Fokker-Planck equation solution methods

Unlike Xi, which in principle cannot be evaluated analytically at arbitrary a [90] for Xjnt an exact solution is possible for arbitrary values of the anisotropy parameter. Two ways were proposed to obtain quadrature formulas for Tjnt. One method [91] implies a direct integration of the Fokker-Planck equation. Another method [58] involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for Tjnt can be expressed in a finite form in terms of hypergeometric (Kummer s) functions. Equivalence of both approaches was proved in Ref. 92. 

H. Risken, The Fokker-Planck equation Methods of solution and applications , Springer- Verlag, Berling, 1984, chapter 3. 

Let us calculate the relaxation time of particles in this potential (escape time over a barrier) which agrees with inverse of the lowest nonvanishing eigenvalue Yj. Using the method of eigenfunction analysis as presented in detail in Refs. 2, 15, 17, and 18 we search for the solution of the Fokker-Planck equation in the... 

From a practical point of view, integrating trajectories for times which are of the order of eP is very expensive. When the reduced barrier height is sufficiently large, then solution of the Fokker-Planck equation also becomes numerically very difficult. It is for this reason, that the reactive flux method, described below has become an invaluable computational tool. 

Analytical solutions of quantum Fokker-Planck equations such as Eq. (63) are known only in special cases. Thus, some special methods have been developed to obtain approximate solutions. One of them is the statistical moment method, based on the fact that the equation for the probability density generates an infinite hierarchic set of equations for the statistical moments and vice versa. 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

B. D. Shizgal and H. Chen,/. Chem. Phys., 107,8051 (1997). The Quadrature Discretization Method (QDM) in the Solution of the Fokker-Planck Equation with Nonclassical Basis Functions. 

This section is based mostly on the results presented in Ref. 78 and is arranged in the following way. In Section III.B.l we note mentioned the problem of superparamagnetic relaxation, which has been already tackled by means of the Kramers method, in the in Section II.A), and show how to obtain the analytical solution (in the form of asymptotic series) for the micromagnetic Fokker-Planck equation in the uniaxial case. In Section III.B.l the perturbative... 

A consistent study of the linear and lowest nonlinear (quadratic) susceptibilities of a superparamagnetic system subjected to a constant (bias) field is presented. The particles forming the assembly are assumed to be uniaxial and identical. The method of study is mainly the numerical solution (which may be carried out with any given accuracy) of the Fokker-Planck equation for the orientational distribution function of the particle magnetic moment. Besides that, a simple heuristic expression for the quadratic response based on the effective relaxation... 

H. Risken, The Fokker—Planck Equation, Methods of Solutions and Applications, Springer-Verlag, Berlin, 1989. 

A. Numerical Solution of the Fractional Fokker—Planck Equation [Eq. (38)] via the Griinwald—Letnikov Method... 

Risken H 1984 The Fokker-Planck Equation, Methods of Solution and Application (Beriin Springer)... 

The method of Ermak and McCammon is consistent with the Fokker-Planck equation, whose solution yields the complete phase-space distribution function directly. Because the sets of particle velocities v(t) and (t+At) are effectively uncorrelated for At the Brownian dynamics algorithm gives no information about instantaneous velocities (the mean square particle velocity (v (t)) over time intervals At is the equilibrium value, 3kT/m ). 

The probability density of the response state vector of a nonlinear system under the excitation of Gaussian white noises is governed by a parabolic partial differential equation, called the Fokker-Planck equation. Exact solutions to such equations are difficult especially when both parametric (multiplicative) and external (additive) random excitations are present. In this paper, methods of solution for response vectors at the stationary state are discussed under two schemes based on the concept of detailed balance and the concept of generalized stationary potential, respectively. It is shown that the second scheme is more general and includes the first scheme as a special case. Examples are given to illustrate their applications. 

