Approximate Mean Value Equations

The equations discussed in the preceding sections of this chapter fully describe-in terms of random forces or probability distributions - the effect of the unpredictability of single individual decisions or actions, even if assumptions about trends are made. But it is difficult to find their solutions, especially in multidimensional cases. Often it is simpler and sufficient to set up and solve equations for the mean values and eventually for the variances. 

If the fundamental equation is a Fokker-Planck equation (3.32) the objective is to derive a closed set of equations for the mean values 

Using (3.32, 33) the equation of motion for xk)t is obtained after partial integration as  

For a uni-modal distribution centered around (x), it follows from the exact equation (3.76) that the approximate closed equation is  

If the xk)t are known by solving (3.77), Eq. (3.80) is a system of non-homogeneous linear differential equations with given time dependent coefficients for the variance matrix Oik(t). 

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Alternatively, the mean value equations for may be derived directly, i.e. without using the Fokker-Planck equation, from the master equation. This can be seen in Chap. 4 and leads from the master equation (4.6) directly to the approximate closed mean value equations (4.14). The result is equivalent to that derived via the Fokker-Planck equation. 

Omitting birth/death processes but taking into account individual transitions between attitudes and between subpopulations as described by transition prob-abiUties and see (3.15, 17), and by putting = 1 for simplicity, the following approximate closed mean value equations in full correspondence to their explicit form (4.14) are obtained... 

In order to estimate how long a uni-modal distribution can survive in the migration process the approximate mean value and variance equations (4.53, 31) are solved, as an example, for parameters chosen as for the results of Fig. 4.8 a, b and for a reasonably rhosen initial variance. 

The mean value of the von Mises stress ean be approximated by substituting in the mean values of eaeh variable in equation 4.78 to give ... 

In order to improve upon the mean-field approximation given in equation 7.112, we must somehow account for possible site-site correlations. Let us go back to the deterministic version of the basic Life rule (equation 7.110). We could take a formal expectation of this equation but we first need a way to compute expectation values of Kronecker delta functions. Schulman and Seiden [schul78] provide a simple means to do precisely that. We state their result without proof... 

Let us now turn our interest to the excited states. The energies Ev E2,. .. of these levels are given by the higher roots to the secular equation (Eq. III.21) based on a complete set, and one can, of course, expect to get at least approximate energy values by means of a truncated set. In order to derive upper and lower bounds for the eigenvalues, we will consider the operator... 

It may be noted that using Underwood s approximation (equation 9.10), the calculated values for the mean temperature driving forces are 41.9 K and 39.3 K for counter- and co-current flow respectively, which agree exactly with the logarithmic mean values. 

For a given pair of ions the average value of the distances d(MX) within a coordination polyhedron, d( MX), is approximately constant and independent of the sum of the p values received by all the anions in the polyhedron. The deviation of an individual bond length from the average value is proportional to Ap = pj -p (p = mean value of the p for the polyhedron). Therefore, the bond lengths can be predicted from the equation ... 

The diffusion layer model satisfactorily accounts for the dissolution rates of most pharmaceutical solids. Equation (43) has even been used to predict the dissolution rates of drugs in powder form by assuming approximate values of D (e.g., 10 5 cm2/sec), and h (e.g., 50 pm) and by deriving a mean value of A from the mean particle size of the powder [107,108]. However, as the particles dissolve, the wetted surface area, A, decreases in proportion to the 2/3 power of the volume of the powder. With this assumption, integration of Eq. (38) leads to the following relation, known as the Hixon-Crowell [109] cube root law ... 

It should be pointed out that Equation (8.6), and its counterpart for thermally thick materials, will hold only for Ts > 7 smm, a minimum surface temperature for spread. Even if we include the heat loss term in Equation (8.4) by a mean-value approximation for the integrand,... 

