Driven-oscillator model

Section 3.1 introduces a number of concepts useful for the discussion of energy transfer by considering the simple driven-oscillator model. Section 3.2 briefly... 

In order to establish certain general features of gas-surface energy transfer it is helpful to consider a simple driven-oscillator model of the collision [Ref.3.1, Chap.10] and [3.6-8]. Although ultimately too simplistic (for example, the model is restricted to col linear collisions), this model provides a qualitatively useful picture of energy transfer and thermal accommodation. The model is shown schematically in Fig.3.1. A gas atom of mass m and energy E is incident upon a surface... 

The driven-oscillator model reveals several interesting points concerning the behavior of thermal-accommodation coefficients. We first note that (3.26) predicts that the accommodation coefficient is a universal function of three dimensionless parameters, the mass ratio, y, the reduced well depth, d, and the adiabaticity parameter, This latter quantity is the ratio of the duration of the collision to the oscillator period. Abrupt, impulsive collisions correspond to c values near zero, while long lasting, languid collisions correspond to large values. For small values we obtain from (3.26)... 

Figure 7. Oscillation and phase-plane diagrams of the externally driven limit cycle of the coherent oscillation model.

Interaction with External Fields. The models considered exhibit cooperative behaviour through nonlinear internal oscillations (models 1, 2, 4) or through nonlinear resonances (model 3). This makes plausible the existence of effects, when the system is driven by weak external fields of appropriate frequency. 

Electron oscillator model For intuition, think about charge displacement in terms of a driven oscillator a negative charge — e of mass me, restrained from moving too far... 

For a weakly perturbed, harmonic damped driven oscillator, the resonance is shifted on the frequency scale depending on the sign of the interaction forces (Fig. 1.13). The availability of analytical expressions facilitates the applications of the weakly perturbed harmonic oscillator models for AM-AFM. Such harmonic models may be useful to illustrate the concepts used in AM-AFM well enough however, in most practical imaging cases, they do not describe the experiments [15]. 

Fig.3.9. aE for a driven Einstein oscillator compared to that for a driven Debye model (3.29,38). The Einstein-os-cillator frequency was taken as v TEojq. Other system parameters were wp = variable, mg = 39.95, m3 = 108, D = 418 K, and a = 1.69 A . Although obscured by the scale, both curves have a shallow minimum near (Dp/co = 0 as a result of the characteristic shape of F(cd)... 

Figures 3.9-11 illustrate the importance of the background-lattice effects in a slightly different way. Shown are various results for the driven Einstein and Debye oscillator models of Sect.3.1. Figure 3.9 illustrates the ratio of the Einstein and Debye energy transfer (equivalently the ratio of the Einstein and Debye accommodation coefficients) as a function of the adiabaticity parameter For...

Noszticzius, Z. Noszticzius, E. Schelly, Z. A. 1983. On the Use of Ion-Selective Electrodes for Monitoring Oscillating Reactions. 2. Potential Response of Bromide-and Iodide-Selective Electrodes in Slow Corrosive Processes. Disproportionation of Bromous and lodous Acid. A Lotka-Volterra Model for the Halate Driven Oscillators, J. Phys. Chem. 87, 510-524. 

In the end, however, we have to stress that the most important feature of the Explodator model is not the chemical identification of its intermediates - they are surely the different oxidation states of the halogens in the case of halate driven oscillators [5,8,9,10,11] - but its structure. That "serial structure [2] contains two consecutive autocatalytic processes just like some of our previous models L5>8,9,l0j or the new revised Oregonator [11,12]. ... 

Figure 3.30. Amplitude and phase versus frequency for an idealized cantilever with Oi = nm Q = 10, and (Oo = 150 kHz. Solid lines denote response for the cantilever driven at resonance according to a damped driven harmonic oscillator model. In (A), dashed lines represent the effect of an attractive force gradient—reduction in amplitude and decrease in phase. In (B), dashed lines represent the effect of a repulsive...

Physically, this formula describes the power dissipated by a harmonic oscillator (the emission dipole with moment t) as it is driven by the force felt at its own location from its own emitted and reflected electric field. PT is calculable given all the refractive indices and Fresnel coefficients of the layered model(12 33)... 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

It is interesting to note that in chaotic regime, the flow rate outlet stream, which is manipulated by the control valve CVl (see Figure 12), and the reactor volume, are driven by the PI controller to the equilibrium point without chaotic oscillations. However, the other variables have a chaotic behavior as shown in Figure 18. So it is possible to obtain a reactor behavior, in which some variables are in steady state and the others are in regime of chaotic oscillations, due to the decoupling or serial connection phenomena. In this case the control system and the volumetric flow limitation of coolant flow rate through the control valve VC2, are the responsible of this behavior. Similar results can be obtained from model. 

D. Gonze and A. Goldbeter, Entrainment versus chaos in a model for a circadian oscillator driven by light-dark cycles. J. Stat. Phys. 101, 649-663 (2000). 

We start our discussion of laser-controlled electron dynamics in an intuitive classical picture. Reminiscent of the Lorentz model [90, 91], which describes the electron dynamics with respect to the nuclei of a molecule as simple harmonic oscillations, we consider the electron system bound to the nuclei as a classical harmonic oscillator of resonance frequency co. Because the energies ha>r of electronic resonances in molecules are typically of the order 1-10 eV, the natural timescale of the electron dynamics is a few femtoseconds to several hundred attoseconds. The oscillator is driven by a linearly polarized shaped femtosecond... 

Let us now formulate the problem of the energy-optimal steering of the motion from a chaotic attractor to the coexisting stable limit cycle for a simple model, a noncentrosymmetric Duffing oscillator. This is the model that, in the absence of fluctuations, has traditionally been considered in connection with a variety of problems in nonlinear optics [166]. Consider the motion of a periodically driven nonlinear oscillator under control... 

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