Auswahl der wissenschaftlichen Literatur zum Thema „Boltzmann-Fermi-Dirac equation“

We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi–Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 × 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.

In this paper we obtain the existence of a radial solution for some elliptic non-local problem with constraints. The problem arises from some mean field equation which models, among other things, a system of self-gravitating particles when one looks for its stationary solutions. We include the cases of Maxwell—Boltzmann, Fermi—Dirac and polytropic statistics.

In this paper we consider a large system of bosons or fermions. We start with an initial datum which is compatible with the Bose–Einstein, respectively Fermi–Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time t, agrees with the analogous expansion for the solution to the Uehling–Uhlenbeck equation. This paper follows the same spirit as the companion work,2 where the authors investigated the weak-coupling limit for particles obeying the Maxwell–Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

Abstract We derive, from first principles and using the Weyl–Wigner formalism, a fully quantum kinetic model describing the dynamics in phase space of Dirac electrons in single-layer graphene. In the limit ℏ → 0, we recover the well-known semiclassical Boltzmann equation, widely used in graphene plasmonics. The polarizability function is calculated and, as a benchmark, we retrieve the result based on the random-phase approximation. By keeping all orders in ℏ, we use the newly derived kinetic equation to construct a fluid model for macroscopic variables written in the pseudospin space. As we show, the novel ℏ-dependent terms can be written as corrections to the average current and pressure tensor. Upon linearization of the fluid equations, we obtain a quantum correction to the plasmon dispersion relation, of order ℏ 2, akin to the Bohm term of quantum plasmas. In addition, the average variables provide a way to examine the value of the effective hydrodynamic mass of the carriers. For the latter, we find a relation in which Drude’s mass is multiplied by the square of a velocity-dependent, Lorentz-like factor, with the speed of light replaced by the Fermi velocity, a feature stemming from the quasi-relativistic nature of the Dirac fermions.

The unsteady shock wave diffraction by a square cylinder in gases of arbitrary particle statistics is simulated using an accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space. The numerical method is based on the discrete ordinate method for discretizing the velocity space of the distribution function and high-resolution method is used for evolving the solution in physical space and time. The specular reflection surface boundary condition is employed. The complete diffraction patterns including regular reflection, triple Mach reflection, slip lines, vortices and their complex nonlinear manifestations are recorded using various flow property contours. Different ranges of relaxation times corresponding to different flow regimes are considered, and the equilibrium Euler limit solution is also computed for comparison. The effects of gas particles that obey the Maxwell–Boltzmann, Bose–Einstein and Fermi—Dirac statistics are examined and depicted.

Nous étudions certaines variantes de l'équation de Boltzmann, cette dernière décrivant via une approche classique les gaz raréfiés simples et monoatomiques à l'échelle mésoscopique. Dans un premier temps, nous proposons un cadre général de modélisation de Boltzmann des gaz polyatomiques, englobant une large classe de modèles pré-existants et permettant d'en construire de nouveaux. D'abord présenté pour un gaz simple, ce cadre est ensuite étendu aux mélanges gazeux, pour lesquels on autorise des réactions chimiques binaires. Dans un deuxième temps, nous nous intéressons à un type de gaz polyatomique singulier, au sein duquel les collisions sont résonantes. Nous prouvons une propriété de compacité pour l'opérateur linéarisé lié à ce modèle. Afin de rendre plus flexible le cadre résonant, nous proposons ensuite un formalisme de Boltzmann pour des collisions quasi-résonantes, étudions ses propriétés-clés et menons des expériences numériques pour étayer notre compréhension de celles-ci. Enfin, dans un troisième temps, nous nous tournons vers une équation de Boltzmann incluant le principe d'exclusion de Pauli, utile notamment pour la description de la distribution d'électrons dans les semi-conducteurs. Nous développons une méthode permettant de transférer certaines inégalités fonctionnelles, liées à l'entropie, connues dans le cas classique, vers ce cas quantique. Par suite, grâce à l'obtention de ces nouvelles inégalités, nous obtenons un taux explicite de relaxation à l'équilibre pour les solutions de l'équation de Boltzmann-Fermi-Dirac homogène pour les potentiels durs avec cut-off
We study some variants of the Boltzmann equation, the latter describing, via a classical approach, single and monatomic rarefied gases at the mesoscopic scale. First, we propose a general framework for Boltzmann modelling of polyatomic gases, encompassing a wide class of pre-existing models and allowing to build new ones. Primarily presented for a single gas, the framework is then extended to mixtures, within which we allow binary chemical reactions. Second, we focus on a singular type of polyatomic gas, the molecules of which undergo resonant collisions, and prove a compactness property of the linearized operator related to this model. In order to make the latter resonant framework more flexible, we then propose a Boltzmann formalism with quasi-resonant collisions, study its key properties and conduct numerical experiences to support our understanding of them. Third, we turn our attention towards a Boltzmann equation which includes Pauli's exclusion principle, notably used in the study of electron distributions in semi-conductors. We develop a method that allows to transfer some functional inequalities, related to entropy, which are known in the classical case, to this quantum case. As a consequence, we use these new inequalities to obtain an explicit rate of relaxation to equilibrium for solutions to the homogeneous Boltzmann-Fermi-Dirac equation with cut-off hard potentials

Abstract We discussed in the previous chapter when we can ignore the coherence effects and treat heat carriers as individual particles without considering their phase information. In the next few chapters, we will describe how to deal with energy transfer under the particle picture. Most constitutive equations for macroscale transport processes, such as the Fourier law and the Newton shear stress laws, are obtained under such particle pictures. These equations are often formulated as laws summarized from experiments. In this chapter, we will see that most of the classical laws governing transport processes can be derived from a few fundamental principles. In chapter 4, we studied systems at equilibrium and developed the equilibrium distribution functions (Fermi-Dirac, Bose-Einstein, and Boltzmann distributions). The distribution function for a quantum state at equilibrium is a function of the energy of the quantum state, the system temperature, and the chemical potential. When the system is not at equilibrium, these distribution functions are no longer applicable. Ideally, we would like to trace the trajectory of all the particles in the system, as in the molecular dynamics approach that we will discuss in chapter 10. This approach, however, is not realistic for most systems, because they have a large number of atoms or molecules. Thus, we resort to a statistical description of the particle trajectory. In the statistical description we use nonequilibrium distribution functions, which depend not only on the energy and temperature of the system but also on positions and other variables. We will develop in this chapter the governing equations for the nonequilibrium distribution functions. In particular, we will rely on the Boltzmann equation, also called the Boltzmann transport equation. From the Boltzmann equation we will derive familiar constitutive equations such as the Fourier law, the Newton shear stress law, and the Ohm law.

Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed