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The functional features of spatial networks depend upon a non-trivial relationship between the topological and physical structure. Here, we explore that relationship for spatial networks with radial symmetry and disordered fractal morphology. Under a geometric graphs approach, we quantify the effectiveness of the exchange of information in the system from center to perimeter and over the entire network structure. We mainly consider two paradigmatic models of disordered fractal formation, the Ballistic Aggregation and Diffusion-Limited Aggregation models, and complementary, the Viscek and Hexaflake fractals, and Kagome and Hexagonal lattices. First, we show that complex tree morphologies provide important advantages over regular configurations, such as an invariant structural cost for different fractal dimensions. Furthermore, although these systems are known to be scale-free in space, they have bounded degree distributions for different values of an euclidean connectivity parameter and, therefore, do not represent ordinary scale-free networks. Finally, compared to regular structures, fractal trees are fragile and overall inefficient as expected, however, we show that this efficiency can become similar to that of a robust hexagonal lattice, at a similar cost, by just considering a very short euclidean connectivity beyond first neighbors.
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We studied the two-step crystallisation process of a magnetic active 2D-granular system placed on different lens concaveness and under the action of an alternating magnetic field which controls its effective temperature. We have observed that the two-step features of the crystallisation process are more evident as the depth of the parabolic potential increases. At the initial formation of the nucleus, as a first step, in the central region of the lens an amorphous aggregate is formed. In an ulterior second step, this disordered aggregate, due to the effective temperature and the perturbations caused by the impacts of free particles moving in the surrounding region, evolves to an ordered crystalline structure. The nucleus size is larger for deeper concaveness of the parabolic potential. However, if the depth of the parabolic potential exceeds a certain value, the reordering process of the second step does not occur. The crystal growth occurs similarly; small disordered groups of particles join the nucleus, forming an amorphous shell of particles which experiments a rearranging while the aggregate grows. In the explored range of the depths of the parabolic potential, crystallisation generally occurs quicker as the deeper parabolic potential is. Also, aggregates are more clearly round-shaped as parabolic potential depth increases. On the contrary, the structures are more branched for a smaller depth of the parabolic potential. We studied the structural changes and features in the system by using the sixth orientational order parameter and the packing fraction.
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It has been shown that a nonvibrating magnetic granular system, when fed by an alternating magnetic field, behaves with most of the distinctive physical features of active matter systems. In this work, we focus on the simplest granular system composed of a single magnetized spherical particle allocated in a quasi-one-dimensional circular channel that receives energy from a magnetic field reservoir and transduces it into a running and tumbling motion. The theoretical analysis, based on the run-and-tumble model for a circle of radius R, forecasts the existence of a dynamical phase transition between an erratic motion (disordered phase) when the characteristic persistence length of the run-and-tumble motion, â_{c}
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We study the crystallisation processes occurring in a nonvibrating two-dimensional magnetic granular system at various fixed values of the effective temperature. In this system, the energy loss due to dissipative effects is compensated by the continuous energy input coming into the system from a sinusoidal magnetic field. When this balance leads to high values of the effective temperature, no aggregates are formed, because particles' kinetic energy prevents them from aggregating. For lower effective temperatures, formation of small aggregates is observed. The smaller the values of the applied field's amplitude, the larger the number of these disordered aggregates. One also observes that when clusters form at a given effective temperature, the average effective diffusion coefficient decreases as time increases. For medium values of the effective temperature, formation of small crystals is observed. We find that the sixth bond-orientational order parameter and the number of bonds, when considering more than two, are very sensitive for exhibiting the order in the system, even when crystals are still very small.
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Applying an unsteady magnetic field to a 2D nonvibrating magnetic granular system induces a random motion in the steel beads with characteristics analogous to that of molecules in a fluid. We investigate the structural characteristics of the solid-like structures generated by different quenching conditions. The applied field is generated by the superposition of a constant field and a collinear sinusoidal field. The system reaches a quasi steady state in which the effective temperature is proportional to the amplitude of the applied field. By reducing the effective temperature at different rates, different cooling rates are produced. A slight inclination of the surface allows us to investigate the effects of small particle concentration gradients. The formation of a wide and rich variety of condensed solid structures, from gel-like and glass-like structures up to crystalline structures, is observed and depends on the cooling rate. We focus our attention on the crystallization process and found this process to be a collective phenomenon. We discuss our results in terms of the measured time evolution of the mean squared displacement, the effective diffusion coefficient, and the radial distribution function.
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Laplacian growth, associated to the diffusion-limited aggregation (DLA) model or the more general dielectric-breakdown model (DBM), is a fundamental out-of-equilibrium process that generates structures with characteristic fractal/non-fractal morphologies. However, despite diverse numerical and theoretical attempts, a data-consistent description of the fractal dimensions of the mass-distributions of these structures has been missing. Here, an analytical model of the fractal dimensions of the DBM and DLA is provided by means of a recently introduced dimensionality equation for the scaling of clusters undergoing a continuous morphological transition. Particularly, this equation relies on an effective information-function dependent on the Euclidean dimension of the embedding-space and the control parameter of the system. Numerical and theoretical approaches are used in order to determine this information-function for both DLA and DBM. In the latter, a connection to the Rényi entropies and generalized dimensions of the cluster is made, showing that DLA could be considered as the point of maximum information-entropy production along the DBM transition. The results are in good agreement with previous theoretical and numerical estimates for two- and three-dimensional DBM, and high-dimensional DLA. Notably, the DBM dimensions conform to a universal description independently of the initial cluster-configuration and the embedding-space.
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Stochastic growth processes give rise to diverse and intricate structures everywhere in nature, often referred to as fractals. In general, these complex structures reflect the non-trivial competition among the interactions that generate them. In particular, the paradigmatic Laplacian-growth model exhibits a characteristic fractal to non-fractal morphological transition as the non-linear effects of its growth dynamics increase. So far, a complete scaling theory for this type of transitions, as well as a general analytical description for their fractal dimensions have been lacking. In this work, we show that despite the enormous variety of shapes, these morphological transitions have clear universal scaling characteristics. Using a statistical approach to fundamental particle-cluster aggregation, we introduce two non-trivial fractal to non-fractal transitions that capture all the main features of fractal growth. By analyzing the respective clusters, in addition to constructing a dynamical model for their fractal dimension, we show that they are well described by a general dimensionality function regardless of their space symmetry-breaking mechanism, including the Laplacian case itself. Moreover, under the appropriate variable transformation this description is universal, i.e., independent of the transition dynamics, the initial cluster configuration, and the embedding Euclidean space.
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In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.
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We study experimentally the cratering process due to the explosion and collapse of a pressurized air cavity inside a sand bed. The process starts when the cavity breaks and the liberated air then rises through the overlying granular layer and produces a violent eruption; it depressurizes the cavity and, as the gas is released, the sand sinks under gravity, generating a crater. We find that the crater dimensions are totally determined by the cavity volume; the pressure does not affect the morphology because the air is expelled vertically during the eruption. In contrast with impact craters, the rim is flat and, regardless of the cavity shape, it evolves into a circle as the cavity depth increases or if the chamber is located deep enough inside the bed, which could explain why most of the subsidence craters observed in nature are circular. Moreover, for shallow spherical cavities, a collimated jet emerges from the collision of sand avalanches that converge concentrically at the bottom of the depression, revealing that collapse under gravity is the main mechanism driving the jet formation.