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

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. The $\alpha =0$ and $\alpha \to \infty $ Limiting Cases

#### 2.2. Equations of Motion

## 3. Results

#### 3.1. Time-Averaged Momenta and Energy One-Particle Distributions

#### 3.2. Nearest-Neighbour Limit

#### 3.3. Size-Scaling of the Largest Lyapunov Exponent

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Stanley, H.E. Introduction to Phase Transitions and Critical Phenomena; Oxford University Press: Oxford, UK, 1971. [Google Scholar]
- Thompson, C.J. Classical Equilibrium Statistical Mechanics; Oxford University Press: Oxford, UK, 1988. [Google Scholar]
- Pathria, R.K.; Beale, P.D. Statistical Mechanics, 3rd ed.; Elsevier: Amsterdam, The Netherlands, 2011. [Google Scholar]
- Fisher, M.E. Magnetism in One-Dimensional Systems-The Heisenberg Model for Infinite Spin. Am. J. Phys.
**1963**, 32, 343–346. [Google Scholar] [CrossRef] - Joyce, G.S. Classical Heisenberg Model. Phys. Rev.
**1967**, 155, 478. [Google Scholar] [CrossRef] - Stanley, H.E. Exact Solution for a Linear Chain in Isotropic Interacting Classical Spins of Arbitrary Dimensionality. Phys. Rev.
**1969**, 179, 570. [Google Scholar] [CrossRef] - Tsallis, C. Classical Statistical Approach to Anisotropic Two-Dimensional XY-Model. Il Nuovo Cimento B
**1976**, 34, 411–435. [Google Scholar] [CrossRef] - Mattis, D.C. Transfer matrix in plane-rotator model. Phys. Lett.
**1984**, 104A, 357–360. [Google Scholar] [CrossRef] - Butera, P.; Caravati, G. Phase transitions and Lyapunov characteristic exponents. Phys. Rev. A
**1987**, 36, 962. [Google Scholar] [CrossRef] - Escande, D.; Kantz, H.; Livi, R.; Ruffo, S. Self-Consistent Check of the Validity of Gibbs Calculus Using Dynamical Variables. J. Stat. Phys.
**1994**, 76, 605–626. [Google Scholar] [CrossRef] - Leoncini, X.; Verga, A.D.; Ruffo, S. Hamiltonian dynamics and the phase transition of the XY model. Phys. Rev. E
**1998**, 57, 6377. [Google Scholar] [CrossRef] - Antoni, M.; Ruffo, S. Clustering and relaxation in Hamiltonian long-range dynamics. Phys. Rev. E
**1995**, 52, 2361. [Google Scholar] [CrossRef] - Firpo, M.-C. Analytic estimation of the Lyapunov exponent in a mean-field model undergoing a phase transition. Phys. Rev. E
**1998**, 57, 6599. [Google Scholar] [CrossRef] - Latora, V.; Rapisarda, A.; Ruffo, S. Lyapunov Instability and Finite Size Effects in a System with Long-Range Forces. Phys. Rev. Lett.
**1998**, 80, 692. [Google Scholar] [CrossRef] - Latora, V.; Rapisarda, A.; Ruffo, S. Superdiffusion and Out-of-Equilibrium Chaotic Dynamics with Many Degrees of Freedom. Phys. Rev. Lett.
**1999**, 83, 2104. [Google Scholar] [CrossRef] - Latora, V.; Rapisarda, A.; Tsallis, C. Non-Gaussian equilibrium in a long-range Hamiltonian system. Phys. Rev. E
**2001**, 64, 056134. [Google Scholar] [CrossRef] [PubMed][Green Version] - Barré, J.; Bouchet, F.; Dauxois, T.; Ruffo, S. Out-of-Equilibrium States and Statistical Equilibria of an Effective Dynamics in a System with Long-Range Interactions. Phys. Rev. Lett.
**2002**, 89, 110601. [Google Scholar] [CrossRef] [PubMed] - Pluchino, A.; Latora, V.; Rapisarda, A. Metastable states, anomalous distributions and correlations in the HMF model. Physica D
**2004**, 193, 315–328. [Google Scholar] [CrossRef][Green Version] - Pluchino, A.; Latora, V.; Rapisarda, A. Glassy phase in the Hamiltonian mean-field model. Phys. Rev. E
**2004**, 69, 056113. [Google Scholar] [CrossRef] [PubMed] - Pluchino, A.; Latora, V.; Rapisarda, A. Dynamics and thermodynamics of a model with lon-range interactions. Contin. Mech. Thermodyn.
**2004**, 16, 245–255. [Google Scholar] - Moyano, L.G.; Anteneodo, C. Diffusive anomalies in a long-range Hamiltonian system. Phys. Rev. E
**2006**, 74, 021118. [Google Scholar] [CrossRef] - Pluchino, A.; Rapisarda, A.; Tsallis, C. Nonergodicity and central-limit behavior for long-range Hamiltonians. EPL
**2007**, 80, 26002. [Google Scholar] [CrossRef][Green Version] - Pluchino, A.; Rapisarda, A.; Tsallis, C. A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model. Physica A
**2008**, 387, 3121–3128. [Google Scholar] [CrossRef] - Anteneodo, C.; Tsallis, C. Breakdown of Exponential Sensitivity to Initial Conditions: Role of the Range of Interactions. Phys. Rev. Lett.
**1998**, 24, 5313. [Google Scholar] [CrossRef] - Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
- Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Campa, A.; Giansanti, A.; Moroni, D. Canonical solution of a system of long-range interacting rotators on a lattice. Phys. Rev. E
