ASPECTS OF NON EQUILIBRIUM

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ASPECTS OF NON EQUILIBRIUM

• Mapping of stochastic equations to Hamilton
  equations of motion

Hans Fogedby Aarhus University and Niels Bohr
Institute Denmark
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Outline

• Equilibrium
• Non equilibrium
• Quantum analogue
• Weak noise - WKB
• Dynamical equations of motion
• Phase space discussion
• Brownian motion
• Oscillator
• Kardar-Parisi-Zhang equation for interface
• Summary and conclusion

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EQUILIBRIUM
Ensemble known - Boltzmann/Gibbs scheme

• P0 Equilibrium distribution
• H(C) Hamiltonian (energy)
• C Configuration
• kBT Temperature
• F Free energy
• S Entropy

Boltzmann factor
Partition function
Thermodynamics
Ising model

• HI Ising Hamiltonian
• si Spin, si 1
• J NN Coupling
• si Configuration
• H Magnetic field
• M Magnetic moment

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NON EQUILIBRIUM
Ensemble unknown - Master/Langevin/Fokker-Planck
scheme
Master equation

• Open systems
• Biophysics
• Soft matter
• Turbulence
• Interface growth
• etc

Langevin equation
Fokker-Planck equation
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QUANTUM ANALOGUE
Fokker Planck equation

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

Diffusion
Drift
Transformation
Schrödinger equation
Kinetic energy
p-dependent potential
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WEAK NOISE
Langevin equation

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

Drift
Noise
Noise correlations
Noise strength
Working hypothesis Weak noise limit captures
interesting physics
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WEAK NOISE
Fokker-Planck equation
Action
WKB ansatz
Variational principle
Hamilton-Jacobi equation

• Principle of least action operational
• in weak noise limit
• Stochastic problem replaced by
• dynamical problem
• Application of dynamical system theory
• Dynamical action S is weight function

Hamiltonian
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DYNAMICAL EQUATIONS
Hamiltonian
Weak noise recipe
Equations of motion

• Solve equations of motion for orbit from xi to x
  in time T
• Momentum p is slaved variable
• Evaluate action S for orbit
• Evaluate transition probability according to WKB
  ansatz
• Stationary distribution obtained in long time
  limit for fixed x

Noise replaced By momentum
Equation for momentum
Action and distribution
WKB
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PHASE SPACE DISCUSSION
Hamiltonian
Equations of motion
Action

• Long time orbits on H0 manifolds
• H0 manifolds yield stationary state
• Saddle point - infinite waiting time
• (Markov behavior)

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BROWNIAN MOTION

• Basic random process
• Model for diffusion
• Wide applications in statistical physics

Solutions
Action and distribution
Langevin equation
Gaussian, width T
Phase space
Hamiltonian (free particle)
Equations of motion
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OSCILLATOR (overdamped)

• Simple random process
• Wide applications in statistical physics and
  biophysics
• E.g. suspended Brownian particle in viscous medium

Mean square displacement
Fokker-Planck equation
Distribution
Langevin equation
Stationary distribution
Gaussian
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Weak noise approach
Langevin equation
Action
Hamiltonian
Distribution
Stationary distribution
Equation of motion
Gaussian
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Phase space discussion
Stationary manifold
Finite time orbit
Infinite time orbit
Transient manifold
Saddle point (long time orbit passes close to SP)
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KARDAR PARISI ZHANG EQUATION

• Generic non equilibrium model
• Describes aspects of growing interfaces
• Field theoretical Langevin equation
• Scaling properties
• Related to turbulence
• Related to disordered systems
• Weak noise method can be applied

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KARDAR PARISI ZHANG EQUATION

• h(r,t) height profile
• ? damping
• ? growth parameter
• F drift
• ?(r,t) noise
• ? noise strength

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KPZ - scaling
Dynamical scaling hypothesis

• Saturation width w
• System size L
• Roughness exponent ?
• Dynamic exponent z
• Scaling function F

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KPZ - scaling
DRG phase diagram

• Dynamical Renormalization
• Group (DRG) calculation
• d2 lower critical dimension
• Expansion in d-2
• Strong coupling fixed point
• in d1, z3/2
• Kinetic phase transition for
• dgt2
• d4 upper critical dimension

Scaling law
DRG equation
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Weak noise approach
KPZ equation
Action
Cole-Hopf transformation to diffusive field w
Distribution
Hamiltonian
Phase space
Field equations of motion
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Pattern formation
Program

• Find localized solutions to static field
  equations
• Boost static solutions to moving growth modes
• Construc dynamical network of dynamical growth
  modes

New parameters
Diffusion equation
Static field equations
Non linear Schrödinger equation
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Diffusion equation
Radial diffusion equation
Solutions for w, height h and slope u
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Non linear Schrödinger equation
Radial NLSE
Solutions for w, height h and slope u
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Galilei transformation
Galilei boost
Growth modes in 2D
Moving growth modes
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Dynamical network
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Growth in d1
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Dipole modes in 2D
Moving height mode
Moving slope mode
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4-monopole height profile in 2D
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SUMMARY AND CONCLUSION

• Nonperturbative asymptotic weak noise approach
• Equivalent to the WKB approximation in QM
• Stochastic problem mapped to dynamical problem
• Stochastic equation replaced by dynamical
  equations
• Principle of least action operational in weak
  noise limit
• Canonical phase space representation
• Application of dynamical system theory
• Application to the KPZ equation - many body
  formulation

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The End
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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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Upper critical dimension
General remarks

• Upper critical dimension usually considered in
  scaling context
• Mode coupling gives d4 above d4 maybe glassy,
  complex behavior (Moore et al.)
• DRG shows singular behavior in d4 (Wiese)
• 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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Scaling in 1D
Dipole mode
Dynamics
Dispersion law

• Stochastic interpretration
• Spectrum of dipole mode
• Gapless dispersion
• Spectral representation
• Comparison with dynamical scaling ansatz
• Dispersion law exponent yields dynamical exponent
  z

Spectral representation
Dynamical scaling
Dynamical scaling exponent
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Scaling in higher D
Dipole mode
Action
Distribution
Pair velocity
Displacement
Distance in time T
Hurst exponent
Dipole random walk
Hurst exponent vs d
Dynamic exponent