{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T10:18:00Z","timestamp":1768645080036,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T00:00:00Z","timestamp":1668643200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Recent mathematical investigations have shown that under very general conditions, exponential mixing implies the Bernoulli property. As a concrete example of statistical mechanics that are exponentially mixing we consider the Bernoulli shift dynamics by Chebyshev maps of arbitrary order N\u22652, which maximizes Tsallis q=3 entropy rather than the ordinary q=1 Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd N. We show that the value of the fine structure constant \u03b1el=1\/137 is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for N=3.<\/jats:p>","DOI":"10.3390\/e24111671","type":"journal-article","created":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T03:24:34Z","timestamp":1668655474000},"page":"1671","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Information Shift Dynamics Described by Tsallis q = 3 Entropy on a Compact Phase Space"],"prefix":"10.3390","volume":"24","author":[{"given":"Jin","family":"Yan","sequence":"first","affiliation":[{"name":"Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4818-6997","authenticated-orcid":false,"given":"Christian","family":"Beck","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK"},{"name":"The Alan Turing Institute, London NW1 2DB, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,17]]},"reference":[{"key":"ref_1","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_2","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"120601","DOI":"10.1103\/PhysRevLett.122.120601","article-title":"Maximum entropy principle in statistical inference: Case for non-Shannonian entropies","volume":"122","author":"Jizba","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0378-4371(03)00019-0","article-title":"Superstatistics","volume":"322","author":"Beck","year":"2003","journal-title":"Physica A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1140\/epjst\/e2020-900210-x","article-title":"Superstatistics and non-Gaussian diffusion","volume":"229","author":"Metzler","year":"2020","journal-title":"Eur. Phys. J. Special Topics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/S0378-4371(00)00368-X","article-title":"A nonextensive thermodynamical equilibrium approach in e+e\u2212\u2192 hadrons","volume":"286","author":"Bediaga","year":"2000","journal-title":"Physica A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/S0378-4371(00)00354-X","article-title":"Non-extensive statistical mechanics and particle spectra in elementary interactions","volume":"286","author":"Beck","year":"2000","journal-title":"Physica A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1140\/epja\/i2009-10806-6","article-title":"Non-extensive approach to quark matter","volume":"40","author":"Biro","year":"2009","journal-title":"Eur. Phys. J. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"054025","DOI":"10.1103\/PhysRevD.91.054025","article-title":"Description of high-energy pp collisions using Tsallis thermodynamics: Transverse momentum and rapidity distributions","volume":"91","author":"Marques","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"035303R","DOI":"10.1103\/PhysRevE.63.035303","article-title":"Measuring nonextensitivity parameters in a turbulent Couette-Taylor flow","volume":"63","author":"Beck","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1080\/00107510902823517","article-title":"Generalised information and entropy measures in physics","volume":"50","author":"Beck","year":"2009","journal-title":"Contemp. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Yoon, P.H. (2009). Thermodynamic, non-extensive, or turbulent quasi-equilibrium for the space plasma environment. Entropy, 21.","DOI":"10.3390\/e21090820"},{"key":"ref_14","unstructured":"Dolgopyat, D., Kanigowski, A., and Rodriguez-Hertz, F. (2021). Exponential Mixing Implies Bernoulli. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"100035","DOI":"10.1016\/j.csfx.2020.100035","article-title":"Distinguished Correlation Properties of Chebyshev Dynamical Systems and Their Generalisations","volume":"5","author":"Yan","year":"2020","journal-title":"Chaos Solitons Fractals X"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0167-2789(01)00265-2","article-title":"Higher-order correlations of Tchebyscheff maps","volume":"56","author":"Hilgers","year":"2001","journal-title":"Physica D"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2354","DOI":"10.1016\/j.chaos.2008.09.010","article-title":"Short chaotic strings and their behaviour in the scaling region","volume":"41","author":"Groote","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.chaos.2004.04.017","article-title":"Periodic orbit theory of two coupled Tchebyscheff maps","volume":"23","author":"Dettmann","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Beck, C. (2002). Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields, World Scientific.","DOI":"10.1142\/9789812778239"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Beck, C., and Schl\u00f6gl, F. (1993). Thermodynamics of Chaotic Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511524585"},{"key":"ref_21","first-page":"483","article-title":"Perturbation theory without gauge fixing","volume":"24","author":"Parisi","year":"1981","journal-title":"Sci. Sin."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"PDamgaard, H., and H\u00fcffel, H. (1988). Stochastic Quantization, World Scientific.","DOI":"10.1142\/0375"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/S0167-2789(02)00540-7","article-title":"Chaotic strings and standard model parameters","volume":"171","author":"Beck","year":"2002","journal-title":"Physica D"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"123515","DOI":"10.1103\/PhysRevD.69.123515","article-title":"Chaotic scalar fields as models for dark energy","volume":"69","author":"Beck","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/S0378-4371(98)00437-3","article-title":"The role of constraints within generalized nonextensive statistics","volume":"261","author":"Tsallis","year":"1998","journal-title":"Physica A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/j.physa.2004.04.041","article-title":"Nonextensive scalar field theories and dark energy models","volume":"340","author":"Beck","year":"2004","journal-title":"Physica A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1140\/epjc\/s10052-021-09486-x","article-title":"Tsallis statistics and generalized uncertainty principle","volume":"81","author":"Luciano","year":"2021","journal-title":"Eur. Phys. J. C"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"045004","DOI":"10.1103\/PhysRevD.104.045004","article-title":"q-generalized Tsallis thermostatistics in Unruh effect for mixed fields","volume":"104","author":"Luciano","year":"2021","journal-title":"Phys. Rev. D"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1140\/epjc\/s10052-019-6740-5","article-title":"Modified cosmology from extended entropy with varying exponent","volume":"79","author":"Nojiri","year":"2019","journal-title":"Eur. Phys. J. C"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1143\/PTP.72.480","article-title":"Period-doubling of kink anti-kink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice","volume":"72","author":"Kaneko","year":"1984","journal-title":"Progr. Theor. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0167-2789(95)90047-0","article-title":"Asymptotic stability of densities in coupled map lattices","volume":"80","author":"Mackey","year":"1995","journal-title":"Physica D"},{"key":"ref_32","unstructured":"(2022, October 25). Available online: https:\/\/sites.google.com\/view\/jinyan-cmls\/home."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1671\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:19:54Z","timestamp":1760145594000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1671"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,17]]},"references-count":32,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["e24111671"],"URL":"https:\/\/doi.org\/10.3390\/e24111671","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,17]]}}}