1
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Ostilli M, Bezerra CG, Viswanathan GM. Spectrum of the tight-binding model on Cayley trees and comparison with Bethe lattices. Phys Rev E 2022; 105:034123. [PMID: 35428099 DOI: 10.1103/physreve.105.034123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2021] [Accepted: 03/03/2022] [Indexed: 06/14/2023]
Abstract
There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley trees. Recent approaches to the problem have relied on the similarity between the Cayley tree and the Bethe lattice. Here, we avoid to make any ansatz related to the Bethe lattice due to fundamental differences between the two lattices that persist even when taking the thermodynamic limit. Instead, we show that one can use a recursive procedure that starts from the boundary and then use the canonical basis to derive the complete spectrum of the tight-binding model on Cayley trees. Our resulting algorithm is extremely efficient, as witnessed with remarkable large trees having hundreds of shells. We also show that, in the thermodynamic limit, the density of states is dramatically different from that of the Bethe lattice.
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Affiliation(s)
- M Ostilli
- Instituto de Física, Universidade Federal da Bahia, Salvador-BA, 40170-115, Brazil
| | - Claudionor G Bezerra
- Departmento de Física, Universidade Federal do Rio Grande do Norte, Natal-RN, 59078-970, Brazil
| | - G M Viswanathan
- Departmento de Física, Universidade Federal do Rio Grande do Norte, Natal-RN, 59078-970, Brazil
- National Institute of Science and Technology of Complex Systems, Universidade Federal do Rio Grande do Norte, Natal-RN, 59078-970, Brazil
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2
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3
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Peng J, Agliari E. Scaling laws for diffusion on (trans)fractal scale-free networks. CHAOS (WOODBURY, N.Y.) 2017; 27:083108. [PMID: 28863489 DOI: 10.1063/1.4997761] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Fractal (or transfractal) features are common in real-life networks and are known to influence the dynamic processes taking place in the network itself. Here, we consider a class of scale-free deterministic networks, called (u, v)-flowers, whose topological properties can be controlled by tuning the parameters u and v; in particular, for u > 1, they are fractals endowed with a fractal dimension df, while for u = 1, they are transfractal endowed with a transfractal dimension d̃f. In this work, we investigate dynamic processes (i.e., random walks) and topological properties (i.e., the Laplacian spectrum) and we show that, under proper conditions, the same scalings (ruled by the related dimensions) emerge for both fractal and transfractal dimensions.
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Affiliation(s)
- Junhao Peng
- School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
| | - Elena Agliari
- Department of Mathematics, Sapienza Università di Roma, 00198 Rome, Italy
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4
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Agliari E, Tavani F. The exact Laplacian spectrum for the Dyson hierarchical network. Sci Rep 2017; 7:39962. [PMID: 28067261 PMCID: PMC5220329 DOI: 10.1038/srep39962] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2016] [Accepted: 11/30/2016] [Indexed: 11/27/2022] Open
Abstract
We consider the Dyson hierarchical graph , that is a weighted fully-connected graph, where the pattern of weights is ruled by the parameter σ ∈ (1/2, 1]. Exploiting the deterministic recursivity through which is built, we are able to derive explicitly the whole set of the eigenvalues and the eigenvectors for its Laplacian matrix. Given that the Laplacian operator is intrinsically implied in the analysis of dynamic processes (e.g., random walks) occurring on the graph, as well as in the investigation of the dynamical properties of connected structures themselves (e.g., vibrational structures and relaxation modes), this result allows addressing analytically a large class of problems. In particular, as examples of applications, we study the random walk and the continuous-time quantum walk embedded in , the relaxation times of a polymer whose structure is described by , and the community structure of in terms of modularity measures.
