Time modeling oscillators

The general linear solid leads to the single relaxation time model the solution of (2.4) for the case of oscillating strain leads to... 

The model thus shows how thresholds in the phosphorylation-dephosphorylation cascade controlling cdc2 kinase play a primary role in the mechanism of mitotic oscillations. The model further shows how these thresholds are necessarily associated with time delays whose role in the onset of periodic behaviour is no less important. The first delay indeed originates from the slow accumulation of cyclin up to the threshold value C beyond which the fraction of active cdc2 kinase abruptly increases up to a value close to unity. The second delay comes from the time required for M to reach the threshold M beyond which the cyclin protease is switched on. Moreover, the transitions in M and X do not occur instantaneously once C and M reach the threshold values predicted by the steady-state curves the time lag in each of the two modification processes contributes to the delay that separates the rise in C from the increase in Af, and the latter increase from the rise in X. The fact that the cyclin protease is not directly inactivated when the level of cyclin drops below C prolongs the phase of cyclin degradation, with the consequence that M and X will both become inactivated to a further degree as C drops well below C. ... 

For the mixture of /vteiphenyi and indole, the single decay-time fit (Figure 4.31, bottom panel) results in residuals which oscillate across the time axis this behavior is characteristic of an incorrect model. Also, the value of s 16.7 is obviously much greater than unity, and according to Table 4.2. there is a less than 0.1% chance that random error could result in such an elevated value of xJ. Additionally, the ratio of xi for this model to that for the two-decay time model is 17.6. which is much larger than the values of the... 

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

However, this simple model of a periodic motion occupied the central position in the theory of oscillations from its very beginning (Galileo) up to the time of Poincar6, when it was replaced by the new model—the limit cycle. 

An interesting improvement from the classical treatment of the bond under stress was proposed by Crist et al, [101], Considering the chain as a set of N-coupled Morse oscillators, these authors determined the elongation and time to failure as a function of the axial stress. The results, reported in Fig. 20, show a decreasing correlation between the total elastic strain before failure and the level of applied force with the chain length. To break a chain within some reasonable time interval (for example <10-3s) requires, however, the same level of stress (a0.7 fb) as found from the simpler de Boer s model. 

If the molecule moves without hindrance in a rigid-walled enclosure (the free enclosure ), as assumed in free volume theories, then rattling back and forth is a free vibration, which could be considered as coherent in such a cell. The transfer time between opposite sides of the cell t0 is roughly the inverse frequency of the vibration. The maximum in the free-path distribution was found theoretically in many cells of different shape [74]. In model distribution (1.121) it appears at a > 2 and shifts to t0 at a - oo (Fig. 1.18). At y — 1 coherent vibration in a cell turns into translational velocity oscillation as well as a molecular libration (Fig. 1.19). 

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