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Journal of Mathematical Physics

Kinetic Equations for Electrons and Phonons

Amiram Ron

Published Sep 1, 1963• 13 citations
Publisher

Abstract

A method is presented for deriving a set of kinetic equations for a system of electrons and phonons in a simplified model of a metal. By employing the second quantization representation for the creation and annihilation operators of the electrons and the phonons, an hierarchy of equations for the distribution functions and correlation functions is introduced. This hierarchy is studied, using an approach originally developed by Bogoliubov, where both truncation of the hierarchy and irreversibility are achieved under general assumptions. A set of kinetic equations is obtained for an homogeneous system, where the electrons dynamically shield both each other and the phonons.

DOI
10.1063/1.1704048

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A method is presented for deriving a set of kinetic equations for a system of electrons and phonons in a simplified model of a metal. By employing the second quantization representation for the creation and annihilation operators of the electrons and the phonons, an hierarchy of equations for the distribution functions and correlation functions is introduced. This hierarchy is studied, using an approach originally developed by Bogoliubov, where both truncation of the hierarchy and irreversibility are achieved under general assumptions. A set of kinetic equations is obtained for an homogeneous system, where the electrons dynamically shield both each other and the phonons.
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.
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.
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.
This chapter generalizes works by N.N. Bogolubov and N.N. Bogolubov, Jr. [1,6], and \nis written with the aim of formulating methods of investigation of the electron-phonon \nsystem and the elimination of phonon operators from the corresponding kinetic equations. In particular, a kinetic equation is derived for the interaction of an electron \nwith a phonon field. Moreover, under the proper approximation, the exact Boltzmann equation for the polaron system follows from this kinetic equation. Methods of \ncalculation of the response functions (impedance and admittance), based on the "approximating" Hamiltonian with linear interaction, are also proposed. The equilibrium \ndensity probability function of the particle is also calculated.
The article outlines key topics related to deriving kinetic equations and, in general, to the relationship between the evolution equations governing quantum many-particle systems. The main issues being considered are the mathematical origin of quantum kinetic equations, novel methods for deriving kinetic equations from hierarchies of evolution equations for quantum systems, specifically focusing on kinetics within the framework of the evolution of observables, and the generalization of the quantum kinetic equation specified by non-separable initial states.
Difference Kinetic Equations are derived quantum mechanically in a plane wavelets representation with account of two-particle correlations. It is shown that the set of plane wavelet orthonormal functions is complete. The set of ket vectors is used as the second quantization basis allowing introducing the positively definite distribution function. It is obtained that inhomogeneous system is described by numbers of particles at quantized positions with quantized momenta. Difference Kinetic Equation for distribution function transforms into the classical Boltzmann Equation in the limit, where expectation value of particles number varies little.