{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:23:00Z","timestamp":1760242980936,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2015,1,19]],"date-time":"2015-01-19T00:00:00Z","timestamp":1421625600000},"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>The time operator and internal age are intrinsic features of entropy producing innovation processes. The innovation spaces at each stage are the eigenspaces of the time operator. The internal age is the average innovation time, analogous to lifetime computation. Time operators were originally introduced for quantum systems and highly unstable dynamical systems. Extending the time operator theory to regular Markov chains allows one to relate internal age with norm distances from equilibrium. The goal of this work is to express the evolution of internal age in terms of Lyapunov functionals constructed from entropies. We selected the Boltzmann\u2013Gibbs\u2013Shannon entropy and more general entropy functions, namely the Tsallis entropies and the Kaniadakis entropies. Moreover, we compare the evolution of the distance of initial distributions from equilibrium to the evolution of the Lyapunov functionals constructed from norms with the evolution of Lyapunov functionals constructed from entropies. It is remarkable that the entropy functionals evolve, violating the second law of thermodynamics, while the norm functionals evolve thermodynamically.<\/jats:p>","DOI":"10.3390\/e17010407","type":"journal-article","created":{"date-parts":[[2015,1,19]],"date-time":"2015-01-19T10:53:54Z","timestamp":1421664834000},"page":"407-424","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Entropy, Age and Time Operator"],"prefix":"10.3390","volume":"17","author":[{"given":"Ilias","family":"Gialampoukidis","sequence":"first","affiliation":[{"name":"School of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece"},{"name":"Information Technologies Institute, Centre for Research and Technology Hellas (CERTH-ITI), 6th km Xarilaou-Thermi, 57001 Thessaloniki, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioannis","family":"Antoniou","sequence":"additional","affiliation":[{"name":"School of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,1,19]]},"reference":[{"key":"ref_1","unstructured":"Flugge, S. 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