Scaling in the diffusion limited aggregation model

We present a self-consistent picture of diffusion limited aggregation (DLA) growth based on the assumption that the probability density P(r,N) for the next particle to be attached within the distance r to the center of the cluster is expressible in the scale-invariant form P[r/R{dep}(N)]. It follows from this assumption that there is no multiscaling issue in DLA and there is only a single fractal dimension D for all length scales. We check our assumption self-consistently by calculating the particle-density distribution with a measured P(r/R{dep}) function on an ensemble with 1000 clusters of 5×10{7} particles each. We also show that a nontrivial multiscaling function D(x) can be obtained only when small clusters (N<10?000) are used to calculate D(x). Hence, multiscaling is a finite-size effect and is not intrinsic to DLA.     

Dynamical scaling and kinetic roughening of single valued fronts propagating in fractal media

We consider the dynamical scaling and kinetic roughening of single-valued interfaces propagating in 2D fractal media. Assuming that the nearest-neighbor height difference distribution function of the fronts obeys Lévy statistics with a well-defined algebraic decay exponent, we consider the generalized scaling forms and derive analytic expressions for the local scaling exponents. We show that the kinetic roughening of the interfaces displays intrinsic anomalous scaling and multiscaling in the relevant correlation functions. We test the predictions of the scaling theory with a variety of well-known models which produce fractal growth structures. Results are in excellent agreement with theory. For some models, we find interesting crossover behavior related to large-scale structural instabilities of the growing aggregates. Received 22 May 2002 Published online 19 November 2002     

Varieties of dynamic multiscaling in fluid turbulence

We show that different ways of extracting time scales from time-dependent velocity structure functions lead to different dynamic-multiscaling exponents in fluid turbulence. These exponents are related to equal-time multiscaling exponents by different classes of bridge relations, which we derive. We check this explicitly by detailed numerical simulations of the Gledzer-Ohkitani-Yamada shell model for fluid turbulence. Our results can be generalized to any system in which both equal-time and time-dependent structure functions show multiscaling.     

二维随机蜂巢网格熔断动力学过程和熔断面标度性质的数值模拟

石墨烯等材料具有典型的二维蜂巢结构,而随机电阻丝模型则是研究非均匀材料断裂十分有效的统计物理学模型.本文尝试对二维蜂巢结构随机电阻丝网络熔断的动力学过程及熔断面性质进行数值模拟分析,以此来研究二维非均质蜂窝材料熔断的动力学性质和熔断面的动力学标度性质.模拟研究表明,二维随机蜂窝网格的熔断动力学过程和熔断面具有明显的标度性质,得到的熔断面整体和局域粗糙度指数分别为α=0.911±0.005和α_(loc)=0.808 ± 0.003,这两者之间的明显差异表明熔断面具有奇异标度性.通过对熔断面极值高度的分析发现,熔断面高度的极值统计分布能很好地满足Asym2sig型分布,而不是最常见的三种极值统计分布.本文的研究表明,随机电阻丝模型在模拟非均匀材料的电流熔断过程和熔断表面标度性的分析中同样适用和有效.     

Universal order statistics of random walks

We study analytically the order statistics of a time series generated by the positions of a symmetric random walk of n steps with step lengths of finite variance σ(2). We show that the statistics of the gap d(k,n) = M(k,n)-M(k+1,n) between the kth and the (k+1)th maximum of the time series becomes stationary, i.e., independent of n as n → ∞ and exhibits a rich, universal behavior. The mean stationary gap exhibits a universal algebraic decay for large k, ~d(k,∞)-/σ 1/sqrt[2πk], independent of the details of the jump distribution. Moreover, the probability density (pdf) of the stationary gap exhibits scaling, Pr(d(k,∞) = δ) ~/= (sqrt[k]/σ)P(δsqrt[k]/σ), in the regime δ~ (d(k,∞)). The scaling function P(x) is universal and has an unexpected power law tail, P(x) ~ x(-4) for large x. For δ> (d(k,∞)) the scaling breaks down and the pdf gets cut off in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual multiscaling behavior.     

