# Nonextensive Footprints in Dissipative and Conservative Dynamical Systems

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## Abstract

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## 1. Introduction

## 2. Few Degrees of Freedom

#### 2.1. Dissipative Models

#### 2.1.1. Sensitivity to Initial Conditions

#### 2.1.2. Relaxation Dynamics

#### 2.1.3. Central Limit Behavior

#### 2.2. Conservative Models

## 3. Many Degrees of Freedom

#### 3.1. Coupled Pendula Models

#### 3.2. The Kuramoto Model

#### 3.3. The HMF Model

#### 3.4. Classical Inertial Rotors in d Dimensions

## 4. Asymptotically Scale-Free Networks

## 5. Clues Concerning the Domains of Validity of BG and q-Statistics

#### 5.1. Clue I—Asymptotically Scale-Free Networks

#### 5.2. Clue II—Momenta and Energy Distributions of Classical Many-Body Hamiltonians

#### 5.3. Clue III—Maximal Lyapunov Exponent of the Classical $\alpha $-Heisenberg Inertial Ferromagnet

#### 5.4. Clue IV—Viscous-Fluid Spherical Capacitor

#### 5.5. Clue V—Overdamped Many-Body Systems

#### 5.6. Clue VI—Kinetics of Point Defects in Short-Range-Interacting Hamiltonians

#### 5.7. The Intriguing Case Of The Lennard–Jones’ Two-Body Potentials For Modeling Real Gases

## 6. Final Remarks and Conclusions

- -
- The spatially averaged two-body potential ${\int}_{1}^{\infty}dr\phantom{\rule{0.166667em}{0ex}}{r}^{d-1}{r}^{-\alpha}$ is finite for $\alpha /d>1$, and diverges for $0\le \alpha /d\le 1$. Such finiteness is necessary but not sufficient for all the BG thermostatistical quantities to be finite. Consistently, the total internal energy is thermodynamically extensive for $\alpha /d>1$, and superextensive for $0\le \alpha /d\le 1$.
- -
- The finiteness of the spatially averaged two-body potential is necessary for BG statistical mechanics to be applicable but it is not sufficient. Its full applicability requires also the finiteness of all the associated momenta, i.e., ${\int}_{1}^{\infty}dr\phantom{\rule{0.166667em}{0ex}}{r}^{d-1}{r}^{-\alpha}\phantom{\rule{0.166667em}{0ex}}{r}^{n}$ must also be finite for $n=1,2,3,\cdots $. Such a strong requirement is satisfied only in the $\alpha /d\to \infty $ limit of the present power-law models, or for Hamiltonians involving interactions only among relatively close neighbors (first, second, and third neighbors, for instance).
- -
- The maximal Lyapunov exponent appears to decay with the number N of elements as ${N}^{-\kappa (\alpha /d)}$ with $\kappa (\alpha /d)\ge 0$. It is possible that roughly $\kappa (\alpha /d)\simeq {q}_{E}(\alpha /d)-1$, for all values of $\alpha /d$. If so, we can guarantee strong chaos (hence, mixing in phase-space, hence ergodicity) in the $N\to \infty $ limit only for $\alpha /d\to \infty $. In all other cases, i.e., $0\le \alpha /d<\infty $, we would have, in the $N\to \infty $, weak chaos, and therefore ergodicity and mixing will not be guaranteed. This is consistent with the failure of the BG theory which is observed (nonexponential energy distribution, and non-Gaussian momenta distribution).
- -
- The fact that a BG partition function, as well as other thermostatistically relevant quantities (e.g., equations of states, energy and velocity distributions) are computable (within analytical mean-field methods, for example) is necessary but not sufficient for the BG theory to satisfactorily describe the system.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Log–log plot of the sensitivity function versus time for the logistic map at the chaos threshold (${a}_{c}=1.4011551890920505$).

**Figure 3.**Normalized probability distribution functions of the logistic map as the chaos threshold point is approached via Huberman–Rudnick law. Numerical convergence to a q-Gaussian with $q\simeq 1.65$ with log-periodic oscillations is appreciated. Figure reproduced from Ref. [36].

**Figure 4.**Phase space portrait for four representative K values of the standard map. As K values increases, the appearance of the chaotic sea is evident.

**Figure 6.**Normalized probability distribution functions of the standard map for $K=0$ case. In this simulation, $N={2}^{22}$ and $M={10}^{8}$. The q-Gaussian fit is realized with $q\simeq 1.94$.

