Upper critical dimension, dynamic exponent, and scaling functions in the mode-coupling theory for the Kardar-Parisi-Zhang equation

Physical Review Letters

Volumn 86, Issue 18, 2001, Pages 3946-3949

  Colaiori, Francesca ^a   Moore, M A ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; COMPUTER SIMULATION; CORRELATION METHODS; INTEGRAL EQUATIONS;

DYNAMIC EXPONENT; KARDAR-PARISI-ZHANG EQUATION; MODE COUPLING THEORY; SCALING FUNCTION; UPPER CRITICAL DIMENSION;

ATOMIC PHYSICS;

EID: 0035971703     PISSN: 00319007     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevLett.86.3946     Document Type: Article

References (40)

1. M. Kardar, G. Parisi, and Y-C. Zhang, Phys. Rev. Lett. 56, 889 (1986).
2. D. Forster, D. R. Nelson, and M. J. Stephen, Phys. Rev. A 16, 732 (1977).
3. D. A. Huse, C. L. Henley, and D. S. Fisher, Phys. Rev. Lett. 55, 2924 (1985); M. Kardar and Y-C. Zhang, Phys. Rev. Lett. 58, 2087 (1987) .
4. D. A. Huse, C. L. Henley, and D. S. Fisher, Phys. Rev. Lett. 55, 2924 (1985); M. Kardar and Y-C. Zhang, Phys. Rev. Lett. 58, 2087 (1987) .
5. J. Z. Imbrie and T. Spencer, J. Stat. Phys. 52, 609 (1988); J. Cook and B. Derrida, Europhys. Lett. 10, 195 (1989); J. Cook and B. Derrida, J. Phys. A 23, 1523 (1990); M. R. Evans and B. Derrida, J. Stat. Phys. 69, 427 (1992).
6. J. Z. Imbrie and T. Spencer, J. Stat. Phys. 52, 609 (1988); J. Cook and B. Derrida, Europhys. Lett. 10, 195 (1989); J. Cook and B. Derrida, J. Phys. A 23, 1523 (1990); M. R. Evans and B. Derrida, J. Stat. Phys. 69, 427 (1992).
7. J. Z. Imbrie and T. Spencer, J. Stat. Phys. 52, 609 (1988); J. Cook and B. Derrida, Europhys. Lett. 10, 195 (1989); J. Cook and B. Derrida, J. Phys. A 23, 1523 (1990); M. R. Evans and B. Derrida, J. Stat. Phys. 69, 427 (1992).
8. J. Z. Imbrie and T. Spencer, J. Stat. Phys. 52, 609 (1988); J. Cook and B. Derrida, Europhys. Lett. 10, 195 (1989); J. Cook and B. Derrida, J. Phys. A 23, 1523 (1990); M. R. Evans and B. Derrida, J. Stat. Phys. 69, 427 (1992).
9. G. Parisi, J. Phys. (Paris) 51, 1595 (1990); M. Mezard, J. Phys. (Paris) 51, 1831 (1990).
10. G. Parisi, J. Phys. (Paris) 51, 1595 (1990); M. Mezard, J. Phys. (Paris) 51, 1831 (1990).
11. For a review, see J. Krug and H. Spohn, in Solids Far from Equilibrium: Growth Morphology and Defects, edited by C. Godreche (Cambridge University Press, Cambridge, UK, 1992); A.-L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, UK, 1995); T. Halpin-Healy and Y-C. Zhang, Phys. Rep. 254, 215 (1995).
12. For a review, see J. Krug and H. Spohn, in Solids Far from Equilibrium: Growth Morphology and Defects, edited by C. Godreche (Cambridge University Press, Cambridge, UK, 1992); A.-L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, UK, 1995); T. Halpin-Healy and Y-C. Zhang, Phys. Rep. 254, 215 (1995).
13. For a review, see J. Krug and H. Spohn, in Solids Far from Equilibrium: Growth Morphology and Defects, edited by C. Godreche (Cambridge University Press, Cambridge, UK, 1992); A.-L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, UK, 1995); T. Halpin-Healy and Y-C. Zhang, Phys. Rep. 254, 215 (1995).
14. M. Kardar, Nucl. Phys. B290, 582 (1987).
15. T. Hwa and E. Frey, Phys. Rev. A 44, R7873 (1991); E. Frey, U. C. Täuber, and T. Hwa, Phys. Rev. E 53, 4424 (1996).
