{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T23:09:18Z","timestamp":1768691358959,"version":"3.49.0"},"reference-count":66,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2015,7,10]],"date-time":"2015-07-10T00:00:00Z","timestamp":1436486400000},"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>In this article, we analyze the interrelationships among such notions as entropy, information, complexity, order and chaos and show using the theory of categories how to generalize the second law of thermodynamics as a law of increasing generalized entropy or a general law of complification. This law could be applied to any system with morphisms, including all of our universe and its subsystems. We discuss how such a general law and other laws of nature drive the evolution of the universe, including physicochemical and biological evolutions. In addition, we determine eliminating selection in physicochemical evolution as an extremely simplified prototype of natural selection. Laws of nature do not allow complexity and entropy to reach maximal values by generating structures. One could consider them as a kind of \u201cbreeder\u201d of such selection.<\/jats:p>","DOI":"10.3390\/e17074863","type":"journal-article","created":{"date-parts":[[2015,7,13]],"date-time":"2015-07-13T03:49:22Z","timestamp":1436759362000},"page":"4863-4890","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Entropy, Information and Complexity or Which Aims the Arrow of Time?"],"prefix":"10.3390","volume":"17","author":[{"given":"George","family":"Mikhailovsky","sequence":"first","affiliation":[{"name":"CALIBRE, 6354 Walker Lane, Alexandria, VA 22310, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander","family":"Levich","sequence":"additional","affiliation":[{"name":"Department of Biophysics, Biology Faculty, Moscow Lomonosov University, Leninskie gory, 1-12, 119991 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,7,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Boltzmann, L. (1970). Weitere Studien Uber das Warmegleichgewicht Unter Gas Molekulen, Springer.","DOI":"10.1007\/978-3-322-84986-1_3"},{"key":"ref_2","unstructured":"Shannon, C.E., and Weaver, W. (1963). The Mathematical Theory of Communication, University of Illinois Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1086\/282106","article-title":"On the Relative Abundance of Species","volume":"94","author":"McArthur","year":"1960","journal-title":"Am. Nat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/0022-5193(70)90124-4","article-title":"A Test for Comparing Diversities Index Based on Shannon Formula","volume":"29","author":"Hutcheson","year":"1970","journal-title":"J. Theor. Biol."},{"key":"ref_5","unstructured":"Wiener, N. (1965). Cybernetics, or the Control and Communication in the Animal and the Machine, The MIT Press. [2nd ed.]."},{"key":"ref_6","unstructured":"Eco, U. (2011). Opera Aperta: Forma e Indeterminazione nelle Poetiche contemporanee, Bompiani."},{"key":"ref_7","unstructured":"Brillouin, L. (1988). La Science et la Theorie de l\u2019Information, Editions Jacques Gabay."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ben-Naim, A. (2012). Entropy and the Second Law: Interpretation and Miss-interpretations, World Scientific.","DOI":"10.1142\/8333"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ben-Naim, A. (2008). Entropy Demystified, World Scientific.","DOI":"10.1142\/6916"},{"key":"ref_10","unstructured":"Gell-Mann, M. (1995). The Quark and the Jaguar, St. Martin\u2019s Press. [3rd ed.]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1109\/MCS.2001.939938","article-title":"Measures of Complexity: A Non-exhaustive List","volume":"21","author":"Lloyd","year":"2001","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0019-9958(64)90223-2","article-title":"A Formal Theory of Inductive Inference. Part I","volume":"7","author":"Solomonoff","year":"1964","journal-title":"Inf. Control."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/S0019-9958(64)90131-7","article-title":"A Formal Theory of Inductive Inference. Part II","volume":"7","author":"Solomonoff","year":"1964","journal-title":"Inf. Control."},{"key":"ref_14","first-page":"1","article-title":"Three Approaches to the Quantitative Definition of Information","volume":"1","author":"Kolmogorov","year":"1965","journal-title":"Probl. Inf. Transm."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1145\/321495.321506","article-title":"On the Length of Programs for Computing Finite Binary Sequences: Statistical Considerations","volume":"16","author":"Chaitin","year":"1969","journal-title":"J. Assoc. Comput. Mach."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1070\/RM1983v038n04ABEH004203","article-title":"Combinatorial Foundations of Information Theory and the Calculus of Probabilities","volume":"38","author":"Kolmogorov","year":"1983","journal-title":"Russ. Math. Surv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1214\/aop\/1176991250","article-title":"Kolmogorov\u2019s Contributions to Information Theory and Algorithmic Complexity","volume":"17","author":"Cover","year":"1989","journal-title":"Ann. Probab."