Spherical harmonics and Cartesian tensor scalar product expansion in the Fokker-Planck equation

On the basis of the expansion of the distribution functionf(v, r,t) in a sum of spherical harmonics, which is equivalent to a Cartesian tensor scalar product expansion of the distribution function, i.e.,f(v, r, t)=f ₀(v,r,t)+v. f ₁(v,r,t)+vvf ₂(v,r,t)+vvvf ₃(v,r,t)+ wheref _k (k=2, 3) arek-th order irreducible tensors, the Rosenbluth potential functions and the Fokker-Planck collision term are expanded in a similar sum. Collisions termsJ _Fk (k=0, 1, 2) and the equations forf _k (k=0, 1, 2) for the case of the Coulomb interactions are also determined.Technická 2, Praha 6, Czechoslovakia.The autor wishes to express his thanks to Prof. J. Kracík, DrSc. for valuable advice and suggestion.     

Magnetization of an electron plasma by microwave pulses: Faraday induction

It has been argued theoretically that the recently proposed vacuum fieldB ⁽³⁾ is not accompanied by a real electric fieldE ⁽³⁾ . Experimental evidence for this interence is available in the data reported by Deschampset al. [10], using microwave magnetization of an electron plasma set up in helium gas. Faraday induction due toB ⁽³⁾ does not occur in the inert gas and is not observed experimentally in the absence of free electrons. WheneverB ⁽³⁾ interacts with free electrons, however, Faraday induction occurs through a pulse of induced magnetization (i.e., induced orbital electronic angular momentum).     

Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on with Hamiltonian H(x)=∑ _i,j J _i,j (x(i)−x(j))²+h∑ _i (x(i)−d(i))², where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ ^d . The configuration d is given and considered as observations. The ‘couplings’ J _i,j are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration m corresponding to the Hamiltonian.     

Quantum condensates classified in superconductivity with topology in the Minkowski space-time

A substantial problem in the macroscopic theory of pure superconductivity has been left forgotten for a long time since London and London in 1935. An impression survived that the Meissner effect is more substantial than the zero-resistivity. But, the London equation [I], the Newtonian equation of motion, was abandoned, whereas the London equation [II], derived from the Maxwell equations, was postulated. The London equation [II] included the logical gap [ α ] in real time, whereas the London equation [I] has been ignored without even noting the logical gap [ β ] in space. Microscopically, after the publication of F. London's book and the discovery of the isotope effect in 1950, the success of the Bardeen--Cooper--Schrieffer (BCS) theory in 1957 was likely to have finally given the definitive explanation on superconductivity by proving only the London equation [II] that claimed the coherent condensation of Cooper pairs in the momentum space. Since then, these arguments have been regarded to be a standard among various preceding theories. Meanwhile, the London equation [I] has faded away and has been long-forgotten. But we must not abandon the London equation [I], and, rather, retrieve it. We later recognized also that the DC-component of a persistent current can never be determined by using the Fourier transform analysis, because of its singularity at ω?=?0 and q ?=?0 with huge differences of space-time domain. Quite recently, in 2003, we first recognized a proper and harmonious view to simultaneously account for (i) the zero-resistivity in an open system with (i-c) the resultant persistent current in a closed system, and (ii) the perfect diamagnetism at T???0?K in the space-time aspects in terms of the gauge field theory. Here, we further clarify where and how we have lost and found a properly perspective view of the superconductivity. Here, we eliminate two logical gaps [ α ] and [ β ] by using the gauge field theory for further clarifying a position of the previous and present works. We especially classify superconductors with topology which eventually leads us such as (ii-2D) magnetic flux quantization in a ring. By projecting the 3-dimensional BCS-theory with the concept of ‘coherence’ among an enormous number of Bosons like Cooper pairs onto the (1?+?3)-dimensional Minkowski space-time [β?=?(v/c)?=?0], we clarify responses of the ground state Ψ _macro at T???0?K in a set of the basic equations, for (i) the zero-resistivity, [E _K ???qφ( R )]?=?0 at ω?=?0 and (ii) the perfect diamagnetism [?K ???qA ( R )]?=?0 at q ?=?0 as an inevitable consequence at the gauge fields in the proper theory of superconductivity.     

