{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T19:55:11Z","timestamp":1770753311433,"version":"3.50.0"},"reference-count":29,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,3,16]],"date-time":"2017-03-16T00:00:00Z","timestamp":1489622400000},"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>A variant of continuous nonequilibrium thermodynamic theory based on the postulate of the scale invariance of the local relation between generalized fluxes and forces is proposed here. This single postulate replaces the assumptions on local equilibrium and on the known relation between thermodynamic fluxes and forces, which are widely used in classical nonequilibrium thermodynamics. It is shown here that such a modification not only makes it possible to deductively obtain the main results of classical linear nonequilibrium thermodynamics, but also provides evidence for a number of statements for a nonlinear case (the maximum entropy production principle, the macroscopic reversibility principle, and generalized reciprocity relations) that are under discussion in the literature.<\/jats:p>","DOI":"10.3390\/e19030126","type":"journal-article","created":{"date-parts":[[2017,3,16]],"date-time":"2017-03-16T11:24:28Z","timestamp":1489663468000},"page":"126","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Nonequilibrium Thermodynamics and Scale Invariance"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7066-8768","authenticated-orcid":false,"given":"Leonid M.","family":"Martyushev","sequence":"first","affiliation":[{"name":"Technical Physics Department, Ural Federal University, 19 Mira St., Ekaterinburg 620002, Russia"},{"name":"Institute of Industrial Ecology, Russian Academy of Sciences, 20 S. Kovalevskaya St., Ekaterinburg 620219, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vladimir","family":"Celezneff","sequence":"additional","affiliation":[{"name":"The Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,16]]},"reference":[{"key":"ref_1","unstructured":"Prigogine, I. (1955). Introduction to Thermodynamics of Irreversible Processes, Thomas."},{"key":"ref_2","unstructured":"De Groot, S.R., and Mazur, P. (1962). Non-Equilibrium Thermodynamics, North-Holland Publishing Company."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gyarmati, I. (1970). Non-Equilibrium Thermodynamics. Field Theory and Variational Principles, Springer.","DOI":"10.1007\/978-3-642-51067-0"},{"key":"ref_4","unstructured":"Kondepudi, D., and Prigogine, I. (1998). Modern Thermodynamics: From Heat Engines to Dissipative Structures, Wiley."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Lebon, G., Jou, D., and Casas-V\u00e1zquez, J. (2008). Understanding Non-Equilibrium Thermodynamics: Foundations, Applications, Frontiers, Springer.","DOI":"10.1007\/978-3-540-74252-4"},{"key":"ref_6","unstructured":"Sneddon, I.N., and Hill, R. (1963). Progress in Solid Mechanics, North-Holland Publishing Company. Chapter 2."},{"key":"ref_7","unstructured":"Ziegler, H. (1983). An Introduction to Thermomechanics, North-Holland Publishing Company."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Dewar, R.C., Lineweaver, C.H., Niven, R.K., and Regenauer-Lieb, K. (2014). Beyond the Second Law Entropy Production and Non-Equilibrium Systems, Springer.","DOI":"10.1007\/978-3-642-40154-1"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2005.12.001","article-title":"Maximum entropy production principle in physics, chemistry and biology","volume":"426","author":"Martyushev","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_10","first-page":"153","article-title":"Nonequilibrium Thermodynamics of the Nonlinear Equations of Chemical Kinetics","volume":"3","author":"Bataille","year":"1978","journal-title":"J. Non-Equilib. Therm."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/0020-7225(74)90050-0","article-title":"On the characterization of fluxes in nonlinear irreversible thermodynamics","volume":"12","author":"Edelen","year":"1974","journal-title":"Int. J. Eng. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1002\/andp.19834950403","article-title":"An Extension of the Governing Principle of Dissipative Processes to Nonlinear Constitutive Equations","volume":"493","author":"Verhas","year":"1983","journal-title":"Annalen der Physik"},{"key":"ref_13","first-page":"184","article-title":"On the Nonlinear Generalizations of Onsager\u2019s Reciprocal Relations","volume":"19","author":"Ciancio","year":"1994","journal-title":"J. Non-Equilib. Therm."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2362","DOI":"10.3390\/e16042362","article-title":"Gyarmati\u2019s Variational Principle of Dissipative Processes","volume":"16","author":"Verhas","year":"2014","journal-title":"Entropy"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0034-4877(75)90004-X","article-title":"Macroscopic and microscopic reversibility","volume":"7","author":"Meixner","year":"1975","journal-title":"Rep. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.physa.2014.05.014","article-title":"The restrictions of the maximum entropy production principle","volume":"410","author":"Martyushev","year":"2014","journal-title":"Physica A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.coche.2014.10.003","article-title":"Maximum entropy production: Application to crystal growth and chemical kinetics","volume":"7","author":"Martyushev","year":"2015","journal-title":"Curr. Opin. Chem. Eng."},{"key":"ref_18","first-page":"779","article-title":"Empiricism or self-consistent theory in chemical kinetics?","volume":"434\u2013435","author":"Cutman","year":"2007","journal-title":"J. Alloys Compd."},{"key":"ref_19","unstructured":"Such nonlocal relations can be obtained from local relations not only by integration with respect to the time, spatial, energy, or other variables, but also by connection (composition) of individual local elements. These are so-called compound systems (see, e.g., [7,16]), where various nonequilibrium processes do not affect each other, existing independently in the system (in particular, these can be combination of local elements in complex electric circuits, complex chemical reactions, etc.). Without loss of generality, such independent processes can be considered separately."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4676","DOI":"10.1063\/1.447402","article-title":"Objections to a proposal on the rate of entropy production in systems far from equilibrium","volume":"81","author":"Andersen","year":"1984","journal-title":"J. Chem. Phys."},{"key":"ref_21","unstructured":"Bridgman, P.W. (1932). Dimensional Analysis, Yale University Press."},{"key":"ref_22","unstructured":"Huntley, H.E. (1967). Dimensional Analysis, Dover."},{"key":"ref_23","unstructured":"All quantities Jj can formally have the same dimension, but they are nevertheless different main (primary) units of measurement (see, e.g., the notion of vector units of measurement in [22])."},{"key":"ref_24","unstructured":"The simplest relation in the form of one of the terms in Equation (1) is traditionally taken [21], but the most general variant has the form of Equation (1) (see Chapter 2, question 6 [22])."},{"key":"ref_25","unstructured":"They can additionally depend on the equilibrium thermodynamic characteristics of the considered element of spacetime, but these quantities can always be included in the expression for the thermodynamic force and\/or flux and can be treated as their part by definition."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"013004","DOI":"10.1103\/PhysRevE.91.013004","article-title":"Morphological stability of an interface between two non-Newtonian fluids moving in a Hele-Shaw cell","volume":"91","author":"Martyushev","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_27","unstructured":"Reynolds, A.J. (1974). Turbulent Flows in Engineering, Wiley."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1016\/0001-6160(64)90130-0","article-title":"The molecular mechanism of solidification","volume":"12","author":"Cahn","year":"1964","journal-title":"Acta Metall."},{"key":"ref_29","unstructured":"The approach developed in this work is not limited by the use of only thermodynamic (Clausius) entropy. The dissipation function can also be introduced in terms of various generalizations of this classical entropy (e.g., information entropy)."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/3\/126\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:30:38Z","timestamp":1760207438000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/3\/126"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,16]]},"references-count":29,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,3]]}},"alternative-id":["e19030126"],"URL":"https:\/\/doi.org\/10.3390\/e19030126","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,16]]}}}