[civilized]

This raises a question I hadn't thought of before. Real-world fluid flow is ultimately well-modeled by the equations of many-body Newtonian mechanics, right (atoms bumping around)? Are those equations vulnerable to blow-ups?

[wwalker3]

Pretty much any mathematical model of a real phenomenon can have some sort of singularity or discontinuity in it.

If you model atoms as dimensionless points (1), then any kind of force law with the distance between atoms in the denominator can lead to a singularity when that distance is zero. In practice, you write the simulator to disallow this, but it's still there in the equations, you're just ignoring it.

If you model your atoms as finite-sized but incompressible billiard balls, then when they hit each other it's a discontinuity, since they instantly change direction when they collide. These collisions conserve total momentum and energy, but they're unphysical because real physical quantities can't jump from one value to another (in classical physics).

Even if you model your atoms as little rubber balls, the model can still be singular. Linear elasticity (the most common choice) allows you to compress a finite-sized object down to zero size with finite energy, which yields infinite energy density. Again, you'd have to disallow that in the simulator, which is very practical, but not theoretically satisfying.

1) https://en.wikipedia.org/wiki/Molecular_dynamics is the typical method of atomistic simulation.

2) https://en.wikipedia.org/wiki/Linear_elasticity

[civilized]

I'm asking about the actual properties of the equations, not if it's hard to do simulations.

[wwalker3]

It's the equations themselves that are singular. When we write simulators, we usually have to paper over the singularities that are inherent in the math.

For example, if you're simulating charged particles moving around, and you use a force equation F = k q1 q2 / d^2 (1), then when d approaches 0 (i.e. when the distance between particles approaches zero), then the force F goes to infinity.

For atoms, it works the same way. If you use a force law like Lennard-Jones (2), it also has the interatomic distance in the denominator, so the equation has a singularity baked right in.

You could always adopt a more complex force equation that doesn't have a singularity in it. But in practice, it's easier to use a simple but singular equation, and then selectively ignore its bad behavior.

1) https://en.wikipedia.org/wiki/Coulomb%27s_law

2) https://en.wikipedia.org/wiki/Interatomic_potential

[civilized]

The presence of a singularity in the force doesn't mean it will cause a blow up in the solution. Two positively charged point particles interacting electrostatically can be shot at each other at any angle or speed and blowup will never occur.

[throwaway81523]

There are all kinds of blow-ups in Newtonian mechanics and in other equations of physics. The singularity at the center of a black hole in general relativity is a famous example. The ultraviolet catastrophe in classical thermodynamics was another. The presumption is that blow-ups in an equation indicate a mismatch between the equation and the true physical world, telling physicists to look for better theories, whose equations don't blow up. For the ultraviolet catastrophe, the mystery was solved through the discovery of quantum mechanics. For GR, it is still unsolved, and the solution is expected to come from a theory of quantum gravity that hasn't yet been invented, but is the target of tons of research.

Here's a cool expository article about blow-ups in classical mechanics and elsewhere: https://arxiv.org/abs/1609.01421

[wwalker3]

That's a great article! 1) I'd seen a lot of other cool stuff from the same author over the years (see https://math.ucr.edu/home/baez/README.html), but had somehow missed this one.

1) "Struggles with the Continuum" by John Baez

[vba616]

This makes me think of:

https://en.wikipedia.org/wiki/Sonoluminescence

I would think that nothing in reality is infinite, but allegedly sound waves collapsing bubbles in a fluid can cause a very small amount of plasma to become hotter than the sun and emit light. Some controversial research claims it might even be possible to create atomic fusion this way.