Three-particle Distribution Function Research Articles

Простейшая аппроксимация интегральных коэффициентов в уравнениях корреляционной магнитодинамики для ферромагнетиков

The equations of correlational magnetodynamics (CMD) describe a magnet in the continuum approximation. The main problem in constructing CMD is the calculation of integral coefficients, in particular, the coefficient describing the production of short-range order, depending on the three-particle distribution functions and the structure of the crystal lattice. The work provides the simplest approximations for the integral coefficients of CMD based on the value of pair correlations at the phase transition point. To ensure an equilibrium solution, the coefficients are additionally determined in the upper part of the phase plane according to the assumption of a helical magnetization structure. The resulting approximation provides qualitative agreement with the simulation results within the framework of the original atomistic model of the magnet, and at the same time it turns out to be simple enough for further analysis.

Structural degeneracy in pair distance distributions.

A traditional basic descriptor of many-particle systems has been the distribution of interparticle pair distances. In the case of structureless particles at thermal equilibrium, with just additive pair interactions, this suffices to determine pressure and mean energy. However, it is usually the case that a given set of pair distances can emerge from a multiplicity of distinguishable many-particle configurations. This paper focuses on the ways in which such a configuration detail can be overlooked. After providing some elementary small-system examples in which full pair distance specification still permits distinct configurational pattern ambiguity, subsequent analysis concentrates on large-system classical canonical ensembles. In that context, configurational degeneracy is analyzed in two-dimensional systems for the shape distribution of triangles, whose chirality occurrence can be controlled by suitable three-particle interactions. For many-particle systems in three dimensions, the possibility is explored that a set of three-particle "pair-invisible" interactions can exist which modify the three-particle distribution function, but which have no effect on the pair distribution function, and thus remain undetected by conventional diffraction experiments. For illustration, a specific mathematical example is presented, applicable to the case where two-particle interactions vanish.

The Use of the Embedded-Atom Method in Statistical Thermodynamics of Metals

An expression for the Helmholtz energy of a metal has been derived using the embedded-atom method (EAM) within the classical perturbation theory. An expression for the effective three-particle distribution function with effective three-particle interactions taken into account within classical statistical thermodynamics has been proposed. Temperature dependences of the atomic density and the internal energy have been calculated for copper and gold. These dependences have been used to analyze the effective EAM-derived potentials.

Recent progress in the theory and simulation of strongly correlated plasmas: phase transitions, transport, quantum, and magnetic field effects

We review recent progress in theory and simulation of strongly correlated classical plasmas, in general, and dusty plasmas, in particular. Using the one-component plasma (OCP) as a model, the structural properties of extended and finite correlated systems are analyzed and criteria for disordering transitions are introduced. These are based on the pair and three-particle distribution functions and the associated reduced entropies and heat capacities. These quantities are computed from the particle positions alone and are, thus, directly accessible in experiments. Further, these quantities are applied to confined systems where disordering proceeds via a sequence of phase transitions which have to be clearly distinguished. In the second part of this review, the transport properties of strongly correlated plasmas in equilibrium are considered, particularly under the influence of an external magnetic field. Examples given are for the diffusion, heat conduction and viscosity. Here, the influence on both fundamental transport coefficients and on particle-resolved dynamical effects is considered. Finally, we give a brief discussion of spin and quantum effects and how they influence the structural and dynamical properties of correlated systems.

Resolving structural transitions in spherical dust clusters.

Finite systems in confining potentials are known to undergo structural transitions similar to phase transitions. However, these systems are inhomogeneous, and their "melting" point may depend on the position in the trap and vary with the particle number. Focusing on three-dimensional Coulomb systems in a harmonic trap a rich physics is revealed: in addition to radial melting we demonstrate the existence of intrashell disordering and intershell angular melting. Our analysis takes advantage of a novel melting criterion that is based on the spatial two- and three-particle distribution functions and the associated reduced entropy which can be directly measured in complex plasma experiments.

Classical non-Markovian Boltzmann equation

The modeling of particle transport involves anomalous diffusion, ⟨x2(t) ⟩ ∝ tα with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.

Collective diffusion in a two-dimensional Brownian fluid

The collective dynamics of a simple model colloidal suspension, namely, a two-dimensional Brownian Yukawa fluid, is studied by Brownian dynamics simulations without hydrodynamic interactions. The results for the intermediate scattering function F(k,t) are employed to assess the quantitative and qualitative accuracy of the single-exponential memory function (SEXP) approximation. We find that, contrary to self-diffusion dynamics, the additional, accesorial approximation of neglecting the three-body correlations in the third moment condition, leads to unphysical predictions of the SEXP approximation. We also show that these spurious features are readily removed by including the contribution of the three-particle distribution function to the third moment of F(k,t), even in a simple approximate manner implemented in this work.

