Asymptotic approximation second scale

With a first-order reaction, the governing equation is linear and could thus be solved without any use of scaling or asymptotic methods. However, we could just as easily assume that the reaction rate is second order in c or add other complications that do not so easily allow an exact analytic solution. The point here is to illustrate the idea of the asymptotic approximation technique, which is easily generalizable to all of these problems. 

Again we can easily calculate the full crossover. As an example Fig. 14.3 shows the scaling function V/s as function of s in the excluded volume limit. In unrenormalized tree approximation this ratio would be a constant proportional to the second virial coefficient. In renormalized theory we see a pronounced variation which rapidly approaches the asymptotic power law. 

As we see, the results of the approximate (asymptotic) analysis are different in the various scales, and cannot be interchanged with each other. The structure of the first correction and remainders allows us to see this feature in detail. The accuracy of the approximation depends on both the small parameter e and the time t. There are terms both in the first correction and in the remainders that grow like r, as t tends to infinity. Such terms, occurring in the asymptotic formulas, are sometimes called secular terms. Due to this result the first formula (2.1) is suitable for times that are not very long t lle. For long times, t-lle, the order of the remainder (et) is the same as that of the leading term. Hence, the approximation turns out to be false for t=lle. The second formula (2.2) is valid for (slow) times that are not very small t e. In this case, the order of the remainder 5 ((E/T) ) is the same as that of the leading term for small times r = e. Hence, the approximation turns out to be false for very small (slow) times t - e. 

Formulas (4.9), (4.10) provide the approximate solution on the slow time scale. From these asymptotics one can see that the second and third components tend to zero, whereas the first one tends to a nonzero constant y = (W/A) at infinity. If we use Taylor expansions of the left sides of Eqs. (4.9) and (4.10), the error estimate of the asymptotic behavior at infinity can easily be derived as follows ... 

With this asymptotic density model (ADM) an advantage over the standard multipole expansion can be achieved, so that the MESP is well approximated not only outside the molecule, but also near the atoms. This scheme has been implemented in SINDOl for first- and second-row elements. The evolving expansion has been truncated after the cumulative dipole moments terms. The three-center integrals have been approximated on the NDDO level. In order to achieve good agreement with ab initio based MESPs, the atomic hybrid moments had to be scaled. In the application to. solvation energies the atomic electronic charges had to be scaled too. 

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