entropy headache
su...@sf6.so-net.ne.jp
What Professor Vakarin says seems very reasonable as long as ergodic systems in nonequilibrium
stationary states are concerned. The Boltzmann-Gibbs-Shannon entropy may work well.
First of all, I am pretty sure that all of us agree that still we do not understand connections between
dynamics and statistics. Therefore, what I am going to say below is also highly unclear. Please keep
this crucial point in mind.
The main purpose of generalizations of conventional statistical mechanics is to describe complex
systems in nonequilibrium stationary states. Here, "complex systems" imply that such dynamical
systems are prepared at the edge of chaos. Using the nonlinear-dynamics terminology, it is stated
that the maximum Lyapunov exponents of such systems vanish. This is in marked contrast with chaotic
dynamical systems, in which there exist positive Lyapunov exponents. In chaotic systems, the associated
dynamical entropy is the celebrated Kolmogorov-Sinai (KS) entropy, which formally has the form of
the Shannon entropy. The crucial point is that the KS entropy has constant entropy-production rate,
that is, it grows linearly in time in the chaotic regime. I wish to call this "temporal extensivity", since
"time" is analog of "the number of particles" in statistical thermodynamics.
Now, at the edge of chaos (i.e., vanishing Lyapunov exponent) where the concept of complexity emerges,
the KS entropy fails to exhibit "temporal extensivity". Instead, generalized entropies including Tsallis'
can grow linearly in time. This has been analytically demonstrated for low-dimensional dissipative
systems (NOT for Hamiltonian systems). This indicates an interesting possibility that at the edge of chaos
the definition of statistical mechanical entropy COULD also be deformed.
What is concerned in statistical mechanics is, however, many-body Hamiltonian systems. And in a
Hamiltonian system, the sum of positive and negative Lyapunov exponents is zero, i.e., cancellation
of positive and negative exponents. Therefore, if there are vanishing Lyapunov exponents, all the
exponents are zero, which implies "freezing of dynamics". This observation makes it highly unclear
if generalized entropies have any physical relevance to Hamiltonian systems.
We do not know yet.
I hope that this clumsy message does not confuse you.
Yours sincerely,
Sumiyoshi Abe