnc quaton as t gna souton c c w a t two oots o t caactstc quaton

3 and t two constants c c a dtnd t two ounda condtons: ; s gvs: 4 ; 4 ; c ;c and c c o A W now ntoduc so s ots o t oots : a ; ; c o s c ca tat aga sows tat q 4 w and q Nxt t us ocus on t stats wt < ω < A sa aoac can tan tatng t aanc quatons as scond-od noognous dnc quatons and tn usng ounda condtons to sov o unnown constants Howv t oowng aoac ss t anass o ana t stats wt ω w gn wt wtng t aanc quaton o t stat : s gvs: Now t aanc quatons o t stats wt ω can wttn as: o tatng wt and gong acwads o < < w av: o A 3

4 A3 ast at o ts st s to ana stats wt aoac s ssnta t sa as o W st gt: Nxt w wt t aanc quatons o a gou o stats { } as: Fo t aov two quatons w gt: A4 Fna wtng t aanc quatons o t gous o stats { } gvs: o A5 t 3: tad-tat Poats o tats wt ω < ; N > Not tat t aanc quatons o stats wt < < ; N n t stctd ocss a: In t unstctd ocss t aanc quaton o t sa stats can wttn as: o < < o souton wc satss t aov quatons s: o < Nxt t us wt t aanc quatons o t st o t stats N : o < ; As n st t gna souton o ts dnc quaton s gvs: c c Usng ounda condtons 4

5 c c c c and as w now tat: w gt: c ; c Fna < ; A6 t 4: tad-tat Poats o tats wt ω ; N > o dtn t oat w w consd t aanc quatons o t agggat stat { < } { } s quatons can wttn as: Not tat a t oats on t gt and sd o t aov xsson can xssd as uts o Fo A6 and t sut c o t oo o o ow w av: Fo stats wt N ts can wttn as: A7 Poo o o : W not tat A dos not dnd on an atca tanston ats dnd o t stctd ocss and to t s tu o t unstctd M Now A toug A5 cuca dnd on t atca tanstons o stats ; < to tsvs wt at s ads to: o < < and o In t unstctd ocss t aanc quatons o ; < can wttn as: o < < and 5

6 o s two sts o quatons a quvant du to A6 wc gvs: o < us A toug A6 a utua consstnt and togt sats t aanc quatons o unstctd M Fna A7 s agan dvd wt no nc to t atca tanston ats o t stctd M o cc tat ts ods n t stctd M not tat t aanc quaton o { } n t stctd M s consstnt wt ts aanc quaton n t unstctd M at s quatons: and a quvant caus o A7: Poo o o : W w st stas sva atonss tat a qud to ov t an asston n t oowng: a n n n o n s can sn dct o wtng t aanc quatons o t st o stats { n} o s can sn dct o wtng t aanc quatons o t st o stats { } n t stctd ocss c o s ts not tat { } 6

7 d s s s du to < ; as n t 3 aov o s ts o t 4 w av: ung u t st o quatons and ong: < ; w otan: { } Addng on ot sds and usng c w gt Now w gn t ast st n t oo wtng t aanc quatons o t st o stats { n}o a n and ong a as oows: ung u ov n n n n n n o n n ds: { } { } { } Not: s not ovds a oua to xss n cton 5 o n ts o s was d to o: 7

8 o aangng and cangng t od o suaton and ong w av: Addng t aov to tos otand n d and w gt: Poo o o 3: W st ov t convxt o o : Rca tat v u v Eang u Eang E u Eang v Eang E v u w and u v and w Eang α as a dnst t w α gvn :! t o w t w t t w α α α α 8

9 o dx d x x x w a a a! s t Posson ass uncton and P Lt dnd as oows: a; a g ; ; ; P P dx x x g x Dnot δ as t st od dnc oato; tat s: δ g g g ; ; ; P ; P o: ; ; d P δ and ; ; > d δ us s convx n In a sa ason w can sow tat s convx n Now ca tat: s s tat > δ 9

10 Poo o o 4: W w us t so Σ to dnot P N Σ In NI P N W nd to sow tat : P N NI P N Fo o w av Σ Usng t agggaton/dsagggaton aoac to ana Maov cans s cwt 99 w can wt: < Σ Σ Σ Σ Atnatv ts can sn s wtng t aanc quaton o t gou o stats { } a w can wt < o Σ Σ wtng t aanc quatons o t gous o stats { ; } Nxt wtng t aanc quaton o t gou o stats { ; } w gt Σ Σ o: P N P N and P N P N o Nxt consd t aanc quaton o t gou o stats ; } wc gvs: a > o Σ Σ Σ Σ > Σ o Σ > Σ o > W w ov t st contadcton Assu t xsts: Mn{ Fo t dnton o : Σ P N ; { } P N > P N Σ

11 Now L P N > P N Σ Σ P N s s oss o no suc xsts and P N P N o a Poo o Pooston 5: Rca o t o 4 tat P N P N o us: E N NI NI P N P N E N Extnsons o t Basc Mod: as o cton : Rta s Poc u oowng gu stcs out t M o ts cas W can oow ssnta t sa anatca sts as w od n t cas o u > can A stctd Maov can w consst o a t stats wt to dvo t stad-stat oats o ts Maov N nt stat stctd can can as sovd cost cacuaton aso oows t sa sts as sntd n cton 4 Rta IP U L Manuactu s nu-n-oducton-sst N 3 3 Dawn o cas L - U 4 Hoonta tanstons owad: Non-oonta tanstons: Hoonta tanstons acwad: not sown

12 Extnsons o t Basc Mod: Manuactu s wo-lv Poc cton 53 oowng M snts t cas wn t anuactu s usng t oc dscd n cton 53 Not tat w av cangd t stat dscton o nu n oducton sst to nsd goods on-and nvnto anuactu nows t ta s nvnto oston I t s gat tan tn t anuactu stos oducton wn on-and nvnto acs otws t oducton contnus unt on-and nvnto acs A stctd ocss can catd xact n t sa wa as dscd n t a souton o t stctd can as on no dncs and t st o t anass oows xact t sa sts Manuactu s nsd-goods on-and U U Rta IP L - L Dawn o cas U - L 3 4 L Hoonta tanstons owad: Non-oonta tanstons: Hoonta tanstons acwad: not sown

The local orthonormal basis set (r,θ,φ) is related to the Cartesian system by:

The local orthonormal basis set (r,θ,φ) is related to the Cartesian system by: TIS in Sica Cooinats As not in t ast ct, an of t otntias tat w wi a wit a cnta otntias, aning tat t a jst fnctions of t istanc btwn a atic an so oint of oigin. In tis cas tn, (,, z as a t Coob otntia an