Poissonian probability

Fig. 5.10 shows theoretical and experimental distribution curves (bars) of the probability for occurrence of m nucleation events for 5 selected time intervals. The theoretical bars are calculated with the average frequency JA = 8.64 x 10 nuclei s The good agreement of the experimental distribution curves with a Poissonian probability distribution is a good evidence that the process of 2D nucleation is random in time [5.20]. 

A laser consists in a medium where stimulated emission dominates over spontaneous emission placed inside an optical cavity which recycles the optical field. Above threshold, the photon number probability density distribution is poissonian, that means that the photon arrival time are a random variable. The probability of obtaining m photons during a given time interval is thus... 

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 ... 

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],... 

For example, in discussing blemishes in objects, the number of blemishes of course has the lower limit zero but can go to very large numbers. Technically, the distribution is probably Poissonian. Under these circumstances it is more appropriate to use lie square root of the number of blemishes as the dependent variablet ). Where the number of blemishes are in many cases less than 10, it is slightly preferable to use, where x is the number of blemishes, Vx -H 1/2... 

The results are very instructive. For the low-value plateau the probability of mutant appearance is essentially Poissonian for the single-mutant chain (i) and only slightly modified for the all-intermediate mutant plateau (ii), indicating that for typical population numbers of 10 °-10 any given mutant that has a Hamming distance from the wild type larger than 4-5 is rarely populated. In contrast, the mountain saddle landscape shows probabilities... 

The direction change is purely random, but the probability of tumbling is constant during a run, so that the run time distribution is Poissonian (6). 

B) Comparison between calculated and experimental distribution of a water-soluble marker (ferritin) inside POPC vesicles. Detailed data analysis shows that in some cases ferritin can be entrapped with efficiency higher than what expected on theoretical basis (Poisson distribution). Data taken from Berciaz et al. (C) Probability of co-entrapment of all macromolecular components of transcription-translation kit inside lipid vesicles of a given radius. The entrapment of each molecule is modelled as a poissonian process, and the cumulative probability is calculated as product of probabilities of independent events. The curve (a) indicates the probability of entrapping at least one copy of each molecular specie inside the same vesicle. The curve (b) indicates the probability of entrapping at least one copy of each molecular species under the hypothesis that their concentrations are all 50 times higher than the nominal (bulk) concentrations. Adapted from Souza et aP ... 

The notation Poi(k,rji Id) is used to denote a Poisson distribution in k with mean rji/p. p(k) is a function of the intensity at the detector and the probability distribution of the intensitypC/r,). The constant rjj is proportional to the detection efficiency (the proportionality constant between the amount of light falling on the detector and the average number of photon counts < k> detected) and incorporates the sampling time. This distribution is thus the Poisson transform of the light intensity distribution at the detector. Thus, for a perfectly steady light source (i.e. one in which pil ) is a delta function), the distribution p(k) will be Poissonian [8,9] and can therefore be described in terms of just the mean count number = t/iZd ... 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

The question arises however to know whether the whole initiation process is Poissonian in time and space, nearly Poissonian or not Poissonian at all. ° As concerns spatial correlations, it is expected that the probability of new pitting events in the vicinity of an existing pit i may be different from the pitting probability in a not pitted region. 

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