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

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## Abstract

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## 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

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**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 ${\left(22\right)}^{3}$.

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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