Specific nonequilibrium states of the quantum harmonic oscillator described
by the Lindblad equation have been hereby suggested. This equation makes it
possible to determine time-varying effects produced by statistical operator or
statistical matrix. Thus, respective representation-varied equilibrium statistical
matrixes have been found. Specific mean value equations have been found
and their equilibrium solutions have been obtained.

Abstract:
Specific nonequilibrium states of the quantum harmonic oscillator described by the Lindblad equation have been hereby suggested. This equation makes it possible to determine time-varying effects produced by statistical operator or statistical matrix. Thus, respective representation-varied equilibrium statistical matrixes have been found. Specific mean value equations have been found and their equilibrium solutions have been obtained.

Abstract:
The mean field method is applied for analysis of valence electrons in metals. It is shown that at low temperatures electrons have two wave-vector distribution patterns. Isotropic distribution refers to the first pattern. Anisotropic distribution refers to another pattern, particularly to specific wave-vector values occurred nearby the Fermi sphere. It is shown that it is the anisotropy that makes the metal obtain its specific superconductor features.

Abstract:
A new variational method for studying the equilibrium states of an interacting particles system has been proposed. The statistical description of the system is realized by means of a density matrix. This method is used for description of conduction electrons in metals. An integral equation for the electron distribution function over wave vectors has been obtained. The solutions of this equation have been found for those cases where the single-particle Hamiltonian and the electron interaction Hamiltonian can be approximated by a quite simple expression. It is shown that the distribution function at temperatures below the critical value possesses previously unknown features which allow to explain the superconductivity of metals and presence of a gap in the energy spectrum of superconducting electrons.

Abstract:
Medium field method is applied for studying valence electron behavior in metals. When different wave-vector electrons are attracted at low temperatures, distribution function gets discontinued. As a result, a specific energy gap occurs.

Abstract:
The variational theory of equilibrium boson system state to have been previously developed by the author under the density matrix formalism is applicable for researching equilibrium states and thermodynamic properties of the quantum Bose gas which consists of zero-spin particles. Particle pulse distribution function is obtained and duly employed for calculation of chemical potential, internal energy and gas capacity temperature dependences. It is found that specific phase transition, which is similar to transition of liquid helium to its superfluid state, occurs at the temperature exceeding that of the Bose condensation.

Abstract:
We consider an open quantum many-particle system in which there are dissipative processes. The evolution of this system is described by a kinetic equation for the density matrix. From the equation describing a random Markov process in this system, we obtain an equation for the single-particle statistical operator. This equation describes the evolution of a system of identical particles in a mean-field approximation. The equation for interacting particles in thermodynamic equilibrium was obtained. The distribution function of a system of interacting electrons in metals has multivalence in a certain region of wave vectors. Among many solutions one is isotropic. Other solutions have the anisotropy of the electron distribution over the wave vectors. The anisotropy arises as a result of repulsion and attraction between electrons.

Abstract:
The distribution function for a system of interacting electrons in metals is multivalent in a certain region of wave vectors. One solution among many is isotropic. For other solutions the distribution of electrons over the wave vectors is anisotropic. In the simplest case, the anisotropy arises as a result of the repulsion between electrons in states with the wave vectors $\bf k$ and $-\hh\bf k$.

Abstract:
A new variational method has been proposed for studying the equilibrium
states of the interacting particle system to have been statistically described
by using the density matrix. This method is used for describing conductivity
electrons and their behavior in metals. The electron energy has been expressed
by means of the density matrix. The interaction energy of two ε_{kk}_{’} electrons dependent on their wave vectors k and k’ has been found. Energy ε_{k}_{ k}_{’} has two summands. The first energy I summand
depends on the wave vectors to be equal in magnitude and opposite in direction.
This summand describes the repulsion between electrons. Another energy I
summand describes the attraction between the electrons of equal wave vectors.
Thus, the equation of wavevector electron distribution function has been
obtained by using the variational method. Particular solutions of the equations
have been found. It has been demonstrated that the electron distribution
function exhibits some previously unknown features at low temperatures. Repulsion
of the wave vectors k and ﹣k electrons
results in anisotropy of the distribution function. This matter points to the
electron superconductivity. Those electrons to have equal wave vectors are attracted
thus producing pairs and creating an energy gap. It is considered the influence
of magnetic field on the superconductor. This explains the phenomenon of
Meissner and Ochsenfeld.

Abstract:
A new variational method has been proposed for studying the equilibrium
states of the interacting particle system to have been statistically described
by using the density matrix. This method is used for describing conductivity
electrons and their behavior in metals. The electron energy has been expressed
by means of the density matrix. The interaction energy of two ε_{kk}_{′}electrons dependent on their wave vectors k and k’ has been found.
Energy ε_{kk}_{′} has
two summands. The first energy I summand depends on the wave vectors to be equal in magnitude and opposite in
direction. This summand describes the repulsion between electrons. Another
energy J summand describes the
attraction between the electrons of equal wave vectors. Thus, the equation of
wave- vector electron distribution function has been obtained by using the variational
method. Particular solutions of the equations have been found. It has been
demonstrated that the electron distribution function exhibits some previously
unknown features at low temperatures. Repulsion of the wave vectors k and –k electrons results in anisotropy of the distribution function.
This matter points to the electron superconductivity. Those electrons to have
equal wave vectors are attracted thus producing pairs and creating an energy
gap. It is considered the influence of magnetic field on the superconductor.
This explains the phenomenon of Meissner and Ochsenfeld. We consider a new possibility of penetration of the external magnetic field into the superconductor.