On Clausius, Boltzmann and Shannon Notions of Entropy

Abstract

Discrete dynamical systems are given by the pair ( X,f ) where X is a compact metric space and f: X → X is a continuous map. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications, other conditions on X and f have been considered. For example, X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper, we are reviewing the origins of the notion of entropy and studying some developing of it leading to modern notions of entropies. At the same time, we will incorporate some mathematical foundations of such old and new ideas until the appearance of Shannon entropy. To this end, we start with the introduction for the first time of the notion of entropy in thermodynamics by R. Clausius and its evolution by L. Boltzmann until the appearing in the twenty century of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn, such notions have evolved to other recent situations where it is necessary to give some extended versions of them adapted to new problems. Of special interest is to appreciate the connexions of the notions of entropy from Boltzmann and Shannon. Since this history is long, we will not deal with the Kolmogorov-Sinai entropy or with topological entropy and modern approaches.