Global Pressure of One-Dimensional Polydisperse Granular Gases Driven by Gaussian White Noise

We study the global pressure of a one-dimensional polydisperse granular gases system for the first time,in which the size distribution of particles has the fractal characteristic and the inhomogeneity is described by a fractal dimension D. The particles are driven by Gaussian white noise and subject to inelastic mutual collisions. We define the global pressure P of the system as the impulse transferred across a surface in a unit of time, which has two contributions,one from the translational motion of particles and the other from the collisions. Explicit expression for the global pressure in the steady state is derived. By molecular dynamics simulations, we investigate how the inelasticity of collisions and the inhomogeneity of the particles influence the global pressure. The simulation results indicate that the restitution coefficient e and the fractal dimension D have significant effect on the pressure.     

Global Pressure of One-Dimensional Polydisperse Granular Gases Driven by Gaussian White Noise

We study the global pressure of a one-dimensional polydisperse granular gases system for the first time, in which the size distribution of particles has the fractal characteristic and the inhomogeneity is described by a fractal dimension D. The particles are driven by Gaussian white noise and subject to inelastic mutual collisions. We define the global pressure P of the system as the impulse transferred across a surface in a unit of time, which has two contributions, one from the translational motion of particles and the other from the collisions. Explicit expression for the global pressure in the steady state is derived. By molecular dynamics simulations, we investigate how the inelasticity of collisions and the inhomogeneity of the particles influence the global pressure. The simulation results indicate that the restitution coefficient e and the fractal dimension D have significant effect on the pressure.     

Dynamic behavior of non-uniform granular gases system with the fractal characteristic in one dimension

A one-dimensional dynamic model of polydisperse granular mixture with a power-law size distribution is presented, in which the particles are subject to inelastic mutual collisions and driven by Gaussian white noise. The particle size distribution of the mixture has the fractal characteristic, and a fractal dimension D as a measurement of the inhomogeneity of the particle size distribution is introduced. We define the global granular temperature and the kinetic pressure of the mixture, and obtain their expressions. By molecular dynamics simulations, we have mainly investigated how the inhomogeneity of the particle size distribution and the inelasticity of collisions influence the steady-state dynamic properties of the system, focusing on the global granular temperature, kinetic pressure, velocity distribution and distribution of interparticle spacing. Some novel results are found that, with the increase of the fractal dimension D, the global granular temperature and the kinetic pressure decrease, the velocity distribution deviates more obviously from the Gaussian one and the particles cluster more pronouncedly at the same value of the restitution coefficient e (0<e<1). On the other hand, as the restitution coefficient e decreases, the dynamic behavior has the similar evolution as above at the fixed fractal dimension D. The dynamic behavior changing with e and D is, respectively, presented.     

Non-Gaussian and Clustering Behavior in One-Dimensional Polydisperse Granular Gas System

We present a one-dimensional dynamic model of polydisperse granular mixture with the fractal characteristic of the particle size distribution, in which the particles are subject to inelastic mutual collisions and are driven by Gaussian white noise. The inhomogeneity of the particle size distribution is described by a fractal dimension D. The stationary state that the mixture reaches is the result of the balance between energy dissipation and energy injection. By molecular dynamics simulations, we have mainly studied how the inhomogeneity of the particle size distribution and the inelasticity of collisions influence the velocity distribution and distribution of interparticle spacing in the steady-state.The simulation results indicate that, in the inelasticity case, the velocity distribution strongly deviates from the Gaussian one and the system has a strong spatial clustering. Thus the inhomogeneity and the inelasticity have great effects on the velocity distribution and distribution of interparticle spacing. The quantitative information of the non-Gaussian velocity distribution and that of clustering are respectively represented.     

一簇金刚石晶格上S~4模型的相变

采用重整化群和累积展开的方法,研究了一簇金刚石晶格上S~4模型的相变,求得了系统的临界点.结果表明:当分支数m=2和m 12时,该系统只存在一个Gauss不动点K~*=b_2/2, u_2~*=0;当分支数3≤m≤12时,该系统不仅有Gauss不动点,还存在一个Wilson-Fisher不动点,并且后一个不动点对系统的临界特性产生决定性的影响.     

