Nonextensive Statistical Mechanics

12 1 Historical Background and Physical Motivations1.3 Symmetry, Energy, and EntropyAt this point, let us focus on some connections between three key concepts ofphysics, namely symmetry, energy, and entropy: see Fig. 1.1. According to Plato,symmetry sits in Topos Ouranos (heavens), where sit all branches of mathematics –science of structures –. In contrast, energy and entropy sit in Physis (nature). Energydeals with the possibilities of the system; entropy deals with the probabilities ofthose possibilities. It is fair to say that energy is a quite subtle physical concept.Entropy is based upon the ingredients of energy, and therefore is, epistemologicallyspeaking, one step further. It is most likely because of this feature that entropyemerged, in the history of physics, well after energy. A coin has two faces, andcan therefore fall in two possible manners, head and tail (if we disconsider the veryunlike possibility that it falls on its edge). This is the world of the possibilities forthis simple system. The world of its probabilities is more delicate. Before throwingthe coin (assumed fair for simplicity), the entropy equals ln 2. After throwing it,it still equals ln 2 for whoever has not seen the result (or just does not know it),whereas it equals zero for whoever has seen the outcome (or knows it). This exampleneatly illustrates the informational nature of the concept.Let us now address the connections. Those between symmetry and energy arelong and well known. Galilean invariance of the equations is central to Newtonianmechanics. Its simplest form of energy can be considered to be the kinetic one of apoint particle, namely p 2 /2m, p being the linear momentum, and m the mass. Thisenergy, although having an unique form, is not universal; indeed it depends on themass of the system. If we replace now the Galilean invariance by the Lorentzian one,this drastically changes the form itself of the kinetic energy, which now becomes(p 2 c 2 + m 2 0 c4 ) 1/2 , c being the speed of light in vacuum, and m 0 the mass at rest. InFig. 1.1 Connections between symmetry, energy, and entropy. QED and QCD respectively denotequantum electrodynamics and quantum chromodynamics.