随机信号双参数脉冲模型的分形特征

为了研究随机信号的统计特性,以幅度、宽度两个相对独立的特征参数为基本概念,建立随机信号的双参数脉冲模型.采用PCI-9812高速数据采集卡,对空气中悬浮颗粒随机性光散射脉冲信号、光电传感器本底噪声随机信号进行计数分布的统计分析.实验结果表明,随机性脉冲信号幅度、宽度子集的计数分布以很高的精度与以自然数为自变量的对数正态分布符合;进一步的计算结果表明,在脉冲信号幅度、宽度定义域内,随机脉冲信号集合的两个特征参数之间存在非线性变换关系,即随机信号的双参数统计分布具有非整数维分形特征. 关键词： 分形 双参数 对数正态分布 随机脉冲信号     

Statistics of a coherent lidar signal in a turbulent atmosphere

Absolute and conditional statistical properties of a pulse coherent Doppler lidar signal in a turbulent atmosphere are studied. Upon coherent receiving of optical fields scattered by a large number of particles, the lidar signal is shown to be a nonstationary non-Gaussian random process with Gaussian conditional statistical characteristics. The appearance of non-Gaussian properties of the signal is caused by correlation of turbulent fluctuations of the wind velocity field within the scattering volume. For the considered signal model, which corresponds to the single scattering approximation and is a sum of a large number of random variables, the central limit theorem is found to be untrue due to the statistical dependence of particles’ positions in a turbulent atmosphere. The results of numerical calculations show that, for a homogeneous and isotropic turbulence, the behavior of the signal statistics significantly depends on the size of the scattering volume and on the state of atmospheric turbulence. A Gaussian statistics is observed at small heights; with an increase in height, the non-Gaussian component becomes considerable in fluctuations of the lidar signal.     

Thermodynamics of rare events

Scaling laws of physics are derived from extreme value distributions. Small jump processes that comprise a compound Poisson distribution generate the asymptotic distributions of stable laws. These extreme value distributions, or their tails, can be expressed in terms of the entropy decrease. As an example, the scaling law for the radius of gyration of a polymer is derived which is comparable to Flory's formula. The entropy is identified by its property of concavity, which is shown to coincide with Boltzmann's probabilistic definition for first passage in a random walk. A more general definition is required for nonintegral dimensions. The relation to mean-field theory of the kinetic Weiss-Ising model is shown and this distribution of the order parameter is governed by an asymptotic distribution for the smallest value rather than a normal distribution. Finally, the logarithm of the sample size is shown to be the yardstick for the decrease in entropy.     

基于超声回波重组相位分析的颗粒粒径测量方法

将超声波作用于沉降的颗粒时,由于颗粒的移动,超声回波会出现相位差异。该文通过对测量杯中某一确定深度处的回波信号进行相位分析和重组,发现重组后信号的频率可以计算出粒径;并分别对两种不同粒径分布的聚甲基丙烯酸甲酯(PMMA)微球悬浮液进行了超声波信号采样重组和去噪的实验,实验结果经小波时频方法分析后,证实了颗粒粒径分布与重组信号频率构成的确存在很高的相关性。     

具有双峰特性的双层超网络模型

随着社会经济的快速发展,社会成员及群体之间的关系呈现出了更复杂、更多元化的特点.超网络作为一种描述复杂多元关系的网络,已在不同领域中得到了广泛的应用.服从泊松度分布的随机网络是研究复杂网络的开创性模型之一,而在现有的超网络研究中,基于ER随机图的超网络模型尚属空白.本文首先在基于超图的超网络结构中引入ER随机图理论,提出了一种ER随机超网络模型,对超网络中的节点超度分布进行了理论分析,并通过计算机仿真了在不同超边连接概率条件下的节点超度分布情况,结果表明节点超度分布服从泊松分布,符合随机网络特征并且与理论推导相一致.进一步,为更准确有效地描述现实生活中的多层、异质关系,本文构建了节点超度分布具有双峰特性,层间采用随机方式连接,层内分别为ER-ER,BA-BA和BA-ER三种不同类型的双层超网络模型,理论分析得到了三种双层超网络节点超度分布的解析表达式,三种双层超网络在仿真实验中的节点超度分布均具有双峰特性.     

