11/21/2023 0 Comments Boltzmann entropyIn Probability Theory, the entropy of a random variable measures the uncertainty over the values which can be reached by the variable. Using the above interpretation, entropy plays a central role in the formulation of the Second Law of Thermodynamics which states that in an isolated physical system, any transformation leads to an increase of its entropy. In Classical Physics, entropy is seen as a magnitude which in every time is proportional to the quantity of energy that at that time cannot be transformed into mechanical work. In such physical systems, only surface waters are the unique which can be used to be transformed into work (for example provoking the rotation of a turbine). The first idea to be considered is that given a physical system the energy contained in it is comparable with the water contained in lakes, rivers or the sea. Introduced by Rudolf Clausius, the notion of entropy appears the first time in the setting of Physics but has been adopted in other fields with different meanings. Received 26 October 2015 accepted 26 January 2016 published 29 January 2016 Since this history is long, we will not deal with the Kolmogorov-Sinai entropy or with topological entropy and modern approaches. Of special interest is to appreciate the connexions of the notions of entropy from Boltzmann and Shannon. 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. Boltzmann until the appearing in the twenty century of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. To this end, we start with the introduction for the first time of the notion of entropy in thermodynamics by R. At the same time, we will incorporate some mathematical foundations of such old and new ideas until the appearance of Shannon entropy. 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. Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. 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). In modern applications, other conditions on X and f have been considered. Among them, one of the most popular is that of topological entropy. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. 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.
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