Access by Xinjiang University

Size distributions of shocks and static avalanches from the functional renormalization group

Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, CNRS, 24 rue Lhomond, 75231 Paris Cedex, France

Phys. Rev. E 79, 051106 – Published 7 May, 2009

DOI: https://doi.org/10.1103/PhysRevE.79.051106

Abstract

Interfaces pinned by quenched disorder are often used to model jerky self-organized critical motion. We study static avalanches, or shocks, defined here as jumps between distinct global minima upon changing an external field. We show how the full statistics of these jumps is encoded in the functional-renormalization-group fixed-point functions. This allows us to obtain the size distribution $P (S)$ of static avalanches in an expansion in the internal dimension $d$ of the interface. Near and above $d = 4$ this yields the mean-field distribution $P (S) \sim S^{- 3 ∕ 2} e^{- S ∕ 4 S_{m}}$ , where $S_{m}$ is a large-scale cutoff, in some cases calculable. Resumming all one-loop contributions, we find $P (S) \sim S^{- τ} \exp (C {(S ∕ S_{m})}^{1 ∕ 2} - \frac{B}{4} {(S ∕ S_{m})}^{δ})$ , where $B$ , $C$ , $δ$ , and $τ$ are obtained to first order in $ϵ = 4 - d$ . Our result is consistent to $O (ϵ)$ with the relation $τ = τ_{ζ} ≔ 2 - \frac{2}{d + ζ}$ , where $ζ$ is the static roughness exponent, often conjectured to hold at depinning. Our calculation applies to all static universality classes, including random-bond, random-field, and random-periodic disorders. Extended to long-range elastic systems, it yields a different size distribution for the case of contact-line elasticity, with an exponent compatible with $τ = 2 - \frac{1}{d + ζ}$ to $O (ϵ = 2 - d)$ . We discuss consequences for avalanches at depinning and for sandpile models, relations to Burgers turbulence and the possibility that the relation $τ = τ_{ζ}$ be violated to higher loop order. Finally, we show that the avalanche-size distribution on a hyperplane of codimension one is in mean field (valid close to and above $d = 4$ ) given by $P (S) \sim K_{1 ∕ 3} (S) ∕ S$ , where $K$ is the Bessel- $K$ function, thus $τ_{hyper plane} = \frac{4}{3}$ .

Article Text

References (96)

