Boltzmann Kinetics Research Articles

Energy-conserving DPD and thermodynamically consistent Fokker–Planck​ equation

The original Fokker–Planck equation does not satisfy most balance equations of continuum mechanics. Dissipative particle dynamics (DPD) is an attempt to remedy the demerits of the Fokker–Planck equation. However, a desirable kinetic equation should satisfy the four balance equations of irreversible thermodynamics, i.e., the balance equations of mass, momentum, energy, and entropy. To the best of the author’s knowledge, only the Boltzmann kinetics fulfills all the conditions. We suggest a modified Fokker–Planck equation that satisfies all the conditions without the assumption of DPD, the pairwise additivity of dissipative and random forces. Our formalism is expected to minimize the number of adjustable functions that must be used in DPD.

Symmetry-related transport on a fractional quantum Hall edge

Low-energy transport in quantum Hall states is carried through edge modes, and is dictated by bulk topological invariants and possibly microscopic Boltzmann kinetics at the edge. Here we show how the presence or breaking of symmetries of the edge Hamiltonian underlie transport properties, specifically d.c. conductance and noise. We demonstrate this through the analysis of hole-conjugate states of the quantum Hall effect, specifically the $\nu=2/3$ case in a quantum point-contact (QPC) geometry. We identify two symmetries, a continuous $SU(3)$ and a discrete $Z_3$, whose presence or absence (different symmetry scenarios) dictate qualitatively different types of behavior of conductance and shot noise. While recent measurements are consistent with one of these symmetry scenarios, others can be realized in future experiments.

Time-dependent calculations of electron energy distribution functions for neon gas in the presence of intense XFEL radiation

Boltzmann electron kinetic simulations are performed to study the time development of the electron energy distribution function (EEDF) in plasma resulting from neon gas subject to a 40 fs x-ray free electron laser radiation source. The simulations are performed without any assumptions of electron temperature. The distributions are calculated as a function of time through 40 fs using Boltzmann kinetics, including the appropriate processes that alter state populations and electron energy. The calculations are also extended into the picosecond regime, after the laser pulse, to track the thermalization of free electrons. Results are presented that predict the evolution of the charge-state distribution, effective temperature, the photon spectrum, as well as the EEDF.

Time-dependent calculations of electron energy distribution functions for cold argon gas in the presence of intense black-body radiation

Boltzmann electron kinetic simulations are performed to study the time development of the electron energy distribution in a plasma resulting from a cold argon gas subject to a black-body radiation source (100–300 eV). The study provides insight into the role of ionized electrons on the kinetics during a period of irradiation. The simulations are performed without any assumptions of electron temperature. The distributions are calculated as a function of time through 1 ps using Boltzmann kinetics, including the appropriate processes that alter state populations and electron energy. The processes included in the electron and atomic kinetics are collisional excitation/de-excitation, photo-excitation/decay, photo-ionization/radiative-recombination, collisional ionization and auto-ionization/di-electronic capture. In addition, terms are included in the electron kinetics to account for electron–electron interactions and for free–free radiation absorption and emission. Results are presented that follow the evolution of the ionization state, effective temperature as well as the electron energy distribution function. The role of inelastic electron collisions, photo-ionization, auto-ionization, resonant radiative excitation and electron–electron interactions is discussed.

Warming reduces metabolic rate in marine snails: adaptation to fluctuating high temperatures challenges the metabolic theory of ecology

The universal temperature-dependence model (UTD) of the metabolic theory of ecology (MTE) proposes that temperature controls mass-scaled, whole-animal resting metabolic rate according to the first principles of physics (Boltzmann kinetics). Controversy surrounds the model's implication of a mechanistic basis for metabolism that excludes the effects of adaptive regulation, and it is unclear how this would apply to organisms that live in fringe environments and typically show considerable metabolic adaptation. We explored thermal scaling of metabolism in a rocky-shore eulittoral-fringe snail (Echinolittorina malaccana) that experiences constrained energy gain and fluctuating high temperatures (between 25°C and approximately 50°C) during prolonged emersion (weeks). In contrast to the prediction of the UTD model, metabolic rate was often negatively related to temperature over a benign range (30-40°C), the relationship depending on (i) the temperature range, (ii) the degree of metabolic depression (related to the quiescent period), and (iii) whether snails were isolated within their shells. Apparent activation energies (E) varied between 0.05 and -0.43 eV, deviating excessively from the UTD's predicted range of between 0.6 and 0.7 eV. The lowering of metabolism when heated should improve energy conservation in a high-temperature environment and challenges both the theory's generality and its mechanistic basis.

Matching of nonthermal initial conditions and hydrodynamic stage in ultrarelativistic heavy-ion collisions

A simple approach is proposed allowing actual calculations of the preequilibrium dynamics in ultrarelativistic heavy-ion collisions to be performed for a far-from-equilibrium initial state. The method is based on the phenomenological macroscopic equations that describe the relaxation dynamics of the energy-momentum tensor and are motivated by Boltzmann kinetics in the relaxation-time approximation. It gives the possibility to match smoothly a nonthermal initial state to the hydrodynamics of the quark gluon plasma. The model contains two parameters, the duration of the prehydrodynamic stage and the initial value of the relaxation-time parameter, and allows one to assess the energy-momentum tensor at a supposed time of initialization of the hydrodynamics.

