ESTUDO DE MODELOS DE CRESCIMENTO DISCRETOS EM … completo.pdf · Ficha catalográfica preparada pela Biblioteca Central da ... sem considerar o ζ lnt, ... Estudo de modelos de crescimento

❯ ❱

❯ ❯ ❯

ssrtçã♦ ♣rs♥t à ❯♥rs

r ❱ç♦s ♦♠♦ ♣rt s ①ê♥s ♦

Pr♦r♠ Pósrçã♦ ♠ ís ♣

♣r ♦t♥çã♦ ♦ tít♦ str

♥t

❱

Ficha catalográfica preparada pela Biblioteca Central daUniversidade Federal de Viçosa - Câmpus Viçosa

T

Carrasco, Ismael Segundo da Silva, 1985-C311e2014

Estudo de modelos de crescimento discretos emsubstratos que crescem lateralmente. / Ismael Segundo daSilva Carrasco. - Viçosa, MG, 2014.

x, 89f. : il. (algumas color.) ; 29 cm.

Inclui apêndices.Orientador : Tiago José de Oliveira.Dissertação (mestrado) - Universidade Federal de

Viçosa.Referências bibliográficas: f.83-89.

1. Simulação matemática. 2. Simulação em MonteCarlo. 3. Modelos de crescimento. 4. Sistemas KPZ.I. Universidade Federal de Viçosa. Departamento de Física.Programa de Pós-graduação em Educação Física. II. Título.

CDD 22. ed. 511.8

FichaCatalografica :: Fichacatalografica https://www3.dti.ufv.br/bbt/ficha/cadastrarficha/visua...

