Undamped oscillation

According to (2.29), dissipation reduces the spread of the harmonic oscillator making it smaller than the quantum uncertainty of the position of the undamped oscillator (de Broglie wavelength). Within exponential accuracy (2.27) agrees with the Caldeira-Leggett formula (2.26), and similar expressions may be obtained for more realistic potentials. 

The (5-part in (2.50) is responsible for elastic scattering, while the second term, proportional to the Fourier transform of C(t), leads to the gain and loss spectral lines. When the system exercises undamped oscillations with frequency Aq, this leads to two delta peaks in the structure factor. 

Assuming that the pj (t) and Qj (t) can be interpreted as a TS trajectory, which is discussed later, we can conclude as before that ci = ci = 0 if the exponential instability of the reactive mode is to be suppressed. Coordinate and momentum of the TS trajectory in the reactive mode, if they exist, are therefore unique. For the bath modes, however, difficulties arise. The exponentials in Eq. (35b) remain bounded for all times, so that their coefficients q and q cannot be determined from the condition that we impose on the TS trajectory. Consequently, the TS trajectory cannot be unique. The physical cause of the nonuniqueness is the presence of undamped oscillations, which cannot be avoided in a Hamiltonian setting. In a dissipative system, by contrast, all oscillations are typically damped, and the TS trajectory will be unique. 

D. E. F. Harrison, Undamped oscillations of pyridine nucleotide and oxygen tension in chemostat cultures of klebsiella aerogenes,/. Cell Biol. 75, 514—521 (1970). 

In the region between to, and to, which for SiC is between about 800 and 1000 cm-1, the reflectance is high not because of large k but because of small n. If n = 0, the normal incidence reflectance is nearly 100% only for the undamped oscillator (y = 0) is the reflectance actually 100%, but solids like SiC approach this rather closely. If the damping constant y in (9.20) is set equal to zero, the real part of the dielectric function becomes... 

If the uncatalysed reaction rate increases with respect to the rate of catalyst decay, so that ku becomes larger than gk2, there are no real solutions to eqn (3.60). The stationary state can no longer become unstable as /i is varied. Damped oscillatory responses can still be observed when we have a stable focus, but undamped oscillations will not be found. 

We have seen in earlier chapters that kinetic systems with two independent concentrations can show additional complexities of dynamic behaviour beyond those of one-variable systems. Of particular interest are undamped oscillations. The cubic autocatalysis with the additional decay step... 

Staying within a class of mono- and bimolecular reactions, we thus can apply to them safely the technique of many-point densities developed in Chapter 5. To establish a new criterion insuring the self-organisation, we consider below the autowave processes (if any) occurring in the simplest systems -the Lotka and Lotka-Volterra models [22-24] (Section 2.1.1). It should be reminded only that standard chemical kinetics denies their ability to selforganisation either due to the absence of undamped oscillations (the Lotka model) or since these oscillations are unstable (the Lotka-Volterra model). 

For y/3 1, one steady state exists and the regime is globally stable for all values of the Lewis number Lw. For 1 < y/3 < (y/3) and for sufficiently low values of Lewis number the system is again globally stable. Evidently for these conditions only one steady state occurs. For Lw > Lw, undamped oscillations exist. For supercritical values of y/3, y/3 > (y/3), and < a single steady state is stable or unstable according to the value of Lewis number. In the domain l>min < < max>... 

While the multiple steady-state phenomena may be, at least qualitatively, explained in terms of a simple one-step kinetic mechanism and interactions of the intraphase and interparticle heat and mass transfer (thermokinetic model), there is no acceptable explanation for the periodic activity (12). Since the values of the Lewis number are at least by a factor of 10 lower than those necessary to produce undamped oscillations, there is no doubt that the instability cannot be viewed in terms of mutual... 

In the homogeneous systems at low temperatures, oscillation phenomena were also observed for this reaction system. Similar observations were made for the CO and 02 system. In order to explain the oscillation mechanism mathematically, a complicated kinetic scheme was devised which is capable of predicting the sustained oscillations [Yang (26, 27)]. Apparently a similar complicated kinetic mechanism governs the undamped oscillations in the heterogeneous system, however, a pertinent analysis has not been performed in the literature so far. 

