Nonextensive Statistical Mechanics

210 6 Generalizing Nonextensive Statistical MechanicsFig. 6.1 Scenario within which nonextensive statistical mechanics is located. At the extreme leftof the q = 1 region we essentially find the noninteracting systems, such as the ideal gas, and theideal paramagnet. At the extreme right of the q = 1 region, we may find the critical phenomenaassociated with standard phase transitions [207]. These systems exhibit, at precisely the criticalpoint, collective correlations which bridge with the q ≠ 1 systems. At the extreme right of theq ≠ 1 region, we cross onto a region of what one may consider as very complex systems. For suchsystems, a statistical mechanics even more general that the nonextensive one might be necessary.Or it just might be impossible to exist. From [200] (see [199] for more details).Fig. 6.2 Log–log plot of ξ ≡ y vs. t ≡ x for q = 2.7, a q = 1, a 1 = 10 −5 , and both r = 1andr = 1.7. The characteristic values tq∗ ≡ x q ∗ ∗∗and tr≡ xr∗∗ are indicated by arrows (the regionscorresponding to short-, intermediate-, and long-abscissa values are clearly exhibited). The slopeof the intermediate region is −1/(q − 1) (from [282]).As we see, this solution makes a crossover from a q-exponential behavior at lowvalues of x, to an exponential one for high values of x (see Fig. 6.2). If x is to beinterpreted as an energy (see Tables 5.1 and 5.2), this constitutes a generalizationof the q-statistical weight. It is from this property that this statistics is sometimesreferred to as crossover statistics.Equation (6.1) can be further generalized as follows: