Statistical properties of the quantum anharmonic oscillator

The random matrix ensembles (RME) of Hamiltonian matrices, e.g. Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applicable to following quantum statistical systems: nuclear systems, molecular systems, condensed phase systems, disordered systems, and two-dimensional electron systems (Wigner-Dyson electrostatic analogy). A family of quantum anharmonic oscillators is studied and the numerical investigation of their eigenenergies is presented. The statistical properties of the calculated eigenenergies are compared with the theoretical predictions inferred from the random matrix theory. Conclusions are derived.Comment: 9 pages; presented as talk on the conference "Polymorphism in Condensed Matter International workshop"; November 13th, 2006 - Novemer 17th, 2006; Max Planck Institute for the Physics of Complex Systems, Dresden, Germany (2006

Quantum fluctuations of systems of interacting electrons in two spatial dimensions

The random matrix ensembles (RME) of quantum statistical Hamiltonian operators, e.g. Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied to following quantum statistical systems: nuclear systems, molecular systems, and two-dimensional electron systems (Wigner-Dyson electrostatic analogy). Measures of quantum chaos and quantum integrability with respect to eigenergies of quantum systems are defined and calculated. Quantum statistical information functional is defined as negentropy (opposite of entropy or minus entropy). The distribution function for the random matrix ensembles is derived from the maximum entropy principle.Comment: 8 pages; presented at poster session of the conference "International Workshop on Critical Stability of Few-Body Quantum Systems"; October 17, 2005 - October 22, 2005; Max Planck Institute for the Physics of Complex Systems, Dresden, Germany (2005). Additional minor corrections and change

Simulations of fluctuations of quantum statistical systems of electrons

The random matrix ensembles (RMT) of quantum statistical Hamiltonian operators, e.g.Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied to following quantum statistical systems: nuclear systems, molecular systems, and two-dimensional electron systems (Wigner-Dyson's electrostatic analogy). The Ginibre ensemble of nonhermitean random Hamiltonian matrices $K$ is considered. Each quantum system described by $K$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. The second difference of complex eigenenergies is viewed as discrete analog of Hessian with respect to labelling index. The results are considered in view of Wigner and Dyson's electrostatic analogy. An extension of space of dynamics of random magnitudes is performed by introduction of discrete space of labeling indices. The comparison with the Gaussian ensembles of random hermitean Hamiltonian matrices $H$ is performed. Measures of quantum chaos and quantum integrability with respect to eigenergies of quantum systems are defined and they are calculated. Quantum statistical information functional is defined as negentropy (opposite of von Neumann's entropy or minus entropy). The probability distribution functionals for the random matrix ensembles (RMT) are derived from the maximum entropy principle.Comment: 7 pages; presented at poster session of the conference "International Workshop on Classical and Quantum Dynamical Simulations in Chemical and Biological Physics"; June 6, 2005 - June 11, 2005; Max Planck Institute for the Physics of Complex Systems, Dresden, Germany (2005

An example of dissipative quantum system: finite differences for complex Ginibre ensemble

The Ginibre ensemble of complex random Hamiltonian matrices $H$ is considered. Each quantum system described by $H$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. For generic $N$-dimensional Ginibre ensemble analytical formula for distribution of second difference $\Delta^{1} Z_{i}$ of complex eigenenergies is presented. The distributions of real and imaginary parts of $\Delta^{1} Z_{i}$ and also of its modulus and phase are provided for $N$=3. The results are considered in view of Wigner and Dyson's electrostatic analogy. General law of homogenization of eigenergies for different random matrix ensembles is formulated.Comment: 5 pages; presented at poster session of the conference "Quantum dynamics in terms of phase-space distributions"; May 22nd, 2000 to May 26th, 2000; Max Planck Institute for the Physics of Complex Systems; Dresden, Germany (2000

Discrete Hessians in study of Quantum Statistical Systems: Complex Ginibre Ensemble