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Affiliation(s)
- Elena Agliari
- Dipartimento di Matematica, Sapienza Università di Roma, P. le A. Moro 5, 00185, Roma, Italy
| | - Flavia Tavani
- Dipartimento SBAI (Ingegneria), Sapienza Università di Roma, via A. Scarpa 16, 00161, Roma, Italy
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5
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Jurjiu A, Galiceanu M, Farcasanu A, Chiriac L, Turcu F. Relaxation dynamics of Sierpinski hexagon fractal polymer: Exact analytical results in the Rouse-type approach and numerical results in the Zimm-type approach. J Chem Phys 2016; 145:214901. [PMID: 28799361 DOI: 10.1063/1.4968209] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this paper, we focus on the relaxation dynamics of Sierpinski hexagon fractal polymer. The relaxation dynamics of this fractal polymer is investigated in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. In the Rouse-type approach, by performing real-space renormalization transformations, we determine analytically the complete eigenvalue spectrum of the connectivity matrix. Based on the eigenvalues obtained through iterative algebraic relations we calculate the averaged monomer displacement and the mechanical relaxation moduli (storage modulus and loss modulus). The evaluation of the dynamical properties in the Rouse-type approach reveals that they obey scaling in the intermediate time/frequency domain. In the Zimm-type approach, which includes the hydrodynamic interactions, the relaxation quantities do not show scaling. The theoretical findings with respect to scaling in the intermediate domain of the relaxation quantities are well supported by experimental results.
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Affiliation(s)
- Aurel Jurjiu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Mircea Galiceanu
- Departamento de Fisica, Universidade Federal do Amazonas, 69077-000 Manaus, Brazil
| | - Alexandru Farcasanu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Liviu Chiriac
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
| | - Flaviu Turcu
- Faculty of Physics, Babes-Bolyai University, Street Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania
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Dolgushev M, Liu H, Zhang Z. Extended Vicsek fractals: Laplacian spectra and their applications. Phys Rev E 2016; 94:052501. [PMID: 27967151 DOI: 10.1103/physreve.94.052501] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2016] [Indexed: 06/06/2023]
Abstract
Extended Vicsek fractals (EVF) are the structures constructed by introducing linear spacers into traditional Vicsek fractals. Here we study the Laplacian spectra of the EVF. In particularly, the recurrence relations for the Laplacian spectra allow us to obtain an analytic expression for the sum of all inverse nonvanishing Laplacian eigenvalues. This quantity characterizes the large-scale properties, such as the gyration radius of the polymeric structures, or the global mean-first passage time for the random walk processes. Introduction of the linear spacers leads to local heterogeneities, which reveal themselves, for example, in the dynamics of EVF under external forces.
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Affiliation(s)
- Maxim Dolgushev
- Institute of Physics, University of Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
- Institut Charles Sadron, Université de Strasbourg and CNRS, 23 rue du Loess, 67034 Strasbourg Cedex, France
| | - Hongxiao Liu
- School of Computer Science, Fudan University, Shanghai 200433, China
- Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Zhongzhi Zhang
- School of Computer Science, Fudan University, Shanghai 200433, China
- Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
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7
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Dolgushev M, Markelov DA, Fürstenberg F, Guérin T. Local orientational mobility in regular hyperbranched polymers. Phys Rev E 2016; 94:012502. [PMID: 27575171 DOI: 10.1103/physreve.94.012502] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2016] [Indexed: 11/07/2022]
Abstract
We study the dynamics of local bond orientation in regular hyperbranched polymers modeled by Vicsek fractals. The local dynamics is investigated through the temporal autocorrelation functions of single bonds and the corresponding relaxation forms of the complex dielectric susceptibility. We show that the dynamic behavior of single segments depends on their remoteness from the periphery rather than on the size of the whole macromolecule. Remarkably, the dynamics of the core segments (which are most remote from the periphery) shows a scaling behavior that differs from the dynamics obtained after structural average. We analyze the most relevant processes of single segment motion and provide an analytic approximation for the corresponding relaxation times. Furthermore, we describe an iterative method to calculate the orientational dynamics in the case of very large macromolecular sizes.
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Affiliation(s)
- Maxim Dolgushev
- Institute of Physics, University of Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany.,Institut Charles Sadron, Université de Strasbourg and CNRS, 23 rue du Loess, 67034 Strasbourg Cedex, France
| | - Denis A Markelov
- St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia.,St. Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University), Kronverkskiy pr. 49, St. Petersburg, 197101, Russia
| | - Florian Fürstenberg
- Institute of Physics, University of Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany
| | - Thomas Guérin
- Laboratoire Ondes et Matière d'Aquitaine (LOMA), CNRS UMR 5798, Talence, France
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8
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Liu H, Lin Y, Dolgushev M, Zhang Z. Dynamics of comb-of-comb networks. Phys Rev E 2016; 93:032502. [PMID: 27078400 DOI: 10.1103/physreve.93.032502] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2015] [Indexed: 11/07/2022]
Abstract
The dynamics of complex networks, a current hot topic in many scientific fields, is often coded through the corresponding Laplacian matrix. The spectrum of this matrix carries the main features of the networks' dynamics. Here we consider the deterministic networks which can be viewed as "comb-of-comb" iterative structures. For their Laplacian spectra we find analytical equations involving Chebyshev polynomials whose properties allow one to analyze the spectra in deep. Here, in particular, we find that in the infinite size limit the corresponding spectral dimension goes as d(s) → 2. The d(s) leaves its fingerprint on many dynamical processes, as we exemplarily show by considering the dynamical properties of polymer networks, including single monomer displacement under a constant force, mechanical relaxation, and fluorescence depolarization.
