Variational principles of magnetostatics of superconductors. I

Fundamental and dual variational principles of magnetostatics of superconductors are formulated. It is shown how test fields can be expressed in terms of vector and scalar potentials. General consequences of the variational magnetostatic formulation are discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 97–102, September, 1991.     

Variational principles of magnetostatics

Variational principles of magnetostatics are formulated, which permit evaluation of the upper and lover bounds on the energy of a magnetic field produced by a known spatial current density distribution with the help of rather broad classes of trial functions. The potentialities for the use of these variational principles as applied to practical problems is demonstrated with an example of the well-known problem of the calculation of the inductance of a straight solenoid of finite length. In particular, it is shown that the formula appearing in all the textbooks for this inductance, in fact, defines the upper bound for the true value of the inductance.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 70–75, July, 1987.     

Variational principles for thermostatted systems

A generalized Lagrangian formalism is proposed for dealing with interacting many body systems subject to time reversible friction forces. The corresponding generalized Hamiltonian and Poissonian formalisms are presented. Explicit connection is made between the new Poisson brackets obtained and the 2-forms of conformally symplectic systems recently investigated. Applications to Nose-Hoover and isokinetic dynamics are treated in detail. (c) 1998 American Institute of Physics.     

Variational principles for relativistic transport coefficients

Variational expressions for the transport coefficients of a one-component, relativistic gas are derived from the linearized relativistic Boltzmann equation for both quantum and classical gases. These expressions depend on functions _χ of the energy of the particles comprising the gas in such a way that: a) if _χ differs from a solution of the linearized Boltzmann equation by ε, then the value of the variational expression calculated with this _χ differs from the true value of the corresponding transport coefficient by ε²; and b) the value of the variational expression is always less than this true value. It is shown that values of the transport coefficients obtained by expanding _χ in a particular set of orthogonal polynomials and keeping only the first nontrivial term in the expansion are equivalent to those obtained using the Grad method of moments. It follows therefore that values obtained using this later method represent lower bounds on the true values. We also show that one can obtain simple, closed-form expressions for the various transport coefficients corresponding to an arbitrary number of terms in an expansion of the trial function _χ in the above-mentioned set of orthogonal polynomials. Finally we point out that all of our results can be carried over to the nonrelativistic case by taking the limit c → ∞.     

Theory of magnetic impurities in superconductors. II

The depression of the superconducting transition temperature due to the pair breaking effect of magnetic impurities is calculated exactly within the Nagaoka-Suhl approach to the Kondo problem in superconductors.     

Variational principles and self-organization in two-fluid plasmas

Self-organization of an ordered structure occurs in a plasma under rather restrictive conditions. A new framework for a variational principle invokes a coercive form that results in a criterion for self-organizing relaxation of a two-fluid plasma. The constraints (constants of motion of the ideal model) are adjusted, through a weakly dissipative process, so that the relaxed state, under well-defined conditions, is a stable equilibrium independent of the direct effects of dissipation.     

Variational wavefunction for multi-species spinful fermionic superfluids and superconductors

We introduce a new fermionic variational wavefunction, generalizing the Bardeen–Cooper–Schrieffer (BCS) wavefunction, which is suitable for interacting multi-species spinful systems and sustaining superfluidity. Applications range from quark matter to the high temperature superconductors. A wide class of Hamiltonians, comprising interactions and hybridization of arbitrary momentum dependence between different fermion species, can be treated in a comprehensive manner. This is the case, as both the intra-species and the inter-species interactions are treated on equally rigorous footing, which is accomplished via the introduction of a new quantum index attached to the fermions. The index is consistent with known fermionic physics, and allows for heretofore unaccounted fermion–fermion correlations. We have derived the finite temperature version of the theory, thus obtaining the renormalized quasiparticle dispersion relations, and we discuss the appearance of charge and spin density wave order.     

Variational principles in the theory of gravitoinertial fields

Variational methods and some features of their application in the theory of gravitoinertial fields are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 110–114, November, 1974.     

Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.     

