Nonextensive Statistical Mechanics

324 8 Final Comments and Perspectivesare scale-invariant (the MTG, TMNT, RST1, and RST3 models strictly, and theRST2 model only asymptotically).(q) Can we have some intuition on what is the physical origin of the nonadditiveentropy S q , hence of q-statistics?Yes, we can. Although rarely looked at this way, a very analogous phenomenonoccurs at the emergence, for an ideal gas, of Fermi–Dirac and Bose–Einstein quantumstatistics. Indeed, their remarkably different mathematical expressions comparedto Maxwell–Boltzmann statistics come from a drastic reduction of the admissiblephysical states. Indeed, let us note E (N)Hthe Hilbert space associated withN particles; the N-particle wavefunctions are of the form |m 1 , m 2 , ..., m N 〉 =i=1 N φ m i(r i ), where φ mi (r i ) represents the wavefunction of the ith particle beingin the quantum state characterized by the quantum number (or set of quantumnumbers) m i . If for any reason (e.g., localization of the particles) we are allowedto consider the N particles as distinguishable, then Boltzmann–Gibbs equalprobabilityhypothesis for an isolated system at equilibrium is to be applied to theentire Hilbert space E (N) . At thermal equilibrium with a thermostat, we consistentlyHobtain, for the occupancy of the quantum state characterized by the wave-vectork and energy E k , fk MB = e −β(E k−μ)= Ne −β E k, where μ is the chemical potential,and MB stands for Maxwell–Boltzmann. If however, the particles are tobe considered as indistinguishable, then only symmetrized (anti-symmetrized) N-particle wavefunctions are physically admissible for bosons (fermions). For example,for N = 2, we have |m 1 , m 2 〉 = 1 √2[φ m1 (r 1 )φ m2 (r 2 ) + φ m1 (r 2 )φ m2 (r 1 )] forbosons, and |m 1 , m 2 〉 = √ 12[φ m1 (r 1 )φ m2 (r 2 ) − φ m1 (r 2 )φ m2 (r 1 )] for fermions. Forthe general case of N particles, let us note, respectively, E (N)(N)H(S) and EH(A) theHilbert spaces associated with symmetrized and anti-symmetrized wavefunctions.We have that E (N)H(S) ⊕ E (N)(N)H(A) ⊆ EH, the equality holding only for N = 2.For increasing N, the reduction of both E (N)(N)H(S) and EH(A) becomes more andmore relevant. It is precisely for this reason that statistics is profoundly changed.Indeed, the occupancy is now given by fkBE = 1/[e β(Ek−μ) − 1] for bosons (BEstanding for Bose–Einstein), and by fkFD = 1/[e β(Ek−μ) + 1] for fermions (FDstanding for Fermi–Dirac). The corresponding entropies are consistently changedfrom S MB /k B =− ∑ k f k ln f k to S BE /k B = ∑ k [− f k ln f k + (1 + f k )ln(1+ f k )]for bosons, and S FD /k B = − ∑ k [ f k ln f k + (1 − f k )ln(1 − f k )] for fermions.The need, in nonextensive statistical mechanics, for an entropy more general thanthe BG one, comes from essentially the same reason, i.e., a restriction of the spaceof the physically admissible states. Indeed, for the classical case for instance, vanishingLyapunov exponents possibly generate, in regions of -space, orbits whichare (multi)fractal-like. Since such orbits are generically expected to have zeroLebesgue-measure, an important restriction emerges for the physically admissiblespace (see also [21]). The basic ideas are illustrated for the microcanonical entropyin Fig. 8.3 for ideal Maxwell–Boltzmann, Fermi–Dirac and Bose–EinsteinN-particle systems (W 1 being the number of states, assumed non-degenerate, ofthe one-particle system), and in Fig. 8.4 for a highly correlated N-body system.