Stochastic analysis of continuum Langevin equation of surface growths through the discrete growth model S. Y. Yoon and Yup Kim Department of Physics, Kyung-Hee.

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Stochastic analysis of continuum Langevin equation of surface growths through the discrete growth model S. Y. Yoon and Yup Kim Department of Physics, Kyung-Hee University Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

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 Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Background of this study 1

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 : Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Background of this study 2

Since W (transition rate) > 0, Probability for the unit Monte-Carlo time Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Background of this study 3

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. Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Calculation Rule 4

For the Edward-Wilkinson equation, Simulation Results  Growth without quenched noise Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 5

For the Kardar-Parisi-Zhang equation, Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 6

 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 Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 7

 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 Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 8

 Our results for the quenched Edward-Wilkinson equation Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 9

original Leschhorn’s model with evaporation allowed  Comparison with Leschhorn’s results Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 10

Near the threshold F c  Our results for the quenched Edward-Wilkinson equation Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 11

Near the threshold p c  Comparison Leschhorn’s results Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 12

 For the quenched Kardar-Parisi-Zhang equation, L = 1024, 2 = 0.1, = 0.1 Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 13

Near the threshold F c Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Simulation results 14

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?). Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems Conclusion 15