Nonextensive Statistical Mechanics

16 1 Historical Background and Physical Motivationsof attraction. One of them leads essentially to half of the same phase-space wherethe system lives in the lim N→∞ lim t→∞ ordering, i.e., the half phase-space whichis associated with a sign for the magnetization which coincides with the sign of theinitial magnetization. The other basin of attraction could well correspond to living ina very complicated, hierarchical-like, geometrical structure. This structure could bea zero Lebesgue measure one (in the full multidimensional phase-space), somewhatsimilar to that of an airlines company, say Air France, whose central hub is located inParis, or Continental Airlines, whose central hub is located in Houston. The specificlocation of the structure in phase-space would depend on the particular initial conditionwithin that special basin of attraction, but the geometrical–dynamical nature ofthe structure would be virtually the same for any initial condition within that basinof attraction. At this point, let us warn the reader that the scenario that we havedepicted here is only conjectural, and remains to be proved. It is however basedon various numerical evidences (see, e.g., [41, 44, 45, 50] and references therein). Itis expected to be caused by a possibly vanishing maximal Lyapunov exponent. Inother words, one would possibly have, instead of strong, only weak chaos.(iv) Now let us focus further on the specific role played by the initial conditions.If the system is strongly chaotic, hence mixing, hence ergodic, this point isirrelevant. We can make or not averages over initial conditions, we can take almostany initial condition, the outcome for sufficiently long times will be the same, in thesense that the set of probabilities in phase-space will be the same. But if the system isonly weakly chaotic, the result can drastically change from initial condition to initialcondition. If two initial conditions belong to the same “basin of attraction,” the differenceat the macroscopic level could be quite irrelevant. If they belong however todifferent basins of attraction, the results can be sensibly different. For some purposeswe might wish to stick to a specific initial condition within a certain class of initialconditions. For other purposes, we might wish to average over all initial conditionsbelonging to a given basin of attraction, or even over all possible initial conditionsof the entire phase-space. The macroscopic result obtained after averaging mightconsiderably differ from that corresponding to a single initial condition.(v) Last but not least, the mathematical form of the entropy functional must beaddressed. Strictly speaking, if we have deduced (from microscopic dynamics) theprobabilities to be associated with every cell in phase-space, we can in principlecalculate useful averages of any physical quantity of interest which is defined inthat phase-space. In this sense, we do not need to introduce an entropic functionalwhich is defined precisely in terms of those probabilities. Especially if we take intoaccount that any set of physically relevant probabilities can be obtained throughextremization (typically maximization) of an infinite number of entropic functionals(monotonically depending one onto the other), given any set of physically andmathematically meaningful constraints. However, if we wish to make contact withclassical thermodynamics, we certainly need to know the mathematical form of suchentropic functional. This functional is expected to match, in the appropriate limits,the classical, macroscopic, entropy ‘a la Clausius. In particular, one expects it tosatisfy the Clausius property of extensivity, i.e., essentially to be proportional to the