Unified Microscopic Foundations of the Quantum Boltzmann Equation

We derive the quantum Boltzmann equation (QBE) of composite fermions at/near\nthe $\\nu = 1/2$ state using the non-equilibrium Green's function technique. The\nlowest order perturbative correction to the self-energy due to the strong gauge\nfield fluctuations suggests that there is no well defined\nLandau-quasi-particle. Therefore, we cannot assume the existence of the\nLandau-quasi-particles {\\it a priori} in the derivation of the QBE. Using an\nalternative formulation, we derive the QBE for the generalized Fermi surface\ndisplacement which corresponds to the local variation of the chemical potential\nin momentum space. {}From this QBE, one can understand in a unified fashion the\nFermi-liquid behaviors of the density-density and the current-current\ncorrelation functions at $\\nu = 1/2$ (in the long wave length and the low\nfrequency limits) and the singular behavior of the energy gap obtained from the\nfinite temperature activation behavior of the compressibility near $\\nu = 1/2$.\nImplications of these results to the recent experiments are also discussed.\n

We present a formula for thermal transport in the bulk of Bose systems based on the quantum Boltzmann equation (QBE). First, starting from the quantum kinetic equation and using the Born approximation for impurity scattering, we derive the QBE of Bose systems and provide a formula for thermal transport subjected to a temperature gradient. Next, we apply the formula to magnons. Assuming a relaxation time approximation and focusing on the linear response regime, we show that the longitudinal thermal conductivity of the QBE exhibits the different behavior from the conventional. The thermal conductivity of the QBE reduces to the Drude-type in the limit of the quasiparticle approximation, while not in the absence of the approximation. Finally, applying the quasiparticle approximation to the QBE, we find that the correction to the conventional Boltzmann equation is integrated as the self-energy into the spectral function of the QBE, and this enhances the thermal conductivity. Thus we shed light on the thermal transport property of the QBE beyond the conventional.

Collective orders and photoinduced phase transitions in quantum matter can evolve on timescales which are orders of magnitude slower than the femtosecond processes related to electronic motion in the solid. Quantum Boltzmann equations can potentially resolve this separation of timescales, but are often constructed by assuming the existence of quasiparticles. Here we derive a quantum Boltzmann equation which only assumes a separation of timescales (taken into account through the gradient approximation for convolutions in time), but is based on a nonperturbative scattering integral, and makes no assumption on the spectral function such as the quasiparticle approximation. In particular, a scattering integral corresponding to nonequilibrium dynamical mean-field theory is evaluated in terms of an Anderson impurity model in a nonequilibrium steady state with prescribed distribution functions. This opens the possibility to investigate dynamical processes in correlated solids with quantum impurity solvers designed for the study of nonequilibrium steady states.

We discuss the equivalence of two non-equilibrium kinetic theories that describe the evolution of a dilute, Bose-Einstein condensed atomic gas in a harmonic trap. The second-order kinetic equations of Walser et al. [PRA 63, 013607 (2001)] reduce to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the low and high temperature limits, respectively. These kinetic equations can thus describe the system in equilibrium (finite temperature) as well as in non-equilibrium (real time). We have found this theory to be equivalent to the non-equilibrium Green's function approach originally proposed by Kadanoff and Baym and more recently applied to inhomogeneous trapped systems by M. Imamovi\'c-Tomasovi\'c and A. Griffin [arXiv:cond-mat/9911402].

Solids facing a plasma are a common situation in many astrophysical systems and laboratory setups. Moreover, many plasma technology applications rely on the control of the plasma-surface interaction. However, presently often a fundamental understanding of them is missing, so most technological applications are being developed via trial and error. In the majority of plasma simulations surface processes are either neglected or treated via phenomenological parameters such as sticking coefficients, sputter rates or secondary electron emission coefficients. However, those parameters are known only in some cases and with very limited accuracy. Similarly, while surface physics simulations have often studied the impact of single ions or neutrals, so far, the influence of a plasma medium and correlations between successive impacts have not been taken into account. Such an approach cannot have predictive power. In this paper we discuss in some detail the physical processes a the plasma-solid interface which brings us to the necessity of coupled plasma-solid simulations. We briefly summarize relevant theoretical methods from solid state and surface physics that are suitable to contribute to such an approach and identify four methods. The first are mesoscopic simulations such as kinetic Monte Carlo (KMC) and molecular dynamics (MD) that are able to treat complex processes on large scales but neglect electronic effects. The second are quantum kinetic methods based on the quantum Boltzmann equation that give access to a more accurate treatment of surface processes using simplifying models for the solid. The third approach are ab initio simulations of surface process that are based on density functional theory (DFT) and time-dependent DFT. The fourths are nonequilibrium Green functions that able to treat correlation effects in the material and at the interface.

Zero sound and plasmon modes for non-Fermi liquids

Nonequilibrium Green functions (NEGF) have been applied to electron transport in semiconductors along several lines, as described, for example, in [116, 184, 228, 295]. In this chapter and the following one, we shall treat two important applications. In this chapter, we shall derive the so-called quantum Boltzmann equation (QBE), derived by Kadanoff and Baym [228] and extended by Mahan and Hansch [179, 180, 298]. This equation is valid for weak applied potentials slowly varying in space and time and it is therefore especially useful for linear transport in homogeneous, stationary systems. In the following, final, chapter we shall report the method developed by Datta [116] for electron transport in open mesoscopic systems, especially useful for the analysis of modern quantum electronic devices.