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Affiliation(s)
- Hongxiao Liu
- School of Computer Science, Fudan University, Shanghai 200433, China.,Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Yuan Lin
- School of Computer Science, Fudan University, Shanghai 200433, China.,Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Maxim Dolgushev
- Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany.,Institut Charles Sadron, Université de Strasbourg and CNRS, 23 rue du Loess, 67034 Strasbourg Cedex, France
| | - Zhongzhi Zhang
- School of Computer Science, Fudan University, Shanghai 200433, China.,Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
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9
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Koda SI. Equivalence between a generalized dendritic network and a set of one-dimensional networks as a ground of linear dynamics. J Chem Phys 2015; 142:204112. [PMID: 26026439 DOI: 10.1063/1.4921730] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
It has been shown by some existing studies that some linear dynamical systems defined on a dendritic network are equivalent to those defined on a set of one-dimensional networks in special cases and this transformation to the simple picture, which we call linear chain (LC) decomposition, has a significant advantage in understanding properties of dendrimers. In this paper, we expand the class of LC decomposable system with some generalizations. In addition, we propose two general sufficient conditions for LC decomposability with a procedure to systematically realize the LC decomposition. Some examples of LC decomposable linear dynamical systems are also presented with their graphs. The generalization of the LC decomposition is implemented in the following three aspects: (i) the type of linear operators; (ii) the shape of dendritic networks on which linear operators are defined; and (iii) the type of symmetry operations representing the symmetry of the systems. In the generalization (iii), symmetry groups that represent the symmetry of dendritic systems are defined. The LC decomposition is realized by changing the basis of a linear operator defined on a dendritic network into bases of irreducible representations of the symmetry group. The achievement of this paper makes it easier to utilize the LC decomposition in various cases. This may lead to a further understanding of the relation between structure and functions of dendrimers in future studies.
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Affiliation(s)
- Shin-ichi Koda
- Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki 444-8585, Japan
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10
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Liu H, Dolgushev M, Qi Y, Zhang Z. Laplacian spectra of a class of small-world networks and their applications. Sci Rep 2015; 5:9024. [PMID: 25762195 PMCID: PMC4356965 DOI: 10.1038/srep09024] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2014] [Accepted: 02/12/2015] [Indexed: 02/05/2023] Open
Abstract
One of the most crucial domains of interdisciplinary research is the relationship between the dynamics and structural characteristics. In this paper, we introduce a family of small-world networks, parameterized through a variable d controlling the scale of graph completeness or of network clustering. We study the Laplacian eigenvalues of these networks, which are determined through analytic recursive equations. This allows us to analyze the spectra in depth and to determine the corresponding spectral dimension. Based on these results, we consider the networks in the framework of generalized Gaussian structures, whose physical behavior is exemplified on the relaxation dynamics and on the fluorescence depolarization under quasiresonant energy transfer. Although the networks have the same number of nodes (beads) and edges (springs) as the dual Sierpinski gaskets, they display rather different dynamic behavior.
