Statistical isotropy

The isotropy properties assumed for the model universe imply that it is statistically spherically symmetric about the chosen origin. If, for the sake of simplicity, it is assumed that the characteristic sampling times over which the assumed statistical isotropies become exact are infinitesimal, then the idea of statistical spherical symmetry, gives way to the idea of exact spherical symmetry thereby allowing the idea of some kind of rotationally invariant radial coordinate to exist. As a first step toward defining such an idea, suppose only that the means exists to define a succession of nested spheres, Sj C S2C ClSp, about the chosen origin since the model universe with infinitesimal characteristic sampling times is stationary, then the flux of particles across the spheres is such that these spheres will always contain fixed numbers of particles, say N, N2,. . . , NP, respectively. 

Fig. 6.3 Illustrates the concept of statistical isotropy (left and right), while the center is anisotropic...

Indirect Fourier Transformation (IFT) Analysis is a method that has been applied in studies of protein stability in a hydrated choline di-hydrogen phosphate ionic liquid [32]. The IFT analysis can also be useful in identilying unknown phases, as well as understanding the link between real space and reciprocal space. This analysis assumes statistical isotropy, as well as the dilute limit [33]. 

In many luminescence centers the intensity is a function of a specific orientation in relation to the crystallographic directions in the mineral. Even if a center consists of one atom or ion, such luminescence anisotropy may be produced by a compensating impurity or an intrinsic defect. In the case of cubic crystals this fact does not disrupt optical isotropy since anisotropic centers are oriented statistically uniformly over different crystallographic directions. However, in excitation of luminescence by polarized fight the hidden anisotropy may be revealed and the orientation of centers can be determined. 

Formally, alignment and orientation follow from the general symmetry properties of statistical tensors pkK introduced in Section 8.4. Spherical symmetry leads to k = k = 0, axial symmetry to k — 0, and alignment requires k = even, and orientation k = odd. Since the dipole approximation in the photoionization process restricts k to k < 2, a photoionized axially symmetric state can only have Poo, p10, and p20 Poo describes isotropy, and the alignment is given by [BKa77]... 

In the following we will consider some basic results of the statistics of such a homogeneous isotropic turbulent field. The consequences of homogeneity and isotropy for the correlation functions were worked out by von Karman and Howarth [179] and the full derivations are available in classical books like [66, 8, 112, 113]. 

Approximately stated Kolmogorov s hypothesis of local isotropy yields ([83] see also [121], p. 184) At sufficiently high Reynolds number, the small scale turbulent motions are statistically isotropic. 

Figure 1.9. Statistical photoorientation of azomolecules, (a) The molecules aligned along the polarization direction of the incident light absorb, isomerize, and reorient. Those aligned perpendicular cannot absorb and remain fixed, (b) Irradiation of an isotropic samples leads to accumulation of chromophores in the perpendicular direction. Circularly polarized light restores isotropy.

The observation of a layer of amorphous carbon at the film surface, in combination with the observation of liquid crystal alignment on this surface, suggested the breakthrough idea to replace the polyimide polymer film with an amorphous carbon layer [34]. The essential requirement for liquid crystal alignment (as stated by our model), namely the presence of an anisotropic distribution of directional bonds, can be fulfilled by an ion beam irradiated amorphous carbon layer. This is demonstrated by the presence of the resonance associated with tt orbitals at 285 eV in the absorption spectrum of amorphous carbon (bottom of Fig, 6.12). Its presence indicates that amorphous carbon contains unsaturated sp2 and sp hybridized carbon atoms. While sps hybridization does not lead to any anisotropy, the directional nature of carbon double and triple bonds formed by sp2 and sp hybridized carbon atoms can lead to a breaking of the isotropy of the molecular distribution. It therefore mainly remains the question whether a statistically significant anisotropy in these carbon bonds can be achieved by ion beam irradiation of an amorphous carbon layer. 

The statistical mechanics of the Curie Weiss mean field or the van der Waals mean field can likewise be discussed by the method of random fields (Siegert (1963) Jalickee (1969)). In these cases the mean field analogous to 0(r) is a position-independent vector. The existence of this mean field, however, implies the destruction of the isotropy of space, i.e., the breaking of a symmetry. As Edwards (1970a, b) notes, therefore, there must also be a breaking of symmetry in order to obtain electron localization in the transla-tionally invariant averaged system. 

In theories of local isotropy, it is assumed that the small eddies are statistically independent of each other. Velocity fluctuations are determined by the local rate of energy dissipation per unit mass of fluid (e) and by the kinematic viscosity (v). 

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