Statistics hierarchy

As is well recognized, various macroscopic properties such as mechanical properties are controlled by microstructure, and the stability of a phase which consists of each microstructure is essentially the subject of electronic structure calculation and statistical mechanics of atomic configuration. The main subject focused in this article is configurational thermodynamics and kinetics in the atomic level, but we start with a brief review of the stability of microstructure, which also poses the configurational problem in the different hierarchy of scale. 

There is a hierarchy of usefulness of data, according to how well it can be statistically manipulated. The accepted order is continuous data > ordinal data > nominal data. 

The primary optimization target of CLs is the effectiveness factor of Pt utilization, Tcl- It includes a factor, that accounts for statistical limitations of catalyst utilization that arise on a hierarchy of scales, as specified in the following equafion. defermines the exchange current density ... 

Clearly it is very important that we get the hierarchy correct. Generally this would be determined by the clinical relevance of the endpoints, although under some circumstances it could be determined, in part, by the likelihood of seeing statistical significance with the easier hits towards the top of the hierarchy. 

This equation for the doublet density if involves the triplet density f. It is a typical problem in statistical mechanics. To make progress, the hierarchy must be broken and this is usually done with a superposition approximation. The manner by which this is done is discussed in fee next sub-section. 

The assumption has been rephrased and set in a variety of forms. See Fundamental Problems in Statistical Mechanics, compiled by E. G. D. Cohen, North-Holland Publ. Co., Amsterdam, 1962, particularly pp. 110-153, where Bogolubov s method of hierarchies and assumption of functionals are set forth. 

The analogy just mentioned with the BBGKI set of equations being quite prominent still needs more detailed specification. To cut off an infinite hierarchy of coupled equations for many-particle densities, methods developed in the statistical theory of dense gases and liquids could be good candidates to be applied. However, one has to take into account that a number of the... 

However, Waite s approach has several shortcomings (first discussed by Kotomin and Kuzovkov [14, 15]). First of all, it contradicts a universal principle of statistical description itself the particle distribution functions (in particular, many-particle densities) have to be defined independently of the kinetic process, but it is only the physical process which determines the actual form of kinetic equations which are aimed to describe the system s time development. This means that when considering the diffusion-controlled particle recombination (there is no source), the actual mechanism of how particles were created - whether or not correlated in geminate pairs - is not important these are concentrations and joint densities which uniquely determine the decay kinetics. Moreover, even the knowledge of the coordinates of all the particles involved in the reaction (which permits us to find an infinite hierarchy of correlation functions = 2,...,oo, and thus is... 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

Equation (1.24) is very similar to that of the single-particle distribution function of classical statistical mechanics. In the limit h—>0 we get the first equation of the BBGKY hierarchy. 

Haines LM, (1998) Interval Judgements in the Analytic Hierarchy Process A Statistical Perspective. In Stewart TJ, van den Honert RC (eds) Trends in Multicriteria Decision Making. Springer, Berlin et al., pp 87-105... 

The regulatory background to extrapolation follows a hierarchy that is dependent on the amount of data available. In general, extrapolations in low-data situations make use of simple uncertainty factors that are believed to be conservative. As the amount of data increases, uncertainty factors may be reduced or replaced by data-derived statistical measures of uncertainty. This hierarchy is illustrated in the approaches to extrapolation for different substances and situations. 

A statistical approach macroscopic equations hierarchy closure. 

In the systems that are far from equilibrium, the stratification into sub systems with fast and slow is also possible, with the subsystem with fast internal variables being characterized by the minimum of the relevant Lyapunov function (provided that such a function exists for the particular process scheme). The ways to describe systems that can be stratified in accordance with the timing hierarchy of the processes involved are under intensive study in modem chemical engineering and biophysics. The methods are based on models that take into account mechanistic (deterministic) and statistical degrees of freedom and their contribution to processes of energy transfer and chemical conversions in the systems with a very complicated process hierarchy (for example, catalytic and biological processes). 

The contributions of Vulpiani s group and of Kaneko deal with reactions at the macroscopic level. The contribution of Vulpiani s group discusses asymptotic analyses to macroscopic reactions involving flows, by presenting the mechanism of front formation in reactive systems. The contribution of Kaneko deals with the network of reactions within a cell, and it discusses the possibility of evolution and differentiation in terms of that network. In particular, he points out that molecules that exist only in small numbers can play the role of a switch in the network, and that these molecules control evolutionary processes of the network. This point demonstrates a limitation of the conventional statistical quantities such as density, which are obtained by coarse-graining microscopic quantities. In other words, new concepts will be required which go beyond the hierarchy in the levels of description such as micro and macro. 

Above we have given the basic strategy for the statistical thermodynamics of the electrical double layer. We shall not discuss the various elaborations In detail, but note that in addition to the BBGYK hierarchy to solve the distribution functions several other methods have been developed. In the Kirkwood hierarchy a coupling constant Is introduced to avoid the spatial Integration... 

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