Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 83for square blocks of area L 2 , i.e., the von Neumann entropy is nonextensive. Thisis so no matter how close the gap energy is to zero. In contrast, when we considerq ≠ 1, it is found [202]S qent (L) ∼ s qent (ω 0 )L 2 , (3.151)i.e., an extensive entropy (see Figs. 3.20 and 3.21). Equation (3.150) can be seen asthe d = 2 case of the so-called areas law, namelyS 1 (L) ∝ L d−1 (d > 1; L →∞) . (3.152)The d = 3 case recovers the celebrated scaling for black holes, namely S 1 (L) ∝L 2 . Equations such as (3.144) and (3.152) can be unified as followsS 1 (L) ∝ Ld−1 − 1d − 1≡ ln 2−d L (d ≥ 1; L →∞) , (3.153)i.e., the Boltzmann–Gibbs–von Neumann entropy is nonextensive. Given the aboveresults for fermionic and bosonic systems, a conjecture is very plausible, namelythat, for such systems, a value of q < 1 exists such that2018161412q = 0.8q = 0.87S q0.84 0.86 0.88 0.90108q = 1642010.999950.99990r 0.999850.999800.999750.99970(ω 0 = 0.01)0 200 400 600 800 1000 1200 1400L 2Fig. 3.20 Block q-entropy S q (ˆρ L ) as a function of the square block area L 2 in a bosonic d = 2array of infinite coupled harmonic oscillators at T = 0, for typical values of q. Only for q =q ent ≃ 0.87, s q is finite (i.e., S q is extensive); for q < q ent (q > q ent ) it diverges (vanishes).Inset: determination of q ent through numerical maximization of the linear correlation coefficient rof S q (ˆρ L ) when using the range 400 ≤ L 2 ≤ 1600.q