Master Equation in Phase Space for a Spin in an Arbitrarily Directed Uniform External Field

The time evolution equation for the probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for the nonaxially symmetric problem of a quantum paramagnet subjected to a uniform magnetic field of arbitrary direction. This is accomplished by first rotating the coordinate system into one in which the polar axis is collinear with the field vector, then writing the reduced density matrix equation in the new coordinate system as an explicit inverse Wigner-Stratonovich transformation so that the phase space master equation may be derived just as in the axially symmetric case [Yu.P. Kalmykov et al., J. Stat. Phys. 131:969, 2008]. The properties of this equation, resembling the corresponding Fokker-Planck equation, are investigated. In particular, in the large spin limit, S→∞, the master equation becomes the classical Fokker-Planck equation describing the magnetization dynamics of a classical paramagnet in an arbitrarily directed uniform external field.     

Theory of one-dimensional and quasi-one-dimensional heat conduction

It is shown that the heat conduction process in a one-dimensional flow of a fluid moving with a velocity V in a constant temperature field follows a law that is considerably more complicated than an “ordinary” exponential law. It is demonstrated that in the quasi-one-dimensional case the heat conduction process in an abstract space of dimension 1+ɛ, where ɛ varies from zero to unity, is described by a modified Fourier equation. Its solution for an infinite space is found. Zh. Tekh. Fiz. 67, 8–12 (July 1997)     

The Dirac System on the Anti-de Sitter Universe

We investigate the global solutions of the Dirac equation on the Anti- de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, a priori, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass M of the field and the cosmological constant Λ > 0: it appears a critical value, Λ/12, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When M ² ≥  Λ/12 there exists a unique unitary dynamics. On the contrary, for the light fermions satisfying M ² < Λ/12, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the hamiltonian is discrete. We also prove a result of equipartition of the energy.     

Calculating coefficients for a system of moment equations used to describe the magnetization kinetics of a superparamagnetic particle in a fluctuating field

A system of equations is derived for moments [averages of spherical harmonics 〈Y _l,m 〉(t)] that determine the dynamics of the magnetization M of a superparamagnetic particle in a fluctuating field. The system is derived by representing the Gilbert equation in a fluctuating field, and the corresponding Fokker-Planck equation for the distribution function of M, in terms of angular momentum operators, which in turn makes it possible to express the coefficients of the system of moment equations in terms of Clebsch-Gordan coefficients. Fiz. Tverd. Tela (St. Petersburg) 41, 2020–2027 (November 1999)     

Quantum Out-States Holographically Induced by Asymptotic Flatness: Invariance under Spacetime Symmetries, Energy Positivity and Hadamard Property

This paper continues the analysis of the quantum states introduced in previous works and determined by the universal asymptotic structure of four-dimensional asymptotically flat vacuum spacetimes at null infinity M. It is now focused on the quantum state λ _M , of a massless conformally coupled scalar field propagating in M. λ _M is “holographically” induced in the bulk by the universal BMS-invariant state λ defined on the future null infinity of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at future null infinity discussed in previous papers. This induction is possible when some requirements are fulfilled, in particular whenever the spacetime M and the associated unphysical one, M͂, are globally hyperbolic and M admits future time infinity i ⁺. λ _M coincides with Minkowski vacuum if M is Minkowski spacetime. It is now proved that, in the general case of a curved spacetime M, the state λ _M enjoys the following further remarkable properties:

Dirac Operators Coupled to the Quantized Radiation Field: Essential Self-adjointness à la Chernoff

We study the Dirac operator D ₀ in an external potential V, coupled to a quantized radiation field with energy H _f and vector potential A. Our result is a Chernoff-type theorem, i.e., we prove, for the operator D ₀+α · A+V +λ H _f with λ ∈{0, 1}, that the essential self-adjointness is not affected by the behavior of V at ∞.      

Distribution function for large velocities of a two-dimensional gas under shear flow

The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degreek⩾4 diverge for shear rates larger than a critical valuea _c ^(k) , which behaves for largek asa _c ^(k) ∼k ⁻¹. This divergence is consistent with an algebraic tail of the formf(V) ∼V ^−4-σ(a), where σ is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.     

