Dynamics of fluctuations in a reactive system of low spatial dimension

We study, using master equation techniques, the time evolution of the average concentration and fluctuations in the two-speciesn-molecule reactionA+(n-1)XnX in one dimension described by a Glauber-type dynamical lattice model for the specific casesn=2 (bimolecular) andn=3 (trimolecular). The evolution is found to be quite different from that described by the Mean-Field equations even for the bimolecular case, where the steady state is meanfield. For the trimolecular process, the values of fluctuation correlations in the nonequilibrium steady state are well predicted by the fixed points of the dynamical equations obtained from the master equation. In addition, three-point fluctuation correlations are found to play an important role in both processes and are accounted for by an extended Bethe-type ansatz. The bimolecular system shows no memory effects of initial conditions, while the trimolecular system is characterized by memory effects in terms of the average concentration, fluctuations as well as the entropy. The spatial decay of fluctuation correlations is found to be short range at the steady state for the trimolecular system.     

Dynamics of the Schlögl models on lattices of low spatial dimension

The steady states and dynamics of the two Schlögl models on one- and two-dimensional lattices are studied using master equation techniques in tandem with simulations. It is found that the classic bistable behavior of model II is modified to monostable behavior at low dimension. An explanation of this modification is proposed in terms of the effective potential that appears in the dynamical equations on considering the significant effect of fluctuation correlations. The behavior can be modeled by replacing the transient average fluctuation correlation by its asymptotic value plus Gaussian white noise and analyzing the resulting effective potential obtained from the Fokker-Planck equation with multiplicative noise. For model I the transcritical bifurcation point is shifted to lower values of the forward ratek ₂ of the second step of the reaction scheme and this shift can also be explained via an effective potential as a function of the average asymptotic fluctuation correlation. Further addition of noise to the asymptotic value is irrelevant for this model since the noise term in the corresponding Fokker-Planck equation turns out to be purely additive.     

The poisson representation. II Two-time correlation functions

Basic formulas for the two-time correlation functions are derived using the Poisson representation method. The formulas for the chemical system in thermodynamic equilibrium are shown to relate directly to the fluctuationdissipation theorems, which may be derived from equilibrium statistical mechanical considerations. For nonequilibrium systems, the formulas are shown to be generalizations of these fluctuation-dissipation theorems, but containing an extra term which arises entirely from the nonequilibrium nature of the system. These formulas are applied to two representative examples of equilibrium reactions (without spatial diffusion) and to a nonequilibrium chemical reaction model (including the process of spatial diffusion) for which the first two terms in a systematic expansion for the two-time correlation functions are calculated. The relation between the Poisson representation method and Glauber-SudarshanP-representation used in quantum optics is discussed.     

Locally Resolving Measurements of PSD,Particle Concentration and Particle Velocity by Means of TFS with Dual Beams

Transmission fluctuation spectrometry with spatial correlation (TFS‐SC) is based on transmitting two beams of radiation through a flowing suspension, whereby the distance of the beam centers is varied. Thus, the spatial correlation of the transmission fluctuations of the two beams is determined as a function of the beam distance. By numerical modeling, the transition functions of the correlation are found as a function of beam distance, beam diameters, particle diameter and beam intersection angle. Experimentally determined spatial correlation spectra can be inverted to obtain the particle concentration and particle size distribution by using the theoretical transition functions for mono‐sized particles. In addition, time correlations can be used to extract information on particle velocity. Some experimental results obtained by TFS‐SC are presented and discussed. This method appears promising for application in the local resolving of measurements of PSD, particle concentration and particle velocity in two‐phase flows, both in the laboratory and in process control.     

有限温度下一维Hubbard模型的化学势泛函理论研究

通过数值方法求解了有限温度下一维均匀Hubbard模型的热力学Bethe-ansatz方程组,得到了在给定温度和相互作用强度情况下,比热c、磁化率χ和压缩比κ随化学势μ的变化图像.基于有限温度下一维均匀Hubbard模型的精确解,利用化学势(μ)-泛函理论研究了一维谐振势下的非均匀Hubbard模型,给出了金属态和Mott绝缘态下不同温度情况时局域粒子密度n_i和局域压缩比_κi随格点的变化情况.     

Numerical integration of the fluctuating hydrodynamic equations

An approach to numerically integrate the Landau-Lifshitz fluctuating hydrodynamic equations is outlined. The method is applied to one-dimensional systems obeying the nonlinear Fourier equation and the full hydrodynamic equations for a dilute gas. Static spatial correlation functions are obtained from computer-generated sample trajectories (time series). They are found to show the emergence of long-range behavior whenever a temperature gradient is applied. The results are in very good agreement with those obtained from solving the correlation equations directly.     

Correlation function of the amplitude and of the intensity fluctuation for a laser model near threshold

In extension of a preceding paper the correlation function of the amplitude and of the intensity fluctuation are calculated in the threshold region. The laser amplitude is treated as a classical random variable obeying a van der Pol equation with a noise term. In order to get correlation functions, the method of distribution functions is employed. The distribution functions are evaluated by the Fokker-Planck equation. The lowest eigensolutions of the Fokker-Planck equation are obtained approximately by a variational method.     

Lyapunov spectra and nonequilibrium ensembles equivalence in 2D fluid mechanics

We perform numerical experiments to study the Lyapunov spectra of dynamical systems associated with the Navier–Stokes (NS) equation in two spatial dimensions truncated over the Fourier basis. Recently new equations, called GNS equations, have been introduced and conjectured to be equivalent to the NS equations at large Reynolds numbers. The Lyapunov spectra of the NS and of the corresponding GNS systems overlap, adding evidence in favor of the conjectured equivalence already studied and partially extended in previous papers. We make use of the Lyapunov spectra to study a fluctuation relation which had been proposed to extend the “fluctuation theorem” to strongly dissipative systems. Preliminary results towards the formulation of a local version of the fluctuation formula are also presented.     

Characteristic correlation between propellers cavitating wake and cavitation noise

The characteristic correlation between the propellers cavitating wake and their cavitation noise has been studied. Based on the viscous multiphase flow theory, the Navier–Stokes equations, a turbulence model and a cavitation model were applied to predict numerically the vapor volume fraction and the pressure in the wake. Compared with experimental results, the numerical predictions of the sheet cavitation and the tip vortex cavitation are in agreement with the corresponding experimental data, which indicates the numerical method in present paper is feasible. The characteristic of the low line spectra of the numerical pressure fluctuation signals and the measured noise signals for uniform inflow and wake flow is analyzed, and the characteristic correlation of the two line spectra is also validated. Therefore, it can be concluded that due to the wake and the radiated noise both modulated by the revolving propeller blades, the features of the low line spectra for noise signals and the pressure signals in the wake are close to each other.     

序参量守恒系统中空间相关的界面生长

用重整化群方法分析了序参量守恒系统在空间关联下的界面生长行为。得到了标度指数x和z作为空间维数d和相关指数ρ的函数。结果表明,序参量守恒条件使生长弛豫时间增长,空间相关使生长表面粗化。并与E.Medina等人和T.Sun等人的结果作了比较。 关键词：     

Gas-Fluid and Fluid-Solid Phase Instability for Restricted Primitive Model

By considering the fluctuation of grand potential Ω around equilibrium with respect to small one-particle density fluctuations δρ_α\vec{r}, the phase instability of restricted primitive model (RPM) of ionic systems is investigated. We use the integral equation theory to calculate the direct correlation functions in the reference hypernetted chainapproximation and obtain the spinodal line of RPM. Our analysis explicitly indicates that the gas-fluid phase instability is induced by k=0 fluctuation mode, while the fluid-solid phase instability is related to k≠0 fluctuation modes. The spinodal line isqualitatively consistent with the result of computer simulations by others.     

A stochastic approach to the kinetic theory of gases

A theory of fluctuations in non-equilibrium diluted gases is presented. The velocity distribution function is treated as a stochastic variable and a master equation for its probability is derived. This evolution equation is based on two processes: binary hard sphere collisions and free flow. A mean-field approximation leads to a non-linear master equation containing explicitly a parameter which represents the spatial correlation length of the fluctuations. An infinite hierarchy of equations for the successive moments is found. If the correlation length is sufficiently short a truncation after the first equation is possible and this leads to the Boltzmann kinetic equation. The associated probability distribution is Poissonian. As to the fluctuation of the macroscopic quantities, an approximation scheme permits to recover the Langevin approach of fluctuating hydrodynamics near equilibrium and its fluctuation-dissipation relations.     

Time-dependent probability density of statistical mechanics

We study fluctuations around nonequilibrium steady states of some model nonlinear chemical systems. A previous result of Nicolis and Prigogine states that the mean square fluctuation computed from a master equation in the space of internal states of the reacting species is identical to that calculated from Einstein's fluctuation formula. Our analysis of fluctuations based on that master equation leads with the assumption of local equilibrium to a result identical to that obtained from a master equation for the total concentration of the reacting species, which is different from the equilibrium (Einstein relation) result. Nicolis and Prigogine approximated one term in their master equation, and a discussion of this approximation is presented. The master equation without this approximation yields at equilibrium the result expected on the basis of Einstein's formula.Work supported in part by the National Science Foundation and Project SQUID, Office of Naval Research.     

Statistical dynamics of conserved densities for the hard-sphere Lorentz gas

The statistical dynamics of a particle scattered by randomly positioned stationary hard spheres are investigated. We examine the somewhat unusual long-wavelength, low-frequency fluctuation spectra resulting from the existence of an infinite set of mutually coupled conserved densities of which the number density and energy density are two members. The analytically soluble infinite-mode correlation functions are compared with the corresponding functions obtained by truncating the set of slow modes at successively increasing orders. Furthermore, we evaluate the long-wavelenght number density autocorrelation function for a fixed speed, υ₀, in terms of a frequency-dependent diffusivity D(ω;υ₀) and obtain the fluctuation spectra of all conserved densities by an additional velocity average with the appropriate canonical weights. The effect of the frequency-dependent diffusivity D(ω;υ₀) on the density fluctuation spectra at frequencies small compared to the mean collision frequency is elucidated.     

