Investigation of Scale Exponents of Complete and Incomplete Aggregation-Annihilation Processes

The complete and incomplete aggregation-annihilation processes are investigated with the method of generating function, and the scale exponents are obtained exactly. We find that the scale exponents of incomplete aggregation-annihilation process are different from the previous exponents obtained by different methods. The time dependence of the total number of clusters and the total mass of clusters are analytically obtained.     

Kinetics of an n-Species Aggregation Chain Model with Complete Annihilation

The kinetic behavior of an n-species (n ≥ 3) aggregation-annihilation chain reaction model is studied. In this model, an irreversible aggregation reaction occurs between any two clusters of the same species, and an irreversible complete annihilation reaction occurs only between two species with adjacent number. Based on the rnean-field theory, we investigate the rate equations of the process with constant reaction rates to obtain the asymptotic solutions of the clustermass distributions for the system. The results show that the kinetic behavior of the system not only depends crucially on the ratio of the aggregation rate I to the annihilation rate J, but also has relation with the initial concentration of each species and the species number's odevity. We find that the cluster-mass distribution of each species obeys always a scaling law. The scaling exponents may strongly depend on the reaction rates for most cases, however, for the case in which the ratio of the aggregation rate to the annihilation rate is equal to a certain value, the scaling exponents are only dependent on the initial concentrations of the reactants.     

全局耦合网络上的两种类集团的聚集-湮没反应动力学

利用Monte-Carlo模拟研究了全局耦合网络上扩散限制的不可逆聚集-湮没过程的动力学行为. 在系统中, 同种类集团相遇, 将发生聚集反应; 不同种类的集团相遇, 则发生部分湮没反应. 模拟结果表明:1) 当两种粒子初始浓度相等时, 系统长时间演化后, 集团浓度c(t)和粒子浓度g(t)呈现幂律形式, c(t)～t^{- α}和g(t)～t^-β, 其中幂指数α 和β 满足α=2β 的关系, 且α=2/(2 + q); 集团大小分布随时间的演化满足标度律, a_kt)=k^-τt^-ω\varPhi (k/t^z), 其中τ≈-1.27q, ω≈(3 + 1.27q)/(2 + q), z=α/2=1/(2 + q); 2) 当两种粒子初始浓度不相等时, 系统经长时间演化后, 初始浓度较小的种类完全湮没, 而初始浓度较大的那个种类的集团浓度c_A(t)仍具有幂律形式, c_A(t)～t^-α, 其中α=1/(1+q), 其集团大小分布随时间的演化也满足标度律, 标度指数为τ≈-1.27q, ω≈(2 + 1.27q)/(1 + q)和z=α=1/(1 + q). 模拟结果与已报道的理论分析结果相符得很好.     

小世界网络上的扩散限制的聚集-湮没反应动力学

通过Monte-Carlo模拟,研究了基于NW网络的两种类集团不可逆聚集-湮没过程的动力学行为.在系统中,两个同种类集团相遇,将不可逆地聚集成一个更大的集团;不同种类的两个集团相遇,则发生部分湮没反应.模拟结果表明,1)当捷径量化参数p相对较大或较小时,系统经较长时间演化后,集团密度c(t)和粒子密度g(t)呈现幂律形式,c(t)∝t^－α和g(t)∝t^－β,其中幂指数α和β满足α=2β的关系;2)当p为其他值时,集团密度和粒子密度随时间按非严格的幂 关键词： 聚集-湮没过程 小世界网络 反应动力学 Monte-Carlo模拟')" href="#">Monte-Carlo模拟     

Novel Biased Aggregation-Annihilation Model

We propose a novel two-species aggregation-annihilation model,in which irreversible aggregation reactions occur between any two aggregates of the same species and biased annihilations occur simultaneously between two different species.The kinetic scaling behavior of the model is then analytically investigated by means of the mean-field rate equation.For the system without the self-aggregation of the un-annihilated species,the aggregate size distribution of the annihilated species always approaches a modified scaling form and vanishes finally; while for the system with the self-aggregation of the un-annihilated species,its scaling behavior depends crucially on t,he details of the rate kernels.Moreover,the results also exhibit that both species are conserved together in some cases,while only the un-annihilated species survives finally in other cases.     

Etched glass surfaces,atomic force microscopy and stochastic analysis

The effect of etching time scale of glass surface on its statistical properties has been studied using atomic force microscopy technique. We have characterized the complexity of the height fluctuation of an etched surface by the stochastic parameters such as intermittency exponents, roughness, roughness exponents, drift and diffusion coefficients and found their widths in terms of the etching time.     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Elastic properties of a disordered continuum of regular fractal structure: Self-consistent approach and finite-element method simulations

