Fokker-Planck equation generalized

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). 

Alternatively, the white-noise processes W(f) could be replaced by colored-noise processes. Since the latter have finite auto-correlation times, the resulting Lagrangian correlation functions for U and would be nonexponential. However, it would generally not be possible to describe the Lagrangian PDF by a Fokker-Planck equation. Thus, in order to simplify the comparison with Eulerian PDF methods, we will use white-noise processes throughout this section. 

Analysis of a physical problem involving Brownian motion can normally determine only the values of the coefficients and that appear in the Fokker-Planck equation. The matrix of coefficients S "" is required only to satisfy Eq. (2.229), which is generally not sufficient to determine a unique value for B ". For L > 1 and M = L, there are generally an infinite number of ways of... 

Special examples involving these boundary conditions have been worked out and it appeared that a systematic expansion in O 1/2 again led to the Fokker-Planck equation with higher order corrections.16 However, a general theory has not yet been developed. 

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. 

The plan of the article is as follows. First, we discuss the phenomenon of hydrodynamic interaction in general terms, and at the same time, we present some convenient notation. Then, we give the usual argument leading to the Fokker-Planck equation. After that we derive the Langevin equation that is formally equivalent to the Fokker-Planck equation, together with a statistical description of the fluctuating force. 

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.)... 

This is the Fokker-Planck equation. Note that the quantity knTC, l plays the role of a generalized diffusion coefficient. 

The Fokker-Planck equation is a special type of master equation, which is often used as an approximation to the actual equation or as a model for more general Markov processes. Its elegant mathematical properties should not obscure the fact that its application in physical situations requires a physical justification, which is not always obvious, in particular not in nonlinear systems. 

We shall meet more general Fokker-Planck equations the special form (1.1) is also called Smoluchowski equation , generalized diffusion equation , or second Kolmogorov equation . The first term on the right-hand side has been called transport term , convection term , or drift term the second one diffusion term or fluctuation term . Of course, these names should not prejudge their physical interpretation. Some authors distinguish between Fokker-Planck equations and master equations, reserving the latter name to the jump processes considered hitherto. 

We have introduced the Fokker-Planck equation as a special kind of M-equation. Its main use, however, is as an approximate description for any Markov process Y(t) whose individual jumps are small. In this sense the linear Fokker-Planck equation was used by Rayleigh 0, Einstein, Smoluchowskin), and Fokker, for special cases. Subsequently Planck formulated the general nonlinear Fokker-Planck equation from an arbitrary M-equation assuming only that the jumps are small. Finally Kolmogorov8 provided a mathematical derivation by going to the limit of infinitely small jumps. 

Thus one obtains the Fokker-Planck equation as an exact result in the limit e -> 0. Yet this derivation is unsatisfactory, because in actual applications one does not have a parameter e that goes to zero. The question is for given a, P (or more generally for given W how good an approximation is provided by the Fokker-Planck equation This question is better answered by Planck s derivation, and in a more systematic way by the derivation in chapter X. 

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. 

The generalization of the Fokker-Planck equation (1.1) to the case that there are r variables yt is... 

Exercise. The most general Fokker-Planck equation is... 

Consider a Brownian particle subject to a force F(X) depending on the position. The obvious generalization of the Fokker-Planck equation (3.5)... 

Warning. The idea of using a nonlinear Fokker-Planck equation as a general framework for describing fluctuating systems has attracted many authors. Detailed balance, in its extended form, was a useful aid, but the link with the deterministic equation caused difficulties. It may therefore be helpful to emphasize three caveats. 

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. 

Equation (71) reduces to the telegrapher s-type equation found in the Brownian limit a = 1 [115]. In the usual high-friction or long-time limit, one recovers the fractional Fokker-Planck equation (19). The generalized friction and diffusion coefficients in Eq. (19) are defined by [75]... 

A generalized Fokker-Planck equation for the unstable state equivalent to the previous GLE (4.202) can be calculated191b 192 ... 

The idea in Kramers theory is to describe the motion in the reaction coordinate as that of a one-dimensional Brownian particle and in that way include the effects of the solvent on the rate constants. Above we have seen how the probability density for the velocity of a Brownian particle satisfies the Fokker-Planck equation that must be solved. Before we do that, it will be useful to generalize the equation slightly to include two variables explicitly, namely both the coordinate r and the velocity v, since both are needed in order to determine the rate constant in transition-state theory. 

II. General Framework The Micromagnetic Fokker-Planck Equation... 

II. GENERAL FRAMEWORK THE MICROMAGNETIC FOKKER-PLANCK EQUATION... 

The general matrix equation of the problem that determines the amplitudes <2 /" is obtained by substitution of the spherical harmonic expansion (4.319) into the Fokker-Planck equation (4.313). After that the result is multiplied from... 

Reference [51] proposes to split the flux term from the Fokker-Planck equation into two terms one accounting for the dynamics of a single neuron independently of all other neurons in the population (the streaming term) and another including the interactions of a neuron with the other neurons in the network (the interaction term). Before going further, let us denote by w the state of a neuron, a vector in the general case. The proposed general expression for these two fluxes is [51] ... 

The Brownian motion of a particle under the influence of an external force field, and its consequent escape over a potential barrier has to be treated, in general, using the Fokker-Planck equation. This equation gives the distribution function W governing the probability that a particle will be after time t at a point x with velocity u (Chandrasekhar, 1943). In one dimension it has the form ... 

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