A test-particle method for the calculation of thethree-particle distribution function of the hard-sphere fluid:density functional theory and simulation

The three-particle distribution function g(3) of the homogeneous hard-sphere fluid is studied by means of the test-particle method of Percus. We consider an inhomogeneous situation in which the position of two particles is fixed and the inhomogeneous density profile that arises is related to g(3). The profile is calculated via Monte Carlo (MC) simulation and density functional theory (DFT). The results of the simulation are compared with standard MC results for the homogeneous fluid in order to check the numerics. The results of DFT are compared with MC data, showing a significant improvement over the well-known superposition approximation of Kirkwood.

Pre-edge fine structure of the 3d atom K x-ray absorption spectra and quantitative atomic structure determinations for ferroelectric perovskite structure crystals

A complete interpretation is proposed for the pre-edge fine structure (PEFS) of the x-ray Ti K-absorption spectra for perovskite structure crystals. The interpretation is based on the results of numerous calculations performed by a modified full multiple scattering method which provides the theoretical spectra for the 3d transition metal oxides in fair agreement with experiment. It is shown that the three main peaks in the PEFS have quite different origin. The first long-wave side peak A is caused mainly by quadrupole transitions. The middle peak B is caused by the p-d mixture effect and the high intensity of it is considered to be a qualitative spectroscopic indication of ferroelectricity in the perovskite structure crystal. A simple formula is obtained which expresses the area under peak B through the lattice constants and mean-square displacement of the absorbing Ti atom from the instantaneous centre of the coordination polyhedron. The peak B area averaged over thermal atomic vibrations is determined by the three-particle atomic distribution function. The short-wave side peak C is caused by the Ti 1s electron transition to the unoccupied 3d states of the neighbouring transition metal atoms. We show that an additional peak on the short-wave side of peak C occurs if there are 4d atoms (for instance Zr atoms in the vicinity of the absorbing Ti atom in the (PZT) solid solution) within the oxygen atom octahedrons surrounding the absorbing 3d atom. The area under peak is directly determined by the average number of 4d atoms in the vicinity of the absorbing Ti one.

Superposition approximation in two-dimensional fluids

Computer simulation results are reported for the triplet distribution function of two model fluids in two dimensions, namely, the repulsive Yukawa and the hard disc fluids. An interesting correspondence between the structure of these two model systems, similar to that observed in the radial distribution function of the three-dimensional version of these systems, is shown to hold here at the level of the three-particle distribution function. The computer simulation results are employed to illustrate in detail the qualitative and quantitative accuracy of the superposition approximation. This comparison is also presented for the four-particle distribution function in the case of the repulsive Yukawa system.

Effectiveness of nonequilibrium closures for the many body forces in concentrated colloidal dispersions

Here we present a theory for predicting the effect of interparticle interactions on the nonequilibrium dynamics of concentrated colloidal dispersions. A configuration-space conservation equation for the pair density P2 provides a fundamental basis for calculating the nonequilibrium microstructure; however, it includes pairwise additive three-body couplings. The resulting forces depend on the three-particle distribution function, necessitating an additional equation to completely specify P2. In this paper nonequilibrium Percus–Yevick and hypernetted chain closures complete the formulation by relating these forces to the interparticle force and pair distribution function. A computational algorithm exploiting Fast Fourier Transforms solves the resulting integro-differential equations for weak perturbations from equilibrium, yielding the perturbed pair density as a function of the volume fraction φ and the interparticle potential. The advantage of a fundamental approach is that clearly defined approximations lead from the characteristics of the individual colloidal particles to the nonequilibrium structure and macroscopic properties. The calculation of all dynamic properties, both rheological stresses and diffusion coefficients, is accomplished with the same approximations. Detailed predictions of the structure provide an additional comparison with simulation and experiment lacking in theories that calculate only bulk properties. The numerical methods demonstrated here allow efficient solution of a class of models more sophisticated than previously attempted. To test the merits of nonequilibrium closures we present predictions of the low-shear viscosity and long-time self-diffusion coefficient as a function of volume fraction for various repulsive potentials without hydrodynamic interactions. Comparison with results available from computer simulations demonstrates that the closures capture the trends in the transport properties with volume fraction and interparticle potential and yield realistic predictions for the nonequilibrium structure. The hypernetted chain closure yields the best agreement with the available data for bulk properties at moderate volume fractions (φ<0.4), but significant quantitative deviations appear at φ⩾0.45.

Three-particle correlations in the optical properties of granular composites

The effective dielectric response of a system of spherical inclusions embedded in an otherwise homogeneous matrix has been calculated using, in the Maxwell Garnett formula, a renormalized polarizability instead of the bare one (Phys. Rev. B 38, 5371 (1988)). This renormalized polarizability was given as a functional of the two- and three-particle distribution functions and results were presented for a very crude approximation of the three-particle distribution function. Here we extend these results and analyze the effects of the three-particle correlations by using the best available form of this functional. We compare our results with other theoretical approaches and recent numerical simulations.