**2000**, 62, 303. [Google Scholar] [CrossRef] - Tamarit, F.; Anteneodo, C. Rotators with Long-Range Interactions: Connection with the Mean-Field Approximation. Phys. Rev. Lett.
**2000**, 84, 208. [Google Scholar] [CrossRef] [PubMed] - Campa, A.; Giansanti, A.; Moroni, D.; Tsallis, C. Classical spin systems with long-range interactions: Universal reduction of mixing. Phys. Lett. A
**2001**, 286, 251–256. [Google Scholar] [CrossRef] - Firpo, M.-C.; Ruffo, S. Chaos suppression in the large size limit for long-range systems. J. Phys. A Math. Gen.
**2001**, 34, L511. [Google Scholar] [CrossRef] - Anteneodo, C.; Vallejos, R.O. Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach. Phys. Rev. E
**2001**, 65, 016210. [Google Scholar] [CrossRef] - Cabral, B.J.C.; Tsallis, C. Metastability and weak mixing in classical long-range many-rotator systems. Phys. Rev. E
**2002**, 66, 065101. [Google Scholar] [CrossRef][Green Version] - Cirto, L.J.L.; Assis, V.R.V.; Tsallis, C. Influence of the interaction range on the thermostatistics of a classical many-body system. Physica A
**2014**, 39, 286–296. [Google Scholar] [CrossRef] - Cirto, L.J.L.; Rodríguez, A.; Nobre, F.D.; Tsallis, C. Validity and failure of the Boltzmann weight. EPL
**2018**, 123, 30003. [Google Scholar] [CrossRef] - Antonopoulos, C.G.; Christodoulidi, H. Weak chaos detection in the Fermi-Pasta-Ulam-α system using q-Gaussian statistics. Int. J. Bifurc. Chaos
**2011**, 21, 2285–2296. [Google Scholar] [CrossRef] - Christodoulidi, H.; Tsallis, C.; Bountis, T. Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics. EPL
**2014**, 108, 40006. [Google Scholar] [CrossRef][Green Version] - Christodoulidi, H.; Bountis, T.; Tsallis, C.; Drossos, L. Dynamics and Statistics of the Fermi-Pasta-Ulam β-model with different ranges of particle interactions. J. Stat. Mech. Theory Exp.
**2016**, 12, 123206. [Google Scholar] [CrossRef] - Bagchi, D.; Tsallis, C. Sensitivity to initial conditions of a d-dimensional long-range interacting quartic Fermi-Pasta-Ulam model: Universal scaling. Phys. Rev. E
**2016**, 93, 062213. [Google Scholar] [CrossRef] [PubMed] - Bagchi, D.; Tsallis, C. Fermi-Pasta-Ulam-Tsingou problems: Passage from Boltzmann to q-statistics. Physica A
**2018**, 491, 869–873. [Google Scholar] [CrossRef] - Brito, S.; da Silva, L.R.; Tsallis, C. Role of dimensionality in complex networks. Sci. Rep.
**2016**, 6, 27992. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nunes, T.C.; Brito, S.; da Silva, L.R.; Tsallis, C. Role of dimensionality in preferential attachment growth in the Bianconi-Barabási model. J. Stat. Mech. Theory Exp.
**2017**, 093402. [Google Scholar] [CrossRef] - Moreira, A.A.; Vieira, C.M.; Carmona, H.A.; Andrade, J.S., Jr.; Tsallis, C. Overdamped dynamics of particles with repulsive power-law interactions. Phys. Rev. E
**2018**, 98, 032138. [Google Scholar] [CrossRef] - Rapaport, D.C.; Landau, D.P. Critical dynamics of a dynamical version of the classical Heisenberg model. Phys. Rev. E
**1996**, 53, 4696. [Google Scholar] [CrossRef] - Nobre, F.D.; Tsallis, C. Classical infinite-range-interaction Heisenberg ferromagnetic model: Metastability and sensitivity to initial conditions. Phys. Rev. E
**2003**, 68, 036115. [Google Scholar] [CrossRef][Green Version] - Nobre, F.D.; Tsallis, C. Metastable states of the classical inertial infinite-range-interaction Heisenberg ferromagnet: Role of initial conditions. Physica A
**2004**, 344, 587–594. [Google Scholar] [CrossRef] - Gupta, S.; Mukamel, D. Quasistationarity in a model of long-range interacting particles moving on a sphere. Phys. Rev. E
**2013**, 88, 052137. [Google Scholar] [CrossRef] [PubMed] - Cirto, L.J.L.; Lima, L.S.; Nobre, F.D. Controlling the range of interactions in the classical inertial ferromagnetic Heisenberg model: Analysis of metastable states. J. Stat. Mech. Theory Exp.
**2015**, 2015, P04012. [Google Scholar] [CrossRef] - Hairer, E.; Lubich, C.; Wanner, G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations; Springer Science and Business Media: Dordrecht, The Netherlands, 2006; Volume 31. [Google Scholar]
- Butusov, D.N.; Karimov, A.I.; Pyko, N.S.; Pyko, S.A.; Bogachev, M.I. Discrete chaotic maps obtained by symmetric integration. Physica A
**2018**, 509, 955. [Google Scholar] [CrossRef] - Pyko, N.S.; Pyko, S.A.; Markelov, O.A.; Karimov, A.I.; Butusov, D.N.; Zolotukhin, Y.V.; Bogachev, M.I. Assessment of cooperativity in complex systems with non-periodical dynamics: Comparison of five mutual information metrics. Physica A
**2018**, 503, 1054–1072. [Google Scholar] [CrossRef] - Benettin, G.; Galgani, L.; Strelcyn, J.-M. Kolmogorov entropy and numerical experiments. Phys. Rev. A
**1976**, 14, 2338. [Google Scholar] [CrossRef]

