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

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
M. Abdel-Mageed ◽  
Ahmed Salim ◽  
Walid Osamy ◽  
Ahmed M. Khedr
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Nearest Neighbor ◽  
Matrix Theory ◽  
Prime Numbers ◽  
Mixed System ◽  
Logarithmic Scale ◽  
Qualitative Properties ◽  
Spacing Distribution ◽  
Mixed Systems

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.

scholarly journals KAPPA-DEFORMED RANDOM-MATRIX THEORY BASED ON KANIADAKIS STATISTICS

2012 ◽  
Vol 26 (10) ◽  
pp. 1250059 ◽  
Author(s):  
A. Y. ABUL-MAGD ◽  
M. ABDEL-MAGEED
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Distribution Functions ◽  
Spacing Distribution ◽  
Initial Stage ◽  
The Matrix ◽  
Gibbs Entropy ◽  
Non Gaussian ◽  
Mixed Systems

We present a possible extension of the random-matrix theory, which is widely used to describe spectral fluctuations of chaotic systems. By considering the Kaniadakis non-Gaussian statistics, characterized by the index κ (Boltzmann–Gibbs entropy is recovered in the limit κ → 0), we propose the non-Gaussian deformations (κ ≠ 0) of the conventional orthogonal and unitary ensembles of random matrices. The joint eigenvalue distributions for the κ-deformed ensembles are derived by applying the principle maximum entropy to Kaniadakis entropy. The resulting distribution functions are base invariant as they depend on the matrix elements in a trace form. Using these expressions, we introduce a new generalized form of the Wigner surmise valid for nearly-chaotic mixed systems, where a basis-independent description is still expected to hold. We motivate the necessity of such generalization by the need to describe the transition of the spacing distribution from chaos to order, at least in the initial stage. We show several examples about the use of the generalized Wigner surmise to the analysis of the results of a number of previous experiments and numerical experiments. Our results suggest the entropic index κ as a measure for deviation from the state of chaos. We also introduce a κ-deformed Porter–Thomas distribution of transition intensities, which fits the experimental data for mixed systems better than the commonly-used gamma-distribution.

scholarly journals Finite-size corrections in random matrix theory and Odlyzko’s dataset for the Riemann zeros

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150436 ◽  
Author(s):  
Peter J. Forrester ◽  
Anthony Mays
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Correction Term ◽  
Finite Size ◽  
Statistical Properties ◽  
Unitary Matrices ◽  
Power Series Method ◽  
Spacing Distribution ◽  
Random Unitary Matrices

Odlyzko has computed a dataset listing more than 10 9 successive Riemann zeros, starting from a zero number to beyond 10 23 . This dataset relates to random matrix theory as, according to the Montgomery–Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and co-workers, have used N × N random unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natural process to apply to the dataset is to minimize it by deleting each member independently with some specified probability, and we proceed to compute empirical two-point correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterize the order 1/ N 2 correction term to the spacing distribution for random unitary matrices in terms of a second-order differential equation with coefficients that are Painlevé transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. In comparison to the Riemann zero data accurate agreement is exhibited.

scholarly journals Modeling of spacing distribution of queuing vehicles at signalized junctions using random-matrix theory

2009 ◽  
Vol 14 (2) ◽  
pp. 252-254 ◽  
Author(s):  
Xuexiang Jin ◽  
Yuelong Su ◽  
Yi Zhang ◽  
Zheng Wei ◽  
Li Li
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Spacing Distribution

scholarly journals Superstatistics in Random Matrix Theory

2012 ◽  
Vol 17 (2) ◽  
pp. 157
Author(s):  
A.Y. Abul-Magd
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Chaotic Dynamics ◽  
Nearest Neighbor ◽  
Matrix Theory ◽  
Quantum Systems ◽  
Classical Counterpart ◽  
Successful Model ◽  
Modeling Systems ◽  
Base Invariance

Random matrix theory (RMT) provides a successful model for quantum systems, whose classical counterpart has chaotic dynamics. It is based on two assumptions: (1) matrix-element independence, and (2) base invariance. The last decade witnessed several attempts to extend RMT to describe quantum systems with mixed regular-chaotic dynamics. Most of the proposed generalizations keep the first assumption and violate the second. Recently, several authors have presented other versions of the theory that keep base invariance at the expense of allowing correlations between matrix elements. This is achieved by starting from non-extensive entropies rather than the standard Shannon entropy, or by following the basic prescription of the recently suggested concept of superstatistics. The latter concept was introduced as a generalization of equilibrium thermodynamics to describe non-equilibrium systems by allowing the temperature to fluctuate. We here review the superstatistical generalizations of RMT and illustrate their value by calculating the nearest-neighbor-spacing distributions and comparing the results of calculation with experiments on billiards modeling systems in transition from order to chaos. 

SSA, Random Matrix Theory, and Noise-Reduced Correlations

2016 ◽  
Author(s):  
Jan W Dash ◽  
Xipei Yang ◽  
Mario Bondioli ◽  
Harvey J. Stein
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory

History—an overview

2018 ◽  
Author(s):  
Oriol Bohigas ◽  
Hans A. Weidenmüller
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Rapid Development ◽  
The Past ◽  
History Of ◽  
And Mathematics

An overview of the history of random matrix theory (RMT) is provided in this chapter. Starting from its inception, the authors sketch the history of RMT until about 1990, focusing their attention on the first four decades of RMT. Later developments are partially covered. In the past 20 years RMT has experienced rapid development and has expanded into a number of areas of physics and mathematics.

Sign in / Sign up

Export Citation Format

Share Document