d-dimensional classical Heisenberg model with arbitrarily-ranged interactions: Lyapunov exponents and distributions of momenta and energies

Abstract

We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model (d = 1, 2, 3) with interactions decaying with the distance rij as 1/rijα (α ≥ 0), where the limit α= 0 (α→ ∞) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α/d > 1 (0 ≤ α/d ≤ 1) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent l in the form λ ~ N-k, where k(α/d) depends only on the ratio α/d; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0 ≤ α/d ≤ 1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0 ≤ α/d ≤ 1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α/d > 1 regime. The universality that we observe for the probability distributions with regard to the ratio α/d makes this model similar to the α-XY and α-Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.