On the derivation of the Boltzmann-Landau equation from the quantum mechanical hierarchy

The general solution of the equation of motion for the quantum mechanical distribu tion functionf ₂(r ₁ P ₁,r ₂ p ₂;t)in the two particle space is given by means of the Schrödinger scattering functions. A special initial condition leads to the usual Boltzmann equation plus density correction terms, which depend on the scattering matrixt(p′,p). In the long wavelength limit and in lowest order oft(p′,p) the Landau corrections to the simple Boltzmann streaming part are obtained.     

Temperature dependence of the surface upper critical field of superconducting alloys

The temperature dependence of the surface upper critical field,H _c3, nearT _c is calculated for arbitrary values of the mean free pathl by taking into account the fourthorder term of the generalized Ginzburg-Landau theory. For finitel the boundary condition is modified such that the normal derivative of the energy gap at the surface becomes positive. The slope of the curveH _c3/H _c2 versust=T/T _c att=1 is found to decrease monotoneously from zero to ?1.040 as one goes from the “dirty” to the “clean” limit.     

Scattering of Josephson plasmons on random quantum jumpers in a disordered <Emphasis Type="Italic">I</Emphasis>-layer of an <Emphasis Type="Italic">S</Emphasis>–<Emphasis Type="Italic">I</Emphasis>–<Emphasis Type="Italic">S</Emphasis> junction

A formula for the relaxation time of Josephson plasmons on random quantum jumpers, i.e., quantum resonant-percolation trajectories (QRPT) in a disordered I-layer of a tunnel S–I–S junction is derived. Domain Ω_r (μ ? E₀, c), in which the strongest plasmon damping takes place, is plotted in the plane of parameters (μ ? E₀, c).     

On Vanishing Theorems for Vector Bundle Valued <Emphasis Type="Italic">p</Emphasis>-Forms and their Applications

Let F : [0, ∞) → [0, ∞) be a strictly increasing C ² function with F(0) = 0. We unify the concepts of F-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce F-Yang-Mills fields, F-degree, F-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds. When \({F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1,}\) and \({1-\sqrt{1-2t},}\) the F-Yang-Mills field becomes an ordinary Yang-Mills field, p-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce the E _F,g ?energy functional (resp. F-Yang-Mills functional) and derive the first variational formula of the E _F,g ?energy functional (resp. F-Yang-Mills functional) with applications. In a more general frame, we use a unified method to study the stress-energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors are naturally linked to F-conservation laws and yield monotonicity formulae, via the coarea formula and comparison theorems in Riemannian geometry. Whereas a “microscopic” approach to some of these monotonicity formulae leads to celebrated blow-up techniques and regularity theory in geometric measure theory, a “macroscopic” version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems for p?forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems for F?harmonic maps (which include harmonic maps, p-harmonic maps, exponentially harmonic maps, minimal graphs and maximal space-like hypersurfaces, etc.), F?Yang-Mills fields, extended Born-Infeld fields, and generalized Yang-Mills-Born-Infeld fields (with the plus sign and with the minus sign) on manifolds, etc. As another consequence, we obtain the unique constant solution of the constant Dirichlet boundary value problems on starlike domains for vector bundle-valued 1-forms satisfying an F-conservation law, generalizing and refining the work of Karcher and Wood on harmonic maps. We also obtain generalized Chern type results for constant mean curvature type equations for p?forms on \({\mathbb{R}^m}\) and on manifolds M with the global doubling property by a different approach. The case p = 0 and \({M=\mathbb{R}^m}\) is due to Chern.     

Generalized Gor’kov relation for two-band superconductor MgB<Subscript>2</Subscript>

A generalized Gor’kov relation between the upper critical magnetic field H _c2, the residual resistivity, and the parameters of the two-band electronic structure from the pure limit to the dirty limit was derived for two-gap superconductor MgB₂. The ratio of the relaxation times τ _π /τ _σ of π-and σ-band electrons for MgB₂ samples with various defect levels was determined from experimental data.     