Long- and short-range order in the Pd<Subscript>6</Subscript>B monoclinic superstructure and M<Subscript>6</Subscript>X<Subscript>5</Subscript> and M<Subscript>6</Subscript>X allied superstructures

Symmetry analysis of the Pd₆B monoclinic superstructure (space group C2/c) formed in the cubic (with the B1 structure) solid solution of boron in palladium (PdB _y ) has been carried out. The formation of this superstructure proceeds as a first-order phase transition via the disorder-order channel including nine nonequivalent superstructure vectors of four stars {k ₁₀}, {k ₄}, {k ₃}, and {k ₀}. For the Pd₆B monoclinic super-structure (space group C2/c), the distribution function for boron atoms is calculated and the interval of admissible values of the long-range order parameters is defined. It is shown that the transition channel determined in this way coincides with the channel in which the M₆X monoclinic superstructure (space group C2) is formed; therefore, the Pd₆B superstructure can also be described in space group C2 to the same degree of accuracy. The higher symmetry of the monoclinic model (space group C2/c) suggests that it describes the structure of the Pd₆B phase (Pd₆B□₅), as well as of mutually inverse phases M₆X□₅ and M₆X₅□, more adequately than the model with space group C2. It is shown that superstructures of the M₆X□₅ type (space groups C2/c, C2, C2/m, and P3₁) and inverse superstructures of the M₆X₅□ type with the same space groups have the positions of the nearest surrounding of metal atoms by two types of nonmetallic sublattice sites located in the first and second coordination spheres.     

Black holes with metric and fields of the same form

We present two rotating black hole solutions with axion ξ, dilaton f{\phi} and two U(1) vector fields. Starting from a non-rotating metric with three arbitrary parameters, which we have found previously, and applying the “Newman–Janis complex coordinate trick” we get a rotating metric g _μν with four arbitrary parameters namely the mass M, the rotation parameter a and the charges electric Q _E and magnetic Q _M . Then we find a solution of the equations of motion having this g _μν as metric. Our solution is asymptotically flat and has angular momentum J = M a, gyromagnetic ratio g = 2, two horizons, the singularities of the solution of Kerr, axion and dilaton singular only when r = a cos θ = 0 etc. By applying to our solution the S-duality transformation we get a new solution, whose axion, dilaton and vector fields have one more parameter. The metrics, the vector fields and the quantity l = x+ie^-2f{\lambda=\xi+ie^{-2\phi}} of our solutions and the solution of: Sen for Q _E , Sen for Q _E and Q _M , Kerr–Newman for Q _E and Q _M , Kerr, Reference Kyriakopoulos [Class. Quantum Grav. 23:7591, 2006, Eqs. (54–57)], Shapere, Trivedi and Wilczek, Gibbons–Maeda–Garfinkle–Horowitz–Strominger, Reissner–Nordstr?m, Schwarzschild are the same function of a, and two functions ρ ² = r(r + b) + a ² cos² θ and Δ = r(r + b) − 2Mr + a ² + c, of a, b and two functions for each vector field, and of a, b and d respectively, where a, b, c and d are constants. From our solutions several known solutions can be obtained for certain values of their parameters. It is shown that our two solutions satisfy the weak the dominant and the strong energy conditions outside and on the outer horizon and that all solutions with a metric of our form, whose parameters satisfy some relations satisfy also these energy conditions outside and on the outer horizon. This happens to all solutions given in the “Appendix”. Mass formulae for our solutions and for all solutions which are mentioned in the paper are given. One mass formula for each solution is of Smarr’s type and another a differential mass formula. Many solutions with metric, vector fields and λ of the same functional form, which include most physically interesting and well known solutions, are listed in an “Appendix”.     

