# Relationship between physical scale and certainty

Partially motivated by this question, I get the impression that it is generally more difficult to make accurate statistical predictions in Physics about "the small" (microscopic phenomena) than "the big" (macroscopic phenomena).

Why? Do we know what explains this relationship between scale and uncertainty?

For example, not only do we seem to have different theories for the small vs the big, they are also often different in what type of predictions they make. For example, classical mechanics is deterministic and quantum mechanics is inherently probabilistic.

One could speculate that predicting physical observations always involves uncertainty, but it was just easier to make more accurate (and thus effectively deterministic) predictions about "the big" than about "the small" because the underlying laws governing the small are intrinsically random, so we naturally ended up developing more probabilistic theories for the small.

But is there a statistical, perhaps simpler mathematical explanation for this difference in "prediction hardness"? (e.g. an aggregation of microscopic random effects that may cancel each other so we can more easily make accurate large-scale predictions). Or even simpler, is this a false premise and we actually have the ability to make predictions with the same level of certainty about the small and the big?

• Are you familiar with the uncertainty principle dominated by the "small" $\hbar$? "Big" systems with $S\gg \hbar$ are not limited by it and appear less riven by randomness in their crude features. Jul 18, 2020 at 20:10
• The probabilistic nature of quantum mechanics is not about precision or accuracy. I would largely just repeat the answer I just wrote to your previous question when trying to answer this one. Jul 18, 2020 at 20:12
• @CosmasZachos I'm not. What's the name of that principle?
– Josh
Jul 18, 2020 at 20:14
• Thanks @ACuriousMind That's a good answer, although I don't quite follow how that would also answer this Q. How would you make the connection that the lack of local hidden variables in the small explains lower uncertainty in the big. What happens as we "zoom out"?
– Josh
Jul 18, 2020 at 20:20
• I meant that this question focuses largely on the notion of "accuracy". As I say in the other answer, it's not about accuracy or our ability to make statistical predictions. If you're trying to ask why the macroscopic world looks "less quantum" than the microscopic world, that's a) only debatably true (e.g. the fusion that sustains the sun is a non-classical feature) and b) has different explanations depending on what exactly you're looking at as "macroscopic" (e.g. actions much larger than $\hbar$ vs. extremely many particles, etc...) Jul 18, 2020 at 20:26

I think the idea that uncertanty in QM is because it deals with very small objects is related to the need of some interaction to know where the object is.

In classical mechanics, the scattering of sunlight on the moon surface is irrelevant to its orbit, while it is very important for us to observe where it is.

Of course it is not the case for an electron, where the change of momentum due to the scattering of EM radiation is not so negligible.

But there are also other effects as trying to know the spin (1/2 ou -1/2) of a particle at an axis with an angle to the orientation where its spin are known with 100% of certainty.

As far I see, there is nothing we can imagine that could let us know more than the probability rule, function of the angle.

The probability rule works, but not due to our lack of knowledge as in the case of a coin. There is nothing more to be known, and still we know the probability.

It is not related to scattering, so it is not related to size, at least in principle.

Macroscopic objects are only immune to randomness in non-dynamic circumstances. Even our solar system is inherently unpredictable further than 5 million years. And this is the case even if we start with perfect knowledge about the system.

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