Depolarizing collisions in ytterbium vapor: Isotropic and anisotropic relaxation

Isotropic depolarizing collisions are studied using a stimulated photon echo with a specific polarization of the excitation radiation pulses in a mixture of ytterbium with krypton for the J = 1 ? J = 0 transition of ¹⁷⁴Yb. The difference between the relaxation rates of orientation and alignment γ _b ⁽²⁾ ? γ _b ⁽¹⁾ of the ³ P ₁(6s6p) ¹⁷⁴Yb level is measured as a function of the krypton pressure. The collision photon echo at the J = 1 ? J = 0 transition induced by the anisotropic relaxation is studied for the Yb + Xe mixture. The power of the collision echo increases from zero with the addition of a buffer gas to ytterbium, reaches an optimal level, and decreases with an increase in the buffer gas pressure. The polarization of this collision-induced echo differs from the polarization of the conventional echo. The experimental results are in qualitative agreement with theoretical predictions.     

Temperature dependence of the magnetization in the mixed state of superconducting alloys

The theory of Abrikosov of the immediate subcritical region in the mixed state is extended by means of the generalized Ginzburg-Landau theory to temperatures belowT _c and arbitrary mean free pathl of the electrons. The results for the magnetization and the free energy can still be presented in the form derived by Abrikosov apart from the fact thatκ is replaced by a new parameterκ ₂(T). The slope of the curveκ ₂/κ versust=T/T _c att=1 is found to decrease monotonically from ?0.105 to — (1.367-0.136κ ^?2) asl increases from zero to infinity. The results for short mean free paths are consistent with experiment, in particular,κ ₂ is always larger thanκ in the vicinity ofT _c . But the experimental temperature dependence ofκ ₂/κ in pure Nb is much higher than the theoretical one.     

Self-similar magnetic structures and magnetic flux “Giant” creep

The penetration of a magnetic flux into a type-II high-T _c superconductor occupying the half-space x > 0 is considered. At the superconductor surface, the magnetic field amplitude increases in accordance with the law b(0, t) = b ₀(1 + t)^m (in dimensionless coordinates), where m > 0. The velocity of penetration of vortices is determined in the regime of thermally activated magnetic flux flow: v = v ₀exp?ub;?(U _0/T )(1-b?b/?x)?ub;, where U ₀ is the effective pinning energy and T is the thermal energy of excited vortex filaments (or their bundles). magnetic flux “Giant” creep (for which U ₀/T? 1) is considered. The model Navier-Stokes equation is derived with nonlinear “viscosity” v ∝ U ₀/T and convection velocity v _f ∝ (1 ? U ₀/T). It is shown that motion of vortices is of the diffusion type for j → 0 (j is the current density). For finite current densities 0 < j < j _c, magnetic flux convection takes place, leading to an increase in the amplitude and depth of penetration of the magnetic field into the superconductor. It is shown that the solution to the model equation is finite at each instant (i.e., the magnetic flux penetrates to a finite depth). The penetration depth x _eff ^A (t) ∝ (1 + t)^{(1 + m/2)/2} of the magnetic field in the superconductor and the velocity of the wavefront, which increases linearly in exponent m, exponentially in temperature T, and decreases upon an increase in the effective pinning barrier, are determined. A distinguishing feature of the solutions is their self-similarity; i.e., dissipative magnetic structures emerging in the case of giant creep are invariant to transformations b(x, t) = β^m b(t/β, x/β^{(1 + m/2)/2}), where β > 0.     

