The mathematical analysis of the Boltzmann equation for a wide range of important models is well developed. It describes physical phenomena which are often of great engineering importance (in aerospace industry, semiconductor design, etc.). For that reason, analytical and computational methods of solving the Boltzmann equation are studied extensively. The idea of describing processes on a scale of the order of the relaxation scales of time and space has been realized. The nonlinear Boltzmann equation possesses the important essence of a physically realistic equation, so it is possible not only to consider the flows of simple media but to formulate new problems due to the ability of this equation to describe nonequilibrium states. In this dissertation, a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation for variable hard potential (VHP) collision kernels with conservative or non-conservative binary interactions is proposed. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in the collision integral computation is reduced to a separate integral over the unit sphere S2. In addition, the conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or to elastic model of slowing down processes. We prove the accuracy, consistency and conservation properties of the proposed conservative spectral method. Existing spectral methods have consistency proofs which are only for elastic collisions, and also such methods do not conserve all the necessary moments of the collision integral. In this dissertation, error estimates for the conservation routine are provided. Such conservation correction is implemented as an extended isoperimetric problem with the moment conservation properties as the constraints. We use and extend an existing bound estimate of Gamba, Panferov and Villani for the inelastic/elastic space homogeneous Boltzmann collision operator. The result is an original extension to the work of Gustaffson. Using these estimates along with projection error estimates and conservation correction estimates, we prove that the conservation correction is bounded by the spectral accuracy. Simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation for both elastic and inelastic VHP interactions. Benchmarking of the self-similar simulations involves the selection of a time rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard-spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods. Recognizing the importance of the Boltzmann equation in the analysis of shock structures and nonequilibrium states, such a study is done for 1D(x) × 3D(v). The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Sone and Aoki, where the wall temperature is suddenly changed, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock. / text

Dissipative time-dependent quantum transport theory with electron-phonon interaction in either weak or strong coupling regime is established. This theory goes beyond the conventional quantum master equation method and Kadanoff-Baym kinetic equations. It provides an efficient method for the simulation of transient quantum transport under arbitrary bias voltage with different electron-phonon coupling strength. First, time-dependent quantum transport theory for non-interacting system and its combination with first-principles method is developed. Based on the Padé expansion to Fermi function, and wide-band limit approximation of lead self-energy, a set of equations of motion is developed for efficient evaluation of density matrix and related quantities. To demonstrate its applicability, this method is employed to study the transient transport through a carbon nanotube based electronic device. Second, a dissipative time-dependent quantum transport theory is established in the weak electron-phonon coupling regime. In addition to the self-energy caused by leads, a new self-energy is introduced to characterize the dissipative effect induced by electron-phonon interaction. In the weak coupling regime, the lowest order expansion is employed for practical implementation. The corresponding closed set of equations of motion is derived, which provides an efficient and accurate treatment of transient quantum transport with electron-phonon interaction in the weak coupling regime. Numerical examples are demonstrated and its combination with first-principles method is also discussed. Next, a dissipative quantum transport theory for strong electron-phonon interaction is established by employing small polaron transformation. The corresponding equation of motions are developed, which is used to study the quantum interference effect and phonon-induced decoherence dynamics. Numerical studies demonstrate the formation of quantum interference effect caused by the transport electrons through two quasi-degenerate states with different couplings to the leads. The quantum interference can be suppressed by phonon scattering, which indicates the importance of considering electron-phonon interaction in these systems with prominent quantum interference effect when the electron-phonon coupling is strong. Last, the dissipative quantum transport theory for weak electron-phonon coupling regime is used to simulate the photovoltaic devices. Within the nonequilibrium Greens function formalism, a quantum mechanical method for nanostructured photovoltaic devices is presented. The method employs density-functional tight-binding theory for electronic structure, which make is possible to simulate the performance of photovoltaic devices without relying on empirical parameters. Numerical studies of silicon nanowirebased devices of realistic sizes with more than ten thousand atoms are performed and the results indicate that atomistic details and nonequilibrium conditions have clear impact on the photoresponse of the devices. / published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy

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published_or_final_version / abstract / toc / Chemistry / Doctoral / Doctor of Philosophy

published_or_final_version / Physics / Master / Master of Philosophy

Integral transport theory is used to compute transfer matrix elements for neutral particle streaming through r-z rectangular voids. The transfer matrix is utilized in a hybrid method; it interfaces with a conventional discrete ordinates calculation outside the void. Since the hybrid interface is accomplished within the inner iteration of an existing discrete ordinates code, standard sweeping schemes used in SN codes are not adversely affected. The resulting transfer matrix is independent of the multigroup energy structure used in a calculation. Further, the transfer matrix is not affected by changes in materials or cross sections outside the void. For these reasons, once a transfer matrix is calculated it can be used in a wide variety of problems containing a similar void. In this study, the transfer matrix hybrid, standard discrete ordinates, and Monte Carlo methods are applied to two sample problems to compare the accuracy of these methods. The Monte Carlo results are considered correct in order to compute errors in the deterministic calculations. The transfer matrix hybrid method is more accurate than conventional SN methods; the ray effect, persistent in discrete ordinates calculations, is substantially reduced. In addition, the transfer matrix hybrid executes faster than higher order SN calculations.

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