Analytic approximations

This may be solved numerically or within some analytic approximation. The Poisson equation is used for obtaining the electrostatic properties of molecules. 

To gain some qualitative insight into the elfect of lateral interactions it is useful to employ simple analytical approximations in the calculation of the chemical potential, of which the quasichemical approximation is the best suited. We split the chemical potential into a non-interacting part, Eq. (10), and a term due to lateral interactions, and get for the... 

Such analytic approximations based on clusters of particles quickly become mathematically intractable with variation in cluster size, geometry, and range of interactions [12,13]. 

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

It seems however that this problem is not fully cleared up. Thus, it was stated in paper [19] on the basis of experimental data obtained earlier that Ye increased with a filler s concentration in proportion to cp4J. Such a law for the concentration dependence of yield stress at the shear Y(universal Value of the YJY ratio. It is quite probable however, that indicated discrepancies follow just from different ways of analytical approximation of particular experimental data. The only unquestionable fact is that Ye as well as Y grow very sharply with an increase in concentration. 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

Results. Values computed on the John von Neumann Center s Control Data Corporation Cyber 205 are displayed and compared to literature results for other model-chains in Figure 3. Squares indicate values obtained in our Monte Carlo simulations, while diamonds are results of Priest s (12) analytical approximation for very narrow pores. 

As has been shown by Sims et al. (1999) an analytical approximation to chromatographic melting can also be obtained by solving Equations (A7) and (A8) while holding the values of and a held constant, respectively. 

Lee S, Feig M, Salsbury FR Jr, Brooks CL III (2003) New analytic approximation to the standard molecular volume definition and its application to generalized Born calculations. J Comput Chem 24 1348-1356. 

After analysis by means of the streak method, a good analytical approximation for the orientation distribution is known. Orientation desmearing becomes possible. For this purpose the method described in Sect. 9.5 can be utilized. 

Figure 3. Principle for analytical approximation of experimental RHR curves.

Fig. 2.10. Eigenvalues of the Fourier component of the dipole-dipole interaction tensor in two-dimensional infinite lattices. The solid lines are for a triangular lattice, the dashed lines are for an analytical approximation (2.2.9), and the dotted lines are for a square lattice.

The new algebraic equation for a is easily derived by differentiation, and can be solved numerically (a simple analytical approximation is outlined in the problems). From Eq. (17.6) the contribution of the metal to the inverse interfacial capacity is ... 

Equation (2.20) is nonlinear and has no analytical solution. However, analytical approximations exist (Harremoes, 1978 Henze et al., 1995). 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

Petersson et al. had earlier proposed [23] an alternative expression E(n) = Eoo + ]CiSn+i A/(l + 1/2)6 in the context of the CBS methods developed in his group. The summation is carried out numerically in that paper, but in fact an elegant analytical approximation exists for... 

Elsum, I. R., and Gordon, R. G. (1982), Accurate Analytic Approximations for the Rotating Morse Oscillator Energies, Wave Functions, and Matrix Elements, J. Chem. Phys. 76, 5452. 

In order to describe the fluorescence radiation profile of scattering samples in total, Eqs. (8.3) and (8.4) have to be coupled. This system of differential equations is not soluble exactly, and even if simple boundary conditions are introduced the solution is possible only by numerical approximation. The most flexible procedure to overcome all analytical difficulties is to use a Monte Carlo simulation. However, this method is little elegant, gives noisy results, and allows resimulation only according to the method of trial and error which can be very time consuming, even in the age of fast computers. Therefore different steps of simplifications have been introduced that allow closed analytical approximations of sufficient accuracy for most practical purposes. In a first... 

Similar sequences of accurate analytical approximations to Cn(t) have been presented for circular DNAs(82) and linear DNAs with both ends clamped,(29,63) but are not reproduced here. At present, no complete sequence of accurate analytical approximations is available for linear DNAs with only one end clamped. 

Moreover, the construction of functionals for the energy that depend explicitly on the one-particle density is, of course, quite feasible, if one uses analytic approximations for the transformed vector/([p] r). Such an alternative is open if, for instance, one resorts to the use of Fade approximants. We discuss this way of dealing with this problem in Sect. 5. 

Wodak, S. J. and Janin, J. (1980) Analytical approximation to the solvent accessible surface area of proteins. Proc. Natl. Acad. Sci. USA 77, 1736-1740. 

A comparison between the values of Nux calculated from Eq. (131) and the exact numerical results of Stewart and Prober [37] is presented in Table VI for m = 0. The comparison shows that, even though the algebraic approach is strictly valid for large Prandtl numbers, it is sufficiently accurate even for Pr->0(1), particularly when the suction rate is large ( K > 1). It may also be noted that Stewart and Prober have provided some rather accurate analytic approximations for the transfer coefficient. 

These equations, for the case of solid diffusion-controlled kinetics, are solved by arithmetic methods resulting in some analytical approximate expressions. One common and useful solution is the so-called Nernst-Plank approximation. This equation holds for the case of complete conversion of the solid phase to A-form. The complete conversion of solid phase to A-form, i.e. the complete saturation of the solid phase with the A ion, requires an excess of liquid volume, and thus w 1. Consequently, in practice, the restriction of complete conversion is equivalent to the infinite solution volume condition. The solution of the diffusion equation is... 

It remains now to solve Eq. (2.3). Here, there are various approaches, depending on the conditions. When a non-steady-state solution is required, one can introduce the decoupling approximation of Sumi and Marcus, if there is the difference in time scales mentioned earlier. Or one can integrate Eq. (2.3) numerically. For the steady-state approximation either Eq. (2.3) can again be solved numerically or some additional analytical approximation can be introduced. For example, one introduced elsewhere [44] is to consider the case that most of the reacting systems cross the transition state in some narrow window (X, X i jA), narrow compared with the X region of the reactant [e.g., the interval (O,Xc) in Fig. 2]. In that case the k(X) can be replaced by a delta function, fc(Xi)A5(X-Xi). Equation (2.3) is then readily integrated and the point X is obtained as the X that maximizes the rate expression. The A is obtained from the width of the distribution of rates in that system [44]. 

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