Fast variable description

The plan of this chapter is as follows. In Section II and the Appendix we review some basic topics in statistical mechanics that underlie questions concerning the regimes of validity of slow and fast variable descriptions of irreversible motion. 

We next briefly overview our fast variable description of liquid chemical dynamics. 

IV. THE FAST VARIABLE DESCRIPTION OF LIQUID PHASE CHEMICAL DYNAMICS... 

Chapter II addresses another fundamental problem under what physical conditions can the Fokker-Planck equation provide a reliable picture of fluctuation-dissipation processes Aware as they are of the technical and conceptual difidculties involved in nonlinear statistics, the authors share Zwanzig s optimistic view that use of the Fokker-Planck equation is practicable and advantageous. Chapter II contains a brief description of rules to construct, via a suitable procedure, equations of Fokker-Planck type for the slow variables of the system under study. This theoretical method eliminates explicit analytic dependence on fast variables and thereby produces a significant simplification of the problem under discussion. 

In Section III, we combine the analyses of Section II and the Appendix with the results of recent experiments [13] and simulations [14] to discuss the slow variable description [7-10] of liquid phase activated barrier crossing. In Section IV, using these same ideas we review our fast variable theory of liquid phase reaction dynamics [19-26] from the standpoint of the recent literature [6,11-17,27,28]. Finally, in Section V we summarize our main points. 

To avoid complexity, we will limit the fast variable evaluation of F[5 x(r)] to times t that are short relative to the fluid s motional response times T. (An outline description of fast particle motion for times r > t is given in Section IV.A and Figure 3.5.)... 

We can now proceed with the analysis as described in Section 6.3. Let us define the stretched, fast time variable r = t/e. On rewriting the model in terms of r and considering the limit e — 0, we obtain a description of the fast dynamics of the process ... 

The descriptions of the fast subsystems are obtained hierarchically, starting from the fastest fast time scale. On introducing a stretched time variable tm = t/eM, the system in Equation (B.l) takes the form... 

In Section 2.2 we mentioned the impossibility to strictly substantiate the equilibrium descriptions for all cases of life and the need to apply equilibrium approximations in some situations. The vivid examples of the cases, where the strongly nonequilibrium distributions of microscopic variables are established in the studied system and the principal difficulties of its description with the help of intensive macroscopic parameters occur, are fast changes in the states at explosions, hydraulic shocks, short circuits in electric circuits, maintenance of different potentials (chemical, electric, gravity, temperature pressure, etc.) in some spatial regions or components of physicochemical composition interaction with nonequilibrium and sharply nonstationary state environment. 

This quantitative description has a number of requirements to be suitable for CombC. It has to be fast to derive or calculate. It has to be relevant, i.e. capture the essential structural properties that influence the biological activities of interest. The structure descriptor variables have to be chemically interpretable. If possible, the description should be reversible, i.e. be possible to also translate backwards from descriptor values to structure. The description should be consistent with similarity dissimilarity, so those compounds that are closely similar have very similar values for the descriptors, and dissimilar compounds have widely different values for at least most ofthe descriptors. 

A simple approach to protein description consists of representing a protein by a sequence of properties of its constituent amino acids. Each amino acid is described by one ore more properties and therefore the total number of protein descriptors is given by the product of the number of amino acids in the protein and the number of selected amino acid properties. As this number of descriptors increases very fast with the size of proteins, this approach is usually applied to small- and medium-size peptides. Moreover, in QSAR studies that require uniform-length descriptors, it can be used only to describe a series of peptide analogues, vhich are peptide sequences with the same length. To enable QSAR studies of peptide sequences with different length, some method is required that is able to translate the peptide sequences into vectorial descriptors with the same number of variables. For example, ACC transforms were applied to compress information about principal properties of amino acids into peptide sequences with different length. 

An overview of the methods used previously in mechanism reduction is presented in Tomlin et al. (1997). The present work uses a combination of existing methods to produce a carbon monoxide-hydrogen oxidation scheme with fewer reactions and species variables, but which accurately reproduces the dynamics of the full scheme. Local concentration sensitivity analysis was used to identify necessary species from the full scheme, and a principle component analysis of the rate sensitivity matrix employed to identify redundant reactions. This was followed by application of the quasi-steady state approximation (QSSA) for the fast intermediate species, based on species lifetimes and quasi-steady state errors, and finally, the use of intrinsic low dimensional manifold (ILDM) methods to calculate the mechanisms underlying dimension and to verify the choice of QSSA species. The origin of the full mechanism and its relevance to existing experimental data is described first, followed by descriptions of the reduction methods used. The errors introduced by the reduction and approximation methods are also discussed. Finally, conclusions are drawn about the results, and suggestions made as to how further reductions in computer run times can be achieved. 

Data and information gathered was exploited within DaCoTA for the estimation of road traffic fatahties based on time-series analysis, as it is important to know in what direction the annual casualties are developing, and how fast this development is expected to go. The methods applied to achieve the forecasts are sophisticated statistical tools, not easily understood by non-experts [THO 13], The forecast resnlts, however, are of direct interest for road safety practitioners with all levels of statistical expertise, therefore it was decided not only to develop a technical description of the forecasting model and of the process that led to its selection for each conntry, bnt also the Country Forecast Fact Sheets pUP 12], The forecast factsheets are meant to give a relatively non-technical description of the past development of the fatalities (and of the exposure if available). The toad traffic fatalities, the traffic volume and the fatality risks are forecasted to 2020 and also forecasts according to mobihty scenarios are carried out for all 30 European countries, with exposure as most important ejqrlaining variable. If known, the (possible) reasons for the developments are shortly described. Forecasts of the road safety situation in every country include a description of the method adopted to produce these forecasts. 

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