The physics of fractals

Bak, Peter; Chen, Kan

doi:10.1016/0167-2789(89)90166-8

Cited by 130 publications

(74 citation statements)

References 12 publications

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“…It is actually a key concept of SOC systems that the spatial structure of avalanches is fractal. Bak & Chen (1989) express this most succintly in their abstract: "Fractals in nature originate from self-organized critical dynamical processes". As it can be seen from the snapshots of an evolving avalanche shown in Fig.…”

Section: The Fractal Geometry Of Instantaneous Energy Dissipationmentioning

confidence: 99%

“…We aim for a universal statistical model of nonlinear energy dissipation processes that is independent of any particular physical mechanism. The fractal structure of selforganized processes has been stressed prominently from beginning (Bak et al 1987(Bak et al , 1988Bak & Chen 1989), but no quantitative theory has been put forward that links the fractal geometry to the size distribution of SOC events. Fractals have been studied…”

Section: Introductionmentioning

confidence: 99%

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A statistical fractal-diffusive avalanche model of a slowly-driven self-organized criticality system

Aschwanden

2012

A&A

118

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Aims. We develop a statistical analytical model that predicts the occurrence frequency distributions and parameter correlations of avalanches in nonlinear dissipative systems in the state of a slowly-driven self-organized criticality (SOC) system. Methods. This model, called the fractal-diffusive SOC model, is based on the following four assumptions: (i) the avalanche size L grows as a diffusive random walk with time T , following L ∝ T 1/2 ; (ii) the energy dissipation rate f (t) occupies a fractal volume with dimension D S ; (iii) the mean fractal dimension of avalanches in Euclidean space S = 1, 2, 3 is D S ≈ (1 + S )/2; and (iv) the occurrence frequency distributions N(x) ∝ x −αx based on spatially uniform probabilities in a SOC system are given by N(L) ∝ L −S , with S being the Eudlidean dimension. We perform cellular automaton simulations in three dimensions (S = 1, 2, 3) to test the theoretical model. Results. The analytical model predicts the following statistical correlations: F ∝ L D S ∝ T D S/2 for the flux, P ∝ L S ∝ T S /2 for the peak energy dissipation rate, and E ∝ FT ∝ T 1+D S /2 for the total dissipated energy; the model predicts powerlaw distributions for all parameters, with the slopes α T = (1 + S )/2, α F = 1 + (S − 1)/D S , α P = 2 − 1/S , and α E = 1 + (S − 1)/(D S + 2). The cellular automaton simulations reproduce the predicted fractal dimensions, occurrence frequency distributions, and correlations within a satisfactory agreement within ≈10% in all three dimensions. Conclusions. One profound prediction of this universal SOC model is that the energy distribution has a powerlaw slope in the range of α E = 1.40−1.67, and the peak energy distribution has a slope of α P = 1.67 (for any fractal dimension D S = 1, ..., 3 in Euclidean space S = 3), and thus predicts that the bulk energy is always contained in the largest events, which rules out significant nanoflare heating in the case of solar flares.

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Section: The Fractal Geometry Of Instantaneous Energy Dissipationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A statistical fractal-diffusive avalanche model of a slowly-driven self-organized criticality system

Aschwanden

2012

A&A

118

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“…For large and medium-sized LS, LS frequency distributions are generally negative power-law (scale-invariant) functions of LS area (Table 1), even though results have been presented (i) as cumulative and non-cumulative frequency distributions, (ii) for LS scar areas, total LS areas, or LS volumes, (iii) for relative complete inventories of LS caused by one triggering event or for historical LS inventories, which are the sum of many single events and which are assumed to be incomplete as morphological characteristics of old shallow LS can be obliterated by erosion and human activities, and (iv) for inventories of LS triggered by rainfall, earthquakes, or rapid snowmelt. Therefore these power-law distributions of LS observed under various conditions have been associated with the concept of self-organized criticality (SOC; [26][27][28][29][30]). According to its definition, selforganized critical behaviour is characterized by steady inputs, and outputs that are a series of 'events' satisfying power-law frequency-size statistics [9].…”

Section: Introductionmentioning

confidence: 99%

“…According to its definition, selforganized critical behaviour is characterized by steady inputs, and outputs that are a series of 'events' satisfying power-law frequency-size statistics [9]. In geomorphological studies, the sand pile model was a first, simple metaphor of a SOC process [26,27]. In this model sand particles are slowly added one by one to a pile.…”

Section: Introductionmentioning

confidence: 99%

Characteristics of the size distribution of recent and historical landslides in a populated hilly region

Eeckhaut

Poesen

Govers

et al. 2007

Earth and Planetary Science Letters

177

155

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Despite the availability of studies on the frequency density of landslide areas in mountainous regions, frequency-area distributions of historical landslide inventories in populated hilly regions are absent. This study revealed that the frequency-area distribution derived from a detailed landslide inventory of the Flemish Ardennes (Belgium) is significantly different from distributions usually obtained in mountainous areas where landslides are triggered by large-scale natural causal factors such as rainfall, earthquakes or rapid snowmelt. Instead, the landslide inventory consists of the superposition of two populations, i.e. (i) small (b 1-2 · 10 − 2 km 2 ), shallow complex earth slides that are at most 30 yr old, and (ii) large (N 1-2 · 10 − 2 km 2 ), deep-seated landslides that are older than 100 yr. Both subpopulations are best represented by a negative power-law relation with exponents of −0.58 and −2.31 respectively. This study focused on the negative power-law relation obtained for recent, small landslides, and contributes to the understanding of frequency distributions of landslide areas by presenting a conceptual model explaining this negative power-law relation for small landslides in populated hilly regions. According to the model hilly regions can be relatively stable under the present-day environmental conditions, and landslides are mainly triggered by human activities that have only a local impact on slope stability. Therefore, landslides caused by anthropogenic triggers are limited in size, and the number of landslides decreases with landslide area.The frequency density of landslide areas for old landslides is similar to those obtained for historical inventories compiled in mountainous areas, as apart from the negative power-law relation with exponent −2.31 for large landslides, a positive power-law relation followed by a rollover is observed for smaller landslides. However, when analysing the old landslides together with the more recent ones, the present-day higher temporal frequency of small landslides compared to large landslides, obscures the positive power-law relation and rollover.

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“…Recently it has been recognized that many interacting dynamical systems naturally evolve into a self-organized critical state, with avalanches of all sizes-large and small others, 1987, 1988;Tang and Bak, 1988;Bak and Chen. 1989).…”

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confidence: 99%