Avalanche-size distribution at the depinning transition: A numerical test of the theory

Alberto Rosso 1, 2, Pierre Le Doussal 1, Kay Joerg Wiese 1

Physical Review B 80 (2009) 144204

We calculate numerically the sizes S of jumps (avalanches) between successively pinned configurations of an elastic line (d=1) or interface (d=2), pulled by a spring of (small) strength m^2 in a random-field landscape. We obtain strong evidence that the size distribution, away from the small-scale cutoff, takes the form P(S) = p(S/S_m) /S_m^2, where S_m:=/(2), proportional to m^(-d-zeta), is the scale of avalanches, and zeta the roughness exponent at the depinning transition. Measurement of the scaling function f(s) := s^tau p(s) is compared with the predictions from a recent Functional RG (FRG) calculation, both at mean-field and one-loop level. The avalanche-size exponent tau is found in good agreement with the conjecture tau = 2- 2/(d+zeta), recently confirmed to one loop via the FRG. The function f(s) exhibits a shoulder and a stretched exponential decay at large s, with ln f(s) proportional to – s^delta, and delta approximately 7/6 in d=1. The function f(s), universal ratios of moments, and the generating function are found in excellent agreement with the one-loop FRG predictions. The distribution of local avalanche sizes, i.e. of the jumps of a subspace of the manifold of dimension d’, is also computed and compared to our FRG predictions, and to the conjecture tau’ =2- 2/(d’+zeta).

  • 1. Laboratoire de Physique Théorique de l’ENS (LPTENS),
    CNRS : UMR8549 – Université Paris VI – Pierre et Marie Curie – Ecole Normale Supérieure de Paris – ENS Paris
  • 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI – Paris Sud
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