Cette thèse avait comme but le développement, l'analyse et l'application de schémas numériques pour la simulation de modèles cinétiques basés sur l'équation de Vlasov, notamment de schémas basés sur le splitting de Strang et une méthode d'interpolation essentiellement non oscillatoire (WENO). Les schémas sont testés sur des cas test de plus en plus compliqués, et finalement sur un modèle Boltzmann-Schrödinger-Poisson qui décrit les états transitoires d'un transistor à l'echelle nanométrique.

In this thesis, two methods to model quantum well lasers will be examined. The first model is based on well-known techniques to determine some of the spectral and dynamical properties of the laser. For the spectral properties, an expression for TE and TM modal amplitude gain is derived. For the dynamical properties, the rate equations are shown. The spectral and dynamical properties can be examined separately for specific operating characteristics or used in conjunction with each other for a complete description of the laser. Examples will be shown to demonstrate some of the analysis and results that can be obtained. The second model used is based on Wigner functions and the quantum Boltzmann equation. It is derived from general non-equilibrium Greens functions with the application of the Kadanoff-Baym ansatz. This model is less phenomenological than the previous model and does not require the separation of physical processes such as the former spectral and dynamical properties. It therefore has improved predictive power for the performance of novel laser designs. To the Author's knowledge, this is the first time such a model has been formulated. The quantum Boltzmann equations will be derived and some calculations will be performed for a simplified system in order to illustrate some calculation techniques as well as results that can be obtained.

An experimental investigation of the electronic transport properties of bismuth telluride nanocomposite materials is presented. The primary transport measurements are electrical conductivity, Seebeck coefficient and Hall effect. An experimental apparatus for measuring Hall effect and electrical conductivity was designed, constructed and tested. Seebeck coefficient measurements were performed on a commercial instrument. The Hall effect and Seebeck coefficient measurements are two of the most important tools for characterizing thermoelectric materials and are widely used in the semiconductor industry for determining carrier types, carrier concentration and mobility. Further, these transport parameters are used to determine the thermal to electrical conversion efficiency of a thermoelectric material. The Boltzmann transport equation was used to analyze the Seebeck coefficient, carrier mobility and electrical conductivity as a function of carrier concentration for eleven samples. The relationship between the electronic transport and material/composite composition is discussed.

Após a excitação de uma amostra semicondutora por um pulso ultracurto, os fotoporadores interagem com a rede excitando modos longitudinais ópticos. Essa interação provoca variações no índice de refração do material, produzindo modulações na resposta óptica do meio (efeito eletro-óptico). Por outro lado, esta dinâmica origina polarizações dependentes do tempo o que gera emissão de radiação terahertz. Experimentos recentes (pump-probe) observaram modulações do campo através de medidas da refletividade resolvidas no tempo. A refletividade e o campo estão relacionados segundo o efeito eletro-óptico. Também se resolve temporalmente o campo irradiado pela amostra, através de antenas que operam na faixa de terahertz. Tanto as medidas eletro-ópticas quanto de emissão terahertz fornecem informações sobre a interação dinâmica do plasma com a rede após a excitação óptica. Nesse trabalho simulamos a interação dinâmica de plasma e fônons em n-GaAs e Telúrio (\"bulk\") após estes serem excitados por um pulso ultracurto. Utilizamos equações hidrodinâmicas para descrever transporte de cargas e uma equação fenomenológica de oscilador harmônico forçado, para descrever oscilações longitudinais ópticas da rede. Complementando nossa descrição temos a equação de Poisson, com a qual calculamos o campo gerado pelo plasma e pela polarização da rede semicondutora. Essas equações constituem um sistema de seis equações diferencias (quatro parciais) acopladas. Para resolvê-las utilizamos o método das diferenças finitas. Do cálculo numérico obtemos a evolução temporal do campo elétrico no interior do material. Com esse campo determinamos as freqüências de oscilação do sistema e calculamos o campo irradiado. Nossos resultados apresentam acordo qualitativo com os experimentos / Above-band-gap optical excitation of semiconductors generates highly non-equilibrium photocarriers which interact with phonons thus exciting vibrational modes in the system. This interaction induces refractive-index changes via the electro-optic effect. Moreover it gives rise to electromagnetic radiation at characteristic frequencies (terahertz). Both effects have been measured by time-resolved ultra fast spectroscopy. Recent pump-probe experiments have found strong modulations of the internal electric field through electro-optic measurements. The emitted electromagnetic radiation has also been detected by a terahertz dipole antenna. Both electro-optic and terahertz emission measurements provide information about the coupled dynamics of photocarriers and phonons. In this work we simulate the dynamics of plasmon-phonon coupled modes in n-GaAs and Tellurium (bulk) following ultrafast laser excitation. The time evolution of the photocarrier densities and currents is described semi classically in terms of the moments of the Boltzmann equation. Phonon effects are accounted for by considering a phenomenological driven-harmonic-oscillator equation, which is coupled to the electron-hole plasma via Poisson\'s equation. These equations constitute a coupled set of differential equations. We use finite differencing to solve these equations. From the numerical results for the evolution of internal fields we can calculate both the characteristic frequencies of system and its terahertz radiation spectrum. Our results are consistent with recent experimental data

