Universality classes in the random-storage sandpile model

The avalanche statistics in a stochastic sandpile model where toppling takes place with a probability p is investigated. The limiting case p=1 corresponds to the Bak-Tang-Wiesenfeld (BTW) model with a deterministic toppling rule. Based on the moment analysis of the distribution of avalanche sizes we conclude that for 0相似文献   

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.     

A method for visualization of invariant sets of dynamical systems based on the ergodic partition

We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al. [J. Stat. Phys. 50, 529 (1988)]. (c) 1999 American Institute of Physics.     

The complete system of algebraic invariants for the sixteen-vertex model: I. Derivation via the three-dimensional orthogonal group

In a previous paper, the partition function of the 16-vertex model was shown to be invariant under a group of linear transformations in the space of the vertex weights. According to a theorem by Hilbert, every algebraic invariant such as the partition function for a finite lattice can be expressed algebraically in terms of a finite set of basic algebraic invariants, which are sums of products of the vertex weights. We construct this set by analysing the structural properties of the transformation group (the direct product of two three-dimensional orthogonal groups). The basic set is found to consist of 21 invariants, ranging from a linear invariant up to invariants of the ninth degree. In particular cases, notably the (general or the symmetric) eight-vertex model, the six-vertex model and the free-fermion model, several invariants vanish and a number of additional algebraic relations between the basic invariants are obtained.     

Phase Transitions and Topology Changes in Configuration Space

The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant (the Euler characteristic) of certain submanifolds in the configuration space of two physical models. The models are the mean-field XY model and the one-dimensional XY model with nearest-neighbor interactions. The former model undergoes a second-order phase transition at a finite critical temperature while the latter has no phase transitions. The computation of this topologic invariant is performed within the framework of Morse theory. In both models topology changes in configuration space are present as the potential energy is varied; however, in the mean-field model there is a particularly strong topology change, corresponding to a big jump in the Euler characteristic, connected with the phase transition, which is absent in the one-dimensional model with no phase transition. The comparison between the two models has two major consequences: (i) it lends new and strong support to a recently proposed topological approach to the study of phase transitions; (ii) it allows us to conjecture which particular topology changes could entail a phase transition in general. We also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two-dimensional surfaces, and a possible direct connection between topological invariants and thermodynamic quantities.     

四维相空间中非中心势动力学系统的Poisson 结构和Casimir 函数

将非中心势动力学系统的运动微分方程写成Ermakov形式，得到Ermakov不变量. 运用Hamilton理论，把Ermakov不变量当作Hamiltonian 函数，在四维相空间中建立了非中心势动力学系统的Poisson 结构。结果表明：此Poisson 结构是一退化的结构，而系统具有四个不变量，即Hamiltonian 函数，Ermakov不变量及两个Casimir函数。     

El-Nabulsi动力学模型下Birkhoff系统Noether对称性的摄动与绝热不变量

基于El-Nabulsi动力学模型,研究了小扰动作用下Birkhoff系统Noether对称性的摄动与绝热不变量问题.首先,将El-Nabulsi提出的在分数阶微积分框架下基于Riemann-Liouville分数阶积分的非保守系统动力学模型拓展到Birkhoff系统,建立El-Nabulsi-Birkhoff方程;其次,基于在无限小变换下El-Nabulsi-Pfaff作用量的不变性,给出Noether准对称性的定义和判据,得到了Noether对称性导致的精确不变量;再次,引入力学系统的绝热不变量概念,研究El-Nabulsi动力学模型下受小扰动作用的Birkhoff系统Noether对称性的摄动与绝热不变量之间的关系,得到了对称性摄动导致的绝热不变量的条件及其形式.作为特例,给出了El-Nabulsi动力学模型下相空间中非保守系统和经典Birkhoff系统的Noether对称性的摄动与绝热不变量.以著名的Hojman-Urrutia问题为例,研究其在El-Nabulsi动力学模型下的Noether对称性,得到了相应的精确不变量和绝热不变量.     

The Rossby wave extra invariant in the physical space

It was found out in 1991 that the Fourier space dynamics of Rossby waves possesses an extra positive-definite quadratic invariant, in addition to the energy and enstrophy. This invariant is similar to the adiabatic invariants in the theory of dynamical systems. For many years, it was unclear if this invariant—known only in the Fourier representation—is physically meaningful at all, and if it is, in what sense it is conserved. Does the extra conservation hold only for a class of solutions satisfying certain constraints (like the conservation in the Kadomtsev-Petviashvili equation)? The extra invariant is especially important because this invariant (provided it is meaningful) has been connected to the formation of zonal jets (like Jupiter’s stripes).In the present paper, we find an explicit expression of the extra invariant in the physical (or coordinate) space and show that the invariant is indeed physically meaningful for any fluid flow. In particular, no constraints are needed. The explicit form also enables us to note several properties of the extra invariant.     

On a new integral formula for an invariant of 3-component oriented links

Using physical considerations we constructed a new invariant of isotropy classes of an arbitrary configuration of three magnetic tubes in the space. The integral expression of this invariant is similar to the Massey product integrals of Milnor invariants of links. We prove that the constructed invariant cannot be expressed from the linking numbers of the configuration of magnetic tubes.     

Construction of exact complex dynamical invariant of a two-dimensional classical system

We present the construction of exact complex dynamical invariant of a two-dimensional classical dynamical system on an extended complex space utilizing Lie algebraic approach. These invariants are expected to play a vital role in understanding the complex trajectories of both classical and quantum systems.     