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Affiliation(s)
- Hongxiao Liu
- 1] School of Computer Science, Fudan University, Shanghai 200433, China [2] Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Maxim Dolgushev
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str.3, D-79104 Freiburg, Germany
| | - Yi Qi
- 1] School of Computer Science, Fudan University, Shanghai 200433, China [2] Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
| | - Zhongzhi Zhang
- 1] School of Computer Science, Fudan University, Shanghai 200433, China [2] Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China
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11
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Jurjiu A, Dockhorn R, Mironova O, Sommer JU. Two universality classes for random hyperbranched polymers. SOFT MATTER 2014; 10:4935-4946. [PMID: 24882064 DOI: 10.1039/c4sm00711e] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We grow AB2 random hyperbranched polymer structures in different ways and using different simulation methods. In particular we use a method of ad hoc construction of the connectivity matrix and the bond fluctuation model on a 3D lattice. We show that hyperbranched polymers split into two universality classes depending on the growth process. For a "slow growth" (SG) process where monomers are added sequentially to an existing molecule which strictly avoids cluster-cluster aggregation the resulting structures share all characteristic features with regular dendrimers. For a "quick growth" (QG) process which allows for cluster-cluster aggregation we obtain structures which can be identified as random fractals. Without excluded volume interactions the SG model displays a logarithmic growth of the radius of gyration with respect to the degree of polymerization while the QG model displays a power law behavior with an exponent of 1/4. By analyzing the spectral properties of the connectivity matrix we confirm the behavior of dendritic structures for the SG model and the corresponding fractal properties in the QG case. A mean field model is developed which explains the extension of the hyperbranched polymers in an athermal solvent for both cases. While the radius of gyration of the QG model shows a power-law behavior with the exponent value close to 4/5, the corresponding result for the SG model is a mixed logarithmic-power-law behavior. These different behaviors are confirmed by simulations using the bond fluctuation model. Our studies indicate that random sequential growth according to our SG model can be an alternative to the synthesis of perfect dendrimers.
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Affiliation(s)
- A Jurjiu
- Leibniz Institut für Polymerforschung Dresden e.V., Hohe Strasse 6, D-01069, Dresden, Germany.
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12
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Galiceanu M. Hydrodynamic effects on scale-free polymer networks in external fields. J Chem Phys 2014; 140:034901. [DOI: 10.1063/1.4861218] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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13
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Liu H, Zhang Z. Laplacian spectra of recursive treelike small-world polymer networks: analytical solutions and applications. J Chem Phys 2013; 138:114904. [PMID: 23534659 DOI: 10.1063/1.4794921] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A central issue in the study of polymer physics is to understand the relation between the geometrical properties of macromolecules and various dynamics, most of which are encoded in the Laplacian spectra of a related graph describing the macrostructural structure. In this paper, we introduce a family of treelike polymer networks with a parameter, which has the same size as the Vicsek fractals modeling regular hyperbranched polymers. We study some relevant properties of the networks and show that they have an exponentially decaying degree distribution and exhibit the small-world behavior. We then study the Laplacian eigenvalues and their corresponding eigenvectors of the networks under consideration, with both quantities being determined through the recursive relations deduced from the network structure. Using the obtained recursive relations we can find all the eigenvalues and eigenvectors for the networks with any size. Finally, as some applications, we use the eigenvalues to study analytically or semi-analytically three dynamical processes occurring in the networks, including random walks, relaxation dynamics in the framework of generalized Gaussian structure, as well as the fluorescence depolarization under quasiresonant energy transfer. Moreover, we compare the results with those corresponding to Vicsek fractals, and show that the dynamics differ greatly for the two network families, which thus enables us to distinguish between them.
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Affiliation(s)
- Hongxiao Liu
- School of Computer Science, Fudan University, Shanghai 200433, China
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14
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Julaiti A, Wu B, Zhang Z. Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications. J Chem Phys 2013; 138:204116. [DOI: 10.1063/1.4807589] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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15
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Lin Y, Zhang Z. Influence of trap location on the efficiency of trapping in dendrimers and regular hyperbranched polymers. J Chem Phys 2013; 138:094905. [DOI: 10.1063/1.4793309] [Citation(s) in RCA: 37] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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16
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Fürstenberg F, Dolgushev M, Blumen A. Dynamics of semiflexible regular hyperbranched polymers. J Chem Phys 2013; 138:034904. [DOI: 10.1063/1.4775584] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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17
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Galiceanu M. Relaxation dynamics of scale-free polymer networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:041803. [PMID: 23214606 DOI: 10.1103/physreve.86.041803] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2012] [Revised: 09/10/2012] [Indexed: 06/01/2023]
Abstract
We focus on polymer networks with a scale-free topology. In the framework of generalized Gaussian structures, by making use of the eigenvalue spectrum of the connectivity matrix, we determined numerically the averaged monomer displacement under external forces and the mechanical relaxation moduli (storage and loss modulus). First, we monitor these physical quantities and additionally the eigenvalue spectrum for structures of different sizes, but with the same γ, which is a parameter that measures the connectivity of the structure. Second, we vary the parameter γ, and we keep constant the size of the structures. This allows us to study in detail the crossover behavior from a simple linear chain to a starlike structure. As expected we encounter a more chainlike behavior for high values of γ, while for small values of γ a more starlike behavior is observed. In the intermediate time (frequency) domain, we encounter regions of constant slope for some intermediate values of γ.