Impact craters in granular media: grains against grains

Impact experiments in granular media are usually performed with solid projectiles that do not fragment at all. Contrastingly, we study here the morphology produced by the impact of spherical granular projectiles whose structure is utterly lost after collision. Simple and complex craters are observed, depending on the packing fraction of the balls. Their diameters D and depths z are analyzed as a function of the drop height h. We find the same power law D ∝ h(1/4) obtained with solid spheres, but a discontinuity at a certain threshold height, related to the cohesive energy of the projectiles, shows up. Counterintuitively, instead of a monotonic increase with the collisional energy, z becomes constant above this threshold.     

Statistical Physics of Fracture Surfaces Morphology

Experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics. One finds scaling exponents associated with correlation and structure functions, indicating a rich phenomenology of anomalous scaling. We argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for. We present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces, successfully reproducing the multiscaling phenomenology at least in 1+1 dimensions. For surfaces in 2+1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the SO(2) symmetry group. It appears that this approach organizes effectively the rich scaling properties. We end up proposing new experiments in which the rotational symmetry is not broken, such that the scaling properties should be particularly simple.     

Towards a realistic model of Higgsless electroweak symmetry breaking

We present a 5D gauge theory in warped space based on a bulk SU(2)L x SU(2)R x U(1)(B-L) gauge group where the gauge symmetry is broken by boundary conditions. The symmetry breaking pattern and the mass spectrum resemble that in the standard model (SM). To leading order in the warp factor the rho parameter and the coupling of the Z (S parameter) are as in the SM, while corrections are expected at the level of a percent. From the anti-de Sitter (AdS) conformal field theory point of view the model presented here can be viewed as the AdS dual of a (walking) technicolorlike theory, in the sense that it is the presence of the IR brane itself that breaks electroweak symmetry, and not a localized Higgs on the IR brane (which should be interpreted as a composite Higgs model). This model predicts the lightest W, Z, and gamma resonances to be at around 1.2 TeV, and no fundamental (or composite) Higgs particles.     

First-order metal-insulator transitions in manganites: are they universal?

Conductivity data for La(2-2x)Sr(1+2xMn2O7 (x = 0.6) show a first-order transition from an orbital- or charge-ordered insulator to a metal as the temperature falls below approximately 160 K. The change in conductivity is 100 times larger than that seen previously in any single-phase manganite in zero field. The metallic low-temperature state is similar to x = 0.58, but x = 0.58 shows no evidence of orbital or charge order. This result supports a conclusion that strongly coupled magnetic-conductive transitions are first order.     

Finite size scaling in BTW like sandpile models

Lattice statistical models of equilibrium critical phenomena generally obey finite size scaling (FSS) ansatz. However, the critical behavior of the prototypical BTW sandpile model demonstrating self-organized criticality at out of equilibrium is described by a peculiar multiscaling behaviour. FSS hypothesis is verified here on two versions (RSM1 and RSM2) of a rotational sandpile model (RSM) with broken mirror symmetry. In these models, sand grains flow only in the forward direction and in a specific rotational direction from an active site after toppling. The toppling rules are such that RSM1 will have less randomness whereas RSM2 will have more randomness with respect to RSM. RSM1 is expected to be more closer to BTW whereas RSM2 is expected to be more closer to Manna’s stochastic model. Both RSM1 and RSM2 are found to belong to the same universality class of RSM. The scaling functions of RSM1 and RSM2 are also found to obey usual FSS hypothesis at out of equilibrium instead of multiscaling as in BTW.     

全向入射条件下一维固流周期结构中低频声裂隙变化特性研究

利用传递矩阵法, 从理论上建立了全向入射条件下一维固-流周期结构中的声传播模型, 在此基础上计算、分析并比较了无限周期结构的声能带结构和有限周期结构中的声传输特性. 研究结果表明, 当声波以一定的入射角入射时, 固-流周期结构的低频通带区域存在一个声裂隙, 该声裂隙所对应的入射角大小与构成周期结构的固体层和流体层的密度或结构尺寸无关, 而仅取决于构成该周期性结构材料的波速. 关键词： 传递矩阵 全向入射 固-流周期结构 声裂隙     

