Tsallis’ statistics in the variability of El Niño/Southern Oscillation during the Holocene epoch

Analysis of the Tsallis q-triplet for the variability of El Niño Southern Oscillation (ENSO) index during the Holocene epoch (last 11,000 years) is presented. Three periods are analyzed, 0-7000, 7000-9700, 9700-11,000 years before the present. During the first and the third periods, the q-index values have the expected usual relations between them (q_sens<1<q_stat<q_rel), and in the second one there is an inversion between q_stat and q_rel (q_stat>q_rel).     

An extended clique degree distribution and its heterogeneity in cooperation-competition networks

After Xiao et al. [W.-K. Xiao, J. Ren, F. Qi, Z.W. Song, M.X. Zhu, H.F. Yang, H.Y. Jin, B.-H. Wang, Tao Zhou, Empirical study on clique-degree distribution of networks, Phys. Rev. E 76 (2007) 037102], in this article we present an investigation on so-called k-cliques, which are defined as complete subgraphs of k (k>1) nodes, in the cooperation-competition networks described by bipartite graphs. In the networks, the nodes named actors are taking part in events, organizations or activities, named acts. We mainly examine a property of a k-clique called “k-clique act degree”, q, defined as the number of acts, in which the k-clique takes part. Our analytic treatment on a cooperation-competition network evolution model demonstrates that the distribution of k-clique act degrees obeys Mandelbrot distribution, P(q)∝(q+α)^−γ. To validate the analytical model, we have further studied 13 different empirical cooperation-competition networks with the clique numbers k=2 and k=3. Empirical investigation results show an agreement with the analytic derivations. We propose a new “heterogeneity index”, H, to describe the heterogeneous degree distributions of k-clique and heuristically derive the correlation between H and α and γ. We argue that the cliques, which take part in the largest number of acts, are the most important subgraphs, which can provide a new criterion to distinguish important cliques in the real world networks.     

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈10⁶) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.     

Tsallis, Rényi and nonextensive Gaussian entropy derived from the respective multinomial coefficients

We explore the generalization of the ordinary multinomial coefficient, based on the deformed q-multiplication and q-division. Aim of this study is to construct the appropriate multinomial coefficients, from which one can obtain the Tsallis, Rényi and nonextensive Gaussian entropy, respectively. We show that for all three above entropies there are two possible ways to define the generalized multinomial coefficient. Its consequence is discussed.     

Evidence for energy regularity in the Mendeleev periodic table

We show that the dependence of the total energy of the atoms on their atomic number can follow a q-exponential (as proposed by C. Tsallis), for practically all elements of the periodic table. The result is qualitatively explained in terms of the way the atomic configurations are arranged to minimize energy.     

Mean wind: Its velocity and temperature fluctuation in low-Prandtl-number thermal convection

Large-scale circulation in the confined thermal turbulence of mercury is studied experimentally. The instantaneous velocity profile at 128 points is directly measured using ultrasonic velocimetry and the temperature fluctuation is simultaneously obtained by a small thermistor at the cell center. The periodic velocity oscillation is observed for aspect ratios Γ=1,2, but for Γ=0.5. The temperature oscillation is confirmed only in the Γ=2 cell. It is found that the flow patterns depending on the aspect ratio are characterized sufficiently by the joint probability density function up to Ra=4×10⁸.     

Nonlinear statistical coupling

By considering a nonlinear combination of the probabilities of a system, a physical interpretation of Tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed. The escort probability is interpreted as the coupled probability, with Q=1−q defined as the degree of nonlinear coupling between the statistical states. Positive values of Q have coupled statistical states, a larger entropy metric, and a maximum coupled-entropy distribution of compact-support coupled-Gaussians. Negative values of Q have decoupled statistical states and for −2<Q<0 a maximum coupled-entropy distribution of heavy-tail coupled-Gaussians. The conjugate transformation between the heavy-tail and compact-support domains is shown to be for coupled-Gaussian distributions. This conjugate relationship has been used to extend the generalized Fourier transform to the compact-support domain and to define a scale-invariant correlation structure with heavy-tail limit distribution. In the present paper, we show that the conjugate is a mapping between the source of nonlinearity in non-stationary stochastic processes and the nonlinear coupling which defines the coupled-Gaussian limit distribution. The effects of additive and multiplicative noise are shown to be separable into the coupled-variance and the coupling parameter Q, providing further evidence of the importance of the generalized moments.     

The status of KATRIN

KATRIN will have the capability to push the limit on the mass of the electron anti-neutrino to 200 meV (90% C.L.) by investigating the kinematics of the electrons from tritium β decay very close to the endpoint of the β spectrum. The importance of this experiment will be discussed in various contributions to this school. KATRIN is currently under construction at KIT (Karlsruhe Institute of Technology). This talk gives an overview over the status of KATRIN with emphasis on the aspects of KATRIN not covered by the talks following this one.     

Spin-dependent nuclear weak processes and nucleosynthesis in stars

Spin dependent nuclear weak processes and nucleosynthesis in stars are investigated based on recent advances in shell model studies of stable and unstable exotic nuclei. Three topics on (1) neutrino-nucleus reactions in supernova explosions and nucleosynthesis of light elements as well as Mn, (2) electron capture reaction rates on Ni and Co isotopes at high densities and temperatures in the core-collapse process, and (3) new β-decay half-lives of N=126 isotones obtained by including both the Gamow-Teller and the first-forbidden transitions, and the effects on the element abundance in the r-process at the third peak region (A∼195), are studied with the use of new shell model Hamiltonians.     

The Kadomtsev-Petviashvili II equation on the half-plane

The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics, where it describes waves in shallow water. It provides a multidimensional generalisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey-Stewartson equation (Fokas (2009) [13]), in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called d-bar formalism, as well as of the so-called global relation. A novel feature of boundary as opposed to initial value problems in 2+1 is that the d-bar formalism now involves a function in the complex plane which is discontinuous across the real axis.     

Frequency-dependent linear and non-linear response properties of single carrier quantum dots: Role of effective mass and anharmonicity in the confinement potential

We explore the pattern of frequency-dependent linear and non-linear optical (NLO) response of one electron quantum dots harmonically confined in two dimensions. For some fixed values of transverse magnetic field strength (ω_c), and harmonic confinement potential (ω₀), the influence of effective mass (m^*) of the system and the symmetry breaking anharmonic interaction on the frequency-dependent linear (α), and the first (β), and second (γ) NLO responses of the dot is computed through linear variational route. The investigation reveals interesting roles played by the anharmonic interaction and effective mass in modulating the response properties.     

Magnetic-field induced quantum critical points of valence transition in Ce- and Yb-based heavy fermions

The mechanism of how the magnetic field controls the critical end point of the first-order valence transition is clarified, which is essentially ascribed to charge degrees of freedom. It is shown that the quantum critical point is induced by applying the magnetic field, which explains a peculiar magnetic response in CeIrIn₅ and sharp contrast between X=Ag and Cd for YbXCu₄. Significance of the proximity of the first-order valence transition in the Ce- and Yb-based heavy fermions is pointed out.     

Proximity effects in a superconductor/ferromagnet junction

A proximity effect in an s-wave superconductor/ferromagnet (SC/F) junction is theoretically studied using the second order perturbation theory for the tunneling Hamiltonian and Green's function method. We calculate a pair amplitude induced by the proximity effect in a weak ferromagnetic metal (FM) and a half-metal (HM). In the SC/FM junction, it is found that a spin-singlet pair amplitude (Ψ_s) and spin-triplet pair amplitude (Ψ_t) are induced in FM and both amplitudes depend on the frequency in the Matsubara representation. Ψ_s is an even function and Ψ_t is an odd function with respect to the Matsubara frequency (ω_n). In the SC/HM junction, we examine the proximity effects by taking account of magnon excitations in HM. It is found that the triplet-pair correlation is induced in HM. The induced pair amplitude in HM shows a damped oscillation as a function of the position and contains the terms of even and odd functions of ω_n as in the case of the SC/FM junction. We discuss that in our tunneling model the pair amplitude of even function of ω_n only contributes to a Josephson current.     

