Fokker - Planck Equation

For a stochastic differential equation, there exists an associated Fokker - Planck equation, which describes the probability that the variable takes the value concerned. The Fokker - Planck equation is also called the forward Kolmogorov equation. To the particular stochastic differential equation (21.13) the following Fokker - Planck equation is associated  

x) is the probability to find the particle at time t at position x. The associated initial condition is 

A more general derivation of this type of equations can be found in the book of Fisz [6]. 

The difficulties in solving the master equation for any but the simplest of systems have been adequately detailed elsewhere (see Further reading). An approximation to the master equation that is capable of solution is the Fokker-Planck equation. 

In order to derive the Fokker-Planck equation, we may write an expansion in the probability 

The expansion in Eq. 13.65 is known as the Kramers-Moyal expansion. Assuming that D (x) = 0 for 2, yields the Fokker-Planck equation 

The Fokker-Planck equation accurately captures the time evolution of stochastic processes whose probahihty distribution can be completely determined by its average and variance. For example, stochastic processes with Gaussian probahihty distributions, such as the random walk, can be completely described with a Fokker-Planck equation. 

A Markov process is a stochastic process, where the time dependence of the probability, P(x, t)dx, that a particle position at time, t, lies between x and x+dx depends only on the fact that x=x(l at t = t0, and not on the entire history of the particle movement. In this regard, the Fokker-Planck equation [11] 

A very important application of the Markov dynamics is random walk. In the special case of random walk/(x) = 0 and g(x) = 1, then the diffusion equation for a random walk in one dimension is 

The solution of this equation with the following initial and boundary conditions 

Therefore, it is very easy to show that the one-dimensional MSD is 

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A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

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

With the Laplace operator V. The diffusion coefficient defined in Eq. (62) has the dimension [cm /s]. (For correct derivation of the Fokker-Planck equation see [89].) If atoms are initially placed at one side of the box, they spread as ( x ) t, which follows from (62) or from (63). 

H. Risken. The Fokker-Planck Equation. Berlin Springer, 1989. 

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... 

Keilson J., Storer J. E. On a Brownian motion, Boltzmann equation and the Fokker-Planck equation, Quart. Appl. Math., 10, 243-53 (1952). 

Barcilon, V, Singular Perturbation Analysis of the Fokker-Planck Equation Kramer s Underdamped Problem, SIAM Journal of Applied Mathematics 56, 446, 1996. 

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows ... 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

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. 

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],... 

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. 

First of all we should mention that the Fokker-Planck equation may be represented as a continuity equation ... 

To calculate the mean escape time over a potential barrier, let us apply the Fokker-Planck equation, which, for a constant diffusion coefficient D = 2kT/h, may be also presented in the form... 

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... 

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). 

In order to achieve the most simple presentation of the calculations, we shall restrict ourselves to a one-dimensional state space in the case of constant diffusion coefficient D = 2kT/h and consider the MFPT (the extension of the method to a multidimensional state space is given in the Appendix of Ref. 41). Thus the underlying probability density diffusion equation is again the Fokker-Planck equation (2.6) that for the case of constant diffusion coefficient we present in the form ... 

It is convenient to present the Fokker-Planck equation in the following dimensionless form ... 

When comparing (5.106) with (5.105), it becomes evident that these expressions coincide to make the interchange xo . This fact demonstrates the so-called reciprocity principle In any linear system, some effect does not vary if the source (at position x = xo) and the observation point (x = ) will be interchanged. The linearity of our system is represented by the linearity of the Fokker-Planck equation (5.72). 

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