Distribution functions independence

The reactants A and B and activated complexes [ABf are at equilibrium and the C and D products that formed have no effect on the equilibrium concentration of the activated [AB] complexes. This is the main assumption of the activated conq>lex theory, which was manifested above in the choice of the equilibrium distribution function independent of the reaction products. The equilibrium constant between the reactants and activated complexes can be expressed through the corresponding partition flmctions (see Table 2.1)... 

The effects due to the finite size of crystallites (in both lateral directions) and the resulting effects due to boundary fields have been studied by Patrykiejew [57], with help of Monte Carlo simulation. A solid surface has been modeled as a collection of finite, two-dimensional, homogeneous regions and each region has been assumed to be a square lattice of the size Lx L (measured in lattice constants). Patches of different size contribute to the total surface with different weights described by a certain size distribution function C L). Following the basic assumption of the patchwise model of surface heterogeneity [6], the patches have been assumed to be independent one of another. 

Distribution functions measure the (average) value of a property as a function of an independent variable. A typical example is the radial distribution function g r) which measmes the probability of finding a particle as a function of distance from a typical ... 

Discussion of the Equation.—The Boltzmann equation describes the manner in which the distribution function for a system of particles, /x = /(r,vx,f), varies in terms of its independent variables r, the position of observation vx, the velocity of the particles considered and the time, t. The variation of the distribution function due to the external forces acting on the particles and the action of collisions are both considered. In the integral expression on the right of Eq. (1-39), the Eqs. (1-21) are used to express the velocities after collision in terms of the velocities before collision the dynamics of the collision process are taken into account in the expression for x(6,e), from Eqs. (1-11) and (1-12), which enters into the k of Eqs. (1-21). Alternatively, as will be shown to be useful later, the velocities before and after collision may be expressed, by Eq. (1-20), in terms of G,g, and g the dynamics of the collision comes into the relation between g and g of Eq. (1-19). 

To see the type of differences that arises between an iterative solution and a simultaneous solution of the coefficient equations, we may proceed as follows. Bor the thirteen moment approximation, we shall allow the distribution function to have only thirteen nonzero moments, namely n, v, T, p, q [p has only five independent moments, since it is symmetric, and obeys Eq. (1-56)]. For the coefficients, we therefore keep o, a, a 1, k2), o 11 the first five of these... 

If the interval r is large compared with the time for a collision to be completed (but small compared with macroscopic times), then the arguments of the distribution functions are those appropriate to the positions and velocities before and after a binary collision. The integration over r2 may be replaced by one over the relative distance variable r2 — rx as noted in Section 1.7, collisions taking place during the time r occur in the volume g rbdbde, where g is the relative velocity, and (6,e) are the relative collision coordinates. Incomplete collisions, or motions involving periodic orbits take place in a volume independent of r when Avx(r) and Av2(r) refer to motion for which a collision does not take place (or to the force-field free portion of the... 

Conditional Distribution Functions and Statistical Independence.—The definition of a conditional distribution function is motivated by the following considerations. Suppose that we have been observing a time function X and that we want to obtain a quanti-... 

Our next result concerns the central limit theorem, which places in evidence the remarkable behavior of the distribution function of when n is a large number. We shall now state and sketch the proof of a version of the central limit theorem that is pertinent to sums of identically distributed [p0i(x) = p01(a ), i — 1,2, ], statistically independent random variables. To simplify the statement of the theorem, we shall introduce the normalized sum s defined by... 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

There are many ways we could assign probability distribution functions to the increments N(t + sk) — N(t + tk) and simultaneously satisfy the independent increment requirement expressed by Eq. (3-237) however, if we require a few additional properties, it is possible to show that the only possible probability density assignment is the Poisson process assignment defined by Eq. (3-231). One example of such additional requirements is the following50... 

