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


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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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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