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A stochastic analysis of continuum Langevin equation for surface growths

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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 equation

1. Renormalization Group theory

2. Numerical Integrations

Numerical Intergration method

Direct method to solve the Langevin equation

Using Euler method,

Dimensionless quantities is defined as following,

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

Motivation of this study

In quantum mechanics,

ShrÖdinger equation

Transition probability between each states

But it has some difficulties to define the prefactor of noise term.

(Ex) Quenched KPZ equation

?

H. Jeong, B. Kahng, and D. Kim, PRL 77, 5094 (1996)

Z. Csahok, K Honda, and T. Vicsek, J. Phys. A 26, L171 (1993)

Evolution Rate

In surface growth problems,

Langevin equation

Evolution rate of an interface

Our Method

(i = integer)

Continuum Lagenvin equation

* F is a driven force.

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.

QM!

Evolution Rate

Evolution Probability

How can we define the time unit?

Our time unit trial

Simulation Results

Edward-Wilkinson equation

L=32, 64, 128, 256, 512

Simulation Results

L=10000

L=10000

Random Deposition

EW universality class

Layer-by-layer growth

EW universality class

Simulation Results

Mullins-Herring equation

L=32, 64, 128

L=10000

Simulation Results

Linear growth equation (MHEW)

The competition between two linear terms

generates a characteristic length scale

Crossover time

L=1000

Simulation Results

Kardar-Parisi-Zhang equation

L=32, 64, 128, 256, 512

Instability comes out as has larger value.

(Intrinsic structures)

C. Dasgupta, J. M. Kim, M. Dutta, and S. Das Sarma

PRE 55, 2235 (1997)

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