Distribution functions asymptotic forms

These expressions contain two orientation-dependent parameters p and ct(N), which can be calculated from the equilibrium orientational distribution function T(a) minimizing AF. When the Onsager trial function (Eq. (17)) is used for f(a), p and ct(N) are given as functions of the unknown parameter a the functional form of ct(N) has already appeared in Eq. (18), while p is expressed asymptotically as [2]... 

A Levy walk, on the other hand, takes into account the fact that longer steps take longer times to complete than do shorter steps. The recognition of this simple fact ties the distribution of step sizes to the distribution of time intervals, which in the case of turbulence was determined by the fluctuations in the fluid velocity [62]. In the present example the continuum form of the Levy walk process is described by Eq. (42), with the autocorrelation function for the random driver being given by the inverse power law Eq. (66) and W is the constant speed of the walker. The asymptotic form of the second moment for this process is... 

A method of solving many coagulation anti agglomeration problems (Chapter 8) has been developed based on the use of a similarity transformation for the size distribution function (Swift and Fricdlander, 1964 Friedlander and Wang. 1966). Solutions found in this way are asymptotic forms approached after long times, and they are independent of the initial size distribution. Closed-form solutions for the upper and lower ends of the distribution can sometime.s be obtained in this way, and numerical methods can be used to match the solutions for intermediate-size particles. Alternatively, Monte Carlo and discrete sectional methods have been used to find solutions. 

Again, if i = j and k = the electron-repulsion integral takes on the simple point-charge asymptotic form of 1/R, where this time R is the distance between the centroids of the two distributions. Since the basis functions are always atom-centred functions, the centroid of the diagonal charge distributions pu) are the relevant atoms and so the distances R are actually inter-atomic distances. 

The other extreme for the amount of electron in a charge distribution is zero when the two basis functions are orthogonal (5y = 0). In this case the asymptotic form for both types of integral is zero. 

It is well known that Zq , the kinetic part of the partition function, is irrelevant in the determination of the equation of state and in the case of equilibrium between two states. In this work we will be interested only in the configurational part of the distribution function Wn/zq ). Asymptotic expansions of the rotational part of for linear rotators and S3mimetric tops can be found elsewhere. We are now in a position to determine the general form of an expansion of Wn/Zq in powers of the quantum parameter A. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

The non-equilibrium particle distribution is clearly observed through the joint correlation functions plotted in Fig. 6.47. Note that under the linear approximation [74] the correlation function for the dissimilar defects Y (r, t) increases monotonically with r from zero to the asymptotic value of unity Y(r —y oo,t) = 1. In contrast, curve 1 in Fig. 6.47 (f = 101) demonstrates a maximum which could be interpreted as an enriched concentration of dissimilar pairs, AB, near the boundary of the recombination sphere, r tq. With increasing time this maximum disappears and Y(r, t) assumes the usual smoothed-step form. The calculations show that such a maximum in Y(r, t) takes place within a wide range of the initial defect concentrations and for a random initial distribution of both similar and dissimilar particles used in our calculations X (r, 0) = Y(r > 1,0) = 1. The mutual Coulomb repulsion of similar particles results in a rapid disappearance of close AA (BB) pairs separated by a distance r < L (seen in Fig. 6.47 as a decay of X (r, t) at short r with time). On the other hand, it stimulates strongly the mutual approach (aggregation) of dissimilar particles leading to the maximum for Y(r, t) at intermediate distances observed in Fig. 6.47. 

Thus, we seek an asymptotic solution of the one-dimensional gasdynamic equations for a given pressure evolution curve, f(t/r), which is characterized by a sufficiently rapid pressure decrease. In a slightly different way, we may formulate the problem thus preserving the form of the dimensionless function f(t/r), we let the pressure duration go to zero and the maximum pressure to infinity, and look for the asymptotic solution—the distribution of the velocity, pressure and other quantities—after a finite time t, at a finite distance x. 

The problem for the first term in an asymptotic solution for the temperature distribution 9 in the outer part of the thermal layer is thus to solve (11 66) subject to the conditions (11 67a), (11 67c), and (11 7c). Again, we see that the geometry of the body enters implicitly through the function ue (x) only. As in the high-Pr limit, a general solution of (11 66) is possible even for an arbitrary functional form for ue (x ). Before we move forward to obtain this solution, however, a few comments are probably useful about the solution (11 69) for the innermost part of the boundary layer immediately adjacent to the body surface. 

The earlier sections of this chapter discuss the mixed electrode as the interaction of anodic and cathodic reactions at respective anodic and cathodic sites on a metal surface. The mixed electrode is described in terms of the effects of the sizes and distributions of the anodic and cathodic sites on the potential measured as a function of the position of a reference electrode in the adjacent electrolyte and on the distribution of corrosion rates over the surface. For a metal with fine dispersions of anodic and cathodic reactions occurring under Tafel polarization behavior, it is shown (Fig. 4.8) that a single mixed electrode potential, Ecorr, would be measured by a reference electrode at any position in the electrolyte. The counterpart of this mixed electrode potential is the equilibrium potential, E M (or E x), associated with a single half-cell reaction such as Cu in contact with Cu2+ ions under deaerated conditions. The forms of the anodic and cathodic branches of the experimental polarization curves for a single half-cell reaction under charge-transfer control are shown in Fig. 3.11. It is emphasized that the observed experimental curves are curved near i0 and become asymptotic to E M at very low values of the external current. In this section, the experimental polarization of mixed electrodes is interpreted in terms of the polarization parameters of the individual anodic and cathodic reactions establishing the mixed electrode. The interpretation then leads to determination of the corrosion potential, Ecorr, and to determination of the corrosion current density, icorr, from which the corrosion rate can be calculated. 

---