Um modelo de mecânica estatística para o estudo de migração com três setores: rural, urbano formal e urbano informal ,e, Transições de fase topológicas em supercondutores : técnicas da informação quântica

UM MODELO DE MEC ˆANICA ESTAT´ISTICA PARA O ESTUDO DE MIGRAC¸ ˜AO COM TRˆES SETORES: RURAL, URBANO FORMAL E URBANO INFORMAL

E TRANSIC¸ ˜OES DE FASE TOPOL ´OGICAS EM SUPERCONDUTORES: T´ECNICAS DA INFORMAC¸ ˜AO

QU ˆANTICA

Tese submetida ao Programa de P´ os-Gradua¸c˜ao em F´ısica para a obten¸c˜ao do Grau de Doutor.

Nilton da Silva Branco

Universidade Federal de Santa Cata-rina: Prof. Dr.

Pedro Domingos Santos do Sacramento Instituto Superior T´ecnico - Universi-dade de Lisboa: Prof. Dr.

Florian´opolis 2014

Ficha de identificação da obra elaborada pelo autor,

através do Programa de Geração Automática da Biblioteca Universitária da UFSC.

Oliveira, Tharnier

Um modelo de Mecânica Estatística para o estudo de migração com três setores: rural, urbano formal e urbano informal e Transições de fase topológicas em

supercondutores: técnicas da informação quântica / Tharnier Oliveira ; orientador, Nilton Branco ; coorientador, Pedro Sacramento. - Florianópolis, SC, 2014.

210 p.

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-Graduação em Física.

Inclui referências

1. Física. 2. urbanização. 3. modelo de três setores. 4. materiais topológicos. 5. informação quântica. I. Branco, Nilton. II. Sacramento, Pedro. III. Universidade Federal de Santa Catarina. Programa de Pós-Graduação em Física. IV. Título.

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ke *TőimHQ kX 1[mBHő#`BQ 2+QMƬKB+Q ;2`H +QK i`āb b2iQ`2b

MȹK2`Q /2 i`#H?/Q`2b 2KT`2;/Qb- Nj

s- T2H j@ûbBK T2[m2M }`KX Ysj = As(Nsj)δ, +QK j = 1, 2, ..., Zs UkXdV b2M/Q δ- 2Mi`2 0 2 1-  2HbiB+B//2 / T`Q/mϽQ /2 b2`pBÏQb 2 As mK 7iQ` /2 2b+HX */ T2[m2M }`K /Q b2iQ` BM7Q`KH #mb+ 2KT`2;` mK MȹK2`Q /2 i`#H?/Q`2b [m2 KtBKBx2 b2m Hm+`Q- Lj s- /2}MB/Q +QKQ Q pHQ` iQiH / T`Q/mϽQ bm#i`ő/Q /Q ;biQ +QK 7mM+BQM€`BQb, Lj s≡ psYsj− wsNsj. UkX3V a2M/Q bbBK- 7x2M/Q mbQ / 2[mϽQ UkXdV-  /2KM/ TQ` i`#H?/Q`2b BM7Q`KBb T2HQ j@ûbBKQ T`Q/miQ` /2 b2`pBÏQb BM7Q`KBb b2`€, Nsj= )<sub>psδAs</sub> ws * 1 1−δ . UkXNV ú BKTQ`iMi2 Q#b2`p` M 2[mϽQ +BK [m2 iQ/b b T2[m2Mb }`Kb 2b+QH?2K Q K2bKQ MȹK2`Q /2 i`#H?/Q`2b- mK p2x [m2 M½Q ?€ +QKT2iBϽQ /2 bH€`BQb 2Mi`2 bBX a2iQ` `m`H

bbBK +QKQ MQ b2iQ` BM7Q`KH- bmTQKQb [m2 Q b2iQ` `m`H b2D iQK/Q` /2 T`2ÏQ- QM/2 iQ/Qb Qb /QMQb /2 i2`` T;K Q K2bKQ bH€`BQ- wj

a = wa, ∀ j-iH [m2 M½Q ?D /2b2KT`2;Q MQ b2iQ`X 1bi2 û 7Q`K/Q TQ` Za T`Q/miQ`2b ;`ő+QHb- [m2 miBHBxK  K2bK i2+MQHQ;B /2 T`Q/mϽQ 2K mK K2`+/Q T2`72BiK2Mi2 +QKT2iBiBpQX  T`Q/mϽQ ;`ő+QH /2 +/ 7x2M/- Yj

a- 2bi€ bmD2Bi ¨ 7mMϽQ /2 T`Q/mϽQ *Q##@.Qm;Hb- +QMbB/2`M/Q T2Mb Q MȹK2`Q /2 i`#H?/Q`2b `m`Bb +QKQ 2Mi`/- TQ`iMiQ

Yaj= Aa(Naj)φ, +QK j = 1, 2, ..., Za UkXRyV b2M/Q φ- 2Mi`2 0 2 1-  2HbiB+B//2 / T`Q/mϽQ ;`ő+QH 2 Aa mK 7iQ` /2 2b+HX

*/ T`Q/miQ` /2 #2Mb ;`ő+QHb #mb+ 2KT`2;` mK MȹK2`Q /2 i`#@ H?/Q`2b [m2 KtBKBx2 b2m Hm+`Q- Lj_{a- /2}MB/Q +QKQ,}

Lj

a2M/Q bbBK- 7x2M/Q mbQ / 2[mϽQ UkXRyV-  /2KM/ TQ` i`#H?/Q`2b `m`Bb T2HQ j@ûbBKQ T`Q/miQ` ;`ő+QH b2`€, Naj= )<sub>paφAa</sub> wa * 1 1−φ UkXRkV M€HQ;Q Q b2iQ` BM7Q`KH- iQ/Qb Qb /QMQb /2 i2`` 2b+QH?2K Q K2bKQ Mȹ@ K2`Q /2 i`#H?/Q`2b `m`Bb- mK p2x [m2 M½Q ?€ +QKT2iBϽQ /2 bH€`BQb 2Mi`2 bBX

kXRXk .2+BbǤ2b /2 +QMbmKQ

S`2+BbKQb /2}MB`  T`iB+BTϽQ /Qb +QMbmKB/Q`2b M 2+QMQKBX S` BbbQ-T`BK2B`Q- bmTQKQb [m2 iQ/Qb Qb Hm+`Qb b½Q +QKTH2iK2Mi2 /Bbi`B#mő/Qb 2 ;biQb T2HQb /QMQb /2 }`Kb T` Q +QMbmKQ /2 #2MbX SQ/2KQb /Bx2` 2Mi½Q [m2 iQ/Qb Qb ;M?Qb Ub2DK 2K bH€`BQb Qm Hm+`QbV ;2`/Qb 2K +/ b2iQ` b½Q +QKTH2iK2Mi2 mb/Qb T` Q +QMbmKQ /2 #2MbX a2;mM/Q- bmTQKQb [m2  /2KM/ TQ` +QMbmKQ /2 #2Mb b2D `2T`2b2Mi/ T2HQ bBbi2K /2 /2bT2@ bb HBM2` (R9)X 6`2KQb mbQ /2 mK +bQ T`iB+mH` /2bb2 bBbi2K-  7mMϽQ miBHB//2 /B`2i /2 *Q##@.Qm;Hb (R9), ˜ Ux=+ k dkHQ;(xk), UkXRjV

+QK dk > 0 2+dk = 1U`2iQ`MQ ¨ 2b+H +QMbiMi2V- 2 xk û  [mMiB//2 /2 #2Mb /Q iBTQ kX 1[mBpH2Mi2K2Mi2- 2K mK 7Q`K KBb 7KBHB` T` ;`M/2xb [m2 T`2b2MiK `2iQ`MQ ¨ 2b+H +QMbiMi2 Up2D- TQ` 2t2KTHQ-  2[X UkXRVV- TQ/2KQb `22b+`2p2`  7mMϽQ /2 *Q##@.Qm;Hb M 7Q`K, Ux=, k (xk)dk_. UkXR9V .Bx2KQb [m2 dk b½Q b 7`ÏǤ2b K`;BMBb /2 Q`ÏK2MiQX 1bbb 7`ÏǤ2b /2}@ M2K +QKQ +/ mMB//2 KQM2i€`B û ;bi 2K +QMbmKQ /2 #2Mb- Qm b2D- dk b½Q b 7`ÏǤ2b / /2bT2b ;bi 2K +/ #2KX a2iQ` m`#MQ 7Q`KH

oKQb bmTQ` [m2 +/ ;2Mi2 j /Q b2iQ` 7Q`KH- b2D 2H2 T`QT`B2i€`BQ /2 +TBiH HQ+/Q Mb }`Kb KMm7im`2B`b Qm i`#H?/Q`- i2K T`272`āM+Bb

