Correlation function induced by a generalized diffusion equation with the presence of a harmonic potential

An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations, which includes short, intermediate and long-time memory effects described by the waiting time probability density function. Analytical expression for the correlation function is obtained and analyzed, which can be used to describe, for instance, internal motions of proteins. The result shows that the generalized diffusion equation has a broad application and it may be used to describe different kinds of systems.     

三维谐振子Wigner函数的星乘解

Wigner函数作为相空间中的一个准概率分布函数,也是密度矩阵的特殊表示形式,具有十分重要的物理意义。首先介绍了Wigner函数的性质及其计算方法,然后利用星本征方程(Moyal方程)计算了三维谐振子的Wigner函数。最后讨论了在相空间中描述声子与电子(或光子)相互作用的方法,并得到了跃迁几率在相空间中所满足的方程。     

Space–time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model

In this paper, we consider a type of continuous time random walk model where the jump length is correlated with the waiting time. The asymptotic behaviors of the coupled jump probability density function in the Fourier–Laplace domain are discussed. The corresponding fractional diffusion equations are derived from the given asymptotic behaviors. Corresponding to the asymptotic behaviors of the joint probability density function in the Fourier–Laplace space, the asymptotic behaviors of the waiting time probability density and the conditional probability density for jump length are also discussed.     

Anomalous transport in fluid field with random waiting time depending on the preceding jump length

Anomalous (or non-Fickian) transport behaviors of particles have been widely observed in complex porous media. To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields, in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced, and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived. As examples, two generalized advection-dispersion equations for Gaussian distribution and lévy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

Anomalous transport in fluid fleld with random waiting time depending on the preceding jump length

Anomalous(or non-Fickian) transport behaviors of particles have been widely observed in complex porous media.To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields,in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced,and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived.As examples,two generalized advection-dispersion equations for Gaussian distribution and levy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

Asymptotic solution of the Wang-Uhlenbeck recurrence time problem

A Langevin particle is initiated at the origin with positive velocity. Its trajectory is terminated when it returns to the origin. In 1945, Wang and Uhlenbeck posed the problem of finding the joint probability density function (PDF) of the recurrence time and velocity, naming it "the recurrence time problem". We show that the short-time asymptotics of the recurrence PDF is similar to that of the integrated Brownian motion, solved in 1963 by McKean. We recover the long-time t(-3/2) decay of the first arrival PDF of diffusion by solving asymptotically an appropriate variant of McKean's integral equation.     

Self-organized criticality in 1D stochastic traffic flow model

Results of computer simulations of a 1D particle hopping model of traffic flow are presented. The model is characterized by parallel update and fully asymmetric stochastic hopping dynamics which allows unbounded series of jumps to empty neighbour sites on the right. The considered case of open boundary conditions can be used to model a “bottleneck” situation in traffic. Evidence for self-organized criticality is found in two aspects: the presence of long-range spatial correlations manifested in the shape of density profiles, and long-time temporal correlations showing up in the low-frequency behaviour of the spectral density of the total particle number and flow. A plausible conjecture is to interpret the observed qualitative changes in these features, as a function of the injection rate and the hopping probability, in terms of a nonequilibrium phase transition between a low-density phase and a maximal current phase. This conjecture is supported by the phase diagram obtained in mean-field approximation.     

Wave motion of incompressible turbulent fluid

Evolution of small disturbances in a fully developed incompressible turbulent flow is considered on the base of the transport equation for the single-point probability density function (PDF) of velocity fluctuations. It is shown that at high frequencies this equation is similar to the Vlasov equation for charged plasma in a self-consistent electromagnetic field having longitudinal wave solutions for turbulent stresses similar to Langmuir waves. It is found that the longitudinal waves of turbulent stresses have a constant phase velocity and can be damped, neutral, or growing waves, depending on the type of undisturbed probability density function of velocity fluctuations. The obtained result differs from the previously published solutions to this problem using the statistical moments closures according to which the wave disturbances should be neutral or damped. The possibilities of experimental observation of longitudinal waves of turbulent stresses are analyzed.     

两相壁面射流PDF方程有限分析方法及数值模拟

提出求解位置-速度相空间中高维两相流PDF(probability density function)方程的有限分析方法,将位置-速度相空间颗粒PDF方程约化到速度空间,并解析求解,颗粒的位置PDF用轨道方法求解.对壁面射流两相流动进行数值模拟,并与颗粒雷诺应力轨道方法进行比较计算,结果优于颗粒雷诺应力轨道方法.     

First-order probability density function of the laser speckle phase

An expression for the first-order probability density function of the laser speckle phase is analytically derived under the assumption that the speckle field obeys a non-circular, complex Gaussian, random process with a certain correlation between the real and imaginary parts of its complex amplitude. The probability density function of the speckle phase is actually evaluated for various cases and shown three-dimensionally as a function of the standard deviation of random object phase variations. The effect of random object phase variations on the probability density function is also investigated in detail.     

Subordinated diffusion and continuous time random walk asymptotics

Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Le?vy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit.     

Linear kinetic equation: long-time behavior of one-particle distribution function

We construct asymptotic (long-time) solution of the linear Boltzmann equation using the time-dependent perturbation theory generalized to non-Hermitian operators. We prove that for times much larger than the relaxation time τ₀, t ≫τ₀, one-particle distribution function separates into spatio-temporal and velocity dependent parts, and provide the explicit expression for the long-time solution of the linear Boltzmann equation. Our analysis does not assume that relative density gradients $n^{-1}(\partial / \partial \mathaccent"017E{r}) n$n^{-1}(\partial / \partial \mathaccent"017E{r}) n are small. It relates the hydrodynamic form of the one-particle distribution function to spectral properties of operators involved in linear Boltzmann equation.     

Time-dependent generalized polynomial chaos

Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan–Orszag problem with favorable results.     

Study of a generalized Langevin equation with nonlocal dissipative force,harmonic potential and a constant load force

We analyze the motion of a particle governed by a generalized Langevin equation with nonlocal dissipative force, linear external force and a constant load force. We consider the dissipative memory kernel consisting of two terms. One of them is described by the Dirac delta function which represents a local friction, whereas for the second one we consider two types: the exponential and power-law functions which represent nonlocal dissipative forces. For these cases, one can obtain exact results for the relaxation function. Then, we obtain the first moments and variances of the displacement and velocity. The long-time behaviors of these quantities are also investigated.     

湍流两相流动有燃烧颗粒相概率密度函数输运方程理论

由有燃烧的湍流气粒两相流动的瞬态方程和统计力学概率密度函数概念出发,推导了有燃烧颗粒相的质量－动量－能量联合概率密度函数（ＰＤＦ）输运方程,并对方程中条件期望项用梯度模拟概念进行了模拟封闭。封闭后的ＰＤＦ方程可作为建立颗粒拟流体模型方程和封闭二阶矩模型的基础,也可以通过Ｍｏｎｔｅ－Ｃａｒｌｏ 法求解用以直接计算颗粒雷诺应力和湍流动能,以便和二阶 矩模型的结果相对照,改善二阶矩模型。     

On the relation between Schrödinger and von Neumann equations

The relation between the density matrix obeying the von Neumann equation and the wave function obeying the Schrödinger equation is discussed in connection with the superposition principle of quantum states. The definition of the ray-addition law is given, and its relation to the addition law of vectors in the Hilbert space of states and the role of a constant phase factor of the wave function is elucidated. The superposition law of density matrices, Wigner functions, and tomographic probabilities describing quantum states in the probability representation of quantum mechanics is studied. Examples of spin-1/2 and Schrödinger-cat states of the harmonic oscillator are discussed. The connection of the addition law with the entanglement problem is considered.     

DSM-LPDF两相湍流模型及旋流两相流动的模拟

本文由流体－颗粒速度的拉氏联合概率密度函数（ＰＤＦ）输运方程出发,用Ｓｉｍｏｎｉｎ建议的Ｌａｎｇｅｖｉｎ模型封闭颗粒所遇到流体瞬时速度的条件期望项,并用Ｍｏｎｔｅ Ｃａｒｌｏ方法直接求解 ＰＤＦ输运方程,将其和求解流体雷诺应力方程模型的有限差分方法结合,建立了雷诺应力－拉氏ＰＤＦ（ＤＳＭ－ＬＰＤＦ,简称ＤＬ）两相湍流模型．用此模型模拟了旋流数为０．４７的突扩旋流气粒两相流动,并与文献中ＰＤＰＡ实验和用类似于单相流动湍流模型封闭方法的时平均统一二阶矩（ＵＳＭ）模型的预报进行了对比．     

Rigorous solution to a quantum statistical mechanical laser model

A quantum mechanical laser model with relaxation and pumping mechanisms is solved rigorously. A basic equation for the density matrix is derived by the damping theory and is transformed into a corresponding c-number equation for a (quasi-) probability density. This is done with the aid of the quantum phase space method. The probability density is expanded in terms of orthogonal polynomials. The expansion coefficients are solved to give a continued fraction. A complete solution is obtained, namely, time evolution of the probability density is determined as well as that for certain physical quantities. The solution is valid even for strong coupling between photons and atoms: it is free from restriction on system parameters. Detailed studies on dynamics are performed for typical values of the system parameters. This is a prototype of interacting quantum nonequilibrium systems. Relevance to systems other than a laser is briefly mentioned.