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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. 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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  1. A stochastic analysis of continuum Langevin equation for surface growths S.Y.Yoon, Yup Kim Kyung Hee University

  2. 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.

  3. 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

  4. 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

  5. Simulation Results  Edward-Wilkinson equation L=32, 64, 128, 256, 512

  6. Simulation Results L=10000 L=10000 Random Deposition  EW universality class Layer-by-layer growth  EW universality class

  7. Simulation Results Mullins-Herring equation L=32, 64, 128 L=10000

  8. Simulation Results Linear growth equation (MHEW) The competition between two linear terms generates a characteristic length scale Crossover time L=1000

  9. 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)

  10. 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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