First Order Approximation

The first order (i.c. ] 1) approximation of the CML system defined by equation 8.44 (using either of the two methods defined above) is given by an elementary fc = 2, r = 1 CA. Since there are only 32 such rules, the particular CA rule corresponding to a CML system with parameters e and s may be found directly by calculating the outcome of each of the five possible local states. Looking at the first-order step function fi x) in equation 8.47, we can identify the absorbing state X = X with the CA state ct = 0, and x = 1/2 with a = 1. 

The other rule-table entries are found is a similar manner. 

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The most important classes of functionalized [60]fullerene derivatives, e.g. methanofullerenes [341, pyrrolidinofullerenes [35], Diels-Alder adducts [34i] and aziridinofullerene [36], all give rise to a cancellation of the fivefold degeneration of their HOMO and tlireefold degeneration of their LUMO levels (figure Cl.2.5). This stems in a first order approximation from a perturbation of the fullerene s 7i-electron system in combination with a partial loss of the delocalization. 

The quantities given by Eq. (56) represent the first-order approximation for the adiabatic bending potentials. If these potentials are known, V can be... 

If S is a single atom or a group of atoms with the bonds attached to the same atom (such as a CHi group), then we have the additivity of bond properties, liie first-order approximation, as given by Eq. (3). 

The next higher order of approximation, the first-order approximation, is obtained by estimating molecular properties by the additivity of bond contributions. In the following, we will concentrate on thermochemical properties only. 

These equations hold if an Ignition Curve test consists of measuring conversion (X) as the unique function of temperature (T). This is done by a series of short, steady-state experiments at various temperature levels. Since this is done in a tubular, isothermal reactor at very low concentration of pollutant, the first order kinetic applies. In this case, results should be listed as pairs of corresponding X and T values. (The first order approximation was not needed in the previous ethylene oxide example, because reaction rates were measured directly as the total function of temperature, whereas all other concentrations changed with the temperature.) The example is from Appendix A, in Berty (1997). In the Ignition Curve measurement a graph is made to plot the temperature needed for the conversion achieved. 

In a similar manner, if an isotropic body is subjected to hydrostatic pressure, p, i.e., ax = ay = 02 = -p, then the volumetric strain, the sum of the three normal or ewensional strains (the first-order approximation to the volume change), is... 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

In the linear or first-order approximation, it is postulated that these activity coefficient terms are directly proportional, as in Eq. (8-92) ... 

X is an acidity function based on the first-order approximation, Eq. (8-92). Values of X have been assigned by an iterative procedure. The data consist of values of Cb/cbh+ as functions of Ch+ for a large number of indicators. For each indicator an initial estimate of pXbh+ and m is made and X is calculated with Eq. (8-94). This yields a large body of X values, which are fitted to a polynomial in acid concentration. From this fitted curve smoothed X values are obtained, and Eq. (8-94), a linear function in X. allows refined values of pXbh + and m to be obtained. This procedure continues until the parameters undergo no further change. Table 8-20 gives X values for sulfuric and perchloric acid solutions. ... 

Let us compare the expressions for pXbh+ under ideal (Eq. 8-95), nonideal zero-order approximation (Eq. 8-96), and nonideal first-order approximation (Eq. 8-97) conditions ... 

This is the Verlet algorithm for solving Newton s equation numerically. Notice that the term involving the change in acceleration (b) disappears, i.e. the equation is correct to third order in At. At the initial point the previous positions are not available, but may be estimated from a first-order approximation of eq. (16.29). 

