The Interaction Energy of Well-Separated Skyrme Solitons

We prove that the asymptotic field of a Skyrme soliton of any degree has a non-trivial multipole expansion. It follows that every Skyrme soliton has a well-defined leading multipole moment. We derive an expression for the linear interaction energy of well-separated Skyrme solitons in terms of their leading multipole moments. This expression can always be made negative by suitable rotations of one of the Skyrme solitons in space and iso-space. We show that the linear interaction energy dominates for large separation if the orders of the Skyrme solitons multipole moments differ by at most two. In that case there are therefore always attractive forces between the Skyrme solitons.     

Dielectric constant of polarizable,nonpolar fluids and suspensions

We study the dielectric constant of a polarizable, nonpolar fluid or suspension of spherical particles by use of a renormalized cluster expansion. The particles may have induced multipole moments of any order. We show that the Clausius-Mossotti formula results from a virtual overlap contribution. The corrections to the Clausius-Mossotti formula are expressed with the aid of a cluster expansion. The integrands of the cluster integrals are expressed in terms of two-body nodal connectors which incorporate all reflections between a pair of particles. We study the two- and three-body cluster integrals in some detail and show how these are related to the dielectric virial expansion and to the first term of the Kirkwood-Yvon expansion.     

Elastic charge density representation of the interaction via the nematic director field

The interaction between particle-like sources of the nematic director distortions (e.g., colloids, point defects, macromolecules in nematic emulsions) allows for a useful analogy with the electrostatic multipole interaction between charged bodies. In this paper we develop this analogy to the level corresponding to the charge density and consider the general status of the pairwise approach to the nematic emulsions with finite-size colloids. It is shown that the elastic analog of the surface electric charge density is represented by the two transverse director components on the surface imposing the director distortions. The elastic multipoles of a particle are expressed as integrals over the charge density distribution on this surface. Because of the difference between the scalar electrostatics and vector nematostatics, the number of elastic multipoles of each order is doubled compared to that in the electrostatics: there are two elastic charges, two vectors of dipole moments, two quadrupolar tensors, and so on. The two-component elastic charge is expressed via the vector of external mechanical torque applied on the particle. As a result, the elastic Coulomb-like coupling between two particles is found to be proportional to the scalar product of the two external torques and does not directly depend on the particles' form and anchoring. The real-space Green function method is used to develop the pairwise approach to nematic emulsions and determine its form and restrictions. The pairwise potentials are obtained in the familiar form, but, in contrast to the electrostatics, they describe the interaction between pairs (dyads) of the elastic multipole moments. The multipole moments are shown to be uniquely determined by the single-particle director field, unperturbed by other particles. The pairwise approximation is applicable only in the leading order in the small ratio particle size-to-interparticle distance as the next order contains irreducible three-body terms.     

Generic flow profiles induced by a beating cilium

We describe a multipole expansion for the low-Reynolds-number fluid flows generated by a localized source embedded in a plane with a no-slip boundary condition. It contains 3 independent terms that fall quadratically with the distance and 6 terms that fall with the third power. Within this framework we discuss the flows induced by a beating cilium described in different ways: a small particle circling on an elliptical trajectory, a thin rod and a general ciliary beating pattern. We identify the flow modes present based on the symmetry properties of the ciliary beat.     

Microscopic quadrupole and hexadecapole moments of rare earth nuclei

The theoretical calculations of multipole moments of even-even rare earth nuclei are presented. The potential energy surface is evaluated by the shell correction method. The condition ensuring the equality of the density distribution of the macroscopic liquid droplet part of the potential energy and the density generated by the single particle potential is added. A single particle Nilsson potential is used. New, less stiff potential surfaces versusε ₄ are obtained while the multipole moments calculated at the equilibrium deformations agree well with experimental data.     

Rheology of polymer suspensions: I. Local flow effects

A systematic study is made of the average local velocity field acting at a selected particle in a fluid suspension. The flow disturbance due to a single particle is analyzed in terms of force multipoles. The theory is developed in close analogy to that for the corresponding problem of the local electric field at a molecule in a polarizable medium. Closed expressions are derived in continuum approximation for the average local velocity, vorticity and strain in terms of the macroscopic average velocity field and force multipole densities. The effect of correlations is discussed briefly.     