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G _D(r,r′, N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r′ . The expression for G(r,r′, s) is found to be the sum of the commonly accepted Dirichlet result G _D(r,r′, N) = G ₀(r,r′, N) - G ₀(r,r”, N) , where r” is the mirror image of r′ , and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.     

Stable potentials I

We discuss a conjecture of Ruelle concerningstable potentials on a group. For the groupsZ ₂,Z ₃, ₄, andZ ₆ any stable potential can be written as the sum of a non-negative function and a function of non-negative type. This is not true for the groupsZ _k (k odd, 5). For the Euclidean groupR ^v the question is open.Supported by NSF GP 7946.     

Nonadiabatic effects in the phonon spectra of superconductors

The nonadiabatic corrections to the self-energy part Σ_s(q, ω) of the phonon Green’s function are studied for various values of the phonon vectors q resulting from electron-phonon interactions. It is shown that the long-range electron-electron Coulomb interaction has no direct influence on these effects, aside from a possible renormalization of the corresponding constants. The electronic response functions and Σ_s(q, ω) are calculated for arbitrary vectors qand energy ω in the BCS approximation. The results obtained for q=0 agree with previously obtained results. It is shown that for large wave numbers q, vertex corrections are negligible and Σ_s(q, ω) possesses a logarithmic singularity at ω=2Δ, where Δ is the superconducting gap. It is also shown that in systems with nesting, Σ_s(Q, ω) (where Q is the nesting vector) possesses a square-root singularity at ω=2Δ, i.e., exactly of the same type as at q=0. The results are used to explain the recently published experimental data on phonon anomalies, observed in nickel borocarbides in the superconducting state, at large q. It is shown, specifically, that in these systems nesting must be taken into account in order to account for the emergence of a narrow additional line in the phonon spectral function S(q, ω)≈−π ⁻¹ Im D _s (q, ω), where D _s (q, ω) is the phonon Green’s function, at temperatures T<T _c . Zh. éksp. Teor. Fiz. 115, 1799–1817 (May 1999)     

Primitive ideals of C q [SL(3)]

The primitive ideals of the Hopf algebraC _q [SL(3)] are classified. In particular it is shown that the orbits in PrimC _q [SL(3)] under the action of the representation groupH C ^*×C ^* are parameterized naturally byW×W, whereW is the associated Weyl group. It is shown that there is a natural one-to-one correspondence between primitive ideals ofC _q [SL(3)] and symplectic leaves of the associated Poisson algebraic groupSL(3,C).Partially supported by a grant from the N.S.A.     

Conditional expectation and commutativity in von Neumann algebras

LetH be a Hilbert space,A the von Neumann algebra of all bounded operators onH,B a von Neumann subalgebra ofA, andw a bounded linear functional onA. The functionalw is said to commute withBA ifw(AB)=w(BA) for allAA. It is shown that the mapBw (BAB) is a complex measure on the orthocomplemented partially ordered set of all orthogonal projections inB for everyAA if and only ifw commutes with all members ofB. For anyAA, the conditional expectation ofA with respect toB andw is defined and it is shown that this expectation exists for an Abelian separableB ifw commutes with all members ofB. Using Gleason's theorem it is shown thatw commutes withB if and only if the density operator ofw commutes withB.     

