Short time approximation

Our more rudimentary approach is basically founded on an ansatz choice for the quantity = of higher quality with respect to the short-time approximation... 

For large P, (3/P is small and it is possible to find a good short-time approximation to the Green function p. This is usually done by employing the Trotter product formula for the exponentials of the noncommuting operators K and V... 

This is called primitive in the sense that the short-time approximation, truncated after the first term, is in its crudest form. Nonprimitive schemes would be those that would improve this approximation, for instance by replacing the bare potential V(x) by an effective quantum potential (see [142, 149]). 

Finally we shall derive the equation used by Bixon and Jortner. Suppose that an intramolecular vibrational mode, say Qi, plays a very important role in electron transfer. To this mode, we can apply the strong-coupling approximation (or the short-time approximation). From Eq. (3.40), we have... 

We then apply the short-time approximation (i.e., the strong coupling) to... 

For unsteady lateral flame spread, Equation (8.25) can be applied in a quasi-steady fashion. We use the short-time approximation for the surface temperature (Equation (7.40)),... 

Thus an element of water 1 micron in size would lose its identity in a very short time, approximately... 

Corresponding values for F, evaluated by finite differences from the governing equations, are shown in Fig. 3.15. As PCp increases, circulation causes F to rise more rapidly, but the effect is not large Tp for a given F decreases by at most a factor of three as Pep/(1 + k) increases from zero to infinity. In fact, the Kronig-Brink curve in Fig. 3.17 is closely approximated by Eq. (3-80) with p replaced by 2.5 p. Thus circulation causes an effective diffusivity at most 2.5 times the molecular value. For negligible external resistance, the short-time approximation given by Eq. (3-72) becomes... 

Figure 1 Vibrational transition probabilities for colinear CO2 in Ar model plotted versus time (ps). The label a isfor the symmetric stretch to ground state transition, the label b is for the asymmetric stretch to ground state transition, and the label c isfor the asymmetric stretch to symmetric stretch transition. The dotted curves employ short time approximations, and the solid curves do not...

By making a short-time approximation with respect to the first delay period r, it is possible to obtain approximate analytical expressions for t T), which turn out to be very accurate for times longer than the bath correlation... 

Some care is needed in deriving a short-time approximation for qF since introduction of eqn. (35a) would make the bracketed term zero. Evidently, the second-order term in the series expansion of exp (l21) erfc (ltv2) should be taken into account. Since this term equals l2t [22], eqn. (36) reduces for ltv2 <0.1 to the simple form... 

There exists another prescription to extend the hydrodynamical modes to intermediate wavenumbers which provides similar results for dense fluids. This was done by Kirkpatrick [10], who replaced the transport coefficients appearing in the generalized hydrodynamics by their wavenumber and frequency-dependent analogs. He used the standard projection operator technique to derive generalized hydrodynamic equations for the equilibrium time correlation functions in a hard-sphere fluid. In the short-time approximation the frequency dependence of the memory kernel vanishes. The final result is a... 

The results presented from vibrational relaxation calculations87 88 97 102 show that the method is numerically very feasible and that the short time approximations are welljustified as long as the energy difference between the initial and final quantum states is not too small. It is also found that the crossover from the early time quantum regime to the rate constant regime can be due to either phase decoherence or due to the loss of correlation in the coupling between the states, or to a combination of these factors. The methodology described in Section n.C has been formulated to account for both of these mechanisms. 

As noted earlier, the fundamental equations of the QCL dynamics approach are exact for this model, however, in order to implement these equations in the approach detailed in section 2 the momentum jump approximation of Eq.(14) is made in addition to the Trotter factorization of Eq.(12). Both approximations become more accurate as the size of the time step 5 is reduced. Consequently, the results presented below primarily serve as tests of the validity and utility of the momentum-jump approximation. For a discussion of other simulation schemes for QCL dynamics see Ref. [21] in this volume. The linearized approximate propagator is not exact for the spin-boson model. However when used as a short time approximation for iteration as outlined in section 3 the approach can be made accurate with a sufficient number of iterations [37]. 

Within the short-time approximation, the center of the wavepacket remains at Re while its center in momentum space, V t, moves outward with constant velocity Vr = —dV/dR. 

For the strong-coupling case, JAS,- 1, the short-time approximation can be used in this case Eq. (4.38) becomes... 

In photo-induced ET the Marcus equation [58-60] is often used and it can be derived from Eq. (96) by using the short-time approximation, i.e.,... 

Derive Marcus equation from Eq. (97) using the short-time approximation and classical limit. 

The solutions of Equation (6.63) under long-time and short-time approximations are given, respectively, by ... 

Equation (4.2) reveals that the fraction of drug released is linearly related to the square root of time. However, (4.2) cannot be applied throughout the release process since the assumptions used for its derivation are not obviously valid for the entire release course. Additional theoretical evidence for the time limitations in the applicability of (4.2) has been obtained [10] from an exact solution of Fick s second law of diffusion for thin films of thickness S under perfect sink conditions, uniform initial drug concentration with cq > cs, and assuming constant diffusion coefficient of drug T> in the polymeric film. In fact, the short-time approximation of the exact solution is... 

Fickian diffusional release form a thin polymer film. Equation (4.3) gives the short-time approximation of the fractional drug released from a thin film of thickness S. 

From the values of A listed in Table 4.1, only the two extreme values 0.5 and 1.0 for thin films (or slabs) have a physical meaning. When A = 0.5, pure Fickian diffusion operates and results in diffusion-controlled drug release. It should be recalled here that the derivation of the relevant (4.3) relies on short-time approximations and therefore the Fickian release is not maintained throughout the release process. When A = 1.0, zero-order kinetics (Case II transport) are justified in accord with (4.4). Finally, the intermediate values of A (cf. the inequalities in Table 4.1) indicate a combination of Fickian diffusion and Case II transport, which is usually called anomalous transport. 

We are interested in supplying a short-time approximation for the solution of the previous equation. There are two ways to calculate this solution. The direct way is to make a Taylor expansion of the solution. The second, more physical way, is to realize that for short initial time intervals the release rate n(t) will be independent of n(t). Thus, the differential equation (4.13) can be approximated by n (t) = — ag (t). Both ways lead to the same result. 

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