Dimensions, maximal growth sites, and optimization in the dielectric breakdown model

Mathiesen, Joachim; Jensen, Mogens H.; Bakke, Jan Øystein Haavig

doi:10.1103/physreve.77.066203

Cited by 8 publications

(11 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As it tries to maximize the flux into its tip, however, its growth sometimes causes the flux to decline. Indeed, substituting the geodesic growth condition (35) into equations (48), (49) and (50), we find that the variation of the flux reads…”

Section: Maximization Of the Flux Entering The Tipmentioning

confidence: 99%

“…More specifically, at each time step, the slit mapping (25) introduces a new singularity in g, since f ∼ ω − a near the pole a. The probability measure thus becomes non-integrable when η gets larger than 2 (this value actually depends on the infinitesimal map [45,46,48]). The walkers then accumulate on existing tips and the aggregate becomes non-fractal.…”

Section: Random Forcingmentioning

confidence: 99%

See 1 more Smart Citation

Laplacian networks: Growth, local symmetry, and shape optimization

et al. 2017

View full text Add to dashboard Cite

Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three formulations of the direction rule, namely geodesic growth, growth that maintains local symmetry, and growth that maximizes flux into tips for a given amount of growth. Surprisingly, we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips.

show abstract

Section: Maximization Of the Flux Entering The Tipmentioning

confidence: 99%

Section: Random Forcingmentioning

confidence: 99%

Laplacian networks: Growth, local symmetry, and shape optimization

et al. 2017

View full text Add to dashboard Cite

show abstract

“…(The latter contains a figure depicting a simulation of the η = 3 DBM.) However, a later study estimates the dimension of η-DBM in more detail and does not find evidence for a phase transition at η = 4, and concludes that the dimension of η-DBM is about 1.08 when η = 4 [MJB08].…”

Section: Interpretation and Conjecture When η Is Largementioning

confidence: 99%

Quantum Loewner evolution

Miller¹,

Sheffield⋆²

2016

Duke Math. J.

124

View full text Add to dashboard Cite

What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the η-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter γ ∈ [0, 2].In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ 2 , η). QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion ν t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of ν t using an SPDE. For each γ ∈ (0, 2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of ν t .We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation.We propose QLE(2, 1) as a scaling limit for DLA on a random spanningtree-decorated planar map, and QLE(8/3, 0) as a scaling limit for the Eden model on a random triangulation. We propose using QLE(8/3, 0) to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE(8/3, 0), up to a fixed time, as a metric ball in a random metric space.

show abstract

“…The criticality of the DBM transition is understood in terms of as a non-thermal order parameter. For example, for , it has been suggested that the full collapse to linear clusters occurs at the “critical” value 46,52–54 . To address this point, the normalized dimension, , can be used as the non-thermal order parameter of the system, where , when at , and , when as .…”

Section: Discussionmentioning

confidence: 99%

“…As function of the parameter , they go from isotropic and compact structures with , for (Eden clusters); through intricate dendritic-like fractals with , for ; to highly anisotropic linear structures () as . For , the collapse to is expected to occur at 46,47,52–54 . Most characterization approaches rely on numerical methods to estimate , with very few theoretical results for 4,5 .…”

Section: Introductionmentioning

confidence: 99%

A universal dimensionality function for the fractal dimensions of Laplacian growth

Nicolás-Carlock

Carrillo-Estrada

2019

Sci Rep

View full text Add to dashboard Cite

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.

show abstract

Dimensions, maximal growth sites, and optimization in the dielectric breakdown model

Cited by 8 publications

References 14 publications

Laplacian networks: Growth, local symmetry, and shape optimization

Laplacian networks: Growth, local symmetry, and shape optimization

Quantum Loewner evolution

A universal dimensionality function for the fractal dimensions of Laplacian growth

Contact Info

Product

Resources

About