Abstract
This note addresses a problem of nineteenth century applied mathematics—is it possible in the context of Hamiltonian mechanics to define a functionS of the generalized coordinates and momenta which is monotonically increasing along orbits? The question is of interest, because, for a sytem not in thermodynamic equilibrium, entropy should increase strictly monotonically along an orbit, and a negative answer implies that mechanical principles different from those of Hamiltonian mechanics must be introduced to explain thermodynamics. This note answers the question rigorously for Hamiltonian systems confined to an invariant region of finite volume in phase space; it is not possible to define a continuous function which increases monotonically along orbits. An appendix gives a translation of an 1889 paper of Poincaré addressing the same issue.
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1. In the papers Poincaré cites [3,4], Helmholtz argued that a thermodynamic system undergoing reversible transformations near a minimum of the free energyU-TS is governed by equations identical to those of a “monocyclic” Hamiltonian system whose potential energy is given by the free energy. A monocyclic system is a system with several generalized coordinates, including a single cyclic one. For terminology, see, e.g., [7,8]. The cyclic coordinate is a thermodynamic variable like the temperature in Helmholtz' framework. Helmholtz was thinking about Hamiltonian systems in which the variables were relatively few phenomenological parameters, and perhaps Poincaré was, too, though with less justification, since his aim was to explore nonequilibrium phenomena, for which it is not so clear that phenomenological parameters like temperature can be defined.
2. Original title: “Sur les Tentatives d'Explication Mécanique des Principes de la Thermodynamique.” Translated into English by K. Feigl and E. T. Olsen, Department of Mathematics, Illinois Institute of Technology. Reproduced by permission of the FrenchAcademie des Sciences.
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Olsen, E.T. Classical mechanics and entropy. Found Phys Lett 6, 327–337 (1993). https://doi.org/10.1007/BF00665652
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DOI: https://doi.org/10.1007/BF00665652