Two-point Distribution Function Research Articles

Correlation function inadequacy in random-sequence entropy measures.

Considering symbolic and numerical random sequences in the framework of the additive Markov chain approach, we establish a relation between their correlation functions and conditional entropies. We express the entropy by means of the two-point probability distribution functions and then evaluate the entropy for the numerical random chain in terms of the correlation function. We show that such approximation gives a satisfactory result only for special types of random sequences. In general case the conditional entropy of numerical sequences obtained in the two-point distribution function approach is lower. We derive the conditional entropy of the additive Markov chain as a sum of the Kullback-Leibler mutual information and give an example of random sequence with the exactly zero correlation function and the nonzero correlations.

Spatial correlations in chaotic nanoscale systems with spin-orbit coupling

We investigate the statistical properties of wave functions in chaotic nanostructures with spin-orbit coupling (SOC), focussing in particular on spatial correlations of eigenfunctions. Numerical results from a microscopic model are compared with results from random matrix theory in the crossover from the gaussian orthogonal to the gaussian symplectic ensembles (with increasing SOC); one- and two-point distribution functions were computed to understand the properties of eigenfunctions in this crossover. It is found that correlations of wave function amplitudes are suppressed with SOC; nevertheless, eigenfunction correlations play a more important role in the two-point distribution function(s), compared to the case with vanishing SOC. Experimental consequences of our results are discussed.

Homogenization-based constitutive models for magnetorheological elastomers at finite strain

This paper proposes a new homogenization framework for magnetoelastic composites accounting for the effect of magnetic dipole interactions, as well as finite strains. In addition, it provides an application for magnetorheological elastomers via a “partial decoupling” approximation splitting the magnetoelastic energy into a purely mechanical component, together with a magnetostatic component evaluated in the deformed configuration of the composite, as estimated by means of the purely mechanical solution of the problem. It is argued that the resulting constitutive model for the material, which can account for the initial volume fraction, average shape, orientation and distribution of the magnetically anisotropic, non-spherical particles, should be quite accurate at least for perfectly aligned magnetic and mechanical loadings. The theory predicts the existence of certain “extra” stresses—arising in the composite beyond the purely mechanical and magnetic (Maxwell) stresses—which can be directly linked to deformation-induced changes in the microstructure. For the special case of isotropic distributions of magnetically isotropic, spherical particles, the extra stresses are due to changes in the particle two-point distribution function with the deformation, and are of order volume fraction squared, while the corresponding extra stresses for the case of aligned, ellipsoidal particles can be of order volume fraction, when changes are induced by the deformation in the orientation of the particles. The theory is capable of handling the strongly nonlinear effects associated with finite strains and magnetic saturation of the particles at sufficiently high deformations and magnetic fields, respectively.

Application of two-point probability distribution functions to predict properties of heterogeneous two-phase materials

An investigation was conducted in order to determine the viability of using two-point probability distribution functions to predict the material properties of two-phase heterogeneous materials. The material of interest in this investigation was Titanium Diboride/Alumina ceramic (TiB 2 + Al 2O 3). In TiB 2 + Al 2O 3 ceramics, it has been noted that the connectivity of the TiB 2 second phase plays a major role in the mechanical properties of the TiB 2 + Al 2O 3 ceramic, spall strength in particular. Two-point distribution functions were obtained for the four types of microstructures of TiB 2 + Al 2O 3 and were compared to experimental results that have been previously reported. The two-point distribution functions showed very good correlation to connectivity and spall strength results.

Ensemble averaging in turbulence modelling

In this Letter we explore a new type of averaging in turbulence modelling, new types of characterization of the internal structure of turbulence (by two-point distribution function of fluid-particles), and consequences of the requirement that the Navier–Stokes equation and the governing equations of the models share a common structure of physical significance (Poisson structure). As a result we arrive at a turbulence model that includes the Reynolds-stress and LANS- α models, unifies them and extends them to models with an active advection.

