Nonextensive Statistical Mechanics

222 7 Thermodynamical and Nonthermodynamical ApplicationsBefore starting with the description of typical applications, let us remind thatthe knowledge of the microscopic dynamics is necessary but not sufficient for theimplementation of the entire theory from first principles. Indeed, it is only in principlethat the microscopic dynamics contains all the ingredients enabling the calculationof the index (or indices) q. It is still necessary to be able to calculate, in the fullphase-space, quantities such as the sensitivity to the initial conditions or the entropyproduction. This calculation can be extremely hard. But, whenever tractable, thenit provides the value(s) of q. Once q is known, it becomes possible to implementthe thermodynamical steps of the theory. This is to say, we can in principle proceedand calculate the partition function of the system, and, from this, calculatevarious important thermodynamical quantities such as specific heat, susceptibility,and others. Naturally, the difficulty of this last step of the calculation should notbe underestimated. It suffices to remember the formidable mathematical difficultiesinvolved in Onsager’ s celebrated solution of the square-lattice spin 1/2 Ising ferromagnet.And he only had to deal with first-neighbor interactions and exponentialthermal weights. Inafullq-statistical calculation, we have to deal typically withinteractions at all distances (or related conditions) and power-law weights!Thisdifficultymight explain why we have, up to now, only partial results for the many-bodylong-range-interacting inertial XY ferromagnet addressed in Section 5.4. This taskwould be hopeless had we not access to approximate solutions based on variationalprinciples, Green-functions, numerical approaches, and others. As a mathematicalexercise, the q-statistics of simple systems such as the ideal gas and the ideal paramagnetare available in the literature. However, these calculations only provide somemathematical hints with modest physical content. Indeed, thermal equilibrium inthe absence of interactions mandates q = 1. Further, and extremely powerful, hintsare also available from the full discussion of simple maps, as shown in Sections 5.1and 5.2. However, these systems, no matter how useful they might be for variousapplications, are nonthermodynamical. In other words, they do not have energy associated,and are therefore useless in order to illustrate the thermostatistical steps ofthe full calculation, and their connection to thermodynamics itself.We present next various applications in various areas of knowledge.7.1 Physics7.1.1 Cold Atoms in Optical LatticesOn the basis of nonextensive statistical mechanical concepts, Lutz predicted in2003 [460] that cold atoms in dissipative optical lattices would have a q-Gaussiandistribution of velocities, withq = 1 + 44E R, (7.1)U 0where E R and U 0 are, respectively, the recoil energy and the potential depth. Theprediction was impressively verified three years later [461], as shown in Fig. 7.1.