k3 *TőimHQ kX 1[mBHő#`BQ 2+QMƬKB+Q ;2`H +QK i`āb b2iQ`2b

`2T`2b2Mi/b T2H b2;mBMi2 7mMϽQ miBHB//2 /2 *Q##@.Qm;Hb,

U_fj= (fmj)dm(fsj)ds(faj)(1−dm−ds), UkXR8V b2M/Q fj

m- fsj 2 faj b [mMiB//2b +QMbmKB/b T2HQ j@ûbBKQ ;2Mi2 /Q b2@ iQ` 7Q`KH /2 #2K KMm7im`/Q /2 b2`pBÏQ BM7Q`KH 2 /2 #2K ;`ő+QH -`2bT2+iBpK2Mi2X /2KBb- dm 2 ds b½Q b 7`ÏǤ2b K`;BMBb /2 Q`ÏK2MiQ /2 T`Q/miQb KMm7im`/Qb 2 /2 b2`pBÏQb- `2bT2+iBpK2Mi2- miBHBx/b TQ` +QMbmKB/Q`2b m`#MQb U7Q`KBb Qm BM7Q`KBbVX LQ ȹHiBKQ i2`KQ / 7mMϽQ miBHB//2 mbKQb  T`QT`B2//2 `2iQ`MQ ¨ 2b+H +QMbiMi2 T` 2b+`2p2`  7`ϽQ K`;BMH /2 #2Mb ;`ő+QHb- (1−dm−ds)- miBHBx/b TQ` +QMbmKB/Q`2b m`#MQbX G2K#`KQb [m2 2bbb 7`ÏǤ2b Q#2/2+2K ¨ `2bi`BϽQ (dm+ ds) < 1X

hQKM/Q +QKQ //Qb Qb T`2ÏQb mMBi€`BQb /Q #2K KMm7im`/Q- pm- /Q b2`pBÏQ BM7Q`KH- ps- 2 /Q #2K ;`ő+QH- pa- Q j@ûbBKQ ;2Mi2 /Q b2iQ` 7Q`KH 2b+QH?2 b2m +QMbmKQ Ufj

m-fsj-fajV iH [m2 KtBKBx2 bm miBHB//2- 2[X UkXR8V-bmD2Bi ¨ b2;mBMi2 `2bi`BϽQ Q`ÏK2Mi€`B,

pmfmj + psfsj+ pafaj = M j

f UkXReV

b2M/Q Mj

f  `2M/ /2bb2 j@ûbBKQ ;2Mi2 /Q b2iQ` 7Q`KHX  `2M/ /2bi2 ;2Mi2 /2T2M/2 /Q b2m TT2H MQ b2iQ` T`Q/miBpQ- KBb T`2+BbK2Mi2- bmTQ`2KQb [m2 mK ;2Mi2 7Q`KH TQbb bbmKB` mK /2 /QBb TTûBb /BbiBMiQb-  b#2`- b2`€ i`#H?/Q` Qm T`QT`B2i€`BQ /2 +TBiH miBHBx/Q T2Hb }`Kb KMm7im`2B`bX .2}MBM/Q Πj

f +QKQ Q Hm+`Q `2+2#B/Q T2HQ ;2Mi2 j +bQ 2H2 b2D T`QT`B2i€`BQ /2 +TBiH- TQ/2KQb bBMi2iBx`  `2M/ /2 mK ;2Mi2 j /Q b2iQ` 7Q`KH +QKQ b2;m2, M<sub>f</sub>j = ⎧ ⎪ ⎨ ⎪ ⎩ 0 b2 7Q` i`#H?/Q` 2 2biBp2` /2b2KT`2;/Q wm b2 7Q` i`#H?/Q` 2 2biBp2` 2KT`2;/Q Πj<sub>f</sub> b2 7Q` T`QT`B2i€`BQ /2 +TBiH KMm7im`2B`Q.

UkXRdV

1K bmK- //Qb Qb T`2ÏQb Upm-ps-paV 2 Q bH€`BQ wm- +/ ;2Mi2 /Q b2iQ` 7Q`KH i2`€ bm K€tBK 7mMϽQ miBHB//2 [mM/Q9_,

fmj = dmMfj pm ; f j s = dsMfj ps ; f j a = (1− dm− ds)Mfj pa . UkXR3V

9 _{o2`B}+@b2 2bi2 `2bmHi/Q mbM/Q KmHiBTHB+/Q`2b /2 G;`M;2 T` +H+mH` Q K€tBKQ}

SQ` bmTQbBϽQ- iQ/  `2M/ /2 +/ i`#H?/Q` b2`€ ;bi MQ +QMbmKQ /2 #2Mb- TQ` M½Q ?p2` 2biQ[m2 /2 +TBiHX a2M/Q bbBK- b 2[mÏǤ2b +BK b½Q +`+i2`őbiB+b / 7mMϽQ miBHB//2 /B`2i /2 *Q##@.Qm;Hb- Mb [mBb Q +QMbmKQ /2 mK #2K û /B`2iK2Mi2 T`QTQ`+BQMH ¨ bm /2bT2bX

a2iQ` m`#MQ BM7Q`KH

oKQb bmTQ` [m2 +/ ;2Mi2 j /Q b2iQ` BM7Q`KH- b2D 2H2 T`QT`B2i€`BQ /2 +TBiH HQ+/Q Mb }`Kb BM7Q`KBb Qm i`#H?/Q`- i2K T`272`āM+Bb `2T`2@ b2Mi/b T2H b2;mBMi2 7mMϽQ miBHB//2,

U_Ij= (Imj)dm(Isj)ds(Iaj)(1−dm−ds), UkXRNV b2M/Q I_{m- I}j j

s 2 Iaj b [mMiB//2b +QMbmKB/b T2HQ j@ûbBKQ ;2Mi2 /Q b2iQ` BM7Q`KH /2 #2K KMm7im`/Q /2 b2`pBÏQ BM7Q`KH 2 /2 #2K ;`ő+QH -`2bT2+iBpK2Mi2X Pb 2tTQ2Mi2b b½Q [m2H2b D€ /2}MB/Qb MQ b2iQ` 7Q`KHX

hQKM/Q +QKQ //Qb Qb T`2ÏQb mMBi€`BQb /Q #2K KMm7im`/Q- pm-/Q b2`pBÏQ BM7Q`KH- ps- 2 pm-/Q #2K ;`ő+QH- pa- Q j@ûbBKQ ;2Mi2 pm-/Q b2iQ` BM7Q`KH 2b+QH?2 b2m +QMbmKQ UI_m-Ij j

s-IajV iH [m2 KtBKBx2 bm miBHB//2- 2[X UkXRNV- bmD2Bi ¨ b2;mBMi2 `2bi`BϽQ Q`ÏK2Mi`B,

pmImj + psIsj+ paIaj = M j

I UkXkyV

b2M/Q Mj

I  `2M/ /2bb2 j@ûbBKQ ;2Mi2 /Q b2iQ` BM7Q`KHX  `2M/ /2bi2 ;2Mi2 /2T2M/2 /Q b2m TT2H MQ b2iQ` T`Q/miBpQ- KBb T`2+BbK2Mi2- bmTQ@ `2KQb [m2 mK ;2Mi2 BM7Q`KH TQbb bbmKB` mK /2 /QBb TTûBb /BbiBMiQb-  b#2`- b2`€ i`#H?/Q` Qm T`QT`B2i€`BQ /2 +TBiH miBHBx/Q T2Hb }`Kb /2 b2`pBÏQ BM7Q`KHX .2}MBM/Q Πj

I +QKQ Q Hm+`Q `2+2#B/Q T2HQ ;2Mi2 j +bQ 2H2 b2D T`QT`B2i€`BQ /2 +TBiH- TQ/2KQb bBMi2iBx`  `2M/ /2 mK ;2Mi2 j /Q b2iQ` BM7Q`KH +QKQ b2;m2,

M_Ij=

-ws b2 7Q` i`#H?/Q`

Πj_I b2 7Q` T`QT`B2i€`BQ /2 +TBiH, UkXkRV MQiM/Q [m2 M½Q ?€ /2b2KT`2;Q M2bi2 b2iQ`X

1K bmK- //Qb Qb T`2ÏQb Upm-ps-paV 2 Q bH€`BQ ws- +/ ;2Mi2 /Q b2iQ` BM7Q`KH i2`€ bm K€tBK 7mMϽQ miBHB//2 [mM/Q, Imj = dmMIj pm ; I j s = dsMIj ps ; I j a = (1− dm− ds)MIj pa . UkXkkV

jy *TőimHQ kX 1[mBHő#`BQ 2+QMƬKB+Q ;2`H +QK i`āb b2iQ`2b

a2iQ` `m`H

SQ` ȹHiBKQ- pKQb bmTQ` [m2 +/ ;2Mi2 j /Q b2iQ` `m`H- b2D 2H2 T`Q@ T`B2i€`BQ /2 +TBiH HQ+/Q Mb 7x2M/b Qm i`#H?/Q`- i2K T`272`āM+Bb `2T`2b2Mi/b T2H b2;mBMi2 7mMϽQ miBHB//2,