16. T. Hwa and E. Frey, Phys. Rev. A 44, R7873 (1991); E. Frey, U. C. Täuber, and T. Hwa, Phys. Rev. E 53, 4424 (1996).
17. L. H. Tang, B. M. Forrest, and D. E. Wolf, Phys. Rev. A 45, 7162 (1992); T. Ala-Nissila, T. Hjelt, J. M. Kosterlitz, and O. Venäläinen, J. Stat. Phys. 72, 207 (1993) ; E. Marinari, A. Pagnani, and G. Parisi, cond-mat/0005105.
18. L. H. Tang, B. M. Forrest, and D. E. Wolf, Phys. Rev. A 45, 7162 (1992); T. Ala-Nissila, T. Hjelt, J. M. Kosterlitz, and O. Venäläinen, J. Stat. Phys. 72, 207 (1993) ; E. Marinari, A. Pagnani, and G. Parisi, cond-mat/0005105.
19. L. H. Tang, B. M. Forrest, and D. E. Wolf, Phys. Rev. A 45, 7162 (1992); T. Ala-Nissila, T. Hjelt, J. M. Kosterlitz, and O. Venäläinen, J. Stat. Phys. 72, 207 (1993) ; E. Marinari, A. Pagnani, and G. Parisi, cond-mat/0005105.
20. M. Marsili, A. Maritan, F. Toigo, and J. R. Banavar, Rev. Mod. Phys. 68, 963 (1996).
21. M. Schwartz and S. F. Edwards, Europhys. Lett. 20, 310 (1992).
22. J. P. Bouchaud and M. E. Cates, Phys. Rev. E 47, R1455 (1993).
23. J. P. Doherty, M. A. Moore, J. M. Kim, and A. J. Bray, Phys. Rev. Lett. 72, 2041 (1994).
24. M. A. Moore, T. Blum, J. P. Doherty, M. Marsili, J. P. Bouchard, and P. Claudin, Phys. Rev. Lett. 74, 4257 (1995).
25. T. Blum and A. J. McKane, Phys. Rev. E 52, 4741 (1995).
26. J. K. Bhattacharjee, J. Phys. A 31, L93 (1998).
27. M. Lässig, Nucl. Phys. B448, 559 (1995); M. Lässig and H. Kinzelbach, Phys. Rev. Lett. 78, 903 (1997).
28. M. Lässig, Nucl. Phys. B448, 559 (1995); M. Lässig and H. Kinzelbach, Phys. Rev. Lett. 78, 903 (1997).
29. K. J. Wiese, J. Stat. Phys. 93, 143 (1998).
30. T. Halpin-Healy, Phys. Rev. Lett. 62, 442 (1989); T. Halpin-Healy, Phys. Rev. A 42, 711 (1990).
31. T. Halpin-Healy, Phys. Rev. Lett. 62, 442 (1989); T. Halpin-Healy, Phys. Rev. A 42, 711 (1990).
32. C. Castellano, M. Marsili, and L. Pietronero, Phys. Rev. Lett. 80, 3527 (1998); C. Castellano, A. Gabrielli, M. Marsili, M. A. Muñoz, and L. Pietronero, Phys. Rev. E 58, R5209 (1998) ; C. Castellano, M. Marsili, M. A. Muñoz, and L. Pietronero, Phys. Rev. E 59, 6460 (1999).
33. C. Castellano, M. Marsili, and L. Pietronero, Phys. Rev. Lett. 80, 3527 (1998); C. Castellano, A. Gabrielli, M. Marsili, M. A. Muñoz, and L. Pietronero, Phys. Rev. E 58, R5209 (1998) ; C. Castellano, M. Marsili, M. A. Muñoz, and L. Pietronero, Phys. Rev. E 59, 6460 (1999).
34. C. Castellano, M. Marsili, and L. Pietronero, Phys. Rev. Lett. 80, 3527 (1998); C. Castellano, A. Gabrielli, M. Marsili, M. A. Muñoz, and L. Pietronero, Phys. Rev. E 58, R5209 (1998) ; C. Castellano, M. Marsili, M. A. Muñoz, and L. Pietronero, Phys. Rev. E 59, 6460 (1999).
35. H. K. Janssen, U. C. Tauber, and E. Frey, Eur. Phys. J. B 9, 491 (1999).
36. Y. Tu, Phys. Rev. Lett. 73, 3109 (1994).
37. 2p in the left-hand side of Eqs. (7) and (8).
38. Matching Eq. (5) at x = 0 with our ansatz for n̂(p) gives AP = 2/J, where J is a convergent integral in the limit z → 2, d → 4.
39. 1 is a convergent integral in the limit z → 2, d → 4, dependent on the specific form of the scaling function f̂.
40. Because of the rescaling invariance of Eqs. (4) and (5), the only relevant quantity is the relative scale of variation of n̂ and ĝ.