Comparison of Boltzmann kinetics with quantum dynamics for a chiral Yukawa model far from equilibrium

Boltzmann equations are often used to describe the nonequilibrium time-evolution of many-body systems in particle physics. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process, which is described by so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the complete Kadanoff-Baym equations. Therefore, we present in this article a detailed comparison of the Boltzmann and the Kadanoff-Baym equations in the framework of a chirally invariant Yukawa-type quantum field theory including fermions and scalars. The obtained numerical results reveal significant differences between both types of equations. Apart from quantitative differences, on a qualitative level the late-time universality respected by the Kadanoff-Baym equations is severely restricted in the case of the Boltzmann equations. Furthermore, the Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. In contrast to this standard Boltzmann equations cannot describe the process of quantum-chemical equilibration, and consequently also cannot feature the above separation of time scales.

Relaxation kinetics for temperature and degeneracy of microcavity polaritons

We apply a semi-classical Boltzmann kinetics for a gas of laser-pulse excited microcavity polaritons taking into account their mutual interaction and their interaction with acoustic phonons. Fitting the temporally evolving polariton distribution above the ground state with a Bose–Einstein distribution, we find the evolution of the temperature and the degeneracy parameter, i.e. the ratio of the chemical potential to the thermal energy. Studying the relaxation in particular for GaAs microcavities we compare our results with recent measurements by Deng et al. In agreement with the experiment we find that the lattice temperature can be reached and that the degeneracy of the condensed gas holds up to 60–80 ps provided a detuning of the cavity mode is applied which increases the exciton component of the lower-branch polaritons and thus their scattering rates.

How can one solve exactly some problems in plasma theory

A review of a technique for the application of a new Lambert function are outlined. The problems discussed include equilibrium electric charge of a solitary body in plasma, nonlinear ion-acoustic waves, the equilibrium configuration of a neutralized electric beam, the dynamics of shaping virtual cathodes in a medium with viscous friction, and the study of the dispersion equation for electron plasma waves in the framework of generalized Boltzmann kinetics.

Theory of granular gases: some recent results and some open problems

One of the goals of the present paper is to present a brief, and admittedly somewhatbiased, review of some recent theoretical advances in the field of granular gases. Anothergoal is to highlight some challenges facing this field. A third goal is to present some newresults concerning the Chapman–Enskog expansion. These include an extension to weaklyfrictional granular gases, and a study of a stationary granular gas subject to a heat flux (anearlier study describing a stationary sheared granular gas is presented as well). In addition,a computer-aided method for inverting the linearized Boltzmann operator, which shouldbe useful for kinetic theorists, is presented in an appendix. A further goal is tobriefly discuss possible extensions beyond Boltzmann kinetics, as well as beyondhydrodynamics, i.e., the moderately dense regime on one hand, and the Knudsenregime on the other hand. An attempt is made throughout this paper to putgranular gases in a general context, by distinguishing between those featuresthat are common to granular and molecular gases and those that are not. Also,stress is put on the physics (and methods) rather than phenomena. While thisis not a review article, parts of it are intended for the uninitiated in the field.

Lattice Boltzmann open boundaries for hydrodynamic models

The mixed initial–boundary value problem for the equations of nonhydrostatic fluid dynamics is mathematically well posed, but is physically and computationally ill posed. The ill-posedness is exacerbated by boundary data taken from integrations of hydrostatic flow equations. The difficulty is found both in fully compressible flow and in Boussinesq flow; the latter is emphasized here. The mixed problem for the Boltzmann equation is well posed in every respect. Inserting lattice Boltzmann kinetics between hydrostatic boundary data and nonhydrostatic dynamics resolves the physical and computational ill-posedness of the dynamics, by relaxing the boundary data to nonhydrostatic dynamical consistency. Simple numerical experiments with boundary data taken from exact and approximate analytical solutions demonstrate the effectiveness of the approach.

Carrier dynamics in the quantum kinetics regime

Theory has predicted that, on a short time scale, the dynamics of optically excited carriers in semiconductors can no longer be described by the semiclassical Boltzmann equation. The carriers rather behave like quantum mechanical waves and the Boltzmann kinetics have to be replaced by quantum kinetics. Here we briefly review recent experiments on the quantum kinetics of electron–LO–phonon scattering. The focus of this review, however, is on more recent experiments at elevated carrier densities which are within the regime of the quantum kinetics of carrier–carrier scattering.

Quantum kinetics and thermalization in an exactly solvable model

We study the dynamics of relaxation and thermalization in an exactly solvable model with the goal of understanding the effects of off-shell processes. The focus is to compare the exact evolution of the distribution function with different approximations to the relaxational dynamics: Boltzmann, non-Markovian and Markovian quantum kinetics. The time evolution of the distribution function is evaluated exactly using two methods: time evolution of an initially prepared density matrix and by solving the Heisenberg equations of motion. There are two different cases that are studied in detail: i) no stable particle states below threshold of the bath and a quasiparticle resonance above it and ii) a stable discrete exact `particle' state below threshold. For the case of quasiparticles in the continuum (resonances) the exact quasiparticle distribution asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes. In the case ii), the distribution of particles does not thermalize with the bath. We study the kinetics of thermalization and relaxation by deriving a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian and Boltzmann approximations are compared to the exact result of the model. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.

