Noninteracting oscillators

The case of noninteracting oscillators takes place when the coupling constant 12 is equal to zero. Then, the systems (46)—(50) with e12 = 0 and (26)—(29) with a = 0 are identical, and their dynamics are considered in Section . .1. 

In the above equations coq is the frequency of a noninteraction oscillator, X(o)) is the eoupling function between the oscillator and the external field, and RHg (ro) is the field density of states. With Eqs (67)-(74) it will now be possible to predict the spectral behavior for localized harmonic oscillators linearly coupled with different model boson continua. 

Two special cases of the theory illustrate the important features. The first is the relaxation of an ensemble of noninteracting harmonic oscillators in contact with a heat bath, and subject to nearest neighbor transitions in the discrete translational energy space. The master equations which describe the evolution of the ensemble can be written13... 

The simple class of models just discussed is of interest because it is possible to characterize the decay of correlations rather completely. However, these models are rather far from reality since they take no account of interparticle forces. A next step in our examination of the decay of initial correlations is to find an interacting system of comparable simplicity whose dynamics permit us to calculate at least some of the quantities that were calculated for the noninteracting systems. One model for which reasonably complete results can be derived is that of an infinite chain of harmonic oscillators in which initial correlations in momentum are imposed. Since the dynamics of the system can be calculated exactly, one can, in principle, study the decay of correlations due solely to internal interactions (as opposed to interactions with an external heat bath). We will not discuss the most general form of initial correlations but restrict our attention to those in which the initial positions and momenta have a Gaussian distribution so that two-particle correlations characterize the initial distribution completely. Let the displacement of oscillator j from its equilibrium position be denoted by qj and let the momentum of oscillator j be pj. On the assumption that the mass of each oscillator is equal to 1, the momentum is related to displacement by pj =. We shall study... 

Fig. 2. Measurement of G(V, B) for a 2 pm junction. Light shows positive and dark negative differential conductance. A smoothed background has been subtracted to emphasize the spectral peaks and the finite-size oscillations. The solid black lines are the expected dispersions of noninteracting electrons at the same electron densities as the lowest ID bands of the wires, ui) and li). The white lines are generated in a similar way but after rescaling the GaAs band-structure mass, and correspondingly the low-voltage slopes, by a factor of 0.7. Only the fines labeled by a, b, c, and d in the plot are found to trace out the visible peaks in G(V,B), with the fine d following the measured peak only at V > —10 mV.

Table 2.42 gives the recommended settings for processes with little or no dead time. As the dead time to time constant ratio rises, the integral setting becomes a smaller percentage of the oscillation period. For noninteracting PID loops, as dead time rises to 20% of the time constant, (I) drops to 45% and (D) to 17% of the period. At 50% dead time, (I) = 40% and (D) = 16%. When the dead time equals the time constant, (I) = 33% and (D) = 13%, finally, if dead time is twice the time constant, (I) = 25% and (D) = 12%. 

For noninteracting control loops with zero dead time, the integral setting (minutes per repeat) is about 50% and the derivative, about 18% of the period of oscillation (P). As dead time rises, these percentages drop. If the dead time reaches 50% of the time constant, I = 40%, D = 16%, and if dead time equals the time constant, I = 33% and D = 13%. When tuning the feedforward control loops, one has to separately consider the steady-state portion of the heat transfer process (flow times temperature difference) and its dynamic compensation. The dynamic compensation of the steady-state model by a lead/lag element is necessary, because the response is not instantaneous but affected by both the dead time and the time constant of the process. 

Similarly, during their effort to understand the thermal energy of solids, Einstein and Debye quantized the lattice waves and the resulting quantum was named phonon. Consequently, it is possible to consider the lattice waves as a gas of noninteracting quasiparticles named phonons, which carries energy, E=U co, and momentum, p = Uk. That is, each normal mode of oscillation, which is a one-dimensional harmonic oscillator, can be considered as a one-phonon state. 

Table 5 Comparison Between the Experimental Oscillator Strength /p and the Sum Rule Computed in the Limit of Noninteracting Electrons I0 and Spinless Fermions / ,...

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