Fokker-Planck Equation of a Stochastic System Driven by Spatially-Related Noises

In this paper we study a general stochastic system driven by the spatially-related Gaussian white noises. The corresponding Fokker-Planck equation is calculated; and some typical cases are analyzed. Finally, by the Fokker-Planck equation derived in the paper we study a single bistable kinetic process with spatially-related noise. The results obtained in the paper provide a correct foundation for the.treatment of the stochastic systems driven by spatially-related noises.     

Looking at Fokker-Planck dynamics with a noisy instrument

We consider the class of experiments which can be characterized by a Fokker-PIanck dynamics corresponding to the overdamped motion of a state point in a suitable stochastic potential. We assume that the general form of the potential is known (or can be guessed with reasonable accuracy), but that its parameters are to be determined experimentally by measurements made with a noisy instrument. This possible method for determining the potential parameters, which exploits the system's own internal stochastic motion in order to explore rapidly its available parameter space, is substantially more efficient than traditional methods involving time averages of single point measurements, and yet does not appear to have been previously considered. The method could be important when, for example, the experiment must be completed in a limited time owing either to the expense of the experimental materials or to the temporary stationarity of the preparation, situations which are commonly encountered in experimental biochemistry and biology.     

Limit cycles and detailed balance in Fokker-Planck equations

A class of non-linear Fokker Planck equations with exactly known steady state solution is investigated and its relation with the models considered earlier in the literature is discussed. A characterisation of the absence of detailed balance and possible existence of limit cycles is given. The implications of detailed balance on the existence and the character of limit cycle behaviour are studied. It is shown that detailed balance does not determine the existence or non-existence of limit cycles but rather their character.     

Limit cycles and detailed balance in Fokker-Planck equations

A class of non-linear Fokker Planck equations with exactly known steady state solution is investigated and its relation with the models considered earlier in the literature is discussed. A characterisation of the absence of detailed balance and possible existence of limit cycles is given. The implications of detailed balance on the existence and the character of limit cycle behaviour are studied. It is shown that detailed balance does not determine the existence or non-existence of limit cycles but rather their character.     

A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker-Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker-Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was offered that a Chapman-Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker-Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker-Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker-Planck pdes and Chapman-Kolmogorov equations for stochastic processes with finite memory.     

Potential Symmetries and Associated Conservation Laws to Fokker-Planck and Burgers Equation

Kara and Mahomed have derived an identity, which does not rely on use of a Lagrangian as needed to obtain conservation laws by Noethers theorem. By using the identity and symbolic computation, conservation laws arising from nonlocal symmetries are obtained for Fokker-Planck equation and burgers equation.     

The Fokker-Planck equation in the second-order pitch angle approximation and its exact solution

The diffusive particle propagation and its pitch angle scattering is studied using kinetic equation of the Fokker-Planck form. The case is considered when charged particles preferable propagate along the strong mean magnetic field direction and undergo the pitch angle scattering with respect to it. The paper deals with solution of the equation for particle distribution function in the second-order approximation in the pitch angle. The exact analytical solution is obtained in an integral form. The well-known solution in the first-order pitch angle approximation can be restored performing the small time limit in the result. Unlike the first-order solution the obtained solution in the second approximation rightly shows that the pitch angle diffusion is closely connected with the particle transport along the mean magnetic field. The expression for particle density for the point instantaneous unidirectional source also has been obtained.     

等离子体中Fokker-Planck方程的有限元模拟

利用有限元方法设计了一套相对简单明了的求解Fokker-Planck方程的方案.这个方案不必严格限制计算格点的步长和时间步长,就可以确保分布函数的非负性和粒子数的守恒.通过一维程序模拟,进一步证实了这个方案的可靠性.对于多维问题的分析和一维问题完全一样,所以非常容易将其推广到多维问题.     

Non-Equilibrium Entropy and Irreversibility in Generalized Stochastic Loewner Evolution from an Information-Theoretic Perspective

In this study, we theoretically investigated a generalized stochastic Loewner evolution (SLE) driven by reversible Langevin dynamics in the context of non-equilibrium statistical mechanics. Using the ability of Loewner evolution, which enables encoding of non-equilibrium systems into equilibrium systems, we formulated the encoding mechanism of the SLE by Gibbs entropy-based information-theoretic approaches to discuss its advantages as a means to better describe non-equilibrium systems. After deriving entropy production and flux for the 2D trajectories of the generalized SLE curves, we reformulated the system’s entropic properties in terms of the Kullback–Leibler (KL) divergence. We demonstrate that this operation leads to alternative expressions of the Jarzynski equality and the second law of thermodynamics, which are consistent with the previously suggested theory of information thermodynamics. The irreversibility of the 2D trajectories is similarly discussed by decomposing the entropy into additive and non-additive parts. We numerically verified the non-equilibrium property of our model by simulating the long-time behavior of the entropic measure suggested by our formulation, referred to as the relative Loewner entropy.     

Quasiclassical Describing of the Bloch Oscillation in a Superlattice by Fokker-Planck Equation

A theoretical study of Bloch electron transport in a superlattice miniband driven by an electric field parallel to the growth axis is carried out, by Fokker-Planck equation (FPE) in momentum space with the averaged momentum relaxation time (γ) approximation. Steadystate drift-velocity/field characteristics exhibit the expected maximum followed by negative differential conductivity (NDC), and then followed by drift-velocity oscillation when γ or electric field is large. The oscillation frequency is an increasing function of γ, and when γ → ∞, the limit of the oscillation frequency is the Bloch frequency as expected.     