2 de 3 30-03-2015 15:48

r♠♥t♦s

①♣rss♦ q ♠s s♥r♦s r♠♥t♦s t♦s s ♣ss♦s q ♠

♦r♠ ♠ r♠ ♦♥qstr ♠s st t♣ r♦ t♦♦s ♠♥

♠í ♣♦ ♣♦♦ ♠ s♣ ♦ ♠ r♠ã♦ ♣ ♥s rsõs ss

t①t♦ r♦ ♦s ♠s ♠♦s ♣♦s ♠♦♠♥t♦s ③r s♦♥trçã♦ r♦

♠r ♣♦r t♦♦ ♠♦r r♥♦ r♥t t♦♦s sts ♥♦s

r♦ ♦ ♦ ♦r♥t♦r ♣ ♣ê♥ ♦♦rçã♦ r♥t sss ♥♦s

r♦ t♠é♠ ♦ í♦ ♦♦r♥t♦r ♣♦s ♦♥s♦s ♥ss tr♦ ♣♦s ♥

♥t♦s r♥t ♠♥ rçã♦

♥♠♥t rç♦ às ♥s ♥♦♥s q ①♠ ♣sqs P

Pq ♠ s♣ P ♣♦ ♥♥♠♥t♦ ♠♥ ♦s

♠ár♦

st rs

s♠♦

strt ①

♥tr♦çã♦

♦♥t♦s ás♦s ♠ ♥â♠ ♥tr

rts t♦♥

s ♥â♠ r♦s

sss ♥rs ♥ ♥â♠ ♥trs

qçõs st♦ásts ♣r♥í♣♦s s♠tr

♣♦sçã♦ tór

ss rs❲♥s♦♥❲

ss rrPrs❩♥P❩

strçõs trs

♥trs rs ♣r♦①♠çã♦ ♦♠í♥♦s rs♥ts

rs♠♥t♦ ♥trs rs

♦♠í♥♦s rs♥ts

♦♦s srt♦s ♣r ♦ rs♠♥t♦ tr rt ♥tr

rs♠♥t♦ tr ♦ sstrt♦

♦♦s srt♦s ss P❩

strt s♦ ♦♥ s♦

t♥

♥st♣

tr♦s ♠♦♦s

st♦s ♠ ♠ ♠♥sã♦

①♣♦♥t rs♠♥t♦

strçõs trs

t♦s s ♣çõs ♦r♠ s ♦rrçõs

r♦ss♦r ♣r ❯

st♦s ♠ s ♠♥sõs

Prâ♠tr♦s ♥ã♦ ♥rss

①♣♦♥t rs♠♥t♦

strçõs trs

♦♠♣rçã♦ ♦♠ ♦ s♦ ♥♠♥s♦♥

♦♥sõs Prs♣ts

♠t ss♥♦ strçã♦ ♥♦♠

❯♠ ♣r♦♣r s♠♣s ♥çõs ♣rós

st rs

strçã♦ ♦ ♣r♦ss♦ rrs♦ ♦r♠çã♦ ♦ tr♥♦ r

♣♥s t♦ ♦t♦ ♣♦s ♠ ♥ú♠r♦ r♥ trçõs

r ①trí ❬❪

♦♠♣♦rt♠♥t♦s tí♣♦s r♦s strçã♦ ♠ ♥çã♦ ♦

t♠♥♦ ♦ sst♠ ♦ t♠♣♦ strçã♦ ♠ ♥çã♦ ♦ t♠♥♦

♦ sst♠

strçã♦ trs tr♦♦♥çã♦ rsts íq♦s ♦♠

♣rs ♦♠ s ♣rsõs tórs r tr é ❯

♣♦♥t s sí♠♦♦s ♣r♥♦s ③s r♠♦s ♦r♠

♦t♦s ♣r ♥trs rrs ♦♠ t♠♣♦s t = 10s t = 30s r

s♣t♠♥t á ♦s sí♠♦♦s ♥ã♦ ♣r♥♦s ③s ♣úr♣r♦s ♦r♠

♦t♦s ♣r rs♠♥t♦s ♣♥♦s ♦♠ t♠♣♦s t = 20s t = 60s r

s♣t♠♥t r ①trí ❬❪ rá♦ s♥ss r

♦s rt♦s r③s ♦ rs♠♥t♦ r♦ ③ ♣♥♦ r♠♦

rá♦ ♠♦str q r♥ç ♥tr s strçõs ♠ é ♠

s♦♠♥t♦ ♥ ♠é ♣♦s s♥ss rt♦s ♦♥r♠ ♣r ♦s

rst♦s ♥ít♦s ❬❪

rá♦ t③♦ ♥ st♠t ♦ ♦r v∞ ss ♦r ♦ ♦t♦

①tr♣♦♥♦ rs♦s sts r③♦s ♥ rã♦ t−2/3 ♣q♥♦s

st♠t ♦ ♦r Γ Pr ss♦ rss s ♣♦tê♥ ♦r♠

sts ♥ rã♦ t♠♣♦s r♥s

①♣♦♥t rs♠♥t♦ ♦s ♠♦♦s t♥ ♣r

r♥ts ♦s trs s ♥s tr sã♦ ♦rrs♣♦♥♥ts

β = 1/3

rá♦s ♥ss rt♦s ♠ ♥çã♦ t−2β s

trçã♦ trs ♦ ♠♦♦ ♥st♣ ♠ sstrt♦s q rs♠

tr♠♥t

♦ ♦rrçã♦ ζt−β−1 s♥♦ ♠é ❯

rá♦s ζ ♣r s r♥ts ♦s ♦s ♠♦♦s

t♥ ♥st♣

á♦ ♦ s♦ strçã♦ s♠ ♦♥srr ♦ ζ ln t ♠

♦♥sr♥♦ ss ♦rrçã♦

rá♦s 〈η〉 ♣r s r♥ts ♦s ♦s ♠♦♦s

t♥ ♥st♣

rá♦ q ♠♦str ♦ rátr tr♠♥íst♦ ζ r q

♠♦str ①stê♥ ♦rrçõs ♦r♠ ♠♦r ♠ q 〈η2〉c 6= 0

rá♦ 〈(~∇h)2〉 ♠ ♥çã♦ ♦ t♠♣♦ ♣r sstrt♦s t♠♥♦

①♦ r ♣rt ír♦s sstrt♦s rs♥ts r r♠

qr♦s ú♠r♦ ♣rtís ♣♦sts ♦ ♣♦

♥ú♠r♦ t♦t ♣rtís q t♥tr♠ sr ♣♦sts r

♠ qr♦s stá ss♦ ♦ s♦ rs♥t ♣rt ír♦s

♣r sstrt♦s ♦♠ t♠♥♦ ①♦ ♥ ③ tr é ♦ ♦r

v∞ ♦ ♠♦♦

rá♦ ♦çã♦ tr ♠é ♣r ♦s ♠♦♦s ♥st♣

t♥ ♦♠ sstrt♦ t♠♥♦ ①♦ r ♣rt sstrt♦

rs♥♦ ♦♠ ♦ vw = 100 r r♠

rá♦ s♥ss rt♦s ♣r r♥ts t♠♥♦s ♥

s r ♣r L0 = 20 ♠ ♥çã♦ L/L0 ♥ã♦ r sí ♥ss

s ♣♦r ss♦ stá ♠ ♥çã♦ t/640 ♥♦s ♦s rá♦s

❱♦ rs♠♥t♦ ♣r r♥ts t♠♥♦s sst♠

♥ tr é ♦ ♦r v∞ rá♦ ∆v ♠ ♥çã♦ 1/L2−2α

♦♥t ♥r ss rt é −λA/2

①♣♦♥t rs♠♥t♦ ♠ ♥çã♦ ♦ t♠♣♦ ♣r r♥ts ♦

s ♦s ♠♦♦s ♥st♣ t♥ ♥

tr r♣rs♥t ♦ ♦r β = 0, 24 q é ♦ ♦r ♦ ①♣♦♥t

ss P❩ ♠ s ♠♥sõs

st♠t s♥ss rt♦s strçã♦ trs

s♥♦ ♦ ♠♦♦ ♥st♣

rá♦s ♣r ♦ ♦ ♦ ♦r g1 r ♦♠ qr♦s ♦

trç s♠ ♦♥srr ♥♥♠ ♦rrçã♦ ♦♠ ír♦s ♣ós ♥♦

♣ss♦s trçã♦ rá♦s ♣r tr♠♥çã♦ ♦rrçã♦

r ♦♠ qr♦s ♦ trç ♦♥sr♥♦ ♦ ♣r♠r♦ g1 ♦

♦♠ ír♦s t③♥♦ ♦ ♦r ♥ g1

rá♦s ζ ♣r s r♥ts ♦s ♦s ♠♦♦s

t♥ ♥st♣

rá♦ q ♠♦str ♠♣♦ss s r ζ ♣r ♦

vw = 2 r g1tβ−1 ♣r s ♦s vw = 2 ír♦s

vw = 6 qr♦s

rá♦s 〈η〉 ♣r s r♥ts ♦s ♦s ♠♦♦s

t♥ ♥st♣

s♠♦

s♠ ♥♦ ❯♥rs r ❱ç♦srr♦ st♦ ♠♦♦s rs♠♥t♦ srt♦s ♠ sstrt♦s

q rs♠ tr♠♥t r♥t♦r ♦ ♦sé r ♦♦r♥t♦rs í♦♦st rrr ú♥♦r ♥② r♦ s

♥â♠ ♥trs rs é ♠ r ♠s í str t♥t♦ ♥ít

q♥t♦ ♦♠♣t♦♥♠♥t q s ♣♥s st♦ t♠ ♠♦t♦ tr♦s ♦♥sr♥♦

♠♦♦s tí♣♦s rs♠♥t♦ ♣♥♦ ♠ sstrt♦s q rs♠ tr♠♥t ♦ ♦♥♦

♦ t♠♣♦ ♦♠♦ ♠ ♣r♠r ♦r♠ ♣r ♥sr ♥trs rr♠♥t r

s ♥trt♥t♦ t♦♦s sts st♦s sr♠s ♠ á♦s ①♣♦♥ts s

r♦s ♣♦rt♥t♦ ♥ã♦ ♠♦♥str♠ ♦♥s♠♥t s st s♠♣çã♦

♠ ♥â♠ s♠r à s s♣rís rs Pr srr st ♣♦♥t♦ ♥ós s

t♠♦s ♠♦♦s ♦♥ ♣♦sçã♦ ♣rtís ♦ rs♠♥t♦ tr ♦ sstrt♦

sã♦ r③♦s st♦st♠♥t ♦r♦ ♦♠ ss rs♣ts ♣r♦s st

♠ét♦♦ ♣r♠t♥♦s str qqr ♠♦♦ srt♦ rs♠♥t♦ ♠ sstrt♦s

q ♠♥t♠ tr♠♥t ♥trt♥t♦ q ♥ós ♥♦s rstr♥♠♦s ♠♦♦s ♥

ss rrPrs❩♥ P❩ ♦♥ s strçõs trs s s♣rís sã♦

r♥ts ♣r ♥trs rs ♣♥s ós ♦t♠♦s q ss♥t♦t♠♥t sts

strçõs sã♦ s ♣s strçõs s ♥trs rs t♥t♦ ♠ sstrt♦s

♥♠♥s♦♥s q♥t♦ ♠♥s♦♥s ♦ út♠♦ s♦ ♦♥ ♦r♠ ♥ít

strçã♦ tr ♥ã♦ é ♦♥ ①t♠♥t ♥ós ♦t♠♦s st♠ts ♣rss

♦s ss ♣r♠r♦s ♠♥ts ♦♥r♠♠♦s s ♥rs r♣r♥♥t♠♥t

♦rrçõs ♦rít♠s ♦r♠ ♥♦♥trs ♥♦ P❩ ♥st③ ♣r s strçõs

trs q ♥ã♦ ①st♠ ♥♦s ♠s♠♦s ♠♦♦s ♠ sstrt♦s stát♦s ♦r♠ sts

♦rrçõs ♦ ①♣ ♦♠♦ ♠ t♦ s ♣çõs ♦♥s ♥♦ rs♠♥t♦ t

r ♦ sstrt♦ ♥♥♦ ♦ rs♠♥t♦ ♠ sstrt♦s r♥s ♠ r♦ss♦r ♦

♥♦♥tr♦ ♥s strçõs trs ♣♥♦ ♣r r♦

strt

s♠ ♥♦ ❯♥rs r ❱ç♦s rr② t② ♦ ♠♦s ♦ srt r♦t ♦♥ sstrts tt r♦ t

r② sr ♦ ♦sé r ♦srs í♦ ♦st rrr ú♥♦r ♥♥② r♦ s

②♥♠s ♦ r ♥trs s ♥ ♥r ♠♦r t t♦ st② ♦t ♥

②t② ♥ ♦♠♣tt♦♥② t♥ t t ♦♥s s s ♠♦tt s♦♠ ♦rs

♦♥sr♥ t②♣ t r♦t ♠♦s ♦♥ sstrts r♦ tr② ♥ t♠ s

rst ♣♣r♦ ♦r ♥②③ tr② r ♥trs ♦r ts sts r

s ♦♥ t t♦♥ ♦ s♥ ①♣♦♥♥ts r♦♠ t r♦♥ss ②♥♠ s♥ ♥

ts t② ♦ ♥♦t s♦ ♦♥s② ts s♠♣t♦♥ s t♦ ②♥♠ s♠r

t♦ t ♦♥ ♦ r srs ♥ ♦rr t♦ r② ts ♣♦♥t st② ♠♦s r

♣rt ♣♦st♦♥ ♥ tr r♦t ♦ t sstrt r st♦st② ♣r♦r♠

♦r♥② t♦ tr rs♣t ♣r♦ts s ♠t♦ ♦s s t♦ st② ♥② srt

r♦t ♠♦ ♦♥ r♦♥ sstrts ♦r r rstrt ♦rss t♦ ♠♦s ♥

rrPrs❩♥ P❩ ss r sr t strt♦♥s r r♥t ♦r

r ♥ t ♥trs ❲ ♦♥ tt s②♠♣t♦t② ts strt♦♥s r ♥

② t ♦♥s ♦ r ♥trs ♥ ♦♥ s s ♥ t♦♠♥s♦♥ sstrts ♥ t

st s r t ♥②t ♦r♠ ♦ t t strt♦♥s r ♥♦t ♥♦♥ ①t②

♦t♥ rt st♠ts ♦ tr rst ♠♥ts ♥ ♦♥r♠ tr ♥rst②

r♣rs♥② ♦rt♠ ♦rrt♦♥s r ♦♥ ♥ t P❩ ♥st③ ♦r t t s

trt♦♥s ♦ ♥♦t ①st ♦r t s♠ ♠♦s ♦♥ stt sstrts ♦r♥ ♦

ts ♦rrt♦♥s s ①♣♥ s ♥ t ♦ t ♣t♦♥ ♦ ♦♠♥s ♥ t tr

r♦t ♦ t sstrt trt♥ t r♦t ♦♥ r sstrts r♦ss♦r s ♦♥

①

♥ t t strt♦♥s r♦♠ t t♦ r ♦♥s

①

♣ít♦

♥tr♦çã♦

♦çã♦ ♠ ♥tr q s♣r ♦s ♠♦s st♥t♦s é ♠ ♠♣♦rt♥t

♣r♦♠ ís sttíst ♦r ♦ qír♦ ss♦ ♣♦rq ss t♣♦ sst♠ é ♠t♦

♦♠♠ ♥ ♥tr③ ①♠♣♦s s♠♣s sr♠ r♥t ♠s ❬❪ ①♦ íq♦

♠ ♠♠♦ ♣♦r♦s♦ ❬❪ é♠ sss ♣♦♠♦s tr ♥ô♠♥♦s ♠♣♦rt♥t ♣çã♦

t♥♦ó ♦♠♦ ♣♦sçã♦ ♠s ♥♦s ❬❪ ♣rís t♠é♠ stã♦ ♣rs♥ts

♠ sst♠s ♦ó♦s ♦♠♦ rs♠♥t♦ ♦ô♥ térs ❬❪ t♠♦rs ❬❪

♠t♦ r♥ts ♦ ♣♦♥t♦ st ♠é♦ trs ♣çõs ♣♦♠ sr ♥♦♥trs

♠ ❬❪

❯♠ ①♠♣♦ ♠s ♥♦s q ♣♦♠♦s tr tít♦ strçã♦ é ♦ trt♦

á♠♦ rs♦ s♦r sí♦ ❬❪ ♣♦ss ♣r♦♣rs q sã♦

♥trss♥ts ♣r ♣çõs t♥♦ós ❯♠ s é ♦ ♣ ♥r rt♦ Eg =

1, 528❱ à 300 ♦♥ ss ♥r é r♦♥ ③ ♦♠ ♦♠♣r♠♥t♦ ♦♥

♦r♠ 827♥♠ é♠ ss♦ ♣♦ss ♠ t♦ ♦♥t s♦rçã♦ ó♣t

♦r♠ 5 × 105cm−1 sss ♣r♦♣rs s♠♦♥t♦rs ó♣ts t♦r♥♠ ss

♦♠♣♦st♦ ♠t♦ út ♥ rçã♦ s♣♦st♦s ♦♣t♦ trô♥♦s ♦t♦♦t♦s

r ♣rs♥t♠♦s s ♠♥s q ♠♦str♠ ♦♠♦ ♠ ♠

♠s r♦s♦ ♠ q ♠s ♣rtís sã♦ ♣♦sts s ts s♦r

ss sst♠ ♣♦♠ sr ♥♦♥tr♦s ♠ ❬❪

♠ sst♠s ♦ó♦s é ♠t♦ ♦♠♠ q s ♥trs s♠ rs ❯♠

①♠♣♦ strt♦ sr ♠ tr és t♠♦rs ♦♠♦ ♥ r ss

♥tr♦çã♦

r ♠♥s ♦ts ♥♦ ♦rtór♦ ♥♥♦s♦♣ ❯❱ rs♠♥t♦ ♦♠ t = 60♠♥ t = 240♠♥ r ①trí ❬❪

t♣♦ ♥tr é ♠s í trtr t♥t♦ ♥t♠♥t q♥t♦ ♠ s♠çã♦

s qçõs r♥s r♦♥s à ♥â♠ sã♦ ♠s ♦♠♣s ♠ ♦♠tr

r t♥♦ ♦ trt♠♥t♦ ♥ít♦ ❬❪ ♦♠♣t♦♥♠♥t ♠♦♦s ás

s♦s ♦♠♦ ♦ ♥ ❬❪ rs♦s ♠ rs ♣r♦③♠ s♣rís q ♣rs♥t♠

♥s♦tr♦♣s ♣r♥♦ s♠tr s♦ r♦tçã♦ ❬❪ P♦r ss♦ ♥ás sttíst

s ♠tr ♥s ♣♦♥t♦s q♥ts ♥tr s♣rç♥♦ rt ♦r♠

♦ t♠♣♦ s♠çã♦ ♥ssár♦ ♣r rr ♠ s♣rí ♥tr Pr ♦♥t♦r♥r ss

♣r♦♠ ♠ ♦♣çã♦ é ③r ♦ rs♠♥t♦ ♦r r ♦♥ ♣sr r árs

♦r♠s s ♠♣♠♥tr ❬❪ ♥♦r♠♠♥t sss rs♠♥t♦s ①♠ ♦rt♠♦s

♦♠♣①♦s q ♠ ③ ♠s ís s ♠♣♠♥tr ♠ q ♠♥t

♠♦s s ♠♥sõs ♥tr ss♠ ♥♦t♠♦s q ♥trs rs sã♦ t♦ ♠s

♦♠♣s q s ♣♥s ♦ ♣♦♥t♦ st tór♦

s rtrísts ①ss s ♥trs rs sã♦ rs♠♥t♦ ár

♥tr rtr ♥ã♦ ♥ s♣rí P♦r s s s ♣rs♥ts

♥♦ st♦ rs♠♥t♦s r♦s t♠ s♦ ♣r♦♣♦st♦ ♠ tr♦s ♥ít♦s ❬❪

♠ s♠çã♦ ❬❪ ♠ ♦r♠ s♠♣ ♥♦♥♦ ♦♠í♥♦s rs♥ts

ss t♣♦ ♥tr ♦♥sr♠♦s ♠ sstrt♦ ♣♥♦ ♦♠ ♦♥çõs ♦♥t♦r♥♦

♣rós q rs tr♠♥t ♦♠ ♦ t♠♣♦ ♦ s t♠♦s ♠ s♣rí ár

rs♥t ♣♦ré♠ s♠ rtr ♠r♦só♣ ss ♦r♠ ♦ ♦♠í♥♦ rs♥t ♣♦

sr st♦ ♦♠♦ ♠ ♠♦ tr♠♦ ♥tr ♦s s♦s ♣♥♦ r♦

♥tr♦çã♦

r ♠♠ ♠ tr és t♠♦rs ♦t ♠r♦s♦♣ ó♣t ♥♦♦rtór♦ ís ♦ó ❯❱ ♦rts ♦ ❱♥③

♥s ♦s ♥ç♦s ♥ç♦s ♥ út♠ é ♠r♠ t♥çã♦ ♣r

ss ♥rs rrPrs❩♥ P❩ ♦♥ s strçõs

trs s ♠ ♦r♠ ♦ts ①t♠♥t ❬❪ ♦ ♠♦str♦ ♣♦r Prä♦r

♣♦♥ q s s stã♦ r♦♥s ♦♠ s strçõs r②❲♦♠ t♦r

♠tr③s tórs s ♦tr♠ q ♥♦s s♦s ♣♥♦s strçã♦ é ♣♦

♥s♠ ss♥♦ ♦rt♦♦♥ ss♥ ♦rt♦♦♥ ♥s♠ ♥♦s r♦s

♣♦ ♥s♠ ss♥♦ ♥tár♦ ❯ss♥ ♥tr② ♥s♠ ss ♦r♠

♣♦♠♦s ③r q ss P❩ s s ♦r♦ ♦♠ ♦♠tr ♥tr

é♠ sss ♥ç♦s ①♣r♠♥t♦s ♣rs♦s ♦r♠ r③♦s t③♥♦ trê♥ ♠

rstíq♦ ♣♦r ❬❪ ♦♥ ♦r♠ ♦♥r♠♦s ♦s rst♦s tór♦s

①st ♠ sssã♦ s♦r ♣ ♥ás s ♠ ♥trs

rs ❬❪ ♥trss♦s ♠ rs♦r ss ♠♣ss ♦s ♣sqs♦rs q str♠

♦s ♦♠í♥♦s rs♥ts s ♣r♦♣r♠ ♣r♥♣♠♥t ♦♠ ♥ás s r

♦s ❬❪ ♦ ♥t♥t♦ ♦♥ ♥ás s ♥♦♥ ♥ã♦ á st♥çã♦

♥tr ♦ s♦ r♦ ♦ ♣♥♦ ♣♦s ♦s ①♣♦♥ts rít♦s sã♦ ♦s ♠s♠♦s ♦♠♦ t♦

♥tr♦r♠♥t ss st♥çã♦ ♣♦ sr t st♥♦ s s ♠♦♦s ♥ ss

P❩ ♣♦s s r♠ ♦ r♦ ♣r ♦ ♣♥♦ ss ♦r♠ sr ♥trss♥t str

s s ♠ ♦♠í♥♦s rs♥ts ss♦ ♠ ♥sr s s sã♦ s ♦ s♦ r♦

♦ ♦ ♣♥♦ ♦ ♥ s r♠ ♠♦s ss♠ sr s ♦ ♦♠í♥♦ rs♥t é

♠ ♣r♦①♠çã♦ r③♦á ♣r ♥trs rs

st ssrtçã♦ st♠♦s rs♦s ♠♦♦s srt♦s rs♠♥t♦ ♠ s

♥tr♦çã♦

strt♦s q rs♠ ♦♠ ♦ t♠♣♦ ♦ s ♥ t♠♣♦ ♦ ♥ú♠r♦ sít♦s

r ♠♥t ❯t③♥♦ ♦rt♠♦s q s♥♦♠♦s ♠ (1 + 1)d (2 + 1)d

st♠♦s ♥â♠ ♦s qtr♦ ♣r♠r♦s ♠♥ts s s ♣r r♥ts t①s

rs♠♥t♦ ♦ss♦s rst♦s ♠♦strr♠ q ♣sr r ♦rts ♦rrçõs ♥

s ♥â♠ ♦s ♠♥ts ♦♥r♠ ss♥t♦t♠♥t ♣r ♦s ♠♥ts ❯

♥♦ s♦ (1 + 1)d ♦ s ♣r strçã♦ ♦♠tr r ♠ s ♠♥sõs

♣sr s strçõs r ♣♥ ♥ ♥ã♦ tr♠ s♦ s ♥t♠♥t

♦s ♦rs ♦s ♣r♠r♦s ♠♥ts s ♦r♠ ♦t♦s ♥♠r♠♥t ❬❪ Pr

sstrt♦s ♠♥s♦♥s t♠é♠ ♦t♠♦s q ♦s ♠♥ts ♦♥r♠ ss♥t♦t

♠♥t ♣r ♦ s♦ r♦

♦s tr♦s ❬❪ ss t♦rs r♠♠ q ♦ t♠♥♦ ♥ ♦ sst♠

♥ã♦ tr ♦r♠ s♥t ♥â♠ ♦s ♦♠í♥♦s rs♥ts ♦ ♥t♥t♦ ♥ós

♦sr♠♦s q s ♦ t♠♥♦ ♥ ♦ sst♠ é rt♠♥t r♥ s s t♥♠

♣r♠r♦ s♦r♠ ♠ r♦ss♦r ♣r ❯ ♦ s t♠♦s ♠ r♦ss♦r

♣♥♦ ♣r r♦ ♣♥♥t ♦ t♠♥♦ ♥ ♦ sst♠

♣ít♦

♦♥t♦s ás♦s ♠ ♥â♠

♥tr

ss ♣ít♦ ♠♦s ♥tr♦③r ♦s ♣r♥♣s ♦♥t♦s ♥ssár♦s ♣r s ♦♠

♣r♥r ♦ tr♦ r③♦ ♥r♠♦s sr♥♦ r♠♥t ♦t♦s rts

♦♥ rá r ♠ s♠tr ♠t♦ ♠♣♦rt♥t ♥♦ st♦ ♥trs ♠ s

str♠♦s s ♥â♠ r♦s q é ♣r♦♣r s♣r ♠s

t③ ♣r s rtr③r ♥â♠ ♥trs ♣♦s ♠♦s ♥tr♦③r ♦

♦♥t♦ ♥rs ♥♦ rs♠♥t♦ s♣rí ♥♦ ♠♦r ♥s ss

P❩ q srá ♦ ♦♦ ♦ ♥♦ss♦ tr♦ Pr ♥③r ♦ ♣ít♦ r♠♦s ♣rs♥tr s

♣r♥♣s ♣r♦♣rs s strçõs trs q t♠é♠ sã♦ ♠t♦ úts ♥

rtr③çã♦ ♥trs é♠ sr♠ ♦ ♣♦♥t♦ ♥♠♥t ♦ ♥♦ss♦ st♦

rts t♦♥

♦♠tr ♥ ①st♠ ♥çõs rss ♦r♠s ♦♠étrs ♦♠♦

♦s ír♦s ♣♥tá♦♥♦s ♦tr♦s ♦ ♥t♥t♦ ♥♠ s♠♣r ♦s ♦t♦s ♥♦♥tr♦s

♥ ♥tr③ ♣♦♠ sr ♣r♦①♠♦s ♣♦r sss ♦r♠s ③s P♦r s ♦♠

♣① ♥♦ ♥ ♦r♠çã♦ ♥s sst♠s ♣♦♠ ♣rr rrrs

q ♥ã♦ sã♦ trtás ♥ ♦♠tr ♥ ♣♦r s sss strtrs ①♠ ♠

♦r♠s♠♦ ♠t♠át♦ ♠s r ♣r sr♠ sts ♥♦t ♥r♦t ♠♦

t♦♥

sss ♦t♦s ♦♠♣①♦s rts ♥♦ ♦ s♥♦♠♥t♦ ss ♦r♠s♠♦ s

ss ♣r♥♣s s ♣♦♠ sr ♥♦♥trs ♠ ❬❪ ♠ rsã♦ ♥s ♦♥t♦s

♠ ❬❪

r t♦ ♥♦ ♦sr♦ ♠ r♥ts ss

♠♥r s♠♣ ♣♦♠♦s r♥r ♦t♦s rts ♦s ♥♦s

♥s♥♦ ♦ t♦ ♠♥çs s ♦srçã♦ s♦r s ♦r♠ ♦♥sr

♠ r♦ ♦♠♦ ①♠♣♦ ♠ q ♦sr♠♦s ♠ ♣♦rçã♦ ♦ r♦ ♦♠

③ ♠s ♠♥çã♦ t♦r♥♥♦s ♠s s ♥t♦ ♠♦r ♠♥çã♦

♠♥♦r ♣r sr rtr ♦ r♦ ♦♠♦ ♥ r ss♦ té ♦ ♠t ♠ q ♦

r♦ s t♦r♥ ♠ rt á s strtrs rts ♥ã♦ s s③♠ à ♠ ♠ q

sã♦ ♦srs ♠ ss ③ ♠♥♦rs sss ♦t♦s sã♦ ♥ r ♥r♥ts

s♦r ♠♥çs s

❯♠ ♠♥ç ♠♥çã♦ ♣♦ sr st ♠t♠t♠♥t ♦♠♦ ♠

tr♥s♦r♠çã♦ s P♥s♥♦ ♠ ♠ sst♠ ♦♦r♥s rts♥♦ ss

tr♥s♦r♠çã♦ sr q♥t ♠t♣r ①♦ ♣♦r ♠ ♦♥st♥t ♠

t♦r s ♥♦ t③♠♦s ♠s♠ ♦♥st♥t ♠ t♦♦s ♦s ①♦s t♠♦s ♠

tr♥s♦r♠çã♦ s♦tró♣ s ♦r ♦ ♦♥trár♦ é t ♥s♦tró♣ t♦s ♥r

♥ts s♦r ♠♥çs s♦tró♣s s sã♦ ♠♦s t♦s♠rs ♥♦ s♦

♥s♦tró♣♦ sã♦ ♦♥♦s ♦♠♦ t♦♥s r t♠♦s ♠ strtr t♦

♠ ♦t ♦♠♦ ♥♦ ♠♣çã♦ ♦ ♦t♦ ♣r ♠s t♦ ss♦ ♣♦rq

① ♠ tr♥s♦r♠çã♦ ♦♠ t♦r s r♥t ♠ ①♦ ♣r q

♠♥t♥ s ♠s♠s rtrísts

Pr strr ♠♦r ♦ q é ♠ rt ♠♦s sr ♦♠♦ ①♠♣♦ ♦ tr♥♦

t♦♥

r t♦ rt ♦sr♦ ♠ r♥ts ss r ①trí ♠t♠t♥♦s♣♦t♦♠r

r♣♥s ss rt é ♦t♦ trés ♠ rr rrs ♦♠♠♦s ♥♠♥t

♠ tr♥♦ qátr♦ ♦ ♠ ♠ s rtr♠♦s ♠ tr♥♦

♦r♠♦ ♣♦s s♠♥t♦s q ♥♠ ♦s ♣♦♥t♦s ♠é♦s ♦s ♦s ♦ tr♥♦ ♥

♣ss♦ r♠♦♠♦s ♠ tr♥♦ qátr♦ ♦ ♥tr♦ tr♥♦ ♦♠♦ ♥

r ♣ós ♠ ♥ú♠r♦ r♥ trçõs ♦t♠♦s r ♦t

q ♦ ♦t♦ é t♦s♠r s ♠♣♠♦s ♠ ♦s trâ♥♦s r r♠♦s

♦tr ♠s♠ r ♥♦♠♥t

r strçã♦ ♦ ♣r♦ss♦ rrs♦ ♦r♠çã♦ ♦ tr♥♦ r♣♥s t♦ ♦t♦ ♣♦s ♠ ♥ú♠r♦ r♥ trçõs r ①trí ❬❪

t♦♥

♥s♥♦ ♦ tr♥♦ r♣♥s ♣♦♠♦s ♥r ♠ rtríst

♠t♦ ♥tr♥t ♦s ♦t♦s rts s ♥ã♦ ♣♦♠ sr ♦♠♣t♠♥t srt♦s

♣s ♠♥sõs ♥s ❱♠♦s ♥♠♥t t♥tr r s ár ♦ ♥í

③r♦ s ár é A0 =√3/4 ♦ ♣ss♦ s♥t rtr♠♦s ♠ tr♥♦ ár 1

22

√34

♦ ♥í ♦s r♠♦s r♠♦r três trâ♥♦s ár 142

√34 s♠♣s ♠♦strr q ♥♦

♣ss♦ N ár é

AN = A0 −√3

12

N∑

i=1

(

3

4

)i

.

N → ∞ s♦♠∑N

i=1

(

34

)i → 3 ss♠ ár t♦t ♦ trâ♥♦ r♣♥s t♥

♣r ③r♦ P♦r ss♦ ♣♦♠♦s ③r q ss ♦t♦ ♥ã♦ ♣♦ss s ♠♥sõs á q

s ár é ♥

♦r ♠♦s r ♦ ♣rí♠tr♦ ♦ ♥í ③r♦ ♦ ♣rí♠tr♦ é três ♦ s♥♦

♣ss♦ tr♠♦s três trâ♥♦s ♦ 1/2 ♦ ♥í ♦s t♠♦s ♥♦ trâ♥♦s ♦

1/4 ss♠ ♣♦♠♦s r ♦ ♣rí♠tr♦ ♥♦ ♣ss♦ N ♠t♣♥♦ ♦ ♥ú♠r♦

trâ♥♦s 3N ♣♦ ♣rí♠tr♦ trâ♥♦ 3/2N rst♥♦ ♠

PN = 3N3

2N= 3

(

3

2

)N

.

ss ♦r♠ q♥♦ N → ∞ ♦ ♣rí♠tr♦ PN → ∞ ss♠ ♦ trâ♥♦ r♣♥s

♥ã♦ ♣♦ sr ♠♦ ♠ ♠ ♠♥sã♦

♦t♠♦s q ♦ trâ♥♦ r♣♥s ♥ã♦ ♣♦ sr srt♦ ♣♦r s ♠♥sõs

♣♦s s ár s ♥ t♠é♠ ♥ã♦ ♣♦ sr srt♦ ♣♦r ♠ ú♥ ♠♥sã♦ ♣♦s

s ♣rí♠tr♦ r ss♠ é ♦♠♦ s ss ♦t♦ ♣♦ssíss ♠ ♥ú♠r♦ ♠♥sõs

r♦♥ár♦ ♥tr ♠ s ♠♥sõs ♥s í ♠ ♦ ♥♦♠ rt

á q ♥ã♦ ♣♦ sr ♠♥sr♦ ♣s ♠♥sõs ♥trs ♥s ♣♦r s

♦ ♦r♠s♠♦ s♥♦♦ ♣♦r ♥r♦t ♣♦♠♦s r ♠♥sã♦ rt ♦

trâ♥♦ r♣♥s ♦r♦ ♦♠ ❬❪

df = liml→0

lnS(l)

ln 1/l,

♦♥ S(l) é ♦ ♥ú♠r♦ trâ♥♦s ♦ l ♥ssár♦s ♣r r♦rr t♦♦ ♦ ♦t♦

s ♥â♠ r♦s

r♣♥s ♦♠♦ á ♠♦s ♣♦s á♦s ár ♦ ♣rí♠tr♦ ♥♦ ♥í N ♦ ♥ú♠r♦

trâ♥♦s é 3N ♦ ♦ s é (1/2)N ❯s♥♦ ss♦ ♥ q tr♠♦s

df = limN→∞

ln 3N

ln 2N=

ln 3

ln 2≈ 1, 5850 .

ss♠ ♥♦t♠♦s q t♦ ♠♥sã♦ rt ♦ tr♥♦ r♣♥s é 1 < df < 2

trâ♥♦ r♣♥s é ♠ ①♠♣♦ rt tr♠♥íst♦ ♣♦s é ♣r♦

③♦ ♣♦r ♠ rr rrs ❯♠ ♠♣çã♦ r ♠ r♣r♦çã♦

ê♥t ♦ ♦t♦ ss♠ s t♦s♠r é ①t ♥tr③ ♥♦♥tr♠♦s

♦t♦s t♦s♠rs ♦ t♦♥s ♠ ♠ s♥t♦ ♠♥♦s rí♦ P♦r ①♠♣♦ s

t♦♠♠♦s ♦ r ♣♦♠♦s ♥♦tr q ♠ r♠♦ ①trí♦ s ♣r

♦♠ ♦ ♥tr ♥♦ ♥t♥t♦ ss ♣q♥♦ r♠♦ ♥ã♦ é ♠ r♣r♦çã♦ ê♥t

♦ ♦t♦ ♦♠♦ ♠ t♦♦ ♠ ♥trs s t♠♦s ♠ s♣rí t♦♠ st♠♦s

③♥♦ q ♠ ♣♦rçã♦ rs ♠♥r ♣r♦♣r ♣♦ss s ♠s♠s ♣r♦

♣rs sttísts q ♥tr ♦♠♦ ♠ t♦♦ P♦r ss♦ ♠ ♥ô♠♥♦s ♥trs

st♠♦s ♥trss♦s ♠ t♦s♠r ♥♦ s♥t♦ sttíst♦

t♦♥ é ♠ s♠tr ♠t♦ út ♥♦ st♦ ♥trs ♦♥s

r♠♦s q h(~x, t) é tr ♣♦♥t♦ ♠ ♥tr ♠ ♠ ♥st♥t t♠♣♦

♠ tr♥s♦r♠çã♦ s ♥s♦tró♣ r srá

~x → ~x′ ≡ b~x , h → h′ ≡ bαh e t → t′ ≡ bzt ,

♦♥ b é ♠ t♦r s rtrár♦ ♦s ①♣♦♥ts α z sã♦ r♦♥♦s à ♥s♦tr♦♣

tr♥s♦r♠çã♦ ss ♥tr ♦r t♦♠ s ♣r♦♣rs sttísts h sã♦

s s h′ ss s♠tr ♣♦ sr t③ ♠ á♦s ♥ít♦s ♦♠♦ r♠♦s

♥ sçã♦

s ♥â♠ r♦s

s rss ♥trs ♥♦♥trs ♥ ♥tr③ sã♦ rt♦ s ♠s st♥ts ♥

trçõs ♥♠♥ts ♥tr ss ♦♥stt♥ts ♦ ♥t♥t♦ ♥♦ts q sst♠s ♠t♦

s ♥â♠ r♦s

st♥t♦s ♣♦♠ ♣rs♥tr ♣r♦♣rs sttísts s♠rs ss ♦♠♣♦rt♠♥t♦

♥rs t♣♠♥t ♣r ♥s ♣r♦♣rs s r♦s Pr ♥r

r♦s ♠♦s ♣r♠r♠♥t ♥r tr ♠é q ♠ ♠ r srt

♦♠ L sít♦s ♣♦ss ♦r♠

h(t) =1

L

L∑

i=1

hi(t) ,

♦♥ hi é tr ♦ és♠♦ sít♦ r s♦♠ é t s♦r t♦♦s ♦s sít♦s

s♣rí r♦s ♥tr q rtr③ s tçõs ♠ t♦r♥♦ ♠é

é s♠♣s♠♥t ♦ s♦ qrát♦ ♠é♦ s trs ♦ ♣♦r

W (L, t) =

√

√

√

√

⟨

1

L

L∑

i=1

(hi(t)− h(t))2

⟩

.

♦♥ 〈. . . 〉 s♥ ♠ ♠é s♦r r♥ts ♠♦strs

♥♦ st♠♦s ♣r♦♠s ♠ ♦♠tr ♣♥ ♥♦r♠♠♥t ♥♠♦s ♦♠

♠ sstrt♦ s♦ ♣♦rt♥t♦ ♦♠ r♦s ♥ ♠ q s ♣rtís ã♦

s♥♦ ♦♥s ♦ sstrt♦ ♠s trs sã♦ ssís ♦ sst♠ ♠♥t♥♦ s

tçõs ♠ t♦r♥♦ ♠é ♠♣♥♦ ♥♠ ♠♥t♦ r♦s ♥♣♥♥t

s ♣rs sst♠ ♦♥t♥t♦ q s ♣rtís ♣♦sts ♥♥♠

♥ ♣♦sçã♦ ♠ sít♦s ③♥♦s r♦s ♦ qtt♠♥t ♦♠♦ ♥ r

ss rá♦ ♣♦♠♦s r q r♦s ♣rs♥t ♦s r♠s rtríst♦s

♥♠♥t rs ♦♠♦ ♠ ♣♦tê♥ ♣♦s á ♠ r♦ss♦r ♣r ♠

r♠ strçã♦ ♦♥ ♣ss tr ♠ t♦r♥♦ ♠ ♦r ♠é♦ st ♥â♠

♥ é ♦♥ ♦♠♦ r♠ rs♠♥t♦ ♥ r♦s s ♦r♠

W (L, t) ∼ tβ ♣r t ≪ tx ,

♦♥ β é ♦ ①♣♦♥t r♦♥♦ ♥â♠ r♦s ♦♥♦ ♦♠♦ ①♣♦♥t

rs♠♥t♦ tx é ♦ t♠♣♦ r♦ss♦r ♥tr ♦s ♦s r♠s ♦r♠♠♥t

♥ t♠♣♦ é ♥ t q ♠ ∆t = 1 t♥t♠♦s ♣♦str ♠ ♠♦♥♦♠

♣rtís

s ♥â♠ r♦s

r ♦♠♣♦rt♠♥t♦ tí♣♦ r♦s ♦♠♦ ♥çã♦ ♦ t♠♣♦ s ♥strs str♠ ♦ ♠ét♦♦ ♦♠♠♥t t③♦ ♣r st♠r ♦ t♠♣♦ strçã♦ tx

♣ós ss r♠ rs♠♥t♦ t♠♦s ♠ ♥tr♦ t♠♣♦ r♦♥♦ ♦

tr♥s♥t ♣r ♦ r♠ strçã♦ ss ♥♦♦ r♠ r♦s t ♠ t♦r♥♦

♠ ♦r ①♦ ss♠ t♠♦s

W (L, t) = Wsat(L) ♣r t ≫ tx .

t

W(L,t)

L

2L

4L

8L

16L

r ♦♠♣♦rt♠♥t♦ tí♣♦ r♦s ♣r sst♠s r♥ts t♠♥♦s

♦♠♦ ♠♦str r ss ♦r strçã♦ ♣♥ ♦ t♠♥♦ ♦

s ♥â♠ r♦s

sst♠ trç♠♦s ♠ rá♦ ♦♠ ♦s r♥ts ♦rs Wsat ♠ ♥çã♦ ♦

t♠♥♦ ♦ sst♠ r ♥♦t♠♦s q ♦ ♦r strçã♦ rs ♦♠♦

♠ ♣♦tê♥ ss♠ t♠♦s

Wsat(L) ∼ Lα ,

♦♥ α é ♦♥♦ ♦♠♦ ①♣♦♥t r♦s

ss♠ ♦♠♦ ♦ ♦r strçã♦ rs ♦♠ ♦ t♠♥♦ ♦ sst♠ ♦ t♠♣♦

strçã♦ tx t♠é♠ ♠♥t ♠ét♦♦ ♥♦r♠♠♥t t③♦ ♣r st♠r ss

♦r ♦ str♦ ♥ r s♠♥t é ♦ t♠♣♦ q ♦ ♣r♦♦♥♠♥t♦ ♦ r♠

rs♠♥t♦ str ♣r ♥çr ♦ ♦r strçã♦ ♥s♥♦ ♣♥ê♥

tx ♦♠ L r ♦t♠♦s

tx ∼ Lz ,

♦♥ z é ♦♥♦ ♦♠♦ ①♣♦♥t ♥â♠♦

L

Wsat

Lα

L

t x Lz

r ♦♠♣♦rt♠♥t♦s tí♣♦s r♦s strçã♦ ♠ ♥çã♦ ♦ t♠♥♦♦ sst♠ ♦ t♠♣♦ strçã♦ ♠ ♥çã♦ ♦ t♠♥♦ ♦ sst♠

strçã♦ r♦s é ♦♥sqê♥ ♠ ♣r♦ss♦ ♠s st q ♦♦rr

♥ ♥tr r♥t ♦ rs♠♥t♦ ss ♣♦ sr ①♣t♦ ♥♦t♥♦ q q♥♦

♦ rs♠♥t♦ ♠ sít♦ ♣♥ ♠ ♦r♠ ♦s ss ③♥♦s ①st ♠

♦rrçã♦ ♥tr s trs ♥tr ♠ q s ♣rtís ã♦ s♥♦


♦♥s ♦rrçõs s ♣r♦♣♠ ♣ s♣rí ss♠ ①st ♠ ♦♠♣r♠♥t♦

♦rrçã♦ tr ξ|| ♥tr♦ ♦ q ♦s sít♦s ♣♦♠ sr t♦s ♦rr♦♥♦s ss

♦♠♣r♠♥t♦ rs ♦r♦ ♦♠

ξ|| ∼ t1/z .

♥♦ ξ|| ≈ L ♦ sst♠ s ♥♦♥tr ♦♠♣t♠♥t ♦rr♦♥♦ ss♠

♥â♠ sít♦ ♣ss str r♦♥ ♦♠ t♦♦s ♦s ♠s ♠♣♥♦ q

s tçõs ♠ t♦r♥♦ ♠é ♠♥t♠ ♣♦r ss♦ r♦s ♣r rsr

P♦♠♦s r ♠ rçã♦ ♥tr ♦s ①♣♦♥ts α β z trés ♠ r♦í♥♦

♠ s♠♣s ♦♠♣♥r♠♦s ♦çã♦ r♦s s ♥♦s ♣r♦①♠r♠♦s

tx ♣♦ r♠ rs♠♥t♦ t♠♦s W (L, t) ∼ tβx á ♣♦ strçã♦ t♠♦s

W (L, t) = Wsat ∼ Lα ss ♦r♠ ♥♦ r♦ss♦r tr♠♦s tβx ∼ Lα ss♠ q

♦♥í♠♦s q Lβz ∼ Lα ♦♦

zβ = α .

sss ♥rs ♥ ♥â♠ ♥tr

s

st♠s ♥♦s qs ♥â♠ ♥♦ ♦çã♦ ♥trs sã♦ ♦♥♣rs♥ts

♥ ♥tr③ ss t♣♦ ♥ô♠♥♦ ♣♦ sr ♥♦♥tr♦ ♥s ♠s st♥ts ss

s sst♠s ♠t♦ r♥s ♦♠♦ ♣r♦ss♦s r♦sã♦ trr♥♦s ♠♦♥t♥s té

sst♠s ♠t♦ ♣q♥♦s ♦♠♦ ♣♦sçã♦ ♠s ♥♦s ss ♦r♠ é ♥t q

sss sst♠s sã♦ ♣r♦③♦s ♣♦s ♠s ♦♠♣①♦s st♥t♦s ♠♥s♠♦s ♥♠♥

ts ♦ ♥t♥t♦ ♦♠♦ á t♦ ♣sr s r♥s r♥çs ♥tr sss sst♠s

♥♦ts q sst♠s ♠t♦ st♥t♦s ♣♦♠ ♣rs♥tr ♣r♦♣rs sttísts s♠

rs sss s♠♥çs sr♠ q ♣♦♠♦s r♣r ♦s sst♠s ♠ sss

♥rs ♥çã♦ ♠s ♦♠♠ ss ♥rs ③ q sst♠s

q ♣♦ss♠ ♦ ♠s♠♦ ♦♥♥t♦ ①♣♦♥ts rít♦s ♣rt♥♠ ♠s♠ ss ♥

rs ♦ ♥t♥t♦ s s♠♥çs ♥ã♦ ♣r♠ ♥sss ①♣♦♥ts ♥tr♦ ♠


♠s♠ ss ♥♦♥trr♠♦s ♥rs ♠ rss ♣r♦♣rs ♣♦r ①♠♣♦

♥s strçõs trs ❬❪ strçõs ①tr♠♦s ❬❪ strçã♦

r♦s ❬❪ é♠ ss♦ sss s♠♥çs t♠é♠ sr♠ q s ♥â♠s

sst♠s ♣rt♥♥ts ♠s♠ ss ♥rs ♣♦♠ sr ♠♦s ♣♦r

♠ ♠s♠ qçã♦ st♦ást qçã♦ ss♠ ♦♠♦ s sss ♥rs

♣♦ss ♦s ♣r♥♣s ♠♥s♠♦s s♠trs ♥♦s ♥♦ t♣♦ ♥tr s

qs stá r♦♥

st sçã♦ ♠♦s str três sss ♥rs ♣♦sçã♦ tór

rs❲♥s♦♥ ❲ rrPrs❩♥P❩ ♣sr ♥♦ss♦s ♣r♥

♣s rst♦s str♠ r♦♥♦s ss P❩ é út str sss ♦trs sss

♣r ♥tr♦③r ♦r♠ r ♥s ♦s ♠♥s♠♦s ♥♦♦s ♥ ♥â♠

♥trs str♠♦s s strçõs trs s♣r♠♥t ♥ ♣ró①♠ sçã♦

♦♥ t③r♠♦s rs♦s ♦♥t♦s ♥tr♦③♦s q

qçõs st♦ásts ♣r♥í♣♦s s♠tr

qçõs st♦ásts sã♦ ♦r♠çõs ♠t♠áts q ♣r♠ ♠ rs♦s

♣r♦♠s ís árs ♥s ♦ ♦♥t①t♦ s♣rís sss ♦r♠çõs sã♦

♣r♦♣♦sts ♣r srr ♥â♠ ♥trs st ♠♦♠ é t ♥♦ ♠t

♦♥tí♥♦ ♣r t♠♥♦s sst♠ t♠♣♦s ♠t♦ r♥s ♦r♠♠♥t t③s

qçõs ♦ t♣♦ ♥♥ ♦♠ rí♦ t♦ ♥ss ♠♦♠ ♠♥r r ss

t♣♦ qçã♦ é ♣♦r

∂h(~x, t)

∂t= G(h(~x, t), ~x, t) + η(~x, t) ,

♦♥ h(~x, t) é ♠ ♣r♦ss♦ st♦ást♦ q ♥♦ ♥♦ss♦ s♦ sã♦ s trs ♥tr

G(h(~x, t), ~x, t) é ♠ ♥çã♦ q r♣rs♥t t♦♦s ♦s tr♠♦s tr♠♥íst♦s qçã♦

η(~x, t) é ♠ rí♦ r♥♦ rs♣♦♥sá ♣ t♦r ♥♦ sst♠ s♥♦ q

〈η(~x, t)〉 = 0


〈η(~x, t)η(~x′, t′)〉 = 2Dδd(~x− ~x′)δ(t− t′) ,

♦♥ D é ♠ ♦♥st♥t q ♠ ♥t♥s ♦ rí♦ δ é ♥çã♦ t r

d é ♠♥sã♦ ♦ sstrt♦ ♣r♠r rçã♦ ♥♦s ③ q ss rí♦ ♥ã♦ tr

tr ♠é s♥ q é t♦t♠♥t s♦rr♦♥♦

❱♦t♥♦ qçã♦ ♦ tr♠♦ G(h(~x, t), ~x, t) tr♠♥ s s♠trs s s

♦♥srçã♦ q ♦ ♣r♦ss♦ rs♠♥t♦ rá sts③r ss♠ ♦♥strçã♦

qçã♦ st♦ást é t r♣rs♥t♥♦ ♠ ♦r♠ r♥ ♦ t♣♦ ♣r♦ss♦

r①çã♦ q é r♥t ♥ s♣rí s s♠trs q ♥♦r♠♠♥t sã♦ ststs

♣s ♥trs sã♦

♥râ♥ s♦r tr♥sçõs s♣ç♦t♠♣♦rs (~x → ~x + ~a, t → t + b) ♦ s

♥tr sr ♥♣♥♥t s♦ ♦r♠ ( ~x0, t0) ♦ ♥♦ss♦ sst♠

♦♦r♥s ss s♠tr ♠♣ q G s ♥çã♦ ①♣t ~x ♦ t

♠tr s♦r tr♥sçã♦ ♥ rçã♦ rs♠♥t♦ ♥á♦♦ ♦ s♦ ♥tr♦r

♥tr sr ♥♣♥♥t s♦ ♦r♠ ♥ ♠ tr ss

s♠tr ♠♣ q G ♣♥ ①♣t♠♥t h

♠tr s♦r r♦tçã♦ ♠ t♦r♥♦ rçã♦ rs♠♥t♦ ❯♠ r♦tçã♦ s♣í

sr t q ~x → −~x r ♦♠ q r ∂h/∂(−x) = −∂h/∂x

ss♠ ♥ss sst♠ r♦ qçã♦ tr s♥s r♥ts ss ♦r♠

s♠tr s♦r r♦tçõs ① q G ♣♥ rs ♦r♠ ♣r h ♦

rs ♦r♠ í♠♣r s ♠ ♣♦tê♥ ♣r ♣♦r ①♠♣♦ ∇nh ♦

(∇h)n ♦♠ n ♣r

♥râ♥ s ♣r♦♣rs sttísts s♦r tr♥s♦r♠çõs s

♥s♦tró♣s ♦♠♦ s ♣á♥ ss s♠tr ① q s strtrs ♦r

♠s s♠ t♦♥s

♦r s s ♦♥srçã♦ ♣♥♥♦ ♦ sst♠ s ♣rtís q ♠

♦ r♦ ♣♦♠ s ♦r♥③r ♦r♠ q ♦çã♦ tr ♠é s ♠♦r

q ♦ ①♦ ♠é♦ ♣rtís ♦ ♠♥♦r s♦ ♠s s ♣rtís ♥ã♦ s r♠


à ♥tr ♦r♠ ♠t♠át ♠ sst♠ ♦♥srt♦ é t q

1

Ld

∫ L1

0

. . .

∫ Ld

0

Gdd~x = F ,

♦♥ é ♦ ①♦ ♠é♦ ♣rtís ♥trçã♦ é t s♦r s d ♠♥sõs ♦

sst♠ ❯♠ sst♠ q stsç ss qçã♦ ♣♦ssrá ∂〈h〉/∂t = F ♦ s

♥ t♠♣♦ tr ♠é ♠♥t ♠ ♦r ♦ ①♦ ♠é♦

♣rtís

♣♦sçã♦ tór

st ss ♥rs é ♦ ♠♦♦ ♠s s♠♣s q ♣♦♠♦s ♠♥r

t♠♦s ♥trs q sã♦ ♦♠♣t♠♥t s♦rr♦♥s ♦ s ♦ rs♠♥t♦

♠ ♠ ♦ é ♥♣♥♥t s ③♥♥ç ss ♦r♠ ♦♠♦ ss♠♦s ♥ ♣á♥

s r♦ss s ♥trs ss ss ♥rs ♥♥ str♠ s♥♦

ss♠ ♥ã♦ tr♠♦s ♦s ①♣♦♥ts α z ♠ ♥♦s

❯♠ ♠♦♦ srt♦ q ①♠♣ ss ss sr ♦ s♥t ♠ ♠ r

srt ♦♠ sít♦s s♦♠♦s t♦r♠♥t ♠ sss sít♦s ♦♥♠♦s ♠

♣rtí ♦♠♦ ♦ rs♠♥t♦ ♠ sít♦ é ♦♠♣t♠♥t ♥♣♥♥t ss

③♥♦s ♥t q ♥ã♦ rá ♣r♦♣çã♦ ♦rrçõs ♥ s♣rí ♣♦r ss♦

s r♦ss s ♥trs ♣r♦③s ♥♥ str♠

P♥s♥♦ ♥ qçã♦ st♦ást r♦♥ ss ss ♥rs

s ①♠♦s q qçã♦ stsç s s♠trs sçã♦ ♥tr♦r G(h(~x, t), ~x, t)

rstrt♦ rs s♣s ♦r♠ ♣r h ♦ rs ♦r♠ í♠♣r s

♠ ♣♦tê♥ ♣r ♦ ♥t♥t♦ ♦♦r rs s♣s ♥♦ tr♠♦ G sr ③r q

♠ ♦r♠ ♦ rs♠♥t♦ ♠ ♠ ♣♦♥t♦ ♥tr ♣♥ s ③♥♥ç

ss ♦r♠ qçã♦ ♣r ss ss é

∂h(~x, t)

∂t= F + η(~x, t) ,

♠r♥♦ q é ♦ ♥ú♠r♦ ♠é♦ ♣rtís ♣♦r sít♦ q ♠ ♦ sstrt♦


♠ ♠ ♥ t♠♣♦

P♦r s s♠♣ ss qçã♦ ♣♦♠♦s ♠♥t r ♦ ①

♣♦♥t rs♠♥t♦ ♥tr♥♦ ♥♦ t♠♣♦ tr♠♦s

h(~x, t) = Ft+

∫ t

0

η(~x, t′)dt′ ,

tr♥♦ ♠é s♣

〈h(~x, t)〉 = Ft+

∫ t

0

〈η(~x, t′)〉dt′ ,

s♥♦ q rst ♠

〈h(~x, t)〉 = Ft .

