6.3.5 Horizons of Predictability
Chaotic processes are not entirely unpredictable. Let us consider, as an example, the
dynamics of the changing atmosphere as reflected in the ever-changing weather.
These changes are commonly believed to behave chaotically, displaying extreme
sensitivity to initial conditions. Still, we are able to predict states such as temperature, pressure and wind speed a few seconds ahead; they are unlikely to change
much. Skilled meteorologists with access to computer models and ‘initial conditions’ sampled worldwide, can predict rather accurately details of the weather a few
days ahead. However, they cannot tell exactly what the temperature will be at a
certain geographic spot in ten days. And better computers, better models and more
accurate initial conditions will improve only marginally on weather forecasts. We
can make predictions on the average, such that winters will be colder than summers,
and possibly forecast the global mean seasonal temperature for the next ten-year
period. Though, it is impossible to predict the particular mean winter temperature
20 year ahead. Similarly, when watching the chaotic Moon’s beam – or a boxing
match, a dripping tap, a baby exploring your computer keyboard – it is possible to
make accurate predictions only small instances of time ahead. Long-term predictions are obviously impossible in these and many other cases.
As we shall see chaotic processes destroy information. As information is
destroyed predictions get worse.
Initial conditions represent information. For example, specifying a particular
initial condition within an accuracy of one part per million amounts to (log10
6 /log2)
% 20 bits of information. For non-chaotic processes this information is preserved in
time, so that from any particular state you may trace your way back to the initial
conditions, or predict the state of the system ad infinitum (or absurdum).
Non-chaotic processes are information preserving.
Chaotic processes destroy information. Any small inaccuracy in initial conditions is repeatedly stretched and folded. Since each stretch and fold represents a loss
of information, sooner or later all information laid down in the initial conditions is
lost.
To illustrate this degrading of information we imagine a baker who wants to
color a clod of pie dough. Onto the dough he slips a drop of red color; we assume
the drop contains 10
5 molecules of red. Initially the baker knows the position of
every molecule to within an accuracy that corresponds to the size of the droplet.
Then he stretches the dough. Knowing the details of the stretch his information is
preserved, since nearby molecules stay near. He is able, at least theoretically, to
trace each molecule back to where it came from. However, when the baker subsequently folds the dough he loses a tiny bit of information, since some nearby
molecules may end up a far distance apart (those at the edge of the fold). Repeating
this process a number of times the baker finds himself with a red clod of dough.
That is, the 10
5 molecules have spread all over. All initial information is lost, and as
for the position of each molecule the baker merely knows they are somewhere in the
dough.
6.3 Tools for Detecting Chaotic Vibrations
337
Chaotic processes are not entirely unpredictable. Let us consider, as an example, the
dynamics of the changing atmosphere as reflected in the ever-changing weather.
These changes are commonly believed to behave chaotically, displaying extreme
sensitivity to initial conditions. Still, we are able to predict states such as temperature, pressure and wind speed a few seconds ahead; they are unlikely to change
much. Skilled meteorologists with access to computer models and ‘initial conditions’ sampled worldwide, can predict rather accurately details of the weather a few
days ahead. However, they cannot tell exactly what the temperature will be at a
certain geographic spot in ten days. And better computers, better models and more
accurate initial conditions will improve only marginally on weather forecasts. We
can make predictions on the average, such that winters will be colder than summers,
and possibly forecast the global mean seasonal temperature for the next ten-year
period. Though, it is impossible to predict the particular mean winter temperature
20 year ahead. Similarly, when watching the chaotic Moon’s beam – or a boxing
match, a dripping tap, a baby exploring your computer keyboard – it is possible to
make accurate predictions only small instances of time ahead. Long-term predictions are obviously impossible in these and many other cases.
As we shall see chaotic processes destroy information. As information is
destroyed predictions get worse.
Initial conditions represent information. For example, specifying a particular
initial condition within an accuracy of one part per million amounts to (log10
6 /log2)
% 20 bits of information. For non-chaotic processes this information is preserved in
time, so that from any particular state you may trace your way back to the initial
conditions, or predict the state of the system ad infinitum (or absurdum).
Non-chaotic processes are information preserving.
Chaotic processes destroy information. Any small inaccuracy in initial conditions is repeatedly stretched and folded. Since each stretch and fold represents a loss
of information, sooner or later all information laid down in the initial conditions is
lost.
To illustrate this degrading of information we imagine a baker who wants to
color a clod of pie dough. Onto the dough he slips a drop of red color; we assume
the drop contains 10
5 molecules of red. Initially the baker knows the position of
every molecule to within an accuracy that corresponds to the size of the droplet.
Then he stretches the dough. Knowing the details of the stretch his information is
preserved, since nearby molecules stay near. He is able, at least theoretically, to
trace each molecule back to where it came from. However, when the baker subsequently folds the dough he loses a tiny bit of information, since some nearby
molecules may end up a far distance apart (those at the edge of the fold). Repeating
this process a number of times the baker finds himself with a red clod of dough.
That is, the 10
5 molecules have spread all over. All initial information is lost, and as
for the position of each molecule the baker merely knows they are somewhere in the
dough.
6.3 Tools for Detecting Chaotic Vibrations
337