Improved phenomenological renormalization schemes

An analysis is made of various methods of phenomenological renormalization based on finite-dimensional scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-dimensional scaling amplitude of the internal energy in the three-dimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-dimensional corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-dimensional equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical index.     

Scaling behavior in a stochastic self-gravitating system

A system of stochastic differential equations for the velocity and density of classical self-gravitating matter is investigated by means of the field theoretic renormalization group. The existence of two types of large-scale scaling behavior, associated with physically admissible fixed points of the renormalization-group equations, is established. Their regions of stability are identified and the corresponding scaling dimensions are calculated in the one-loop approximation (first order of the epsilon expansion). The velocity and density fields have independent scaling dimensions. Our analysis supports the importance of the rotational (nonpotential) components of the velocity field in the formation of those scaling laws.     

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.     

Excited TBA equations II: massless flow from tricritical to critical Ising model

We consider the massless tricritical Ising model perturbed by the thermal operator _1,3 in a cylindrical geometry and apply integrable boundary conditions, labelled by the Kac labels (r,s), that are natural off-critical perturbations of known conformal boundary conditions. We derive massless thermodynamic Bethe ansatz (TBA) equations for all excitations by solving, in the continuum scaling limit, the TBA functional equation satisfied by the double-row transfer matrices of the A₄ lattice model of Andrews, Baxter and Forrester (ABF) in Regime IV. The resulting TBA equations describe the massless renormalization group flow from the tricritical to critical Ising model. As in the massive case of Part I, the excitations are completely classified in terms of (m,n) systems but the string content changes by one of three mechanisms along the flow. Using generalized q-Vandermonde identities, we show that this leads to a flow from tricritical to critical Ising characters. The excited TBA equations are solved numerically to follow the continuous flows from the UV to the IR conformal fixed points.     

Atoms in flight and the remarkable connections between atomic and hadronic physics

Atomic physics and hadron physics are both based on Yang Mills gauge theory; in fact, quantum electrodynamics can be regarded as the zero-color limit of quantum chromodynamics. I review a number of areas where the techniques of atomic physics provide important insight into the theory of hadrons in QCD. For example, the Dirac-Coulomb equation, which predicts the spectroscopy and structure of hydrogenic atoms, has an analog in hadron physics in the form of light-front relativistic equations of motion which give a remarkable first approximation to the spectroscopy, dynamics, and structure of light hadrons. The renormalization scale for the running coupling, which is unambiguously set in QED, leads to a method for setting the renormalization scale in QCD. The production of atoms in flight provides a method for computing the formation of hadrons at the amplitude level. Conversely, many techniques which have been developed for hadron physics, such as scaling laws, evolution equations, and light-front quantization have equal utility for atomic physics, especially in the relativistic domain. I also present a new perspective for understanding the contributions to the cosmological constant from QED and QCD.     

Towards the Evaluation of the Relevant Degrees of Freedom in Nonlinear Partial Differential Equations

We investigate an operator renormalization group method to extract and describe the relevant degrees of freedom in the evolution of partial differential equations. The proposed renormalization group approach is formulated as an analytical method providing the fundamental concepts of a numerical algorithm applicable to various dynamical systems. We examine dynamical scaling characteristics in the short-time and the long-time evolution regime providing only a reduced number of degrees of freedom to the evolution process.     

Generalized Central Limit Theorem and Renormalization Group

We introduce a simple instance of the renormalization group transformation in the Banach space of probability densities. By changing the scaling of the renormalized variables we obtain, as fixed points of the transformation, the Lévy strictly stable laws. We also investigate the behavior of the transformation around these fixed points and the domain of attraction for different values of the scaling parameter. The physical interest of a renormalization group approach to the generalized central limit theorem is discussed.     

The β function,scaling, and improved action forSU(3) lattice gauge theory

We present a detailed analysis of the nonperturbativeβ function along the Wilson axis for theSU(3) pure gauge theory using the Monte Carlo renormalization group method. The scaling behavior of the string tension, the deconfinement transition temperature, and the O⁺⁺ glueball mass obtained from published data is compared. The results show that there is no asymptotic scaling forK _F=(6/g ²)<6.1. We also estimate the renormalized action generated by the √3 block transformation for use in future calculations.     

A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows

We propose a new model and a solution method for two-phase compressible flows. The model involves six equations obtained from conservation principles applied to each phase, completed by a seventh equation for the evolution of the volume fraction. This equation is necessary to close the overall system. The model is valid for fluid mixtures, as well as for pure fluids. The system of partial differential equations is hyperbolic. Hyperbolicity is obtained because each phase is considered to be compressible. Two difficulties arise for the solution: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. We propose robust and accurate discretisation of these terms. The method solves the same system at each mesh point with the same algorithm. It allows the simulation of interface problems between pure fluids as well as multiphase mixtures. Several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible. The method provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.     

