{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T15:39:58Z","timestamp":1773157198269,"version":"3.50.1"},"reference-count":92,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2010,5,7]],"date-time":"2010-05-07T00:00:00Z","timestamp":1273190400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant \u201cadditivity\u201d properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally \u201cmost random\u201d distributions.<\/jats:p>","DOI":"10.3390\/e12051145","type":"journal-article","created":{"date-parts":[[2010,5,7]],"date-time":"2010-05-07T10:41:03Z","timestamp":1273228863000},"page":"1145-1193","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":75,"title":["Entropy: The Markov Ordering Approach"],"prefix":"10.3390","volume":"12","author":[{"given":"Alexander N.","family":"Gorban","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Leicester, Leicester, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pavel A.","family":"Gorban","sequence":"additional","affiliation":[{"name":"Institute of Space and Information Technologies, Siberian Federal University, Krasnoyarsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"George","family":"Judge","sequence":"additional","affiliation":[{"name":"Department of Resource Economics, University of California, Berkeley, CA, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2010,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1002\/andp.18652010702","article-title":"\u00dcber vershiedene f\u00fcr die Anwendungen bequeme Formen der Hauptgleichungen der W\u00e4rmetheorie","volume":"125","author":"Clausius","year":"1865","journal-title":"Poggendorffs Annalen der Physic und Chemie"},{"key":"ref_2","unstructured":"Gibbs, J.W. (1875\u20131876). On the equilibrium of heterogeneous substance. Trans. Connect. Acad., 108\u2013248. 1875\u20131876, 343\u2013524."},{"key":"ref_3","first-page":"275","article-title":"Weitere Studien \u00fcber das W\u00e4rmegleichgewicht unter Gasmolek\u00fclen","volume":"66","author":"Boltzmann","year":"1872","journal-title":"Sitzungsberichte der keiserlichen Akademie der Wissenschaften"},{"key":"ref_4","unstructured":"Gibbs, J.W. (1981). Elementary Principles in Statistical Mechanics, Ox Bow Press."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Gallavotti, G., Reiter, W.L., and Yngvason, J. (2008). Boltzmann\u2019s Legacy, EMS Publishing House.","DOI":"10.4171\/057"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1119\/1.1971557","article-title":"Gibbs versus Boltzmann entropy","volume":"33","author":"Jaynes","year":"1965","journal-title":"Am. J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.physd.2004.01.008","article-title":"On the (Boltzmann) entropy of non-equilibrium systems","volume":"193","author":"Goldstein","year":"2004","journal-title":"Physica D"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6620","DOI":"10.1103\/PhysRevE.56.6620","article-title":"Dynamics and thermodynamics of complex fluids. I. Development of a general formalism","volume":"56","author":"Grmela","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"\u00d6ttinger, H.C. (2005). Beyond Equilibrium Thermodynamics, Wiley-Interscience.","DOI":"10.1002\/0471727903"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1002\/j.1538-7305.1928.tb01236.x","article-title":"Transmission of information","volume":"July","author":"Hartley","year":"1928","journal-title":"Bell Syst. Tech. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Statist."},{"key":"ref_13","first-page":"144","article-title":"Proof of the uniqueness of the solution of the equations of the law of mass action","volume":"Volume 1","author":"Ostriker","year":"1996","journal-title":"Selected Works of Yakov Borisovich Zeldovich"},{"key":"ref_14","unstructured":"Yablonskii, G.S., Bykov, V.I., Gorban, A.N., and Elokhin, V.I. (1991). Kinetic Models of Catalytic Reactions, Series: Comprehensive Chemical Kinetics, Elsevier."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"772","DOI":"10.3390\/e12040772","article-title":"Engineering model reduction and entropy-based lyapunov functions in chemical reaction kinetics","volume":"12","author":"Hangos","year":"2010","journal-title":"Entropy"},{"key":"ref_16","unstructured":"Childers, D.G. (,  1967). Maximum entropy spectral analysis. Proceedings of the 37th meeting of the Society of Exploration Geophysicists, Oklahoma City, OK, USA. Reprinted in Modern Spectrum Analysis."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1190\/1.1440265","article-title":"The relationship between maximum entropy spectra and maximum likelihood spectra","volume":"37","author":"Burg","year":"1972","journal-title":"Geophysics"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1103\/PhysRev.108.171","article-title":"Information theory and statistical mechanics. II","volume":"108","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"191","DOI":"10.3390\/e3030191","article-title":"Maximum entropy fundamentals","volume":"3","year":"2001","journal-title":"Entropy"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1080\/00107510902823517","article-title":"Generalized information and entropy measures in physics","volume":"50","author":"Beck","year":"2009","journal-title":"Contemp. Phys."