Prescribed diffusion

In the spirit of prescribed diffusion, Samuel and Magee write the normalized distribution of radicals at time t as... 

Equation (7.24) can be compared with the corresponding prescribed-diffusion equation, namely d(N)/dt = (l/2)pi(t)(N)((N) - 1) (Clifford et al., 1982a). These two equations would be equivalent if (N2) = (N)2—that is, the variance of N would be zero. This implies that all spurs would have exactly the same number of radicals at a given time. Since stochasticity denies this, a considerable difference is expected between the results of these two methods however, this difference tends to decrease with the spur size Ng (Clifford et al, 1982a Pimblott and Green, 1995). 

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

Mozumder (1971) calculated F(t) by the prescribed diffusion method. For the isolated ion-pair case, the solution appears in (7.28) for the multiple ion-pair case, further approximation was introduced in the nature of mean force acting on an electron, by which the problem was reduced to that of a collection of isolated... 

The limitation of the prescribed diffusion approach was removed, for an isolated ion-pair, by Abell et al. (1972). They noted the equivalence of the Laplace transform of the diffusion equation in the absence of scavenger (Eq. 7.30) and the steady-state equation in the presence of a scavenger with the initial e-ion distribution appearing as the source term (Eq. 7.29 with dP/dt = 0). Here, the Laplace transform of a function/(t) is defined by... 

In the following two sections, deterministic (prescribed diffusion and FACSIMILE) and stochastic (random flights and IRT) approaches for the modeling of radiation chemical kinetics will be described. Then representative calculations for simple aqueous systems will be shown The stochastic approach to modeling radiolysis kinetics is more physically realistic than the primitive deterministic models however, it is also more conceptually advanced, requiring a more detailed (fuller) knowledge of the system under consideration. 

This connection is further discussed in Sect. 2.3. (The prescribed diffusion method of solution of this problem has been used by Mozumder [76], but it was shown to be unsatisfactory by Hong and Noolandi [72].) Recently, Clifford et al. [322] have shown that the diffusion and drift equation can be re-cast in a form which is symmetric to the sign of the coulomb potential. Consequently, the care necessary to define the sign of the Onsager distance, rc, is no longer required and it is sufficient to solve for an attractive potential. This is particularly valuable when performing numerical studies, as only attractive potentials need be considered (and such situations are more easily solved numerically than repulsive cases). 

Miller [441] has combined the prescribed diffusion approximation for modelling the decay of spurs with the Monte Carlo model of spur formation developed by Wilson and Paretzke [435]. This allows the position of and energy deposited in spurs to be included rather more satisfactorily, but it does not remove the inherent imperfections of the prescribed diffusion approximation. 

See also Chap. 9, Sect. 6.3.) This is very similar to the stochastic prescribed diffusion approximation [eqn. (182a)]. The solution of eqn. (183) is identical in form to that shown in eqn. (182b) but with p(f) = — 1/2 In IIa(f), though this identification is not valid since n and X are different quantities. 

Considering the diffusion-recombination stage below, we neglect an interaction between the thermalized positron and its blob. This approximation, as we discussed above, assumes that the appearance of a positive potential in the blob, caused by outdiffusion of electrons, is nearly cancelled by the negative potential caused by e+ screening inside the blob. In this case we can apply the prescribed diffusion method to obtain the solution of Eq. (17). Let us write Cj(r,t) in the following form ... 

Numerical Experiments with the Classical Finite Difference Scheme Principles for Constructing Special Finite Difference Schemes Special Finite Difference Schemes for Problems (2.12), 2.13) and (2.14), (2.15) Numerical Experiments with the Special Difference Scheme Numerical Solutions of the Diffusion Equation with Prescribed Diffusion Fluxes on the Boundary... 

III. NUMERICAL SOLUTIONS OF THE DIFFUSION EQUATION WITH PRESCRIBED DIFFUSION FLUXES ON THE BOUNDARY... 

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