Fluid molecule oscillations and the continuous spectrum near the rayleigh line

Of late there have been considerable developments in the theoretical study of the contour of spectral lines and the Rayleigh line by general statistical methods in the theory of random processes. On the basis of the general equations describing the rotational motion of molecules [1,2], and also of the equation which describes the change in the projection of the dipole moment onto the laboratory coordinate axes [3], we seek a correlation function whose Fourier transform leads finally to the required spectral distribution. In the present article we solve the problem of the spectral distribution by a direct analysis of the change in the projection onto the laboratory coordinate axes of the dipole moment induced in a molecule by an incident light wave. We consider a specific model for the rotational motion of fluid molecules.     

Electrostatic solution and rayleigh approximation for small nonspherical particles in a spheroidal basis

The electrostatic problem for the case of axially symmetric particles is analyzed in a spheroidal basis. In this case, the wavenumber is zero and Maxwell’s equations are reduced to the Laplace equation for scalar potentials. An alternative approach involves solving integral equations that are similar to those obtained within the framework of the extended boundary conditions method. The scalar potentials are represented as expansions in terms of eigenfunctions of the Laplace equation in a spheroidal frame of reference, and unknown expansion coefficients are determined from an infinite set of linear algebraic equations (the separation of variables method). These two approaches yield exact solutions of the problem in the case of axially symmetric particles, which coincide with known solutions in particular cases. Investigation of infinite systems allowed finding the boundaries where these algorithms are valid. Numerical calculations showed that, for spheroidal Chebyshev particles (i.e., perturbed spheroids), the Rayleigh approximation based on the electrostatic solution is applicable in a wide range of the problem parameters and is in fair agreement with the results obtained using the discrete dipole approximation.     

Axisymmetric wave propagation of thermoelastic transversely isotropic half-space under buried loading using potential functions

By virtue of a new scalar potential function and Hankel integral transforms, the wave propagation analysis of a thermoelastic transversely isotropic half-space is presented under buried loading and heat flux. The governing equations of the problem are the differential equations of motion and the energy equation of the coupled thermoelasticity theory. Using a scalar potential function, these coupled equations have been uncoupled and a six-order partial differential equation governing the potential function is received. The displacements, temperature, and stress components are obtained in terms of this potential function in cylindrical coordinate system. Applying the Hankel integral transform to suppress the radial variable, the governing equation for potential function is reduced to a six-order ordinary differential equation with respect to z. Solving that equation, the potential function and therefore displacements, temperature, and stresses are derived in the Hankel transformed domain for two regions. Using inversion of Hankel transform, these functions can be obtained in the real domain. The integrals of inversion Hankel transform are calculated numerically via Mathematica software. Our numerical results for displacement and temperature are calculated for surface excitations and compared with the results reported in the literature and a very good agreement is achieved.     

Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity

Stress-strain relation in Eringen's nonlocal elasticity theory was originally formulated within the framework of an integral model. Due to difficulty of working with that integral model, the differential model of nonlocal constitutive equation is widely used for nanostructures. However, paradoxical results may be obtained by the differential model for some boundary and loading conditions. Presented in this article is a finite element analysis of Timoshenko nano-beams based on the integral model of nonlocal continuum theory without employing any simplification in the model. The entire procedure of deriving equations of motion is carried out in the matrix form of representation, and hence, they can be easily used in the finite element analysis. For comparison purpose, the differential counterparts of equations are also derived. To study the outcome of analysis based on the integral and differential models, some case studies are presented in which the influences of boundary conditions, nonlocal length scale parameter and loading factor are analyzed. It is concluded that, in contrast to the differential model, there is no paradox in the numerical results of developed integral model of nonlocal continuum theory for different situations of problem characteristics. So, resolving the mentioned paradoxes by means of a purely numerical approach based on the original integral form of nonlocal elasticity theory is the major contribution of present study.     

Initialized fractional differential equations with Riemann-Liouville fractional-order derivative

The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.     

