Production rate of the coarse-grained Gibbs entropy and the Kolmogorov-Sinai entropy: A real connection?

Observational Modeling of the Kolmogorov-Sinai Entropy

In this paper, Kolmogorov-Sinai entropy is studied using mathematical modeling of an observer $ Theta $. The relative entropy of a sub-$ sigma_Theta $-algebra having finite atoms is defined and then   the ergodic properties of relative  semi-dynamical systems are investigated.  Also,  a relative version of Kolmogorov-Sinai theorem  is given. Finally, it is proved  that the relative entropy of a...

Kolmogorov - Sinai Entropy Rate versus Physical Entropy

We elucidate the connection between the Kolmogorov-Sinai entropy rate k and the time evolution of the physical or statistical entropy S. For a large family of chaotic conservative dynamical systems including the simplest ones, the evolution of Sstd for far-from-equilibrium processes includes a stage during which S is a simple linear function of time whose slope is k. We present numerical confir...

A Characterization of the Entropy--Gibbs Transformations

Let h be a finite dimensional complex Hilbert space, b(h)+ be the set of all positive semi-definite operators on h and Phi is a (not necessarily linear) unital map of B(H) + preserving the Entropy-Gibbs transformation. Then there exists either a unitary or an anti-unitary operator U on H such that Phi(A) = UAU* for any B(H) +. Thermodynamics, a branch of physics that is concerned with the study...

Bridging Conceptual Gaps: The Kolmogorov-Sinai Entropy

Kolmogorov-Sinai entropy from the ordinal viewpoint

In the case of ergodicity much of the structure of a one-dimensional time-discrete dynamical system is already determined by its ordinal structure. We generally discuss this phenomenon by considering the distribution of ordinal patterns, which describe the up and down in the orbits of a Borel measurable map on a subset of the real numbers. In particular, we give a natural ordinal description of...