Symmetry analysis of the monoclinic Pd<Subscript>6</Subscript>B superstructure: Long- and short-range orders

A symmetry analysis of the monoclinic (space group C2/c) Pd₆B superstructure formed in the cubic (with structure B1) boron solid solution PdB _y in palladium has been performed. The formation of this superstructure occurs as a first-order phase transition via the disorder-order transition channel including nine nonequivalent superstructure vectors of four stars {k ₁₀}, {k ₄}, {k ₃}, and {k ₀}. For the monoclinic (space group C2) Pd₆B superstructure, the distribution function of boron atoms has been calculated and the interval of admissible values of long-range order parameters has been determined. It has been shown that the found transition channel is identical to the channel of the formation of the monoclinic (space group C2/c) M ₆ X superstructure; therefore, the Pd₆B superstructure can be described with the same accuracy in space group C2. The higher symmetry of the monoclinic (space group C2/c) model suggests that it more accurately describes the structure of the phase Pd₆B (Pd₆B□₅) and mutually inverse phases M ₆ X□₅ and M ₆ X ₅□ than the model with space group C2. It has been demonstrated that there are two types of the nearest environment of metal atoms with non-metal sublattice sites arranged in the first and second coordination spheres in M ₆ X□₅-type superstructures (space groups C2/c, C2, C2/m, and P3₁) and in inverse M ₆ X ₅□-type superstructures with the same space groups.     

Search for one large extra dimension with the DELPHI detector at?LEP

Single photons detected by the DELPHI experiment at LEP2 in the years 1997–2000 are reanalysed to investigate the existence of a single extra dimension in a modified ADD scenario with slightly warped large extra dimensions. The data collected at centre-of-mass energies between 180 and 209 GeV for an integrated luminosity of ∼650 pb⁻¹ agree with the predictions of the Standard Model and allow a limit to be set on graviton emission in one large extra dimension. The limit obtained on the fundamental mass scale M _D is 1.69 TeV/c ² at 95% CL, with an expected limit of 1.71 TeV/c ². Deceased.     

Dynamic magnetic permeability of a thin, high-T c superconducting wafer

The field dependence of the vibrational contribution to the dynamic magnetic permeability μ _V(H) is calculated for a thin (of thickness d∼λ) high-T _c superconducting wafer in a magnetic field parallel to the surface. The resulting curves are plotted on the basis of an exact numerical analysis of the vortex structures both for the thermodynamic-equilibrium vortex lattice and in the presence of pinning forces and the Bean-Livingston surface barrier. It is shown that the μ _V(H) curves are highly sensitive to the size factor (d/λ) and exhibit abrupt changes corresponding to a change in the number of vortex rows. The equilibrium μ _V(H) curve is found to be similar in its general behavior and absolute value (obtained with allowance for the distribution of grain sizes and with appropriate values of λ and ϰ) to the experimental μ _V(H) curve plotted at nitrogen temperature for fine-grained YBa₂Cu₃O_x with grain diameters 〈D〉∼λ in an increasing magnetic field. It is established that the main cause of the experimentally observed irreversible behavior of the μ _V(H) curves during cyclic variation of the applied magnetic field is the existence of a surface barrier to the exit of vortices from the superconductor. The lower limit H _min(B) of stability of the mixed state in the presence of an ideal surface barrier in a thin, high-T _c superconducting wafer (d∼λ) is determined, along with the range of the vortex state (H _max-H _min) for a fixed number of vortices in micrometer-size grains of the investigated YBaCuO samples. Fiz. Tverd. Tela (St. Petersburg) 39, 1943–1947 (November 1997)     

Conformal Vector Fields and Eigenvectors of Laplacian Operator

In this paper, we consider an n-dimensional compact Riemannian manifold (M,g) of constant scalar curvature and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ together with a condition on Ricci curvature of M, that the Ricci curvature in the direction of a certain vector field is greater than or equal to (n − 1)λ, forces M to be isometric to the n-sphere S ⁿ (λ).     

Thermomagnetic history effects in niobium and its implication for H c1 in highT c superconductors