Functional integral equation for the complete effective action in quantum field theory

Based on a methodological analysis of the effective action approach, certain conceptual foundations of quantum field theory are reconsidered to establish a quest for an equation for the effective action. Relying on the functional integral formulation of Lagrangian quantum field theory, we propose a functional integral equation for the complete effective action which can be understood as a certain fixed-point condition. This is motivated by a critical attitude toward the distinction, artificial from an experimental point of view, between classical and effective action. While for free field theories nothing new is accomplished, for interacting theories the concept differs from the established paradigm. The analysis of this new concept concentrates on gauge field theories, treating QED as the prototype model. An approximative approach to the functional integral equation for the complete effective action of QED is exploited to obtain certain nonperturbative information about the quadratic kernels of the action. As a particular application the approximate calculation of the QED coupling constant α is explicitly studied. It is understood as one of the characteristics of a fixed point given as a solution of the functional integral equation proposed. Finally, within the present approach the vacuum energy problem is considered, as are possible implications for the concept of induced gravity.     

Formulation of the theory of turbulence in an incompressible fluid

The theory of turbulence in an incompressible fluid is formulated using methods similar to those of quantum field theory. A systematic perturbation theory is set up, and the terms in the perturbation series are shown to be in one to one correspondence with certain diagrams analogous to Feynman diagrams. From a study of the diagrams it is shown that the perturbation series can be rearranged and partially summed in such a way as to reduce the problem to the solution of three simultaneous integral equations for three functions, one of which is the second order velocity correlation function. The equations have the form of infinite power series integral equations, and the first few terms in the power series are derived from an analysis of the diagrams to sixth order. Truncation of the integral equations at the lowest nontrivial order yields Chandrasekhar's equation, and truncation at a higher order yields the equations discussed by Kraichnan.     

Transport Equations of Plasma in a Curvilinear Magnetic Field

The problem of transport equations of a collisional plasma in a curvilinear magnetic field is studied. Two main approaches to this problem are presented: that based on using the Boltzmann kinetic equation and the drift kinetic equation approach. In the frame of the first approach a multimoment transport equation set is found which is more general than the transport equation sets of Braginskii and Grad. The tensor equations of this set are described in an arbitrary curvilinear coordinate system. This allow to use these equations in problems of a plasma confined in toroidal magnetic configurations. Simplification of the multimoment transport equation set in the case of high magnetic field is performed. In the frame of the drift kinetic equation approach, a generalization of the drift transport equations derived earlier by the authors (Zh. Eksp. Teor. Fiz. 83 (1982) 139) is given.     

Thermal and reactive nonequilibrium of magnetoactive plasma flows

Based on multifluid equations derived from the Boltzman equation under inclusion of ionization and recombination reactions, the following convective processes in magnetoactive plasma flows transverse to homogeneous magnetic fields are investigated:The build-up of the nonuniform plasma state, the intercomponent thermal nonequilibrium and the reactive nonequilibrium in dependence of the flow coordinate. Due to the complexity of the problem, a numerical approach is used, parallel to which an approximate analytical theory is developed.     

Green's function for arbitrarily shaped dielectric bodies of revolution

In this paper, a solution is developed to calculate the electric field at one point in space due to an electric dipole exciting an arbitrarily shaped dielectric body of revolution (BOR). Specifically, the electric field is determined from the solution of coupled surface integral equations (SIE) for the induced surface electric and magnetic currents on the dielectric body excited by an elementary electric current dipole source. Both the interior and exterior fields to the dielectric BOR may be accurately evaluated via this approach. For a highly lossy dielectric body, the numerical Green's function is also obtainable from an approximate integral equation (AIE) based on a surface boundary condition. If this equation is solved by the method of moments, significant numerical efficiency over SIE is realized. Numerical results obtained by both SIE and AIE approaches agree with the exact solution for the special case of a dielectric sphere. With this numerical Green's function, the complicated radiation and scattering problems in the presence of an arbitrarily shaped dielectric BOR are readily solvable by the method of moments.     