♦r ♥♦ q ♦ qr♦ s♥♦ s qçõs t♠♦s

〈h2(~x, t)〉 = F 2t2 + 2Dt .

r♦s srá ♣♦r

W =√

〈h2〉 − 〈h〉2 =√2Dt .

ss♠ ♦ ①♣♦♥t rs♠♥t♦ srá β = 1/2

ss rs❲♥s♦♥❲

ss ss ♦ ♣r♦♣♦st ♣♦r rs ❲♥s♦♥ ❬❪ ♥♥♦ ♠ tr♠♦

r♦♥♦ t♥sã♦ s♣r ♥tr ♥ qçã♦ ss tr♠♦ t♥

rstrr ♦ ♠tr ♥ ♥tr ♦r♠ s③á Pr♠r♦ ♠♦s ♦tr

qçã♦ st♦ást trés r♠♥t♦s s♠tr ♣♦s ♥sr♠♦s ♠ t

♦ t♦ ss t♥sã♦ s♣r

qçã♦ rs❲♥s♦♥ ❲ ♣♦ sr ♦t ①♥♦ ♠s ♠

s♠tr é♠ s sts ♥ ♣á♥ ♦♥♥♦ ♥râ♥ s♦r r①ã♦

♠ t♦r♥♦ tr ♠é ♦ s s♠tr s♦ tr♥s♦r♠çã♦ h → −h ♥♦ rr♥


q s s♦ ♥t♦ à tr ♠é ss tr♥s♦r♠çã♦ ∂h/∂t → −∂h/∂t ♣♦r

ss♦ ♣rs♠♦s q G → −G q ♣r♦í ①stê♥ tr♠♦s (∇h)n ♦♠ n ♣r

s♦r♥♦ ♣♥s ∇nh ♦♠ n ♣r qçã♦ ♠s s♠♣s q rá sts③r sss

♦♥çõs srá∂h(~x, t)

∂t= ν∇2h+ η(~x, t) ,

♦♥ ν é ♠ ♣râ♠tr♦ qçã♦ ♦ s r ♦r♦ ♦♠ s ♥s

sst♠ ①♦ ♠é♦ ♣rtís F ♥ã♦ ♣r ♣♦s st♠♦s ♥♦ rr♥

tr ♠é

♦♠♦ á ♦ ♥t♦ ♦ tr♠♦ ν∇2h r♣rs♥t t♥sã♦ s♣r ♥ ♥

tr ss tr♠♦ rstr ♦ ♠tr ♥ s♣rí ♠♥r ♦♥srt ♥ã♦

♠♦♥♦ ♦ ♦r tr ♠é ♥t♥s ss t♦ stá r♦♥ ♦

♣râ♠tr♦ ν ♦ s ss ♣râ♠tr♦ r♦♥ qçã♦ st♦ást sst♠

♠ ♣rtr r ♣rs♥t♠♦s ♠ ①♠♣♦ ♥♠♥s♦♥ ♦ t♦ ss

tr♠♦ s♦r ♠ ♥tr ♦t ♦♠♦ t♥ s③r ♥tr ♠♥

♥♦ ♣r♦♥ ♠♥t♥♦ rr ♦

0 50 100 150x

0

0,2

0,4

0,6

0,8

1

1,2

h(x)

d2h/dx

2

h(x) + d2h/dx

2

r t♦ ♦ tr♠♦ ν∇2h s♦r ♠ ♠ ♠ ♥tr ♥♠♥s♦♥ ❯t③♠♦s ν = 8 ♥ss strçã♦

P♦r s s♠♣ qçã♦ ❲ ♦s ①♣♦♥ts rít♦s ♣♦♠ sr

♦s ♠♥t ss♦ ♣♦ sr t♦ rs♦♥♦ rt♠♥t qçã♦ ♠

tr♥s♦r♠ ♦rr ♥♦ s♣ç♦ ♥♦ t♠♣♦ ♦♠♦ t♦ ♥ rrê♥ ❬❪ ♦ ♥t♥t♦


é ♠t♦ ♠s s♠♣s t③r ♦ t♦ q s ♥trs ♣r♦③s ♣ qçã♦ ❲

sã♦ ♥r♥ts s♦r ♠ tr♥s♦r♠çã♦ s ♥s♦tró♣ ♣r♦♣r ♦ s

sã♦ strtrs t♦♥s ss ♦r♠ ③♠♦s

~x → ~x′ ≡ b~x , h → h′ ≡ bαh e t → t′ ≡ bzt ,

♦♥ b é ♠ t♦r s

t♦ sss tr♥s♦r♠çõs s s♦r ♦ r♦ η ♣♦ sr ♥t♦

trés ♥çã♦ ♠r♥♦ q δd(a~x) = a−dδ(~x) ss♠ tr♠♦s

〈η(b~x, bzt)η(b~x′, bzt′)〉 = 2Db−(d+z)δd(~x− ~x′)δ(t− t′) .

ss ♦r♠ ♣♦s s tr♥s♦r♠çõs qçã♦ s t♦r♥

bα−z ∂h(~x, t)

∂t= νbα−2∇2h+ b−(d+z)/2η(~x, t) ,

q ♥♦ ♣♦r bα−z rst ♠

∂h(~x, t)

∂t= νbz−2∇2h+ b−(d−z)/2−αη(~x, t) .

t♦♥ ① q ♦s rst♦s ss qçã♦ s♠ ♦s ♠s♠♦s q ♦s

ss♠ ♦s ①♣♦♥ts b ♠ s ♥r sr♠♦s t♠é♠ rçã♦ ♥tr ♦s

①♣♦♥ts ♠♦s ♦tr

z = 2 , α =2− d

2e β =

2− d

4

sss sã♦ ♦s ①♣♦♥ts s r♦♥♦s ss ❲ ♦t q s ♣♥♠ ♦

♥ú♠r♦ ♠♥sõs d ♥tr

strçõs trs

ss rrPrs❩♥P❩

♠♦♦ ♦♥tí♥♦ ♦ qçã♦ st♦ást ss♦♦ st ss ♦ ♣r♦♣♦st♦

♣♦r rr Prs ❩♥ ❬❪ ss tr♦ s ♦♥r♠ ♠ tr♠♦ à qçã♦

❲ q é rs♣♦♥sá ♣♦r ♠ rs♠♥t♦ ♦♠♥t ♥♦r♠ à ♥tr ss

tr♠♦ ♠♦ ♣♦r ①♠♣♦ sst♠s ♥♦s qs s ♣rtís t♥♠ s rr ♥♦

♣♦♥t♦ ♦♥ s t♦♠ ♥tr

qçã♦ ♠s s♠♣s q sts③ s s♠trs ♣á♥ ♦ s qr

s♠tr (h → −h) ss ❲ é ♣♦r

∂h(~x, t)

∂t= ν∇2h+

λ

2(∇h)2 + η(~x, t) ,

♦♥ λ é ♠ ♣râ♠tr♦ r♦♥♦ ♥t♥s ♦ t♦ s♦ ♣♦ tr♠♦ ♥ã♦

♥r ♦♠♦ á ♦ t♦ ♦ tr♠♦ λ2(∇h)2 é rs♣♦♥sá ♣♦r ♠ rs♠♥t♦ ♦♠♥t

♥♦r♠ à ♥tr ss♠ ♣r ♠ ♦♠♣♦♥♥t tr ♥♦ rs♠♥t♦ s♣rí

ss tr♠♦ s ♠ rs♠♥t♦ ♥ã♦ ♦♥srt♦ r♦♥♦ ♠ ①ss♦ ♦

rçã♦ ♣♥♥♦ ♦ s♥ λ ♥ t① ♠♥t♦ ♥tr ♦♥trçã♦

ss tr♠♦ é t ♥ã♦ ♦♥srt ♣♦r ♠♦r ♥â♠ tr ♠é

r ♠♦str♠♦s ♦ t♦ ♦ tr♠♦ ♥ã♦♥r s♦r ♠ ♥tr ♦t

♦♠♦ ♥t ♦♠♣♦♥♥t tr q ♣r ♥♦ rs♠♥t♦ ♦♠♦ ♦

♠s strt♦ ♦♠ s ss ♠s í♥r♠s

P♦r s ♥ã♦♥r qçã♦ P❩ ♥ã♦ ♣♦ss s♦çã♦ ♥ít

♦♥ é♠ ss♦ ♦s ①♣♦♥ts ♥ã♦ ♣♦♠ sr ♦s trés tr♥s♦r

♠çõs s ♦♠♦ ♦ t♦ ♣r ❲ ss♦ ♣♦rq ♦s ♣râ♠tr♦s ν λ D ♥ã♦

r♥♦r♠③♠ ♦r♠ ♥♣♥♥t ♥♦ ♦♣♦s ♣ós tr♥s♦r♠çã♦ ♦ q

três qçõs ♥♦♥sst♥ts ♣r ♦s ♦s ①♣♦♥ts ♦ ♥t♥t♦ ♣r ♠

♠♥sã♦ ♦s ①♣♦♥ts ♣♦♠ sr ♦t♦s trés ♦ r♣♦ r♥♦r♠③çã♦ ♦♠♦

é t♦ ♥ rrê♥ ❬❪ ss trt♠♥t♦ α = 1/2 β = 1/3 z = 3/2

st♦s ♥♠ér♦s ♣♦♥t♠ q ♣r s ♠♥sõs ♦s ①♣♦♥ts sã♦ α ≈ 0, 38

β ≈ 0, 24 z ≈ 1, 67 ❬❪

strçõs trs

0 50 100 150 200x

0

0,2

0,4

0,6

0,8

1

1,2

h(x)

(dh/dx)2

h(x)+(dh/dx)2

r t♦ ♦ tr♠♦ λ2 (∇h)2 s♦r ♠ ♠ ♥tr ♥♠♥s♦♥

❯t③♠♦s λ = 100 ♥ss strçã♦

strçõs trs

♥♦ ♥♠♦s s sss ♥rs ♦ t♦ q ♥trs s

♥â♠s sã♦ rs ♣♦s ♠s♠♦s ①♣♦♥ts rít♦s ♣rt♥♠ à ♠s♠ ss

♦ ♥t♥t♦ s s s r♦s ♥ã♦ sã♦ s ú♥s ♣r♦♣rs sttíst

s ♥rss ♥♦ rs♠♥t♦ ♥trs ♥tr ♦trs s strçõs tr

st strt♦♥ s ♥trs t♠é♠ ♣rs♥t♠ ♥rs ss♠

♣♦♠ sr t③s ♣r ssás

♦♠♦ ♦ ♣ró♣r♦ ♥♦♠ sr é ♠ ♥s ♣r♦ ♦ s

é ♠ ♥çã♦ p(h) ♦♥ p(h)dh é ♣r♦ ♥♦♥trr♠♦s ♠ tr ♥tr h

h+dh Pr ①r ♠s r♦ ♦ q s s♣r sss strçõs ♠♦s ♥sás

♣r s três sss q ♥tr♦③♠♦s ♥ sçã♦ ♣ss

♥♥♦ ♣ ss ♦♥sr♠♦s ♠ ♠♦♦ srt♦ ♦♥ t♠♦s ♠

r L sít♦s ♥tr é r s♦rt♥♦ sít♦s t♦r♠♥t ♦♥♥♦

♠ ♥ s tr ♥s♥♦ ♠ ♦♥ ♥tr ♣r♦

tr sr h é ♣r♦ ♦ sít♦ ♦rrs♣♦♥♥t sr s♦rt♦ h ③s ♥tr♦

♦ ♥ú♠r♦ t♦t N ♣rtís ♣♦sts ♥♦ 1/L ♣r♦ ♦ sít♦

♥trss sr s♦rt♦ 1− 1/L ♣r♦ ♥ã♦ sr ♣♦♠♦s ♥♦tr q p(h)

strçõs trs

srá ♠ strçã♦ ♥♦♠ ♦♦

p(h) =N !

h!(N − h)!

(

1

L

)h(L− 1

L

)N−h

♦ ♠t ♠ q N → ∞ q é s♠♥t t → ∞ ♣♦♠♦s ♠♦strr q ss

strçã♦ t♥ ♣r ♠ ss♥ ss ♠♦♥strçã♦ é t ♥♦ ♣ê♥

♥♦ ♦s á♦s ss ♣ê♥ t♠♦s q ss strçã♦ ♣r ss é

♣♦r

pRD(h) =1√2πt

exp

(−(h− t)2

2t

)

.

♥♣♥♥t♠♥t ss ♥rs ♠♣t râ♥

t① ♠♥t♦ tr ♠é ♣♥♠ ♦s ♣râ♠tr♦s s qçõs st♦ásts

ss♠ ss♠♠ r♥ts ♦rs ♣r sst♠ ♠ ♣rtr ss ♦r♠

♣r ♦♠♣rr strçõs ♦ts ♠ t♠♣♦s r♥ts ♦ sst♠s st♥t♦s é ♠s

♥trss♥t rsás ♣r ♠é ♥ râ♥ ♥tár ss tr♥s♦r♠çã♦ rá

♦♣sr s strçõs r♥ts sst♠s ♣rt♥♥ts ♠ ♠s♠ ss ♠

♠ r ♥rs q é ♠ rtríst ♥trí♥s ss ♥rs

ss ♦r♠ é ♠s ♥trss♥t str s s rss ♣ s♥t tr♥s♦r

♠çã♦

p(h) −→ p′(h) ≡ Wp(h) e h → h′ ≡ h− 〈h〉w

á♦ ♣r ss ❲ ♣♦ sr ♥♦♥tr♦ ♥ rrê♥ ❬❪

t♠é♠ é ss♥ ♥♠♥t ♦çã♦ strçã♦ é ♥á♦ à ss

♥♦ ♥t♥t♦ q♥♦ r♦s ♥tr str râ♥ strçã♦ ♣r

♠♥tr ss ♦♥t♥rá s s♦r ♣♦r s ♦çã♦ tr ♠é

♠s ♦♠ rr ① ♣ós strçã♦

♥ts str♠♦s s s ss P❩ ♠♦s ♣rs♥tr ♦ ♠♦♦ P

P♦②♥r r♦t ♠♦ ♣♦rq ♦ ♠♣♦rt♥t ♥♦ á♦s ①t♦ sss s

trçõs ❯♠ s ③çõs ♥s ss ♠♦♦ ♣♦ sr ♥♦♥tr ♥ rr

ê♥ ❬❪ s ♦rt♠♦ ♣r s♠çõs ♦♥t r♦ ♣♦ sr ♥♦♥tr♦ ♥

rrê♥ ❬❪ ss ♠♦♦ s sã♦ ♥s ♠ t① J ♣♦r sít♦ r ♠

strçõs trs

♣♦sçõs tórs s s ♦♠ç♠ ♦♠ r♦ ♥♦ s ①♣♥♠ ♦♠ ♠ ♦

r v ♥t♦ t① rs♠♥t♦ q♥t♦ ♥çã♦ sã♦ ♥♣♥♥ts

♦ t♠♣♦ ♣♦sçã♦ ♥ ♥tr ♥♦ s s s ♥♦♥tr♠ s ♦s♠

♦r♠♥♦ ♠ ú♥ ♠♦r r str♠♦s ♦r♠ rs♠♥t♦

ss ♠♦♦ ♠ ♠ ♠♥sã♦

r strçã♦ rr rs♠♥t♦ ♦ ♠♦♦ P st rt ♥ ♣♦sçã♦ ♠ ré♠ ♥ r ①trí ❬❪

♣♦ssí t③r ♦ ♠♦♦ P ♣r ♣r♦③r ♥trs rs Pr ss♦

st ♥r ♦ rs♠♥t♦ ♦♠ ♠ ♥çã♦ ♥ ♦r♠ só ③r ♥♦s ♥çõs

s♦r rã♦ ♦♠♣r♥ ♣♦r ss ♣r♠r ♥trs rs ss ♠♥r

t♠ ♦r♠ ♠ ♦t ❯♠ ①♠♣♦ é ♣rs♥t♦ ♥ r

r ♣rí ♣r♦③ ♣♦ ♠♦♦ P ♥ ♦r♠ ♦t r ①trí ❬❪

Pr ss P❩ ♣♦r s ♥ã♦♥r qçã♦ st♦ást ♦tr

strçõs trs

é ♠ tr ♠t♦ ♦♠♣① ❯♠ ♦s ♣r♠r♦s ♣ss♦s ♥ss s♥t♦ ♦ ♦

♣♦r ♦♥ss♦♥ ❬❪ q ♦ ♥t♠♥t strçã♦ ss♥tót ♠ ♠♦

♦ s♠r ♦ ♥t♣ ♣rs♥t♦ ♥ sçã♦ t♦r ♠♦str♦ q ss

strçã♦ ♦♥r ♣r strçã♦ r②❲♦♠ ♦ ♠♦r t♦ ♦r ♦ ♥s♠

ss♥♦ ♥tár♦ ❯ss♥ ♥tr② ♥s♠ q é ♠ strçã♦

t♦r ♠tr③s tórs P♦♦ ♣♦s Prä♦r ♣♦♥ ❬❪ ♦♥sr♠

r strçã♦ ♦ ♠♦♦ P ♥ ♦r♠ ♣♥ ♥ ♦t

s ♠♦strr♠ q ♦ ♦♥trár♦ ♦ ①♣♦♥t rs♠♥t♦ q é ♥♣♥

♥t rtr ss P❩ ♣♥ ♦♠tr ♥tr ♦ s♦

♣♥♦ ss stá r♦♥ ♦ ♥s♠ ss♥♦ ♦rt♦♦♥ ss♥ ♦r

t♦♦♥ ♥s♠ ♥♦ s♦ r♦ ♦ ♥s♠ ss♥♦ ♥tár♦ ❯ ss♠

ss ♥rs P❩ s ♠ ssss ♦♠ ♣r♦♣rs ♣♥♥ts

♦♠tr ♥tr

P♦♠♦s srr s trs ♠ s♣rí ss P❩ ♦♠♦

h(x, t) = v∞t+ s(λ)(Γt)βχ ,

♦♥ s(λ) = +1 ♦ − 1 ♥♦s á ♦ s♥ λ ♥ q v∞ Γ sã♦ ♣râ♠tr♦s

♥ã♦♥rss ♣♥♥ts ♦ ♠♦♦ ♥q♥t♦ χ é ♠ rá tór ♣♦r

♠ strçã♦ ♥rs β é ♦ ①♣♦♥t rs♠♥t♦ ♣r♠r♦ tr♠♦ ♦ ♦

rt♦ r♣rs♥t ♠ ♣r tr♠♥íst ♦♠♥♥t ♥♦ ♠♥t♦ tr ♠é

á ♦ s♥♦ é ♠ tr♠♦ tór♦ r♦♥♦ às tçõs ♣rs♥ts ♥ ♥tr

st tr♠♦ χ é ♣ ♦ ❯ ♣♥♥♦ ♦♠tr Γ é ♠

♦♥st♥t r♦♥ à ♠♣t sss tçõs

♦ s♦ ♣♥♦ ♥♠♥s♦♥ ♣ós strçã♦ r♦s ss

P❩ s♦r ♠ r♦ss♦r ♣r ♠ ss♥ ❬❪ ss♠ ♣ss sr

s♠étr ♣ós strçã♦ ♦ s ♥tr ♣ss tr s♠tr s♦ r①ã♦ ♠

t♦r♥♦ tr ♠é ♦♠♦ ♥ ss ❲

sss rst♦s ♥ít♦s s♦r sã♦ ss P❩ ♠ ♣♥♦ r♦ ♦r♠

♦♥r♠♦s ①♣r♠♥t♠♥t ♥ rrê♥ ❬❪ ss♦ ♦ t♦ t③♥♦ tr♦

strçõs trs

♦♥çã♦ rst íq♦ ♥♠át♦ ♦♥♥♥♦ ss rst íq♦ ♠ ♠ rã♦

strt ♥tr ♦s ♣♥♦s ♣♥♦ ♠ t♥sã♦ tr♥ ♥♦ sst♠ ♣rs♥t

s ss tr♥ts s♥♦ ♠ ♠tstá ♦tr stá ①♣r♠♥t♦ ♥

♦♠ ♦ rst íq♦ ♥ s ♠tstá ♥tã♦ r ♣ ♠ ♣rtrçã♦ q ♦

③ tr♥str ♣r s stá ♦♠♦ sss st♦s s♣♠ ③ ♦r♠ r♥t

é ♣♦ssí rstrr ♦çã♦ ♥tr q s♣r s s ss s ts

s♦r ss ①♣r♠♥t♦ ♣♦♠ sr ♥♦♥tr♦s ♥ rrê♥ ❬❪ ♥tr ♦t é

r s ♣rtrçã♦ ♦r ♣ ♠ ú♥♦ ♣♦♥t♦ ♣♥ s♦ ♣ ♠ ♠

♥ é♠ ♥♦♥trr♠ ♦s ①♣♦♥ts ss P❩ ♦s t♦rs ♠♦str♠ q ♥ss

sst♠ s s sã♦ ♦♥sst♥ts ♦♠ qçã♦ r ♠♦str♠♦s

♦♠♣rçã♦ ♦s rst♦s ①♣r♠♥ts ♦♠ ♦s ♥ít♦s

r strçã♦ trs tr♦♦♥çã♦ rsts íq♦s ♦♠♣rs♦♠ s ♣rsõs tórs r tr é ❯ ♣♦♥t s sí♠♦♦s♣r♥♦s ③s r♠♦s ♦r♠ ♦t♦s ♣r ♥trs rrs ♦♠ t♠♣♦s t = 10s t = 30s rs♣t♠♥t á ♦s sí♠♦♦s ♥ã♦ ♣r♥♦s ③s ♣úr♣r♦s ♦r♠ ♦t♦s♣r rs♠♥t♦s ♣♥♦s ♦♠ t♠♣♦s t = 20s t = 60s rs♣t♠♥t r ①trí ❬❪ rá♦ s♥ss r♦s rt♦s r③s ♦ rs♠♥t♦ r♦ ③ ♣♥♦ r♠♦ rá♦ ♠♦str q r♥ç ♥tr s strçõs ♠ é ♠s♦♠♥t♦ ♥ ♠é ♣♦s s♥ss rt♦s ♦♥r♠ ♣r ♦s rst♦s ♥ít♦s ❬❪

♣sr s♥ss rt♦s s strçõs ♠ ♦♥♦rr♠

♦♠ ♣rsã♦ tór ♦♠♦ ♣♦♠♦s r ♥ r ♣♦♠♦s ♦srr q

♦s rst♦s ①♣r♠♥ts s ♥♦♥tr♠ r♠♥t s♦♦s ♠ rçã♦ às rs

strçõs trs

tórs ♦♠♦ ♠♦str♦ ♥ rrê♥ ❬❪ ss r♥ç ♥♦ ♦r ♠é♦ ♦♠

t−1/3 ss s♦♠♥t♦ t♠é♠ ♦ ♥♦♥tr♦ ♠ s♠çõs ❬❪ ♥ s♦çã♦

qçã♦ P❩ ❬❪ ss♦ sr ♦ rés♠♦ tr♠♦s r♦♥♦s ♦rrçõs ♥ q

s t ❬❪ ♣r♦♣sr♠ qçã♦

h(x, t) = v∞t+ s(λ)(Γt)βχ+ η + ζt−γ + . . . ,

♦♥ η ζ ♣♦♠ sr rás tórs ♦ tr♠♥ísts η ♣r ♦ ♠

s♦♠♥t♦ ♥ ♠é strçã♦ ♦ s ♥ ♠é s trs ❱ rsstr

q ss η ♥ã♦ é ♦ r♦ r♥♦ q ♣r ♥s qçõs st♦ásts sçã♦

s♥♦ tr♠♦ rs♥t♦ stá r♦♥♦ ♦ t♠♣♦ ♦♥rê♥ ♦s ♠♦♠♥t♦s

strçã♦ s♠ trr ♦s ss ♦rs ss♥tót♦s s rtê♥s ♥♦ ♥

qçã♦ r♣rs♥t♠ ♦trs ♦rrçõs q ♦♥r♠ ♣r ③r♦ ♠s rá♣♦ q t−γ

♦♦s sss rst♦s r♦♥♦s às s ss P❩ ♦r♠ ♦t♦s ♣r ♦

s♦ ♥♠♥s♦♥ ♠ s ♠♥sõs ♥ã♦ ①st♠ rst♦s ♥ít♦s r♦♥♦s

às strçõs trs ♦ ♥t♥t♦ rst♦s ♥♠ér♦s ❬❪ sr♠ q

♠ qçã♦ s♠r à ♥♦♥ ♠ ♠ sst♠s ♠♥s♦♥s ss s♦ ♦s

♣râ♠tr♦s v∞ Γ sã♦ r♥ts ♦ s♦ ♥♠♥s♦♥ é♠ ss♦ strçã♦

♥á♦ ♦ χ t♠é♠ é r♥t s strçõs r②❲♦♠ ♦ ♥t♥t♦

ss♠ ♦♠♦ ♠ ♠ ♠♥sã♦ χ ♣♥ ♦♠tr ♥tr

í q♥tr s r♥çs ♥tr strçõs ♥s♥♦ ♣♥s r

p(h) ♣♦r h P♦r ss♦ é ♥trss♥t str s strçõs trés ss ♠

♥ts ♠♦♠♥t♦ ♦r♠ ♥ ♠ strçã♦ trs p(h) é ♦ ♣♦r

〈hn〉 = 1

L

L∑

i=1

hni .

s ♠♥ts strçã♦ ♣♦♠ sr srt♦s ♦♠♦ ♦♠♥çõs ♥rs ♦s ♠♦

strçõs trs

♠♥t♦s ❬❪ té ♦ ♠♥t qrt ♦r♠ t♠s

〈h〉c = 〈h〉 ;

〈h2〉c = 〈h2〉 − 〈h〉2 ;

〈h3〉c = 〈h3〉 − 3〈h2〉〈h〉+ 2〈h〉3 ;

〈h4〉c = 〈h4〉 − 4〈h3〉〈h〉 − 3〈h2〉2 + 12〈h2〉〈h〉2 − 6〈h〉4 .

♣♦ssí ♥r rss r③õs ♠♥s♦♥s sss ♠♥ts rês s

sã♦ s♣♠♥t ♥trss♥ts rt♦sK ♦ ♦♥t ss♠tr S ♦ s♥ss

♠ r③ã♦ ♥tr ♦s ♦s ♣r♠r♦s ♠♥ts R sss r♥③s sã♦ ♥s ♣♦r

R =〈h〉c

〈h2〉1/2 , S =〈h3〉c〈h2〉3/2c

K =〈h4〉c〈h2〉2c

.

