Itinerant oscillator

Fig. 1. Energy distribution of neutrons scattered by hydrogen in liquid argon at 100"K and 26.5 atm, as a function of time of flight for various scattering angles, 6. Solid curve, interpolation model. Dashed curve, Sears itinerant oscillator model. Jagged curve, Experiment. Abscissa is time of flight in usec/m and ordinate is cross section in mb sr-1 usee-1.

Exercise. Consider the following simplified version of the itinerant oscillator model. A body moves in a fluid and contains in its interior a damped oscillator (fig. 22). The equations of motion are... 

Exercise. Another type of itinerant oscillator is given by two angles (j>1, (j>2 obeying... 

The powerful continued fraction procedure (CFP) described by Grosso and Pastori Parravicini in Chapter III may be used to solve Eqs. (3) and (4). An alternative approach has been provided by Ferrario et al., who have computed a variety of numerically derived orientation and velocity acFs for a simple cosine potential and the more comphcated cosinal itinerant oscillator, another RMT-allowed structure. In Chapter VI, Ferrario et al. describe deexcitation effects from the two-dimensional disk-annulus itinerant oscillator also studied by Brot and coworkers. ... 

Both vibrational and rotovibrational relaxation can be described analyti-caDy as multiplicative stochastic processes. For these processes, RMT is equivalent to the stochastic Liouville equation of Kubo, with the added feature that RMT takes into account the back-reaction from the molecule imder consideration on the thermal bath. The stochastic Liouville equation has been used successfully to describe decoupling in the transient field-on condition and the effect of preparation on decay. When dealing with liquid-state molecular dynamics, RMT provides a rigorous justification for itinerant oscillator theory, widely applied to experimental data by Evans and coworkers. This implies analytically that decoupling effects should be exhibited in molecular liquids treated with strong fields. In the absence of experimental data, the computer runs described earlier amount to an independent means of verifying Grigolini s predictions. In this context note that the simulation of Oxtoby and coworkers are semistochastic and serve a similar purpose. 

The argument by Zwanzig links the violation of condition (24) with a well-defined nonlinearity in the stochastic equations describing the system under consideration (an ensemble of molecules). One way of introducing this type of nonlinearity is to use the nonlinear itinerant oscillator... 

The details of how F(t) is related to /(/) are given elsewhere in this volume. Equation (29) may be used to describe the deexdtation effects described in Section II. A more accurate analytical description is provided by the numerical solution of the set of equations making up the nonlinear itinerant oscillator using continued fraction analysis. 

Section VI is devoted to describing the details of this experiment, which will be widely used to monitor equilibrium and nonequilibrium properties. These will prove to be in excellent agreement with the predictions of the nonlinear itinerant oscillator, diereby providing a convincing account for the effects discussed by Evans in Chapter V (which are recovered in this two-dimensional case). Section VII is devoted to a critical discussion of the RMT in the light of the experimental results rq>orted here. 

In Fig. 2 we compare the itinerant oscillator with the exact autocorrelation function Jg UQt). 

Figure 2. The timebehavi< of ttecorrelationfuiictioii (/),Eq. (2.6), with/t>l,

Another piece of information that we wanted to extract from our experiments was connected with the dynamic behavior of spatial variables. If we consider three successive particles in the chain and we denote by the distance of the middle one from the center of mass of the other two and by the distance between these two, we can compute the normalized autocorrelation function of these two variables. They are shown in Fig. 9 as can be immediately observed, they decay to zero on a time scale which is much greater than that of the velocity variable. Also, the center of mass decays faster than R . In the next section we shall argue that this suggests that the virtual potential characterizing the itinerant oscillator model has to be assumed to be fluctuating around a mean shape, which, moreover, will be shown to be nonlinear and softer than its harmonic approximation. 

We note here that in the case of a linear microscopic interaction, P jiRa) does not depend on R, and this potential simply reduces to a linear one. The effective potential, in this case, turns out to be evirtual potential of the well-known linear itinerant oscillator (see Section II). In the more general case, the virtual potential is given by Eq. (4.9), and the Liouvillian reads... 

As a final remark we wish to note that the result of the one-dimensional computer simulation shown in Fig. 9 suggests that the nonlinear version of the itinerant oscillator model should be improved by giving a fluctuating character to the virtual potential, since Rq does not turn out to be a very fast variable. Therefore, the potential of (4.9) should be interpreted as an average potential around which an effective potential would fluctuate. 

It will be shown that the effective dynamical operator of the preceding section, obtained by averaging on R, v and Rg being regarded the slowest variables of the system, is nothing but a nonlinear version of the celebrated itinerant oscillator (ref. 10 see also Chapter V). 

The long-time properties of the nonlinear itinerant oscillator are studied. 

In the linear case the itinerant oscillator model is well known, and its rigorous foundation has been illustrated in Section II. Its nonlinear version... 

Figure 11. Theoretical calculations via the CFP for a nonlinear itinerant oscillator with eflective potential harder than the harmonic one. The equilibrium autocorrelation function (—) and the excited autocorrelation function corresponding to an excited distribution P2 (—) are shown in the case of (F -1, 0 0.01, F - 0.03, and r — 3. For comparison with the linear case, we have also plotted the equilibrium autocorrelation function at (F - 0 (- -).

In the nonlinear case, the function ij t does not vanish (at times intermediate between t = 0 and t = oo). The itinerant oscillator with an effective potential harder than the linear one is shown to result in ri(t) < 0 in accordance with the results of CFP calculations which show its decay after excitation to be faster than the corresponding equilibrium correlation function (see Fig. 11). 

In spite of the fact that the decay after excitation of the hard-potential itinerant oscillator is similar to the experimental computer simulation result of Figs. 7 and 8, we do not believe that it is the reduced model equivalent to the one-dimensional many-particle model under study. As remarked above, indeed, the e(r) function is not correctly reproduced by this reduced model. The choice of a virtual potential softer than the linear one seems also to be in line with the point of view of Balucani et al. They used an itinerant oscillator with a sinusoidal potential, which is the simplest one (to be studied via the use of CFP) to deal with the soft-potential itinerant oscillator. Note that the choice... 

The nonlinear itinerant oscillator implies a slower decay of angular velocity as the intensity of the rotational kinetic energy is increased, thereby speeding up the decay of the dipole vector. 

This experimental result has to be compared with the prediction of the model of (5.69). This model can be studied by using the CFP. The results of ref. 14 are illustrated by Figs. 19 and 20. This figures show that the qualitative agreement between the theory based on the nonlinear itinerant oscillator of Eq. (5.69) and the experiment of Fig. 18 is indeed satisfactory. 

A somewhat surprising result of Chapter VI is that the foundation of a nonlinear version of the itinerant oscillator is unavoidably accompanied by a fluctuating character of the virtual potential itself. In this chapter we... 

It is found, in particular, that when using the well-known model of the itinerant oscillator, one cannot give up the assumption that the interaction between real and virtual variables is linear without also making this interaction fluctuate randomly in time. This establishes the link with Chapter VII. This fluctuating process can be used to model the influence of hydrogen bond dynamics, the long-time effects of which are then carefully explored and... 

Itinerant Oscillator. The itinerant oscillator model of dielectric response was introduced by Hill." It has considerable attraction as a model of the liquid state, and the name has become well known since Sears " used... 

II. Application of the Itinerant Oscillator (Cage) Model to the Calculation of the Dielectric Absorption Spectra of Polar Molecules... 

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