{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T07:11:23Z","timestamp":1763968283506,"version":"build-2065373602"},"reference-count":89,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,2,26]],"date-time":"2018-02-26T00:00:00Z","timestamp":1519603200000},"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>We consider nonequilibrium (NEQ) states such as supercooled liquids and glasses that are described with the use of internal variables. We classify the latter by the state-dependent hierarchy of relaxation times to assess their relevance for irreversible contributions. Given an observation time     \u03c4 obs    , we determine the window of relaxation times that divide the internal variables into active and inactive groups, the former playing a central role in the NEQ thermodynamics. Using this thermodynamics, we determine (i) a bound on the NEQ entropy and on the residual entropy and (ii) the nature of the isothermal relaxation of the entropy and the enthalpy in accordance with the second law. A theory that violates the second law such as the entropy loss view is shown to be internally inconsistent if we require it to be consistent with experiments. The inactive internal variables still play an indirect role in determining the temperature     T ( t )     and the pressure     P ( t )     of the system, which deviate from their external values.<\/jats:p>","DOI":"10.3390\/e20030149","type":"journal-article","created":{"date-parts":[[2018,2,27]],"date-time":"2018-02-27T03:36:12Z","timestamp":1519702572000},"page":"149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Hierarchy of Relaxation Times and Residual Entropy: A Nonequilibrium Approach"],"prefix":"10.3390","volume":"20","author":[{"given":"Purushottam","family":"Gujrati","sequence":"first","affiliation":[{"name":"Department of Physics, The University of Akron, Akron, OH 44325, USA"},{"name":"Department of Polymer Science, The University of Akron, Akron, OH 44325, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,2,26]]},"reference":[{"unstructured":"Goldstein, M., and Simha, R. (1976). The Glass Transition and the Nature of the Glassy State, New York Academy of Sciences.","key":"ref_1"},{"unstructured":"Nemilov, S.V. (1995). Thermodynamic and Kinetic Aspects of the Vitreous State, CRC Press.","key":"ref_2"},{"doi-asserted-by":"crossref","unstructured":"Debenedetti, P.G. (1996). Metastable Liquids, Concepts and Principles, Priceton University Press.","key":"ref_3","DOI":"10.1515\/9780691213941"},{"doi-asserted-by":"crossref","unstructured":"Gutzow, I.S., and Schmelzer, J.W.P. (2013). The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization, Springer. [2nd ed.].","key":"ref_4","DOI":"10.1007\/978-3-642-34633-0"},{"unstructured":"There are several interpretation of the IEQ state in the literature such as by Simon [6] and by Landau and Lifshitz [7]. There is some difference with our usage here. These authors do not use any internal variables and restrict themselves to using the state variable X. However, Simon allows the use of IEQ for a part of the degrees of freedom such as the vibrations that have come to equilibrium with the medium, even though other degrees of freedom have not. He then speaks of the third law being applicable to the entropy of the IEQ degrees of freedom such as the vibrational entropy. We on the other hand use \u201cequilibrated\u201d for vibrations (as emphasized by the inactive \u03beE introduced later) and \u201cnonequilibrated\u201d for others (as emphasized by the active \u03beN introduced later) under this condition. We reserve the use of the IEQ state for the entire system and do not make any claim about the third law being applicable to all IEQ states, which may include glasses.","key":"ref_5"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1016\/S0031-8914(37)80205-0","article-title":"On the third law of thermodynamics","volume":"4","author":"Simon","year":"1937","journal-title":"Physica"},{"unstructured":"Landau, L.D., and Lifshitz, E.M. (1986). Statistical Physics, Pergamon Press. [3rd ed.].","key":"ref_7"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1088\/0034-4885\/55\/3\/001","article-title":"Relaxation processes in supercooled liquids","volume":"55","author":"Gotze","year":"1992","journal-title":"Rep. Prog. 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A simple example that is very useful in our analysis is the kinetic energy K, which together with the potential energy U, determines the energy E = K + U. While the sum K + U can be controlled, the value of K (or U) cannot be controlled. As is commonly known [3], the vibrational modes in glasses or supercooled liquids come to equilibrium with the medium at its temperature T0, the configurational degrees of freedom do not. Another example of an internal variable is the possible inhomogeneity of the system consisting of two parts of energy E1 and E2. The total E1 + E2 is an observable, but the \u201cinhomogeneity\u201d E1 \u2212 E2 is an internal variable that cannot be controlled from the outside.","key":"ref_13"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1515\/zna-1949-0809","article-title":"Thermodynamik und Relaxationserscheinungen","volume":"4","author":"Meixner","year":"1949","journal-title":"Z. 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