Thursday, September 5, 2013

macroscopically describable

If we repeat an experiment 2^k times, and define an event to be “simply describable” (macroscopically describable) if it can be described in m or fewer bits (so that there are 2^m or fewer such events), and “extremely improbable” when it has probability 1/2^n or less, then the probability that any extremely improbable, simply describable event will ever occur is less than (2^(k+m))/(2^n). Thus we just have to make sure to choose n to be much larger than k + m. If we flip a billion fair coins, any outcome we get can be said to be extremely improbable, but we only have cause for astonishment if something extremely improbable and simply describable happens, such as “all heads,” or “every third coin is tails,” or “only every third coin is tails.” Since there are 10^23 molecules in a mole of anything, for practical purposes anything that can be described without resorting to an atom-by-atom accounting (or coin-by-coin accounting, if there are enough coins) can be considered “macroscopically” describable.
    Granville Sewell, "Entropy, Evolution and Open Systems", note 5

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