Liouville and Fokker–Planck dynamics for classical plasmas and radiation

We consider a nonequilibrium statistical system formed by many classical non‐relativistic particles of opposite electric charges (plasma) and by the classical dynamical electromagnetic (EM) field. The charges interact with one another directly through instantaneous Coulomb potentials and with the dynamical degrees of freedom of the transverse EM field. The system may also be subject to external influences of: i) either static, but spatially inhomogeneous, electric and magnetic fields (case 1)), or ii) weak distributions of electric charges and currents (case 2)). The particles and the dynamical EM field are described, for any time t > 0, by the classical phase‐space probability distribution functional (CPSPDF) f and, at the initial time (t = 0), by the initial CPSPDF f_in. The CPSPDF f and f_in, multiplied by suitable Hermite polynomials (for particles and field) and integrated over all canonical momenta, yield new moments. The Liouville equation and f_in imply a new nonequilibrium linear infinite hierarchy for the moments. In case 1), f_in describes local equilibrium but global nonequilibrium, and we propose a long‐time approximation in the hierarchy, which introduces irreversibility and relaxation towards global thermal equilibrium. In case 2), the statistical system, having been at global thermal equilibrium, without external influences, for t ≤ 0, is subject to weak external charge‐current distributions: then, new hierarchies for moments and their long‐time behaviours are discussed in outline. As examples, approximate mean‐field (Vlasov) approximations are treated for both cases 1) and 2).     

On the Zeros of Polynomials Satisfying Certain Linear Second-Order ODEs Featuring Many Free Parameters

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather explicitly known) polynomials satisfying a (recently identified) second-order ODE which features many free parameters and only polynomial solutions. Some of these formulas have a Diophantine connotation. Techniques to manufacture infinite sequences of second-order ODEs featuring only polynomial solutions are also reported.     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.     

Yang-Mills fields of nonstationary spherical objects with big charges

We study the nonlinear electrodynamics suggested in our paper in [Russ. J. Math. Phys. 12 (3), 379–385 (2005)] by using the Yang-Mills equations. This theory is intended to describe strong fields generated by objects with large electric charges. For nonstationary sources with spherical symmetry, we find formulas for the electric field strengths. This solution generalizes our previous result related to the stationary case. The corresponding formulas for nonlinear electric fields are used to explain some puzzling properties of the Earth atmosphere.     

Confluent Hypergeometric Orthogonal Polynomials Related to the Rational Quantum Calogero System with Harmonic Confinement

Two families (type A and type B) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri-type recurrence formulas for these families. In the one-variable case, the polynomials in question reduce to the Hermite polynomials (type A) and the Laguerre polynomials (type B), respectively. The multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum Calogero models with harmonic confinement (for the classical root systems) and are closely connected to the (symmetric) generalized spherical harmonics investigated by Dunkl. Received:Received: 20 October 1996 / Accepted: 3 March 1997     

Electrostatic field of a thin, unclosed spherical shell and a torus

The electrostatic problem for a thin, unclosed spherical shell and a torus is reduced to paired summation equations in the Legendre polynomials by means of formulas relating the spherical and toroidal harmonic functions. The paired equations are transformed to a Fredholm integral equation of the second kind. Formulas are obtained for computing the charges of the conductors in the form of a series in a small parameter. The capacitance is computed for certain geometrical parameters of the conductors. Zh. Tekh. Fiz. 68, 1–6 (July 1998)     

点电荷在拓扑绝缘体和导体中感应磁单极

利用电势和磁标势的第一类零阶贝塞尔函数的公式及拓扑绝缘体材料的本构关系, 推导了点电荷在电介质、 拓扑绝缘体和接地导体三个区域的感应电势及感应磁标势. 研究表明: 点电荷 在电介质、 拓扑绝缘体和接地导体中感应了像电荷和像磁单极; 感应像电荷和感应像磁单极的大小和正负除了与场源电荷、 拓扑绝缘体材料参数等因素有关外, 还与像电荷和像磁单极所处的空间位置有关.     

Jack Polynomials in Superspace

This work initiates the study of orthogonal symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given explicitly.     

