Abstract
We study the mean-field version of a model proposed by Leschhorn to describe the depinning transition of interfaces in random media. We show that evolution equations for the distribution of forces felt by the interface sites can be written directly for an infinite system. For a flat distribution of random local forces the value of the depinning threshold can be obtained exactly. In the case of parallel dynamics (all unstable sites move simultaneously), due to the discrete character of the interface heights allowed in the model, the motion of the center of mass is non-uniform in time in the moving phase close to the threshold, and the mean interface velocity vanishes with a square-root singularity.
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REFERENCES
D. S. Fisher, Collective transport in random media: from superconductors to earthquakes, Physics Reports 301:113–150 (1998).
R. Bruinsma and G. Aeppli, Phys. Rev. Letters 52:1547 (1984).
J. Koplik and H. Levine, Phys. Rev. B 32:280 (1985).
D. Kessler, H. Levine, and Y. Tu, Phys. Rev. A 43:4551 (1991).
E. Rolley, C. Guthmann, R. Gombrowicz, and V. Repain, Phys. Rev. Letters 80:2865 (1998).
T. Nattermann, S. Stepanow, L.-H. Tang, and H. Leschhorn, J. Physique (France) II 2:1483 (1992).
H. Leschhorn, T. Nattermann, S. Stepanow, and L.-H. Tang, Ann. Physik 6, 1–34 (1997).
M. Kardar, Nonequilibrium dynamics of interfaces and lines, Physics Reports 301:85–112 (1998).
O. Narayan and D. S. Fisher, Phys. Rev. B 46:11520 (1992).
O. Narayan and D. S. Fisher, Phys. Rev. B 48:7030 (1993).
L. Balents and D. S. Fisher, Phys. Rev. B 48:5949 (1993).
D. Ertas and M. Kardar, Phys. Rev. E 49:2532 (1994).
P. Chauve, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 62:6241 (2000).
A. Hazareesing and M. Mezard, Phys. Rev. E 60:1269 (1999).
H. Leschhorn, J. Physics A 25:L255 (1992).
H. Leschhorn, Physica A 195:324 (1993).
M. Jost, Phys. Rev. E 57:2634 (1998).
J.-P. Nadal, R. M. Bradley, and P. Strenski, J. Physics A 19:L505 (1986).
S. Roux and A. Hansen, J. Physique I (France) 4:515 (1994).
H. Leschhorn and L.-H. Tang, Phys. Rev. E 49:1238 (1994).
Y. Pomeau and J. Vannimenus, J. Colloid Interface Sci. 104:477 (1985).
P. G. de Gennes, Rev. Mod. Phys. 57:827 (1985).
A. A. Middleton and D. S. Fisher, Phys. Rev. B 47:3530 (1993).
H. Eisfeller and M. Opper, Phys. Rev. Letters 68:2094 (1992).
G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw–Hill, New York, 1961), pp. 16–25.
M. Paczuski, S. Maslov, and P. Bak, Phys. Rev. E 53:414 (1996).
O. Narayan, Phys. Rev. E 62:R7563 (2000).
Y. Pomeau and P. Manneville, J. Physique Lettres 40:L-609 (1979).
E. Raphael and P. G. de Gennes, J. Chem. Phys. 90:7577 (1989).
J. F. Joanny and M. O. Robbins, J. Chem. Phys. 92:3206 (1990).
D. S. Fisher, Phys. Rev B 31:1396 (1985).
J. W. Cahn, Acta Metall. 8:554–562 (1960).
A. Paterson and M. Fermigier, Phys. Fluids 9:2210 (1997).
A. Prevost, M. Poujade, E. Rolley, and C. Guthmann, Physica B 280:80 (2000).
J. P. Bouchaud and A. Georges, Phys. Rev. Letters 68, 3908 (1992).
T. Emig and T. Nattermann, Eur. Phys. J. B 8:525–546 (1999).
A. Hazareesing and J. P. Bouchaud, Phys. Rev. Letters 81, 5953 (1998).
E. T. Seppälä, M. J. Alava, and P. M. Duxbury, Phys. Rev. E 62:3230 (2000).
M. C. Marchetti, A. A. Middleton, and T. Prellberg, Phys. Rev. Letters 85, 1104 (2000).
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Vannimenus, J., Derrida, B. A Solvable Model of Interface Depinning in Random Media. Journal of Statistical Physics 105, 1–23 (2001). https://doi.org/10.1023/A:1012278408260
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DOI: https://doi.org/10.1023/A:1012278408260