Fokker-Planck equation stochastic differential equations

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. 

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. 

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.)... 

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)... 

Connection between the Fokker-Planck Equation and Stochastic Differential 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 ... 

We exploit the spherical symmetry inherent in this situation to consider relative displacement between the two particles only along the radial coordinate. Accordingly, we let r" be the radial displacement of the particle of volume x relative to that of volume x and p(r", t + r r — r) be the probability density for the spatial distribution of the particle of volume x given that their initial separation is r — r. This density function must satisfy the Fokker-Planck equation " associated with the stochastic differential equation (3.3.35), viz.,... 

The Fokker-Planck equation for the stochastic differential equation (3.3.47) will differ from Equation (3.3.36) because of the drift term. However, since the calculation of the aggregation frequency depends on the function P(r", t + t) as defined in Section 3.3.5.2, we will directly proceed to the differential equation in P(r", t + t). Recognizing spherical symmetry, we have... 

Contents A Historical Introduction. - Probability Concepts. -Markov Processes. - The Ito Calculus and Stochastic Differential Equations. - The Fokker-Planck Equatioa - Approximation Methods for Diffusion Processes. - Master Equations and Jump Processes. - Spatially Distributed Systems. - Bistability, Metastability, and Escape Problems. - Quantum Mechanical tokov Processes. - References. - Bibliogr hy. - Symbol Index. - Author Index. - Subject Index. 

We now summarize the equations that govern the dynamical evolution of the probability that the system is in state S at time t, which we denote by P S, t). Typically, such equations are written in one of three formalisms the master equation (ME), the Fokker-Planck equation (FPE), or the stochastic differential equation (SDE) each is summarized below in turn. Details of the derivations can be found in Refs. 2-4. 

The stochastic differential equation (3.2), together with the noise properties (3.4) may be transformed to the Fokker-Planck equation... 

In recent years, there has been much interest in the nature of the fluctuations in nonequilibrium systems [ ll. Most of the work in this field has consisted of studying the composition fluctuations for a given system through the Master Equation, the Fokker-Planck Equation or a Stochastic Differential Equation. Recently, these methods have been applied to the study of thermal systems by Nicolis, Baras, and Malek Mansour [ 2l In this paper, we review their analysis of the two reservoir model. We discuss a computer simulation which has been developed to study this system and present a confirmation of their thermal fluctuation predictions. 

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) ... 

Exploiting the relation between this stochastic differential equation and its Fokker-Planck equation, it can be shown that the fluctuation-dissipation theorem holds [46], and that the method therefore simulates a canonical ensemble. DPD can be extended to thermalize the perpendicular component of the interparticle velocity as well, thereby allowing more control over the transport properties of the model [49,57]. 

Equation 13.76 is a stochastic differential equation that can describe the time evolution of variable X, which is under the influence of noise. Generally, the Langevin equation can be used to sample the probability distribution P(X, t). Indeed it can be proven that the solutions of the Fokker-Planck equation and of the Langevin equation are equivalent (see MoyaTs text in Further reading). 

A CLE is an Ito stochastic differential equation (SDE) with multiplicative noise terms and represents one possible solution of the Eokker-Planck equation. From a multidimensional Fokker-Planck equation we end up with a system of CLEs ... 

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. 

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. 

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. 

In this section, we formulate the dynamical description of Levy flights using both a stochastic differential (Langevin) equation and the deterministic fractional Fokker-Planck equation. For the latter, we also discuss the corresponding form in the domain of wavenumbers, which is a convenient form for certain analytical manipulations in later sections. 

As far as the present work is concerned, the relevance of numerical stochastic methods for polymer dynamics in micro/macro calculations resides in their ability to yield (within error bars) exact numerical solutions to dynamic models which are insoluble in the framework of polymer kinetic theory. In addition, and mainly as a consequence of the correspondence between Fokker Planck and stochastic differential equations, complex polymer dynamics can be mapped onto extremely efficient computational schemes. Another reason for the efficiency of stochastic dynamic models for polymer melts stems from the reduction of a many-chain problem to a single-chain or two-chain representation, i.e., to linear computational complexity in the number of particles. This circumstance permits the treatment of global ensembles consisting of several tens of millions of particles on current hardware, corresponding to local ensemble sizes of O(IO ) particles per element. 

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)... 

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. 

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