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Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This investigation is concerned with the evaluation and interpretation of the propagator, or conditional probability distribution function P(y,t| y0), of the Linear Boltzmann or Master equation from the "central limit viewpoint". We have obtained what is to our knowledge the first evaluation in series of the propagator for the typical kinetic-theoretical processes studied here, which are those underlying the problems usually studied in the approximation of the Fokker-Planck equation. We have been able to put the successive terms of this series in closed form; and have shown that the series can be interpreted as a generalized solution of the central limit problem of mathematical probability theory, the generalization consisting in the extension to a process in continuous time with non-independent increments. [TRUNCATED] / 2031-01-01

Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The differential-operator approximation to the linear Boltzmann operator (the Master equation operator) has been studied by several authors. In 1960 Siegel proposed a systematic approach called the CD expansion. He represents the approximating series in terms of creation and destruction operators for Hermite functions. In this thesis we study the physical meaning of a small parameter which usually exists in the CD expansion and which ensures the convergence of the series. We also establish the CD-expansion formalism for N-dimensional processes and initiate the study of the CD-expansion of the linear Boltzmann collision operator in the kinetic theory of gases. In the case of one-dimensional processes; the models we study are the density fluctuations with a particle reservoir of finite volume, Alkemade's diode model, and Rayleigh disk. We find that the expansion parameter is the ratio of the average microscopic agitation interval to the macroscopic relaxation time. We further prove that this ratio is equal to the ratio of the average variance of the discontinuity of the random process determined by the linear. Boltzmann operator to the variance of the macroscopic observable at equilibrium. Since the CD expansion is an expansion with respect to the parameter of discontinuity, the expansion series reduces to the Fokker-Planck operator in the limiting case where the parameter becomes zero. In the N-dimensional formalism, we use tensor Hermite polynomials and find a formalism valid for processes of any finite dimensionality. In extending the study to the kinetic theory of gases, we establish a method of obtaining derivate moments directly from the collision operator, and obtain a formula for the Hermite coefficients of derivate moments for an arbitrary force field. We propose the CD hypothesis: The terms of the CD expansion are homogeneous and of successively increasing order in the parameter of discontinuity of the process. This hypothesis I holds for all the models we study in the one-dimension case. In three-dimensional collision processes it holds for an intermolecular force field obeying an inverse power law and for rigid spheres. Beyond these cases, the necessary and sufficient conditions for the hypothesis are rather complicated. A sufficient condition is, however, that the scattering cylinder (scattering cross section multiplied by the magnitude of the relative velocity) be a homogeneous function of the magnitude of the relative velocity. In addition to the general results mentioned in the above, we obtain a number of particular results. Special cases of our density fluctuation model are the density fluctuations studied by van Kampen (1961), and Ehrenfest's urn model. We also introduce an isotropic Maxwellian particle which corresponds to s-wave scattering in wave mechanics, and which yields the CD expansion in a diagonal form. / 2031-01-01

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Aurorae often break down into elongated filaments parallel to the geomagnetic field lines (B) with cylindrically symmetric structures. The object of this thesis is to study the ion distribution function and transport properties in response to the sudden introduction of a radial electric field (E) in such a cylindrical geometry. Both collision-free and collisional situations are considered.<p> The thesis starts by solving a collision-free problem where the electric field is constant in time but increases linearly with radius, while the initial ion density is uniform in space. The attendant Boltzmann equation is solved by tracking the ions back in time, thereby using the temporal link between the initial position and velocity of an ion and its position and velocity at an arbitrary time and place. Complete analytical solutions show that the ion distribution function is a pulsating Maxwellian in time, and all transport parameters (e.g., bulk speed, temperature, etc.) oscillate in time but independent of radius. If the ion-neutral collisions are taken into account by employing a simple relaxation model, analytical solutions are also obtained. In this case, the ion distribution function can be driven to horseshoe shapes which are symmetric with respect to the ExB direction. The bulk parameters evolve in a transition period of the order of one collision time as they go from oscillating to the non-oscillating steady state.<p> In more realistic electric field structures which are spatially inhomogeneous but still constant in time, a generalized semi-numerical code is developed under collision-free conditions. This code uses a backmapping approach to calculate the ion velocity distribution and bulk parameters. With arbitrarily selected electric field rofiles, calculations reveal various shapes of ion velocity distribution functions (e.g., tear-drop, core-halo, ear-donut, etc). The associated transport properties are also obtained and discussed.<p> Under both collision-free and collisional conditions, the effect of the density inhomogeneities at the initial time is studied in an electric field which is proportional to radius and constant in time. With two profiles of the initial ion density for the collision-free case, and one profile for the collisional case, complete analytical solutions are obtained. The results reveal that the distribution function and the bulk properties are now strongly dependent on radial position.<p> If the radial electric field is unable to stay constant with time but modulated by in-coming charged particles, a fluid formalism is used to study the excitation of several plasma waves under different kinds of initial conditions. These identified waves include the ion cyclotron oscillation, the ion and electron upper-hybrid oscillations, and the lower-hybrid oscillation.<p> The results of this thesis are expected to be applicable to high-resolution observations. Future work should also include the mirror effect and the formation of conics in velocity space. Finally, the velocity distributions obtained in this thesis could trigger various plasma instabilities, and this topic should also be looked at in the future.

