Towards a Fluid computer (2405.20999v1)

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.

• The paper demonstrates that Euler flows in contact geometry can simulate Turing machines, establishing a framework for universal computation in fluid systems.
• It reveals that complex hydrodynamic behaviors lead to undecidable fluid particle paths, challenging traditional computational predictability.
• The study introduces innovative construction techniques using Beltrami fields and Reeb flows to achieve Turing complete solutions in both Riemannian spheres and Euclidean space.

Overview of "Towards a Fluid Computer"

The paper "Towards a Fluid Computer," authored by Robert Cardona, Eva Miranda, and Daniel Peralta-Salas, explores the intriguing intersection between hydrodynamics and computation. Specifically, it investigates whether hydrodynamics can perform universal computation akin to a Turing machine. This investigation builds upon foundational work by Moore and seeks to address the undecidability and computational complexity inherent in physical systems.

The central focus of this paper is the construction of a Turing complete system using ideal fluid dynamics. The authors leverage the mathematical intricacies of the Euler equations, particularly through the lens of Beltrami fields and Reeb flows in contact geometry, to demonstrate how a system of this nature can simulate Turing machines. In essence, the research substantiates that certain fluid particle paths are undecidable, which has profound implications for understanding the computational aspects of physical systems.

Highlights and Claims

1. Euler Equations and Contact Geometry: The paper elucidates the relationship between Euler flows and contact geometry. By referencing techniques from symbolic dynamics and the achievements of Etnyre and Ghrist, the authors offer constructions of stationary solutions to Euler equations on a 3-dimensional Riemannian sphere capable of simulating a Turing machine.
2. Undecidability in Fluid Dynamics: Highlighting the undecidability of fluid particle paths, the paper suggests that fluid systems can exhibit highly complex behaviors with computational and algorithmic implications. The presence of undecidable trajectories implies that certain long-term behaviors of these fluid systems cannot be computed.
3. Construction Methodologies: Two significant construction techniques are discussed. The first method utilizes Euler flows in a contact-geometric context, while the second establishes Turing complete solutions in Euclidean space—a groundbreaking achievement that extends the capacity for computational simulation to classical physical environments.
4. Turing Complete Systems: The notion of Turing completeness is extended to dynamical systems. The paper details how specific stationary solutions to the Euler equations can essentially act as computational systems, reinforcing the conceptual potential of hydrodynamics to perform computations.

Implications and Future Directions

The results presented in this paper underscore a need to reevaluate certain aspects of fluid dynamics through the computational lens. They raise important questions about the nature of physical laws and their ability to perform computations that could be broadly applicable in computational complexity theory.

The constructions provided open up avenues of research into the computational capabilities of other physical phenomena, such as the n-body problem in celestial mechanics. By demonstrating the undecidability inherent in these systems, this work pushes the boundaries of what is understood by Turing completeness in natural systems.

Future efforts may concentrate on refining these mathematical models and constructions, possibly examining their applicability to solutions of the Navier-Stokes equations. Furthermore, the pursuit of robust Turing complete systems within fixed Riemannian structures or other geometrically constrained environments would present a substantial advancement in the field.

In summary, "Towards a Fluid Computer" makes a significant scholarly contribution to the understanding of fluid dynamics within the broader context of computational theory. It engages with classical mathematical questions and presents insightful connections between hydrodynamics and computation, offering a fertile ground for theoretical exploration and practical applications.