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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
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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
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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
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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
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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
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( 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
w´
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
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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
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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
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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
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7
Scaling Collapse to in 2D equilibrium state.
WzG L
tLg
KtLW ln
2
1),(2
176.02
1a
KG , Z = 2.5
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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
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9
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)
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,5.0 ,33.0 5.1z
Normal RSOS Model (KPZ)
10
z 0
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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
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12
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
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Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5)
zL
tfLW
12-1
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7
0.175
0.162Dimer
0.175Monomer
Slope aModel
Extremal 0.174
2-site
Monomer & Extremal & Dimer & 2-site