Mean squares

The ternary diagrams shown in Figure 22 and the selectivi-ties and distribution coefficients shown in Figure 23 indicate very good correlation of the ternary data with the UNIQUAC equation. More important, however, Table 5 shows calculated and experimental quarternary tie-line compositions for five of Henty s twenty measurements. The root-mean-squared deviations for all twenty measurements show excellent agreement between calculated and predicted quarternary equilibria. 

This subroutine also prints all the experimentally measured points, the estimated true values corresponding to each measured point, and the deviations between experimental and calculated points. Finally, root-mean-squared deviations are printed for the P-T-x-y measurements. 

To estimate the quality of restoration having both, simulated and reconstructed images, using some kind of quality criterion For this last purpose the following mean square measure was applied ... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

If this condition is not satisfied, there is no unique way of calculating the observed value of ff, and the validity of the statistical mechanics should be questioned. In all physical examples, the mean square fluctuations are of the order of 1/Wand vanish in the thennodynamic limit. 

Both (E) and Cy are extensive quantities and proportional to N or the system size. The root mean square fluctuation m energy is therefore proportional to A7 -, and the relative fluctuation in energy is... 

The leading order quantum correction to the classical free energy is always positive, is proportional to the sum of mean square forces acting on the particles and decreases with either increasing particle mass or mcreasing temperature. The next tenn in this expansion is of order This feature enables one to independently calculate the leading correction due to quanmm statistics, which is 0(h ). The result calculated in section A2.2.5.5 is... 

The average value and root mean square fluctuations in volume Vof the T-P ensemble system can be computed from the partition fiinction Y(T, P, N) ... 

Since 5(A /5 j. (N), tlie fractional root mean square fluctuation in N is... 

Comparing the two results and substituting the relation of the mean square number fluctuations to isothennal compressibility, equation (A2.2.128) one has... 

For free particles, the mean square radius of gyration is essentially the thennal wavelength to within a numerical factor, and for a ID hamionic oscillator in the P ca limit. 

Now for a particle undergoing difflision, it is also known that its mean square displacement grows linearly in time, for long times, as... 

For the system in thennal equilibrium, one can compute the time-dependent mean square displacement (ICr)... 

The intensity of light scattering, 7, for an isolated atom or molecule is proportional to the mean squared amplitude... 

One of the most important fiinctions in the application of light scattering is the ability to estimate the object dimensions. As we have discussed earlier for dilute solutions containing large molecules, equation (B 1.9.38) can be used to calculate tire radius of gyration , R, which is defined as the mean square distance from the centre of gravity [12]. The combined use of equation (B 1.9.3 8) equation (B 1.9.39) and equation (B 1.9.40) (tlie Zimm plot) will yield infonnation on R, A2 and molecular weight. 

If we consider the scattering from a general two-phase system (figure B 1.9.10) distinguished by indices 1 and 2) containing constant electron density in each phase, we can define an average electron density and a mean square density fluctuation as ... 

The above radius of gyration is for an isotropic system. If the system is anisotropic, the mean square radius of gyration is equal to... 

The relationship between mean squared phase shift and mean squared displacement can be modelled in a simple way as follows This motion is mediated by small, random jumps in position occurring with a mean interval ij. If the jump size in the gradient direction is e, then after n jumps at time the displacement of a spin is... 

Thus, by plotting as a function of in tire limit of small q tire mean square of tire end-to-end distance can be... 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

A polymer chain can be approximated by a set of balls connected by springs. The springs account for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

In the collapse phase the monomer density p = N/R is constant (for large N). Thus, the only confonnation dependent tenn in (C2.5.A1) comes from the random two-body tenn. Because this tenn is a linear combination of Gaussian variables we expect that its distribution is also Gaussian and, hence, can be specified by the two moments. Let us calculate the correlation i,) / between the energies and E2 of two confonnations rj ]and ry jof the chain in the collapsed state. The mean square of E is... 

---