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Metropolis-type evolution rules for surface growth models with the global constraints

Metropolis-type evolution rules for surface growth models with the global constraints on one and two dimensional substrates. Yup Kim, H. B. Heo, S. Y. Yoon KHU. 1. 1. Motivation. In equilibrium state, Normal restricted solid-on-solid model : Edward-Wilkinson universality class

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Metropolis-type evolution rules for surface growth models with the global constraints

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  1. Metropolis-type evolution rules for surface growth models with the global constraints on one and two dimensional substrates Yup Kim, H. B. Heo, S. Y. Yoon KHU

  2. 1 1. Motivation • In equilibrium state, • Normal restricted solid-on-solid model • : Edward-Wilkinson universality class • Two-particle correlated surface growth • - Yup Kim, T. S. Kim and H. Park, PRE 66, 046123 (2002) • Dimer-type surface growth • J. D. Noh, H. Park, D. Kim and M. den Nijs, PRE 64, 046131 (2001) • - J. D. Noh, H. Park and M. den Nijs, PRL 84, 3891 (2000) • Self-flattening surface growth • -Yup Kim, S. Y. Yoon and H. Park, PRE(RC) 66, 040602(R) (2002)

  3. 2 Steady state or Saturation regime , Partition function, nh: the number of columns with height h 1. Normal RSOS (z =1) Normal Random Walk (1D) (EW) 2. Two-particle correlated (dimer-type) growth (z = -1) nh=even number, Even-Visiting Random Walk (1D)

  4. 3 ? ? ? z z= 1 z= 0 z=-1 Even-Visiting Random Walk Self-attracting Random Walk Normal Random Walk Phase diagram (2D) ? z z= 1 z= 0 z=-1 Normal Random Walk Even-Visiting Random Walk Self-attracting Random Walk 3. Self-flattening surface growth (z = 0) Self-attracting random walk (1D) Phase diagram (1D)

  5. 4 Evaluate the weight ( nh : the number of sites which have the same height h ) in a given height configuration 2. Generalized Model Choose a column randomly. Decide the deposition (the evaporation) attempt with probability p (1-p) Calculate for the new configuration from the decided deposition (evaporation) process Acceptance parameter P is defined by

  6. 5 ( a primitive lattice vector in the i – th direction ) p =1/2 L = 10 z = 0.5 hmax hmin w w´ n´+2 = 2 n´+1 = 2 n´0 = 2 n´-1 = 2 n´-2 = 2 n+2 = 1 n+1 = 3 n 0 = 2 n-1 = 2 n-2 = 2 If P  1 , then new configuration is accepted unconditionally. If P < 1 , then new configuration is accepted only when P  R. where R is generated random number 0< R < 1 (Metropolis algorithm) Any new configuration is rejected if it would result in violating the RSOS contraint P  R

  7. 6 3. Simulation Results  Equilibrium model (1D, p=1/2)

  8. 7 z 1.5 1.1 1 0.9 0.5 0 -0.5 -1  (L)  0.33 0.33 1/2 0.33  0.33  0.33  0.34  0.33   0.22 0.22 1/4 0.22  0.22  0.22  0.22  0.19

  9. 7  Equilibrium model (2D, p=1/2)

  10. 7 , Z = 2.5 Scaling Collapse to in 2D equilibrium state.

  11. 7 Phase diagram in equilibrium (1D) z = 1 -1/2 1/2 3/2 z = 0 z =-1 z = 0.9 z = 1.1 2-particle corr. growth Self-flattening surface growth Normal RSOS Phase diagram in equilibrium (2D) z z= 1.5 z= 1 z= 0 z= 0.5 z=-1 z= -0.5 Normal Random Walk Even-Visiting Random Walk Self-attracting Random Walk

  12. 9 • Growing (eroding) phase (1D, p=1(0) ) z  0 : Normal RSOS model (Kardar-Parisi-Zhang universality class)

  13. 10 z  0 Normal RSOS Model (KPZ)

  14. 11 z  0 z=-0.5 p=1 L=128

  15. 12 4. Conclusion  Equilibrium model (1D, p=1/2) z 3/2 0.9 1.1 1 1/2 0 -1 -1/2 2-particle corr. growth (EVRW) Normal RSOS (Normal RW) Self-flattening surface growth (SATW) • Growing (eroding) phase (1D, p= 1(0) ) 1. z  0 : Normal RSOS model (KPZ universality class) 2. z  0 : Groove phase ( = 1) Phase transition at z=0 (?)

  16. Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5) 12-1

  17. 7 Slope a Model Monomer 0.175 Extremal 0.174 Dimer 0.162 2-site 0.175 Monomer & Extremal & Dimer & 2-site

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