Power law distributions and dynamic behaviour of stock markets

A simple agent model is introduced by analogy with the mean field approach to the Ising model for a magnetic system. Our model is characterised by a generalised Langevin equation = F ϕ + G ϕ t where t is the usual Gaussian white noise, i.e.: t t ^′ = 2Dδ t-t ^′ and t = 0. Both the associated Fokker Planck equation and the long time probability distribution function can be obtained analytically. A steady state solution may be expressed as P ϕ = exp{ - Ψϕ - ln G(ϕ)} where Ψϕ = - F/ G dϕ and Z is a normalization factor. This is explored for the simple case where F ϕ = Jϕ + bϕ² - cϕ³ and fluctuations characterised by the amplitude G ϕ = ϕ + ɛ when it readily yields for ϕ≫ɛ, a distribution function with power law tails, viz: P ϕ = exp{2bϕ-cϕ² /D}. The parameter c ensures convergence of the distribution function for large values of ϕ. It might be loosely associated with the activity of so-called value traders. The parameter J may be associated with the activity of noise traders. Output for the associated time series show all the characteristics of familiar financial time series providing J < 0 and D≈ | J|. Received 25 July 2000     

Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the (n) vector ϕ⁴ model, with long-range interaction decaying algebraically with the interparticle distance r like r ^{-d - σ}, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature T _c. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. 0 < σ < 2 and it turns out to be exponential in case of short-range interaction i.e.σ = 2. The results are valid for arbitrary dimension d, between the lower ( d _< = σ) and the upper ( d _> = 2σ) critical dimensions. Received 2 July 2001 and Received in final form 4 Septembre 2001     

Potentials of pointlike particles in gauge theory with a dilaton

I examine the potential of a pointlike particle carrying SU (N _c) charge in a gauge theory with a dilaton. The potential depends on boundary conditions imposed on the dilaton: For a dilaton that vanishes at infinity the resulting potential is a regulatized Coulomb potential of the form (r+r _ϕ)⁻¹, withr _ϕ, inversely proportional to the decay constant of the dilaton. Another natural constraint on the dialaton ϕ is independence of (1/g ²) exp(ϕ/f_ϕ) from the gauge couplingg. This requirement yields a confining potential proportional tor.     

Current-voltage characteristics for a Peierls-conducting point contact

Current-voltage (J-V) and differential-conductivity-voltage ( dJ/dV-V) characteristics are analytically calculated at zero temperature for a point contact consisting of: two Peierls conductors P ( = 1, 2) separated by an insulator (I). Here P is a conductor with charge density wave (CDW). The J-V and dJ/dV-V characteristics depend on the CDW phases ( = 1, 2) in the mean field approximation. To calculate them analytically we assumed, = ≡Δ where ( = 1, 2) are the energy gaps of P ( = 1, 2). The current J has a discontinuous jump at eV = 2Δ for ϕ ₁ = ϕ ₂≠ 0. The differential conductivity dJ/dV has a singularity at eV = 2Δ for ϕ ₁ = ϕ ₂≠ 0. The relation J(V,ϕ ₁,ϕ ₂) = - J(- V,ϕ ₁ + π,ϕ ₂ + π) is obtained. Received 4 July 2001 and Received in final form 13 September 2001     

Two-point correlation function in systems with van der Waals type interaction

The behavior of the bulk two-point correlation function G(;T| d ) in d-dimensional system with van der Waals type interactions is investigated and its consequences on the finite-size scaling properties of the susceptibility in such finite systems with periodic boundary conditions is discussed within mean-spherical model which is an example of Ornstein and Zernike type theory. The interaction is supposed to decay at large distances r as r ^{- (d + σ)}, with 2 < d < 4, 2 < σ < 4 and d + σ≤6. It is shown that G(;T| d ) decays as r ^{- (d - 2)} for 1 ≪r≪ξ, exponentially for ξ≪r≪r ^*, where r ^* = (σ - 2)ξlnξ, and again in a power law as r ^{- (d + σ)} for r≫r ^*. The analytical form of the leading-order scaling function of G(;T| d ) in any of these regimes is derived. Received 28 May 2001     

The multiphoton ionization rate and the energy shift of atoms interacting with weak dichromatic fields with commensurate frequencies are simple functions of the phase difference

