# Browsing by Subject "Mathematical physics"

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Item The dynamics of bose gases(2015-05) Taliaferro, Kenneth William; Chen, Thomas (Ph. D. in mechanical engineering and Ph. D. in mathematical physics); Maggi, Francesco; Pavlovic, Natasa; Tzirakis, Nikolaos; Vasseur, AlexisShow more We study the Gross-Pitaevskii (GP) hierarchy, which is an infinite sequence of coupled partial differential equations that models the dynamics of Bose gases and arises in the derivation of the cubic and quintic nonlinear Schrödinger equations from an N-body linear Schrödinger equation. In Chapter 2, we consider the cubic case in R³ and derive the GP hierarchy in the strong topology corresponding to the spaces used by Klainerman and Machedon in (82). We also prove that positive semidefiniteness of solutions is preserved over time and use this result to prove global well-posedness of solutions to the GP hierarchy. This is based on a joint work with Thomas Chen (24). In Chapters 3 and 4, we prove uniqueness of solutions to the GP hierarchy in R[superscript d] in a low regularity Sobolev type space in the cubic and quintic cases, respectively. These chapters are an extension of the work of Chen-Hainzl-Pavlović-Seiringer (17) and are based on joint works with Younghun Hong and Zhihui Xie (70,71).Show more Item Hamiltonian and Action Principle formulations of plasma fluid models(2015-05) Lingam, Manasvi; Morrison, Philip J.; Hazeltine, Richard; Waelbroeck, Francois; Breizman, Boris; Gamba, Irene MShow more The Hamiltonian and Action Principle (HAP) formulations of plasmas and fluids are explored in a wide variety of contexts. The principles involved in the construction of Action Principles are presented, and the reduction procedure to obtain the associated noncanonical Hamiltonian formulation is delineated. The HAP formulation is first applied to a 2D magnetohydrodynamics (MHD) model, and it is shown that one can include Finite Larmor Radius effects in a transparent manner. A simplified 2D limit of the famous Branginskii gyroviscous tensor is obtained, and the origins of a powerful tool - the gyromap - are traced to the presence of a gyroviscous term in the action. The noncanonical Hamiltonian formulation is used to extract the Casimirs of the model, and an Energy-Casimir method is used to derive the equilibria and stability; the former are shown to be generalizations of the Grad-Shafranov equation, and possess both flow and gyroviscous effects. The action principle of 2D MHD is generalized to encompass a wider class of gyroviscous fluids, and a suitable gyroviscous theory for liquid crystals is constructed. The next part of the thesis is devoted to examining several aspects of extended MHD models. It is shown that one can recover many such models from a parent action, viz. the two-fluid model. By performing systematic orderings in the action, extended MHD, Hall MHD and electron MHD are recovered. In order to obtain these models, novel techniques, such as non-local Lagrange-Euler maps which enable a transition between the two fluid frameworks, are introduced. A variant of extended MHD, dubbed inertial MHD, is studied via the HAP approach in the 2D limit. The model is endowed with the effects of electron inertia, but is shown to possess a remarkably high degree of similarity with (inertialess) ideal MHD. A reduced version of inertial MHD is shown to yield the famous Ottaviani-Porcelli model of reconnection. Similarities in the mathematical structure of several extended MHD models are explored in the Hamiltonian framework, and it is hypothesized that these features emerge via a unifying action principle. Prospects for future work, reliant on the HAP formulation, are also presented.Show more Item On using physical properties to make mathematical choices in quantum mechanical scattering(2015-08) Hebert, Joshua Russell; Bohm, Arno, 1936-; Dicus, Duane; Morrison, Phil; Li, Elaine X; Pavlovic, NatasaShow more Traditional Quantum Scattering theory is built upon a Hilbert space formalism. The nature of the Hilbert space has led to a number of ideas about the nature of scattering systems, chief among them being the understanding that the decay of unstable quan- tum systems cannot be purely exponential and that there can only be an approximate relationship between the width Γ of a scattering resonance and the lifetime τ of the resonant state whose formation gives rise to it. That such physical conclusions are de- rived from purely mathematical properties of the underlying topological vector space inform the notion that, in choosing a particular space, one is choosing the mathemat- ical representation of the physical universe for the system. With this notion in mind, purely physical characteristics of quantum systems are used to answer the question: “What is the proper mathematical space for quantum mechanical calculations?” In the particular case of quantum scattering, it is shown that this approach casts the choice of the Hilbert space into doubt and motivates instead the use of rigged Hilbert spaces employing Hardy spaces; in these Hardy rigged Hilbert spaces, exponential decay is permissible and the time evolution of decaying states is governed by semigroups. The original theories concerning deviation from exponential decay in the Hilbert space are revisited to reveal how their conclusions flow from a mathematical property whose physical interpretation is problematic. Finally, an application of a Hardy rigged Hilbert space formalism is presented: Time Asymmetric Quantum Mechanics, as developed by A. Bohm, M. Gadella, and S. Wickramasekara.Show more