♠ strçõs s♠étrs S = 0 ♠é é ♠♦ ♦♥ st út♠♦ é

♦ ♦r ♠s ♣r♦á strçã♦ ♦s s♦s ♠ q S > 0 ♠♦ é ♠♥♦r q

♠é ♦ ♦ rt♦ t♥ ③r♦ ♠s r q ♦ ♦ sqr♦

♥♦ S < 0 ♦♥t ♦ ♦♥trár♦ str♠♦s ss♦ ♥ r

r ♥ss strçõs ss♠étrs

rt♦s ♥♦s á ♥♦r♠çã♦ s♦r ♦ ♣s♦ s s strçã♦ ♦♠♦

strçã♦ é ♥♦r♠③ q♥t♦ ♠♦r ♦ ♣s♦ s s ♠s strt♦ é ♦ ♣♦

♥♦ K > 0 t♠♦s ♠ ♣♦ ♠s strt♦ s ♠s ♣ss q ♠ ss♥

q♥♦ K < 0 ♦♦rr ♦ ♦♥trár♦ ss♦ é str♦ ♥ r

s ♦rs ♦s ♣r♠r♦s ♠♥ts strçã♦ trs ss P❩ ♠

♠ s ♠♥sõs stã♦ rs♠♦s ♥ t s ♦rs ♣r ♠ ♠♥sã♦

strçõs trs

r ♣rs♥tçã♦ qtt rt♦s

♦r♠ ①trí♦s rrê♥ ❬❪ ♣r s ♠♥sõs tr♠♦s ♠é ♦s ♦rs

♦t♦s ♣r ♦s r♥ts ♠♦♦s ♣♥♦s ❬❪ r♦s ❬❪ st♦s

❯ P♥♦ r♦ 〈χ〉c 〈χ2〉c

❱♦rs s s ♣♥ r ss P❩ ♠ ♠ s ♠♥sõs

Pr ♥③r st sçã♦ ♠♦s rsstr ♦♠♦ ssã♦ ss P❩

é ♣r♦♣í ♣r ♥♦ss♦ tr♦ ♦♠♦ ♦ t♦ ♥ ♥tr♦çã♦ sstrt♦s rs♥ts

tê♠ s♦ t③♦s ♦♠♦ ♣r♦①♠çõs ♥trs rs st♠♦s ♥trss♦s

♠ r s sss ♦♠í♥♦s rs♥ts sã♦ t♦ ♠ ♦ ♣r♦①♠çã♦ ♦♠♦

strçã♦ trs ss P❩ ♣♥ ♦♠tr ♥ss ss ♣♦♠♦s

str ♦ ♦♠í♥♦ rs♥t ♣r rr s é ♦♥sst♥t ♦♠ r

♦ ♣♥ ♦♠ ss♦ ♣♦♠♦s t♠é♠ tr♠♥r s rtr ♦ é ♦ ♥ã♦

♦ t♦r ♣r♥♣ ♥ ♣r♦çã♦ r♥ts strçõs

♣ít♦

♥trs rs ♣r♦①♠çã♦


❱♠♦s ♦♠çr st ♣ít♦ st♥♦ ♥trs rs r♠♦s str ss

♣r♥♣s rtrísts r♥çs ♠ rçã♦ ♦ s♦ ♣♥♦ é♠ ss♦ ♠♦s

♣♦♥tr s s q ♣r♠ ♥ss ♦♠tr t♥t♦ ♦ ♣♦♥t♦ st ♥ít♦

q♥t♦ ♦♠♣t♦♥ t♦ ss♦ ♠♦s ♥tr♦③r ♦s ♦♠í♥♦s rs♥ts r♠♦s

♣rs♥tr ♦♥ s ♦r♠ s♦s ♦♠♦ ♠ ♣r♦①♠çã♦ ♣r ♦ s♦ r♦ rs♠r

♥s ♦s ♣r♥♣s rst♦s s♦r ♦ ss♥t♦


rs♦s ♥ô♠♥♦s ♥trs ♥♦♠ ♥â♠ ♥trs rs rs

♠♥t♦s ♦♠ ss ♦♠tr sã♦ s♣♠♥t ♦♠♥s ♠ sst♠s ♦ó♦s ①♠

♣♦s q ♣♦rí♠♦s tr sr♠ ♦ô♥s térs t♠♦rs ♥♦s ❬❪

♥♦ ♦♠♣r♠♦s s♣rís rs ♦♠ ♣♥s ♥♦t♠♦s s r♥çs

♣r♥♣s ♣r♠r é q ♠ sst♠s r♦s ♥tr ♠♥t ♦♠ ♦ t♠♣♦

♦ s♦ ♥♠♥s♦♥ ♣♦r ①♠♣♦ s♠♣r q ♦ r♦ ♦ r♦ rs ♠

♥ ♦ ♣rí♠tr♦ ♥tr rs 2π ♦tr r♥ç é rtr ♠ s

s rs s♣s ♥s qçõs st♦ásts r♦♥♠ ♥â♠ ♠ ♣♦♥t♦

♦♠ s ③♥♥ç ❯♠ rtr ♦ ♥ ♥tr tr ♦ rst♦ sss


rs s♣s ♦ q ♠ t♦ ♥â♠ ♦ sst♠

♠s♠ ♠♥r q é t♦ ♠ rs♠♥t♦s ♣♥♦s ♣♦♠♦s ssr ♥

trs rs trés ♦s ①♣♦♥ts rít♦s q r♠ ♥â♠ r♦s

P♦ré♠ ♦ ♦♥trár♦ ♦s sst♠s ♣♥♦s ♥♦s r♦s r♦s ♥ã♦ str ss♦

♣♦rq ♥tr ♠♥t ♦♠ ♦ t♠♣♦ ♣sr s ♦rrçõs s ♣r♦♣r♠ ♣♦

sst♠ ♦ rs♠♥t♦ st ♠♣ q t♦ s♣rí q ♦rr♦♥ ss♠

①st ♣♥s ♦ r♠ rs♠♥t♦ ♦♥ r♦s s ♦♠ tβ s♥♦ ♦ ①

♣♦♥t β ♦ ♥♦♥tr♦ ♥♦ s♦ ♣♥♦ s ts s♦r ss ss♥t♦ srã♦

trt♦s ♥ sçã♦

str ♥t♠♥t ♥trs rs r♠♥t é ♠s í ♦ q ♦ s♦

♣♥♦ ss♦ ♣♦rq é♠ s rs r♠ ♠s ♦♠♣①s q♥♦ srts ♠

♦♦r♥s rí♥s ♣ró♣r r♣rs♥tçã♦ ♦s t♦s é ♠s ♦♠♣ ♦♠♦

à t♥sã♦ s♣r ♣♦r ①♠♣♦ ❬❪ Pr strr ss t♦ ♣♦♠♦s ♥sr

qçã♦ P❩ ♥♠♥s♦♥ q ♦r♦ ♦♠ ❬❪ s t♦r♥

∂R

∂t=

Ω

R2

∂2R

∂θ2− Ω

R+Ψ

[

1 +1

2R2

(

∂R

∂θ

)2]

+η(θ, t)√

R.

♦♥ Ω Ψ sã♦ ♦s ♣râ♠tr♦s qçã♦ srt♦s ♦♠ ♦trs trs ♣r rsr q

♥ã♦ sã♦ ♦s ♠s♠♦s ♦rs ♦ s♦ ♣♥♦ ss qçã♦ é r♠♥t ♠s í

trr q P❩ ♣♥ q ♣♦s ♦ tr♠♦ r♦♥♦ r s♥

q r ♥r ♥♦ s♦ ♣♥♦ ♣ss sr ♥ã♦♥r

①st ♠ ♦♥tr♦érs ♠ rçã♦ ♣ ♥s s ♥

♦r♠ s♥♦ ♣r ♥trs ♣♥s ♠ s♦s r♦s ♦r♦ ♦♠ ❬❪

♥s s é s♥♦ s♦r s ♣ótss q ♥tr ♣♦ sr srt

♣♦r ♠ rr♥ ♥♦ q ♥ã♦ ♠ t♠♥♦ r♥t ♥â♠ ♦

sst♠ ss ♠s♠♦ tr♦ sr♦ ♠♦str q ♣r ♣♦sçã♦ tór

♥ ♦♠tr r ♠ ♠ ♠♥sã♦ r♦s ♠ t♠♣♦s ♠t♦ ♦♥♦s ♣r

W = (ln(t))0.5 q ♣r s ♦ ♠s ♠♥sõs s♣rí s t♦r♥ ♣♥ ♣♦rq

♦s rí♦s s t♦r♥♠ rr♥ts sss rst♦s sr♠ ♠ r③♦á sssã♦

❬❪ é♠ ♦♥tr③r rst♦s ♥tr♦rs ❬❪ ss ♦r♠ ♣


♥ás s tr♦♥ à rs♠♥t♦s r♦s é ♠ ♣r♦♠ ♠ rt♦ ♥

♥â♠ ♥trs

♥trs rs t♠é♠ sã♦ ♠s ís r ♠ s♠çã♦ ♦♠♦ ①

♠♣♦ ♠♦s t♦♠r ♦ ♠♦♦ ♥ ❬❪ ss ♠♦♦ ♦ ♣r♦♣♦st♦ ♣r s♠r

♦ s♥♦♠♥t♦ ♠ tr és ♦♥sst ♠ ♥♥♦ ♦♠ ♠ s

♠♥t ♥ ♦r♠ ♥♠ r qr ♠♥s♦♥ rsr ♦ r♦ ♦♥♥♦

t♦r♠♥t ♣rtís s ③♥♥ç ♦r r♦ r♦ ♣♦r ss ♠♦

♦ é ♥s♦tró♣♦ ♦ ♦ t♦ tr♠♥s rçõs rsr♠ ♠s rá♣♦ q

♦trs ss t♦ é ♠ r①♦ ♥s♦tr♦♣ ♣ró♣r r qr r

♣rs♥t♠♦s ♠ ①♠♣♦ ♥tr ♦t ♣♦ ♠♦♦ ♥ ♦♥ ♣♦

♠♦s ♥♦tr q ♦ ♣rí♠tr♦ ♠r ♠ ♦s♥♦ ♥ã♦ ♠ r♥rê♥ ss ♦r♠

♥s♦tró♣ tr③ ♣r♦♠s ♥♦ st♦ ♥â♠ r♦s ❬❪

r é ♥♦ ♣rí♠tr♦s ♦t♦s ♦♠ rsã♦ ♦ ♠♦♦ ♥ r♥çã♦ ♠ ❬❪ ♣rí♠tr♦ ♦ rs♦ té qtr♦ ♠õs ♣rtís ①trí ❬❪

♦r♠ ♥s♦tró♣ é s ♣♦ rs♠♥t♦ ♥ rçã♦ r sr ♠s

rá♣♦ ♣♦s ♣♦r ①♠♣♦ q♥♦ s s♦rt ♠ ③♥♦ s♠♥t ♣♥s qtr♦

rçõs rs♠♥t♦ sã♦ ♣♦ssís s ♦♥s ♠ ♣♥♥ts ♠ s♥♦

s♦rt♦ ♣r rsr♠ ss r♥ç ♥ ♦ ♠r♥♦ qçã♦ ♥♦s

♥ q ♦ v∞ rr ♥r♠♥t Γ t♠é♠ sr ♥çã♦ ♦ â♥♦

♣♦s ♥s rçõs r ♦s r♦s sã♦ ♠♦rs q ♦ r♦ ♠é♦ ♥s ♦♥s ♦s


r♦s sã♦ ♠♥♦rs ♦♥ ss r♥ç ♠ rçã♦ ♦ ír♦ r♦ ♠é♦ r

♠ Γ ♠♦r ♥tr sss rõs t♠♦s ♥trsçõs ♦ ♣rí♠tr♦ ♦ r♦ ♦♠

♦ ír♦ ♦♥ ♦ Γ sr ♠♥♦r ss ♦r♠ q♥♦ st♠♦s strçã♦ ♦s

r♦s ♦ ♠♦♦ ♥ ♠♦s ♥♦s ♠tr rçõs s♣ís r ss♦ ♣r

r♥tr q st♠♦s st♥♦ ♣♦♥t♦s ♦♠ ♦s ♠s♠♦s ♦rs ♥♦s ♣râ♠tr♦s ♥ã♦

♥rss ss ♣r♦♠ ♣r ♠t♦ sttíst ♣♦s t③♠♦s ♣♦♦s ♣♦♥t♦s

♣♦r r♦ rs♦ ①♥♦ ♠ r♥ ♥ú♠r♦ ♠♦strs ♣r ♦tr rst♦s

r③♦ás é♠ ss♦ t♠♦s ♠ r♥ s♣rí♦ rrs♦s ♦♠♣t♦♥s ♣♦s

♠♦r ♣rt ♥tr é s♠♣s♠♥t srt

❯♠ ♦r♠ ♦♥t♦r♥r ss ♣r♦♠ é ♠♣♠♥t♥♦ ♦ rs♠♥t♦ ♦r

r ♥ ♦♥sr♥♦ ♦ ♠♦♦ ♥ ♣♦♠♦s t③r ♦ ♦rt♠♦ ♥tr♦③♦

♠ ❬❪ ♦♠♦ ①♠♣♦ ss ♦rt♠♦ s ♣rtís sã♦ s♦s ♦♠ â♠tr♦ a q sã♦

rs♥ts ♠ ♣♥♦ rs♠♥t♦ ♥♥♦ ♦♠ ♠ s♠♥t ♦ rs♠♥t♦

é t♦ s♦rt♥♦ ♠ ♣rtí ♦ ♣rí♠tr♦ ♦ r♦ rs♥t♥♦ ♠ ♥♦

♣rtí à s ③♥♥ç q é ♦♦ t♦♥♦ ♦r q ♦ s♦rt

♣rtí é ♦♥ ♠ ♠ rçã♦ tór ♦♥t♥t♦ q ♥ã♦ s s♦r♣♦♥

♦trs á ①st♥ts ❯♠ ①♠♣♦ r♦ r♦ ♣♦r ss ♦rt♠♦ é ♣rs♥t♦

♥ r

r r♦ ♦♠ ss ♠ ♣rtís r♦ ♦♠ ♦ ♠♦♦ ♥ ♦r r ♣rí♠tr♦ stá rç♦ ♠ r♠♦ ①trí ❬❪


①st♠ rss ♦r♠s s ♠♣♠♥tr rs♠♥t♦s ♦r r ♦trs

♦♣çõs ♣r ♥ ♣♦♠ sr ♥♦♥trs ♠ ❬❪ ♦ ♥t♥t♦ ♣♦♠♦s ③r q

♠♥r r s♠çã♦ rs♠♥t♦s ♦r r é ♠s ♦♠♣ ss♦

♣♦rq ♦s ♦rt♠♦s s t♦r♥♠ ③ ♠s ♦♠♣①♦s ♠ q ♠♥t♠♦s s

♠♥sõs ♥tr é♠ ss♦ ♥♦r♠♠♥t ss t♣♦ rs♠♥t♦ ① ♠s

♦♣rçõs ♦♠♣t♦♥s ♣♦r ♣rtí sr rs♥t ③♥♦ q ♦ rs♠♥t♦

r♦s r♥s ♠♥♠ ♠t♦ t♠♣♦ ♦♠♣tçã♦

♦s sss ♦♠♣çõs ♥♦♥♦ ♦ st♦ ♠ ♦♠tr r ♠♦tr♠

♠ t♥tt s♠♣r ss ♣r♦♠ ♦♠♦ ss♠♦s ♥trs rs ♣♦ss♠

♣rí♠tr♦s rs♥ts rtrs ♦s ♣r♦①♠çã♦ q t♠ s♦ ♣r♦♣♦st

♦♥sst ♠ srtr rtr ♦♥srr sst♠s ♣♥♦s ♦♠ ♥trs q

rs♠ tr♠♥t ♣ró①♠ sçã♦ ♠♦s str ♦s rst♦s á ♥ç♦s

♥ss ss♥t♦


s ♣r♠r♦s rst♦s s♦r ♦♠í♥♦s rs♥ts ♣♦♠ sr ♥♦♥tr♦s ♠

❬❪ ss tr♦ ♦s t♦rs str♠ ♦ ♠♦♦ ♥ ♠ sst♠s ♣♥♦s q

rs♠ tr♠♥t ♦r♦ ♦♠ ♠ ♣♦tê♥ rs♠♥t♦ tr ♦♦rr

♦r♠ tr♠♥íst ③♥♦ s♥t ♦♠♣rçã♦

L < L0

(

N

L

)a

♦♥ L é ♦ t♠♥♦ ♥st♥tâ♥♦ ♦ sst♠ L0 ♦ t♠♥♦ ♥ N é ♦ ♥ú♠r♦

♣rtís ♣♦sts a ♥ t① rs♠♥t♦ L ∼ ata−1 ♦♥çã♦

qçã♦ ♦r stst ♠ ♦♥ ♦ r♦ s♦rt ♣ ♥♦ ♦♥

é ♦♥ ♦ ♦ ♦r♥

s r♦s ss ♠♦♦ é t ♣♦ ♦r a a > 1/z ♦♥

z é ♦ ①♣♦♥t ♥â♠♦ ①st ♣♥s ♠ r♠ ♥♦ q r♦s rs ♦♠ tβ

♦♥ β é ♦ ①♣♦♥t rs♠♥t♦ ♦ s ♥♥ str ss♦ ♣♦rq ♣r♦♣çã♦

s ♦rrçõs é ♠s ♥t q ♦ rs♠♥t♦ ♦ sst♠ ss♠ ♥♥


♦♠♣t♠♥t ♦rr♦♥♦ P♦r ♦tr♦ ♦ s a < 1/z ♠ ♠ ♠♦♠♥t♦ rá

♠ r♦ss♦r ss r♠ rs♠♥t♦ s ♣r ♦tr♦ r♦s rs ♦♠

aα ♦♥ α é ♦ ①♣♦♥t r♦s ss♦ ♦♦rr ♣♦rq ♣sr s ♦rrçõs á

tr♠ s s♣♦ ♣ s♣rí r♦s ♥ã♦ str ♣♦rq ♦ sst♠ ♦♥t♥

rs♥♦ ss♠ r♦s ♣ss sr ♦♠♦ W (L, t → ∞) ∼ Lα ♠s ♦♠♦

L ∼ ta t♠♦s W ∼ taα r ♠♦str ♦ rst♦ ♣r a = 0.2 ♠ r♥ts

t♠♥♦s ♥s ♦ sst♠ ♣r♠r rt é ♠ ♣♦tê♥ ♦♠ ①♣♦♥t

β1 = 0.33 ♣ós tr♥sçã♦ ♣♦tê♥ s t♦r♥ β2 = 0.1 = aα

r ♥â♠ r♦s ♣r ♦ ♠♦♦ ❬❪ ♦♠ a = 0.2 r♥tst♠♥♦s ♥s r♦s sã♦ ♣r L0 = 128 ír♦s L0 = 64 trâ♥♦s L0 = 32 ♦s♥♦s L0 = 8 rt só t♠ ♥♥çã♦ β1 = 0.33 rt tr β2 = 0.1 ①tr♦ ❬❪

P♦str♦r♠♥t ♣rts sss rst♦s ♦r♠ ♦t♦s ♥t♠♥t ♣♦r s

r♦ ❬❪ ♦♥sr♥♦ ♠ qçã♦ r ♦♠ ♠ ♦♣r♦r sã♦ ♦r♠ ζ

q ♣r P❩ ❲ é ♦r♠ ♦s sr♦ ♠♦str q s a > 1/ζ s♣rí

♥♥ str ♦♠♦ ♦ ♦t♦ ♠ ❬❪ ♦ ♥t♥t♦ ♣r a < 1/ζ ♦ t♦r ♣♥s ♦♥

q ♥tr s t♦r♥ ♦♠♣t♠♥t ♦rr♦♥ s♠ ♦tr s

♣rs♥t ♥♦s rst♦s ❬❪


sssã♦ ♠ rçã♦ ♣ ♥s s tr♦♥ à ♥tr

s rs ♦t à t♦♥ ♦♠ ♦ tr♦ s♦ t ❬❪ sss t♦rs ③r♠

á♦s ♥ít♦s ♣r ♦♠tr r ♠ ♠ ♠♥sã♦ q ♦♥tr③♠ ♦s rs

t♦s sr♦ ♠♦str♥♦ ♥ás s s Pr ♦rr♦♦rr

ss rst♦s ♥ít♦s s ♣rs♥t♠ t♠é♠ rst♦s ♥♠ér♦s ♦♥ ♥ás

s ♥♦♥ ♠ ♦ ♥t♥t♦ s s♠çõs s ♥ã♦ sã♦ sst♠s ♦♠♣r♦

♠♥t r♦s ♠s ♥ r trt♠s ♦♠í♥♦s rs♥ts ss ♦r♠

♣r♥í♣♦ ♥ã♦ á rçã♦ rt ♥tr ss rst♦s ♥ít♦s s s♠çõs

♦ ♦rt♠♦ t③♦ ♠ ❬❪ ♦ ♦♥sr♦ ♠ sstrt♦ ♣♥♦ ♠♦♦s

srt♦s á ♦♥♦s s♥♦ s ♦s ♠♦♦s ❬❪ ❬❪ s sss

❲ P❩ rs♣t♠♥t ♠ ♥ t♠♣♦ ♥♠♥t ♣♦sts

♠ ♥ú♠r♦ ♣rtís ♦ t♠♥♦ tr r t♦ ss♦ ♥

t♠♣♦ tr♠♥ ♣ós ss ♦♥s sr♠ ♦♥s ♦ sstrt♦ ♠♥r ♠♥t

s♣ç Pr ♥r ♦ ♦r sss trs sr♠ ♦♥s strçã♦

trs ♥tr é ♦♥strí ♥ sã♦ s♦rt♦s ♦s ♦rs s trs ss s♦rt♦

é rt♦ s ♠ tr ♥ã♦ sts③ s rstrçõs ♦ ♠♦♦ ♥sã♦ s ♦♥s

♦r♠ tr♠♥íst ♥♦ ♥ s ♥s t♠♣♦ é ♠ st♥t ♠ r♦

rr♠♥t r♦ ♣♦s ss♠ ♦♠♦ s r♦ rs t♦r♠♥t ♦ ♣rí♠tr♦

♥tr t♠é♠ sr ♠ ♥â♠ st♦ást é♠ ss♦ s s♣rís

♣r♦③s ♣♦r ss ♠♦♦ ♥ã♦ ♣♦ss♠ rtr ♦ ss♠ ss ♦rt♠♦ é ♥

r ♠ ♦♠í♥♦ rs♥t ♥ã♦ ♥ssr♠♥t s♥♦ ♠s♠ ♥â♠ q

sst♠s r♦s

Ps sssõs ♣rs♥ts ♥ss sçã♦ ♥t q sr ♥trss♥t

str ♠s ♥♦ ♠♦♦s rs♠♥t♦ s♦r sstrt♦s q ♠♥t♠ tr

♠♥t tr♠♥r s ss ♣r♦①♠çã♦ t♦ ♣r♦③ ♥trs ♦♥sst♥ts ♦♠ ♦

s♦ r♦

♣ít♦

♦♦s srt♦s ♣r ♦ rs♠♥t♦

tr rt ♥tr

st ♣ít♦ ♣rs♥tr♠♦s ♦s ♠♦♦s t③♦s ♥♦ ♥♦ss♦ tr♦ ❱♠♦s

♥r st♥♦ ♦ rs♠♥t♦ tr ♥tr sr♥♦ ♦ ♠♦♦ q ♥ós

t③♠♦s ♣r ♦♠í♥♦s rs♥ts ♣♦s ♣rs♥tr♠♦s s rrs rçã♦

♠♦♦ st♦


ós ♥♠♦s ♠ ♠♦♦ ♦♠ ♦rt♠♦ ♣r♦ ♦s ❬❪ ♣♦ré♠ ♠s

st♦ást♦ ♦ ♥♦ss♦ ♠♦♦ s ♣♦sçõs ♦ rs♠♥t♦ ♦ sstrt♦ ♦♥t♠

s♠t♥♠♥t ♦r♠ tór ♦rt♠♦ q ♥♠♦s é ♦ s♥t

• ♥♠♦s ♠ t① rs♠♥t♦ ♦♥st♥t δ q srá ♦ ♥♠r♦ ♠é♦

♦♥s ♦♥s à r ♣♦r ♥ t♠♣♦

• ♦♠ç♠♦s ♦♠ ♠ sstrt♦ t♠♥♦ L0 = δ ♦ s ♥♠♦s ♦ t♠♥♦

♥ ♦ ♠ó♦ ♦ rs♠♥t♦ tr vw = δ

• ♥♠♦s ♠ ♣r♦ ♣r ♥t♦ q ♣♦ ♦♦rrr ♥ s♣rí

③♠♦s Pg =L

L+δsr ♣r♦ s ♣♦str ♠ ♣rtí Pw = δ

L+δ

♦ sstrt♦ rsr tr♠♥t


• rs♠♥t♦ tr é t♦ ♣♥♦ ♠ ♦♥ s♦ t♦r♠♥t

♥♦ ♦♥ é ♦♥ rt s♦rt

• ♥t♦ ♦ t♠♣♦ é ♥r♠♥t♦ ∆t = 1L+δ

ss♠ ♠ ♥

t♠♣♦ é ♣♦st♦ ♠ ♠é ♠ ♥ú♠r♦ ♣rtís ♦ t♠♥♦

♠é♦ ♦ sstrt♦ ♥ss ♥tr♦ ♣s δ ♦♥s ♠ ♠é

r strçã♦ ♦ ♠♦♦ ♣r ♦ rs♠♥t♦ tr ♦ sstrt♦

sstrt♦ rs ♦♠ ♠ t① ♦♥st♥t ♣♦rq ♦♥sr♥♦ ♠ rs

♠♥t♦ r♦ s ♦ r♦ rs ♦♠ ♠ t① ♦♥st♥t ♣♦r ①♠♣♦ R = vt ♦

♣rí♠tr♦ P = 2πR = 2πvt t♠é♠ rsrá ♦♠ ♠ t① ① ♦♠♦ s trs

♥♦ ♥♦ss♦ ♠♦♦ rs♠ ♦♠ ∂th = v∞+βΓβtβ−1χ q é ♣r♦①♠♠♥t ♦♥st♥t

♠ t♠♣♦s r♥s ♠ t① rs♠♥t♦ tr ♦♥st♥t é ♦ ♠s ♥tr

♦r ♦ t♠♥♦ ♥ ♦ sst♠ ♥ós s♦♠♦s L0 = δ ♣♦rq é ♦ ♠♥♦r

t♠♥♦ ♦♥ rá ♠ ♥ú♠r♦ ♣rá ♣♦sçõs ♣♦s s L0 ≪ δ ♥ ♣r♠r

♥ t♠♣♦ ♣rt♠♥t só ♦♦rrr♠ ♣çõs s♦♠♦s ♦ t♠♥♦

♥ ss ♦r♠ ♣♦rq ♥♦t♠♦s q ss ♣râ♠tr♦ ♣♦ss ♠ t♦ ♠♣♦rt♥t

♥ ♥â♠ s strçõs trs ♦ t♠♥♦ ♥ ♦r ♠t♦ r♥ s

s P❩ t♥♠ ♥♠♥t ♣r ♣♥♦ ♣♦s ♦♥r♠ ♣r ❯

str♠♦s ss r♦ss♦r ♠ t ♥ sçã♦


100 1000 10000t

10Wβ=0,33

β=0,20

r ♦♠♣rçã♦ s r♦s ♣r ♣♦sçã♦ íst ♠ sstrt♦srs♥ts ♦r♦ ♦♠ ♦ ♥♦ss♦ ♠♦♦ ♣♦♥t♦s ♣rt♦s ♦ ❬❪ ♣♦♥t♦s r♠♦s

