Algorithmic Reduction of Poincaré–Dulac Normal Forms and Lie Algebraic Structure

The Poincaré–Dulac normal form of a given resonant system is in general nonunique; given a specific normal form, one would like to further reduce it to a simplest normal form. In this Letter we give an algorithm, based on the Lie algebraic structure of the set of normal forms, to obtain this. The algorithm can be applied under some condition, nongeneric but often met in applications. When applicable, it is only necessary to solve linear equations, and is more powerful than the one proposed in previous work by the same author [Lett. Math. Phys. 42 (1999), 103–114; and Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), 461–514].     

Higher-order <Emphasis Type="Bold">?</Emphasis> corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order ? contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of ?. In different theoretical work the trace formulae have been extended to higher orders of ?. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with ? corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order ? corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system. Received 10 September 2001 and Received in final form 3 January 2002     

Universality of a dynamical percolative approach to 1/f γ noise

A dynamical percolative model explaining the universality of 1/ f ^γ noise is reported. Exponents γ ranging from 0 to 2 are obtained under the hypothesis that noise originates from random switching events between two ON-OFF states in elemental parts (switchers) of a physical system. The usual noise behaviour with γ very close to 1 in an arbitrarily wide frequency range is obtained assuming a statistical distribution of switcher relaxation time τ proportional to τ ^-1 , as in McWhorter's model. The impact of these results with respect to recent self-organised criticality models is discussed. Received 6 November 2000 and Received in final form 22 May 2001     

Universal incommensurate structures

Certain phase transitions in quasiperiodic systems are characterized byuniversal structures. In these cases the functional form of the order parameter corresponding to the modulated phase,P(r), is determined by the symmetry properties of the system and is independent of the details of the associated Landau-Ginzburg model. Here we consider a simple one-dimensionalXY-like model corresponding to this type of phase transition. The universal modulated structure of this model is calculated numerically at various points along the critical line.     

Ewald method for polytropic potentials in arbitrary dimensionality

The Ewald summation technique is generalized to power-law 1/| r | ^k potentials in three-, two- and one-dimensional geometries with explicit formulae for all the components of the sums. The cases of short-range, long-range and ‘marginal’ interactions are treated separately. The jellium model, as a particular case of a charge-neutral system, is discussed and the explicit forms of the Ewald sums for such a system are presented. A generalized form of the Ewald sums for a non-cubic (non-square) simulation cell for three- (two-) dimensional geometry is obtained and its possible field of application is discussed. A procedure for the optimization of the involved parameters in actual simulations is developed and an example of its application is presented.     

Scaling in the ab resistivity of TmBaCuO single crystals in the normal state

For single crystals of the system TmBa₂Cu₃O_x with 6.37<x<6.71, a scaling transformation is found which reduces the temperature dependences of the derivative of the resistivity in the ab plane in the normal state to a universal form which is independent of oxygen content. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 11, 699–703 (10 December 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.     

Measurement of the thermal diffusivity of the surface layers of opaque solid objects

A method is discussed for the layer-by-layer determination of the thermal diffusivity a and thickness d of the subsurface layers of opaque solid objects from measurements of the phase difference of signals of two centered ring-shaped p-n photodiode junctions by successively varying the coefficient of angular magnification of the optical system of the device in which a radiation source with Gaussian cross-sectional beam power density creates a thermal object on the surface of the object in the form of harmonically varying concentric heat waves. Estimates are found for the frequency range and minimum angular magnification corresponding to the limits of validity of the relations used to calculate a and d. Zh. Tekh. Fiz. 67, 128–130 (August 1997)     

Theory and computer simulation of tweed texture

Abstract

An analytical theory of the ordering interaction J(R _ij ) in structural phase transitions mediated by elastic relaxation in the material is outlined. The ordering process in cell i sets up a local stress field due to the sizes, shapes or displacements of atoms or atomic groups, which is propagated elastically to a distant cell j. The atomistic theory for ferro- and antiferro-elastic transitions takes into account two types of singularity, one due to elastic anisotropy and the other to the Zener interaction J _z of infinite range in ferroelastic transitions. The form of J _k in Fourier space is highly anisotropic with a few “soft” directions coinciding with the orientation of twin boundaries. The asymptoptic J(R) at large R is shown to be very anisotropic as well and decays as R ^?3 in ferroelastic and R ^?5 in antiferroelastic systems.

Computer simulations for a three-dimensional model of about 29,000 particles show a strong tendency to form tweed texture, as observed experimentally. Well above the structural phase transition temperature, the strain fluctuations show well-developed embryos of the tweed texture. On quenching to below the transition temperature, a pronounced micro-twinning appears which follows almost exactly the shape of the embryos and then develops towards a stripe texture. After a certain time needle-shaped domains are formed and a peculiar step-wise process of generating new stripes is observed.     

Homogeneous Manifold,Loop Algebra,Coupled KdV System and Generalised Miura Transformation

Abstract

Coupled KdV equations are deduced by considering the homogeneous manifold corresponding to the homogeneous Heisenberg subalgebra of the Loop group (L(S ¹, SL(2,C)). Utilisation of Birkhoff decomposition and further subalgebra consideration leads to a new generalised form of Miura map and two sets of modified equations. A second set of Miura transformation can also be generated leading to complicated form of coupled integrable systems.     

Concentration dependence of the compositional order-disorder phase-transition temperature in the solid solution PbSc0.5Nb0.5O3-PbSc0.5Ta0.5O3

The degree of compositional order S of crystals of the complete series of solid solutions (1−x)PbSc_0.5Nb_0.5O₃-xPbSc_0.5Ta_0.5O₃ grown under identical conditions is determined by x-ray examination. The form of the dependence S(x) and the variations of S for different compositions, after high-temperature annealing, indicate that the concentration dependence of the compositional order-disorder phase-transition temperature is close to linear in the given system of solid solutions. Fiz. Tverd. Tela (St. Petersburg) 41, 1828–1830 (October 1999)     

On the ferromagnetism of high-spin states

It is established on the basis of the idea of strong interaction within the unit cell that ferromagnetic instability is possible in a system with hopping between d-electron states of atoms that are in high-spin states and form a bcc lattice. The phase diagram of the existence of ferromagnetic ordering as a function of the degree of filling of the d shell is constructed. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 4, 275–280 (25 August 1998)     

The thermodynamics of energy-storing photoprocesess

A solution is found for the most general formulation of the problem of the thermodynamic limitations on the rate of increase of the free energy of a closed photochemical system interacting with a heat bath and a radiation field. A reaction of the form A ₁ ^* +A ₂→A ₃+A ₄ is used to compare the thermodynamically maximum energy yield of the process of storing radiant energy with the “kinetic” limit of the same quantity. Zh. Tekh. Fiz. 67, 54–58 (June 1997)     

Low-temperature metal-insuslator transition and magnetic properties in the VxMn1−x S disordered system

A study of the structure and electrical and magnetic properties of the V_xMn_1−x S disordered system is reported. The existence of a low-temperature metal-insulator transition for Fermi-glass 0.4<x<0.5 compositions in paramagnetic phase, which is accompanied by a change in the structure and magnetic properties, has been established. An analysis of the magnetic properties permits a conjecture that current carriers become delocalized in these solid solutions at the metal-insulator transition temperature to form small ferromagnetically ordered regions (ferrons). Fiz. Tverd. Tela (St. Petersburg) 39, 1428–1431 (August 1997)     

DNA molecule as an elastic Heisenberg chain

A DNA molecule is simulated by an anisotropic elastic fiber which defines the configuration of the molecule central line and is supplemented with a chain of quantum two-level systems imitating hydrogen bonds between two polynucleotide chains in the DNA double helix. The system Hamiltonian consists of Kirchhoff’s classical elastic energy and the energy of a quantum anisotropic chain of “spins” 1/2. The two-level systems and macroscopic vector variables which determine the conformation of the central line are coupled by a classical vector field q, which is introduced to take into account the existence of two polynucleotide strands. Averaging over fast (microscopic) variables yields an effective potential U(q). In the approximation of weak coupling between the systems, the spectrum of elementary excitations and effective potential U(q) have been calculated in explicit form. The relation between elementary excitations in the “magnetic” subsystem and so-called breathing modes [C. Mandel, N. R. Kallenbach, and S. W. Englander, J. Mol. Biol. 135, 391 (1980); G. Manning, Biopolymers 22, 689 (1983)] corresponding to low-frequency excitations in DNA molecules is discussed. Zh. éksp. Teor. Fiz. 111, 1833–1844 (May 1997)     

Monitoring the pH-Dependent Oximation of Methyl Ethyl Ketone and Benzaldehyde by in situ Fourier Transform IR Spectroscopy in a Heterogeneous Liquid–Liquid Two-Phase System

In situ Fourier transform IR spectroscopy was a useful tool in monitoring the pH-dependent and two-step oximation of methyl ethyl ketone and benzaldehyde in a liquid–liquid two-phase system, one phase of which was water. Carrying out the oximations at pH 8, the oximation was complete within 45 min and the corresponding carbonyl compound–hydroxylamine adduct (2) could be detected as an intermediate, but at pH 10, the corresponding oxime (3) was formed faster, practically without the appearance of the intermediate. At pH ～ 2.5 using methyl ethyl ketone, the protonated form of the corresponding oxime (3'a) developed gradually without the transient appearance of the intermediate (2). pH dependence of the 3a → 3'a and 3'a → 3a interconversions were studied in separate experiments.     

Limiting Behavior of Eigenvectors of Large Wigner Matrices

A new form of empirical spectral distribution of a Wigner matrix W _n with weights specified by the eigenvectors is defined and it is then shown to converge with probability one to the semicircular law. Moreover, central limit theorem for linear spectral statistics defined by the eigenvectors and eigenvalues is also established under some moment conditions, which suggests that the eigenvector matrix of W _n is close to being Haar distributed.     

Magnetic susceptibility of multilayered carbon nanotubes

A theoretical study of the magnetic susceptibility of multilayered carbon nanotubes in fields both parallel, H _∥, and perpendicular, H _⊥, to the tube axis has been carried out disregarding electron-electron interaction. The temperature dependences of the magnetic susceptibility obtained exhibit a nontrivial form for H _∥, which is related to the quasi-one-dimensionality of such a system as the nanotube. The dependences of the magnetic susceptibility on chemical potential χ(μ) have also been derived. At low temperatures, χ(μ) has sharp peaks in fields H _∥, which is connected with the presence of singularities in the density of states of nanotubes. The effect of interlayer coupling on magnetic susceptibility of small-radius tubes has been investigated numerically. Fiz. Tverd. Tela (St. Petersburg) 40, 1950–1954 (October 1998)