Fully Implicit Iterative Solving Method for the Fokker-Planck Equation in Tokamak Plasmas

A three dimensional bounce-averaged Fokker-Planck （FP） numerical code has been newly developed based on fully implicit iterative solving method, and relativistic effect is also included in the code. The code has been tested against various benchmark cases： Ohmic con ductivity in the presence of weak Ohmic electric field, runaway losses of electrons in the presence of strong Ohmic electric field, lower hybrid current drive and electron cyclotron current drive via two- or three-dimensional simulation. All the test cases run fast and correctly during calculations. As a result, the code provides a set of powerful tools for studying radio frequency wave heating and current drive in tokamak plasmas.

On New Invariant Solutions of Generalized Fokker-Planck Equation

The generalized one-dimensional Fokker-Planck equation is analyzed via potential symmetry method and the invariant solutions under potential symmetries are obtained. Among those solutions, some are new and first reported.

Immersed Interface Method for Fokker-Planck Equation with Discontinuous Drift

The Immersed Interface Method (IIM) is derived to solve the corresponding Fokker-Planck equation of Brownian motion with pure dry friction, which is one of the simplest models of piecewise-smooth stochastic systems. The IIM is capable of treating a discontinuity in the drift of Fokker-Planck equation and it is readily extended to the dry and viscous friction model. Analytic results of the considered model are used to confirm the effectiveness and design accuracy of the scheme.

Some dynamical property of the Tsallis distribution from a Fokker-Planck equation

This paper studies the possible dynamical property of the Tsallis distribution from a Fokker--Planck equation. For the Langevin dynamical system with an {arbitrary} potential function, Markovian friction and Gaussian white noise, it shows that the current form of Tsallis distribution cannot describe any nonequilibrium dynamics of the system, and it only stands for a simple isothermal situation of the system governed by a potential field. So the form of Tsallis distribution and many existing applications using the Tsallis distribution need to be reconsidered.

STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS

In this article, we establish the exponential time decay of smooth solutions around a global Maxwellian to the non-linear Vlasov–Poisson–Fokker–Planck equations in the whole space by uniform-in-time energy estimates. The non-linear coupling of macroscopic part and Fokker–Planck operator in the model brings new difficulties for the energy estimates, which is resolved by adding tailored weighted-in-v energy estimates suitable for the Fokker–Planck operator.

From Chemical Langevin Equations to Fokker-Planck Equation: Application of Hodge Decomposition and Klein-Kramers Equation

没有详细平衡的随机的系统在各种各样的化学反应系统是普通的，例如新陈代谢的网络系统。在这些系统的研究，潜在的风景的概念是有用的。然而，是潜在的功能的存在的足够、必要的条件的，仍然是一个开的问题。在微分表格理论的使用农夫分解定理，我们集中于一般化学 Langevin 方程，它反映复杂化学反应系统。我们为潜力的存在的条件系统美化的分析。由印射随机的微分方程到一个 Hamiltonian 机械系统，我们获得化学反应系统的 FokkerPlanck 方程。获得的 FokkerPlanck 方程能在复杂化学反应系统的另外的稳定的性质的进一步的研究被使用，例如他们的稳定的州的熵。

Nanofluids Transport Model Based on Fokker-Planck Equation and the Convection Heat Transfer Calculation

In current research about nanofluid convection heat transfer, random motion of nanoparticles in the liquid distribution problem mostly was not considered. In order to study on the distribution of nanoparticles in liquid, nanofluid transport model in pipe is established by using the continuity equation, momentum equation and Fokker-Planck equation. The velocity distribution and the nanoparticles distribution in liquid are obtained by numerical calculation, and the effect of particle size and particle volume fraction on convection heat transfer coefficient of nanofluids is analyzed. The result shows that in high volume fraction （ 0 _-- 0.8% ）, the velocity distribution of nanofluids characterizes as a ＂cork-shaped＂ structure, which is significantly different from viscous fluid with a parabolic distribution. The convection heat transfer coefficient increases while the particle size of nanoparticle in nanofluids decreases. And the convection heat transfer coefficient of nanofluids is in good agreement with the experimental result both in low （0 ~〈 0.1% ） and high （ q = 0.6% ） volume fractions. In presented model, Brown motion, the effect of interactions between nanoparticles and fluid coupling, is also considered, but any phenomenological parameter is not introduced. Nanoparticles in liquid transport distribution can be quantitatively calculated by this model.

Dissipaton Equation of Motion Theory versus Fokker-Planck Quantum Master Equation

The quest of exact and nonperturbative methods on quantum dissipation with nonlinear coupling environments remains in general a great challenge.In this review we present a comprehensive account on two approaches to the entangled system-and-environment dynamics,in the presence of linear-plus-quadratic coupling bath.One is the dissipaton-equation-ofmotion(DEOM)theory that has been extended recently to treat the nonlinear coupling environment.Another is the extended Fokker-Planck quantum master equation(FP-QME)approach that will be constructed in this work,based on its DEOM correspondence.We closely compare these two approaches,with the focus on the underlying quasi-particle picture,physical implications,and implementations.

Complex-Variable Fokker-Planck Equation Solved in Entangled State Representation

We find that the Fokker-Planck equation in complex variables can be conveniently solved in the context of bipartite entangled state representation and its relationship with SU(2) Lie algebraic generators' new realization {(1/4)[(Q_1-Q_2)~2+(P_1+P_2)~2], (1/4)[(Q_1+Q_2)~2+(P_1-P_2)~2], and-(i/2)(Q_1P_2+Q_2P^1)}, the quadratic combination of canonical operators.

