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Discrete scale invariance, which corresponds to a partial breaking of the scaling symmetry, is reflected in the existence of a hierarchy of characteristic scales l₀,l₀λ,l₀λ²,…, where λ is a preferred scaling ratio and l₀ a microscopic cut-off. Signatures of discrete scale invariance have recently been found in a variety of systems ranging from rupture, earthquakes, Laplacian growth phenomena, "animals" in percolation to financial market crashes. We believe it to be a quite general, albeit subtle phenomenon. Indeed, the practical problem in uncovering an underlying discrete scale invariance is that standard ensemble averaging procedures destroy it as if it was pure noise. This is due to the fact, that while λ only depends on the underlying physics, l₀ on the contrary is realization-dependent. Here, we adapt and implement a novel so-called "canonical" averaging scheme which re-sets the l₀ of different realizations to approximately the same value. The method is based on the determination of a realization-dependent effective critical point obtained from, e.g., a maximum susceptibility criterion. We demonstrate the method on diffusion limited aggregation and a model of rupture.

This paper presents a computer model of diffusion limited aggregation (DLA) in percolation cluster. Simulation of the aggregation clusters in percolation cluster with varying occupancy probability is performed, and their fractal dimension and multifractal spectrum are obtained. The simulation results show that the percolation cluster has stronger effects on the aggregation clusters' pattern structure when occupancy probability is smaller. The dimension D_f of aggregation clusters increases together with the increase of occupancy probability. Furthermore, the multifractal spectra f(α) curve becomes higher and the range of singularity α wilder. The bigger the occupancy probability is, the more irregular and non-uniform the aggregation clusters becomes.

A modified diffusion-limited aggregation (DLA) model for two-dimensional (2D), three-dimensional (3D) fractal growth and 3D island growth was established based on the DLA model in this paper. The number of particles $N$ and the size of the box size $V$ (related to side length $L$), which are related to film thickness, are considered in the study. The simulation results are a good reflection of the actual experimental results. The results show that the particle number and simulation box size can affect the fractal morphology and fractal dimension of the film, and also the 2D to 3D transformation. In addition, the critical particle number $N_{\mathrm{\text{critical}}}$ and the critical box size $V_{\mathrm{\text{critical}}}$ during the transformation process are also given.

In this paper, the membership function in fuzzy systems is used in the Diffusion Limited Aggregation (DLA) model to investigate the fractal diffusion of soot particles from diesel engine emissions. The transformation of the morphology of soot particle aggregates and the control of fractal diffusion of soot particles are investigated by analyzing the nonlinear relationship between the motion steps and angles of diffusing particles. The simulation results demonstrate that the morphology of the aggregates varies from loose to compact by changing the particles’ motion steps and angles in membership functions. Meanwhile, the Ballistic Aggregation (BA)-like aggregates are obtained. Furthermore, the control of the morphology of soot particle aggregates is realized, which makes the settlement of the aggregates become easier. This will provide a reference for further understanding the growth mechanism of soot particle diffusion and enhancing the purification technology of the soot particles.

The development of shape and form is intrinsic to the structure and function of many biological macromolecules including tubulin, actin and collagen. Type I collagen is a major structural protein in the body, providing mechanical strength for tissues such as bone and skin. It is present in the form of fibrils which display a regular banding pattern known as D-periodicity (where D = 67 nm). Type I collagen is a long rod-like molecule (300 nm ×1.5 nm) consisting of a triple helix formed from three polypeptide chains. In vivo and in vitro studies have shown that collagen molecules self-assemble in a regular D-staggered array to form striated fibrils. Further studies have shown that the process, termed fibrillogenesis, is entropy driven. A model based on diffusion limited aggregation was used to investigate the properties of rod self-assembly. This simple model reproduced several experimentally observed features of collagen fibril morphology including a linear mass/unit length profile and a preference for tip growth.

Historically, fractal analysis has been remarkably successful in describing wide ranging kinetic processes on (idealized) scale invariant objects in terms of elegantly simple universal scaling laws. However, as nanostructured materials find increasing applications in energy storage, energy conversion, healthcare, etc., one must reexamine the premise of traditional fractal scaling laws as it only applies to physically unrealistic infinite systems, while all natural/engineered systems are necessarily finite. In this article, we address the consequences of the 'finite-size' problem in the context of time dependent diffusion towards fractal surfaces via the novel technique of Cantor-transforms to (i) illustrate how finiteness modifies its classical scaling exponents; (ii) establish that for finite systems, the diffusion-limited reaction is decelerated below a critical dimension and accelerated above it; and (iii) to identify the crossover size-limits beyond which a finite system can be considered (practically) infinite and redefine the very notion of 'finiteness' of fractals in terms of its kinetic response. Our results have broad implications regarding dynamics of systems defined by the same fractal dimension, but differentiated by degree of scaling iteration or morphogenesis, e.g. variation in lung capacity between a child and adult.

Theoretical and computer simulation analysis of clusters growing by diffusion limited aggregation under rotation around a germ is presented. The theoretical model allows to study statistical properties of growing clusters in two different situations: in the static case (the cluster is fixed), and in the case when the growing structure has a nonzero rotation velocity around its germ. By the direct computer simulation the growth of rotating clusters is investigated. The fractal dimension of such clusters as a function of the rotation velocity is found. It is shown that for small enough velocities the fractal dimension is growing, but then, with increasing rotation velocity, it tends to unity.