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

Entropy, Maximum Entropy Priciple and quantum statistical information for various random matrix ensembles

The random matrix ensembles (RME) of quantum statistical Hamiltonians, e.g. Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied in literature to following quantum statistical systems: molecular systems, nuclear systems, disordered materials, random Ising spin 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 "Dynamics Days 2003; XXIII annual conference - 4 decades of chaos 1963-2003"; September 24, 2003 - September 27, 2003; University of the Balearic Islands, Palma de Mallorca, Spain (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

Dynamics of complex quantum systems with energy dissipation

A complex quantum system with energy dissipation is considered. The quantum Hamiltonians $H$ belong the complex Ginibre ensemble. The complex-valued eigenenergies $Z_{i}$ are random variables. The second differences $\Delta^{1} Z_{i}$ are also complex-valued random variables. The second differences have their real and imaginary parts and also radii (moduli) and main arguments (angles). For $N$=3 dimensional Ginibre ensemble the distributions of above random variables are provided whereas for generic $N$- dimensional Ginibre ensemble second difference distribution is analytically calculated. The law of homogenization of eigenergies is formulated. The analogy of Wigner and Dyson of Coulomb gas of electric charges is studied.Comment: 4 pages; presented at poster session of the conference "Dynamics and Statistics of Complex Systems"; May 17th, 2000 to May 18th, 2000; Max Planck Institute for the Physics of Complex Systems; Dresden, Germany (2000

Stabilization of highly dimensional statistical systems: Girko ensemble

A quantum statistical system with energy dissipation is studied. Its statisitics is governed by random complex-valued non-Hermitean Hamiltonians belonging to complex Ginibre ensemble. The eigenenergies are shown to form stable structure in thermodynamical limit (large matrix dimension limit). Analogy of Wigner and Dyson with system of electrical charges is drawn.Comment: 5 pages; presented at poster session of the conference "Symposium on Entropy"; June 25th, 2000 to June 28th, 2000; Max Planck Institute for the Physics of Complex Systems; Dresden, Germany (2000

Gasdynamic model of street canyon

A general proecological urban road traffic control idea for the street canyon is proposed with emphasis placed on development of advanced continuum field gasdynamical (hydrodynamical) control model of the street canyon. The continuum field model of optimal control of street canyon is studied. The mathematical physics approach (Eulerian approach) to vehicular movement, to pollutants' emission and to pollutants' dynamics is used. The rigorous mathematical model is presented, using gasdynamical (hydrodynamical) theory for both air constituents and vehicles, including many types of vehicles and many types of pollutant (exhaust gases) emitted from vehicles. The six optimal control problems are formulated and numerical simulations are performed. Comparison with measurements are provided. General traffic engineering conclusions are inferred.Comment: 2 pages; appeared in: "Proceedings of the XIV Polish Conference on Computer Methods in Mechanics (PCCMM'99)", 26th May 1999- 28th May 1999, Rzeszow, Poland, 85-86 (1999

Statistical properties of the quantum anharmonic oscillator in one spatial dimension

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 in one spatial dimension 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: 6 pages; presented at poster session of the conference "Polymorphism in Condensed Matter International workshop"; November 13th, 2006 - November 17th, 2006; Max Planck Institute for the Physics of Complex Systems, Dresden, Germany (2006