Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

The Discrete Spectrum in the Gaps of the Continuous One for Non-Signdefinite Perturbations with a Large Coupling Constant

Given two selfadjoint operators A and V=V ₊ -V _-, we study the motion of the eigenvalues of the operator A(t)=A-tV as t increases. Let α>0 and let λ be a regular point for A. We consider the quantities N ₊(λ,α), N _-(λ,α), N ₀(λ,α) defined as the number of the eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α>0. An abstract theorem on the asymptotics for these quantities is presented. Applications to Schr?dinger operators and its generalizations are given. Received: 9 April 1997 / Accepted: 26 August 1997     

On the Existence of Force-Free Magnetic Fields¶with Small Nonconstant α in Exterior Domains

The existence of force-free magnetic fields in the exterior domain of some compact simply connected surface S is proved via an iteration scheme. The iteration starts with an arbitrary exterior vacuum field, which contains flux tubes originating and ending on S. At one cross-section of such a flux tube with S an arbitrary function α is prescribed. For small values of α (in the H?lder-norm 1, λ; 0 < λ < 1) the iteration is shown to converge to a force-free field with the prescribed values of α in a flux tube which is close to the vacuum flux tube and α≡ 0 outside. The force-free field is close (in the H?lder- norm 1,λ) to the starting vacuum field, in particular, it has the same field line topology, the same boundary values on S and satisfies the same decay conditions in spatial infinity. It is in general three-dimensional and requires no continuous symmetries. Received: 20 June 1999 / Accepted: 25 October 1999     

BD-FRW models in the framework of Israel-Stewart-Hiscock theory

BD-FRW universe filled with imperfect fluid having bulk viscosity is investigated under the framework of Israel-Stewart-Hiscock causal theory. The field equations have been solved by using the relationφ=KR ^α whereK andα are constants, between the Brans-Dicke scalar fieldϕ and the scale factorR. This relation, in fact, leads to a constant deceleration parameterq. It is shown that the constancy of the deceleration parameter permits only two possibilities i.e. eitherH=constant withm=1 orm=(1+b −α)/(2(1+b) −α), irrespective of the value ofɛ.     

The Discrete Spectrum¶of the Perturbed Periodic Schrödinger Operator¶in the Large Coupling Constant Limit

Let A be a periodic Schr?dinger operator and let V ₀≥ 0 be a decaying potential. We study the number of the eigenvalues of the operator A(α) =A−αV ₀ inside a fixed interval (λ₁,λ₂). We obtain an asymptotic formula for as α→∞. Received: 12 September 2000 / Accepted: 22 November 2000     

Julia Sets in Parameter Spaces

Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {f _b (z)=λz+b z ²+z ³} _{b ∈ ℂ} contains quasi-conformal copies of the quadratic Julia set J(λz+z ²). As a corollary, we show that when the Julia set J(λz+z ²) is not locally connected (for example when z↦λz+z ² has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense G _δ subset of S ¹. Received: 22 September 2000 / Accepted: 16 January 2001     

The magnetic field and energy of an Abrikosov vortex in an anisotropic London superconductor

The behavior of a straight Abrikosov vortex in an anisotropic uniaxial London superconductor is studied. Analytical expressions are derived that approximately describe the magnetic field in three regions: the asymptotic region, where the distance r from the vortex line is greater than λΓ (λ is the London length and Γ is the anisotropy constant), the intermediate region λ<r<λΓ, and the region r<λ. It is found that in the intermediate region with high anisotropy the component of the magnetic field along the vortex line changes sign for a certain interval of angles between the vortex line and the anisotropy axis. Because of this the interaction of parallel vortices whose plane is parallel to the anisotropy axis has a minimum and a maximum. This means that numerous metastable vortex lattices can exist. Additional terms in the vortex self-energy are obtained, and although they are smaller than the leading logarithmic term, they display a different dependence on the angle between the vortex line and the anisotropy axis. Zh. éksp. Teor. Fiz. 111, 954–963 (March 1997)     