Kinetic and mass mixing with three abelian groups

We present the possible mixing effects associated with the low-energy limit of a Standard-Model extension by two abelian gauge groups U₁(1)×U₂(1). We derive general formulae and approximate expressions that connect the gauge eigenstates to the mass eigenstates. Applications using the well-studied groups UB(1), U(1)_B−L, U(1)_Lα−Lβ (Lα being lepton flavor numbers), and UDM(1) (a symmetry acting only on the dark matter sector) are discussed briefly.     

Phase transition in dually weighted colored tensor models

Tensor models are a generalization of matrix models (their graphs being dual to higher-dimensional triangulations) and, in their colored version, admit a 1/N expansion and a continuum limit. We introduce a new class of colored tensor models with a modified propagator which allows us to associate weight factors to the faces of the graphs, i.e. to the bones (or hinges) of the triangulation, where curvature is concentrated. They correspond to dynamical triangulations in three and higher dimensions with generalized amplitudes. We solve analytically the leading order in 1/N of the most general model in arbitrary dimensions. We then show that a particular model, corresponding to dynamical triangulations with a non-trivial measure factor, undergoes a third-order phase transition in the continuum characterized by a jump in the susceptibility exponent.     

Alpha decay chains from superheavy nuclei

Magic islands for extra-stable nuclei in the midst of the sea of fission-instability were predicted to be around Z=114, 124 or, 126 with N=184, and Z=120, with N=172. Whether these fission-survived superheavy nuclei with high Z and N would live long enough for detection or, undergo α-decay in a very short time, remains an open question. α-decay half lives of nuclei with 130≥Z≥100 have been calculated in a WKB framework using density-dependent M3Y interaction with Q-values from different mass formulae. The results are in excellent agreement with the experimental data. Fission survived Sg nuclei with Z=106, N=162 is predicted to have the highest α-decay half life (∼3.2 h) in the Z=106-108, N=160-164 region called small island/peninsula. Superheavy nuclei with Z>118 are found to have α-decay half lives of the order of microseconds or less.     

The non-perturbative renormalization group in the ordered phase

We study some analytical properties of the solutions of the non-perturbative renormalization group flow equations for a scalar field theory with Z₂ symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility χ(M) at M=±M₀ (M₀ spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space d<4.By generalizing a method proposed earlier by Bonanno and Lacagnina [Nucl. Phys. B 693 (2004) 36] to any kind of cut-off we propose to solve numerically the renormalization group flow equations for the threshold functions rather than for the local potential. It yields an algorithm sufficiently robust and precise to extract universal as well as non-universal quantities from numerical experiments at any temperature, in particular at sub-critical temperatures in the ordered phase. Numerical results obtained for the φ⁴ potential with three different cut-off functions are reported and compared. The data confirm our theoretical predictions concerning the analytical behavior of χ(M) at M=±M₀.Fixed point solutions of the adimensioned renormalization group flow equations are also obtained in the same vein, that is by solving the fixed points equations and the associated eigenvalue problem for the threshold functions rather than for the potential. We report high precision data for the odd and even spectra of critical exponents for different cut-offs obtained in this way.     

Variational treatment of Faraday waves in inhomogeneous Bose-Einstein condensates

Motivated by the recent experimental progress on the collective modes of a Bose-Einstein condensate whose atomic scattering length is tuned via Feshbach resonances, we analyze by variational means the dynamics of Faraday waves in trapped Bose-Einstein condensates. These waves can be excited by modulating periodically either the strength of the magnetic trap or the atomic scattering length. To study their dynamics, we develop a variational model that describes consistently both the bulk part of an inhomogeneous, low-density, cigar-shaped condensate and small-amplitude, small-wavelength Faraday waves. The main ansatz used in the variational treatment is tailored around a set of Gaussian envelopes and we show extensions for the high-density regime using a q-Gaussian function. Finally, we show explicitly that for drives of small amplitude, the two methods of obtaining Faraday waves are equivalent, and we discuss the existing experimental results.