Establishment of stochastic discrete models for continuum Langevin equation of surface growths

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

2. 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) equa- tion and quenched Kardar-Parisi-Zhang(KPZ) equation as well as the thermal EW and KPZ equations. Abstract

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

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

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

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

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

8. For the Kardar-Parisi-Zhang equation,

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

10.  Question? Is the evaporation process accepted, when the rate Wie>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 pin- ning force i, h which takes the value 1 with probability p and -1 with probability q = 1-p. 2. For a given time t the value is determined for all i . 3. The interface configuration is updated simultaneously for for all i :

11. Our results for the quenched Edward-Wilkinson equation

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

13. Our results for the quenched Edward-Wilkinson equation Near the threshold Fc

14. Comparison Leschhorn’s results Near the threshold pc

15. For the quenched Kardar-Parisi-Zhang equation, L = 1024, 2 = 0.1 ,  = 0.1

16. Near the threshold Fc

17. 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?) .