Towards a Fluid computer

TL;DR

This paper reviews the construction of a 3D fluid computer that demonstrates undecidable fluid particle paths, combining symbolic dynamics and contact geometry, and discusses the implications for fluid computation and open problems.

Contribution

It presents a novel construction of a fluid computer in three dimensions that exhibits undecidable particle paths, linking fluid dynamics with computational theory.

Findings

Existence of undecidable fluid particle paths.

Construction of a 3D fluid computer using symbolic dynamics.

Discussion of metric properties affecting fluid computability.

Abstract

In 1991, Moore [20] raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao [25] asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in [8] 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 [9]. We also sketch the completely different construction for the Euclidean metric in $\mathbb R^3$ as given in [7]. These results reveal the existence of undecidable fluid particle paths. We conclude the article with a list of open problems.