A stochastic analysis of continuum Langevin
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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.

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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

A stochastic analysis of continuum Langevin equation

for surface growths

S.Y.Yoon, Yup Kim

Kyung Hee University


A stochastic analysis of continuum langevin equation for surface growths

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.


A stochastic analysis of continuum langevin equation for surface growths

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


A stochastic analysis of continuum langevin equation for surface growths

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


A stochastic analysis of continuum langevin equation for surface growths

Simulation Results

 Edward-Wilkinson equation

L=32, 64, 128, 256, 512


A stochastic analysis of continuum langevin equation for surface growths

Simulation Results

L=10000

L=10000

Random Deposition

 EW universality class

Layer-by-layer growth

 EW universality class


A stochastic analysis of continuum langevin equation for surface growths

Simulation Results

Mullins-Herring equation

L=32, 64, 128

L=10000


A stochastic analysis of continuum langevin equation for surface growths

Simulation Results

Linear growth equation (MHEW)

The competition between two linear terms

generates a characteristic length scale

Crossover time

L=1000


A stochastic analysis of continuum langevin equation for surface growths

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)


A stochastic analysis of continuum langevin equation for surface growths

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


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