On the state space of the dipole ghost

A particular representation of SO(4, 2) is identified with the state space of the free dipole ghost. This representation is then given an explicit realization as the solution space of a 4th-order wave equation on a spacetime locally isomorphic to Minkowski space. A discrete basis for this solution space is given, as well as an explicit expression for its SO(4, 2) invariant inner product. The connection between the modes of dipole field and those of the massless scalar field is clarified, and a recent conjecture concerning the restriction of the dipole representation to the Poincaré subgroup is confirmed. A particular coordinate transformation then reveals the theory of the dipole ghost in Minkowski space. Finally, it is shown that the solution space of the dipole equation is not unitarizable in a Poincaré invariant manner.     

Post-Newtonian approximation for a rotating frame of reference

The field equations of general relativity are solved to post-Newtonian order for a rotating frame of reference. A new method of approximation is used based on a 3+1 decomposition of the equations. The results are expressed explicitly in terms of the gravitational potentials. The space-time is asymptotically flat but not locally flat. The space-time metric contains gravitational terms, inertial terms, and coupled gravitational-inertial terms. The inertial terms in the equation of motion are in agreement with terms obtained by other authors using kinematic methods. The metric and equation of motion reduce to those for an inertial frame of reference under a simple coordinate transformation. The total energy of a particle is given. For the restricted three-body problem this represents the relativistic extension of Jacobi's integral to post-Newtonian order.This article received an honorable mention from the Gravity Research Foundation for the year 1984—Ed.     

Diffuse waves in a random medium layer with rough boundaries

Abstract

The problem of scattering from a random medium layer with rough boundaries is formulated as an integral equation in which the random fluctuations are represented as a zero-mean random operator. The analysis for the diffuse fields is based on the ladder-approximated Bethe–Salpeter equation. An integral equation for the diffuse intensities thus derived displays the various multiple-scattering processes involved in our problem. Transport equations are also derived and several special cases are considered to illustrate the characteristics of the results and to compare them with those in the literature.     

Quasiconfigurations and the theory of effective interactions

Perturbation theory is reformulated. Schrödinger's equation is recast as a non linear integral equation which yields by Neumann expansion a linked cluster series for the degenerate, quasi degenerate or non degenerate problem. An effective interaction theory emerges that can be formulated in a biorthogonal basis leading to a non Hermitian secular problem. Hermiticity can be recovered in a clear and rigorous way. As the mathematical form of the theory is dictated by the request of physical clarity the latter is obtained naturally. When written in diagrammatic many body language, the integral equation produces a set of linked coupled equations for the degenerate case. The classic summations (Brueckner, Bethe-Faddeev and RPA) emerge naturally. Possible extensions of nuclear matter theory are suggested.     

Scattering of electromagnetic radiation by a two-layer nonconfocal spheroid that is small compared to the wavelength

We have considered the electrostatic problem for a two-layer nonconfocal spheroid. The approach is based on surface integral equations that are similar to equations in terms of the extended boundary condition method for wave problems. Electrostatic fields are related to scalar potentials, which are represented as expansions in terms of eigenfunctions of the Laplace equation in two spheroidal coordinate systems, while unknown expansion coefficients are determined from infinite systems of linear algebraic equations. The constructed rigorous solution to the problem coincides with the known solution in a particular case of a confocal two-layer spheroid. In addition, for the nonconfocal two-layer spheroid, we have constructed an explicit approximated solution assuming that the field in the particle core is constant. This solution coincides with the rigorous solution if the scatterer shells are confocal. The formula found for the polarizability of the two-layer nonconfocal spheroid has a very simple form compared to the previously proposed cumbersome algorithm (B. Posselt et al., Measur. Sci. Technol. 13, 256 (2002)) and is more efficient numerically.     

