Nonextensive Statistical Mechanics

58 3 Generalizing What We Learntg(T ⋆ , p ⋆ , H ⋆ ) = u(T ⋆ , p ⋆ , H ⋆ ) − T ⋆ s(T ⋆ , p ⋆ , H ⋆ ) + p ⋆ v(T ⋆ , p ⋆ , H ⋆ )− H ⋆ m(T ⋆ , p ⋆ , H ⋆ ) , (3.73)where the definitions of T ⋆ and all the other variables are self-explanatory (e.g.,T ⋆ ≡ T/N ⋆ ). In other words, in order to have finite thermodynamic equations ofstates, we must in general express them in the (T ⋆ , p ⋆ , H ⋆ ) variables. If α/d >1, thisprocedure recovers the usual equations of states, and the usual extensive(G, U, S, V, M) and intensive (T, p, H) thermodynamic variables. But, if 0 ≤α/d ≤ 1, the situation is more complex, and we realize that three, instead of thetraditional two, classes of thermodynamic variables emerge. We may call themextensive (S, V, M, N), pseudo-extensive (G, U), and pseudo-intensive (T, p, H)variables. All the energy-type thermodynamical variables (G, F, U) giverisetopseudo-extensive ones, whereas those which appear in the usual Legendre thermodynamicalpairs give rise to pseudo-intensive ones (T, p, H,μ) and extensive ones(S, V, M, N) (See Figs. 3.10 and 3.11).The possibly long-range interactions within Hamiltonian (3.65) refer to the dynamicalvariables themselves. There is another important class of Hamiltonians,where the possibly long-range interactions refer to the coupling constants betweenlocalized dynamical variables. Such is, for instance, the case of the following classicalHamiltonian:H = K + V =N∑i=1L 2 i2I − ∑ i≠ jJ x s x i sx j+ J y s yi s y j+ J z s z i sz jr α ij(α ≥ 0) , (3.74)where {L i } are the angular momenta, I the moment of inertia, {(si x, s yi , sz i)} are thecomponents of classical rotators, (J x , J y , J z ) are coupling constants, and r ij runsover all distances between sites i and j of a d-dimensional lattice. For example, forFig. 3.11 For long-range interactions (0 ≤ α/d ≤ 1) we have three classes of thermodynamicvariables, namely the pseudo-intensive (scaling with N ⋆ ), pseudo-extensive (scaling with NN ⋆ ),and extensive (scaling with N) ones. For short-range interactions (α/d > 1) the pseudo-intensivevariables become intensive (independent from N), and the pseudo-extensive variables merge withthe extensive ones, all being now extensive (scaling with N), thus recovering the traditional twotextbook classes of thermodynamical variables.