Fokker-Planck model

Improved Fokker-Planck model for the joint scalar, scalar gradient PDF. Physics of Fluids 6, 334—348. 

P. Hanggi, H. Grabert, P. Talkner, and H. Thomas. Bistable systems Maser equation versus Fokker-Planck modelling. Phys. Rev. A, 29 371—378, 1984. 

Expressions of this form for D have also been derived earlier by Skinner and Wolynes for BGK and Fokker-Planck models of chemically reacting systems. In those circumstances, one could use the knowledge of the eigenfunctions of these simple collision operators to reduce the result for D further. This is not possible for the more complex collision operator L (12 z), and one must resort to more approximate methods. We may, for example, evaluate the by inserting a complete set of velocity states, which we... 

The TF theory and electronic description is considered as the referential for the uniform distribution of electrons in atoms and molecules, respecting which the electronic accumulation in bonding is further described, usually as a perturbation - as in DFT when density gradient expansions are considered, or by general reformulation of the problem in terms of localization - in which case special quantum treatment as provided by stochastic Fokker-Planck modeling is needed these issues will be in next addressed and imfolded. 

Haenggi, P., Grabert, H., Talkner, P. Thomas, H. (1984). Bistable systems master equation versus Fokker-Planck modeling, Phys. Rev., 29, 371-8. 

Figxire 4 shows the predicted cxirves from these models for an intermediate value of T j, together with the second cumulant approximation to these models. The difference between them is striking, with no deviation between the different L values for the Fokker-Planck model, whose curves lie below the lowest cumulant approximation, and a distinct fanning out of the J diffusion curves. Comparing these with the results from a low torque CBr simulation we found reasonable agreement with the J diffusion model (6). 

Results from simulations show that the next step is not easy. Compressed gases and low torque liquids may reasonably be modelled by Steele s torque approximation Powles information theory expression or by the J diffusion models, although the evidence suggests that the Fokker-Planck model is not very successful. But commonly occurring high torque liquids cannot be successfully modelled by these techniques and there seems to be no simple alternative. It is likely that at least two parameters will be needed even for spherical and linear molecules. Further work to compare two parameter approximations with simulation results will show the most satisfactory next approximation. 

H. Grabert, W. Weidlich Renormalization of transport equations in Fokker-Planck models. Phys. Rev. A21, 2147 (1980)... 

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 main disadvantage of the perfect sink model is that it can only be applied for irreversible deposition of particles the reversible adsorption of colloidal particles is outside the scope of this approach. Dahneke [95] has studied the resuspension of particles that are attached to surfaces. The escape of particles is a consequence of their random thermal (Brownian) motion. To this avail he used the one-dimensional Fokker-Planck equation... 

This problem is very difficult to solve in general however, we have to keep conditions (303) in mind, which we used in order to obtain the Fokker-Planck collision term (304). With this approximation, it is expected that the ions will exhibit random Brownian motion instead of free particle motion between two successive coulombic interactions. We shall thus refer to this model as the Brownian-static model (B.s.). 

When the linear model is inserted into (6.46), a linear Fokker-Planck equation (Gardiner 1990) results ... 

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 term A2Pr is a direct result of employing the IEM model. If a different mixing model were used, then additional terms would result. For example, with the FP model the right-hand side would have the form /3 = A3PC + A2Pr + AsPra, where rd results from the diffusion term in the Fokker-Planck equation. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

The Fokker-Planck method was set forth in a series of papers by Kirkwood and collaborators.3 After taking into account a certain error in the original formulation of this method,4 the theory may be regarded as complete, in the sense that it provides a well-defined method of calculation. (There are reasons, however, for questioning the correctness of the model, i.e., point sources of friction in a hydrodynamic continuum, for which the theory was constructed. These reasons will be discussed in another place.)... 

The Fokker-Planck equation is a special type of master equation, which is often used as an approximation to the actual equation or as a model for more general Markov processes. Its elegant mathematical properties should not obscure the fact that its application in physical situations requires a physical justification, which is not always obvious, in particular not in nonlinear systems. 

The knowledge of the two-minima energy surface is sufficient theoretically to determine the microscopic and static rate of reaction of a charge transfer in relation to a geometric variation of the molecule. In practice, the experimental study of the charge-transfer reactions in solution leads to a macroscopic reaction rate that characterizes the dynamics of the intramolecular motion of the solute molecule within the environment of the solvent molecules. Stochastic chemical reaction models restricted to the one-dimensional case are commonly used to establish the dynamical description. Therefore, it is of importance to recall (1) the fundamental properties of the stochastic processes under the Markov assumption that found the analysis of the unimolecular reaction dynamics and the Langevin-Fokker-Planck method, (2) the conditions of validity of the well-known Kramers results and their extension to the non-Markovian effects, and (3) the situation of a reaction in the absence of a potential barrier. 

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