Generalized Smoluchowski equation

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... 

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. 

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 ... 

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 . ... 

The steady-state distribution can be obtained from the generalized Smoluchowski equation by imposing a detailed balance condition, i.e. by replacing Ns(t) by N,(oo) and setting the derivative equal to zero [16]. There are different options to actually solve the problem. In this case we will perform a scaling analysis of Equation 8 and hence obtain information about the... 

Family et al. [15] proposed that the aggregation and fragmentation kernels satisfied the scaling forms given in Equations 10 and 11 with a = k = X and P = 2-co. It was also assumed that the Smoluchowski equation was invariant under a scaling transformation k —> X-k and s -> V-5. As a result, in the steady state limit of the generalized Smoluchowski equation, i.e. if fragmentation is included, it follows that... 

X is the crossover time when the balance between the aggregation processes is established. The validity of this approach was confirmed by Montecarlo simulations of the full equation using a constant coagulation kernel and a breakup probability equal to k(i+j) where a and k were adjustable parameters [15]. Spatial fluctuations were compensated by cluster breakup and the generalized Smoluchowski equation had a critical dimension dc < 1. 

The propagator in (9.17) may then be expanded in powers of L. The analysis is similar to that used in the reduction of the Fokker-Planck equation to a generalized Smoluchowski equation. We obtain... 

Here v = v — (v) is the fluctuating component of the velocity and the overbar or angular brackets denote an average over a normalized velocity probability density. Thirdly, by using the analytical solution of the FPE, in conjuction with its first two-velocity moment equations given above, a convective-diffusion equation, or generalized Smoluchowski equation (GSE), may be derived without the use of non-perturbative approximations in the diffusive limit. 

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... 

Kinetic methods describing the evolution of distributions of molecules by systems of kinetic differential equations (obeying either the classic mass action law of chemical kinetics or the generalized Smoluchowski coagulation process). 

Alternatively, one can make the reactivity of groups dependent on the size and shape of the reacting molecule. In such a way, for instance, the effect of steric hindrances, cyclization, and diffusivities of the molecules can be modeled using generalized Smoluchowski coagulation differential equations. 

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 ... 

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). 

When there are two or more reactants diffusing throughout space, the motion of each reactant influences that of all the others due to the solvent being squeezed from between the approaching reactants. The effect of this hydrodynamic repulsion on the rate of a diffusion-limited reaction was discussed in Chap. 8, Sect. 2.5. In this section, this discussion is amplified. First, the nature of the hydrodynamic repulsion is discussed further and then a general diffusion equation for many particles is derived. The two-particle diffusion equation is selected and solved subject to the usual Smoluchowski initial and boundary conditions to obtain the rate coefficient. Finally, this is compared with the rate coefficients in the absence of hydrodynamic repulsion and from experiments. 

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.)... 

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. 

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... 

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. 

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... 

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... 

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