♥♦ tst♠♦s ♦ ♦rt♠♦ ❬❪ ♥♦ ♠♦♦ ♣♦sçã♦ íst ♠ d =

1 + 1 ♥♦ ♥ sçã♦ q ♥ã♦ ♣♦ss rstrçõs tr ♦t♠♦s ①♣♦♥ts

rs♠♥t♦ ♠t♦ st♥ts ♦ β = 1/3 ss P❩ ♦ ♥t♥t♦ s ♣♥s

♣♠♦s s ♦♥s ♦♠♦ ♦ t♦ ♠ ❬❪ ♦ ①♣♦♥t ♠t♦ ♠s ♣ró①♠♦

♦♠♦ ♠♦str♦ ♥ r ss♦ ♦♦rr ♣♦rq ♦ rs♥tr ♠ sít♦ ♦♠ ♠

tr qqr ♥tr ♣♦ ♦♥tr ss ♥♦ ♦♥ ♣♦ssr tr

♠t♦ r♥t ♦s ss ③♥♦s ❯♠ ♦♥ ♦♠ tr ♠t♦ ♠♦r q ♦s ③♥♦s

♣♦r ①♠♣♦ ♣r♦③rá ♦ rs♠♥t♦ tr ♠ strtr s♦♠r♥♦ t♦ s

①t♥sã♦ ss♦ ♥tr♦③r ♦rts ♦rrçõs s ♥♦ sst♠

♦ ♠♦♦ t③♦ ♠ ❬❪ s r♥çs tr sã♦ ♥♦ ♠á①♠♦ ±1

Pr ♠ ♦♥ sr ♦♥ sts③r ss ♦♥çã♦ ♣♦r ss♦ ♦s rst♦s

❬❪ ♥ã♦ ♣♦ss♠ ♦ ♣r♦♠ q ♥♦♥tr♠♦s ♥ ♣♦sçã♦ íst P♦ré♠ ♠

♣r♦♠ ♦s ♠♦♦s ♦♠ rstrçã♦ tr é q ♦ t♠♣♦ ♦♠♣tçã♦ ♣♦

sr ♠t♦ r♥ ♣♦s ♣♦♠ sr ♥ssár♦s ár♦s s♦rt♦s té q ♠ ♦♥ q

stsç rstrçã♦ s ♥♦♥tr

P♦r s s ♦♥srçõs ♥tr♦rs ♥♦ ♥♦ss♦ ♦rt♠♦ çã♦ ♦♥s

é t ♣♥♦ tr♠♥t tr ♠ sít♦ s♦♦ t♦r♠♥t ♦♠♦ é

t♦ ♠ ❬❪


r♥t♠♥t ♦s ♠♦♦s ❬❪ q ♠♣♠♥t♠♦s ♥â♠

♦r♠ ♠s st♦ást ♣♦ssí ♣r ♥ã♦ ♥tr♦③♠♦s ♦rrçõs ♦♥s ♥♦ ss

t♠ Pr t♥t♦ ♥♠♦s ♦ t♠♣♦ ♦r♠ r♥tr q ♠ ♠é ♥tr♦ ♠

♥ t♠♣♦ δ ♦♥s ♦ss♠ ♣s q L ♣rtís ♦ss♠ ♣♦sts

ós t♠é♠ st♠♦s sst♠s ♠♥s♦♥s ♦♠ ♠ r♥t ♦ ♠♦♦

♣r♦♣♦st♦ Pr ss♦ t♠♦s q ③r ♠ s♠♣s ♥r③çã♦ ♦ ♥és ♣

r♠♦s ♠ sít♦ ♦♠♦ ♥ r ♥♠♥s♦♥ ♣ss♠♦s ♣r ♠ ♥ ♦

♦♥ ♥tr sít♦s ss♠ ♠♦s tr três ♥t♦s ♣♦ssís ♥ r ♣♦str

♠ ♣rtí ♣r ♠ ♥ ♦ ♠ ♦♥ ♥tã♦ ♦ ♥ssár♦ r♥r s

♣r♦s ♥ r ♠ ♣♦sçã♦ ♦♥t♥ s♥♦ Pg = L1L2

L1L2+2δ ♦♥

L1 L2 sã♦ ♦s ♦s ♦ sstrt♦ δ é ♦ rs♠♥t♦ ♠ rçã♦

♦ s ♦ ♥ú♠r♦ ♠é♦ ♦♥s ♦ ♥s q srã♦ ♣s ♣♦r ♥

t♠♣♦ ♠♦s ♠♥t ♣r♦ ♣çã♦ ♥tr ♥s ♦♥s

ss♠ ♠s sã♦ Pw1 = Pw2 =δ

L1L2+2δ str♠♦s ss ♥r③çã♦ ♥ r

r ♥r③çã♦ ♦ ♠♦♦ ♣r ♥trs ♠♥s♦♥s



st sçã♦ ♠♦s ♣rs♥tr s rrs rçã♦ t♦♦s ♦s ♠♦♦s q

st♠♦s ♦♠♦ st♠♦s ♥trss♦s ♥ strçã♦ trs ss P❩ t♦♦s

♦s ♠♦♦s ss sçã♦ ♣rt♥♠ ss ss ós ♠♦s ♣rs♥tr ♣♥s s rrs

♥♠♥s♦♥s ♦s ♠♦♦s ♣♦s s ♥r③çã♦ ♣r s ♠♥sõs é ♠t

strt s♦ ♦♥ s♦

ss ♠♦♦ ♦ ♣r♦♣♦st♦ ♣♦r ♠ ♦strt③ ❬❪ rr ♦♥sst ♠ r

str♥r s r♥çs tr s♣rí ♦ q é t♦ ♥♥♦ ♠ ♣râ♠tr♦ M

♦♠♦ ♠♦r r♥ç tr ♣r♠t ♥tr sít♦s ♣r♠r♦s ③♥♦s ♥♥♦

♦♠ ♠ sstrt♦ s♦ t♠♥♦ L ♦ rs♠♥t♦ tr♥s♦rr t♥t♥♦ s ♣♦str

L ♣rtís ♣♦r ♥ t♠♣♦ ♠ sít♦s s♦rt♦s t♦r♠♥t ♣♦sçã♦

♠ ♣rtí rá ♣r♦③r r♥ç tr ♠♦r q M ♣rtí é rt

rs ♣rtís ♥ss ♠♦♦ é rs♣♦♥sá ♣♦ ①ss♦ ♦

rtríst♦ ss P❩ ss s♦ ♠ ③ ♠ rs♠♥t♦ ♥♦r♠ ♥tr

①st ♠ rs rs♠♥t♦ ♠ ♣♦rçõs ♥♥s ♥tr ss rçã♦

é q♥t ♠ λ ♥t♦ ♥ qçã♦ P❩ q ss ♦r♠ t①

rs♠♥t♦ tr ♠é é ♠♥♦r q ♦ ①♦ ♣rtís ss♠ ♦ ①ss♦

♦ é ♥t♦

♥♦ rstrçã♦ tr é t ♠ ♦ ♠♦♦ é ♦ q ♣♦ss

♦rrçõs ♠s rs ♥tr ♦s ss ss P♦r ss♦ é ♦ ♠♦♦ srt♦ ♠s

t③♦ ♣r r♣rs♥tr ss P❩ ♦♦s ♦s rá♦s sçã♦ ♦r♠ t♦s

t③♥♦ ss ♠♦♦

♠ ♠ ♠♥sã♦ ♦s ♣râ♠tr♦s ♥ã♦ ♥rss ss ♠♦♦ ♦r♠ ♦t♦s

♥♠r♠♥t ♠ ❬❪ ♥♦ v∞ = 0, 419030(3) Γ = 0, 252(1) 〈η〉 = −0, 32(4)

♠é♠ ①st♠ rst♦s ♥♠ér♦s ♣r s ♠♥sõs q ♦r♦ ♦♠ ❬❪

sã♦ v∞ = 0, 31270 Γ = 0, 66144 〈η〉 = −0, 5(1) sss út♠♦s rst♦s ♥ã♦

♣♦ss♠ rrs rr♦ ♣♦rq ♦ t♦r ♦ ♣♥s ♦ ♦r ♠é♦


r strçã♦ rr rs♠♥t♦ ♦ ♠♦♦ ♦♠ rstrçã♦ M = 1

t♥

ss ♠♦♦ ♦ ♣r♦♣♦st♦ ♣♦r ♦ t ♠ ❬❪ ♦ rs♠♥t♦ ♥

♦♠ ♠ sstrt♦ s♦ t♠♥♦ L ♦♥ ♣♦sts L ♣rtís ♣♦r ♥

t♠♣♦ ♠ sít♦s s♦rt♦s t♦r♠♥t ♦ ♣♦str ♠ ♣rtí ♠ ♠ sít♦ i

r ♥ss s trs ♦s sít♦s ♣r♠r♦s ③♥♦s ♦ sít♦ q ♣♦ssr tr

♠♥♦r q h(i)− 1 rá sr rs♦ té ss ♦r

rr rs♠♥t♦ ss ♠♦♦ é str ♥ r ss ♠♦♦

s trs ♦ ♣óst♦ rs♠ ♠ ♦r ♠♦r q ♦ ♥ú♠r♦ t♦t ♣rtís

♦♥s ♦ sst♠ ♦ q ① ♠ r♦ ♣rs♥ç ♠ ①ss♦ ♦

♦♥♥♦ ♦♠ qçã♦ P❩ q ss rs♠♥t♦ ①tr str ss♦♦

♠ λ ♣♦st♦ ♦ ♦♥trár♦ ♦ ♠♦♦

r strçã♦ rr rs♠♥t♦ ♦ ♠♦♦ t♥


Pr s ♠♥sõs ♦s ♣râ♠tr♦s ♥ã♦ ♥rss ss ♠♦♦ ♦r♠

♦s ♠ ❬❪ ♦ ♥♦♥tr♦ v∞ = 3, 3340(1) (Γ)β〈χ〉 = −2, 348(3) 〈η〉 = 0, 6(1)

ss tr♦ ♦s t♦rs ♥ã♦ r♠ ①♣t♠♥t ♦ ♦r Γ ♣♦r ♥ã♦ r

á♦s ♥ít♦s strçã♦ trs ♠ s ♠♥sõs

ós ♥ã♦ ♥♦♥tr♠♦s ♥ trtr rst♦s ♣r ♦s ♣râ♠tr♦s ♥ã♦ ♥rss

♠ ♠ ♠♥sã♦ Pr st♠á♦s ♥ós rs♠♦s s♣rís ♦♠ 220 sít♦s té ♠

t♠♣♦ ♥ t = 5× 103 ♦ t♦t ♦r♠ t③s ♠♦strs ♣r r③çã♦

sttíst Pr ♦tr v∞ ♥ós r♠♦s qçã♦ ♠ rçã♦ ♦ t♠♣♦ q

rst ♠∂〈h〉∂t

= v∞ + β Γβtβ−1〈χ〉

ss ♦r♠ s trç♠♦s ♠ rá♦ ∂h/∂t ♠ ♥çã♦ tβ−1 ♠♦s ♦tr ♠

rt ♦♠♦ ♠♦str♦ ♥ r ①tr♣♦♥♦ ss rt ♣r t → ∞ ♦t♠♦s ♦

♦r v∞ = 2, 13995(8)

Pr ♦tr♠♦s ♦ ♦r Γ ♥ós s♦♠♦s ♦ tr♠♦ Γβ tβ−1 r

trç♠♦s ♠ rá♦ ∂th−v∞β〈χ〉 ♠ ♥çã♦ t Pr tr♠♥r ♦ ♦r Γ st♠♦s

♠ ♣♦tê♥ ♥ rã♦ t♠♣♦s r♥s ♦♥ s ♦rrçõs ♠ str ♠s

♣ró①♠s ③r♦ rst♦ q ♦t♠♦s ♦ Γ = 4, 90(9)

0 0,02 0,04

t -2/3

2,12

2,135

d<

h>

/dt

Etching

10 1000t

0,01

1

(Γ)βtβ-1

Etching

r rá♦ t③♦ ♥ st♠t ♦ ♦r v∞ ss ♦r ♦ ♦t♦ ①tr♣♦♥♦ rs♦s sts r③♦s ♥ rã♦ t−2/3 ♣q♥♦s st♠t ♦ ♦r Γ Pr ss♦ rss s ♣♦tê♥ ♦r♠ sts ♥ rã♦ t♠♣♦s r♥s


♥st♣

♥çã♦ ss ♠♦♦ ♣♦ sr ♥♦♥tr ♠ ❬❪ ss ♠♦♦ é rt

♦r♠ s♠r ♦ ♦♠ M = 1 ♣♦r t♠é♠ ♣r♦③r s♣rís ♠ ss

♦♠ ♦rrçõs rt♠♥t rs s r♥çs tr sã♦ s♠♣r s

♠ ♦ q r ♦ q ♣r♠t trs s ss ♠♦♦ ♦♠ç♠♦s

♦♠ ♠ sstrt♦ q t♥ ♠♥♦r r♦s stsç ♦♥çã♦ r♥çs

tr s♠♣r ♠ ♦ s sít♦s tr ♠ ③r♦ tr♥♦s ♠ t♦♦

♦ sstrt♦ s ♣rtís ss ♠♦♦ sã♦ í♠r♦s rts sã♦ ts ♣♥s

q♥♦ ♣♦sts ♠ ♠í♥♠♦s ♦s rr rs♠♥t♦ é str ♥ r

r strçã♦ rr rs♠♥t♦ ♦ ♠♦♦ ♥st♣

ss ♠♦♦ é s♠r ♦ ♥♦ s♥t♦ ♥ã♦ ♣♦str ♠ ♣♦rçõs ♥

♥s ♥tr ss♠ t♠♦s ♠ rs rs♠♥t♦ tr ♦♠♦ ♥♦

♦ λ é ♥t♦ r♦♥♦ ♠ ①ss♦ ♦ q r③ ♦çã♦ tr

♠é

①ê♥ q s r♥çs tr s♠ s♠♣r s 1 ♥♦s ♠♣

♣r rr rs♠♥t♦ tr ♦ sstrt♦ ♥ ♦r♠ ♦♠♦ ♥♠♦s ♥

sçã♦ ss♦ ♣♦rq ♦ ♣r ♠ ♦♥ r♠♦s ♠ r♥ç tr ♥

♥ s♣rí Pr ♦♥t♥r sts③♥♦ ♦♥çã♦ ♦ ♥st♣ ♥ós t♠♦s q

♣r s ♦♥s s♦rt s ③♥♦ sqr♦ ♦♠♦ str♦ ♥ r

tr♦s ♠♦♦s

ss ♠♥r ♦ rs♠♥t♦ tr ♣ss sr vw = 2δ

r r rs♠♥t♦ tr ♦ sstrt♦ ♣r ♦ ♠♦♦ ♥st♣

Pr ♠ ♠♥sã♦ ss ♠♦♦ ♣♦ sr ♠♣♦ ♥♦ ♣r♦♠ ♠ r

♥r ♦♠ ♣rtís s♣♥ ♠♦ trés ss ♠♥ç r♣rs♥tçã♦ ♦s

♣râ♠tr♦s ♥ã♦ ♥rss ♦r♠ ♦s ♥t♠♥t ♠ ❬❪ ♦♥ ♦ ♦t♦

v∞ = 1/2 Γ = 1/2 Pr s ♠♥sõs sss ♣râ♠tr♦s ♦r♠ tr♠♥♦s ♥

♠r♠♥t ♠ ❬❪ s♥♦ v∞ = 0, 341368(3) (Γ)β〈χ〉 = −0, 881(1) 〈η〉 = −0, 6(1)

tr♦s ♠♦♦s

st sçã♦ ♠♦s ♣♥s ♥r rr rs♠♥t♦ ♦s ♠♦♦s q

♦r♠ t③♦s ♠ ♥s tsts q r③♠♦s ❯♠ s é ♦ ♠♦♦ ♣♦sçã♦

íst q ♣rt♥ ss P❩ ♣rtí s r ①t♠♥t ♥ ♣♦sçã♦

♠ q t♦ ♥tr r s ♣♦st♠♦s ♠ ♣rtí ♥♦ sít♦ i tr♠♦s

h(i) = max[h(i−1), h(i)+1, h(i+1)] ♦♠♦ ♠♦str♦ ♥ r s ♥♦r♠çõs

s♦r ss ♠♦♦ ♣♦♠ sr ♥♦♥trs ♠ ❬❪

s♥♦ ♠♦♦ q ♠♦s ♥r é ♦ ♠② q é ♣rt♥♥t ss

❲ ss ♠♦♦ ♣rtí ré♠ ♣♦st ♣♦ ♥r tr♠♥t s♦ ♦s

③♥♦s ♦ sít♦ s♦♥♦ t♥♠ trs ♠♥♦rs s rrá ♥ ♣♦sçã♦

♠♥♦r tr ♥tr ♦s ss ③♥♦s ♦♥ ♥ê♥ i s trs ♦s

tr♦s ♠♦♦s

r strçã♦ rr rs♠♥t♦ ♦ ♠♦♦ ♣♦sçã♦ íst

③♥♦s ♦r♠ s ♠♥♦rs q hi ♠ s♦rt♦ é r③♦ ♣r ♥r ♠ q

sít♦ ♣rtí s r ♦♠♦ ♠♦str♦ ♥ r s ♥♦r♠çõs s♦r ss

♠♦♦ ♣♦♠ sr ♥♦♥trs ♠ ❬❪

r strçã♦ rr rs♠♥t♦ ♦ ♠♦♦ ♠②

♣ít♦

st♦s ♠ ♠ ♠♥sã♦

st ♣ít♦ ♠♦s ♣rs♥tr ♦s rst♦s s ♥♦sss s♠çõs ♦♠í♥♦s

rs♥ts ♠ ♠ ♠♥sã♦ ♦♠♦ t♦ ♥♦ ♣ít♦ ♣ss♦ st♠♦s ♦s ♠♦♦s

t♥ ♥st♣ Pr ♠♦♦ rs♠♦s s♣rís ♦♠ ♦s

trs vw = 6, 12, 20 100 ♦♥s ♣♦r ♥ t♠♣♦ Pr ♦

rs♠♦s ♥tr ③ ♠ ♥t ♠ ♠♦strs ♣r♠r sçã♦ ♠♦s ♠♦strr ♦s

①♣♦♥ts rs♠♥t♦ ♦t♦s ♥♥♦ ♦ sstrt♦ ♦♠ t♠♥♦ L0 = vw ♠

s ♠♦s str strçã♦ tr sss ♠♦♦s ♠ sstrt♦s rs♥ts

♣♦s ♠♦s str ♦♠♦ s ♣çõs ♣r♦③♠ s ♦rrçõs ♦rít♠s q

♣r♠ ♥sss rs♠♥t♦s P♦r út♠♦♠♦s ♠♦strr ♦ t♦ ♦ t♠♥♦ ♥

♦ sstrt♦ q ♣r♦③ ♠ r♦ss♦r ♣r ❯ q♥♦ L0 é ♠t♦ r♥

①♣♦♥t rs♠♥t♦

♦♥♦r♠ st♦ ♥♦ ♣ít♦ ♠ tr♦s ♥tr♦rs ♦♥sr♥♦ s

strt♦s rs♥ts ❬❪ ♦ ♦♦ ♣r♥♣ ♦ ♦ ①♣♦♥t rs♠♥t♦ ós ♥

s♠♦s ♦çã♦ t♠♣♦r ♦ ①♣♦♥t rs♠♥t♦ t♦ ♦t♦ ♥♦

r r lnW ♣♦r ln t ss ♦r♠ t♠♦s ①♣♦♥ts t♦s q ♣♥

♠ ♦ t♠♣♦ r ♣rs♥t♠♦s ts ①♣♦♥ts ♣r ♦ ♠♦♦ ♠

sstrt♦s q ♠♥t♠ tr♠♥t ss rá♦ ♣♦♠♦s ♥♦tr q ♦♥rê♥

♦ ①♣♦♥t rs♠♥t♦ ♣♥ ♦ tr ♦ s s ♦rrçõs

strçã♦ tr

s ♣♥♠ vw ss ♦♠♣♦rt♠♥t♦ é s♠r ♥♦ ♠♦♦ ♥st♣ ♦

♠♦♦ t♥ ♦♥rê♥ ♦ ①♣♦♥t t♠é♠ ♣♥ ♦ tr

♣♦ré♠ ♠ ♠♥r r♥t ♦♠♦ ♣♦ sr ♥♦t♦ ♥ r ♦ ♥t♥t♦

♣r t♦s s ♦s t♦♦s ♦s ♠♦♦s ♥♦t♠♦s q ♦ ①♣♦♥t ♦♥r ♣r

β = 1/3 ♠ t♠♣♦s ♦♥♦s ss ♦r♠ ♦ ①♣♦♥t rs♠♥t♦ sss sst♠s

q rs♠ tr♠♥t ♦♥t♥♠ ♦♥sst♥ts ♦♠ ss P❩ ♦♠♦ t♠é♠ ♦

♥♦♥tr♦ ♠ ❬❪

2 4 6 8 10ln(t)

0,32

0,34

0,36

0,38

0,4

β

vw=100

vw=12

vw=2

RSOS

2 4 6 8 10ln(t)

0,32

0,34

0,36

0,38

0,4

β

vw=100

vw=20

vw=6

Etching

r ①♣♦♥t rs♠♥t♦ ♦s ♠♦♦s t♥ ♣r r♥ts♦s trs s ♥s tr sã♦ ♦rrs♣♦♥♥ts β = 1/3

strçõs trs

❱♠♦s ♥r ♥♦ss sssã♦ s♦r s strçõs trs ♦s ♦♠í♥♦s

rs♥ts ♣rs♥t♥♦ ♦s ♥♦ss♦s rst♦s ♣r s♥ss rt♦s ❱♠♦s r♣tr

q qçã♦ ♣r tr tr s♥♦

h(x, t) = v∞t+ s(λ)(Γt)βχ+ η + ζt−γ + . . . .

tr rçã♦ ♥tr s♥ss strçã♦ tr rá χ é s♠♣s

♣♦s ♦ tr♠♦ v∞t ♠s tr♠♦s tr♠♥íst♦s qçã♦ s ♥♠ ♥♦

á♦ ♦s ♠♥ts ♦r♠ ♠♦r q ♠ ss♠ ♦♠♦ ♣♦ sr ♠♥t

strçã♦ tr

♠♦str♦ ♦ tr♠♦ ♦♠♥♥t ♥sss ♠♥ts é 〈hn〉c = (Γt)nβ〈χn〉c + . . . ♥♦

♠♦s s♥ss t♠♦s

S =〈h3〉c〈h2〉3/2c

=(Γt)3β〈χ3〉c + . . .

((Γt)2β〈χ2〉c + . . . )3/2≈ 〈χ3〉c

〈χ2〉3/2c

+ . . . .

ss ♦r♠ rçã♦ ♥tr s♥ss strçã♦ rá χ é

rt ❯♠ á♦ ♥á♦♦ ♠♦str q ♦ ♠s♠♦ é ♦ ♣r rt♦s

s rs ♣rs♥t♠♦s ♥â♠ s♥ss r

t♦s ♣r ♦ ♠♦♦ ♥st♣ ♠ sstrt♦s rs♥ts sss rá♦s ♠♦str♠ q

t♥t♦ s♥ss q♥t♦ rt♦s sã♦ ♠t♦ ♠s ♣ró①♠s ❯ r♦ ♦ q

sss rs ①♠ ♥t q ♥♦ r♠ ss♥tót♦ t → ∞ sss

♠♥ts t♥♠ ♣r ❯ é♠ ss♦ ♦♥rê♥ sss r♥③s ♣♥

♦ rs♠♥t♦ tr ♥s♥♦ sss rá♦s ♥t q ♥ ♠

♦rs q st♠♦s q♥t♦ ♠♦r ♦ tr ♠s rá♣♦ strçã♦

s ♣r♦①♠ ❯ ss ♦♠♣♦rt♠♥t♦ s r♣t ♥♦s ♠♦♦s t♥

0 0,002 0,004 0,006

t(-2β)

0,18

0,22

0,26

0,3

Skew

nes

s

GOEGUEv

w=100

vw

=20

vw

=12

Single-step

0 0,002 0,004

t-2β

0

0,1

0,2

Curtose

GOEGUEvw=100

vw=20

vw=12

Single-step

r rá♦s ♥ss rt♦s ♠ ♥çã♦ t−2β strçã♦ trs ♦ ♠♦♦ ♥st♣ ♠ sstrt♦s q rs♠ tr♠♥t

♥â♠ s♥ss rt♦s ♥♦s ♣r♠t r♠r q ♦ s♥♦

trr♦ ♦ qrt♦ ♠♥ts strçã♦ trs t♥♠ ss♥t♦t♠♥t ♣r

strçã♦ ♦ s♦ r♦ ♥ás ♦ ♣r♠r♦ ♠♥t é ♠ ♣♦♦ ♠s tr

♦s ♣♦s t♦♦s ♦s tr♠♦s qçã♦ sã♦ r♥ts Pr str ss ♠♥t

strçã♦ tr

♠♦s sr ♦s ♠s♠♦s ♣râ♠tr♦s v∞ Γ q ♦r♠ ♦s ♣r ♦s sstrt♦s ♦♠

t♠♥♦ ①♦ ♦rê♥ ♦s á♦s q ♠♦s r③r ♥ss sçã♦ ♥ q

sss ♣râ♠tr♦s t♦ ♥ã♦ ♠♠ s♦ ♦♥trár♦ t♦s ♥á♦♦s ♦s q ♠♦s

str ♥♦ ♣ró①♠♦ ♣rár♦ ♦♦rrr♠ ♠ t♦s s ♥áss q r③áss♠♦s

sçã♦ ♠♦s str ♠s s♦r ♥râ♥ sss ♣râ♠tr♦s

10 100 1000t

0,01

0,1

-ζ t

α

0.446 t-0.64

Etching vw

=20

10 100t

0,001

0,01

0,1

-ζ t

α

0.231 t -1.012

Etching vw

=20

r ♦ ♦rrçã♦ ζt−β−1 s♥♦ ♠é ❯

♦♥sr♥♦ q v∞ Γ sã♦ ♦s ♠s♠♦s ♣r str 〈h〉 ♣rs♠♦s ♦tr

η ζ P♦♠♦s ♠♥r ♦ η tr♥♦ ♠é r♥♦ qçã♦ ♦ q rst

♠∂〈h(x, t)〉

∂t= v∞ + s(λ)βΓβtβ−1〈χ〉 − γζt−γ−1 + . . . .

ss ♦r♠ ♣♦♠♦s r ♦ ♦r ζ s s♦r♠♦s ♦ ♦r 〈χ〉 ♦♠♦ ♦

tr♠♦ s(λ)βΓβtβ−1〈χ〉 t♥ ③r♦ ♠t♦ ♠s ♥t♠♥t q ♦rrçã♦ ζt−γ−1 s

t③r♠♦s ♠ ♦r rr♦ ♥ 〈χ〉 ♠♦s ♦tr ♠ r r♦♥ r♥ç ♥tr

ss ♦r ♦ ♦rrt♦ s♦♥♦ ♦ tr♠♦ ζt−γ−1 s t③♠♦s ♠é ♦♠♦

♥ r ♦t♠♦s ♠ ♥çã♦ q ♦♠ t−0.64 ♦ ♥t♥t♦ −0.64 ≈ β−1

q é ♣♦t♥ ♦ tr♠♦ q ♠t♣ 〈χ〉 ♦♦ ♠é ♥ã♦ é ♦ ♦r

♦rrt♦ P♦r ♦tr♦ ♦ s t③r♠♦s ♦ ♦r ♠é ❯ ♦♠♦ ♥ r

♦t♠♦s ♠ r q ♠s rá♣♦ q tβ−1 ss ♦r♠ ♣♦♠♦s ③r q ♦

♣r♠r♦ ♠♥t strçã♦ tr t♠é♠ ♦♥r ss♥t♦t♠♥t ♣r♦t q s ♦ ♦r Γ ♦ss r♥t ♦♥t♥rí♠♦s t♥♦ ♠ r q ♣r♦①♠

♠♥t ♦♠ tβ−1

strçã♦ tr

♦ s♦ r♦ é♠ ss♦ r ♦rrçã♦ ♣r♦①♠♠♥t ♦♠ t−1 ♦

s t♠♦s q ζt−γ → ζ ln t ♥ qçã♦ ss ♦rt♠♦ ♣r ♠ t♦s s

♦s trs ♦s ♠♦♦s q st♠♦s ss♠ ♦t♠♦s q s trs sss

sstrt♦s q rs♠ tr♠♥t sã♦ ♣♦r

h(x, t) = v∞t+ s(λ)(Γt)βχ+ η + ζ ln t+ . . . .

s ♦rs ζ ♦t♦s ♣r ♦s ♠♦♦s t♥ ♥st♣ sã♦ ♣r

s♥t♦s ♥♦s rá♦s rs♣t♠♥t ♦♠♦ sss rá♦s

♠♦str♠ ♣r ♠♦♦ ζ ♥ã♦ ♣♥ ♠t♦ ♦ tr ss ♦r é

♣r♦①♠♠♥t ♦♥st♥t ♥tr♦ s rrs rr♦

10 100vw

-0,2

-0,18

-0,16

ζ

RSOS

10vw

0,15

0,25

ζ

Etching

10vw

-0,25

-0,15

-0,05

ζ

Single-step

r rá♦s ζ ♣r s r♥ts ♦s ♦s ♠♦♦s t♥

♥st♣

P♦♠♦s ♦♥r♠r ♣rs♥ç ss ♦rt♠♦ t♥t♥♦ r ♦ s♦♠♥t♦

strçã♦ tr

〈η〉 strçã♦ Pr ss♦ s♦♠♦s ss tr♠♦ s♥t ♦r♠

〈h〉 − v∞t− ζ ln t

(Γt)β− 〈χ〉 = 〈η〉

(Γt)β.

