### 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 | |

State | Published - Mar 1994 |

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### 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

**Brownian motion with restoring drift : The petit and micro-canonical ensembles.** / McKean, H. P.; Vaninsky, K. L.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 160, no. 3, pp. 615-630. https://doi.org/10.1007/BF02173433

}

TY - JOUR

T1 - Brownian motion with restoring drift

T2 - The petit and micro-canonical ensembles

AU - McKean, H. P.

AU - Vaninsky, K. L.

PY - 1994/3

Y1 - 1994/3

N2 - Let f(Q) be odd and positive near +∞. Then the non-linear wave equation ∂2Q/∂t2-∂2Q/∂x2-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-Hd∞Qd∞P; and for L↓∞, Q settles down to the stationary diffusion with infinitesimal operator 1/2 ∂2/∂Q2+m(Q)∂/∂Q, m being the logarithmic derivative of the ground state of -d2/dQ2{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)/Q2→∞, the same type of diffusion appears, but with drift arising from the modified potential F(Q)+cQ2, c being chosen so that the mean of Q2 is the assigned number D. The proof employs Döblin's method of "loops" [1937] and steepest descent. The same is true for F(Q)=m2Q2, only now the proof is elementary. The outcome is also the same if F(Q)/Q2→0, provided D is smaller than the petit canonical mean of Q2; for D larger than this mean, the matter is more subtle and the outcome is unknown.

AB - Let f(Q) be odd and positive near +∞. Then the non-linear wave equation ∂2Q/∂t2-∂2Q/∂x2-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-Hd∞Qd∞P; and for L↓∞, Q settles down to the stationary diffusion with infinitesimal operator 1/2 ∂2/∂Q2+m(Q)∂/∂Q, m being the logarithmic derivative of the ground state of -d2/dQ2{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)/Q2→∞, the same type of diffusion appears, but with drift arising from the modified potential F(Q)+cQ2, c being chosen so that the mean of Q2 is the assigned number D. The proof employs Döblin's method of "loops" [1937] and steepest descent. The same is true for F(Q)=m2Q2, only now the proof is elementary. The outcome is also the same if F(Q)/Q2→0, provided D is smaller than the petit canonical mean of Q2; for D larger than this mean, the matter is more subtle and the outcome is unknown.

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U2 - 10.1007/BF02173433

DO - 10.1007/BF02173433

M3 - Article

VL - 160

SP - 615

EP - 630

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -