Abstract
If one demystifies entropy the second law of thermodynamics comes out as an emergent property entirely based on the simple dynamic mechanical laws that govern the motion and energies of system parts on a micro-scale. The emergence of the second law is illustrated in this paper through the development of a new, very simple and highly efficient technique to compare time-averaged energies in isolated conservative linear large scale dynamical systems. Entropy is replaced by a notion that is much more transparent and more or less dual called ectropy. Ectropy has been introduced before but we further modify the notion of ectropy such that the unit in which it is expressed becomes the unit of energy. The second law of thermodynamics in terms of ectropy states that ectropy decreases with time on a large enough time-scale and has an absolute minimum equal to zero. Zero ectropy corresponds to energy equipartition. Basically we show that by enlarging the dimension of an isolated conservative linear dynamical system and the dimension of the system parts over which we consider time-averaged energy partition, the tendency towards equipartition increases while equipartition is achieved in the limit. This illustrates that the second law is an emergent property of these systems. Finally from our large scale linear dynamic model we clarify Loschmidt’s paradox concerning the irreversible behavior of ectropy obtained from the reversible dynamic laws that govern motion and energy at the micro-scale.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Willems, J.C.: Book review. IEEE Trans. Autom. Control 51(7), 1217 (2006)
Haddad, W.M., Chellaboina, V.S., Nersesov, S.: Thermodynamics: A Dynamical Systems Approach. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2005)
Bernstein, D.S., Bhat, S.P.: Energy equipartition and the emergence of damping. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, December, p. 2913 (2002)
Bhat, S.P., Bernstein, D.S.: Average-preserving symmetries and equipartition in linear Hamiltonian systems. In: 43rd Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, 14–17 December, p. 2155 (2004)
Rapisarda, P., Willems, J.C.: Conserved- and zero-mean quadratic quantities in oscillatory systems. Math. Control Signals Syst. 17, 173 (2005)
Patrascioiu, A.: The Ergodic Hypothesis: A Complicated Problem in Mathematics and Physics. Los Alamos Science Special Issue, vol. 263 (1987)
Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. 51(7), 1529 (2002)
Wrang, G.M., Sevick, E.M., Mittag, E., Searles, D.J., Evans, D.J.: Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. Phys. Rev. Lett. 89(5), 50601 (2002)
Lamb, J.S.W., Roberts, J.A.G.: Time-reversal symmetry in dynamical systems: a survey. Physica D 112, 1 (1998)
Von Bayer, H.C.: Warmth Disperses and Time Passes. The Modern Library, New York
Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems. In: Ulam, S. (ed.) Sets, Numbers and Universes. MIT Press, Cambridge (1974). (See also Los Alamos Scientific Laboratory report LA-1940 (1955))
Duncan, T.L., Semura, J.S.: Information loss as a foundational principle for the second law of thermodynamics. Found. Phys. 37, 1767 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
W.L. De Koning retired from the Department of Mathematics of Delft University of Technology.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Van Willigenburg, L.G., De Koning, W.L. Emergence of the Second Law out of Reversible Dynamics. Found Phys 39, 1217 (2009). https://doi.org/10.1007/s10701-009-9341-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10701-009-9341-6