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On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff

In: The Mathematics of Time

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  • Steve Smale

    (University of California at Berkeley, Department of Mathematics)

Abstract

I would like to develop the idea that by introducing a dissipation/forcing term into Hamilton’s equations of physics, one might be able to revive the ergodic hypothesis of Boltzmann and Birkhoff. The compelling fact is that with one main exception (end of section 1), Hamiltonian systems are not ergodic; furthermore other qualitative features of these systems don’t seem amenable to study. On the other hand, large classes of general ordinary differential equations have ergodic structurally stable attractors with mixing and other properties. Ergodic theory was developed originally in response to problems arising from Hamilton’s ordinary differential equations; but it seems now to have a more natural home in general dynamical systems (at least to the extent it relates to ordinary differential equations).

Suggested Citation

  • Steve Smale, 1980. "On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff," Springer Books, in: The Mathematics of Time, pages 137-144, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-8101-3_9
    DOI: 10.1007/978-1-4613-8101-3_9
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