Smoluchowski equation general form

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. 

Smoluchowski s equation, like the fragmentation equation, can be written in terms of the scaling distribution. Furthermore, general forms may be determined for the tails of the scaling distribution—limits of small mass, xls(t) < 1, and large mass, x/s(t) > 1. The details can be found in van Dongen and Ernst (1988). 

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. 

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... 

It was shown by D. Henry that for particles of arbitrary shape at any ratio of particle size to the ionic atmosphere thickness, 8 = 1/k, the Helmholtz-Smoluchowski equation can be written in a generalized form as ... 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

With this definition of the mobility matrix, the general form of the Smoluchowski equation is written as... 

Smoluchowski,1 treating the problem from a much more general point of view, obtained the same equation (28), and all subsequent workers have found an equation similar in form, notw ithstanding that it is universally agreed now that the double layer is not of the simple, plane parallel type used as an illustration by Helmholtz. All the formulae agree except as to the exact value of the constant. Smoluchowski found Debye and Huckel2 6 for spherical particles ... 

This confirmation of the Smoluchowski derivation also illustrates why it is generally valid at high Ka (outside the relatively thin double layer, general hydrodynamics applies with zero electric field) and why the outcome is independent of a ( ( is independent of a). Smoluchowski already anticipated that the equation therefore remains valid for other than spherical geometries (Including hollow and irregularly formed surfaces) provided xa 1. This was later confirmed by Morrison ),... 

The Smoluchowski type equation (6.22) is of the same formal form as the equivalent equation (5.48) of the one-dimensional case. Its general steady state... 

This form is useful in the generalization of Eq. (485) to fractional diffusion. The investigation of the diffusion equation (485) began when Louis Bachellier (Jules Poincare s student) wrote his thesis in 1900. It was called The Theory of Speculations and was devoted to the evolution of the stock market. Many of the most famous scientists have contributed to our knowledge of diffusion processes, amongst them Einstein, Langevin, Smoluchowski, Fokker, Planck, Levy, and others. 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

Equation (99) differs by a factor of 2/3 from Eq. (102). The latter equation was first derived by Smoluchowski and the former by Hiickel. The latter equation is used for values of ka greater than 10, and the former for values of ka smaller than 1.0. ° Both equations are limiting forms of a more general expression which was derived by Henryk... 

According to the general rules of physicochemical kinetics the slowest process is rate controlling. If flie coagulation step is rate controlling, namely, when condition (34) is valid, then the coalescence is rapid and flie general equation of the theory in Ref 38 is reduced to second-order kinetics, i.e., to Smoluchowski s equation [Eq. (27)]. Floes composed of three, four, etc., droplets cannot be formed, because of rapid coalescence within the floe. In this case the structure of the floes becomes irrelevant... 

If the depth of the primary minimum (that on the left from the maximum in Fig. 6a) is not so great, i.e., the attractive force which keeps the drops together is weaker, then the floes formed are labile and can disassemble into smaller aggregates. This is the case of reversible flocculation (3). For example, a floe composed of i+j drops can be split into two floes containing i and j drops. We denote the rate eonstant of this reverse process by (see Fig. 20a). In the present case bofli the straight process of flocculation (Fig. 19) and the reverse process (Fig. 20a) take simultaneously plaee. The kinetics of aggregation in this more general and eomplex case is described by the Smoluchowski set of equations, Eq. (96), where one is to substitute ... 

Given the central role of the expulsion rate constant for micellar stability, formation, and dissociation, it is essential to determine the physical governing factors and functional form. Aniansson and Wall based their calculations [54] on a general diffusion in an external potential. In this approach, the diffusion coefficient, D(r) is dependent on the position, r, due to the potential V(r). In a sphero-symmetric system, we can imagine that the diffusion of a unimer only depends on the distance, r, from the origin and this problem can be summarized in a Einstein-Smoluchowski type equation ... 

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