Urj = (rjm)(1−da)(rja)da, UkXkjV b2M/Q rj

m2 raj b [mMiB//2b +QMbmKB/b T2HQ j@ûbBKQ ;2Mi2 /Q b2iQ` `m`H /2 #2K KMm7im`/Q 2 /2 #2K ;`ő+QH- `2bT2+iBpK2Mi2X /2KBb- da û  7`ϽQ K`;BMH /2 Q`ÏK2MiQ /2 #2Mb ;`ő+QHb miBHBx/ TQ` +QMbmKB/Q`2b `m`BbX LQ T`BK2B`Q i2`KQ / 7mMϽQ miBHB//2 mbKQb  T`QT`B2//2 `2iQ`MQ ¨ 2b+H +QMbiMi2 T` 2b+`2p2`  7`ϽQ K`;BMH /2 T`Q/miQb KMm7im`@ /Qb- (1−da)- miBHBx/ TQ` +QMbmKB/Q`2b `m`BbX *#2 bHB2Mi` [m2 Q b2`pBÏQ BM7Q`KH M½Q 7x T`i2 /Q +QMbmKQ /Qb ;2Mi2b HQ+HBx/Qb MQ b2iQ` `m`H-mK p2x [m2 b2`pBÏQb BM7Q`KBb b½Q `2HBx/Qb T2Mb 2K bmb T`QtBKB//2b-Qm b2D- M €`2 m`#MX

hQKM/Q +QKQ //Qb Qb T`2ÏQb mMBi€`BQb /Q #2K KMm7im`/Q- pm- 2 /Q #2K ;`ő+QH- pa- Q j@ûbBKQ ;2Mi2 /Q b2iQ` `m`H 2b+QH?2 b2m +QMbmKQ Urj<sub>m-r</sub>j<sub>aV iH [m2 KtBKBx2 bm miBHB//2- 2[X UkXkjV- bmD2Bi ¨ b2;mBMi2 `2bi`BϽQ</sub>

Q`ÏK2Mi€`B,

pmrjm+ parja= Mrj UkXk9V b2M/Q Mj

r  `2M/ /2bb2 j@ûbBKQ ;2Mi2 /Q b2iQ` `m`HX  `2M/ /2bi2 ;2Mi2 /2T2M/2 /Q b2m TT2H MQ b2iQ` T`Q/miBpQ- KBb T`2+BbK2Mi2- bmTQ`2KQb [m2 mK ;2Mi2 `m`H TQbb bbmKB` mK /2 /QBb TTûBb /BbiBMiQb-  b#2`- b2`€ i`#H?/Q` Qm T`QT`B2i€`BQ /2 +TBiH miBHBx/Q T2Hb 7x2M/bX .2}MBM/Q Πjr+QKQ Q Hm+`Q `2+2#B/Q T2HQ ;2Mi2 j +bQ 2H2 b2D T`QT`B2i€`BQ /2 +TBiH-TQ/2KQb bBMi2iBx`  `2M/ /2 mK ;2Mi2 j /Q b2iQ` `m`H +QKQ b2;m2,

Mj r = -wa b2 7Q` i`#H?/Q` Πj r b2 7Q` T`QT`B2i€`BQ /2 +TBiH, UkXk8V MQiM/Q [m2 M½Q ?€ /2b2KT`2;Q M2bi2 b2iQ`X

1K bmK- //Qb Qb T`2ÏQb Upm-ps-paV 2 Q bH€`BQ wa- +/ ;2Mi2 /Q b2iQ` `m`H i2`€ bm K€tBK 7mMϽQ miBHB//2 [mM/Q,

rjm= (1− da)Mj r pm ; r j a= daMj r pa . UkXkeV

kXk 1[mBHő#`BQ /2 K2`+/Q

iû ;Q`- /2i2`KBMKQb b +QM/BÏǤ2b /2 T`Q/mϽQ 2 /2 +QMbmKQ- /2pB/b ¨ T`iB+BTϽQ /Qb ;2Mi2b M 2+QMQKBX ;Q`- /Q TQMiQ /2 pBbi 2+QMƬKB+Q-/2}MB`2KQb b +QM/BÏǤ2b /2 2[mBHő#`BQ MQb K2`+/Qb /2 #2Mb 2 /2 i`#H?QX

kXkXR J2`+/Q /2 #2Mb

JMm7im`/Qb LQ 2[mBHő#`BQ- iQ/b b }`Kb /Q b2iQ` m`#MQ 7Q`KH 2b+QH?2K Q K2bKQ bH€`BQ (Rk- R8)- BX2X- wj m = wm,∀ jX SQ`iMiQ- / 2[mϽQ UkX8V Q#i2KQb mK it /2 /2b2KT`2;Q /2 2[mBHő#`BQ }t, u = η b. UkXkdV

L2bb2 `2;BK2- Q 7iQ` /2 2b7Q`ÏQ /Qb i`#H?/Q`2b- 2[X UkXkV- b2`€ Q K2bKQ 2K [mH[m2` }`K- //Q TQ`, εj _{≡ ε =} ) _η 1_{− η} *η . UkXk3V

*QKQ +QMb2[māM+B- iQ/b b }`Kb +QMi`i`½Q Q K2bKQ MȹK2`Q /2 i`#@ H?/Q`2b Nj

m- /2 KQ/Q [m2 Q MȹK2`Q iQiH /2 i`#H?/Q`2b MQ b2iQ` m`#MQ 7Q`KH û Nm= ZmN_{mX a2M/Q bbBK-  T`Q/mϽQ iQiH /2 #2Mb KMm7im`@}j /Qb- Ym= ZmYmj- û // TQ` Up2D 2[X UkXRVV,

Ym= ξm(Nm)α _UkXkNV

M [mH

ξm≡ AmZm1−αεα UkXjyV

SQ` /2}MBϽQ-  /2KM/ ;`2;/ /2 #2Mb KMm7im`/Qb û B;mH Q +QMbmKQ iQiH /2 #2Mb KMm7im`/Qb T2HQb i`āb b2iQ`2b,

Dm≡+ j fmj + + j Imj + + j rjm. UkXjRV

AMb2`BM/Q Q +QMbmKQ /2 #2Mb KMm7im`/Qb /2 +/ b2iQ` Up2D 2[bX UkXR3V-UkXkkV 2 UkXkeVV M 2[mϽQ +BK `2bmHi,

pmDm=+ j dmM_fj++ j dmM_Ij++ j (1− da)Mrj. UkXjkV

jk *TőimHQ kX 1[mBHő#`BQ 2+QMƬKB+Q ;2`H +QK i`āb b2iQ`2b

*QMbB/2`M/Q b `2bi`BÏǤ2b Q`ÏK2Mi€`Bb UkXRdV- UkXkRV 2 UkXk8V-  2[mϽQ Mi2`BQ` `22b+`2p2@b2 +QKQ, pmDm = dm ⎛ ⎝ Nm + j=1 wm++ j Πj<sub>f</sub> ⎞ ⎠ + dm ⎛ ⎝ Ns + j=1 ws++ j Πj<sub>I</sub> ⎞ ⎠ +(1− da) ⎛ ⎝ Na + j=1 wa++ j Πjr ⎞ ⎠ = dm ⎛ ⎝wmNm++ j Πjf ⎞ ⎠ + dm ⎛ ⎝wsNs++ j ΠjI ⎞ ⎠ +(1<sub>− d</sub>a) ⎛ ⎝waNa++ j Πj r ⎞ ⎠ , UkXjjV b2M/Q + j Πj<sub>f</sub>- + j ΠjI 2 + j Πjr Qb Hm+`Qb iQiBb /Qb b2iQ`2b m`#MQ 7Q`KH-m`#MQ BM7Q`KH 2 `m`H- `2bT2+iBpK2Mi2X lK p2x [m2  bQK / Kbb /2 bH€`BQ 2 / Kbb /2 Hm+`Q 2K mK b2iQ` T`Q/miBpQ M/ KBb û /Q [m2 Q pHQ` iQiH / T`Q/mϽQ /2bi2 b2iQ`- p2` TQ` 2t2KTHQ 2[X UkXjV- b2;m2 [m2,

pmDm= dm(pmYm+ psYs) + (1− da)paYa. UkXj9V SQ`iMiQ- Q 2[mBHő#`BQ MQ K2`+/Q /2 #2Mb KMm7im`/Qb Q+Q``2`€ [mM/Q  /2KM/ ;`2;/ B;mH`@b2  Q72`i ;`2;/- Dm= Ym- D€ [m2  iQ/Q KQK2MiQ bmTQKQb [m2 M½Q ?€ 2biQ[m2 /2 #2Mb Qm T`Q/miQ- Qm b2D,

pmYm= dm

(1_{− d}m)psYs+

(1_{− d}a)

(1_{− d}m)paYa. UkXj8V a2`pBÏQb BM7Q`KBb

C€ pBKQb [m2 MQ b2iQ` m`#MQ BM7Q`KH iQ/b b }`Kb +QMi`iK Q K2bKQ MȹK2`Q /2 i`#H?/Q`2b Nj

s- TQ`iMiQ- Q iQiH /2 i`#H?/Q`2b M2bi2 b2iQ` û Ns = ZsNsX . K2bK 7Q`K-  T`Q/mϽQ û  K2bK T` iQ/b bj T2[m2Mb }`Kb- b2M/Q bbBK-  Q72`i ;`2;/ /2 b2`pBÏQb BM7Q`KBb- Ys= ZsYsj- }+,

M [mH ξs≡ AsZs1−δ. UkXjdV  /2KM/ ;`2;/ /2 b2`pBÏQb BM7Q`KBb û B;mH Q +QMbmKQ iQiH /2 b2`pBÏQb BM7Q`KBb MQ b2iQ` m`#MQ- mK p2x [m2- TQ` /2}MBϽQ- Q b2iQ` `m`H M½Q miBHBx 2bb2b b2`pBÏQb, Ds≡+ j fsj+ + j Isj. UkXj3V

