Multifractal analysis of the branch structure of diffusion-limited aggregates

Quantum multifractality as a probe of phase space in the Dicke model

We study the multifractal behavior of coherent states projected in the energy eigenbasis of the spin-boson Dicke Hamiltonian, a paradigmatic model describing the collective interaction between a single bosonic mode and a set of two-level systems. By examining the linear approximation and parabolic correction to the mass exponents, we find ergodic and multifractal coherent states and show that they reflect details of the structure of the classical phase space, including chaos, regularity, and features of localization. The analysis of multifractality stands as a sensitive tool to detect changes and structures in phase space, complementary to classical tools to investigate it. We also address the difficulties involved in the multifractal analyses of systems with unbounded Hilbert spaces.

Optical properties of soot aggregates with different monomer shapes

The monomer of soot fractal aggregate is usually considered to be sphere, but the monomer shapes are cube and hexagon by some transmission electron microscope (TEM) and scanning electron microscope (SEM) observation. In this paper, the fractal soot models of different monomer shapes (sphere, cube, ellipsoid, hexagonal prism) were established. And the optical properties of models are calculated by discrete dipole approximation (DDA). After systematically comparing the Muller matrix and optical cross section properties between the models, we find that monomer deviation from sphericity does not necessarily lead to further decline of F₂₂(π)/F₁₁(π) even at shorter wavelengths. In other words, the non-sphericity of monomers does not necessarily affect the non-sphericity of whole soot particle. This can provide some implication for lidar remote sensing observation. However, other light scattering matrix elements can keep good consistency. The maximum deviation of extinction cross section of hexagonal prism model is 11.2%. The more the monomer shape deviates from the sphere, the more the optical integral properties of the non-spherical monomer model deviates from the optical integral properties of sphere monomer model. Hence, the difference in optical properties caused by different monomer shapes cannot be neglected when the monomer deviates significantly from a spherical shape. This work is helpful to evaluate the optical properties of soot aggregates more precisely.

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.

Random deposition with a power-law noise model: Multiaffine analysis

We study the random deposition model with power-law distributed noise and rare-event dominated fluctuation. In this model instead of particles with unit sizes, rods with variable lengths are deposited onto the substrate. The length of each rod is chosen from a power-law distribution P(l)∼l^{-(μ+1)}, and the site at which each rod is deposited is chosen randomly. The results show that for μ<μ_{c}=3 the log-log diagram of roughness, W(t), versus deposition time, t, increases as a step function, where the roughness in each interval acts as W_{loc}(t)≈t^{β_{loc}}. The local growth exponent, β_{loc}, is zero for μ=1. By increasing the μ exponent, the value of β_{loc} is increased. It tends to the growth exponent of the random distribution model with Gaussian noise, β=1/2, at μ_{c}=3. The fractal analysis of the height fluctuations for this model was performed by multifractal detrended fluctuation analysis algorithm. The results show multiaffinity behavior for the height fluctuations at μ<μ_{c} and the multiaffinity strength is greater for smaller values of the μ exponent.

How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth

We study the fractal structure of diffusion-limited aggregation (DLA) clusters on a square lattice by extensive numerical simulations (with clusters having up to 10^{8} particles). We observe that DLA clusters undergo strongly anisotropic growth, with the maximal growth rate along the axes. The naive scaling limit of a DLA cluster by its diameter is thus deterministic and one-dimensional. At the same time, on all scales from the particle size to the size of the entire cluster it has a nontrivial box-counting fractal dimension which corresponds to the overall growth rate, which, in turn, is smaller than the growth rate along the axes. This suggests that the fractal nature of the lattice DLA should be understood in terms of fluctuations around the one-dimensional backbone of the cluster.

Interplay of model ingredients affecting aggregate shape plasticity in diffusion-limited aggregation

We analyze the combined effect of three ingredients of an aggregation model--surface tension, particle flow and particle source--representing typical characteristics of many aggregation growth processes in nature. Through extensive numerical experiments and for different underlying lattice structures we demonstrate that the location of incoming particles and their preferential direction of flow can significantly affect the resulting general shape of the aggregate, while the surface tension controls the surface roughness. Combining all three ingredients increases the aggregate shape plasticity, yielding a wider spectrum of shapes as compared to earlier works that analyzed these ingredients separately. Our results indicate that the considered combination of effects is fundamental for modeling the polymorphic growth of a wide variety of structures in confined geometries and/or in the presence of external fields, such as rocks, crystals, corals, and biominerals.

Barycentric fixed-mass method for multifractal analysis

We present a method to estimate the multifractal spectrum of point distributions. The method incorporates two motivated criteria (barycentric pivot point selection and nonoverlapping coverage) in order to reduce edge effects, improve precision, and reduce computation time. Implementation of the method on synthetic benchmarks demonstrates the superior performance of the proposed method compared with existing alternatives routinely used in the literature. Finally, we use the method to estimate the multifractal properties of the widely studied growth process of diffusion-limited aggregation (DLA) and compare our results with recent and earlier studies. Our tests support the conclusion of a genuine but weak multifractality of the central core of DLA clusters, with D(q) decreasing from 1.75±0.01 for q=-10 to 1.65±0.01 for q =+10.