Fokker-Planck equation solution

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

The Fokker-Planck equation is a partial differential equation. In most cases, its time-dependent solution is not known analytically. Also, if the Fokker-Planck equation has more than one state variable, exact stationary solutions are... 

The second way is to obtain the solution of Eq. (2.6) for one-dimensional probability density with the initial distribution (2.9). Indeed, multiplying (2.6) by VT(xo, to) and integrating by xo while taking into account (2.4), we get the same Fokker-Planck equation (2.6). 

For obtaining the solution of the Fokker-Planck equation, besides the initial condition one should know boundary conditions. Boundary conditions may be quite diverse and determined by the essence of the task. The reader may find enough complete representation of boundary conditions in Ref. 15. 

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... 

Let us now suppose that each ion in the electrolyte solution may be described by a distribution function — (R, u4 t) which obeys the Fokker-Planck equation (203). 

Homogeneous, linear Fokker-Planck equations are known to admit a multi-variate Gaussian PDF as a solution.33 Thus, this closure scheme ensures that a joint Gaussian velocity PDF will result for statistically stationary, homogeneous turbulent flow. 

In die literature on stochastic processes, the above Fokker-Planck equation describes a multi-variate Omstein— Uhlenbeck process. For a discussion on the existence of Gaussian solutions to this process, see Gardiner (1990). 

We will denote the positive solution of this equation as f. As shown in Refs. 39,83,84 one may consider the parabolic barrier problem in terms of a Fokker-Planck equation, whose solution is known analytically. One may then obtain... 

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. 

VTST has also been applied to systems with two degrees of freedom coupled to a dissipative bath." Previous results of Berezhkovskii and Zitserman which predicted strong deviations from the Kramers-Grote-Hynes expression in the presence of anisotropic friction for the two degrees of freedom " were well accounted for. Subsequent numerically exact solution of the Fokker-Planck equation further verified these results. 

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 s terms in Eq. (80) contribute only the term E,2 in Eq. (97). Thus, the term represents the quantum diffusional. v-terms in the Fokker-Planck equation. The other terms in Eqs. (93)-(100) originate in the drift terms of the Fokker-Planck equation. The terms B12 and C in Eqs. (93)-(94) play the role of feedback terms that pump quantum fluctuations into the classical Bloembergen equations. If the s terms in Eq. (80) do not appear (the classical case), the term in Eq. (97) does not appear, either. In this case the subset (95)—(100) with zero initial conditions has zero solutions and in consequence leads to the first truncation [171]. 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

Adelman [530] and Stillman and Freed [531] have discussed the reduction of the generalised Langevin equation to a generalised Fokker— Planck equation, which provides a description of the probability that a molecule has a velocity u at a position r at a time t, given certain initial conditions (see Sect. 3.2.). The generalised Fokker—Planck equation has important differences by comparison with the (Markovian) Fokker— Planck equation (287). However, it has not proved so convenient a vehicle for studies of chemical reactions in solution as the generalised Langevin equation (290). 

In a stochastic approach, one replaces the difficult mechanical equations by stochastic equations, such as a diffusion equation, Langevin equation, master equation, or Fokker-Planck equations.5 These stochastic equations have fewer variables and are generally much easier to solve than the mechanical equations, One then hopes that the stochastic equations include the significant aspects of the physical equations of motion, so that their solutions will display the relevant features of the physical motion. 

Now we present the standard derivation of the Fokker-Planck equation for polymers in solution. (Terminology can often be confusing in the present instance, the equation of interest is also called the Smoluchowski equation, and may be regarded as a limiting case of a more general Fokker-Planck equation, or a Kramers equation.)... 

If A < 0 the stationary solution (1.4) is Gaussian. In fact, in that case it is possible by shifting y and rescaling, to reduce (1.5) to (IV.3.20), so that one may conclude the stationary Markov process determined by the linear Fokker-Planck equation is the Ornstein-Uhlenbeck process. For Al 0 there is no stationary probability distribution. 

This is a linear Fokker-Planck equation. Apart from constants which can be scaled away, it is identical with the equation (IV.3.20) obeyed by the transition probability of the Ornstein-Uhlenbeck process. The stationary solution of (4.6) is the same as the Pl given in (IV.3.10). Thus, in equilibrium V(t) is the Ornstein-Uhlenbeck process. 

Suppose one is faced with a one-step problem in which the coefficients rn and g are nonlinear but can be represented by smooth functions r(n), g(n). Smooth means not only that r(n) and g(n) should be continuous and a sufficient number of times differentiable, but also that they vary little between n and n+ 1. Suppose furthermore that one is interested in solutions pn(t) that can similarly be represented by a smooth function P(n, t). It is then reasonable to approximate the problem by means of a description in which n is treated as a continuous variable. Moreover, since the individual steps of n are small compared to the other lengths that occur, one expects that the master equation can be approximated by a Fokker-Planck equation. The general scheme of section 2 provides the two coefficients, but we shall here use an alternative derivation, particularly suited to one-step processes. 

In chapter X we shall encounter a linear Fokker-Planck equation of the form (6.4) whose coefficients Aij9 Bare given functions of time. The solution is again Gaussian, and can be obtained in much the same way as before. 

The Gaussian (6.11) with this E and with the averages (6.15) constitutes the solution of the linear Fokker-Planck equation (6.4) with time-dependent coefficients. 0... 

Exercise. Find the explicit solution of the one-variable time-dependent Fokker-Planck equation... 

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. 

E is an electric field and V the interaction potential. When Ll,L2 are independent Langevin forces, write the corresponding Fokker-Planck equation. Find the equilibrium solution and determine and r2. ... 

We call this equation quasilinear to express the fact that the coefficient of L(t) is still a constant. Although its solution cannot be given explicitly it can still be argued that it is equivalent to the quasilinear Fokker-Planck equation... 

Before embarking on this discussion one fact must be established. In many applications the autocorrelation function of L(t) is not really a delta function, but merely sharply peaked with a small rc > 0. Accordingly L(t) is a proper stochastic function, not a singular one. Then (4.5) is a well-defined stochastic differential equation (in the sense of chapter XVI) with a well-defined solution. If one now takes the limit rc —> 0 in this solution, it becomes a solution of the Stratonovich form (4.8) of the Fokker-Planck equation. This theorem has been proved officially, but the result can also be seen as follows. 

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