r trç♠♦s ♦ rá♦ 〈η〉/(Γt)β s♠ ♦♥srr ♦rrçã♦ ♦rít♠

ζ ln t ♥s♥♦ ss r ♠ s ♦rít♠ ♥t q ♥ã♦ s ♠

♣♦tê♥ ♠ ♥ ♠♣♦sst♥♦ ♦ á♦ 〈η〉 P♦ré♠ s ♦♥sr

♠♦s ♦ tr♠♦ ζ ln t ♦♠♦ ♠ ♦t♠♦s ♠ r q ♣r♦①♠♠♥t

♦♠ t−β q é ♦♥sst♥t ♦♠ 〈η〉/(Γt)β ♦ q ♥♦s ♣r♠t r ♦s s♦ ♥

strçã♦ ss ♦r♠ ♦ ♦rt♠♦ t♦ ③ ♣rt qçã♦ q sr s

trs ♥tr

1 10 100 1000 10000t

1

-<η

>/(

Γt)

β

RSOS vw

= 20

1 10 100 1000 10000t

0,1

1

-<η

>/(

Γt)

β

1.36 t -0.34

RSOS vw

=20

r á♦ ♦ s♦ strçã♦ s♠ ♦♥srr ♦ ζ ln t ♠ ♦♥sr♥♦ ss ♦rrçã♦

ós ♠♦s ♦ ♦r ss s♦ ♣r s r♥ts ♦s trs ♦s

♠♦♦s q st♠♦s ♣rs♥t♠♦s sss ♦rs ♥♦s rá♦s

ss♠ ♦♠♦ ζ sss rá♦s ♠♦str♠ q 〈η〉 t♠é♠ ♥ã♦ r s♥t♠♥t ♦♠

♦

♦♠♦ á ♦ ♠♣ít♦ ♣ ♥♦tçã♦ q t③♠♦s té q ζ é ♠ rá

tr♠♥íst η é ♠ rá tór Pr ①♣tr ♥tr③ sss rás

♦♥sr♠♦s qçã♦ ♣r r ♦ s♥♦ ♠♥t strçã♦

strçã♦ tr

10 100vw

-0,9

-0,88

-0,86

-0,84

<η

>

RSOS

10 100t

2

3

4

<η

>

Etching

10vw

-0,48

-0,44

-0,4

<η

>

Single-step

r rá♦s 〈η〉 ♣r s r♥ts ♦s ♦s ♠♦♦s t♥

♥st♣

trs ♦ q rst ♠

〈h2〉c = (Γt)2β〈χ2〉c + 〈η2〉c + 〈ζ2〉c ln2 t+ . . . ,

♦♥ s♣♦♠♦s q ♥ã♦ ①st ♦rrçã♦ ♥tr s rás χ η ζ Pr ♠♦strr q

ζ é tr♠♥íst♦ r♠♦s ss qçã♦ ♠♥♣♠♦s ♣r ♦tr

∂t〈h2〉c − 2βΓ2βt2β−1〈χ2〉c = 2〈ζ2〉cln t

t+ . . . .

r trç♠♦s ♠ rá♦ ♥çã♦ ♦ ♦ sqr♦ qçã♦ ♥tr♦r

s♥♦ 〈χ2〉c ❯ ♦♠♦ ss r ① r♦ ss r t♥ ③r♦ ♠t♦ ♠s

strçã♦ tr

rá♣♦ q ln t/t ♦♦ râ♥ ζ é ♥ ss♠ é ♠ rá tr♠♥íst

♦♠ rçã♦ η ♦t♠♦s ♥ qçã♦ s♦♠♦s 〈η2〉c r

trç♠♦s ♠ rá♦ 〈η2〉c + . . . ♠ ♥çã♦ t ♣sr ss r ♥ã♦ ♥♦s

♣r♠tr r ♦ ♦r 〈η2〉c ♦ t♦ ♥ã♦ t♥r ③r♦ ♠♦str q 〈η2〉c 6=0 ♦ s η é ♠ rá tór ♦♠♦ tr♠♦s tr♠♥íst♦s s ♥♠ ♥

qçã♦ rçã♦ t♠♣♦r r ♠ ♠♦str q ①st♠ ♦rrçõs

st♦ásts ♦r♠ s♣r♦r ♥♦ t♠♣♦ é♠ ss♦ sss ♦rrçõs ♦r♠ s♣r♦r

♠ sr ♥çã♦ ♦ rs♠♥t♦ tr ♣♦s s sr♠ rs♣♦♥sás

♣ ♦♥rê♥ strçã♦ tr ♣♥r ss ♦ ♦♠♦ s rs

♠♦str♠

0 10000t

-0,0004

0

0,0004

0,0008

d<

h2>

c/d

t -

2β

Γ2β<

χ2>

c t2

β−

1

RSOS vw

= 20

0 10000 20000t

0

0,2

0,4

< h

2>

c -

(Γ

t)2β<

χ2>

c

RSOS vw

= 20

r rá♦ q ♠♦str ♦ rátr tr♠♥íst♦ ζ r q ♠♦str ①stê♥ ♦rrçõs ♦r♠ ♠♦r ♠ q 〈η2〉c 6= 0

r♦r ♥ã♦ ♣♦♠♦s ③r q ζ ln t é ♠ ♦rrçã♦ ♣♦s ss tr♠♦ r

q♥♦ t → ∞ ♦ ♥t♥t♦ é s♠♣r ♠t♦ ♠♥♦r q ♦s tr♠♦s v∞t (Γt)βχ

ss ♦r♠ ♣♦♠♦s ③r q ζ ln t r♣rs♥t ♠ s♣é ♦rrçã♦ ♦rt s

♣ ♣çã♦ s ♦♥s sçã♦ ♠♦s ♦♠♥tr ♠s s♦r ss ♦rt♠♦

ss ♦r♥s

♥tr♥t rt ♦r♠ q s strçõs trs ♦s ♦♠í♥♦s rs

♥ts t♦ ♦♥r♠ ♣r ♦ s♦ r♦ ♥tr é ♣♥ ♥sss ♠♦♦s q

st♠♦s ♥ã♦ á rtr ♦ ♦ sstrt♦ ♣♥s rs tr♠♥t ♥♦

♦♥sr♠♦s q ♠ ♥tr r é s♠♥t ♦♠♣♦st ♣♦r ♠ ♣rí♠tr♦ rs


♥t ♠ rtr ♦ ♣r♠r st rtr é ♦♠♣♦♥♥t ♠s r

tríst ♣ró①♠ sçã♦ ♠♦s ①r r♦ ♦ q s ♠♥ç χgoe → χgue

♥♦s sst♠s sstrt♦ rs♥t


r ♣rs♥tçã♦ ♦s t♦s s ♣çõs s♦r s ♣r♥♣s rtrísts r

❱♠♦s ♥r ♥♦ss sssã♦ ♦♥sr♥♦ r ♦♥ ♦♥sr♠♦s

t♦♦s ♦s t♣♦s ③♥♥çs q ♣♦♠ ①str ♣r ♠ sít♦ ♦ ♠♦♦ ♠

d = 1+1 r ①st♠ ♠s três t♣♦s sít♦s ♣♦ré♠ s sã♦ ♣♥s r①õs

♠ t♦r♥♦ ♦ sít♦ ♥tr ♦s t♣♦s D E F ss♠ ♣r s♠♣r ♦♥sr♠♦s

q sss três t♣♦s t♠é♠ r♣rs♥t♠ ss r①õs st♠♦s ♥trss♦s ♠

♥tr ♦ q ♠ ♥sss ♣q♥s ♣♦rçõs ♥tr q♥♦ ♣♠♦s ♦s sít♦s

st♦s ♠ ③ ♦♥ ♥tt r♥sçõs st♠♦s ♠♦str♥♦ q t♣♦s

sít♦s sã♦ ♣r♦③♦s q♥♦ ♦♦rr ♠ ♣çã♦ P♦r ①♠♣♦ ♥ út♠ ♥


t♠♦s F → D,E q s♥ q ♦ ♣r♠♦s ♠ sít♦ ♦ t♣♦ F s tr♥s♦r♠

♠ ♠ sít♦ ♦ t♣♦ D ♣r♦③ ♠ ♦tr♦ sít♦ ♦ t♣♦ E s ♦♥s s♥ts

♠♦str♠♦s ♦♠♦ árs ♣r♦♣rs s♣rs sã♦ trs ♣ ♣çã♦

♦♥ ∆〈h〉 ♠♦str♠♦s ♦♠♦ ♣çã♦ ♠ ♦♠♥t tr ♠é

♦ s tr ♠é ♦ ♦♥♥t♦ três ♦ qtr♦ sít♦s Pr strr ♠♦s ♦♥

srr s♥ ♥ ♦♠♥♦ tr ♠s ① ③r♦ ♥ts ♣çã♦

tr ♠é sss três sít♦s é 1/3 ♣ós ♣çã♦ t♠♦s qtr♦ sít♦s ♦♠ tr

♠é 1/2 Pr r♠♦s ♦ t♦ t♦t s ♣çõs s♦r tr ♠é ♥ã♦

st s♠♣s♠♥t s♦♠r ss ♦♥ ss♦ ♣♦rq sss sít♦s ♥ã♦ ♣r♠ ♥ r

♦♠ ♠s♠ rqê♥ ♦ ♣r♦ r ♠♦str ♦♠ q ♣r♦

t♣♦ sít♦ é ♥♦♥tr♦ ♥ r t♠♥♦ ①♦ rs trs ♥

rs♥t ♥s sós ♦t ♦♠♦ ♦s sít♦s ♦ t♣♦ F s t♦r♥♠ ♠♥♦s rq♥ts

q♥♦ ③♠♦s s ♣çõs ♥tã♦ ♣r♦ ♦s ♠s sít♦s ♠♥

tr r ① r♦ q ♦ ♠♦r ♠♥t♦ ♦♦rr ♥♦s sít♦s sít♦s A D E

q sã♦ qs ♦♠ ③♥♦s ♠s♠ tr ♦♠♦ r s s♣rr ♣♦r s trtr

♣çõs

100 300t

0,1

0,2

probabilidade

FE

D

AC

B

r rá♦ ♦♠ s ♣r♦s t♣♦ sít♦ sr ♥♦♥tr♦ ♥ r s♥s trs sã♦ r♦♥s sst♠s ♦♠ t♠♥♦ ①♦ s sós ♦s rs♥ts


♥s♥♦ s r♥çs ♣r♦ ♥tr ♦ sstrt♦ t♠♥♦ ①♦ ♦

rs♥t ♣♦♠♦s rr s rá rés♠♦ íq♦ ♥ tr ♠é s♦ ♣s

♣çõs ♦♠♦ A F ♥ã♦ ♣r♦③♠ ∆〈h〉 s r♥çs ♠ ss ♣r♦s

♥ã♦ é ♠♣♦rt♥t ♣r tr ♠é s sít♦s ♦ t♣♦ B C qs ♥ã♦ tr♠ s

♣r♦ ss♠ t♠é♠ ♥ã♦ srã♦ ♥♥♠ t♦ á ♥♦s sít♦s D E s

♣r♦s ♠♠ ♠ q♥t ♣r♦①♠♠♥t ♦♠♦ ♦s ∆〈h〉s♥tr♦③♦ ♣♦r s ♣♦ss♠ ♠s♠♦ ♠ó♦ ♣♦ré♠ s♥s ♦♥trár♦s ts ♣çõs

t♠é♠ ♥ã♦ ♠ sr t♦s íq♦s

Pr ♦♥r♠r q ♦ t♦ íq♦ s ♣çõs ♥ ♠é s ♥♠ ♥ós

r③♠♦s s♠çõs rstr♥♦ rçã♦ ♥ tr ♠é ♥tr s ♣♦r

♣çã♦ ss s♦ st♠♦s ♥s♥♦ r♥ç ♥tr♦③ ♥ ♥tr

♦♠♦ ♠ t♦♦ ♥ã♦ ♣♥s ♥♦ sít♦ ♣♦ s ③♥♥ç ós tr♠♦s

♠é ♥tr rss ♠♦strs ♦ ∆〈h〉 t♦t ♥tr♦③♦ ♣♦r ♥ t♠♣♦ ♦♠♦

♣♦♠♦s r ♥ r sss ♦rs t♠ ♠ t♦r♥♦ ③r♦ ♦♠ ♠♣t q

r♣♠♥t ♠♥♦r q 10−6 ♠♦s ♠é t♦t ss r ♥♦ t♠♣♦ ♦

rst♦ é ♦r♠ 10−9 sss rst♦s ♥♦s ♥♠ q t♦ ♦ ∆〈h〉 sr③r♦ P♥s♥♦ ♠s ♥♦ ♥♦t♠♦s q ss♦ r s s♣rr ♣♦s ♦ s♦rtr♠♦s

♦♥s r ♣r ♣r ♥ós st♠♦s ♥tr♦③♥♦ trs t♦r♠♥t

♦r♦ ♦♠ strçã♦ trs ss♠ ♥tr♠♥t ♠é sss ♦♥trçõs

♦♥ ♦♠ ♠é strçã♦ ss ♦r♠ ♣♦♠♦s ③r q ♥ ♠é ♦s

∆〈h〉s ♥tr♦③♦s ♣s ♣çõs sã♦ ♥♦s

t♦r♥♥♦ r ♥ ♦♥ ∆〈~∇2h〉 ♥s♠♦s ♦ t♦ s ♣çõs

s♦r ♠é r s♥ ♠♦s r s♥ ♦r♠

♥♠ér ♠ sít♦ ③ ♦ q é t♦ ♦♥sr♥♦ ∆2h/∆x2 = h(x+ 1) + h(x−1) − 2h(x) P♦♠♦s r♣tr ♠s♠ sssã♦ s♦r rçã♦ s ♣r♦s

♦s sít♦s ♥ r r♦♥ à r ♦♠♥t s ♦♥trçõs D

E B C s ♥r♠ ♠t♠♥t Pr ♦♥r♠r ss♦ ♥ós r③♠♦s s♠çõs

q ♠♦strr♠ q 〈~∇2h〉 é ①t♠♥t ③r♦ ♠ t♦♦s ♦s ♥st♥ts t♠♣♦ ♠s♠♦

♣ós ♣♦sçã♦ ♦ ♣çã♦ ss♦ ♦♥r♠ q ♥ã♦ á t♦ íq♦ ♥ r

s♥ é♠ ss♦ ♦♠♦ ∂2h∂x2 stá r♦♥ rtr ♥tr ss rst♦


100 1000

t

-1e-05

-5e-06

0

5e-06

1e-05

∆<

h>

RSOS vw

=20

r é ♣♦r ♥ t♠♣♦ ♦s ∆〈h〉 ♥tr♦③♦s ♣s ♣çõs

① r♦ q ♣çõs ♥ã♦ ♣r♦③♠ ♠ rtr ♦ ♥ r ss rst♦

é s♣r♦ ♦ ♣♦♥t♦ st ♥ít♦ ♣♦s ♦♠♦ é tr s ♠♦strr ♣ê♥

qqr ♥çã♦ s ♦♠ ♦♥çã♦ ♦♥t♦r♥♦ ♣ró ♣♦ss⟨

dnhdxn

⟩

= 0

út♠ ♦♥ r ♥ós ♠♦s ♦ t♦ s♦r r ♣r♠r

♦ qr♦ Pr ss♦ s♦♠♠♦s ♦ ♠ó♦ s r♥çs tr ♠♦s ♣♦r

♦s ♥ts ♣çã♦ ♣♦s st♠♦s ♦♥sr♥♦ ♦ ♥ú♠r♦ rs ♣♦ssís

♠♦s ♣♦r três ♣ós ♣çã♦ ♥♣♥♥t♠♥t sss ♦rs t♦s s

♦♥trçõs sã♦ ♥ts ♦ q ① ♥t q rá ♠ trçã♦ íq

♥ss r♥③ t♦ ♦ rá♦ r ♠♦str q ♦ 〈(~∇h)2〉 ♥♦ sstrt♦

rs♥t qr♦s é ♠♥♦r q ♥♦ ♦♠ t♠♥♦ ①♦ ír♦s ss rst♦

é rt ♦r♠ ♥tt♦ ♣♦s s ♦♥sr♠♦s ♠ r ♦♥tí♥ q ♣♦ss ♠

♠♦rr♦ tr ∆h rr ∆x ♥ss s♦ ♦♥tí♥♦ s ♣çõs sr♠ ♠

s♣é ♦♥♠♥t♦ ♥♦ ①♦ ♦r③♦♥t ♥ts ss ♦♥♠♥t♦ trí♠♦s ∆h/∆x

♣♦s ∆h/∆x′ ♦♠ ∆x′ > ∆x ♦♦ ∆h/∆x > ∆h/∆x′ ♦ q ♥ rçã♦

♥♦ r♥t ♠é♦


10 1000t

0,6

0,64

< (

dh

/dx

)2>

RSOS vW

=20

10 100 1000t

0,45

Na /

Nt

RSOS vw

=20

r rá♦ 〈(~∇h)2〉 ♠ ♥çã♦ ♦ t♠♣♦ ♣r sstrt♦s t♠♥♦ ①♦r ♣rt ír♦s sstrt♦s rs♥ts r r♠ qr♦s ú♠r♦ ♣rtís ♣♦sts ♦ ♣♦ ♥ú♠r♦ t♦t ♣rtís q t♥tr♠ sr ♣♦sts r♠ qr♦s stá ss♦ ♦ s♦ rs♥t ♣rt ír♦s ♣r sstrt♦s ♦♠ t♠♥♦ ①♦ ♥ ③ tr é ♦ ♦r v∞ ♦ ♠♦♦

♥♦ qçã♦ st♦ást ss P❩ ♣♦r

∂h(~x, t)

∂t= ν∇2h+

λ

2(∇h)2 + η(~x, t) ,

st♠♦s ♥ sçã♦ ♥ q ♥tr♦③♠♦s ♦ ♠♦♦ q ♥ss ♠♦♦

♦ λ é ♥t♦ ♦ q rt ♠ ♠ rs ♣♦sçã♦ ♠ ♣♦rçõs ♥♥s

s♣rí ss ♦r♠ rçã♦ ♦ 〈(~∇h)2〉 r ♠ ♠♥t♦ ♥ t①

♣♦sçã♦ ♣rtís r ♣rs♥t♠♦s ♠ rá♦ ♠♦str♥♦ q

♥♦ sstrt♦ rs♥t qr♦s t① ♣♦sçã♦ é ♠♦r q ♥♦ sstrt♦ ♦♠

t♠♥♦ ①♦ ír♦s ss♠ ♥♦t♠♦s q s ♣çõs t♠ ♦♥trçã♦ ♦

tr♠♦ ♥ã♦♥r qçã♦ P❩

♦♥♥♦ ♦ t♦ s ♣çõs s♦r ♥tr ♣♦♠♦s ♠♦strr ♦♥

♠ ♦rrçã♦ ζ ln t qçã♦ Pr ss♦ ♠♦s ♦♥srr q ♠ ♠ ♥st♥t

t♠♣♦ t ♦ sstrt♦ t♥ t♠♥♦ L ♦ ♣♦r ♠ ♦♥♥t♦ SL = S1, S2, . . . , SL sít♦s ♥ã♦ ♦r♠ ♣çõs tr♠♦s

〈(~∇h)2〉Ft+1 =1

L

∑

SL

(~∇h)2F ≡ K

L.


P♦ré♠ s ♣♠♦s vw ♦♥s ♥♦ t♠♣♦ t ♠ ♦♥♥t♦ Svw =

SL+1, SL+2, . . . , SL+vw srá ♥tr♦③♦ ♥♦ sst♠ í

〈(~∇h)2〉Gt+1 =1

L+ vw

∑

SL

(~∇h)2G +∑

Svw

(~∇h)2

.

L ≫ vw ♥tã♦∑

SL(~∇h)2F ≈

∑

SL(~∇h)2G ≈ K ♥♥♦ ~∇h ≡ hi − hi−1

t♠♦s q ~∇h ≈ 0 ♣r t♦s s ♦♥s ♦♥s ♦♦∑

Svw (~∇h)2 ≈ 0 ♥tã♦

〈(~∇h)2〉Gt+1 ≈K

L+ vw< 〈(~∇h)2〉Ft+1 =

K

L

P♦♠♦s srr

K

L+ vw=

K

L− a ⇒ a =

K

L

(

1− 1

1 + ε

)

,

♦♥ ε ≡ vw/L ε ≪ 1 ♣♦♠♦s ①♣♥r (1 + ε)−1 ♦tr

a =K

L

(

ε+O(ε2))

⇒ a ≈ KvwL2

.

♦♦K

L+ vw≈ K

L

(

1− vwL

)

.

♦♠♦ L = vw(t+ 1) ♣r t ≫ 1 t♠♦s L ≈ vwt ss♠

〈(~∇h)2〉Gt+1 ≈ 〈(~∇h)2〉Ft+1

(

1− 1

t

)

♥♠♥t ♦♠♦

∂〈h〉∂t

=λ

2〈(~∇h)2〉G ≈ λ

2〈(~∇h)2〉F − λµ

t,

♦♥ µ é ♠ ♣râ♠tr♦ q ♦♥♠♦s ♣♦rq ♠♣t ss ♦rrçã♦ ♣♦r s

s ♣r♦①♠çõs q ③♠♦s ♥ã♦ é s♠♣s♠♥t λ ss♠ ♥♦t♠♦s q t♦

♠♦s tr ♠ ♦rrçã♦ q s ♦♠ ♦ ♦rt♠♦ ♦ t♠♣♦ ♥ qçã♦ ♣r s


trs

té q ♥s♠♦s ♣♥s ♦ ♠♦♦ ♦ ♥t♥t♦ ♣♦♠♦s ♦♥r♠r

♦♥sstê♥ ss sssã♦ ♥tr♦③♥♦ ♦s ♠s ♠♦♦s ♦r♦ ♦♠ ♦ q

♦ t♦ ♦♠♦ λ ♦ ♠♦♦ ♥st♣ é ♥t♦ ♠♦s tr ♠ ♠♥t♦ ♥ t①

♣♦sçã♦ ss ♠♦♦ ss♦ é ♦♥r♠♦ ♣ r q ♠♦str q

tr ♠é ♦ sstrt♦ rs♥t qr♦s ♠♥t r♠♥t ♠s rá♣♦ q

♦ s♦ ♦♠ t♠♥♦ ①♦ ír♦s á ♦ ♠♦♦ t♥ ♣♦ss λ ♣♦st♦ ss♠

s③çã♦ s ♣s ♣çõs r③r t① rs♠♥t♦ ♦ q é

♦♥r♠♦ ♣ r ♦♥ tr ♠é ♦ sst♠ rs♥t qr♦s

é ♠♥♦r q ♦ ♦♠ t♠♥♦ ①♦ ír♦s ss ♦r♠ ♥♦t♠♦s q ♥♦ss

sssã♦ é t♦ ♦♥sst♥t

450 550 650t

220

260

300

340

<h>

Single-step

400 550t

900

1200

<h>

400 550

900

1200

Etching

r rá♦ ♦çã♦ tr ♠é ♣r ♦s ♠♦♦s ♥st♣ t♥ ♦♠ sstrt♦ t♠♥♦ ①♦ r ♣rt sstrt♦ rs♥♦ ♦♠ ♦vw = 100 r r♠

♦r q ♦♠♣r♥♠♦s ♦s t♦s s ♣çõs s♦r s♣rí ♠♦s

♦♥srr ♦ tr♠♦ ζ ln t qçã♦ ♦♠♦ st♠♦s st♥♦ trçã♦ ♥ t①

rs♠♥t♦ ♠♦s ♥sr r qçã♦ q é

∂〈h(x, t)〉∂t

= v∞ + s(λ)βΓβtβ−1〈χ〉+ ζt−1 .

♦♠♦ ♦ ♠♦str♦ ♥ sçã♦ ζ ♣♦ss s♥ ♣♦st♦ ♣r ♦ ♠♦♦

t♥ s♥ ♥t♦ ♣r ♦s ♠♦♦s ♥st♣ q ♣r ♦♥tr③r


t♦ sssã♦ q ③♠♦s ♥ss sçã♦ ♦ ♥t♥t♦ s ♣çõs ③♠ ♦ tr♠♦

s(λ)(Γt)βχgoe → s(Γt)βχgue s♥♦ 〈χ〉goe = −0.76007 〈χ〉gue = −1.77109 ss

♦r♠ s♥♦ λ ♣♦st♦ ♥♦ ♠♦♦ t♥ ss tr♠♦ rá ♠s ♥t♦ q♥♦

♣♠♦s ♦♥s ♦♠♦ ♦rrçã♦ ζ/t t♥ ③r♦ ♠s rá♣♦ ♣♦♠♦s ③r

q ss tr♠♦ rá ♠♥③r r♥ç s ♣ ♠♥ç strçã♦

♣r ❯ Pr ♦s ♠♦♦s ♥st♣ ♦♠♦ t♥t♦ λ q♥t♦ ζ ♠♠

s♥ é ♥t q ♦♥sã♦ srá ♥á♦

10 1000t

0,39

0,41

v∞

+ ...

RSOS vw

=20

r rá♦ ∂t〈h(x, t)〉 − s(λ)βΓβtβ−1〈χ〉goe ♦ ♠♦♦ ♠ sstrt♦ t♠♥♦ ①♦ ír♦s ∂t〈h(x, t)〉−s(λ)βΓβtβ−1〈χ〉gue trâ♥♦s ∂t〈h(x, t)〉−s(λ)βΓβtβ−1〈χ〉gue − ζ/t qr♦s ♥♦ sstrt♦ rs♥♦ ♦♠ vw = 20

♥t♥♦ ♦♠♣r♥r ♠s ♥♦ ♦ ♣♣ ♦ tr♠♦ ζ/t ♦♥sr♠♦s ♠

rs♠♥t♦ ♠ sstrt♦ q é ♣♥♦ ♥♠♥t ♦s ♥st♥ts ♥s

t① ♣♦sçã♦ ♥ã♦ sr ♠t♦ ♥♥ ♣s ♣çõs ♣♦s ♣rt♠♥t

t♦s s ♣rtís sã♦ ts á q ①st♠ ♣♦s r♥çs tr ♥ r Pr

t♠♣♦s r♥s tr♠♦s ♦♠♣tçã♦ ♥tr ♦s t♦s ❯♠ s é q q♥t♦ ♠♦r

r♦s ♠s r♥çs trs tr♠♦s ♥ r ss♠ ♠♦r sr ♦ t♦

s ♣çõs s♦r t① ♣♦sçã♦ ♦ ♥t♥t♦ ♦♠♦ ♥♦ss♦ sst♠ é rs♥t

q♥t♦ ♠s ♦ t♠♣♦ ♣ss ♠♦r é ♦ t♠♥♦ tr s♣rí t♦r♥♥♦ ♠♥♦s

r♥t ♦ t♦ ♣çã♦ ♠s ♦♥s ss♠ ♣sr sr ♠s í

③r r♠çõs ♣r t♠♣♦s r♥s ♥♦s ♥st♥ts ♥s ♥â♠ ♦ sst♠


rs♥t r sr s♠r ♦ sst♠ t♠♥♦ ①♦ ♦ s

s(λ)βΓβtβ−1〈χ〉goe ≈ s(λ)βΓβtβ−1〈χ〉gue + ζ/t .

