{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:32:33Z","timestamp":1760243553067,"version":"build-2065373602"},"reference-count":52,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2012,2,28]],"date-time":"2012-02-28T00:00:00Z","timestamp":1330387200000},"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>In this paper, we combine the two universalisms of thermodynamics and dynamical systems theory to develop a dynamical system formalism for classical thermodynamics. Specifically, using a compartmental dynamical system energy flow model we develop a state-space dynamical system model that captures the key aspects of thermodynamics, including its fundamental laws. In addition, we establish the existence of a unique, continuously differentiable global entropy function for our dynamical system model, and using Lyapunov stability theory we show that the proposed thermodynamic model has finite-time convergent trajectories to Lyapunov stable equilibria determined by the system initial energies. Finally, using the system entropy, we establish the absence of Poincar\u00e9 recurrence for our thermodynamic model and develop clear and rigorous connections between irreversibility, the second law of thermodynamics, and the entropic arrow of time.<\/jats:p>","DOI":"10.3390\/e14030407","type":"journal-article","created":{"date-parts":[[2012,2,28]],"date-time":"2012-02-28T11:11:05Z","timestamp":1330427465000},"page":"407-455","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Temporal Asymmetry, Entropic Irreversibility, and Finite-Time Thermodynamics: From Parmenides\u2013Einstein Time-Reversal Symmetry to the Heraclitan Entropic Arrow of Time"],"prefix":"10.3390","volume":"14","author":[{"given":"Wassim M.","family":"Haddad","sequence":"first","affiliation":[{"name":"The School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,2,28]]},"reference":[{"unstructured":"Haddad, W.M., Chellaboina, V., and Nersesov, S.G. (2005). Thermodynamics: A Dynamical Systems Approach, Princeton University Press.","key":"ref_1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/BF01450409","article-title":"Untersuchungen \u00fcber die grundlagen der thermodynamik","volume":"67","year":"1909","journal-title":"Math. Ann."},{"unstructured":"Carath\u00e9odory, C. (, 1925). \u00dcber die Bestimmung der Energie und der absoluten Temperatur mit Hilfe von reversiblen Prozessen. Proceedings of the 1925 Sitzungsberichte der Preu\u03b2ischen Akademie der Wissenschaften, Math. Phys. Klasse, Berlin, Germany.","key":"ref_3"},{"unstructured":"Carath\u00e9odory\u2019s definition of an adiabatic process is nonstandard and involves transformations that take place while the system remains in an adiabatic container. For details see [2,3].","key":"ref_4"},{"unstructured":"Bridgman, P. (1941). The Nature of Thermodynamics, Harvard University Press. Reprinted by Peter Smith: Gloucester, MA, USA, 1969.","key":"ref_5"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/S1355-2198(01)00016-8","article-title":"Bluff your way in the second law of thermodynamics","volume":"32","author":"Uffink","year":"2001","journal-title":"Stud. Hist. Philos. Mod. Phys. B"},{"unstructured":"Perhaps a better expression here is the geodesic arrow of time, since, as Einstein\u2019s theory of relativity shows, time and space are intricately coupled, and hence one cannot curve space without involving time as well. Thus, time has a shape that goes along with its directionality.","key":"ref_7"},{"unstructured":"Planck, M. (, 1925). \u00dcber die Begrundung des zweiten Hauptsatzes der Thermodynamik. Proceedings of the 1925 Sitzungsberichte der Preu\u03b2ischen Akademie der Wissenschaften, Math. Phys. Klasse, Berlin, Germany.","key":"ref_8"},{"doi-asserted-by":"crossref","unstructured":"Reichenbach, H. (1956). The Direction of Time, University of California Press.","key":"ref_9","DOI":"10.1063\/1.3059791"},{"unstructured":"Gold, T. (1967). The Nature of Time, Cornell University Press.","key":"ref_10"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"543","DOI":"10.2307\/2024363","article-title":"Irreversibility and temporal asymmetry","volume":"64","author":"Earman","year":"1967","journal-title":"J. Philos."},{"doi-asserted-by":"crossref","unstructured":"Kroes, P. (1985). Time: Its Structure and Role in Physical Theories, Reidel.","key":"ref_12","DOI":"10.1007\/978-94-009-6522-5"},{"unstructured":"Horwich, P. (1987). Asymmetries in Time, MIT Press.","key":"ref_13"},{"unstructured":"In statistical thermodynamics the arrow of time is viewed as a consequence of high system dimensionality and randomness. However, since in statistical thermodynamics it is not absolutely certain that entropy increases in every dynamical process, the direction of time, as determined by entropy increase, has only statistical certainty and not an absolute certainty. Hence, it cannot be concluded from statistical thermodynamics that time has a unique direction of flow.","key":"ref_14"},{"doi-asserted-by":"crossref","unstructured":"Lamb, J.S.W., and Roberts, J.A.G. (1998). Time reversal symmetry in dynamical systems: A survey. Physica D, 112.","key":"ref_15","DOI":"10.1016\/S0167-2789(97)00199-1"},{"unstructured":"Eddington, A. (1935). The Nature of the Physical World, Dent & Sons.","key":"ref_16"},{"unstructured":"Prigogine, I. (1980). From Being to Becoming, Freeman.","key":"ref_17"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/j.nonrwa.2006.10.002","article-title":"Time-reversal symmetry, Poincar\u00e9 recurrence, irreversibility, and the entropic arrow of time: From mechanics to system thermodynamics","volume":"9","author":"Haddad","year":"2008","journal-title":"Nonlinear Anal. R. World Appl."