The Critical Behavior of Partially Directed SAW on Sierpinski Carpets

Using the fractal-cell generation method we perform a numerical simulation study for partially directed self-avoiding walks (PDSAW) on Sierpinski carpets. The obtained critical exponents v_H is found to be independent of the fractal dimension of Sierpinski carpet d_f, but v_⊥ is dependent on d_f . This result indicates that PDSAW on different Sierpinski carpets belong to different universality classes. Compared with the fully directed self-avoiding walks (FDSAW) on the same carpets, the obtained results indicate that PDSAW and FDSAW belong to the same universality class.     

Dynamic Properties of Two-Dimensional Polydisperse Granular Gases

We propose a two-dimensional model of polydisperse granular mixtures with a power-law size distribution in the presence of stochastic driving. A fractal dimension D is introduced as a measurement of the inhomogeneity of the size distribution of particles. We define the global and partial granular temperatures of the multi-component mixture. By direct simulation Monte Carlo, we investigate how the inhomogeneity of the size distribution influences the dynamic properties of the mixture, focusing on the granular temperature, dissipated energy, velocity distribution, spatial clusterization, and collision time. We get the following results： a single granular temperature does not characterize a multi-component mixture and each species attains its own ＂granular temperature＂; The velocity deviation from Gaussian distribution becomes more and more pronounced and the partial density of the assembly is more inhomogeneous with the increasing value of the fractal dimension D; The global granular temperature decreases and average dissipated energy per particle increases as the value olD augments.     

Dynamic Properties of Two-Dimensional Polydisperse Granular Gases

We propose a two-dimensional model of polydisperse granular mixtures with a power-law size distribution in the presence of stochastic driving. A fractal dimension D is introduced as a measurement of the inhomogeneity of the size distribution of particles. We define the global and partial granular temperatures of the multi-component mixture. By direct simulation Monte Carlo, we investigate how the inhomogeneity of the size distribution influences the dynamic properties of the mixture, focusing on the granular temperature, dissipated energy, velocity distribution, spatial clusterization, and collision time. We get the following results: a single granular temperature does not characterize a multi-component mixture and each species attains its own "granular temperature"; The velocity deviation from Gaussian distribution becomes more and more pronounced and the partial density of the assembly is more inhomogeneous with the increasing value of the fractal dimension D; The global granular temperature decreases and average dissipated energy per particle increases as the value of D augments.     

Steady State Properties of One-Dimensional Non-uniform Granular Gases Subjected to Gaussian White Noise Driving

We present a model of non-uniform granular gases in one-dimensional case, whose granularity distribution has the fractal characteristic. We have studied the nonequilibrium properties of the system by means of Monte Carlo method. When the typical relaxation time T of the Brownian process is greater than the mean collision time To, the energy evolution of the system exponentially decays, with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state in which the velocity distribution strongly deviates from the Gaussian one. Three other aspects have also been studied for the steady state： the visualized change of the particle density, the entropy of the system and the correlations in the velocity of particles. And the results of simulations indicate that the system has strong spatial clustering; Furthermore, the influence of the inelasticity and inhomogeneity on dynamic behaviors have also been extensively investigated, especially the dependence of the entropy and the correlations in the velocity of particles on the restitute coefficient e and the fractal dimension D.     

2+1维刻蚀模型生长表面等高线的共形不变性研究

为了更全面、有效地研究刻蚀模型(etching model)涨落表面的统计性质,基于Schramm Loewner Evolution(SLEκ)理论,对2+1维刻蚀模型饱和表面的等高线进行了数值模拟分析.研究表明,2+1维刻蚀模型饱和表面的等高线是共形不变曲线,可用Schramm Loewner Evolution理论进行描述,且扩散系数κ=2.70±0.04,属κ=8/3普适类.相应的等高线分形维数为df=1.34±0.01.     