A comparative investigation of an AB- and AA-stacked bilayer graphene sheet under an applied electric field: A density functional theory study

An AB- and AA-stacked bilayer graphene sheet(BLG) under an electric field is investigated by ab initio calculation.The interlayer distance between the two layers,band structures,and atomic charges of the system are investigated in the presence of different electric fields normal to the BLG.The AB- stacked BLG is able to tune the bandgap into 0.234 eV with the increase of the external electronic field to 1 V/nm,however,the AA-stacked BLG is not sensitive to the external electric field.In both the cases,the spacing between the BLG slightly change in terms of the electric field.The charges in the AB- stacked BLG are increased with the increase of the electric field,which is considered to be the reason that causes the bandgap opening in the AB- stacked BLG.     

Chebyshev polynomials and Fourier transform of <Emphasis Type="Italic">SU</Emphasis>(2) irreducible representation character as spin tomographic star-product kernel

Spin tomographic symbols of qudit states and spin observables are studied. Spin observables are associated with the functions on a manifold whose points are labeled by the spin projections and sphere S ² coordinates. The star-product kernel for such functions is obtained in an explicit form and connected with the Fourier transform of characters of the SU(2) irreducible representation. The kernels are shown to be in close relation to the Chebyshev polynomials. Using specific properties of these polynomials, we establish the recurrence relation between the kernels for different spins. Employing the explicit form of the star-product kernel, a sum rule for Clebsch–Gordan and Racah coefficients is derived. Explicit formulas are obtained for the dual tomographic star-product kernel as well as for intertwining kernels which relate spin tomographic symbols and dual tomographic symbols.     

On the Relation Between Orthogonal,Symplectic and Unitary Matrix Ensembles

For the unitary ensembles of N×N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w/w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w/w. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations     

Effect of electrodes on the properties of a thin ferroelectric film

The effect of various metal electrodes on the properties of thin ferroelectric films is considered using the phenomenological Ginzburg-Landau theory. The electric field produced by charges on electrodes is taken into account (with allowance for the screening of the charges in the metals) in the free energy functional and in the Euler-Lagrange equation for the film polarization. This equation is solved using the variational method, and the free energy functional is reduced to the conventional free energy with a renormalized coefficient of P ². This coefficient is dependent on the properties and thickness of the film and the properties of the electrode. Therefore, the physical characteristics of the size effect can be found by merely substituting the renormalized coefficient into the usual formulas from phenomenological theory. The calculations are shown to be in good agreement with the experimental data for Pt, Ir, IrO₂, and SrRuO₃.     

A Method for q-Calculus

Abstract

We present a notation for q-calculus, which leads to a new method for computations and classifications of q-special functions. With this notation many formulas of q-calculus become very natural, and the q-analogues of many orthogonal polynomials and functions assume a very pleasant form reminding directly of their classical counterparts.

The first main topic of the method is the tilde operator, which is an involution operating on the parameters in a q-hypergeometric series. The second topic is the q-addition, which consists of the Ward–AlSalam q-addition invented by Ward 1936 [102, p. 256] and Al-Salam 1959 [5, p. 240], and the Hahn q-addition.

In contrast to the the Ward–AlSalam q-addition, the Hahn q-addition, compare [57, p. 362] is neither commutative nor associative, but on the other hand, it can be written as a finite product.

We will use the generating function technique by Rainville [76] to prove recurrences for q-Laguerre polynomials, which are q-analogues of results in [76]. We will also find q-analogues of Carlitz’ [26] operator expression for Laguerre polynomials. The notation for Cigler’s [37] operational calculus will be used when needed. As an application, q-analogues of bilinear generating formulas for Laguerre polynomials of Chatterjea [33, p. 57], [32, p. 88] will be found.     

