Simple Models of Motions

However, this simple model of a periodic motion occupied the central position in the theory of oscillations from its very beginning (Galileo) up to the time of Poincar6, when it was replaced by the new model—the limit cycle. 

Kramers [67], Northrup and Hynes [103], and also Grote and Hynes [467] have considered the less extreme case of reaction in the liquid phase once the reactants are in collision where such energy diffusion is not rate-limiting. Let us suppose we could evaluate the (transition state) rate coefficient for the reaction in the gas phase. The conventional transition state theory needs to be modified to include the effect of the solvent motion on the motion of the reactants as they approach the top of the activation barrier. Kramers [67] used a simple model of the... 

In Chap. 2, Sect. 6.6 and Chap. 6, Sect. 2.3, the phenomenon of hydro-dynamic repulsion was referred to, but not discussed in any detail. While this effect is strictly a many-body effect, it can be approximated very well by a simple model. The motion of one solute, A, necessarily requires that the surrounding solvent moves aside to let the A molecule pass. The motion of the solvent near A in turn requires more distant solvent molecules to move. This action is transmitted by collision, but effectively the solute A entrains solvent molecules to move in the same direction as it is doing itself. The degree of the entrainment of solvent decreases as the... 

The main problem of the boundary motion, however, remains the description of relaxation processes that take place when supersaturated point defects are pumped into the boundary region A R. Outside the relaxation zone Asimple model of a relaxation box is shown in Figure 10-14c. The four exchange reactions 1) between the crystals a and /3, and 2) between their sublattices are... 

An electronic transition involves the motion of an electron from one orbital to another. In the simple model of a one-electron transition it is assumed that all other particles remain in their initial states. If the initial and final orbitals of the electron are separated in space there will be a change in dipole moment, so long as the molecule has no centre of symmetry. When this change is very large the excited state is described as a charge transfer ... 

Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle 61... 

The problem of cross-section calculation for various inelastic collisions is mathematically equivalent to the solution of a set (in principle, infinite) of coupled wave equations for nuclear motion [1]. Machine calculations have been done recently to obtain information about nonadiabatic coupling in some representative processes. Although very successful, these calculations do not make it easy to interpret particular transitions in terms of a particular interaction. It is here that the relatively simple models of nonadiabatic coupling still play an important part in the detailed interpretation of a mechanism, thus contributing to our understanding of the dynamic interaction between electrons and nuclei in a collision complex. 

The motion of electrons in a magnetic field in a situation in which inhomogeneity of some kind exists remains of considerable interest at the time of writing. Therefore, in this Appendix, we shall first summarize some results of Freeman and March [49] for the current density in a simple model of independent harmonically confined electrons in a constant magnetic field. Then we shall go on to discuss the semiclassical theory of current density in atoms. 

A simple model of an elastomer network is depicted in Fig. 7.1.8. The segmental motion of inter-cross-link chains is fast but anisotropic at temperatures of 100-150 K above the glass transition temperature The end-to-end vector R of such a chain reorients on a much slower timescale because it appears fixed between seemingly static cross-link points. As a result of the fast but anisotropic motion, the dipolar interaction between spins along the cross-link chains is not averaged to zero, and a residual dipolar coupling remains [Cohl, Gotl, Litl]. 

Here Ef is the amplitude, t the duration, and co the frequency of the ith pulse. This scheme has been applied in Ref [46] to a generic two-dimensional HT model which incorporated a H-atom reaction coordinate as well as a low-frequency H-bond mode. In a subsequent work [47] the approach has been specified to a simple model of HT in thioacetylacetone. The Hamiltonian was tailored to the form of Eq. (4.1) based on the information available for the stationary points, that is, the energetics as well as the normal modes of vibration. From these data an effective two-dimensional potential was constructed including the H-atom coordinate as well as a coupled harmonic oscillator, which describes the 0-S H-bond motion. Although perhaps oversimplified, this model allowed the study of some principle aspects of laser-driven H-bond motion in an asymmetric low-barrier system. 

The Brillouin spectra of molecular liquids are more complicated than the spectra of simple liquids. Molecular internal degrees of freedom generally couple to the translational motion of the molecules, thereby leading to additional relaxation mechanisms for the density fluctuations. In this section we explore a simple model of molecular liquids first proposed by Mountain (1966) in which the density fluctuations are weakly coupled to the relaxing molecular internal degrees of freedom. 

An atomic nucleus with a large number of component nucleons is a very complicated structure indeed. But in some situations an extraordinarily simple model of it will do for predictive and explanatory purposes. When we are dealing with many aspects of nuclear fission, it is adequate to treat the nucleus as if it were a blob of fluid. Indeed, only the way such a fluid would behave when set into oscillatory motion and as described by classical mechanics is needed to account for many aspects of the fission process. Just think of the blob of fluid as bounded by its surface, a surface that is characterized by tensional forces parallel to itself. Then think of the nucleus into which a neutron has just been injected to tri er the fission process, say, as such a liquid put into a higher energy state and forced to oscillate subject to the constraint of its own surface tension. Many of the important features of the fission process can be predicted and explained using this simple model. 

We have discussed only the equilibrium properties of polymers. Of course, in many real systems, the time scales for equilibriation can be very large. It is thus of interest to study non-equilibrium properties of statistical mechanical systems on fractals. A simple prototype is the study of kinetic Ising model on fractals. Closer to our interests here, one can study, say, the reptation motion of a polymer on the fractal substrate. This seems to be a rather good first model of motion of a polymer in gels. 

Computational studies of equilibrium and transport properties of simple models of molecular systems have become an important part of statistical-mechanical research. Such studies include Monte Carlo calculations, which are described by Valleau in Volume 5 and the molecular dynamics calculations described in this chapter and in Chapter 2 by Kushick and Berne. Here we consider the molecular dynamics (MD) method for systems of hard-core particles. Because of the simplicity of the intermolecular interaction, the integration of the classical equations of motion is trivial and the methods used for the study of various material properties are frequently different from those for soft potentials. 

Neat liquids are, in a way, difficult objects for NMR relaxation studies. The simple modelling of reorientational motion as small-step rotational diffusion is based on hydrodynamics (large body immersed in continuum solvent) and becomes problematic if we deal with a liquid consisting of molecules of a single kind. Deviations from the models based on few discrete correlation times can therefore be expected. 

Fortunately, results of MD studies [27, 28, 42-49] demonstrate (see also Sect. 4 below) that the MD trajectories of small gases in atomistic microstructures of dense polymers are consistent with those expected from the hopping mechanism for gas motion and the structural relaxation of the polymer chains does not seem to contribute appreciably to the rate constants of the solute hops, thus supporting the adequacy of the hopping mechanism for the motion of a light solute in dense polymers and lending credibility to the simple modeling of the transport processes. 

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