Steady-State Properties of Driven Polydisperse Granular Mixtures

We represent a two-dimensional model of polydisperse granular mixtures with a power-law size distribution. The model consists of smooth hard disks in a rectangular box with inelastic collisions, driven by a homogeneous heat bath at zero gravity. The width of particle size distribution is characterized by the only
parameter, namely, the fractal dimension D. The energy dissipation of the mixture is increased as D increases or as e decreases. Furthermore, it is found that the steady-state properties of the mixture such as the collision rate, granular temperature, kinetic pressure and velocity distribution depend sensitively on size distribution parameter D.     

Grayscale projection of finite size fractal clusters

Microscopy techniques are suitable to obtain structural information of colloidal clusters with high resolution, but yield only a two dimensional projection of the objects. When imaging finite size objects with fractal properties, such as clusters of colloidal particles, this projection process has to be taken into account for the calculation of the fractal dimension. In this paper we present a technique to calculate the fractal dimension of finite size clusters with fractal properties using grayscale projections such as images obtained by X-ray microscopy. The grayscales are interpreted as different occupation counts within a projection. It is shown, that the radial distribution of these occupation counts varies with the fractal dimension d of the cluster. Using the radius of maximum occupation probability the fractal dimension up to 2.2 of finite size clusters can be calculated. The theoretical predictions are verified by test calculations employing numerically generated clusters.     

Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

Two-dimensional random Lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap. In systems of finite spatial extension, the asymptotic decay of the survival probability is exponential and characterized by an escape rate γ, which can be related to the average positive Lyapunov exponent and to the dimension of the fractal repeller of the system. For infinite systems, the survival probability obeys a stretched exponential law of the form P(c,t)∼exp(−Ct^1/2). The transition between the two regimes is studied and we show that, for a given trap density, the non-integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space. Nevertheless, the repeller remains fractal. We determine the special scaling properties of this fractal.     

Statistical analysis of the Perseus-Pisces redshift survey: spatial and luminosity properties

We analyze the spatial and the luminosity properties of the Perseus-Pisces redshift survey. We find that the two point correlation function (CF) Γ(r) is a power law up to the sample effective depth ( 30 h⁻¹ Mpc), showing the fractal nature of the galaxy distribution in this catalog. The fractal dimension turns out to be D 2. We also consider the CF ξ(r) and in particular the behavior of the “correlation length” r₀ (ξ(r₀)1) as function of the sample size. In this respect we find, unambiguously, that the luminosity segregation effect is not supported by any experimental evidence. In addition we have studied the galaxian number-density (n(r)) and number-counts (N(m)) in the VL subsamples finding a good agreement with the properties of a fractal distribution. In particular our conclusion is that the n(r) relation permits to extend the analysis of the fractal nature up to a deeper depth than that reached by the CF analysis, and, we find evidence for fractal properties up to the limiting depth of 130 h⁻¹ Mpc. We clarify the role of the small-scale fluctuations in the determination of the galaxy counts. Even in this case the results are in agreement with the previous ones. Finally we have considered the correlations between galaxy positions and luminosities by means of the multifractal analysis. We find clear evidence for self-similar behavior of the whole luminosity-space distribution. These results confirm and extend those of Coleman and Pietronero (1992).     

Fractal Character of Dynamic Light Scattering of Particles

Dynamic light scattering signals from particles, exhibit fractal characteristics. This feature can be used to determine the particle size. The use of the fractal dimension, as a quantitative method to analyze the properties of dynamic light scattering signals from submicron particles, is presented. The analysis is performed directly on the time‐resolved scattered intensity, and the Box Dimensions of light scattering signals of particles with diameters 100, 200, 500 and 1000 nm. The experimental results show that the fractal dimensions of light scattering signals correlate well with particle size. In the submicron size range, the smaller the particles, the larger their fractal dimensions. Compared with the PCS technique, only several hundreds of samples are required in the fractal method. Therefore, the data processing is easily accomplished. However, this method only provides the mean particle size, but not the particle size distribution.     