Multivariate pressure effects on an electron hopping process in ferroelectric KTa<Subscript>1−x</Subscript>Nb<Subscript>x</Subscript>O<Subscript>3</Subscript>

The effect induced by the presence of a polaron related relaxation process on the dielectric properties of a ferroelectric KTa_1?x Nb _x O₃ (KTN) crystal was investigated (10^-2?10⁶ Hz, at 300?375 K) using broadband dielectric spectroscopy. Characterization of the process using just the standard frequency domain dielectric parameters can nonetheless provide penetrating insight into its nature and origins. The three parameters, namely: relaxation time (τ), Cole-Cole loss broadening (α), and dielectric strength (Δ?) provide each one in its own way, much useful and often overlooked information. The Activation Energy along with the Meyer-Neldel dependance, both extracted from τ serve to illuminate the dynamic properties. At the same time, α and especially the combined α(lnτ) relationship, expose the fractal structure of the underlying landscape. Finally, the static parameter Δ?, enables quantification of the dipolar correlations. Hydrostatic pressure (up to 7.5 kbar) was applied to gently perturb the system and observe the outcome on all of the various parameters. This additional degree of freedom allows for a much more comprehensive exploration of the phase space behavior of the system.     

Dual structures and transition regions in <Emphasis Type="Italic">x</Emphasis>PbTiO<Subscript>3</Subscript>-(1−<Emphasis Type="Italic">x</Emphasis>)Pb(Zn<Subscript>1/3</Subscript>Nb<Subscript>2/3</Subscript>)O<Subscript>3</Subscript> crystals

Model concepts of dual structures are developed as applied to crystals of xPbTiO₃-(1?x)Pb(Zn_1/3Nb_2/3)O₃ solid solutions in the range 0 ≤ x ≤ 0.08. The conditions of the formation of dual structures upon partial and complete relaxation of internal mechanical stresses are considered. A new model of transition regions is proposed for analyzing several variants of changing the unit cell parameters that satisfy the condition of complete stress relaxation inside the transition regions in crystals at concentrations x = 0.045 and 0.08. The influence of the intermediate phase P4mm and stability of its 90° domain structure on the formation of dual structures at x ≥ 0.045 is discussed.     

Superconducting properties of superlattices containing ferromagnetic layers

Using the microscopic theory formulated by de Gennes and extended by Takahashi and Tachiki, we calculate the transition temperatureT _c and the pair functionF for the superlattices consisting of superconducting and ferromagnetic layers. Superconducting layers. (s) and ferromagnetic layers (f) are modeled byV _s ≠0 andI _m,s =0 andV _f =0 andI _m,f ≠0, whereV _s .(V _f ) is the BCS coupling constant andI _m,s (I _m,f ) is the molecular field fors (f) layers.     

Effect of cation substitution on the lattice dynamics and ferroelectric instability of cubic BaTiO<Subscript>3</Subscript> and BaZrO<Subscript>3</Subscript> doped with Bi and La ions

The spectra of lattice vibration frequencies of solid solutions Ba_{1 ? x} Bi_2x/3? _x/3Ti(Zr)O₃ and Ba_{1 ? x} La _x Ti(Zr)_{1 ? x/4}? _x/4O₃ are calculated in terms of a generalized Gordon-Kim model with inclusion of the dipole and quadrupole polarizabilities. Over the entire concentration range, the calculated phonon spectra contain a ferroelectric soft mode. The effect of various interactions on the ferroelectric instability of these solid solutions is studied. It is shown that the character of ferroelectric instability is largely determined by the mechanism of charge compensation.     

The universal dependence of the relative voltage of static breakdown in gases on a generalized variable

In terms of the Townsend model of a static electrical gas discharge, the ratio of breakdown voltage U _B to its minimal value U _min is expressed as a function of generalized variable Ξ _B ? f(Ξ _b ), such that the function depends on neither the gas type nor the cathode material.     

