Natural Science Forum / Physics / Research / January 2008

Lou,

I realize that the underlying math of a chaotic system could be
deterministic but the point of the theorem, if I remember correctly,
is that it assumed chaotic behavior and concluded the existence of a
deterministic property.

Shubee

What you have described for systems of low dimensionality is indeed
the case. Liovilles theoem does indeed imply volume conservation, but
this volume may be multidimensional. What we have in general is a
Hausdorff set - a topological space, which because of the properties
of the Hamiltonian has a lower dimensionality than the number of
degrees of freedom. In a complex topology return to an initial state
is virtually impossible.

Chaos implies that a small disturbance is amplified "A butterfly in
Japan causes a hurricane in the Gulf". Shubee also talks about the
"ergodic theorem". The ergodic theorem applies when we have a large
number of states. It says (in effect) I don't know what each
individual particle is doing but collectively I can make
generalizations. We say the second law of thermodynamics is obeyed.
There is a second law of thermodynamics because the number of states
is so large that the probability of a group of particles being in a
particlar state is effectively zero.

 - Ian Parker

Thus spake Igor Khavkine <igor.kh@gmail.com>

You're on the wrong track. This was to do with a discussing between
myself and Shubee on s.p.f. We are thinking in terms of statistical
properties of a chaotic system, e.g. mean behaviours are predictable
while individual behaviours are not. Examples in physics would include
black body radiation, the second law of thermodynamics, the gas laws,
but I think the theorem, if it exists, is much more general and applies
to any statistical system.

Regards