### Abstract

McKean and Vaninsky proved that the canonical measure e^{-H}d^{∞}Q d^{∞}P based upon the Hamiltonian {Mathematical expression} of the wave equation ∂^{2}Q/∂t^{2} - ∂^{2}Q/∂x^{2} +f(Q) = 0 with restoring force f(Q)=F'(Q) is preserved by the associated flow of Q and P =Q^{{dot operator}}, and they conjectured that metric transitivity prevails, always on the whole line, and likewise on the circle unless f(Q)=Q or f(Q)=sh Q. Here, the metric transitivity is proved for the whole line in the second case. The proof employs the beautiful "d'Alembert formula" of Krichever.

Original language | English (US) |
---|---|

Pages (from-to) | 731-737 |

Number of pages | 7 |

Journal | Journal of Statistical Physics |

Volume | 79 |

Issue number | 3-4 |

DOIs | |

State | Published - May 1995 |

### Fingerprint

### Keywords

- ergodic theory
- Partial differential equations
- statistical mechanics

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*79*(3-4), 731-737. https://doi.org/10.1007/BF02184878

**Statistical mechanics of nonlinear wave equations. 3. Metric transitivity for hyperbolic sine-gordon.** / McKean, H. P.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 79, no. 3-4, pp. 731-737. https://doi.org/10.1007/BF02184878

}

TY - JOUR

T1 - Statistical mechanics of nonlinear wave equations. 3. Metric transitivity for hyperbolic sine-gordon

AU - McKean, H. P.

PY - 1995/5

Y1 - 1995/5

N2 - McKean and Vaninsky proved that the canonical measure e-Hd∞Q d∞P based upon the Hamiltonian {Mathematical expression} of the wave equation ∂2Q/∂t2 - ∂2Q/∂x2 +f(Q) = 0 with restoring force f(Q)=F'(Q) is preserved by the associated flow of Q and P =Q{dot operator}, and they conjectured that metric transitivity prevails, always on the whole line, and likewise on the circle unless f(Q)=Q or f(Q)=sh Q. Here, the metric transitivity is proved for the whole line in the second case. The proof employs the beautiful "d'Alembert formula" of Krichever.

AB - McKean and Vaninsky proved that the canonical measure e-Hd∞Q d∞P based upon the Hamiltonian {Mathematical expression} of the wave equation ∂2Q/∂t2 - ∂2Q/∂x2 +f(Q) = 0 with restoring force f(Q)=F'(Q) is preserved by the associated flow of Q and P =Q{dot operator}, and they conjectured that metric transitivity prevails, always on the whole line, and likewise on the circle unless f(Q)=Q or f(Q)=sh Q. Here, the metric transitivity is proved for the whole line in the second case. The proof employs the beautiful "d'Alembert formula" of Krichever.

KW - ergodic theory

KW - Partial differential equations

KW - statistical mechanics

UR - http://www.scopus.com/inward/record.url?scp=21844484872&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21844484872&partnerID=8YFLogxK

U2 - 10.1007/BF02184878

DO - 10.1007/BF02184878

M3 - Article

VL - 79

SP - 731

EP - 737

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -