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Journal of Mathematical Physics

Quantum Kinetic Equations for Electrons in a Periodic System

Amiram Ron

Published Jun 1, 1964• 1 citation
Publisher

Abstract

The approach to equilibrium of a system of electrons in a periodic system is studied by deriving a kinetic equation in the self-consistent field approximation. The derivation based on the introduction of an hierarchy of equations for the s-body density matrices and employing a truncation scheme, valid in the self-consistent field approximation. The irreversibility is introduced by the adiabatic hypothesis of Bogoliubov. The kinetic equation takes proper account of the collective aspects of the electron system and includes the dynamic shielding.

DOI
10.1063/1.1704174

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The approach to equilibrium of a system of electrons in a periodic system is studied by deriving a kinetic equation in the self-consistent field approximation. The derivation based on the introduction of an hierarchy of equations for the s-body density matrices and employing a truncation scheme, valid in the self-consistent field approximation. The irreversibility is introduced by the adiabatic hypothesis of Bogoliubov. The kinetic equation takes proper account of the collective aspects of the electron system and includes the dynamic shielding.
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Some aspects of the correlations between interacting electrons in a magnetic field are studied by a quantum kinetic approach, taking into account the quantization of the orbital motion of the electrons. The treatment is based on the introduction of an hierarchy of equations of motion for the sss-body density matrices and employing a decoupling scheme, valid in the self-consistent field approximation. In this way the dielectric function is introduced "naturally," and the response of the system to an applied field is calculated. The dielectric function reflects the two main effects of the correlations, i.e., the collective excitations of the system and the dynamical shielding. The dielectric function plays a role also in the "dressing" of the interacting electrons. This is shown by calculating the spectrum of the density fluctuations and the equation which governs the approach to equilibrium of the system. The method is not limited to systems in thermal equilibrium, but also applies to systems in other stationary equilibrium states.
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Abstract Exact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.
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