Good piece

Posted Nov 20, 2012 22:41 UTC (Tue) by man_ls (guest, #15091)
In reply to: Good piece by wolfgang
Parent article: LCE: Don't play dice with random numbers

Yes, my examples are mostly based on statistical mechanics, which provides a nice framework to generate randomness. But I disagree about this part:

Systems may /seem/ random, but this randomness is only epistemological, caused by insufficient knowledge about the (initial conditions of the) system.

In fact the second law of thermodynamics (which incidentally was postulated before the first) ensures that knowledge about the system will degrade in time, so even if perfect information is held at the start, any system will quickly degrade into random patterns. It is not an artifact of our limitations, but an essential principle of nature. We will never be able to predict the minute variations in thermal noise, no matter how much we know about the system.

As for macroscopic randomness, the humble three-body problem in gravitation generates a chaotic system using only classical equations: minor deviations cause major, unpredictable changes in the system.

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Good piece

Posted Nov 20, 2012 23:10 UTC (Tue) by wolfgang (subscriber, #5382) [Link] (5 responses)

> In fact the second law of thermodynamics (which incidentally was
> postulated before the first) ensures that knowledge about the system will
> degrade in time, so even if perfect information is held at the start, any
> system will quickly degrade into random patterns. It is not an artifact of
> our limitations, but an essential principle of nature. We will never be
> able to predict the minute variations in thermal noise, no matter how much
> we know about the system.
Thermodynamics implies that a reservoir is involved in the modelling of a physical process, and that this reservoir is only treated effectively (any degrading system implies such a reservoir; otherwise, there would be nothing it could degrade into). If all parts of the system were treated with the dynamical laws of classical mechanics, it would not be necessary to fall back to a thermodynamic approximation. There would also be no reservoir.

So using a reservoir to model a physical system amounts to a lack of knowledge, which in turn causes the perceived randomness.

> As for macroscopic randomness, the humble three-body problem in
> gravitation generates a chaotic system using only classical equations:
> minor deviations cause major, unpredictable changes in the system.
A chaotic system is by its very definition a deterministic system. Initially close states may diverge arbitrarily far by temporal propagation (loosely spearking; things become more involved when the associated phase space volume can shrink), but the individual trajectories are still governed by deterministic dynamics.
They are very well predictable. If the initial conditions could be determined with infinite accuracy -- which, in the framework of classical mechanics, is theoretically possible -- there is no randomness involved.

But admittedly, these considerations are for the /really/ paranoid. Solving coupled dynamical equations of 10**23 or so particles certainly presents a /tiny/ practical problem ;)

Perfect information

Posted Nov 21, 2012 9:08 UTC (Wed) by man_ls (guest, #15091) [Link]

The second law of thermodynamics is not a consequence of reservoirs; they are just an artifact introduced to model and simplify the outside world, since the second law works just as well without them.

It goes a bit further than that: perfect information about any system is unattainable in classical mechanics. Infinite bits would be needed just to keep the state of a single particle, let alone 10^23 of them, let alone placing them in a non-linear system, let alone computing anything beyond a perfect gas. No matter how paranoid you are, modeling a real world gas particle by particle is not feasible.

In fact, a modern information-theoretic formulation of the second law would say that "to extract one bit of information from a system you have to reduce its entropy in one bit", which explains why perfect information about systems cannot be achieved. Many people believe that only quantum mechanics say that "observation modifies the system being observed", but it is not so.

To find a formulation to which both of us can agree, we might say that "classical mechanics has an element of randomness unless we have perfect information about a system; perfect information cannot be achieved; therefore classical mechanics is random in nature".

Good piece

Posted Nov 25, 2012 5:58 UTC (Sun) by rgmoore (✭ supporter ✭, #75) [Link] (3 responses)

A chaotic system is by its very definition a deterministic system. Initially close states may diverge arbitrarily far by temporal propagation (loosely spearking; things become more involved when the associated phase space volume can shrink), but the individual trajectories are still governed by deterministic dynamics. They are very well predictable. If the initial conditions could be determined with infinite accuracy -- which, in the framework of classical mechanics, is theoretically possible -- there is no randomness involved.

Sure, but that doesn't describe the real world. The real world exhibits quantum behaviors, and classical mechanics is just a simplifying assumption. We can't know the exact initial position and momentum for every particle in a system; our knowledge is limited by the Uncertainty Principle. Even if we could somehow bypass the Uncertainty Principle and determine objects' initial states perfectly, it wouldn't necessarily get us anything. There are other quantum effects that are truly random, like spontaneous emission of infrared photons from molecules that are in excited vibrational states. As long as a system is chaotic by classical mechanics, those real world quantum effects will guarantee that it is truly unpredictable.

Good piece

Posted Nov 27, 2012 0:12 UTC (Tue) by wolfgang (subscriber, #5382) [Link]

Precisely! I'm absolutely not arguing that it is not possible to generate randomness, and it is actually much simpler for all practical purposes than in theory. The point is that it is most important to state under which assumptions, especially under the assumed validity of which theory, one is trying to generate randomness. For instance, if you subscribe to the Bohmian interpretation of quantum mechanics, you're back to the same trouble as with classical mechanics, whereas you can safely relax with the Kopenhagen interpretation. And even if you assume the latter, a careful analysis is still required -- for instance, an enemy that also happens to be an experimental god could be entangled with the atom-photon system you mentioned in an apt way, and thus obtain perfect information about your measured state.

Uncertainty principle

Posted Nov 28, 2012 12:45 UTC (Wed) by tialaramex (subscriber, #21167) [Link] (1 responses)

It doesn't mean anything to "bypass the Uncertainty Principle".

The principle is often misunderstood as a limitation on our observations of a system that really has properties like "exact momentum" and "precise location" - just like the observer effect in classical mechanics. Indeed it is sometimes explained this way in high school or in pop science books. That's not what's going on! The Uncertainty Principle actually says that these properties _do not exist_ not that we have some trouble measuring them. We can do experiments which prove that either an electron does not actually _have_ a specific position when its momentum is known or else that position is somehow a hidden property of the entire universe and not amenable to our pitiable attempts to discover it in the locale of the actual electron. The Uncertainty Principle says that the former is the more plausible explanation (and certainly the only one that's consistent with the remainder of our understanding about how the universe "works").

Uncertainty principle

Posted Nov 29, 2012 18:02 UTC (Thu) by davidescott (guest, #58580) [Link]

AFAIK no experiment has shown that a non-local hidden variable theory is inconsistent with experimental evidence.

For a descriptive theory I would personally prefer determinism and non-locality. For a predictive theory non-determinism and locality are clearly better.