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

Rodríguez, Antonio; Nobre, Fernando D.; Tsallis, Constantino

doi:10.3390/e21010031

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

Antonio Rodríguez

^1,2,*

Fernando D. Nobre

³ and

Constantino Tsallis

^3,4,5

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

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

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.

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

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}

1 / r_{i j}^{α}

(

α \geq 0

), where the limit

α = 0

(

α \to \infty

) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio

α / d > 1

(

0 \leq α / d \leq 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 \sim 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 \leq α / d \leq 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 \leq α / d \leq 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

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MDPI and ACS Style

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. https://doi.org/10.3390/e21010031

AMA Style

Rodríguez A, Nobre FD, Tsallis C. d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies. Entropy. 2019; 21(1):31. https://doi.org/10.3390/e21010031

Chicago/Turabian Style

Rodríguez, Antonio, Fernando D. Nobre, and Constantino Tsallis. 2019. "d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies" Entropy 21, no. 1: 31. https://doi.org/10.3390/e21010031

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

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