Neighbor Spacing Distribution Research Articles

Prediction of Solar Ultraviolet Radiations Using Random Matrix Theory

Purpose: The correlation of different samples can be described by analytical models such as random matrix theory. In this study, we tried to describe the correlation of different types of ultraviolet values in different months, weeks, and hours to get a significant relationship of special times, which one needs to get enough intensity of the sun or avoid getting sunburn.
 Materials and Methods: To this aim, we focused on the hourly and daily mean amounts of ultraviolet A, B, and C intensities of solar radiation in Tabriz urban area were measured during a full year of 2017-2018. We used such ultraviolet values which are measured at the same hour of the day to satisfy the same symmetry criteria which are necessary in random matrix theory. These data are unfolded and classified in different sequences to analyze in the nearest neighbor spacing distribution framework via the maximum likelihood estimation technique.
 Results: Strong correlation is yielded for daily values of UVA in comparison with the other types of ultraviolet radiations. Also, we considered the dependence of correlation degrees of these three types of ultraviolet to average temperature and humidity at different months.
 Conclusion: The results propose more correlation of UVA indices in August while such correlation of UVC radiations are yielded in December.

Classical and quantum elliptical billiards: Mixed phase space and short-range correlations in singlets and doublets

Billiards are flat cavities where a particle is free to move between elastic collisions with the boundary. In chaos theory, these systems are simple prototypes. The conservative dynamics of a billiard may vary from regular to chaotic, depending only on the shape of the boundary. This work aims to shed light into the quantization of classically chaotic systems. We present numerical results on classical and quantum properties in two bi-parametric families of billiards, namely Elliptical Stadium Billiard (ESB) and Elliptical-C3 Billiards (E-C3B). Both families entail elliptical perturbations of chaotic billiards with originally circular sectors on their borders. Our numerical calculations provide evidence that these elliptical families may exhibit a mixed classical dynamics, identified by the chaotic fraction of the phase space, the parameter χc<1. We use this quantity to guide our analysis of quantum spectra. We explore the short-range correlations through nearest neighbor spacing distribution p(s), revealing that in the mixed region of the classical phase space, p(s) is well described by the Berry–Robnik–Brody (BRB) distributions for the ESB. In agreement with the expectation from the so-called ergodic parameter α=tH/tT, the ratio between the Heisenberg time and the classical diffusive-like transport time. Our findings indicate the possibility of quantum dynamical localization when α<1. For the E-C3B family, the eigenstates can be split into singlets and doublets. BRB successfully describes p(s) for singlets as the previous family in the mixed region. However, for doublets, new distributions recently introduced in the literature come into play, providing descriptions for p(s) with a focus on cases where χc=1. We observed that as χc decreases, the p(s) distributions simultaneously deviate from the Gaussian Orthogonal Ensemble (GOE) for singlets, and Gaussian Unitary Ensemble (GUE) for doublets.

Spacing ratio statistics of multiplex directed networks

Eigenvalues statistics of various many-body systems have been widely studied using the nearest neighbor spacing distribution under the random matrix theory framework. Here, we numerically analyze eigenvalue ratio statistics of multiplex networks consisting of directed Erdős-Rényi random networks layers represented as, first, weighted non-Hermitian random matrices and then weighted Hermitian random matrices. We report that the multiplexing strength rules the behavior of average spacing ratio statistics for multiplexing networks represented by the non-Hermitian and Hermitian matrices, respectively. Additionally, for both these representations of the directed multiplex networks, the multiplexing strength appears as a guiding parameter for the eigenvector delocalization of the entire system. These results could be important for driving dynamical processes in several real-world multilayer systems, particularly, understanding the significance of multiplexing in comprehending network properties.

Investigation of the statistical fluctuations of the electric quadrupole transition rates of spherical nuclei by kernel density estimation method‎

In this study, the statistical correlation of the electric quadrupole transition probabilities in the spherical nuclei is investigated. To this aim, the spherical even-even nuclei in the 100 < A < 126 mass region are selected and the electric quadrupole transition rates of different levels in the ground band are determined by the interacting boson model in the U(5) dynamical limit. A non-parametric kernel density estimation method is used in the framework of nearest neighbor spacing distribution of random matrix theory to consider the statistical correlation of these quadrupole transition rates. Also, the Kullback-Leibler divergence is used to describe the regular or chaotic behavior of the considered sequences. The results show the correlation of the electric quadrupole transition probabilities between different levels of the ground band. Also, the correlation increased for such transitions that the angular momentum of the initial state of quadrupole transitions increased.

