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It has been shown that spectral statistics techniques, based on random matrix theory, can be applied to study correlations at ground level between the secondary particles of simulated extensive air showers. The statistical description of shower fronts provided by appropriate spectral measures makes it possible to separate them into different classes depending on the type of the primary cosmic ray. Using a suitable combination of spectral statistics in the framework of discriminant analysis, we introduce a new statistic which separates shower fronts according to the primary type with improved efficiency.

Spectral statistics techniques based on random matrix theory are applied to study the correlations at ground level among secondary particle density distributions of simulated extensive air showers. High-energy interactions of TeV cosmic γ -rays and protons with Earth atmosphere and the resulting extensive air showers have been simulated by means of the CORSIKA Monte Carlo code. The statistical description of shower fronts provided by this new type of analysis allows to distinguish events generated by primary protons from those initiated by primary γ -rays, and leads to a powerful discrimination method whose efficiency is evaluated using the standard quality factors.

8 pages, 2 tables, 7 figures.--PACS nrs.: 21.60.-k; 05.45.Mt; 24.60.Lz.--Available on line on Nov 13, 2006., Sequences of energy levels in nuclei are often plagued with missing levels whose number and position are unknown. It is also quite usual that all the quantum numbers of certain levels cannot be experimentally determined, and thus levels of different symmetries are mixed in the same sequence. The analysis of these imperfect spectra (from the point of view of spectral statistics) is unavoidable if one wants to extract some statistical information. The power spectrum of the δq statistic has emerged in recent years as an important tool for the study of quantum chaos and spectral statistics. We derive analytical expressions for the observed power spectrum in terms of the fraction of observed levels and the number of mixed sequences. These expressions are tested with large shell model spectra simulating realistic experimental situations. A good estimation of the number of mixed symmetries and the fraction of missing levels is obtained by means of a least-squares fit in a wide set of different situations., This work is supported in part by Spanish Government grants BFM2003-04147-C02 and FTN2003-08337-C04-04.

6 pags., 4 figs. -- 11th International Spring Seminar on Nuclear Physics: Shell Model and Nuclear Structure – achievements of the past two decades 12–16 May 2014, Ischia, Italy, Shell-model calculations with realistic empirical interactions constitute an excellent tool to study statistical properties of nuclei. Using large-scale shell-model calculations in pf- shell nuclei, we study how the onset of chaos depends on different properties of the nuclear interaction and on excitation energy. We make use of classical random matrix theory and other theoretical developments based on information theory and time series analysis. We show that besides energy-level statistics, other statistical properties like the complexity of wave functions are fundamental for a proper determination of the dynamical regime of nuclei. Important deviations from GOE are observed in level fluctuations and in the complexity of wave functions.

We perform a power spectrum analysis of energy level fluctuations in experimental quantum Sinai billiards. In spite of experimental limitations we find the 1/f behavior expected for quantum chaotic systems. Moreover, the deviations observed at low frequencies are easily explained within the semiclassical periodic orbit theory. In conclusion, it is shown that the power spectrum analysis of energy level fluctuations is a robust signature of quantum chaos.

High-energy interactions of cosmic rays with Earth atmosphere and the resulting extensive air showers have been simulated by means of the CORSIKA Monte Carlo code. Statistical fluctuations of the two-dimensional secondary Cerenkov photon density distributions at ground level produced by GeV cosmic γ rays and protons are studied by means of principal components analysis (PCA). This provides a decreasing sequence of covariance matrix eigenvalues that may be fitted to an analytical expression allowing to distinguish among different primary cosmic rays. A very efficient discrimination method of gamma ray induced showers from proton simulated extensive air showers is proposed as a result of this analysis. A cutting parameter is calculated, and the efficiency of the cutting procedure for γ–proton separation is evaluated under experimental conditions.

High-energy interactions of cosmic γ rays and protons with the earth atmosphere have been simulated by means of the CORSIKA Monte Carlo code, and the secondary particle density distributions at ground level in the resulting extensive air showers have been studied. It is shown that the power spectrum of the bidimensional particle density fluctuations have features typical of a 1/ f noise in the bidimensional frequency domain, and is found to have different features for different primary cosmic rays. This property is applied to the separation of electromagnetic from proton simulated extensive air showers and it is proposed as a new discrimination method that can be used experimentally for real-time γ–proton separation. A cutting parameter related to the bidimensional power spectrum is calculated and the efficiency of the cutting procedure for γ–proton separation is evaluated.

