universality and criticality of two-particle correlated evolution model s. y. yoon and yup kim...
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![Page 1: Universality and criticality of two-particle correlated evolution model S. Y. Yoon and Yup Kim Department of Physics, Kyung-Hee University Satellite Meeting](https://reader036.vdocuments.net/reader036/viewer/2022062803/56649f3b5503460f94c5a322/html5/thumbnails/1.jpg)
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
![Page 2: Universality and criticality of two-particle correlated evolution model S. Y. Yoon and Yup Kim Department of Physics, Kyung-Hee University Satellite Meeting](https://reader036.vdocuments.net/reader036/viewer/2022062803/56649f3b5503460f94c5a322/html5/thumbnails/2.jpg)
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
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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
Satellite Meeting STATPHY 22 in Seoul, KoreaNonequilibrium Statistical Physics of Complex Systems
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
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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 !!
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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
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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
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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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Satellite Meeting STATPHY 22 in Seoul, KoreaNonequilibrium Statistical Physics of Complex Systems
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
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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
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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 ,
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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