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▼❆❘❈❖❙ P❆❯▲❖ ❉❊ ❖▲■❱❊■❘❆ ▲❖❯❘❊■❘❖

P❘❖P❆●❆➬➹❖ ❉❊ ❉❆◆❖❙ ◆❖

▼❖❉❊▲❖ ❉❊ P❖❚❚❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐❣ê♥✲ ❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❆♣❧✐❝❛❞❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

❱■➬❖❙❆

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Ficha catalográfica preparada pela Seção de Catalogação e Classificação da Biblioteca Central da UFV

T

Loureiro, Marcos Paulo de Oliveira, 1978-

L892p Propagação de danos no modelo de Potts / Marcos 2006 Paulo de Oliveira Loureiro. – Viçosa : UFV, 2006.

xiv, 69f. : il. ; 29cm.

Orientador: José Arnaldo Redinz.

Dissertação (mestrado) - Universidade Federal de Viçosa.

Referências bibliográficas: f. 65-69.

1. Física estatística. 2. Monte Carlo, Método de. 3. Ising, Modelo de. 4. Transformações de fase (Física estatística). I. Universidade Federal de Viçosa. II.Título.

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MARCOS PAULO DE OLIVEIRA LOUREIRO

PROPAGAÇÃO DE DANOS NO MODELO DE POTTS

Dissertação apresentada à

Universidade Federal de Viçosa, como parte

das exigências do Programa de

Pós-Graduação em Física Aplicada, para obtenção

do título de Magister Scientiae.

APROVADA: 01 de setembro de 2006.

Prof. Marcelo Lobato Martins

Prof. João Antônio Plascak

(Co-orientador)

Prof. Ladário da Silva

Prof. Sílvio da Costa Ferreira Júnior

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➚ ♠✐♥❤❛ ♠ã❡ ▼❛r✐❛ ▲❡♦♥♦r✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❏♦sé ❆r♥❛❧❞♦ ❡ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ▲✐③✐❛♥❡✳

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❆●❘❆❉❊❈■▼❊◆❚❖❙

✲ ➚ ♠✐♥❤❛ ♠ã❡ ▼❛r✐❛ ▲❡♦♥♦r q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❡ ♠❡ ❛♣♦✐♦✉✱ s❡♠ ❡❧❛ ❡✉ ♥ã♦ t❡r✐❛ ❝♦♥❞✐çõ❡s ❞❡ ❡st✉❞❛r✳

✲ ❆♦ ❛♠✐❣♦ ❡ ♣r♦❢✳ ❏♦sé ❆r♥❛❧❞♦ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ♥❡st❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛ ❛❥✉❞❛ ♥♦ ♠❡✉ ♣r♦❝❡ss♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣❡ss♦❛❧✳

✲ ➚ ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ▲✐③✐❛♥❡ ♣❡❧♦ s❡✉ ❛♠♦r ❡ s✉❛ ❝♦♠♣r❡❡♥sã♦✳

✲ ❆ ❯❋❱ ❡ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉P❋ ♣❡❧♦s ❛♥♦s ❞❡ ❛♣r❡♥❞✐③❛❞♦ ❡ ❝♦♥✈í✈✐♦✳ ✲ ➚ ❈❛♣❡s ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

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❇■❖●❘❆❋■❆

▼❛r❝♦s P❛✉❧♦ ❞❡ ❖❧✐✈❡✐r❛ ▲♦✉r❡✐r♦✱ ✜❧❤♦ ❞❡ ❏♦sé ❊✉stáq✉✐♦ ▲♦✉r❡✐r♦ ❡ ▼❛r✐❛ ▲❡♦♥♦r ❞❡ ❖❧✐✈❡✐r❛ ❡ ❙✐❧✈❛✱ ❜r❛s✐❧❡✐r♦ ♥❛s❝✐❞♦ ❡♠ ✷✹ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✶✾✼✽ ♥♦ ♠✉♥✐❝í♣✐♦ ❞❡ ❇❡❧♦ ❍♦r✐③♦♥t❡✱ ♥♦ ❊st❛❞♦ ❞❡ ▼✐♥❛s ●❡r❛✐s✳

◆♦ ❛♥♦ ✶✾✾✾ ✐♥❣r❡ss♦✉✲s❡ ♥♦ ❝✉rs♦ ❞❡ ❣r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ♦♥❞❡ ❣r❛❞✉♦✉✲s❡ ♥♦ ❛♥♦ ✷✵✵✹ ❝♦♠♦ ❇❛❝❤❛r❡❧ ❡ ▲✐❝❡♥❝✐❛❞♦ ❡♠ ❋ís✐❝❛✳ ❊♠ ✷✵✵✹ ✐♥✐❝✐♦✉✲s❡ ♥♦ ❝✉rs♦ ❞❡ ♠❡str❛❞♦ ❡♠ ❋ís✐❝❛ ❆♣❧✐❝❛❞❛ ♥❛ ♠❡s♠❛ ✐♥st✐t✉✐çã♦ ♦♥❞❡ s❡ ❣r❛❞✉♦✉ ❡ ✈❡✐♦ ❛ ♦❜t❡r ♦ tít✉❧♦ ❞❡ ♠❡str❡ ❛♣ós ❛ ❞❡❢❡s❛ ❞❡ s✉❛ ❞✐ss❡rt❛çã♦ ❡♠ ♣r✐♠❡✐r♦ ❞❡ s❡t❡♠❜r♦ ❞❡ ✷✵✵✻✳

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❙❯▼➪❘■❖

▲■❙❚❆ ❉❊ ❚❆❇❊▲❆❙ ✈✐✐

▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✈✐✐✐

❘❊❙❯▼❖ ①✐✈

❆❇❙❚❘❆❈❚ ①✈

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❖ ▼♦❞❡❧♦ ❞❡ P♦tts ✺

✸ ❖ ▼ét♦❞♦ ❞❡ Pr♦♣❛❣❛çã♦ ❞❡ ❉❛♥♦s ✶✶

✸✳✶ ▼♦❞❡❧♦ ❞❡ ■s✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✶✳✶ ❋❡rr♦♠❛❣♥❡t♦ ❞❡ ■s✐♥❣ ♥❛ ❘❡❞❡ ◗✉❛❞r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✶✳✷ ❋❡rr♦♠❛❣♥❡t♦ ❞❡ ■s✐♥❣ ♥❛ ❘❡❞❡ ❈ú❜✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✷ ❆♣❧✐❝❛çõ❡s ❞♦ ♠ét♦❞♦ ❡♠ ♠♦❞❡❧♦s ♠❛✐s ❝♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷✳✶ ▼♦❞❡❧♦ ❘❡❧ó❣✐♦ ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷✳✷ ▼♦❞❡❧♦ ❳❨ ❇✐❞✐♠❡♥s✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✷✳✸ ▼♦❞❡❧♦ ❳❨ ❜✐❞✐♠❡♥s✐♦♥❛❧✿ ❡✈✐❞ê♥❝✐❛ ❞❡ ❛♣❡♥❛s ❞✉❛s ❢❛s❡s ❞✐✲

♥â♠✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✸ ▼♦❞❡❧♦ ❞❡ P♦tts ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✸✳✶ ▼♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q= 3 ❡st❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✸✳✷ ▼♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q ❡st❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✸✳✸ ▼♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q ❡st❛❞♦s ✉s❛♥❞♦ ❞✐♥â♠✐❝❛ ❞❡ ❛❣r❡❣❛❞♦s ✳ ✸✺

✹ ❘❡s✉❧t❛❞♦s ✸✽

✹✳✶ ❉❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ ❞❛♥♦D(t) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✷ ❙✐♠✉❧❛çõ❡s ❝♦♠ ❛ ❞✐♥â♠✐❝❛ ❞❡ ❇❛♥❤♦ ❚ér♠✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✸ ❙✐♠✉❧❛çõ❡s ❝♦♠ ❛ ❞✐♥â♠✐❝❛ ❞❡ ▼❡tr♦♣♦❧✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

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✺ ❈♦♥❝❧✉sõ❡s ❡ ♣❡rs♣❡❝t✐✈❛s ✻✸

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▲■❙❚❆ ❉❊ ❚❆❇❊▲❆❙

✹✳✶ ❱❛❧♦r❡s ❛ss✐♥tót✐❝♦s ❞♦ ❞❛♥♦ ♠é❞✐♦ < DΦ(t, T > Td) > ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ ♦s ♥ú♠❡r♦s ❞❡ ❡st❛❞♦s q = 2✱ ✸✱ ✳✳✳ ❡ ✼ ♦❜t✐❞♦s ♣♦r ♠❡✐♦ ❞❡ s✐♠✉❧❛çõ❡s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✉s❛♥❞♦ ❛ ❞✐♥â♠✐❝❛ ❞❡ ▼❡tr♦♣♦❧✐s✳ ❖ ❞❛♥♦ ❢♦✐ ♠♦♥✐t♦r❛❞♦ ♣♦r t = 20000 ♣❛ss♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❛♣ós ♦ s✐st❡♠❛ ❡♥tr❛r ❡♠ ❡q✉✐❧í❜r✐♦ t❡r♠♦❞✐♥â♠✐❝♦ ♥❛ t❡♠♣❡r❛t✉r❛ T = 3.0✳ ❆ s✐♠✉❧❛çã♦ ❢♦✐ r❡♣❡t✐❞❛ ♣❛r❛ M = 100 ❛♠♦str❛s ❡ ❡♠ s❡❣✉✐❞❛ ❢❡✐t❛ ❛ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❞♦ ❞❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

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▲■❙❚❆ ❉❊ ❋■●❯❘❆❙

✷✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ♣❧❛♥♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♥t❡♥❞♦ ❛❧❣✉♥s ❞♦s q ✈❡t♦r❡s ♣r♦♣♦st♦s ♣♦r ❉♦♠❜✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞♦sq ✈❡t♦r❡s ♥♦ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦ q1❞✐♠❡♥s✐♦♥❛❧✳ ✳ ✳ ✽ ✸✳✶ ❉❡♣❡♥❞ê♥❝✐❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ψ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♣❛r❛

✉♠❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ✈✐❛ ❞✐♥â♠✐❝❛ ❞❡ ●❧❛✉❜❡r✳ ✭❋✐❣✉r❛ r❡t✐✲ r❛❞❛ ❞❛ r❡❢✳ ❬✶✵❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✷ ❉❛♥♦ ♠é❞✐♦ s♦❜r❡ t♦❞❛s ❛s ❛♠♦str❛s ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❡♠ d= 3

❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✷✻❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✸ ❉❛♥♦ ♠é❞✐♦ s♦❜r❡ ❛s ❛♠♦str❛s q✉❡ s♦❜r❡✈✐✈❡r❛♠< d(t)>❡♠ ❢✉♥çã♦ ❞❛

t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ♦ ♠♦❞❡❧♦Z4✳ P❛r❛ ♦s ❝ír❝✉❧♦s ❛❜❡rt♦s ❡ ❢❡❝❤❛❞♦s