Collisional Relaxation of Luminescence Anisotropy in Rarefied Gases in the Condensed Phase

The orientational relaxation of optically induced anisotropy in rarefied gases and at a damped rotation has been investigated. It has been found that the anisotropy relaxation in rarefied gases is described by a reduced kinetic equation depending only on free rotation integrals. The behavior of the integral anisotropy of luminescence for free symmetric and asymmetric top molecules has been elucidated. The law of luminescence depolarization has been obtained for asymmetric top molecules in the Gordon J-diffusion model. It represents the sum of two Stern–Volmer-type dependences, whose relative contribution is determined by the orientation of the dipole moments of transitions with absorption and emission of light in the molecular coordinate system and by the principal moments of inertia of the molecular top. It has been established that in the limit of a strongly damped rotation, kinetic equations of the general form reduce to equations of rotational diffusion. A number of modified diffusion equations correctly describing the contribution of inertial effects to the orientational relaxation of anisotropy have been obtained.     

Diffuse waves in a random medium layer with rough boundaries

The problem of scattering from a random medium layer with rough boundaries is formulated as an integral equation in which the random fluctuations are represented as a zero-mean random operator. The analysis for the diffuse fields is based on the ladder-approximated Bethe-Salpeter equation. An integral equation for the diffuse intensities thus derived displays the various multiple-scattering processes involved in our problem. Transport equations are also derived and several special cases are considered to illustrate the characteristics of the results and to compare them with those in the literature.     

Bethe-Salpeter Equation with Self-Energy Graphs I-Generalized Fredholm Theory,Parameter Imbedding Method

In this paper the classical Fredholm theory is generalized. The conceptions of the generalized fredholm denominator (GFD) and generalized Fredholm numerator (GFN) are defined. A set of parameter imbedding equations for GFD and GFN is deduced. In this way, the eigenvalue problem of the BS equation in ladder approximation with self-energy graphs, and the eigenvalue problem of nonlinear parameter integral equation, are carried over into an initial-value problem of a set of ordinary differential equations.     

Generator coordinate amplitude for scattering

In the generator coordinate method for scattering the proper boundary condition is accomplished by requiring the GC amplitude to satisfy an integral equation of the first kind. Attempts to solve this problem are first reviewed and then an improved approximation is proposed which is applicable to a wider class of scattering problems in addition to the Coulomb scattering.A better approximation is obtained in the asymptotic region, where the generator coordinate, i.e., the distance between two shell-model wells of the fragments, is larger than the touching distance of the colliding nuclei, by deriving partial differential equations of first order for the terms of an asymptotic series in $\text{1}\text{E}$, where E is the scattering energy.Extracting the information on the GC amplitude for small values of the generator parameter from the integral equation of the first kind is an ill-posed problem. It is shown that the method of statistical regularization offers a powerful and controllable procedure to uncover the GC amplitude. The unknown GC amplitude is treated as a random function with an a priori distribution of probability which is based on the assumption that the amplitude is bounded and that the errors in the input are random with zero expectation value. A useful procedure is found for fixing parameters of the a priori distribution. The solution for small values of the GC parameter is expressed in the form of a Dini series.The method is applied to the calculation of the GC amplitude for scattering of two α-particles at 15 MeV c.m. energy. The measure of the accuracy is the difference between the input wave function of relative motion and the result of folding of the GC amplitude with the kernel of the integral equation. The prescribed accuracy is reached with this method on a much larger interval than with any previously proposed method.     

Light scattering on nanowire antennas: A semi-analytical approach

Two semi-analytical approaches to solve the problem of light scattering on nanowire antennas are developed and compared. The derivation is based on the exact solution of the plane wave scattering problem in case of an infinite cylinder. The original three-dimensional problem is reduced in two alternative ways to a simple one-dimensional integral equation, which can be solved numerically by a method of moments approach. Scattering cross sections of gold nanowire antennas with different lengths and aspect ratios are analyzed for the optical and near-infrared spectral range. Comparison of the proposed semi-analytical methods with the numerically rigorous discrete dipole approximation method demonstrates good agreement as well as superior numerical performance.