Separation distance Gaussian distribution

A basic theoretical model for flexible polymers is the Gaussian chain which assumes N ideal beads with intramolecular distance between them following a Gaussian distribution, so that the mean quadratic distance between two beads separated by n-1 ideal and not correlated bonds is given by [ 15,20]... 

Second, we will consider that the degree of correlation between the fluctuating bilayers depends on their separation distance, with total correlation as z — - 0 and no correlation as z - °o. This provides an asymmetric distribution (Figure lb), which will be described using an asymmetry coefficient a in a truncated Gaussian distribution... 

Figure 4. Root-mean-square fluctuation a vs the separation distance a, for (1) eq 15 and (2-4) a calculated for truncated Gaussian distribution with various asymmetrycoefficients (2) a = 1 (3) a = 1.4 (4) a = 2.0 (5) HelMch proportionality relation, a2=tta2 (ii = 0.183) (6) approximate solution for small distances, eq 16.

In Figure 7 we present the free energy for an asymmetric Gaussian distribution (a = 1.4) as a function of distance for various values of the Hamaker constant (with all the other parameters unchanged). For H > 3.825 x 10-21 J, a stable minimum is obtained at a finite distance. For H < 3.825 x 10 21 J, the stable minimum is at infinite distance however, for 3.825 x 10-21c7 > H > 3.45 x 10-21 J, a local (unstable) minimum is still obtained at finite distance. For H = 3.825 x 10-21 J, a critical unbinding transition occurs, since the minima at finite and infinite distances become equal. However, these two minima are separated by a potential barrier, with a maximum height of 1.68 x 10 7 J/m2, located at a separation distance of 90 A. The results remained qualitatively the same for any combination of the interaction parameters. 

Resolution. An important quantitative parameter frequently used to evaluate the performance of the separation methods is the resolution, Rs. This parameter is calculated as the ratio of the relative distance of two totally or partially resolved zones to the average standard deviation of their widths expressed in the same units. The resolution of the focusing process that is described approximately by Gaussian distribution is given by... 

The mean squared end-to-end distance of a linear unperturbed chain of N monomers is proportional to N. As R is a sum of a large number of independent random vectors, the probability density for the end-to-end separation vector to have a certain value R is given by a Gaussian distribution ... 

The y-radiation-induced conductivity of n-hexane under high hydrostatic pressure was studied by Holroyd (1994) and by Holroyd, Chen et al. (1996). The free ion yield at T = 295 K, Gfi(E = 0), decreased from 0.14 at 1 bar to 0.1 at 2 kbar. Evaluation of the data by Equations 39 and 40 with a Gaussian distribution of separation distances showed that the product b x d (separation distance times density) is essentially constant with pressure. This finding complements the data of Schmidt and Allen (1970) (Section 5.6, Table 7), who found that b x d remained constant when the density was varied by change of temperature. 

We have seen that at a distance r from a monomer the monomers within a range n ia/r) of m along the chain make the major contribution to On occasion, especially for photophysical processes where two monomers must come to proximity, one needs to know the behavior of a more detailed correlation function, p(r,/i), the probability that two monomer units n monomers apart on the chain are separated in space by a vector r. For an ideal chain this is the well-known Gaussian distribution... 

As indicated previously, an elastomer can be identified with an assembly of chains connected through randomly distributed cross-links that are separated from one another by a quadratic average end-to-end distance ((> o) satisfying a Gaussian distribution function P(n, r) (see Chapter 5). In the following treatment, the network is considered ideal, without dangling chains and entanglements. 

First, consider a Gaussian chain, which is a model for an ideal flexible chain with no excluded-volume effects. The Gaussian chain is defined as such a chain that the probability density for the distance, r, separating the ath and /9th segments in the chain is given by the Gaussian distribution. 

Resolution, on the other hand, is a more technical term. It refers to the distance between adjacent bands relative to their bandwidths and acknowledges the fact that proteins are distributed in Gaussian profiles with overlapping distributions. The numerical expression for resolution is obtained by dividing the distance between the centers of adjacent bands by some measure of their average bandwidths. It expresses the distance between band centers in units of bandwidth and gives a measure of the overlap between two adjacent bands. For preparative applications, when maximal purity is desired, two proteins to be isolated should be separated by at least a bandwidth. In many applications it is sufficient to be able to simply discern that two bands are distinct. In this case bands can be less than a bandwidth apart. 

In the independent pair model, the average or expectation value of N is the survival probability of the radical pairs when each is distributed as a Gaussian. It can be calculated by convolution and will not be unity even at time t = 0 because some pairs are formed with r0 pair survival probability was called IIa(f). Assuming that the distances of separation of the reactant pairs is independent of all other distances (this is not strictly true) and that only one pair reacts at any one time, Clifford et al. developed the probability that a spur initially containing N0 reactants [and hence M0 = (N0/2) (N0 — 1) pairs] and IV at time t [and hence M — (AT/2) (IV —1) pairs] does not react further for a time r and this was shown to be... 

---