Anomalous scaling in depinning transitions

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 62, Issue 6, 2000, Pages

  Narayan, Onuttom ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FUNCTIONS; INTERFACES (MATERIALS); MATHEMATICAL MODELS; PHASE TRANSITIONS;

DEPINNING TRANSITIONS; RENORMALIZATION GROUP;

STATISTICAL MECHANICS;

EID: 0034505051     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.62.R7563     Document Type: Article

References (45)

1. D.S. Fisher, Phys. Rev. Lett. 50, 1486 (1983)
2. D.S. FisherPhys. Rev. B 31, 1396 (1985).
3. O. Narayan and D.S. Fisher, Phys. Rev. Lett. 68, 3615 (1992)
4. O. NarayanD.S. FisherPhys. Rev. B 46, 11 520 (1992).
5. T. Nattermann, S. Stepanow, L.-H. Tang, and H. Leschhorn, J. Phys. II 2, 1483 (1992)
6. O. Narayan and D.S. Fisher, Phys. Rev. B 48, 7030 (1993).
7. D. Ertas and M. Kardar, Phys. Rev. E 49, 2532 (1994)
8. K. Dahmen and J.P. Sethna, Phys. Rev. Lett. 71, 3222 (1993)
9. K. DahmenJ.P. SethnaPhys. Rev. B 53, 14 872 (1996)
10. Phys. Rev. BL. Balents, M.C. Marchetti, and L. Radzihovsky, 57, 7705 (1998)
11. P. Chauve, T. Giamarchi, and P. le Doussal, Europhys. Lett. 44, 110 (1998)
12. P. ChauveT. GiamarchiP. le Doussale-print cond-mat/0002299
13. P. Chauve, P. le Doussal, and K. Wiese, e-print cond-mat/0006056.
14. O. Narayan and A.A. Middleton, Phys. Rev. B 49, 244 (1994).
15. For (Formula presented), the phase transition, describing the CDW as an elastic manifold, is preempted by “tearing” of the CDW [S.N. Coppersmith, Phys. Rev. Lett. 65, 1044 (1990)]. In this paper we discuss connections between the elastic model and SOC models; for the systems for which the latter are used, there is no analog of tearing.
16. P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).
17. M. Paczuski, S. Maslov, and P. Bak, Phys. Rev. E 53, 414 (1996), and references therein.
18. C. Tang and P. Bak, Phys. Rev. Lett. 60, 2347 (1988)
19. Phys. Rev. Lett.C. Tang, K. Wiesenfeld, P. Bak, S.N. Coppersmith, and P.B. Littlewood, 58, 1161 (1987)
20. S.N. Coppersmith and P.B. Littlewood, Phys. Rev. B 36, 311 (1987).
21. The mapping 5 is from a CDW with periodic boundary conditions to a similar sandpile 7, instead of open boundary conditions. This creates an “above threshold” sandpile phase, where grains tumble around forever without additional injection. However, avalanches below threshold are unaffected for a sufficiently large system. Also, ramping up the force below threshold for a CDW corresponds to dropping sand on sites in a random order, and cycling through this sequence repeatedly, instead of dropping it spatially and temporally randomly. This does not matter in the large system limit.
22. M. Paczuski and S. Boettcher, Phys. Rev. Lett. 77, 111 (1996).The mapping is from a SOC model to a (1+1)-dimensional interface pulled at one end at a constant speed. In steady state, this is equivalent (after averaging over randomness) to driving the interface with a uniform applied force, with one end constrained to move at the (spatial and temporal) average velocity of the entire interface. The constraint is inconsequential for large systems.
23. S. Majumdar and D. Dhar, Physica A 185, 129 (1992).
24. K. Sneppen, Phys. Rev. Lett. 69, 3539 (1992), model B.
25. H. Leschhorn and L.-H. Tang, Phys. Rev. E 49, 1238 (1994).
26. A. Tanguy, M. Gounelle, and S. Roux, Phys. Rev. E 58, 1577 (1998).This paper considers long ranged as well as (effectively) short ranged elastic interactions.
27. K. Sneppen and M.H. Jensen, Phys. Rev. Lett. 71, 101 (1993).
28. S. Krishnamurthy, A. Tanguy, and S. Roux, Eur. Phys. J. B 15, 149 (2000).
29. In Ref. 17, the model of Ref. 15 is started with a flat interface, but the same results are obtained starting from steady state [A. Tanguy (private communication)]. For the model of Ref. 13, these results are only obtained starting from steady state 16. This is probably because the model of Ref. 13 is a “hybrid” model, where only a single site moves at a time step, but can then trigger a brief avalanche (at the same time step). This is neither constant velocity nor constant force driving. In steady state, the avalanches have a characteristic size, and the model effectively has constant velocity driving. However, starting from a flat interface, the mean avalanche size rises from unity to its saturated value; in fact, all scaling in the short time regime for this case is quite imperfect. [The connection between the changing mean avalanche size and the peculiar short time behavior of the roughness is seen better in the original version of the model 13, where the initial pinning strengths are Gaussian, but the new pinning strengths when sites move are uniform over (Formula presented).]
30. For an alternative view, see Ref. 16 and M. Paczuski, Phys. Rev. E 52, R2137 (1995).
31. The scaling form looks slightly different than that in Ref. 17, because the time t is used and not the scaled time (Formula presented). The dynamic exponent (Formula presented) is the (Formula presented) of Ref. 17.
32. Numerical estimates (Formula presented) and (Formula presented) for interfaces with short range elasticity [Eq. (4)] in (Formula presented) violate this equation, probably due to pathologies for (Formula presented). Estimates for long range elastic models in (Formula presented) satisfy Eq. (3) 15.
33. R. Bruinsma and G. Aeppli, Phys. Rev. Lett. 52, 1547 (1984)
34. J. Koplik and H. Levine, Phys. Rev. B 32, 280 (1985)
35. D.A. Kessler, H. Levine, and Y. Tu, Phys. Rev. A 43, 4551 (1991).
36. M. Kardar, G. Parisi, and Y-C. Zhang, Phys. Rev. Lett. 56, 889 (1986).Such terms exist and are relevant for an interface moving in an anisotropic medium, where the pinning has different strength in different directions 23.
37. L.A.N. Amaral, A.-L. Barabasi, and H.E. Stanley, Phys. Rev. Lett. 73, 62 (1994)
38. Phys. Rev. Lett.L.-H. Tang, M. Kardar, and D. Dhar, 74, 920 (1995).
39. P.C. Martin, E. Siggia, and H. Rose, Phys. Rev. A 8, 423 (1973)
40. H.K. Janssen, Z. Phys. B 23, 377 (1976).
41. Avalanches can actually not be clearly separated in this regime, but it may be possible for continuum models to do this approximately by separating out fast and slow motion. In automaton models, this is much more problematic 26, but the discussion here should be valid for time scales for any measure of activity.
42. A. Corral and M. Paczuski, Phys. Rev. Lett. 83, 572 (1999).For the model of Ref. 11, the avalanches merge for strong driving. The scale found for the crossover from separate to merged avalanches is as expected by requiring (Formula presented) and using numerical exponents for (Formula presented) depinning: (Formula presented), compared to 27 (Formula presented) and (Formula presented). Other exponents in this paper also agree with results for (Formula presented) depinning 27.
43. H. Leschhorn, Physica A 195, 324 (1993).
44. A.A. Middleton, Ph.D. thesis, Princeton University, 1990 (unpublished);see also Ref. 5.
45. H. Leschhorn and L.-H. Tang, Phys. Rev. Lett. 70, 2973 (1993).