Are chaotic systems really determinitistic?

[SW VandeCarr]

Yeah, like the n-body problem?

I don't understand your question. What does it have to do with what you quoted?

 

[atyy]

The thought sometimes bugged me that if you are uncertain of the period of the Earth in its orbit by say 1 day (only Earth and sun existing) then after 360 years you have no idea where it is. But I don't think of that as a chaotic system. Having said that I now think I see the answer but could someone spell it out? Else not only Newtonian but even Ptolemaic dynamics would be chaotic!

Newtonian systems can be chaotic. The difference between chaotic and non-chaotic Newtonian systems is that in the former the error increases exponentially with time.

 

[SW VandeCarr]

Newtonian systems can be chaotic. The difference between chaotic and non-chaotic Newtonian systems is that in the former the error increases exponentially with time.

Epenguin was postulating an isolated system. Can you have Newtonian chaos if a two-body gravitational system is not perturbed?

 

[denisv]

I don't understand your question. What does it have to do with what you quoted?

It's not true. Many Newtonian systems are chaotic.

A simple example is the double pendulum.

 

[SW VandeCarr]

It's not true. Many Newtonian systems are chaotic.

A simple example is the double pendulum.

Well yes. I did read the that the solar system appears to exhibit chaotic behavior, but the quote refers to the central limit theorem as it applies to measuring the state of reasonably stable systems.

Again, epenguin was referring to an isolated system. I don't know if an isolated gravitational two body system can exhibit chaotic behavior. It's a theoretical question since, as I've been arguing, no system in the real word is isolated. My original post asked if we can really assert that chaos obeys classical deterministic laws in the real world since we are dealing with a system that involves the entire universe.

 

[denisv]

My original post asked if we can really assert that chaos obeys classical deterministic laws in the real world since we are dealing with a system that involves the entire universe.

I'm not sure what you mean. Knowing one point in the phase space uniquely determines the time evolution forever. In that sense, physical systems (classical and quantum without measurement) are deterministic.

 

[atyy]

My original post asked if we can really assert that chaos obeys classical deterministic laws in the real world since we are dealing with a system that involves the entire universe.

http://www.mpipks-dresden.mpg.de/~tisean/

See eg. their comments on surrogate time series.

 

[SW VandeCarr]

http://www.mpipks-dresden.mpg.de/~tisean/

See eg. their comments on surrogate time series.

Thank you for the links. I'll do more work on this.

 

[SW VandeCarr]

I'm not sure what you mean. Knowing one point in the phase space uniquely determines the time evolution forever. In that sense, physical systems (classical and quantum without measurement) are deterministic.

When you say "quantum without measurement" I assume you mean the universe (and any subsystem) exists in all states in superposition. Yes, but we can only observe one state at a time. The wave function evolves deterministically in a space of probabilities and this gets into the whole issue of interpretations of "quantum reality" which I want to avoid in this thread.

Chaos theory deals mostly with macrosystems and I obviously have a problem with the clockwork universe. When you say "knowing one point uniquely determines the time evolution forever" I cannot imagine you're making a statement about the real world. What kind of experimental evidence supports this?

 

[D H]

I don't understand your question. What does it have to do with what you quoted?

The two body problem is not chaotic. The orbit of a single pair of bodies can be described by 5 constant orbital elements plus one time varying element: M=M₀+Mdot*t. The two body problem is linear.

The n-body problem is highly non-linear and most certainly can be chaotic. Think of it this way: Jupiter has already expelled most of the asteroids from the solar system.

 

[SW VandeCarr]

I'm not sure what you mean. Knowing one point in the phase space uniquely determines the time evolution forever. In that sense, physical systems (classical and quantum without measurement) are deterministic.

If this is the model for the world, then we have scientifically embraced fatalism. There is no free will, and the future is completely determined down to finest detail. The quantum measurement problem is an artifact of our inability to relate to this level of scale. All possible states exist in superposition and decoherence protects us from quantum "weirdness".

Now let's consider human history as chaotic. It seems reasonable that small changes in the initial conditions of some historical process can alter the evolution of that process considerably. (ie if Hitler died in the poison gas attack that hospitalized him in 1918.)

Now suppose some whacko with power decides either to go to war or not go to war based on the outcome of a quantum measurement. A spin up photon means war, a spin down photon means peace. In effect, the course of history is determined by a quantum measurement. You can't say that the outcome of the measurement was predetermined.

 

[atyy]

If this is the model for the world, than we have scientifically embraced fatalism. There is no free will, and the future is completely determined down to finest detail.

Yes, that is interesting. Strictly speaking, there is no problem with complete determinism. The problem is in practice scientists talk about designing experiments, and engineers talk about designing structures. What is the true meaning of such statements?

 

[SW VandeCarr]

Yes, that is interesting. Strictly speaking, there is no problem with complete determinism. The problem is in practice scientists talk about designing experiments, and engineers talk about designing structures. What is the true meaning of such statements?

Wait a minute. I just presented a problem with complete determinism. Can you address it?

 

[atyy]

Wait a minute. I just presented a problem with complete determinism. Can you address it?

Maybe the Bohmian interpretation of quantum mechanics. But I don't think there is any consensus whether it applies to relativistic quantum mechanics, and we are still lacking a quantum theory of gravity. However, despite a possible problem with determinism from quantum mechanics, I find it hard to believe that that provides enough "free will" for the common sense interpretations of statements such as "I am going to design this experiment to test that hypothesis." There is a "free will theorem" by Conway and Kochen which is interesting, but I'm not sure what to make of it.

 

[SW VandeCarr]

Maybe the Bohmian interpretation of quantum mechanics. But I don't think there is any consensus whether it applies to relativistic quantum mechanics, and we are still lacking a quantum theory of gravity. However, despite a possible problem with determinism from quantum mechanics, I find it hard to believe that that provides enough "free will" for the common sense interpretations of statements such as "I am going to design this experiment to test that hypothesis." There is a "free will theorem" by Conway and Kochen which is interesting, but I'm not sure what to make of it.

This really is not example of free will. The whacko with power is allowing the outcome of a quantum measurement to decide his course of action.

 

[atyy]

Many-Worlds is also deterministic.

 

[atyy]

This really is not example of free will. The wacko with power is allowing the outcome of a quantum measurement decide his course of action.

What does "allowing" mean?

 

[SW VandeCarr]

Many-Worlds is also deterministic.

Is it? How is it decided which parallel reality you experience. Isn't it random?

 

[SW VandeCarr]

What does "allowing" mean?

The whacko doesn't make a decision other than to follow a course based on the outcome of a random event.

 

[atyy]

Is it? How is it decided which parallel reality you experience. Isn't it random?

My understanding is that every "you" experiences one of every possible reality.

 

[SW VandeCarr]

My understanding is that every "you" experiences one of every possible reality.

Frankly, interpretations of quantum reality don't even rank as theories. "Many Worlds" is inherently untestable and the picture it paints is beyond absurd (although that alone doesn't prove it's wrong). I've described an intrusion of quantum weirdness into the classical world which I think should be addressed directly in terms of the world we actually experience. Even the Schrodinger's Cat example is not particularly consequential since it just involves just one poor cat (no offense to cat lovers). The Whacko with Power example is potentially very consequential and sadly not unrealistic. I admit I don't like the fatalistic clockwork universe but I'd be more willing to accept it (and have more respect for the scientists involved) if my arguments could be dealt with in a more straightforward manner. There are many undesirable aspects to fatalism, not least of which is how we view the responsibility we have for our decisions and actions.

 

[denisv]

Chaos theory deals mostly with macrosystems and I obviously have a problem with the clockwork universe. When you say "knowing one point uniquely determines the time evolution forever" I cannot imagine you're making a statement about the real world. What kind of experimental evidence supports this?

This is how the mathematical models work.

Now let's consider human history as chaotic. It seems reasonable that small changes in the initial conditions of some historical process can alter the evolution of that process considerably. (ie if Hitler died in the poison gas attack that hospitalized him in 1918.)

I don't think anyone knows what happens on extremely large scales like that. It's possible that things simply 'average out'.

