Diffusion approximations

Strekalov M. L. Semiclassical theory of rotational energy relaxation in diffusion approximation, Khim. Fiz. 7, 1182-92 (1988). 

In the fluid model the momentum balance is replaced by the drift-diffusion approximation, where the particle flux F consists of a diffusion term (caused by density gradients) and a drift term (caused by the electric field ) ... 

In order to be able to explain the observed results plasma modeling was applied. A one-dimensional fluid model was used, which solves the particle balances for both the charged and neutral species, using the drift-diffusion approximation for the particle fluxes, the Poisson equation for the electric field, and the energy balance for the electrons [191] (see also Section 1.4.1). 

An intuitive explanation of biofilm drug resistance is that antimicrobial compounds are physically excluded from the community by the barrier properties of the glycocalyx. Such intuition however envisages that the glycocalyx functions as a biocide-impermeable umbrella, but since it generally possesses a diffusivity approximating that... 

The kinetic-diffusion approximation predicts an attachment coefficient similar to the hybrid theory for all CMDs and for both Og m 2 and 3 (Figs. 3 and 4). The advantage of this theory is that the average attachment coefficient can be calculated from an analytical solution numerical techniques are not required. 

Unattached fractions of RaA (at t = °°) for two mine aerosols and for a typical room aerosol are shown in Table III. It is usually assumed that the attachment of radon progeny to aerosols of CMD < 0.1 ym follows the kinetic theory. In Table III it is apparent that the hybrid and kinetic theories predict similar unattached fractions for monodisperse aerosols. However, for more polydisperse aerosols, the kinetic theory predicts lower unattached fractions than the diffusion theory and thus the diffusion theory is the more appropriate theory to use. It is also evident that the kinetic-diffusion approximation predicts unattached fractions similar to those predicted by the hybrid theory in all cases. 

The light fluxes are now linear functions of the depth coordinate z as it is predicted also by Fick s first law for steady-state diffusion without sink. For weak absorption, the equations for Td and Ro of the Kubelka-Munk formalism are also directly equivalent to the results of the diffusion approximation. Comparing Eqs. (8.22) and (8.23) with Eqs. (8.11), (8.12), and (8.14) under diffuse irradiation or under //o = 2/3, the Kubelka-Munk coefficients can be expressed by<31 34)... 

A coarse bubble aeration system for McCook Reservoir requires only 1,160 diffusers, approximately one-third of the 3,438 diffusers needed for a fine bubble aeration system. However, significantly less air flow is needed for the fine bubble diffusers in comparison with the coarse bubble diffusers. At the more common depth of 10 m, the air flow required by the fine bubble diffusers was 39% of that required by the coarse bubble diffusers. These considerations and the mixing requirements of the reservoir are a part of the aeration system design. 

A useful comparison may be made between bent, flat, and linear molecules by considering the diffusion coefficients for ethane, ethene, and ethyne. In the sinusoidal channel, ethane diffuses the slowest, ethene approximately 30% faster, and ethyne 3 times as fast. In the straight channel and parallel to the z-axis, ethene and ethyne both diffuse approximately 3 times faster than ethane. These ratios are consistent with the relative cross-sectional area of the three C2 hydrocarbons. 

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. 

Comparison with (X.2.16) shows the following drastic differences. The dominant term of (X.2.16) is absent and therefore no equation for the macroscopic part of X can be extracted. In other words, on the macroscopic scale the system does not evolve in one direction rather than the other. The remaining evolution of P is merely the net outcome of the fluctuations. Accordingly the time scale of the change is a factor slower than in the preceding case, compare (X.2.14). Since P is not sharply peaked the coefficients a(x) cannot be expanded around some central value but they remain as nonlinear functions in the equation. The first line of (1.4) contains the main terms and is called the diffusion approximation... 

The diffusion approximation (1.5) is the nonlinear Fokker-Planck equation (VIII.2.5). In fact, we have now justified the derivation in VIII.2 by demonstrating that it is actually the first term of a systematic expansion in Q 1 for those master equations that have the property (1.1). Only under that condition is it true that the two coefficients... 

All this is true regardless of the specific form of the transition probabilities or w(r r ). If they are isotropic and the jumps are small compared with the distance over which u(r) varies one may replace the operator by its diffusion approximation ... 

With the diffusion approximation, integration over v, gR becomes ... 

Up to now we have been considering defect diffusion in continuous approximation, despite the fact that crystalline lattice discreteness was explicitly taken into account defining the initial distribution for geminate pairs. Note, however, that such continuous diffusion approximation is valid only asymptotically when defects (particles) before recombination made large number of hops (see Kotomin and Doktorov [50]). This condition could be violated for recombination of very close defects which can happen in several hops. The lattice statement of the annihilation kinetics has been discussed in detail by Schroder et al. [3, 4, 83], Dederichs and Deutz [34]. Let us consider here just the most important points of this problem. 

These results yield an impressive justification for the use of the continuous diffusion approximation even for short distances and small recombination regions. 

As it follows from equation (4.3.29), in another extreme case, A — 00, ftr tends to r, as it is indicated by a broken line in the insert of Fig. 4.14. It is also demonstrated in [85] that the continuous diffusion approximation, equation (4.1.63), gives quite reasonable reaction rates up to A 5ro. This comes from the fact that K(t) is a convolution of the correlation function and the exponentially decaying reaction probability o(r), that is, the essential deviation of the reaction profile from the diffusion limit does not affect the reaction rate considerably. 

The results obtained in these computer simulations of the hopping reactions were applied in [86] to the centre recombination in KC1-T10 stimulated by step-like temperature increase. As it is clearly seen from Fig. 4.15, taking directly into account finite hop lengths (A = ao/(2 /2) 2.2 A to be compared with ro < 1 A of the electron Tl° centre) permits us to obtain a much better agreement with the experimental data than the standard continuous diffusion approximation (curves 1 and 3, respectively). 

Fig. 4.15. A comparison of experimental delayed kinetics of an increase of tunnelling luminescence intensity after sudden change of their mobility (temperature increase from 175 to 180 K) in KC1 with theory [86], 1 - hopping kinetics for A = 2n> obtained by means of equation (4.4.1), 2 - experimental curve, 3 - results of continuous diffusion approximation...

It is of importance to point out that if the right-hand side is truncated after two terms (diffusion approximation), the last relation leads to an expression similar to the familiar Fokker-Planck equation (4.116). The approximation of a master equation of a birth-death process by a diffusion equation can lead to false results. Van Kampen has critically examined the Kramers-Moyal expansion and proposed a procedure based on the concept of system size expansion.135 It can be stated that any diffusion equation can be approximated by a one-step process, but the converse is not true. 

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