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국문초록 12

Abstract 13

1. Introduction 14

2. Scaling theory and universality class 17

2.1 Brief introduction to scaling concepts 17

2.2 The second-order growth equations: EW and KPZ 21

2.3 The fourth-order growth equations: MH and NMH 23

3. Anisotropic model in two dimensions 26

3.1 Dynamic rules 27

3.2 Numerical results 28

3.3 Results on a rectangular substrate 32

3.4 Conclusion 34

4. Restricted curvature model with suppression in two dimensions 36

4.1 Restricted curvature model with suppression 37

4.2 Numerical Results for d=2+1 39

4.3 Conclusion 44

5. Depinning transition by quenched disorder 46

5.1 Basic phenomena 47

5.2 Critical behaviors and scaling relations 49

6. Depinning transitions of quenched directed polymer 51

6.1 Recursion relation in zero temperature limit 52

6.2 Directed polymer model with columnar noise 53

6.3 Quenched directed polymer model 59

6.4 conclusion 64

7 Depinning transitions of QKPZ and QEW equation 66

7.1 Quenched KPZ equation 67

7.2 Quenched EW equation 75

7.3 Depinning transition without an external driving force 81

7.4 Conclusion 85

8. Mullins-Herring equation with quenched noise 87

8.1 Numerical results 88

8.2 Conclusion 89

References 94

표목차

Table 7.1: Summary of exponents for our work. All values are approximate quantities, rounding up at three decimal places 85

Table 8.1: Summary of exponents for the QMH equation. All values are approximate quantities, rounding up at three decimal places 89

그림목차

Figure 2.1: Log-log plot of the time evolution of the surface width for the RSOS model 20

Figure 2.2: According to the scaling relation (2.5), this presents the log-log plot of W²(L; t)=L²α against t=L z by using α = 0:50 and z = 1.50 20

Figure 3.1: Log-log plot of W?L; t) as a function of time t for (N rsos = 1, N rc = 4) 29

Figure 3.2: Log-log plot of W?L; t), W?rc(L; t), and W?rsos(L; t) as a function of time t, with L = 128 30

Figure 3.3: Log-log plot of the rescaled width W�rsos=W�rsos, versus the rescaled time 31

Figure 3.4: Rescaled plot of W²(L; t) . We use α = 0:95 and z = 4.0 32

Figure 3.5: Log-log plot of W?for systems on rectangular substrates, L rc X L rsos 33

Figure 3.6: The time evolution of the surface width for each axes 33

Figure 4.1: Log-log plot of the saturated width W?as a function of L for L = 12, 16,24, 32, 48, 64, 96 40

Figure 4.2: Rescaled data collapse to a single curve with α = 0:646 and z = 2:760, for P out = 0.10, where L = 16, 24, 32, 48, 64, and 96 41

Figure 4.3: Log-log plot of the rescaled height-height correlation function G(r; t)=r² α' versus rescaled distance r/t¹/z' with α' = 0.95 and z' = 4.04 for P out = 0:10 with 42

Figure 4.4: Log-log plot of the rescaled saturated height-height correlation function Gs(r)=L²α versus rescaled distance r/Lδ with α = 0.64 and δ = 0:68, for P out = 0:10 43

Figure 6.1: log-log plot of average velocity v(t) for F = 0.0498, 0.0500, and 0.0502 for L = 10⁴. At Fc = 0.0500, we get δ = 0.50 ± 0:01 for L = 10 5 55

Figure 6.2: Log-log plot of W?as a function of time t at Fc = 0.05, for L = 32, 64,128, 256, 512, 1024, and 10 5 from the bottom to the top 56

Figure 6.3: Plot of v as a function of ｜F― Fc｜. We get θ ∼1.00 from the relation v ~ ｜F― Fc｜θ.｜ 57

Figure 6.4: Scaling plot of log v(L, t)θ/vz versus log t=L/zv with θ = 1.0, vx = 1.0, and zv = 2.0 at Fc 58

Figure 6.5: The log-log plot of v(t) as a function of t at the transition point F = 0:05 in d = 1 + 1 59

Figure 6.6: The plot of log [ △E ]₂against log(t) for various value of L at F = 0:05 in d = 1+1 60

Figure 6.7: The plot of [ x² ] as a function of t in d = 1 + 1 for F = 0:10 (top) and 0.05 (bottom) with △ = 0:05 62