The classical problem of multiple solutions and undamped oscillations occurring in a continuous stirred-tank reactor, dealt with in the papers by Aris and Amundson (39), involved a single homogeneous exothermic reaction. Their theoretical analysis was extended in a number of subsequent theoretical papers (40, 41, 42). The present paragraph does not intend to report the theoretical work on multiplicity and oscillatory activity developed from analysis of governing equations, for a detailed review the reader is referred to the excellent text by Schmitz (3). To understand the problem of oscillations and multiplicity in a continuous stirred-tank reactor the necessary and sufficient conditions for existence of these phenomena will be presented. For a detailed development of these conditions the papers by Aris and Amundson (39) and others (40) should be consulted. 

If steam flow is linear with controller output, as it is in Fig. 8-50, undamped oscillations will be produced when the flow decreases by... 

Undamped oscillations had been reported by Adlhoch et al. (160) for a Pt ribbon operated in the 10 5 torr range and by Schiith and Wicke (124) for a supported Pt catalyst working near atmospheric presure. An estimate for the former case yields temperature variations of the order of 10 K due to the exothermicity of the reaction in the latter case even periodic changes by 25 K were measured—quite obviously heat conductance is efficient enough to synchronize the oscillatory behavior of these systems. 

Then the approximate solutions of (1.33) are generally complex. However, in the case of the polariton, these solutions are still real, owing to the geometry of the system A photon emitted by the infinite 3D lattice is necessarily reabsorbed in other words, the exciton is not coupled to a continuum of photons (in which it is irreversibly diluted), but it is coupled to discrete photons with well-defined wave vectors, with subsequent undamped oscillations, which are the essence of the polariton. The situation is dramatically different when emission occurs in a... 

For a chemical reaction system, the characteristics of the periodic solutions are uniquely determined by the kinetic constants as well as by the concentrations of the reactants and final products. Starting from the neighborhood of steady state as an initial condition, the system asymptotically attains a closed orbit or limit cycle. Therefore, for long times, the concentrations sustain periodic undamped oscillations. The characteristics of these oscillations are independent of the initial conditions, and the system always approaches the same asymptotic trajectory. Generally, the further a system is in the unstable region, the faster it approaches the limit cycle. 

Homogeneous dissipation within a viscoelastic solid propellant can produce significant damping if the propellant participates in the oscillations. Usually the solid does not participate appreciably, but under special conditions it does. To see what those special conditions are we may first consider undamped oscillations in a one-dimensional, two-medium system having... 

The situation becomes much more complicated in the presence of two or more connected bands the band gaps cause asymptotically undamped oscil-lations We refer to the papers of Magnus and Turchi et al. fm a very detailed theoretical and numerical analysis of the asymptotic behavior of the continued fraction coefficients. 

The presence of the gap produces undamped oscillations in the continued fraction parameters. In particular, a oscillates between (E + 4) G and b between G), while the multiplicity of the period of oscillations de-... 

At approximately the same time, Lotka proposed his famous models of oscillating chemical reactions based on irreversible autocatalytic processes. The first model included one autocatalytic step and gave damped oscillations. The second model became a paradigm in oscillating chemistry. It consists of two consecutive autocatalytic steps, resulting in undamped oscillations. The Lotka models attracted great attention from theoretical biologists, because... 

An understanding of the transient hehavior of continuous reactors is important for start-up and reactor control considerations. Continuous oscillations have been observed by a number of workers. Figures 10 and 11 show data for styrene and methyl methacrylate. Gerrens and Ley (1974) reported continuous, undamped oscillations in surf e tension during a styrene emulsion polymerization run which lasted for more than 50 mean residence times. Nearly five complete cycles were observed during this run. Berens (1974) conducted experiments with vinyl chloride in whidi the measured panicle size changed with time. No steady state was achieved with the data shown in Fig. 12. 

Oscillating reactions In these coupled autocatalytic reactions, repeated undamped oscillations occur in concentrations of reactants. These reactions have been reviewed by Field. For example, in the cerium-catalyzed reaction of bromate with malonic acid, changes in concentration of bromide ion and in the Ce(rV)-Ce(III) ratio over several orders of magnitude occur repeatedly. These systems as yet are not well understood. For the bromate-malonic acid reaction a 10-step process for the mechanism has been proposed. ... 

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