By implementing a time-independent, nonperturbative many-electron, many-photon theory (MEMPT), cycle-averaged complex eigenvalues were obtained for the He atom, whose real part gives the field-induced energy shift, Δ(ω ₁, F ₁;ω ₂, F ₂,ϕ), and the imaginary part is the multiphoton ionization rate, Γ(ω ₁, F ₁;ω ₂, F ₂,ϕ), where ω is the frequency, F is the field strength and ϕ is the phase difference. Through analysis and computation we show that, provided the intensities are weak, the dependence of Γ(ω ₁, F ₁;ω ₂, F ₂,ϕ) on ϕ is simple. Specifically, for odd harmonics, Γ varies linearly with cos(ϕ) whilst for even harmonics it varies linearly with cos(2ϕ). In addition, this dependence on ϕ holds for Δ(ω ₁, F ₁;ω ₂, F ₂,ϕ) as well. These relations may turn out to be applicable to other atomic systems as well, and to provide a definition of the weak field regime in the dichromatic case. When the combination of (ω ₁, F ₁) and (ω ₂, F ₂) is such that higher powers of cos(ϕ) and cos(2ϕ) become important, these rules break down and we reach the strong field regime. The herein reported results refer to Γ(ω ₁, F ₁;ω ₂, F ₂,ϕ) and Δ(ω ₁, F ₁;ω ₂, F ₂,ϕ) for He irradiated by a dichromatic ac-field consisting of the fundamental wavelength λ = 248 nm and its 2nd, 3rd and 4th higher harmonics. The intensities are in the range 1.0×10¹²-3.5×10¹⁴ W/cm², with the intensity of the harmonics being 1-2 orders of magnitude smaller. The calculations incorporated systematically electronic structure and electron correlation effects in the discrete and in the continuous spectrum, for ¹S, ¹P, ¹D, ¹F, ¹G, and ¹H two-electron states of even and odd parity. Received 9 July 2000 and Received in final form 2 November 2000     

Two-particle dispersion in model two-dimensional velocity fields

We consider two-particle dispersion in a velocity field, where the relative two-point velocity scales according to v ²(r) ∝r ^α and the corresponding correlation time scales as τ(r) ∝r ^β, and fix α = 2/3, as typical for turbulent flows. We show that two generic types of dispersion behavior arize: For α/2 + β < 1 the correlations in relative velocities decouple and the diffusion approximation holds. In the opposite case, α/2 + β > 1, the relative motion is strongly correlated. The case of Kolmogorov flows corresponds to a marginal, nongeneric situation. In this case, depending on the particular parameters of the flow, the dispersion behavior can be rather diffusive or rather ballistic. Received 13 March 2001     

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     

On the Poisson Limit Theorems of Sinai and Major

Let f(ϕ) be a positive continuous function on 0 ≤ϕ≤Θ, where Θ≤ 2 π, and let ξ be the number of two-dimensional lattice points in the domain Π _R (f) between the curves r=(R+c ₁/R)f(ϕ) and r=(R+c ₂/R)f(ϕ), where c ₁<c ₂ are fixed. Randomizing the function f according to a probability law P, and the parameter R according to the uniform distribution μ _L on the interval [a ₁ L,a ₂ L], Sinai showed that the distribution of ξ under P×μ _L converges to a mixture of the Poisson distributions as L→∞. Later Major showed that for P-almost all f, the distribution of ξ under μ _L converges to a Poisson distribution as L→∞. In this note, we shall give shorter and more transparent proofs to these interesting theorems, at the same time extending the class of P and strengthening the statement of Sinai. Received: 15 June 1999 / Accepted: 11 February 2000     

Annihilation of nematic point defects: Pre-collision and post-collision evolution

The annihilation of the nematic hedgehog and anti-hedgehog within an infinite cylinder of radius R is studied. The semi-microscopic lattice-type model and Brownian molecular dynamics are used. We distinguish among the i) early pre-collision, ii) late pre-collision, iii) early post-collision, and iv) late post-collision stages. In the pre-collision stage our results agree qualitatively with the existing experimental observations and also continuum-type simulations. The core of each defect exhibits a ring-like structure, where the ring axis is set perpendicular to the cylinder symmetry axis. For ξ⁽⁰⁾_d/(2R) > 1 the interaction between defects is negligible, where ξ⁽⁰⁾_d describes the initial separation of defects. Consequently, the defects annihilate within the simulation time window for ξ⁽⁰⁾_d/(2R) < 1. For close enough defects their separation scales as ξ_d (t_c - t)^0.4±0.1, where t_c stands for the collision time. In elastically anisotropic medium the hedgehog is faster than the anti-hedgehog. In the early pre-collision stage the defects can be treated as point-like particles, possessing inherent core structure, that interact via the nematic director field. In the late pre-collision stage the cores reflect the interaction between defects. After the collision a charge-less ring structure is first formed. In the early post-collision stage the ring adopts an essentially untwisted circular structure of the radius ξ_r. In the late post-collision stage we observe two qualitatively different scenarios. For μ = ξ_r/R < μ_c ∼ 0.25 the ring collapses leading to the escaped radial equilibrium structure. For μ > μ_c the chargeless ring triggers the nucleation growth into the planar polar structure with line defects.     