✈❛❧❡ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ✭✶✮❀ ♣❛r❛ ♦s tr✐â♥❣✉❧♦s ❛❜❡rt♦s✱ ✈❛❧❡ ❛ ❝♦♥❞✐çã♦ ✐♥❝✐❛❧ ✭✷✮ ✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✸✵❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✹ ❉❛♥♦ ♠é❞✐♦ ❞❛s ❛♠♦str❛s s♦❜r❡✈✐✈❡♥t❡s < d(t) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠✲

♣❡r❛t✉r❛ T ♣❛r❛ ♦ ♠♦❞❡❧♦ Z10✳ ❖s ❝ír❝✉❧♦s ❛❜❡rt♦s ✭L = 10✮ ❡ ❢❡❝❤❛✲

❞♦s ✭L= 20✮ r❡♣r❡s❡♥t❛♠ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ✭✶✮❀ ♦s tr✐â♥❣✉❧♦s ❛❜❡rt♦s ✭L = 10✮ r❡♣r❡s❡♥t❛♠ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ♦♥❞❡ ♦s s♣✐♥s ❞❛s ❞✉❛s ré✲ ♣❧✐❝❛s ❡♥❝♦♥tr❛♠✲s❡ ❛❧✐♥❤❛❞♦s ❡ ❣✐r❛❞♦s ❣❧♦❜❛❧♠❡♥t❡ ♣♦r π/5✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✸✵❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✺ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ P(t) ❛♣ós ♦ s✐st❡♠❛ ❡✈♦❧✉✐r ❛té ✉♠

t❡♠♣♦tmax❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛T ♣❛r❛ ♦ ♠♦❞❡❧♦Z10✳ ❖s ❝ír❝✉❧♦s

❛❜❡rt♦s ✭L= 10✮ ❝♦rr❡s♣♦♥❞❡♠ ❛ ✉♠ t❡♠♣♦ Tmax = 1000❡ ♦s ❝ír❝✉❧♦s ❢❡❝❤❛❞♦s ✭L = 20✮ ❝♦rr❡s♣♦♥❞❡♠ ❛ ✉♠ t❡♠♣♦ Tmax = 4000✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✸✵❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

(11)

✸✳✻ ❆ ✢✉t✉❛çã♦ ❞♦ ❞❛♥♦σd ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ Z10✳

✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✸✵❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✼ ❉❛♥♦ ♠é❞✐♦ s♦❜r❡ ❛s ❛♠♦str❛s q✉❡ s♦❜r❡✈✐✈❡r❛♠< d(t)>❡♠ ❢✉♥çã♦ ❞❛

t❡♠♣❡r❛t✉r❛ T ♥✉♠ t❡♠♣♦ t = 500 ♣❛r❛ ❛s ❝♦♥❞✐çõ❡s✿ ✭❛✮ nθ(iA)o = 0 ❡ nθi(B)o=π❀ ✭❜✮ nθi(A)o❛❧❡❛tór✐♦ ❡ nθi(B)o=nθ(iA)o+π❀ ✭❝✮ nθ(iA)o ❡ nθ(iB)o ❛❧❡❛tór✐♦s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s❀ ✭❞✮ nθi(A)o ❛❧❡❛tór✐♦ ❡ nθi(B)o =

n

θi(A)o ❡①❝❡t♦ ♣♦r ✉♠ ú♥✐❝♦ s♣✐♥ θi(B) ♦♣♦st♦ ❛ θ(iA)✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✺❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✽ ❉❛♥♦ ♠é❞✐♦ < D(t) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ❞✐❢❡r❡♥t❡s

❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✳ ❆ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100 ❛♠♦str❛s ❛♣ós ❝❛❞❛ ❛♠♦str❛ ❡✈♦❧✉✐r ❞✉r❛♥t❡ ✉♠ t❡♠♣♦ t= 1500✳ ✭❋✐✲ ❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✻❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✾ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛P(t)❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛T ♣❛r❛

❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ t❡♠♣♦ t ♣❛rt✐♥❞♦ ❞❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ✭❝✮✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✶✶❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✶✵ ❉❛♥♦ ♠é❞✐♦ < D(t) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ❞✐❢❡r❡♥t❡s

❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✿ ✭❛✮ D(0) = 1❀ ✭❜✮ D(0) = 0,5❀ ✭❝✮ D(0) = 0,05✳ ❆ ♠❡❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100 ❛♠♦str❛s ❡♠ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❞❡ t❛♠❛♥❤♦ L = 64 ❛ ❝❛❞❛ t❡♠♣♦ t = 10000✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✶✶❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✶✶ ●rá✜❝♦ ❧♦❣✲❧♦❣ ❞♦ ❞❛♥♦ ♠é❞✐♦ < D(t) > ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ t ♣❛r❛

t❡♠♣❡r❛t✉r❛s ❡♠ t♦r♥♦ ❞❡ Tc(q= 2)✳ ❆ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100 ❛♠♦str❛s ❡♠ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❞❡ t❛♠❛♥❤♦ L = 256 ❝♦♠ ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ♦♥❞❡ D(0) = 1 ✭

σA

i ❝♦♠♣❧❡t❛♠❡♥t❡ ❛❧❡❛tór✐❛ ❡

σB i 6=

σA

i ∀i✮✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✶✶❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

(12)

✸✳✶✷ ❉❛♥♦ ♠é❞✐♦ < D(t) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q❂✸✱✳✳✳✱✽ ❡st❛❞♦s✳ ❆ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 64 ❛♠♦str❛s ♥✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ t = 2000✳ ❖ ❞❛♥♦ ✐♥✐❝✐❛❧ D(0) = 1/N ❢♦✐ ❛♣❧✐❝❛❞♦ ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ❞❡ t❛♠❛♥❤♦ L= 50✳ ◆♦t❡ q✉❡ ♥❛ r❡❣✐ã♦ ❝❛ót✐❝❛ ♦ ❞❛♥♦ < D(t) > (q1)/q✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✶✷❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✶✸ ❉❛♥♦ ♠é❞✐♦< D(t)>❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ r❡❞✉③✐❞❛T /Tc ♣❛r❛ ♦

♥ú♠❡r♦ ❞❡ ❡st❛❞♦s q = 3 ❡ ❞✐❢❡r❡♥t❡s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ✭✈❡r ✜❣✉r❛✮✳ ❖ ❞❛♥♦ < D(t)>❝♦♥✈❡r❣❡ ♣❛r❛ (q1)/q ♥❛ ❢❛s❡ ❞❡ ❝❛ót✐❝❛ ♠❛s ♠♦str❛ ✉♠ ❞❡♣❡♥❞ê♥❝✐❛ ❡♠ ❢✉♥çã♦ ❞❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♥❛ ❢❛s❡ ♦r❞❡♥❛❞❛✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✶✷❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✶✹ ❉❛♥♦ ♠é❞✐♦ < D(t) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ♦s ♥ú♠❡r♦s

❞❡ ❡st❛❞♦s q = 2, ...,8 ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❞❡ t❛♠❛♥❤♦ L = 201✳ ❆ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 10 ❛♠♦str❛s ❡ ✸✵✵✵✵ ♣❛ss♦s ❞❡ t❡♠♣♦ ❛♣ós ♦ s✐st❡♠❛ t❡r t❡r♠❛❧✐③❛❞♦✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✶✸❪✮ ✸✻ ✸✳✶✺ ❉❛♥♦ ♠é❞✐♦< D(t)>❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦t✭▼❈❙✮ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s

❞❡ t❡♠♣❡r❛t✉r❛T ❡♠ t♦r♥♦ ❞❡T1 ♣❛r❛ ♦s ♥ú♠❡r♦s ❞❡ ❡st❛❞♦s ✭❛✮q = 2

❡ ✭❜✮ q = 8✳ ❆ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ♠♦str❛ ❛ ✐♥❝❧✐♥❛çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ à ❝❧❛ss❡ ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡ ❞❛ ♣❡r❝♦❧❛çã♦ ❞✐r❡❝✐♦♥❛❞❛✳ ✭❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❛ r❡❢✳ ❬✶✸❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞♦s â♥❣✉❧♦s ❡ ✈❡t♦r❡s ♥♦ ♣❧❛♥♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦rr❡s♣♦♥✲

❞❡♥t❡s à ❞❡✜♥✐çã♦ ❞❡ ❞❛♥♦ D(t)♣r♦♣♦st❛ ♣♦r ❈❤✐✉ ❡t ❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ P(t) ❞♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q = 3

❡st❛❞♦s✱ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛T ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ t❡♠♣♦ t✳ ❊st❡ r❡s✉❧t❛❞♦ s❡ r❡❢❡r❡ ❛ ✉♠❛ s✐♠✉❧❛çã♦ ❞❡M = 100❛♠♦str❛s ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ L= 64 ✉s❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ✭❝✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

(13)

✹✳✸ ❙✐♠✉❧❛çã♦ ❞♦ ❞❛♥♦ ♠é❞✐♦ < Dδ(t) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q = 3 ❡st❛❞♦s ✉s❛♥❞♦ ❛ ❞✐♥â♠✐❝❛ ❞❡ ❜❛♥❤♦ tér♠✐❝♦✳ ❖s s♣✐♥s ❞❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ L = 64 ❡✈♦❧✉✐r❛♠ ❞✉r❛♥t❡ t = 10000 ♣❛ss♦s ❞❡ t❡♠♣♦ ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ❞❡ t❡♠♣❡r❛t✉r❛ T ❡ ❛ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100❛♠♦str❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✹ ❉❛♥♦ ♠é❞✐♦ < Dδ(t) > ❞♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q = 3 ❡st❛❞♦s✱ ❡♠

❢✉♥çã♦ ❞♦ t❡♠♣♦ t ♣❛r❛ ❛s t❡♠♣❡r❛t✉r❛s T = 0,20 ❡ T = 0,99✳ ❙✐♠✉✲ ❧❛çã♦ s♦❜r❡ M = 100 ❛♠♦str❛s ✉s❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ✭❜✮ ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ L= 64✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✺ ❉❛♥♦ ♠é❞✐♦ < Dδ > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ♦ ♠♦❞❡❧♦❞❡

P♦tts ❝♦♠ q = 2❡st❛❞♦s✳ ❆ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100 ❛♠♦str❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✻ ❉❛♥♦ ♠é❞✐♦ < Dδ(t = 10000) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ♦

♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ ♦s ♥ú♠❡r♦s ❞❡ ❡st❛❞♦s✿ ✭✐✮ q = 2✱ ✭✐✐✮ q= 3✱ ✭✐✐✐✮ q = 4✱ ✭✐✈✮ q = 5✱ ✭✈✮ q = 6 ❡ ✭✈✐✮ q = 7✳ ❆ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100 ❛♠♦str❛s ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ L = 64✳ ❆s ❧✐♥❤❛s tr❛❝❡❥❛❞❛s ✐♥❞✐❝❛♠ ❛s t❡♠♣❡r❛t✉r❛s ❞❛ tr❛♥s✐çã♦ T1 ❡ T2✳ ✳ ✳ ✳ ✳ ✹✽