AMb2`BM/Q Q +QMbmKQ /2 b2`pBÏQb /2 +/ b2iQ` Up2D 2[bX UkXR3V 2 UkXkkVV M 2[mϽQ +BK 2 +QMbB/2`M/Q b `2bi`BÏǤ2b Q`ÏK2Mi`Bb UkXRdV 2 UkXkRV i2K@b2 mK /2KM/ ;`2;/ /2 b2`pBÏQb BM7Q`KBb // TQ`, psDs= ds ⎛ ⎝wmNm++ j Πj_f ⎞ ⎠ + ds ⎛ ⎝wsNs++ j Πj_I ⎞ ⎠ . UkXjNV A/2MiB}+M/Q  bQK / Kbb /2 bH€`BQ 2 / Kbb /2 Hm+`Q 2K mK b2iQ` T`Q/miBpQ +QKQ Q pHQ` iQiH / T`Q/mϽQ /2bi2 b2iQ`- i2KQb,

psDs= ds(pmYm+ psYs). UkX9yV

P 2[mBHő#`BQ MQ K2`+/Q /2 b2`pBÏQb BM7Q`KBb Q+Q``2`€- TQ`iMiQ- [mM/Q  /2KM/ ;`2;/ B;mH`@b2  Q72`i ;`2;/- Ds= Ys- Qm b2D,

psYs= ds

(1− ds)pmYm. UkX9RV

;`ő+QHb

LQ b2iQ` `m`H iQ/Qb Qb T`Q/miQ`2b ;`ő+QHb iāK K2bKQ MȹK2`Q /2 i`#H?@ /Q`2b- Nj

a- /2 KQ/Q [m2 Q iQiH /2 i`#H?/Q`2b M2bi2 b2iQ` û Na = ZaNajX a2M/Q bbBK-  Q72`i ;`2;/ /2 #2Mb ;`ő+QHb- Ya= ZaYj

a- }+,

Ya = ξa(Na)φ_{, UkX9kV}

M [mH

j9 *TőimHQ kX 1[mBHő#`BQ 2+QMƬKB+Q ;2`H +QK i`āb b2iQ`2b

SQ` /2}MBϽQ-  /2KM/ ;`2;/ /2 #2Mb ;`ő+QHb û B;mH Q +QMbmKQ iQiH /2bb2b #2Mb MQb i`āb b2iQ`2b- Qm b2D- M½Q ?€ 2biQ[m2 /2 #2Mb ;`ő+QHb,

Da_≡+ j faj+ + j Iaj+ + j raj. UkX99V

AMb2`BM/Q Q +QMbmKQ /2 #2Mb ;`ő+QHb /2 +/ b2iQ` Up2D 2[bX UkXR3V- UkXkkV 2 UkXkeVV M 2[mϽQ +BK 2 +QMbB/2`M/Q b `2bi`BÏǤ2b Q`ÏK2Mi`Bb UkXRdV-UkXkRV 2 UkXk8V i2K@b2 mK /2KM/ ;`2;/ /2 #2Mb ;`ő+QHb // TQ`,

paDa = (1_{− d}m− ds) ⎛ ⎝wmNm++ j Πj_f+ wsNs++ j Πj_I ⎞ ⎠ +da ⎛ ⎝waNa+ + j Πjr ⎞ ⎠ . UkX98V A/2MiB}+M/Q bQK / Kbb /2 bH€`BQ 2 / Kbb /2 Hm+`Q 2K mK b2iQ` T`Q/miBpQ +QKQ Q pHQ` iQiH / T`Q/mϽQ /2bi2 b2iQ` i2KQb,

paDa= (1_{− d}m− ds)(pmYm+ psYs) + dapaYa. UkX9eV P 2[mBHő#`BQ MQ K2`+/Q /2 #2Mb ;`ő+QHb Q+Q``2`€ [mM/Q  /2KM/ ;`2@ ;/ B;mH`@b2  Q72`i ;`2;/- Da= Ya- Qm b2D,

paYa= (1− dm− ds)

(1_{− d}a) (pmYm+ psYs). UkX9dV

S`2ÏQb /2 2[mBHő#`BQ

lK p2x [m2 +QMbB/2`KQb +QKQ pHQ` mMBi€`BQ Q T`2ÏQ pm/Qb #2Mb KMm7@ im`/Qb- /BbTQM/Q /b 2[mÏǤ2b UkX9RV 2 UkX9dV /2i2`KBMKQb Qb T`2ÏQb /Qb #2Mb ;`ő+QHb 2 /Qb b2`pBÏQb BM7Q`KBb- 2K mMB//2b /2 #2Mb KMm7im`/Qb-b2DK 2H2b, ps= ds (1_{− d}s) Ym Ys UkX93V 2 pa= (1− dm− ds) (1_{− d}a)(1_{− d}s) Ym Ya, UkX9NV

b2M/Q Ym- Ys 2 Ya +H+mH/Qb  T`iB` /b 2[mÏǤ2b UkXkNV- UkXjeV 2 UkX9kV-`2bT2+iBpK2Mi2X

kXkXk J2`+/Q /2 i`#H?Q

1K /2i2`KBM/Q T2`őQ/Q /2 i2KTQ ě HQM;Q Q bm}+B2Mi2 T` [m2  2+Q@ MQKB TQbb b2MiB` Qb 272BiQb +mb/Qb TQ` KB;`ÏǤ2b BMi`@b2iQ`BBb- Kb [m2 KMi2M?@b2 T`QtBK/K2Mi2 +QMbiMi2 M2bi2 T2`őQ/Q U/BMi2 `2H@ +BQM/Q +QK mK TbbQ /2 bBKmHϽQV ě bmTQKQb [m2 Qb i`#H?/Q`2b /2 [mH[m2` b2iQ` Q72`iK mK mMB//2 /2 i`#H?Q- /2 KQ/Q [m2  Q72`i iQiH /2 i`#H?Q 2K +/ b2iQ` 2[mBpH2 Q MȹK2`Q `2bT2+iBpQ /2 i`#H?/Q`2bX 1K i2`KQb 2+QMƬKB+Qb-  Q72`i /2 i`#H?Q û T2`72BiK2Mi2 BM2H€biB+  +m`iQ T`xQX

l`#MQ 7Q`KH

.2Mi`Q /b +QM/BÏǤ2b /2 2[mBHő#`BQ /2b+`Bib M b2ϽQkXkXR- b2 BMp2`i2`KQb  2[mϽQ UkXeV Ĝ [m2 `2H+BQM Q MȹK2`Q /2 i`#H?/Q`2b [m2 +/ }`K K@ Mm7im`2B` 2b+QH?2 +QMi`i` +QK Q bH€`BQ [m2 2bi 2b+QH?2 T;` Ĝ Q#i2KQb mK bH€`BQ }tQ- T;Q T2Hb }`Kb KMm7im`2B`b- B;mH ,

wm= ζm

(Nm)1−α, UkX8yV

+QK

ζm≡ αAm(Zm)1−αεα. UkX8RV

oH2 H2K#`` [m2  /2KM/ TQ` i`#H?Q MQ b2iQ` m`#MQ 7Q`KH û 2b@ i`BiK2Mi2 K2MQ` [m2 MȹK2`Q /2 i`#H?/Q`2b- TQ` BbbQ-  [mMiB//2 /2 i`#H?/Q`2b 2KT`2;/Qb Nmû mK 7`ϽQ / 7Q`Ï /2 i`#H?Q M2bb2 b2iQ`,

Nm= (1_{− u)N}f, UkX8kV

b2M/Q Nf Q MȹK2`Q iQiH /2 ;2Mi2b HQ+HBx/Qb MQ b2iQ` m`#MQ 7Q`KH 2 uû  it /2 /2b2KT`2;Q T2`KM2Mi2- 2tBbi2Mi2 bQK2Mi2 MQ b2iQ` 7Q`KH-// T2H 2[mϽQ UkXkdVX

l`#MQ BM7Q`KH

1K#Q` KMi2M?KQb  T`2KBbb /2 [m2  Q72`i /2 i`#H?Q M2bi2 K2`+/Q û T2`72BiK2Mi2 BM2H€biB+- /B72`2Mi2K2Mi2 /Q K2`+/Q /2 i`#H?Q

7Q`KH-je *TőimHQ kX 1[mBHő#`BQ 2+QMƬKB+Q ;2`H +QK i`āb b2iQ`2b

+QMbB/2``2KQb Q K2`+/Q /2 i`#H?Q BM7Q`KH +QKQ T2`72BiK2Mi2 +QKT2@ iBiBpQ- Qm b2D- Qb bH€`BQb T;Qb b½Q B;mBb 2K iQ/b b T2[m2Mb }`Kb 2 DmbiK@b2 /2 KQ/Q [m2 M½Q ?D /2b2KT`2;Q M2bi2 b2iQ`X

