Niemeyer, Pietronero, and Wiesmann Respond:

Niemeyer, L.; Pietronero, L.; Wiesmann, H. J.

doi:10.1103/physrevlett.57.650

Cited by 83 publications

(132 citation statements)

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“…July 1995 nomenological analysis or modeling can be found in various other fields. The structure of velocity field in fully developed turbulence, for example, can be described phenomenologically by a simple model of multifractal cascade (Benzi et al, 1984;Paladin and Vulpiani, 1987). This interesting observation, however, does not explain how scaling behavior actually arises from the NavierStokes equation.…”

Section: Introduction: Physics Of Fractalsmentioning

confidence: 94%

“…The first part of this program was accomplished a few years ago with the introduction of the model of diffusion-limited aggregation (DLA; Witten and Sander, 1981) and the dielectric breakdown model (DBM; Niemeyer et al, 1984). These models have a direct relation with several phenomena like dendritic growth, dielectric breakdown, and viscous fingering, and they are considered prototypical fractal-growth models in a more general perspective, too.…”

Section: Introduction: Physics Of Fractalsmentioning

confidence: 99%

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The fixed-scale transformation approach to fractal growth

1995

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Irreversible fractal-growth models like diffusion-limited aggregation (DLA) and the dielectric breakdown model (DBM) have confronted us with theoretical problems of a new type for which standard concepts like field theory and renormalization group do not seem to be suitable. The fixed-scale transformation (FST) is a theoretical scheme of a novel type that can deal with such problems in a reasonably systematic way. The main idea is to focus on the irreversible dynamics at a given scale and to compute accurately the nearest-neighbor correlations at this scale by suitable lattice path integrals. The next basic step is to identify the scale-invariant dynamics that refers to coarse-grained variables of arbitrary scale. The use of scaleinvariant growth rules allows us to generalize these correlations' to coarse-grained cells of any size and therefore to compute the fractal dimension. The basic point is to split the long-time limit (t -+ 00 ) for the dynamical process at a given scale that produces the asymptotically frozen structure, from the large-scale limit (r--+ 00 ) which defines the scale-invariant dynamics. In addition, by working at a fixed scale with respect to dynamical evolution, it is possible to include the fluctuations of boundary conditions and to reach a remarkable level of accuracy for a real-space method. This new framework is able to explain the self-orgauized critical nature and the origin of fractal structures in irreversible-fractal-growth models. It also provides a rather systematic procedure for the analytical calculation of the fractal dimension and other critical exponents. The FST method can be naturally extended to a variety of equilibrium and nonequilibrium models that generate fractal structures.

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Section: Introduction: Physics Of Fractalsmentioning

confidence: 94%

Section: Introduction: Physics Of Fractalsmentioning

confidence: 99%

The fixed-scale transformation approach to fractal growth

1995

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“…Small DB clusters also appear to be self-similar fractals with D 1.7 [2,5]. However, we are unaware of any studies of large () 10 particles) DB clusters and the on-lattice DLA studies suggest that large clusters are necessary to reveal the full behavior of the model [14].…”

mentioning

confidence: 99%

“…We [2] are among the most widely studied models for generating fractal growth patterns (for recent reviews, see, e.g. , [3,4]).…”

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Comparative study of large-scale Laplacian growth patterns

1995

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We investigate the scaling of cluster size with mass for our simulations of diffusion-limited aggregation (DLA) clusters and dielectric-breakdown (DB) clusters of 10 particles grown on a square lattice, and DLA clusters of 10 particles grown off lattice. We [2] are among the most widely studied models for generating fractal growth patterns (for recent reviews, see, e.g. , [3,4]). Each of these models is a Laplacian growth process in which difFusing particles released from a distant boundary attach to the perimeter of a growing cluster. The essential difFerence between the models is a boundary condition [5].In on-lattice versions the diffusing particle undergoes a random walk on a lattice and particles in the cluster are represented by occupied (aggregate) sites on the lattice.In the DLA model the diffusing particles are terminated at the first surface site (unoccupied site adjacent to an aggregate site) that they contact. This surface site is then converted to an aggregate site. In the DB model the diffusing particles can diffuse through surface sites but are terminated at the first aggregate site they contact. The surface site immediately preceding this termination site in the difFusion process is then converted to an aggregate site. The DB boundary condition introduces an inherent surface tension [6,7]. In this sense, the DB model is more physically appealing than the DLA model, which has zero surface tension [8].An "ofF-lattice" Laplacian growth model has also been introduced [9]. In this model, the diffusing particle slides over fixed step lengths in free space until it first intersects the cluster. The difFusing particle is then moved back and is attached at its first point of contact with the cluster. We will follow convention and refer to this growth process as "off-lattice DLA. " However, this offlattice growth process more closely realizes the boundary conditions for the DB model, since the difFusing particle can difFuse along the "surface" of the cluster in each of these models.Much of the theoretical interest in the above models has been directed at determining precise numerical and algebraic estimates for the &actal dimension D [3,4]. OfFlattice DLA clusters are homogeneous self-similar fractals, with fractal dimension D 1.7 in d = 2, independent of cluster size (see, e.g. , [10,11] DB clusters and the on-lattice DLA studies suggest that large clusters are necessary to reveal the full behavior of the model [14]. It is generally believed that the differences between large-scale on-lattice and off-lattice DLA patterns are due to the underlying lattice anisotropy. However, as noted above, the kinetics of attachment implied by the boundary conditions are also difFerent in these two models. The role played by this difFerence has not been fully explored.Large-scale studies of on-lattice DB patterns are also useful for this purpose because the kinetics of attachment are similar to off-lattice DLA, but the diffusion process is confined to the underlying lattice. The possibility of signatures of the micros...

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Prediction of Energy Storage Performance in Polymer Composites Using High‐Throughput Stochastic Breakdown Simulation and Machine Learning

et al. 2022

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Polymer dielectric capacitors are widely utilized in pulse power devices owing to their high power density. Because of the low dielectric constants of pure polymers, inorganic fillers are needed to improve their properties. The size and dielectric properties of fillers will affect the dielectric breakdown of polymer‐based composites. However, the effect of fillers on breakdown strength cannot be completely obtained through experiments alone. In this paper, three of the most important variables affecting the breakdown strength of polymer‐based composites are considered: the filler dielectric constants, filler sizes, and filler contents. High‐throughput stochastic breakdown simulation is performed on 504 groups of data, and the simulation results are used as the machine learning database to obtain the breakdown strength prediction of polymer‐based composites. Combined with the classical dielectric prediction formula, the energy storage density prediction of polymer‐based composites is obtained. The accuracy of the prediction is verified by the directional experiments, including dielectric constant and breakdown strength. This work provides insight into the design and fabrication of polymer‐based composites with high energy density for capacitive energy storage applications.

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Niemeyer, Pietronero, and Wiesmann Respond:

Cited by 83 publications

References 3 publications

The fixed-scale transformation approach to fractal growth

The fixed-scale transformation approach to fractal growth

Comparative study of large-scale Laplacian growth patterns

Prediction of Energy Storage Performance in Polymer Composites Using High‐Throughput Stochastic Breakdown Simulation and Machine Learning

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