},{"key":"ref_22","unstructured":"Mittelhammer, R., Judge, G., and Miller, D. (2000). Econometric Foundations, Cambridge University Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/S0304-4076(01)00116-6","article-title":"Generalized moment based estimation and inference","volume":"107","author":"Judge","year":"2002","journal-title":"J. Econom."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1111\/j.1467-9280.1992.tb00012.x","article-title":"On the psychology of ampliative inference","volume":"3","author":"Myers","year":"1992","journal-title":"Psychol. Sci."},{"key":"ref_25","first-page":"337","article-title":"A summary of entropy statistics","volume":"31","author":"Esteban","year":"1995","journal-title":"Kybernetica"},{"key":"ref_26","first-page":"547","article-title":"On measures of entropy and information","volume":"Volume 1","year":"1961","journal-title":"Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 1960"},{"key":"ref_27","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_28","doi-asserted-by":"crossref","unstructured":"Abe, S., and Okamoto, Y. (2001). Nonextensive Statistical Mechanics and its Applications, Springer.","DOI":"10.1007\/3-540-40919-X"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1111\/j.2517-6161.1984.tb01318.x","article-title":"Multinomial goodness of fit tests","volume":"46","author":"Cressie","year":"1984","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Read, T.R., and Cressie, N.A. (1988). Goodness of Fit Statistics for Discrete Multivariate Data, Springer.","DOI":"10.1007\/978-1-4612-4578-0"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3230","DOI":"10.1016\/j.physleta.2009.07.014","article-title":"On the way towards a generalized entropy maximization procedure","volume":"373","author":"Bagci","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1268","DOI":"10.1126\/science.297.5585.1268","article-title":"A fresh take on disorder, or disorderly science?","volume":"297","author":"Cho","year":"2002","journal-title":"Science"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/e1010001","article-title":"Diversity and entropy","volume":"1","author":"Lin","year":"1999","journal-title":"Entropy"},{"key":"ref_34","unstructured":"Gorban, A.N. (1984). Equilibrium Encircling. Equations of Chemical kinetics and their Thermodynamic Analysis, Nauka."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"016104","DOI":"10.1103\/PhysRevE.67.016104","article-title":"Family of additive entropy functions out of thermodynamic limit","volume":"67","author":"Gorban","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"067104","DOI":"10.1103\/PhysRevE.67.067104","article-title":"The additive generalization of the Boltzmann entropy","volume":"67","author":"Gorban","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_37","unstructured":"Nonextensive statistical mechanics and thermodynamics: bibliography. Available online: http:\/\/tsallis.cat.cbpf.br\/TEMUCO.pdf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"304","DOI":"10.3390\/e12030304","article-title":"From f-divergence to quantum quasi-entropies and their use","volume":"12","author":"Petz","year":"2010","journal-title":"Entropy"},{"key":"ref_39","first-page":"193","article-title":"Smooth entropy and R\u00e9nyi entropy","volume":"Volume 1233","author":"Fumy","year":"1997","journal-title":"Proceedings of EUROSCRIPT\u201997"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1145\/1273496.1273523","article-title":"Information-theoretic metric learning","volume":"227","author":"Davis","year":"2007","journal-title":"ICML"},{"key":"ref_41","unstructured":"R\u00e9nyi, A. (1970). Probability Theory, North-Holland."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/S0375-9601(00)00337-6","article-title":"Axioms and uniqness theorem for Tsallis entropy","volume":"271","author":"Abe","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0306-4573(84)90069-4","article-title":"Measuring information beyond communication theory","volume":"20","author":"Aczel","year":"1984","journal-title":"Inf. Process. Manage."},{"key":"ref_44","unstructured":"Aczel, J., and Daroczy, Z. (1975). On Measures of Information and Their Characterizations, Academic Press."},{"key":"ref_45","first-page":"73","article-title":"Information measures: A critical survey","volume":"Volume B","year":"1978","journal-title":"Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions and the Eighth European Meeting of Statisticians"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1016\/S0378-4371(03)00578-8","article-title":"Monotonically equivalent entropies and solution of additivity equation","volume":"328","author":"Gorban","year":"2003","journal-title":"Physica A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1103\/RevModPhys.50.221","article-title":"General properties of entropy","volume":"50","author":"Wehrl","year":"1978","journal-title":"Rev. Mod. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1143\/JPSJ.18.328","article-title":"Markov processes and the H-theorem","volume":"12","author":"Morimoto","year":"1963","journal-title":"J. Phys. Soc. Jap."},{"key":"ref_49","first-page":"85","article-title":"Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten","volume":"8","year":"1963","journal-title":"Magyar. Tud. Akad. Mat. Kutato Int. Kozl."