Journal of Statistical Physics

, Volume 79, Issue 3–4, pp 731–737 | Cite as

Statistical mechanics of nonlinear wave equations. 3. Metric transitivity for hyperbolic sine-gordon

  • H. P. McKean
Articles
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Abstract

McKean and Vaninsky proved that the canonical measureeHdQ dP based upon the Hamiltonian\(H = \smallint [\tfrac{1}{2}P^2 + \tfrac{1}{2}(Q')^2 + F(Q)]dx\) of the wave equation ∂2Q/∂t2 - ∂2Q/∂x2 +f(Q) = 0 with restoring forcef(Q)=F'(Q) is preserved by the associated flow ofQ andP =Q, and they conjectured that metric transitivity prevails,always on the whole line, and likewise on the circleunless f(Q)=Q orf(Q)=shQ. Here, the metric transitivity is proved for the whole line in the second case. The proof employs the beautiful “d'Alembert formula” of Krichever.

Key Words

Partial differential equations statistical mechanics ergodic theory 

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References

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    H. Dym and H. P. McKean,Gaussian Processes, Function Theory, and the Inverse Spectral Problem (Academic Press, New York, 1976).Google Scholar
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    H. P. McKean,Stochastic Integrals (Academic Press, New York, 1965).Google Scholar
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    H. P. McKean and K. L. Vaninsky, Statistical mechanics of nonlinear wave equations (1): the petit and microcanonical ensembles,in Trends and Perspectives in Applied Mathematics, L. Sirovich, ed. (Springer-Verlag, Berlin, 1994), pp. 239–264.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • H. P. McKean
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York

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