Fokker-Plank equation

In a case where F would contain a stochastic term (e.g., Brownian motion, noise), this equation would lead to the celebrated Fokker-Plank equation with a diffusion (second-order) term. 

A cornerstone of condensed phase reaction theory is the Kramers-Grote-Hynes theory. In a seminal paper Kramers solved the Fokker-Plank equation in two limiting cases, for high and low friction, by assiuning Markovian dynamics y(t) 5(t). He foimd that the rate is a non-monotonic function of the friction ( Kramers turnover .) Further progress was made by Grote and Hynes - who... 

Sitarski and Seinfeld (6) were the first to provide a theoretical basis for Fuch s semi-empirical formula, by solving the Fokker-Plank equation by means of Grad s (7) 13-moment method. Their solution was further improved by Mork et al. (8). The Brownian coagulation coefficient predicted by these models agrees fairly well with the Fuchs interpolation formula. However, the model does not predict the proper free molecular limit. The validity of the Fuchs semiempirical formula was further reinforced, by the Monte Carlo simulations of Brownian coagulation, by Nowakowski and Sitarski (9). 

Let us now consider the relative motion of two particles of the same radius Rp and mass mp, and denote by W(r, Ci r2, c2)dr dcidr2dc2 the probability of finding the first particle between r and n + drt, with the velocity between c and Ci + dc, and the second particle between r2 and r2 + t/r2, with the velocity between c2 and c2 + r/c2. The distribution function W satisfies the steady-state Fokker-Plank equation... 

As pointed out earlier, the Fokker-Plank equation [18] describes the motion of the fictitious particle only outside a sphere of radius 7 s + Xr. where RS(=2RV) is the radius of the sphere of influence. The motion of the fictitious particle in the region of thickness... 

Introducing the expansion [25] into the Fokker-Plank equation [24] and using the orthogonal properties of the Hermite polynomials, the following moment equations are obtained ... 

If inertia is not negligible, the problem must l>e solved by using the more general Fokker-Plank equation (Chandrasekhar, 1943, p. 65). 

The second chapter examines the deposition of Brownian particles on surfaces when the interaction forces between particles and collector play a role. When the range of interactions between the two (which can be called the interaction force boundary layer) is small compared to the thickness of the diffusion boundary layer of the particles, the interactions can be replaced by a boundary condition. This has the form of a first order chemical reaction, and an expression is derived for the reaction rate constant. Although cells are larger than the usual Brownian particles, the deposition of cancer cells or platelets on surfaces is treated similarly but on the basis of a Fokker-Plank equation. 

Oh, Y.M., Lee, S. H., Park, S., Lee, J.S. (2004) A Numerical Study on Ultra-short Pulse Laser-induced Damage on Dielectrics Using the Fokker-Plank Equation, Int. J. Heat Mass Transfer, to appear. 

The state of the system is described by a probability distribution P (s, (p, t), which is a function of the walker position s and a functional of the field p. P s,(p,t) satisfies a Fokker-Plank equation that can be directly derived from (22) using standard techniques [57]. In [32] we show that, for large t, P s, functional space and centered around a function po (s ) that is the field corresponding to the free energy of the system F (s) ... 

Jumarie, G. 1992. A Fokker-Plank equation of fractional order with respect to time. J. Math. Phys. 33 3536-3542. 

Finally, there are some constitntive models that cannot be expressed in closed form as differential or integral equations, but require the solution of a Fokker-Plank equation (or an equivalent set of stochastic differential eqnations) for the orientation distribution of chain segments in order to compnte the stress (Reference 4, pp. 338 ff Reference 10 and references therein). This technique may become more useful as computing power increases, but to date it has been used only for viscometric flows and a few very simple non-viscometric geometries. 

Propagation of the fast subsystem - chemical Langevin equations The fast subset dynamics are assumed to follow a continuous Markov process description and therefore a multidimensional Fokker-Planck equation describes their time evolution. The multidimensional Fokker-Plank equation more accurately describes the evolution of the probability distribution of only the fast reactions. The solution is a distribution depicting the state occupancies. If the interest is in obtaining one of the possible trajectories of the solution, the proper course of action is to solve a system of chemical Langevin equations (CLEs). 

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