Nonextensive Statistical Mechanics

194 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics0.500.48N = 20000 - U = 0.69 (M 0 = 1)Single events of class 2Equilibrium0.46Temperature0.440.42QSS0.400.3810 1 10 2 10 3 10 4 10 5timeFig. 5.43 Different events of class 2 are plotted for the case N = 20, 000. A large variability isobserved for this class, at variance with the other two (from [46]).where Z qstat ≡ ∑ Wj=1 e−β qstat E jis the partition function. The various interpretationsare summarized in Table 5.2. The set (q sen , q rel , q stat ) constitutes what we shallrefer to as the q-triplet (occasionally referred also to as the q-triangle). In theBG particular case, we recover q sen = q rel = q stat = 1. The existence of thesethree q-exponentials characterized by the q-triplet was predicted in 2004 [190]and confirmed in 2005 [361]: see Fig. 5.63, where the observations done (throughprocessing the data sent to Earth by the spacecraft Voyager 1) in the solar wind aredepicted (more along these lines can be found in [368]). 77 If we have a triplet (x, y, z) of real numbers such that one of them, say x, is the arithmetic averageof the other two (i.e., x = y+z ), and one of the other two, say y, is the harmonic average of the other2two (i.e., y −1 = x−1 +z −1), then, remarkably enough, the third number necessarily is the geometric2average of the other two (i.e., z = √ xy). If we define now ɛ ≡ 1 − q, we have, from [199], that(ɛ sen ,ɛ rel ,ɛ stat ) = (3/2, −3, −3/4). By identifying (x, y, z) ≡ (ɛ stat ,ɛ rel ,ɛ sen ), it can be checkedthat they satisfy the just mentioned remarkable relationships! [369]. In fact, these relations admitonly one degree of freedom. In other words, we can freely choose only one number, say x; theother two (y and z) are automatically determined. If x ≥ 0, the solution is x = y = z; ifx < 0,the solution is x = y/4 =−z/2. The set (ɛ stat ,ɛ rel ,ɛ sen ) = (−3/4, −3, 3/2) belongs to this lattercase.