Topological Invariants of Vapor–Liquid,Vapor–Liquid–Liquid and Liquid–Liquid Phase Diagrams

The study of topological invariants of phase diagrams allows for the development of a qualitative theory of the processes being researched. Studies of the properties of objects in the same equivalence class may be carried out with the aim of predicting the properties of unexplored objects from this class, or predicting the behavior of a whole system. This paper describes a number of topological invariants in vapor–liquid, vapor–liquid–liquid and liquid–liquid equilibrium diagrams. The properties of some invariants are studied and illustrated. It is shown that the invariant of a diagram with a miscibility gap can be used to distinguish equivalence classes of phase diagrams, and that the balance equation of the singular-point indices, based on the Euler characteristic, may be used to analyze the binodal-surface structure of a quaternary system.     

Exact invariants and adiabatic invariants of dynamical system of relative motion

Based on the theory of symmetries and conserved quantities, the exact invariants and adiabatic invariants of a dynamical system of relative motion are studied. The perturbation to symmetries for the dynamical system of relative motion under small excitation is discussed. The concept of high-order adiabatic invariant is presented, and the form of exact invariants and adiabatic invariants as well as the conditions for their existence are given. Then the corresponding inverse problem is studied.     

Basin of attraction of cycles of discretizations of dynamical systems with SRB invariant measures

Computer simulations of dynamical systems arediscretizations, where the finite space of machine arithmetic replaces continuum state spaces. So any trajectory of a discretized dynamical system is eventually periodic. Consequently, the dynamics of such computations are essentially determined by the cycles of the discretized map. This paper examines the statistical properties of the event that two trajectories generate the same cycle. Under the assumption that the original system has a Sinai-Ruelle-Bowen invariant measure, the statistics of the computed mapping are shown to be very close to those generated by a class of random graphs. Theoretical properties of this model successfully predict the outcome of computational experiments with the implemented dynamical systems.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Complete classification of simple current modular invariants for RCFT's with a center (Z p ) k

Simple currents have been used previously to construct various examples of modular invariant partition functions for given rational conformal field theories. In this paper we present for a large class of such theories (namely those with a center that decomposes into factors Z _p ,p prime) thecomplete set of modular invariants that can be obtained with simple currents. In addition to the fusion rule automorphisms classified previously forany center, this includes all possible left-right combinations of all possible extensions of the chiral algebra that can be obtained with simple currents, for all possible current-current monodromies. Formulas for the number of invariants of each kind are derived. Although the number of invariants in each of these subsets depends on the current-current monodromies, the total number of invariants depends rather surprisingly only onp and the number ofZ _p factors.     

Asymmetric abeiian sandpile models

In the Abelian sandpile models introduced by Dhar, long-time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system. Dhar proposed a simple procedure, theburning algorithm, as a possible test of whether a configuration is recurrent, and later with Majumdar verified the correctness of this test when the toppling rules of the sandpile are symmetric. We observe that the test is not valid in general and give a new algorithm which yields a test correct for all sandpiles; we also obtain necessary and sufficient conditions for the validity of the original test. The results are applied to a family of deterministic one-dimensional sandpile models originally studied by Lee, Liang, and Tzeng.     

非线性热弹耦合Sine-Gordon型系统的整体吸引子

研究受Peierls-Nabarro力作用的非线性热弹耦 合Sine-Gordon型系统的动力行为.利用算子半群理论证明了 在一定的初边界条件下系统存在连续解, 利用算子半群分解技巧构造了渐近紧的不变吸收集, 进而证明了系统存在整体吸引子.     

Thermodynamics and topology of disordered systems: Statistics of the random knot diagrams on finite lattices

The statistical properties of random lattice knots, the topology of which is determined by the algebraic topological Jones-Kauffman invariants, was studied by analytical and numerical methods. The Kauffman polynomial invariant of a random knot diagram was represented by a partition function of the Potts model with a random configuration of ferro-and antiferromagnetic bonds, which allowed the probability distribution of the random dense knots on a flat square lattice over topological classes to be studied. A topological class is characterized by the highest power of the Kauffman polynomial invariant and interpreted as the free energy of a q-component Potts spin system for q→∞. It is shown that the highest power of the Kauffman invariant correlates with the minimum energy of the corresponding Potts spin system. The probability of the lattice knot distribution over topological classes was studied by the method of transfer matrices, depending on the type of local junctions and the size of the flat knot diagram. The results obtained are compared to the probability distribution of the minimum energy of a Potts system with random ferro-and antiferromagnetic bonds.     

Embedded graph invariants in Chern-Simons theory

Chern-Simons gauge theory, since its inception as a topological quantum field theory, has proved to be a rich source of understanding for knot invariants. In this work the theory is used to explore the definition of the expectation value of a network of Wilson lines — an embedded graph invariant. Using a generalization of the variational method, lowest-order results for invariants for graphs of arbitrary valence and general vertex tangent space structure are derived. Gauge invariant operators are introduced. Higher order results are found. The method used here provides a Vassiliev-type definition of graph invariants which depend on both the embedding of the graph and the group structure of the gauge theory. It is found that one need not frame individual vertices. However, without a global projection of the graph there is an ambiguity in the relation of the decomposition of distinct vertices. It is suggested that framing may be seen as arising from this ambiguity — as a way of relating frames at distinct vertices.