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NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis, three parts of my work are reported. The first part of the work was done with Prof. Rudy Marcus, the second and third parts of the work were done with Prof. Bill Goddard. Here I would like to summarize the results from each part briefly. In the first part, we propose doing Scanning Tunneling Microscopy(STM) current calculations with a new model in the spheroidal coordinate system. The tip is modeled as a hyperboloid. The electrostatic potential part of this model is solved exactly. The free electron model of the whole system is also solved exactly. In the second part, we found that the Nose Canonical Molecular Dynamics, the most commonly used CMD method, leads to the wrong heat capacity for the system and hence is inconsistent with the thermodynamics. To solve this problem, we propose Two Nose variable Dynamics by changing the dimensionality of the Nose variable s and its conjugate [...] from one- to two-dimension. In the third part, The exact expression for the quantum statistical partition function in the canonical ensemble is given. The physical interpretation of each term for N-particle system is discussed. The new formula is applied to the weakly degenerate quantum ideal case. By analyzing each term in the expansion, it is possible that this canonical partition function can be applied to cases where both quantum correlation effects and particle interactions are important.

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis, three parts of my work are reported. The first part of the work was done with Prof. Rudy Marcus, the second and third parts of the work were done with Prof. Bill Goddard. Here I would like to summarize the results from each part briefly. In the first part, we propose doing Scanning Tunneling Microscopy(STM) current calculations with a new model in the spheroidal coordinate system. The tip is modeled as a hyperboloid. The electrostatic potential part of this model is solved exactly. The free electron model of the whole system is also solved exactly. In the second part, we found that the Nose Canonical Molecular Dynamics, the most commonly used CMD method, leads to the wrong heat capacity for the system and hence is inconsistent with the thermodynamics. To solve this problem, we propose Two Nose variable Dynamics by changing the dimensionality of the Nose variable s and its conjugate [...] from one- to two-dimension. In the third part, The exact expression for the quantum statistical partition function in the canonical ensemble is given. The physical interpretation of each term for N-particle system is discussed. The new formula is applied to the weakly degenerate quantum ideal case. By analyzing each term in the expansion, it is possible that this canonical partition function can be applied to cases where both quantum correlation effects and particle interactions are important.

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis, three parts of my work are reported. The first part of the work was done with Prof. Rudy Marcus, the second and third parts of the work were done with Prof. Bill Goddard. Here I would like to summarize the results from each part briefly. In the first part, we propose doing Scanning Tunneling Microscopy(STM) current calculations with a new model in the spheroidal coordinate system. The tip is modeled as a hyperboloid. The electrostatic potential part of this model is solved exactly. The free electron model of the whole system is also solved exactly. In the second part, we found that the Nose Canonical Molecular Dynamics, the most commonly used CMD method, leads to the wrong heat capacity for the system and hence is inconsistent with the thermodynamics. To solve this problem, we propose Two Nose variable Dynamics by changing the dimensionality of the Nose variable s and its conjugate [...] from one- to two-dimension. In the third part, The exact expression for the quantum statistical partition function in the canonical ensemble is given. The physical interpretation of each term for N-particle system is discussed. The new formula is applied to the weakly degenerate quantum ideal case. By analyzing each term in the expansion, it is possible that this canonical partition function can be applied to cases where both quantum correlation effects and particle interactions are important.

The purpose of this work is to examine the stress distribution caused by the bending of a thin elastic plate containing a line discontinuity. Specifically, the plate under consideration is of constant thickness and occupies a whole plane exterior to the line discontinuity. The line discontinuity is either a crack or a rigid inclusion. The loading is applied to the plate at infinity by certain combinations of tractions which leave the plate in equilibirum. The analysis of the problems considered here is based on an approximate theory which is more refined than the classical theory ordinarily applied to problems of bending of plates. This is because results based on the classical theory may be incorrect, even in first approximation for thin plates, near a boundary, and it is precisely the region near a boundary (in this case, the line discontinuity) which is of primary interest in these problems. In fact one of the principal objectives in this work is to compare the stress distributions near the line discontinuity as predicted by the two theories. The principal techniques used in this work are based on integral equations and the calculus of variations. Results based on the two theories are found to agree for thin plates away from the line discontinuity, but differ significantly in the vicinity of the discontinuity, even for very thin plates.

The purpose of this work is to examine the stress distribution caused by the bending of a thin elastic plate containing a line discontinuity. Specifically, the plate under consideration is of constant thickness and occupies a whole plane exterior to the line discontinuity. The line discontinuity is either a crack or a rigid inclusion. The loading is applied to the plate at infinity by certain combinations of tractions which leave the plate in equilibirum. The analysis of the problems considered here is based on an approximate theory which is more refined than the classical theory ordinarily applied to problems of bending of plates. This is because results based on the classical theory may be incorrect, even in first approximation for thin plates, near a boundary, and it is precisely the region near a boundary (in this case, the line discontinuity) which is of primary interest in these problems. In fact one of the principal objectives in this work is to compare the stress distributions near the line discontinuity as predicted by the two theories. The principal techniques used in this work are based on integral equations and the calculus of variations. Results based on the two theories are found to agree for thin plates away from the line discontinuity, but differ significantly in the vicinity of the discontinuity, even for very thin plates.

The purpose of this work is to examine the stress distribution caused by the bending of a thin elastic plate containing a line discontinuity. Specifically, the plate under consideration is of constant thickness and occupies a whole plane exterior to the line discontinuity. The line discontinuity is either a crack or a rigid inclusion. The loading is applied to the plate at infinity by certain combinations of tractions which leave the plate in equilibirum. The analysis of the problems considered here is based on an approximate theory which is more refined than the classical theory ordinarily applied to problems of bending of plates. This is because results based on the classical theory may be incorrect, even in first approximation for thin plates, near a boundary, and it is precisely the region near a boundary (in this case, the line discontinuity) which is of primary interest in these problems. In fact one of the principal objectives in this work is to compare the stress distributions near the line discontinuity as predicted by the two theories. The principal techniques used in this work are based on integral equations and the calculus of variations. Results based on the two theories are found to agree for thin plates away from the line discontinuity, but differ significantly in the vicinity of the discontinuity, even for very thin plates.