@ARTICLE{10.21494/ISTE.OP.2023.0985, 

TITLE={The winding path to entropy}, 
  
AUTHOR={Constantino Tsallis, }, 
	
JOURNAL={Entropy: Thermodynamics – Energy – Environment – Economy }, 
	
VOLUME={4}, 

NUMBER={Special issue LILA}, 
	
YEAR={2023}, 

URL={http://openscience.fr/The-winding-path-to-entropy}, 
	
DOI={10.21494/ISTE.OP.2023.0985}, 
	
ISSN={2634-1476}, 
	
ABSTRACT={The concept of entropy has traveled along winding roads since it was introduced in 1865 by Clausius as a key piece to complete thermodynamics and its Legendre transforms structure. Boltzmann, followed by Gibbs, has then revealed its microscopic interpretation, thus leading to the additive expression $$$S_{BG}=k\sum_{i=1}^W p_i \ln (1/p_i)$$$. A few decades later von Neumann provided its quantum form, and Shannon connected it to the theory of communications. Some time later, in 1961, Rényi generalized the Boltzmann-Gibbs-von Neumann-Shannon form, though preserving its additivity. There was then a real explosion of nonadditive entropic functionals, close to fifty nowadays. Inspired by multifractals, we postulated in 1988 the form $$$S_q\equiv k \sum_{i=1}^W p_i \ln_q (1/p_i)$$$ [with $$$\ln_q z \equiv (z^{1-q}-1)/(1-q);\, \ln_1z=\ln z$$$] as a basis for generalizing the Boltzmann-Gibbs statistical mechanics itself. Along the present brief perspective, we present the foundations and main applications of this theory, currently referred to as nonextensive statistical mechanics.}}