Relative zeta-determinants of Dirac Laplacians on a half-infinite cylinder with boundary conditions in the smooth,self-adjoint Grassmannian

In this paper we compute the relative zeta-determinants of Dirac Laplacians on a half-infinite cylinder when the Dirichlet boundary condition and the boundary condition belonging to the smooth, self-adjoint Grassmannian are imposed on the bottom of the half-infinite cylinder. We next apply this result to compute the relative zeta-determinants of Dirac Laplacians on a manifold with cylindrical end when the boundary condition belonging to the smooth, self-adjoint Grassmannian is imposed on the bottom of the half-infinite cylinder.     

Effect of the character of conduction electron reflection on the electromagnetic properties of an inhomogeneous cylindrical particle

The cross section of magnetic absorption of a small elongated cylindrical particle with a dielectric core and metallic shell is calculated. The general case of an arbitrary value of the ratio of the dielectric core radius to the radius of the particle is considered. The condition of mixed (mirror-diffuse) reflection of conduction electrons from the boundaries of the metal layer of the particle is chosen as the boundary condition to the problem. The limiting cases are considered, and the results obtained are discussed.     

Boundary condition model for diffractive elastic scattering

Diffractive elastic scattering is studied by a boundary condition model that does not utilize a potential model.S-matrix elements are calculated and compared to those found from an optical potential model calculation. The unitarity of the model is related to the boundary condition imposed. A radially ingoing boundary condition is imposed on the wave function at one angle only, at a scattering angle of 180 °. This condition is required to hold in the vicinity of the nuclear radius, but not for all radii. Elastic scattering peaks at forward and backward angles are reproduced and discussed. The model is applied to composite particle scattering above the Coulomb barrier.     

Kinetic theory of hydrodynamic flows. I. The generalized normal solution method and its application to the drag on a sphere

We consider the flow of a dilute gas around a macroscopic heavy object. The state of the gas is described by an extended Boltzmann equation where the interactions between the gas molecules and the object are taken into account in computing the rate of change of the distribution function of the gas. We then show that the extended Boltzmann is equivalent to the usual Boltzmann equation, supplemented by boundary conditions imposed on the distribution function at the surface of the object. The remainder of the paper is devoted to a study of the solution of the extended Boltzmann equation in the case that the mean free path of a gas molecule is small compared to some characteristic dimension of the macroscopic object. We show that the Chapman-Enskog normal solution of the ordinary Boltzmann equation is not in general a solution of the extended equation near the surface of the object and must be supplemented by a boundary layer term. We then introduce a projection operator method which allows us to decompose the solution of the extended equation into a normal solution part and a boundary layer part when the gas flow is sufficiently slow. As a specific example of the method we consider the flow around a sphere, and derive the Stokes-Boussinesq form for the frequency-dependent force on the sphere for arbitrary slip coefficient. This derivation is the first one that starts from the Boltzmann equation for a general dilute gas and incorporates the effect of the boundary layer on the drag force.Work supported by the National Science Foundation.     

The effect of the character of electron reflection on the electromagnetic properties of an inhomogeneous spherical particle

The magnetic absorption cross section of a small spherical particle with a dielectric core and a metallic shell is calculated. The general case is considered when the ratio of the radius of the dielectric core to the total radius of the particle may take arbitrary values. The condition of specular-diffuse reflection of conduction electrons from the surfaces of the metal layer of the particle is chosen as the boundary conditions of the problem. The limit cases are considered, and the results are discussed.     

Interaction of electromagnetic radiation with a cylindrical particle of finite length

The magnetic absorption cross section of a cylindrical metal particle of finite length is calculated. A general case is considered when the ratio of the transversal dimension of a particle to its length may take arbitrary values. Diffuse reflection of electrons from the internal surface of a particle is chosen as the boundary condition for the problem. Limiting cases are considered, and the results are discussed.     

A three dimensional ferromagnetic Ising ‘fluid’ model

The three dimensional ferromagnetic spin-half Ising model with an arbitrary external magnetic field is considered in the spatial continuum limit and under a certain tempering condition to be imposed on the pair-wise spin-spin interaction. An expression for the partition function has been obtained for a tempered RKKY type interaction. The solution predicts the classical mean-field behaviour above a critical temperature below which the spontaneous magnetization jumps discontinuously from zero to the saturation value.     

Integral form of the time-dependent radiation transfer equation—III. Moving boundaries

The theory of characteristics is used to give the general time-dependent integral form of the transfer equation in a time-dependent, inhomogeneous medium, submitted to arbitrary boundary and initial conditions. The medium emits and absorbs radiations, but scattering is neglected. The boundary condition is applied to a moving surface. The general solution is given analytically in Cartesian and spherical coordinates for a 1-D configuration.     

