Fokker-Planck kinetic equation

Fokker-Planck Kinetic Equation for Determination of EEDF... 

EEDF /(e) can be considered in this approach as the number density of electrons in energy space and canbe found from the continuity equation in the energy space, called the Fokker-Planck kinetic equation ... 

At steady-state conditions, the Fokker-Planck kinetic equation (3-121) yields, after integration, the quasi-equilibrium exponential Boltzmaim distribution with temperature To ... 

The Treanor distribution function (see Section 3.1.8) makes the W flux (3-122) equal to zero. Thus, the Treanor distribution is a steady-state solution of the Fokker-Planck kinetic equation (3-116), if W exchange is a dominating process and the vibrational temperature Tv exceeds the translational temperature Tq ... 

Vibrational distributions in non-equilibrium plasma are mostly controlled by W-exchange and VT-relaxation processes, while excitation by electron impact, chemical reactions, radiation, and so on determine averaged energy balance and temperatures. At steady state, the Fokker-Planck kinetic equation (3-116) gives J(E) = const. At E oo = 0,... 

Consider evolution of the translational energy distribution function / E) of a small admixture of alkaline atoms in non-equilibrium diatomic molecular gas (TV > Tq). The distribution is determined by competition of fast VT-relaxation energy exchange between the alkaline atoms and diatomic molecules and Maxwelhzation translational-translational (TT) processes in collisions of the same partners. It can be described by the Fokker-Planck kinetic equation for diffusion of the atoms along the translational energy spectrum (Vakar etal., 1981a,b,c,d) ... 

Other inverse temperature parameters in relation (3-163) are Po = T, p i = (Tv, ). Isotope mixtures usually consist of a larger fraction of a light gas component (higher frequency of molecular oscillations, concentration nj) and only a small fraction of a heavy component (lower oscillation frequency, concentration n ). In this case, the steady-state solution of the Fokker-Planck kinetic equation (3-160) gives... 

Macrokinetic reaction rates of vibrationaiiy excited molecules are self-consistent with the influence of the reactions on vibrational distribution functions / E), which can be taken into account by introducing into the Fokker-Planck kinetic equation (3-130) an additional flux related to the reaction ... 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

Fokker-Planck Equation. The preceding identification of the kinetic SDE with an equivalent Ito SDE yields a Fokker-Planck equation of the desired form... 

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]. 

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]... 

This paper by Ya.B. helped lay the foundation for the study of the kinetics of phase transitions of the first kind. It considers the fluctuational formation and subsequent growth of vapor bubbles in a fluid at negative pressures. It is assumed that the fluid state is far from the boundary of metastability and that the volume of the bubbles formed is still small in comparison with the overall volume of the fluid. The first assumption ensures slowness of the process the time of transition to another phase is large compared to the relaxation times of the fluid per se. This allows the application of the Fokker-Planck equation in the space of embryo dimensions to describe the growth of the embryos. 

Fractional dynamics is a made-to-measure approach to the description of temporally nonlocal systems, the kinetics of which is governed by a selfsimilar memory. Fractional kinetic equations are operator equations that are mathematically close to the well-studied, analogous Brownian evolution equations of the Klein-Kramers, Rayleigh, or Fokker-Planck types. Consequently, methods such as the separation of variables can be applied. More-... 

The experimental results were analyzed using an integrated approach. To obtain the temporal evolution of the temperature and the density profiles of the bulk plasma, the experimental hot-electron temperature was used as an initial condition for the 1D-FP code [26]. The number of hot electrons in the distribution function were adjusted according to the assumed laser absorption. The FP code is coupled to the 1-D radiation hydrodynamic simulation ILESTA [27]. The electron (or ion) heating rate from hot electrons is first calculated by the Fokker-Planck transport model and is then added to the energy equation for the electrons (or ions) in ILESTA-1D. Results were then used to drive an atomic kinetics package [28] to obtain the temporal evolution of the Ka lines from partially ionized Cl ions. 

To evaluate the heating, a relativistic 1-D Fokker-Planck code was used. The configuration space is 1-D but the momentum space is 2-D, with axial symmetry. This code is coupled to a radiation-hydrodynamic simulation in order to include energy dissipation via ionization processes, hydrodynamic flow, the equation-of-state (EOS), and radiation transport. The loss of kinetic energy from hot electrons is treated through Coulomb and electromagnetic fields. 

Rubi and Perez-Madrid (2001) derived some kinetic equations of the Fokker-Planck type for polymer solutions. These equations are based on the fact that processes leading to variations in the conformation of the macromolecules can be described by nonequilibrium thermodynamics. The extension of this approach to the mesoscopic level is called the mesoscopic nonequilibrium thermodynamics, and applied to transport and relaxation phenomena and polymer solutions (Santamaria-Holek and Rubi, 2003). 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

The approach of Nigmatullin and Ryabov [28] is, however, entirely phenomenological because no underlying kinetic equation is involved. Nevertheless, their method may also be applied to the Fokker-Planck equation [Eq. (88)] so that a kinetic equation and thus a microscopic model is involved. Indeed, we can rewrite the normal Fokker-Planck equation [Eq. (88)] as an... 

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