Mechanism for giant electro-optic response of porphyrin J-aggregates in polymer film and aqueous solution

We report the dependence of the polarizability difference Δα on the orientational distribution of porphyrin J-aggregates. For quasi-one-dimensionally oriented aggregates in a polymer film, the red shift in Y ∥ F configuration was about 2 to 3 times smaller than that in Y ∥ F configuration, where Y is the orientational direction of the J-aggregates and F is the applied AC electric field vector. For an aqueous solution where J-aggregates were three-dimensionally oriented, the exciton band showed an electric-field-induced broadening. In addition, a red-shifted signal for two-dimensionally oriented aggregates in the polymer film was reduced by one order of magnitude at 77K compared with that at room temperature. These results were explained reasonably well by the molecular rearrangement model, which was applied to a variety of orientational distributions.     

Projective indecomposable modules over the small quantum osp(1|2)

In this paper, we construct the finite dimensional Hopf superalgebra u _q (osp(1|2)) arising from U _q(osp(1|2)) when q is a root of unity and describe the projective objects and the irreducible morphisms in a category of Z-graded u _q (osp(l|2))-modules.     

Non-abelian electrodynamics and the vacuumB (3)

By using an 0(3) gauge group, a non-Abelian theory of vacuum electrodynamics is developed in which the newly discovered longitudinal vacuum fieldsB ⁽³⁾ andi E ⁽³⁾ appear self-consistently with the usual plane wavesB ⁽¹⁾,B ⁽²⁾,E ⁽¹⁾, andE ⁽²⁾ in the circular basis (1), (2), (3), a complex representation of space. Using the charge quantization condition the vacuum Maxwell equations are given in the non-Abelian representation.     

Mechanism of giant magnetoresistance in intermetallic compounds of rare-earth ions and actinides

Giant positive or negative magnetoresistance is calculated in a band model. The spectra of the band electrons in a two-sublattice antiferromagnetic intermetallic compound depend on the antiferromagnetism vector L(T,H). The metamagnetic transition to the ferromagnetic phase is accompanied by splitting with respect to the spin σ, displacement of the energy bands, and a decrease in the effective masses of the band electrons. This mechanism of giant negative magnetoresistance is also accompanied by an increase in the relaxation time τ_jσ. Scattering by chemical-bond fluctuations is considered as the main relaxation mechanism. Giant positive magnetoresistance results from a four-subband model of 4f and 5f intermetallic compounds. The electron effective masses m _jσ(J _jT ) of the (j,σ) bands increase with the mean angular momentum J _1T (T,H) of an ion in the jth sublattice of 4(5)f ions. The thermodynamics of such a four-sublattice model, the nonlinear magnetization and magnetoresistance curves, and the nonmonotonic dependence of the specific heat C _m(T,H) on the field H are calculated. Fiz. Tverd. Tela (St. Petersburg) 39, 1806–1814 (October 1997)     

On the linearized relativistic Boltzmann equation

The linearized relativistic Boltzmann equation inL ²(r,p) is investigated. The detailed analysis of the collision operatorL is carried out for a wide class of scattering cross sections.L is proved to have a form of the multiplication operatorv(p) plus the compact inL ²(p) perturbationK. The collisional frequencyv(p) is analysed to discriminate between relativistic soft and hard interactions. Finally, the existence and uniqueness of the solution to the linearized relativistic Boltzmann equation is proved.     

Magnetic properties of erbium iron garnet in high magnetic fields up to 150kOe

The magnetic properties of single crystals of erbium iron garnet (ErIG) were studied in applied fields up to 150kOe between 1.4 and 300K. At low temperature, the macroscopic easy direction of the bulk magnetization is [100]; below the compensation temperature (80±2K), the magnetization presents non-linear field evolution. On the assumption of an isolated ground doublet, the anisotropy constantsK _i (i=1,2) of ErIG are given byK _i (Er)+K _i (YIG); theK _i are calculated as a function of theG andg tensor components. It is worthwhile noting that theK _i (Er) are strongly temperature dependent; so at low temperature the anisotropy of the garnet is determined by the rare earth ions, while in the 50 K regionK ₁(Er) becomes comparable toK ₁(YIG) with the opposite sign which results in a very weak anisotropy of the garnet. Above 50 K,K ₁(YIG) is predominant and the Fe³⁺ ions determine the garnet anisotropy.     

