Avalanche dynamics in evolution, growth, and depinning models.

In a recent paper Paczuski, Maslov and Bak present a comprehensive theory of avalanche dynamics in models of growth, interface depinning and evolution. One of their main results is the so-called gamma equation, which is claimed to be exact. In this note it is shown that this equation requires a numerical correction factor in order to become exact. The exactness is needed when using the equation to determine the exponent γ. The correct equation is tested against numerical results for the Bak-Sneppen evolution model and two closely related models, and it turns out that it improves the description of data in a statistically significant way. PACS numbers: 05.40+j,64.60.Lx [email protected] [email protected] In the Bak-Sneppen (BS) evolution model [1] each site on a lattice is assigned a random number between zero and one. At each update step, the smallest random number, fmin, is located. That site and its nearest neighbors are then assigned new random numbers, drawn independently from the uniform distribution between zero and one. Avalanches may be defined in terms of fmin(s), which is the value of the smallest random number at time s. An f0-avalanche starts when fmin(s) passes the level f0 from above, and ends at the first return to this level. The average temporal size of f0-avalanches, 〈S〉f0 , diverges as 〈S〉f0 ∼ (fc − f0) −γ (1) as f0 approaches the critical value fc. The gamma equation was derived by Refs. [2, 3] and states that d ln〈S〉f0 df0 = 〈ncov〉f0 1− f0 (2) where ncov denotes the number of sites covered by an avalanche. It was claimed to be exact for the BS evolution and Sneppen interface [4] models. Assuming the scaling ansatz Eq. 1, it implies that γ = lim f0→fc 〈ncov〉f0(fc − f0) 1− f0 (3) If correct, this relation provides a useful alternative method for determining fc and γ; a plot of 1−f0 〈ncov〉f0 against f0 yields fc as the intersection with the f0 axis and −γ −1 as the asymptotic slope close to fc. However, the derivation of Eq. 2 neglects one important detail. In deriving Eq. 2 one relates (f0 + df0)-avalanches to f0-avalanches for small df0. In doing that, one needs the probability that, at the end of a given f0-avalanche, fmin(s) hits the band between f0 and f0 + df0 and then turns downwards again (in that case one (f0+df0)-avalanche corresponds to two f0-avalanches). In Ref [3] this probability is incorrectly taken as 〈ncov〉f0df0 (1−f0) , which is the probability that f0 < fmin(s) < f0 + df0 at the end of the avalanche. This number has to be multiplied by the probability C(f0) that fmin(s) turns downwards, which happens unless all the new random numbers are greater than f0. It follows that C(f0) = 1− (1 − f0) 2d+1 (4) for the BS model on a hypercubic lattice in d dimensions. Taking this factor into account, we find that d ln〈S〉f0 df0 = C(f0) 〈ncov〉f0 1− f0 (5) and γ = lim f0→fc C(f0) 〈ncov〉f0(fc − f0) 1− f0 (6) These changes do not affect the scaling relations between different exponents discussed by Ref. [3]. However, they are important when extracting γ; the use of the erroneous Eq. 3 leads to an overestimate by a factor of C(fc) .