What Can One Learn About Self-Organized Criticality from Dynamical Systems Theory?

We develop a dynamical system approach for the Zhang model of self-organized criticality, for which the dynamics can be described either in terms of iterated function systems or as a piecewise hyperbolic dynamical system of skew-product type. In this setting we describe the SOC attractor, and discuss its fractal structure. We show how the Lyapunov exponents, the Haussdorf dimensions, and the system size are related to the probability distribution of the avalanche size via the Ledrappier–Young formula.     

Dynamically driven renormalization group

We present a detailed discussion of a novel dynamical renormalization group scheme: the dynamically driven renormalization group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady state. The method is based on a real-space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open nonlinear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scalling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed.     

非线性动力学方法在气候突变检测中的应用

快速、准确的检测气候突变, 对于我们认识气候系统的变化和对未来气候系统演变趋势的预测有着重要的现实意义和社会经济价值. 本文主要回顾了近年来非线性突变检测技术的主要研究进展及其在实际观测资料中的应用, 其中包括基于气候系统长程相关性的检测方法, 如滑动去趋势波动分析方法、滑动移除去趋势波动分析方法、滑动移除重标极差方法和指纹法等; 以及基于时间序列复杂性的检测方法, 如近似熵方法, Fisher信息和小波Fisher信息等. 此外, 本文还指出发展针对空间场的突变检测技术是未来一个可能的发展方向. 由于空间场所包含的气候系统的演变信息远高于单点时间序列, 空间场的突变检测技术将会使得对气候突变的检测时间大大缩短, 从而使得人们有足够的时间去采取行动, 以便为适应气候突变所带来的新挑战做好准备.     

Spatial correlations in permeability distributions due to extreme dynamics restructuring of unconsolidated sandstone

We propose a model for porous sandstone formation from unconsolidated sand based on a series of restructuring events where the local pressure difference due to flow in the sand is the largest. We investigate the local and global permeability distributions after steady state has been reached. Whereas we find no spatial correlations in the local permeability distribution, the distribution of inverse permeability shows spatial correlations consistent with a fractional Brownian noise characterized by a Hurst exponent of 0.88(9). The global permeability of the system shows time fluctuations as restructuring proceeds consistent with self-affinity characterized by a Hurst exponent of 0.25(3), crossing over to white noise at larger time scales.     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

基于位错动力学方法的动态塑性变形研究

金属材料的动态塑性变形行为是一个多尺度的瞬变动力学过程,是物理学、力学以及材料科学等学科的交汇点,相关研究对工程应用具有重要的指导意义。动态载荷作用下,微观层面单个缺陷行为与介观层面缺陷群的集体演化行为交织在一起,导致金属材料呈现复杂的宏观力学现象。已有研究表明,金属材料的动态塑性变形与准静态变形存在显著差异,并且受到诸多内部及外部因素的影响。近几十年来,人们发展了位错动力学方法研究金属材料的动态塑性变形。但是由于动态变形问题的复杂性,对动态塑性变形的认识仍然存在不足。本文从计算方法和变形理论两个方面对该领域国内外发展历史及重要进展进行了回顾,以期为动态塑性变形研究提供有益的参考。

     

Convergence of dynamical zeta functions

I study poles and zeros of zeta functions in one-dimensional maps. Numerical and analytical arguments are given to show that the first pole of one such zeta function is given by the first zero ofanother zeta function: this describes convergence of the calculations of the first zero, which is generally the physically interesting quantity. Some remarks on how these results should generalize to zeta functions of dynamical systems with pruned symbolic dynamics and in higher dimensions follow.     

A dynamical system approach to SOC models of Zhang's type

We discuss Zhang's model of SOC in the framework of hyperbolic dynamical systems with singularities. The fractal structure of the invariant energy distribution, correlation decay-like phenomena, and symbolic coding are discussed.     

Identification of intrinsic gapped behavior in spin-1/2 ladder with staggered dimerization

The zero- and low-temperature behaviors of spin-1/2 two-leg ladder with staggered dimerization are investigated by the Green’s function theory. At zero temperature, the ground state phase diagram is explored, wherein the leg-dimer and rung-singlet phases are revealed, which reflect two different intrinsic gapped behaviors. The former is attributed to the bond alternation along the legs, while the latter is due to the strong rung coupling. It is found that the quantum phase transition from one to another is of the first order, which can be clearly signaled by the rung entanglement entropy. At finite temperatures, the temperature dependence of thermodynamic quantities such as the magnetic susceptibility, specific heat, thermal Drude weight and rung entanglement entropy are calculated to characterize the corresponding quantum phases. It is shown that the magnetic behaviors clearly manifest a typical antiferromagnetism at low temperature, which is in accordance with the experimental results. It is also found that the intrinsic gapped low-lying excitations are responsible for the observed thermodynamic behaviors.     

Divergence of the bulk resistance at criticality in disordered media

Consider the electrical resistancer _n (p) of a hypercubic bond lattice [O,n] ^d inZ ^d , where the bonds have resistance 1 with probabilityp or with probability 1-p. Letp _n (p)=n ^2-d r _n (p) andp(p)=lim_np_n(p). It is well known thatp(p)< ifp>p _c andp(p)= ifp<p _c , wherep _c is the percolation threshold. Here we show thatp(p _c )=, and .     

Chaotic and self-organized critical behavior of a generalized slider-block model

The dynamical behavior of two-dimensional arrays of slider blocks is considered. The blocks are pulled across a frictional surface by a constant-velocity driver; the blocks are connected to the driver and to each other by springs. Only one block is allowed to slip at a time and its displacement can be obtained analytically; the system is deterministic with no stochastic inputs. Studies of a pair of slider blocks show that they exibit periodic, limit-cycle, or choatic behavior depending upon parameter values and initial conditions. Studies of large, two-dimensional arrays of blocks show self-organized criticality. Positive Lyapunov exponents are found that depend upon the stiffness and size of the array.