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

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

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.

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