### Abstract

Let f(Q) be odd and positive near +∞. Then the non-linear wave equation ∂^{2}Q/∂t^{2}-∂^{2}Q/∂x^{2}-f(Q)=0, considered on the circle 0≤x<L, can be written in Hamiltonian form Q^{⊙}=∂H/∂P, P^{⊙}=-∂H/∂Q with {Mathematical expression} the corresponding flow preserves the (suitably interpreted) "petit ensemble"e^{-H}d^{∞}Qd^{∞}P; and for L↓∞, Q settles down to the stationary diffusion with infinitesimal operator 1/2 ∂^{2}/∂Q^{2}+m(Q)∂/∂Q, m being the logarithmic derivative of the ground state of -d^{2}/dQ^{2}{norm of matrix}F(Q). This diffusion is the "Brownian motion with restoring drift"; see McKean-Vaninsky [1993(1)]. For reasons suggested by the paper of Lebowitz-Rose-Speer [1988] on NLS, it is interesting to study the "micro-canonical ensemble" obtained by restricting to the sphere {Mathematical expression} and making L↓∞ with fixed D=N/L. Now, for F(Q)/Q^{2}→∞, the same type of diffusion appears, but with drift arising from the modified potential F(Q)+cQ^{2}, c being chosen so that the mean of Q^{2} is the assigned number D. The proof employs Döblin's method of "loops" [1937] and steepest descent. The same is true for F(Q)=m^{2}Q^{2}, only now the proof is elementary. The outcome is also the same if F(Q)/Q^{2}→0, provided D is smaller than the petit canonical mean of Q^{2}; for D larger than this mean, the matter is more subtle and the outcome is unknown.

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

Pages (from-to) | 615-630 |

Number of pages | 16 |

Journal | Communications in Mathematical Physics |

Volume | 160 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 1994 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Communications in Mathematical Physics*,

*160*(3), 615-630. https://doi.org/10.1007/BF02173433