Fokker-Planck equation particles

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. 

Let us calculate the relaxation time of particles in this potential (escape time over a barrier) which agrees with inverse of the lowest nonvanishing eigenvalue Yj. Using the method of eigenfunction analysis as presented in detail in Refs. 2, 15, 17, and 18 we search for the solution of the Fokker-Planck equation in the... 

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... 

Furthermore, since (6.159) does not depend on y, if the notional particles are uniformly distributed the Fokker-Planck equation for f is... 

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... 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

If the velocity dependence of the rate of a reaction could be assumed to be constant and equal to k for velocities in excess of u0 and zero below 0, then reaction could be regarded as bleeding-off those reactant (Brownian) particles which have an energy in excess lmti02. This perturbs the velocity distribution of reactants and hence of solvent molecules [446]. Under such circumstances, the Fokker—Planck equation should be used to describe the chemical reaction. If this simple form of representing reaction is incorrect, there is little that can be done currently. The Fokker—Planck equation contains too much information about Brownian motion. In particular, the velocity dependence of the Brownian particles distribution is relatively unimportant. Davies [447] reduced the probability... 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

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). 

Exercise. A high-speed particle traverses a medium in which it encounters randomly located scatterers, which slightly deflect it with differential cross-section o(6). Find the Fokker-Planck equation for the total deflection, supposing that it is small. 

Consider a Brownian particle subject to a force F(X) depending on the position. The obvious generalization of the Fokker-Planck equation (3.5)... 

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

The equivalence of the Langevin equation (1.1) to the Fokker-Planck equation (VIII.4.6) for the velocity distribution of our Brownian particle now follows simply by inspection. The solution of (VIII.4.6) was also a Gaussian process, see (VIII.4.10), and its moments (VIII.4.7) and (VIII.4.8) are the same as the present (1.5) and (1.6). Hence the autocorrelation function (1.8) also applies to both, so that both solutions are the same process. Q.E.D. 

The stochastic differential equation (2.2.15) could be formally compared with the Fokker-Planck equation. Unlike the complete mixing of particles when a system is characterized by s stochastic variables (concentrations the local concentrations in the spatially-extended systems, C(r,t), depend also on the continuous coordinate r, thus the distribution function f(Ci,..., Cs]t) turns to be a functional, that is real application of these equations is rather complicated. (See [26, 34] for more details about presentation of the Fokker-Planck equation in terms of the functional derivatives and problems of normalization.)... 

The formulation of the Fokker-Planck equation is due to Fokker s and Planck s independent works on the description of the Brownian motion of particles [17, 18]. Commonly, an N variables equation of the type... 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

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... 

We now specialize the Fokker-Planck equation to the case of Brownian motion in Section 11.1. In this case, the variable y is the velocity v of the Brownian particle. We also note that the average of a function of the velocity v at time t, given that v = vo at t = t0, is simply expressed in terms of the transition probability by... 

The Fokker-Planck equation for the Brownian particle system is then... 

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. 

The rotary diffusion (Fokker-Planck) equation for the distribution function W(e,t) of the unit vector of the particle magnetic moment was derived by Brown [47]. As shown in other studies [48,54], it may be reduced to a compact form... 

A consistent study of the linear and lowest nonlinear (quadratic) susceptibilities of a superparamagnetic system subjected to a constant (bias) field is presented. The particles forming the assembly are assumed to be uniaxial and identical. The method of study is mainly the numerical solution (which may be carried out with any given accuracy) of the Fokker-Planck equation for the orientational distribution function of the particle magnetic moment. Besides that, a simple heuristic expression for the quadratic response based on the effective relaxation... 

Taking thermal fluctuations into account, the motion of the particle magnetic moment is described by the orientational distribution function W(e,t) that obeys the Fokker-Planck equation (4.90). For the case considered here, the energy function is time-dependent ... 

The Fokker-Planck equation governing the evolution of the orientational distribution function W(e, n. t) for arbitrary i/ and i/j, that is, when the particle... 

A Markov process is a stochastic process, where the time dependence of the probability, P(x, t)dx, that a particle position at time, t, lies between x and x+dx depends only on the fact that x=x(l at t = t0, and not on the entire history of the particle movement. In this regard, the Fokker-Planck equation [11]... 

The Fokker-Planck equation for the conditional probability density / ( i 0 0) can be obtained by averaging Eq. [20] with respect to the particle position to yield... 

The relative Brownian motion between the constituents of doublets consisting of sufficiently small equal-size aerosol particles is described by a one-dimensional Fokker-Planck equation in the particle energy space. A first passage time approach is employed for the calculation of the average lifetime of the doublets. This calculation is based on the assumption that the initial distribution of tire energy of the relative motion of the constituent particles is Maxwellian. The average dissociation time of doublets, in air at 1 atm and 298 K, for a Hamaker constant of 10 12 erg has been calculated for different sizes of the constituent particles. The calculations are found to be consistent with the assumption that the... 

Equations [13], [14], and [15] involve the assumption that the time scale of the process is large compared to the relaxation time t of the velocity distribution of particles, hence that this distribution reaches equilibrium rapidly In each of the points of the system. A measure of this relaxation time is the reciprocal of the friction coefficient obtained from the Langevin equation for the Brownian motion of a free particle (t = M/Ctoi/ ), where Mis the mass of the particle. If this condition is not satisfied, the Fokker-Planck equation (8) should be the starting point of the analysis. 

The Brownian motion of a particle under the influence of an external force field, and its consequent escape over a potential barrier has to be treated, in general, using the Fokker-Planck equation. This equation gives the distribution function W governing the probability that a particle will be after time t at a point x with velocity u (Chandrasekhar, 1943). In one dimension it has the form ... 

It is important to notice that both the original and the modified Fokker-Planck equations give the probability distribution of a particle as a function of time, position and velocity. However, if we are interested in time intervals large enough compared to jS 1, the Fokker-Planck equation, equation (3), can be reduced to a diffusional equation for the distribution function w, frequently called the Smoluchowski equation (Chandrasekhar, 1943) ... 

In order to establish the validity condition of a diffusion like equation for the probability of escape of a particle over a potential barrier, the solution of the modified Fokker-Planck equation is compared to the solution of the modified Smoluchowski equation. Since the main contribution to the determination of the escape probability comes from the neighborhood of the maximum in the potential energy (x = x J, the potential energy function was approximated by a parabolic function and the original Fokker-Planck equation was approximated at the vicinity of xmax by (Chandrasekhar, 1943) ... 

---