Fine structure of splitting of the separatrix of a nonlinear resonance

This report is a continuation of an analysis, initiated elsewhere V.V. Vecheslavov and B. V. Chirikov, Zh. éksp. Teor. Fiz. 114, 1516 (1998) [JETP 86, 823 (1998)], of the effect of splitting of the separatrix of a nonlinear resonance for the model of standard mapping, based on results of direct measurements of the splitting angle α(K), where K is the system parameter. Measurements were made in the previously used wide range 0.1≳α≳10⁻²⁰⁸ (1⩾K⩾0.0004), but with significantly higher relative (better than 10⁵⁰) and average (∼10⁻⁵⁵) accuracy. This procedure made it possible to substantially refine the effects observed in Ref. 1 and construct qualitatively new empirical dependences providing reliable extrapolation of the data obtained for the angle and the invariant in the intermediate asymptotic limit K≲10⁻² beyond the limits of the investigated region. The results obtained by us can be useful for further development of the theory of separatrix splitting and formation of the stochastic layer of a nonlinear resonance. Zh. éksp. Teor. Fiz. 116, 336–346 (July 1999)     

Random Homogenization and Singular Perturbations¶in Perforated Domains

The paper considers the singularly perturbed Dirichlet problem −ɛΔu ^ɛ+u ^ɛ=f in a randomly perforated domain Ω^ɛ, which is obtained from a bounded open set Ω in R ^N after removing many holes of size ɛ ^q . The perforated domain is described in terms of an ergodic dynamical system acting on a probability space. Imposing certain conditions on the domain, the behaviour of u ^ɛ when ɛ→ 0 in Lebesgue spaces L ⁿ (Ω) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size ɛ ^p with the intensity λɛ^{− r} is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach. Received: 15 December 1999 / Accepted: 14 April 2000     

The influence of a magnetic field on the inelastic properties of LiF crystals

The effect of weak magnetic fields (0.1–0.8 T) on the internal friction and Young’s-modulus defect of LiF crystals is investigated over a range of relative strain amplitudes ɛ ₀ from 10⁻⁶ to 10⁻⁴ at frequencies of 40 and 80 kHz. Experiments with these fields show that the internal friction increases and the effective elastic modulus decreases, indicating an increase in the plasticity of the samples. Plots are obtained of the internal friction versus the magnitude of the magnetic field at various values of the strain amplitude ɛ ₀. Fiz. Tverd. Tela (St. Petersburg) 41, 1035–1040 (June 1999)     

Exact Analytical Solution of the Schrödinger Equation for an N-Identical Body-Force System

A mathematical method is presented for solving the Schr?dinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated within the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyperradius x = (ξ₁² + ξ₂² + ⋯ + ξ_N−1²)^1/2 only, where ξ₁,ξ₂,…,ξ_N−1 are Jacobi relative coordinates which are functions of N-particle relative positions r₁₂,r₂₃,…,r_N1. The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax² + bx − (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schr?dinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles.     

Stimulated brillouin scattering of electromagnetic ion cyclotron waves in a plasma

Using the fluid model for the nonlinear response of ions, we have studied the nonlinear scattering of an electromagnetic ion cyclotron wave off the ion acoustic wave in a plasma. The low frequency nonlinearity arises through the parallel ponderomotive force on ions and the high frequency nonlinearity arises through the nonlinear current density of ions. For a typical nonisothermal plasma (T _e/T _i∼10) the threshold for this instability in a uniform plasma is ∼1mW/cm². At power densities ≳10² W/cm², the growth rate for backscatter turns out to be ∼10⁴s⁻¹.     

