Correlation long range, absence

When the system contains more than one component it is important to be able to explore the distribution of the different components both locally and at long range. One way in which this can be achieved is to evaluate the distribution function for the different species. For example in a binary mixture of components A and B there are four radial distribution functions, g (r), g (r), g (r) and g (r) which are independent under certain conditions. More importantly they would, with the usual definition, be concentration dependent even in the absence of correlations between the particles. It is convenient to remove this concentration dependence by normalising the distribution function via the concentrations of the components [26]. Thus the radial distribution function of g (r) which gives the probability of finding a molecule of type B given one of type A at the origin is obtained from... 

Mesomorphic forms characterized by conformationally ordered polymer chains packed in lattices with different kinds of lateral disorder have been described for various isotactic and syndiotactic polymers. For instance, for iPP,706 sPP,201 sPS,202 syndiotactic poly(p-methylstyrene) (sPPMS),203 and syndiotactic poly(m -methylstyrene),204 mesomorphic forms have been found. In all of these cases the X-ray fiber diffraction patterns show diffraction confined in well-defined layer lines, indicating order in the conformation of the chains, but broad reflections and diffuse haloes on the equator and on the other layer lines, indicating the presence of disorder in the arrangement of the chain axes as well as the absence of long-range lateral correlations between the chains. 

LamIso, LamN, and LamSm phases, where the layers of the rod-like molecular cores are separated by the layers formed by the disordered semiperfluorinated lateral chains. In the LamSm and LamN phases there is orientational correlation of the aromatic cores in adjacent layers, leading to birefringence and typical optical textures. The absence of a long range positional correlation of the in-plane periodicity of the layers in the LamSm phase (LamSm/dis phase) indicates only a weak coupling of these layers by the layers of the fluid chains. 

One of the main uncertainties and difficulties in assessing the role of atmospheric C02 in climate change has to do with the absence of adequate information about its temporal variability, and, in particular, whether C02 observations remain residually correlated with one another even after many years (long-range dependence). 

It is worthwhile to note that since the information content of a sequence can be represented as a mathematically defined quantity, the whole process of evolution of biopolymer sequences can be specified in exact mathematical terms. The formulated fundamental problem is extremely difficult because of the absence of direct information on the early prebiological evolution. Therefore, of particular interest are toy models of evolution of sequences that show different possibilities for appearance of statistical complexity and of long-range correlations in the sequences. 

Unlike topological torsions, atom pairs are sensitive to long-range correlations between the atoms in molecules and therefore to small changes in one part of even large molecules. Atom pair descriptors usually are Boolean variables encoding the presence or absence of a particular atom pair in each molecule. 

---