Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1301.4712

Quantum master equation for a system of identical particles

Boris Volodumurovich Bondarev

Published Jan 21, 2013• 1 citation
arXivHTMLPDF

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.

DOI
10.48550/arxiv.1301.4712

Related papers

Kinetic equations for open quantum system in the occupation number representation

Elmar G. Petrov

Master equation for density matrix of an open many–particle system is derived in the occupation number representation. The Born approximation with respect to system–bath interaction is utilized and the fast relaxation within the system is assumed to be fulfiled. The reduction of a linear master equation to a nonlinear set of kinetic equations for one–particle distribution functions is carried out at the condition of strong particle–particle interaction. As an example, the procedure of derivation of kinetic equations for description of electron transfer through specific molecular nanostructures like molecular wires is demonstrated with taking into consideration the strong Coulomb repulsion between the transferred electrons.

•Jan 1, 2001•0 citations

Master Equation for State Occupancies of an Open Quantum System

Elmar G. Petrov

Based on the non-equilibrium density matrix method and using the Bogolyubov’s approach for the decoupling of many-particle distribution functions, a closed set of kinetic equations is derived for the state occupancies of closed and open quantum systems. It is shown that the transformation of a non-Markovian master equation into a Markovian one becomes possible at a high-frequency stochastic field alternating the energy levels of a quantum system. At a weak interaction between quantum subsystems composing the open quantum system, the set of linear balance-like kinetic equations for multistate occupancies of the system is transformed into a set of nonlinear kinetic equations for the occupancies of separate quantum subsystems.

•Jan 1, 2010•0 citations

Evolution of open many-electron systems : from a quantum statistical description towards the semi-classical Boltzmann equation

A. Gliesche

Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.

•Jan 1, 2007•0 citations

Irreversible evolution of many-electron systems coupled to a statistical environment

A. Gliesche, K. Maschke, F. Reuse

We study the evolution of a many-electron system that is confined in a finite spatial region and coupled to a statistical environment. The latter may be composed of several independent bath subsystems, which are held at some statistical equilibrium. From the master equations describing the evolution of the coarse-grained $N$-particle density matrix, we obtain the equations, which describe the evolution of the $n$-particle density operators ${\overline{D}}^{(n)}$ for $n<N$. These equations are hierarchically coupled through the electron-electron interaction. We show that the hierarchy can be truncated under the assumption that the residual interaction of the electrons in the considered system with the environment introduces a memory loss, which hinders the electronic system to build up more than two-particle correlations. We first consider a weakly excited electronic system, which looses memory but where energy exchange with the statistical environment can be neglected. This is the quantum analog of the classical Boltzmann gas in a box. We derive the master equations, which describe the irreversible evolution of the coarse-grained one-particle density matrix. Based on this result, we show that, in accord with the second law of thermodynamics, the corresponding von Neumann entropy either increases with time or it remains constant. Finally, allowing also for energy exchange with one or more bath subsystems provided, e.g., by phonons or photons, we obtain the corresponding general master equations that describe the evolution of a spatially confined interacting electron gas of metallic density.

Physical Review B•Jun 4, 2008•1 citation

On Operators Generated by Density Matrix

V. I. Gerasimenko

In this survey the possible approaches to the description of the evolution of states of quantum many-particle systems by means of the possible modifications of the density operator which kernel known as density matrix are considered. In addition, an approach to the description of the evolution of states by means of the state of a typical particle of a quantum system of many particles is discussed or in other words, the foundations of describing the evolution of states by kinetic equations are considered.

arXiv:2001.01180•Jan 5, 2020•0 citations

Toward Dissipative Quantum Mechanics

Roumen Tsekov

A new approach to dissipative quantum mechanics is proposed. The irreversible processes occurring in an isothermic system are treated on the basis of statistical and linear non-equilibrium thermodynamics. The equilibrium probability density provided by the present theory coincides with the quantum canonical Gibbs distribution. In the classical limit the obtained kinetic equation reduces to a Markov theory result.

•Feb 1, 2001•0 citations
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.
Master equation for density matrix of an open many–particle system is derived in the occupation number representation. The Born approximation with respect to system–bath interaction is utilized and the fast relaxation within the system is assumed to be fulfiled. The reduction of a linear master equation to a nonlinear set of kinetic equations for one–particle distribution functions is carried out at the condition of strong particle–particle interaction. As an example, the procedure of derivation of kinetic equations for description of electron transfer through specific molecular nanostructures like molecular wires is demonstrated with taking into consideration the strong Coulomb repulsion between the transferred electrons.
Based on the non-equilibrium density matrix method and using the Bogolyubov’s approach for the decoupling of many-particle distribution functions, a closed set of kinetic equations is derived for the state occupancies of closed and open quantum systems. It is shown that the transformation of a non-Markovian master equation into a Markovian one becomes possible at a high-frequency stochastic field alternating the energy levels of a quantum system. At a weak interaction between quantum subsystems composing the open quantum system, the set of linear balance-like kinetic equations for multistate occupancies of the system is transformed into a set of nonlinear kinetic equations for the occupancies of separate quantum subsystems.
Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.
We study the evolution of a many-electron system that is confined in a finite spatial region and coupled to a statistical environment. The latter may be composed of several independent bath subsystems, which are held at some statistical equilibrium. From the master equations describing the evolution of the coarse-grained NNN-particle density matrix, we obtain the equations, which describe the evolution of the nnn-particle density operators D‾(n){\overline{D}}^{(n)}D(n) for n<Nn<Nn<N. These equations are hierarchically coupled through the electron-electron interaction. We show that the hierarchy can be truncated under the assumption that the residual interaction of the electrons in the considered system with the environment introduces a memory loss, which hinders the electronic system to build up more than two-particle correlations. We first consider a weakly excited electronic system, which looses memory but where energy exchange with the statistical environment can be neglected. This is the quantum analog of the classical Boltzmann gas in a box. We derive the master equations, which describe the irreversible evolution of the coarse-grained one-particle density matrix. Based on this result, we show that, in accord with the second law of thermodynamics, the corresponding von Neumann entropy either increases with time or it remains constant. Finally, allowing also for energy exchange with one or more bath subsystems provided, e.g., by phonons or photons, we obtain the corresponding general master equations that describe the evolution of a spatially confined interacting electron gas of metallic density.
In this survey the possible approaches to the description of the evolution of states of quantum many-particle systems by means of the possible modifications of the density operator which kernel known as density matrix are considered. In addition, an approach to the description of the evolution of states by means of the state of a typical particle of a quantum system of many particles is discussed or in other words, the foundations of describing the evolution of states by kinetic equations are considered.