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Rockafellar, R.T. (1970). Convex Analysis, Princeton University Press. reprint 1997.","DOI":"10.1515\/9781400873173"},{"key":"ref_51","unstructured":"Liese, F., and Vajda, I. (1987). Convex Statistical Distances, Teubner."},{"key":"ref_52","first-page":"163","article-title":"Central limit theorem for non-stationary Markov chains I, II","volume":"1","author":"Dobrushin","year":"1956","journal-title":"Theory Probab. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Seneta, E. (1981). Nonnegative Matrices and Markov Chains, Springer.","DOI":"10.1007\/0-387-32792-4"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1090\/conm\/149\/01262","article-title":"Majorization, Monotonicity of relative entropy and stochastic matrices","volume":"149","author":"Cohen","year":"1993","journal-title":"Contemp. Math."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/0024-3795(93)90331-H","article-title":"Relative entropy under mappings by stochastic matrices","volume":"179","author":"Cohen","year":"1993","journal-title":"Linear. Alg. Appl."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s00440-003-0270-6","article-title":"On contraction properties of Markov kernels","volume":"126","author":"Ledoux","year":"2003","journal-title":"Probab. Theory Relat. Field"},{"key":"ref_57","unstructured":"Leung, C.S., Lee, M., and Chan, J.H. (December, January 1). Divergence, Optimization, Geometry\u00a0. Proceedings of the 16th International Conference on Neural Information Processing, Bangkok, Thailand. LNCS 5863."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/0041-5553(67)90040-7","article-title":"The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming","volume":"7","author":"Bregman","year":"1967","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_59","unstructured":"Itakura\n\t\t\t\t\t\t\t\t\t              F.\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              Saito\n\t\t\t\t\t\t\t\t\t              S.\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t          Analysis synthesis telephony based on the maximum likelihood method\n\t\t\t\t\t\t\t\t\t          Proceedings of the 6th International Congress on Acoustics\n\t\t\t\t\t\t\t\t\t          Tokyo, Japan\n\t\t\t\t\t\t\t\t\t          1968\n\t\t\t\t\t\t\t\t\t          C17\n\t\t\t\t\t\t\t\t\t          C20\n\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t  Reprinted in\n\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t  Speech Synthesis\n\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              Flanagan\n\t\t\t\t\t\t\t\t\t              J.L.\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              Rabiner\n\t\t\t\t\t\t\t\t\t              L.R.\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t  Dowden, Hutchinson & Ross\n\t\t\t\t\t\t\t\t\t\t\t  Stroudsburg, PA, USA\n\t\t\t\t\t\t\t\t\t\t\t  1973\n\t\t\t\t\t\t\t\t\t          289\n\t\t\t\t\t\t\t\t\t\t\t  292"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/BF02075991","article-title":"Invariant sets for kinetic equations","volume":"10","author":"Gorban","year":"1979","journal-title":"React. Kinet. Catal. Lett."},{"key":"ref_61","unstructured":"Gorban, A.N., Kaganovich, B.M., Filippov, S.P., Keiko, A.V., Shamansky, V.A., and Shirkalin, I.A. (2006). Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibria, Springer."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/BF02124750","article-title":"On the Lambert W function","volume":"5","author":"Corless","year":"1996","journal-title":"Adv. Comput. Math."},{"key":"ref_63","unstructured":"Edwards, R.E. (1995). Functional Analysis: Theory and Applications, Dover Publications."},{"key":"ref_64","first-page":"74","article-title":"Sulla teoria di Volterra della lotta per l\u2019esistenza","volume":"7","author":"Kolmogorov","year":"1936","journal-title":"Giornale Instituto Ital. Attuari"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1137\/0129022","article-title":"Nonlinear aspects of competition between three species","volume":"29","author":"May","year":"1975","journal-title":"SIAM J. Appli. Math."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Bazykin, A.D. (1998). Nonlinear Dynamics of Interacting Populations, World Scientific Publishing.","DOI":"10.1142\/9789812798725"},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Charpentier, \u00c9., Lesne, A., and Nikolski, N.K. (2007). Kolmogorov\u0092s Heritage in Mathematics, Springer.","DOI":"10.1007\/978-3-540-36351-4"},{"key":"ref_68","unstructured":"Horn, F. (1964). The Third European Symposium on Chemical Reaction Engineering, Pergamon Press."},{"key":"ref_69","first-page":"1803","article-title":"A geometric approach to steady flow reactors: the attainable region and optimisation in concentration space","volume":"26","author":"Glasser","year":"1987","journal-title":"Am. Chem. Soc."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/0098-1354(89)89005-2","article-title":"Batch reactor optimization by use of tendency models","volume":"13","author":"Bordet","year":"1989","journal-title":"Comput. Chem. Eng."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"2161","DOI":"10.1016\/0009-2509(90)80091-R","article-title":"The attainable region and optimal reactor structures","volume":"45","author":"Hildebrandt","year":"1990","journal-title":"Chem. Eng. Sci."