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/0003-4916(88)90094-2","article-title":"Complexity as Thermodynamic Depth","volume":"188","author":"Lloyd","year":"1988","journal-title":"Ann. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1606","DOI":"10.3390\/e17041606","article-title":"A Fundamental Scale of Descriptions for Analyzing Information Content of Communication Systems","volume":"17","author":"Febres","year":"2015","journal-title":"Entropy"},{"key":"ref_20","unstructured":"Grunwald, P., and Vitanyi, P. (2004). Shannon Information and Kolmogorov Complexity, arXiv, cs\/0410002."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s13194-012-0056-8","article-title":"What is a Complex System?","volume":"3","author":"Ladyman","year":"2013","journal-title":"Eur. J. Philos. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Lineweaver, C.H., Davis, P.C.W., and Ruse, M. (2013). Complexity and the Arrow of Time, Cambridge University Press.","DOI":"10.1017\/CBO9781139225700"},{"key":"ref_23","unstructured":"Sant\u2019Anna, A.S. (2004). Entropy is Complexity, arXiv, math-ph\/0408040."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"595","DOI":"10.3390\/e13030595","article-title":"Entropy Measures vs. Kolmogorov Complexity","volume":"13","author":"Teixeira","year":"2011","journal-title":"Entropy"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lineweaver, C.H., Davis, P.C.W., and Ruse, M. (2013). Complexity and the Arrow of Time, Cambridge University Press.","DOI":"10.1017\/CBO9781139225700"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Levich, A.P. (1995). On the Way to Understanding the Time Phenomenon: The Constructions of Time in Natural Science, Part 1, World Scientific.","DOI":"10.1142\/9789812832092_0001"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1080\/019697299125118","article-title":"Category-Functor Modelling of Natural Systems","volume":"30","author":"Levich","year":"1999","journal-title":"Cybern. Syst."},{"key":"ref_28","unstructured":"Levich, A.P. (2012). Art and Method of Systems Modeling: Variational Methods in Communities Ecology, Structural and Extremal Principles, Categories and Functors, Computing Investigation Institute."},{"key":"ref_29","first-page":"1035","article-title":"Problems and Theorems of Variational Modeling in Ecology of Communities","volume":"8","author":"Levich","year":"2002","journal-title":"Fundam. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Lineweaver, C.H., Davis, P.C.W., and Ruse, M. (2013). Complexity and the Arrow of Time, Cambridge University Press.","DOI":"10.1017\/CBO9781139225700"},{"key":"ref_31","unstructured":"Levich, A.P. (2004). Systems Approach in Modern Science, Progress-Tradition."},{"key":"ref_32","unstructured":"Eddington, A.S. (2012). The Nature of the Physical World: Gifford Lectures 1927, Cambridge University Press."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1038\/scientificamerican0971-179","article-title":"Energy and Information","volume":"225","author":"Tribus","year":"1971","journal-title":"Sci. Am."},{"key":"ref_34","unstructured":"Cooper, L.N. (1968). An Introduction to the Meaning and Structure of Physics, Harper."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Goldreich, O. (2010). P, NP, and NP-Completeness: The Basics of Computational Complexity, Cambridge University Press.","DOI":"10.1017\/CBO9780511761355"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4489","DOI":"10.3390\/e16084489","article-title":"Complexity and the Emergence of Physical Properties","volume":"16","author":"Fuentes","year":"2014","journal-title":"Entropy"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Lineweaver, C.H., Davis, P.C.W., and Ruse, M. (2013). Complexity and the Arrow of Time, Cambridge University Press.","DOI":"10.1017\/CBO9781139225700"},{"key":"ref_38","unstructured":"Lloyd, S. (1988). Black Holes, Demons and the Loss of Coherence: How Complex Systems Get Information, and What They Do with It. [Ph.D. Thesis, The Rockefeller University]."},{"key":"ref_39","unstructured":"Wolchover, N. Available online: http:\/\/www.wired.com\/2014\/04\/quantum-theory-flow-time\/."},{"key":"ref_40","first-page":"3","article-title":"February 1927: Heisenberg\u2019s uncertainty principle","volume":"17","author":"Wolchover","year":"2008","journal-title":"Am. Phys. Soc. News."