Reduction of Chandrasekhar's planetary problem in the case of a specularly reflecting boundary to the standard problem

The reduction of Chandrasekhar's planetary problem to the standard problem has been made by several authors (cf. Chandrasekhar⁽¹⁾; Sobolev;⁽²⁾Van de Hulst;⁽³⁾Coulsonet al.; ⁽⁴⁾Ueno; ⁽⁵⁾Kagiwada and Kalaba;⁽⁶⁾. However, the diffuse reflection and transmission problem in a finite anisotropically scattering atmosphere bounded by a specular reflector has not been reduced to the standard problem. In the present paper, starting with an integral equation governing the source function for Chandrasekhar's planetary problem in the case of a specular reflector, we show how to express the required scattering and transmission functions in terms of the Chandrasekhar's scattering and transmission functions. So far as we know, the reduction formulae are new. They will be useful for the numerical study of Chandrasekhar's planetary problem in the case of a specular reflector with the aid of high-speed digital computer.     

Radiation field of an optically finite homogeneous atmosphere with internal sources

The equation of radiative transfer in an optically finite homogeneous atmosphere with different internal sources is solved using the method of kernel approximation the essence of which is to approximate the kernel in the equation for the Sobolev resolvent function by a Gauss-Legendre sum. This approximation allows to solve the equation exactly for the resolvent function while the solution is a weighted sum of exponents. Since the resolvent function is closely connected with the Green function of the integral radiative transfer equation, the radiation field for different internal sources can be found by simple integration. In order to simplify the obtained formulas we have defined the x and y functions as the generalization of the well-known Ambarzumian-Chandrasekhar X and Y functions.For some types of internal sources the package of codes in Fortran-77 can be found at http://www.aai.ee/∼viik/HOMOGEN.FOR.     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

A study of partial frequency redistribution of monochromatic source radiation

The consequences of angle-dependent partial frequency redistribution are investigated with a Monte Carlo model for a monochromatic source of radiation incident upon a uniform isothermal slab of resonantly-scattering atoms. The frequency redistribution probability is characterized by Hummer's R_II function. Emergent line profiles are computed for optical depths of up to 10⁵ at line-center.     

Numerical results for finite order X and Y functions of radiative transfer

It is well known that, in the theory of radiative transfer, Chandrasekhar's X and Y functions play an important role in the diffuse reflection and transmission problem (cf. Chandrashekhar⁽¹⁾). In a preceding paper (cf. Bellmanet al.⁽¹⁰⁾), graphs and selected tables of these functions covering wide ranges of slab thickness and albedos for single scattering have been provided. In this paper, making use of a system of coupled integral recurrence relations for finite order X and Y functions (cf. Bellmanet al.⁽¹⁴⁾), numerical results for these basic functions are tabulated up to optical thickness τ = 2.0 from τ = 0.1, assuming the conservative case of isotropic scattering. The maximum order of these functions is taken to be fifteenth. It is shown that the accuracy obtained is satisfactory in the domain under consideration. Furthermore, numerical results for Chandrasekhar's approximation for X and Y functions are also tabulated for stabs of small optical thickness.     

Polarizabilities and susceptibilities of nematic azoxybenzenes

The mean polarizabilities of homologous series of nematic liquid crystals of p,p' -di-n-alkyl azoxybenzenes are evaluated by using a modified Lippincott-δ-function potential model.¹ From these polarizabilities, the diamagnetic susceptibilities of the liquid crystals are calculated by using the method of Rao and Murthy.² The merits and limitations of the method are discussed.     

Biorthogonality and radiative transfer in finite slab atmospheres

It is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation.The adjoints are obtained in terms of Case's eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green's functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green's functions and their adjoints. Linear and non-linear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.     

Evolution of Wigner's phase-space density under a nonintegrable quantum map

A nonintegrable area-preserving map for a system with one freedom is quantized, and the evolution of Wigner's function W(q,p) illustrated by contour plots of W in the paase plane. In the classical limit, propagation is governed by Liouville's equation and the contours of W rapidly develop an intricate structure of whorls and tendrils. When Planck's constant ? is not zero, the quantum map smooths out classical detail in phase-space areas smaller than ?. The quantum-mechanical distributions spread more slowly than their classical counterparts.     

