### Abstract

Dyson's model is a one-dimensional system of Brownian motions with longrange repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant β/2. We give sufficient conditions for initial configurations so that Dyson's model with β = 2 and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of ℤ is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.

Original language | English |
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Pages (from-to) | 469-497 |

Number of pages | 29 |

Journal | Communications in Mathematical Physics |

Volume | 293 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2009 Sep 18 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Non-equilibrium dynamics of Dyson's model with an infinite number of particles.** / Katori, Makoto; Tanemura, Hideki.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 293, no. 2, pp. 469-497. https://doi.org/10.1007/s00220-009-0912-3

}

TY - JOUR

T1 - Non-equilibrium dynamics of Dyson's model with an infinite number of particles

AU - Katori, Makoto

AU - Tanemura, Hideki

PY - 2009/9/18

Y1 - 2009/9/18

N2 - Dyson's model is a one-dimensional system of Brownian motions with longrange repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant β/2. We give sufficient conditions for initial configurations so that Dyson's model with β = 2 and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of ℤ is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.

AB - Dyson's model is a one-dimensional system of Brownian motions with longrange repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant β/2. We give sufficient conditions for initial configurations so that Dyson's model with β = 2 and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of ℤ is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.

UR - http://www.scopus.com/inward/record.url?scp=78651354043&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651354043&partnerID=8YFLogxK

U2 - 10.1007/s00220-009-0912-3

DO - 10.1007/s00220-009-0912-3

M3 - Article

VL - 293

SP - 469

EP - 497

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -