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Old 2019-04-25, 05:59   #1
jasong
 
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Default Ordered disorder

Sorry if this is deemed to be deserving of Bloggorhea, but anyway...

How do you define order so that you could make something as disordered as possible? For example, take the numbers 0-9, what order would be the most disordered, and can you explain it, or generate it, deterministically? Could prime numbers ultimately be explained as a disorder of a particular type? What abouts binary, what's the most disordered binary series you can come up with, and what the rules to generate it again?

If you guys can recommend Kindle books or web pages about this, that would be awesome.
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Old 2019-04-25, 06:13   #2
retina
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I'm unaware of any formal definition of disorder.

But a reasonable proxy could be randomness. And from there you could measure compressibility (as in zip or 7z etc.). Generally the more randomness you have the less compressible is the input.

So just run your data through 7z and examine the file size. A larger ratio of output size to input size could be a measure of disorder.

For the digits 0-9 appearing only once, a compressor probably wouldn't be a good measure, you'd need more data to get a good approximation.
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Old 2019-04-25, 07:55   #3
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There is a concept known as information entropy
Maybe that is what you are looking for?
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Old 2019-04-25, 09:14   #4
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Also its cousin Kolmogorov complexity
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Old 2019-04-25, 15:27   #5
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If it's a physical system, I'd go with entropy.

If it's a sequence of numbers, say decimal or binary digits, I would think instead of (apparent) randomness. For binary digits, a "rule" like "flip a coin, 1 for heads, 0 for tails" might do. There are statistical tests for (apparent) randomness.

Sequences of digits may be viewed as representing real numbers. There are what are called normal numbers, whose (e.g. decimal) expansions contain all digits and all blocks of 2, 3, or any finite number of digits in the correct proportions. AFAIK there is no good way to test known numbers (like pi) for being "normal" (to base ten). It is however very easy to show that "almost all" numbers are normal to every base.

Another notion would seem to be related to the "Kolmogorov complexity" mentioned above. namely, can you compute the n-th digit of the number without computing all the previous digits? For the decimal expansion of pi, the answer is AFAIK not known, but in binary the answer is "yes."

However, if memory serves, it makes no sense to say, "This number is random."
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Old 2019-04-26, 07:15   #6
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Defining a "most disordered thing" (to keep the language) is a known paradox, and it is not possible. If you have some "disordered" thing to which you attach "the most", then it is not as disordered as other things (you already ordered it, put it in the "top" of some list with disordered things).
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