We consider a gas of indistinguishable bosons, confined to a ring of radius R, and
interacting via a pair-wise cosine potential. This may be thought of as the quantized
Hamiltonian Mean Field (HMF) model for bosons originally introduced by Chavanis
as a generalization of Antoni and Ruffo’s classical model.
This thesis contains three parts: In part one, the dynamics of a Bose-condensate are
considered by studying a generalized Gross-Pitaevskii equation (GGPE). Quantum
effects due to the quantum pressure are found to substantially alter the system’s
dynamics, and can serve to inhibit a pathological instability for repulsive interactions.
The non-commutativity of the large-N , long-time, and classical limits is discussed.
In part two, we consider the GGPE studied above and seek static solutions. Exact
solutions are identified by solving a non-linear eigenvalue problem which is closely
related to the Mathieu equation. Stationary solutions are identified as solitary waves
(or solitons) due to their small spatial extent and the system’s underlying Galilean
invariance. Asymptotic series are developed to give an analytic solution to the non-
linear eigenvalue problem, and these are then used to study the stability of the solitary
wave mentioned above.
In part three, the exact solutions outlined above are used to study quantum fluctuations
of gapless excitations in the HMF model’s symmetry broken phase. It is found that
this phase is destroyed at zero temperature by large quantum fluctuations. This
demonstrates that mean-field theory is not exact, and can in fact be qualitatively
wrong, for long-range interacting quantum systems, in contrast to conventional wisdom. / Thesis / Doctor of Philosophy (PhD) / The Hamiltonian Mean Field (HMF) model was initially proposed as a simplified
description of self-gravitating systems. Its simplicity shortens calculations and makes
the underlying physics more transparent. This has made the HMF model a key tool in
the study of systems with long-range interactions.
In this thesis we study a quantum extension of the HMF model. The goal is to
understand how quantum effects can modify the behaviour of a system with long-range
interactions. We focus on how the model relaxes to equilibrium, the existence of
special “solitary waves”, and whether quantum fluctuations can prevent a second order
(quantum) phase transition from occurring at zero temperature.
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/24814 |
| Date | January 2019 |
| Creators | Plestid, Ryan |
| Contributors | O'Dell, Duncan H.J., Physics and Astronomy |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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