Finite-size scaling for mean-field percolation

By studying transfer matrix eigenvalues, correlation lengths for a mean field directed percolation model are obtained both near and far from the critical regime. Near criticality, finite-size scaling behavior is derived and an analytic technique is provided for obtaining the finite-size scaling function. Our methods involve the generating function, matched asymptotic expansions, and certain formulas developed for the study of eigenvalues of the transfer matrix for metastability.     

Measures of critical exponents in the four-dimensional site percolation

Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-to-scaling exponent and, with great accuracy, the critical density.     

Hard-hexagon model: Calculation of anisotropic interfacial tension from asymptotic degeneracy of largest eigenvalues of row-row transfer matrix

To find the directional dependence of the interfacial tension of the hard-hexagon model, an inhomogeneous system is studied. This system is defined on a square lattice with (1+v)M columns so that the lhs of the (M+1)th column is the hard-hexagon model and the rhs of the (M+1)th column works as the operator which shifts the particle configuration of a column downward. A triplet of the largest eigenvalues of the row-row transfer matrix are asymptotically degenerate asM under the conditions that (1–v)M0 (mod 3), withv being fixed to be constant. The interfacial tension of a tilted interface is calculated from the finite correction terms in this limit.     

Advection of an impurity in percolation media with a finite correlation length

The impurity transport regimes in percolation media with a finite correlation length, which are caused by advection and diffusion mechanisms, have been analyzed. It has been shown that the change in the transport characteristics of a medium from the self-similar type to the statistically homogeneous type occurs through two stages because of the structural features of percolation clusters (presence of a backbone and dead ends). As a result, new anomalous transport regimes appear in the system. The quasi-isotropic and moderately and strongly anisotropic media have been considered.     

Trap-size scaling of finite Bose systems within an exact canonical ensemble

Based on an exact canonical partition function, we investigate the trap-size scaling for ideal Bose gases with a finite number of particles N confined in a cubic box or in a harmonic trap. We study the trap-size scaling behaviors of condensate fraction 〈n₀〉/N and specific heat C_N around the transition temperature T_c (i.e., t = T/T_c − 1 → 0) for the three different traps, where a trap exponent θ in dependence of the trapping potential and the universality class of transition are introduced. In the box trap with periodic and Dirichlet boundary conditions, where θ → 1, we find that the scaling functions governing the various critical behaviors are universal but respective of the boundary conditions. The calculated critical exponents are in nice agreement with analytical scaling predictions. The borders of universality validity are obtained numerically. In the case of the harmonic trap, the critical behavior of the system is also found to be universal, and the trap exponent is obtained as θ ? 0.068.     

Different self-avoiding walks on percolation clusters: A small-cell real-space renormalization-group study

We calculate the average number of stepsN for edge-to-edge, normal, and indefinitely growing self-avoiding walks (SAWs) on two-dimensional critical percolation clusters, using the real-space renormalization-group approach, with small H cells. Our results are of the formN=AL ^D _SAW+B, whereL is the end-to-end distance. Similarly to several deterministic fractals, the fractal dimensionsD _SAW for these three different kinds of SAWs are found to be equal, and the differences between them appear in the amplitudesA and in the correction termsB. This behavior is atributed to the hierarchical nature of the critical percolation cluster.     

Scaling behavior of permeability and conductivity anisotropy near the percolation threshold

We use the finite-size scaling method to estimate the critical exponent that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdp _c . Here is defined by the scaling lawk _l /k _t –1(p–p _c ), wherek _t andk _t are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction, respectively. The results are (d=2)0.819±0.011 and (d=3)0.518±0.001. We interpret these results in terms of the structure of percolation clusters and their chemical distance. We also compare our results with the predictions of a scaling theory for due to Straley, and propose that (d=2)=t- _B , wheret is the critical exponent of the conductivity or permeability of the system, and _B is the critical exponent of the backbone of percolation clusters.     

The correlation length of commodity markets 2. Theoretical framework

The second of this series of two papers is devoted to a theoretical analysis of spatial interaction between commodity markets. The theoretical framework that we present is referred to as the stochastic spatial arbitrage model (SSAM); it accounts for most of the empirical regularities observed in the first paper. Two basic mechanisms are found to be responsible for spatial inter-market interaction, namely (i) spatial arbitrage and hedging conducted by traders, (ii) spatial correlation between local shocks; the latter is favored by a similar economic and cultural environment. The SSAM includes both effects and offers a wide range of predictions about price volatility, trade, price correlations, price differentials. Statistical tests display a convergent array of evidence in favor of the model. However several predictions cannot be tested by lack of statistical evidence, a circumstance which shows that yet additional “experimental” work is r! equired. Received 17 May 1999 and Received in final form 31 May 1999     

The correlation length of commodity markets 1. Empirical evidence

It is a common belief nowadays that the world economy is fairly well “integrated”. Yet, this belief often turns out to be in contradiction with empirical evidence. As a matter of fact the way distant markets interact is a question that has largely been ignored by economists. In this series of two papers we examine the role that space, that is to say geographical distance, plays in the economics of commodity markets. The first of these papers presents the empirical evidence while the second develops a theoretical framework. The empirical enquiry discloses several noteworthy features, e.g. (i) with respect to spatial interaction there is a sharp contrast between stock markets and commodity markets. While there is almost perfect spatial arbitrage in the first case, this is not true for commodity markets. (ii) In spite of their chaotic behavior in the course of time commodity prices display well defined spatial patterns, (iii) as in statistical physics and fluid dynamics interactions can be described in terms of correlation length. The correlation length of a set of markets is seen to increase along with the number of transactions; it also increases when transport costs decline as was the case during the “transportation revolution” of the mid-nineteenth century. Using the notion of correlation length one is able to give a quantitative meaning to the otherwise ill-defined concept of market integration. Received 17 May 1999 and Received in final form 31 May 1999     

Fluctuation scaling in complex systems: Taylor's law and beyond1

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form ‘fluctuations ≈ constant × average^α’, where the exponent α is predominantly in the range [1/2, 1]. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names Taylor's law or fluctuation scaling. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling.     

Monte Carlo studies of percolation phenomena for a simple cubic lattice

The site-percolation problem on a simple cubic lattice is studied by the Monte Carlo method. By combining results for periodic lattices of different sizes through the use of finite-size scaling theory we obtain good estimates forp _c (0.3115±0.0005), (0.41±0.01), (1.6±0.1), and(0.8±0.1). These results are consistent with other studies. The shape of the clusters is also studied. The average surface area for clusters of sizek is found to be close to its maximal value for the low-concentration region as well as for the critical region. The percentage of particles in clusters of different sizesk is found to have an exponential tail for large values ofk forP <p _c. Forp >p _c there is too much scatter in the data to draw firm conclusions about the size distribution.Work supported in part by USAFOSR Grant #73-2430B and by ERDA #E(11-1)-3077.     

Global correlation matrix spectra of the surface temperature of the oceans from random matrix theory to Poisson fluctuations

In this work we use the random matrix theory (RMT) to correctly describe the behavior of spectral statistical properties of the sea surface temperature of oceans. This oceanographic variable plays an important role in the global climate system. The data were obtained from National Oceanic and Atmospheric Administration (NOAA) and delimited for the period 1982 to 2016. The results show that oceanographic systems presented specific β values that can be used to classify each ocean according to its correlation behavior. The nearest-neighbors spacing of correlation matrix for north, central and south of the three oceans get close to a RMT distribution. However, the regions delimited in the Antarctic pole exhibited the distribution of the nearest-neighbors spacing well described by the Poisson model, which shows a statistical change of RMT to Poisson fluctuations.     

Reactivity of acenes: mechanisms and dependence on acene length

Acenes are polycyclic aromatic hydrocarbons consisting of linearly fused benzene rings. These compounds are currently the subject of great interest from both fundamental and applied perspectives, particularly for use in organic electronics. This review highlights the computational studies carried out to understand acene reactivity, their reaction mechanisms, and the relationship of the latter to acene length. Generally, as acene length increases, the reactivity of acenes increases, and a greater tendency towards a biradical mechanism is observed. For example, an interesting change in mechanism (from concerted to biradical) is observed for the reaction of acenes (benzene through pentacene) with molecular oxygen. A computational study of the addition of HCl (which behaves as an electrophile) and water (which behaves mostly as a nucleophile) to acenes revealed that acene reactivity increases along the series up to hexacene and remains constant from hexacene and above because of the biradical character of the ground state of higher acenes. Although the exothermicity of the addition of water and HCl to acene is similar, the activation barrier for the addition of water is significantly higher than that for HCl. There is a substantial substituent effect on the energy barriers for electrophilic and nucleophilic reactions. Phenyl substitution at the most reactive meso‐carbon atoms of the central ring of acene blocks dimerization through this ring but does not efficiently prevent dimerization through other rings. With respect to the self‐reactivity (such as dimerization, etc.) of acenes, a computational study showed that the preferred pathways are the formation of acene dimers via a central benzene ring and the formation of acene‐based polymers. Interestingly, longer acenes are predicted to self‐react to form polymers, rather than acene dimers, as the thermodynamically preferred product. These findings invite experimentalists to reinvestigate the dimerization of long unsubstituted acenes. Given that computational studies directly relate to the gas phase reactivity of acenes, whereas long acenes find application in organic semiconductor devices as solids, the next research challenge is to obtain an understanding of the solid state reactivity of long acenes. Copyright © 2012 John Wiley & Sons, Ltd.     