Stochastic calculus in physics

The relationship of the Ito-Stratonovich stochastic calculus to studies of weakly colored noise is explained. A functional calculus approach is used to obtain an effective Fokker-Planck equation for the weakly colored noise regime. In a smooth limit, this representation produces the Stratonovich version of the Ito-Stratonovich calculus for white noise. It also provides an approach to steady state behavior for strongly colored noise. Numerical simulation algorithms are explored, and a novel suggestion is made for efficient and accurate simulation of white noise equations.     

A classification of Fokker-Planck models and the small and large noise asymptotics

Then-variable Fokker-Planck equation can be written in an equivalent form as a system ofn + 1 first-order equations by introducing as auxiliary variables the components of the drift velocityR _l. The stationary state defines a stationaryR uniquely, which allows an intrinsic classification of the stationary states in terms of the properties ofR, without reference to detailed balance. This representation is very appropriate for the study of questions such as the existence of stationary states and their small and large noise asymptotics, as well as for the construction of models having some specified behavior.R provides also a classification of the dynamics, which corresponds to the hermiticity properties of the associated eigenvalue problem.     

Noise-intensity fluctuation in Langevin model and its higher-order Fokker-Planck equation

In this paper, we investigate a Langevin model subjected to stochastic intensity noise (SIN), which incorporates temporal fluctuations in noise-intensity. We derive a higher-order Fokker-Planck equation (HFPE) of the system, taking into account the effect of SIN by the adiabatic elimination technique. Stationary distributions of the HFPE are calculated by using the perturbation expansion. We investigate the effect of SIN in three cases: (a) parabolic and quartic bistable potentials with additive noise, (b) a quartic potential with multiplicative noise, and (c) a stochastic gene expression model. We find that the existence of noise-intensity fluctuations induces an intriguing phenomenon of a bimodal-to-trimodal transition in probability distributions. These results are validated with Monte Carlo simulations.     

HIRFL–CSRm Betatron振荡随机冷却

对拟在HIRFL CSRm上建造的横向自由振荡随机冷却进行了数值优化计算和设计,得到了最佳带宽、冷却时间、频谱上束流谱密度分布函数随时间的变化,以及在冷却过程中束流横向位移的分布等值,并且对功率限定情况作了讨论,从而为冷却系统的设计、优化、建造和运行提供了理论依据     

Stochastic evolution of cosmological parameters in the early universe

We develop a stochastic formulation of cosmology in the early universe, after considering the scatter in the redshift-apparent magnitude diagram in the early epochs as an observational evidence for the non-deterministic evolution of early universe. We consider the stochastic evolution of density parameter in the early universe after the inflationary phase qualitatively, under the assumption of fluctuating w factor in the equation of state, in the Fokker-Planck formalism. Since the scale factor for the universe depends on the energy density, from the coupled Friedmann equations we calculated the two variable probability distribution function assuming a flat space geometry.     

Projector formalism of generalized Brownian motion theory applied to dissipative and noisy systems

The projector formalism of Zwanzig-Mori type is extended to obtain generalized Fokker-Planck and generalized nonlinear Langevin equations for coarse-grained variables when the underlying microscopic dynamics is dissipative and noisy (stochastic).     

基于Fokker-Planck方程的电介质材料短脉冲激光破坏机制分析

基于Pokker-Planck方程,建立一个描述短脉冲激光作用下电介质材料中导带电子能量分布随时间变化的模型,用数值方法计算电子能量分布与电子数密度随时间的演化过程,根据临界等离子体密度准则得到了不同激光脉冲宽度和波长下电介质材料(以SiO<,2>为例)的破坏阈值.结果发现,尽管激光波长通过3种途径对材料破坏阈值的确定...     

基于三维Fokker-Planck方程的电子回旋波加热与电流驱动模拟

利用反弹平均的三维Fokker-Planck方程,对电子回旋波加热和电流驱动进行数值模拟.考虑超热电子径向扩散对电流驱动的影响,在方程中加入径向扩散输运项,采用九点格式的中心差分对方程进行数值离散得到系数矩阵,采用不完全LU分解对系数矩阵进行预处理,利用双共轭梯度稳定法求解得到分布函数.在不考虑电子径向扩散输运条件下,得到电子回旋波驱动电流密度与功率沉积密度的分布;考虑径向扩散输运的计算结果与BANDIT3D进行比较,驱动电流分布的趋势基本一致.     

Stochastic solution of the linearized Boltzmann equation

For the linearized Boltzmann equation with finite cross section, the solution is represented as an integral over the paths of a Markov jump process. The integral is only shown to converge conditionally, where the limiting process is defined by an increasing sequence of stopping times. The notion of local martingale plays an important role. A number of related kinetic models are also mentioned.Supported by NSF Grant GP 28576.     

储存环内纵向Palmer方式随机冷却过程模拟

基于Fokker–Planck方程,模拟了兰州重离子加速器冷却储存环(HIRFL–CSR)实验环内重离子束的Palmer方式纵向动量冷却过程,在模拟过程中,得到并应用使冷却达到最快的系统最优增益曲线.对模拟结果进行了讨论.