Fluid dynamicist here. If a simulation is too symmetric, one can insert random noise into the initial conditions. I've done this in the past, and it's fine, even physically justified as reality is rarely so perfect. There are measures of the noise that one can match.

How do you ensure the initial conditions are divergence-free after introducing the noise?

Great question. I don't recall precisely what the research code I used then did. The velocity fields for constant mass density incompressible flow need to be divergence-free. But other variables and other situations do not have the same restrictions (though they might need to satisfy some other conditions). The code might have introduced noise in the the temperature field.

I recall that there are algorithms for generating divergence-free random fields, but I'm not particularly familiar with them. In another code I worked with more when I was younger, I recall discussing with a much more senior researcher than myself this issue, and if I recall correctly he said to not worry about the initial condition not satisfying the divergence constraint (the divergence was non-zero but known), as the numerical method will enforce the divergence constraint for all future time steps. Presumably one could modify the algorithm (this was a "pressure projection scheme") to take an arbitrary velocity field along with the desired divergence field and correct the initial velocity field to be divergence free. This may not preserve the desired statistical properties of the noise, however.

Small perturbations in the velocity field are totally fine even if they are not divergence-free. Most fluid dynamics codes used a discrete Poison equation to enforce the conservation of mass (i.e. divergence-free velocity field). Here is some background: https://en.wikipedia.org/wiki/Discrete_Poisson_equation#Appl...

One trick would be to generate a random field and then take its curl. Perhaps there are better ways though.

Yes, numerical error breaks the symmetry. It could be as small as the rounding error due to finite precision, but most likely a few orders of magnitude bigger thanks to errors in the numerical method.

In reality, even if you can manage perfectly symmetric initial and boundary conditions, thermal noise would still cause this symmetry breaking.

Since these systems are physically unstable, the source of initial perturbation is unimportant as long as it is small.

As an addition to my other comment, I've read and heard some fluid dynamics researchers/engineers rely on numerical error to break symmetries and lead to turbulence, etc., but it would be hard to call this physical. I think a better approach would be as I said in my other comment, adding noise to the initial condition. This is particularly problematic for studies of the transition to turbulence, as presumably such a phenomena would be strongly dependent on the details of the accumulation of errors/noise, and relying on numerical errors would essentially be like adding noise without quantifying it.

To be clear, it absolutely is correct to say that this would lead to symmetry breaking. Depending on the numerical method and the computer, it might take a long time, if I recall correctly.

Why would you expect symmetry? Smoke from a real flame does not have symmetry.

I think they're asking how to create the asymmetry and if the imperfections in the calculations are exploited to do so.

Wouldn't that physical analogy be caused by factors unknown to this algorithm though? Air flux, non-homogeneous air, air impurity etc.

Would flames not be symmetrical if all of the above plus everything imaginable like quantum physics, solar activity, dark matter, wtv, are all controlled for?

If we're going to the level of QM, then no. Because QM is inherently a random process. Like radioactive decay is a great way to make true random number generators. The real world has truly random events, and while it bothers many people, physicists don't generally argue about this point (anymore).

Maybe if you had a perfectly homogeneous air density (which you'd never see, even in a really good vacuum and near absolute zero temperature. You can thank QM for that), that constantly stayed homogeneous. Problem in real world conditions is that nothing is perfectly homogeneous (entropy exists), so things will heat up unevenly, certain parts will gain more velocity than others, and just a small offset can create a large change in outcome. Short answer is that these systems are chaotic in nature, so they are not stable.

But that doesn't have anything to do with simulations (except what you are trying to emulate). As far as simulations, numerical accuracy will play a role, but really what you look for is if it is realistic, because the real world has random events. That's more what I was trying to get at. You want something that represents reality, not an overly simplified example that you can't use in a meaningful way. Even with these inaccuracies you can get representative models (they will reflect what happens in a physical experiment). And I say representative, because you aren't going to account for all those factors in a simulation, but you are accurate enough to make extremely effective conclusions. I'll even note that some people will add random noise into their simulations (I don't know if this author did, but numerics can play that role).

> The real world has truly random events, and while it bothers many people..

Replace the word random with unpredictable and it shows why some people are bothered.

To say something is fundamentally unpredictable means that it is immeasurable and not merely complex.

Random is the word we use in physics. It isn't wrong. You're kind of just giving the definition of random. True random. Not pseudo random, which computers give you (and is usually called pseudo random explicitly). And you can measure the events, you just can't PREDICT it with certainty. Though you can predict it stochastically.

> To say something is fundamentally unpredictable means that it is immeasurable.

No, it just means that the measure cannot be predicted in advance.

OTOH, the observer effect can make measurement problematic.

This is true (Heisenberg uncertainty more says you can only get so much resolution in your measurement), but I was staying more general because there are other events that are random. Radioactive decay is completely random and isn't affected by an observer.

That is because you don't actually observe the radioactive decay event; you observe it's byproducts. And the presence of an observer of those byproducts certainly changes their behavior (attenuation/scattering, energy, absorption depending on detection method) as opposed to if there were no observer looking at that byproduct.

I was more referencing the actual event of decay. Sure, the byproducts exhibit the standard quantum effects. the byproducts are particles after all.

And just to be clear, you do not think an observer has to be a conscious being, right? I only ask because pop culture science gives this impression. A photon can be an observer.