Hard-hexagon model: Anisotropy of correlation length and interfacial tension

The directional dependence of the correlation length of the hard-hexagon model is calculated by a new method which introduces the shift operator into the usual transfer matrix method. This method is also applied to the calculation of the interfacial tension of the hard-hexagon model, which is anisotropic. In addition, the equilibrium droplet shape of one phase embedded inside another is obtained from the analysis of the interfacial tension by the use of Wulff's construction.     

Modeling ternary mixtures by mean-field theory of polyelectrolytes: Coupled Ginzburg-Landau and Swift-Hohenberg equations

The purpose of this work is to model ternary mixtures using the theory of pattern formation and of polyelectrolytes, with mean-field approximations. The model has two local, non-conserved order parameters. In the free energy short-range and long-range nonlocal interactions between elements of the mixture are considered. The spatiotemporal dynamics of the system is described by coupling the time-dependent Ginzburg-Landau equation and the Swift-Hohenberg equation. These non-linear partial differential equations are solved with numerical methods to study the emergent spatially stable configurations. The model shows a large diversity of patterns, which permit an interpretation of the behavior of some biological systems and presents different growth lengths within its spatial structures.     

On the genre-fication of music: a percolation approach

We analyze web-downloaded data on people sharing their music library. By attributing to each music group usual music genres (Rock, Pop ...), and analysing correlations between music groups of different genres with percolation-idea based methods, we probe the reality of these subdivisions and construct a music genre cartography, with a tree representation. We also discuss an alternative objective way to classify music, that is based on the complex structure of the groups audience. Finally, a link is drawn with the theory of hidden variables in complex networks.     

空气中声源激发水下声场的水平纵向相关性

声场的空间相关特性是声场的重要特征,对水下探测、水声通讯等各种设备在实际海洋环境中应用的参数选择具有重要意义,是水声工程技术研究的重要基础之一。相比于水中声源激发声场的相关特性研究,空气中声源的相关研究很少。本文推导了空气中声源激发水下声场的水平纵向相关的简正波表达式,并通过数值仿真分析,比较了声源分别位于空气中和水中时水下声场的水平纵向相关特性。对南海海域进行的一次悬挂汽笛空气声源、海底水平阵接收信号海上实验获得的数据进行分析,结果表明:空气中声源位于不同距离时,其发射的声信号激发水下声场的水平纵向相关均存在明显的起伏结构,基于本文提出的空气中声源激发水下声场的水平纵向相关系数的简正波表达式能够较好地解释该现象.     

Finite data-size scaling of clustering in earthquake networks

An earthquake network is known to be of the small-world type. The values of the network’s characteristics, however, depend not only on the cell size (i.e., the scale of coarse graining needed for constructing the network) but also on the size of a seismic data set. Here, discovery of a scaling law for the clustering coefficient in terms of the data size, which is referred to here as finite data-size scaling, is reported. Its universality is shown to be supported by the detailed analysis of the data taken from California, Japan and Iran. Effects of setting a threshold of magnitude are also discussed.     

Neutron depolarization study on the magnetic correlation length of nickel ferrite with different packing densities

The magnetic correlation length of a mixed nickel ferrite powder was studied by a newly commissioned depolarized neutron beamline at the W3 port of Tsing Hua Open Pool Reactor (THOR). In this work, Ni ferrite powder samples with different packing densities were studied. The magnetic correlation lengths of the sample were observed to be 2 μm at virgin state and about 3.1 μm at remanent state from the packing density of 20–60%. This magnetic domain size is smaller than particle size. No significant change of domain size at this packing density implies the domain wall motion is hindered by the porosity effectively up to at least 60% of packing density.