Microstructure design of a two phase composite using two-point correlation functions

Two-point distribution functions are used here as to introduce “Microstructure Sensitive Design” in two-phase composites. Statistical distribution functions are commonly used for the representation of microstructures and also for homogenization of materials properties. The use of two-point statistics allows the composite designer to include the morphology and distribution in addition to the properties of the individual phases and components. Statistical continuum mechanics is used to make a direct link between the microstructure and properties (elastic and plastic) in terms of these two-point statistical functions. An empirical form of the two-point statistical function is used which allows the construction of a composite hull. Two different composites (isotropic and anisotropic) are considered and the effect of anisotropy for the prediction of the elastic properties is discussed

Exact statistical properties of the zeros of complex random polynomials

The zeros of complex Gaussian random polynomials, with coefficients such that the density in the underlying complex space is uniform, are known to have the same statistical properties as the zeros of the coherent state representation of one-dimensional chaotic quantum systems. We extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest Landau level. A study of the statistical properties of the zeros is undertaken using exact formulae for the one- and two-point distribution functions. Attention is focused on the moments of the two-point correlation in the bulk, the variance of a linear statistic, and the asymptotic form of the two-point correlation at the boundary. A comparison is made with the same quantities for the eigenvalues of complex Gaussian random matrices.

Computer simulation study of phase separation in a binary mixture with a glass-forming component

We present a computer simulation study of spinodal decomposition with one of the two phases freezing in a glassy state during phase separation. As a model we used the Cahn-Hilliard equation with a concentration-dependent mobility coefficient which decreases rapidly with increasing concentration of the glass-forming component. We solved the Cahn-Hilliard equation numerically for two dimensions. The domain growth depends crucially on the volume fraction of the glassy phase. For high volume fractions, when the glassy phase forms a percolating matrix, a novel coarsening mechanism is discovered, which arises from the migration and coalescence of liquid droplets within the glassy matrix. Various quantities characterizing the time-dependent domain pattern, like droplet size distribution, one- and two-point distribution function and structure factor of the concentration field, are computed. We checked the validity of the dynamic scaling hypothesis.

Extended kinetic theory

The domain of applicability of one-point kinetic theory is enlarged by admitting two-point distribution functions or some of their moments as extra state variables. Kinetic equations governing the time evolution in the extended state space are introduced as particular realizations of the nonlinear Onsager–Casimir equation. This abstract equation collects the common structure of equations arising in the analysis of compatibility of two levels of description (e.g., kinetic theory and equilibrium thermodynamics or kinetic theory and hydrodynamics). The structure consists of a Poisson bracket and two potentials. The Poisson bracket expresses kinematics, the first potential, called the thermodynamic potential, generates the time evolution and links the theory with thermodynamics. The second potential, called the dissipative potential, introduces dissipation into the time evolution that is compatible with the kinematics and thermodynamics. To find a particular realization of the nonlinear Onsager–Casimir equation means to identify kinematics (Poisson bracket) and the two potentials, that correspond to a particular system and situations under consideration. This way of introducing kinetic equations has two advantages. First, the kinetic equations are guaranteed to be intrinsically consistent and compatible with more macroscopic theories. Second, the clear physical meaning of kinematics and the two potentials, together with the possibility of discussing them separately, allows one to use a broad range of physical insights and considerations to specify them for particular systems and situations of interest.

Distribution functions in the statistical theory of convective MHD turbulence of an incompressible fluid

In this paper, we define a hierarchy of distribution functions for simultaneous velocity, magnetic, and temperature fields. Some properties of the constructed distribution functions such as reduction, separation, and coincidence are discussed. Equations for the evolution of one- and two-point distribution functions have been derived. Finally, a comparison of the equation for the single-point distribution function in case of zero viscosity, negligible diffusivity, and infinite electrical conductivity is made with first equation of BBGKY hierarchy in the kinetic theory of gases.

Microscopic theory of dynamics in an orientationally ordered fluid

A variety of dynamical properties are studied in a model complex liquid composed of hard parallel spherocylinders. For this fluid a revised Enskog kinetic theory (RET) is introduced. In the long wavelength and long time limit the RET leads to liquid crystal nematic-like hydrodynamic equations with explicit expressions for all transport coefficients in terms of an assumed known equilibrium two-point distribution function. The transport process of self-diffusion in this fluid is also studied in the Enskog approximation. This model has a nematic to smectic phase transition (N→A) where a one-dimensional translational order occurs. Near this phase transition the RET is shown to lead to a van Hove-like theory of critical dynamics. The critical mode near the N→A transition is analogous to the so-called soft extended heat mode recently discussed for simple fluids. Near the N→A transition the critical singularities in the nematic viscosities are discussed. It is shown that the singularities arise from the same mode coupling mechanism that is responsible for the anomalously slow relaxation of the stress-tensor autocorrelation function in dense simple liquids.