✹✳✼ ●r❛✜❝♦s ❧♦❣✲❧♦❣ ❞♦ ❞❛♥♦ ♠é❞✐♦< Dδ(t)>❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦t♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q = 3 ❡ q= 7 ♥ú♠❡r♦s ❞❡ ❡st❛❞♦s✳ ❖s ✈❛❧♦r❡s ❞❡ t❡♠♣❡r❛t✉r❛ ❡stã♦ ❡♠ t♦r♥♦ ❞❛ t❡♠♣❡r❛t✉r❛ ❞❛ s❡❣✉♥❞❛ tr❛♥s✐çã♦ T1

❞❡ ❝❛❞❛ ♠♦❞❡❧♦✳ ❆s r❡t❛s tr❛❝❡❥❛❞❛s s❡r✈❡♠ ❞❡ ❣✉✐❛ ♣❛r❛ ❛ ✐♥❝❧✐♥❛çã♦ ✵✱✹✻✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✽ ❉❛♥♦ ♠é❞✐♦< DΦ(t)>❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛T ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡

P♦tts ❝♦♠ q = 3 ❡st❛❞♦s✱ ♣❛rt✐♥❞♦ ❞❛s ❝♦♥❞✐çõ❡s ✐♥❝✐❛✐s ✭❛✮✱ ✭❜✮ ❡ ✭❝✮✳ ❖s s♣✐♥s ❞❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ L = 64 ❡✈♦❧✉✐r❛♠ ❞✉r❛♥t❡ t = 10000 ♣❛ss♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡ ❛ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡M = 100 ❛♠♦str❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

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✹✳✾ ❉❛♥♦ ♠é❞✐♦ < DΦ(t) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ♦ ♠♦❞❡❧♦

❞❡ P♦tts ❝♦♠ q= 4✱ ✺✱ ✻ ❡ ✼ ❡st❛❞♦s r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s s♣✐♥s ❞❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ L = 64 ❡✈♦❧✉✐r❛♠ ❞✉r❛♥t❡ t = 10000 ♣❛ss♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡ ❛ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100❛♠♦str❛s✳ ✳ ✳ ✳ ✺✷ ✹✳✶✵ ❉❛♥♦ ♠é❞✐♦< DΦ(t)> ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦t ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts

❝♦♠ q = 3 ❡st❛❞♦s ♥❛s t❡♠♣❡r❛t✉r❛sT = 0,20❡ T = 0,99♣❛rt✐♥❞♦ ❞❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ✭❜✮✳ ❖s s♣✐♥s ❞❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠L= 64 ❡✈♦❧✉✐r❛♠ ❞✉r❛♥t❡ t= 10000♣❛ss♦s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡ ❛ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100❛♠♦str❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✶✶ ❉❛♥♦ ♠é❞✐♦< DΦ(t)>❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛T ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡

P♦tts ❝♦♠ q= 3 ❡st❛❞♦s ♣❛rt✐♥❞♦ ❞❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ✭❜✮✳ ❈❛❞❛ ❝✉r✈❛ r❡♣r❡s❡♥t❛ ♦ t❡♠♣♦ ♥♦ q✉❛❧ ❛s ❝♦♥✜❣✉r❛çõ❡s ❡✈♦❧✉✐r❛♠✳ ❆ ♠é❞✐❛ t❡r✲ ♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100❝♦♥✜❣✉r❛çõ❡s ❞❡ r❡❞❡s q✉❛❞r❛❞❛s ❝♦♠ L= 64✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✶✷ ❉❛♥♦ ♠é❞✐♦ < Dδ(t)>❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛T ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡

P♦tts ❝♦♠ q = 3 ❡st❛❞♦s✳ ❖s q✉❛❞r❛❞♦s✱ ❝✐r❝✉❧♦s ❡ tr✐â♥❣✉❧♦s ✈❛③✐♦s ❝♦rr❡s♣♦♥❞❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ às ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ✭❞✮✱ ✭❡✮ ❡ ✭❢✮✳ ❆ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡ M = 100 ❝♦♥✜❣✉r❛çõ❡s ❞❡ r❡❞❡s q✉❛❞r❛❞❛s ❝♦♠ L= 64✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✶✸ ❉❛♥♦ ♠é❞✐♦ < Dδ(t)>❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛T ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡

P♦tts ❝♦♠ ♦s r❡s♣❡❝t✐✈♦s ♥ú♠❡r♦s ❞❡ ❡st❛❞♦s✿ ✭✐✮ q = 2❀ ✭✐✐✮ q= 3❀ ✭✐✐✐✮ q = 4❀ ✭✐✈✮ q = 5❀ ✭✈✮q = 6❀ ✭✈✐✮ q = 7✳ ❖s q✉❛❞r❛❞♦s✱ ❝✐r❝✉❧♦s ❡ tr✐â♥✲ ❣✉❧♦s ✈❛③✐♦s ❝♦rr❡s♣♦♥❞❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ às ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ✭❞✮✱ ✭❡✮ ❡ ✭❢✮✳ ❆ ♠é❞✐❛ t❡r♠♦❞✐♥â♠✐❝❛ ❢♦✐ ❢❡✐t❛ s♦❜r❡M = 100❝♦♥✜❣✉r❛çõ❡s ❞❡ r❡❞❡s q✉❛❞r❛❞❛s ❝♦♠ L= 64✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✶✹ ❉❛♥♦ ♠é❞✐♦< DΦ(t)>❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛T ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡

P♦tts ❝♦♠ q❡st❛❞♦s ❡✈♦❧✉✐♥❞♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐♥â♠✐❝❛ ❞❡ ▼❡tr♦♣♦❧✐s✳ ✺✽

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✹✳✶✺ ❉❛♥♦ ♠é❞✐♦ < DΦ(t) > ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ T ♣❛r❛ ♦ ♠♦❞❡❧♦

❞❡ P♦tts ❝♦♠ ♦s r❡s♣❡❝t✐✈♦s ♥ú♠❡r♦s ❞❡ ❡st❛❞♦s✿ ✭✐✮ q = 2❀ ✭✐✐✮ q = 3❀ ✭✐✐✐✮ q = 4❀ ✭✐✈✮ q = 5❀ ✭✈✮ q = 6❀ ✭✈✐✮ q = 7✳ ❖s q✉❛❞r❛❞♦s✱ ❝✐r❝✉❧♦s ❡ tr✐â♥❣✉❧♦s ✈❛③✐♦s ❝♦rr❡s♣♦♥❞❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ às ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ✭❞✮✱ ✭❡✮ ❡ ✭❢✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✶✻ ◆♦ ♣r✐♠❡✐r♦ ❣rá✜❝♦✱ ♠♦str❛♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ✈❛❧♦r ❛ss✐♥tót✐❝♦

❞♦ ❞❛♥♦ ♠é❞✐♦ < DΦ(t) > ♥❛ ❢❛s❡ ♣❛r❛♠❛❣♥ét✐❝❛ ❡♠ ❢✉♥çã♦ ❞♦s q

❡st❛❞♦s ❞♦ ♠♦❞❡❧♦ ❞❡ P♦tts✳ ◆♦ s❡❣✉♥❞♦ ❣rá✜❝♦✱ ✐❞❡♥t✐✜❝❛♠♦s ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❞❛♥♦ ♠é❞✐♦ ❝♦♠♦ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❡♠ t❡r♠♦s ❞❡ q✱ ♦✉ s❡❥❛✱ D(q)q−γ ❝♦♠ γ = 1,386±0,001✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✶✼ ●rá✜❝♦ ❧♦❣✲❧♦❣ ❝♦♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❞❛♥♦ ♠é❞✐♦ < DΦ(t) > ❡♠

❢✉♥çã♦ ❞♦ t❡♠♣♦ t ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q = 3 ❡ q = 7 ❡st❛❞♦s r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ■❞❡♥t✐✜❝❛♠♦s ❛ t❡♠♣❡r❛t✉r❛ ❞❡ tr❛♥s✐çã♦ ♣❛r❛ q = 3 ❡st❛❞♦s ❡♠ Td= 0,995±0,003 ❡ ♣❛r❛ q= 7 ❡st❛❞♦s ❡♠ Td = 0,772± 0,002✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

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❘❊❙❯▼❖

▲❖❯❘❊■❘❖✱ ▼❛r❝♦s P❛✉❧♦ ❞❡ ❖❧✐✈❡✐r❛✱ ▼✳❙❝✳✱❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❙❡t❡♠❜r♦ ❞❡ ✷✵✵✻✳ Pr♦♣❛❣❛çã♦ ❞❡ ❞❛♥♦s ♥♦ ♠♦❞❡❧♦ ❞❡ P♦tts✳ ❖r✐❡♥✲ t❛❞♦r✿ ❏♦sé ❆r♥❛❧❞♦ ❘❡❞✐♥③✳ ❈♦✲❖r✐❡♥t❛❞♦r❡s✿ ▼❛r❝❡❧♦ ▲♦❜❛t♦ ▼❛rt✐♥s ❡ ❘✐❝❛r❞♦ ❘❡✐s ❈♦r❞❡✐r♦

❊st✉❞❛♠♦s ❛s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ♥♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q ❡st❛❞♦s ✭q = 2✱ ✳✳✳✱ ✼✮ ♣♦r ♠❡✐♦ ❞❡ s✐♠✉❧❛çõ❡s ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡ ♣r♦♣❛❣❛çã♦ ❞❡ ❞❛♥♦s✳ ❊st❡ ♠♦❞❡❧♦ ❛♣r❡s❡♥t❛ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ r✐❝♦✱ ❝♦♠ tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ❞❡ s❡❣✉♥❞❛ ✭q≤4✮

❡ ♣r✐♠❡✐r❛ ♦r❞❡♥s ✭q5✮✳ ❋♦❝❛♠♦s ♥♦ss♦s ❡st✉❞♦s ♥❛ q✉❡stã♦ ❞❛ ❞✐♥â♠✐❝❛ ❡ ❞♦ ❞❛♥♦ q✉❡ r❡s♣❡✐t❛♠ ❛ ✐♥✈❛r✐â♥❝✐❛ r♦t❛❝✐♦♥❛❧ ❞♦ s✐st❡♠❛✳ ▼♦str❛♠♦s q✉❡✱ s❡ ❞❡✈✐❞❛♠❡♥t❡ ❡s❝♦❧❤✐❞❛s ❛ ❢✉♥çã♦ ❞❛♥♦ ❡ ❛ ❞✐♥â♠✐❝❛ ❞❡ ❡✈♦❧✉çã♦✱ ♦ ❞❛♥♦ ❛♣r❡s❡♥t❛ ❛♣❡♥❛s ❞✉❛s ❢❛s❡s ♣❛r❛ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ q✳ ◆❡st❡ s❡♥t✐❞♦✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞❛♥♦ ❡ ❞❛ ❞✐♥â♠✐❝❛ ❡stã♦ ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛s tr❛♥s✐çõ❡s ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ♠♦❞❡❧♦✳ ◆ós ✐♥✈❡s✲ t✐❣❛♠♦s t❛♠❜é♠ ❛ ❝♦✐♥❝✐❞ê♥❝✐❛✱ ♦✉ ♥ã♦✱ ❞❛s t❡♠♣❡r❛t✉r❛s ❞❡ tr❛♥s✐çã♦ ❞♦ ❞❛♥♦ ❝♦♠ ❛s t❡♠♣❡r❛t✉r❛s ❝r✐t✐❝❛s ❡stát✐❝❛s ❞♦ ♠♦❞❡❧♦ ❞❡ P♦tts✳