GLOBAL CLASSICAL SOLUTIONS FOR QUANTUM KINETIC FOKKER-PLANCK EQUATIONS

We consider a class of nonlinear kinetic Fokker-Planck equations modeling quantum particles which obey the Bose-Einstein and Fermi-Dirac statistics, respectively. We establish the existence and convergence rate to the steady state of global classical solution to such kind of equations around the steady state.

The solution of the time-dependent Fokker-Planck equation of non-degenerate optical parametric amplification and its application to the optimum realization of EPR paradox

In this paper, the solution of the time-dependent Fokker-Planck equation of non-degenerate optical parametric amplification is used to deduce the condition demonstrating the Einstein-Podolsky-Rosen （EPR） paradox. The analytics and numerical calculation show the influence of pump depletion on the error in the measurement of continuous variables. The optimum realization of EPR paradox can be achieved by adjusting the parameter of squeezing. This result is of practical importance when the realistic experimental conditions are taken into consideration .

On the Application of Fokker-Planck Equation to Psychological Future Time

This paper tries to make a comparison and connection between Fokker-Planck or forward Kolmogorov equation and psychological future time which is based on quantum mechanics. It will be showed that in quantum finance forward interest rate model can be further improved by noting that the predicted correlation structure for field theory models depends only on variable where t is present time and x is future time. On the other side, forward Kolmogorov equation is a parabolic partial differential equation, requiring international conditions at time t and to be solved for . The aforementioned equation is to be used if there are some special states now and it is necessary to know what can happen later. It will be tried to establish the connection between these two equations. It is proved that the psychological future time if applied and implemented in Fokker-Planck equation is unstable and is changeable so it is not easily predictable. Some kinds of nonlinear functions can be applied in order to establish the notion of psychological future time, however it is unstable and it should be continuously changed.

A New Approach to Time-Dependent Solutions to the Non-Linear Fokker-Planck Equations Related to Arbitrary Functions of Tsallis Entropy: A Mathematical Study and Investigation

The following article has been retracted due to the investigation of complaints received against it. Mr. Mohammadali Ghorbani (corresponding author and also the last author) cheated the authors’ name: Alireza Heidari and Seyedali Vedad. The scientific community takes a very strong view on this matter and we treat all unethical behavior such as plagiarism seriously. This paper published in Vol.3 No.5 420-429, 2012, has been removed from this site.

The Hydrodynamic Limit of Nonlinear Fokker-Planck Equation

The non-linear Fokker-Planck equation arises in describing the evolution of stochastic system, which is a variant of the Boltzmann equation modeling the evolution of the random system with Brownian motion, where the collision term is replaced by a drift-diffusion operator. This model conserves mass, momentum and energy;the dissipation is much weaker than that in a simplified model we considered before which conserved only mass, thus more difficult to analyze. The macro-micro decomposition of the solution around the local Maxwellian introduced by T.-P. Liu, T. Yang and S.-H. Yu for Boltzmann equation is used, to reformulate the model into a fluid-type system incorporate viscosity and heat diffusion terms, coupled with an equation of the microscopic part. The viscosity and heat diffusion terms can give dissipative mechanism for the analysis of the model.

Numeric Solution of the Fokker-Planck-Kolmogorov Equation

The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested;2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.

A Method to Solve the Fokker-Planck Equation With Coordinate-dependent Mass,Friction and Temperature

An analytical expression of the propagator is obtained by using Lie algebramethod.The fission rates at the saddle point in both constant and coordinate-dependentmass,friction and temperature cases are calculated based on the expression of thepropagator and local approximation.The numerical calculation for 240pu shows thatthe fission rates from our method are reasonable.

一种三维Poisson-Nernst-Planck方程的虚单元计算

利用虚单元方法在多面体网格上求解一种三维稳态Poisson-Nernst-Planck(PNP)方程,并给出PNP方程的虚单元离散形式,推导电势方程及离子浓度方程的刚度矩阵与荷载向量的矩阵表达式.数值实验结果表明,在3种不同的多面体网格下实现了PNP方程的虚单元计算,数值解在L^(2)和H^(1)范数下均达到最优阶.

Investigating Initial Conditions of the WdW Equation in Flat Space in a Transition from the Pre-Planckian Physics Era to the Electroweak Regime of Space-Time

This document is due to reviewing an article by Maydanyuk and Olkhovsky, of a Nova Science conpendium as of “The big bang, theory assumptions and Problems”, as of 2012, which uses the Wheeler De Witt equation as an evolution equation assuming a closed universe. Having the value of k, not as the closed universe, but nearly zero of a nearly flat universe, which leads to serious problems of interpretation of what initial conditions are. These problems of interpretations of initial conditions tie in with difficulties in using QM as an initial driver of inflation. And argue in favor of using a different procedure as far as forming a wave function of the universe initially. The author wishes to thank Abhay Ashtekar for his well thought out criticism but asserts that limitations in space-time geometry largely due to when is formed from semi classical reasoning, i.e. Maxwell’s equation involving a close boundary value regime between Octonionic geometry and flat space non Octonionic geometry is a datum which Abhay Ashekhar may wish to consider in his quantum bounce model and in loop quantum gravity in the future.