Itinerant oscillator nonlinear

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. 

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. 

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). 

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... 

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