Relaxation Oscillators

HOPF BIFURCATIONS, THE GROWTH OF SMALL OSCILLATIONS, RELAXATION OSCILLATIONS, AND EXCITABILITY... 

The validity of Eq. (15.11) even at the limit Na Nm means in fact that for harmonic oscillators relaxing in a heat bath there exists a closed equation for the mean energy. This is intimately connected with the linear dependence of transition probabilities on the vibrational quantum number (see Eq. (14.1)) and implies that the energy relaxation rate is independent of the initial distribution of harmonic oscillators over vibrational states. Still another peculiarity of this system is known the initial Boltzmann distribution corresponding to the vibrational temperature Tq =t= T relaxes to the equilibrium distribution via the set of the Boltzmann distributions with time-dependent temperatures. If (E ) is explicitly expressed by a time-dependent temperature, this process is again described by Eq. (15.11). 

As we have seen so far, the intrinsic membrane potentials of the neurons is the main cause of neuronal oscillations. However, there also exist other mechanisms that can cause oscillations in the neurons. In general, every single neuron can be classified into two major types of oscillators, i.e., relaxation oscillators and conditional oscillators. Relaxation oscillators are spontaneous and oscillate in the absence of inputs, whereas conditional oscillators are those that fire as a consequence of the inputs it receives. 

It was determined, for example, that the surface tension of water relaxes to its equilibrium value with a relaxation time of 0.6 msec [104]. The oscillating jet method has been useful in studying the surface tension of surfactant solutions. Figure 11-21 illustrates the usual observation that at small times the jet appears to have the surface tension of pure water. The slowness in attaining the equilibrium value may partly be due to the times required for surfactant to diffuse to the surface and partly due to chemical rate processes at the interface. See Ref. 105 for similar studies with heptanoic acid and Ref. 106 for some anomalous effects. 

In many materials, the relaxations between the layers oscillate. For example, if the first-to-second layer spacing is reduced by a few percent, the second-to-third layer spacing would be increased, but by a smaller amount, as illustrated in figure Al,7,31b). These oscillatory relaxations have been measured with FEED [4, 5] and ion scattering [6, 7] to extend to at least the fifth atomic layer into the material. The oscillatory nature of the relaxations results from oscillations in the electron density perpendicular to the surface, which are called Eriedel oscillations [8]. The Eriedel oscillations arise from Eenni-Dirac statistics and impart oscillatory forces to the ion cores. 

Treanor C E, Rich J W and Rehm R G 1968 Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions J. Chem. Phys. 48 1798-807... 

McMorrow D and Lotshaw WT 1991 Dephasing and relaxation in coherently excited ensembles of intermolecular oscillators Cham. Phys. Lett. 178 69-74... 

For sufficiently long times (index n ), the exponential can be neglected, leaving an oscillation of the turnover variable phase shifted with respect to the sound wave and with its amplitude reduced by the finite relaxation... 

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

The relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

Fig. 4. Temporal pulse characteristics of lasers (a) millisecond laser pulse (b) relaxation oscillations (c) Q-switched pulse (d) mode-locked train of pulses, where Fis the distance between mirrors and i is the velocity of light for L = 37.5 cm, 2L j c = 2.5 ns (e) ultrafast (femtosecond or picosecond) pulse.

Another resonant frequency instmment is the TA Instmments dynamic mechanical analy2er (DMA). A bar-like specimen is clamped between two pivoted arms and sinusoidally oscillated at its resonant frequency with an ampHtude selected by the operator. An amount of energy equal to that dissipated by the specimen is added on each cycle to maintain a constant ampHtude. The flexural modulus, E is calculated from the resonant frequency, and the makeup energy represents a damping function, which can be related to the loss modulus, E". A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

Equation (2.41) describes either damped oscillations (at tls < 2do) or exponential relaxation OItls > 2do). Since tls grows with increasing temperature, there may be a cross-over between these two regimes at such that 2h QoJ Ao)coth P hAo) = 2Aq. If the friction coefficient... 

Coupling to these low-frequency modes (at n < 1) results in localization of the particle in one of the wells (symmetry breaking) at T = 0. This case, requiring special care, is of little importance for chemical systems. In the superohmic case at T = 0 the system reveals weakly damped coherent oscillations characterised by the damping coefficient tls (2-42) but with Aq replaced by A ft-If 1 < n < 2, then there is a cross-over from oscillations to exponential decay, in accordance with our weak-coupling predictions. In the subohmic case the system is completely localized in one of the wells at T = 0 and it exhibits exponential relaxation with the rate In k oc - hcoJksTY ". 

The coiled cable is not acceptable for low-speed (i.e., less than 300 rpm) applications or where there is a strong electromagnetic field. Because of its natural tendency to return to its relaxed length, the coiled cable generates a low level frequency that corresponds to the oscillation rate of the cable. In low-speed applications, this oscillation frequency can mask real vibration that is generated by the machine. 

Tobolsky, A. V. and DuPre, D. B. Macromolecular Relaxation in the Damped Torsional Oscillator and Statistical Segment Models. Vol. 6, pp. 103 — 127. 

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