Mean Field Approximation as a first order approximation, we will ignore all correlations between values at different sites and parameterize configurations purely in terms of the average density at time t p. The time evolution of p under an arbitrary rule [Pg.73]

Table 8.6 traces the evolution of the approximating CA rules as e increases from 0 to 1 for 2 < s < 3. We see that as e increases, only 7 of the 32 possible rules are actually visited. Nonetheless, even at this crude first order approximation, a remnant of the CML s transition to spatiotemporal intermittency remains. In particular, there is a threshold value of e, = 2 — 4/s, that acts as a boundary point below which the approximating CA-rule is simple-periodic (class 1 or 2) and above which it is complex (class 3 or 4). The surprising fact is not that the CML s transition appears to bo approximated by the CA-rule path - after all, allowing for finite-length computer words, the CML itself is essentially just a very-high order CA... 

The recovery of the Navier-Stokes also follows from momentum conservation but requires that a first-order approximation be made to the full solution to the Boltzman-equation. We sketch the main steps of the recovery below (see [huangk63]). 

In order to obtain a first-order approximation to the exact solution of the Boltz-man equation, we first write... 

Using this expression for A, we now calculate the first-order approximation to the pressure-tensor Pij ... 

The general heat-conduction equation, along with the familiar diffusion equation, are both consequences of energy conservation and, like we have just seen for the Navier-Stokes equation, require a first-order approximation to the solution of Boltz-man s equation. 

Eq. (12.14) is recovered. The presence of traps lowers ihe mobility as expected. The essential message of Figures 12-17 and 12-18 is that, to a first order approximation, Eq. (12.14) maintains the icmperalurc dependency of the mobility if one replaces the disorder parameter by an effective disorder parameter ocJj or, equivalently, an effective width of the DOS that depends on both the concentration and the depth of the traps. Deviations from the behavior predicted by Eq. (12.14) become important for ,>0.3 eV, notably at lower temperatures. It is noteworthy, though, that the T- oo intercepts of p(7), if plotted as In p versus 7 2, vary by no more than a factor of 2 upon varying trap depth and concentration. 

Furthermore, LandS s theory only represents a first-order approximation, and the L and S quantum numbers only behave as good quantum numbers when spin-orbit coupling is neglected. It is interesting to note that the most modem method for establishing the atomic ground state and its configuration is neither chemical nor spectroscopic in the usual sense of the word but makes use of atomic beam techniques (38). 

The simplest form of the quasi-chemical theory that is of interest in the present connection is the first-order approximation, which considers the distribution of nearest-neighbor pairs as affected by nonzero values of the linear combination,... 

The first order approximation may be found by assuming p and A to be small, but not zero. If Eq. (1-86) is multiplied by p, all coefficients on the left side, being proportional to some power of i, give terms... 

For a. first-order approximation, a straight line is fit between the points A = 0 and A to get the first-order, forward difference approximation... 

Using a first-order approximation for the derivative in Equation (9.4), the wall boundary condition becomes... 

To conclude, we think that valuable information can ce obtained from such relaxation experiments. They could provide a direct, kinetic proof of the conjecture that the Berry mechanism is the most probable one, as is indicated by some recent experimental and theoretical work. The applicability of this model is however restricted to situations where the energy of the molecule does not depend on the distribution of the ligands on the skeleton and where, as a consequence, there is one rate constant for each process. If this is not true, the present description could be the first-order approximation of a perturbation calculation. Such a work will be undertaken soon. 

The heat loss to the melting polymer was assumed (for a first order approximation) to be negligible compared to the heat loss by convection. This is one area of the model which could profit from more study to determine the exact magnitude of energy exchange with the polymer. 

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

Equation 6.80 is exact rather than a first order approximation as Equation 6.74 is. This is simply because Equation 6.80 is Equation 6.74 evaluated at the point of linearization, k0). Thus Equation 6.80 can be used to compute g(t) as... 

To a first order approximation, the scattering potential of a crystal may be represented as a sum of contributions from isolated atoms, having charge distributions of spherical symmetry around their nuclei. In a real crystal the charge distribution deviates from the spherical symmetry around the nucleus and the difference reflects the charge redistribution or bonding in the crystal. The problem of experimental measurement of crystal bonding is therefore a problem of structure factor refinement, i.e. accurate determination of the difference between the true crystal structure factors... 

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