A New Scaling Theory for the Damage Spreading in Surface Growth Models

C.K. Lee, Yup Kim

초청 강연이 절대 아님10 분 Talk 로 투고 된 논문임제도나 분과 위원장의 실수 (?)

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I Introduction Damage Spreading Dynamics

1. 초기에 작은 차이 ( 결함 ) 가 있는 2 개의 동일한 dynamical system 을 동일한 random number 와 동일한 dynamic rule 로 전개시키면서 결함이 얼마나 퍼지는가를 봄 .

Damage Spreading in Ising Model Dynamics

1. Ising phase transition & Damage spreadings(DSs) (Stanley et al. ,1987)

sitesdamagedoffraction

cTT /0.00 1.00 1.04 1.08

0.50.40.30.20.1

0.00 1.00 1.04 1.08

1.0

0.0

m

cTT /

2. Correlated-nonlinear system 들의 dynamical property를 연구하는데 powerful 한 이론적 방법으로 알려져있음 .

3. Biological system (Kauffman, 1969), cellular automata ( Jan and Arcangelis, 1994), spin glass (Derrida and Weisbuch, 1987),Ising model (Derrida,1987…. Thomas, 1998).

m=magnetization

1

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z

z

z

LtL

LtttLW

L

tfLthhW

~),(

~])]([[ 2/12

z

z

zjj

trt

trrtrG

t

rgrtxhtrxhtrG

/12

/12

/122

~),(

~)],(),([),(

Surface width

Height-Height Correlation Function

I Scaling relations

DSRGDSRG

ln (W / L

ln (t / Lz)

scaling

L1

L2

L3

t

ln W

ln t

2

I Damage Spreading

1. Consider two systems A and B 2. Start from two different initial conditions, which are the

same except one point at r0

3. Evolve under the same growth rules and under the same sequence of random numbers

4. The surface configurations of them evolve differently due to the different initial conditions

5. A damage site is defined as the point where the surface heights hA(r,t) and hB(r,t) are not the same

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t

r0

Damaged Siteh

t

A

d

d||

r0

B

DSRGDSRG 4

z

zz

LtL

LttLtfLdtLD

/),(

ln L

3 4 5

ln D

0

1

ln t

4 6

ln D

0

1

I RSOS Model

ln t

4 6

ln D

||

1

2

3

4

ln L

3 4 5

ln D

||

1

2

3

4

= 0.52

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z

zzz

LtL

LttLtLfdtLD

/1

|||||| )/(),(

1/z = 0.69 기울기 = 1.0

I Real Physics of the Damage Spreadings in Surface Growth Models

zi Ltt z

i Ltt

L L

0r

||d

L

{ztd /1

|| : no damaged site No Informations for physics

ztd /1|| :dynamical self-affine property 를 만족함 .

zi dt || : )( ||dti = the time at which the damage first touch

the site at ||d

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I A New Scaling Theory

0)( || dd 0|| zcdt

z

z

L

cdtfLdd ||

||)(

zz

zzz

cdtLL

Lcdtcdtdd

||

||||

||;

0;)()(

(self-affine property)

5

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)7.0;(|| cRSOScdt zi

6

ln d||

0 2 4

ln t i( d

||)

0

2

4

6

X

X

X

XXXXXXXXXXXXXXXXXXXXXXXXXXXX

XXXXXXXXXXXXXXXXXXXXXXXX

RSOSDTX

4||d

2/3||d

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zz Lcdt ||0

t

0 50 100 150 200

d

0

1

2

3

XX X X

XX

XXX

X

d||=0

d||=8

d||=16

d||=20X

t0 50 100 150 200

d

0

1

2

3

)()( ||||zcdtAdd

1024L

7

A 는 Atd )0( 로 부터 얻음 425.0A

c 는 )()( ||||zcdtAdd 로부터 얻음 . 5.0c

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t-cd||z

0 50 100 150 200

d

0

1

2

X

XX

X

XX

XX

XX

X

d||=0

d||=8

d||=16

d||=20X

)5.0(),( || Ldd

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ln L(L=128,256,512,1024)

5 6 7

ln d(

)

1

2

X

X

X

X

d||=0

d||=8

d||=16

d||=20X

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I Super Roughening Models

I Das Sarma-Tamborenea(DT) Model

11

1111

11

,2

,1

,0

iiii

iiiiii

iiii

i

hhandhh

hhhorhhh

hhandhh

k

DSRGDSRG

1) >1 , W sat /L diverges in the thermodynamic limit

2) Height-Height Correlation Function in Super Roughening

z

zk

zzkk

trt

tLrforLr

Ltrfortr

trG/12

/1''2

/1/''2

~),(

10

(DT, LC, RC, CRSOS,…)

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t

0 50 100 150 200

d

0

1

2

3

4

X X X

X

X

X

X X

X X

X

X

XX

XX

X

XX X

XX X

X X

XXX

X XX

X XX

X

X

X

XX X X X X

X

X

X

XX X X

d||=0

d||=8

d||=16

d||=20X

zz Lcdt )(0 || AtcdtAdd z )()( ||||

256L

11

예상보다 훨씬 빨리 damage 의 전파가 일어남 .그러나 dynamical self-affinity 는 만족함 .

A 는 Atd )0( 로 부터 얻음 61.0A

I DT Model

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t-cd||z

0 50 100 150 200

d

0

2

4

X

X

X

X

X X

X X

X

X

XX

XX

X

XX X

XX X

X X

XXX

X XX

X XX

X

X

X

XX X X X X

X

X

X

XX X X

d||=0

d||=8

d||=16

d||=20X

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01.053.1

13

ln L(L=32,64,128,256)

4 5

ln d

1

2

3

4

5

X

X

X

X

d||=0

d||=4

d||=8

d||=8

d||=16X