Abstract
The concept of entropy constitutes, together with energy, a cornerstone of contemporary physics and related areas. It was originally introduced by Clausius in 1865 along abstract lines focusing on thermodynamical irreversibility of macroscopic physical processes. In the next decade, Boltzmann made the genius connection—further developed by Gibbs—of the entropy with the microscopic world, which led to the formulation of a new and impressively successful physical theory, thereafter named statistical mechanics. The extension to quantum mechanical systems was formalized by von Neumann in 1927, and the connections with the theory of communications and, more widely, with the theory of information were respectively introduced by Shannon in 1948 and Jaynes in 1957. Since then, over fifty new entropic functionals emerged in the scientific and technological literature. The most popular among them are the additive Renyi one introduced in 1961, and the nonadditive one introduced in 1988 as a basis for the generalization of the Boltzmann–Gibbs and related equilibrium and nonequilibrium theories, focusing on natural, artificial and social complex systems. Along such lines, theoretical, experimental, observational and computational efforts, and their connections to nonlinear dynamical systems and the theory of probabilities, are currently under progress. Illustrative applications, in physics and elsewhere, of these recent developments are briefly described in the present synopsis.
Highlights
(1839–1903) further discussed and extended the physical meaning of this connection [5,6,7]. Their efforts culminated in the formulation of a powerful theory, currently known as statistical mechanics
We are able to macroscopically calculate the entropy of the composed system without any need of entering into the knowledge of the microscopic states of the subsystems. This property appears to be a natural one for an entropic form if we desire to use it as a basis for a statistical mechanics which would naturally connect to thermodynamics
It is worthy to emphasize that the BG entropy and associated statistical mechanics appear to be sufficient but not necessary for the validity of classical thermodynamics and the Einstein likelihood principle, see [94]
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. T integrating factor, which transforms the differential transfer of heat (dependent on the specific path of the physical transformation) into the exact differential quantity of entropy (path-independent) This relation was thereafter generalized by Clausius into its celebrated inequality dS ≥ δQ/T, the equality corresponding to a reversible process. The American physicist, chemist and mathematician Josiah Willard Gibbs (1839–1903) further discussed and extended the physical meaning of this connection [5,6,7] Their efforts culminated in the formulation of a powerful theory, currently known as statistical mechanics. We mention here an important step concerning entropy, namely its extension to quantum mechanical systems It was introduced in 1927 [8] by the Hungarian-American mathematician, physicist and computer scientist János Lajos Neumann (John von Neumann; 1903–1957). Very few among them have found neat applications in physics and elsewhere
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