The Ginibre ensemble of nonhermitean random Hamiltonian matrices $K$ is considered. Each quantum system described by $K$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. The second difference of complex eigenenergies is viewed as discrete analog of Hessian with respect to labelling index. The results are considered in view of Wigner and Dyson's electrostatic analogy. An extension of space of dynamics of random magnitudes is performed by introduction of discrete space of labeling indices.Comment: 6 pages; "QP-PQ: Quantum Probability and White Noise Analysis - Volume 13, Foundations of Probability and Physics, Proceedings of the Conference, Vaxjo, Sweden, 25 November - 1 December 2000"; A. Khrennikov, Ed.; World Scientific Publishers, Singapore, Vol. 13, 115-120 (2001

Complex-valued second difference as a measure of stabilization of complex dissipative statistical systems: Girko ensemble

A quantum statistical system with energy dissipation is studied. Its statistics is governed by random complex-valued non-Hermitean Hamiltonians belonging to complex Ginibre ensemble. The eigenenergies are shown to form stable structure. Analogy of Wigner and Dyson with system of electrical charges is drawn.Comment: 6 pages; "Space-time chaos: Characterization, control and synchronization; Proceedings of the International Interdisciplinary School, Pamplona, Spain, June 19-23, 2000"; S. Boccaletti, J. Burguete, W. Gonzalez-Vinas, D. L. Valladares, Eds.; World Scientific Publishers, Singapore, 45-52 (2001

Quantum statistical information, entropy, Maximum Entropy Principle in various quantum random matrix ensembles

Random matrix ensembles (RME) of quantum statistical Hamiltonian operators, {\em e.g.} Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), found applications in literature in study of following quantum statistical systems: molecular systems, nuclear systems, disordered materials, random Ising spin systems, quantum chaotic systems, and two-dimensional electron systems (Wigner-Dyson electrostatic analogy). Measures of quantum chaos and quantum integrability with respect to eigenergies of quantum systems are defined and calculated. Quantum statistical information functional is defined as negentropy (opposite of entropy or minus entropy). Entropy is neginformation (opposite of information or minus information. The distribution functions for the random matrix ensembles are derived from the maximum entropy principle.Comment: 10 pages; presented at poster session of the conference "Motion, sensation and self-organization in living cells"; Workshop and Seminar: October 20, 2003 - October 31, 2003; Max Planck Institute for the Physics of Complex Systems Dresden, Germany (2003

Quantum information and entropy in random matrix ensembles

The random matrix ensembles (RME), especially Gaussian random matrix ensembles GRME and Ginibre random matrix ensembles, are applied to following quantum systems: nuclear systems, molecular systems, and two-dimensional electron systems (Wigner-Dyson electrostatic analogy). Measures of quantum chaos and quantum integrability with respect to eigenergies of quantum systems are defined and calculated. The distribution function for the random matrix ensembles is derived from the maximum entropy principle. Information functional is defined as negentropy (opposite of entropy or minus entropy).Comment: 9 pages; presented at poster session of the conference "Collective Phenomena in the Low Temperature Physics of Glasses"; June 16, 2003 - June 20, 2003; Max Planck Institute for the Physics of Complex Systems, Dresden, Germany (2003

Quantum anharmonic oscillator and its statistical properties in the first quantization scheme

A family of quantum anharmonic oscillators is studied in any finite spatial dimension in the scheme of first quantization and the investigation of their eigenenergies is presented. The statistical properties of the calculated eigenenergies are compared with the theoretical predictions inferred from the Random Matrix theory. Conclusions are derived.Comment: 10 page

Fluctuations of Quantum Statistical Two-Dimensional Systems of Electrons

The random matrix ensembles (RME) of quantum statistical Hamiltonian operators, {\em e.g.} Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied to following quantum statistical systems: nuclear systems, molecular systems, and two-dimensional electron systems (Wigner-Dyson electrostatic analogy). Measures of quantum chaos and quantum integrability with respect to eigenergies of quantum systems are defined and calculated. Quantum statistical information functional is defined as negentropy (either opposite of entropy or minus entropy). The distribution function for the random matrix ensembles is derived from the maximum entropy principle.Comment: 7 page