Nonextensive Statistical Mechanics

3.9 Various Other Entropic Forms 1053.9 Various Other Entropic FormsFor simplicity, we shall assume k = 1 in all the following definitions.The Renyi entropy is defined as follows [108]:S R q≡ ln ∑ Wi=1 pq i1 − q= ln[1 + (1 − q)S q]1 − q. (3.246)The Curado entropy is defined as follows [120]:S C bW ≡ ∑(1 − e −bp i) + e −b − 1(b ∈ R; b > 0) . (3.247)i=1The entropy introduced in [383], and which we shall from now on refer to asexponential entropy, is defined as follows:S E =W∑i=1p i(1 − e p i −1p i). (3.248)The Anteneodo–Plastino entropy is defined as follows [121]:S APη≡W∑ [( η + 1η , − ln p 1) − p i ( η + 1 ]η ) (η ∈ R; η>0) , (3.249)i=1where(μ, t) ≡∫ ∞tdy y μ−1 e −y =∫ e −t0dx (− ln x) μ−1 (μ>0) (3.250)is the complementary incomplete Gamma function, and (μ) = (μ, 0) is theGamma function.The Landsberg–Vedral–Rajagopal–Abe entropy, orjustnormalized S q entropy,is defined as follows [397, 398]:LV RASq ≡ Sq N ≡ S q∑ Wi=1 pq i[ ∑W1 −i=1 pq i=1 − q] −1=S q1 + (1 − q)S q. (3.251)The so-called escort entropy is defined as follows [60]:S E q≡ 1 −[ ∑W] −qi=1 p1/q) iq − 1=1−q1 − [1 −qS 1/q] −q. (3.252)q − 1The Kaniadakis entropy, also called the κ-entropy, is defined as follows [399]: