Sierpinski 地毯上S4模型的临界特性

通过键移动重整化群的方法, 分析了Sierpinski 地毯上S⁴模型的临界行为, 得到了系统的临界点. 由得到的结果可知, 本系统不仅有一个高斯不动点, 而且还存在着一个Wilson Fisher不动点, 把它与Sierpinski 地毯上的高斯模型相互对比, 发现本系统的临界点变化很大. 这说明这两个系统隶属于两个不同的普适类.     

基于分形特征和双层光子带隙结构的宽阻带低通滤波器

提出一种新颖的基于分形特征和双层光子带隙（PBG）结构的宽阻带低通滤波器. 该滤波器在接地板上刻蚀一阶Sierpinski carpet PBG结构，在顶层微带线与接地板之间增加一层具 有三阶Sierpinski gasket PBG结构的金属层，该金属层经过打通孔与接地板连通. 这种双 层PBG结构的低通滤波器，具有良好的宽阻带特性，且电路尺寸小、结构紧凑. 对比了单层 普通方孔PBG结构的低通滤波器、单层一阶Sierpinski carpet PBG结构的低通滤波器和双层 分形PBG结构低通滤波器的传 关键词： 低通滤波器 双层PBG结构 分形 宽阻带特性     

NaCl稀溶液结晶的分形形态研究

通过光学显微镜观察NaCl稀溶液在玻璃基底上快速蒸发结晶形成了完整的立方晶体、DLA凝聚体和形状复杂的凹凸状结晶3种形貌.对复杂的凹凸状结晶进行分析,认为不完整的NaCl小晶粒构成了一个个微小的生成元,以随机Koch曲线的生长方式自组织地顺次相互连接起来,指出此类结晶符合随机Koch曲线,是远离平衡态下的自组织现象.通过改变观测尺度求维数的方法,得到分形维数是1.15±0.06,与Koch曲线相近.     

自仿射Sierpinski地毯中的粘滞指进

构造了自仿射Sierpinski地毯.在自仿射Sierpinski地毯中,认为喉管半径服从截断瑞利分布,采用逐次超松弛技术,模拟了自仿射Sierpinski地毯中的粘滞指进.计算了粘滞指进分维,结果表明,当粘滞比M→∞时,指进图样与在自仿射Sierpinski地毯中DLA模拟的结果类似：当M=1时,驱替流体在长标度范围内具有紧凑的结构,且具有稳定的位移. 关键词：     

一种推广的混合自旋模型的临界温度曲线

本文提出了一种union jack晶格上推广的混合自旋模型。文中分别用平均场近似、自由费密近似及同普适类等标度变换理论对该模型进行了研究,分别得到了相互之间符合较好的临界温度曲线,并对不同处理方法进行了比较。在简化为特定可解模型时,得到与严格解一致的临界点。 关键词：     

超导交叉膜隧道结的电流-电压滞迴

本文指出对于超导交叉膜隧道结,跨越结的隧道电流除引起结区温升外,同时在结区产生不均匀的自场。这将使结区超导膜进入中间态。用这个物理模型不仅可以解释超导Pb膜的不可逆非平衡电压曲线的滞迴以及实验上出现的阈值电流I_(t3)的存在,而且还可以得出通常的隧道I-V曲线中出现滞迴的奇异现象。 关键词：     

键电导宽分布的等级网络

研究了等级模型分形网络在键电导指数型宽分布g=g₀e^Wx情形下的一些标度行为,分析了无序宽度w对总电阻和分布矩的影响,得到了相应的普适曲线.可以认为这是无序宽度和网络大小之间竞争效应的一个渡越行为.本文的结果和Tong等对Sierpinski蜂巢网络的数值模拟是一致的。     

推广伊辛自旋模型的临界温度曲线

本文运用作者所发展的同普适类等标度变换理论,研究推广的伊辛自旋模型,得到了这种模型的临界温度曲线。 关键词：     

无标度立体Koch网络上随机游走的平均吸收时间

作为一种基本的动力学过程,复杂网络上的随机游走是当前学术界研究的热点问题,其中精确计算带有陷阱的随机游走过程的平均吸收时间(mean trapping time,MTT)是该领域的一个难点.这里的MTT定义为从网络上任意一个节点出发首次到达设定陷阱的平均时间.本文研究了无标度立体Koch网络上带有一个陷阱的随机游走问题,解析计算了陷阱置于网络中度最大的节点这一情形的网络MTT指标.通过重正化群方法,利用网络递归生成的模式,给出了立体Koch网络上MTT的精确解,所得计算结果与数值解一致,并且从所得结果可以看出,立体Koch网络的MTT随着网络节点数N呈线性增长.最后,将所得结果与之前研究的完全图、规则网络、Sierpinski网络和T分形网络进行比较,结果表明Koch网络具有较高的传输效率.     

