Ensemble mean value

First, to demonstrate the operational procedure in LES/FMDF, the instantaneous number density of the Monte-Carlo elements or particles is shown in Fig. 4.1a. This figure shows the total number of particles within the computational domain. However, the weighted distribution of the particles is not, and should not be, uniform. Rather, it must be proportional to the flow density. This is demonstrated in Fig. 4.16 in which the ensemble-mean values of the weighted Monte-Carlo particle number density are compared with the fluid density as obtained from finite difference solution of the filtered density field. The very good correlation attained in this way verifies the consistency of the stochastic procedure and its Lagrangian Monte-Carlo solver. 

For simplicity we drop the bracket notation on the concentration (c,j, which has been used in Chapter 18 to denote the ensemble mean concentration. All concentrations in this chapter are, however, understood to represent theoretical ensemble mean values. Likewise, we drop the overbars on the wind velocity components, but they are also understood to represent mean values. 

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... 

MD runs for polymers typically exceed the stability Umits of a micro-canonical simulation, so using the fluctuation-dissipation theorem one can define a canonical ensemble and stabilize the runs. For the noise term one can use equally distributed random numbers which have the mean value and the second moment required by Eq. (13). In most cases the equations of motion are then solved using a third- or fifth-order predictor-corrector or Verlet s algorithms. 

In an ensemble of crystals, each of thickness /, the mean value of / is given by ... 

Rumpf (R4) has derived an explicit relationship for the tensile strength as a function of porosity, coordination number, particle size, and bonding forces between the individual particles. The model is based on the following assumptions (1) particles are monosize spheres (2) fracture occurs through the particle-particle bonds only and their number in the cross section under stress is high (3) bonds are statistically distributed across the cross section and over all directions in space (4) particles are statistically distributed in the ensemble and hence in the cross section and (5) bond strength between the individual particles is normally distributed and a mean value can be used to represent each one. Rumpf s basic equation for the tensile strength is... 

The unperturbed Hamiltonian 3 is the same for all systems and is time-independent. The time-dependent perturbation G(t), different for each system, is considered as a stationary stochastic variable. We may, without loss of generality, suppose that the mean value of G(t) over the ensemble is equal to zero. We denote by a,p,y,. . . the eigenstates of supposed to be non-degenerate, and by fix, the corresponding energies. 

What is really desired, of course, is not the random concentration resulting from one realization of the flow, but the expected (mean) value resulting from an entire ensemble of flows with identical macroscopic conditions. Letting (c(x, y, z, /)) = E c x, y, z, r) and then taking the expected value of Eq. (2.3) leads to... 

Because the velocity u contains the random component u, the concentration c is a stochastic function since, by virtue of Eq. (2.2), c is a function of u. The mean value of c, as expressed in Eq. (2.5), is an ensemble mean formed by averaging c over the entire ensemble of identical experiments. Temporal and spatial mean values, by contrast, are obtained by averaging v ues from a single member of the ensemble over a period or area, respectively. The ensemble mean, which we have denoted by the angle brackets ( ), is the easiest to deal with mathematically. Unfortunately, ensemble means are not measurable quantities, although under the conditions of the ergodic theorem they can be related to observable temporal or spatial averages. In Eq. (2.7) the mean concentration (c) represents a true ensemble mean, whereas if we decompose c as... 

We have so far limited ourselves to a classical description, the natural requirement for which is the condition /, /" —> oo. In order that the description is valid for any angular momentum value, it is necessary to employ the quantum mechanical approach. We presume that the reader is acquainted with the density matrix (or the statistic operator) introduced into quantum mechanics for finding the mean values of the observables averaged over the particle ensemble. Under the conditions and symmetry of excitation considered here one must simply pass from the prob-... 

Equation 23 is a form of the Stirling equation, only valid if Nepartition function of S becomes that of a canonical ensemble without mean values ... 

Each individual measurement of any physical quantity yields a value A. But, independently of any possible observation errors associated with imperfect experimental measurements, the outcomes of identical measurements in identically prepared microsystems are not necessarily the same. The results fluctuate around a central value. It is this collection or Spectrum of values that characterizes the observable A for the ensemble. The fraction of the total number of microsystems leading to a given A value yields the probability of another identical measurement producing that result. Two parameters can be defined the mean value (later to be called the expected value ) and the indeterminacy (also called uncertainty by some authors). The mean value A) is the weighted average of the different results considering the frequency of their occurrence. The indeterminacy AA is the standard deviation of the observable, which is defined as the square root of the dispersion. In turn, the dispersion of the results is the mean value of the squared deviations with respect to the mean (A). Thus,... 

Equation (4.3.5) is the Fourier transform of the linear correlation (4.3.1). This relationship is referred to as the correlation theorem. If x(t) is white noise, then ensemble averages have to be incorporated into equations (4.3.5)-(4.3.7), because the power spectrum of white noise is again white noise, but with a variance as large as its mean value [Beni]. When the linear correlation theorem (4.3.5) is applied to the same functions, then Ci((o) is the power spectrum of Y((o) = X co). Conversely, Ci(w) is then the Fourier transform of the auto-correlation function of y(f) = x(t) (cf. eqn 4.3.1). 

One way to see that a transition is discontinuous is to detect a coexistence of two phases, in this case the orientationally ordered and disordered phases, in a temperature interval. This is revealed by time variation of the potential energy of the cluster. In the temperamre region of phase coexistence, each cluster dynamically transforms between the phases, and its potential energy fluctuates around two different mean values (Fig. 4). In an ensemble of clusters, the coexistence of different phases is observable insofar as a fraction of the clusters (e.g., in a beam [17]) can exhibit the structure of one phase, while another fraction takes on the stmcture of another phase. 

In the case of turbulent advection velocity, the transported quantity in the PBE (i.e. the NDF) fluctuates around its mean value. These fluctuations are due to the nonlinear convection term in the momentum equation of the continuous phase. In turbulent flows usually the Reynolds average is introduced (Pope, 2000). It consists of calculating ensemble-averaged quantities of interest (usually lower-order moments). Given a fluctuating property of a turbulent flow f>(t,x), its Reynolds average at a fixed point in time and space can be written as... 

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