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### A variational formula for the free energy of an interacting many-particle system

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Adams, S. (Stefan), Collevecchio, Andrea and König, Wolfgang.
(2011)
*A variational formula for the free energy of an interacting many-particle system.*
Annals of Probability, Volume 39
(Number 2).
pp. 683-728.
ISSN 0091-1798

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1214/10-AOP565

## Abstract

We consider N bosons in a box in R(d) with volume N/rho under the influence of a mutually repellent pair potential. The particle density p is an element of (0, infinity) is kept fixed. Our main result is the identification of the limiting free energy, f(beta, p), at positive temperature 1/beta, in terms of an explicit variational formula, for any fixed rho if beta is sufficiently small, and for any fixed beta if rho is sufficiently small.

The thermodynamic equilibrium is described by the symmetrized trace of e-beta H(N), where H(N) denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of N interacting Brownian bridges. Due to the symmetrization, the bridges are organized in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process.

In our proof of the lower bound for the free energy, we drop all interaction involving "infinitely long" cycles, and their possible presence is signalled by a loss of mass of the "finitely long" cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the "finitely long" cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyze it further in future.

Item Type: | Journal Article |
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Many-body problem, Gibbs' free energy, Bose-Einstein condensation, Large deviations, Brownian bridges (Mathematics) |

Journal or Publication Title: | Annals of Probability |

Publisher: | Institute of Mathematical Statistics |

ISSN: | 0091-1798 |

Official Date: | March 2011 |

Volume: | Volume 39 |

Number: | Number 2 |

Page Range: | pp. 683-728 |

Identification Number: | 10.1214/10-AOP565 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Deutsche Forschungsgemeinschaft (DFG), Italy. Ministero dell'istruzione, dell'università e della ricerca (MUIR) |

Grant number: | 718 (DFG), 2007TKLTSR (MUIR) |

References: | [1] ADAMS, S. (2009). Large deviations for empirical path measures in cycles of integer partitions. |

URI: | http://wrap.warwick.ac.uk/id/eprint/41502 |

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