universality and criticality of two-particle correlated evolution model s. y. yoon and yup kim...
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Universality and criticality of two-particle correlated evolution
model
S. Y. Yoon and Yup Kim
Department of Physics, Kyung-Hee University
Satellite Meeting STATPHY 22 in Seoul, KoreaNonequilibrium Statistical Physics of Complex Systems
Background of this study
1D Roughening Transition
(U. Alon, M.R. Evans, H. Hinrichsen and D. Mukamel, Phys. Rev. E. 57 ,4997 (1998))
Normal deposition : p Allow evaporation only at the edges of terraces : 1- p
= the density of vacancies on bottom layer
Active state Absorbing state
RoughSmooth pC
Absorbing stateActive state
( W ~ L )
Monomer deposition/evaporation Model
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orp 1-p
Background of this studyBackground of this study
1
Dimer deposition/evaporation Model (Modulo 2 conservation) ( H. Hinrichsen and G. Odór, Phys. Rev. Lett. 82,1205 (1999) , J. D. Noh, H. Park, M den Nijs, Phys. Rev. Lett. 84, 3891 (2000) )
Directed Ising (DI) type Transition with
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5.0,285.0||
Background of this study 2
Directed Percolation (DP) Class with
zz
z
c L
tfL
LtL
LtttLp
/
/
/ ||
),,(
||z
252.0,159.0||
r = 0, p = pC
rp
r = 1
r = 0pC
p=1/2 ( = 1/3)
Smooth
facet
Rough
facetr
r : Digging probability of the particle inside the terraces
Two-particle correlated growth Model
1. r = 1 (Yup Kim,T.S. Kim, and Hyunggyu Park, Phys. Rev. E 66,046123 (2002))
groove = 1
p = 1/2 groove = 1
rp
r : Digging probability of the particle inside the terraces
r = 1
r = 0pc
= 1/3
Smooth ?(rougheing)
p
1-p
p = q (q=1-p), L → ∞ )
p q , L → ∞ )
zL
tfLW
)(
)(z
z
LtL
Ltt
Dynamical Scaling Law for Kinetic Surface Roughening
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Background of this study 3
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Background of this study 4
2. r = 0
1) Is there Roughening Transition for r = 0 ?
2) What is the Critical Phenomena at Critical Point ?
• Monomer Deposition - Evaporation Model DP
• Dimer Deposition - Evaporation Model(Modulo-2 conservation) DI
• Two-particle correlated growth Model (Modulo-2 conservation) ?
To answer the questions, we should first study the two particles correlated monolayer model !!
q
Model ( Model ( Two-particle correlated monolayer Model )Two-particle correlated monolayer Model )
i) Annihilation
ii) Branching
10
1 q
Most general model with modulo-2 conservation of particles.
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Model 5
Simulation resultsSimulation results
)(
)(1
),,(ts
tnLtLq
i i
c
s(t) : number of survival samples at t
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zz
z
c L
tfL
LtL
LtttLq
/
/
/ ||
),,(
||z
Simulation results 5
159.0||
L = 105, T=10713796.0cq
1 10 100 1000 10000 100000 1000000 1E7
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
t0.15
9
t
1 10 100 1000 10000 100000 1000000 1E70.01
0.1
1
p=0.13785 p=0.13794 p=0.13795 p=0.13796 p=0.13797 p=0.13798 p=0.13820
t
1 10 100 1000 10000 100000 1000000 1E70.01
0.1
1
p=0.13785 p=0.13796 p=0.13820
t1 10 100 1000 10000 100000 1000000 1E7
1
p=0.13785 p=0.13796 p=0.13820
t0.15
9
t
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LL
tLtLLeff ln)2ln(
),(ln),2(ln)(/
Simulation results 6
3 4 5 6 7 8
-3.5
-3.0
-2.5
-2.0
-1.5
p=0.1375 p=0.13796 p=0.1385
sat
L0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
p=0.1375 p=0.13796 p=0.1385
?
eff
1/L
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
-2.8
-2.6
-2.4
-2.2
-2.0
-1.8
-1.6
-1.4
-1.2
/¤Ç
=8
0.49ln
ln L
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2 3 4 5 6 7 8 9 10
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
/||=0.28
L=32 L=64 L=128 L=256 L=512
ln
ln t
Dimer type (DI class)
q2
1 qi) Annihilation ii) Branching
Simulation results
3588.0cq
7
75.1,49.0,28.0||
z
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Case 1
i) Annihilation
ii) Branching
Simulation results
q
6
1 q
8
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1586.0||
16296.0cq
Simulation results 9
L = 105, T=107
1 10 100 1000 10000 100000 1000000 1E7
0.1
1
Type-1, T=107, L=105
p=0.16290 p=0.16292 p=0.16294 p=0.16295 p=0.16296 p=0.16298 p=0.16300
t
1 10 100 1000 10000 100000 1000000 1E70.6
0.7
0.8
0.9
1.0
1.1
1.2 Type-1, T=107, L=105
t0.15
9
t
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016
-0.8
-0.6
-0.4
-0.2
0.0
p=0.161 p=0.16296 p=0.165
/?
eff
1/L
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Simulation results
Case 2
i) Annihilation
ii) Branching
q
4
1 q
10
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Simulation results
152.0||
08422.0cq
11
L = 105, T=107
1 10 100 1000 10000 100000 1000000 1E70.01
0.1
1
Type-2, T=107, L=105
p=0.08400 p=0.08410 p=0.08420 p=0.08422 p=0.08424 p=0.08426 p=0.08428 p=0.08430 p=0.08440
t
1 10 100 1000 10000 100000 1000000 1E70.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
type-2, T=107, L=105
t0.15
9
t
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 p=0.083 p=0.08422 p=0.086
?
eff
1/L
3.5 4.0 4.5 5.0 5.5 6.0 6.5
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
8
W
ln L
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Two-particle correlated type growth model
z
z
LtL
LttW
ln
ln
1044.0c
p
zLtgatLW ln),(
57.1z
Simulation results (Preliminary results) 12
L = 32, 64, 128, 256, 512
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284.0,166.0||
Simulation results (Preliminary results) 13
L = 29
At pc=0.1044 ,
Conclusion
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Conclusion Conclusion
14
1. Critical Phenomena at Critical Point
• Monomer Deposition - Evaporation Model DP • Dimer Deposition - Evaporation Model (Modulo 2 conservation) DI • Two-particle correlated growth Model (Modulo 2 conservation) DP?
Class /|| / z
DI 0.285 0.5 1.75
DP 0.159 0.252 1.58
Two-particle Model 0.159 0.25
PCPD ~0.20