Fokker-Planck equation moment equations

In the case of weak collisions, the moment changes in small steps AJ (1 — y)J < J, and the process is considered as diffusion in J-space. Formally, this means that the function /(z) of width [(1 — y2)d]i is narrow relative to P(J,J, x). At t To the latter may be expanded at the point J up to terms of second-order with respect to (/ — /). Then at the limit y -> 1, to — 0 with tj finite, the Feller equations turn into a Fokker-Planck equation... 

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

To our knowledge, the first paper devoted to obtaining characteristic time scales of different observables governed by the Fokker-Planck equation in systems having steady states was written by Nadler and Schulten [30]. Their approach is based on the generalized moment expansion of observables and, thus, called the generalized moment approximation (GMA). 

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. 

The Fokker-Planck equation (80) generates an infinite and hierarchic set of equations for the statistical moments (see Section IV.A. 1). Below, we restrict ourselves to a Gaussian approximation. The cumulants are defined by the following relations ... 

Exercise. For the linear Fokker-Planck equation (1.5), the equation (1.7) is a closed equation for the mean value , which can be solved. Show that in this way all moments can be obtained successively. 

Exercise. For the decay process in (IV.6) construct the Fokker-Planck equation using (1.6). Show that it gives the first and second moments correctly, but not Ps. 

Show also that these are exact consequences of the M-equation itself. But the analogous equations for the higher moments are not correctly reproduced by the Fokker-Planck equation. (Compare V.8.)... 

One expects that the Langevin equation (1.1) is equivalent to the Fokker-Planck equation (VIII.4.6). This cannot be literally true, however, because the Fokker-Planck equation fully determines the stochastic process V(t), whereas the Langevin equation does not go beyond the first two moments. The reason is that the postulates (i), (ii), (iii) in section 1 say nothing about... 

The equivalence of the Langevin equation (1.1) to the Fokker-Planck equation (VIII.4.6) for the velocity distribution of our Brownian particle now follows simply by inspection. The solution of (VIII.4.6) was also a Gaussian process, see (VIII.4.10), and its moments (VIII.4.7) and (VIII.4.8) are the same as the present (1.5) and (1.6). Hence the autocorrelation function (1.8) also applies to both, so that both solutions are the same process. Q.E.D. 

This is a linear Fokker-Planck equation, whose coefficients depend on t through ( ). It has been solved in VIII.6 and the result was that 77 is Gaussian. It therefore suffices to determine the first and second moments of , which contain the most important information anyway. By the usual trick one obtains from (1.1)... 

This is a linear Fokker-Planck equation whose coefficients depend on time through . This approximation was christened in section 1 linear noise approximation . The solution of (4.1) was found in VIII.6 to be a Gaussian510, so that it suffices to determine the first and second moments of On multiplying (4.1) by and 2, respectively, one obtains... 

This is a multivariate linear Fokker-Planck equation of the type solved in VIII.6. We use it to determine the moments of c and i/. 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

In the literature, this relation is commonly called the Fokker-Planck equation. It is important and instructive to point out that the derivation of the Fokker-Planck relation requires the existence of the first two moments. For the Levy processes, there does not exist a Fokker-Planck equation. 

The rotary diffusion (Fokker-Planck) equation for the distribution function W(e,t) of the unit vector of the particle magnetic moment was derived by Brown [47]. As shown in other studies [48,54], it may be reduced to a compact form... 

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. 

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... 

Taking thermal fluctuations into account, the motion of the particle magnetic moment is described by the orientational distribution function W(e,t) that obeys the Fokker-Planck equation (4.90). For the case considered here, the energy function is time-dependent ... 

Let us proceed to the derivation of the pertinent Fokker-Planck equation. The probability density W(e,n,t) of various orientations of the magnetic moment and the easy magnetization axis of a ferroparticle must satisfy the conservation law... 

In the Markovian case, where is the shortest time scale, it is usually found that (1) moments of the form <(AJ)"(A ) > with m -I- k > 2 are of order t", n > 2, and therefore do not contribute to Eq. (5.41), and (2) all the relevant terms (that is, terms of order t) which contribute to the first and second moments (m -H k = 1 or 2) are obtained at the second iteration stage. This leads to the standard Fokker-Planck equation. 

We skip the technical details, which are straightforward but very cumbersome, and note only that, as in the Markovian case, only first and second moments yield terms that are not negligible by these criteria. Unlike in the Markovian case, three iteration steps are needed to collect all relevant contributions to these moments. The final result is the Fokker-Planck equation for... 

Assume for a moment that the hidden state is fixed, i.e. jt = j. Then, the evolution of a probability density p t,Y j) under the d3mamics given by (11) can be obtained as the solution of the corresponding Fokker Planck equation ... 

This potential has two potential minima on the sites at <(> = 0 and = n as well as two energy barriers located at < ) = jt/2 and <[) = 3n/2. This model has been treated in detail for normal diffusion in Refs. 8,61, and 62. Here we consider the fractional Fokker-Planck equation [Eq. (55)] for a fixed axis rotator with dipole moment p moving in a potential [Eq. (163)]. 

We have mentioned that the question posed above was answered in part by Shliomis and Stepanov [9]. They showed that for uniaxial particles, for weak applied magnetic fields, and in the noninertial limit, the equations of motion of the ferrofluid particle incorporating both the internal and the Brownian relaxation processes decouple from each other. Thus the reciprocal of the greatest relaxation time is the sum of the reciprocals of the Neel and Brownian relaxation times of both processes considered independently that is, those of a frozen Neel and a frozen Brownian mechanism In this instance the joint probability of the orientations of the magnetic moment and the particle in the fluid (i.e., the crystallographic axes) is the product of the individual probability distributions of the orientations of the axes and the particle so that the underlying Fokker Planck equation for the joint probability distribution also... 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

In Section II it is shown how the effects of thermal agitation may be included in Gilbert s equation and how the Fokker-Planck equation for the density of orientations of the magnetic moments on the unit sphere may be written down in an intuitive manner from Gilbert s equation. (The rigorous derivation of the Fokker-Planck equation from Gilbert s equation is given in Appendix D). We coin the term Brown s equation for this particular form of the Fokker-Planck equation. 

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