Lb K2bKb +QM/BÏǤ2b /2 2[mBHő#`BQ /Q b2iQ` m`#MQ 7Q`KH- b2 BMp2`i2`@ KQb  2[mϽQ UkXNV Q#i2KQb mK bH€`BQ `2H U//Q 2K i2`KQb / BM~ϽQV-T;Q T2HQb T`QT`B2i€`BQb /2 +TBiH MQ b2iQ` BM7Q`KH- B;mH ,

ws ps = ζs (Ns)1−δ, UkX8jV +QK ζs_{≡ δA}s(Zs)1−δ. UkX89V _m`H bbBK +QKQ Q K2`+/Q /2 i`#H?Q m`#MQ BM7Q`KH- pKQb /KBiB` [m2 Q K2`+/Q /2 i`#H?Q `m`H iK#ûK b2D T2`72BiK2Mi2 +QKT2iBiBpQX M€HQ;Q Qb Qmi`Qb /QBb b2iQ`2b-  T`iB` / 2[mϽQ UkXRkV- Q#i2KQb Q bH€`BQ `2H-T;Q T2HQb T`QT`B2i€`BQ /2 +TBiH MQ b2iQ` `m`H- B;mH ,

wa pa =

ζa

(Na)1−φ, UkX88V

+QK

ζa_{≡ φA}a(Za)1−φ. UkX8eV

aBbi2K 2+QMƬKB+Q

*QKQ TQ/2@b2 MQi` Mb b2ÏǤ2b Mi2`BQ`2b-  /Bbi`B#mBϽQ /2 TQTmHϽQ 2K +/ b2iQ` û  #b2 /Q +€H+mHQ T`  2+QMQKB /2 mK Tőb 2K /2b2MpQHpB@ K2MiQX ./ mK /Bbi`B#mBϽQ b2iQ`BH /2 i`#H?/Q`2b UNm-Ns-NaV 2 b +QMbiMi2b T`Kûi`B+b UAm- α- Zm- dm- η- b- As- δ- Zs- ds- Aa- φ- Za- daV-TQ/2KQb +H+mH` b T`Q/mÏǤ2b ;`2;/b /2 +/ b2iQ` UYm-Ys-YaV- //b T2Hb 2[mÏǤ2b UkXkNV- UkXjeV 2 UkX9kV- `2bT2+iBpK2Mi2X 1bb2b pHQ`2b- TQ` bm p2x- TQ/2K b2` BMb2`B/Qb Mb 2[mÏǤ2b UkX93V 2 UkX9NV T` Q#i2`KQb Qb T`2ÏQb /Qb b2`pBÏQb 2 /Qb #2Mb ;`ő+QHb- `2bT2+iBpK2Mi2X G2K#`M/Q [m2 iQKKQb Q T`2ÏQ /Q #2K KMm7im`/Q +QKQ MmK2`€`BQ (j)- i2KQb 2Mi½Q Qb T`2ÏQb /Qb T`Q/miQb Upm≡ 1-ps-paVX 6BMHK2Mi2- 2bb2b T`2ÏQb TQ/2K b2`

bm#iBimő/Qb Mb 2[mÏǤ2b UkX8yV- UkX8jV 2 UkX88V T` +H+mH`KQb Qb bH€`BQb T;Qb T2Hb }`Kb /2 +/ b2iQ`- `2T2+iBpK2Mi2 Uwm-ws-waVX

_2bmKBM/Q- //b b +QMbiMi2b T`Kûi`B+b UAm- α- Zm- dm- η- b-As- δ- Zs- ds- Aa- φ- Za- daV- TQ/2KQb +H+mH` Q K+`Q@2bi/Q 2+QMƬKB+Q UYm-Ys-Ya-pm-ps-pa-wm-ws-waV /2 mK Tőb 2K T`Q+2bbQ /2 m`#MBxϽQ +Q@ M?2+2M/Q T2Mb  bm /Bbi`B#mBϽQ b2iQ`BH /2 i`#H?/Q`2b UNm-Ns-NaVX

JB;`ϽQ BMi2`b2iQ`BH, mK KQ/2HQ /2 K2+MB+ 2biiőbiB+

L2bi2 +TőimHQ /2b+`2p2`2KQb Q KQ/2HQ /2 BMi2`ϽQ 2Mi`2 Qb ~mtQb KB;`@ iƦ`BQb 2 Qb T`Q+2bbQb /2 Km/MÏb M 2+QMQKBX G2pKQb 2K +QMbB/2`ϽQ /mb BM~māM+Bb /2i2`KBMőbiB+b T`  /2+Bb½Q /Qb ;2Mi2b, 2+QMƬKB+ 2 bQ+BHX .2 +Q`/Q +QK  T`QTQbi / `272`āM+B (Re)- +?KKQb /2 miBHB//2

/2i2`KBMőbiB+ T`Bp/  BM~māM+B 2+QMƬKB+ 2 +?KKQb /2 miBHB//2 /2@ i2`KBMőbiB+ bQ+BH  BM~māM+B bQ+BHX HûK /2bbb- H2pKQb 2K +QMbB/2`ϽQ

 T`i2 B/BQbbBM+`€iB+ UM½Q@Q#b2`p€p2HV MQ T`Q+2bbQ /2 KB;`ϽQ- /2}MB/ +QKQ miBHB//2 T`Bp/ H2iƦ`Bc /2b+`2p2`2KQb K2H?Q` 2bi miBHB//2 /B@ Mi2X

 T`QTQbi û [m2 +/ ;2Mi2- MQ T`Q+2bbQ /2 2b+QH? /2 b2iQ`- T`Q+m` KtBKBx` bm miBHB//2 iQiH U- Qm b2D-  bQK /2 bmb miBHB//2b /2i2`@ KBMőbiB+b- T`Bp/ Up _{2 bQ+BH U}s_{- 2 bm miBHB//2 H2iƦ`B ε}R_,

U(σi) =_Up(σi) +_Us(σi) + ε(σi), UjXRV b2M/Q [m2 σi`2T`2b2Mi Q b2iQ` MQ [mH Q ;2Mi2 i 2bi€ HQ+HBx/QX

Lb b2ÏǤ2b b2;mBMi2b p2`2KQb [m2  KtBKBxϽQ / miBHB//2 TQ/2 b2` 2b+`Bi +QKQ  KBMBKBxϽQ /2 mK ?KBHiQMBM 2 [m2- TQ` +mb / ?2i2`Q@ ;2M2B//2 /Qb ;2Mi2b- Q T`Q#H2K û 2[mBpH2Mi2 Q /2 mK bBbi2K 7őbB+Q /2 bTBMb 2K +QMiiQ +QK mK `2b2`piƦ`BQ iû`KB+QX 1bb +QKT`ϽQ 7QB 7mM/@ K2MiH T` b BMi2`T`2iÏǤ2b /Qb `2bmHi/QbX 1K b2;mB/- /2b2MpQHp2`2KQb Ki2KiB+K2Mi2  miBHB//2 M½Q@Q#b2`p€p2H 2- /2pB/Q ¨ bm H2iQ`B2//2-HMÏKQb K½Q /2 mK `iB7ő+BQ +QKTmi+BQMH +QM?2+B/Q +QKQ bBKmHϽQ /2

R _{L½Q +QM7mM/B`  miBHB//2 H2iƦ`B +QK Q 2b7Q`ÏQ /2 +/ i`#H?/Q` /2}MB/Q MQ}

9y *TőimHQ jX JB;`ϽQ BMi2`b2iQ`BH, mK KQ/2HQ /2 K2+MB+ 2biiőbiB+

JQMi2 *`HQX SQ` }K- /Bb+miB`2KQb +QK /2iH?2b Qb T`Q+2bbQb [m2 +QKTǤ2K iH bBKmHϽQX

jXR liBHB//2b /2i2`KBMőbiB+b

jXRXR liBHB//2 T`Bp/

1K i2`KQb 2+QMƬKB+Qb-  miBHB//2 /2 mK ;2Mi2 2K /2i2`KBM/Q b2iQ` û T`QTQ`+BQMH ¨b +QM/BÏǤ2b /2 K2`+/Q /2 i`#H?Q- 2K Qmi`b THp`b-[mMiQ KBQ` 7Q`  +?M+2 /2 b2` 2KT`2;/Q 2 THp`b-[mMiQ KBQ` 7Q` Q bH€`BQ `2+2#B/Q 2Mi½Q KBQ` b2`€ bm miBHB//2 M[m2H2 b2iQ`X .2}M2K@b2 b miBHB@ //2b T`Bp/b /Qb b2iQ`2b 7Q`KH- BM7Q`KH 2 `m`H- `2bT2+iBpK2Mi2- +QKQ, Uf ormalp ≡ k ¯wm ; U p inf ormal ≡ k ¯ws ; U p rural≡ k ¯wa UjXkV Mb [mBb k û mK +QMbiMi2 T`Kûi`B+ 2 ( ¯wm, ¯ws, ¯wa) b½Q +?K/Qb /2 bH€`BQb 2bT2`/Qb-  b#2`- +QKQ bƦ ?€ /2b2KT`2;Q MQ b2iQ` 7Q`KH- 2K MQbbQ KQ/2HQ i2KQb ¯wm = (1_{− bu)w}m Up2` 2[mϽQ UkXkVV- ¯ws = ws 2

¯ wa = waX *QM?2+2M/Q b miBHB//2b T`Bp/b /2 +/ b2iQ`- /2 KQ/Q ;2`H- TQ/2@ KQb 2b+`2p2`  miBHB//2 T`Bp/ M 7Q`K, Up(σi) = k ' ¯ wmσi(σi+ 1) 2 + ¯ws σi(σi<sub>− 1)</sub> 2 + ¯wa(1− σ 2 i) ( , UjXjV +QK σi= ⎧ ⎪ ⎨ ⎪ ⎩ +1 T` b2iQ` 7Q`KH −1 T` b2iQ` BM7Q`KH 0 T` b2iQ` `m`H, UjX9V