A relativistic model of the composite π meson

The pion is treated as a fermion-antifermion composite state, described by the Euclidean Bethe-Salpeter equation. The kernel of the equation is a local potential having an exponentially infinite spectral function, connected with the empirical mass spectrum of resonances. For the simplest potential U(r) = f²(r²?a²)^?1 the equation for the massless pion is solved by using WKBJ method, and the parameters f, a and the fermion mass M are estimated.     

Generalized density expansion in the kinetic theory and the resistivity of liquid metals

For the system of electrons and immovable interacting centers an exact equation for averaged electron Green's function is formulated. The expansion of self-energy part over the one-particle t-matrices and explicit Green's functions is derived. It represents a kind of a generalized density series containing the correlation functions of the centres. In the low approximation over t-matrix, the transition probability (t)²S in the kinetic equation is obtained (S = the structure factor of centers).     

Problèmes mathématiques de l'équation de Boltzmann relativiste

This work deals with relativistic Boltzmann equation and more particulary with integral operator of complete equation and integral operator of linearized equation. These operators depend on the differential cross sectionh(〈p, q〉, cos θ) which is a fonction of energy 〈p, q〉 and of the deviation angle θ. The only hypothesis is thath is a symetric function of cosθ. The second part deals essentially with linearized equation in Special Relativity. We take for the distribution function: $$F\left( {x,p} \right) = a e^{ - \frac{{\lambda p}}{2}} \left( {e^{ - \frac{{\lambda p}}{2}} + \varepsilon f\left( {x,p} \right)} \right)$$ wherea is a constant, λ a constant vector and ? a small constant so that ?² can be neglected. We obtain the equation: $$\frac{{p^\alpha }}{{p^0 }}\frac{{\partial f}}{{\partial x^\alpha }} = - K\left( p \right) \cdot f + G\left( f \right)$$ whereK(p) is a positive function andG an Hilbert-Schmidt operator. Then we resolve the Cauchy's problem by taking the Fourier's transformation off, and in the last part by investigating properties of the resolvent of ?K+G we establish that asx ⁰→+∞ the solution of this problem has for limit the equilibrium distributiona e ^?λp .     

A statistical approach to quantum mechanics

A Monte Carlo method is used to evaluate the Euclidean version of Feynman's sum over particle histories. Following Feynman's treatment, individual paths are defined on a discrete (imaginary) time lattice with periodic boundary conditions. On each lattice site, a continuous position variable x_i specifies the spacial location of the particle. Using a modified Metropolis algorithm, the low-lying energy eigenvalues, |ψ₀(x)|², the propagator, and the effective potential for the anharmonic oscillator are computed, in good agreement with theory. For a deep double-well potential, instantons were found in our computer simulations appearing as multi-kink configurations on the lattice.     

A discussion of the relativistic equal-time equation

Ruan Tu-nan et al. [1] have proposed an equal-time equation for composite particles which is derived from Bethe-Salpeter (B-S) equation. Its advantage is that the kernel of this equation is a completely definite single rearrangement of the B-S irreducible kernel without any artificial assumptions. In this paper we shall give a further discussion of the properties of this equation. We discuss the behaviour of this equation as the mass of one of the two particles approaches the limitM ₂→∞ in the ladder approximation of single photon exchange. We show that up to orderO(α⁴) this equation is consistent with the Dirac equation. If the crossed two photon exchange diagrams are taken into account the difference between them is of orderO(α⁶).     

Anisotropic scattering in inhomogeneous media—1. A new representation formula for the solution of the auxiliary integral equation for the source function

A new representation formula for the solution of the auxiliary integral equation for the source function in inhomogeneous, anisotropically scattering media is presented. It involves two new functions Φ and ψ of two variables instead of the original five variables. This generalizes earlier results of Kagiwada et al. (1969) and Sobolev (1972) applicable to homogeneous atmospheres. The corresponding Bellman-Krein formula for the resolvent kernel is also derived. The present representation for the solution of Fredholm integral equations of second kind with unsymmetric kernels provides a new approach to radiative transfer in anisotropic inhomogeneous media.     