Leading-order 2πγ exchange NN interaction: Central potentials proportional to g<Subscript>A</Subscript><Superscript>0</Superscript> and g<Subscript>A</Subscript><Superscript>2</Superscript>

We calculate at two-loop order in chiral perturbation theory the electromagnetic corrections to the leading-order 2π exchange NN interaction proportional to g _A ⁰ and g _A ². The resulting 2πγ exchange potential contains isospin-breaking components which reach up to about -2% of the corresponding isovector 2π exchange potential. With a value of only -17keV at r = m _π ^-1 = 1.4fm the charge-independence breaking central potential obtained here is negligibly small in comparison to the one generated by the isoscalar c₃ contact vertex. Our calculation confirms that the largest long-range isospin-violating NN potentials arise from the 2πγ exchange diagrams involving the large low-energy constants c ₄ ≃ - c ₃ ≃ 3.3GeV^-1 representing the important Δ(1232) dynamics.     

Once more about gauge invariance in <Emphasis Type="Italic">ϕ</Emphasis> → <Emphasis Type="Italic">γπ</Emphasis><Superscript>0</Superscript><Emphasis Type="Italic">π</Emphasis><Superscript>0</Superscript>

It is argued that the realization of gauge invariance condition as a consequent of cancellation between the ϕ → γf ₀ → γπ ⁰ π ⁰ resonance contribution and the ϕ → γπ ⁰ π ⁰ background one, suggested by A.V. Anisovich et al., Yad. Fiz. 68, 1614 (2005), is misleading. The text was submitted by the author in English.     

Hole doping by Li substitution and antiferromagnetism in YBa Cu O studied by neutron powder diffraction measurements

The magnetic structure of tetragonal insulating YBa₂Cu_3-xLi_xO_y has been studied as a function of x and y. The Néel temperature and the mean ordered magnetic moment on the Cu2 sites were determined by neutron powder diffraction measurements. The decrease of these two parameters as compared to YBa₂Cu₃O₆ is much stronger for lithium than for zinc substitution. The difference is quantitatively explained by the presence of holes created in the CuO₂ planes. These holes arise from the substitution of plane Cu²⁺ by Li⁺. We suggest an explanation why such holes are not seen for the same substitution of plane Cu²⁺ by Li⁺ in orthorhombic superconducting YBa₂Cu_3-xLi_xO _{7 - δ} . Received 31 October 2001 and Received in final form 6 March 2002 Published online 25 June 2002     

Hadron diffractive processes: the structure of soft pomeron and colour screening

On the basis of the experimental data on diffractive processes in πp, pp and pˉp collisions at intermediate, moderately high and high energies, we restore the scattering amplitude related to the t-channel exchange by vacuum quantum numbers by taking account of the diffractive s-channel rescatterings. At intermediate and moderately high energies, the t-channel exchange amplitude turns, with a good accuracy, into an effective pomeron which renders the results of the additive quark model. At superhigh energies the scattering amplitude provides a Froissart-type behaviour, with an asymptotic universality of cross sections such as σ^tot _πp/σ^tot _pp→ 1 at s→∞. The quark structure of hadrons being taken into account at the level of constituent quarks, the cross sections of pion and proton (antiproton) in the impact parameter space of quarks, σ_π(r _1⊥, r _2⊥; s) and σ_p(r _1⊥, r _2⊥, r _3⊥; s), are found as functions of s. These cross sections implicate the phenomenon of colour screening: they tend to zero at |r _i⊥−r _k⊥|→ 0. The effective colour screening radius for pion (proton) is found for different s. The predictions for the diffractive cross sections at superhigh energies are presented. Received: 15 December 1998     

Effect of magnetic field on the sound velocity of a dilute Kramers-ion glass at low temperature

Recent results on the effect of magnetic field on the sound velocity V in aluminosilicate glasses doped with dysprosium are analyzed on the basis of a minimal model for the ground state of Dy³⁺ (Kramers ion with J=15/2) described by a wave function ϕ _± = ϕ _± J _m + ηϕ _{± 1/2}. The first term represents a state with a large J projection on the local crystal field axis and the random parameter η(〈η〉=0, 〈η ²〉≪1) introduces a small admixture of the state ϕ _±1/2 into the ground state. The relative variation of V due to the resonance interaction of sound waves with this state split by H is determined as a function of H and T. It possesses a universal asymptotic behavior. Our results are in reasonable agreement with the experiment. A possible structure of the crystal fields that can induce this state is discussed. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 4, 341–346 (25 February 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Quasi-long-range order in the random anisotropy Heisenberg model