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Affiliation(s)
- M Galiceanu
- Departamento de Física, Universidade Federal do Amazonas, 69077-000 Manaus, Brazil.
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18
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Wu B, Lin Y, Zhang Z, Chen G. Trapping in dendrimers and regular hyperbranched polymers. J Chem Phys 2012; 137:044903. [DOI: 10.1063/1.4737635] [Citation(s) in RCA: 51] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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19
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Jurjiu A, Volta A, Beu T. Relaxation dynamics of a polymer network modeled by a multihierarchical structure. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:011801. [PMID: 21867199 DOI: 10.1103/physreve.84.011801] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2011] [Revised: 05/27/2011] [Indexed: 05/31/2023]
Abstract
We numerically analyze the scaling behavior of experimentally accessible dynamical relaxation forms for polymer networks modeled by a finite multihierarchical structure. In the framework of generalized Gaussian structures, by making use of the eigenvalue spectrum of the connectivity matrix, we determine the averaged monomer displacement under local external forces as well as the mechanical relaxation quantities (storage and loss moduli). Hence we generalize the known analysis for both classes of fractals to the case of multihierarchical structure, for which even though we have a mixed growth algorithm, the above cited observables still give information about the two different underlying topologies. For very large lattices, reached via an algebraic procedure that avoids the numerical diagonalizations of the corresponding connectivity matrices, we depict the scaling of both component fractals in the intermediate time (frequency) domain, which manifests two different slopes.
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Affiliation(s)
- A Jurjiu
- Faculty of Physics, Simulation Laboratory of Nanostructured Systems, Universitatea Babes-Bolyai, Str Mihail Kogalniceanu, nr 1, RO-400084 Cluj-Napoca, Romania.
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20
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Zhang Z, Wu B, Zhang H, Zhou S, Guan J, Wang Z. Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:031118. [PMID: 20365708 DOI: 10.1103/physreve.81.031118] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2010] [Indexed: 05/29/2023]
Abstract
The family of Vicsek fractals is one of the most important and frequently studied regular fractal classes, and it is of considerable interest to understand the dynamical processes on this treelike fractal family. In this paper, we investigate discrete random walks on the Vicsek fractals, with the aim to obtain the exact solutions to the global mean-first-passage time (GMFPT), defined as the average of first-passage time (FPT) between two nodes over the whole family of fractals. Based on the known connections between FPTs, effective resistance, and the eigenvalues of graph Laplacian, we determine implicitly the GMFPT of the Vicsek fractals, which is corroborated by numerical results. The obtained closed-form solution shows that the GMFPT approximately grows as a power-law function with system size (number of all nodes), with the exponent lies between 1 and 2. We then provide both the upper bound and lower bound for GMFPT of general trees, and show that the leading behavior of the upper bound is the square of system size and the dominating scaling of the lower bound varies linearly with system size. We also show that the upper bound can be achieved in linear chains and the lower bound can be reached in star graphs. This study provides a comprehensive understanding of random walks on the Vicsek fractals and general treelike networks.
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Affiliation(s)
- Zhongzhi Zhang
- School of Computer Science, Fudan University, Shanghai 200433, China.