The concept of detailed balance has the origin from thermodynamics. It describes a state not necessarily in thermal equilibrium where microscopic reversibility is permissible. On the other hand, the concept of generalized stationary potential is based on the pattern of probability flow. The two concepts are unrelated, and it is remarkable that the procedures developed from the two to obtain exact solutions for the Fokker-Planck equations are essentially the same. However, since one of the conditions for detailBd balance, which places a restriction on the type of diffusion coefficients, is not required in the method of generalized stationary potential, the latter method is more general. 

The parameters Cq, C, Ce, and are model constants and need to be specified [30,31]. The same goes for the pressure dilatation term lid [31,32]. The transport equations for all of the SGS moments are readily obtained by integration of this Fokker-Planck equation. This provides a complete statistical description of turbulence. The idea is to find methods that could take advantage of quantum resources in order to speed up these calculations, at least polyno-mially in the number of variables. Because of the size of the problem typically considered, such a speedup could transform the way these problems are treated in engineering providing solutions to problems many orders of magnitude faster than are possible with classical computers. 

As seen above, a solution of the Langevin equation (Equation 6.50) (which is a nonlinear partial differential equation with random noise) consists of constructing the correlation functions of f (t) from the equation and then averaging the expressions with the help of the properties of the noise r(t). An alternative method of solution is to find the probability distribution function P(x, t) for realizing a situation in which the random variable f (t) has the particular value X at time t. P(x, t) is an equivalent description of the stochastic process f (0 and is given by the Fokker-Planck equation (Chandrasekhar 1943, Gardiner 1985, Risken 1989, Redner 2001, Mazo 2002)... 

If (4.17, 34, 35) are incorporated into the master equation (4.18) and by using (4.22, 23, 28) in the mean value and variance equations (4.29, 31) as well as in the Fokker-Planck equation (4.32), the explicit form for all kinds of equations of motion is obtained. Relevant cases for which solutions can be obtained either analytically or by numerical methods will now be considered. 

As is well known, dynamic properties of polymer molecules in dilute solution are usually treated theoretically by Brownian motion methods. Tn particular, the standard approach is to use a Fokker-Planck (or Smoluchowski) equation for diffusion of the distribution function of the polymer molecule in its configuration space. 

As far as the present work is concerned, the relevance of numerical stochastic methods for polymer dynamics in micro/macro calculations resides in their ability to yield (within error bars) exact numerical solutions to dynamic models which are insoluble in the framework of polymer kinetic theory. In addition, and mainly as a consequence of the correspondence between Fokker Planck and stochastic differential equations, complex polymer dynamics can be mapped onto extremely efficient computational schemes. Another reason for the efficiency of stochastic dynamic models for polymer melts stems from the reduction of a many-chain problem to a single-chain or two-chain representation, i.e., to linear computational complexity in the number of particles. This circumstance permits the treatment of global ensembles consisting of several tens of millions of particles on current hardware, corresponding to local ensemble sizes of O(IO ) particles per element. 

ABSTRACT The paper presents a probabilistic method to assess lifetimes of devices/components that operate under conditions typical of ageing processes. It has been assumed that the random rate of the component s wear is of the form taken by the failure rate function for the Weibull distribution, or approximately follows the linear pattern. From the point of view of mathematics, it has been based on the difference equation that, after some rearrangements, results in a partial differential equation of the Fokker-Planck type. From the particular solution to this equation one gets density function of the wear-and-tear in the form of normal distribution. Having found the density function of the wear-and-tear, one can formulate a relationship for reliability for the assumed permissible value of the wear-and-tear. With the normal distribution normalized and the required level of reliabUity reached, one can then compute the lifetime of a device or component under consideration. 

Quantum mechanics and statistical mechanisms did not change. Starting from the basic equations - Schrodinger, Langevin, Fokker-Planck - one has developped very good methods of solution as the simulation methods which take into account most of the observed features and, perhaps, some imaginary ones. 

Brownian dynamics is nothing but the numerical solution of the Smoluchowski equation. The method exploits the mathematical equivalence between a Fokker-Planck type of equation and the corresponding Langevin... 

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