Self-Similar Distribution Functions

Size distributions measured at different locations and/or times, either in the atmosphere or in processes gases, can sometimes be scaled by introducing a characteristic particle diameter based on moments of the distribution function. For example, suppose the distribution function depends only on two of its moments, say Noo and f , in addition to the particle diameter  

The four quantities—nj, N, 0, and dp— thai appear in (1.31) have two dimensions, L and /, corresponding to the coordinate space and particle size. Thus on dimensional grounds, there are two dimensionless groups that can be formed from these quantities. The groups can be expressed as follows  

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FIGURE 6.2.1 The self-similar distribution function for (i) the constant aggregation frequency (dotted line), (ii) the sum frequency (continuous line), and (hi) the Brownian aggregation frequency (dot-and-dash line). (From Wright and Ram-krishna, 1992. Reprinted with permission from Elsevier Science.)... 

The choice of the basis functions depends crucially on the behavior of the self-similar distribution (see footnote 5). For example, suppose that the self-similar distribution z6 (z) has the asymptotic behavior (ju < 1) in the region of z close to zero. Then it is possible to show (see Appendix of Sathyagal et al, 1995) that the function g u) is approximated by for u close to zero. In other words, g(u) is of order 0(m ). Consequently, g u) is not analytic at w = 0, and a very large number of basis functions in the expansion (6.1.9) are required to describe adequately the behavior near the origin. This problem can be overcome by choosing basis functions that have the same dependence on u near w = 0, as g u) does. Incorporating as much known analytical information as possible about the nature of the solution is an important aspect of the solution of inverse problems. Let us see how... 

Finally, we inquire into the inverse problem for the determination of the function g u). Since the inverse problem is in terms of the function pg u we examine the results with respect to it. In Fig. 6.1.4 is shown the solution to the inverse problem as a function of the number of basis functions. Plotted alongside is the actual function used in the simulation. It is interesting to note that = 3 or 4 provides the best solution to the inverse problem while the solution for 5, 6, and 7 show progressive deterioration. The demand on accuracy goes up with an increasing number of functions, and even minor errors in the self-similar distribution will cause significant deviations of the inverted solution from the actual function as seen in Fig. 6.1.4. 

Our choice of basis functions is designed so that the trial solution (6.2.9) has the desired asymptotic properties of the aggregation frequency. Such properties are often available through knowledge of their relationship to the asymptotic properties of the self-similar distribution. What we mean by asymptotic properties and how those of the frequency are related to those of the self-similar distribution are elucidated in the discussion that follows. 

We now return to the issue of the choice of basis functions for solution of the inverse problem (6.2.8). The behavior of the aggregation frequency that relates to the small- / behavior of the function (j)(rj) is the issue of specific interest. We choose to fit with y-distributions that can accommodate either a singular or nonsingular nature of the self-similar distribution near the origin and accordingly set... 

The median diameter of limestone pellets in the ball growth region as a function of the granulation time on log-log scales is shown in Fig. 20. The corresponding size distributions are self-similar as illustrated in Fig. 21. 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

The question of whether proteins originate from random sequences of amino acids was addressed in many works. It was demonstrated that protein sequences are not completely random sequences [48]. In particular, the statistical distribution of hydrophobic residues along chains of functional proteins is nonrandom [49]. Furthermore, protein sequences derived from corresponding complete genomes display a distinct multifractal behavior characterized by the so-called generalized Renyi dimensions (instead of a single fractal dimension as in the case of self-similar processes) [50]. It should be kept in mind that sequence correlations in real proteins is a delicate issue which requires a careful analysis. 

The resulting steady-state size distribution functions were compared with experiments, and it was demonstrated that size distributions were indeed self-similar, and the functionality deduced in the development of the model was observed. 

For instance, in the SLSP model, a HSR may be obtained by taking into account both the self-similarity of the percolating cluster and the scale invariance of the cluster size distribution function (71). By utilizing the renormalization procedure [213], in which the size L of the lattice changes to a new size Lc with a scaling coefficient b = Lc/L, the relationship between the distribution function of the original lattice and the lattice with the adjusted size can be presented as w(s, sm) = bd nDw(s, sm), where s = s D and sm = smD. 