J. S. Urbach, R. C. Madison, and J. T. Markert, Phys. Rev. Lett. 75, 276 (1995).
V. Repain et al., Europhys. Lett. 68, 460 (2004).
S. Field, J. Witt, F. Nori, and X. Ling, Phys. Rev. Lett. 74, 1206 (1995).
S. Moulinet, C. Guthmann, and E. Rolley, Eur. Phys. J. A 8, 437 (2002).
S. Moulinet, A. Rosso, W. Krauth, and E. Rolley, Phys. Rev. E 69, 035103(R) (2004).
T. Emig, P. Claudin, and J. P. Bouchaud, Europhys. Lett. 50, 594 (2000).
D. Cule and T. Hwa, Phys. Rev. B 57, 8235 (1998).
D. S. Fisher, Phys. Rep. 301, 113 (1998).
D. S. Fisher, K. Dahmen, S. Ramanathan, and Y. Ben-Zion, Phys. Rev. Lett. 78, 4885 (1997).
P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).
Deepak Dhar, e-print arXiv:cond-mat/9909009.
E. V. Ivashkevich and V. B. Priezzhev, Physica A 254, 97 (1998).
Deepak Dhar, Physica A 263, 4 (1999).
M. Stapleton and K. Christensen, e-print arXiv:cond-mat/0510626.
D. Dhar and S. N. Majumdar, J. Phys. A 23, 4333 (1990).
D.-S. Leea, K.-I. Goha, B. Kahng, and D. Kima, Physica A 338, 84 (2004).
I. Jensen, Phys. Rev. E 47, R1 (1993).
H. Agrawal and D. Dhar, Phys. Rev. E 63, 056115 (2001).
D. Dhar and A. Dhar, Phys. Rev. E 55, R2093 (1997).
S. Banerjee, S. B. Santra, and I. Bose, Z. Phys. B: Condens. Matter 96, 571 (1995).
K. Dahmen and J. P. Sethna, Phys. Rev. B 53, 14872 (1996).
Y. Liu and K. A. Dahmen, e-print arXiv:cond-mat/0609609.
A. A. Middleton and D. S. Fisher, Phys. Rev. B 47, 3530 (1993).
O. Narayan and A. A. Middleton, Phys. Rev. B 49, 244 (1994).
S. Lübeck and K. D. Usadel, Phys. Rev. E 56, 5138 (1997).
P. Le Doussal, A. A. Middleton, and K. J. Wiese, Phys. Rev. E 79, 050101(R) (2009).
O. Narayan and D. S. Fisher, Phys. Rev. B 48, 7030 (1993).
S. Zapperi, P. Cizeau, G. Durin, and H. E. Stanley, Phys. Rev. B 58, 6353 (1998).
D. S. Fisher, Phys. Rev. B 31, 7233 (1985).
D. S. Fisher, Phys. Rev. Lett. 56, 1964 (1986).
L. Balents and D. S. Fisher, Phys. Rev. B 48, 5949 (1993).
P. Chauve, P. Le Doussal, and K. J. Wiese, Phys. Rev. Lett. 86, 1785 (2001).
S. Scheidl and Y. Dincer, e-print arXiv:cond-mat/0006048.
P. Le Doussal and K. J. Wiese, Phys. Rev. Lett. 89, 125702 (2002).
P. Chauve and P. Le Doussal, Phys. Rev. E 64, 051102 (2001).
P. Le Doussal, K. J. Wiese, and P. Chauve, Phys. Rev. E 69, 026112 (2004).
P. Le Doussal and K. J. Wiese, Phys. Rev. B 68, 174202 (2003).
P. Le Doussal and K. J. Wiese, Nucl. Phys. B 701, 409 (2004).
P. Le Doussal and K. J. Wiese, Phys. Rev. E 72, 035101(R) (2005).
L. Balents and P. Le Doussal, Europhys. Lett. 65, 685 (2004).
L. Balents and P. Le Doussal, Phys. Rev. E 69, 061107 (2004).
L. Balents and P. Le Doussal, Ann. Phys. 315, 213 (2005).
A. A. Fedorenko, P. Le Doussal, and K. J. Wiese, Phys. Rev. E 74, 061109 (2006).
K. J. Wiese, J. Phys.: Condens. Matter 17, S1889 (2005).
T. Nattermann, S. Stepanow, L. H. Tang, and H. Leschhorn, J. Phys. II 2, 1483 (1992).
O. Narayan and D. S. Fisher, Phys. Rev. B 46, 11520 (1992).
P. Chauve, T. Giamarchi, and P. Le Doussal, Europhys. Lett. 44, 110 (1998).
P. Chauve, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 62, 6241 (2000).
P. Le Doussal, K. J. Wiese, and P. Chauve, Phys. Rev. B 66, 174201 (2002).
P. Le Doussal and K. J. Wiese, Phys. Rev. E 67, 016121 (2003).
P. Le Doussal and K. J. Wiese, Phys. Rev. E 68, 046118 (2003).
P. Le Doussal, K. J. Wiese, E. Raphael, and R. Golestanian, Phys. Rev. Lett. 96, 015702 (2006).
A. Fedorenko, P. Le Doussal, and K. J. Wiese, Phys. Rev. E 74, 041110 (2006).
D. E. Feldman, Phys. Rev. B 61, 382 (2000).
D. E. Feldman, Int. J. Mod. Phys. B 15, 2945 (2001).
D. E. Feldman, Phys. Rev. Lett. 88, 177202 (2002).
G. Tarjus and M. Tissier, Phys. Rev. Lett. 93, 267008 (2004).
P. Le Doussal and K. J. Wiese, Phys. Rev. Lett. 96, 197202 (2006).
P. Le Doussal and K. J. Wiese, Phys. Rev. Lett. 98, 269704 (2007).
M. Tissier and G. Tarjus, Phys. Rev. Lett. 96, 087202 (2006).
M. Tissier and G. Tarjus, Phys. Rev. B 74, 214419 (2006).
P. Le Doussal, Europhys. Lett. 76, 457 (2006).
