Physically coupled Oscillators

Figure 12.2 Schematic diagram of an apparatus consisting of two CSTRs for studying physically coupled oscillating reactions. A needle valve controls the flow between the reactors. Inputs to the reactors are independently controlled. Drop detectors ensure that liquid flows out of the two reactors at the same rate so that there is no net mass transfer from one to the other. Reprinted, in part, with permission from Crowley, M. F. Epstein, I. R. 1989. Experimental and Theoretical Studies of a Coupled Chemical Oscillator Phase Death, Multistability, and In-Phase and Out-Of-Phase Entrainment, J. Phys. Chem. 93, 2496-2502. CC 1989 American Chemical Society.)...

The first theoretical model of optical activity was proposed by Drude in 1896. It postulates that charged particles (i.e., electrons), if present in a dissymmetric environment, are constrained to move in a helical path. Optical activity was a physical consequence of the interaction between electromagnetic radiation and the helical electronic field. Early theoretical attempts to combine molecular geometric models, such as the tetrahedral carbon atom, with the physical model of Drude were based on the use of coupled oscillators and molecular polarizabilities to explain optical activity. All subsequent quantum mechanical approaches were, and still are, based on perturbation theory. Most theoretical treatments are really semiclassical because quantum theories require so many simplifications and assumptions that their practical applications are limited to the point that there is still no comprehensive theory that allows for the predetermination of the sign and magnitude of molecular optical activity. 

The Hamiltonian of Eq. (9.2) couples the reaction coordinate to the environmental oscillator degrees of freedom by terms linear in both reaction coordinate and bath degree of freedom. This is derived in Zwanzig s original approach by an expansion of the full potential in bath coordinates to second order. This innocuous approximation in fact conceals a fair amount of missing physics. We have shown [16a] that this collection of bilinearly coupled oscillators is in fact a microscopic version... 

More testing of these approaches is certainly required before any final pronouncements on their worth can be made. One important contribution to assessing the ideas was that of Sewell et al. (53). For a system of coupled oscillators, they showed that the constraints of Miller et al. and Bowman et al. (45,46) may induce physically undesirable effects. In particular, they gave an example of a trajectory in the quasiperiodic regime being transformed into a chaotic one by the action of the constraints. 

The factor b may be determined from physical theories of optical rotations (75). In particular, coupled oscillator (75), polarizability (76) or free-electron theories (77) of molar rotations in the transparent region for helical line models of molecules consisting of N interacting units i, j,. .. involve relations of the... 

On the basis of physical properties, the coupled oscillators can be divided in three categories ... 

The equivalent electrical circuit, rearranged under the influence of an apphed physical field, is considered as a parallel resonant circuit coupled to another circuit such as an antenna output circuit Thus, in Figure 15.4c, Wj, Cd, La, and Ra correspond to the circuit elements each Wd represents active emitter-coupled oscillator and Cd, Ld, and Rd, represent passive capacitive, inductive, and resistive elements respectively. The subscript d is related to the particular droplet diameter, that is, the droplet under consideration. Now, again the initial electromagnetic oscillation is represented by... 

FIGURE 15.4 Definition sketch for understanding the theory of electroviscoelasticity (a) rigid droplet (b) incident physical field, for example, electromagnetic (c) equivalent electrical circuit-antenna output circuit. Wd represents the emitter-coupled oscillator and Cd, and i d are capacitive, inductive, and resistive elements of the equivalent electrical circuit, respectively. Subscript d is related to the particular diameter of the droplet under consideration. (Courtesy of Marcel Dekker, Inc.) Spasic, A.M. Ref. 3., p. 854. 