Now suppose some whacko with power decides either to go to war or not go to war based on the outcome of a quantum measurement. A spin up photon means war, a spin down photon means peace. In effect, the course of history is determined by a quantum measurement. You can't say that the outcome of the measurement was predetermined.

Highly unlikely situation :-) People who find themselves in such situations are invariably neurotic or otherwise mentally ill and could hardly be expected to understand or rely on quantum physics. As for measurement in quantum physics, that's a tricky subject.

Personally I find indeterminism more troubling than determinism. What meaning does life have if the universe is flipping a coin?

 

[atyy]

Frankly, interpretations of quantum reality don't even rank as theories. "Many Worlds" is inherently untestable and the picture it paints is beyond absurd (although that alone doesn't prove it's wrong). I've described an intrusion of quantum weirdness into the classical world which I think should be addressed directly in terms of the world we actually experience. Even the Schrodinger's Cat example is not particularly consequential since it just involves just one poor cat (no offense to cat lovers). The Whacko with Power example is potentially very consequential and sadly not unrealistic. I admit I don't like the fatalistic clockwork universe but I'd be more willing to accept it (and have more respect for the scientists involved) if my arguments could be dealt with in a more straightforward manner. There are many undesirable aspects to fatalism, not least of which is how we view the responsibility we have for our decisions and actions.

Sure. Actually, I personally like determinism, but the most standard "interpretation" of QM is probabilistic, and Bohmian and MWI are still controversial.

 

[SW VandeCarr]

This is how the mathematical models work.

Highly unlikely situation :-) People who find themselves in such situations are invariably neurotic or otherwise mentally ill and could hardly be expected to understand or rely on quantum physics. As for measurement in quantum physics, that's a tricky subject.

Personally I find indeterminism more troubling than determinism. What meaning does life have if the universe is flipping a coin?

It really doesn't matter what you or I think or like. Human beings have the technical ability to measure quantum states in terms of obtaining a single random outcome and this fact alone allows quantum indeterminacy to affect the "classical" world. My example is one variation of many that could be put forward. We don't even need a human actor. The launch of a missile, or many missiles, could be directly tied electronically to obtaining a particular quantum state. This clearly establishes that the classical world is potentially subject to random, unpredictable (in principle) events which can alter the evolution of a given system (chaotic or stable). I don't think we can escape this conclusion simply by resorting to some interpretation like Many Worlds (which is untestable by definition).

As for how mathematical models work, I believe I've made clear that I don't have a problem with the models provided we accept them as approximations to reality under a specified set of largely artificial assumptions (isolated systems, etc). Most posters who have taken issue with me seem to regard the models as reality and the world as actually deterministic.

 

[SW VandeCarr]

Sure. Actually, I personally like determinism, but the most standard "interpretation" of QM is probabilistic, and Bohmian and MWI are still controversial.

Yes, and I'd come to accept the Copenhagen interpretation as reasonable despite its weirdness. It's like taking a long auto trip. Many routes are possible, but you end up only experiencing one route and that experience is the reality of record. I discuss the "Random Roam" in a book I wrote (see 'About Me' under my user name). The "new determinism" baffles me, given what we've learned from QM. It seems like a throwback to eighteenth century rationalism. It appears Copenhagen is now out of favor. That leaves MW and Bohm's pilot wave. MW is constructed in a way that can't be falsified which means it's metaphysical (not that there's anything wrong with that, it's just not science). I'm less sure about Bohm.

In any case, I think it has been shown (beginning with Schrodinger) that human agency can potentially falsify complete determinism as a description of the world (see my previous post).

 

[denisv]

Human beings have the technical ability to measure quantum states in terms of obtaining a single random outcome and this fact alone allows quantum indeterminacy to affect the "classical" world.

This is why quantum measurement is problematic (to my mind, at least).

As for how mathematical models work, I believe I've made clear that I don't have a problem with the models provided we accept them as approximations to reality under a specified set of largely artificial assumptions (isolated systems, etc). Most posters who have taken issue with me seem to regard the models as reality and the world as actually deterministic.