The effect of crystal equilibrium condition on the lattice dynamical model parameters of a body centred cubic metal

Using barium, for which results that compare favourably with experiment have been obtained, the effect on the lattice dynamical model parameters of a body centred cubic metal of explicitly imposing the crystal equilibrium condition on various ion-ion coupling schemes when used with the modified electron-ion interaction scheme of Bhatia has been studied. When the equilibrium condition is not used, the calculated model parameters lose their physical significance with the result that at equilibrium, the sum of the ionic and electronic pressures is not zero. This inconsistency is removed when the equilibrium condition is explicitly imposed on the models. It is not meaningful to compare the two results with and without the equilibrium condition imposed, not only because the imposition of the equilibrium condition reduces the number of independent parameters of the models by one, but also because the results apply to different states of the crystal. Also the comparison of calculated model parameters and measured quantities such as phonon frequencies make sense only when both are determined for the same state of the crystal. This condition is satisfied only when the equilibrium condition, which has unfortunately been ignored in most lattice dynamical studies, is explicitly imposed on lattice dynamical models.     

On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain

We consider a mixed boundary value problem for a second-order strongly elliptic equation in a Lipschitz domain. The boundary condition on a part of the boundary is of the first order and contains a weight function and the spectral parameter, while on the remaining part the homogeneous Dirichlet condition is imposed. The aim is to weaken the conditions sufficient for justifying the classical asymptotic formula for the eigenvalues. We show that it suffices to assume the boundary to be C ¹ in a neighborhood of the support of the weight outside a closed subset of zero measure. The work of the author is supported by the RFBR grant no. 04-01-00914.     

The Bohm criterion and boundary conditions for a multicomponentsystem

In the asymptotic limit λ_D→O (small Debye length) the boundary layer problem of a plasma is split up into the separate problems of a: quasineutral presheath and a collision free planar sheath. The formation of a stationary sheath requires that the ions fulfil a sheath condition (“Bohm criterion”). In this paper, the boundary layer problem of a multicomponent system is investigated. Both in the frames of a hydrodynamic and of a kinetic theory, a generalized sheath condition is derived accounting for various ion components and arbitrary electron- and ion distributions. It is shown that the sheath condition is usually fulfilled marginally (in the equality form) and provides a boundary condition for the plasma analysis     

The Effect of Interface Potential Barriers on Boundary Conditions in the Theory of Dirty Inhomogeneous Superconducting Systems Near Transition Temperature

A theoretical treatment of the superconductor-normal metal contact with interface potential barrier is presented in the dirty limiting case. Our result for the kernel of the linearized self-consistency condition allows near transition temperature the determination of boundary conditions and the behaviour of Usadel's function at interfaces. This represents a microscopic derivation of the so-called “extended” de Gennes boundary condition. With the help of a variational procedure an expression is derived for the Ginsburg-Landau extrapolation length of dirty SN systems with interface potential barrier.     

Absorption of electromagnetic radiation by an inhomogeneous spherical particle

The cross section of absorption of electromagnetic radiation by an inhomogeneous spherical particle is calculated. The general case of an arbitrary value of the ratio of the particle radius to the radius of the dielectric core is considered. The condition of diffuse reflection of electrons from the inner and outer surfaces of the metal layer of the particle is taken as the boundary condition of the problem. The limiting cases are considered and the results obtained are discussed.     

On Implementation of Boundary Conditions in the Application of Finite Volume Lattice Boltzmann Method

A new implementation of boundary condition based on the half-covolume and bounce-back rule for the non-equilibrium distribution function for the finite volume LBM is proposed here. The numerical simulation results for the expansion channel flow and driven cavity problem indicate that this method is workable for arbitrary meshes. In addition, the fourth order Runge–Kutta scheme is found to be a practical way in the LBM to accelerate the calculation speed.     

Influence of the Prandtl and Knudsen numbers on heat-transfer process in the problem of planar Poiseuille flow

The analytic solution (in the form of the Neumann series) has been derived for the problem of computing the heat flux in a planar channel in the presence of a pressure gradient parallel with the walls (in the problem of planar Poiseuille flow) within the framework of the kinetic approach for arbitrary values of the Prandtl number. The ellipsoidal-statistical model of the Boltzmann kinetic equation is used as the governing equation, and the model of diffuse reflection is used as the boundary condition. The conducted numerical analysis of final expressions obtained in the present work showed a substantial dependence of the heat flux on the value of the Prandtl number of gas for channels whose thickness is comparable with the mean free path of gas molecules.     