Qm b2D- [mM/Q iQKKQb σi = +1 M 2[X UjXjV `2iQ`MKQb ¨ miBHB//2 T`Bp/ 7Q`KH- T` σi=<sub>−1 i2KQb  miBHB//2 T`Bp/ BM7Q`KH 2 T` σ</sub>i= 0i2KQb  miBHB//2 T`Bp/ `m`HX  }K /2 bBKTHB}+` MQbb BMi2`T`2iϽQ-7+BHK2Mi2 KQbi`@b2 [m2  miBHB//2 T`Bp/- 2[X UjXjV- TQ/2 b2` `22b+`Bi M 7Q`K, Up_{(σi) = k} ') ¯ wm+ ¯ws 2 − ¯wa * σ2i + ) ¯ wm− ¯ws 2 * σi+ ¯wa ( . UjX8V

.Q TQMiQ /2 pBbi 2+QMƬKB+Q- Q T`BK2B`Q i2`KQ / 2[mϽQ +BK H2p 2K +QMi mK +QKT`ϽQ 2Mi`2 Qb bH€`BQb 2bT2`/Qb / €`2 `m`H 2 / €`2 m`#MX P b2;mM/Q i2`KQ û `2H2pMi2 T`  /2+Bb½Q /2 2b+QH? /Q ;2Mi2 2Mi`2 Qb b2iQ`2b 7Q`KH 2 BM7Q`KHX P i2`+2B`Q i2`KQ û mK +QMbiMi2 B``2H2@ pMi2- mK p2x [m2 2bi`2KQb BMi2`2bb/Qb M /B72`2MÏ /2 miBHB//2b- 2 b2`€ QKBiB/ /[mB TQ` /BMi2X

jXRXk liBHB//2 bQ+BH

LQ KQ/2HQ /2 KB;`ϽQ-  BM~māM+B bQ+BH û `2T`2b2Mi/ T2H miBHB//2 bQ+BH 2 H2p Q ;2Mi2  KB;`` 2Mi`2 /B72`2Mi2b €`2b- Qm b2D- 2H `2T`2@ b2Mi  i2M/āM+B /Q ;2Mi2 2K KB;`` / €`2 `m`H T`  €`2 m`#M 2 pB+2@p2`bX JBb T`2+BbK2Mi2-  miBHB//2 bQ+BH M½Q /BbiBM;m2 Qb b2iQ`2b 7Q`KH 2 BM7Q`KH- Qm b2D-  BM~māM+B bQ+BH M½Q 7x +QK [m2 mK ;2Mi2 `2bQHp Km/` b2m b2iQ` /2 i`#H?Q /2Mi`Q / €`2 m`#MX P +ő`+mHQ bQ+BH /2 mK ;2Mi2 û [m2 7`€ +QK [m2 bm miBHB//2 bQ+BH b2D KBQ` Qm K2MQ` 2K /2i2`KBM/ €`2- BX2X- 2H b2`€ KBQ` QM/2 2biBp2`2K  KBQ`B /2 b2mb +QMiiQb T2bbQBb Ub2DK 2H2b T`2Mi2b- KB;Qb- pBxBM?Qb- 2i+XVX Ji2KiB+@ K2Mi2- 2bb miBHB//2 bQ+BH i`/mx@b2 +QKQ,

Us_{(σi) = J}+ j

δσ2

i,σ2j, UjXeV

M [mH J (> 0)k _{û mK +QMbiMi2 T`Kûi`B+-  bQK 2K j û bQ#`2 Qb}

T`BK2B`Qb pBxBM?Qb /Q ;2Mi2 i- δa,b û Q /2Hi /2 E`QM2+F2` ě δa,b = 1 b2 a = b 2 δa,b = 0 +bQ +QMi`€`BQ ě 2- TQ` }K- σi b2;m2 b `2;`b / 2tT`2bb½Q UjX9VX ú BKTQ`iMi2 MQi` [m2 T`BK2B`Qb pBxBM?Qb M½Q bB;MB}+K M2+2bb`BK2Mi2 pBxBM?Qb ;2Q;`€}+Qb- Kb bBK  `2/2 /2 +QMiiQb /Q ;2Mi2 iX L2bb2 b2MiB/Q- TQ` 2t2KTHQ- T` mK ;2Mi2 HQ+HBx/Q M €`2 m`#M- +/ ;2Mi2 T2`i2M+2Mi2  bm €`2 /2 BM~māM+B- +QMi`B#mB +QK mK mMB//2 J T`  miBHB//2 bQ+BHX

k _{ +QMbiMi2 J /2p2 b2` TQbBiBp mK p2x [m2 +QMiiQb T2bbQBb HQ+HBx/Qb MQ K2bKQ}

9k *TőimHQ jX JB;`ϽQ BMi2`b2iQ`BH, mK KQ/2HQ /2 K2+MB+ 2biiőbiB+

lK KQ/2HQ /2 K2+MB+ 2biiőbiB+

iû ;Q`- pBKQb /2}MBÏǤ2b 2 +QM+2BiQb `2H2pMi2b ¨ T`i2 2+QMƬKB+ /2 mK Tőb 2K /2b2MpQHpBK2MiQX Jb QM/2 b2 2M+Bt  7őbB+ MBbbQ im/Q\ SQBb #2K- +QK2Ï`2KQb  DmMi` b T2Ïb MHBbM/Q Q +QM+2BiQ /2 miBHB//2X 1K 2+QMQKB- mK ;2Mi2 b2KT`2 T`Q+m` KtBKBx` bm miBHB//2- 2[mBpH2M@ i2K2Mi2- 2K i2`KQ/BMKB+- Q 2bi/Q /2 2[mBHő#`BQ /2 mK bBbi2K û //Q T2H KBMBKBxϽQ /2 bm 2M2`;B HBp`2- /2b+`Bi Ki2KiB+K2Mi2 i`pûb / 7mMϽQ /2 T`iBϽQ U2- +QMb2[m2Mi2K2Mi2- /2 bm ?KBHiQMBMVX 1K i2`@ KQb 2biiőbiB+Qb- TQ/2KQb bbQ+B` Q M2;iBpQ / miBHB//2 ¨ ?KBHiQMBMX oKQb 2Mi½Q 2b+`2p2`  miBHB//2 /2 KM2B` [m2 TQbbKQb +QKT`€@H +QK H;mK ?KBHiQMBM D€ +QM?2+B/X *QK2ÏKQb `22b+`2p2M/Q  miBHB//2 bQ@ +BH /2 KM2B`  2HBKBM` Q /2Hi /2 F`QM2+F2`- bbBK  2[mϽQ UjXeV TQ/2 b2` `22b+`Bi M 7Q`K, Us(σi) = 2J+ j ' σi2σj2− 1 2(σ 2 i + σj2− 1) ( . UjXdV

P i2`KQ −1 /2Mi`Q /Q T`āMi2b2b `2bmK2@b2 2K mK +QMbiMi2 [m2 b2`€ QKBiB/ /[mB TQ` /BMi2- mK p2x [m2 2bi`2KQb BMi2`2bb/Qb M /B72`2MÏ /2 miBHB//2bX .BbTQM/Q /b 2[mÏǤ2b UjX8V 2 UjXdV- 2biKQb TiQb  2b+`2p2`  miBHB//2 /2i2`KBMőbiB+- Ud ≡ Up₊ Us_{- +QKQ,} Ud(σi) = k )_{wm¯ + ¯ws} 2 − ¯wa * σi2+ k )_wm¯ − ¯ws 2 * σi +2J+ j ' σi2σ2j− 1 2(σ 2 i + σ2j) ( . UjX3V

 }K /2 7+BHBi` MQbb BMi2`T`2iϽQ /2 ?KBHiQMBM- 7`2KQb H;mKb KMBTmHÏǤ2b H;û#`B+b M 2[mϽQ UjX3VX S`BK2B`Q- iQKKQb Q M2;iBpQ / miBHB//2 /2i2`KBMőbiB+- /2TQBb bQKKQb bQ#`2 iQ/Qb Qb ;2Mi2b i 2 bbBK Q#i2KQb Q M2;iBpQ / miBHB//2 /2i2`KBMőbiB+ /2 iQ/Qb Qb ;2Mi2b UQm  ?KBHiQMBM /2 iQ/Q Q bBbi2KVX 62BiQ BbbQ- TQ/2KQb bm#biBimB`  `2T`2b2M@ iϽQ Ud <sub>TQ` H 2 }+KQb +QK,</sub> H(σi) = k1+ i σ2i + k2 + i σi+ 4J + <i,j> ' 1 2(σ 2 i + σ2j)− σi2σj2 ( . UjXNV

i2M/Q /2}MB/Q k1≡ k ) ¯ wa−wm¯ ₂+ ¯ws * 2 k2≡ k )_ws¯ − ¯wm 2 * X P HBKBi2 BM@ 72`BQ` /Q bQKiƦ`BQ- < i, j >- bB;MB}+  bQK bQ#`2 iQ/Qb Qb T`2b /2 T`BK2B@ `Qb pBxBM?Qb-  [mH 7QB Q#iB/ mbM/Q  `2HϽQ + <i,j> (_{· · · ) =} 1 2 + i + j (_{· · ·} )-b2M/Q [m2  bQK 2K j û bQ#`2 Qb T`BK2B`Qb pBxBM?Qb /2 iX S2`+2#2M/Q BM/ [m2 Qb i2`KQb σ2

i 2 σ2j- /Q ȹHiBKQ bQKiƦ`BQ / 2[X UjXNV- +QMi`B#m2K B;mH@ K2Mi2 T` Q ?KBHiQMBMQ- TQ/2KQb `22b+`2pā@HQ MQpK2Mi2 +QKQ,