Macromolecular structures, as well as aggregation of filler in polymer-based composites, often may be described properly as fractals. Scaling behavior of the elastic moduli of a modeled fractal, the Sierpinski carpet, was the subject of this study. Sheng and Tao [1] and Patlazhan [2] found that, in the case of voids in on elastic host, axial and shear moduli exhibit distinct scaling dependencies on the size of the system. Nevertheless, it is widely accepted that moduli of random isotropic fractals (percolation clusters) scale with the same exponents. Explanation of the discrepancy is one of the main targets of the paper. The self-consistent approach and position space renormalization group technique (PSRG) have been applied for this goal. The mapping, corresponding to PSRG, was constructed numerically using the finite-element method (FEM) in the cases of voids and rigid inclusions. The self-consistent approach gives scaling behavior with exponents of values of about 0.11, independent of the modulus and type of inclusion, at developed stages of the fractal. It has been shown that mappings of PSRG on the plane, for two ratios of three independent moduli, have stable fixed points. This means that different elastic moduli exhibit scaling behavior with the same exponents (0.29 for voids and 0.17 for rigid squares) for developed fractal structure. The discrepancy in the exponent values obtained in the previous simulations is caused by the analysis of the initial stages of the structure. We believe that analogous results are valid for the wide class of self-similar fractals, and the dimension is the main parameter that governs the exponents and fixed point values.     

Breakdown of universality in directed spiral percolation

Directed spiral percolation (DSP), percolation under both directional and rotational constraints, is studied on the triangular lattice in two dimensions (2D). The results are compared with that of the 2D square lattice. Clusters generated in this model are generally rarefied and have chiral dangling ends on both the square and triangular lattices. It is found that the clusters are more compact and less anisotropic on the triangular lattice than on the square lattice. The elongation of the clusters is in a different direction than the imposed directional constraint on both the lattices. The values of some of the critical exponents and fractal dimension are found considerably different on the two lattices. The DSP model then exhibits a breakdown of universality in 2D between the square and triangular lattices. The values of the critical exponents obtained for the triangular lattice are not only different from that of the square lattice but also different form other percolation models.Received: 12 March 2004, Published online: 23 July 2004PACS: 02.50.-r Probability theory, stochastic processes, and statistics - 64.60.-i General studies of phase transitions - 72.80.Tm Composite materials     

Moment Analysis of a Rice-Pile Model

Large scale simulations of a rice-pile model are performed. We use moment analysis techniques to evaluate critical exponents and data collapse method to verify the obtained results. The moment analysis yields well-defined avalanche exponents, which show that the rice-pile model can be coherently described within a finite size scaling framework. The general picture resulting from our analysis allows us to characterize the large scale behavior of the present model with great accuracy.     

Fractal geometry of critical Potts clusters

Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the recently derived corrections-to-scaling theory. A new method for thermalization of spin systems is presented. The method allows a speed up of an order of magnetization for large lattices. We also show snapshots of the Potts clusters for different values of q, which clearly illustrate the fact that the clusters become more compact as q increases, and that this affects the fractal dimensions in a monotonic way. However, the approach to the asymptotic region is slow, and the present range of the data does not allow a unique identification of the exact correction exponents.Received: 2 June 2003, Published online: 9 September 2003PACS: 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) - 05.45.Df Fractals - 75.10.-b General theory and models of magnetic ordering - 75.40.Cx Static properties (order parameter, static susceptibility, heat capacities, critical exponents, etc.)     

Length scales and power laws in the two-dimensional forest-fire model

We re-examine a two-dimensional forest-fire model via Monte-Carlo simulations and show the existence of two length scales with different critical exponents associated with clusters and with the usual two-point correlation function of trees. We check resp. improve previously obtained values for other critical exponents and perform a first investigation of the critical behaviour of the slowest relaxational mode. We also investigate the possibility of describing the critical point in terms of a distribution of the global density. We find that some qualitative features such as a temporal oscillation and a power law of the cluster-size distribution can nicely be obtained from such a model that discards the spatial structure.     

Critical Exponents Taking into Account Dynamic Scaling for Adsorption on Small-Size One-Dimensional Clusters

Adsorption on small-size one-dimensional clusters is investigated using the Monte Carlo method. The effect of temperature and system size variations on adsorption is studied. Critical exponents of the correlation length and dynamic critical exponent z are calculated taking into account the hypothesis of dynamic scaling. The results obtained demonstrate that non-equilibrium adsorption in nanosystems can occur in a much different fashion than in macrosystems. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 82–87, August, 2005.     

天体光谱信号去噪的小波域复合阈值新算法

利用谱线和噪声在小波域内的不同相关特性,提出了一种小波域复合阈值去噪算法。首先将小波系数作NeighShrink阈值处理,然后对得到的小波系数进行二值化,在此基础上定义了每一小波系数所对应的横向相关性指数和纵向相关性指数,最后确定出决定小波系数取舍的决策系数。由于该决策系数是通过多重判据得到的,因此该方法克服了简单阈值方法过保留或过扼杀的缺点,同时可以去除大脉冲噪声,实验结果表明了该方法的有效性。     

Discrete-scale invariance and complex dimensions

We discuss the concept of discrete-scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the 1970s, complex exponents have been studied in the 1980s in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete-scale invariance and its associated complex exponents may appear “spontaneously” in Euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete-scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete-scale invariance. The main motivation to study discrete-scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes.