Exact probability distribution for soluble model with quadratic noise

A one-dimensional evolution equation transformable into a linear one coupled to a quadratic Smoluchowski (an Ornstein-Uhlenbeck) noise is considered. A one-dimensional probability distribution is obtained by way of a characteristic function which is expressed by functionals of the Smoluchowski process. It is shown that in the frame of the presented approach the probability density can be found only for a particular value of the damping constant in the linear-type relaxation equation. It is also shown that in a special case the white noise limit may be performed.Supported in part by the Polish Academy of Sciences under Contract No. MR 1-9.     

Steady-State Analysis of a Two-Mode Laser System with Cross-Correlation Between Real and Imaginary Parts of Quantum Noise

An inhomogeneously broadened two-mode laser system with cross-correlations between the real and imag- inary parts of quantum noise is considered. The Fokker-Planek equation of the system is derived by the phase-locking method. The steady-state probability distribution, the mean light intensity, the normalization autocorrelation function, and cross correlation function are calculated. The results indicate that： （i） The cross-correlation between the real and imaginary parts of quantum noise can cause the stationary probability distribution from one peak structure to two extrema structure when the laser system is operated above threshold; （ii） The cross-correlation between the real and imaginary parts of quantum noise enhance the light intensity fluctuation and decrease the laser output when the laser system is operated below or near threshold; （iii） The effect of the cross-correlation between the real and imaginary parts of quantum noise is very weak on the stationary properties when the laser system is operated far above threshold.     

Verhulst model with Lévy white noise excitation

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise; (ii) noise with a probability density of increments expressed in terms of Gamma function; and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.     

随机系统的概率密度函数形状调节

针对受高斯白噪声激励的非线性随机系统,提出了使状态响应的概率密度函数形状跟踪期望形状的调节方法.首先,确立了非线性随机系统的多项式反馈机制,同时对系统中的非线性部分进行多项式展开;然后,以Fokker-Planck-Kolmogorov方程为工具,导出了与控制增益相关的各阶矩递推方程,并根据跟踪问题的要求,构造了矩逼近优化问题,用梯度搜索法求解该优化问题,获得了调节函数;再依据特征函数与概率密度函数构成Fourier对的关系,对状态响应的概率密度函数进行重构;最后,通过两个例子仿真,验证了本文方法的有效性.     

Asymptotic laws of superdiffusion

The dynamic and statistical characteristics of diffusion trajectories are studied by introducing the notion “fractional drift.” An equation for the random walk probability density for cases of subdiffusion and superdiffusion is derived. A solution to this equation is constructed on the basis of the Mittag-Leffler function properties.     

Probability distribution function for systems driven by superheavy-tailed noise

We develop a general approach for studying the cumulative probability distribution function of localized objects (particles) whose dynamics is governed by the first-order Langevin equation driven by superheavy-tailed noise. Solving the corresponding Fokker-Planck equation, we show that due to this noise the distribution function can be divided into two different parts describing the surviving and absorbing states of particles. These states and the role of superheavy-tailed noise are studied in detail using the theory of slowly varying functions.     

Photon-Number Distributions of Lasers Generated via Pump-Noise Suppression

The photon-number distribution of lasers generated via pump-noise suppression is derived in the photon-number representation. The derivation shows that the negative probability emerging in the calculation of the photon-number distribution in the photon-number representation is not caused by the representation itself but by the truncation of the master equation. Here without truncating the master equation we derive the characteristic function and the distribution function of the photon-number distribution of the laser system with regular pump and with spontaneous atomic decays being eliminated. It is convenient to use the characteristic function to calculate any order of the moment of the photon-number distribution. Such a method can be used to some more general cases, in which the laser system has imperfectly regular pump and the spontaneous atomic decays are also considered.     

Noise signal propagation in soft biological tissues

Mathematical models are formulated that discribe linear and nonlinear wave propagation in biological tissues. The basis of the method is evolutionary integro-differential equations with a kernel that takes into account the specific properties of tissue. An equation is obtained for the correlation function of acoustic noise in a medium with memory. The procedure for calculating the correlation function by the given type of kernel and noise spectrum at the entrance to the medium is described. It is shown that in different tissue, there is a difference in the laws of decrease in full intensity of a wideband signal with distance. It is demonstrated that the nonlinear equation in the limiting cases of ??short-?? and ??long-term?? memory reduces to equations that have been well studied in statistical nonlinear acoustics.     

Exact solution of a Langevin equation with nonlinear periodic noise

An exact stochastic average of a Langevin equation with a multiplicative nonlinear periodic noise is performed. The noise is described by an arbitrary periodic function of the diffusion Wiener-Lévy stochastic process. The solution of this stochastic equation is given by periodic solutions of the Hill equation.     

The kinetic equation and the equation of momentum transfer for a plasma

An attempt is made to derive a simple form of the collision integral of the kinetic equation for a plasma, by using Rostoker’s equation which expresses the pair correlation function in terms of the distribution functions of the particles, and the conditional probability of one particle shielding the other. The conditional probability function is assumed to be given by its equilibrium value. By taking first order velocity-moment of the resulting kinetic equation, the equation of momentum transfer has been obtained.     

