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Growth and Pattern Formation in the KPZ equation

Guwahati - 2008. KPZ equation. 2. Outline. IntroductionWeak noise OscillatorInterfacePatternsConclusionExtras. Guwahati - 2008. KPZ equation. 3. Introduction. Non equilibrium processes is a fundamental issue in modern condensed matter and statistical physicsNo ensemble available - processes d

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Growth and Pattern Formation in the KPZ equation

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    1. Growth and Pattern Formation in the KPZ equation Hans Fogedby Aarhus University and Niels Bohr Institute

    2. Guwahati - 2008 KPZ equation 2 Outline Introduction Weak noise Oscillator Interface Patterns Conclusion Extras

    3. Guwahati - 2008 KPZ equation 3 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

    4. Guwahati - 2008 KPZ equation 4 Weak noise (1)

    5. Guwahati - 2008 KPZ equation 5 Weak noise (2)

    6. Guwahati - 2008 KPZ equation 6 Weak noise (3)

    7. Guwahati - 2008 KPZ equation 7 Oscillator (1)

    8. Guwahati - 2008 KPZ equation 8 Oscillator (2)

    9. Guwahati - 2008 KPZ equation 9 Interface (1)

    10. Guwahati - 2008 KPZ equation 10 Interface (2)

    11. Guwahati - 2008 KPZ equation 11 Interface (3)

    12. Guwahati - 2008 KPZ equation 12 Interface (4)

    13. Guwahati - 2008 KPZ equation 13 Patterns (1)

    14. Guwahati - 2008 KPZ equation 14 Patterns (2)

    15. Guwahati - 2008 KPZ equation 15 Patterns (3)

    16. Guwahati - 2008 KPZ equation 16 Patterns (4)

    17. Guwahati - 2008 KPZ equation 17 Patterns (5)

    18. Guwahati - 2008 KPZ equation 18 Patterns (6)

    19. Guwahati - 2008 KPZ equation 19 Patterns (7)

    20. Guwahati - 2008 KPZ equation 20 Conclusion

    21. Extras

    22. Guwahati - 2008 KPZ equation 22 General stochastic description

    23. Guwahati - 2008 KPZ equation 23 General weak noise approximation (1)

    24. Guwahati - 2008 KPZ equation 24 General weak noise approximation (2)

    25. Guwahati - 2008 KPZ equation 25 Bound state solution for the NLSE

    26. Guwahati - 2008 KPZ equation 26 Dynamical network

    27. Guwahati - 2008 KPZ equation 27 Growth in d=1

    28. Guwahati - 2008 KPZ equation 28 KPZ equation Height profile of interface: h(r,t) Damping coefficient: n Growth parameter: l Constant drift: F Noise representing environment: h Noise strength: D

    29. Guwahati - 2008 KPZ equation 29 KPZ scaling properties Dynamical Renormalization group calculation (DRG) Expansion in d-2 d=2 lower critical dimension Strong coupling fixed point in d=1, z=3/2 Kinetic phase transition for d>2 zL Lssig (operator expansion) zWK Wolf-Kertesz (numerical) zKK Kim-Kosterlitz (numerical) d=4 upper critical dimension

    30. Guwahati - 2008 KPZ equation 30 Overdamped oscillator

    31. Guwahati - 2008 KPZ equation 31 Phase space description

    32. Guwahati - 2008 KPZ equation 32 Stochastic Quantum analogue

    33. Guwahati - 2008 KPZ equation 33 Upper critical dimension Upper critical dimension usually considered in scaling context Mode coupling gives d=4; above d=4 maybe glassy, complex behavior DRG shows singular behavior in d=4 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

    34. Guwahati - 2008 KPZ equation 34 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 d<4 Above d=4 no bound state no growth

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