scholarly journals Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 1023
Author(s):  
Vito Dario Camiola ◽  
Liliana Luca ◽  
Vittorio Romano
Keyword(s):  
Wigner Function ◽  
Natural Generalization ◽  
Bloch Equation ◽  
Equilibrium Density ◽  
Diagonal Form ◽  
Energy Dispersion ◽  
Unitary Evolution ◽  
Von Neumann ◽  
General Energy

The approach based on the Wigner function is considered as a viable model of quantum transport which allows, in analogy with the semiclassical Boltzmann equation, to restore a description in the phase-space. A crucial point is the determination of the Wigner function at the equilibrium which stems from the equilibrium density function. The latter is obtained by a constrained maximization of the entropy whose formulation in a quantum context is a controversial issue. The standard expression due to Von Neumann, although it looks a natural generalization of the classical Boltzmann one, presents two important drawbacks: it is conserved under unitary evolution time operators, and therefore cannot take into account irreversibility; it does not include neither the Bose nor the Fermi statistics. Recently a diagonal form of the quantum entropy, which incorporates also the correct statistics, has been proposed in Snoke et al. (2012) and Polkovnikov (2011). Here, by adopting such a form of entropy, with an approach based on the Bloch equation, the general condition that must be satisfied by the equilibrium Wigner function is obtained for general energy dispersion relations, both for fermions and bosons. Exact solutions are found in particular cases. They represent a modulation of the solution in the non degenerate situation.

scholarly journals Correction: Camiola, V.D., et al. Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation. Entropy 2020, 22, 1023

Entropy ◽  
2021 ◽  
Vol 23 (4) ◽  
pp. 417
Author(s):  
Vito Dario Camiola ◽  
Liliana Luca ◽  
Vittorio Romano
Keyword(s):  
Dispersion Relation ◽  
Wigner Function ◽  
Energy Dispersion ◽  
General Energy

In Section 5 of Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation [...]

UNIFIED VARIATIONAL PRINCIPLES FOR THE VON NEUMANN, MASTER AND BOLTZMANN-BLOCH EQUATIONS AND THE T-MATRIX

1995 ◽  
Vol 09 (10) ◽  
pp. 1227-1242
Author(s):  
MASUMI HATTORI ◽  
HUZIO NAKANO
Keyword(s):  
Density Matrix ◽  
Variational Principles ◽  
Matrix Theory ◽  
Bloch Equation ◽  
Irreversible Processes ◽  
Collision Term ◽  
Maximum Condition ◽  
Von Neumann ◽  
Maximum Problem ◽  
T Matrix

The variational principle of irreversible processes, which was previously presented for the von Neumann equation as a stationarity problem and then converted into a maximum problem by contracting the density matrix perturbatively, is reinvestigated w.r.t. the contraction of the density matrix. The present contraction relies on the T-matrix theory of scattering, where no perturbational consideration enters. By taking the electron transport in solids as a typical example, the contraction is performed in two steps: the even component of the density matrix as to time reversal is eliminated first and then the off-diagonal elements in the scheme of diagonalizing the unperturbed Hamiltonian. The maximum problem thus obtained is for the diagonal elements of the odd component of the density matrix. The maximum condition gives the master equation, which is reduced to the Boltzmann-Bloch equation in the scheme of one-body picture. It is noticeable in this equation that the collision term is given in terms of the T-matrix in scattering theory.

Suppression of turbulence in the drift-resistive plasma where zonal flow changes direction

2020 ◽  
Vol 86 (3) ◽  
Author(s):  
Chang-Bae Kim
Keyword(s):  
Electric Potential ◽  
Zonal Flow ◽  
Equilibrium Density ◽  
Turbulence Level ◽  
Edge Region ◽  
Transport Barrier ◽  
Large Gradient ◽  
Resistive Plasma ◽  
Flow Changes

The edge region of quasi-adiabatic plasma is pedagogically simulated by the dynamics between the electric potential $\unicode[STIX]{x1D711}$ and the electron density $n$ whose equilibrium density gradient is negative and held fixed. The zonal flow (ZF) $V$ is either enforced sinusoidally or generated self-consistently from the turbulence. Cross-phase $\unicode[STIX]{x1D6FF}$ between $\unicode[STIX]{x1D711}$ and $n$ , which is important in the determination of the turbulence level and the transport, is strongly influenced by the ZF. In the region near $V=0$ , $\unicode[STIX]{x1D6FF}$ becomes negative due to the large gradient of the ZF. It is found that the instabilities are quenched there, and the fluctuations decay. The ZF thus works as a transport barrier in the region where the ZF changes direction with large gradient.

Direct Determination of Lifetime and Energy Dispersion for the EmptyΔ1Conduction Band of Copper

1978 ◽  
Vol 41 (12) ◽  
pp. 825-828 ◽  
Author(s):  
D. E. Eastman ◽  
J. A. Knapp ◽  
F. J. Himpsel
Keyword(s):  
Direct Determination ◽  
Energy Dispersion

scholarly journals Petrographic structures: Khibiny ijolites and urtites

Vestnik MGTU ◽  
2021 ◽  
Vol 24 (2) ◽  
pp. 160-167
Author(s):  
Yuri Leonidovich Voytekhovsky ◽  
Alena Alexandrovna Zakharova
Keyword(s):  
Crystalline Rocks ◽  
Diagonal Form ◽  
Standard Description ◽  
Khibiny Massif ◽  
Constitutional Principle ◽  
Hardy Weinberg Equilibrium ◽  
Definition Of ◽  
Canonical Diagonal Form ◽  
Strict Definition

In addition to the standard description of the structures and textures of crystalline rocks the mathematical approaches have been proposed based on a rigorous determination of the petrographic structure through the probabilities of binary intergrain contacts. In general, the petrographic structure is defined as an invariant aspect of rock organization, algebraically expressed by the canonical diagonal form of the symmetric Pij matrix and geometrically visualized by structural indicatrices - surfaces of the 2nd order. The agreed nomenclature of possible petrographic structures for an n-mineral rock is simple: the symbol Snm means that there are exactly m positive numbers in the canonical diagonal form of the Pij matrix. New types of barycentric diagrams have been proposed. To describe the massive texture, the concept of Hardy - Weinberg equilibrium has been proposed. This boundary classifies barycentric diagrams into areas within which canonical types of Рij matrices and topological types of structural indicatrices are preserved. The change in the organization of the rock within a type is quantitative, the transition from one type to another means structural restructuring. The methods are used to describe ijolites and urtites of the Khibiny massif, the Kola Peninsula. In the modern taxonomy of rocks, the boundaries between them are mostly conditional and are drawn according to the contents of rock-forming minerals, for example, between ijolites and urtites - according to the contents of nepheline and pyroxene. The strict definition of the petrographic structure proposed by the authors makes it possible to introduce into petrography the constitutional principle (structure + composition), which is successfully acting in mineralogy.

scholarly journals Petrographic structures and Hardy – Weinberg equilibrium

2020 ◽  
Vol 242 ◽  
pp. 133
Author(s):  
Yury VOYTEKHOVSKY ◽  
Alena ZAKHAROVA
Keyword(s):  
Quadratic Forms ◽  
Dimensional Space ◽  
Complete Classification ◽  
Algebraic Expression ◽  
Diagonal Form ◽  
Hardy Weinberg Equilibrium ◽  
Algebraic Forms ◽  
Canonical Diagonal Form

The article is devoted to the most narrative side of modern petrography – the definition, classification and nomenclature of petrographic structures. We suggest a mathematical formalism using the theory of quadratic forms (with a promising extension to algebraic forms of the third and fourth orders) and statistics of binary (ternary and quaternary, respectively) intergranular contacts in a polymineralic rock. It allows constructing a complete classification of petrographic structures with boundaries corresponding to Hardy – Weinberg equilibria. The algebraic expression of the petrographic structure is the canonical diagonal form of the symmetric probability matrix of binary intergranular contacts in the rock. Each petrographic structure is uniquely associated with a structural indicatrix – the central quadratic surface in n-dimensional space, where n is the number of minerals composing the rock. Structural indicatrix is an analogue of the conoscopic figure used for optical recognition of minerals. We show that the continuity of changes in the organization of rocks (i.e., the probabilities of various intergranular contacts) does not contradict a dramatic change in the structure of the rocks, neighboring within the classification. This solved the problem, which seemed insoluble to A.Harker and E.S.Fedorov. The technique was used to describe the granite structures of the Salminsky pluton (Karelia) and the Akzhailau massif (Kazakhstan) and is potentially applicable for the monotonous strata differentiation, section correlation, or wherever an unambiguous, reproducible determination of petrographic structures is needed. An important promising task of the method is to extract rocks' genetic information from the obtained data.

scholarly journals Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1292
Author(s):  
Florio M. Ciaglia ◽  
Fabio Di Di Cosmo ◽  
Alberto Ibort ◽  
Giuseppe Marmo
Keyword(s):  
Composite System ◽  
Classical System ◽  
Quantum Systems ◽  
Entangled State ◽  
Countable Number ◽  
Unitary Evolution ◽  
Von Neumann ◽  
New Family ◽  
Abelian Von Neumann Algebra ◽  
Totally Disconnected

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.

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