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1637","DOI":"10.1016\/S0009-2509(96)00471-X","article-title":"Optimal reactor design from a geometric viewpoint\u2013I. Universal properties of the attainable region","volume":"52","author":"Feinberg","year":"1997","journal-title":"Chem. Eng. Sci."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1016\/j.compchemeng.2008.11.013","article-title":"Chemical product engineering\u2014the third paradigm","volume":"33","author":"Hill","year":"2009","journal-title":"Comput. Chem. Eng."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"1076","DOI":"10.1021\/ie960358x","article-title":"Attainable regions for polymerization reaction systems","volume":"36","author":"Smith","year":"1997","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"3766","DOI":"10.1016\/j.ces.2009.05.012","article-title":"Use of the attainable region analysis to optimize particle breakage in a ball mill","volume":"64","author":"Metzger","year":"2009","journal-title":"Chem. Eng. Sci."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1021\/ie980380l","article-title":"The attainable region and Pontryagin\u2019s maximum principle","volume":"38","author":"McGregor","year":"1999","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"2015","DOI":"10.1016\/S0009-2509(02)00101-X","article-title":"Linear programming formulations for attainable region analysis","volume":"57","author":"Kauchali","year":"2002","journal-title":"Chem. Eng. Sci."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"1563","DOI":"10.1016\/j.compchemeng.2003.12.005","article-title":"The Shrink-Wrap algorithm for the construction of the attainable region: an application of the IDEAS framework","volume":"28","author":"Manousiouthakis","year":"2004","journal-title":"Comput. Chem. Eng."},{"key":"ref_79","first-page":"42","article-title":"Methods for qualitative analysis of chemical kinetics equations","volume":"Volume 10","author":"Gorban","year":"1979","journal-title":"Numerical Methods of Continuum Mechanics"},{"key":"#cr-split#-ref_80.1","unstructured":"Gorban, A.N., Yablonskii, G.S., and Bykov, V.I. (1980). Mathematical Problems of Chemical Thermodynamics, Nauka. (in Russian)"},{"key":"#cr-split#-ref_80.2","unstructured":"English translation: Int. Chem. Eng. 1982, 22, 368-375."},{"key":"ref_81","first-page":"1171","article-title":"On one unused possibility in planning of kinetic experiment","volume":"250","author":"Gorban","year":"1980","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/BF00529057","article-title":"A note on the attainability of states by equalizing processes","volume":"68","author":"Zylka","year":"1985","journal-title":"Theor. Chim. Acta"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/0009-2509(84)85079-4","article-title":"Accessible composition domains for monomolecular systems","volume":"39","author":"Krambeck","year":"1984","journal-title":"Chem. Eng. Sci."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1021\/i100018a005","article-title":"Structure of complex catalytic reactions: thermodynamic constraints in kinetic modeling and catalyst evaluation","volume":"24","author":"Shinnar","year":"1985","journal-title":"Ind. Eng. Chem. Fundam."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"2303","DOI":"10.1016\/0009-2509(88)87119-7","article-title":"Thermodynamic analysis in chemical process and reactor design","volume":"43","author":"Shinnar","year":"1988","journal-title":"Chem. Eng. Sci."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"1943","DOI":"10.1021\/ie00069a038","article-title":"Comments on \u201cStructure of complex catalytic reactions: thermodynamic constraints in kinetic modeling and catalyst evaluation\u201d","volume":"26","author":"Bykov","year":"1987","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_87","unstructured":"Alberti, P.M., and Uhlmann, A. (1982). Stochasticity and Partial Order\u2014Doubly Stochastic Maps and Unitary Mixing (Mathematics and its Applications 9), D. Reidel."},{"key":"ref_88","unstructured":"Plath, P.J., and Hass, E.-Chr. (2008). Vernetzte Wissenschaften\u2014Crosslinks in Natural and Social Sciences, Logos Verlag."},{"key":"ref_89","unstructured":"Harremo\u00ebs, P. (,  2004). A new look on majorization. Proceedings ISITA 2004, Parma, Italy."},{"key":"ref_90","doi-asserted-by":"crossref","unstructured":"Harremo\u00ebs, P., and Tishby, N. (,  2007). The information bottleneck revisited or how to choose a good distortion measure. Proceedings ISIT 2007, Nice, France.","DOI":"10.1109\/ISIT.2007.4557285"},{"key":"ref_91","unstructured":"Acz\u00e9l, J. (1966). Lectures on Functional Equations and Their Applications, Academic Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/12\/5\/1145\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:02:22Z","timestamp":1760220142000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/12\/5\/1145"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,5,7]]},"references-count":92,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2010,5]]}},"alternative-id":["e12051145"],"URL":"https:\/\/doi.org\/10.3390\/e12051145","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,5,7]]}}}