Fokker-Planck equation related equations

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... 

In order to go beyond the simple description of mixing contained in the IEM model, it is possible to formulate a Fokker-Planck equation for scalar mixing that includes the effects of differential diffusion (Fox 1999).83 Originally, the FP model was developed as an extension of the IEM model for a single scalar (Fox 1992). At high Reynolds numbers,84 the conditional scalar Laplacian can be related to the conditional scalar dissipation rate by (Pope 2000)... 

The Fokker-Planck equation (80) generates an infinite and hierarchic set of equations for the statistical moments (see Section IV.A. 1). Below, we restrict ourselves to a Gaussian approximation. The cumulants are defined by the following relations ... 

Kramers and Fokker-Planck equations can be expressed in terms of its Brownian analogue, Wi, according to Eq. (49). Application of relation (49) to the Laplace transform p(u) = (rK + u) l of the exponential survival probability, Eq. (62a), produces... 

In the literature, this relation is commonly called the Fokker-Planck equation. It is important and instructive to point out that the derivation of the Fokker-Planck relation requires the existence of the first two moments. For the Levy processes, there does not exist a Fokker-Planck equation. 

It is of importance to point out that if the right-hand side is truncated after two terms (diffusion approximation), the last relation leads to an expression similar to the familiar Fokker-Planck equation (4.116). The approximation of a master equation of a birth-death process by a diffusion equation can lead to false results. Van Kampen has critically examined the Kramers-Moyal expansion and proposed a procedure based on the concept of system size expansion.135 It can be stated that any diffusion equation can be approximated by a one-step process, but the converse is not true. 

Converting the Fokker-Planck equation (4.160) into a diffusion equation for the energy, Kramers has obtained the following approximate relation for the rate kQ 16 (see also Eqs. 2.8 and 2.9) ... 

It is easy to see that the probability P(y t) itself also satisfies a Fokker-Planck equation, when we use the relation from proposition (c) ... 

Unlike Xi, which in principle cannot be evaluated analytically at arbitrary a [90] for Xjnt an exact solution is possible for arbitrary values of the anisotropy parameter. Two ways were proposed to obtain quadrature formulas for Tjnt. One method [91] implies a direct integration of the Fokker-Planck equation. Another method [58] involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for Tjnt can be expressed in a finite form in terms of hypergeometric (Kummer s) functions. Equivalence of both approaches was proved in Ref. 92. 

The sets of relaxation times x and weight coefficients vc entering Eqs. (4.233) and (4.234) were evaluated numerically. Substitution of expansions (4.228) into the Fokker-Planck equation (4.225) yields a homogeneous tridiagonal recurrence relation... 

The potential, U(x), in the barrier region is approximated to an inverted parabola with a frequency Wf> related to the barrier curvature (tu , = [/x 1 d2U 0)/d2x Y 2). Actually, Kramers solves the steady-state Fokker-Planck equation associated with eq.(19) to find the following rate constant ... 

The second model of Debye or the Debye-Frohlich model may also be generalized to fractional diffusion [8,25] (including inertial effects [26]). Moreover, it has been shown [25] that the Cole-Cole equation arises naturally from the solution of a fractional Fokker-Planck equation in the configuration space of orientations derived from the diffusion limit of a CTRW. The broadening of the dielectric loss curve characteristic of the Cole-Cole spectrum may then be easily explained on a microscopic level by means of the relation [8,24]... 

A convenient way to formulate a dynamical equation for a Levy flight in an external potential is the space-fractional Fokker-Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Levy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k — 0 and u > 0, we have X(k) 1 — CTa fe and /(w) 1 — uz, whence [52-55]... 

In Section I.B we discuss how to devise a general MFPKE to describe complex liquids. A three-body model will be presented as a description of a system in which at least two significant additional sets of solvent degrees of freedom are introduced. In Section I.C we show the relation between some of the previously cited approaches and particular cases of our model. In particular, augmented Fokker-Planck equations (AFPE) of Stillman and Freed are seen to be directly related to the MFPK formal-... 

It is impossible to do justice within the limited extent of one chapter to all theoretical developments. For example, I will omit methods relating to the Fokker-Planck equation representation of the dynamics. This includes the method of adiabatic elimination discussed extensively in Ref. 33 or the approach based on the Rayleigh quotient, developed by Talkner (34,35). There are a number of reviews, monographs, and special journal issues devoted to the theory of activated rate processes (5,13,14,36-40), the interested reader is urged to consult them. I will also omit any quantum theory of activated rate processes. The thread which connects the material presented in this chapter will be the use of the Hamiltonian equivalent form of the STGLE and more general forms to derive the classical theory of activated rate processes. 

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