Showing papers on "Fractal dimension published in 1985"

Introduction to percolation theory

Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques

Self-Affine Fractals and Fractal Dimension

Abstract: Evaluating a fractal curve's approximate length by walking a compass defines a compass exponent. Long ago, I showed that for a self-similar curve (e.g., a model of coastline), the compass exponent coincides with all the other forms of the fractal dimension, e.g., the similarity, box or mass dimensions. Now walk a compass along a self-affine curve, such as a scalar Brownian record B(t). It will be shown that a full description in terms of fractal dimension is complex. Each version of dimension has a local and a global value, separated by a crossover. First finding: the basic methods of evaluating the global fractal dimension yield 1: globally, a self-affine fractal behaves as it if were not fractal. Second finding: the box and mass dimensions are 1.5, but the compass dimension is D = 2. More generally, for a fractional Bownian record BH(t), (e.g., a model of vertical cuts of relief), the global fractal dimensions are 1, several local fractal dimensions are 2-H, and the compass dimension is 1/H. This 1/H is the fractal dimension of a self-similar fractal trail, whose definition was already implicit in the definition of the record of BH(t).

Broad bandwidth study of the topography of natural rock surfaces

Abstract: The mechanical and hydraulic behavior of discontinuities in rock, such as joints and faults, depends strongly on the topography of the contacting surfaces and the degree of correlation between them. Understanding this behavior over the scales of interest in the earth requires knowledge of how topography or roughness varies with surface size. Using two surface profilers, each sensitive to a particular scale of topographic features, we have studied the topography of various natural rock surfaces from wavelengths less than 20 microns to nearly 1 meter. The surfaces studied included fresh natural joints (mode I cracks) in both crystalline and sedimentary rocks, a frictional wear surface formed by glaciation, and a bedding plane surface. There is remarkable similarity among these surfaces. Each surface has a “red noise” power spectrum over the entire frequency band studied, with the power falling off on average between 2 and 3 orders of magnitude per decade increase in spatial frequency. This implies a strong increase in rms height with surface size, which has little tendency to level off for wavelengths up to 1 meter. These observations can be interpreted using a fractal model of topography. In this model the scaling of the surface roughness is described by the fractal dimension D. The topography of these natural rock surfaces cannot be described by a single fractal dimension, for this parameter was found to vary significantly with the frequency band considered. This observed inhomogeneity in the scaling parameter implies that extrapolation of roughness to other bands of interest should be done with care. Study of the increase in rms height with profile length for two extreme cases from our data provides an idea of the expected variation in mechanical and hydraulic properties for natural discontinuities in rock. This indicates that in addition to the scaling of topography, the degree of correlation between the contacting surfaces is important to quantify.

Limits of the Fractal Dimension for Irreversible Kinetic Aggregation of Gold Colloids

Abstract: We show that there are two regimes of irreversible, kinetic aggregation of aqueous colloids, determined by the short-range interparticle potential, through its control of the sticking probability upon approach of two particles. Each regime has different rate-limiting physics, aggregation dynamics, and cluster-mass distributions, and results in clusters with different fractal dimensions. These results set limits on the fractal dimension, ${d}_{f}$, for gold aggregates of $1.75l~{d}_{f}l~2.05$ (\ifmmode\pm\else\textpm\fi{}0.05).

Random Fractal Forgeries

Abstract: Mandelbrot’s fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension, a number that agrees with our intuitive notion of dimension but need not be an integer. In section I, a series of computer generated and rendered random fractal shapes provide a visual introduction to the concepts of fractal geometry. These complex images, with details on all scales, are the result of the simplest rules of fractal geometry. Their success as forgeries of the natural world has played an important role in the rapid establishment of fractal geometry as a new scientific discipline and exciting graphic technique. Section II presents a brief mathematical characterization of these forgeries, as variations on Mandelbrot’s fractional Brownian motion. The important concepts of fractal dimension and exact and statistical self-similarity and self-affinity will be reviewed. Finally, section III will discuss independent cuts, Fourier filtering, midpoint displacement, successive random additions, and the Weierstrass-Mandelbrot random function as specific generating algorithms.

Fractal dimension of vegetation and the distribution of arthropod body lengths

Abstract: Following Mandelbrot1, recent studies2–6 demonstrate that some natural surfaces are fractal. Here we show that transects across vegetation are fractal, and consider one possible consequence of this observation for arthropods (mainly insects) living on plant surfaces. An important feature of a fractal curve or surface is that its length or area, respectively, becomes disproportionately large as the unit of measurement is decreased1. This suggests that if vegetation has a fractal structure, there is more usable space for smaller animals living on vegetation than for larger animals. Hence, there should be more individuals with a small body length than a large body length. We show that this is the case, and that relative numbers of small and large individual arthropods collected from vegetation are broadly consistent with theoretical predictions originating from the fractal nature of vegetation7 and individual rates of resource utilization.

