Re: Various(Dynamical systems & Dynamical (classical and quantum) Entropy)

[Sonnet Nguyen <Sonnet.Nguyen@fuw.edu.pl>]

Hi friends,
First of all, Thanks to N T Zung for simple introducing to
Kolmogorov-Shannon entropy.

Few weeks ago I had a pleasure to study a discussion about ENTROPY in VNSA
forum.  Many interesting things one could learn reading Aiviet's thoughts.
In this discussion, only "statical entropy" was reminded.  In fact,
according to the reductionism phylosophy one can define statical entropy
by only one line of algebraic symbols:
Entropy S:= k ln (SIGMA), 
here k=1 (or k = Boltzman's constant for historical reason; 
i.e. according to thermodynamical interpretation of entropy.  For me k=1) 
and SIGMA is a statistical sum.  Because the function ln is just a
homomorfism from (R^{+},.) to (R,+), then ENTROPY defined by this way
is a homomorphism from the set (class) of independent systems to (R,+).     

Classical dynamical entropy (was introduced by Kolmogorov (the best
russian mathematician in XX-century) and independently by Shinai) is a
measure of chaos of system.  This entropy is called by Kolmogorov-Shinai
Entropy.
Usually when the dynamical system has a compact configurational space
(and dim>2), one could get a bifurcation, stranger attractors, etc.
The problem of generalization of classical version of dynamical entropy to
quantum dynamical entropy is not easy.  I know at least three (may be
four) unequivalent quantum versions of dynamical entropy.   (Of course, if
someone believes in the existence of classical limits of quantum effects, 
then it's easy to verify which of them is the best or at least to compare 
them together. Probably, from two of them we can get Kolmogorov-Shinai
entropy in the classical limit, etc.)

Hope someday I can present here short simple definition of quantum
dynamical entropy which is a natural generalization of K-S entropy...
enjoys,
SN  
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|Sonnet Nguyen,   Polish Academy of Sciences            |
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