Renormalized field theory of critical dynamics

We formulate a Gell'Mann-Low-type renormalization group approach to the critical dynamics of stochastic models described by Langevin or Fokker-Planck equations including mode-coupling terms.Dynamical correlation and response functions are expressed in terms of path integrals, which are investigated by well-known methods of renormalized perturbation theory.Dynamical scaling laws and relations between static and dynamic critical exponents are derived. The leading temperature-dependence of correlation and response functions is obtained from the Kadanoff-Wilson short-distance expansion. We also consider corrections to dynamic scaling which are due to a finite lattice constant.     

Scaling of a Slope: The Erosion of Tilted Landscapes

We formulate a stochastic equation to model the erosion of a surface with fixed inclination. Because the inclination imposes a preferred direction for material transport, the problem is intrinsically anisotropic. At zeroth order, the anisotropy manifests itself in a linear equation that predicts that the prefactor of the surface height–height correlations depends on direction. The first higher order nonlinear contribution from the anisotropy is studied by applying the dynamic renormalization group. Assuming an inhomogeneous distribution of soil substrate that is modeled by a source of static noise, we estimate the scaling exponents at first order in an ε-expansion. These exponents also depend on direction. We compare these predictions with empirical measurements made from real landscapes and find good agreement. We propose that our anisotropic theory applies principally to small scales and that a previously proposed isotropic theory applies principally to larger scales. Lastly, by considering our model as a transport equation for a driven diffusive system, we construct scaling arguments for the size distribution of erosion “events” or “avalanches.” We derive a relationship between the exponents characterizing the surface anisotropy and the avalanche size distribution, and indicate how this result may be used to interpret previous findings of power-law size distributions in real submarine avalanches.     

Anomalous scaling exponents in nonlinear models of turbulence

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by "statistically preserved structures" which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.     

Scaling,Renormalization and Statistical Conservation Laws in the Kraichnan Model of Turbulent Advection

We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of ξ, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.     

Scaling in the ideal Bose gas

We have made a detailed study of scaling in the ideal Bose gas in order to resolve the apparent inconsistencies that occur in the scaling laws when the dimensionality of the system is greater than four. We have found that there are not one, but two critical exponents associated with the specific heat singularity that appear in the scaling laws. We have proposed a modification of the scaling laws which is correct in any dimension.     

Complex networks renormalization: flows and fixed points

Recently, it has been claimed that some complex networks are self-similar under a convenient renormalization procedure. We present a general method to study renormalization flows in graphs. We find that the behavior of some variables under renormalization, such as the maximum number of connections of a node, obeys simple scaling laws, characterized by critical exponents. This is true for any class of graphs, from random to scale-free networks, from lattices to hierarchical graphs. Therefore, renormalization flows for graphs are similar as in the renormalization of spin systems. An analysis of classic renormalization for percolation and the Ising model on the lattice confirms this analogy. Critical exponents and scaling functions can be used to classify graphs in universality classes, and to uncover similarities between graphs that are inaccessible to a standard analysis.     

Scaling properties of geometric parallelization

We present a universal scaling law for all geometrically parallelized computer simulation algorithms. For algorithms with local interaction laws we calculate the scaling exponents for zero and infinite lattice size. The scaling is tested on local (cellular automata, Metropolis Ising) as well as cluster (Swendsen-Wang) algorithms. The practical aspects of the scaling properties lead to a simple recipe for finding the optimum number of processors to be used for the parallel simulation of a particular system.     

Biorthogonal quantum criticality in non-Hermitian many-body systems

We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of the ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility.     

Numerical investigations of discrete scale invariance in fractals and multifractal measures

Fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. Here, we focus on lattice multifractals which exhibit complex exponents associated with observable log-periodicity. We perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments. A novel numerical approach is proposed to extract the log-frequencies. In the non-lattice case, there is no visible log-periodicity, i.e., no preferred scaling ratio since the set of complex exponents spreads irregularly within the complex plane. A non-lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non-lattice one. An algorithm for the construction of the lattice sequence is proposed explicitly.     

Inhomogeneous systems with unusual critical behaviour

The phase transitions and critical properties of two types of inhomogeneous systems are reviewed. In one case, the local critical behaviour results from the particular shape of the system. Here scale-invariant forms like wedges or cones are considered as well as general parabolic shapes. In the other case the system contains defects, either narrow ones in the form of lines or stars, or extended ones where the couplings deviate from their bulk values according to power laws. In each case the perturbation may be irrelevant, marginal or relevant. In the marginal case one finds local exponents which depend on a parameter. In the relevant case unusual stretched exponential behaviour and/or local first-order transitions appear. The discussion combines mean field theory, scaling considerations, conformal transformations and perturbation theory. A number of examples are Ising models for which exact results can be obtained. Some walks and polymer problems are considered, too.     

Efficient method of finding scaling exponents from finite-size Monte-Carlo simulations

Monte-Carlo simulations are routinely used for estimating the scaling exponents of complex systems. However, due to finite-size effects, determining the exponent values is often difficult and not reliable. Here, we present a novel technique of dealing with the problem of finite-size scaling. This new method allows not only to decrease the uncertainties of the scaling exponents, but makes it also possible to determine the exponents of the asymptotic corrections to the scaling laws. The efficiency of the technique is demonstrated by finding the scaling exponent of uncorrelated percolation cluster hulls.     