High-energy interactions of cosmic γ rays and protons and also helium, oxygen and iron nuclei with the Earth atmosphere have been simulated by means of the CORSIKA Monte Carlo code, and the secondary-particle density distributions at ground level in the resulting extensive air showers (EAS) have been studied. It is shown that the fluctuations of the particle density distributions have features typical of a 1/f noise. The multifractal spectrum of the samples is obtained and is found to have different features for different primary cosmic rays. This property is applied to the separation of electromagnetic from hadronic simulated EAS and it is proposed as a discrimination method when real data are available. A cutting parameter related to the multifractal spectrum is calculated and the efficiency of the cutting procedure for γ-hadron separation is evaluated.

High-energy interactions of gamma rays and protons with the Earth atmosphere have been simulated by means of the CORSIKA Monte Carlo code, and the secondary-particle density distributions in the resulting extensive air showers, at ground level, have been studied. It is shown that the fluctuations of the density distributions as a function of the polar angle have features typical of a 1/f noise. The sample is then analysed in order to study its scaling behaviour and we find that it can be parametrized by means of a universal multifractal approach.

We have simulated the interactions of high energy rays and protons with the terrestrial atmosphere, and studied the ground level behaviour of the distributions of secondary . The fluctuation seen in these distributions show the typical aspect of a flicker noise. We show how deconvolution of this noise leads to an improved /hadron discrimination in a simple separation scheme.

A precise knowledge of the angular resolution of scintillator arrays used to observe extended air showers (EAS) is of key importance in the search for VHE/UHE γ point sources. Four independent methods have been used to determine the mean resolution for which a value of 〈ΔΘ63〉 of 0.8°(1.0°) at a proton threshold of 50 (40) TeV has been found for the HEGRA EAS-array.

The HEGRA collaboration has searched for γ-ray emission from 36 potential cosmic point sources with the wide angle air Cerenkov detector AIROBICC which is part of the HEGRA cosmic ray detector complex on La Palma. AIROBICC has a γ-ray energy threshold of around 24 TeV and a mean angular resolution of 0.29 °. None of the sources could be found. Upper limits to their fluxes are given. For the VHE γ-ray emitters, the Crab nebula and Mrk 421, these flux limits are below those extrapolated from lower energy observations. Our lower limit for Mrk 421 might be interpreted either as being due to an energy cut-off in the acceleration mechanism or to the onset of γ-interaction with photons of the universal infrared background.

A survey of chaotic dynamics in atomic nuclei is presented, using on the one hand standard statistics of quantum chaos studies, and on the other a new approach based on time series analysis methods. The study of shell-model spectra in the pf shell shows that nuclear chaos is strongly isopin dependent and increases with excitation energy. On the other hand, it is found that chaotic quantum systems exhibit 1/f noise and regular systems exhibit 1/f 2 behaviour. It is shown that the time series approach can be used to calculate quite accurately the fraction of missing levels and the existence of mixed symmetries in experimental level spectra.

6 pags., 2 figs. -- XXXIII Symposium on Nuclear Physics 5–8 January 2010, Hacienda Cocoyoc, Morelos, Mexico, The power spectrum of the 5 statistic of quantum spectra presents 1/f(alpha) noise. For chaotic systems alpha = 1 while for regular systems alpha = 2. Although the transition from regularity to chaos is non universal, for a wide variety of systems with a mixed phase space the value of a is intermediate between 1 and 2 and can be related to the fraction of regular or chaotic orbits in the total phase space. This statistic can be a very useful tool for the analysis of experimental spectra, specially in the case of missing levels or spectral sequences with mixed symmetries., This research has been supported in part by Spanish Government grant No. FIS2009- 07277 and No. FIS2006-12783-C03-02, CSPD-2007-00042-Ingenio2010, and by the Universidad Complutense de Madrid grant UCM-910059. A.R. is supported by the CPAN-Consolider program.

11 pages, 3 figures.-- PACS nrs.: 05.45.Mt; 02.10.Yn; 24.60.-k.-- PMID: 18517325 [PubMed]., The continuous Gaussian ensemble, also known as the v-Gaussian or v-Hermite ensemble, is a natural extension of the classical Gaussian ensembles of real (v=1), complex (v=2), or quaternion (v=4) matrices, where v is allowed to take any positive value. From a physical point of view, this ensemble may be useful to describe transitions between different symmetries or to describe the terrace-width distributions of vicinal surfaces. Moreover, its simple form allows one to speed up and increase the efficiency of numerical simulations dealing with large matrix dimensions. We analyze the long-range spectral correlations of this ensemble by means of the δn statistic. We derive an analytical expression for the average power spectrum of this statistic, [overline Pk[sup δ]], based on approximated forms for the two-point cluster function and the spectral form factor. We find that the power spectrum of δn evolves from [overline Pk[sup δ]] proportional to 1/k at v=1 to [overline Pk[sup δ]] proportional to 1/k^2 at v=0. Relevantly, the transition is not homogeneous with a 1/fα noise at all scales, but heterogeneous with coexisting 1/f and 1/f^2 noises. There exists a critical frequency kc proportional to v that separates both behaviors: below kc, [overline Pk[sup δ]] follows a 1/f power law, while beyond kc, it transits abruptly to a 1/f^2 power law. For v>1 the 1/f noise dominates through the whole frequency range, unveiling that the 1/f correlation structure remains constant as we increase the level repulsion and reduce to zero the amplitude of the spectral fluctuations. All these results are confirmed by stringent numerical calculations involving matrices with dimensions up to 10^5., This work was supported in part by Spanish Government Grants Nos. FIS2006-12783-C03-01 and FIS2006-12783-C03-02 and by Comunidad de Madrid–CSIC Grant No. 200650M012. A.R. was supported by the Spanish program "Juan de la Cierva".