Fractal growth viscous fingers: quantitative characterization of a fluid instability phenomenon

Abstract: What happens when one attempts to push water through a fluid of higher viscosity? Under appropriate experimental conditions, the water breaks through in the form of highly branched patterns called viscous fingers. Water was used to push a more viscous but miscible, non-newtonian fluid in a Hele-Shaw cell. The resulting viscous finger instability was found to be a fractal growth phenomenon. Reproducible values of the fractal dimension df were found and were interpreted using a modification of the diffusion limited aggregation model.

Fractal model for the ac response of a rough interface

Abstract: A fractal model is proposed for a rough interface between two materials of very different conductivities, e.g., an electrode and an electrolyte. The equivalent circuit of the model, which takes into consideration the resistance in the two substances and the capacitance of the interface, has the property of the so-called constant-phase-angle element, i.e., a passive circuit element whose complex impedance has a power-law singularity at low frequencies. The exponent of the frequency dependence is related to the fractal dimension. The model also provides insight into the conducting properties of the percolating cluster and the source of the 1/f noise in electronic components.

Viscous fingering fractals in porous media.

Abstract: Gas displacing a high-viscosity fluid in a two-dimensional porous disk intrudes in the form of ramified fingers similar to the structures obtained in diffusion-limited aggregation. We find that the resulting finger structures are described by a fractal dimension $D=1.62\ifmmode\pm\else\textpm\fi{}0.04$ consistent with $D$ for diffusion-limited aggregation clusters. This result confirms the analogy between diffusion-limited aggregation and two-fluid displacement in porous media introduced by Paterson.

Invasion percolation in an etched network: Measurement of a fractal dimension.

Abstract: A wetting fluid is displaced at very low flow rate by a nonwetting fluid in a 250 000-duct, transparent, etched network. The structure formed by the injected fluid is ramified and the scale invariance is described by a measured fractal dimension $1.80lDl1.83$. This value agrees with theoretical results of invasion percolation with trapping.

Determining modes and fractal dimension of turbulent flows

Abstract: Research on the abstract properties of the Navier–Stokes equations in three dimensions has cast a new light on the time-asymptotic approximate solutions of those equations. Here heuristic arguments, based on the rigorous results of that research, are used to show the intimate relationship between the sufficient number of degrees of freedom describing fluid flow and the bound on the fractal dimension of the Navier–Stokes attractor. In particular it is demonstrated how the conventional estimate of the number of degrees of freedom, based on purely physical and dimensional arguments, can be obtained from the properties of the Navier–Stokes equation. Also the Reynolds-number dependence of the sufficient number of degrees of freedom and of the dimension of the attractor in function space is elucidated.

Fractal surfaces of proteins.

Abstract: Fractal surfaces can be used to characterize the roughness or irregularity of protein surfaces. The degree of irregularity of a surface may be described by the fractal dimension D. For protein surfaces defined with probes in the range of 1.0 to 3.5 angstroms in radius, D is approximately 2.4 or intermediate between the value for a completely smooth surface (D = 2) and that for a completely space-filling surface (D = 3). Individual regions of proteins show considerable variation in D. These variations may be related to structural features such as active sites and subunit interfaces, suggesting that surface texture may be a factor influencing molecular interactions.

Fractals in fractography

Abstract: The fractal characteristics of a series of fractured AISI 4340 steel specimens were studied experimentally by means of vertical sections through the fracture surface. Extensive fractal data have been generated for both the fracture surfaces and their profiles. Fractal plots for the irregular profile curves differ significantly from those of the infinitely subdivisible curves with “self-similarity” as postulated by Mandelbrot. Instead of a linear fractal curve and constant fractal dimension D , we find a reversed sigmoidal curve (RSC) and variable D . Because of the inadequacies of the linear Richardson fractal equation, we have developed an alternative procedure whereby constant slopes β and γ are obtained from the RSCs for both profiles and surfaces respectively. Since a fractal equation for surfaces apparently does not exist, we propose one that parallels the Richardson equation for profiles. Slopes from the RSCs are related to the new constant fractal dimensions D β and D γ in the modified fractal equations for profiles and fracture surfaces. Some insight into the physical nature of the fractal dimension is afforded by its close similarity to fracture roughness parameters that have simple physical meanings. It is believed that the results of the extensive study may have general validity for the irregular complex curves and surfaces of nature.

Computer simulation of chemically limited aggregation

Abstract: Cluster-cluster aggregation is studied via a computer simulation in the chemically limited regime. True polydisperse flocculation is compared with idealised monodisperse aggregation in two and three space dimensions in terms of fractal dimensions and the exponents associated with the scaling of the reaction rates with cluster size.

Fractal geometry of two-dimensional fracture networks at Yucca Mountain, southwestern Nevada: proceedings

Abstract: Fracture traces exposed on three 214- to 260-m{sup 2} pavements in the same Miocene ash-flow tuff at Yucca Mountain, southwestern Nevada, have been mapped at a scale of 1:50. The maps are two-dimensional sections through the three-dimensional network of strata-bound fractures. All fractures with trace lengths greater than 0.20 m were mapped. The distribution of fracture-trace lengths is log-normal. The fractures do not exhibit well-defined sets based on orientation. Since fractal characterization of such complex fracture-trace networks may prove useful for modeling fracture flow and mechanical responses of fractured rock, an analysis of each of the three maps was done to test whether such networks are fractal. These networks proved to be fractal and the fractal dimensions (D) are tightly clustered (1.12, 1.14, 1.16) for three laterally separated pavements, even though visually the fracture networks appear quite different. The fractal analysis also indicates that the network patterns are scale independent over two orders of magnitude for trace lengths ranging from 0.20 to 25 m. 7 refs., 7 figs.