Aurorae often break down into elongated filaments parallel to the geomagnetic field lines (B) with cylindrically symmetric structures. The object of this thesis is to study the ion distribution function and transport properties in response to the sudden introduction of a radial electric field (E) in such a cylindrical geometry. Both collision-free and collisional situations are considered.<p> The thesis starts by solving a collision-free problem where the electric field is constant in time but increases linearly with radius, while the initial ion density is uniform in space. The attendant Boltzmann equation is solved by tracking the ions back in time, thereby using the temporal link between the initial position and velocity of an ion and its position and velocity at an arbitrary time and place. Complete analytical solutions show that the ion distribution function is a pulsating Maxwellian in time, and all transport parameters (e.g., bulk speed, temperature, etc.) oscillate in time but independent of radius. If the ion-neutral collisions are taken into account by employing a simple relaxation model, analytical solutions are also obtained. In this case, the ion distribution function can be driven to horseshoe shapes which are symmetric with respect to the ExB direction. The bulk parameters evolve in a transition period of the order of one collision time as they go from oscillating to the non-oscillating steady state.<p> In more realistic electric field structures which are spatially inhomogeneous but still constant in time, a generalized semi-numerical code is developed under collision-free conditions. This code uses a backmapping approach to calculate the ion velocity distribution and bulk parameters. With arbitrarily selected electric field rofiles, calculations reveal various shapes of ion velocity distribution functions (e.g., tear-drop, core-halo, ear-donut, etc). The associated transport properties are also obtained and discussed.<p> Under both collision-free and collisional conditions, the effect of the density inhomogeneities at the initial time is studied in an electric field which is proportional to radius and constant in time. With two profiles of the initial ion density for the collision-free case, and one profile for the collisional case, complete analytical solutions are obtained. The results reveal that the distribution function and the bulk properties are now strongly dependent on radial position.<p> If the radial electric field is unable to stay constant with time but modulated by in-coming charged particles, a fluid formalism is used to study the excitation of several plasma waves under different kinds of initial conditions. These identified waves include the ion cyclotron oscillation, the ion and electron upper-hybrid oscillations, and the lower-hybrid oscillation.<p> The results of this thesis are expected to be applicable to high-resolution observations. Future work should also include the mirror effect and the formation of conics in velocity space. Finally, the velocity distributions obtained in this thesis could trigger various plasma instabilities, and this topic should also be looked at in the future.

In this thesis, we develop and test a fundamentally new linear-discontinuous least-squares (LDLS) method for spatial discretization of the one-dimensional (1-D) discrete-ordinates (SN) equations. This new scheme is based upon a least-squares method with a discontinuous trial space. We implement our new method, as well as the lineardiscontinuous Galerkin (LDG) method and the lumped linear-discontinuous Galerkin (LLDG) method. The implementation is in FORTRAN. We run a series of numerical tests to study the robustness, L2 accuracy, and the thick diffusion limit performance of the new LDLS method. By robustness we mean the resistance to negativities and rapid damping of oscillations. Computational results indicate that the LDLS method yields a uniform second-order error. It is more robust than the LDG method and more accurate than the LLDG method. However, it fails to preserve the thick diffusion limit. Consequently, it is viable for neutronics but not for radiative transfer since radiative transfer problems can be highly diffusive.

