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Stochastic analysis of continuum Langevin equation of surface growths through the

Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems. 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. 1.

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Stochastic analysis of continuum Langevin equation of surface growths through the

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  1. Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems 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

  2. 1 Background of this study 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 : Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  3. 2 Background of this study 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. ) Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  4. 3 Background of this study 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

  5. 4 Calculation Rule 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. Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  6. 5 Simulation results Simulation Results  Growth without quenched noise For the Edward-Wilkinson equation , Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  7. 6 Simulation results For the Kardar-Parisi-Zhang equation, Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  8. 7 Simulation results  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 Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  9. 8 Simulation results  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 : Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  10. 9 Simulation results Our results for the quenched Edward-Wilkinson equation Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  11. 10 Simulation results Comparison with Leschhorn’s results original Leschhorn’s model with evaporation allowed original Leschhorn’s model Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  12. 11 Simulation results Our results for the quenched Edward-Wilkinson equation Near the threshold Fc Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  13. 12 Simulation results Comparison Leschhorn’s results Near the threshold pc Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  14. 13 Simulation results 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

  15. 14 Simulation results Near the threshold Fc Satellite Meeting STATPHY 22 in Seoul, Korea Nonequilibrium Statistical Physics of Complex Systems

  16. 15 Conclusion 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

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