κ-generalized statistics in personal income distribution

Starting from the generalized exponential function , with exp ₀(x)=exp (x), proposed in reference [G. Kaniadakis, Physica A 296, 405 (2001)], the survival function P_>(x)=exp _κ(-βx^α), where x∈R⁺, α,β>0, and , is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function P_> ⁰(x)=exp (-βx^α) to which reduces as κ approaches zero behaving in very different way in the x→0 and x→∞ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law P_>(x)∼(2βκ)^-1/κx^-α/κ. This makes the κ-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range.     

Statistical model of fluorescence blinking

Blinking of single molecules and nanocrystals is modeled as a two-state renewal process with on (fluorescent) and off (non-fluorescent) states. The on and off-times may have power-law or exponential distributions. A fractional generalization of the exponential function is used to develop a unified treatment of the blinking statistics for both types of distributions. In the framework of the two-state model, an equation for the probability density p(t _on|t) of the total on-time is derived. As applied to power-law blinking, the equation contains derivatives of fractional orders α and β equal to the exponents of the on and off-time power-law distributions, respectively. In the limit case of α = β = 1, the distributions become exponential and the fractional differential equation reduces to an integer order differential equation. Solutions to these equations are expressed in terms of fractional stable distributions. The Poisson transform of p(t _on|t) is the photon number distribution that determines the photon counting statistics. It is shown that the long-time asymptotic behavior of Mandel’s Q parameter follows a power law: M(t) ∝ t ^γ . The function γ(α, β) is defined on the (α, β) plane. An analysis of the relative variance of the total on-time shows that it decays only when α = β = 1 or α < β. Otherwise, relative fluctuations either exhibit asymptotic power-law growth or approach a constant level. Analytical calculations are in good agreement with the results of Monte Carlo simulations.     

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.     

Quark momentum-space charge distribution in deuteron and neutron/proton structure functions ratio

The electromagnetic polarizabilities of the nucleon are shown to be essentially composed of the nonresonant α _p(E ₀₊) = + 3.2, α _n(E ₀₊) = + 4.1, the t-channel α ^t _{p, n} = - β ^t _{p, n} = + 7.6 and the resonant β _{p, n}(P ₃₃(1232)) = + 8.3 contributions (in units of 10^-4fm^3). The remaining deviations from the experimental data Δα _p = 1.2±0.6, Δβ _p = 1.2±0.6, Δα _n = 0.8±1.7 and Δβ _n = 2.0±1.8 are contributed by a larger number of resonant and nonresonant processes with cancellations between the contributions. This result confirms that dominant contributions to the electric and magnetic polarizabilities may be represented in terms of two-photon coupling to the σ-meson having the predicted mass m _σ = 666MeV and two-photon width Γ _γγ = 2.6keV.     

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ _{β ,p}  > 0, such that for all exponential time scales, exp(γ N), with γ < γ _{β ,p} , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β ² < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.     

Stretched exponential relaxation in the Coulomb glass

The relaxation of the specific heat and the entropy to their equilibrium values is investigated numerically for the three-dimensional Coulomb glass at very low temperatures. The long time relaxation follows a stretched exponential function, f (t) = f ₀exp - (t/τ)^β , with the exponent β increasing with the temperature. The relaxation time diverges as an Arrhenius law when T→ 0. Received 24 May 2001 and Received in final form 12 September 2001     

Coarsening process in one-dimensional surface growth models

Surface growth models may give rise to instabilities with mound formation whose typical linear size L increases with time (coarsening process). In one dimensional systems coarsening is generally driven by an attractive interaction between domain walls or kinks. This picture applies to growth models for which the largest surface slope remains constant in time (corresponding to model B of dynamics): coarsening is known to be logarithmic in the absence of noise ( L(t) ∼ ln t) and to follow a power law ( L(t) ∼t ^1/3) when noise is present. If the surface slope increases indefinitely, the deterministic equation looks like a modified Cahn-Hilliard equation: here we study the late stages of coarsening through a linear stability analysis of the stationary periodic configurations and through a direct numerical integration. Analytical and numerical results agree with regard to the conclusion that steepening of mounds makes deterministic coarsening faster : if α is the exponent describing the steepening of the maximal slope M of mounds ( M ^α∼L) we find that L(t) ∼t ⁿ: n is equal to for 1≤α≤2 and it decreases from to for α≥2, according to n = α/(5α - 2). On the other side, the numerical solution of the corresponding stochastic equation clearly shows that in the presence of shot noise steepening of mounds makes coarsening slower than in model B: L(t) ∼t ^1/4, irrespectively of α. Finally, the presence of a symmetry breaking term is shown not to modify the coarsening law of model α = 1, both in the absence and in the presence of noise. Received 28 September 2001 and Received in final form 21 November 2001     

Local magnetic susceptibility in γ- and α-Ce

The local susceptibility, below and above the γ-α transition, was measured with the DPAC method using the¹⁴⁰Ce probe. The ratio β between the local and the applied field was β_γ=1.20(2) and β_α=1.13(1) at 295 K. The latter value is larger than expected for a completely delocalized 4f electron.     

