Nonextensive Statistical Mechanics

5.4 Many-Body Long-Range-Interacting Hamiltonian Systems 185(see details in [177]) into H ′ = H/N ⋆ , whereH ′ = 1 2N∑i=1(p ′ i )2 + 12N ⋆ ∑i≠ j1 − cos(θ i − θ j )r α ij(α ≥ 0) . (5.46)It is in this form, and omitting the “primes,” that this system is usually presentedin the literature. Although physically meaningless (since it involves microscopiccoupling constants which, through N ⋆ , depend on N), it has the advantage of being(artificially) extensive, such as the familiar short-range-interacting ones. Unless explicitlydeclared otherwise, we shall from now on conform to this frequent use. Forthe α = 0 instance, it will present the widespread mean-field-like form, frequentlyreferred to in the literature as the HMF model [833] (see also [834–836]),H = 1 2N∑i=1p 2 i + 12N∑[1 − cos(θ i − θ j )]. (5.47)i≠ jThis model, as well as its generalizations and extensions, are being intensivelystudied (see [372, 373, 376, 377] and references therein) in the literature throughvarious procedures. A particularly interesting one is the molecular dynamical approachof an isolated N-sized system. Its interest comes from the fact that this is afirst-principles calculation, since it is exclusively based on Newton’s law of motion,and therefore constitutes a priviledged viewpoint to try to understand in depth themicroscopic dynamical foundations of statistical mechanics 5 (both the BG and thenonextensive theories). The time evolution of the system depends on the class ofinitial conditions that are being used. Two distinct such classes are frequently used,namely thermal-equilibrium-like ones (characterized by a initial Gaussian distributionof velocities) and the water-bag-ones (characterized by a initially uniformdistribution of velocities within an interval compatible with the assumed total energyU(N) of the system). The initial angle distribution ranges usually from allspins being aligned (say to the θ i = 0 axis), which corresponds to maximal averagemagnetization (i.e., m = 1), to angularly completely disordered spins, whichcorresponds to minimal average magnetization (i.e., m = 0). The simplest model(HMF) presents, in its microcanonical version, a second order phase transitionat the scaled total energy u c = 0.75, where u ≡ U(N)/NN ⋆ or u ≡ U(N)/N,depending on whether we are adopting Hamiltonian (5.44) or (5.46), respectively.For 0 ≤ u ≤ u c , the system tends to be ordered in a ferromagnetic phase, whereasfor u > u c it is in a disordered paramagnetic phase.5 This is sometimes referred to as the Boltzmann program. Boltzmann himself died without havingaccomplished it, and rigorously speaking it so remains until today!