Effect of molecular nitrogen on the electron mobility in a mixture of argon and optically excited sodium vapor

A parametric study of the electron energy distribution function (EEDF) and the electron mobility in the mixture Na + Ar + N₂ is carried out. An analysis is made of the conditions that obtain in a photoplasma when the detachment of the mean electron energy from the neutral gas temperature is due to superelastic collisions (collisions of the second kind) with excited sodium atoms. The case of low ionization of the medium at low vibrational temperatures of the ground state of the nitrogen molecules is considered. To find the EEDF a numerical solution of the Boltzmann transport equation is carried out. It is found that in the indicated mixture the presence of nitrogen leads to a depletion of the EEDF in the region of efficient vibrational excitation of the molecules and promotes the formation of inversion in the EEDF ∂f(ɛ)/∂ɛ>0 in the energy range corresponding to the Ramsauer minimum in the cross section of elastic collisions of electrons with the argon atoms. It is shown that the nonequilibrium character of the EEDF leads to a complicated dependence of the electron mobility on the partial ratios of the components of the mixture, the degree of ionization of the medium, and the population of the resonantly excited sodium atoms. Zh. Tekh. Fiz. 69, 14–19 (April 1999)     

SCQGP in quark stars

From an analogy with non-relativistic degenerate QED plasma we make an estimate of the coupling strength of QGP hypothesized to be present in compact star interiors. At densities ranging from 3ρ ₀–10ρ ₀ (normal nuclear density ρ ₀=0.16 fm⁻³), quark matter is found to be strongly to intermediately coupled. The equation of state for QED plasma obtained via Pade approximation, modified to QGP, yields stable stellar sequences with maximum mass ≳2M _⊙ for B ^1/4≲215 MeV.     

The ground state problem in the infinite-U Hubbard model

The problem of the ground state of the electronic system in the Hubbard model for U=∞ is discussed. The author investigates the normal (singlet or nonmagnetic) N state of the electronic system over the entire range of electron densities n⩽1. It is shown that the energy of the N state ɛ ₀ ⁽¹⁾ (n) in a one-particle approximation, such as (e.g.) the extended Hartree-Fock approximation, is lower than the energy of the saturated ferromagnetic FM state ɛ _FM(n) for all n. The dynamic magnetic susceptibility is calculated in the random phase approximation, and it is shown that the N state is stable over the entire range of electron densities: The static susceptibility (ω=0) does not have a band singularity in the zero-wave vector limit q→0. A formally exact representation is obtained for the mass operator of the one-particle Green’s function, and an approximation of this operator is proposed: M _k(E)⋍λF(E), where λ=n(1−n)/(1−n/2)z is the kinematic interaction parameter, z is the number of nearest neighbors, and F(E) is the total single-site Green’s function. For an elliptical density of states the integral equation for F(E) is solved exactly, ad it is shown that the spectral intensity rigorously satisfies the sum rule. The calculated energy of the strongly correlated N state ɛ ₀(n)<ɛ _FM(n) for all n, and in light of this relationship the author discusses the hypothesis that the ground state of the system is the normal (singlet) state in the thermodynamic limit. The electron distribution function at T=0 differs significantly from the Fermi step; it is “smeared” along the entire energy spectrum, and discontinuities do not occur in the region of the chemical potential m. Fiz. Tverd. Tela (St. Petersburg) 39, 193–203 (February 1997)     

Internal waves in a compressible two-layer model atmosphere: Hamiltonian description

We consider slow, compared to the speed of sound, motions of an ideal compressible fluid (gas) in a gravitational field in the presence of two isentropic layers with a small specific-entropy difference between them. Assuming the flow to be potential in each of the layers (v _{1, 2} = ▿ϕ_{1, 2}) and neglecting the acoustic degrees of freedom (div($ \bar \rho $ \bar \rho (z)▿ϕ_{1, 2}) ≈ 0, where $ \bar \rho $ \bar \rho (z) is the average equilibrium density), we derive the equations of motion for the boundary in terms of the shape of the surface z = η(x, y, t) itself and the difference between the boundary values of the two velocity field potentials: ψ(x, y, t) = ψ₁ − ψ₂. We prove the Hamilto nian structure of the derived equations specified by a Lagrangian of the form ℒ = ∫$ \bar \rho $ \bar \rho (η)η _t ψdxdy − ℋ{η, ψ}. The system under consideration is the simplest theoretical model for studying internal waves in a sharply stratified atmosphere in which the decrease in equilibrium gas density due to gas compressibility with increasing height is essentially taken into account. For plane flows, we make a generalization to the case where each of the layers has its own constant potential vorticity. We investigate a system with a model dependence $ \bar \rho $ \bar \rho (z) ∝ e ^−2αz with which the Hamiltonian ℋ{η, ψ} can be represented explicitly. We consider a long-wavelength dynamic regime with dispersion corrections and derive an approximate nonlinear equation of the form u _t + auu _x − b[−$ \hat \partial _x^2 $ \hat \partial _x^2 + α²]^1/2 u _x = 0 (Smith’s equation) for the slow evolution of a traveling wave.     