The generation parameter defining the generation of ionizing trajectories in the self-similar structure in Fig. 10 is related to the number w of encounters of the two electrons at ri = T2 rather than to the ionization time. This interpretation is confirmed in Fig. 11 which shows the density n of trajectories starting with initial conditions uniformly distributed in the middle panel of Fig. 10 as function of the number w of encounters of the two electrons and of the ionization time T. The density n is proportional to minus the derivative of the survival probability with respect to the relevant variable (w or T). The logarithmic plot in Fig. 11a reveals an exponential decay of the density, n(w)ocexp(—0.27w), and hence also of the survival probability, as a function of the number of encounters of the two electrons, just as expected for a self-similar fractal set of trapped trajectories. The doubly logarithmic plot of the density of trajectories in Fig. 11b reveals a power-law decay of the density, (T) oc and hence... 

Figure 34. Cartoon showing the SAXS region where self-similarity within aggregates can be detected. Fitting of this region can reveal the fractal dimension of the aggregate, and if combined with analysis of other regions, yield information on the complete distribution function. From Hiemenz and Rajagopalan (1997), used by permission of Marcel-Dekker.

However, there are essential differences between the Lorentz gas model and IS structures in many-dimensional molecular systems. The most obvious difference is the size distribution of basins. In the Lorentz gas model, the size of unit cells is identical but the basin size, as well as the depth of potentials, of molecular systems is believed to range quite broadly, and possibly distributed in a self-similar way, reflecting that local potential minima increase exponentially as a function of the system size. As a result, one expects that the diffusion among multibasin structures bears different characters. The situation of the latter is... 

Then, as the first step of approaching the study of self-similarity, we investigate two features that must appear if self-similarity exists power-type distribution function of momenta and anomalous diffusion. We are particularly... 

Anomalous diffusion was first investigated in a one-dimensional chaotic map to describe enhanced diffusion in Josephson junctions [21], and it is observed in many systems both numerically [16,18,22-24] and experimentally [25], Anomalous diffusion is also observed in Hamiltonian dynamical systems. It is explained as due to power-type distribution functions [22,26,27] of trapping and untrapping times of the orbit in the self-similar hierarchy of cylindrical cantori [28]. 

Abstract Generalization of the self-similar solution for ultrarelativistic shock waves (Bland-ford McKee, 1976) is obtained in presence of losses localized on the shock front or distributed in the downstream medium. It is shown that there are two qualitatively different regimes of shock deceleration, corresponding to small and large losses. We present the temperature, pressure and density distributions in the downstream fluid as well as Lorentz factor as a function of distance from the shock front. 

Figure 1. The distributions of the Lorentz factor, pressure and particle density of the shocked gas as functions of self-similar variable All quantities are normalized to unity at the shock front where x = L Long-dashed lines correspond to the losses uniformly distributed in the downstream medium and decreasing with time as f 1. Short-dashed lines are for the case of localized losses with 5= rj= 0. The solid line — solution of Blandford and McKee (1976).

Inside the Debye sphere, strong electron acceleration takes place. The electrical field that surrounds the ion current channel accelerates the electrons toward the filaments where they are deflected on the induced magnetic field. The scenario is depicted in fig. 2. It have previously been shown that the ion filaments are generated in a self-similar coalescence process (Medvedev et al., 2004) which implies that a spatial Fourier decomposition exhibits power law behavior. As a result, the electrons are accelerated to a power law distribution function (fig. 3) as shown by Hededal et al., 2004. 

These results point towards a universality in the particle radial distribution function from r = 0 to 2R. This is shown in Fig. 8 where the first contact peaks, which are self-similar, are collapsed for all packing fractions onto the master Gaussian curve ... 

Substituting the self-similar form for the size distribution function, (5.23), we obtain... 

The problem is to determine the velocity distribution in the fluid as a function of time. In this problem, the fluid motion is due entirely to the motion of the boundary - the only pressure gradient is hydrostatic, and this does not affect the velocity parallel to the plate surface. At the initial instant, the velocity profile appears as a step with magnitude Uat the plate surface and magnitude arbitrarily close to zero everywhere else, as sketched in Fig. 3 11. As time increases, however, the effect of the plate motion propagates farther and farther out into the fluid as momentum is transferred normal to the plate by molecular diffusion and a series of velocity profiles is achieved similar to those sketched in Fig. 3-11. In this section, the details of this motion are analyzed, and, in the process, the concept of self-similar solutions that we shall use extensively in later chapters is introduced. 

J. Sukhatme and R.T. Pierrehumbert. Decay of passive scalars under the action of single scale smooth velocity fields in bounded two-dimensional domains From non-self-similar probability distribution functions to self-similar eigenmodes. Phys. Rev. E, 66 056302, 2002. 

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