Nonextensive Statistical Mechanics

22 2 Learning with Boltzmann–Gibbs Statistical Mechanicsalready-mentioned property lim x→0 (x ln x) = 0. So, we have thatS BG (p 1 , p 2 ,...,p W , 0) = S BG (p 1 , p 2 ,...,p W ) . (2.9)2.1.2.4 AdditivityLet O be a physical quantity associated with a given system, and let A and B betwo probabilistically independent subsystems. We shall use the term additive if andonly if O(A + B) = O(A) + O(B). If so, it is clear that if we have N equal systems,then O(N) = NO(1), where the notation is self-explanatory. A weaker conditionis O(N) ∼ N for N → ∞, with 0 < || < ∞, i.e., lim N→∞ O(N)/N isfinite (generically ≠ O(1)). In this case, the expression asymptotically additivemight be used. Clearly, any observable, which is additive with regard to a givencomposition law, is asymptotically additive (with = O(1)), but the opposite isnot necessarily true.It is straightforwardly verified that, if A and B are independent, i.e., if the jointprobability satisfies p A+Bij= pi A p B j(∀(ij)), thenTherefore, the entropy S BG is additive.S BG (A + B) = S BG (A) + S BG (B) . (2.10)2.1.2.5 ConcavityLet us assume two arbitrary and different probability sets, namely {p i } and {pi ′},associated with a single system having W states. We define an intermediate probabilityset as follows:p ′′i = λp i + (1 − λ)p ′ i (∀i; 0 0) satisfies−p i ′′ ln p i ′′ >λ(−p i ln p i ) + (1 − λ)(−p i ′ ln p′ i ) (∀i; 0