Establishment of stochastic discrete models for continuum Langevin equation of surface growths Yup Kim and Sooyeon Yoon Kyung-Hee Univ. Dept. of Physics.

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Establishment of stochastic discrete models for continuum Langevin equation of surface growths Yup Kim and Sooyeon Yoon Kyung-Hee Univ. Dept. of Physics

Based on the relations among Langevin equation, Fokker- Planck equation, and Master equation for the surface growth phenomena. It can be shown that the deposition (evaporation) rate of one particle to(from) the surface is proportional to. Here, and are from the Langevin. From these rates, we can construct easily the discrete stochastic models of the corresponding continuum equation, which can directly be used to analyze the continuum equation. It is shown that this analysis is successfully applied to the quenched Edward-Wilkinson(EW) equation and quenched Kardar-Parisi-Zhang(KPZ) equation as well as the thermal EW and KPZ equations.Abstract

Continuum Langevin Equation : Discretized version : Master Equation : Fokker-Planck Equation : is the transition rate from H’ to H. White noise : A stochastic analysis of continnum Langevin equation for surface growths

If we consider the deposition(evaporation) of only one particle at the unit evolution step. ( a is the lattice constant. ) (deposition) (evaporation) Including quenched disorder in the medium :

Since W (transition rate) > 0, Probability for the unit Monte-Carlo time

Calculation Rule 1. For a given time  the transition probability 2. The interface configuration is updated for i site : compare with new random value R. is evaluated for i site.

For the Edward-Wilkinson equation, Simulation Results  Growth without quenched noise

For the Kardar-Parisi-Zhang equation,

 Growth with quenched noises pinned phase : F < F c critical moving phase : F  F c moving phase : F > F c Near but close to the transition threshold F c, the important physical parameter in the regime is the reduced force f average growth velocity

 Question? Is the evaporation process accepted, when the rate W ie >0 ? ( Driving force F makes the interface move forward. ) (cf) Interface depinning in a disordered medium numerical results ( Leschhorn, Physica A 195, 324 (1993)) 1. A square lattice where each cell (i, h) is assigned a random pinning force  i, h which takes the value 1 with probability p and -1 with probability q = 1-p. 3. The interface configuration is updated simultaneously for for all i : is determined for all i. 2. For a given time t the value

 Our results for the quenched Edward-Wilkinson equation

original Leschhorn’s model with evaporation allowed  Comparison with Leschhorn’s results

Near the threshold F c  Our results for the quenched Edward-Wilkinson equation

Near the threshold p c  Comparison Leschhorn’s results

 For the quenched Kardar-Parisi-Zhang equation, L = 1024, 2 = 0.1, = 0.1

Near the threshold F c

Conclusion and Discussions 1. We construct the discrete stochastic models for the given continuum equation. We confirm that the analysis is successfully applied to the quenched Edward-Wilkinson(EW) equation and quenched Kardar- Parisi-Zhang(KPZ) equation as well as the thermal EW and KPZ equations. 2. We expect the analysis also can be applied to Linear growth equation, Kuramoto-Sivashinsky equation, Conserved volume problem, etc. 3. To verify more accurate application of this analysis, we need Finite size scaling analysis for the quenched EW, KPZ equations, 2-dimensional analysis (phase transition?).