This thesis studies the so-called “criticality problem”, an important generalised eigenvalue problem arising in neutron transport theory. The smallest positive real eigenvalue of the problem contains valuable information about the status of the fission chain reaction in the nuclear reactor (i.e. the criticality of the reactor), and thus plays an important role in the design and safety of nuclear power stations. Because of the practical importance, efficient numerical methods to solve the criticality problem are needed, and these are the focus of this thesis. In the theory we consider the time-independent neutron transport equation in the monoenergetic homogeneous case with isotropic scattering and vacuum boundary conditions. This is an unsymmetric integro-differential equation in 5 independent variables, modelling transport, scattering, and fission, where the dependent variable is the neutron angular flux. We show that, before discretisation, the nonsymmetric eigenproblem for the angular flux is equivalent to a related eigenproblem for the scalar flux, involving a symmetric positive definite weakly singular integral operator(in space only). Furthermore, we prove the existence of a simple smallest positive real eigenvalue with a corresponding eigenfunction that is strictly positive in the interior of the reactor. We discuss approaches to discretise the problem and present discretisations that preserve the underlying symmetry in the finite dimensional form. The thesis then describes methods for computing the criticality in nuclear reactors, i.e. the smallest positive real eigenvalue, which are applicable for quite general geometries and physics. In engineering practice the criticality problem is often solved iteratively, using some variant of the inverse power method. Because of the high dimension, matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix-free inneriterations. This leads to inexact iterative methods for criticality computations, for which there appears to be no rigorous convergence theory. The fact that, under appropriate assumptions, the integro-differential eigenvalue problem possesses an underlying symmetry (in a space of reduced dimension) allows us to perform a systematic convergence analysis for inexact inverse iteration and related methods. In particular, this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge. The theory is illustrated with numerical examples on several test problems of physical relevance, using GMRES as the inner solver. We also illustrate the use of Monte Carlo methods for the solution of neutron transport source problems as well as for the criticality problem. Links between the steps in the Monte Carlo process and the underlying mathematics are emphasised and numerical examples are given. Finally, we introduce an iterative scheme (the so-called “method of perturbation”) that is based on computing the difference between the solution of the problem of interest and the known solution of a base problem. This situation is very common in the design stages for nuclear reactors when different materials are tested, or the material properties change due to the burn-up of fissile material. We explore the relation ofthe method of perturbation to some variants of inverse iteration, which allows us to give convergence results for the method of perturbation. The theory shows that the method is guaranteed to converge if the perturbations are not too large and the inner problems are solved with sufficiently small tolerances. This helps to explain the divergence of the method of perturbation in some situations which we give numerical examples of. We also identify situations, and present examples, in which the method of perturbation achieves the same convergence rate as standard shifted inverse iteration. Throughout the thesis further numerical results are provided to support the theory.

In this thesis, two methods to model quantum well lasers will be examined. The first model is based on well-known techniques to determine some of the spectral and dynamical properties of the laser. For the spectral properties, an expression for TE and TM modal amplitude gain is derived. For the dynamical properties, the rate equations are shown. The spectral and dynamical properties can be examined separately for specific operating characteristics or used in conjunction with each other for a complete description of the laser. Examples will be shown to demonstrate some of the analysis and results that can be obtained. The second model used is based on Wigner functions and the quantum Boltzmann equation. It is derived from general non-equilibrium Greens functions with the application of the Kadanoff-Baym ansatz. This model is less phenomenological than the previous model and does not require the separation of physical processes such as the former spectral and dynamical properties. It therefore has improved predictive power for the performance of novel laser designs. To the Author's knowledge, this is the first time such a model has been formulated. The quantum Boltzmann equations will be derived and some calculations will be performed for a simplified system in order to illustrate some calculation techniques as well as results that can be obtained.

The distribution of particles in a rarefied gas in a vessel can be described by the Boltzmann equation. As an approximation of the solution to this equation, Caprino, Pulvirenti and Wagner [3] constructed a random N-particle system. In the equilibrium case, they prove in [3] that the L1-distance between the density function of k particles in the N-particle process and the k-fold product of the solution to the stationary Boltzmann equation is of order 1/N. They do this in order to show that the N-particle system converges to the system described by the stationary Boltzmann equation as the number of particles tends to infinity. This is different from the standard approach of describing convergence of an N-particle system. Usually, convergence in distribution of random measures or weak convergence of measures over the space of probability measures is used. The purpose of the present thesis is to compare different modes of convergence of the N-particle system as N tends to infinity assuming stationarity.

