The Phase Space Model of Nonrelativistic Quantum Mechanics

We focus on several questions arising during the modelling of quantum systems on a phase space. First, we discuss the choice of phase space and its structure. We include an interesting case of discrete phase space. Then, we introduce the respective algebras of functions containing quantum observables. We also consider the possibility of performing strict calculations and indicate cases where only formal considerations can be performed. We analyse alternative realisations of strict and formal calculi, which are determined by different kernels. Finally, two classes of Wigner functions as representations of states are investigated.

TRANSITION AND EVOLUTION OF INFORMATION AT THE MICROSCOPIC LEVEL

A simultaneous development of the fundamental research areas of the information theory is needed for efficient development in the information technologies. It is known that for the complicated macroscopic systems information evolution may be shaped on the basis of the principal thermodynamics laws (the second law of thermodynamics, etc). At the same time it is not known whether the fundamentals of the information theory for the macroscopic systems may be applicable to the microscopic systems. The study works out a mathematic model of the discrete phase space adapted to describing the evolution of information (entropy) of the microscopic systems. The discrete phase-space model rests on the indeterminacy principle and fundamental properties of the discrete continuous-time Markovian systems. The Kolmogorov equations represent the main mathematical tools technique. The suggested model refers to the smallest metric scale when the external macroscopic observation is possible. This scale can be viewed as a quasiclassical level. The research results are the following. The structure of the phase space of the elementary signal is revealed. It is demonstrated that the entropy of the microscopic systems increases, i.e. for the microscopic systems the second law of thermodynamics is true. There has been demonstrated transition from the microscopic model to the macroscopic one thus proving the former’s adequacy. The discrete phase-space model is promising in the aspect of further development. For example, it can be applied to the physical systems “particle – field”. The approach represented by the model will allow to study electromagnetic and gravity fields at the quasiclassical level. The above model of the discrete phase space and its application in the study of the evolution of the microscopic systems is a proprietary design of the authors.

Phase-Space Model of a Collisionless Stellar Cylinder Embedded in a Rotating Halo

The phase-space model of a stellar cylindrical bar embedded in a rotating triaxial halo is constructed. The equations of motion of an individual star in the bar are derived and solved. The model has three integrals of motion and the condition of the cylinder boundary conservation is derived. The model is found to represent a four-dimensional ellipsoid in six-dimensional phase space. The phase-space distribution function of stars is derived, which depends on isolating integrals of motion. The centroid velocity field describes longitudinal shear averaged flows in the cylinder. Two non-zero components of the velocity dispersion tensor depend quadratically on coordinates and vanish at the surface of the cylindrical bar.