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Neither asymptotic stability nor nonrecurrence is a sufficient basis for the second law

1990; Brown University; Volume: 48; Issue: 3 Linguagem: Inglês

10.1090/qam/1074959

1552-4485

W. A. Day,

Control and Stability of Dynamical Systems

1. Introduction.This paper is concerned with two assertions about the behaviour of isolated thermodynamic systems, viz., Asymptotic stability.Isolated systems seek a dead level.Nonrecurrence.Un systeme isole ou se sont exercees des influences thermiques ne revient pas a un etat anterieur.The first of these is due to Bridgman [1, p. 116], while the second is due to Perrin [2, p. 63],The question at issue here is whether asymptotic stability or nonrecurrence will serve as surrogate for any of the more familiar statements of the second law of thermodynamics that are associated with Clausius, Kelvin, or Planck.It would be a most interesting state of affairs if either assertion would so serve, for neither is framed directly in terms of the concepts of heat and work which occur in traditional statements of the second law.Perrin certainly believed nonrecurrence to be equivalent to the second law for he said, "Je dois a M. Langevin la conviction ou je suis que, sous l'une ou l'autre de ces formes, l'idee que je viens d'enoncer contient mieux que les enonces ordinaires ce qu'il y a d'essentiel dans le second principe de la Thermodynamique, ou principe de Carnot..."[loc.cit.].Bridgman did not claim so much for asymptotic stability, and he recognised that various formulations of the second law are not likely to be exactly equivalent.Nonetheless, he regarded asymptotic stability as the background against which the second law should be viewed, and, in the light of Perrin's claims for nonrecurrence, it is natural to ask if a hypothesis of asymptotic stability is strong enough to imply the second law.The expectation that an asymptotically stable system will have a Lyapunov function, from which its entropy might be constructed, suggests that the answer to this question may be an affirmative one.My purpose is to show that, in fact, neither asymptotic stability nor nonrecurrence, whether taken separately or together, is sufficient to imply the second law.I shall do so by constructing constitutive relations which describe a thermoelastic fluid which exhibits both asymptotic stability and nonrecurrence but whose efficiency in a cyclic process exceeds what the second law permits.