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.     

Stratified quantization approach to dissipative quantum systems: Derivation of the Hamiltonian and kinetic equations for reduced density matrices

A technique for describing dissipative quantum systems that utilizes a fundamental Hamiltonian, which is composed of intrinsic operators of the system, is presented. The specific system considered is a capacitor (or free particle) that is coupled to a resistor (or dissipative medium). The microscopic mechanisms that lead to dissipation are represented by the standard Hamiltonian. Now dissipation is really a collective phenomenon of entities that comprise a macroscopic or mesoscopic object. Hence operators that describe the collective features of the dissipative medium are utilized to construct the Hamiltonian that represents the coupled resistor and capacitor. Quantization of the spatial gauge function is introduced. The magnetic energy part of the coupled Hamiltonian is written in terms of that quantized gauge function and the current density of the dissipative medium. A detailed derivation of the kinetic equation that represents the capacitor or free particle is presented. The partial spectral densities and functions related to spectral densities, which enter the kinetic equations as coefficients of commutators, are evaluated. Explicit expressions for the nonMarkoffian contribution in terms of products of spectral densities and related functions are given. The influence of all two-time correlation functions are considered. Also stated is a remainder term that is a product of partial spectral densities and which is due to higher order terms in the correlation density matrix. The Markoffian part of the kinetic equation is compared with the Master equation that is obtained using the standard generator in the axiomatic approach. A detailed derivation of the Master equation that represents the dissipative medium is also presented. The dynamical equation for the resistor depends on the spatial wavevector, and the influence of the free particle on the diagonal elements (in wavevector space) is stated.     

Dynamical properties of colloidal systems: II. Correlation- and response functions

For systems of interacting Brownian particles a Fokker-Planck equation is derived for the probability distribution function of the concentration fluctuations, using assumption of a Gaussian static distribution function. The drift- and the diffusion term are determined by static correlation functions. By this approach specific properties of different systems as e.g., suspensions of charged spherical particles or chain polymers are taken into account. Although the diffusion term is fluctuation dependent the properties of detailed balance and both fluctuation dissipation theorems are satisfied. Using the formalism of Martin, Siggia and Rose, Dyson- and vertex-equations for the two-particle correlation functions are derived. An explicit calculation of these functions, together with related quantities as the dynamic structure factor, and of the diffusion coefficients, is given in a mean-field approximation. The results are compared with several earlier theories, which were developed for specific systems.     

HL—1装置边缘扰动谱的初步分析

一、引言目前许多实验都已证实托卡马克边缘扰动对等离子体的反常输运有重要的影响,由于扰动而产生的反常粒子输运流大约在玻(王母)扩散量级,而且,边缘扰动的大小、湍流频谱和波数谱的宽度对约束时间的影响很大。用探针对边缘等离子体进行湍流扰动分析已在许多托卡马克实验中进行过,用静电探针得到的电子密度扰动n和悬浮电位扰动,可以估计出扰动产生的粒子输运通量。本文主要给出了一套静电探针系统及其数据获取和频谱分析方法,     

Correlation functions for restricted Brownian motion from the ballistic through to the diffusive regimes

We calculate the spectra of the correlation functions for fields with arbitrary spatial dependence as seen by Brownian particles in bounded geometries from knowledge of the spectra of the conditional probability density functions in the infinite domain. Our results show a significant difference for the spectra for 1D, 2D and 3D motions. To our knowledge this is the first demonstration of the influence of dimensionality on the form of the correlation functions. Our results also show the different power dependence on frequency for the ballistic and diffusive cases and the treatment of the crossover is unique.     

X-Ray Tube Voltage Dependence of Wiener Spectra of Screen Structure Mottle Recorded on Front and Back Emulsions

A low-speed screen/film system was used to investigate structure mottle, i.e., the density fluctuation of an x-ray film due to inhomogeneous screen structure. The tube voltage dependence of the Wiener spectra of the structure mottle was separately determined for front and back film emulsions. The Wiener spectral values of the structure mottle of the front emulsion were greater than those of the back emulsion at lower tube voltages. The spectral values of the structure mottle of the front emulsion decreased with tube voltage, while those of the back emulsion increased. We explain these phenomena by the behavior of the Wiener spectra of the relative spatial fluctuation of fluorescence intensity due to following reasons: (1) spatial thickness fluctuation of the screen produces a relative spatial fluctuation of x-ray energy absorbed in the screen, and (2) as the distance between the emulsion of the film and the average position of the x-ray absorption in the screen lengthens, the number of random scattering and absorption of light photons increase.Presented at the International Commission for Optics Topical Meeting, Kyoto, 1994.