Foundations of Physics

, 39:1217

Emergence of the Second Law out of Reversible Dynamics

Open Access

DOI: 10.1007/s10701-009-9341-6

Cite this article as:
Van Willigenburg, L.G. & De Koning, W.L. Found Phys (2009) 39: 1217. doi:10.1007/s10701-009-9341-6


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.


Ectropy Entropy fluctuations Hamiltonian systems Reversible dynamics Emergent properties Second law of thermodynamics 

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Systems & Control GroupWageningen UniversityWageningenThe Netherlands
  2. 2.LeidenThe Netherlands

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