# Non-linear Differential Equations and Psuedo-randomness

I was thinking about the non-linear Navier-Stokes equation this morning and was briefly browsing a text on the subject. I'm aware that one popular approach to dealing with turbulence is to take averages and look at correlators (which, in turn, can be related to field theory.) Now, one thing which strikes me as odd about this statistical approach to dealing with turbulence is that the Navier-Stokes equation is fully deterministic; given some set of initial and boundary conditions, the entire time evolution of the system is determined.

So, I have a question about the Navier-Stokes equation: is it known that the solutions are psuedo-random for high Reynold's number? Of course, one can extend this more generally to dynamical systems with positive Lyapunov exponents -- are the solutions to such systems (e.g., chaotic differential equations) known to be psuedo-random? If so, it would appear to be natural to attack them statistically since they would pass tests for randomness.

## Answers and Replies

Many solutions having a global attractor in 3D are pseudo-random. Howver, I haven't heard of a general theorem.

Interesting. Do you have any examples? I'd like to know more about this subject.

Check the Lorenz attractor ,for instance.
There are a few theorems connecting ergodicity & randomness, but the general case of the Navier-Stokes equation is monstrous. It's not known if a general
solutions exists ( a million dollar problem) , let alone the contingent property of pseudorandomness.

Yes, I'm aware of the related Millenium Problem. At any rate, does any one have any information connecting ergodicity and psuedo-randomness for other systems? That seems intriguing to me.