Time-Scales to Equipartition in the Fermi–Pasta–Ulam Problem: Finite-Size Effects and Thermodynamic Limit

We investigate numerically the common α+β and the pure β FPU models, as well as some higher order generalizations. We consider initial conditions in which only low-frequency normal modes are excited, and perform a very accurate systematic study of the equilibrium time as a function of the number N of particles, the specific energy ε, and the parameters α and β. While at any fixed N the equilibrium time is found to be a stretched exponential in 1/ε, in the thermodynamic limit, i.e. for N→∞ at fixed ε, we observe a crossover to a power law. Concerning the (usually disregarded) dependence of T _eq on α and β, we find it is nontrivial, and propose and test a general law. A central role is played by the comparison of the FPU models with the Toda model.     

Computer simulation of the frustrated antiferromagnetic Heisenberg model on a layered triangle lattice

Critical properties of the 3D frustrated Heisenberg model on a triangle latticeare investigated using a replica Monte-Carlo method that considers the interaction between next nearest neighbors. Static magnetic and chiral critical indices for heat capacity α, susceptibility γ, γ _k , magnetization β, β _k , and correlation radius ν are calculated using the theory of finite-size scaling.     

Properties of the πη,$ \eta{^\prime}$,σ, f0(980) and a0(980) mesons and their relevance for the polarizabilities of the nucleon

The signs and values of the two-photon couplings F _Mγγ of mesons (M) and their couplings g_MNN to the nucleon as entering into the t -channel parts of the difference of the electromagnetic polarizabilities (α - β) and the backward angle spin polarizabilities γ_π are determined. The excellent agreement achieved with the experimental polarizabilities of the proton makes it possible to make reliable predictions for the neutron. The results obtained are α_n = 13.4±1.0 , β_n = 1.8±1.0 (10^-4fm^3), and γ⁽ⁿ⁾ _π = 57.6±1.8 (10^-4fm^4). New empirical information on the flavor wave functions of the f ₀(980) - and the a ₀(980) -meson is obtained.     

The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

We have investigated the proof of the H theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a κ power law generalization of the exponential Juttner distribution, e.g., , with θ=α(x)+β_μp^μ, where α(x) is a scalar, β_μ is a four-vector, and p^μ is the four-momentum. As a simple example, we calculate the relativistic κ power law for a dilute charged gas under the action of an electromagnetic field F^μν. All standard results are readly recovered in the particular limit κ→0.     

A frustrated heisenberg antiferromagnet on a triangular lattice with next-nearest neighbor interactions

Using the Monte Carlo method, we study the critical properties of the three-dimensional frustrated Heisenberg model on a triangular lattice with allowance for next-nearest neighbor interactions. Using the theory of finite-size scaling, we calculate the static magnetic and chiral critical exponents of heat capacity α, susceptibility γ, γ _k , magnetization β, β _k , and correlation length ν.     

Entanglement and Mixedness of Locally Cloned Non-Maximal W-State

In this work we describe a protocol by which two of three parties generate two bipartite entangled state among themselves without involving third party, from a non maximal W-state or W-type state |X〉=α|001〉₁₂₃+β|010〉₁₂₃+γ|100〉₁₂₃,α ²+β ²+γ ²=1 shared by three distant partners. Also we have considered the case β=γ, to obtain a range for α ², for which the local output states are separable and non local output states are inseparable. We also find out the dependence of the mixedness of inseparable states with their amount of inseparability, for that range of α ².     

Electric-field dependence of dielectric properties of sol-gel derived Ba(Zr0.2Ti0.8)O3 ceramics

The effect of a dc bias field on the diffuse phase transition and nonlinear dielectric properties of sol-gel derived Ba(Zr_0.2Ti_0.8)O₃ (BZT) ceramics are investigated. Diffuse phase transitions were observed in BZT ceramics and the Curie–Weiss exponent (CWE) was γ∼2.0. The dielectric constant versus temperature characteristics and the γ in the modified Curie–Weiss law, ε ⁻¹=ε _m ⁻¹[1+(T−T _m ) ^γ /C₁](1≤γ≤2), as a function of the dc bias field was obtained for BZT ceramics. The results indicated that γ is a function of dc bias field, and the γ value decreased from 2.04 to 1.73 with dc bias field increasing from 0 to 20 kV/cm. The dielectric constant decreases with increasing dc bias field, indicating a field-induced phase transition. The dc bias field has a strong effect on the position of the dielectric peak and affects the magnitude of the dielectric properties over a rather wide temperature range. The peak temperature of the dielectric loss does not coincide with the dielectric peak and an obvious minimum value for the dielectric loss at the temperature of the dielectric peaks is observed. At room temperature, 300 K, the high tunability (K=80%), the low loss tangent (≈0.01) and the large FOM (74), clearly imply that these ceramics are promising materials for tunable capacitor-device applications.     

