Scaling Laws for KPZ Surface Growth Nicholas Chia

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Scaling Laws for KPZ Surface GrowthNicholas Chia
Committee MembersRalf BundschuhGustavo
LeoneJunko ShigemistuDavid StroudDongping
Zhong
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Outline

• The KPZ Equation
• Purpose
• Scaling Laws
• Properties
• Analytical Solutions
• Perturbation Theory
• Renormalization Group
• d 1, d ? 8, and higher dimensionalities
• Difficulties
• Fixed point
• Numerical models
• Eden model
• Ballistic deposition
• Experiment
• Flame fronts
• Biophysical example bacteria growth
• Universality and the continuum limit

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Why Bother?

• The Prize
• Understanding the dynamics of every physical
  interface imaginable
• Examples tumor cells, flame fronts, forest
  formation, geological formation, avalanches,
  shorelines, crystal defects, every sort of
  deposition
• The Problem
• Difficult equation to solve
• Perturbation theory results in diverging integral
• Renormalization Group (RG)
• Lack of a proper fixed point for d ? 2
• Universality does not apply to non-equilibrium
  systems
• Microscopic systems matter noise
• Different actions govern different interfacial
  surface perhaps no universal solution
• The Compromise
• Work around d 2
• Model microscopic behavior for specific problems
  universality classes

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Background

• Scaling Law

• Interfacial Width

• The EW equation

• The KPZ equation

• Noise

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General Properties

• Tilt and Translational Invariance

• Hopf Transformation

• Vortex Free Burgers Equation

• Stationary State Skewness
• Lack of particle-hole symmetry

• Relationship between ? and z

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Solving the KPZ Equation

• Perturbation Theory
• One loop calculations
• result in divergent integral
• Two loop diagrams cancel exactly for d 1

• Renormalization Group
• Momentum cutoff

• Examples of Other Approaches
• Functional renormalization
• Non-perturbative renormalization (C Castellano)
• Replica renormalization (M Lassig)
• Dimensional renormalization (K Wiese)

• Solutions
• Success for d 1, ? ½
• Upper-critical dimensionality?? d gt dc, ? 0,
  dc 4?
• High dimensionality ?upper ? 1/d
• Lässig conjecture ?2/(d2)
• Infinite dimensionality d ? ?, ? 0

• Problems
• d 2 ??
• Fixed point for d ? 2

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Fixed Point

• Fixed point for d 1 enables us to find a
  solution via perturbation theory.
• Lack of KPZ region fixed point for d ? 2
• Complete lack of nonlinear fixed point for
  the critical dimension, d 2

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Eden Model
Plischke and Rácz
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Ballistic Deposition
Family and Viscek
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Flame Fronts

• Experimental Works
• Zhang et al. ?0.71(7) lens paper 5.5 mm/s
• Maunuksela et al. ?0.48(1) copier paper 0.51
  mm/s
• Maunuksela et al. ??0.5 cigarette paper 1.64
  mm/s
• Report short range irregularities ??0.7-0.8
• Short range dynamics remain unexplained

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Bacterial Growth

• Highly nutrient dependent
• Rate limited reaction KPZ
• Nutrient limited reaction DLA
• EW behavior observable depends on agar
  hardness, nutrient level, inhibitor
  concentration, presence of pollutants.

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Universality

• Universality
• All similar processes can be described by the
  same equations, i.e. surface growth of any source
  should be described by KPZ. This because the
  details at the microscopic level does not effect
  the gross properties of the macroscopic level
• But
• Experiment and theoretical work show that
  microscopic details do matter, and that different
  surface growth systems exhibit different scaling
  behavior.
• All studies that verify the concept of
  universality are, in fact, equilibrium studies.
  So it might be best to limit universality to
  equilibrium processes.
• Zhang found that adjusting the noise term
  effected the scaling behavior of the KPZ system.
  This flies in the face of the concept of
  universality.

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Power Law Noise
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References

• J López and H Jensen, Phys. Rev. Lett. 81 1734
  (1998).
• M Kardar, G Parisi, and Y Zhang, Phys. Rev.
  Lett. 56 889 (1986).
• T Haplin-Healy and Y Zhang, Phys. Rep. 254 215
  (1995), and references therein.
• P Meakin, Phys. Rep. 235 189 (1993). and
  references therein.
• J Krug, Advances in Phys. 46 139 (1997), and
  references therein.
• M Nijs, lecture notes from the 6th APCTP Winter
  School (2002), and references therein.
• N Provatas, T Ala-Nissila, M Grant, K Elder, and
  Luc Piché, Phys. Rev. E 51 4232 (1995).
• J Maunuksela, M Myllys, O Kahkonen, J Timonen,
  N Provatas, M Alava, and T Ala-Nissila, Phys.
  Rev. Lett. 79 1515 (1997).
• M Myllys, J Maunuksela, M Alava, T Ala-Nissila,
  and J Timonen, Phys. Rev. Lett. 84 1946 (2000).
• M Kuittu, M Haataja, and T Ala-Nissila, Phys.
  Rev. E 59 2677 (1999).
• http//www.phys.washington.edu/dennijs/
• M Lässig, cond-mat9501094
• C Castellano, cond-mat9802284
• K Wiese, cond-mat9406009