Statistical description

In the statistical description of ununolecular kinetics, known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory [4,7,8], it is assumed that complete IVR occurs on a timescale much shorter than that for the unimolecular reaction [9]. Furdiemiore, to identify states of the system as those for the reactant, a dividing surface [10], called a transition state, is placed at the potential energy barrier region of the potential energy surface. The assumption implicit m RRKM theory is described in the next section. 

Rather than using transition state theory or trajectory calculations, it is possible to use a statistical description of reactions to compute the rate constant. There are a number of techniques that can be considered variants of the statistical adiabatic channel model (SACM). This is, in essence, the examination of many possible reaction paths, none of which would necessarily be seen in a trajectory calculation. By examining paths that are easier to determine than the trajectory path and giving them statistical weights, the whole potential energy surface is accounted for and the rate constant can be computed. 

Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. 

In the next section we shall take up the statistical description of various possible sequences. 

From appropriate ratios of these sequence lengths, what conclusions can be drawn concerning terminal versus penultimate control of addition The following are experimental tacticity fractions of polymers prepared from different monomers and with various catalysts. On the basis of Fig. 7.9, decide whether these preparations are adequately described (remember to make some allowance for experimental error) by a single parameter p or whether some other type of statistical description is required ... 

This report is a statistical description of pipe system failures. The characteristics of these failures have been derived from reports submitted by the utilities to the Nuclear Regulatory Commission. The bulk of the data is from Licensee Event Reports supplemented as necessary by plant outage and maintenance data. 

Simulation models aim to replicate the workings and logic of a real system by using statistical descriptions of the activities involved. For example, a line may run at an average rate of 1000 units per hour. If we assume that this is always the case, we lose the understanding of what happens when, say, there is a breakdown or a halt for routine maintenance. The effect of such a delay may be amplified (or absorbed) when we consider the effect on downstream units. 

To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t) ... 

Standard deviation, 227—228 Standard error of the difference, 230 Standard values, 249—251 Statistical power, 253 Statistical significance, 227 Statistics descriptive... 

STATISTICAL MECHANICS AS THE SECOND SUBMODEL. The interaction potential obtained via quantum mechanics constitutes the input necessary to obtain a statistical description of many water molecules interacting at a given pressure and temperature. 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

Analysis of most (perhaps 65%) pharmacokinetic data from clinical trials starts and stops with noncompartmental analysis (NCA). NCA usually includes calculating the area under the curve (AUC) of concentration versus time, or under the first-moment curve (AUMC, from a graph of concentration multiplied by time versus time). Calculation of AUC and AUMC facilitates simple calculations for some standard pharmacokinetic parameters and collapses measurements made at several sampling times into a single number representing exposure. The approach makes few assumptions, has few parameters, and allows fairly rigorous statistical description of exposure and how it is affected by dose. An exposure response model may be created. With respect to descriptive dimensions these dose-exposure and exposure-response models... 

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

Kuhs, W.F. (1983) Statistical description of multimodal atomic probability densities, Acta Cryst. A, 39, 148-158. 

FUNDAMENTALS THE STATISTICAL DESCRIPTION OF COMPLEX MULTICOMPONENT SEPARATIONS... 

The basis of the transition path sampling method is the statistical description of dynamical pathways in terms of a probability distribution. To define such a distribution consider a molecular system evolving in time and imagine that we take snapshots of this system at regularly spaced times fj separated by the time step At. Each of these snapshots, or states, consists of a complete description z of the system in terms of the positions q = <71, <72, , [Pg.252]

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