damage spreading phase transitions in a themal roughening model

DAMAGE SPREADING PHASE TRANSITIONS IN A THEMAL

ROUGHENING MODEL

Yup Kimwith C. K. Lee

Kyung Hee Univ.

Ref.: 1. Yup Kim and C. K. Lee, Phys. Rev E 62, 3376 (2000). 2. Yup Kim, Phys. Rev E 64, 027101 (2001).

I Introduction Damage Spreading Dynamics

1. Two identical systems, which are initially the same except for a small subset of the system (damages), are simulated by the same dynamical rules and by the same sequence of random numbers and it is observed how damages are spreading during the dynamical evolution by a detailed comparision of the two systems.

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

Characteristics of Damage Spreading Dynamics in Ising Model

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 /

Ordered Phase=Damage-Frozen PhaseDisordered Phase = Damage-Spreading Phase

1 Kyung Hee Univ.Kyung Hee Univ.DSRGDSRG

)/(~),(

~])]([[ 2/12

zLtL

LtttLW

L

tfLthhW

z

z

z

Dynamical Self-affinity

Surface width W (root-mean-square fluctuation of surface height h(x, t))

I Kinetic Surface Roughening and Its Scaling relations

ln (W / L

ln (t / Lz)

scaling

L1

L2

L3

t

ln W

ln t

2

~zLtfL

z

z

LtL

Ltt z

:

:1

Kyung Hee Univ.Kyung Hee Univ.DSRGDSRG

)( LL

( = correlation length of the surface growth parellel to substrate)

I Algorithm for Damage-Spreadings for Kinetic Roughenings

1. Consider two surface growth systems A and B 2. Initial conditions, which are the same except one point

at r0.

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

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

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

t >

r0

Damaged Siteh

t

A

d

d||

r0

B

d : lateral damage spreading distance d : vertical damage spreading distance


I Physics of the Damage-Spreadings in Surface Growth Models

(Anisotropy between lateral and vertical direction.)

4

zLt zLt

L L

0r

||d

L

{ztd /1

|| : no damaged column. No Informationsfor physics.

ztd /1|| : dynamical self-affine property

zi dt || )( ||dti, ( = the time at which the damage

first touch the site at )||d


I A New Scaling Theory

~),( |||| zLtfLtLD

2.

ztd

fdtdP p 1||1

|||| ),(

( or ) .)0( constf p 1|||| ),( dtdP

2

0 |||||||||| ),(),(L

ddtdPddtLD

2

1

2

1

0 ||

0

/1||

||

0

)()0(

)(

)(

)1(

L

z

L

z

zp

z

pz

p

z

LtLddf

LtL

dxxftddt

df

Ltt

( J. M. Kim,

Y. K. Lee, I. M. Kim, 1996)

),( || tdd

zz

zzz

z

LcdtifL

LcdtifcdtA

cdtif

||

||||

||

1)(

00

zz

z

L

tfLtd

L

cdtfLd ),0(||

1. ),( || tdd Average vertical DS distance at ||d


3.

z

z

LtL

Ltt

ddtdPtdddDL

;

1;

),(),(2

0 ||||||

We confirmed our theory of DS dynamics for the kinetic surface roughening phenomena using surface growth models like Restriced Solid-on-Solid model. (J. M. Kim, J. M. Kosterlitz, 1989)

One of efficient way to probe kinetic roughening phenomena by D. S. Dynamics.

zL

tfLdtdd 0

|| ),0( z

z

LtL

Ltt

:

:

zL

tfLtLD |||| ),(

zLtt z :1


zLtL :

7

I Thermal Surface Roughening Transition

Rough Phase

1. Dynamics Scaling for kinetic surface roughening.

zL

tfLW

z

z

LtL

Ltt

2. Equilibrium Saturation regime

)ln,0;0(),,( LWLLtTW

Smooth Phase

finiteLtTWL ),,(

Transition from rough phase to smooth phase ( TR ) ( Ising Model : Tc > TR )

L

W


?)or( tLt z

8

I Transition from Damage-spreading phase to Damage-frozen phase

Damage-spreading phase ( Rough phase? )

1. Dynamics

Damage-frozen phase ( Smooth phase? )

zL

tfLtTLd ),,(0

z

z

LtLL

Ltt

)0;ln(

zL

tfLtTLD |||| ),,(

z

z

LtL

Ltt z

1

2. Equilibrium (?)

)ln,0;0(),,(0 LWLTtLd

LTtLD ),,(||

finiteTtLdL ),,(0

finiteTtLDL ),,(||

(?)RFS TT

Transition from Damage-speading phase to Damage-frozen phase


3. Dynamics for Roughening Phase.

)(ln~)(

ln~)(~:2

4/1,2/1~:1

Rz

z

TTtLtW

LLtWd

d

1. 1d : No roughening phase transition.

2. 2d : Roughening phase transition exists.

8061.0)/( RBR TkJK

Edwards-Wilkinson(EW) universality class (z = 2)

ih jh

n

ijji hhJH

(n=1)

9

I SOS Model


10

I Dynamics (Simple Glauber Dynamics)

))1()1(

))1()1(

11

110

11

iiii

iiii

iiii

hhhhJH

hhhhJH

hhhhJH

)/(

)/(

)/(

0

0

0

0

HHHH

HHHH

HHHH

eeeeP

eeeeP

eeeeP

)1(.Prob

)(.Prob

)1(.Prob

0

ii

ii

ii

hhP

hhP

hhP


11

I Simulation Results 1d Only Damage-Spreading Phase Exists. (T > 0)


5.01 z

L = 32, 64, 128, 256, 512

2

1

),,(0 LttLd

LttLD ),,(||

12

2d : Transition Damage-Spreading phase to Damage-frozen Exists.


LTtLd

TT R

ln~),0,(

;0

finiteTtLd

TTL

R

),0,(

;0

LTtLD

TT R

),,(

;

||

finiteTtLD

TTL

R

),,(

;

||

RSF TT

damage spreading phase transitions in a themal roughening model

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