Averages, temporal evolution

M 4] [P 3] The ensemble-averaged temporal evolution of voxel-averaged spatial intensity PDFs was followed in the improved EKI mixer device, both in the mixing... 

Three general approaches lead to values for gp q,r). First, scatterers can be treated microscopically as objects having hydrodynamic and direct, e.g., excluded volume, interactions, and dg]p q, r)/dr can be calculated. Second, because ag t) = exp[iq (rj(f)] is the q spatial Fourier component of the scatterer density, semicontinuum hydrodynamics and the Onsager regression hypothesis predict the average temporal evolution of fluctuations. Third, scatterers could be assumed to perform simple Brownian motion. 

It was found [1] that the values of and a, obtained in minimizing the error of fitting experimental conversion-time data, satisfactorily described the temporal evolutions of the molecular weight averages. Also, the model performed better in the description of the experimental data when a value of 3 = 1/2 was used. 

Our aim is to derive a model that describes the temporal evolution of the ratio of the average concentration per unit particle mass measured on the dried macroparticle, Cs(0, and the dissolved concentration outside the macroparticles, C° ... 

Figure 24.6 Temporal evolution of a concentration cloud along a river. The curves show cross-sectional averaged dye concentration measured at six sites in the Waikato River (New Zealand) below an instantaneous transverse line source. From Rutherford (1994).

Ensemble-averaged temporal PDF evolution for EKI mixing under EOF conditions... 

Lastly, let us consider wavepacket c in Table 5.1, which has a bond-directed initial momentum Pi = 12H/a0 along the R direction. Its average energy measured from the bottom of Vg is about 1.1 eV, and the temporal evolution of Pe is shown in the right-hand column of Fig. 5.9. Pe is created on the excited state in the same domain as that of the wavepacket a, which is located on the steep inner wall of Ve. It moves rapidly downhill along the... 

By carrying out the above procedure from time 0 to time /,Mm, we evidently obtain only one possible realization of the stochastic process. In order to get a statistically complete picture of the temporal evolution of the system, we must actually carry out several independent realizations or runs. These runs must use the same initial conditions of the problem but different starting numbers for the uniform random number generator in order for the algorithm to result in different but statistically equivalent chains. If we make K runs in all, and record the population sizes (k, t) in run k at time t (i = 1,..., m and k = 1,...,K), then we may assert that the average number of particles at time t is... 

The temporal evolution of the total (i.e. spatially integrated) power deposited by the spark is presented on Fig. 10.10, along with the total heat release LOT and the spatially averaged temperature. After a heating phase due to the source term on the energy equation (up to approximately 0.08 ms), the temperature is sufficient to initiate the reaction between fuel vapour and air, leading to a sudden increase of the heat release of the exothermic... 

Fig. 12. Temporal evolution of the relative concentrations and spectra of the five electronic species including in the given mechanism for electron solvation in neat methanol at room temperature the spectrum labeled e is an average assigned to all thermalizing species form e to e i. [Reprinted from Ref. 104, Copyright 1999, with permission from Elsevier.]...

Temporal evolutions of M(t) are shown in Fig. 2. In order to suppress fluctuations, we have calculated averages over realizations. Throughout this chapter, unless no comments appear, the numbers of realizations are n = 1000,100, and 8 for AT = 100,1000, and 10,000, respectively. We divide... 

Because of the substantial reduction of information associated with the transition between the micro- and macroworlds, one may readily conclude that the overwhelming amount of information buried in the detailed description of the spatio-temporal evolution of a inanyTpartielc system must be largely irrelevant for the thermod3mamics of an equilbrium system. In fact, one may suspect that it is the on-average behavior of the many-particle system what matters for its macroscopic properties, which, in turn, immediately suggests to employ statistical concepts. 

One of the outcomes of these calculations is the ability to characterize the kinetics of grain growth. In particular, it is possible to address the question of the mean grain size (/ ) as a function of time as well as the distribution of grain sizes. The temporal evolution of the average grain area is shown in fig. 10.47 An outcome of this analysis is that it is found that the mean grain size scales with time with an exponent of 1 /2. 

Fig. 10.47. Temporal evolution of mean grain size in phase field model of grain growth (adapted from Chen and Yang (1994)). Plots are of logarithm of average grain area as a function of time, with the two curves corresponding to four (crosses) and thirty-six (circles) different order parameter fields.

Figure 4 Temporal evolution of scattered intensities IA(Q) averaged according to three Q ranges to distinguish large, medium and small sized particles as determined by SALS on BLG/AG dispersions at 0.1 wt% total biopolymer concentration, pH 4.2 and Pr.Ps weight ration of 2 1. [A] large particles (Q 0.008-0.042 pm ), [ ] medium particles (Q 0.05-0.67 pm1), [O] small particles (Q 0.8-10.4 pm1). Bars are standard deviation based on duplicate experiments...

If, for example the isotropic and anisotropic distributions fQ(U,z,t) and f U,z,t) have been determined by solving the equation system of the two-term approximation, Eqs. (12), adapted to a specific kinetic problem, the steady-state values, the temporal evolution, or the spatial alteration of the macroscopic quantities can be calculated by appropriate energy space averaging over these distribution functions according to the corresponding representation given in Eqs. (15) to (30). 

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