Permanent and interaction-induced far-infrared spectra of CO in dense Ar: a molecular dynamics approach

Permanent and electric multipole induced contributions to the far-infrared absorption spectrum of CO in Ar for dense gas and liquid densities have been calculated by means of classical molecular dynamics simulations. The comparison of the simulated spectra with experiments let us to give an estimation of some multipole moments of CO: quadrupole, octupole and hexadecapole. Although the experimental profiles lack of a fine rotational structure, the results of the simulations and their comparison with previous theoretical works, indicate that at low temperatures, the consideration of a quantum time correlation function for the dipole moment is necessary to get a good agreement with experiments. Finally, it is shown that the permanent-induced and induced-induced cross terms in the absorption coefficient, that usually are difficult to calculate from a theoretical viewpoint, are not very relevant for this system.     

The definition of multipole moments for extended bodies

There is considerable freedom in the definition of multipole moments of the energy-momentum tensor of an extended body in general relativity. By studying the corresponding Newtonian theory we obtain guidelines which enable us to choose the most suitable definitions in the relativistic theory. In this way we find two sets, the complete moments and the reduced moments, of which the latter are the most natural choice for studying the dynamics of extended bodies. Expressions as explicit integrals are give for both sets, and the multipole equations of motion of the body are given in a form exact to all orders. Proofs of the relativistic results will appear elsewhere. Presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

Approaches to the Monopole-Dynamic Dipole Vacuum Solution Concerning the Structure of Its Ernst Potential on the Symmetry Axis

The FHP (Fodor, Hoenselaers, Perjés) algorithm [1] allows to obtain the relativistic multipole moments of a vacuum stationary axisymmetric solution in terms of coefficients which appear in the expansion of its Ernst potential on the symmetry axis. First of all, we will use this result in order to determine, at a certain approximation degree, the Ernst potential on the symmetry axis of the metric whose only multipole moments are mass and angular momentum. By using Sibgatullin's method [2] we then analyse a series of exact solutions with the afore mentioned multipole characteristic; besides, we present an approximate solution whose Ernst potential is introduced as a power series of a dimensionless parameter. The calculation of its multipole moments allows us to understand the existing differences between both approximations to the proposed pure multipole solution.     

Force density induced on a sphere in linear hydrodynamics: I. Fixed sphere,stick boundary conditions

We evaluate the surface force density induced on a sphere placed in an arbitrary nonstationary flow field of a viscous incompressible fluid for stick boundary conditions. The calculation leads to a generalization of Faxén's theorem to force multipole moments of arbitrary order.     

A unified treatment of electromagnetic decays,electroproduction, and pair production in e+e− annihilation for hadrons of arbitrary spin

We gather together in a unified notation formulas for electromagnetic decay rates of resonances and cross sections for electroproduction and e⁺e^? annihilation for all those processes which can be expressed in terms of matrix elements of the electromagnetic current between single particle (or resonant) helicity states. We show in complete generality how to decompose such helicity matrix elements into form factors which are free of all kinematic singularities and constraints simultaneously at both physical thresholds, and relate them to the familiar multipole moments and to some others which have been used in data analysis. Tables are given of the form factor decomposition of the helicity matrix elements for many cases of actual and potential practical interest.     

Realizable high-order finite-volume schemes for quadrature-based moment methods

Dilute gas–particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag and particle–particle collisions. However, direct numerical solution of kinetic equations is often infeasible because of the large number of independent variables. An alternative is to reformulate the problem in terms of the moments of the velocity distribution. Recently, a quadrature-based moment method was derived for approximating solutions to kinetic equations. The success of the new method is based on a moment-inversion algorithm that is used to calculate non-negative weights and abscissas from the moments. The moment-inversion algorithm does not work if the moments are non-realizable, which might lead to negative weights. It has been recently shown [14] that realizability is guaranteed only with the 1st-order finite-volume scheme that has an inherent problem of excessive numerical diffusion. The use of high-order finite-volume schemes may lead to non-realizable moments. In the present work, realizability of the finite-volume schemes in both space and time is discussed for the 1st time. A generalized idea for developing realizable high-order finite-volume schemes for quadrature-based moment methods is presented. These finite-volume schemes give remarkable improvement in the solutions for a certain class of problems. It is also shown that the standard Runge–Kutta time-integration schemes do not guarantee realizability. However, realizability can be guaranteed if strong stability-preserving (SSP) Runge–Kutta schemes are used. Numerical results are presented on both Cartesian and triangular meshes.     

