Mean-statistical fluctuations

Mean-statistical fluctuations Region of solution stability 

Debye and Bueche (1960) have employed Einstein s (1910) expression for the turbidity of a two-component solution to calculate the turbidity of a monomolecular polymer solution over the whole concentration range (c in g/cm ) (sec Equations 2.1- 32, 2.4-7,-2d) 

To being the turbidity induced by density fluctuations (the solvent turbidity) while the second summand is due to the concentration fluctuations 

Experimental data, in general, confirm Equation 3. but there are relatively few papers on this subject (Tager ct al., 1964b, 1968 Aiidreycvaet al., 1970 llj dc, 1972). 

The functional dependence of turbidity in polymer systems is also based on Einstein s equation (Equations 2.4-24, 2.1-32 Table 2.3), but calculation of the mean. square fluctuation of refractive index (Ar ) in 

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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. 

Noise. So fat, as indicated at the beginning of this section on semiconductor statistics, equihbtium statistics have been considered. Actually, there ate fluctuations about equihbtium values, AN = N— < N >. For electrons, the mean-square fluctuation is given by < ANf >=< N > 1- ) where (Ai(D)) is the Fermi-Dirac distribution. This mean-square fluctuation has a maximum of one-fourth when E = E-. These statistical fluctuations act as electrical noise and limit minimum signal levels. 

The relaxation time r of the mean length, = 2A Loo, gives a measure of the microscopic breaking rate k. In Fig. 16 the relaxation of the average length (L) with time after a quench from initial temperature Lq = 1.0 to a series of lower temperatures (those shown on the plot are = 0.35,0.37, and 0.40) is compared to the analytical result, Eq. (24). Despite some statistical fluctuations at late times after the quench it is evident from Fig. 16 that predictions (Eq. (24)) and measurements practically coincide. In the inset is also shown the reverse L-jump from Tq = 0.35 to = 1.00. Clearly, the relaxation in this case is much ( 20 times) faster and is also well reproduced by the non-exponential law, Eq. (24). In the absence of laboratory investigations so far, this appears the only unambiguous confirmation for the nonlinear relaxation of GM after a T-quench. 

As previously discussed, we expect the scaling to hold if the polydisper-sity, P, remains constant with respect to time. For the well-mixed system the polydispersity reaches about 2 when the average cluster size is approximately 10 particles, and statistically fluctuates about 2 until the mean field approximation and the scaling break down, when the number of clusters remaining in the system is about 100 or so. The polydispersity of the size distribution in the poorly mixed system never reaches a steady value. The ratio which is constant if the scaling holds and mass is conserved,... 

Adjust the constant r so that the stationary solution reproduces the correct mean square fluctuations as known from statistical mechanics, or find r from other considerations, see, e.g., (5.1) below. 

Poisson distribution of mean n =0.0121 for events with NHIT 2 30. Hence the rate of occurrence of 6 events in a 10 second time interval due to a statistical fluctuation is less than one in 7 x 10 years in our experiment. 

Clearly these large fluctuations are due to cyclic variations not turbulent fluctuations. The dashed curve is an attempt to remove this cyclic variation effect by using the most probable density value as the mean value of a normal distribution. The standard deviation of the distribution is determined from fitting the data to the side of the new mean that has not been distorted by flame arrival. The reduction of the apparent fluctuations near the flame arrival crank angle is dramatic. Both curves of Figure 5 have had the Poisson statistical fluctuations subtracted. 

One cannot expect a molecule that follows a random migration path, full of frivolous excursion, to arrive after a fixed time at exactly the same point as its equally frivolous companions. There will be a mean distance of migration X, but the individual molecules will exhibit fluctuations about this mean due to the peculiarities of their own migration. These statistical fluctuations will lead to zone broadening. The statistical (stochastic) theory of zone broadening was first developed by Giddings and Eyring [4] and has been expanded subsequently by a number of authors [5-8]. 

Statistical methods are the most popular techniques for EN analysis. The potential difference and coupling current signals are monitored with time. The signals are then treated as statistical fluctuations about a mean level. Amplitudes are calculated as the standard deviations root-mean-square (rms) of the variance according to (for the potential noise)... 

One sees that in arbitrarily dense media the isotropic variations Aeg are due to the presence of statistical fluctuations of the mean optical... 

When for a sample with a given concentration a number of signal measurements n are obtained with a standard deviation single value, P will be the probability within which limits it deviates from the true value fts as a result of statistical fluctuations. For the mean fis of n measurements ... 

The second reason for potential deviations between analysis and bulk mass is the inevitable statistical fluctuation of any analytical value. Even in an ideal mixture the characteristic to be determined (e.g. the percentage passing at a particular dimension) will result in different values for each sample. The results of measurements will fluctuate around a mean value whereby results near the mean value are most frequently obtained. There is a probability smaller than unity that a result will be within a certain range around the mean value. Often, the confidence interval for 95% probability is used, which means that 95 out of 100 measurements will produce values within the range. The relative width of this interval depends critically on sample size and the nature of the characteristic to be measured—the larger the sample the narrower the interval. Detailed information is provided in textbooks on statistics and quality control.For example, for the determination of number populations a rule of thumb requires that the sample must consist of at least 1000 or better 10 000 particles. 

In engineering practice only very simple statistics of turbulence is used based on the average (mean), the fluctuation (standard deviation) and the second moment (fluxes) taken at one point in space (i.e., one point time correlation functions). 

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