Analytical representations of non-Gaussian laws of random walks

An analytical representation of a random process with independent increments in some space (random walks introduced by Pearson) is considered. The law of random walk distribution in space is derived from the general representation of stochastic elementary hops (distribution law of hop probability) using Kadanoff’s concept of the unit increment as one hop. For limited hop laws and laws of hop distributions with all moments there naturally arises Chandrasekhar’s result that describes ordinary physical diffusion. For laws of hop distributions without the second and highest moments there also arise known Lévy walks (flights) sometimes treated as superdiffusion. For the intermediate case, where the distributions of hops have at least the second moment and not all finite moments (these hops are sometimes called truncated Lévy walks), the asymptotic form of the random walk distribution was obtained for the first time. The results obtained are compared with the experimental laws known in econophysics. Satisfactory agreement is observed between the developed theory and the empirical data for insufficiently studied truncated Lévy walks.     

微粒形体特征概率分布函数的光散射频谱分析法

针对微粒(包括生物微粒)光散射测量中微粒形体对测量结果的影响以及形体分布不能实时测量的问题,本文根据光散射测量原理,从微粒群的统计性出发,对散射脉冲的脉宽、脉峰分布及其与微粒大小和形体分布物理动态特征的相互关系进行了系统的分析,得到了微粒形体、微粒大小分布与其散射谱分布的特征关系,提出了等值概率面积确定法则,并据此建立了一种实时由散射谱一体分析获取微粒形体特征分布的方法,得到了较好的实验验证,由此可有效地修正因微粒形体变化而对测量精度的影响,从而为粒度分析仪以及以光散射原理为测量理论的流式细胞仪技术的提升提供了有力的基础。     

Broad distribution effects in sums of lognormal random variables

The lognormal distribution describing, e.g., exponentials of Gaussian random variables is one of the most common statistical distributions in physics. It can exhibit features of broad distributions that imply qualitative departure from the usual statistical scaling associated to narrow distributions. Approximate formulae are derived for the typical sums of lognormal random variables. The validity of these formulae is numerically checked and the physical consequences, e.g., for the current flowing through small tunnel junctions, are pointed out. Received 8 November 2002 / Received in final form 17 March 2003 Published online 7 May 2003     

噪声信号特征量分布函数相似特性的研究

应用高速数据采集卡,记录悬浮颗粒计数光电传感器本底噪声信号序列值,研究噪声信号独立特征量, 如信号幅度极值、上升沿幅值、下降沿幅值、极值时序间隔,以及特征量乘积值等统计分布规律. 结果表明,上述独立特征量均能够与以自然数l为自变量的对数正态分布函数高度符合, 噪声信号特征量统计分布呈现出高度相似特征.基于这种统计相似性,还可推出噪声信号不同特征量对应的统计自变量之间的幂函数变换关系.噪声信号特征量统计分布函数相似性可理解为噪声信号集合统计分形特性的一种表现.     

Betweenness centrality in finite components of complex networks

We use generating function formalism to obtain an exact formula of the betweenness centrality in finite components of random networks with arbitrary degree distributions. The formula is obtained as a function of the degree and the component size, and is confirmed by simulations for Poisson, exponential, and power-law degree distributions. We find that the betweenness centralities for the three distributions are asymptotically power laws with an exponent 1.5 and are invariant to the particular distribution parameters.     

二维随机蜂巢网格熔断动力学过程和熔断面标度性质的数值模拟

石墨烯等材料具有典型的二维蜂巢结构,而随机电阻丝模型则是研究非均匀材料断裂十分有效的统计物理学模型.本文尝试对二维蜂巢结构随机电阻丝网络熔断的动力学过程及熔断面性质进行数值模拟分析,以此来研究二维非均质蜂窝材料熔断的动力学性质和熔断面的动力学标度性质.模拟研究表明,二维随机蜂窝网格的熔断动力学过程和熔断面具有明显的标度性质,得到的熔断面整体和局域粗糙度指数分别为α=0.911±0.005和α_(loc)=0.808 ± 0.003,这两者之间的明显差异表明熔断面具有奇异标度性.通过对熔断面极值高度的分析发现,熔断面高度的极值统计分布能很好地满足Asym2sig型分布,而不是最常见的三种极值统计分布.本文的研究表明,随机电阻丝模型在模拟非均匀材料的电流熔断过程和熔断表面标度性的分析中同样适用和有效.     

