You are on the right track. You have reinvented the basic idea behind the "Reynolds-averaged Navier–Stokes" (RANS) equations, dating back to 1895. The goal is to predict an average velocity field, and this requires modeling unclosed terms (the "turbulence problem" in one respect). RANS and statistical methods in general are the most popular turbulence modeling approaches and much of turbulence theory is based around these ideas. My opinion is that RANS answers are typically what you want, but the models don't work that well. LES (large eddy simulation; applying a low pass filter to NS) is a newer approach that is gaining popularity, and makes physical sense, but is a lot more computationally expensive. And direct simulation (DNS) is an option too, but the computational costs are prohibitive outside of some relatively simple academic problems.

If you are going to describe the fluid motion as probabilities, are you going to assume those probabilities to be independent or not?

An independence assumption really helps with computation. However, considering the `eddies' in turbulence, I'd suppose there is large and complex coupling between the probabilities of flow even for positions that are separated.

You might deal with the coupling, but at a first guess that feels like it requires tracking many branching paths, which would have exponential memory requirements.

Also FTA :

>>> Nonunique Leray solutions would mean that, according to the rules of Navier-Stokes, the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense and implies that the equations aren’t really describing what they’re supposed to describe.

This basically say that from one given starting conditions one only expect one (and only one) outcome. Doesn't this conflate a model with the actual physical reality ?

I'm not sure what you're getting at. Classically we expect that given the same initial conditions, the outcome of a fluids experiment should always be the same. If an equation meant to describe fluid flows doesn't have this property, it is probably a bad model.

Not really. We know that the idea of a "fluid" has to break down eventually (at least due to finite particle sizes, possibly due to discretisations of space or whatever). An equation which describes "fluids" at sufficiently low resolution and breaks down once a resolution is reached where the idea of a "fluid" breaks down as well is then perfectly sufficient and, combined with the idea of a fluid, gives us as good model for reality.

Contrary to what the article claims here:

> the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense

it makes quite a lot of "physical sense" to get two possible outcomes out of one initial state, though we would not expect the Navier-Stokes equations to describe that situation.

The article is simply wrong in arguing that because we expect classical fluids to behave classically and because Navier-Stokes may break down in certain limits, NS may be a bad model. I actually struggle to put together a coherent sentence which comes close to what the article tries to say regarding the relation between "physical sense" and our expectations for the results of Navier-Stokes.

Yeah, I wondered about this too since the typical NS BVP, velocity vector is assumed to be confined to the domain of real numbers[0].

[0] https://warwick.ac.uk/fac/sci/statistics/staff/academic-rese...

   The second is 
   fantastical but mathematically permissible: 
   The water starts still, erupts in the middle
   of the night, then returns to stillness.

Big bang in the middle of the night.

Some quirling around until all matter and energy is equally distributed. What a journey!

Then returns to stillness.

> Also permissible, and also vanishingly unlikely, under quantum theory.

Navier-Stokes has nothing to do with Quantum Mechanics..

You missed the point of their comment.

The responders' point was that Quantum Mechanics is a different framework for modeling physical phenomena which takes a probabilistic framework, and so if fluid were to be modeled in a similar framework, you could work more naturally with these "vanishingly unlikely" events.

Quantum Mechanics is entirely deterministic and linear.

(It only become non-deterministic, when you muck around with collapse of the wave function.)

Yes well the Copenhagen interpretation is the most widely accepted interpretation of Quantum mechanics, so it's not crazy incorrect to equate the two

Navier-Stokes is only incidentally related to real life. It's pure math.

Right, smooth vector fields aren’t actually physically realizable.

Navier Stokes is probably Turing Complete.

Probably, yes. But the gymnastics required to make it Turing complete are unlikely to be in the regime that describes real fluids.

In the sixties some people experimented with fluidic logic gates.

Those probably relied on friction?

No one said it does. But given that statistics based on complex probabilities are effective in one domain, maybe they're worth investigating in another.

No need for quantum mechanics, it's permissible under statistical mechanics. It is incredibly unlikely to happen though (and any way of making it happen necessarily takes a lot of effort).