Equation for the Distribution Function

In this section we present a development of the partial differential equation for the distribution function ip(R). This important equation is obtained by combining the equation of motion describing the relative motion of the beads, and an equation of continuity for the distribution function. Before discussing these, however, we need to make a few comments regarding the representation of the velocity field. 

We shall consider here only velocity fields which are homogeneous, in the sense that the rate-of-deformation tensor is the same at all points in space. Hence we represent the velocity field by the expression =(K-r] where k is a traceless tensor which is independent of position but may be dependent on the time t, and r is the position vector. The tensor k must be traceless, since we are considering only incompressible 

For such a flow field the rate of deformation tensor4 y = F + (P )t (here means transpose ) is just y = + k, a quantity which may be 

For steady, homogeneous, potential flows, v is derivable from a velocity potential ( = — P P), with = — (k rr) where k = Kf and tric=0. 

For each bead of the dumbbell we may write an equation of motion, indicating that the mass-times-acceleration equals the sum erf the hydro-dynamic drag force, the Brownian motion force, and the force through the connector5  

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If only one type of particle is present, mx = m2 however, the expressions relating the velocities before and after collision do not simplify to any great extent. If several types of particles are present, then there results one Boltzmann equation for the distribution function for each type of particle in each equation, integrals will appear for collisions with each type of particle. That is, if there are P types of particles, numbered i = 1,2,- , P, there are P distribution functions, ft /(r,vt, ), describing the system ftdrdvt is the number of particles of type i in the differential phase space volume around (r,v(). The set of Boltzmann equations for the system would then be ... 

Each factor in square brackets arises from an exclusion factor, E, in the defining equation for the distribution function G, Eq. (170). The factor (1 — 1/ ) by which the result differs from the simple association theory arises from the term in square brackets in Eq. (172) and has already been commented on. 

The kinetic equation for the distribution function f(r, a t) must include all these effects. Doi and Edwards [4,99] proposed for it the generalized Smoluchowski equation... 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

As discussed in Section IV, we may derive from (5.9) a kinetic equation for the distribution function of the electrons, which takes into account reactions and large-scale fluctuations. From such an equation we get a Langevin-type rate equation for the concentrations of electrons, ions, and atoms, which reads in the case of local equilibrium35... 

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... 

Equations (F.12) and (F.13) are followed an equation for the distribution function of the distance between the centres of resistance of the relaxator... 

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) ... 

Fokker-Planck(F-P) equation for the distribution functional /[n(r),(] and discuss genered properties of the TD-DFT. By combining a number-conservation law, the DFT and theory of Brownian motion, we derived the following equations for the density n(r,t) and the momentum density g(r,t) ... 

Recently, Yamakawa (42) tried also to evaluate the (smotic pressure by using Eq. (3.8). His aim was to obtain the osmotic pressure for moderately high concentrations. However, as one can expect, he had to introduce crude approximations in order to break BBGKJY-type hierarchy of equations for the distribution functions. Such approximations do not yield generally good results at higher concentrations, even for low molecular weight molecules. 

Ding and Gidaspow [16], for example, derived a two-phase flow model starting with the Boltzmann equation for the distribution function of particles and incorporated fluid-particle interactions into the macroscopic equations. The governing equations were derived using the classical concepts of kinetic theory. However, to determine the constitutive equations they used the ad hoc distribution functions proposed by Savage and Jeffery [65]. The resulting macroscopic equations contain both kinetic - and collisional pressures but only the collisional deviatoric stresses. The model is thus primarily intended for dense particle flows. 

In the preceding sections many results have been presented for the bead-rod (rigid dumbbell) suspensions these results were obtained by solving the equation for the distribution function and then calculating the components of the stress tensor. It was pointed out in Eq. (4.23) for bead-spring (Hookean dumbbell) suspensions, that there is a constitutive equation which can be used to calculate the stresses directly without any need for finding the distribution function. Hence obtaining the Hookean dumbbell suspension results presents no difficulty. 

In order to answer this question one has to find out what modifications are necessary in (a) the diffusion equation for the distribution function, and (b) the expression for the stress tensor. Kirkwood and coworkers (39,40,67) and Kotaka (42)w studied this problem for multibead dumbbells including complete hydrodynamic interaction. If one neglects the hydrodynamic interaction entirely, then from the articles cited above one concludes that all the results for rigid dumbbells can be taken over for the multibead dumbbells by replacing X — (,I / 2kT by XN — XN(N + l)/6(iV — 1) everywhere. For the case of complete hydro-dynamic interaction no such simple replacement is possible. 

In order to refine these theories, we need to include the second equations in the corresponding hierarchy of the integral equations for the distribution functions. For the PY g(r) and the CHNC g(r), this procedure is equivalent to the inclusion of more complicated rooted graph integrals in calculating the g( ) 16,17 por the BGY hierarchy, one possible way of the procedure is to truncate the quadruplet distribution function 1234 occurring in the second equation describing the triplet correlation function [= ( "12. is> 23) ra = ki - -il] by a proper combination of [= and g,. , s. The... 

This calculation is quite similar to that for the derivation of Stokes law from kinetic theory, where one has an equation for the distribution function similar to (10.23) for. To obtain Stokes law, one must project the kinetic equation onto the hydrodynamic eigenfunctions, and it is essential to retain terms to first order in the gradients if the proper numerical factor (f = 47TTj/ for specular reflection and = 6 rrr)R for diffuse reflection) is to be obtained. In our calculation, it is also essential to retain the 6(V) terms in the Z)> eigenfunction. If these 6(V) terms are dropped, the result for the rate coefficient ky(z = 0) still has the form of (3.7), kf =kj + k but the zf O result does not agree with (3.6) and (3.8). The gradient terms are essential if one is to obtain the simple z-dependence given by kp z) = k + ooab). How this comes about is clearly demonstrated by the calculation in Appendix E. 

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