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Classification of Two-dimensional Surface Growth Models using Schramm -Loewner Evolution

Dashti Naserabadi, Hor | 2013

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 44301 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Rohani, Shahin; Saberi, Abbas Ali
  7. Abstract:
  8. Rough surfaces and growth process are the important and significant problems of theoretical and condensed matter physics to model phenomena ranging from the extremely small (biological phenomena) to largest one (Earth’s relief). Although the equations that describe these processes are well defined, but the question of how to characterize these surfaces which display large fluctuations, is open. The existence of (often) scale invariant clusters and very large fluctuations in surface growth process are reminiscent of critical fluctuations in equilibrium systems. Therefore, it is natural to try characterizing the surface by means of critical exponents, i.e., by the scaling dimensions of various macroscopic quantities. The quantities such as surface width and correlation length have scaling exponents, e.g., , and z. In the most cases one cannot derive these exponents analytically. The usual way is simulation of surface growth models which often contains large numeric errors. In this thesis, we have introduced another way to study different surface growth models that is based on iso-height lines and classification of interface by statistical properties of these lines. Also, we have shown that if the boundary conditions have been defined nicety and the lines have been extracted accurately, then these contours have conformal symmetry and are described by the Schramm-Loewner evolution (SLE). The Schramm-Loewner evolution has provided the possibility of investigation of conformal symmetry in some geometrical patterns of complex systems in continuum limit. This evolution reduces the median of two dimensional critical systems to one dimensional Brownian motion problem. Futhermore, the diffusion constant of SLE () characterizes the universality classes of some of the two dimensional critical systems. We have shown that the statistics of iso-height lines for the three discrete models, i.e., ballistic deposition (BD), Eden and restricted solid-on-solid (RSOS) are in selfavoiding walk (SAW) universality class with = 8=3. Moreover, we have studied single-step model (SSM) with tunable parameter p and saw that contrary to previous reports, there exists a critical value pc 0:25, such that for p > pc (p < pc), the SSM lies in the EW (KPZ) universality class
  9. Keywords:
  10. Surface Growth ; Conformal Invariance ; Conformal Fields Theory ; Iso Height Lines ; Schramm-Loewner Evolution (SLE)

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