Flocculation and percolation in reversible cluster-cluster aggregation

Off-lattice dynamic Monte-Carlo simulations were done of reversible cluster-cluster aggregation for spheres that form rigid bonds at contact. The equilibrium properties were found to be determined by the life time of encounters between two particles (t_e). t_e is a function not only of the probability to form or break a bond, but also of the elementary step size of the Brownian motion of the particles. In the flocculation regime the fractal dimension of the clusters is d_f=2.0 and the size distribution has a power law decay with exponent τ=1.5. At larger values of t_e transient gels are formed. Close to the percolation threshold the clusters have a fractal dimension d_f=2.7 and the power law exponent of the size distribution is τ=2.1. The transition between flocculation and percolation occurs at a characteristic weight average aggregation number that decreases with increasing volume fraction.     

Surface structures of equilibrium restricted curvature model on two fractal substrates

With the aim to probe the effects of the microscopic details of fractal substrates on the scaling of discrete growth models, the surface structures of the equilibrium restricted curvature(ERC) model on Sierpinski arrowhead and crab substrates are analyzed by means of Monte Carlo simulations. These two fractal substrates have the same fractal dimension df, but possess different dynamic exponents of random walk zrw. The results show that the surface structure of the ERC model on fractal substrates are related to not only the fractal dimension df, but also to the microscopic structures of the substrates expressed by the dynamic exponent of random walk zrw. The ERC model growing on the two substrates follows the well-known Family–Vicsek scaling law and satisfies the scaling relations 2α + df≈ z ≈ 2zrw. In addition, the values of the scaling exponents are in good agreement with the analytical prediction of the fractional Mullins–Herring equation.     

Property of robustness to size and its realization on fractal dimension

Images in one class often have varied sizes due to different imaging system. Thus it will provide convenience to image classification if the indicator used in the classification is robust to the size of images. We regard the robustness to size of image as a property of image indicator. The property means that images from one class have small variance with the sizes, and is different from such traditional properties as the robustness to scale, rotation and illumination. Fractal dimension is an indicator which has the three traditional properties. We realize the property on fractal dimension in the statistical sense by modifying differential-box counting method. Tests on two classes of images demonstrate the effectiveness of the modifications. Tests on scaling process give a standard of FD’ robustness as 0.0611, and experiments on both the two class and four sets of images show the statistical validity of the standard and verify the realization. An indicator with this property can be a tool for the classification.     

Resistive state of superconducting structures with fractal clusters of a normal phase

The effect of morphologic factors on magnetic flux trapping and critical currents in a superconducting structure, which presents a type II percolation superconductor with pinning centers, is considered. The role of pinning centers is played by fractal clusters of the normal phase. The properties of these clusters are analyzed in detail: their statistics is studied, the distribution of critical currents of depinning is found, and the depen-dences of the main statistical parameters on the fractal dimension are obtained. The effect of fractal clusters of the normal phase on the electric field caused by the motion of the magnetic flux after the vortices have been broken away from pinning centers is considered. The current-voltage characteristics of superconducting structures in a resistive state are obtained for an arbitrary fractal dimension. It is found that the fractality of the boundaries of normal-phase clusters forces magnetic flux trapping, thereby increasing the critical current.     

Effects of Fractal Size Distributions on Velocity Distributions and Correlations of a Polydisperse Granular Gas

By the Monte Carlo method, the effect of dispersion of disc size distribution on the velocity distributions and correlations of a polydisperse granular gas with fractal size distribution is investigated in the same inelasticity. The dispersion can be described by a fractal dimension D, and the smooth hard discs are engaged in a two- dimensional horizontal rectangular box, colliding inelastically with each other and driven by a homogeneous heat bath. In the steady state, the tails of the velocity distribution functions rise more significantly above a Gaussian as D increases, but the non-Gaussian velocity distribution functions do not demonstrate any apparent universal form for any value of D. The spatial velocity correlations are apparently stronger with the increase of D. The perpendicular correlations are about half the parallel correlations, and the two correlations are a power-law decay function of dimensionless distance and are of a long range. Moreover, the parallel velocity correlations of postcollisional state at contact are more than twice as large as the precollisional correlations, and both of them show almost linear behaviour of the fractal dimension D.