Determination of low-energy parameters of neutron-proton scattering in the the shape-parameter approximation from present-day experimental data

On the basis of the total cross sections for neutron-proton scattering in the region of laboratory energies below 150 keV, the value of σ₀ = 20.4288(146) b was obtained for the total cross sections for neutron-proton scattering at zero energy. This value is in very good agreement with the experimental cross sections obtained by Houke and Hurst, but it is at odds with Dilg’s experimental cross section. By using the value that we found for σ₀ and the experimental values of the neutron-proton coherent scattering length f, the deuteron binding energy ɛ _t , the deuteron effective radius ρ _t (−ɛ _t , −ɛ _t ), and the total cross section in the region of energies below 5 MeV, the following values were found in the shape-parameter approximation for the low-energy parameters of neutron-proton scattering in the spin-triplet and spin-singlet states: a _t = 5.4114(27) fm, r _0t = 1.7606(35) fm, v _2t = 0.157 fm³, a _s = −23.7154(80) fm, r _0s = 2.706(67) fm, and v _2s = 0.491 fm³.     

Dielectric nonlinearity of crystalline Li2−x NaxGe4O9 (x≈0.23)

The dielectric nonlinearity of ferroelectric Li_2−x Na_xGe₄O₉ (x≈0.23) crystals is measured in the neighborhood of the phase transition temperatures. The magnitude of the nonlinear coefficient β is estimated from the shift in T _c and the reduction in ɛ _max under the influence of E ₌, from the dielectric nonlinearity in the paraphase, and from the temperature dependence of P _s in crystalline Li_2−x Na_xGe₄O₉ (x≈0.23). The resulting values of β are 1.87, 1.26, 2.17, and 1.17×10⁻⁹ (CGSE cm²)⁻², respectively. The mechanism for the phase transition in crystalline Li_2−x Na_xGe₄O₉ (x≈0.23) is discussed. Fiz. Tverd. Tela (St. Petersburg) 41, 1070–1072 (June 1999)     

Spectral Properties of Hypoelliptic Operators

 We study hypoelliptic operators with polynomially bounded coefficients that are of the form K=∑ _i=1 ^m X _i ^T X _i +X ₀+f, where the X _j denote first order differential operators, f is a function with at most polynomial growth, and X _i ^T denotes the formal adjoint of X _i in L ². For any ɛ>0 we show that an inequality of the form ||u||_δ,δ≤C(||u||_0,ɛ+||(K+iy)u||_0,0) holds for suitable δ and C which are independent of yR, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of Hérau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy|x≥|y|^τ−c,τ(0,1],cR}. Received: 30 July 2002 / Accepted: 18 October 2002 Published online: 25 February 2003 RID="*" ID="*" Present address: Mathematics Research Centre of the University of Warwick Communicated by A. Kupiainen     

Atmospheric gamma rays in the energy region 40–1000 GeV

A large stack of lead-emulsion sandwich detector assembly was flown over Hyderabad, India. High energy gamma rays at the float altitude were unambiguously identified from the cascades they induced, and their energies reliably determined by improved methods. From an analysis of 163 gamma rays of energy ≳ 30 GeV, it is found that the differential energy spectrum is represented by the power lawJ _r (E)= 129·4E ^{−2·62±0·12} photons m⁻² sr⁻¹sec⁻¹ GeV⁻¹ at an effective atmospheric depth of 14·3 g cm⁻²; this is the first reliable balloon measurement of atmospheric gamma rays in the energy range 40–1000 GeV. After correcting for the gamma rays radiated by the primary cosmic ray electrons, the production spectrum of gamma rays, resulting from the collisions of cosmic ray nuclei with air nuclei, at the top of the atmosphere isP _r (E, 0)=8·2 × 10⁻⁴ E^2.60±0.09 photons g⁻¹sr⁻¹sec⁻¹ GeV⁻¹. The atmospheric propagation of the electromagnetic component due to the cascade process is also derived from the gamma ray production spectrum.     

