{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T16:46:50Z","timestamp":1773679610397,"version":"3.50.1"},"reference-count":72,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,22]],"date-time":"2018-01-22T00:00:00Z","timestamp":1516579200000},"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>Internal and mesoscopic variables differ fundamentally from each other: both are state space variables, but mesoscopic variables are additionally equipped with a distribution function introducing a statistical item into consideration which is missing in connection with internal variables. Thus, the alignment tensor of the liquid crystal theory can be introduced as an internal variable or as one generated by a mesoscopic background using the microscopic director as a mesoscopic variable. Because the mesoscopic variable is part of the state space, the corresponding balance equations change into mesoscopic balances, and additionally an evolution equation of the mesoscopic distribution function appears. The flexibility of the mesoscopic concept is not only demonstrated for liquid crystals, but is also discussed for dipolar media and flexible fibers.<\/jats:p>","DOI":"10.3390\/e20010081","type":"journal-article","created":{"date-parts":[[2018,1,22]],"date-time":"2018-01-22T13:40:39Z","timestamp":1516628439000},"page":"81","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Macroscopic Internal Variables and Mesoscopic Theory: A Comparison Considering Liquid Crystals"],"prefix":"10.3390","volume":"20","author":[{"given":"Christina","family":"Papenfuss","sequence":"first","affiliation":[{"name":"Department of Engineering 2, Hochschule f\u00fcr Technik und Wirtschaft Berlin, 12459 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wolfgang","family":"Muschik","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik, Technische Universit\u00e4t Berlin, 10623 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1515\/jnet.1990.15.2.127","article-title":"Internal variables in non-equilibrium thermodynamics","volume":"15","author":"Muschik","year":"1990","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_2","first-page":"250","article-title":"Thermodynamics with internal variables. I. General Concepts, II. Applications","volume":"19","author":"Maugin","year":"1994","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Maugin, G.A. (1992). The Thermomechanics of Plasticity and Fracture, Cambridge University Press.","DOI":"10.1017\/CBO9781139172400"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Truesdell, C., and Noll, W. (1965). Non-Linear Field Theories of Mechanics. Encyclopedia of Physics, Volume III\/3, Sect. 98, Springer.","DOI":"10.1007\/978-3-642-46015-9_1"},{"key":"ref_5","unstructured":"Cosserat, E., and Cosserat, F. (1909). Th\u00e9orie des Corps D\u00e9formables, Hermann and Fils. (In French)."},{"key":"ref_6","first-page":"1139","article-title":"Sur la mechanique g\u00e9n\u00e9rale","volume":"145","author":"Cosserat","year":"1907","journal-title":"Acad. Sci. Paris"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/BF00281389","article-title":"Anisotropic fluids","volume":"4","author":"Ericksen","year":"1960","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/BF00251810","article-title":"Some constitutive equations for liquid crystals","volume":"28","author":"Leslie","year":"1965","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1515\/zna-1975-6-704","article-title":"Irreversible thermodynamics of nonequilibrium alignment phenomena in molecular liquids and in liquid crystals","volume":"30","author":"Hess","year":"1975","journal-title":"Z. Naturforsch."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Axelrad, D.R., and Muschik, W. (1988). Thermodynamic modelling of polymers in solution. Constitutive Laws and Microstructure, Springer.","DOI":"10.1007\/978-3-642-83303-8"},{"key":"ref_11","unstructured":"Verhas, J. (1997). Thermodynamics and Rheology, Akad\u00e9miai Kiad\u00f3."},{"key":"ref_12","first-page":"311","article-title":"The thermodynamic theory of non-Newtonian flows","volume":"18","author":"Verhas","year":"1993","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1016\/0017-9310(87)90018-4","article-title":"Irreversible thermodynamics for rheological properties of colloids","volume":"30","author":"Verhas","year":"1987","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/BF01177946","article-title":"A thermodynamic approach to viscoanelasticity and plasticity","volume":"53","author":"Verhas","year":"1984","journal-title":"Acta Mech."},{"key":"ref_15","first-page":"149","article-title":"On symmeric irreducible tensors in d dimensions","volume":"51","author":"Ehrentraut","year":"1998","journal-title":"Int. J. Phys. Eng. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/S0377-0257(00)00141-5","article-title":"A sketch of continuum thermodynamics","volume":"96","author":"Muschik","year":"2001","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_17","first-page":"807","article-title":"Objectivity and frame indifference of acceleration-sensitive materials","volume":"50","author":"Muschik","year":"2012","journal-title":"J. Theor. Appl. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Jou, D., Casas-V\u00e1zquez, J., and Lebon, G. (1993). Extended Irreversible Thermodynamics, Springer.","DOI":"10.1007\/978-3-642-97430-4"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"M\u00fcller, I., and Ruggeri, T. (1993). Extended Thermodynamics, Springer.","DOI":"10.1007\/978-1-4684-0447-0"},{"key":"ref_20","unstructured":"Bridgman, W. (1943). The Nature of Thermodynamics, Harvard University Press. reprinted in Harper and Brothers: New York, USA, 1961."