Electronic structure of high-temperature superconductors II. La-Ba-Cu-O systems

Electronic structure of La_2–x Ba _x CuO₄ superconductors has been studied. Model systems La₁₀CuO ₂₂ ^–12 , La₁₀cuO ₁₄ ⁺⁴ and La_gBaCuO ₁₄ ⁺³ with various Ba positions have been investigated using the quasirelativistic INDO molecular orbital method. The oxidation state of copper atoms corresponds to Cu(II). The influence of Ba/La substitution on copper atoms is much lower than on their oxygen ligands sphere.     

Variational principles and spatially-homogeneous universes,including rotation

The validity of imposing spatial homogeneity on the variations in the usual action principle for Einstein's equations is studied. It is proved that with this procedure the standard and ADM Lagrangians give correct Einstein equations if and only if the space belongs to Class A of Ellis and MacCallum [1], i.e., the structure constants of the simply transitive group satisfy C _fg ^f =0. The possibility of overcoming this difficulty in the Class B spaces is examined.This work was supported in part by the United States Atomic Energy Commission under Contract Number AT 104-37-39 Project Agreement No. 125.     

Variational principles of electrostatics and higher polarizabilities of conductors

Based on variational principles, new concepts of higher polarizability of an isolated conductor are introduced that generalize well-known concepts of capacity and polarizability. It is shown that the lth-order polarizability in a spherical coordinate system is represented by a (2l+1) × (2l+1) matrix. The corresponding (2l+1) potentials can be considered as a system of lth-order multipoles typical of a conductor and more general than systems of spherical multipoles known in electrodynamics. It is exemplified what new results can be obtained using the higher polarizability technique for solving problems in electrostatics. Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 61–65, October, 1999.     

Variational approach to strong correlation in the photoemission of electron-doped superconductors

We use a variational approach with strictly strong-correlated constraint to gain insight into low-energy states of t-t^′-t^″-J model in the electron-doped regime. Compared with the recent results on the electron-doped cuprates obtained by angle-resolved photoemission spectroscopy (ARPES), we show that based on the long-range ordered antiferromagnetic metallic state prohibiting vacant sites, our results lead to qualitatively similar trends in ARPES spectra and Fermi surface topology. Additionally, the results about the evolution of the energy gap and spectral weight as a function of doping will be discussed.     

Variational principles for the calculation of the response function

Several variational principles for inclusive processes are presented and illustrated by simple examples. By choosing appropriate trial functions, the doorway-state, moment- and cumulant-expansion of the response functions are derived from them.     

Variational principles for coupled nonlinear Schrödinger equations

Via He's semi-inverse method, a variational principle is established for coupled nonlinear Schrödinger equations arising in nonlinear directional coupler. This Letter shows the effectiveness and convenience of the semi-inverse method.     

Variational principles for scattering in the generator coordinate method

The generator coordinate method (GCM) wave function is used as a trial function in a Kohn type variational principle for scattering phase shifts. It is shown that a GCM trial function is a solution of the variational equations if the Hill-Wheeler integral equation is satisfied subject to an appropriate boundary condition. A new method for introducing the scattering boundary condition is presented. There is a uniqueness theorem for the phase shift.     

Muonic study of type II superconductors

A theory of μSR method is developed for uniaxial anisotropic high-T _c superconductors. In two extreme cases ofH _ext‖c andH _ext⊥c analytical formulas are obtained which makes it possible to determine from the position of van Hove singularities in the Fourier-spectrum of muonium polarization, the type and parameters of the vortex lattice, such as λ _ab (λ _c ). In caseH _ext≲H _c2 we obtained the shape of the Fourier-spectrum in analytical form and the simple method of determining the Ginsburg-Landau parameter κ. Convenient expressions for numerical calculations are obtained for the arbitrary orientation of the external field, and an algorithm is provided to compute the mean fieldB, the vortex lattice parameters and the bulk field distribution in anisotropic superconductor. The Fourier spectrum of polarization based on these calculations can be used to independently check the validity of the high-T _c parameters determination for “appropriate” orientations.