Mean fluctuations

The statistical nature of the homogeneity problem can be treated theoretically and it is clear that it is a matter of large numbers. Either a large number of individual particles in the investigated unit or a large number of individual analyses will produce consistent mean results, whereas the deviations from the mean (fluctuation of individual results) are an indication of the element distribution in the matrix (homogeneity). 

Note that (7.61) is second-order accurate in time. Also, by definition, the estimated mean fluctuating velocity should be null u X = 0. This condition will not be automatically satisfied due to numerical errors. Muradoglu et al. (2001) propose a simple correction algorithm that consists of subtracting the interpolated value of u X (e.g., the LOME found as in (7.45)) from uin) t + At). 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

Mean fluctuation path (8h) = rsin P0)> A Mean reduced transverse amplitude (E,0) Mean transverse amplitude (yo), A ... 

Figure 3.17 shows the evolution of the permeate flow rate when we work at constant pressure. We can observe (curves FI and FIS) that controlling the pressure pump with a precision of 0.1 bar (for instance see Table 3.2) produces a mean fluctuation of the flow rate that begins with 20% and progressively decreases to as little as 5% when we approach the total clogged state. In this case of slow surface clogging, it must be mentioned that the operating time before the total... 

Rotta (Rl) studied the Poisson equation in some detail, and proposed Eq. (49) for the portion of P, independent of the mean-fluctuation inter-... 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

X) and (1/2)V x u - The bulk viscosity is rjy. Note that the numerical values of these viscosities are different from those in Section 4. Therefore we have primed these viscosities. In order to find simple fluctuation relations for them we have to use an ensemble where the thermodynamic forces are constants of motion and the fluxes are zero mean fluctuating phase functions. This can be done if both the director angular velocity and the streaming angular... 

These fluctuations are illustrated in Fig. 9.95 in two different ways the time-dependent electric field E t) and its mean fluctuations of the amplitude 0 and phase (p are shown in an E t) diagram and in a polar phase diagram with the axes E and 2- In the latter, amplitude fluctuations cause an uncertainty of the radius r = o, whereas phase fluctuations cause an uncertainty of the phase angle (p (Fig. 9.95b). Because of Heisenberg s uncertainty relation it is not possible that both uncertainties of amplitude and phase become simultaneously zero. 

X Variation means fluctuation in a process that directly affects the desired or expected output. 

Gillespie (1979, 1980) has emphasised the possibility of using stochastic simulation methods for identifying the quasistable stationary states and calculating the associated mean transition times, mean fluctuation periods, and effective fluctuation ranges. 

The first hypothesis is quite plausible for concentrated suspensions, and the fluctuation temperature may be defined in terms of the mean fluctuation kinetic energy associated with one translational degree of freedom of a particle in the following form ... 

In the same suspension with fluctuations, the interphase interaction force also fluctuates. Representing the variables involved in Equation 3.2 as sums of their mean values and zero-mean fluctuations (marked with a prime), for the mean value of this force we obtain... 

Figure 3.8 Mean fluctuation of the average nearest-neighbor distance calculated for Na clusters at different temperatures (solid triangles = iow temperature T = 200, 240, 260, 260 and 340 K for n =6, 8, 10, 18 and 20 solid squares = high temperature T = 600, 470, 440, 520, 570 and 640 K for n = 6, 8, 10, 13, 18 and 20). 5 = 0.10 indicates the onset of enhanced diffusion. Reprinted with permission from [123]. Copyright 1991 American Institute of Physics...

In any case, rotation means fluctuating hydrodynamic forces on the surface elements of the particle. The maximum stress for spherical particles can be calculated as (Raasch 1962) ... 

The mean fluctuation time is determined by counting the local maxima in a section of the equilibrium graph a section containing 20 or so local maxima produces a reliable evaluation of At. For our calculations in solid and fluid sodium, fluctuation times are in the range 30At to 60At. Important properties of the signals X(t), for different dynamical variables in the same MD calculation, are as follows. 

There are substantial velocity fluctuations, especially near the wall, with the time mean fluctuating component typically reaching a maximum near the edge of the wall layer (Yang et ah, 1992 Caloz et ah, 1999). Flows near the wall undergo reversals, i.e., the particle velocity there is sometimes upward and sometimes downward. The fraction of individual particles... 

Within the fi-ameworks of the cluster model [55] it has been assumed, that the segment joined to cluster means fluctuation fi-ee volume microvoid shrinkage and vice versa. In this case the microvoids number AA/j, forming in polymer dilation process, should be approximately equal to segments number AN subjected to partial melting process. These parameters can be estimated by following methods. The value AN is equal to [51] ... 

The turbulent viscosity p, is considered proportional to the density p, the mean fluemating velocity (expressed by and the fluctuating characteristic length the latter being proportional to both the mean fluctuating velocity and the time (expressed by -). Combining these relationships, p, oc k - k , we have... 

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