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1515\/zpch-1943-5320","article-title":"Zur Thermodynamik der irreversiblen Prozesse","volume":"538","author":"Meixner","year":"1943","journal-title":"Z. Phys. Chem."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Maugin, G.A. (1999). The Thermomechanics of Nonlinear Irreversible Behaviors\u2014An Introduction, World Scientific.","DOI":"10.1142\/9789812796271"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1221","DOI":"10.1080\/02678299308027831","article-title":"A continuum theory for liquid crystals describing different degrees of orientational order","volume":"14","author":"Blenk","year":"1993","journal-title":"Liq. Cryst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1080\/00268949108046600","article-title":"Orientation balances for liquid crystals and their representation by alignment tensors","volume":"204","author":"Blenk","year":"1991","journal-title":"Mol. Cryst. Liq. Cryst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1515\/JNETDY.2000.011","article-title":"Concepts of mesoscopic continuum physics with application to liquid crystals","volume":"25","author":"Muschik","year":"2000","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1515\/jnet.1991.16.1.67","article-title":"Orientational balances for nematic liquid crystals","volume":"16","author":"Blenk","year":"1991","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.jnnfm.2004.01.011","article-title":"Sketch of the mesoscopic description of nematic liquid crystals","volume":"119","author":"Muschik","year":"2004","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF00380413","article-title":"Liquid crystals with variable degree of orientation","volume":"113","author":"Ericksen","year":"1991","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1093\/qjmam\/19.3.357","article-title":"Some constitutive equations for anisotropic fluids","volume":"19","author":"Leslie","year":"1966","journal-title":"Q. J. Mech. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1515\/jnet.1995.20.1.92","article-title":"Ericksen-Leslie Liquid Crystal Theory Revisited from a Mesoscopic Point of View","volume":"20","author":"Muschik","year":"1995","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1224","DOI":"10.1515\/zna-1975-1002","article-title":"Irreversible thermodynamics of nonequilibrium alignment phenomena in molecular liquids and in liquid crystals ii","volume":"30","author":"Hess","year":"1975","journal-title":"Z. Naturforsch."},{"key":"ref_32","unstructured":"Landau, L.D., and Lifschitz, E.M. (1984). Statistische Physik. Lehrbuch der Theoretischen Physik, Akademie-Verlag. [6th ed.]. translated into German."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Vertogen, G., and deJeu, W.H. (1988). Thermotropic Liquid Crystals, Fundamentals. Chemical Physics, Number 45, Springer.","DOI":"10.1007\/978-3-642-83133-1"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1080\/02678298708086335","article-title":"An extension of the Landau-Ginzburg-de Gennes theory for liquid crystals","volume":"2","author":"Longa","year":"1987","journal-title":"Liq. Cryst."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"De Gennes, P.G. (1974). The Physics of Liquid Crystals, Clarendon Press.","DOI":"10.1016\/B978-0-12-460350-9.50040-0"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"De Gennes, P.G., and Prost, J. (1995). The Physics of Liquid Crystals, Monographs on Physics, Clarendon Press. [2nd ed.].","DOI":"10.1063\/1.2808028"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Virga, E.G. (1994). Variational Theories for Liquid Crystals, Chapman and Hall.","DOI":"10.1007\/978-1-4899-2867-2"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"De Gennes, P.G. (1992). Simple Views on Condensed Matter. Series in Modern Condensed Matter Physics, World Scientific.","DOI":"10.1142\/1606"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3264","DOI":"10.1103\/PhysRevA.30.3264","article-title":"Landau theory of cholesteric blue phases: The role of higher harmonics","volume":"30","author":"Grebel","year":"1984","journal-title":"Phys. Rev. A"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1114","DOI":"10.1103\/PhysRevA.28.1114","article-title":"Landau theory of cholesteric blue phases","volume":"28","author":"Grebel","year":"1983","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/0378-4371(91)90420-H","article-title":"Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation","volume":"174","author":"Blenk","year":"1991","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1016\/0378-4371(80)90166-1","article-title":"On the theory of irreversible processes in molecular liquids and liquid crystals, nonequilibrium phenomena assosiated with the second and fourth rank alignment-tensors","volume":"100","author":"Pardowitz","year":"1980","journal-title":"Physica"},{"key":"ref_43","first-page":"79","article-title":"Exploiting the second law in weakly non-local continuum physics","volume":"49","year":"2005","journal-title":"Period. Polytech. Ser. Mech. Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s00161-004-0195-z","article-title":"The Ginzburg-Landau equation as a consequence of the second law","volume":"17","year":"2005","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_45","first-page":"142","article-title":"Weakly nonlocal irreversible thermodynamics","volume":"12","year":"2003","journal-title":"Ann. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/S0927-0256(00)00138-5","article-title":"Theory of liquid crystals as an example of mesoscopic continuum mechanics","volume":"19","author":"Papenfuss","year":"2000","journal-title":"Comput. Mater. Sci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1080\/10587259908025632","article-title":"Orientational order in free standing liquid crystalline films and derivation of a closure relation for higher order alignment tensors","volume":"330","author":"Papenfuss","year":"1999","journal-title":"Mol. Cryst. Liq. Cryst."