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Fluctuation and Noise Letters

FLUCTUATIONS IN NON-EQUILIBRIUM ELECTRON GAS: EFFECT OF QUANTUM STATISTICS

R. Katilius, S. V. Gantsevich, V. D. Kagan, M. I. Muradov

Published Jul 13, 2010• 4 citations
Publisher

Abstract

Here we develop a mathematical apparatus to describe quasi-classical fluctuations in a non-equilibrium electron gas with electron-electron collisions. We substantiate the method by deriving, from general principles of quantum kinetics, an equation recently proposed by us for an equal-time electron-electron correlation function. The derivation is performed using the kinetic diagram technique. In degenerate non-equilibrium gas, the theory predicts that there exists a specific equal-time correlation between electrons. Due to the prevalence of small-angle electron-electron scattering, the equation in question takes a rather simple and treatable form (the Coulomb-type electron-electron interaction stands out against the background of all other types of interaction as one that does not generate, in the framework of quasi-classical approach, any direct exchange effects).

DOI
10.1142/s0219477510000290

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Here we develop a mathematical apparatus to describe quasi-classical fluctuations in a non-equilibrium electron gas with electron-electron collisions. We substantiate the method by deriving, from general principles of quantum kinetics, an equation recently proposed by us for an equal-time electron-electron correlation function. The derivation is performed using the kinetic diagram technique. In degenerate non-equilibrium gas, the theory predicts that there exists a specific equal-time correlation between electrons. Due to the prevalence of small-angle electron-electron scattering, the equation in question takes a rather simple and treatable form (the Coulomb-type electron-electron interaction stands out against the background of all other types of interaction as one that does not generate, in the framework of quasi-classical approach, any direct exchange effects).
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.
Correlation effects of an electron gas in an external potential are derived using an Effective Action functional method. Corrections beyond the random phase approximation (RPA) are naturally incorporated by this method. The Effective Action functional is made to depend explicitly on two-point correlation functions. The calculation is carried out at imaginary time. For a homogeneous electron gas, we calculate the effect of exchange on the ring diagrams at zero temperature and show how to include some of the ladder diagrams. Our results agree well with known numerical calculations. We conclude by showing that this method is in fact a variant of the time dependent density functional method and suggest that it is suitable to be applied to the study of correlation effects in the non-homogeneous case.
Quantum statistical aspects of the anharmonic crystal in equilibrium, the approach to equilibrinm, and the zero temperature properties of a conduction electron in a crystal lattice are treated. A description of the Bravais lattice without imperfections is given. The reduction of the harmonic part of H to normal modes is demonstrated. The quantum mechanical problem of the harmonic crystal is investigatad, including the effect of the anharmonic terms. Calculations of the free energy of a crystal, the lowtemperature limit, and other macroscopic quantities and microscopic averages were made. The development of a Hamiltonian for studying the equilibrium of a system composed of electrons in an insulating crysta is discussed. A rough calculation of B'. the interaction constant, is presented. The physical effects of the interaction of one electron with the photon gas are described. Diagram calculations for the partition function are presented and partial summation of diagrams is discussed. A mathematical description of the dissipation process is given, including derivation of the master equation which refers to states of the entire system. The Boltzmann equation for governing the momentum distribution of the electron as a function of time is derived. The effects of virtual emission in the electron- photon interactionsmore » were studied. (C.J.G.)« less
We investigate the potential of a quantum Boltzmann equation without momentum conservation, i.e., a kinetic equation for the local distribution function, for describing correlated electron systems out of equilibrium. The equation can be derived either for the dynamics of well-defined quasi- particles, neglecting the momentum conservation of the scattering in the spirit of dynamical mean field theory (DMFT), or, equivalently, as the quantum Boltzmann equation for the local occupied density of states in non-equilibrium DMFT. The approach may thus be applied even in cases where well-defined quasiparticles do not exist, with the main assumption that the spectral function remains rigid under the non-equilibrium evolution. We compare the result of the Boltzmann equation to non-equilibrium DMFT for the case of photo-carrier relaxation in Mott insulators, where processes on very different timescales emerge, i.e., impact ionization, intra-Hubbard-band thermalization, and full thermalization. Since the Boltzmann without momentum conservation is much simpler, it may allow us to simulate more complicated systems, whole devices, and much longer times.
Abstract Correlated classical and quantum many‐particle systems out of equilibrium are of high interest in many fields, including dense plasmas, correlated solids, and ultracold atoms. Accurate theoretical description of these systems is challenging both, conceptionally and with respect to computational resources. While for classical systems, in principle, exact simulations are possible via molecular dynamics, this is not the case for quantum systems. Alternatively, one can use many‐particle approaches such as hydrodynamics, kinetic theory, or nonequilibrium Green functions (NEGF). However, NEGF exhibit a very unfavorable cubic scaling of the CPU time with the number of time steps. An alternative is the G1–G2 scheme [N. Schlünzen et al., Phys. Rev. Lett. 124, 076601 (2020)] which allows for NEGF simulations with time linear scaling, however, at the cost of large memory consumption. The reason is the need to store the two‐particle correlation function. This problem can be overcome for a number of approximations by reformulating the kinetic equations in terms of fluctuations – an approach that was developed, for classical systems, by Yu.L. Klimontovich [JETP 33, 982 (1957)]. Here, we present an overview of his ideas and extend them to quantum systems. In particular, we demonstrate that this quantum fluctuations approach can reproduce the nonequilibrium GW approximation [E. Schroedter et al., Cond. Matt. Phys. 25, 23401 (2022)] promising high accuracy at low computational cost which arises from an effective semiclassical stochastic sampling procedure. We also demonstrate how to extend the approach to the two‐time exchange‐correlation functions and the density response properties. [E. Schroedter et al., Phys. Rev. B 108, 205109 (2023)]. The results are equivalent to the Bethe–Salpeter equation for the two‐time exchange‐correlation function when the generalized Kadanoff‐Baym ansatz with Hartree‐Fock propagators is applied [E. Schroedter and M. Bonitz, phys. stat. sol. (b) 2024, 2300564].