We explore the capability of lattice Boltzmann equation (LBE) method for complex fluid flows involving turbulence, mixing, and reaction. In the first study, LBE schemes for binary scalar mixing and multi-component reacting flow with reactions are developed. Simulations of initially non-premixed mixtures yield scalar probability distribution functions that are in good agreement with numerical data obtained from Navier-Stokes (NS) equation based computation. One-dimensional chemically-reacting flow simulation of a premixed mixture yields a flame speed that is consistent with experimentally determined value. The second study involves direct numerical simulation (DNS) and large-eddy simulation (LES) of decaying homogenous isotropic turbulence (HIT) with and without frame rotation. Three categories of simulations are performed: (i) LBE-DNS in both inertial and rotating frames; (ii) LBE-LES in inertial frame; (iii) Comparison of the LBE-LES vs. NS-LES. The LBE-DNS results of the decay exponents for kinetic energy k and dissipation rate ε, and the low wave-number scaling of the energy spectrum agree well with established classical results. The LBE-DNS also captures rotating turbulence physics. The LBE-LES accurately captures low-wave number scaling, energy decay and large scale structures. The comparisons indicate that the LBE-LES simulations preserve flow structures somewhat more accurately than the NS-LES counterpart. In the third study, we numerically investigate the near-field mixing features in low aspect-ratio (AR) rectangular turbulent jets (RTJ) using the LBE method. We use D3Q19 multiple-relaxation-time (MRT) LBE incorporating a subgrid Smagorinsky model for LES. Simulations of four jets which characterized by AR, exit velocity, and Reynolds number are performed. The investigated near-field behaviors include: (1) Decay of mean streamwise velocity (MSV) and inverse MSV; (2) Spanwise and lateral profiles of MSV; (3) Half-velocity width development and MSV contours; and (4) Streamwise turbulence intensity distribution and spanwise profiles of streamwise turbulence intensity. The computations are compared against experimental data and the agreement is good. We capture both unique features of RTJ: the saddle-back spanwise profile of MSV and axis-switching of long axis from spanwise to lateral direction. Overall, this work serves to establish the feasibility of the LBE method as a viable tool for computing mixing, combustion, and turbulence.

In this dissertation, we study the evolution and properties of the relic (or cosmic) neutrino distribution from neutrino freeze-out at T=O(1) MeV through the free-streaming era up to today, focusing on the deviation of the neutrino spectrum from equilibrium and in particular we demonstrate the presence of chemical non-equilibrium that continues to the present day. The work naturally separates into two parts. The first focuses on aspects of the relic neutrinos that can be explored using conservation laws. The second part studies the neutrino distribution using the full general relativistic Boltzmann equation. Part one begins with an overview of the history of the Universe, from just prior to neutrino freeze-out up through the present day, placing the history of cosmic neutrino evolution in its proper context. Motivated by the Planck CMB measurements of the effective number of neutrinos, we derive those properties of neutrino freeze-out that depend only on conservation laws and are independent of the details of the scattering processes. Part one ends with a characterization of the present day neutrino spectrum as seen from Earth. The second part of this dissertation focuses on the properties of cosmic neutrinos that depend on the details of the neutrino reactions, as is necessary for modeling the non-thermal distortions from equilibrium and computing freeze-out temperatures. We first develop some geometry background concerning volume forms and integration on submanifolds that is helpful in computations. We then detail a new spectral method for solving the Boltzmann equation, based on a dynamical basis of orthogonal polynomials. Next, we detail an improved procedure for analytically simplifying the corresponding scattering integrals for subsequent numerical computation. Using this, along with the spectral method mentioned above, we solve the Boltzmann equation through the neutrino freeze-out period. Finally, we conclude by using our novel solution methods to perform parametric studies of the dependence of the neutrino freeze-out standard model parameters. This exploration is performed with the aim of recognizing mechanisms in the neutrino freeze-out process that are capable of leading to the measured value of the effective number of neutrinos.

In this thesis we consider mathematical problems related to different aspects of hard sphere systems. In the first part we study planar billiards, which arise in the context of hard sphere systems when only one or two spheres are present. In particular we investigate the possibility of elliptic periodic orbits in the general construction of hyperbolic billiards. We show that if non-absolutely focusing components are present there can be elliptic periodic orbits with arbitrarily long free paths. Furthermore, we show that smooth stadium like billiards have elliptic periodic orbits for a large range of separation distances. In the second part we consider hard sphere systems with a large number of particles, which we model by the Boltzmann equation. We develop a new approach to derive hydrodynamic limits, which is based on classical methods of geometric singular perturbation theory of ordinary differential equations. This method provides new geometric and dynamical interpretations of hydrodynamic limits, in particular, for the of the dissipative Boltzmann equation.