The projection approach to the Fokker-Planck equation. I. Colored Gaussian noise

It is shown that the Fokker-Planck operator can be derived via a projection-perturbation approach, using the repartition of a more detailed operator into a perturbation ₁ and an unperturbed part ₀. The standard Fokker-Planck structure is recovered at the second order in ₁, whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure. To get rid of these higher order terms, a key approximation, local linearization (LL), is made. In general, to evaluate at the second order in ₁ the exact expression of the diffusion coefficient which simulates the influence of a Gaussian noise with a finite correlation time, a resummation up to infinite order in must be carried out, leading to what other authors call the best Fokker-Planck approximation (BFPA). It is shown that, due to the role of terms of higher order in ₁, the BFPA leads to predictions on the equilibrium distributions that are reliable only up to the first order in t. The LL, on the contrary, in addition to making the influence of terms of higher order in ₁ vanish, results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in t provided by the Fox theory. The corresponding diffusion coefficient in turn is shown to lead in the limiting case to exact results for the steady-state distributions. Therefore, over the whole range 0 the LL turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order in ₁ vanish. In the short- region the LL leads to results virtually coincident with those of the BFPA. In the large- region the LL is a more accurate approximation than the BFPA itself. These theoretical arguments are supported by the results of both analog and digital simulation.     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator excited by colored noise

In Rodríguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity of the forcing function. In Weinstein and Benaroya, the effect of the order of expansion is investigated by carrying the expansion to a higher order. The effect of varying the system parameters is also investigated. All results are verified by comparison to Monte Carlo experiments. In this paper, the van Kampen expansion is modified and applied to the case of a Duffing oscillator excited by colored noise. The effect of the correlation time is investigated. Again the results are compared to those of Monte Carlo experiments. It is found that the expansion compares closely with those of the Monte Carlo experiments as the correlation time _c is varied from 0.001 to 10 sec. Examination of the results reveals that the colored noise can be categorized in one of four ways: (1) for the noise can be considered as white for all intents and purposes, (2) for the noise can be considered white for some purposes, (3) for the correlated nature of the noise must be considered in an analysis, and (4) for the noise can be considered as deterministic.     

Multiplicative non-Gaussian noise and additive Gaussian white noise induced transition in a piecewise nonlinear model

We study the transition problems in a piecewise nonlinear model induced by correlated multiplicative non-Gaussian noise and additive Gaussian white noise. Firstly, applying the path integral approach, the unified colored noise approximation, the analytical expression of the steady-state probability density function (SPD) is derived. Then the change regulation of the SPD is analyzed with the change of the strength and relevance of multiplicative noise and additive noise. From numerical computations we obtain some new nonlinear phenomena: the transition can be induced by the cross-correlation strength between noises, the non-Gaussian noise intensity and the Gaussian noise intensity as well as the non-Gaussian noise deviation parameter. This indicates that the effect of the non-Gaussian noise intensity on SPD is the same as that of the Gaussian noise intensity. Moreover, we also find the correlation time of the non-Gaussian noise can not induce the transition.     

Solvable models of the Fokker-Planck equation: An approach based on the Gel'fand-Levitan method

From a given solvable Fokker-Planck equation one can construct a number of other solvable models for diffusion in a stable or bistable potential fields using the Gel'fand-Levitan method of the inverse scattering theory. The simplest way of achieving this is to change the lowest eigenvalue and/or the normalization of the lowest eigenfunction of the ordinary differential equation obtained by separating the time-dependent part. For these cases it is shown that the new probability distribution is expressible in terms of integrals involving the original probability distribution and the Gel'fand-Levitan kernel. The possibility of changing the lowest eigenvalue enables one to find bistable potential fields which would correspond to a well-defined long time relaxation rate for the probability current.     

