Unified statistical theory

In the case that the same physical approximations are applied to fluxes in a canonical ensemble, we call this canonical unified statistical theory (CUS)" and the recrossing factor is given by... 

The next step in the unified statistical theory is to look for minima in the flux with respect to s and to calculate the cumulative reaction probability N(E). This is defined exactly as... 

For examples of applications of the unified statistical model employing this kind of ad hoc quantization, see B. C. Garrett and D. G. Truhlar, Generalized transition state theory. Quantum effects for collinear reactions of hydrogen molecules, J. Chem. Phys. 83 1079 (1979) 84 682(E) (1980) Application of variational transition state theory and the unified statistical model to H + CI2 -> HCl + Cl, J. Phys. Chem. 84 1749 (1980) B. C. Garrett, D. G. Truhlar, R. S. Grev, and R. B. Walker, Comparison of variational transition state theory and the unified statistical theory with vibrationally adiabatic transmission coefficients to accurate collinear rate constants for T + HD TH + D, J. 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

Statistical theories of macromolecules in solutions have recently attracted considerable attention of theorists because of remarkable and wide-ranged properti of macromolecules, of their close connection to the theories of phase transitions in lattices, and relations to ferromagnetism and adsorption problems and of the discoveries in the structures and functions of DNA and other biological macromolecules. Needless to say, a great many papers and books have been pubUshed recently, but we confine our attention to statistical theories of macromolecules in solutions. In spite of the great number of papers in this field, however, the development of rigorous statistical theories of macromolecular solutions has been rather slow, and there have been presented many different approaches some of which have probably confused readers. Therefore, in this paper we aim at a rather unified and simplified theory of macromolecular solutions and at the same time we discuss some of the feattues of various other macromolecular solution theories and elucidate the present situation. In so doing we hope to attract attention of more theoretical chemists and physicists whose participation in this field is certainly needed. 

Summarizing, one can say that the lattice theories need improvement and compact macromolecules need more refined treatment. We shall develop in this paper a refined and unified theory of macromolecular solutions with special emphasis on dilute solutions. We shall put our standpoint on the general theory of solutions developed by McMillan and Mayer in 1945 and Kirkwood and Buff in 1951 (9). TTiese theories do not use the lattice model and are more natural for application especially to dilute solutions. The theories extend statistical theories on gases and this is the reason why we used the name gas theories (70) in the beginning of this Introduction. 

It is not the aim of this book to give a full account of the present stage of the collision and statistical theory by considering all various approaches to the solution of the dynamic problems involved. Instead, it attempts to present a detailed discussion of the relations between both theories from a unified point of view. Therefore, attention is paid not so much to computational techniques as to the fundamental aspects of the problem. Their complete elucidation is possible only by means of exact definitions of the concepts and by accurate formulations of the theories. Computational approaches are certainly of great importance for the practical application of any physical theory. In particular, the physical chemist is much interested in how to calculate the reaction velocities, which requires an estimation of various parameters entering the rate equations. Very often, however, we ask about the procedure of evaluating some quantities which are not well defined, for instance, the quantum correction to a classical (or semiclassical), collision or statistical theory. As a consequence, large discrepancies between the results of different approaches arise mainly... 

We shall deal here with a Lattice Gas (LG) model and a model of Unified Gas-Adsorbate Layer (UGAL). These two models correspond to two alternative approaches in the statistical theory of equilibrium adsorbates (Flood 1967) and can be considered as mutually complementary ones. Which of them should be used depends on the adsorbate properties (ideal, weakly non-ideal, liquid, polycrystalline, localized, partially localized, delocalized), external conditions (isothermaJ, non-isothermal), meclianisms of elementai y processes (adiabatic, non-adiabatic), etc. 

The "unified" statistical model was introduced originally in order to reconcile, or unify, two different kinds of statistical theories, transition state theory which is appropriate for reactions proceeding via a direct" reaction mechanism, and the phase space theory of Lightand Nikitin which is designed to describe reactions proceeding via a long-lived collision complex. It is particularly straightforward to apply this theory within the framework of the reaction path Hamiltonian. 

B. C. Garrett, D. G. Truhlar, R. S. Grev, A. W. Magnuson, and J. N. L. Connor, Variational transition state theory, vibra-tionally adiabatic transmission coefficients and the unified statistical model tested against accurate quantal rate constants for collinear F + H2, H + F2, and isotopic analogs, J. Chem. 

A unified approach to the glass transition, viscoelastic response and yield behavior of crosslinking systems is presented by extending our statistical mechanical theory of physical aging. We have (1) explained the transition of a WLF dependence to an Arrhenius temperature dependence of the relaxation time in the vicinity of Tg, (2) derived the empirical Nielson equation for Tg, and (3) determined the Chasset and Thirion exponent (m) as a function of cross-link density instead of as a constant reported by others. In addition, the effect of crosslinks on yield stress is analyzed and compared with other kinetic effects — physical aging and strain rate. 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

STATISTICAL PHYSICS, Gregory H. Wannier. Classic text combines thermodynamics, statistical mechanics and kinetic theory in one unified presentation of thermal physics. Problems with solutions. Bibliography. 532pp. 55 x 85. 

After the seminal work of Guggenheim on the quasichemical approximation of the lattice statistical-mechanical theory[l], various practical thermodynamic models such as excess Gibbs energies[2-3] and equations of state[4-5] were proposed. However, the quasichemical approximation of the Guggenheim combinatory yields exact solution only for pure fluid systems. Therefore one has to resort to numerical procedures to find the solution that is analytically applicable to real mixtures. Thus, in this study we present a new unified group contribution equation of state[GC-EOS] which is applicable for both pure or mixed state fluids with emphasis on the high pressure systems[6,7]. 

We have reviewed the recent development of a nonequilibrium statistical mechanical theory of polymeric glasses, and have provided a unified account of the structural relaxation, physical aging, and deformation kinetics of glassy polymers, compatible blends, and particulate composites. The specific conclusions are as follows ... 

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