Diffusion coefficient Fokker-Planck equation

With the Laplace operator V. The diffusion coefficient defined in Eq. (62) has the dimension [cm /s]. (For correct derivation of the Fokker-Planck equation see [89].) If atoms are initially placed at one side of the box, they spread as ( x ) t, which follows from (62) or from (63). 

To calculate the mean escape time over a potential barrier, let us apply the Fokker-Planck equation, which, for a constant diffusion coefficient D = 2kT/h, may be also presented in the form... 

In order to achieve the most simple presentation of the calculations, we shall restrict ourselves to a one-dimensional state space in the case of constant diffusion coefficient D = 2kT/h and consider the MFPT (the extension of the method to a multidimensional state space is given in the Appendix of Ref. 41). Thus the underlying probability density diffusion equation is again the Fokker-Planck equation (2.6) that for the case of constant diffusion coefficient we present in the form ... 

Fokker-Planck Equation. A Stratonovich SDE obeys a Fokker-Planck equation of the form given in Eq. (2.222) with the drift velocity V (X) given in Eq. (2.243), and the diffusivity given in Eq. (2.229). The resulting diffusion equation may be written in terms of the drift coefficient... 

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. 

This is the Fokker-Planck equation. Note that the quantity knTC, l plays the role of a generalized diffusion coefficient. 

The diffusion approximation (1.5) is the nonlinear Fokker-Planck equation (VIII.2.5). In fact, we have now justified the derivation in VIII.2 by demonstrating that it is actually the first term of a systematic expansion in Q 1 for those master equations that have the property (1.1). Only under that condition is it true that the two coefficients... 

The monovariate Fokker-Planck equation with a position dependent diffusion coefficient D x),... 

Equation (71) reduces to the telegrapher s-type equation found in the Brownian limit a = 1 [115]. In the usual high-friction or long-time limit, one recovers the fractional Fokker-Planck equation (19). The generalized friction and diffusion coefficients in Eq. (19) are defined by [75]... 

We consider the pdf of the displacements of a Brownian particle in a process characterized by an equation like (532). The normal Fokker - Planck equation would now be (here the diffusion coefficient is denoted by B)... 

Having illustrated the problem associated with multiplicative noise we will now illustrate how the procedure is applied to obtain the drift and diffusion coefficients for the two-dimensional Fokker-Planck equation in phase space for a free Brownian particle and for the Brownian motion in a one-dimensional potential. This equation is often called the Kramers equation or Klein-Kramers equation [31]. 

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

In our most recent work, we investigated the impact of thermal fluctuations mi the driven translocation dynamics, theoretically and by means of extensive MD simulation [88]. Indeed, the role of thermal fluctuations is by no means self-evident. Our theoretical consideration is based on the Fokker-Planck equation (FPE) Eq. (6), which has a nonlinear drift term and a diffusion term with a time-dependent diffusion coefficient... 

In order to explain the dissipation of large amounts of relative kinetic energy into internal excitation and shape degrees of freedom, microscopic transport theories based on statistical nucleon exchange have proven to be of broadest utility. Based upon a master-equation approach (Norenberg et al. 1974, 1976 Randrup 1978), the macroscopic variables are accounted for qualitatively via a Fokker-Planck equation in which a drift coefficient describes the net flow of nucleons across the target-projectile interface and a diffusion coefficient that accounts for nuclear friction effects. 

Equation (27) is the Fokker-Planck equation and describes diffusion in the presence of an external force, plays a role similar to that of the diffusion coefficient in the Pick equation for the special case oi C = C° = constant, substituting Un for D, we obtain Pick s law. 

The effects of flow on the diffusion of particles suspended in a simple liquid is an important problem which was analyzed many times along the last 30 years. Several approaches have been followed in order to understand and quantify the effects of the presence of a velocity gradient on the diffusion coefficient D of the particles. Usually, these are restricted to the case of shear flow because this case is more manageable from the mathematical point of view, and because there are several experimental systems that allow the evaluation and validity of the corresponding results. The approaches followed vary from kinetic theory and projector operator techniques, to Langevin and Fokker-Planck equations. Here, we summarize some recent contributions to this subject and their results. 

A different formahsm in which the diffusion of a Brownian gas in a fluid under stationary and non-stationary flow has been analyzed is mesoscopic nonequilibrium thermodynamics (MNET) (Perez-Madrid, 1994 Rubi Mazur, 1994 Rubi P rez-Madrid, 1999). This theory uses the nonequUibrium thermodynamics rules in the phase space of the system, and allows to derive Fokker-Planck equations that are coupled with the thermodynamic forces associated to the interaction between the system and the heat bath. The effects of this coupling on system s dynamics are not obvious. This is the case of Brownian motion in the presence of flow where, as we have discussed previously, both the diffusion coefficient and the chemical potential become modified by the presence of flow (Reguera Rubi, 2003a b Santamaria Holek, 2005 2009 2001). 

The main fact to stress here is that the fluctuation term in the Fokker-Planck equation (23) contains the diffusion coefficient in velocity space Dz>, Eq. (24), that is similar to the one obtained through kinetic theory, see Eq. (8). This coefficient contains the expected thermal contribution hgT /m and, more interesting, a non-thermal contribution coming from the cross effect coupling the diffusion of probability in configuration and velocity subspaces Vuo- This particular dependence of D, leads to a modification of the spatial diffusion coefficient D(r, f) of the Brownian particles which is in agreement with experimental and simulation results (Pine, 2005 Sarman, 1992). However, before discussing this point, some comments on the diffusion or Smoluchowski equation are in turn. 

The concept of detailed balance has the origin from thermodynamics. It describes a state not necessarily in thermal equilibrium where microscopic reversibility is permissible. On the other hand, the concept of generalized stationary potential is based on the pattern of probability flow. The two concepts are unrelated, and it is remarkable that the procedures developed from the two to obtain exact solutions for the Fokker-Planck equations are essentially the same. However, since one of the conditions for detailBd balance, which places a restriction on the type of diffusion coefficients, is not required in the method of generalized stationary potential, the latter method is more general. 

The stationary solution of the Fokker-Planck equation, which includes the friction force F=— /3v, and the momentum diffusion coefficient (eqn 5.22), is a 3D Gaussian distribution... 

If all monomers of the chain are the same, as in a homopolymer, with uniform diffusion coefficient, km is independent of m and the Fokker-Planck equation for the translocation kinetics becomes... 

The mapping of the time-evolution equation for the translocation kinetics to the Fokker-Planck equation allows immediate deduction of the various properties of polymer translocation, directly from the equations presented in Chapter 6. The inputs in obtaining the results are the free energy landscapes derived in Chapter 5 and the diffusion constants km. We give below the key results for polymer translocation by copying the general solutions presented in Chapter 6. We shall take the diffusion coefficient of the monomer km to be uniform (ko) in the following sections. 

For this limit, the linear Fokker-Planck equation assumes the form of the Smoluchowski equation givenby Eq. (248) for the case of a single particle with an isotropic diffusion tensor and in the absence of a force field due to other particles undergoing Brownian motion. One can also show that the diffusion coefficient Z) = T / wf is given by... 

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