The Short-Time Approximation

This approximation assumes that T is much less than the decay times of the viscoelastic medium. It is the assumption underlying the work of Hunter (1960) and Graham (1978). The viscoelastic functions are expanded in a Taylor expansion about t = 0, and only linear terms are retained. Therefore, from (5.3.9)  

In this approximation, the creep function has the form associated with a Maxwell s solid, as given by (1.6.15). Equation (5.3.27) becomes 

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

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)),... 

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... 

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. 

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. 

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. 

When only smooth electronic spectra can be obtained, no vibronic structure is available as a check on the Raman determined displacements and the highest accuracy will be obtained from analysis of the full Raman excitation profiles by using Eqs. (S)-(7). Pre-resonance Raman data and the short-time approximation can also be used as a first estimate of the displacements. Although obtaining full experimental profiles is time-consuming, these profiles will provide the most accurate data for calculating the distortions. 

The most efficient way of using Raman data is to take one spectrum in preresonance with the absorption band of interest so that the short time approximations are valid. The pre-resonance Raman data for W(CO)spyr-idine are given in Table 3. The relative intensities of the peaks were determined by integrating the peaks. All of the peaks in the experimental spectrum having intensities greater than three percent of that of the most intense peak were measured and used in the calculations. 

We can thus expect from the short-time approximation that quantum noise does not significantly affect the classical solutions when the initial pump field is strong. We will return to this point later on, but now let us try to find the short-time solutions for the evolution of the quantum noise itself—let us take a look at the quadrature noise variances and the photon statistics. Using the operator solutions (94) and (95), one can find the solutions for the quadrature operators Q and P as well as for Q2 and P2. It is, however, more convenient to use the computer program to calculate the evolution of these quantities directly. Let us consider the purely SHG process, we drop the terms containing b and b+ after performing the normal ordering and take the expectation value in the coherent... 

Let us start with the short-time approximation in which we can use the symbolic manipulation computer program described in Appendix A to find the corrections coming from the quantum fluctuations of the fields. The operator formulas (94) and (95) are valid also for the degenerate downconversion because the two processes are governed by the same Hamiltonian, but now initially the second-harmonic mode is populated while the fundamental mode is initially in the vacuum state. Assuming that the pump mode at the frequency 2oo is in a coherent state fi0) (p0 = /Ni,exp(k )h)), we have... 

The time, f,-, for the induction period (region I) to end is an important factor in determining the surface tension as a function of time, since only when that period ends does the surface tension start to fall rapidly. The value of f,- has been shown (Gao, 1995 Rosen, 1996) to be related to the surface coverage of the air-aqueous solution interface and to the apparent diffusion coefficient, Dap, of the surfactant, calculated by use of the short-time approximation of the Ward-Tordai equation (Ward, 1946) for diffusion-controlled adsorption (equation 5.6) ... 

As mentioned above, the value of t, has been shown to be related to the coverage of the air-aqueous solution interface by the surfactant and to its apparent diffusion coefficient, Dap (equation 5.7). To calculate the values of Dap at short times, equation 5.8 (Bendure, 1971), based upon the short-time approximation equation of Ward and Tordai (equation 5.6), and using dynamic short-time surface tension data, may be used ... 

As a note, it should be mention that Parr and Yang (1989) have shown, how the integral formulation of the Kohn-Sham DFT arrives to the electronic density expression performing Wigner semiclassical expansion combined with the short time approximation regarding to the P parameter. However, the common tool between their and the actual electronic density... 

Barsov et al. (1986c) directly use the short-time approximation to evaluate the field correlations. This leads to an exponential time dependence of the different components of the field correlations... 

Franck-Condon approximation.The short-time approximation... 

Table 5.3 Characteristic decoherence times Toveriap (fs) in the short time approximation (eqn (5.44)). The inter-Kthium distances (Ru-Li) are given in A.

There is, however, a much better pair of solutions, obtained by Mahon and Oldham [47] and simplified a little a year later [48]. They used what they call the Cope-Tallman method, involving the Green function, to find much improved short-time and long-time solutions for the current at a disk electrode. Their formulae express currents at T values as defined above (12.13) (previously designated by t), and normalised by itnFDac, rather than the steady-state value. Here they are converted to the present scale by the simple expedient of a multiplication factor. The short-time approximation is then ... 

It is straightforward to show, along the lines of the short-time approximation equation (9), that the intensity Io->l(i) into the fundamental of the th mode is... 

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