Quasi-free Compton scattering and the polarizabilities of the neutron

Differential cross-sections for quasi-free Compton scattering from the proton and neutron bound in the deuteron have been measured using the Glasgow/Mainz photon tagging spectrometer at the Mainz MAMI accelerator together with the Mainz [48]cm ;SPMOslash; × [64]cm NaI(Tl) photon detector and the G?ttingen SENECA recoil detector. The data cover photon energies ranging from [200]MeV to [400]MeV at θ^LAB _γ = 136.2^°. Liquid deuterium and hydrogen targets allowed direct comparison of free and quasi-free scattering from the proton. The neutron detection efficiency of the SENECA detector was measured via the reaction p(γ,π⁺ n). The “free” proton Compton scattering cross-sections extracted from the bound proton data are in reasonable agreement with those for the free proton which gives confidence in the method to extract the differential cross-section for free scattering from quasi-free data. Differential cross-sections on the free neutron have been extracted and the difference of the electromagnetic polarizabilities of the neutron has been determined to be α_n - β_n = 9.8±3.6(stat)^+2.1 _-1.1(syst)±2.2(model) in units of [10^-4]fm ³. In combination with the polarizability sum α_n + β_n = 15.2±0.5 deduced from photoabsorption data, the neutron electric and magnetic polarizabilities, α_n = 12.5±1.8(stat)^+1.1 _-0.6(syst)±1.1(model) and β_n = 2.7±1.8(stat)^+0.6 _-1.1(syst)±1.1(model) are obtained. The backward spin polarizability of the neutron was determined to be γ⁽ⁿ⁾ _π = (58.6±4.0)×10^-4 fm ⁴. Received: 21 August 2002 / Accepted: 16 October 2002 / Published online: 11 February 2003 RID="a" ID="a"Part of the Doctoral Thesis. RID="b" ID="b"Present address: Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig. RID="a" ID="a"Part of the Doctoral Thesis. RID="b" ID="b"Present address: Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig. RID="c" ID="c"Part of the Habilitation Thesis. RID="d" ID="d"e-mail: schumacher@physik2.uni-goettingen.de Communicated by Th. Walcher     

Low frequency fluctuations of laser beam intensity run through the system of cavitation water clusters

Experiments on water cavitation in ultrasonic field have been carried out. Low frequency fluctuations of the intensity of laser beam run through the cavitation area have been studied. Experiments have proved presence of low-frequency random fluctuations with frequency dependence of power spectra, S ∼ 1/f ^α where the exponent α ranged within 0.8 ≤ α ≤ 1.2. From experimental realizations, large-scale low frequency pulsations characterized by scale invariance, which duration is distributed according to the power law, have been distinguished. The results are explained on the basis of mathematical model for the rise of scale invariant fluctuations with 1/f ^α power spectrum in the system of two nonlinear stochastic differential equations describing interaction of heterogeneous phase transitions. Distribution of extreme low frequency emissions obtained from numerical solutions to stochastic equations takes the power series form. Correlations of dynamic scaling between critical indicators determining frequency dependence of pulsations power spectra α and distribution functions of extreme low frequency pulsation amplitudes β have been determined. It is shown that both in the experiments on acoustic cavitation of water and in the theoretical model of interacting phase transitions critical indicators are bound with the correlation α + β = 2. Spectra of fluctuation power are determined in the experiments simpler and more accurately than the function of extreme amplitudes distribution. In case when only one frequency dependence of fluctuation capacity spectra is known correlations between indicators serve to obtain information on the distribution of large scale emissions and to estimate critical amplitudes.     

Strange quark mass from pseudoscalar sum rule with O(\alpha_s^4) accuracy

We include the new, five-loop, O(α_s⁴) correction into the QCD sum rule used for the s-quark mass determination. The pseudoscalar Borel sum rule is taken as a study case. The OPE for the correlation function with N⁴LO, O(α_s⁴) accuracy in the perturbative part, and with dimension d≤6 operators reveals a good convergence. We observe a significant improvement of stability of the sum rule with respect to the variation of the renormalization scale after including the O(α_s⁴) correction. We obtain the interval m̄_s(2 GeV)=105±6±7 MeV, which exhibits about 2 MeV increase of the central value, if the O(α_s⁴) terms are removed.