{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T20:17:36Z","timestamp":1770495456069,"version":"3.49.0"},"reference-count":98,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,7]],"date-time":"2023-02-07T00:00:00Z","timestamp":1675728000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Despite its centennial successes in describing physical systems at thermal equilibrium, Boltzmann\u2013Gibbs (BG) statistical mechanics have exhibited, in the last several decades, several flaws in addressing out-of-equilibrium dynamics of many nonlinear complex systems. In such circumstances, it has been shown that an appropriate generalization of the BG theory, known as nonextensive statistical mechanics and based on nonadditive entropies, is able to satisfactorily handle wide classes of anomalous emerging features and violations of standard equilibrium prescriptions, such as ergodicity, mixing, breakdown of the symmetry of homogeneous occupancy of phase space, and related features. In the present study, we review various important results of nonextensive statistical mechanics for dissipative and conservative dynamical systems. In particular, we discuss applications to both discrete-time systems with a few degrees of freedom and continuous-time ones with many degrees of freedom, as well as to asymptotically scale-free networks and systems with diverse dimensionalities and ranges of interactions, of either classical or quantum nature.<\/jats:p>","DOI":"10.3390\/sym15020444","type":"journal-article","created":{"date-parts":[[2023,2,8]],"date-time":"2023-02-08T03:57:16Z","timestamp":1675828636000},"page":"444","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Nonextensive Footprints in Dissipative and Conservative Dynamical Systems"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7235-417X","authenticated-orcid":false,"given":"Antonio","family":"Rodr\u00edguez","sequence":"first","affiliation":[{"name":"GISC, Departamento de Matem\u00e1tica Aplicada a la Ingenier\u00eda Aeroespacial, Universidad Polit\u00e9cnica de Madrid, Plaza Cardenal Cisneros s\/n, 28040 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9368-5800","authenticated-orcid":false,"given":"Alessandro","family":"Pluchino","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Catania, 95123 Catania, Italy"},{"name":"Dipartimento di Fisica e Astronomia \u201cE. Majorana\u201d, University of Catania, 95123 Catania, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1104-0847","authenticated-orcid":false,"given":"Ugur","family":"Tirnakli","sequence":"additional","affiliation":[{"name":"Department of Physics, Faculty of Arts and Sciences, Izmir University of Economics, Izmir 35330, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8290-8183","authenticated-orcid":false,"given":"Andrea","family":"Rapisarda","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia \u201cE. Majorana\u201d, University of Catania, 95123 Catania, Italy"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dterstrasse 39, A-1090 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9387-9194","authenticated-orcid":false,"given":"Constantino","family":"Tsallis","sequence":"additional","affiliation":[{"name":"Complexity Science Hub Vienna, Josefst\u00e4dterstrasse 39, A-1090 Vienna, Austria"},{"name":"Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Penrose, O. (1970). Foundations of Statistical Mechanics: A Deductive Treatment, Pergamon.","DOI":"10.1016\/B978-0-08-013314-0.50011-X"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"264","DOI":"10.3390\/encyclopedia2010018","article-title":"Entropy","volume":"2","author":"Tsallis","year":"2022","journal-title":"Encyclopedia"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_4","unstructured":"(2023, January 10). See This Website for a Regularly Updated Bibliography. Available online: http:\/\/tsallis.cat.cbpf.br\/biblio.htm."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"114027","DOI":"10.1103\/PhysRevD.91.114027","article-title":"From QCD-based hard-scattering to nonextensive statistical mechanical descriptions of transverse momentum spectra in high-energy pp and pp\u00af collisions","volume":"91","author":"Wong","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"034019","DOI":"10.1103\/PhysRevD.101.034019","article-title":"Fractals, non-extensive statistics, and QCD","volume":"101","author":"Deppman","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1764","DOI":"10.1038\/s41598-018-20036-6","article-title":"Generalized statistical mechanics of cosmic rays: Application to positron-electron spectral indices","volume":"8","author":"Yalcin","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"110601","DOI":"10.1103\/PhysRevLett.96.110601","article-title":"Tunable Tsallis distributions in dissipative optical lattices","volume":"96","author":"Douglas","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1038\/nphys2751","article-title":"Beyond Boltzmann-Gibbs statistical mechanics in optical lattices","volume":"9","author":"Lutz","year":"2013","journal-title":"Nat. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"238301","DOI":"10.1103\/PhysRevLett.115.238301","article-title":"Experimental validation of nonextensive scaling law in confined granular media","volume":"115","author":"Combe","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"15377","DOI":"10.1073\/pnas.0503807102","article-title":"Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive","volume":"102","author":"Tsallis","year":"2005","journal-title":"Proc. Natl. Acad. Soc. USA"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1038\/261459a0","article-title":"Simple mathematical models with very complicated dynamics","volume":"261","author":"May","year":"1996","journal-title":"Nature"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/BF01020332","article-title":"Quantitative universality for a class of nonlinear transformations","volume":"19","author":"Feigenbaum","year":"1978","journal-title":"J. Stat. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/BF01107909","article-title":"The Universal Metric Properties of Nonlinear Transformations","volume":"21","author":"Feigenbaum","year":"1979","journal-title":"J. Stat. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Beck, C., and Schl\u00f6gl, F. (1993). Thermodynamics of Chaotic Systems: An Introduction, Cambridge University Press.","DOI":"10.1017\/CBO9780511524585"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0370-1573(79)90023-1","article-title":"A universal instability of many-dimensional oscillator systems","volume":"52","author":"Chirikov","year":"1979","journal-title":"Phys. Rep."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1016\/S0960-0779(96)00167-1","article-title":"Power-law sensitivity to initial conditions\u2014New entropic representation","volume":"8","author":"Tsallis","year":"1997","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1103\/PhysRevE.56.245","article-title":"Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity","volume":"56","author":"Costa","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1103\/PhysRevLett.80.53","article-title":"Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems","volume":"80","author":"Lyra","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/BF03219171","article-title":"Circular-like maps: Sensitivity to the initial conditions, multifractality and nonextensivity","volume":"11","author":"Tirnakli","year":"1999","journal-title":"Eur. Phys. J. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"036207","DOI":"10.1103\/PhysRevE.65.036207","article-title":"Asymmetric unimodal maps at the edge of chaos","volume":"65","author":"Tirnakli","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/S0378-4371(01)00649-5","article-title":"Dissipative maps at the chaos threshold: Numerical results for the single-site map","volume":"305","author":"Tirnakli","year":"2002","journal-title":"Phys. A"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Beck, C., and Schlogl, F. (1993). Thermodynamics of Chaotic Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511524585"},{"key":"ref_24","unstructured":"Molteni, A. (2015). An Efficient Method for the Computation of the Feigenbaum Constants to High Precision. arXiv, Available online: https:\/\/arxiv.org\/pdf\/1602.02357.pdf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0375-9601(00)00484-9","article-title":"The rate of entropy increase at the edge of chaos","volume":"273","author":"Latora","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/S0960-0779(01)00029-7","article-title":"Time evolution of thermodynamic entropy for conservative and dissipative chaotic maps","volume":"13","author":"Baranger","year":"2002","journal-title":"Chaos Solitons Fractals"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0375-9601(01)00570-9","article-title":"Generalization of the Kolmogorov\u2013Sinai entropy: Logistic-like and generalized cosine maps at the chaos threshold","volume":"289","author":"Tirnakli","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"6361","DOI":"10.1103\/PhysRevE.62.6361","article-title":"Convergence to the critical attractor of dissipative maps: Log-periodic oscillations, fractality, and nonextensivity","volume":"62","author":"Tirnakli","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1140\/epjb\/e2006-00064-6","article-title":"Numerical study of the oscillatory convergence to the attractor at the edge of chaos","volume":"2006 50","author":"Tonelli","year":"2006","journal-title":"Eur. Phys. J. B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"140601","DOI":"10.1103\/PhysRevLett.95.140601","article-title":"Temporal Scaling at Feigenbaum Points and Nonextensive Thermodynamics","volume":"95","author":"Grassberger","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1016\/j.physa.2006.06.003","article-title":"Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points","volume":"370","author":"Robledo","year":"2006","journal-title":"Physica A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"032613","DOI":"10.1103\/PhysRevE.77.036213","article-title":"q-deformed statistical-mechanical property in the dynamics of trajectories en route to the Feigenbaum attractor","volume":"77","author":"Robledo","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_33","unstructured":"Billingsley, P. (1968). Convergence of Probability Measures, Wiley."