周期场时间导数Ornstein-Uhlenbeck噪声Fokker-Planck方程的小参数展开求解

通过引入变量,周期场中内部时间导数Ornstein-Uhlenbeck噪声驱动的布朗运动可用高维Fokker-Planck方程来描述. 上述系统不能直接应用通常的小参数展开和势谷中心展开近似求解. 用一种变通的小参数展开方法近似求解了系统的Fokker-Planck方程,结果适用于小势垒高度、中等关联时间和较大的相空间区域,近似解析解可获得系统的改进. 关键词： Fokker-Planck方程 周期势 时间导数Ornstein-Uhlenbeck噪声 小参数展开     

From the Perron-Frobenius equation to the Fokker-Planck equation

We show that for certain classes of deterministic dynamical systems the Perron-Frobenius equation reduces to the Fokker-Planck equation in an appropriate scaling limit. By perturbative expansion in a small time scale parameter, we also derive the equations that are obeyed by the first- and second-order correction terms to the Fokker-Planck limit case. In general, these equations describe non-Gaussian corrections to a Langevin dynamics due to an underlying deterministic chaotic dynamics. For double-symmetric maps, the first-order correction term turns out to satisfy a kind of inhomogeneous Fokker-Planck equation with a source term. For a special example, we are able solve the first- and second-order equations explicitly.     

Brownian motion of nonlinear oscillators: The projection operator method and its application to a double-well-potential oscillator

A projection operator method is presented, which provides the most efficient way for calculating the stationary behavior of nonlinear Brownian motion. A continued-fraction expansion of the Fourier-Laplace transform of the displacement correlation function or the spectral density is used. This method utilizes a successive optimization procedure on the nonlinear terms and includes the method of statistical linearization as the lowest order approximation. A systematic way to calculate the continued fraction numerically up to sufficient order for convergence is developed, which enables us to obtain the spectral density of a system previously uncomputable.Numerical computations of the spectral density of a nonlinear oscillator with a double-well potential are presented and compared with the results obtained by statistical linearization.This work was supported in part by the National Science Foundation under Grant CHE 75-20624.     

External noise and its interaction with spatial degrees of freedom in nonlinear dissipative systems

A brief introduction to the field is given together with an overview of the lectures given at the workshop on External Noise and its Interaction with Spatial Degrees of Freedom in Nonlinear Dissipative Systems organized by the Center for Nonlinear Studies at Los Alamos, March 28–31, 1988. It is hoped that the publication of papers presented at the workshop in a single issue of theJournal of Statistical Physics will help draw attention to the recent developments in this rapidly area of nonequilibrium phenomena.     

Asymptotic properties of coupled nonlinear langevin equations in the limit of weak noise. II: Transition to a limit cycle

We apply the singular perturbation technique, developed in the companion paper, to the study of the fluctuations at the onset of a limit cycle, both for the cases of a soft and a hard transition. The technique and results are illustrated on the Poincaré model (soft transition) and on the Van der Pol oscillator (hard transition).     

带源项的变系数非线性反应扩散方程的精确解

本文利用广义条件对称方法对带源项的变系数非线性反应扩散方程 f(x)u_t=(g(x)D(u)u_x)_x+h(x)P(u)u_x+q(x)Q(u)进行研究. 当扩散项D(u)取u^m (m≠-1,0,1)和e^u两种重要情形时, 对该方程进行对称约化,得到了具有广义泛函分离变量形式的精确解. 这些精确解包含了该方程对应常系数情况下的解. 关键词： 广义条件对称 精确解 非线性反应扩散方程     

The quasiclassical Langevin equation and its application to the decay of a metastable state and to quantum fluctuations

We report on investigations on the consequences of the quasiclassical Langevin equation. This Langevin equation is an equation of motion of the classical type where, however, the stochastic Langevin force is correlated according to the quantum form of the dissipation-fluctuation theorem such that ultimately its power spectrum increases linearly with frequency. Most extensively, we have studied the decay of a metastable state driven by a stochastic force. For a particular type of potential well (piecewise parabolic), we have derived explicit expressions for the decay rate for an arbitrary power spectrum of the stochastic force. We have found that the quasiclassical Langevin equation leads to decay rates which are physically meaningful only within a very restricted range. We have also studied the influence of quantum fluctuations on a predominantly deterministic motion and we have found that there the predictions of the quasiclassical Langevin equations are correct.     

物理力学公式、定律在"和平号"空间站坠毁过程中的运用

利用物理力学定律对“和平号”空间站坠毁过程作了比较详细的计算和描述，旨在使读者在航天器有关知识有更多了解。     

Exact and explicit solutions to the discrete nonlinear Schrödinger equation with a saturable nonlinearity

We analyze the discrete nonlinear Schrödinger equation with a saturable nonlinearity through the (G^′/G)-expansion method to present some improved results. Three types of analytic solutions with arbitrary parameters are constructed; hyperbolic, trigonometric, and rational which have not been explicitly computed before.