**Figure 7.**(

**Lower panel**) The asymptotic order parameter r of the Kuramoto model is plotted as a function of the coupling K for a system of N = 20,000 oscillators and for several bounded Gaussian distributions $g\left(\omega \right)$, with increasing standard deviations $\sigma $ and $\omega \in [-3,3]$. (

**Upper panel**) The largest Lyapunov exponent is plotted as function of K. Increasing $\sigma $ its behavior changes continuously from the characteristic peak of the Gaussian $g\left(\omega \right)$ to the sharper peak characteristic of the uniform one. Each dot represents an average over 10 runs. See text. Figure reproduced from Ref. [54].

**Figure 8.**Temporal evolution of the order parameter $r\left(t\right)$ near the phase transition for several runs and different distributions $g\left(\omega \right)$: the Gaussian one (panels

**a**–

**c**) and the uniform one (panels

**d**–

**f**). Metastable states are visible in panels (

**a**,

**f**). See text. Figure reproduced from Ref. [54].

**Figure 9.**CLT PDFs for a Kuramoto system with N = 20,000 and K below the critical threshold for both uniform (left panels, $K=0.1$) and Gaussian (right panels, $K=0.6$) velocity distributions. In both cases, $\delta =200$ and $n=220$: due to the weak chaos regime, robust fat-tailed attractors appear (bottom panels), well fitted by q-Gaussians (full lines) functions with $q=1.7$ and different values of $\beta $. A standard Gaussian with unitary variance (dashed line) is reported for comparison in each plot. In the upper rows of the figure, the order parameter $r\left(t\right)$ (top panels) and the LLE (middle panels) are also reported as function of time, after a transient of 100 time-steps. See text. Figure reproduced from Ref. [60].

**Figure 10.**The temperature T and the magnetization M are reported as a function of the energy per particle U in the ferromagnetic case. Symbols refer to equilibrium molecular dynamics simulations for $N={10}^{2}$ and ${10}^{3}$, whereas the solid lines refer to the canonical equilibrium prediction obtained analytically (see text). The vertical dashed line indicates the critical energy density ${U}_{c}=0.75$ and ${\beta}_{c}=\frac{1}{{T}_{c}}=2$. Figure reproduced from Ref. [48]; see also text.

**Figure 11.**Microcanonical numerical simulations for $N=500$ and energy density $U=0.69$. In the central part of the figure, we twice plot the average kinetic energy per particle (which gives the temperature) as a function of time (filled red triangles). One can see a long matastable plateau (QSS regime) preceeding the relaxation toward the Boltzmann–Gibbs equilibrium temperature (BG regime). In the BG regime, one finds as expected, a very good agreement with the equilibrium thermodynamics value for the temperature, panel (

**d**). In this regime, the velocity PDFs reported in panel (

**b**) are Gaussians. At variance, in the QSS region, we can see strong deviations from the expected equilibrium temperature. Here the specific heat becomes negative, (panel

**c**), and the velocity PDFs (reported in panel

**a**), are very different from the Gaussian equilibrium curve, reported as a full line for comparison. Figure reproduced from Ref. [48].

**Figure 12.**Numerical simulations for the HMF model for N = 50,000, U = 0.69 and water bag initial conditions in the QSS regime. (

**a**) PDFs of single-rotor velocities at the times t = 200 and t = 250,000 (ensemble average over 100 realizations). (

**b**) Time average PDF for the variable y calculated over only one single realization in the QSS regime and after a transient time of 200 units. In this case we used $\delta =100$ and $n=5000$. A q-Gaussian reproduces very well the calculated PDF both in the tails and in the central part (see inset). Figure reproduced from Ref. [66] (see also text for further details).

**Figure 13.**Inertial $\alpha $-XY d-dimensional model (for $d=1,2,3$) for $\alpha /d=0.9$.

**Left**: ${q}_{p}$-Gaussian distribution of momenta (for comparison, a Maxwellian distribution is indicated in dashed line).

**Right**: ${q}_{E}$-exponential distribution of energies (for comparison, a BG distribution is indicated in dashed line). Both distributions are averaged along the very long-time interval indicated in the insets. Figure reproduced from Ref. [72].

**Figure 15.**Inertial $\alpha $-Heisenberg d-dimensional model (for $d=1,2,3$) for $\alpha /d=0.9$.

**Top**: ${q}_{L}$—Gaussian distribution of momenta.

**Bottom**: ${q}_{E}$—exponential distribution of energies. Both distributions are averaged after a very long time has elapsed. Figure reproduced from Ref. [76].

**Figure 16.**

**Top**: $\alpha /d$-dependence of the $\kappa $ exponent for the maximal Lyapunov exponent $\lambda \left(N\right)\sim {N}^{-\kappa (\alpha /d)}$ of the d-dimensional model $\alpha $-Heisenberg. This figure is reproduced from Ref. [76].

