A General Nonlinear Evolution Equation for Irreversible Conservative Approach to Stable Equilibrium

  • Gian Paolo Beretta
Part of the NATO ASI Series book series (NSSB, volume 135)


The problem of understanding entropy and irreversibility has been tackled by thousands of physicists during the past century. Schools of thought have formed and flourished around different perspectives of the problem. But a definitive solution has yet to be found.1


Equilibrium Solution Nonlinear Evolution Equation Linear Manifold Escape Time Quantum Statistical Mechanics 
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    For recent reviews of the problem, see: R. Jancel, Foundations of Classical and Quantum Statistical Mechanics, Pergamon Press, Oxford, 1969; A. Wehrl, Rev. Mod. Phys., Vol. 50, 221 (1978); O. Penrose, Rep. Mod. Phys., Vol. 42, 129 (1979); J.L. Park and R.F. Simmons, Jr., in Old and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology, ed. A. van der Merwe, Plenum Press, 1983.Google Scholar
  2. 2.
    G.P. Beretta, Thesis, MIT, 1981, unpublished.Google Scholar
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    G.P. Beretta, E.P. Gyftopoulos, J.L. Park and G.N. Hatsopoulos, Nuovo Cimento B, Vol. 82, 169 (1984).MathSciNetADSCrossRefGoogle Scholar
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    Throughout this lecture we proceed heuristically and disregard all questions of purely technical mathematical nature.Google Scholar
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    E.T. Jaynes, Phys. Rev., Vol. 106, 620 (1957); Vol. 108, 171 (1957).MathSciNetADSCrossRefGoogle Scholar
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    Here D (F) is the domain of definition of function F, which does not necessarily coincide with the set S Google Scholar
  8. 8.
    Equilibrium solution ρe is stable according to Liapunoff if and only if for every ɛ > 0 there is a δ (ɛ) > 0 such that any solution ρ (t) with | | ρ (0) - ρe | | < δ (ɛ) remains with | | ρ (t) - ρe | | < ɛ for every t > 0, where | | • | | denotes the norm on L defined by | | A | | = (A | A).Google Scholar
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    G.P. Beretta, Int. J. Theor. Phys., Vol. 24, 119 (1985).MathSciNetCrossRefGoogle Scholar
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    G.P. Beretta, to be published.Google Scholar
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    G.P. Beretta, E.P. Gyftopoulos and J.L. Park, Nuovo Cimento B, in press.Google Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • Gian Paolo Beretta
    • 1
    • 2
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Politecnico di MilanoMilanoItaly

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