Investigation of statistical correlation of negative parity energy levels in Tm nucleus with A=170

In this paper, the statistical properties of 1- to 7- negative parity levels in the 170Tm nucleus are considered with emphasis on the nearest neighbor spacing distribution in the framework of random matrix theory. The latest available experimental data for different negative parity energy levels in the E≤ 3000 keV region are used to classifiy in different sequences and are analysed. The Berry – Robnik distribution function and the least square extraction method have been used to determine the parameter of distribution function in considered sequences. The results show the adaptation of the statistical behavior of all considered negative parity levels with the Wigner distribution function and therefore, the statistical correlation of such levels. Also, the strong correlations are yield for the sequences prepared by levels with odd spin values while the 2- levels show a weak correlation and a Poisson-like behavior. The dependence of the statistical behavior of the negative parity levels to the energy was investigated which suggests the maximum correlation in the 1300 <E < 2100 keV energy region.

On the Robin spectrum for the equilateral triangle* *Dedicated to Michael Berry for his 80th birthday.

The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lamé in 1833, despite not admitting separation of variables. In this paper, we study the Robin spectrum of the equilateral triangle, which was determined by McCartin in 2004 in terms of a system of transcendental coupled secular equations. We give uniform upper bounds for the Robin–Neumann gaps, showing that they are bounded by their limiting mean value, which is hence an almost sure bound. The spectrum admits a systematic double multiplicity, and after removing it we study the gaps in the resulting desymmetrized spectrum. We show a spectral gap property, that there are arbitrarily large gaps, and also arbitrarily small ones, moreover that the nearest neighbour spacing distribution of the desymmetrized spectrum is a delta function at the origin. We show that for sufficiently small Robin parameter, the desymmetrized spectrum is simple.

From integrability to chaos in quantum Liouvillians

The dynamics of open quantum systems can be described by a Liouvillian, which in the Markovian approximation fulfills the Lindblad master equation. We present a family of integrable many-body Liouvillians based on Richardson-Gaudin models with a complex structure of the jump operators. Making use of this new region of integrability, we study the transition to chaos in terms of a two-parameter Liouvillian. The transition is characterized by the spectral statistics of the complex eigenvalues of the Liouvillian operators using the nearest neighbor spacing distribution and by the ratios between eigenvalue distances.

Time-reversal-invariant hexagonal billiards with a point symmetry.

A biparametric family of hexagonal billiards enjoying the C_{3} point symmetry is introduced and numerically investigated. First, the relative measure r(ℓ,θ;t) in a reduced phase space was mapped onto the parameter plane ℓ×θ for discrete time t up to 10^{8} and averaged in tens of randomly chosen initial conditions in each billiard. The resulting phase diagram allowed us to identify fully ergodic systems in the set. It is then shown that the absolute value of the position autocorrelation function decays like |C_{q}(t)|∼t^{-σ}, with 0<σ⩽1 in the hexagons. Following previous examples of irrational triangles, we were able to find billiards for which σ∼1. This is further evidence that, although not chaotic (all Lyapunov exponents are zero), billiards in polygons might exhibit a near strongly mixing dynamics in the ergodic hierarchy. Quantized counterparts with distinct classical properties were also characterized. Spectral properties of singlets and doublets of the quantum billiards were investigated separately well beyond the ground state. As a rule of thumb, for both singlet and doublet sequences, we calculate the first 120000 energy eigenvalues in a given billiard and compute the nearest neighbor spacing distribution p(s), as well as the cumulative spacing function I(s)=∫_{0}^{s}p(s^{'})ds^{'}, by considering the last 20000 eigenvalues only. For billiards with σ∼1, we observe the results predicted for chaotic geometries by Leyvraz, Schmit, and Seligman, namely, a Gaussian unitary ensemble behavior in the degenerate subspectrum, in spite of the presence of time-reversal invariance, and a Gaussian orthogonal ensemble behavior in the singlets subset. For 0<σ<1, formulas for intermediate quantum statistics have been derived for the doublets following previous works by Brody, Berry and Robnik, and Bastistić and Robnik. Different regimes in a given energy spectrum have been identified through the so-called ergodic parameter α=t_{H}/t_{C}, the ratio between the Heisenberg time and the classical diffusive-like transport time, which signals the possibility of quantum dynamical localization when α<1. A good quantitative agreement is found between the appropriate formulas with parameters extracted from the classical phase space and the data from the calculated quantum spectra. A rich variety of standing wave patterns and corresponding Poincaré-Husimi representations in a reduced phase space are reported, including those associated with lattice modes, scarring, and high-frequency localization phenomena.

