Oscillating case

Fig. 16.6 Transient temperature histories in a reactor whose volume oscillates as illustrated in Fig. 16.5. In addition to the no-oscillation case, three oscillation frequencies are shown. The lower panel is an enlargement of the period just around the ignition of the a> = 2500 Hz...

It should be noted that due to er(Af)a a the Fourier transform of % (Lorentzian band-shape with the band-width determined by the dephasing constant of different Fourier transform bands are approximately proportional to the FCFs (0aij 0gg 2 (0ai/ 0go) 2- For the single displaced oscillator case, for example, I I2 = S e 2S, where S is the Huang-Rhys factor of this mode. 

We first consider the intermolecular modes of liquid CS2. One of the details that two-dimensional Raman spectroscopy has the potential to reveal is the coupling between intermolecular motions on different time scales. We start with the one-dimensional Raman spectrum. The best linear spectra are based on time domain third-order Raman data, and these spectra demonstrate the existence of three dynamic time scales in the intermolecular response. In Fig. 3 we have modeled the one-dimensional time domain spectrum of CS2 for 3 cases (A) a single mode represented by the sum of three Brownian oscillators, (B) three Brownian oscillators, and (C) a distribution of 20 arbitrary Brownian oscillators. Case (A) represents the fully coupled, or isotropic case where the liquid is completely homogeneous on the time scales of the simulation. Case (B) deconvolutes the linear response into the three time scales that are directly evident in the measured response and is in the limit that the motions associated with each of the three timescales are uncoupled. Case (C) is an example where the liquid is represented by a large distribution of uncoupled motions. 

Combining equations (A51) and (A52) leads immediately, for the harmonic oscillator case, to the differential equation for the density... 

To see the significance of the effective temjjerature more explicitly, we may consider the harmonic-oscillator case together with the simple choices... 

Going beyond the periodic oscillations case, a more complex situation is that of synchronization of a chaotic local dynamics in a mixing flow, as was considered by Straube et al. (2004). When the flow is time-independent one can look for solutions of the form... 

In the second step we account for the essential physical effect of intermode couplings, as explained in the two-oscillator case. With three equal oscillators, such interactions will result in (1) the splitting of degeneracies associated with the purely local behavior and (2) the correspondingly specific symmetry of wavefunctions, under bond permu-... 

As already discusf d for the single-oscillator case in Section II.C.2, it is possible to obtain Dunham-like form for the local-mode eigenspectrum (4.33). Given the anharmonic character of the algebraic formulation, we expect to find a certain number of relations between the Dunham coefficients similar to those of the x-K relations of Lehmann [73, 74]. In fact, we can compare the traditional Dunham series for a triatomic molecule... 

When an upward loop is not vented (Fig. 5.16, c, e with a closed vent valve or no vent line), inerts may accumulate at the high point, causing intermittent siphon action and reflux flow oscillation. Cases were reported (68) where hot vapor was sucked back from the column into the high point and caused hammering upon contacting the subcooled reflux. This ruptured the reflux line and nozzle. The phenomenon is most troublesome in vacuum, where slight air leaks occur. 

Therefore R(r) will become infinite as r goes to infinity and will not be quadrati-cally integrable. The only way to avoid this infinity catastrophe (as in the harmonic-oscillator case) is to have the series terminate after a finite number of terms, in which case the e factor will ensure that the wave function goes to zero as r goes to infinity. Let the last term in the series be Then, to have 6 +, b +2,... all vanish, the fraction multiplying bj in the recursion relation (6.86) must vanish when j = k we have... 

However, the seeond (explicit) level of quantum theory in the Heisenberg matrix approach regards the evaluation of the diagonal components of the total eneigy matrix through employment of the commutation rule in the matrix forms specialized to actual harmonic oscillator case it looks equivalently like ... 

In the harmonic-oscillator case, Ef, [Pg.13]

The limit is no longer saturated in the harmonic-oscillator case, S r] with equal strengths, but it is for strengths proportional to the product of the masses, S [86]. 

Thermal event start System delay time Temperature difference between set value and actual value Heating power Creep case Oscillating case Aperiodic limit... 

Note that all coefficients on the right-hand sides of these equations are independent of 0, in contrast to the previous oscillator case, and this saves us cumbersome averaging procedures like those in Sect. 4.2. Equations (4.3.19, 20) may now be decomposed into finer balance equations ... 

For the quartic anharmonic oscillator case, the (approximate) form of a depends on whether c b, m) is positive or negative. In the positive case, the term (in the exponential of the second and third modes given above) dominates. If c(6, m) is negative, then the a terms (with a real factor) dominates. As such we have ... 

The model of the anharmonic oscillator or local mode approximation more closely follows the actual condition for molecular absorption than that of the harmonic oscillator. Figure 1.15 illustrates the differences between the ideal harmonic oscillator case that has been discussed in detail for this chapter vs. the anharmonic oscillator model (better representing the actual condition of molecules). Unlike the ideal model illustrated by the harmonic oscillator expression, the anharmonic oscillator involves considerations such that, when two atoms are in close proximity (minimum distance), they repel each other when two atoms are separated by too large a distance, the bond breaks. The anharmonic oscillator potential energy curve is most useful to predict behavior of real molecules. 

In the polyatomic harmonic oscillator case, the semiclassical wave-packet methods are exact, while method (iii) offers some advantages of computational ease in the handling of nonlinear dynamics. For the displaced wavepackets shown in the potential surface of Fig. 8, Fig. 7 shows I <(j) I (j) (t) > I computed by ab initio quantum mechanical methods (Fig. 7a), and by classical trajectory method (iii) (Fig. 7b). 

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