Poisson’s distribution

As mentioned earlier, although we cannot directly observe the local breakdown process of passive film, according to Shibata and Takeyama,21,22 the stochastic breakdown of passive film follows Poisson s distribution. 

Figure 13 shows the relationship between the time interval At of passive film breakdown of stainless steel with chloride ions and the logarithms of cumulative probability P(Af) for breakdown at time intervals longer than At. From these results, it is clear that the logarithm of the probability is almost proportional to the time interval, and therefore the cumulative probability for film breakdown follows Poisson s distribution, i.e., the following equation is obtained,... 

Transformation based on square root from data X = /X is applied when the test values and variances are proportional as in Poisson s distribution. If the data come from counting up and the number of units is below 10 transformation form X --fX + 1 and text X =s/X + /X I 1 is used. If the test averages and their standard deviations are approximately proportional, we use the logarithm transformation X =log X. If there are data with low values or they have a zero value, we use X =log (X+l). When the squares of arithmetical averages and standard deviations are proportional we use the reciprocal transformation X =l/X or X =1/(X+1) if the values are small or are equal to zero. The transformation arc sin [X is used when values are given as proportions and when the distribution is Binomial. If the test value of the experiment is zero then instead of it we take the value l/(4n), and when it is 1, l-l/(4n) is taken as the value and n is the number of values. Transforming values where the proportion varies between 0.30 and 0.70 is practically senseless. This transformation is done by means of special tables suited for the purpose. 

The experimental results were expressed as linear dependences 1 g(Niml) = am + b (where m is fraction number), confirming the application of Poisson s distribution but the physical sense of a and b coefficients and their dependence on time were not clarified in [10],... 

This type of distribution is called a Poisson s distribution... 

The function V(( ) ) is however an ixnknown fxxnction, and we can only say that V(fg) = 0 at = 0. If we introduce the assvmption that the number or defect parts in a cell is distributed obeying a Poisson s distribution and the strength of a cell is decreased in proportion to the number of the defects contained in the cell, then the following equation can be obtained assuming a Poisson s distribution of the number of the defects in a cell (3). 

The general expression [Eq. (8.53)] for the mole fraction of x-mer in the polymer is Poisson s distribution formula [6j. From the nature of the problem it is obvious, without the above derivation, that the numbers of molecules of various sizes must be represented by Poisson s distribution. Since >P = i oo + 1 at the completion of reaction [cf. Eq. (8.36)], the number average molecular weight will be given by... 

One of the common types of information available to safety advisers is the number of accidents sustained by employees in a factory or department. Statistically, this information is referred to as the distribution of accidents over the group, and it is usual to find that most of the group have no accidents, some have only one, while certain individuals have four, five or more. Where a high number is found, the obvious action is to investigate the reasons for them, but to save effort it is useful to have a method for determining whether or not an observed distribution of accidents comes within the compass of chance variation. The method makes use of Poisson s distribution ° which enables the expected distribution of the number of... 

Thus the mole fraction of j -mer in the polymer is Poisson s distribution formula (Hory, 1940). Substituting for Voo from Eq. (8.32), and using [M]sp = [M] for high polymers (i.e., high ratio of initial monomer to initiator), one obtains... 

Expressions (116), (120), and (122) correspond to the recurrent correlations binding two neighboring terms in Poisson s distribution. Hence, in the early stages of the process and/or at a low polymer concentration, the distribution of probabilities of macromolecule number j... 

The number of initial particles in the system, N, was calculated according to Eq. (215), obtained from Poisson s distribution equation. It can be used for discrete processes such as flocculation [81]. The Poisson s distribution allows one to calculate the distribution of aggregates with the number of initial particles in them during flocculation ... 

ACA contains insufficient particles, there is of course a certain probability that no particle exists in the joint and an open will result. On the other hand, bridging is possible due to too many particles in a short spacing, causing a short circuit between neighboring pads. Accurate probability estimates of open and bridging are needed to define the limiting pitch of ACA interconnects. An analytical method to estimate the open probability was proposed (Ref 47). Assume that the number of particles on a pad obeys Poisson s distribution ... 