Thermally activated conductivity and current-voltage characteristic of dielectric phase in granular metals

Models of thermally activated linear and high-field nonlinear conductivity of a dielectric phase in granular metals (nanocomposites), i.e., aggregates of small metallic grains in a dielectric matrix, have been suggested. Given a sufficiently large spread of grain sizes, the temperature dependence of the nanocomposite conductivity should be described by a universal “power-1/2” law: G∝exp[−(T ₀/T)^1/2]. An analytical expression for T ₀ has been obtained. It is found that there are two regimes of nonlinear conductivity in a high electric field, namely, a low-field regime, when both the number and mobility of carriers change with the field strength, and a high-field regime, when only the mobility of carriers is variable. Analytical expressions for the nonlinear conductance in both regimes have been obtained. Zh. éksp. Teor. Fiz. 115, 1484–1496 (April 1999)     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

Spin-orbit coupling,spin-lattice relaxation,and spin-memory studies ofF-center-molecular-ion pairs in alkali halides

Summary The spin-orbit parameter Δ, spin-lattice relaxation timeT ₁, and spin-mixing parameter ɛ ofF _H(OH⁻) andF _H(CN⁻) centers in several alkali halides have been studied with magnetic circular dichroism at ∼2 K. A close comparison of the experimental results before and after optically induced association of theF center with the molecular ion has been made. In crystals of NaCl structure the negative spin-orbit parameter Δ changes little betweenF andF _H centers in the same host. For CsCl and CsBr two values of Δ have been derived forF _H(CN⁻) centers with axis parallel or perpendicular to the magnetic field. In all studied systems, the spin-lattice relaxation timeT ₁ is already shorter before aggregation compared toF formation. The spin-mixing parameter ɛ decreases slightly forF _H(OH⁻) compared toF centers, while it increases drastically forF _H(CN⁻) defects and reaches its maximum possible value ɛ=0.5 in cesium halides. First attempts to interpret these magneto-optical results will be presented. In honour of Prof. Fausto Fumi on the occasion of his retirement from teaching.     

On the longitudinal magnetic susceptibility of fractal ferrodielectrics

The longitudinal component of the magnetic susceptibility tensor χ _zz of a fractal ferrodielectric with dimension d _F = 3 − ɛ, where ɛ > 0, is calculated. The magnon dispersion in this case is shown to be a strongly anisotropic function of the parameter ɛ which, in turn, leads to nontrivial frequency dependences of χ _zz .     

Cross sections for single and double ionization of atomic helium under impact by fast multiply charged ions

Expressions for the cross sections for single and double ionization of atomic helium in collisions with fast multiply charged ions are obtained in the collision parameter range υ ²≫Z≳υ, υ ₀<c, where Z and υ are, respectively, the ion charge and ion velocity, υ ₀ is the characteristic velocity of electrons in the atomic helium ground state, and c is the velocity of light. Zh. Tekh. Fiz. 68, 1–7 (March 1998)     

Dielectric spectroscopy of PbZr1 ? xTixO3 solid solutions (0.495 ≤ x ≤ 0.51) in a temperature range of 100–300 K at frequencies from 1 × 10?2 to 2 × 107 Hz

The behavior of the relative permittivity ɛ/ɛ₀ of PbZr_{1 − x} Ti _x O₃ (PZT) solid solutions (0.495 ≤ x ≤ 0.51) in the temperature range of 100–300 K at frequencies from 1 × 10⁻² to 2 × 10⁷ Hz was investigated. Diffuse, strongly relaxing maxima at T = 230−260 K (x = 0.495−0.505) and 150–160 K (x = 0.510) were observed in the PZT studied. The relaxation processes are well described by the Vogel-Fulcher law, and the dielectric spectra are approximated by the Cole-Cole formula.