Statistics symmetry

The mathematical expression of N(6, q>, i//) is complex but, fortunately, it can be simplified for systems displaying some symmetry. Two levels of symmetry have to be considered. The first is relative to the statistical distribution of structural units orientation. For example, if the distribution is centrosymmetric, all the D(mn coefficients are equal to 0 for odd ( values. Since this is almost always the case, only u(mn coefficients with even t will be considered herein. In addition, if the (X, Y), (Y, Z), and (X, Z) planes are all statistical symmetry elements, m should also be even otherwise = 0 [1]. In this chapter, biaxial and uniaxial statistical symmetries are more specifically considered. The second type of symmetry is inherent to the structural unit itself. For example, the structural units may have an orthorhombic symmetry (point group symmetry D2) which requires that n is even otherwise <>tmn = 0 [1], In this theoretical section, we will detail the equations of orientation for structural units that exhibit a cylindrical symmetry (cigar-like or rod-like), i.e., with no preferred orientation around the Oz-axis. In this case, the ODF is independent of t/z, leading to n — 0. More complex cases have been treated elsewhere [1,4]... 

Here we have used the statistical symmetry between the second and third directions in velocity phase space to express the granular temperature in terms of the bivariate moments mj k- The NDF n is the equilibrium (Maxwellian) distribution with the conservation properties m QQ = mo,o, tnl = mi,o, wJq i = Physically, these equalities result... 

The remarkable phenomenon of statistical symmetry was noted by Loeb [1-41]. There are some apparently totally asymmetrical structures in which characteristic parameters are, however, subject to certain well-defined constrained patterns when averaged according to some system. 

In the simplest possible case, where the sample has uniaxial statistical symmetry and where oci = as = a the five independent /o J]ayap, are given by... 

The non-consen>ed variable (.t,0 is a broken symmetry variable, it is the instantaneous position of the Gibbs surface, and it is the translational synnnetry in z direction that is broken by the inlioinogeneity due to the liquid-vapour interface. In a more microscopic statistical mechanical approach 121, it is related to the number density fluctuation 3p(x,z,t) as... 

It is beyond the scope of these introductory notes to treat individual problems in fine detail, but it is interesting to close the discussion by considering certain, geometric phase related, symmetry effects associated with systems of identical particles. The following account summarizes results from Mead and Truhlar [10] for three such particles. We know, for example, that the fermion statistics for H atoms require that the vibrational-rotational states on the ground electronic energy surface of NH3 must be antisymmetric with respect to binary exchange... 

Symmetry considerations have long been known to be of fundamental importance for an understanding of molecular spectra, and generally molecular dynamics [28-30]. Since electrons and nuclei have distinct statistical properties, the total molecular wave function must satisfy appropriate symmehy... 

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

The modeling of amorphous solids is a more dilhcult problem. This is because there is no rigorous way to determine the structure of an amorphous compound or even dehne when it has been found. There are algorithms for building up a structure that has various hybridizations and size rings according to some statistical distribution. Such calculations cannot be made more efficient by the use of symmetry. 

As a consequence of these various possible conformations, the polymer chains exist as coils with spherical symmetry. Our eventual goal is to describe these three-dimensional structures, although some preliminary considerations must be taken up first. Accordingly, we begin by discussing a statistical exercise called a one-dimensional random walk. 

The probabihty-density function for the normal distribution cui ve calculated from Eq. (9-95) by using the values of a, b, and c obtained in Example 10 is also compared with precise values in Table 9-10. In such symmetrical cases the best fit is to be expected when the median or 50 percentile Xm is used in conjunction with the lower quartile or 25 percentile Xl or with the upper quartile or 75 percentile X[j. These statistics are frequently quoted, and determination of values of a, b, and c by using Xm with Xl and with Xu is an indication of the symmetry of the cui ve. When the agreement is reasonable, the mean v ues of o so determined should be used to calculate the corresponding value of a. 

Another recent database, still in evolution, is the Linus Pauling File (covering both metals and other inorganics) and, like the Cambridge Crystallographic Database, it has a "smart software part which allows derivative information, such as the statistical distribution of structures between symmetry types, to be obtained. Such uses are described in an article about the file (Villars et al. 1998). The Linus Pauling File incorporates other data besides crystal structures, such as melting temperature, and this feature allows numerous correlations to be displayed. 

It is easy to invent rules that conserve particle number, energy, momentum and so on, and to smooth out the apparent lack of structural symmetry (although we have cheated a little in our example of a random walk because the circular symmetry in this case is really a statistical phenomenon and not a reflection of the individual particle motion). The more interesting question is whether relativistically correct (i.e. Lorentz invariant) behavior can also be made to emerge on a Cartesian lattice. Toffoli ([toff89], [toffSOb]) showed that this is possible. 

In contrast to phthalocyanines (tetra- or octasubstituted) in which the isoindoline units carry all the same substituents, reports of phthalocyanines with lower symmetry, which have been prepared by using two different phthalonitriles, have rarely appeared. This is due to the problems which are associated with their preparation and separation. For the preparation of unsymmetrical phthalocyanines with two different isoindoline units four methods are known the polymer support route,300 " 303 via enlargement of subphthalocyanines,304 " 308 via reaction ofl,3,3-trichloroisoindoline and isoindolinediimine309,310 and the statistical condensation followed by a separation of the products.111,311 319 Using the first two methods, only one product, formed by three identical and one other isoindoline unit, should be produced. The third method can be used to prepare a linear product with D2h symmetry formed by two identical isoindoline units. For the synthesis of the other type of unsymmetrical phthalocyanine the method of statistical condensation must be chosen. In such a condensation of two phthalonitriles the formation of six different phthalocyanines320 is possible. 

The structural and magnetochemistry of RE3Ni7B2 phases confirms an earlier structure for Mr = Gd, Tb, Dy, Ho, Er, Tm, Lu (U3CO7B2 type, P63/mmc). For Mre = Sm, Eu, Yb, however, a symmetry reduction is claimed (P6/mmm) corresponding to a different site occupation, which actually represents an occupation variant of the CeCo4B type insofar as Eu + Ni statistically occupy the 2c sites . 

The presence of substituents on the bidentate ligands often degenerate their local C2-symmetry and, as a consequence, further isomerism occurs. This results in trisbidentate derivatives of type P(ab)3 in the presence of facial or meridional isomers, as depicted in Fig. 17 [102]. Usually, the meridional isomer is preferred over the facial and a statistical 3 1 ratio is observed. Compounds 6,9,12 and 30... 

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