Fokker-Planck-Kolmogorov

There is an important analogy between the Fokker-Planck-Kolmogorov equation and the property transport equation. Indeed, the term which contains A(t,x) describes the particle displacement by individual processes and the term which contains D(t,x) describes the left and right movement in each individual displacement or diffusion. We can notice the very good similarity between the transport and the Kolmogorov equation. In addition, many scientific works show that both... 

The Fokker-Planck-Kolmogorov approximation of the master equation is based on the assumption that all the terms greater than second order, which are extracted from the Taylor expansion of P (2 Az, t), vanish. This is rarely true in practice, however, and a more rational way of approximating the master equation is to systematically expand it in powers of a small parameter, which can be chosen approximately. This parameter is usually chosen in order to have the same size as the system. 

Most of the analytical solutions for nonlinear systems are based on the Fokker-Planck-Kolmogorov (FPK) diffusion equation ... 

They can be classified into analytical and simulation methods. Among the first ones are considered in the following the Fokker-Planck-Kolmogorov (FPK) equation, the equivalent linearization, the perturbation method, and the... 

Solution of the Fokker-Planck-Kolmogorov Equation for an SDOF Elastic Nonlinear Second-Order System... 

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],... 

These are the so-called forward and backward Kolmogorov equations for the Ornstein-Uhlenbeck process. Their paramount importance will appear in VIII.4 under the more familiar name of Fokker-Planck equation. 

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. 

This is called the Kramers-Moyal expansion. ) Formally (2.6) is identical with the master equation itself and is therefore not easier to deal with, but it suggests that one may break off after a suitable number of terms. The Fokker-Planck approximation assumes that all terms after v = 2 are negligible. Kolmogorov s proof is based on the assumption that av = 0 for v>2. This, however, is never true in physical systems In the next chapter we shall therefore expand the M-equation systematically in powers of a small parameter and find that the successive orders do not simply correspond to the successive terms in the Kramers-Moyal expansion. 

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. 

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]. 

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. 

The n-variable version of Kolmogorov s forward equation (or Fokker-Planck equation) can be written as... 

The corresponding backward Fokker-Planck equation (Kolmogorov equation) for the conditional density / ( " / b t ) is given by (5)... 

We also need some background material about (19). If m(x) denotes the equilibrium probability density function of x(t), i.e. the probability density to find a trajectory (reactive or not) at position x at time t, m(x) satisfies the (steady) forward Kolmogorov equation (also known as Fokker-Planck equation)... 

The investigation of Brownian particle motion led to the development of the mathematical theory of random processes which has been widely adopted. A great contribution to the mathematical theory of Brownian motion was made by Wiener and Kolmogorov for example, Wiener proved that the trajectory of Brownian motion is almost everywhere continuous but nowhere differentiable, Kolmogorov introduced the concept of forward and backward Fokker-Planck equations. 

The derivation of the Fokker-Planck (FP) equation described above is far from rigorous since the conditions for neglecting higher-order terms in the expansion of exp( 9/9x) were not established. Appendix 8A outlines a rigorous derivation of the FP equation for a Markov process that starts from the Chapman-Kolmogorov equation... 

In Appendix 8A we show that when these conditions are satisfied, the Chapman-Kolmogorov integral equation (8.118) leads to two partial differential equations. The Fokker-Planck equation describes the future evolution of the probability distribution... 

The forward equation The Fokker-Planck equation is now derived as the differential form of the Chapman-Kolmogorov equation For any function f (x)... 

This is the Fokker-Planck equation that corresponds, under the conditions specified, to the Chapman-Kolmogorov equation (8.118). 

For the quantities W(x z,t) equal to zero, the differential Chapman-Kolmogorov equation takes the form of the 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 ... 

The forward Kolmogorov equation (5.10) is referred to in physical literature as the Fokker-Planck equation. For the absolute density function g(x, t) (which contains less information than/) the following Kolmogorov-like equation holds ... 

The transition probability p(s,x t,y) satisfies the Fokker-Planck equation (also known as Kolmogorov s forward equation) [ l]... 

By using the same cmicepts, a very large niun-ber of other problems may be solved. Such an example the probability density function of a random variable may be obtained with the same technique here used for representing cross-correlations in terms of FSMs. It follows that Fokker-Planck equation, Kolmogorov-Feller equation, Einstein-Smoluchowski equation, and path integral solution (Cottone et al. 2008) may be solved in terms of FSM. Moreover, wavelet transform and classical or fractional differential equations may be easily solved by using fractional calculus and Mellin transform in complex domain. 

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