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/0378-4371(90)90173-P","article-title":"Brownian motion from deterministic dynamics","volume":"169","author":"Beck","year":"1990","journal-title":"Phys. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"040106","DOI":"10.1103\/PhysRevE.75.040106","article-title":"Central limit behavior of deterministic dynamical systems","volume":"75","author":"Tirnakli","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"056209","DOI":"10.1103\/PhysRevE.79.056209","article-title":"Closer look at time averages of the logistic map at the edge of chaos","volume":"79","author":"Tirnakli","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"20003","DOI":"10.1209\/0295-5075\/101\/20003","article-title":"Generalized Huberman-Rudnick scaling law and robustness of q-Gaussian probability distributions","volume":"101","author":"Ozgur","year":"2013","journal-title":"EPL"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1103\/PhysRevLett.45.154","article-title":"Scaling Behavior of Chaotic Flows","volume":"45","author":"Huberman","year":"1980","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Lichtenberg, A.J., and Lieberman, M.A. (1992). Regular and Chaotic Dynamics, Springer.","DOI":"10.1007\/978-1-4757-2184-3"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"23644","DOI":"10.1038\/srep23644","article-title":"The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics","volume":"6","author":"Tirnakli","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"126659","DOI":"10.1016\/j.physleta.2020.126659","article-title":"Cauchy distributions for the integrable standard map","volume":"384","author":"Bountis","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Ruiz, G., Tirnakli, U., Borges, E.P., and Tsallis, C. (2017). Statistical characterization of the standard map. J. Stat. Mech., 063403.","DOI":"10.1088\/1742-5468\/aa728b"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"042158","DOI":"10.1103\/PhysRevE.96.042158","article-title":"Statistical characterization of discrete conservative systems: The web map","volume":"96","author":"Ruiz","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Tirnakli, U., Tsallis, C., and Cetin, K. (2020). Dynamical robustness of discrete conservative systems: Harper and generalized standard maps. J. Stat. Mech., 063206.","DOI":"10.1088\/1742-5468\/ab8117"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"8575","DOI":"10.1038\/s41598-022-12213-5","article-title":"A generalization of the standard map and its statistical characterization","volume":"12","author":"Cetin","year":"2022","journal-title":"Sci. Rep."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Dauxois, T., Ruffo, S., Arimondo, E., and Wilkens, M. (2002). Dynamics and Thermodynamics of Systems with Long Range Interactions, Springer. Lecture Notes in Physics.","DOI":"10.1007\/3-540-45835-2"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Dauxois, T., Latora, V., Rapisarda, A., Ruffo, S., and Torcini, A. (2002). The Hamiltonian Mean Field Model: From Dynamics to Statistical Mechanics and Back, Springer. Lectures Notes in Physics.","DOI":"10.1007\/3-540-45835-2_16"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s00161-003-0170-0","article-title":"Dynamics and thermodynamics of a model with long-range interactions","volume":"16","author":"Pluchino","year":"2004","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.physd.2004.01.029","article-title":"Metastable states, anomalous distributions and correlations in the HMF model","volume":"193","author":"Pluchino","year":"2004","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1051\/epn:2005607","article-title":"Nonextensive thermodynamics and glassy behaviour","volume":"36","author":"Rapisarda","year":"2005","journal-title":"Europhys. News"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence, Springer.","DOI":"10.1007\/978-3-642-69689-3"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2730","DOI":"10.1103\/PhysRevLett.68.2730","article-title":"Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping","volume":"68","author":"Strogatz","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.physa.2006.01.039","article-title":"Metastability in the Hamiltonian Mean Field model and Kuramoto model","volume":"365","author":"Pluchino","year":"2006","journal-title":"Phys. A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"10007","DOI":"10.1209\/0295-5075\/85\/10007","article-title":"Phase Transitions and Chaos in Long-Range Models of Coupled Oscillators","volume":"85","author":"Miritello","year":"2009","journal-title":"Europhys. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1103\/RevModPhys.77.137","article-title":"The Kuramoto model: A simple paradigm for synchronization phenomena","volume":"77","author":"Bonilla","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"R67","DOI":"10.1088\/0951-7715\/28\/3\/R67","article-title":"Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators","volume":"28","author":"Panaggio","year":"2015","journal-title":"Nonlinearity"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"094822","DOI":"10.1063\/1.4961435","article-title":"Bistability of patterns of synchrony in Kuramoto oscillators with inertia","volume":"26","author":"Belykh","year":"2016","journal-title":"Chaos Interdisc. J. Nonlinear Sci."