Three Body Problem (and Chaos Theory, Determinism and Prigogine)

[nonequilibrium]

Hello, I've used the search button but the topic I found on it didn't quite answer my questions, so here is my go at it: (I have done one year of maths at college, to know my level in case it's important for formulating your answer)

So I've started reading "The End of Certainty" by Prigogine dealing with the enthralling area of non-equilibrium Thermodynamics, coming to the discussion of determinism and chaos theory.

Is the three body problem important for two things? As I understand it:

• No Exact Method Known: but it can be approximated numerically -- am I thus right to assume it has a solution, in the way that any given, exact, mathematical starting point will result in only one possible evolution?
• Chaotic Behaviour: in general chaos emerges, and this could mean two things (I don't know which, maybe both, in relation to each other)
  1. in the Approximation: you can approximate the exact solution numerically, but the approximation and the exact solution always diverge for $$t \to \infty$$ (as an analogy I'm thinking of any finite polynomial approximation (and thus unbounded) to the bounded sine function)
  2. in the Starting Points: two very near starting points can have very different evolutions; to make this exact I'd try to say "there is a certain number M so that for all ensembles of starting points (in the phase space), there will always be at least two starting points in that (small) ensemble with their evolutions in phase space, for $$t \to \infty$$, a greater distance apart than M" (it's just an attempt to grasp it, don't shoot me, not claiming this is correct).

And then I also wonder: why is this of such importance in the discussion of determinism?

Is it that in making the jump to physics, one has to rely on measurements to find out the starting point, and we know (practically in classical mechanics, theoretically in quantum mechanics) that the 'error' in measurement is never non-zero; this in combination to the fact it's a chaotic system and we have no access to the analytical solution, this system becomes indeterminate (practically in classical mechanics, theoretically in quantum mechanics)?

Thank you,
mr. vodka

 

[Pythagorean]

A) I don't think the importance pertains to the three body problem particularly as much as the >2 problem:

The N-body problem—that of describing the motion of
a system of N particles interacting through specified
forces—is one of the oldest problems in physics. Even
now it continues to be a problem in fields ranging from
nuclear physics to stellar evolution and cosmology.

from
http://www.math.ucdavis.edu/~svenbac/Chaos.pdf

B) Chaos is sensitivity to initial conditions (your 2).C) It's important to determinism because it demonstrates that we can have a rich set of dynamics in which it appears that the universe is stochastic, while it is actually deterministic. So a determinist would have a defense against criticisms about the apparent randomness of nature. What you're saying is valid too.

By QM in theory, I'm guessing you mean the UP. Interesting thought.

 

[alxm]

No Exact Method Known: but it can be approximated numerically -- am I thus right to assume it has a solution, in the way that any given, exact, mathematical starting point will result in only one possible evolution?

The specific case of the three-body problem was shown by Sundman to have a power series solution. It's an extremely slowly converging series. Which illustrates an easily-overlook aspect of analytical solutions - they are exact, but only in principle. It's not necessarily the case that the analytical solution can be calculated to a given degree of accuracy faster than a numerical solution can. So an analytical solution isn't necessarily of any practical use. (This is the case with Sundman's theorem)

Chaotic Behaviour: in general chaos emerges, and this could mean two things (I don't know which, maybe both, in relation to each other)

Chaotic behavior usually means an exponential dependence on initial conditions.

And then I also wonder: why is this of such importance in the discussion of determinism?

Well I think you've run into a popular misconception here, which is the equating of 'predictable' and 'deterministic'. A deterministic theory isn't necessarily predictable, even in principle, because there can be in-principle limitations on how accurately you may know the initial conditions of the system. (deBB interpretation of QM falling into this category)

Quantum mechanical systems aren't usually chaotic though (e.g. the QM three-body problem, the Helium atom isn't), and classical chaos is largely an 'emergent phenomenon'.

 

[Pythagorean]

Quantum mechanical systems aren't usually chaotic though (e.g. the QM three-body problem, the Helium atom isn't), and classical chaos is largely an 'emergent phenomenon'.

Though it's an interesting question how the GUP ultimately contributes to chaos in classical systems.

 

[nonequilibrium]

Thank you for the replies. (As a small note: I do not know what the abbreviations deBB and GUP stand for.)

I understand determinism doesn't imply you actually have to be able to know the full evolution, but simply that the evolution is determined.

So as I get: chaos doesn't really manifest itself in QM (H-atom is non-chaotic), but it has inherent randomness (by general consensus, as I believe?); chaos does manifest itself in CM (classical), but it has no randomness. This seems to result in the fact that QM is indeterministic (if radioactive decay, for example, is truly random), and that CM is deterministic. But then in this youtube link: Prigogine quotes the president of the international union of pure and applied mechanics having "apologized for his colleagues having propogated for 3 centuries that Newtonian systems are deterministic". What could this refer to? Newtonian mechanics is not inherently deterministic?

 

[alxm]

Thank you for the replies. (As a small note: I do not know what the abbreviations deBB and GUP stand for.)

deBB is the de Broglie-Bohm interpretation of QM, which is deterministic. It is, however, a minority position. I'm not sure what Pythagorean is referring to with GUP. Generalized Uncertainty Principle? Or a typo for the more common "HUP"? (Heisenberg U.P.)

So as I get: chaos doesn't really manifest itself in QM (H-atom is non-chaotic), but it has inherent randomness (by general consensus, as I believe?); chaos does manifest itself in CM (classical), but it has no randomness. This seems to result in the fact that QM is indeterministic (if radioactive decay, for example, is truly random), and that CM is deterministic.

Essentially yes. I believed (and it was a previously long-held position) that quantum mechanical systems cannot be chaotic. But I was recently corrected on this. I'm still not sure if there are any real examples of it, though.

Anyway, the point is that classical mechanics is the limiting case of QM as $$\hbar \rightarrow 0$$, so in the modern viewpoint, CM is an approximation of QM which happens to be deterministic as well. So if you believe that QM is fundamentally indeterministic, as the majority do (and Prigogine certainly does), then CM is also fundamentally indeterministic.

 

[Galap]

This is important in the discussion of determinism in that it demonstrates that deterministic systems can behave chaotically and sometimes can't have an analytic solution. The important part there is the part where such systems are deterministic.

It would be easy to say the chaotic nature and impredictabilty of such things is due to nondeterminism, whereas it is merely due to a limitation in our mathematics and our use of it as a tool to describe reality.

 

[Pythagorean]

GUP = generalized uncertainty principal

Quantum chaos would have to be defined slightly differently than classical chaos because sensitivity to initial conditions doesn't really work with GUP. But the study of quantum chaos also entails investigating what quantum processes give rise to classical chaos.

 

[sophiecentaur]

Would it be true to say that we could do without 'randomness' if the basic model (sub - quantum, if you like) gave chaotic outcomes? This could deal with Einstein's problem with 'God' playing dice.