**Bottom**: Top panel in Log-linear scale. For $\alpha /d=2$, we obtain $\kappa \simeq 0.001$.

**Figure 17.**Sample of a $N=100$ network for $(d,{\alpha}_{A},\eta ,{w}_{0})=(2,1,5,1,1)$. As can be seen, for this choice of parameters, hubs (highly connected nodes) naturally emerge in the network. Each link has a specific width ${w}_{ij}$ and the total energy ${\epsilon}_{i}$ associated to the site i will be given by half of the sum over all link widths connected to the site i (see zoom of site i). This figure is reproduced from Ref. [84].

**Figure 18.**The index q is associated with ${e}_{q}^{-{\beta}_{q}\phantom{\rule{0.166667em}{0ex}}\epsilon}$, as a function of ${\alpha}_{A}/d$, where ${\alpha}_{A}$ characterizes the distance range of the preferential attachment and d is the dimension of the growing network. All the points corresponding to typical values of ${w}_{c}$ have been satisfactorily fitted with the analytical expression indicated inside the figure. Figure reproduced from Ref. [85] where further details can be seen.

**Figure 19.**Dependence on $\alpha $ (exponent characterizing the range of the two-body interactions) of the index ${q}_{p}$ corresponding to the ${q}_{p}$-Gaussian momenta distributions of the $\alpha $-Fermi–Pasta–Ulam–Tsingou one-dimensional $\beta $ model.

**Top**: Figure reproduced from Ref. [87].

**Bottom**: Figure reproduced from Ref. [88].

**Figure 20.**Conjectural $(\alpha /d)$-dependence of each momenta and energy indices ${q}_{p}$ and ${q}_{E}$, respectively, for classical d-dimensional many-body Hamiltonian models; ${q}_{p}\left(0\right)=5/3$ and ${q}_{E}\left(0\right)=4/3$ are plausible values. Furthermore, a more general conjecture emerges, namely $\frac{2}{{q}_{p}(\alpha /d)-1}=\frac{1}{{q}_{E}(\alpha /d)-1}$; hence $\frac{{q}_{p}(\alpha /d)-1}{{q}_{E}(\alpha /d)-1}=2\phantom{\rule{0.166667em}{0ex}},\forall (\alpha /d)$ (see Figure 21); the BG instance (${q}_{p}={q}_{E}=1$) is attained only in the $\alpha /d\to \infty $ limit.

**Figure 21.**A conjectural scaling, namely $\frac{{q}_{p}(\alpha /d)-1}{{q}_{E}(\alpha /d)-1}=2$, $\forall (\alpha /d)$, cannot be excluded from the low-precision data in [72]; such scaling is in fact analytically verified for the Coherent Noise Model, as shown in Equation (4.85) of [89]. Clearly, more sensibly precise data are needed to draw conclusions. In any case, this conjecture is consistent with the hypothesis that both the distribution probability of the energy and that of the kinetic energy (which is proportional to ${p}^{2}$) behave proportionally to ${e}_{{q}_{E}}^{-{\beta}_{q}\phantom{\rule{0.166667em}{0ex}}energy}$.

**Figure 22.**Alternative conjecture for the $(\alpha /d)$-dependence of each momenta and energy indices ${q}_{p}$ and ${q}_{E}$, respectively, for classical d-dimensional many-body models. In this case, the three long-range [$0\le \alpha /d\le 1$], intermediate [$1<\alpha /d<{(\alpha /d)}_{c}$], and short-range [$\alpha /d\ge {(\alpha /d)}_{c}$] regions for the two-body interactions are apparent.

**Figure 23.**

**Top**: Spherical capacitor with equally charged and equally massive small particles in overdamped motion between two oppositely charged conductive spheres.

**Bottom**: Radial density distribution at the stationary state (for comparison, the $q=1$ solution, i.e., Debye–Huckel or Yukawa, is indicated as well in dashed line). Figure reproduced from Ref. [90].

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**MDPI and ACS Style**

Rodríguez, A.; Pluchino, A.; Tirnakli, U.; Rapisarda, A.; Tsallis, C.
Nonextensive Footprints in Dissipative and Conservative Dynamical Systems. *Symmetry* **2023**, *15*, 444.
https://doi.org/10.3390/sym15020444

**AMA Style**

Rodríguez A, Pluchino A, Tirnakli U, Rapisarda A, Tsallis C.
Nonextensive Footprints in Dissipative and Conservative Dynamical Systems. *Symmetry*. 2023; 15(2):444.
https://doi.org/10.3390/sym15020444

**Chicago/Turabian Style**

Rodríguez, Antonio, Alessandro Pluchino, Ugur Tirnakli, Andrea Rapisarda, and Constantino Tsallis.
2023. "Nonextensive Footprints in Dissipative and Conservative Dynamical Systems" *Symmetry* 15, no. 2: 444.
https://doi.org/10.3390/sym15020444