We’ve done “chaos is not randomness” before. Here’s another interesting property to do with mixing.

Mixing is a property of dynamical systems whereby the state of the system in the distant future cannot be predicted from its initial state (or any given state a long way in the past). This is pretty much the same as the kind of mixing you get when you put milk in a cup of tea and swirl it around: obviously when you first put the milk in, it stays roughly where you put it, but after time it spreads out evenly. The even spread of the milk will be the same no matter where you put the milk in originally. More formally, if

is a “distribution” or density function of where the “particles” of milk are when you have just put them in the tea, and

is the distribution after seconds. “Mixing” is formally defined as

You don’t have to think about these distributions as probability distributions, but I find it easier if you do. For those that know probability, it is obvious that what the above is saying is that the distribution of milk after a long time is probabilistically independent of its distribution at the start.

In cups of tea, this happens (mostly) because of the “random” Brownian motion of the milk (possibly enhanced by someone swirling it with a spoon).

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