Fokker-Planck theory

In Kirkwood s original formulation of the Fokker-Planck theory, he took into account the possibility that various constraints might apply, e.g., constant bond length between adjacent beads. This led to the introduction of a chain space of lower dimensionality than the full 3A-dimen-sional configuration space of the entire chain and it led to a complicated machinery of Riemannian geometry, with covariant and contravariant tensors, etc. 

Equations (19)—(23) provide a complete translation of Kirkwood s Fokker-Planck theory into Langevin terms, when the entire 3JV-dimen-sional configuration space is used. 

Becke-Edgecombe Chemical bonding electronic localization Fokker-Planck Theory Hartree-Fock... 

The roots of QSM theory lie in Mori s statistical mechanical theory of transport processes and Kubo s theory of thermal disturbances. The version of QSM theory given here with its refinements and modem embellishments was used by the author to develop an irreversible thermodynamic theory for photophysical phenomena and a quantum stochastic Fokker-Planck theory for adiabatic and nona-diabatic processes in condensed phases. 

A fiill theory of micleation requires a dynamical description. In the late 1960s, the early theories of homogeneous micleation were generalized and made rigorous by Langer [47]. Here one starts with an appropriate Fokker-Planck... 

The dependence of the electron ion recombination rate constant on the mean free path for electron scattering has also been analyzed on the basis of the Fokker Planck equation [40] and in terms of the fractal theory [24,25,41]. In the fractal approach, it was postulated that even when the fractal dimension of particle trajectories is not equal to 2, the motion of particles is still described by difihsion but with a distance-dependent effective diffusion coefficient. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. For the... 

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]. 

Special examples involving these boundary conditions have been worked out and it appeared that a systematic expansion in O 1/2 again led to the Fokker-Planck equation with higher order corrections.16 However, a general theory has not yet been developed. 

In recent articles1,2 on the dynamics of stiff polymer chains, the Langevin version of Brownian motion theory was used instead of the more common Fokker-Planck approach. These investigations were made, however, only in the free-draining limit. 

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.)... 

Because of the formal equivalence of Fokker-Planck and Langevin methods, there is no intrinsic difficulty in translating Kirkwood s theory into Langevin terms. As far as 1 am aware, this has not yet been done. The main purpose of this article is to perform the translation. 

The Landau equation in plasma theory is a nonlinear variant, but there P is a particle density rather than a probability. L.D. Landau, Physik. Z. Sovjetunion 10, 154 (1963) = Collected Papers (D. ter Haar ed., Pergamon, Oxford 1965) p. 163. The same is true for the nonlinear Fokker-Planck equation in M. Shiino, Phys. Rev. A 36, 2393 (1987). 

Planck equation. In this way the actual equations of motion need be solved only during At, which can be done by some perturbation theory. The Fokker-Planck equation then serves to find the long-time behavior. This separation between short-time behavior and long-time behavior is made possible by the Markov assumption. 

Kinetics is concerned with many-particle systems which require movements in space and time of individual particles. The first observations on the kinetic effect of individual molecular movements were reported by R. Brown in 1828. He observed the outward manifestation of molecular motion, now referred to as Brownian motion. The corresponding theory was first proposed in a satisfactory form in 1905 by A. Einstein. At the same time, the Polish physicist and physical chemist M. v. Smolu-chowski worked on problems of diffusion, Brownian motion (and coagulation of colloid particles) [M. v. Smoluchowski (1916)]. He is praised by later leaders in this field [S. Chandrasekhar (1943)] as a scientist whose theory of density fluctuations represents one of the most outstanding achievements in molecular physical chemistry. Further important contributions are due to Fokker, Planck, Burger, Furth, Ornstein, Uhlenbeck, Chandrasekhar, Kramers, among others. An extensive list of references can be found in [G.E. Uhlenbeck, L.S. Ornstein (1930) M.C. Wang, G.E. Uhlenbeck (1945)]. A survey of the field is found in [N. Wax, ed. (1954)]. 

Kramers theory is based on the Fokker-Planck equation for the position and velocity of a particle. The Fokker-Planck equation is based on the concept of a Markov process and in its generic form it contains no specific information about any particular process. In the case of Brownian motion, where it is sometimes simply called the Kramers equation, it takes the form... 

The idea in Kramers theory is to describe the motion in the reaction coordinate as that of a one-dimensional Brownian particle and in that way include the effects of the solvent on the rate constants. Above we have seen how the probability density for the velocity of a Brownian particle satisfies the Fokker-Planck equation that must be solved. Before we do that, it will be useful to generalize the equation slightly to include two variables explicitly, namely both the coordinate r and the velocity v, since both are needed in order to determine the rate constant in transition-state theory. 

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