Fokker-Planck-Langevin model

McClung RED (1980) The Fokker-Planck-Langevin model for rotational Brownian-motion. 1. General-theory. J Oiem Phys 73(5) 2435-2442. doi 10.1063/1.440394... 

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 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. 

The stochastic model of ion transport in liquids emphasizes the role of fast-fluctuating forces arising from short (compared to the ion transition time), random interactions with many neighboring particles. Langevin s analysis of this model was reviewed by Buck [126] with a focus on aspects important for macroscopic transport theories, namely those based on the Nernst-Planck equation. However, from a microscopic point of view, application of the Fokker-Planck equation is more fruitful [127]. In particular, only the latter equation can account for local friction anisotropy in the interfacial region, and thereby provide a better understanding of the difference between the solution and interfacial ion transport. 

Velocity relaxation effects can be accounted for in an approximate fashion by going to a phase-space description in terms of Fokker-Planck or Langevin equations. Perhaps the best known study of this type is due to Kramers, who studied the escape of particles over potential barriers as a model for certain types of isomerization or dissociation reaction. 

However, the transport equation for Fl is not in a closed form and must be modelled. For this, as indicted earlier, we consider the general diffusion process, [26,27] given by the system of SDEs. The modeling of the SDEs must be in such a way that is amenable to QC. The starting point will be our simplified Langevin model (SLM) and linear mean-square estimation (LMSE) [28] coupled with an equation of state and obeying the first law of thermodynamics. With construction of the SDEs, the corresponding Fokker-Planck equation [29] will essentially be the modelled FDF transport equation. Our proposed model is under construction and is of the form ... 

In the examples that we have considered thus far, we have used a microscopic model to motivate the growth law, and then used it to find the FPT distribution. A complementary approach is the phenomenological one in which one proposes Langevin dynamics consistent with the observed mean growth law, and assumes an ansatz for the noise term for cell-size fluctuations. By then going to the corresponding Fokker-Planck description, one can use standard techniques [5] (see Section II.D) to compute the FPT distribution. 

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. 

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