Tuesday, January 7, 2020

The Real Butterfly Effect

Chaos and fractals were the youngest fields of math that I studied in college (operations research was a close runner up). But, it was also among the most fascinating and profound. 
The name “Butterfly Effect” was popularized by James Gleick in his 1987 book “Chaos” and is usually attributed to the meteorologist Edward Lorenz. But I recently learned that this is not what Lorenz actually meant by Butterfly Effect. . . . 
Lorenz, in this [1969] paper, does not write about butterfly wings. He instead refers to a sea gull’s wings, but then attributes that to a meteorologist whose name he can’t recall. The reference to a butterfly seems to have come from a talk that Lorenz gave in 1972, which was titled “Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?” 
The possibility that a small change in initial conditions can lead to a massively different outcome somewhere (the contemporary meaning of the phrase), was actually not the strongest claim made in the 1969 paper.
How well can you make predictions using the data from your measurements? You have data on that finite grid. But that does not mean you can generally make a good prediction on the scale of that grid, because errors will creep into your prediction from scales smaller than the grid. You expect that to happen of course because that’s chaos; the non-linearity couples all the different scales together and the error on the small scales doesn’t stay on the small scales. 
But you can try to combat this error by making the grid smaller and putting in more measurement devices. 
For example, Lorenz says, if you have a typical grid of some thousand kilometers, you can make a prediction that’s good for, say, 5 days. After these 5 days, the errors from smaller distances screw you up. So then you go and decrease your grid length by a factor of two. 
Now you have many more measurements and much more data. But, and here comes the important point: Lorenz says this may only increase the time for which you can make a good prediction by half of the original time. So now you have 5 days plus 2 and a half days. Then you can go and make your grid finer again. And again you will gain half of the time. So now you have 5 days plus 2 and half plus 1 and a quarter. And so on. 
Most of you will know that if you sum up this series all the way to infinity it will converge to a finite value, in this case that’s 10 days. This means that even if you have an arbitrarily fine grid and you know the initial condition precisely, you will only be able to make predictions for a finite amount of time. 
And this is the real butterfly effect. That a chaotic system may be deterministic and yet still be non-predictable beyond a finite amount of time.
Sabine Hossenfelder tells the entire story in more depth at her blog, Backreaction. 

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