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A variational formula for the free energy of an interacting manyparticle system
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Adams, S. (Stefan), Collevecchio, Andrea and König, Wolfgang (2011) A variational formula for the free energy of an interacting manyparticle system. Annals of Probability, Volume 39 (Number 2). pp. 683728. doi:10.1214/10AOP565 ISSN 00911798.
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Official URL: http://dx.doi.org/10.1214/10AOP565
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 ebeta H(N), where H(N) denotes the corresponding Hamilton operator. The wellknown FeynmanKac 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 largedeviations analysis of the stationary empirical field. The resulting variational formula ranges over random shiftinvariant 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 BoseEinstein condensation and intend to analyze it further in future.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics Q Science > QC Physics 

Divisions:  Faculty of Science, Engineering and Medicine > Science > Mathematics  
Library of Congress Subject Headings (LCSH):  Manybody problem, Gibbs' free energy, BoseEinstein condensation, Large deviations, Brownian bridges (Mathematics)  
Journal or Publication Title:  Annals of Probability  
Publisher:  Institute of Mathematical Statistics  
ISSN:  00911798  
Official Date:  March 2011  
Dates: 


Volume:  Volume 39  
Number:  Number 2  
Page Range:  pp. 683728  
DOI:  10.1214/10AOP565  
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) 
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