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Maciej M. Duras

Publications and source records attributed to Maciej M. Duras.

At least 37 records · Page 2Linked to original sources
arXiv2003

Applications of the random matrix ensembles to the quantum statistical systems

The random matrix ensembles are applied to the quantum statistical systems. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The linear operators describing the systems act on these Hilbert spaces and they are treated as random matrices in generic bases of the eigenfunctions. The random eigenproblems are presented and solved. Examples of random operators are presented with connection to physical problems.

cond-mat.stat-mech↗
arXiv2003

Application of hermitean and nonhermitean random matrices to quantum statistical systems

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.

cond-mat.stat-mech↗
arXiv2003

Various applications of the random matrix ensembles to the quantum chaotic systems

The random matrix ensembles are applied to the quantum chaotic systems. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The linear operators describing the systems act on these Hilbert spaces and they are treated as random matrices in generic bases of the eigenfunctions. The random eigenproblems are presented and solved. Examples of random operators are presented with connection to physical problems.

cond-mat.stat-mech↗
arXiv2003

Extension of the Ginibre Ensembles of Random Matrices

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.

cond-mat.stat-mech↗
arXiv2003

Hydrodynamical Approach to Vehicular Flow in the Urban Street Canyon

The vehicular flow in the urban street canyon is considered. The classical field description is used in the modelling of the vehicular movement and of gaseous mixture in generic urban street canyon. The dynamical variables include vehicular densities, velocities, and emissivities: of pollutants, heat and exhaust gases, as well as standard mixture components' variables: densities, velocities, temperature, pressures. The local balances' equations predict the dynamics of the complex system. The automatic control of the vehicular flow is attained by the sets of coordinated traffic lights. The automatic control is aimed at minimization of traffic ecological costs by the application of variational calculus (Lagrange's and Bolz's problems). The theoretical description is accompanied by numerical examples of computer fluid dynamics based on real traffic data.

cond-mat.dis-nn↗
arXiv2003

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.

cond-mat.stat-mech↗
arXiv2003

Continuum Field Model of Street Canyon: Theoretical Description; Part I

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.

cond-mat.dis-nn↗
arXiv2003

Continuum Field Model of Street Canyon: Numerical Examples; Part II

Continuum field control model of a street canyon is considered. The six separate monocriterial optimal control problems consist of minimization of functionals of the total travel time, of global emissions of pollutants, and of global concentrations of pollutants, both in the studied street canyon, and in its two nearest neighbour substitute canyons, respectively. The six optimization problems for the functionals are solved numerically. General traffic control issues are inferred. The discretization method, comparison with experiment, mathematical issues, and programming issues, are presented.

cond-mat.dis-nn↗
arXiv2003

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 $Δ^{1} Z_{i}$ 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.

cond-mat.stat-mech↗
arXiv2003

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 $Δ^{1} Z_{i}$ of complex eigenenergies is presented. The distributions of real and imaginary parts of $Δ^{1} Z_{i}$ 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.

cond-mat.stat-mech↗
arXiv2003

Quantum symmetrical statistical system: Ginibre-Girko 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 $Δ^{1} Z_{i}$ of complex eigenenergies is presented. The distributions of real and imaginary parts of $Δ^{1} Z_{i}$ 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.

cond-mat.stat-mech↗
arXiv2003

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.

cond-mat.stat-mech↗
arXiv2003

Statistics of infinite dimensional random matrix ensembles

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.

cond-mat.stat-mech↗
arXiv2003

Statistics of complex dissipative systems

A quantum statistical random system with energy dissipation is studied. Its statistics is governed by random complex-valued non-Hermitean Hamiltonians belonging to complex Ginibre ensemble of random matrices. The eigenenergies of Hamiltonians are shown to form stable structure. Analogy of Wigner and Dyson with system of electrical charges is drawn.

cond-mat.stat-mech↗
arXiv2003

Applications of methods of random differential geometry to quantum statistical systems

We apply concepts of random differential geometry connected to the random matrix ensembles of the random linear operators acting on finite dimensional Hilbert spaces. The values taken by random linear operators belong to the Liouville space. This Liouville space is endowed with topological and geometrical random structure. The considered random eigenproblems for the operators are applied to the quantum statistical systems. In the case of random quantum Hamiltonians we study both hermitean (self-adjoint) and non-hermitean (non-self-adjoint) operators leading to Gaussian and Ginibre ensembles Refs. [1], [2], [3]. [1] M. M. Duras, ``Finite-difference distributions for the Ginibre ensemble,'' {\em J. Opt. B: Quantum Semiclass. Opt.} {\bf 2}, 287-291, 2000. [2] M. M. Duras, K. Sokalski, ``Finite Element Distributions in Statistical Theory of Energy Levels in Quantum Systems,'' {\em Physica} {\bf D125}, 260-274, 1999. [3] M. M. Duras, K. Sokalski, ``Higher-order finite-element distributions in statistical theory of nuclear spectra,'' {\em Phys. Rev. E} {\bf 54}, 3142-3148, 1996.

cond-mat.stat-mech↗
arXiv2003

Quantum fluctuations and random matrix theory

The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The linear operators describing the systems act on these Hilbert spaces and they are treated as random matrices in generic bases of the eigenfunctions. The random eigenproblems are presented and solved. Examples of random operators are presented with connection to physical problems.

cond-mat.stat-mech↗
arXiv2003

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).

cond-mat.stat-mech↗