Application of Volterra–Wiener Series for Bounding the Overall Conductivity of Heterogeneous Media. II. Suspensions of Equi-sized Spheres

The variational procedure of Part I of this paper [this Journal, 47 (1987), pp. 831–849] is employed for bounding the overall conductivity $\kappa^*$ of a suspension of equi-sized spheres. The procedure uses classes of trial functions in the form of truncated Volterra–Wiener series generated here by the random density function of the system of sphere centers. Integrodifferential equations are derived whose solutions yield bounds, upper and lower, on $\kappa^*$ for given two- and three-point distribution functions $f_2 $, $f_3 $ of the system of spheres. The bounds are explicitly calculated to the order $c^2 $, where c is the volume fraction of the spheres; they include a statistical parameter $m_2 $ that can be simply evaluated by means of the two-point distribution function $f_2 $ and that reflects the tendency of the spheres to form clusters at higher c. The predictions of various approximate models of composite mechanics are checked against the $c^2 $-bounds. The check yields a number of curious results concerning the random constitution of suspensions of impenetrable spheres that can be described by means of these models.

Two-point distribution function for a dispersion of impenetrable spheres in a matrix

Results are reported for the two-point distribution function G2(r) of two-phase random media composed of equisized impenetrable spheres in a matrix. G2 arises in various expressions for transport and mechanical properties of heterogeneous materials.

D-dimensional interpenetrable-sphere models of random two-phase media: Microstructure and an application to chromatography

For a class of models of random two-phase media we derive and numerically evaluate expressions for the porosity (matrix volume fraction) φ and the specific surface (interface area per unit volume) s, as well as certain generalizations of these quantities. In these models the medium is considered as a suspension of mutually interpenetrable D-dimensional spheres of radius R, embedded in a uniform matrix. The models and quantities considered have applications to a variety of problems concerning transport, mechanical, electromagnetic, and chemical properties of composite media. Spatial dimensionality D=3 is appropriate for a wide range of composite media, whereas D = 2 describes certain fiber composites (parallel cylindrical inclusions), and D = 1 is appropriate for layered structures. The models considered here contain a hardness parameter ϵ, such that for ϵ = 1 they reduce to mutually impenetrable (hard) spheres, and for ϵ = 0 they reduce to randomly placed spheres. We consider three specific models in this class. The first of these, the permeable—sphere model, is considered in the Percus-Yevick approximation to obtain the two-point distribution function g 2(r 1, r 2). We obtain the n-point distribution functions g n (r 1, …, r n ) for n ⩾ 3 in the generalized superposition approximation. We also evaluate the lowest-order corrections to this approximation, finding that for all φ and ϵ it reliably gives φ as a function of inclusion number density. For the second model, the concentric-shell model, φ and s are evaluated in the scaled-particle approximation. The generalizations φ( R̂) and s( R̂), which are the volume fraction and specific surface available to a probe particle of finite radius R ̂ , are also derived and evaluated. For randomly placed spheres (ϵ = 0) these quantities are also considered for polydisperse media. An application to gel size-exclusion chromatography is considered. We find that varying the hardness parameter ϵ or the polydispersivity provides better agreement between theory and experiment than is obtained by using a monodisperse randomly placed sphere model of the gel. A comparison of the density expansions of φ and s for the permeable-sphere and concentric-shell models confirms that the considerable quantitative differences we find between these two conceptually distinct models are not simply the result of the approximation procedures that we use in evaluating their properties. Finally we introduce a new class of models, based on mixtures of spheres of different mutual interpenetrability. For one particular such model we obtain exact expressions for φ and s for general ϵ, as well as the two-point matrix function S 2(r) in the Percus—Yevick and Verlet—Weis approximations.

The total and direct correlation function in the interface of the lattice gas in two dimensions

The interface between two co-existing phases of a lattice gas in two dimensions in a certain non-zero external potential is studied by a computer simulation, which produces numerically accurate results by the use of a transfer matrix which is constructed numerically. Strips of dimensions 7 x ∞ and 11 x ∞ were studied at T ≼0.5 T c . The two-point distribution function, the two-point correlation function, the direct correlation function of Ornstein–Zernike and their Fourier transforms were computed and are shown and discussed.