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Levich, A.P. (1995). On the Way to Understanding the Time Phenomenon: The Constructions of Time in Natural Science, Part 1, World Scientific.","DOI":"10.1142\/9789812832092_0001"},{"key":"ref_42","first-page":"72","article-title":"Organization of Time in Biological Systems","volume":"50","author":"Mikhailovsky","year":"1989","journal-title":"J. Gen. Biol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1098\/rsif.2008.0018","article-title":"Top-down Causation by Information Control: From a Philosophical Problem to a Scientific Research Program","volume":"5","author":"Auletta","year":"2008","journal-title":"J. R. Soc. Interface."},{"key":"ref_44","unstructured":"Ellis, G. (2013). Questioning the Foundations of Physics, Springer."},{"key":"ref_45","unstructured":"Prigogine, I. (1980). From Being to Becoming: Time and Complexity in the Physical Sciences, W.H. Freeman & Co."},{"key":"ref_46","unstructured":"Shulman, M. Kh. Available online: http:\/\/www.timeorigin21.narod.ru\/eng_time\/Entropy_and_evolution_eng.pdf."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/cpa.3160130102","article-title":"The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Richard Courant Lecture in Mathematical Sciences Delivered at New York University, May 11, 1959","volume":"13","author":"Wigner","year":"1960","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_48","unstructured":"Yanofsky, N.S. (2013). Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us, The MIT Press."},{"key":"ref_49","unstructured":"Thomson, W. (1851). On the Dynamical Theory of Heat. Trans. R. Soc. EdinburghExcerpts, \u00a7\u00a71\u2013100."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1086\/148307","article-title":"A Measurement of Excess Antenna Temperature at 4080 Mc\/s","volume":"142","author":"Penzias","year":"1965","journal-title":"Astrophys. J"},{"key":"ref_51","unstructured":"Freedman, W.L. (2004). Measuring and Modeling the Universe, Cambridge University Press."},{"key":"ref_52","unstructured":"Newton, I. (1684). Philosophiae Naturalis Principia Mathematica, Royal Society."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Liddle, A.R., and Lyth, D.H. (2000). Cosmological Inflation and Large-scale Structure, Cambridge University Press.","DOI":"10.1017\/CBO9781139175180"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Mukhanov, V. (2005). Physical Foundations of Cosmology, Cambridge University Press.","DOI":"10.1017\/CBO9780511790553"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"237901","DOI":"10.1103\/PhysRevLett.88.237901","article-title":"Computational Capacity of the Universe","volume":"88","author":"Lloyd","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1534\/genetics.113.151555","article-title":"Stabilizing Selection, Purifying Selection, Mutational Bias in Finite Populations","volume":"194","author":"Charlesworth","year":"2013","journal-title":"Genetics"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1152","DOI":"10.3390\/e15041152","article-title":"Entropy and Entropy Production: Old MMisconceptions and New Breakthroughs","volume":"15","author":"Martyushev","year":"2013","journal-title":"Entropy"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"658","DOI":"10.3390\/e17020658","article-title":"Entropy Production of Main-Sequence Stars","volume":"17","author":"Martyushev","year":"2015","journal-title":"Entropy"},{"key":"ref_59","unstructured":"Prigogine, I. (1977). Time, structure and fluctuations, 23. Nobel lecture."},{"key":"ref_60","unstructured":"Zeigler, H. (1962). Some Extremum Principles in Irreversible Thermodynamics with Application to Continuum Mechanics, Swiss Federal Institute of Technology."},{"key":"ref_61","unstructured":"Prigogine, I. (1961). Introduction to Thermodynamics of Irreversible Processes, Interscience."},{"key":"ref_62","unstructured":"Haken, H. (2006). Information and Self-organization: A Macroscopic Approach to Complex Systems, Springer."},{"key":"ref_63","unstructured":"Nicolis, G., and Prigogine, I. (1977). Self-organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley."},{"key":"ref_64","unstructured":"Sharov, A.A., and Gordon, R. (2013). Life before Earth, arXiv, 1304.3381."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"5428","DOI":"10.3390\/e16105428","article-title":"Directionality Theory and the Entropic Principle of Natural Selection","volume":"16","author":"Demetrius","year":"2014","journal-title":"Entropy"},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Gregersen, N.H. (2003). From Complexity to Life (on the Emergence of Life and Meaning), Oxford University Press.","DOI":"10.1093\/oso\/9780195150704.001.0001"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/7\/4863\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:49:02Z","timestamp":1760215742000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/7\/4863"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,7,10]]},"references-count":66,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2015,7]]}},"alternative-id":["e17074863"],"URL":"https:\/\/doi.org\/10.3390\/e17074863","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,7,10]]}}}