近独立子系系统的统计规律(一)——二维系统统计规律的计算机模拟

利用刚球模型,根据力学规律,对二维近独立子系系统粒子的运动进行了计算机模拟;模拟中,在不同时刻对系统中各粒子的能量和速率进行多次测量,并对测量结果进行统计平均,得到了粒子数不同的力学系统中粒子按能量和速率的分布图;根据所得的分布图形曲线,给出了任意数目的粒子系统的统一的分布函数.从而完全证明了少量粒子构成的力学系统,其长时间行为也具有确定的统计规律.     

近独立子系系统的统计规律（一）——二维系统统计规律的计算初席封以

利用刚球模型，根据力学规律，对二维近独立子系系统粒子的运动进行了计算机模拟；模拟中，在不同时刻对系统中各粒子的能量和速率进行多次测量，并对测量结果进行统计平均，得到了粒子数不同的力学系统中粒子按能量和速率的分布图；根据所得的分布图形曲线，给出了任意数目的粒子系统的统一的分布函数．从而完全证明了少量粒子构成的力学系统，其长时间行为也具有确定的统计规律．     

相控阵天线电磁发射特性的统计分析

相控阵天线波束指向高度动态变化,其电磁发射特性呈现出显著的统计规律,分析和测试所需的资源巨大。采用多项式混沌展开（PCE）探究二维平面相控阵天线发射特性的统计特性,根据方向图乘积定理等确定代表相控阵天线电磁发射特性的目标函数,利用PCE建立目标函数的等效代理模型。从理想点源构成的相控阵天线着手,分别考虑了主波束指向服从均匀分布和正态分布两种典型情况,通过计算机仿真模拟得到等效代理模型的概率密度函数和累积分布函数,并使用传统的蒙特卡罗方法结果作为参照来评估PCE方法的有效性和可靠性,最后对小型偶极子相控阵天线的波束指向服从两种典型分布的情况进行讨论。仿真对比结果表明,PCE方法在保证结果准确度的同时可以大大减少采样样本点数目,大幅提升相控阵天线电磁发射特性分析和测试的效率。     

Probability characteristics of the absolute maximum of generalized rayleigh stochastic process

We obtain the limiting laws of distribution of the absolute maximum of the generalized Rayleigh random process. Using the methods of statistical modeling, we show that the asymptotic approximations are in good agreement with the actual distributions over a wide range of parameters of the random process. Voronezh State University, Voronezh, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 12, pp. 1213–1222, September 1999.     

Time-Inhomogeneous Fokker-Planck Equation for Wave Distributions in the Abelian Sandpile Model

The time and size distributions of the waves of topplings in the Abelian sandpile model are expressed as the first arrival at the origin distribution for a scale invariant, time-inhomogeneous Fokker-Plank equation. Assuming a linear conjecture for the time inhomogeneity exponent as a function of a loop-erased random walk (LERW) critical exponent, suggested by numerical results, this approach allows one to estimate the lower critical dimension of the model and the exact value of the critical exponent for LERW in three dimensions. The avalanche size distribution in two dimensions is found to be the difference between two closed power laws.     

非一致随机分布复合材料力学参数预测的统计二阶双尺度方法

通过建立统计的二阶双尺度计算方法,预测了非一致随机分布复合材料结构的力学参数,包括刚度参数和弹性极限强度参数.所谓非一致随机分布复合材料结构,是指在整个结构中夹杂随机分布,但分布特征并不是处处相同,而是逐渐变化的,从而导致材料在宏观上具有随着位置连续变化的力学性能.描述了一致和非一致随机分布复合材料结构的特征及其细观表征方法,并建立了统计的二阶双尺度计算公式,讨论了材料的弹性极限强度准则.最后,针对不同的非一致随机分布复合材料,预测了材料的力学参数并与实验数据进行了对比.结果表明,统计的二阶双尺度方法对于预测非一致随机分布复合材料的力学参数是有效的. 关键词： 统计二阶双尺度方法 非一致随机分布 复合材料结构     

