A New Scaling Theory for the Damage Spreading in Surface Growth Models C.K. Lee, Yup Kim 초청...
Transcript of A New Scaling Theory for the Damage Spreading in Surface Growth Models C.K. Lee, Yup Kim 초청...
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