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❆❇❙❚❘❆❈❚

▲❖❯❘❊■❘❖✱ ▼❛r❝♦s P❛✉❧♦ ❞❡ ❖❧✐✈❡✐r❛✱ ▼✳❙❝✳✱❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❙❡♣t❡♠❜❡r✱ ✷✵✵✻✳ ❉❛♠❛❣❡ s♣r❡❛❞✐♥❣ ✐♥ t❤❡ P♦tts ♠♦❞❡❧✳ ❆❞✈✐s❡r✿ ❏♦sé ❆r♥❛❧❞♦ ❘❡❞✐♥③✳ ❈♦✲❆❞✈✐s❡rs✿ ▼❛r❝❡❧♦ ▲♦❜❛t♦ ▼❛rt✐♥s ❛♥❞ ❘✐❝❛r❞♦ ❘❡✐s ❈♦r❞❡✐r♦

❲❡ st✉❞② t❤❡ ♣❤❛s❡ tr❛♥s✐t✐♦♥s ✐♥ t❤❡ q✲st❛t❡ P♦tts ♠♦❞❡❧ ✭✇✐t❤ q ❢r♦♠ ✷ t♦ ✼✮ t❤r♦✉❣❤ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❛♥❞ t❤❡ ❞❛♠❛❣❡ s♣r❡❛❞✐♥❣✳ ❚❤✐s ♠♦❞❡❧ ♣r❡s❡♥ts ❛ r✐❝❤ ❝r✐t✐❝❛❧✐t② ✇✐t❤ s❡❝♦♥❞✲ ✭q 4✮ ❛♥❞ ✜rst✲♦r❞❡r ✭q 5✮ ♣❤❛s❡ tr❛♥s✐t✐♦♥s✳ ❲❡ ❛❞❞r❡ss ❤❡r❡ ♠❛✐♥❧② t❤❡ q✉❡st✐♦♥ ♦❢ t❤❡ ❞②♥❛♠✐❝s ❛♥❞ ❞❛♠❛❣❡ ❞❡✜♥✐t✐♦♥s✱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ r♦t❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡ s②♠♠❡tr② ♦❢ t❤❡ s②st❡♠✳ ❲❡ s❤♦✇ t❤❛t ✇✐t❤ ♣r♦♣❡r ❞❡✜♥✐t✐♦♥s ♦❢ ❞❛♠❛❣❡ ❛♥❞ ❞②♥❛♠✐❝s✱ t❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ♣❤❛s❡s ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ❤✐❣❤t t❡♠♣❡r❛t✉r❡ r❛♥❞♦♠ ♣❤❛s❡ ❛♥❞ ❛ ❧♦✇ t❡♠♣❡r❛t✉r❡ ♦r❞❡r❡❞ ♣❤❛s❡ ❢♦r ❛❧❧ ✈❛❧✉❡s ♦❢ q✳ ❚❤❡r❡❢♦r❡✱ t❤❡s❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❞❛♠❛❣❡ ❛♥❞ ❞②♥❛♠✐❝s ❢♦r t❤❡ P♦tts ♠♦❞❡❧ ❛r❡ ♠♦r❡ ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ ❡q✉✐❧✐❜r✐✉♠ tr❛♥s✐t✐♦♥s ✐♥ t❤✐s ♠♦❞❡❧✳ ❲❡ ✐♥✈❡st✐❣❛t❡ ❛❧s♦ t❤❡ ❝♦✐♥❝✐❞❡♥❝❡✱ ♦r ♥♦t✱ ♦❢ t❤❡ ❞❛♠❛❣❡ tr❛♥s✐t✐♦♥ t❡♠♣❡r❛t✉r❡s ✇✐t❤ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝r✐t✐❝❛❧ t❡♠♣❡r❛t✉r❡s ♦❢ t❤❡ P♦tts ♠♦❞❡❧✳

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❉❡s❞❡ t❡♠♣♦s ♠✉✐t♦ ❛♥t✐❣♦s✱ ♦ ❤♦♠❡♠✱ ♣♦r ♠❡✐♦ ❞❡ ✜❧♦s♦✜❛ ♦✉ ❝✐ê♥❝✐❛✱ ✈❡♠ t❡♥t❛♥❞♦ ❡①♣❧✐❝❛r ♦✉ ♠❡s♠♦ ❡♥t❡♥❞❡r ♦ ❝♦♠♣❧❡①♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♥❛t✉r❡③❛✳ ❉❡t❡r♠✐♥❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛s ✈❛r✐á✈❡✐s ✭♦✉ ♣❡❧♦ ♠❡♥♦s ❛q✉❡❧❛s ♠❛✐s ✐♥✢✉❡♥t❡s✮ q✉❡ ❣♦✈❡r♥❛♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ s✐st❡♠❛✱ ♣♦❞❡♥❞♦ ❛ss✐♠ ♣r❡✈❡r s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ❢✉t✉r♦✱ é✱ ❢♦✐ ❡ s❡rá s❡♠♣r❡ ✉♠ ❡♠♣♦❧❣❛♥t❡ ❡stí♠✉❧♦ ♣❛r❛ ♦ ❤♦♠❡♠✳ ❙❛❜❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠♦ ♦s ♥❡✉rô♥✐♦s ✐♥t❡r❛❣❡♠ ♥♦ s✐st❡♠❛ ♥❡r✈♦s♦✱ ❝♦♠♦ s❡ ♣r♦♣❛❣❛ ✉♠ ❝â♥❝❡r✱ ❝♦♠♦ ❛♥✐♠❛✐s ❞❡t❡❝t❛♠ s✉❛s ♣r❡s❛s ♦✉ ♣r❡❞❛❞♦r❡s ❡♠ s❡✉s ❡❝♦ss✐st❡♠❛s✱ s❛❜❡r ❛té ♠❡s♠♦ ❛ tr❛❥❡✲ tór✐❛ ❞❡ ❣rã♦s ❞❡ ❛r❡✐❛ ❞✉r❛♥t❡ ❛ ♠♦✈✐♠❡♥t❛çã♦ ❞❛s ❞✉♥❛s ♦✉ ❞❡s❡♥✈♦❧✈❡r t❡❝♥♦❧♦❣✐❛s ♠❛❣♥❡t♦✲❡❧❡trô♥✐❝❛s ❝❛♣❛③❡s ❞❡ tr❛♥s♣♦rt❛r ♦✉ ❛r♠❛③❡♥❛r ❞❛❞♦s✱ sã♦ ♠♦t✐✈❛çõ❡s ♠❛✐s q✉❡ s✉✜❝✐❡♥t❡s ♣❛r❛ ❝♦❧♦❝❛r ✉♠❛ ❣❛♠❛ ❞❡ ❝✐❡♥t✐st❛s ❛trás ❞❡st❛s r❡s♣♦st❛s✳ ▼❛s s✐s✲ t❡♠❛s r❡❛✐s sã♦✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ ✐♠♣♦ssí✈❡✐s ❞❡ s❡r❡♠ ❝♦♠♣❧❡t❛♠❡♥t❡ ❡st✉❞❛❞♦s ❞❡✈✐❞♦ ❛♦ ❛❧t♦ ❣r❛✉ ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❛s ✈❛r✐á✈❡✐s q✉❡ ♦ ✐♥✢✉❡♥❝✐❛♠ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡✲ t❛♠❡♥t❡✳

❆ ❢ís✐❝❛✱ ♣♦r s❡r ❡♠ ❡ssê♥❝✐❛ ✉♠❛ ❝✐ê♥❝✐❛ q✉❡ ❡st✉❞❛ ♦s ❢❡♥ô♠❡♥♦s ❞♦ ✉♥✐✈❡rs♦✱ ♦❜✈✐❛♠❡♥t❡ ❡♥❝♦♥tr❛ sér✐❛s ❞✐✜❝✉❧❞❛❞❡s q✉❛♥❞♦ s❡ ❞❡♣❛r❛ ❝♦♠ s✐t✉❛çõ❡s r❡❛✐s ❡♠ q✉❡ ♦s ♠♦❞❡❧♦s ♠❛✐s s✐♠♣❧❡s✱ ❞❡ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛✱ ♥ã♦ s❡ ❛♣r♦①✐♠❛♠ ❞❛ r❡❛❧✐❞❛❞❡✳

❉❡✈✐❞♦ à ❡♥♦r♠❡ ❞✐✜❝✉❧❞❛❞❡ ❡♠ s❡ ❡st✉❞❛r ♦s ♠♦❞❡❧♦s r❡❛✐s ❛♥❛❧✐t✐❝❛♠❡♥t❡✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s t❡♦r✐❛s ❝❛♣❛③❡s ❞❡ r❡❛❧✐③❛r ❛♣r♦①✐♠❛çõ❡s✳ ❊st❛s t❡♦r✐❛s✱ ❝♦♠✉✲ ♠❡♥t❡ ❛ss♦❝✐❛❞❛s à ❋ís✐❝❛ ❊st❛tíst✐❝❛✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ t♦r♥❛r❛♠ ♦ tr❛❜❛❧❤♦ ❞♦s ❢ís✐❝♦s ♠❛✐s s✐♠♣❧❡s✱ ✉♠❛ ✈❡③ q✉❡ ❛s ♣ró♣r✐❛s ❛♣r♦①✐♠❛çõ❡s ❡♥✈♦❧✈❡♠ ✉♠❛ sér✐❡ ❞❡

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á❧❣❡❜r❛s ♠✉✐t❛s ✈❡③❡s ❞✐❢í❝❡✐s ❞❡ s❡r❡♠ r❡s♦❧✈✐❞❛s✳

❈♦♠ ♦ ❛❞✈❡♥t♦ ❞♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♥❢♦r♠❛çõ❡s ♣♦r ♠❡✐♦ ❞❡ ♠áq✉✐♥❛s ♦✉ ♠✐❝r♦❝♦♠♣✉t❛❞♦r❡s✱ ❡st❛s t❡♦r✐❛s ❞❡ ❛♣r♦①✐♠❛çã♦ ♣✉❞❡r❛♠ s❡r ♠✉✐t♦ ♠❛✐s ❜❡♠ ❡st✉✲ ❞❛❞❛s ❡ ❝♦♠ ✐ss♦✱ ♦s s✐st❡♠❛s r❡❛✐s ♣✉❞❡r❛♠ s❡r ♠❡❧❤♦r ❝♦♠♣r❡❡♥❞✐❞♦s✳