An exact solution to the problem of current transport through the grain boundary in a metal

The problem of electrical resistance of a planar boundary between two crystalline grains of a polycrystalline metal is solved analytically. The solution is derived using the Case method. The electric field and the electron distribution function are represented in the form of an eigenfunction expansion of the corresponding characteristic equation. It is demonstrated that the electric field at the interface increases drastically when the Debye frequency substantially exceeds the frequency of electron collisions in the bulk of the metal. It is found that the resistance of the grain boundary does not depend on the frequency of electron collisions in the bulk, i.e., on their mean free path.     

Non-Gibbsian stochastic light-mode dynamics of passive mode locking

We study a stochastic light-mode system with non-Gibbsian steady state statistics, unravelling global nonequilibrium phase transition properties. It relates to the onset of passive mode-locking in the general case of lasers with arbitrary dispersion and Kerr nonlinearity that includes the nonsolitonic regime. The solution is facilitated by a special stationarity criterion imposed by the system gain balance. We show that the mode-locking phase transition is generic, and give exact expressions for the pulse power and its stability map. We find that at the boundary of the mode-locking stability the pulse power is exactly one half of the total intracavity power, and that the parameter region for the most resistant pulses against noise destabilization is not at the soliton condition.     

Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties

This study is devoted to the derivation of some properties of the von Kármán equations for geometrically nonlinear models of plates, with a boundary of arbitrary shape, for applications to nonlinear vibration and buckling. An intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an Airy stress function as unknowns is provided. Classical homogeneous boundary conditions—with vanishing prescribed forces and displacements—are derived in terms of the Airy stress function in the case of a boundary of arbitrary geometry. A special property of this operator, crucial for some energy-conserving numerical schemes and called “triple self-adjointness”, is derived in the case of an edge of arbitrary shape. It is shown that this property takes a simple form for some classical boundary conditions, so that the calculations in some practical cases are also simplified. The applications of this work are either semi-analytical methods of solution, using an expansion of the solution onto an eigenmode basis of the associated linear problem, or special energy-conserving numerical methods.     

An Efficient Algorithm for Truncating Spatial Domain in Modeling Light Scattering by Finite-Difference Technique

The finite-difference time domain technique is one of the most robust and accurate numerical methods for the solution of light scattering by small particles with arbitrary composition and geometry. In practice, this method requires that the spatial domain for the computation of near-field be truncated. An absorbing boundary condition must be imposed in conjunction with this truncation. The performance of this boundary condition is essential to the stability of numerical computations and the reliability of results. In the present study, a new boundary condition, referred to as the mixed T algorithm, has been developed, which is a generalization of the transmitting boundary condition originally developed by Liao and co-workers. The present algorithm does not require spatial interpolation for wave values at interior grid points. In addition, it produces two minima of spurious reflections at small and large incident angles, allowing efficient absorption of the scattered waves at the boundary for large incident angles. When the third-order mixed T algorithm is used, the reflection coefficient of the boundary is less than 1% for incident angles from 0° to about 70°. We find that the numerical instability associated with the transmitting boundary condition is caused by the location-dependent amplitude of outgoing waves in the vicinity of the boundary. For this reason, the mixed T algorithm is stabilized by consistently introducing diffusive coefficients into the boundary equation. When the stabilized algorithm is applied, the near-field within the truncated domain can be computed by using single-precision arithmetic without overflows for more than 10⁵steps in the time-marching iteration. Finally, the new absorbing boundary condition is validated by carrying out numerical experiments involving the propagation of a TM wave excited by a sinusoidal point source, simultaneous simulation of the wave propagation in small and large domains, and the scattering of a TM wave by an infinite circular cylinder.     

An eigenvalue-based approach for the design of reflector systems and prisms with a specified image orientation change

The image orientation change (IOC) of an object following its reflection by a system comprising an arbitrary number of flat boundary surfaces can be described using a merit function (Γ) expressed in the form of a 3×3 matrix. The present study proposes a design methodology for stable-IOC reflector and prism systems in which the merit function is solved using an eigenvalue-based approach. It is shown that a reflector system remains IOC-stable following its rotation about the eigenvector of the IOC merit function, provided that the image can still physically enter the system’s aperture. Furthermore, it is shown that an IOC-stable prism can be obtained by adding two refracting flat boundary surfaces at the entrance and exit positions of the light ray in an optical system comprising multiple reflectors provided that the condition n _n =Γn ₁ is maintained. Illustrative examples are provided to demonstrate the validity of the proposed design approach.