Nonextensive Statistical Mechanics

1.4 A Few Words on the Foundations of Statistical Mechanics 13other words, this change of symmetry is far from innocuous; it does nothing less thanchanging Newtonian mechanics into special relativity! Maxwell electromagnetismis, as well known, deeply related to this same Lorentzian invariance, as well as togauge invariance. The latter plays, in turn, a central role in quantum electrodynamicsand quantum field theory. Quantum chromodynamics also is deeply related to symmetryproperties. And so is expected to be quantum gravity, whenever it becomesreality. Summarizing, the deep connections between symmetry and energy are standardknowledge in contemporary physics. Changes in one of them are concomitantlyfollowed by changes in the other one.What about the connections between energy and entropy? Well, also these arequite known. They naturally emerge in thermodynamics (the possibility and mannersfor transforming work into heat, and the other way around). This obviouslyreflects on BG statistical mechanics itself.But, what can we say about the possible connections between symmetry (its natureand evolution) and entropy? This topic has remained basically unchanged andvirtually unexplored during more than one century! Why? Hard to know. However,it is allowed to suspect that this intellectual lethargy comes, on one hand, from the“sloppiness” of the concept of entropy, and, on the other one, from the remarkablefact that the unique functional form that has been used in thermal equilibrium-likephysics is the BG one (Eq. (1.8) and its continuum or quantum analogs), which dependsonly on one of the universal constants, namely Boltzmann constant k B . Withinthis intellectual landscape, generation after generation, the idea installed in the mindof very many physicists that the physical entropy must be universal, and that it is ofcourse the BG one. In the present book, we try to convince the reader that it is notso, that many types of entropy can be physically and mathematically meaningful.Moreover, we shall argue that dynamical concepts such as the time-dependence ofthe sensitivity to the initial conditions, mixing, and the associated occupancy andvisitation networks they may cause in phase-space, have so strong effects, that eventhe functional form of the entropy must, in some occasions, be modified. The BGentropy will then still have a highly priviledged position. It surely is the correct onewhen the microscopic nonlinear dynamics is controlled by positive Lyapunov exponents,hence strong chaos. If the system is such that strong chaos is absent (typicallybecause the maximal Lyapunov exponent vanishes), then the physical entropy to beused appears to be a different one.1.4 A Few Words on the Foundations of Statistical MechanicsA mechanical foundation of statistical mechanics from first principles should essentiallyinclude, in one way or another, the following main steps [40].(i) Adopt a microscopic dynamics. This dynamics typically is deterministic, i.e.,without any phenomenological noise or stochastic ingredient, so that the foundationmay be considered as from first principles. This dynamics could be Newtonian, orquantum, or relativistic mechanics (or some other mechanics to be found in fu-