The large-N limit in statistical physics and the one-particle Schrödinger equation

Nonlinear d-dimensional vector σ models, such as O(N), SU(N), and CP ^N, are considered in the limit of an infinite number of components N. It is shown that the equation for the two-point correlation function in these models is similar to the Schrödinger equation for a quantum particle moving in a δ-function potential well (?T)δ(x), where T is the temperature. This equation adequately describes the systems under study both above and below the Curie point. Within this approach, the critical behavior of the SU(N)-invariant Ginzburg-Landau model in an external uniform magnetic field is determined in the vicinity of the upper critical magnetic field. The critical indices in this case are the same as in the spherical model in a random magnetic field. An exact equation describing the H _c2(T) curve of continuous phase transitions is derived, which allows one to determine the asymptotes of this curve in strong and weak fields. The relation between the one-particle Schrödinger equation and critical phenomena is analyzed, and applications of this method to various models in solid state physics and statistical mechanics are discussed.     

Non-Markovian diffusion equations and processes: Analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.     

Radiation field in a semi-infinite homogeneous atmosphere with internal sources

The equation of radiative transfer in a semi-infinite homogeneous atmosphere with different internal sources is solved by the method of kernel approximation—the kernel in the equation for the Sobolev resolvent function is approximated by a Gauss-Legendre sum. Then the obtained approximate equation can be solved exactly and the solution is a weighted sum of exponentials. All the necessary coefficients of the solutions may be easily found. Since the resolvent function is closely connected with the Green function of the integral radiative transfer equation, the radiation field for different internal sources can be found by simple integration. For the considered cases the formulas for the radiation field are obtained and the respective accuracy estimated. The package of codes in Fortran-77 is given at http://www.aai.ee/∼viik/homogen.for.     

Kinetic theory of diffusion in liquids: A hydrodynamic approximation

Diffusion in simple classical liquids is analyzed in terms of the test-particle phase-space density, with emphasis upon its long-time behavior. The Green's function of the generalized Fokker-Planck equation is used to define auxiliary quantities, in particular the transport mean path that enters solutions of the Chapman-Enskog type. Approximations for the lowest eigenvalues and eigenfunctions of the Fourier- and Laplace-transformed F.-P. operator σ_ks are constructed, and an expansion for the resolvent operator (s + ik · v ? σ_ks)^-1 proposed. With the additional assumption that branch-points on the negative real axis of s are the only singularities of the transformed F.-P. operator, a Laplace inversion is tentatively carried out, so that the general form of the solution is obtained. This is found to agree with the solution derived by hydrodynamic arguments. Only in a limited sense is the latter method equivalent to that of mode-mode coupling.     

Klein paradox in the Breit equation

We demonstrate that in the Breit equation with a central potentialV(r) having the propertyV(r ₀)=E there appears a Klein paradox atr=r ₀. This phenomenon, besides the previously found Klein paradox arr→∞ appearing ifV(r)→∞ atr→∞, seems to indicate that in the Breit equation valid in the singleparticle theory the sea of particle-antiparticle pairs is not well separated from the considered two-body configuration. We conjecture that both phenomena should be absent from the Salpeter equation which is consistent with the hole theory. We prove this conjecture in the limit ofm ⁽¹⁾→∞ andm ⁽²⁾→∞, where we neglect the terms ～1/m ⁽¹⁾ and 1/m ⁽²⁾. In Appendix I we show that in the Breit equation the oscillations accumulating atr=r ₀ in the case ofm ⁽¹⁾≠m ⁽²⁾ are normalizable to the Dirac δ-function. In Appendix II the analogical statement is justified for the nonoscillating singular behaviour appearing atr=r ₀ in the case ofm ⁽¹⁾=m ⁽²⁾.