},{"unstructured":"In the terminology of [6], state irreversibility is referred to as time-reversal noninvariance. However, since the term time reversal is not meant literally (that is, we consider dynamical systems whose trajectory reversal is or is not allowed and not a reversal of time itself), state reversibility is a more appropriate expression.","key":"ref_19"},{"doi-asserted-by":"crossref","unstructured":"Bhat, S.P., and Bernstein, D.S. (2003, January 4\u20136). Arc-length-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria. Proceedings of the 2003 American Control Conference, Denver, CO, USA.","key":"ref_20","DOI":"10.1109\/ACC.2003.1243775"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1137\/S0363012902407119","article-title":"Nontangency-based Lyapunov tests for convergence and stability in systems having a continuum of equilibra","volume":"42","author":"Bhat","year":"2003","journal-title":"SIAM J. Control Optim."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1137\/S0363012997321358","article-title":"Finite-time stability of continuous autonomous systems","volume":"38","author":"Bhat","year":"2000","journal-title":"SIAM J. Control Optim."},{"unstructured":"Hale, J.K. (1980). Ordinary Differential Equations, Wiley. [2nd ed.]. Reprinted by Krieger: Malabar, FL, USA, 1991.","key":"ref_23"},{"doi-asserted-by":"crossref","unstructured":"Liberman, P., and Marle, C.M. (1987). Symplectic Geometry and Analytical Mechanics, Reidel.","key":"ref_24","DOI":"10.1007\/978-94-009-3807-6"},{"unstructured":"Here we assume that the system Lagrangian is hyperregular [24] so that the map from the generalized velocities q . to the generalized momenta p is bijective (i.e., one-to-one and onto).","key":"ref_25"},{"doi-asserted-by":"crossref","unstructured":"Arnold, V.I. (1989). Mathematical Models of Classical Mechanics, Springer-Verlag.","key":"ref_26","DOI":"10.1007\/978-1-4757-2063-1"},{"key":"ref_27","first-page":"1","article-title":"Sur le probl\u00e9me des trois corps et les \u00e9quations de la dynamique","volume":"13","year":"1890","journal-title":"Acta Math."},{"unstructured":"A Lie group is a topological group that can be given an analytic structure such that the group operation and inversion are analytic. A Lie pseudogroup is an infinite-dimensional counterpart of a Lie group.","key":"ref_28"},{"unstructured":"Apostol, T.M. (1974). Mathematical Analysis, Addison-Wesley.","key":"ref_29"},{"unstructured":"We say that V is dense in N if and only if N is contained in the closure of V ; that is, V \u2286 N is dense in N if and only if N \u2286 V \u00af .","key":"ref_30"},{"unstructured":"A key distinction between thermodynamics and mechanics is that thermodynamics is a theory of open systems, whereas mechanics is a theory of closed systems. The notions, however, of open and closed systems are different in thermodynamics and dynamical system theory. In particular, thermodynamic systems exchange matter and energy with the environment, and hence, interact with the environment. Such systems are called open systems in the thermodynamic literature. Systems that exchange heat (energy) but not matter with the environment are called closed, whereas systems that do not exchange energy and matter with the environment are called isolated. Alternatively, in mechanics it is always possible to include interactions with the environment (via feedback interconnecting components) within the system description to obtain an augmented closed system in the sense of dynamical system theory. That is, the system can be described by an evolution law with, possibly, an output equation wherein past trajectories define the future trajectory uniquely and the system output depends on the instantaneous (present) value of the system state.","key":"ref_31"},{"unstructured":"Bhat, S.P., and Bernstein, D.S. (1999, January 2\u20134). Lyapunov analysis of semistability. Proceedings of the 1999 American Control Conference, San Diego, CA, USA.","key":"ref_32"},{"doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., and Lakshmikantham, V. (1993). Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, World Scientific.","key":"ref_33","DOI":"10.1142\/1988"},{"unstructured":"Yoshizawa, T. (1966). Stability Theory by Liapunov\u2019s Second Method, Math. Soc. Japan.","key":"ref_34"},{"unstructured":"Coddington, E.A., and Levinson, N. (1955). Theory of Ordinary Differential Equations, McGraw-Hill.","key":"ref_35"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s00498-005-0151-x","article-title":"Geometric homogeneity with applications to finite-time stability","volume":"17","author":"Bhat","year":"2005","journal-title":"Math. Control Signals Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/0167-6911(92)90078-7","article-title":"Homogeneous Lyapunov function for homogeneous continuous vector field","volume":"19","author":"Rosier","year":"1992","journal-title":"Syst. Control Lett."},{"unstructured":"In a geometric, coordinate-free setting, the only link between homogeneity of functions and vector fields is that the Lie derivative of a homogeneous function along a homogeneous vector field is also a homogeneous function. In the special case where the coordinate functions are homogeneous functions, the fact mentioned above can be used to relate the homogeneity of a vector field with that of the components (considered as functions) of its coordinate representation. Such a relation is very familiar in the case of conventional dilations seen in the homogeneity literature [37].","key":"ref_38"},{"unstructured":"The domain of semistability (with respect to R \u00af + n ) is the set of points x0 \u2208 R \u00af + n such that if x(t), t \u2265 0, is a solution to Equation (30) with x(0) = x0, then x(t) converges to a Lyapunov stable (with respect to R \u00af + n ) equilibrium point in R \u00af + n .","key":"ref_39"},{"doi-asserted-by":"crossref","unstructured":"Haddad, W.M., and Chellaboina, V. (2008). Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach, Princeton University Press.","key":"ref_40","DOI":"10.1515\/9781400841042"},{"unstructured":"Mizutani, T. (2011). Thermodynamics, InTech.","key":"ref_41"},{"unstructured":"It can be argued here that a more appropriate terminology is assumptions rather than axioms since, as will be seen, these are statements taken to be true and used as premises in order to infer certain results, but may not otherwise be accepted. However, as we will see, these statements are equivalent (within our formulation) to the stipulated postulates of the zeroth and second laws of thermodynamics involving transitivity of a thermal equilibrium and heat flowing from hotter to colder bodies, and as such we refer to them as axioms.","key":"ref_42"},{"doi-asserted-by":"crossref","unstructured":"Berman, A., and Plemmons, R.J. (1979). Nonnegative Matrices in the Mathematical Sciences, Academic Press.","key":"ref_43","DOI":"10.1016\/B978-0-12-092250-5.50009-6"},{"unstructured":"It is important to note that our formulation of the second law of thermodynamics as given by Axiom ii) does not require the mentioning of temperature nor the more primitive subjective notions of hotness or coldness. As we will see later, temperature is defined in terms of the system entropy after we establish the existence of a unique, continuously differentiable entropy function for G .","key":"ref_44"},{"unstructured":"Since in our formulation we are not considering work performed by and on the system, the notions of an isolated system and an adiabatically isolated system are equivalent.","key":"ref_45"},{"unstructured":"Stuart, E.B., Gal-Or, B., and Brainard, A.J. (1970). A Critical Review of Thermodynamics, Mono Book Corp.","key":"ref_46"},{"doi-asserted-by":"crossref","unstructured":"Lavenda, B. (1978). Thermodynamics of Irreversible Processes, Macmillan. Reprinted by Dover: New York, NY, USA, 1993.","key":"ref_47","DOI":"10.1007\/978-1-349-03254-9"},{"unstructured":"Diestel, R. (1997). Graph Theory, Springer-Verlag.","key":"ref_48"},{"doi-asserted-by":"crossref","unstructured":"Godsil, C., and Royle, G. (2001). Algebraic Graph Theory, Springer-Verlag.","key":"ref_49","DOI":"10.1007\/978-1-4613-0163-9"},{"doi-asserted-by":"crossref","unstructured":"Lozano, R., Brogliato, B., Egeland, O., and Maschke, B. (2000). Dissipative Systems Analysis and Control, Springer-Verlag.","key":"ref_50","DOI":"10.1007\/978-1-4471-3668-2"},{"doi-asserted-by":"crossref","unstructured":"Haddad, W.M., Chellaboina, V., and Hui, Q. (2010). Nonnegative and Compartmental Dynamical Systems, Princeton University Press.","key":"ref_51","DOI":"10.1515\/9781400832248"},{"unstructured":"The differential operator notation in \u03bd(x) is standard differential geometric notation used to write coordinate expressions for vector fields. This notation is based on the fact that there is a one-to-one correspondence between first-order linear differential operators on real-valued functions and vector fields.","key":"ref_52"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/3\/407\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:49:06Z","timestamp":1760219346000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/3\/407"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2,28]]},"references-count":52,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2012,3]]}},"alternative-id":["e14030407"],"URL":"https:\/\/doi.org\/10.3390\/e14030407","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2012,2,28]]}}}