Poissonian distribution

If nonreactive MPC collisions maintain an instantaneous Poissonian distribution of particles in the cells, it is easy to verify that reactive MPC dynamics yields the reaction-diffusion equation,... 

Eckhardt, 1988). For systems which are regular in the classical limit its behaviour is close to a Poissonian distribution function. 

Let us consider a fluorescent probe and a quencher that are soluble only in the micellar pseudo-phase. If the quenching is static, fluorescence is observed only from micelles devoid of quenchers. Assuming a Poissonian distribution of the quencher molecules, the probability that a micelle contains no quencher is given by Eq. (4.22), so that the relationship between the fluorescence intensity and the mean occupancy number < > is... 

Assuming a Poissonian distribution, the probability Pn that a micelle contains n quenchers is given by Eq. (4.21). The observed fluorescence intensity following d-pulse excitation is obtained by summing the contributions from micelles with different numbers of quenchers ... 

This distribution is called a Poissonian. Fig. 4.3 shows how the proba-bihty distributions Pn x), n = 2,3,... tend to Poo x) in the limit n —> 00. The Poissonian distribution is peaked and finite at a = 0. This means that the Xj values in the unit interval have a tendency to cluster. Thus, if the spacing statistics of a sequence of energy levels is Poissonian, the spectrum is characterized by level clustering. 

Shalitin and Katchalski [79] derived analytical expressions for the molecular weight and compositional distributions of the copolymers. Poissonian distributions were calculated for the over-all molecular weights and for the concentrations of molecules containing a specified number of A units regardless of the number of B present. 

In the treatment above, a random, Poissonian distribution of the nucleation events in the time domain has been assumed. This assumption can be verified by the following simple experiment ... 

As can be seen in Fig. 2.10 this is a decidedly asymmetrical distribution. Moreover, since the outcome can only assume discrete values, the Poissonian distribution is a collection of points rather than a curve. Yet another difference between the Gaussian and Poissonian distribution is that (2.10-1) contains only one parameter, A, whereas the Gaussian distribution (2.1-1) has two the average value x and the standard deviation a. The average value (N) of the Poissonian distribution is... 

Fortunately, for a sufficiently large cohort, the Poissonian distribution approaches the Gaussian one, a general limit more carefully described by the central limit theorem of statistics. Because of the large individual fluctuations in the human life span, insurance companies must operate with a large number of subscribers. Under those circumstances they can use statistics to set their premiums so as to provide a useful service to society while also making a profit. 

Fig. 3. P,(t) for a coherent state driven by the first blue sideband interaction, showing collqjse and revival behavior. The data are fitted to a coherent state distribution, yielding (n) = 3.1(1). The inset shows the results of inverting the time-domain data by employing a Fourier cosine transform at the known Rabi firequencies Poissonian distribution, yielding (n) = 2.9(1). Each data...

It should be pointed out that the gain mechanism in a PMT tube operates as a random process. The number of secondary electrons is different for each primary electron. The relative width of the distribution can be expected at least of the size of the standard deviation of a poissonian distribution, n, of the secondary emission coefficient, n, at the first dynode. Therefore the single-photon pulses obtained from a PMT have a considerable amplitude jitter, see Sect. 6.2.5, page 226. 

Poissonian distribution, and should be size-dependent, as shown in Figure 17.6C, curve a. Large vesicles, in fact, may contain all required ingredients (and be therefore functional), whereas small ones are expected to be not functional. Since the experimental results clash with theoretical predictions, a possible physical explanation suggests that local (internal) enzymes and ribosomes concentrations are enhanced by a factor >10 (Figure 17.6C, curve b). Further studies are in progress in order to clarify the nature of such effect. 

The discussions above show that each component of the energy spectrum as a poissonian distribution, and the sum of any number of components also has a poissonian distribution. This also means that the i-th component can be considered to be defined by a channel at energy Ei in the multichannel analyzer memory. If the ni counts in each of s adjacent channels are summed to arrive at the total counts within a peak associated with a particular characteristic x-ray line, then the number of counts in the peak can be reported as... 

Particular results can be derived for certain classes of the Markov population process (Kingman, 1969). Necessary and sufficient criteria were given for polynomial simple Markov population processes to have Poissonian stationary distributions when the detailed balance condition holds. Special relations among the coefficients are necessary to get Poissonian distribution (Toth Torok, 1980 Toth, 1981 Toth et al., 1983). 

Since white noise is qualified by a single parameter, namely by its intensity, the characterisation of real noise requires at least another parameter, its correlation time. The theoretical treatment of nonlinear systems subject to real, i.e. coloured , external noise has two newer difficulties. First, only the white noise idealisation leads to a Markov process. Second, from the practical point of view the Gaussian and Poissonian distributions are relevant only to describe white noise, and real noise can have a richer description. The disadvantage due to the loss of Markov property is partially compensated by the fact that non-Markovian processes have smoother realisations than Markov processes. Therefore no particular (Ito or Strato-... 

The one-atom maser can be used to investigate the statistical properties of nonclassical light [14.106,107]. If the cavity resonator is cooled down to T < 0.5 K the number of thermal photons becomes very small and can be neglected. The number of photons induced by the atomic fluorescence can be measured via the fluctuations in the number of atoms leaving the cavity in the lower level n-l). It turns out that the statistical distribution does not follow a Poissonian statistics as in the output of a laser with many photons per mode, but shows a sub-Poissonian distribution with photon-number fluctuations 70% below the vacuum-state limit [14.108]. 

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