Glassy behaviour in simple systems
<p>In this thesis we study several different models which display glassy behaviour. Firstly, we investigate a simple, purely topological, cellular model for which the Hamiltonian is non-interacting but the dynamics are constrained. We find a non-thermodynamic transition to a glassy phase in w...
| Päätekijät: | , |
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| Muut tekijät: | |
| Aineistotyyppi: | Opinnäyte |
| Kieli: | English |
| Julkaistu: |
2001
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| Aiheet: |
| _version_ | 1826315658114105344 |
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| author | Davison, L L. Davison |
| author2 | Sherrington, D |
| author_facet | Sherrington, D Davison, L L. Davison |
| author_sort | Davison, L |
| collection | OXFORD |
| description | <p>In this thesis we study several different models which display glassy behaviour. Firstly, we investigate a simple, purely topological, cellular model for which the Hamiltonian is non-interacting but the dynamics are constrained. We find a non-thermodynamic transition to a glassy phase in which the energy fails to reach the equilibrium value below a characteristic temperature which is dependent on the cooling rate. This model involves activated processes and displays two-step relaxation in both the energy and the correlation functions; the latter also exhibit signs of aging. The relaxation time can be well-fitted at all temperatures by an offset Arrhenius law. Some predictions of Mode-coupling Theory are tested with some agreement found, but no convincing evidence that this description is the most fitting. By defining a suitable response function, we find that the equilibrium Fluctuation-Dissipation Theorem (FDT) is upheld for all but very short waiting-times, despite the fact that the system is not in equilibrium.</p> <p>This topological model is simplified to a hexagonally-based spin model, which also displays glassy behaviour, involves activated processes and exhibits two-step relaxation. This is a consequence of reaction-diffusion processes on two different time-scales, one temperature-independent and the other an exponential function of inverse temperature. We study two versions of this model, one with a single absorbing ground state, and the other with a highly degenerate ground state. These display qualitatively similar but quantitatively distinct macroscopic behaviour, and related but different microscopic behaviour. We extend this work to a square lattice, and find that the geometry of the lattice has a considerable impact on the behaviour, and to three dimensions, which provides support for the reaction-diffusion classification of the early behaviour. We find observable-dependent FDT plots; the observable can be chosen such that FDT is upheld for a region whilst the system is out of equilibrium — this observation is supported by some preliminary results for one-dimensional kinetically-constrained Ising chains.</p> |
| first_indexed | 2024-03-06T18:52:38Z |
| format | Thesis |
| id | oxford-uuid:10c594d7-1fa5-45f5-bba4-0fefb837aadf |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:30:07Z |
| publishDate | 2001 |
| record_format | dspace |
| spelling | oxford-uuid:10c594d7-1fa5-45f5-bba4-0fefb837aadf2024-12-01T13:28:08ZGlassy behaviour in simple systemsThesishttp://purl.org/coar/resource_type/c_db06uuid:10c594d7-1fa5-45f5-bba4-0fefb837aadfGlassesMathematical modelsIsing modelEnglishPolonsky Theses Digitisation Project2001Davison, LL. DavisonSherrington, DSherrington, D<p>In this thesis we study several different models which display glassy behaviour. Firstly, we investigate a simple, purely topological, cellular model for which the Hamiltonian is non-interacting but the dynamics are constrained. We find a non-thermodynamic transition to a glassy phase in which the energy fails to reach the equilibrium value below a characteristic temperature which is dependent on the cooling rate. This model involves activated processes and displays two-step relaxation in both the energy and the correlation functions; the latter also exhibit signs of aging. The relaxation time can be well-fitted at all temperatures by an offset Arrhenius law. Some predictions of Mode-coupling Theory are tested with some agreement found, but no convincing evidence that this description is the most fitting. By defining a suitable response function, we find that the equilibrium Fluctuation-Dissipation Theorem (FDT) is upheld for all but very short waiting-times, despite the fact that the system is not in equilibrium.</p> <p>This topological model is simplified to a hexagonally-based spin model, which also displays glassy behaviour, involves activated processes and exhibits two-step relaxation. This is a consequence of reaction-diffusion processes on two different time-scales, one temperature-independent and the other an exponential function of inverse temperature. We study two versions of this model, one with a single absorbing ground state, and the other with a highly degenerate ground state. These display qualitatively similar but quantitatively distinct macroscopic behaviour, and related but different microscopic behaviour. We extend this work to a square lattice, and find that the geometry of the lattice has a considerable impact on the behaviour, and to three dimensions, which provides support for the reaction-diffusion classification of the early behaviour. We find observable-dependent FDT plots; the observable can be chosen such that FDT is upheld for a region whilst the system is out of equilibrium — this observation is supported by some preliminary results for one-dimensional kinetically-constrained Ising chains.</p> |
| spellingShingle | Glasses Mathematical models Ising model Davison, L L. Davison Glassy behaviour in simple systems |
| title | Glassy behaviour in simple systems |
| title_full | Glassy behaviour in simple systems |
| title_fullStr | Glassy behaviour in simple systems |
| title_full_unstemmed | Glassy behaviour in simple systems |
| title_short | Glassy behaviour in simple systems |
| title_sort | glassy behaviour in simple systems |
| topic | Glasses Mathematical models Ising model |
| work_keys_str_mv | AT davisonl glassybehaviourinsimplesystems AT ldavison glassybehaviourinsimplesystems |