带强迫项变系数组合KdV方程的无穷序列复合型类孤子新解

为了获得变系数非线性发展方程的无穷序列复合型新解,研究了G′(ξ)G(ξ)展开法.通过引入一种函数变换,把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题.在此基础上,利用Riccati方程解的非线性叠加公式,获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解.借助这些复合型新解与符号计算系统Mathematica,构造了带强迫项变系数组合KdV方程的无穷序列复合型类孤子新精确解.     

CONSERVATION LAWS FOR SELF-ADJOINT FIRST-ORDER EVOLUTION EQUATION

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.     

Statistical symmetries and its impact on new decay modes and integral invariants of decaying turbulence

Classical decay laws of isotropic turbulence usually derived from the von Kármán–Howarth equation are essentially based on two paradigms. First, scaling symmetries of space and time, both tracing back to the Navier–Stokes equations in the limit of large Reynolds numbers (or r?η), give rise to a temporal power-law decay for the turbulent kinetic energy and at the same time an algebraic growth of the integral length scale at an exponent that is uniquely coupled to the latter energy decay. Second, global invariants such as Birkhoff or Loitsianskii integrals determine the exponent of both power laws. We presently show that this class of decay laws may be considerably extended considering the entire set of multi-point correlation equations that admit a much wider class of symmetries. It was recently shown that these new symmetries are of paramount importance, e.g. in deriving the logarithmic law of the wall being an analytic solution of the multi-point equations. For the present case, it is particularly an additional scaling group, which we call statistical scaling group, that gives rise to two additional families of ‘canonical’ decay laws including those with an exponential characteristic for both the kinetic energy and the integral length scale. Finally, a second rather generic group admitted by all linear differential equations corresponding to the superposition principle induces an infinite set of scaling laws of rather complex form that may match rather generic initial conditions. All scaling laws are analyzed in the light of the above-mentioned integral invariants that have been further extended in the present contribution to an exponential-type invariant.     

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker-Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t₀ to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t₀, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker-Planck equation are also performed in a unified way in the present method.     

Variational principles,renormalization group,and Kadanoff's universality

Variational principles satisfied by various thermodynamic functionals are used to set up a completely renormalized scheme for the analysis of critical phenomena. Different aspects of Kadanoff's universality can be expressed in a simple fashion in our language. The main result of the paper is a unified derivation of the scaling laws combining the variational principles with renormalization group techniques which are especially simple in this formalism. A suitable choice of the normalization point has led to a new renormalization group transformation. The corresponding differential equation can be solved even in the nonasymptotic region. The discussion of the asymptotic theory and of the approach to it is therefore simpler. The connection with the more traditional approaches is discussed. The calculation of the critical indices is reduced to only two of them which are directly expressed in terms of renormalized quantities. From this point onwards their evaluation proceeds along standard lines. Special emphasis has been given to the illustration of the power and conceptual simplicity of the method.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

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.     

含有广义守恒律的生长方程标度奇异性的直接标度分析

利用直接标度分析方法研究一个含有广义守恒律生长方程的标度奇异性,得到强弱耦合区域的奇异标度指数.作为其特殊情况,这个方程包含Kardar-Parisi-Zhang（KPZ）方程、 Sun-Guo-Grant（SGG）方程以及分子束外延（MBE）生长方程,并能对其进行统一的研究.研究发现, KPZ方程和SGG方程,无论在弱耦合还是在强耦合区域内都遵从自仿射Family -Vicsek正常标度规律;而MBE 方程在弱耦合区域内服从正常标度,在强耦合区域内能呈现内禀奇异标度行为.这里所得到生长方程的奇异标度性质与利用重正化群理论、数值模拟以及实验相符很好. 关键词： 标度奇异性 强耦合 弱耦合     

Parameter renormalization of maps based on potential function

A systematic way for deriving the parameter renormalization group equation for one-dimensional maps is presented and the critical behavior of periodic doubling is investigated. Introducing a formal potential function in one-parameter cases, it is shown that accumulation points correspond to local potential maxima and universal constants are easily determined. The estimates of accumulation points and universal constants match the known values asymptotically when the order of potential grows large. The potential function shows scaling in the parameter space with the universal convergent rate at the accumulation point similar to the Feigenbaum universal function. For two-parameter cases, a parameter reduction transformation is found to be useful to determine some important fixed points. A locally defined potential function is introduced and its scaling property is discussed. (c) 1997 American Institute of Physics.     

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.