Statistics, Gaussian

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing Y particles within a molecular size volume v is a Gaussian fimction of N. The moments of the probability distribution fimction are related to the n-particle correlation functions and... 

Crooks G E and Chandler D 1997 Gaussian statistics of the hard sphere fluid Phys. Rev. E 56 4217... 

For normal statistics, the mean and the variance are completely sufficient to characterize the process all the other moments are zero. For standard normal or Gaussian statistics (i.e., normal statistics with zero mean), the variance p,2... 

The intensity obeys a gaussian statistics so that its variance is equal to the mean intensity... 

A polymer coil does not only possess a structure on the atomistic scale of a few A, corresponding to the length of covalent bonds and interatomic distances characteristic of macromolecules are coils that more or less, obey Gaussian statistics and have a diameter of the order of hundreds of A (Fig. 1.2) [17]. Structures of intermediate length scales also occur e. g., characterized by the persistence length. For a simulation of a polymer melt, one should consider a box that contains many such chains that interpenetrate each other, i. e., a box with a linear dimension of several hundred A or more, in order to ensure that no artefacts occur attributable to the finite size of the simulation box or the periodic boundary conditions at the surfaces of the box. This ne-... 

Assuming Gaussian statistics one expects (R ) oc ( 2)Mi and hence the number average becomes ... 

Since Gaussian statistics contain only L and /K in the combination Edwards scaling formula for a = 3 and is also followed by the packing models. 

Since A is roughly proportional to vTz according to the Gaussian statistics valid approximately before the quench, we expect... 

In both examples discussed in this section, the second-order approximation to AA turned out to be satisfactory. We, however, do not want to leave the reader with the impression that this is always true. If this were the case, it would imply that probability distributions of interest were always Gaussian. Statistical mechanics would then be a much simpler field. Since this is obviously not so, we have to develop techniques to deal with large and not necessarily Gaussian-distributed perturbations. This issue is addressed in the remainder of this chapter. 

Pab is the mutual concentration of reactive groups separated by a chain of v bonds, assuming the chain obeys Gaussian statistics. 

Probabilities of equilibrium conformations of polymer chains are describ-able by the Gaussian statistics. 

A satisfactory theory of macrocyclisation equilibria for chains obeying Gaussian statistics was presented by Jacobson and Stockmayer (1950) long before the availability of suitable experimental data for the proper testing of the theory. According to this theory, the macrocyclisation equilibrium constant Kx (see p. 10) is related to the density W(0) of the distribution of the end-to-end vector r in the region r = 0 through relation (57), where NA is... 

There are several reasons why the behaviour of the shorter chains deviate from the original formulation of the Jacobson and Stockmayer theory (Flory, 1969). First, if the ring size is small enough to induce strain, the enthalpy change for cyclisation (16) will differ from that for the intermolecu-lar process (17). In terms of the 0 operator (39), 0AH° will differ from zero and, presumably, be positive. Secondly, (57) is based on the implicit assumption that the relative orientation of the reacting bonds, when they come in close proximity in the cylisation reaction, is random. This independence of orientation and proximity, which leads to the absence of any factor referring to orientation in (57), must fail for short chains. Thirdly, short chains may not follow Gaussian statistics. When this occurs, an appropriate expression for the density of end-to-end vectors is required. 

Polymeric chains in the concentrated solutions and melts at molar-volumetric concentration c of the chains more than critical one c = (NaR/) ] are intertwined. As a result, from the author s point of view [3] the chains are squeezed decreasing their conformational volume. Accordingly to the Flory theorem [4] polymeric chains in the melts behave as the single ones with the size R = aN112, which is the root-main quadratic radius in the random walks (RW) Gaussian statistics. 

Two main approaches for osmotic pressure of polymeric solutions theoretical description can be distinguished. First is Flory-Huggins method [1, 2], which afterwards has been determined as method of self-consistent field. In the initial variant the main attention has been paid into pair-wise interaction in the system gaped monomeric links - molecules of solvent . Flory-Huggins parameter % was a measure of above-said pair-wise interaction and this limited application of presented method by field of concentrated solutions. In subsequent variants such method was extended on individual macromolecules into diluted solutions with taken into account the tie-up of chain links by Gaussian statistics [1]. 

This result is appHcable to semi-dilute and concentrated solutions [21], and is also useful to check many simulations that do not include HI. For non-draining chains, introducing Gaussian statistics in Eq. (35), and transforming the summations over a large number of units in Eq. (34) into integrals, the translational friction coefficient can finally be written as [ 15]... 

The introduction of branching in the Kirkwood formula and the KR calculations can be accomplished in a relatively easy way if Gaussian statistics corresponding to ideal chains are maintained. This description cannot, however, be very accurate in molecules with centers of high functionality because of the presence of cores with a high density of polymer units, which profoundly perturbs the internal distribution of distances. Stockmayer and Fixman [81 ] employed the Kirwood formula and Gaussian statistics to calculate h in the case of uniform stars, obtaining an analytical formula. They also performed a KR evaluation of the viscosity and proposed that g could be evaluated from the approximation... 

If the rays possess ideal flexibility to allow application of Gaussian statistics, the resultant structure will resemble a soft sphere. This was the reason why the present author introduced the soft sphere model [38]. This model reduces to dendrimers in the narrow sense when no spacer chains between the branching units are present. 

For a polyelectrolyte chain that has non-Gaussian statistics, exact analytical expression for B is not feasible. To get some insight, we notice that the static structure factor has the limiting behavior. 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

It was assumed that the experimental uncertainties followed Gaussian statistics with equal standard deviation a for all points. Then the standard deviation was determined as ct = (x /v), where v is the number of degrees of freedom in the fit. v is equal to the number of the experimental points less the number of parameters used in the minimization. The best fit with the Frumkin prediction has one free parameter (j6) and gives p =- 2.051 and CT = 0.41 mN/m. The standard deviation in surface tension is small, indicating that the fit with the Frumkin prediction is statistically significant. Similar best fits are obtained for the other surfactants of the homologue series of... 

IPC [292]. The data are normalized to the transit time and shifted along the current axis for clarity. However, as in the case of CT, there are some difficulties in formulating a simple interpretation of the results obtained due to the impracticability of the Gaussian statistics because the necessary superlinear dependence of the transit time on the sample thickness was not observed. 

Cyclic oligomers with x - 2-9 are found to be present in poly(1,3-dioxolane) samples prepared by monomer-polymer-equilibrations using boron trifluoride diethyl etherate as catalyst. The molecular cyclization equilibrium constants 7fx are measured and the values are in agreement with those calculated by the Jacobson-Stockmayer theory, using an RIS model to describe the statistical conformations of the corresponding chains and assuming that the chains obey Gaussian statistics. 

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