Understanding finite size effects in quasi-long-range orders for exactly solvable chain models

In this paper, we investigate how much of the numerical artefacts introduced by finite system size and choice of boundary conditions can be removed by finite size scaling, for strongly correlated systems with quasi-long-range order. Starting from the exact ground-state wave functions of hardcore bosons and spinless fermions with infinite nearest-neighbor repulsion on finite periodic chains and finite open chains, we compute the two-point, density-density, and pair-pair correlation functions, and fit these to various asymptotic power laws. Comparing the finite-periodic-chain and finite-open-chain correlations with their infinite-chain counterparts, we find reasonable agreement among them for the power-law amplitudes and exponents, but poor agreement for the phase shifts. More importantly, for chain lengths on the order of 100, we find our finite-open-chain calculation overestimates some infinite-chain exponents (as did a recent density-matrix renormalization-group (DMRG) calculation on finite smooth chains), whereas our finite-periodic-chain calculation underestimates these exponents. We attribute this systematic difference to the different choice of boundary conditions. Eventually, both finite-chain exponents approach the infinite-chain limit: by a chain length of 1000 for periodic chains, and >2000 for open chains. There is, however, a misleading apparent finite size scaling convergence at shorter chain lengths, for both our finite-chain exponents, as well as the finite-smooth-chain exponents. Implications of this observation are discussed.     

A Non-Stationary Poisson Model for the Scaling of Urban Traffic Fluctuations

We investigate the traffic flow volume data on the time dependent activity of Beijing＇s urban road network. The couplings between the average flux and the fluctuations on individual links are shown to follow certain scaling laws and yield a wide variety of scaling exponents between 1/2 and 1. To quantitatively explain this interesting phenomenon, a non-stationary Poisson arriving model is proposed. The scaling property is interpreted as the result of the time- variation of the arriving rate of flux over the network, which nicely explicates the effect of aggregation windows, and provides a concise model for the dependence of scaling exponent on the external/internal force ratio.     

Diffusion in a random medium: A renormalization group approach

We investigate through a continuous random diffusion equation the long-distance properties of the general non-symmetric hopping model. The lower and upper critical dimensionalities are d = 1 and d = 2 respectively. A renormalization group analysis shows that the velocity and the diffusion constant obey scaling laws with non-classical exponents, which are computed to first order in ε = 2 ? d. Similar scaling laws, based on heuristic arguments, are conjectured for the AC conductivity.     

Power-Law Behavior in Geometric Characteristics of Full Binary Trees

Natural river networks exhibit regular scaling laws in their topological organization. Here, we investigate whether these scaling laws are unique characteristics of river networks or can be applicable to general binary tree networks. We generate numerous binary trees, ranging from purely ordered trees to completely random trees. For each generated binary tree, we analyze whether the tree exhibits any scaling property found in river networks, i.e., the power-laws in the size distribution, the length distribution, the distance-load relationship, and the power spectrum of width function. We found that partially random trees generated on the basis of two distinct types of deterministic trees, i.e., deterministic critical and supercritical trees, show contrasting characteristics. Partially random trees generated on the basis of deterministic critical trees exhibit all power-law characteristics investigated in this study with their fitted exponents close to the values observed in natural river networks over a wide range of random-degree. On the other hand, partially random trees generated on the basis of deterministic supercritical trees rarely follow scaling laws of river networks.     

Scaling laws for all Liapunov exponents: Models and measurements

We introduce simple diamond models of random symplectic matrices in order to study the scaling laws of all Liapunov exponents. These universal properties appear in physical problems that are modeled by transfer matrices: dynamical systems, random potentials, random fields, etc. Numerical experiments for the general case are in agreement with the results derived from the models.     

CRITICAL SCALING THEORY OF GENERALIZED PHASE TRANSITION AND ITS UNIVERSALITY

Generalized phase transition (GPT) refers to the transition process of material systems from one steady-state to another. It includes equilibrium phase transition (EPT) and nonequilibrium phase transition (NPT), and phase transitions intermediate between them. In this paper some results on the study of critical scaling relations of the NPT and EPT are obtained. We developed the critical scaling theory of EPT and advanced a universal critical scaling theory of GPT. The critical scaling relations(scaling laws) has more niversality. The critical exponents calculated from our theory are identical with the results of experiments and other theories about EPT and NPT systems. Because the basic model of the theory does not depend on the concrete material system, it has a certain unversality. Its results thus can be applied to generlized phase transition systems, such as the electrorheological fluid and magnetorheological fluid systems.     

Scaling properties in the production range of shear dominated flows

In large Reynolds number turbulence, isotropy is recovered as the scale is reduced and homogeneous-isotropic scalings are eventually observed. This picture is violated in many cases, e.g., wall bounded flows, where, due to the shear, different scaling laws emerge. This effect has been ascribed to the contamination of the inertial range by the larger anisotropic scales. The issue is addressed here by analyzing both numerical and experimental data for a homogeneous shear flow. In fact, under strong shear, the alteration of the scaling exponents is not induced by the contamination from the anisotropic sectors. Actually, the exponents are universal properties of the isotropic component of the structure functions of shear dominated flows. The implications are discussed in the context of turbulence near solid walls, where improved closure models would be advisable.