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  Riddled attraction basins and determinism in classical physics

+ 1 like - 0 dislike

A riddled basin is a “basin of attraction [with] the property that every point in the basin has pieces of another attractor’s basin arbitrarily nearby,” explains the review paper “Fractal structures in nonlinear dynamics” (2009), by J. Aguirre et al.

In other words, a riddled basin is a region of the phase space that can be thought of as a bulky “fat fractal” boundary between different attraction basins. Every neighborhood of a point in a riddled basin, no matter how small, contains points that will eventually reach different attractors. Therefore, no matter how accurate is the specification of the starting point, the attractor that the system will eventually reach is undetermined.

Riddled basins, which have been found in many dissipative systems, “show that totally deterministic systems might present in practice an absolute lack of predictability,” note Aguirre at al. See also the book “Transient Chaos: Complex Dynamics on Finite Time Scales” (2011), by Ying-Cheng Lai and Tamás Tél.

I suspect that the fractal depth of riddled basins might be widespread in real-world, dissipative dynamical systems, and perhaps be the rule rather than the exception. If so, chaotic evolution is really nondeterministic.

Nature “knows” the starting point of the system as an infinitely precise real number. But we can’t know the starting point with infinite precision, and any finitely precise starting point contains the possibility of different outcomes. 

In view of this, does it even make sense to claim that classical physics is deterministic in principle (though nondeterministic in practice)? Or shouldn't we, instead, follow Born and accept that the inevitably finite precision of initial conditions makes classical (non-quantum) physics nondeterministic in principle?

“As a mathematical tool the concept of a real number represented by a nonterminating decimal fraction is exceptionally important and fruitful. As the measure of a physical quantity it is nonsense… concepts which correspond to no conceivable observation should be eliminated from physics… the determinism of classical physics turns out to be an illusion, created by overrating mathematico-logical concepts.” - Max Born's Nobel Prize lecture.

asked May 9, 2018 in Chat by Giulio Prisco (190 points) [ revision history ]
recategorized May 9, 2018 by Arnold Neumaier
Most voted comments show all comments

Every experimentalist knows that his measurements must be sufficiently representative and for that one applies a statistical treatment in order to arrive at a deterministic result. Thus uncertainties and inequalities are an essential part of physics in general and in creating a deterministic picture in CM or elswhere. In other words, a deterministic (simplified) picture in physics must be accompanied with certain inequalities and uncertainties; then there is no temptation to apply a theory outside its region of applicability.

''Nature “knows” the starting point of the system as an infinitely precise real number.'' But not the dynamical equations, as there are always interactions with the worlds outside the system. not accounted for by any dynamical equation for the system alone. This already introduces enough indeterminism for any proper subsystem of the universe. 

Thus for systems different from the whole universe, determinism is always an idealization - just as classical mechanics itself. 

I moved this to chat since the question posed is a philosophical one, not one of theoretical physics. 

@ArnoldNeumaier re "Clearly a question asking ''Does it really make sense to claim...?" is philosophical and invites personal opinions, not physical theory or experiment. Thus there cannot be clear answers, but just discussion. This is what chat is for."

OK, fair enough.

However, I think personal opinions are useful in formulating/interpreting physical theories. Can you name one physical theory that didn't start as a personal opinion (and often a controversial one)?

I don't care about "earning points," but about getting answers (OK, opinions). Chat is not on the front page, so most people don't see chat threads without searching.

I agree that personal opinions can be useful, but on PO, chat is the place for it.  

Anyone interested in chat can simply press the chat button.  

Most recent comments show all comments

@ArnoldNeumaier No objection to that, but my point is that even a simplified classical toy universe regulated by simple mathematical laws can be nondeterministic. This seems to suggest that the universe as a whole might be nondeterministic as well.

@ArnoldNeumaier - this is the second time you move my question to chat because "the question posed is a philosophical one."

While in this case I disagree, it's true that the topics that I am really interested in often fall in the (missing) category "Philosophy of Physics" (or something like that). So how about adding "Philosophy of Physics" to the main Q/A categories? How should I make the proposal?

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