Second-order approximation, time-dependent

The last step is to find a symplectic, second order approximation st to exp StL ). In principle, we can use any symplectic integrator suitable for time-dependent Schrddinger equations (see, for example, [9]). Here we focus on the following three different possibilities corresponding to special properties of the spatially truncated operators H q) and V q). 

As before, we make the fundamental assumption of TST that the reaction is determined by the dynamics in a small neighborhood of the saddle, and we accordingly expand the Hamiltonian around the saddle point to lowest order. For the system Hamiltonian, we obtain the second-order Hamiltonian of Eq. (2), which takes the form of Eq. (7) in the complexified normal-mode coordinates, Eq. (6). In the external Hamiltonian, we can disregard terms that are independent of p and q because they have no influence on the dynamics. The leading time-dependent terms will then be of the first order. Using complexified coordinates, we obtain the approximate Hamiltonian... 

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

We now resort to the crucial approximation that a it) varies slower than either e(t) or 0(t). This approximation is justifiable in the weak-coupling regime (to second order in Hj) as discussed below. Under this approximation, Eq. (4.41) is transformed into a differential equation describing relaxation at a time-dependent rate ... 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

Bakale et al. [397] pulse irradiated the hydrocarbons cyclopentane, cyclohexane and n-hexane with 0.9 MeV electrons of duration 10 or 100 ns. The transient conductivity decreased approximately exponentially with time for low doses of radiation. The first-order decay of the conductance is probably due to electrons reacting with impurities. With higher doses, the conductance decays approximately as inverse time, characteristic of a second-order recombination of free ions. No evidence for time-dependent geminate ion-pair recombination effects was observed. 

This concludes a discussion of exactly solvable second-order processes. As one can see, only a very few second-order cases can be solved exactly for their time dependence. The more complicated reversible reactions such as 2Apt C seem to lead to very complicated generating functions in terms of Lame functions and the like. This shows that even for reasonably simple second- and third-order reactions, approximate techniques are needed. This is not only true in chemical kinetic applications, but in others as well, such as population and genetic models. The actual models in these fields are beyond the scope of this review, but the mathematical problems are very similar. Reference 62 contains a discussion of many of these models. A few of the approximations that have been tried are discussed in Ref. 67. It should also be pointed out at this point that the application of these intuitive methods to chemical kinetics have never been justified at a fundamental level and so the results, although intuitively plausible, can be reasonably subject to doubt. 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

Utilization of both ion and neutral beams for such studies has been reported. Toennies [150] has performed measurements on the inelastic collision cross section for transitions between specified rotational states using a molecular beam apparatus. T1F molecules in the state (J, M) were separated out of a beam traversing an electrostatic four-pole field by virtue of the second-order Stark effect, and were directed into a noble-gas-filled scattering chamber. Molecules which were scattered by less than were then collected in a second four-pole field, and were analyzed for their final rotational state. The beam originated in an effusive oven source and was chopped to obtain a velocity resolution Avjv of about 7 %. The velocity change due to the inelastic encounters was about 0.3 %. Transition probabilities were calculated using time-dependent perturbation theory and the straight-line trajectory approximation. The interaction potential was taken to be purely attractive ... 

Finally, we like to mention that equivalent to the conventional energy frame KHD formulation, the time-dependent theory of Raman scattering is free from any approximations except the usual second order perturbation method used to derive the KHD expression. When applied to resonance and near resonance Raman scattering, the time-dependent formulation has shown advantages over the static KHD formulation. Apparently, the time-dependent formulation lends itselfs to an interpretation where localized wave packets follow classical-like paths. As an example of the numerical calculation of continuum resonance Raman spectra we show in Fig. 6.1-7 the simulation of the A, = 4 transitions (third overtone) of D excited with Aq = 488.0 nm. Both, the KHD (Eqs. 6.1-2 and 6.1-18) as well as the time-dependent approach (Eqs. 6.1-2 and 6.1-19) very nicely simulate the experimental spectrum which consists mainly of Q- and S-branch transitions (Ganz and Kiefer, 1993b). 

In the theoretical analysis of shock instability, shock waves that are not too strong are presumed to propagate axially back and forth in a cylindrical chamber, bouncing off a planar combustion zone at one end and a short choked nozzle at the other [101], [102]. The one-dimensional, time-dependent conservation equations for an inviscid ideal gas with constant heat capacities are expanded about a uniform state having constant pressure p and constant velocity v in the axial (z) direction. Since nonlinear effects are addressed, the expansion is carried to second order in a small parameter e that measures the shock strength discontinuities are permitted across the normal shock, but the shock remains isentropic to this order of approximation. Boundary conditions at the propellant surface (z = 0) and at the... 

Clerc and Barat " flash-photolyzed CO2 in the vacuum ultraviolet and watched CO formation by kinetic spectroscopy. They found [CO] to increase rapidly to a maximum (at about 25 i sec when [CO2] = 3 torr, [Ar] = 300 torr) and then remain constant. They attributed the observation to the reaction of excess 0( /)) with the CO produced. They calculated a rate coefficient for 0( Z)) + CO recombination of 10 l. mole . sec or 2x 10 l.mole sec depending whether or not the reaction required a third body. They noted that the third-order rate coefficient was too large for a normal three-body reaction, and suggested a long-lived intermediate complex. In two later papers " ° ° they reported the value of the second-order rate coefficient to be first 6x 10 and then 1.2 x 10 ° l.mole" sec As mentioned above, Clerc and Reiffsteck " recently reassessed the relative rates of addition of 0( J9) to CO2 and CO, and found the latter reaction to be approximately 55 times faster than the former. 

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