Chaos VIII : Statistics

Lorenz' water mill

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The dependance on initial conditions for the future of a system can look discouraging. However, there is a positive and constructive approach. In fact, this Lorenz' real message, but it is not that well known by the general public.

« More generally, I am proposing that over the years, miniscule disturbances neither increase nor decrease the frequency of occurrence of various weather events such as tornadoes. The most they may do is to modify the sequence in which these events occur.  »

Take three regions on the Lorenz attractor (they could represent conditions of hurricane, drought or snow). If we measure the proportions of the time that trajectories with different initial conditions spend within these regions, then we find that for all trajectories, these proportions converge to the same numbers, even if the order in which the trajectories encounter the three regions is incomprehensible.

Lorenz developed a real physical system with the help of physicists Howard and Markus. This system mimics the dynamics of the Lorenz system and so again does not have very much to do with real meteorology. It is a mill driven by the weight of buckets of water. Three parameters can describe its situation at any time: the two coordinates of its center of gravity, and the angular velocity. This mill is also very sensible to initial conditions, and a 3D graph of the three parameters has the shape of a butterfly, just like the Lorenz attractor. 

We set in motion two identical mill wheels, under different initial conditions, and we note down the angular velocity as often as we can. If we put the data into a bar chart, the graphs for both wheels, after a certain time, become quasi identical.

When  a statistical measure of a trajectory is insensible to initial conditions, then the associated dynamic has a Sinaï-Ruelle-Bowen measure: an SRB measure. If the dynamics involve the weather, then it is up to the weatherman to determine what these statistics are.

The Lorenz attractor does have an SRB measure. Its variables apparently change at random, but with well-defined probabilities!

By refocusing on statistical issues, science can still make predictions!

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