Ensemble average value

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... 

To find the component Xdyn( ) we aPPly the average perturbation (AP) theorem, formulated and proved in GT, pp. 373-376 (see also VIG, pp. 82-87). This theorem allows us to express the ensemble average value for 8p , induced by a.c. electric field, through an integral, including unperturbed time dependence p (f). The formulation of the theorem is8... 

Figure 12. PDF of C (T t ) for different r - t /T and a = 0.8. Diamonds are numerical simulations. Curves are analytical results without fitting For r = 0, Eq. (32) is used (full line) for r = 0.01, 0.1 and 0.5 Eq (35) is used (dashed) and for r = 0.9 and 0.99, Eq. (38) is used (full lines). If compared with the cases a = 0.3 and 0.5, the distribution function exhibits a weaker nonergodic behavior, namely for r = 0 the distribution function peaks on the ensemble average value of 1/2.

Figure 4.2 shows the radial variations of the ensemble-averaged values of CH4 and NO mass fractions. The good agreement as observed here (and also... 

To identify tbe model parameters that are most significant in affecting biomolecular properties, one needs to calculate many sensitivity coefficients relating these properties to the parameters. For any property characterized by the variable O, whose ensemble-averaged value is O, the derivative of O with respect to a parameter can be shown to be ... 

In the usual computer simulations, the trend that Ecv+Ey is high when Ec is low and EcY+Ey is low when Ec is high, is no more present (notice that these quantities are not the ensemble-averaged values but the instant ones). However, during sufficiently many steps, the peptide molecule can change its conformation more readily in water than in vacuum. The same is true in the real systems, and peptide conformations are stabilized quite rapidly due to the presence of water. 

For the coupling energy operators H p and Hp in Eqs. (32), this amounts to regarding the matrix elements <.y V py or

as independent random variables with equally probable positive or negative values (or random phases). For the ensemble averaged values, one then has, in particular. 

From our discussions of the Boltzmann equation and the Enskog equation, we know that a quantity of special interest to us is/(r, v, i), which we now interpret as the ensemble average value of the number of particles in dr d about r, v at time t. Since the F-space is properly formulated in r and p instead of... 

Standard methods for assessing permeabilities, PDf, yield ensemble average values, reflecting an average film thickness and impKcitly ascribing to the film a uniform permeability. For microcrystalline... 

By considering the above values, equations of motion of the building are solved and the results are compared for both fixed base, and base isolated building supported on isolators with linear behavior in Tables 2 and 3, respectively (Pourzeynali Zarif, 2008). It should be mentioned that, for brevity, only the results of the four most important earthquakes (e. g., Kobe, El Centro, Loma Prieta, and Northridge Earthquakes), as well as the ensemble average values of the responses for 18 reference earthquakes are shown in the tables. 

By calculating the parameters of the linear base-isolation system using GAs, a reduction of 64.5% is obtained for the ensemble average value (calculated for 18 worldwide earthquakes) of the building s top story horizontal displacement response. 

In NMR experiments, the observables of interest are generally the components of the macroscopic nuclear magnetization, which are proportional to the ensemble average values of the components of the nuclear spin operator I, (ly), and If. The magnetization in the X-direction of an ensemble of nuclear spins, for example, is given by ... 

The mean value of any configurational property determined from a sufficiently large number of configurations provides an estimate of the ensemble-average value of that quantity the nature of the ensemble average depends upon the chosen probability density. 

The ensemble average value of is (neglecting any correction prefactors)... 

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