A stochastic analysis of continuum Langevin equation

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A stochastic analysis of continuum Langevin
equation for surface growths
S.Y.Yoon, Yup Kim Kyung Hee University
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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.
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Motivation of this study
?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
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Our Method
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
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Simulation Results
? Edward-Wilkinson equation
L32, 64, 128, 256, 512
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Simulation Results
L10000
L10000
Random Deposition ? EW universality class
Layer-by-layer growth ? EW universality
class
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Simulation Results
?Mullins-Herring equation
L32, 64, 128
L10000
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Simulation Results
?Linear growth equation (MH?EW)
The competition between two linear terms
generates a characteristic length scale
Crossover time
L1000
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Simulation Results
?Kardar-Parisi-Zhang equation
L32, 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)
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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