Nuclear processes are statistically determined and can be described by Poisson s distribution. For Poisson s distribution, the standard deviation cr is related to the number of measured counts n... 

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

These four equations, using the appropriate boundary conditions, can be solved to give current and potential distributions, and concentration profiles. Electrode kinetics would enter as part of the boundary conditions. The solution of these equations is not easy and often involves detailed numerical work. Electroneutrahty (eq. 28) is not strictly correct. More properly, equation 28 should be replaced with Poisson s equation... 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

Besides, the potential and its first derivatives are continuous at boundaries where a volume density is discontinuous function. It is obvious that in this case the solution of the forward problem is unique. Now consider a completely different situation, when a density 5 q) is given only inside some volume V surrounded by a surface S, Fig. 1.8a. Inasmuch as the distribution of masses outside V is unknown, it is natural to expect that Poisson s equation does not uniquely define the potential U, and in order to illustrate this fact let us represent its solution as a sum ... 

This means that Poisson s equation defines the potential with an uncertainty of a harmonic function 14. Regardless of a distribution of masses outside the volume the potential C4 remains harmonic function inside V and, correspondingly, there are an infinite number of potentials U which satisfy Equation (1.70), and they can be represented as ... 

This first case vividly illustrates the importance of the boundary condition. Indeed, Poisson s equation or the system of field equations have an infinite number of solutions corresponding to different distributions of masses located outside the volume. Certainly, we can mentally picture unlimited variants of mass distribution and expect an infinite number of different fields within the volume V. In other words, Poisson s equation, or more precisely, the given density inside the volume V, allows us to find the potential due to these masses, while the boundary condition (1.83) is equivalent to knowledge of masses situated outside this volume. It is clear that if masses are absent in the volume V, the potential C7 is a harmonic function and it is uniquely defined by Dirichlet s condition. 

The relation between the spatial potential distribntion and the spatial distribution of space-charge density can be stated, generally, in terms of Poisson s dilferential equation. 

The trace vanishes because only p- and /-electrons contribute to the EFG, which have zero probability of presence at r = 0 (i.e. Laplace s equation applies as opposed to Poisson s equation, because the nucleus is external to the EFG-generating part of the electronic charge distribution). As the EFG tensor is symmetric, it can be diagonalized by rotation to a principal axes system (PAS) for which the off-diagonal elements vanish, = 0. By convention, the principal axes are chosen such that... 

The convenience of Eq. (6) is realizable only in the rather unrealistic situation where the charge distribution exhibits cylindrical or spherical symmetry. For storage silos, blenders, fluidized bed reactors, and other real vessel geometries, integral solutions are usually not possible, necessitating an alternate problem formulation. Poisson s equation serves this need, relating the volume charge distribution to the electrostatic potential. 

A second relation between p(r) and 4>(r) may be obtained by noting that the sources of 0(r) are the point charge Ze of the nucleus, located at the origin and the charge distribution due to the N electrons. Treating the charge density —ep(r) of the electrons as continuous, Poisson s equation of electrostatics may be used to write... 

From Poisson s equation in one dimention, the charge distribution is defined by... 

The distribution of excess charge of hydrated ions in the diffuse layer can be derived by using Poisson s equation, d% dx = - o(jc)/e, and Boltzmann s distribution equation, Ci(x) = Cys) exp -Zie )/ 7 , to obtain the relationship in Eqn. 5-3 between the interfacial charge, om, and the diffuse layer potential, ohp ... 

In the same way as described in Sec. 5.2 for a diifiise layer in aqueous solution, the differential electric capacity, Csc, of a space charge layer of semiconductors can be derived from the Poisson s equation and the Fermi distribution function (or approximated by the Boltzmann distribution) to obtain Eqn. 5-69 for intrinsic semiconductor electrodes [(Serischer, 1961 Myamlin-Pleskov, 1967 Memming, 1983] ... 

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