Abstract
We review a recent asymptotic weak noise approach to the Kardar-Parisi-Zhang equation for the kinetic growth of an interface in higher dimensions. The weak noise approach provides a many-body picture of a growing interface in terms of a network of localized growth modes. Scaling in 1d is associated with a gapless domain wall mode. The method also provides an independent argument for the existence of an upper critical dimension.
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References
A-L Barabasi and H E Stanley, Fractal concepts in surface growth (Cambridge University Press, 1995)
J Krug and H Spohn, Solids far from equilibrium; Kinetic roughening of growing surfaces: Fractal concepts in surface growth (Cambridge University Press, 1992)
J Krug, Adv. Phys. 46, 139 (1997)
M Kardar, G Parisi and Y C Zhang, Phys. Rev. Lett. 56, 889 (1986)
E Medina, T Hwa, M Kardar and Y C Zhang, Phys. Rev. A39, 3053 (1989)
T Halpin-Healy and Y C Zhang, Phys. Rep. 254, 215 (1995)
D Forster, D R Nelson and M J Stephen, Phys. Rev. Lett. 36, 867 (1976); Phys. Rev. A16, 732 (1977)
M Kardar and Y C Zhang, Phys. Rev. Lett. 58, 2087 (1987)
M Kardar, Nucl. Phys. B290, 582 (1987)
D A Huse, C L Henley and D S Fisher, Phys. Rev. Lett. 55, 2924 (1985)
E Frey and U C Täuber, Phys. Rev. E50, 1024 (1994)
E Frey, U C Täuber and T Hwa, Phys. Rev. E53, 4424 (1996)
K J Wiese, J. Stat. Phys. 93, 143 (1998)
F Colaiori and M A Moore, Phys. Rev. Lett. 86, 3946 (2001)
E Katsav and M Schwartz, Physica A309, 69 (2002)
M Lässig, Phys. Rev. Lett. 80, 2366 (1998); Nucl. Phys. B448, 559 (1995)
M Lässig and H Kinzelbach, Phys. Rev. Lett. 78, 903 (1997)
P Le Doussal and K J Wiese, Phys. Rev. E72, 035101 (2005)
H C Fogedby, Phys. Rev. Lett. 94, 195702 (2005); Phys. Rev. E73, 031104 (2006); Phys. Rev. E68, 026132 (2003); Phys. Rev. E59, 5065 (1999); Phys. Rev. E57, 49431 (1998); Phys. Rev. Lett. 80, 1126 (1998)
L Onsager and S Machlup, Phys. Rev. 91, 1505 (1953), ibid 1512
P C Martin, E D Siggia and H A Rose, Phys. Rev. A8, 423 (1973)
R Baussch, H K Janssen and H Wagner, Z. Phys. B24, 113 (1976)
H C Fogedby, J Hertz and A Svane, Europhys. Lett. 62, 795 (2003)
H C Fogedby and V Poutkaradze, Phys. Rev. E66, 021103 (2002)
H C Fogedby and Ralf Metzler, Phys. Rev. Lett. 98, 070601 (2007); Phys. Rev. E76, 061915 (2007)
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Fogedby, H.C. Patterns in the Kardar-Parisi-Zhang equation. Pramana - J Phys 71, 253–262 (2008). https://doi.org/10.1007/s12043-008-0158-1
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DOI: https://doi.org/10.1007/s12043-008-0158-1
Keywords
- Scaling
- weak noise
- growth modes
- dynamical network
- upper critical dimension
- nonlinear Schrödinger equation
- domain walls
- solitons
- dispersion
- diffusive modes