# d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

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GISC, Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros s/n, 28040 Madrid, Spain

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Department of Physics, University of Warwick, Coventry CV4 7AL, UK

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Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Dr. Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil

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Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA

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Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria

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Author to whom correspondence should be addressed.

Received: 27 November 2018 / Revised: 22 December 2018 / Accepted: 2 January 2019 / Published: 4 January 2019

(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)

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 ${r}_{ij}$ as $1/{r}_{ij}^{\alpha}$ ( $\alpha \ge 0$ ), where the limit $\alpha =0$ ( $\alpha \to \infty $ ) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio $\alpha /d>1$ ( $0\le \alpha /d\le 1$ ) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size $\mathcal{N}$ of the maximum Lyapunov exponent $\lambda $ in the form $\lambda \sim {\mathcal{N}}^{-\kappa}$ , where $\kappa (\alpha /d)$ depends only on the ratio $\alpha /d$ ; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime $0\le \alpha /d\le 1$ (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 $0\le \alpha /d\le 1$ (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 $\alpha /d>1$ regime. The universality that we observe for the probability distributions with regard to the ratio $\alpha /d$ makes this model similar to the $\alpha $ -XY and $\alpha $ -Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.