Equation (28) is still exact. To introduce the classical-path approximation, we assume that the nuclear dynamics of the system can be described by classical trajectories that is, the position operator x is approximated by its mean value, namely, the trajectory x t). As a consequence, the quantum-mechanical operators of the nuclear dynamics (e.g., Eh (x)) become classical functions that depend parametrically on x t). In the same way, the nuclear wave functions dk x,t) become complex-valued coefficients dk x t),t). As the electronic dynamics is evaluated along the classical path of the nuclei, the approximation thus accounts for the reaction of the quantum DoE to the dynamics of the classical DoF. 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

The boundary layer equations for an axisymmetric body, Eqs. (1-55), (10-17), and (10-18) have been solved approximately for arbitrary Sc (L4). For Sc oo the mean value of Sh can be computed from Eq. (10-20). Solutions have also been obtained for Sc oo for some shapes without axial symmetry, e.g., inclined cylinders (S34). Data for nonspherical shapes are shown in Fig. 10.3 for large Rayleigh number. The characteristic length in Sh and Ra is analogous to that used in Chapters 4 and 6 ... 

Substitution of equation (7.1.19) into equation (7.1.17) leads to the problem equivalent to the calculation of statistical integrals in a system of m particles placed in volume vq and interacting by the pair-wise forces. It is clear that such a problem could be solved only approximately. The main approximation we use hereafter is the replacement of the mean value of a product of functions for a product of two mean values. Thus... 

For main group metallophthalocyanines, the ring centred redox is the only process to occur. The separation between the first oxidation and reduction potentials corresponds to the energy difference of the HOMO and LUMO, hence to the Q(0,0) absorption band at 670 nm, and is about 1.56 V. Deviation from the mean value becomes large when the size of the metal significantly exceeds the cavity of the Pc ring. The first reduction and oxidation potentials themselves (E° vs. NHE) depend on the polarizing power of the metal ion (Zejy) and are approximated by equation (32). 

Suppose now that the model yh = 0 + ru does not adequately describe the true behavior of the system. Then we would not expect replicate experiments to have a mean value of zero (yv = 0 see Equation 6.11) and the sum of squares due to purely experimental uncertainty would not be expected to be approximately equal... 

A simplifying assumption is that all members of a particular category of alkene, e.g. the 1-n-alkenes, will have about the same enthalpy of hydrogenation and, after enthalpies for a few representative compounds have been precisely measured, the mean value is applicable to other alkenes of the same structural type. This assumption is only true, or approximately so, under well-defined circumstances if applied indiscriminately it can lead to errors in interpreting the energetics of alkenes. The basis for the assumption is clear after recasting equation 5 in the linear form of equation 7. 

As mentioned in Section 2.1, the usual Boltzmann equation conserves the kinetic energy only. In this sense the Boltzmann equation is referred to as an equation for ideal systems. For nonideal systems we will show that the binary density operator, in the three-particle collision approximation, provides for an energy conservation up to the next-higher order in the density (second virial coefficient). For this reason we consider the time derivative of the mean value of the kinetic energy,12 16 17... 

Equation (2.57) represents the negative time derivative of the mean value of the potential energy in the approximation of the second virial coefficient (binary collision approximation). Therefore, we have from (2.44) and (2.57)... 

To illustrate this approximation, let us consider a pressure flow in which the driving-force pressure drop varies with time. We set dp/dt and d /dt in the equations of continuity and motion, respectively, equal to zero and proceed to solve the problem as if it were a steady-state one, that is, we assume AP to be constant and not a function of time. The solution is of the form v = v(x, A P t), geometry, etc.). Because AP was taken to be a constant, v is also a constant with time. The pseudo-steady-state approximation pretends that the foregoing solution holds for any level of AP and that the functional dependence of v on time is v(x,-, t) = v(x,-, AP(t), geometry, etc.). The pseudo-steady state approximation is not valid if the values of A(pv)/At (At being the characteristic time of fluctuation of AP) obtained using this approximation contribute to an appreciable fraction of the mean value of the applied AP. 

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