Expansion moments for the local field distribution that involve the three-particle distribution function

The second moment of the Lennard-Jones local field distribution in a hard-sphere fluid is evaluated using the PY3 three-particle distribution function. An approximation due to Lado that avoids the explicit calculation of the latter is shown to be accurate. Partial results are also given for certain cavity-hard-sphere radial distribution functions that occur in a closest particle expansion for the local field.

Three-particle correlations in a hard-sphere fluid

New theoretical results for the three-particle equilibrium probability distribution function g 3(123) for classical hard-sphere fluids are given on the basis of the Percus—Yevick approximation applied to be the inhomogeneous pair distribution function in the presence of a third particle, g 2( 12 3 ). In particular, comprehensive results are given for three spheres in rolling contact and partial results are given for two spheres in contact with a third particle in arbitrary position. When compared with available simulation data, our results are found to be accurate, and the inhomogeneous Percus—Yevick approach appears to be the first quantitatively successful one for the three-particle configurations considered here.

An improved kirkwood superposition approximation for three atoms in rolling contact

A surprisingly simple ansatz is given which greatly improves the Kirkwood superposition approximation for the three-particle distribution function when two atoms are in contact with a third. This geometry occurs in a number of theories of the liquid state, and is the one for which the Kirkwood approximation is most inaccuate. Results are given for hard sphere and for Lennard-Jones fluids.

Universal expansion of three-particle distribution function

In this paper, we consider classical systems with two-body interaction and obtain a universal (independent of the Hamiltonian) expansion of the three-particle distribution function with respect to the two-particle correlation functions. In particular, we establish that allowance for the first few orders of the expansion in the absence of longrange correlations leads to the Kirkwood approximation. We find the first corrections to the Kirkwood approximation both in the presence and the absence of long-range correlations. Our point of departure is the construction of a quasi-equilibriu

Molecular dynamics studies on molten alkali hydroxides. I. Static properties of molten LiOH

Molecular dynamics calculation for molten LiOH has been executed using a set of effective pair potentials obtained by a small modification of ab initio ones. Some experimental data on static and dynamic properties of the melt were successfully reproduced by the calculation. Anisotropic coordination structures of Li+ and OH− ions around an OH− ion have been elucidated from two-dimensional pair correlation functions, three-particle angular distribution functions, and four-particle distance distribution functions. The most probable location of a Li+ ion is on the O atom side of the OH− ion, being somewhat different from the pair potential minimum position. The correlation is strikingly large irrespective of the libration of the OH− ion. The OH− ion was found to coordinate to the neighboring OH− ion in two ways. The first one consists of a direct contact between ions on their H atom sides in spite of the large pair repulsion. The second is an indirect one on the O atom side through intervening Li+ ions which have strong Coulombic attraction to the OH− ion. This feature has a good correspondence to that of the LiOH crystal. These anisotropic structures result in an approximately isotropic coordination of the O atom around the OH− ion and give rise to a gOO(r) with a considerably sharp and large first peak and long range oscillatory correlations. This is the origin of the sharp and high first peak of the neutron structure function characteristic of the LiOH melt. It was also found that, on the average, OH− ions around a Li+ ion form a regular tetahedron, whereas the Li+ ions are located around the apexes of a square which is slightly folded downward along a diagonal just under the O atom of the OH− ion. These short range framework structures of the liquid are nearly the same with those of the crystal except that the four Li+ ions just under the OH− ion form only one plane in the crystal. Space–time correlation functions indicate that the above structures dissipate substantially in about 1 ps.

Effective dielectric response of polydispersed composites.

We generalize the concept of renormalized polarizability introduced in a recently developed effective-medium theory to the case of polydispersed composites. We deal with a homogeneous and isotropic ensemble of spherical inclusions with a continuous distribution of sizes. The theory accounts for the correlated fluctuations among the induced dipole moments and the induced internal field within the long-wavelength dipolar approximation. The statistical properties of the ensemble enter in the final result through the two- and three-particle distribution functions. Numerical results are presented for a system of silver spheres in gelatin with a log-normal distribution of radii. The roles played by the dipolar fluctuations and by the size-dependent relaxation time, due to the surface scattering of the electrons, are analyzed and compared.

An “exact” integral equation approach to the inverse problem in two-dimensional fluids

It is shown that the Born-Green-Yvon integral equation can be used to extract interparticle pair potentials from two- and three-particle distribution functions in certain two-dimensional fluids. The approach was tested against Monte Carlo data from two pairs of closely related inverse-power-law potentials and found satisfactory over a density regime from infinite dilution to near crystal packing.

An improved description of three-particle correlations in liquids and a modified Born-Green equation

The application of the superposition approximation for the three-particle distribution function in a liquid, within the context of the Born-Green equation, is re-examined. It is shown how to correct this approximation for configurations in which one particle is well separated from the other two, and then demonstrated explicitly for the dipole-induced dipole pair potential that this correction is consistent with the known asymptotic expression for the radial distribution function. A modified form of the Born-Green equation is proposed which contains this improved description of triplet correlations.