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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2302.07326 |
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| _version_ | 1866913889610891264 |
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| author | Gayral, Léo Sablik, Mathieu Taati, Siamak |
| author_facet | Gayral, Léo Sablik, Mathieu Taati, Siamak |
| contents | Chaotic dependence on temperature refers to the phenomenon of divergence of Gibbs measures as the temperature approaches a certain value. Models with chaotic behaviour near zero temperature have multiple ground states, none of which are stable. We study the class of uniformly chaotic models, that is, those in which, as the temperature goes to zero, every choice of Gibbs measures accumulates on the entire set of ground states. We characterise the possible sets of ground states of uniformly chaotic finite-range models up to computable homeomorphisms.
Namely, we show that the set of ground states of every model with finite-range and rational-valued interactions is topologically closed and connected, and belongs to the class $Π_2$ of the arithmetical hierarchy. Conversely, every $Π_2$-computable, topologically closed and connected set of probability measures can be encoded (via a computable homeomorphism) as the set of ground states of a uniformly chaotic two-dimensional model with finite-range rational-valued interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_07326 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models Gayral, Léo Sablik, Mathieu Taati, Siamak Mathematical Physics Statistical Mechanics Computational Complexity Combinatorics Dynamical Systems 82B20, 37D35, 68Q17 (Primary) 68Q04, 68Q87, 37B51, 05B45 (Secondary) Chaotic dependence on temperature refers to the phenomenon of divergence of Gibbs measures as the temperature approaches a certain value. Models with chaotic behaviour near zero temperature have multiple ground states, none of which are stable. We study the class of uniformly chaotic models, that is, those in which, as the temperature goes to zero, every choice of Gibbs measures accumulates on the entire set of ground states. We characterise the possible sets of ground states of uniformly chaotic finite-range models up to computable homeomorphisms. Namely, we show that the set of ground states of every model with finite-range and rational-valued interactions is topologically closed and connected, and belongs to the class $Π_2$ of the arithmetical hierarchy. Conversely, every $Π_2$-computable, topologically closed and connected set of probability measures can be encoded (via a computable homeomorphism) as the set of ground states of a uniformly chaotic two-dimensional model with finite-range rational-valued interactions. |
| title | Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models |
| topic | Mathematical Physics Statistical Mechanics Computational Complexity Combinatorics Dynamical Systems 82B20, 37D35, 68Q17 (Primary) 68Q04, 68Q87, 37B51, 05B45 (Secondary) |
| url | https://arxiv.org/abs/2302.07326 |