},{"key":"ref_48","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_49","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_50","unstructured":"M\u00fcller, I., and Ruggeri, T. (1987). Kinetic Theory and Extended Thermodynamics, Pitagora Editrice Bologna."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"6505","DOI":"10.1088\/0305-4470\/20\/18\/047","article-title":"Maximization of the entropy in non-equilibrium","volume":"20","author":"Dreyer","year":"1987","journal-title":"J. Phys. A"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"M\u00fcller, I., and Ruggeri, T. (1993). Extended Thermodynamics, Springer. Springer Tracts in Natural Philosophy.","DOI":"10.1007\/978-1-4684-0447-0"},{"key":"ref_53","unstructured":"Cerciniani, C. (1975). Theory and Application of the Boltzmann Equation, Elsevier."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1007\/s00419-007-0193-2","article-title":"Systematic remarks on objectivity and frame-indifference, liquid crystal theory as an example","volume":"78","author":"Muschik","year":"2008","journal-title":"Arch. Appl. Mech."},{"key":"ref_55","first-page":"152","article-title":"Changing the Observer and Moving Materials in Continuum Physics: Objectivity and Frame-Indifference","volume":"22","author":"Muschik","year":"2002","journal-title":"Tech. Mech."},{"key":"ref_56","unstructured":"Papenfuss, C., V\u00e1n, P., and Muschik, W. (2000, January 5\u20139). Mesoscopic theory of microcrack dynamics. Proceedings of the SolMech 2000, Volume of Abstracts, Zakopane, Poland."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"6206","DOI":"10.1103\/PhysRevE.62.6206","article-title":"Mesoscopic dynamics of microcracks","volume":"62","author":"Papenfuss","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_58","first-page":"481","article-title":"Mesoscopic theory of microcracks","volume":"55","author":"Papenfuss","year":"2003","journal-title":"Arch. Mech."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/S0378-4371(03)00551-X","article-title":"A closure relation for the higher order alignment tensors in liquid crystal theory and the alignment-Fabric tensors in damage mechanics from a statistical background","volume":"331","author":"Papenfuss","year":"2003","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"5315","DOI":"10.1088\/0305-4470\/37\/20\/005","article-title":"Griffith cracks in the mesoscopic microcrack theory","volume":"37","author":"Papenfuss","year":"2004","journal-title":"J. Phys. A"},{"key":"ref_61","unstructured":"Maruszewski, B.T., Muschik, W., and Radowicz, A. (2004). Damage evolution in micro-cracked materials under load. Trends in Continuum Physics, World Scientific."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/JNETDY.2007.005","article-title":"Dynamics of the size and orientation distribution of microcracks and evolution of macroscopic damage parameters","volume":"32","author":"Papenfuss","year":"2007","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"132","DOI":"10.3176\/proc.2008.3.03","article-title":"Scalar, vectorial, and tensorial damage parameters from the mesoscopic background","volume":"57","author":"Papenfuss","year":"2008","journal-title":"Proc. Eston. Acad. Sci."},{"key":"ref_64","first-page":"132","article-title":"Application of the mesoscopic theory to dipolar media","volume":"22","author":"Papenfuss","year":"2002","journal-title":"Tech. Mech."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1515\/JNETDY.2005.028","article-title":"A mesoscopic approach to diffusion phenoma in mixtures","volume":"30","author":"Palumbo","year":"2005","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.1007\/s10910-017-0778-0","article-title":"Mesoscopic continuum thermodynamics for mixtures of particles with orientation","volume":"30","author":"Florindo","year":"2017","journal-title":"J. Math. Chem."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1515\/JNETDY.2008.011","article-title":"A mesoscopic continuum description of dry granular materials","volume":"33","author":"Chen","year":"2008","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1098\/rspa.2005.1609","article-title":"On the macroscopic-mesoscopic mixture of a magnetorheological fluid","volume":"462","author":"Chen","year":"2006","journal-title":"Proc. R. Soc. A"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/j.compstruct.2014.11.018","article-title":"An orthotropic material model for steel fibre reinforced concrete based on the orientation distribution of fibres","volume":"121","author":"Eik","year":"2014","journal-title":"Compos. Struct."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.compstruct.2016.03.046","article-title":"The effect of approximation accuracy of the orientation distribution function on the elastic properties of short fibre reinforced composites","volume":"148","author":"Eik","year":"2016","journal-title":"Compos. Struct."},{"key":"ref_71","unstructured":"Krajcinovic, D. (1996). Damage Mechanics, Elsevier."},{"key":"ref_72","unstructured":"Griffith, A.A. (1924, January 22\u201326). The theory of rupture. Proceedings of the First International Congress of Applied Mechanics, Delft, The Netherlands."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/81\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:52:09Z","timestamp":1760194329000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/81"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,22]]},"references-count":72,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010081"],"URL":"https:\/\/doi.org\/10.3390\/e20010081","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201712.0190.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1,22]]}}}