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Proceedings of the Physical Society

The boltzmann equations in electron-phonon systems

H. Fröhlich, A. W. B. Taylor

Published May 1, 1964• 15 citations
Publisher

Abstract

The present paper derives conditions on which the Boltzmann equations for electron-phonon systems (two coupled equations) can be derived without the use of the objectionable random phase approximation. The treatment uses von Neumann's density matrix based on the Hamiltonian introduced some time ago by one of us in connection with superconductivity. Formally exact expressions for the rate of change of coarse-grained one-particle distribution functions are derived. They become identical with the collision terms in the Boltzmann equations if (i) the influence of an external field on them is negligible, (ii a ) particle correlation is absent at an initial time, and (ii b ) the relevant series in terms of the interaction parameter converges and is at all times dominated by a certain term. (ii b ) excludes the possibility of extensive correlations as they occur in superconductivity.

DOI
10.1088/0370-1328/83/5/305

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The present paper derives conditions on which the Boltzmann equations for electron-phonon systems (two coupled equations) can be derived without the use of the objectionable random phase approximation. The treatment uses von Neumann's density matrix based on the Hamiltonian introduced some time ago by one of us in connection with superconductivity. Formally exact expressions for the rate of change of coarse-grained one-particle distribution functions are derived. They become identical with the collision terms in the Boltzmann equations if (i) the influence of an external field on them is negligible, (ii a ) particle correlation is absent at an initial time, and (ii b ) the relevant series in terms of the interaction parameter converges and is at all times dominated by a certain term. (ii b ) excludes the possibility of extensive correlations as they occur in superconductivity.
Starting from the quantum-Boltzmann equation derived in a previous paper, we study the irreversible evolution of an electron gas in the one-particle phase space. The connection with phase space is established by expressing one-electron states in terms of the overcomplete and nonorthogonal generating system of coherent states. By using the generalized closure relation for coherent states, as well as the fact that a one-particle operator is completely determined by the ensemble of expectation values for all coherent states, we obtain the master equations in a form that allows us to follow the evolution in phase space. This form of the master equations provides a direct link between the quantum-statistical approach and the semi-classical Boltzmann equation. The latter is obtained after a coarse-graining procedure in the one-particle phase space and by using the fact that the electron-electron interaction, as well as the interactions between the electron gas and the bath subsystems provided by phonons or photons, are local in real space.
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.
A derivation of the quantum-mechanical analog of the Maxwell-Boltzmann equation of transport for gases at low density is presented. The analysis requires: the definition of a time differential which is necessary for the selection of only secular variations in the time dependence of the distribution functions (coarse-graining in time); a binary collision approximation; the connection between the phase-space transformation function, relating the Wigner distribution function at different times, and the transition matrix in the theory of scattering of two particles; and a coarse graining in configuration space leading to a kinetic equation for spatially nonuniform systems. A consequence of the binary collision approximation is the exclusion of the effects of symmetrization (Bose-Einstein or Fermi-Dirac statistics) resulting in density-dependent terms.
An alternative approach to the quantum statistics of interacting particles is proposed. It consists of calculating the equilibrium thermodynamical quantities of the many-body system via the pair distribution function with the assumption that the particles interact with each other only through pair central forces. The proposed approach has some advantage over the usual treatment via the partition function in that the pair distribution function is easier to deal with than the partition function in certain circumstances particularly when the collective motion description of the system is desirable. This is because only the pair distribution function can be expressed directly in terms of the collective interaction which is closely connected with collective elementary excitation, such as a plasmon in the electron gas and a phonon in the hard-sphere Boson gas. The equation of state as well as the internal energy are obtained in the form of integrals of the pair distribution function. The close analogy between the pair distribution function and the two-body propagator, which appears in the quantum field theory, makes it possible to analyze the former by the use of Feynman diagrams identical with those usually introduced for the latter. The collective interaction, which is defined by the sum of the direct and the indirect interactions, is introduced as a particular partial sum of the perturbation series of the pair distribution function. This is used in rewriting the pair distribution function in terms of the collective interaction.It is shown that, while the simple chain approximation to the collective interaction in the electron gas is responsible for the transfer of a plasmon, the same approximation to the collective pseudo-interaction in the hard-sphere Boson system has a relation similar to that of transfer of a phonon, both cases occuring at low temperatures. The explicit calculation of the pair distribution functions for these systems at the absolute zero temperature is carried out up to the first order in the collective interaction (simple chain approximation.) These results are used to calculate the ground-state energies. For the electron gas the energy thus obtained confirms Gell-Mann-Bruckner's calculation of the correlation energy. For the hard-sphere Bosons the calculated energy reproduces the result of Lee, Huang, and Yang. The extension of the calculation to the finite temperature case is also indicated. In particular the classical Debye-Huckel equation of state for the electron gas is briefly discussed.
The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.
By means of the time-smoothing method developed in the previous papers of this series, the quantum kinetic equation is derived from the principles of statistical mechanics. In this paper, we avoid the BBGKY hierarchy; instead we use the Liouville-von Neumann equation for the time-smoothed density matrix. The Boltzmann distribution function is defined in terms of this time-smoothed density matrix. The present scheme may be applied to strong-coupling systems even if they are not in near equilibrium.