Pr rr s ss♦ é r ♥ r trç♠♦s ♦s rá♦s ∂t〈h(x, t)〉 −s(λ)βΓβtβ−1〈χ〉 ♠ ♥çã♦ ♦ t♠♣♦ ♣r ♦ s♦ q rs trâ♥♦s ♦ q ♥ã♦

rs ír♦s s♥♦ 〈χ〉gue 〈χ〉goe rs♣t♠♥t r r♠ qr♦s

é ♦ s♦ rs♥t ♦♥sr♥♦ ♦ tr♠♦ ζ/t ♥s♥♦ r ♥♦t♠♦s q

♦rr♦♦r ♦♠ ♣r♦①♠çã♦ ss ♦r♠ t♦ ♥çã♦ ♦ tr♠♦ ζ/t é

♠♥③r ♥♦s t♠♣♦s ♥s r♥ç s ♣♦ sstrt♦ rs♥t sr

strçã♦ ❯ ♦ ♥és

❯♠ ♣♦ss q ♣♦r sr ♦♥sr é q ♦ ♥és χgoe → χgue

q♠ s♦r ♠ trçã♦ é ♦ ♣râ♠tr♦ Γ ♥♦ Γ → Γ′ ♦ ♥t♥t♦ ♦♠♦

♠♦str♠♦s ♥ sçã♦ ♦ Γ s ♥ ♥♦ á♦ s♥ss ♥♦ rt♦s

ss ♦r♠ Γ ♥ã♦ ♣♦r sr rs♣♦♥sá ♣ ♦♥rê♥ s♥ss

rt♦s ♣r ❯ ss♠ ♦♥í♠♦s q ss ♣óts q Γ → Γ′ ssttr

χgoe → χgue é s

0 10000t

0,985

1

1,015

d<h>/dt

Family

r ① rs♠♥t♦ tr ♠é ♦ ♠♦♦ ♠② ♠ sstrt♦ ♦♠t♠♥♦ ①♦ ♥ tr ♥♦ rs♥t r só

♦r♦ ♦♠ sssã♦ s♥♦ ♥ss sçã♦ ♦ rs♠♥t♦ s♣rí

r♦ss♦r ♣r ❯

t ♦ (~∇h)2 ♦r♠ t q strçã♦ trs ♥tr ♠

♣r ❯ Pr ♥③r ♣rs♥t♠♦s ♥ r ♦ rst♦ ♦ rs♠♥t♦

tr ♠ ♠♦♦ ss ❲ ♦ ♠♦♦ ♠② ♥♦ ♥ sçã♦ ss

r r r♠ só stá r♦♥ ∆t〈h〉 ♦ sstrt♦ rs♥t

♣rt ♥ tr ♦ ♦♠ t♠♥♦ ①♦ ♦♠♦ ♥ss ss (~∇h)2 ♥ã♦ t

♥â♠ s ♣çõs ♣♥s ♥tr♦③♠ tçõs ♥ r tr ♠é s♠

trr ♦♠♥t ♦çã♦ tr ♠é ♦♠♦ á ♦ st♦ sss tçõs

ê♠ s ♦♥s rs♥ts ♥ã♦ ♣♦ssír♠ tr à tr ♠é

r♦ss♦r ♣r ❯

♦s rst♦s sçã♦ ♥tr♦r t③♠♦s ♦ t♠♥♦ ♥ ♦ sstrt♦

♦ ♥ú♠r♦ ♦♥s q r♠ ♣s ♣♦r ♥ t♠♣♦ ♦♠♦ ♦ ①♣♦

♥ sçã♦ st♠♦s s♥ss rt♦s ♣r r♥ts t♠♥♦s ♥s

♥♦t♠♦s q s ♥â♠ ♣♥ ss ♣râ♠tr♦ r ♠♦str♠♦s ♦

rá♦ s♥ss ♥ ♦ rt♦s ♣r r♥ts t♠♥♦s ♥s ♦ ①♦

s ssss ♦ t♠♣♦ ♦ tr♦♦ ♣♦ t♠♥♦ ♥tr ♦ ♣♦ t♠♥♦

♥ s♥♦ L = L0 + vwt ♥s♥♦ sss rs ♣♦♠♦s r q ♣r L0 ≫vw s strçõs trs ♥♠♥t s ♣r♦①♠♠ ♣♦s t♥♠

ss♥t♦t♠♥t ♣r ❯ sss strçõs t♥r♠ ♣r ♥♠♥t

r rt ♦r♠ s♣r♦ ♣♦s ♠ ♠ sstrt♦ ♠t♦ r♥ ♣çã♦

♠s ♦♥s tr ♠ t♦ ♥ ♣q♥♦ s♦r ♠é ♠s ♠♥ts

strçã♦ P♦ré♠ é ♥trss♥t q ♠s♠♦ ♣r t♠♥♦s ♥s ♠t♦ r♥s

sss strçõs t♥♠ ss♥t♦t♠♥t ♣r ❯ ♠ t♠♣♦s ♦♥♦s

s rá♦s ♠♦str♠ q s rs ♣r♠ s ♣r♦①♠r

♦♠ç♠ s rr ♣r ❯ ♠ Lc/L0 ∼ 2 ♦♥ Lc é ♦ t♠♥♦

r♦ss♦r ♥♦ Lc = L0 + vwtc t♠♦s q ♦ t♠♣♦ r♦ss♦r s ♦♠

tc ∼ L0/vw ss♠ ♣♦♠♦s ♠♥♠③r ss t♠♣♦ r♦ss♦r ③♥♦ L0 = vw

❯t③r t♠♥♦s ♠♥♦rs r ♦♠ q ♥ ♣r♠r ♥ t♠♣♦ ♣r♦

♣r ♦ss ♠♦r q ♣♦str ss♠ r♠ rss ♣çõs

r♦ss♦r ♣r ❯

0 10 20 30

L/L0

0,2

0,24

0,28

Sk

ew

ness

8x104

4x104

2x104

104

5x103

20

RSOS vw

=20

GOE

GUE

0 10 20 30

L/L0

0,05

0,1

0,15

Curt

ose

8x104

4x104

2x104

104

5x103

20

RSOS vw

=20

GUE

GOE

r rá♦ s♥ss rt♦s ♣r r♥ts t♠♥♦s ♥s r ♣r L0 = 20 ♠ ♥çã♦ L/L0 ♥ã♦ r sí ♥ss s ♣♦r ss♦ stá ♠♥çã♦ t/640 ♥♦s ♦s rá♦s

♦♥s s ♥♦ ♥ ss ♣r♠r ♥ t♠♣♦ trí♠♦s ♠ sstrt♦ ♦♠

t♠♥♦ ♦r♠ vw ♦♠ ♣♦s ♣rtís ♣♦sts ♦ q r ♣rt♠♥t

♥♦ ♠s♠♦ q ♥r ♦♠ vw sít♦s ss ♦r♠ s t③♠♦s L0 = vw ♠♥♠③♠♦s

♦ t♠♣♦ r♦ss♦r ③♥♦ sr ♦r♠ ♠ ♥ t♠♣♦

♥s♥♦ ♠s ♥♦ r ♥♦t♠♦s q s rs L0 =

5× 103, 104 2× 104 trss♠ ♦ ♦r ❯ ♥♦ ♠♣rssã♦ q s rt♦ss

ã♦ ♦♥rr ♣r ♠ ♦r ♠♥♦r q ♦ ❯ ♦ ♥t♥t♦ ♥ sçã♦ ♥ós

♠♦str♠♦s q r ♣r L0 = 20 ♦♥r ♣r ❯ ♥♦ r♠ ss♥tót♦

ss♠ ♦♠♦ ♦♥rê♥ ss ♠♥t ① ♠ t♠♣♦ ♠t♦ ♠♦r q ♦ t♠♣♦

r♦ss♦r s rs L0 = 5× 103, 104 2× 104 ♠ s ♣r♦①♠r ♦♠ L0 = 20

s ♠ ♦♥rr ♥ts ♣r ❯

♣ít♦

st♦s ♠ s ♠♥sõs

st ♣ít♦ ♠♦s ♣rs♥tr ♦s rst♦s ♦t♦s ♣r sstrt♦s rs♥ts

♠ s ♠♥sõs P♦ré♠ ♦♠♦ ♠♦s ♥♦ ♣ít♦ ♥tr♦r s ♥áss ♦s rst

♦s ♣♥♠ ♦s ♣râ♠tr♦s ♥ã♦ ♥rss ♦♠♦ ♥s ♦s ♣râ♠tr♦s q ♠♦s

t③r ♥ ♥ã♦ ♦r♠ ♦t♦s ♥ trtr ♥ ♣r♠r sçã♦ ♠♦s ♣rs♥tr ♦s

á♦s sts ♠ s ♠♥sõs ♠ s ♠♦s str ♦s ①♣♦♥ts rs

♠♥t♦ ♦s ♦♠í♥♦s rs♥ts ♠♥s♦♥s t♦ ss♦ r♠♦s ♣rs♥tr ♥♦ss♦s

rst♦s ♣r strçã♦ trs ♦♠♦ sss strçõs ♥ã♦ ♦r♠

s ♥t♠♥t ♠ s ♠♥sõs ♥ós ♠♦s t③r ♥♦ss♦s rst♦s ♣r

st♠r ♣rs♠♥t ss ♠♥ts ós t♠é♠ tr♠♥♠♦s s ♦rrçõs ♠

s ♠♥sõs Pr ♥③r ♠♦s ♠♦strr ♦♠♦ s sssõs sçã♦ s

st♥♠ ♣r ♦ s♦ ♠♥s♦♥

♦♦s ♦s ♠♦♦s ♦r♠ st♦s ♦♠ ♦ rs♥♦ ♦♠ ♦s

vw = 2, 4, 6 10 rs♠♦s ♥tr ♥t ♠ tr♥t ♠ ♠♦strs ♣r ♦


♥♦ ♠♦s s ♦rrçõs ♥♦ ♣ít♦ ♥tr♦r ③♠♦s s♦ ♦s ♣râ♠tr♦s

♥ã♦ ♥rss v∞ Γ s ♦rs v∞ ♦s ♠♦♦s q st♠♦s á ♦r♠ ♦s

♠ s ♠♥sõs ♥♦ tr♦ ❬❪ sss ♦rs ♣♦♠ sr ♦t♦s ♠s♠ ♦r♠

q ♦ t♦ ♠ ♠ ♠♥sã♦ sçã♦ ♦ ♥t♥t♦ ♦s ♦rs Γ ♠ s


♠♥sõs sã♦ ♠s ís s ♦tr ♣♦s ♥ss s♦ ♥ã♦ ♦♥♠♦s ①t♠♥t

strçã♦ rá χ ss♠ qçã♦ só ♥♦s ♣r♠t ♦tr ♦s ♣r♦t♦s

Γnβ〈χn〉cst sçã♦ ♠♦s ♣rs♥tr ♦s rst♦s rss s♠çõs q ♦r♠

♥ssárs ♣r ♦ á♦ Γ ♠ t♦s s ♦ sstrt♦ ♥ã♦ rs tr♠♥t ♦♠

♦ t♠♣♦

Pr r ♦s ♦rs Γ ♥ós s♠♦s ♦ ♠s♠♦ ♣r♦♠♥t♦ t③♦

♠ ❬❪ ①♣çõs ♠s ♣r♦♥s s♦r s qçõs q ♠♦s t③r ♣♦♠ sr

♥♦♥trs ♠ ❬❪ ♦r Γ ♣♦ sr ♦ ♣ s♥t qçã♦

Γ = A1/αλ ,

♦♥ λ é ♦ ♦♥t ♦ tr♠♦ ♥ã♦♥r qçã♦ P❩ A é ♠♣t s

♦rrçõs trtr ♥♦ s♣ç♦ ♦rr 〈|h(~k)|2〉 = Ak2−2α ss ♦r♠ ♣r

r ♦s ♦rs Γ ♣rs♠♦s ♣r♠r♦ ♦tr sss ♦tr♦s ♣râ♠tr♦s

0 0,2 0,4m

0,28

0,3

0,32

v(m

)

RSOS

r rá♦ ♦ rs♠♥t♦ ♠ ♥çã♦ ♥♥çã♦ ♦ sstrt♦♣r ♦ ♠♦♦ ♦♠ L = 2048

λ ♣♦ sr ♦ ♠♥♣♥♦ ♦ t♦ ♦ tr♠♦ ♥ã♦ ♥r qçã♦

P❩ ❬❪ tr♠♦s ♠é ss qçã♦ ♥♦ rr♥ stát♦ ♦ sstrt♦


t♠♦s

v = F +λ

2L

∫ L

0

d2x(~∇h)2 ,

♦♥ ♠♠♦s 〈∂th〉 = v F é ♦ ①♦ ♠é♦ ♣rtís ♣♦r sít♦ ♦♠♦ á ♦

st♦ rss ③s ♦ tr♠♦ ♥ã♦ ♥r é rs♣♦♥sá ♣♦r ♠ rs♠♥t♦ ♥♦r♠

à s♣rí ♥t♥s ss t♦ é ♠♥sr ♣♦r λ ss ♦r♠ s ♥srr

♠♦s rt♠♥t ♠ ♥♥çã♦ ♦ ♥♦ sstrt♦ 〈~∇h〉 = m ♦

rs♠♥t♦ sr tr ♦r♦ ♦♠ tr♠♦s

v(m) = v(0) +λ

2m2 .

ss♠ ♣♦♠♦s str ♠ ♣rá♦ à r v(m) ♣♦r m ♣r ♦tr ♦ ♦r

λ r ♠♦str♠♦s ♠ rá♦ ♦ rs♠♥t♦ ♠ ♥çã♦

♥♥çã♦ ♣r ♦ ♠♦♦ ♦r v(m) é ♦ ♠s♠ ♦r♠ q ♦

♦t♦ ♦ v∞ ♦ ♠♦♦ t♥ ♥ sçã♦

Pr r ♦ ♦r A ♠♦s t③r ♦ t♦ t♠♥♦ ♥t♦ ♥ ♦

rs♠♥t♦ ♦♠♦ ♥t ♥ r ss ♦ ♥ç ♠ ♦r

♠t♥t q ♣♥ ♦ t♠♥♦ ♦ sst♠ ♦r♦ ♦♠ ❬❪ ss t♦

t♠♥♦ ♥t♦ s s♥t qçã♦

∆v = v(L)− v∞ = − Aλ

2L2−2α.

♦♥♥♦ ♦ ♦r λ ♣♦♠♦s t③r ss qçã♦ ♣r tr♠♥r ♦ ♦r

A r ♣rs♥t♠♦s ♠ rá♦ ∆v ♣♦r 1/L2−2α s♥♦ ♦ ①♦

s ssss ss ♠♥r ♦t♠♦s ♠ rt ♦♥t ♥r −λA/2 ♥tã♦

t③♥♦ ♦ ♦r λ st♠♦ ♠ ♦t♠♦s ♦ ♦r A

♣♦ss ♦s ♦rs A λ ♣♦♠♦s r ♦ ♦r Γ ♣r ♠♦♦

♣rs♥t♠♦s ♦s ♦rs sss ♣râ♠tr♦s ♥ t s rr♦s ♦r♠ ♣r♦♣♦s

t③♥♦ ♦ ♠ét♦♦ r Pr ♦ ♠♦♦ sss ♣râ♠tr♦s á ♠

s♦ ♦s ♠ ❬❪ ♦♥ ♦ ♥♦♥tr♦ λ = −0, 414 A = 1, 2005 Γ = 0, 66144

♦ ♥♦ss♦s rst♦s λ é ♦ ú♥♦ q ♥ã♦ é ♦♥sst♥t ♦♠ ♦ ♦r ♦t♦ ♠ ❬❪

①♣♦♥t rs♠♥t♦

0,1 0,3

t-0,24

0,315

0,325

d<h>/dt

163264128256512

RSOS

0 0,02

L-(2+2α)

0

0,004

0,008

v(L

) -

v∞

RSOS

r ❱♦ rs♠♥t♦ ♣r r♥ts t♠♥♦s sst♠ ♥tr é ♦ ♦r v∞ rá♦ ∆v ♠ ♥çã♦ 1/L2−2α ♦♥t ♥rss rt é −λA/2

♥♦ ♥t♥t♦ s ♦ t♦r tss ♦ s rrs rr♦ ♣r♦♠♥t ♦s rst♦s

tr♠ ♠ s♦r♣♦sçã♦

❱ rsstr q ss ♣râ♠tr♦s ♦r♠ ♦s ♠ sst♠s q ♥ã♦

rs♠ s t③r♠♦s ♣♥s ♦ Γ

v∞ λ A Γ

t♥

♥st♣

❱♦rs ♥s ♦s ♣râ♠tr♦s ♥ã♦ ♥rss ♦s ♠♦♦s t♥ ♥st♣ s ♦rs v∞ ♦r♠ ①trí♦s ❬❪

①♣♦♥t rs♠♥t♦

♠s♠ ♦r♠ q ♦ t♦ ♥♦ ♣ít♦ ♥tr♦r ♠♦s ♥sr ♦♥rê♥

♦ ①♣♦♥t rs♠♥t♦ ♣r rr s ♦ rs♠♥t♦ tr t ss

♥rs ♥tr s rs ♣rs♥t♠♦s sss

rst♦s ♣r ♦s ♠♦♦s ♥st♣ t♥ rs♣t♠♥t ♦♠♦ ♦♦rr

♠ ♠ ♠♥sã♦ ♦♥rê♥ ♦ ①♣♦♥t ♣♥ ♦ tr Pr

♦ ♦ sstrt♦ t♠ ♠ t♠♥♦ r♥t ♠ ♠ ♠s♠♦ ♥st♥t

strçã♦ tr

t♠♣♦ ss♠ ♦♠♦ ss ①♣♦♥t s♦r t♦s t♠♥♦ ♥t♦ é ♥tr q

♦♥rê♥ ss ①♣♦♥t ♣♥ ♦ é♠ ss♦ ss♠ ♦♠♦ ♠ ♠

♠♥sã♦ ♠ ①str ♦rrçõs ♦r♠ s♣r♦r q ♣♥♠ ♦

rs♠♥t♦ tr s rá♦s ♠♦str♠ q ♦s ①♣♦♥ts t♥♠ ♣r ♠ ♦r ♣ró①

♠♦ β ≈ 0, 24 ss P❩ ss s♦ ♦♠♦ ♦ sst♠ rs ♦♠ ♦ qr♦

♦ t♠♣♦ t♦r♥s ♥á ♥çr t♠♣♦s ♠t♦ r♥s ♣♦r ss♦ ♥ss ♠♥s♦

♥ ♣r ♥s vws ♦s ①♣♦♥ts s ♥♦♥tr♠ ♦♥ ♦ s♣r♦ ♠s♠♦ ♥♦s

♠♦rs t♠♣♦s s♠♦s P♦♠♦s ♥♦tr q ♣r ♦ ♠♦♦ t♥ sss ①

♣♦♥ts s ♥♦♥tr♠ ♠s st♥ts q ♥♦s ♦tr♦s ♠♦♦s ♦ q ♦♦rr ♣♦rq ss

♠♦♦ s♦r ♠s t♦s t♠♥♦ ♥t♦ q ♦s ♠s

3 4 5 6 7ln(t)

0,225

0,235

0,245

0,255

β

vw=2

vw=4

vw=6

vw=10

Single-step

3 5ln(t)

0,24

0,3

β

vw=10

vw=6

vw=4

vw=2

Etching

3 5 7ln(t)

0,235

0,245

0,255

β

vw=2

vw=4

vw=6

vw=10

RSOS

r ①♣♦♥t rs♠♥t♦ ♠ ♥çã♦ ♦ t♠♣♦ ♣r r♥ts ♦s ♦s♠♦♦s ♥st♣ t♥ ♥ tr r♣rs♥t ♦ ♦r β = 0, 24q é ♦ ♦r ♦ ①♣♦♥t ss P❩ ♠ s ♠♥sõs

strçã♦ tr

strçõs trs

♥s♥♦ s strçõs trs ♥ss ♠♥s♦♥ ♥♦t♠♦s q

ss♠ ♦♠♦ ♠ ♠ ♠♥sã♦ ♦s ♠♥ts s ♦s ♦♠í♥♦s rs♥ts t♥♠

♣r ♦rs ♣ró①♠♦s ♦s rst♦s ♥♠ér♦s r♦♥♦s à ♥trs rs ♦♠♦

♥ã♦ ①st♠ rst♦s ♥ít♦s s♦r strçã♦ χ ♠ s ♠♥sõs ♥ós

t♥t♠♦s st♠r ♥♠r♠♥t ♦s ♠♥ts ss strçã♦ ♦♠ s ♠ ♥♦ss♦s

rst♦s st♠t s♥ss rt♦s ♦ t trç♥♦ ♦ rá♦ sss

r♥③s ♠ ♥çã♦ t−2β ①tr♣♦♥♦ ♣r t → ∞ s rs

str♠♦s sss ①tr♣♦çõs ♣r s♥ss rt♦s ♦ ♠♦♦ ♥st♣

r♥♦ ♠ ♠é s ①tr♣♦çõs ♠ r♥ts ♦s ♠♦s ♦ ♦r

s♥ss rt♦s strçã♦ ♦rrs♣♦♥♥t ♠♦♦ t

♠♦str♠♦s sss rst♦s s rrs rr♦ ♦r♠ st♠s ♣♦ s♦ ♣rã♦ ♦s

rst♦s s ①tr♣♦çõs ♣r r♥ts ♦s

0 0,04 0,08 0,12

t(-2x0.24)

0,25

0,35

Skew

nes

s

KPZ 2d planoKPZ 2d curvov

w=10

vw

=6

vw

=4

Single-step

0 0,1 0,2

t(-2x0,24)

0,1

0,3

Curt

ose

KPZ 2d planoKPZ 2d curvov

w=10

vw

=6

vw

=4

Single-step

r st♠t s♥ss rt♦s strçã♦ trs s♥♦♦ ♠♦♦ ♥st♣

Pr ♥sr ♠é strçã♦ ♦♠♦ ♥ã♦ s♠♦s ①t♠♥t ♦ ♦r

〈χ〉 ♠ s ♠♥sõs ♥ós ♥♠♥t ♦♥sr♠♦s ♦ ♣r♦t♦ Γβ〈χ〉 = g1 Pr

strçã♦ tr

st♠r ss ♣r♦t♦ ♥ós r♠♦s qçã♦ ♠♥♣♠♦s ♣r ♦tr

∂t〈h〉 − v∞β

= s(λ)Γβ〈χ〉tβ−1 .

ss ♦r♠ s trç♠♦s ♠ rá♦ (∂t〈h〉 − v∞)/β ♠ ♥çã♦ tβ−1 ♠♦s

♦tr ♠ rt ♦♠ ♦♥t ♥r g1 = Γβ〈χ〉 ♦ ♥t♥t♦ ♦♠♦ ♣♦♠♦s r

♥ r ♦♠ ír♦s r ♥ã♦ ♦t♠♦s ♠ ♦ rt ♦ q ♥♦s ♥

q ♠ s ♠♥sõs t♠é♠ ①st♠ ♦rrçõs ♦rts P♦ré♠ ♣r ♦tr sss

♦rrçõs ♣rs♠♦s ♦ ♦r g1 ♦♠♦ ♦ t♦ ♥ sçã♦ ♦ q ♥♦s ① ♠

♠ ♠♣ss

0 0,2 0,4

tβ-1

0

0,4

0,8

g1

tβ-1

RSOS vw

= 6

1 100t

0,001

0,1

-ζtγ /β

RSOS vw

= 6

r rá♦s ♣r ♦ ♦ ♦ ♦r g1 r ♦♠ qr♦s ♦ trçs♠ ♦♥srr ♥♥♠ ♦rrçã♦ ♦♠ ír♦s ♣ós ♥♦ ♣ss♦s trçã♦ rá♦s♣r tr♠♥çã♦ ♦rrçã♦ r ♦♠ qr♦s ♦ trç ♦♥sr♥♦ ♦ ♣r♠r♦g1 ♦ ♦♠ ír♦s t③♥♦ ♦ ♦r ♥ g1

s♦çã♦ q ♥♦♥tr♠♦s ♣r ss ♣r♦♠ ♦ t③r ♠ ♠ét♦♦ t♦

♦♥sst♥t ós st♠♦s ♠ rt ♥ rã♦ t → ∞ ♦ rá♦ ♦t♠♦s

♠ ♦r ♣r♦sór♦ ♣r g1 t③♥♦ ss ♦r ♥ós ♠♦s ♦rrçã♦ ζt−γ

♦♠♦ ♦ ♦r g1 ♥ã♦ é ♦ ♦rrt♦ ♥tr♠♥t r ♦rrçã♦ t♠é♠ ♥ã♦

♣rs♥t ♠ ♣♦tê♥ ♠ ♥ ♦♠♦ ♠♦str♠♦s ♥ r ♣♦♥t♦s

rrs r s♠♦ ss♠ ♥ós st♠♠♦s ♠ ♦r♠ ♣r♦sór ♣r

♦rrçã♦ st♥♦ ♠ ♣♦tê♥ ♠ t♠♣♦s ♣q♥♦s ♠ s t③♠♦s

ss ♦rrçã♦ ♣r r ♠ ♥♦♦ ♦r g1 ♦♠ ss ♥♦♦ ♦r r♠♦s

strçã♦ tr

♦rrçã♦ ♦♥t♥♠♦s ♥♦ sss q♥ts té s ♣rr♠ rr

s♥t♠♥t ♥s trçõs Pr s ♦s q st♠♦s ♣r♦①♠♠♥t

♥♦ ♣ss♦s ♦r♠ ♦ s♥ts ♣r sss ♦rs ♦♥rr♠ ♦s rá♦s

♠♦str♠♦s r ♥ g1 ♦rrçã♦ ♠s sã♦ r♣rs♥ts ♣s

rs ♦♠ ♣♦♥t♦s qr♦s ♦t ♦♠♦ t♦ ♦ rá♦ qr♦s

♣♦ss ♠ ♦ rt qr♦s s ♠ ♣♦tê♥ ♠ ♥

♣ós s trçõs q ss♠ ♦♠♦ ♠ ♠ ♠♥sã♦ é ♣r♦①♠♠♥t t−1

♠ s ♠♥sõs ♦t♠♦s ♥♦♠♥t ♠ ♦rrçã♦ ♦ t♣♦ ζ ln t ♠

q ♦s rá♦s s rs ♥ós ♠♦str♠♦s q ss♠

♦♠♦ ♠ ♠ ♠♥sã♦ ζ ♥ã♦ r s♥t♠♥t ♦♠ ♦ tr ♣r

♥♥♠ ♦s ♠♦♦s é♠ ss♦ ss ♦rrçã♦ ♥ã♦ ♣r ♥ râ♥ ♣♦r ss♦

st♠♦s ♦♥sr♥♦ q ζ é ♠ rá tr♠♥íst

0 4 8vw

-0,38

-0,34

-0,3

-0,26

ζ

RSOS

4 8vw

0,35

0,4

0,45

ζ

Etching

0 4 8

-0,45

-0,35

-0,25

ζ

Single-step

r rá♦s ζ ♣r s r♥ts ♦s ♦s ♠♦♦s t♥

♥st♣

♦s rá♦s ♣r ♦ ♠♦♦ t♥ ♥ã♦ ♦♦♠♦s ♦s rst♦s ♦rrs♣♦♥

strçã♦ tr

♥ts às ♦s vw = 1 2 ss♦ ♣♦rq ♦♠♦ ♠♦str♦ ♥ r s

♦rrçõs ζ ln t s ♦r♠ s♣r♦r q ♥ós s♦♥sr♠♦s té q ♣r♠

♦♠♣tr ③♥♦ r ♠ r③r ♦ ③r♦ rss ③s ss ♣r♦♠ ♦♥

t ♠s♠♦ s t③r♠♦s ♦ ♦r ♥ g1 ♦t♦ ♦♠ ♦trs ♦s ss t♦

♥♦s ♠♣ r ♦ ♦r ζ 〈η〉 ♣r sss ♦s P♦♠♦s ♥♦tr q

t♦ s ♦rrçõs ♠ ♣q♥s ♥sss ♦s q♥♦ t♥t♠♦s r ♦s

♦rs g1 r ♠♦str♠♦s ♦ rá♦ ss á♦ ♣r vw = 2 ír

♦s vw = 6 qr♦s s♠ ♦♥srr ♥♥♠ ♦rrçã♦ ss r ♥♦t♠♦s

q ♦ rá♦ ♣r vw = 2 ♠ ♠s ♥r q ♦ ♣r vw = 6 ♦♠♦ s ♦rrçõs

sã♦ ♣q♥s ♥s ♦s vw = 1 2 ♥ã♦ t♠♦s ♣r♦♠s ♣r r ♠é

strçã♦ s♠ ♦♥srás

0 200 400t

-0,002

0

0,002

0,004

ζ t

-1

Etching vw

= 2

0 0,2 0,4

tβ−1

-3

-2

-1

0

(d<

h>

/dt

- v

∞)/

β

Etching

r rá♦ q ♠♦str ♠♣♦ss s r ζ ♣r ♦vw = 2 r g1t

β−1 ♣r s ♦s vw = 2 ír♦s vw = 6 qr♦s

ós t♠é♠ ♠♦s ♦ s♦ 〈η〉 strçã♦ ss♦ ♦ t♦ ♠♥r

♥á♦ ♦ s♦ ♥♠♥s♦♥ srt♦ ♥ sçã♦ ss♠ ♦♠♦ ♠ ♠ ♠♥sã♦

sss ♦rs ♥ã♦ r♠ s♥t♠♥t ♦♠ ♦ ♦♠♥t ♥ã♦ s♠♦s

s η é tr♠♥íst♦ ♦ st♦ást♦ ss r♥③ ♦r st♦ást ♦ á♦ q ♥ós

r③♠♦s tr♠♥ ♣♥s ♦ ♦r ♠é♦ ss rá

♥♦ ♦ ♦r Γ ♥ós ♠♦s ♦ ♦r 〈χ〉 s♥♦ ♦s rst♦s ♣r

g1 sss ♦rs stã♦ s♣♦st♦s ♥ t ♦♠♥t ♦ tr ♠é ♦

s♦ ♣rã♦ ♣r s r♥ts ♦s ♠ ♠ ♠s♠♦ ♠♦♦

strçã♦ tr

0 4 8vw

-2,2

-1,8

<η

>

RSOS

4 8vw

4,6

5

<η

>

Etching

0 4 8vw

-1,6

-1,2

<η

>

Single-step

r rá♦s 〈η〉 ♣r s r♥ts ♦s ♦s ♠♦♦s t♥

♥st♣

♣♦ss s ♦rrçõs ♣♦♠♦s r t♠é♠ r③ã♦ R = 〈χ〉c/〈χ2〉0,5c

ss st♠t ♦ t ♠s♠ ♦r♠ q s♥ss rt♦s ♣♦ré♠ t③♥♦

t−β ♣r ①tr♣♦çã♦ s rst♦s stã♦ ♥ t

♥s♥♦ t ♥♦t♠♦s q ♦s ♥♦ss♦s rst♦s stã♦ ♦r♦

♦♠ ♦s ♦t♦s ♠ ❬❪ ♣r ♦♠tr r ss P❩ ♠ s ♠♥sõs

♥s♥♦ ♠r♠ ♠ q ♦s rst♦s ❬❪ r♠ ♥♦t♠♦s q 〈χ〉c r♥tr −2, 22 −2, 41 〈χ2〉c ♥tr 0, 304 0, 377 ♥tr 3, 86 4, 09 ♥ss ♥tr

0, 303 0, 348 rt♦s ♥tr 0, 201 0, 230 á ♠ ❬❪ ①st♠ rst♦s ♣r

♦s ♠♦♦s ♠ ♦♠tr r ♦s ♦rs ♥ss ♥♦♥tr♦s ♦r♠ 0, 32

0, 339 ♦s rt♦s 0, 21 0, 20 ♦♠♣r♥♦ ♦s rst♦s ♥tr♦rs ♦♠ ♦s

♦♠♣rçã♦ ♦♠ ♦ s♦ ♥♠♥s♦♥

〈χ〉c 〈χ2〉c ♥ss rt♦s

t♥

♥st♣

é râ♥ ♠ó♦ ♥ss rt♦s strçã♦ trs♦s ♦♠í♥♦s rs♥ts ♠ s ♠♥sõs sss ♦rs ♦r♠ ♦s tr♥♦ ♠é♦ rst♦ ♣r s r♥ts ♦s

t ♥♦t♠♦s q ♥♦ss♦s rst♦s ♣r ♦s r♥ts ♠♦♦s r♠ ♥tr♦

♠ ♠r♠ ♠♥♦r q ♦s st♠♦s ♠ ❬❪ ♠ ❬❪ ss ♦r♠ t③♥♦

sss ♠♦♦s srt♦s ♦♠tr ♣♥ ♠ sstrt♦s rs♥ts ♥ós ♦♥s♠♦s

st♠r ♦s ♠♥ts strçã♦ r ♠♥s♦♥ ss P❩ ♠♥r

♠s ♣rs q ♦s rst♦s trtr ♦t♦s t③♥♦ ♥trs ♦♠ rtr

♦♠♣rçã♦ ♦♠ ♦ s♦ ♥♠♥s♦♥

♦♠♦ ♦ ♠♦str♦ ♥ss ♣ít♦ s strçõs trs ♦s ♦♠í♥♦s rs

♥ts ♠ s ♠♥sõs t♠é♠ ♦♥r♠ ♣r ♦ s♦ r♦ é♠ ss♦ s

t♠é♠ ♣♦ss♠ ♠ ♦rrçã♦ ♦rít♠

♥t q ♠ s ♠♥sõs s ♣çõs t♠é♠ trã♦ ♦ tr♠♦

〈(~∇h)2〉 s③♥♦ s♣rí ss♠ sssã♦ r♦♥ ♦s t♦s s ♣

çõs r③ ♥ sçã♦ t♠é♠ é á ♥♦ s♦ ♠♥s♦♥

❯♠ á♦ ♥á♦♦ ♦ t♦ ♥ sçã♦ ♣♦ sr t③♦ ♣r ♠♦strr

♦r♠ ♦rrçã♦ ♦rít♠ ♥♦ q ♦ ♥ú♠r♦ sít♦s ♠ s ♠♥sõs é

L1L2 q vw(L1 + L2) sít♦s sã♦ ♦♥♦s ♠ ♠é ♠ ♠ ♥ t♠♣♦

♣♦♠♦s ③r s ♠s♠s ♦♥srçõs ♥♦ r qçã♦ ♦tr

K

L1L2 + vw(L1 + L2)=

K

L1L2

(

1− vw(L1 + L2)

L1L2

)

.

