Reduced distribution functions

For the simplest type of uniaxial orientation the distribution function reduces to N(6), where N(6)do) is the fraction of units for which OX3 lies within any small solid angle dco at angle 6 to 0X3. By characterising the distribution is meant finding out as much as possible about N 6) for the various types of structural unit that may be present in the polymer. 

Weakly Heterogeneous Surfaces.—In this case, qM Qm ksTp, the expression of the energy distribution function is given by equation (29). In addition, and in order to simplify the computations, it is supposed here that lateral interactions are negligible (xo very high). The energy distribution function reduces to the exponential distribution (30), which under equilibrium conditions leads to Freundlich behaviour. 

In conclusion, we have presented a new formulation of the CVM which allows continuous atomic displacement from lattice point and applied the scheme to the calculations of the phase diagrams of binary alloy systems. For treating 3D systems, the memory space can be reduced by storing only point distribution function f(r), but not the pair distribution function g(r,r ). Therefore, continuous CVM scheme can be applicable for the calculations of phase diagrams of 3D alloy systems [6,7], with the use of the standard type of computers. 

Viewed in a somewhat different light, the point here is that we are not trying to fix, once and for all, the class of functions X(t) for which all of our results will be valid but, rather, we limit our attention in any specific situation to those time functions for which certain time averages (such as a distribution function) assume specified values. This being the case, the only existence question of importance reduces to making sure that the underlying class of functions is not vacuous. 

Three ranges of values of n were considered, >1, 0.7—1.0 and <0.7. When n> 1, and particularly when 3 < n < 4, the Weibull distribution readily reduces to a normal distribution if the Erofe ev function is symmetrical about a = 0.5. [The Weibull distribution is symmetrical for n = 3.26, i.e. (1 — In 2)-1, and the inflection point varies only slowly with n.] Thus, under these conditions (3 < n < 4 and symmetry about a = 0.5), we may derive the parameters of the corresponding normal distribution (where p defines the half-life of the reaction and the dispersion parameter, a, is a measure of the lack of homogeneity of the surface centres), viz. 

The same is observed if the maximum speed of the speed distribution function is varied and the acceleration is kept constant. The tire life is drastically reduced and the rating of the silica compound reverses in relation to the OESBR as the speed is increased. 

The expression in brackets in Eq. (87) is of the form (sin x/x)2t where x = (tom — o>)tf 2. Thus, for a given time t for the duration of the perturbation, the spectrum, e.g. the transition probability as a function of the ahgular frequency w is as shown in Fig. 2. The width at half-maximum of this spectral feature is represented by A for a given value of the time, t. If, for example, the perturbation time is increased by a factor of four, the width of the spectral distribution is reduced by the same factor, as shown by (he solid tine in Fig. 2. Equation (87) expresses the probability that the system, initially in die state k = n, will be in the state m after a sinusoidal perturbation over a relatively short period of time t. 

Thus, the allowance for the dependence of the resonance integral on qsk may not be reduced in general to averaging the transition probability over the distribution function in Eq. (102). The function s(qk) plays the role of the distribution function for the coordinates qk in the case of the symmetric transition. In the classical limit, the results of Flynn and Stoneham62 can be obtained from Eq. (103), and in the low-temperature limit, the result of Kagan and Klinger64 can be obtained. 

X p Zxxp is diminished from that calculated for an isotropic distribution and approaches zero as the nematic distribution function becomes more tightly squeezed around 0=0 and 8=ir. This is related to the fact that the projection of DANS z-axes onto the laboratory X-axis is reduced. 

For two particles having masses mi, m2, the distribution function of relative velocity is obtained by substituting for m the reduced mass m mil m + m2) (see Problem 6 at the end of this chapter). 

Prove the assertion in the text that the relative velocity of two sets of particles having individual Maxwellian velocity distribution functions also has a Maxwellian distribution with the masses replaced by the reduced mass. 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

The reduced distribution functions at the instant t — 0, f are factorized when the s particles are distributed among several groups separated from each other by a distance greater than the range of the forces. 

Often the reduced radial distribution function G(r) is deduced which is given by ... 

Figure 4.11. Reduced chi-square for fitting a single Gaussian distribution function of decays with either a discrete single or double exponential model as a function of dis tribud on width (/f)...

For simplicity, we shall commonly refer to the Q-electron distribution function as the 2-density and the 2-electron reduced density matrix as the 2-ntatrix. In position-space discussions, the diagonal elements of the 2-ntatrix are commonly referred to as the 2-density. In this chapter, we will also refer to the diagonal element of orbital-space representation of the Q-vaatnx as the 2-density. 

Accordingly, the catalytic activity in a given catalytic reaction depends on only four factors. Two of them are specific for the system as a whole the activation energy and the reaction order. The latter may be reduced to the heat of adsorption, as b0 is a nearly universal constant. The other two factors are, at least in first approximation, properties of the catalyst its surface area F and its energy distribution function. Future work will have to answer the question of which parameters control, qualitatively and quantitatively, these four factors. 

To construct a diffusion equation for the soft coordinates alone, we consider a reduced distribution function... 

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