This book is a collection of survey articles covering many of the most recent developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals.
Fractal to Non-Fractal Morphological Transitions in Stochastic Growth Processes
Fractal Geometry and Stochastics IV. Selfsimilarity and Random Walks. Conformal Dynamics and SchrammLoewner Evolution. Random Fractal Processes. Emergence of Fractals in Complex Systems. A Survey of Dynamical Percolation. Schematic diagram of the fundamental aggregation models top row used in this work, where particles, that are launched one-by-one into the system from r L with uniform probability in position and direction, a follow straight-line trajectories before aggregation in BA, b perform a random walk in DLA, and c get radially attached to the closest particle in the cluster as a result of an infinite-range radial interaction in MF.
The corresponding characteristic cluster with its fractal dimension D 0 is shown in the bottom row. However, one of the most challenging aspect of the theory comes when the growth is not purely limited by diffusion, e. In this scenario, the generalized HTM equation [ 17 , 27 ], given as,. Nonetheless, as shown here, Eq.
With regard to the latter, there are two physical mechanisms related to these interactions and two models that are able to reproduce their effects. Second, the model referred here to as the p -model [ 12 ], incorporates anisotropy effects coming from surface-tension-like interactions by means of a Monte Carlo approach to aggregation using fundamental stochastic and energetic models as explained below and illustrated in Figure 2d. Therefore, by controlling the interplay of any of these two mechanisms with a pure stochastic model in this case the DLA or BA models , one is able to generate fractal to nonfractal morphological transitions.
Schematic diagrams of the energetic aggregation schemes. This is determined by the position of the aggregated particles projected onto the direction of motion of the incoming particle. On the other hand, for the MF model, particles always aggregate to the closest particle in the cluster. This is determined by projecting the position of the aggregated particles along the direction of motion of the incoming particle see Figure 1c.
Further on, as typically done, their mean free path is modified as the particles wander beyond a distance larger than r L or in-between branches. The evaluation of the aggregation scheme to be used is only updated once and the particle has been successfully aggregated to the cluster under a given dynamics. In all measurements, we performed an ensemble average over clusters containing 1.
In first place, we measured the fractal dimension from the two-point density correlation function,. In second place, we also measured the radius of gyration given by. In particular, for the p -model, linear-fits at different scales were performed in order to capture the main local fractal features. In addition, we averaged the results of 10 linear fits, distributed over a given interval, in order to improve the precision of the measurements. The squares display the multiscaling evolution of the structure.
Clusters of 1. These transitions exhibit fast morphological transformations as p increases, from unstable tip-splitting DLA or dense branching BA , through inhomogeneous dendritic, to needle-like growth MF. To do so, let us start by showing that Eq. It easily follows that,. Additionally, the first-order approximation of Eq.
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Its first-order approximation is then,. In the first approach to morphological transitions, we will consider the case when long-range attractive interactions are introduced in the growth dynamics. In this case, the way to obtain self-similar clusters, that is, clusters with a single fractal dimension, is to maintain a proper balance between the energetic and entropic contributions to the growth process.
It can be appreciated that this growth presents three well-defined stages as illustrated in Figures 3d and 4d. In consequence, the individual interaction regions of the aggregated particles are highly overlapped, forming an almost circular envelope or effective boundary of aggregation around the cluster. This makes the last aggregated particle the most probable aggregation point in the cluster for the next incoming particle. This structural feature is reminiscent of a mean-field MF behavior. In the second stage, clusters exhibit a transition in growth dynamics.
Here, the envelope starts to develop small deviations from its initial circular form, with typically three main elongations or growth instabilities associated with the main branches.
Then, when the interactive envelope develops a branched structure itself, particles are able to penetrate into the inner regions of the aggregate and another transition in growth dynamics takes place, from interaction-limited to trajectory-limited. That is, even though interactions leave a strong imprint in the local structure and fractality of the clusters, the stochastic nature of the particle trajectories will ultimately determine their global characteristics.
In the second approach, a general stochastic aggregation process can be model under a Monte Carlo scheme involving three fundamental and simple off-lattice models of particle-cluster aggregation. On the other, we introduce a mean-field MF model of long-range interactive particle-cluster aggregation [ 11 , 12 ] that provides the most energetic and noiseless aggregation dynamics that, simultaneously, acts as the main source of anisotropy.
We must remark that this anisotropy is purely generated by the growth dynamics and not from lattice effects [ 28 ] see Figure 1c.
vestbucontidi.gq These multiscaling features reveal a crossover behavior that can be properly quantified by measuring a local or effective, D p , at different scales [ 7 ], as shown in Figure 7a details for the values of the parameter used to produce Figure 7 are presented in Table 1. Analytically, all measurements can be described by Eqs. Indeed, the data for D p as obtained through to C r are very well described by Eq. Thus, the data obtained are then described by Eq.
This analysis is presented in Figure 7d. Parameters for the plots of D p and D q in Figure 7. In the second block for Eqs. All of the fittings to the numerical data were performed using the gnuplot embedded algorithms. The curves for the DBM given by Eqs. In this case, Eqs. For further details about the parameter values used, see Table 1.
See, for example, Figure 7c and the bottom pane in Figure 7d. From Eqs.