Smoluchowski equation generalization

We shall meet more general Fokker-Planck equations the special form (1.1) is also called Smoluchowski equation , generalized diffusion equation , or second Kolmogorov equation . The first term on the right-hand side has been called transport term , convection term , or drift term the second one diffusion term or fluctuation term . Of course, these names should not prejudge their physical interpretation. Some authors distinguish between Fokker-Planck equations and master equations, reserving the latter name to the jump processes considered hitherto. 

In the general case, whether the e-ion pair is isolated or not, the probability density P(r, t) that an electron will remain extant at time t is given by the Smoluchowski equation... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

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... 

It should also be noted that in the limit of kRs - 0, Equation (47a) reduces to the Hiickel equation, and in the limit of kRs - oo, it reduces to the Helmholtz-Smoluchowski equation. Thus the general theory confirms the idea introduced in connection with the discussion of Figure 12.1, that the amount of distortion of the field surrounding the particles will be totally different in the case of large and small particles. The two values of C in Equation (40) are a direct consequence of this difference. Figure 12.5a shows how the constant C varies with kRs (shown on a logarithmic scale) according to Henry s equation. 

Friauf et al. [326] have presented the general solution for the survival probability of an ion-pair in the presence of scavengers and an external electric field based on the analysis of the Debye—Smoluchowski equation by Hong et al. [72, 323—326] (Chap. 3, Sect. 1). Further comments have been made by Tachiya [357], Tachiya and Sano [358a], and Sano [358b]. The expression is only valid at low concentrations of scavenger, that is cs alkane solvents (see Sect. 2.5). 

It is clear from Eqs. (1) and (2) that calculation of p and F is essentially a matter of obtaining averages of the forms and <[(/(ro) /(ri)]2>- These may sometimes be obtained without solving the Smoluchowski equation explicitly, by means of a method we have used previously.4 We now restate this method in the slightly more general form needed for our present purpose. 

Now we present the standard derivation of the Fokker-Planck equation for polymers in solution. (Terminology can often be confusing in the present instance, the equation of interest is also called the Smoluchowski equation, and may be regarded as a limiting case of a more general Fokker-Planck equation, or a Kramers equation.)... 

Instead of the one-step process we now consider the generalized diffusion or Smoluchowski equation (1.9). Take a finite interval Lsplitting probabilities nL(y) and nR(y). They obey a differential equation with the adjoint operator... 

Exercise. Find for the general Smoluchowski equation (5.1) between two regular boundaries L, R the condition that two mixed boundary conditions at L and R are compatible. 

New difficulties arise when we try to take into account the dynamical interaction of particles caused by pair potentials U(r) mutual attraction (repulsion) leads to the preferential drift of particles towards (outwards) sinks. This kind of motion is described by the generalization of the Smoluchowski equation shown in Fig. 1.10. In terms of our illustrative model of the chemical reaction A + B —> B the drift in the potential could be associated with a search of a toper by his smell (Fig. 1.12). An analogy between Schrodinger and Smoluchowski equations is more than appropriate indeed, it was used as a basis for a new branch of the chemical kinetics operating with the mathematical formalism of quantum field theory (see Chapter 2). 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

Modeling with the Smoluchowski-like equation generalized to take into account FSSE is not limited to the simple RAf polymerization. A kinetics approach similar to that described in this section have been used to study crosslinking reactions of epoxy resins with components introduced into the system at different times [17]. Kinetic equations analogous to Eq. (101) have been derived [48] for an RA2 + R B2 system as well as for systems containing 3-functional monomers having functional groups of intrinsically different reactivities [49]. 

The Smoluchowski equation was thought of as failing beyond the gel point [22]. Dusek [53] and Ziff [54] have demonstrated this not to be true for the Flo-ry-Stockmayer model. At the same time Leyvraz and Tschudi [32] presented an elegant and more general derivation of the solutions of the Smoluchowski equation both prior to and beyond the gel point. 

Of interest also are the results of applications of the Smoluchowski equation for systems with more complex aggregation physics than that provided by bilinear kernels. Leyvraz and Tschudi [32] conjectured that for the kernel ICy-fi) ) gelation occurs only when co > 1/2. The post-gelation behavior of a general system with a multiplicative kernel given by Eq. (60) has been analyzed by van Don-gen et al. [56]. By assuming that the distribution past the gel point could be expressed through that at t=tc, thus... 

An interesting modification of the Smoluchowski equation leading to the so called generalized Smoluchowski equation [73] can be obtained by adding the terms describing the rates at which clusters or polymer molecules undergo fragmentation ... 

Various analytical solutions to the von Smoluchowski equation set have been developed over the years, originating with the solution presented by von Smoluchowski coinciding with the expression of coagulation theory in 1917 [1]. No general analytical solutions to the von Smoluchowski equation are available, but many expressions have been developed with simplifying assumptions. This section will review the available analytical solutions and discuss the uses and limitations of the solutions. 

K.M. Hong and J. Noolandi, Solution of the Smoluchowski equation with a Coulomb potential. I. General results. J. Chem. Phys. 68(11) 5163-5171 (1978). 

The numerical coefficient 5 has entered here only because the Einstein-Smoluchowski equation = 2Dt for a one-dimensional random walk was considered. In general, it is related to the probability of the ion s jumping in various directions, notjust forward and backward. For convenience, therefore, the coefficient will be taken to be unity, in which case... 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

As a later generalization we present a more rigorous derivation of the Helmholtz-Smoluchowski equation for high yq, in which the curvature of the field is explicitly taken into account. This method, given by Anderson and Prleve l, may be considered as an elaboration of the Smoluchowski theorem for the case of electrophoresis of large spheres. Surface conduction is still Ignored. 

According to general experience the Influence of sol concentration Is absent when the volume fraction remains of the order of a few percent. This may be concluded from older experiments and from recent electroacoustic studies, discussed in sec. 4.5d. Experiments involving mlcro-electrophoresls are not suited to stud3dng the volume fraction effect because the required extreme dilution may readily lead to spurious adsorption on the particles. Reed and Morrison studied theoretically the hydrodynamic interactions between pairs of different particles in electrophoresis they corrected the Helmholtz-Smoluchowski equation for various distances between the (spherical) particles and values of the electroklnetlc potentials of the particles, and... 

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