Network efficiency of spatial systems with fractal morphology: a geometric graphs approach

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.

Two-step crystallisation in a 2D active magnetic granular system confined by a parabolic potential

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.

Magnetized granular particles running and tumbling on the circle S^{1}

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, ℓ_{c}<R/2, to a persistent motion (ordered phase) when ℓ_{c}>R/2. It is found that the limiting behaviors of these phases correspond to Brownian motion on the circle and a simple uniform circular motion, respectively. Furthermore, it is qualitatively shown that the smaller the magnetization of a particle, the larger the persistence length. It is so at least within the experimental limit of validity of our experiments. Our results show a very good agreement between theory and experiment.

Crystallization processes in a nonvibrating magnetic granular system with short range repulsive interaction

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.

A universal dimensionality function for the fractal dimensions of Laplacian growth

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.

Universal fractality of morphological transitions in stochastic growth processes

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.

Penetration of granular projectiles into a water target

The penetration of low-speed projectiles into a water target has been studied in the last several years to understand the physics behind the formation and collapse of cavities. In such studies, the projectiles employed were solid bodies or liquid drops. Here we report similar impact experiments using granular projectiles, with the aim to investigate how the morphology of the cavities is determined by the balance between the dynamic pressure exerted by the fluid and the cohesive strength of the impactors. From the results we present and discuss in this manuscript, we speculate on the dynamics of meteorite disintegration in the atmosphere of our planet.

Nucleation, aggregation, annealing, and disintegration of granular clusters

The processes of nucleation, aggregation, annealing, and disintegration of clusters of non-Brownian paramagnetic beads in a vibrofluidized system are experimentally investigated. The interaction among the beads is induced by a magnetic seed composed of two dipoles allocated outside the container cell. We observe a clearly differentiated nucleation stage, whose evolution (nucleation time versus acceleration strength) follows a power law. Thereafter, the beads aggregate forming 2D disordered clusters around the nucleus. Both processes (nucleation and aggregation) are determined by the competition between magnetic forces and the drag produced by a thermal bath created by glass particles. Once the agglomerates reach a final state (shape and length), they are annealed by increasing and decreasing the granular temperature. We found that the fractal dimension and the lacunarity index clearly describe the structural variations of the clusters. Our discussion on this phenomenon is addressed, making a rough analogy with the glass transition in a super-cooled liquid. Finally, we study the disintegration of the clusters as a function of time and the density of the surrounding gas. The question is not if, but how they disintegrate upon removing the external field; we find that the disintegration follows an exponential decay.

Free-energy landscapes of granular clusters grown by magnetic interaction

We experimentally study the aggregation of small clusters made of non-Brownian dipolar beads in a vibro-fluidized system. The particles are paramagnetic spheres that add around a fixed magnetic seed inside a granular gas of glass beads. We observe that under appropriate physical conditions symmetric and asymmetric cluster configurations are created and, as the number of particles increases, the aggregation time obeys a power law. We use an ensemble statistics to evaluate the free-energies and entropies landscapes of the granular clusters. The correspondence between such landscapes shows that, even if the system is of macroscopic scale and not in strict equilibrium, our approach to understand the relationship between the cluster structures and the interactions that create them is reliable.