Why are computer/game physics engines often non-deterministic? - unity3d

Question

Developing with various game physics engines over the years, I've noticed that on the same machine I observe widely different results in physics simulations between runs. Most recently, the Unity engine does this, even though physics are calculated at set intervals of time (FixedUpdate) -- as far as I can determine it should be completely independent of frame-rate.
I've asked this question on game forums before, and was told it was due to chaotic motion: see double pendulum. But, even the double pendulum is deterministic if the starting conditions are exactly controlled, right? On the same machine, shouldn't floating point math behave the same way?
I understand that there are problems with floating point math accuracy, but I understand those problems (as outlined here) to not be problems on the same hardware -- isn't floating point inaccuracy still deterministic? What am I missing?
tl;dr: If running a simulation on the same machine, using the same floating point math(?), shouldn't the simulation be deterministic?
Thank you very much for your time.

Answer 1

Yes, you are correct. A program executed on the same machine will give identical results each time (at least ideally---there might be cosmic rays or other external things that affect memory and what not, but I would say these are not of our concern). All calculations on a computer are deterministic, and so all algorithms of a computer will necessarily be deterministic (which is the reason it's so hard to make random number generators)!
Most likely the randomness you see is implemented in the program with some random number generator, and the seed for the random numbers varies from run to run. Should you start the simulation with the same seed, you will see the same result.
Edit: I'm not familiar with Unity, but doing some more research seems to indicate that the FixedUpdate routine might be the problem.
Except for the audio thread, everything in unity is done in the main thread if you don't explicitly start a thread of your own. FixedUpdate runs on the very same thread, at the same interval as Update, except it makes up for lost time and simulates a fixed time step.
source
If this is the case, and the function itself looks somewhat like:
void physicsUpdate(double currentTime, double lastTime)
{
double deltaT = currentTime - lastTime;
// do physics using `deltaT`
}
Here we will naturally get different behaviour due to deltaT not being same from two different runs. This is determined from what other processes are running in the background, as they could delay the main thread. This function would be called irregularly and you would observe different results from runs. Note that these irregularities will mostly not be due to floating point inprecision, but due to inaccuracies when doing integration. (E.g. velocity is often calculated by v = a*deltaT, which assumes a constant acceleration since last update. This is in general not true).
However, if the function would look like this:
void physicsUpdate(double deltaT)
{
// do physics using `deltaT`
}
Every time you do simulations using this you will always get the exact same result.

Answer 2

I've not got much experience with Unity or its physics simulations, but I've found the following forum post which also links to an article which seems to indicate it's down to precision with the floating point calculations.
As you've mentioned, a lot of people seem to keep rehashing this question!
The forum also links to this blog post which may shed some light on the issue.