New computational method in the theory of spinodal decomposition

We present a new series of calculations in the theory of spinodal decomposition. The computational scheme is based on a simple ansatz for the two-point distribution function which leads to closure of the hierarchy of equations of motion for the high-order correlation functions. The resulting theory is accurate throughout the spinodal region of the phase diagram, including at the boundaries of this region where the spinodal mechanism is difficult to distinguish from nucleation and growth. The computational scheme is worked out in detail for parameters approximating those of the three-dimensional, kinetic, spin-exchange Ising model with nearest-neighbor interactions. Numerical agreement with recent Monte Carlo data appears to be satisfactory.

Approach to the origin of turbulence on the basis of two-point kinetic theory

Equations for the fluctuation correlation in an incompressible shear flow are derived on the basis of kinetic theory, utilizing the two-point distribution function which obeys the BBGKY hierarchy equation truncated with the hypothesis of “ternary” molecular chaos. The step from the molecular to the hydrodynamic description is accomplished by a moment expansion which is a two-point version of the thirteen-moment method, and which leads to a series of correlation equations, viz., the two-point counterparts of the continuity equation, the Navier-Stokes equation, etc. For almost parallel shearing flows the two-point equation is separable and reduces to two Orr-Sommerfeld equations with different physical implications. Solution of an eigenvalue problem for the Blasius boundary layer is obtained in a certain parallelism to the classical stability theory, and is used for predicting the transition Reynolds number of a “quiescent” Blasius flow in which thermodynamic fluctuations alone are the initiating mechanism. Also, the calculated spatial growth rate of fluctuation agrees with the Schubauer-Klebanoff experiment, which gives an account of unexplained experimental evidence that the fluctuation complex (turbulence bursts plus the Tollmien-Schlichting wave), as a whole, obeys a certain linear theory.

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.

On the van der Waals Theory of the Vapor-Liquid Equilibrium. III. Discussion of the Critical Region

The discussion of the properties of the Kac one-dimensional fluid model presented in Parts I and II of this series of papers breaks down near the critical point. In Sec. II of the present paper we develop a new successive-approximation method for the eigenvalues and eigenfunctions of the Kac integral equation which is valid in the critical region and which connects smoothly with the developments in the one- and two-phase regions given in Part I. The perturbation parameter is (γδ)½ where γδ is the ratio of the ranges of the repulsive and attractive forces. The main physical consequence is that in the critical region the long-range behavior of the two-point distribution function is represented by an infinite series of decreasing exponentials with ranges all of order 1/γ(γδ)½, and with amplitudes of order (γδ)⅔. This leads to deviations from the Ornstein-Zernike theory and to a specific heat anomaly which are discussed in Sec. V. We conclude with some comments on the possible relevance of our results for the three-dimensional problem.

On the van der Waals Theory of the Vapor-Liquid Equilibrium. II. Discussion of the Distribution Functions

For the same one-dimensional fluid model discussed in Part I, we have derived general expressions for the two- and three-particle distribution functions. It is seen that these distribution functions depend on all the eigenvalues and eigenfunctions of the basic Kac integral equation, and the dependence is so transparent that the generalization to s particles is obvious. The fluctuation and virial theorems are discussed and shown to be consequences of our general formula. In the van der Waals limit, the behavior of the two-point distribution function is discussed, both for distances of the order of the hard core and for distances of the order of the range of the attractive force. The long-range behavior is, in first approximation, equivalent to the one-dimensional version of the Ornstein-Zernike theory, but only in the one-phase region and not too near the critical point. In the two-phase region, all distribution functions are linear combinations of the two corresponding distribution functions of the saturated vapor and liquid, with coefficients proportional to the mole fractions of vapor and liquid. This is shown for our model; we also give arguments for our belief that these relations are general, and express the geometrical separation of the two phases. The relation to the Ornstein-Zernike theory is discussed in more detail, especially in connection with a recent formulation of this theory by Lebowitz and Percus. We conclude with some comments on the relevance of our results for the three-dimensional problem.