H(σi) = (k1+ 2zJ)+ i σ2 i + k2 + i σi_{− 4J} + <i,j> σ2 iσ2j UjXRyV b2M/Q z Q MȹK2`Q /2 T`BK2B`Qb pBxBM?QbX 6BMHK2Mi2- TQ/2KQb B/2MiB}+`  2[mϽQ +BK +QKQ mK ?KBHiQMBM /2 "HmK2@1K2`v@:`B{i?b U"1:V (Rd)- +QK Q i2`KQ #BHBM2` UT`QTQ`+BQMH  σiσjV MmHQ 2 /B+BQM/ /2 mK +KTQ K;MûiB+Q 2ti2`MQ Ui2`KQ [m2 +QTH +QK σiVX "1:- MQ `272`B/Q `iB;Q- /2b+`2p2K mK KBbim` /2 >23<sub>@>2</sub>4 <sub>2 KQbi`K- Q K2MQb [mHBiiB@</sub> pK2Mi2- b +`+i2`őbiB+b /2 b2T`ϽQ /2 7b2 2 bmT2`~mB/2x /2bb KBbim`X 1Mi2M/2M/Q K2H?Q`  2[mϽQ UjXRyV- Q i2`KQ 2K σi 2bi€ `2H+BQM/Q  mK +KTQ 2ti2`MQ THB+/Q 2  bm Kû/B < σi > MQb /€- TQ` 2t2KTHQ-  K;M2iBxϽQ /2 mK bBbi2K /2 bTBMbX LQ KQ/2HQ /2 "1:-  bmT2`~mB/2x / KBbim` >23_@>24 <sub>û `2T`2b2Mi/ T2HQ Q`/2MK2MiQ K;MûiB+Q /2bb2b bTBMbX</sub> *?KKQb /2 +KTQ +`BbiHBMQ Q +Q2}+B2Mi2 /2 σ2 i- MQ +bQ- (k1+ 2zJ)X 1K mK bBbi2K /2 bTBMb  Kû/B < σ2 i >`272`2@b2 Q KQK2MiQ /2 [m/`m@ TQHQ 2 MQ KQ/2HQ /2 "1: Q Q`/2MK2MiQ /2bb2b KQK2MiQb /2 [m/`mTQHQ `2T`2b2Mi  b2T`ϽQ /2 7b2 / KBbim` >23<sub>@>2</sub>4<sub>X SQ` }K- /Bx@b2 b2` Q</sub> i2`KQ /2 BMi2`ϽQ #B[m/`€iB+ [m2H2 [m2 +QTH +QK σ2

iσ2jX S` "1:-2bi2 i2`KQ TQ/2 b2` pBbiQ +QKQ mK T2[m2M T2`im`#ϽQ +mb/ T2HQb /B72`2Mi2b pHQ`2b /b i`āb TQbbőp2Bb BMi2`ÏǤ2b 2Mi`2 Qb €iQKQb U>23_@>23 ->23_@>24 _{2 >2}4_@>24_VX

jXk liBHB//2 M½Q Q#b2`p€p2H UH2iƦ`BV

a#2KQb [m2 mK ;2Mi2 /2+B/2 KB;`` /2 €`2 Qm Km/` /2 b2iQ` #mb+M/Q bm K€tBK miBHB//2 2 pBKQb M b2ϽQ Mi2`BQ`  miBHB//2 /2i2`KBMőbiB+ /2 mK ;2Mi2- Qm b2D- /2+BbǤ2b +QKmMb  iQ/Qb Qb ;2Mi2b- iH [m2 mK

99 *TőimHQ jX JB;`ϽQ BMi2`b2iQ`BH, mK KQ/2HQ /2 K2+MB+ 2biiőbiB+

Q#b2`p/Q` 2ti2`MQ TQbb /2i2+i€@HbX 1Mi`2iMiQ- Q T`Q+2bbQ /2 KB;`ϽQ iK#ûK 2MpQHp2 /2+BbǤ2b #b2/b 2K BMi2`2bb2b T2bbQBb /2 +/ ;2Mi2-b2K [m2 mK Q#b2`p/Q` TQbb B/2MiB}+€@Hb, bbQ+BKQb  miBHB//2 H2iƦ`B  2bbb /2+BbǤ2bX .2pB/Q  miBHB//2 H2iƦ`B-  KB;`ϽQ /2 +/ ;2Mi2 iQ`M@b2 mK T`Q+2bbQ 2biQ+€biB+Q- Q [mH /BbTQKQb /2 mK bBKmHϽQ T` 2bim/€@HQX L2+2bb`BK2Mi2- +/ ;2Mi2 KB;` T` Q b2iQ` [m2 H?2 7Q`M2+2  KBQ` miBHB//2 iQiH U/2i2`KBMőbiB+ 2 H2iƦ`BVc bbBK Q ;2Mi2 i /`€ T`272`āM+B Q b2iQ` σi 2K `2HϽQ Q b2iQ` σ′

i b2, Ud_{(σi) + ε(σi) >}

Ud_(σ′_{i) + ε(σ}′_{i), ∀ σ}′

i̸= σi UjXRRV H2K#`M/Q [m2 Ud<sub>(σi)- 2[X UjX3V- û  miBHB//2 /2i2`KBMőbiB+ /Q ;2Mi2 i MQ</sub> b2iQ` σi 2 ε(σi)- p2D 2[X UjXRV- û  miBHB//2 H2iƦ`B /Q ;2Mi2 i MQ b2iQ` σi UQ K2bKQ pH2 T` iQ/Qb Qb Qmi`Qb b2iQ`2b σ′iVX

T2b` /Q MQK2- bmTQM/Q [m2 b miBHB//2b H2iƦ`Bb εi ≡ ε(σi),∀i, b2DK B/2MiB+K2Mi2 2 BM/2T2M/2Mi2K2Mi2 /Bbi`B#mő/b- 2Mi½Q 2bbb 2b+QH?b M½Q Q#b2`p€p2Bb b½Q #2K /2b+`Bib TQ` mK /Bbi`B#mBϽQ /2 pHQ` 2ti`2KQ Qm- /2 7Q`K 2[mBpH2Mi2- T2H 7mMϽQ HQ;Bi (R3),

f (εi) = e−εi2tT(−e−εi_). UjXRkV lK p2x +QM?2+B/b b /Bbi`B#mBÏǤ2b /2 T`Q##BHB//2b /b p`B€p2Bb H2@ iƦ`Bb U//b T2H 2[mϽQ +BKV- T`Q+m`KQb  T`Q##BHB//2 /2 [m2  +QM/BϽQ UjXRRV Q+Q``- Qm bBKBH`K2Mi2

ε(σ′i) < εi+_Ud(σi)_{− U}d(σ′i), _{∀ σ}′i̸= σi, UjXRjV T` //Qb pHQ`2b /2 εi 2 σ′<sub>iX 1bi T`Q##BHB//2- /2}MB/ +QKQ,</sub> Pεi,σ′i(σi)≡ S`Q#[ ε′i< εi+ ∆U ] , -∆_{U ≡ U}d_(σi) − Ud_(σ′_i) ε′i ≡ ε(σ′i) UjXR9V û +H+mH/ T2H /Bbi`B#mBϽQ /2 T`Q##BHB//2 +mKmH/ / p`B€p2H H2@ iƦ`B ε′_{i- Qm} b2D-Pεi,σ′i(σi) = 2 εi+∆U −∞

*QM7Q`K2 7Q` bmTQbiQ- b p`B€p2Bb H2iƦ`Bb ε′

i b½Q BM/2T2M/2Mi2b 2- TQ` BbbQ-  T`Q##BHB//2 T` [m2  +QM/BϽQ UjXRjV Q+Q`` T` iQ/Qb Qb TQbbő@ p2Bb pHQ`2b /2 σ′

i iQ`M@b2 Q T`Q/miQ /b /Bbi`B#mBÏǤ2b +mKmH/b /2 +/ p`B€p2H,

Pεi(σi) = , σ′_i̸=σi

Pεi,σ′i(σi). UjXReV *QKQ M½Q [m2`2KQb +H+mH`  T`Q##BHB//2 /Q ;2Mi2 i 2b+QH?2` Q b2iQ` σi T` mK //Q pHQ` /2 εi- Kb bBK T` [mH[m2` pHQ` /Q K2bKQ- 2Mi½Q  T`Q##BHB//2 [m2 T`Q+m`KQb û // T2H BMi2;`H /2 Pεi(σi)bQ#`2 iQ/Qb Qb pHQ`2b /2 εi T2b/Qb TQ` bmb 7mMÏǤ2b /2MbB//2b UjXRkV- iH [m2,

P (σi)_≡ 2 ∞

−∞

Pεi(σi)f (εi)dεi. UjXRdV _2bQHp2M/Q  BMi2;`H +BK- TƦb H;mKb KMBTmHϽQ H;û#`B+b Up2D TāM/B+2V- +?2;@b2 2K, P (σi) = eU d_(σ i) + σ′_i eUd(σ′i) . UjXR3V 1bb T`Q##BHB//2 /2 2b+QH? /2 b2iQ` TQ/2 b2` /Ti/ T` mK M€@ HBb2 2KTő`B+ 2bT2+B}+M/Q  7Q`K 7mM+BQMH / miBHB//2X oKQb bmTQ` [m2  miBHB//2 /2i2`KBMőbiB+ p`B2 HBM2`K2Mi2 +QK bmb K2/B/b Q#b2`p€p2Bb-BX2X- Ud _{= β}