Notice that those distribution functions that satisfy Eq. (4-179) still constitute a convex set, so that optimization of the E,R curve is still straightforward by numerical methods. It is to be observed that the choice of an F(x) satisfying a constraint such as Eq. (4-179) defines an ensemble of codes the individual codes in the ensemble will not necessarily satisfy the constraint. This is unimportant practically since each digit of each code word is chosen independently over the ensemble thus it is most unlikely that the average power of a code will differ drastically from the average power of the ensemble. It is possible to combine the central limit theorem and the techniques used in the last two paragraphs of Section 4.7 to show that a code exists for which each code word satisfies... 

The characteristic function and the moment generating function are important tools for computing moments of distributions, studying limits of sequences of distributions, and finding the distribution of sums of independent variables. If Z — X + F, where X and T are independently distributed according to the distribution functions F(x) and 6( y) respectively, the distribution function of Z is given by... 

Consider the following inventory problem. There are p time periods at the start of each of which an order of n items is made at a cost A(n), which is an increasing function of % (e.g., A(n) = a + bn). The length of each period is a random variable, and, hence, there are p random variables Xt (i = 1,- -, p) that are assumed to be independently and identically distributed according to the distribution function Fn(x)—for each period, it is the probability that there is a demand for... 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

Hence use of the normalized distribution function allows a reduction of the number of independent parameters in the kernel (253). 

In the Danekwens model of mass transfer it is assumed that the fractional rate of surface renewal s is constant and independent of surface age. Under such conditions the expression for the surface age distribution function is = If the fractional rate of surface renewal were proportional to surface age (say s — bt. where b is a constant), show that the surface age distribution function would then assume the form ... 

Danckwerts assumed a random surface renewal process in which the probability of surface renewal is independent of its age. If s is the fraction of the total surface renewed per unit time, obtain the age distribution function for the surface and show that the mean mass transfer rate Na over the whole surface is ... 

Stochastic equation (A8.7) is linear over SP and contains the operators La and V.co of differentiation over time-independent variables Q and co. Therefore, if we assume that the time fluctuations of the liquid cage axis orientation Z(t) are Markovian, then the method used in Chapter 7 yields a kinetic equation for the partially averaged distribution function P(Q, co, t, E). The latter allows us to calculate the searched averaged distribution function... 

The monomer conversion in this seeded polymerization system is independent of the degree of segregation as long as an exponential residence time distribution function is maintained. 

When the system contains more than one component it is important to be able to explore the distribution of the different components both locally and at long range. One way in which this can be achieved is to evaluate the distribution function for the different species. For example in a binary mixture of components A and B there are four radial distribution functions, g (r), g (r), g (r) and g (r) which are independent under certain conditions. More importantly they would, with the usual definition, be concentration dependent even in the absence of correlations between the particles. It is convenient to remove this concentration dependence by normalising the distribution function via the concentrations of the components [26]. Thus the radial distribution function of g (r) which gives the probability of finding a molecule of type B given one of type A at the origin is obtained from... 

Thus, one can be far from the ideal world often assumed by statisticians tidy models, theoretical distribution functions, and independent, essentially uncorrupted measured values with just a bit of measurement noise superimposed. Furthermore, because of the costs associated with obtaining and analyzing samples, small sample numbers are the rule. On the other hand, linear ranges upwards of 1 100 and relative standard deviations of usually 2% and less compensate for the lack of data points. 

A time-independent wave function is a function of the position in space (r = x,y,z) and the spin degree of freedom, which can be either up or down. The physical interpretation of the wave function is due to Max Born (25, 26), who was the first to interpret the square of its magnitude, > /(r)p, as a probability density function, or probability distribution function. This probability distribution specifies the probability of finding the particle (here, the electron) at any chosen location in space (r) in an infinitesimal volume dV= dx dy dz around r. I lu probability of finding the electron at r is given by )/(r) Id V7, which is required to integrate to unity over all space (normalization condition). A many-electron system, such as a molecule, is described by a many-electron wave function lF(r, r, l .I -.-), which when squared gives the probability den-... 

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