Statistical theory of extreme values

Aziz PM. Application of the statistical theory of extreme values to the analysis of maximum pit depth for aluminium alloys. Corrosion, 12, 1980 35-46. 

Gumbel, E. J., Statistical Theory of Extreme Values and Some Practical Applications, Vol. 33, 1954, Washington, D.C., National Bureau of Standards, Mathematics Series. 

Aziz, P. M., "Application of Statistical Theory of Extreme Values to the Analysis of Maximum Pit Depth Data for Aluminum, Corrosion, Vol. 12, 1956, pp. 495-506t. 

Gumbel E.J., Statistical theory of extreme values and some practical applications, US Department of Commerce, Applied Mathematics Series 33, 1954. 

The simple collision theory for bimolecular gas phase reactions is usually introduced to students in the early stages of their courses in chemical kinetics. They learn that the discrepancy between the rate constants calculated by use of this model and the experimentally determined values may be interpreted in terms of a steric factor, which is defined to be the ratio of the experimental to the calculated rate constants Despite its inherent limitations, the collision theory introduces the idea that molecular orientation (molecular shape) may play a role in chemical reactivity. We now have experimental evidence that molecular orientation plays a crucial role in many collision processes ranging from photoionization to thermal energy chemical reactions. Usually, processes involve a statistical distribution of orientations, and information about orientation requirements must be inferred from indirect experiments. Over the last 25 years, two methods have been developed for orienting molecules prior to collision (1) orientation by state selection in inhomogeneous electric fields, which will be discussed in this chapter, and (2) bmte force orientation of polar molecules in extremely strong electric fields. Several chemical reactions have been studied with one of the reagents oriented prior to collision. ... 

The univariate statistical theory is used, for example, for rejecting one extreme value in a set of scattered results in a given sample. For this purpose, the extreme value x is temporarily eliminated from the sample. Then, from the sample Xi, X2...Xn - Xe there are calculated m, s and the value ... 

Statistical theory teaches that under the assumption that the population means of the two groups are the same (i.e. if Hq is true), the distribution of variable T depends only on the sample size but not on the value of the common mean or on the measurements population variance and thus can be tabulated independently of the particulars of any given experiment. This is the so-called Student s f-distribution. Using tables of the f-distribution, we can calculate the probability that a variable T calculated as above assumes a value greater or equal to 4.7, the value obtained in our example, given that H0 is true. This probability is <0.0001. Thus, if H0 is true, the result obtained in our experiment is extremely unlikely, although not impossible. We are forced to choose between two possible explanations to this. One is that a very unlikely event occurred. The second is that the result of our experiment is not a fluke, rather, the difference Mb — Ma is a positive number, sufficiently large to make the probability of this outcome a likely event. We elect the latter explanation and reject H0 in favor of the alternative hypothesis Hx. 

The Boltzmann integro-differential kinetic equation written in terms of statistical physics became the foundation for construction of the structure of physical kinetics that included derivation of equations for transfer of matter, energy and charges, and determination of kinetic coefficients that entered into them, i.e. the coefficients of viscosity, heat conductivity, diffusion, electric conductivity, etc. Though the interpretations of physical kinetics as description of non-equilibrium processes of relaxation towards the state of equilibrium are widespread, the Boltzmann interpretations of the probability and entropy notions as functions of state allow us to consider physical kinetics as a theory of equilibrium trajectories. These trajectories as well as the trajectories of Euler-Lagrange have the properties of extremality (any infinitesimal part of a trajectory has this property) and representability in the form of a continuous sequence of states of rest. These trajectories can be used to describe the behavior of (a) isolated systems that spontaneously proceed to final equilibrium (b) the systems for which the differences of potentials with the environment are fixed (c) and non-homogeneous systems in which different parts have different values of the same intensive parameters. 

For this purpose, we reanalyze all the available static EOS data for Th, as shown in table 1, with a set of three different EOS forms, and compare the effect of the different EOS forms with the effects resulting from different data sets. As EOS forms we use the Birch equation (Birch 1978) in second order, BE2, and two recently proposed forms (Holzapfel 1990,1991) in second-order form, H02 and HI2, which are related to the Thomas-Fermi theory and are distinguished by the fact that H12 is bound to approach the Fermi-gas limit at infinite compression. A close inspection of table 1 shows very clearly that most of the data are fitted almost equally well by any of these forms, without any significant difference in the fitted parameters Kq and K g or in the minimized standard deviation of the pressure, Tp. In contrast to many other publications, table 1 presents the unrestricted standard deviations of Kq and K, which correspond to the extreme values of the error ellipsoids presented in fig. 11, and not just to the width of the error ellipsoids along and K at the center points, which are usually given as (restricted) statistical errors. Thus, it becomes obvious that... 

At this point it is worthwhile to review the possible failures of RRKM theory [9, 14] within a classical framework. First, the dynamics in some regions of phase space may not be ergodic. In this instance, which has been termed intrinsic non-RRKM behaviour [38], the use of the statistical distribution in Eq. (2.2) is inappropriate. In the extreme case of two disconnected regions of space, with one region nonreactive, the lifetime distribution is still random with an exponential decay of population to a non-zero value. However, the averaging of the flux must then be restricted to the reactive part of the phase space, and the rate coefficient is then increased by a factor equal to the reciprocal of the proportion of the phase space that is reactive. 

The trend observed in the experimental data is in line with the theoretical modelling presented in Section 2.3.2. The conduction moves from a mechanism controlled by the presence of the depletion layer (theoretical value 1 experimental value 1.0 0.1) to one controlled by transport through the accumulation layer where the Boltzmann statistics are still valid (theoretical value 2 experimental value 2.35 0.12) to the extreme case in which the Fermi level extends deep into the conduction band on the surface. It could be demonstrated in the theory (BSrsan et al, 2011) that, in this area, the value should increase above 2, reflecting that the influence of the surface band bending on the resistance decreases. This decrease is also supported by the experimental/phenomenological parameter showing a value of 3.80 0.05. 

In the permissible stress approach, the loads are specified exactly, the response analysis is carried out on the basis of elastic theory, and the structure is assessed safe, if the calculated stresses are less than the specified permissible stress. There is no separate consideration of system and parameter uncertainty or the nature of the structure, nor the consequences of failure. The loads are specified usually by other codes of practice which recommend, for example in Britain, a mixture of fair average estimates for dead loads in B. S. 648, extreme maximal estimates for imposed loads in C.P. 3 Chapter V Part 1, and statistical estimates for wind load in C.P.3 Chapter V Part 2. The uncertainty is catered for informally by the safe conservative assumptions of the designer s theoretical model and formally by an appropriate choice of loads and permissible stress values. 

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