Astrophysically significant solutions of the Einstein and Einstein-Maxwell equations from the Laplace’s seed

The stationary solutions given by Amenedo and Manko generated from known solutions of Laplace’s equation as seed have been generalised to include the electromagnetic field. Further, the exterior solution of an axially symmetric rotating body with higher multipole moments and a solution corresponding to a Kerr object embedded in a gravitational field are given. We also give a method for constructing stationary vacuum solutions from static magnetovac solutions and vice versa and discuss a specific application of this method.     

Quantum Phase for an Electric Multipole Moment in Noncommutative Quantum Mechanics

We study the noncommutative nonrelativistic quantum dynamics of a neutral particle, which possesses an electric multipole moment, in the presence of an external magnetic field. First, by introducing a shift for the magnetic field we give the Schrödinger equations in the presence of an external magnetic field both on a noncommutative space and a noncommutative phase space, respectively. Then by solving the Schrödinger equations, we obtain quantum phases of the electric multipole moment both on a noncommutative space and a noncommutative phase space. We demonstrate that these phase are geometric and dispersive.     

On the virtual photon spectrum for electromagnetic dissociation of relattvistic nuclei in peripheral collisions

We give the virtual photon spectrum appropriate to electromagnetic reactions between relativistic nuclei and stationary massive targets, for all multipoles, in terms of the classical trajectories of the ions. Within the approximation that projectile and target charges do not overlap, the dissociation cross section can be written directly in terms of the various multipole photoabsorption cross sections. The dominant multipole is E1. M1 contributions are shown to be negligible, while E2 effects become significant for heavy projectiles and could be measured in coincidence experiments. The effects of the curvilinear trajectory are also small and can be included with a minor modification of the results for straight-line trajectories.     

Proof of a multipole conjecture due to Geroch

A result, first conjectured by Geroch, is proved to the extent, that the multipole moments of a static space-time characterize this space-time uniquely. As an offshoot of the proof one obtains an essentially coordinate-free algorithm for explicitly writing down a geometry in terms of it's moments in a purely algebraic manner. This algorithm seems suited for symbolic manipulation on a computer.     

Simulation study on microstructure formations in magnetic fluids

We propose the Langevin-type microscopic equations of motion for magnetic fluids. Magnetic fluids are modeled as an ensemble of interacting ferromagnetic nanoparticles suspended in a viscous fluid. The present model is described in terms of position vectors of nanoparticles and orientation vectors of their magnetic dipole moments. In this model, forces and torques arising from the magnetic origin and the surrounding fluid flow are included. Effects of non-spherical particle shape are also taken into account. From the Brownian dynamics simulations of the model, it is found that the present model exhibits various microstructure formation processes in magnetic fluids.     

Spherical magnetic multipole moments system of currents

Concepts of spherical magnetic multipoles that represent distributions of electric currents over a spherical surface are introduced. Vector potentials of magnetic multipoles meet solenoidal- and harmonic-field conditions outside of the spherical surface and are continuous on it. Within the sphere, the vector potential of currents flowing outside of it is represented by the sum of vector potentials of basis magnetic multipoles with coefficients expressed by spherical multipole moments of system of currents. This expansion of the vector potential is in many respects analogous to the multipole expansion known from electrostatics. The first three terms of the expansion represent components of the well-known magnetic moment, the next five terms represent components of the magnetic quadrupole moment, etc. Possible applications of the magnetic spherical multipole technique are discussed. Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 66–72, October, 1999.     

Multipole expansion of the retarded interatomic potential energy: Derivation from relativistic electron theory

The inductive and dispersive retarded interaction energies of two ground state hydrogen atoms described by Dirac theory are obtained up to all multipole orders. The long range terms are given as symmetric expression in the electric and magnetic dipole moments.