P. Le Doussal, e-print arXiv:0809.1192.
P. Le Doussal and K. J. Wiese, EPL 77, 66001 (2007).
A. A. Middleton, P. Le Doussal, and K. J. Wiese, Phys. Rev. Lett. 98, 155701 (2007).
A. Rosso, P. Le Doussal, and K. J. Wiese, Phys. Rev. B 75, 220201 (2007).
A. A. Fedorenko, P. Le Doussal, and K. J. Wiese (unpublished).
M. Alava, J. Phys.: Condens. Matter 14, 2353 (2002).
S. N. Majumdar and D. Dhar, Physica A 185, 129 (1992).
V. B. Priezzhev, e-print arXiv:cond-mat/9904054.
V. S. Poghosyan, S. Y. Grigorev, V. B. Priezzhev, and P. Ruelle, Phys. Lett. B 659, 768 (2008).
S. Moghimi-Araghi, M. A. Rajabpour, and S. Rouhani, Nucl. Phys. B 718, 362 (2005).
G. Piroux and P. Ruelle, Phys. Lett. B 607, 188 (2005).
M. Jeng, Phys. Rev. E 71, 016140 (2005).
A. A. Fedorenko, P. Le Doussal, and K. J. Wiese, J. Stat. Phys. 133, 805 (2008).
H. Bucheli, O. S. Wagner, V. B. Geshkenbein, A. I. Larkin, and G. Blatter, Phys. Rev. B 57, 7642 (1998).
T. Nattermann and S. Scheidl, Adv. Phys. 49, 607 (2000).
K. J. Wiese and P. Le Doussal, Markov Processes Relat. Fields 13, 777 (2007).
A. A. Middleton, Phys. Rev. E 52, R3337 (1995).
Noh Dong Jae and H. Rieger, Phys. Rev. Lett. 87, 176102 (2001).
A. A. Middleton, Phys. Rev. Lett. 68, 670 (1992).
Note however that even if the weight in probability of avalanches of the size of the UV cutoff is negligible as $m \to 0$ , these may still control some (negative) moments of the probability distribution.
It depends on it subdominantly, i.e., in ${(S_{\min} ∕ S_{m})}^{2 - τ}$ .
We use that $\int_{- \infty}^{\infty} d u Δ^{″} (u) = 0$ to conclude that $- \lim_{ϵ \to 0} \int_{- ϵ}^{ϵ} d u Δ^{″} (u) = \lim_{ϵ \to 0} [\int_{- \infty}^{- ϵ} d u Δ^{″} (u) + \int_{ϵ}^{\infty} d u Δ^{″} (u)] = - 2 Δ^{'} (0^{+})$ .
A. N. Kolmogorov, C. R. Acad. Sci. URSS 30, 301 (1941).
A. Rosso, W. Krauth, P. Le Doussal, J. Vannimenus, and K. J. Wiese, Phys. Rev. E 68, 036128 (2003).
We thank A. Fedorenko for this observation.
H. W. Watson and Francis Galton, Journal of the Anthropological Institute of Great Britain 4, 138 (1875).
B. Alessandro, C. Beatrice, G. Bertotti, and A. Montorsi, J. Appl. Phys. 68, 2901 (1990).
P. Le Doussal and K. J. Wiese, Phys. Rev. E 79, 051105 (2009).
To be precise, this means in powers of the local part $R (u)$ of the functional $R [u]$ since higher multilocal parts can themselves be expressed as a function of the local part $R (u)$ (see, e.g., [63]).
P. Le Doussal and K. J. Wiese (unpublished).
The case of a vanishing $\int_{x} ϕ (x)$ has to be considered separately.
P. Le Doussal, M. C. Marchetti, and K. J. Wiese, Phys. Rev. B 78, 224201 (2008).
The integration path for the inverse Laplace transform should be chosen such that the integrand remains real. For $ϵ = 0$ this is achieved by taking $λ = - u ∕ 2 + i s$ , $s ∊ R$ . For $ϵ > 0$ , two cuts appear, from $- 1$ to $- \infty$ and from 1 to $\infty$ . Taking $u$ large, only the left one matters. At leading order in $ϵ$ , one can still follow almost the same contour, except that the integral from $- u ∕ 2$ to $- 1$ below the cut and from $- 1$ to $- u ∕ 2$ above the cut has to be added. Its nonvanishing part comes from the $2 π i$ discontinuity of the ln across the cut, thus does not give a ln contribution. The final result at leading order is then obtained by replacing $λ$ by $λ - u ∕ 2$ , doing the (almost) Gaussian integral in $λ$ in imaginary direction, with the result that $P (u) \approx {(G 4) ∣}_{λ \to - u ∕ 2}$ . Exponentiating the ln leads to the quoted result for $δ$ .
Note that the above result is for $m L ⪢ 1$ : the result for arbitrary $m L$ can in principle be obtained by modifying the integrals into periodic sums. The opposite limit $m L \to 0$ was studied in [53].

Physical Review E

Export citation

Export citation

Size distributions of shocks and static avalanches from the functional renormalization group

Abstract

Article Text

References (96)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation