Macroscopic Approximation

These two approaches, the supermolecular and the statistical, have up to now been used separately to study distinct manifestations of solvation effects. The model methods for their description may be divided into the discrete (microscopic) and the continual (macroscopic) ones. 

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This is no longer a closed equation for , but higher moments enter as well. The evolution of < Y> in the course of time is therefore not determined by itself, but is influenced by the fluctuations around this average. The macroscopic approximation consists in ignoring these fluctuations, and keeping only the first term in the expansion (8.5). With this approximation therefore (8.4) is valid even when a y) is nonlinear. Thus one obtains as macroscopic equation... 

It is clear that for the very existence of a macroscopic approximation it is necessary that the fluctuations are small. So far we have appealed to experience to argue that this is the case, but it may now be linked to the properties of the master equation by using (8.9). In this equation a2> 0 by definition, and al(K)<0 at y— ye and hence in some neighborhood. (The case of a[(ye) = 0 is treated in ch. XI.) It follows that o2 tends to increase at a rate a2, but this tendency is kept in check by the second term. Hence [Pg.125]

Actually all eigenvalues are imaginary, but we also know that in a suitable macroscopic approximation the bath tends to the equilbrium pB. That amounts to replacing J fB with an operator having only negative eigenvalues. The highest one determines tc. 

This is known as a commutation relation and in algebra, of course, the result would be zero. Classical physics would also predict that the result is zero, but in quantum mechanics it isn t. We can think about this in the following manner. Classical physics is a macroscopic approximation to quantum physics in the limit of large dimensions, quantum physics goes over to classical physics. The commutator, defined as xp - px, is small, but... 

In this section we discuss in some detail a macroscopic approximate model for adsorption kinetics on fractal interfaces. Further details can be found in [8. In diffusion-limited conditions, the balance equations for adsorption on flat surfaces take the form... 

Adsorption kinetics on fractal surfaces can be described with sufficient accuracy by macroscopic approximate models based on a constitutive equation (fiux/concentration gradient) of Riemann-Liouville type. 

The complex Hamaker coeflBcients are predicted from individual self-interacting Hamaker coefficients (for example. An) evaluated by Visser (30) from direct Lifshitz solutions or Ninham and Parsegian s (36) macroscopic approximations. We used combining rules derived from thermodynamics and the Lifshitz theory by Bargeman and Van Voorst Vader (30, 37)... 

Initially, we will focus on the mesoscopic description associated with the radiative transfer equation. Then, we will introduce the single-scattering approximation and two macroscopic approximations the PI approximation and two-flux approximation. AH of these discussions are based on the configuration shown in Fig. 6. Collimated emission and Lambertian emission wiU also be considered in the discussion later they correspond to the direct component and the diffuse component of solar radiation, respectively. Throughout our study, the biomass concentration Cx is homogeneous in the reaction volume V (assumption of perfect mixing), and the emission phenomena in V are negligible. The concentration Cx is selected close to the optimum for the operation of the photobioreactor the local photon absorption rate. 4 at the rear of the photobioreactor is close to the compensation point A.C (see Section 5 and chapter Industrial Photobioreactors and Scale-up Concepts by Pruvost et al.). 

The BBB model is a means of macroscopic approximation to the system on an exclusively electrostatic basis it describes the solvation effects with the aid of classical field theories, primarily electrostatics and hydrodynamics. The nomenclature BBB, an abbreviation of the expression Brass Balls in a Bathtub , originates from Frank [Fr 65]. The more important classical theories included in this group have led to a result only for dilute solutions nevertheless, their refinement has continued up to the present [Ab 79, Be 78, Kr 79, Li 79]. 

Transition states, optimized at macroscopic approximation using the COSMO procedure [40] are characterized by lower degree of O2-C bond formation and greater degree of the C-O, bond cleavage, as compared with the gas-phase calculations, i.e. the transition states are looser in the former case. In this case, the endo isomer of epoxynorbomanes turns out to be more reactive however, the variation of... 

The complete continuum approach was employed in the Kirkwood model on an ab initio level with the basis set of the floating Gauss functions in 1976 [17]. Around that time, a similar formalism for taking the solvent into account was included in the CNDO/2 method [18]. However, such calculational schemes did not gain wide acceptance by reason of excessive expenditure of computer time, difficulties in evaluating some integrals and overall drawbacks inherent in the macroscopic approximation. Eventually some simplified techniques were developed, each of which takes usually one of the components in Eq. (3.8) into account. Next the simplest of these will be considered. 

Introduction Macroscopic Approximations Allometry Orders of Magnitude and Characteristic Time Constants Time Constant Ratios Systems of Multiple Time Constants Pseudocontinuum Models More Complex Situations References... 

Hamaker (1937) first calculated the dispersion (van der Wsials-London) interaction energy for larger bodies by a pair-wise summation of the properties of the individual molecules (assuming these properties to be additive, and non-retarded). Using this macroscopic approximation, the total dispersion energy for two semi-infinite flat parallel bodies (of material i), separated by a distance , in air or in vacuo, becomes (for greater than a few atomic diameters) ... 

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