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Is it accurate to suggest that in chaos theory, information is in practice lost due to the impossibility of characterizing the system's state with infinite precision, making it unfeasible to run the governing equations backwards in time to find previous states?

Or is it an in-principle information loss due to mathematical requirement of infinite precision?

To clarify, in chaos theory, while it's always possible to run the governing equations backwards in time, the lack of a perfect characterization of the state means we may end up at incorrect previous states, primarily due to the excessive sensitivity of the dynamics to the initial state. When we fail to arrive at the correct previous states, it signifies a loss of information.

Information here means all the details needed to fully specify the state of a physical system.

Edit: I asked this question because I believe the practical loss of information in chaos theory differs from what people typically expect to lose during dissipative processes within deterministic mechanics. In the latter, if one improves the modeling of lossy processes to trace energy conversion or transfer deeply into the microscopic world, then there is no loss and thus no loss of information. However, in chaos theory, losing information is more fundamental as one can never know everything about a state with infinite precision in the mathematical sense. Therefore, we lose information in chaos theory, and we can never retrieve it due to a mathematical barrier. It seems to me like an in-principle information loss. If this is the case, then why are we so concerned with information loss during quantum measurements and black hole evaporation? why not during chaotic evolution?

Omid
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    The question seems framed for classical mechanics (there's no chaos as in sensitivity to initial conditions in quantum mechanics) but then why the tag quantum-information? – lcv Mar 23 '24 at 08:08
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    Define information. – Tobias Fünke Mar 23 '24 at 08:31
  • It's time-symmetric, so neighbouring states may deviate when you run the equations either forwards or backwards in time, beyond some short time span. – PM 2Ring Mar 23 '24 at 08:40
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    I do agree with your description of the inability to reconstruct even an approximation to a more distant past state based on limited information about the present state of a chaotic system. Now I do not exactly understand, what kind of answer you expect. It seems reasonable to me to claim that “information is lost in practice”, but that also seems like a rather semantic statement, not containing any physical information beyond the ones mentioned in your question. So did you have any question regarding the actual physical processes happening or would this commend already suffice as answer? – Zaph Mar 23 '24 at 10:06
  • I added the purpose of my question. Thanks for reminding me @Zaph. – Omid Mar 23 '24 at 15:47
  • Is this useful? Entropy in chaos dynamics. See also the references in this answer. – Quillo Mar 23 '24 at 21:17
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    Voting to reopen. A perfectly clear question with a good answer below. – gandalf61 Mar 24 '24 at 11:47
  • Thank you, everyone, for voting to reopen the question. – Omid Mar 24 '24 at 15:38
  • I still love to hear your thoughts on this question. – Omid Mar 24 '24 at 15:43
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    @Quillo, yes, it was helpful, thanks for your suggestion. – Omid Mar 24 '24 at 15:45

1 Answers1

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In short: Using the OP's terms, information is "lost" in practice and not in principle — but I would contend that there is no information loss at all, since we either never had it to start with or we still 'have it'.

In detail

To obtain the system state at another time point we have to describe the system with a precision that grows exponentially [1] with the time length $\Delta t$. This is impossible to fulfill in practice for long enough $\Delta t$ (and at some point we hit the validity limits of our model or even of classical physics), but in principle infinite precision is only needed for infinitely long time lengths.

As for the connection with information loss, perhaps one could talk instead about "information irrecoverability" (in practice) to refer to the practical impossibility of recovering past states due to the demand of exponentially increasing precision. But this is no information loss, since the information still exists in the system. If we were to define irrecoverable information as lost, then lots of current information about the universe is lost, since we can't access it, without invoking chaos or even time evolution at all — and, without time evolution, information would be simply not there, without a process that destroys it.

[1] Developments using machine learning, see for instance this, can improve chaotic systems predictability by more than an order of magnitude, which is remarkable but doesn't change our basic reasoning here.

stafusa
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    Thanks for the response! The idea of information being irrecoverable totally makes sense! – Omid Mar 23 '24 at 19:16