Transcript
Page 1: Metropolis-type evolution rules for surface growth models  with the global constraints

Metropolis-type evolution rules for surface growth models with the global constraints

on one and two dimensional substrates

Yup Kim, H. B. Heo, S. Y. Yoon

KHU

Page 2: Metropolis-type evolution rules for surface growth models  with the global constraints

1. Motivation

In equilibrium state,

Normal restricted solid-on-solid model: Edward-Wilkinson universality class

Two-particle correlated surface growth- Yup Kim, T. S. Kim and H. Park, PRE 66, 046123 (2002)

Dimer-type surface growth- J. D. Noh, H. Park, D. Kim and M. den Nijs, PRE 64, 046131 (2001) - J. D. Noh, H. Park and M. den Nijs, PRL 84, 3891 (2000)

Self-flattening surface growth-Yup Kim, S. Y. Yoon and H. Park, PRE(RC) 66, 040602(R) (2002)

1

Page 3: Metropolis-type evolution rules for surface growth models  with the global constraints

1. Normal RSOS (z =1)

2. Two-particle correlated (dimer-type) growth (z = -1)

)1(}{

21

max

min

h

RSOS

n

h

h

hh

zZ

2

3

1

nh=even number,

Even-Visiting Random Walk (1D)

0)}({ RSOShP

Z

z

hP

h

hh

n

RSOS

h

max

min

)1(

)}({21

RSOSh

RSOS ZZ

hP}{

1,1

)}({Normal Random Walk (1D)

4

1,

2

11

RWz

Partition function,

(EW)

)( zLt Steady state or Saturation regime ,

nh : the number of columns with height h

Page 4: Metropolis-type evolution rules for surface growth models  with the global constraints

3. Self-flattening surface growth (z = 0)

3

)(

}{ 2

1cH

h RSOSr

Z

)1)(( minmax hhcH 3

1

Self-attracting random walk (1D)

Phase diagram (1D)

z = 0 z = 1z =-1Normal

Random WalkEven-Visiting Random Walk

2

1

3

1

3

1 ??

Self-attracting Random Walk

?z

Phase diagram (2D)

z = 0 z = 1z =-1Normal

Random WalkEven-Visiting Random Walk

?

Self-attracting Random Walk

z

LK

WG

ln2

12

Page 5: Metropolis-type evolution rules for surface growth models  with the global constraints

Choose a column randomly.x

2. Generalized Model

4

,1)()( xhxh

1)()( xhxh

Acceptance parameter P is defined by

)})(({

)})('({

rhw

rhwP

Decide the deposition (the evaporation) attempt with probability p (1-p)

Calculate for the new configuration from the decided deposition(evaporation) process

)})('({ rhw

)}('{ rh

)1(2

1)})(({

max

min

hnh

hh

zrhw

( nh : the number of sites which have the same height h ) r

Evaluate the weight in a given height configuration )}('{ rh

Page 6: Metropolis-type evolution rules for surface growth models  with the global constraints

( a primitive lattice vector in the i – th direction ) ie

Any new configuration is rejected if it would result in violating the RSOS contraint

1)ˆ()( ierhrh

5

n+2 = 1n+1 = 3n 0 = 2n-1 = 2n-2 = 2

wn´ +2 = 2

n´ +1 = 2

n´ 0 = 2

n´ -1 = 2

n´ -2 = 2

hmax

0)( ixh

hmin

p =1/2L = 10z = 0.5

P R

If P 1 , then new configuration is accepted unconditionally. If P < 1 , then new configuration is accepted only when P R.where R is generated random number 0< R < 1 (Metropolis algorithm)

9259.0)5.01()5.01(

)5.01()5.01(

213

21

2212

21

)})(({

)})('({

rhw

rhwP

Page 7: Metropolis-type evolution rules for surface growth models  with the global constraints