The fluctuations of the particle density distributions in extensive air showers have been studied at ground level. In order to achieve meaningful statistics, the interaction of cosmic rays with the earth atmosphere has been simulated by means of the CORSIKA Monte Carlo code. It is shown that the fluctuations of the particle density distributions as a function of the polar angle have features typical of a 1/f noise. The sample is then analysed in order to study its scaling behaviour and we find that it can be parametrized by means of a universal multifractal approach.

Many complex systems in nature and in human society exhibit time fluctuations characterized by a power spectrum S(f) which is a power function of the frequency f. Examples with this behavior are the Sun spot activity, the human heartbeat, the DNA sequence, or Bach’s First Brandenburg Concert. In this work, we show that the energy spectrum fluctuations of quantum systems can be formally considered as a discrete time series, with energy playing the role of time. Because of this analogy, the fluctuations of quantum energy spectra can be studied using traditional methods of time series, like calculating the Fourier transform and studying the power spectrum. We present the results for paradigmatic quantum chaotic systems like atomic nuclei (by means of large scale shell‐model calculations) and the predictions of random matrix theory. We have found a surprising general property of quantum systems: The energy spectra of chaotic quantum systems are characterized by 1/f noise, while regular quantum systems exhibit...

6 pages, 1 table, 4 figures.--PACS nrs.: 05.45.Mt; 05.45.Tp; 05.40.-a, It was recently pointed out that the spectral fluctuations of quantum systems are formally analogous to discrete time series, and therefore their structure can be characterized by the power spectrum of the signal. Moreover, it is found that the power spectrum of chaotic spectra displays a 1/f behavior, while that of regular systems follows a 1/f2 law. This analogy provides a link between the concepts of spectral rigidity and antipersistence. Trying to get a deeper understanding of this relationship, we have studied the correlation structure of spectra with high spectral rigidity. Using an appropriate family of random Hamiltonians, we increase the spectral rigidity up to hindering completely the spectral fluctuations. Analyzing the long range correlation structure a neat power law 1/f has been found for all the spectra, along the whole process. Therefore, 1/f noise is the characteristic fingerprint of a transition that, preserving the scale-free correlation structure, hinders completely the fluctuations of the spectrum., This work is supported in part by Spanish Government Grants No. BFM2003-04147 and No. FTN2003-08337-C04-04.

14 pages, 3 tables, 13 figures.--PACS nrs.:05.45.Mt; 05.45.Pq; 24.60.-k, It is widely accepted that the statistical properties of energy level spectra provide an essential characterization of quantum chaos. Indeed, the spectral fluctuations of many different systems like quantum billiards, atoms, or atomic nuclei have been studied. However, noninteracting many-body systems have received little attention, since it is assumed that they must exhibit Poisson-like fluctuations. Apart from a heuristic argument of Bloch, there are neither systematic numerical calculations nor a rigorous derivation of this fact. Here we present a rigorous study of the spectral fluctuations of noninteracting identical particles moving freely in a mean field emphasizing the evolution with the number of particles N as well as with the energy. Our results are conclusive. For N2 the spectra of these systems exhibit Poisson fluctuations provided that we consider sufficiently high excitation energies. Nevertheless, when the mean field is chaotic there exists a critical energy scale Lc; beyond this scale, the fluctuations deviate from the Poisson statistics as a reminiscence of the statistical properties of the mean field., This work is supported in part by Spanish Government Grants No. BFM2003-04147 and No. FTN2003-08337-C04-04.