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"19621","DOI":"10.1038\/s41598-019-54769-9","article-title":"Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs","volume":"9","author":"Odor","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1098\/rsos.210122","article-title":"New forms of structure in ecosystems revealed with the Kuramoto model","volume":"8","author":"Vandermeer","year":"2021","journal-title":"R. Soc. Open Sci."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"4818","DOI":"10.1016\/j.physa.2009.08.023","article-title":"Central limit behavior in the Kuramoto model at the \u2018edge of chaos\u2019","volume":"388","author":"Miritello","year":"2009","journal-title":"Phys. A"},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Benkadda, S., Elskens, Y., and Doveil, F. (1994). Transport, Plasma Physics, World Scientific.","DOI":"10.1142\/9789814534857"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"2361","DOI":"10.1103\/PhysRevE.52.2361","article-title":"Clustering and relaxation in Hamiltonian long-range dynamics","volume":"52","author":"Antoni","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1103\/PhysRevLett.80.692","article-title":"Lyapunov instability and finite size effects in a system with long-range forces","volume":"80","author":"Latora","year":"1998","journal-title":"Phy. Rev. Lett."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"2104","DOI":"10.1103\/PhysRevLett.83.2104","article-title":"Superdiffusion and out-of-equilibrium chaotic dynamics with many degrees of freedom","volume":"83","author":"Latora","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"056134","DOI":"10.1103\/PhysRevE.64.056134","article-title":"Non-Gaussian equilibrium in a long-range Hamiltonian system","volume":"64","author":"Latora","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"26002","DOI":"10.1209\/0295-5075\/80\/26002","article-title":"Nonergodicity and central-limit behavior for long-range Hamiltonians","volume":"80","author":"Pluchino","year":"2007","journal-title":"EPL"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"056113","DOI":"10.1103\/PhysRevE.69.056113","article-title":"Glassy phase in the Hamiltonian mean-field model","volume":"69","author":"Pluchino","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"5313","DOI":"10.1103\/PhysRevLett.80.5313","article-title":"Breakdown of Exponential Sensitivity to Initial Conditions: Role of the Range of Interactions","volume":"80","author":"Anteneodo","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/0375-9601(90)90092-3","article-title":"Construction of higher order symplectic integrators, Phys","volume":"150","author":"Yoshida","year":"1990","journal-title":"Lett. A"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/S0375-9601(01)00440-6","article-title":"Classical spin systems with long-range interactions: Universal reduction of mixing","volume":"286","author":"Campa","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"6599","DOI":"10.1103\/PhysRevE.57.6599","article-title":"Analytic estimation of the Lyapunov exponent in a mean-field model undergoing a phase transition","volume":"57","author":"Firpo","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"30003","DOI":"10.1209\/0295-5075\/123\/30003","article-title":"Validity and failure of the Boltzmann weight","volume":"123","author":"Cirto","year":"2018","journal-title":"EPL"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"042110","DOI":"10.1103\/PhysRevE.103.042110","article-title":"Quasi-stationary-state duration in the classical d-dimensional long-range inertial XY ferromagnet","volume":"103","author":"Nobre","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"014144","DOI":"10.1103\/PhysRevE.104.014144","article-title":"Criticality in the duration of quasistationary state","volume":"104","author":"Nobre","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Cirto, L.J.L., Lima, L.S., and Nobre, F.D. (2015). Controlling the range of interactions in the classical inertial ferromagnetic Heisenberg model: Analysis of metastable states. J. Stat. Mech. Theory Exp., P04012.","DOI":"10.1088\/1742-5468\/2015\/04\/P04012"},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Rodriguez, A., Nobre, F.D., and Tsallis, C. (2019). d-dimensional classical Heisenberg model with arbitrarily-ranged interactions: Lyapunov exponents and distributions of momenta and energies. Entropy, 21.","DOI":"10.3390\/e21010031"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"023153","DOI":"10.1103\/PhysRevResearch.2.023153","article-title":"Quasi-stationary-state duration in d-dimensional long-range model","volume":"2","author":"Nobre","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1209\/epl\/i2004-10467-y","article-title":"Preferential attachment growth model and nonextensive statistical mechanics","volume":"70","author":"Soares","year":"2005","journal-title":"Europhys. Lett."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1209\/epl\/i2005-10221-1","article-title":"Nonextensive aspects of self-organized scale-free gas-like networks","volume":"72","author":"Thurner","year":"2005","journal-title":"Europhys. Lett."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"27992","DOI":"10.1038\/srep27992","article-title":"Role of dimensionality in complex networks","volume":"6","author":"Brito","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Nunes, T.C., Brito, S., da Silva, L.R., and Tsallis, C. (2017). Role of dimensionality in preferential attachment growth in the Bianconi-Barabasi model. J. Stat. Mech., 093402.","DOI":"10.1088\/1742-5468\/aa8198"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"012305","DOI":"10.1103\/PhysRevE.99.012305","article-title":"Scaling properties of d-dimensional complex networks","volume":"99","author":"Brito","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"043404","DOI":"10.1088\/1742-5468\/ab75e6","article-title":"A generalised model for asymptotically-scale-free geographical networks","volume":"2020","author":"Cinardi","year":"2020","journal-title":"J. Stat. Mech."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"1130","DOI":"10.1038\/s41598-020-80939-1","article-title":"Connecting complex networks to nonadditive entropies","volume":"11","author":"Brito","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_85","doi-asserted-by":"crossref","unstructured":"de Oliveira, R.M., Brito, S., da Silva, L.R., and Tsallis, C. (2022). Statistical mechanical approach of complex networks with weighted links. JSTAT, 063402.","DOI":"10.1088\/1742-5468\/ac6f51"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"053126","DOI":"10.1063\/5.0090864","article-title":"Complex network growth model: Possible isomorphism between nonextensive statistical mechanics and random geometry","volume":"32","author":"Tsallis","year":"2022","journal-title":"Chaos"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"40006","DOI":"10.1209\/0295-5075\/108\/40006","article-title":"Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics","volume":"108","author":"Christodoulidi","year":"2014","journal-title":"EPL"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1016\/j.physa.2017.09.098","article-title":"Fermi-Pasta-Ulam-Tsingou problems: Passage from Boltzmann to q-statistics","volume":"491","author":"Bagchi","year":"2018","journal-title":"Phys. A"},{"key":"ref_89","doi-asserted-by":"crossref","unstructured":"Tsallis, C. (2023). Introduction to Nonextensive Statistical Mechanics\u2014Approaching a Complex World, Springer. [2nd ed.].","DOI":"10.1007\/978-3-030-79569-6"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"10005","DOI":"10.1209\/0295-5075\/126\/10005","article-title":"New type of equilibrium distribution for a system of charges In a spherically-symmetric electric field","volume":"126","author":"Casas","year":"2019","journal-title":"EPL"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"260601","DOI":"10.1103\/PhysRevLett.105.260601","article-title":"Thermostatistics of overdamped motion of interacting particles","volume":"105","author":"Andrade","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"032138","DOI":"10.1103\/PhysRevE.98.032138","article-title":"Overdamped dynamics of particles with repulsive power-law interactions","volume":"98","author":"Moreira","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1103\/PhysRevLett.78.401","article-title":"Vortex velocities in the O(n) symmetric time-dependent Ginzburg-Landau model","volume":"78","author":"Mazenko","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"021109","DOI":"10.1103\/PhysRevE.68.021109","article-title":"Vortex dynamics in a coarsening two-dimensional XY model","volume":"68","author":"Qian","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1002\/andp.19033160802","article-title":"Zur kinetischen Theorie der einatomigen Korper","volume":"316","author":"Mie","year":"1903","journal-title":"Ann. Phys."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1098\/rspa.1924.0081","article-title":"On the determination of molecular fields.\u2014I. From the variation of the viscosity of a gas with temperature","volume":"106","author":"Jones","year":"1924","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1098\/rspa.1924.0082","article-title":"On the determination of molecular fields. \u2014II. From the equation of state of a gas","volume":"106","author":"Jones","year":"1924","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1088\/0959-5309\/43\/5\/301","article-title":"Cohesion","volume":"43","year":"1931","journal-title":"Proc. Phys. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/444\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:26:42Z","timestamp":1760120802000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/444"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,7]]},"references-count":98,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020444"],"URL":"https:\/\/doi.org\/10.3390\/sym15020444","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,7]]}}}