Stochastic analysis of continuum Langevin equation of surface growths through the

Stochastic analysis of continuum Langevin equation of surface growths through the

discrete growth 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

),(),(),( )1( txtxhK

t

txh

Continuum Langevin Equation :

Discretized version : )()( )1( tHK

t

thii

i

NiihH 1

,0)( ti )'(2)'()'()( )2( ttDttKtt ijijijji

Master Equation :

''

),()',(),'(),'(),(

HH

tHPHHWtHPHHWt

tHP

Fokker-Planck Equation :

),(2

1),(

),( )2(2

,

)1( tHPKhh

tHPKht

tHPij

jijiii

i

'

')1( )',()()(H

iii HHWhhHK

)',()()()( '

'

')2( HHWhhhhHK jjH

iiij

HHW ,' is the transition rate from H’ to H.

White noise :

A stochastic analysis of continnum Langevin equation for surface growthsA stochastic analysis of continnum Langevin equation for surface growths


Background of this study 1

),(),(

),()(),()()()1(

ahhaWahhaW

ahhWhahahhWhahHK

iiii

iiiiiiiii

If we consider the deposition(evaporation) of only one particle at the unit evolution step.

:)()(

)()(

,2

,2

ahhWhah

ahhWhah

iiii

iiii

ji

:0 ji

( a is the lattice constant. )

ahi

ahi

(deposition)

(evaporation)'ih

)(),,,()(242)1( hhhhHK ii

Including quenched disorder in the medium :

)'(2)'()( hhhh ijji ,0)( hi

ijij DK 2)2(



D

HKD

HKW ii

id 2

)(

2

)( )1()1(

D

HKD

HKW ii

ie 2

)(

2

)( )1()1(

Since W (transition rate) > 0 ,

,),( ahhWW iiid ),( ahhWW iiie

2

)1(

2

)(

a

D

a

HKW i

id 2

)1(

2

)(

a

D

a

HKW i

ie

)1( a

• Probability for the unit Monte-Carlo time

)1( idid WP

)1( ieie WP



Calculation RuleCalculation Rule

1. For a given time the transition probability

2. The interface configuration is updated for i site :

)1( idid WP

)1( ieie WP

otherwise

RPthRPthth ieiidi

i ,0

)if,1)((if,1)()1(

1)()1( thth ii

,)1(1If ieid PP

,)1(1if else ieid PP compare with new random value R.

is evaluated for i site.

)1)()1(( thth ii


Calculation Rule 4

For the Edward-Wilkinson equation ,

2.0,0.25,0.50 z

iiiii hhhhK 2][ 1122

2)1(

Simulation ResultsSimulation Results

Growth without quenched noise

zL

tfLtLW ),(

z


Simulation results 5

For the Kardar-Parisi-Zhang equation,

)(2][ 1121

11222

2)1(

iiiiiii hhhhhhhK

,),(

zL

tfLtLW

z

5.1,0.34,0.51 z



Growth with quenched noises

• pinned phase : F < Fc

• critical moving phase : F Fc

• moving phase : F > Fc

• Near but close to the transition threshold Fc, the important physical parameter in the regime is the reduced force f

c

c

F

FFf

• average growth velocityfv

dt

hd~



Question? Is the evaporation process accepted, when the rate Wie>0 ? ( Driving force F makes the interface move forward. )

(cf) Interface depinning in a disordered medium numerical results ( Leschhorn, Physica A 195, 324 (1993))

1. A square lattice where each cell (i , h) is assigned a random pin- ning force i, h which takes the value 1 with probability p and -1 with probability q = 1-p.

3. The interface configuration is updated simultaneously for for all i :

hiiiii thththv ,11 )(2)()(

is determined for all i .

2. For a given time t the value

otherwiseth

vifthth

i

iii ,)(

0,1)()1(



Our results for the quenched Edward-Wilkinson equation

FhhhhFhhK iiiiii )(2])([ 1122

2)1(

on)distributi(uniform]1,1[)( ih

fvdt

hd~

24.0,608.0cF



fvdt

hd~

original Leschhorn’s modeloriginal Leschhorn’s model with evaporation allowed

Comparison with Leschhorn’s results

22.0,8004.0cp



,608.0At cF

Near the threshold Fc

Our results for the quenched Edward-Wilkinson equation

,),(

zL

tfLtLW

z

001.0883.0 ,02.025.1



Near the threshold pc

,8004.0At cp

Comparison Leschhorn’s results

,),(

zL

tfLtLW

z

02.088.0 ,01.025.1



For the quenched Kardar-Parisi-Zhang equation,

Fhhhhhh

FhhhK

iiiiii

ii

)()(2

])([

1121

112

222

)1(

on)distributi(uniform]1,1[)( ih

fvdt

hd~

L = 1024, 2 = 0.1 , = 0.1

6529.0,162.0cF



,),(

zL

tfLtLW

z

Near the threshold Fc

0 2 4 6 8-1

0

1

2

3

4 L = 4096, 2 = 0.1 , = 0.1

= 0.6347(1)

ln W

ln t

,162.0At cF

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.5

1.0

1.5

2.0

2.5

3.0

L = 32, 64, 128, 256, 512, 1024

= 0.61(1)

ln W

s

ln L

,635.0,61.0 96.0z



Conclusion and DiscussionsConclusion and Discussions

1. We construct the discrete stochastic models for the given continuum equation. We confirm that the analysis is successfully applied to the quenched Edward-Wilkinson(EW) equation and quenched Kardar- Parisi-Zhang(KPZ) equation as well as the thermal EW and KPZ equations.

2. We expect the analysis also can be applied to

• Linear growth equation , • Kuramoto-Sivashinsky equation , • Conserved volume problem , etc.

3. To verify more accurate application of this analysis, we need

• Finite size scaling analysis for the quenched EW, KPZ equations , • 2-dimensional analysis (phase transition?) .


Conclusion 15

Stochastic analysis of continuum Langevin equation of surface growths through the

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