Figure 6.8: In d = 1 + 1, the plot of v(m, f) as a function of m for various values of fwith △ = 0:05 63

Figure 6.9: The plot of v(m, f)/fθ as a function of m² according to Eq.(6.14) 64

Figure 6.10: In d = 2 + 1, the plot of log [ △E ]² against log(t) for L = 10, 20, 40, 60, and 80 (from the bottom to the top) 65

Figure 7.1: Log-log plot of v(t) as a function of time t. A strong line indicates Fc =0:049 68

Figure 7.2: Log-log plot of vtδ as a function of time t, with δ = 0:36 68

Figure 7.3: Log-log plot of vtδ against t=Lz at Fc, with δ = 0.36 and z = 0:97. Log-log plot of v against t is shown in the inset 70

Figure 7.4: At Fc = 0.049, log-log plot of the scaled W² (L; t) with α = 0.64 and z = 0.97 70

Figure 7.5: Log-log plot of the stationary state velocity v sat as a function of (F-Fc)-=Fc with a system size L = 4096 71

Figure 7.6: The scaling plot according to Eq. (7.5) with δ = 0:36 and vt = 1.73 with 0:03 ≤ F≤ 0.07 with L = 16384 72

Figure 7.7: Scaling plot of vLθ/vz against t=Lz with θ/vx = 0.36 and z = 1 at Fc 72

Figure 7.8: Log-log plot of W²sat as a function of (Fc - F)/Fc for L = 4096 and 0.001 ≤ F 0.048 74

Figure 7.9: Log-log plot of H sat as a function of (Fc - F)/Fc for L = 4096 and 0.001 ≤ F 0.048 74

Figure 7.10: Log-log plot of v(t) as a function of time t near Fc = 0.9306 77

Figure 7.11: Log-log plot of vtδ against t near Fc with δ = 0.1425 77

Figure 7.12: Log-log plot of vtδ against t=Lz with δ = 0:143 and z = 1.4 at Fc. The inset shows log-log plot of v(t) 78

Figure 7.13: At Fc = 0.9306, log-log plot of the time evolution the surface width. We get β = 0.839(2) for L = 16384 78

Figure 7.14: Log-log plot of v sat as a function of (F - Fc)/Fc for L = 4096 and 0.933 ㅎ≤ F ≤ 0.97 79

Figure 7.15: Log-log plot of vtδ against t｜F - Fc｜vt with δ = 0.143 and vt = 1.97 for L = 16384 and 0.9286 ≤ F ≤ 0.9324 79

Figure 7.16: Log-log plot of vLθ/vz against t/L z at Fc with θ = 0.28, vx = 1.4 and z = 1.4 80

Figure 7.17: Log-log plot of W²sat as a function of (Fc - F)/Fc for L = 4096 and 0.1 ≤ F ≤ 0.929 80

Figure 7.18: Log-log plot of H sat as a function of (Fc - F)/Fc for L = 4096 and 0.1 ≤ F ≤ 0.929 81

Figure 7.19: Log-log plot of v(t) as a function of time t near λc = 6.944 for L = 16384 84

Figure 7.20: At λ = 6.944, log-log plot of the time evolution of the surface width 84

Figure 7.21: Log-log plot of vtδ against t/Lz with δ = 0.34 and z = 0.97 at λc 85

Figure 8.1: Log-log plot of v(t) as a function of time t near Fc = 0.988 90

Figure 8.2: Log-log plot of vtδ against t near Fc with δ = 0.16 90

Figure 8.3: Log-log plot of vtδ against t/Lz with δ = 0.16 and z = 1.6 at Fc 91

Figure 8.4: At Fc = 0.988, log-log plot of the time evolution the surface width 91

Figure 8.5: Log-log plot of vsat as a function of (F - Fc)/Fc for L = 4096 and 0.99 ≤ F≤ 1.095 92

Figure 8.6: Log-log plot of vtδ against t｜F - Fc｜vt with δ = 0.16 and vt = 1.81 for L = 16384 and 0.985 ≤ F ≤ 0.99 92

Figure 8.7: Log-log plot of vLθ/vz against t/Lz at Fc with θ = 0.29 , vz = 0.95 and z = 1.6 93

Figure 8.8: Log-log plot of W²sat and H sat as a function of (Fc - F)/Fc for L = 4096 and 0.001 ≤ F≤ 0.987 93

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