The quasiparticle approximation and corrections beyond it are derived from expansion in the spirit of the virial corrections. This way, the Boltzmann equation for electrons in semiconductors is recovered from the nonequilibrium Green's functions without unjustified neglect of the former theory. The present approach gives higher-order contributions to the scattering rates and provides observables as functionals of the quasiparticle distribution. The application scheme is of the same simplicity as the one based on the quasiparticle approximation. Five applications to phenomena beyond the scope of the quasiparticle approximation are presented: two-phonon scattering rates, compensation of wave-function remormalization for isotropic elastic scattering, quasiparticle backflow, backflow current of waves, and discussion of the collisional broadening.

Phonon-phonon interactions are systematically studied by nonequilibrium Green's function (NEGF) formalism in momentum space at finite temperatures. Within the quasiparticle approximation, phonon frequency shift and lifetime are obtained from the retarded self-energy. The lowest-order NEGF provides the same phonon lifetime as Fermi's golden rule. Thermal conductance is predicted by the Landauer formula with a phenomenological transmission function. The main advantage of our method is that it covers both ballistic and diffusive limits, and thermal conductance of different system sizes can be easily obtained once the mode-dependent phonon mean-free path is calculated by NEGF. As an illustration, the method is applied to two one-dimensional atom chain models [the Fermi-Pasta-Ulam (FPU)-$\ensuremath{\beta}$ model and the ${\ensuremath{\phi}}^{4}$ model] with an additional harmonic on-site potential. The obtained thermal conductance is compared with that from a quasiclassical molecular-dynamics method. The harmonic on-site potential is shown to remove the divergence of thermal conductivity in the FPU-$\ensuremath{\beta}$ model.

A simple and efficient approximation scheme to study electronic transport characteristics of strongly correlated nano devices, molecular junctions or heterostructures out of equilibrium is provided by steady-state cluster perturbation theory. In this work, we improve the starting point of this perturbative, nonequilibrium Green's function based method. Specifically, we employ an improved unperturbed (so-called reference) state $\hat{\rho}^S$, constructed as the steady-state of a quantum master equation within the Born-Markov approximation. This resulting hybrid method inherits beneficial aspects of both, the quantum master equation as well as the nonequilibrium Green's function technique. We benchmark the new scheme on two experimentally relevant systems in the single-electron transistor regime: An electron-electron interaction based quantum diode and a triple quantum dot ring junction, which both feature negative differential conductance. The results of the new method improve significantly with respect to the plain quantum maste equation treatment at modest additional computational cost.

Hot electrons in superlattices: quantum transport versus Boltzmann equation

The quasiclassical approximation for degenerate fermion systems that uses momentum-integrated nonequilibrium Green's functions is introduced into the theory of highly excited semiconductors. As an example, the complex density-dependent retarded self-energies and the spectral functions are evaluated analytically for a thermal electron-hole plasma. Furthermore, the renormalizations of the bands are calculated. The resulting band shrinkage extends in momentum space only between one and two Fermi wave numbers around the band extrema. The quasiclassical approach particularly emphasizes dynamical correlations that are neglected in the conventional quasistatic and quasiparticle approximations. It is a valuable method, if dynamical effects are important, as, for example, in line-shape theories.

By revising the application of the open quantum system approach to the early universe and extending it to the conditions beyond the Markovian approximation, we obtain a new non-Markovian quantum Boltzmann equation. Throughout the paper, we also develop an extension of the quantum Boltzmann equation to describe the processes that are irreversible at the macroscopic level. This new kinetic equation is, in principle, applicable to a wide variety of processes in the early universe. For instance, using this equation one can accurately study the microscopic influence of a cosmic environment on a system of cosmic background photons or stochastic gravitational waves. In this paper, we apply the non-Markovian quantum Boltzmann equation to study the damping of gravitational waves propagating in a medium consisting of decoupled ultra-relativistic neutrinos. For such a system, we study the time evolution of the intensity and the polarization of the gravitational waves. It is shown that, in contrast to intensity and linear polarization which are damped, the circular polarization (V-mode) of the gravitational wave (if present) is amplified by propagating through such a medium.

In this paper, we propose quantum mechanical operator formalism for Touchard polynomials whose generating function is [Formula: see text] That is replacing [Formula: see text] by [Formula: see text] where [Formula: see text] is the number operator, and using the method of integration within ordered product we find that [Formula: see text] is just the normal ordering form [Formula: see text]. Then by virtue of the Weyl ordering form of quantum mechanical operator, we also introduce a new special polynomial whose generating function is [Formula: see text] With use of the Weyl ordering form of [Formula: see text] we prove [Formula: see text] where [Formula: see text] denotes Weyl ordering. It seems that quantum mechancal operator formalism presents a new and simpler approach for generalizing Touchard polynomial theory.

We study the effects of kinetic helicity fluctuations in a turbulence with large-scale shear using two different approaches: the spectral tau approximation and the second-order correlation approximation (or first-order smoothing approximation). These two approaches demonstrate that homogeneous kinetic helicity fluctuations alone with zero mean value in a sheared homogeneous turbulence cannot cause a large-scale dynamo. A mean-field dynamo is possible when the kinetic helicity fluctuations are inhomogeneous, which causes a nonzero mean alpha effect in a sheared turbulence. On the other hand, the shear-current effect can generate a large-scale magnetic field even in a homogeneous nonhelical turbulence with large-scale shear. This effect was investigated previously for large hydrodynamic and magnetic Reynolds numbers. In this study we examine the threshold required for the shear-current dynamo versus Reynolds number. We demonstrate that there is no need for a developed inertial range in order to maintain the shear-current dynamo (e.g., the threshold in the Reynolds number is of the order of 1).