Nonextensive Statistical Mechanics

326 8 Final Comments and PerspectivesWe verify in Fig. 8.3 that, in the limit of large systems (N →∞and W 1 →∞),the MB, FD, and BE systems yield a BG entropy which is extensive, i.e., thermodynamicallyadmissible. This is not the case for the highly correlated N-body system.Indeed, the BG entropy asymptotically becomes independent from N, whereas thenonadditive entropy S q exhibits extensivity for a special value of q, and is thereforethermodynamically admissible. In other words, when the reduction of the (physically)admissible number of states is inexistent (MB model), or moderate (FD andBE models), the BG entropy is extensive. But if this reduction is very severe (presenthighly correlated model), then we are obliged to introduce a different entropy inorder to satisfy thermodynamics. Obviously this point is most important, since itbasically makes legitimate the use of virtually all general formulas of textbooks ofthermodynamics.8.3 Open QuestionsAs in any physical theory in intensive development, a large amount of open questionsstill exist within nonextensive statistical mechanics. Since we do not intendhere to make a lengthy description, we will simply mention some of those few pointsthat we find particularly intriguing and fruitful.(a) What are the q-indices relevant to the stationary-state associated with a d-dimensional classical many-body Hamiltonian including (say attractive) interactionsthat are not singular (or are, at least, integrable) at the origin and decay withdistance r like 1/r α (α ≥ 0)?We know that, for α/d > 1 (i.e., short-range interactions), q = 1 (hence q sen =q rel = q stat = 1). What happens for 0 ≤ α/d ≤ 1 (i.e., long-range interactions)What would be the possible (α, d)-dependences (perhaps (α/d)-dependences) ofindices such as (q sen , q rel , q stat )?(b) Compatibility between the (presumably) scale-invariant correlations leading toan extensive S q and the q-exponential form for the stationary-state distribution ofenergy for many-body Hamiltonian systemsMore precisely, what must be satisfied by the interaction Hamiltonian H ABwithin the form H A+B = H A + H B + H AB when A and B aretwolargesystems?Let us be more concrete and discuss the q = 1 case. Assume that we are dealingwith short-range interactions, and that A and B are two equally sized d-dimensionalsystems. Let L be the linear size of each of them. Then the energy corresponding toH A increases like L d , and the same happens with system B. Let us also assume thatA and B are in contact only through a common (d − 1)-dimensional surface. Thenthe energy corresponding to H AB increases like L d−1 . In the limit L →∞, we canneglect the interaction energy, i.e., consider H AB = 0. Then H A+B = H A + H Bis clearly compatible with piA = e −β E i A/Z A , piB = e −β E iB/Z B and p A+Bij= pi A p B j .The question we would like to answer is what exactly happens for q ≠ 1?(c) What is the geometrical-dynamical interpretation of the escort distribution?