0.00 0.01 0.02 0.030.2

0.3

0.4

0.5

z = - 1

z = 1.5 z = 0.5 z = 0

z = - 0.5

eff

1/L

3. Simulation Results

6

Equilibrium model (1D, p=1/2)

zL

tfLW

z

z

Ltt

LtL

,

,

LL

tLWtLWLeff ln)2ln(

),(ln),2(ln)(

z

0.00 0.01 0.02 0.030.2

0.3

0.4

0.5

1/3

z=0.9

z=1.1

eff

1/L

Page 8: Metropolis-type evolution rules for surface growth models  with the global constraints

0 1 2 3 4 5

-0.5

0.0

0.5

1.0

=0.22

z=-0.5

z=0

z=0.5

z=0.9

z=1.1

z=1.5

ln W

ln t

7

0.22

0.33

1.1

0.22

0.33

0.9

0.19 0.22 0.22 0.22 1/4 0.22

0.33 0.34 0.33 0.331/2

0.33 (L)

-1-0.500.511.5z

Page 9: Metropolis-type evolution rules for surface growth models  with the global constraints

2.5 3.0 3.5 4.0 4.5 5.00.4

0.5

0.6

0.7

0.8

0.9

1.0

z = -1

z = -0.5

z = 0

z = 0.5

z = 1

z = 1.5

W2

ln L

7

WzG L

tLg

KtLW ln

2

1),(2

z

G

z

G

LtLK

LttK

,ln2

1

,ln2

1

Equilibrium model (2D, p=1/2)

z -1 -0.5 0 0.5 1 1.5

a 0.176 0.176 0.176 0.175 0.176 0.179

176.02

1a

KG

Page 10: Metropolis-type evolution rules for surface growth models  with the global constraints

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Scaling Collapse to in 2D equilibrium state.

WzG L

tLg

KtLW ln

2

1),(2

176.02

1a

KG , Z = 2.5

Page 11: Metropolis-type evolution rules for surface growth models  with the global constraints

Phase diagram in equilibrium (1D)

z = 0

z = 1z =-1

Normal RSOS

2-particle corr. growth

2

1

3

1

3

1

Self-flattening surface growth

3

1

3

1

3

1

-1/2 1/2 3/2

z = 0.9 z = 1.1

3

1

3

1

7

z = 0 z = 1z =-1Normal

Random WalkEven-Visiting Random Walk

Self-attracting Random Walk

z

Phase diagram in equilibrium (2D)

LK

WG

ln2

12

176.0

2

1a

KG

z = -0.5 z = 0.5 z = 1.5

Page 12: Metropolis-type evolution rules for surface growth models  with the global constraints

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Growing (eroding) phase (1D, p=1(0) )

p (L)

1.5 0.52

0.5 0.51

0 0.49

zL

tfLW

z

z

Ltt

LtL

,

,

z 0

)1,31,2

1( zz

: Normal RSOS model (Kardar-Parisi-Zhang universality class)

Page 13: Metropolis-type evolution rules for surface growth models  with the global constraints

,5.0 ,33.0 5.1z

Normal RSOS Model (KPZ)

10

z 0

Page 14: Metropolis-type evolution rules for surface growth models  with the global constraints

11

z 0

z=-0.5 p=1 L=1280.00 0.01 0.02 0.03

0.5

1.0

1.5

2.0

z = -0.1 z = -0.5

z = -1

eff

1/L

Page 15: Metropolis-type evolution rules for surface growth models  with the global constraints

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4. Conclusion

Equilibrium model (1D, p=1/2)

0 1-1

Normal RSOS

(Normal RW)

2-particle corr. growth

(EVRW)

2

1

3

1

3

1

Self-flattening surface growth

(SATW)

3

1

3

1

3

1

3/21/2-1/2

Growing (eroding) phase (1D, p = 1(0) )

1. z 0 : Normal RSOS model (KPZ universality class)

2. z 0 : Groove phase ( = 1)

Phase transition at z=0 (?)

z0.9 1.1

3

1

3

1

Page 16: Metropolis-type evolution rules for surface growth models  with the global constraints

Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5)

zL

tfLW

12-1

Page 17: Metropolis-type evolution rules for surface growth models  with the global constraints

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0.175

0.162Dimer

0.175Monomer

Slope aModel

Extremal 0.174

2-site

Monomer & Extremal & Dimer & 2-site


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