Sufficient theory state-space formulation

Modularity. Since we would like to use the sufficient theory in a variety of contexts and problems, we need a theory that was easy to extend and modify depending on the context. In our state-space formulation the sufficient theory is couched in terms of constraints on variables. This theory gives us the opportunity to modularize its representation, partitioning the information necessary to prove the looseness of one type of constraint from that required to prove the looseness of a different constraint type. The ability to achieve modularity is a function not only of the theory but also of the representation, which should have sufficient granularity to support the natural partitioning of the components of the theory. 

The Boltzmann equation is considered valid as long as the density of the gas is sufficiently low and the gas properties are sufficiently uniform in space. Although an exact solution is only achieved for a gas at equilibrium for which the Maxwell velocity distribution is supposed to be valid, one can still obtain approximate solutions for gases near equilibrium states. However, it is evident that the range of densities for which a formal mathematical theory of transport processes can be deduced from Boltzmann s equation is limited to dilute gases, since this relation is reflecting an asymptotic formulation valid in the limit of no coUisional transfer fluxes and restricted to binary collisions only. Hence, this theory cannot without ad hoc modifications be applied to dense gases and liquids. 

Until the discovery of high-temperature superconductivity of cuprates by Bednorz and Muller in 1986 [2] and synthesis of first 90 K superconductor [3] in 1987, understanding of microscopic mechanism of superconducting (SC) state transition formulated within the BCS theory in 1957 [4] was generally accepted as a firm theoretical basis behind the physics of this phenomenon. The idea of Cooper-pair formation, i.e. formation of boson-like particles in momentum space, which are stable in a thin layer above the Fermi level and drive the system into more stable - superconducting state, is crucial in this case. Sufficient condition of pair formation is a weak, but attractive interaction between electrons. The possibility of effective attractive electron-electron (e-e) interactions was derived by Frdhlich [5,6] as a consequence of electron-phonon (e-p) interactions. 

For these reasons we cannot use (7(R) as a rigid support for dynamical studies of trajectories of representative points. G(R) has to be modified, at every point of each trajectory, and these modifications depend on the nature of the system, on the microscopic properties of the solution, and on the dynamical parameters of the trajectories themselves. This rather formidable task may be simplified in severai ways we consider it convenient to treat this problem in a separate Section. It is sufficient to add here that one possible way is the introduction into G (R) of some extra coordinates, which reflect the effects of these retarding forces. These coordinates, collectively called solvent coordinates (nothing to do with the coordinates of the extra solvent molecules added to the solute ) are here indicated by S, and define a hypersurface of greater dimensionality, G(R S). To show how this approach of expanding the coordinate space may be successfully exploited, we refer here to the proposals made by Truhlar et al. (1993). Their formulation, that just lets these solvent coordinates partecipate in the reaction path, allows to extend the algorithms and concepts of the above mentioned variational transition state theory to molecules in solution. 

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