Fractal空间上自迴避行走的吸收问题

本文运用严格重整化群理论研究了两种Fractal空间上自迴避行走的吸收问题。计算所得的交跨指数φ和相关长度指数ν的严格解结果表明,de Gennes的推测公式φ=1—ν并不成立。相变流图与欧几里德空间中相应问题的流图定性相同。唯一而重要的差别在于Fractal空间中的吸收相变在无穷小吸收作用的硬壁影响下就可发生,而在欧几里德空间只是在一定大小的吸收作用下才会发生。 关键词：     

Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk—Monte Carlo Tests

Simulations of the two-dimensional self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm, and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's stochastic Loewner evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance implies that if we map infinite length walks in the cut-plane into the half plane using the conformal map $z \to \sqrt z$ , then the resulting walks will have the same distribution as the SAW in the half plane. The simulations show excellent agreement between the distributions.     

Transforming Fixed-Length Self-avoiding Walks into Radial SLE<Subscript>8/3</Subscript>

We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE_8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE_8/3. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE_8/3, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values.     

Self-avoiding walk model for proteins

We study a model for the backbone of proteins on a square lattice which consists of the path traced out by a self-avoiding walk (SAW) on the lattice and bridges not belonging to sites on the SAW but connecting nearest neighbor sites of the SAW. We calculated the fractal dimensiond _w for random walk on this model and found thatd _w2.6, in disagreement with a recent suggestion thatd _w should be 2.     

Monte Carlo tests of stochastic Loewner evolution predictions for the 2D self-avoiding walk

The conjecture that the scaling limit of the two-dimensional self-avoiding walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE) with kappa = 8/3 leads to explicit predictions about the SAW. A remarkable feature of these predictions is that they yield not just critical exponents but also probability distributions for certain random variables associated with the self-avoiding walk. We test two of these predictions with Monte Carlo simulations and find excellent agreement, thus providing numerical support to the conjecture that the scaling limit of the SAW is SLE(8/3).     

Study of Monte Carlo methods for generating self-avoiding walks

The concept of fractal dimensionality is used to study different statistical methods for generating self-avoiding walks (SAWs). The reliability of SAWs traced by the enrichment technique and the dynamic Monte Carlo technique is verified. The number of dynamic cycles which represent a single independent SAW ofN ₀ steps is found to be about 0.1N ₀ ³ . We show that the enrichment process for generating SAWs may be presented as a critical phenomenon.     

Kolmogorov dispersion for turbulence in porous media: A conjecture

We will utilise the self-avoiding walk (SAW) mapping of the vortex line conformations in turbulence to get the Kolmogorov scale dependence of energy dispersion from SAW statistics, and the knowledge of the effects of disordered fractal geometries on the SAW statistics. These will give us the Kolmogorov energy dispersion exponent value for turbulence in porous media in terms of the size exponent for polymers in the same. We argue that the exponent value will be somewhat less than for turbulence in porous media.     

Self-Avoiding Walk is Sub-Ballistic

We prove that self-avoiding walk on ${\mathbb{Z}^d}$ is sub-ballistic in any dimension d ≥ 2. That is, writing ${\| u \|}$ for the Euclidean norm of ${u \in \mathbb{Z}^d}$ , and ${\mathsf{P_{SAW}}_n}$ for the uniform measure on self-avoiding walks ${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$ for which γ ₀ = 0, we show that, for each v > 0, there exists ${\varepsilon > 0}$ such that, for each ${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big\{\| \gamma_k \| : 0 \leq k \leq n\big\} \geq vn \big) \leq e^{-\varepsilon n}}$ .     

Whole-Plane Self-avoiding Walks and Radial Schramm-Loewner Evolution: A Numerical Study

We numerically test the correspondence between the scaling limit of self-avoiding walks (SAW) in the plane and Schramm-Loewner evolution (SLE) with κ=8/3. We introduce a discrete-time process approximating SLE in the exterior of a small disc and compare the distribution functions for an internal point in the SAW and a point at a fixed fractal variation on the SLE, finding good agreement. This provides numerical evidence in favor of a conjecture by Lawler, Schramm and Werner. The algorithm turns out to be an efficient way of computing the position of an internal point in the SAW.     

Exact linear renormalization of a one-dimensional system with phase transition

We study by real-space renormalization a class of one-dimensional self-avoiding walks (SAWs) exhibiting a nonzero critical temperature. A linear renormalization transformation is carried out in closed form in a three-parameter subspace of SAW Hamiltonians. We find lines of fixed points along which the degree of localization of the fixed-point interactions varies. The role of the spin rescaling factor in the transformation is explicitly demonstrated.     

Bridge Decomposition of Restriction Measures

Motivated by Kesten’s bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of SLE(8/3), and as a result can be generalized to the wider class of restriction measures. Specifically we show that the restriction hulls with index less than one can be decomposed into a Poisson Point Process of irreducible bridges in a way that is similar to Itô’s excursion decomposition of a Brownian motion according to its zeros.