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PATTERNS IN THE KPZ EQUATION

PATTERNS IN THE KPZ EQUATION. Mapping of stochastic equations to Hamilton equations of motion. Hans Fogedby Aarhus University and Niels Bohr Institute Denmark. Outline. Equilibrium Non equilibrium Quantum analogue Weak noise - WKB Dynamical equations of motion Phase space discussion

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PATTERNS IN THE KPZ EQUATION

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  1. PATTERNS IN THE KPZ EQUATION Mapping of stochastic equations to Hamilton equations of motion Hans Fogedby Aarhus University and Niels Bohr Institute Denmark

  2. Outline • Equilibrium • Non equilibrium • Quantum analogue • Weak noise - WKB • Dynamical equations of motion • Phase space discussion • Brownian motion • Oscillator • Finite-time-singularity • Kardar-Parisi-Zhang equation for interface • Summary and conclusion NON EQUILIBRIUM

  3. EQUILIBRIUM Ensemble known - Boltzmann/Gibbs scheme • P0: Distribution • H(C): Hamiltonian • C: Configuration • kBT: Temperature • F: Free energy • S: Entropy Boltzmann factor Partition function Thermodynamics Example: • HI: Ising Hamiltonian • σi : Spin, σi =±1 • J: NN Coupling • {σi}: Configuration • H: Magnetic field • M: Magnetic moment Ising model NON EQUILIBRIUM

  4. Open systems Biophysics Soft matter Turbulence Interface growth etc NON EQUILIBRIUM Ensemble unknown - Master/Langevin/Fokker-Planck scheme Master equation Langevin equation Fokker-Planck equation NON EQUILIBRIUM

  5. 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 QUANTUM ANALOGUE Fokker Planck equation Diffusion Drift Transformation Schrödinger equation Kinetic energy p-dependent potential NON EQUILIBRIUM

  6. WEAK NOISE Langevin equation • Noise drives x into a stationary • stochastic state • Noise strength singular parameter • Δ=0, relaxational behavior • Δ~0, stationary behavior • Cross over time diverges as Δ → 0 Drift Noise Noise correlations Noise strength Working hypothesis: Weak noise limit captures interesting physics NON EQUILIBRIUM

  7. WEAK NOISE - WKB 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 NON EQUILIBRIUM

  8. 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 DYNAMICAL EQUATIONS Hamiltonian Langevin equation Equations of motion Weak noise recipe Noise replaced by momentum Equation for momentum Action and distribution WKB NON EQUILIBRIUM

  9. PHASE SPACE DISCUSSION Hamiltonian Equations of motion Action • Long time orbits on H=0 manifolds • H=0 manifolds yield stationary state • Saddle point - infinite waiting time • (Markov behavior) NON EQUILIBRIUM

  10. Basic random process Model for diffusion Wide applications in statistical physics BROWNIAN MOTION Solutions Action and distribution Langevin equation Gaussian, width ~T Phase space Hamiltonian (free particle) Equations of motion NON EQUILIBRIUM

  11. 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 NON EQUILIBRIUM

  12. Weak noise approach Langevin equation Action Hamiltonian Distribution Stationary distribution Equations of motion Gaussian NON EQUILIBRIUM

  13. Phase space discussion Stationary manifold Finite time orbit Infinite time orbit Transient manifold Saddle point (long time orbit passes close to SP) NON EQUILIBRIUM

  14. FINITE-TIME-SINGULARITIES Generic model with singularity FTS encountered in … • Stellar structure • Turbulent flow • Bacterial growth • Euler flow • Free surface flow • Econophysics • Geophysics • Material physics Finite-time solution Sharp event horizon NON EQUILIBRIUM

  15. NOISY FINITE-TIME-SINGULARITY Absorbing state distribution W(t) • Noise smears out singularity • Event horizon becomes fuzzy • e.g. hydrodynamical flow on nano scale near finite-time singularity Generic model with noise Noise intrinsic or thermal, Δ ≈ T Power law behavior Scaling exponent NON EQUILIBRIUM

  16. Weak noise approach FTS model with noise Phase space Hamiltonian Equations of motion Long time analysis yields absorbing state distribution and scaling exponent Action and distribution NON EQUILIBRIUM

  17. KARDAR PARISI ZHANG EQUATION • Generic non equilibrium model • Describes aspects of growing interface • Field theoretical Langevin equation • Scaling properties • Related to turbulence • Related to disordered systems • Weak noise method can be applied NON EQUILIBRIUM

  18. KARDAR PARISI ZHANG EQUATION • h(r,t): height profile • ν: damping • λ: growth parameter • F: drift • η(r,t): noise • Δ: noise strength NON EQUILIBRIUM

  19. KPZ - scaling Dynamical scaling hypothesis • Saturation width: w • System size: L • Roughness exponent: ς • Dynamic exponent: z • Scaling function: F NON EQUILIBRIUM

  20. KPZ - scaling DRG phase diagram • Dynamical Renormalization • Group (DRG) calculation • d=2 lower critical dimension • Expansion in d-2 • Strong coupling fixed point • in d=1, z=3/2 • Kinetic phase transition for • d>2 • d=4 upper critical dimension Scaling law DRG equation NON EQUILIBRIUM

  21. Weak noise approach KPZ equation Action Cole-Hopf transformation to diffusive field w Distribution Hamiltonian Phase space Field equations of motion NON EQUILIBRIUM

  22. Pattern formation Program • Find localized solutions to static field equations • Boost static solutions to moving growth modes • Construct dynamical network of dynamical growth modes New parameters Diffusion equation Static field equations Non linear Schrödinger equation NON EQUILIBRIUM

  23. Diffusion equation Radial diffusion equation Solutions for w, height h and slope u NON EQUILIBRIUM

  24. Non linear Schrödinger equation Radial NLSE Solutions for w, height h and slope u NON EQUILIBRIUM

  25. Galilei transformation Galilei boost Growth modes in 2D Moving growth modes NON EQUILIBRIUM

  26. Dynamical network NON EQUILIBRIUM

  27. Growth in d=1 NON EQUILIBRIUM

  28. Dipole modes in 2D Moving slope mode Moving height mode NON EQUILIBRIUM

  29. 4-monopole height profile in 2D NON EQUILIBRIUM

  30. 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 NON EQUILIBRIUM

  31. The End NON EQUILIBRIUM

  32. Bound state solution for the NLSE Solution of radial NLSE by Runge-Kutta (matlab) In d=1 Bound states (numerical) w_(r) 1D domain wall In higher d bound state narrows, amplitude increases In d≥4 bound state disappears! r NON EQUILIBRIUM

  33. Upper critical dimension General remarks • Upper critical dimension usually considered in scaling context • Mode coupling gives d=4; above d=4 maybe glassy, complex behavior (Moore et al.) • DRG shows singular behavior in d=4 (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 d=4 DRG phase diagram NON EQUILIBRIUM

  34. NLSE from variational principle yields Identity 1 Scale transformation yields Identity 2 Identity 2 involves dimension d Demanding finite norm of bound state implies d<4 Above d=4 no bound state – no growth Proof by Derrick’s theorem NON EQUILIBRIUM

  35. 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 NON EQUILIBRIUM

  36. 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 NON EQUILIBRIUM

  37. Stationary state Langevin level Fokker Planck level Weak noise method Phase space (one degree of freedom) NON EQUILIBRIUM

  38. Stationary state1D Noisy Burgers case Phase space (1D Burgers) Hamiltonian manifolds Stationary state Comments: • Stationary state given by zero-energy manifold • Zero-energy manifold not identified in D>1 • Hamiltonian density has form: NON EQUILIBRIUM

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