XFEM modeling of hydraulic fracture in porous rocks with natural fractures

Hydraulic fracture (HF) in porous rocks is a complex multi-physics coupling process which involves fluid flow, diffusion and solid deformation. In this paper, the extended finite element method (XFEM) coupling with Biot theory is developed to study the HF in permeable rocks with natural fractures (NFs). In the recent XFEM based computational HF models, the fluid flow in fractures and interstitials of the porous media are mostly solved separately, which brings difficulties in dealing with complex fracture morphology. In our new model the fluid flow is solved in a unified framework by considering the fractures as a kind of special porous media and introducing Poiseuille-type flow inside them instead of Darcy-type flow. The most advantage is that it is very convenient to deal with fluid flow inside the complex fracture network, which is important in shale gas extraction. The weak formulation for the new coupled model is derived based on virtual work principle, which includes the XFEM formulation for multiple fractures and fractures intersection in porous media and finite element formulation for the unified fluid flow. Then the plane strain Kristianovic-Geertsma-de Klerk (KGD) model and the fluid flow inside the fracture network are simulated to validate the accuracy and applicability of this method. The numerical results show that large injection rate, low rock permeability and isotropic in-situ stresses tend to lead to a more uniform and productive fracture network.     

Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.     

d x 2 - y 2 superconductivity and the Hubbard model

The numerical studies of d x 2 - y 2 -wave pairing in the two-dimensional (2D) and the 2-leg Hubbard models are reviewed. For this purpose, the results obtained from the determinantal Quantum Monte Carlo and the Density-Matrix Renormalization-Group calculations are presented. These are calculations which were motivated by the discovery of the high- T c cuprates. In this review, the emphasis is placed on the microscopic many-body processes which are responsible for the d x 2 - y 2 -wave pairing correlations observed in the 2D and the 2-leg Hubbard models. In order to gain insight into these processes, the results on the effective pairing interaction as well as the magnetic, density and the single-particle excitations will be reviewed. In addition, comparisons will be made with the other numerical approaches to the Hubbard model and the numerical results on the t - J model. The results reviewed here indicate that an effective pairing interaction which is repulsive at ( ~ , ~ ) momentum transfer, and enhanced single-particle spectral weight near the ( ~ ,0) and (0, ~ ) points of the Brillouin zone, create optimum conditions for d x 2 - y 2 -wave pairing. These are two effects which act to enhance the d x 2 - y 2 -wave pairing correlations in the Hubbard model. Finding additional ways is an active research problem.     

多标度分形上的输运方程

 利用质量守恒定律,得到在三维情况下多标度无序分形介质中的一般输运方程,并在讨论布朗运动和分数布朗运动以及在分形介质上的标准扩散方程的基础上,得到多标度无序分形介质中的分数阶输运方程。     

高能核–核碰撞Drell–Yan过程K因子常数性的研究

在高能核–核碰撞Drell-Yan过程考虑湮没项和康普顿散射项的基础上,利用双重Q²重标度模型,分别计算了碳、钙、铁、锡与其自身的核–核碰撞.结果表明,对于不同的x₁值,K因子随x₂的变化很不相同;同时,K因子变化的幅度是随核素A不同而发生微弱变化的,在特定的x₁,x₂取值有限范围内它不能近似取为常数.计算结果与实验的对比是对所用模型及QCD理论适用性的一种检验.     

Dynamics of passive-scalar turbulence

We present the first study of the dynamic scaling or multiscaling of passive-scalar turbulence. For the Kraichnan version of passive-scalar turbulence we show analytically, in both Eulerian and quasi-Lagrangian frameworks, that simple dynamic scaling is obtained but with different dynamic exponents. By developing the multifractal model we show that dynamic multiscaling occurs in passive-scalar turbulence only if the advecting velocity field is itself multifractal. We substantiate our results by detailed numerical simulations in shell models of passive-scalar advection.     

Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.     

D(A) steps and 2D islands of double layer height in the SiGe(001) system

The surfaces of step graded, partially relaxed Si(1-x)Ge(x)/Si(001) buffers were studied by scanning tunneling microscopy. The surface slips along <110> forming the crosshatch pattern, consisting of bunches of D(A) steps of double layer height. The D(A) steps are present in regions of large surface gradients close to the slips, as well as in planar regions between the slips. These regions are also characterized by the appearance of 2D islands of double layer height. The observations can be explained by assuming the strain due to the misfit dislocations to be locally anisotropic. Anisotropic misfit strain and efficient strain relaxation by the ( 2x8) Ge reconstruction were identified as the main factors causing the unusual step structure.