❆t✉❛❧♠❡♥t❡✱ ♦ ❡st✉❞♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❡ s✐st❡♠❛s ❝♦♠♣❧❡①♦s t♦r♥♦✉✲s❡ q✉❛s❡ q✉❡ ✉♠❛ ❝✐ê♥❝✐❛ ❞✐st✐♥t❛✱ ❡♠ q✉❡ s✐st❡♠❛s ❝♦♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ♣♦✲ ❞❡♠ s❡r ❛♥❛❧✐s❛❞♦s ❡ ♣r❡✈✐sõ❡s ❜❛st❛♥t❡ ♣r❡❝✐s❛s ♣♦❞❡♠ s❡r ❢❡✐t❛s ❡♠ r❡❧❛çã♦ ❛ ❡st❡s s✐st❡♠❛s✳

P❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❡st❡s s✐st❡♠❛s✱ ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s ♠é✲ t♦❞♦s ✉t✐❧✐③❛❞♦s é ♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❬✶✱ ✷❪✳ ❊❧❡ ❝♦♥s✐st❡ ❜❛s✐❝❛♠❡♥t❡ ❡♠ ❣❡r❛r r❡❝✉r✲ s✐✈❛♠❡♥t❡✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❞❛❞❛ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧✱ ♥♦✈❛s ❝♦♥✜❣✉r❛çõ❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ✉♠❛ ❞✐♥â♠✐❝❛ ♣ré✲❡st❛❜❡❧❡❝✐❞❛ q✉❡✱ ♥♦ ❝❛s♦ ❞❡ s✐st❡♠❛s ❡♠ ❡q✉✐❧í❜r✐♦✱ ❝♦♥❞✉③ à ❞✐str✐❜✉✐çã♦ ❞❡ ❇♦❧t③♠❛♥♥✳ ■st♦ é ♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ✉♠❛ ✏❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈✑✱ ♦✉ s❡❥❛✱ ❛ tr❛❥❡tór✐❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝♦♥✜❣✉r❛çã♦ ♥♦ t❡♠♣♦ t+ 1❞❡♣❡♥❞❡ ❞✐r❡t❛♠❡♥t❡ ❞♦ ❡st❛❞♦ ❞❛ ❝♦♥✜❣✉r❛çã♦ ♥♦ t❡♠♣♦t✳ ❖ t❡♠♣♦ ❛ q✉❡ ♥♦s r❡❢❡r✐♠♦s é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✏♣❛ss♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✑ ✭▼❈❙✮ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡✱ ♥❛ ♠é❞✐❛✱ ❝❛❞❛ s♣✐♥ s❡❥❛ ✏✈✐s✐t❛❞♦✑ ❡ ❡✈❡♥t✉❛❧♠❡♥t❡ ❛t✉❛❧✐③❛❞♦ ✉♠❛ ✈❡③✳ ❆❧❣✉♠❛s ❞✐♥â♠✐❝❛s ❞❡ ❡✈♦❧✉çã♦ s❡rã♦ ✈✐st❛s ♥♦ ❞❡❝♦rr❡r ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦✳ ❊①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çõ❡s ❞♦s ❡st✉❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣♦❞❡♠ s❡r ✈✐st♦s ❡♠ ♣r❛✲ t✐❝❛♠❡♥t❡ t♦❞❛s ❛s ár❡❛s ❞❛ ❝✐ê♥❝✐❛✳ ❊st❛s ❛♣❧✐❝❛çõ❡s ✈ã♦ ❞❡s❞❡ ❡st✉❞♦s s♦❜r❡ t❡r✲ r❡♠♦t♦s✱ ❢♦♥t❡s ❞❡ ❡♥❡r❣✐❛s✱ ❞✐str✐❜✉✐çã♦ ♣♦♣✉❧❛❝✐♦♥❛❧✱ tr❛❥❡tór✐❛s ❞❡ ❝♦r♣♦s ❝❡❧❡st❡s✱ ❡t❝✱ ✐♥❝❧✉✐♥❞♦ ♦s ❡①❡♠♣❧♦s ❝✐t❛❞♦s ♥♦ ✐♥í❝✐♦ ❞❡st❡ ❝❛♣ít✉❧♦✳

❯♠❛ ❞❛s ár❡❛s q✉❡ ❞❡s♣❡rt❛r❛♠ ❡ ❛✐♥❞❛ ❞❡s♣❡rt❛♠ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♥❛ ❢ís✐❝❛ ❞❡ s✐st❡♠❛s ❝♦♠♣❧❡①♦s✱ é ❛ q✉❡ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❛♦s ❢❡♥ô♠❡♥♦s ♠❛❣♥ét✐❝♦s✳ ❊st❡ ❢❛t♦ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ ❡♥t❡♥❞✐❞♦ ✉♠❛ ✈❡③ q✉❡ ✉♠ ❞♦s ♣♦✉❝♦s ♠♦❞❡❧♦s ❛♥❛❧ít✐❝♦s ♥ã♦ tr✐✲ ✈✐❛✐s ❝♦♠ s♦❧✉çã♦ ❡①❛t❛ ❡stá ❛ss♦❝✐❛❞♦ ❛♦ ♠❛❣♥❡t✐s♠♦✳ ❊st❡ ♠♦❞❡❧♦ é ♦ ❢❡rr♦♠❛❣♥❡t♦ ❞❡ ■s✐♥❣✱ r❡s♦❧✈✐❞♦ ❛♥❛❧✐t✐❝❛♠❡♥t❡ ♣♦r ❊r♥st ■s✐♥❣ ❬✸❪ ❡♠ ✉♠❛ ❞✐♠❡♥sã♦ ❡ ♣♦r ▲❛rs ❖♥s❛❣❡r ❬✹❪ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s ♥❛ r❡❞❡ q✉❛❞r❛❞❛✳

❆♦ ♠❡s♠♦ t❡♠♣♦✱ ♦ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❡♠ ❡st✉❞❛r ♦s s✐st❡♠❛s ❝♦♠♣❧❡①♦s ❡stá ♥❛ r❡❣✐ã♦ ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦✱ ❞✉r❛♥t❡ ❛ ❡t❛♣❛ q✉❡ ❝❤❛♠❛♠♦s ❞❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✳ ❖

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❡①❡♠♣❧♦ ♠❛✐s ❝♦♠✉♠ ❞❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ♦❝♦rr❡ ❞✉r❛♥t❡ ❛ ♠✉❞❛♥ç❛ ❡♥tr❡ ♦s três ❡st❛❞♦s ❞❛ á❣✉❛ ✭só❧✐❞♦✱ ❧✐q✉✐❞♦ ❡ ❣❛s♦s♦✮✳ ❖✉tr♦s ❡①❡♠♣❧♦s✱ t❛❧✈❡③ ♠❡♥♦s ❝♦♠✉♥s✱ s❡r✐❛♠ ❛s ❞✉❛s ❢❛s❡s ✭❧íq✉✐❞❛ ❡ ❣❛s♦s❛✮ ❡♥❝♦♥tr❛❞❛s ♣♦r ❍❡✐❦❡ ❑❛♠❡r❧✐♥❣❤ ❖♥♥❡s ♣❛r❛ ♦ ❤é❧✐♦ ✭❍❡✮ ❛ ❜❛✐①❛s t❡♠♣❡r❛t✉r❛s ✭♣rê♠✐♦ ◆♦❜❡❧ ❡♠ ✶✾✶✸✮ q✉❡ ❡stã♦ ❛ss♦❝✐❛❞❛s ❛♦ ❢❡♥ô♠❡♥♦ ❞❛ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡ ❡ ❛ tr❛♥s✐çã♦ ❢❡rr♦♠❛❣♥ét✐❝❛ q✉❡ ♦❝♦rr❡ ❡♠ ♠❛t❡r✐❛✐s ❝♦♠♦ ❢❡rr♦ ✭❋❡✮ ♦✉ ♥íq✉❡❧ ✭◆✐✮✳

❖ ♣r✐♠❡✐r♦ tr❛❜❛❧❤♦ ❝♦♥s✐st❡♥t❡ ❛ r❡s♣❡✐t♦ ❞❛ t❡♦r✐❛ ❞❛s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ✈❛♥ ❞❡r ❲❛❛❧s ❡♠ ✶✽✼✸ ❡ ❛♣r❡s❡♥t♦✉ ❛ t❡♦r✐❛ q✉❡ ❞❡s❝r❡✈❡✉ ❛ ✏❝♦♥t✐♥✉✐✲ ❞❛❞❡ ❞♦s ❡st❛❞♦s ❧íq✉✐❞♦ ❡ ❣❛s♦s♦ ❞❛ ♠❛tér✐❛✑✳ P♦st❡r✐♦r♠❡♥t❡✱ P✐❡rr❡ ❈✉r✐❡ ❡ P✐❡rr❡ ❲❡✐ss ♣r♦♣✉s❡r❛♠ ❡ ❞❡s❡♥✈♦❧✈❡r❛♠ ✉♠❛ t❡♦r✐❛ ♣❛r❛ ❛ tr❛♥s✐çã♦ ❢❡rr♦♠❛❣♥ét✐❝❛✳

❉❡♥tr♦ ❞❛ t❡♦r✐❛ ❞❡ ▲❛♥❞❛✉ ♣❛r❛ ♦ ❢❡rr♦♠❛❣♥❡t✐s♠♦✱ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ é ❞❡s❝r✐t❛ ❛tr❛✈és ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ ❞❡♥♦♠✐♥❛❞❛ ✏♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠✑✱ ✐st♦ é✱ ✉♠ ♣❛râ♠❡tr♦ q✉❡ ❛♣r❡s❡♥t❛ ✈❛❧♦r ♥ã♦ ♥✉❧♦ ♥❛ ❢❛s❡ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ♦r❞❡♥❛❞❛ ❡ ♥✉❧♦ ♥❛ ❢❛s❡ ❞❡s♦r❞❡♥❛❞❛✳ ❖ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ é ❞❡✜♥✐❞♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ s✐st❡♠❛ ❢ís✐❝♦ ❡st✉❞❛❞♦✳ ◆♦ ❝❛s♦ ❞♦ ❢❡rr♦♠❛❣♥❡t♦✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ♥❛t✉r❛❧ s❡r✐❛ ❛ ♠❛❣♥❡t✐③❛çã♦ ❡s♣♦♥tâ♥❡❛ ✭❛ ❝❛♠♣♦ ❡①t❡r♥♦ ♥✉❧♦✮✳ P❛r❛ ✉♠ s✐st❡♠❛ ❧íq✉✐❞♦✲ ❣ás✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ s❡r✐❛ ❛ ❞✐❢❡r❡♥ç❛ ❞❛s ❞❡♥s✐❞❛❞❡s ❞❛ ❢❛s❡ ❧íq✉✐❞❛ ❡ ❞❛ ❢❛s❡ ❣❛s♦s❛✳

