One-point Distribution Function Research Articles

The one-point cluster distribution function and its moments

We derive the one-point probability density function (pdf) of the smoothed Abell-ACO cluster density field and we compare it with that of artificial cluster samples, generated as high peaks of a Gaussian field in such a way that they reproduce the low-order (two- and three-point) correlation functions and the observed cluster selection functions. We find that both real and simulated pdfs are well approximated by a log-normal distribution, even when the Gaussian smoothing radius is as large as 40 |$h^{-1}$| Mpc. Furthermore, the low-order moments of the pdf are found to obey a relation |$\gamma\approx1.8(\pm0.2)\sigma^4$|⁠, with γ being the skewness. Since clusters have not had enough time to depart significantly from their original birth-place positions, these results are consistent with the theory that clusters are high peaks of an underlying initial Gaussian density field. A by-produce of our analysis is that when we re-scale the pdf cluster moments to those of the QDOT-IRAS galaxies, using linear biasing with bcI∼4.5 and for the common smoothing radius of 20 |$h^{-1}$| Mpc, we find them to be significantly smaller than those directly estimated from the QDOT data by Saunders et al.

Signatures of chaos in the modulus and phase of time-dependent wave functions

For a classically chaotic two-dimensional bound system, based on the assumption that an initially-well-localized, semiclassical wave packet can be represented by a superposition of a large number of random plane waves at fixed times, we show that the modulus and phase of the wave function are independent random functions having a Rayleigh and a uniform one-point spatial distribution function, respectively. These predictions are confirmed through our numerical wave-packet study for one-quarter of the Sinai billiard. Streamline vortices can form around wave-function nodes, a fact first discovered by Dirac in 1931. For a classically chaotic billiard, the random plane-wave superposition approximation predicts that both the number of nodal points and the maximum number of vortices in a wave packet with initial wave number k is N, where N refers to the Nth eigenstate with wave number ${\mathit{k}}_{\mathit{N}}$ which is closest to k.

A Simple Derivation of the Neutron Transport Equation

In this paper, a new and simple derivation of the neutron transport equation is given. The approach is similar to that used in the Liouville equation and its applications to the Boltzmann equation in that it is formulated in terms of the one-particle or one-point density function, as opposed to the traditional reactor physics approach of counting neutrons in a volume of the phase-space. It makes use of the recognition that the expected number of particles in a phase cell dV is the same as the probability of finding one particle in dV. A novelty of the derivation here is that because of the linear Markovian property of the process, it is possible to derive a master (Chapman-Kolmogorov) equation for the one-particle density, that is, for the neutron density or neutron flux of the traditional transport equation. This way, the forward and the backward (adjoint) equations of neutron transport can be derived from a single master equation. The variance of the one-point distribution function is also derived, and an explicit solution is given.

The Einstein-Smoluchowski promeasure and the stochastic Boltzmann-Peierls equation

The essential properties of a spatially homogeneous gas consisting of quasiparticles (phonons, magnons, rotons, etc.) are usually described in terms of the one-point distribution function f. This is the last in a series of papers (Physica A 180 (1992) 309, 336), the overall objective of which is the construction of the Einstein-Smoluchowski promeasure μ ɛ in order to formulate and develop a theory dealing with the equilibrium fluctuations of f. The basic idea was the inversion of Boltzman's relation connecting entropy and probability. As a consequence, the derivation of μ ɛ started by considering an infinite-dimensional analog of Einstein's thermodynamic theory of equilibrium fluctuations. Now, investigating a situation in which the time evolution of f is governed by the linear stochastic differential Boltzman-Peierls equation, the object here is to find the conditions under which the stationary measure for the aforesaid equation can be obtained from the Einstein-Smoluchowski promeasure μ ɛ . In agreement with such a general program, the fluctuation-dissipation theorem is analysed and some evidence of the necessity for an algebraic approach to the stochastic kinetic equations is also presented. And finally, the paper discusses the time-correlation function formalism and shows that this formalism is consistent with the universal ideas of Einstein. To summarize, the present paper demonstrates how the Einstein-Smoluchowski promeasure μ ɛ can be used to study some problems in the extended context of the time-dependent theory of equilibrium fluctuations.