Statistical fluctuations of negative parity levels in even mass nuclei

This study presents the first statistical analysis on the correlation of negative parity states. All the available experimental data of 1−&2− levels of the even mass nuclei in the 16≤A≤212 mass region are used to prepare the sequences and classified as their masses. Random matrix theory together with a usual least - square procedure are used to extract the chaos measures of Berry – Robnik distribution function for different sequences in the framework of nearest neighbor spacing distribution. Also, we considered the chaotic degrees of these levels in relation to different modes of calculated deformations values and the shell model configuration of the last protons and neutrons of even mass nuclei. The results of this consideration suggest the most correlation of 1− levels in the A<50, mass area, but similar statistics of 2− levels are yield in the 150<A<200 region. The most GOE-like behavior of 1− levels is reported for sequences that the last protons lie in the d3/2 orbitals, and the neutrons filled the h9/2 orbitals, but for 2− levels, nuclei with 1h11/2 and 1i13/2 configurations for respectively, protons and neutrons, show the maximum correlation. Also, the results of this study suggest the most correlation of 1− levels for some mass regions which have the maximum values of the averaged quadrupole and hexacontatetrapole deformations.

Random matrix analysis of multiplex networks

We investigate the spectra of adjacency matrices of multiplex networks under random matrix theory (RMT) framework. Through extensive numerical experiments, we demonstrate that upon multiplexing two random networks, the spectra of the combined multiplex network exhibit superposition of two Gaussian orthogonal ensemble (GOE)s for very small multiplexing strength followed by a smooth transition to the GOE statistics with an increase in the multiplexing strength. Interestingly, randomness in the connection architecture, introduced by random rewiring to 1D lattice, of at least one layer may govern nearest neighbor spacing distribution (NNSD) of the entire multiplex network, and in fact, can drive to a transition from the Poisson to the GOE statistics or vice versa. Notably, this transition transpires for a very small number of the random rewiring corresponding to the small-world transition. Ergo, only one layer being represented by the small-world network is enough to yield GOE statistics for the entire multiplex network. Spectra of adjacency matrices of underlying interaction networks have been contemplated to be related with dynamical behavior of the corresponding complex systems, the investigations presented here have implications in achieving better structural and dynamical control to the systems represented by multiplex networks against structural perturbation in only one of the layers.

A Study on the Statistical Properties of the Prime Numbers Using the Classical and Superstatistical Random Matrix Theories

The prime numbers have attracted mathematicians and other researchers to study their interesting qualitative properties as it opens the door to some interesting questions to be answered. In this paper, the Random Matrix Theory (RMT) within superstatistics and the method of the Nearest Neighbor Spacing Distribution (NNSD) are used to investigate the statistical proprieties of the spacings between adjacent prime numbers. We used the inverse χ 2 distribution and the Brody distribution for investigating the regular-chaos mixed systems. The distributions are made up of sequences of prime numbers from one hundred to three hundred and fifty million prime numbers. The prime numbers are treated as eigenvalues of a quantum physical system. We found that the system of prime numbers may be considered regular-chaos mixed system and it becomes more regular as the value of the prime numbers largely increases with periodic behavior at logarithmic scale.

Superposition and higher-order spacing ratios in random matrix theory with application to complex systems

The statistical properties of spectra of quantum systems within the framework of random matrix theory is widely used in many areas of physics. These properties are affected, if two or more sets of spectra are superposed, resulting from the discrete symmetries present in the system. Superposition of spectra of $m$ such circular orthogonal, unitary and symplectic ensembles are studied numerically using higher-order spacing ratios. For given $m$ and the Dyson index $\beta$, the modified index $\beta'$ is tabulated whose nearest neighbor spacing distribution is identical to that of $k$-th order spacing ratio. For the case of $m=2$ ($m=3$) in COE (CUE) a scaling relation between $\beta'$ and $k$ is given. For COE, it is conjectured that for $k=m+1$ ($m\geq2$) and $k=m-3$-th ($m\geq5$) order spacing ratio distribution the $\beta'$ is $m+2$ and $m-4$ respectively. Whereas in the case of CSE, for $k=m+1$ ($m\geq2$) and $k=m-1$-th ($m\geq3$) the $\beta'$ is $2m+3$ and $2(m-2)$ respectively. We also conjecture that for given $m$ ($k$) and $\beta$, the sequence of $\beta'$ as a function of $k$ ($m$) is unique. Strong numerical evidence in support of these results is presented. These results are tested on complex systems like the measured nuclear resonances, quantum chaotic kicked top and spin chains.