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21
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Galiceanu M, Blumen A. Spectra of Husimi cacti: Exact results and applications. J Chem Phys 2007; 127:134904. [DOI: 10.1063/1.2787005] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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22
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Generalized Gaussian Structures: Models for Polymer Systems with ComplexTopologies. POLYMER ANALYSIS POLYMER THEORY 2005. [DOI: 10.1007/b135561] [Citation(s) in RCA: 104] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
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23
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Koslowski T, Jurjiu A, Blumen A. Polaron Formation and Hopping Conduction in Hyperbranched Polymers: A Theoretical Approach. J Phys Chem B 2004. [DOI: 10.1021/jp037263a] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Thorsten Koslowski
- Institut für Physikalische Chemie II and Theoretische Polymerphysik, Universität Freiburg, D-79104 Freiburg im Breisgau, Germany
| | - Aurel Jurjiu
- Institut für Physikalische Chemie II and Theoretische Polymerphysik, Universität Freiburg, D-79104 Freiburg im Breisgau, Germany
| | - Alexander Blumen
- Institut für Physikalische Chemie II and Theoretische Polymerphysik, Universität Freiburg, D-79104 Freiburg im Breisgau, Germany
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24
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Blumen A, von Ferber C, Jurjiu A, Koslowski T. Generalized Vicsek Fractals: Regular Hyperbranched Polymers. Macromolecules 2003. [DOI: 10.1021/ma034553g] [Citation(s) in RCA: 91] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- A. Blumen
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany, and Institut für Physikalische Chemie, Universität Freiburg, Albertstrasse 23a, D-79104 Freiburg, Germany
| | - Ch. von Ferber
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany, and Institut für Physikalische Chemie, Universität Freiburg, Albertstrasse 23a, D-79104 Freiburg, Germany
| | - A. Jurjiu
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany, and Institut für Physikalische Chemie, Universität Freiburg, Albertstrasse 23a, D-79104 Freiburg, Germany
| | - Th. Koslowski
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany, and Institut für Physikalische Chemie, Universität Freiburg, Albertstrasse 23a, D-79104 Freiburg, Germany
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Jurjiu A, Koslowski T, von Ferber C, Blumen A. Dynamics and scaling of polymer networks: Vicsek fractals and hydrodynamic interactions. Chem Phys 2003. [DOI: 10.1016/j.chemphys.2003.07.006] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Blumen A, Jurjiu A, Koslowski T, von Ferber C. Dynamics of Vicsek fractals, models for hyperbranched polymers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:061103. [PMID: 16241195 DOI: 10.1103/physreve.67.061103] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2003] [Indexed: 05/04/2023]
Abstract
We consider the dynamics of Vicsek fractals of arbitrary connectivity, models for hyperbranched polymers. Their basic dynamical properties depend on their eigenvalue spectra, which can be determined iteratively. This paves the way for theoretical studies to very high precision for regular, finite, arbitrarily large hyperbranched structures.
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Affiliation(s)
- A Blumen
- Theoretische Polymerphysik, Universität Freiburg, Germany
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Hu Y, Tian DC, You JQ. Spectral properties of the Vicsek fractal. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:5070-5073. [PMID: 9984092 DOI: 10.1103/physrevb.53.5070] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Kara A, Jayanthi CS, Wu SY, Ercolessi F. Structure and dynamics of the reconstructed Au(511) surface. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:17046-17062. [PMID: 9978717 DOI: 10.1103/physrevb.51.17046] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Jayanthi CS, Wu SY, Cocks JA. Real-space Green's-function approach to study the effects of disorder on the frequency spectrum of Si/Ge alloys. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:7992-7995. [PMID: 9974793 DOI: 10.1103/physrevb.50.7992] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Jayanthi CS, Wu SY. Dynamics of a Vicsek fractal: The boundary effect and the interplay among the local symmetry, the self-similarity, and the structure of the fractal. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:897-906. [PMID: 9975755 DOI: 10.1103/physrevb.50.897] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Kara A, Jayanthi CS, Wu SY, Ercolessi F. Local analysis of the dynamics of the relaxed and reconstructed Au(511) surface using the real space Green's function method. PHYSICAL REVIEW LETTERS 1994; 72:2223-2226. [PMID: 10055820 DOI: 10.1103/physrevlett.72.2223] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Wu SY, Cocks J, Jayanthi CS. General recursive relation for the calculation of the local Green's function in the resolvent-matrix approach. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:7957-7963. [PMID: 10009558 DOI: 10.1103/physrevb.49.7957] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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You JQ, Lam CH, Nori F, Sander LM. Exact renormalization-group approach to the generating function of the Vicsek fractal. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:R4183-R4186. [PMID: 9961183 DOI: 10.1103/physreve.48.r4183] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Wu SY, Xie ZL, Potoczak N. Feasibility study on the application of the method of the resolvent matrix to complex systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:14826-14832. [PMID: 10008014 DOI: 10.1103/physrevb.48.14826] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Jayanthi CS, Wu SY. Frequency spectrum of a general Vicsek fractal. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:10188-10198. [PMID: 10007294 DOI: 10.1103/physrevb.48.10188] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Jayanthi CS, Wu SY. Exact calculation of the dynamics of a general Vicsek fractal. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:10199-10203. [PMID: 10007295 DOI: 10.1103/physrevb.48.10199] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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