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Entropy 2019, 21(1), 31; https://doi.org/10.3390/e21010031

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

1
GISC, Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros s/n, 28040 Madrid, Spain
2
Department of Physics, University of Warwick, Coventry CV4 7AL, UK
3
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
4
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
5
Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria
*
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)
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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 r i j as 1 / r i j α ( 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 λ in the form λ N κ , where κ ( 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. View Full-Text
Keywords: complex Hamiltonian systems; nonextensive statistical mechanics; long-ranged-interacting thermostatistics; lyapunov exponents complex Hamiltonian systems; nonextensive statistical mechanics; long-ranged-interacting thermostatistics; lyapunov exponents
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Rodríguez, A.; Nobre, F.D.; Tsallis, C. d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies. Entropy 2019, 21, 31.

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