Universality and criticality of two-particle correlated evolution model

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


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


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





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.


Model 5

Simulation resultsSimulation results

)(

)(1

),,(ts

tnLtLq

i i

c

s(t) : number of survival samples at t


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


LL

tLtLLeff ln)2ln(

),(ln),2(ln)(/


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


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


Case 1

i) Annihilation

ii) Branching

Simulation results

q

6

1 q

8


1586.0||

16296.0cq


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


Simulation results

Case 2

i) Annihilation

ii) Branching

q

4

1 q

10


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


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


284.0,166.0||

Simulation results (Preliminary results) 13

L = 29

At pc=0.1044 ,

Conclusion


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

Universality and criticality of two-particle correlated evolution model

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