A thermochemical instability. II. Inhomogeneous fluctuations

The behavior of coupled stochastic fields near a cusp bifurcation is studied. The results are applied to a thermochemical model and the possibility of observing nonclassical critidal behavior is discussed.     

Response of an elastic Bingham fluid to oscillatory shear

The response of an elastic Bingham fluid to oscillatory strain has been modeled and compared with experiments on an oil-in-water emulsion. The newly developed model includes elastic solid deformation below the yield stress (or strain), and Newtonian flow above the yield stress. In sinusoidal oscillatory deformations at low strain amplitudes the stress response is sinusoidal and in phase with the strain. At large strain amplitudes, above the yield stress, the stress response is non-linear and is out of phase with strain because of the storage and release of elastic recoverable strain. In oscillatory deformation between parallel disks the non-uniform strain in the radial direction causes the location of the yield surface to move in-and-out during each oscillation. The radial location of the yield surface is calculated and the resulting torque on the stationary disk is determined. Torque waveforms are calculated for various strains and frequencies and compared to experiments on a model oil-in-water emulsion. Model parameters are evaluated independently: the elastic modulus of the emulsion is determined from data at low strains, the yield strain is determined from the phase shift between torque and strain, and the Bingham viscosity is determined from the frequency dependence of the torque at high strains. Using these parameters the torque waveforms are predicted quantitatively for all strains and frequencies. In accord with the model predictions the phase shift is found to depend on strain but to be independent of frequency.Notation A plate strain amplitude (parallel plates) - A _R plate strain amplitude at disk edge (parallel disks) - G elastic modulus - m torque (parallel disks) - M normalized torque (parallel disks) = 2m/R ³₀ - N ratio of viscous to elastic stresses (parallel plates) =µ A/ ₀ ratio of viscous to elastic stresses (parallel disks) =µ A _R/₀ - r normalized radial position (parallel disks) =r/R - r radial position (parallel disks) - R disk radius (parallel disks) - t normalized time = t — /2 - t time - _E elastic strain - _P plate strain (displacement of top plate or disk divided by distance between plates or disks) - _PR plate strain at disk edge (parallel disks) - ₀ yield strain - _E normalized elastic strain = _E/₀ - _P normalized plate strain = _P/₀ - _PR normalized plate strain at disk edge (parallel disks) = _PR/₀ - ₀ normalized plate strain amplitude (parallel plates) =A/ ₀ — normalized plate strain amplitude at disk edge (parallel disks) =A _R/₀ - phase shift between _P andT (parallel plates) — phase shift between _PR andM (parallel disks) - µ Bingham viscosity - stress - ₀ yield stress - T normalized stress =/ ₀ - frequency     

High-density percolation: Exact solution on a Bethe lattice

A new percolation problem is posed where the sites on a lattice are randomly occupied but where only those occupied sites with at least a given numberm of occupied neighbors are included in the clusters. This problem, which has applications in magnetic and other systems, is solved exactly on a Bethe lattice. The classical percolation critical exponents=gg=1 are found. The percolation thresholds vary between the ordinary percolation thresholdp _c (m=1)=l/(z – 1) andp _c(m=z) =[l/(z – 1)]¹/(z–1). The cluster size distribution asymptotically decays exponentially withn, for largen, p p _c .Supported in part by National Science Foundation grant DMR78-10813.     

Generalized nonlinear master equations for local fluctuations: Dimensionality expansions and augmented mean field theory

By application of a projection operator technique we derive a formally exact generalization of the nonlinear mean field master equation introduced recently for the study of local fluctuations in a reacting medium. Our starting point is a phenomenological cell master equation. The results of our theory are applicable to the theory of a fluctuating hydrodynamic reacting system. The mean field equation is placed on a firm theoretical foundation by showing it to be the lowest order approximation in an expansion in the dimensionality of the physical space keeping the product of the number of nearest neighbors (an increasing function of dimensionality) and the typical diffusion coefficient constant. A more accurate nonlinear master equation that allows for the correlation and fluctuations in the environment of a given volume element is derived in the form of an augmented mean field equation.Work supported in part by a grant from the National Science Foundation.     

Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

Scaling properties of ℤ k- 1 actions on the circle

We derive universal scaling properties for ^k–1 actions on the circle whose generators have rotation numbers algebraic of degreek. As fork=2 these properties can be explained for arbitraryk in terms of a renormalization group transformation. It has at least one trivial fixed point corresponding to an action whose generators are pure rotations. The spectrum of the linearized transformation in this fixed point is analyzed completely. The fixed point is hyperbolic with a (k–1)-dimensional unstable manifold. In the casek=2 the known results are therefore recovered.     

基于随机矩阵理论的城市人群移动行为分析

移动通信应用为人类移动规律的研究提供了独特的数据来源. 本文通过城市手机用户的分布数据,研究城市移动人群的整体动力学行为. 借助随机矩阵理论的方法,通过比较移动人群数据与随机数据在互相关矩阵谱分布上的差异,发现移动人群数据互相关矩阵的相关系数均值、最大本征值及其对应的本征向量明显偏离于随机互相关矩阵的分布,指出这种差异体现了城市移动人群的整体行为特性,且这种差异在不同时间段也会有所不同. 研究结果体现出相关系数的均值和最大本征值的波动趋势,并指出本征向量成员权重的时空模式与城市移动人群整体行为特征的波动过 关键词： 随机矩阵理论 移动人群 宏观行为     

Decay Rates in Probability Metrics Towards Homogeneous Cooling States for the Inelastic Maxwell Model

We quantify the long-time behavior of solutions to the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures. Existence, uniqueness, and precise estimates of overpopulated high energy tails of the self-similar profile proved in ref. 9 are revisited and derived from this new Liapunov functional. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as t → ∞ to the self-similar solution in this distance, which metrizes the weak convergence of measures. Moreover, we can relate this Fourier distance to the Euclidean Wasserstein distance or Tanaka functional proving also its exponential convergence towards the homogeneous cooling states. The findings are relevant in the understanding of the conjecture formulated by Ernst and Brito in refs. 15, 16, and complement and improve recent studies on the same problem of Bobylev and Cercignani⁽⁹⁾ and Bobylev, Cercignani and one of the authors.⁽¹¹⁾     

Dynamic Behavior of Lambda-Type Three-Level Atoms and Two-Mode Cavity Field

A system comprising of Lambda-type three-level atoms and the two-mode cavity field is considered in this paper. Under the adiabatical approximation and the large detuning condition, the effective Hamiltonian of the system in the interaction picture can be given out. If the two identical three-level atoms pass through the cavity in turn, the entangled state atoms can be generated. When the interaction time is taken to an appropriate value, the maximally entangled states are created. At the same time, the dynamic behaviors of the system are studied in detail.     

Anomalous diffusion in a random velocity field

We construct diffusions in random velocity fields which present anomalous superdiffusive behavior. The mean square displacement can be made to have any power lawt for 1<2. Higher moments and characteristic functions are also investigated.