Integral equation theory of the exchange potential,HOMO–LUMO properties,and sum rules for the exchange-correlation force

Recently, a formally exact integral equation for the exchange potential V _x(r) has been presented by the authors. In the admittedly simplistic limit in which Slater–Kohn–Sham and Hartree–Fock determinants become equal, this integral equation reduces to that given by Della Sala and Görling. Here, a proposal is made to relate, but now approximately, the formally exact equation for V _x(r) to HOMO–LUMO properties. The addition of a correlation contribution V _c(r) to V _x(r), the sum being the exchange-correlation potential V _xc(r), is finally considered, some exact properties and especially sum rules for the force ??V _xc/?r being the focus.     

Large-scale amplification of a magnetic field by turbulent helicity fluctuations

In this paper the procedure of large-scale averaging of the magnetic-field diffusion equation with the α-term curlα(r,t)B(r,t) is used to show that a nonuniform distribution of the turbulent helicity fluctuations (more precisely, the fluctuations of the coefficient α) with a zero average value gives rise to large-scale amplification of the initial magnetic field. A detailed study is carried out of the dependence of the resulting large-scale α effect on the characteristics of the correlator 〈〈α(r, t)α(r″,t″)〉〉 in a rotating medium with a nonuniform distribution of the angular velocity ω=ω(ρ,z) (ρ is the distance for the rotation axis z). The effect of helicity fluctuations and the diffusion coefficient on the turbulent diffusion process is also investigated. Zh. éksp. Teor. Fiz. 116, 85–104 (July 1999)     

Noether's theorem and the fieldB (3)

It is shown that the longitudinal, magnetic flux density,B ⁽³⁾ , of vacuum electromagnetic radiation can be accommodated rigorously within Noether's theorem, which relates fundamental spacetime symmetries to fundamental conservation laws. This demonstration linksB ⁽³⁾ to the canonical energy-momentum tensorT _µv that appears in Einstein's field equations of general relativity. Thus,B ⁽³⁾ provides a link between electromagnetism and gravitation which might eventually lead to an unified understanding of field theory.     

从流场分解角度改进Q矢量分析方法及其在暴雨动力识别中的应用

依据水平风场分解思路,运用调和-余弦算法,准确分解有限区域Q矢量.根据已有的关于Q矢量散度和新发展的Q矢量涡度与天气系统对应关系的研究,运用分解得到的Q矢量旋转分量场和辐散分量场,改进传统Q矢量分析方法中单纯依靠Q矢量水平散度场进行的诊断分析.通过对一次暴雨个例的比较研究表明,在雨带形势和暴雨中心位置的动力识别方面,改进的Q关键词： 矢量分析 调和-余弦算法 暴雨 动力识别     

Packing of molecules and hole formation in lyotropic systems

A vector field q (the order parameter of the molecular packing) describing the packing (specifically, the orientation) of membrane-forming amphiphilic molecules is introduced to describe the structures of lyotropic phases constructed from membranes. In the general case q·n≠0 (where n is the unit normal vector) and therefore the singularities of the vector field q are not determined uniquely by the topology of the surface. The condition q·n=0 signifies disruption of the packing of the molecules. This corresponds to holes, which can form in membranes when lyotropic systems are diluted. As an illustration, the simplest type of such singularities, in which the distribution of the field q around a hole is described by a part of an instanton with unit topological charge, is studied. It is shown that such a distribution guarantees the existence of a local minimum under the condition that the tension per unit length λ of the hole boundary is small compared with the deformation energy of the field q: λh/K≪l (K is the modulus of the orientational elasticity of the field q and h is the thickness of the membrane). The radius of the hole which is formed equals L≈2.52(K/λh)^1/3 and the energy E≈59.79K(λh/K)^1/3. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 8, 575–580 (25 October 1996)