The memory of persistence

Now that my school year is winding down somewhat, I have been turning back to research issues. One project that has been moving slowly for me is on so-called “persistent patterns”.

These arise in situations where you’ve got chaotic dynamics interacting with some sort of noise or diffusive behavior, with applications ranging from chemical reactors to ocean modeling. Think about putting cream in coffee: As you stir, you get tendrils growing out of the original blob, and combined with the diffusion, this results in a homogeneous mixture very soon. You would think that if the stirring dynamics was chaotic, the mixing would go even more efficiently and quickly — and this is often true.

However,  work over the last decade or so, both theoretical and experimental (see the superb work by Jerry Gollub’s group, for example), shows that it is not always true: In certain incompressible time-periodic fluid flows long-lived patterns emerge. Once these patterns emerge, the mixing process is completely determined by the rates imposed by the slowest decaying structures. Basically, these patterns show up that hang around effectively forever.

Getting a little technical, these persistent regions of high concentration of the passive scalar have been shown to be associated with the stable and unstable manifolds of the underlying chaotic dynamics. But we are far from figuring out the conditions for their emergence and other details about their properties.

I’ve been trying to understand this on and off for years, and there’s a current preprint with my friends Bala Sundaram and Drew Poje where I think we’ve nailed down some critical issues. Almost. Sigh.

Why would someone interested in quantum mechanics and the quantum-classical transition and decoherence care about fluid mixing? It turns out that the behavior of these fluids in real space is identical to that of the phase-space behavior of classical probability densities in Hamiltonian systems with added noise. So understanding the behavior of these fluid dynamics systems is a way of building intuition about the classical limit of the quantum systems I have been thinking about for years. Which is how I got into this problem. And it’s an excellent mathematical physics problem in its own right — with experimental tests of claims available through ‘table-top’ experiments. Wonderful.

Now if we could only finish the last couple of things we need to …