Figure 12.7 Experimental traces of platinum (Pt) electrode potential in a physically coupled BZ oscillator experiment. Bottom trace is potential VI from reactor 1 middle trace is potential V2 from reactor 2, shifted up by 200 mV. Top trace is (VI — V2)/2, and is shifted so that 0 mV corresponds to 450 mV on the millivolt axis. 0-20 min uncoupled oscillations, p 0 23-45 min out-of-phase entrainment, p = 0.5 45-60 min in-phase entrainment, p = 0.75. (Adapted from Crowley and Epstein, 1989.)...

Figure 12.8 Experimental traces of Pt electrode potential in a physically coupled BZ oscillator experiment. Bottom trace is potential VI from reactor 1 upper trace is potential V2 from reactor 2, shifted up by 200 mV.

Figure 12.9 Experimental stability diagram for physically coupled BZ system showing ranges of p for which different behaviors are stable. IP = in-phase entrained oscillations, SS = complementary pair of steady states, OP = out-of-phase entrained oscillations. (Adapted from Crowley and Epstein, 1989.)...

In addition to the compound oscillation depicted in Figures 12.16 and 12.17, the collision of two limit cycles may lead to other scenarios. One possibility is complex periodic oscillation in which one cycle of one type is followed by several of another type. This type of behavior is analogous to the bursting mode of oscillation of neural oscillators, in which a period of relative quiescence is followed by a series of action potentials. In Figure 12.18, we compare a membrane potential trace from a crab neuron with potential oscillations in a pair of physically coupled chlorine dioxide-iodide oscillators. 

Figure 12.20 Bifurcation diagram for two physically coupled Degn-Harrison oscillators. (Reprinted with permission from Lengyel, I. Epstein, I. R. 1991. Diffusion-Induced Instability in Chemically Reacting Systems Steady-State Multiplicity, Oscillation, and Chaos, Chaos /, 69-76. 1991 American Institute of Physics.)...

Synchronization or entrainment is a key concept to the understanding of selforganization phenomena occurring in the fields of coupled oscillators of the dissipative type. We may even say that Part II is devoted to the consideration of this single mode of motion in various physical situations. Specifically, Chap. 6 is concerned with wave phenomena and pattern formation, which may be viewed as typical synchronization phenomena in distributed systems. In contrast, we shall study in Chap. 7 turbulence in reaction-diffusion systems, which is caused by desynchronization among local oscillators. Chapter 5 deals with self-synchronization phenomena in the discrete populations of oscillators where the way they are distributed in physical space is not important (for reasons stated later). We shall introduce some kind of randonmess by assuming that the oscillators are either different in nature from each other or at best statistically identical. One may then expect phase-transition-like phenomena, characterized by the appearance or disappearance of collective oscillations in the oscillator communities. In describing such a new class of phase transitions. Method I turns out to be very useful. 

One of the most widely used and helpful forms of spectroscopy, and a technique that has transformed the practice of chemistry, biochemistry, and medicine, makes use of an effect that is familiar from classical physics. When two pendulums are joined by the sameslightly flexible support and one is set in motion, the other is forced into oscillation by the motion of the common axle, and energy flows between the two. The energy transfer occurs most efficiently when the frequencies of the two oscillators are identical. The condition of strong effective coupling when the frequencies are identical is called resonance, and the excitation energy is said to resonate between the coupled oscillators. 

We present a derivation of the broadening due to the solvent according to a system/ bath quantum approach, originally worked out in the field of solid-state physics to treat the effect of electron/phonon couplings in the electronic transitions of electron traps in crystals [67, 68]. This approach has the advantage to treat all the nuclear degrees of freedom of the system solute/medium on the same foot, namely as coupled oscillators. The same type of approach has been adopted by Jortner and co-workers [69] to derive a quantum theory of thermal electron transfer in polar solvents. In that case, the solvent outside the first solvation shell was treated as a dielectric continuum and, in the frame of the polaron theory, the vibrational modes of the outer medium, that is, the polar modes, play the same role as the lattice optical modes of the crystal investigated elsewhere [67,68]. The total Hamiltonian of the solute (5) and the medium (m) can be formally written as... 

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

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