Boltzmann-gibbs Entropy versus Tsallis Entropy: Recent Contributions to Resolving the Argument of Einstein concerning " Neither Herr Boltzmann nor Herr Planck Has given a Definition of W " ?

Classical statistical mechanics of macroscopic systems in equilibrium is based on Boltzmann's principle. Tsallis has proposed a generalization of Boltzmann-Gibbs statistics. Its relation to dynamics and nonextensivity of statistical systems are matters of intensive investigation and debate. This essay review has been prepared at the occasion of awarding the 'Mexico Prize for Science and Technology 2003' to Professor Constantino Tsallis from the Brazilian Center for Research in Physics. Mathematical structures entered the development of physics, and problems emanating from physics influenced developments in mathematics. Examples are the role of Riemann's differential geometry in Einstein's general relativity, the dynamical theory of space and time, and the influence of Heisenberg's quantum mechanics in the development of functional analysis built on the understanding of Hilbert spaces. A prospective similar development occurred more than a decade ago when non-Abelian gauge theories emerged as the quantum field theories for describing fundamental-particle interactions (Greene, 2000). Recently , attention has turned to the applications of Riemann-Liouville fractional calculus and Mandelbrot's fractal geometry of nature to physics, including the The latter is the subject of this essay review. Statistical mechanics concerns mechanics (classical, quantum, special or general relativistic) and the theory of probabilities through the adoption of a specific entropic functional. Connection with thermodynamics and its macroscopic laws is established through this functional. The entropy concept of statistical