Virtual potential

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 eitinerant oscillator (see Section II). In the more general case, the virtual potential is given by Eq. (4.9), and the Liouvillian reads... 

This result confirms the point of view of Balucani et al. on the need of using a virtual potential softer than the harmonic one. On the other hand, this potential restricts the motion of the particle in a region around Ri, = 0, which contrasts with the idea of a sinusoidal potential used in ref. 7. This... 

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. 

Predictions are made about the results of ideal excitation experiments, preparing the system of tagged particles in an unstable initial distribution. The qualitative behavior of the system after this preparation phase is traced back to the form of the virtual potential, that is, the interaction between real and virtual particles. 

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... 

Figure 10 shows that our system is significantly nonlinear, thereby making the results largely dependent on the actual form of the virtual potential. 

The results of computer experiments indicate that the virtual potential must be softer than the linear approximation. However, a simple analytical expression for this, which is also sound from a physical point of view, can be obtained only in the rotational case, where complete agreement between theory and experiment can be achieved. Comparably good agreement in the translational case requires further investigation. 

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... 

The stability of the atmosphere can be related directly to the net gradient of the virtual potential temperature in the vertical direction as follows ... 

A distinction has to be made here between chemical and virtual potentials. As discussed at some length in Chap. 5, since the activity or chemical potential of an individual ion or charged defect cannot be defined, it follows that its chemical potential is also undefined. The distinction, however, is purely academic, because defect reactions are always written so as to preserve site ratios and electroneutrality, in which case it is legitimate to discuss their chemical potentials. 

Although the effective potential is replaced with a virtual potential in Eq. (2.4), this does not necessarily require such a replacement. The effective potential corresponding to the wavefunction in Eq. (2.5) is derived as... 

This equality allows one to define a virtual potential as the common value of both members... 

The virtual potential differs from the tme chemical potential by an undefined constant which is incorporated in °. For this reason the absolute value of the virtual chemical potential can not be determined experimentally. 

Gibbs free energies. Real changes may thereby be described in terms of virtual potentials. Kroger et al. pointed out that a similar situation occurs in aqueous electrolytes. In this case, arbitrary thermodynamic potentials are assigned to separate ions, although it is impossible to determine experimentally the separate potentials. 

The electric contributions from the principle of virtual potential, as derived in Section 3.4.4 and given by Eq. (3.53), still have to be incorporated. This can be achieved equivalently by the addition or subtraction of Eqs. (3.60b) and (3.53). In conformance with Allik and Hughes [4] and in view of the symmetry properties of the not yet introduced constitutive relation, the electrostatic expressions will be subtracted from the mechanical ones. The virtual work of external contributions takes the following form ... 

We may conclude from this example that the effect of curvature of the reaction path is to introduce a virtual potential barrier at which there will be a reflection probability given by equation 16-44 or 16-45. However, it is important to note that for a hyperbolic channel there can be no interaction of vibration and translation, since the Schrodinger equation is exactly separable in the coordinates corresponding to these motions. This, as we have seen, is never completely realized in actual reactions, although for many reactions it may be a good approximation. 

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