Material parameters for thermoelectric performance

The thermoelectric performance of a thermoelement is ideally defined in terms of the so-called figure-of-meritZ = α²σ/λ, where α,σ and λ refer respectively to the Seebeck coefficient, electrical conductivity and thermal conductivity of the thermoelement material. However, there are other parameters which are fairly good indicators of a material’s thermoelectric ‘worth’. A simple yet useful performance indicator is possible with only two parameters — energy gap and lattice thermal conductivity. This indicator can outline all potentially useful thermoelectric materials. Thermal conductivity in place of lattice thermal conductivity can provide some additional information about the temperature range of operation. Yet another performance indicator may be based on the slope of α vs. ln σ plots. α plotted against ln σ shows a linear relationship in a simplified model, but shows a variation with temperature and carrier concentration. Assuming that such a relationship is true for a narrow range of temperature and carrier concentration, one can calculate the slope m of α vs. ln σ plots against temperature and carrier concentrations. A comparison between the variation ofZT and slopem suggests that such plots may be useful to identify potential thermoelectric materials.     

A Classical Cosmological Model for Triviality

The aim of this paper is to study the triviality of λ ϕ⁴ theory in a classical gravitational model. Starting from a conformal invariant scalar tensor theory with a self-interaction term λ ϕ⁴, we investigate the effect of a conformal symmetry breaking emerging from the gravitational coupling of the large-scale distribution of matter in the universe. Taking in this cosmological symmetry breaking phase the infinite limit of the maximal length (the size of the universe) and the zero limit of the minimal length (the Planck length) implies triviality, i.e. a vanishing coupling constant λ. It suggests that the activity of the self-interaction term λ ϕ⁴ in the cosmological context implies that the universe is finite and a minimal fundamental length exists.     

Magnetostriction of the spin-Peierls compound CuGeO3

The longitudinal and transverse magnetostriction λ of the spin-Peierls compound CuGeO₃ in the b-c plane is measured in magnetic fields up to 20 T both above and below the transition temperature T _sp=14.3K. It is found that for a given crystallographic direction the value of magnetostriction is weakly dependent on the magnetic field direction. In the uniform U phase at T⩾T _sp, λ is negative and approximately equal in the b and c directions, while in the dimerized D phase at T<T _sp, λ is positive and λ _b >λ _c . At low temperatures, λ increases sharply at the magnetic-field-induced transition from the dimerized to the magnetic M phase. The experimental data allow estimation of the stress derivatives of the antiferromagnetic intrachain exchange interaction parameter and of the stress dependence of the critical field of the D-Mphase transition. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 3, 156–159 (10 August 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

On a class of three-dimensional corner eddies

Consider Stokes flow in a viscous fluid filling a corner, of angle 2α, bounded by two infinite plane walls. Assume that the flow is symmetrical about some plane which is normal to the walls bounding the corner. Since superposition is valid we may consider flows that are symmetrical about the plane bisecting the comer and those that are antisymmetrical about this plane. In either case it is shown that for a class of corner eddies, the corner flow is made up of an infinite sequence of eddies asr → 0, wherer is the radial distance from the corner. Moreover, the eigenvalues λ which determine the structure of the corner eddy fields satisfy the same equation, sin λα = ± λ sin 2α, that arises in the corresponding plane case. The three-dimensional velocity fields are, however, quite different from those seen in the plane case. In particular, in the symmetric case the streamlines are not closed and foci, rather than elliptic stagnation points, are the centres of the eddies in the plane of symmetry. These results represent, in this special context, a generalization to three-dimensions of Moffatt’s classical result for planar corner eddies.     

Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers

The phase φ of any wave is determined by the ratio x/λ consisting of the distance x propagated by the wave and its wavelength λ. Hence, the dependence of φ on λ constitutes an analogue system for the mathematical operation of division, that is to obtain the hyperbolic function f(ξ)≡1/ξ. We take advantage of this observation to decompose integers into primes and implement this approach towards factorization of numbers in a multi-path Michelson interferometer. This work is part of a larger program geared towards unraveling the connections between quantum mechanics and number theory. We briefly summarize this aspect.     