Colloidal aggregation revisited: new insights based on fractal structure and surface-enhanced raman

Abstract: We have examined both the structure and surface chemlstry of gold clusters formed by the kinetic aggregation of colloidal gold particles. The highly disordered. ramified aggregate?, can be very accurately described as self-similar or fractal objects with a fractal dimension equal to 1.75. Spectroscopic studies performed with surface-enhanced Raman scattering (SERS), clearly indicate that colloidal gold surfaces are highly heterogeneous. consisting both of donor and acceptor sites which can be identified as Au(O) and Au(I), respectively. Aggregation occurs when negatively charged species are displaced from the gold surface by more strongly bound molecular adsorhateb, with the rate determined by the nature and concentration of the displacing species. The new insights afforded by the fractal description of the structure of the aggregates and the SERS probe of the chemical nature of the colloid surface should lead to a more complete understanding of the basic mechanisms of colloid aggregation. This potential is illustrated with a quantitative description of the dynamics of aggregate growth measured by dynamic light scattering.

Fractal Viewpoint of Fracture and Accretion

Abstract: A model from a point of view of the fractal is presented to describe fracture and accretion, It is shown to give a power-law dependence of the cumulative number of similar objects in their collection whose sizes are larger than r , N ( r )∼ r - x , which has been observed in many phenomena associated with fracture and accretion. The case of drift ice is shown as an example. A relationship is also derived among exponents of cumulative number and mass, and a fractal dimension of similar objects.

The fractal nature of the inhomogeneities in the lithosphere evidenced from seismic wave scattering

Abstract: In this paper we show evidences of the fractal nature of the 3-D inhomogeneities in the lithosphere from the study of seismic wave scattering and discuss the relation between the fractal dimension of the 3-D inhomogeneities and that of the fault surfaces. Two methods are introduced to measure the inhomogeneity spectrum of a random medium: 1. the coda excitation spectrum method, and 2. the method of measuring the frequency dependence of scattering attenuation. The fractal dimension can be obtained from the inhomogeneity spectrum of the medium. The coda excitation method is applied to the Hindu-Kush data. Based on the observed coda excitation spectra (for frequencies 1–25 Hz) and the past observations on the frequency dependence of scattering attenuation, we infer that the lithospheric inhomogeneities are multiple scaled and can be modeled as a bandlimited fractal random medium (BLFRM) with an outer scale of about 1 km. The fractal dimension of the 3-D inhomogeneities isD3=31/2–32/3, which corresponds to a scaling exponent (Hurst number)H=1/2–1/3. The corresponding 1-D inhomogeneity spectra obey the power law with a powerp=2H+1=2–5/3. The intersection between the earth surface and the isostrength surface of the 3-D inhomogeneities will have fractal dimensionD1=1.5–1.67. If we consider the earthquake fault surface as developed from the isosurface of the 3-D inhomogeneities and smoothed by the rupture dynamics, the fractal dimension of the fault trace on the surface must be smaller thanD1, in agreement with recent measurements of fractal dimension along the San Andreas fault.

Surfaces, interfaces, and screening of fractal structures

Abstract: Fractal objects strongly screen external fields; only a small ``surface'' portion of the object is exposed appreciably to the field. We have studied this exposed surface of several random fractals as measured by random walkers and by ballistic particles launched from outside and absorbed by the fractal. The number of absorbing sites weighted by their rate of absorption shows an apparent power-law scaling with fractal mass. For diffusion-limited aggregates, ballistically generated aggregates, and screened-growth clusters in two dimensions, this power-law relationship is for the most part in accord with mean-field predictions of previous work. This accord is poorest for the objects of lowest fractal dimensionality. We have confirmed that this scaling is different from that of the old-growth--new-growth interface studied previously. We also find that a ``hierarchy'' of fractal dimensions describes the external surface of diffusion-limited aggregates.

Anisotropy and cluster growth by diffusion-limited aggregation.

Abstract: We describe a simple theory of diffusion-limited-aggregation cluster growth which relates the large-scale shape of the cluster to its fractal dimension. We present results of computer simulation for DLA clusters grown with anisotropic sticking rules which provide strong confirmation of our model in two dimensions. New universal exponents are predicted and found. We are also able to obtain good estimates for the fractal dimension of ordinary DLA.

Renormalization, unstable manifolds, and the fractal structure of mode locking.

Abstract: The apparent universality of the fractal dimension of the set of quasiperiodic windings at the onset of chaos in a wide class of circle maps is described by construction of a universal one-parameter family of maps which lies along the unstable manifold of the renormalization group. The manifold generates a universal ``devil's staircase'' whose dimension agrees with direct numerical calculations. Applications to experiments are discussed.