Nonextensive Statistical Mechanics

184 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics10 7α = 0.0α = 0.310 6t c ~ a –1.84α = 0.6α = 0.9α = 1.510 5α = 2.0α = 3.0t c 10 410 310 210 110 –3 10 –2 10 –1aFig. 5.35 Temperature dependence on a. Fixed constants are N = 100 and b = 2. Initial conditionscorrespond to θ 0 = 0.5, δθ = 0.5, p 0 = 0.3, and δp = 0.05. We averaged over 100realizations (from [359]).where the rotors are localized on a lattice (e.g., a translationally invariant Bravaislattice, a quasi-crystal, a hierarchical network). If the lattice is a d-dimensionalhypercubic one (with periodic boundary conditions) we have r ij = 1, 2, 3,... ifd = 1, r ij = 1, √ 2, 2,... if d = 2, and r ij = 1, √ 2, √ 3, 2,... if d = 3. Thepotential energy has been written in this particular manner so that its value forthe ground state (i.e., θ i = θ j ∀(i, j)) vanishes in all cases. We have consideredunit momenta of inertia and unit first-neighbor coupling constant without loss ofgenerality, and (p i ,θ i ) are conjugate canonical pairs. Due to the periodic boundaryconditions, the model is defined on a torus in d dimensions (i.e., a ring for d = 1).Consequently, between any (i, j) pair of spins, there are more than one distances;in every case we consider as r ij in the Hamiltonian the minimal of those distances.The model basically is a classical inertial XY ferromagnet (coupled rotators), andthe limiting cases α →∞and α = 0 correspond to the first-neighbor and meanfield-likemodels, respectively. Clearly, the α = 0 case does not depend on theparticular lattice on which the spins are localized. This Hamiltonian is extensive (inthe thermodynamical sense) if α/d > 1, and nonextensive if 0 ≤ α/d ≤ 1. Indeed,in contrast with its kinetic energy, which scales like N, the potential energy scaleslike NN ⋆ , whereN ⋆ ≡N∑ 1r α j=1 ij. (5.45)See also Eq. (3.69). For instance, for α = 0, N ⋆ = N , and for α/d ≥ 1 andN →∞, N ⋆ → constant. Since the variables {p i } involve a first derivative withrespect to time, if we define t ′ = √ N ⋆ t, Hamiltonian H in Eq. (5.44) is transformed