Mixing and sorting of bidisperse two-dimensional bubbles

We have examined a number of candidates for the minimum-surface-energy arrangement of two-dimensional clusters composed of N bubbles of area 1 and N bubbles of area λ ( λ≤1). These include hexagonal bubbles sorted into two monodisperse honeycomb tilings, and various mixed periodic tilings with at most four bubbles per unit cell. We identify, as a function of λ, the minimal configuration for N → ∞. For finite N, the energy of the external (i.e., cluster-gas) boundary and that of the interface between honeycombs in “phase-separated” clusters have to be taken into account. We estimate these contributions and find the lowest total energy configuration for each pair (N,λ). As λ is varied, this alternates between a circular cluster of one of the mixed tilings, and “partial wetting” of the monodisperse honeycomb of bubble area 1 by the monodisperse honeycomb of bubble area λ. Received 1 August 2002 RID="a" ID="a"e-mail: paulo@ist.utl.pt     

Stochastic resonance induced by a multiplicative periodic signal in a logistic growth model with correlated noises

The stochastic resonance (SR) phenomenon induced by a multiplicative periodic signal in a logistic growth model with correlated noises is studied by using the theory of signal-to-noise ratio (SNR) in the adiabatic limit. The expressions of the SNR are obtained. The effects of multiplicative noise intensity α and additive noise intensity D, and correlated intensity λ on the SNR are discussed respectively. It is found that the existence of a maximum in the SNR is the identifying characteristic of the SR phenomena. In comparison with the SR induced by additive periodic signal, some new features are found: (1) When SNR as a function of λ for fixed ratio of α and D, the varying of α can induce a stochastic multi-resonance, and can induce a re-entrant transition of the peaks in SNR vs λ; (2) There exhibits a doubly critical phenomenon for SNR vs D and λ, i.e., the increasing of D (or λ) can induce the critical phenomenon for SNR with respect to λ (or D); (3) The doubly stochastic resonance effect appears when α and D are simultaneously varying in SNR, i.e., the increment of one noise intensity can help the SR on another noise intensity come forth.      

Generalized Two-Mode Squeezed States and Some Nonclassical Characteristics

A class of generalized two-mode squeezed states |φ〉 is presented, which are generated from the generalized two-mode squeezing operator U(γ,λ) acting on the two-mode coherent state |α ₁,α ₂〉. We first investigate some mathematical properties of U(γ,λ) including the squeezing transformation under U(γ,λ), ket-bra integral form in the coordinate representation, normally ordered form. Then we evaluate some nonclassical characteristics of the state |φ〉 such as higher-order squeezing behavior, entanglement analysis and analytical expression of the Wigner function.     

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 ∞.      

Elementary Quantum Mechanics in a Space-Time Lattice

Studies of quantum fields and gravity suggest the existence of a minimal length, such as Planck length. It is natural to ask how the existence of a minimal length may modify the results in elementary quantum mechanics (QM) problems familiar to us. In this paper we address a simple problem from elementary non-relativistic quantum mechanics, called “particle in a box”, where the usual continuum (1+1)-space-time is supplanted by a space-time lattice. Our lattice consists of a grid of λ ₀×τ ₀ rectangles, where λ ₀, the lattice parameter, is a fundamental length (say Planck length) and, we take τ ₀ to be equal to λ ₀/c. The corresponding Schroedinger equation becomes a difference equation, the solution of which yields the q-eigenfunctions and q-eigenvalues of the energy operator as a function of λ ₀. The q-eigenfunctions form an orthonormal set and both q-eigenfunctions and q-eigenvalues reduce to continuum solutions as λ ₀→0. The corrections to eigenvalues because of the assumed lattice is shown to be O(l₀²)O(\lambda_{0}^{2}). We then compute the uncertainties in position and momentum, Δx, Δp for the box problem and study the consequent modification of Heisenberg uncertainty relation due to the assumption of space-time lattice, in contrast to modifications suggested by other investigations.     

Critical temperature of Bose-Einstein condensation for weakly interacting bose gas in a potential trap

It has been a long history to study Bose-Einstein condensation (BEC) of weakly in-teracting Bose gas, and several theoretical models have been developed to research uni-form and weakly interacting Bose gas. Ref. [1] summarized all of these models and the corresponding results, which gave a derivation of critical temperature from ideal case 1/30Tc c n,?T = α (1) with a wide spread of parameter c from 0.7 to 2.33, where α is the scattering length of s wave and n is atom number density. Due…     

Quantum Brownian Motion in a Simple Model System

We consider a quantum particle coupled (with strength λ) to a spatial array of independent non-interacting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at large times for small, but finite λ. Our proof relies on an expansion around the kinetic scaling limit ( l\searrow 0{\lambda \searrow 0}, while time and space scale as λ⁻²) in which the particle satisfies a Boltzmann equation. We also show an equipartition theorem: the distribution of the kinetic energy of the particle tends to a Maxwell-Boltzmann distribution, up to a correction of O(λ²).