A new method has been developed to incorporate particles with internal structure into the framework of the Variance Reduction method [17] for solving the discrete velocity Boltzmann Equation. Internal structure in the present context refers to physical phenomena like rotation and vibration of molecules consisting of two or more atoms. A gas in equilibrium has all modes of internal energy at the same temperature as the translational temperature. If the gas is in a non-equilibrium state, translational temperature and internal temperatures tend to proceed towards an equilibrium state during equilibration, but they all do so at different relaxation rates. In this thesis, rotational energy of a distribution of molecules is modeled as a single value at a point in a discrete velocity space; this represents the average rotational energy of molecules at that specific velocity. Inelastic collisions are the sole mechanism of translational and rotational energy exchange, and are governed by a modified Landau-Teller equation. The method is tested for heat bath simulations, or homogeneous relaxations, and one dimensional shock problems. Homogeneous relaxations demonstrate that the rotational and translational temperatures equilibrate to the correct final temperature, which can be predicted by conservation of energy. Moreover, the rates of relaxation agree with the direct simulation Monte Carlo (DSMC) method with internal energy for the same input parameters. Using a fourth order method for convecting mass along with its corresponding internal energy, a one dimensional Mach 1.71 normal shock is simulated. Once the translational and rotational temperatures equilibrate downstream, the temperature, density and velocity, predicted by the Rankine-Hugoniot conditions, are obtained to within an error of 0.5%. The result is compared to a normal shock with the same upstream flow properties generated by the DSMC method. Internal vibrational energy and a method to use Larsen Borgnakke statistical sampling for inelastic collisions is formulated in this text and prepared in the code, but remains to be tested. / text

Orientador: Augusto Agostinho Neto / Orientador: Elso Drigo Filho / Coorientador: Tereza Pereira de Souza / Banca: Iolanda Midea Cuccovia / Banca: Marcelo Andres Fossey / Resumo: Proteínas e peptídeos são constituídos por subunidades estruturalmente mais imples chamadas aminoácidos. Uma importante propriedade destes é que, dependendo das características do meio (tais como pH e concentração iônica), os seus grupos onizáveis podem ceder prótons e, assim, adquirir carga elétrica não nula. Tal carga nfluenciará na eficiência da formação de ligações peptídicas e em interações proteína- igante, por exemplo. Partindo da hipótese de que a diferença entre os valores de pK dos rupos ionizáveis isolados e destes como partes constituintes de um aminoácido é devida, principalmente, à interação eletrostática adicional que se atribui à presença de rupos vizinhos, elaborou-se um modelo que emprega a forma linearizada da equação de Poisson-Boltzmann para o estudo de propriedades físico-químicas de moléculas com rês grupos ionizáveis. Neste trabalho tal modelo foi aplicado aos aminoácidos: Aspartato, Glutamato, Cisteína, Tirosina, Arginina, Lisina e Histidina. Calcularam-se os valores de pK e as respectivas cargas elétricas médias de tais moléculas. Como os esultados obtidos concordaram com aqueles oriundos de trabalhos experimentais, o modelo teórico foi expandido para tratar de di, tetra, pentapeptídeos e de resíduos de isina e glutamato da proteína Staphylococcal Nuclease. Os valores do Fator de Correlação de Pearson calculados para ambos proteínas e peptídeos são superiores a 0,99, fato este que evidencia a eficiência e versatilidade do modelo ao reproduzir alores de pK reportados por outros autores / Abstract: Proteins and peptides are composed of subunits structurally simpler called amino acids. An important property of these is that, depending on the medium characteristics (such pH and ionic concentration), its ionizable groups may provide protons and thereby acquire a nonzero electric charge. Such charge will affect the formation of peptide bond and protein-ligand interactions, for example. Assuming that the difference between pK values of the isolates ionizable groups and of these as constituents parts of an amino acid is mainly due to the extra electrostatic interaction attributed to the presence of neighboring groups, was developed a structure-based model that employs the linearized form of the Poisson-Boltzmann equation for the study of physicochemical properties of molecules with three ionizable groups. In this work it was applied to the amino acids: aspartate, glutamate, cysteine, tyrosine, arginine, lysine and histidine. The pK values and respective mean electric charges were calculated. As the calculated values agreed with those from experimental studies, the theoretical model has been expanded to the treatment of di, tetra, pentapeptides and Staphylococcal Nuclease residues. The Pearson Correlation Factor calculated for both proteins and peptides are above 0.99, what points to the effectiveness and versatility of the model to reproduce pK values reported by other works / Mestre