Nonextensive Footprints in Dissipative and Conservative Dynamical Systems
Abstract
: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 -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 is finite for , and diverges for . 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 , and superextensive for .
- -
- 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., must also be finite for . Such a strong requirement is satisfied only in the 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 with . It is possible that roughly , for all values of . If so, we can guarantee strong chaos (hence, mixing in phase-space, hence ergodicity) in the limit only for . In all other cases, i.e., , we would have, in the , 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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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
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 StyleRodríguez, 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