❆♥❛❧✐t✐❝❛♠❡♥t❡✱ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ s❡ ❝❛r❛❝t❡r✐③❛ ♣♦r s✐♥❣✉❧❛r✐❞❛❞❡s ❡♠ s✉❛s ❢✉♥çõ❡s t❡r♠♦❞✐♥â♠✐❝❛s ✭❡♥❡r❣✐❛ ❧✐✈r❡ ❡ ❞❡r✐✈❛❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♠♦ ♠❛❣♥❡t✐③❛çã♦ ❡ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡✮✳ ❙❡ ✉♠❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ❞❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❢♦r ❞❡s❝♦♥tí♥✉❛ ♥❛ t❡♠♣❡r❛t✉r❛ ❞❡ tr❛♥s✐çã♦ Tc✱ ❛ tr❛♥s✐çã♦ é ❞✐t❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ P❛r❛ tr❛♥s✐çõ❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❡ss❛s ♣r✐♠❡✐r❛s ❞❡r✐✈❛❞❛s sã♦ ❝♦♥tí♥✉❛s✳

◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ❛s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ♥♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q ❡st❛❞♦s ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ❛ ❝❛♠♣♦ ♥✉❧♦ ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ❞❛♥♦s ✈✐❛ s✐♠✉❧❛çã♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ❖ ♠♦❞❡❧♦ ❞❡ P♦tts é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ♣❛r❛ ♠❛✐s ❞❡ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ s♣✐♥✳ ❊❧❡ é ✉♠ ♠♦❞❡❧♦ r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s s❡ ❝♦♠♣❛r❛❞♦ ❛♦ ♠♦❞❡❧♦ ❳❨ ❬✺✱ ✻❪ ♦✉ ❆◆◆◆■ ❬✼❪✱ ♠❛s ❛♣r❡s❡♥t❛ tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ❞❡ ♦r❞❡♥s ❞✐❢❡r❡♥t❡s ❬✽✱ ✾❪ ♣❛r❛ ❝❡rt♦s ✈❛❧♦r❡s ❞❡q ✭♣❛r❛q 4❛ tr❛♥s✐çã♦ s❡ ❝❛r❛❝t❡r✐③❛ ❝♦♠♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❡ ♣❛r❛q >4 ❛ tr❛♥s✐çã♦ é ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣❛r❛ ♦ ♠♦❞❡❧♦

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❜✐❞✐♠❡♥s✐♦♥❛❧ ♥❛ r❡❞❡ q✉❛❞r❛❞❛✮✳ ❖ ♠ét♦❞♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ❞❛♥♦s ❝♦♥s✐st❡ ❜❛s✐❝❛✲ ♠❡♥t❡ ♥♦ ♠♦♥✐t♦r❛♠❡♥t♦ ❞❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ❞✉❛s ♦✉ ♠❛✐s ❝♦♥✜❣✉r❛çõ❡s ❞❡ ✉♠ ♠❡s♠♦ s✐st❡♠❛✱ ❝♦♠ ❞✐❢❡r❡♥t❡s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱ s✉❥❡✐t❛s ❛ ✉♠❛ ❞✐♥â♠✐❝❛ ❡s♣❡❝í✜❝❛ ❡ ❛ ✉♠ ♠❡s♠♦ r✉í❞♦ tér♠✐❝♦✳ ❆ ❢✉♥çã♦ ✐♠♣♦rt❛♥t❡ ❛ s❡r ❛♥❛❧✐s❛❞❛ ♥❡st❛ té❝♥✐❝❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞✐stâ♥❝✐❛ ❞❡ ❍❛♠♠✐♥❣ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❞❛♥♦✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❞❛♥♦ ❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ♦ t❡♠♣♦✱ t❡♠♣❡r❛t✉r❛✱ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱ ♦✉ q✉❛❧q✉❡r ♦✉tr♦ ♣❛râ♠❡tr♦ r❡❧❡✈❛♥t❡✱ ♥♦s ❢♦r♥❡❝❡♠ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ ❝r✐t✐❝❛❧✐❞❛❞❡ ❞♦ s✐st❡♠❛✳

➱ ❢❛t♦ ♥♦tór✐♦ q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❞❛♥♦ ❞❡♣❡♥❞❡♠ ❢♦rt❡♠❡♥t❡ ❞❛ ❞✐♥â♠✐❝❛ ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❡ ❞❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ❞❡ ❞❛♥♦✳ ❯♠ ❡①❡♠♣❧♦ ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞♦ é ♦ ❞♦ ❢❡rr♦♠❛❣♥❡t♦ ❞❡ ■s✐♥❣ ♥❛ r❡❞❡ q✉❛❞r❛❞❛✳ ◆♦ ❡st✉❞♦ r❡❛❧✐③❛❞♦ ♣♦r ❙t❛♥❧❡② ❡t ❛❧ ❬✶✵❪✱ ❢♦r❛♠ ❡♥❝♦♥tr❛❞❛s ❞✐❢❡r❡♥ç❛s ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❞❛♥♦ q✉❛♥❞♦ ❛ ❞✐♥â♠✐❝❛ ❞❡ ❡✈♦❧✉çã♦ ❢♦✐ ❛❧t❡r❛❞❛✳ ❘❡s✉❧t❛❞♦s ♠❛✐s s✉r♣r❡❡♥❞❡♥t❡s ♣♦❞❡♠ s❡r ✈✐st♦s ❡♠ ♠♦❞❡❧♦s ♠❛✐s ❝♦♠♣❧❡①♦s ✭♠♦❞❡❧♦ ❆◆◆■✱ ♠♦❞❡❧♦ ❘❡❧ó❣✐♦ ✭Zp✮✱ ♠♦❞❡❧♦ ❳❨✱ ♠♦❞❡❧♦ ❞❡ P♦tts✮✱ ♦♥❞❡ ❢♦r❛♠ ❡♥❝♦♥tr❛❞❛s ❢❛s❡s ❞✐♥â♠✐❝❛s s❡♠ ❝♦rr❡s♣♦♥❞❡♥t❡s t❡r♠♦❞✐♥â♠✐❝♦s✳ ❊ss❡s r❡s✉❧t❛❞♦s ✐♠♣❧✐❝❛♠ ❡♠ ✉♠❛ r❡❞✉çã♦ ♥♦ ✐♥t❡r❡ss❡ ✐♥✐❝✐❛❧ ❡ ♥❛ ❡s♣❡r❛♥ç❛ q✉❡ ❤❛✈✐❛ ❝♦♠ r❡❧❛çã♦ ❛♦ ♠ét♦❞♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❞❛♥♦s ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❞❡ ❞❡t❡❝çã♦ ❞❡ ❢❛s❡s ❝❛ót✐❝❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠ ♠♦❞❡❧♦s s❡♠ ✉♠❛ ❞✐♥â♠✐❝❛ ✐♥trí♥s❡❝❛✳

◆❡st❡ tr❛❜❛❧❤♦ ✐r❡♠♦s ❡st✉❞❛r ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❞❛♥♦s ♥♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♠ q❡st❛❞♦s✳ ▼♦str❛♠♦s q✉❡✱ s❡ ❞❡✈✐❞❛♠❡♥t❡ ❡s❝♦❧❤✐❞❛ ❛ ❢✉♥çã♦ ❞❛♥♦D(t)❡ ❛ ❞✐♥â♠✐❝❛ ❞❡ ❡✈♦❧✉çã♦ ❞❡ ❢♦r♠❛ ❛ ♣r❡s❡r✈❛r ❛ ✐♥✈❛r✐â♥❝✐❛ r♦t❛❝✐♦♥❛❧ ❞♦ s✐st❡♠❛✱ ♦ ❞❛♥♦ ❛♣r❡s❡♥t❛ ❡♥tã♦ ❛♣❡♥❛s ❞✉❛s ❢❛s❡s ♣❛r❛ q✉❛❧q✉❡r ✉♠ ❞♦s q ❡st❛❞♦s ❞♦ ♠♦❞❡❧♦✱ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ q✉❡ ❢♦✐ ♣r♦♣♦st♦ ❡♠ tr❛❜❛❧❤♦s ❛♥t❡r✐♦r❡s ❬✶✶✱ ✶✷✱ ✶✸❪ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts✳ ▼♦str❛♠♦s t❛♠❜é♠ q✉❡✱ ♣❛rt✐♥❞♦ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❞❛♥♦ ❡ ❞❛ ❞✐♥â♠✐❝❛ ♣r♦♣♦st❛ ♥❡st❡ tr❛❜❛❧❤♦✱ ♦ s✐st❡♠❛ ♥ã♦ ❛♣r❡s❡♥t❛ ♥❡♥❤✉♠ ❡❢❡✐t♦ ❞❡ ♠❡♠ór✐❛✱ ♦✉ s❡❥❛✱ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✳ ❱❛♠♦s ❞✐s❝✉t✐r t❛♠❜é♠ ❛ ❝♦✐♥❝✐❞ê♥❝✐❛✱ ♦✉ ♥ã♦✱ ❞❛s t❡♠♣❡r❛t✉r❛s ❞❡ tr❛♥s✐çã♦ ❞❡ ❞❛♥♦ ❝♦♠ ❛s t❡♠♣❡r❛t✉r❛s ❝rít✐❝❛s ❡stát✐❝❛s ❞♦ ♠♦❞❡❧♦ ❡ ❛ q✉❡stã♦ ❞❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛ ♦r❞❡♠ ❞❛ tr❛♥s✐çã♦ ❛tr❛✈és ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❞❛♥♦✳

◆♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s ❢❛r❡♠♦s ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ♠❛✐s ❞❡t❛❧❤❛❞❛ ❞♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❜❡♠ ❝♦♠♦ ❞♦ ♠ét♦❞♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ❞❛♥♦s ❡ ❛♣r❡s❡♥t❛r❡♠♦s ♥♦ss♦s r❡s✉❧t❛❞♦s ❡ ❞✐s❝✉ssõ❡s✳

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❈❛♣ít✉❧♦ ✷

❖ ▼♦❞❡❧♦ ❞❡ P♦tts

❖ ▼♦❞❡❧♦ ❞❡ P♦tts ❝♦♠q❡st❛❞♦s ❬✶✹❪ é ✉♠ s✐st❡♠❛ ❝❧áss✐❝♦ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ s♣✐♥s ❡♠ r❡❞❡s ❝r✐st❛❧✐♥❛s ♥♦t❛✈❡❧♠❡♥t❡ r✐❝♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦ ❡ ❝♦♠ ✈ár✐❛s ❛♣❧✐❝❛çõ❡s ❡①♣❡r✐♠❡♥t❛✐s✳ ❖ ♠♦❞❡❧♦✱ q✉❡ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ♣❛r❛ ♠❛✐s ❞❡ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ s♣✐♥✱ ❢♦✐ ♣r♦♣♦st♦ ♣❡❧♦ ❡♥tã♦ ♣r♦❢❡ss♦r ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❖①❢♦r❞✱ ❈②r✐❧ ❉♦♠❜✱ ❛♦ ❡st✉❞❛♥t❡ ❘❡♥❢r❡② ❇✳ P♦tts ❡♠ ✶✾✺✶ ❬✶✺❪ ❝♦♠♦ t❡♠❛ ❞❡ s✉❛ t❡s❡ ❞❡ P❤❉✳ ❆ ✐❞é✐❛ ❞♦ ♠♦❞❡❧♦ s✉r❣✐✉ ❛♣ós ❍✳ ❆✳ ❑r❛♠❡rs ❡ ●✳ ❍✳ ❲❛♥♥✐❡r ❡♠ ✶✾✹✶ ❬✶✻❪ ❣❡♥❡r❛❧✐③❛r❡♠ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ✐♥t❡r❛çõ❡s ❡♥tr❡ s♣✐♥s ♣❛r❛❧❡❧♦s ❡ ❛♥t✐♣❛r❛❧❡❧♦s✱ ❝♦♠♦ ✉♠ s✐st❡♠❛ ❞❡ s♣✐♥s ❝♦♥✜♥❛❞♦s ❡♠ ✉♠ ♣❧❛♥♦ q✉❡ ♣♦❞✐❛♠ ❛ss✉♠✐r ✸ ❡st❛❞♦s ❞✐❢❡r❡♥t❡s ✭0✱ 2π/3 ❡ 4π/3✮✳ ❆ ✐❞é✐❛ ❞❡ ❉♦♠❜ é q✉❡ s❡r✐❛ ♣♦ssí✈❡❧ ❡st❡♥❞❡r ♦ r❡s✉❧t❛❞♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❑r❛♠❡rs✲❲❛♥♥✐❡r ♣❛r❛q ✈❡t♦r❡s ✭❡st❛❞♦s✮ ❧♦❝❛❧✐③❛❞♦s ❡♠q❞✐r❡çõ❡s s✐♠étr✐❝❛s ♥♦ ♣❧❛♥♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❡s♣❡❝✐✜❝❛❞❛s ♣❡❧♦s â♥❣✉❧♦s

