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<p>In 1991, Moore (Nonlinearity 4:199–230, 1991) raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao (J Am Math Soc 29(3):601–674, 2016) asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in&#xa0;Cardona et al. (Proc Natl Acad Sci 118(19):e2026818118, 2021) of a “Fluid computer” in dimension 3 that combines techniques in symbolic dynamics with the connection between steady Euler flows and contact geometry unveiled by Etnyre and Ghrist. In addition, we argue that the metric that renders the vector field Beltrami cannot be critical in the Chern-Hamilton sense&#xa0;(Chern and Hamilton in On Riemannian metrics adapted to three-dimensional contact manifolds, Springer, Berlin, 1985). We also sketch the completely different construction for the Euclidean metric in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> as given in&#xa0;Cardona et al. (J Math Pures Appl 169:50–81, 2023). These results reveal the existence of undecidable fluid particle paths. We conclude the article with a list of open problems.</p>

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Towards a Fluid Computer

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月13日

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  • Robert Cardona,
  • Eva Miranda,
  • Daniel Peralta-Salas

摘要

In 1991, Moore (Nonlinearity 4:199–230, 1991) raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao (J Am Math Soc 29(3):601–674, 2016) asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in Cardona et al. (Proc Natl Acad Sci 118(19):e2026818118, 2021) of a “Fluid computer” in dimension 3 that combines techniques in symbolic dynamics with the connection between steady Euler flows and contact geometry unveiled by Etnyre and Ghrist. In addition, we argue that the metric that renders the vector field Beltrami cannot be critical in the Chern-Hamilton sense (Chern and Hamilton in On Riemannian metrics adapted to three-dimensional contact manifolds, Springer, Berlin, 1985). We also sketch the completely different construction for the Euclidean metric in \(\mathbb {R}^3\) R 3 as given in Cardona et al. (J Math Pures Appl 169:50–81, 2023). These results reveal the existence of undecidable fluid particle paths. We conclude the article with a list of open problems.