Growth and Pattern Formation in the KPZ equation

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Growth and Pattern Formation in the KPZ equation

• Hans Fogedby
• Aarhus University
• and
• Niels Bohr Institute

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Outline

• Introduction
• Weak noise
• Oscillator
• Interface
• Patterns
• Conclusion
• Extras

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Introduction

• Non equilibrium processes is a fundamental issue
  in modern condensed matter and statistical
  physics
• No ensemble available - processes defined by
  dynamics, e.g. Master equation, Langevin
  equation, Fokker-Planck equation
• Search for new methods in non equilibrium physics
• Present work Weak noise approach

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Weak noise (1)
Langevin equation

• Noise ? drives x into stationary stochastic state
  
• Noise strength ? singular parameter
• ?0, relaxational behavior
• ?0, stationary behavior
• Crossover time diverges as ??0

Drift Noise
Noise correlations
Working hypothesis Weak noise limit
??0 captures interesting physics
Noise strength
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Weak noise (2)
Fokker-Planck equation

• ? is effective Planck constant
• ?d/dx is the momentum operator
• Distribution P is the wave function
• Weak noise corresponds to classical limit (??0)

Diffusion Drift
Schrödinger equation in imaginary time
WKB ansatz
Classical action
Kinetic energy
Potential energy (velocity dependent)
Time evolution
Noise strength
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Weak noise (3)
Classical limit ??0
Hamiltonian
Weak noise recipe

• Solve equations of motion for orbit from initial
  xi to final x in time T
• Momentum p is a slaved variable
• Evaluate action S(xi,x,T) for orbit
• Evaluate transition probability according to WKB
  ansatz
• P(xi,x,T)exp- S(xi,x,T)/?

Equations of motion
Noise replaced by momentum
Equation for momentum
Comments
Action

• Stochastic problem replaced by dynamical problem
  in weak noise limit
• Dynamical action is generic weight function
  (compare with Boltzmann factor exp(-E/kT))

Details
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Oscillator (1)
Langevin equation
Action
Hamiltonian
Distribution
Equations of motion
Text book result
Stationary distribution
Gaussian distribution
Details
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Oscillator (2)
Canonical phase space
Stationary manifold
Finite time orbit
Infinite time orbit
Transient manifold
Saddle point (long time orbits pass close to
saddle point)
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Interface (1)
Growing interfaces some examples
Deposition of snow
Bacterial growth
Molecular beam epitaxi
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Interface (2)
The KPZ equation

• The Kardar-Parisi-Zhang (KPZ) equation is a
  generic continuum nonequilibrium growth model
• The KPZ equation is a nonlinear Langevin equation
  driven by locally correlated white noise
• The KPZ equation is at a critical point and has
  strong coupling scaling properties
• The KPZ equation is amenable to the weak noise
  analysis

Diffusion Growth Drift Noise
Noise correlations
Noise strength
White noise
Height field
Details
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Interface (3)
Nonlinear Cole-Hopf transformation
maps the KPZ equation

• Langevin equation with multiplicative noise
• Weak noise scheme can be extended to the case of
  multiplicative noise

to the Cole-Hopf equation
Multiplicative noise
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Interface (4)
Weak noise scheme
Action
Cole-Hopf Hamiltonian
Parameters
Transition probability
Phase space
Field equations
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Patterns (1)
Program

• Seek localized solutions to static field
  equations
• Boost static solutions to propagating growth
  modes
• Construct dynamical network of dynamical growth
  modes

Static field equations
Linear diffusion equation
Nonlinear Schrödinger equation
consistent for p0 (or w0) identical for pw
Nonlinear
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Patterns (2)
Diffusion equation

• Cole-Hopf field exponentially growing
• Height field linearly growing cone (pit)
• Slope field constant amplitude monopole field -
  positive charge
• k continuous charge, k2F, F drift in KPZ
  equation
• Drift F fixes charge k

Cone
Radial Cole-Hopf, height and slope fields
Positive monopole Radial vector field
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Patterns (3)
NLSE bound state
Nonlinear Schrödinger equation (NLSE)

• Cole-Hopf field exponentially damped
• Height field linearly growing inverted cone (tip)
• Slope field fixed amplitude monopole field -
  negative charge
• k continuous charge, k2F, F drift in KPZ
  equation
• Drift F fixes charge k

Inverted cone
Radial Cole-Hopf, height and slope fields
Negative monopole Radial vector field
Details
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Patterns (4)
Galilei boost
Growth modes in 2D
Moving frame
Constant slope
Constant shift
Tilted height cone
Propagating growth modes
Shifted vector slope field
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Patterns (5)
Dynamical network

• Localized modes as building blocks
• Boost modes by means of Galilei transformation
• Treat amplitudes k as charges
• Use charge language positive and negative
  charges
• Growing mode, positive charge
• Decaying mode, negative charge
• Impose flat interface at infinity zero slope
• Form self-consistent dynamical network from nodes
• Convenient to implement for slope field
• Flat interface at infinity corresponds to charge
  neutrality
• Evolving network corresponds to growing interface