θn= 2πn

q , n= 0,1, ..., q−1 ✭✷✳✶✮ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✷✳✶✳

❊st❛ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧ ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❝♦♥s✐❞❡r❛ ❛♣❡♥❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ s♣✐♥s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✱ s❡♥❞♦ q✉❡ ❡st❛s ✐♥t❡r❛çõ❡s ❞❡♣❡♥❞❡♠ ❞♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ❡♥tr❡ ♦s s♣✐♥s ✭tr❛t❛❞♦s ❝♦♠♦ ✈❡t♦r❡s✮✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ❍❛♠✐❧t♦♥✐❛♥❛ ❣❡♥❡r❛❧✐③❛❞❛

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❋✐❣✉r❛ ✷✳✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ♣❧❛♥♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♥t❡♥❞♦ ❛❧❣✉♥s ❞♦s q ✈❡t♦r❡s ♣r♦♣♦st♦s ♣♦r ❉♦♠❜✳

♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦

H = X <i,j>

J(θij) ✭✷✳✷✮

❡♠ q✉❡ ❛ ❢✉♥çã♦J(θ) é2π✲♣❡r✐ó❞✐❝❛ ❡ θij =θi−θj é ♦ â♥❣✉❧♦ ❡♥tr❡ ♦s s♣✐♥s ✈✐③✐♥❤♦s ❧♦❝❛❧✐③❛❞♦s ♥♦s sít✐♦s ✐ ❡ ❥ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❖ ♠♦❞❡❧♦ ❡s♣❡❝í✜❝♦ s✉❣❡r✐❞♦ ♣♦r ❉♦♠❜ ❬✶✺❪✱ q✉❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ P❧❛♥❛r✱ ❞❡✜♥❡ ❛ ❢✉♥çã♦ J(θ) ❝♦♠♦

J(θ) =ǫ1cos(θ) ✭✷✳✸✮

❯s❛♥❞♦ ❛ ♠❡s♠❛ ❛❜♦r❞❛❣❡♠ ❞❡ ❑r❛♠❡rs✲❲❛♥♥✐❡r ❬✶✻❪✱ P♦tts ♣ô❞❡ ❞❡t❡r♠✐♥❛r ❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ✭♦✉ t❡♠♣❡r❛t✉r❛ ❞❡ ❈✉r✐❡ ♦✉ ♣♦♥t♦ ❞❡ ❈✉r✐❡✮ ♣❛r❛ ♦ ♠♦❞❡❧♦ P❧❛♥❛r ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠q❂✷✱ ✸ ❡ ✹ ♥ú♠❡r♦s ❞❡ ❡st❛❞♦s ❬✽❪✳ ❉❡✈✐❞♦ à ✐♠♣♦ss✐❜✐✲ ❧✐❞❛❞❡ ❞❡ ❡st❡♥❞❡r ♦s r❡s✉❧t❛❞♦s ♣❛r❛ ✉♠ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s q❃✹✱ P♦tts s✉❣❡r✐✉ ✉♠❛ ♥♦✈❛ ❢♦r♠❛ ❞❡ ❞❡✜♥✐r ❛ ❢✉♥çã♦J(θ)✱ ❡♠ q✉❡ ♦s ❡st❛❞♦s sã♦ ❞✐s❝r❡t✐③❛❞♦s ❞❛ ❢♦r♠❛

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❡♠ q✉❡ δKr r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦ ❉❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r ✭δKr(α, α) = 1✱ δKr(α, β) = 0 ❝♦♠ α 6= β✮✳ ❆s ❝♦♥st❛♥t❡s ǫ1 ❡ ǫ2 sã♦ ❝♦♥st❛♥t❡s ❞❡ ❛❝♦♣❧❛♠❡♥t♦✱ s❡♥❞♦ q✉❡✱ ♣❛r❛

ǫ2❃✵ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts é ❢❡rr♦♠❛❣♥ét✐❝♦ ❡ ♣❛r❛ǫ2❁✵ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦✶✳

❖ ♠♦❞❡❧♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r P♦tts s❡ ❞❡st❛❝♦✉ ♥♦ ❡st✉❞♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ s❡♥❞♦ ♠❛✐s r✐❝♦ ❡ ♠❛✐s ❣❡r❛❧ q✉❡ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣✳ ❊❧❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ❞❡ P♦tts P❛❞rã♦ ✭♦✉ ♠♦❞❡❧♦ ❞❡ ❆s❤❦✐♥✲❚❡❧❧❡r✲P♦tts✮✳ ◆♦t❡ q✉❡ ❛♠❜♦s ♦s ♠♦❞❡❧♦s ❝♦✐♥❝✐❞❡♠ ❝♦♠ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s éq= 2✳

❯s❛♥❞♦ ❛ s✐♠❡tr✐❛ ❞♦ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦sq1❞✐♠❡♥s✐♦♥❛❧✱ ♣♦❞❡♠♦s ❞❡✜♥✐r✱ ❞❡ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛✱ ❛ ❢✉♥çã♦ ❉❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r ❡♠ t❡r♠♦s ❞❡ q ✈❡t♦r❡s ✉♥✐tár✐♦s ♥♦ ❡s♣❛ç♦q1❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♦

δKr(a, b) = 1

q[1 + (q−1) ˆSa•Sˆb] ✭✷✳✺✮ ❡♠ q✉❡SˆaSˆb✱ ❝♦♠ab= 0,1, ..., q1✱ r❡♣r❡s❡♥t❛♠q✈❡t♦r❡s ✉♥✐tár✐♦s q✉❡ ❛♣♦♥t❛♠ ♣❛r❛ ❛sq ❞✐r❡çõ❡s s✐♠étr✐❝❛s ❞♦ ❤✐♣❡rt❡tr❛❡❞r♦✷ ❞❡ ❞✐♠❡♥sã♦ q1✳

P♦❞❡♠♦s ♦❜s❡r✈❛r ❡①❡♠♣❧♦s ❞♦sq✈❡t♦r❡s ♥♦ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦sq1❞✐♠❡♥s✐♦♥❛❧ ♥❛ ✜❣✉r❛ ✷✳✷✳

❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ♦♥❞❡ ♦ ♠♦❞❡❧♦ P❧❛♥❛r ❡ ♦ ♠♦❞❡❧♦ P❛❞rã♦ t♦r♥❛♠✲s❡ ✐❞ê♥✲ t✐❝♦s é q✉❛♥❞♦ ❛s ❝♦♥st❛♥t❡s ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ♠❛❣♥ét✐❝♦ ♣❛r❛ ♦s ❡st❛❞♦sq= 2 ❡q = 3 ❛ss✉♠❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿ ǫ2 = 2ǫ1 ❡ ǫ2 = 3ǫ1/2✳ ❆q✉✐ ❛♠❜♦s

♦s ♠♦❞❡❧♦s ♣❛ss❛♠ ❛ t❡r ♦ ♠❡s♠♦ ❣r❛✉ ❞❡ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ❡ ♦ ♠❡s♠♦ ✏❣❛♣✑ ❞❡ ❡♥❡r❣✐❛✳ ❆❧é♠ ❞❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✱ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❣❡♥❡r❛❧✐③❛❞♦ ❞♦ ♠♦❞❡❧♦ ❞❡ P♦tts ❝♦♥té♠ ♦s t❡r♠♦s ❞❡ ✐♥t❡r❛çã♦ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ✭❡♥tr❡ s❡❣✉♥❞♦s✱ t❡r✲ ❝❡✐r♦s✱ ✳✳✳✱ ♥✲és✐♠♦s ✈✐③✐♥❤♦s✮ ❜❡♠ ❝♦♠♦ ♦ t❡r♠♦ ❞❡ ✐♥t❡r❛çã♦ ❝♦♠ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✶▼❛t❡r✐❛✐s ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦s ✭M nO2✮ sã♦ ❛q✉❡❧❡s ❝✉❥❛ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ❢♦rç❛ át♦♠♦s ✈✐③✐♥❤♦s

❛ ❛ss✉♠✐r❡♠ ♦r✐❡♥t❛çõ❡s ❞❡ s♣✐♥s ❛♥t✐♣❛r❛❧❡❧♦s✳ ❚❛✐s ♠❛t❡r✐❛✐s ❛♣r❡s❡♥t❛♠ ✉♠ ♠❛❣♥❡t✐s♠♦ ❡❢❡t✐✈♦ ❡①t❡r♥♦ ♠✉✐t♦ ♣❡q✉❡♥♦ ♦✉ ♠❡s♠♦ ♥✉❧♦✳

❚❡tr❛❡❞r♦ é ✉♠ só❧✐❞♦ ❣❡♦♠étr✐❝♦ ❝✉❥❛ s✉♣❡r❢í❝✐❡ é ❝♦♠♣♦st❛ ♣♦r ✹ ❢❛❝❡s tr✐❛♥❣✉❧❛r❡s✳ ❖ ❤✐♣❡r✲

t❡tr❛❡❞r♦ é ✉♠ t❡tr❛❡❞r♦ ❡♠ ♠❛✐s ❞❡ três ❞✐♠❡♥sõ❡s✳

(25)

❋✐❣✉r❛ ✷✳✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦s q ✈❡t♦r❡s ♥♦ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦ q1 ❞✐♠❡♥s✐♦♥❛❧✳

❡①t❡r♥♦✱ ❝♦♠♦ ♠♦str❛ ❛ ❡q✉❛çã♦ ❛❜❛✐①♦✿

−βH =BX i

δKr(σi,0) +ξ

X

(i,j)

δKr(σi, σj) +ξ3

X

(i,j,k)