Whenever you represent a system with fewer degrees of freedom than it has in reality, you are making a (deterministic) approximation. I think this is commonly accepted.

Yes, and I'd come to accept the Copenhagen interpretation as reasonable despite its weirdness.

What about an interpretation of the Copenhagen interpretation? :-)

It's like taking a long auto trip. Many routes are possible, but you end up only experiencing one route and that experience is the reality of record.

It's more like taking a trip where you take a fuzzy-wavy route (defined by a deterministic equation of motion) and then at the end you randomly teleport to one of the possible destinations. Putting it like that, I think most would agree that it's quite a bizarre way of travelling.

The "new determinism" baffles me, given what we've learned from QM. It seems like a throwback to eighteenth century rationalism.

I think the idea that the world is deterministic is a matter of faith in human reasoning. The human concept of randomness is based on what I would call practical knowledge. Eg. a coin flip is random to us, but ultimately there is an underlying deterministic system. It is a belief that the ultimate system underlying reality is deterministic. In practice this system may be incomprehensibly complex and we may be stuck with quantum physics forever, but surely a system exists.

 

[SW VandeCarr]

This is why quantum measurement is problematic (to my mind, at least).

I think the idea that the world is deterministic is a matter of faith in human reasoning. The human concept of randomness is based on what I would call practical knowledge. Eg. a coin flip is random to us, but ultimately there is an underlying deterministic system. It is a belief that the ultimate system underlying reality is deterministic. In practice this system may be incomprehensibly complex and we may be stuck with quantum physics forever, but surely a system exists.

No. The world is not deterministic. The models are deterministic. I think I made the case explicitly. Through human agency, we can alter the evolution of a macroscopic system based on the outcome of a quantum measurement. We can inject the randomness of outcomes of quantum measurements into the classical world. This has nothing to do with free will (which may or may not exist). Schrodinger gave the first example although few have understood that his thought experiment falsified complete determinism as a viable view of the world (unless you want to invalidate QM). QM is deterministic in terms of superposition of quantum states, but in terms of what we actually observe, it is random (according to a strict probability model). The essence of science is based on observation and testing. The outcomes of quantum measurements are random. Yes it's a problem. It's a problem for complete determinism.

 

[denisv]

I would say indeterminism is a much bigger problem for quantum mechanics than quantum mechanics is a problem for determinism.

 

[SW VandeCarr]

I would say indeterminism is a much bigger problem for quantum mechanics than quantum mechanics is a problem for determinism.

Fair enough, but your burden is much greater than mine. You are going to need to show that the outcomes of quantum measurements are not random.

 

[xshell]

It seems to me that the physical definitions are confused with literal meanings of terms.

Deterministic systems are those whose equation of motion is specified. Now within such formalism, the case of positive Lyapunov exponent means that the system is called "chaotic"; in other words systems, whose equation of motion leads to diverging trajectories which differ a 'little' in initial conditions, are said to be chaotic !

Now the end result of 'chaos' is "space filling" (mathematically) for any physical system whose phase space is confined. For ex, water confined in beaker. For our purposes consider only the surface of water. Disturb the surface a little at point X, a set ripples form. Disturb it at X +dX, a different pattern arises. But due to confinement, in the end ripples create crests and troughs at all points of surface without any bias i..e full phase space allowed for excitations is covered. (Incidentally a analogous situations leads to equilibrium in molecular dynamics; here the confining parameter is external pressure or temperature etc.)

But for those systems whose phase space is not confined, the net effect seems to be totally erratic. Imagine water not confined in the above situation i.e. water at the tip of beaker. A small change in the place where u disturb water surface wud lead to completely different behaviour ! (how different i can't say w/o invoking dynamics.)

You might wonder now, which dynamics have i associated with these situations ? I believe one can invoke dynamics of diffusion kind and be able to simulate both these phenomena. In effect we can "write down" equations of motion; which means our dynamics are deterministic. If invoked and simulated properly, am sure the chaotic nature of dynamics can be ascertained..

Thus, chaos is only defined by the property of equation of motion generically i.e its a dynamical property; the end observation might vary acc' to other external constraints imposed on the phase space of the system !