>it's all deterministic and there is no opposing player making choices
It's not deterministic, it's chaotic. That is the nature of the N-body problem. We can only approximate trajectories in such a system using numerical methods, within a certain margin of error. In principle, the object is gravitationally interacting with everything else in the solar system. But for the most part, most interactions are negligible and could be ignored (eg, other small objects far away), except of the large bodies. But there are many unknowns (as stated before), where initial conditions will affect the outcome of the trajectory simulation, and errors will certainly amplify over time. I'm guessing Monte Carlo is used to "fuzz" the simulations with randomised initial conditions to account for the range of unknowns, and see what the outcome is under these different scenarios.
Chaotic doesn't mean non-deterministic, it just means that small changes in initial conditions result in a large change in the trajectory. The system itself can be both chaotic and deterministic.
It's also a reasonable question to ask, because the simulations are deterministic. It's just that because the system is also chaotic and there's noise in the measurement, that can result in a large spread of deterministic trajectory simulations.
It's only deterministic in the sense of the mathematical constructs that models the system, like differential equations that drive the simulations at each finite time step. But the information or the state which the simulation is applied on is always chaotic. That is because delta at each time step is an approximation with some error. It's impossible to make the state in the system behave deterministically, because that requires time deltas to approach to zero (or infinite amount of infinitely small differential steps).
Energy drift doesn't make the system non-deterministic, it just means that the time evolution has some error. The error is still deterministic. If you simulate a deterministic but chaotic system like n-body orbitals with a non-symplectic integrator, you'll always get the same result for the same initial conditions. The drift created by the finite timestep will also be the same.
It’s the error with the ground truth that you can’t predict. Otherwise you would just be able to cancel it out. You can only predict probability distributions..
If you're saying that it's the uncertainty in the initial measurement, then we're in agreement. If the initial measurement were perfect, the only source of error would be the finite timestep. N-body simulation itself is deterministic, and so the only source of randomness is our uncertainty about the object's true mass, size, shape, position, velocity, etc.
The N-body _reality_ _might_ be deterministic. The N-body simulation using digital computers will technically still introduce errors because of the time steps even if you had perfect knowledge of initial conditions.
The errors are deterministic. Determinism has nothing to do with the existence of errors, it's about uncertainty. They're different things. A system that is deterministic will produce the same results every time given the same initial conditions. If there are numerical errors, they will be identical for each run. A non-deterministic system will give you different results every time given the same initial conditions, with some variance. You can still have numerical errors in such a system.
Ironically, reality probably isn't deterministic. It definitely isn't at small scales (e.g. radioactive decay). If it's non-deterministic at a macro scale, the effect is small enough that we don't see it.
That’s the point, reality isn’t deterministic,so you can’t really use deterministic math to describe it. That’s just an approximation, regardless of errors in the simulation. That’s also why you run Montecarlo simulations, not to even out simulation errors, but to compute as many probable outcomes as possible and then have a probability distribution that represents your best bet at guessing the non deterministic reality that you are trying to predict. If you “run” reality twice your not gonna get the same result
We don't know the configutation it's in precisely. We don't know the initial conditions. Small unobservable differences will lead to large difference in outcome. That's the chaotic part.
I get that. I'm pointing out that these are separate factors. Chaotic does not imply non-deterministic, and vice versa. The only source of randomness here is the uncertainty in the observation of the object, because (as you point out) multiple combinations of parameters could produce the same observation, and each one will have a different trajectory. The randomness doesn't come from the chaotic nature of the system, it comes from noise in our measurements. It also doesn't (as other posts are claiming) come from energy drift in the simulation, because that's also deterministic.
It's not deterministic, it's chaotic. That is the nature of the N-body problem. We can only approximate trajectories in such a system using numerical methods, within a certain margin of error. In principle, the object is gravitationally interacting with everything else in the solar system. But for the most part, most interactions are negligible and could be ignored (eg, other small objects far away), except of the large bodies. But there are many unknowns (as stated before), where initial conditions will affect the outcome of the trajectory simulation, and errors will certainly amplify over time. I'm guessing Monte Carlo is used to "fuzz" the simulations with randomised initial conditions to account for the range of unknowns, and see what the outcome is under these different scenarios.