Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations

Nakayama, Tsuneyoshi; Yakubo, Kousuke; Orbach, Raymond Lee

doi:10.1103/revmodphys.66.381

Cited by 521 publications

(377 citation statements)

References 461 publications

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“…The total number of spins in the cluster is regarded as its "mass", m, and its radius of gyration, R g , is evaluated via the definition R 2 g = 1/(2m 2 ) ij | r i − r j | 2 where r i is the position of spin i [74]. The fractal dimension d f of the clusters is defined from the scaling of the "mass" m with the radius of gyration R g , m ∝ R d f g .…”

Section: A Fractal Analysismentioning

confidence: 99%

Spatially heterogeneous ages in glassy systems

et al. 2003

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We construct a framework for the study of fluctuations in the nonequilibrium relaxation of glassy systems with and without quenched disorder. We study two types of two-time local correlators with the aim of characterizing the heterogeneous evolution in these systems: in one case we average the local correlators over histories of the thermal noise, in the other case we simply coarse-grain the local correlators obtained for a given noise realization. We explain why the noise-averaged correlators describe the fingerprint of quenched disorder when it exists, while the coarse-grained correlators are linked to noise-induced mesoscopic fluctuations. We predict constraints on the distribution of the fluctuations of the coarse-grained quantities. In particular, we show that locally defined correlations and responses are connected by a generalized local out-of-equilibrium fluctuation-dissipation relation. We argue that large-size heterogeneities in the age of the system survive in the long-time limit. A symmetry of the underlying theory, namely invariance under reparametrizations of the time coordinates, underlies these results. We establish a connection between the probabilities of spatial distributions of local coarse-grained quantities and the theory of dynamic random manifolds. We define, and discuss the behavior of, a two-time dependent correlation length from the spatial decay of the fluctuations in the two-time local functions. We characterize the fluctuations in the system in terms of their fractal properties. For concreteness, we present numerical tests performed on disordered spin models in finite and infinite dimensions. Finally, we explain how these ideas can be applied to the analysis of the dynamics of other glassy systems that can be either spin models without disorder or atomic and molecular glassy systems.

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Section: A Fractal Analysismentioning

confidence: 99%

Spatially heterogeneous ages in glassy systems

et al. 2003

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“…Generation of disordered aggregated structures by silver filaments and particles is obviously related to the stochastic nature of the development process in specific conditions, i.e., the local excess of the active volume developer and the high overpotential of the anode step of the reaction, the occurrence of both direct chemical and solution physical development mechanisms, and the presence of a protective polymer layer on the crystal surfaces [16]. Moreover, the obtained results suggest that the networks arise in some type of critical process, where such features as self-similarity, scaling and universality are displayed, and which, in principle, may be suitably described by percolation [17]. The power-law behaviour of the silver mass, N, (the number of occupied sites) vs. the square size, S, shown in Fig.…”

Section: Ag Colloidsmentioning

confidence: 80%

Study of Quasi-Fractal Many-Particle-Systems and Percolation Networks by Zero-Loss Spectroscopic Imaging, Electron Energy-Loss Spectroscopy and Digital Image Analysis

Oleshko

Kindratenko

Gijbels

et al. 1996

Microbeam and Nanobeam Analysis

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Abstract. Submicron colloidal Ag particles and nano-sized filaments forming a statistical percolation network during "in situ" development of double structure tabular microcrystals of AgBr(I) emulsions have been studied by electron energy-loss spectroscopy and zero-loss electron spectroscopic imaging (EELS/ ZLESI). Image analysis has shown that random quasi-fractal clusters were formed in the colloid. ZLESI has been applied to characterise the morphology and defect structure of aggregated particles and filaments. Their energy-loss spectra revealed plasmon excitations and interband 4d electron transitions between 4-32 eV energy-loss. To study the cluster structure and its relation to the physical properties, fractal analysis including estimations of cluster fractal dimensions and of density autocorrelation functions has been performed. Mechanisms of fractal aggregation based on known models of diffusion limited aggregation, cluster-cluster aggregation and percolation are discussed.

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“…3 we report the C DID (ω) coupling coefficients for bond percolators with concentrations ranging from c B = 0.249 to c B = 0.7. As the concentration increases, the low frequency "scaling" [24] part of the spectrum at threshold concentration is more and more covered from both sides: from low frequency by the "phononic" slope and from high frequency by a roundish spectrum to which no slope can reasonably be associated. Already at a bond concentration c B = 0.325, corresponding to a mass concentration c M ≈ 0.79, the threshold slope (S(c T ) ≈ 0.2) has completely disappeared.…”

Section: ) This Is Contrary To Whatmentioning

confidence: 99%

Low-frequency Raman scattering in model disordered solids: percolators above threshold

Pilla

Viliani

Dell’Anna

et al. 1997

Physica A: Statistical Mechanics and its Applications

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The Raman coupling coefficients of site-and bond-percolators at concentration higher than percolation threshold are computed for two scattering mechanisms: Bond Polarizability (BPOL) and Dipole-Induced-Dipole (DID). The results show that DID does not follow a scaling law at low frequency, while in the case of BPOL the situation is less clear. The numerically computed frequency dependence in the case of BPOL, which can be considered a good scattering mechanism for a wide class of real glasses, is in semiquantitative agreement with experimental results.

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Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations

Cited by 521 publications

References 461 publications

Spatially heterogeneous ages in glassy systems

Spatially heterogeneous ages in glassy systems

Study of Quasi-Fractal Many-Particle-Systems and Percolation Networks by Zero-Loss Spectroscopic Imaging, Electron Energy-Loss Spectroscopy and Digital Image Analysis

Low-frequency Raman scattering in model disordered solids: percolators above threshold

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