Distribution function universality

Molecular weight distribution function for the case where the length of the growth stage is short compared to the residence time in reactor. (Reprinted with permission from Chemical Reactor Theory, by K. G. Denbigh and J. C. R. Turner. Copyright 1971 by Cambridge University Press.)... 

Stribeck N (1980) Computation of the Lamellar Nanostructure of Polymers by Computation and Analysis of the Interface Distribution Function from the Small-Angle X-ray Scattering. Ph.D. thesis, Phys. Chem. Dept., University of Marburg, Germany... 

The complex scattering wave function can be specified by nodal points at which u = 0,v = 0. They have great physical significance since they are responsible for current vortices. We have calculated distribution functions for nearest distances between nodal points and found that there is a universal form for open chaotic billiards. The form coincides with the distribution for the Berry function and hence, it may be used as a signature of quantum chaos in open systems. All distributions agree well with numerically computed results for transmission through quantum chaotic billiards. 

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. 

D. J. Tannor, unpublished (1984) D. Kohen, Phase-Space Distribution Function Approach to Molecular Dynamics in Solutions, Ph.D. Thesis, University of Notre Dame, 1995. 

However, Waite s approach has several shortcomings (first discussed by Kotomin and Kuzovkov [14, 15]). First of all, it contradicts a universal principle of statistical description itself the particle distribution functions (in particular, many-particle densities) have to be defined independently of the kinetic process, but it is only the physical process which determines the actual form of kinetic equations which are aimed to describe the system s time development. This means that when considering the diffusion-controlled particle recombination (there is no source), the actual mechanism of how particles were created - whether or not correlated in geminate pairs - is not important these are concentrations and joint densities which uniquely determine the decay kinetics. Moreover, even the knowledge of the coordinates of all the particles involved in the reaction (which permits us to find an infinite hierarchy of correlation functions = 2,...,oo, and thus is... 

But any complete description of the evolution of perturbations in the universe will link all of these terms initial velocity and density perturbations to the various components (baryons, dark matter, photons) evolve prior to last scattering as discussed above, and so photon overdensities occur in potential wells, and velocity perturbations occur in response to gravitational and pressure forces. Indeed, to solve this problem in its most general form, we must resort to the Boltzmann equation. The Boltzmann equation gives the evolution of the distribution function, fi(xp,Pp) for a particle of species i with position Xp, and momentum p/(. In its most general form, the Boltzmann equation is formally... 

Answer 2 given above invites, of course, another question Where do the fundamental thermodynamic relation h = h x) and the relation y = y x) come from An attempt to answer this question makes us to climb more and more microscopic levels. The higher we stay on the ladder the more detailed physics enters our discussion of h = h(x) and y = y(x). Moreover, we also note that the higher we are on the ladder, the more of the physics enters into y = y(x) and less into h = h x). Indeed, on the most macroscopic level, i.e., on the level of classical equilibrium thermodynamics sketched in Section 2.1, we have s = s(y) and y y. All the physics enters the fundamental thermodynamic relation s s(t/), and the relation y = y is, of course, completely universal. On the other hand, on the most microscopic level on which states are characterized by positions and velocities of all ( 1023) microscopic particles (see more in Section 2.2.3) the fundamental thermodynamic relation h = h(x) is completely universal (it is the Gibbs entropy expressed in terms of the distribution function of all the particles) and all physics (i.e., all the interactions among particles) enters the relation y = y(x). 

Different functions are used in the analytical presentation of the distribution curves. One of the most universal distribution functions is the gamma-function. T-function of bubble distribution by radius can be expressed as... 

FIGURE llJS Radial distribution function for suspensions of hard spheres (a) in the disordered state and (b) in the ordered state. Taken from Russel [30, pp. 339—340]. Copjni t 1989 by Ceunbridge University Press. Reprinted with the permission of Cambridge University Press. 

A different type of analysis has now provided this information (20) The dimension distributions p(a) of independent spherical scatterers with uniform density and diameter a which produces each of the terms in the sum in Equation (3) can be calculated (19) After obtaining the constants in the sum in Equation (3) by least-squares fits of this equation to the scattering curve measured for Beulah lignite at the University of North Dakota, we used these constants to evaluate the sum of the pore-dimension distribution functions for uniform spheres that are obtained (19) from the terms in the sum in Equation (3) The sum of these pore-dimension distributions was very similar to the power-law distribution given by Equation (4) The fact that we could obtain almost the same power-law dimension distribution by two independent methods suggests that such a distribution may be a good approximation to the pore-... 

Figure 2.3. Radial distribution function hs( ) for suspensions of hard spheres in the disordered state at various volume fractions

Fig. 28 Distribution functions Wox and Erecj at equilibrium and under application of a cathodic overpotential r). These distribution functions represent the contributions to the current by the oxidized/reduced states in the solution, originated by electron transfer from/to the energy state e in the metal. (Reprinted with permission from Ref 22, Copyright 1996 by Oxford University Press).

Similar methods can be used to construct universal plots for molar mass distributions of linear and hyperbranched condensation polymers. The number distribution function n p, N) for linear condensation polymers is obtained from the number fraction distribution [Eq. (1.66)] ... 

Using the proportionality between the characteristic degree of polymerization N and a universal distribution function for hyperbranched polymers can be constructed by plotting n p, M jNl, against N/N. The facr that the cutoff function contains the power law in z makes the apparent power law exponent 3/2 in the molar mass distribution different from r = 2. The cutoff function for gelation has no power law so the apparent exponent is equal to t. 

The universality class of the randomly branched polymers can be determined by constructing a universal molar mass distribution plot, like the ones shown in Fig. 6.26. First, the number density distribution function n(p, N) is determined from the concentration and weight-average molar... 

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