The existence of a remanent magnetization (M _rem) on switching off the field of a field cooled (FC) sample of a highT _c superconductor is often reported. It has recently been argued thatM _rem should equal the difference in FC and zero field cooled (ZFC) magnetizations (M _FC —M _ZFC) in hard superconductors and this has been demonstrated to hold in single crystals of YBCO at 4.2K over a limited range ofH values. We report the detailed magnetization measurements under various thermomagnetic histories (of whichM _rem is one special case) on two specimens of Nb, which show different extents of flux trapping. We find that there are in general three regions inH, T space, corresponding toM _rem+M _ZFC−M _FC=0,M _rem<(M _FC−M _ZFC) andM _rem>(M _FC−M _ZFC). At anyT, the equality holds forH<H _c1(T), and forH→H _c2 (M _FC−M _ZFC) asymptotically vanishes and thereM _rem>(M _FC−M _ZFC). We show that there exists an intermediate region in all hard superconductors, whereM _rem<(M _FC−M _ZFC). The range over which this situation persists, however, depends on the degree of irreversibility in a sample. We can explain qualitatively all the history dependent magnetization data in terms of the critical state model. We point out an inconsistency in an earlier analysis to determineH _c1(T) from such data in YBCO. We also propose a new criterion for putting limits onH _c1(T) in hard superconductors.     

Magnetoplastic effect and spin-lattice relaxation in a dislocation-paramagnetic-center system

A magnetic induction threshold B _c above which the magnetoplastic effect — depinning of dislocations from paramagnetic pinning centers — can be observed in samples placed in a magnetic field is predicted and observed in Al, NaCl, and LiF crystals. The existence of a threshold is associated with the fact that for B<B _c the spin-lattice relaxation time τ_sl in a dislocation-paramagnetic-center system is less than the time required for spin evolution in a magnetic field resulting in the removal of the spin forbiddenness of an electronic transition that “switches off” the dislocation-pinning-center interaction. It is shown that the threshold field B _c is sensitive to temperature and x-ray irradiation of the samples. A new method for measuring the spin-lattice relaxation time in paramagnetic centers on dislocations is proposed on the basis of the data obtained. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 8, 628–633 (25 April 1996)     

Exact solutions in gravity with a sigma model source

We consider a D-dimensional model of gravity with non-linear “scalar fields” as a matter source. The model is defined on the product manifold M, which contains n Einstein factor spaces. General cosmological type solutions to the field equations are obtained when n − 1 factor spaces are Ricci-flat, e.g. when one space M ₁ of dimension d ₁ > 1 has nonzero scalar curvature. The solutions are defined up to solutions to geodesic equations corresponding to a sigma model target space. Several examples of sigma models are presented. A subclass of spherically symmetric solutions is studied and a restricted version of “no-hair theorem” for black holes is proved. For the case d ₁ = 2 a subclass of latent soliton solutions is singled out.     

Hidden Grassmann Structure in the XXZ Model II: Creation Operators

In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space of all quasi-local operators, which are products of the disorder field with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ), c(ζ) which mutually anti-commute and kill the “primary field”. Here we construct the creation counterpart b*(ζ), c*(ζ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t*(ζ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b*(ζ), c*(ζ), t*(ζ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants. Membre du CNRS On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992 Moscow, Russia Dedicated to the memory of Alexei Zamolodchikov     

Schrödinger Operators and the Zeros of Their Eigenfunctions

In n-dimensional Euclidean space let us be given an infinitely differentiable real valued function V that is bounded below. We associate with the formal operator that sends a complex valued function ψ into −div(grad ψ) + V ψ a uniquely defined self adjoint operator which we will denote by −Δ + V.     

Gravity waves accompanying supernova explosions

Gravitational radiation arising during the formation of a protoneutron star is studied. Here it is mainly large-scale nonuniformities that develop inside the star. The entropy and density profiles of such nonuniformities resemble the “mushroom cloud” of a nuclear explosion. A bubble of hot neutron matter floats to the surface of the star, like the “mushroom cloud” of an explosion in the earth’s atmosphere. Depending on the symmetry of the problem, from two to six bubbles can float upward at the same time. The characteristic masses of such bubbles are 0.01M _⊗ and the radial velocities reach ∼0.1c. The energy radiated in the form of gravitational waves in one cycle of bubbles floating to the surface is ∼10⁻² M _⊗ c ²−10⁻¹⁰ M _⊗ c ². Such cycles occur repeatedly as the neutron star cools. This phase can last up to seconds. The total energy radiated in the form of gravitational radiation can reach 10⁻¹ M _⊗ c ². Pis’ma Zh. éksp. Teor. Fiz. 64, No. 12, 817–822 (25 December 1996)