A stochastic analysis of continuum Langevin equation for surface growths

S.Y.Yoon, Yup Kim

Kyung Hee University

Motivation of this study

To solve the Langevin equation1. Renormalization Group theory2. Numerical Integrations

Numerical Intergration method

Direct method to solve the Langevin equation

),(),( 22

2 txFhht

txh

Using Euler method,

)(2)(~

')(~

')()(2

22 iiiii Dhhvhh

Dimensionless quantities is defined as following,

000,, th

hhxxx

r0, t0, and h0 are appropriately chosen units of length, time, and height.

)(2)()(2

')()()(')()( 2

2 iiiiiiii Dxxhxxhxhxxhxxhvhh

)'()'(2)','(),( ttxxDtxtx d

Langevin equation

In quantum mechanics,

ShrÖdinger equation Transition probability between each states

In surface growth problems,

Evolution rate of an interface

But it has some difficulties to define the prefactor of noise term. (Ex) Quenched KPZ equation

),(),( 22

2 hxFhht

txh

)'()'(2)','(),( hhxxDhxhx d

)]),([,()()()(2

')()()(')()( 3

222 xhxxxhxxhxhxxhxxhvhh iiiiiiii

H. Jeong, B. Kahng, and D. Kim, PRL 77, 5094 (1996)Z. Csahok, K Honda, and T. Vicsek, J. Phys. A 26, L171 (1993)

)]),([,(')( xhxi

?

Evolution Rate

Motivation of this study

Our Method

Fhhht

hiii

i ),][,][,]([

242

max

2

2

),]([

),]([

i

i

h

hP

Ftxhhht

txh

),(),,,(),( 242

)}({)},({ thtxh i

(i = integer)

1 ii hh

Evolution Rate

Evolution Probability RateEvolutionP

How can we define the time unit?

max

1

i

QM!

Continuum Lagenvin equation

In our method, we can present by selecting i in random. This is the easy way to use the numerical integration concept without complicated prefactor of noise term.

,1

,1

maxi

ii

Our time unit trial

* F is a driven force.

Simulation Results

Edward-Wilkinson equation

Fht

hi

i ][ 2

2

iii hhh 2112

-12 -10 -8 -6 -4 -2 0 2-4.0

-3.6

-3.2

-2.8

-2.4

-2.0

-1.6

0 2 4 6 8 10 12-1

0

1

2

L=32 L=64 L=128 L=256 L=512

=1.0, F=1.0

ln(W

/L )

ln(t/Lz)

ln W

ln t

3.5 4.0 4.5 5.0 5.5 6.0 6.5

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

=1.0, F=1.0

=0.5

ln W

ln L

L=32, 64, 128, 256, 512

zL

tfLtLW ),( 008.2,247.0,496.0 z

ih][ 22

Fhv

FhvP

i

max

22

22

][

][

0 1 2 3 4 5 6 7-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

=0.5

=0.25

=0.25

ln W

ln t

F=1.0 F=100.0

Random Deposition EW universality class

Layer-by-layer growth EW universality class

0 1 2 3 4 5 6 7

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

=0.24

=0.25

=0.42

ln W

ln t

2=1.0

2=10.0

L=10000L=10000

Simulation Results

Fht

hi

i

][ 22

Mullins-Herring equation

3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

2.0

2.5

3.0

3.5

4.0 4=1.0, F=1.0

=1.5

ln W

ln L

Fht

hi

i ][ 4

4

ih][ 44

L=32, 64, 128

Simulation Results

0 1 2 3 4 5 6 7-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

4=1.0 , F=1.0

=0.372

ln W

ln t

]6)(4)[( 11224 iiiii hhhhhv

L=10000

Fhv

FhvP

i

max

44

44

][

][

zL

tfLtLW ),( 952.3,372.0,47.1 z

Linear growth equation (MHEW)

Fhht

hii

i ][][ 4

42

2

Simulation Results

iii hhh 2112 ih][ 22

ih][ 44 ]6)(4)[( 11224 iiiii hhhhhv

Fhvhv

FhvhvP

i

max

44

22

44

22

][

][

21

2

4~

l

22

4~v

tcCrossover time

The competition between two linear terms generates a characteristic length scale

0 1 2 3 4 5 6 7

-1.0

-0.5

0.0

0.5

1.0

1.5

4=1.0,

2=1.0, F=1.0

=0.2653(2)

=0.374(4)

ln W

ln t

L=1000

Kardar-Parisi-Zhang equation

Fhht

hii

i ][][

222

3.5 4.0 4.5 5.0 5.5 6.0 6.5

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2=1.0, =1.5, F=1.0

=0.5

ln W

ln L

495.1,333.0,498.0 z

L=32, 64, 128, 256, 512

Simulation Results

Instability comes out as has larger value.(Intrinsic structures)

C. Dasgupta, J. M. Kim, M. Dutta, and S. Das Sarma PRE 55, 2235 (1997)

iii hhh 2112 ih][ 2

2 ih ][2

2

11 )(2

1

ii hh

0 1 2 3 4 5 6 7-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6 7

-1

0

1

2

2=1.0, F=1.0, =1.5

~ 0.25

~ 0.33

ln W

ln t

=10.0

~0.33

~0.7

ln W

ln t

Fhhv

FhhvP

i

max

222

222

][

][

Conclusions

We confirmed that the stochastic analysis of Langevin equations for the surface growth is simple and useful method.

We will check for another equations

• Kuramoto-Sivashinsky equation

• Quenched EW & quenched KPZ equation