Partial dynamical symmetry versus quasi dynamical symmetry examination within a quantum chaos analyses of spectral data for even\u2013even nuclei

Statistical analyses of the spectral distributions of rotational bands in 51 deformed prolate even–even nuclei in the 152 ≤ A ≤ 250 mass region R_{{4_{1}^{ + } /2_{1}^{ + } }} ge 3.00 are examined in terms of nearest neighbor spacing distributions. Specifically, the focus is on data for 0+, 2+, and 4+ energy levels of the ground, gamma, and beta bands. The chaotic behavior of the gamma band, especially the position of the 2_{gamma }^{ + } band-head compared to other levels and bands, is clear. The levels are analyzed within the framework of two models, namely, a SU(3)-partial dynamical symmetry Hamiltonian and a SU(3) two-coupled quasi-dynamical symmetry Hamiltonian, with results that are further analyzed using random matrix theory. The partial and quasi dynamics both yield outcomes that are in reasonable agreement with the known experimental results. However, due to the degeneracy of the beta and gamma bands within the simplest SU(3) picture, the theory cannot be used to describe the fluctuation properties of excited bands. By changing relative weights of the different terms in the partial and quasi dynamical Hamiltonians, results are obtained that show more GOE-like statistics in the partial dynamical formalism as the strength of the pairing term is increased. Also, in the quasi-dynamical symmetry limit, more correlations are found because of the stronger couplings.

Territorial behaviour of buzzards versus random matrix spacing distributions

A deeper understanding of the processes underlying the distribution of animals in space is crucial for both basic and applied ecology. The Common buzzard (Buteo buteo) is a highly aggressive, territorial bird of prey that interacts strongly with its intra- and interspecific competitors. We propose and use random matrix theory to quantify the strength and range of repulsion as a function of the buzzard population density, thus providing a novel approach to model density dependence. As an indicator of territorial behaviour, we perform a large-scale analysis of the distribution of buzzard nests in an area of 300 square kilometres around the Teutoburger Wald, Germany, as gathered over a period of 20 years. The nearest and next-to-nearest neighbour spacing distribution between nests is compared to the two-dimensional Poisson distribution, originating from uncorrelated random variables, to the complex eigenvalues of random matrices, which are strongly correlated, and to a two-dimensional Coulomb gas interpolating between these two. A one-parameter fit to a time-moving average reveals a significant increase of repulsion between neighbouring nests, as a function of the observed increase in absolute population density over the monitored period of time, thereby proving an unexpected yet simple model for density-dependent spacing of predator territories. A similar effect is obtained for next-to-nearest neighbours, albeit with weaker repulsion, indicating a short-range interaction. Our results show that random matrix theory might be useful in the context of population ecology.

Normal mode analysis of spectra of random networks

Several spectral fluctuation measures of random matrix theory (RMT) have been applied in the study of spectral properties of networks. However, the calculation of those statistics requires performing an unfolding procedure, which may not be an easy task. In this work, network spectra are interpreted as time series, and we show how their short and long-range correlations can be characterized without implementing any previous unfolding. In particular, we consider three different representations of Erdős–Rényi (ER) random networks: standard ER networks, ER networks with random-weighted self-edges, and fully random-weighted ER networks. In each case, we apply singular value decomposition (SVD) such that the spectra are decomposed in trend and fluctuation normal modes. We obtain that the fluctuation modes exhibit a clear crossover between the Poisson and the Gaussian orthogonal ensemble statistics when the average degree of ER networks changes. Moreover, by using the trend modes, we perform a data-adaptive unfolding to calculate, for comparison purposes, traditional fluctuation measures such as the nearest neighbor spacing distribution, number variance Σ2, as well as Δ3 and δn statistics. The thorough comparison of RMT short and long-range correlation measures make us identify the SVD method as a robust tool for characterizing random network spectra.

Manifestation of crossover from RMT fluctuation to Poisson fluctuation in BSE sensex, a prototype of financial systems

We use the nearest neighbour spacing distribution (NNSD) of random matrix theory (RMT) as a tool to identify the type of spectral fluctuations of matrix ensemble constructed using high frequency BSE sensex values. NNSD shows Wigner distribution for ensemble formed with detrended index values. The detrended raw index values has the universal properties of the Gaussian orthogonal ensemble of random matrices. In contrast, NNSD shows Poisson distribution for ensemble formed using log transformed return of the index values which is corroborated by computation of the number variance. We have also shown the intermediate transitions connecting RMT and Poissonian fluctuations applying NNSD on the data sets generated using power transformation on the return index values. Log transformation is obtained in the asymptotic limit of the power transformation. This gradual changes in the spectral fluctuations from RMT to Poissonian type is reminiscent of localization theory results obtained in simulated experiments. On the contrary, the BSE Sensex values, approximately 2 million data for a period of 2006–2010, has been taken as a prototype of real systems representing an emerging economy of the world.

Manifestations of isospin in nearest neighbor spacing distributions for the f-p model space

The strong interactions are charge independent. If we limit ourselves to the strong interactions, we have the isospin [Formula: see text] as a good quantum number. Here, we consider the lack of level repulsion of states of different isospin and how this effect manifests in nearest neighbor spacing (NNS) histograms, which provide a visual and statistical context in which to study distributions of energy level spacings. In particular, we study nucleons in the [Formula: see text] model space for the nucleus [Formula: see text]. We also compare NNS distributions produced in the isospin formalism to distributions produced in the proton–neutron formalism.

Test for a universal behavior of Dirac eigenvalues in the complex Langevin method

We apply the complex Langevin (CL) method to a chiral random matrix theory (ChRMT) at non-zero chemical potential and study the nearest neighbor spacing (NNS) distribution of the Dirac eigenvalues. The NNS distribution is extracted using an unfolding procedure for the Dirac eigenvalues obtained in the CL method. For large quark mass, we find that the NNS distribution obeys the Ginibre ensemble as expected. For small quark mass, the NNS distribution follows the Wigner surmise for correct convergence case, while it follows the Ginibre ensemble for wrong convergence case. The Wigner surmise is physically reasonable from the chemical potential independence of the ChRMT. The Ginibre ensemble is known to be favored in a phase quenched QCD at finite chemical potential. Our result suggests a possibility that the originally universal behavior of the NNS distribution is preserved even in the CL method for correct convergence case.

STRUCTURAL STUDIES OF EIGHT BRIGHT RIMMED CLOUDS IN THE SOUTHERN HEMISPHERE

ABSTRACT We carried out deep- and wide-field near- and mid-infrared observations for a sample of eight bright-rimmed clouds (BRCs). Supplemented with the Spitzer archival data, we have identified and classified 44 to 433 young stellar objects (YSOs) associated with these BRCs. The Class I sources are generally located toward the places with higher extinction and are relatively closer to each other than the Class II sources, confirming that the young protostars are usually found in regions with denser molecular material. On the other hand the comparatively older population, Class II objects, are more randomly found throughout the regions, which can be due to their dynamical evolution. Using the minimal sampling tree analyses, we have extracted 13 stellar cores of eight or more members, which contain 60% of the total YSOs. The typical core is ∼0.6 pc in radii and somewhat elongated (aspect ratio of 1.45), of relatively low stellar density (surface density 60 pc−2), consisting of a small (35) number of YSOs of relatively young sources (66% Class I), and partially embedded (median A K = 1.1 mag). But the cores show a wide range in their mass distribution (∼20 to 2400 M ⊙), with a median value of around 130 M ⊙. We found the star-formation efficiencies in the cores to be between 3% and 30% with an average of ∼14%, which agrees with the efficiencies needed to link the core mass function to the initial mass function. We also found a linear relation between the density of the clouds and the number of YSOs. The peaked nearest neighbor spacing distributions of the YSOs and the ratio of Jeans lengths to the YSO separations indicates a significant degree of non-thermally driven fragmentation in these BRCs.

Supersymmetric analogue of BCN type rational integrable models with polarized spin reversal operators

We derive the exact spectra as well as partition functions for a class of BCN type of spin Calogero models, whose Hamiltonians are constructed by using supersymmetric analogues of polarized spin reversal operators (SAPSRO). The strong coupling limit of these spin Calogero models yields BCN type of Polychronakos–Frahm (PF) spin chains with SAPSRO. By applying the freezing trick, we obtain an exact expression for the partition functions of such PF spin chains. We also derive a formula which expresses the partition function of any BCN type of PF spin chain with SAPSRO in terms of partition functions of several AK types of supersymmetric PF spin chains, where K⩽N−1. Subsequently we show that an extended boson–fermion duality relation is obeyed by the partition functions of the BCN type of PF chains with SAPSRO. Some spectral properties of these spin chains, like level density distribution and nearest neighbor spacing distribution, are also studied.