Up+ β_Us _{→ βU}dX 1K mK T`BK2B`Q KQK2MiQ TQ/2KQb `;m@ K2Mi` [m2- 2K +2`iQ HBKBi2- [mH[m2` 7mMϽQ TQ/2 b2` T`QtBK/ TQ` mK 7mMϽQ HBM2` /2 H;mMb T`K2i`QbX JBb /Q [m2 BbiQ- J+6//2M (RN) KQb@ i`Qm [m2  7mMϽQ /2 T`Q##BHB//2 HQ;`őiKB+ UdzHQ;@HBF2HB?QQ/ 7mM+iBQMǴV û +ƬM+p 2K `2HϽQ  β- Q [m2 7+BHBi Qb T`Q+2/BK2MiQb /2 KtBKBxϽQ MmKû`B+X HûK /Q KBb- H;mMb T+Qi2b MmKû`B+Qb /2 KtBKBxϽQ /2 7mM@ ÏǤ2b dzHQ;BiǴ mbK T`K2i`Qb HBM2`2b T` `2T`2b2Mi`  miBHB//2X a2M/Q bbBK-  2b+QH? /Q b2iQ` σiT2HQ ;2Mi2 i- /2pB/Q  /2+BbǤ2b M½Q Q#b2`p€p2Bb-b2;m2  7mMϽQ /2 T`Q##BHB//2, P (σi) = e βUd (σi) + σ′_i eβUd(σ′i). UjXRNV

9e *TőimHQ jX JB;`ϽQ BMi2`b2iQ`BH, mK KQ/2HQ /2 K2+MB+ 2biiőbiB+

1K i2`KQb 2+QMƬKB+Qb- Q T`K2i`Q β 2bi€ `2H+BQM/Q ¨ ?2i2`Q;2M2B//2 /Qb ;2Mi2b- b2M/Q BMp2`bK2Mi2 T`QTQ`+BQMH  2H UBX2X- [mMiQ K2MQ` 7Q` Q T`K2i`Q β- KBQ` b2`€  ?2i2`Q;2M2B//2 /Qb ;2Mi2b 2- TQ`iMiQ- KBb H2iƦ`Bb b2`½Q b /2+BbǤ2b iQK/bVX 1K T`iB+mH`- MQbbQ KQ/2HQ i`i i`āb b2iQ`2b- [m2 b2`½Q `2T`2b2Mi/Qb TQ` σi- σ′

i2 σ′′iX *QKQ  `2T`2b2MiϽQ [mMiBiiBp / miBHB//2 M½Q i2K 2K ;2`H bB;MB}+/Q 2+QMƬKB+Q- TQ/2KQb 2b+`2p2`  2[mϽQ UjXRNV M 7Q`K,

P (σi) = 1

1 + eβ[Ud_(σ′i)−Ud(σi)]_{+ e}β[Ud(σ′′i)−Ud(σi)]. UjXkyV

H;Q`BiKQ /2 JQMi2 *`HQ

LQ +QMi2tiQ / 7őbB+- b bBKmHÏǤ2b /2 72MƬK2MQb 7őbB+Qb- #b2/b 2K KQ@ /2HQb /2 K2+MB+ 2biiőbiB+- mbK H;Q`BiKQb /2 JQMi2 *`HQ T` `2T`Q@ /mxB` 272BiQb H2iƦ`BQbX hBb H;Q`BiKQb b2;m2K #bB+K2Mi2 /mb `2;`b,

2`;Q/B+B//2 2 #HMÏQ /2iH?/Q (ky)X  T`BK2B` ;`Mi2 [m2 M½Q 2biKQb BbQHM/Q [mH[m2` KB+`Q@2bi/Q TQbbőp2H [m2 mK bBbi2K TQbb H+MÏ`X C€ Q #HMÏQ /2iH?/Q MQb ;`Mi2 [m2 Q bBbi2K +?2;`€  mK 2[mBHő#`BQ-//Q T2H /Bbi`B#mBϽQ /2 T`Q##BHB//2b /2b2D/X .B72`2Mi2b /Bbi`B#mBÏǤ2b /2 T`Q##BHB//2b /2}M2K /B72`2Mi2b H;Q`BiKQbX *QKT`M/Q MQbbQ KQ/2HQ +QK bBKmHÏǤ2b /2 JQMi2 *`HQ- TQ/2KQb H2`  2[mϽQ UjXRNV UQm- KBb T`2+BbK2Mi2-  2[mϽQ UjXkyVV +QKQ  T`Q##BHB//2 /2 2M+QMi`KQb mK /2i2`KBM/Q ;2Mi2 i MQ 2bi/Q σi T` Q H;Q`BiKQ /2 #M?Q iû`KB+Q (ky)X

jXj .BMKB+ /2 KB;`ϽQ

P [m2 }x2KQb iû ;Q` 7QB /2b2MpQHp2` mK KQ/2HQ T`  2+QMQKB /2 mK Tőb 2K /2b2MpQHpBK2MiQ- /BpB/B/ 2K i`āb b2iQ`2b /2 i`#H?Q- 2 Q +QK@ TQ`iK2MiQ ;2`H / TQTmHϽQ M 2b+QH? /Q b2iQ` /2 i`#H?QX oBKQb M b2ϽQ jXk [m2 2bbb 2b+QH?b iQ`MK@b2 T`Q+2bbQb 2biQ+€biB+Qb /2pB/Q  7@ iQ`2b T2bbQBb UM½Q Q#b2`p€p2BbV 2MpQHpB/QbX SQ` BbbQ- T` /2b+`2p2`KQb Q 72MƬK2MQ /2 Km/MÏ /2 b2iQ`2b 2K mK Tőb 2K /2b2MpQpBK2MiQ- /m`Mi2 +2`iQ T2`őQ/Q /2 i2KTQ- TQ/2KQb `2+Q``2`  mK bBKmHϽQ /2 JQMi2 *`HQX a2M/Q bbBK- /2b+`2p2`2B ;Q` /2iH?/K2Mi2  /BMKB+ 2 H;mKb 2b@

+QH?b UiBb +QKQ Q iBTQ /2 `2/2- Q MȹK2`Q /2 pBxBM?Qb- b +QM/BÏǤ2b /2 +QMiQ`MQ- 2Mi`2 Qmi`bV +2`+ /2bb bBKmHϽQX

1b+QH?2KQb `2T`2b2Mi` Q Tőb TQ` mK `2/2 [m/`/- M [mH +/ ;2Mi2 2bi€ HQ+HBx/Q 2K T2Mb mK bőiBQ /2bb `2/2 2 /QBb ;2Mi2b M½Q Q+mTK Q K2bKQ bőiBQ- b2K [m2 ?DK bőiBQb pxBQbX S` mK //Q MȹK2`Q /2 ;2Mi2b- /Bx2KQb [m2  `2/2 i2K iKM?Q HBM2` L- Qm b2D- Q iQ/Q  `2/2 [m/`/ i2K N = L2 <sub>bőiBQb U;2Mi2bVX ./Q mK BMbiMi2 /2</sub> i2KTQ-+/ ;2Mi2 TQ/2 2bi` 2K T2Mb mK /Qb i`āb 2bi/Qb TQbbőp2Bb- b2DK 2H2b, σi=<sub>−1- 0 Qm +1- `2T`2b2MiM/Q Qb b2iQ`2b BM7Q`KH- `m`H Qm 7Q`KH- `2bT2+@</sub> iBpK2Mi2- p2` };m` RX ú BKTQ`iMi2 /2bi+` [m2 ;2Mi2b HQ+HBx/Qb 2K bőiBQb pBxBM?Qb M½Q `2T`2b2MiK M2+2bb`BK2Mi2 mK pBxBM?MÏ ;2Q;`€}+-Kb bBK b BM~māM+Bb bQ+BBb UQm +ő`+mHQ bQ+BHV /2bb2b ;2Mi2bX BM/ M2bb `2/2- QTiKQb TQ` mb` +QM/BÏǤ2b /2 +QMiQ`MQ ?2HB+QB/BbX S` bBbi2Kb bm@ }+B2Mi2K2Mi2 ;`M/2b- 2bT2`@b2 [m2 /B72`2Mi2b +QM/BÏǤ2b /2 +QMiQ`MQ M½Q BM~m2M+B2K MQ 2[mBHő#`BQ /Q bBbi2K Kb T2Mb MQ i2KTQ /2 +?2;/ Q 2[mBHő#`BQX 1K MQbb bBKmHϽQ- Q bBbi2K +?2; Q 2[mBHő#`BQ `TB/K2Mi2-bbBK b2M/Q-  2b+QH? / +QM/BϽQ /2 +QMiQ`MQ iQ`M@b2 T`iB+K2Mi2 B`@ `2H2pMi2X SQ` }K- /2+B/BKQb [m2 b BMi2`ÏǤ2b b2`BK /Q iBTQ T`BK2B`Qb

pBxBM?Qb- Qm b2D- mK ;2Mi2 i2K BMi2`ϽQ bQ+BH bQK2Mi2 +QK b2mb pBxB@

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