Quantum Coherence in Ultrafast Semiconductor Spectroscopy

Coherent optical phenomena such as the optical Stark effect, Rabi flopping, photon echo and quantum beating which are well-known in atomic spectroscopy can also be observed in semiconductors by using femtosecond laser pulses. On these short time scales, the quantum coherence of the optical excitations in the solid do not only influence the optical properties but change at the same time the relaxation and dephasing kinetics. The quasi-classical Boltzmann kinetics has to be replaced by quantum kinetics. Coherence leads to the appearance of memory in the scattering integrals. For femtosecond four-wave mixing and pump-and-probe spectroscopy the use of quantum kinetics for LO-phonon and for carrier-carrier scattering will be reviewed.

Quantum kinetics regime during and immediately after laser excitation of semiconductors

We investigated carrier dynamics in the semiconductor GaAs, during and just after creation by ≈ 30 fs laser pulses, much shorter than the phonon and plasma time scales. We observed a quasi-instantaneous spread of the carrier population in momentum-energy space, that is not even qualitatively consistent with Boltzmann kinetics. The observations are, however, in qualitative agreement with quantum kinetic theories of carrier-carrier and carrier-phonon scattering.

Calculation of the gain saturation in cw semiconductor lasers with Boltzmann kinetics for Coulomb and LO phonon scattering

The characteristics of the photon and carrier densities versus pump current are calculated for a bulk GaAs semiconductor laser including Coulomb and LO phonon scattering in terms of the Boltzmann collision integral. From these characteristics the nonlinear gain compression epsilon is determined as a function of the pump rate. Various Coulomb screening models and their influence on epsilon are discussed. Well above threshold a constant value of epsilon approximately=10-17 cm-3+or-20% is found. The validity of the relaxation time approximations is investigated. Only for high pump powers do the resulting epsilon values of the full kinetics and of relaxation time approximations agree well. Furthermore, the spectral hole burning and the carrier heating in the steady state distributions are calculated.

Thermodynamic parameterization

A new method of succesive construction of a solution is developed for problems of strongly nonequilibrium Boltzmann kinetics beyond normal solutions. Firstly, the method provides dynamic equations for any manifold of distributions where one looks for an approximate solution. Secondly, it gives a successive procedure of obtaining corrections to these approximations. The method requires meither small parameters, nor strong restrictions upon the initial approximation; it involves solutions of linear problems. It is concordant with the H-theorem at every step. In particular, for the Tamm-Mott-Smith approximation, dynamic equations are obtained, an expansion for the strong shock is introduced, and a linear equation for the first correction is found.

Study of the Coulomb Boltzmann kinetics in a quasi-twodimensional electron gas by eigenfunction expansions and Monte Carlo simulations

For carrier-carrier scattering the relaxation kinetics of a quasi-twodimensional degenerate electron gas is studied in the framework of the Boltzmann equation using a static RPA screening. The time development of an initial nonequilibrium distribution is calculated both by ensemble Monte Carlo simulations and by expansions in terms of eigenfunctions of the linearized collision operator. A systematic comparison of the results of both methods yields a rather detailed understanding of the Coulomb relaxation kinetics and of the strengths and limitations of both methods. The dependence of the relaxation rate on the position and width of the initial population deviation is investigated and conditions for nonexponential decay are given.

Interband Quantum Kinetics with LO‐Phonon Scattering in a Laser‐Pulse‐Excited Semiconductor I. Theory

AbstractNonequilibrium Green's function theory is employed to derive with the use the generalized Kadanoff‐Baym ansatz quantum kinetic equations for the interband polarization and the densities of electrons and holes. These equations are valid in the regime of ultrafast spectroscopy, where that usual Boltzmann kinetics with its individual energy‐conserving collisions no longer holds. The memory kernel of the retarded quantum kinetic equations is determined by retarded and advanced Green's functions. It is stressed and explicitly shown by numerical solutions in an adjacent paper that these Green's functions have to be approximated consistently with the coupling to phonons.

Measurement of HgXe (30+) fluorescence efficiency in an electron-beam-preionized, volume-stable discharge and comparison with Boltzmann kinetics code calculations

A HgXe(30+) fluorescence efficiency of 12±2% was measured in an electron-beam-preionized, volume-stable discharge with an E/N value of 2×10−17 V cm2. This efficiency was obtained for gas mixtures with mercury pressures in the range of 2–10 Torr and a constant xenon pressure of 3000 Torr. This efficiency was compared with Boltzmann-kinetics code calculations. It was found that processes involving molecular mercury formation had to be included in the kinetics code in order to obtain agreement between the measured and calculated HgXe (30+) fluorescence efficiency.