Nonextensive Statistical Mechanics

192 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanicsfrequencyfrequency10.80.60.40.2010.80.60.40.2010.8U = 0.69 (M 0 = 1)events of class 1events of class 2(a)(b)(c)frequency0.60.40.2events of class 3010 3 10 4 N10 5 10 6Fig. 5.41 Relative frequency of occurrence for the three classes of events shown in Fig. 1 as afunction of N. A total of 20 realizations for each N was considered. The three curves add up tounity (from [46]).where O is some dynamical observable essentially related to the evolution of thesystem in phase-space (e.g., the time evolution of entropy while the system approachesequilibrium). We typically expect(t) = e −t/τ 1, (5.52)where τ 1 is the relaxation time. Finally, in the third interpretation, we have Eq. (2.64)with Eq. (2.65), i.e.,Z 1 p i = e −β E i, (5.53)where Z 1 ≡ ∑ Wj=1 e−β E jis the partition function. The various interpretations aresummarized in Table 5.1.Let us now generalize these statements. The solution of the differentialequationdydx = ayq (y(0) = 1) (5.54)