Slow macroscopic variable

The most obvious variables of interest are the slow macroscopic variables represented by the operators A,B,C,... Because the statistical mechanics provides the macroscopic properties from the microscopic world, we also need to deal with fast microscopic variables most often the flux variables. Expressions (91) and (92) show that, at short time, the dynamics is dominated by the first few terms of the expansion... 

Under this term we understand the regularities of the processes developing in the system in time on long-term scale, from tens minutes to several days or more, which results in relatively slow changes of macroscopic variables. A similar term is dynamics it is used more often in physical papers and will be used in this book as well. 

The sometimes contradictory results from different workers in relation to the elements mentioned above extends to other elements . Some of these differences probably arise from variations in test methods, differences in the amounts of alloying additions made, variations in the amounts of other elements in the steel and the differing structural conditions of the latter. Moreover, the tests were mostly conducted at the free corrosion potential, and that can introduce further variability between apparently similar experiments. In an attempt to overcome some of these difficulties, slow strain-rate tests were conducted on some 45 annealed steels at various controlled potentials in three very different cracking environments since, if macroscopic... 

Simplification of the solution or complete exclusion of the problem of dividing the variables into fast and slow is a great computational advantage of MEIS in comparison with the models of kinetics and nonequilibrium thermodynamics. The problem is eliminated, if there are no constraints in the equilibrium models on macroscopic kinetics. Indeed, the searches for the states corresponding to final equilibrium of only fast variables and states including final equilibrium coordinates of both types of variables with the help of these models do not differ from one another algorithmically. With kinetic constraints the division problem is solved by one of the three methods presented in Section 3.4, which are applied in the majority of cases to slow variables limiting the results of the main studied process. 

In my opinion, an active centre of alkene polymerization in the liquid phase is not a single chemical entity to be visualized by a single (and simple) chemical formula. Probably a set of compounds, of complexes with variable composition, a dynamic system where the effects of individual components are mutually complementary or overlapping is really in play. The same macroscopic effect (centres of equal activity and iso-specific regulating ability) can be obtained with various starting organometals and donors. In such a system, subsystems may exist each of which is externally manifested as an individual active centre (rapid or slow, isotactic, with a tendency to transfer or termination, or living, etc.) [225],... 

In Section II we compare particle mechanics in the slow and fast variable timescale regimes. We start the discussion by showing the following. For damped macroscopic particles, the potential energy function whose minima locate the particle s points of static equilibrium also produces the forces which drive its dynamics. For damped microscopic particles, in contrast, the potential that determines the particle s statics may or may not produce the forces that drive its dynamics. 

The particle s slow variable equations of motion turn out to be identical in form to the phenomenological equations of damped motion of a macroscopic particle immersed in, say, a viscous fluid. For example, compare the slow variable Eqs. (3.24) and (3.25) with the phenomenological eqs. (A.30) and (A.31) for macroscopic sedimentation processes. 

Eor this reason, in Section 11 and the Appendix we have presented a critical review of the standard Langevin, Onsager, Mori slow variable model of irreversible dynamics [1-5] that identifies the physical assumptions that underlie the model. Especially emphasized in this discussion is that the standard model emerged from attempts to explain purely macroscopic phenomena and is ultimately founded on fully macroscopic measurements. 

At first glance a review of Onsager s theory might appear to have little or no relevance to the topics of this chapter. This, however, is not the case. This is because Onsager s theory, as extended to include memory by Mori [5] is the most general slow variable theory of irreversible motion. Thus, our examination of Onsager s work exposes limitations inherent in all slow variable models. Especially, it shows that the limitations of the Kramers-type models for reactions merely reflect the macroscopic scope of the general theory of irreversible processes [1,3-5]. 

We next turn to a discussion of the physical content of Onsager s theory. In this discussion we will develop Onsager s Eq. (A.15) from slow variable assumptions, thus showing the close link between slow variable models and macroscopic phenomenology. 

We next develop the Machlup-Onsager equation from Eq. (A.19) by making the familiar assumption that the macroscopic parameters A(f) are slow variables. This assumption is usually justified by the idea that a timescale separation exists between the parameters A(t) and their bath variables, due to the macroscopic nature of the former and microscopic nature of the latter. 

In summary, Onsager did succeed in finding a nonequilibrium extension of equilibrium thermodynamics. However, to resolve the dual problem of formulating a thermodynamic equation of motion and of choosing the thermodynamic forces, he was obliged to make limiting slow variable assumptions. Thus his central Eq. (A.53) models actual macroscopic parameters motions in a highly simplified way, namely, as coupled diffusive motions in the equilibrium potential oc S Tp A) of Eq. (A.36). 

The slow variable models are ultimately based on purely macroscopic measurements. For example Onsager s diffusional model of Eq. (A.53) derives from Eq. (A. 15), which in turn is based on macroscopic phenomenology. Similarly, Onsager s Langevin model of Eq. (A.52) derives from Eq. (A.28), which in turn is suggested by the empirical friction law —for a macroscopic particle. 

As discussed in Section II the macroscopic roots of slow variable equations like Onsager s clearly render such equations questionable as frameworks for studying microscopic processes. 

In summary, the macroscopic nature of the unconstrained A s does not guarantee that they relax like classical slow variables. Thus, even for fully macroscopic processes F Fp, open questions about kinetics remain. 

In the case of a nonreacting fluid, where one is usually interested in macroscopic equations for conserved (in the limit k O) variables, the origin and region of validity of this approximation is clear. In the small k limit the conserved fields do decay much more slowly than other variables in the system, and the limit z- 0, k O has the effect of extracting the decay on this slow time scale. (Mode coupling contributions spoil some of these arguments, but it is now known how to account for these effects. We discuss this aspect of the problem in Section VII.)... 

Out of equilibrium there is no such rigorous principle. However, macroscopicaUy one can find a large variety of phenomenological equations for the time evolution which are based on macroscopic quantities alone, e.g., the diffusion equation, the heat transport equation, and the Navier-Stokes equations for hydrodynamics. A microscopic dynamical theory for the time evolution of slow variables such as the momentum density or the particle density with molecular spatial resolution is highly desirable. 

As in previous chapters we work in the continuum limit employing quantities averaged over macroscopically infinitesimal volume elements and disregarding microscopic local variations associated with the molecular structure (see Brown 1956). These considerations will be limited to processes sufficiently slow to restrict the treatment to time independent or quasistatic fields. The validity of Maxwell s equations of electrostatics is presupposed. The basic electric state variables are the electric field strength vector E, the electric flux density (or electric displacement) vector D, and the electric polarization vector P, related by... 

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