Details
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Patterns (6)
Dipole mode in 2D
Propagating height mode
Propagating slope mode
Velocity
Monopoles
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Patterns (7)
Four-monopole height profile in 2D
Asymptotically flat interface with height
offset As monopoles propagate subject to
periodic boundary conditions interface grows
Height field
x-y plane
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Conclusion

• Nonperturbative asymptotic weak noise approach
• Equivalent to WKB approximation in QM
• Stochastic problem mapped to dynamical problem
• Stochastic equation replaced by dynamical
  equations
• Canonical (symplectic) scheme
• Scheme captures strong coupling features
• For KPZ equation localized propagating growth
  modes
• Dynamical network represents strong coupling
  aspects
• Method yields a picture of stochastic pattern
  formation
• Only NLSE localized mode below d4, upper
  critical dimension for KPZ equation ! (details)

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Extras
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General stochastic description
Generic Langevin equation for stochastic field
wn(t) (Stratonovich formulation)
Generic Fokker-Planck equation for distribution
P(wn,t)
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General weak noise approximation (1)
Equations of motion
WKB ansatz
Hamilton-Jacobi equation
Action
Hamiltonian
Reduced action
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General weak noise approximation (2)
Generic canonical phase space
Stationary manifold (zero energy, H0)
Finite time orbit (on H?0 manifold
Infinite time orbit (on H0 manifolds)
Saddle point (Markov behavior)
Transient manifold (zero energy, H0)
Back
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Bound state solution for the NLSE
Solution of radial NLSE by Runge-Kutta
(matlab) In d1
Bound states (numerical)
w_(r)
1D domain wall
In higher d bound state narrows, amplitude
increases In d4 bound state disappears
r
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Dynamical network
Construct network of static modes in terms of
vector slope field
Assign velocities to modes according to Galilean
invariance and matching conditions
Boost modes to assigned velocities
Construct self-consistent dynamical network in
terms of slope field and height field
Back
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Growth in d1
Growing interface
Positive charge Right hand domain wall
Negative charge Left hand domain wall
Back
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KPZ equation
Growing interface

• Height profile of interface h(r,t)
• Damping coefficient n
• Growth parameter l
• Constant drift F
• Noise representing environment h
• Noise strength D

Nonlinear term
Lateral growth
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KPZ scaling properties

• Dynamical Renormalization group calculation (DRG)
• Expansion in d-2
• d2 lower critical dimension
• Strong coupling fixed point in d1, z3/2
• Kinetic phase transition for dgt2
• zL Lässig (operator expansion)
• zWK Wolf-Kertesz (numerical)
• zKK Kim-Kosterlitz (numerical)
• d4 upper critical dimension

Strong coupling fixed point ??2/?3
DRG phase diagram
Spatial dimension d
Back
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Overdamped oscillator
Langevin equation for overdamped oscillator
elementary step
Example
white noise
Mean square displacement
Suspended Brownian particle of size R in viscous
medium with viscosity ?
Fokker-Planck equation for overdamped oscillator
Stokes law
Time-dependent distribution
Stationary distribution
Back
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Phase space description

• Stochastic Langevin equation replaced by coupled
  deterministic Hamilton equations of motion
• Noise replaced by canonical momentum p
• Solutions of equations of motion determine orbits
  in canonical phase space spanned by x and p
• Energy is conserved orbits lie on energy
  surfaces
• Long time orbits through saddle point (SP)
• Orbits on zero-energy surface yield stationary
  state
• Action evaluated for orbit from x0 to x in time t
  yields transition probability from x0 to x in
  time t

Canonical Phase Space
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Stochastic Quantum analogue

• Distribution corresponds to wave function
• Fokker-Planck equation corresponds to
  Schrödingers equation
• Noise yields kinetic energy
• Drift yields (v-dependent) potential energy
• Small noise strength corresponds to small Planck
  constant
• Weak noise corresponds to small quantum
  fluctuations, i.e., the correspondence limit

• WKB approximation yields in both cases a
  classical action
• Principle of least action is operational
• Classical orbits and phase space discussion
• Structure of energy manifolds different
• Stationary zero-energy state corresponds to
  saddle point in stochastic case
• Method goes back to Onsager

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Upper critical dimension
General remarks

• Upper critical dimension usually considered in
  scaling context
• Mode coupling gives d4 above d4 maybe glassy,
  complex behavior
• DRG shows singular behavior in d4
• Numerics inconclusive!
• Issue of upper critical dimension unclear and
  controversial
• In present context we interprete upper critical
  dimension as dimension beyond which growth modes
  cease to exist
• Numerical computation of bound state shows
• d4

DRG phase diagram
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Proof by Derricks theorem

• NLSE from variational
• principle yields Identity 1
• Scale transformation
• yields Identity 2
• Identity 2 involves
• dimension d
• Demanding finite norm of
• bound state implies dlt4
• Above d4 no bound state
• no growth

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