Exactly Solvable Cases

The analytical solutions for the following three exactly solvable cases are presented (1) delta-function barrier, (2) parabolic potential barrier, and (3) Eckart potential barrier. These solutions may be useful for some analyses. 

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In general, it can be very difficult to determine the nature of the boundary terms. A specific result in an exactly solvable case is discussed in Section IV.A.2. Equation (55) is the Gallavotti-Cohen FT derived in the context of deterministic Anosov systems [28]. In that case, Sp stands for the so-called phase space compression factor. It has been experimentally tested by Ciliberto and co-workers in Rayleigh-Bemard convection [52] and turbulent flows [53]. Similar relations have also been tested in athermal systems, for example, in fluidized granular media [54] or the case of two-level systems in fluorescent diamond defects excited by light [55]. 

B. Derrida, J. L. Lebowitz, and E. R. Speer, Free energy functional for nonequilibrium systems an exactly solvable case. Phys. Rev. Lett. 87, 150601 (2001). 

So, our main purpose is attained, since we have solved the equations of motion for the case, which is one of the most general cases of exactly solvable models. This aim can be enlarged in the sense, that all quadratic cases will be solved in the following papers. 

This concludes a discussion of exactly solvable second-order processes. As one can see, only a very few second-order cases can be solved exactly for their time dependence. The more complicated reversible reactions such as 2Apt C seem to lead to very complicated generating functions in terms of Lame functions and the like. This shows that even for reasonably simple second- and third-order reactions, approximate techniques are needed. This is not only true in chemical kinetic applications, but in others as well, such as population and genetic models. The actual models in these fields are beyond the scope of this review, but the mathematical problems are very similar. Reference 62 contains a discussion of many of these models. A few of the approximations that have been tried are discussed in Ref. 67. It should also be pointed out at this point that the application of these intuitive methods to chemical kinetics have never been justified at a fundamental level and so the results, although intuitively plausible, can be reasonably subject to doubt. 

Now we see clear the problem while the new dot Hamiltonian (154) is very simple and exactly solvable, the new tunneling Hamiltonian (162) is complicated. Moreover, instead of one linear electron-vibron interaction term, the exponent in (162) produces all powers of vibronic operators. Actually, we simply remove the complexity from one place to the other. This approach works well, if the tunneling can be considered as a perturbation, we consider it in the next section. In the general case the problem is quite difficult, but in the single-particle approximation it can be solved exactly [98-101]. 

To conclude this chapter, we present the quantum mechanical results of the particle-in-a-triangle problem. Before going into details, we first need to note that, if the box is a scalene triangle, no analytical solution is known. Indeed, the Schrodinger equation is exactly solvable for only a few triangular systems. In addition, for all these solvable (two-dimensional) cases, the wavefunctions are no longer the simple products of two functions each involving only one variable. 

In the case of a free particle (V(x) = 0), the model described by the Caldeira-Leggett Hamiltonian (1) is exactly solvable.1... 

The application of this method to systems described by one-dimensional potentials is particularly simple (15, 18). Therefore, in this paper, the feasibility and the accuaracy of the approach has been illustrated by considering transitions between states described by several exactly solvable onedimensional models (20) and between X and B1 states of Henergy curves (21). It results, that with a proper choice of the functional form of the envelope, already three-moment curves give a very accurate description of the band shape. For harmonic oscillators (22), the Gram-Charlier-type expansions (23) are very accurate. They axe also rather good for the cases reasonably well approximated by harmonic-oscillator-type potentials (15,18). However, if the departure from harmonicity is considerable, these kinds of expansions are inappropriate. 

Both of the models just discussed present features which we want to remark upon. First, both are reported to be exactly solvable in some limit (i.e., thermodynamic limit for the BCS model, and point-source limit for the van Hove model). Second, in both cases the physics seems to come out right (i.e., energy-gap in BCS, and 5 = 7 in van Hove). Third, both models show a strange mathematical behavior. Finally, both involve an infinite number of degrees of freedom. 

All in all, the Rouse model provides a reasonable description of polymer dynamics when the hydrodynamic interactions, excluded volume effects and entanglement effects can be neglected a classical example of its applicability is short-chain polymer melts. Since the Rouse model is exactly solvable for polymer chains, it represents a basic reference frame for comparison with more involved models of polymer dynamics. In particular, the decouphng of the dynamics of the Rouse chain into a set of independently relaxing normal modes is fundamental and plays an important role in other cases, such as more complex objects of study, or in other models, such as the Zimm model. 

This EOM is very useful for discussing the exactly solvable limits of the AIM. In section 3.2.1 we discuss the atomic limit and in section 3.2.2 the noninteracting case U = 0. 

Fig. 2.2 Similarly to the above case, the exactly solvable potential of Eq. 2.29, their eigenfunctions and eneigy spectra, as well as the corresponding position-dependent mass distribution. The mass parameter is P = 1...

Shapes of molecular electronic bands are studied using the methods of the statistical theory of spectra. It is demonstrated that while the Gram-Charlier and Edgeworth type expansions give a correct description of the molecular bands in the case of harmonic-oscillator-like potentials, they are inappropriate if departure from harmonicity is considerable. The cases considered include a set of analytically-solvable model potentials and the numerically exact potential of the hydrogen molecule. 

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