The crossover from classical to quantum regime in the problem of the decay of the metastable state

The probability of the decay of the metastable state has been found as a function of viscosity and temperature. At low temperatures, a classical overbarrier transition changes to the quantum tunneling. At low viscosity in classical and transitive semiquantum region of temperatures a depopulation of the distribution function is significant. The distribution function is shown to satisfy the integral kinetic equation, the kernel of which equals the transition probability. The probability of transitions, induced by the red noise, with the frequency comparable to the transition frequency, is found.     

Stationary Response of Lotka-Volterra System with Real Noises

A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original system is firstly transformed to a pair of It o stochastic differential equations. The Ito formula is then carried out to obtain the It o stochastic differential equation for the period orbit function. The orbit function is considered as slowly varying process under reasonable assumptions. By applying the stochastic averaging method to the orbit function in one period, the averaged Ito stochastic differential equation of the motion orbit and the corresponding Fokker-Planck equation are derived. The probability density functions of the two species are thus formulated. Finally, a classical real noise model is given as an example to show the proposed approximate method. The accuracy of the proposed procedure is verified by Monte Carlo simulation.     

Stationary states in bistable system driven by Lévy noise

We study the properties of the probability density function (PDF) of a bistable system driven by heavy tailed white symmetric Lévy noise. The shape of the stationary PDF is found analytically for the particular case of the Lévy index α = 1 (Cauchy noise). For an arbitrary Lévy index we employ numerical methods based on the solution of the stochastic Langevin equation and space fractional kinetic equation. In contrast to the bistable system driven by Gaussian noise, in the Lévy case, the positions of maxima of the stationary PDF do not coincide with the positions of minima of the bistable potential. We provide a detailed study of the distance between the maxima and the minima as a function of the depth of the potential and the Lévy noise parameters.     

基于伴随方程法的材料热传导系数反演方法

建立利用材料内部温度场的测量结果反演材料热传导系数随时间和空间位置变化函数的伴随方程法,参考迭代正则化的思想在优化过程中给目标函数设置停止准则.对典型算例计算表明,方法在测量噪声较小情况下能得出较为合理的反演结果.但在有测量噪声的情况下,反演结果与真值在边界x=0和L处存在着一定偏差.当测量噪声较大时,反演结果与真值的偏差较为明显,且初值选取会对反演结果有相当大的影响.     

The study on a stochastic system with non-Gaussian noise and Gaussian colored noise

In this paper, a stochastic system with correlation between non-Gaussian noise and Gaussian colored noise is investigated. We carry out the functional methods to derive the approximate Fokker-Planck equation, and the expressions of stationary probability density function and mean first-passage time are presented. Also we explore the effects of correlation between non-Gaussian and Gaussian noise for the mean first-passage time.     

Probability Density and Joint Probability Density of SDE with General Nonlinear Gaussian Noise

We consider a stochastic differential equation with a general nonlinearity in Gaussian noise; With both D (fluctuation intensity) and γ (correlation time) small quantities with D/γ < 1/10, approximate equations for the probability density p(q, t) and the joint probability density p(q, t, q^t, t^p) are derived. As applications of our general equations, quadratic noise, exponential noise and triangle function noise are studied.     

Relaxation problem with a quadratic noise: Analysis

The properties of a linear differential equation with an additive quadratic noise are analyzed. The graphs of the probability distribution of the process are presented for various values of the noise strength and the damping constant. The time evolution of the distribution is also shown. An infinitesimal generator of the evolution operator of the process is constructed. A diffusion-type approximation is considered and a comparison of the exact solution with the approximate solution is carried out.This paper is dedicated to the memory of Prof. A. Pawlikowski.     

An exactly soluble relaxation problem

The problem of a particle moving in a two-valued random potential occurred in a recent paper by Pomeau. The exact time-dependent solution is here obtained for a quadratic potential by two different methods. The first method treats the problem as a stochastic differential equation and leads to the characteristic function of the probability distribution of the particle coordinate. In the second method the equation for the joint probability density of particle and potential is solved, which leads to the temporal Laplace transform of the distribution. The spectral properties of the evolution operator are examined.     

Analytical theory of DC SQUID with a resistively shunted inductance driven by thermal noises

An analytical expression for the stationary probability distribution of the DC superconducting quantum interference device (SQUID) with a resistively shunted inductance driven by thermal noise is derived from the two-dimensional Fokker—Planck equation. The effects on the SQUID characteristics subject to a large thermal fluctuation with a noise parameter Γ>0.20 are discussed by taking into account the thermal noise in the accuracy of numerical simulation. This theory is valid for a reduced inductance β≤1. The analytical formulae for the SQUID characteristics, e.g. the circulating current, the average voltage and the voltage modulation, are obtained and discussed. The theory shows that the voltage modulation increases with the shunted inductance more efficiently for a large inductance parameter β and small fluctuation parameter Γ.     

Statistics of defect-mediated turbulence influenced by noise

The influence of white noise on defect-mediated turbulence which is modeled by the complex Ginzburg-Landau equation is investigated. We show that the dynamics of defects in the noise-driven spatiotemporal chaos can be described by a simple statistical model. The noise enhances significantly the ability of the turbulent background to advocate new defects with a constant rate, and at the same time it increases the vanishing of defects in the system by introducing an additional annihilation rate that is proportional to the number of defects. A universal probability distribution function is derived for the number of defect pairs.