Statistical mechanics of random paths on disordered lattices

The dependence of the universality class on the statistical weight of unrestricted random paths is explicitly shown both for deterministic and statistical fractals such as the incipient infinite percolation cluster. Equally weighted paths (ideal chain) and kinetically generated paths (random walks) belong, in general, to different universality classes. For deterministic fractals exact renormalization group techniques are used. Asymptotic behaviors for the end-to-end distance ranging from power to logarithmic (localization) laws are observed for the ideal chain. In all these cases, random walks in the presence of nonperfect traps are shown to be in the same universality class of the ideal chain. Logarithmic behavior is reflected insingular renormalization group recursions. For the disordered case, numerical transfer matrix techniques are exploited on percolation clusters in two and three dimensions. The two-point correlation function scales with critical exponents not obeying standard scaling relations. The distribution of the number of chains and the number of chains returning to the starting point are found to be well approximated by a log-normal distribution. The logmoment of the number of chains is found to have an essential type of singularity consistent with the log-normal distribution. A non-self-averaging behavior is argued to occur on the basis of the results.     

Exact electronic spectra and inverse localization lengths in one-dimensional random systems : I. Random alloy, liquid metal and liquid alloy

Analytic continuations into the complex energy plane of Dyson-Schmidt type of equations for the calculation of the density of states are constructed for a random alloy model, a liquid metal and for a liquid alloy. In all these models the characteristic function follows from the solution of this equation. Its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions. The equations have been solved exactly for some distributions of the random variables. In the random alloy case the strengths of the delta-potentials have an exponential distribution. They may also have finite, exponentially distributed values with probability 0 p 1 and be infinite with probability q = 1 −p. In the liquid metal the liquid particles are assumed to behave like hard rods. This implies an exponential distribution of the distances between the particles. The common electronic potential may be arbitrary, but is assumed to vanish outside the rods. In the one-dimensional liquid alloy there is, apart from positional randomness of the liquid particles, a distribution of the strengths of the electronic delta-potentials. For Cauchy distributions an argument of Lloyd is extended to obtain the characteristic function from the one in the model with equal strengths. For the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above. For several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed.

New results are found: exponential decay of the density of states near special energies in the random alloy and liquid metal; divergence of the density of states at certain energies with non-classical exponent 1/3 in the random alloy if the average of the potential strengths vanishes; exponentially small broadening of the bound-state levels for low concentrations of the liquid particles; peak in the localization length at the bound-state energies, which becomes exponentially narrow for low concentrations; different exponent in the decay of the inverse localization length at large energies for delta potentials and square-well potentials. Further an expression for the grand potential is given, involving a sum over the characteristic function at certain points and divergence of the zero-point energy is found for Cauchy distributions of the delta potential strengths.     

Determining aerosol particle size distributions using time-resolved laser-induced incandescence

The particle size distribution within an aerosol containing refractory nanoparticles can be inferred using time-resolved laser-induced incandescence (TR-LII). In this procedure, a small volume of aerosol is heated to incandescent temperatures by a short laser pulse, and the incandescence of the aerosol particles is then measured as they return to the ambient gas temperature by conduction heat transfer. Although the cooling rate of an individual particle depends on its volume-to-area ratio, recovering the particle size distribution from the observed temporal decay of the LII signal is complicated by the fact that the LII signal is due to the incandescence of all particle size classes within the sample volume, and because of this, the particle size distribution is related to the time-resolved LII signal through a mathematically ill-posed equation. This paper reviews techniques proposed in the literature for recovering particle size distributions from TR-LII data. The characteristics of this problem are then discussed in detail, with a focus on the effect of ill-posedness on the stability and uniqueness of the recovered particle size distributions. Finally, the performance of each method is evaluated and compared based on the results of a perturbation analysis. PACS  44.05.+e; 47.70.Pq; 78.70.-g; 65.80.+n; 78.20.Ci