Self-similar distribution in “giant” magnetic flux creep

The problem of magnetic field penetration into the half-space is considered in parallel geometry in an external magnetic field increasing with time in accordance with the law B(0, t, τ₀ = B _{c 1} (1 + t/τ₀) ^m , m ≥ 0, t ≥ 0 (τ ₀ is the time of magnetic flux redistribution and B _{c 1} is the lower critical field). It is assumed that the flow of vortices is thermally activated in the “giant” creep mode (i.e., for weak pinning creep and high temperatures). A model equation is derived for describing the magnetic flux evolution. Analytic formulas are obtained for the depth and velocity of magnetic field penetration. It is shown that the giant creep regime is stable for 0 ≤ m ≤ 1/2.     

Ferroelectric phase transitions in ionic conductors based on bismuth vadanate

Ceramic solid solutions (Bi_{1 ? x} La _x )₄V₂O_{11 ? z} (I), Bi₄(V_{1 ? x} Fe _x )₂O_{11 ? y} (II), and (Bi_{1 ? x} La _x )₄(V_0.96Fe_0.04)₂O_{11 ? y} (III) (x = 0–0.3, step Δx = 0.02) are prepared using solid-phase synthesis. The concentration and temperature ranges of stabilization of different polymorphic modifications, including the ranges of concentrations x corresponding to the stabilization of the ferroelectric phase, are established. It is revealed that an increase in the concentration x in the region of existence of the pseudoorthorhombic phase α of the solid solutions studied leads to a decrease in the transition temperature, smearing of the transition, and an increase in the width of the thermal hysteresis of the ferroelectric phase transition. The effect of compressing of the domain walls by oxygen vacancies was revealed in the samples from the region of existence of the ferroelectric α phase, and the effect of dielectric relaxation was detected in the samples from the region of existence of the orthorhombic phase β.     

Evolution of an optically pumped ensemble of cold ground-state atoms in weak light fields

The evolution of multipole moments is analyzed for optically pumped cold ground-state atoms in the limit of low saturation of a closed j₀ → j₁ dipole transition. The longest multipole-moment relaxation times are analyzed as functions of ellipticity and frequency detuning from resonance for transitions with j₀ ? 5. The qualitative difference between the evolution toward steady-state Zeeman sublevel populations and dynamics of transient spontaneous emission is demonstrated for transitions of the following types: j → j?1, j → j with integer j, j → j with half-integer j, and j → j + 1.     

Anomalous temperature dependence of magnetic relaxation in YBa<Subscript>2</Subscript>Cu<Subscript>2</Subscript>O<Subscript>7 − δ</Subscript> oxygen-deficient single crystals

The dependence of the magnetization relaxation rate S = ?d lnM/dlnt on temperature T is measured in YBa₂Cu₃O_{7 ? δ} samples with various oxygen concentrations. It is found that the S(T) curve changes qualitatively when oxygen deficit δ exceeds the threshold value δ_th = 0.37. For δ < δ_th (T _c > 60 K, where T _c is the superconducting transition temperature), function S(T) has the well-known peak at T/T _c = 0.4. For δ > δ_th (at T _c < 51 K), this peak transforms into a plateau and a new sharp peak appears at T/T _c = 0.1. The threshold value δ_th of the oxygen deficit corresponds to the transition of the sample from the disordered state into the ordered state of oxygen vacancies. We consider the change in the shape of the S(T) curve as a macroscopic manifestation of this transition.     

Spectral representation for singlet and triplet green's functions

Spectral representations of special Green functions are given explicitly. We consider the density correlation functionG ₂(x ₁ η ₁ x ₂ η ₂,x ₁ ⁺ η ₁ x ₂ ⁺ η′₂) and the functionG ₂(x ₁ η ₁ x ₁ ^? η₂,x ₂ η ₁ ^′ x ₂ ^? η ₂ ^′ Coupling the field operators Ψ^? (x, η), Ψ(x, η) to singlet and triplet operatorsA _{SMs TT3} (x), we obtain spectral representations for theseG-functions. The formulae derived may be of use when studying the system of equations for the Green functions, which describe many particle systems from a microscopic point of view.