Temporal distribution of clusters

In the very initial stage of a phase transition, the appearance of nuclei on the electrode surface can be considered as a flux of random independent events along the time axis [3.1-3.16]. Therefore the probability Pm to form exactly m nuclei within a time interval [0,t] can be expressed by the Poisson distribution law  

Equation (3.1) allows us to derive an expression for the probability Pa to form at least m, i.e. m or more than m, nuclei within the interval [0,t]  

Correspondingly for the probability dP to form the mth nucleus within the infinitesimal time interval [t, t + rfr] it results  

In particular, equation (3.6) defines the probability of formation of at least one nucleus within the time interval [0,r] and is a very frequently studied 

As seen the probabilities P, dP and t contain essential information on the most important kinetic characteristics of the nucleation process the nucleation rate J(f) = dN(t)ldt and the average number of nuclei N(t) = 

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Probabilistic approach to nucleation — Basic stochastic properties of assemblies of clusters randomly distributed in space or appearing in time as a nonstationary flux of random independent events can be examined in terms of the Poisson theory [i]. See - temporal distribution of clusters -> spatial distribution of clusters. 

To assess homogeneity, the distribution of chemical constituents in a matrix is at the core of the investigation. This distribution can range from a random temporal and spatial occurrence at atomic or molecular levels over well defined patterns in crystalline structures to clusters of a chemical of microscopic to macroscopic scale. Although many physical and optical methods as well as analytical chemistry methods are used to visualize and quantify such spatial distributions, the determination of chemical homogeneity in a CRM must be treated as part of the uncertainty budget affecting analytical chemistry measurements. 

The temporal evolution of spatial correlations of both similar and dissimilar particles for d = 1 is shown in Fig. 6.15 (a) and (b) for both the symmetric, Da = Dft, and asymmetric, Da = 0 cases. What is striking, first of all, is rapid growth of the non-Poisson density fluctuations of similar particles e.g., for Dt/r = 104 the probability density to find a pair of close (r ro) A (or B) particles, XA(ro,t), by a factor of 7 exceeds that for a random distribution. This property could be used as a good aggregation criterion in the study of reactions between actual defects in solids, e.g., in ionic crystals, where concentrations of monomer, dimer and tetramer F centres (1 to 3 electrons trapped by anion vacancies which are 1 to 3nn, respectively) could be easily measured by means of the optical absorption [22], Namely in this manner non-Poissonian clustering of F centres was observed in KC1 crystals X-irradiated for a very long time at 4 K [23],... 

The temporal evolution of the cluster size distribution was analyzed using Jhe following proposed scaling relation for cluster-cluster aggregation ... 

The expression obtained by Roux and Guyon in their study of temporal development of invasion percolation [4], to determine the bursts distribution is only valid in d=2, and the expression (1), correct in d = 1 and in d = 2 and in d > 6 is expected to be general (its proof does not involve any restriction on the dimensionality). A precise numerical confumation remains necessary in d = 3. In d = 2, numerical calculations have been performed in the bond percolation case [4,6,8], and in the site percolation case [3,5], leading to results compatible with equation (1). This supposes that for site percolation, when removal a site, the contribution due to many daughter clusters remains marginal [8]. 

The presensillum clusters form in a characteristic pattern of three half elliptical domains in the antennal disk, in a temporal sequence from out to in (Figure 23.IE). The early expression patterns of ato and amos in the antennal disk outline domains that determine the distribution patterns of the developing sensillum categories. On the palp ato specifies B sensilla (Gupta and Rodrigues 1997) which illustrates the fact that these genes influence sensillum type dependent on... 

In the following section the power of the fractional derivative technique is demonstrated using as example the derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of the microstructure of disordered media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. 

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