Fokker-Planck differential equation

In closing this section, we note that the stochastic master equation, Eq. (16), can be used to study the effect of boundary conditions on transport equations. If a(x, y, t) is sufficiently peaked as a function of x — y, that is if transitions occur from y to states in the near neighborhood of y, only, then the master equation can be approximated by a Fokker-Planck equation. The effects of the boundary on the master equation all appear in the properties of a(x, y, t). However, in the transition to the Fokker-Planck differential equation, these boundary effects appear as boundary conditions on the differential equation.7 These effects are prototypes for the study of how molecular boundary conditions imposed on the Liouville equation are reflected in the macroscopic boundary conditions imposed on the hydrodynamic equations. 

The procedure is to write logZy as a sum of N terms recursively related, and then express the sum as an integral over an appropriate distribution function. The distribution function was obtained, after several approximations, from a Fokker-Planck differential equation. The... 

The dynamical behavior of particles whose mass and size are much larger than those of the solvent particles can be explained by the theory of Brownian motion. Two approaches, Fokker-Planck and Chandrasekhar, have generally been used to solve the Langevin equation to describe Brownian motion. The Fokker-Planck differential equation is the diffusion equation in velocity space, while the Chandrasekhar equation is... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

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... 

Equation (2.5) is a stochastic differential equation. Some required characteristics of stochastic process may be obtained even from this equation either by cumulant analysis technique [43] or by other methods, presented in detail in Ref. 15. But the most powerful methods of obtaining the required characteristics of stochastic processes are associated with the use of the Fokker-Planck equation for the transition probability density. 

The diffusion term in this expression differs from the Fokker-Planck equation. This difference leads to a fT-dependent term in the drift term of the corresponding stochastic differential equation (6.177), p. 294. 

In order to go beyond the simple description of mixing contained in the IEM model, it is possible to formulate a Fokker-Planck equation for scalar mixing that includes the effects of differential diffusion (Fox 1999).83 Originally, the FP model was developed as an extension of the IEM model for a single scalar (Fox 1992). At high Reynolds numbers,84 the conditional scalar Laplacian can be related to the conditional scalar dissipation rate by (Pope 2000)... 

In a purely formal sense, Langevin and Fokker-Planck approaches to a problem are equivalent but, as is often the case, one approach or the other may be preferable for practical reasons. In the Fokker-Planck method, one has to solve a partial differential equation in many variables. In the Langevin method, one has to solve coupled equations of motion for the same variables, under the influence of a random force. It is likely that this second approach will be most useful for performing computer experiments to simulate the actual motion of individual polymer molecules. 

The Fokker-Planck equation is a master equation in which W is a differential operator of second order... 

Exercise. A high-speed particle traverses a medium in which it encounters randomly located scatterers, which slightly deflect it with differential cross-section o(6). Find the Fokker-Planck equation for the total deflection, supposing that it is small. 

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. 

So far the Fokker-Planck approximation has only been formulated for cases where there is no boundary, or where the boundary is too far away to bother about it. The question now is how a boundary with certain physical properties is to be translated into a boundary condition for the differential equation. In the case of a reflecting boundary the answer is clear the probability flow (1.3) has to vanish, as in (3.6),... 

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. 

That no differential equation of higher order than Fokker-Planck can describe the evolution of a probability density rigorously has been proved by R.F. Pawula, Phys. Rev. 162, 186 (1967). 

The traditional derivation of the Fokker-Planck equation (1.5) or (VIII. 1.1) is based on Kolmogorov s mathematical proof, which assumes infinitely many infinitely small jumps. In nature, however, all jumps are of some finite size. Consequently W is never a differential operator, but always of the type (V.1.1). Usually it also has a suitable expansion parameter and has the canonical form (X.2.3). If it then happens that (1.1) holds, the expansion leads to the nonlinear Fokker-Planck equation (1.5) as the lowest approximation. There is no justification for attributing a more fundamental meaning to Fokker-Planck and Langevin equations than in this approximate sense. 

Note that the differentiation d/du T acts on everything behind it hence the right-hand side involves first and second derivatives of P and has the form of a Fokker-Planck equation. 

The stochastic differential equations and the Fokker—Planck equation... 

Use of the stochastic differential equation (2.2.2) as the equation of motion instead of equation (2.1.1) results in the treatment of the reaction kinetics as a continuous Markov process. Calculations of stochastic differentials, perfectly presented by Gardiner [26], allow us to solve equation (2.2.2). On the other hand, an averaged concentration given by this equation could be obtained making use of the distribution function / = f(c, ..., cs t). The latter is nothing but solution of the Fokker-Planck equation [26, 34]... 

The stochastic differential equation (2.2.15) could be formally compared with the Fokker-Planck equation. Unlike the complete mixing of particles when a system is characterized by s stochastic variables (concentrations the local concentrations in the spatially-extended systems, C(r,t), depend also on the continuous coordinate r, thus the distribution function f(Ci,..., Cs]t) turns to be a functional, that is real application of these equations is rather complicated. (See [26, 34] for more details about presentation of the Fokker-Planck equation in terms of the functional derivatives and problems of normalization.)... 

Since a number of particles involved in any reaction event are small, a change in concentration is of the order of 1 /V. Therefore, we can use for the system with complete particle mixing the asymptotic expansion in this small parameter 1 /V. The corresponding van Kampen [73, 74] procedure (see also [27, 75]) permits us to formulate simple rules for deriving the Fokker-Planck or stochastic differential equations, asymptotically equivalent to the initial master equation (2.2.37). It allows us also to obtain coefficients Gij in the stochastic differential equation (2.2.2) thus liquidating their uncertainty and strengthening the relation between the deterministic description of motion and density fluctuations. 

Brownian transport processes and the related relaxation dynamics in the presence and absence of an external potential are most conveniently described in terms of partial differential equations of the Fokker-Planck (Smo-luchowski) [13, 14, 17-19], Rayleigh [13, 20], and Klein-Kramers [13, 14,... 

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. 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

The picture offered by the Fokker-Planck equation is, of course, in complete agreement with the Langevin equation and the assumptions made about the process. If we can solve the partial differential equation we can determine the probability density or eventually the transition probabilities at any time, and thereby determine any average value of functions of v by simple quadratures. 

Employing the stochastic differential equations [12] and [14], the Fokker-Planck equation for the evolution of the conditional joint probability density w(r, t r0, E0 0) has the form (6)... 

In order to relate the system of Eqs. (77) to a time-independent Fokker-Planck formalism, we replace that set of stochastic differential equations with the equivalent one. 

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