The random field and random anisotropy N-vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg (N=3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈m(r ₁)m(r ₂)〉∼|r ₁−r ₂|^{− 0.62ε}. The magnetic susceptibility diverges at low fields as χ∼H ^−1+0.15ε. In the random field N-vector model the correlation length is finite at arbitrarily weak disorder for any N>3. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 2, 130–135 (25 July 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Laser-induced fluorescence measurements of a 3 Π metastable carbon monoxide in a high enthalpy flow

Excitation of carbon monoxide molecules has been carried out in a cold cell and in a low-pressure plasma jet using an ArF narrow-band excimer laser. The different excitation models are discussed and the relevance of atomic carbon absorption into the laser cavities is pointed out. Excitation spectra of Cameron bands have been obtained in a room-temperature cell and compared with calculated spectra. A value of the constant σ related to the interaction strength between a ³ Π( v = 2) state and its neighbouring singlet states is derived: 0≤σ≤0.05. The fluorescence spectrum following broad band excitation of CO has been observed both in UV and visible. Similar experiments carried out in a high enthalpy flow have allowed to point out the presence of a ³ Π metastable carbon monoxide. A method for relative measurements of this species concentration is proposed. Received 9 April 2001 and Received in final form 13 June 2001     

New phase transition in an easy-axis “triangular” antiferromagnet

The NMR of ⁵⁵Mn in the quasi-one-dimensional noncollinear anti-ferromagnet CsMnI₃ is investigated at T=1.3 K in magnetic fields up to ∼80 kOe and angles between the field and C ₆ axis ϕ≈ 0.5° and ϕ=7°. A new reorientational magnetic phase transition is observed in a field H _c1≈39.0 kOe. The magnetic structure for H>H _c1 is determined. The average Mn²⁺ spins of the magnetic sublattices in the new phase are determined from an analysis of the NMR spectrum to be 〈 S _C 〉=1.63 and 〈S _D 〉=1.72. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 12, 988–993 (25 June 1998)     

CP-violation in B → π+π- and the unitarity triangle

We analyze the extraction of weak phases from CP-violation in B → π⁺π^- decays. We propose to determine the unitarity triangle by combining the information on mixing-induced CP-violation in B → π⁺π^-, S, with the precision observable sin2β obtained from the CP-asymmetry in B → ψK_S. It is then possible to write down exact analytical expressions for and as simple functions of the observables S and sin2β and of the penguin parameters r and ϕ. As an application clean lower bounds on and can be derived as functions of S and sin2β, essentially without hadronic uncertainty. Computing r and ϕ within QCD factorization yields precise determinations of and since the dependence on r and ϕ is rather weak. It is emphasized that the sensitivity to the phase ϕ enters only at second order and is extremely small for moderate values of this phase, predicted in the heavy-quark limit. Transparent analytical formulas are further given and discussed for the parameter C of direct CP-violation in B → π⁺π^-. Predictions and uncertainties for r and ϕ in QCD factorization are examined in detail. It is pointed out that a simultaneous expansion in 1/m_b and 1/N leads to interesting simplifications. At first order infrared divergences are absent, while the most important effects are retained. Independent experimental tests of the factorization framework are briefly discussed. Received: 16 April 2005, Revised: 6 August 2005, Published online: 21 November 2005 PACS: 11.30.Er, 12.15.Hh, 13.25.Hw     

Classical spin liquid properties of the infinite-component spin vector model on a fully frustrated two dimensional lattice

Thermodynamic quantities and correlation functions (CFs) of the classical antiferromagnet on the checkerboard lattice are studied for the exactly solvable infinite-component spin-vector model, D↦∞. In contrast to conventional two-dimensional magnets with continuous symmetry showing extended short-range order at distances smaller than the correlation length, r ξ _c∝ exp(T ^*/T), correlations in the checkerboard-lattice model decay already at the scale of the lattice spacing due to the strong degeneracy of the ground state characterized by a macroscopic number of strongly fluctuating local degrees of freedom. At low temperatures, spin CFs decay as < >∝ 1/r ² in the range a ₀≪r≪ξ _c∝T ^-1/2, where a₀ is the lattice spacing. Analytical results for the principal thermodynamic quantities in our model are very similar with MC simulations, exact and analytical results for the classical Heisenberg model (D = 3) on the pyrochlore lattice. This shows that the ground state of the infinite-component spin vector model on the checkerboard lattice is a classical spin liquid. Received 16 November 2001 and Received in final form 12 February 2002