Fokker-Planck equation stochastic processes

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. 

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). 

One expects that the Langevin equation (1.1) is equivalent to the Fokker-Planck equation (VIII.4.6). This cannot be literally true, however, because the Fokker-Planck equation fully determines the stochastic process V(t), whereas the Langevin equation does not go beyond the first two moments. The reason is that the postulates (i), (ii), (iii) in section 1 say nothing about... 

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]... 

ANOMALOUS STOCHASTIC PROCESSES IN THE FRACTIONAL DYNAMICS FRAMEWORK FOKKER-PLANCK EQUATION, DISPERSIVE TRANSPORT, AND NON-EXPONENTIAL RELAXATION... 

Diffusion can be considered as a stochastic or random process and described by the so-called Fokker-Planck equation adapted to Brownian motion. This equation is also known as the Smoluchowski equation. We consider the description of stochastic processes and Brownian motion in more detail in Section 11.1 and Appendix H. 

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 rigorous analysis of the same problem has been given by San Miguel and Sancho. They emphasized that the reduction process produces unavoidably non-Markovian statistics. Nevertheless they noted that the existence of Fokker-Planck equations does not conflict with the non-Markovian character of the stochastic process, since the corresponding solution, in harmony with ref. 19, is valid only to evaluate one-time averages and is of no use in multitime averages. 

Transport problems of fields can be investigated in terms of particles motion, which is an interesting aspect per se. This is clear if we note that Eq. (2) is nothing but the Fokker-Planck equation of the stochastic process describing the motion of test particles ... 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

A precise derivation of the Fokker-Planck equation requires the introduction of the notion of a stochastic process. This is given in Appendix B. 

The equation above is the conventional linear Fokker-Planck equation for stochastic processes with additive noise and the noise strength measured in terms of the temperature T. 

A great amount of stochastic physics investigates the approximation of jump processes by diffusion processes, i.e. of the master equation by a Fokker-Planck equation, since the latter is easier to solve. The rationale behind this procedure is the fact that the usual deterministic (CCD) and stochastic (CDS) models differ from each other in two aspects. The CDS model offers a stochastic description with a discrete state space. In most applications, where the number of particles is large and may approach Avogadro s number, the discreteness should be of minor importance. Since the CCD model adopts a continuous state-space, it is quite natural to adopt CCS model as an approximation for fluctuations. 

One of the most extensively discussed topics of the theory of stochastic physics is whether the evolution equations of the discrete state-space stochastic processes, i.e. the master equations of the jump processes, can be approximated asymptotically by Fokker-Planck equations when the volume of the system increases. We certainly do not want to deal with the details of this problem, since the literature is comprehensive. Many opinions about this question have been expressed in a discussion (published in Nicolis et ai, 1984). However, some comments have to be made. 

Other kinds of Fokker-Planck equations can be also derived. The continuous state-space stochastic model of a chemical reaction, which considers the reaction as a diffusion process , neglects the essential discreteness of the mesoscopic events. However, some shortcomings of (5.65) have been eliminated by using a direct Fokker-Planck equation obtained by means of nonlinear transport theory (Grabert et al., 1983). 

Given this qualitative description of the transition process, the rate of the reaction L F can be computed. A complete calculation along the lines presented in Sec. 3 gives rise to a number of problems associated with the solution of the multi-dimensional Fokker-Planck equation for this system. While this is a promising route for future developments in this area, we instead show that many features of the rate can be accounted for by a simple stochastic model. 

As seen above, a solution of the Langevin equation (Equation 6.50) (which is a nonlinear partial differential equation with random noise) consists of constructing the correlation functions of f (t) from the equation and then averaging the expressions with the help of the properties of the noise r(t). An alternative method of solution is to find the probability distribution function P(x, t) for realizing a situation in which the random variable f (t) has the particular value X at time t. P(x, t) is an equivalent description of the stochastic process f (0 and is given by the Fokker-Planck equation (Chandrasekhar 1943, Gardiner 1985, Risken 1989, Redner 2001, Mazo 2002)... 

Let us consider a particle undergoing a drift-diffusion stochastic process given by the Fokker-Planck equation (Equations 6.56 and 6.57)... 

The above equations are known as the forward and backward Fokker-Planck equations. For a particular stochastic process (namely, prescribed A x) and p(x,t xo,0) needs to be calculated with appropriate boundary conditions (Gardiner 1985). Substituting this result in Equation 6.86, g xo, r) is then obtained. 

The theory of stochastic processes began in the nineteenth century when physicists were trying to show that heat in a medium is essentially a random motion of the constituent molecules. At the end of that century, some researches began to adopt more direct mathematical models of random disturbances instead of considering random motion as due to collisions between objects having a random distribution of initial positions and velocities. In this context several physicists, among which Fok-ker (1914) and Planck (1915), developed partial differential equations, which were versions of what was subsequently called the Fokker-Planck equation, to study the theory of Brownian motion. 

In this section, we first introduce the most frequently used definitions of static properties, which will be used throughout this chapter and in some later chapters, and then discuss stochastic processes in the motion of macromolecular chains that is. Brownian motion that leads to the well-known Fokker-Planck equation, which further reduces to the Smoluchowski equation and Langevin equation. These two equations play a very important role in describing the motion of macromolecular chains. Owing to the limited space available here, we do not present rigorous derivations of various expressions. 

The Fokker-Planck equation is a differential equation describing various stochastic processes. It concerns the probability p(x, t) of finding our system at the point x at time t in the potential field U(x) ... 

We then generalize the discussion to other stochastic processes, deriving appropriate modeling formalisms such as the master equation, the Fokker-Planck equation and the Langevin 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. 

An equivalent approach to the Fokker-Planck equation for determining the time evolution of a stochastic process is based on the Langevin equation. 

Salis and Kaznessis proposed a hybrid stochastic algorithm that is based on a dynamical partitioning of the set of reactions into fast and slow subsets. The fast subset is treated as a continuous Markov process governed by a multidimensional Fokker-Planck equation, while the slow subset is considered to be a jump or discrete Markov process governed by a CME. The approximation of fast/continuous reactions as a continuous Markov process significantly reduces the computational intensity and introduces a marginal error when compared to the exact jump Markov simulation. This idea becomes very useful in systems where reactions with multiple reaction scales are constantly present. 

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