Optimal-path cracks in correlated and uncorrelated lattices

Abstract

The optimal path crack model on uncorrelated surfaces, recently introduced by Andrade et al. (Phys. Rev. Lett. 103, 225503, 2009), is studied in detail and its main percolation exponents computed. In addition to β/ν = 0.46 ± 0.03 we report, for the first time, γ/ν = 1.3 ± 0.2 and τ = 2.3 ± 0.2. The analysis is extended to surfaces with spatial long-range power-law correlations, where non-universal fractal dimensions are obtained when the degree of correlation is varied. The model is also considered on a three-dimensional lattice, where the main crack is found to be a surface with a fractal dimension of 2.46 ± 0.05.

Percolation with long-range correlated disorder

“…Here, we study the case where the heights h have long-range spatial correlations. Such a power-law correlated disorder can be generated using the Fourier filtering method (Ffm) [6,16,[25][26][27][28][29][30][31][32]43], which is based on the Wiener-Khintchine theorem (WKt) [25,44]. The WKt states that the auto-correlation of a time series equals the Fourier transform of its power spectrum, i.e.…”

Watersheds in disordered media

“…However real landscapes are characterized by spatial long-range correlated height distributions. Numerically, such distributions can be generated from fractional Brownian motion (fBm) [36,59], using the Fourier filtering method [19,55,[60][61][62][63][64][65][66][67][68][69][70]. This method allows to control the nature and the strength of correlations, which are characterized by the Hurst exponent H. The uncorrelated distribution of heights is solely obtained for H = −d/2, i.e., H = −1 and H = −3/2 in two and three dimensions, respectively.…”

“…It was recently shown that watersheds can be described in the context of percolation theory in terms of bridges and cutting bond models [15], with numerical evidence that they are Schramm-Loewer Evolution (SLE) curves in the continuum limit [16]. This association explains why watersheds on uncorrelated landscapes as well as other statistical physics models, such as, optimal path cracks [17][18][19], fuse networks [20], and loopless percolation [15], belong to the same universality class of optimal paths in strongly disordered media.…”

Multi-scale approach to invasion percolation of rock fracture networks

“…However any type of aperture field can be treated analogously. The utilized disorder is given by a power-law probability density distribution such that the aperture h i of each site i can be generated as h i =exp(B(x i -1)) with the disorder parameter B and x i being a uniformly distributed random number in [0,1] (Oliveira, et al, 2011;Braunstein et al, 2002). For each realization, an aperture h i , i=1,…,L d , (where L is the size of the aperture field, e.g.…”