Irreducible tensor description. I. A classical gas

A classical, moderately rarefied, simple, monatomic gas is considered, and it is supposed that its behavior is governed by Boltzmann’s equation. Then, after a brief review of the fundamental properties of the symmetric traceless tensors, the irreducible equations of transfer for the relative symmetric traceless moments of the one-point distribution function f are systematically derived and related to those of Johnston. Subsequently, Grad’s expansion of the distribution function in terms of reducible three-dimensional Hermite ‘‘polynomials,’’ which does not fit in together and with the equations of transfer just mentioned, is rigorously transformed into its irreducible counterpart fashioned by mathematical apparatus such as one-dimensional Laguerre polynomials and Ikenberry’s tensorial harmonics. Finally, some useful conversion formulas between the relative symmetric traceless moments and the tensorial Laguerre–Ikenberry expansion coefficients of f are deduced and the irreducible variant of Grad’s moment truncation procedure is discussed. The conclusions that have so far been reached concerning Grad’s method are quite specific in that they apply to one-dimensional (classical or quasiparticle) gases. In many cases of interest, the treatment of certain aspects of the kinetic theory of ‘‘actual’’ gases requires, as a prerequisite, a comprehensive discussion of some complicated tensorial problems. In this work the so-called irreducible tensor description of three-dimensional gaseous systems is exploited, and this subject is developed only insofar as it relates to Grad’s moment procedure and to those universal questions which have already been formulated in previous papers. [Z. Banach, J. Stat. Phys. 48, 813 (1987); Arch. Mech. (to be published); Physica A 129, 95 (1984); 145, 105 (1987).]

Early-time spinodal decomposition in the surface layer.

The early stages of spinodal decomposition in the surface layer of a semi-infinite Ising model are studied by using the theory of Langer, Bar-on, and Miller and the self-consistent method. It is shown that the speed of phase separation in the surface layer is faster than that in the bulk because of the surface enhancement, and the one-point distribution function in the surface is different from that of the bulk. It is inferred that the surface with a negative enhancement (Cg0) acts as a two-dimensional ``nucleus'' for the phase separation and for forming a ``wetting layer.''

Fracture of parallel elements

A new approach to modeling of progressive damage of interphases in random particulate composites is proposed. In the model the interphases are represented by layers of springs whose stiffness parameters change as a result of damage associated with gradually increasing loading. Consequently, due to local degradation of the interphase, variation of these parameters over the inhomogeneity surface becomes progressively non-uniform. This, in turn, causes the effective properties of the composite to become increasingly anisotropic. To describe the effects of interphase damage on the overall properties of the composite three ideas proposed in the past with the authors’ participation are utilized. One is the Method of Conditional Moments (MCM), a statistical homogenization technique developed in the past to analyze the effective properties of random composites without interphases. The second idea is the recently introduced notion of Energy Equivalent Inhomogeneity (EEI), which replaces the original inhomogeneities and their interphases with uniform inhomogeneities; the notion of EEI allows methods devised to examine composites without interphases (such as MCM) to analyze composites with interphases. The most important, and novel in application to problems considered in this work, is the third idea concerning the statistical interphase damage model. In the proposed approach local micro-damage of interphases is modelled by an increasing fraction of randomly distributed destructed springs in the spring layers surrounding the inhomogeneities of the composite. The model assumes that the local interphase micro-strength at the points located on the surface of different inhomogeneities and specified by the same spherical coordinates is described by one-point Weibull distribution function. In view of the strong nonlinearity of the problem, an incremental/iterative scheme is used in practical evaluation of effective properties. The approach is illustrated considering unidirectional and triaxial stretching of a composite.

Semiempirical theory of turbulent transfer

A closed system of equations is constructed for flow of a nonisotropic character on the assumption that the mixing path length is not small compared with the characteristic dimension of the stream. It is assumed that the velocity pulsation field can be characterized by a multipoint distribution function, which satisfies the continuity equation. This enables equations to be obtained for the one-point and two-point distribution function. A series of assumptions is made concerning the nature of the forces acting on the turbulent formation (“turbule” or vortex) in the stream and concerning the correlation time of the random force with the scale and intensity of the turbulence. Assumptions are also made concerning the expression of the integral in the equation for the one-point distribution function and the expression for the correlation tensor in the isotropic case. After the moments are calculated, a system of Reynolds' equations is obtained in which approximations, usually acceptable from dimensional considerations, follow for a series of terms. Here, this is a consequence of the approximations for the forces in the equation for the distribution function. Closing the system of equations for the moments reduces to solving the equation for the distribution function. It turns out that the integral character of the transfer (diffusion of a nongradient type) is connected with taking third-order moments into account. A series of examples of flow is considered, and values of the empirical constants are determined. The system of equations obtained enables us to consider flow with strong anisotropy of turbulent transfer.

Model Equation for Nonhomogeneous Turbulence

The equation for the one-point turbulent distribution function is closed by assuming a relaxation model for the pressure fluctuation term and several other less critical approximations. The resulting equation is solved for a class of rectilinear flows and for decay of a periodic wake. Good agreement with available experiments is found. A theory of turbulent transport coefficients is derived in a certain approximation.