Computability and Beltrami fields in Euclidean space
In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solution...
| Autores: | , , |
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| Tipo de recurso: | informe técnico |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/365660 |
| Acceso en línea: | https://hdl.handle.net/2117/365660 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Partial Computational Complexity Mathematical Physics Dynamical Systems Equacions en derivades parcials Classificació AMS::35 Partial differential equations::35Q Equations of mathematical physics and other areas of application Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics |
| Sumario: | In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of R3, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on T3 (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) arbitrarily high computational complexity. We also quantify the energetic cost for a Beltrami field on T3 to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability 1. Finally, our proof also yields Turing complete flows and maps on S2 with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity. |
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