The existence of a formal analogy between quantum energy spectra and discrete time series has been recently pointed out. When the energy level fluctuations are described by means of the δ_(n) statistic, it is found that chaotic quantum systems are characterized by 1/f noise, while regular systems are characterized by 1/f(2). In order to investigate the correlation structure of the δ_(n) statistic, we study the qth-order height-height correlation function C-q(tau), which measures the momentum of order q, i.e., the average qth power of the signal change after a time delay tau. It is shown that this function has a logarithmic behavior for the spectra of chaotic quantum systems, modeled by means of random matrix theory. On the other hand, since the power spectrum of chaotic energy spectra considered as time series exhibit 1/f noise, we investigate whether the qth-order height-height correlation function of other time series with 1/f noise exhibits the same properties. A time series of this kind can be generated as a linear combination of cosine functions with arbitrary phases. We find that the logarithmic behavior arises with great accuracy for time series generated with random phases.

Many complex systems in nature and in human society exhibit time fluctuations characterized by a power spectrum S(f) which is a power function of the frequency f . Examples with this behavior are the Sun spot activity, the human heartbeat, the DNA sequence, or Bach’s First Brandenburg Concert. In this work, we show that the energy spectrum fluctuations of quantum systems can be formally considered as a discrete time series, with energy playing the role of time. Because of this analogy, the fluctuations of quantum energy spectra can be studied using traditional methods of time series, like calculating the Fourier transform and studying the power spectrum. We present the results for paradigmatic quantum chaotic systems like atomic nuclei (by means of large scale shell-model calculations) and the predictions of random matrix theory. We have found a surprising general property of quantum systems: The energy spectra of chaotic quantum systems are characterized by 1= f noise, while regular quantum systems exhibit 1= f^2 noise. Some other interesting applications of this time series analogy are a test of the existence of quantum chaos remnants in the nuclear masses, and the study of the order to chaos transition in semiclassical systems. In this case, it is found that the energy level spectrum exhibits 1= f^α noise with the exponent changing smoothly from α = 2 in regular systems to α= 1 in chaotic systems.

The power law $1/{f}^{\ensuremath{\alpha}}$ in the power spectrum characterizes the fluctuating observables of many complex natural systems. Considering the energy levels of a quantum system as a discrete time series where the energy plays the role of time, the level fluctuations can be characterized by the power spectrum. Using a family of quantum billiards, we analyze the order-to-chaos transition in terms of this power spectrum. A power law $1/{f}^{\ensuremath{\alpha}}$ is found at all the transition stages, and it is shown that the exponent $\ensuremath{\alpha}$ is related to the chaotic component of the classical phase space of the quantum system.

We apply a principal components analysis (PCA) to the secondary particle density distributions at ground level produced by cosmic gamma-rays and protons. For this purpose, high-energy interactions of cosmic rays with Earth's atmosphere and the resulting extensive air showers have been simulated by means of the CORSIKA Monte Carlo code. We show that a PCA of the two-dimensional particle density fluctuations provides a decreasing sequence of covariance matrix eigenvalues that have typical features of a polynomial law, which are different for different primary cosmic rays. This property is applied to the separation of electromagnetic showers from proton simulated extensive air showers, and it is proposed as a new discrimination method that can be used experimentally for gamma-proton separation. A cutting parameter related to the polynomial behavior of the decreasing sequence of covariance matrix eigenvalues is calculated, and the efficiency of the cutting procedure for gamma-proton separation is evaluated.

The power law 1/f(alpha) in the power spectrum characterizes the fluctuating observables of many complex natural systems. Considering the energy levels of a quantum system as a discrete time series where the energy plays the role of time, the level fluctuations can be characterized by the power spectrum. Using a family of quantum billiards, we analyze the order-to-chaos transition in terms of this power spectrum. A power law 1/f(alpha) is found at all the transition stages, and it is shown that the exponent alpha is related to the chaotic component of the classical phase space of the quantum system.

It was recently conjectured that 1/f noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the behavior of the power spectrum of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory we derive theoretical expressions that explain the power spectrum behavior at all frequencies. These expressions reproduce to a good approximation the power laws of type 1/f (1/f^2) characteristics of chaotic (integrable) systems, observed in almost the whole frequency domain. Although we use random matrix theory to derive these results, they are also valid for semiclassical systems., Comment: 5 pages (Latex), 3 figures

The main signature of chaos in a quantum system is provided by spectral statistical analysis of the nearest-neighbor spacing distribution P(s) and the spectral rigidity given by the Delta(3)(L) statistic. It is shown that some standard unfolding procedures, such as local unfolding and Gaussian broadening, lead to a spurious saturation of Delta(3)(L) that spoils the relationship of this statistic with the regular or chaotic motion of the system. This effect can also be misinterpreted as Berry's saturation.

The HEGRA collaboration has collected 225 million air shower events with a scintillator array distributed over 3.2×104 m2 in the time between 1989 and 1992. We have searched the data for steady state and sporadic emission from point sources. The median energy of γ showers registers by our detector is about 75 TeV and the average angular resolution 0.7°. No significant steady state or sporadic excess has been found. Upper limits are presented.