δKr(σi, σj, σk) +... ✭✷✳✻✮

❝♦♠β = 1/kBT✱ ❡σi = 0,1, ..., q−1❡s♣❡❝✐✜❝❛ ♦ ❡st❛❞♦ ❞♦ s♣✐♥ ❞♦ ✐✲és✐♠♦ sít✐♦✳ ◆♦t❡ q✉❡ ❛ ❢✉♥çã♦ ❉❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r é ✉♥✐tár✐❛ ♣❛r❛ t♦❞❛ ❝♦♠❜✐♥❛çã♦ ❞❡ s♣✐♥s ♥♦ ♠❡s♠♦ ❡st❛❞♦✱ ♦✉ s❡❥❛✱

δKr(σi, ..., σn) = 1, s❡ σi =...=σn

= 0, s❡ ❤♦✉✈❡r ✉♠ ♦✉ ♠❛✐s ❡st❛❞♦s ❞✐❢❡r❡♥t❡s

❆q✉✐✱ ξ = ǫ2✱ ξn ❝♦♠ n > 3 é ❛ ❝♦♥st❛♥t❡ ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ♠❛❣♥ét✐❝♦ ❞❛ ✐♥t❡r❛çã♦ ❞❡ ❜❧♦❝♦s ❞❡ns♣✐♥s ❡B r❡♣r❡s❡♥t❛ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦ ♥❛ ❞✐r❡çã♦ ❞♦ ❡st❛❞♦ ✵✳

❉❡✜♥✐❞♦ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞♦ ♠♦❞❡❧♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❢✉♥çã♦ ❞❡ ♣❛rt✐çã♦ ❝♦♠♦✿

Z(q;B, ξ, ξn) =

X

{σ}

e−βH ✭✷✳✼✮

❊st❛ ❢✉♥çã♦✱ q✉❡ ♥♦s ♣❡r♠✐t❡ ❞❡t❡r♠✐♥❛r t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s r❡❧❡✈❛♥✲

(26)

t❡s ❞❡ ✉♠ s✐st❡♠❛✱ ❡stá ❛ss♦❝✐❛❞❛ à ♥♦r♠❛❧✐③❛çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡Pi ❞♦ s✐st❡♠❛ ❡st❛r ❡♠ ✉♠ ❞♦s s❡✉s ❡st❛❞♦s ❛❝❡ssí✈❡✐s ❝♦♠ ❡♥❡r❣✐❛Ei ❞❛❞❛ ♣♦r

Pi = e−βEi

Z ✭✷✳✽✮

❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦ ✭♦✉ s❡❥❛✱ ❧✐♠✐t❡ ♥♦ q✉❛❧ ♦ ✈❛❧♦r ❞❛s ❣r❛♥❞❡③❛s t❡r♠♦❞✐♥â♠✐❝❛s✱ ❝♦♠♦ ❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡ ❡ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡ ♠❛❣♥❡t✐❝❛✱ ❡♥❡r❣✐❛ ❧✐✈r❡✱ ❝❛❧♦r ❡s♣❡❝í✜❝♦✱ ❡t❝✱ s♦❢r❡♠ ❞❡s✈✐♦s r❡❧❛t✐✈♦s ❡①tr❡♠❛♠❡♥t❡ ♣❡q✉❡♥♦s ♣♦✐s ♦ s✐st❡♠❛ é ❡①tr❡♠❛♠❡♥t❡ ❣r❛♥❞❡ ✭N → ∞✮✮ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ♠❛❣♥ét✐❝❛ ✭♦✉

❡♥❡r❣✐❛ ❧✐✈r❡ ❞❡ ●✐❜❜s✮ ♣♦r sít✐♦ ❝♦♠♦

−βg(q;B, ξ, ξn) = lim N→∞

1

Nln(Z(q;B, ξ, ξn)) ✭✷✳✾✮ ❉❡st❡ ♠♦❞♦✱ ♣♦❞❡♠♦s ❡♥tã♦ ❡s❝r❡✈❡r ❛❧❣✉♠❛s ❞❛s ❢✉♥çõ❡s t❡r♠♦❞✐♥â♠✐❝❛s ❝♦♠♦ ❛ ❡♥❡r❣✐❛ ✐♥t❡r♥❛ ♣♦r sít✐♦

u(q;B, ξ, ξn) = − ∂

∂βg(q;B, ξ, ξn) ✭✷✳✶✵✮ ❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ♣♦r sít✐♦

m(q;B, ξ, ξn) = − ∂

∂Bg(q;B, ξ, ξn) ✭✷✳✶✶✮ ❆s ❢✉♥çõ❡s t❡r♠♦❞✐♥â♠✐❝❛s sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛ ♦r❞❡♠ ❞❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ♥♦ ♠♦❞❡❧♦ ❞❡ P♦tts✳ ❖ ♠♦❞❡❧♦ ❞❡ P♦tts ❛♣r❡s❡♥t❛ tr❛♥s✐çõ❡s ❞❡ ♦r❞❡♥s ❞✐❢❡r❡♥t❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s✳ P❛r❛q4♦ ♠♦❞❡❧♦ ❛♣r❡s❡♥t❛ ✉♠❛ tr❛♥s✐çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❡ ♣❛r❛q >4 ♦ ♠♦❞❡❧♦ ❛♣r❡s❡♥t❛ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❝❛r❛❝t❡r✐③❛❞❛ ♣❡❧❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❞❛s ❢✉♥çõ❡s t❡r♠♦❞✐♥â♠✐❝❛s ❡♠ Tc = 1/ln(1 +√q)✳

❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡①♣❡r✐♠❡♥❛❧✱ ♦ ♠♦❞❡❧♦ ❞❡ P♦tts ♣♦ss✉✐ r❡❧❛çã♦ ❝♦♠ ❞✐✈❡rs♦s ♠♦❞❡❧♦s ❡♠ ❢ís✐❝❛ ❡st❛tíst✐❝❛ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ♠♦❞❡❧♦ ❞❡ ✈ért✐❝❡s ❝♦♠ r❡❣r❛ ❞❡ ❣❡❧♦✱ ♣❡r❝♦❧❛çã♦✱ r❡❞❡s ❞❡ r❡s✐st♦r❡s✱ ❡t❝ ❬✽❪✳ ❖✉tr❛s ❛♣❧✐❝❛çõ❡s ♣♦❞❡♠ s❡r ✈✐st❛s ❡♠

(27)

s✐st❡♠❛s ❞❡ ❛❞❡sã♦ ❝❡❧✉❧❛r ❬✶✼✱ ✶✽❪✱ ♠♦❞❡❧♦s ❞❡ ❡s♣✉♠❛s ❬✶✾❪✱ ❡t❝✳

(28)

❈❛♣ít✉❧♦ ✸

❖ ▼ét♦❞♦ ❞❡ Pr♦♣❛❣❛çã♦ ❞❡ ❉❛♥♦s

❖ ♠ét♦❞♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ❞❛♥♦s✱ ❛♣❡s❛r ❞❡ ❡st❛r ❢♦rt❡♠❡♥t❡ ✐♥❝♦r♣♦r❛❞♦ à ❋ís✐❝❛ ♥♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s✱ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❙t✉❛rt ❑❛✉✛♠❛♥ ❬✷✵❪ ♥✉♠ ❝♦♥t❡①t♦ ❜✐♦❧ó❣✐❝♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞✐♥â♠✐❝♦ ❞❡ ❣❡♥❡s ❡ s✉❛s ♠✉t❛çõ❡s✳ ❆ ✐❞é✐❛ ✐♥✐❝✐❛❧ ❝♦♥s✐st✐❛ ❡♠ ❡st✉❞❛r ❛ ✐♥✢✉ê♥❝✐❛ ❞❡ ♣❡q✉❡♥❛s ♠✉t❛çõ❡s ✭❞❛♥♦s✮ ❞✉r❛♥t❡ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ss❡s ❣❡♥❡s✳ ❊♠ s❡✉ ♠♦❞❡❧♦✱ ❑❛✉✛♠❛♥ ❞❡✜♥✐❛ s❡✉s ❣❡♥♦♠❛s ❝♦♠♦ ✉♠❛ s❡qüê♥❝✐❛ ❞❡ ❜✐ts q✉❡ ❡✈♦❧✉í❛♠ s❡❣✉♥❞♦ r❡❣r❛s s✐♠♣❧❡s✱ ❞♦ t✐♣♦ ❛✉tô♠❛t♦ ❝❡❧✉❧❛r✳ ❊❧❡ ❝♦♥s✐❞❡r❛✈❛ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ s❡qüê♥❝✐❛ {σi(A)(t)} ❝♦♠♦ s❡♥❞♦ ✉♠ ❣❡♥♦♠❛ ✏♥♦r♠❛❧✑✱ ❡♥q✉❛♥t♦ q✉❡ ✉♠❛ ♦✉tr❛ s❡qüê♥❝✐❛{σi(B)(t)}r❡♣r❡s❡♥t❛✈❛ ❡ss❡ ♠❡s♠♦ ❣❡♥♦♠❛ ❛♣ós s♦❢r❡r ✉♠❛ ♠✉t❛çã♦ ✭✉♠ ❞❛♥♦✮✳ ❑❛✉✛♠❛♥ ❝♦♥s✐❞❡r♦✉ ❡♥tã♦ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ❞✉❛s s❡qüê♥❝✐❛s ❜✐♥ár✐❛s ❝♦♠♦ s❡♥❞♦ ♦ ❞❛♥♦ ✭♦✉ ❞✐stâ♥❝✐❛ ❞❡ ❍❛♠♠✐♥❣✮✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ❞❛♥♦ ❞❡ ❑❛✉✛♠❛♥✱ ♥✉♠ t❡♠♣♦t✱ ❢♦✐ ❞❡✜♥✐❞♦ ❝♦♠♦

❉✐stâ♥❝✐❛(t) =D(t) 1 N

N

X

i=1

σAi (t)−σiB(t)

✭✸✳✶✮

❡♠ q✉❡N r❡♣r❡s❡♥t❛ ♦ t❛♠❛♥❤♦ ❞❛s s❡qüê♥❝✐❛s✳

❊st✉❞❛♥❞♦ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦s ❣❡♥❡s ❛tr❛✈és ❞❛ ♠❡❞✐❞❛ ❞♦ ❞❛♥♦D(t)✱ ❢♦✐ ♣♦ssí✈❡❧ ♦❜s❡r✈❛r ♦ q✉❛♥t♦ ✉♠❛ ♠✉❞❛♥ç❛ ♥❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s✱ ✐♥✢✉❡♥❝✐❛✈❛ ❛ tr❛❥❡tór✐❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ◗✉❛♥❞♦ ❞✉❛s ❝♦♥✜❣✉r❛çõ❡s ✐♥✐❝✐❛✐s ❡✈♦❧✉❡♠ ♥♦ t❡♠♣♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❢❛✐①❛ ❞❡ ♣❛râ♠❡tr♦s✱ ❡ ❡st❛s tr❛❥❡tór✐❛s s❡❣✉❡♠ ♣ró①✐♠❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❛té ❝♦❛❧❡s❝❡r❡♠✱ ♦ s✐st❡♠❛ é ❞✐t♦ ❡♥❝♦♥tr❛r✲s❡ ❡♠

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