**Figure 1.**Probability distribution functions of time-averaged angular momenta $P\left(\langle {L}_{i}\rangle \right)$ are represented in conveniently rescaled variables, for one-, two-, and three-dimensional systems with $\mathcal{N}=$ 262,144 rotators and $\alpha /d=0.9$. The three data sets collapse into a q-Gaussian (full line), $P\left(\langle {L}_{i}\rangle \right)={P}_{0}{exp}_{{q}_{L}}(-{\beta}_{L}{\left(\langle {L}_{i}\rangle {P}_{0}\right)}^{2})$ (cf. Equation (4)), with ${q}_{L}(\alpha /d)=1.42$ and ${\beta}_{L}=4.44$; the maximum values ${P}_{0}\equiv P(\langle {L}_{i}\rangle =0)$ varied slightly with the lattice dimensionality (see text). In the left inset we preset the evolution in time of the kinetic temperature. A first plateau at a temperature close to ${T}_{\mathrm{QSS}}(\alpha /d)=U-\frac{1}{2}$, whose duration depends on d, is observed. After the transition to the second plateau, the temperature reaches the equilibrium value ${T}_{\mathrm{BG}}(\alpha /d)=0.3118$ predicted by the BG caloric curve [33]; the time window [60,000, 100,000] used for the time averages is indicated. In the inset on the right we represent the same data in the q-logarithm $({q}_{L}=1.42)$ of Equation (2) versus ${\left(\langle {L}_{i}\rangle {P}_{0}\right)}^{2}$, where the slope of the full straight line yields the value of ${\beta}_{L}$.

