Abstract

Suppose you find yourself face-to-face with Navier-Stokes or Young-Mills or a nonlinear PDE or a funky metamaterial or a cloudy day. And you ask yourself, is this thing “turbulent”? What does that even mean? Our goal is to answer this question pedagogically, as a sequence of pencil and paper calculations. First I will explain what is ‘deterministic chaos’ by walking you through its simplest example, the coin toss or Bernoulli map, but reformulated as problem enumerating admissible global solutions on an integer-time lattice. Then I will do the same with the ‘kicked rotor’, the simplest mechanical system that is chaotic. Finally, I will take an infinity of `rotors’ coupled together on a spatial lattice to explain what `chaos’ or `turbulence’ looks like in the spacetime. What emerges is a spacetime which is very much like a big spring mattress that obeys the familiar continuum versions of a harmonic oscillator, the Helmholtz and Poisson equations, but instead of being “springy”, this metamaterial has an unstable rotor at every lattice site, that gives, rather than pushes back, with the theory formulated in terms of Hill determinants and zeta functions. We call this simplest of all chaotic field theories the `spatiotemporal cat’.In the spatiotemporal formulation of turbulence there is no evolution in time, there are only a repertoires of admissible spatiotemporal patterns, or `periodic orbits’, very much as the partition function of the Ising model is a weighted sum formed by enumerating its lattice states. In other words: throw away your integrators, and look for guidance in clouds’ repeating patterns. That’s `turbulence’. And if you don’t know, now you know.No actual cats, graduate or undergraduate, have shown interest in, or were harmed during this research.