♦♠♦ L1 ≈ L2 ≈ vwt s t ≫ 1 t♠♦s

vw(L1 + L2)

L1L2

≈ 2v2wt

v2wt2∼ t−1 .

♦♠♣rçã♦ ♦♠ ♦ s♦ ♥♠♥s♦♥

♦♥t♥r♠♦s ♦ á♦ t♠é♠ ♠♦s r

∂〈h〉∂t

=λ

2〈(~∇h)2〉G ≈ λ

2〈(~∇h)2〉F − λµ

t.

ss♠ ♠ s ♠♥sõs t♠é♠ ♠♦s tr ♠ ♦rrçã♦ q s ♦♠ ♦ ♦

rt♠♦ ♦ t♠♣♦ ♥ qçã♦ ♣r s trs q

r é ♥t q ♦ tr♠♦ ζ ln t ♣rr ♠ qqr ♠♥s♦♥

♥t♦ ♥ q q♥t♦ ♠ ♥ rçã♦ ♥tr♦ ♦ ♣rê♥ts ♦ ♦ rt♦

qçã♦ t♠♦s ♦ ♥ú♠r♦ ♦♥s ♦♥s ♠ ♠ ♥ t♠♣♦ ♦

♣♦ t♠♥♦ ♦ sst♠ ♦ s t① rs♠♥t♦ tr ♦ sst♠

♣♦ t♠♥♦ ♦ sst♠ ♠ ♠ sst♠ d ♠♥s♦♥ ♦ ♥ú♠r♦ sít♦s s

♦♠ td ss♠ t① rs♠♥t♦ s ♦♠ td−1 ss ♦r♠ é ♥tr q

rçã♦ sss r♥③s s ♦♠ t−1 ♠ qqr ♠♥s♦♥

♣ít♦

♦♥sõs Prs♣ts

st ssrtçã♦ ♦r♠♦s ♦r♠ t ♥â♠ ♥trs ♠

sstrt♦s q rs♠ tr♠♥t ♦♠♦ sss sst♠s ♣♥♦s q rs♠ t

r♠♥t tê♠ s♦ s♦s ♦♠♦ ♠ ♣r♦①♠çã♦ ♣r ♥trs rs ❬❪ ♦

♦t♦ ♣r♥♣ ♦ ♥♦ss♦ tr♦ ♦ tr♠♥r s ss t♣♦ sst♠ é ♠ ♦

♣r♦①♠çã♦ ♣r ♥trs rs

ós ♣r♦♣s♠♦s ♠ ♠♦♦ ♦♠♣t♠♥t st♦ást♦ ♣r ♠♣♠♥tr

♣♦sçõs ♠ sstrt♦s q rs♠ tr♠♥t ♦♠ ss ♠♦♦ ♥ós rs♠♦s

♥trs ♠ ♠ s ♠♥sõs

st♥♦ ♠♦♦s srt♦s ss P❩ ♥sss ♦♠í♥♦s rs♥ts

♠♦str♠♦s q strçã♦ tr sss sst♠s t♥ ss♥t♦t♠♥t ♣r

strçã♦ ♦rrs♣♦♥♥t ♥trs rs t♥t♦ ♠ ♠ q♥t♦ ♠ s

♠♥sõs é♠ ss♦ st♠♠♦s ♠♥r r③♦♠♥t ♣rs ♦s ♣r♠r♦s ♠

♥ts strçã♦ rá χ ♦rrs♣♦♥♥t ♥trs rs ♠♥s♦♥s

sss rst♦s ♠♦str♠ q rtr ♦ ♥ã♦ é ss♥ ♣r ♣r♦çã♦

♥trs ♦♠ strçõs trs q sã♦ ss♦s rs♠♥t♦s r♦s

st♥♦ ♦s t♦s ♦ rs♠♥t♦ tr ♥tr ♥ós ♠♦str♠♦s q

s ss rs♠♥t♦ é r③♦ s③♥♦ ♥tr ss s③çã♦ t ♦ tr♠♦

♥ã♦ ♥r (~∇h)2 qçã♦ P❩ ❯♠ ♦s t♦s ss ♥ô♠♥♦ é trr t①

rs♠♥t♦ s trs ③♥♦ ♦ tr♠♦ (Γt)β〈χ〉goe + η → (Γt)β〈χ〉gue + η′ + ζ ln t

qçã♦ é♠ ss♦ ♠♦str♠♦s ♥t♠♥t q ♠ tr♠♦ ζ ln t

♦♥sõs Prs♣ts

♣rr ♥sss sst♠s q rs♠ tr♠♥t

♥s♠♦s ♠ ts s ♦rrçõs ♥♦s ♥ ♥â♠ sss sst♠s

rs♥ts ♠ ♠ s ♠♥sõs ♠♦s ♦ s♦ 〈η〉 strçã♦ ♦ ♦r ζ ♦rrçã♦ ζ ln t q ♣r♠ ♥sss sst♠s ♦t♠♦s q sss ♦rrçõs

♥ã♦ ♣♥♠ ♦r♠ s♥t ♦ rs♠♥t♦ tr ♦ ♥t♥t♦

rsst♠♦s q ♠ r ♦rrçõs ♦r♠ ♠♦r q é♠ st♦ásts

♠ ♣♥r ♦ ♠♥t♦ ♦ sstrt♦

♦str♠♦s t♠é♠ q ♠ ♠ ♠♥sã♦ s ♦ t♠♥♦ ♥ ♦r r♥

strçã♦ tr t♥ ♥♠♥t ♣r ♦ ♥t♥t♦ ♠s♠♦ s ss

t♠♥♦ ♥ ♦r ♠t♦ r♥ ♣ós ss tr♥s♥t ♥ ♦♥ t♥ ♣r

strçã♦ trs ♦♥r ss♥t♦t♠♥t ♣r ❯

♠♦s ♦♠♦ ♣rs♣ts trs ♣r ♦ ♥♦ss♦ ♠♦♦ rs♠♥t♦s ss

P❩ ♠ ♠♥sõs s♣r♦rs ③ s ♣♦ssí st♠r ♦♠ ♣rsã♦ ♦s ♣r♠r♦s

♠♥ts s strçõs trs s ♥trs rs ♠ três ♦ ♠s ♠♥sõs

tr ♣♦ss sr t③r ♦ ♠♦♦ ♣r♦♣♦st♦ ♠ rs♠♥t♦s ♦trs

sss ♥rs ♥ã♦ ♥rs ♣♦r ①♠♣♦ ss ❱♥s r♠

❱ ss ♦r♠ ♠♦s ♣♦r ♥tr s ①st ♠ r♥ç ♥tr s

strçõs ♦s sst♠s sstrt♦ rs♥t ♦s t♠♥♦ ①♦ ♦r

♣♦♠♦s t♥tr r ♥♠r♠♥t ss strçã♦ r♦♥ ♦s sstrt♦s

rs♥ts ♣♦ss♠♥t ♦♥r q st r♦♥ ♥trs rs

rs♣t ss ♥rs

♣ê♥

♠t ss♥♦ strçã♦

♥♦♠

♦♥sr♥♦ strçã♦

p(h) =N !

h!(N − h)!

(

1

L

)h(L− 1

L

)N−h

,

♥♦ r♠ ♠ q N → ∞ s trs t♠é♠ sã♦ ♠t♦ r♥s ss♠ ♣sr

s trs rr♠ srt♠♥t ∆h = 1 é ♠ rçã♦ ♠t♦ ♣q♥ q♥♦

♦♠♣r à h ♣♦r ss♦ ♣♦♠♦s ♦♥srr q p(h) é ♠ ♥çã♦ ♦♥tí♥ h

Pr r ♦♠ ♦s t♦rs ♥ss r♠ ss♥tót♦ é ♦♥♥♥t ♣r ♦ ♦

rt♠♦ ♦s ♦s ♦s ♦ q rst ♠

ln p(h) = lnN !− lnh!− ln(N − h)! + h ln

(

1

2

)

+ (N − h) ln

(

L− 1

L

)

P♦♠♦s ♠♥r ♦s t♦rs t③♥♦ ①♣♥sã♦ ss♥tót tr♥

lnN ! = N lnN −N +O(lnN)

çã♦ ss ①♣♥sã♦ ♣♦ sr ♥♦♥tr ♠ ❬❪


❯s♥♦ ss ♦r♠ ♠ t♠♦s

ln p(h) = N lnN−h lnh−(N−h) ln(N−h)+h ln

(

1

2

)

+(N−h) ln

(

L− 1

L

)

.

Pr ♦tr♠♦s ♦ ♠t ss♥♦ strçã♦ ♥♦♠ ♠♦s ①♣♥r ss

♦rt♠♦ ♠ t♦r♥♦ s ♦r ♠á①♠♦ q ♦♥ ♦♠ ♦ ♦r ♠á①♠♦ p(h)

♣♦s ♦ ♦rt♠♦ é ♠ ♥çã♦ ♠♦♥♦t♦♥♠♥t rs♥t ♥♦

∂ ln p(h)

∂h= − lnh+ ln(N − h) + ln

(

1

2

)

− ln

(

L− 1

L

)

,

♥♦s ♣♦♥t♦s rít♦s t♠♦s

ln(N − hm)− lnhm + ln

(

1

2

)

− ln

(

L− 1

L

)

= 0 ,

ln

(

N − hm

hm

)

= ln(L− 1) ⇒ hm =N

L.

Pr tr♠♥r s ss ♣♦♥t♦ rít♦ é ♠ ♠á①♠♦ ③♠♦s

∂2 ln p(h)

∂h2

∣

∣

∣

∣

∣

h=hm

= − 1

h+m− 1

N − hm

< 0 ,

ss♠ hm é t♦ ♠ ♠á①♠♦

♦♠♦ é ♦ ♥ú♠r♦ t♦t ♣rtís ♣♦sts t♠♦s q N =∑L

i=0 hi

ss♠

hm =N

L=

1

L

L∑

i=0

hi = h .

♦♦ ♦ ♠á①♠♦ ♦♥ ♦♠ ♠é ♦♠♦ ♠ ♠ strçã♦ ss♥

♦r ♠♦s ①♣♥r ln p(h) ♠ sér ②♦r ♠ t♦r♥♦ ♦ ♠á①♠♦

ln p(h) = ln p(hm)+∂ ln p(h)

∂h

∣

∣

∣

∣

∣

h=hm

(h−hm)+∂2 ln p(h)

∂h2

∣

∣

∣

∣

∣

h=hm

(h− hm)2

2+ . . . .


♦ ♠á①♠♦ r ♣r♠r s ♥ ♠♦s ♦♠

ln p(h) = ln p(hm)−L2

2N(L− 1)(h− hm)

2 .

srt♥♦ ♦s tr♠♦s ♦r♠ ♠♦r t♠♦s

p(h) = C exp

( −L2

2N(L− 1)(h− hm)

2

)

.

r♦♠♦s p(hm) ♣♦r C ♣♦rq s rss ♣r♦①♠çõs q ③♠♦s ♥♦s ♦r♠

r♥♦r♠③r strçã♦ ss ♦r♠ ♠♦s

∫ ∞

−∞C exp

( −L2

2N(L− 1)(h− hm)

2

)

dh =

√

π2N(L− 1)

L2,

q ♠♣ ♠ C = L/√

2πN(L− 1) ss ♦r♠ strçã♦ ♥♦ ♠t ss♥tót♦

s t♦r♥

pRD(h) =

√

L

2πt(L− 1)exp

( −L

2t(L− 1)(h− hm)

2

)

,

♦♥ s♠♦s q N = Lt ♦♥sr♠♦s q ♦ sst♠ é ♠t♦ r♥ tr♠♦s

pRD(h) =1√2πt

exp

(−(h− t)2

2t

)

,

♦♥ t③♠♦s q 〈h〉 = N/L = t

♣ê♥

❯♠ ♣r♦♣r s♠♣s ♥çõs

♣rós

♦♥sr ♠ ♥çã♦ s h(x) ♥ ♠ ♠ ♥tr♦ ♦♠♣r♠♥t♦ L

q stsç ♦♥çõs ♦♥t♦r♥♦ ♣rós ts q h(x) = h(x + L) P♦♠♦s

rr ss ♦♥çã♦ rss ③s ♦♥r q ♣r qqr ♦r n s♥t

qçã♦ é ádnh(x)

dxn=

dnh(x+ L)

dx.

P♦♠♦s ♠♦strr q ♠é⟨

dnhdxn

⟩

é ♥ ③♥♦

⟨

dnh

dxn

⟩

=1

L

∫ a+L

a

dnh

dxndx =

1

L

(

dn−1h

dxn−1

∣

∣

∣

∣

∣

a+L

− dn−1h

dxn−1

∣

∣

∣

∣

∣

a

)

,

q ♦r♦ ♦♠ é ③r♦

rê♥s ♦rás

❬❪ ❩♥ ❨ ❩♥ P str♦♠ ♥s♥♦♥ ♦rst r ②

♣♣rr♥♥ ①♣r♠♥t r③t♦♥ ♦ t ♥tr r♦t ♠♥s♠

P②s

❬❪ r♥ ♥③s♦r♥♦ ♦t♦♣♦♥t♥♦s ♠t♦♥

①♣r♠♥t ♥ s♣♣r ts P②s ttrs

❬❪ ♠♥t♦ rrr rrrt ♥♦♠♦s s♥ ♥ t

♣t① r♦t ♦ s♠♦♥t♦r ♠s r♦♣②ss ttrs

❬❪ tsst s♦♥♠t r♦t ♥ tr ♦♦♥② ♦r♠

t♦♥ P②s

❬❪ ♠rrt ♣ ♦ ♥r ♠♦r r♦t ♦♠str② ♥ ♦

♣②ss

❬❪ rs t♥② rt ♦♥♣ts ♥ r r♦t ♠r

♠r ❯♥rst② Prss

❬❪ ❳ t ❲ ♦♠♣s♦♥ ❱ P ♥ r ❱♠♥

ts P st♥♦♣♠♥t ♦ t♥ ♠s ♦♥ ① sstrts

r P②s

❬❪ ♠ rrr s ♥ô♠ t ♥í♦s ❯♥

rs é♠ ♥râ♥ s ♠ ♠s ♥♦s

♦♥♦r ❯♥rs r ❱ç♦s

rê♥s ♦rás

❬❪ ♠ rrr r s ❯♥rs

tt♦♥s ♥ t r♦t ♦ s♠♦♥t♦r t♥ ♠s r❳

❬❪ ♥ Pr♦ t r② ②♠♣♦s♠ ♦♥ t♠t ttsts ♥ Pr♦

t② ②♠♥ r② ❯♥rst② ♦ ♦r♥ Prss

❬❪ Prä♦r ♣♦♥ ❯♥rs strt♦♥s ♦r r♦t Pr♦sss ♥

♠♥s♦♥s ♥ ♥♦♠ trs P②s ttrs

❬❪ r♦ss♦r r♦♠ r♦♥ t♦ tt♦♥r② ♥trs ♥ t rr

Prs❩♥ ss P②s ttrs

❬❪ sr♦ ②♥♠s ♦ r ♥trs ♥♥s ♦ P②ss

❬❪ P r tr t♥② r strtr ♥ ♥s♦tr♦♣② ♦ ♥

strs♦r♥ ♦ P②ss

❬❪ rs ❲♦ ♥s♦tr♦♣② ♥ s♥ ♦ ♥ strs ♥ t♦ ♥ tr

♠♥s♦♥s ♦r♥ ♦ P②ss

❬❪ ❲ ♥♥♥ ❨ ❲♥ tt r ♥ str r♦t ♥ t♦ ♥

tr ♠♥s♦♥s ♦r♥ ♦ ttst ♥s P

❬❪ ❨ ❲♥ P ss♥tt tt ♥ str r♦t

♠♦ ♦r♥ ♦ P②ss

❬❪ ♥ P ♥ ♠♣ tr♠♥s♦♥ ♠♦s ♦r st ♣♦st♦♥

t rstrtr♥ r♦♣②ss ttrs

❬❪ sr♦ t♦st r♦t qt♦♥s ♦♥ r♦♥ ♦♠♥s ♦r♥ ♦ t

tst ♥s P

❬❪ s♦ ♦ss♥ ♦ ♦rr♠ ♦♥ tts

t ♥②ss ♦ r ♥tr r♦t ♦r♥ ♦ ttst ♥s

rê♥s ♦rás

❬❪ Pst♦r ♥♦ ②♥♠ s♥ ♥ ♥rs♥ s②st♠s ♥tr

r♦♣♥ ♦r♥ ♦ P②ss

❬❪ ♥r♦t rt ♦♠tr② t s t ♥ t ♦s t ♦ Pr♦

♥s ♦ t ♦② ♦t②

❬❪ sss ❱ r♥ ♦t ♥r st♦

♦♠tr rt ♣r♦♣rs rtrísts rts s st

rsr ♥s♥♦ ís

❬❪ P ♥ P ♠♥ ♥r st ♣♦st♦♥ ♦♥ srs

P②s

❬❪ ♦ s r srt t♦♠st ♠♦ t♦ s♠t

t♥ ♦ r②st♥ s♦ P②s

❬❪ ♣♥② ♠♥s♦♥ rt P♦②♠r ♥ ♥♦♠ ♠

♥ P♥♦♠♥ ♥ ❯♥rs strt♦♥s P②s ttrs

❬❪ ♣♥② ①tr♠ ♣ts t st♦st t qt♦♥ ♥ t tr

♠♥s♦♥ rrPrs❩♥ ♥rst② ss P②s

❬❪ r s rrr rrPrs❩♥ ♥rst②

ss ♥ ♠♥s♦♥s ❯♥rs ♦♠tr②♣♥♥t strt♦♥s ♥

♥tt♠ ♦rrt♦♥s P②s

❬❪ sr♦ ②♥♠ ♥ ♦ ♦♥♥ ♥trs P②s

ttrs

❬❪ r ♦♠♠♥t ♦♥ ②♥♠ ♥ ♦ ♦♥♥ ♥trs P②s

ttrs

❬❪ sr♦ ♦♠♠♥t ♦♥ ②♥♠ ♥ ♦ ♦♥♥ ♥trs P②s

ttrs

rê♥s ♦rás

❬❪ ♥ Prsst♥ ♦ sr tt♦♥s ♥ r② r♦♥ srs

♦r♥ ♦ ttst ♥s P

❬❪ ♠② ❱s ♥ ♦ t t ③♦♥ ♥ t ♥ ♣r♦ss ♦♥ ♣r♦t♦♥

♥t♦rs ♥ t st ♣♦st♦♥ ♠♦ ♦r♥ ♦ P②ss t♠t

♥ ♥r

❬❪ rr Prs ❨ ❩♥ ②♥♠ ♥ ♦ r♦♥ ♥trs

P②s ttrs

❬❪ s r rrr ♦♥♥rs ♣r♠trs ♦rrt♦♥s

♥ ♥rst② ♥ rrPrs❩♥ r♦t ♦r♥ ♦ ttst ♥

s P

❬❪ ♥ ♦tt ♥ ♣r♦♣rts ♦ t sr ♦ t ♥ ♠♦ ♥

♦r♥ ♦ P②ss

❬❪ ♠② ♥ ♦ r♦ srs ts ♦ sr s♦♥ ♦r♥ ♦

P②ss

❬❪ ♠ ♦strt③ r♦t ♥ strt ♦♦♥♦ ♦

P②s ttrs

❬❪ ❨ ❨ P♥ ♣♥②♦♥s t♦♥ ♦ t s♥ ♥

t♦♥ ①♣♦♥♥ts ♥ ♣r♦t② ♥t♦♥ ♦ t rs❲♥s♦♥ tt♦♥

t ♦rrt ♥♦s P②s

❬❪ r♥ t♦♥♦♥tr♦ r♦t ♦♥ ♦♥♠♥s♦♥ r♦t ♦ ♥t

♥t ♦r♥ ♦ r②st r♦t

❬❪ ♠r r♥s♥ts ♥ t rt ♦ r②st r♦t ♦r♥ ♦ r②st

r♦t

❬❪ P rrr Prä♦r ♥♠♥s♦♥ st♦st r♦t ♥ ss♥ ♥

s♠s ♦ r♥♦♠ ♠trs ♥ ♣r♦♥s ♦ ♥♦♠♦♥♦s ♥♦♠ ②st♠s

r♦ Pr♦sss t s

rê♥s ♦rás

❬❪ ♦♥ss♦♥ ♣ tt♦♥s ♥ r♥♦♠ ♠trs♦♠♠♥t♦♥s ♥

t♠t P②ss

❬❪ Prä♦r ♣♦♥ ttst ss♠rt② ♦ ♦♥♠♥s♦♥ r♦t ♣r♦

sss P②s

❬❪ ②r r♥st♦♥ Pr③②② ❨ ♣r ①♠ t ♥

♦ ♥t② r♦♥ rs P②s ttrs

❬❪ ♠r ♦♠tt ①t ①♠ t strt♦♥ ♦ tt♥

♥trs P②s ttrs

❬❪ ❩ á③ Ps ❲t strt♦♥ ♦r ♠♥s♦♥ r♦t ♥

♣♦st♦♥ ♣r♦sssP②s

❬❪ r s ♥ts③ ts ♥ r♦♥ss strt♦♥

s♥P②s

❬❪ rs ❲♥s♦♥ sr sttsts ♦ r♥r r

tPr♦♥s ♦ t ♦② ♦t②

❬❪ ♥♦ ❯♥rs tt♦♥s ♦ r♦♥ ♥trs

♥ ♥ r♥t q r②sts P②s ttrs

❬❪ ♥♦ s♠♦t♦ ♣♦♥ r♦♥ ♥trs

♥♦r ♥rs tt♦♥s ♥ s ♥r♥ ♥t ♣♦rts

♦sr♣

❬❪ ♥♦ ♥ ♦r ♦♠tr②♣♥♥t ❯♥rs

tt♦♥s ♦ t rrPrs❩♥ ♥trs ♥ qr②st r♥

♦r♥ ♦ ttst P②ss

❬❪ r rrr s ❯♥rs tt♦♥s ♥ rr

Prs❩♥ r♦t ♦♥ ♦♥♠♥s♦♥ t sstrts P②s

rê♥s ♦rás

❬❪ s♠♦t♦ ♣♦♥ ♥♠♥s♦♥ rrPrs❩♥ qt♦♥ ♥

①t ♦t♦♥ ♥ ts ❯♥rst② P②s ttrs

❬❪ rr ttst P②ss ♦ Prts ♠r ♠r ❯♥rst②

Prss

❬❪ tsr ②③♠ ♥ rt r♦t r♦♥t ♦ s♣rs ♦r②③

P②s

❬❪ t♦r ♥r② ❲tt ♦♥t♥♠ ♠♦ ♦r r ♥tr

r♦tP②s

❬❪ ♣r ♣♣♦ P♦t ②♥♠s ♦ ♦♠♣① ♥tr

sP②s

❬❪ rrr s Pts ♥ t tr♠♥t♦♥ ♦ t ♥rs

t② ss ♦ r strs♦r♥ ♦ ttst ♥s ♦r② ♥

①♣r♠♥t♦P

❬❪ ❱♦ ♥♠r ♣♣r♦ t♦ t ♣r♦♠ ♦ s♠♥t ♦♠♦r♥

♦ ♦♦ ♥ ♥tr ♥

❬❪ tr♥ ♦♠♠♥ts ♦♥ ❱♦s s♠t♦♥ ♦ ♦ ♦r♠t♦♥♦r♥ ♦

♦♦ ♥ ♥tr ♥

❬❪ ♠② ♥ ♦ r♦ srs ts ♦ sr s♦♥♦r♥ ♦

P②ss t♠t ♥ ♥r

❬❪ r P ♥ ♣♥② ②♥♠s ♦ ♦♠♣① ♥trsP②s

♥ú♠r♦

❬❪ rrr r ♣str♥s ♦♥ ♥trt♥ ♣rt s②st♠s

P❩ ♥rst② ♥ r♥♦♠ ♠trs♦r♥ ♦ P②ss t♠t ♥

♦rt

❬❪ r ♣♦♥ ♥s♠ ♦r r♦t♦r♦ tr♥st♦♥s ♥ sr r♦t

P②s ttrs

rê♥s ♦rás

❬❪ r P ♥ ❯♥rs ♥ts③ ts ♥ t rt ♦ r♦t ♣r♦sss

♦r♥ ♦ P②ss t♠t ♥ ♥r

❬❪ ♥s ♥tr♦çã♦ à ís sttíst t♦r ❯♥rs ã♦

P♦

Documents

ESTUDO DE MODELOS DE CRESCIMENTO DISCRETOS EM … completo.pdf · Ficha catalográfica preparada pela Biblioteca Central da ... sem considerar o ζ lnt, ... Estudo de modelos de crescimento