Field-induced giant static dielectric constant in nano-particle aggregates at room temperature

The analogy between magnetism and electricity was established by Maxwell in the 19th century, despite the subtle difference. While magnetic materials display paramagnetism, ferromagnetism, antiferromagnetism and diamagnetism, only paraelectricity, ferroelectricity and antiferrolelectricity have been found in dielectric materials. The missing ‘diaelectricity’ may be found if there exists a material that has a dc-polarization opposing the electric field or a negative dielectric susceptibility ?′???1, with ?′ being the real part of the relative dielectric constant. Both of these properties have been observed in nano-particle aggregates under a dc electric bias field at room temperature. A possible collective effect in the nano-particle aggregates is proposed to account for the observation. ‘Diaelectricity’ implies overscreening by polarization to the external charges. Materials with a negative static ?′ are expected to provide attraction to similar charges and unusual scattering to electromagnetic waves with possible profound implications for high temperature superconductivity and communications.     

Modern Diffraction Methods,edited by E.J. Mittemeijer and U. Welzel

By considering simple examples from the viewpoint of both classical electromagnetism and the special theory of relativity, it is shown that the magnetic forces between moving charges are of order v ²/c ² times the electric forces between the charges, where v is the velocity of the charges. The relevance of electric forces in the definition of the ampere is discussed.     

Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials

It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald- Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced; the combinatorial formulas for them are given.     

Calculation of vibrational matrix elements of the CO2 molecule in the formalism of polynomials of the quantum numbers

The vibrational energies and the matrix elements of the electric dipole moment function and the rotation function are calculated for an arbitrary linear polyatomic molecule in the first order of perturbation theory using the formalism of polynomials of quantum numbers. The formulas obtained are tested for the example of the CO₂ molecule. The experimental data available at present are briefly analyzed. It is shown that the calculated matrix elements are in good agreement with the corresponding experimental data even in the first order of the theory.     

A hypothetical model of confinement in a dia-electric medium

The assumption of a perfect color dia-electric vacuum in quantum chromodynamics (QCD) led to a reasonable explanation of quark confinement. Along the same lines, a hypothetical dia-electric medium in electromagnetism is shown to lead to charge confinement; although such a medium does not exist, the analysis of this hypothetical problem illuminates in a very simple and straightforward way how this dia property (negative susceptibility) of a medium leads to confinement. It is shown that adding a dipole (particle-antiparticle) charge distribution to such a dia-electric medium would lead the creation of a cavity surrounding the charges; by a simple analysis, using the image problem along with a series expansion in Legendre polynomials, it is shown that the polarization charge distribution on the cavity's surface would lead to an increase in the dipole energy by a finite amount; on the other hand, the increase in the electric energy of a single charge inserted in such a medium would be infinite. It is thus concluded that in such a dia-electric medium, separating the charges of a dipole would require an infinite amount of energy, and thus illustrating in a simple way the confinement concept.     

载流子在金属/聚对苯乙炔/金属结构中注入及输运的动力学研究

采用紧束缚的Su-Schrieffer-Heeger模型,利用非绝热动力学方法研究了载流子在金属/聚对苯乙炔(poly(p-phenylene vinylene,简记为PPV))/金属三明治结构中注入与输运的动力学过程.发现由于强的电子-晶格相互作用,注入的电荷在PPV链中形成波包,波包的形成与施加在金属电极上的偏压、PPV链上的电场强度及金属电极与PPV之间的界面耦合强度密切相关.在无外电场的情况下,当偏压达到临界值时电荷能够从金属电极注入到PPV链中并形成波包.随着电场强度的增大,波包能 关键词： 金属/聚对苯乙炔/金属结构 载流子输运 波包     

Can the macroscopic Maxwell equations be obtained from the microscopic Maxwell-Lorentz equations by performing averages?

It is shown that the usual procedures of obtaining the macroscopic Maxwell equations from the microscopic Maxwell-Lorentz equations by performing averages contain an arbitrary choice of gauge. By a suitable different choice of the gauge the so-obtained Maxwell equations can be cast back to the form of the starting Maxwell-Lorentz equations. Therefore one cannot consider the Maxwell equations to be obtainable from the Maxwell-Lorentz equations by simply performing averages. The implication of this result is that besides the electromagnetic fields produced by the moving electric charges, as given by the Maxwell-Lorentz equations, there may be some other agents that cannot be identified as some kind of motion of the electric charges and that participate in the production of the electromagnetic fields.