A complexity perspective on fluid mechanics: proteinoids as fluid computers

Clay Mathematical Institute, in the year 2000, formulated a list of seven unsolved mathematical problems which might influence and direct the course of the research in the 21st century. Navier-Stokes regularity problem is one of the seven problems and has not been solved till date. These equations govern the motion of the viscous fluids, such as liquids, gases, gels, polymers. The present article is a spotlight on the NSE problem: a review of the past and current efforts to solve it, and our changing perspectives and understanding of these equations. The major debate is to either prove that the solutions to Navier-Stokes equations (NSE) are smooth or to prove that the solutions reach singularity at a finite-time (blowup); given some initial conditions. My talk will discuss the recent developments that are pointing towards that the idea that the fluid equations should be viewed as a "computer program". Given this is the case, the article then argues to see "continuum fluid" from a complexity notion, and thus, as a Turing machine. A case study of proteinoids as solidified gels made from poly(amino acids) based polymers that exhibit oscillatory electrical activity is presented. It has been proposed that proteinoids are capable of performing analog computing as their electrical activity can be converted into a series of Boolean gates. To quantify the complexity of proteinoid ensembles, we measure nine complexity metrics (to name a few: average degrees Degav, maximum number of independent cycles u, average connections per node Connav, resistance reseff, percolation threshold perct ) which shine light on the morphological, functional complexity of the proteinoids, and the information transmission that happens across the undirected graph abstraction of the proteinoid microspheres ensembles. With this work, we hope to provide a complexity toolkit for hardware designers of analog computers to design their systems with the right set of complexity ingredients guided one-to-one by the protocol chosen at the first place. On a more fundamental note, this study also sets forth the need to treat gels, microspheres, and fluidic systems as fundamentally information-theoretic in nature, rather than continuum mechanical, a perspective emerging out from recent program by Terence Tao to treat fluids as Turing-complete (programmable).