Cluster densities time fluctuations

Consider now the opposite case of p 1. Since the creation rate of new particles A is very small, it does not compensate the decay of B s whose density decreases down to a very small value. (However, the existence itself of B particles, even in small concentrations, is of vital importance for the future waves of A particle transformations.) If particles B were absent, the concentration of A s after time t would be pt. It is also clear that even at small densities n large cluster of these particles could be formed as a result of the statistical fluctuation. Touch of this cluster by some particle B will create a local wave of the A —> B transformations which is propagated for a finite distance and thus retains finite number of particles B in the system. It is important to note that for small densities tia a number of such clusters is also small. 

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],... 

Whereas FCS measures fluorescence fluctuations over time, a related technique, ICS, measures fluorescence fluctuations over space, in particular from images collected using a laser-scanning microscope (56). ICS analysis of pixels within a single image provides information about protein clustering and density. A variation of ICS known as image cross-correlation spectroscopy evaluates the interactions of molecules labeled with different fluorescent probes. ICS can also be performed on stacks of... 

Let us first ask the question how can a species exist in clusters of molecules at equilibrium with its own monomers We necessarily must consider the fluctuation of any system about its equilibrium. Under normally liquid conditions, the density fluctuations are small in extent (localized) and persist for short times. A microscopic description of the phenomenon may be that of temporary formation of weak bonds, which are then quickly dissipated. If, however, these fluctuations lead to clusters that have a nontransient existence, then... 

The cross-correlation technique measures the time of flight of an inherent flow tag passing through two sensors separated by a known distance. The technique has been used successfully to monitor single-phase fluid flows in which turbulent eddies modulate the interrogating ultrasonic beams. This type of correlation flowmeter has also been developed for solid/liquid and gas/liquid flows, in which the density fluctuation, caused by clusters of solids and by gas bubbles, is the prime inherent flow tag. 

The first assumption in deriving the Boltzmann equation is the so-called molecular chaos, or the velocities of coUiding particles are uncorrelated. This assumption makes Boltzmann equation irreversible in time. And clusters which reflect strongly correlated density fluctuations cannot be considered in this way. The second assumption is that the gas is dilute and only binary. 

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