Short-time region

Examination of the Long-Richman solution to Eq. (1) indicates that the quantity defined previously, i. e., the time for the inflection point on the sigmoid second stage portion, is very nearly equal to 1/2/8, provided D/L2 is not too small compared with /8. If, as has been discussed in section 3.3, it is plausible to interpret / as being related to the rate of relaxation motions of polymer molecules, then the second stage portion should shift to the short time region as the initial concentration of the experiment becomes higher, since as the solid contains more diluent it is more plasticized and thus the chain relaxation becomes more rapid [Fujita and Kishimoto (1958)]. This expectation is borne out by the data shown in Fig. 9. 

This result is very attractive because it describes the transition from a short-time region dominated by ballistic transport to a time asymptotic behavior indistinguishable from ordinary diffusion. In conclusion, all this suggests that the assumption that the operator Ljnt acting in the exponent of Eq. (24) can be neglected is not an approximation, insofar as it yields an equation of motion, Eq. (133), that is exact under the DF assumption on the higher-order correlation function. 

In this problem, in ad tion to discussed in Section IV, we have another time scale, Let us assume that our experimental apparatus only allows us to observe long-time regions corresponding to r yr/, where is the mechanical time scale introduced in provision (i) of Section V.A. In consequence, the dynamics induced by the interplay of inertia and external noise belongs to the short-time region. In such a limit T,y - Ty, both and v play the role of fast-relaxing variable while x is our variable of interest. By applying our AEP, we then recover the corrections to the diffusional approximation due to the non-white external noise. 

Due to the way we followed to arrive at Eq. (5.59), the elTective damping at 1 = 0 [i.e., virtually that of Eq. (5.44)] is the result of a sort of coarse-grained measurement on the short time region of the corresponding equilibrium correlation function. In accordance with the linear response theory,(y(l)) as given by Eq. (5.59) should tend to coincide with the corresponding equilibrium correlation function as (v )cxc... 

The third group of models succeeds in accounting for some thermodynamic properties in the stable region of water (0 C starting point for describing the short-time region by properly defining a continuous stochastic variable that could be made to depend suitably on the stochastic parameters of Eq. (4.1). 

Further results have been obtained by using the electric circuit of Section III. This experiment shows that after the first threshold, T exhibits a rapid increase with increasing Q. This increase is still more rapid than predicted by the CFP. The experin tal behavior of as a function of Q at several values of is shown in Fig. 14. These results show that the basic assumptions of the AEP are invalidated with increasing Q. This points out also the importance of a technique such as the current version of the CFP which allows us to get information on the short-time region, which turns out to be the most significant region within the present context. 

The calculated results agreed qualitatively with that by molecular dynamics simulations (Fig. 1). In the long-time region, solvent relaxation for a change in a solute charge from 0 to e (z=0- l) was slower than that obtained by the linearized equation. In addition, relaxation for z=0- 1 was slower than that for z= 1 - 0. On the other hand, in the short-time region, solvent relaxation for z=0—1 was similar to that by the linearized equation. 

The intensity of a Raman peak at a given excitation wavenumber cU is related to the area under the curve in the overlap versus time plot. However, only the overlap in the short time region of the plot will be important in determining the Raman spectra of large molecules in condensed media because the damping factor is always nonzero. For example, when the damping factor is 300 cm , most of the recurrences die and the first peak will... 

It is natural to conceive that this short-time behavior should be due to some time interval for a trajectory to spend to look for exit ways to the next basins in the complicated structure of phase space. In the next section, we will propose a geometrical view that shows what this complexity is. Hence we consider that the hole of Na- b(t) in the short-time region should be a reflection of chaos, which is just opposite to the behavior arising from nonchaotic direct paths as observed in Hj" dynamics. The present effect is therefore expected to be more significant as the molecular size increases or the potential surface and corresponding phase-space structure become more complicated. Another important aspect of the hole in Na-,b t) is an induction time for a transport of the flow of trajectories in phase space. It is of no doubt that the RRKM theory does not take account of a finite speed for the transport of nonequilibrium phase flow from the mid-area of a basin to the transition states. Berblinger and Schlier [28] removed the contribution from the direct paths and equate the statistical part only to the RRKM rate. One should be able to do the same procedure to factor out the effect of the induction time due to transport. We believe that the transport in phase space is essentially important in a nonequilibrium rate theory and have reported a diffusion model to treat them [29]. 

More recently, Martin and Unwin simulated chronoamperometric feedback allowing for unequal diffusion coefficients of the oxidized and reduced forms of the redox mediator (6). Unlike steady-state SECM response, the shape of the tip current transients is sensitive to the ratio of the diffusion coefficients, y = DG/DR (Fig. 3). When D0 = DR, the tip current attains a steady-state value much faster than for any -y A 1. At -y < 1, a characteristic minimum appears in the short-time region, which is quite unusual for potentiostatic transients. 

The presence of forces and spatial dependence of D will, of course, modify these results (cf. Section XII). In addition, according to the arguments presented throughout this chapter, we expect these results to lose all validity in the short-time region. Here the nonhydrodynamic states, discussed in Section X.C, will play a crucial role. 

In the short-time region where the process AS 0)AS t)) is dominant, the approximation as used in Eq. (17.8) is expected to be good. By contrast, over a long period of time, as the nonvanishing residual fluctuations in S t) are small and more comparable in (relative) magnitude to the slow fluctuations in bx t)by t), the approximation expressed as a product of two separate terms S and bx 0)by 0))(bx t)by t))) in Eq. (17.11) may not be well justified. Nevertheless, the approximate form as given by Eq. (17.8) helps us understand the coexistence of the fast and slow modes of motion as distinctly observed in the simulation results. To illustrate the results and at... 

Gs t) and in the fast-mode region, the short-time region may be... 

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