**Figure 2.**Probability distribution functions of time-averaged angular momenta $P\left(\langle {L}_{i}\rangle \right)$ are represented in conveniently rescaled variables $({P}_{0}\equiv P(\langle {L}_{i}\rangle =0))$ for one-, two-, and three-dimensional systems with $\mathcal{N}=$ 262,144 rotators and $\alpha /d=1.6$. The data cannot be adequately fitted by a single q-Gaussian; a Gaussian (dashed line) is exhibited for comparison, showing a good fitting around the central region only, i.e., for small values of $\langle {L}_{i}\rangle $. The left inset exhibits the time evolution of the kinetic temperature: one sees that the QSS is not present in the short-range regime and the system rapidly reaches its equilibrium temperature which depends on $\alpha $ and d separately; the same time window shown in Figure 1 has been used for the time averages.

**Figure 3.**Probability distribution functions of time-averaged individual energies $P\left(\langle {E}_{i}\rangle \right)$ are represented versus $\langle {E}_{i}\rangle $ for one-, two-, and three-dimensional systems. The parameters are the same as in Figure 1, i.e., $\mathcal{N}=$ 262,144 rotators, $\alpha /d=0.9$, and the same time window. All data are well fitted by a shifted q-exponential (cf. Equation (5)) of the form $A{e}_{{q}_{E}}^{-{\beta}_{E}(\langle {E}_{i}\rangle -\mu )}$, with the values of A, ${q}_{E}$, ${\beta}_{E}$, and $\mu $ specified. In the inset we exhibit the same data in the q-logarithm $({q}_{E}=1.3)$ of Equation (2) versus $\langle {E}_{i}\rangle $, which, as expected, approaches a straight line.

**Figure 4.**Analytic caloric curve for the Boltzmann-Gibbs (BG) long-range (full line) and one-dimensional nearest-neighbour (black dashed line) models compared with nearest-neighbour (NN) molecular dynamics simulation results for dimensions $d=1$, 2 and 3 and system size $\mathcal{N}=$ 46,656 $={\left(216\right)}^{2}={\left(36\right)}^{3}$. Fully magnetized initial state has been considered in all cases. The dashed lines connecting $d=2$ and $d=3$ numerical results for $U>{U}_{c}$ correspond to Equation (12) with $\mathcal{L}$ evaluated at $1/2dT$.

**Figure 5.**Analytic $M-U$ curve for the BG long-range (full line) model compared with nearest-neighbour (NN) molecular dynamics simulation results for dimensions $d=1$, 2 and 3 and system size $\mathcal{N}=$ 46,656 $={\left(216\right)}^{2}={\left(36\right)}^{3}$. Fully magnetized initial state has been considered in all cases.

**Figure 6.**Maximum Lyapunov exponent as a function of the system size for $U=5.0$, $d=1$ and different values of $\alpha $. The considered system sizes are $\mathcal{N}={10}^{3},2\times {10}^{3},3\times {10}^{3}\cdots ,9\times {10}^{3}$ and ${10}^{4}$.

**Figure 7.**Maximum Lyapunov exponent as a function of the system size for $U=5.0$, $d=2$ and different values of $\alpha $. The considered system sizes are $\mathcal{N}={\left(30\right)}^{2},{\left(40\right)}^{2},{\left(50\right)}^{2}\cdots ,{\left(100\right)}^{2}$.

**Figure 8.**Maximum Lyapunov exponent as a function of the system size for $U=5.0$, $d=3$ and different values of $\alpha $. The considered system sizes are $\mathcal{N}={\left(10\right)}^{3},{\left(12\right)}^{3},{\left(14\right)}^{3},{\left(16\right)}^{3},{\left(18\right)}^{3},{\left(20\right)}^{3}$ and ${\left(22\right)}^{3}$.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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