Nonextensive Statistical Mechanics

3.2 Nonadditive Entropy S q 41We shall from now on refer to these two functions as the q-exponential and theq-logarithm, respectively [104]. They will play an important role through the entiretheory. We may, in fact, anticipate that virtually all the generic expressions associatedwith BG statistics and its (nonlinear) dynamical foundations will, remarkablyenough, turn out to be generalized essentially just by replacing the standard exponentialand logarithm forms by the above q-generalized ones. Let us add that,whenever the 1 + (1 − q)x argument of the q-exponential is negative, the functionis defined to vanish. In other words, the definition is eqx ≡ [1 + (1 − q)x]1/(1−q) + ,where [z] + = max{z, 0}. However, for simplicity, we shall, most of the time, avoidthis notation. Typical representations of the q-exponential function are illustrated inFigs. 3.1, 3.2, 3.3, and 3.4. It is immediately verified that the q →−∞, q = 0, andq = 1 particular instances precisely recover the cases presented in Eqs. (3.1), (3.4),and (3.7) respectively.3.2 Nonadditive Entropy S q3.2.1 DefinitionThrough the metaphor presented above, and because of various other reasons thatwill gradually emerge, we may postulate the following generalization of Eq. (1.3):S q = k ln q W (S 1 = S BG ) . (3.16)See Fig. 3.5 for the illustration of this generalization of the celebrated formulafor equal probabilities. Let us address next the general case, i.e., for arbitrary {p i }.We saw in Eq. (2.8) that S BG can be written as the mean value of ln(1/p i ). Thisquantity is called surprise [105] or unexpectedness [106] by some authors. This isquite appropriate, in fact. If we have certainty (p i = 1forsomevalueofi) thatsomething will happen, when it does happen we have no surprise. On the oppositeextreme, if something is very unexpected (p i ≃ 0), if it eventually happens, weare certainly very surprised! Along this line, it is certainly admissible to considerthe quantity ln q (1/p i ) and call it q-surprise or q-unexpectedness. It then appears asquite natural to postulate the simultaneous generalization of Eqs. (2.8) and (3.16) asfollows:S q = k 〈ln q (1/p i )〉 . (3.17)If we use the definition (3.14) in this expression, we straightforwardly obtainS q = k 1 − ∑ Wi=1 pq iq − 1. (3.18)